Contemporary Ideas on Ship Stability and Capsizing in Waves

Contemporary Ideas on Ship Stability and Capsizing in Waves FLUID MECHANICS AND ITS APPLICATIONS Volume 96 Series Edi...

Author: Marcelo Almeida Santos Neves | Vadim L. Belenky | Jean Otto de Kat | Kostas Spyrou | Naoya Umeda

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Contemporary Ideas on Ship Stability and Capsizing in Waves

FLUID MECHANICS AND ITS APPLICATIONS Volume 96 Series Editor:

R. MOREAU MADYLAM Ecole Nationale Supérieure d’Hydraulique de Grenoble Boîte Postale 95 38402 Saint Martin d’Hères Cedex, France

Aims and Scope of the Series The purpose of this series is to focus on subjects in which fluid mechanics plays a fundamental role. As well as the more traditional applications of aeronautics, hydraulics, heat and mass transfer etc., books will be published dealing with topics which are currently in a state of rapid development, such as turbulence, suspensions and multiphase fluids, super and hypersonic flows and numerical modeling techniques. It is a widely held view that it is the interdisciplinary subjects that will receive intense scientific attention, bringing them to the forefront of technological advancement. Fluids have the ability to transport matter and its properties as well as to transmit force, therefore fluid mechanics is a subject that is particularly open to cross fertilization with other sciences and disciplines of engineering. The subject of fluid mechanics will be highly relevant in domains such as chemical, metallurgical, biological and ecological engineering. This series is particularly open to such new multidisciplinary domains. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of a field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

For further volumes: http://www.springer.com/series/5980

Marcelo Almeida Santos Neves • Vadim L. Belenky Jean Otto de Kat • Kostas Spyrou • Naoya Umeda Editors

Contemporary Ideas on Ship Stability and Capsizing in Waves

Editors Marcelo Almeida Santos Neves Department of Naval Architecture and Oce Federal University of Rio de Janeiro Parque Tecnologico, Quadra 7 21941-972 Rio de Janeiro Brazil [email protected] Jean Otto de Kat A. P. Moeller - Maersk A/S Esplanaden 50 1098 Copenhagen Denmark [email protected] Naoya Umeda Department of Naval Architecture and Oce Graduate School of Eng Osaka University Yamadaoka Suita 2-1 565-0971 Osaka Japan [email protected]

Vadim L. Belenky Naval Surface Warfare Center Carderock D MacArthur Blvd 9500 20817-5700 West Bethesda Maryland USA [email protected] Kostas Spyrou School of Naval Architecture and Marine National Technical University of Athens Iroon Polytechnion, Zographou 9 157 73 Athens Greece [email protected]

Stability Criteria Evaluation and Performance Based Criteria Development for Damaged Naval Vessels, by Andrew James Peters © Copyright QinetiQ Limited 2010, by permission of QinetiQ Limited.

ISSN 0926-5112 ISBN 978-94-007-1481-6 e-ISBN 978-94-007-1482-3 DOI 10.1007/978-94-007-1482-3 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2011929340 © Springer Science+Business Media B.V. 2011 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Cover design: eStudio Calamar S.L. Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

Table of Contents Preface .................................................................................................................... xi List of Corresponding Authors ..........................................................................xiii

1 Stability Criteria Review of Available Methods for Application to Second Level Vulnerability Criteria ............................................................................................... 3 Christopher C. Bassler, Vadim Belenky, Gabriele Bulian, Alberto Francescutto, Kostas Spyrou and Naoya Umeda A Basis for Developing a Rational Alternative to the Weather Criterion: Problems and Capabilities ...................................................................................... 25 K.J. Spyrou Conceptualising Risk.............................................................................................. 47 Andrzej Jasionowski and Dracos Vassalos Evaluation of the Weather Criterion by Experiments and its Effect to the Design of a RoPax Ferry .............................................................................. 65 Shigesuke Ishida, Harukuni Taguchi and Hiroshi Sawada Evolution of Analysis and Standardization of Ship Stability: Problems and Perspectives ..................................................................................................... 79 Yury Nechaev SOLAS 2009 – Raising the Alarm ....................................................................... 103 Dracos Vassalos and Andrzej Jasionowski

2 Stability of the Intact Ship Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas ........................................................................................................ 119 A. Yucel Odabasi and Erdem Uçer Historical Roots of the Theory of Hydrostatic Stability of Ships ........................ 141 Horst Nowacki and Larrie D. Ferreiro The Effect of Coupled Heave/Heave Velocity or Sway/Sway Velocity Initial Conditions on Capsize Modeling .............................................................. 181 Leigh S. McCue and Armin W. Troesch

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The Use of Energy Build Up to Identify the Most Critical Heeling Axis Direction for Stability Calculations for Floating Offshore Structures ................. 193 Joost van Santen Some Remarks on Theoretical Modelling of Intact Stability............................... 217 N. Umeda, Y. Ohkura, S. Urano, M. Hori and H. Hashimoto

3 Parametric Rolling An Investigation of Head-Sea Parametric Rolling for Two Fishing Vessels.................................................................................................................. 231 Marcelo A.S. Neves, Nelson A. Pérez, Osvaldo M. Lorca and Claudio A. Rodríguez Simple Analytical Criteria for Parametric Rolling............................................... 253 K.J. Spyrou Experimental Study on Parametric Roll of a Post-Panamax Containership in Short-Crested Irregular Waves ................................................. 267 Hirotada Hashimoto, Naoya Umeda and Akihiko Matsuda Model Experiment on Parametric Rolling of a Post-Panamax Containership in Head Waves .............................................................................. 277 Harukuni Taguchi, Shigesuke Ishida, Hiroshi Sawada and Makiko Minami Numerical Procedures and Practical Experience of Assessment of Parametric Roll of Container Carriers ............................................................. 295 Vadim Belenky, Han-Chang Yu and Kenneth Weems Parametric Roll and Ship Design ......................................................................... 307 Marc Levadou and Riaan van’t Veer Parametric Roll Resonance of a Large Passenger Ship in Dead Ship Condition in All Heading Angles......................................................................... 331 Abdul Munif, Yoshiho Ikeda, Tomo Fujiwara and Toru Katayama Parametric Rolling of Ships – Then and Now...................................................... 347 J. Randolph Paulling

4 Broaching-to Parallels of Yaw and Roll Dynamics of Ships in Astern Seas and the Effect of Nonlinear Surging .................................................................... 363 K.J. Spyrou

Table of Contents vii

Model Experiment on Heel-Induced Hydrodynamic Forces in Waves for Realising Quantitative Prediction of Broaching............................. 379 H. Hashimoto, Naoya Umeda and Akihiko Matsuda Perceptions of Broaching-To: Discovering the Past ............................................ 399 K.J. Spyrou

5 Nonlinear Dynamics and Ship Capsizing Use of Lyapunov Exponents to Predict Chaotic Vessel Motions ........................ 415 Leigh S. McCue and Armin W. Troesch Applications of Finite-Time Lyapunov Exponents to the Study of Capsize in Beam Seas ...................................................................................... 433 Leigh S. McCue Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas .................... 449 Marcelo A.S. Neves, Jerver E.M. Vivanco and Claudio A. Rodríguez

6 Roll Damping A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships on the Basis of Ikeda’s Method and Its Limitation ......................... 465 Yuki Kawahara, Kazuya Maekawa and Yoshiho Ikeda A Study on the Characteristics of Roll Damping of Multi-Hull Vessels............. 487 Toru Katayama, Masanori Kotaki and Yoshiho Ikeda

7 Probabilistic Assessment of Ship Capsize Capsize Probability Analysis for a Small Container Vessel ................................ 501 E.F.G. van Daalen, J.J. Blok and H. Boonstra Efficient Probabilistic Assessment of Intact Stability.......................................... 515 N. Themelis and K.J. Spyrou Probability of Capsizing in Beam Seas with Piecewise Linear Stochastic GZ Curve ............................................................................................ 531 Vadim Belenky, Arthur M. Reed and Kenneth M. Weems Probabilistic Analysis of Roll Parametric Resonance in Head Seas.................... 555 Vadim L. Belenky, Kenneth M. Weems, Woei-Min Lin and J. Randolf Paulling

viii Table of Contents

8 Environmental Modeling Sea Spectra Revisited ........................................................................................... 573 Maciej Pawłowski On Self-Repeating Effect in Reconstruction of Irregular Waves......................... 589 Vadim Belenky New Approach To Wave Weather Scenarios Modeling ...................................... 599 Alexander B. Degtyarev

9 Damaged Ship Stability Effect of Decks on Survivability of Ro–Ro Vessels ............................................ 621 Maciej Pawłowski Experimental and Numerical Studies on Roll Motion of a Damaged Large Passenger Ship in Intermediate Stages of Flooding................................... 633 Yoshiho Ikeda, Shigesuke Ishida, Toru Katayama and Yuji Takeuchi Exploring the Influence of Different Arrangements of Semi-Watertight Spaces on Survivability of a Damaged Large Passenger Ship............................. 643 Riaan van’t Veer, William Peters, Anna-Lea Rimpela and Jan de Kat Time-Based Survival Criteria for Passenger Ro-Ro Vessels............................... 663 Andrzej Jasionowski, Dracos Vassalos and Luis Guarin Pressure-Correction Method and Its Applications for Time-Domain Flooding Simulation ............................................................................................. 689 Pekka Ruponen

10 CFD Applications to Ship Stability Applications of 3D Parallel SPH for Sloshing and Flooding............................... 709 Liang Shen and Dracos Vassalos Simulation of Wave Effect on Ship Hydrodynamics by RANSE........................ 723 Qiuxin Gao and Dracos Vassalos A Combined Experimental and SPH Approach to Sloshing and Ship Roll Motions......................................................................................................... 735 Luis Pérez-Rojas, Elkin Botía-Vera, José Luis Cercos-Pita, Antonio Souto-Iglesias, Gabriele Bulian and Louis Delorme

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11 Design for Safety Design for Safety with Minimum Life-Cycle Cost.............................................. 753 Romanas Puisa, Dracos Vassalos, Luis Guarin and George Mermiris

12 Stability of Naval Vessels Stability Criteria Evaluation and Performance Based Criteria Development for Damaged Naval Vessels........................................................... 773 Andrew J. Peters and David Wing A Naval Perspective on Ship Stability ................................................................. 793 Arthur M. Reed

13 Accident Investigations New Insights on the Sinking of MV Estonia........................................................ 827 Andrzej Jasionowski and Dracos Vassalos Experimental Investigation on Capsizing and Sinking of a Cruising Yacht in Wind ...................................................................................................... 841 Naoya Umeda, Masatoshi Hori, Kazunori Aoki, Toru Katayama and Yoshiho Ikeda Author Index....................................................................................................... 855

Preface During the last decade significant progress has been made in the field of ship stability: the understanding of ship dynamics and capsize mechanisms has increased through improved simulation tools and dedicated model tests, design regulations have been adapted based on better insights into the physics of intact and damaged vessels, and ships can be operated with better knowledge of the dynamic behaviour in a seaway. Yet in spite of the progress made, numerous scientific and practical challenges still exist with regard to the accurate prediction of extreme motion and capsize dynamics for intact and damaged vessels, the probabilistic nature of extreme events, criteria that properly reflect the physics and operational safety of an intact or damaged vessel, and ways to provide relevant information on safe ship handling to ship operators. This book provides a comprehensive review of the above issues through the selection of representative papers presented at the unique series of international workshops and conferences on ship stability held between 2000 and 2009. The first conference on the stability of ships and ocean vehicles was held in 1975 in Glasgow (STAB ‘75) and the first international workshop on ship stability was held in 1995 in Glasgow. The aim of the workshop was to address issues related to ship capsize and identify ways to accelerate relevant developments. Since then the workshops have been held each year in succession, whereby each third year a STAB conference has been held as a cap of the preceding workshops and to present the latest developments. The editorial committee has selected papers for this book from the following events: STAB 2000 Conference (Launceston, Tasmania), 5th Stability Workshop (Trieste, 2001), 6th Stability Workshop (Long Island, 2002), STAB 2003 Conference (Madrid), 7th Stability Workshop (Shanghai, 2004), 8th Stability Workshop (Istanbul, 2005), STAB 2006 Conference (Rio de Janeiro), 9th Stability Workshop (Hamburg, 2007), 10th Stability Workshop (Daejeon, 2008), and STAB 2009 Conference (St. Petersburg). The papers have been clustered around the following themes: Stability Criteria, Stability of the Intact Ship, Parametric Rolling, Broaching, Nonlinear Dynamics, Roll Damping, Probabilistic Assessment of Ship Capsize, Environmental Modelling, Damaged Ship Stability, CFD Applications, Design for Safety, Naval Vessels, and Accident Investigations. We would like to express our sincere gratitude to all of the authors who have presented papers at the conferences and workshops; to the organizers and all of the participants of these events; to the members of the International Standing Committee for the Stability of Ships and Ocean Vehicles; and to Professor Marcelo Neves for his role as main editor of this book. Dr. Jan Otto de Kat Chairman of the International Standing Committee for the Stability of Ships and Ocean Vehicles (on behalf of the editorial committee)

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List of Corresponding Authors 1 Stability Criteria Review of Available Methods for Application to Second Level Vulnerability Criteria Vadim Belenky [email protected] A Basis for Developing a Rational Alternative to the Weather Criterion: Problems and Capabilities K.J. Spyrou [email protected] Conceptualising Risk Dracos Vassalos [email protected] Evaluation of the Weather Criterion by Experiments and its Effect to the Design of a RoPax Ferry Shigesuke Ishida [email protected] Evolution of Analysis and Standardization of Ship Stability: Problems and Perspectives Yury Nechaev [email protected] SOLAS 2009 – Raising the Alarm Dracos Vassalos [email protected] 2 Stability of the Intact Ship Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas E. Uçer [email protected] Historical Roots of the Theory of Hydrostatic Stability of Ships Horst Nowacki [email protected]

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xiv List of Corresponding Authors

The Effect of Coupled Heave/Heave Velocity or Sway/Sway Velocity Initial Conditions on Capsize Modeling Leigh S. McCue [email protected] The Use of Energy Build Up to Identify the Most Critical Heeling Axis Direction for Stability Calculations for Floating Offshore Structures Joost van Santen [email protected] Some Remarks on Theoretical Modelling of Intact Stability Naoya Umeda [email protected] 3 Parametric Rolling An Investigation of Head-Sea Parametric Rolling for Two Fishing Vessels Marcelo de Almeida Santos Neves [email protected] Simple Analytical Criteria for Parametric Rolling K.J. Spyrou [email protected] Experimental Study on Parametric Roll of a Post-Panamax Containership in Short-Crested Irregular Waves Hirotada Hashimoto [email protected] Model Experiment on Parametric Rolling of a Post-Panamax Containership in Head Waves Harukuni Taguchi [email protected] Numerical Procedures and Practical Experience of Assessment of Parametric Roll of Container Carriers Vadim Belenky [email protected] Parametric Roll and Ship Design Riaan van’t Veer [email protected]

List of Corresponding Authors xv

Parametric Roll Resonance of a Large Passenger Ship in Dead Ship Condition in All Heading Angles Toru Katayama [email protected] Parametric Rolling of Ships – Then and Now J. Randolph Paulling [email protected] 4 Broaching-to Parallels of Yaw and Roll Dynamics of Ships in Astern Seas and the Effect of Nonlinear Surging K.J. Spyrou [email protected] Model Experiment on Heel-Induced Hydrodynamic Forces in Waves for Realising Quantitative Prediction of Broaching Hirotada Hashimoto [email protected] Perceptions of Broaching-To: Discovering the Past K.J. Spyrou [email protected] 5 Nonlinear Dynamics and Ship Capsizing Use of Lyapunov Exponents to Predict Chaotic Vessel Motions Leigh S. McCue [email protected] Applications of Finite-Time Lyapunov Exponents to the Study of Capsize in Beam Seas Leigh S. McCue [email protected] Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas Marcelo A.S. Neves, [email protected] 6 Roll Damping A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships on the Basis of Ikeda’s Method and Its Limitation Yoshiho Ikeda [email protected]

xvi List of Corresponding Authors

A Study on the Characteristics of Roll Damping of Multi-hull Vessels Toru Katayama [email protected] 7 Probabilistic Assessment of Ship Capsize Capsize Probability Analysis for a Small Container Vessel E.F.G. van Daalen [email protected] Efficient Probabilistic Assessment of Intact Stability K.J. Spyrou [email protected] Probability of Capsizing in Beam Seas with Piecewise Linear Stochastic GZ Curve Vadim Belenky [email protected] Probabilistic Analysis of Roll Parametric Resonance in Head Seas Vadim L. Belenky [email protected] 8 Environmental Modeling Sea Spectra Revisited Maciej Pawłowski [email protected] On Self-Repeating Effect in Reconstruction of Irregular Waves Vadim Belenky [email protected] New Approach To Wave Weather Scenarios Modeling Alexander B. Degtyarev [email protected] 9 Damaged Ship Stability Effect of Decks on Survivability of Ro–Ro Vessels Maciej Pawłowski [email protected]

List of Corresponding Authors xvii

Experimental and Numerical Studies on Roll Motion of a Damaged Large Passenger Ship in Intermediate Stages of Flooding Yoshiho Ikeda [email protected] Exploring the Influence of Different Arrangements of Semi-Watertight Spaces on Survivability of a Damaged Large Passenger Ship Riaan van’t Veer [email protected] Time-Based Survival Criteria for Passenger Ro-Ro Vessels Dracos Vassalos [email protected] Pressure-Correction Method and Its Applications for Time-Domain Flooding Simulation Pekka Ruponen [email protected] 10 CFD Applications to Ship Stability Applications of 3D Parallel SPH for Sloshing and Flooding Dracos Vassalos [email protected] Simulation of Wave Effect on Ship Hydrodynamics by RANSE Dracos Vassalos [email protected] A Combined Experimental and SPH Approach to Sloshing and Ship Roll Motions Luis Pérez-Rojas [email protected] 11 Design for Safety Design for Safety with Minimum Life-Cycle Cost Dracos Vassalos [email protected]

xviii List of Corresponding Authors

12 Stability of Naval Vessels Stability Criteria Evaluation and Performance Based Criteria Development for Damaged Naval Vessels Andrew J. Peters [email protected] A Naval Perspective on Ship Stability Arthur M. Reed [email protected] 13 Accident Investigations New Insights on the Sinking of MV Estonia Dracos Vassalos [email protected] Experimental Investigation on Capsizing and Sinking of a Cruising Yacht in Wind Naoya Umeda [email protected]

1 Stability Criteria

Review of Available Methods for Application to Second Level Vulnerability Criteria Christopher C. Bassler*, Vadim Belenky*, Gabriele Bulian**, Alberto Francescutto**, Kostas Spyrou***, Naoya Umeda**** *Naval Surface Warfare Center Carderock Division (NSWCCD) - David Taylor Model Basin, USA; **Trieste University; Italy; ***Technical University of Athens, Greece; ****Osaka University, Japan Abstract The International Maritime Organization (IMO) has begun work on the development of next generation intact stability criteria. These criteria are likely to consist of several levels: from simple to complex. The first levels are expected to contain vulnerability criteria and are generally intended to identify if a vessel is vulnerable to a particular mode of stability failure. These vulnerability criteria may consist of relatively simple formulations, which are expected to be quite conservative to compensate for their simplicity. This paper reviews methods which may be applicable to the second level of vulnerability assessment, when simple but physics-based approaches are used to assess the modes of stability failure, including pure-loss of stability, parametric roll, surf-riding, and dead-ship condition.

1 Introduction Current stability criteria have been used for several decades to determine the level of safety for both existing and novel ship designs. However, their current state is not representative of the level of our understanding about the mechanisms of dynamic stability failure of ships. Acknowledging this deficiency, the IMO has begun work on the development of next generation intact stability criteria to address problems related to dynamic phenomena and expand the applicability of criteria to current and future ship designs. The new generation criteria should include a range of fully dynamic aspects in their formulation. A thorough critical analysis of the existing situation at the beginning of the revision of the Intact Stability Code and a description of the present stateof-the-art can be found in a series of works by Francescutto (2004, 2007). The current framework of new generation intact stability criteria started to take shape at the 50th session of IMO Sub-committee on Stability and Load lines and M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_1, © Springer Science+Business Media B.V. 2011

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on Fishing Vessels Safety (SLF) with SLF-50/4/4 (2007), submitted by Japan, the Netherlands, and the United States and SLF-50/4/12 (2007), submitted by Italy. A significant contribution was also made in SLF-50.INF.2 (2007), submitted by Germany. These discussions resulted in a plan of development of new generation of criteria (see Annex 6 of SLF 50/WP.2). Following this plan, the intersessional Correspondence Group on Intact Stability developed the framework in a form of the working document that can be found in Annex 2 of SLF 51/4/1 (2008), submitted by Germany. Annex 3 of this paper contains the draft terminology agreed upon for the new criteria that was also developed by the Correspondence Group. The most current version of the framework document can be found in Annex 1 of the report of the Working Group on Intact Stability (SLF 51/WP.2). To facilitate these discussions, Belenky, et al. (2008) presented a broad review of issues related with development of new criteria, including the physical background, methodology, and available tools. It is the intention of the authors of this paper to provide a follow-up review, with a particular focus on methods for application to second level vulnerability criteria. The views and opinions expressed in this paper are solely and strictly those of the authors, mainly for facilitating wider and deeper discussion inside the research community. They do not necessarily reflect those of the delegations that involve the authors, the intersessional correspondence group on intact stability, or the working group on intact stability of the International Maritime Organization. The contents of this paper also do not necessarily indicate a consensus of opinion among the authors, but the authors agree on the need for further discussion of the contents.

2 Overview of New Generation of Intact Stability Criteria The major modes of stability failures were listed in section 1.2 of the 2008 IS Code, part A. They include:  Phenomena related to righting lever variations, such as parametric roll and pure loss of stability;  Resonant roll in dead ship condition defined by SOLAS regulation II-1/3  Broaching and manoeuvring-related problems in waves A multi-tier structure of new criteria has been identified from the ongoing discussion at IMO. Each of the four identified stability failure modes: pure-loss of stability, parametric roll, surf-riding and broaching, and dead ship conditions is evaluated using criteria with different levels of sophistication. The intention of the vulnerability criteria is to identify ships that may be vulnerable to dynamic stability failures, and are not sufficiently covered by existing regulation (SLF 51/WP.2, Annex 1).

Review of Available Methods for Application to Second Level Vulnerability Criteria

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A first level of criteria (vulnerability criteria) could be followed by other stability assessment methods with increasing levels of sophistication. The first level of the stability assessment is expected to be relatively simple, preferably geometry based if possible. The function of this first level is to distinguish conventional ships, which are obviously not vulnerable to stability failure. The second tier is expected to be more sophisticated and assess vulnerability to dynamic stability with enough confidence that the application of direct calculation could be justified. The expected complexity of the second level vulnerability criteria should consist of a spreadsheet-type calculation. An outline of the process is shown in Fig. 1. The methods will most likely be different for each stability failure mode. Ship Design

Stability Failure Mode

Vulnerability Criteria Level 1

For each mode

Fail Pass IMO IS Code

Performance-Based Criteria

Vulnerability Criteria Level 2

Fail Pass IMO IS Code

Fig. 1. The proposed assessment process for next generation intact stability criteria

3 Pure Loss of Stability Differences in the change of stability in waves, in comparison with calm water, were known to naval architects since late 1800s (Pollard & Dudebout, 1892; Krylov 1958). However, it was uncommon until the 1960s that attempts to calculate the change of stability in waves (Paulling, 1961) and evaluate it with a series of model tests (Nechaev, 1978; available in English from Belenky & Sevastianov, 2007) were made. As a distinct mode of stability failure, pure-loss of stability was identified during the model experiments in San-Francisco Bay (Paulling, et al., 1972, 1975; summary available from Kobylinski & Kastner, 2003). Accurate calculation of the change of stability in waves presents certain challenges, especially at high speed, as the pressure around the hull and waterplane shape are influenced by the nonlinear interaction between encountered, reflected, and radiated waves (Nechaev, 1989).

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The physical mechanism of pure-loss of stability is rather simple: if the stability is reduced for a sufficiently long time, a ship may capsize or attain large roll angle (see Fig, 2, taken from Belenky, et al., 2008).

200

Time history of roll and capsizing

Righting / Heeling Arm, m 1 Wave trough Magnitude of heeling Calm water moment 0.5 Wave crest

150 0 100 50

10 20 30 40 50

 deg 60

-0.5 Phase of heeling moment Phase of encounter wave

0

50

100

150

t, s

50

Fig. 2. Capsizing due to pure-loss of stability

This phenomenon is presently addressed, although in a very short and qualitative form, by the MSC.1/Circ.1228 (2007). MSC.1/Circ.1228 does not provide any ship-dependent information concerning the considered failure mode. The difficulty of developing criteria for pure-loss of stability is not limited to the calculation of the GZ curve in waves. A probabilistic characterization of the associated stability changes is also difficult, due to nonlinear nature of stability changes in waves. Boroday (1967, 1968) developed a method for the assessment of statistical characteristics for restoring moments in waves, followed by an energy balancebased method for evaluation of the probability of capsizing (Boroday & Netsevatev, 1969). Using energy balance methods for changing stability in waves was also the focus of Kuo & Vassalos (1986). Dunwoody (1989a, 1989b), Palmquist (1994), and Bulian & Francescutto (2006a) considered statistical characteristics of GM and other elements of the righting moment as a stochastic process. The alternative approach of the “effective wave” was proposed by Grim (1961). In this method, the length of the wave is equal to ship length— the wave is unmovable. However, its amplitude is a random process (Fig. 3).

Fig. 3. A concept of an effective wave

Review of Available Methods for Application to Second Level Vulnerability Criteria

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The amplitude of the effective wave is calculated in order to minimize the difference between the effective and encountered waves. Umeda, et al. (1993) performed comparative calculations and suggested the effective wave to be of satisfactory accuracy for engineering calculations. The effective wave approach was used in a number of works; of special interest are those with a “regulatory perspective.” Helas (1982) discussed the reduction of the righting arm in waves. To consider the duration where stability is reduced, the amplitude of the effective wave was averaged over a time sufficient to roll to a large angle, about 60% of natural roll period. As a result, the average effective wave amplitude becomes a function of Froude number. Analyzing results of the calculation performed for different vessels in different sea conditions, recommendations were formulated for a stability check in waves. Small ships and ships with “reduced stability” were meant to be checked using the GZ curve— evaluated for the effective wave with averaged amplitude against conventional calm water stability criteria. Umeda and Yamakoshi, (1993) used the effective wave to evaluate the probability of capsizing due to pure-loss of stability in short-crested irregular stern-quartering waves with wind. The capsizing itself was associated with departure from the time-dependent safe basin. The effect of initial conditions was taken into account using statistical correlation between roll and the effective wave. Instead of the Grim effective wave, Vermeer (1990) assumed waves to be a narrow-banded stochastic process. Capsizing is associated with the appearance of negative stability during the time duration, and it is sufficient for a ship to reach a large roll angle under the action of a quasi-static wind load. The probability of capsizing is considered as a ratio of encountered wave cycles, where the wave amplitude is capable of a significant and prolonged deterioration of stability. Themelis and Spyrou (2007) demonstrated the use of the concept of a “critical wave” to evaluate probability of capsizing. Because pure-loss of stability is a one wave event, it is straightforward to separate the dynamical problem from the probabilistic problem. First, the parameters that are relevant to the waves capable of causing pure-loss of stability are searched. Then, the probability of encountering such waves is assessed. To complete the probabilistic problem, the initial conditions are treated in probabilistic sense. Bulian, et al. (2008) combined the effective wave approach with upcrossing theory, where the phenomenon of pure-loss of stability is associated with an upcrossing event for the amplitude of the effective wave. Such an approach allows the time of exposure to be considered directly and produces a time-dependent probability of stability failure. An additional advantage of upcrossing theory is the possibility to characterize the “time above the level”, which in this context is the duration of decreased stability. The mean value of this time can be evaluated by a simple formula, but evaluation of other characteristics is more complex (Kramer and Leadbetter, 1967). In summary, it seems that the second level vulnerability criteria for pure-loss of stability are likely to be probabilistic, due to the very stochastic nature of the

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phenomenon. For criteria related to this mode of stability failure, the time duration while stability is reduced on a wave could be included. Methods useful for addressing pure-loss of stability may include, but are not limited to, the effective wave, the narrow-banded waves assumption, the critical wave approach, and upcrossing theory. As Monte-Carlo simulations may be too cumbersome for vulnerability criteria, at this time, simple analytical and semi-analytical models seem to be preferable.

4 Parametric Roll Parametric roll is the gradual amplification of roll amplitude caused by parametric resonance, due to periodic changes of stability in waves. These stability changes are essentially results of the same pressure and underwater geometry changes that were the cause in the pure-loss of stability mode. However, in contrast to pure-loss of stability, parametric roll is generated by a series of waves of certain frequency. Therefore, parametric roll cannot be considered as a one wave event (Figure 4). a)

20

, deg

0

100

200

300

400

500

100

200

300

400

500

t, s

-20 b)

, deg 200 150 100 50 t, s 0 -50

Fig. 4. Partial (a) and total (b) stability failure caused by parametric resonance

Research on parametric roll began in Germany in the 1930s (Paulling, 2007). In the 1950s, the study of parametric roll for a ship was continued by Paulling & Rosenberg (1959). Paulling, et al. (1972, 1975) observed parametric roll in following waves during model tests in San-Francisco Bay (summary available from Kobylinski & Kastner, 2003). Later it was discovered that this phenomenon could occur in head or near head seas as well (Burcher, 1990; France, et al., 2003).

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Parametrically excited roll motion is at present widely recognized as a dangerous phenomenon, where the cargo and the passengers and the crew may be at risk. Hull forms have evolved from those considered in the development of the 2008 IS Code, as MSC Res. 267(85)— basically the same ship types used 30-50 years ago for the definition of Res. A. 749(18) (2002). For certain ship types, present hull forms appear to have resulted in the occurrence of large amplitude rolling motions associated with parametric excitation. According to this evidence, there is a compelling need to incorporate appropriate stability requirements specific to parametric roll into the new generation intact stability criteria. The simplest model of parametric resonance is the Mathieu equation. It can describe the onset of instability in regular waves. Because the Mathieu equation has a linear stiffness term, it is incapable of predicting the amplitude of roll, and not applicable in irregular waves. Nevertheless, it was used by ABS (ABS, 2004; Shin, et al., 2004) to establish estimates for initial susceptibility to parametric excitation, which corresponds more with intended use for the first level vulnerability criteria. More complex 1(.5)-DOF models include nonlinearity in the time-dependent variable stiffness term. This nonlinearity leads not only to stabilization of roll motion, e.g. establishing finite roll amplitude, but also to a significant alteration of instability boundary in comparison of Ince-Strutt diagram— with unstable motions found outside of the Mathieu instability zone (Spyrou, 2005). This should cause concern about relying on the standard linear stability boundaries. Inclusion of additional degrees of freedom, primarily pitch and heave (Neves & Rodríguez, 2007) results in additional realism. These are especially important for parametric roll in head seas, where pitch and heave may be large. Spyrou (2000) included surging as well. These models, even with their low dimensional character, seem to give reasonable agreement with model tests (Bulian, 2006). Evaluation of the steady-state amplitude of parametric roll can be carried out using asymptotic methods, such as the method of multiple scales (Sanchez & Nayfeh, 1990). Approximate analytical expressions exist for the determination of the nonlinear roll response curve (e.g. ITTC, 2006). The potential of these methods is discussed in Spyrou et al. (2008). The introduction of irregular waves results in drastic changes in the way parametric roll develops. Large oscillations may not be persistent anymore: it may stay, but it may also be transitory (Figure 5). Such behavior has significant influence on the ergodicity of the process— the ability to obtain statistical characteristics from one realization if it is long-enough. Theoretically, the process remains ergodic, as the integral over its autocorrelation function still remains finite, but the time necessary for convergence may be very large. As a result, the term “practical non-ergodicity” is used (Bulian, et al., 2006; Hashimoto, et al., 2006).

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Roll Angle, deg

Roll Angle, deg

40 20 0 -20 -40

Time, s 0

200

400

600

800 1000 1200 1400

0

200

400

600

Time, s 800 1000 1200 1400

40 20 0 -20 -40

Fig. 5. Sample records of simulated parametric roll in long-crested seas (Shin, et al., 2004)

Distribution of parametric roll may be different from Gaussian. Hashimoto, et al. (2006) obtained distributions with large excess of kurtosis from model tests in long- and short-crested irregular seas (Fig. 6). a)

b)

Fig. 6. Distribution of roll angles during parametric resonance in (a) long-crested and (b) short crested seas

Roll damping has more influence on roll amplitude in irregular waves. The damping threshold defines if a wave group can contribute to parametric excitation and therefore, to results in an increase of the roll amplitude. Characteristics of wave groups, such as number, length, and the height of waves in the group, may also have an influence on how fast the roll amplitude increases. In a sense, a regular wave can be considered as an infinite wave group. Therefore, using regular waves could be too conservative for an effective vulnerability criterion, due to the transient nature of this phenomenon (SLF48/4/12). Focusing on the instability boundaries in irregular waves, Francescutto, et al., (2002) used the concept of “effective damping,” also mentioned in Dunwoody (1989a); that is meant to be subtracted from roll damping. Bulian, et al. (2004) used the Fokker-Planck equation and the method of multiple scales for the same purpose. They utilized Dunwoody’s spectral model of GM changes in waves (1989a, 1989b). Furthermore, Bulian (2006) made use of Grim’s effective wave

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concept. Despite overestimating the amplitude of roll, the boundary of instability was well identified. Application of the Markov processes for parametric roll in irregular waves was considered by Roberts (1982). To facilitate Markovian properties, (dependence only on the previous step), white noise must be the only random input of the dynamical system. Therefore, the forming filter, usually a linear oscillator, must be part of the dynamical system to ensure irregular waves have a realistic-like spectrum. These complexities allow the distribution of time before the process reaches a given boundary to be obtained— the first passage method. This method is widely used in the field of general reliability. Application of this approach to parametric roll was considered by Vidic-Perunovic and Jensen, (2009). Time-domain Monte-Carlo simulations (Brunswig, et al., 2006) are increasingly used as computers become more powerful and are integral part of any naval architect’s office. At the same time, caution has to be exercised when using MonteCarlo simulations, due to statistical uncertainty and other methodological challenges that may limit its use for second level vulnerability criteria. Nevertheless, having in mind the difficulties associated with prediction of amplitude for parametric resonance in irregular waves, Monte-Carlo method should not be dismissed. Themelis & Spyrou (2007) used explicit modelling of wave groups as a way to separate problems related to ship dynamics from the probabilistic problem. This method exploited the growth rate of parametric roll due to the encounter of a group, i.e. by solving the problem in the transient stage. Thereafter, the probability of encounter of a wave group can be calculated separately. In principle, this may allow enough simplification to make Monte-Carlo methods into feasible and robust tool for the second level vulnerability criteria for parametric roll. In summary, it seems that the second level vulnerability criteria for parametric roll may be related to the size of the instability area in irregular seas. The group wave approach seems to be promising, as a method to build a criterion based on amplitude.

5 Surf-Riding and Broaching Broaching is the loss of controllability of a ship in astern seas, which occurs despite maximum steering effort. Stability failure caused by broaching may be “partial” or “total”. It can appear as heeling during an uncontrollable, tight turn. This is believed to be intimately connected with the so called surf-riding behaviour. Moreover, it can manifest itself as the gradual, oscillatory build-up of yaw, and sometimes roll, amplitude. The dynamics of broaching is probably the most dynamically complex phenomenon of ship instability. Well known theoretical studies on broaching from the earlier period include Davidson (1948), Rydill (1959), Du Cane & Goodrich (1962), Wahab & Swaan (1964), Eda (1972); Renilson & Driscoll (1981), Motora, et al.(1981); and more recently, e.g. by Umeda &

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Renilson (1992), Ananiev & Loseva (1994), and Spyrou (1996, 1997, 1999). Broaching has also been studied with experimental methods. Tests with scaled radio-controlled physical models have taken place in large square ship model basins; e.g. Nicholson (1974), Fuwa (1982), De Kat & Thomas (1998). Surf-riding is a peculiar type of behaviour where the ship is suddenly captured and then carried forward by a single wave. It often works as predecessor of broaching and its fundamental dynamical aspects have been studied by Grim (1951), Ananiev (1966), Makov (1969), Kan (1990), Umeda (1990), Thomas & Renilson (1990), and Spyrou (1996). Key aspects of the phenomenon are outlined next. In a following or quartering sea, the ship motion pattern may depart from the ordinary periodic response. Figure 7 illustrates diagrammatically the changes in the geometry of steady surge motion, under the gradual increase of the nominal Froude number (Spyrou 1996). An environment of steep and long waves has been assumed. Surf-riding equilibria

Periodic surging

cos(2xG/

At wave trough

At wave crest a

xG Fncr1

b

c Fncr2

d

Fn

Fig. 7. Changing of surging/surf-riding behavior with increasing nominal Froude number

For low Fn, there is only a harmonic periodic response (plane (a)) at the encounter frequency. However, as the speed approaches the wave celerity, the response becomes asymmetric (plane (c)). The ship stays longer in the crest region and passes quickly from the trough. In parallel, an alternative stationary behavior begins at some instance to coexist. This results from the fact that the resistance force which opposes the forward motion of the ship may be balanced by the sum of the thrust produced by the propeller and the wave force along the ship’s longitudinal axis. This is surf-riding and the main feature is that the ship is forced to advance with a constant speed equal to the wave celerity. It has been shown that surf-riding states appear in pairs: one nearer to wave crest and the other nearer to trough. The one nearer to the crest is unstable in the surge direction. Stable surfriding can be realised only in the vicinity of the trough and, in fact, only if sufficient rudder control has been applied.

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Surf-riding is characterised by two speed thresholds: the first is where the balance of forces becomes possible (plane(b)). The second threshold indicates its complete dominance in phase-space, with sudden disappearance of the periodic motion. It has been pointed out that, this disappearance is result of a global bifurcation phenomenon, known in the nonlinear dynamics literature as homoclinic saddle connection (Spyrou, 1996). It occurs when a periodic orbit collides in statespace with an unstable equilibrium— the surf-riding state near to the wave crest. A dangerous transition towards some, possibly distant, undesirable state is the practical consequence. To be able to show the steady motion of a ship overtaken by waves as a closed orbit, one should opt to plot orbits on a cylindrical phase-plane. On the ordinary flat phase-plane the bifurcation would appear as a connection of different saddles (“heteroclinic connection”, see Figures 8 and 9 based on Makov). This is characterised by the tautochronous touching of the “overtaking wave” orbits, running from infinity to minus infinity, with the series of unstable equilibria at crests. This has been confirmed numerically by Umeda (1999). Therefore, no matter what terminology is used, there is consensus and clear understanding, rooted in dynamics, about the nature of phenomena. 20

.

G, m/s

10 Surf-riding

Surf-riding

300

200

Surging

100

0

100

G, m

Surging 10

Fig. 8. Phase plane with surging and surf-riding, Fncr1
The initial attraction into surf-riding, caused by the unstable surf-riding equilibrium near the crest, should be followed by capture of the ship at the stable surf-riding equilibrium near the trough. However, this is a violent transition and, if the rudder control is insufficient, it will end in broaching. The effectiveness of rudder control in relation to the inception of this behaviour, and also characteristic simulations, can be found in the literature (Spyrou, 1996, 1999).

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.

20

G, m/s Surf-riding

10

Surf-riding

G, m 300

200

100

0

100

Surf-riding 10

Fig. 9. Phase plane with surf-riding only Fn>Fncr2

From the above, one sees that, susceptibility to this type of broaching could be assessed by targeting three sequential events: the condition of attraction to surfriding; the condition defining an inability to stay in stable surf-riding, thus creating an escape; and the condition of reaching dangerous heeling during this escape— manifested as tight turn. For the first condition, we already have deterministic (analytical and numerical) criteria based on Melnikov and other methods (Ananiev, 1966; Kan, 1990; Spyrou, 2006). For the second, criteria have been discussed, linking ship manoeuvring indices with control gains (Spyrou, 1996, 1998). The third condition can be relatively easily expressed as the balance of the roll restoring versus the moment of the centrifugal force that produces the turn. A proper probabilistic expression of these conditions is still wanting. However, the critical wave approach may provide a very practical way of dealing with this (Themelis & Spyrou, 2007). Efforts in similar spirit were reported by Umeda (1990) and Umeda, et al. (2007). In reviews of broaching incidents, those that had happened at low to moderate speeds (i.e. far below the wave celerity of long waves) are usually distinguished from the more classical high-speed events. These incidents could have some special importance, because they seem to be relevant also to vessels of larger size, which otherwise are unlikely to broach through surf-riding. Let’s consider now the nature of such a broaching mechanism, which we will call “direct broaching” since the ship is not required to go through surf-riding. It has been shown that, a fold bifurcation is possible to arise for a critical value of the commanded heading, This phenomenon can have a serious influence on the horizontal plane dynamics, creating a stable sub-harmonic yaw (Fig. 10). Further increase of the commanded heading was found to cause a rapid increase of the amplitude of yaw oscillation, leading shortly to a turn backwards of the steady yaw response curve and, inevitably, a sudden and dangerous jump to resonant yaw (Spyrou, 1997). The transient behaviour looks like a growing oscillatory yaw which corresponds closely to what has been described as cumulative-type broaching.

Review of Available Methods for Application to Second Level Vulnerability Criteria

One period motion

One period motion

Fold bifurcation

Rudder angle

Amplitude of periodic motion

One period Stable submotion harmonic yaw

15

Yaw angle

C commanded heading

Flip bifurcation

Stable subharmonic yaw

Fig. 10. Direct broaching

Instability could be an intrinsic feature of the yaw motion of a ship overtaken by very large waves. It has been pointed out that, starting from Nomoto’s equation, the equation of yaw in following waves becomes Mathieu-type (in Spyrou (1997) the derivation and discussion on this and its connection with the above scenario was presented). The amplitude of the parametric forcing term is the ratio of the wave yaw excitation to the hull’s own static gain— maneuvering index K. The damping ratio in this equation receives large values, unless there is no differential control. Due to the large damping, the internal forcing that is required in order to produce instability is much higher (steeper waves) than that at the near zero encounter frequency (i.e. for the mechanism of broaching through surf-riding). Simple criteria connecting the hull with rudder control have accrued from this formulation and could be directly in the vulnerability criteria. A probabilistic formulation has not yet been attempted.

6 Dead Ship Condition Both the term and the concept of dead ship conditions took its root during the steamer era. When a ship looses power, it becomes completely uncontrollable— a significant disadvantage in comparison with ship equipped with sails. The worst possible scenario is a turn into beam seas, where the ship is then subjected to a resonant roll combined with gusty wind. Traditional ship architecture, with a superstructure amidships, made this scenario quite possible. Significant changes in marine technology have led to more reliable engines and a variety of architectural

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types. Nevertheless the dead ship condition— zero speed, beam seas, resonant roll, and wind action, maintains its importance. The present Intact Stability Code (MSC.267(85), 2008), as well as its predecessor (IMO Res. A749(18), 2002), contains the “Severe Wind and Rolling Criterion (Weather Criterion).” This criterion contains a simple mathematical model for ship motion under the action of beam wind and waves. However, some parameters of this model are based on empirical data. Therefore, it may not be completely adequate for unconventional vessels. Thus MSC.1/Circ.1200 (2006) and MSC.1/Circ. 1227 (2007) were implemented, to enable alternative methodologies to be used for the assessment of the Weather Criterion on experimental basis. The problem of capsizing represents the ultimate nonlinearity: the dynamical system transits to motion about another stable equilibrium. When the wave excitation amplitude approaches a critical value, a number of nonlinear phenomena take place. The boundary of the safe basin becomes fractal (Kan & Taguchi, 1991) as a result of the self-crossing branches of an invariant manifold (Falzarano, et al., 1992). Considering the deterioration of the safe basin (Figure 11), Rainey and Thompson (1991) proposed a transient capsize diagram. It uses the dramatic change of the measure of integrity of the safe basin as a boundary value. FINAL CRISIS

FRACTAL BOUNDARY

SMOOTH BOUNDARY 1

d a

Area of SAFE BASIN

b

0

c

e

f

0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized amplitude of excitation

Fig. 11. Deterioration of safe basin with increasing of wave amplitude

Another obvious way to improve criterion is to consider more realistic, e.g. stochastic, models for the environment. This does not necessarily means that the new criterion must be probabilistic. The improvement will be made by adjusting parameters of deterministic criterion to reach similar level of safety for all the vessels which meet it. Studies of probabilistic methods for these criteria were carried out starting in the 1950s (Kato, et al., 1957). As the conventional weather criterion uses an

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energy balance approach, an attempt to consider it from probabilistic point of view seems logical (Dudziak & Buczkowski 1978; a brief description is available from Belenky & Sevastianov, 2007). Similar ideas are behind the Blume criterion (Blume, 1979, 1987), which uses a measure of the remaining area under the GZ curve. Bulian & Francescutto (2006b) and Bulian, et al. (2008) developed a method where the righting arm is locally linearized at the equilibrium, due to the action of a steady beam wind. The probability of large roll is calculated using an “equivalent area” to account for the nonlinearities of the righting arm. The piecewise linear method (Belenky, 1993; Paroka & Umeda, 2006) separates probability of capsizing into two less challenging problems: upcrossings through maximum of the GZ curve and capsizing if such upcrossing has occurred (Figure 12). The probability of upcrossing is well defined in upcrossing theory. While the probability of capsizing after upcrossing can be found using the probability of roll rate at upcrossing that lead to capsizing; this is a linear problem in the simplest case. Statistical linearization can be used to evaluate parameters of the piecewise linear model. Simplicity and a direct relation with time of exposure are among the strengths of the piecewise linear method. 

Upcrossing not leading to capsizing Capsized equilibrium

Threshold

Upcrossing leading to capsizing t

Fig. 12. Separation principle in a piecewise linear method

In principle, a critical wave approach can be used for dead ship conditions as well. However, the outcome of capsizing vs. non-capsizing strongly depends on initial conditions. Therefore, a critical wave group approach similar to Themelis and Spyrou (2007) may be also considered as a candidate for vulnerability criteria.

7 The Choice of Environmental Conditions If vulnerability criterion is deterministic and based on a regular wave assumption, the wave is characterized with height and length. These characteristics must then be related to a corresponding sea state and further on with a safety level. This formulation is not new— this approach is used for evaluation of extreme loads. A typical scheme for calculation of extreme loads is based on long-term assumption, so a number of sea states are considered. An operational profile is

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usually assumed based on existing experience; it includes the fraction of time that a ship is expected to spend in each sea state. Short-term probability of exceedance is calculated for each sea state; then the formula for the total probability is used to determine the life-time probability of exceedance of the given level. This level is typically associated with significant wave height and zero-crossing, or mean period. These data are then directly used to define a regular wave that is expected to cause extreme loads. Similar ideas underlie the reference regular wave in the Weather Criterion in the 2008 IS Code, which was determined from the significant wave steepness, based on the work of Sverdrup and Munk (1947). With all of the similarities, a direct transfer of methods used for extreme loads to stability applications may be difficult. Structural failure is associated with exceeding certain stress levels, actual physical failure is not considered in Naval Architecture. Stresses do not have their own inertia, they follow the load without time lag. The same can be observed about the loads, they occur simultaneously with a wave. They do not depend on initial conditions; even in a nonlinear formulation and do not have multiple responses for the given position of a ship on a wave. Nevertheless, these motion nonlinearities affect only short-term probability, while the scheme as a whole may be applicable. Determining the equivalence between regular and irregular waves represents a problem by itself. The use of significant wave height or steepness has been a conventional assumption, but it does not have a theoretical background behind it. The general problem of equivalency between regular and irregular roll motions was considered by Gerasimov (1979; a brief description in English is available from Belenky & Sevastianov, 2007) in the context of statistical linearization with energy conservation, resulting in an energy statistical linearization. The idea of a regular wave as some sort of equivalent for sea states is attractive because of its simplicity. However, the physics of some stability failures may be sufficiently different in regular and irregular waves. As it was mentioned in the section regarding parametric roll, a regular wave is an infinitely long wave train. Roll damping in this case has a very small influence on the response amplitude, while in irregular seas damping has a significant influence on the variance of response. Therefore, the regular equivalent of irregular waves cannot be applied universally. The application of different environmental models is possible for different stability modes. Surf-riding/broaching is an example of a one-wave event where the concept of a critical wave may fit well. If some of vulnerability criteria are made probabilistic, the next important choice is time scale: long-term or short term. Here, short-term refers to a time interval where an assumption of quasi-stationarity can be applied. Usually this is an interval of three to six hours, where changes of weather can be neglected. Long-term consideration covers a larger interval: a season, a year, or the life-time of a vessel. The short-term description of the environment is simpler and can be characterized with just with one sea state or spectrum. However, if chosen for vulnerability criteria, justification will be required as to why a particular sea must be used. This

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choice is important because sea states which are too severe may make the criterion too conservative and diminish its value. Therefore, special research is needed in order to choose the sea state “equivalent” or “representative” for a ship operational profile. This may naturally result in ship-specific sea-state to use for assessment. An alternative to the selection of a limited set of environmental conditions may be the use of long-term statistics considering all the combinations of weather parameters available from scatter diagrams, or appropriate analytical parametric models (see, e.g. Johannessen, et al., 2002).

8 Summary and Concluding Comments Vulnerability to pure-loss of stability is caused by prolonged exposure to decreased stability on a wave crest, and is likely to be characterized by probabilistic criteria. Methods for a first attempt for vulnerability criteria include the effective wave, a narrow-band or envelope presentation, the critical wave approach, and upcrossing theory. Vulnerability criteria for parametric roll would be more useful if defined in irregular seas. This criterion may be based on size of instability area, while the wave group approach seems to be a good candidate for amplitude-based criterion. Vulnerability to broaching through surf-riding may be judged by the vulnerability to surf-riding. The critical wave approach seems to be the best fit; however, other methods are not excluded. Vulnerability criterion for direct broaching is likely to be deterministic. Vulnerability to stability failure in dead ship conditions may be judged using a variety of methods, as it was the main area of application for both probabilistic and deterministic methods. The modified weather criterion, piecewise linear method, and critical wave or wave group approach may be among the first to be considered. A rational choice of environmental conditions for vulnerability criteria is at least as important as the criteria. An unrealistic environmental condition may lead to incorrect results, even if the criteria are technically correct. However, several possibilities do exist: a regular wave equivalent for life-time risk, a short-term sea state deemed “representative” for a specific ship operational profile, and a longterm approach using a scatter diagram for a representative part of the World Ocean.

9 Acknowledgments The authors are grateful for the funding support from Mr. James Webster (NAVSEA) for this work. The authors are grateful for the guidance and encouragement they received from Mr. William Peters (USCG). The authors also appreciate Dr. Arthur Reed and Mr. Terrence Applebee (NSWCCD) for their help and support.

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References ABS (2004) American bureau of shipping guide for the assessment of parametric roll resonance in the design of container carriers. Houston, Texas Ananiev D M (1966) On surf-riding in following seas. Trans of Krylov Soc 73. Russian Ananiev D M, Loseva L (1994) Vessel’s heeling and stability in the regime of manoeuvring and broaching in following seas. Proc. 5th Intl Conf on Stab of Ships and Ocean Veh. Melbourne, Florida Belenky VL (1993) A capsizing probability computation method. J. Ship Res 37:200-207 Belenky VL, Sevastianov NB (2007) Stability and safety of ships - risk of capsizing (second edition). The Soc of Nav Archit and Mar Eng (SNAME). Jersey City, NJ Belenky V, de Kat JO, Umeda N (2008) Toward performance-based criteria for intact stability. Mar Technol 45:101-123 Blume P (1979) Experimentelle bestimmung von koeffizienten der wirksamen rolldaempfung und ihre anwendung zur abschaetzung extremer rollwinkel. Schiffstechnik, 26 Blume P (1987) Development of new stability criteria for modern dry cargo vessels. Proc. 5th Intl. Symp. on Pract Des of Ships and Other Float Struct. Trondheim, Norway Boroday IK (1967) Statistical characteristics of stability and the probability of capsize of a ship running on any course in irregular seas. Doc of the USSR Expert of the IMCO, Working Gr on Stab of Fish Vessels, IMO. London Boroday IK (1968) Statistical characteristics and probability of capsizing of ship heading with arbitrary course in irregular seas. Trans. of russian register of shipping— theoretical and practical problems of stability and survivability. Transp Publ. Moscow, Russian Boroday IK, Netsvetaev YA, (1969) Ship motion in a seaway. Sudostroenie. Leningrad, Russian Brunswig J, Pereira R, Kim D (2006) Validation of parametric roll motion predictions for a modern ship containership design Proc. 9th Intl. Conf. on the Stab of Ships and Ocean Veh. Rio de Janeiro, Brazil 157-168 Bulian G (2006) Development of analytical nonlinear models for parametric roll and hydrostatic restoring variations in regular and irregular waves. PhD Thesis. Univ of Trieste Bulian G, Francescutto A (2006) On the effect of stochastic variations of restoring moment in long-crested irregular longitudinal sea. Proc 9th Int Conf. on Stab of Ships and Ocean Veh. Rio de Janeiro, Brazil. 1:131-146 Bulian G, Francescutto, A Lugni C (2004) On the nonlinear modeling of parametric rolling in regular and irregular waves. Int Shipbuild Prog. 51 2/3:173-203 Bulian G, Francescutto, A Lugni C (2006) Theoretical, numerical and experimental study on the problem of ergodicity and ‘practical ergodicity’ with an application to parametric roll in longitudinal long crested irregular sea. Ocean Eng, 33:1007-1043 Bulian G, Francescutto A, Maccari A (2008) Possible simplified mathematical models for roll motion in the development of performance-based intact stability criteria– extended and revised version. University of Trieste Burcher RK (1990) Experiments into the capsize of ships in head seas. Proc 4th Int Conf on Stab of Ships and Ocean Veh, Naples, 82-89 Davidson KSM (1948) A note on the steering of ships in following seas. Proc 7th Int Congr of Appl Mech. London, 554-556 DeKat JO, Thomas WL (1998) Extreme rolling, broaching and capsizing— model tests for validation of numerical ship motion predictions. Proc 22nd Symp on Nav Hydrodyn. Washington, DC DuCane P, Goodrich GJ (1962) The following sea, broaching and surging. Trans. RINA 104 Dunwoody AB (1989a) Roll of a ship in Astern Seas—metacentric height spectra. J Ship Res, 33 3:221-228 Dunwoody AB (1989b) Roll of a ship in Astern Seas –response to GM fluctuations. J Ship Res, 33 4:284-290

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Dudziak J, Buczkowski A (1978) probability of ship capsizing under the action of the beam wind and sea as a background of stability criteria. Pol Regist of Ships. Prace StudialnoRoswojowe, Zeszyt 13, Gdansk Eda H (1972) Directional stability and control of ships in waves. J Ship Res 16 3 Falzarano JM, Shaw SW, Troesch AW (1992) Application of global methods for analyzing dynamical systems to ship rolling motion and capsizing Int J of Bifurcation and Chaos, 2 1:101-115 France WG, Levandou M, Treakle TW, Paulling JR, Michel RK, Moore C (2003) An investigation of head seas parametric rolling and its influence on container lashing systems. Mar Technol, 40 1:1-19 Francescutto A (2004) Intact ship stability: the way ahead. Mar Technol, 41:31–37 Francescutto A (2007) Intact stability of ships– recent developments and trends. Proc 10th Int Symp on Pract Des of Ships and Other Float Struct PRADS2007. Houston 1:487-496 Francescutto A., Bulian, Lugni C. (2002) Nonlinear and stochastic aspects of parametric rolling modelling. Proc of 6th Intl. Ship Stab Workshop, Webb Institute, N.Y. also available from Mar Technol, 41 2:74-81 Fuwa T, Sugai K, Yoshino T, Yamamoto T (1982) An experimental study on broaching of a small high speed boat. Papers of Ship Res Inst 66 Gerasimov AV (1979) Energy-statistical theory of nonlinear irregular ship motion. Sudostroenie. Leningrad Russian Grim O (1951) Das schiff in von achtern auflaufender see. Jahrb Schiffbautech Ges 4: 264-287 Grim O (1961) Beitrag zu dem problem der sicherheit des schiffes im seegang schiff und hafen 6:490-497 Helas G (1982) Intact stability of ships in following waves. Proc of Int Conf on the Stab of Ships and Ocean Veh. Tokyo, Japan International Towing Tank Conference - ITTC (2006) Predicting the occurrence and magnitude of parametric rolling— recommended procedure 7.5-02-07-04.3. Rev.01. Effective Date: 2006 Johannessen K, Meling TS, Haver S (2002) Joint distribution for wind and waves in the northern north sea. Int J of Offshore and Polar Eng 12 1 :1-8 Hashimoto H, Umeda N, Matsuda A, Nakamura S (2006) Experimental and numerical studies on parametric roll of a post-panamax container ship in irregular waves. Proc 9th Int Conf on Stab of Ships and Ocean Veh. Rio de Janeiro, Brazil 1:181-190 Kan M (1990) Surging of large amplitude and surf-riding of ships in following seas. Sel Pap Naval Archit Ocean Eng. Soc Nav Archit. Japan 28 Kan M, Taguchi H (1991) Chaos and fractal in capsizing of a ship. Proc of Int Symp on Hydroand Aerodyn in Mar Eng HADMAR’91. Varna, Bulgaria 1:81-88 Kato H, Motora S, Ishikawa K, (1957) On the rolling of a ship in irregular wind and wave. Proc Symp on Beh of Ships in a Seaway, September. Wageningen 1:43-58 Kuo C, Vassalos D, Alexander JG (1986) Incorporating theoretical advances in usable ship stability criteria. RINA Int conf of SAFESHIP project. Ship Stab and Saf. London Kobylinski LK, Kastner S (2003) Stability and safety of ships: regulation and operation. Elsevier, Amsterdam Kramer H, Leadbetter MR (1967) Stationary and related stochastic processes. John Wiley. NY Krylov AN (1958) Selected papers. USSR Acad of Science. Moscow Makov YL (1969) Some results of theoretical analysis of surf-riding in following seas. Trans of Krylov Soc “Maneuverability and Seakeeping of Ships”. Leningrad 126:124-128. Russian Motora S, Fujino M, Koyonagi M, Ishida S, Shimada K, Maki T (1981) A consideration on the mechanism of occurrence of broaching-to phenomena. Trans JSNA 150 MSC.1/Circ.1200, 2006, “Interim guidelines for alternative assessment of the weather criterion. 24 May IMO. London UK MSC.1/Circ.1227, 2007, “Explanatory notes to the interim guidelines for alternative assessment of the weather criterion” 11 January IMO. London UK

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MSC.1/Circ.1228, 2007, “Revised guidance to the master for avoiding dangerous situations in adverse weather and sea conditions” 11 January, IMO. London UK MSC. Res. .267(85), 2008, “adoption of the international code on intact stability, 2008 (2008 IS Code)” 4 Dec IMO. London UK Nechaev YI (1978) Stability of ships in following seas. Sudostr. Leningrad Russian Nechaev YI (1989) Modelling of ship stability in waves. Sudostr. Leningrad Russian Neves MAS, Rodríguez CA (2007) Nonlinear aspects of coupled parametric rolling in head seas. Proc 10th Int Symp on Prac Des of Ships and Other Float Struct. Houston Nicholson K (1974) Some parametric model experiments to investigate broaching-to. Int Symp Dynam Mar Veh and Struct in Waves. London. Palmquist M (1994) On the statistical properties of the metacentric height of ships in following seas. Proc 5th Int Conf on Stab of Ships and Ocean Veh. Melbourne Florida Paroka D, Umeda N (2006) Capsizing probability prediction for a large passenger ship in irregular beam wind and waves: comparison of analytical and numerical methods. J. Ship Res 50:371–377 Paulling JR, Rosenberg RM (1959) On unstable ship motions resulting from nonlinear coupling. J Ship Res 3(1):36-46 Paulling JR (1961) The transverse stability of a ship in a longitudinal seaway j ship research, 4 4 :37-49 Paulling JR, Kastner S, Schaffran S (1972) Experimental studies of capsizing of intact ships in heavy seas. U.S. Coast Guard, Tech Rep (Also IMO Doc. STAB/7, 1973). Paulling JR, Oakley OH, Wood PD (1975) Ship capsizing in heavy seas: the correlation of theory and experiments. Proc of 1st Int Conf on Stab of Ships and Ocean Veh. Glasgow Paulling JR (2007) On parametric rolling of ships. Proc 10th Int Sym on Pract Des of Ships and Other Float Struct. Houston Pollard J, Dudebout A (1892), Theorie du navire. 3. Paris Rainey RCT, Thompson JMT (1991) The transient capsize diagram— a new method of quantifying stability in waves. J Ship Res 35 1 :58-62 Renilson MR, Driscoll A (1982) Broaching – an investigation into the loss of directional control in severe following seas. Trans RINA 124 Res. A. 749(18) as amended by Res. MSC.75(69), 2002, “Code on intact stability for all types of ships covered by imo instruments” IMO. London UK Roberts JB (1982) Effect of parametric excitation on ship rolling motion in random waves. J Ship Res 26:246-253 Rydill L J (1959) A linear theory for the steered motion of ships in waves. Trans RINA 101 Sanchez NE, Nayfeh AH (1990) Nonlinear rolling motions of ships in longitudinal waves. Int Shipbuild Prog 37 411:247-272 Shin YS, Belenky VL, Paulling JR, Weems KM, Lin WM (2004) Criteria for parametric roll of large containerships in longitudinal seas trans. SNAME 112 SLF48/4/12 (2005) On the development of performance-based criteria for ship stability in longitudinal waves. Submitt by Italy, IMO, London SLF50/4/4 (2007) Framework for the development of new generation criteria for intact stability. Submitt by Japan, Netherlands, and USA, IMO, London SLF50/4/12 (2007) Comments on SLF 50/4/4. Submitt by Italy, IMO, London SLF50/INF.2 (2007) Proposal on additional intact stability regulations. Submitt by Germany, IMO, London SLF50/WP.2 (2007) Revision of the intact stability code - report of the working group (part I). IMO, London SLF51/WP.2 (2008) Revision of the intact stability code - report of the working group (part I). IMO, London SLF51/4/1 (2008) Report of the intersessional correspondence group on intact stability. Submitt by Germany, IMO, London

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Spyrou KJ (1996) Dynamic instability in quartering seas: The behaviour of a ship during broaching J. Ship Res 40 1 :46-59 Spyrou KJ (1997) Dynamic instability in quartering seas-Part III: Nonlinear effects on periodic motions. J Ship Res 41 3:210-223 Spyrou KJ (1999) Annals of Marie–Curie fellowships, edited by the European commission. http://www.mariecurie.org/annals/volume1/spyrou.pdf Spyrou KJ (2000) On the parametric rolling of ships in a following sea under simultaneous nonlinear periodic surging. Phil Trans R Soc London, A 358:1813-1834 Spyrou KJ (2005) Design criteria for parametric rolling. Ocean Eng Int, Canada 9 1:11-27 Spyrou KJ (2006) Asymmetric surging of ships in following seas and its repercussions for safety. Nonlinear Dyn, 43:149-172 Spyrou KJ, Tigkas I, Scanferla G, Themelis N (2008) Problems and capabilities in the assessment of parametric rolling. Proc 10th Int Ship Stab Workshop. Daejeon, Korea 47-55 Sverdrup HU, Munk WH (1947) Wind, sea and swell, theory of relations for forecasting. Hydrogr Off Publ 601 Themelis N, Spyrou KJ (2007) Probabilistic assessment of ship stability trans. SNAME, 115:181-206 Thomas G, Renilson M (1992) Surf-riding and loss of control of fishing vessels in severe following seas. Trans RINA 134:21-32 Umeda N (1990) Probabilistic study on surf-riding of a ship in irregular following seas. Proc of 4th Int Conf onn Stab of Ships and Ocean Veh. Naples 336-343 Umeda N, Renilson MR (1992) Broaching— A dynamic analysis of yaw behavior of a vessel in a following sea. In Wilson PA (ed) Maneuvering and Control of Mar Craft, Comput Mech Publ. Southampton 533-543 Umeda N, Yamakoshi Y (1993) Probability of ship capsizing due to pure loss of stability in quartering seas. Nav Archit and Ocean Eng: Sel Pap Soc of Naval Arch of Japan 30:73-85 Umeda N, Shuto M, Maki A (2007) Theoretical prediction of broaching probability for a ship in irregular astern seas. Proc 9th Int Ship Stab Workshop. Hamburg Umeda N, Hori M, Hashimoto H (2007) Theoretical prediction of broaching in the light of local and global bifurcation analysis. Int Shipbuild Prog 54 4:269-281 Vermeer H (1990) Loss of stability of ships in following waves in relation to their design characteristics. Proc of 4th Int Conf on Stab of Ships and Ocean Veh. Naples Vidic-Perunovic J, Jensen JJ (2009) Estimation of parametric rolling of ships – comparison of different probabilistic methods. Proc 2nd Int Conf on Mar Struct. Lisbon, Portugal Wahab R, Swaan WA (1964) Course-keeping and broaching of ships in following seas. J Ship Res 7 4

A Basis for Developing a Rational Alternative to the Weather Criterion: Problems and Capabilities K.J. Spyrou School of Naval Architecture and Marine Engineering, National Technical University of Athens, 9 Iroon Polytechneiou, Zographos, Athens 15780, Greece

Abstract

The feasibility of developing a practical ship dynamic stability criterion based on nonlinear dynamical systems’ theory is explored. The concept of “engineering integrity” and Melnikov’s method are the pillars of the current effort which can provide a rational connection between the critical for capsize wind/wave environment, the damping and the restoring characteristics of a ship. Some discussion about the dynamical basis of the weather criterion is given first before describing the basic theory of the new method. Fundamental studies are then carried out in order to ensure the validity, the potential and the practicality of the proposed approach for stability assessment of ships in a wider scale.

1 Introduction The weather criterion adopted as Resolution A.562 by IMO’s Assembly in 1985 was a leap beyond the “statistical approach” of the earlier Rahola-type general intact ship stability criteria of 1969 where the safety limits were based empirically on GZ characteristics of vessels lost even a 100 years ago (IMO 1993). Despite the criticism the weather criterion has a rational basis in the sense that there is some account of ship roll dynamics integrated within the stability assessment process. Nonetheless this analysis has a simplified character and in the light of recent research advances on the one hand and new trends in design on the other, it seems that the time has come for looking more seriously into the potential of alternative approaches. It is no surprise that at IMO the discussion about the development of new criteria has been re-opened. In the current paper a few steps are taken towards clarifying whether the concept of “engineering integrity” of dynamical systems could be used for setting up an improved and “general purpose” stability criterion (Thompson 1997). This corresponds basically to implementing a M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_2, © Springer Science+Business Media B.V. 2011

25

26 K.J. Spyrou

geometrical approach addressing global behavior and going beyond the ordinary simulation (see for example Falzarano et al 1992). The principal aim is to characterise system robustness to critical environmental excitations. Whilst these ideas have attracted the interest of researchers in the field of ship stability in most cases they have relied on simple generic forms of roll restoring. Hence the question whether such an approach could be used for practical stability assessment remains still unanswered.

2 Basics of the Weather Criterion The IMO version of the weather criterion follows closely Yamagata’s “Standard of stability adopted in Japan” which was enacted as far back as 1957 (Yamagata 1959) while the basic idea had appeared in Watanabe (1938) or perhaps even earlier. The ship is assumed to obtain a stationary angle of heel θ0 due to side wind loading represented by a lever lw1 which is not dependent on the heel angle and is the result of a 26 m/s wind. “Around” this angle the ship is assumed to perform due to side wave action resonant rolling motion as a result of which it reaches momentarily on the weather side a maximum angle θ1 (Fig. 1). As at this position the ship is most vulnerable to excitations from the weather-side it is further assumed that it is acted upon by a gust wind represented by a lever lw2 = 1.5 lw1. This is translated into a 1.5  1.2247 increase of the experienced wind velocity assumed to affect the ship for a short period of time but at least equal with half natural period under the assumption of resonance. The choice of wind corresponds to extreme storm situation. In fact the criterion corresponds to a kind of average between the centre of a typhoon where the wind is very strong but the rolling is assumed that it is not so violent due to the highly irregular nature of the sea; and the ensuing situation which prevails right after the centre of the typhoon has moved away where the wind velocity is lower but the rolling becomes more intense as the waves obtain a more regular form. As pointed out by Rachmanin in his discussion of Vassalos (1986) the criterion is based on a roll amplitude with 2% probability of exceedance which perhaps corresponds to Yamagata’s choice of roll amplitude θ1 as 70% of the resonant roll amplitude in regular waves. The requirement for stability is formulated as follows: should the ship roll freely from the off-equilibrium position with zero angular velocity the limiting angle θ2 to the lee-side should not be exceeded during the ensuing half-cycle. This limiting angle is either the one where significant openings are down-flooded the vanishing angle θν or the angle of 50 degrees which can be assumed as an explicit safety limit whichever of the three is the lowest. Expressed as an energy balance: the work done by the wind excitation as the ship rolls from the wind-side to the leeside should not exceed the potential energy at the limiting angle θ2.

A Basis for Developing a Rational Alternative to the Weather Criterion

Restoring and heeling lever

a

b lw2

lw1 θ2

0 θ1

27

θc

θ

θ0 Fig. 1 The IMO weather criterion

θ2

θ

Fig. 2 The structure of the weather criterion

A number of extra comments pertain to the modeling of system dynamics: although the input of energy from the waves is taken into account for the calculation of the attained angle θ 1, it is not considered during the final half-cycle and the ship is like being released in still-water from the angle θ1 (Fig. 2). The only energy balance performed concerns potential energies in the initial and final positions. An advantage of the criterion is that the damping is somehow present in the calculation of the resonant roll angle θ 1 both in terms of hull dimensions, form

28 K.J. Spyrou

and fittings. Also, the fact that the energy dissipated through damping during the half-roll is not accounted is not so important because it does not reduce seriously the maximum attained angle and in any case the result is conservative. However, the damping function is taken as a pure quadratic which is today an unnecessary simplification. Finally, as in most critical cases the nonlinear part of the restoring curve “participates” in the dynamics, it appears unreasonable this effect to be left unaccounted in the calculation of θ1 (the only nonlinearity considered concerns the damping function).

c GZC 0

GZmax

θ θdownflooding

25 deg Fig. 3 The naval version of the weather criterion

In 1962 Sarchin & Goldberg presented the “naval version” of the weather criterion which whilst more stringent than the above it adheres to the same principle with only few minor differences. This work is the basis for the stability standards applied from most western Navies nowadays as evidenced by documents such as N.E.S 109 (2000) of the British Navy, DTS 079-1 of the U.S. Navy, NAV-04-A013 of the Italian etc. For ocean-going naval vessels the wind speed is assumed to be 90 knots and it is varying with the square of the heel angle (seemingly an improvement over the straight-line shown in Fig. 1; yet perhaps an equally crude approximation of reality, see the discussion of Prohaska in Yamagata 1959). The amplitude of resonant roll due to beam waves is prescribed to 25 degrees which means that the important connection with the roll damping characteristics of the ship that is found in IMO’s and Yamagata’s criterion is lost. Here however is required that the equilibrium angle (point c in Fig.3) does not exceed 20 degrees and also that the GZ at that point is less than 60% of the maximum GZ. The energy requirement of the criterion prescribes that a substantial margin of potential energy should be available at the limiting angle position in excess of the overturning energy. This is expressed through the well-known relationship A2  1.4 A1 .

A Basis for Developing a Rational Alternative to the Weather Criterion

29

3 The Concept of “Engineering Integrity” Several articles have been published about the concept of “loss of engineering integrity” of dynamical systems. The reader who is not acquainted with the basic theory and terminology is referred for example to Thompson (1997). The basic idea is that the safety robustness of an engineering system can be represented by the integrity of its “safe basin” comprised by the set of initial conditions (in our case pairs of roll angle and velocity) which lead to bounded motion. It is known that the basin could suffer some serious reduction of area (“basin erosion”) once some critical level of excitation is exceeded given the system’s inertial damping and restoring characteristics. The reason for this is the complex intersection of “manifolds” i.e. those special surfaces which originate from, or end on, the unstable periodic orbits corresponding to the vanishing angles which become timedependent when there is wave-forcing. Detailed analysis about these phenomena can be found in various texts see for example Guckenheimer & Holmes (1997). The initiation of basin erosion can be used as a rational criterion of system integrity which in our case and for a naval architectural context would be translated as sufficient dynamic stability in a global sense. The methods that exist for predicting the beginning of basin erosion are the following:

a) Use of the so-called Melnikov analysis which provides a measure of the “closeness” of the critical pair of manifolds and hence can produce the condition under which this distance becomes zero. For capsize prediction the method is workable when the damping is relatively low (strictly speaking it is accurate for infinitesimal damping - however several studies have shown that the prediction is satisfactory for the usual range of ship roll damping values). Ideally the method is applied analytically; or it may be combined with a numerical part when the required integrations cannot be performed analytically. b) Direct numerical identification of the critical combination of excitation damping and restoring where the manifolds begin to intersect each other. This method is more accurate but requires the use of specialised software. c) Indirect identification of the critical condition by using repetitive safe basin plots until the initiation of basin erosion is shown. The same comment as in (b) applies here. In the next Section is provided an outline of Melnikov’s method and application for a family of restoring curves parameterised with respect to bias intensity (e.g. produced by the wind excitation). The critical for capsize wave slope as a function of the wind-induced heel and the damping is determined. In the remaining Sections is targeted the concept of engineering integrity itself. More specifically the following two topics are addressed:

30 K.J. Spyrou

 The value of the concept for higher order restoring curves: the practically interesting point here is to find whether it is workable for arbitrary types of restoring. Towards this the family of 5th order restoring functions is analysed. In addition, the concept is applied for the assessment of an existing ship thus giving an idea of how the method would work for “real-world” stability assessment.  Possible use of the concept in a framework of design optimisation: a simple parameterised family of ship-like hull forms has been considered with main objective to determine whether the method can discriminate meaningfully between good and poor designs.

4 Outline of Melnikov’s Method Assume the dynamical system

x τ   f x   ε g x ,τ 

(1)

Where:

x is in our case the vector x, dx / d  of roll displacement (normalised with respect to the vanishing angle) and roll velocity Ha m

ilto nia n

x f (x)

d(t)

Fig. 4 The safe basin and the distance of manifolds

f x  represents the “unperturbed” Hamiltonian part of the dynamical system; and gx ,  is the “perturbation” which includes the damping and the explicitly time-dependent wave-forcing.  is a parameter that represents the smallness of gx ,  . It can be shown that the closeness of manifolds is expressed as (Fig. 4): d τ 0  

ε Μ τ 0   O ε2 f x 

 

(2)

A Basis for Developing a Rational Alternative to the Weather Criterion

31

Where τ 0 is a phase angle in the range 0  τ 0  2 /  . Since the denominator

is of the order of 1 the function M τ 0  (which is called Melnikov function) is to first order a good measure of the distance of manifolds:

M τ 0  







f x τ   gx τ , τ  τ 0 dτ

(3)

g The wedge symbol  means to take the cross product of the vectors f and . The main intention when this method is applied is to identify the critical combinations of design/operational parameters where the Melnikov function admits real zeros which means that the manifolds begin to touch each other (it is essential the crossing of manifolds to be transversal but we do not discuss this here further).

5 Melnikov’s Method for Ship Rolling With a Wind-Induced Bias We have applied the method for the following family of restoring curves which have the bias as free parameter:

Rx   x1  x (1  ax   x  1  a x 2  ax 3

(4)

The parameter a indicates the strength of the bias assumed here to be due to beam wind loading. Letting x  x1 and x  x 2 the roll motion could be described by the following pair:

x1  x2

x2   x1  1  a x12  ax13  F sin Ωτ  τ 0   β x2 

(5)

If the bias is relatively strong the unperturbed part is represented by the vector:

x2   f x    2 3  x1  1  a x1  ax1  Where

(6)

32 K.J. Spyrou

x  x   1   x2 

(7)

While the perturbation is:

0   g x     F sin Ωτ β x 2 

(8)

After substitution into (3) the Melnikov function takes the form of the following integral:

M  0  





-

x2 F sinΩτ  τ 0   βx2 dτ  M E τ 0   M D

(9)

where M E τ 0  is the part due to wave forcing that is explicitly time-dependent; and M D is the part due to the damping. 0.055 F

Ω=1 0.05 0.045

Ω =0.9

0.04 Ω =0.8 0.035 0.8

0.82

0.84

0.86

0.88

0.9

0.92

a

0.94

Fig. 5 Critical F as function of the bias. To be noted the initial “sharp” fall of the critical F as the bias a departs from 1 which corresponds to a symmetric system

For the calculation of (9) we need to have available an explicit expression in terms of time of the roll velocity x 2 corresponding to the unperturbed system. This is determined as follows: the homoclinic orbit goes through the corresponding saddle point at x = 1. With the coordinate change s  x  1 the equation of the unperturbed system becomes:

A Basis for Developing a Rational Alternative to the Weather Criterion

d 2s dτ 2

 as 3  2a  1s 2  a  1s

33

(10)

With some manipulation the above can be written as:

ds   hs d

s  p 2  q 2

(11)

The parameters that appear in (11) are defined as follows:

h

21  a 2  a  3a

22a  1 a ,p and q  2 3a

(12)

The solution of (11) is:

sτ   xτ   1  ∓

p

2

 q2



(13)

p  qcosh 1  a τ

Differentiation of (13) yields:

 



ds dx q 1  a p 2  q 2 sinh 1  aτ   x2   2 d d p  qcosh 1  aτ







(14)

With substitution of (14) into (9) we can calculate the integral (9) with the method of residues. After some algebra, one obtains that the Melnikov function becomes zero when:





 p h  p 2  q 2 p 2  2q 2  3 pq 2 arccosh  q sinh μπ F β   p 6ππ p 2  q 2 sin  μ arccosh  q 

(15)

The sustainable wave slope is:

 Ak crit. 

Fφ ν ν Ω2

(16)

34 K.J. Spyrou

where φ ν is the vanishing angle and ν 

Ix . Ix  δIx

The critical wave slope obtained with application of the above method is shown in Fig. 5 as a function of the frequency ratio for different values of the bias. Notably Melnikov’s analysis produces an identical result with energy balance where the energy influx due to the forcing is equated with the energy dissipated through damping around the remotest orbit of bounded roll of the corresponding “unperturbed” system. This formulation based on energy balance is intuitively quite appealing. It will be discussed in more detail in Appendix I. Melnikov’s method has been applied also for irregular wave excitation (e.g. Hsieh et al. 1994). In Appendix II we outline the main issues involved in the formulation of Melnikov’s method for a stochastic wave environment.

6 Concept Evaluation for Realistic GZ Here our objective is twofold: firstly to clarify whether the engineering integrity concept is meaningful when applied to higher-order and thus more practical GZ curves. Secondly to apply the new method for an existing ship which satisfies the weather criterion (this work was part of a diploma thesis at NTUA (Papagiannopoulos 2001).

a) Higher-order GZ In the first instance we considered the family of fifth-order restoring polynomials

Rx   x  cx 3  c  1 x 5

(17)

Characteristic GZ shapes produced by (17) as c is varied are shown in Fig. 6. A feature of this family is that it covers hardening ( c  0 ) as well as softening ( c  0 ) type levers. It must be noted that the curves shown in Fig. 5 are scaled which means that there is no need the increase of c to lead to a higher value of the real GZ max . The main purpose of the study was to confirm that there is indeed

A Basis for Developing a Rational Alternative to the Weather Criterion

35

sudden reduction of basin area beyond some critical wave forcing for all the members of this family i.e. that the concept is “robust” with respect to the characteristics of the restoring function.

Fig. 6 Scaled “quintic” restoring curves

In Fig. 7 are shown some typical results of this investigation for an initially hardening GZ at four different levels of damping. The points on the curves are identified with measurement of the basin area and scaling against the area of the corresponding unforced system. The curves confirm that for the fifth-order levers the concept of engineering integrity is applicable without any problem i.e. there is indeed a quick fall of the curve beyond some critical forcing. This critical forcing should be approximately equal to the forcing determined from the Melnikov method. An advantage however of the direct basin plot is that it allows to determine fractional integrities also like 90% 80% etc. as function of damping and excitation (90% integrity means that the remaining basin area is 90% of the initial). This is quite important for criteria development since it provides the necessary flexibility for setting the boundary line; for example one could base the maximum sustainable wave slope at 90% integrity rather than at 100% which might be too stringent.

36 K.J. Spyrou

c =1, Ω=1

integrity 1

0.8

0.6

0.4

0.2 0

0.05

0.1

0.15

0.2

0.25

Forcing amplitude F integrity

c =1, Ω=0.88

1

b 0.01 0.05 0.1 0.2

0.8

0.6

0.4

0.2 0

0.05

0.1

0.15

0.2

0.25

Forcing amplitude F

Fig. 7 Integrity diagrams

b) Application for an existing ship A ferry was selected which had operated in Greek waters with LBP  153 m B = 22.8 m T = 6.4 m Cb = 0.548 and KG  10.004 m. The GZ curve was known while the damping as a function of frequency was determined using the wellknown method of Himeno (1981). For the wave-making part we used the panel code Newdrift (1989) which is available at the Ship Design Laboratory of NTUA. In Fig. 8 is shown the sustainable wave slope for a range of frequencies around resonance on the basis of repetitive basin plotting. From this diagram it is possible to determine the range where survivability in a beam sea condition is ensured. This graph could be contrasted against Fig. 11 of Yamagata (1959) which gives the

A Basis for Developing a Rational Alternative to the Weather Criterion

37

wave steepness considered in the weather criterion as function of natural period under the assumption of resonance condition.

Fig. 8 Critical wave environment for an existing ship on the basis of the new concept

7 Use of the Concept of Engineering Integrity for Design Optimisation The final study reported here was based on another NTUA diploma Thesis (Sakkas 2001). It includes application of the method for a family of simplified hull forms whose offsets are given by the following simple equation:

y   f x , z    X x   Z x , z 

(18)

The same family was used in a paper by Min & Kang (1998) of the Hyundai Heavy Industries for optimising the resistance of high-speed craft. The shape of waterlines is described by the function X x  which depends only on the x nondimensionalised position x  where x is the longitudinal position measured l from the middle of the ship and l is the entrance length with 1  x  1 . The function X x  is expressed as the 4th order polynomial

38 K.J. Spyrou

X x  



B 3 1  a 2 x 2  a3 x  a 4 x 4 2



(19)

Fig. 9 Characteristic hull-shape of the considered family

The transverse hull sections depend on both longitudinal position and height. They are expressed by the following simple function

Z x , z   1  z n  x 

(20)

where

n x   s  t  x

(21)

with s, t free parameters. After a few trials we found that for somewhat realistic hull shapes the parameters a 2 , a3 and a 4 should take values between –1 and 1 while the parameters s and t should lie between 0 and 0.4. As s and t increase the hull becomes more slender. A characteristic hull-shape can be seen in Fig. 9.

A Basis for Developing a Rational Alternative to the Weather Criterion

39

0,08 0,07 0,06 0,05 0,04 0,03 0,02 0,01 0 1

3

5

7

9 11 13 Ship No

15

17

19

Fig. 10 Roll damping for each examined hull

Although originally the height z is non-dimensionalised with respect to the draught, in our case this is done with respect to the depth because the hull-shape above the design waterline is very important in stability calculations ( 1  z  0 ). To narrow further on the free parameters of this investigation we have fixed the length, the beam and the depth; respectively to 150m, 27.2m and 13.5m. The displacement was also kept constant at 23710 t which means that only the block coefficient and the draught were variable. In addition we have considered only hulls fore-aft symmetric. KG was linked to the depth while the windage area was set to be 4.1 L x T. 0,035 0,03

(H / λ) critical

0,025 0,02 0,015 0,01 0,005 0 1

3

5

7

9 11 13 Ship No

15

17

19

Fig. 11 Dynamic stability performance: Left bars are for waves only; while the right bars are for wind and waves.

40 K.J. Spyrou

The free parameters were then varied in a systematic way so that a 2  a3  a 4  1 and s  t . In total 20 ships were collected for further stability investigation. For each one of these ships the corresponding GZ curve was determined by using the commercial programme Autoship while the damping was calculated as described in the previous Section. The roll radius of gyration was calculated with a well-established empirical formula. The nondimensional damping value β at roll resonance for each ship is shown in Fig. 10. For all these simplified ships we applied the weather criterion and we verified that it is fulfilled. Thereafter knowing the restoring damping and inertial characteristics for each hull we run the programme Dynamics of Nusse & Yorke (1998) in order to determine the critical wave slope Ak at which basin erosion is initiated. The wind loading was calculated as prescribed in the weather criterion. The performance of each ship as determined from this procedure is shown in the bar chart of Fig. 11. In the same figure is shown the stability performance of the ships when they were subjected to beam wind loading of the same type and intensity as the considered in the IMO weather criterion. The best hull determined is shown in Fig. 12.

Area (m-rad) 1.5 1.0 GZ (m) 0.5 0 0

50 Heel angle (degrees)

100

Fig. 12 Best hull and the corresponding restoring curve

8 Concluding Remarks An investigation on the concept of engineering integrity and whether it can be used for developing a practical ship stability criterion has been presented. In the spirit of the weather criterion we have confined ourselves to a study of system robustness to combined wind and wave excitations coming from abeam and under the assumption that there are no phenomena such as wave breaking on the ship

A Basis for Developing a Rational Alternative to the Weather Criterion

41

side or accumulation of water on the deck. Also, we have left out of the present context any discussion about stability in following seas, a matter which deserves a separate approach due to the different nature of the dynamics involved. Although the following-sea is the most dangerous environment of ship operation, it was not seriously addressed up to now in the IMO Regulations. As a matter of fact, ships are designed with no requirement for checking their stability at the most critical condition that they may encounter although the primary modes of ship capsize in a following sea environment are now well understood. Of course such a matter is not raised for the first time and in the academic world several approaches have been discussed in the past: some are based on the area under the time-varying GZ curve (butterfly diagram see e.g. Vassalos 1986) while more recently there is a trend for taking into account “more fully” system dynamics. For parametric instability already exist general criteria that have been developed in the field of mechanics. However it is doubted whether these are known to ship designers (for an attempt to summarize these criteria see for example Spyrou et al. 2000). Furthermore the clarification of the dynamics of broaching-to has opened up new possibilities for the development of a simple criterion for avoiding the occurrence of this phenomenon too. It is emphasized that the proposed methodology is a platform for developing also formulations targeting the following-sea environment. Furthermore since this approach is a global method and is not based on ordinary simulation it presents also distinctive advantages over the so-called “performance-based methods” a term which in an IMO-style vocabulary is assumed to mean the execution of a limited number of experimental or simulation runs. Our analysis up to this stage shows that although the proposed method can produce meaningful results there are a few practical problems whose solution is uncertain and would require further research. A deeper look into the Melnikovbased approach is presented in a companion paper (Spyrou et al 2002). Some questions that immediately come to mind are: • How to deal with down-flooding angles or more generally limits of stability that are lower than the vanishing angle? The problem arises from the fact that the method targets the phenomena affecting the basin boundary while the amplitude of rolling may be well below that level if there is flooding of non-watertight compartments. The formulation of Melnikov’s method as an energy balance might give the idea for a practical solution.

• How to combine the deterministic and the stochastic analysis? This happened quite meaningfully in the weather criterion and some comparable yet not obvious in terms of formulation approach is required. • How to adjust the level of stringency of the criterion? This could be achieved perhaps by setting the acceptable level of integrity to a fractional value such as 90% rather than the 100% that is currently considered in research studies.

42 K.J. Spyrou

On the other hand, the method seems that it can easily fulfil the prerequisites of a successful stability criterion which in our own opinion are the following:

• It should have an unambiguous direction of stability improvement i.e. should be formulated in such a way that the designer can maximise the stability margin of a ship under consideration (not in isolation of course but in parallel with other performance and safety matters) and not simply check for conformance to limiting values. • The criterion should be representative of stability in a global sense i.e. should not be based on a prescribed and narrowly defined condition where stability should be ensured. • It should always make clear to the designer the connection with the limiting environmental conditions. • The method should take into account the transient nature of the ship capsize process and it should account with sufficient accuracy for system dynamics. • It should be flexible in order to accommodate changes in design trends i.e. should not be overly dependent on existing ship characteristics which means that the degree “empiricism” intrinsic to the method should be minimal.

References Falzarano J Troesch AW and Shaw SW (1992) Application of global methods for analysing dynamical systems to ship rolling and capsizing. Int J of Bifurc and Chaos 2:101-116. Frey M and Simiu E (1993) Noise-induced chaos and phase-space flux Physica D 321-340. Guckenheimer J and Holmes P (1997) Nonlinear oscillations dynamical systems and bifurc of vector fields. Springer-Verlag N.Y. Himeno Y (1981) Prediction of roll damping – state of the art. Report No. 239. Dep of Naval Archit and Marine Eng The Univ of Michigan Ann Arbor MI. Hsieh SR Troesch AW and Shaw SW (1994) A nonlinear probabilistic method for predicting vessel capsizing in random beam seas. Proc of the Royal Soc A446 1-17. IMO (1993) Code on Intact Stability for All Types of Ships Covered by IMO Instruments. Resolut A.749(18) London. Min KS and Kang SH (1998) Systematic study on the hull form design and resistance prediction of displacement-type-super-high-speed ships. J of Marine Sci and Technol 3: 63-75. NES 109 (2000) Stability Standard for Surface Ships. U.K. Ministry of Defence Issue 1 Publication date 1st April 2000. Nusse HE and Yorke JA (1998) Dynamics: Numerical explorations Spinger-Verlag N.Y. Papagiannopoulos S (2001) Investigation of ship capsize in beam seas based on nonlinear dynamics theory. Diploma Thesis, Dep of Naval Archit and Marine Eng, Nat Technical Univ of Athens Sept. Newdrift Vers. 6 (1989) User’s Manual. Ship Design Laboratory, Nat Technical Univ of Athens. Sakkas D (2001) Design optimisation for maximising dynamic stability in beam seas. Diploma Thesis Dep of Naval Archit and Marine Eng, Nat Technical Univ of Athens Sept.

A Basis for Developing a Rational Alternative to the Weather Criterion

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Spyrou KJ (2000) Designing against parametric instability in following seas. Ocean Eng, 27 625653. Spyrou KJ, Cotton B and Gurd B (2002) Analytical expressions of capsize boundary for a ship with roll bias in beam waves. J of Ship Res 46 3 167-174. Thompson JMT (1997) Designing against capsize in beam seas: Recent advances and new insights. Appl Mech Rev. 50 5 307-325. Vassalos D (1986) A critical look into the development of ship stability criteria based on work/energy balance. R.I.N.A. Transactions, 128, 217-234. Watanabe (1938) Some contribution to the theory of rolling I.N.A. Transactions, 80, 408-432. Wiggins S. (1992) Chaotic transport in dynamical systems Springer-Verlag N.Y. Yamagata, M. (1959) Standard of stability adopted in Japan, RINA Transactions, 101, 417-443.

Appendix I: Melnikov’s Method as Energy Balance Consider again the roll equation with bias:

x  β x  x1  x 1  ax   F cos τ

(I1)

The corresponding Hamiltonian system is:

x  x 1  x 1  ax   0

(I2)

with kinetic energy

KE 

1 2 x 2

(I3)

and potential energy

Fig. I1 Energy balance around the homoclinic orbit

44 K.J. Spyrou

PE  x 1  x 1  ax   c 



1 2 1  a  3 a 4 x  x  x 2 3 4

(I4)

It is obvious that if we take as initial point the saddle (x=1 x  0 ) and as final the same point after going over the homoclinic orbit (“saddle loop”) the difference in potential energy will be zero. Zero will be also the difference in kinetic energy (at the saddle point the roll velocity is obviously zero). The energy lost due to the damping “around” the homoclinic orbit is represented by the area A inside the loop times the damping coefficient which is expressed as (Fig. I1):





DE  β xdx  β x 2 dτ

(I5)

The input of energy on the other hand due to the wave excitation is the integral (in terms of the roll angle) of the external roll moment taken around the homoclinic orbit. Mathematically this is expressed as

WE  F sin  τ  τ 0  dx  F sin  τ  τ 0  x dτ





(I6)

The time constant τ 0 allows to vary the phase between forcing and velocity x . The balance of energies requires therefore that i.e. 

DE  WE  β





x 2 d  F cos  τ  τ 0  x dτ





(I7)



which is identical with (9) when set equal to zero i.e. we have obtained a condition identical to the condition M τ 0   0 described by (3). Despite the clarity of the above viewpoint a note of caution is essential: the above energy balance “works” also for other nonlinear global bifurcation phenomena (e.g. a saddle connection) and application should be attempted only when it is already ensured that the underlying phenomenon is an incipient transverse intersection of manifolds which indeed generates basin erosion.

Appendix II: Formulation for Irregular Seas In the “stochastic” version of Melnikov’s method the objective is to determine the time-average of the rate of phase-space flux that leaves the safe basin. The basic theory can be found in Wiggins (1990) and Frey & Simiu (1993). However the

A Basis for Developing a Rational Alternative to the Weather Criterion

45

specific formulation and adaptation for the ship capsize problem is owed to Hsieh et al (1994). The key idea is that the probability of capsize in a certain wave environment is linked to the rate of phase-flux. The formulation is laid out as follows: Let’s consider once more the roll equation, this time however with a stochastic wave forcing F (τ ) at the right-hand-side. It is noted that the equation should be written in relation to the absolute roll angle because the presentation based on angle relatively to the wave in this case is not practical. It can be shown that the flux rate i.e. the rate at which the dynamical system loses safe basin area scaled by the area A of the basin of the unperturbed system is given by the following expression (Hsieh et al 1994):

⎛ M Φ ε ⎡ = ⎢ H s σ p ⎜⎜ D A Α ⎢⎣ ⎝ Hs σ

⎞ ⎛ M ⎟ + M D P⎜ D ⎟ ⎜H σ ⎠ ⎝ s

( )

⎤ ⎞ ⎟ − M D ⎥ + O ε2 ⎟ ⎠ ⎦⎥

(II1)

where H s is the significant wave height σ is the RMS value of the Melnikov function M (τ ) = M E (τ 0 ) − M D for unity H s and M D is the damping part of the Melnikov function which is commonly taken as non explicitly time-dependent. For wave forcing with zero mean the mean value of the wave part of the Melnikov function is zero thus the mean E [M (τ 0 )] is − M D . Also p, P are respectively the standard Gaussian probability density and distribution functions. The assumption can be made, supported however by simulation studies (Hsieh et al 1994 Jiang et al. 2000), that the flux rate reflects reliably the capsize probability. Conceptually this appears as a direct extension for a probabilistic environment of the connection between basin area and capsizability of a ship under regular wave excitation. The “behaviour” of equation (II1) is that the flux rate increases quickly once some critical significant wave height is exceeded gradually approaching an asymptote as H s tends to infinity. Therefore, someone could set a desirable environment of ship operation in terms of a certain wave spectrum, significant wave height etc. and then through proper selection of the damping and restoring characteristics he could try ensure that his design “survives” in this environment. This is translated in keeping the flux rate as determined by (II1) below a certain limit. Hsieh et al (1994) have suggested to use the point where the asymptote intersects the H s axis as the critical one. The various quantities that appear in (II1) are calculated as follows: For the damping term M D of the Melnikov function it is entailed to determine the integral of some power of the time expression of roll velocity at the homoclinic orbit. In some cases it is possible this to be done analytically. For example for the system described by equations (5) it becomes after calculation:

46 K.J. Spyrou





1  q M D  β h  p 2  q 2 p 2  2q 2  3 pq 2 arccosh  3  p

(II2)

where h, p, q are function of the bias parameter a and they are defined in Section 5.  connects with the spectrum of wave elevation as follows:



 

 2  E M E2 τ 0  



0

   d  2 SM





0

S x  S F   d

(II3)

The spectral density function of the wave forcing is linked to the wave spectrum

S F

   F   2 S   

(this is perhaps the main weakness of the method as this

relationship is valid only for a linear process). The final quantity that needs to be calculated is the power spectrum of the scaled roll velocity

S x   

1 2π





-

xe iτ dτ

(II4)

For the homoclinic loop shown in Fig. I1 this is easily calculated numerically although it can be proven that it receives also an exact analytical solution in terms of hypergeometric functions. With the above we can plot the significant wave height versus, for example, the characteristic wave period which gives a straightforward platform for setting a level of acceptability (e.g. survival up to a certain H s ).

Conceptualising Risk Andrzej Jasionowski, Dracos Vassalos SSRC, NAME, University of Strathclyde / Safety At Sea Ltd

Abstract

Ever present jargon and colloquial notion of risk, coupled with obscuring of the simplicity inherent in the concepts of risk by the sophistication of computer software applications of techniques such as event trees, fault trees, Bayesian networks, or others, seem to be major factors inhibiting comprehension and systematic proliferation of the process of safety provision on the bases of risk in routine ship design practice. This paper attempts to stimulate consideration of risk at the fundamental level of mathematical axioms, by proposing a prototype of a comprehensive yet plain model of risk posed by the activity of ship operation. A process of conceptualising of substantive elements of the proposed risk model pertinent to the hazards of collision and flooding is presented. Results of tentative sensitivity studies are presented and discussed.

1 Introduction It is well recognised that the engineering discipline entails constant decision making under conditions of uncertainty, that is decisions are made irrespective of the state of completeness and quality of supporting information, as such information is never exact, it must be inferred from analogous circumstances or derived through modelling, and thus be subject to various degrees of approximation, or indeed, the information often pertains to problems involving natural processes and phenomena that are inherently random. No better example of such decision making process in engineering can be invoked, than that of development of regulations for the provision of safety. The process of rules development is well recognised to rely on both, (a) qualified “weighting” of a multitude of solutions derived on the basis of scientific research, engineering judgement, experiential knowledge, etc, and (b) debates accommodating the ubiquitous political agendas at play and governing societal concerns. The result of such complex and predominantly subjective pressures is often a simplistic compromise normally in the form of deterministic rules which does not reflect the underlying uncertainty explicitly.

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_3, © Springer Science+Business Media B.V. 2011

47

48

A. Jasionowski and D. Vassalos

Although such standards have formed the main frame of safety provision over a century or so, the acknowledgement of existence of these uncertainties, brought about by the occurrences of accidents with undesirable consequences on one hand, and realisation that economic benefit could be compromised by inbuilt but potentially irrelevant margins, on the other hand, instigated a search such as (SAFEDOR 2005) for better methodologies for dealing with the issues of safety. Notably, such search has been enhanced, or possibly even inspired, by the tremendous progress in computational technologies, which by sheer facilitation of fast calculations have allowed for development of numerical tools modelling various physical phenomena of relevance to engineering practice, at levels of principal laws of nature. Such tools present now the possibility to directly test physical behaviour of systems subject to any set of input parameters, and thus also allow derivation of an envelope of the inherent uncertainty. Existence of such methods, however, is not sufficient for progressing with the development of more efficient means of safety provision, without a robust framework for reasoning on the results of any such advanced analyses undertaken for safety verification processes. Such means evolve naturally from the principles of inductive logic; a field of mathematics dealing with reasoning in a state of uncertainty, (Jaynes 1995) (Cox 1961). In particular the branch comprising probability theory forms a suitable framework for such reasoning in the field of engineering. However, while the concept of probability is known to engineers very widely, it does not seem that the many unresolved to this day subtleties associated with either, its fundamental principles or relevant linguistics, are recognised by the profession. This is the situation especially when a special case of application of the probability theory, namely that of risk, is considered. The uptake of risk assessment process, or more recently strife to develop and implement risk-based design paradigms, (SAFEDOR 2005) (Konovessis 2001) (Guarin 2006), are examples of a new philosophy that could alleviate the aforementioned problems of safety provision, if they can become suitable for routine practice. This in turn, can only be achieved if fundamental obstacles such as lack of common understanding of the relevant concepts are overcome. Therefore, this paper sets to examine some lexical as well as conceptual subtleties inherent in risk, offers some suggestions on more intuitive interpretations of this concept, and puts forward a proposal for risk modelling process as a contributory attempt to bringing formalism in this development.

2 The Risk Lexicon As has been the case with the emergence of probability in the last 350 years or so, (Hacking 1975), the terminology on risk has not evolved into anything that can be regarded as intuitively plausible lexicon, as neither have the pertinent syntax nor semantics been universally endorsed.

Conceptualising Risk 49

Any of tautological phrases, such as: “level of safety standard”, “hazard threatening the safety”, semantically imprecise statements such as: “collision risk”, “risk for collision”, “risk from collision”, “risk of collision”, ”risk from hazard aspects of …”, simple misnomers such as “safety expressed as risk”, “safety risk”, or philosophically dubious slogans such as “zero risk1”, can be found in the many articles discussing or referring to the concepts of risk. A few other, easily misconceiveable terms used in this field can be mentioned: (a) (b) (c) (d) (e) (f) (g)

likelihood, chance, probability frequency, rate uncertainty, doubt, randomness risk, hazard, danger, threat analysis, assessment, evaluation risk control measures, risk control options safety goals, safety objectives, safety functional statements, safety performance, relative safety

Although some preamble definitions of these and other terms are offered from article to article, lack of generic coherence in discussions of the risk concepts among the profession is not helpful in promulgating this philosophy as of routinely quality. This article makes no pretence of having resolved all these subtleties. It merely attempts to adopt a scheme of thinking, which appears to allow for systematic understanding of the concepts of safety and risk.

2.1 Safety By way of introduction of this scheme, the term underlying all this discussion, namely safety, shall be presented to formally set the terms of reference. Without going into intricacies of how complex the definition of the term “safety” can be, it is hereby emphasised that for all engineering purposes and currently emerging wider acceptance of risk concepts, the following definition of safety, (Vassalos 1999), should be adopted widely:

1 The Cournot’s lemma stating that “an event of small probability will not happen” has been initially endorsed by many mathematicians, even acclaimed as the physical “logic of probable” by some, however, the concept has been dismissed for the latter part of the last century on the grounds of Bayesian interpretations of probability, i.e. that “zero” is subject to “personal” interpretation. Hence a statement of “zero risk” remains a philosophical conjecture with little, or indeed, no physical justification.

50

A. Jasionowski and D. Vassalos

safety is the state of acceptable risk This definition is unambiguous and it is practical. It emphasizes explicitly that safety is a state. It thus cannot be calculated, it can only be verified by comparison of the actually established quantity of risk with the quantity of risk that is considered acceptable by relevant authoritative bodies; a criterion or standard, in other words. Thus, it follows that statements such as “higher safety” are incompatible with this definition, as the safety is either attained or it is not and no grades of safety can be distinguished. Appropriate statement would be “higher safety standard”, which directly implies lowered levels of risk as acceptable. Also, expressions such as “safety performance” become semantically imprecise unless, contrary to intuitive perception, the word “performance” implies a binary mode of compliance/no compliance with the set standard, rather than a scale, which in turn, relates to risk not safety. This definition also establishes clear distinction between risk and safety. Risk is the benchmark vehicle to demonstrate a state of safety or lack of it. Thus, safety cannot be “expressed as risk”. What the definition does not resolve is the consistency of the relevant syntax which has come to be used in the every day communication on safety. For instance, expressions such as “fire safety” and “ship safety” imply fundamentally different meanings despite use of the same syntax. While the interpretation analogous to that of the colloquial speech examples such as “orange juice” and “baby juice”, could be accepted or taken for granted, as it is done today, such subtleties should be addressed and resolved for avoidance of any ambiguities in engineering applications, especially when many new concepts such as safety goals or others are being introduced. This, however, is beyond the scope of the discussion of this article. Another worthwhile note should be made here, that this definition is also fully compatible with the universally accepted in today’s world mechanism of safety provision through compliance with regulations. Irrespective of the deterministic or more complex nature of the regulation, safety is said to have been attained, when an acceptable criterion has been met. Although the inherent risk is not disclosed, it is said to have been brought to an acceptable level. Very often expressions, such as “relative safety” or “conditional safety” are used to describe the limited scope of such regulations and undisclosed nature of the risk, though neither of the words “relative” nor “conditional” has any quantifiable explanation. But, as mentioned earlier, since simplistic and disparate regulations could compromise commercial gain or undermine societal approval of the standards proposed for safety provision, when critical accidents happen, new methods are sought after. The direct application of the concept of risk is one such method, especially since the above definition of safety implies the “presence of risks”. The key question now is what is risk?

Conceptualising Risk 51

2.2 Risk An ISO 8402:1995 / BS 4778 standard offers the following definition of risk: “Risk is a combination of the probability, or frequency, of occurrence of a defined hazard and the magnitude of the consequences of the occurrence” A similar definition is put forward at IMO, MSC Circ 1023/MEPC Circ 392: “Risk is a combination of the frequency and the severity of the consequence”. Although rather widely endorsed, these definitions are so ambiguous that they hardly qualify as useful for any engineering purposes. The ambiguity derives from the following: (a) the word “combination” is open to any interpretation whatsoever, both in terms of relevant mathematics as well as underlying physical interpretations (b) the words “probability” and “frequency” have, in common engineering practice, different physical interpretations, which itself can be arbitrary, (c) the words “magnitude” or “severity” are unspecific, (d) the “consequence” is again open to any interpretation. For instance, the syntax of the above definition implies differentiation of the semantics of the phrases such as “hazard” and “consequence”, without any qualification how such differentiation is attained. Thus, the sinking of the vessel can be considered as a hazard, which can lead to the loss of human life, but it can also be considered as an ultimate consequence, or in other words as the loss. See here Fig. 1 and the subsequent discussion. While the merit of proposal of such definitions of risk could be its generic nature, overly lack of any specificity deprives it of practical significance. The following alternative definition is suggested, which is equally generic yet it is comprehendible more intuitively and, apparently, more pragmatic:

risk is a chance of a loss The word “chance” is one of the most fundamental phrases referring to, or being synonymous with the natural randomness inherent in any of human activities and which, in this case, can bring about events that are specifically undesirable, the “loss”. The engineering aspects of the concept of chance are effectively addressed by the fields of probability and statistics, both of which offer a range of instruments to quantify it, such as e.g. expectation, and hence there is no need for introduction of unspecific terms such as “combination”. These well known mathematical tools will be discussed in section 3. This definition of risk resolves also, to some extent, the manner of its classification. Namely reference to risk should be made in terms of the loss rather than in terms of any of the intermediate hazards in the chain of events leading to the loss, a nuance just mentioned.

52

A. Jasionowski and D. Vassalos

Fig. 1 The concept of a chain of events in a sample scenario leading to the loss. Each of the events is a hazard after it materialises. A scenario can be identified by a principal hazard, see Table 1.

For instance, semantics of a statement “risk to life” or “risk of life loss” seem to indicate sufficiently precisely the chance that an event of “fatality” can take place, without much space for any other interpretations. On the other hand, a statement such as “risk of collision” can be interpreted as a chance of an event of two ships colliding; however, given a further set of events that are likely to follow, such as flooding propagation, heeling, capsizing, evacuation/abandoning and fatalities, see Fig. 2, it becomes unclear what is the “loss”, and thus what is the risk? Furthermore, the above definition supports lexically the recently more pronounced efforts to introduce the concept of holism in risk assessment. Namely, rather than imposing a series of criteria for acceptable risk levels in a reductionism manner, whereby separate criteria pertaining to different hazards are introduced, such as e.g. criteria on fire hazards or criteria on flooding hazards (e.g. probabilistic concept of subdivision, (SLF 47/17 2004) ), a unique criterion should be proposed that standardises the acceptable risk level of the ultimate loss, e.g. criteria for loss of life, environmental pollution, etc, and which risks account for each hazard that can result in the loss, a view already mentioned in (Skjong et al. 2005) or (Sames 2005). This concept underlines the risk model proposed in the following section 3.

3 Risk Modelling As mentioned above, the fields of probability and statistics provide all the essential instruments to quantify the chance of a loss.

3.1 Frequency Prototyping To demonstrate the process of application of these instruments for risk modelling, perhaps it will be useful to stress the not necessarily trivial difference between the concepts of probability and frequency. Namely, to assess a “probability of an event”, a relevant random experiment, its sample space and event in question on that sample space must all be well defined. The relative frequency of occurrence of this event, whereby relative implies relative to the sample space, is a measure of probability of that event, since such relative frequency will comply with the axioms of probability.

Conceptualising Risk 53

In the case of “an experiment” of operating a ship, however, whereby events involving fatalities can occur in the continuum of time, the pertinent sample space is customarily not adopted, and rather, the rate of occurrence of these events per unit time is used. The rate can be greater than unity, and obviously can no longer be referred to as probability. Thus, also use of the term “likelihood” would seem to be displaced here since it is synonymous with probability rather than frequency. Therefore, application of the term frequency per ship per year is customary, also since its quantification relies on the historical data for the relevant fleet. Lack of a well defined sample space, however, does not prevent one from using concepts of probability, such as e.g. distribution, expectation, etc, in analyses and analytical modelling. Thus, following from this preamble, it is proposed here that for the purpose of risk modelling the following assumptions are made: (a) The loss is assumed to be measured in terms of an integer number of potential fatalities among passengers that can occur as a result of activity relating to ship operation (b) The loss is a result of occurrence of a set of scenarios which can lead to this loss (c) Scenarios are intersections of a set of events (all must occur) and are identified by principal hazards2, e.g. fire (d) The scenarios are disjoint (if one scenario occurs then the other does not) These assumptions are shown graphically in Fig 2 below.

Fig. 2 Illustration of the concept of a compound event referred to as “fatality” by Venn’s diagram, (Jasionowski 2005), which is a union of mutually exclusive scenarios, whereby each scenario is an intersection of a set of relevant events and is identified by principal hazards, such as collision, fire, etc.

2

an event of a loss scenario is a hazard that materialised

A. Jasionowski and D. Vassalos

54

Deriving from the above, it is hereby proposed that the frequency frN (N) of occurrence of exactly N fatalities per ship per year is derived based on Bayes theorem on total probability, as follows:

frN N    frhz hz j   prN N hz j  n hz

(1)

j 1

Where

nhz is the number of loss scenarios considered, and hzj represents an

event of the occurrence of a chain of events, (a loss scenario), identifiable by any of the following principal hazards: Table 1. Principle hazards

1

Principal hazards, hzj

Average historical frequency of its occurance, frhz(hzj)

Collision and flooding

2.48e-3, (Vanem and Skjong 2004)

2

Fire

-

3

Intact Stability Loss

-

4

Systems failure

-

…, etc.

-

Furthermore, frhz(hzj) is the frequency of occurrence of a scenario hzj per ship per year, and prN(N|hzj) is the probability of occurrence of exactly N fatalities, given loss scenario hzj occurred. In principle, there is nothing new to this proposal, except, perhaps, for the emphasis of the need to estimate the probability for occurrence of exactly N fatalities, prN(N|hzj), conditional on the occurrence of any of the principal scenarios j, an essential element, conceived during (Jasionowski et al. 2005), and often unaccounted for in risk concept proposals, such as e.g. (Sames 2005). Modelling of either of the elements of the equation (1) requires an in-depth analysis of historical data as well as thorough analytical considerations of all reallife processes affecting them, practical attainment of which remains a challenge, requiring inter-disciplinary approach. For a demonstration purposes, a prototype model of the probability of an event of occurrence of exactly N fatalities conditional on the occurrence of a scenario of collision ∩ flooding, that is scenario corresponding to j=1 in Table 1, will be discussed. Namely, it can be shown that prN(N|hzj=1) can be expressed as (2), (Jasionowski et al. 2005). 3 n flood

prN N hz1     wi  p j   ek  ci , j , k  N  i

j

n Hs k

(2)

Conceptualising Risk 55

Where the terms wi and pj are the probability mass functions of three, i=1 ..3, possible ship loading conditions and a j=1 … nflood flooding extents, respectively, assigned according to the harmonised probabilistic rules for ship subdivision, (SLF 47/17 2004), or (Pawlowski 2004). The term ek is the probability mass function 1 derived from (15) for the sea state Hsk, where 0  Hsk  4m , Hsk  k  4  n Hs and nHs is the number of sea states considered. The term ci,j,k(N) is the probability mass function of the event of capsizing in a time within which exactly N number of passengers fail to evacuate, conditional on events i,j and k occurring, and can be tentatively assigned with model (3). t fail  N   t fail  N   ci , j , k ( N )    ln  i , j , k    i , j , k  30   30  

(3)

The term i,,j,k (with y) represents the phenomenon of the capsize band shown in Fig. 3 that is the spread of sea states where the vessel might capsize. These can be estimated as follows:

 Hsk  Hscrit ( sij )         Hs s crit ij  y 

 i , j , k  1  

(4)

 y Hscrit   0.039  Hscrit  0.049

(5)

Where  is the cumulative standard normal distribution. The Hscrit(s) is calculated from equation (16) of Appendix 2. The sij is the probability of survival, calculated according to (SLF 47/17 2004). Its interpretation is discussed in Appendix 2. The concept of a capsize band

6

1 0.9

5

0.7

pdf(Hs)

4

0.6

3

0.5 0.4

2

0.3 0.2

1 0

pcap = cdf(Hs)

0.8

0.1 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0

Hs crititcal

Fig. 3 The concept of “capsize band”, (Jasionowski et al. 2004), for critical sea states of 0.5 and 4.0m.

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t fail  N   t fail  N   t fail  N  1

(6)

t fail  N   N fail1 t 

(7)

N fail t   N max  N evac t 

(8)

Finally, the term Nevac(t) is the number of passengers evacuated within time t , and is referred to as an “evacuation completion curve”, see Fig. 4. Such a curve can effectively be estimated on the basis of numerical simulations, (Vassalos et al. 2001).

Fig. 4 Evacuation completion curve.

It is worthwhile to now offer discussion of some preliminary results from a sample case studies demonstrating risk sensitivity to some key input parameters. The study is discussed below, after introducing the final elements of the risk model in the next section section 3.2.

3.2 Risk as a Summary Statistic The frequency of occurrence of exactly N fatalities per ship year, that is the information provided by the model (1), once all its elements are calculable, allows for direct application of fundamental concepts of statistics for describing the statistical properties of the random events in question, in this case fatalities. The most obvious one is use of a graphical plot of a relationship of this frequency with the number of fatalities. For example, it is very common to graphically plot the cumulative frequency of N or more fatalities, so referred to FN plot, given by equation (9).

Conceptualising Risk 57

FN  N  

N max

 fr i 

iN

(9)

N

Where Nmax is the total number of persons considered (e.g. number of crew, or number of passengers, or both). An example of such relationship derived on the basis of historical data, rather than analytical/numerical modelling, is shown in Figure 5 below.

Fig. 5 Example of an FN plot derived from historical data, (Lawson 2004).

However, while conceptually useful and, indeed, accepted widely as an expression of risk especially when plotted together with related criteria lines, (Skjong et al. 2005), it has been well known that for the purposes of any consistent decision making, (Evans 2003), some form of aggregate information, derived on the basis of such distributions, is required. Commonly used summary statistics, such as expected value, are examples of such aggregate information. It is hereby emphasised that this form of information be used to quantify the “chance” of a loss, or the risk. Unsurprisingly, the expected number of fatalities, E(N), given by (10) and often referred to as the potential loss of life, PLL, has been used among the pertinent profession routinely. See Appendix 1 on the form of equation (10).

Risk PLL  E N  

N max

 F i  i 1

N

(10)

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A. Jasionowski and D. Vassalos

As it is known in the field of statistics, it is always desirable to find a statistic which is not only consistent and efficient, but ideally is sufficient, as then all relevant information is contained in one such number. In case of a statistic which is not sufficient, additional information is required to convey the characteristics of the underlying random process. Since the expectation is hardly ever a sufficient statistic, it is necessary to examine what additional information is required additionally to (10) to quantify risk sufficiently comprehensively. Namely, it is well known that the public tolerability for large accidents is disproportionally lesser than the tolerance for small accidents even if they happen at greater numbers. This tendency is not disclosed in the model (10), as simply the same expected number of fatalities can result from many small accidents or a few larger ones. It is for this reason, that model (10) is also referred to as a risk-neutral or accident-size-neutral, (Evans 2003). Thus, public aversion to large accidents cannot be built into any relevant criteria based on PLL in its current form and without recurs to some form of utility function. Although there are strong suggestions negating such a need and supporting the view that a risk-neutral approach is sufficient, e.g. (Skjong et al. 2005), it does not seem that the above simple fact of real life can be ignored. Indeed, recently proposed criteria for stability based on the so called “probabilistic concept of subdivision” defy this risk-neutral philosophy, as ships with more passengers are required higher index of damage stability. Therefore, a proposal by (Vrijling and Van Gelder 1997) shown as equation (11) seems worth serious consideration as an alternative to (10), as it allows for controlled accommodation of the aversion towards larger accidents through stand. dev.  and a risk-aversion index k.

Risk  E  N   k    N 

(11)

Which form of a statistic is to be used to quantify risk should be the important next step in research in this field. To summarise this chapter, as one can see, risk can be shown to be a statistic of the loss, which can be systematically modelled and used for better informed decision making during any design process. To demonstrate potential benefits of some elements of the concept of a holistic approach to risk modelling and contained in model (1) and (10), simple case studies are discussed next.

4 Sensitivity Studies As has been mentioned earlier, there are no readily available elements related to even a handful of loss scenarios listed in Table 1 and necessary to complete model (1).

Conceptualising Risk 59

However, since it is known that collision and flooding is one of the major risk contributors, a sample study aiming to demonstrate the usefulness of the proposed risk model is undertaken here based on only this loss scenario. Consider a RoRo ship, carrying some 2200 passengers and crew, and complying with the new harmonised rules on damaged ship stability by meeting the required index of subdivision of 0.8 exactly, see Figure 6. Consider also that the evacuation arrangements allow for two different evacuation completion curves, as shown in Figure 7. The question is what is the effect of given evacuation curve on risk to life posed by this ship? The risk is calculated for nhz=1 in (1), using constant frhz(hz1)=2.48•10–3 per ship year. Element (2) is estimated based on the information shown in Figure 6 and Figure 7. Fig. 8 demonstrates the effect of the evacuation completion curve on the conditional probability prN(N|hz1). It seems that the majority of this scenario variants are either no fatalities at all or a very large number of fatalities. Note that because of this trend, the function estimates at these limits determine the ultimate risk, and that the impression of higher overall frequencies for 60-min evacuation curve seen in Fig. 8, must be viewed with care. 1 0.9 0.8 0.7

si

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.002 0.004 0.006 0.008

0.01

0.012 0.014 0.016 0.018 0.02

pi

Number of passengers evacuated [-]

Fig. 6 Elements of index A, sample case.

2000 60 minutes

120 minutes

1500

1000

500

0

0

20

40

60 80 100 time to evacuaute [minutes]

Fig. 7 Hypothetical evac. completion curve

120

140

A. Jasionowski and D. Vassalos conditional probability mass function of exactly N fatalities prN (N/hz1)

60

1.00E-01 0

500

1000

1500

2000

1.00E-02 1.00E-03

1.00E-04

1.00E-05

1.00E-06 Number of fatalities evacuation complete 60min evacuation complete 120min

Fig. 8 Prob. prN(N|hz1) for sample case

The resultant FN curve derived on the basis of (9) can be seen in the Figure 9. It seems that the distribution is not affected significantly by different evacuation completion curves, though when comparing the risk of 0.881 and 0.939 fatalities per ship year, estimated according to equation (10) for 60 minutes and 120 minutes evacuation completion curves, respectively, a difference of some 6% can be seen as a considerable design/operation target worth achieving.

Fig. 9 FN plot, effect of evacuation completion characteristics on risk

Another interesting test was undertaken, whereby a hypothetical ship with only two flooding extents possible was used together with the 60minutes evacuation completion curve shown in Figure 7. Two cases were assumed for the characteristics of flooding extents, such that in both cases the resultant index A remained the same. Namely in case 1, p11  0.1, s11  0.3000, p12  0.9, s12  0.8560, and for case 2, p12  2.1, s121  0.6000, p22  0.8, s22  0.8505, were assumed. The resultant FN curves curves are shown in Figure 10 below.

Conceptualising Risk 61

Fig. 10 FN plot, effect of subdivision on risk, 60-min evacuation completion curve

As can be seen, contrary to the commonly expressed view, two ships with different subdivision but the same index A, seem unlikely to have the same level of risk.

5 Further Work The process of conceptualising risk discussed in this article is one of the many needed before the vision of Risk Based Design, a design paradigm utilising the concept of risk for the purpose of safety verification, can become a reality. The fundamental elements that need to be in place are (a) an explicit risk model capable of accommodating a number of loss scenarios, sufficient to meaningfully quantify risk, and (b) the criteria, which will reflect both, the societal risk tolerability, and as importantly, the epistemic and the aleatory uncertainties of the proposed risk model.

6 Conclusions This paper discussed the concepts of safety and risk. Many lexical issues, often ignored in the communication on these concepts, have been pointed out. Some suggestions on more intuitive definitions have been made. A process of risk modelling has been explained to some extent, and a simple yet comprehensive model has been presented. A tentative model for probability of exactly N fatalities and conditional on the occurrence of a collision ∩ flooding loss scenario has been presented. Results of calculations of contribution to risk from this scenario have been presented. Preliminary results seem to indicate that the common notion that two different ships with the same index of subdivision correspond to the same level of risk is not justified.

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Acknowledgements The present research was funded by the EC 6th Framework Programme (20022006), project EC FP6 IP 516278 SAFEDOR SP2.1 www.safedor.org. The support received is hereby gratefully acknowledged.

References Comstock JP, Robertson JB (1961) Survival of collision damage versus the 1960 Convention on Safety of Life at Sea. SNAME Transactions, Vol 69, pp.461-522 Cox TR (1961) The algebra of Probable Inference. Oxford Univ Press Evans A W (2003) Transport fatal accidents and F-N curves: 1967-2001. Univ College London, Centre for Transport Studies, Report 073 for HSE Guarin L (2006) Risk Based Design Concept. SAFEDOR, D5.1.1 Hacking I (1975) The emergence of probability. Cambridge Univ Press Jasionowski A (2005) Damage Stability and Survivability. Training Course on Risk-Based Ship Design, TN SAFER EURORO II, Design For Safety: An Integrated Approach to Safe European Ro-Ro Ferry Design, Univ of Glasgow and Strathclyde Jasionowski A, Vassalos D, Guarin L (2004) Theoretical Developments On Survival Time PostDamage. Proc 7th Int Ship Stab Works, Shanghai Jasionowski A, Bulian G, Vassalos D et al (2005) Modelling survivability. SAFEDOR, D2.1.3 Jaynes E T (1995) Probability Theory: The Logic of Sci Washington Univ Konovessis D (2001) A Risk-Based Design Framework For Damage Survivability Of Passenger Ro-Ro Ships. PhD thesis, Univ of Strathclyde, Glasgow Lawson D (2004) Engineering disasters – Lessons to be learned. John Wiley, ISBN 1860584594 MSC 72/16 (2000) Decision parameters including risk acceptance criteria Pawlowski M (2004) Subdivision and damage stability of ships. Euro-MTEC series, Technical Univ of Gdansk SAFEDOR (2005) Risk-Based Design, Operation and Regulation for Ships. EC FP6 IP 516278, www.safedor.org Sames P (2005) Risk Based Design Framework: Design Decision Making. SAFEDOR, D5.1.2, Rev 7 Skjong R, Vanem E, Oyvind E (2005) Risk Evaluation Criteria. SAFEDOR, D4.5.2, DNV SLF 47/17 (2004) Sub-Committee On Stability And load Lines And On Fishing Vessels Safety Tagg R, Tuzcu C (2002) A Performance-based Assessment of the Survival of Damaged Ships – Final Outcome of the EU Research Project HARDER. Proc 6th Int Ship Stab Workshop, Webb Inst Vanem E, Skjong R (2004) Collision and Grounding of Passenger Ships – Risk Assessment and Emergency Evacuations. Proc 3rd Int Conf on Collision and Grounding of Ships, Izu Vassalos D (1999) Shaping ship safety: the face of the future. J. Marine technol36 (2):61-74 Vassalos D, Kim H, Christiansen G, Majumder J (2001) A Mesoscopic Model for Passenger Evacuation in a Virtual Ship-Sea Environment and Performance Based Evaluation. Pedestrian and Evacuation Dynamics, Duisburg Vassalos D, Jasionowski A, Guarin L (2005) Passenger Ship Safety – Science Paving the Way. Proc 8th Int Ship Stab Workshop, Istanbul Vrijling JK, Van Gelder PHAJM (1997) Societal Risk And The Concept Of Risk Aversion. Dep of Civil Eng, Delft Univ Of Technol, Delft

Conceptualising Risk 63

Appendix 1 A Note on the Expected Rate of Fatalities It can be shown that the area under the FN curve represents the expected number of fatalities per ship year, (Vrijling and Van Gelder 1997): N max

N max N max

i 1

i 1 j  i

 FN i  





fN  j 

N max i

 f i  i 1 j 1

N

N max

(12)

 i  f i   E N   PLL i 1

N

Appendix 2 A Note on the Interpretation of the Probability of Survival The survival factor “si” in (SLF 47/17), represents the probability of an event that a vessel survives after flooding of a set of compartments i due to a collision with another vessel and subsequent flooding. The question is what does “survive” mean? The following is the proposed interpretation. The relationship between the survival factors, considering only the final stage of flooding, and the ship parameters are: 1

si , final

Range  4  GZ  K   max  16   0.12

(13)

This has been derived from (14), i.e. the relationship between the parameters GZmax and Range and the critical sea state3, Hscrit, established through physical model experiments, (Tagg and Tuzcu 2002); and the cumulative distribution function F Hscollision of the sea states recorded at the instant of collision and given by equation (15), see also Fig. 11. Note, that for the purpose of derivation of (13), it was implied that Hscrit ~ Hscollision, whereby the relationship between (14) and (15) was obtained through regression analyses rendering directly formula (13).

 GZ max Range  Hscrit  4     16   0.12

(14)

3 a sea state causing the vessel capsizing during about half of the 30minutes scaled model tests, the damage opening modelled was that known as SOLAS damage

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A. Jasionowski and D. Vassalos

FHscollision  e  e

0.16 1.2 Hscollision

(15)

It is worth now offering the form of hard numbers in explaining the meaning of the probability (13), given the underlying relationships (14) and (15). One such form could read as follows: should the vessel suffer 100 collision incidents, say in her lifetime, always leading to flooding of the same spaces “ i ” with a survival factor of, say, si,final = 0.9, then 90 of these incidents will take place in a sea state with significant wave height below Hs(si, final)=2m, see equation (16), with the vessel remaining afloat for at least 30minutes at a relevant damage-state equilibrium (in other words the vessel will “survive”). The remaining 10 of these incidents will take place in a sea state with the significant wave height above Hs(si, final)=2m, and the vessel will not be able to sustain this for the 30 minutes after collision (the vessel will not “survive”), see the note on critical sea state3.

Fig. 11 CDF of significant wave heights at instants of collisions, eq.(15), (Tagg and Tuzcu 2002).

This interpretation follows the “philosophy” behind the formulation for the si,final, discussed in (Tagg and Tuzcu 2002), currently underlying the new harmonised Chapter II Part B of the SOLAS Convention, (SLF 47/17 2004). The relationship between the significant wave heights at the instant of collision, here equal to the critical sea state3, and the “ s ” factors is given by equation (16), (Tagg and Tuzcu 2002), which is derived from equation (15).

Hscrit ( s)  Hscollision ( s ) 

0.16  ln  ln s  1.2

(16)

The aforementioned time of 30 minutes derives from the re-scaled model test duration assumed during the campaign of physical model experiments of the HARDER project, (Tagg and Tuzcu 2002), which underlines the relationship between ship parameters and the sea states3 leading to capsize.

Evaluation of the Weather Criterion by Experiments and its Effect to the Design of a RoPax Ferry Shigesuke Ishida, Harukuni Taguchi, Hiroshi Sawada National Maritime Research Institute

Abstract

The guidelines of experiments for alternative assessment of the weather criterion in the intact stability code were established in IMO/SLF48, 2005. Following the guidelines, wind tunnel tests and drifting tests for evaluating wind heeling lever, lw1, and roll tests in waves for evaluating the roll angle, 1 , were conducted. The results showed some difference from the current estimation, for example the wind heeling moment depended on heel angle and the centre of drift force existed higher than half draft. The weather criterion was assessed for allowable combinations of these results and the effect of experiment-supported assessment on the critical KG and so forth was discussed.

1 Introduction In 2005, the IMO Sub-Committee on Stability, Load Lines and Fishing Vessels Safety (SLF) restructured the Intact Stability Code (IS Code, IMO, 2002), and the weather criterion (Severe wind and rolling criterion), defined in section 3.2 of the code, was included in the Mandatory Criteria (Part A) of the revised code (IMO, 2006). The necessity of the criterion has been recognized to ensure ship stability safety in “dead ship condition”, in which the ability to control the ship is lost. However, the applicability of the criterion to some types of ships (e.g. modern large passenger ships), which did not exist at the time of development of the criterion, have been questioned. In order to solve the problem, the alternative assessment with model experiments is mentioned in the revised code. To ensure uniform applicability of model experiments, which evaluate the wind heeling lever and the resonant roll angle, the guidelines were developed and included as Annex 1 in the revised code. However, they were set as “interim guidelines” because the feasibility, reliability and so forth are not fully clarified and it is

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_4, © Springer Science+Business Media B.V. 2011

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66 S. Ishida et al.

recognized that a considerable accumulation of the experimental experience is required to correctly evaluate the safety. Some effects of this assessment were already discussed (Bulian et al., 2004, Francescutto et al., 2004). However, they were not based on full experiments included in the guidelines. With this background the authors conducted experiments with a Ro-Pax ferry model following the guidelines and examined the above mentioned items. The previous paper (Taguchi et al, 2005, hereafter just referred as “previous paper”) reported the results except the wind tunnel tests. In this paper, the effects of this experiment-supported assessment by full experiments are reported. In the following chapters the items explained in the previous paper are mentioned concisely.

2 The Weather Criterion and its Alternative Assessment

Lever

The weather criterion evaluates the ability of a ship to withstand the combined effects of beam wind and waves. The criterion requires that area “b” should be equal to or greater than area “a” (see Fig. 1), where lw1 : steady wind heeling lever at wind speed of 26 m/s lw2 : gust wind heeling lever (lw2 = 1.5 lw1) 1 : roll amplitude in beam waves specified in the code 2 : downflooding angle or 50 degrees or angle of second intercept between lw2 and GZ curves, whichever is less.

GZ

b

lw2

lw1 a

2 0

Angle of heel

1

Fig. 1 Weather criterion

In the revised code, lw1 and 1 can be evaluated by model experiments in the following conditions and it is allowed to consider the heeling lever as dependent on the heeling angle like the broken line in Fig. 1, lw1 : to the satisfaction of the Administration 1 : when the parameters of the ship are out of the following limits or to the satisfaction of the Administration;

Evaluation of the Weather Criterion 67

- B/d smaller than 3.5 - (KG/d-1) between -0.3 and 0.5 - T smaller than 20 seconds where B, d and KG are the breadth, draft and the height of CG above keel of the ship respectively, and T is the natural rolling period.

3 The Subject Ship The subject ship is a Japanese Ro-Pax ferry. Table 1, Figs. 2 and 3 show the principal particulars, the general arrangement and GZ curve respectively. Compared to general European Ro-Pax ferries this ship has finer shape. From Fig. 3 it is found that up to about 40 degrees the GZ curve has a very small nonlinearity to heel angle. Table 1 Principal particulars

Length between perpendiculars Breadth Depth Draft Displacement Blockage coefficient Area of bilge keels Vertical center of gravity Metacentric height Flooding angle Natural rolling period Lateral projected area Height of center of AL above WL

Lpp B D D W Cb B/d Abk KG G0M

f

Tr AL Hc

Fig. 2 General arrangement

[m] [m] [m] [m] [ton] [m2] [m] [m] [deg] [sec] [m2] [m]

170.0 25.0 14.8 6.6 14,983 0.521 3.79 61.32 10.63 1.41 39.5 17.90 3433.0 9.71

68 S. Ishida et al.

1.4 1.2 GZ [m]

1 0.8 0.6 0.4 0.2 0 0

20

40

60

80

Angle of Heel [deg]

Fig. 3 GZ curve

4 Wind Heeling Lever lW1 The wind heeling lever is estimated from the heeling moment when the ship is drifting laterally by beam wind. Therefore, wind tunnel tests and drifting tests are necessary.

4.1 Wind Tunnel Tests The wind tunnel tests were conducted at “Pulsating wind tunnel with water channel” of NMRI (National Maritime Research Institute), with wind section of 3m in width and 2m in height. The test arrangement is shown in Fig. 4. The connection between the model and load cell had a rotating device for testing the model in heeled conditions. In heeled conditions, the height of the model was adjusted by the adjusting plate to keep the displacement constant when floating freely. By using the model of 1.5m in length, the blockage ratio was kept less than 5%, which is requested by the guidelines. The gap between the model and the floor plate was kept within approximately 3mm and covered by soft sheets for avoiding the effect of downflow through the gap.

Evaluation of the Weather Criterion 69

Fig. 4 Arrangement for wind tunnel tests

The wind speed was varied from 5m/s to 15m/s in upright condition and confirmed that the drag coefficient is almost constant in this speed range. For the full tests a wind speed of 10m/s was used, corresponding to the Reynolds’ number of 1.52  105, as defined by the following equation:

Re 

U B



(1)

where U  is the uniform wind speed outside the boundary layer, B is the breadth of the model and  is the kinematic viscosity coefficient of air. The horizontal force Fwind, the heeling moment M and the lift force L were measured by the load cell. The heeling moment M was converted to the one with respect to point O, defined as Mwind in the guidelines, by the following equation:

M wind  M  Fwind l cos   L  l sin 

(2)

where l is the distance from the centre of the load cell to the point O, which is defined as the cross point of the centre line of the ship and waterline in upright condition.

4.2 Results of Wind Tunnel Tests The measured drag coefficient CD, lift coefficient CL and heeling moment coefficient CM are shown in Fig. 5. They are nondimensionalized by the following equations:

 CD   Fwind   1  2  C    L   2  airU AL      L 

(3)

70 S. Ishida et al.

CM  M wind

2  1 2 AL   airU  L pp  2

CD, CL, CM

CD CL CM CM (standard criterion)

(4)

1.25 1 0.75 0.5 0.25 0

-35 -30 -25 -20 -15 -10 -5

0

5

10

15 20

25

30

35

Angle of heel (deg.)

Fig. 5 Results of wind tunnel tests

In the figure the angle of heel is defined as positive when the ship heels to lee side as shown in Fig. 4. The broken line is the heeling moment coefficient calculated from the current weather criterion (IMO, 2002, called standard weather criterion hereafter). It is characteristic in Fig. 5 that at all the quantities (CD, CL and CM) vary significantly with heel angle. As for the heeling moment, it is smaller than the standard criterion and further reduces when the ship heels, especially to lee side. The lift force is not so small and close to the drag force when the heeling angle is 5 degrees (weather side). However, the adjustment of the vertical position of the model is not necessary since the lift force is 0.7% of the displacement of the ship in the assumed wind speed of 26m/s. The result is also shown in Fig. 6 as the height of the centre of wind force above waterline , lwind, by the following equation. It can be observed that the centre of wind force is also a function of heel angle.

lwind  M wind / Fwind

(5)

Evaluation of the Weather Criterion 71 0.150 0.125

lwind (m)

0.100 0.075 Centre of lateral projected area

0.050 0.025

0.000 -35 -30 -25 -20 -15 -10 -5 0

5

10 15 20 25 30 35

Angle of heel (deg.)

Fig. 6 Height of the centre of wind force above waterline (model scale)

Although it is not requested in the guidelines, the effect of encounter angle,  , was investigated. Fig. 7 shows the wind heeling moment coefficient, CM. Here  <90 means following wind. The figure shows that the wind heeling moment is almost at the maximum in beam wind condition (  =90 degrees). This fact supports the assumption of existing regulations. However, for developing performance based, physics based criteria, the information on the effect of encounter angle to heeling moment might be necessary.

0.75

15 (deg.) 30 45 60

CM

0.50

75 90 105

0.25

120 135 150

0.00 -35 -30 -25 -20 -15 -10 -5 0

5

10 15 20 25 30 35

165

Angle of heel [deg.]

Fig. 7 Wind heeling moment coefficient for various encounter angles

72 S. Ishida et al.

4.3 Drifting Tests The detail of drifting tests was reported in the previous paper. Here, the height of the centre of drift force above waterline, lwater, is shown in Fig. 8. lwater was calculated in the same manner as equation (3). The angle of heel is positive when the ship heels to the drift direction. The drift speed (towing speed) was decided, as requested in the guidelines, to make the measured drift force equal to the wind force at the wind speed of 26m/s in ship scale. Because drifting tests were conducted before wind tests, the wind drag coefficient, CD, was assumed to be from 0.5 to 1.1. In upright condition and CD=0.8 the drift speed was 0.195m/s (1.80m/s in ship scale). CD=0.5 CD=0.7 CD=0.9 CD=1.1

0.6 0.4

CD=0.6 CD=0.8 CD=1.0

Iwater / draft

0.2 0.0 −20 −15 −10 −5 0 −0.2 −0.4

5

10

15

20

25

30

35

Angle of Heel (deg.)

Half draft

−0.6

Fig. 8 Height of the centre of drift force above waterline

Fig. 8 shows that the centre of drift force is above half draft (which is assumed in the standard criterion) and is generally above the waterline for this ship. This phenomenon was reported by Hishida and Tomi (1960), Ishida (1993), Ishida and Fujiwara (2000) and referred in IMO/SLF (2003). This is due to the more dominant effect of the bottom pressure distribution than the side pressure when breadth/draft ratio is large. For the cross sections with this proportion, high position of the centre of sway force can be easily found in hydrodynamic tables of Lewis Form. This fact suggests that potential theory would explain this phenomenon. However, the effect of separated flow, e.g. from bilges, was also pointed out (Ishida and Fujiwara, 2000). It was confirmed experimentally in the previous paper that lwater reduces when the draft is enlarged.

Evaluation of the Weather Criterion 73

4.4 Determination of lw1 The heeling moments by wind, Mwind, and by drifting, Mwater, both around point O, were divided by the displacement,  , and the wind heeling lever, l w1, was calculated as a function of heel angle (equation (6)). Figs. 9 and 10 show the results.

M wind  M water

(6)

 Standard Criterion Wind test Drift test Wind + Drift tests

0.15 0.125 0.1 IW1 [m]

0.075 0.05 0.025 0 −0.025 −0.05 −20

−10

0

10

20

30

40

Angle of Heel (deg.)

Fig. 9 Wind heeling lever, lw1, evaluated by the tests

1

0.5 GZ, Iw1 [m]

lw1 

0 −20

−10

0

−0.5

10

20

30

40

50

GZ Standard Criterion Wind + Drift tests

−1 Angle of Heel [deg]

Fig. 10 Wind heeling lever, lw1, compared with the GZ curve

74 S. Ishida et al.

In Fig. 9, the heeling levers due to wind ( M wind  ) and drift motion ( M water  ) are also included. In both figures, lw1 at angles greater than 30 degrees (tested range) is assumed to keep the same value as at 30 degrees as prescribed in the guidelines. Figs. 9 and 10 show that, in the considered case, the wind heeling lever estimated by wind and drift tests is sensibly smaller than that required by the standard weather criterion.

5 Roll Angle 1 The formula of roll angle 1 in the weather criterion implies the maximum amplitude out of 20 to 50 roll cycles in beam irregular waves. And 1 is related to the resonant roll amplitude, 1r , in regular waves, whose height and period are equal to the

significant wave height and mean wave period of the assumed irregular waves (IMO, 2006, Watanabe et al., 1956). The reduction factor is 0.7 (see equation (7)) and this alternative assessment estimates 1r instead of 1 by model experiments.

1  0.71r

(7)

5.1 Direct Measurement Procedure In the guidelines, this procedure is called “Direct measurement procedure” because the resonant roll angle, 1r , is measured directly in waves with the steepness specified in the IS Code and the period equal to the natural roll period. The results of experiments were mentioned in the previous paper. Here, Fig.11 is shown again. In the figure, “s” is the wave steepness, which is tabled in the Code as a function of the natural roll period. For this ship s=0.0383 (1/26.1), but lower steepnesses were also used. Due to the linearity of the GZ curve the amplitudes reach the maxima at the vicinity of the natural roll frequency in all steepnesses. From this result, 1 was decided as 19.3 degrees (=0.7 1r ).

Roll Amplitude / Wave slope

Evaluation of the Weather Criterion 75 7

S=1/60 =1/40

6

=1/26.1 5 4 3 2 1 0 0.6

0.8

1.0

1.2

1.4

Wave frequency / Natural roll frequency

Fig. 11 Roll amplitude in regular waves

5.2 Alternative Procedures In the guidelines, alternatives procedures are included, i.e. “Three steps procedure” and “Parameter identification technique (PIT)”. In the “Three steps procedure”, the roll damping coefficient N is estimated by roll decay tests. And, the effective wave slope coefficient r is estimated by roll motion tests in waves with smaller value of s. Finally, 1r (degrees) is calculated by the following equation:

1r 

90 rs N 1r 

(8)

This method was adopted when the standard weather criterion was developed and is based on linear theory except roll damping. The previous paper showed that the estimated value of 1 is 19.5 degrees when the resonant roll amplitude at s=1/60 is used. This value of 1 is very close to that by the “Direct measurement procedure” due to the linear feature of GZ curve. The PIT is a methodology to determine the parameters included in the equation of roll motion. Once all the parameters are decided by test data at small wave steepness, the roll amplitude at prescribed s can be extrapolated. In the guidelines, an equation with 9 parameters is presented, in which nonlinear features of roll damping, GZ curve and wave exciting moment are included. PIT analysis is not carried out in this paper. However, the difference of 1 by PIT from other 2 procedures is expected to be limited because of the linearity of GZ curve.

76 S. Ishida et al.

6 Alternative Assessment of the Weather Criterion In the guidelines, simplified procedures on the wind heeling lever, lw1, are also mentioned for making the assessment practically easier. For wind tunnel tests, the lateral horizontal force Fwind and the heeling moment Mwind can be obtained for the upright condition only and considered as constants (not depending on heeling angle). Instead of drifting tests, the heeling moment Mwater due to drift can be considered as given by a force equal and opposite direction to Fwind acting at a depth of half draft in upright condition, as assumed in the standard criterion. And the combinations of complete procedures and simplified procedures are to the satisfaction of the Administration. The comparison of the assessments of the weather criterion using experimental results is summarized in Table 2. In the table all the possible combinations of the wind tests and the drifting tests, complete procedures and simplified procedures are included. As for 1 , the standard criterion and the result of “Direct measurement procedure” are included. The results of “Three steps procedure” can be omitted here since the estimated 1 was almost equal to the one of “Direct measurement procedure” for this ship. Table 2 Assessments of weather criterion by experiments

The last line of Table 2 shows the critical values of the vertical centre of gravity (KG), in which the ratio of area b/a=1 (see Fig. 1). These last results are to be taken with some caution, since the effects of changing the vertical centre of gravity on Tr (natural roll period) and on the other quantities related to roll motion, including 1 , have been neglected. Table 2 shows that the alternative assessment by model experiments can change b/a significantly with respect to the standard criterion. For this ship, b/a’s of the right side of the table are smaller than those of the left side. This tendency comes from the increased 1 obtained by experiments and it was suggested in the previous

Evaluation of the Weather Criterion 77

paper that 1 was enlarged by the small damping coefficient (N=0.011 at 20 degrees). On the other hand, the lw1 evaluated through all the combinations of the wind tests and drifting tests, complete procedures and simplified procedures, tends to make b/a larger than the standard criterion. It has to be noted that the leading cause of the fluctuations is the large variation in the vertical centre of hydrodynamic pressure when evaluated through the drifting tests. Fig. 12 shows the critical values of KG. From Table 2 and Fig. 12, it is recognized that the changes in the critical value of the vertical centre of gravity are more contained than b/a, but as much as 0.4m at the maximum from the standard criterion.

Fig. 12 Variation of critical KG

7 Conclusions In this paper, the results of the alternative assessment of the weather criterion by model experiments are reported. Almost full tests, included in the interim guidelines, were conducted. Considerable fluctuations of the assessment and their reasons are clarified. In order to make the assessment more uniform and to remove the word of “interim” from the guidelines, more extensive confirmation from the experience gained through the application of the guidelines is needed.

Acknowledments Some parts of this investigation were carried out as a research activity of the SPL research panel of the Japan Ship Technology Research Association in the fiscal year of 2005, funded by the Nippon Foundation. The authors express their sincere gratitude to the both organizations.

78 S. Ishida et al.

References Bulian, G, Francescutto, A, Serra, A and Umeda, N (2004) The Development of a Standardized Experimental Approach to the Assessment of Ship Stability in the Frame of Weather Criterion. Proc. 7th Int Ship Stab Workshop 118-126 Francescutto, A, Umeda, N, Serra, A, Bulian, G and Paroka, D (2004) Experiment-Supported Weather Criterion and its Design Impact on Large Passenger Ships. Proc. 2nd Int Marit Conf on Des for Saf 103-113 Hishida, T and Tomi, T (1960) Wind Moment Acting on a Ship among Regular Waves. J of the Soc of Nav Archit of Japan 108:125-133 (in Japanese) IMO (2002) Code on Intact Stability for All Types of Ships Covered by Instruments. Int Marit Organ IMO (2003) SLF46/6/14 Direct Estimation of Coefficients in the Weather Criterion: submitt by Japan IMO (2006) SLF49/5 Revised Intact Stability Code prepared by the Intersessional Correspondence Group (Part of the Correspondence Group’s report): submitt by Germany Ishida, S (1993) Model Experiment on the Mechanism of Capsizing of a Small Ship in Beam Seas (Part 2 On the Nonlinearity of Sway Damping and its Lever). J of the Soc of Nav Archit of Japan 174: 163170 (in Japanese) Ishida, S and Fujiwara, T (2000) On the Capsizing Mechanism of Small Craft in Beam Breaking Waves. Proc. 7th Int Conf on Stab of Ships and Ocean Veh (STAB2000) B:868-877 Taguchi, H, Ishida, S and Sawada, H (2005) A trial experiment on the IMO draft guidelines for alternative assessment of the weather criterion. Proc. 8th Int Ship Stab Workshop Watanabe, Y, Kato, H, Inoue, S et al (1956) A Proposed Standard of Stability for Passenger Ships (Part III: Ocean-going and Coasting Ships). J of Soc of Nav Archit of Japanese 99:29-46

Evolution of Analysis and Standardization of Ship Stability: Problems and Perspectives Yury Nechaev St.Petersburg State Marine Technical University, Russia

Abstract

This paper describes the research and development in the field of ship stability in waves, including works from more than 50 years of the author’s experience. The first works of the author on stability were submitted to IMO in 1965, regarding development of stability regulations for fishing vessels.

1 Introduction Analysis of the behavior of a ship under action of external forces in various operational conditions is one of the most complex problems of dynamics of nonlinear systems. Extreme situations present an especially difficult challenge. All of the mathematical models describing the behavior of a ship in waves have a similar structure; all of them are nonlinear. Weak nonlinearity is a relatively wellstudied area of a ship dynamics. General methods for obtaining solutions and effective algorithms for practical applications exist. The methods for the analysis of ship stability in waves are based on more complex mathematical models and advanced computational methods, due to the nonlinearity of these problems. The analysis of nonlinearity allows insight into the problem and the ability to construct a general theory describing ship behavior as a nonlinear dynamical system for the control and forecasting of stability changes in various conditions of operation. The in-depth research of ship dynamics in waves requires understanding of physical laws, effects, and phenomena.

2 Phases of Stability Research The research of ship stability in waves is complemented by the development of intelligent decision support systems. More frequently these applications use artificial intelligence. These technologies are focused on the process chain: M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_5, © Springer Science+Business Media B.V. 2011

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modeling-forecasting-decision-making using complex integrated intelligent systems (IS). The analysis of “informational snapshots” of the investigated situations allows one to formulate the basic principles determining “latent” information processes. Currently new approaches for intelligent technologies, including non-algorithmic control which is naturally parallel and non-deterministic, are under development. The period of development of the studies about ship stability is marked by a change of paradigms as shown in Fig. 1. Theoretical-experimental paradigm marks the beginning of research on ship stability in waves. At this stage, the basis of experimental and theoretical methods occur within the framework А. N. Кrylov’s hypothesis- the ship does not affect the wave (Lugovsky 1966). The main attention here is given to the development of theoretical models for the description of the righting moment and estimation of dynamic roll of a ship in waves (Boroday and Netsvetaev 1969, 1982). Physical pictures of roll and capsizing ships from experimental research with radio-controlled models of ships with natural waves were also used (Nechaev 1978, 1989). The basic scientific result of the research was the determination of critical situations for a ship in waves (complete loss of stability, low-frequency resonance, broaching) and justification of the necessity for theoretical models of dynamics to account for nonlinear wave interference components — for ships with small L/B ratios having high Froude numbers. The generalization of the results using physical models allowed the author to develop linear regression formula (submitted to IMO in 1965 (Nechaev 1965)), describing the function of the righting moment for a preliminary estimation of stability in waves. The main works of Russian researchers on this paradigm, were performed within 1960-1970. The theoretical-experimental paradigm was implemented on the basis of digital and analog computers with serial processing. Computer paradigm, where the majority of works on ship dynamics in waves is currently carried out in Russia, is connected to the further development of stability theory using methods and models of research of nonlinear dynamical systems (ex. Fokker-Plank-Коlmogorov equation, method of Monte-Carlo, method of moments, phase plane, catastrophe theory), and also using modern methods of data processing of physical experimental data with fast algorithms. Most of the principle advancements in ship dynamics in waves were achieved within the frames of this paradigm (works performed during 1971-1995). Development of software and analysis procedures for ship stability in waves was the most important part of these works; with the particular emphasis on probabilistic models and criteria (Boroday et al. 1989, Nechaev 1978, 1989). Availability of high-performance computers allowed the author to carry out robust analysis of experimental results and develop a nonlinear regression model describing the spatial function of the righting moment on waves (Nechaev 1978, 1989). Different types of bifurcations, including strange attractors and the phenomenon of deterministic chaos were also studied within the framework of the computer paradigm; leading to a better understanding of the behavior of nonlinear dynamical systems describing large-amplitude motions of a ship in waves. Implementation of

Evolution of Analysis and Standardization of Ship Stability 81

the computer paradigm is related with processing of information using parallel and network computing.

Fig. 1 The approaches to paradigms for information processes for research of ship stability in waves

The phenomenon of artificial intelligence; its application in ship dynamics marked a new paradigm. Artificial intelligence is a principle component of onboard intelligent systems (IS) with intended applications for prediction and control of ship motions in waves as well as monitoring of safe navigation. Development of these systems involves significant difficulties related to formalization of knowledge and very complex computation algorithms. This concept considers stability in waves as a complex, open set of coordinated dynamical procedures that are carried out for analysis of extreme situations with an objective of safe navigation. Implementation of new technology for ship control and the prediction of stability on waves requires extensive use of mathematical modeling and supercomputers (Nechaev et al. 2002, Nechaev and Zavialova 2002). The development of new generation IS with a dynamic knowledge base and a system of adaptation requires reconsideration of general principles of organization and formalization of knowledge of interaction between a ship and its environment, as well as with hardware and software (Nechaev and Petrov 2005). The requirements of self-training and self-organizing in a continuously changing and fuzzy environment

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lead to re-examination of the contents of adaptive components of the knowledge base. These requirements also have increased a role of mathematical modeling for prediction of ship behavior in waves as a complex dynamic object (DO). Most of the information here comes from sensors of a measuring system; analysis, prediction and interpretation of extreme situations is carried out on the basis of this information. As a result, problems of integration of information and an analysis of alternatives were revealed. Complimenting the knowledge base with solvers for numerical, logical, and combinatorics problems, enabled the author to achieve significant progress in one of the most complex areas of knowledge engineering – development of methods for processing of information in dynamic environment. The development of new methods of control of information processing gave birth to new generation onboard IS for motion and stability control. The associative approach of this technology allows organizing the process of making decisions for control of a dynamical object by integration of information and algorithms, taking into account restrictions and under-determinate of models. This new paradigm of the control and prediction of ship stability in waves with on-board IS is focused on extreme situations while using a wide variety of mathematical models (Nechaev et al. 2001, 2006). The model of an extreme situation is based on direct interaction with the environment; in this context the application of parallel computing is both necessary and natural. This new type of control is based on the measurements from sensors and results of simulation; such an approach considerably changes organization of data processing by the onboard IS, making it distributed and independent of number of processors. A new conceptual approach is underway to substitute non-random and consecutive computational processes; this process is based on models of the current situation and associative self-organizing in parallel computational processes. Thus, the research and development in the area of ship stability in waves allows consideration of a model of ship/environment interaction as a complex holistic open system. One of the main focuses of the analysis is the development of integrated on-board IS, using models based on fuzzy logic and neural networks. It allows development of more flexible systems and adequate description of a phenomenon of ship behavior as complex dynamic object in conditions of uncertainty and incompleteness of the initial information.

3 Analysis and Criteria of Stability of Ship in Waves. Background Theory and Validation Dynamics of the interaction of a ship with environment as a problem of stability in waves generally can be described as follows:

Evolution of Analysis and Standardization of Ship Stability 83

dx  f ( X , Y , t ), x(t0 )  X 0 , dt F ( X , t )  0, t  [t0 , T ]

(1)

where Х is a n-dimensional vector of phase coordinates; Y is a m-dimensional vector stochastic excitations; F(X, t) is the area of phase coordinates corresponding to a safe operation; f ( X , Y , t ) is a vector-valued function describing the dynamical system and x(t0 )  X 0 are initial conditions. The solution of a problem (1) results in allowable values of output, used for formulation of stability criteria:

R1  Q j ,

j  1,…, J ;

Q1  1 | z1*  FG1  z *j* , "

(2)

Q1   J | z  FG1  z , * 1

** j

KGCR  KGmin . Here 1 , … ,  J are the area of possible values of criteria, taking into account uncertainty and incompleteness of the initial information; KGCR — is the critical elevation of a centre of gravity of a ship. Using a concept of KGCR, it is possible to associate probability of capsizing with a probability of a parameter exceeding some allowable value

P Z G  Z CR   P C1 ,…, Ck  dC1 ,…, dCk



(3)



where  is the area where variables C1 , … , Ck satisfy the following inequality

Z G (C1 , … , Ck )  Z CR

(4)

An alternative formulation of a problem is that the value of KGCR can be exceeded with the given probability P0 :

R( Z G  Z CR )  P0

(5)

Therefore, the problem formulation (1), (2) consists of developing an algorithm for the analysis and judgment of stability in waves with an estimate of correctness

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of formulation of criteria in the conditions of uncertainty and incompleteness of the initial information. The following theoretical principles are formulated for the research of ship stability in waves and further development of stability standards. Consider an information operator processing information on dynamics of interaction of a ship with environment. A structure of this operator should reflect the following properties of information flows (Nechaev 1995):  Relatively small uncertainty in input may be related with significant uncertainty of the output. This uncertainty must be accounted correctly, as the output represents vital stability information.  Formulation of stability criteria should take into account properties of a ship as dynamical system under external excitation. A normal operating area is around stable equilibria with a corresponding safe basin parameter space. The behavior of system on and in vicinity of the boundaries depends on the physics of the problem. Uncertainty leads to appearance of a “gray” area limited by “safe” and “unsafe” boundaries. Small excursions through the “safe” boundary lead to equally small changes in the state of the dynamical system, while similar excursions through the “unsafe” boundary lead to a qualitatively different state of the system that is incompatible with safe operation. Practical implementation of these theoretical principles for setting boundary values for criteria results in difficulties caused by the uncertainty and incompleteness of the input data. The generalized model developed within the framework of this approach allows formal consideration of ship stability in waves, including both analysis and criteria. Research of ship stability in waves includes consideration of direct and inverse problems. The solution of a direct problem is the transition from a known structure and inputs of a dynamical system to the characteristics of output and stability criteria. The objective of the inverse problem is synthesis; it is transition from the desirable characteristics and known criteria to unknown structure of the dynamical system and characteristics of its components. These models characterize the theoretical formalized core of stability regulations. Formulation of the problems of the control and prediction of stability in waves is inherently related with conditions of uncertainty and incompleteness of the input data. Therefore, the problem of validity of the mathematical models becomes the most important. Let Fi (a), (i  1, … , n) be a criterion of validity defined from the analysis of a mathematical model describing a certain type (or types) of ship motions. This criterion is either a function of parameters a j , ( j  1, … , k ) , or a function of the solution of differential equations. The coefficients of mathematical model a j , satisfying given parametrical, functional and criteria restrictions, make an allowable area Ea in a space of criteria F ( Ea ) . Setting the accuracy of approximation of parameters a j as  j , ( j  1, … , k ) , and the accuracy of criteria

Evolution of Analysis and Standardization of Ship Stability 85

 i , (i  1, … , n) , it is possible to present criterion for the proximity of the data from mathematical modeling to data from physical experiments, which is the criterion of validity of the mathematical model:



F  F1P  F1E , … , FnP  FnE



(6)

where index P stands for calculation and E for experiment. The approximation of area Ea is necessary for the evaluation of coefficients of a mathematical model with the given accuracy. It is achieved by finding values such as:

min F (a)  F (a r ), a  F (a ), r  1, … , N

(7)

under conditions for determining allowable area Ea

Fi P  Fi E  Fi**

(8)

Here a1 parameters that are within the boundaries a*j  a j  a*j* , N – number of experiments; Fi** - allowable criteria limits (level of validity), assigned with account for accuracy of the experiment. Attempt to make dependences (6)-(8) specific, leads to the following formulation. Let's present mathematical model (1) as local discrete transformation:

Si   m (Wi , Si1 , Vi ) ,

(9)

where Si is the modeled state of a ship in the i-th instant of time; while Si 1 is the state corresponding to the previous instant of time i – 1; Wi and Vi are variables discriminating state of environment and internal state of the dynamical system in the i-th instant of time. The condition of validity is determined as

Y S

 c  Cm ,

(10)

where S is a domain limiting meaningful behavior of model; Cm is a set of objectives for modeling. For example, if a model of ship motions is linear and its domain of meaningful behavior is limited to relatively small-amplitude motions, it is valid if the objective is a conventional seakeeping analysis. Development of the model, indeed, includes a choice of the type of the model and its domain. To quantify deviation from a penalty function is introduced:

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 0 if Y  S .  Y , S     Y , S  if Y  S

(11)

The discussion at IMO during 1965-67 was focused on the problem of validity of mathematical models of stability and the correct design of physical experiments. Observed differences between the theoretical and experimental data may be used as criteria of validity with the peak data of motion time history (deterministic or probabilistic expressions for linear or angular motions, velocities and accelerations), as well as the limitation of divergence between time histories of phase trajectories. Alternatively, the applicability of theoretical distributions (Gaussian, Rayleigh, Weibull, etc) can be tested against the empirical data. The improvement of mathematical models of ship stability in waves can be facilitated by improvement of the theoretical background and experimental technology of hydro-aerodynamic research, the use of effective methods for processing and analyzing experimental results. The validity of mathematical models can be improved by the development of more reliable ways of evaluation of components of these models by better accounting for distortions that a moving vessel introduces into the wave field (interference and diffraction of waves, change of a field of pressure), as well as a more complete description of the spatial and temporary structure of wind air-flows and roll damping forces. Mathematical models for stability in waves (Boroday and Netsvetaev 1969, 1982, Lugovsky 1966, 1971, 1980, Nechaev 1978, 1989) are usually developed for extreme situations. The background of these models is a description of the ship interaction with environment, so it is a nonlinear dynamic system with six degrees of freedom (Кrylov , 1958). A specific type of mathematical model is determined by the physics which are observable during experiments. For example, when considering stability in following waves or stability in breaking waves it is enough to use the differential equations of drift and roll. While considering broaching, it is necessary to use a system of at least four differential equations, including the surge, sway, roll and yaw. In more complex situations, it is necessary to formulate hypotheses with all six degrees of freedom included. It is important to include a term describing continually changing stability in waves in the equation of roll motions. The nonlinear spatial function of the righting moment in waves is represented by the following formula (Nechaev, 1978,1989):

M W  M ( ,  , t )  D[l ( ,  )   l ( ,  ) cos( k t   )]; l ( ,  )  0.5[l ( ,  ) max  l ( ,  ) min ],  l ( ,  )  0.5[ l ( ,  ) max   l ( ,  ) min ]; (12) M W   ( ,  k , t )  D l ( ,  , t ), where  l ( ,  ) max and  l ( ,  ) min are the magnitude of stability changes while the ship is on a wave crest or a wave trough, at various course angles ï ; l ( ,  ) is

Evolution of Analysis and Standardization of Ship Stability 87

the righting arm determined by interpolation on  and  for the various moments of time,  is the phase of a ship relative to a wave. 0,3

M( , ,t)/D 1

2

0.3

3

0,2

,t)/D M( ,

2



1

0.2

0,1

0.1

0

0 -0,1 0



80



60

30

40



o

-0.1 0



20

M( , ,t)/D 3

40

20

0

90

60

80

o



M( , ,t)/D

2

1

60

k

o



0

0.3

1

0,2

2



0.2

0 0.1 0

-0,2

80

60 o



40

20

0

0







o



-0.1 0 30

k

60 90

0

20

40

60

80

o



Fig. 2 Nonlinear function, describe of righting component: 1 – initial function; 2 – transform function; 3 – wave shape

The geometric interpretation of function M ( ,  , t ) is given in Fig. 2.

4 Methods for the Analysis of Nonlinear Dynamic Systems in the Modeling of Ship Stability in Waves The research on ship stability in waves as a complex nonlinear dynamic system requires understanding of the physics governing the phenomena. These problems not always can be resolved within the frames of existing concepts and approaches and may require development of the new approaches, methods and models. Nonlinearity is, probably, one of the most important properties of the system, which defines the methods applicable for analysis, such as: method of MonteCarlo, method of the moments, method of action functional, equation of FokkerPlank-Кhоlmogorov, method of a phase plane and catastrophe theory. These methods are helpful in both the qualitative and quantitative interpretation of complex physical phenomena associated with stability of ship in waves. The author applied the Monte-Carlo method for stability and roll of a ship in waves in 1975; it is described in the monograph (Nechaev, 1978). This method plays a role of a bridge between theory and experiment. Being quite a powerful tool of research in statistical dynamics, the Monte-Carlo method is related with handling rather extensive volumes of data. It requires high speed of computation

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and a large volume of memory. For the solution of such problems, high-performance computers with parallel data processing technologies are best. The mathematical model for the application of the Monte-Carlo method is transformed into a dynamical system containing stochastic nonlinear function representing the righting moment. This function can be presented as

M ( ,  , t )  {1   (t ) cos[k t   0 (t )]}(2  a  2 sgn  ). Jx   Jx

(13)

Parameter (t) and phase (t) are considered random variables with normal and uniform distribution, respectively.

 (t )  {M *[  (t )], D*[  (t )]},  0 (t )  [0, 2 ]

(14)

The hypothesis of Gaussian distribution of the process (t) is checked by the analysis of the time history of process. The correlation function R* (t ) of the process (t) is estimated using the data of real sea waves. Calculations show, that the function R* (t ) can be presented with the following approximation:

R* ( )  D *[  (t )] exp[ (  ) | t   |] cos  (  )(t   ),

(15)

where () and () — parameters to be found by the calculations. For modeling the stochastic process (t), the method of forming filters is used in time domain. The parameters of the filter can be found from variance D *  (t ) and autocorrelation function R* ( ) . The filter is described by system of the differential equations:

d    (  )  (t )   (  ) (t )  2 (  ) D  * W1 (t ); dt d    (  )  (t )   (  ) (t )  2 (  ) D * W2 (t ), dt

(16)

where M *[W1 (t )W1 ( )  * (t   ); M *[W1 (t )W2 ( ) 0; M *[W2 (t )W2 ( )  * (t   ); M *[ (t ) 0, M *[  2 (0) D  * ; M *[ 2 (0) D * , D   D ; M * [  (0) (0) 0.

(17)

Evolution of Analysis and Standardization of Ship Stability 89

Here W1 (t ) and W2 (t ) are white noise stochastic processes; () and () are parameters of the autocorrelation function; (t) is the auxiliary stochastic process; M * and D* are operators of mean and variance;  * is the delta function. This mathematical model is a part of practical procedure of modeling the system with the given initial conditions. An important point here is the necessity of formation of the random initial conditions determined by expressions (17). As shown by calculations, the Monte-Carlo method has a number of advantages over other methods of solution (method statistical linearization, theory of Markov processes) for problems of statistical dynamics of nonlinear systems. The main advantages are compactness of the calculation scheme, stability of results in a case of hardware failure, and a rather simple way to estimate the accuracy of results. Another convenient feature is that there is no limitation on the structure of differential equations as any type on nonlinear terms can be included. A traditional approach of classical dynamics considers either deterministic (regular) or stochastic (irregular) processes described by the appropriate nonlinear differential equations of roll motions. However, the nonlinear deterministic dynamical system can exhibit chaotic behavior, when phase trajectories that were initially close, become divergent in the limited area of phase space. This phenomenon of the nonlinear system is known as deterministic chaos. Nonlinear dynamics makes a clear distinction between simple (usual) and strange (chaotic) attractors. The simple attractors are encountered quite frequently when describing nonlinear roll and capsizing of a ship on waves. The geometrical interpretation of a simple attractor describing a dissipative system represents on a phase plane either a focus or a limit cycle. For both these cases, all phase trajectories have a shape of spirals that converge to a stable equilibrium or to steady-state stable cycle. Strange attractors have more complex structure, as they contain both stable (attractors) and unstable (repeller) trajectories. Essentially, they are saddletype trajectories that are being attractors in one dimension and repellers in another; they form a set of layers connecting to each other in quite complex ways.

Fig. 3 Formation of a limiting cycle caused by of group waves

The papers (Degtyarev and Boukhanovsky 1995, 1996) describe the development of parametric resonant motions of a ship caused by passage of group of waves. It is shown that complex structures of oscillatory roll modes are formed in this case. These structures are sets of attractors looking like unstable limit cycles. The cycleattractor can lose stability through several scenarios. A passing group of large

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waves stabilizes the limit cycle described due to the nonlinearity of large-amplitude response (Fig. 3). This cycle appears when the wave height in a sequence of waves in a group exceeds the critical value of creating enough parametric excitation to reach practically constant amplitude of response. However, the subsequent gradual reduction of wave heights breakes the conditions for parametric resonance and the cycle disappears. A more complex scenario is realized when the stable cycle collides with an unstable cycle (Fig. 4). Such a situation is much rarer and it can be generated by a consecutive passage of waves groups containing wave of significantly different height. The first wave group with small heights results in a relatively smallamplitude limit cycle, while the second group generates large-amplitude cycle. Appearance and loss of stability of an oscillatory mode (“birth and death of a cycle” using a terminology by А.А. Аndronov) occur due to limitedness of a rolling resonant zone on a rather small time interval where large-amplitude motions are generated by a passage of wave groups.

Fig. 4 Appearance (A) and loss of stability of a cycle (B) (C)

The problems and methods of controlled chaos is an area of intensive research in recent decades. In the beginning, the concept of deterministic chaos was considered as an exotic phenomenon that is of interest to mathematicians only; the very possibility of occurrence of chaotic response in practical applications was doubtful. However, later on, chaotic dynamics was found to be present in many dynamical systems across disciplines: in mechanics, laser physics and radiophysics, chemistry and biology, economy, and health science. Systems with chaos demonstrate both good controllability and surprising flexibility: the system quickly reacts to external excitations, while keeping a mode of motions. The combination of controllability and flexibility in part is the reason why chaotic dynamics is a characteristic type of behavior for many dynamical systems describing ship motions in waves. From the point of view of the synergetic approach, chaotic systems represent possibilities for realization of self-organizing processes. One of the most typical scenarios for the transition to chaos is through a sequence of period-doubling bifurcations, which is observed for systems with viscous friction under action of excitation forces. The research of chaotic behavior of a nonlinear dynamical system describing roll motions was carried out by systematic variation of control parameters. The result was shown on Poincare map, allowing to clearly see the

Evolution of Analysis and Standardization of Ship Stability 91

period-doubling and sub-harmonic bifurcation (Nechaev 1993). Another scenario is the mechanism of transition to chaotic response, known as intermittency, where intervals of deterministic chaos alternate with almost periodic motions (Fig. 5).

t

1

2

A

B

Fig. 5 Occurrence of chaos of an alternated type: А – Duffing model; IB – Mathieu model at presence of an initial heel angle.

The Lyapunov exponent is an important tool for study of behavior of chaotic systems. It can be demonstrated with the Duffing equation, which is one of the interpretations of nonlinear rolling (Nechaev, 2007). Introduction of the third dimension for the independent variable allows presentation of the Duffing equation as a system of the third order: x2   f1 ( x)      3 f ( x)   f 2 ( x)     x1  k x2  B cos x3    f ( x)   1  3   

(18)

This system is defined with two positive parameters: coefficient of linear damping k and amplitude of excitation B. Negative divergence is determined by damping (dissipation)

 f1  f 2  f 3    k  0  x1  x2  x3

(19)

The linearized system leads to the following matrix

 0  A    3x12  0 

1 k 0

0    B sin x3   0 

(20)

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The results of calculations for В = 10, k =0.1 are shown in a Fig. 6. These parameters lead to the appearance of chaotic behavior of the system (18), which is confirmed by time histories, phase trajectories and the Lyapunov exponent. Theoretical development of chaotic dynamics has revealed a series of possible practical applications, including ship motions and stability in waves, where the chaotic responses may be encountered as a result of complex interaction of the ship and environment. Moreover, there are possible practical applications, where the control over nonlinear system is realized by changing a degree of its “chaoticness”. Methods for the solution of similar problems have been developed recently, especially for application of different control algorithms for chaotic systems.

Fig. 6 Attractor and Lyapunov exponents for the Duffing equation

5 Analysis of Extreme Situations and Control of Ship Stability in Waves The evaluation and prediction of stability changes for motion control of a ship in waves are carried out using the information from acting sensors (Nechaev (ed.) 2001). In this case, the extreme situation caused by deterioration of stability can be described in compact set with variable z  (1, … , m) . The analysis produces scenarios for the possible development of extreme situations, taking into account physics of the interaction between a ship and environment. When the probabilities of encounter of extreme situations are estimated, measures of statistical uncertainty have to be evaluated as well. Let number of situations be N, while M(w) is an average

Evolution of Analysis and Standardization of Ship Stability 93

estimate, and D(w) is the variance estimate of a value representing possible consequences of the situation.

N 0j  q j N 0

( j  1, … , m)

(21)

where N 0 is the predicted total number of extreme situations for the considered period of time; q j is the fraction of situations for the j-th class in the distribution F(w), determined by the formula

q j  P ( wmin j  w  wmax j )

(22)

The value q j is calculated with known distribution F(w) and its parameters M(w) and D(w). The accuracy of the forecast under the formula (21) is characterized by the standard deviation:



 [ N10 ]  q j [ N 0 ]  N 0 [q j ] 





(23)

2 1/ 2

2



2





2

random deviation q j [N 0 ] and N 0 [q j ] included in the formula (23) are determined by statistical processing of results from the mathematical modeling. The development of an algorithm for the extreme situation prediction is carried out using the information from a measuring system. This information is preprocessed and represents a time-series reconstructed on the basis of the Takens’ theorem. According to the Takens’ theorem, a phase portrait restored as:

Z (t )  [ y (t )]  [ y (t ), y (t   ), … , ] y[t  (m  1) ]  [ z1 (t ), …, zm (t )]

(24)

is equivalent topologically to the attractor of initial dynamic system.

6 Method of Action Functional The search of a universal principle that could help to describe behavior of a dynamic system, has resulted in discovery of a principle “of least action”, similar to a principle of “minimum potential energy” that determines the position of equilibria. According to a principle of the least action, the variation problem is a search of a stationary value of the action integral, defined on the interval (t1 , t2 ) . Responses of the dynamical system corresponding to all possible initial conditions are compared.

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If the interval of time (t2  t1 ) is small enough, the action integral has a minimal value as well as the stationary value. The implementation of the algorithm of functional action for ship dynamics in waves is considered in (Nechaev and Dubovik 1995). The advantages of the method can be summarized as follows:

 use of the information for an external disturbance, received during real-time measurements;  taking into account significant nonlinearity of a problem for any kind of function – possibility of the solution without use of a priori data on behavior of system and its initial conditions. Functional action is defined by an asymptotic expression (Nechaev (ed.) 2001, Nechaev and Dubovik 1995):





lim  2 ln Px X t  D   min S0T  

  C0T R

r n

(25)

,   x, ,    D 0

t 1

It allows estimating probability Px that critical set D (capsizing) will be achieved. This estimation is carried out using optimum control for the stability criteria formulated above. The solution of such problem (25) is given by the main term of logarithmic asymptotic expansion of the probability at   0 .

уз

Fig. 7 Probability of capsizing as a function of time (А), dynamics of roll of a ship on waves (B) and polar diagram (С)

This approach makes possible the development of new methods for analysis and prediction of behavior of nonlinear dynamical systems in real-time, including stability control in waves. The results of a method of functional action are presented as a probability of capsizing as a function of time (Fig. 7A, В). Then interpretation of the results for the given external conditions allows “drawing a picture” of capsizing for a ship in a considered extreme situation. Additionally information on heeling moment leading to capsizing can be made available on the screen. Using the method of the functional action it is possible to evaluate combinations of dangerous

Evolution of Analysis and Standardization of Ship Stability 95

speeds and headings relative to waves and present those in a form of a polar diagram (see Fig. 7C).

7 Study of Ship Stability Using Climatic Spectra and Scenarios of Extreme Situations Modelling the environment is one of the most complex problems that need to be solved while developing procedures for onboard IS. Contemporary approaches to the solution of these problems require new presentations of wind-wave fields in the ocean. These include the conception of climatic officially accepted on 18th Assembly of IMO in 1993 along with the concept “wave climate” (Boukhanovsky et al. 2000). These concepts open the opportunities for more detailed description of waves in different parts of the World Ocean. The hydrodynamic model of waves in the spectral form is represented as the equation of the balance of wave energy:

N N N N N N     k    G t   k  

(26)

Here N is the spectral density of wave action; it is function of latitude , longitude , wave number k, and the angle between a wave direction and a parallel , as well as frequency  and time t. The equation (26) relates inflow of energy from wind, dissipation and redistribution, as well as nonlinear interaction between frequency components of the process. Often, the source function G is written as a sum of three components G  Gin  Gnl  Gds (incoming energy from wind to waves, weak nonlinear interaction in a wave spectrum, and dissipation of wave energy). Examples of classification of wave spectra are shown in Fig. 8. There are six classes of spectra: A – swell; B – wind waves; C – combined swell and wind waves systems with prevalence of swell; D – combined swell and wind waves systems with prevalence of wind waves; E – combined swell and wind waves systems without clear division and with prevalence of swell; F – combined swell and wind waves systems without clear division and with prevalence of wind waves.

Y. Nechaev

96

А

B

C

D

E

F

Fig. 8 The typical normalized spectra of sea waves; vertical axis is value spectral density S()/Smax and horizontal axis is circular frequency , in sec–1

It is important to note that the standard calculation of dynamics of complex objects in real sea waves is done with a one-peak-spectrum, which may result in a type II error in comparison with climatic spectrum. Therefore, climatic spectra must be used for development of onboard IS, as type II errors may result in catastrophic consequences.

8 Paradoxes of Stability Standards The concepts of complexity and randomness are closely related. Simple criteria are often preferred over the complex ones. However, while gaining better understanding of physics of ship/environment interaction, it becomes clear that the problem of a choice of criteria is a very complex one. That’s why sometimes there are unexpected results and paradoxes (Nechaev, 1997). Paradox of “ideal” norms. It is known that the guarantee (defined as a probability that failure will not occur if an object satisfies the norms, see also (Belenky and Sevastianov 2007, Belenky et al. 2008)) for any, even most primitive, norms can be made very close to unity (ideal norms), if

    2  exp   d  0, 2 y x2  y 2   0

1



 KG  KGCR   100; y  KG  

i  



y0

i   

2

N 1

(27)

, 

1 i N

where KG, KGCR are the actual and critical position of center of gravity.

Evolution of Analysis and Standardization of Ship Stability 97

It can only be achieved by making norms stricter. Simple calculations show that considerably more physically valid stability criteria may result in further deviations from the ideal, compared to simplified IMO criteria. The paradox is that the result is achieved by the extremely uneconomical way, at the expense of making a norm stricter than the average norms, even when it is not really necessary. Paradox of zero probability. The paradox of zero-probability is directly related to probabilistic stability criteria. Practically, the probability of capsizing equals to zero (based on casualty statistics, probability of capsizing is about 10–4, i.e. corresponds to a range of risk (1-10) 10–4 similar to landing helicopters, horse races with obstacles and sports car races), but this event is not impossible. There is a paradox, whether it is possible to compare “chances” of events having zeroprobability. Also whether it is real when the conjunction of events having zeroprobability can result in finite-value probability, i.e. the addition of many “nothings” results in “something”. Probabilistic analysis of stability failures leads to a concept of rare events coming from “unusual occurrences” (combination of external conditions for assumed situations, encounter with extreme waves, etc.). The small probability of such events may make a false impression that simultaneous combination of many adverse factors is practically impossible, like conditions that lead to large-amplitude rolling (synchronous and parametrical resonance). This paradox is also closely related to the problem of rarity. Paradoxes of the distributions. Many paradoxes are related with the application of theoretical distributions. One such peculiarity is caused by asymptotic properties of theoretical distributions and results in insensitivity of the probability of stability failure to the large changes of the centre of gravity in the vicinity P0  1 . Other problems are related to the change of distributions by a nonlinear dynamical system, depending on the level of excitation. It is known that for the normal distribution, the average  *   i / N of a



random variable  is an unbiased estimate

E ( * )  0

and converges

P(|  *   |   )  1  at the large values N. However if the distribution is not known a priori, the estimate  * may end up being biased with the minimum variance; in case of multi-dimensional distributions, such an estimate may simply not exist for quadratic loss function (   * ) 2 . If such a check was not



performed during the statistical analysis the resulting criteria may be not valid. Paradox of a choice of boundaries for a criterion. The paradox of rational choice boundaries for a stability criterion is very important. Use of fuzzy boundaries for stability criteria makes sense due to the random nature and uncertainty of the input data that is used as a basis for calculation scheme of evaluation of stability (Fig. 9) (Nechaev, 1997).

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The errors of an inclining experiment, limited accuracy of data on loading conditions of a ship, as well as the uncertainty of other factors, may result in the true value of the parameter Х actually belonging to the interval [ X 0   ; X 0   ] . Then according to the requirements when X *  X 0 , the norm is observed exactly, while in reality X 0   . The area of acceptance of a hypothesis X  X 0 , i.e. the interval ( X 1 , X  ) equals  ( - significance value). Then the probability of a type II error for deviation from hypothetical value X q equals ; the value 1 –  characterizes “power of the criterion”. Reducing  leads to a reduction in the probability of a type I error (the zero hypothesis is rejected, when it is correct). However, the probability  is increased of the type II error and the “power” of criterion is reduced.

X min

X max

Fig. 9 Errors of practical use of stability criteria: X 0 – parameter; ( X max , X min ) - area of actual change Х; A – area zero-hypothesis (1) and alternative hypotheses (2); B – areas corresponding to a type II error

Paradoxes of computer implementation. Many paradoxes are related to computer implementation of stability problems. Any statistical solution, which can be implemented on the computer, now becomes available. As a result, “stable” and multi-dimensional methods, requiring huge number of operations, have entered engineering practice without a sufficient theoretical justification. Meanwhile, it is possible to justify many empirical collisions, using robust statistical methods. Unfortunately this is a common practice in stability assessment and regulation.

Evolution of Analysis and Standardization of Ship Stability 99

9 Future Directions The further developments of the analysis and criteria of stability can be achieved by advancements in the following strategic directions: 1. Development of hydrodynamic model of the interaction of a ship with external environment while under action of extreme waves. 2. Study of physical response of roll and capsizing of a ship in extreme situations using climatic spectra. 3. Development of a hydrodynamic model for nonlinear interaction of ship waves with incident waves in storm conditions. 4. Further development of criteria for ship stability in waves using new approaches for description of uncertainty in complex dynamic environments. 5. Development of effective means of the control of ship stability in waves using algorithms, which take into account the change of ship dynamics in time.

10 Conclusion The considered problems of the analysis and criteria of ship stability in waves reflects only a small portion of the research applications in which the ideas developed by the author, his students and colleagues have found a place. The foundation for these works combines stability and the theory of oscillators and offers a methodology for better understanding how both these concepts are applied for modeling. Thus the author emphasizes applications that attract both scientific and practical interest while researching complex behavior of ships in various extreme situations. Making decisions for the control of these situations may benefit from the new approaches based on formulations of uncertainty and construction of system of knowledge using artificial intelligence and fuzzy mathematics. Advancements in cybernetics, information technology and the general theory of systems led to a new scientific picture of the world. This picture has found its reflection in new approaches to the problem of stability of a ship in waves. The leading role belongs to nonlinear dynamics and is related to the concept of synergy that has changed the understanding of the relation between chaos and order, entropy and information. The theory of synergy came from works on the theory of bifurcation of dynamical systems and was further developed by new generations of researchers. Complexity of structures and processes plays a key role for modeling of the stability of a ship in waves. Their instability, randomness, and transitive nature led to new scientific paradigm. Changes of the structure of a dynamical system are studied within this paradigm, rather than a behavior of a system with unchangeable structure. The problem of analysis and development of stability criteria is one of important avenues for ensuring safety of operation. Uncertainty and incompleteness of the input

100 Y. Nechaev

information are inherent for the complex problem of ship stability. Information technology for monitoring ship dynamics in waves being developed by the author and his colleagues may be implemented into on-board intelligent systems. Such technology should not be reflected in the final form of criteria, as a mathematical formulation allows more flexible analysis of stability in complex situations.

Acknowledgments The author is grateful to his advisor prof. N.B. Sevastianov, who formulated the general approach to study the very difficult problem of ship stability in waves, and also to professors S.N. Blagoveshchensky, I.К. Boroday, N.N. Rakhmanin, V.V. Semenov-Тyan-Shansky and V.G. Sizov – for their support and advice.

References Ananiev DM (1964) On directional stability of a ship in waves. Trans of Krylov Soc 54. Leningrad (in Russian) Ananiev DM (1981) On stability of forced rolling of vessel with given GZ curve. Trans of Kaliningrad Inst of technol 93 Seakeeping of ships, Kaliningrad (in Russian) Ananiev D (1994) Determination the boundaries of surf-riding domain analyzing surging stability. Proc. of the 5th Int conf STAB-94 5 Melbourne, Florida, USA Belenky V, Sevastianov N (2007) Stability and safety of ships: Risk and capsizing. Second Edition SNAME, Jersey City, NJ. Belenky V, de Kat J, Umeda N (2008) Toward performance-based criteria for intact stability. Marine Tech 45(2):122–123 Blagovescensky SN (1951) On stability regulations of sea vessels. Trans of Central Mar Inst. 8, Leningrad (in Russian) Blagovescensky S (1962) Theory of ship motions. In two volumes, Dover Publ, New York Blagovescensky SN (1965) On the wind load on vessels. Trans of Russian Register of Shipping, Theor and Pract probl of stab and survivability, Transp Publish Leningrad 100–146 (in Russian) Blagovescensky SN, Kholodilin AN (1976) Seakeeping and stabilization of ships on wave. Sudostroenie, Leningrad (in Russian) Boroday IК, Netsvetaev YA (1969) Ship motion on sea waves. Sudosroenie, Leningrad (in Russian) Boroday IК, Netsvetaev YA (1982) Seakeeping. Sudosroenie, Leningrad (in Russian) Boroday IK, Morenshildt VA, Vilensky GV et al. (1989) Applied problems of ship dynamic on waves. Sudostroenie, Leningrad (in Russian) Boukhanovsky A, Degtyarev A, Lopatoukhin L, Rozhkov V (2000) Stable states of wave climate: applications for risk estimation. Proc of the 7th Int Conf STAB’2000, Launceston, Tasmania, Australia 831–846 Degtyarev A (1993) Distribution of nonlinear roll motion. Proc of Int Workshop. On the probl of phys and math modeling. OTRADNOYE-93 2 Paper 9 Degtyarev A, Boukhanovsky A (1995) On the estimation of the ship motion in real sea. Proc of Int symp. Ship safety in seaway: stability, maneuverability, nonlinear approach. Sеvastyanov Symposium), Kaliningrаd, 2, Paper 8

Evolution of Analysis and Standardization of Ship Stability 101 Degtyarev A, Boukhanovsky A (1996) Nonlinear Stochastic Ship Motion Stability in Different Wave Regimes. Trans of 3rd Int conf CRF-96, St.Petersburg, 2: 296-306. Degtyarev A, Boukhanovsky A (2000) Pecularities of motions of ship with low buoyancy on asymmetrical random waves. Proc of the Int conf STAB’2000, Launceston. Tasmania. Australia, 2: 665-679. Nechaev YI (ed) (2001) Intelligence systems in mar res and technol. SMTU, St.-Petersburg (in Russian) Кrylov АN (1958) Selected papers. Publ Аcad of Sci USSR, Мoskow (in Russian) Lugovsky VV (1966) Nonlinear problems of seakeeping of ships. Sudostroenie, Leningrad (in Russian) Lugovsky VV (1971) Theoretical problems background of stability standards of sea. Sudostroenie, Leningrad (in Russian) Lugovsky VV (1980) Hydrodynamics of ship non-linear motions. Sudostroenie, Leningrad (in Russian) Makov YL (2005) Stability. What is this? Sudostroenie St.-Petersburg (in Russian) Nechaev Y (1965) Method for determining the decrease of righting arms of stability on wave crest. IMCO. PFV/11.1965 Nechaev YI (1978) Ship stability on following waves. Sudostroenie, Leningrad (in Russian) Nechaev YI (1989) Stability on wave. Modern tendency. Sudostroenie, Leningrad (in Russian) Nechaev Y (1993) Determined chaos in the phase portrait of ships dynamic in a seaway. Proc of Int Workshop OTRADNOYE-93 2: 143-145 Nechaev Y (1996) Problem of uncertainty in hydrodynamic experiment planning. Proc of Int symp “Marine intelligence technology” CRF-96, St.-Petersburg: 453-457. Nechaev Y (1995) The algorithm a correct estimation on the stability under conditions exploitation. Proc of Int symp. Ship safety in seaway: stability, maneuverability, nonlinear approach. (Sеvastyanov Symposium), Kaliningrаd, 2: Paper 17 Nechaev Y Standardization of stability: Problems and perspectives. Proc of 6th Int conf STAB’97, Varna, Bulgaria 2: 39-45. Nechaev Y, Degtyarev A, Boukhanovsky A (1997) Chaotic dynamics of damaged ship in waves Proc of 6th Int conf STAB’97, Varna, Bulgaria 1: 281-284. Nechaev Y, Degtyarev A, Boukhanovsky A (1997) Analysis of extreme situations and ship dynamics on seaway in intelligence system of ship safety monitoring. Proc of 6th Int conf STAB’97, Varna, Bulgaria 1: 351-359. Nechaev Y, Degtyarev A, Boukhanovsky A (1999) Adaptive forecast in real-time intelligence systems. Proc. of 13th Int conf on hydrod in ship design, Hydronaf-99 and Manoeuvring-99, Gdansk – Ostroda, Poland: 225-235 Nechaev Y (2000) Problems of experiments design and choice of regression structure for estimation of ship stability in exploitation conditions. Proc of Int conf STAB'2000, Launceston. Tasmania, Australia 2: 965-972. Nechaev Y, Degtyarev A (2000) Account of peculiarities of ship’s non-linear dynamics in seaworthiness estimation in real-time intelligence systems. Proc of Int conf STAB'2000, Launceston, Tasmania, Australia 2: 688-701 Nechaev Y, Zavyalova O (2001) Analysis of the dynamic scenes at architecture of the interacting in real-time intelligence systems. Proc оf 3rd Int conf on marine ind MARIND-2001, Varna. Bulgaria 2: 205-209. Nechaev Y, Zavyalova O (2001a) The broaching interpretation in learning intelligence systems. Proc of 14th Int Conf on Hydr in Ship Design, Szcecin-Miedzyzdroje, Poland: 253-263. Nechaev Y, Degtyarev A (2001) Knowledge formalization and adequacy of ships dynamics mathematical models in real time intelligence systems. Proc of 14th Int Conf on Hydr in Ship Design, Szcecin-Miedzyzdroje, Poland: 235-244 Nechaev Y, Degtyarev A, Boukhanovsky A. (2001) Complex situation simulation when testing intelligence system knowledge base. Computational Science – ICCS 2001, Part 1, LNCS. Springer: 453 – 462

102 Y. Nechaev Nechaev Y (2002) Principle of competition at neural network technologies realization in onboard real-time intelligence systems. Proc of 1st Int cong on mech and elect eng and tech «MEET-2002» and 4th Int conf on marine ind «MARIND-2002», 3 Varna, Bulgaria: 51-57 Nechaev Y, Degtyarev A, Kirukhin I, Tikhonov D (2002a) Supercomputer technologies in problems of waves parameters and ship dynamic characteristics definition. Proc of 1st Int cong on mech and elect eng and tech «MEET-2002» and 4th Int conf on marine ind «MARIND-2002», 3 Varna, Bulgaria: 59-64 Nechaev Y, Zavialova O (2002) Analysis and interpretation of extreme situation in the learning and decision support systems on the base of supercomputer. Proc of 1st Int cong on mech and elect eng and tech «MEET-2002» and 4th Int conf on marine ind «MARIND-2002», 3 Varna, Bulgaria: 23-29 Nechaev Y, Zavyalova O (2003) Criteria basis for estimation of capsizing danger in broaching extreme situation for irregular following waves. Proc of 8th Int conf STAB’2003, Madrid. Spain: 25-34 Nechaev Y, Degtyarev A (2003) Scenarios of extreme situations on a ship seaworthiness Proc of 8th Int conf STAB’2003, Madrid. Spain: 1-10 Nechaev Y, Zavyalova O (2003) Manoeuvrability loss in a broaching regime the analysis and control of extreme situation. Proc of 15th Int conf on hydrod in ship design, safety and operation, Gdansk, Poland: 201-208 Nechaev Y, Dubovik S (2003) Probability-asymptotic methods in ship dynamic problem. Proc of 15th Int conf on hydrod in ship design, safety and operation, Gdansk Poland: 187-199 Nechaev Y, Petrov O (2005) Fuzzy knowledge system for estimation of ship seaworthiness in onboard real time intelligence systems. Proc of 16th Int conf on hydrod in ship design, 3rd Int symp on ship manoeuvring, Gdansk – Ostroda Poland: 356 – 366 Nechaev Y, Degtyarev A, Anischenko O (2006) Ships dynamic on wave-breaking condition. Proc of 9th Int conf STAB’2006, Rio de Janeiro Brazil 1: 409 – 417 Nechaev Y, Makov Y (2006) Strategy of ship control under intensive icing condition. Proc of 9th Int conf STAB’2006, Rio de Janeiro Brazil 1: 435 – 446 Nechaev Y (2007) Nonlinear dynamic and computing paradigms for analysis of extreme situation. Proc of Int conf «Leonard Euler and modern science». Russian Academy of Science, St-Petersburg: 385 – 390 (in Russian) Nekrasov VA (1978) Probabilistic problem of seakeeping. Sudostroenie, Leningrad (in Russian) On-board intelligence systems (2006) Radiotechnik, Moskow (in Russian) Rakhmanin NN (1966) On dynamic stability of ship with water on deck. Trans of Krylov Soc. 73 Sudostroenie, Leningrad (IMCO STAB/INF. 27.10.66. Submitted by USSR) (in Russian) Rakhmanin NN (1971) Approximate assessment of safety of small vessel in seas. Trans of Russian Register of Shipping, Leningrad: 12 – 24 (in Russian) Rakhmanin N (1995) Roll damping when deck is submerging into seawater. Proc of Int Symp «Ship Safety in Seaway Stability, manoeuvrability, nonlinear approach» (Sеvastyanov Symp) Kaliningrad 2: Paper 1 Sevastianov NB (1963) On probabilistic approach to stability standards. Trans of Kaliningrad Inst of Techn 18: 3–12 (in Russian) Sevastianov NB (1968) Comparison and evaluation of different systems of stability standards. Sudostroenie 6: 3–5 (in Russian) Sevastianov NB (1970) Stability of fishing vessels. Sudostroenie, Leningrad (in Russian) Sevastianov NB (1977) Mathematical experiment on ship capsizing under combined action wind and waves. Rep of Kaliningrad Inst of Tech, No77-2.1.6 (in Russian) Sevastianov NB (1978) On possibility of practical implementation of probabilistic stability regulations. Sudostroenie 1: 13–17 Sevastianov NB (1993) Theoretical and practical models for probabilistic estimation of vessels stability. Proc of Int Workshop OTRADNOYE-93 1: Paper 5

SOLAS 2009 – Raising the Alarm Dracos Vassalos* Andrzej Jasionowski** *The Ship Stability Research Centre (SSRC), Dept of Naval Architecture and Marine Engineering, University of Strathclyde, UK** Safety at Sea Ltd (SaS), Glasgow, UK

Abstract

In anticipation of the new harmonised probabilistic rules for damage stability being adopted in January 2009, a number of ship owners are opting to follow these rules in advance, somewhat hesitantly and reluctantly considering the general lack of experience and understanding but also the confusion that prevails. In this climate, the authors have found a fertile ground for introducing a methodology to deal optimally with this problem, deriving from the arsenal of tools and knowledge available at SSRC and SaS. As a result, the authors are involved in some way or other with most ships currently being designed in accordance with the new probabilistic rules for damage stability. This involvement has revealed a somewhat more serious problem with probabilistic damage stability calculations, which is hidden in the detail of the rules, but one ht at matters most. The authors are highlights this problem and recommends a way forward.

1 Introduction In January 2009, the new harmonised probabilistic rules for ship subdivision will become mandatory, are rules initiating rule making? in the maritime industry, in line with contemporary developments, understanding and expectations. This will be the culmination of more than 50 years of work, one of the longest gestation periods of any other rule. Considering that this is indeed a step change in the way safety is being addressed and regulated, “taking our time” is well justified. However, the intention to provide a qualitative assessment of safety (a safety index) might have been enough at the time the probabilistic framework for damage stability was conceived (indeed for as recent as a few years ago) but this is not the case today. With the advent of Design for Safety and of Risk-Based Design, quantification of safety, consistently and accurately, is prerequisite to treating safety as a design objective. This, in turn, entails that the level of detail in the method used to quantify safety carries a much bigger weight. With this in mind and in the knowledge that the Attained Index of Subdivision in the probabilistic rules is a weighted summation of survival factors, calculating survival factors consistently M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_6, © Springer Science+Business Media B.V. 2011

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and accurately is paramount. Unfortunately, a close scrutiny of the work that led to the current formulation of the s-factor revealed that it is simply the result of a series of unjustified compromises, inadvertently creeping in during the rule-making process. In the form, due to be adopted in 17 months time, the s-factor derives from a regression analysis of only a filtered set of old cargo ships and as such it will be unsuitable even for this category of ships, considering the evolutionary changes in most cargo ship types. More importantly, recent research results indicate that, on average, the formulation to be adopted seriously underestimates the inherent survivability of cruise ships whilst it drastically undermines the survivability of Ro-Ro passenger ships. What is of crucial significance, there is little consistency between the actual survivability of both these vessel types (the most safety-critical of ships) and that postulated by the currently proposed formulation. The problem does not end here: since this formulation was used to calculate Index-A for a representative sample of various vessel types so as to evaluate the Required Index of Subdivision (in principle, the safety level to be adopted), these results will also be in disarray. In short, SOLAS 2009 is in need of major introspection! This paper attempts to lay the bare facts and in so doing to raise the alarm, hoping that in the time remaining no effort will be spared in ensuring that the best ever regulatory achievement in our industry will not fall flat on its face before it gets started.

2 The Probabilistic Concept of Ship Subdivision The first probabilistic damage stability rules for passenger vessels, deriving from the work of Kurt Wendel on “Subdivision of Ships”, (Wendel 1968) were introduced in the late sixties as an alternative to the deterministic requirements of SOLAS ‘60. Subsequently and at about the same time as the 1974 SOLAS Convention was held, the International Maritime Organisation (IMO), published Resolution A.265 (VIII). The next major step in the development of damage stability standards came in 1992 with the introduction of SOLAS part B-1 (Chapter II-1), containing a probabilistic standard for cargo vessels, using the same principles embodied in the 1974 regulations. This, in turn, necessitated launching at IMO the regulatory development of “Harmonisation of Damage Stability Provisions in SOLAS, based on the Probabilistic Concept of Survival” in the belief that this represented a more rational approach to addressing damage stability. However, the slowly changing world of IMO engulfed this process and it almost stagnated. In this state of affairs, the compelling need to understand the impact of introduction of probabilistic damage stability regulations in the design of cargo and passenger ships and the growing appreciation of deeply embedded problems in both the rules and the harmonisation process itself, provided the motivation for the adoption of an in-depth evaluation and re-engineering of the whole probabilistic framework through the EC-funded €4.5M, 3-year project HARDER (HARDER

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2003) The overriding goal of this project was to develop a rational procedure for probabilistic damaged ship stability assessment, addressing from first principles all relevant aspects and underlying physical phenomena for all types of ships and damage scenarios. In this respect, HARDER became an IMO vehicle carrying a major load of the rule-development process and fostering international collaboration at its best – a major factor contributing to the eventual success in achieving harmonisation and in proposing a workable framework for damage stability calculations in IMO SLF 47. A stage has now been reached where the draft text of the major revision to the subdivision and damage stability sections of SOLAS Chapter II-1 based on a probabilistic approach has been completed, following final amendments in January 2005 to Regulation 7-1 involving calculation of the “p” factor. The revised regulations were adopted in May 2005 by the IMO MSC and will be entering into force for new vessels with keels laid on or after 1 January 2009. One of the fundamental assumptions of the probabilistic concept of ship subdivision in the proposed regulations is that the ship under consideration is damaged, that is the ship hull is assumed breached and there is (large scale) flooding. This implies that the cause of the breach, the collision event with all the circumstances leading to its occurrence, are disregarded, and hence the interest focuses on the conditional probability of the loss of stability. In other words, risk to life is assumed to be irrelevant on the likelihood of occurrence of a collision event that ends in hull breaching and flooding. For this reason, the regulations imply the same level of “safety” irrespective of the mode of operation that can e.g. take place in area of varying density of shipping (congestion of traffic), or indeed can be so different depending on ship type, or can involve vastly different consequences, etc, all of which might imply considerably different levels of actual risk. This said, all risk-related factors (e.g. size of ship, number of persons on board, life saving appliances arrangement, and so on) are meant to indirectly be accounted for by the Required Index of Subdivision, R. Summarizing, the probability of ship surviving collision damage is given by the Attained Index of Subdivision, A, and is required not to be lesser than the Required Index of Subdivision, R, as given by expression (1): J

I

A   w j  pi  s ji ; A  R

(1)

j 1 i 1

Where: A/R Attained/Required Index of Subdivision j loading condition (draught) under consideration Draught and GM J number of loading conditions considered in the calculation of A (normally 3 draughts) i represents each compartment or group of compartments under consideration for each j

106 D. Vassalos and A. Jasionowski

I wj pi sij

set of all feasible flooding scenarios comprising single compartments or groups of adjacent compartments for each j probability mass function of the loading conditions (draughts) probability mass function of the extent of flooding (that the compartments under consideration are flooded) probability of surviving the flooding of the compartment(s) under consideration, given loading (draught) conditions j occurred

The index A can thus be considered as the expected value of the “s-factor”, with “p- and w-factors” being the respective likelihoods, reflected in worldwide ship collision statistics:

A  E (s )

(2)

Consequently, (1-A) can be considered as the marginal probability of (sinking/ capsize) in these scenarios, and as such it can be used for deriving the collisionrelated risk contribution, (Vassalos 2004). The required Index of subdivision, R (derived principally from regression on A-values of representative samples of existing ships) represents the “level of safety” associated with collision∩flooding events that is deemed to be acceptable by society, in the sense that it is derived using ships that society considers fit for purpose, since they are in daily operation. The new regulations represent a step change away from the current deterministic methods of assessing subdivision and damage stability. Old concepts such as floodable length, criterion numeral, margin line, 1 and 2 compartment standards and the B/5 line will be disappearing. With this in mind, there appears to be a gap in that, whilst development of the probabilistic regulations included extensive calculations on existing ships which had been designed to meet the current SOLAS regulations, little or no effort has been expended into designing new ships from scratch using the proposed regulations. However, attempting to fill this gap through research and though participation in a number of new building projects revealed a more serious problem that affects consistency and validity of the derived results and hence of the whole concept. These findings constitute the kernel of this paper and are described next, following a brief reminder on the derivation in project HARDER and the eventual adoption at IMO of the survival factor “s”.

3 Probability of Survival – “S-Factor” It has to be pointed out that despite the wide-raging investigation undertaken in HARDER, the survival factor “s” adopted in the new harmonised probabilistic rules designates simply the probability of a damaged vessel surviving the dynamic effects of waves once the vessel has reached final equilibrium, post-damage. All other

SOLAS 2009 – Raising the Alarm 107

pertinent factors and modes/stages of flooding are not accounted for. Moreover, whilst considering 7 ship models in the test programme (HARDER 2001), representing a range of different types, sizes and forms, namely: 3 Passenger ships (2 Ro-Ro’s and one cruise liner) and 3 dry cargo ships (Ro-Ro Cargo Ship, Containership, and Bulk Carrier - a Panamax Bulk Carrier was added later), the formulation for the s-factor focused on Ro-Ros and non-Ro-Ros. This categorisation derives primarily from the capsize mechanisms pertinent to these two categories. Therefore, and without labouring the detail of the formulation (for which SSRC was responsible), the following two approaches were followed:

3.1 s-Factor for Low Freeboard Ro-Ro Ships It was always the intention to base the survival factor for Ro-Ro ships on the SEM methodology, (Vassalos et al. 1996). In brief, the method statically develops the volume of water that will reduce the damage GZ curve to exactly zero and at the ensuing critical heel angle, θ, two parameters: h – the dynamic water head and f – the freeboard are calculated as shown in Figure 1. The statistical relationship between parameters (h) and (f) and the significant wave height of the critical survivable sea state (HS) is then established.

GZmax

θ critical

G h f

Fig. 1 The SEM Methodology

108 D. Vassalos and A. Jasionowski

Accounting finally for the likelihood of occurrence of this sea state at the time of casualty led to the following relationship (Figure 2):

s = exp(-exp (f –8h +1.3)

Fig. 2 SEM-Based s-Factor for Ro-Ro Ships (25 Cases – 4 Ship Models)

3.2 s-Factor for non-Ro-Ro Ships Having applied the SEM methodology to non-Ro-Ro ships and achieved what was considered unsatisfactory results, the focus shifted towards the more traditional damage stability criteria employing the use of properties of the GZ curve, such as GZ max, GZ Range, and GZ Area. In simple terms, the correlation of these three traditional parameters with the observed survival sea states from the non-Ro-Ro model tests was first established, shown here in Figures 3 to 5. Range vs. Hs 10 9 8

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SOLAS 2009 – Raising the Alarm 109 Area vs. Hs 10 9 8

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A review of these results led to discarding the GZ area parameter due to lack of correlation as well as most points not falling naturally on the observed trends. Adopting finally an existing SLF format and with Hs limited to 4m, similar to RoRo vessels, yielded the following final formulation (Figure 6):

110 D. Vassalos and A. Jasionowski

s = [ (GZmax / 0.12) (Range / 16) ]1/4

Fig. 6 GZ-Based “s”-Factor for Conventional Ships (25 Cases – 3 Ship Models)

It may be noticed that 25 damage cases were used in the formulations described in the Figure 5 and another 25 damage cases in Figure 6. Thus, the whole data set used in project HARDER, and shown here in Figure 7, comprises 51 data points (the one point not accounted for relates to the cruise ship). 10 Conventional

9

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1

3 2

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1 0

0

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44

0

GZmax [m]

Fig. 7 Data set of 51 measured points of survivability in the HARDER project

Based on the aforementioned developments the following points are noteworthy:  The formulation for the survival factor in the new harmonised probabilistic regulations for damage stability, due to be adopted in January 2009, is based solely on conventional cargo ships, Figure 6. OOOPS!!!  The SEM methodology was never adopted; hence there is no formulation for survival factor in SOLAS 2009 that could be applied to anything other than non-Ro-Ro ships.

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 Examining also Figure 7, leads to some additional and even more disturbing observations:  The single result on the cruise ship, which was never used, shows that even with marginal static stability characteristics the vessel survives sea states with Hs in excess of 4m. Several tests since then clearly demonstrate that the current formulation underestimates the survivability for cruise ships; as a general trend the bigger the vessel the larger the error!  On the other end of the spectrum, the formulation due to be adopted could overestimate the survivability of Ro-Ro ships. In other words, survivability of Ro-Ro ships can be lower than what the formulation implies. Having made these sobering observations, there is yet another blunder to consider: the development of the Required Index, R, for passenger ships. The original results, shown in Figure 8, represent the A-values calculated on the basis of the new formulation for a representative sample of ships. Considering the arguments made above, it may justifiably be proposed that the observed trends simply reflect the gross oversight of the fact that the s-factors considered in the new harmonised rules had nothing to do with passenger ships.

All PASS ships

1,1 1

Rnew

0,9 Rnew (SLF) Rnew (HARDER) Linear (Rnew (SLF)) Linear (Rnew (HARDER))

0,8 0,7 0,6 0,5 0,4

0

25

50

75 100 125 150 175 200 225 250 275 300

Ls (m)

Fig. 8 Rnew for all passenger ships

Not realising this fact, rather than questioning the method of calculation, it was attempted instead to make sense of the results at hand, using an approach questionable at best, which was surprisingly adopted at IMO; the tidal wave of the new probabilistic rules managed to push an excellent framework through the door but it seems that a lot of rumble went through with it.

112 D. Vassalos and A. Jasionowski

4 Emerging Evidence for Raising the Alarm on Solas 2009

4.1 SAFENVSHIP Project In the proposed formulation for the s-factor, described above, the still water GZ curve characteristics are calculated by assuming that the lowest-most spaces within the damaged zone flood instantly and that static balance between vessel mass, displaced water and floodwater describe fully the vessel stability. To gain better understanding of the meaning of the above approximation and, in general, of the stability of modern passenger ships, a study was undertaken, (Jasionowski, 2005), including probabilistic damage stability calculations, numerical time-domain simulations and scaled-model tank testing of three cruise ship designs, considered representative of modern passenger ships. Figure 9 shows a sample result of the study. Altogether 33 different damage cases of the three vessels were investigated, chosen to provide a range of s-factors and distribution of damages along the whole length of the vessel; damage openings were chosen with a level of “high severity” in mind. Deriving from this study the following points are noteworthy:  Of the 33 cases considered, 16 were found to lead to vessel capsizing within two hours, sometimes rapidly. Of the 16 “capsizing” cases, 10 had an s-factor between 0 and 1, with some having s=1.0. Of the remaining 17 “surviving” cases, 7 had an s-factor equal to zero!  The study demonstrates that traditional GZ-curve characteristics cannot adequately describe the behaviour (and hence the destiny) of a damaged ship with the complexity in watertight subdivision of internal spaces found on a typical cruise vessel, and consequently, that the formulation for the s-factor in its present form cannot meaningfully represent the average resistance of such ships to capsize when subjected to flooding, following collision damage. Another reason for this deficiency derives from explicitly neglecting the presence of multiple free surfaces (MFS? effect), a phenomenon specific to ships with complex watertight subdivision, which substantially weakens or indeed leads to complete erosion of stability at any of the stages of flooding (this is a finding postHARDER).

SOLAS 2009 – Raising the Alarm 113 C1 - STD Opened 1 0.9 0.8 0.7 0.6 0.5

A=0.638

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Fig. 9 A map of “s” factors (upper plot) and the expression p.(1-s) for all damage cases considered as a function of the damage locations. The red boxes in the lowest-most plot indicate cases where the ship capsized within 2 hours simulation time, whereas green boxes indicate survival cases

4.2 Newbuilding Design Experience With the introduction of the new probabilistic rules, the dilemma of prescriptive SOLAS-minded designers must be overcome. Using an approach, known as platform optimisation, SSRC/SaS developed a fully automated optimisation process leading to ship designs with an acceptable level of flooding-related “risk” (i.e., compliance with statutory requirements) that can be optimised along with cost-effectiveness, functionality and performance objectives. Alternatively, high survivability internal ship layouts can be developed, without deviating much from the current SOLAS practice, thus making it easier for ship designers to relate to the proposed procedure. This procedure has been extended to address in full the damage survivability of new building cruise and RoPax vessels from first principles in a systematic and all embracing way, as explained in (Vassalos 2007). Moreover, a direct link between Index-A and flooding risk has been established, thus facilitating a better understanding of the meaning of all the factors involved and of their significance in affecting safety. This also led to new evidence concerning the inconsistency and inaccuracy in the details of the new probabilistic rules, as summarised in Figure 10.

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Fig. 10 Cumulative probability distributions for time to capsize for a traditional Ro-Ro vessel and a large new building cruise ship. Deepest draft loading conditions.

The cumulative distributions for time to capsize, for the ships shown in Figure 10, are derived using two approaches: (a) an analytical expression based on the formulation of the new probabilistic rules and (b) time-domain numerical simulations of 500 damage cases derived my means of Monte Carlo sampling that represent the statistics of damage characteristics used in these rules. Both ships comply with the new rules. Moreover, as explained in (Vassalos 2007), the values of these distributions tend to 1-A within a reasonable time period of several hours. Therefore, based on these results which seem to be typical for these groups of vessels, the following observations can be made:  For the Ro-Ro vessel, (Jasionowski 2007), approximately one in three collision ∩flooding events would lead to capsize within an hour as inferred from either analytical solution or numerical simulation (this is not a good standard!)  Application of the new rules indicates that the cruise ship will follow a similar fate in approximately one every 10 events whilst using first-principles timedomain simulation tools the rate becomes approximately 1:100. This is a big difference! These and other similar results have alerted the industry to the extent that a new investigation is being launched in an attempt to sort out the details in the new rules before they are adopted in January 2009, in the belief that the probabilistic rules constitute one of the most significant developments in the rule-making history of the maritime profession.

5 Conclusions Based on the results and arguments presented in this paper the following conclusions can be drawn:

SOLAS 2009 – Raising the Alarm 115

 The index A of new probabilistic rules on ship subdivision can be interpreted as a value that reflects the average survivability of a vessel following collision damage and flooding in a seaway. As such, an accurate calculation of the survival probability in these rules is of paramount importance.  This being the case, there is new evidence mounting that indicates gross errors in the derivation of survival factors, demanding swift action by the profession to avert “embarrassment” in global scale.  Such action is already being planned but it needs a wider participation to ensure that results will be available on time to re-engineer a correct formulation before the new rules are enforced.  More importantly, in the knowledge that the probabilistic rules are inconsistent and inaccurate, due care is needed in designing new ships; in particular, the use of time-domain simulation tools or physical model tests must be fully exploited.

Acknowledgments The financial support of the UK Maritime and Coastguard Agency and of the European Commission in the research undertaken is gratefully acknowledged. Special thanks are due to Royal Caribbean International and to Carnival for providing the inspiration needed to delve beyond trodden paths, even when it is not called for.

References HARDER (2001) Task 3.1 – Main Report of Model Tests. Report Number 3-31-D-2001-13-1 HARDER (2003) Harmonisation of Rules and Design Rationale. U Contact No. GDRB-CT1998-00028. Final Technical Report Jasionowski A, (2005) Survival Criteria For Large Passenger Ships. SAFENVSHIP Project, Final Report, Safety at Sea Ltd Jasionowski A, York A, (2005) Investigation Into The Safety Of Ro-Ro Passenger Ships Fitted With Long Lower Hold. UK MCA RP564, Draft Report MCRP04-RE-001-AJ, Safety at Sea Ltd Vassalos D (2004) A risk-based approach to probabilistic damage stability. 7th IntWorkshop on the Stab and Oper Safety of Ships, Shanghai, China Vassalos D (2007) Risk-Based Design: Passenger Ships. SAFEDOR Mid-Term Conf, the Royal Inst of Naval Archit, Brussels Vassalos D, Pawlowski M, Turan O (1996) A Theoretical Investigation on the Capsizal Resistance of Passenger/Ro-Ro Vessels and Proposal of Survival Criteria. Joint Northwest European Project Wendel K (1968) Subdivision of Ships. Diamond Jubilee Int Meeting, N.Y.

2 Stability of the Intact Ship

Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas A. Yucel Odabasi and Erdem Uçer Istanbul Technical University, Faculty of Naval Architecture and Ocean Engineering

Abstract This study intended to evaluate the behavior of a biased ship under main, sub and super harmonic resonant excitation. For the sake of simplicity Duffing’s equation is used for un-biased roll equation. In the first part of this study, first and second order approximate solutions of biased roll equation have been obtained for different equation parameters in main, sub and super harmonic resonance regions by using method of multiple scales and Bogoliubov-Mitropolsky asymptotic method. It was found that second order approximate solutions had a better compliance with numerical results when the solution is stable. In the second part, stable solution bounds of the biased roll equation were obtained for different equation parameters by using numerical methods. It was found that the size of the bounds are highly dependent on initial bias angle, linear damping coefficient, amplitude and frequency of wave excitation and phase angle of the excitation force and also the symmetry of the bounds were dependent on the magnitude of the initial bias angle. From the results obtained it can be concluded that Lyapunov’s stability theory provides the most reliable results.

1 Introduction Resonance which is an important phenomenon can lead to capsize of ships because it causes amplitude of rolling motion to take large values after a few cycles. Thus, many scientists and researchers investigated resonance case in the rolling motion by either making scaled model test or using asymptotic, perturbation and numerical methods and also Melnikov Method. Wright and Marshfield (1980) presented experimental and theoretical study on ship roll response and capsize behaviour in beam seas. They made experiments to measure the steady state sinusoidal roll amplitudes for three type of models (Low, Medium and High freeboard model) at a range of frequencies with constant wave slope amplitude inputs. In their theoretical study, they used regular perturbation M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_7, © Springer Science+Business Media B.V. 2011

119

120 A.Y. Odabasi and E. Uçer

method, harmonic balance method and the method of averaging to solve roll equation with or without bias. Another important experimental study was made by Grochowalski (1989). He presented experimental study on the mechanism of ship capsizing in quartering and beam waves. He used a 1:14 scaled GRP model which was fitted with bulwarks, freeing ports, superstructure and stern ramp and also had a large centreline skeg and a single four blade propeller. He made experiments for two load conditions (light and full load condition). He used free model runs to give an insight into the kinematics of ship rolling and also captive tests to identify the composition of wave exciting hydrodynamic forces and moments. In his study, he indicated the danger created by bulwark submergence and accumulation of water on deck and also stability reduction on a wave crest. In his experiments he also found that the quartering waves could be as dangerous as beam waves. Cardo, Francescutto and Nabergoj (1981) used Bogoliubov-Krylov-Mitropolsky asymptotic method to determine first order steady state solution of roll equation without bias in ultra harmonic and sub harmonic resonance regions. (Nayfeh and Khdeir 1986) used method of multiple scales to determine a second order approximate solution in the main resonance region for the nonlinear harmonic response of biased ships in regular beam waves. Nayfeh (1979) obtained the first and second order solutions of Equation (1) in main, sub harmonic and super harmonic resonance regions with the method of multiple scales. This equation is similar to the equation used in this article. However, the solutions obtained in this article are different from Nayfeh’s solutions due to transformations made in Equation 1.

~ x  2 ˆ x  02 x   2 x 2   3 x 3  K Cos  t

(1)

Jiang et al. (1996) investigated the influence of bias on capsizing both in terms of the reduction of the safe basin and in terms of phase space flux. Their conclusions were based on Melnikov analyses and also supported by extensive numerical solution. Macmaster and Thompson (1994) examined capsize of ships excited by the naturally propagating wavefront created by a laboratory wave maker that is suddenly switched on. They used transient capsize diagrams and showed the importance of bias. Cotton et al. (2000) used the same equation shown below to examine the sensitivity of capsize to a symmetry breaking bias in beam seas and stated that the case α=0 to be a sensible model of ship roll motion.

x   x  x (1  x ) (1   x )  F Sin  t

(2)

Spyrou et al. (1996) used Melnikov method to obtain analytical formulas which can characterize capsizability of ships taking into account the presence of some roll bias. They have tested these formulas against numerical simulations targeting

Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 121

the process of safe basin erosion. Their mathematical model for rolling motion was also Equation (2). The aim in the first part of this study is to present available methods in the solution of roll equation (method of multiple scales, Bogoliubov-Mitropolsky asymptotic method and numerical methods), compare the obtained results and also show the sensitivity of solution to the effect of equation parameters. It was also observed from the solutions that neglecting the effect of cubic coefficient of restoring term is not sensible. In the second part of this study, stable solution bounds of the biased roll equation were obtained for different equation parameters and the effect of these parameters on the size and symmetry of the bounds were also examined. In the concluding part of the paper it is stated that boundedness of solution, in the sense of Lyapunov, constitute a sound basis for the intact stability research.

2 Methodology of the Investigation

2.1 Equation of Motion and Preliminary Considerations In this paper, while the rolling motion of the ship was being modelled, interactions between rolling and other modes of motion have been ignored and the ship was considered to have a rigid body and sea water was ideal and incompressible. Under these assumptions, rolling motion of a ship was written as in Equation (3).

 

I   D  ,   M R    E t   M wind    

(3)

θ : Rolling angle with respect to calm sea surface  : Roll angular velocity I : Mass moment of inertia including the added mass moment of inertia. MR(θ) : Restoring moment. Its coefficients can be obtained either from a computed righting arm curve by curve fitting or from the characteristics of the righting arm curve, such as GM, GZmax, area under the righting arm curve, by constrained curve fitting.

 D(θ,  ) : Damping moment. In this study, the linear damping moment associated with wave produced by the hull was only taken care of.  E(t) : Roll exciting moment, due to external force. When the wave elevation is induced by regular transversal waves, the exciting term may be written as E t   Ew Cos t  where Ew is the amplitude and Ω the angular frequency Mwind : Wind Moment

122 A.Y. Odabasi and E. Uçer

In this study, for the sake of simplicity, the following approximations have been used M R ( )   k1  k 2 3 and D( , )  s  where Δ is the displacement and s is the linear damping coefficient Under the above assumptions Equation (3) can be written as









I   s    k1  k 2 3  Ew Cos  t  M wind

(4)

Dividing both sides of the equation by the mass moment of inertia, the expression given in Equation (5) is obtained.

  ˆ   c1  c2 3  F Cos t  Fruz

(5)

This type of roll equation was also examined from other author’s earlier studies. If θ = x + θs transformation is made, Equation (5) can be re-written as

x  ˆ x  02 x  d1 x 2  d 2 x 3  F Cos t 2

2

(6) 3

where 0  c1  3 c 2  s , d1  3  s c 2 , d 2  c 2 , Fruz  c1 s  c 2  s In the following parts of this paper Equation (6) were used as simplified rolling equation.

2.2 Main, Sub and Super Harmonic Resonance Investigation In this study, wave excitation was assumed to be approximately half, twice and equal to the natural frequency to make an investigation in sub, super and main harmonic resonance regions. Then biased roll equation was solved in these regions by using the method of multiple scales, Bogoliubov-Mitropolsky asymptotic method and numerical methods. The effects of parametric resonance have not been investigated. Finally, the stable solution bounds of roll motion were obtained for different bias angles, linear damping coefficients, and amplitudes and frequencies of wave excitation in the sub, super and main harmonic resonance regions by using Mathematica NDSolve routine and the Fortran program written by the authors. The necessary data used in numerical computation were shown in Table 1.

Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 123 Table 1 The data used in numerical calculations. θs (rad.) ω0

d1

ˆ

d2

F

0.005

0.610257 0.001975

0.100

0.607020 0.039499 0.131664 0.04,0.07,0.10,0.13 0.08,0.10,0.12

0.200

0.597179 0.078998

2.3 Solution Methods 2.3.1 Method of Multiple Scales Method of multiple scales assumes that the expansion representing the response to be a function of multiple independent variables, or scales instead of a single variable. In this process, first the new independent variables are introduced.

Tn   n t For n=0,1,2…… where ε is a small parameter Time derivatives then can be re-written as

d d T0  dT    1  ........  D0   D1  ........ dt d t  T0 d t  T1

(7a)



(7b)



d2  D02  2  D0 D1   2 D12  2 D0 D2  .......... d t2

To illustrate the implementation of the method first order approximate solution of Equation (6) in the main resonance region is provided below. In the main resonance region where the excitation frequency is near the natural frequency of the ship (    0    ), the linear damping coefficient, quadratic and cubic restoring moment coefficients and finally amplitude of the excitation frequency were considered to have order of є. Thus, Equation (6) was written in as;





x  02 x    x  d x 2  g x 3 

 f 2

e 

i  0T0  T1 

 e  i  0T0  T1 



(8)

If the expressions shown in Equation (7a) and (7b) are substituted into Equation (8), Equation (9) is obtained.

124 A.Y. Odabasi and E. Uçer

D02 x  2  D0 D1 x   2 D12 x  2 2 D0 D2 x  02 x    D0 x   2  D1 x   d x 2   g x3 

 f 2

e 

i  0 T0    T1 

(9)



 e  i  0T0    T1   ......

The solution of above equation can be represented by an expansion as

x  x0 (T0 , T1 , T2 ,....)   x1 (T0 , T1 , T2 ,....)

(10)

  2 x2 (T0 , T1 , T2 ,....)  ...........

If the expansion shown in Equation (10) is substituted into Equation (9) and coefficients of like powers of ε are equated, we obtain the following equations.

 0 : D02 x0  02 x0  0

(11a)

 1 : D02 x1  02 x1  2 D0 D1 x0   D0 x0  d x02  g x03 





(11b)

 D12 x0   D0 x1   D1 x0

(11c)

f i0T0 iT1 e e  e i0T0 e iT1 2

 2 : D02 x2  02 x2  2 D0 D1 x1  2 D0 D2 x0  2d

x0 x1  3 gx02

x1

The solution of Equation (11a) is

x0  A(T1 ) ei 0T0  A (T1 ) e i  0T0 where A(T1 ) 

(12)

1 1 a(T1 ) ei  (T1 ) and A (T1 )  a(T1 ) e  i  (T1 ) , A and A are the 2 2

only functions of T1 as we are searching for first approximate solution. If x0 solution shown in Equation (12) is substituted into Equation (11b), a new differential equation is obtained. The right hand side of this equation includes secular terms. These secular terms are eliminated by equating the coefficients of e i 0 T0 and e  i 0 T0 to zero.

Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 125

f iT1 e 0 2

(13a)

f iT1 e 0 2

(13b)

 2 i 0 D1 A  i  0 A  3 A A2 g  2 i 0 D1 A   i  0 A  3 A A 2 g 

If A and A are substituted into the above equations, after some transformations which will not be shown here due to limited space, Equation (14a) and (14b) are obtained.

a 

a 2

a  



f 2 0

Sin T1     0

3g 3 f a  Cos T1     0 8 0 2 0

(14a)

(14b)

where (΄) symbol represents derivative taken with respect to T1. Defining  T1     and        , the above equation becomes

a  

a 2



f 2 0

Sin  

(15)

The above equations can also be written in the form of the equations below.

 a  f  a    Sin   2 0  2 

(16a)

 3g 3  f a    a     a  Cos   2 0  8 0 

(16b)

The first order approximate solution of biased roll equation is written as:

x  a Cos   t       

(17)

where a and γ can be found from Equation (16a) and (16b). This method had been used to obtain an approximate solution of roll equation from many scientist and researchers (Nayfeh 1979, Nayfeh and Khdeir 1986).

126 A.Y. Odabasi and E. Uçer

Thus, second order approximate solution and also solution in sub and super harmonic resonance regions weren’t explained here. Due to decrement in linear damping coefficient or increment in initial bias angle and amplitude of wave excitation, the magnitude of coefficient “a” in the steady state solution increases in main, sub and super harmonic resonance regions. As can be seen from Fig. 1 and 2, when the wave excitation frequency is approximately half of the natural frequency, coefficient “a” takes its value in negative small values of σ (deviation between excitation frequency and natural frequency). The jump phenomenon was also observed when linear damping coefficient µ was equal to 0.04 and amplitude of the wave excitation (F) taken value higher or equal to 0.12. m=0.04 F=0.12 d2=0.131664 0.6 0.5

Bias=0.20 rad. Bias=0.15 rad. Bias=0.10 rad.

a

0.4 0.3 0.2 0.1 0 -0.15 -0.12 -0.09 -0.06 -0.03

0 s

0.03 0.06 0.09 0.12 0.15

Fig. 1 a-σ graph The effect of initial bias angle w0=0.597179 d1=0.078998 d2=0.131664 m=0.04 0.6

F=0.12 F=0.1 F=0.08

0.5

a

0.4 0.3 0.2 0.1 0 -0.15 -0.12 -0.09 -0.06 -0.03

0 s

0.03 0.06 0.09 0.12 0.15

Fig. 2 a-σ graph The effect of amplitude of excitation

As can be seen from Fig. 3, if the wave excitation frequency is approximately twice of the natural frequency, coefficient a in the first and second approximate solution of rolling equation goes to directly to zero without any oscillation when time goes to infinity. If the linear damping coefficient is increased, coefficient “a” goes to zero faster.

Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 127

Fig. 3 a-t graph (µ=0.04, F=0.12, σ =-0.05,

  2  0 , θs=0.2 rad.)

As can be seen from the Fig. 4 and Fig. 5, while the other parameters of the equation are fixed if we take σ =-0.1 instead of σ =0.1 in the main resonance region, the coefficient “a” gets much larger values (nearly twice of the former).

Fig. 4 a-t graph (µ=0.04, F=0.12, σ =0.1,

   0 , θs=0.2 rad.)

Fig. 5 a-t graph (µ=0.04, F=0.12, σ =-0.1,

   0 , θs=0.2 rad.)

2.3.2 Bogoliubov-Mitropolsky Asymptotic Method While examining the main resonance region by Bogoliubov-Mitropolsky asymptotic method, ω02=Ω2+εΔ, linear damping coefficient, quadratic and cubic term of restoring

128 A.Y. Odabasi and E. Uçer

moment and finally amplitude of excitation force have been assumed to be order of ε. Under these assumptions biased roll equation were written as



x   2 x     x  d x 2  g x 3  f Cos t   x



(18)

The above equations can be written in more efficient form as

   x  d x 2  g x 3   x   2 x      f Cos Cos  f Sin Sin   x   

(19)

where ψ = Ω t+β The solution is assumed to have the form (Bogoliubov and Mitropolsky 1961),

x  a Cos   u1  a,  t ,   ...........

(20)

and a and β in the above equation must satisfy the following expressions

da   A1 (a,  )   2 A2 (a,  )  ......... dt

(21a)

d   B1 (a,  )   2 B2 (a,  )  .......... dt

(21b)

To obtain first order approximate solution of biased roll equation, only first terms were taken in the right hand side of the Equation (20), (21a) and (21b). If the first term of expression shown in Equation (20) is put into the left hand side of the Equation (19), Equation (22) is obtained. If it is put into right hand side of the Equation (19), Equation (23) is obtained.

x   2 x    2 A1 Sin  2 a  B1Cos   a cos





   x  d x 2  g x 3  f CosCos  f Sin Sin   x  d a2   a Cos   a  Sin  2   2 3 d a g a   2 Cos 2  4 Cos 3 

 3 g a3  f CosCos  f Sin Sin  4 

(22)

 (23)

Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 129

If the right hand sides of the Equation (22) to Equation (23) are equated and the terms which have Cosψ and Sinψ are brought together, the following expressions are obtained, and with the help of other terms, u1 solution can be found as:

 2  A1   a   f Sin 

 2 a  B1    a 

u1 

3 g a3  f Cos  4

(24a)

(24b)

d a2 d a2 g a3  Cos 2   Cos3 2 2 6 2 32  2

(25)

If A1 and B1 are substituted into the Equation (21a) and Equation (21b), the following expressions are obtained:

da ˆ a F   Sin  dt 2 2 d  02   2 3 d 2 a 2 F    Cos  dt 2 8 2a

(26a)

(26b)

The first order approximate solution of biased roll equation is the given as:

x  a Cos  t       

(27)

where a and β coefficients are determined from Equation (26a) and (26b). In sub and super harmonic resonance regions instead of using Equation (18), F Equation (28) is used. Equation (28) is obtained by making x  y  2 Cos  t 0   2 transformation in the Equation (6).

  y     Sint  d  y   Cos  t 2  y  02 y    0 3  g  y   Cos  t   where  

F   2 2 0

(28)

130 A.Y. Odabasi and E. Uçer

2.3.3 Numerical Methods Biased roll equation was solved by Mathematica 5-NDSolve with 50000 steps in sub, super and main harmonic resonance regions for different initial conditions and parameters of the equation. The results also controlled with Fortran IVPAG routine. Stable bounds for different equation parameters were obtained by Mathematica 5-NDSolve and the program (“Rk.f90”) written in Fortran 90 language by the authors. “Rk.f90” uses Runge Kutta Fourth order method with time step 0.001 second to solve biased roll equation and finds out the initial conditions cause to solution goes to infinity when time goes to infinity. Solutions of “Rk.f90” program are the same as the solutions obtained by Mathematica 5-NDSolve.

3 Compliance of Perturbation & Asymptotic Results with Numerical Solution The solutions obtained by using method of multiple scales (MSM) shown in Fig. 6-11 in and Bogoliubov-Mitropolsky asymptotic method solutions shown in Fig. 12-17. As can be seen from the figures, the second order approximate solutions obtained by using MSM and asymptotic method in sub, super and main harmonic resonance regions, have better compliance with numerical solution rather than the first order approximate solutions. Second order approximate solutions only have difference with numerical solutions in transient region. However, the first order approximate solutions also have differences in steady state region. Although this good compliance of perturbation and asymptotic results with numerical solution in stable bounds, perturbation and asymptotic methods could not explain unstable cases. 1.5

ω=0.597179 d=0.078998 σ=0.1 F=0.12

MSM O(ε^2) Numeric

1

x(t)

0.5 0

0

50

100

150

200

250

300

350

400

-0.5 -1 -1.5 t

Fig. 6 Comparison of second order approximate MSM and numeric solution when    0

Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 131 1.50

x(0)=0 dx/dt=0.56 ω=0.59718 Ω=1.29436 d=0.078998

MSM O(ε^2) Numeric

1.00

x(t)

0.50

0.00 50

0

100

150

200

250

300

350

400

-0.50

-1.00

Fig. 7 Comparison of second order approximate MSM and numeric solution when   20 1

MSM O(ε^2) Numeric

x(0)=0.08045 dx/dt=0.21482 ω=0.59718 Ω=0.28359 d=0.078998

0.8 0.6 0.4

x(t)

0.2 0

0

50

100

150

200

250

300

350

400

-0.2 -0.4 -0.6 -0.8 -1 -1.2 t

Fig. 8 Comparison of second order approximate MSM and numeric solution when    0 2 1.5

x(0)=0 dx/dt=0.459 ω=0.59718 Ω=0.69718 d=0.078998

MSM O(ε) Numeric

1

x(t)

0.5

0

0

50

100

150

200

250

300

350

400

-0.5

-1

-1.5 t

Fig. 9 Comparison of first order approximate MSM and numeric solution when    0

132 A.Y. Odabasi and E. Uçer 1.5

x(0)=0 dx/dt=0.56 ω=0.59718 Ω=1.29436 d=0.078998

MSM O(ε) Numeric

1

x(t)

0.5

0

0

50

100

150

200

250

300

350

400

-0.5

-1

t

Fig. 10 Comparison of first order approximate MSM and numeric solution when   2 0 1

x(0)=0.1 dx/dt=0.2 ω=0.59718 Ω=0.2835 d=0.078998

0.8

MSM O(ε) Numeric

0.6 0.4

x(t)

0.2 0 -0.2

0

50

100

150

200

250

300

350

400

-0.4 -0.6 -0.8 -1 -1.2

t

Fig. 11 Comparison of first order approximate MSM and numeric solution when    0 2 1.5

μ=0.1,F=0.12, Ω=0.697179, ω=0.597179 x(0)=0 dx/dt=0.535

Numeric Asymptotic O(ε^2)

1

x(t)

0.5

0

0

50

100

150

200

250

300

350

400

-0.5

-1 t

Fig. 12 Comparison of second order approximate asymptotic and numeric solution when    0

Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 133 1.5

μ=0.04,F=0.12, Ω=1.294358,ω=0.597179 x(0)=0 dx/dt=0.56 Bias=0.2 rad.

Numeric Asymptotic O(ε^2)

1

x(t)

0.5

0

0

50

100

150

200

250

300

400

350

-0.5

-1 t

Fig. 13 Comparison of second order approximate asymptotic and numeric solution when   20 1.5

Ω=0.285755 F=0.12 x(0)=0.25 y(0)=0.268 Bias=0.2 rad.

Asymptotic O(ε^2) Numeric

1

x(t)

0.5 0 0

50

100

150

200

250

300

350

400

-0.5 -1 -1.5 t

Fig. 14 Comparison of second order approximate asymptotic and numeric solution when 1.5

  0 2

μ=0.1,F=0.12, Ω=0.697179,ω=0.597179 x(0)=0 dx/dt=0.535

Asymptotic O(ε) Numeric

1

x(t)

0.5 0

0

50

100

150

200

250

300

350

400

-0.5 -1 -1.5 t

Fig. 15 Comparison of first order approximate asymptotic and numeric solution when    0

134 A.Y. Odabasi and E. Uçer 1.5

x(0)=0 y(0)=0.56 ω=0.597179 Ω=1.29436 Bias=0.2 rad.

Asymptotic O(ε) Numeric

1

x(t)

0.5

0

0

100

50

150

200

250

300

350

400

-0.5

-1 t

x(t)

Fig. 16 Comparison of first order approximate asymptotic and numeric solution when   20 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 0 -0.4 -0.6 -0.8 -1 -1.2 -1.4

ω=0.597179 Ω=0.285755 Bias=0.2 rad.

50

100

150

200

250

300

Asymptotic O(ε) Numeric

350

400

t

Fig. 17 Comparison of first order approximate asymptotic and numeric solution when    0 2

4 Existence of Bounds for Initial Conditions and its Dependence to Parameters As can be seen from Table 2, in the main resonance region, when σ = -0.1 (σ: Deviation of excitation frequency from natural frequency) and |F| ≥ 0.1 (F: Amplitude of wave excitation), there is no stable bound. If amplitude of wave excitation is decreased to 0.08, there are stable bounds for very large linear damping coefficients. The effect of sign and magnitude of σ can easily be seen from Table 3 When σ = 0.1, there are stable bounds even for |F|=0.12. The increment of the amplitude and initial bias angle or decrement of linear damping coefficient causes the narrowing of stable solution bounds.

Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 135 Table 2 Stable bounds in main resonance region for σ = -0.1 Amplitude of

Initial Bias Angle

Initial Bias Angle

Wave excitation

θs=0.2 rad.

θs=0.1 rad.

Initial Bias Angle θs=0.005 rad.

|F|=0.12

-

-

-

|F|=0.10

-

-

-

|F|=0.08

-

Stable Bound

Stable Bound

for µ ≥0.13

for µ ≥0.1

Table 3 Stable bounds in main resonance region for σ = 0.1 Amplitude of

Initial Bias Angle Initial Bias Angle Initial Bias Angle

Wave excitation θs=0.2 rad. F=±0.12 F=±0.10 F=±0.08

θs=0.1 rad.

θs=0.005 rad.

Stable Bound

Stable Bound

Stable Bound

For µ > 0.10

For µ ≥ 0.07

For µ ≥ 0.04

Stable Bound

Stable Bound

Stable Bound

For µ > 0.07

For µ > 0.04

For µ ≥ 0.04

Stable Bound

Stable Bound

Stable Bound

For µ > 0.04

for µ ≥0.04

for µ ≥0.04

There is another factor which effects size of stable bounds. That is the phase angle which will be given to excitation force. The stable bound obtained for μ=0.1, d1=0.039499, d2=0.131664, ω0=0.60702, F=±0.12 and σ = 0.1 shown in Table 4. Table 4 Boundaries of stable bound for μ=0.1 |F|=0.12 θs=0.1 rad. σ =0.1 F=0.12

µ=0.1

F=-0.12

µ=0.1

x (rad.)

x (rad/s)

x (rad.)

x (rad/s)

0.000

0.634

0.000

0.844

0.200

0.605

-0.200

0.854

0.400

0.558

-0.400

0.842

0.600

0.491

-0.600

0.806

0.800

0.404

-0.800

0.742

1.000

0.298

-1.000

0.636

1.200

0.170

-1.065

0.587

1.427

0.000

-1.065

0.000

Let’s take a point which is inside the stable bound shown in Table 4. For example,

x (0)  0 and x (0)  0.634 . If this point is taken as the initial condition, some points on the solution orbit in x- x phase space are out of the stable bound (see Table 5 and Fig. 18).

136 A.Y. Odabasi and E. Uçer Table 5 Some points out of the stable bound tt5.24 t (sn.)

x (rad.)

x

19.2

-0.033582

0.764827

19.3

0.042842

0.763051

19.4

0.118912

0.757785

19.5

0.194286

0.749140

19.6

0.268632

0.737270

(rad/s)

Fig. 18 Solution orbit in x- x phase space for initial condition x(0)=0 x (0)=0.634

As it can be seen from Fig. 19, if x(0)=0.194286 x (0)=0.749140 are taken as initial condition and solve the biased roll equation, the solution is unstable (when (when time goes to infinity, solution also goes to infinity). y 1.8 1.6 1.4 1.2 0.8

1

2

3

4

x

0.6

Fig. 19 Solution orbit in x- x phase space for initial condition x(0)=0.194286 x (0)=0.749140

The solution is stable for x(19.5)=0.194286 and y(19.5)=0.74914 initial condition, Fig. 20.

Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 137

phase space for initial condition x(19.5)=0.19428 Fig. 20 Solution orbit in x- x

x (19.5)=0.74914

The solution was showed unstable for x(0)=0.194286 x (0)=0.74914 initial condition. If δ=1.220519 rad. phase angle is put into the excitation force and the biased roll equation is solved for x(0)=0.194286 x (0)=0.749140 initial condition again, the solution is stable (see Fig. 21). Hence, the effect of phase angle which will be given the excitation force were proved.

Fig. 21 Solution orbit in x- x phase space for initial condition x(0)=0.194286

x (0)=0.74914 δ=1.220519 rad. In sub and super harmonic resonance regions, the increment of the amplitude and initial bias angle or decrement of linear damping coefficient causes to narrow of stable solution bounds and also the phase angle which will be given to excitation force effects size of stable bounds like main resonance region. Some of the stable bounds obtained when the excitation frequency is approximately half of the natural frequency are shown in Fig. 22-24. As can be seen from these figures, the symmetry of the stable bounds corrupts after the initial bias angle increases. Fig. 25 shows sample of orbit for frequency equal to half the natural frequency.

138 A.Y. Odabasi and E. Uçer

Fig. 22 Stability bound for µ=0.04 θs=0.20 rad. |F|=0.12 σ =-0.078

Fig. 23 Stability bound for µ=0.04 θs=0.10 rad. |F|=0.12 σ =-0.04

Fig. 24 Stability bound for µ=0.04 θs=0.005 rad. |F|=0.12 σ =-0.03

Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 139

Fig. 25 Sample solution orbit in x- x phase space for    0 2

5 Concluding Remarks Due to decrement in linear damping coefficient or increment in initial bias angle and amplitude of wave excitation, the magnitude of coefficient a in the steady state solution increases in main, sub and super harmonic resonance regions. When the wave excitation frequency is approximately half of the natural frequency, coefficient “a” takes its maximum value in negative small values of σ (Deviation between excitation and natural frequency) and also there is jump phenomenon when linear damping coefficient is equal to 0.04 and amplitude of the excitation is higher or equal to 0.12. If the wave excitation frequency is approximately twice of the natural frequency, coefficient “a” in the first and second approximate solution of rolling equation goes to directly to zero without any oscillation when time goes to infinity. If the linear damping coefficient, is increased coefficient “a” goes to zero faster. While the other parameters of the equation are fixed if σ =-0.1 is taken instead of σ =0.1 in the main resonance region, the coefficient “a” gets much bigger values (nearly twice of the former). The second order approximate solutions of asymptotic method and the method of multiple scales have better compliance with numerical solutions in main, sub and super harmonic resonance rather than first order approximate solution of the asymptotic method and the method of multiple scales. The second order approximate solutions only have differences with numerical solutions in transient part of the solutions. However, the first approximate solutions also have differences with numerical solutions in steady state part of the solutions. Although this good compliance of perturbation and asymptotic results with numerical solution in stable bounds, perturbation and asymptotic methods could not explain unstable cases well.

140 A.Y. Odabasi and E. Uçer

The magnitude of linear damping coefficient, initial bias angle, amplitude and frequency of wave excitation and the phase angle which are given to excitation force effects the size of stable solution bounds in main, sub and super harmonic resonance regions. The increment of the amplitude and initial bias angle or decrement of linear damping coefficient causes the narrowing of stable solution bounds. The phase angle of the excitation force can make an unstable initial condition stable initial condition. Thus, importance of this parameter shouldn’t be neglected. The initial bias angle is also affected on the symmetry of the stable solution bounds. The increment of initial bias angle causes to increase asymmetries.

References Bogoliubov N N, Mitropolsky Y A (1961) Asymptotic methods in the theory of nonlinear oscillations. Intersciense NY Cardo A, Francescutto A et al. (1981) Ultra harmonics and subharmonics in the rolling motion of a ship: steady state solution. Int Shipbuild Prog 28:234-251 Cotton B, Bishop S R et al. (2000) Sensitivity of capsize to a symmetry breaking bias. In: Vassalos D, Hamamato M et al. (eds). Contemp Ideas on Ship Stab. Elsevier Oxford Grochowalski S (1989) Investigation into the physics of ship capsizing by combined captive and free running model tests. Trans of Soc of Nav Archit and Mar Eng 97:169-212 Jiang C, Troesch A W et al. (1996) Highly nonlinear rolling motion of biased ships in random beam seas. J of Ship Res 40 2:125-135 Macmaster A G, Thompson J M T (1994) Wave tank testing and the capsizability of hulls. Proc of the R Soc London A 446:217-232 Nayfeh A H (1979) Nonlinear oscillations. John Wiley NY Nayfeh A H, Khdeir A A (1986) Nonlinear rolling biased ships in regular beam waves. Int Shipbuild Prog 33:84-93 Spyrou K J, Cotton B et al. (1996) Analytical expressions of capsize boundary for a ship with roll bias in beam waves. J of Ship Res 46 3:125-135 Wright J H G, Marshfield W B (1980) Ship roll response and capsize behavior in beam seas. Trans of RINA 122:129-149

Historical Roots of the Theory of Hydrostatic Stability of Ships Horst Nowacki* Larrie D. Ferreiro** * Technical University of Berlin (Germany), ** Defense Acquisition University and Catholic University of America (United States)

Abstract The physical principles of hydrostatic stability for floating systems were first pronounced by ARCHIMEDES in antiquity, although his demonstration examples were limited to simple geometrical shapes. The assessment of stability properties of a ship of arbitrary shape at the design stage became practically feasible only about two millennia later after the advent of infinitesimal calculus and analysis. The modern theory of hydrostatic stability of ships was founded independently and almost simultaneously by Pierre BOUGUER (“Traité du Navire”, 1746) and Leonhard EULER (“Scientia Navalis”, 1749). They established initial hydrostatic stability criteria, BOUGUER’s well-known metacenter and EULER’s restoring moment for small angles of heel, and defined practical procedures for evaluating these criteria. Both dealt also with other aspects of stability theory. This paper will describe and reappraise the concepts and ideas leading to these historical landmarks, compare the approaches and discuss the earliest efforts leading to the practical acceptance of stability analysis in ship design and shipbuilding.

1 Introduction Human awareness of the significance of ship stability for the safety of ocean voyages is probably as ancient as seafaring. An intuitive, qualitative understanding of stability and of the risks of insufficient stability must have existed for millennia. The foundations for a scientific physical explanation and for a quantitative assessment of hydrostatic stability were laid by ARCHIMEDES in antiquity (Archimedes 2002, “On Floating Bodies”). Yet despite many important contributions and partially successful attempts by scientists in the early modern era like STEVIN, HUYGENS, and HOSTE among others it took until almost the mid-eighteenth century before a mature scientific theory of ship hydrostatic stability was formulated and published. Pierre BOUGUER (Bouguer 1746, “Traité du Navire”) and Leonhard EULER (Euler 1749, “Scientia Navalis”) were the founders of modern ship stability M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_8, © Springer Science+Business Media B.V. 2011

141

142 H. Nowacki and L.D. Ferreiro

theory, who quite independently and almost simultaneously arrived at their landmark results for hydrostatic stability criteria. BOUGUER developed the theory and introduced the terminology of the metacenter and the metacentric curve. EULER defined the criterion of the initial restoring moment, which for hydrostatic stability amounts to an equivalent concept. The full implementation of computational methods for evaluating these criteria and their acceptance by practitioners took even several decades longer. Paradoxically this chain of events raises two nearly contradictory questions, which this paper will address:  Why did it take so long for these formalized, quantitative criteria for the stability of ships of arbitrary shape to be pronounced?  When at last the discoveries were made, why then did two independent, but equivalent solutions suddenly appear almost at the same time? The knowledge required to evaluate the stability of a ship rests on many concepts and requires many autonomous discoveries to be made and to be brought into concerted application. These include:  The idea of conceptual experiments in dealing with mechanical systems.  The abstraction of thinking in terms of resulting forces and moments of weight and buoyancy (“lumped effects”) substituted for distributed effects of gravity and pressure.  The axiom of force equilibrium, here between weight and buoyancy force (Principle of ARCHIMEDES).  The axiom of moment equilibrium in a system at rest.  A definition and a test for system stability.  A method for finding the combined weight and center of gravity (CG) for several weight components.  A principle for finding the resultant buoyancy force and its line of action (through the center of buoyancy, CB).  A method for calculating volumes and their centroids, first for simple solids, then for arbitrary ship shapes.  An analytical formulation of a stability criterion (for infinitesimal and for finite angles of inclination). Although the physical principles of hydrostatic stability were already established by ARCHIMEDES, it took a long time before the analytical formulation of stability criteria could be pronounced for the general case of ships, essentially by means of calculus, and before numerical evaluations became feasible. BOUGUER and EULER were the first to find an analytical criterion for initial stability, BOUGUER in terms of the metacenter. EULER in terms of the initial restoring moment. BOUGUER went beyond this in several practical aspects. The intriguing question remains: How did two scientific minds work independently to come to rather equivalent results? What were their sources, their background, their approach, their logic, their justification and verification? Which were their unique original thoughts and how did they differ?

Historical Roots of the Theory of Hydrostatic Stability of Ships 143

To answer these questions it is not sufficient to look only at their final results and conclusions, but it is necessary to examine more closely the methodical approaches taken to the ship stability issue and to compare the trains of thought by which the individual authors arrived at their results. In this article we will review the developments that led to this historical stage when modern hydrostatic stability theory was founded.

2 Precursors 2.1 Archimedes ARCHIMEDES of Syracuse (ca. 287-212 B.C.), the eminent mathematician, mechanicist and engineering scientist in antiquity, is also the founder of ship hydrostatics and hydrostatic stability, which he established as scientific subjects on an axiomatic basis. ARCHIMEDES was brought up in the early Hellenistic era in the tradition of Greek philosophy, logical rigor and fundamental geometric thought. ARCHIMEDES was well educated in these subjects in Syracuse and very probably also spent an extended study period in Alexandria, the evolving Hellenistic center of science, at the Mouseion (founded in 286 B.C.). There he met many leading contemporary scientists, e.g., DOSITHEOS, ARISTARCHOS and ERATOSTHENES, with whom he maintained lifelong friendships and scientific correspondence. Steeped in this tradition of Greek geometry and mechanics, ARCHIMEDES learned how to excel in the art of deductive proofs from first premises usually based on conceptual models, i.e., models of thought, by which physical reality was idealized. But ARCHIMEDES was also unique, as many legends on his engineering accomplishments tell us, in applying practical observation to test his scientific hypotheses and to develop engineering applications, although these achievements are not mentioned in his own written work. We are fortunate that many, though not all, of ARCHIMEDES’ treatises are preserved, all derived from a few handwritten copies made in the Byzantine Empire during the 9th and 10th centuries and transmitted to posterity on circuitous routes to resurface essentially through the ARCHIMEDES revival during the Renaissance. Luckily, one Greek manuscript, which was later lost, had been translated into Latin by the Dominican monk Willem van MOERBEKE. This translation, which was published in 1269 and was later named Codex B, also contained ARCHIMEDES’ treatise “On Floating Bodies”, Books I and II. It became the only accessible reference to ARCHIMEDES’ work on hydrostatics for many centuries until in 1906 most surprisingly an old 10th century palimpsest was rediscovered in a Greek monastery in Constantinople by J. HEIBERG (Heiberg 1906-07). This palimpsest, called Codex C by HEIBERG, under the writing of a 12th century monk of a Greek prayerbook had retained significant traces of the

144 H. Nowacki and L.D. Ferreiro

rinsed off text of a Greek manuscript from ARCHIMEDES including the Greek version of “On Floating Bodies”. ARCHIMEDES’ texts were soon reconstructed from this source, transcribed and translated into modern languages (Archimedes 2002). Meanwhile this palimpsest, which had disappeared in private possession in the aftermath of the Greek-Turkish war in 1920-22, has turned up again recently and is under new scientific evaluation at the Walters Art Museum in Baltimore (Netz and Noel 2007). It is on these sources primarily that we can base a reliable evaluation of ARCHIMEDES’ contributions to ship hydrostatics today. ARCHIMEDES preceded his work on hydrostatics by a number of other treatises establishing certain axioms of mechanics: 2.1.1 The Law of the Lever: In his treatise “The Equilibrium of Planes” ARCHIMEDES first deals with the equilibrium of moments about a fulcrum in a lever system. Although he claims to deduce this principle from geometric reasoning alone, it is actually understood today that the law of the lever is equivalent to the axiom of moment equilibrium in mechanics. Second, he introduces the concept of “centroids” of quantities (areas, volumes, weights) into which the quantities can be “lumped” as concentrated effects so that moment equilibrium is retained. Third, he proposes a method for finding the “compound centroid” of a system of components, e.g., a center of gravity. Finally, he proves the critical “centroid shift theorem” i.e., a rule for the shift of the system centroid when some quantity is added to, removed from or shifted within the system. All of these concepts and results are essential physical principles as prerequisites for his work on hydrostatics. 2.1.2 Quadrature: In his treatise “The Quadrature of the Parabola” ARCHIMEDES illustrates by the example of the parabola how the Greeks determined areas and volumes of elementary shapes without the availability of calculus. Here he uses a method, well-known since EUDOXUS (Boyer 1939, 1949), based on an inscribed polygonal approximant which is continually refined by interval halving until under the given premises it converges to the given curve within a specified error tolerance. This type of deduction was later called “method of exhaustion”. Thus the quadrature problem is reduced to that of evaluating the sum of a finite, truncated or sometimes even infinite series of approximations. This method of quadrature generally is not equivalent to calculus for lack of a limiting process to infinitesimal step width, but has nonetheless inspired many future developments until this day.

Historical Roots of the Theory of Hydrostatic Stability of Ships 145

2.1.3 The Method of Mechanical Theorems: This famous treatise, which was actually also discovered in the palimpsest of 1906, explains how ARCHIMEDES used a reasoning based on principles of mechanics (like moment equilibrium of volume quantities) in deriving geometrical results (like volume centroids). He regarded such findings as propositions for later rigorous deduction by strictly geometrical proofs. 2.1.4 On Floating Bodies: In this treatise ARCHIMEDES makes use of all these prior results and proceeds to lay the foundations for ship hydrostatics and stability in the following steps (Archimedes 2002, Czwallina-Allenstein 1996): Book I of this treatise begins with Postulate 1 describing the properties of a fluid at rest axiomatically. “Let it be supposed that the fluid is of such character that, its parts lying evenly (N.B.: at the same level) and being continuous (N.B.: coherent), that part which is thrust the less is driven along by that which is thrust the more and that each of its parts is thrust by the fluid which is above it in a perpendicular direction, unless the fluid is constrained by a vessel or anything else”. Although ARCHIMEDES does not use the word “pressure” and the Greeks did not know that concept in antiquity, he does infer that parts under more pressure would drive parts under less pressure so that a fluid cannot be at rest unless the pressure is uniform at a given depth, while the weight of a vertical column of fluid rests on the parts below it. From these very simple axiomatic premises, which do not permit evaluating the local pressure anywhere in the fluid, ARCHIMEDES is able to derive the principles of hydrostatic equilibrium and stability of floating bodies. This is achieved by considering the equilibrium of the resultant buoyancy and gravity forces and of their moments. ARCHIMEDES’ famous Principle of Hydrostatics is stated in Book I, Proposition 5: “Any solid lighter than a fluid will, if placed in the fluid, be so far immersed that the weight of the solid will be equal to the weight of the fluid displaced” (Archimedes 2002). The proof of this law, usually pronounced today as Δ = γ V, is brilliantly brief and conclusive. In all brevity it rests on the argument that in equilibrium the solid is at rest in a fluid at rest, thus if the body is removed from the fluid and the cavity left by its underwater volume is filled with fluid matter, then the fluid can only remain at rest if the replacing fluid volume weighs as much as the solid, else the fluid would not remain in equilibrium and hence at rest.

146 H. Nowacki and L.D. Ferreiro

In Book II, mainly Proposition 2, ARCHIMEDES deals with the stability of hydrostatic equilibrium by treating the special case of a solid of simple shape, viz., a segment of a paraboloid of revolution of homogeneous material whose specific gravity is less than that of the fluid on whose top it floats. In equilibrium it floats in an upright condition. The stability is tested by inclining the solid by a finite angle to the vertical, but so that the base of the segment is not immersed. The equilibrium is defined as stable if the solid in the inclined position has a restoring moment tending to restore it to the upright condition. For the homogeneous solid this stability criterion is readily evaluated geometrically by examining the lever arm between the buoyancy and the gravity force resultants (Fig. 1). The buoyancy force acts through the centroid of the underwater volume B, which ARCHIMEDES finds for the inclined paraboloid from theorems proven earlier. The gravity force or weight acts through the Center of Gravity R of the homogeneous solid. Our conventional righting arm, the projection of BR on the horizontal, is positive. Instead of using this stability measure, ARCHIMEDES takes a shortcut for this homogeneous solid by splitting off and removing the weight of the submerged part of the solid Δ1 and the corresponding equal share of the buoyancy force, which have no moment about B because they both act through B. Thus only the weight of the abovewater section of the solid Δ2, acting through C, and the equal and opposite buoyancy force increment, acting through B, are taken into account. The centroid C is found from B and R by applying the centroid shift theorem when removing the underwater part from the system. This yields a positive “incremental righting arm” for the force couple of Δ2, acting through B and C respectively. The restoring moment is thus positive and the solid will return to the upright position. This application of the hydrostatic stability criterion by ARCHIMEDES is limited to the special case of a homogeneous solid of simple parabolical shape. It demonstrates the physical principles of the hydrostatic stability problem for a finite angle of inclination. It does not extend to floating bodies of arbitrary shape and of non-homogeneous weight distribution, hence actual ships. Still the foundations were laid to enable others much later to treat the generalized case of the ship on the same fundamental grounds. A more detailed appraisal of ARCHIMEDES’ contributions to the hydrostatic stability of floating systems and a thorough historical account is given by NOWACKI (Nowacki 2002).

Historical Roots of the Theory of Hydrostatic Stability of Ships 147

Fig. 1 Restoring moments, righting arms for inclined homogeneous paraboloid, based on ARCHIMEDES’ “On Floating Bodies”, Book II (Archimedes 2002) (figure from (Nowacki 2002)).

2.2 Stevin and Pascal In the beginning of the modern era of science the manuscripts of ARCHIMEDES had been rediscovered by several humanists during the Renaissance and were made accessible in print after 1500 by several editions in Greek, Latin and later by translations into modern languages (Archimedes 2002, Czwallina-Allenstein 1996, Dijksterhuis 1987). Simon STEVIN (born 1548 in Bruges, died 1620 in Leyden), the celebrated Flemish/Dutch mechanicist and hydraulic engineer, who knew and admired ARCHIMEDES’ works, was probably the first modern scientist who resumed and resurrected the study of hydrostatics and applied it to hydraulics, but also to ships. In his important work “De Beghinselen des Waterwichts” (Stevin 1955), first published in Dutch in 1586 and translated into Latin by Willibrord Snellius in 1605, he deals with the principles of hydrostatics and hydraulics. He adopts the idea of “specific gravity” of a fluid from ARCHIMEDES and introduces the concept of “a hydrostatic pressure distribution”, as we would call it today, proportional to the weight of a water prism reaching down to the depth in question. This enables him to calculate water loads on walls in a fluid, an important foundation in hydraulic engineering. He also rederives ARCHIMEDES’ Principle of Hydrostatics. However when he proceeds to examine the hydrostatic stability of a ship in his later note “Van de Vlietende Topswaerheyt” (Stevin 1586), he correctly reconfirms that for equilibrium buoyancy and weight force resultants must act in the same

148 H. Nowacki and L.D. Ferreiro

vertical line through the centers of buoyancy (CB) and gravity (CG). But he erroneously concludes that for stability the CB must always lie above the CG. This error occurred because – unlike ARCHIMEDES - he neglected the centroid shifts resulting from the volume displacement from the emerging to the immersing side in heel. Despite this flaw in an application STEVIN deserves high recognition for founding modern hydrostatics on the concept of hydrostatic pressure. Blaise PASCAL (born 1623 in Clermont, died 1662 in Paris) is also counted among the founders of modern hydrostatics. He was familiar with ARCHIMEDES’ and probably with STEVIN’s work. In his “Traité de l’ équilibre des liqueurs”, Paris (1663), he arrives at similar conclusions and justifications regarding the fundamentals of hydrostatics in a fluid as STEVIN did, though he did not deal with ships. The assertion and experimental verification of hydrostatic laws also being applicable to air belong to his original contributions to this subject.

2.3 Huygens Christiaan HUYGENS (born 1629 in The Hague, died 1695 in The Hague), the eminent Dutch physicist, at the youthful age of 21 made an excursion into hydrostatic stability, which is not well known. In 1650 he wrote a three volume treatise “De iis quae liquido supernatant” (Huygens 1967a), in which he applies the methodology of ARCHIMEDES to the stability of floating homogeneous solids of simple shape, reconfirming ARCHIMEDES’ results and extending the applications to floating cones, parallelepipeds, cylinders etc., at the same time studying the stability of these solids through a full circle of rotation. He never published this work in his lifetime because he felt it was incomplete or “of small usefulness if any” or in any case not sufficiently original in comparison to ARCHIMEDES. He wanted the manuscript to be burnt, but it was found in his legacy and at last published in 1908. HUYGENS must be admired for his deep insights into ARCHIMEDES’ fundamentals of hydrostatics and for his own creative extensions. He recognized e.g. that for homogeneous, prismatic solids their specific weight and their aspect ratio are the essential parameters of hydrostatic stability. In his derivations he used a formulation based on virtual work as a principle of equilibrium. These examples of famous physicists between 1500 and 1700 working in hydrostatics and on hydrostatic stability of floating solids and ships demonstrate that the foundations inherited from ARCHIMEDES were understood by certain specialists, but had not yet been extended or applied to the design and stability evaluation of actual ships. Although the physical principles of stability were understood, the practical evaluation of volumes, volume centroids, weights and centers of gravity for floating bodies of arbitrary shape and non-homogeneous weight distribution de facto still posed substantial difficulties. These were not overcome before the advent and application of calculus during the 18th century.

Historical Roots of the Theory of Hydrostatic Stability of Ships 149

2.4 Hoste The French mathematics professor Paul HOSTE, s.j., (1652-1700) was the first to attempt to quantify the problem of ship stability. He did so in his 1697 treatise of naval architecture, Théorie de la construction des vaisseaux, which was appended to his book on naval tactics, written at the behest of Admiral TOURVILLE (Hoste 1697). However, he did not apply calculus to the problem, which was not yet widely known. HOSTE assumes, without citing STEVIN, that the CG and CB were in a vertical line; but he also allows without proof that the CG could be above the CB. However, to explain how this could be possible without the ship tipping over, he also assumes that the buoyancy force was equally divided between the two halves of the ship, forming the base of a triangle. He continues this error by stating: “If the center of gravity of the ship is known, the force with which it has to carry sail is easily known, which is no other thing than the product of the weight of the ship by the distance between these centers (of weight and displacement)” (Hoste 1697, p.54). In modern terms, the righting moment is equal to Δ x ( KG – KB ). In other words, the higher the center of gravity, the more stable the ship. Although he does not provide a theoretical means for determining this “power to carry sail”, HOSTE does describe a procedure which could empirically demonstrate this: The inclining experiment. HOSTE asserts that, by measuring the angle of inclination due to suspending a weight from a boom at a certain height, that the “force to carry sail” can be determined (Hoste 1697, p. 56). Although HOSTE’s argument contains several fundamental errors, it was the first attempt to express the stability of a ship in mathematical terms, and his book remained the only published inquiry into stability for almost half a century.

2.5 La Croix César Marie de LA CROIX (1690-1747) was the head of administration and finance for the Rochefort dockyard, and maintained the records for the galley fleet. He was not a scientist or engineer. Yet he developed some fundamental concepts on ship motions and hydrostatic stability, derived for a floating parallelepiped (La Croix 1736), so certainly before BOUGUER’s and EULER’s work appeared. LA CROIX was interested in the motions, but also in the hydrostatic restoring moments for this parallelepiped when heeled. He correctly understood the role of weight and buoyancy forces acting in the same vertical line in equilibrium. He also followed ARCHIMEDES in his stability criterion by examining the heeled body and requiring positive righting arms, accounting for the wedge volume shifts. But he falsely determined the influence of the wedge shift moments and

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hence the righting arms. Besides, since he lacked integration methods based on calculus, he was not able to generalize his results for arbitrary section and waterplane shapes, hence to make them applicable to ships. Yet although LA CROIX’s work was flawed, it may have triggered EULER’s renewed interest in ship hydrostatics. In 1735 EULER was asked by the Russian Imperial Academy of Sciences to review the treatise (La Croix 1736), which LA CROIX had submitted there prior to publication. EULER quickly responded and appraised the merits of LA CROIX ’s problem formulation, but also pointed out the weaknesses of the solution. He then put on record his own correct solution for the initial restoring moment, hence stability criterion, for the parallelepiped (or any other body of constant cross sectional shape), which he claimed to have found earlier (Euler 1972). This dates EULER’s earliest written mention of the criterion for this special case to 1735. In 1736 in reply to LA CROIX ’s rebuttal EULER also provided the complete explicit derivation of this result.

2.6 Sailing Vessel Propulsion and Maneuvering Questions of ship propulsion, ship motions and maneuvering, but also of ship stability have always aroused an acute interest, not only among seafarers and shipbuilders, but also in the scientific community. The propulsion of sailing vessels by the forces of wind, e.g., was investigated by ARISTOTLE in antiquity and later is associated with such celebrated names of scientists as Francis BACON, Thomas HOBBES and Robert HOOKE, among others. Toward the end of the 17th century the subject of the propulsion of sailing vessels had gained new scientific interest, also under the influence of the navies in applying scientific principles to ship design and operation. Treatises and monographs on the theory of maneuvering and on the equilibrium of aerodynamic propelling forces and hydromechanic response had appeared, notably by IgnaceGaston PARDIES (1673), Bernard RENAU d’ELISSAGARAY (1689), Christiaan HUYGENS (1693), Jakob BERNOULLI (1695) and Johann (I) BERNOULLI (1714). This had led to lively controversies, but also to an increasing depth of understanding of the mechanics of the sailing vessel. In fact through the various treatises the principles for applying hydrodynamic underwater and aerodynamic sail forces to the vessel in order to establish its equilibrium position had become well understood. At the same time the importance of allowing for trim and heel (as well as yaw) under these forces and moments had been recognized and dealt with, although the quantitative assessment of the hydrostatic restoring effects remained largely an open issue. Both BOUGUER and EULER would begin their investigations into the nature of hydrostatic stability with the study of propulsion and maneuvering of sailing ships.

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2.7 Displacement Estimates Before Calculus The infinitesimal calculus was important to the development of stability for several reasons, including the practical evaluation of underwater volume and volume centroids. Yet a number of methods to calculate underwater volumes (and therefore displacements) were developed in the 1500s and 1600s, well before any theoretical framework for stability was available to make use of it. Why would shipbuilders go through the trouble of making such calculations? It appears that there were two separate reasons for this: The development of the gunport, and the measurement of cargo tonnage. The gunport was introduced in the early 1500s, and came into wide use by the middle of the century. This greatly increased the firepower of naval ships, but brought two problems; ships got much heavier, and with large holes in the side of the ship, the available freeboard got dramatically smaller. Thus, the margin for error in estimating the waterline was considerably reduced. To handle this problem, shipbuilders like Anthony DEANE (1638-1721) developed methods to calculate how much armament, ballast and stores should be loaded on a ship to bring it to the correct waterline below the gunports. DEANE and others made use of ship’s plans, which were just beginning to be employed by shipbuilders as a construction guide. In his manuscript “Doctrine of Naval Architecture” (Deane 1981), never published but widely circulated, DEANE demonstrates two methods to calculate the area underneath waterlines at each “bend” or frame of the hull; using either (1) an approximation for the area of a quarter-circle or (2) by dividing the area into rectangles and triangles. DEANE then sums the areas for each frame, multiplies by the frame spacing and multiplies the volume by the specific weight of water to obtain the displacement. He does this for several different waterlines, including the desired waterline below the gunports. When a ship is launched, he can immediately determine the light displacement, and then calculate how much weight should be added to arrive at the design waterline. A second reason for introducing displacement calculations was to more accurately measure the cargo capacity of a ship. For example, from 1646 to 1669 the Dutch and Danish governments carried on a series of negotiations on cargo measurement. In 1652 a Dutch mathematician Johannes HUDDE identified the basic issue of measuring cargo deadweight by determining the difference between the weight of the ship empty and fully-laden, using a difference-in-drafts method. He suggested to the Dutch authorities that they measure the waterplane areas at each draft of an actual ship in the water (not from plans), by taking measurements to the hull from a line extended at the side of the ship parallel to the centerline. The space between hull and an overall rectangle formed by the length and beam was then divided into trapezoids and triangles, the areas calculated and summed, multiplied by the difference in drafts and multiplied by seawater density, to obtain cargo tonnage. Although the suggestion was never used, HUDDE’s cousin

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Nicolas WITSEN reported it in his 1671 shipbuilding manuscript “Aeloude en hedendaegsche scheepsbouw en bestier” (Witsen 1671; Cerulus 2001).

3 Development of Stability Theory 3.1 Bouguer 3.1.1 Biographical Sketch Pierre BOUGUER (Fig. 2) was born on 10 February 1698 in the French coastal town of Le Croisic, near Saint Nazaire at the mouth of the Loire. Educated in the Jesuit school in Vannes, he quickly showed himself a child prodigy. When his father Jean BOUGUER, a royal hydrographer and mathematician, died when Pierre was only 15, he applied for his father’s position. After initial hesitation by the authorities, he passed the rigorous exam and was given the post of Royal Professor of Hydrography. BOUGUER won several Royal Academy of Sciences prizes for masting, navigation and astronomy before he was 30. He moved to the port of Le Havre in 1731, about the same time he became a member of the Academy. His work caught the attention of the French Minister of the Navy MAUREPAS (1701-1781), who, like COLBERT before him, was convinced of the strategic benefit of ship theory as a way of compensating by quality against the quantitative superiority of the British navy. MAUREPAS supported BOUGUER’s research and sponsored his publications. In 1734, BOUGUER became involved in a controversy over the Earth’s shape; those who believed in DESCARTES’ vortex theories of physics held that the Earth was elongated at the poles, while those who accepted NEWTON’s theories of gravitational attraction argued that the Earth was wider at the equator due to centrifugal force. MAUREPAS also held the view that a full understanding of the Earth’s shape was essential to navigation, so he sent BOUGUER, along with several other members of the Academy of Sciences (accompanied by two Spanish naval officers) on a Geodesic Mission to Peru to determine the length of a meridian arc for a degree of latitude at the equator. One of his companions on the Mission was the young Spanish lieutenant Jorge JUAN Y SANTACILIA, who would later become a prominent naval constructor and author of a highly recognized treatise on naval architecture. BOUGUER spent ten years away (1735-1744), during which time he not only surveyed and calculated a meridian arc length of three degrees of latitude, he also performed various experiments on gravity and astronomy. It was during this Mission, in the peaks and valleys of the Andes far from the ocean, that he wrote much of his monumental work “Traité du navire”, the first comprehensive synthesis of naval architecture.

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Fig. 2 Portrait of Pierre BOUGUER.

“Traité du navire” was published in 1746, shortly after BOUGUER’s return. He remained unmarried, lived in Paris and devoted himself to revising his geodesic work, and carrying out further studies of naval architecture, astronomy, optics, photometry and navigation. BOUGUER died in Paris on 15 August 1758, aged 60. Further details on BOUGUER’s biography and scientific work are presented by FERREIRO (Ferreiro 2007) and DHOMBRES (Dhombres 1999). 3.1.2 Early Work on Ship Theory In 1721 BOUGUER was asked by the Academy of Sciences in Paris to compare the accuracy of two methods of calculating cargo tonnage being proposed to the Council of the Navy for port fees. Although the mathematician Pierre VARIGNON proposed estimating the volume of a ship as a semi-ellipsoid, BOUGUER found that the best results came from a proposal of Jean-Hyacinthe HOCQUART of

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Toulon, who used a difference-of-waterplanes method that employed equal-width trapezoids to estimate waterplane areas. This was similar to HUDDE’s approach but allowed a direct calculation of areas from ship’s plans. BOUGUER would later adopt and refine this “method of trapezoids” for his own stability work (Mairan 1724). In 1727 the Academy offered a Prize for the best treatise on masting, which BOUGUER’s entry won. He postulated that a “point vélique”, the intersection of the sail force and the resistance of water against the bow, should be directly above the center of gravity to minimize trimming by the bow. In his treatise he relied on HOSTE’s theories to explain stability, although where HOSTE implied that the advantage of doubling is through an increase in the center of gravity, BOUGUER invoked ARCHIMEDES to point out that the buoyancy of the added portion of the ship moves the center of buoyancy laterally when heeling, thus increasing the righting moment. Still, this treatise did not yet show any insights into the evaluation of trim or heel angles or restoring moments that he would develop five years later (Bouguer 1727). 3.1.3 The Metacenter BOUGUER probably began formulating his theory of stability around 1732, after he had moved to Le Havre, for he tested it using the little 18-gun frigate Gazelle, laid down in that dockyard in May 1732 and delivered in January 1734 (Demerliac 1995). However, no letters or manuscripts from that period survive to confirm this, and his derivation of the metacenter would not appear in print until 14 years later in “Traité du navire”. The following steps illustrate BOUGUER’s derivation of the metacenter in his “Traité du navire” (Bouguer 1746, pp. 199-324). STEP 1: Premises and axioms BOUGUER implicitly defines the hydrostatic properties of fluids, based on the principle of hydrostatics that weight and buoyancy are equal, opposite and act in the same vertical line. He does so without proof, without the use of equations and without mentioning ARCHIMEDES by name, although he shows general familiarity with his work. He also implies through geometrical arguments that the pressure of the fluid follows a hydrostatic distribution with depth and is everywhere normal to the surface of the submerged body.

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Fig. 3 BOUGUER’s diagram of the metacenter (Bouguer 1746, fig. 54).

He denotes: p = submerged volume Γ = upright CB γ = inclined CB AB = upright waterline ab = inclined waterline I = CG of an unstable ship G = CG of a stable ship g = metacenter 1 = centroid of immersed wedge 2 = centroid of emerged wedge 3 = centroid of body without wedges STEP 2: Magnitude of buoyancy force BOUGUER resolves the submerged surface into very small elements (though without using any calculus notations at this point) and equates the vertical components of the hydrostatic pressure forces to the weight of the water column resting on top of the element in the interior of the submerged volume conceptually. Hence, the total pressure resultant is equal to the total weight of water filling the submerged volume. He thus reconfirms the law of ARCHIMEDES by pressure integration. STEP 3: Measurement of volume and centroids BOUGUER first suggests two methods for calculating the volume of the ship as a regular solid. The first is to model the ship as an ellipsoid, as originally proposed by VARIGNON, and the second is to divide it into prisms. The set of quadrilateral prisms then forms a polyhedral approximant of the hull surface for volume summation. BOUGUER sees clearly that, by analogy, the same principle of

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polygonal approximation also holds for evaluating the area of planar curves. He thus quickly homes in on a quadrature rule for curves based on equally wide trapezoids, his favorite rule, also known as trapezoidal rule. He uses it to measure areas of waterline “slices” and then combining those two-dimensional slices to obtain a three-dimensional volume. Although he borrowed the idea from HOCQUART ’s 1717 proposal, BOUGUER refined the method, first by dividing each waterplane into many sections (HOCQUART only proposed three sections), and second by taking the areas of several waterlines to develop the entire volume of the hull (HOCQUART took only one “slice”). BOUGUER then arrives at the conclusion that his quadrature rule is suitable for evaluating the integral of any continuous function of one variable or, if applied recursively, for continuous functions of any number of independent variables. Interestingly he thus interprets the analytical formulation of stability criteria, which he introduces later, as being equivalent to the discretized quadrature rules presented here earlier. The ancestry of his concepts from both practical shipbuilding traditions and modern calculus is still leaving some visible traces. BOUGUER then digresses for many pages into using the trapezoidal rule to calculate incremental waterlines for estimating a ship’s payload capacity, and gives various rules for tunnage admeasurement. For finding the centroid of the underwater volume, the center of buoyancy CB, BOUGUER then begins by explaining, in very simplistic terms, how to use the sum-of-moments method to determine the centroid of an object. He then derives the area centroid of a planar figure (2D case), then the volume centroid of a solid (3D case), which for a ship he calls the “center of gravity of the hull”. He then discusses by example how to evaluate these expressions numerically by the method of trapezoids. STEP 4: Stability criterion The center of buoyancy having been determined, BOUGUER next explains why he chooses the metacenter as the initial stability criterion, using the geometrical argument shown in Fig. 3. The ship’s center of gravity g is always in the same vertical line as the center of buoyancy Γ, but this geometry is not constant due to the ship’s movement. If the ship has a very high center of gravity I, and moves even a little from the upright position (waterline A-B) to another position (waterline a-b), it is no longer statically stable; the center of buoyancy moves from Γ to γ, i.e., away from the vertical of the center of gravity, and the vertical force of buoyancy shifts from Γ-Z to γ-z. The ship’s weight, centered at I and on the opposite side of the inclination from the new center of buoyancy, tends to push the ship even further over, rendering it unstable. However, if the center of gravity of the ship is at G, below the intersection g of the upright and inclined vertical forces of buoyancy, then the center of gravity is on the same side as the new center of buoyancy and the resulting force always tends to restore the ship to the horizontal. BOUGUER states in his definition of the metacenter (Bouguer 1746, pp. 256257):

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“Thus one sees how important it is to know the point of intersection g, which at the same time it serves to give a limit to the height which one can give the center of gravity G, it [also] determines the case where the ship maintains its horizontal situation from that where it overturns even in the harbor without being able to sustain itself a single instant. The point g, which one can justly title the metacenter [BOUGUER’s italics] is the term that the height of the center of gravity cannot pass, nor even attain; for if the center of gravity G is at g, the ship will not assume a horizontal position rather than the inclined one; the two positions are then equally indifferent to it: and it will consequently be incapable of righting itself, whenever some outside cause makes it heel over.” BOUGUER does not use the terms “stable” or “unstable”, but rather states that G is either lower or higher than g. Step 5: Evaluation of the criterion The determination of this point of intersection, BOUGUER says, reduces to the question of the distance between the centers of buoyancy Γ and γ of the submerged body upright and just slightly inclined. BOUGUER employed the allimportant shifting of equal immersed and emerged wedges to determine this distance, by drawing a triangle between points 1, 2 and 3. Since the centers of buoyancy Γ and γ lie on the legs of that triangle, the distance Γ – γ must be proportional to the distance between points 1 and 2, and the distance between Γ and point 3 must be proportional to the ratio of the volume of the underwater hull and the wedges. This geometrical explanation sets the stage for the second half of the explanation, the mathematical analysis of the mechanics of stability. Step 6: Results In the next section, BOUGUER uses calculus to determine the three unknowns: The distance from point 1 to point 2; the volume of the wedges; and the volume of the hull. BOUGUER imagines that Fig. 3 represents the largest section of the ship, which actually extends through the plane of the page in the longitudinal direction x, with the immersed and emerged wedges actually an infinite sequence of triangles of width y (the largest being b = F-B, or the half-breadth of the hull at the waterline) and height e ( = H-B) going through the length of the hull at a distance dx from each other; integrating, the volume of the wedge is Vwedge =

e ∫ y2 dx 2b

(1)

The second unknown is the distance between the centers of the wedges, points 1 to 2; but since the center of a triangle is two-thirds the height from apex to base, it is straightforward to obtain this distance as

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3 Distance 1- 2 = 4  y dx 3  y 2 dx

(2)

The third unknown, the volume of the hull p, is derived using the trapezoidal rule. Putting the three together, the distance is:

3 Γγ = 2 e  y dx 3 bp

(3)

Finally, observing that the triangle Γgγ (i.e., between the centers of buoyancy and the intersection of their vertical lines of force) is similar to the triangles formed by the immersed and emerged wedges, the height of the metacenter g above the center of buoyancy Γ is found via Euclid to be: 3 Γg = 2  y dx 3p

(4)

This is the now-famous equation for the height of the metacenter. 3.1.4 Implications of the Metacenter BOUGUER develops in thorough detail both the theoretical and practical aspects of the metacenter. He derives the metacenter for various solids (ellipsoid, parallelepiped, prismatic body) and presents the procedures for its practical, numerical calculation for ships. These explanations were detailed enough for practical applications and became the foundation for later textbooks. But BOUGUER also charged ahead beyond the initial metacenter for infinitesimal angles of heel when he introduced the concept of the “métacentrique”, i.e., the metacentric curve for finite angles of heel. First, he clearly recognized that the same physical principles and stability criteria apply to an inclined position of the ship as they do for the upright case. Second, he recognized that the metacentric curve for finite angles is in fact the locus of the centers of curvature of the curve of the centers of buoyancy. This was a brilliant, original insight. The locus of the centers of curvature of a curve was known since Christiaan HUYGENS’s work on the pendulum clock (Huygens 1967b) under the name of “développée” (or evolute). Third, BOUGUER also knew how to construct the evolute of a given curve. For a wall-sided ship (or parallelepiped) the metacentric curve is a cusp shaped curve composed of two hyperbolas lying above the metacenter, as BOUGUER showed. His demonstration examples for the “métacentrique” do document that he understood

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how to approach stability for finite angles of heel, though he never used a “righting arm” criterion. On practical aspects of initial stability he recommends that the widest point of the ship be no lower than where the maximum heel would be before it begins to tumble-home, or even that ship sides be straight or flared throughout. In other words, BOUGUER was advising to avoid tumble-home altogether, although in somewhat oblique terms. BOUGUER next provides a numerically-worked example of the application of the metacenter to a real ship, underlining the importance of Gg, the distance between the center of gravity and the metacenter. Using the Gazelle as his model, BOUGUER explains how to account for the weight of each part of the ship, including the frames, planking, nails, etc., how to measure the center of gravity of each piece using the keel as the reference point, and how to sum the moments to obtain the overall center of gravity G. He then calculates displacement and center of buoyancy of the ship using the trapezoidal rule, which enables him to find the metacenter g. His calculations confirmed that, once properly ballasted, the Gazelle would be stable. BOUGUER continues for almost 50 pages to outline the practical implications of the metacenter on hull design and outfitting. In many cases the implications are re-statements of what constructors already knew; but BOUGUER provided for the first time a rigorous analysis of why they were true. Some of the main points he brings out are:  The greatest advantage to stability lies in increasing the beam. BOUGUER claims that the “stability” varies as the cube of the beam, by which he must have been referring to the transverse moment of inertia I ,T while the metacentric height requires closer scrutiny. But he wanted to emphasize the rapid increase of stability with beam. This also explains for the first time why doubling a ship improves stability, although BOUGUER does not explicitly state so.  Stability is improved by diminishing the weight of the topsides; though this was well-understood, BOUGUER details how to accurately assess the effects. BOUGUER also correctly describes for the first time how to evaluate the inclining experiment, for whose basic idea he credits HOSTE, in order to determine Gg, the distance between the center of gravity and the metacenter. In Fig. 4, he demonstrates how to use a known weight suspended from a mast to incline the ship and measure the angle of heel. Using the law of similar triangles, he shows that the ratio of the ship’s displacement and the suspended weight is proportional to the ratio of Gg to the angle of heel and distance the weight moves; on the assumption of small angles of heel this allows Gg (i.e., the reserve of stability) to be calculated directly without knowing the exact position of the metacenter. BOUGUER demonstrates how to use his metacenter by detailing the calculations of weights and centers of gravity for Gazelle while it was still being framed out. It is therefore curious that he did not verify this by performing an inclining experiment on Gazelle. It would appear that, while BOUGUER was completing his initial stability work in 1734, he was caught up in the events that led to the

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Geodesic Mission, and would not return to the subject until he was in the mountains of Peru.

Fig. 4 BOUGUER’s diagram of an inclining experiment, (Bouguer 1746, fig. 55).

3.2 Euler 3.2.1 Biographical Sketch Leonhard EULER (Fig. 5) was born in Basel, Switzerland on 15 April 1707. He was the son of Paulus EULER, parson of the Reformed Church, and his wife Margaretha née BRUCKER. He went to a Latin grammar school in Basel and, as his father recognized his talent early, took private lessons in mathematics. In 1720 he enrolled at the University of Basel as a student and later as a young scientist, in the first three years in philosophy where he received a Magister’s degree, then in theology. But he soon turned his main interest to mathematics and mechanics, which he studied under Johann (I) BERNOULLI, who was recognized as a leading mathematician of that era. In fact Johann (I) BERNOULLI, who was 40 years EULER’s senior, saw EULER’s maturing genius and invited him to join the Saturday afternoon “privatissimum” in mathematics, a private circle held in the BERNOULLI’s home, where he also met and made friends with younger members of the BERNOULLI family, notably Niklaus (II), Daniel (I) and Johann (II) BERNOULLI. In this period Johann (I) BERNOULLI laid the foundations from which EULER’s later mathematical fame developed. In 1727 EULER, who had been looking for an academic position, received an offer from the Russian Imperial Academy of Sciences in St. Petersburg, which Tsar Peter had initiated and which had been opened in 1725, where Niklaus (II) and Daniel (I) BERNOULLI held appointments as professors from this beginning. EULER accepted this prestigious offer, starting as an Adjunct (élève) for a modest

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salary, and arrived there – after a three month voyage by coach and by ship – in June 1727. EULER began his scientific career in St. Petersburg very successfully, advanced to a professorship in physics and full membership in the Academy in 1731, and spent his “First St. Petersburg Period” (1727-1741) very productively, working on a wide range of scientific subjects and publishing more than 50 treatises and books. In 1741, amidst political transitions and uncertainties in Russia, EULER received an attractive offer from King FREDERICK II of Prussia to come to Berlin and to work for the Royal Academy of Sciences, which was to be founded. EULER moved there in 1741, continued his illustrious scientific work on an increasing variety of subjects, and when the Academy at last opened in 1746 soon became recognized as one of its most eminent members and as the leader of the mathematical class. He remained immensely productive and published some of his most famous books and treatises during his period in Berlin from 1741 to 1766. Despite EULER’s scientific fame and the considerable merits he earned in the Berlin Academy during its formative first two decades he never developed a relationship with King FREDERICK that made him feel recognized, appreciated and understood. This was one contributing factor why after 25 years in Prussian service he decided to return to St. Petersburg. There he was welcomed and honored at all levels, and spent his “Second St. Petersburg Period (1766-1783)” in relative comfort, peace and recognition despite his weakening health and fading eyesight. His scientific creativity never ceased. A large share of his more than 800 treatises and books also stems from this second period in Russia. His work encompasses the full gamut of scientific topics in his era, not only in mathematics and mechanics, but also in other branches of physics, astronomy, ballistics, music theory, philosophy and theology. He died in St. Petersburg in 1783. More detailed accounts of EULER’s life and scientific vita are given by BURCKHARDT et al. (Burckhardt et al 1983), FELLMANN (Fellmann 1995) and CALINGER (Calinger 1996). 3.2.2 Early Work in Ship Theory In 1726 the Académie des Sciences in Paris invited contributions to a prize competition on the optimum placement, number and height of masts in a sail propelled vessel. EULER as a student and protégé of Johann (I) BERNOULLI was of course familiar with the earlier treatises and disputes that had arisen about the forces acting on a sail propelled maneuvering vessel between RENAU, HUYGENS, Johann (I) BERNOULLI a.o. So he felt well enough prepared, also encouraged by Johann (I) BERNOULLI, to submit a treatise (“De Problemate Nautico…”) (Euler 1974) in 1727, which gained honorable mention, an “accessit” equivalent to a second prize. Pierre BOUGUER was awarded the first prize. In this treatise EULER, who still treads in the foot-steps of Johann (I) BERNOULLI, does not yet show deeper insights into ship hydrostatics although he does draw

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attention to the requirement that for a ship before the wind the propelling sail force is limited by the acceptable forward trim angle. But he has no reliable physical basis for estimating that angle. Yet this early experience in his life at the age of 20 gave EULER a first thorough acquaintance with the mechanics of ships, familiarized him with nautical and ship construction terminology and created in him a lasting interest in ships as topics to be studied by methods of mathematical analysis and engineering mechanics.

Fig. 5 Portrait of Leonhard EULER.

In St. Petersburg EULER soon had ample opportunities to return to these subjects. It is not clear when he first occupied himself more deeply with ship hydrostatics. But evidently the appearance of LA CROIX ’s treatise on the hydrostatic stability of a parallelepiped in 1735 found EULER well prepared, when the Academy asked for his review, to detect certain flaws in LA CROIX ’s derivations and to present the correct result for this simple shape, which first established EULER’s initial restoring moment criterion on hydrostatic stability. In his second review in 1736, in response to LA CROIX ’s rebuttal, EULER went beyond his first results, dealt with some other prismatic shapes (trapezoidal and triangular prisms) and

Historical Roots of the Theory of Hydrostatic Stability of Ships 163

alluded to his general approach to stability (Euler1972). This evidence is sufficient to date EULER’s earliest written mention of the hydrostatic stability criterion. There is also some indirect evidence of EULER’s earliest results on hydrostatic stability in his correspondence with the Danish naval constructor and naval attaché in London, Friderich WEGERSLØFF, which he maintained between 1735 and 1740. In a letter of 14 September 1736 WEGERSLØFF acknowledges receipt of a solution for the hydrostatic stability problem, on 22 May 1738 EULER replies and gives a derivation of his stability results and also mentions some experimental verification (Euler 1963). It is not clear from the context what sort of tests he had performed. By 1737 EULER’s interests in ship theory were well known at the Imperial Academy. Thus - probably not without his own prior knowledge or suggestion - he was commissioned by the Academy to write a book on this subject, resulting in his monumental two-volume “Scientia Navalis”. This work encompasses a presentation of the complete scope of ship theory according to the state of the art at that time and includes many new findings and derivations by EULER. The chapters in the two parts deal with ship hydrostatics, ship resistance and propulsion, maneuvering and ship motions. Many results have been of lasting value and have served as foundations for the growing scientific body of ship hydromechanics. According to EULER’s own report in the Preface of “Scientia Navalis”, he had worked on this manuscript from 1737 to 1740 (Euler 1967, pp. 15/16). When he left St. Petersburg in 1741 he had completed the first part, which contains all four chapters on hydrostatic equilibrium and stability, and half of the second part. The remaining sections of Part 2 were finished by 1741 in Berlin. Unfortunately, he had much difficulty finding a publisher for this voluminous opus, which thus was not published until 1749 by the Academy in St. Petersburg. Comprehensive appraisals of EULER’s overall contributions to ship theory and related matters are given by HABICHT (Habicht 1974), MIKHAILOV (Mikhailov 1983) and TRUESDELL (Truesdell 1983). 3.2.3 The Initial Stability Criterion To understand the history of EULER’s involvement in ship theory it is important to read the Preface to his “Scientia Navalis”. Here he explains that his work will go beyond the established disciplines of hydrography or nautical science and will concentrate on a physical and analytical investigation of the mechanics of ships, at rest and in motion, for which fundamental theoretical works were still missing at that time. In hydrostatics EULER departs from the Principle of ARCHIMEDES, to whom in the preface of “Scientia Navalis” he gives credit and praise. But he adds that the hydrostatic stability of ships must be newly approached and quantified in order to be able to distinguish between stable and unstable equilibrium of ships at the design stage. The experience of naval architects (“architecti navales”) alone, long

164 H. Nowacki and L.D. Ferreiro

established as it may be, will not be sufficient to prevent unexpected stability accidents. EULER acknowledges the motivation he received from reviewing LA CROIX’s treatise in 1735-1736, which prompted him to investigate more profoundly the transverse and longitudinal stability of ships. EULER also gives very favorable credit to BOUGUER’s “Traité du Navire”, which had appeared 3 years before “Scientia Navalis”, but he takes much care also to avert any suspicion of plagiarism by calling on the Imperial Academy as witnesses that he had written “Scientia Navalis” essentially between 1737 and 1741 during which period there was no communication with BOUGUER who was in Peru. These statements were never disputed between the two authors, who corresponded in a respectful and amicable fashion on other subjects later. EULER’s derivation of his stability criterion in “Scientia Navalis” proceeded in the following steps (Euler 1967, pp. 3-166): STEP 1: Premises and axioms In the first chapter of Book I, EULER deals with the equilibrium of bodies floating in water at rest. In essence he rederives the Hydrostatic Principle of ARCHIMEDES from the modern viewpoint of infinitesimal calculus by integration of the hydrostatic pressure distribution prevailing in a fluid over the surface of the body. The properties of the fluid at rest and the use of calculus for this purpose both were new at the time when EULER wrote these passages. As for the pressure distribution he states axiomatically in the opening paragraph of his book: “Lemma: The pressure which the water exerts on the individual points of a submerged body is normal to the body surface; and the force which any surface element sustains is equal to the weight of a straight cylinder of water whose base is equal to the same surface element and whose height is equal to the depth of the element under the water surface”. These brief axioms, viz. the normality of pressure to a surface and the inferred equality of the pressure at a point in a given depth in all directions, were the first analytical formulation for the properties of the fluid and are regarded as the necessary and sufficient conditions for the foundation of hydrostatics (Truesdell 1954). STEP 2: Magnitude of buoyancy force From these premises EULER proceeds to define by integral calculus the buoyancy force as the pressure resultant and the center of buoyancy (EULER calls the CB: “centrum magnitudinis”), through which it acts, as the volume centroid of the submerged part. He reconfirms also that for equilibrium the buoyancy and weight forces must act in the same vertical line and must be equal in magnitude and opposite in direction. He then illustrates these principles by examples of simple shapes like parallelepipeds and prisms of triangular and trapezoidal cross section. For each of these solids he finds the possible equilibrium conditions over a full circle of rotation as a function of the specific weight of these homogeneous solids, not unlike HUYGENS in his unpublished treatise of 1650.

Historical Roots of the Theory of Hydrostatic Stability of Ships 165

In Chapter II, briefly digressing from hydrostatics, EULER discusses the resulting motions of a floating body if it is temporarily displaced from its “upright” equilibrium position. Here he explains the “lumping” of masses in their center of gravity (CG), introduces the definition of the principal axes of inertia for ships, underscores the significance of the CG as a reference point, e.g., for decomposing a resulting motion into the translation of the CG and the rotation about it. EULER will adhere to the CG as his system reference point, also when returning to ship motions later. STEP 3: Measurement of volumes and volume centroids In stark contrast to BOUGUER, EULER confines himself to analytical definitions of stability criteria, volumes, centroids, areas, moments of inertia etc. He does not address their numerical evaluation at all. He is taking it for granted that once the shape of the ship or body is defined by some function the integrations can be readily performed. In his examples he usually deals with simple shapes where the integrations can be performed in closed form. STEP 4: Stability criterion In Chapter III, the main chapter on hydrostatic stability, EULER defines his stability criterion right away (Proposition 19) by: “The stability which a body floating in water in an equilibrium position maintains, shall be assessed by the restoring moment if the body is inclined from equilibrium by an infinitesimally small angle”. EULER illustrates this principle by discussing cases of unstable, neutral and stable equilibrium of ships and adds that it is necessary to quantify stability in terms of the restoring moment because even a stable ship may be in danger by external heeling moments and may require righting moments of greater magnitude. (The issue is not only whether the ship is initially stable or not, but also how much “stability capacity” it has). STEP 5: Evaluation of the criterion EULER then enters into the determination of the restoring moments by first examining a planar cross section of arbitrary shape (or thin disk) floating upright (Fig. 6). The Figure AMFNB = “AFB” is inclined by a small angle dw so that the new floating condition has the waterline ab. Since the center of gravity G remains his reference point, also for later purposes, he draws the parallel lines MN and mn to the waterlines before and after inclination, also through G. The center of buoyancy of the cross section is designated by O. A normal VOo to the inclined waterline is drawn through O. The equality of the immersed and emerging wedges requires that ab intersect AB at the center point C so that AC = BC.

166 H. Nowacki and L.D. Ferreiro

Fig. 6 EULER’s figure for centroid shift in inclined cross section, 2D case, (Euler 1967, fig. 39).

Let the displacement per unit length and the equal buoyancy force of the cross section before inclination be denoted by M = γ (AFB). For the inclined position the restoring moment is composed of three contributions: 1. The effect of the original submerged volume forming a positive restoring couple of forces through G and V: M GV = M GO dw

(5)

2. The effect of the submerged wedge CBb whose cross section area is:

BC 2 dw AB 2 dw = 2 8

(6)

and whose restoring moment about G hence is:

AB 2 dw γ (qo + GV) 2 where

qo =

(7)

2 1 Cb = AB 3 3

3. Likewise for the emerging wedge ACa the restoring moment is

AB 2 dw (po - GV) -γ 8 where

po =

(8)

2 1 Ca = AB 3 3

For all moments combined, replacing γ by M/ (AFB):

Historical Roots of the Theory of Hydrostatic Stability of Ships 167

M (AB 2 dw )(po  qo) M REST = M GO dw + 8AFB AB 3 ] Mdw [GO + 12AFB

=

(9)

The expression in square brackets has the dimension of a length and is the now well known result, corresponding to, in BOUGUER’s later terminology, the metacentric height of the cross section. This is how EULER’s restoring moment and BOUGUER’s metacentric height are connected. Note that the GO term reverses sign if G lies above the center of buoyancy O, as is common in cargo ships. EULER discusses this result at some length and for several simple shapes. In Proposition 29 EULER then arrives at the general three-dimensional case of a floating body of arbitrary, in particular asymmetrical shape, whose waterplane is drawn in Fig. 7. He denotes: M = displacement or weight of body V = submerged volume GO = distance between CG and CB, positive for G above B CD = reference axis of waterplane through centroid of waterplane area, parallel to axis through G CX = x = abscissa from origin C XY = y = ordinate in upper part of waterplane XZ = z = ordinate in lower part of waterplane p and q = area centroids of upper and lower parts of waterplane P and Q = equivalent pendulum lengths of waterplane area parts w.r.t. axis CD With

 y dx 2  ydx 2

pr =

3 2  y dx PR = 3 y 2 dx 

 z dx qs = 2  zdx 2

3 2  z dx QS = 3 z 2 dx 

168 H. Nowacki and L.D. Ferreiro

Fig. 7 EULER’s figure for the derivation of the stability criterion for a body of arbitrary shape, 3D case (Euler 1967, fig. 48).

EULER derives, since the axis CD runs through the waterplane centroid: ∫ y2 dx = ∫ z2 dx, and hence

 ( y  z )dx PR + QS = 3  y dx 3

2

3

2

(10)

from which in analogy to the planar case the restoring moment, divided by the angle dw, for the ship becomes:

M (GO +

 (y

3

 z 3 )dx 3V

)

(11)

In the special case of port/starboard symmetry (y = z), with IT =

I 2 3 ∫ y dx; and T = OM V 3

In our familiar notation (GO= GB , OM= BM ) the restoring moment simplifies into: M ( GB + BM ) = M GM

(12)

Historical Roots of the Theory of Hydrostatic Stability of Ships 169

STEP 6: Results This summarizes the course of steps EULER took to derive the initial restoring moment. EULER never used the word metacenter. EULER’s result thus is the initial restoring moment, divided by (M dw), which he uses as his stability criterion. Again EULER illustrates this result by many examples for simple shapes of solids, even by an analytical formulation for a shiplike body with parabolic section shapes. But he does not present any numerical calculations for an actual ship although he discusses many practical implications of his results. 3.2.4 Implications of the Stability Criterion

EULER interprets his results for practical applications to ships, mainly in Part II of “Scientia Navalis”. For EULER ships are “floating objects, carrying a cargo or payload, symmetrical starboard and port, propelled by rowing and/or sails”. Among the major conclusions he draws, we quote: Stability is judged by the restoring moment divided by the angle dw. Transverse and longitudinal stability are clearly distinguished and both addressed. EULER also derives an expression for combined heel and trim under oblique sail force moments, deriving the oblique restoring effects by rotation of the reference axis from the principal axes. To assess stability you need to know the ship’s displacement, the centroids CB and CG, the waterplane area and shape, which infers the restoring moment (without using the metacenter). To improve stability, lower the CG, raise the CB and/or widen the beam. The addition, removal and internal shifts of weight as well as the role of ballast for stability are discussed. The use of doubling (“soufflage”) with its pros and cons is mentioned. In Chapter IV of Part 1 EULER addresses practical problems of inclining ships under external moments or by internal weight shifts, though again only analytically, not numerically. He also addresses the issue of external wind loads and required stability, especially for small ships vs. big ships, cargo ships vs. war ships. EULER finally also makes the interesting “philosophical” remark that sailors knew about measures of stability all along. When the sailors say, “This ship can sustain a strong wind in its sails”, then EULER claims they mean the same thing as he expresses by his restoring moments.

3.3 Comparison of Approaches Based on the facts presented above it is now possible and in fact intriguing to compare the approaches taken by BOUGUER and EULER in their creative work on hydrostatic stability. This will bring out certain commonalities, but also underscore the differences. We have examined several aspects.

170 H. Nowacki and L.D. Ferreiro

3.3.1 Chronology:

BOUGUER worked on the subject of ship hydrostatic stability from about 1732 to at least 1740, and his treatise was published in 1746. EULER was engaged with this topic between about 1735 to at least 1740, “Scientia Navalis” appeared in 1749. During these formative periods of their new concepts there existed no contacts and no communications between them. Thus their work originated independently. But after their first scientific acquaintance through their participation in the 1727 Paris Academy award competition it is likely that they both underwent a common fermentation period on related subjects. Probably they were both looking for the missing pieces in the mosaic of sailing ship maneuvering, among which the hydrostatic response was foremost. It appears that their deeper familiarity with ARCHIMEDES originated during the period between 1727 and 1735. 3.3.2 Sources:

BOUGUER writes that he was familiar with the work of PARDIES, HOSTE, RENAU, most likely also with relevant publications by HUYGENS and the BERNOULLIs. He shows close acquaintance with ARCHIMEDES’ principles, although he does not mention his name. On calculus he quotes the well-known treatise by NEWTON and Roger COTES, but very likely he also thoroughly read de l’HÔPITAL, whose notation in the LEIBNIZ style he preferably uses. His knowledge of calculus was mostly self-taught, based on occasional tutorials and recommendations from mathematics professor Charles René REYNEAU, who wrote the basic textbook on the subject, “Analyse démontrée” in 1708. EULER also knew ARCHIMEDES, HOSTE, RENAU, whom he mentions. He learnt calculus first-hand from the BERNOULLIs, who stood firmly in the LEIBNIZ tradition. EULER’s book on mechanics, “Mechanica” in 2 volumes, where he displays exquisite knowledge of the principles of equilibrium and motions of mechanical systems, had appeared in 1736. Although both authors are not very generous in giving references on their sources it is possible to a certain extent to trace their intellectual ancestry by looking at their scientific and technical terminologies, even if they wrote in different languages, French and Latin. We have analyzed their vocabularies in their main treatises on hydrostatic stability and identified commonalities and distinct differences. They shared much common ground by using such established words as equilibrium, stability, weight, buoyancy, specific weight, centers of gravity and buoyancy, inclination (derived from ARCHIMEDES) and pressure, hydrostatics (from STEVIN and PASCAL). BOUGUER is unique in inventing the metacenter and using the evolute (from HUYGENS), but never the word “restoring moment”, while EULER’s vocabulary was vice versa. This indicates where their approaches differed.

Historical Roots of the Theory of Hydrostatic Stability of Ships 171

3.3.3 Stability Criteria:

Sections 3.1 and 3.2 have shown that BOUGUER and EULER derived equivalent stability criteria, but used different approaches. To sum up the comparison: Premises and axioms: BOUGUER’s work is founded on the premises of ARCHIMEDES, though augmented by a hydrostatic pressure law (STEVIN). EULER defines the axioms of hydrostatics a bit more strictly by postulating the direction independence of pressure and its normality to a surface. Magnitude of buoyancy force: Both agree and reconfirm the law of ARCHIMEDES by pressure integration and calculus. Measurement of volumes and centroids: BOUGUER takes the route of establishing numerical procedures, based on quadrature rules, for measuring volumes, centroids, areas etc. before giving analytical definitions of the stability criterion. In passing he skillfully adapts the “method of trapezoids” to shipbuilding applications. EULER skips this step entirely and drives directly at analytical formulations for his stability criterion. Stability criterion: BOUGUER’s invention and choice for a measure of stability is “metacentric height”, a geometric quantity representative of initial stability capacity. EULER chooses the “initial restoring moment”, a physical quantity, also measuring the ship’s initial stability capacity. Although the two ideas are closely connected, their meanings are conceptually different. BOUGUER never refers to “restoring moments”, EULER never uses “metacenter”. Neither author recurs to ARCHIMEDES’ idea of a “righting arm”. Evaluation of the criterion: BOUGUER derives the metacenter mainly from geometric arguments and, allowing for the shift of wedge volumes and hence CB, determines the metacenter as the point of intersection of the upright and inclined buoyancy directions. EULER uses physical arguments of moment equilibrium and shift of volumes and finds the restoring moment simply by a summation of moments. Results: BOUGUER’s result is metacentric height, a distance. EULER’s result is a restoring moment. The two results are equivalent in practice, but not equal in approach.

3.3.4 Audience, Style and Language:

BOUGUER’s “Traité du navire”, the first modern synthesis of theoretical naval architecture, is written in French for a readership of scientists and constructors who are to be introduced to the use of theory in practical ship design and shipbuilding. BOUGUER benefited from both his experience as a hydrographer and his collaboration with ship constructors. His style is clear and logical, explaining many practical details, almost as in a textbook. He is concerned about how this methodology can be implemented. This style does not cause any sacrifices in rigor. His language is lucid, his own new ideas come across crisply.

172 H. Nowacki and L.D. Ferreiro

EULER in his Latin text of “Scientia Navalis” writes as an applied mechanicist/ physicist and addresses mainly the scientific community of his era. Among those who were educated to read scientific treatises of this sort in Latin, which was still a lingua franca in the academic world, EULER’s brilliance of style, his inimitable logic and clarity were highly praised. These attributes also apply to his “Scientia Navalis”. His work on hydrostatic stability is still valid today. But at the same time he did not have any experience with practical shipbuilding applications. Thus he left many details for implementation by later successors, which delayed the spreading of his methods, certainly in comparison to BOUGUER. Yet in the final balance the scientific and engineering community must be grateful to both BOUGUER and EULER.

3.4 Synopsis Table 1 gives a synopsis of important milestones and ideas in the development of the theory of hydrostatic stability by indicating where certain elements of this theoretical knowledge first occurred. The importance of ARCHIMEDES ’s physical insights and the relevance of BOUGUER’s and EULER’s parallel discoveries described by calculus become clearly visible in this condensed tabular summary of historical steps. Table 1. Historical development of stability concepts

Historical Roots of the Theory of Hydrostatic Stability of Ships 173

4 Further Work 4.1 Practical Applications Stability theory quickly found direct applications in the day-to-day practice of ship design. This occurred in two ways: first, by the increasing sophistication of calculations for weights, centers of gravity and the metacenter within the design process; and second, by the use of inclining experiments to validate stability. These applications were surprisingly widespread; stability theory was quickly incorporated in navies where there was already a strong institutional development of scientific naval architecture, notably in France, Denmark, Sweden and Spain. Those countries also created schools of naval architecture during the 18 th century, where students were weaned on stability theory and naturally used it when they became constructors at the dockyards. This was not the case for the British or Dutch navies, which had provided little direct support for scientists working on ship theory, and which did not establish permanent schools of naval architecture until the 19th century. 4.1.1 Stability Calculations During Design

The most famous of these schools of naval architecture was the Ecole du Génie Maritime, established in 1741 in France by the scientist Henri Louis DUHAMEL DU MONCEAU (1700-1782), whom MAUREPAS had appointed to InspectorGeneral of the Navy. DUHAMEL worked with BOUGUER to create the first comprehensive textbook for the students, based on “Traité du navire”. DUHAMEL’s genius was to take BOUGUER’s complex mathematics and render them into step-by-step instructions on how to calculate the metacenter. The textbook, “Elémens de l’architecture navale” (Duhamel 1752) became the standard reference for both students and constructors. DUHAMEL DU MONCEAU was also involved in developing the Ordinance of 1765 under Navy minister Etienne François, Duke of CHOISEUL, which created the Corps du Génie Maritime and formalized the data to be included on ship’s plans: “centers of gravity and resistance, and height of the metacenter” as well as accompanying calculations and a tabulation of hull materials. Later, the standardized plans introduced in 1786 by DUHAMEL’s successor as InspectorGeneral, Jean-Charles de BORDA, listed the specific immersion of the hull at full and light load in tonnes-per-cm equivalent, which allowed a rapid estimate of the effect of loading weights on the ship.

174 H. Nowacki and L.D. Ferreiro

4.1.2 Inclining Experiments

Another facet of rapid adoption of the metacenter was that the first recorded inclining took place in 1748, just two years after BOUGUER published his work. FrancoisGuillaume CLAIRIN-DESLAURIERS, then a junior constructor at Brest, performed the experiment on the newly-built 74-gun ship Intrépide, apparently out of curiosity to test the new theory. He hung two 24-pound cannons (each weighing over 2 tonnes) from a buttress built on the side of the ship (Fig. 8), and although the buttress broke, he was able to take enough measurements to ascertain that the ship’s GM was 1.8m (Clairin-Deslauriers 1748).

Fig. 8 CLAIRIN-DESLAURIERS’ drawing of the inclining of Intrépide (Clairin-Deslauriers 1748)

4.1.3 CHAPMAN and SIMPSON’s Rule

Another important contribution to the promulgation of hydrostatic stability calculations came from the Swedish naval constructor and innovative ship designer Fredrik Henrik CHAPMAN. CHAPMAN (1721-1808), the son of the superintendent of the Göteborg dockyard, was endowed by this background with much practical shipbuilding knowledge. He visited Britain in 1750 to improve his knowledge in mathematics and other fundamentals. He took classes from Thomas SIMPSON (1710-1761), a gifted mathematician trained in NEWTON’s tradition and an instructor at the Royal Military Academy in Woolwich near London. SIMPSON reportedly taught CHAPMAN a quadrature rule for planar curves, which yields a numerical approximation of the integral of a continuous function using at least one equally spaced double interval, hence at least three offset points per double interval, or any desired number of further double intervals of the same spacing. This quadrature rule later became known as “SIMPSON’s Rule”, especially in shipbuilding, although the rule has several historical precursors and SIMPSON never claimed to have invented it. CHAPMAN, after further visits to France and Holland and having picked up much valuable background including

Historical Roots of the Theory of Hydrostatic Stability of Ships 175

the recent findings by BOUGUER on the metacenter, returned in 1757 to Sweden to continue his career as an important ship constructor. In this process he soon made it his habit to calculate ship displacements (1767), also as a function of draft, and assessing the metacenter, applying SIMPSON’s Rule for numerical integration. His engineering methods applied in ship design became well known internationally soon after the publication of his book Tractat om Skepps-Byggeriet (Chapman 1775), where he explained his calculations in much detail (Harris 1989). Thus CHAPMAN helped to promulgate the use of the metacenter and to make its calculation a routine matter among naval architects. At the same time SIMPSON’s Rule, which offers certain efficiency advantages over the trapezoidal rule for an equal number of intervals, was popularized and is still much favored today.

4.2 Extensions of Stability Theory 4.2.1 Hydrostatic Stability About Other Axes

In their work on the masting of sailing ships in 1727 it must have occurred to both BOUGUER and EULER that in general the external wind moments would act in an oblique direction to the ship’s centerplane so that heeling and trimming would result simultaneously. EULER remembered this open question and in “Scientia Navalis”, Part 2, returned to this issue. He derived an expression for the restoring moment about an oblique axis between the axes of heel and trim. The moments of inertia about this axis could be constructed, when the moments of inertia IT and IL for transverse and longitudinal stability were known, by using an ellipse of inertia moment construction. Thus this method was able to predict at least the initial hydrostatic response (small angles) to a wind moment in an oblique plane in terms of simultaneous heel and trim angles. EULER mentioned on the side, too, that longitudinal stability was always much greater than transverse stability. It was some time later that the Spanish naval constructor and engineering scientist Jorge JUAN Y SANTACILIA (1713-1773), who knew BOUGUER and his work very well from the Peruvian expedition and also was in correspondence with several European Academies, concerned himself not only with ship stability, but also with ship oscillations, especially with pitching motions. In this context he extended BOUGUER’s concepts of the metacenter to longitudinal inclinations and first introduced the definition of the longitudinal metacentric height GML. He pointed out that it by far exceeds the transverse metacentric height. His book “Examen Maritimo”, which appeared in 1771, combined the theoretical insights of his days with much practical design experience and became a widely used reference and textbook (Juan y Santacilia 1771).

176 H. Nowacki and L.D. Ferreiro

4.2.2 Large Angles of Heel

A broader extension of stability theory occurred not on the European continent but in Britain. In two papers presented to the Royal Society of London, the British mathematician George ATWOOD examined the inclination of ships at large angles of heel. The first paper (Atwood 1796) based on a thorough knowledge of ARCHIMEDES, BOUGUER and EULER, reviews the fundamental physical principles of hydrostatic stability, applied to finite angles of heel. Here ATWOOD investigates the stability properties of homogeneous solids of simple shape (parallelepiped, cylinder, paraboloid) through 360 degrees of rotation as a function of body draft at rest (or specific weight). He finds numerous equilibrium positions (8 or 16), only some of them stable. He develops a shifted wedge volume method for finite heeling angles and reviews the numerical quadrature rules, settling for STIRLING’s 3 interval rule. This paper already drew full attention to the fact that stability must be judged over a range of finite, practical heel angles; initial stability alone is inadequate as a stability measure. In the second paper (Atwood and Vial du Clairbois 1798), which ATWOOD coauthored with the French constructor Honoré-Sébastien VIAL DU CLAIRBOIS, the investigation was extended to actual ships, and for the first time a numerical analysis of the “righting moments” of ships over a large range of heeling angles was performed. ATWOOD and VIAL DU CLAIRBOIS introduced the term GZ for the “righting arm”, which was again numerically evaluated by a wedge volume shift method. This successfully demonstrated the feasibility of numerical stability analysis for actual ships over a range of finite heeling angles. The necessity of performing such calculations rather than just relying on initial stability measures was again underscored. Yet in shipbuilding practice it still took several more decades before the initial difficulties in numerical integration could be overcome by more robust methods and instruments. Even half a century later such stability evaluation had not yet become a routine matter. It was not before the early 19th century that the knowledge on the metacenter progressed one further step by the work of French constructor and mathematician Charles DUPIN from metacentric curves to metacentric surfaces (Dupin 1813). DUPIN stated that the ship for finite angles of inclination in whatever direction possesses a surface of centers of buoyancy. This surface at every point has two principal curvatures, thus one can construct two sheets of metacentric surfaces, i.e., evolute surfaces of the buoyancy surface. The two metacenters for the upright condition, MT and ML, each is a special point on one of the surfaces. It is probably fair to say that at this level of perception the issue of hydrostatic stability had been fully resolved in terms of the relationships between ship geometry, physical responses to inclinations, and differential geometry of metacentric surfaces for finite angles of heel.

Historical Roots of the Theory of Hydrostatic Stability of Ships 177

5 Conclusions ARCHIMEDES laid the foundations for the stability of floating systems, introduced a stability measure similar to the righting arm and presented an approach for assessing the ability of a floating inclined solid to right itself. But his applications were limited to simple geometrical shapes. Fortunately his manuscript “On Floating Bodies” survived in a few copies and became more accessible again by a Latin translation in the 13th c. and also in print after 1500. Yet it took a few more centuries before modern hydrostatic stability was established and could be applied to actual ships, in particular also at the design stage. The interest in scientific solutions for ship stability gained new momentum for practical and scientific reasons by about 1700: The leading navies were getting more concerned about stability risks with increasing ship sizes, gunports that were close to the water and increasingly heavier armament. In the beginnings of the Age of Enlightenment, new expectations were raised with regard to the capabilities of science to predict physical phenomena and the performance of technical systems. Mathematical breakthroughs occurred in infinitesimal calculus, analytical and numerical methods. The modern age of engineering science made rapid progress in mechanics and hydromechanics, including the study of equilibrium and stability of mechanical systems. All of this contributed to an atmosphere in the scientific community in the early 18th c. that was open to new challenges, also in application to ships. Both BOUGUER and EULER were actively involved in these general developments and certainly exposed to this unique zeitgeist. BOUGUER responded to the recognized problem of hydrostatic stability more as an engineering scientist, EULER reacted rather more like an applied mechanicist and mathematician. Both were able to reformulate and solve this problem in their own unique and original ways. As our comparisons in this paper have reconfirmed, BOUGUER’s and EULER’s nearly simultaneous work was not only performed quite independently, which was never doubted, but was also distinctly different in approach and justification. Both investigated the ability of the ship to right itself after an infinitesimal heeling displacement. BOUGUER reasoned mainly geometrically and for the inclined ship derived a length measure, the metacentric height, as the decisive geometric measure of initial stability. EULER argued mainly on mechanical grounds and deduced the initial restoring moment as his measure of hydrostatic stability. Despite these contrasting styles of justification, as we appreciate today of course, both stability measures are in fact equivalent and can be converted into each other. We may still benefit from both lines of thought and should be grateful for the insights we owe to these two congenial pioneers.

178 H. Nowacki and L.D. Ferreiro

We have explained why it took several more decades before these landmark discoveries in ship stability were fully accepted and widely applied in shipbuilding practice. But the foundations laid by BOUGUER and EULER have remained of lasting value as secure cornerstones in our knowledge for designing safe and stable ships.

Acknowledgments The authors are pleased to acknowledge the support and encouragement they received during the preparation of this work from many colleagues, in particular from Jean DHOMBRES, Emil A. FELLMANN, Jobst LESSENICH and Gleb MIKHAILOV. The first author also expresses his gratitude to the Max Planck Institute for the History of Science Berlin, where he is a Visiting Scientist and where his work was much promoted by personal communications and the excellent library resources.

References Archimedes (2002) The Works of Archimedes, edited and translated by T.L. Heath, repub by Dover Publ, Mineola, N.Y. Atwood G (1796) The Construction and Analysis of Geometrical Propositions, Determining the Positions Assumed by Homogeneal Bodies which Float Freely, and at Rest, on a Fluid’s Surface; Also Determining the Stab of Ships and of Other Float Bodies, Philos. Trans. of the R. Soc. of London, Vol. 86: 46-278. Atwood G, Vial du Clairbois HS (1798) A Disquisition on the Stability of Ships, Philos. Trans. of the R. Soc. of London, Vol. 88, vi-310 Bouguer P (1727) De la mâture des vaisseaux, pièce qui a remporté le prix de l’Acad. R. des Sci. de 1727, (On the masting of ships) Jombert, Paris. Bouguer P (1746) Traité du Navire, de sa Construction et de ses Mouvemens (Treatise of the Ship, its Construction and its Movements), Jombert, Paris Boyer CB (1949) The History of Calculus and its Conceptual Development, 1939, reprint Dover Publ, N.Y. Burckhardt JJ, Fellmann EA, Habicht W, eds (1983) Leonhard Euler 1707-1783, Mem. Monogr., issued by the Canton Basel-City. Birkhäuser Verlag, Basel. Calinger R, Leonhard Euler (1996) The First St. Petersburg Years (1727-1741), Hist. Math., vol. 23, 121-166, Academic Press. Cerulus FA (2001-Jan) Un mathématicien mesure les navires (A mathematician measures the ships), in Géomètre; édition des Histoires de la mesure, Publi-Topex editions, Paris, 34-37. Chapman FH af (1968) Tractat om Skeppsbyggeriet (Treatise on Shipbuilding), Johan Pfeffer, Stockholm (1775); translation and facsimile with commentary in Architectura Navalis Mercatoria (Merchant Nav Archit), Praeger Publ, N.Y. Clairin-Deslauriers G (1748-May 8) Letter to Duhamel du Monceau, Peabody-Essex Museum, collection Duhamel du Monceau, MH 25 box 2 folder 2 item 4.8; Bres. Czwallina-Allenstein A (ed.) (1996) Treatises of Archimedes, translated and annotated by the editor in 1922, republished in Ostwald’s Klassiker der exakten Wissenschaften, vol. 201, Verlag Harri Deutsch, Frankfurt am Main.

Historical Roots of the Theory of Hydrostatic Stability of Ships 179 Deane Sir A (1981) Deane’s Doctrine of Naval Architecture (1670), edited by Brian Lavery, Conway Maritime Press, London, 71-73. Demerliac A (1995) La Marine de Louis XV: Nomenclature des navires français de 1715 à 1774, (The Navy of Louis XV: nomenclature of French ships from 1715 to 1774), Editions Omega, Nice, p. 68. Dhombres J (1999) Mettre la géométrie en Crédit: Découverte, Signification et Utilisation du Métacentre Inventé par Pierre Bouguer (Giving Credit to Geometry: Discovery, Significance and Use of the Metacenter invented by Pierre Bouguer), in Sci. et Tech. en Perspect.”, série 3, fasc. 2, 305-363, Paris. Dijksterhuis EJ (1987) Archimedes, translated from Dutch by C. Dikshoorn, Princeton Univ Press, Princeton, N.J. Duhamel du Monceau HL (1752) Elémens de l’architecture navale (Elements of naval archit), Jombert, Paris. Dupin C (1813) Développements de géometrie: avec des applications à la stabilité des vaisseaux, aux déblais et remblais, au défilement, à l’optique, etc. pour faire suite a la Géometrie descriptive et a la Géometrie analytique de M. Monge (Developments in geometry; with applications to the Stab of vessels, to trenches and embankments, to ramparts, to optics, etc. following the Descriptive Geometry and Analytic Geometry of M. Monge), Mme Ve Courcier, Paris (1813). Euler, L. Leonhard Euler. Letters on Scientific Subjects, letters R 2701 (20 June 1737) and R 2705 (12 April 1740), from Euler to Wegersløff, in German with Russian translation, ed by the Acad. of Sci. of the USSR, Moscow-Leningrad (1963). Euler L. Scientia Navalis seu Tractatus de Construendis ac Dirigendis Navibus (Science of Ships or Treatise on How to Build and Operate Ships), 2 vols., St. Petersburg, (1749). Republ in Euler’s Collected Works, Series II, vols. 18 and 19, issued by the Euler Comm of the Swiss Acad of Natural Sci, Zurich and Basel (1967 and 1972). Euler L (1972) Judicium de Libello Domini de LA CROIX ex Gallia Huc Transmisso (Review of the Booklet by M. de LA CROIX from France Submitted Here) (1735); and: Notae ad Responsiones Viri Illustris de LA CROIX Factae ab L. EULER (Notes on the Response by the Famous M. de LA CROIX Made by L. EULER) (1736), Appendices I and II in vol. 19, Series II, Euler’s Collected Works, issued by the Euler Comm of the Swiss Acad of Natural Sci, Basel. Euler L (1974) Meditationes super Problemate Nautico, quod Illustrissima Regia Parisiensis Academia Scientiarum Proposuit (Thoughts on the Nautical Problem Proposed by the Most Illustrious Paris Royal Acad of Sci, vol. 20, Series II, Euler’s Collected Works, issued by the Euler Comm of the Swiss Acad of Natural Sci, Commentationes Mech et Astronomicae, Basel. Fellmann EA: Leonhard Euler, in German, Rowohlts Monographien, volume 387, Rowohlt Taschenbuch Verlag, Reinbek (1995), translated by E. Gautschi and W. Gautschi, in English, Birkhäuser Verlag, Basel Boston Berlin (2007). Ferreiro LD (2007) Ships and Science: The Birth of Naval Architecture in the Scientific Revolution, 1600-1800, MIT Press, Camb. Habicht W (1974) Introduction to Vol. 20 of the 2nd Series, in German, pp. VII-LX in Euler’s Collected Works, Series II, vol. 20. Harris DG and FH Chapman (1989) The First Naval Architect and His Work, Conway Marit Press, London, 20, 215-220. Heiberg JL and Zeuthen HG (1906-07) A New Treatise by Archimedes, in German, Bibliotheca Mathe, B.G. Teubner Verlag, Leipzig. Hoste P (1697) Théorie de la construction des vaisseaux (Theory of the construction of vessels), Arisson & Posule, Lyon. Huygens C (1967a) De iis quae liquido supernatant libri tres (Three books on bodies floating on top of a liquid) (1650). Published in Œuvres Complètes de Christiaan Huygens, vol. XI (1908). Repub Swets and Zeitlinger, Amsterdam.

180 H. Nowacki and L.D. Ferreiro Huygens C (1967b) Horologium Oscillatorium…, (Pendulum Clockwork…), Part III, F. Muguet, Paris, 1673. In Œuvres Complètes de Christiaan Huygens, vol. 18 (1934). Swets and Zeitlinger, Amsterdam. Juan y Santacilia J (1771) Examen marítimo, Theórico Práctico (Maritime examination, theoretical and practical), Manuel de Mena, Madrid. La Croix CM de (1736) Eclaircissemens sur l’Extrait du méchanisme des mouvemens des corps flotans (Clarifications on the extract of the mech of the movements of floating bodies), Robustel, Paris. Netz R, Noel W (2007) The Archimedes Codex. Revealing the Secrets of the World’s Greatest Palimpsest, Weidenfeld & Nicholson, Great Britain. Mairan JJ d’Ortous de (1724) “Instruction abregée et méthode pour le jaugeage des navires” (Abridged instruction and method for admeasuring ships), in Hist. et Mem. de l’Acad. des Sci. 227-240 Mikhailov GK (1983) Leonhard Euler and the Development of Theoretical Hydraulics in the 2nd Quarter of the 18th Century, in German. 229-241, Burckhardt et al. Nowacki H (2002) Archimedes and Ship Stability, Proc. Intl. Multi-Conf./Euroconf. on Passenger Ship Design, Operation and Safety, Anissaras/Crete, edited by A. Papanikolaou and K. Spyrou, NTU Athens (Oct. 2001). Republ as Preprint No. 198, Max Planck Inst for the Hist of Sci, Berlin. Stevin S (1586) The Principal Works of Simon Stevin, 5 vols, ed. E.J. Dijksterhuis, C.V. Swets and Zeitlinger, Amsterdam (1955). In Vol I (General Introd, Mech): De Beghinselen des Waterwichts (Elements of Hydrostatics) and in the supplement to Beghinselen der Weegkonst (Elements of the Art of Weighing) the short note: Van de Vlietende Topswaerheit (On the Floating Top-Heaviness), first publ in Dutch. Truesdell C (1954) Rational Fluid Mechanics, 1687-1765, in Euler’s Collected Works, Series II, vol. 12, pp. VII-CXXV, issued by the Euler Comm of the Swiss Acad of Natural Sci, Lausanne. Truesdell C (1983) Euler’s Contributions to the Theory of Ships and Mechanics, an essay review, in Centaurus, vol 26, 323-335. Witsen N (1994) Aeloude en hedendaegsche scheeps-bouw en bestier (Ancient and modern shipbuilding and handling), Casparus Commelijn; Broer en Jan Appelaer Amsterdam (1671). Facsimile by Canaletto-Alphen Aan Den Rijn, Amsterdam (1979). Reprinted with commentary in Hoving, A.: Nicolaes Witsens Scheeps-Bouw-Konst open gestelt, (Nicolaes Witsen’s System of Ship Building), Uitgeverij Van Wijnen, Franeker.

The Effect of Coupled Heave/Heave Velocity or Sway/Sway Velocity Initial Conditions on Capsize Modeling Leigh S. McCue* and Armin W. Troesch** *A erospace and Ocean Engineering, Virginia Tech **Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, Michigan

Abstract

Obar et al. (2001) described a series of model tests and a numerical model of a rectangular barge in regular beam seas with three degrees of freedom in roll, heave, and sway used to investigate the onset of capsize. The model had minimal freeboard resulting in significant water-on-deck. This paper employs the numerical model to examine specifically the impact of the coupling in heave with heave velocity initial conditions or in sway with sway velocity initial conditions on ultimate stability. Direct comparison with experimental results was used to validate the numerical model and can be found in Lee et al (Lee, 2001; Lee et al, 2006).

1 Introduction One of the greatest threats to vessel safety is capsize. While capsize stability has been experimentally studied in great detail for one degree of freedom, see for example (Soliman and Thompson, 1991; Thompson, 1997), at the time of the 2003 Ship Stability conference, little had been done with multiple degree of freedom experiments in which great attention was paid to initial conditions. This work extends the experimentation conducted by Obar et al. (2001) and the numerical simulation and experimental structure developed by Lee et al. (Lee, 2001; Lee et al, 2006) to the case of coupled state variables. Vassalos and Spyrou (1990) demonstrated that combinations of effects (in their case the effect of directional instabilities on transverse stability) can dramatically alter the ultimate stability of a vehicle. Thompson and de Souza (1996) show the dependence of stability on nonlinear coupling between roll and heave. While Soliman and Thompson (1991) and Thompson (1997) as well as the work of Taylan (2003)

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_9, © Springer Science+Business Media B.V. 2011

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182 L.S. McCue and A.W. Troesch

thoroughly investigate the one degree of freedom problem, Vassalos and Spyrou (1990) and Thompson and de Souza (1996) make clear the importance of studying combined effects. Murashige and Aihara (1998) investigate the case of a forced flooded ship experimentally and mathematically. While they improve upon the traditional onedimensional model through inclusion of the effects of water on deck, their model is based upon assumptions that cause one to neglect sway and heave degrees of freedom. The work presented in this paper seeks to expand upon the one degree of freedom models previously discussed, and the computationally intensive models developed for fully nonlinear simulation and/or water on deck models which are currently being developed and validated (see for example Huang et al, 1999 and Belenky et al, 2002). This paper discusses results from a quasi-nonlinear time domain model developed by Lee (2001) which is far less computationally intensive than its fully nonlinear counterparts allowing for investigation of far greater quantities of data. Additionally, the numerical model is validated by three degree of freedom experimental results conducted at the University of Michigan Marine Hydrodynamics Laboratory (Obar et al, 2001; Lee, 2001, Lee et al, 2006). Over one hundred eighty separate experiments were used in the study to validate the numerical model (Obar et al, 2001; Lee, 2001, Lee et al, 2006). For each test case a simple box barge was excited at a frequency of 6.8 rad/s. The waves acting upon the model had a wave height to wave length ratio of 0.02. Each release, however, featured different initial conditions in roll, roll velocity, sway, sway velocity, heave, and heave velocity. These values were measured in 1/30th of a second increments for the duration of the experimental run. Approximately 37% of the test runs resulted in capsize suggesting that initial conditions were a significant contributor to subsequent dynamics. While roll initial conditions (i.e. initial roll angle and roll velocity) are well known to influence capsize (e.g. Thompson, 1997), little has been said about heave initial conditions. A reducedmodel nonlinear time-domain computer program is used in this work to provide guidance in interpreting the experimental results. Qualitative behavior of the capsize dynamics is presented in a number of formats. The roll phase space is characterized by “safe/unsafe basins” (e.g. Thompson, 1997). These figures show capsize boundaries based upon roll initial conditions. The evolution or change of these boundaries is investigated for varying sets of heave and sway initial conditions. From similar analyses, “integrity curves” are generated for systematic variations in the initial heave displacement and heave velocity as well as sway displacement and velocity. Each point on an “integrity curve” represents the ratio of safe (i.e. non-capsize) basin area for a given wave amplitude to a similar safe basin area derived from zero wave amplitude. In this way, the integrity curves show the relative influence of incident wave excitation on capsize relative to vessel safety in the absence of incident waves. Results from the simulation are presented here with discussions of accuracy and time of computation. Qualitative guidelines are suggested and recommendations are

Sway/Sway Velocity Initial Conditions on Capsize Modeling 183

made for determining whether experimental programs may be sensitive to test initial conditions.

2 Numeral Analysis 2.1 Definition of Numerical Model The works of Lee (2001) and Lee et al (2006) develop a quasi-nonlinear three degree of freedom ‘blended’ hydrodynamic model to simulate highly nonlinear roll motion of a box barge. The model uses an effective gravitational field to account for the centrifugal forces due to the circular water particle motion in addition to the earth’s gravitational field. This results in a local time dependent gravitational field normal to the local water surface. Additionally the model assumes long waves. Equation 1 below gives the equations of motion for the numerical model in global coordinates where a and b are added mass and damping coefficients, m and I are the mass and moment of inertia of the body, ge is the time dependent effective gravitation, f D are diffraction forces, is the time dependent displaced volume of water, and GZ is the ship’s time dependent roll righting arm.

m  a 22   0  a 42 

0 m  a33

b22 0  b42

b24   x g  0 0 0   x g    0   y g   0 0 0   y g b1  g  0 0 b2  g g 

0 b33 0

0

a 24   xg    0   yg   I cg  a 44  g      

(1)

 g e2   f 2D      g e3   mg  f 3D   D   g e4 GZ  f 4  The global and body fixed coordinate systems for the model are defined in Figure 1.

184 L.S. McCue and A.W. Troesch



Fig. 1 Coordinate system definition, Model scale definitions: 18.25 cm draft, 1.12 cm freeboard, 66 cm length, and 30.48 cm beam.

While the numerical model cannot account for the dynamics of water on deck, such as sloshing, it does account for hydrostatic effects due to deck immergence. The forcing due to the waves is modeled as a cosine wave. To simulate laboratory transients, the wave “ramps” from zero until steady state. An experimentally determined envelope curve was generated to create a ramped cosine wave that closely resembles the waves generated in the tank (Obar et al, 2001). All release times for the model are defined relative to the maximum transient wave crest as shown in Figure 2 below and indicated by the notation “to”.

Fig. 2 Sample wave profile (top) with enlarged region showing definition of points relative to to (bottom).

Sway/Sway Velocity Initial Conditions on Capsize Modeling 185

While this ‘blended’ model has admitted weaknesses due to the simplifications employed, it has the distinct advantage of computational efficiency. Other, multidegree of freedom models that simulate various dynamics of water on deck run significantly slower than real time (Belenky et al, 2002). However, this model, whose results are qualitatively experimentally validated (Lee et al, 2006), runs in a fraction of the actual run time. The work that makes up this numerical study represents years worth of real time data; using this quasi-nonlinear model such data can be collected in a matter of days.

2.2 Numerical results As discussed in the work of Soliman and Thompson (1991), one can consider a set of model releases in which all initial conditions, save roll and roll velocity, are held constant. A sweep is made of roll and roll velocity initial condition pairs with conditions resulting in capsize marked in black and non capsize left white yielding a ‘safe basin’. Figure 3 shows a sample safe basin generated with this numerical model (note e denotes excitation frequency, n represents vessel natural frequency in roll, similarly te and tn represent excitation and natural periods respectively).

Fig. 3 Safe basin for release time of to-2 with sway, sway velocity, heave, and heave velocity initial conditions set equal to zero at release. en≈3, h/=0.019.

186 L.S. McCue and A.W. Troesch

Fig. 4 Safe basin for release times starting from to-2 in 30 degree increments with en≈3, h/=0.019, roll angle from -30 to 30 deg and roll velocity from -30 to 30 deg/s; i.e. the left hand column represents safe basins at to-2, to-1.5, to-1, to-.5, to, to+.5, to+1, and to+1.5. All heave and sway displacements and velocities initially set equal to zero. x and y axes represent roll and roll velocity respectively.

Sway/Sway Velocity Initial Conditions on Capsize Modeling 187

One can then consider a set of safe basins with some third parameter varied. For example, Figure 4 shows a series of safe basins for different release times. The figure shows safe basins for 48 different release times starting from to-2 until to+2. Thus a shift between two safe basins represents a 30 degree shift in phase location along the wave profile (i.e. 1/12 ). This figure illustrates how crucial accurate accounting of time, and implicitly sway location, is on predicting vessel stability. Additionally, this plot demonstrates the impact transients in the incident wave profile have upon the safe basins. The safe basins before steady state, such as from to-2 to to-1, exhibit no periodicity (i.e. to-2 and to-1’s safe basins are quite different), while basins from to to to+2 produce a periodicity in that every 360 degrees of phase (or one wave length) the safe basin is virtually identical (i.e. to+.5 and to+1.5 are nearly the same). It should be noted that a change in sway displacement can be explicitly written as a shift in time. For example, upon reaching steady state, a sway displacement equal to one wave length at time tx could instead correspond to a sway displacement of zero at a time tx+1, that is time moved forward through one wave period. Thus a shift in time corresponds to a change in sway displacement if time is referred to by a particular location in the wave profile, e.g. the maximum wave crest, to (Figure 2). As described by Thompson (1997), one can then introduce the idea of integrity values. Integrity is a ratio of safe area for given initial conditions to safe area for some reference case. For a typical integrity curve, wave frequency and all non-roll initial conditions would be held constant while wave amplitude is varied. As an example, Figure 5 presents an integrity curve for the box barge numerical model described in the previous section. In Figure 5 sway, sway velocity, heave, and heave velocity are all set to zero at the instant of release. A safe basin is generated for varying roll and roll velocity initial conditions (between -20 to 20 degrees and –15 to 15 degrees/sec respectively). This is done for multiple wave amplitudes with integrity values being calculated as the ratio of safe area at a particular wave height to safe area for a wave height of zero. Thus the curve begins at one. The resulting curve is an integrity curve as defined by Thompson (1997) in which one notes the crucial characteristic ‘Dover cliff ’ (1997) at which point there is a distinct, rapid loss of stability. One can note that in Figure 5, integrity values are greater than one for wave amplitude to wave length ratios of 0 to approximately 0.007. Physically, integrity values greater than 1.0 indicate that the presence of waves at certain wave amplitudes can have a stabilizing effect upon the vessel relative to the zero wave condition.

188 L.S. McCue and A.W. Troesch

1.2

1

Integrity

0.8

0.6

0.4

0.2

0 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

Wave Amplitude per Wave Length (  o/ )

Fig. 5 Integrity Curve. Sway, sway velocity, heave, and heave velocity 0 at time of release. Te/Tn~3.

In Lee et al (2006) a detailed study is conducted considering the ways in which these curves shift for varying initial conditions of sway, sway velocity, heave, and heave velocity. This results in series of curves such as those found in Figure 6. In this case one sees the variations in the location of the cliff; however, direct comparison is difficult as the normalization factor for each curve varies. This is analysed in detail for the 6 state variables in Lee et al (2006). Heave (normalized by beam) 1.4

1.2

-0.09 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

-0.09

1

Integrity

0.8

0.6

0.4

0.2 0.09

0 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

Wave Amplitude per Wave Length (  / )

Fig. 6 Heave Integrity Curves. Sway, sway velocity, and heave velocity 0 at time of release. Te/Tn~3. Note significant change with respect to heave in location of cliff on curve but not in xaxis intercept (location where integrity is zero).

Sway/Sway Velocity Initial Conditions on Capsize Modeling 189

While this lends great insight into anticipated behaviour from individual state variable variations, it still cannot fully capture the idiosyncrasies of experiments as there is no graphic coupling amongst the remaining state variables, sway, sway velocity, heave, and heave velocity. Therefore, instead of viewing the parameters as integrity curves, one can consider a surface, in which integrity (defined on the roll and roll velocity phase space plane) for a fixed wave amplitude is calculated for pairings of two other state variables such as sway and sway velocity (Figure 7) or heave and heave velocity (Figure 8). Each of these surfaces is for a fixed wave height and frequency (h/=.019 and en≈3) and is normalized such that the integrity value is 1 when sway, sway velocity, heave, and heave velocity are zero. Figure 7 presents a three dimensional integrity surface where each point is an integrity value representing the ratio of safe area to unsafe area for a roll/roll velocity grid as a function of sway and sway velocity. Heave and heave velocity are defined as zero at the time of release. Additionally excitation frequency and incident wave amplitude are constants. This figure clearly shows the relative independence of stability on sway velocity initial conditions. If one were to pick a fixed value of sway the plot collapses to a two-dimensional curve representing integrity as a function of sway velocity. This curve is relatively horizontal and constant thus showing the independence of integrity on sway velocity. For example, for sway equal to one wave length (or correspondingly zero wave length), the maximum integrity value as a function of sway velocity is 1.24 and the minimum is 0.70. While this is non-trivial, comparatively speaking, this is a relatively small change in contrast to the effects seen by changes in sway displacement, heave displacement, and heave velocity. If one were to instead consider a constant sway velocity, we see integrity values as a function of sway vary from zero to 2.2. Therefore one can see that for any fixed sway velocity value, there is a strong dependence of the integrity value upon sway position. In fact, for values of sway from approximately .15 to .65the integrity values are zero irregardless of sway velocity. This indicates for some initial release positions on a wave (the region 15%-65% of a wave length beyond a peak) the vessel shall always capsize regardless of initial roll angle, initial roll velocity, or sway velocity for a wave amplitude to wave length ratio of 0.0096 and an excitation frequency to natural frequency ratio of approximately 3. This is due to the fact that this wave amplitude is in a particularly steep region of the integrity curve. Referring to Figure 5, one should note the steep drop off at approximately o/=.01. For a set of sway integrity curves the location of this ‘cliff ’ is relatively invariant. More details on this are given by Lee et al (2006). Figure 8 demonstrates the intricate interdependence between heave and heave velocity state variables in determining ultimate vessel stability. For the surface presented in Figure 8 sway and sway velocity are zero at the time of release and

190 L.S. McCue and A.W. Troesch

wave height and frequency are constants for each point determining the surface. It should be noted though that after the time of release the vessel is unconstrained in all three degrees of freedom, thus sway and sway velocity are allowed to evolve in time just as heave, heave velocity, roll and roll velocity evolve. This is true for the data comprising both Figures 7 and 8. Unlike Figure 7 we note there are no similar statements we can make regarding a relative independence. Both heave and heave velocity have substantial influence on the ultimate state of the vessel and a coupling between the two conditions of heave and heave velocity can significantly alter the location of the vessel on such an integrity surface indicating a dramatic shift in vessel safety. Only generally can we say that maximum integrity values occur when heave velocity initial conditions are near zero for all heave.

3 Conclusions This paper presents the results of a thorough computational study into the interdependence of initial conditions in multiple degrees of freedom on predicting ultimate state of a simple box barge. While the hydrodynamic model is highly simplified it serves as a useful tool having demonstrated the influence coupled degrees of freedom can have on capsize prediction. The final two figures, 7 and 8, demonstrate the dependence of vessel stability on heave, heave velocity, sway and to a lesser extent sway velocity. While many models used in the present day literature focus upon one degree of freedom or a reduced order model, this work has shown that such models can be vulnerable to neglecting crucial dynamics. It is not sufficient to consider simply one state variable such as roll angle nor two state variables models coupling roll with roll velocity. This work demonstrates that elaborate couplings of multiple state variables in roll, sway and heave can significantly alter predictions of safe versus capsize and thus one must exercise care in the use of reduced order models.

Acknowledgements The authors would like to acknowledge the support of the National Defense Science and Engineering Graduate Fellowship program for sponsoring this research. Additionally the authors would like to thank Young-Woo Lee and Michael Obar whose work laid the basis for this paper.

Sway/Sway Velocity Initial Conditions on Capsize Modeling 191

Fig. 7 Sway/sway velocity integrity surface. Heave and heave velocity 0 at time of release, to. en≈3, h/=0.019.

Fig. 8 Heave/heave velocity integrity surface. Sway and sway velocity 0 at time of release, to. en≈3, h/=0.019.

192 L.S. McCue and A.W. Troesch

References Belenky VD, Liut KW, Shin YS. Nonlinear ship roll simulation with water-on-deck. Proc of the 2002 Stab Workshop. Webb Inst. 2002 Huang J, Cong L, Grochowalski S, Hsiung CC. Capsize analysis for ships with water shipping on and off the deck. Twenty-Second Symp on Naval Hydrodyn. Washington D.C. 1999 Lee YW. Nonlinear ship motion models to predict capsize in regular beam seas. Doctoral dissertation, Dep of Naval Archit and Mar Eng. University of Michigan Ann Arbor MI 2001 Lee YW, McCue LS, Obar MS, Troesch AW. Experimental and numerical investigation into the effects of initial conditions on a three degree of freedom capsize model. J of Ship Res 2006 50:1 63-84 Murashige S, Aihara K. Coexistence of periodic roll motion and chaotic one in a forced flooded ship. Int J of Bifurcation and Chaos 8:3 1998 619-626 Obar M, Lee YW, Troesch A. An experimental investigation into the effects initial conditions and water on deck have on a three degree of freedom capsize model. Proc of the Fifth Int Workshop on the Stab and Oper Saf of Ships. September 2001 Soliman, Thompson. Transient and steady state analysis of capsize phenomena. Appl Ocean Res 1991 Taylan M. Static and dynamic aspects of a capsize phenomenon. Ocean Eng. 30: 2003 331-350 Thompson J. Designing against capsize in beam seas: Recent advances and new insights. Appl Mech Rev. 50:5 May 1997 Thompson JMT, de Souza JR. Suppression of escape by resonant modal interactions: in shell vibration and heave-roll capsize. Proc of the Royal Soc of London. A452 1996 2527-2550 Vassalos D, K Spyrou. An investigation into the combined effects of transverse and directional stabilities on vessel safety. Fourth Int Symp on the Stab of Ships and Ocean Veh. Naples, Italy. 519-527 1990

The Use of Energy Build Up to Identify the Most Critical Heeling Axis Direction for Stability Calculations for Floating Offshore Structures Joost van Santen GustoMSC the Netherlands

Abstract For offshore structures like semi submersibles and jack-ups, hydrostatic stability is to be determined for what is called the weakest axis, which is not necessarily the same as the longitudinal axis of symmetry of the structure. When allowing trim to take place, the determination of the critical axis is complicated as free trimming leads to multiple solutions regarding the position for a given heel angle. It will be shown that for a freely floating structure, looking at the increase in potential energy can be used to identify those axis directions which are critical as well as realistic. The theoretical results will be illustrated with detailed data obtained for a two typical offshore structures using a standard stability program.

Nomenclature     Cxy CoB CoG E GML

= = = = = = = = =

displacement (t) heel angle axis direction (twist angle) trim angle cross moment of the waterline plane for rotation around its centre ≈∫x.y.dA Centre of Buoyancy Centre of gravity potential energy divided by displacement metacentric height for trim at a particular heel angle

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_10, © Springer Science+Business Media B.V. 2011

193

194 J. van Santen

GMT Mx My VCB x xcob, ycob y

= metacentric height for heel at a particular heel angle = moment around the longitudinal horizontal axis divided by displacement = moment around the transverse horizontal y axis divided by displacement = vertical centre of buoyancy above the centre of gravity = horizontal axis along the initial heeling axis, forward positive = location centre of buoyancy relative to CoG = horizontal axis perpendicular to the x axis (right handed system)

1 Introduction Historically, determination of the stability offshore rigs is seen as extension of stability for ships. For ships the longitudinal axis is taken as the heeling axis, where trim is allowed to remove the trimming moment. Early on, the offshore industry has recognized that the most critical axis is not necessarily the longitudinal axis. So, the wording critical axis was introduced, but as a kind of heritage one also had to consider free trim. This directly leads to a problem. This problem is that a given position can be defined by infinite combinations of axis directions, heel angle and trim. Two angles suffice to uniquely define the position. Using three angles means that one is superfluous. In the 80’s several papers appeared introducing the use of pressure integration to replace the conventional way of slicing up the structure and calculation of the contribution of each slice, (Witz and Patel, 1985, van Santen, 1986). In the publication of van Santen (1986), the problems mentioned above were raised and a way to avoid them by using free twist was introduced. In this paper, also the concept of energy build up was touched upon. More recently (Breuer and Sjölund, 2006), the problem was looked at again and using the build up of energy was proposed as a solution. In the underlying publication, the increase in potential energy of a structure due to forced heel with and without trim will be looked at and examples will be shown. This paper deals with freely floating structures. Effects due to mooring or dynamic positioning are specifically excluded.

Axis Direction for Stability Calculations for Floating Offshore Structures 195

2 Example with a Barge For a rectangular barge (Figure 1) with a raised forecastle, we can construct the righting arm curves for a range of axis directions (without trim) as shown in Figure 2. 25.0 m

Volume 25200 m3 LCG 70.0 m TCG 0.0 m VCG 17.0 m Intact draft 5.0 m Seawater density 1025 kg/m3

8.0 m

8.5 m

140.0 m

36.0 m

Fig. 1 Barge dimensions

40 90

35

80

Arm (m)

30 25 60

20 15

40

10 5

0

0 0

10

20

30

Heel angle (deg)

Fig. 2 Righting arm curves, no trim

Applying free trim has dramatic consequences as is shown in Figure 3. For some axis directions the righting arm curves stop prematurely. Why is this? In order to analyze this, the trimming moment as a function of trim angle is to be studied, see

196 J. van Santen

Figure 4. The trim angle for which this moment is zero is the free trim angle belonging to a particular heeled situation.

Fig. 3 Righting arm curves, free trim

Trimming arm for an axis direction of 80 deg zero value indicates a free trim position Trimming arm (m)

6

-50

Heel angle

7° No solution

6° 5° 4° 3°

7 3

6 5 4 3

0 -40

-30

Unstable solution

-20

-10

0

Stable solution -3

Trim angle (deg)

Fig. 4 Trimming arm curves

For an axis direction of 80 deg, when heeled beyond 5.6 deg heel, the zero crossings disappear, meaning that there are no solutions with zero trim moment. When passing a certain position, continuation of the righting arm curve can only

Axis Direction for Stability Calculations for Floating Offshore Structures 197

be achieved by reducing the heel whilst at the same time increasing the trim angle. This leads to the righting arm curve as shown in Figure 5 and the trim angle as shown in Figure 6. 15 Continuous path 10

in ble

trim

Righting arm

Sta

5 0 0

1

2

3

-5 n tr

i ble sta Un

-10

4

5

6

im

-15 Heel angle (deg)

Fig. 5 Continuous righting arm curves, free trim 0 0

1

2

3

4

5

6

-5

Trim angle [deg]

-10 -15 -20

Continuous path

-25 -30 -35 Heel angle (deg)

Fig. 6 Trim angle for continuous heel

For intact offshore rigs, some authorities require a range of positive stability of 30 (or 36) deg. For a longitudinal axis direction this criterion would be met, but for an axis direction of 80 deg, the structure would fail to meet this criterion. So, one could say that 80 deg axis direction is the most critical one and the rig would fail to comply with the range requirement.

198 J. van Santen

Looking at the structure (figure 1) it is obvious that one should select an almost longitudinal axis as the most critical axis direction and not 80 deg. But for a more generic structure it is not that easy to identify the critical axis direction. Why is it that from visual observations we reject 80 deg and accept 0 deg ? The key can be found in the rate at which the potential energy in the system increases with heel. For 0 deg axis direction this is far less than for 80 deg. The potential energy is given by the negative of the vertical distance (VCB) between the centre of gravity CoG and the centre of buoyancy CoB.

Energy  VCB  displacement,

(1)

where E will be used to denote energy divided by displacement, so E = -VCB. At the equilibrium position the energy is taken as the reference value, E0. For a change in heel, the increase in energy is given as

E  E     E 0

(2)

 VCB0  VCB   

The input of energy is the work done by the overturning heeling moment given by 

 M   d

(3)

0

where M() is the overturning moment divided by displacement. This results in: 

  VCB     VCB0    M   d 0



    righting arm  d 0

(4)

Note that for a free floating structure, a positive overturning moment results in a positive righting arm, but this means in fact that the counteracting restoring moment given by ycob is negative. The most critical axis can be viewed as the axis direction for which a given heel angle is reached with the least effort. In this way, the least energy is to be fed into the system in order to reach that particular heel angle. Figure 7 shows the amount of energy fed into the barge depending on axis direction and heel angle.

Axis Direction for Stability Calculations for Floating Offshore Structures 199

Fig. 7 Energy surface for the barge

The trim is fixed to zero. From this figure it is seen that for an axis direction of about 0 deg the lowest amount of energy is needed to reach a given heel angle. This agrees with the subjective feeling that the most critical axis is almost longitudinal. The problem now is how to determine the axis direction for which the energy increase is lowest. But before we do so, another example will be given, as found in the publication by Breuer and Sjöland (2006), see also Figure 8.

3 ABS Jack-Up Following discussions between MSC and ABS on the approval of a particular MSC jack-up, related to the Range of Stability (RoS) requirement (ABS rules 2008), Breuer and Sjölund (2006) published their paper in which they showed energy contour plots for a range of trim and heel angles. For the damage as indicated in Figure 8, the increase in energy depending on axis direction and heel angle is given in Figure 9. This figure shows that the lowest rate of increase is found for an axis direction of about 320 deg.

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Fig.8 Dimensions of the studied jack

Fig. 9 Energy surface of the studied jack up, damaged condition

For this direction, the righting arm is shown in Figure 10 (free trim). Selecting another axis direction (280 deg) results in a vanishing righting arm curve, as shown in Figure 11. Note the large trim angles related to the vanishing arm curve. Using an axis direction of 280 deg may lead to the false impression that the RoS criterion is not. Actually it is met, but it is for an axis direction of 320 deg.

Axis Direction for Stability Calculations for Floating Offshore Structures 201 2.5 Righting arm (m)

2

Trim angle (deg) 1.5 1 0.5 0 0

5

10

-0.5

15

Heel angle

-1 RoS=7+1.5*2.6 deg=10.9 deg -1.5

Fig. 10 Righting arm and trim angle, axis direction 320 deg

20 Righting arm (m) 15

Trim angle (deg)

10

5

0 0 -5

2

4

6

8

10

Heel angle

Fig. 11 Righting arm and trim angle, axis direction 280 deg

4 Axis Convention When using heel, trim and twist, it is important to have a clear understanding of their meaning. For the remainder of this publication, the following convention is used when positioning a structure in a heeled condition (see Figure 12):

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  

first the heeling axis direction is chosen, the structure is heeled around this axis the structure is trimmed around a transverse axis which was initially horizontal but changes direction due to heel.

5 Free Trim Versus Free Twist As is seen above, looking at energy increase during heel is a clear indicator for the determination of the critical axis direction. The next question is how to deal with trim. For ships, introducing free trim is clearly meant to obtain the lowest righting arm. This is easily seen when going from the free trim to the situation with nil trim. When going from the trimmed to the free trim situation, energy is dissipated, thus the righting arm is less than for the fixed axis direction and zero trim situation. The question is if this also works when varying the axis direction. For this purpose consider a heeled position with both a zero trimming moment and a zero moment around the twist axis which is the initially vertical axis. When applying a small change in the trim angle (d) and in the axis direction (d) reactive moments will result. For a constant displacement, using conventional hydrostatic considerations, these moments can be determined by looking at the horizontal and vertical components of the rotations. The horizontal component () causes a change in waterline shape and in CoB position. The combined effect translates into a reactive moment -GML . The rotation () around the vertical axis causes a transverse movement of the CoB which causes a reactive moment ycob . Thus, for a change of trim, d, the externally imposed moment needed for this change in trim becomes:

Axis Direction for Stability Calculations for Floating Offshore Structures 203 Select axis direction

Shift to CoG

z

axis system through CoG

y

x

Axis direction

Heel angle input axis system

Heel angle

Position after structure is heeled around a horizontal axis

z

y

Mx

x

My Heel angle

waterline

Trim angle

Position after structure is trimmed around an inclined transverse axis

z

x

y

Fig. 12 Application of twist, heel and trim

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My t   GM L cos   y cob sin   d

(5)

Note that the value of GM is the instantaneous metacentric height for the given heeled position. Similarly, due to a change of axis direction d the externally imposed moment becomes:

My a   GM L sin   y cob cos   d

(6)

These moments are around a horizontal axis. When considering energy input, the moment is to be taken along the axis around which the rotation takes place. For instance, for trim, the moment around the trim axis is Myacos. Thus, for trim the energy input starting from the free twist, zero trim, position is:

1 dE t  cos   GM L cos   y cob sin   d2 2

(7)

Due to a change in axis direction the energy input starting from free twist is:

1 dE a  sin   GM L sin   y cob cos   d 2 2

(8)

The term between the brackets in Equation (7) is in general positive. In equation 8, for small heel angles the term between the brackets can be approximated by GML  - GMT  cos. So, for small heel angles, it is in general small but also positive. For larger angles it is in general positive. So, in general, when starting from the free twist position with zero trim and applying either a small change in twist or in trim an increase in energy in the system is found. Thus, the free twist position is the position with the lowest potential energy for a given heel angle. The question arises if there is a reduction in energy when going from a free twist position with zero trim to a new position (with a change in twist of d) and letting the rig free to trim with angle d. The energy input needed to twist is given by dEa. By letting the rig free to trim, the energy recovered is – dEt. So the total energy input is dEa - dEt. The twist angle is obtained by imposing a moment on the rig. This moment is nullified by letting it free to trim. So:

Axis Direction for Stability Calculations for Floating Offshore Structures 205

My a  My t

(9)

Using Equations (5) and (6) gives the relation between d and d:

d 

GM L cos   y cob sin   d GM L sin   y cob cos 

(10)

By substituting Equation (10) in Equation (8) the difference in energy input is found to be

dE  dE a  dE t GM L cos   y cob sin  1   d2 y cob 2 GM L sin   y cob cos 

(11)

Generally, the following applies:  GML is positive,  ycob is negative for a positive righting arm Thus, for the weakest axis, the numerator GML cos -ycob sin is in general positive. The denominator is also in general positive. So, when going from a free twist to a free trim situation the energy input is positive. This leads to the conclusion that the free twist situation contains less energy than fixing the axis direction combined with free trim. The free twist position equals the situation with a local minimum energy as both moments Myt and Mya are zero. For the jack-up shown in Figure 8, the numerical values for small variations in trim and twist around the free twist point are shown in Figure 13. These are based on the calculations with a stability program as well as on the approximations given above. Also, the calculated relation according to Equation (10) between axis and trim for which energy Ea is exchanged with Et is shown (“lowest envelope”). This figure shows the validity of the theoretical formulations. It is important to be aware that in the free twist method, trim is always zero. Hence, the heel angle is always equal to the steepest slope of an initial horizontal plane (like a deck).

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energy (m)

Energy depending on trim for varying axis directions damaged rig, 10 deg heel 0.005 0.004 0.003

325,3 324,3

0.002

323,3

0.001

322,3

SE-10 -1

-0.5

-0.001

Approximation axis 324,3 deg 326,3

321,3 0

0.5

1

trim angle (deg)

Calculated lowest envelope

Fig. 13 Energy depending on trim

6 Change in Axis Direction Due to a Change in Heel Angle In section 5 it is shown that the free twist method results in the lowest amount of potential energy build up for a given heel angle. For a free twist situation, the moment around the initial vertical axis is zero. When making a small step in the heel angle, the change in axis direction can be estimated using this condition. For a constant displacement a heel increase d results in moment around the twist axis of -Cxy · d, where Cxy = cross moment of inertia of the waterline for a rotation around its centre of floatation. Making this equal but opposite to the moment due to a small change in twist (Mya, Equation (6)) the change in axis direction results

d 

C xy / volume GM L sin   y cob cos 

d

(12)

It is seen that the axis direction is influenced by both the metacentric height (GML) and the righting arm (which equals -ycob).

Axis Direction for Stability Calculations for Floating Offshore Structures 207

7 Stability of the Solution The free twist position is characterized by the twist moment Mya being zero. It is not necessarily a position with the lowest energy, but it can also be a position with the highest energy. For a given heel angle, there are several axis directions for which the energy is either minimum or maximum. The maximum is by definition an unstable position, the minimum is a stable position. Stability is indicated by the energy input Equations (8) (for twist) and (7) (for trim). If the term GMLsin+ycobcos is negative, the position is unstable in twist. For an intact structure at a small heel angle, the following approximations apply: ycob ~ -GMT  sin() ~ cos() ~ 1.0

stability term   0 means stable 

 GM L  GM T  

(13)

Clearly, for small angles the position is stable when heeling is around the axis with the smallest GM value. For larger angles, the actual values of GML and ycob are to be considered. When the structure is in free to trim, also the stability for trim should be looked at. For trim to be unstable, the term GMLcos-ycobsin is to be negative. Keeping in mind that -ycob is the righting arm, it is seen that for a positive righting arm this is the case as long as GML, being the metacentric height for the given heeled position, is positive.

8 Case Studies For two structures, analyses have been done into the increase in potential energy during progressive heeling. These structures are:

 

ABS jack-up, a simplified semi submersible

8.1 ABS Jack-Up Intact. This structure, see Figure 8 has been analyzed for both the intact and damaged condition. For the intact condition, Figure 14 shows the energy surface for a range of heel angles (0-24 deg) and a range of axis directions (0-360 deg).

208 J. van Santen

Fig.14 Energy surface intact

It is seen that for larger heel angles 3 axis directions are present following the local minimum path. These are 900, 2100 and 3300. The 210 and 330 deg directions are in fact identical due to the symmetry of the structure. From the graph, it is also seen that the 330 (or 210) degree yields the lowest energy increase. For angles exceeding about 6 degrees heel there are 3 maxima and 3 minima in the plot. What is not clear is that for smaller heel angles there are only 2 maxima and 2 minima. The extremes at around 35 deg and 145 deg disappear for small heel angles. This is shown in Figure 15 which shows the axis directions which have a zero twist moment depending on the heel angle.

Fig. 15 Axis directions for minimum energy build up, intact

Axis Direction for Stability Calculations for Floating Offshore Structures 209

Damaged. The damage case has a damaged compartment as indicated in Figure 8. For a VCG of 23.45 m, the range of stability criterion of ABS is just met. This range should be 7 deg + 1.5 x steady heel (2.59 deg) = 10.89 deg. The plot of the energy (Figure 9) shows a minimum value for an axis direction of around 320 deg. Figure 16 shows the axis direction for which free twist is satisfied, i.e. where the moment around the y axis is indeed zero. The figure also shows the estimated axis direction based on Equation (12). In this estimate, the start value at zero heel is taken from the calculations with the stability program. The other values are based on the summed increments. A very good fit is seen between the estimated and the actual axis directions. For the damaged case, at larger heel angles, there are two paths following a local extreme in the energy surface. For small heel angles there is only a single path. 18 16

Heel angle (deg)

14 12

Unstable 10

Stable

8 6

Estimated path (eq13)

4

Minimum energy path

2 0 200

250

300

350

Axis direction (deg)

Fig. 16 Axis directions for minimum energy build up, damaged

8.2 Semi-Submersible Figure 17 shows a semi-submersible which at the given displacement has an intact draft of 9.106 m. Both the intact and damage condition pose a challenge when calculating the stability. Intact. Figure 18 shows the energy surface plot for the intact condition. Note that because of symmetry, the data for an axis direction of α is also found for axis direction α + 180. It is seen that for moderate heel angles, the minimum energy is

210 J. van Santen

found for an axis direction of 180 deg (and 0 deg). This is even better seen in the righting arm plot, Figure 19. When for 180 deg axis direction, heel is increased beyond 25.5 deg, the path follows a local unstable maximum instead of a local stable minimum. Instead, for larger heel angles, the minimum energy is found for axis directions of about 135 and 225 deg. A more detailed analysis can be done by looking at the twist moment depending on heel and axis direction.

Fig. 17 Shape of the semi submersible

Fig. 18 Energy surface, intact

Axis Direction for Stability Calculations for Floating Offshore Structures 211

Fig. 19 Surface plot of righting arm

Fig. 20 Surface plot of the trimming arm

Figure 20 shows this as a surface plot of xcob versus heel and axis direction. The free twist locations are those for which xcob equals 0. The combination of heel and axis directions for which the twist moment (or xcob) is zero are identified by the change in color in Figure 20. The stable and unstable positions can be identified by looking at the slope dxcob/ d for a given heel angle. Using this figure, the stable and unstable paths for which there is an extreme in the energy build up are as given in Figure 21 for axis directions between 120 and 240 degrees.

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Intact, VCG=40 m 50 Heel angle (deg)

Stable branches

Unstable branches

40 30 20 10 0 120

170 Axis direction (deg)

220

Fig. 21 Stable and unstable paths

Starting with zero heel, a gradual increase in the overturning moment will result in the rig to follow a seemingly erratic path: heel axis direction 0 – 1.6 180 deg 1.6 – 4.6 151-154 deg or 209 – 206 deg 4.7 – 25.4 180 deg 25.5 –50 133-138 deg or 227 – 222 deg At hindsight, the unstable area for very small heel angles is also seen in Figure 19 by a barely noticeable hill for axis 180 and heel 4 deg. Damaged. When damaging the rig by removing the pontoon corner part as indicated in Figure 17, the energy surface looks more simple than for the intact rig. The axis direction for minimum energy increase is around 140 deg, see Figure 22. Still, also in this case, the axis direction changes considerably for increasing heel as shown in Figures 22 and 23 When actually increasing the heel angle in a stepwise manner, the rig will suddenly change position when passing the 12 and 23 degrees heel, see Figures 23 and 24. At 12 degrees heel, it will change from axis direction about 140 deg to about 166 degree. At 23 degree heel it will change back to about 130 deg axis direction. Further study showed that this behavior hardly depends on the VCG.

Axis Direction for Stability Calculations for Floating Offshore Structures 213

Fig 22 Energy plot, damaged 40

Heel angle (deg)

35

Jump in axis direction for increasing heel

30 25 20 15 10

Jump in axis direction for decreasing heel

5 0 120

130

140

150

160

170

Axis direction (deg) Fig. 23 Relation between heel angle and axis direction, free twist

180

214 J. van Santen 5 4.5

Jump in righting arm

Righting arm (m)

4 3.5 3 2.5 2 1.5 1 0.5 0 0

10

20

30

40

Heel angle (deg)

Fig. 24 Righting arm following free twist

9 Use of Minimum Energy Path in Evaluating Stability Criteria When evaluating stability criteria, a distinction can be made between those without and with external influence. An example of the first group is ABS’s range of stability criterion for jack-ups. An example of the second group is the well known requirement on ratio between the area under the restoring moment and the wind overturning moment curves. For the first group, the minimum energy path can be followed as being the most critical, i.e. it requires the least effort to reach a particular heel angle. For the second group, both the magnitude of the restoring moment and of the overturning moment is to be considered. This highly complicates the task of selecting the most critical heeling axis direction. When following the minimum energy path, it is relatively easy to adapt the wind overturning moment to the instantaneous axis direction. But, this still assumes that the direction of the wind overturning moment is the same as that of the restoring moment. In general, it is well possible that this is not the case and that the wind overturning moment has a trimming component as well. But, this raises the question if one should absorb this by letting the unit trim or if the axis direction should be modified. It is also possible that the wind overturning moment for other directions is higher and thus more governing. Another way would be to look directly at the ratio between energy inputs versus energy build up for all axis directions without considering trim. When including possible downflooding, the complexity of the calculation increases further.

Axis Direction for Stability Calculations for Floating Offshore Structures 215

Apart from the calculation difficulties, the major issue is the nature of the calculation. Traditionally, the effect of wind for an otherwise calm sea is looked at. The effect of waves and resulting motions can be important, especially for a damaged structure (van Santen, 1999). Also, the calculation of the wind overturning effect is usually simplified in that the reactive force is assumed to work at the lateral center of resistance and that the forces due wind and the reactive forces do not introduce a yawing moment. Including the effect of mooring or DP makes the analysis even more complex. In view of this, there seems to be no justification to focus on one detail whilst ignoring other effects which may be much more important. In the end, we should not forget that the criteria are quite abstract. Being abstract, they lend themselves to consistency and reproducibility. The end results are not so much influenced by the details of the method used. As such, they fulfill the requirement of a criterion of which the evaluation is clearly defined and can be reproduced independently. By Couser (2003) the difficulty in programming the calculation of the AVCG curves as limited by the various requirements was mentioned. The examples shown here indicate that apart from the vagueness in the criteria, the structure itself can cause problems. For the semi-submersible case at transit draft it is almost impossible to do a proper AVCG calculation. A possible way out is to use a fixed axis direction with nil trim. For a righting arm curve constructed in this way to be acceptable, it should show only modest trimming moments as this indicates that the energy depletion by letting it free to twist or trim is small.

10 Conclusions 1. Extending the free trim approach as used for ships to offshore units, in combination with a varying heeling axis direction, may lead to severe interpretation problems. 2. A free trim or free twist approach is sensible as this generally leads to the slowest build up of potential energy for increasing heel angle. Thus the lowest righting arm curve is achieved. A pre-requisite is that the rig remains stable in trim and twist for the relevant range of heel angles. 3. The free twist approach with zero trim leads to the lowest gain in potential energy during heeling. This is based on both theory and data obtained from direct calculations. Thus when left free to trim and free to twist, the result is that the twist angle varies and that the trim angle remains zero. 4. Application of the lowest gain in energy approach is an unambiguous way to define the most critical or weakest axis. 5. For the determination of the critical axis direction, other effects are to be considered as well. When wind overturning is in the criterion, axes other than the weakest based on energy, may have to be considered. Also when looking at openings, other axis directions have to be looked at.

216 J. van Santen

For these complex cases, it is strongly suggested to perform the calculations for any axis direction and with the trim fixed at zero.

Postscript Since the original publication in 2009, some errors have been removed. Most important is the change in width of the compartment in figure 17 from 7 to 6 m. This has an important influence on the results. Also later at the City University conference 2009, an additional paper was published (Santen 2009) looking into more detail at the various methods including the Steepest Descent Method as proposed by Breuer and Sjölund, 2006. It explains and analyzes the SDM, gives details and shows results obtained with this method. The author performed some tests with scale models of the jack up and semisubmersible which confirmed the findings reported in this paper in a qualitative way. I owe many thanks to Andy Breuer for having many open minded and fruitfull discussions about the problems met when doing stability calculations for offshore structures. His encouragement to publish my views resulted in this and in the City University paper.

References ABS rules for building and classing Mobile Offshore Drilling Units 2008. Breuer JA and Sjölund KG (2006) “Orthogonal Tipping in Conventional Offshore Stability Evaluations”, Proc of the 9 th Int conf on Stab of Ships and Ocean Veh (STAB 2006), Rio de Janeiro. Santen JA (1986) “Stability calculations for jack-ups and semi-submersibles”, Conf on computer Aided. Design, Manuf and Oper in the Mar and Offshore Ind CADMO 1986, Washington. Santen JA (1999) “Jack-up model tests for dynamic effects on intact and damaged stability”, Seventh Int Conf The Jack-up Platform: Design, Constr and Oper, City Univ, London, Sept. Santen JA (2009) The use of energy build up to identify the most critical heeling axis direction for stability calculations for floating offshore structures, review of various methods. 12th Jack-Up Conf 2009 City Univ, London. Witz JA, Patel MH (July, 1985) “A pressure integration technique for hydrostatic analysis”, RINA suppl pap, Vol. 127. Couser P (2003) “A Software Developer’s Perspective of Stability Criteria”, 8th Int Conf on the Stab of Ships and Ocean Veh, STAB 2003, Madrid.

Some Remarks on Theoretical Modelling of Intact Stability N. Umeda*, Y. Ohkura*, S. Urano*, M. Hori* and H. Hashimoto* *Dept. of Naval Architecture and Ocean Engineering, Osaka University, 2-1, Yamadaoka, Suita, Osaka, 565-0871, JAPAN

Abstract It is essential for realising risk-based design or performance-based criteria to establish theoretical modelling of extreme behaviours of an intact ship in waves up to capsizing. For this purpose, this paper summarises latest progresses in the researches conducted by the authors at Osaka University for improving theoretical modelling of intact stability in waves. Firstly, the methodology for calculating capsizing probability in beam wind and waves with piece-wise linear assumption is examined. Here a new formula covering both capsizing towards windward and leeward directions is provided. Secondly, for calculating restoring moment in longitudinal waves with sufficient accuracy, a hydrodynamic prediction method with wave-making and lifting components taken into account is proposed, and is reasonably well compared with captive model tests of a post Panamax container ship in head and following waves. This can be used in place of the Froude-Krylov prediction or captive tests for assessing the danger of parametric rolling for a practical purpose.Thirdly, for more efficiently identifying dangerous conditions in following and quartering waves, a numerical method for determining a heteroclinic bifurcation is proposed with its successful example to applied to surfriding thresholds with nonlinear factors taken into account.

1 Introduction At the International Maritime Organisation (IMO), the review of Intact Stability Code started in 2002 and could open a door to use direct stability assessment with physical and theoretical modelling as an alternatives to prescriptive rules in the near future. Here quantitative accuracy of these modelling is essential. The benchmark testing of the 23rd International Towing Tank Conference (ITTC), however, revealed that existing theoretical models for predicting capsizing do have only qualitative accuracy. This means that the existing best models can predict whether a ship capsizes or not and how she does but they can not predict M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_11, © Springer Science+Business Media B.V. 2011

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218 N. Umeda et al.

exactly when she does after the specified initial conditions. (Vassalos et al., 2002) Therefore, improvement of theoretical modelling is indispensable for its practical use. After the 23rd ITTC, the research team of Osaka University continues to improve its own theoretical models from various aspects. (Umeda, Hashimoto et al., 2003, Umeda et al., 2004, Hashimoto et al., 2004A, Hashimoto et al., 2004B) In this paper, some of their latest progress are presented to initiate the discussion at the stability workshop. These theoretical modelling to be investigated at Osaka University covers the dead ship condition, parametric rolling and broaching. The dead ship condition is related to the revision of the weather criterion and the others were the capsizing scenarios that the 23rd ITTC benchmark testing specified.

2 Capsizing Probability in Beam Wind and Waves Although a ship could capsize in following and quartering seas with much smaller wave steepness than in beam seas, it is still important to assess stability in beam seas. This is because dangerous situations in following and quartering seas can be avoided by ship operation but those in beam seas cannot be. For example, if all operational means such as propeller thrust and rudder control are lost, a ship that is almost longitudinally symmetric suffers bean wind and waves. Here, since a ship cannot escape from a severe sea state by herself, she should survive for sufficiently long duration. For guaranteeing survivability of such dead ship condition (IMO, 2001), the weather criterion was adopted by several administrations (Yamagata, 1959) and then by IMO (IMO, 2002). Since this criterion utilises some empirical methods to predict aero- and hydrodynamic coefficients, applicability of this criterion to new ship types such as a large passenger ship is rather questionable. However, because the probabilistic safety level of the weather criterion was adjusted by selecting the average wind velocity with casualty statistics, (Yamagata 1959) it is necessary to evaluate the safety level that the current weather criterion implicitly guarantees for conventional ships. (Umeda et al. 1992, Umeda & Yoshinari 2003) For this purpose, analytical methods to calculate capsizing probability in beam wind and waves are indispensable. This is because numerical or experimental methods require prohibitively many realisations to obtain the reliable value of capsizing probability for a practical ship, which should have very small capsizing probability. Among some existing analytical methods, a piece-wise linear approach proposed by Belenky (1993) seems to be the most promising because it utilises analytically-obtained exact but simple solutions of linear equations only. The authors investigate Belenky’s method by executing numerical calculation with his exact and simplified methods and extend his methods to the case in beam wind and waves with both capsizing in windward and leeward directions taken into account. As a result, some problems and their measures are found. Among

Some Remarks on Theoretical Modelling of Intact Stability 219

them this paper shows one problem that capsizing probability could exceed 1 when a sea state is extremely severe. According to Belenky (1993), the formula of capsizing probability for the duration T without beam wind is as follows:

PT   2  [1  exp(u l T )]  PA  A  0;    mo 

(1)

where

A







1 1  p1  2  p1  1  1  2

(2)

: apparent roll angle, m0: border of two roll angle regions, 1, 2: eigenvalues of the roll system in the second range, 1. 1 : initial values of roll angle and roll angular velocity in the second range, p1, p1 : initial values of forced roll angle and roll angular velocity in the second range, ul: expected number of up-crossing at mo for a unit time and PA: probability of positive value of the coefficient A when the up-crossing of  occurs. On the other hand, the authors propose the following formula for the case of wind and waves.

PT   1  exp[{ul PA  A  0;   mo   u w PA  A  0;   mo }  T ]

(3)

where uw is expected number of down-crossing at -mo for a unit time. If we ignore wind here, the following formula can be obtained:

PT   1  exp[2u l PA  A  0;   mo   T ]

(4)

Therefore, P(T) defined here cannot be greater than 1. The authors applied the above new formula to a car carrier, whose principal particulars are shown in Table 1. Numerical results are presented in Fig. 1. Here the righting arm is approximated with two lines by keeping the metacentric height, the angle of vanishing stability and the dynamic stability from upright to the angle of vanishing stability because a time-varying external force induces capsizing in this scenario. The wind velocity, UT, is assumed to be constant and waves are done to be fully developed with this wind velocity and to be modelled with ITTC spectrum. PA are calculated with both exact and simplified methods. The former is derived from 3-dimensional Gaussian probabilistic density function; the latter is based on the assumption that no resonance occurs in the second region. The results demonstrate that capsizing probability tends to 1 when the sea state becomes

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Capsizing Probability

extremely severe and Belenky’s simplified method slightly overestimates capsizing probability obtained by the exact method. Further discussion is available in the work of Kubo et al. (2010). Moreover, the effect of fluctuating wind has been incorporated for the capsizing probability calculation by the authors. (Francescutto, Umeda et al. 2004)

15.00 1.00E+00 1.00E-04 1.00E-08 1.00E-12 1.00E-16 1.00E-20 1.00E-24 1.00E-28 1.00E-32 1.00E-36

25.00

35.00

45.00

U (m/s) 55.00

Simplified Method Exact Method

Fig. 1 Capsizing probability of the car carrier in beam wind and waves by using the simplified and exact methods of the piece-wise linear approach. (Paroka Ohkura & Umeda 2006)

Table 1. Principal dimensions of the car carrier

Items

Car carrier

Length overall: Loa

190 m

Length between perpendiculars: Lpp

180 m

Breadth: B

32.20 m

Draft: T

8.925 m

Vertical centre of gravity: KG Metacentric height: GM Lateral projected area: AL Height to centre of lateral projected area: HC

14.105 m 1.300 m 4327.860 m2 11.827 m

Some Remarks on Theoretical Modelling of Intact Stability 221

3 Hydrodynamic Effect on the Change in GM Due to Waves At the 23rd ITTC benchmark testing even the most reliable mathematical model can provide only qualitative agreements with free-running model experiments for capsizing due to parametric rolling. As well known, parametric rolling is induced by the change in GM due to waves. Thus the accurate prediction of the change in GM due to waves is essential but all mathematical model in the benchmark testing predict the change in GM due to waves with the Froude-Krylov assumption. The captive model experiment for the ITTC A-1 Ship, as a follow-up of the ITTC benchmark testing, revealed that the Froude-Krylov calculation significantly overestimates the change in GM due to waves. (Umeda et al., 2004) Possible reason of the difference between the experiment and the Froude-Krylov calculation are hydrodynamic forces such as a wave-making effect and a hydrodynamic lift due to a heeled hull. For modelling such hydrodynamic effects on the change in GM due to waves, Boroday (1990) attempted to use a strip theory with the heel angle taken into account. He reported that good agreement with model experiment is obtained by ignoring wave-making damping components but the wave-making damping deteriorates the agreement. For finding a final conclusion on this issue, the authors reformulate a strip theory to calculate the change in GM due to waves for a heeled ship in longitudinal waves. Here all radiation components in the roll moment due to heave and pitch motions and the heel effect of the roll diffraction moment are consistently taken into account; the end terms are also included to explain hydrodynamic lift components. Two-dimensional hydrodynamic forces are estimated with an integral equation method with the Green function. The subject ship used here is a 6600TEU postPanamax container ship, whose principal particulars are shown in Table 2. Its numerical results are shown together with the captive model test results in Fig. 2. While the Froude-Krylov calculation does not depend on the Froude number, the present calculation depends on the Froude number because of hydrodynamic effects. In following seas the Froude-Krylov calculation significantly overestimates the amplitude of GM variation but the present calculation well explains the reduction of the amplitude. In head seas the Froude-Krylov calculation overestimates the amplitude of the model runs with the Froude number of 0.1 but underestimates those with the Froude numbers of 0.2 and 0.3. The present calculation predicts the amplitude in the order obtained from the experiment. This improvement indicates that the hydrodynamic effect cannot be ignored when we accurately estimate the change in GM due to waves.

222 N. Umeda et al.

Fig. 2 Amplitude of GM variation in following and head seas for the post Panamax container ship with the wave length to ship length ratio of 1.0. Here h/ and Fn indicates the wave steepness and the Froude number, respectively. And “cal. (FK)” means the Froude-Krylov calculation.

Some Remarks on Theoretical Modelling of Intact Stability 223 Table 2. Principal dimensions of the post Panamax container ship

Items length : Lpp breadth : B depth : D draught at FP : Tf mean draught : T draught at AP : Ta block coefficient : Cb pitch radius of gyration : yy/Lpp longitudinal position of centre of gravity from the midship : xCG metacentric height : GM natural roll period : T 

Containert Ship 283.8m 42.8m 24.0m 14.0m 14.0m 14.0m 0.630 0.239 5.74m aft 1.08m 30.3 s.

4 Surf-Riding Threshold Broaching associated with surf-riding was also the capsizing scenario used in the 23rd ITTC benchmark testing for intact stability and quantitative disagreement with free-running model experiments was pointed out. In particular, the surf-riding threshold estimated with the existing model usually underestimates that from the experiment. Then some of the authors continue their effort to improve theoretical modelling by examining various nonlinear factors that had been ignored. As a result, they found that nonlinearity in the wave-induced surge force acting on a hull and the wave effect on propeller thrust are essential to improve the prediction accuracy of surf-riding threshold. (Hashimoto et al, 2004A) It is well established that the prediction of surf-riding threshold could depend on initial conditions if we simply apply numerical simulation in time domain but the surf-riding threshold for a self-propelled ship starting from sufficiently low propeller revolution does not depend on initial conditions. This is because onset of surf-riding or broaching can be regarded as a heteroclinic bifurcation. (Makov, 1969, Umeda, 1990, Spyrou, 1996, Umeda, 1999) That is, surf-riding or broaching occurs when an unstable invariant manifold of a saddle-type equilibrium in a phase space coincides with a stable invariant manifold of neighbouring saddletype equilibrium. Although a perturbation method (Ananiev, 1966 ) and an exact method (Spyrou, 2001) are available for a simplest mathematical model, the introduction of nonlinear factors and coupling could prevent their application. A numerical technique to identify surf-riding threshold as a heteroclinic bifurcation can be applied to more complicated theoretical model. The technique reported so far, however, is not suitable for a computer algorithm. (Umeda, 1990, Umeda, 1999) Here it is necessary to calculate time series starting from a saddle-type equilibrium

224 N. Umeda et al.

to other saddle-type one with the information from locally linearised systems. Thus, in this paper, a mathematical procedure suitable for a computer program is proposed with a successful example for following sea case and perspectives towards quartering sea case is added. For a simplicity sake, the uncoupled surge motion in following waves is investigated with the following mathematical model.

(m  mx )G  T (u ,  G ;  a , n)  R (u )  X W ( G ;  a )

(5)

where G: horizontal displacement of the ship centre to a wave trough, m: ship mass, mx: ship added mass, u: ship velocity ( G  u  c ), n: propeller revolution number, a: wave amplitude, T: propeller thrust, R: ship resistance and Xw: waveinduced surge force acting on a hull. The proposed procedure can be summarised as follows. First, by assuming a certain value of n, the equilibria from Equation (5) are identified and their eigenvalues and eigenvectors of the locally linearised systems at the equilibria are calculated. Here the equilibria correspond to surf-riding; one is stable and the other is unstable. Then we integrate Equation (5) with time from the point that is slightly shifted from an unstable equilibrium in the unstable direction of its eigenvector. And we also integrate Equation (5) with time backwards from the point slightly shifted from the neighbouring unstable equilibria in the stable direction of its eigenvector. Then we quantify the difference between two obtained values of u at a specified value of G as (n). Next, we apply the Newton method to find the parameter n* that  (n*)  0 . The obtained n* can be regarded as a heteroclinic bifurcation point. By using the relationship between the propeller revolution and ship speed in calm water, the nominal Froude number as the bifurcation point is determined. The above numerical method is used to determine surf-riding threshold for the ITTC A-2 ship with and without the nonlinear surge force and the wave effect on propeller thrust taken into account. Figs. 3-4 illustrate rapid convergence of the Newton method applied to their problem. The results shown in Fig. 5 show that the nonlinear surge force and the wave effect on propeller thrust significantly increase critical velocity for surf-riding. This means that these two nonlinear factors are essential to improve prediction accuracy for capsizing associated with surf-riding.

Some Remarks on Theoretical Modelling of Intact Stability 225

Fig. 3 Time domain simulation from saddle-type equilibria with time forwards and backwards with the wave steepness of 1/15 and the wave length to ship length ratio of 1.5. Here c and  are wave celerity and wave length, respectively: Fn indicates the nominal Froude number.

Fig. 4 Convergence of the procedure for determining the surf-riding threshold. Here the wave steepness is 1/15 and the wave length to ship length ratio is 1.5.

For predicting the broaching threshold, it is necessary to use an coupled surgesway-yaw-roll-rudder motion. As a result, the stable invariant manifold becomes 7-dimensional while the unstable invarioant manifold is 1-dimensional. (Umeda, 1999) Thus the additional condition to be satisfied is that the vector normal to the stable invariant manifold should be orthogonal to the unstable invariant manifold. Numerical works based on this procedure were published later (Umeda et al., 2007)

226 N. Umeda et al. 0.45 0.40

present method conventional method

Fn 0.35 0.30 0.25 0.8

1.0

1.2

1.4 l/L

1.6

1.8

2.0

Fig. 5 Surf-riding threshold for the ITTC A-2 ship in following seas predicted by numerical procedure to identify a heteroclinic bifurcation with the wave steepness of 1/10. Here Fn and /L mean the nominal Froude number and the wave length to ship length ratio, respectively. The present method is one that takes nonlinearity in wave-induced surge force and the wave effect of propeller thrust into account and the conventional method ignores them.

5 Conclusions As follow-ups of the 23rd ITTC benchmark testing, the research team of Osaka University provides some improvements in theoretical modelling of intact stability from the viewpoints of probabilistic theory, hydrodynamics and nonlinear dynamics. The major outcomes are as follows: 1. The formula to calculate capsizing probability in a dead ship condition is presented as an extension of Belenky’s piece-wise linear approach. 2. Hydrodynamic modelling of the change in GM due to waves could improve agreements with captive model experiments. 3. A numerical algorithm to determine the surf-riding threshold is presented with some important nonlinear elements taken into account. The newly-obtained results provide much larger critical velocity for surf-riding than the conventional results.

Acknowledgements This work was carried out as a research activity of the RR-S202 and RR-SP4 research panel of the Shipbuilding Research Association of Japan in the fiscal years of 2003 and 2004, funded by the Nippon Foundation. The work is partly supported by a Grant-in-Aid for Scientific Research of the Ministry of Education, Culture, Sports,

Some Remarks on Theoretical Modelling of Intact Stability 227

Science and Technology of Japan (No. 15360465). The authors express their sincere gratitude to the above organizations. The authors also thank Dr. Gabriele Bulian for his effective advice on the formula of capsizing probability, which enabled us to correct Equations (3) and (4) in this version.

References Ananiev DM (1966) “On Surf-Riding in Following Seas”, (in Russian) Transactions of Kryloc Soc, 13:169-176. Belenky VL (1993) “A Capsizing Probability Computation Method” J of Ship Res, 37:3: 200-207. Boroday IK (1990) “Ship Stability in Waves: On the Problem of Righting Moment Estimations for Ships in Oblique Waves”, Proc of the 4th Int Conf Stab of Ships and Ocean Veh, Naples, II: 441-451. Francescutto A, N Umeda et al (2004) “Experiment-Supported Weather Criterion and Its Design Impact on Large Passenger Ships”, Proc of the 2nd Int Maritime Conf on Design for Safety, Osaka Pref Univ, pp. 102-113. Hashimoto H, N Umeda and A Matsuda (2004A) “Importance of several nonlinear factors on broaching prediction”, J of Marine Sci and Technol, 9: 2: 80-93. Hashimoto H, N Umeda and A Matsuda (2004B) “Model Experiment on Heel-Induced Hydrodynamic Forces in Waves for Broaching Prediction”, Proc of the 7th Int Workshop on Stab and Oper Safety of Ships, Shanghai Jiao Tong Univ, pp. 144-155. IMO (2001) “SOLAS : consolidated edition, 2001”, 44. IMO (2002) “Code on Intact Stability for All Types of Ships Covered by IMO Instruments”. Kubo T, E Maeda and N Umeda. (2010). “Theoretical Methodology for Quantifying Probability of Stability Failure for a Ship in Beam Wind and Waves and its Numerical Validation”, Proceedings of 4th International Maritime Conference on Design for Safety, Fincantieri, pp. 1-8. Makov Y (1969) “Some Results of Theoretical Analysis of Surf-Riding in Following Seas” (in Russian) Transactions of Krylov Soc, 126: 124-128. Paroka D, Y Ohkura and N Umeda (2006) “Analytical Prediction of Capsizing Probability of a Ship in Beam Wind and Waves”, J of Ship Res, 50:187-195. Spyrou KJ (1996) “Dynamic Instability in Quartering Seas: the Behaviour of a Ship During Broaching” J of Ship Res, 40: 46-59. Spyrou KJ (2001) “Exact Analytical Solutions for Asymmetric Surging and Surf-Riding”, Proceedings of the 5th Int Workshop on Stab and Oper Safety of Ships, Univ of Trieste, 4:4:1-3. Umeda N (1990) “Probabilistic Study on Surf-riding of a Ship in Irregular Following Seas”, Proc of the 4th Int Conf on Stab of Ships and Ocean Veh, Naples, I: 336-343. Umeda N (1999) “Nonlinear Dynamics on Ship Capsizing due to Broaching in Following and Quartering Seas”, J of Marine Sci of Technol, 4: 16-26. Umeda N, H Hashimoto and A Matsuda (2003) “Broaching Prediction in the Light of an Enhanced Mathematical Model with Higher Order Terms Taken into Account”, J of Marine Sci and Technol, 7:145-155. Umeda N, H Hashimoto, D Vassalos, S Urano and K Okou (2004) “Nonlinear Dynamics on Parametric Roll Resonance with Realistic Numerical Modelling”, Int Shipbuilding Prog, 51: 2/3: 205-220. Umeda N, Y Ikeda and S Suzuki (1992) “Risk Analysis Applied to the Capsizing of High-Speed Craft in Beam Seas”, Proc of the 5th Int Symposium on the Practical Design of Ships and Mobile Units, Elsevier, 2:1131-1145.

228 N. Umeda et al. Umeda N and K Yoshinari (2003) “Examination of IMO Weather Criterion in the Light of Capsizing Probability Calculation, Conf Proc of the Soc of Naval Archit of Japan, 1: 65-66. Umeda N, M Hori and H Hashimoto (2007), “ Theoretical Prediction of Broaching in the Light of Local and Global Bifurcation Analysis”, Int Shipbuilding Prog, 54:4, 269-281. Vassalos D et al (2002) “The Specialist Committee on Prediction of Extreme Ship Motions and Capsizing –Final Report and Recommendations to the 23rd ITTC”, Proc of the 23rd ITTC, INSEAN (Rome) II: 611-649. Yamagata M (1959) “Standard of Stability Adopted in Japan”, Transactions of the Inst of Naval Archit, 101: 417-443.

3 Parametric Rolling

An Investigation on Head-Sea Parametric Rolling for Two Fishing Vessels Marcelo A.S. Neves*, Nelson A. Pérez**, Osvaldo M. Lorca***, Claudio A. Rodríguez* *LabOceano, COPPE/UFRJ, Brazil **Universidad Austral de Chile ***InterMoor Brazil

Abstract

The paper describes an experimental and numerical investigation on the relevance of parametric resonance for two typical fishing vessels in head seas. The investigation is aimed at assessing the influence of the incorporation of a transom stern to the design. Results for different Froude numbers are discussed. The first region of resonance is investigated. Distinct metacentric heights and wave amplitudes are considered. Quite intense resonances are found to occur. These are associated with specific values of metacentric height, ship speed and stern shape. In order to analyse the experimental/numerical results, analytic consideration is given to distinct parameters affecting the dynamic process of roll amplification. The influences of heave, pitch, wave passage effect, speed and roll restoring characteristics are discussed.

1 Introduction Parametric resonance of ships in longitudinal regular waves is discussed in the paper, as this is an important particular situation. Results for two typical fishing vessels are compared. Head seas conditions are contemplated in the discussions. Recently, some papers have called attention to the problem of parametric excitation in head seas. (France et al. 2003) reported on strong roll amplification in the case of a container ship. (Dallinga et al. 1998) and (Luth and Dallinga 1999) reported on the development of head seas parametric resonance in cruise vessels. Previous investigations conducted by the authors, Pérez and Sanguinetti (1995), Neves, Pérez and Valerio (1999), indicated that in longitudinal regular waves, at zero speed of advance, resonant parametric amplification may result in large and dangerous rolling motions, particularly for fishing vessels with a transom stern in low metacentric height loading conditions. In the present paper, experimental results for different Froude numbers in head seas are discussed. This is a condition often encountered by fishing vessels, as while trying to maintain position in rough weather. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_12, © Springer Science+Business Media B.V. 2011

231

232 M.A.S. Neves et al.

For the present study, comprehensive testing of two typical fishing vessels has been carried out, aiming at investigating the relevance of parametric resonance of the roll motion for the two ships in head seas under distinct speeds. In the experiments, the first region of resonance was investigated. This is defined as the condition corresponding to the encounter frequency coinciding with twice the roll natural frequency. The two vessels have very similar characteristics but their sterns are different. One is a round stern vessel; the other one has a transom stern. For each hull, two metacentric heights were tested. Additionally, different speeds and wave amplitudes were considered. The transom stern configuration had already shown large roll amplifications in the case of zero speed of advance (Neves et al. 1999). One relevant question that the present paper intends to answer is whether a transom stern arrangement defines strong parametric resonance when different speeds in head seas are considered. Some conditions show intense amplification. The influence of hull stern shape is discussed. Its influence on the dynamics of roll parametric resonance is shown to be relevant when low metacentric height conditions are considered. Influence of damping level is also discussed. For some tuning conditions the transom stern hull may display at high speeds stronger roll amplifications than those registered in the low speed range. Results are interpreted having into account the main terms affecting the energy balance of coupled modes. For this purpose, the distinct influence of the heave and pitch motions are discussed. The paper shows that similar hulls tested in similar conditions may display very distinct responses at specific testing conditions. The experiments clearly demonstrate that in some cases strong parametric resonance in head seas can take place in quite few cycles. It is demonstrated that the effect of speed on parametric resonance is strongly dependent on stern shape. A transom stern, incorporating longitudinal asymmetry in flare, may exert a significant influence in establishing the tendency of a fishing vessel hull to display strong parametric amplification in head seas, particularly in a condition of low metacentric height. These conclusions are relevant in practice for hull design and operational considerations in rough seas. A coupled mathematical model capable of describing roll parametric amplification in waves is introduced. The connection between the coupling terms and parametric excitation is discussed. An assessment is made of the influence of some relevant parameters directly related to practical aspects of ship design for improved stability. Finally, the dynamics of motion around one particular tuning frequency is analysed. Considering steady roll responses around the ωe  2ωn tuning condition for a given reference speed, it is observed that for speeds slightly lower than this reference speed, due to reduced damping and non-linear restoring characteristics of the hull, roll responses are larger than at the exact tuning. It is shown that the roll response curve has a marked asymmetry with respect to the reference speed. Jump effect is identified as a potentially relevant dynamic characteristic to be taken into account in the design of ships subjected to roll parametric amplification.

An Investigation on Head-Sea Parametric Rolling for two Fishing Vessels 233

2 Tested Conditions The tests were conducted at the Ship Model Basin of the Austral University, Valdivia, Chile. The tank main dimensions are: 45.0m of total length, 3.0m of breadth and 1.8m of depth. A flap-type wave generator positioned at one extreme of the tank generates regular waves. More details about the towing tank and experimental arrangements may be encountered in (Neves et al. 1999). Whenever a ship model is to be towed in waves, a major difficulty has to be overcome: to perform the towing of the model without undesirable interference of the towing arrangement with the wave excited bodily motions of the model. This is particularly complicated when large perturbations are associated with the translational average motion of the hull. The satisfactory solution to the problem that was implemented in the present series of tests is illustrated in Fig. 1. Two auxiliary lines were respectively fixed to the bow and stern of the model at calm water level, with these two lines tied to the towing wire. The resulting elasticity of the set was found in all cases to be appropriate in order to secure free evolution of the different (symmetric and anti-symmetric) modes of motion at a controlled speed. In particular, large yaw motions could develop without noticeable interference in practically all runnings.

Fig. 1 Towing arrangement adopted in the experiments

The main characteristics of the ships used in this paper are listed in Table I. Figure 2 shows their lines plans. From Table I and Figure 2 it may be observed that the two hulls have very similar dimensions and characteristics but different stern arrangements. The first one will be called RS. It is a quite typical fishing vessel hull form with conventional round stern. The second hull, which will be called TS, corresponds to a typical transom stern fishing vessel. The tested models were built to a scale 1:30. As mentioned previously, the two hulls had already been tested under parametric resonance in the case of zero speed of advance, (Pérez and Sanguinetti 1995, Neves et al. 1999).

234 M.A.S. Neves et al. Table I Principal Particulars of Ships

Denomination Length (m) Length between perp. (m) Beam (m) Depth (m) Draught (m) Displacement (tons) Waterplane area (m2) Radius of gyration ryy (m)

RS 24.36 21.44 6.71 3.35 2.49 162.6 102.5 5.35

TS 25.91 22.09 6.86 3.35 2.48 170.3 121.0 5.52

Fig. 2 Body plans of tested vessels: RS (left); TS (right)

In the experiments, the first region of parametric resonance was investigated. This is defined as the condition corresponding to the encounter frequency coinciding with twice the roll natural frequency (Kerwin 1955, Paulling and Rosenberg 1959). In other words, the conditions were defined associated with one single encounter

An Investigation on Head-Sea Parametric Rolling for two Fishing Vessels 235

frequency e    kU cos  for each loading condition. For each hull two loading conditions were defined, one with low GM and the other one with high GM. Table II Tested Conditions for RS

GM

Fn

(m)

 (m)

0.10

39.0

0.14

43.2

0.20

49.2

0.34

62.4

0.10

28.7

0.14

32.1

0.20

37.0

0.34

47.8

0.34

0.48

a (m) 0.59 0.69 0.84 0.51 0.80 0.84 0.75 0.87 0.96 0.90 1.02 0.48 0.66 0.90 0.54 0.66 0.90 0.48 0.66 0.90 1.02 1.02

Wave steepness



1/33 1/28 1/23 1/42 1/27 1/26 1/33 1/28 1/26 1/35 1/31 1/30 1/22 1/16 1/30 1/25 1/18 1/39 1/28 1/21 1/18 1/23

(deg.) 7 14 30 3 20 24 5 7 17 0 5 10 20 32 9 18 30 5 16 20 21 0

hw / 

In the above definition,  is the wave frequency, U is the ship speed, k is the wave number and  is the angle of wave incidence with respect to the ship. For head seas,   180 0 . The tested conditions were those defined in Tables II and III. In these tables, wavelength.

a is wave amplitude, hw is the wave height and  is

236 M.A.S. Neves et al. Table III Tested Conditions for TS

GM (m)

Fn



(m) 0.11

31.6

0.15

35.1

0.20

39.4

0.30

47.6

0.11

24.5

0.15

27.5

0.20 0.30

31.2 38.1

0.37

0.50

a (m) 0.30 0.66 0.45 1.02 0.45 0.60 0.60 0.78 0.39 0.63 1.02 0.39 0.60 1.08 1.02 1.02

Wave steepness



1/53 1/24 1/39 1/17 1/44 1/33 1/40 1/31 1/31 1/19 1/12 1/35 1/23 1/13 1/15 1/19

(deg.) 15 27 18 28 4 19 5 38 19 22 27 2 13 16.5 0 0

hw / 

The last columns of Tables II and III present, for each speed and wave amplitude, for the tested conditions corresponding to e  2n (where  n is the natural frequency in the roll mode) the registered values of  (in degrees), the final (steady) roll amplitude observed in each test after all transients had died out. Clearly, for constant encounter frequency, this final roll amplitude is a measure of the level of parametric amplification in each tested condition. In summary, the experimental test programme involved two hulls with distinct stern arrangements to be compared, each vessel with two metacentric heights, varying speeds and wave amplitudes.

3 Equations of Motion A derivative model is adopted. Following the nomenclature previously used in (Neves and Rodríguez 2006) equations of motion in heave, roll and pitch, with hydrostatic and wave passage nonlinearities defined up to third order may be written as:

An Investigation on Head-Sea Parametric Rolling for two Fishing Vessels 237

m  Z z z  Z z z  Z   Z    Z z z  Z    1 Z zz z 2  1 Z   2  1 Z   2  Z z z 

2 2 2 1 1 1 1 1 1 Z zzz z 3  Z zz z 2   Z z  2 z  Z  2   Z z  2 z  Z   3  Z z t z  6 2 2 2 2 6 Z  t   Z z t z  Z zz t z 2  Z  t   Z z t z  Z  t  2  Z  (t ) 2  Z w t 

(1)

J



1  K    K    K      K     3  K    K z z  K    K zz z 2   2 1 1 3 2 K    K     K z z  K  t   K  t   K z t z  K  t   K w (t ) 6 2

xx

(2)

J

1 1 1  M    M    M z z  M z z  M z z  M    M zz z 2  M   2  M   2  2 2 2 1 1 1 1 1 1 3 2 2 2 2 M z z  M zzz z  M zz z   M z  z  M     M z  z  M   3  6 2 2 2 2 6 2 M z t z  M  t   M z t z  M zz t z  M  t   M z t z  M  t  2 

yy

M  (t ) 2  M w t 

(3) On the right hand side of these equations, Z w (t ) , K w (t ) and M w (t ) describe the wave external excitation in the heave, roll and pitch modes, respectively. These wave actions are assumed to linearly related to the wave amplitude, a. In the left hand side of the equations, nonlinear restoring terms include dependence on all body modes ( z ,  ,  ) and wave passage ( ) . Dots refer to velocities; double dots to accelerations. In all modes, coefficients with dotted and double dotted subscripts are damping and added masses coefficients, respectively. A set of linear equations with time-dependent coefficients is obtained when the variational equations are obtained. Defining the motions as:

z (t )  zˆ(t )   (t )  z a cos( e t   z )   (t )  (t )  ˆ(t )   (t )   cos( t   )   (t ) a

e



 (t )  ˆ(t )   (t )   a cos( e t    )   (t )

(4)

 (t ), (t ), (t ) are perturbations superimposed to the steady solutions ˆ ˆ zˆ, , , then, inserting expressions (4) into equations (1), (2) and (3), the

where

equations of motion relative to the perturbed motions become:

238 M.A.S. Neves et al.

1 1 1 (m  Zz )  Zz  Zθ  Zθ  Zz  Z  (Zzz zˆ  Zzˆ  Zzzzzˆ2  Zzˆ2  Zzˆ2  Zzz zˆˆ)  2 2 2 1 1 1 2 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ (Z  Zz zˆ  Z )  (Zz zˆ  Z  Zzz zˆ  Z  Z  Zz zˆˆ)  2 2 2 [Zz (t)  Z z (t)  2Z zz (t)zˆ  Z z (t)ˆ]  Zφ (t)  [Z (t)  Z (t)  Z z (t)zˆ  2Z (t)ˆ]  0

(5)

1 1 (Jx  Kφ )  K  Kˆ  K (Kzˆ  Kzz zˆˆ  Kzˆˆ) (Kz zˆ  Kˆ  Kˆ  Kzz zˆ2  2 2 1 Kθˆ 2  Kz zˆˆ ) (Kˆ  Kˆˆ  Kz zˆˆ)  2 Kz (t) [K (t)  K(t)  K z (t)zˆ  K (t)ˆ ]  Kθ (t)  0 (6) 1 1 1 (Jy  Mθ )  Mθ  Mz Mz  Mz Mθ (Mzz zˆ  Mzθθˆ  Mzzz zˆ2  Mzˆ2  Mzˆ 2 Mzz zˆˆ )  2 2 2 1 1 1 2 2 2 (M  Mz zˆˆ  M θˆ θˆ) (Mz zˆ M θθˆ  Mzz zˆ  Mˆ  Mˆ  Mz zˆˆ )  2 2 2 ˆ [Mz (t) M z (t) 2 M zz(t)zˆ  M z (t) ]  Mφ (t) [M (t) M (t) M z (t)zˆ 2M (tˆ) ]  0

(7) Equations (5 - 7) form a set of time-dependent coupled equations, assumed to regulate, to first approximation, the stability of the dynamical system defined by the non-linear equations defined previously, equations (1 - 3).

4 Roll Instabilization In the case of longitudinal waves, roll motions will not be externally excited, thus

ˆ  0 . The roll variational equation may then be expressed as:

1 1  ( J xx  K  )  K    K   K z zˆ  K  ˆ  K zz zˆ 2  K  ˆ 2  K z zˆˆ  K  t   K z t zˆ  2 2   K  t ˆ  K  t    0 

(8) In equation (8) the terms within brackets are restoring terms displaying the complexity of the parametric excitation, as determined by derivatives and the linear and squared vertical motions. The derivative coefficient K  corresponds to the linear roll restoring moment, that is, K   g . G M ; the derivative

An Investigation on Head-Sea Parametric Rolling for two Fishing Vessels 239

coefficients

K z , K , K zz , K and K z are associated with parametric

excitation due to nonlinear hydrostatic pressure terms, as shown in (Neves and Rodríguez 2006) and are related to derivatives of geometric characteristics of the waterplane area:

K z 

I xx I xx  2 I xx  2 I xx  2 I xx ; K   ; K zz  ; ; K  K   z z  z z 2  2

which, in turn, are dependent on the longitudinal distributions of breadth and flare of the hull. The following expressions have been derived (Neves and Rodríguez 2006):

I xx y  2 y 2 dx z z I y K   xx  2 xy 2 dx  z 2  2 I xx  y  K zz  4 y    z  dx  A0 z 2 K z 

2

K z

 2 I xx  y   4  xy  dx  A0 x f 0  z  z 

K 

 2 I xx  y   4  x 2 y  dx  I yy 0  2   z 

2

The oscillatory coefficients K

t , Kz t , K t 

(9) and

K t  describing

second and third order terms associated with parametric excitation due to nonlinear wave pressure terms are related to longitudinal distributions of breadth and flare of the hull and incident wave characteristics:

K  (t )  2g  y 2 L

y dx z

  y  2  K z (t )  g  4 y    2 y  dx   z  L   2    y  K  (t )  g  4 x y   2 x y dx  z   L 

240 M.A.S. Neves et al.

  y  2  K  (t )  g  2 y    y   2 dx   z   L 

(10)

where   x, t   Aw coskx  et  is the undisturbed wave elevation along the ship hull. More details of the mathematical model may be found in Neves and Rodríguez (2006). Equation (8) is a time-dependent equation describing parametric amplification of the roll motion (to third order), which contains contributions from the heave and pitch motions and the wave passage effect. Considering the harmonic character of the functions zˆ (t ),ˆ(t ) and Hill’s differential equation:

J

xx



 K   K 

K



 ( x, t ) , equation (8) typically corresponds to a



 a 2 R0  a R1C Coset  R1S Sinet   a 2 R2C Cos 2et  R2 S Sin 2et    0 (11)

It is pointed out that the time-varying terms are a compound result of volumetric changes in the submerged part of the hull due to the heave and pitch motions and the cyclic passage of the longitudinal progressive wave along the hull. It is well known that this type of equation may have strong instabilities at encounter frequencies close to ω e  2 ω n , where the roll natural frequency, squared, is defined as: 2

ωn 

Kφ J x  K φ

.

In the following sessions, the influence of speed on this problem will be discussed for the condition ω e  2 ω n and other conditions close to it. A relevant result at this stage is that the coefficients R 0 , R1S , R1C , R 2 S and R 2C which are all defined in terms of expressions (9, 10) define the intensity of the parametric excitation in equation (8) (or its equivalent equation (11)) are all dependent on the longitudinal distributions of sectional flare and breadth. These are analytical results that allow interesting conclusions with respect to the influence of hull form on the occurrence of parametric rolling, as will be discussed ahead. On the other hand, the intensity of the vertical motions are also relevant to the combined results of the terms appearing within brackets in equations (8) and (11). These will be examined in the next section, as they will be helpful for the objective of interpreting the experimental results.

An Investigation on Head-Sea Parametric Rolling for two Fishing Vessels 241

5 Vertical Motions Figures 3 to 6 present the computed heave and pitch linear responses (amplitude and phase) in the frequency domain for the two hulls. The tested frequencies are indicated by vertical dotted lines. It is observed that the heave responses for the RS and TS hulls, Figures 3 and 4, are practically the same, for zero speed and all other speeds tested. As expected, higher responses are obtained for higher speeds. RAO - Heave phase

RAO - Heave Amplitude Ship RS: ksi = 180°

1.20 1.00

Fn = 0.00 Fn = 0.10

135

Fn = 0.14 Fn = 0.20

90

0.80

Phase [deg]

Amplitude [m/m]

Ship RS: ksi = 180°

180

0.60 0.40

Fn = 0.00 Fn = 0.10 Fn = 0.14 Fn = 0.20

45 0 -45 -90

0.20 0.00 0.00

-135 0.50

1.00 1.50 2.00 We [rad/s]

2.50

-180 0.00

3.00

0.50

1.00 1.50 2.00 We [rad/s]

2.50

3.00

Fig. 3 Heave transfer functions, RS hull

For the pitch mode, Figures 5 and 6, some characteristics become apparent. First of all, one notices that for zero speed the two hulls respond in practically identical conditions. As speed increases, the responses become distinct. It may be observed that for the RS hull the pitch amplitudes have a general trend to increase with speed. In the case of the TS hull, there is a much more limited increase of the pitch amplitudes with speed. It is apparent, then, that if in the zero speed case the two hulls would show similar levels of internal excitation, as speed increases the RS hull would have to cope with higher levels of internal excitation due to the vertical motions. RAO - Heave Amplitude Ship TS: ksi= 180°

1.20 1.00

Fn = 0.00 Fn = 0.11 Fn = 0.15 Fn = 0.20

90

0.80 0.60 0.40

45 0 -45 -90

0.20 0.00 0.00

Ship TS : ksi= 180°

135

Fn = 0.15 Fn = 0.20

phase [deg]

Amplitude [m/m]

RAO - Heave phase

180

Fn = 0.00 Fn = 0.11

-135 0.50

1.00

1.50 2.00 We [rad/s]

2.50

3.00

-180 0.00

0.50

1.00

Fig. 4 Heave transfer functions, TS hull

1.50 2.00 We [rad/s]

2.50

3.00

242 M.A.S. Neves et al.

Yet, as will be made clear below, due to hull characteristics, this is not strictly the case. Another relevant aspect is that the influence of speed is effective not only in inducing a compounded effect on the amount of parametric excitation being transferred from the wave excited vertical motions to the rolling motion. It should also be taken into account that as different levels of internal excitation are fed into the dynamic system, there is also a higher level of damping incorporated into the process, its main mechanism being lift damping due to the higher speed. In fact, ship speed tends to (nonlinearly) reduce eddy damping and increase (linearly) lift damping (Himeno 1981). RAO - Pitch Amplitude

RAO - Pitch phase

Ship RS: ksi = 180°

12.00

315 270

8.00 Phase [deg]

Amplitude [deg/m]

10.00

Ship RS: ksi = 180°

360

Fn = 0.00 Fn = 0.10 Fn = 0.14 Fn = 0.20

6.00 4.00

225 180 135 90

2.00 45 0.00 0.00

0.50

1.00

1.50

2.00

2.50

0 0.00

3.00

Fn = 0.00 Fn = 0.10 Fn = 0.14 Fn = 0.20

0.50

1.00 1.50 We [rad/s]

We [rad/s]

2.00

2.50

3.00

Fig. 5 Pitch transfer functions, RS hull RAO - Pitch Amplitude Ship TS: ksi = 180°

12.00 10.00

Fn = 0.20

315

8.00

270

6.00 4.00

225 180 135 90

2.00

45 0.00 0.00

Ship TS: ksi = 180°

360

Phase [deg]

Amplitude [deg/m]

RAO - Pitch phase

Fn = 0.00 Fn = 0.11 Fn = 0.15

0.50

1.00 1.50 We [rad/s]

2.00

2.50

3.00

0 0.00

Fn = 0.00 Fn = 0.11 Fn = 0.15 Fn = 0.20

0.50

1.00 1.50 2.00 We [rad/s]

2.50

3.00

Fig. 6 Pitch transfer functions, TS hull

The net effect of speed on the balance of internal transfer of energy versus damping is not evident at all, and should be in any case carefully investigated. This observation leads to important considerations directly related to practical aspects of ship design for improved stability.

An Investigation on Head-Sea Parametric Rolling for two Fishing Vessels 243

At this stage, it is important to note that the RS hull is less damped than the TS hull. This was demonstrated by the roll decrement tests (conducted for the zero speed case). This may be observed in Figure 7, in which the following nondimensional forms are defined for roll damping and frequency of oscillation, B44

'

respectively: B44 

L   2

5

;   e

L g

.

Fig.7 Equivalent damping at zero speed

6 Results for RS Hull In graphical form, the experimental results for the RS hull given in Table II may be summarized as in Figure 8. The graphs show roll amplitude against wave amplitude at the tested Froude numbers and display all tested conditions. φ

40

φ

RS GMt = 0.34 m

40 RS GMt = 0.48 m

Fn = 0.10

30

Fn = 0.14 Fn = 0.20

30

Fn = 0.10 Fn = 0.14 Fn = 0.20

Fn = 0.34

20

20

10

10

0 0.20

0.40

0.60

a

0.80

1.00

1.20

0 0.20

0.40

0.60

a

0.80

1.00

1.20

Fig. 8 Variation of roll amplitude with wave amplitude for different Fn. RS hull: GM  0.34 m (left); GM  0.48 m (right)

In Figure 8 it is observed that for both values of GM and for each Fn there is in general a linear tendency in the growth of the roll amplitude for larger wave amplitudes. Additionally, this tendency is kept almost the same as larger speeds

244 M.A.S. Neves et al.

are considered. There is a clear tendency, when the same wave amplitude is considered, for roll amplitudes to become smaller at higher speeds. Figure 9 illustrates this tendency. The figure displays time series of roll motion obtained for the RS hull, GM  0.48m for three Froude numbers and a = 0.90m. Clearly, there is a marked reduction in roll amplification as speed increases. It is noted that for this value of metacentric height no roll amplification was observed for Fn  0.34 . 40

(a)

RS GMt=0.48 m Fn=0.10 ha=0.90 m

φ

20

0

-20

-40 0

20

40

60 t[s]

80

100

40

120 (b)

RS GMt=0.48 m Fn=0.14 ha=0.90 m

φ

20

0

-20

-40 0

20

40

60 t[s]

80

100

40 RS

120 (c)

GMt=0.48 m Fn=0.20 ha=0.90 m

φ

20

0

-20

-40 0

20

40

60

80

100

120

t[s]

Fig. 9 Time series of roll motion for RS hull, GM  0.48 m , a  0.90m , (a) Fn = 0.10; (b) Fn = 0.14 and (c) Fn = 0.20, respectively

The implication here is that with increased speed of advance, more damping (mainly lift damping) now acts against a higher level of vertical motions, but resulting in less roll parametric amplification. To help in the assessment of the balance between parametric excitation and damping, Figure 10 shows how internal inside the unstable zone of the stability diagram are the corresponding tested

An Investigation on Head-Sea Parametric Rolling for two Fishing Vessels 245

conditions for Fn = 0.10. The stability diagram is defined here in terms of encounter frequency versus wave height. GMt = 0.48 m

3.0

Fn=0.10

RS We=1.784 [rad/s]

2.5

hw/2 [ m ]

2.0

1.5

1.0

0.5

0.0 0.00

0.40

0.80

1.20

1.60

2.00

We [ r/s ]

Fig. 10 Stability diagram illustrating how internal the experimental points are for Fn = 0.10

Another way of viewing this roll reduction with speed is as shown in Figure 11. In this figure, final (steady) roll amplitudes are plotted against Froude number for one particular wave amplitude. For the lower metacentric height condition, Figure 11 (left), the tested wave amplitudes are not exactly the same for all speeds, thus linear interpolation has been adopted. But the important aspect to be observed from the two graphs is the common tendency for roll motion amplification to attenuate in face of higher speeds, irrespective of the frequency tuning. 40 40

RS GM=0.34m a=0.84m

RS GM=0.48m a=0.90m

30 Phi (deg.)

Phi (deg.)

30

20

10

0 0.00

20

10

0 0.10

Fn

0.20

0.30

0.00

0.10

0.20 Fn

0.30

Fig. 11 Roll amplitudes against Fn for RS hull: [ GM  0.34 m , a  0.84 m , e  1.49 rad / s ] (left); [ GM  0.48 m , a  0.90 m , e  1.78rad / s ](right)

246 M.A.S. Neves et al.

7 Results for TS Hull Consider now the results for TS hull given in Table III. These are shown in graphical form in Figure 12. It may be observed from Fig. 12 (left) that for the lower GM and small Froude numbers there is a tendency for roll angles to change in a similar way as discussed previously to the RS hull. Yet, for higher speeds, the tendency is reversed, and it is observed that very large roll angles are obtained for Fn=0.30.

30

φ

φ 40

TS GMt = 0.37 m

Fn = 0.11 Fn = 0.15 Fn = 0.20 Fn = 0.30

40

Fn = 0.11 Fn = 0.15

30

20

20

10

10

0 0.20

0.40

0.60

TS GMt = 0.50 m

a

0.80

1.00

1.20

0 0.20

0.40

0.60

a

0.80

1.00

1.20

Fig. 12 Variation of roll amplitude with wave amplitude for different Fn. TS hull, GM  0.37 m (left); GM  0.50 m (right)

It may be observed that for the lower GM and small Froude numbers there is a tendency for roll angles to change in a similar way as discussed previously for the RS hull. Yet, for higher speeds, the tendency is reversed, and it is observed that very large roll angles are obtained for Fn=0.30. In fact, considering the complete test program, the largest roll response was obtained for this condition, corresponding to angles of the order of 38 degrees being reached in few cycles, wave amplitude being a=0.78m. The time series corresponding to this impressive TS GMt = 0.37m Fn = 0.30 a = 0.78m

40

20

0

-20

-40 0

40

t[s]

80

120

Fig.13 Roll angle time series for TS hull at Fn=0.30, low GM, a=0.78m

An Investigation on Head-Sea Parametric Rolling for two Fishing Vessels 247

40

40

30

30 Phi (deg.)

Phi (deg.)

parametric roll amplification in head seas is given in Figure 13. This intense resonance for Fn=0.30 is larger than the roll amplitudes registered for zero speed, reported in (Pérez and Sanguinetti 1995, Neves et al. 1999). For zero speed, in less favourable conditions (GM=0.35m and a=0.90m), roll angles of 34 degrees were obtained. Still considering this low GM for TS testing, it is observed that at Fn=0.20 there are intense responses, though not as intense as in the Fn = 0.30 case. For the higher metacentric height, GM=0.50m, a tendency similar to what was observed for the RS hull is back: higher speeds now imply lower amplifications. It may be noticed that for TS hull at this high GM condition, for Fn>0.15 practically no amplification was observed. It may be observed that the situation for TS hull at the low GM condition is completely different from the RS hull responses in the high-speed range. Interpolating in Table III for the lower Froude numbers one gets Fig. 14 (left). When the same procedure is applied in the case of higher GM, for a=1.02m, one gets Figure 14 (right).

20

20

TS GM=0.37m a=0.78m

10

0 0.00

TS GM=0.50m a=1.02m

10

0.10

0.20 Fn

0.30

0 0.00

0.10

0.20 Fn

0.30

Fig. 14 Roll amplitudes against Fn for TS hull: [ GM  0.37 m , a  0.78 m , e  1.72 rad / s ] (left); [ GM  0.50 m , a  1.02 m , e  2.00 rad / s ](right)

It is seen in Figure 14 (left) that for TS hull in the range of low speeds corresponding to 0.11  Fn  0.15, roll amplitudes decay, as was the case with RS hull. But for higher speeds, instead of progressively lower responses, there is now an increase in roll response. In fact, responses are all high, but what is striking is the difference in trend. Now, it is seen that for the higher metacentric height GM=0.50m, Figure 14 (right), there is the same previous trend for roll amplitudes to decay with increased speeds. In fact, for Fn=0.20 and above this value, no amplification was observed for this condition.

248 M.A.S. Neves et al.

8 Summary of Comparisons One notices at first that the TS hull form is in general a more damped hull form than the RS hull, see Figure 7. Secondly, it is noticed that reports in general point out to less parametric amplification in head seas with speed (France et al. 2003). This is the observed trend in the present investigation for the RS hull in the two tested loading conditions. It is also the situation with the TS hull at the high metacentric height condition, GM=0.50m. What would be the explanation for the distinct trend for TS hull for the low GM condition? Given the coupling of the modes and the terms in the Hill’s equation, expressions (8) and equation (11), it may be concluded that the internal transfer of energy from the vertical modes to the roll motion is regulated essentially by hull form parameters such as the longitudinal distribution of local breadth b(x) and dy flare at waterline ( x) . dz 0 It is shown in Figure 15 that TS hull, due to its transom stern configuration displays much larger longitudinal asymmetry in flare distribution than the RS hull. Thus it is clearly a more efficient converter of energy from vertical modes to roll motion. In this context, it may be argued that for such critical dynamic characteristic to take place, metacentric height must be low. If this is not the case, then increased damping at high speeds will prevail against the unstabilizing effect of parametric excitation. In summary, low GM (GM=0.37m) for transom stern TS hull produces intense parametric resonances at all speeds at wave amplitudes of the order of a=0.78m. The other hull, which is less damped, even at a slightly lower metacentric height, GM=0.34m, and larger wave amplitude, a=0.90m, tends to respond less and less for higher speeds. The same TS hull for a high GM condition (GM=0.50m) does not amplify parametric excitation at high speeds, indicating that a transom stern shape together with a low GM has a relevant effect on parametric amplification. 5.00

y-coordinates at water line, RS hull dy/dz at water line, RS hull y-coordinates at water line, TS hull

breadth and flare at water line

4.00

dy/dz at water line, TS hull

3.00

2.00

1.00

0.00 -8.00

-4.00

0.00

Stations

4.00

8.00

12.00

Fig. 15 Longitudinal distribution of breadth and flare at waterline for the two hulls

An Investigation on Head-Sea Parametric Rolling for two Fishing Vessels 249

9 Roll Responses Around a Given Tuning Condition In the preceding sections the responses of roll motion in parametrically excited conditions have been examined. In all cases, for each metacentric height, as speed was modified, the ratio of the encounter frequency to the roll natural frequency remained the same, always corresponding to ω e  2ω n . In practice, it meant that in each case, for higher Froude numbers, longer waves excited the hull in the resonant condition. Different wave amplitudes allowed the analysis of the responses in a vertical sense in the stability diagrams, see Figure 9. In other words, the previous analysis is related to the investigation of speed effect on the tuning ωe  2ωn for a fixed metacentric height condition. It will be the next step to complementarily examine roll responses in parametric resonance when one single wave is considered (defining the tuning ωe  2ωn ) and distinct speeds are simulated, thus considering hull responses above and below one given tuning condition. we/ wn 1.25

1.50

1.75

2.00

2.50

RS GMt=0.48 m Wn=0.89 [rad/s] Fn=0.10

2.00

1.50

a [m] 1

2

3

4

5

6

1.00

0.50

0.00 1.20

1.60

2.00

we [rad/ s]

Fig. 16 Points corresponding to different speeds plotted in the stability diagram for Fn=0.10

In this sense, different speeds give the responses in a horizontal sense in the stability diagrams, see Figure 16. For the present purpose the Fn  0.10 condition will be examined. Thus, the tuning condition reported in Figure 8 (a) will be taken as reference for an assessment of the amplification of roll motion when ship speed is lower or higher than the one defining the exact tuning ωe  2ωn . Clearly, for

250 M.A.S. Neves et al.

the same wave, different speeds will define distinct values for the tuning ratio

ωe . ωn

Considering the limits of stability corresponding to Figure 9, defined in terms of wave amplitude against encounter frequency, points corresponding to different ship speeds in the same wave conditions (wave amplitude and length) will appear as indicated in Figure 16. Numbered points represent six different conditions (speeds) in head seas, in a wave with amplitude a  1.0 m and frequency ω  1.46 rad/s. For clarity, a second horizontal scale is displayed in the upper part of the figure, showing the ratio

ωe . One observes that point number 4 corresponds to the ωn

tuning ωe  2ωn . Time series obtained from numerical integration for constant conditions and varying speeds were analysed and the steady roll amplitudes (after transients) were plotted in a non-dimensional roll diagram in Figure 17, that is, Fn 

U gL

φ versus ka

. Therefore, roll amplitudes corresponding to the numbered points of

Figure 16 may be observed in Figure 17, plotted against a scale of Froude numbers. So, again, point number 4 corresponds to the Fn=0.10 condition, defining the exact tuning. It is seen that for very low speeds, corresponding to large detunings, the result is no roll amplification. Point 1 is in this category. Analogously, point 6 is a case of no roll amplification at the high-speed side. Maximum amplification in the response diagram takes place for Fn=0.07, between points 2 and 3. Roll amplification at the exact tuning is of the order of ¾ of the maximum amplification. There is a marked asymmetry in the response diagram. For speeds corresponding to Fn<0.065 (to the left of point 2) there is the appearance of a jump effect, with abrupt changes in roll amplitude, from large to very small roll amplitudes. On the other hand, the right hand side of the response curve displays a smooth reduction in amplitude as speed increases with corresponding larger detuning. Clearly, for all the other conditions related to the other tuning conditions tested, for instance, Fr=0.14, Fn=0.20, etc, similar curves will be obtained. Time series corresponding to conditions between points 1 and 2 effectively may display instabilities associated with jump phenomenon, as may occur in dynamic systems mathematically represented by the so-called externally excited Duffing equation. Yet, it is pointed out that the excitation in the present case is in effect an internal transfer of energy from the vertical modes to the roll mode, with speed effects as discussed in the previous session on the components of parametric excitation.

An Investigation on Head-Sea Parametric Rolling for two Fishing Vessels 251

RS We(Fn=0.10)=1.784 r/s GMt=0.48m

10 9

2

8

3

7 4

6

φ /ka

5 5

4 3 2 1 0 0.00

1

0.02

0.04

6

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

Fn

Fig. 17 Non-dimensional steady roll responses (

 ka

) for different Froude numbers

10 Conclusions Results of a series of experiments on parametric rolling undertaken for two fishing vessels in head seas have been presented. The first region of Mathieu instability was investigated. The parameters varied in the test programme were: wave amplitude, metacentric height and ship speed. Similar hulls tested in similar conditions displayed very distinct responses at specific testing conditions. The experiments demonstrate that in some cases strong parametric resonances in head seas can take place in quite few cycles. Angles of the order of 38 degrees have been reached for wave conditions often met by fishing vessels at sea. This is indicative that for this type of vessel, head seas conditions may be a source of real risk of ship capsize. A third order coupled non-linear mathematical model in the restoring modes was presented. The time-dependent linear equations defining the essential stability aspects of the nonlinear system were also given. The analytical aspects of the mathematical model were taken as a basis to explore different dynamical characteristics and explain distinct responses obtained in the experiments by the two hulls. The effect of speed on parametric resonance is strongly dependent on stern shape, but there seems to exist a complex interaction with other effects. A transom stern design, usually incorporating longitudinal asymmetry in the flare distribution, may exert a significant influence in establishing the tendency of a fishing vessel hull to display strong parametric amplification in head seas, particularly in a condition of low metacentric height. These conclusions are relevant in practice for hull design and operational considerations in rough seas. If a longitudinal distribution of flare is considered as

252 M.A.S. Neves et al.

a design solution, then proper account of the potentially higher parametric excitation should be given. It is pointed out that, as shown in the analysis, an increase in metacentric height is a possible via of solution to the problem, but this is not always available for some vessel types. This is clearly the situation with typical fishing vessels. Finally, considering steady roll responses around a given speed corresponding to the ωe  2ωn condition, it is observed that for speeds slightly lower than this reference speed, due to reduced damping and non-linear restoring characteristics of the hull, roll responses are larger than at the exact tuning. The response curve has a marked asymmetry with respect to the reference speed. Jump effect may be an associated dynamic feature in a roll amplification phenomenon due to parametric resonance.

Acknowledgement This investigation was partially supported by CNPq and FAPERJ of Brazil and CONICYT of Chile. The authors express their thanks for this financial support.

References Dallinga RP, Blok JJ, Luth HR (1998) Excessive rolling of cruise ships in head and following waves. RINA Int Conf on Ship Motions & Manoeuvrability. London. Feb France WN, Levadou M, Treakle TW, Paulling JR, Michel RK, Moore C (2001) An investigation of head-sea parametric rolling and its influence on container lashing systems. Mar Technol 2003 40:1 1-19 Himeno Y (1981) Prediction of ship roll damping - state of art. Dep of Nav Archit and Mar Eng. The Univ of Michigan 239 Kerwin JE (1955) Notes on rolling in longitudinal waves. Int Shipbuild Prog 16 Luth HR, Dallinga RP (1999) Prediction of excessive rolling of cruise vessels in head and following waves. Proc of PRADS Conf Neves MAS, Pérez NA, Valerio L (1999) Stability of small fishing vessels in longitudinal waves. Ocean Eng 26: 12 Neves MAS, Rodríguez CA (2006) On unstable ship motions resulting from strong non-linear coupling. Ocean Eng 33: 14-15 1853-1883 Paulling JR, Rosenberg RM (1959) On unstable ship motions resulting from nonlinear coupling. J of Ship Res 3: 1 Pérez NA, Sanguinetti CFO (1995) Experimental results of parametric resonance phenomenon of roll motion in longitudinal waves for small fishing vessels. Int Shipbuild Prog 42: 431 Sept

Simple Analytical Criteria for Parametric Rolling K.J. Spyrou School of Naval Architecture and Marine Engineering, National Technical University of Athens, 9 Iroon Polytechneiou, Zographos, Athens 157 80, Greece

Abstract Analytical design criteria are sought for predicting the parametric rolling behaviour of ships, taking into account: the growth rate during the inception stage of parametric rolling; and the steady amplitude that is reached when regular parametric excitation is assumed. The connection with the shape of the restoring curve and the effect of roll damping are shown. The results of the current deterministic treatment are transferred to the domain of probabilistic analysis by use of the concept of wave groups which seems to provide a powerful basis for criteria development.

1 Introduction The phenomenon of parametric rolling is known for at least half a century (Grim 1952; Kerwin 1995; Arndt & Roden 1958; Pauling & Rosenbersg 1959). Nonetheless, no specific design requirements referring to parametric rolling have found their way yet into the IMO stability regulations. A plausible explanation is that, whilst it is often the cause of intensive rolling, it is rarely documented to lurk behind a specific capsize accident. However, since the “nineties” the shipping world has become alert to this problem, realising that such exotic and in principle ‘non capsizal’ instabilities could incur tremendous effects in terms of loss or damage of property and business interruption (Gray 2001; Tinslay 2003). A classification society took the lead and published a guide for the parametric rolling of containerships (ABS 2004). By-and-large, the dynamics of the parametric rolling of ships should be considered in our days to be well understood (e.g Spyrou 2000, Neves 2002, Bulian et al 2003; Shin et al 2004). Nonetheless, some modern ship hull-forms, like post panamax containerships and probably some of the new large passenger ships, that are characterised by their heavily flared bow and flat stern with wide transom, may be sailing without having examined their tendency for parametric rolling in a longitudinal seaway. Besides issues of ship design and/or operation, currently, there is M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_13, © Springer Science+Business Media B.V. 2011

253

254 K.J. Spyrou

also discussion about the effectiveness of our physical model testing techniques for verifying that a hull form does not show tendency for parametric rolling (Belenky et al 2003). Whilst these issues should not be regarded as separate, here the focus will be set only on the question of design criteria. Specifically, we intend to put forward a new concept for assessing, at an early design stage, the tendency of a ship for parametric rolling, which combines the deterministic and probabilistic sides of the problem and encompasses the following three principles: Vulnerability:  The probability of exhibiting parametric rolling due to the encounter of a dangerous wave group should be kept lower than the acceptable level. Assessment of Post-Critical Behaviour: 

The amplitude of parametric rolling should not exceed a predefined, ship specific, limit of safe operation. Abrupt growth of roll within a small number of critical wave encounters should not be allowed.



We shall expand on these three principles, taking the last one first:

2 A Transient Criterion for Initial Parametric Roll Growth It can be shown that for a Mathieu system the unstable motion for exact resonance in the first region of instability builds-up according to the following approximate general law (Hayashi 1985):

 t   c1e  0 t sin  0 t     c 2 e   0 t sin  0 t   

where a 

4 02

 e2

(1)

(  0 is the natural roll frequency and  e is the wave encounter

frequency) and h

isthe scaled amplitude of GM variation. Also,  ,  are

functions of a , h determined, to first-order, from the expressions:

cos 2 

a 2 h 2  4a  12 2 a  1  (     0 ),    ah 4 2

(2)

Simple Analytical Criteria for Parametric Rolling 255

At a  1 the coefficient  obtains its maximum value  max   h 4 where    4 . From (1) and after substitution of the initial conditions  0    0 and  0   0 , we can extract the growth of amplitude after p roll cycles:

  p T0  e  0

p h 2

e 2



p h 2

(3)

In the presence of linear damping the growth rate is reduced:

 T0   wT0  e



h

2k

0



e

2

e 2



h 2



e





2k

0

4k  h 2  0

w0 

e

4k  h 2  0

   

e 2





4k  h 2  0

   

0

(4)

   

From the exponential term e we can extract the well-known condition of stability (that rules out the possibility of motion growth):

hcrit 

4k

(5)

0

The above is a condition of asymptotic stability; i.e. in principle a very long sequence of tuned waves having the ‘right’ height is required for instability. Apparently, there is an advantage in exploiting equation (4), which corresponds to transient rolling, rather than (5), which refers to asymptotic behaviour and leads to an unnecessarily more stringent requirement. On the basis of (4), roll growth at exact resonance, ( a  1 ) after p roll cycles, should be predicted from the following expression:

  pT0   e



2 pk

0

p h  p h  e 2  e 2  2  

   0  

(6)

256 K.J. Spyrou

A q-fold increase of roll amplitude from some initial angle value should entail p roll cycles:

ln q  

2p k

0

p h  p h  e 2  e 2  ln  2  

    

(7)

Since, after one or two roll cycles the exponential term of (7) with positive sign becomes dominant, the above may be written further, approximately, as:

ln q  

2p k

0



p h p  ln 2  2 2

 4k h    0 

   0.693  

(8)

It accrues that the number of cycles p required for a q-fold increase of amplitude may be obtained from the expression:

h

4k

0



0.693  ln q 1.571 p

(9)

The corresponding time t m should be p times the natural period T0 :

t q  p T0

(10)

To demonstrate the value of (9), let us think in terms of the following criterion: a 10-fold increase of roll amplitude should not come about in less than 4 roll cycles (which means 8 wave encounters – it is possible to link this to the probability of encountering a dangerous wave group). Let us for example consider waves with  L  1.0, H   1 20 . These translate into the following requirement for h, k :

h

4k

0



0.693  ln 10 3   0.477 1.57  4 6.282

(11)

In summary, by focusing on transient response, one can determine the critical parametric amplitude h , or equivalently the critical damping, for any wave group run length, thus achieving a meaningful interface with the probabilistic nature of

Simple Analytical Criteria for Parametric Rolling 257

ocean waves. The condition of asymptotic stability is recovered from (9) by setting p   . The criterion should be probably supplemented by a time requirement based on (10). For example, the 4 roll cycles should take 25.7  4  102.8 s. For very low natural frequencies, the required time becomes excessive. This leaves plenty of time for reaction (i.e. change of speed or heading) if the initiation of the phenomenon is promptly recognised.

3 Roll Growth Ends Nonlinearly on a Limit Cycle As is well known, there is no reason for this growth to persist up to infinity and thus lead by necessity to capsize. The detuning due to the nonlinear character of the GZ curve together with the increased dissipation due to the mild nonlinearity of damping, create the prospect of capture into steady oscillatory rolling with moderate amplitude. In effect, for a typical parametric growth with nonlinear restoring the boundary curves of stability discussed earlier represent loci of bifurcations giving birth to stable and unstable oscillatory behaviour (see for example Skalak & Yarymovych 1960; Soliman & Thompson 1992). The instability boundary curves of the upright state of a ship do not contain entirely the domain where parametric oscillations are realisable. The emerging stable roll oscillations need not be confined inside the “tongues” of the linear system and stable oscillations exist also well outside these regions (Scalak & Yarymovich 1961, Thompson & Soliman 1993). Should anyone worry about these nonlinear oscillations whose domain of existence extends outside the “tongues” of the linear parametric oscillator? The answer is, probably yes. In an idealised environment of a periodic seaway that is free from other external disturbances, a ship should find no dynamical reason for leaving the upright state as long as the combination of frequency ratio and parametric amplitude corresponds to some point in the region of stability. However, should the stable upright condition be sufficiently disturbed, this oscillatory behaviour can be incurred in an abrupt way. It may be said of course that, by moving away from the condition of exact resonance, the probability of occurrence of parametric rolling decreases. However, it is doubted whether it is safe to assume that this probability is sufficiently low so that the phenomenon can be completely ignored. Perhaps one should place less emphasis on the necessity of fulfilling the condition of exact principal resonance for the occurrence of parametric rolling. To explain these points further, let us consider a Mathieu-type roll equation with a single, cubic nonlinear term:

258 K.J. Spyrou

  2k    02 1  h cos e t   n 02  3  0

(12)

The constant n could be negative, in which case we are examining the oscillations corresponding to the initial part of the GZ   curve which is often of “hardening spring” type. Or we could consider it as positive, in which case we refer generically to the entire GZ   curve up to the angle of vanishing stability. Application of a perturbation method like harmonic balance or averaging leads to the following explicit formula for the amplitude:

A2 

4  1  1    3 n  a  

h 2 4k 2  4 a 02

   

(13)

Setting A  0 we find the curve whereon the oscillations are created. It comes to no surprise that this curve is independent of the nonlinear coefficient n and it coincides with the boundary of linear stability. Also, the term inside the square root, as well as the whole expression of A 2 , should be non-negative. For an initially hardening restoring ( n  0 ) these yield,

h

4k

0 a

and a  1

(14)

In essence, (14) defines the locus of “saddle-node” (or so-called “fold”) bifurcations. On this curve, the unstable periodic states that are shed from the left boundary of the instability region undergo a U turn and become stable (this can be understood by considering that, for negative n , at the ‘lower a ’ boundary of the instability region a subcritical bifurcation takes place, creating unstable oscillations; whereas the boundary at a  1 is a supercritical one i.e. stable oscillations are created). Condition (14) determines the true boundary of periodic response. For a certain level of h , the region where stable oscillations are encountered is wider than the predicted from the linear analysis. In Fig. 1 is seen the development of steady roll 4 amplitude as function of a and h ( A  A *  ) for a container with the 3n dimensions of APL China and  0  0.2448 s 1 , k  0.015 s . The stability of the emerging steady roll oscillations is indicated (dashed line represents unstable states).

Simple Analytical Criteria for Parametric Rolling 259

A* 0.8 0.8 0.9

0.6

1.1 1.2

0.4 0.2 0 0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 h

Fig. 1 Amplitude of response for hardening restoring

The domain of oscillatory behaviour (bounded by the thick continuous line) can easily be found with some manipulation: 2

 3n 16k 2  1  h  4  A 2  1     2 0 a  a  4

(15)

The combinations of h, a  that give rise to oscillations of predefined A* is shown in Fig. 2. It is not difficult to prove that the descending part of each iso- A* curve corresponds to stable rolling and the ascending to unstable. The boundary of stable rolling is reconfirmed. As we have multiple coexisting stable responses, the initial conditions and the availability of sufficiently strong external disturbances determine whether the ship can stay upright; or she should adopt the one (desired) or the other (undesired and possibly dangerous) way of behaviour. In Fig. 3 is shown the variation of the roll amplitude A for n=-2.35, as a function of the linear damping and the amplitude of parametric forcing. For small (yet realistic for several operating ships) damping, the effect on the response amplitude is relatively small. The same applies for h. A more influential parameter is the coefficient of the cubic stiffness term n which, at first approximation, is linear to the response amplitude.

260 K.J. Spyrou

1

0.7

h

0.6 0.8

0.5

0.6

0.4

0.4

0.3 0.2 0.1 h = 4k/(w0 a) 0.6

0.7

0.8

0.9

1

1.1

a

Fig. 2 Iso-A curves (from 0.1 to 0.7) for k=0.015, n=-2.35 (hardening),

1.2

 0 = 0.2448 s-1

A

/

1 2

0.8

x

1x

0.6 2x 0.4 3x 0.2 0 0

0.01

0.02

0.03

0.04

k

0.05

Fig. 3 Effect of damping on the amplitude of periodic response with parameter the coefficient of nonlinear stiffness (“hardening”)

3 Effect of the Fifth Order Term (Initially Hardening, then Softening): Consider again the roll equation, this time with a fifth-order polynomial for restoring which can take better into account the details of the GZ curve up to large inclinations:

Simple Analytical Criteria for Parametric Rolling 261

d 2 d

2



2 k a d  a 1  h cos 2   na  3  ma 5  0  0 d

(16)

The approximate steady-state solution is:

A2  

2 1 3 n 8  3 n  1       5m 5m a 5 m 

h 2 4k 2  4  02 a

   

(17)

We select a GZ curve (see Fig. 4) very close to that of the post-panamax containership of France et al (2001). The selected values for the coefficients n and m are, respectively, -0.14 and 0.25. The amplitudes as functions of the parametric term h, for the frequency ratios examined earlier, are shown in Fig. 5. Several changes of stability are noted on each one of these curves. An interesting feature of this diagram is that it shows the behaviour at large angles where the fifth order term of the restoring function becomes influential. Contrast of Fig. 5 versus Fig. 1 is enlightening in this respect.

4 Requirement of Limited Amplitude of Steady Parametric Rolling A supplementary criterion based on the steady roll amplitude may thus be introduced at this stage, as a second requirement concerning post-critical behaviour: For a critical wave, say with  L  1.0, H   1 20 , the max roll amplitude should not exceed, say, 15 deg (this value is proposed by ABS; perhaps this value can be vessel specific depending e.g. on lashings’ strength). The combination of restoring and damping coefficients that satisfy the requirement of no exceedance of this limiting angle can be obtained with some manipulation of (13); or of (17) if judged that the later part of the GZ curve should also be considered (for a 15 deg limit it is unlikely this to be necessary).

262 K.J. Spyrou GZ(j) 1.5 1.25 1 0.75 0.5 0.25

0.2

0.6

0.4

0.8

j

1

Fig. 4 The exact (dots) and the polynomial fit (line) of the considered GZ curve.

A 1 0.8 0.6

1.1

0.4

1.2 0.9

0.2 0.8 0 0.3

0.4

0.5

0.6

0.7

0.8

h Fig. 5 Amplitude of roll oscillation for 5th order restoring, assuming time-dependence only for the linear term. The different curves are according to the value of the frequency parameter a.

5 Interfacing with the Probabilistic Seaway: The Key Is in the Groupiness of High Waves It is well known that higher waves tend to arise in groups. As the nearly regular characteristics of waves in a group are essential for giving rise to fundamentally resonant motions like parametric rolling, there is a meaningful link between the

Simple Analytical Criteria for Parametric Rolling 263

probabilistic nature of ocean weaves and the deterministic analysis. The probability of occurrence of parametric rolling could be assumed to be equal to the probability of encountering longitudinally a wave group with sufficient run length and exceeding the threshold height (determined from the deterministic analysis), given that the frequency falls in the critical range (which however has to be very wide). One may use theoretical or parametric models for joint distributions of wave parameters. In practical terms one has to make certain assumptions about the correlation of key parameters in a wave group and opt to use available bivariate distributions either of wave height and period, or of successive wave periods, or finally of successive wave heights. Ocean wave statistics suggest that height H and period T are correlated (e.g. Longuet-Higgins 1975). The following bivariate distribution proposed by Longuetm m  m2 Higgins (1983) is based on the spectral width parameter   0 2 2 1 (which m1 has the advantage to depend only on the first three moments):



p H , T  

 f    H *  4

where H * 

 H*

2

  e  T* 

H H

, T* 

4

2

2   1 1  T*  1  2  

T , Tz

f   

      

(18)



2 1  2







is the spectral width

1  2 parameter, m 0 , m1 , m 2 are respectively zeroth, first and second moment of the wave spectrum, H is the mean wave height; and Tz is the mean zero-upcrossing period that is determined from spectral moments, T z  2

m0 . m2

The use of joint distributions of successive wave periods for assessing probabilistically resonant ship rolling in beam seas was investigated by Myrhaug et al. (2000). However, unlike parametric rolling, the wave frequency where resonance occurs is well defined in a beam sea because the speed of the ship does not influence the encounter frequency. Moreover, to be initiated, it does not require a threshold wave height, like in the case of parametric rolling. Hence, an approach based on the statistics of successive wave heights, i.e. the condition of having a wave group with heights exceeding a known critical level, seems to be more relevant for the investigation of parametric rolling. The probability to encounter a sequence of waves with height above the critical level H c was considered by Blocki (1980) using the approach of Goda (1976). He

264 K.J. Spyrou

had assumed the occurrence of successive heights above H c as independent events. This means that practically, the probability of encountering a certain run length was underestimated. The degree of correlation of successive wave weights depends on the sharpness of the spectral peak. For the effect of the spectral bandwidth on the distribution of wave height see for example Kimura (1980), Tayfun (1983) and Longuet-Higgins (1984). Stansell et al (2002) found that, as bandwidth increases, there is a rather slight reduction in the mean run and group length, up to a bandwidth   0.6 beyond which they become rather insensitive (to obtain a sense of magnitude we note that   0.425 for a Pierson Moskowitz and   0.389 for a JONSWAP spectrum). According to Tayfun, the sharpness of the spectral peak reflects the variability of height between successive waves; and spectral peakedness is best represented by the correlation coefficient of the wave envelope R HH which could be calculated as follows:

R HH 



E    1   2 1

 K2   4



4



4 6    2      16  4  16 64   



(19)

E  , K   are complete elliptic integrals, respectively of the first and second kind. The correlation parameter  could be calculated as follows (see Stansell et al. 2002 for an extensive discussion on alternative methods of calculation):

 T  

1 m0

A2  B 2 , A 





0

E  f  cos 2f T df , B 





0

E  f  sin 2f T df (20)

Goda (1976) has found that for swells the correlation coefficient R HH is about 0.6 while for wind waves it is only about 0.2. Assuming that successive wave heights follow a Rayleigh distribution, Kimura (1980) derived the following bivariate probability density function pH 1 , H 2  for consecutive wave heights:

p HH H 1 , H 2  



4H 1 H 2

1   H 2

2 rms

e

H  H  1 H 2 1 2

2 2 2 rms

 2  H1 H 2 I0  1 2 H 2 rms 





   

(21)

where H rms is the root mean square wave height and I 0 is the modified Bessel function of zeroth order. The probability of having two consecutive wave heights above the critical height H c will be then:

Simple Analytical Criteria for Parametric Rolling 265

PH i 1  H c H i  H c  =



 Hc





p HH H 1 , H 2  dH 1 dH 2

Hc





Hc

p H H  dH

(22)

where p H H  is the marginal probability density which is Rayleigh type:

p H H  

2H 2 H rms



e

H2 2 H rms

(23)

The assumption of Markov chain for successive wave heights leads to the following probability function for the occurrence of a group with length j and peaks higher than H c which is in fact the probability of occurrence of parametric rolling:

Ppr  P j 1 1  P 

(24)

The above value should be multiplied by a factor indicating whether the speed range of the ship produces encounter frequencies that overlap with the frequencies of principal resonance.

6 Concluding Remarks The transient growth of parametric roll could be regarded as the result of a ship’s encounter with some wave group having suitable characteristics to incite such behaviour in a realistic seaway. Then, the probability of occurrence of parametric roll could be obtained from the probability of encountering longitudinally a wave group with sufficient run length and exceeding the threshold wave height, as determined from the deterministic analysis, given that the frequency falls in the critical range. One may exploit the well developed literature of theoretical or parametric models for the joint distributions of wave parameters of a Gaussian sea. Such a framework seems to be physically meaningful enough and practical for criteria development.

References ABS 2004 Guide for the parametric roll for the design of container carriers (draft), Houston, USA, 40 pages. Arndt B and Roden, S. 1958 Stabilität bei vor- und achterlichen Seegang. Schiffstechnik, 5 (29), 192-199.

266 K.J. Spyrou Belenky VL, Weems KM, Lin WM, Paulling JR (2003) Probabilistic analysis of roll parametric resonance in head seas. Proc, 8th Int Conf on the Stab of Ships and Ocean Veh, Madrid, 325340. Bulian G, Francescutto A and Lugni C (2003) On the nonlinear modeling of parametric rolling in regular and irregular waves, Proc, 8th Int Conf, Stab of Ships and Ocean Veh, 305-324, Madrid. Blocki W (1980) Ship safety in connection with parametric resonance of the roll. Int Shipbuilding Prog, 27, 36-53. Esparza I and Falzarano JM (1993) Nonlinear rolling motion of a statically biased ship under the effect of external and parametric excitations. Proc, Symposium on Dyn and Vibration of Time-Varying Systems, OE, 56, ASME, 111-122. Francescutto A and Dessi (2001) Some remarks on the excitation of parametric rolling in nonlinear modeling. Proc, 5th Int Workshop, Stab and Oper Safety of Ships, Trieste, 4.9.1-4.9.8. Goda Y (1976) On wave groups. Proc, BOSS’76, Vol. 1, 115-128. Gray M (2001) Rolling case for more research, Lloyd’s List (Section: Insight and Opinion), February 18, London. Grim O (1952) Rollschwingungen, Stabilität und Sicherheit im Seegang. Schiffstechnik, 1 (1), 10-21. Hayashi C (1985) Nonlinear Oscillations in Physical Systems, Princeton Univ Press, ISBN 0-691-08383-5, Princeton, New Jersey. Kerwin JE (1955) Notes on rolling in longitudinal waves. Int Shipbuilding Prog, 2 (16) 597-614. Kimura A (1980) Statistical properties of random wave groups. Proc, 1st Coastal Eng Conf, Vol. 2, 2955-2973. Longuet-Higgins MS (1975) On the joint distribution of the periods and amplitudes of sea waves. J of Geophys Res, 80, 6778-6789. Longuet-Higgins MS (1983) On the joint distribution of wave period and amplitudes in a random wave field. Proc of the Royal Soc, London, Series A, 310, 219-250. Longuet-Higgins MS (1984) Statistical properties of wave groups in a random sea state. Proc of the Royal Soc, London, Series A, 389, 241-258. Myrhaug D, Dahle EA, Slaattelid OH (2000) Statistics of successive wave periods with application to rolling of ships. Int Shipbuilding Prog, 47, 451, 253-266. Neves M (2002) On the excitation of combination modes associated with parametric resonance in waves. Proc, 6th Int Ship Stab Workshop, Webb Inst, Long Island. Paulling JR and Rosenberg RM (1959) On unstable ship motions resulting from nonlinear coupling. J of Ship Res, 3, 1, 36-46. Shin YS, Belenky VL, Paulling JR, Weems KM, Lin WM (2004) Criteria for parametric rolling of large containerships in longitudinal seas, SNAME Annual Meeting, Sept 30 – October 1, Washington D.C. (early copy). Skalak R and Yarymovych MI (1960) Subharmonic oscillations of a pendulum. J of Appl Mech, 27, 159-164. Soliman M and Thompson JMT (1992) Indeterminate sub-critical bifurcations in parametric resonance. Proc of the Royal Soc of London, Series A, 438, 433-615. Spyrou KJ (2000) Designing against parametric instability in following seas, Ocean Eng, 27, 625-653. Stansell P, Wolfram J and Linfoot B (2002) Statistics of wave groups measured in the northern North Sea: comparisons between time series and spectral predictions. Appl Ocean Res, 24, 91-106. Tayfun A (1983) Effects of spectrum bandwidth on the distribution of wave heights and periods. Ocean Eng, 10, 107-118. Tinsley, D. 2003 DNV Project thinks about the box. Lloyd’s List (Section: Shipbuilding and Shiprepair) . November 18, London.

Experimental Study on Parametric Roll of a Post-Panamax Containership in Short-Crested Irregular Waves Hirotada Hashimoto*, Naoya Umeda* and Akihiko Matsuda** *Osaka University, 2-1, Yamadaoka, Suita, Osaka, 565-0871, Japan **National Research Institute of Fisheries Engineering,7620-7, Hasaki, Kamisu, Ibaraki, 314-0408, Japan

Abstract Free running model experiments of a post-Panamax containership were conducted to examine the danger of parametric roll in regular waves, long-crested and shortcrested irregular waves. As a result, parametric roll in head and bow waves were clearly recorded even in long-crested and short-crested irregular waves. From the experimental results, the effects of wave steepness, forward velocity, heading angle, irregularity of waves on parametric roll were systematically examined.

1 Introduction Recent accidents of a post-Panamax containership due to parametric roll in head waves (France et al, 2003) forced the International Maritime Organization (IMO) to start to revise the Intact Stability Code (2002) and the guidance for the master in following and quartering waves (MSC Circ. 707, 1995). Although several experimental data were published for parametric roll even in irregular waves (Umeda et al, 1995), only limited outcomes are available for that in head and bow waves. In this revision work at the IMO, the following questions were now raised to be urgently solved; does the parametric roll in head waves lead to not only the cargo damage but also ship capsize, does the danger of parametric roll decrease in irregular waves, especially in short-crested waves, in comparison with an ideal regular waves, and how can we avoid and predict parametric roll in realistic head or bow waves? Responding to this situation, a free running model experiment mainly focusing on irregular head and bow waves was conducted to directly obtain the answers for the above questions. From the experimental results, the effect of wave height on parametric roll in regular waves, the effects of heading angle and forward velocity on parametric roll in regular waves, the effect of irregularity of waves on parametric roll in head waves, the effects of heading angle and forward velocity on parametric roll in short-crested irregular waves were systematically examined. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_14, © Springer Science+Business Media B.V. 2011

267

268 H. Hashimoto et al.

2 Free Running Model Experiment 2.1 Outline of Model Experiments A free running model experiment was conducted in the basin of National Research Institute of Fisheries Engineering (length: 60m, width: 25m, depth: 3.2m) with an 80-segmented wave maker. A scaled ship model of a post-Panamax containership was used, and her principal particulars and body plan are shown in Table.1 and Figure 1, respectively. The ship model had no deck house, but was watertight, and was propelled by an electric motor with a constant revolution control system. An onboard computer realised an autopilot system of the constant gain of 1.0 and stored data of roll, pitch and yaw motions obtained by a fiber-optic gyroscope. In addition, surge and sway velocity were obtained from data of ship position recorded by an optical tracking sensor fixed to the basin. In regular wave runs, several sets of propeller revolution number, nc, wave height, H, wave length, , autopilot course, c, were used but the wave length to ship length ratio was fixed to be 1.6. Furthermore vertical displacement of incident waves at the centre of ship gravity was obtained in consideration of wave phase velocity, ship position and wave height records. In long-crested irregular wave runs, significant wave height H1/3 of 0.221m in model scale and mean period T01 of 1.3s, which correspond to the wave height and period of regular waves, were used with the ITTC spectrum. In short-crested wave runs, H1/3 and T01 were the same as the long-crested irregular wave runs, but with cosine to the 2nd or 4th power as the directional distributions function were used. By utilising the single summation method for generating wave signals, uniformity of short-crested waves Table 1. Principal particulars of the subject ship Item length between perpendiculars: L

Ship

Model

283.8m

2.838m

breadth: B

42.8m

0.428m

depth: D

24.0m

0.24m

draught at FP: Tf

14.0m

0.14m

mean draught: T

14.0m

0.14m

draught at AP: Ta

14.0m

0.14m

block coefficient: Cb

0.630

0.630

longitudinal position of centre of

5.74m

0.0574m

pitch radius of gyration: Kyy/Lpp

0.258

gravity from the midship: XCG

aft

Aft

metacentric height: GM

1.06m

0.0106m

natural roll period: T

30.3s

natural pitch period: T

3.20s 0.86s

Experimental Study on Parametric Roll of a Post-Panamax Containership 269

in space is realised within 5% error (Sera and Umeda, 2000). In case of the irregular wave, model runs were repeated so that number of encounter waves is about 150 in average. A photograph of the model runs in short-crested irregular waves is shown in Figure 2 as an example.

Fig. 1 Body plan of the subject ship

Fig. 2 A photograph of the free running model experiment in short-crested head waves

2.2 Experimental Results Examples of measured time history of head sea parametric roll in regular waves, long-crested irregular waves and short-crested irregular waves are shown in Figs. 3-5. Here propeller revolution number is the same and H1/3 and T01 are corresponding to the wave height and period of regular waves. In the Fig.3, parametric roll with its steady amplitude of about 18 degrees can be clearly recorded after 10 seconds. In long-crested irregular waves, parametric roll was also observed with the maximum roll angle of about 20 degrees as shown in Fig.4. Even in short-crested waves shown in Fig. 5, parametric roll does not disappear and the maximum roll angle of 22 degrees can be found. These results demonstrate that wave irregularities or short-crestedness does not exclude danger of parametric roll in head waves.

270 H. Hashimoto et al. 30

angle (degrees)

20 10 0 0

10

20

30

40

50

60

roll pitch

-10 -20 -30

time (s)

Fig. 3 Time history of parametric roll in regular waves (H/=1/20.6, /L=1.6, c=180 degrees and nc=15rps) 30

angle (degrees)

20 10 0 0

20

40

60

80

100

120

roll pitch

-10 -20 -30

time (s)

Fig. 4 Time history of parametric roll in long-crested irregular waves (H1/3=0.221m, =1.32s, c=180 degrees and nc=15rps) 30

angle (degrees)

20 10 0 0

20

40

60

80

100

120

roll pitch

-10 -20 -30

time (s)

Fig. 5 Time history of parametric roll in short-crested irregular waves (H1/3=0.221m, =1.32s, cos2 distribution, c=180 degrees and nc=15rps)

Experimental Study on Parametric Roll of a Post-Panamax Containership 271

The effect of wave height on parametric roll in regular head waves is shown in Fig. 6. Here, wave length to ship length ratio is 1.6 and propeller revolution number was set to realise ship forward velocity satisfying the ratio of 2.0 between encounter frequency and roll natural frequency. Parametric roll can be found even in small wave steepness. With increase of wave steepness, parametric roll with its amplitude around 20 degrees was observed at H/=0.0176, and the amplitude does not significantly decrease up to H/=0.0528. Finally parametric roll disappears at H/=0.0726. This can be explained that a ship condition deviates from a parametric roll condition with change in mean of restoring variation. Because H/ of 0.072 corresponds to the wave height of 32m in full scale and the maximum roll angle remains about 23 degrees, it is difficult to find direct relationship between the observed parametric roll and capsizing. The effects of heading angle and forward velocity on parametric roll in regular waves are shown in Fig. 7. Here, the heading angle of 180 degrees means a pure head sea condition, and 270 degrees does a beam sea condition. The Froude number is estimated with the speed loss due to wave taken into account. In head waves, several runs of parametric roll with its maximum roll angle of about 20 degrees were found in smaller Froude number. At the Froude number of 0.1, parametric roll suddenly disappears. When the heading angle becomes larger the maximum roll angle is smaller than the case of 180 degrees of heading angle, but parametric roll itself does not disappear even in relatively high forward velocity while it does in head waves. This is because the area that satisfies the parametric roll condition shifts to higher forward velocity with increase of heading angle. However, if we focus on severe parametric roll whose maximum angle is larger than 15 degrees, the danger decreases with increase of heading angle. The effects of irregularity of incident waves on parametric roll in head waves are shown in Fig. 8. In cases of smaller forward velocity, roll angles both in longcrested and short-crested irregular waves are slightly smaller than that in regular waves. Here some statistical fluctuation due to non-ergodicity (Belenky, 2004) can be found. When the Froude number is larger than 0.1, however, parametric roll in irregular waves still occurs while no parametric roll occurs in regular waves. This tendency is more conspicuous in the case of short-crested irregular waves. The effect of heading angle on the maximum roll angle of parametric roll in short-crested irregular waves is shown in Fig. 9. Here cosine to the 2nd power was used as the directional distribution function. This figure shows that the ship cannot avoid parametric roll only by changing its course in short-crested irregular waves, and also shows that increasing the speed is effective to reduce danger of parametric roll at least in the heading angle ranging from 180 degrees to 240 degrees.

272 H. Hashimoto et al.

amplitude of parametric roll (deg.)

25 20 15 10 5 0 0

0.02

0.04

0.06

0.08

wave steepness

Fig. 6 Effect of wave height on parametric roll in regular waves (/L=1.6 and c=180 degrees) amplitude of parametric roll (deg.)

25 180 deg. 203 deg.

20

225 deg. 15

248 deg. 270 deg.

10 5 0 0

0.05

0.1

0.15

0.2

0.25

0.3

Froude number

Fig. 7 Effect of heading angle and forward velocity on parametric roll in regular waves (H/=1/20.6 and /L=1.6)

amplitude of parametric roll (deg.)

25

regular waves long-crested waves

20

short-crested waves (cos^4) short-crested waves (cos^2)

15 10 5 0 0

0.05

0.1

0.15 0.2 Froude number

0.25

0.3

0.35

Fig. 8 Effect of irregularity of waves on parametric roll in head waves (H1/3=0.221m, =1.32s and c=180 degrees)

Experimental Study on Parametric Roll of a Post-Panamax Containership 273

amplitude of parametric roll (deg.)

25 180 deg. 210 deg.

20

240 deg. 15 10 5 0 0

0.05

0.1

0.15 0.2 Froude number

0.25

0.3

Fig. 9 Effect of heading angle on parametric roll in short-crested irregular waves (H1/3=0.221m, =1.32s and cos 2  distribution)

2.3 Statistical Analysis Statistical analysis focusing on instantaneous angles and amplitudes of roll and pitch was carried out with recorded time histories. Probability density functions were obtained for both instantaneous angles and amplitudes of roll and pitch, and compared them with the Gaussian distribution and the Rayleigh distribution, respectively. Examples of the results in long-crested irregular waves are shown in Figs. 1011. Here the experimental condition is H1/3=0.221m, T01=1.32s, and autopilot course c=180 degrees. As shown in Figs. 10-11, the instantaneous pitch angle agrees with the Gaussian distribution and the pitch amplitude agrees with the Rayleigh distribution. Thus a linear theory can explain experimental results of the pitch motion even with quite high wave height of 22.1m in full-scale. On the other hand, probability density functions of instantaneous angle and amplitude of roll around 0 degrees are conspicuously large, and does not follow the Gaussian and Rayleigh distributions at all. These results clearly indicate that parametric roll is a strongly nonlinear phenomenon. Pitch motion immediately starts whenever a ship meets incident waves while a ship starts to roll only when a ship meets the wave group that exceeds a parametric roll threshold. Therefore the probability density around 0 degrees becomes large. For this reason, a conventional seakeeping theory based on a linear or weakly nonlinear assumption cannot be applied to parametric roll problems. Similar results have been reported by Belenky et al. (2003) and Ribeiro e Silva et al. (2005) based on their numerical simulations, however, no publication based on model experiments can be found so far. Examples of the results in short-crested irregular waves are shown in Figs 12-13. Here the experimental condition is H1/3=0.221m, T01=1.32s, autopilot course c=180

274 H. Hashimoto et al.

0.16

0.16

0.14

0.14

0.12

0.12

probability density

probability density

degrees and cos2  distribution. The instantaneous pitch and the pitch amplitude agree with the Gaussian and the Rayleigh distributions, respectively. Instantaneous roll angle and roll amplitude do not agree with each distribution like long-crested irregular wave runs, however, its difference becomes smaller to some extent.

0.1 0.08 0.06 0.04 0.02 0 -30

0.1 0.08 0.06 0.04 0.02

-20 -10 0 10 20 elevation of roll angle (degrees)

0 -20

30

-10 0 10 elevation of pitch angle (degrees)

20

Fig. 10 Probability density function of instantaneous roll and pitch angle in long-crested irregular waves (histogram: experiment; dashed line: Gaussian distribution) 0.25 0.2

0.15

probability density

probability density

0.2

0.1

0.05

0 0

0.15 0.1 0.05

5 10 15 20 25 amplitude of roll angle (degrees)

0 0

30

5 10 15 amplitude of pitch angle (degrees)

20

0.16

0.16

0.14

0.14

0.12

0.12

probability density

probability density

Fig. 11 Probability density function of roll and pitch amplitude in long-crested irregular waves (histogram: experiment; dashed line: Rayleigh distribution)

0.1 0.08 0.06 0.04

0.06 0.04 0.02

0.02 0 -30

0.1 0.08

-20 -10 0 10 20 elevation of roll angle (degrees)

30

0 -20

-10 0 10 elevation of pitch angle (degrees)

20

Fig. 12 Probability density function of instantaneous roll and pitch angle in short-crested irregular waves (histogram: experiment; dashed line: Gaussian distribution)

Experimental Study on Parametric Roll of a Post-Panamax Containership 275 0.2

0.3 0.25 probability density

probability density

0.15

0.1

0.05

0.2 0.15 0.1 0.05

0 0

5 10 15 20 25 amplitude of roll angle (degrees)

30

0 0

5 10 15 amplitude of pitch angle (degrees)

20

Fig. 13 Probability density function of roll and pitch amplitude in short-crested irregular waves (histogram: experiment; dashed line: Rayleigh distribution)

3 CONCLUSIONS Free running model experiments of a post-Panamax containership were systematically conducted to directly examine the danger of parametric roll in head and bow waves in regular waves, long-crested and short-crested irregular waves. As a result, the following conclusions are obtained: 1. 2. 3. 4. 5. 6. 7.

Capsizing due to parametric roll in head or bow waves was not observed for the post-Panamax containership model with the designed GM value. Maximum roll amplitude of parametric roll does not always increase with wave steepness in regular waves. It is recommended to avoid ±45degrees of heading angle from head sea condition to prevent severe parametric roll exceeding 15 degrees of roll amplitude in regular waves. Maximum roll angle of parametric roll in long-crested irregular waves or shortcrested irregular waves is almost the same as large as the steady amplitude in regular waves. Maximum roll angle of parametric roll does not depend on heading angle in short-crested irregular bow waves so much. Increasing ship forward velocity could be recommended to reduce the parametric roll danger in irregular head and bow waves, as an alternative to decreasing it. Instantaneous roll angle and amplitude of parametric roll cannot be explained as the Gaussian and Rayleigh distributions, respectively.

Acknowledgements This work was supported by a Grant-in Aid for Scientific Research of the Japan Society for Promotion of Science (No. 15360465). The work was partly carried out as a research activity of the RR-SP4 research panel of the Shipbuilding Research

276 H. Hashimoto et al.

Association of Japan in the fiscal year of 2004, funded by the Nippon Foundation. The authors express their sincere gratitude to the above organisations.

References Belenky VL (2004) On Risk Evaluation at Extreme Waves. Proceedings of the 7th Int Ship Stab Workshop 188-202 Belenky VL, Weems KM, Lin WM et al (2003) Probabilistic Analysis of Roll Parametric Resonance in Head waves. Proc of the 8th Int Conf on Stab of Ships and Ocean Veh 325-340 France WL, Levadou M, Treakle TW et al (2003) An investigation of head-sea parametric roll and its influence on container lashing system. Marine Technol 40(1) 1-19 IMO (2002) Code on Intact Stability for All Types of Ships Covered by IMO Instruments IMO (1995) Guidance to the Master for Avoiding Dangerous Situations in Following and Quartering Waves. MSC Circular 707 Ribeiro e Silva S, Santos TA, Guedes Soares C (2005) Parametrically Excited Roll in Regular and Irregular Head waves. Int Shipbuilding Prog 52(1) 29-56 Sera W, Umeda N (2000) Homogeneity of Directional Irregular Waves with Side-Wall Reflections. J of the Soc of Naval Archit of Japan 188 251-256 (in Japanese) Umeda N, Hamamoto M, Takaishi Y et al (1995) Model Experiments of Ship Capsize in Astern Waves. J of the Soc of Naval Archit of Japan 177 207-217

Model Experiment on Parametric Rolling of a Post-Panamax Containership in Head Waves Harukuni Taguchi, Shigesuke Ishida, Hiroshi Sawada and Makiko Minami National Maritime Research Institute (NMRI), 6-38-1 Shinkawa, Mitaka, Tokyo, 181-0004, Japan

Abstract Parametric rolling resonance in head waves was investigated experimentally. An experiment using a scale model of a post-Panamax containership was carried out in regular head waves at the 80 metres square basin of NMRI. In the experiment the wavelength, the wave height, the model speed and the encounter angle were widely varied to clarify overall property of parametric rolling resonance in head waves. As a result, conditions in which the head wave parametric rolling resonance is likely to occur and effects of the encounter period, the wave height and the encounter angle on parametric rolling amplitude were clarified. Moreover numerical simulations of roll motion of a ship in pure head waves were also carried out to investigate the property of parametric rolling resonance further.

1 Introduction A post-Panamax containership accident in the north Pacific Ocean, in which that ship suffered extensive loss and damage to onboard containers due to severe parametric rolling in head wave condition (France et al. 2003), led the International Maritime Organization (IMO) to start work to revise the Intact Stability Code (IS Code) in this aspect. Regarding parametric rolling resonance, phenomenon in following waves is well known, while that in head waves has been less studied so far (e.g. Dallinga et al. 1998, Neves and Valerio 2000, France et al. 2003, Neves et al. 2003, Bulian et al. 2003). Therefore the property of behaviour of ships in parametric rolling resonance in head waves has not been clarified thoroughly. In order to establish appropriate safety measures against severe parametric rolling in head wave condition, influence of hull forms, operating condition (ship speed and course in waves), and wave condition (wave length and wave height) on the occurrence of parametric rolling resonance and the resultant magnitude of roll motion should be clarified. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_15, © Springer Science+Business Media B.V. 2011

277

278 H. Taguchi et al.

In this context an experiment with a scale model of a post-Panamax containership was carried out in regular head waves at the 80 metres square basin of NMRI. As a result, some property of the head wave parametric rolling resonance, e.g. conditions in which the phenomenon is likely to occur, have been clarified. In order to examine the property of parametric rolling resonance, some numerical investigation was also carried out.

2 Outline of Model Experiment The experiment was carried out with a free running model in regular waves of various length and height. Using an autopilot device and a motor controller, the encounter angle and the propeller revolution were kept constant for each run.

F.P.

A.P １

9 8

2

7

3

6

4

5

Fig. 2.1 Body plan of model ship

Table 2.1 Principal particulars

Lpp (m) B (m) D (m) d (m) 3 V (m ) Cb GM (m) T (sec.)

Ship 283.8 42.8 24.4 14.0 106,970 0.629 1.08 30.26

Model 3.700 0.558 0.318 0.183 0.237 0.629 0.014 3.460

Model Experiment on Parametric Rolling of a Post-Panamax Containership 279

1.4 1.2 1.0

GZ (m)

0.8 0.6 0.4 0.2 0.0 0.2 0

10

20

30

40

50

60

70

80

90

φ (deg.) Fig. 2.2 Stability curve (GM = 1.08 m)

The model ship is a 1/76.7 scale model of a 6600TEU containership. The main hull up to the upper deck and the forecastle were reproduced in the model, but the deckhouse was ignored. Fig. 2.1 shows the body plan of the model and its principal particulars are shown in Table 2.1. Fig. 2.2 shows the stability curve in the test condition, which was calculated with only the main hull and the forecastle taken into account. In the experiment six degrees of freedom motion, relative water heights at stem, S.S.5 (weather side) and A.E., lateral acceleration beneath the upper deck at S.S.8, rudder angle, number of propeller revolution and speed and trajectory of the model were measured. Table 2.2 Waves used in the experiment / L 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

5cm

8cm

　　 ○

○

　　　

○ 　　

Hw 11cm ○ ○ ○ ○ ○ ○ ○ ○ ○

15cm

20cm

○ ○ 　

○

In order to investigate the overall property of parametric rolling resonance, the wavelength , the wave height Hw, the model speed Vm, and the encounter angle  were varied. Table 2.2 summarizes the length and height of waves used in the experiment. The wave height of 11 cm in the model scale corresponds to that of 8.4 m in the actual scale. The encounter angle was varied at every 15 degrees from 135 degrees (bow wave) to 180 degrees (pure head wave). The model speed was

280 H. Taguchi et al.

varied in a wide range so that the critical condition for parametric rolling resonance could be clarified.

3 Experimental Results 3.1 Examples of Parametric Rolling Typical time series of the ship motion and the lateral acceleration at bow in parametric rolling resonance are shown in Figs. 3.1 and 3.2.

φ (deg.)

20 10 0 10 20 0

10

20

30 sec.

40

50

60

40

50

60

50

60

θ (deg.)

4 2 0 2 4 0

10

20

30 sec.

0.4 η (g)

0.2 0.0 0.2 0.4 0

10

20

30 sec.

40

Fig. 3.1 Measured roll angle (top), pitch angle (middle) and lateral acceleration at bow (bottom). /L = 1.6, Hw = 11 cm,  = 180°, and Vm = 0.48 m/s (Fn = 0.08)

Fig. 3.1 shows the measured roll angle  (top), pitch angle  (middle) and lateral acceleration at bow  (bottom) under the conditions of the wavelength to the ship length ratio /L = 1.6, the wave height Hw = 11 cm, the encounter angle = 180 degrees, and the model speed Vm = 0.48 m/s (the corresponding Froude number Fn = 0.08). The ratio of the measured encounter period Te to the natural rolling period T, Te/T is about 0.48. Parametric rolling resonance, where there are two

Model Experiment on Parametric Rolling of a Post-Panamax Containership 281

φ (deg.)

pitch cycles for each one roll cycle, occurs from the beginning. The roll motion immediately becomes in steady state where the amplitude of parametric rolling reaches 19 degrees. And the steady state amplitude of the lateral acceleration beneath the upper deck at S.S.8 is about 0.29 g (including the gravity component).

10 5 0 5 10 15 0

20

40 sec.

60

80

60

80

60

80

θ (deg.)

4 2 0 2 4 20

0

40 sec.

0.4 η (g)

0.2 0.0 0.2 0.4 0

20

40 sec.

Fig. 3.2 Measured roll angle (top), pitch angle (middle) and lateral acceleration at bow (bottom). /L = 1.6, Hw = 8 cm,  = 135°, and Vm = 0.63 m/s (Fn = 0.11)

Time histories of the measured data under the conditions of /L = 1.6, Hw = 8 cm,  = 135 degrees, and Vm = 0.63 m/s (Fn = 0.11) are shown in Fig. 3.2. In this case Te/T is about 0.49. At the beginning the model exhibits a large and small roll responses to every two encounter waves and gradually transits to parametric rolling response. But even at the end of the run, the rolling amplitude seems not to reach steady state.

282 H. Taguchi et al.

3.2 Occurrence of Parametric Rolling Occurrence of parametric rolling resonance in waves of Hw = 11 cm are summarised in Figs. 3.3 and 3.4. χ = 180 deg. 0.65

0.60

Te/ T φ

0.55

0.50 0.45 0.40

0.35 0.00

0.05

0.10

0.15

0.20

0.25

Fn

λ /L = 0.9

χ = 150 deg.

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

0.65

0.60

Te/T φ

0.55

0.50

0.45

0.40

0.35 0.00

0.05

0.15

0.10

0.20

0.25

Fn

Fig. 3.3 Occurrence of parametric rolling resonance. Hw = 11 cm and  = 180° (upper),  = 150° (lower)

Fig. 3.3 shows the occurrence of parametric rolling resonance with the wavelength to the ship length ratio /L as parameter, where the encounter angle  = 180 degrees for the upper figure and  = 150 degrees for the lower one. The horizontal axis is the model speed in Froude number and the vertical axis is the ratio between the encounter period and the natural rolling period in each figure.

Model Experiment on Parametric Rolling of a Post-Panamax Containership 283

The experimental results are indicated on the lines that show the relation between the model ship speed and the encounter period. The symbol of black circle means condition where the steady state parametric rolling resonance (see Fig. 3.1) was observed, the cross means one where parametric rolling resonance was not observed, and the white triangle means one where parametric rolling resonance was observed but did not reach the steady state (see Fig. 3.2). λ/L = 1.2 0.25

0.20

Fn

0.15

0.10

0.05

0.00 135

150

165

180

χ (deg.) λ/L = 1.6 0.25

0.20

Fn

0.15

0.10

0.05

0.00 135

150

165

180

χ (deg.)

Fig. 3.4 Occurrence of parametric rolling resonance. Hw = 11 cm and /L = 1.2 (upper), /L = 1.6 (lower)

Fig. 3.4 summarizes the occurrence of parametric rolling resonance at various encounter angels ( = 135°, 150°, 165°, and 180°), where /L = 1.2 in the upper figure and /L = 1.6 in the lower one. In each figure the horizontal axis is the encounter angle and the vertical axis is the model speed in Froude number. Symbols mean as the same as in Fig. 3.3.

284 H. Taguchi et al.

From these figures we see the followings. (1) Parametric rolling resonance occurs in relatively wide range of the encounter period to the natural rolling period ratio, Te/T, namely Te/T = 0.4 ~ 0.6. (2) In the same waves parametric rolling resonance is more likely to occur in pure head wave condition ( = 180 degrees) than in bow wave condition (= 135 ~ 165 degrees).

3.3 Amplitude of Parametric Rolling 3.3.1 Influence of Encounter Period Fig. 3.5 shows the steady state amplitude of parametric rolling for various encounter periods under the condition of the wave height Hw = 11cm and the encounter angle  = 180 degrees. The horizontal axis is the encounter period to the natural rolling period ratio and the vertical axis is the parametric rolling amplitude s normalised by the wave slope, s/kHw (k is the wave number). 7.0

λ/L =1.0 =1.2 =1.4 =1.6 =1.8 =2.0 =2.2

6.0

φs/kHw

5.0 4.0 3.0 2.0 1.0 0.0 0.30

0.35

0.40

0.45 0.50 Te / Tφ

0.55

0.60

0.65

Fig. 3.5 Influence of encounter period on parametric rolling amplitude. Hw = 11 cm and  = 180°

From this figure we see the followings. (1) The amplitude of parametric rolling changes largely with the variation of the encounter period. (2) For waves of the same length the parametric rolling amplitude becomes the maximum at about Te/T = 0.43 ~ 0.48. (3) The normalised rolling amplitudes for the same encounter period are almost the same except for waves of /L = 2.0 and 2.2.

Model Experiment on Parametric Rolling of a Post-Panamax Containership 285

3.3.2 Influence of Wave Height and Encounter Angle As indicated in Table 2.2, for waves of /L = 1.0, 1.2, and 1.6 measurements with different wave heights were carried out. Fig. 3.6 shows the steady state parametric rolling amplitudes measured under the conditions of  = 180 degrees and Te/T = 0.40 for /L = 1.0, Te/T = 0.45 for /L = 1.2 and 1.6. The horizontal axis is the wave steepness Hw/, and the vertical axis is the normalised rolling angle. In the experiment the rolling amplitude did not become larger as the wave height increased beyond 11 cm. Therefore as indicated in Fig. 3.6 the normalised rolling amplitude becomes smaller as the wave height increases. 8.0 λ/L=1.0 =1.2 =1.6

7.0

φs/kHw

6.0 5.0 4.0 3.0 2.0 1.0 0.0 0.00

0.01

0.02

0.03

0.04

0.05

Hw/λ

Fig. 3.6 Influence of wave height on parametric rolling amplitude.  = 180° and Te/T = 0.40 for /L = 1.0, Te/T = 0.45 for /L = 1.2 and 1.6

Fig. 3.7 shows the influence of the encounter angle on the parametric rolling amplitudes under the conditions of Te/T = 0.45 and /L = 1.2. The normalised rolling amplitudes for bow wave condition ( =150 and 165 degrees) also become smaller as the wave height increases. And the rolling amplitudes for bow wave conditions are smaller than that for pure head wave condition ( = 180 degrees). Investigation into the influence of the wave height on the parametric rolling amplitude is important for judging whether a parametric rolling resonance leads to capsizing in severe waves. In order to draw conclusion of this issue it seems that more data in various conditions (encounter period, wavelength wave height) is necessary.

286 H. Taguchi et al. 7.0

χ =180 (deg.) =165 (deg..) =150 (deg..)

6.0

φs/kHw

5.0 4.0 3.0 2.0 1.0 0.0 0.00

0.01

0.02

0.03 Hw/λ

0.04

0.05

Fig. 3.7 Influence of encounter angle on parametric rolling amplitude. /L = 1.2 and Te/T = 0.45

4 Numerical Investigation In order to investigate the property of parametric rolling resonance further, we carried out numerical simulations with an equation of roll motion of a ship with stability variation in waves. In this paper, as the first step of the numerical investigation, we examine the parametric rolling resonance in pure head waves, where no roll exciting moment acts, and investigate effects of ship speed and wave height on the occurrence of parametric rolling resonance and the resultant rolling amplitude.

4.1 Equation of Roll Motion Roll motion of a ship with stability variation in pure head wave can be generally expressed with equation (4.1)

 I xx  J xx    A  B    W  GZ  ; t   0

(4.1)

where  is the roll angle, Ixx is the moment of inertia, Jxx is the added moment of inertia, A and B are the linear and the quadratic damping coefficients, W is the displacement, GZ(; t) is the time dependent restoring arm in waves. The dots over the symbol represent differentiation with respect to time t. By dividing all terms in equation (4.1) by (Ixx + Jxx), this equation is rewritten to following form

Model Experiment on Parametric Rolling of a Post-Panamax Containership 287

GZ  ; t    2       2 0 GM

(4.2)

where GM is the metacentric height in still water,  = A/2(Ixx + Jxx),  = B/(Ixx + Jxx), and  = WGM/(Ixx + Jxx). The stability variation in waves arises from the relative position of ship to wave G(t), and the vertical ship motions, heave (t) and pitch (t) in waves. Therefore the time dependent restoring arm in waves can be expressed as equation (4.3).

GZ  ; t   GZ  ; G (t ),  (t ),  (t )

(4.3)

GZ is calculated by integrating the hydrostatic pressure over the instantaneous submerged hull under the undisturbed wave profile, which is determined by the relative position of ship to wave and heave and pitch motions. With this calculation method the instantaneous variation of displacement due to the vertical motions is included. In this paper, the vertical motions are assumed to be linear to the wave height and computed with the strip theory. In equation (4.2), the linear and the quadratic damping coefficients obtained from roll decay tests with forward velocity were used. Moreover we examine only steady state response of equation (4.2). Therefore numerical investigation was carried out with calculation method for obtaining bifurcation diagrams (Taguchi et al. 2003).

4.2 Occurrence of Parametric Rolling Occurrence of parametric rolling resonance is directly detected with bifurcation diagram. 4.2.1 Influence of Encounter Period To draw bifurcation diagrams, numerical simulations were conducted for /L = 1.2, 1.6, and 2.0 with the ship speed gradually changed from Fn = 0 to 0.25 (service speed) but the wave height kept constant as Hw = 11 cm. Fig. 4.1 shows the obtained bifurcation diagrams expressed as the function of the resultant encounter period to the natural rolling period ratio, Te/T. In the diagrams the normalised steady state roll responses at Poincaré section, (nT)/kHw, are plotted as much as 20 cycles for every Te/T calculated. Two points at one Te/T means that parametric rolling resonance occurs at such encounter period. In each diagram experimental results are also shown with symbols of the same meaning as in Fig. 3.3.

288 H. Taguchi et al. λ/L=1.2

φ(nT)/kHw

4.0 2.0 0.0 −2.0 −4.0 0.30

0.40

0.50

Te / Tφ

0.60

λ/L=1.6

φ(nT)/kHw

4.0 2.0 0.0 −2.0 −4.0 0.30

0.40

0.50

Te / Tφ

0.60

λ/L=2.0

φ(nT)/kHw

4.0 2.0 0.0 −2.0 −4.0 0.30

0.40

0.50

0.60 Te / Tφ

Fig. 4.1 Bifurcation diagram with changing Te/T. Hw = 11 cm,  = 180°, and /L = 1.2 (top); /L = 1.6 (middle); /L = 2.0 (bottom)

From the top diagram, it is found that for /L = 1.2 the numerical simulation predicts that parametric rolling resonance occurs in wide range of the encounter period to the natural rolling period ratio, namely Te/T = 0.32 (Fn= 0.23) ~ 0.49 (Fn = 0). However as indicated in the diagram, in the model experiment parametric rolling resonance was not observed at Te/T = 0.38. So the numerical simulation with equation (4.2) overestimates occurrence of parametric rolling resonance in short encounter period range. For /L = 1.6 (the middle diagram) it is found that parametric rolling resonance occurs in all the calculated range of parameter. In this case good agreement between the numerical simulation and the experimental results is found. For /L = 2.0 (the bottom diagram) the parametric rolling resonance is predicted to occur in the range of Te/T = 0.45 ~ 0.57. Compared with the experimental results, it is found that the numerical simulation well predicts the lower limit of the encounter periods for the occurrence of parametric rolling resonance, while it underestimates the upper limit.The discrepancies between the numerical simulation and the experimental results in the range of encounter period

Model Experiment on Parametric Rolling of a Post-Panamax Containership 289

which leads to parametric rolling resonance for /L = 1.2 and 2.0 seem to be mainly caused by the prediction of the stability variation in waves. 4.2.2 Influence of Wave Height λ/L = 1.2

6.0

φ(nT)/kHw

4.0 2.0 0.0 2.0 4.0 6.0

0

0.01

0.02

0.03

0.04

0.05

0.06

0.04

0.05

0.06

0.04

0.05

0.06

Hw /λ λ/L = 1.6

φ(nT)/kHw

6.0 4.0 2.0 0.0 2.0 4.0 6.0

0

0.01

0.03

0.02

Hw /λ λ/L = 2.0

φ(nT)/kHw

6.0 4.0 2.0 0.0 2.0 4.0 6.0

0

0.01

0.02

0.03

Hw /λ

Fig. 4.2 Bifurcation diagram with changing Hw/. Te/T = 0.45,  = 180°, and /L = 1.2 (top); /L = 1.6 (middle); /L = 2.0 (bottom)

Fig. 4.2 shows bifurcation diagrams for /L = 1.2, 1.6, and 2.0 with the wave height gradually changed but the ship speed kept constant as the resultant encounter period satisfies that Te/T = 0.45. In each diagrams the horizontal axis is the wave steepness, while the vertical axis is the roll responses at Poincaré section normalised by the wave slope. The experimental results are also shown with the same manner as in Fig. 4.1. From Fig. 4.2 it is found that the numerical simulations are almost consistent with the experimental results for all the wavelengths examined. And it is also found by the numerical simulations that there are two threshold wave heights, the lower limit and the higher limit of wave heights, for occurrence of parametric

290 H. Taguchi et al.

rolling resonance. For /L = 1.2 (the upper diagram), the numerical simulation predicts that as the wave height increases from 0, the parametric rolling resonance appears at about Hw/ = 0.006 and continue to exist up to about Hw/ = 0.045. As the wave height increases further, the parametric rolling resonance disappears. As shown in Fig. 3.6, for /L = 1.2 the model experiment was carried out in waves of up to 20 cm in height (Hw/ = 0.045), and parametric rolling resonance was observed in all the wave height tested. In order to confirm the existence of the higher threshold of wave height for parametric rolling resonance, additional model experiment in higher waves seems necessary. Fig. 4.2 also shows that as the wavelength increases, the range of wave height, which leads to parametric rolling resonance, becomes smaller. As the occurrence of parametric rolling is directly related to the stability variation in waves, further investigation including quantitative examination of the stability variation seems necessary to confirm the effects of the wave height on the occurrence of parametric rolling resonance.

4.3 Amplitude of Parametric Rolling In this section, steady state parametric rolling amplitudes obtained in the calculation of bifurcation diagrams are examined. 4.3.1 Influence of Encounter Period Fig. 4.3 shows the parametric rolling amplitude predicted by the numerical simulations with changing the ship speed as parameter. This figure corresponds to Fig. 4.1. The horizontal axis is the encounter period to the natural rolling period ratio and the vertical axis is the steady state rolling amplitude normalised by the wave slope. In the diagrams the experimental results are shown with the white circles. From Fig. 4.3 the numerical simulation seems to capture the tendency of parametric rolling amplitude found in the model experiment. But except for /L = 2.0, the numerical simulation underestimates the maximum amplitude of parametric rolling and there is also some discrepancy in the encounter period leading to the maximum response. As the stability curve of this ship is hard spring type (see Fig. 2.2), the natural rolling period for large rolling amplitude tends to become shorter than that for small amplitude rolling. The numerical simulation may overestimate this effect and the resultant peak encounter period to the natural rolling period ratio becomes smaller than that in the model experiment.

Model Experiment on Parametric Rolling of a Post-Panamax Containership 291

φs/kHw

φs/kHw

φs/kHw

λ /L=1.2 6.0 5.0 4.0 3.0 2.0 1.0 0.0 0.30

6.0 5.0 4.0 3.0 2.0 1.0 0.0 0.30

6.0 5.0 4.0 3.0 2.0 1.0 0.0 0.30

0.50

0.40

Te / Tφ

0.60

λ /L=1.6

0.50

0.40

Te / Tφ

0.60

λ /L=2.0

0.40

0.60

0.50

Te / Tφ

Fig. 4.3 Influence of encounter period on parametric rolling amplitude. Hw = 11 cm,  = 180°, and /L = 1.2 (top); /L = 1.6 (middle); /L = 2.0 (bottom).

4.3.2 Influence of Wave Height Fig. 4.4 shows the predicted parametric rolling amplitude with changing the wave height as parameter. Compared to the experimental results, which are shown with white circles in the diagrams, the numerical simulation is found to predict the parametric rolling amplitude well. It is also found that the numerical simulation reproduces the tendency shown in the model experiment that the normalised parametric rolling amplitude becomes smaller as the wave height gets larger. In the numerical investigation, as shown in Fig. 4.5, this tendency was observed at different encounter periods too.

292 H. Taguchi et al. λ

1.2

φs

k w

12.0 10.0 8.0 6.0 4.0 2.0 0.0

0

0.01

0.02

0.03

λ

0.05

0.06

0.04 w λ

0.05

0.06

0.04

0.05

0.06

1.6

φs

k w

12.0 10.0 8.0 6.0 4.0 2.0 0.0

0.04

w λ

0

0.01

0.02

0.03

λ

2.0

φs

k w

12.0 10.0 8.0 6.0 4.0 2.0 0.0

0

0.01

0.02

0.03

w λ

Fig. 4.4 Influence of wave height on parametric rolling amplitude. Te/T = 0.45,  = 180°, and /L = 1.2 (top); /L = 1.6 (middle); /L = 2.0 (bottom)

e

φ 0.40

φs

k w

12.0 10.0 8.0 6.0 4.0 2.0 0.0 0.00

0.01

0.02

0.03

e

0.05

0.06

φ 0.50

φs

k w

12.0 10.0 8.0 6.0 4.0 2.0 0.0

0.04 w λ

0

0.01

0.02

0.03

w λ

0.04

0.05

0.06

Fig. 4.5 Influence of wave height on parametric rolling amplitude at Te/T = 0.40 (upper) and 0.50 (lower).  = 180° and /L = 1.6

Model Experiment on Parametric Rolling of a Post-Panamax Containership 293

5 Conclusions The behaviour of a ship in parametric roll resonance in head waves was investigated experimentally with the free running model of a post-Panamax containership in regular waves. And some numerical investigation using an equation of roll motion of a ship with stability variation in waves was also carried out to examine the property of parametric rolling resonance. As a result the followings are clarified. (1) The parametric rolling resonance occurs in relatively wide range of the encounter period, where the ratio of it to the natural roll period, Te/T ranges from about 0.4 to 0.6. (2) In the same wave parametric rolling resonance is more likely to occur in head wave condition ( = 180 degrees) than in bow wave condition ( = 135 ~ 165 degrees). (3) The amplitude of parametric rolling changes largely with the variation of the encounter period and it becomes the maximum at about Te/T = 0.43 ~ 0.48. (4) The parametric rolling amplitude normalised by the wave slope becomes smaller as the wave height increases. (5) The parametric rolling amplitudes for bow wave conditions become smaller than that for the head wave condition. (6) The numerical simulation using equation (4.2) can predict the property of parametric rolling resonance qualitatively. (7) Discrepancies found in both the occurrence range and the resultant amplitude of parametric rolling implies that some refinement for the stability variation model in waves, equation (4.3), is necessary to get more precise prediction.

Acknowledgements Some parts of this investigation were carried out as a research activity of the RRSP4 research panel of the Shipbuilding Research Association of Japan in the fiscal year of 2004, funded by the Nippon Foundation. The authors express their sincere gratitude to the both organizations.

References Bulian G, Francescutto A and Lugni C (2003) On the Nonlinear Modeling of Parametric Rolling in regular and Irregular Waves. Proc of the 8th Int Conf on the Stab of Ships and Ocean Veh 305-323.

294 H. Taguchi et al. Dallinga RP, Blok JJ and Luth HR (1998) Excessive Rolling of Cruise Ships in Head and Following Waves. Proc of RINA Int Conf on Ship Motions & Manoeuvrability, London 1-16. France WN, Levadou M and Treakle T (2003) An Investigation of Head-Sea Parametric Rolling and Its Influence on Container Lashing Systems. Mar Technol 40 1: 1-19. Neves MAS and Valerio L (2000) Parametric Resonance in Waves of Arbitrary Heading. Proc of the 7th Int Conf on the Stab of Ships and Ocean Veh 680-687. Neves MAS, Perez N and Lorca O (2003) Analysis of Roll Motion and Stability of a Fishing Vessel in Head Seas. Ocean Eng 30: 921-935. Taguchi H, Sawada H and Tanizawa K (2003) A Study on Complicated Roll Motion of a Ship Equipped with an Anti-Rolling Tank. Proc of the 8th Int Conf on the Stab of Ships and Ocean Veh 607-616.

Numerical Procedures and Practical Experience of Assessment of Parametric Roll of Container Carriers Vadim Belenky, Han-Chang Yu and Kenneth Weems

Abstract The paper examines several aspects related to practical assessment of parametric roll of container carriers. The numerical procedure is based on application of the Large Amplitude Motion Program (LAMP), the nonlinear potential flow code. Viscous roll damping is included from the roll decay test. The rational choice of the loading conditions is considered. Practical non-ergodicity of the response is taken into account.

1 Introduction The problem of large amplitude roll motions caused by parametric resonance, though known for quite some time, recently made its reappearance in relation to an accident with a large container carrier, (France et al 2003). This paper describes the development of on-board information required by the Guide on Parametric Roll (ABS, 2004). Below is a brief literature review on the related subject. The physical phenomenon of parametric roll was one of the focuses of the 8th International Ship Stability Workshop that was hosted by Istanbul Technical University on October 6th and 7th, 2005 in Istanbul, Turkey. The paper by Neves and Rodríguez (2005) looking into a new mathematical model with nonlinearities defined up to the third order in terms of the heave, roll and pitch couplings is introduced in order to simulate strong roll parametric amplifications in head seas. The influence of hull stern shape is discussed. A theoretical analysis discloses some essential dynamical characteristics associated with the proposed coupled third order mathematical model. The paper by Umeda et al (2005) reviews the latest developments at Osaka University and, among others, includes theoretical prediction techniques for magnitude of parametric rolling. Poincaré mapping and averaging method were then used to obtain the magnitude of parametric roll.

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_16, © Springer Science+Business Media B.V. 2011

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A paper by Ikeda et al (2005) describes a model test carried out to evaluate the effect of bilge keels on parametric roll in beam seas. In the considered case, the amplitude of parametric roll decreased from 25 degrees to almost zero. The paper SLF 48/4/12 considered parametric roll in relation to the new Intact Stability Code. Several important points were made: (1) regular wave case may be more complicated due to nonlinear behavior, (2) regular wave case appears to be more severe than irregular wave case, (3) for the irregular wave case, the probability distribution is not Gaussian. Other relevant papers at SLF-48 were focused on updating of the MSC Circular 707. The paper SLF 48/4/4 contains guidelines for all types of ships, while definition of parametric roll conditions are based on TR≈2TE (TR natural period of roll, TE encounter period of waves) while wave length is between 0.8L and 1.2LThe paper SLF 48/4/8 also suggests that the MSC Circ. 707 guidelines be made applicable to all types of ships as far as parametric roll is concerned. The paper addresses both wave pass effect and pitch coupling and recommends changing speed and heading in order to avoid the encounter period being twice the natural roll period. The papers SLF 48/4/16 and SLF 48/4/17 describe diagrams which are meant to be present on board a vessel. Parametric roll criterion is based on the frequency range 1.9 TE < TR <2.1 TR. Spyrou (2004) examines an approximation of transient solution of a Mathieu equation to estimate how quickly parametric roll rises. The influence of restoring nonlinearity is studied and parametric roll response is found outside of the asymptotic instability zone. The probabilistic aspect includes an envelope presentation, which leads to a formula for joint probability of height of two consecutive waves above the given threshold. The Markov chain could be used to obtain the result for the entire wave group. The paper by Bulian et al (2004) presents the preliminary results of an analytical, numerical and experimental study regarding the problem of non-ergodicity of parametric roll in longitudinal irregular long-crested waves. A series of numerical simulations is performed with 1.5-DOF model of parametrically excited roll motion. Qualitative indications given by the numerical simulations were then compared with experimental tests, showing good agreement.

2 Polar Diagrams as a Form of On-board Information Polar diagrams in wave heading/speed coordinates are a convenient way to present any numerical information for on-board use (see e.g., MSC Circ. 707). Polar diagrams for parametric roll were discussed by Shin, et al, (2003, 2004) and were recommended by ABS (2004). A Generic Polar Diagram for Parametric Roll was proposed by SLF 48/4/4. Polar diagrams for parametric roll considered in this paper are a further development of format (ABS, 2004). They represent the maximum roll angle observed during a one-hour simulation limited by lashing (ABS, 1988) and engine

Numerical Procedures and Practical Experience of Assessment 297

criteria (ABS, 2006). The area of the diagram where maximum roll angle did not exceed these criteria remains blank The area where these criteria were exceeded is color-coded in different types of yellow and red (from 22 to 40 degrees ) in accordance with the value of observed angle. See Figure 1.

Fig. 1 Sample polar diagram, Bretsshnieder spectrum, with significant wave height of 12.5 m and zero-crossing period of 11.5 seconds

3 Intended Use of Polar Diagrams These polar diagrams have been developed as a planning tool and are not intended for use when a vessel is already experiencing extreme roll motions. Real-time use of a polar diagram would be difficult. An experienced sailor is capable of evaluating significant wave height, but nearly no one could tell the exact wave heading with the naked eye. Decreased visibility and nighttime makes estimate of wave parameters even more difficult. Most large container carriers nowadays use a weather routing service that usually includes a weather map that is updated every six hours. When a storm is on the route and a master is considering options, the polar diagrams are expected to be most useful. A weather map would include wave direction, so determining wave heading when the vessel encounters a storm will not be difficult. The speed will be also known as speed loss in waves is also known from operation experience. These two figures will determine the position of the point on the polar diagram corresponding most closely to current loading condition and expected significant wave height and zero-crossing periods of waves.

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4 Numerical Tools for Development of Polar Diagrams Parametric roll resonance is a result of changing stability in waves, so a minimum requirement for any numerical simulation tool is a capability to compute hydrostatic forces on instantaneous submerged surface. Such software would be capable of reproducing the phenomenon, but in order to achieve numerical accuracy sufficient for practical purposes, roll damping has to match experimental results (France, et al 2003). With computational capability currently available for mass use, application of potential hydrodynamic methods seems to be a viable, practical solution. Viscous and vortex-induced forces could be added as external components to facilitate matching of roll damping to experimental results. Two numerical tools which meet these requirements for evaluating parametric roll are the Large Amplitude Motions Program (LAMP) and NLOAD3D codes. Both of these codes are time-domain ship motions and loads prediction codes that are built on a 3-D potential flow solution of the wave-body hydrodynamic interaction problem. Both incorporate an approximate body-nonlinear approach (Weems, et al, 2000), which computes the incident wave forcing and hydrostatic restoring pressure on the actual wetted hull surface while solving for the hydrodynamic disturbance potential on the mean wetted surface using the instantaneous ship velocity and a combined body boundary condition. This approach is sophisticated enough to capture the critical hydrodynamic phenomena associated with parametric roll while remaining fast enough to run the large number of simulations required to build the polar diagrams. Of the two, the LAMP code is the more general implementation of the approach and suitable for multi-hull and high-speed displacement ships as well as conventional monohulls. NLOAD3D has been specifically developed for large commercial ships. Key features of NLOAD3D include an automated procedure for calibrating roll damping based on experimental roll decay data and an integrated module for setting up and running dozens of simulations over a range of speeds, headings, and wave realizations for the polar diagram.

5 Environmental and Loading Conditions Parametric roll, as with any other resonance phenomenon, is quite sensitive to excitation frequency range as well as to natural frequency of the dynamical system. Excitation frequency range is defined by a geographical region of ocean, season and other meteorological and oceanographic factors. The most precise way to present such a range is to use measured or hindcasted wave spectrum in the region of interest. Although precise, such a method is quite expensive, as the set of diagrams then becomes route-specific and unusable for other routes.

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The next best way is to use approximated spectra. Most of them have two parameters related to significant wave height and mean zero-crossing period. These figures could be obtained from wave statistics-based scatter diagrams. Here, an averaged scatter diagram from IACS Recommendation 34 was used. In order to develop set route-independent diagrams, simulations have to be performed for a series of significant wave heights and zero-crossing periods. In lieu of any other data, the simulations are performed for three most probable mean zero-crossing periods for each significant wave height. The calculations start from survival condition with 14.5 m of significant wave height and are repeated for smaller significant wave height (2 meters as a step was used in this case) until large roll motions no longer present a practical problem. The natural roll frequency is most sensitive to GM, therefore, the choice of loading conditions has to cover the entire range of operational values of GM. In lieu of sensitivity analysis, using a smaller step for GM, where the most frequent loading conditions are, seems to be a quite practical idea.

6 Roll Damping Calibration Being potential flow time-domain simulation codes, LAMP and NLOAD3D calculate only the wave component of roll damping, so the rest of the roll damping moment (vortex and friction components) has to be added. At the same time, there are no reliable methods of extracting these contributions from the result of the roll decay test. Based on previous experience (France, et al 2003), the best way to get a realistic simulation with respect to roll damping is to calibrate the code to produce the same results as the roll decay test. Non-potential roll damping moment is accepted in quadratic form:

R( )  B1  B2 |  | 

(1)

Here, B1 is the linear damping coefficient and B2 is the quadratic damping coefficient. Following the standard procedure of processing of results of the roll decay test, a straight line is fitted, using points on the roll decrement chart

D  k  Amp  f

(2)

Here, D is a relative decrement, Amp is roll amplitude, k is a slope and f is an intercept of the fitted line. The calibration procedure for roll damping is essentially a numerical “roll decay test” being repeated for different values of damping coefficients B1 and B2 with the following standard roll decay processing until numerical slope and intercept would match those from the model test. This procedure could be expressed in the form of

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a system of two nonlinear algebraic equations and solved with any appropriate numerical method

k , f   Damp( B1 ( f ), B2 (k ))  0

(3)

Here, Damp is a symbolic expression for a run of numerical simulation followed by a standard procedure of roll decay processing. Symbols B1(f) and B2(k) express that the roll linear damping coefficient depends on intercept while the quadratic roll damping coefficient depends on the slope. Practical experience has shown that the calculations converge after 5-6 iterations, on average.

7 Consideration of Practical Non-ergodicity and Length of Record Practical non-ergodicity of nonlinear rolling has been addressed by a number of authors since the late 1990s (Belenky et al, 1997, 1998, 2001, 2003, Shin, et al, 2004, Bulian, et al 2004) and was found to be especially strong in parametric roll (Belenky 2004). The reason is that waves capable of exciting parametric roll may appear in a different sequence, one after another in one wave record or separated by “harmless” waves in another one. In the last case, kinetic energy may be dispersed before the next portion of parametric excitation will come and, as a result, the roll variance may significantly differ between records. The practical implication of the absence of ergodicity means that several statistically independent wave records have to be used for numerical simulations. Statistical independence of wave records is achieved by using different initial phases. Another aspect to be addressed is the length of a single record. As is well known (see, for example, Belenky and Sevastianov, 2007), the length of a wave record modeled by the inverse Fourier Transform is related to the frequency step. If this length is exceeded, the record can no longer be considered as statistically representative. The reason for the effect is an error in the numerical integration (Belenky 2005, se also the Chapter “On Self-Repeating Effect in Reconstruction of Irregular Waves” in this book). Evidently, this length could be increased by decreasing the frequency step. However, it will lead to an increasing number of wave components necessary for correct presentation of the wave spectrum. Increasing the number of wave components may lead to a significant increase in the cost of the simulation as more components have to be added up at every time step. Based on the observation, an attempt to increase the length of the simulation twice leads to a four times increase in calculation time. At the same time, several relatively short records do have an advantage in comparison with a fewer long ones. They not only save computation time but take care of the practical non-ergodicity problem. Obviously, statistical methods should

Numerical Procedures and Practical Experience of Assessment 301

be used to estimate the accuracy that could be achieved by different combinations of number and length of records.

8 Some Aspects of Topology of the Polar Diagram As the polar diagram (such as shown in Fig. 1) is a result of numerical simulation, all large amplitude roll motions are included (coming from both parametric and synchronous resonance). However, a simple analysis of frequencies could reveal motions apparently coming from synchronous resonance. One polar diagram is generated for a specific sea state defined with significant wave height and mean zero-crossing period. Each of the two-dimensional grid points of the polar diagram represent wave heading and ship speed. The encounter wave period can be determined using the mean zero-crossing period, wave heading and ship speed for each grid point. Large roll angles are often observed around the grid points where the encounter wave period is close to the roll natural period. This is typically due to the roll synchronous resonance rather than parametric roll response, which is mostly to occur near the grid points where the encounter period is one half of the roll natural period. For example, the large roll angle peaks around 20-25 knots speed near 60° and 300° headings in Fig, 1 are typically due to roll synchronous resonance. The most evident feature related to parametric roll is that, generally, the maximum angle is decreased with the increase of speed. This could be explained by the fact that roll damping increased with the speed raises the threshold of sensitivity to parametric excitation, so a lesser number of waves can transfer energy. It is noticeable that for many such diagrams, the maximum of the parametric roll does not necessarily fall exactly in head seas. At the same time, head and following seas are expected to cause the most difference in stability between crest and trough of the wave. Shifting of maxima is also less noticeable for smaller speeds. Such a shift of maximum can probably be explained by changes in encounter spectrum. As is well known, in head encounter, the spectrum becomes wider (see Fig. 2a, where four encountered spectra are shown, corresponding to 90, 120, 150, 180 degrees of wave heading). As the encounter spectrum becomes broader, less energy is concentrated in the frequency range where the dynamical system is susceptible to parametric excitation. There are two competing factors; with 180 degrees of heading approaching, the dynamical system becomes more susceptible to parametric excitation due to increased wave influence on stability, and there is less parametric excitation due to the form of the encounter wave spectrum.

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20

b)

a) Encounter Spectrum

200

o

90

Encounter Spectrum 0o

10

150o

o

30o

120

100 180o

60o

, 1/s 0

0.5

1

1.5

0.2

0.3

0.4

90o 0.5

0.6

, 1/s 0.7

Fig. 2 Transformation of encounter spectrum at 15 knots significant wave height 8.5 m

Wave direction has a smaller influence on encounter spectrum for smaller speeds; therefore, there is not as much of a shift of maximum for slower speeds. A similar physical mechanism is responsible for topology in following seas. For the slower speeds and headings close to 90 degrees, the encounter spectrum becomes narrower (Fig. 2b), but the system already can take parametric excitation. As the concentration of energy is high (the encounter spectrum is narrow), severe parametric roll is observed. However, these conditions do not exist in the large area in the sample diagram in Fig. 1 because decreasing heading and increasing speed leads to the appearance of negative encounter frequencies and even more narrow encounter spectrum. Apparently, the range of susceptible frequencies of the dynamical system is different from the frequency range of encounter spectrum, and the parametric roll mode very quickly ceases to exist. For the cases of smaller GM, (see sample diagram in Fig. 3), a significant area with parametric roll is also possible in following seas. Topology of such a diagram is generally similar to the one observed above and could be explained by the same physical reasons. Parametric roll in following seas is more likely because the low GM range of sensitivity to parametric excitation is shifted towards lower frequencies and following seas can match it.

Numerical Procedures and Practical Experience of Assessment 303

Fig. 3 Polar diagram - relatively small GM Bretschnieder spectrum, with significant wave height of 8.5 m and zero-crossing period of 11.5 seconds

9 Summary The polar diagram is based on the maximum roll angle observed during a one-hour simulation. Criteria for polar diagrams are based on lashing strength and engine conditions. Exceeding of these criteria is color-coded, depending on the value of the roll angle. Simulations for polar diagrams are carried out using potential codes calibrated for roll damping with roll decay tests. Polar diagrams have complex topology, primarily influenced by encounter spectrum of waves and natural frequency. The choice of loading conditions should be determined by sufficient coverage of the range of natural frequency.

Acknowledgements The authors wish to express their appreciation and gratitude to the management of the American Bureau of Shipping and Science Application International Corporation. The development of the LAMP System has been supported by the U.S. Navy, the Defence Advanced Research Projects Agency (DARPA), the U.S. Coast Guard, ABS, and SAIC. LAMP development has been supported by the Office of Naval Research (ONR) under Dr. Patrick Purtell. The development of NLOAD3D program has been supported by ABS.

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References American Bureau of Shipping (1988) Guide for Certification of Container Securing Systems, N. Y. American Bureau of Shipping (2004) Guide for the Assessment of Parametric Roll Resonance in the Design of Container Carriers, Houston, Texas American Bureau of Shipping (2006) Rules for Building and Classing Steel Vessels, Houston, Texas Belenky V (2004) On Risk Evaluation at Extreme Seas. Proc of 7th Int Ship Stab Workshop, Shanghai Belenky V (2005) On Long Numerical Simulations at Extreme Seas. Proc of 8th Int Ship Stab Workshop, Istanbul Belenky V, Degtyarev AB, Boukhanovsky AV (1998) Probabilistic Qualities of Nonlinear Stochastic Rolling Ocean Eng 25(1):1-25 Belenky V, Suzuki S, Yamakoshi Y (2001) Preliminary Results of Experimental Validation of Practical Non-Ergodicity of Large Amplitude Rolling Motion, Proc of 5th Int Stab Workshop¸ Trieste Belenky V, Sevastianov NB, (2007) Stability and safety of ships. Risk of capsizing, 2nd ed. SNAME, Jersey City Belenky V, Weems KM, Lin WM, Paulling JR (2003), Probabilistic analysis of roll parametric resonance in head seas, Proc of 8th Int Conf on Stab of Ships and Ocean Veh, Madrid Bulian G, Lugni C, Francescutto A (2004) A contribution on the problem of practical ergodicity of parametric roll in longitudinal long crested irregular sea Proc of 7th Int Ship Stab Workshop, Shanghai France WM, Levadou M, Treakle TW, Paulling JR, Michel K, Moore C (2003) An Investigation of Head-Sea Parametric Rolling and its Influence on Container Lashing Systems. Marine Technol¸ 40(1):1-19 IACS Recommendation No 34, Standard Wave Data Ikeda Y, Munif A, Katayama A, Fujiwara T (2005) Large Parametric Rolling of a Large Passenger Ship in Beam Seas and Role of Bilge Keel in Its Restraint Proc of 8th Int Ship Stab Workshop, Istanbul Neves MA, Rodríguez CA (2005) Stability Analysis of Ships Undergoing Strong Roll Amplifications in Head Seas Proc of 8th Int Ship Stab Workshop, Istanbul Shin YS, Belenky VL, Lin WM, Weems KM, Engle AH (2003) Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design SNAME Trans 111 Shin YS, Belenky VL, Paulling JR, Weems KM, Lin WM (2004) Criteria for Parametric Roll of Large Containerships in Longitudinal Seas, SNAME Trans 112 SLF 48/4/4 Review of MSC/Circ.707 to include parametric rolling in head seas (Australia and Spain) SLF 48/4/8 Proposed Revision of MSC/Circ.707 (to include parametric rolling in head seas) (Germany) SLF 48/4/12 On the development of performance-based criteria for ship stability in longitudinal waves (Italy) SLF 48/4/16 Proposal on MSC/Circ.707 revision (Russian Federation) SLF 48/4/17 Proposal on MSC/Circ.707 revision (Russian Federation) Spyrou KJ (2004) Criteria for parametric rolling, Proc of 7th Int Ship Stab Workshop, Shanghai

Numerical Procedures and Practical Experience of Assessment 305 Tikka KK, Paulling JR (1990) Prediction of Critical Wave Conditions for Extreme Vessel Response in Random Seas. Proc of 4th Int Conf on Stab of Ships and Ocean Veh, Naples Umeda N, Hashimoto H, Paroka D, Hori M (2005) Recent Developments of Theoretical Prediction on Capsizes of Intact Ships in Waves. Proc of 8th Int Ship Stab Workshop, Istanbul Weems KM, Lin WM, Zhang S, Treakle T (2000) Time Domain Prediction for Motions and Loads of Ships and Marine Structures in Large Seas Using a Mixed-Singularity Formulation, Proc of the Fourth Osaka Colloquium on Seakeeping Performance of Ships, Osaka

Parametric Roll and Ship Design Marc Levadou and Riaan van’t Veer Maritime Research Institute Netherlands (MARIN)

Abstract The variety of vessels which can suffer from parametric roll is large. It has been observed on small fishing vessels sailing in following waves but also on large cruise and container vessels in head and following waves. The occurrence of parametric roll, with high roll angles, is governed by a complex combination of main dimensions, loading condition, hull form, appendage configuration, speed and encountered wave conditions. In this paper the influence of main dimension and of variations on the fore and aft body on the occurrence of parametric roll are investigated. A single degree of freedom motion method and a nonlinear time domain simulation method were used. The results were validated with model tests on a C11 container ship. Also, the influence of different roll damping devices on the occurrence of parametric roll is evaluated.

Nomenclature Axx

bc B Bxx Btotal Bcrit Cxx

C Dbk C FH C FH C L  g GMT

: roll added mass : damping gain : ship beam : roll damping : total damping : critical damping : restoring : effective drag coefficient : magnification of the flow over the bilge coefficient : fin to the hull coefficient : lift slope : gravity : transverse metacentre height

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_17, © Springer Science+Business Media B.V. 2011

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hBK

: bilge keel height H : wave height Hs : significant wave height Ixx : ship inertia in roll kxx : radius of gyration of roll L or L pp : length between perpendiculars

l BK Mx

rBK rfin

: bilge keel length : first order wave excitation : arm of bilge keel to centre of gravity : arm of fin stabilizer to centre of gravity

SGM S e

: spectral density of the wave encounter

T

: roll natural period

Tp

: wave peak period : encounter wave peak period

T pe Vs

 

ρ



 a  

: spectral density of GM fluctuations

: ship speed : roll motion : roll velocity : density of water : displacement : natural roll frequency wave : amplitude : non dimensional damping reduction : wave length

1 Introduction In 1998 a post-Panamax, C11 class containership lost 1/3 of her deck containers and damaged another 1/3 in a severe storm. The incident was analyzed by means of numerical simulations and model tests. The results confirmed that the vessel suffered from a severe case of parametric roll during the storm (France 2003). Since the publication of these results, ship operators and ship designers have become more aware of the fact that this phenomenon can occur for larger vessels in confused seas and not only for small vessels in regular waves which was thought for many years.

Parametric Roll and Ship Design 309

New designs of large vessels, in particular container vessels, are since then more and more checked on their tendency for parametric roll behaviour. Class societies acknowledged the problem and have started incorporating parametric roll in their guides (Shin 2004). Still, model tests are considered as the way to assess the sensitivity for parametric roll and such tests are expensive in a design parameter space. Analytical tools that can predict the phenomenon are not easy to use, not always reliable or available. Therefore hull lines variations or optimization with regard to parametric roll is seldom performed in an early stage of new designs. In order to incorporate parametric roll in ship design an understanding of the phenomenon is required. A greater understanding is needed in how main dimensions, hull form changes and appendages configuration alter the probability of parametric roll. In this paper result of a study on the effect of main dimension variations, hull form variations and different appendages configuration on the occurrence of parametric roll will be presented. In Chapter 2 the phenomenon is explained and a discussion of the minimal requirements for parametric roll to occur is given. In the third chapter a discussion will be given of how the main dimensions and the loading condition of a vessel can result in conditions where parametric roll can occur. Different graphs are presented showing the “critical” combinations of vessel length and loading condition. In Chapter 4 results of the effect of hull from variations on a C11 post-panamax container vessel on the occurrence of parametric roll are presented. Variation of bow flare and stern configuration is investigated. In the last Chapter the effect of different roll reduction devices such as bilge keels, anti roll tanks and active fin stabilizers is discussed.

2 Background 2.1 Theory The theory behind parametric roll and its consequence has been studied and described by many investigators, see for example Kempf (1938), Graff and Heckscher (1941), Paulling (1959 and 1961), Oakley (1974), Dunwoody (1998), Dallinga (1998), or Luth (1998). Therefore, in this paper only the principles of parametric roll will be described. In “normal” sailing conditions the ship motions of a vessel are caused by direct wave excitation. Resonant roll motion often occur in beam waves and stern quartering waves, when the combination of wave period, vessel speed and heading leads to an encounter frequency wave period close to the natural roll period of the vessel. In pure head seas condition, the first order roll wave excitation is zero. Nevertheless, under certain conditions of encounter period, roll motion can be excited in head seas, via a different phenomenon. This phenomenon is referred to as “auto

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parametrically excited motion” which is usually shortened to “parametric motion” or “parametric roll”. The term describes a state of motion that results not from direct excitation by a time-varying external force or moment but from the periodic variation of certain parameters of the oscillating system. The roll motion, once started, may grow to large amplitude, limited by roll damping and, in extreme conditions, may result in danger to the ship or its contents. For a ship in head or stern seas the uneven wave surface together with the pitch-heave motion of the ship results in a time-varying underwater hull geometry. This varying geometry, in turn, results in time-varying changes in the metacentric height, i.e., in the static roll stability. The variation of the static roll stability can cause instability if it occurs in the appropriate period. From theory and as validated by model tests (Dallinga 1998, Luth 1998, France 2003), parametric roll occurs when the following requirements are satisfied:  The natural period of roll is equal to approximately twice the wave encounter period.  The wavelength is in the order of the ship length (between 0.8 and 1.2 times LBP).  The wave height (thus the GM variations) exceeds a critical value.  The roll damping is low (lower threshold wave height).

2.2 Prediction of Parametric Roll Nonlinear, time domain seakeeping computer codes are able to predict the phenomenon of parametric roll (see below). These computations are however not easy to use or available for everyone. In a preliminary design stage a simple and fast method is desirable. For parametric roll Dunwoody (1989) proposed such a method, which is used in this paper. His method is based on a single degree of freedom motion equation for roll, using a time varying restoring coefficient. This motion equation is known as the Mathieu equation, which is presented first. 2.2.1 Modeling in One Degree of Freedom The equation for one degree of freedom roll motions is given by equation (1): d 2 d (1)  I xx  Axx  2  Bxx  Cxx  M x dt dt In head seas condition the roll moment excitation will be zero, similar as in a roll decay test. The roll motion equation reduce in these situations to: d 2 d  2  2  0 (2) 2 dt dt where the roll period is defined by:

Parametric Roll and Ship Design 311

 

Cxx  I xx  Axx ( )

 g GM I xx  Axx ( )

and where the damping ratio is defined as: Bxx ( )  2  I xx  Axx ( ) 

(3)

(4)

The restoring moment or static stability GM of the ship when sailing in waves will vary in time and the variation is a function of the actual wetted surface contour and thus depends on the hull lines around the calm water line. The largest variations in restoring occurs when the ship has large pitch motions, so when it sails in a wave length equal to about the ship length. A formula for the time varying restoring force, for the upright ship, in regular waves is given by: Cxx (t )   g   GM m  GM a cos( t )  (5) Where GM m is the mean GM, that is the GM in calm water GM a is the amplitude of the GM variation and  is the wave frequency. When equation (5) is used in equation (1) a one-dimensional Mathieu equation is obtained; a linear second order differential equation with periodic coefficients. The damped Mathieu equation is now written as: d 2 d  2  2  a cos( t )    0 (6) 2 dt dt Parametric resonance conditions for this equation can be found when:   1 (7) n  2 2 2 According to Francescutto (2002) the threshold value for parametric roll in regular waves is:  GM 4 (8)  GM n This threshold has been used and compared with simulations and model test. The results are presented in section 4.4. Furthermore in this paper a one degree of freedom based method proposed by Dunwoody (1989) will be used. The method is shortly described below. 2.2.2 Response to GM Fluctuations (Dunwoody, 1989) The method proposed by Dunwoody is based on the following assumptions: 1. The roll motion can be expressed by the differential equation for a single degree of freedom oscillator with parametric excitation of the stiffness. 2. That there is a linear relation between roll stiffness excitation (GM fluctuations) and wave height.

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3. That a relation can be made between the spectrum of the stiffness fluctuations with the incident wave height spectrum and the speed of the ship. 4. That GM fluctuations produce an effect analogous to a reduction in the roll damping. 5. If the reduction in roll damping is larger than the roll damping from the hull and appendages an unstable situation occurs and the vessel is subject to parametric roll. Furthermore, Dunwoody demonstrates that for a wide band random process the spectral density can be estimated for a frequency of twice the natural roll frequency. Given the assumptions above the spectral density of the GM fluctuations can be expressed as the product of the spectral density of the wave encounter spectrum (at twice the natural roll frequency) and the square of the transfer function of the GM fluctuations (at twice the natural roll frequency). This is expressed in equation (9). 2

 GM a ( )  (9) SGM    Se ( ) a   According to Dunwoody the non dimensional damping reduction follows from:  g 2 SGM (10)   43 k xx4 Parametric roll will occur if the non dimensional roll damping reduction exceeds the total roll damping Btotal (made dimensionless by dividing with the critical roll damping Bcrit ), see equation (11).

Btotal    0 Bcrit Bcrit  2

(11)

 Axx  I xx   g GM

The threshold wave height (the wave height for which parametric roll will start) can be determined by varying the wave spectrum in equation (9) and determining when equation (11) becomes zero. 2.2.3 Nonlinear, Time Domain Seakeeping Code PRETTI The development of a 3D panel code for seakeeping motion prediction has been a point of interest for the Co-operative Research Ships (CRS) since many years. In this joint-industry project a large group of different companies, such as class societies, ship-yards, ship operators, navies and research/engineering companies are actively involved in research related to many aspects in the design and operation of ships. In recent years the CRS has developed a time-domain seakeeping code (Pretti) based on the hydrodynamics as calculated in the frequency domain by a 3D panel code (Precal). The current time-domain code incorporates nonlinear excitation by pressure integration over the actual wetted surface. Diffraction forces are considered

Parametric Roll and Ship Design 313

linear. Hydrodynamic coefficients and oscillatory (manoeuvring) derivatives are specific to sinusoidal motions and in a general theoretical model a ship manoeuvring in a seaway, the ship motion cannot be considered simply sinusoidal. The motion equation must account for transient and random motions. This problem was initially discussed by Cummins (1962) and this approach of impulse response functions is adopted in Pretti. The behaviour or a ship travelling in a seaway ‘integrates’ in practice two areas which are traditionally studied as separate problems and have lead to two different mathematical approaches: a seakeeping theory assuming small motion amplitudes and the theory of manoeuvring assuming calm water and thus frequency independent hydrodynamics. In Pretti the two theories are combined as discussed by Bailey (1997) or Fossen (2004). This is a challenging area of research since there is an overlap between the two models, which require a careful implementation of manoeuvring coefficients. Ideally the aim is that with vanishing wave height the manoeuvring capabilities of the ship are found, and that the seakeeping hydrodynamics are captured in moderate wave conditions. In large wave conditions with large amplitude motions the assumptions behind both the seakeeping and manoeuvring theory are violated since large variations in wetted surface are not accounted for when the basic coefficients in the models are calculated. Model tests are an essential guidance for the user of nonlinear time domain simulation tools to gain experience in the use of a unified model. The current nonlinear time domain code of CRS focuses on course-keeping of the ship in 6 degrees of freedom (6DOF), in which the interaction with the manoeuvring model is already essential. Especially the sway hydrodynamics will influence roll motions and this means that seakeeping (roll damping) and manoeuvring is to be combined. Yaw manoeuvring and a PID controlled rudder are further more essential to keep the ship at its track and course.

3 Influence of Main Dimensions on Parametric Roll In the theoretical background four criteria for parametric roll to occur were given. The first two (the natural period of roll is equal to approximately twice the wave encounter period and the wavelength is on the order of the ship length) can be described with the following equations: T  2Tpe (12)

  LPP

(13) Equation (14) gives an approximation for the natural roll period (assuming 10% added mass). Equation (15) gives a relation between wavelength (λ) and wave period for deep water.

314 M. Levadou and R. van’t Veer

T 

2.2 k xx

(14)

gGM 2 g

Tp 

(15)

For zero speed equations (14) and (15) can be substituted in (12). GM can then be expressed in the following way (for zero speed).

GM  0.605

k xx2

(16)



For non zero speed the following relation between peak wave period and peak encounter wave period should be used (Vs being the ship speed and μ the wave direction, 180 being head waves).

Tpe 

Tp 2 Vs cos  1 Tp g

(17)

Equation (16) gives a relation between GM, radius of gyration and wave length. By substituting also the second criteria for parametric roll (equation (13)) the relation becomes purely dependent of geometry and vessel main dimensions. When the radius of gyration is not known, standard values for kxx/B can be used. By also using standard values for L/B equation (16) can be rewritten in the following way:

GM  0.605

L2  k xx    L 

2

(18)

In Table 1 the values for kxx/L are given for a range of L/B and kxx/B values. Table 1 kxx/L for a range of L/B and kxx/B values L/B kxx /B

5.6

5.8

6

6.2

6.4

0.38

0.0679 0.0696 0.0714 0.0732 0.0750

0.0655 0.0672 0.0690 0.0707 0.0724

0.0633 0.0650 0.0667 0.0683 0.0700

0.0613 0.0629 0.0645 0.0661 0.0677

0.0594 0.0609 0.0625 0.0641 0.0656

0.39 0.40 0.41 0.42

Using equation (18) and the above table, two graphs, Fig. 1 for zero speed and Fig. 2 for 10 knots, can be made. They show the combinations of GM and Lpp for which parametric roll instability might occur. In other words where the first two criteria for parametric roll to occur are met.

Parametric Roll and Ship Design 315

Fig. 1 Combinations of GM and vessel length resulting in parametric instability at 0 knots

Fig. 2 Combinations of GM and vessel length resulting in parametric instability at 10 knots

Fig.1 and Fig 2. show the relation between GM and Lpp for an upper limit of wavelength/Lpp = 1.2 (dashed lines) and a low limit of wavelength/Lpp = 0.8 (solid lines). For a certain kxx/Lpp ratio the area between the dashed and solid lines give the GM range for a certain Lpp for which the criteria are met. With these graphs it is easy to check in an early stage if the vessel can be subject to parametric roll. To further illustrate how to use these graphs an example is given below. C11 class container vessel: Lpp = 262 m; B = 40 m; kxx/B = 0.40; Thus: kxx/L = 0.061 In Fig. 1 and Fig. 2 the range of GM’s which will give parametric instability is indicated. At zero speed the range will be between GM = 1.5 m and GM = 2.2 m and at 10 knots speed between GM = 2.2 m and GM = 3.8 m. This doesn’t mean the vessel will actually be subject to parametric roll. This depends on other factors

316 M. Levadou and R. van’t Veer

like the amount of parametric excitation (GM fluctuations in waves) and the amount of roll damping. These factors depend on the hull form and the size of the appendages.

4 Hull Form Variations 4.1 Selected Case In order to investigate the influence of the hull form on the occurrence of parametric roll a hull form needed to be selected. As a case the post-Panamax, C11 containership which encountered the storm, as described in the introduction, is used. It is a logical starting point since model tests were previously performed on the same vessel at MARIN.

Fig. 3 C11 Hull Form

A rendering of the C11 hull form is presented in Fig. 3. Typical of postPANAMAX containerships, the C11 has extensive bow and stern flare. The ship’s natural roll period as used in this study was 25.7 seconds. This corresponds to the estimated roll period of the ship at the time of the incident. The GM in calm water was about 2 m. The original hull form and variations at the bow and aft of the hull form were investigated. In the aft a Pram type hull form and a hull form with higher deadrise were taken. At the bow hull lines with a less pronounced bow flare (55 deg) and a more pronounced bow flare (45 deg) were used. This resulted in four variations of the original hull form. In order to have a fair comparison between the different variations it is necessary to keep the draft, GM and natural roll period the same. Because of the different under water hull form it means that the displacement and KG will vary slightly between the different hull variations.

Parametric Roll and Ship Design 317

4.2 Model Tests In order to have validation material model tests were performed. These were performed in the Seakeeping and Manoeuvering Basin (SMB) at MARIN. The basin measures 170 x 40 x 5 m in length, width and depth respectively. Not all hull form variations were tested. The original hull form (model 8004-1) and the pram aft body (model 8004-2) variation were tested.

Fig. 4 Original C11 (upper) and Pram type (lower) aft ship

318 M. Levadou and R. van’t Veer

The models were self propelled during the tests and completely free sailing. The only connection between the model and the carriage consisted of an umbilical for power and data transmission. The model tests were performed for an average speed in waves of 5 knots. The tests were done for head sea conditions and a variation of wave heights and wave periods were tested. For each model the tests started with a test in high waves (Hs = 7.5 m) for which several wave periods were tested. This was done to determine the most critical wave period. For this critical wave period a series of tests with increasing wave heights were done. For all these tests the realization of the encountered waves was kept the same. Only the height of the waves was increased. The same realization of the waves was used for both the models. This means that wave group effect do not influence the comparison of the two vessels. From the model test results it is possible to determine the wave heights for which parametric roll will start (threshold wave height).

4.3 Model tests Results In Fig. 5 the roll damping of the two hull forms (determined from roll decay tests) is compared. One can see that the roll damping of the pram aft shape is slightly higher, which is according to expectations. Both models were equipped with 40 cm high 76.54 m long bilge keels.

Fig. 5 Roll damping comparison

In Fig. 6 a sample of roll motion time traces are shown for the C11 original hull form in head seas at 5 knots speed. Results for two wave heights are given. They show the roll time trace for the vessel within the same wave realization (same wave

Parametric Roll and Ship Design 319

Fig. 6 Time trace roll motion of C11 Original hull form

but only different amplitude). As can be seen, although the difference in wave height is small (Hs = 4.3 m versus Hs = 3.65 m) the difference in roll response is big. The results illustrate quite well the threshold behaviour of parametric roll. The model tests results are summarized in Fig. 4-5. The figure shows the mean of the 1/10 highest roll motions (A1/10+) as function of the wave height. Each dot in the figure represents a test in irregular head seas.

Fig. 7 Model test results

The model tests results show that the pram hull form results in higher roll angles. It also shows that the roll motions due to parametric roll start earlier meaning that the threshold wave height is lower. From Fig. 5 it could be seen that the damping of the pram is slightly higher than the original v-shape. The results presented in Fig. 7 however show that the roll response is higher. It means that the excitation (GM variations) of the pram hull

320 M. Levadou and R. van’t Veer

form is higher which is also according to expectations. The difference in GM variations will be discussed in the next section. Using a criterion for parametric roll of 10 deg A1/10 one can determine the threshold wave height from Fig. 7. For the C11 original hull form the threshold wave height becomes Hs = 4.3 m and for the pram Hs = 3.5 m.

4.4 Approximation Methods The first step of the calculations was to validate the different methods. If a good enough agreement was found with the model test the second step was to perform calculations for the other hull form variations which were not tested. Calculations were performed using a simplified one degree of freedom method (Dunwoody) and non linear time domain program PRETTI (see also section 2.2). GM variations As discussed earlier in section 2.2 the GM variations in head waves represent the excitation for parametric roll. The static stability in “frozen” longitudinal waves is a good indication of the GM variations of a vessel sailing in waves. These were calculated for the original C11 hull form and the different variations. For this purpose the program SHCP (NSSC 2003) was used. The input for the program are the hull form, loading condition, wave height, wave length and the longitudinal position of the wave crest with respect to the hull. A range of wave conditions and roll angles can be entered. For each condition the pitch-heave static equilibrium is solved (thus preserving equilibrium of weight, buoyancy and trim moments). The righting arms calculated for each condition can be used to determine the GM and thus the GM difference between the sagging and hogging conditions. Using this method one assumes that the dynamic pitch and heave motion do not influence the GM variations. Sample results are given in the Fig. 8. From Fig. 8 the difference in GM between the sagging in hogging conditions can be determined. This can be done for different wave lengths.

Fig. 8 Curve of static stability C11 class container vessel

Parametric Roll and Ship Design 321

In Fig. 9 the GM variations for a range of wave lengths are given for a wave height of H = 5.0 m. The figure shows the GM in sagging condition, hogging condition, the difference between the two (Delta GM) and the difference divided by the wave height (Delta GM/H) which is plotted on the second y-axis (right).

Fig. 9 GM and GM variations versus wave length for C11 class container vessel

In Fig. 10 the linearized GM variations (DeltaGM/H) is given as function of wave frequency for the C11 hull form and the four variations. The frequency corresponding to a wave height equal to the length of the vessel is indicated in the figure.

Fig. 10 Influence of hull shape of GM variations

The following observation can be made from the results presented in Fig. 10. First it can be observe that the changes to the bow flare do not seem to influence the GM variations. Secondly, the V shape aft hull form shows smaller GM variations compared to the original C11 hull form and the pram aft shape shows larger GM variations as the C11 hull form. The latter is in accordance with the model tests results presented in Section 4-3.

322 M. Levadou and R. van’t Veer

Threshold value for parametric roll in regular waves Using equation (8) an estimate of the required GM fluctuations in regular waves can be made. This was done using the following data at 5 knots forward speed: GM = 2.0 m; Bxx = 2.5 e5 Nms/rad, Ixx = 2.12 e7 kNms2; Axx= 3.0 E5 kNms2, which leads to δGM=0.16 m in regular waves of 12.8 s. This is the threshold value at zero roll amplitude. Model tests in regular were not performed with the C11, so that these values cannot be verified. However, compared to the observations in irregular seas, the GM fluctuations are about a factor 10 larger before the C11 shows a steep increase in roll. Response to GM fluctuations (Dunwoody) The results presented in Fig. 10 were used in Dunwoody’s method (see formula 7, 8 and 9). For the damping the results from roll decay tests (see Fig. 5) were used. It was assumed that the different hull form variations had the same roll damping as the original C11 hull form. For each hull form variation the threshold wave height was determined for a speed of 5 knots and a wave period of Tp = 14 s. The results are given in Table 2. Table 2 Comparison threshold wave height Dunwoody and Model tests. Threshold wave height for parametric roll Dunwoody

Model tests

C11 original

4.3 m

4.3 m

Pram aft

3.2 m

3.5 m

V shape aft

5.3 m

-

Large bow flare (45 deg)

4.3 m

-

Small bow flare (55 deg)

4.3 m

-

The results using Dunwoody’s method show very good comparison with the model tests performed. Furthermore the results from Dunwoody’s method confirm the results found from the GM variation calculation. The bow flare does not seem to have any influence on the threshold wave height. The shape of the aft of the vessel has some influence. A very pronounced V shape aft gives the best performance with regard to parametric roll. It must be mentioned that although the bow shape does not directly influence the occurrence of parametric roll it can indirectly influence it. A more pronounced bow flare will results in more slamming events which in turn will make a vessel master decide to reduce the speed earlier than a vessel with a less pronounced bow flare. Because the roll damping at low speed is smaller it is generally more vulnerable to parametric roll than at high speed. So, indirectly the bow flare does have influence o the occurrence of parametric roll. With Dunwoody’s method it is very easy to perform variations in wave conditions. The threshold wave height can then be determined for a variation of wave periods. By combining these lines with a wave scatter diagram it is then possible to evaluate

Parametric Roll and Ship Design 323

the effect of the hull form on the probability of occurrence of parametric roll. This is shown in Fig. 11.

Fig. 11 Threshold wave heights within North Atlantic scatter diagram

In Fig. 11 the limiting wave height is shown for the original C11 hull form and the two aft shape variations (pram and sharp V). The probability of each combination of Hs and Tp is given in thousands. The wave scatter diagram used was area 9 (North Atlantic) all directions from the Global Wave Statistics (Hogben 1986). From Fig. 11 it can be determined that the V shape aft hull form has a 10% smaller number of occurrences of parametric roll (at 5 knots speed in head waves) in the North Atlantic. The results presented in the figure cannot be used to estimate the exact number of occurrences of parametric roll for a given ship. In order to make that estimation the probability of the speed, wave heading and loading condition need to be incorporated. A method to determine this was presented by the author in 2003 (Levadou 2003).

4.5 Simulations Methods The nonlinear time domain simulations have been performed in 5DOF (sway, heave, roll, pitch, yaw) for the lower sea states and in 3 DOF (heave, roll, pitch) for the more extreme sea states. It was verified that the results for intermediate sea states were comparable. The reason to limit the number of DOF in large sea states is a practical one: the Pretti code does not account for large yaw variations. Besides, in larger sea states the numerical model is more difficult to control, in particular due to the relative low forward speed which makes the rudder rather inefficient.

324 M. Levadou and R. van’t Veer

Future enhancements are expected to overcome these limitations. Surge degree of freedom was neglected as well, for similar practical reason. The surge balance requires a dedicated implementation of added resistance, ship resistance and propulsion plant control, which is currently in development. The viscous roll damping in Pretti is currently based on a time-domain implementation of Ikeda’s method. For the bilge keel damping the water velocities at the bilge keel are assessed at each time instance and with a Keulegan-Carpenter number the drag on the bilgekeels are calculated. Lift en eddy damping follows Ikeda’s empirics. All simulations were performed in the same ‘relative’ sea state realisation. This means that is the wave component phase and relative amplitudes were kept identical when the significant sea state was varied. To obtain reliable statistics the simulations were performed for a duration of 3 hours. The computer time requires (CPU) is about half the simulation time. This means that a parametric study as presented in this paper is feasible in a design stage, although significant computer CPU is required for several days. A mesh of the C11 container ship as used in the simulations is presented in Fig. 12.

Fig. 12 C11 container ship mesh for time domain simulations

Model tests were carried out for two different hull shapes, the original C11 container ship and a modified one with the pram aft-body, denoted SIM Pram aft in Fig. 13. The significant roll amplitude compares very well between simulations and model tests for the lower sea states. In the higher sea states the simulations predict larger parametric roll motions then the model tests show. The trend is however well predicted and the significant wave height at which parametric roll starts agree well. The pram aft body might be beneficial for calm water resistance, but makes the design more sensitive for parametric roll. Striking is the fact that the original design shows less parametric roll in the higher sea states than the modified design, while this is opposite in the simulations. We consider this an effect of the not fully developed numerical models and the assumptions made in PRETTI. But,

Parametric Roll and Ship Design 325

most likely nonlinear hydrodynamics or more sophisticated nonlinear (viscous) damping is required.

Fig. 13 Significant roll amplitude from model experiments (EXP) and from nonlinear simulations (SIM) for the C11 and Pram modified hull

Following these results three other hull shape variations were investigated. The results are summarized in the Fig 14.

Fig. 14 Significant roll amplitude in Jonswap wave spectra for the C11 and 4 modified hull shapes

326 M. Levadou and R. van’t Veer

The numerical results are summarized in Table 3, which compares the threshold value with Dunwoody. A threshold significant roll amplitude of about 5 degrees was used. The comparison between Dunwoody and direct simulations is good, which is a striking conclusion. The dynamic GM variations in the simulations are to comparable but to some extent different from the GM variations in an assumed static wave due to the fact that the equilibrium buoyancy condition in waves is not obtained; heave and pitch motions will change the buoyancy of the ship. The numerical simulations show a stronger influence of the bow flare then the static simulations, which could be due to the mentioned reason; the position of the ship in waves will be different in simulations than in a static approach. Table 3 Comparison Dunwoody and Pretti simulations. Threshold wave height for parametric roll Dunwoody

Simulations

C11 original

4.3 m

4.2 m

Pram aft

3.2 m

3.5 m

V shape aft

5.3 m

5.1 m

Large bow flare

4.3 m

4.2 m

Small bow flare

4.3 m

4.6 m

5 Roll Stabilization In the previous chapter the influence of the hull form on the parametric roll threshold wave height has been demonstrated. One other way to influence the threshold wave height is to change the roll damping of the vessel. Usually vessels have a small potential damping. Therefore, adding appendages (bilge keels, fin stabilizers) or anti roll tanks can increase the roll damping drastically. The influence of bilge keels and active fin stabilizers on the threshold wave height is investigated by using Dunwoody’s method.

5.1 Bilge Keel The bilge keel damping is often associated with the energy dissipated by the drag forces of the bilge keel (Dallinga 1998). Within this concept the damping is proportional with the roll velocity amplitude. Equation (19) gives the increase of roll damping per roll velocity amplitude change.

Parametric Roll and Ship Design 327

BBK 4 2   2lBK hBK  CHF rBK  rBK CD , BK (19)  3  According to Ridjanonic (1962) the effective drag coefficient depends on the amplitude of the transverse flow and the bilge keel height, as per equation (20). hBK CD , BK  22.5  2.4 (20)  rBK CHF  Using these equations the bilge keel damping was calculated for several bilge keel heights. The contribution as a fraction of the total damping linearized for 10 deg roll amplitude is given in the first row of Table 4 for 5 knot speed. Table 4 Roll damping contribution from the bilge keel. Bilge keel height 0 cm

20 cm

40 cm

60 cm

Fraction

0

0.256

0.428

0.547

Total Damping [MNms/rad]

145

195

253

320

As can be seen the roll damping contribution of the bilge keel is very large. Using these fractions on the total roll damping determined from the roll decay tests, the damping for a variation of bilge keels can be determined. These values are indicated in the second row of Table 4. Using Dunwoody’s method the threshold wave height was determined for the different bilge keel heights. The results are given in Fig 15. As can be seen, a 40 cm bilge keel raises the threshold wave height by Hs = 1 m. Adding 20 cm raises the threshold by about 0.6 m.

Fig. 15 Influence of bilge keel height on threshold wave height

328 M. Levadou and R. van’t Veer

5.2 Active Fin Stabilizer The influence of active fin stabilizers can be estimated using the same approach. The roll damping (passive and active part) was estimated using the following equation (Dallinga 1993, 1998).  CL rFIN CHF CL  1 (21)  BFIN  2rFIN VS2 AFIN CFH  bC   act  2   pas VS Using these equations the damping of fin stabilizers was calculated for several fin stabilizers area. The damping linearized for a 10 deg roll amplitude is given in Table 5. Table 5 Roll damping contribution of fin stabilizers. Fin stabilizer area 0 m2

10 m2

21 m2

Fin Stab damping [MNms/rad]

0

87

173

Total Damping [MNms/rad]

253

340

426

Using Dunwoody’s method the threshold wave height was determined for the different fin stabilizers area. The results are given in Fig 16 for 5 knot speed. As can be seen a 10 m2 fin stabilizer raises the threshold wave height by Hs = 0.7 m. Adding 21 m2 fin stabilizer raises the threshold by about 1.0 m. The results show that the fin stabilizers give approximates the same increase in threshold wave height as bilge keels for 5 knot speed. However, for speeds higher the influence of the fin stabilizers will be higher. This is due to the fact that the fin stabilizer damping increasing with the square of the speed.

Fig. 16 Influence o fin stabilizers on threshold wave height

Parametric Roll and Ship Design 329

5.3 Results Summary The results of the simulations and model tests have shown that the C11 original hull form results in a lesser probability of parametric roll to occur than the pram aft body. The pram aft body is however beneficial with respect to calm water resistance. Using the results presented in the previous chapters an estimation can be made of the bilge keel height or fin stabilizer area needed on the pram aft body in order to have a comparable threshold wave height as the C11 hull form. This estimation is given in Table 6. The results show that in order to achieve the same threshold wave height the bilge keel must be twice as big or the pram aft body must be equipped with a 21 m2 fin stabilizer. These results are valid for 5 knots speed. For higher speeds the fin stabilizer will give more roll damping than the 0.80 m bilge keel. Table 6 Appendage size, bilge keel or fin stabilizer, needed for same threshold wave height. Appendage size Hull shape

BK height (m)

Fin stabilizer area (m2)

C11 original

0.40

-

Parm aft body

0.80

-

Pram aft body

0.40

21

6 Conclusions In the paper the influence of main dimensions, hull form and roll stabilization on the occurrence of parametric roll are discussed. A one degree of freedom motion method and nonlinear time domain simulations were used and validated with model tests. Regarding the results the following conclusion may be drawn. A relatively simple one degree of freedom motion method can be used in the preliminary stage of a design. An important factor on the results is the used roll damping. Empirical models for the roll damping, tuned with model tests, can be used in a preliminary stage. The method gives an idea of the threshold wave height for which parametric roll will start but not the actual roll angles associated with the rolling. Nonlinear time domain simulations can be used to determine the threshold wave height and to determine the roll angles associated with the parametric roll. Here also tuning of the roll damping is needed in order to get reliable results. The model tests and calculations on a C11 type container vessel have demonstrated that the aft body configuration has more influence on the occurrence of parametric roll than the bow flare. A V-shaped aft body is preferable to a pram type aft body.

330 M. Levadou and R. van’t Veer

Finally, the influence of bilge keel height on the occurrence of parametric roll has been shown. It has also been shown that active fin stabilizers can be used at low speed in order to increase the threshold wave height.

References Bailay PA, Price WG and Temarel P (1997) A unified mathematical model describing the manoeuvring of a ship travelling in a seaway, RINA Cummins WE (1962) The impulsive response function of ship motions, Schifftechnik, Band 7 Dallinga RP (1993) Hydromechanic Aspects of the Design of Fin stabilizers, RINA Spring Meet, Lond Dallinga RP, Blok JJ and Luth HR (1998) Excessive rolling of cruise ships in head and following waves, RINA Int Conf on ship Motions & Maneuverability, Lond Dunwoody AB (1989) Roll of a ship in astern Seas – Metacentric height spectra, J of Ship Res. 33: Nº 3: 221-228 Dunwoody AB (1989) Roll of a ship in astern Seas – Response to GM fluctuations, J of Ship Res. 33: Nº 4 : 284-290 Fossen TI and Smogeli ON (2004) Nonlinear Time-Domain Strip Theory Formulation for LowSpeed Manoeuvring and Station-Keeping, Modeling, Identification and Control. 25: 4 France WN, Levadou M, Treakle TW, Paulling JR, Michel RK and Moore C (2003) An Investigation of Head-Sea Parametric Rolling and its influence on Container Lashing Systems, Mar Technol 40: 1 Graff W and Heckscher E (1941) Widerstands und Stabilitäts versuche mit drei Fischdampfermodellen, Werft-Reederei-Hafen, 22 (also DTMB Translation 75, June 1942) Hogben N, Dacuhna NMC and Olliver GF (1986) Global Wave Statistics, BMT, Lond Ikeda Y, Himeno Y, Tanaka Y (1978) A prediction method for ship roll damping, Technical Report 00405, Univ of Osaka Kempf G (1938) Die Stabilitätsbeanspruchung der Schiffe durch Wellen und Schwingungen, Werft-Reederei-Hafen, 19 Levadou M, Palazzi L (2003) Assessment of Operational Risks of Parametric roll, World Maritime Technol Conf, San Francisco Luth HR and Dallinga RP (1998) Prediction of excessive rolling of cruise vessels in head waves and following waves, PRADS, The Hague Netherlands Naval Sea Systems Command: Ship Hull Characteristics Program User Manual version 4.3 (2003) Paulling JR and Rosenberg RM (1959) On Unstable Ship Motions Resulting from Nonlinear Coupling, J of Ship Res. 3: 1 Paulling JR (1961) The Transverse Stability of a Ship in a Longitudinal Seaway, J of Ship Res. 4:4 Ridjanovic M (1962) Drag coefficients of Flat Plates Oscillating Normally to their Planes, Schifftechnik, Band 9, Heft 45 Shin YS, Belenky VL, Paulling JR, Weems KM, Lin WM (2004) Criteria for Parametric Roll of Large Containerships in Longitudinal Seas, SNAME

Parametric Roll Resonance of a Large Passenger Ship in Dead Ship Condition in All Heading Angles Abdul Munif*, Yoshiho Ikeda*, Tomo Fujiwara**, Toru Katayama* *Osaka Prefecture University; ** National Maritime Research Institute Abstract Although it has been widely believed that a ship in dead ship condition has the largest roll motion at resonance condition, the authors found that, because of parametric roll resonance, a kind of ships has significant heavy rolling in beam waves with slightly smaller period of the half value of its natural roll period if the roll damping is small. In the present paper, model experiments to measure ship motions of a large passenger ship in waves with various heading angle are carried out to confirm the region of heading angles where large parametric rolling appears. The results demonstrate that parametric rolling appears at wide region of heading angles and disappears in certain heading angles.

1 Introduction As well known, occurrence of small parametric rolling of ships in beam seas has been pointed out by many researchers. In the previous paper proposed by Ikeda et al. (2005), however, the authors found experimentally that heavy roll motion with much larger angle than that in resonance appears for a large passenger ship with flat stern and large bow flare in heavy beam seas due to parametric rolling. The measured results demonstrated that the large parametric rolling appears only when the ship has no bilge keel or smaller one. As wave height increases, the parametric rolling suddenly appears at higher wave height than a certain value. The authors also did a simulation of roll motion, and confirmed that similar parametric rolling appears for the ship in beam waves (Munif et al. 2005). In the present study, some additional experimental works to clarify effects of wave height, size of bilge keels and heading angles of waves on occurrence of large parametric rolling are carried out to reveal the characteristics of the parametric rolling.

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_18, © Springer Science+Business Media B.V. 2011

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332 A. Munif et al.

2 Experimental Setup The ship used in the experiments is the 110,000GT passenger ship designed by Fincantieri for an international cooperated research on damage stability of large passenger ships in IMO. The body plan and the principal particulars are shown in Fig. 1 and Table 1, respectively. The bilge keels designed for the ship are divided into two parts, short forward one and long aft one. The roll damping can be changed by selecting these bilge keels.

Fig. 1 Body plan of the ship. Table 1 Principle particulars -

Full Scale

Model

Scale

1/1

1/125.32

LOA

290 m

2.200 m

LPP

242.24 m

1.933 m

Breadth

36 m

0.287 m

Draft

8.4 m

0.067 m 8

Displacement

5.20 × 10 N

264.67 N

GM

1.579 m

0.0126 m

T roll

23 sec

2.05 sec

Bilge keel : width

1.1 m

0.0088 m

Bilge keel : location

s.s.3.0－5.0, s.s.5.25－6.0

In the previous study, a scale-model of the large passenger ship was located in transverse direction in the towing tank of Osaka Prefecture University, and the ship motions, roll, heave, pitch, sway and drift motions, were measured in regular beam waves. In the experiment, yaw and surge motions were fixed.

Parametric Roll Resonance of a Large Passenger Ship 333

In the present study, the same model is located in regular waves for freely yawing conditions or various fixed heading angles for incident waves. Only transverse motion of the model is fixed by a guide. This means that drift motion in incident-wave direction is free but the motion perpendicular to it is fixed. The experimental conditions are shown in Table 2. Table 2 Experimental conditions -

Full scale

Model

Wave period

6.7 – 24.6 sec

0.6 – 2.2 sec

Wave length

70 – 944 m

0.56 – 7.55 m

1.25 – 10 m

0.01 – 0.08 m

Wave height

without , full ( front + aft ) , front , aft Bilge keel

front : s.s. 5.25 – 6.0 aft : s.s. 3.0 – 5.0

3 Experimental Results As mentioned before, in the previous paper, the authors experimentally showed that the effect of wave height on parametric rolling in regular beam waves. To confirm the results in wider wave height region, parametric rolling of the ship without bilge keels is measured for various wave heights. The maximum amplitude of the parametric rolling at each wave height is shown in Fig. 2. The results demonstrate that the parametric rolling appears just over 30mm of wave height, rapidly increases with wave height, and reaches the maximum amplitude that is about 27 degree. It should be noted that the roll amplitude does not proportionately increase with increasing the wave height but seems to saturate to the maximum one. In Fig. 3, effect of area of bilge keels on maximum amplitude of parametric rolling is shown. The results demonstrate that the parametric rolling significantly depend on the area of bilge keels. It is safely said that large bilge keels can completely erase parametric rolling in beam seas. To investigate the effect of heading angle on the parametric rolling of the dead ship, the models are released in head wave condition (χ=180°) or following wave condition (χ=0°) in regular waves with 0.04m of wave height and 0.95 seconds of wave period which is the period when the large parametric rolling in beam seas appears. The time histories of the heave, pitch, roll and yaw motions are measured and showed in Figs. 4 and 5. The model has no bilge keels in these measurements.

334 A. Munif et al.

From the time histories of the yaw angles, we can see that, in both cases, the heading angle of the ship slowly changes to beam sea condition. The time histories of roll motion in these figures show that the period of the roll motion is twice to the heave and pitch motions. The results demonstrate that the parametric rolling occurs in wide heading angles as well as in beam seas. 30 25

Φ 0 (deg)

20 15 10 5 0

0

20

40

H w (mm)

60

80

100

Fig. 2 Effect of wave height on maximum amplitude of parametric rolling of the ship in regular beam seas. 30 25

Φ 0 (deg)

20 15 10 5 0

0

0.2

0.4

0.6

0.8

1

Area of Bilge Keel / Area of Designed Full Bilge Keel

Fig. 3 Effect of area of bilge keels on maximum amplitude of parametric rolling in regular beam seas.

Parametric Roll Resonance of a Large Passenger Ship 335

Fig. 4 Time histories of motions of the ship without bilge keel released from head sea condition in regular waves at Tw=0.95sec and Hw= 0.04m.

336 A. Munif et al.

Fig. 5 Time histories of motions of the ship without bilge keel released from following wave condition in regular waves at Tw= 0.95sec and Hw= 0.04m.

Parametric Roll Resonance of a Large Passenger Ship 337

Since it may take time to lead to the parametric rolling and reach its steady condition, a measurement of roll motion at some fixed heading angle; 0, 90, 180 degree are carried out. And using the results of above-mentioned measurement and of previous ones shown in Figs. 4 and 5, the roll amplitude of the parametric rolling for each heading angle is plotted in a polar diagram as shown in Fig. 6. The results show that amplitudes of the parametric rolling are significant in following and head seas as well as beam seas when no bilge keel is attached.

Fig. 6 Effect of wave direction on amplitude of parametric rolling of the ship without bilge keel in regular waves. 2 1.5 1

GZ(m)

0.5 0

-0.5 0

10

20

30

40

50

60

-1

-1.5 -2 -2.5

Still water Wave trough wave crest

-3

heel angle (deg) Fig. 7 Variation of GZ curves in head waves of λ/L=1.5 and Hw/λ=0.033.

338 A. Munif et al.

2 1.5 1 GZ(m)

0.5 0

0

10

20

30

40

50

- 0.5 -1 - 1.5 -2

still water wave trough wave crest φ (deg)

Fig. 8 Variation of GZ curves in beam waves of λ/L= 0.47 and Hw/λ=0.04.

Figs. 7 and 8 show variation of GZ-curves of the ship on the basis of FroudeKrylov assumption in heading and beam seas, respectively. We can see that the GZ-curves significantly vary between wave crest and wave trough in head and beam waves. These large variations of the GZ-curve create the large amplitude of parametric rolling. To investigate the effect of roll damping on occurrence of parametric rolling in various heading angles, motions of the model with bilge keels are measured in regular waves (Hw = 0.08m, Tw = 0.95 sec). In Fig 9 the time histories of motions of the ship with smaller front bilge keels released from following wave condition in the regular waves are shown. Smaller parametric rolling occurs in all heading angles of waves. When larger aft bilge keels or full bilge keels are attached, no parametric rolling appears as shown in Figs. 10 and 11.

Parametric Roll Resonance of a Large Passenger Ship 339

Fig. 9 Time histories of motions of the ship with short (front) bilge keels released from following wave condition at Tw=0.95sec and Hw= 0.08m.

340 A. Munif et al.

Fig. 10 Time histories of motions of the ship with long (aft) bilge keels released from head wave condition at Tw= 0.95sec and Hw= 0.08m.

Parametric Roll Resonance of a Large Passenger Ship 341

Fig. 11 Time histories of motions of the ship with full (front and aft) bilge keels released from head wave condition at Tw=0.95sec and Hw= 0.08m.

342 A. Munif et al.

To search the possibility of occurrence of parametric rolling of the ship with bilge keels in head waves, measurements of ship motions of the model with full bilge keels in fixed yaw condition are carried out in regular head waves with 0.08m of wave height and various wave periods. The results are shown in Fig. 12. The results suggest that parametric rolling with about 10 degrees can occur for the ship with full bilge keels in head waves at 1.2 second of wave period, which is longer by 26% than that when parametric rolling in beam seas occurs. In Fig. 13, the time histories of motions of the case that the maximum parametric rolling occurs are shown. It should be noted it takes long time to develop parametric rolling. In Fig. 14, the effect of heading angle on parametric rolling of the ship with full bilge keels in head waves is shown in a polar diagram. We can see that in heading angles of  20 degrees in head and following waves parametric rolling occurs even if size of bilge keels is enough to erase it in beam seas.

12 Tw = wave period

10

Troll = roll natural period

φ 0 (deg)

8 6 4 2 0

0

0.5 Tw / Troll

1

Fig. 12 Parametric rolling of the ship with full bilge keels in regular head waves with Tw= 1.2sec and Hw= 0.08m.

Parametric Roll Resonance of a Large Passenger Ship 343 wave hight (mm)

50 30 10 - 10 0

10

20

30

40

50

60

70

80

50

60

70

80

50

60

70

80

50

60

70

80

- 30 - 50

time (sec)

heave (mm)

030

0 0

10

20

30

-030

40

time (sec)

pitch (deg)

4 2 0 0

10

20

30

40

-2 -4

time (sec)

roll (deg)

30 20 10 0 - 10

0

10

20

30

40

- 20 - 30

time (sec)

Fig. 13 Time histories of motions of the ship with full bilge keels released from head wave condition in regular waves with Tw= 1.2sec and Hw= 0.08m.

344 A. Munif et al.

Fig. 14 Effect of wave direction on amplitudes of rolling and its period of the ship with full bilge keels in regular waves.

In Fig. 15, difference of wave periods for which parametric rolling appears in beam and head waves. The ship has no bilge keels and wave height is 0.04m. The results suggest that parametric rolling appears in different wave periods in beam and head waves. This may be because of differences of drift speed and amplitude of stability variation in beam and head waves. 30 Hw = 0.04m 180deg head sea 90deg beam sea

20

without Bilge Keel

φ 0 (deg)

25

15 10 5 0

0

0.5

λ / Lpp

1

1.5

Fig. 15 Difference of wave periods for which parametric rolling appears in beam and head waves.

Parametric Roll Resonance of a Large Passenger Ship 345

4 Conclusion Following the previous paper of the authors in which large parametric rolling in beam seas is confirmed for a large passenger ship with a flat and shallow sternbottom and large flare in bow in no bilge keel condition, parametric rolling in all heading angles of the same ship is experimentally investigated. Following conclusions are obtained. 1. When the ship has no bilge keels, parametric rolling occurs in wide heading angles including head and following seas as well as beam seas. Large variation of GZ value in waves due to a flat and shallow stern-bottom and large flare in bow induces the parametric rolling. 2. The effect of bilge keels on occurrence of large parametric rolling is significant. Large bilge keels can completely erase parametric rolling of the large passenger ship in dead ship condition in any heading angles of waves except head and following waves. 3. Parametric rolling of the ship in head seas can occur even if large bilge keels enough to erase it in beam seas are attached. 4. Wave periods when parametric rolling occurs in beam and head waves are different.

Acknowledgements This work was a part of the first author’s post-doctoral research supported by Grand-in-Aid funded by the Japanese Society for the Promotion of Science (JSPS). Just after the present study, Dr. A. Munif was died on the 9th, February 2006 because of sickness. The other authors express their deepest regret for him and his family.

References Ikeda Y, Munif A, Katayama T et al (2005) Large Parametric Rolling of a Large Passenger Ship in Beam Seas and Role of Bilge Keel in Its Restraint. Proc. of 8th International Ship Stability Workshop. Munif A, Katayama T, Ikeda Y (2005) Numerical Prediction of Parametric Rolling of a Large Passenger Ship in Beam Sea. Conference Proc. of the Japan Society of Naval Architects and Ocean Engineers, No.1.

Parametric Rolling of Ships – Then and Now J. Randolph Paulling Professor Emeritus, University of California

Abstract Modern research on parametric rolling of ships was first conducted in Germany in the late 1930s. This work was initiated in an effort to explain the capsizing of some small ships such as coasters and fishing vessels in severe following seas. The work included experiments with models in open water as well as numerical and theoretical computations. The phenomenon was thought to be of concern principally in following seas and for small, low freeboard ships. In the 1990s, however, there were reports of containerships and even some cruise ships experiencing heavy rolling in head seas. These were ships having a hull form characterized by great flare forward and wide flaring stern sections, features that are known to lead to significant stability variations in waves. The APL CHINA casualty in October 1998 focused attention on parametric rolling in head seas. The investigation of this casualty included theoretical computations of GZ variations, model experiments and numerical simulations as well as meteorological studies of the wind and sea conditions prevailing at the time. Results of the investigation received wide dissemination in the technical press. Since the APL CHINA casualty and other similar incidents, much theoretical and experimental work has been focused on head seas parametric roll. IMO and many of the Class Societies now have recommendations to designers and masters for avoiding head seas parametric rolling situations.

1 Introduction The phenomenon of parametric roll has been recognized by naval architects for more than fifty years. It may be described as a spontaneous rolling motion of the ship moving in head or following seas that is not directly excited by the waves but comes about as a result of a dynamic instability of the motion. In pure head or following seas the transverse symmetry of the ship would imply that no waveinduced roll exciting moment should be present. Nevertheless, for certain frequencies of wave encounter, it is found that a small initial disturbance in roll can trigger an oscillatory rolling that can grow to appreciable amplitude after only a few cycles. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_19, © Springer Science+Business Media B.V. 2011

347

348 J.R. Paulling

This is explained in terms of a dynamic instability or bifurcation in the motion characteristics. In 1863, William Froude identified a ship roll-heave coupling phenomenon akin to parametric roll consisting of heave oscillations at the heave natural frequency excited by the small roll induced variations in the buoyant force which occur at twice the roll frequency. The occurrence of parametric roll is directly related to variations in the transverse stability of the ship as it moves through head or following waves. As the ship encounters successive waves, the geometry of the immersed portion of the hull varies as the waves move along the ship length. The underwater geometry changes with time due to the pitch-heave motion combined with the wave profile. This results in a time varying transverse stability as measured by both GM for small angles of heel and the GZ curve at large angles. Figure 1 contains such righting arm curves for a general cargo ship of 1960s vintage and a modern post Panamax containership, both in waves of length equal to ship length and height equal to L/20. The general character of the time varying curves is illustrated here consisting of diminished stability when the wave crest is amidships and enhanced stability when a trough is amidships. For intermediate positions of the waves the stability values vary accordingly.

2 A Simple Dynamic Model of Parametric Roll The dynamic rolling motion of a ship undergoing these stability variations in head or following seas is analogous to that of a spring-mass system in which the spring constant varies sinusoidally in time. For the ship the frequency of these variations equals the frequency of wave encounter. Let us now examine the small amplitude roll dynamics of a ship with periodic stability variations using a simplified model as follows. As our ship moves through head or following seas the time-varying GM may be expressed as Equation (1).

GM (t )  GM o (1  C cos  t )

(1)

Here, GMo = still water GM, C = fractional variation of GM due to waves, heave and/or pitch,  = encounter frequency. The equation of motion for uncoupled roll motion without excitation is now given by Equation (2).

Ix

d 2   (GM o  CGM o cos t )  0 dt 2

(2)

Parametric Rolling of Ships – Then and Now 349 C4-S-57a IN L/20 WAVES

1,6 1,4 1,2 1

SW

GZ - M

0,8

CRST MID TRGH MID

0,6 0,4 0,2 0 -0,2

0

10

20

30

40

50

60

70

80

-0,4

HEEL - DEG

C11 CONTAINERSHIP IN L/20 WAVES

2,5 2 1,5

GZ - M

1

SW CRST

0,5

TRGH

0 -0,5

0

10

20

30

40

50

60

70

80

-1 -1,5

HEEL - DEG

Fig. 1 Righting arm curves for two ships in calm water and following seas: C4 – AMERICAN CHALLENGER Cargo Ship of 1960s vintage and C11 Post Panamax Containership (Body plans not to scale).

350 J.R. Paulling

Here, = angle of roll, Δ = ship displacement, Ix = mass moment of inertia in roll, including added mass effect. Now divide both sides of (2) by Ix and make the change of variable, 2 d2 2 d    t for which 2   . dt d 2 GM o 2 We note that  n  , Ix where  n  natural frequency of roll.

We define,



GM o  n2  2,  2Ix 



C GM o  n2  C . I x 2 2

The equation of roll motion now becomes,

d 2  (   cos  )  0 . d 2

(3)

This ordinary differential equation with sinusoidally varying spring constant is known as the Mathieu equation and the behavior of its solutions have been determined. Figure 2, known as the Ince-Strutt diagram illustrates the stability of solutions for the Mathieu equation. In this diagram the shaded regions represent stable solutions to the equation and unshaded regions correspond to unstable solutions. Thus, if the two parameters (δ,ε) for the system lie in a stable region, an arbitrarily small initial disturbance will die out with increasing time, while if they lie in an unstable region, the disturbance will grow with time. We see that  is equal to the square of the ratio of the natural frequency of roll to the frequency of variation of GM and  is proportional to the fractional change in GM. The first unstable region is centered on a value =1/4 or a ratio of natural frequency to frequency of GM variation of ½. If the frequency of GM variation does not exactly satisfy this value, unstable motion can still occur for a sufficiently large value of C, i.e., if the amplitude of the variation in GM is sufficiently large. The effect of linear damping is to merely raise the threshold value of C at a given frequency of variation, . The unstable motion will still take place if C is sufficiently large and, in general, will grow without bound. In order for a limit on the ultimate amplitude to exist, there must be nonlinear damping in the system of which quadratic roll damping is an example. From the foregoing simplified analysis we can expect that, if the ship encounters regular head or following seas at a frequency near one-half the natural frequency of roll, a small disturbance in roll will grow to appreciable amplitude depending on the amplitude of the stability variation and the roll damping. In real situations, the initial disturbance is almost always present and supplied by some external effect such as wind or oblique wave components.

Parametric Rolling of Ships – Then and Now 351

Fig. 2 Ince-Strutt diagram illustrating stability of solutions of the Mathieu Equation.

3 Early Parametric Roll Research Much of the earliest work on parametric roll was conducted in Germany starting in the late 1930s and focused on smaller working ships such as fishing boats, seagoing tugs and small coastal cargo craft which had experienced a number of casualties. Some of the research involved experiments with free running models which demonstrated the phenomenon quite convincingly. The reasons for concentrating on smaller ships, aside from their casualty records, were two in number. First, smaller ships are more likely than large ships to encounter waves that are high and steep in relation to the ship dimensions. Second, they generally operate at Froude numbers that are more likely to result in a frequency of encounter within the critical range for the ship of marginal stability. Such ships are, of course, the ones in greatest danger of capsize. Further, it was felt that the transient behavior of the roll motion was rather gradual, i.e., a large number of cycles would be required in order for the rolling motion to grow to appreciable amplitude. This was due in part to observations during the aforementioned model tests and in part to some early numerical simulations of the phenomenon which showed a slow growth of amplitude of the roll motion. This slow transient growth was interpreted to mean that in order for the rolling motion to grow to appreciable amplitude, a rather large number of nearly regular waves would have to be encountered.

352 J.R. Paulling

Fig.3 Unidirectional random waves observed from moving and stationary points.

Waves in the real ocean, however, are random in character although in a system of random waves there will, from time to time, be found groups of several waves that appear to be almost regular. Furthermore, the waves as observed from a moving point appear to be somewhat different from the waves at a fixed point. To a moving ship, therefore, following waves appear to have distinctly different characteristics compared to head seas. This is a consequence of the dispersion relation, i.e., the dependence of wave velocity on wave length. When observed from a point moving in the same direction as the waves, as in following seas, the wave spectrum is compressed into a narrower band of frequencies than that for a fixed point, while head seas result in a spreading of the spectrum over a wider band of frequencies. The resulting appearance of the wave time history is shown in Figure 3. Here, the upper graph contains the wave profile observed from a point moving in the same direction as the waves, the center graph is the profile observed from a fixed point and the lower graph corresponds to a point moving opposite to the wave motion.

4 The APL CHINA Casualty In the early 1990s several containerships reported the occurrence of heavy rolling in head seas and the description of the incidents indicated the possibility of parametric rolling. Some of the reports indicated that the rolling motion built very quickly, during the encounter of three or four waves not several tens of waves as would have been expected. These containerships had hull forms characterized by great flare forward and wide flaring stern sections. The reason for these form features was a desire to obtain more container space on deck while keeping a fine underwater form to minimize resistance at the high operating speeds of the ships. These form features, however, lead to large stability changes as the ship moves through waves from the crest amidships to trough amidships positions. Furthermore, the typical metacentric heights at which these ships operate put their natural roll frequencies into the range of one-half the head sea encounter frequencies for wave lengths approximately equal

Parametric Rolling of Ships – Then and Now 353

to ship length. These are typically the waves causing the most pronounced stability variations. The effect on stability of the flared form can be seen in Figure 1. At the smaller angles of heel, the righting arm curves, trough versus crest amidships, diverge much more strongly for the C11 containership than for the older, more wall-sided dry cargo ship of traditional form. This implies a larger value for C, the coefficient of variation of GM in Equation (1) and would lead to increased susceptibility to occurrence of parametric roll, all other things being equal.

6 TH International Ship Stability W ork sh op. A n Investigation of H ea d-Sea Param etric R olling and its Influence on C ontain er L ashing System s, O ctob er 200 2.

Fig. 4 APL CHINA container damage.

The APL CHINA casualty in October 1998 focused worldwide attention on head seas parametric roll. While eastbound in the North Pacific Ocean, the ship found itself in a rapidly moving weather system that was formed by the convergence of two low pressure systems and, despite weather routing advice, was unable to avoid the rapidly changing situation. The resulting combined system was described as an “explosively intensifying low” or meteorological ‘bomb” and referred to by some as a Pacific version of the “perfect storm”. At the height of the storm, the ship’s deck log recorded estimated wind of Beaufort force 11 and sea state 9. A weather and sea state hindcast performed after the accident gave peak sustained winds of 30.8 m/s or 60 knots and a significant wave height of 14.9 m During the height of the storm, the master attempted to maintain a heading into the seas at a speed just sufficient to retain control. The ship reported rolls of over 40 degrees and violent pitching in the very heavy and confused seas. As a result of the violent motions combined with boarding seas, heavy damage to- and loss of a large number of containers was sustained and this is illustrated in Figure 4. In order to determine the causes of this casualty, extensive studies including weather analyses, model experiments, numerical simulations, and theoretical analyses were undertaken. At the conclusion, it was felt that head seas parametric roll was clearly established as the major cause. One significant outcome of this and related work is the development by classification societies as well as IMO of rules and guidelines for the avoidance of parametric roll situations both in the design and operation of ships. Much more fundamental research on parametric roll has been conducted since the APL CHINA studies. Many model tests have been conducted, simulation codes have been developed and exercised, and theoretical work has been conducted on

354 J.R. Paulling

the probabilistic nature of nonlinear phenomena. A considerable body of literature on the subject exists, much of it the product of the STAB symposia. Rather than attempting to summarize it we will instead take a look at two specific side issues, one concerning the nature of the transient roll response in regular waves and wave groups and the second concerning the difference in behavior of the stability variations in head seas in contrast to following seas.

5 A Somewhat More Sophisticated Model Let us consider a slightly modified form of Equation (2) as follows,

d 2 d  B( )   GZ ( , t )  0 (4) 2 dt dt d Here, B ( ) is a general roll damping which may include linear and nonlinear dt (quadratic) terms. GZ ( , t ) is the time varying restoring moment function. Ix

The most suitable method of solving equation (4) is stepwise numerical integration and a number of suitable methods are available. The forth order Runge-Kutta method was used in the present examples. The function GZ ( , t ) may be visualized as a tabulated family of curves similar to those in Figure 1 computed for successive closely spaced positions of the wave along the ship length. The damping term may be estimated from data found in the literature and many ship hydrostatics codes are available for constructing the family of GZ curves. A simple interpolation scheme is used for finding the instantaneous value of GZ for the angle of heel and wave position at each time step of the integration. Having coded this numerical integrator, it is easy to play “what if” games that may lead to understanding of some of the basic features of parametric roll response. Two examples are presented here, the first consisting of the response time history for the ship in regular head seas where the computation is started with different initial conditions in each case. The second looks at the response in head seas consisting of a series of repeated wave groups. The groups are formed by modulating the fundamental sinusoidal wave train by a sinusoidal envelope function of lower frequency chosen to produce groups somewhat resembling those encountered in a random seaway. The C11 at its designed draft having a GM of 0.05B is used as the example in both of these computations. The family of GZ curves was constructed for 20 equally spaced positions of the wave crest along the ship length and the nonlinear roll damping was estimated using the method embedded in the US Navy’s SMP code. The natural undamped roll period of the ship was estimated to be about 23 seconds assuming a mass radius of gyration in roll of 0.4B. In head seas of length

Parametric Rolling of Ships – Then and Now 355

equal to the ship length, the critical frequency ratio of ½ occurs at a speed of about 5.5 knots. In constructing the wave groups for the second set of computations, we first examine Figure 3 for some impression of the behavior of random seas. The middle graph of Figure 3 corresponds to observation from a fixed point in space and shows wave groups at about 20, 400 and 700 seconds. Each group consists of four to six waves. At the higher speed of the lower graph, there are groups at about 10, 320, 500 and 650 seconds. The latter groups appear to have fewer component waves and persist for shorter time periods. For the present example, we have constructed a wave system in which each group contains five principal waves formed by multiplying the fundamental ship length sine wave whose period is about 12.9 seconds by a modulating sine wave with period of 129.5 seconds. This is found to produce individual wave groups having an appearance similar to that which might be found in a real wave system.

6 Initial Condition Sensitivity Figure 5 presents results in regular waves for an initial roll of 0.1, 1.0 and 5.0 degrees. As one would expect, this demonstrates that the transient build up of the motion to steady state is strongly dependent on the initial disturbance. Reported cases of unexpected head seas parametric roll have indicated that the motion built up rapidly over the occurrence of five or six extreme rolls and then subsided. In heavy head seas some component waves from directions other than dead ahead will always exist and these could very likely set up roll motions of five degrees or so that would then trigger the rapid build up of parametric response.

Fig. 5 Sensitivity of transient response to initial disturbance for initial roll angle = 0.1, 1.0, 5.0 deg.

356 J.R. Paulling

7 Wave Group Response The transient response in group waves appears somewhat more sluggish than in regular waves with initial conditions of both one and five degrees as shown in Figure 6. There is some sensitivity to the time of application of the initial roll disturbance with respect to the peaks and nodes of the envelope, but computations in which the time of application was varied showed it to be of secondary importance to the magnitude of the initial roll angle.

Fig. 6 Response of C11 to wave groups, head seas, initial roll angle = 1.0 and 5.0 degrees.

8 Sensitivity of GZ Values to Forward Speed Several of the hydrostatic stability computer codes in common use by naval architects today are capable of computing the GZ curves for the ship assumed poised in static equilibrium on the wave. It has been shown in some experiments by Paulling (1961) that this assumption of static equilibrium is usually satisfactory for moderate speeds in following seas. In these experiments, a model was towed at a fixed angle of heel using an apparatus that allowed the model freedom in pitch and heave. As the model was towed on a course at right angles to the wave crests the righting moment was continuously measured and recorded. Results from experiments conducted at various model speeds in following seas revealed very little variation with model speeds in the range of the normal speeds for seagoing ships. These experiments were carried out on a family of models representing the DTMB Series 60 constructed with variations in beam and freeboard. The experiments were conducted in following seas only and did not explore the possible effects that might be associated with the greater pitch-heave motions to be expected when the ship operates in head seas. Examples of results obtained in following sea experiments compared with hand computations are shown in Figure 7.

Parametric Rolling of Ships – Then and Now 357

The computations assumed static equilibrium of the ship on wave and the agreement is seen to be good for the following seas case. In head seas, the assumption of static equilibrium of pitch and heave is probably not as valid an approximation as it is in following seas. This is especially true at higher speeds for which the encounter frequencies may approach resonance in the pitch-heave modes.

Fig. 7 Model measurements of GZ in following seas. Series 60 Cb=0.60, Varied freeboard.

In order to test the sensitivity of the GZ curve to forward speed, an approximate computation procedure was adopted which would simulate the model experiments. It was assumed that the pitch-heave motion in head seas of the heeled ship would differ only slightly from the corresponding motion of the upright ship. With this assumption, these motions could be computed using a standard linear ship motions code. The pitch-heave motion time history so obtained was then input to the wave stability computations to give a closer approximation to the exact dynamic attitude of the ship in waves than given by the usual static equilibrium assumption. The results could then be regarded as an approximation to the GZ for the dynamically pitching and heaving ship. The dynamic pressure distribution on the ship due to the motions would be neglected as would the effect of heel on the pitch-heave motion. Since the Froude-Krylov pressure terms are usually dominant in determining the forces on the ship in waves, this procedure can be thought of as providing a first order correction for the exact dynamic attitude of the ship in the waves. An example of the ship’s changing attitude with passage of the waves is shown in Figure 8.

358 J.R. Paulling

Fig. 8 C11 Containership in L/20 head seas at 18 knots.

Figure 9 contains results of these computations in the form of maximum and minimum GZ curves. These correspond to a wave trough and a crest near amidships, for the ship moving at 18 knots in L/20 head and following seas. Also shown are GZ curves for an assumed static pitch-heave attitude of the ship which simulates following seas. The dynamic motion curves are labeled “F” for following and “H” for head seas. From these results we conclude that the effect of pitch-heave dynamics is to reduce somewhat the variation of the GZ curves about the still water curve. Further, we see that the assumed static pitch-heave attitude is quite close to the dynamic attitude in following seas.

9 Conclusions A very brief background is given of some of the practical implications of parametric roll for ships in following and head seas. Early work on the phenomenon had focused almost entirely on rolling and capsize in following seas but the APL CHINA casualty brought out quite dramatically the possibility and consequences of head seas parametric roll. Much research has come about as a result and we now find that regulatory bodies and shipowners are beginning to take steps to avoid a repetition of the incident both in the ship design process and in the operation of ships.

Parametric Rolling of Ships – Then and Now 359

C11 CONTAINERSHIP GM=0.05B L/20 WAVES

2,5 2 1,5

GZ - M

1 0,5 0

-0,5

0

10

20

30

40

50

60

70

80

-1 -1,5

HEEL - DEG Fig. 9 Max – min GZ curves for C11 containership – dynamic pitch-heave attitude at 18 knots.

Acknowledgement The work reported upon here has been supported over a period of many years principally by the US Coast Guard, the American Bureau of Shipping and the Society of Naval Architects and Marine Engineers

References Arndt B, Rodin S (1958) Stabilität bei vor-und-achterlichem seegang. Schiffstechnik. V 5: 29, Nov 192-199. DeKat JO, Paulling JR (1989) The simulation of ship motions and capsizing in severe seas. Transactions. The Soc of Nav Archit and Mar Eng V 97 France WN, Levadou M, Treakle TW, Paulling JR, Michel RK, Moore C (2003) An investigation of head-sea parametric rolling and its influence on Container Lashing Systems. Mar Technol. V 40:1 Jan 1:19 Froude W (1863) Remarks on Mr. Scott-Russell’s paper on rolling. Transactions. The Inst of Nav Archit Graff W, Heckscher E (1942) Widerstands und Stabilitäts versuche mit drei Fischdampfermodellen. Werft-Reederei-Hafen V 22 1941:115-120 (also DTMB Translation 75 June) Grim O (1952) Stabilität und sicherheit im Seegang. Schiffstechnik. V 1:10-21 IMO (1995) Guidance to the master for avoiding dangerous situations in following and quartering seas. MSC Circ. 707 dated 19 Oct Kempf G (1938) Die stabilitätsbeanspruchung der schiffe durch Wellen und Schwingungen. Werft-Reederei-Hafen. V 19:200-202 Kerwin J E (1955) Notes on rolling in longitudinal waves. Int Shipbuild Prog V2:597-614 Oakley OH, Paulling JR, Wood PD (1974) Ship motions and capsizing in Astern Seas. Proc Tenth ONR Symp on Nav Hydrodyn ONR ACR 204

360 J.R. Paulling Paulling JR, Rosenberg R M (1959) On Unstable ship motions resulting from nonlinear coupling. J Ship Res. V 3:1 Paulling JR (1961) The transverse stability of a ship in a longitudinal Seaway. J of Ship Res. V 4:4 Shin YS, Belenky VL, Paulling JR, Weems KM, Lin WM (2004) Criteria for parametric roll of large containerships in longitudinal seas. Transactions. SNAME. V 112:14-47 Tikka K, Paulling JR (1990) Prediction of critical wave conditions for extreme vessel response in random seas. Proc Stab90 Naples Wendel K (1954) Stabilitätseinbussen im Seegang und durch Koksdeckslast. Hansa 2016-2022 Wendel K (1960) Safety from capsizing. Fishing boats of the world 2 - J O Traung Ed. Fishing News Books Ltd. London 496-504

4 Broaching-to

Parallels of Yaw and Roll Dynamics of Ships in Astern Seas and the Effect of Nonlinear Surging K.J. Spyrou School of Naval Architecture and Marine Engineering, National Technical University of Athens, 9 Iroon Polytechneiou, Zographos, Athens 15780, Greece

Abstract In the first part of this paper we consider in parallel the roll and the yaw dynamics in astern seas and we point out a number of interesting fundamental dynamical analogies between the two modes. In the second part we focus on the roll dynamics, especially the parametric and the pure-loss mechanisms; however taking into account nonlinear surging effects. Analytical solution for the nonlinear surge motion is proposed. Moreover, characteristic graphs showing the quantitative effect of surging on capsize tendency are presented. Other aspects considered are, the effect of restoring modulation based on two frequencies and the dynamic effect of an initially hardening restoring.

1 Introduction As is well-known, fluctuations of the roll righting-arm in large following waves can result in ship roll instability and capsize according to the pure-loss or the parametric mechanism (Grim 1952). Fluctuations of a similar nature, concerning the motion’s stiffness term, may take also place in actively controlled yaw, originating from the combined effect of rudder control with the wave induced yaw moment. This could give rise to course instability resulting in deviation from the desired heading and possibly in broaching-to (Spyrou 1996). Consider a ship operating in long following sinusoidal waves. In order to avoid coupling complications let us assume further that, due to high natural frequencies in heave and in pitch compared to the encounter frequency, the ship can maintain a state of quasi-static equilibrium on the vertical plane. If the waves are relatively steep, the geometry of the submerged part of the hull will vary according to ship’s position on the wave. This is likely to be reflected in the roll righting-arm, with a reduced or even negative roll restoring arising when the middle of the ship is near to a wave crest. If roll restoring remains negative for sufficient time, so that heel finds the time to develop unopposed from some small initial angle up to levels M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_20, © Springer Science+Business Media B.V. 2011

363

364 K.J. Spyrou

well beyond the vanishing angle, then capsize due to the so-called pure-loss of stability mechanism will be realised (Paulling 1961). In this case the magnitude of roll damping affects insignificantly ship survivability. Capsize can occur of course also in a typical parametric resonance fashion and here the magnitude of damping should be a much more critical factor (Kerwin 1955). Practically, the variation of restoring must be however quite intensive, so that the large-amplitude roll motion could build-up within a small number of wave cycles (Spyrou 2000a). The onset of yaw instability in a similar environment represents a slightly more complex process; because yaw is always coupled with sway and also the control law of the rudder bears a serious influence on the dynamics. Unlike with roll, in the absence of active rudder control no restoring yaw moment can be produced in still water. In waves however, the movement of the rudder is intended to bring the ship back to her course. If waves with a length equal to the ship length or longer meet the ship from behind, they will create a yaw moment which will be dependent upon the angle between the direction of wave propagation and the ship’s heading. This wave yaw moment works as a positive restoring component when the ship passes from a wave crest (stabilizing effect). The opposite will be happening in the vicinity of a trough because the wave will tend to bring the ship vertical to the direction of wave propagation. As a result, the comparative strengths of the rudder and wave yaw moments will give rise to a restoring fluctuation of a certain amplitude which may sometimes have the potential to destabilize the horizontalplane motion of the ship. The commonality of the underlying fundamental dynamics of yaw and roll becomes therefore prevalent. Our first objective in this paper is to identify the correspondence between key yaw and roll motion parameters from the perspective of these Mathieu-type phenomena. As is well known, when the waves are large, the nonlinearity of surge during “following-sea” operation cannot be neglected (Kan 1990). A manifestation of this nonlinearity is a virtual rescaling of time as the ship is spending longer time on the crests than on the troughs of the waves. Despite the significance of this mechanism for the yaw and roll motions, there has been no earlier assessment of its effect for ship survivability and safety. The second objective therefore is to shed some light on the effect of surging on well-known capsize mechanisms.

2 A Simple Model of Yaw Motion in Astern Seas Consider the linear differential equations of sway and yaw (Clarke et al. 1983), with the addition of wave excitation terms at their right-hand side: Sway:

m  Yv  v  Yv v  mxG  Yr  r  m  Yr  r   Y  Ywave 

(1)

Parallels of Yaw and Roll Dynamics of Ships in Astern Seas 365

 ) Yaw: mxG  N v  v  N v v  I z  N r  r  mxG  N r  r   N    N (wave

(2)

In the above v, r  are respectively sway velocity and yaw angular velocity,  is the rudder angle, m is ship mass and xG is the longitudinal position of ship’s centre of gravity; Yv, Yr , N v , N r are acceleration coefficients (added masses/

moments of inertia) and Yv, Yr , N v , N r , Y are velocity coefficients (hydrodynamic damping terms). Y(wave) , N (wave) are respectively the wave’s sway force and yaw

moment. The prime indicates nondimensionalised quantity and the dot differentiation 3 over time. We use ship length L for length scaling, L for mass and L U for time. At first instance, we shall assume that the yaw and sway velocities are restrained from building-up to high values (thus they remain small and the resulting damping forces can be considered as linear) through use of appropriate rudder control. Additionally, ship behaviour is examined at “some distance” from the region of surf-riding, so that, for this first part of the paper, surge velocity is taken as nearly constant. Then, we express the wave terms Ywave  , N (wave) in respect with the frequency of encounter (rather than as functions of absolute wave frequency and position):

Y(wave)  Yw  sin e t 

(3)

N (wave)  N w  cos e t 

(4)

The following notation is applied: Yw , N w are wave force/moment coefficients;  is the ship’s heading relatively to the wave (  0 when the sea is exactly following – generally,  is assumed small). Consider further rudder control with a linear law based on two gains, k1 and k 2 :

k1 multiplies the instantaneous heading deviation from the desired course  r , while k 2′ multiplies yaw’s angular velocity:

  k1 (  r )  k 2 r 

(5)

Substituting (3), (4) and (5) in (1) and (2), uncoupling yaw from sway and using well known expressions for system gain and time constants, K , T1 , T2 , T3, the following differential equation of heading angle is obtained:

  b  p1  f cose t     q 2 1  hcose t      j

(6)

366 K.J. Spyrou

The above third-order differential equation has time-dependent coefficients in two places. As is well known however, if T1 is much greater than T2 and T3 , we can use the so-called simplified yaw response model of Nomoto et al (1957). In that case the order of equation (6) is reduced by one:

T      K   A cos(e t   a)

(7)

K , T  are respectively system gain and time constants,  is relative heading

angle (assumed small),  is rudder angle, A is wave excitation amplitude, e is the encounter frequency that is calculated here with reference to the desired heading and is thus independent of the oscillatory yaw motion; and a is a phase angle. Coupling (7) with the autopilot (5) and drop for simplicity of the phase angle a , leads to:

      0 ( yaw) 2 [1  h cos(e t )]  j In the above formulation,  0 ( yaw ) 

(8)

k 1 K  T  ,   1  k 2 K  T  (damping),

h  A k1 K  (amplitude of parametric variation of restoring), j  k1 K  r T  . It is easily recognized that (8) is Mathieu’s equation with the addition however of biaslike external static forcing term, j . For stability, positive T  is required as 1 T  is the inverse of the damping of the unsteered vessel. However, large positive T  implies slow convergence towards the corresponding steady rate-of-turn which is determined by the value of the static gain K  . A trend exists for large T  to appear in conjunction with large K  which gives a nearly straight-line spiral curve. The effect of active control on damping is represented by the quantity k 2 K  T  . It depends thus on the yaw rate (“differential”) gain term in the autopilot. If T   0 , suitable choice of k 2 can turn the damping of the system positive since k 2 multiplies the positive quantity K  T  , thereby yielding stability for the steered ship in calm sea. The wave effects are lumped into the restoring and independent-periodic-forcing terms since the quantities K  and T  were assumed to be, at first approximation, unaffected by the wave. If the amplitude of wave excitation A exceeds k1 K  then, on the basis of (8), negative yaw restoring will arise around the trough. Should the duration of operation under negative restoring be long enough, undesired turning motion will be initiated (“broaching-to”). From a dynamics perspective there is complete equivalence with a capsize event of the so-called “pure-loss” type. It can be avoided if the proportional gain k1 is chosen to be always greater than A K  even for the most extreme wave environment where the ship will operate (it should be a matter of further investigation to what extend this is technically feasible). A notable difference between the manifestation of this instability in roll

Parallels of Yaw and Roll Dynamics of Ships in Astern Seas 367

and in yaw is that in roll it arises near the crest of the wave, whereas in yaw the ship becomes vulnerable around the trough. We may rewrite (8) on the basis of heading error  1    r and then apply the time transformation s   0 ( yaw) t  :

d 2 1 ds

2

 2

d 1  [1  h cos(s )]  f cos s ds

Where   e  0 ( yaw) ,

h  A / k1 K  and

(9)

f  A  r (T   0 ( yaw) 2 ) . The

damping ratio is given by the expression: 2  1  k 2 K 

k1 K  T  (the presence

of k1 should be noted in  ).

3 Comparison with the Roll Equation It is obvious from (9) that parametric instability of yaw may also arise, very much like that of roll. To establish the analogy we remind that the generic equation of roll in a “following” sea linearised in terms of the roll angle is:

d 2  d

2

 2

d   1  h cos    0 d

(10)

  is the scaled roll angle,      v , with  the true roll angle and  v the angle of vanishing stability. Also,   0( roll) t . The damping ratio is given in this

case by, 2  B 0( roll ) M g (GM )  where B is the dimensional linear damping coefficient, M is ship mass and GM  is the metacentric height. Roll’s natural frequency is 0( roll )  Mg (GM ) ( I  I ) . With substitution of 0( roll ) we

obtain further that 2  B ( I  I ) M g (GM ) . Also, the amplitude of the parametric term is h   GM  GM  where  GM  is the difference of metacentric height at the crest from the still water value. This is a common assumption but of course it results in a highly idealised formulation because the average (GM ) has no reason to be identical with the still water GM  . In addition, the variation from trough to crest may not be harmonic.

368 K.J. Spyrou

3.1 Conditions for “Exact” Resonance Let us neglect for a while the damping terms of (9) and (10) in order to find out how the Froude number of the vertex of the principal resonance varies between roll and yaw. For overtaking waves the frequency of encounter, e , is positive and the condition of exact resonance is written as: e 0  2 n , n  1,2,3,... where 0 can be the frequency of encounter of either yaw or roll. Thus with increasing n the vertices will accumulate near to zero frequency of encounter. For  r  0 we

can write, e  (2  )c  U  . 

Fn 

 2 L 

(11a)

   1  0( yaw )  n   L   

Considering the domain of variation of the natural frequency of yaw,  0 ( yaw )  k 1 K  T  , for conventional ships the ratio K  T  is usually within

the range 0.3  1.4 [see for example Barr et al. (1981)]. As a matter of fact

 0 ( yaw) should lie in the range [0.55 k1 - 1.18 k1 ]. With a proportional gain k1 between 1.0 and 2.0 ,  0 ( yaw) should be between 0.55 and 1.67 . For roll on the other hand, the natural frequency is nondimensionalised on the basis of ship length and acceleration of gravity, 0  0 L g . The different nondimensionalisation results in a different parametric expression of the critical Froude number:

 Fn 

L  2

 0 ( roll )  

n

L

(11b)

3.2 Relative Magnitudes and Effect of Damping Even when a ship is equipped with bilge keels, the damping ratio is usually quite low. Noted that the roll damping ratio will change if the position of the centre of gravity is modified. As far as yaw is concerned, the damping ratio depends strongly on the autopilot’s gains. Common values are in the range 0.8    1.0 Fossen (1994). Here lies therefore another significant difference between the roll and yaw equations: The yaw damping ratio is normally very large. As a matter of fact, to be placed in one of the regions of resonance there should be very considerable loss of yaw restoring at the trough region. Usually this means that very steep waves will be required. Unfortunately, for such highly damped motion

Parallels of Yaw and Roll Dynamics of Ships in Astern Seas 369

it is not easy to derive simple expressions for the critical parametric amplitude. Even the expression of Gunderson, Rigas & Van Vleck (1974), which is applicable also for large damping values, would not work as  approaches 1.0 :

 02  h  1   2 tanh   2  e2   





(12)

3.2.1 Nonlinearity The dynamic behaviour considering the nonlinearities that in reality exist in the restoring (strong) and in damping (mild) is relatively well understood as it has been studied by a number of investigators, for example Blocki (1980); Zavodney et al. (1990); Soliman & Thompson (1992); Kan (1992); Hamamoto et al. (1995). For the yaw equation, if the autopilot is relatively effective, nonlinearity will reside mainly in the damping.

4 Some Other Aspects of Parametric Rolling 4.1 Bi-Chromatic Waves An interesting and rather little known subject is the effect for parametric rolling occurrence of a quasi-periodic variation of (GZ) (instead of the often considered in a deterministic context periodic one). The consideration of a Mathieu-type equation with internal forcing based on a single frequency represents of course a highly idealised scenario. In Fig. 1 are shown the stability transition curves when two independent frequencies are present. This particular graph was drawn with the second frequency being 75% of the first. Also, the amplitude of parametric forcing at the second frequency was 3 times that of the first. It is noticed that secondary “spikes” have grown on each primary resonance region. Further research has shown that their number tends to increase as the second frequency keeps aloof from the first.

370 K.J. Spyrou

Fig. 1 Boundaries of parametric resonance with two incommensurate frequencies. The inserted graph shows a time realisation of the time-dependent restoring.

4.2 Hardening Restoring Another matter considered was the effect of an initially hardening restoring on the stability transition curves. We have assumed a quintic restoring curve which can be parameterised on the basis of a single parameter  . In scaled form the expression of restoring is: R( )       3  (1   )  5 where   is the scaled roll angle with respect to the vanishing angle. Increase of  means basically stronger initial hardening. The parametric variation was applied only on the linear term. In Fig. 2 are shown the transition curves for restoring which is moderately or strongly hardening. Generally, a process of transformation of the boundary from sharp to ‘brittle’ is in place as  is increased. Fractal-like stability boundaries have been presented earlier for a generic cubic restoring and for a more exact restoring curve (Spyrou 2000a).

Parallels of Yaw and Roll Dynamics of Ships in Astern Seas 371

Fig. 2 Capsize boundaries for increasingly hardening restoring. The left graph is with moderately hardening restoring (   2 ) and the right one is with strongly (   17 ).

5 The Fundamental Nonlinear Surge Dynamics It is well-known that the surging behaviour of a ship in following waves of substantial steepness can become strongly nonlinear for a certain range of speeds (normally for Froude numbers higher than 0.3). This nonlinear behaviour is manifested with a gradual transformation of ship response towards an asymmetric pattern of surging even if the considered wave is of a simple harmonic form. Typically, the ship spends more time on the crests and less on the troughs, a tendency which becomes more pronounced as the speed is increased further. Furthermore, unusual types of behaviour with a stationary nature and in competition with the periodic pattern arise, featuring a forced motion at a speed equal to the wave celerity and with the ship “locked” between two consecutive wave crests. These phenomena have already been studied on the basis of numerical models for a following as well as for a quartering sea environment and specific explanations of their dynamics have been produced (Spyrou 1997). A typical phase-plane plot of nonlinear surging is shown in Fig. 3. We have recently endeavoured to developing a purely analytical description for the surging behaviour, even for the strongly nonlinear range (Spyrou 2000b). This would be very useful for design where closed-form expressions are always preferred. It would benefit also advanced investigations on phenomena such as broaching-to and loss of transverse stability. The differential equation of surging motion has a strong nonlinearity in the stiffness term since the wave force is a sinusoidal function of position. There is

372 K.J. Spyrou

also a weak nonlinearity in the damping (=difference between resistance and thrust). A general form for the equation of surge on a sinusoidal wave is:

m  X u  u  [ Ru, c   T u, c, n ] 

f sin kx   0

(13)

The velocity relatively to the wave is x  u  c. With substitution of suitable polynomial expressions for the thrust and the resistance, eq. (1) becomes,

m  X u  x  3r3 c 2  2r2   2 c  r1    1 n x  3r3 c  r2   2  x 2  r3 x 3  f sin kx  



 

(14)



  2 c 2   1 c n   0 n 2  r1c  r2 c 2  r3 c 3     T c ; n 

R c 

The right-hand-side of (14) reveals the polynomial expressions used for T and R.

Surf-riding: This calculation is straightforward: If  1 

T c; n   R c   1 then 1  sin kx  1 . f

Stationary solutions become possible (surf-riding), located at, x

 T c; n   R c    T c; n   R c   2 1 (2  1) 1  arcsin   arcsin   , x  (15) k k f k k f    

Asymmetric surging: Equation (14) can be brought into the following form, by postulating the substitution of the damping terms with an equivalent quadratic on the basis of a leastsquare fit,

(m  X u ) x   c; n x x  f sin(kx)  T (c; n)  R(c)

(16)

Parallels of Yaw and Roll Dynamics of Ships in Astern Seas 373

For an overtaking wave dx/dt < 0 and the above leads to the following expression for the orbits of the phase plane x, x  ,

2qcos kx  2 psin kx  r dx 1  x   c2 qe 2 pkx   dt k p 1 4 p2



where p 



k m  X u 

, q



(17)

T c; n   Rc k . fk and r  m  X u  m  X u 

The constants c1 and c 2 depend on the initial conditions. The term c 2 qe 2 pkx represents the transient part of the solution and it vanishes gradually since x   (the ship is trailing behind the waves). Therefore the expression for the steady periodic motion is, dx 1  x   dt k

2qcos kx  2 psin kx 

1  4 p  2



r p

(18)

With suitable transformations it can be shown that (18) can be solved explicitly for t, t

m  kx    F , m a  2 

(19)

Where F is the elliptic integral of the first kind, F 





0

1 1  m cos 2 

d with

modulus m . In Fig. 4 is shown the relation between time and position which shows clearly the distortion from the linear pattern. With inversion of (19) we obtain further the following expression for x in terms of t (see also Fig. 5),

 a coskx     1  2 sn 2   t, m 

 m  

(20)

When the ship operates away from the strongly nonlinear regime, the right-handside of (20) tends to obtain the cyclic form cos e t  .

374 K.J. Spyrou

Condition for Homoclinic Connection The surface of a cylinder is the “natural” phase plane for the dynamics of our problem. On such a plane one should note the possibility of a homoclinic connection event evoking the disappearance of the periodic motion. In physical space this could correspond to the realization of global surf-riding which is further linked with the occurrence of “broaching-to”. It could happen as soon as the unstable equilibrium near to the wave crest “collides” with the periodic orbit. Unstable stationary point:

x  0 x

2  1 k



1 T c; n   Rc  sin 1 k f

(21)

The steady periodic orbit is given by eq. (18). Substitution of (21) into (18) yields the following expression of critical amplitude for the surge wave force,

f crit 

Rc; n   T n  2 k m  X u 2





2

 4 2

(22)

. x

x

Fig. 3 Typical phase-portrait of surf-riding

Parallels of Yaw and Roll Dynamics of Ships in Astern Seas 375 2

4

6

8

10

kx

-2 -4 -6 -8

t Fig. 4 During asymmetric surging the relation between b time and position is distorted from the straight line. 1

cos(kx)

0.5 t 2

4

6

8

10

12

14

-0.5

-1 Fig. 5 Nonlinear surging according to eq. (20).

6 The Effect of Nonlinear Surging on Roll As is well known, an implicit assumption made in the analysis of pure-loss of stability and parametric rolling is that the forward speed is constant. Such an assumption is not always valid. In principle, for dangerous roll behavior, steep and long waves are required. Waves of this kind will incur however also significant nonlinear effects on surge. The characteristic of large-amplitude surging is that it is asymmetric and the ship stays longer near the crests than near the troughs. This effect is imported into the yaw and roll dynamics through the restoring terms of the corresponding equations. The nonlinearity of surge is detrimental for roll stability because around the crest (where the ship stays longer) restoring capability is reduced. For yaw on the other hand, the effect is opposite. Yaw stability is not

376 K.J. Spyrou

worsened because the passage of the ship from the trough is quicker. The danger arises in steeper waves and especially during the process of capture in surf-riding. The three main forces acting in the surge direction are the resistance, the wave and the propulsion force. As is well known these result in a pendulum-like equation for surge the exact form of which may be found for example in Spyrou (2000b). This surge equation should be solved simultaneously with the following equation of roll (or that of yaw for broaching-to):

  2     02 1  h coskx    q 3  0

(23)

Note that the cosine is written in terms of the position on the wave, x , rather than in terms of time. The coupling to the surge equation exists because of the presence of x in the restoring term of the above roll equation. We can show now how the transition curves are modified when roll is coupled with surge.

Fig. 6 Coupled surging and rolling leading to capsize

The calculations were based on a ship with  0( roll )  0.84 (  0 ( roll )  1.577 ) and

  0.0585 . We have examined whether the normalised roll angle   exceeds the value of 1 (from an initial perturbation 0.01 and with zero initial velocity) within a specific amount of time ( t  200 s). In Fig. 6 is presented an example of time realisation for the coupled surging and rolling leading to capsize when strong nonlinear effects in surge are present. Surge motion can have a very profound effect on the “capsize” domains as realised from the stability diagram of Fig. 7. Rather than plotting the frequency ratio versus the parametric amplitude, we found more relevant to present the information in terms of the Froude number. For the considered ship, the principal resonance cannot be realised in following waves because the required Froude number is negative (the ship should be backing rather than going forward). The lower part of the fundamental is the only place where there is some commonality with the conventional (‘damped’) Ince-Strutt diagram. The upper part of the fundamental tends to become considerably wider. The

Parallels of Yaw and Roll Dynamics of Ships in Astern Seas 377

immediately next resonance occupies an enlarged domain; but the following two seem to degenerate. This may relate with the emergence of a surf-riding domain where the behaviour of the ship is stationary and travels with the wave with its centre located near a trough. 1.5 h 1.4 1.3 1.2 1.1 1 0.9

0.1

0.15

0.2

0.25 0.3 0.35 Nominal Fn

Fig. 7 Capsize boundary with nonlinear surging

7 Concluding Remarks When roll stability in a following sea is examined, it is common to distinguish between two mechanisms of capsize: pure-loss of stability, where the ship departs from the state of upright equilibrium due to negative restoring on a wave crest. Then heel increases monotonically until the ship is overturned. In this mode the magnitude of damping plays little role. In parametric instability on the other hand, which is near to the classical Mathieu-type mechanism, the build-up is oscillatory and the magnitude of damping is very important. Instabilities of a similar nature are possible in yaw, resulting in the behaviour known as broaching-to: The parallel to the pure-loss mechanism can be termed as broaching-to due to surfriding. It can happen at Froude numbers relatively near to the wave celerity. The parametric-type mechanism of broaching-to is relevant for lower Froude numbers but requires higher wave steepness. Extra forcing terms and couplings sometimes alter significantly the domains of instability. Surging dynamics can affect both mechanisms of roll instability and this effect deserves further investigation.

378 K.J. Spyrou

References Barr R, Miller ER, Ankudinov V and Lee FC (1981) Technical basis for manoeuvring performance standards. Technical Report 8103-3, Hydronautics, Inc, submitted by the USA to the Int Marit Organ (IMO). Blocki W (1980) Ship safety in connection with parametric resonance of the roll, Int Shipbuilding Prog, 27:36-53. Clarke D, Gendling P and Hine G (1983) Application of manoeuvring criteria in hull design using linear theory, Trans. RINA, 125:45-68. Fossen TI (1994) Guidance and Control of Ocean Vehicles, John Wiley and Sons, Chichester, UK. Grim O (1952) Rollschwingungen, Stabilität und Sicherheit im Seegang, Schiffstechnik, 1, 1:10-21. Gunderson H, Rigas H and Van Vleck FS (1974) A technique for determining stability regions for the damped Mathieu equation. SIAM J of Appl Math, 26, 2:345-349. Hamamoto M, Umeda N, Matsuda A and Sera W (1995) Analysis of low-cycle-resonance in astern seas. J of the Soc of Naval Archit of Japan, 177:197-206. Kan M (1990) Surging of large amplitude and surf-riding of ships in following seas. Selected Papers in Naval Archit and Ocean Eng, The Soc of Naval Archit of Japan. Kan M (1992) Chaotic capsizing. Proceedings Osaka Meeting on Seakeeping Performance, 20 th ITTC Seakeeping Committee and Kansai Fluid-Dynamics Res Group, 155-180. Kerwin JE (1955) Notes on rolling in longitudinal waves, Int Shipbuilding Prog, 2, 16:597-614. Nomoto K, Taguchi K, Honda K and Hirano S (1957) On the steering qualities of ships. Int Shipbuilding Prog, 4, 35, July, 354-370. Paulling JR (1961) The transverse stability of a ship in a longitudinal seaway, J of Ship Res 4:37-49. Soliman MS and Thompson JMT (1992) Indeterminate sub-critical bifurcations in parametric resonance, Proc of the Royal Soc of London, A, 438, 511-518. Spyrou KJ (1996) Dynamic instability in quartering waves: the behaviour of a ship during broaching. J of Ship Res, 40, 1, 46-59. Spyrou KJ (1997) On the nonlinear dynamics of broaching-to. Proc, Int Conf on Design for Abnormal Waves, The Royal Instof Naval Archit, Glasgow, October, 12 pages. Spyrou KJ (2000a) Designing against parametric instability in following seas, Ocean Eng, 27, 6, 625-654. Spyrou KJ (2000b) On the parametric rolling of ships in a following sea under simultaneous nonlinear periodic surging. Philos Transactions of the Royal Soc of London, Α 358:1813-1834. Zavodney LD, Nayfeh AH and Sanchez (1990) Bifurcations and chaos in parametrically excited single-degree-of-freedom systems. Nonlinear Dyn 1, 1:1-21.

Model Experiment on Heel-Induced Hydrodynamic Forces in Waves for Realising Quantitative Prediction of Broaching Hirotada Hashimoto*, Naoya Umeda* and Akihiko Matsuda** *Osaka University, Yamadaoka, Suita, Osaka, 565-0871, Japan ** National Research Institute of Fisheries Engineering, Hasaki, Kamisu, Ibaraki, 314-0408, Japan

Abstract Mathematical models for capsizing due to broaching, one of the great threat to ships running in following and quartering seas, have been developed by many researchers. However these models can predict it only qualitatively. For realising quantitative prediction of broaching, we have conducted systematic captive model experiments with large heel angles up to 50 degrees in severe following waves. As a result, the details of nonlinear heel-induced hydrodynamic forces with respect to heel angle in waves are presented. Then comparisons between a free running model experiment and the numerical simulation with direct use of the measured heel-induced hydrodynamic forces are carried out. As a result, the mathematical model for realising quantitative prediction of broaching in severe following and quartering seas is presented. Finally the simplified mathematical model is also proposed for more practical uses with less experimental efforts.

Nomeclature c Fn GZ H Ixx Izz Jxx Jzz KMNL KRNL KRWNL Kp Kr KrW KR Kv

wave celerity nominal Froude number righting arm wave height moment of inertia in roll moment of inertia in yaw added moment of inertia in roll added moment of inertia in yaw nonlinear manoeuvring forces in roll nonlinear heel-induced hydrodynamic roll moment in still water nonlinear heel-induced hydrodynamic roll moment in waves derivative of roll moment with respect to roll rate derivative of roll moment with respect to yaw rate wave effect on the derivative of roll moment with respect to yaw rate rudder gain derivative of roll moment with respect to sway velocity

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_21, © Springer Science+Business Media B.V. 2011

379

380 H. Hashimoto et al. KvW Kw K KW K’ L m mx my n NMNL NRNL NRWNL Nr NrW Nv NvW Nw N NW N’ p r R t T TD TE TW u v XMNL XRNL X’w Xrud X’ YMNL YRNL YRWNL Yr YrW Yv YvW Yw Y YW Y’ zH

 c   

wave effect on the derivative of roll moment with respect to sway velocity wave-induced roll moment derivative of roll moment with respect to rudder angle wave effect on the derivative of roll moment with respect to rudder angle K’=K/(1/2Ld2u2) ship length between perpendiculars ship mass added mass in surge added mass in sway propeller revolution number nonlinear manoeuvring forces in yaw nonlinear heel-induced hydrodynamic yaw moment in still water nonlinear heel-induced hydrodynamic yaw moment in waves derivative of yaw moment with respect to yaw rate wave effect on the derivative of yaw moment with respect to yaw rate derivative of yaw moment with respect to sway velocity wave effect on the derivative of yaw moment with respect to sway velocity wave-induced yaw moment derivative of yaw moment with respect to rudder angle wave effect on the derivative of yaw moment with respect to rudder angle N’=N/(1/2L2du2) roll rate yaw rate ship resistance time propeller thrust time constant for differential control time constant for steering gear wave effect on propeller thrust surge velocity sway velocity nonlinear manoeuvring forces in surge nonlinear heel-induced hydrodynamic surge force in still water wave-induced surge force with empirical correction rudder-induced surge force X’=X/(1/2Ldu2) nonlinear manoeuvring forces in sway nonlinear heel-induced hydrodynamic sway force in still water nonlinear heel-induced hydrodynamic sway force in waves derivative of sway force with respect to yaw rate wave effect on the derivative of sway force with respect to yaw rate derivative of sway force with respect to sway velocity wave effect on the derivative of sway force with respect to sway velocity wave-induced sway force derivative of sway force with respect to rudder angle wave effect on the derivative of sway force with respect to rudder angle Y’=Y/(1/2Ldu2) vertical position of centre of sway force due to lateral motions heading angle from wave direction desired heading angle for auto pilot rudder angle roll angle wave length

Model Experiment on Heel-Induced Hydrodynamic Forces 381   G G

pitch angle water density longitudinal position of centre of gravity from a wave trough vertical distance between centre of gravity and still water plane

1 Introduction Recent model experiments (e.g. Umeda et al. 1999) demonstrate that a ship complying with the current Intact Stability Code (IS Code) of the International Maritime Organisation (IMO) rarely capsizes in non-breaking beam waves but could occasionally capsize when she runs in following and quartering seas. Although the IMO circulated a simple guidance applicable to all ships for avoiding danger in following and quartering seas, real capsizing boundaries might depend on detailed particulars of each ship. By responding to these situations the Sub-Committee on stability, load lines and on fishing vessel safety has started to review the IS Code. On this revision, an alternative approval of safety with direct assessment by physical or numerical tests is considered. This performance-based assessment of safety is especially expected to prevent ship capsizing due to broaching in following and quartering seas. This is because the existing stability criteria cannot deal with this phenomenon which depends on hull form details as well as operational practices. At this stage numerical models are required to provide not only qualitative agreement but also quantitative one with help of model experiments. Toward this direction, the International Towing Tank Conference (ITTC) had conducted a benchmark test of several numerical models by comparing them with existing capsizing model experiments in following and quartering seas, which cover ship capsizing due to parametric rolling and broaching. As a result, it was confirmed that only a few numerical models could qualitatively predict capsizing due to broaching and none could do it quantitatively (Umeda et al. 2001). Therefore, existing numerical modelling techniques should be upgraded to realise the quantitative prediction of broaching. For this purpose, it is necessary to systematically examine all factors relevant to capsize due to broaching in following and quartering seas further. It has been pointed out that the effect of heel-induced manoeuvring forces in calm water is significant for broaching prediction (Renilson et al. 2000). This effect was taken into account as a linear function of roll angle and their derivatives were obtained from a captive model test for small heel angle in calm water. However the relationship between heel-induced hydrodynamic forces and a roll angle could be nonlinear because under-water hull form becomes quite asymmetric with increasing of roll angle. Therefore it is important to conduct a model experiment to measure heel-induced hydrodynamic forces especially for large heel angle. However this kind of experiment is quite difficult because of the limitation of the experimental setup. Within a framework of ordinary experimental procedure and setup, problems of waterinflow and water-accumulation in a hull cannot be avoided. To overcome these problems,

382 H. Hashimoto et al.

using a complete watertight ship model or devices to prevent a dynamometer to be flooded is required. In past Nakato et al. conducted a captive model experiment with special setup that a dynamometer is settled quite higher position than the centre of ship gravity to measure restoring moments in waves with a wide range of roll angle. (Nakato et al. 1982) However their experimental method and results were not available to practical use because a dynamometer settled at high position provides significant trim moment during towing. For these reasons, a captive model experiment to measure hydrodynamic forces induced by large heel is quite difficult even in calm water. Therefore it is important to develop an experimental procedure to directly measure these forces in both calm water and waves. In our previous work, we had proposed a new experimental method with a purpose-built ship model and an experimental setup, which can realise captive tests with up to 90 degrees of heel. (Hashimoto et al. 2004) Then captive tests were conducted to measure heel-induced hydrodynamic forces in calm water to discuss its importance on broaching prediction. In this paper captive model tests to measure heel-induced hydrodynamic forces in heavy following waves were conducted. Based on these results, the mathematical model was upgraded by introducing a nonlinear model of the hydrodynamic forces as a function of roll angle, and was applied to the prediction of ship motions in following and quartering waves. Then comparisons of the 4 DOF mathematical models with and without this effect were conducted, together with the existing free-running experiments, to examine the importance of nonlinear heelinduced hydrodynamic forces in waves on the prediction of ship capsizing due to broaching in following and quartering seas.

2 Captive Model Experiments with Large Heel 2.1 Procedure of a Model Experiment For measuring heel-induced hydrodynamic forces with large heel angles, a new 1/25 scaled model of the 135GT Japanese purse seiner known as the ITTC Ship A-2 was built. The Body plan and general arrangement of the ship model and the principal particulars are shown in Fig. 1, Fig. 2 and Table. 1, respectively. Here small model size was selected by considering a maximum loading allowance of the dynamometer in case whole of a ship model is in the air. The ship model is completely watertight and has bulwarks, freeing ports and the super structures that are similar to the ship model used in the free-running model experiment (Umeda et al. 1999). There are two poles, having a scale, that fix the model and adjust a model attitude. Each pole has gimbals at their bottom to realize a specified heel angle. Captive model experiments were conducted at the seakeeping and manoeuvring basin of National Research Institute of Fisheries Engineering (NRIFE). The model was towed by a main towing carriage in long-crested regular waves, and model was equipped with

Model Experiment on Heel-Induced Hydrodynamic Forces 383

a rudder but without a propeller. The model was completely fixed by two poles in all directions. Since a model attitude is fixed, only one data for certain relative position of a wave can be obtained per one run of towing carriage. In this experiment heel-induced hydrodynamic forces in waves are identified by measuring eight relative positions of a wave with the same heel angle. The surge force, sway force, heave force, roll moment, pitch moment and yaw moment acting on the towed model were detected by a dynamometer located in vertically high position where no watersplash happens.

Fig. 1 Body plan of the subject ship

Fig. 2 General arrangement of the purpose-built ship model Table 1 Principal particulars of the subject ship Item

Values

length : Lpp

34.5 m

breadth : B

7.60 m

depth : D

3.07 m

mean draught : d

2.65 m

block coefficient : Cb

0.597

longitudinal position of centre of

1.31 m

gravity from the midship : xCG

aft

metacentric height : GM

1.00 m

natural roll period : T

7.4 s

rudder area : AR

3.49 m2

time constant of steering gear : TE

0.63 s

proportional gain: KP

1.0

time constant for differential control: TD

0.0 s

maximum rudder angle: max

 35 deg

384 H. Hashimoto et al.

The procedure of the experiment is as follows. Firstly instantaneous heave and pitch with no heel are estimated for certain relative position of a wave from the model tests with conventional setup in moderate wave steepness. Since wave steepness is different from a desired value, we estimate a ship instantaneous heave and pitch by assuming linearity of them with respect to wave steepness. The effect of heel is added with the Froude-Krylov assumption. Here the calculation is obtained by integrating the undisturbed wave pressure up to the wave surface with the Smith effect for free heave and pitch. By adjusting length of two poles and angles of gimbals, the sinkage, trim and heel angle are set to be equal to the estimated values. An example photograph of the experimental setup is shown in Fig. 3. Here the estimated ship attitude does not usually coincide with the real one. However it is quite difficult to accurately estimate a ship attitude with large heel at high Froude number in steep waves. To correct the difference of a running attitude, additional experiments are required for obtaining the derivatives of heel-induced hydrodynamic forces with respect to heave and pitch. If we conduct this additional experiment to all relative positions and heel angles, the number of model runs could be prohibitively large. To avoid this problem, we apply the following method. As mentioned we repeat eight model runs to measure heel-induced hydrodynamic forces for one wave cycle. Since a ship model is slowly overtaken by waves, the other seven data measured with undesired ship attitude is available to obtain the derivatives of hydrodynamic forces with respect to heave and pitch at specified relative position. Using such derivatives we can reasonably estimate all forces and moments under the accurate attitude in heave and pitch by solving the simultaneous equations satisfying the heave force and pitch moment equal to zero.

Fig. 3 Setup of the captive model experiment

2.2 Procedure of an Analysis The surge force, X, sway force, Y, heave force, Z, roll moment, K, pitch moment, M and yaw moment, N, were measured as functions of the heave, pitch, roll, Froude number and horizontal position of ship to a wave trough, G/. Wave height at the ship centre of gravity was also measured by a servo-needle wave probe. The

Model Experiment on Heel-Induced Hydrodynamic Forces 385

definition of directions of the measured forces and moments is shown in Fig. 4. All forces and moments were measured in horizontal axes. Each moment is geometrically converted from the measured value around the centre of the dynamometer to that of the ship gravity. By assuming the additional change in heave and pitch are small the measured forces and moments can be expanded as Eqs. (1)-(6).   N

X

Y

K

Z

M

N Z

Fig. 4 Definition of directions of measured forces and moments

X ( G , ; , Fn , G /  )  X ( G 0 ,0 ;  , Fn , G /  )  X  ( G 0 ,0 ;  , Fn , G /  )   G *  X  ( G 0 ,0 ;  , Fn , G /  )   *

Y ( G , ; , Fn , G /  )  Y ( G 0 ,0 ; , Fn , G /  )  Y  ( G 0 ,0 ; , Fn , G /  )   G *  Y  ( G 0 ,0 ; , Fn , G /  )  * Z ( G , ; , Fn , G /  )  Z ( G 0 ,0 ;  , Fn , G /  )  Z  ( G 0 ,0 ;  , Fn , G /  )   G *  Z  ( G 0 ,0 ; , Fn , G /  )   * K ( G , ;  , Fn , G /  )  K ( G 0 ,0 ;  , Fn , G /  )  K  ( G 0 ,0 ;  , Fn , G /  )   G *  K  ( G 0 , 0 ;  , Fn , G /  )   * M ( G , ;  , Fn , G /  )  M ( G 0 ,0 ; , Fn , G /  )  M  ( G 0 , 0 ; , Fn , G /  )   G *  M  ( G 0 ,0 ; , Fn , G /  )   * N ( G , ; , Fn , G /  )  N ( G 0 ,0 ;  , Fn , G /  )  N  ( G 0 ,0 ;  , Fn , G /  )   G *  N  ( G 0 ,0 ; , Fn , G /  )   *

(1)

(2)

(3)

(4)

(5)

(6)

Here  G*   G   G 0

(7)

 *   0

(8)

where G0 and 0 ni dicate the heave and pitch angle, respectively, initially estimated. If a ship model is free in heave and pitch, the heave force and pitch moment should be zero. Therefore G*,  * can be obtained by solving the following simultaneous equations.

386 H. Hashimoto et al.

Z ( G 0 ,0 ;  , Fn , G /  )  Z  ( G 0 ,0 ;  , Fn , G /  ) G *  Z  ( G 0 ,0 ;  , Fn , G /  ) *  0

(9)

M ( G 0 ,0 ; , Fn , G /  )  M  ( G 0 ,0 ; , Fn , G /  ) G *  M  ( G 0 ,0 ; , Fn , G /  ) *  0 (10)

Then, the surge force and sway force, roll moment and yaw moment for the case in which model is free in heave and pitch can be estimated with the Equations (1), (2), (4) and (6).

3 Experimental Results Captive model experiments to measure heel-induced hydrodynamic forces in severe following waves were conducted. The wave condition was selected to correspond to that of the ITTC Benchmark Testing Programme, H/=1/10 and /L=1.637. The captive tests were conducted for various sets of heel angle at Froude number of 0.4. The here heel angles were 0, 10, 20, 30, 40 and 50 degrees. The heel-induced hydrodynamic surge force, sway force, yaw moment and roll moment in waves can be obtained by subtracting a constant value obtained in calm water with each heel angle. Finally these results are identified by the Fourier expansion with encounter frequency. An example photograph of the experiment is shown in Fig. 5.

Fig. 5 A photograph of the captive test in waves

The experimental results of non-dimensional heel-induced surge force, X’, sway force, Y’, yaw moment, N’, and roll moment, K’, in following waves at Fn=0.4 are shown in Figs. 6-9. The amplitude of X’ linearly increases with respect to roll angle and the phase of X’ drastically changes. Since the transverse hull asymmetry has a secondary effect on a longitudinal force, the amplitude of X’ itself is much smaller than the amplitude of wave-induced surge force for upright condition. Therefore the heel-induced surge force in waves can be neglected for numerical prediction. The amplitude of Y’ linearly increases up to 20 degrees but does nonlinearly over 20 degrees. The phase of Y’ also changes with respect to roll angle. The N’ and K’ have similar tendency in its change as the Y’. Since the

Model Experiment on Heel-Induced Hydrodynamic Forces 387

K’ could be important for capsizing prediction, a comparison of roll restoring moment in waves among the experimental result, the Froude-Krylov calculation on its own and the sum of the Froude-Krylov calculation and the experimental formula of lift effect (Umeda et al. 2002). Here the Froude-Krylov calculation is done by integrating the undisturbed wave pressure up to the wave surface with the Smith effect for free heave and pitch. Fig. 10 shows a comparison of GZ curve between the experiment and the two calculations. The Froude-Krylov calculation overestimates the amplitude of GZ variation. There is significant difference in the average value and the phase of GZ variation. The experimentally obtained relative position of the minimum GZ is not a wave crest but a wave down-slope. The calculation by the Froude-Krylov assumption corrected by the experimental formula can predict these tendencies. However the difference from the experimental result is not negligibly small from a viewpoint of realising the quantitative prediction. Therefore a proposed captive model experiment could be recommended to accurately predict the wave effect on roll restoring moment for large heel angle in severe waves. 0.08 0.06 0.04 o

 = 10

0.02  X'

o

 = 20

0.00

o

 = 30

o

 = 40

-0.02

o

 = 50

-0.04 -0.06 -0.08 0.00

0.25

0.50

0.75

1.00

 G /

Fig. 6 Experimental result of non-dimensional heel-induced surge force with H/=1/10, /L=1.637, =0 degrees and Fn=0.4 0.3 0.2 0.1

o

 = 10

o

 Y'

 = 20

0.0

o

 = 30

o

 = 40

o

-0.1

 = 50

-0.2 -0.3 0.00

0.25

0.50

0.75

1.00

 G /

Fig. 7 Experimental result of non-dimensional heel-induced sway force with H/=1/10, /L=1.637, =0 degrees and Fn=0.4

388 H. Hashimoto et al. 0.100 0.075 0.050 o

 = 10

0.025  N'

o

 = 20

0.000

o

 = 30

o

 = 40

-0.025

o

 = 50

-0.050 -0.075 -0.100 0.00

0.25

0.50

0.75

1.00

 G /

Fig. 8 Experimental result of non-dimensional heel-induced yaw moment with H/=1/10, /L=1.637, =0 degrees and Fn=0.4 0.4 0.3 0.2 o

 = 10

0.1  K'

o

 = 20

0.0

o

 = 30

o

 = 40

-0.1

o

 = 50

-0.2 -0.3 -0.4 0.00

0.25

0.50

0.75

1.00

 G /

Fig. 9 Experimental result of non-dimensional heel-induced roll moment with H/=1/10, /L=1.637, =0 degrees and Fn=0.4 0.4 0.3

GZ (m)

0.2 0.1 0.0 0.00

0.25

0.50

0.75

-0.1 -0.2 -0.3

1.00 G/ 

EXP. F.K. F.K. with lift

Fig. 10 Comparison of righting arm in waves between the experiment, the Froude-Krylov calculation and the sum of Froude-Krylov calculation and the experimental formula of lift effect with H/=1/10, /L=1.637, =0 degrees, Fn=0.4 and =20 degrees

Model Experiment on Heel-Induced Hydrodynamic Forces 389

4 Mathematical Modelling The mathematical model of surge-sway-yaw-roll motions was developed by Umeda and Renilson (Umeda et al. 1992) and Umeda (Umeda 1999) for capsizing associated with surf-riding in following and quartering waves. The details of this model can be found in the literature (Umeda et al. 2002). Since this model had consistently ignored higher order terms of waves, several higher order terms were added for improving the prediction accuracy (Hashimoto et al. 2004). Then the heel-induced hydrodynamic forces measured in waves are newly added in this research. Two coordinate systems used here are shown in Fig. 11: (1) a wave fixed with its origin at a wave trough, the  axis in the direction of wave travel; and (2) an upright body fixed with its origin at the centre of ship gravity. 

x Ship



G O



y



G

wave trough



O

y

G



Z 

Fig. 11 Coordinate systems

The state vector, x and control vector, b , of this system are defined as follows. x  ( x1 , x 2 ,  , x8 ) T  { G /  , u , v,  , r ,  , p,  }T

(11)

b  {n,  c }T

(12)

The dynamical system can be represented by the following state equation. x  F(x;b)  { f1 (x;b), f 2 (x;b),, f 8 (x;b)}T

(13)

where f1 (x;b)  (u cos   v sin   c) / 

(14)

f2 (x;b)  {T (u ; n )  T W (G / , u ,  ; n )  R (u )  X R N L (u ,  )  X M N L (u , v , r )  X rud (G /  , u ,  ,  ; n )  X w (G / ,  )}/ (m  m x )

(15)

390 H. Hashimoto et al. f3 (x;b)  {(m  m x )u r  Y v (u ; n )v  Y v W (G /  , u ,  )v  Y r (u ; n )r  Y r W (G /  , u ,  )r Y

M NL

(u , v , r )  Y

R NL

(u ,  )  Y

RW NL

(G /  ,  ,  )  Y  (u ; n )  Y  W (G /  , u ,  ; n )

(16)

Y w (G /  , u ,  ; n )} / (m  m y )

f 4 ( x; b )  r f5 (x;b)  {N v (u ; n )v  N v W (G /  , u ,  )v  N r (u ; n )r  N r W (G /  , u ,  )r N

M NL

(u , v , r )  N R N L (u ,  )  N R W N L (G /  ,  ,  )  N  (u ; n )  N  W (G /  , u ,  ; n )

(17) (18)

 N w (G /  , u ,  ; n )} / ( I ZZ  J Z Z ) f 6 ( x; b )  p

f7 ( x;b)  [m x z H (u , v )u r  K v (u , v ; n )v  K v W (G /  , u , v ,  )v  K r (u , v ,; n )r  K r W (G /  , u , v ,  )r  K M N L (u , v , r )  K P (u ) p  K R N L (u ,  )  K R W N L (G /  ,  ,  )

(19)

 K  (u ; n )  K  W (G /  , u ,  ; n )  K w (G /  , u ,  ; n )  m gGZ ( )] / ( I xx  J xx )

f 8 (x;b)  {  K R (    C )  K R TD r} / TE

(20)

The underlined parts indicate the wave effect on heel-induced hydrodynamic forces newly added. Here the Grim’s effective wave concept (Grim 1961) is used for the estimation of heel-induced hydrodynamic forces in waves with heading angle. In a numerical simulation, newly measured heel-induced hydrodynamic forces in calm water are also taken into account. Although the experimental results shown in Figs. 6-9 are obtained for only one Froude number, we assume that these results are applicable to all Froude numbers because of the limitation of available experimental data. In the previous work, the wave effect on roll restoring moment is considered as the sum of the Froude-Krylov calculation and the experimental formula of lift effect (Umeda et al. 2002). However the estimation accuracy is not satisfactory as shown in Fig.10. Therefore the measured heel-induced hydrodynamic roll moment in waves is directly used in the mathematical model. Throughout this paper, we call this model including all higher order terms “enhanced model”.

5 Numerical Results Firstly comparisons between the numerical results with nonlinear heel-induced hydrodynamic forces in waves and without them (Hashimoto et al., 2004) as well as the existing free running model experiments (Umeda et al., 1999) were conducted. The comparison in the case that the ship experiences a periodic motion is shown in Fig.12. The difference between the calculated results with and without the wave effect

Model Experiment on Heel-Induced Hydrodynamic Forces 391

on heel-induced hydrodynamic forces is not so significant because its maximum roll angle is not so large. The comparison in the case that the ship suffers surf-riding, broaching and capsizing is shown in Fig. 13. The mathematical model with the nonlinear heel-induced hydrodynamic forces in waves provides better agreement with the model experiment to some extent. The comparison in boundaries of ship motion modes for control parameters of auto pilot course and nominal Froude number is shown in Fig.14. The procedure of this calculation can be found in the literature (Umeda et al. 2002). In the numerical results, a region of capsizing due to broaching becomes smaller and that of stable surf-riding does larger like the experimental result. These comparisons demonstrate that the prediction accuracy becomes better by taking the heel-induced hydrodynamic forces in waves into account, and the quantitative prediction of broaching and capsizing in heavy following and quartering waves is realised by the enhanced model. Secondly a comparison between the enhanced model and that without heelinduced hydrodynamic sway force and yaw moment in waves was conducted to examine the importance of roll restoring variation for large heel angle on capsizing prediction. The comparison in boundaries of ship motion modes for control parameters is shown in Fig. 15. In the numerical result, a region of capsizing due to broaching becomes larger and that of stable surf-riding does smaller when the wave effect on heel-induced hydrodynamic sway force and yaw moment are neglected. However the difference between two calculations is not so significant as compared to Fig. 14. This result indicates that the wave effect on heel-induced roll moment at large heel angle is important for accurate prediction of capsizing boundary. 20

Pitch(degrees)

20

0 0 -10

5

10

15

20

t(s)

-20

5

10

15

20

t(s)

-40 -60

Yaw(degrees)

60 40 20 0 -200

5

10

15

20

t(s)

Rudder(degrees)

5

5

10

15

20

t(s)

-20 60 40 20 0 -200

5

10

15

20

t(s)

-40 -60

Yaw(degrees)

-40 -60

45 30 15 0 -150

0 0 -10

Roll(degrees)

Roll(degrees)

60 40 20 0 -200

-30 -45

Pitch(degrees)

10

10

60 40 20 0 -200

5

10

15

20

t(s)

-40 -60

Rudder(degrees)

10

15

20

t(s)

45 30 15 0 -150 -30 -45

5

10

15

20

t(s)

Experiment Calculation

Fig. 12 Comparison of numerical results with H/=1/10, /L=1.637, Fn=0.3, c=-30 degrees. (left: without heel-induced hydrodynamic forces in waves, right: with them)

392 H. Hashimoto et al. Pitch(degrees)

20 10 0 0 -10 -20 90 60 30 0 -300 -60 -90

Pitch(degrees)

5

10

15

20

t(s) Roll(degrees)

10

15

20

t(s)

Yaw(degrees)

10

15

20

t(s)

90 60 30 0 -300 -60 -90

5

10

15

20

t(s)

Yaw(degrees)

5

10

15

20

t(s)

Rudder(degrees)

45 30 15 0 -150 -30 -45

5

Roll(degrees)

5

60 40 20 0 -200 -40 -60

20 10 0 0 -10 -20

5

60 40 20 0 -200 -40 -60

5

10

15

20

t(s)

Rudder(degrees)

10

15

20

t(s)

45 30 15 0 -150 -30 -45

5

10

15

20

t(s)

Experiment Calculation

Fig. 13 Comparison of numerical results with H/=1/10, /L=1.637, Fn=0.43, c=-10 degrees. (left: without heel-induced hydrodynamic forces in waves, right: with them) 0.5

nominal Froude number

nominal Froude number

0.5 0.4 0.3 0.2 0.1 0

10

20

30

40

0.4 0.3 0.2 0.1

auto pilot course (degrees) capsizing due to broaching capsizing without broaching broaching without capsizing

stable surf-riding periodic motion not identified

0

10

20

30

auto pilot course (degrees)

40

EXP.(capsizing due to broaching) EXP.(capsizing without broaching) EXP.(near stable surf-riding) EXP.(periodic motion)

Fig. 14 Comparison of boundaries of ship motion modes with H/=1/10, /L=1.637 and the initial periodic state for Fn=0.1 and c=0 degrees. (left: without heel-induced hydrodynamic forces in waves, right: with them)

Model Experiment on Heel-Induced Hydrodynamic Forces 393 0.5

nominal Froude number

nominal Froude number

0.5 0.4 0.3 0.2 0.1 0

10

20

30

40

0.4 0.3 0.2 0.1

auto pilot course (degrees) capsizing due to broaching capsizing without broaching broaching without capsizing

0

10

20

30

auto pilot course (degrees)

stable surf-riding periodic motion not identified

40

EXP.(capsizing due to broaching) EXP.(capsizing without broaching) EXP.(near stable surf-riding) EXP.(periodic motion)

Fig. 15 Comparison of boundaries of ship motion modes with H/=1/10, /L=1.637 and the initial periodic state for Fn=0.1 and c=0 degrees. (left: with all higher order terms, right: without heel-induced hydrodynamic sway force and yaw moment in waves)

6 Simplification of Mathematical Model To improve the prediction accuracy of broaching, the authors had investigated higher order terms and executed numerical calculations with and without these terms one by one. Numerical prediction results on the higher order terms were compared with the captive model tests and, when a prediction method for a certain term is not available, the captive test data were directly used for the comparative studies. As a result, quantitative prediction of broaching associated with surf-riding in severe following and quartering waves can be realized by the enhanced model which takes account of higher order terms listed below. a) b) c) d) e) f) g) h) i)

nonlinear manoeuvring forces in calm water (Umeda & Hashomoto, 2002) wave effect on linear manoeuvring forces (Umeda et al., 2003) wave effect on roll restoring moment (Umeda et al., 2003) wave effect on rudder force (Umeda et al., 2003) nonlinear wave forces including mean values (Hashimoto et al., 2004) nonlinear sway-yaw coupling (Hashimoto et al., 2004) wave effect on propeller thrust (Hashimoto et al., 2004) heel-induced hydrodynamic forces for large heel angle in calm water (Hashimoto et al., 2004) wave effect on heel-induced hydrodynamic forces for large heel angle (in this research)

394 H. Hashimoto et al.

By carrying out systematic comparisons of numerical simulations, it was confirmed the following terms are essential among them to realise accurate prediction of dynamic behaviours related to surf-riding and broaching. i) ii) iii) iv) v)

nonlinear manoeuvring forces in calm water. wave effect on linear manoeuvring forces nonlinear wave-induced surge force nonlinear sway-yaw coupling heel-induced hydrodynamic forces for large heel angle in calm water

For the terms i) and iv), conventional captive model tests in manoeuvrability are sufficient. The term ii) can be estimated with a potential theory. For the terms iii) and v), captive mode tests are required but a seakeeping and manoeuvring basin is not necessary. Thus this theoretical prediction method with the minimum captive test data is still practical. Although the heel-induced hydrodynamic forces for large heel angle in waves are also important for quantitative prediction particularly for the capsizing boundary, the captive model test for measuring them is complicated to execute from a practical point of view. Based on the above, the simplified mathematical model for predicting broaching in following and quartering waves are proposed for a practical purpose, which is called “proposed model”. The state equations replaced with the Eqs.(14)-(20) are expressed as follows: f1 (x;b)  (u cos   v sin   c) / 

(21)

f2 (x;b)  {T (u ; n )  T W (G / , u ,  ; n )  R (u )  X R N L (u ,  )  X M N L (u , v , r )

(22)

 X w (G / ,  )}/ (m  m x ) f3 (x;b)  {(m  m x )u r  Y v (u ; n )v  Y v W (G /  , u ,  )v  Y r (u ; n )r  Y r W (G /  , u ,  )r (23) Y M N L (u , v , r )  Y R N L (u ,  )  Y  (u ; n )  Y w (G /  , u ,  ; n )} / (m  m y )

f 4 ( x; b )  r

(24)

f5 (x;b)  {N v (u ; n )v  N v W (G /  , u ,  )v  N r (u ; n )r  N r W (G /  , u ,  )r

(25)

 N M N L (u , v , r )  N R N L (u ,  )  N  (u ; n )  N w (G /  , u ,  ; n )} / ( I Z Z  J Z Z ) f 6 ( x; b )  p

Model Experiment on Heel-Induced Hydrodynamic Forces 395 f7 ( x;b)  [m x z H (u , v )u r  K v (u , v ; n )v  K v W (G /  , u , v ,  )v  K r (u , v ,; n )r  K r W (G /  , u , v ,  )r  K M N L (u , v , r )  K P (u ) p  K R N L (u ,  )  K  (u ; n )

(26)

 K w (G /  , u ,  ; n )  m gGZ ( )] / ( I xx  J xx )

(27)

f 8 (x;b)  {  K R (    C )  K R TD r} / TE

Here the underlined parts indicate the higher order terms added to the original model (Umeda, 1999). The comparison in the case that the ship experiences a periodic motion is shown in Fig. 16. No significant difference is found but the average of yaw angle is different to some extent. The comparison in the case that the ship suffers surf-riding, broaching and capsizing is shown in Fig. 17. The difference can be only found in the final stage of capsizing. The comparison in boundaries of ship motion modes for control parameters is shown in Fig. 18. Although the region of stable surf-riding decreases significantly and the region of broaching appears at high speed zone in the proposed model, it can reasonably predict broaching and capsizing around the critical Froude number where surf-riding occurs. From these comparisons, it could be concluded that the proposed model has sufficient accuracy for practical use but with less experimental efforts as compared to the enhanced model. Pitch(degrees)

20 10 0 0 -10 -20

60 40 20 0 -200 -40 -60

5

Pitch(degrees)

10

15

20

t(s) Roll(degrees)

5

10

15

20

t(s)

Yaw(degrees)

60 40 20 0 -200 -40 -60

5

10

15

20

t(s)

5

60 40 20 0 -200 -40 -60

5

10

15

20

t(s) Roll(degrees)

5

10

15

20

t(s)

Yaw(degrees)

Rudder(degrees)

45 30 15 0 -150 -30 -45

20 10 0 0 -10 -20

60 40 20 0 -200 -40 -60

5

10

15

20

t(s)

Rudder(degrees)

10

15

20

t(s)

45 30 15 0 -150 -30 -45

5

10

15

20

t(s)

Experiment Calculation

Fig. 16 Comparison of numerical results with H/=1/10, /L=1.637, Fn=0.3, c=-30 degrees. (left: enhanced model, right: proposed model)

396 H. Hashimoto et al. Pitch(degrees)

20 10 0 0 -10 -20 90 60 30 0 -300 -60 -90

5

Pitch(degrees)

10

15

20

t(s) Roll(degrees)

5

10

15

20

t(s)

Yaw(degrees)

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7 Conclusions Nonliear heel-induced hydrodynamic forces in severe following waves were measured by a new experimental method. Then the mathematical model taking these forces into account is developed, and is compared with the existing free running model experiment. As a result, the following conclusions are obtained:

Model Experiment on Heel-Induced Hydrodynamic Forces 397

1. 2. 3. 4. 5.

The proposed experimental method is applicable to measure heel-induced hydrodynamic forces for large heel angle even in severe waves. Heel-induced hydrodynamic surge force in waves is small and can be neglected for capsizing prediction. Heel-induced hydrodynamic sway force, yaw moment and roll moment in waves have certain nonlinearity with respect to roll angle especially over 20 degrees of heel angle. Heel-induced hydrodynamic forces in waves are important for accurate prediction of capsizing. Quantitative prediction of broaching and capsizing in severe following and quartering waves can be realised when the heel-induced hydrodynamic forces in waves as well as other higher order terms are consistently included.

Furthermore, the simplified model with taking only essential higher order terms into account is proposed, and it is confirmed that the proposed model has sufficient prediction accuracy of broaching and capsizing in following and quartering waves for a practical purpose.

Acknowledgements This research was supported by a Grant-in Aid for Scientific Research of the Ministry of Education, Culture, Sports, Science and Technology of Japan (No. 15360465) and a Fundamental Research Developing Association for Shipbuilding and Offshore of the Shipbuilders’ Association of Japan. The authors express their sincere gratitude to the above organisations.

References Grim O (1961) Beitrag zu dem Problem der Sicherheit des Schiffes im Seegang. Shiff und Hafen 6: 490-497 (in German) Hashimoto H, Umeda N, Matsuda A (2004) Importance of Several Nonlinear Factors on Broaching Prediction. J of Marine Science and Technol 9:2: 80-93 Nakato M and Kohara S (1982) Stability Experime\ts in the Following Sea with Ship Speed –An Utilization of Circulating Water Channel–. Proceedings of the 2nd Int Conf on Stab and Ocean Veh 399-410 Renilson MR, Manwarring T (2000) An Investigation into Roll/Yaw Coupling and Its Effect on Vessel Motions in Following and Quartering Seas. Proceedings of the 7th Int Conf on Stab and Ocean Veh 452-459 Umeda N (1999) Nonlinear dynamics of ship capsizing due to broaching in following and quartering seas. J of Marine Science and Technol 4:1: 16-26 Umeda N, Hashimoto H (2002) Qualitative Aspects of Nonlinear Ship Motions in Following and Quartering Seas with High Forward Velocity. J of Marine Science and Technol 6:2 111-121

398 H. Hashimoto et al. Umeda N, Hashimoto H, Matsuda A (2002) Broaching Prediction in the Light of an Enhanced Mathematical Model with Higher Order Terms Taken into Account. J of Marine Science and Technol 7:3: 145-155 Umeda N, Matsuda A, Hamamoto M et al. (1999) Stability Assessment for Intact Ships in the Light of Model Experiments. J of Marine Science and Technol 4:2: 45-57 Umead N, Renilson MR (1992) Broaching -a dynamic behaviour of a vessel in following seas-. Wilson PA (ed) Manoeuvring and control of marine craft, Computational Mechanics Publications 533-543 Umeda N, Renilson MR (2001) Benchmark Testing of Numerical Prediction on Capsizing of Intact Stability in Following and Quartering Seas. Proceedings of the 5th Int Stab Workshop 6.1.1-6.1.10

Perceptions of Broaching-To: Discovering The Past K.J. Spyrou School of Naval Architecture and Marine Engineering, National Technical University of Athens, 9 Iroon Polytechneiou, Zographos, Athens 15780, Greece

Abstract

The article summarizes the key findings of an investigation into old bibliographical sources, aiming to determine how broaching-to was considered in the past; especially in terms of the magnitude of the problem and its perception in the older days. Several relevant citations have been identified: in voyager’s records, in nautical journals, in training manuals, in encyclopaedias and even in literary sources. The fear of broaching-to is referred-to in a famous old British poem, inspired by a real 18th century disaster.

From Falconer (1804)

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_22, © Springer Science+Business Media B.V. 2011

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1 Introduction Whilst an essential grasp of the scientific basis of the broaching-to behaviour of ships has been achieved only after the middle of the 20th century, the subject has been alive in the nautical bibliography since much longer. In the Merriam-Webster dictionary, next to the definition of the word broach [“to veer or yaw dangerously so as to lie broadside to the waves – often used with to”], one is informed that, use of the term in a nautical context is traced as far back as the year 1705. Indeed, in several book excerpts from the 19th, 18th and even the 17th century one can read captivating narratives of real occurrences of broaching-to; attempts of defining the phenomenon formally; and steering practices devised specifically for dealing with it in the “difficult hour”. A glimpse into the days when navigation was, in its own capacity, an adventure reveals that, the prevention of broaching-to was a central issue of ship operation. For a very long time, and more for the past than for our days, the maritime community have associated the term with one of the most dreadful conditions that a ship could be engaged during operation in a rough sea. Nonetheless, current professionals of the maritime field maintain limited recollection of the past of the subject beyond, say, a 50-year horizon. Vivid descriptions of relatively recent broaching-to occurrences can be found, for example, in the review article of Conolly (1972). It is a fair conjecture that, the broaching-to behaviour of sailing ships should have been influenced by phenomena that could not play any role for modern motor ships. Especially, the arrangement of sails, and the incurred significant destabilising wind forces, should have been very influential for turning, despite all efforts, a ship broadside to the weather. The combination of wind and waves should have placed more heavy demands upon the Master of the old days. From our investigation comes out that, broaching-to was mostly feared when a vessel was heading towards the shore; especially when being very near to it. Moreover, it was often discussed within a scenario of ship sailing at high speed, “rushing before the weather”. The current paper is an attempt to shed some light about how broaching-to was perceived in that, almost forgotten, era. And, to possibly trace some seeds of scientific thinking, let pre-mature, about the nature and the mechanics of the phenomenon.

2 Deeping Into the Past 2.1 William Dampier (1699) “A New Voyage Round the World” At the dusk of 17th century, Captain William Dampier describes his experiences of navigating around the globe:

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“... and besides, often very violent and fierce, so that a Ship with her sails loose, would be in danger to be over-set by them, or at least lose Masts or Yards, or have the Sails split, besides the Consternation that all Men must needs be in at such a time, especially if the Ship, by any unforeseen accident, should prove unruly, as by the mistake of the Man at Helm, or he that Conns, or by her broaching to (author’s note: it appears as “too” but it is corrected in an Errata Table that is included) against all endeavours, which often happens when a fierce gust comes; which though it does not last long, yet would do much damage in a short time, and tho’ all things should fall out well, yet the benefit of it would not compensate the danger:”

2.2 A Sailor Turned Famed Poet - William Falconer’s “The Shipwreck” [Robert Anderson (1795)] A celebrated poem written by a sailor, William Falconer, firstly published in 1762 under the title ”The shipwreck” describes quite movingly what should have been the poet’s stunning personal experience of broaching-to, leading to loss of men and the ship. The drama unfolded off the shore of Attica; at the very spot where, according to the myth, the ancient king of Athens Aegeus had jumped into the sea (henceforth giving his name to the Sea in front), disillusioned by a wrong sign alluding to the loss of his son, the mythical hero Theseus, in his battle against the Minotaur in Minoan Crete. William Falconer was a native of Scotland and “bred to the sea in which he spent the greatest part of his life”. Unable to steer clear of his destiny, he went down deep in the Mozambique Chanel with the frigate Aurora, in the winter of 1769 (Knight 1837). The true story and proceedings surrounding his poem come out from the following extract: “… he served on board the Britannia, a merchantman, bound from Alexandria to Venice, which touched at the Island of Candia (author’s note: this is today’s Crete), whence, proceeding on her voyage, she met with a violent storm, that drove her on the coast of Greece, where she suffered shipwreck near Cape Colonne (author’s note: nowadays Cape Sounion, in Attica), three only of the crew being left alive. The dangers which he really experienced he feelingly described in his next performance intituled The Shipwreck, a Poem, in three Cantos, by a Sailor, 4to. 1762. He inscribed it to the Duke of York and prefixed a chart of the ship’s way and a section of the ship itself, in order to render the poem completely intelligible.”

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From Canto III: “While shoreward now the bounding vessel flies, Full in her van St. George’s cliffs arise; High o’er the rest a pointed crag is seen, That hung projecting o’er a moffy green. Nearer and nearer now the danger grows, And all their skill relentless fates oppose. For, while more eastward they direct the prow, Enormous waves the quiv’ring deck o’erflow. While, as she wheels, unable to subdue Her sallies, still they dread her broaching-to.” Amazing is the truly complete definition of broaching-to, supplied in a footnote of Anderson’s (1795) book and found also in several later publications of the poem (it could not be confirmed that this definition had appeared also in the 1762 original publication of Falconer’s poem): “Broaching-to is a sudden and involuntary movement in navigation, wherein a ship, whilst scudding or sailing before the wind, unexpectedly turns her side to windward. It is generally occasioned by the difficulty of steering her, or by some disaster happening to the machinery of the helm.” Reference to broaching appears also earlier in the poem, however what is implied there is the placing of the ship beam to the wave. It is unclear if this is assumed to be voluntary or not: Canto II: Forbid it, Heaven, that, in this dreadful hour, I claim the dangerous reins of purblind power! But should we now resolve to bear away, Our hopless state can suffer no delay. Nor can we, thus berest of every fail, Attempt to steer obliquely on the gale. For then, if broaching sideward to the sea, Our dropsy’d ship may founder by the lee; No more obedient to the pilot’s power, Th’o’oerwhelming wave may soon her frame devour.

2.3 John Hamilton Moore (1791) “The Practical Navigator” A definition of “broach-to” appears in this 18th century book by J. Hamilton Moore, who portrays himself as “Teacher of Navigation, Hydrographer and Chart Seller”. Although more concise, his definition conveys an identical meaning:

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“Broach to: Is when a Ship, on a sudden, lays her Broadside to the Sea, and is dangerous in bad Weather.”

3 Broaching-To In The 19th Century 3.1 “The Life-boat” (1855), Extract from the Article “Management of Ships in a Surf and Broken Water” (Vol. II, No 18). (Author’s note: According to the Merriam-Webster Dictionary, surf – date 1685 – “is the swell of the sea which breaks upon the shore; and the foam, splash, and sound of breaking waves”). In this 1855 article one finds quite a lengthy discussion on operational practices for avoiding broaching-to and also about what is the cause of the phenomenon: “On the second point, running before a broken sea, an equal variety of management is observable, as practised on the coast, yet all alike intended to meet the one great risk of “broaching-to”, which nearly all agree in considering to be the greatest danger to which a boat can be exposed, and to be that which calls for the most skill and management to obviate it. As before observed, the greater number of skilful boatmen on the coast are in the habit of checking a boat’s way through the water or of backing her against a heavy sea on its approach. Their practice is to stop the boat’s way by backing their oars until the crest of the wave has struck the boat’s stem and passed her midship part, then to give way again, running in on the back of that wave, as far as they may be able to, then watching for the approach of the next, and repeating the same operation until they arrive at the beach, being careful, by steering with oars at the quarter or stern to keep the boat as far as possible, end on to the direction in which the sea is running. It must be here observed, that this management is so far varied according to the character of the boat; that in cobles, and other square sterned boats which have their bows better formed for meeting a sea than their sterns are, their position is reversed before entering the broken water, and they are taken in stern foremost and bow outwards, but the same principle being acted on о rowing back to meet each heavy wave instead of running from it. In a sailing boat this principle can only be so far acted on as to diminish the boat’s speed through water by taking her in under a very educed amount of sail, which is commonly one and by towing weights or instruments lade for the purpose. The advocates of this system of management have certainly reason on their side, in addition to experience; for as all acknowledge that there is greater danger in running before a broken sea than in going off against it, it is obvious that the more the former operation be assimilated to the latterm the safer it must be also, which is therefore effected by an alternate progressive and retrogressive movement, the

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latter being effected at the particular moments when the progressive motion would be dangerous. The true theory on which this practice is founded we will endeavour presently to explain. Proceeding then to the opposite practice of giving a boat speed, in fact running away from the sea, which constitutes the other important distinction in practice. The principle then acted on is to escape from the danger as fast as possible and other expedients are then resorted to prevent the risk of broaching-to. The most common of these is to trim the boat by the stern, by bringing all moveable weights aft (this supposes her stern to be outwards; if she were being taken in stern foremost, she would then be trimmed by the head). The force of the sea or wind on either quarter has then less power to beat it off, and cause the boat to broach-to, than it would have if the stern were light. Another expedient is to tow a pig of ballast, or a basket, or other instrument, which by its weight or hold on the water has the effect of a drag on the rear end of the boat, and prevents its being beat to leeward by the sea, thus keeping her end on to it. On the coast of Norfolk the following ingenious plan is commonly practised. The boatmen there employ an instrument for the above purpose called a “drogue”; it is a conical shaped canvas bag, of the form of a common candle extinguisher, about 2 feet diameter at the base or mouth of the bag, and 6 feet long, having a small opening at the other end or apex of the cone When running before a heavy sea in broken water, the drogue is thrown over from the stern, and towed by a stout rope with the large end foremost, when it instantly fills, and from the resistance it opposes to the water, holds the stern back and prevents the boat’s broaching-to: as soon as the danger is past the large tow rope is let go, and the drogue then towed with the smaller end foremost by a small line attached to that end, it then immediately collapses is emptied of water and offers but little resistance. The steering with an oar on each quarter is another expedient, employed to prevent broaching-to, as when running, a boat will not answer her helm on being overtaken by a sea. The recommendation to watch for an opportunity and avoid a sea, equally in running as on going off, could only be practised at those localities where the beach is steep. In reply to the question, as to whether any particular kinds of boats are more liable than others to broach-to, the answers given are so conflicting and contradictory as to afford no information on the point. In reply to another question as to the cause of a boat’s broaching-to, the almost invariable answer is, “because the stern is thrown out of the water and the rudder therefore ceases to act”. From our own observation we have formed the opinion that this is not the case, although it is quite true that at the moment of broachingto, a bout will not answer her helm. The phenomenon of broaching-to, we believe, may be correctly accounted for as follows: - on a boat encountering a heavy broken sea or roller end on, if she be stationary or is being profiled in a contrary direction to the wave, she will receive its blow, and it will quickly pass by her, her own inertia preventing her being

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carried away by it. If however she is being propelled in the same direction as the waves, and running rapidly through the water with her stern towards them, on a wave overtaking her, its first effect is to throw her stern up and to depress her bow, but so far from her rudder being out of water, both it and her stern are buried in the crest of the wave; in consequence, however, of her previous motion being in the same direction as that of the wave, she now offers so slight resistance to it, that instead of its passing her, she is hurried along with it at a rapid rate over the ground, her stem high up still immersed in the crest of the sea, and her bow low down at its base; as the wave approaches shoaler water, its inshore surface approaches more and more nearly to a perpendicular, and the tendency of the boat to run down this steep inclination added to the momentum she has already from her previous motion, causes her to run her bow under water, when her buoyancy at that end being destroyed her stern still light is pressed onward by the summit of the wave, and the undercurrent from the last receding wave at the same time acting on her bow, she is instantly, if a short boat, turned “end over end,” or if a long one, capsized quarter wise If she have so high a bow that it does not become altogether immersed, or if, as in a life-boat, the end of the boat is occupied by a water-tight air-case to the height of the gunwale, so as to prevent the admission of the water over the bow, the effect then is that the boat is instantaneously turned round broadside to the sea, when again, unless she be a life boat of a superior description, she is almost certain to be upset. In the circumstances thus described, the sole cause of a boat’s running herself under water or broaching-to is that of running from a sea instead of awaiting it, and suffering it to pass by; and the cause of the rudder being useless to keep the boat end on to the wave, is not that it is thrown out of the water, (although at other times it doubtless is so), for it is actually buried in it as is also the stern of the boat up to her gunwale, but it is because it is stationary in it the crest of the wave having acquired an actually progressive motion equal to that of the boat. If on the other hand the wave passes the boat, as its crest advances from the stern to the fore part, the rudder and stern are thrown out of the water; steering oars are therefore a most valuable auxiliary aid when running before a sea, but we would recommend the use of a rudder as well. We have been rather prolix in our account of the phenomenon of broaching-to, because it is a very interesting one, which it is important should be understood in order to arrive at the proper management and to obviate its disastrous effects, which have been more fatal to the lives and property of boatmen on our coasts than those proceeding from any other cause whatever.” Further exchanges appear in subsequent issues of the same journal. In one case (Vol II, No 19) an interesting observation is presented about frequent occurrence of broaching-to events near the English coast town Deal, that is located to the NorthEast of Dover: “… nevertheless, we are positively informed that boats have been lost by broaching-to when running for the shore at Deal; and we have known a Suffolk yawl of 18 or 20 tons burden, broach-to, upset, and drown the greater part of her crew, when running under sail for the shore on as steep a beach as that at Deal.”

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3.2 Henry Coleman Folkard (1863) “The Sailing Boat” In the section “Causes of ship capsizing” of this book one finds a description of broaching. But more enlightening for the reader is the discussion that ensues, on the “drogues”. These were (and still are) used for helping the ship to keep her course and avoid broaching (and bow diving): “The action of the sea upon a boat running into a heavy surf, may be thus described: - when on the top of a heavy wave or roller, the bows are lifted high out of the water; then, as the sea recedes, the boat is hurled forward, and the bows are buried under water; when the sea acting powerfully on her head and fore gripe, twists her round, broadside to the waves, called “broaching to;” and the sea then runs over the gunwale into the boat; the next motion that inevitably follows, is a heavy lurch on the other side and another sea breaks completely over, and fills or capsizes the boat. This may happen either under sail or oars. There is considerable difficulty in preventing a boat from broaching to, when stem and stern are alternately lifted out of the water by the waves; and should the boat broach to and meet a very heavy roller, broadside on, the chances are fifty to one that she will be swamped. Drogues* are now a good deal used on the Eastern Coast, in both sailing and rowing boats; they serve to check the boat’s way, and keep her end on to the waves; and are, therefore, of great assistance to the crew, in preventing the boat from broaching to. Experience teaches, that when a heavy breaker follows the boat up astern, it is useless to attempt running away from it: then a question naturally arises, what must be done on the impulse of the moment. “For your lives men! back her astern; hard at it every one of you! and let the man in the stern-sheets creep forward a moment, to lighten the boat’s stern!” By this effort the wave strikes the boat kindly, and passes on; but if allowed to follow her up astern, so surely as such an experiment is tried, the sea will either curl over the stern, or the boat will broach-to and take it over the gunwale. * A drogue is a conical shaped collapsible bag, about two feet in diameter at the mouth, and four feet six inches in length. When towed by the mouth, the drogue fills with water, and draws heavily; thereby checking the progress of the boat. A tripping-line is made fast to the apex or pointed end, and by slacking the towingrope and hauling on the tripping-line, the drogue collapses, and may be drawn on board very easily.“ Even in our days, drogues [and indeed multiple (“series”) drogues] are considered as indispensable drag devices onboard sailing boats, intended to be used in heavy weather (consult for example Hinz 2003).

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3.3 John McNeill Boyd (1860) “A Manual for Naval Cadets”. Drawing upon his experience from the Royal Navy, Captain Boyd advises in his book young cadets of the time about broaching-to and how to avoid it through operation. In the first paragraph he is probably alluding to the condition of “surfriding”. His discussion on the drogues should also be noted. “The great danger when running before a broken sea, is that of broaching-to. The cause of a boat’s broaching-to when running before a broken sea or surf is, that her own motion being in the same direction as that of the sea, whether it be given by the force of oars or sails, or by the force of the sea itself, she opposes no resistance to it, but is carried before it. Thus if a boat be running with her bow to the shore and her stern to the sea, the first effect of a surf or roller on its overtaking her, is to throw up the stern, and as a consequence to depress the bow; if she then has sufficient inertia (which will be proportional to weight) to allow the sea to pass her, she will in succession pass through the descending, the horizontal, and the ascending positions, as the crest of the wave passes successively her stern, her midships, and her bow, in the reverse order in which the same positions occur to a boat propelled to seaward against a surf. This may be defined as the safe mode of running before a broken sea. But if a boat on being overtaken by a heavy surf, has not sufficient inertia to allow it to pass her, the first of the three positions above enumerated alone occurs, her stern is raised high in the air and the wave carries the boat before it, on its front, or unsafe side, sometimes with frightful velocity, the bow all the time deeply immersed in the hollow of the sea, where the water, being stationary or comparatively so, offers a resistance, whilst the crest of the sea, having the actual motion which causes it to break, forces onward the stern, or rear end of the boat. A boat will in this position sometimes, aided by careful oar steerage, run a considerable distance until the wave has broken and expended itself. But it will often happen that, if the bow be low it will be driven under water, when the buoyancy being lost forward, whilst the sea presses on the stern, the boat will be thrown (as it is termed) end over end; or if the bow be high, or it be protected, as in some life-boats, by a bow air chamber, so that it does not become submerged, that the resistance forward acting on one bow will slightly turn the boat’s head, and the force of the surf, being transferred to the opposite quarter, she will in a moment be turned round broadside by the sea and be thrown by it on her beam-ends, or altogether capsized. It is in this manner that most boats are upset in a surf, especially on flat coasts, and in this way many lives are annually lost amongst merchant seamen when attempting to land after being compelled to desert their vessels. Hence it follows that the management of a boat, when landing through a heavy surf, must as far as possible be assimilated to that when proceeding to seaward against one, at least, so far as to stop her progress shoreward at the moment of being overtaken by a heavy sea, and thus enabling it to pass her. There are different ways of effecting this object:-

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1st. By turning a boat’s head to the sea before entering the broken water, and then backing in stern foremost, pulling a few strokes ahead to meet each heavy sea and then again backing astern. If a sea be really heavy and a boat small, this plan will be generally the safest, as a boat can be kept more under command when the full force of the oars can be used against a heavy surf than by backing them only. 2nd. If rowing to shore with the stern to seaward, by backing all the oars on the approach of a heavy sea, and rowing ahead again as soon as it has passed to the bow of the boat, thus ,rowing in on the back of the wave; or, as is practised in some lifeboats, placing the after-oarsmen, with their faces forward, and making them row back at each sea on its approach. 3rd. If rowed in bow foremost, by towing astern a pig of ballast or large stone, or a large basket, or a canvass bag termed a “drogue” or drag, made for the purpose, the object of each being to hold the boat’s stern back and prevent her being turned broadside to the sea or broaching-to. Drogues are in common use by the boatmen on the Norfolk coast; they are conical-shaped bags of about the same form and proportionate length and breadth as a candle extinguisher about two feet wide at the mouth and four and a half feet long. They are towed with the mouth foremost by a stout rope or a small line termed a tripping-line, being fast to the apex or pointed end. Whеn towed with the mouth foremost they fill with water, and offer a considerable resistance, thereby holding back the stern; by letting go the stouter rope and retaining the smaller line, their position is reversed, when they collapse, and can be readily hauled into the boat. Drogues are chiefly used in sailing-boats, when they both serve to check a boat’s way and to keep her end on the sea. They are however a great source of safety in rowing-boats, and many rowing life-boats are now provided with them.”

4 Other References to Broaching-to With the opportunity of the establishment of the “Society for the Improvement of Naval Architecture” (forerunner of RINA) in London, Hutchinson (1791) advises that, in bad weather the best helmsman should be employed: “… and if the waves run high, when carrying a pressing sail, large, by such bad steering there is great danger of broaching the ship to; therefore none but the best helmsman should be permitted to steer at such times.” Later on, Hutchinson advises how to avoid broaching-to even when the foremast is lost. He refers to broaching-to in connection to scudding, a condition quite uniquely connected with sailing boats: The following entry appeared in one of the oldest editions of Encyclopædia Britannicca (1797): “Scudding, the movement by which a ship is carried precipitately before a tempest. As a ship flies with amazing rapidity through the water whenever this expedient is put in practice, it is never attempted in a contrary wind unless when her condition renders her incapable of sustaining the mutual effort of the wind and

Perceptions of Broaching-To: Discovering The Past 409

waves any longer on her side, without being exposed to the most imminent danger of being overset. A ship either feuds with a sail extended on her foremast, or, if the storm is excessive, without any sail; which in the sea phrase, is called scudding under bare poles. In sloops and schooners and other small vessels, the sail employed for this purpose is called the square sail. In large ships, it is either the foresail at large, reefed, or with its goose-wings extended, according to the degree of the tempest; or it is the fore-top sail, close reefed, and lowered on the cap; which last is particularly used when the sea runs to high as to becalm the foresail occasionally, a circumstance which exposes the ship to the danger of broaching-to. The principal hazards incident to scudding are generally a pooping sea; the difficulty of steering, which exposes the vessel perpetually to the risk of broaching-to; and the want of sufficient sea-room A sea striking the ship violently on the stern may dash it inwards, by which the must inevitably founder. In broaching-to (that is, inclining suddenly to windward) she is threatened with being immediately overturned; and, for want of sea-room, she is endangered by shipwreck on a lee-shore, a circumstance too dreadful to require explanation.” In another encyclopaedia was found a section offering advice for selecting the shape of a life-boat, with specific reference to the avoidance of broaching-to (Encyclopædia Perthensis 1816). We are informed also that, the invention of the lifeboat was claimed by William Wouldhave and that his proposed design was firstly built in 1789: “The curvature of the keel has however been demonstrated to be the principal or only error in the construction of the vessel and we would recommend to those who in future may construct such vessels to preserve the spheroidal form of the body of the boat, yet, so as to leave a straight keel and a sufficient quantity of gripe to hinder the boat from broaching to, on receiving the stroke of the waves on her ends.” In an American navigation textbook (Bowditch 1821) we learn about the difference between “broaching-to” and “bringing by the lee”: “To broach to. To incline suddenly to windward of the ship’s course, so as to present her side to the wind, and endanger her oversetting. The difference between broaching to and bringing by ht e lee may be thus defined: Suppose a ship under great sail is steering south, having the wind at N.N.W. then west is the weather side, and east the lee-side. If, by any accident, her head turns round to the westward, so that her sails are all taken a-back on the weather side, she is said to broach to. If, on the contrary, her head declines so far eastward as to lay her sails aback on that side which was the lee side, it is called bringing by the lee.” At the same time (and side of the Atlantic), Guest (1824) narrates his experiences of sea travel: “On the 25th, at 3, A.M. finding we could not, with any degree of safety, lay to any longer, we commenced scudding, under a close reefed foresail but soon found we could only scud under our bare poles, which we continued to do until 6 A.M.. The vessel then broached to, and lay with her broad side to the wind, in the trough of the sea (which is the most dangerous situation that a vessel can be in:) a most

410 K.J. Spyrou

tremendous sea was rolling down upon us, which we expected would soon swallow us up. I shall never forget the frightful looks and behaviour of the man who was at the helm when the vessel broached to. Although he was a very experienced seaman, he exclaimed calling on his God and Saviour -we are gone! we are gone! nothing can save us!” Fishbourne (1846) in his lectures on naval architecture argues for selecting a less fine stern compared to the bow: “The reason why short vessels do not run well.- It is notorious that vessels with fine after bodies, particularly if they be short, run badly. It is because there is so little action in the after body in such case, for the water cannot turn in upon it, there is even a danger in such vessels of their broaching to against their helm. The after body then should be greater than the fore body, in some ratio inverse of this action of the water.” In the shipbuilders guide of Partington (1826) we are advised that the use of sliding keels could save a ship: “… no misfortune, similar to that of broaching to, can ever befall a vessel furnished with sliding keels.” About these keels we learn from another passage referring to a 66 ft vessel: “ She has three sliding keels inclosed in a case or well; they are each 14 ft in length; the fore and the after keels are 3 ft broad each, and the middle keel is 6 feet broad. The keels are movable by means of a winch, and may be let down 7 feet below the real keel; and they work equally well in a storm as in still water.” Short references expressing the fear of broaching-to appear in numerous narratives of legendary expeditions around the globe. Two of them are mentioned below: 

In the search for a North-West passage from the Atlantic to the Pacific (Parry 1824).

 In the tales of the voyages of the ship H.M.S. Samarang in the islands of the Eastern Archipelago (Belcher 1848).

References Anderson R (1795) The Works of the British Poets: With Prefaces, Biographical and Critical, Printed for John & Arthur Arch; and for Bell & Bradfwte; and J. Mundell & Co. Edinburgh. Belcher E (1848) Narrative of the voyage of H.M.S. Samarang, during the years 1843-46; in the Islands of the Eastern Archipelago; Pub by Reeve, Benham, and Reeve, King William Street, Strand. Bowditch N(1821) The New American Practical Navigator. Published by E. M. Blunt, N. Y. McNeill Boyd J (1860) A Manual for Naval Cadets. Published by Longman, Green, Longman, and Roberts, 548 pp. Conolly JE (1972) Stability and control in waves: A survey of the problem. Journal of Mechanical Eng Science, Vol. 14, No 7. Dampier W (1699) A New Voyage Round the World, Published by J. Knapton. Encyclopædia Britannica: Or, A Dictionary of Arts, Sciences, and Miscellaneous Literature (1797), Edinburgh, Printed for A. Bell and C. MacFarquhar.

Perceptions of Broaching-To: Discovering The Past 411 Encyclopædia Perthensis; or Universal Dictionary of the Arts, Sciences, Literature &c. (1816) 2nd edition. Edinburgh, Printed by John Brown, Anchor Close. Falconer W (1804) The Shipwreck: a poem in three cantos. With a life of the author by James Stanier Clarke. Printed for William Miller, Old Bond Street, London. Fishbourne EG (1846) Lectures on Naval Architecture, Being the Substance of Those Delivered at the United Service Institution. Pub by John Russell Smith, London. Folkard HC (1863) The Sailing Boat: A Treatise on English and Foreign Boats , Descriptive of the Various Forms of Boats and Sails of Every Nation ; with Practical Directions for Sailing, Management &c , Pub from Longman, Green, Longman, and Roberts, 317 pp. Hinz E (2003) Heavy Weather Tactics Using Sea Anchors and Drogues, Paradise Cay Publications, 2nd Edition, 164 pages, ISBN-10 0939837374. Hutchinson W (1791) A treatise founded upon philosophical and rational principles: towards establishing fixed rules, for the best form and proportional dimensions in length, breadth and depth of merchsant ships in general; and also the management of them to the greatest advantage by pratical seamanship. Printed in Liverpool by Thomas Billinge, Castle Street. Knight C (1837) Penny Cyclopaedia of the Society for the Diffusion of Useful Knowledge, Vol. IX, Dionysius-Erne, Charles Knight and Co, 22 Ludgate Street, London. Moore HJ (1791) The Practical Navigator, and Seaman’s New Daily Assistant: Being an Epitome of Navigation: Including the Different Methods of Working the Lunar Observations. With Every Particular Requisite for Keeping a Complete Journal at Sea . Printed for and sold by R. Law and Son, London, 296 pages. Parry E (1824) Journal of a Second Voyage for the Discovery of a Northwest Passage from the Atlantic to the Pacific: Performed in the Years 1821- 22-23, in His Majesty’s Ships Fury and Hecla, Under the Orders of Captain William Edward Parry. Pub by J. Murray, London. The Life-boat, Or, Journal of the National Life-Boat Institution (1853). Pub by the Royal National Life-Boat Inst for the Preservation of Life from Shipwreck (Great Britain).

5 Nonlinear Dynamics and Ship Capsizing

Use of Lyapunov Exponents to Predict Chaotic Vessel Motions Leigh S. McCue* and Armin W. Troesch** *Aerospace and Ocean Engineering, Virginia Tech, **Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, Michigan

Abstract It is the aim of this paper to further the use of Lyapunov and local Lyapunov exponent methods for analyzing phenomena involving nonlinear vessel dynamics. Lyapunov exponents represent a means to measure the rate of convergence or divergence of nearby trajectories thus denoting chaos and possibly leading to the onset of conditions that produce capsize. The work developed here makes use of Lyapunov exponent methodologies to study capsize and chaotic behavior in vessels both experimentally and numerically using a multi-degree of freedom computational model. Since, the Lyapunov exponent is defined in the limit as time approaches infinity, one encounters fundamental difficulties using Lyapunov exponents on the capsize problem, which is inherently limited to a finite time. This work also incorporates the use of local Lyapunov exponents, which do not require an infinite time series, to demonstrate their usefulness in analyzing finite time chaotic vessel phenomena. The objective is to demonstrate the value of the Lyapunov exponent and local Lyapunov exponent as a predictive tool with which to indicate regions with crucial sensitivity to initial conditions. Through the intelligent use of Lyapunov exponents in vessel analysis to indicate specific regions of questionable stability, one may significantly reduce the volume of costly simulation and experimentation.

1 Background The method of Lyapunov characteristic exponents serves as a useful tool to quantify chaos. Specifically, Lyapunov exponents measure the rates of convergence or divergence of nearby trajectories (Haken, 1981; Wolf, 1986). Negative Lyapunov exponents indicate convergence, while positive Lyapunov exponents demonstrate

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_23, © Springer Science+Business Media B.V. 2011

415

416 L.S. McCue and A.W. Troesch

divergence and chaos. The magnitude of the Lyapunov exponent is an indicator of the time scale on which chaotic behavior can be predicted or transients decay for the positive and negative exponent cases respectively (Wolf, 1986). Physically, the Lyapunov exponent is a measure of how rapidly nearby trajectories converge or diverge. If one considers a ball of points in N-dimensional phase space, in which each point follows its own trajectory based upon the system’s equations of motion, over time, the ball of points will collapse to a single point, will stay a ball, or will become ellipsoid in shape (Glass & Mackey, 1988). The measure of the rate at which this infinitesimal ball collapses or expands is the Lyapunov exponent. For a system of equations written in state-space form x  u x  , small deviations from the trajectory can be expressed by the equation

    then defined by Equation 11. Often times only the maximal Lyaponov exponent is discussed since the maximal exponent is simplest to calculate from a numerical time series and yields the greatest insight into the dynamics of the system. However, for a space with dimension N, there are N Lyapunov exponents which make up the Lyapunov spectrum and correspond to the rate of expansion or contraction of the principal axes of the infinitesimal N-dimensional ball. For example, after ordering Lyapunov exponents with 1 being the largest and N being the smallest, the length of the most rapidly growing principal axis is proportional to e1t, the area of the two most rapidly growing principal axes is proportional to e1+2 t etc... (Wolf et al., 1985). Other useful quantities are the short time Lyapunov exponent and the local Lyapunov exponent. A short time Lyapunov exponent is simply a Lyapunov exponent defined over some finite time interval. The local Lyapunov exponent is a short time Lyapunov exponent in the limit where the time interval approaches zero. Both are dependent on starting points, and the short time Lyapunov exponent is also dependent on the magnitude of the time interval. Equations for short and local Lyapunov exponents are presented in Equations 2 and 3 respectively (Eckhardt & Yao, 1993). Since it is not practically possible in a numeric or experimental sense to take the limit T → 0, for the purposes of this paper the phrases ‘short time’ and ‘local’ Lyapunov exponents will be used interchangeably in reference to the approach given by Equation 2.

 xi   ui x   x j (Eckhardt & Yao, 1993). The maximal Lyapunov exponent is j

1The Lyapunov exponent is defined with the logarithm base e. Depending on the nature of the application, at times the exponent is calculated base 2 in order to allow the output to be expressed in terms of bits per second. This lends physical insight as to a rate at which information about the state of the system is created or destroyed (Schuster, 1984). For consistency, in this work, the exponent is always calculated using the true definition base e and is therefore expressed nominally with units of 1/time. ‘Nominally’ is due to the mixing of dimensions in the sections 3 and 4 of this work. It is important for the practitioner to give careful thought towards how they wish to treat multiple degree of freedom problems in calculation of Lyapunov exponents. Arguments can be made for maintaining dimensionality, and thus ‘weighting’ some degrees of freedom more heavily than others, while in other circumstancees it may make more sense to nondimensionalize all degrees of freedom in some consistent manner, such as nondimensionalizing all terms by the individual term’s standard deviation.

Use of Lyapunov Exponents to Predict Chaotic Vessel Motions 417

Experimental data was collected with a time step of 0.033 s and thus for consistency the short time Lyapunov exponent is calculated at this interval in the numerical model.

1 t  t

  lim log

xt  x0

T xt , x0 

(1)

xt  T  1 log T xt 

(2)

xt  T  xt 

(3)

1 T 0 T

local xt   lim log

One can draw conclusions about the nature of the dynamical system from the spectra assembled. For a one-dimensional system a positive Lyapunov exponent indicates chaos, a negative exponent defines a periodic orbit, and a zero value represents an orbit with marginal stability (Wolf et al., 1985). For the case of a three-dimensional system there are three Lyapunov exponents making up the spectra. A fixed point consists of all negative exponents. A limit cycle’s spectrum would have two negative values and a zero. A two-torus has two zeros and a negative value in its spectrum, and a strange attractor has one each of positive, negative, and zero values.(Wolf et al., 1985) This pattern can be extended to higher dimensional spaces. Since the Lyapunov spectrum is derived from a long-time evolution of an infinitesimal sphere, it is not a local quantity either spatially or temporally (Wolf et al., 1985). Therefore, one of the potential benefits to examining the capsize problem through the method described by Lyapunov exponents, is the number of numerical or physical experiments necessary to draw quantitative conclusions can be drastically reduced from the quantity which would be necessary by brute force simulation. The use of Lyapunov exponents to study capsize appears in the literature for both naval architecture and nonlinear dynamics. For example, the asymptotic Lyapunov exponent has been calculated from equations of motion for the mooring problem (Papoulias, 1987), single-degree-of-freedom capsize models (Falzarano, 1990; Murashige and Aihara, 1998a and Murashige and Aihara, 1998b; Murashige et al., 2000 and Arnold et al., 2003), work studying the effects of rudder angle while surf-riding as it leads to capsize (Spyrou, 1996), applied to parametric roll problems (McCue & Bulian, 2007, Neves et al., 2009), and real-time capsize warning for single and multi-DOF systems (McCue, 2004; McCue & Troesch, 2006; McCue et al, 2006; Story, 2009).

418 L.S. McCue and A.W. Troesch

The work of the following sections provides a detailed study into the use of Lyapunov exponents for analyzing stable, marginally unstable, and unstable dynamics leading to capsize. The goal is to evaluate Lyapunov exponents as a predictive tool with which to indicate regions of crucial sensitivity to initial conditions. To allow for detailed investigation into regions of questionable stability, one may reduce the necessary volume of simulations and experimentation of the entire phase space through intelligent use of Lyapunov exponents. Additionally, since the Lyapunov exponent provides a quantitative value, it can be used as a tool to directly compare the accuracy of a numerical model to experimental runs. Rather than attempting to qualitatively evaluate a numerical tool through predicting capsize with a heads or tails type comparison of capsize versus non-capsize, the Lyapunov exponent can be used to demonstrate that a numerical model is in fact simulating the same type, or magnitude of chaos as experiments.

2 Lyapunov Exponents from Experimental Time Series 2.1 Theory A large volume of work has been dedicated to the problem of calculating Lyapunov exponents from experimental time series. A number of researchers have developed methods which can be divided into two distinct approaches, direct methods and tangent space methods. Direct methods consist of searching the time series for neighbors at any given point and calculating expansion rates through comparison to these neighboring points. The first such method was that of Wolf et al (1985). Wolf et al (1985) developed a methodology in which one can calculate the largest positive Lyapunov exponent from a data set by following the long term evolution of one principal axis, a ‘fiducial trajectory’, progressively reorthonormalized maintaining phase space orientation (Wolf et al., 1985). Wolf’s method is highly sensitive to inputs, however, and can easily lead to an erroneous result. In the early 1990’s two separate research groups produced a new method (Rosenstein et al., 1993; Kantz, 1994). The approach eliminates the requirement Wolf imposes upon maintaining phase-space orientation stating it is unnecessary for calculating the largest Lyapunov exponent (Rosenstein et al., 1993). Additionally, rather than following one trajectory, the full data set is used, and in essence a trajectory for every pair of nearest neighbors is calculated. For details refer to Rosenstein et al (1993) and Kantz (1994). Both methods are substantively similar. The Kantz algorithm (and similarly the Rosenstein algorithm)

Use of Lyapunov Exponents to Predict Chaotic Vessel Motions 419

calculates the largest Lyapunov exponent by searching for all neighbors within a neighborhood of the reference trajectory and computes the average distance between neighbors and the reference trajectory as a function of time (or relative time scaled by the sampling rate of the data) (Kantz, 1994; Rosenstein et al., 1993). The algorithm computes values for Equation 4 with parameters defined as follows: xt, arbitrary point in time series; Ut, neighborhood of xt; xi neighbor of xt; , relative time scaled by sampling rate; T length of time series; S() stretching factor with region of robust linear increase showing slope equal to Lyapunov exponent ie et  eS() (Kantz, 1994; Kantz & Schreiber, 2004). However, this post-processing requirement of a robust linear increase in slope introduces new errors. While the method is useful and accurate for systems with known values for the Lyapunov exponent, the choice of region and parameters over which a ‘robust linear increase’ are found is somewhat arbitrary. It is the opinion of the authors that this tool is useful only if one knows what value of Lyapunov exponent is desired and can thus choose the region exhibiting a slope equal to that value.

S   

1 T  1  ln T t 1  U t



iU t

 xt   xi   

(4)

Tangent space methods, developed simultaneously by the separate research teams of Sano and Sawada (1985) and Eckmann and coauthors (1985; 1986) allow for calculation of the full spectrum of Lyapunov exponents through local predictions of the Jacobian along the time series trajectory. For example, for a given trajectory x(t) defined by Equation 5, the tangent vector  is given by the linearized form of Equation 5 presented in Equation 6 where J is the Jacobian matrix of f, J = f/x (Sano & Sawada, 1985). Sano and Sawada (1985) solve Equation 6 through a least squares estimate of the time dependent linear operator Aj which approximates the map from (0) to (t). The Lyapunov exponents are then computed using Equation 7 where  is a flow scale time increment, n is then number of data points, and e is an orthonormal basis maintained using a GramSchmidt renormalization process (Sano & Sawada, 1985). For details of this process refer to Sano and Sawada (1985) or the similar works of Eckmann et al (1985; 1986).

x  f x 

(5)

  J xt   

(6)

1 n  n

i  lim

n

 ln A e j 1

j j i

(7)

420 L.S. McCue and A.W. Troesch

The weakness of this approach is in its sensitivity to choice of embedding dimension. Too small an embedding dimension outputs erroneous Lyapunov exponents while too large an embedding dimension creates spurious exponents (Kantz, 1994). However, for this application with careful attention paid to the choice of embedding dimension, the tangent space method was found to be more robust then the direct methods as it was not dependent on any form of arbitrary postprocessing. To check for spurious exponents, the technique first suggested by Parlitz (1992) of analyzing both the original time series and the reversal of the original time series was used. No significant errors were noted for the largest Lyapunov exponents presented in this work. The implementation of the Sano and Sawada method included in the TISEAN (Hegger et al., 2000) package was used to calculate Lyapunov exponents for the experimental time series in the following subsections.

2.2 Application to the Capsize Problem The Sano and Sawada algorithm (1985) contained in TISEAN (Hegger et al., 2000) was applied to roll time series for the capsize and non-capsize experiments detailed in works by Obar and Lee (2001) and Lee et al (2006) with an embedding dimension equal to 6 to represent the six state-space variables. The physical model used in the experimental analysis and replicated numerically was a box barge featuring the following principal parameters: beam, B = 30.48 cm; beam/draft, B/T = 1.67; depth/draft, D/T = 1.06; beam/wave length, B/ = 0.23; wave amplitude/wave length, 0/= 0.01; roll natural frequency n = 2.28 rad/s; excitation frequency/roll natural frequency, e/n = 3.0; and angle of vanishing stability v = 11.4 degrees. The results of completing the process of using Sano and Sawada’s algorithm to calculate the largest Lyapunov exponent for all capsize and non-capsize runs are presented in Figure 1. As one might expect, there is relatively little variation in Lyapunov exponent values. The mean Lyapunov exponent for all runs is 1.76 s-1. The mean exponent for capsize runs, 1.83 s-1 is somewhat larger than that of noncapsize runs, 1.57 s-1 with significantly larger outliers. The histograms in Figure 2 show the distribution of Lyapunov exponents for both the capsize and non-capsize cases. Due to the brevity of the time series leading to capsize, it is impossible to confirm convergence of the Lyapunov exponent. However, it is feasible to discern an order of magnitude of the Lyapunov exponent with which to compare the numerical model as well as to conclude that runs leading to capsize are more chaotic in nature than those not leading to capsize as is apparent in Figures 1-2. The Lyapunov exponent is, by definition, an infinite time parameter, but by nature capsize is a finite event. For this reason greater insight can be gleaned through

Use of Lyapunov Exponents to Predict Chaotic Vessel Motions 421

calculation of the local Lyapunov exponent as will be discussed in Section 4. It should be noted that all runs result in a positive, chaotic, Lyapunov exponent. Recall that under the definition of the Lyapunov exponent a stable limit cycle would have a zero Lyapunov exponent. Therefore, the motions measured in the six dimensional state-space are growing rather that oscillating sinusoidally. Chaotic behavior is detected even in the non-capsize case.

3 Lyapunov Exponents from Ordinary Differential Equations 3.1 Theory As discussed in the previous section, the Lyapunov exponents measure the evolution of an infinitesimal sphere. However, infinitesimal quantities are not computationally feasible, and for a chaotic system with a finite initial separation between the principal axes of the sphere, it is impossible to assure convergence of solution for Lyapunov exponents before infinite values are encountered (Wolf,1986; Wolf et al., 1985). Therefore, other means of evaluation are necessary. To overcome this difficulty, Benettin et al proposed a method in which the trajectory of the center of the infinitesimal sphere, the fiducial trajectory, is defined by the nonlinear equations based upon initial conditions (Benettin et al., 1980). Principal axes are then calculated from the linearized form of the equations of motion about the fiducial trajectory (Wolf, 1986). These axes will, by definition, be infinitesimal relative to the attractor (Wolf et al., 1985). In implementation two computational difficulties arise, namely, computational limitations prohibit calculation of exponential growth andall basis vectors will have a tendency towards the direction of most rapid growth allowing for calculation of only the largest Lyapunov exponent (Wolf et al., 1985). To overcome these computational difficulties a Gram-Schmidt orthonormalization procedure is used (Wolf et al., 1985; Bay, 1999). Through noting the magnitude of vectors prior to re-normalization, growth rates are calculated. Additionally, by maintaining an orthonormal basis all Lyapunov exponents can be calculated. The first vector will naturally tend towards the largest rate of growth, the second vector towards the second most rapid growth rate, and so on (Wolf et al., 1985). Through the use of a Gram-Schmidt reorthonormalization the rapidly growing axes can be renormalized to represent an orthonormal basis maintaining the volume’s phase-space orientation (Wolf, 1986; Wolf et al., 1985). For further details on this process refer to (Benettin et al., 1980; Lichtenberg & Lieberman, 1983; Parker & Chua, 1989; Seydel, 1988; Wolf, 1986; Wolf et al., 1985).

422 L.S. McCue and A.W. Troesch

Fig. 1 Lyapunov exponents based on experimental time series. Phase space showing Lyapunov exponents versus roll and roll velocity initial conditions sampled at time t0. Model release time and length of time series variable between runs

Fig. 2 Histograms of Lyapunov exponent values for runs leading to capsize and non-capsize

Use of Lyapunov Exponents to Predict Chaotic Vessel Motions 423

3.2 Application to the Capsize Problem 3.2.1 Implementation Calculating the Jacobian of the equations of motion used in the numerical simulator (Obar et al., 2001; McCue & Troesch, 2003; Lee et al., 2006), given by Equation 8 is non-trivial. While the mass and linear damping terms are easily treated, the quadratic damping and forcing terms require extra consideration. m  a 22   0  a 42 

0

 xg  b22     yg    0  a 44    b42

a 24

0 m  a33

0

I cg

0 b33 0

b24  x g    0  y g  b1   

(8)

D  0 0 0  0   g e 2   f 2     D  0 0 0  0    g e 3   mg  f 3    0 0 b2      g e 4 GZ  f 4D   

Two approaches to treat the quadratic damping term are as follows. One method is to replace, in the linearized model, the term

  with  2 for   0 ,

  2 for   0 , and assume that the precise singularity at   0 will never be encountered due to double precision computational accuracy. The second approach is to use Dalzell’s (1978) treatment for quadratic damping. Dalzell

 k   (1978) fits an odd function series of the form     . Solving  k 1, 3,... k  k  2  c

5   35  3   over some c  for k the truncated third order fit becomes    16 48 c range  c    c (Dalzell, 1978). Basic testing indicated both treatments yield similar results, therefore the Dalzell treatment, with  = 10 degrees, was used for c

the results presented herein to avoid any difficulties due to the singularity associated with the first method. The linearized influence of the forcing side of the equation is calculated using a simple differencing scheme. Forces are calculated as the difference between their values on the fiducial trajectory and their values at the offset from the trajectory. Due to linear superposition this can be calculated in a more computationally

424 L.S. McCue and A.W. Troesch

efficient manner for the differential at (x+x, y+y, +, t) rather than conducting the summation of force differentials at (x+x, y, , t), (x, y+y, , t), and (x, y, +, t). Therefore, the linearized form of the equations of motion about the fiducial trajectory are written as Equation 9:

a24  xg  b22 0 b24     0  yg    0 b33 0 2   5 35     I  a44    b42 0 b1  b2 16 c  16 0 c   g e 2   f 2D   g e 2   f 2D        g e3  mg  f 3D    g e 3  mg  f 3D   g GZ  f D   g GZ  f D  e4 4 e4 4     m  a22  0   a42

0 m  a33



 x x , y y ,  ,t 

 x g     y g      



(9)

 x , y , ,t 

Six sets of these linearized equations are integrated in time to calculate the ith Lyapunov exponent for i=1,6 by measuring the logarithm of the rates of growth of the six systems. Numerically this corresponds to the discrete representation of Equation 1 as Equation 10 in which ‘m’ represents the number of renormalization steps conducted and ‘L’ denotes the length of each element. The methodology employed is derived from that published by Wolf and coauthors (Wolf et al., 1985; Wolf, 1986). m

L(t j 1 )

j 1

L(t j )

1 m  1t  log

(10)

3.2.2 Results Figure 3 presents a three-dimensional phase space portrayal with the value of the first Lyapunov exponent plotted on the z-axis. Ten minutes of simulated data was collected for non-capsize runs and capsize runs were simulated until the point of capsize. Initially this plot could appear enigmatic. Graphically, it is apparent that all non-capsize runs result in an equal valued positive Lyapunov exponent of approximately 1.77 s-1 after 600 simulated seconds. Intuitively it is expected that capsize runs would have larger Lyapunov exponent values than non-capsize, however capsize runs all feature small, even near-zero Lyapunov exponents.

Use of Lyapunov Exponents to Predict Chaotic Vessel Motions 425

Fig. 3. Lyapunov exponents based on numerical simulation. Phase space showing Lyapunov exponents versus roll and roll velocity initial conditions with numerical model released at t0

Fig. 4 Numerically calculated Lyapunov exponent as a function of time for neighboring capsize and non-capsize cases. Top panel shows full time series from t = 0 to t = 600. Bottom panel shows identical data over critical region from t = 25 to t = 40. Both runs released with zero initial roll velocity, sway, sway velocity, heave, or heave velocity. Initial roll for non-capsize and capsize runs equal to 0 and 1 degree respectively

426 L.S. McCue and A.W. Troesch

To make physical sense of this counter-intuitive result consider Figure 4 which shows the convergence, or lack thereof, of Lyapunov exponents for neighboring capsize and non-capsize cases. In Figure 4 it is evident that the time series resulting in capsize is too short to allow for convergence of the Lyapunov exponent. While the Lyapunov exponent of both the non-capsize and capsize cases follow closely, it is only immediately before capsize that the value of the Lyapunov exponent for the capsize case rises somewhat above the value of the Lyapunov exponent for the non-capsize case. Yet capsize occurs sufficiently rapidly as to prevent convergence of the exponent and instead results in small, non-converging, Lyapunov exponents. This explains the small values of Lyapunov exponents for capsize runs shown in Figure 3. As is evident, hundreds of cycles are necessary for exponent convergence. Even with a lack of convergence of the Lyapunov exponent for capsize cases, key information can be derived from the result. Primarily, since the Lyapunov exponent is an indicator of the magnitude of the chaos of the system, by way of comparison of experimental and numerical results for large amplitude motions near capsize this methodology serves as a validation tool for the numerical model. Consider Figures 1 and 3. In Section 2 it was shown that the Lyapunov exponent for experimental runs not leading to capsize was 1.57 s-1. This value is close to the Lyapunov exponent of approximately 1.77 s-1 for non-capsize runs using the numerical model. Additionally, due to the finite nature of the experimental tests, it is anticipated that if it were feasible to collect data over longer time interval, such as the ten minute interval over which the numerical model was simulated, the experimental Lyapunov exponent would converge to a somewhat larger value thus reducing the difference between the two mean exponents. It is impossible then to calculate an accurate value of the Lyapunov exponent for a capsize run. To understand the chaotic behavior leading to capsize it is important to consider a short term Lyapunov exponent instead as is discussed in Section 4.

4 Short Time Lyapunov Exponents from Ordinary Differential Equations 4.1 Implementation The short time Lyapunov exponent from ordinary differential equations is calculated in much the same manner as the Lyapunov exponent. Again, ‘n’ sets of differential equations linearized about the fiducial trajectory are calculated to measure incrementally stretching and shrinking principal axes. The ‘n’ linearized sets, where ‘n’ is the dimension of the phase space, are reorthonormalized after each step. Numerically, Equation 2 is calculated as Equation 11.

Use of Lyapunov Exponents to Predict Chaotic Vessel Motions 427

1 xt , t  

1 Lt  t  log t Lt 

(11)

4.2 Results Consider Figure 5 in comparison to Figure 4. In both figures the numerical model is released at to, the location of the maximum transient wave peak, or 27.1705 seconds (Lee et al., 2006). Figure 5 shows the short time Lyapunov exponent as a function of time for the same neighboring cases leading to capsize and noncapsize. The non-capsize case rises to a maximum value of 4.37 s-1 then, similarly to Figure 4, converges to oscillatory behavior about a short time Lyapunov exponent of 1.85 s-1. However, for the capsize case useful predictive information is now visible. For the last few steps prior to capsize the short time Lyapunov exponent is larger than the short time Lyapunov exponent for non-capsize although they initially feature similar behavior. At capsize the short time Lyapunov exponent rapidly increases to a value an order of magnitude larger than that of the noncapsize case.

Fig. 5 Numerically calculated short time Lyapunov exponent as a function of time for neighboring capsize and non-capsize cases. Top panel shows full time series from t = 0 to t = 600. Bottom panel shows identical data over critical region from t = 25 to t = 40. Both runs released with zero initial roll velocity, sway, sway velocity, heave, or heave velocity. Initial roll for non-capsize and capsize runs equal to 0 and 1 degree respectively

428 L.S. McCue and A.W. Troesch

Figure 6 presents the values of short time Lyapunov exponent as a function of initial roll and roll velocities. While Figure 3 featured relatively invariant values of Lyapunov exponent based upon capsize or non-capsize, Figure 6 yields a range of short time Lyapunov exponents for runs leading to capsize. In Figure 6 runs not leading to capsize all feature peak short time Lyapunov exponent values near 4.2 s-1 as shown by the histogram plot of the same data given in Figure 7. However, as seen in Figure 7 runs leading to capsize have a wide range of values for short time Lyapunov exponent with a mean of 8.3 s-1 and a standard deviation of 6.1 s-1, an order of magnitude larger than the standard deviation of the peak first short time Lyapunov exponent for runs not leading to capsize.

Fig. 6 Short time Lyapunov exponents based on numerical simulation. Phase space showing short time Lyapunov exponents versus roll and roll velocity initial conditions with numerical model released at to

It should be noted that peak short time Lyapunov exponent for capsize runs ranges from near zero to 33 s-1. It is the hypothesis of the authors that capsize runs featuring small short time Lyapunov exponents are a result of a ‘blue sky catastrophe’ or a process with hysteresis as discussed by Thompson and coauthors (1987) and Lee et al (2006). In such cases the rapid growth of the capsize attractor results in a capsize which is potentially difficult to detect unless the time interval of measurement is exceedingly small. This is consistent with the top panel of Figure 8 which indicates that the bulk of the runs with both the smallest and largest local Lyapunov exponents capsize within the first time steps immediately after the maximal exponent and the bottom panel which shows that the smallest

Use of Lyapunov Exponents to Predict Chaotic Vessel Motions 429

Fig. 7 Histograms indicating range of short time Lyapunov exponent values for runs leading to capsize and non-capsize

Fig. 8 (top) Peak value of largest local Lyapunov exponent as a function of the number of cycles from peak exponent to capsize. (bottom) Peak value of largest local Lyapunov exponent as a function of the number of cycles from release to capsize

Lyapunov exponents are found for runs which capsize within the first two cycles after release. Analysis of Figure 6 also indicates that that those capsize runs with

430 L.S. McCue and A.W. Troesch

smallest local Lyapunov exponent tend to be found at large negative initial roll angles. Many of these roll angles are statistically unlikely to occur and therefore choosing initial limits of phase space using a statistically validated methodology, such as that described in McCue and Troesch (2004; 2005) is a relevant first step in the analysis of real vessels operating in real seaways.

5 Conclusions From the study presented herein the usefulness of the Lyapunov exponent and short time Lyapunov exponent as analytical tools for studying vessel capsize is investigated. To summarize:  The Lyapunov exponent is a fundamental system parameter taking on fixed values in the non-capsize region with minimal transitional areas. It can be used to isolate regions of questionable stability to aid in the reduction of test matrices.  The Lyapunov exponent is shown effective as a validation tool for capsize simulators. Through consistency between the magnitude of a system parameter such as the Lyapunov exponent for large amplitude roll motions measured from an experimental time series and those measured from a numerical simulator, it is possible to demonstrate that the numerical model is capturing the inherent physics of the problem.  The inherent ineffectiveness of the Lyapunov exponent to yield informative results for the capsize problem is demonstrated. Rather, the importance of calculating a finite time value, such as a short time Lyapunov exponent is discussed.  From the calculation of the short time Lyapunov exponent further system information is gained which can be used as a predictive tool to indicate lost stability potentially leading to capsize. However, more work must be done to further identify the different types of capsize as characterized by small and large short time Lyapunov exponents. For more current work along these lines, the interested reader is referred to McCue and Troesch (2006) and Story (2009).

Acknowledgements The authors wish to express their gratitude for funding on this project from the Department of Naval Architecture and Marine Engineering at the University of Michigan and the National Defense Science and Engineering Graduate Fellowship program.

Use of Lyapunov Exponents to Predict Chaotic Vessel Motions 431

References Arnold L, Chueshov I & Ochs G (2003) Stability and capsizing of ships in random sea-a survey. Tech. rept. 464. Universität Bremen Institut für Dynamicsche Systeme Bay John S (1999) Fundamentals of Linear State Space Systems. McGraw-Hill Benettin G, Galgani L, Giorgilli A, Strelcyn JM (1980) Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: a method for computing all of them. Mecc. 9–20 Dalzell JF (1978) A Note on the Form of Ship Roll Damping. J of Ship Res, 22 3 178–185 Eckhardt B, Yao D (1993) Local Lyapunov exponents in chaotic systems. Phys D 65:100–108 Eckmann JP, Ruelle D (1985) Ergodic theory of chaos and strange attractors. Rev of Mod Phys 57 3: 617–656 Eckmann JP, Kamphorst S, Oliffson, Ruelle D, Ciliberto S (1986) Liapunov exponents from time series. Phys RevA 34 6 Falzarano JM (1990) Predicting complicated dynamics leading to vessel capsizing. Ph.D. thesis, Univ of Michigan Glass L, Mackey M (1988) From Clocks to Chaos. Princet, NJ: Princet Univ Press Haken H (ed) (1981) Chaos and Order in Nature. N.Y. Springer-Verlag Hegger R, Kantz H, Schreiber T et al (2000) TISEAN 2.1, Nonlinear time series analysis. http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.1/index.html Kantz H (1994) A robust method to estimate the maximal Lyapunov expon of a time series. Physi Lett A, 185:77–87 Kantz H, Schreiber T (2004) Nonlinear time series analysis. Second edn. Camb Univ Press Lee YW, McCue L, Obar M, Troesch A (2006) Experimental and numerical investigation into the effects of initial conditions on a three degree of freedom capsize model. J of Ship Res. 50:1 63-84 Lichtenberg AJ, Lieberman MA (1983) Regular and stochastic motion. N.Y.: SpringerVerlag McCue LS (2004) Chaotic vessel motions and capsize in beam seas. Ph.D. thesis, Univ of Michigan McCue LS, Bassler C, Belknap W (2006) Real-time identification of behavior leading to capsize. 9th Int Conf on Stab of Ships and Ocean Veh (STAB 2006), Rio J. Braz, Sept McCue LS, Bulian G (2007) A numerical feasibility study of a parametric roll advance warning system. J of Offshore Mech and Arct Eng, 129:3 165–175 McCue LS, Troesch AW (2003-sept). The effect of coupled heave/heave velocity or sway/sway velocity initial conditions on capsize modelling. In: 8th Int Conf on the Stab of Ships and Ocean Veh McCue LS, Troesch AW (2004-july). Overlay of probability density functions on multidegree of freedom integrity curves for statistical prediction of critical capsize wave height. Soc for Ind and Appled Math, Annual Meet, Portland, Oregon McCue LS, Troesch AW (2005) Probabilistic determination of critical wave height for a multi-degree of freedom capsize model. Ocean Eng. 32:13 1608–1622 McCue LS, Troesch AW (2006) A combined numerical-empirical method to calculate finite time Lyapunov exponents from experimental time series with application to vessel capsizing. Ocean Eng. 33:13 1796–1813 Murashige S, Aihara K (1998) Coexistence of periodic roll motion and chaotic one in a forced flooded ship. Int J of Bifurc and Chaos, 8:3 619–626 Neves MAS, Vivanco JEM, Rodríguez CA (2009) Nonlinear dynamics onparametric rolling of ships in head seas. 10th Int Conf on Stab of Ships and Ocean Veh., St. Petersburg, Russia, June 2009

432 L.S. McCue and A.W. Troesch Obar MS, Lee YW, Troesch AW (2001-Sept). An experimental investigation into the effects initial conditions and water on deck have on a three degree of freedom capsize model. In: Fifth Int Workshop on the Stab and Oper Saf of Ships Ott E, Sauer T, Yorke J (eds) (1994). Coping with Chaos. NY.: John Wiley and Sons Papoulias FA (1987). Dynamic analysis of mooring systems. Ph.D. thesis, Dep of Nav Archit and Marine Eng, Univ of Michigan, Ann Arbor, MI Parker TS, Chua LO (1989) Practical numerical algorithms for chaotic systems. NY. Springer-Verlag Parlitz U (1992) Identification of true and spurious Lyapunov exponents from time series. Int J of Bifurc and Chaos 2:155–165. Reprinted in (Ott et al., 1994) Rosenstein MT, Collins JJ, De Luca CJ (1993) A practical method for calculating largest Lyapunov exponents from small data sets. Phys D, 65:117–134 Sano M, Sawada Y (1985) Measurement of Lyapunov spectrum from a chaotic time series. Phys Rev Lett. 55 10 Schuster HG (1984) Deterministic Chaos. Ger: Phys-Verlag Seydel R (1988) From Equilibrium to Chaos. N.Y.: Elsevier Science Publ Spyrou K (1996) Homoclinic connections and period doublings of a ship advancing in quartering waves, Chaos 6 2 Story WR (2009) Application of Lyapunov exponents to strange attractors and intact & damaged ship stability. MS Thesis. Dep of Aerosp and Ocean Eng. Virginia Tech Blacksbg VA Thompson JMT, Bishop SR, Leung LM (1987). Fractal basins and chaotic bifurcations prior to escape from a potential well. Phys Lett A 121 3 Wolf A (1986) Quantifying chaos with Lyapunov exponents. Chaos. Princet, NJ: Princet Univ Press 13 Wolf A, Swift J, Swinney H, Vastano J (1985) Determining Lyapunov exponents from a time series. Phys D, 16 285–317

Applications of Finite-Time Lyapunov Exponents to the Study of Capsize in Beam Seas Leigh S. McCue* *Aerospace and Ocean Engineering, Virginia Tech

Abstract This paper demonstrates the use of finite-time Lyapunov exponents (FTLE) for the detection of large amplitude roll motions and capsize. The study is conducted for a simple, single degree of freedom model of the Edith Terkol in regular and random beam seas. The effectiveness of the FTLE technique is qualitatively compared to a scalar ship motion metric based upon the ‘energy index (EI)’ concept used for real-time identification of quiescence.

1 Introduction For centuries chaotic vessel motions and capsize have resulted in lost cargo, ships, and lives. For this reason it is of the utmost importance to develop tools which can provide mariners with indicators of inclement ship motions. Lyapunov exponents provide a measure of the rate of convergence or divergence of nearby trajectories thus indicating the level of sensitivity of a system to initial conditions. They have been used to demonstrate exponentially divergent, chaotic behavior in vessel motions as demonstrated by references such as Papoulias (1987), Falzarano (1990), Spyrou (1996), Murashige and collaborators (1998a; 1998b; 2000), Arnold et al. (2003), and McCue and Troesch (2004; 2004; 2006). The Lyapunov exponent, however, is defined in the limit as time approaches infinity; therefore to detect instabilities in time leading to an event, such as capsize, the finite-time form of the Lyapunov exponent is necessary. For a system of   ux  , small deviations from the equations written in state-space form x trajectory can be expressed by the equation Yao,1993).

x is

 

 j 

x i   u i x x j (Eckhardt

&

a vector representing the deviation from the trajectory with

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_24, © Springer Science+Business Media B.V. 2011

433

434 L.S. McCue

components for each state variable of the system. From this, the equation for the finite-time Lyapunov exponent can be written as Equation 1 (Eckhardt & Yao, 1993).

T xt , x0 

xt  T  1 log T xt 

(1)

For a multi-dimensional state space a spectrum of finite-time Lyapunov exponents can be calculated with the same number of exponents as dimension of the state-space. Methods for computing Lyapunov exponents and finite-time Lyapunov exponents from equations of motion are well developed in the literature with details available in references such as Benettin et al. (1980), Wolf et al. (1985), and Eckhardt and Yao (1993). In this work, the results generated via finite-time Lyapunov exponents are compared to results from a path-independent scalar ship motion metric known as the energy index (EI). The energy index, initially proposed by O’Reilly (1987), is an experimentally-tested, empirically-based indicator of quiescent periods suitable for sea-based aviation operations in which an energy-like scalar value is used as a measure of ship motions (O’Reilly, 1987; Ferrier & Manning, 1998; Ferrier et al., 2000a; Ferrier et al., 2000b). While it is desirable for current ship-air operations to indicate motions in all six degrees of freedom, particularly vertical accelerations (Colwell, 2002b), this study compares finite-time Lyapunov exponents to a simple formulation of the energy index based upon the two state space variables considered in this work, namely roll and roll velocity. The intention of this work is to demonstrate the capabilities of FTLE algorithms for detecting instabilities without the dependence more complicated energy index algorithms have upon empirically defined coefficients (Ferrier & Manning, 1998). It should be clear, however, that this is not the traditional use of the energy index, rather that the energy index concept in this case is being used as a benchmark. Because the energy index has been experimentally tested and shown to accurately indicate regions of quiescence (O’Reilly, 1987), it provides a useful comparison for this proof-of- concept study. This work aims to illustrate that a mathematical approach, such as the FTLE, can yield as much, or greater information into the fundamental physics driving the motions of the system to capsize or quiescence, than a method with ship and condition-dependent parameters.

Study of Capsize in Beam Seas 435

2 Regular Seas 2.1 Regular Seas: Modelling Details For the study presented in this work, the simple single degree of freedom roll model for the Edith Terkol established by Soliman and Thompson (1991) and given in Equation 2 was simulated.

  b1  b2    c1  c 2    c3 3  c 4   3  c5 5 

M t  WM  (2) I I

In Equation 2, bi and ci terms are linear and nonlinear damping and stiffness coefficients respectively with specific values: b1 = 0.0043, b2 = 0.0225, c1 = 0.384, c2 = 0.1296, c3 = 1.0368, c4 = -4.059, c5 = 2.4052, I = 1174, and WM = 0 (Soliman & Thompson, 1991). The term M(t) in regular seas is defined by Equation 3 (Senjanović et al., 2000), where 0, the effective wave slope, is taken to equal 0.73 (Francescutto & Serra, 2002; Gu, 2004), n and e equal 0.62 and 0.527 rad/s respectively (Soliman & Thompson, 1991), and the terms H = 4.94m and  = 221.94m are chosen to be consistent with results published by Soliman and Thompson in which interesting behavior is detected (1991). The general form of Equation 3 is widely available in the literature (Nayfeh & Khdeir, 1986; Nayfeh & Sanchez, 1990). Each simulation was allowed to run for 100 wave encounters or to the point of capsize. For the purposes of this work capsize was defined as exceeding the vessel’s angle of vanishing stability, v = 0.88 rad (Soliman & Thompson, 1991).

M t   I 0 n2

H



sin  e t

(3)

2.2 Regular Seas: Finite-Time Lyapunov Exponents Finite-time Lyapunov exponents were calculated via simulation for the numerical model of the Edith Terkol given by Equations 2–3. Particular attention was paid to the values and timing of maxima in the FTLE time series. Figure 1 presents histograms illustrating the values of maximum FTLE values for both capsize (top) and non-capsize (bottom) cases. Additionally, by consideration of the middle panel of Figure 1, it is apparent that those capsize cases with small FTLE values correspond to cases which capsize within one excitation cycle, i.e. those runs which capsize, for all practical purposes, immediately. Figure 2 illustrates that elimination of

436 L.S. McCue

those cases which capsize prior to one wave cycle essentially results in removal of the cases at the extrema of roll and roll velocity initial conditions.

Fig. 1 Histograms of maximum FTLE1 for capsize and non-capsize cases in regular seas

Fig. 2 Safe basins indicating non-capsize and capsize for all cases (left) and those cases capsizing after one excitation period (right) in regular seas

To demonstrate that instabilities are detectable in a time frame suitable for corrective measures to be taken, Figure 3 presents histograms of the number of wave cycles encountered between the occurrence of the maximum FTLE to the actual time of capsize. Again, the top panel includes all capsize runs while the bottom panel is restricted to those runs which capsize after one wave period. Considering only those runs which capsize after one encounter cycle, the majority capsize more

Study of Capsize in Beam Seas 437

than half a wave cycle after the maximum FTLE, with many capsizing multiple cycles after the location of the maxima. Detecting trends towards increasing values prior to the occurrence of maxima could lead to even greater time windows in which stabilizing controls changes are executed. This is discussed further in Section 2.3.

Fig. 3 Histograms of wave encounter cycles from the occurrence of the maximum FTLE1 to the time of capsize in regular seas

2.3 Regular Seas: Finite-Time Lyapunov Exponents in Comparison to the Energy Index For the single degree of freedom model, a simple definition of the energy index (EI) was used as given in Equation 4 (O’Reilly, 1987). Equation 4 emulates the form of the equation for the EI of O’Reilly’s for motions in multiple degrees of freedom (O’Reilly, 1987). More complicated EI models exist which rely upon empirically determined weighting coefficients for each degree of freedom (Ferrier & Manning, 1998; Ferrier et al., 2000a; Ferrier et al., 2000b). The purpose of this research is to demonstrate the ability to warn of large amplitude motions without a reliance upon empirical coefficients, thus justifying the use of the definition in Equation 4.

EI   2   2

(4)

438 L.S. McCue

Figure 4 presents a comparison of roll, FTLE, and EI values for a single ship motion simulation of the Edith Terkol with initial conditions (  rad,  rad/s) = (0, 0.2269). While the energy index reflects the instantaneous behavior of the system, the FTLE time series exhibits odd behavior, deviating from a fairly regular periodic response, 20 seconds prior to capsize and encounters peak values one cycle before capsize. Of particular interest is the middle plot, which displays the sum of the two FTLE for an entropy rate-like measure (Falkovich & Fouxon, 2004). Specifically, one cycle prior to capsize, this value rises above zero. Similar behavior is identified when examining a range of capsize and non-capsize cases in Figure 5 which present simulations near to the initial conditions of Figure 4. Those cases leading to capsize begin to exhibit unique FTLE time series behavior multiple cycles prior to capsize in addition to phase differences and peaks relative to the non-capsize runs. Additionally, phase differences in the FTLE time series are present in the non-capsize runs as time progresses. Some runs experience increasing roll motions while others encounter decreasing motions. Based upon this model, there is promise that measurement of these differences and detection of peaks can indicate variations in magnitude of ship motion in addition to capsize. Quantitatively, consideration of the sum of the FTLE values for those neighboring capsize and non-capsize cases in Figure 5 indicates that all capsize cases have positive FTLE sums and non-capsize cases have negative FTLE sums. A summary of initial conditions and values of peak FTLE1+FTLE2 displayed in Figure 5

is as follows: (  rad,  rad/s, FTLE1+FTLE2 1/s) = (0, -0.2618, 0.0025), (0, -0.2443, 0.0012), (0, -0.2269, 5.1439e-004), (0, -0.2094, -0.0030), (0, -0.1920, -0.0041), (0, 0.1745, -0.0049).

Fig. 4 Roll plotted with first and second FTLE (top), sum of FTLE (middle), and EI (bottom) for Edith Terkol in regular beam seas,



= 0 rad,



= -0.2269 rad/s

Study of Capsize in Beam Seas 439

Fig. 5 FTLE1 (top), sum of FTLE (middle top), EI (middle bottom), and roll (bottom) for Edith

 = 0 rad,  =-0.2618 rad/s;  = 0 rad,  = -0.2443 rad/s;  = 0 rad,  = -0.2269 rad/s;  = 0 rad,  = -0.2094 rad/s;  = 0 rad,  = -0.1920 rad/s;  = 0 rad,  = -0.1745 rad/s. Dotted and Terkol in regular beam seas. Six cases shown corresponding to initial conditions:

solid lines denote capsize and non-capsize runs respectively

Figure 6 shows the range of values for maximum FTLE1 + FTLE2 for non-capsize cases and those capsize cases in which capsize occurs after one excitation period. The mean value for capsize cases is 0.00098 (1/s) while the mean value fornoncapsize cases is -0.0037 (1/s). Positivity of the entropy rate-like value FTLE1 + FTLE2 is an indicator of potential danger. As apparent in Figure 7, those noncapsize runs with positive values for peak FTLE1 + FTLE2 represent simulations in the regions of greatest sensitivity to initial conditions. Thus, in addition to those areas resulting in capsize, it is not illogical to identify such conditions as necessitating precautionary measures and that a definition based upon FTLE methods could provide a conservative indicator of capsize.

440 L.S. McCue

Fig. 6 Histograms of maximum FTLE1 + FTLE2 for capsize and non-capsize cases in regular seas

Fig. 7 3-dimensional plot of initial conditions and maximum FTLE1 + FTLE2 for capsize and non-capsize cases in regular seas

Study of Capsize in Beam Seas 441

3 Random Seas 3.1 Random Seas: Modelling Details In random seas, the term M(t) in Equation 2 is written as Equation 5 (Senjanović et al., 2000; Gu, 2004):

M t   I 0 n2

2 d g

N

 i 1

2 i

S i  sin i t   i 

(5)

where N = 180 is the number of wave components, d = 0.0083 rad/s is the size of the wave frequency increment, i is the encounter frequency for each wave component, i is a random phase angle between 0 and 2π for each wave component, and S() is a wave energy spectrum. For this study the ISSC two-parameter spectrum, Equation 6 was used with a significant wave height Hs= 4.94m and characteristic frequency z = 0.527 rad/s for consistency with Section 2.



     

   S    0.11H s2 z5 exp  0.44 z   4





4

(6)

Fig. 8 Safe basin indicating non-capsize and capsize in random seas for those cases capsizing after one excitation period

442 L.S. McCue

Details as to the derivation of Equation 5 can be found in Senjanović et al. (2000), Jiang et al. (1996), and Chapter VIII of Principles of Naval Architecture (1989). A sample safe basin representing the simulation of Equations 2 and 5 is presented in Figure 8. In Figure 8 the same random seed is used for each simulation, therefore each initial condition pair is subject to the same random wave time series.

3.2 Random Seas: Finite-Time Lyapunov Exponents in Comparison to the Energy Index In random seas, finite-time Lyapunov exponents and energy index values were calculated via simulation for the numerical model of the Edith Terkol given by Equations 2 and 5. Figure 9 compares data for two time series with identical initial conditions in random seas in which one case leads to capsize, and one non-capsize. As in Figures 4-5, EI values serve as a reflection of the roll time series behavior. Conversely, the finite-time Lyapunov exponent time series demonstrates distinct variations in behavior well before the point of capsize.

Fig. 9 FTLE1 (top), sum of FTLE (middle top), EI (middle bottom), and roll (bottom) for Edith Terkol in random beam seas. Both cases correspond to initial conditions



= 0 rad,



= 0 rad/s.

Dotted and solid lines denote capsize and non-capsize respectively

For example, after approximately 540 seconds of simulation, there is a marked phase change between capsize and non-capsize FTLE time series in which the run leading to an ultimate state of capsize changes from leading to lagging the noncapsize time series. Additionally, the FTLE1 values increase substantially at

Study of Capsize in Beam Seas 443

approximately 550 seconds of simulation. In consideration of the sum of the two FTLE for both capsize and non-capsize runs, it is found that over the entire simulation time the maximum value for FTLE1 + FTLE2 for the capsize case is positive, 0.0028 while it is -0.0028 for the non-capsize case. This again lends validity to the belief that quantitative indicators of capsize can be formulated via the use of the FTLE as well as that the FTLE holds promise as an indicator of periods of coming quiescence. Figure 10 shows histograms of the maximum first FTLE for capsize and noncapsize cases in random seas. Only those runs capsizing after one excitation cycle are considered as established in Section 2. In Figure 10, similar to Figure 1, those runs resulting in capsize typically experienced larger maximum FTLE. Also shown in Figure 10 is the cycles from the occurrence of the peak FTLE to capsize. While in random seas, the cycles to capsize is dependent upon the random seed in the phase term of the wave train, it is apparent for the sample set presented herein, that for many cases capsize occurs sufficiently long after the peak in the FTLE time series that corrective measures could be implemented.

Fig. 10 Histograms of maximum FTLE1 for capsize (top) and non-capsize (middle) cases in random seas as well as cycles from maximum FTLE1 to capsize (bottom)

Figures 11–12 plot histograms and state space of the entropy rate-like value summing the two finite-time Lyapunov exponents. Those runs which capsize after one excitation cycle have, on average, a positive maximum FTLE1 +FTLE2 while those runs which do not capsize tend toward negative values as shown in Figure 11. As seen in Figure 12 those non-capsize cases with positive maximum FTLE1 + FTLE2

444 L.S. McCue

often lie on the fractal boundaries between capsize and non-capsize. Measures detecting positive, or small negative values of the sum of all Lyapunov exponents, in addition to detecting peaks, phase differences, and other oddities in the FTLE time series, can indicate impending capsize.

Fig. 11 Histograms of maximum FTLE1 +FTLE2 for capsize and non-capsize cases in random seas

Fig. 12 3-dimensional plot of initial conditions and maximum FTLE1 + FTLE2 for capsize and non-capsize cases in random seas

Study of Capsize in Beam Seas 445

4 Conclusions Based upon numerical simulation of Soliman and Thompson’s (1991) simplified single degree of freedom roll motion model, this work demonstrates that finitetime Lyapunov exponent calculations provide viable indicators of impending capsize in both regular and random seas. Instabilities leading to capsize are detected through: 1. peaks in the FTLE time series, 2. time series phase changes and/or other qualitative idiosyncratic behavior, and 3. positivity of the sum of the FTLE spectrum in time. This further supports the conclusions arrived at by McCue and Troesch in their studies of three-degree of freedom numerical and experimental results (2004; 2006). To implement this theory in a practical, on-board manner, a number of areas for future research arise. Specifically, it would be of use to further study the feasibility of this technique in detecting quiescence as applied to ship based aviation applications via comparison of results from both methods for ship motions in six degrees of freedom. Colwell expressed concern that a scalar representation of the ship motion environment obscures some of the understanding of the limiting aspects of sea-air operations (2002a). It is the author’s belief that a mathematical approach, such as that presented by finite-time Lyapunov exponents, enables understanding of the fundamental physics of the system while still reaping the benefits of a scalar output. Additionally, more quantitative means to detect variations in the FTLE time series are required. It is not always practical, feasible, or computationally efficient to calculate the entire FTLE spectrum, and thus measuring for positivity of the sum of the finite-time exponents may not be the ideal quantitative indicator. Instead, benchmarks for peak values and techniques to quantify phase and other behavior changes in the FTLE1 time series must be developed through further simulation and analysis of experimental data for a range of hull forms in realistic seas. For further research along these lines the interested reader is referred to Story (2009).

Acknowledgements This research was conducted while funded by the ONR-ASEE Summer Faculty Research Program. Additionally, the author wishes to thank William Belknap (Seakeeping Division, Code 5500, NSWC-CD), Judah Milgram (Sea-Based Aviation, Code 5301, NSWC-CD), and Armin Troesch (University of Michigan) for their assistance and insights on this topic.

446 L.S. McCue

References Arnold L, Chueshov I and Ochs G (2003) Stability and capsizing of ships in random sea-a survey. Tech rept 464. Universität Bremen Institut für Dynamicsche Systeme Beck RF, Cummins WE, Dalzell JF, Mandel P and Webster WC (1989) Motions in Waves. Chap. VIII of: Lewis EV. (ed) Princip of Nav Archit Soc of Nav Archit and Mar Eng Benettin G, Galgani L, Giorgilli A and Strelcyn JM (1980) Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: a method for computing all of them. Mecc 9-20 Colwell JL (2002-a May) Maritime Helicopter Ship Motion Criteria-Challenges for Operational Guidance. In: NATO RTO SCI-120 Symp on \Chall.... Colwell JL (2002-b) Real time ship motion criteria for maritime helicopter operations. In: ICAS 2002 Congr Eckhardt B, Yao D (1993) Local Lyapunov exponents in chaotic systems. Phys D 65: 100-108 Falkovich G and Fouxon A (2004) Entropy production and extraction in dynamical systems and turbulence. New J of Phys 6 50 Falzarano JM (1990) Predicting complicated dynamics leading to vessel capsizing. Ph.D. thesis, Univ of Michigan Ferrier B and Manning T (1998) Simulation and testing of the landing period designator (LPD) helicopter recovery aid. Nav Eng J 110 1: 189-205 Ferrier B, Baitis A E and Manning A (2000a) Evolution of the Landing Period Designator (LPD) for Shipboard Air Ope. Nav Eng J 112 4 297-315 Ferrier B, Applebee T, Manning A and James D (2000b-May) Landing period designator visual helicopter recovery aide: theory and real-time application. In: Proc of the 56th Annual Forum of the American Helicopter Soc Francescutto A and Serra A (2002) Experimental tests on ships with large values of B/T, OG/T and roll period. In: 6th Int Ship Stab Workshop Gu JY (2004) Nonlinear rolling motion of ship in random beam seas. J of Marine Sci and Technol, 12(4): 273-279. Nat Taiwan Ocean Univ, publ Jiang C, Troesch AW and Shaw SW (1996) Highly nonlinear rolling motion of biased ships in random beam seas. J of Ship Res 40 2: 125-135 McCue LS (2004) Chaotic vessel motions and capsize in beam seas. Ph.D. thesis, Univ of Michigan McCue LS and Troesch AW (2004-Nov) Use of Lyapunov exponents to predict chaotic vessel motions. In: 7th Int Ship Stab Workshop McCue LS and Troesch AW (2006) A combined numerical-empirical method to calculate finite time Lyapunov exponents from experimental time series with application to vessel capsizing. Ocean Eng. 33 Issue 13:1796-1813 Murashige S, Yamada T and Aihara K (2000) Nonlinear analyses of roll motion of a flooded ship in waves. Philos Transactions of the Royal Soc of Lond A 358:1793-1812 Murashige S and Aihara K (1998a) Co-existence of periodic roll motion and chaotic one in a forced flooded ship. Int J of Bifurc and Chaos 8 3: 619-626 Murashige S and Aihara K (1998b) Experimental study on chaotic motion of a flooded ship in waves. Proc of the Royal Soc of Lond A 454: 2537-2553 Nayfeh AH and Khdeir AA (1986) Nonlinear rolling of ships in regular beam seas. Int Shipbuilding Prog 33:379 Nayfeh AH and Sanchez NE (1990) Stability and complicated rolling responses of ships in regular beam seas. Int Shipbuilding Prog 37 412: 331-352 O’Reilly PJF (1987) Aircraft/deck interface dynamics for destroyers. Marine Technol 24 1: 1525 Papoulias FA (1987) Dynamic analysis of mooring systems. Ph.D. thesis, Dep of Nav Archit and Marine Eng, Univ of Michigan, Ann Arbor, MI

Study of Capsize in Beam Seas 447 Senjanović I, Ciprić G and Parunov J (2000) Survival analysis of fishing vessels rolling in rough seas. Philos Transactions of the Royal Soc of Lond A 358:1943-1965 Soliman MS and Thompson JMT (1991) Transient and steady state analysis of capsize phenomena. Appl Ocean Res 13 2 Spyrou KJ (1996) Homoclinic connections and period doublings of a ship advancing in quartering waves. Chaos 6 2 Story WR (2009) Application of Lyapunov Exponents to Strange Attractors and Intact & Damaged Ship Stability. MS Thesis. Dep of Aerosp and Ocean Eng Virginia Tech Blacksburg VA Wolf A, Swift J, Swinney H and Vastano J (1985) Determining Lyapunov exponents from a time series. Phys D 16: 285-317

Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas Marcelo A.S. Neves*, Jerver E.M. Vivanco*, Claudio A. Rodríguez** *LabOceano, COPPE/UFRJ , **COPPE/UFRJ

Abstract The present paper employs nonlinear dynamics tools in order to investigate the dynamical characteristics governing the complex coupling of the heave, roll and pitch modes in head seas at some regions of the numerical stability map of a fishing vessel. Bifurcation diagrams and Poincaré mappings are computed and employed to investigate the appearance of multistability and chaos associated with increased values of the selected control parameter, the wave amplitude. The connection between these nonlinear characteristics and the coupled nature of the mathematical model are analyzed. Lyapunov exponents corresponding to the three coupled models are computed.

1 Introduction It is well known that parametric rolling in head seas may lead to large roll angles and accelerations in few cycles. Even though many studies on the subject simplify the analysis to a single degree of freedom, there is nowadays a wide acceptance of the relevance of the nonlinear coupling of the roll mode with heave and pitch. In previous studies Neves and Rodríguez (2005, 2006) have introduced a mathematical model in which the heave, roll and pitch motions are nonlinearly coupled to each other. Using this model they investigated the occurrence of head seas parametric rolling on a small fishing vessel. They showed, by means of numerical simulations, comparable to experimental results, the occurrence of strong dependence of the roll responses in head seas conditions on initial conditions, Neves and Rodríguez (2007). In order to investigate the quantitative and qualitative changes of parametric rolling with respect to the encounter frequency tuning and wave amplitude, Neves and Rodríguez (2007a,b) proposed the computation of analytical and numerical

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_25, © Springer Science+Business Media B.V. 2011

449

450 M.A.S. Neves et al.

maps representing the boundaries of stability. The numerical maps aggregate information not only on the boundaries of stability, but also on the amplitude of roll response in the whole region of parametric amplification. The present paper investigates the dynamical characteristics governing the complex coupling of modes at some regions of the numerical stability map. Bifurcation diagrams and Poincaré mappings, Guckenheimer and Holmes (1983), Seydel (1988) are employed in order to investigate the appearance of multistability and chaos associated with increased values of the control parameter wave amplitude. The connection between these nonlinear characteristics and the coupled nature of the model are analyzed. Finally, Lyapunov exponents corresponding to the three coupled models are computed.

2 Mathematical Model Employing Taylor series expansions up to third order, Neves and Rodríguez (2005, 2006) expressed restoring actions in the heave, roll and pitch modes in a coupled way. Wave actions are taken into consideration not only in the Froude-Krilov plus diffraction first order forcing functions, but also in second and third order terms resulting from volumetric changes of the submerged hull due to vertical motions and wave passage effects. The model corresponds to an extension, both in the order of non-linearities and in the levels of coupling, of the model introduced by Paulling and Rosenberg (1959) and Paulling (1961). The equations are taken here in the explicit form described in detail in Neves and Rodríguez (2005, 2006). Thus, the non-linear heave, roll and pitch equations are introduced as:

m Zz z Zz z  Z Z  Zz z  Z  1 Zzz z2  1 Z2  2

2 1 1 1 1 1 Z 2  Zz z  Zzzzz3  Zzz z2  Zz 2 z  Z 2  2 6 2 2 2 1 1 Zz 2 z  Z 3  Zz t z  Z t   Zz t z  Zzz t z2  2 6 Zz t z  Z t  2  Z (t) 2  Zw t 

J

xx

(1)



 K    K   K      K   K z z  K   

1 1 1 K zz z 2  K  3  K  2  K z z  K  t   2 6 2 K  t   K z t z  K  t   0

(2)

Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas 451

J

yy

 M    M   M zz  M z z  M z z  M  

1 1 1 1 M zz z 2  M  2  M  2  M z z  M zzz z 3  2 2 2 6 1 1 1 1 M zz z 2  M z 2 z  M  2  M z 2 z  2 2 2 2 1 M  3  M z t z  M  t   M z t z  6 M zz t z 2  M  t   M z t z  M  t  2 

(3)

M  (t ) 2  M w t  On the left hand side of Eqs. (1-3), added masses and wave damping terms are assumed linear. A quadratic roll damping is considered in equation (2). The terms associated with variables z ,  ,  and wave elevation  (t ) correspond to the non-zero linear and non-linear (up to third order) coefficients due to hydrostatic and wave pressure effects analytically derived by Neves and Rodríguez (2005, 2006), dependent on hull characteristics and on wave amplitude, frequency and T time. On the right hand side of Eqs. (1-3), Z W (t ) K W (t ) M W (t ) represent linear wave excitation. Due to the particular wave incidence considered,

KW  0

has been assumed in Eq. (2). Once all the coefficients in Eqs. (1-3) are known, this set of three equations may be numerically integrated without difficulty.

3 Numerical Limits of Stability Numerical simulations have been performed in the case of a fishing vessel denominated TS. Details of the ship are given in Fig. 1 and Table 1. Parametric rolling of this fishing vessel has been experimentally and numerically examined in detail in Neves and Rodríguez (2005, 2006), Neves et al. (2002). Numerical simulations performed using equations (1-3) have been successfully compared to experimental results for different wave conditions and ship speeds. It was verified that the fishing vessel employed in the present investigation is quite prone to strong parametric rolling in head seas.

452 M.A.S. Neves et al.

Fig. 1 Hull form of transom stern (TS) fishing vessel

Table 1 Ship main characteristics

Denomination

Ship TS

Overall length

[m]

25.91

Length between perpendiculars

[m]

22.09

Breadth

[m]

6.86

Depth

[m]

3.35

Draft

[m]

2.48

Displacement

[ton] 170.3

Longitudinal radius of gyration

[m]

5.52

Metacentric height

[m]

0.37

Table 2 Two sets of initial conditions

z0 ( m ) z0 (m / sec) 0 (deg)  (deg/ sec) 0

 0 (deg) 0 (deg/ sec)

I.C.#01 0.00

I.C.#02 0.00

0.01

0.01

2.00

2.50

-0.50

-0.80

0.00

0.00

0.01

0.01

Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas 453

It is important to investigate parametric rolling not only at the exact encounter frequency tuning we / wn 4  2.0 . In fact, large amplifications may take place in a quite broad spectrum of excitation frequencies. In order to comprehensively investigate the unstable regions Neves and Rodríguez (2007b) proposed the computation of numerical maps representing the boundaries of stability but containing information on the amplitude of roll response in the whole region of parametric amplification. Figure 3 shows the limits of stability of the fishing vessel in head seas at Fn = 0.30, corresponding to the first region of instability. The mapping is constructed by numerically computing the roll amplitude for different encounter frequencies and wave amplitudes. All points of the map are computed for the set of initial conditions I.C.#01 defined in Table 2. The intensity of the final roll amplitude is indicated by the color scale displayed on the right hand side of the figure. Four important features of the new limits of stability are:  The appearance of upper boundaries, indicating that for increased wave amplitudes, parametric rolling may not necessarily increase; in fact, it tends to disappear.  A general tendency of the unstable area to bend to the right, indicating that the exact tuning we / wn 4  2.0 is not necessarily the tuning with stronger amplifications.  Smooth growth of roll amplitude at lower level of boundaries, abysmal (sudden) decrease in the upper boundaries.  Upper boundaries with fractal geometry.  Larger area of instability as the roll initial conditions were modified.

Fig. 2 Limits of stability, ship TS, Fn=0.30, I.C.#01

454 M.A.S. Neves et al.

4 Bifurcation Diagrams In Figure 2 the whole spectrum of exciting frequencies was explored. It has been observed that distinct characteristics are revealed. It is now desirable to get an in depth knowledge of dynamical characteristics as the parameter wave amplitude AW is increased. So, if previously we have been more interested in the limits of stability as a whole, now we wish to have a closer look at some domains inside the unstable area. For this purpose, we will investigate the changes in dynamic characteristics as we cross the area inside the limits. A limited region of the map of limits of stability will be explored, that is, we will follow a vertical line defined at the tuning we / wn 4  2.0 in Figure 2. Aiming at demonstrating the influence of initial conditions on the solutions corresponding to points inside the area of the limits of stability we developed a brute-force algorithm for capturing the branching of solutions for a specified set of initial conditions. Thus, using the AW parameter, the type of bifurcation diagrams as shown in Figure 3 for the heave, roll and pitch motions are developed for the same set of initial conditions used in the mapping of the limits of stability. These diagrams reveal the branching structure for distinct ranges of wave amplitude; qualitative and quantitative types of responses are noticed. These distinct results may be summarized as shown in Table 3.

Fig. 3 Heave, roll and pitch bifurcation diagrams, Fn=0.30, I.C.#01

Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas 455 Table 3 Roll solutions for Fn=0.30

Range of Aw (m) 0.0000 - 0.6036 0.6037 - 0.6129 0.6130 - 0.6626 0.6627 - 0.6758 0.6759 - 0.6782 0.6783 - 0.67881 0.67882 - 0.7000

Type of roll response Typically linear period-3 Multistability, period-1 Multistability, period-2 Multistability, period-4 Multistability, period-8 Chaos

Two interesting characteristics, not observable in the numerical limits of stability, are revealed by this bifurcation analysis. First, in the short second range of AW one observes the appearance of a solution with 3 periods that ends with a sudden appearance of a burst of non-periodic solutions. The period-3 solutions of heave, roll and pitch motions are illustrated in Figures 4-6, respectively. In each of them time history, phase plane and Poincaré map are shown. The appearance of non-periodic solutions is illustrated in Figure 7 which shows the roll time series, the corresponding phase diagram and Poincaré map for Aw=0.6129 m.

Fig. 4 Heave motion, phase plane and Poincaré map, period-3 solution, Aw=0.605 m, I.C.#01

Fig. 5 Roll (max=24.18°), phase plane and Poincaré map, period-3 solution, Aw=0.61 m, I.C.#01

456 M.A.S. Neves et al.

Fig. 6 Pitch motion, phase plane and Poincaré map, period-3 solution, Aw=0.605 m, I.C.#01

Fig. 7 Roll motion, phase plane and Poincaré map, Aw=0.6129 m, I.C.#01. Non-periodic solutions

The second interesting characteristic encountered is multistability with associated alternance of values. This dynamical feature arises immediately after the occurrence of a burst of non-periodic solutions, as shown in Figure 3. In the bifurcation diagram one may get the impression that the motion has migrated to a period-2 solution, but a detailed analysis will show that this is not the case. In fact the roll solutions in the third range of Aw are period-1, but as illustrated in Figures 8 and 9, the solutions continuously alternate from one attractor to another one which is situated close by, at each new value of the parameter Aw. In other words, roll motion either lives in one attractor or in the other, but always with a single period. Subsequently, for higher wave amplitudes, flip bifurcation will take place together with multistability: period-2, 4 and 8 solutions will appear in sequence ending in chaos.

Fig. 8 Roll motion: (a) Aw=0.639 m, (b) Aw=0.6391 m. Multistability for two neighboring points, I.C.#01

Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas 457

It is interesting to observe that in this third range of Aw (0.6130 - 0.6626) the roll motion undergoes multistability with period-1 solutions, as shown in Figures 8 and 9. But in this same range, the vertical motions have already undergone a period doubling bifurcation. This is shown in Figures 10 and 11 for heave and pitch, respectively. Another aspect worth noting is that the alternating process does not contaminate these modes, Vivanco (2009). Subsequently, in the fourth range of Aw (0.6627 - 0.6758) the roll motion continues with multistability but responding with period-2 solutions, as shown in Figure 12, whereas the heave and pitch motions now respond with period-4 solutions. The sequence of flip bifurcations soon leads the coupled system to respond with chaotic motions. Figure 13 illustrates the period-4 roll motion and finally, Figure 14 shows the chaotic behaviour for Aw=0.683 m. The region with chaotic behaviour ends abruptly at the wave amplitude corresponding to the upper limit of stability of Figure 2.

Fig. 9 Roll phase planes: (a) Aw=0.639 m, (b) Aw=0.6391 m. Multistability for two neighboring points, I.C.#01

Fig. 10 Heave time history, phase plane and Poincaré map, period-2 solution, Aw=0.64 m, I.C.#01

458 M.A.S. Neves et al.

Fig. 11 Pitch time history, phase plane and Poincaré map, period-2 solution, Aw=0.64 m, I.C.#01

Fig. 12 Roll motion (max=24.27°), phase plane and Poincaré map, period-2 solution for Aw=0.67 m, I.C.#01

Fig. 13 Roll motion ( 

max=24.73°),

phase plane and Poincaré map, period-4 solution for

Aw=0.678 m, I.C.#01

Fig. 14 Roll motion ( 

max=25°),

phase diagram and Poincaré map, chaotic behaviour for Aw=0.683 m, I.C.#01

Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas 459

Fig. 15 Roll bifurcation diagram, Fn=0.30, I.C.#02

To illustrate the dependency of the limits of stability on initial conditions, a second set of initial conditions (I.C.#02) defined in Table 2 is considered. When initial conditions are changed, the general bifurcation set is preserved, but it may be observed in the new bifurcation diagram that, as shown in Figure 15, the range of responses with period-3 solution simply is not reached. At AW  0.617 m the motion jumps to a higher value. Instead of alternating continuously, the roll motion now undergoes period-1 motions with jumps, the first one producing a jump from 17 to 21 degrees. A second jump takes place at AW  0.648m , from 24 to 16 degrees. At AW  0.663m the motion begins a flip bifurcation at the same point as obtained in Figure 3. Period-2, 4, 8 solutions appear in cascade, at all stages featuring multistability. Finally, chaotic motions take place again for AW  0.6788m .

5 Lyapunov Exponents Lyapunov exponents offer a quantitative measure of the sensitivity of a nonlinear dynamic system to initial conditions. As such, they characterize the chaotic behaviour of the system. The Lyapunov exponents provide information on the global stability of the system: values obtained for these exponents, after the transients, are negative for stable zones, zero at bifurcation and positive in the chaotic zones. The method of trajectories developed by Wolf and Vastago (1985) is employed here in order to compute all the exponents relative to the heave, roll and pitch coupled motions for the tuning we / wn 4  2.0 and different wave

460 M.A.S. Neves et al.

amplitudes. Figures 16 and 17 show the six exponents for two wave amplitudes, Aw=0.62 m and Aw=0.63 m, respectively. In Figure 16 all exponents are negative, with the larger one being quite close to zero, whereas in Figure 17 this larger exponents is practically zero. Compiling the values of the Lyapunov exponents obtained for varying wave amplitudes at t=800 sec, Figure 18 gives a complete picture of the evolution of the first three larger exponents, with the control parameter Aw, showing that at Aw=0.63 m the system undergoes chaos. This indicates that other attractors may be competing with those observed in Figures 3 and 15. In any case, this result demonstrates that the motions shown in Figure 7, corresponding to Aw=0.6129 m are non-periodic, not chaotic.

Fig. 16 Lyapunov exponents, Aw=0.62 m

6 Conclusions Numerical limits of stability for a fishing vessel at Fn=0.30 undergoing strong parametric rolling in head seas have been computed for a range of encounter frequencies. The main dynamical characteristics of these limits have been discussed.

Fig. 17 Lyapunov exponents, Aw=0.63 m

Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas 461

Fig. 18 Lyapunov exponents (t=800 sec) for different wave amplitudes

For the encounter frequency tuning corresponding to the first region of instability of the Mathieu stability map, bifurcation diagrams for the heave, roll and pitch motions have been computed considering wave amplitude as control parameter. Interesting phenomena such as coexistence of attractors with period-3 solutions, appearance of a burst of non-periodic solutions, multistability with alternance, fold and flip bifurcation and chaos have been identified. The phase planes and Poincaré mappings showed that the period-3 solutions and burst of non-periodic solutions are common to the three modes of motion considered. On the other hand, multistability with alternance only takes place for the roll motion. It was observed that when a different set of initial conditions was considered, both the non-periodic motions and alternance disappeared. Finally, Lyapunov exponents have been computed for the same encounter frequency tuning, again taking the wave amplitude as control parameter.

Acknowledgements The present investigation is supported by CNPq within the STAB project (Nonlinear Stability of Ships). The Authors also acknowledge financial support from CAPES, FAPERJ and LabOceano. Thanks are due to Prof. Marcelo A. Savi for many fruitful discussions.

462 M.A.S. Neves et al.

References Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Appl Math Sci 42 Springer-Verlag Neves M A S, Rodríguez C (2005) A nonlinear mathematical model of higher order for strong parametric resonance of the roll motion of ships in waves. Mar Syst & Ocean Tech – J Sobena 1 2:69-81 Neves M A S, Rodríguez C (2006) Unstable ship motions resulting from strong nonlinear coupling. Ocean Eng 33:99-108 Neves M A S, Rodríguez C (2007a) Influence of nonlinearities on the limits of stability of ships rolling in head seas. Ocean Eng 34:1618-1630 Neves M A S, Rodríguez C (2007b) An investigation on roll parametric resonance in regular waves. Int Shipbuild Prog 54:207-225 Neves M A S, Pérez N A, Lorca O (2002) Experimental analysis on parametric resonance for two fishing vessels in head seas. 6th Int Ship Stab Workshop Webb Inst NY Paulling J R, Rosenberg R M (1959) On unstable ship motions resulting from nonlinear coupling. J of Sh Res 3 1:36-46 Paulling J R (1961) The transverse stability of a ship in a longitudinal seaway. J of Ship Res 4 4Mar:37-49 Seydel R (1988) From equilibrium to chaos: practical bifurcation and stability analysis. Elsevier Sci Publ NY Vivanco J E M (2009) Parametric rolling of a fishing vessel - nonlinear dynamics. M.Sc. Diss COPPE/UFRJ Jan (Port) Wolf A, Vastago A (1985) Determining Llyapunov exponents from a time series. Phys Rev 16:25-317

6 Roll Damping

A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships on the Basis of Ikeda’s Method and Its Limitation Yuki Kawahara, Kazuya Maekawa, Yoshiho Ikeda Osaka Prefecture University, Sakai, Japan

Abstract Since the roll damping of ships has significant effects of viscosity, it is difficult to calculate it theoretically. Therefore experimental results or some prediction methods are used to get the roll damping in design stage of ships. Among some prediction methods, Ikeda’s one is widely used in many ship motion computer programs. Using the method, the roll damping of various ship hulls with various bilge keels are calculated to investigate its characteristics. Ship hull forms are systematically changed by changing length, beam, draft, midship sectional coefficient and prismatic coefficient. On the basis of these predicted roll damping of various ships, a very simple prediction formula of the roll damping of ships is deduced. It is found, however, that this formula cannot be used for ships that have higher position of the center of gravity.

1 Introduction In 1970’s, strip methods for predicting ship motions in 5-degree of freedoms in waves have been established. The methods are based on potential flow theories (Ursell-Tasai method, source distribution method and etc), and can predict pitch, heave, sway and yaw motions of ships in waves in fairly good accuracy. In roll motion, however, the strip methods do not work well because of significant viscous effects on the roll damping. Therefore, some empirical formulas or experimental data are used to predict the roll damping in the strip methods. To improve the prediction of roll motions by these strip methods, one of the authors carried out a research project to develop a roll damping prediction method which has the same concept and the same order of accuracy as the strip methods which are based on hydrodynamic forces acting on strips, or cross sections of a ship [Ikeda et al. (1976), Ikeda et al. (1977a), Ikeda et al. (1977b), Ikeda et al. (1978): All papers in English versions were published as Reports of Dept. of Naval M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_26, © Springer Science+Business Media B.V. 2011

465

466 Y. Kawahara et al.

Architecture, Univ. of Osaka Pref.]. The review of the prediction method was made by Himeno (1981) and Ikeda (1982) with the computer program. The prediction method, which is now called Ikeda’s method, divides the roll damping into the frictional (BF), the wave (BW), the eddy (BE) and the bilge keel (BBK) components at zero forward speed, and at forward speed, the lift (BL) is added. Increases of wave and friction components due to advance speed are also corrected on the basis of experimental results. Then the roll damping coefficient B44 (=Roll damping moment (kgfm)/Roll angular velocity (rad/sec)) can be expressed as follows.

B44  B F  BW  B E  B L  B BK

(1)

At zero forward speed, each component except the friction component is predicted for each cross section with unit length and the predicted values are summed up along the ship length. The friction component is predicted by Kato’s formula for a three-dimensional ship shape. Modification functions for predicting the forward speed effects on the roll damping components are developed for the friction, wave and eddy components. The computer program of the method was published, and the method has been widely used. For these thirty years, the original Ikeda’s method developed for conventional cargo ships has been improved to apply many kinds of ships, for examples, more slender and round ships, fishing boats, barges, ships with skegs and so on. The original method is also widely used. However, sometimes, different conclusions of roll motions were derived even though the same Ikeda’s method was used in the calculations. Then, to check the accuracy of the computer programs of the same Ikeda’s method, a more simple prediction method with the almost same accuracy as the Ikeda’s original one has been expected to be developed. It is said that in design stages of ships, Ikeda’s method is too complicated to use. To meet these needs, a simple roll damping prediction method was deduced by using regression analysis [Kawahara et al. (2008)].

2 Previous Prediction Formula The simple prediction formula proposed in the previous paper [Kawahara (2008)] cannot be used for modern ships that have high position of center of gravity or long natural roll period such as large passenger ships with relatively flat hull shape. In order to investigate its limitation, the authors compared with the result of this prediction method and original Ikeda’s one out of its calculating limitation. Figure 1 shows the result of the comparison with these method of the roll damping. The upper one is on the condition that the center of gravity is low and the lower one on the condition that the center of gravity is high.

A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships

467

(a) OG/d = -0.2

(b) OG/d = -1.5 Fig. 1 Comparison between Ikeda’s method and proposed one of roll damping at L/B=6.0 B/d=4.0, Cb=0.65, Cm=0.98, φa=10°, bBK/B=0.025 and lBK/Lpp=0.2. (OG denotes the distance between water surface and center of gravity, and defined plus when the center of gravity is below water surface.)

From this figure, the roll damping estimated by this prediction formula are in good agreement with the roll damping calculated by the Ikeda’s method for low position of center of gravity, but the error margin grows for the high position of center of gravity. The results suggest that the previous prediction formula is necessary to be revised.

3 Methodical Series Ships Modified prediction formula will be developed on the basis of the predicted results by Ikeda’s method using the methodical series ships. This series ships are constructed based on the Taylor Standard Series and its hull shapes are methodically changed by changing length, beam, draft, midship sectional coefficient and longitudinal prismatic coefficient. The geometries of the series ships are given by the following equations.

468 Y. Kawahara et al.

y1  Q( x)  C P P ( x)  f11tT ( x)  nN ( x)  f12 F ( x)

(2)

y2  Q( x)  CW P( x)  f 21tT ( x)  nN ( x)  f 22 F ( x)

(3)

where

Q ( x )  30 x 2  100 x 3  105 x 4  36 x 5

(4)

P ( x )  60 x 2  180 x 3  180 x 4  60 x 5

(5)

T ( x )  x  6 x 2  12 x 3  10 x 4  3 x 5

(6)

N ( x)  0.5 x 2  2 x 3  2.5 x 4  x 5

(7)

F ( x)  26.562 x 6  105.74 x 5  162.71x 4  116.58 x 3  34.532 x 2  0.6998 x  0.7923 t  3969.7C P6  16664.6C P5  28230C P4  24951C P3  12205C P2  3147.7C P  335.67

(8) (9)

(at stern side and CP<0.73)

t  113.64C P2  149.68C P  50.221

(10)

t  96.339C P5  173.59C P4  159.75C P3

(11)

(at stern side and CP ≥0.73)

 113.4C P2  54.123C P  10.686

(at bow side and CP <0.72)

t  41.667C P2  49.167C P  14.9

(12)

n  5.7035C P3  30.16C P2  33.471C P  10.606

(13)

n  10.417C P2  13.458C P  4.305

(14)

n  1664.6C P5  5596.7C P4  7413.2C P3

(15)

(at bow side and CP ≥0.72)

(at stern side and CP <0.74) (at stern side and CP ≥0.74)

 4815.6C P2  1525.3C P  186.79

(at bow side and CP <0.73)

n  0.625C P  0.4312 (at bow side and CP ≥0.73)

(16)

A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships

469

where y1 is the non-dimensional value obtained by sectional-immersed-area divided by maximum-immersed-area, y2 is the non-dimensional value obtained by sectional-water-line-breadth divided by maximum-water-line-breadth, x is the non-dimensional value of longitudinal position when x is measured from the extremity of either the bow or stern, CP is the longitudinal prismatic coefficient, CW is the water-plane coefficient, f11 is constant and equal to 0.6 for the stern and 1.0 for the bow, f12 is constant and equal to 0.05, f21 is constant and equal to 2.0 for the stern and 1.0 for the bow, f22 is constant and equal to 0.15 for the stern and 0.1 for the bow. However, occasionally, corrections of the results obtained from the Eqs. 2, 3 are required. For example, if maximum of y1 exceeds 1.0, it is adjusted to 1.0. Figures 2 and 3 show the sectional area curves obtained from Eq. 2 and a body plan of a ship with Cb=0.84 (Cb : block coefficient) and Cm=0.98 (Cm : midship section coefficient), respectively. Since the hull shapes used here are conventional ones as shown in Figure 3, an application of the deduced prediction method to modern unconventional hull shapes, for example buttock-flow hull, should be careful.

Fig. 2 Sectional-area curves of series model used in calculation

Fig. 3 An example of body plan of ship with Cb=0.84 and Cm=0.98

470 Y. Kawahara et al.

4 Proposal of New Prediction Method of Roll Damping In this chapter, the characteristics of each component of the roll damping, the frictional, the wave, the eddy and the bilge keel components at zero advanced speed, are discussed, and a simple prediction formula of each component is developed. The roll damping coefficient (B44) and circular frequency (ω=2π/Tw) are defined as follows,

B Bˆ44  44 2 ∇ B

ˆ  

B 2g

(17)

B 2g

(18)

where ρ denotes water density, ∇ displacement volume, B beam and g is gravity acceleration, respectively. The relationship between B44 and N coefficient (Bertin) is as follows. Bˆ N  Bˆ 44 ( a in Eq. 19 is in deg .) GM a

(19)

4.1 Frictional Component (BF) In Ikeda’s method, the friction damping at Fn=0 is given by Kato’s formula as follows,

BF 

4 s f r f3 a c f 3

(20)

where cf is frictional coefficient, rf is average radius from the axis of rolling and sf is wetted surface area. These parameters in the equation are given by the following equations, 1

 3.22r f2 a2  2  c f  1.328   T   0.887  0.145Cb 1.7d  Cb B   2OG rf 



s f  LPP 1.75d  Cb B 

(21)

(22) (23)

A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships

471

where φa denotes roll amplitude, T roll period, ν dynamic coefficient of viscosity, OG distance from calm water surface to the axis of rolling (downward direction is positive) and d draft, respectively. In the present study, since the frictional component of roll damping is already given for a whole ship as Eqs. 20-23, the formula is used without any modification in the simple prediction method developed in the present study. It should be noted, however, the frictional component is negligible for a full-scale ship although it takes about 5-10% of the total roll damping for a small scale model.

4.2 Wave Component (Bw) As well known, the wave component of the roll damping for a two-dimensional cross section can be calculated by potential flow theories in fairly good accuracy. In Ikeda’s method, the wave damping of a strip section is not calculated and the calculated values by any potential flow theories are used as the wave damping. 4.2.1 Characteristics of the Wave Component The reason why viscous effects are significant in only roll damping can be explained as follows. Figure 4 shows the wave component of the roll damping for 2-D sections calculated by a potential flow theory. We can see that the component is very small for sections with half breadth/draft ratio, H0=05-1.5, particularly when area coefficient, σ is large. Usually, conventional ships have such H0 at parallel middle body part. Then ships have relatively small wave roll damping, and viscous effects play an important role in the roll damping. In Figures 5 and 6, calculated distributions of the wave component of the roll damping are shown. The results for a full ship (Cb=0.8) shown in Figure 5 demonstrate that the damping created by the mid-ship body is very small, and that created by the stern body is large. The results for a slender ship (Cb=0.5) shows two peaks at the cross sections of SS=3 and SS=7. The reason why the peaks appear on bow and stern parts can be easily understood from the calculated results shown in Figure 4. Dependencies of the wave component on location of roll axis or center of gravity are shown in Figure 7.

472 Y. Kawahara et al.

Fig. 4 Characteristics of roll damping coefficient of wave component for two-dimensional section at OG/d=0 and ˆ =1.25

Fig. 5 Longitudinal distribution of σ, H0 and roll damping coefficient of wave component for a ship with L/B=5, B/d=3.5, Cm=0.94, Cb=0.8,OG/d=0 and ˆ =1.25

A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships

473

Fig. 6 Longitudinal distribution of σ, H0 and roll damping coefficient of wave component for a ship with L/B=5, B/d=3.5, Cm=0.94, Cb=0.5, OG/d=0 and ˆ =1.25

Fig. 7 Characteristics of roll damping coefficient of wave component for a whole ship at L/B=5, B/d=3.5, Cm=0.94, Cb=0.8

4.2.2 Proposed Formula of the Wave Roll Damping By fitting these predicted wave components a simple prediction formula is deduced as follows.

474 Y. Kawahara et al. x1  B d , x2  Cb , x3  Cm , x4  1  OG d , x5  ωˆ BˆW  A1 x5  exp (-A2(LOG(x5 )-A3 )2 / 1.44 ) 2

A1=(A11 x 4 +A12 x4+A13 )AA1 3

2

A2=-1.402 x4 +7.189 x4 -10.993 x4+9.45 6

5

4

3

2

A3=A31 x4 +A32 x4 +A33 x4 +A34 x4 +A35 x4 +A36 x 4+A37+AA3 x6=x4-AA32 AA1  ( AA11 x3  AA12 )  (1  x4 )  1.0 9

8

7

AA3=AA31(-1.05584 x6 +12.688 x6 -63.70534 x6 + 6

5

4

172.84571x6 -274.05701x6 +257.68705 x6 -141.40915 x6 2

3

2

+44.13177 x6 -7.1654 x6-0.0495 x1 +0.4518 x1-0.61655 ) 3

2

AA31=(-0.3767 x1 +3.39 x1 -10.356 x1+11.588 )  AA311 2

AA32=-0.0727 x1 +0.7 x1-1.2818 3

2

AA311=(-17.102 x2 +41.495 x2 -33.234 x2+8.8007 )  x4+ 3

2

36.566 x2 -89.203 x2 +71.8 x2-18.108 6

5

4

A31=-7686.0287 x2 +30131.5678 x2 -49048.9664 x2  3

2

42480.7709 x2 -20665.147 x2 +5355.2035 x2-577.8827 6

5

4

A32=61639.9103 x2 -241201.0598 x2 +392579.5937 x2  3

2

340629.4699 x2 +166348.6917 x2 -43358.7938 x2+4714.7918 6

5

A33=-130677 .4903 x2 +507996.2604 x2 -826728.7127 x2 3

 722677.104 x2 -358360.7392 x2

4

2

+95501.4948 x2-10682.8619 6

5

A34=-110034.6584 x2 +446051.22 x2 -724186.4643 x2 3

 599411.9264 x2 -264294.7189 x2

4

2

+58039.7328 x2-4774.6414 6

5

A35=709672.0656 x2  2803850.2395 x2 +4553780.5017 x2 3

 3888378.9905 x2 +1839829.259 x2

2

 457313.6939 x2+46600.823 6

5

A36=-822735.9289 x2 +3238899.7308 x2 -5256636.5472 x2 3

 4500543.147 x2  2143487 .3508 x2 +538548.1194 x2  55751.1528

2

4

4

A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships 6

5

A37=299122.8727 x 2 -1175773.1606 x 2 +1907356.1357 x 2 3

 1634256.8172 x 2 +780020.9393 x 2

475

4

2

 196679.7143 x 2+20467 .0904 2

AA11=AA111 x 2 +AA112 x 2+AA113 2

AA12=AA121 x 2 +AA122 x 2+AA123 3

2

AA111=17.945 x1 -166.294 x1 +489.799 x1-493.142 3

2

AA112=- 25.507 x1 +236.275 x1 -698.683 x1+701.494 3

2

AA113=9.077 x1 -84.332 x1 +249.983 x1- 250.787 3

2

AA121=-16.872 x1 +156.399 x1 - 460.689 x1+463.848 3

2

AA122=24.015 x1 -222.507 x1 +658.027 x1-660.665 3

2

AA123=-8.56 x1 +79.549 x1 - 235.827 x1+236.579 2

A11=A111 x 2 +A112 x 2+A113 3

2

A12=A121 x 2 +A122 x 2 +A123 x 2+A124 3

2

A13=A131 x 2 +A132 x 2 +A133 x 2+A134 3

2

A111=-0.002222 x1 +0.040871x1 -0.286866 x1+0.599424 3

2

A112=0.010185 x1 -0.161176 x1 +0.904989 x1-1.641389 3

2

A113=-0.015422 x1 +0.220371x1 -1.084987 x1+1.834167 4

3

A121=-0.0628667 x1 +0.4989259 x1 +0.52735 x1

2

-10.7918672 x1+16.616327 4

3

A122=0.1140667 x1 -0.8108963 x1 -2.2186833 x1

2

+25.1269741x1-37.7729778 4

3

A123=-0.0589333 x1 +0.2639704 x1 +3.1949667 x1

2

-21.8126569 x1+31.4113508 4

3

A124=0.0107667 x1 +0.0018704 x1 -1.2494083 x1

2

+6.9427931x1-10.2018992 3

2

A131=0.192207 x1 - 2.787462 x1 +12.507855 x1-14.764856 3

2

A132=-0.350563 x1 +5.222348 x1 -23.974852 x1+29.007851 3

2

A133=0.237096 x1 -3.535062 x1 +16.368376 x1-20.539908 3

2

A134=-0.067119 x1 +0.966362 x1 -4.407535 x1+5.894703

 0.5  C b  0.85, 2.5  B / d  4.5, ˆ  1.0      1.5  OG / d  0.2, 0.9  C m  0.99 

(24)

476 Y. Kawahara et al.

4.3 Eddy Component (BE) The eddy component of the roll damping is created by small separation bubbles or small shedding vortices generated at the bilge part of midship section and large vortices generated at the relatively sharp bottom of bow and stern sections. Although vortex shedding flow from oscillating bluff bodies is usually govern by Keulegan-Carpenter number, Kc, it was found by Ikeda et al. (1978b) that the viscous forces created by such small separation bubbles or small shedding vortices do not significantly depend on Kc. In Ikeda’s prediction method, the distribution of the pressure created on a hull surface by such separation bubble is assumed as a simple shape for each shape of cross sections on the basis of experimental results of pressure distribution on hull surfaces. The pressure value was determined as the calculated eddy components of the roll damping for various cross sections are in good agreement with measured ones. Then, the eddy damping of a strip section is calculated by following formulas.

 R  OG R  1  f1 d 1  d  f1 d    4  2 LPP d 2rmax a  BE  CP 0 2 3   f  H  f R    2  0 1 d  (25)

1 1  tanh20  0.7  2 1 f 2  1  cos    1.5 1  e 51  sin 2  2 1 C P 0  0.87 e   4e  0.1877  3 2 B H0  2d f1 





 

S Bd





(26) (27) (28) (29) (30)

where, rmax is maximum distance from center of gravity (roll axis) to hull surface, R is bilge radius and σ is sectional area coefficient. The eddy roll damping of a whole ship is calculated by integrating BE’ to longitudinal direction.

A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships

477

4.3.1 Characteristics of Eddy Component In Figures 8 and 9, the longitudinal distribution of the predicted eddy component of the roll damping are shown for a slender ship (Cb=0.5) and a full ship (Cb=0.8). The eddy component of a slender ship is large at bow and stern parts, and has a small peak at mid-ship, as shown in Figure 8. For a full ship, however, the eddy damping at parallel middle body part becomes larger and longer as shown in Figure 9. The component at stern is large as a slender ship, but becomes small at bow part. These may be caused by that both ships has similar V shape stern but the full ship has U shape bow section which is not thin and flat shape as a slender ship. Figure 10 shows the effect of midship-section coefficient (Cm ) on the eddy damping. The results demonstrate that the eddy component of the roll damping of parallel body parts of ships is significantly sensitive to Cm. This is because of larger flow separation occurs at bilge corners of ships with large midship-section coefficient.

Fig. 8 Longitudinal distribution of σ, H0 and roll damping coefficient of eddy component for a ship with L/B=5, B/d=3.5, Cm=0.94, Cb=0.5,OG/d=0 and ˆ =1.25

478 Y. Kawahara et al.

Fig. 9 Longitudinal distribution of σ, H0 and roll damping coefficient of eddy component for a ship with L/B=5, B/d=3.5, Cm=0.94, Cb=0.8, OG/d=0 and ˆ =1.25

Fig. 10 Effects of midship-section coefficient, Cm on eddy component of roll damping for slender ships (Cb=0.5) and full ships (Cb=0.8)

4.3.2 Proposed Formula of the Eddy Roll Damping By fitting these predicted eddy components a simple prediction formula is deduced as follows.

A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships

479

x1  B d , x 2  C b , x 3  C m , x 4  OG d Bˆ E 

4 L pp d 4 ˆ  a 3  B

2

CR 

4 ˆ  a 3  x 2  x1

C R  AE  exp( B E1  B E 2  x 3

BE 3

3

CR

)

AE  (0.0182 x 2  0.0155)  ( x1  1.8) 3  79.414 x 2 3

4

2

 215.695 x 2  215.883x 2  93.894 x 2  14.848 B E1  (0.2 x1  1.6)  (3.98 x 2  5.1525)  x 4



2



 (0.9717 x 2  1.55 x 2  0.723)  x 4  (0.04567 x 2  0.9408) B E 2  (0.25 x 4  0.95)  x 4 3

2

 219.2 x 2  443.7 x 2  283.3x 2  59.6 B E 3  (46.5  15 x1 )  x 2  11.2 x1  28.6  0.5  C b  0.85, 2.5  B / d  4.5     1 . 5  /  0 . 2 , 0 . 9   0 . 99 OG d C m  

(31)

4.4 Bilge Keel Component (BBK) The bilge keel component is usually the largest one in the roll damping. The component creates 50-80% of the total roll damping. The component is created by shedding vortices from the sharp edges of bilge keels due to roll motion. The component can be divided into two components, the normal force component (BN) and the hull pressure component (BS). Both components are created by the same vortices from the edge of bilge keels. The former one is created by the force acting a bilge keel, and the latter by the pressure over the hull surfaces in front and back sides of the bilge keel. In Ikeda’s method, the pressure distributions in front and back of a bilge keel are assumed on the basis of the measured ones, and are integrated over the hull surface. This means that the method may be available for any shape of cross section. Ikeda et al. (1976) experimentally found that the magnitude and distribution of the pressure created by a bilge keel significantly depends on Kc. The roll damping due to the normal force acting on bilge keel is given by following expressions.

480 Y. Kawahara et al.

BN 

8 r 3l BK bBK  af 3

2

  b  22.5 BK  2.4  rf a  

f  1  0.3 exp 1601   

(32) (33)

where r is distance from the axis of rolling to bilge keel, bBK is width of bilge keel, lBK is length of bilge keel and f is a correction factor on bilge radius to take the velocity increase there into account. The roll damping due to the hull pressure created by bilge keels is calculated by following equations. BS 

I

4 r 2 d 2  a f 2 I 3

1 C P1l BK ds d2 

(34) (35)

where CP1 is pressure coefficient on hull surface. The positive pressure coefficient (CP1+) of face of a bilge keel and the negative pressure coefficient (CP1-) of back side of a bilge keel are given by following formulas.

C P1  1.2 C P1  

22.5bBK  1.2 rf a

(36) (37)

4.4.1 Characteristics of the Bilge Keel Component Calculated bilge keel components of the roll damping by Ikeda’s method are shown in Figs. 11-13 to demonstrate the characteristics. In Figure 11 the bilge keel component of the roll damping of slender ships (Cb=0.58) and full ships (Cb=0.81) with different midship-section coefficients are shown. In the prediction the area of bilge keels is systematically changed. The results show that the roll damping component increases with increasing area of bilge keels. For full ships the increase of the component is almost linear, but for slender ships it shows non-linearly increases. The magnitude of the component significantly depends on midship-section coefficients, but not so sensitive on block coefficient Cb.

A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships

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Fig. 11 Bilge keel component of roll damping of slender and full ships for various bilge keel lengths and constant breadth (bBK)

Fig. 12 Effect of location of center of gravity, or roll axis on bilge keel component of roll damping.

Fig. 13 Bilge keel component of roll damping of slender and full ships with bilge keel of the same area (SBK) but different lengths.

In Figure 12, effects of location of center of gravity on the bilge keel component of the roll damping are shown for slender and full ships. As height of center of gravity decreases, or OG/d increases, the bilge keel component decreases. This is because

482 Y. Kawahara et al.

relative flow speed at bilge corner decreases and flow separation at the edge of bilge keels weakens. In Figure 13, effects of aspect ratio (=length/breadth of a bilge keel) on the roll damping of ships with the same area bilge keels are shown. It can be seen that the component is not so sensitive with aspect ratio, but increases with increasing aspect ratio, or more slender bilge keels for full ships with large midship-section coefficients. 4.4.2 Proposed Formula of the Bilge Keel Roll Damping By fitting these predicted bilge keel components a simple prediction formula is deduced as follows. x1  B d , x 2  C b , x3  C m , x 4  OG d , x5  ωˆ x 6   a (deg) , x 7  bBK B , x8  l BK LPP B Bˆ BK  ABK  exp( B BK 1  B BK 2  x3 BK 3 )  x5 ABK  f 1 ( x1 , x 2 )  f 2 ( x 6 )  f 3 ( x 7 , x8 ) f 1  ( 0.3651x 2  0.3907)  ( x1  2.83) 2  2.21x 2  2.632 2

f 2  0.00255 x 6  0.122 x 6  0.4794 2

f 3  ( 0.8913 x 7  0.0733 x 7 )  x8

2

2

 (5.2857 x 7  0.01185 x 7  0.00189)  x8 B BK 1  {5 x 7  0.3 x1  0.2 x8 2

 0.00125 x 6  0.0425 x 6  1.86}  x 4 B BK 2  15 x 7  1.2 x 2  0.1x1 2

 0.0657 x 4  0.0586 x 4  1.6164 B BK 3  2.5 x 4  15.75  0.5  C b  0.85, 2.5  B / d  4.5      1.5  OG / d  0.2, 0.9  C m  0.99    0 . 01  /  0 . 06 , 0 . 05  /  0 . 4 b B l L BK BK pp  

(38)

5 Validation of Proposed Method To verify the validity of the proposed method, the values of the roll damping calculated by the original Ikeda’s method and the proposed method are compared on two kinds of ships. The results of these comparisons are shown in Figure 14.

A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships

483

These results indicate that the calculated values of the roll damping by the proposed formulas are in good agreement with the roll damping calculated by the Ikeda’s method, although the estimated values by using the proposed method have some errors about 10% when ˆ becomes larger for the ship with OG/d=-0.2 as shown in Figure 14. However, the error margin becomes small in the lowfrequency range because the bilge keel component is predominant.

(a) OG/d = -0.2

(b) OG/d =-1.5 Fig. 14 Comparison between Ikeda’s method and proposed one of roll damping at L/B=6.0, B/d=4.0, Cb=0.65, Cm=0.98, φa=10°, bBK/B=0.025 and lBK/Lpp=0.2.

The results in Figure 14 suggest that the errors come from the discrepancy of the wave component between the proposed method and the potential theory. Especially, this discrepancy grows larger on condition of B/d=4.5 and OG/d=-0.2 ~ -0.5, so it should be noted for the use of the proposed method on this condition. This may be because the wave component of the roll damping intricately depends on frequency and locations of roll axis. Therefore, if more accurate prediction is needed, the calculated wave damping by any potential theory should be used.

484 Y. Kawahara et al.

6 Limitation of Ikeda’s Method In recent years, the number of ships that have buttock flow stern, such as large passenger ship or pure car carrier, has been increasing. In such a type of ships, however, prediction accuracy of the roll damping calculated by Ikeda’s method might decrease remarkably. In order to investigate its limitation, free roll decay tests are carried out by using three types of model ships, large passenger ship (LPS), modern pure car carrier (PCC) and wide breadth and shallow draft car carrier (WSPCC). The principal particulars of these ships and the body plan of a large passenger ship are shown in Table 1 and Figure 15, respectively. As shown in Figure 15, the ship has shallow stern-bottom with something like a skeg. Table 1. Principal particulars of the ships Scale LOA(m) LPP(m) Breadth(m) Draft(m) Displacement(kg) GM(m) Tnr(sec) OG/d BBK/Breadth LBK/LPP

LPS 1/125 2.200 1.933 0.287 0.067 26.98 0.0126 1.88 -1.13 -

PCC 1/96 2.083 2.00 0.336 0.0938 33.43 0.0194 2.06 -0.071 0.0217 0.225

WSPCC 1/96 2.083 2.00 0.378 0.0772 28.05 0.063 1.05 -1.35 0.0217 0.225

Fig. 15 Body Plan of large passenger ship.

In Figure 16, the obtained results of the extinction coefficient N are shown. The results demonstrate that the accuracy of Ikeda’s prediction method decrease remarkably when the roll angle is small. However, the error of roll damping becomes smaller for the large passenger ship and the pure car carrier in condition of large roll angle. For the wide breadth and shallow draft car carrier, however, the roll damping calculated by Ikeda’s prediction method is overestimated from the experimental result for whole roll angle. This is pointed out by Tanaka et al. (1981) that the

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485

effect of the bilge keel component on the roll damping significantly decreases for the wide breadth and shallow draft ships. Also, the discrepancies between the simple prediction formula and Ikeda’s method are attributed to differences between methodical series model and real model ship and to using out of its possible calculating condition. Then, the extinction coefficient N of a large passenger ship is relatively large even for naked hull because of relatively small bilge radius. Thus, Ikeda’s prediction method is valid only in large roll angle for a modern type of ships with buttock flow stern, and overestimate the roll damping for a very flat ship with large bilge keels.

(a) Large passenger ship

(b) Pure car carrier

(c) Wide breadth and shallow draft car carrier Fig. 16 Comparison of roll damping between Ikeda’s method and experimental result

7 Conclusions A simple prediction method of the roll damping of ships is developed on the basis of the Ikeda’s original prediction method which was developed in the same concept as a strip method for calculating ship motions in waves. Using the data of a ship, B/d,

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Cb, Cm, OG/d, ˆ , bBK/B, lBK/LPP, φa, the roll damping of a ship can be approximately predicted. Moreover, the limit of application of Ikeda's prediction method to modern ships that have buttock flow stern is demonstrated by the model experiment. The computer program of the developed method can be downloaded from the Home Page of Ikeda’s Labo. (http://www.marine.osakafu-u.ac.jp/~lab15/roll_damping.html). The computer program of Ikeda’s roll damping prediction method can be downloaded from the Home page of Katayama’s Labo. (http://www.marine.osakafu-u.ac.jp/~lab07/katalaboHP/New_program_of_roll_damping/Roll%20Damping%20Program.html).

Acknowledgements This work was supported by a Grant-in Aid for Scientific Research of the Japan Society for Promotion of Science (No. 18360415). The authors wish to express sincere appreciation to Prof. N. Umeda of Osaka University for valuable suggestions to this study.

References Himeno Y (1981) Prediction of ship roll damping, state of the art. Rep of Dept of Nav Archit & Mar Eng, Univ of Michigan 239 Ikeda Y, Himeno Y, Tanaka N (1976) On roll damping force of ship, effects of friction of hull and normal force of bilge keels. J of the Kansai Soc of Nav Archit, Japan 161 41:49 Ikeda Y, Komatsu K, Himeno Y, Tanaka N (1977a) On roll damping force of ship, effects of hull surface pressure created by bilge keels, J of the Kansai Soc of Nav Archit, Japan 165 31:40 Ikeda Y, Himeno Y, Tanaka N (1977b) On eddy making component of roll damping force on naked hull, J of the Soc of Nav Archit, Japan 142 59:69 Ikeda Y, Himeno Y, Tanaka N (1978) Components of roll damping of ship at forward speed, J of the Soc of Nav Archit, Japan 143 121:133 Ikeda Y, Fujiwara T, Himeno Y, Tanaka N Velocity field around ship hull in roll motion, J of the Kansai Soc of Nav Archit Japan 171 33:45 Ikeda Y (1982) Prediction method of roll damping, Rep of Dept of Nav Archit, Univ of Osaka Prefect Ikeda Y. (1984) Roll damping, 1ST Symp of Mar Dyn Res Group Japan 241:250 Kawahara Y, (2008) Characteristics of roll damping of various ship types and a simple prediction formula of roll damping on the basis of Ikeda’s method, The 4th Asia-Pacific Workshop on Mar Hydrodymics, Taipei 79:86

A Study on the Characteristics of Roll Damping of Multi-Hull Vessels Toru Katayama, Masanori Kotaki, Yoshiho Ikeda Osaka Prefecture University

Abstract In this study, for a catamaran and a trimaran as multi-hull vessels, the characteristics of the roll damping are investigated experimentally. A free roll decay test and a forced roll motion test with and without forward speed are carried out. The results show that the roll damping of them is much larger than that of conventional monohull vessels, and the component created by side-hull accounts for a significant rate for trimaran. Especially, at the condition without forward speed, the interference of waves created by hulls are significant, the measured roll damping values by different experiments are different on the basis of different water surface condition created by hulls. Moreover, the simplified prediction method is proposed.

1 Introduction They are very important to understand the characteristics of the roll damping for any kind of vessels and to estimate it adequately, because it is significantly affect on the occurrence of parametric rolling, the amplitude of resonances and so on. However, it is very complicated to calculate the roll damping theoretically, because of significant viscous component depending on vortex shedding. It is known that there is a prediction method of the roll damping proposed by Ikeda (one of authors) for conventional displacement type of mono-hull vessels, barge ships and a small hard chine fishing boat. It is composed of wave making, friction, transverse lift and eddy making prediction component those are developed with theoretically and experimentally backgrounds based on the hydrodynamic characteristics of the roll damping for the above-mentioned types of vessels. Therefore, it is difficult to apply the method to the resent vessels that have large different hull form from above-mentioned types of vessels. In this study, for catamaran and trimaran as multi-hull vessels, the characteristics of the roll damping are investigated experimentally. And based on the results, the simplified prediction method is proposed. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_27, © Springer Science+Business Media B.V. 2011

487

488 T. Katayama et al.

2 Models The photographs of two model ships are shown in Figs. 1 and 2, and their principle particulars are shown in Tables 1 and 2. The model of catamaran in Fig. 1 is wave piercing type high-speed craft, and its loading conditions are decided by

Fig. 1 Model of catamaran.

Fig. 2 Model of trimaran and its hull arrangement.

A Study on the Characteristics of Roll Damping of Multi-Hull Vessels 489

based on over-100m class wave piercing catamaran. On the other hand, the model of trimaran in Fig. 2 is stabilized slender mono-hull type high-speed craft, and its distributions of displacement for main-hull and side-hulls are decided by based on a real high speed vehicle-passenger trimaran ferry. Table 1 Principal particulars of trimaran. -

Main-hull

L/B

12

Side-hull 10.2

LOA (m)

1.5

0.888

Lpp (m)

1.42

0.808

breadth (m)

0.125

0.0874

depth (m)

0.1

0.07

draft (m)

0.037

0.012

displacement (N)

30.8

0.778×2

GM (m)

0.163

0.163

Tn (sec)

1.04

0.104

Table 2 Principal particulars of catamaran. Scale

1/80

LOA (m)

1.408

LPP (m)

1.32

breadth( m)

0.38

depth (m)

0.07

draft (m)

0.04

displacement (N)

20.9×2

GM (m)

0.85

Tn (sec)

0.46

3 Characteristics of Roll Damping without Forward Speed In order to investigate the characteristics of roll damping of the models, at first, a free roll decay test is carried out without forward speed. In this test, for model, heaving, pitching, rolling, swaying and yawing are free, and it is initially heeled by fixing only rolling at arbitrary angle. From the condition with which model’s attitude is balancing, motions are measured after freeing rolling. The measured damping curves of rolling for each model are shown in Figs. 3 and 4. The result for trimaran in Fig. 3 seems to continuously damp, on the other hand, the result for catamaran in Fig. 4 shows that the damping curve has some different periods and it damps irregularly.

490 T. Katayama et al.

Therefore, in this study, the results of the free roll decay test with catamaran, only the first swing is analyzed. And the results of the same test with trimaran are analysed for the first swing and other swings separately. ROLL 4 deg. t(sec)

0 0 –4

Fig. 3 Time histories of roll motion of the trimaran.

ROLL 4 deg. 0 0

4

t(sec) 8

–4 –8

Fig. 4 Time histories of roll motion of the catamaran.

As another measurement of the roll damping, a forced roll motion test is also carried out. In the analysis, one degree of freedom of rolling motion equation is used, in which nonlinear terms are replaced by equivalent linear terms

( I 44  a 44 )  B44  C 44  M R ,

(1)

where (I44 + a44), B44, C44 and MR denote the inartia term, damping term, restoring term and roll excitation moment, respectively. In this test, model is forced to sinusoidal rolling around the center of gravity as

   a sin t ,

(2)

A Study on the Characteristics of Roll Damping of Multi-Hull Vessels 491

The measured roll moment is represented with Fourier series. Using the Fourier coefficient of the fundamental period for measured MR, the amplitude MRF in phase of roll angular velocity of measured MR is obtained, and B44 is written as follows

B44 

M RF

a

,

(3)

where ω and a are roll circular frequency and amplitude. The roll damping coefficient B44 can be represented by Bertin’s N-coefficient on the condition that the energy losses of them in one period are equal,

B44  2 g

GM  a



N,

(4)

where  is displacemental volume, and the unit of a is degree. And B44 is nondimensionalized as follow B44 Bˆ 44  B 2

B , 2g

(5)

where B is breadth of vessel.

a=2(deg)

a=5(deg)

a=3(deg)

a=5(deg)

exposed

a=6(deg)

a=10(deg)

a=10(deg)

a=16(deg)

Fig. 5 Relative position of the heeling model to the water surface.

492 T. Katayama et al.

In Fig. 5, the relative positions of the models to the water surface when the model has heel angle. In the free roll decay test, the initial heel angles, where hull of model is not exposed the air, are selected. And the experimental conditions in forced roll motion test are shown in Table 3. Table 3 Experimental conditions in forced roll motion test. -

trimaran

center height of roll motion (m)

0.156

Catamaran 0.144

trim angle (deg)

0

0

forced rolling period: T (sec)

0.4, 0.8, 1.04, 1.3

0.3, 0.46, 0.7, 1.0

roll amplitude: a (deg)

2, 5, 10

3, 5, 10

In Figs. 6 ~ 9, B44 and N-coefficient for two models are shown. From the results of N-coefficient for both models in Figs. 6 and 7, it is confirmed that their roll damping are sensuously large in comparison with it for common mono-hull vessels and the N10 is about 0.08. On the other hand, from the results of B44, the effect of amplitude is observed. And, from Figs. 8 and 9, the effect of roll period is also clearly found. On the comparisons among different tests and analysis, quantitatively slightly different results are obtained. The results for trimaran show that the results of the first swing of the free roll decay test indicate lager value than those of the other swing of the same test and the forced roll motion test. For the results for catamaran, the same N coefficient free roll decay test(first swing) free roll decay test(2nd～5th swings)

0.4

forced roll motion test Wave making component +Eddy making component Wave making component Main hull roll damping

0.2

0 0

5

10 15 Roll amplitude(deg)

5

10 15 Roll amplitude(deg)

0.06 B44 hat 0.04 0.02 0 0

Fig. 6 Roll damping of the trimaran.

A Study on the Characteristics of Roll Damping of Multi-Hull Vessels 493

tendency is also observed. As a reason of this, it is supposed that the free surface conditions in these tests are different and there is the disturbed water surface by previous swing excepting the results of the first swing of free roll decay test. N coefficient

free roll decay test(first swing) free roll decay test(2nd～5th swings)

0.4

forced roll motion test Wave making component +Eddy making component Wave making component

0.2

0 0

5

10 15 Roll amplitude(deg)

5

10 15 Roll amplitude(deg)

0.06 B44hat 0.04 0.02 0 0

Fig. 7 Roll damping of the catamaran.

Forced roll motion test amplitude(deg) :2 :5 : 10

0.06 B44hat

: Wave making component Wave making component +Eddy making component amplitude(deg) :2 :5

0.04 0.02 0 0

0.4

0.8

1.2 roll period(sec)

Fig. 8 Roll damping of the trimaran.

: 10

494 T. Katayama et al. Forced roll motion test amplitude(deg) :3 :5 : 10 : Wave making component Wave making component +Eddy making component amplitude(deg) :3 :5

: 10

0.06 B44hat 0.04 0.02 0 0

0.4

0.8

1.2 roll period(sec)

Fig. 9 Roll damping of the catamaran.

As reasons for the effect of amplitude on B44, the nonlinearity of wave component caused by large amplitude motion, the interference of wave among hulls and the viscosity effects are occurred to. In Figs. 8 and 9, the effect is also clearly observed for small amplitude. Therefore the viscosity component is considered in this study. Fig. 10 shows significant roll damping components acting on catamaran. One of them is the wave making component BW which is created by the almost vertically oscillating demi-hull. This component can be roughly estimated by using the heave potential damping of demi-hull B33 as following equation BW   BW  a  2l B 33 l a  2l 2 B 33 ,

(6)

where l is the distance of demi-hull from the centre line shown in Fig. 10. It is noted that the B43 is not include the interference of wave among hulls, in this study. Another component is the eddy making component and this component acts on the demi-hull moving upward shown in Fig. 10. This component can be expressed by using drag coefficient CDA

1 SC DA l a 2 2 8 a  l2 SC DA  a  , 3 8 a  l2 SC DA 3

B E  

(7)

where  is density of fluid, S is hull projected area from below. In this study, CDA is obtained from the data in Fig. 11 and following equation

A Study on the Characteristics of Roll Damping of Multi-Hull Vessels 495

C DA  C D  C DF ,

(8)

In Fig. 11, CD is the total drag coefficient of a 2-D section for a c/t, and CDF and CDA are the drag coefficients acting on forward-body and aft-body of the 2-D section, respectively. And CDF is the drag coefficient at c/t=∞. Moreover, CDA is including roughly KC number effects by using the ratio of CD for a KC number shown in Fig. 12. In this study, KC number is assumed the following equation KC 

2l a  2l a d   Rn  ,  b  

(9)

where b is the breadth of cross section of demi-hull. For catamaran, the total roll damping is calculated by adding these two components. On the other hand, for trimaran, the total roll damping is calculated by adding these two components and the main-hull’s roll damping calculated by Ikeda’s prediction method1). In Figs. 6 ~ 9, the estimated B44 is also shown as lines. The estimation can express the effects of roll amplitude and period for the trimaran, however, can not always for the catamaran. As one of reasons for the difference, it is supposed that the interference of waves among hulls is significant for the catamaran.

Fig. 10 Roll damping components for catamaran

Fig. 11 Drag coefficients of 2-D section with several aspect ratio c to t. In this study, c is draft and t is breadth of demi-hull. (S.F. Hoerner, 1965)

496 T. Katayama et al.

Fig. 12 KC number‘s effects of drag coefficient for cylinder. (Kuelegan G.H., Carpenter L.H., 1958)

4 Characteristics of Roll Damping with Forward Speed In order to investigate the characteristic of the roll damping with forward speed, the forced roll motion test is carried out. In this test, the cases of forward speed are decided by based on the maximum speed of some real multi-hull high-speed craft. The experimental condition is shown in Table 4. Table 4 Experimental conditions in forced roll motion test

Fn trim angle (deg) forced rolling period: T (sec) roll amplitude: a (deg)

trimaran

catamaran

0.148, 0.296

0.136, 0.272

0.443, 0.591

0.407, 0.543

0

0

0.4, 0.8

0.3, 0.46

1.04, 1.3

0.7, 1.0

2, 5, 10

3, 5, 10

In Figs. 13 and 14, B44 for two models are shown, and their forced roll periods are their roll natural periods, respectively. For the results of the trimaran in Fig. 13, the effect of forward speed on B44 is small. On the other hand, for the catamaran in Fig. 14, the effect is significant. In Figs. 15 and 16, the effects of roll period on B44 for two models are shown, and their Fn are their service speed, respectively. The result of the trimaran in Fig. 15 shows the same tendency as the results at Fn=0. On the other hand, the result of the catamaran in Fig. 16 shows the different tendency from the results at Fn=0.

A Study on the Characteristics of Roll Damping of Multi-Hull Vessels 497

0.06 B44hat

roll amplitude(deg) :2 :5 : 10

0.04 0.02 0 0

Fn 0.2

0.4

0.6

Fig. 13 Roll damping of the trimaran at the roll natural period

0.06 B44hat

roll amplitude(deg) :3 :5 : 10

0.04 0.02 0 0

Fn 0.2

0.4

0.6

Fig. 14 Roll damping of the catamaran at the roll natural period

0.06 B44hat

roll amplitude(deg) :2 :5 : 10

0.04 0.02 0 0

0.4

0.8

1.2 roll period(sec)

Fig. 15 Roll damping of the trimaran at Fn=0.591

498 T. Katayama et al.

0.06 B44hat

roll amplitude(deg) :3 :5 : 10

0.04 0.02 0 0

0.4

0.8

1.2 roll period(sec)

Fig. 16 Roll damping of the catamaran at Fn=0.543

5 Conclusion In this study, in order to investigate the characteristics of the roll damping for multihull vessel, the roll damping of a catamaran and a trimaran is measured. Some conclusions can be remarked as follow. 1. In the result of free roll decay tests, time histories of roll motion of the catamaran damp irregularly. This is caused by the interference of waves created by two hulls. 2. The roll damping of a multi-hull has effects of roll amplitude and period. The effects for a trimaran can be created by the eddy making component created by the demi-hull moving upward. However, for a catamaran, it may be necessary to take into account the interference of waves created by hulls. 3. The effect of forward speed on the roll damping for a trimaran is small. On the other hand, effect for a catamaran is significant. 4. The characteristics of the roll damping of a catamaran is different from that of a trimaran at Fn=0.

References Ikeda Y (1984) Roll Damping. The 1th Symp of Mar Dyn Res Group 241-249 Motora S, Koyama T, Fujino M et al (1997) Dyn of Ships and Offshore Struct. S F Hoerner (1965) FLUID-DYNAMIC DRAG. Publ by the Author Kuelegan G H, Carpenter L H (1958) Forces on cylinders and plates in an oscillating fluid. J. Res Nat Bur Stand LX N 5:423-440

7 Probabilistic Assessment of Ship Capsize

Capsize Probability Analysis for a Small Container Vessel E.F.G. van Daalen*, J.J. Blok* H. Boonstra** *MARIN, Wageningen **Delft University of Technology, The Netherlands

Abstract In this paper we investigate the long term capsize probability for a small container vessel which is operated in a regular service schedule on the North Sea and the north-east part of the Atlantic Ocean. The numerical simulation techniques involved are a ship route scenario simulation method and a time domain simulation method for large amplitude ship motions. From the combined statistics of the scenario simulations and the ship motion simulations we derive the capsize probability. Finally, the effect of avoiding bad weather conditions on the short term and long term capsize probability is discussed.

1 Introduction In this paper we investigate the long term capsize probability for a small container vessel which is operated in a regular service schedule on the North Sea and the north-east part of the Atlantic Ocean. The numerical simulation techniques involved are a ship route scenario simulation method and a time domain simulation method for large amplitude ship motions. The scenario simulation tool GULLIVER was developed at MARIN as a product for clients interested in the performance of their ship(s) in service conditions, with respect to safety, economy and reliability. The large amplitude ship motion program FREDYN was developed at MARIN within the framework of the Cooperative Research Navies project. Both methods have been applied successfully in many projects and have proven their value in both a research and a commercial context. Our primary goal is to find a way to combine these two tools into a method for calculating the long term capsize probability. With GULLIVER we are able to account for the effect of involuntary speed loss due to waves, wind and current and to quantify the encountered weather conditions in terms of (long term) scatter diagrams. This part of the approach is described in Section 2. With FREDYN we are able to identify the conditions in which the ship is prone to capsize. This part of the work is described in Section 3. Combining the two gives an impression of the average short term capsize risk. The final step is to translate this short term capsize risk into a long term capsize risk, which is described in Section 4. As for the captain’s influence on the safety of the ship, Gulliver provides us some means to quantify the effect of the captain’s decisions based on the ship behaviour. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_28, © Springer Science+Business Media B.V. 2011

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502 E.F.G. van Daalen et al.

However, at the time of writing this paper, these measures are restricted to reducing speed whenever a certain criterium is exceeded; they do not (yet) include the possibility to change course during a trip. In Section 5 we will present a method to mimic course-deviations a posteriori, in order to estimate the effect of the captain’s decisions on the capsize risk.

start: set initial position

environment databases

get local environment

calculate sustained speed

RAO database

calculate behaviour

scenario

criteria satisfied ?

no

adapt sustained speed

yes update ship position

destination reached ?

no

yes stop

Fig. 1 Scenario simulation procedure

Capsize Probability Analysis for a Small Container Vessel 503

2 GULLIVER Simulations Figure 1 shows a schematic representation of the scenario simulation procedure followed in GULLIVER. At each time stage the sustained speed is calculated from the balance between the total resistance (i.e. the sum of the calm water resistance, the added wave resistance and the wind resistance) and the available thrust. Voluntary speed loss may occur whenever a certain criterium is exceeded, e.g. roll angle or relative motion. The ship position is updated and this process is repeated until the destination has been reached. We simulated a two-weekly container service between The Netherlands, the United Kingdom, the Faroer Islands and Iceland. The routes are shown in Figure 2 and the schedule is presented in Table 1. The selected ship is a small size container vessel. Table 2 lists the key characteristics. The selected loading conditions are shown in Table 3. We applied a distribution of 25%, 50% and 25% for the conditions 100%, 80% and 50% loaded respectively. The vessel’s calm water resistance and effective thrust were calculated for the 3 loading conditions using MARIN’s program DESP. The results are presented in Figure 3. Not surprisingly, the resistance increases with increasing draft. The effective thrust is plotted for the 80% loaded condition only, since it hardly depends on the draft.

Fig. 2 Ship routes

504 E.F.G. van Daalen et al. Table 1. Service schedule Port

Day

Reykjavik

Monday

Reydarfjordur

Wednesday

Klaksvík

Thursday

Immingham

Saturday

Rotterdam

Monday

Reydarfjordur

Friday

Table 2. Ship key characteristics Description

Value

Length between perpendiculars 93.80 m Maximum breadth

15.85m

Draught

6.40 m

Speed

15.5 knots

Container capacity

364 TEU

Deadweight

4,820 tons

Main engine

3,960 kW

Table 3. Loading conditions Loading condition Displacement (tons) Mean draft [m] Trim [m] KG [m] GM [m] 5567

4.87

1.00

6.13

0.85

80%

4826

4.29

1.50

6.36

0.71

50%

3715

3.40

2.00

6.06

1.48

resistance and effective thrust in calm water [kN]

100%

450 Rcalm (50% loaded)

400

Rcalm (80% loaded) Rcalm (100% loaded)

350

Teff (80% loaded)

300 250 200 150 100 50 0 0

2

4

6 8 10 12 ship speed [knots]

14

Fig. 3 Calm water resistance and effective thrust

16

18

Capsize Probability Analysis for a Small Container Vessel 505

The vessel’s added resistance in waves was calculated in terms of quadratic transfer functions for the 3 loading conditions listed in Table 3. For each loading condition, the added resistance in head waves was calculated at service speed (15.5 knots) using our frequency domain strip theory program SHIPMO. With this result, the added wave resistance in all other speed-heading combinations was calculated using the (Jinkine and Ferdinande 1974) formulation. Figure 4 shows the added wave resistance for the 80% loaded condition at 10 and 15 knots. The added resistance is shown in the form of polar diagrams with the wave frequency along the radial axis and with the wave direction in the angular direction. From these diagrams, we observe that the added wave resistance reaches its maximum in bow quartering to head seas conditions, it increases with speed.

200

RAO DRIFT FORCE FX [kN per m2] 180 150 210 120

240 150

120

240 150

0.5 1

270 100

90 1.5 100 60

300 50

200

RAO DRIFT FORCE FX [kN per m2] 180 150 210

330

30 0

80% loaded: U=10kn

0.5 1

270

60

300 50

90 1.5

330

30 0

80% loaded: U=15kn

Fig. 4 Added wave resistance as function of wave frequency and wave direction

The resistance due to wind is calculated from the frontal area and the relative wind speed which includes the ship’s forward speed. Figure 5 shows the wind resistance coefficient as a function of the relative wind direction. Wave and wind data are obtained from ECMWF and are available for the North Atlantic, specifically between (60N, 90W) and (30N, 3E). The wave and wind parameters comprise:  Significant wave height, mean wave direction and mean wave period for both wind sea and swell;  Wind speed at 10m above mean sea level and wind direction. The parameters are given on a 1.5 deg by 1.5 deg grid with a time step of 6 hours, i.e. 4 samples per day.

506 E.F.G. van Daalen et al. 0.8

wind drag coefficient [-]

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 0

90

180 270 relative wind direction [deg]

360

Fig. 5 Wind drag coefficients

The current consists of a global (ocean-scale) current field and a local (tidal) current field. These components are superimposed and the resulting current is accounted for in the calculation of the sustained speed over ground. The results represent the encountered weather conditions and the ship behaviour during 5 years of operation in service. Figure 6 shows the distribution of the vessel’s sustained speed. The top figure shows the probability of occurrence in histogram form: For instance, in about 47% of the operational time the sustained speed is about 15 knots. The bottom figure shows the probability of exceedance: For instance, starting at 15 knots on the horizontal (sustained speed) axis, the vertical grid line intersects the graph at about 1.5x10-1, which is the corresponding probability of exceedance. This means that in 15% of the time the sustained speed is 15 knots or higher or, equivalently, in 85% of the time the sustained speed is 15 knots or lower. Figure 7 shows the distributions of the significant wave height and the peak wave period. The range 1-5m covers about 90% of the waves. The range 8-10s covers about 50% of the waves. Figure 8 shows the distribution of the peak wave direction relative to the vessel’s course. This is a more or less uniform distribution. Figure 9 shows the distribution of the wind force. The most frequent range is 3-5 Beaufort, in about 2% of the time the wind force of Beaufort 8 or higher.

Capsize Probability Analysis for a Small Container Vessel 507 100

probability of exceedance

10-1

10-2

10-3

10-4

10-5

10-6

2

3

4

5

6

7 8 9 10 11 12 13 14 15 16 sustained speed [knots]

Fig. 6 Distribution of sustained speed 40

probability of occurrence [%]

35 30 25 20 15 10 5 0

0

5 10 significant wave height [m]

15

Fig. 7 Distributions of significant wave height (top) and peak wave direction (bottom)

508 E.F.G. van Daalen et al.

probability of occurrence [%]

20

15

10

5

0

0

5

10 peak wave period [s]

15

20

Fig. 8 Distribution of relative wave direction 180° 150°

210°

0.12 0.09

240°

120°

0.06 0.03 0

270°

90°

300°

60°

330°

30° 0°

Fig. 9 Distribution of wind force

Figure 10 shows the scatter diagram of waves encountered during 5 years of service. This scatter diagram was compared with other available scatter diagrams (NOAA, GWS) and the overall impression was good.

Capsize Probability Analysis for a Small Container Vessel 509 30

probability of occurrence [%]

25

20

15

10

5

0

0

2

4 6 wind force [BF]

8

10

Fig. 10 Encountered wave scatter diagram

3 FREDYN Simulations The purpose of the FREDYN simulations is to quantify the capsize probability for each combination of (1) loading condition, (2) speed, (3) wave direction, (4) peak wave period and (5) significant wave height, as encountered by the ship during the GULLIVER simulations. To this end, we distribute the encountered combinations (or states, see Section 5) into 5-dimensional bins and for each bin we calculate the corresponding capsize probability. The capsize probability calculation procedure is as follows: for each state bin we perform a number (say, 10) of time domain simulations, each with a different seed for the random phase distribution of the irregular wave system. In case of a high capsize risk, the number of simulations (seeds) is extended to 30. The capsize probability is then based on the number of capsizes.

510 E.F.G. van Daalen et al.

Fig. 11 Ship geometry

roll angle [deg]

wave height [m]

Figure 11 shows the vessel geometry as used in the FREDYN simulations. The hull description includes the fore castle, the coamings and the poop deck, which is needed for the calculation of the nonlinear Froude-Krylov (wave excitation) forces. 5 0 -5 0

50

100

150

200

250

150

200

250

150

200

250

time [s] 30 0 -30 -60 -90 0

50

100

yaw angle [deg]

time [s] 40 35 30 25 0

50

100 time [s]

Fig. 12 Sample of FREDYN output

Capsize Probability Analysis for a Small Container Vessel 511

Figure 12 shows an example of a simulation where the ship sails in stern quartering seas at design speed. Course deviations up to 8 degrees occur and the ship rolls at angles up to 10 degrees. Finally, the ship capsizes due to a combination of large yaw motions and loss of roll stability. Figure 13 shows a combination of results from GULLIVER, FREDYN and model tests. From GULLIVER the scatter diagram of the encountered waves is shown; the area representing 90% of the most occurring waves is boxed. From FREDYN the capsize probability is shown in color, ranging from 0% (blue) to 100% (red). Note that the capsize risk is not calculated for sea states which do not occur in the GULLIVER simulations, since their contribution to the overall capsize probability will be zero anyway. The dashed line represents the Hs/Tp2 = 0.05 wave steepness limit. On top of all this, the model test results are plotted as “S” (save, or no capsize) and “C” (capsize). There is a good correlation between the numerically predicted capsize probabilities and the model test results.

Fig. 13 Wave and capsize probability scatter diagram, results from simulations and model tests

4 Capsize Probability Calculation The long term capsize probability is defined as the probability of capsizing during a long period (say, 5 years) of operation under a certain scenario in a specified area or on a specified route. The long term capsize risk depends on many factors:  the average loading condition and the loading condition variation;  the encountered wave, wind and current conditions;  the captain’s decisions.

512 E.F.G. van Daalen et al.

The long term probability that the ship will capsize is, mathematically speaking, equal to the probability that the ship will capsize at least once during a long time. If we consider a long time range (say, 1 year) as a concatenation of a large number N of short time ranges (say, 1 hour), then we are able to express the long term capsize probability in terms of the short term capsize probability: Pcap,LT = 1 – ( 1 – Pcap,ST )N

(1)

As we have seen in Section 2, a GULLIVER simulation represents a continuous service of 5 years in which the ship makes about 130 round trips, i.e. about 780 single trips in total. Each single trip yields a registration of the ship behaviour and of the encountered waves, wind and current, at a sample rate of typically 1 hour. At this point, we introduce a composed variable, which we call state: The state is a 5-dimensional array build from the loading condition (index) LC , the sustained speed U, the wave heading μ (relative to the ship course), the significant wave height Hs and the peak wave period Tp. A GULLIVER simulation generates a state series which can be sorted into an well-chosen set of 5dimensional bins, so as to obtain the state distribution. The short term capsize probability is calculated as follows: Pcap,ST = Σ Pcap,ST(s=s*) x PST(s*)

(2)

This formula expresses the short term capsize probability as a sum of products of probabilities; for each state bin, the conditional capsize probability is calculated. This means that we will have to calculate the capsize probability under the assumption that each of the 5 ship state components fall within the state bin limits. The calculation of these conditional probabilities is done with FREDYN, as explained in Section 3.

5 Results A number of Matlab® routines were written for the processing of the GULLIVER and FREDYN results and the calculation of the short term and long term capsize risks. As already mentioned in Section 2, the scenario adopted in GULLIVER corresponds to a constant power setting of 85% of MCR. Hence, only involuntary speed loss due to adverse weather (waves, wind) is accounted for. If we follow the calculation procedure outlined in Section 4, we obtain the following values for the short term and long term capsize risks: Pcap,ST =1.4% and Pcap,LT =100%

(3)

These numbers may seem alarming, but one has to keep in mind that they correspond to a “passive captain scenario”: no matter what the conditions are, the ship will be kept on its original course at a fixed engine power setting.

Capsize Probability Analysis for a Small Container Vessel 513

A common on board procedure to survive in extreme weather conditions is to change course such that a head seas condition is attained. Clearly, this is only possible if the vessel’s steering capacity is sufficient to move away safely from a stern (quartering) or beam seas condition. Once in a head seas condition, the roll motion amplitude will decrease and the capsize risk will be reduced correspondingly. In our calculation procedure, the effect of this change-to-head-seas procedure on the overall capsize risk can be estimated roughly by selecting all states corresponding to a capsize risk higher than a certain critical level. Each selected state represents an unfavourable condition in which the captain would have decided to change course to head seas, until the conditions have improved such that the capsize risk is on or below the critical level. Therefore, for each selected state we replace the capsize risk value by the critical level value. This will reduce the short term capsize risk and, as a direct consequence, the long term capsize risk. Table 4. Effect of “change to head seas” procedure on capsize risk Pcrit,ST

fstate

ftime

Pcap,ST

Pcap,LT

10-3

8.3%

3.4%

3.7e-5

60%

10-4

8.9%

3.9%

4.3e-6

10%

10-5

9.9%

4.6%

5.6e-7

1.4%

-6

11%

6.3%

7.3e-8

0.18%

10-7

13%

7.9%

8.9e-9

0.022%

10-8

14%

9.5%

1.1e-9

0.0026%

10-9

15%

11%

1.2e-10

0.0030%

10

Table 4 summarizes the results for different values of the critical capsize level Pcrit,ST. For instance, if we set Pcrit,ST to 10-4, then in 8.9% of the states the capsize risk is too high, which corresponds to 3.9% of the service time. Changing to head seas in these states reduces the short term capsize probability Pcap,ST to 4.3x10-6 and the long term capsize probability Pcap,LT to 10%. A further reduction is achieved when the critical capsize level is lowered: for 10-7 we obtain a long term capsize probability of 0.022%, which requires the captain to take appropriate action in 7.9% of the service time. These numbers indicate the possible impact of the captain on the vessel’s safety.

6 Conclusions In this paper we have demonstrated the feasibility of a quantitative approach towards the capsize probability of a ship ni service. By combining scenario simulations with time domain large amplitude motion simulations, we obtained meaningful numbers for the short term and long term capsize probability. Instead of selecting a wave scatter diagram for a particular area and making assumptions

514 E.F.G. van Daalen et al.

on the distribution of the ship speed and heading with respect to the waves, wind and current, we simulated a large number of round trips to obtain the really encountered conditions. The capsize probability was calculated for each state, i.e. for each combination of loading condition, speed, wave direction, peak wave period and significant wave height. The results show a good correlation with the findings from model tests. However, since only one specific situation was considered, further investigations into the quality of the numerically predicted ship behavior are recommended. Specific points of interest are the autopilot model and the manoeuvring model. The influence of the captain on the short term and long term capsize probabilities was modelled in an approximative way by modification of the encountered wave scatter diagram. Applying a head seas scenario in case of significant capsize risk, the captain may be able to reduce the capsize probability drastically. It should be noted, however, that the adopted approach does not account for the effect of delays due to these scenarios and therefore the numbers are only indicative. Therefore, we recommend the further development of the GULLIVER code to incorporate a true captain’s scenario, including realistic measures as course changing and speed reduction.

Acknowledgements The research reported in this paper is part of a large study into the safety of small container ships, jointly sponsored by the Dutch Ministry of Transport, Public Works and Water Management, Directorate-General for Freight Transport and the Maritime Knowledge Centre (MKC). The MKC is a cooperation of The Delft University of Technology (TUD), The Institute for Applied Physical Research (TNO), The Royal Netherlands Navy (RNLN), and the Maritime Research Institute Netherlands (MARIN) Any opinions expressed in this paper are those of the individual authors.

References Jinkine V, Ferdinande V (1974) A method for predicting the added resistance of fast cargo ships in head waves, Int Shipbuilding Prog, Vol. 21, No. 238, pp. 149-167

Efficient Probabilistic Assessment of Intact Stability N. Themelis & K.J. Spyrou National Technical University of Athens, School of Naval Architecture and Marine Engineering, 9 Iroon Polytechneiou, Zographos, Athens 157 80, Greece

Abstract Near regularity of excitation is conducive to large amplitude responses. Moreover, higher waves tend to appear in groups (Draper 1971). These observations are the drivers of a new approach for the probabilistic assessment of intact stability. The intention is to maintain the rigour and breadth of the deterministic approach while taking fully into account the probabilistic character of the seaway. Critical wave groups are specified on the basis of deterministic analysis. A procedure is put forward for calculating the probability of encountering these wave groups. A containership’s tendency for instability in a specific North-Atlantic route is used as the showcase for demonstrating the feasibility of the approach.

1 Introduction The mechanics that govern extreme ship behaviour and could host loss of intact stability have been studied for several years in a primarily deterministic context. This analysis has improved our understanding of the various types of ship instability, sometimes supplying also simple criteria to guide design. However, none could disregard that potentially destabilizing environmental excitations are of a probabilistic nature. A method to interface the deterministic analyses of ship dynamics with wind/wave models and statistics has been outlined recently, exploiting the groupiness characteristic of high waves and the idea that the probability of occurrence of a certain instability could be assumed as equal to the probability of encountering the critical (or “worse”) wave groups that generate the instability (Spyrou, 2005; Spyrou and Themelis 2005). The method builds upon certain ideas that have been around in the field of ship stability for a number of years: Tikka and Paulling (1990) for example, discussed the calculation of the probability of encountering a high run of waves in astern seas and determined combinations of ship’s speed and heading that could favour such an encounter. DeKat (1994) pointed out the M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_29, © Springer Science+Business Media B.V. 2011

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516 N. Themelis and K.J. Spyrou

importance of considering wave groups and he referred to the use of joint distributions of wave length and steepness in order to determine such wave groups, given a significant wave height and period. Myrhaug et al. (1999) investigated synchronous rolling using joint distributions of successive wave periods, targeting essentially the encounter of a wave group having critical period. In the current paper the new methodology is applied for an existing large containership. Hence, probabilities of “instability” due to beam-sea resonance, parametric rolling and pure-loss of stability are calculated with reference to a specific North-Atlantic route during specific days of bad weather, obtained on the basis of a hindcast study. The result is presented in the form of time that the ship is exposed to critical excitation, normalized with respect to the duration of the voyage.

2 Basics of the New Methodology Given a ship, the methodology can be deployed for “short” or “long-term” assessments, depending on the intended period of exposure to the weather. In the current context, as “short-term” is meant an assessment gauging safety during a single trip. It is thus fed by the “few hours” forecast of weather parameters. Such an assessment could serve as a decision-making tool in connection with a system of departure control like the one used in Greece for passenger ships (Spyrou et al. 2004); or with other operational measures like weather routing. On the other hand, long-term assessments could be performed for a variety reasons. Probabilities of instability on a seasonal or annual basis for reference routes can be determined with obvious utility at corporal and national administration levels. Moreover, by projecting the annual statistics to the ship’s life-span, a long-term assessment could be tied to ensuring a satisfactory safety level by design. In long-term assessments, the anticipated service profile should be specified beforehand. A restricted service, referring to specific routes, should lead to a different assessment result, compared to one of unrestricted service that sets no narrow limits to the navigational area (e.g. North Atlantic). Different ship types are prone to instabilities of a different nature. In a general sense, an effective portfolio of criteria should cover against resonance phenomena (beam-sea resonance and parametric rolling in longitudinal seas); pure - loss of stability on a wave crest in following seas; instability due to breaking waves from abeam and “water-on-deck”; and finally broaching, including the so-called cumulative type. To become these criteria meaningful, norms of unsafe behaviour should be associated with each one of them. The setting of warning and failure levels per criterion and ship type has been proposed, on the basis of threshold angular and linear displacements and accelerations, referring to the safety of the ship and her cargo (Spyrou and Themelis 2005). The setting of warning level should play a cautionary role and its exceedence could be allowed with a controlled probability.

Efficient Probabilistic Assessment of Intact Stability 517

The probability of occurrence of dangerous ship motion, loosely referred-to from here on as “instability”, could be assumed as equal to the probability of encounter of the critical (or worse) wave group that gives rise to this instability. Certain instabilities entail some regularity in the excitation, i.e. the amplitude is built-up gradually; whereas others represent the outcome of the encounter of a single critical wave that, in a dynamics vocabulary, could ‘kick’ the system out of its (safe) potential well. One might think therefore to disassemble the problem into two parts: one deterministic, for deducing the specification of the critical wave group (represented by the height, period and run-length) focused purely on ship motion dynamics; and one probabilistic centered upon seaway statistics, in order to determine the probability of encounter of such a wave group. For the specification of critical wave groups one could exploit the strengths of deterministic analyses: numerical simulation tools based on detailed models; and analytical techniques capturing key system dynamics (growth of amplitude per cycle, Melnikov’s method etc.) represent the two ends of the spectrum, each having their particular strengths and weaknesses. By disassembling (methodologically) the ship dynamics part from the probabilistic treatment of the seaway, informed decisions based on more than one tools are enabled; i.e. results from different simulation codes or combinations of simulation with independent stability analysis techniques can be utilised. A flow-chart of the methodology is presented in Fig. 1. A significant element is the specification of the complete set of critical wave groups. As these are based on transient ship response, it is natural to wonder whether their specification depends significantly on initial conditions. The simplest scenario is of course to assume that the ship is initially upright with zero roll velocity and in vertical equilibrium condition when approached by the wave group. This idea has some background from ship roll dynamics investigations (Rainey and Thomson 1991). However, one might prefer to treat as probabilistic quantity the initial state, integrating it with the subsequent process of elicitation of critical wave groups. A short analysis about the effect of initial conditions is presented in the Appendix. The groupiness characteristic of high waves coupled with the magnification of safety threats due to near regularity of the excitation, allow the conceptual bridging of the deterministic and probabilistic viewpoints. Given a distribution of “weather nodes” (along the route or in a wider navigational area), individual probabilities per node are calculated. In summing up these probabilities, the duration of ship stay in the influence area of each node, as well as the principal direction of wave field encounter, are taken into account (Fig. 2). To facilitate decision-making, the probabilistic treatment should be embedded upon a risk-based platform of assessment. In implementing this however, a proper currency for quantifying the consequences needs to be established.

518 N. Themelis and K.J. Spyrou Type of assessment

• Short Term • Long Term

Type of service

• Restricted • Unrestricted •Parametric rolling in longitudinal seas •Pure – loss of stability • Resonant rolling in beam wind/ waves Water – on – deck •Wave breaking on side •Broaching (direct/cumulative)

Portfolio of stability criteria

• Ship • Cargo

Norms of unsafe behaviour

Specification of critical wave groups

Calculation of the probability to encounter the critical wave groups

• Numerical simulations • Analytical methods

=

Probability of “instability”

Fig. 1 Flow chart of proposed methodology Route of the ship

weather node Influence area of weather node

Navigational area of interest

Fig. 2 Weather nodes with their areas of influence

Efficient Probabilistic Assessment of Intact Stability 519

3 Probability of Wave Groups For a brief introduction to wave groups see for example Medina and Hudspeth (1990) and Masson and Chandler (1993). Application of the proposed methodology entails calculation of the probability of encounter of wave groups with successive periods in the critical range (related to the ship’s roll natural period), given run length; and heights consistently above the critical height as determined from the deterministic analysis. A variety of parametric models might be useful in this respect. Bivariate distributions of wave height and period have been proposed by Longuet – Higgins (1975, 1983); Cavanié et al. (1976); Tayfun (1993); and others. The probability density function (pdf) proposed by Tayfun (1993) is:

 1  f  h,   CT h  1  e 32  h2   2

2 1  4 h 4     / h         2 1     / h    

where:  / h  1   2 (1   2 ) 3/ 2 and  / h 

CT 

2 2 2 4 (1   ) / h

(1)

2 8h(1   2 )

(2)

(3)

 / h and  / h are the conditional mean and standard deviation, CT a H T normalizing factor, h  ,  the dimensionless wave height and Hrms Tm m th period, Tm  2 0 the mean spectral period, m j the j ordinary moment of m1 wave spectrum. According to Longuet – Higgins (1975) the spectral bandwidth v is given by: v

m2 m0 m12

1

(4)

The parameter  depends on Tm and the frequency spectrum. According to (Stansell et al 2002) it can be calculated as follows:

520 N. Themelis and K.J. Spyrou

2 

1 m0



 S ( )e

it

d  , t  Tm

(5)

0

Tayfun (1993) approximated the conditional distribution of successive wave periods given the wave height on the basis of the Gaussian distribution for one wave period. Wist et al. (2004) noted that, for three wave periods at least, the multivariate Gaussian distribution is a satisfactory model of the conditional T

distribution. Their conditional pdf of p successive wave periods T  T1 ,..., Tp  , given that each wave height in the group exceeds the threshold H cr , is given by equation (6). A general formula for variable threshold

H cr ,i per crest in the group

could also be derived.

f T/H  τ hi  hcr  

e





1 τ μ / hcr 2

 2 



T

p 2



1/ hcr τ μ / hcr

 / hcr

 (6)

12

where the covariance matrix is given by:

 / hcr

  2/ hcr Cov T1 , T2 / H cr  Cov T1 , T p / H cr      ...    2 ...   / hcr  Cov T1 , T p / H cr  

(7)

and Cov Ti , T j / H cr   ij2/ h . The mean values μ / h and the standard cr cr  deviations  / hcr are calculated from equations (2).Assuming the Markov chain property for the waves, the correlation coefficients calculated by: 1 j  12j 1 The correlation coefficient 12 of two successive wave heights is calculated as

follows (Stansell et al 2002). E 

, K  

are complete elliptic integrals of the

first and second kind, respectively. The above, rather lengthy, procedure is essential because for resonance phenomena, one needs to determine the probability, a specified number of successive wave periods to lie in some narrow interval 1 , 2  , given that the corresponding wave heights exceed the critical level hcr .

Efficient Probabilistic Assessment of Intact Stability 521

12 



E    1   2 1





K   2



4

 4 

 2 4 6      16  4  16 64 



(8)

4 Application Basic data concerning the assessed containership are shown in Table 1. Unfortunately, no information of her bilge keels was available, so a bare hull was only considered. The selected route between Hamburg and New York is a rather popular one for containerships, although here the unusual choice of going over the Shetlands has been made. The length of the route was about 3422.86 nautical miles, covered in 142.82 hours if the service speed of 24 kn could be maintained. In total 28 “weather nodes” were cast along this route, where part of it is shown in Fig. 3. Their density was decided by ensuring that wave characteristics, in terms of significant wave height H S and peak period TP , do not change significantly while the ship is still inside the influence area of any particular node. Wave hindcast data for the North Atlantic referring to the period between 1990 and 1999 has been consulted (Behrens 2006). As the intention was to perform a “short-term” assessment, the data was searched in order to find specific days of bad weather at places near to the ship’s route. It was found that waves of significant height exceeding 10 m should have been realised in some part of the route, in the period between 13/01/1991 and 18/01/1991. The variation of H S and

TP in the vicinity of the defined route, are presented in Figs. 4 and Fig.5. Table 1. Ship data LBP (m)

264.4 m

B (m)

40

T0 (s)

39.12

D (m)

24.3

KG(m)

18,79

Cb

0.600

Td (m)

13.97

GM (m)

0.61

VS (kn)

24

TEU

5048

The percentage of the ship’s scaled time of exposure to beam, head and following seas per node had then to be worked out on the basis of ship heading (as defined by the route) and the distribution of mean direction of the local wave field around each node, weighted by the time spent in its area of influence (Fig. 6).

522 N. Themelis and K.J. Spyrou

Fig. 3 Part of route showing weather nodes and their areas of influence

4.1 Norms of Unsafe Response: “Capsize” Threshold and Shift- of-Cargo Threshold These norms are defined respectively as: a critical roll angle for the ship and a critical acceleration for the cargo.To determine a roll angle as threshold of “capsize” the principle of the weather criterion was adopted. The critical angle should then be the minor of: the angle of vanishing stability c  520 ; the flooding angle  f and the prescribed value a  500 . The flooding angle was assumed to correspond to the least transverse inclination (with submerged volume preserved) at which the highest point of a hatch coaming is immersed. According to the drawings, hatch coamings rise 1.7 m above the deck. A rendered view of the hull (with some key deck structures) inclined to that angle is shown schematically in Fig. 7. From hydrostatic calculations it should be  f  350 . H S [m] 12 10 8 6 4 2 0 1

3

5

7

9

11

13

15

17

19

21

23

weather node

Fig. 4 Variation of significant wave height along the route

25

27

Efficient Probabilistic Assessment of Intact Stability 523 TP [s] 17.5 15.5 13.5 11.5 9.5 7.5 5.5 1

3

5

7

9

11

13

15

17

19

21

23

25

27

weather node

Fig. 5 Variation of peak period

On the other hand, the shift-of-cargo threshold was identified by the critical transverse acceleration that could result in damage of the containers’ lashings. The acceleration due to rolling motion has been estimated for tiers of 4, 5 and 6 TEUs, placed on the deck. The relevant calculations have been carried out according to the Cargo Securing Manual (DNV 2002). Specifically, the sufficiency of lashings’ in terms of transverse sliding and tipping of the tier has been checked. More details for the relevant calculations and data can be found in Themelis and Spyrou (2008). The most critical condition was identified to correspond to transverse sliding for a tier of 6 TEUs and the specific value of this critical acceleration was calculated as a y  4.02 m/s 2 . Percentage of exposure 7.00% 6.00% 5.00%

Following-Seas

4.00%

Head-Seas

3.00%

Beam-Seas

2.00% 1.00% 0.00% 1

3

5

7

9

11

13 15 17 weather node

19

21

Fig.6 Exposure to beam, head and following seas

23

25

27

524 N. Themelis and K.J. Spyrou

Fig. 7 Critical heel angle for immersion of hatch coaming

4.2 Critical Waves Critical wave groups have been specified for the following types of instability: a) beam-sea resonance, b) parametric rolling in longitudinal seas; and c) pure-loss of stability. Their characteristics were found from numerical simulations, using the well-known panel code SWAN2 (2002). For beam-seas resonance, in order to determine the critical combinations of wave height, period and group run length that could generate exceedence of a stability norm, deterministic numerical simulations have been carried out. The ship was assumed with no initial rolling. The practical range of wave periods that could be realised in the specific sea region has been scanned. Fig. 8 presents the key characteristics of identified critical wave groups referring to ship responses. We have to remind that no bilge keels have been considered Nonlinear FroudeKrylov force has been included in the calculation. For the assumed speed of VS  24 kn the ship could be prone to head-seas parametric rolling. Specifically, we target the principal mode of parametric instability can be realized, however the required wavelengths are extremely long. An uncertain initial roll disturbance range was considered in order to realize growth of roll amplitude. Up to a sequence of 8 wave encounters has been examined; because having more waves in a group is of truly negligible probability when the waves are high. The characteristics of critical wave groups were determined from repetitive numerical simulations, taking record whenever roll growth up to the critical norm was realised, within the allowed number of wave encounters. An example is shown in Fig. 9. The variation of critical height and run length in the vicinity of exact principal resonance can be seen in Fig. 10.

Efficient Probabilistic Assessment of Intact Stability 525 18

16 n=2

14

Hcr (m) n=3

12 n=4

n=5

10

n=6

8 9.5

12.5

15.5

18.5

T(s)

Fig. 8 Critical wave groups with reference to the limiting roll angle for beam seas (ship) j (deg)

40 30 20 10 0 0

10

20

30

40

50

60

70

80

90

100

-10 -20 -30 -40

t (s)

Fig. 9 Parametric roll growth in head waves 80 % off principal resonance and for H=14 m 20

α = 0.7

15

0.8 0.9

Hcr (m)

1

10

5

3

4

5

6

7

8

n (number of waves)

Fig. 10 Required wave heights from an initial roll disturbance “around” 4.50 (ship)

526 N. Themelis and K.J. Spyrou

As the panel code is not suitable for use at very low frequencies of encounter, an analytical criterion of pure loss of stability was used. The key idea exploited was that the critical fluctuation of GZ could be identified on the basis of the following condition: the time of experiencing negative restoring in the vicinity of a crest should be, at least, equal to the time that is necessary for developing capsizal inclination, assuming an initial roll disturbance. The respective critical wave heights were calculated taking into account the restoring variation on the waves using Maxsurf. However their values were extremely high and so had very small probability to be met.

5 Calculation of Probabilities In the presentation of the results we have introduced the concept of “critical time ratio”. Rather than using probability figures that refer essentially to number of wave encounters irrespectively of their periods, we considered as more meaningful to convert probabilities of encountering wave groups to the scaled time ratio of t T experiencing these wave groups according to the formula ti  i  Pi i where ttot Tm

Pi is the calculated probability of a wave group i having wave period around the value Ti ; ttot is the duration of the part of the voyage inside the rectangle of the considered node and Tm is the mean spectral period associated with the same node. The obtained results the scaled critical time per node for each type of instability along the route, thus present one can easily deduce which type of instability is more likely to occur at any specific stage of the journey, hence providing useful information for weather routeing. Fig. 11 shows “critical time ratio” for beam seas resonance instability, both for the ship and cargo norm, while in Fig. 12 are overlaid the obtained “critical time ratio” curves for her cargo. Due to the low probability values for pure loss of stability, the respective curve is not presented in the diagram, while for the same reason only a part of the relevant curve for head seas parametric rolling is shown. In Table 2 are presented the total probabilities and critical time ratios for the complete voyage taking into account the percentage of exposure to beam, head and following seas. It could be perhaps enlightening if we presented an example of the calculation of the probability of “instability” with reference to a specific part of the route. Take for example node 5 whereabouts the time spent is 3.83 hr. The sea state is characterized by H S  7.6m and TP  16.4 s . The probability of critical waves for that node and for cargo shifting is 7.23 x 10-5. For the assumed speed, the mean encounter wave period is 12.66 s and the number of waves encountered by the ship in one hour should be 13777(s)/12.66(s)=1089. Hence the probability of instability for this time of exposure should be 7.88% which is quite a high value

Efficient Probabilistic Assessment of Intact Stability 527

(one recalls here of course that the bilge-keels were not considered, which would reduce this number substantially). Beam seas resonance 1.0E+00

cargo

ti

1.0E-05 ship

1.0E-10 1

3

5

7

9

11

13

15

17

19

21

23

25

27

nodes

Fig. 11 Collective view of “critical time ratio” diagrams for beam seas resonance mode of instability (ship and cargo)

ti

cargo

1.0E+00 beam seas resonance 1.0E-05 1.0E-10 head seas parametric rolling

1.0E-15 1.0E-20 1

3

5

7

9

11

13

15

17

19

21

23

25

27

nodes

Fig. 12 Collective view of “critical time ratio” diagrams for cargo (upper) Table 2. Summed probability of instability and associated “critical times”

Ship: (  > 35 ) 0

2

Cargo: ( a y > 4 m/s )

Pi

ti

3.33E-04

3.95E-05

9.38E-04

1.11E-04

6 Concluding Remarks Risk-based approaches gradually pervade all facets of naval architecture and almost naturally, the question of a solid and yet practical probabilistic approach is being heard more loudly than ever. An effort to fill this gap has been presented in the

528 N. Themelis and K.J. Spyrou

current paper. Practical application of the new probabilistic methodology of ship stability assessment has been presented for a modern containership. A survey of literature on probabilistic intact stability assessment would reveal that other existing methods are: either focusing narrowly on the problem (e.g. study of beam-sea resonance or parametric rolling only, with dubious assumptions regarding the nature of excitation and/or type of response); or, due to the rush for practicality, the widening of scope is accompanied by paying little attention to the true dynamical nature of the phenomena. As a result, until recently only limited confidence could be gained that stability standards could be based on the state-ofthe-art of probabilistic approaches. The proposed methodology gives proper account for system dynamics without disregarding the probabilistic nature of the wave field. Wind effects could be handled in a similar spirit and their incorporation represents a future goal.

Acknowledgements The research work has been supported by the integrated project SAFEDOR funded by the European Commission.

References Behrens A (2006) Environmental data: Inventory and new data sets. Safedor S.P. 2.3.2 Deliverable Report Cavanié A, Arhan M and Ezrtaty R (1976) A statistical relationship between individual heights and periods of storm waves. Proc, Conf on Behaviour of Offshore Structures, Trondheim: 354-360 De Kat J O (1994) Irregular waves and their influence on extreme ship motions. Report No. 208883-OP1 – ONR’94. Det Norske Veritas (2002) Cargo Securing Manual, Model manual. Version 3.1, Oslo. Draper L (1971) Severe wave conditions at sea. J of ‘of Navigation 24, 3, pp. 273-277 Hasselmann, K (1973) Measurements of wind - wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Dt. Hydrogr. Z. Reihe A (8) 12 Longuet-Higgins M S (1975) On the joint distribution of the periods and amplitudes of sea waves. J of Geophys Res 80:6778–6789 Longuet-Higgins MS (1983) On the joint distribution of wave periods and amplitudes in a random wave field. Proc of the Royal Soc A 389:241–258 Masson D, Chandler P (1993) Wave groups: a closer look at spectral methods. Coastal Eng 20:249–275 Medina JR, Hudspeth RT (1990) A review of the analyses of ocean wave groups. Coastal Eng 14: 515 – 542 Myrhaug D, Dahle EA, Rue H, Slaattelid OH (1999) Statistics of successive wave periods with application to rolling of ships. Int Shipbuilding Prog 47:253-266 Rainey RCT, Thompson JMT (1991) Transient capsize diagram – a new method of quantifying stab in waves. J of Ship Research 41:58–62

Efficient Probabilistic Assessment of Intact Stability 529 Spyrou KJ (2005) Design criteria for parametric rolling. Oceanic Eng Int 9:11-27 Spyrou KJ, Politis K, Loukakis T, Grigoropoulos G (2004) Towards a risk – based system for the departure control of passengers ships in rough weather in Greece. Proc 2nd Int Maritime Conf on Design for Safety, Sakai, Japan:255 – 261. Spyrou KJ, Themelis N (2005) Probabilistic assessment of intact stability. Proc of 8th Int Ship Stab Workshop, Istanbul, Oct. 6-7. Stansell P, Wolfram J, Linfoot B (2002) Statistics of wave groups measured in the northern North Sea; comparisons between time series and spectral predictions. Appl Ocean Res 24:91–106 SWAN2 (2002) Ship Flow Simulation in Calm Water and in Waves. Boston Marine Consulting, User Manual Tayfun, MA (1993) Joint distributions of large wave heights and associated periods. J of Waterway, Port, Coastal and Ocean Eng 119:261–273 Themelis N, Spyrou K (2005) A coupled heave-sway-roll model for the analysis of large amplitude ship rolling and capsize of ships in beam seas on the basis of a nonlinear dynamics approach. Proc 16th Int Conf on Hydrodyn in Ship Design, Gdansk, Poland, Sept. 7 – 11 Themelis N, Spyrou K (2008) Probabilistic assessment of ship stability based on the concept of critical wave groups. Proc of 10th Int Ship Stab Workshop, Daejeon, Korea 2008 Tikka KK, Paulling JR (1990) Prediction of critical wave conditions for extreme vessel response in random seas. Proc 4th Int Conf on Stab of Ships and Ocean Veh STAB’90, Naples: 386–394 Wist H, Myrhaug D, Rue H (2004) Statistical properties of successive wave heights and successive wave periods. Appl Ocean Res 26:114–136

Appendix The process is divided in two sub-processes; the first one considers the ship’s state before the encounter of the wave group. This sub-process provides the (probabilistic) initial condition for the second sub-process when the wave group excitation is applied. The probability to lie at a certain neighbourhood in system’s state space is apparently connected to the weather. Any given state of a dynamical system can be regarded as initial condition for any state zi that belongs to z0’s later time evolution. Thus, a system’s safe basin could be realistically taken as the appropriate continuum of initial conditions that should be targeted for probabilistic treatment. Given a ship and a sea state, one could sensibly assume that each infinitesimal sub-region within it could be associated with a probability of being “visited” at the moment when the wave group excitation is applied. For a linear oscillator, lines of constant potential-plus-kinematic energy are approximately cyclic. For the first sub-process a linear oscillator could be a rational choice of generic model as, for non-critical conditions, the ship rolling motion is not expected to deviate too far from the upright equilibrium. Thus a grid of initial conditions could be created, spanning up to the energy level represented by the critical inclination zcr . For each initial state z ij , i.e. a point of the grid, the critical forcing can be identified for the assumed oscillator, in terms of characteristics of critical wave groups (number of waves, height and period). Then, given the assumption of a



“Gaussian sea”, the probability P Cij z ij



to encounter these wave groups is

530 N. Themelis and K.J. Spyrou

easily determined according to the methodology described earlier in this paper. On the other hand, the ship’s state at the moment of wave group encounter could be derived by a multivariate pdf f  z  accounting for ship state and the relative position to the wave group when the encounter begins. For the purpose of this preliminary analysis the joint pdf of roll angle-roll velocity response has been considered. The corresponding mathematical formulation for the calculation of the probability of initial conditions can be found in Themelis and Spyrou (2008). Assuming independence, the total probability can be derived by multiplying the



probability of wave group P Cij z ij

 

P z ij



with the probability of the initial state

and then summing up for all initial states. In our investigation we have

considered various values of the ratio r / zcr where r is the radius of the circle of initial conditions. In Fig. 13 is shown a characteristic result of this investigation. It seems that the assumption of initial upright state influences the actual probability, yet the effect does not change the order of magnitude which, in a risk assessment framework and given the rarity of capsize, is the decisive factor. r/zcr = 0.4, TP = 15s

P 1.00E - 02

1.00E - 04

1.00E - 06

1.00E - 08

1.00E - 10

probabilistic upright

1.00E - 12 4

6

8

10

HS(m)

Fig. 13 Total probabilities for the “quiescent” and for the joint case ( r / zcr  0.4 ) varying H S

Probability of Capsizing in Beam Seas with Piecewise Linear Stochastic GZ Curve Vadim Belenky*, Arthur M. Reed*, Kenneth M. Weems** *Naval Surface Warfare Center Carderock Division (NSWCCD) - David Taylor Model Basin, **Science Application International Corporation (SAIC) Abstract The probability of capsizing for a dynamical system with time-varying piecewise linear stiffness is presented. The simplest case is considered, in which only the angle of the maximum of the restoring curve is changing. These changes are assumed to be dependent on wave excitation; such a system can be considered as a primitive model of a ship in beam seas, where all changes in stability are caused by heave motions. A split-time approach is used, in which capsizing is considered as a sequence of two random events: upcrossing through a certain threshold (nonrare problem) and capsizing after upcrossing (rare problem). To reflect the timevarying stability, a critical roll rate is introduced as a stochastic process defined at any instant of time. Capsizing is then associated with an upcrossing when the instantaneous roll rate exceeds the critical roll rate defined for the instant of upcrossing. A self-consistency check of the method, in which a statistical frequency of capsizing was obtained by time-domain evaluation of the response of the piecewise linear dynamical system and favorably compared with the theoretical prediction is described.

1 Introduction The calculation of the probability of capsizing for an intact ship in irregular seas is a formidable task, first of all because capsizing is an extremely rare event. Capsizing is also an extremely nonlinear phenomenon; it is, essentially, a transition from ship motions near a stable equilibrium to another stable equilibrium. The combination of nonlinearity and rarity severely limits the set of available methods that can be applied to capsizing assessment. A number of methods are available to treat rare events: in fact, the entire statistics of extremes is essentially focused on rarity (see for example Gumbel, 1962). These methods are based on the asymptotic properties of the tails of probability distributions and essentially rely on an extrapolation of the observed data.

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_30, © Springer Science+Business Media B.V. 2011

531

532 V. Belenky et al.

The behavior of a nonlinear dynamical system under deterministic excitation has been addressed in a quite comprehensive manner (see for example Guckenheimer and Holms, 1983). The essence of the nonlinearity of roll motion is that the physics of the dynamical system changes with roll, leading to phenomena that are impossible for a linear system. The nonlinear qualities of roll motion have been the subject of study within the stability community for decades making, and an important part of program of the STAB conferences since 1975. The change of the physical properties of the dynamical system makes the direct application of statistical extrapolation difficult as the probability may change along with the physics. This positions the probability of capsizing among the very special problems of stochastic dynamics — a very wide discipline stretching from mathematics (Arnold, 1998) to applied mechanics (Roberts and Spanos, 2003). As it is the combination of severe nonlinearity and extreme rarity that makes the problem so difficult, it seems natural to separate it in two problems and consider dynamics separately from probability. This approach was used by Themelis & Spyrou (2007) and Umeda, et al. (2007) for different scenarios of capsizing. The complex nonlinear behavior was considered with a sinusoidal wave or a deterministic wave group and probability of encounter was then evaluated using oceanographic statistics. Further development of this approach seems quite promising, especially with the consideration of its application with model tests (Bassler, et al., 2009).

GZ

v

 

 

f  () m0 Range 0 Range 1

m1 Range 2

Fig. 1. Phase plane topology and piecewise linear stiffness (Belenky, 1993)

The piecewise linear approach (Belenky, 1993; Paroka, et al., 2006; Paroka and Umeda 2006) represents another way to separate the problem into two. Since the change of physics presents the most significant problem, the separation of the problem comes at the points where the physics changes. These are the peaks of the roll righting arm (GZ) curve, which plays the role of the stiffness of the dynamical

Probability of Capsizing in Beam Seas with Piecewise Linear Stochastic GZ Curve 533

system. With even a linear approximation for the GZ curve in each range, it is possible to preserve the topology of phase plane (see Fig. 1) and therefore to reflect the most important physical properties of roll motion. Once the problem is separated into two sub-problems, different solutions can be built for each of the ranges. These two solutions are connected at the separation point through initial conditions. The main advantage of the piecewise linear method for the problem of ship roll is that closed-form solutions can be presented for both ranges (Belenky, 1993). It should be noted, however, that general problem with a piecewise linear stiffness term does not necessarily have a closed-form solution. While the ordinary differential equation for roll with piecewise linear stiffness has been shown to capture the key physical phenomena of nonlinear roll motion including capsizing, it is hardly a complete model for a ship rolling in waves. However, the piecewise linear system provides a theoretical model for studying nonlinear roll and building a bridge to a practical analysis method involving more complete models of roll. With the development of sophisticated methods and tools for numerical simulations of ship motions (Beck and Reed, 2001), the numerical analog of the piecewise linear method seems to be the logical next step. The split-time method described by Belenky, et al. (2008) is just such a method. It is a direct generalization of piecewise linear method allowing the application of hydrodynamic simulation codes instead of linear ordinary differential equations. The key elements of method are illustrated in Fig. 2. 

 04

 03

Threshold



 01

Upcrossing not leading to capsizing

 02 t

Non-rare problem

Rare problem

Capsized equilibrium

Threshold

Upcrossing leading to capsizing

t

Fig. 2. Summary of time-split method: separation principle and critical roll rate

As shown in Fig. 2, capsizing is considered as an upcrossing of a threshold roll angle leading to the transition to another stable equilibrium. The first part of the method is the solution of a “non-rare” problem that provides enough roll motion data for the upcrossing probability to be evaluated. The second part is the solution of a “rare” problem, which consists of a series of short simulations starting at the

534 V. Belenky et al.

threshold level, and is aimed at finding the “critical” roll rate at upcrossing that will lead to capsizing (see Fig.2 insert). The capsizing probability can then be evaluated as the probability of upcrossing with a roll rate exceeding the “critical” value. In the initial development of the piecewise linear model of roll and its numerical application as the split-time method, the separation of the problem has assumed that the stiffness (GZ curve) was constant with time. The next step in the development of an approach is to consider changes of stiffness as the ship moves in waves. In order to provide a sound theoretical background for the development of this approach, it is first desirable to consider the solution of the basic piecewise linear method if the stiffness is random.

2 Dynamical System Consider a dynamical system incorporating a piecewise linear restoring (stiffness) term with a random time-dependent decreasing part. The decreasing part, however, remains parallel to itself all the time, see Fig. 3. Such a scheme is necessary to keep a linear relation and therefore normality throughout the problem. The stiffness term, indeed, is represented as a function of two variables, the roll angle and time; a surface illustrating such a function is shown in Fig. 4. Values with zero in a subscript (e.g. m0) are related to “calm-water” terms. The time-varying decreasing part of stiffness is assumed to be dependent on the heave motion only. While this model, as a whole, hardly describes the motion of a ship, even in beam seas, it nevertheless still possesses the key characteristics of the change of stability with ship motion. f*()

mt v0 m1





m0 vt Fig. 3. Piecewise linear stiffness term with time-varying decreasing part

Probability of Capsizing in Beam Seas with Piecewise Linear Stochastic GZ Curve 535



f*(t)

t Fig. 4. Piecewise linear stiffness term with time-varying decreasing part as a surface

The value of angle of maximum of the GZ curve is assumed to have the following form.

 m (t ) 

 m0 02

k d (t )  d   bd 

(1)

Here 0 is natural frequency of roll in calm water, m0 is the initial boundary between two linear segments of the piecewise linear stiffness term (the angle of maximum of the GZ curve in calm water), d is the draft,  is the heave displacement, and kd and bd are linear fit coefficients, see Fig. 5.

Square of roll frequency

1.5 1

0.5 Draft, m 0

2

4

6

8

10

12

Fig. 5. Dependence of square of natural frequency of roll on draft for a schematic ship (see the inset)

536 V. Belenky et al.

The entire piecewise linear stiffness term is expressed as:

 |  |  m (t ) f * ()   02  k1  b1 (t ) |  |  m (t ) ,

(2)

where linear coefficients k1 and b1 are calculated as follows:

k1  

 m0 v 0   m0

b1 (t )  v 0 k1 1  k d (t )  .

(3)

The dynamical system is described by two ordinary differential equations:

  2   f * ()  f (t )  E   2   2   f (t ) .   E

(4)

Here fE and fE are stationary ergodic stochastic processes describing wave excitation. The coupling between the equations is realized only through the stiffness term in the roll equation, and there is no influence of roll on heave.

3 Non-Rare Problem The objective of the non-rare problem is the evaluation of the probability of upcrossing the threshold roll angle, which is now changing with time. Below the threshold, the equations become decoupled and the solution for roll and heave are trivial:

 (t )  (t ) 

N



ai

i 1 N



cos(i t   i  i ) (5)

ai

sin(i t   i   i ) .

i 1

Here i is the frequency of the components; ai and ai are roll and heave component amplitudes; i are random phases; and i and i are roll and heave phase shifts respectively. These components can be calculated in the frequency domain without difficulty. Similar expression can be written for the time history of the angle of the maximum of the GZ curve:

Probability of Capsizing in Beam Seas with Piecewise Linear Stochastic GZ Curve 537

 m (t )   m0 

N



mai

cos(i t   i   i )

i 1

 mai 

 m0 02

(6)

k d  ai .

As the relation between the angle of the maximum and the heave motion is assumed to be deterministic and linear, the phases of the maximum are identical to those of heave motions. To identify the transition to rare problem, it is convenient to introduce a so-called “carrier” process representing an instantaneous difference between maximum and instantaneous roll angles:

x(t )  (t )   m (t )   m0 .

(7)

The transition to the rare problem can be now associated with the upcrossing of the level of the angle of maximum restoring in calm water by the carrier process. A time history of the carrier process can be expressed in a form of the Fourier series:

x(t )   m 0 

N

x

ai

cos(i t   xi  i ) .

(8)

i 1

The amplitudes xai and phases xi of the carrier process can be evaluated trivially as the roll and angle of maximum are presented with Fourier series containing the same frequency discertization:

x ai   2mai   2ai  2 mai  ai sin(  i   i )

 xi   arctan

 ai cos  i   mai sin  i  ai sin  i   mai cos  i

.

(9)

(10)

The carrier process x(t) is differentiable; its derivative is expressed as:

x (t )  

N

x

ai i

sin(i t   xi  i ) .

(11)

i 1

The probability of upcrossing of the level m0 by the carrier process then can be calculated as (assuming applicability of Poisson flow):

538 V. Belenky et al.

PU (T )  1  exp(T ) .

(12)

With the upcrossing rate equal to:



 2 V x exp  m 0  2V Vx x 

 ,  

(13)

where the variance of the carrier, V x , and its derivative, V x , can be derived from their Fourier presentations: N

x

V x  0.5

i 1

2 ai

N

x

; V x  0.5

2 2 ai i

(14)

i 1

4 Rare Problem 4.1 Equation for Rare Problem The essence of the rare problem is the characterization of the roll motion after an upcrossing occurs and the evaluation of the probability of capsizing after upcrossing. Consider the roll equation (4) for the post-upcrossing range |  |  m (t ) after substitution of the time dependent stiffness (3):

  2  k   k  (t )  k   f (t ) 1 b 1 v0 E

k b  k1k d .

(15)

Equation (15) describes a dynamical system with a repelling force; following the usual assumption for the rare solution with a piecewise-linear system (Belenky, et al., 2008), the influence of wave excitation is neglected. The influence of the time-variant stiffness expressed by the term kb (t) will be kept in the solution.

  2  k   k  (t )  k  . 1 b 1 v0

(16)

As will be seen in the subsequent analysis, neglecting the influence of excitation no longer provides a significant simplification. Expression (16) is an ordinary linear heterogeneous differential equation with a constant coefficient, allowing a close-form solution. Its total solution consists of

Probability of Capsizing in Beam Seas with Piecewise Linear Stochastic GZ Curve 539

the general solution of the homogeneous equation and a particular solution of the original heterogeneous equation.

(t )   H (t )  G (t ) .

(17)

The solution of the homogeneous equation is:

 H (t )  A exp(1t )  B exp( 2 t ) .

(18)

The constants A and B depend on initial conditions and will be defined later; the eigenvalues are

1, 2     2  k1

(19)

The particular solution must be similar to excitation; i.e. a sum of the heave motions plus a constant:  G (t ) 

N

p

ai

cos(i t   pi   i )   v 0  p(t )   v 0 .

(20)

i 1

The components of the particular solutions are defined in the usual way for a linear equation:

p ai 

 ai k b (i2

 pi  arctan

 k1 ) 2  4 2 i2 20 i2  k1

(21)

.

The arbitrary constants then can be expressed as:

A

( 1  p 1 )   2 1   v 0  p1  1   2

B

( 1  p 1 )  1 1   v 0  p1  . 1   2

(22)

(23)

540 V. Belenky et al.

Here 1 and  1 are initial conditions while p1 and p 1 are the values of the particular solution (20) and its derivative at the initial instant t1. The initial instant is essentially the upcrossing instant, so:

1   m (t1 )

(24)

The complete general solution and its first derivative are

(t )  A exp(1t )  B exp( 2 t )  p (t )   v 0

(25)

 (t )  1 A exp(1t )   2 B exp( 2 t )  p (t )

(26)

4.2 Process of Critical Roll Rate It is now necessary to introduce the concept of the critical roll rate,  cr , which is defined as the value of the initial roll rate at the up-crossing of the first threshold  1 , which leads to crossing of the 2nd threshold and therefore capsizing. For every given instant of time and set of initial phases i, there is a pair of initial conditions,  m (t );  cr , that deterministically lead to having the roll angle reach the second threshold, m1, beyond which the ship will capsize. It can be directly concluded from this definition that the critical roll rate is a stochastic process taking different values at different instant of time; it is also different for different realizations at the same instant of time. The critical roll rate can be calculated numerically by an iterative algorithm, as described in (Belenky, et al., 2008). Alternatively, the critical roll rate can be calculated from the condition:

A(  cr )  0 .

(27)

Equation (27) can be solved using formula (22); for any instant of time, t,

 cr (t )   2  m (t )   v 0  p (t )   p (t ) .

(28)

Thus, the critical roll rate is a stochastic process defined at any instant of time. As formula (28) represents a linear combination of three normal processes, the distribution of roll rates is normal as well. The process of critical roll rate can then be presented in the form of a Fourier series. Technically, it is convenient to

Probability of Capsizing in Beam Seas with Piecewise Linear Stochastic GZ Curve 541

calculate the difference between the first two processes and then add the third one. Consider an auxiliary process w(t) defined by the following formula:

w(t )   2  m (t )   v 0  p (t )  .

(29)

This process is presented using a Fourier series as follows:

w(t )   2 ( m 0   v 0 ) 

N

w

ai

cos(i t   wi   i ) .

(30)

i 1

The amplitudes of the components of the auxiliary process w are expressed as: 2 wai   2  mai  p ai2  2 mai 1ai cos(  mi   pi )

(31)

The phase shift for the auxiliary process, w, is expressed as:  zi  arctan

 mai sin(  i )  1ai sin(  pi )

(32)

 mai cos(  i )  1ai cos(  pi )

The process of critical roll rates is now expressed as a simple sum of the auxiliary process, w, and the derivative of the particular solution, p 1 :

 cr (t )  w(t )  p (t )

(33)

The process can be presented in the form of a Fourier series

 cr (t )   2 ( m 0   v 0 ) 

N



crai

cos(i t   cri   i ) .

(34)

i 1

The components of the Fourier presentation (34) are as follows:

 crai  wai2  i2 p ai2  2wai p ai i sin(  wi   pi )

 cri  arctan

wai sin(  wi )  p ai i cos(  pi ) wai cos(  wi )  p ai i sin(  pi )

.

(35)

(36)

542 V. Belenky et al.

4.3 Process of Difference between Critical and Instantaneous Roll Rates The condition of capsizing after upcrossing can be formulated as follows: when an upcrossing occurs, the roll rate at upcrossing should exceed the critical roll rate:

 (t1 )   cr (t1 ) .

(37)

It makes sense, therefore, to consider a process of the difference between the instantaneous and critical roll rates

 d (t )   cr (t )   (t )

(38)

 (t ) , is normal as both The process of the difference, further identified as  d critical and instantaneous roll rates are normal. An assumption that upcrossing follows Poisson flow infers that upcrossings are rare; it allows the use of a particular solution of the non-rare problem, because the general solution of homogeneous equation generated after each down-crossing will not be statistically significant. Therefore,

 (t ) 

N



ai  i

cos(i t   i   i ) .

(39)

i 1

The process of the difference of rates can now be trivially presented as a Fourier series as it is already defined for both processes on the right-hand-side of equation (38):  d (t )   2 ( m 0   v 0 ) 

N

a

rdi

cos( i t   rdi   i ) .

(40)

i 1

The components of this formulation are defined as follows: 2 a rdi   crai  i2  2ai  2 crai  ai i cos(  cri   i )

 rdi  arctan

crai sin(  cri )  ai i sin(  i ) crai cos( cri )  ai i cos( i )

.

(41)

(42)

Probability of Capsizing in Beam Seas with Piecewise Linear Stochastic GZ Curve 543

4.4 Probability of Capsizing Following the formulation of the condition of capsizing after upcrossing (37), its probability can be expressed as: 0

PC|U 

 f  d cr

d

d

.

(43)



 

Here f cr  d is the distribution of the distance between critical and instantaneous roll rates at the instance of upcrossing. This distribution can be expressed as (see derivation in Appendix 1): 

f cr ( d ) 

 xf ( x   0



 , d )dx m0 , x .



(44)



f ( x  m 0 ) xf ( x )dx 0

Here f (x) and f (x ) are the probability density distributions of the carrier and its derivative. These distributions are known to be normal. The term, f ( x, x ,  ) , d

expresses a joint distribution of the carrier process, its derivative and the distance between instantaneous and critical roll rates. As marginal distributions of all these processes are normal, it is logical to assume that their joint distribution is also normal. Then their mutual dependence can be fully characterized by correlation moments. Evaluation of these correlation moments does not present any difficulties as Fourier presentations are available for all of them:





N

x

M x,  d  0.5

ai a rdi

cos(  xi   rdi ) ,

(45)

i 1





N

x

M x ,  d  0.5

ai ardi i

sin(  xi   rdi ) ,

(46)

i 1

and

M x, x   0 .

(47)

544 V. Belenky et al.

Respectively, correlation coefficients are:

rxd 









M x,  d M x,  d . ; rxd   xd  x  d

(48)

Standard deviations are:

 x  Vx ;  x  Vx ;

 d  Vd .

(49)

The variances of the carrier process and its derivative are defined by formulae (14). The variance of the distance between instantaneous and critical roll rates can be found from its Fourier presentation (40): N

a

Vd  0.5

2 rdi

.

(50)

i 1

As a formula for tri-variate normal distribution is rather cumbersome, it is more convenient to consider it as a product of marginal bi-variate and conditional distributions:

f ( x, x,  d )  f ( x | x,  d ) f ( d , x) .

(51)

The first term in (51) is a conditional distribution of the derivative of the carrier if two other processes have taken particular values. As the tri-variate distribution is normal, the conditional distribution is normal too. Its parameters are defined through the parameters of the tri-variate distribution in the following way: m x| xd 

 x  ( d  md )rxd  m 0 rxd rxd   2  d x 1  rxd 

   

(52)

and

 x| xd   x

2 1  rxd  rx2d 2 1  rxd

.

(53)

Here md is a mean value of the distance between instantaneous and critical roll rate. It is readily available from (40):

Probability of Capsizing in Beam Seas with Piecewise Linear Stochastic GZ Curve 545

md   2 (  m 0   v 0 ) .

(54)

The standard deviation (53) of the conditional distribution is a constant, while the mean value is a function of  d ; therefore, as expected, the conditional distribution in (51) is a function of two variables:

f ( x | x  m 0 ,  d )  f m 0 ( x |  d ) 

1  x| xd





 x  m ( ) 2  x | xd d . exp    22x| xd 2  

(55)

The second term in (51) is just a bi-variate normal distribution of the carrier and the distance between the instantaneous and critical roll rates, evaluated at the angle of the maximum of the restoring curve in calm water, x   m 0 . It is convenient to present it in the form of a conditional distribution:

f ( d , x   m 0 ) f ( d | x   m 0 )  . f ( x   m0 )

(56)

Distribution (56) is also normal with mean value and standard deviation defined as:

md | x   m 0

d rxd x

(57)

and 2  d | x  d 1  rxd .

(58)

The conditional distribution (56) can then be expressed as follows:

f ( d | x  m 0 )  f m 0 ( d ) 

1 d | x



   m d d |x exp  2  2 d | x 2 

2 .  

(59)

As the distribution of the derivative of the carrier is also normal, the integral in the denominator in (44) can be evaluated in closed form: 

 xf ( x)dx  0

 x

2

.

(60)

546 V. Belenky et al.

The substitution of formulae (51), (55), (59) and (60) into (44) gives the final formula for the distribution of the distance between instantaneous and critical roll rate at the upcrossing of the carrier: 

f cr ( d ) 

2 f m0 ( d ) xf m0 ( x |  d )dx .  x



(61)

0

Finally, the probability of capsizing during time T is expressed by combining (12) and (43),

PC (T )  1  exp(PC |U T ) .

(62)

5 Self-Consistency Check In order to ensure that the theoretical solution is correct, a self-consistency check was performed. In this check, the dynamical system described by (4), with stiffness defined by (2) and (3), was evaluated for a large number of records in the time domain and the number of observed capsizes was counted in order to get a statistical probability of capsize. The results of the study, which are presented below, demonstrate clear convergence of the statistical probability of capsize to the theoretical solution (62). In addition to the final probabilities, some intermediate results are also presented in order to demonstrate how the theory works. Numerical data for the self-consistency check, including ship and wave properties, are shown in Table 1. The irregular seaway was derived using a Bretschneider open ocean wave spectrum. Fig. 6 shows the autocorrelation function of waves evaluated from the spectrum using cosine Fourier transform on the accepted discretization. It shows no sign of a self-repeating effect and therefore demonstrates the statistical representativeness of the accepted spectrum discretization for the desired duration of the time records (30 min). The initial set of calculations consisted of 200 records, each representing a 30 minutes realization of the same irregular seaway, and was used to check intermediate results such as the characteristics of upcrossing as well as the final capsizing result. Two sets of 6000 records each were then used to check the convergence of the statistical probability of capsizing. Fig. 7 shows one of the capsizing episodes observed in this calculation set. The upper graph shows the time histories of the ship roll angle () and the instantaneous angle of the maximum of the restoring curve (m). At approximately t = 1425 s, the roll angle exceeds the instantaneous angle of maximum of the restoring curve. The lower graph shows the time histories of the critical, ( cr ) ,

Probability of Capsizing in Beam Seas with Piecewise Linear Stochastic GZ Curve 547

 ) , rates. At that time, the instantaneous roll rate and instantaneous roll, ( exceeded the critical roll rate. As a result, the dynamical system experiences capsizing and transits to another equilibrium state. Table 1. Numerical data for self-consistency test Variable

Symbol

Unit

Value

Roll natural frequency,

0

1/s

0.65

Heave natural frequency



1/s

1.3

Angle of maximum in calm water

m0

Rad

0.5

Roll damping (fraction of critical)



-

0.2

Heave damping (fraction of critical)



-

0.4

Slope of decreased part of stiffness

k1

-

1.0

Angle of vanishing stability in calm water

v0

Rad

1.0

Slope coefficient for stability change

kd

1/m

0.0256

Duration of each record

T

s

1800

Time step

t

s

0.2

Number of frequencies

N

-

245

Significant wave height

Hs

m

11.5

Modal wave period

Tm

s

16.4

Autocorrelation (-) 1 0 -1 0

Time, s 200

400

600

800 1000

1200

1400 1600

Fig. 6. Autocorrelation function of waves calculated using Fourier cosine transform

Applicability of the Poisson flow to the upcrossing events of the carrier process is one of the key assumptions in the method. This can be checked by comparing the cumulative distribution of the time before upcrossing with the theoretical distribution and using the Kholmogorov-Smirnov (K-S) test to judge the goodnessof-fit. The test gives a 95% probability that the difference between the two is a result of random causes, which confirms the fit. Fig. 8 shows the theoretical curve calculated with equation (12) along with statistical points, representing a probability that at least one upcrossing has been observed during that time. The inset in Figure 8 compares the theoretical rate of upcrossing (13) to the mean number of observed upcrossings per unit of time shown with a 95% confidence interval. Justification and more details on these procedures can be found in (Belenky, et al., 2008).

548 V. Belenky et al.

Angles, rad

4

 3

2

1

m

0



Upcrossing through timedependent threshold

Angular velocity, rad/s 1.0

 cr

0.5 0



Time, s

   cr 1410

1420

1430

1440

1450

Fig. 7. Capsizing episode: the roll crosses instantaneous angle of maximum, and instantaneous roll rate exceeded critical roll rate

1

F(T)

0.9

, s-1

0.8

0.0065

Statistics

0.7 0.0060

Theory

0.6 0.0055 0.5

0

500

T, s 1000

1500

2000

Fig. 8. Cumulative distribution of the time with at least one upcrossing. Probability that the fit is good is 0.95 (K-S Test); inset: upcrossing rate

Probability of Capsizing in Beam Seas with Piecewise Linear Stochastic GZ Curve 549

Another important point to check is the distribution of the difference between instantaneous and critical roll rate at upcrossing of the carrier (61) as well as probability of capsizing after upcrossing (43). Fig. 9 shows a histogram of the values of the difference between the instantaneous and critical roll rate taken at the instant of upcrossing. The theoretical curve is calculated with formula (61). The importance of the difference between theoretical results and observed statistics was judged using Pierson chi-square criterion that has shown a finite probability of 7.7% that the observed difference is caused by random reasons. The inset in Fig. 9 shows comparison of the theoretical probability of capsizing after upcrossing (43) with the statistical estimate (Out of 200 runs, 70 ended up with capsizing and there were 1783 upcrossings all in total). 0.06

Theory

0.04

Probability of capsizing after upcrossing

Statistics

0 02 0

 

f cr  d

4

2

-0 1

-0 2

0

0.1

02

03

 d

, rad/s

Fig. 9. Distribution of the distance between critical and instantaneous roll rate at upcrossing. Probability that the fit is good is 0.077

The boundaries of the confidence interval were evaluated using the standard formula for statistical frequency:

P*  Pu ,l 

K 2 2N

 K 1





K 2 P* 1  P*  N 4N 2 2 K

(63)

N

Here P* is the statistical frequency, N is the volume of the sample, and K is the half-breadth is expressed in terms of standard deviations; for the accepted confidence probability of 95%, K = 1.95996. Fig. 10 shows a comparison between the theoretical and statistical estimates of probability of capsizing during a period of 30 minutes. As can be seen from this figure, the theoretical probability of 0.32 is within of the confidence interval of the statistical frequency of capsizing 0.35 ± 0.07 estimated over 200 independent realizations.

550 V. Belenky et al.

Probability of Capsizing in 30 min 0.4 0.35

Theory

0.3 Statistics

Fig. 10. Probability of Capsizing in 30 min estimated from statistics over 200 records and calculated with formula (62)

To complete the self-consistency check, the convergence of the probability was tested. As shown in Fig. 11, the statistical probability of capsizing was evaluated for an increasing number of records and plotted along with their confidence interval and the theoretical solution (63). The upper and lower graphs correspond to two independent sets of initial phases for the waves. These two sets were used to see different patterns of convergence. a

0.40 Probability of capsizing in 30 min

0.35

0.30 Theoretical value 0.25 Number of records (runs) 0.20

b

1000

2000

3000

4000

5000

6000

0.40 Probability of capsizing in 30 min 0.35

0.30 Theoretical value 0.25 Number of records (runs) 0.20

1000

2000

3000

4000

5000

6000

Fig. 11. Convergence of the statistical frequency to theoretical solution as number of runs increases. Upper (a) and lower (b) graphs differ in sets of initial phases.

Probability of Capsizing in Beam Seas with Piecewise Linear Stochastic GZ Curve 551

As it can be seen from both graphs in Figure 11, the statistical probability of capsizing does converge and the theoretical result stays within the confidence interval of the statistical value. The convergence is relatively fast in the first 1000 records but slows down afterwards. This may be partially caused by imperfect phase generation as capsizing phenomenon is known to be very sensitive to phases. Taking into account both the analysis of intermediate results such as upcrossing rate and a verification of the convergence of the statistical probability of capsizing to the theoretical value, the self-consistency check for the present method may be considered as an affirmation in the sense that it does not disapprove the theory.

6 Conclusions This paper presents the application of a split-time approach to evaluating the probability of a ship capsizing in waves with consideration of the ship’s change in stability in waves. The change in stability for a ship in waves can be modeled in a dynamical system with piecewise linear stiffness by considering the boundary between ranges as a stochastic process correlated with the excitation. This process may be introduced as a deterministic function of heave motions. If the increasing range remains linear and the decreasing part remains parallel to itself all the time, the solution remains linear within each range, and an analytic solution to the roll equation can be presented. The concept of critical roll rate as a stochastic process has been introduced. The critical roll rate is defined as the roll rate at which a ship upcrossing the maximum of the GZ curve will capsize. Since the GZ curve is changing with the motion in waves, the critical roll rate is a function of time. At each instant of time, if an upcrossing occurs and the roll rate at upcrossing exceeds the critical roll rate, then a capsizing is imminent. The capsizing probability can therefore be associated with the probability of upcrossing the maximum of the GZ curve with a roll rate at the instant of upcrossing exceeding the critical roll rate. Upcrossings are assumed to follow Poisson flow. The self-consistency of the method was demonstrated by evaluating the response of the dynamical system in time domain, counting the observed capsizes to get a direct statistical estimate of the probability of capsizing, and demonstrating that this value converges to the theoretical result as the volume of statistical data is increased. In general, the viability of a split-time approach considering the change of stability in waves was demonstrated using a simple example. This paves the way to the application of the split-time approach with more sophisticated tools including advanced numerical simulations of ships in waves.

552 V. Belenky et al.

Acknowledgements The work described in this paper has been funded by the Office of Naval Research under Dr. Patrick Purtell. Discussions of this work with Prof. Pol Spanos of Rice University, Prof. Kostas Spyrou of the National Technical University of Athens and Prof. Naoya Umeda of the University of Osaka were very helpful.

References Arnold L (1998) Random dynamical systems. Springer, New York Bassler CC, Dipper MJ, Lang GE (2009) Formation of large-amplitude wave groups in an experimental model basin. Proc 10th Int Conf on Stab of Ships and Ocean Veh, St. Petersburg Beck RF, Reed AM (2001). Modern computational methods for ships in seaway. Trans SNAME, 109: 1-48. Belenky VL (1993) A capsizing probability computation method. J Ship Res 37:200- 207 Belenky VL Weems K M, Lin W M (2008) Numerical procedure for evaluation of capsizing probability with split time method. Proc. 27th Symp. Nav Hydrodyn, Seoul Gumbel E (1962) Statistics of extremes, Colombia Univ Press, New York Guckenheimer J, Holmes, P. (1983). Nonlinear oscillations, dynamical system and bifurcation of vector fields, Springer-Verlag, New York Paroka D, Okura Y, Umeda N (2006) Analytical prediction of capsizing probability of a ship in beam wind and waves. J Ship Res, 50 2:187-195 Paroka D, Umeda N (2006) Capsizing probability prediction of the large passenger ship in irregular beam wind and waves: comparison of analytical and numerical methods. J Ship Res 50 4:371-377 Roberts JB, Spanos P (2003) Random vibration and statistical linearization. Dover Publ Themelis N, Spyrou K J (2007) Probabilistic assessment of ship stability. Trans SNAME 115 Umeda N, Shuto M, Maki A (2007), Theoretical prediction of broaching probability for a ship in irregular astern seas. Proc. 9th Int. Ship Stab Workshop, Germanischer Lloyd, Hamburg

Appendix 1 Consider a stationary stochastic process, x(t), that crosses a level a at an arbitrary instant of time t. Consider another stochastic process, y(t), that depends on the process x(t). The objective is to find the probability density distribution of the instantaneous value of the process y(t) when the process x(t) up-crosses the level a.

Probability of Capsizing in Beam Seas with Piecewise Linear Stochastic GZ Curve 553

A random event of upcrossing is defined as:

 x(t )  a U   x(t  dt )  a .

(A1)

By definition, the cumulative probability distribution is:

Fcr ( y )  P( y  b | U ) .

(A2)

The conditional probability in formula (A2) can be expressed as:

P( y  b | U ) 

P( y  b  U ) . P (U )

(A3)

Here P ( y  b  U ) is the probability of occurrence of an upcrossing with the value of the process, y(t), not exceeding an arbitrary number b. This random event can expressed through the following system of inequalities:  x(t )  a  x(t )  a   y  b  U   x(t  dt )  a   x(t )  a  xdt y  b y  b .  

(A4)

Probability of the random event defined by equation (A4) can be expressed trivially through a joint distribution of the process x(t), its derivative and the process y(t): b 

a

   f ( x, x, y)dxdxdy .

P( y  b  U ) 

(A5)

 0 a  xdt

The most internal integral in the formula (A5) has limits that are infinitely close to each other. Application of the Integral Mean Value Theorem yields: b 

P ( y  b  U )  dt

  x f a , x , y d x dy .

(A6)

 0

The probability of upcrossing P(U) can be expressed in similar way: 



P(U )  dt f (a ) xf ( x )dx . 0

(A7)

554 V. Belenky et al.

The cumulative distribution of the value of y(t) at upcrossing can be expressed by substituting (A6) and (A7) into (A3) and (A2): b 

Fcr ( y ) 

  xf a, x, y dxdy

 0



.

(A8)



f (a ) xf ( x )dx 0

The probability density is obtained from (A8) by taking a derivative with respect to y: 

f cr ( y ) 

 xf (a, x, y)dx 0



.

(A9)



f (a) xf ( x )dx 0

Equation (A9) represents the final result for the distribution of the dependent process at upcrossing. This result may be known in the Upcrossing Theory; however the authors were unable to locate appropriate references.

Probabilistic Analysis of Roll Parametric Resonance in Head Seas Vadim L. Belenky*, Kenneth M. Weems**, Woei-Min Lin**, J. Randolf Paulling*** *Naval Surface Warfare Center Carderock Division (NSWCCD) - David Taylor Model Basin; **Science Application International Corporation (SAIC); ***Professor Emeritus, University of California at Berkeley

Abstract The paper presents some background for an analysis of the risk of severe parametric roll motion for a ship operating in head seas. This background includes a consideration of basic probabilistic qualities of parametric roll in head seas: ergodic qualities and distributions, since these results are necessary to establish a method of prediction of extreme values. The ship motions that influence on parametric roll, heave and pitch, have also been studied in this analysis. The postPanamax C11 class container carrier was chosen for analysis, since a vessel of this type is known to have suffered significant damage in an incident attributed to severe parametric roll. It was shown that despite large-amplitude of motion, pitch and heave retain their ergodic qualities and normal character of distribution, while the roll motions are clearly non-ergodic and do not have a normal distribution. The analysis is built upon the numerical simulation of ship motion in head seas using Large Amplitude Motion Program (LAMP).

1 Introduction and Background The phenomenon of parametrically induced roll has been known to naval architects for over fifty years (Paulling and Rosenberg, 1959). Initially, it was thought to be a phenomenon of following seas and was of significance for smaller, high-speed displacement vessels such as some fishing boats and seagoing tugs. In recent years, however, parametric roll has been observed in large seagoing ships, particularly container ships operating in head seas. A significant recent casualty is described in (France et al. 2003) in which a large number of containers were lost from a postPanamax container ship caught in a severe storm in the North Pacific. A number of other container losses have occurred and are thought to be attributable to the same cause. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_31, © Springer Science+Business Media B.V. 2011

555

556 V.L. Belenky et al.

When a ship sails in head or following seas, the geometry of the underwater hull is constantly changing with time as a result of the wave surface along the hull as well as the pitching and heaving motions. In general, stability is greater than that in still water when a wave crest is at bow and stern, and it is diminished relative to still water when a crest is amidships. The effects are most pronounced in waves of length about equal to the ship length, and increase with increasing wave steepness. In pure head or following seas, there is no direct wave-induced rolling moment such as would exist if the waves approached the ship from any other direction. Nevertheless, if the period of wave encounter is approximately one-half the natural period of roll, a rolling motion can exist even in the absence of a direct roll exciting moment. This is a kind of dynamic motion instability and is a consequence of the periodically varying stability. Specifically, rolling motion can be set up if the stability variations occur at the critical period ratio of ½ and there is some arbitrarily small initial disturbance. The disturbance always exists in natural waves because of directional spreading. Under most conditions, the roll is quickly damped out and is of no consequence. In some conditions of high seas, however, the proper ship speed and heading and for certain hull form characteristics, the rolling motion can grow to large proportions. Capsizings have been recorded, particularly of small high-speed fishing vessels when heavily loaded in following seas. The head sea parametric roll is a more recently identified phenomenon and seems especially likely to occur in the case of large container ships as a result of certain features of their hull form that lead to especially pronounced variations in stability as the ship sails through head or following seas. This paper describes an ongoing work carried out and sponsored by American Bureau of Shipping. Since the work is ongoing, it is not meant to offer a complete solution but rather to continue the discussion started in (France et al. 2003).

2 Numerical Simulations Since the parametric roll phenomenon is caused by time variation of transversal stability, the numerical simulation method must be able to adequately model the changes of geometry of immersed part of the hull due to large waves and ship motions. Following (France et al. 2003), the simulations were made using the Large Amplitude Motion Program (LAMP) with its “approximate body nonlinear” formulation. In this formulation, which is also referred to as the LAMP-2 approach, the hydrostatics and Froude-Krylov forces are computed over the instantaneous wetted hull surface while the perturbation potential, which includes radiation and diffraction effects, is computed over the mean wetted surface. Since it is the nonlinear hydrostatics that is “responsible” for the parametric roll phenomenon, the use of the mean waterline formulation is justified. This formulation provides a

Probabilistic Analysis of Roll Parametric Resonance in Head Seas 557

substantial speed-up of calculations as compared to the fully body-nonlinear approach, which is especially critical when working with stochastic processes. In brief, LAMP is a time domain simulation system based on a 3-D potential flow panel solution of the wave-body interaction problem and incorporating flexible models for control systems, green-water-on-deck, viscous forces, and other effects. The latter feature is very important for adequate modeling of parametric resonance, since the viscous roll damping model can be tuned based on the results of the model test (France, et al., 2003). As was shown in (France, et al., 2003), roll damping, including nonlinear terms in roll, is a key factor in predicting roll amplitude in the regime of parametric resonance. Lin and Yue (1990, 1993) contain details on its theoretical background. Shin, et al. (2003) describes recent developments and applications of the LAMP System.

3 Ship Configuration The ship chosen for the present analysis is the C11 class container carrier that lost many containers in a casualty involving large roll motions in severe head seas, and was the subject of the experimental and numerical study described in (France et al. 2003). Fig. 1 presents a general view of her hull geometry. The ship has large bow flare and an overhanging stern, exactly the geometry features that can invoke parametric roll. The large bow flare and overhanging stern result in significant changes to the waterplane as the ship pitches and heaves in waves. When the encounter period is close to ½ of the natural roll period, these changes provide the parametric excitation leading to a rise of roll motions (France, et al., 2003). How large the roll becomes is determined by nonlinear factors, including the shape of the GZ curve and nonlinear roll damping. That is why body nonlinear hydrostatics and a correct model of roll damping are critical for evaluating extreme parametric roll motions. Here an empirical model of roll damping, tuned and validated with experimental results, is used (France et al. 2003).

Fig. 1 Hull of post-Panamax container carrier

558 V.L. Belenky et al.

4 Parametric Roll in Regular Waves Before proceeding with parametric roll of a ship, we would like to consider the Mathieu equation, which is the simplest mathematical model of parametric resonance. It is a linear ordinary differential equation with periodic coefficient:

d 2   p  q cos     0 d 2

(1)

where p is related to the square of the ratio of the forcing frequency to the natural frequency and the parameter q plays the role of the amplitude of the parametric excitation. Depending on the values of parameters p and q, the solution of Mathieu equation might be decaying, periodic (also known as Mathieu function) or rising infinitely. An example of the rising (or “unstable”) solution is illustrated by phase trajectory shown in Fig. 2. An Ince-Strutt diagram in fig. 3 shows zones where combinations of the parameters p and q in the Mathieu equation result in such an unstable solution. There are several zones of instability: the first zone starts from p=0.25 and corresponds to a natural period exactly twice the period of parametric excitation.

First derivative of solution

0.6 0.4 0.2 0.2

0.1

0

0.1

0.2

-0.2 -0.4 -0.6 Solution of Mathieu equation

Fig. 2 Phase trajectory of unstable solution of the Mathieu equation (p=2.5, q=0.2, damping factor 0.05, initial displacement 0.01)

Adding a linear damping to Mathieu equation does not limit the amplitude of its solution. Instead it “lifts” the zone boundaries and creates a “threshold” for the amplitude of parametric excitation (q) which results in a rising solution.

Probabilistic Analysis of Roll Parametric Resonance in Head Seas 559

2 Threshold

Zone 4

Zone 3 Zone 2

4

Zone 1

6

Boundary with damping

q

2

0

p

4

p=0.25

Fig.3 Ince-Strutt diagram

This is why it is possible to use Mathieu equation for modeling the occurrence raised motions caused by parametric excitation, but not for evaluating how large the parametric oscillations might develop. To do so, nonlinear damping or stiffness terms must be added to “stabilize” the rising oscillations. Numerical simulations for the ship in head, regular waves can be used to illustrate the mechanics of parametric roll. For waves with an encounter period away from ½ the roll natural period or with small amplitude, the roll motion will remain small even if a fairly large roll perturbation is introduced. For waves in the proper frequency range, a sufficiently large wave amplitude will induce large rolling motion with an arbitrarily small roll perturbation. The development of roll motion shown in Fig. 4 is typical for the parametric resonance regime: the roll motion takes relatively long time to start, increases rapid, and finally stabilizes at steady-state amplitude. 40

Roll, deg

20 0

20

40

60

80

100

20 40

Time, s

Fig. 4 Development of parametric roll in regular waves (wave amplitude 4.2 m, -1

frequency 0.44 s , speed 10 knots)

560 V.L. Belenky et al.

Fig. 5 shows a steady-state mode of parametric roll along with timesynchronized heave and pitch motions. It is very clear that roll period is twice that of heave and pitch, which corresponds to the first zone of instability in the InceStrutt diagram (Fig. 4). It is peculiar that the steady-state roll motions shown in Fig. 5 are not sinusoidal. This can be seen especially clearly in Fig. 5, which plots the phase trajectory of the roll in the parametric roll mode.

Pitch, deg

5 0

Roll, deg

-5 50 0

Heave, m

-50 2 0 -2

Time, s 800

820

840

860

880

Fig. 5 Steady state parametric roll, pitch, and heave

The deviation from the sinusoidal form can be explained by two factors. First, there is the nonlinearity of the damping and restoring terms with roll: at an amplitude close to 30 degrees, their influence is likely to be significant (see GZ curve in France et al. 2003). Secondly, the parametric response, even in the simplest case described with the Mathieu equation, might be far from sinusoidal (see Fig. 6).

Roll Velocity, deg/s

10 5 30

20

10

0

10

20

30

5 10 Roll Angle, deg

Fig. 6 Phase trajectory of parametric roll

Some deviation from sinusoidal form can be observed in the heave and pitch motions as well, but they are visually much smaller. The phase trajectories of heave and pitch are shown in Fig. 7.

Probabilistic Analysis of Roll Parametric Resonance in Head Seas 561 2

0.5 2

1.5

1

0.5

0

0.5

1

1.5

0.5

Pitch velocity, deg/s

Vertical velocity, m/s

1

1 Heave, m

1 4

2

0

2

4

1 2 Pitch angle, deg

Fig. 7 . Phase trajectory of heave and pitch in the regime of steady state parametric roll

5 Parametric Roll in Irregular Waves The large amplitude of the roll response encountered in a parametric resonance may significantly influence the probabilistic characteristics of rolling. The deviation from sinusoidal form of the steady state parametric roll, as discussed above, might also be considered a sign of nonlinearity. The conventional models of ship behavior in irregular seas used by most seakeeping and stability applications are not always valid. These models assume ergodicity (a quality of stochastic process that allows estimation of statistics using one long realization) and a normal distribution of rolling. However, these assumptions do not always have a solid background. Observations and records have validated the assumption of normal distribution and ergodicity of waves at sea. If a ship is considered to be a linear system (St. Denis and Pierson 1953), the Weiner-Khinchin theorem states that the ship response will also be normal and ergodic. If, however, nonlinearity is involved, then this assumption no longer holds. Numerical simulations (Belenky et al. 1997, 1998, Belenky 2000) and model testing (Belenky et al. 2001) have shown that large amplitude roll cannot be considered ergodic. Roll distribution, however, may be assumed normal for lowbuilt ships; if a ship has high freeboard and GZ has S-shape, roll distribution might not be Gaussian.

5.1 Model of Irregular Waves In the irregular wave analysis, only head long-crested seas are considered so that roll is excited and coupled with pitch and heave. Following (France et al. 2003), a JONSWAP spectrum, shown in Fig. 8, was created for a wind velocity of

562 V.L. Belenky et al.

30 m/s, a fetch of 100nm, and peak enhancement factor ( of 1.39. This spectrum produces a significant wave height of about 9 m. To create a discrete wave model, 200 wave components were created with -1 -1 equal frequency intervals over a range from 0.245 s to 1.04 s ; providing 26 minutes of simulation time before the second peak of the correlation function. The incident wave elevation is defined by the well-known form of a Fourier series: N

 w (t ) 

 a cos t    i

i

(2)

i

i 1

where i is the frequency set, amplitudes ai are defined from the spectrum, and phase shift i are random numbers with uniform distribution. Each realization of waves is generated with a new set of random phase shift. A total of 50 wave realizations were created and analyzed for the present study. Spectral Density, m2s

50 40 30 20 10 0.2

0.4

0.6 0.8 Wave Frequency, 1/s

1

1.2

Fig. 8 JONSWAP Spectrum (=1.39, Wind velocity 30 m/s, Fetch 100 nm)

5.2 Predicted Roll Response For each of the wave realizations, a 26 minute LAMP simulation was made to evaluate the ship’s response while running into the waves at 10 knots. Fig 9a shows the predicted roll response for the first realization. The above result is similar to those published in (France, et al., 2003) and shows a highly pronounced group structure. The roll response to the 2nd realization (Fig. 9b) shows a change in the sequence of groups and an interval with low roll angle, where parametric roll is not observed. After some time, however, parametric roll it is again excited.

Probabilistic Analysis of Roll Parametric Resonance in Head Seas 563

b)

Roll Angle, deg

40

Roll Angle, deg

a)

20 0 -20 -40

Time, s 0

200

400

600

800 1000 1200 1400

0

200

400

600

Time, s 800 1000 1200 1400

40 20 0 -20 -40

Fig. 9 Roll response in irregular long-crested waves

5.3 Stationarity The statistical characteristics of waves at sea change with time. Formally, this means that waves cannot be considered as a stationary stochastic process. The changes of these characteristics, however, are typically slow in comparison with the wave period. This allows a hypothesis of quasi-stationarity to be introduced, assuming that the waves could be considered as a stationary process within a certain time, during which the changes of the statistics can be neglected. This time, commonly referred as “period of quasi-stationarity,” lasts from half an hour to several hours. Extreme values observed during this time are defined in seakeeping analysis as “short term extremes”. Considering parametric roll within the period of quasi-stationarity, we have a good reason to assume that it is a stationary process, as long as the speed, heading, and loading conditions of the ship are not altered.

5.4 Ergodicity: Visual Check Ergodicity is only applicable to stationary stochastic processes. If the process is ergodic, its statistical characteristics can be estimated from one sufficiently long realization, rather than the whole set of realizations required for non-ergodic processes. This means that for an ergodic process, the following equality of mean values takes place: 

mx 





T

1 x(t )dt T  T

xf ( x)dx  lim

 0

where f(x) is probability density. The same could be written for variance estimates:

(3)

564 V.L. Belenky et al. 

Vx 



T

1 ( x  m x ) f ( x)dx  lim ( x(t )  m x ) 2 dt T  T



2



(4)

0

Consider a number of realizations for the same stochastic process. As a consequence of equations (3) and (4), statistical characteristics estimated for different realizations of the ergodic process must be essentially the same. So, if the statistical characteristics are evaluated cumulatively over time, they would form a set of converging curves. Such a set is shown in Fig. 10a, which plots the variance estimate of wave elevation at the fixed origin for a number of different realizations. Fig. 10a clearly shows the tendency of convergence. Analogous behavior was observed for the encounter waves, heave and pitch motions, and heave and pitch velocities. For example, Fig 10b shows the behavior of variance estimates for pitch. A very strong tendency to converge has been found for mean values of all studied processes, including roll angle and roll velocity in the regime of parametric resonance. Estimates of variances in roll angle and velocity, however did not converge within observed duration (see Fig. 10c), which may be considered as non-ergodicity for practical purposes. a)

b)

35

c) 2 140 Variance of roll deg

2

Variance of wave elevations, m2

Variance of pitch, deg

9

30

120

8 7

25

100

6 20

80

5

15

4

10

3

60 40

2 5 Time, s 0

500

1000

1500

20

1

Time, s 0

500

1000

1500

Time, s 0

500

1000

Fig. 10 Variance estimates for wave elevations (a), pitch (b) and roll (c)

1500

Probabilistic Analysis of Roll Parametric Resonance in Head Seas 565

5.5 Ergodicity: Confidence Intervals The mean values and variances calculated with any finite data volume are only estimates. These estimates actually are random numbers, because deviation from the theoretical value is random. Therefore, there is always a possibility that a difference between roll variances estimated on different realizations is caused by finiteness of available data. To evaluate the likelihood that observed behavior is (or is not) caused by statistical uncertainty, confidence intervals were used, following a similar analysis applied in (Belenky et al. 2001 and Belenky 2000). The confidence interval is a range that contains the true value of mean value, variance, or any other probabilistic characteristic, with a given confidence probability . Here =0.95 is used. Following standard statistical procedure, the distribution of the random deviation of the estimated value from the true value is assumed to be Gaussian (this does not imply Gaussian distribution for the analyzed stochastic process). Then the confidence interval half-width for estimate Z (mean value or variance) can be calculated as: (5)

Z  Pinv (, m[ Z ],V [ Z ])

where Pinv is the inverse Gaussian cumulative probability, m[Z] is the mean value of the estimate, and V[V] is the variance of the estimate As can be seen from equation (5), in order to calculate the width of the confidence intervals, it is necessary to evaluate the mean value and variance of the statistical characteristics, which are the mean values and variances of each realization. To avoid confusion, all estimates of realizations bear a subscript r and all estimates of estimates are marked with a tilde (~) above. The variance of the mean value is related to the estimated autocorrelation function and the number of points of the realization (Priestley, 1981): ~[m ]  m m r

r

~ 1 V [ mr ]  Nr

N r 1

 |i | ~ 1   R| i| N r  i  ( N r 1) 



(6)

where mr and Vr are mean value and variance estimated for one realization and Nr is the number of points in the realization, while R is autocorrelation function estimated as: ~ Wi Ri  Nr  i

N r i

 x  m x i

r

j

 mr



j 1

Here W is statistical weight taken as:

(7)

566 V.L. Belenky et al.

 N i  Wi   r  Nr 

4

(8)

The mean value and variance of the variance estimate can be calculated as follows: ~[V ]  V m r r ~ 2 V [Vr ]  Nr

N r 1

 | i |  ~2 1   R| i| N r  i  ( N r 1) 

(9)



The confidence intervals for each realization are shown in Fig. 11, a through d. Each figure contains two diagrams: one for mean value and one for variance. Each diagram shows an estimate for each realization with corresponding confidence interval. When the confidence intervals for different realizations have an overlap, the difference between realization estimates might well be treated as statistical error. a) Wave elevations at fixed origin 0.05

b) Heave motions

Mean value estimates, m

5.08

Mean value estimates, m

5.06 0

5.04 5.02

-0.05

5 2

Variance estimates m2

Variance estimates m 14

2.5

12

2

10

1.5

8

1

c) Pitch motions 0.05

Mean value estimates, deg

d) Roll motions

Mean value estimates, deg

0.2 0 0 -0.05

-0.2

Variance estimates deg

2

5

150

4

100

3

50

2

0

Variance estimates deg2

Fig. 11 Realization estimates for mean value and variance

Probabilistic Analysis of Roll Parametric Resonance in Head Seas 567

Comparing overlaps of variances of waves, heave, pitch, and roll in Fig. 11 or converging tendency of waves, pitch, and roll in Fig. 10, it is natural to observe that the wave elevations are “most ergodic”, roll motions are “less ergodic” or “practically non-ergodic”

5.6 Probability Distributions The next issue to address is probability distribution. It is widely accepted that wave elevation has normal distribution; it is attributed to the fact that many different factors influence wave generation. The Fourier series model of irregular wave (Equation 2) reproduces normal distribution very well. Fig. 12a shows a histogram of wave elevation at the fixed origin, calculated from all 50 realizations used in this study. On top of this is plotted the theoretical Gaussian distribution calculated from the mean value and variance estimate for the whole set of realizations. As shown in Fig. 12a, the theoretical and statistical distributions are so close that the curve can barely be seen. Previous studies of large-amplitude roll motion in beam seas (Belenky et al. 1997, 1998) (see also review by Belenky, 2000) have shown that the nonlinear roll response might be normal or not normal, depending on the shape of the nonlinear term in the most statistically significant range. The heave and pitch motions, despite being calculated with nonlinear hydrostatic and Froude-Krylov forces (LAMP-2 formulation), do not show any visible deviation from Gaussian distribution (see Figs. 12b and 12c). The fact that pitch and heave in this case are nonlinear is clear: the very reason for parametric roll is the changing the geometry of the hull’s wetted portion, which also causes nonlinear heave and pitch. However, this nonlinearity did not create significant non-ergodicity nor a deviation from Gaussian distribution. The distribution of the roll response from these realizations is shown in Fig. 12d, and it is quite far from normal: the peak of the distribution is significantly sharper. The reason why rolling does not follow Gaussian distribution may be sought in two directions. First, it could be just inherent roll nonlinearity expressed in damping and GZ curve; a similar shape of roll distribution was observed in beam seas (Belenky et al. 1997, 1998). Secondly, it could be the shape of parametric roll time history (see Figs. 2 and 5) and the stronger-than-usual group structure that contributes to the parametric roll distribution’s deviation from Gaussian. Working with non-Gaussian and practically not ergodic process certainly presents a challenge. At the same time, methods based on the group structure of waves and envelope presentation of parametric excitation, might be very promising and deserve special attention (Blocki, 1980, Tikka and Paulling, 1990, Francescutto, et al. 2002).

568 V.L. Belenky et al.

a)

b)

p.d.f.

p.d.f. 0.3

0.1

0.2 0.05

15

10

0.1

5 0 5 Wave elevations, m

10

15

d)

0.2

10

4

2 0 2 Heave motions, m

6

0.08

0.15

0.06

0.1

0.04

0.05

0.02

5 0 Pitch motions, deg

4

p.d.f.

p.d.f.

c)

-6

5

10

40 30

20 10 0 10 20 Roll motions, deg

30

40

Fig. 12 Distributions for wave elevations at fixed origin (a), heave motions (b), pitch motions (c) and roll motions (d)

6 Conclusions and Comments While this and other efforts analyzing the parametric roll phenomena are far from being completed, some preliminary conclusions can be drawn. With the recognition of parametric roll in head seas as a significant danger for the stability of large container ships and for the safety of the cargo and people aboard, reliable methods are needed to evaluate the risk of operation. While model tests and nonlinear numerical simulations remain the best tools for evaluating parametric roll in particular situations, approximate methods should be developed for predicting the likelihood and magnitude of large roll events for everyday engineering analysis. Certain caution, however, has to be exercised, as far as modeling assumptions are concern, since parametric roll is practically nonergodic and not necessarily a Gaussian stochastic process.

Probabilistic Analysis of Roll Parametric Resonance in Head Seas 569

Acknowledgements The development of the LAMP System has been supported by the U.S. Navy, the Defense Advanced Research Projects Agency (DARPA), the U.S. Coast Guard, the American Bureau of Shipping (ABS), and SAIC. The anti-rolling tank study was performed by Mr. Thomas Treakle. The authors wish to especially thank Dr. Yung Shin, the manager of ABS Department of Research, for his support.

References Belenky VL, Degtyarev AB and Boukhanovsky AV (1997) Probabilistic qualities of severe ship motions, Proc of 6th Int Conf on Stab of Ships and Ocean Veh, Varna, Bulgaria, 1:163-172. Belenky V, Degtyarev AB, Boukhanovsky AV (1998) Probabilistic Qualities of Nonlinear Stochastic Rolling, Ocean Eng 25(1):1-25. Belenky VL (2000) “Probabilistic Approach for Intact Stability Standards: State of the Art Review and Related Problems”, Trans SNAME, 108:123-146. Belenky VL Suzuki S, Yamakoshi Y (2001) “Preliminary Results of Experimental Validation of Practical Non-Ergodicity of Large Amplitude Rolling Motion”, Proc of 5th Int Ship Stab Workshop¸ Trieste, Italy. Blocki W (1980) Ship safety in connection with parametric resonance of the roll, Int Shipbuilding Progress, 27(306):36-53. France WM, Levadou M, Treakle TW, Paulling JR, Michel K, Moore C (2003) An Investigation of Head-Sea Parametric Rolling and its Influence on Container Lashing Systems,. Marine Technol, 40(1):1-19. Francescutto A, Bulian G, Lugni C (2002) Nonlinear and Stochastic Aspects of Parametric Rolling Modelling. Proc of 6th Int Ship Stab Workshop, Webb Institute, N.Y. also available from Marine Technol, 41(2):74-81. Lin WM, Yue DKP (1990) Numerical Solutions for Large-Amplitude Ship Motions in the Time-Domain, Proc of the 18th Symp of Naval Hydrodyn, Michigan, U.S.A. Lin WM, Yue DKP (1993) Time-Domain Analysis for Floating Bodies in Mild-Slope Waves of Large Amplitude, Proc of the 8th Int Workshop on Water Waves and Floating Bodies, Newfoundland, Canada. Paulling JR, Rosenberg RM (1959) On Unstable Ship Motions Resulting from Nonlinear Coupling. J Ship Res, 3(1):36-46. Pristley MB (1981) Spectral Analysis and Time Series, Academic Press, London. Shin YS, Belenky VL, Lin WM, Weems KM, Engle AH (2003) Nonlinear Time Domain Simulation Technology for Seakeeping and Wave-Load Analysis for Modern Ship Design Trans SNAME 111. St. Denis M, Pierson WJ (1953) “On the Motions of Ships in Confused Seas”, Trans SNAME, 61. Tikka KK, Paulling JR (1990) Prediction of Critical Wave Conditions for Extreme Vessel Response in Random Seas. Proc of 4th Int Conf on Stab of Ships and Ocean Veh, Naples, Italy.

8 Environmental Modeling

Sea Spectra Revisited Maciej Pawłowski School of Ocean Engineering and Ship Technology, TU of Gdansk, 80-952, Poland,

Abstract The paper demonstrates that the sea spectra recommended by ITTC, based on the Bretschneider formulation, can be reduced to a nondimensional spectrum, the same for all the spectra, with unit area. In other words – these well-known spectra have geometrical affinity. This fact has been unknown in literature. Any ITTC sea spectrum, described by two parameters A and B, can be generated using the nondimensional spectrum. The same also applies to JONSWAP spectra. The latter requires in addition a third parameter, termed as the peak-shape parameter. The paper explores this possibility. The fact that all the sea spectra used by naval architects can be reduced to a common dimensionless spectrum of unit area opens up the possibility of approximating them by probability density functions of certain types. Such spectra, contrary to ITTC ones, are narrow-banded, with the bandwidth parameter less than 1, and have moments of any order.

1 Introduction The ITTC spectral formulation for fully developed seas, derives from Bretschneider, and are given by the following equation: 

(1)

S() = (A/) e –B/ ,

where A and B are constants. It is convenient to apply a substitution t  B/ for calculating the spectral moments. For the nth moment, we get: mn = ¼ AB ¼

n –  –¼ n –t t e dt 0



 ¼ AB ¼

n –

 ( ¼ n ),

(2)

where (x) is the gamma function, defined for positive x by the integral  (x)   x–1 –t   t e dt. As the argument of the function gamma has to be positive, the moments exist only for n . The 4th and higher moments are infinite. Therefore the bandwidth parameter  is unity, which implies the ITTC spectrum is wide-banded.

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_32, © Springer Science+Business Media B.V. 2011

573

574 M. Pawłowski

Substituting n and, and making use of the well-known feature of the gamma function:  (x  1)  x (x), the following results for the first moments: m  ¼ A/B, m2  ¼   A/B

m1  ⅓  ()A/B3/4, m4 = .

(3)

For large , S decays as . Due to this reason, the 4th and higher spectral moments do not exist, which is far from reality. For real seas all moments exist and the bandwidth parameter  is less than 1, from the region           .

2 Standard Sea Spectra ITTC spectra are defined by two parameters A and B, as seen in equation (1). In order to define these parameters, we have to use two characteristic values describing wave intensity. The most important is the significant wave height hs , where  is the standard deviation of wave elevation. Hence, hs   1 m   A/B. The constant A is then given by the equation: A = ¼ Bhs2,

(4)

depending on the other constant B, related to one of the wave periods. Most frequently, the characteristic period is used T1 = 2 m0/m1. Making use of equations (3), we get 4

 1.5  1 691.18 B  4  T4   1.75  T1 1

(5)

A spectrum with the above constants A and B is called the ITTC spectrum. It is easy to show that all the characteristic frequencies, such as the modal frequency –, and the average zero-crossing frequency  are m, the mean (average) frequency   1/4 in proportion to B , which means they are in the same proportions relative themselves. This in turn suggests that ITTC spectra have geometrical affinity, which can be proved rigorously. It is worth noting that the average peak frequency  does not exist for ITTC spectra. According to Pierson and Moskowitz the constants A and B are as follows: A  c1 g 2 ,

B  c2 (g /U)4,

(6)

where c1 , c2 , g is the acceleration due to gravity, and U is the mean wind speed at 19.5 m above the sea surface. A spectrum with such constants is called the Pierson–Moskowitz spectrum. As can be seen, it is a one-parameter spectrum, solely dependent on wind speed, which is not very convenient. In the applications, it is more convenient to utilise the significant wave height rather than the wind speed.

Sea Spectra Revisited 575

To do so, the constant B has to be related to the significant wave height hs . Since   m0, therefore (hs /)   ¼ A/B. Hence, B  A /hs. 30 m2s

S(ω)

25 hs =11 m

20

15 9m

10 7m

5 5m

0 0,2

0,4

0,6

0,8

1

ω (1/s)

1,2

Fig. 1 Pierson–Moskowitz spectra as a function of significant wave height hs

The Pierson–Moskowitz spectra, depending on the significant wave height hs, are shown in Figure 1. As can be seen, the modal frequency m decreases with the significant wave height hs, which can also be deduced from equation (7) for the modal frequency: m  (B),

(7)

which results from: S'() B . Equation (7) yields B  m. Inserting it to equation (6) yields: m  (c) (g /U)  (g /U).

(8)

Equating B  A/hs to B given by equation (6) yields a standard relation between wind speed and sea severity: U 4  (c/c)(ghs)2. Hence, U  (c/c)(ghs)  2.186 (ghs).

(9)

For instance, for hs m, the standard wind speed U   m/s.

3 JONSWAP Spectrum The JONSWAP formulation is based on an extensive wave measurement programme known as the Joint North Sea Wave Project carried out in the years 1968–69. The spectrum concerns wind-generated seas with fetch limitation, and wind speed and fetch length are inputs to this formulation, which is as follows:

576 M. Pawłowski

S    / c1  S PM   expm 

2

where

   m

x– x

/ 2  m 2



(10)

 parameter,  as an average   –x  – scale parameter   for   m, and  for   m    (g /U) x–   gx/U  – dimensionless fetch  fetch length (in m)

The scale parameter   c, if x–   , which is quite large. Therefore, in most cases   c. The parameter  is called the peak-shape parameter and it represents the ratio of the maximum spectral energy density to the maximum of the corresponding Pierson–Moskowitz spectrum. The term associated with the exponential power of  is called the peak enhancement factor, and the JONSWAP spectrum is the product of the Pierson–Moskowitz spectrum (with B   m) and the peak enhancement factor. The effect of the peak-shape parameter on the JONSWAP spectrum for wind speed U  m/s and fetch length x  km is shown in Figure 2. The modal frequency in this case m s and the ratio /c , which means that the area under the original Pierson–Moskowitz spectrum for  is increased by %. -value increases area under the spectrum, hence – sea severity. 70 m2s S(ω) 60

6 50

40

4 30

20

2 10

γ=1

ω (1/s)

0 0.2

0.4

0.6

0.8

1

1.2

1.4

Fig. 2 Effect of peak-shape parameter  on JONSWAP spectra

Assuming A g, and B  m, the first equation in (3) yields for the area under the JONSWAP spectrum with  the value: m   g/ m.

(11)

Sea Spectra Revisited 577

The parameter  is actually a random variable, approximately normally distributed from 1 to 6, with mean 3.3 and variance 0.62, as shown by Ochi (1998). Similarly to the scale parameter , also the peak-shape parameter can be presented as a function of dimensionless fetch:    x– .

(12)

The JONSWAP spectral formulation, as given by equation (10), is a function of wind speed and fetch length, resulting in a spectrum of certain significant wave height, unknown beforehand, which is not very convenient. Ochi, in his book, provides a relationship between the resultant significant wave height, wind speed and fetch length, as follows: U  k x  hs,

(13)

where U is in m/s, x in km, hs in m, and k is a constant depending on -value. Its reasonable quadratic approximation is as follows: k     .

(14)

Ochi derived equation (13) using equation (10) for various combinations of fetch length and wind speed. With the help of equation (13) the JONSWAP spectrum can be presented now for a specified significant wave height hs and fetch length x.

4 Non-Standard Spectra ITTC spectra do not describe best real seas, as they are wide, with the bandwidth parameter , whereas for real seas this parameter is from the range . This results from too slow decay of ITTC spectra for large ; they should decay exponentially, whereas they decay as . A question arises here, whether seas spectra could be approximated better to allow for an exponential decay, yielding all spectral moments. Before we answer this question, first we prove that ITTC spectra have geometrical affinity.

4.1 Nondimensional ITTC Spectrum Dividing the ITTC spectrum, given by equation (1), by the area m0 = ¼A/B, yields a spectrum of unit area, as follows: 4

S1() = (4B/5) e –B/

.

(15)

578 M. Pawłowski

It is handy to introduce a new constant b in place of B  (1/b), having the dimension of time. The above unit-area spectrum now takes the form: S1()   [b/(b)]e –1/(b) , which can be presented shortly, as follows: S1()  bs(b),

(16)

where s (x  b) is a unit-area nondimensional ITTC spectrum, given by: 4

s(x)  (4/ x5) e –1/ x

(17)

.

Equation (16) is a mathematical statement that between the unit-area spectrum S1() and the nondimensional spectrum there is an affinity. The scale of transformation along the  axis is 1/b. If b  s the graph is diminished linearly b times along the  axis, and increased b times along the vertical axis, to keep the area constant. When b  s, it is the opposite – the graph is increased 1/b times along the  axis, and reduced b times along the vertical axis. The general ITTC formulation can be presented with the help of the nondimensional spectrum. Multiplying the unit-area spectrum, given by equation (16), by the area m0, we get: S() = ¼Ab s(x),

(18)

where b  B–1/4, x  b, whereas s(x) is the nondimensional (universal) ITTC spectrum, given by equation (17) and shown in Figure 3, common for all the spectra. 1.6 s (x)

1.2

0.8

0.4

x = bω 0 0

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 3 Nondimensional ITTC spectrum

The largest energy density equals sm e  and occurs at the nondimensional modal frequency xm  . The spectrum begins

Sea Spectra Revisited 579

practically at x . The nondimensional spectrum is a generic spectrum – to obtain any ITTC spectrum, the abscissa axis is divided by b = B , and the ordinate axis is multiplied by ¼Ab . In particular, maximum of spectrum occurs at m  (B) and equals Sm ¼Ab  Ab5 . Spectral moments of the nondimensional spectrum, denoted by sn, are given by equations (3), in which A , and B . Applying substitution x = bin the integral  mn =  0 n S()d, for n = , ,  and using for spectrum equation (18), it is easy to express spectral moments by moments of the nondimensional spectrum: mn = ¼ Ab 4 – n sn,

for n = 0, 1, 2, …

(19)

It follows from equation (18) that in order to carry out calculations, it is sufficient to approximate the nondimensional spectrum s(x), given by equation (17). Note that this function can be considered as if it were a probability density function since it satisfies all the conditions required for the probability density function. Hence, for the approximation any probability density function that diminishes exponentially can be used. There are many possibilities. Four of them will be discussed here: the log-normal distribution, a generalised gamma distribution, the gamma distribution and the Weibull one. The Log-normal Distribution Probability density is given by the equation (DNV, 1996): f x  

  ln x  a     2  , exp 12     2   x  a      1

(20)

where a,  and  are constants to be fixed; a is a lower bound. The modal value occurs at a point x  a  exp(  ). Moments of any order n  , , , …with respect to the lower bound are as follows: sn' = exp[½(n)2 + n ].

(21)

To get moments with respect to the origin, they have to be suitably transformed, which is not difficult. We get these: s  s'  a, s  s'  2as  a , s  s'  3as  a  s1  a, s  s'  4as  a  s2  as1  a, etc.

(22)

Knowing the nondimensional moments sn, moments of the spectrum itself can be obtained with the help of equation (19). The coefficients in equation (22) follow the pattern of the Pascal triangle. The best method of finding the constants a,  and , describing the log-normal distribution (20) is the least squares method. Minimising the sum of squared

580 M. Pawłowski

deviations between the functions s(x) and f (x) at the range x  yields: a  ,   – ,   . The differences between the two curves are at the second decimal place, i.e., around the thickness of a line, as can be seen in Figure 4. Generalised Gamma Distribution Probability density function is given by (DNV, 1996): (23)

c

f (x) = [c / ()][(x – a)]c  –1 e – [ (x– a)] , 1.6 s (x)

1.2

0.8

0.4

x = bω

0 0

1

2

3

4

Fig. 4 Nondimensional ITTC spectrum and log-normal distribution

where the four constants: a, c,  and  are to be fixed; a is the lower bound, whereas the inverse of  is the scale of distribution   . The constant c governs the speed of tail decay. The modal frequency occurs at a point: x  a   [(c 1)/c]1/c. The spectral moments with respect to the lower bound are these: sn' = n  ( + n/c)/ ().

(24)

The moments with respect to the origin are given by equations (22). The four constants, obtained with the help of the least squares method are as follows: a , c ,   and  . As can be seen in Figure 5, the two curves differ insignificantly but the fit is slightly worse than in the case of the log-normal distribution.

Sea Spectra Revisited 581 1.6 s (x)

1.2

0.8

0.4

x = bω

0 0

1

2

3

4

Fig. 5 Nondimensional ITTC spectrum and generalised gamma distribution

It can be shown that, depending on the selection of the constants a, c,  and  the generalised gamma distribution becomes a variety of different distributions, such as chi square, gamma, exponential, Rayleigh, Maxwell and Weibull, a fact which is not very well known. In other words, the above mentioned distributions are particular cases of the generalised gamma distribution, therefore they cannot get better results, which will be clearly seen below. Gamma Distribution Probability density function is given by (DNV, 1996): f(x) = [ / ()][(x – a)]  –1 e – (x– a),

(25)

where the three constants: a,  and  are to be fixed; as before a is the lower bound, the inverse of  is the scale of distribution   , and  is called the shape parameter. The modal frequency occurs at a point x = a +  ( – 1). The spectral moments with respect to the lower bound are as follows: sn' = n  ( + n)/ ().

(26)

The moments with respect to the origin are given by equations (22). The three constants, obtained with the help of the least squares method are as follows: a ,    and  . As expected, the gamma distribution fits somewhat worse than in the previous case, which is well seen in Figure 6.

582 M. Pawłowski 1.6 s (x)

1.2

gamma

0.8

0.4

ITTC

x = bω

0 0

1

2

3

4

Fig. 6 Nondimensional ITTC spectrum and gamma distribution

Weibull Distribution Probability density function is given by (DNV, 1996): 

f(x) =  (x – a)  –1 e –  (x – a) ,

(27)

where the three constants: a,  and  are to be fixed; as before a is the lower bound. The modal frequency occurs at a point x = a   [( – 1)/] 1/, where  =  –1/ is the scale of distribution, and  is the shape parameter. The spectral moments with respect to the lower bound are as follows: sn' = n  (1 + n/),

(28)

The moments with respect to the origin are given by equations (22). The three constants, obtained with the help of the least squares method are as follows: a ,   and  = 2,8; the scale  . Both curves differ yet more than in the two previous cases, which can be seen in Figure 7. This spectrum has the smallest bandwidth parameter , as its tail has the fastest decay.

Sea Spectra Revisited 583 1.6

s (x)

1.2 Weibull 0.8

0.4

ITTC x = bω 0 0

1

2

4

3

Fig. 7 Nondimensional ITTC spectrum and Weibull distribution

Statistical parameters of the discussed spectra are compiled in Table 1 according to the bandwidth parameter . It helps to select a spectrum according to this parameter. The constant c at the generalised gamma distribution (23), governing the tail decay, can always be chosen in such a way to get a spectrum with a given bandwidth . As can be seen from Table 1, all the approximations of ITTC spectrum have practically the same modal frequency xm. The best approximation is the log-normal one. The mean frequency x is identical with the centre of gravity of the area under the spectrum, whereas the zero-crossing frequency is the same as radius of inertia of the spectrum area. Therefore, for any spectrum the following holds x0  x1, supported by Table 1. The nondimensional frequency is understood as x b, where b  B. Table 1. Statistical parameters for nondimensional spectra of various approximations

xm x x x 

ITTC

log-normal

general 

gamma

Weibull

    

    

    

    

    

where xm is the nondimensional modal frequency, x is the zero-crossing frequency, x is the mean (characteristic, visual) frequency, and x is the peak frequency.

584 M. Pawłowski

4.2 Nondimensional JONSWAP Spectrum The nondimensional ITTC spectrum s(x), shown in Figure 3, refers to fully developed seas at open sea, whereas the JONSWAP spectrum represents windgenerated seas with fetch limitation. Contrary to ITTC spectra, for given wind speed and fetch length the resulting sea severity (in terms of the significant wave height hs) is random, having, however, a determinate modal frequency. In literature there is no explanation provided for this randomness, which probably results from the time elapsed from the previous storm. The random sea severity is governed by the peak-shape parameter , of random nature, whose mean value equals 3.3. Dividing JONSWAP spectrum, given by equation (10), by the area m   g/m for spectrum with , and introducing, as before, the nondimensional frequency x  b, leads to a nondimensional JONSWAP spectrum s(x):

s   x   s x   exp x xm 

2

/ 2  xm 2

,

(29)

where s(x) is the nondimensional ITTC spectrum, given by equation (17) and shown in Figure 3, whereas xm is the nondimensional modal frequency xm . Graphs of s(x), shown in Figure 8, illustrate the effect of -parameter on the nondimensional JONSWAP spectra. Since spectrum with  has a unit area, other spectra, with , have areas clearly greater than  (see Table 2). 10 * s (x)

γ=6

8

6 4 4 2 2 1 bω

0 0

0.5

1

1.5

2

2.5

3

3.5

Fig. 8 Nondimensional JONSWAP spectra s (x) as a function of -parameter 

Table 2. Area under the nondimensional spectrum s(x) for specified -value















m













Sea Spectra Revisited 585 

m varies almost linearly with -parameter. Its linear and quadratic approximations are these: m   , m       ,

(30)

For the mean value of  m . With the help of the above quantity, the area under the JONSWAP spectrum can now be easily calculated as the product of the area for , given by equation (11), and the quantity m: (31)

m  m g/  m.

The -related m factor has the meaning of a coefficient of amplification for the JONSWAP spectrum. Since m  hs, and substituting for  and m the expressions given at equation (10), the following is obtained for sea severity: hs   x x–   m .

(32)

For fetch length x   m and wind speed U  m/s, x–  . Assuming , the above yields for hs      m  m. Solving equation (32) with respect to wind speed, yields: (33)

U   m  x  hs,

where U is in m/s, x in km, hs in m, and m is a constant depending on -value. Contrary to equation (13), showing some degree of approximation, equation (33) is strict. Normalising the nondimensional spectra s(x) with respect to the coefficient of amplification m, a unit-area nondimensional JONSWAP spectra sJP(x) is obtained, shown in Figure 9. 2 sJP(x)

6

1.6

1.2

0.8

0.4

γ=1 bω

6 0 0

0.5

1

1.5

2

2.5

3

3.5

Fig. 9 Normalised JONSWAP spectra as a function of -parameter

586 M. Pawłowski

The ratio between the maximum density for specified -value and , denoted by , has the meaning of a real peak-shape parameter. As is seen in Figure 9, the greater the -value the greater the -value. Values of  are shown in Table 3. Table 3. -value for specified -value for the JONSWAP spectrum



lg 



       

       

       

The above -value can be very well approximated relative to the logarithm of :      lg    lg,

(34)

Assuming that the modal density is  times greater relative to the nondimensional ITTC spectrum and applying affinity transformation we get that the latter spectrum has to be  times reduced along the abscissa axis with the centre of transformation at the modal frequency. The area and the modal frequency are then unchanged. The following results in such a case for the normalised JONSWAP spectra: sJP(x')   s [x  xm   (x' – xm)],

(35)

where s(x) is the generic nondimensional ITTC spectrum, given by equation (17), x  b is the nondimensional frequency, the constant b equals B   xm /m, xm  , and x' is a new x after transformation. The peak-shape parameter  governs the concentration of the spectrum around the modal value. Comparison between the normalised JONSWAP spectrum for the extreme value of  (curve ) and a spectrum obtained through the affinity transformation (curve ) is shown in Figure 10. As can be seen, there are some modest differences between the two curves, particularly in regions away from the modal frequency. A perfect approximation for the normalised JONSWAP spectra sJP(x) can be achieved by applying the log-normal distribution, given by equation (20). Minimising the sum of squared deviations between the functions sJP(x) and f(x) at the range x   yields: a  ,   – ,   . As can be seen in Figure 11, computed for the extreme value of , differences between the two curves are invisible in the scale of this figure.

Sea Spectra Revisited 587 2

sJP(x) 1.6

1.2

0.8 1 0.4 2

2 1

bω

0 0

0.5

1

1.5

2

2.5

2

2.5

3

3.5

Fig. 10 2

sJP(x) 1.6

1.2

0.8

0.4

bω 0 0

0.5

1

1.5

3

3.5

Fig. 11

5 Conclusions The paper demonstrated that the ITTC and JONSWAP spectra can be reduced through the affinity transformation to a common unit-area nondimensional spectrum that can be precisely approximated by a log-normal distribution. Such approximated spectra, contrary to the original, are narrow-banded, with the bandwidth parameter less than 1, and have moments of any order.

References Seria OMK Ochi, M., K., (1998) Ocean waves, Cambridge University Press, ISBN 052156378 X, pp. 46–47 DNV, 1996, Proban distributions, Det Norske Veritas, Report No 94-7089/Rev. 1

On Self-Repeating Effect in Reconstruction of Irregular Waves Vadim Belenky Naval Surface Warfare Center Carderock Division (NSWCCD) - David Taylor Model Basin

Abstract The paper is focused on the previously known effect of “statistical self-repetition” encountered when a wave record is reconstructed from spectrum with a constant frequency step. A correlation function calculated for such a record shows a repetitive pattern of high values. It means an unrealistically strong probabilistic dependence between certain time sections of the process. The effect remains even if the autocorrelation function is calculated directly from spectrum without actual reconstruction of the wave record. An attempt to work the effect around using a variable frequency step leads to a spread of the error. It appears as a series of unrealistically high values instead of discrete peaks. The previous work of the author also found that the presence or absence of the effect strongly depends on the method of numerical integration. The paper concludes that the nature of this effect is a numerical error caused by the highly oscillatory character of an integrand in cosine Fourier transform for large values of time. The paper also discusses the practical implications of this conclusion and is based on earlier publications (Belenky 1994, 1995).

1 Reconstruction of the Irregular Waves: Visual Patterns Following (St. Denis and Pierson, 1953), the inverse Fourier transformation is the most common way to present irregular waves defined with their spectral density s(): N

 W (t )   rWi cos(i t  i )

(1)

i 1

Here, the set of initial phases i is comprised of random numbers, distributed uniformly from 0 to 2; this is the only stochastic figure in the model. The amplitude of the component rWi is calculated with the spectral density:

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_33, © Springer Science+Business Media B.V. 2011

589

590 V. Belenky i 0.5 i 1

rWi  2 SWi  2

 s()d ;

i  i  i 1

(2)

i 0.5 i

Wave elev. m

The set of frequencies i has to cover a significant part of the spectrum. However, it is not the only consideration when it comes to numerical simulation. It is known from the simulation experience that it is preferred not to distribute these frequencies evenly to avoid generation of the self-repeating process. Figure 1 shows a realization of the wave elevation process calculated with formula (1) using a Bretschneider type of spectrum for a significant wave height of 7 meter; 21 evenly distributed frequencies were used.

0 Time, s 0

100

200

300

400

500

600

700

Fig. 1 Time history of wave elevations

A closer look at the time history shown in Fig. 1 reveals that there are four wave groups looking suspiciously similar, these groups are shown separately in Figures 2a, 2b and 2c a) Wave elevation m 0 -5

0

50

100

150

Time, s 200

b) 0 -5 200

250

300

350

Time, s 400

450

500

550

Time, s 600

c) 0 -5 400

Fig. 2 Zoomed-in repeated patterns

Figure 2 shows that the first wave group in Figure 2a is very close to the third wave group in Figure 2c, the second wave group in Figure 2b resembles the first group taken but with the sign inverted. Figure 3a shows the first and the second wave group. The latter one is taken with the sign inverted and is superimposed over the first one with a 4 time step shift. The third curve in Figure 3a is a difference

On Self-Repeating Effect in Reconstruction of Irregular Waves 591

between them: it can be clearly seen that these fragments are similar, but not the same. The smaller solid curve shown in Fig. 3a indicates the difference between them. Figure 3b shows a similar picture for the first and the third wave group, with the latter group shifted 4 time steps for better visualization. The smaller solid curve in Fig. 3b shows the difference between the first and the third wave groups. a)

Wave elevation m 5 0 -5

b)

0

50

100

150

0

50

100

150

Time, s 200

5 0 Time, s

-5

200

Fig. 3 Patterns and difference between them (a) the first pattern (solid) and inverted second pattern (dashed) (b) The first pattern (solid) and the third pattern (dashed)

The similarities between the wave groups are reflected in a character of the autocorrelation function, taking into account that the zero mean value equals:

Rj 

1 Nt  j

Nt  j

x x

(3)

i i j

i 1

The result of calculations of autocorrelation function is presented in Figure 4. The autocorrelation function in this figure has several characteristic features. Intervals with the high amplitude oscillation are followed with low-amplitude intervals in the form of repeating patterns. The patterns are repeated as the group similarities are encountered. The second and the fourth patterns however, have an inverted sign in comparison with the first and the third one. Amplitude of oscillations in low-amplitude patterns rises: slightly for most of the duration and significantly at the end. The latter feature is eventually the result of decreasing volume of available data for formula (1) at the end of realization. 5 Autocorrelation, m2 0 -5

0

100

200

300

400

500

600

700

Time, s 800

Fig. 4 Autocorrelation function for irregular waves calculated for realization in Fig. 1

592 V. Belenky

As the spectral density of this process is known, cosine Fourier transform can be used (methods of rectangles used for numerical integration): 



R( j )  s () cos( j )d 

Nt

S

Wi

cos(i  j ) ;

 j  t  j

(4)

i 1

0

Here s is value of spectral density, SWi is the power spectrum, tj is current moment of time and t is a time step. The results are presented in Figure 5. The autocorrelation function presented there is generally similar to the previous one, but does not experience an increase of amplitude the statistics-related accuracy consideration is no longer relevant. 5 Autocorrelation, m2 0 -5

0

100

200

300

400

500

600

Time, s 800 700

Fig. 5 Autocorrelation function calculated by cosine Fourier transform with rectangular for numerical integration

Using the adaptive quadrature method for numerical integration dramatically changes the form of the autocorrelation function, see Figure 6. The Romberg method yields results identical to those shown in Figure 6. 5 Autocorrelation, m2 0

Time, s

50

100

200

300

400

500

600

700

800

Fig. 6 Autocorrelation function calculated by cosine Fourier transform with adaptive integration

Repeating patterns have disappeared completely from the autocorrelation function, shown in Figure 6. As the only difference between the results in Figure 5 and Figure 6 is the method of integration, it would be logical to state sensitivity of the calculation result of autocorrelation function is due to method of numerical integration applied. The presence of very similar repeating patterns in a process reconstructed with the constant frequency step is generally known. The period of the repeating is

TR 

2 

(5)

The periodicity of the autocorrelation function compromises validity of the model when the length of realization is longer than the time TR. To avoid such inadequacy for evenly distributed frequencies, their number should be increased with corresponding decreasing of frequency step it solves the problem but

On Self-Repeating Effect in Reconstruction of Irregular Waves 593

increases computational cost, as more components have to be dealt with at every time step.

2 Non-Uniformed Frequency Distribution: Spread Error Conventional way to avoid repeating patterns is to use non-uniformly distributed frequencies: for example, to increase the frequency step by 10% for each following (subsequent) frequency outward of the spectrum peak. This technique results in the spectrum shown in Fig.7 with the realization time history in Fig. 8. As can be seen from Fig. 8, the similar wave groups are no longer present, which is also confirmed by the autocorrelation function estimated with this realization by formula (3) and shown in Fig. 9. This autocorrelation function, however, exhibits a noticeable amplitude increase, past the time 110 seconds; these oscillations become visually stationary after approximately 200 seconds and exhibit significant increases of amplitude after approximately 700 seconds. The latter increase is probably caused by diminishing volume of statistics closer to the end of the time range. Similar behavior was observed in the previous case, see Fig. 4. Also, as in the previous case the end-of-the range amplitude increase is not reproduced when the correlation function is calculated directly from the spectrum with formula (4), see Fig. 10. The noticeable increase of the amplitude of oscillation observed after 110 seconds in the case with non-uniformly distributed frequencies can be considered as the similar kind of error that has been seen in the first case. The similarity between these two cases can be seen as different discretizations present the same integral in formula (3) or Fourier integral in rectangular discretization (4). Application of a different method of numerical integration (adaptive method with the result shown in Fig. 6) eliminates both errors. The difference is that the first one is seen as a repeating pattern, while in the second case the error is “spread” along most of the time interval. Spectral Density, m2 s 10

5

0.2

0.4

0.6

0.8

Frequency, 1/s

Fig. 7 Spectral density with non-uniform frequency set

594 V. Belenky Wave elevation, m

5 0 -5

0

100

200

300

400 500 Time, s

600

700

800

Fig. 8 Time history reconstructed with non-uniformly distributed frequencies 5

Autocorrelation, m2

0 -5

0

100

200

300

400

500

600

Time, s 700 800

Fig. 9 Statistical estimate of autocorrelation (3) evaluated foe the realization at Fig. 8 5 Autocorrelation, m2 0 Time, s -5

0

100

200

300

400

500

600

700

800

Fig. 10 Autocorrelation function calculated with cosine Fourier transform (4)

3 Adaptive Integration Method The adaptive quadrature method of numerical integration is based on the idea that smooth function requires less point to integrate accurately while quickly varying (especially oscillatory) would take more point to reach the same accuracy (Heath, 2002). In the first step, the integration is performed on the whole range (Simpson method typically is used). Then the range is divided in half and the integration is performed for both halves. The sum of these integrals then is compared with the result of integration over the whole range. If the difference is below the given threshold, the procedure is complete. To achieve better accuracy, Gauss-Legendre Quadrature (Bronshtein and Semendyayev, 1997) was used instead of Simpson rule. b

 a

7

f ( x)dx 

 0.5  A  (b  a)  f ( y ) i

i

(6)

i 1

where Ai are weights for the values of function f, calculated in points (nodes) specified as:

On Self-Repeating Effect in Reconstruction of Irregular Waves 595

yi  0.5(b  a ) zi  a  b 

(7)

Numerical values are given in Table 1 (Booth 1957). The Gauss-Legendre quadrature was used along with adaptive algorithm primary because the former is known for its outstanding accuracy with only a few points. Therefore, a combination of the Gauss-Legendre quadrature with the adaptive technique may be expected to yield the best accuracy. However, the number of points required to maintain accuracy rises quite significantly, as shown in Fig. 11. Table 1. Numerical Values for Gauss-Legendre Quadrature Ordinates, z

Weights, A

-0.9491079

0.129485

-0.74153119

0.27970539

-0.40584515

0.38183005

0

0.41795918

0.40584515

0.38183005

0.74153119

0.27970539

0.9491079

0.129485

Actually, if a number of frequencies will be increased, so will the time while the wave reconstruction is still valid. To achieve a duration of simulation of one hour, a conventional method with rectangular integration would require about 1150 frequencies. An adaptive method would need only about 600, which is also quite significant number. It means that the better accuracy of the autocorrelation function in Fig. 6 requires an increasing amount of frequencies, but due to the specific way adaptive algorithm works, the number of points is simply not known in advance.

Fig.11 Number of points of Adaptive / Gauss Legendre Quadrature vs. time of integration for autocorrelation function

596 V. Belenky

4 Oscillatory Character of the Integrand in Cosine Fourier Transform The increasing number of points needed to keep a given accuracy of the adaptive numerical integration procedure is a clear indication of the oscillatory character of the integrand in cosine Fourier transform. Actually, it could be seen from the structure of the formula (4). The time  plays a role of “frequency”, while the real physical frequency  is a variable of integration. Once the time increases, the integrand becomes quite oscillatory, which is illustrated in Figure 12. a) τ =0

b) τ =20 s 5 s(ω)cos(ωτ)

s(ω)cos(ωτ)

10

5

ω 0

0.5

1

1.5

ω

0

0.5

1

5

1.5

c) τ =40 s

d) τ =60 s 10

10

5 s(ω)cos(ωτ)

s(ω)cos(ωτ)

5 ω 0

0.5

1

1.5

5

ω

0

0.5

1

1.5

5

10 10 e) τ =100 s

f) τ =300 s

10

10 ω 0

0.5

1

1.5

s(ω)cos(ωτ)

s(ω)cos(ωτ)

10 ω 0

0.5

1

1.5

10

Fig. 12 Oscillatory behavior of the integrand of cosine Fourier Transform

Plots on Figure 12 also show how quickly the integrand becomes oscillatory. Integration of such an oscillatory function, indeed, presents a challenge and requires using many points on the frequency axis.

On Self-Repeating Effect in Reconstruction of Irregular Waves 597

Numerical errors, which are inevitable when time is large and the number of frequencies is limited, will cause unrealistic close probabilistic dependence of an otherwise distant section of the stochastic process. As it has been demonstrated above, these errors will be transferred to the time history records, making results of the simulation questionable from a statistical point of view.

5 Concluding Comments In general, the self-repetition effect of a stochastic process presented with the inverse Fourier Transform is related with numerical error. This error is caused by the highly oscillatory nature of the integrand in the cosine Fourier transform and therefore cannot be eliminated by rearranging frequencies. In the case of a constant frequency step, the inverse Fourier Transform is equivalent to Fourier series and the self-repetition effect is also a direct result of periodicity of Fourier series defined at the finite interval. What is the practical meaning of the above statement? It means that if a long period of simulation is needed, a large number of frequencies must be used. For example, for the Bretschinder spectrum for significant wave height 12.5 m with zero crossing period 11.5 s, about 2050 frequencies would be needed to reconstruct a six hours wave record. It means that 2050 components have to be summed at every time step of the simulation which significantly increases the computational cost of such a simulation. Alternatively, only 80 frequencies are necessary for the correct reconstruction of a 12 minute long wave record. Keeping in mind that only one minute is needed for complete decay of wave autocorrelation function (in this sample), it would be quite practical to use 30 such wave records instead of one six-hour-long. Each of these wave records must have its own independent set of initial phases. It would be reasonable, however, to check and compare confidence intervals for main statistical characteristics of on six-hour-long record with the set of 30 twelveminutes records.

References Belenky V (2004) Risk evaluation at extreme seas. Proc of 7th Int Ship Stab Workshop, Snaghai Belenky V (2005) On Long Numerical Simulations at Extreme Seas. Proc of 8th Int Ship Stab Workshop, Istanbul Booth AD (1957) Numerical methods. 2nd ed, Butterworths Sci Publ, London Bronshtein IN, Semendyayev KA (1997) Handbook of Mathematics. Springer, Berlin Heath MT (2002) Scientific Computing an Introductory Survey. 2nd ed, McGraw-Hill, N.Y. St. Denis M, Pierson WJ (1953) On the motions of ships in confused seas, Trans SNAME 61

New Approach to Wave Weather Scenarios Modeling Alexander B. Degtyarev St.Petersburg State University, St.Petersburg, Russia

Abstract In the paper problems of wave climate description and modelling are considered. Wave climate is considered as ensemble of conditions of spatio-temporal wave fields characterized by frequency-directional spectra. Such approach with reference to shipbuilding is caused by expansion of the nomenclature of characteristics of wave and a wind, which are necessary for construction and operation of engineering offshore constructions, and operation of ships. Using of expanded set of wave and wind characteristics takes possibility to introduce term “scenario” of wave weather much more correctly and to use it for estimation of navigation safety.

1 Introduction Initially the concept of “scenario” of wave weather was associated with ideas of “mission” or “assumed situations”. First such approach was suggested by N.B. Sevastianov in the field of stability in the beginning of 60s (Sevastianov 1970). One of the main ideas proposed by N.B. Sevastianov was assumption that the uncountable infinite set of situations may be substituted with sufficient accuracy by the countable and finite sets of discrete situations. Investigation of each element from such set and consideration of each situation in integral estimation with correspondent weight give possibility, as a result, to calculate risk function with the help of full probability formula (Sevastianov 1994). Thus reference of assumed situations to some patterns is an important procedure of risk assessment. Considered vector of assumed situations, in which each element is a set of situation parameters, includes both parameters of ship and characteristics of wind and wave excitations. Characteristic set is defined by state-of-the-art of method applied for assumed situations investigation. Problem of involvement of new knowledge about waves, their spectrum, variability, alternation of storms and slack sea arises with methods accuracy improvement. As was mentioned another approach to considered problem is related with “mission” concept. In 80s some authors proposed to consider ship’s ability to M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_34, © Springer Science+Business Media B.V. 2011

599

600 A.B. Degtyarev

carry posed task from the complex point of view including seaworthiness, region of operation, etc. Traditional consideration of only evident extreme stormy situations conditions using of simple approximate expressions for waves spectral density. It restricts shipbuilders’ requirements down to knowledge of wave height (hs) or set (hs,  ) (Comstock and Keane 1980, Boroday and Netsvetaev 1982, etc.). Neither complex sea, nor storm duration or character of alternation of bad and good weather are not explicitly used in applied methods for risk estimation. Information about geographic region where ship will operate is rarely used for risk estimation (see, for example (McCreight and Stahl 1985)). Static character of considered phenomena a priori excludes concept of representative voyage, assumed trip, and operation scenario. We can improve risk estimation method if additional parameters characterising waves in assumed situations (different wave systems, wave directions for considered region) are taken into account. But lack of situation evolution in computational method inevitably makes it inexact. The author in co-authorship with scientists in oceanology field has considered such methods in a series of papers (Boukhanovsky et al. 2000, 2001, etc.).

2 Climatic Wave Spectra. Wave Climate Development and perfection of methods of wave measurements (e.g. ocean weather ships, buoys, platforms, etc.) allow one to gather hundreds (Ochi 1978) or even thousands (Buckley 1988, 1993) of wave spectrum in separate points. It takes possibility to introduce new conception of “climatic wave spectrum”. Initially it was proposed at the XVIII IMO Assembly (1993) as “A wave spectrum which results from averaging the energy density ordinates of a number of measured wave spectrum. It is determined for specified significant wave height class intervals, typically 1 meter”. At the same time at this Assembly wave climate was defined: “A seaway characterization determined primarily from climatic wave spectra obtained at a particular geographic location”. It is characterized by parametric and other properties of the local climatic wave spectra and by the associated probability density distribution of significant wave height. However, the problem of climatic wave spectra calculation is still open because operation of averaging is acceptable only for physical homogeneous phenomena. It involves necessity of climatic wave spectra classification. So Buckley proposed to classify wave spectra in formal way. He divided all measured spectra into classes with different significant wave height. Then he averaged them in each class at each frequency (Buckley 1988, 1993). However, wind waves of the same height could result from various weather conditions (rising sea, decaying wind waves, swell, etc.). They are characterised by different spectra. We cannot consider introduced terms as settled because some terms used in definitions are not defined. So definition of climatic wave spectrum is acceptable if we shall keep in mind during averaging procedure probabilities of wave making conditions

New Approach to Wave Weather Scenarios Modeling 601

corresponding to different kinds of spectra (i.e. wind speed, duration, fetch, etc.) (Boukhanovsky et al. 1997). Let us consider example of wind wave spectrum variations during a storm passage. Fig. 1 shows an example of variation of frequency spectrum S(,t) at a point located in the North Atlantic as measured by RV “Weather Reporter” of the UK from 15th to 19th December of 1959 (Wilson 1965). The upper panel of the figure also shows data on wind speed u, wave height variance D, and mean wave period . Wind strengthening up to 30 knots was taking place from 6 to 18h of December 16, 1959. The wind wave spectrum during that period of time did not change significantly. From 18h of December 16th to 03h of December 17th the wind veered and strengthened up to 62 knots. The wave spectrum was growing quickly and reached its peak by 18h of December 17. Then it weakened a little following corresponding decrease in the wind speed and again strengthened reaching the maximum at 0h of December 18th. Subsequent variations of the spectrum correspond to the storm wave decay.

Fig. 1 An example of variations of frequency spectrum S(,t) (m2s). The upper panel: variations of total variance D(t) (m2), mean wave period (s)  and wind speed u (knots).

This example shows very good, that spectral density character even at wind and waves of equal intensity depends on waveformation prehistory. It does not take right to group wave weather states using trivial principle for unification (e.g. significant wave height) in determination of average for some class of spectra. Similar examples are included in chapter 11 of (Lopatoukhin et al. 2001). In any case linear wind waves, as a stochastic hydrodynamic process, are characterized by a directional spectrum S(, , r, t), depending on the frequency , the wave direction , the spatial location r, and the time t. So for detailed description of any wind wave process including climatic wave states it is reasonable to apply this term, and Buckley’s idea about climatic wave spectrum is good. Synoptic, annual and year-to-year variability produce, in turn, polymodulation of

602 A.B. Degtyarev

the associated wave field ζ(r, t). By an ensemble of climatic wave spectra we mean a collection {S(, , r, t)} for r  R , t  T, where R is a spatial region, and T a time span of the order of decades. The main objectives of spectral wave climate investigations are in broad terms  to select classes of wave spectra and estimate their probability;  to propose a parameterization which allows one to justify a choice of difference between the various classes (i.e., the selection of discriminant variables);  to approximate the ensemble {S(,)} in terms of its probabilistic characteristics;  to elaborate a stochastic model of the spectral wave climate;  to estimate climatic wave spectra for extreme waves (design waves).

3 Parameterization of Climatic Directional Wave Spectra The main approach to presentation and calculation of climatic wave spectra is posed in papers (Boukhanovsky et al. 2000, Lopatoukhin et al. 2001, 2002, etc.). It consists of wave spectra typification in frequency range in accordance with genetic set of patterns so as spectra in each class are geometrically similar each other and differ only in parameters correspondent to different wave formation. Such parameterisation of considered random functions (spectral density) is caused by wishes to simplify simulation of considered process. Standard approach to such simplification is presentation of random function with the help of deterministic function with a set of random parameters . The parameterisation allowed one to write wind wave spectra, Sww, and swell spectra, Ssw, as non-random functions S  S ( , , )

(1)

dependent on a set of random arguments, . The feasibility of an approach like (1) obviously depends on to which accuracy the spectrum Sp(,may be specified by the parameters p taken from their multidimensional distribution F() In the present study, parameters of the spectrum related to wave height, spectral shape, the frequency of the spectral peak, max, and the main wave direction, max, are selected as parameters in . The single field model spectrum may be written Sp(/max, -max, r), where r signifies the rest of the parameters. More general spectra S(,θ are obtained as

S ( ,  )  m00

N

 p 1

p

 Sp ,    max ,  r  p 

p

   

(2)

New Approach to Wave Weather Scenarios Modeling 603

where m00, the 0th moment of the spectrum, is equal to the total variance of wave field, N is the number of wave fields (peaks in the spectrum), and p are weight factors for each system so that,



N p 1

 p 1.

The following presentation is rather good for wind waves and swells:

S ( , max , n) 

n    max  max

n   n    exp       n  1  max   

1n

   

 ,  

(3)

Similarly, a well known directional distribution is

Q0  ,  max , m   Cm cos m    max ,

   max 

 2

,

(4)

where Cm is a normalizing parameter such that integral of function Q0 from 0 to 2 is equal one, and the m parameter determines the width of the angular distribution. Thus application of formulas (2)-(3) allows to describe each of N wave systems in wave spectrum (1) by the set of five parameters {p, max p, max p, n, m} that determine its shape and location. At that m00, max p, max p could be found directly from the spectra and we can apply Monte-Carlo method for n, m, p estimation (see (Lopatoukhin et al., 2002)). The results of general procedure of spectra classification for Barents Sea carried are shown below. For calculations measurement data and data of hydrodynamic modeling were used. The following classes were select: One-peaked spectra (I). One wave system prevails – either wind waves (I-1) or swell (I-2). In relation (2), N=1, 1=1, and only one extreme (max, max) occurs in the marginal S() and Q(θ). The separation between wind waves and swell is based on non-dimensional steepness defined as



2 g max  hs

 2g 2 m00 max

If , then spectrum belongs to a swell, otherwise to wind waves. Two-peaked spectra (II). Two wave systems exist simultaneously. In relation (2) N = 2, 1= , 2=1–. We can recognize three subclasses (depending on the number of maxima in the marginal spectra): II-1. Mixed spectra with separation both in frequency and direction. In this case, there are two pronounced maxima (max1,max1), and (max2,max2), both in the frequency spectrum and marginal angular distribution. II-2. Mixed spectra with separation only by direction. In this case, there is one peak in the frequency spectrum and two peaks in the angular distribution. As a

604 A.B. Degtyarev

result in the two-dimensional spectra, there will be peaks at the same frequency (max1,max1) and (max1,max2). II-3. Mixed waves with separation only by frequency. The angular distribution has only one peak and the frequency spectrum is broad with a not so pronounced second peak. Multipeaked spectra (III). Complicated wave fields with two or more swell fields. In this case, the angular distribution has more than two pronounced peaks. The estimated probability for belonging to each class of spectra in different months and during the year is presented in Table 1. Table 1. The probability of classes of climatic wave spectra in the western part of the Barents Sea. Probability, % Сlasses

Subclass

January

April

July

October

I

I–1

45

42

32

48

42

I–2

17

18

24

20

20

II–1

10

6

3

3

6

II

III

Annual

II–2

22

25

30

25

26

II–3

1

3

3

2

2

III–1

5

6

8

2

4

Examples of input one, two and multi-peaked spectra and their approximation by the model spectra are shown in the Fig. 2. In fig. 3 the example of calculation of an average spectrum S(,,  ) of complex waves (Northeast part of the Black Sea), and also correspondent probabilistic intervals are shown.

4 Parameterization of Storms and Good Weather Time series of wind wave heights in mid-latitudes and subtropical areas of the World Oceans make alternating sequences of storms and weather windows. We define a storm of duration  and intensity h+ as a situation when random function h(t) exceeds a predefined value Z. The period  during which the wave height is less than this threshold will be called a weather window of intensity h– . The parameter  shows the asymmetry of the storm: =(tp–tb)/tb, tp , te are times of storm start, maximum development and end, respectively. Fig. 4 clarifies these definitions. Random functions  and  represent duration of over-shots and under-shots. Therefore their distributions should asymptotically tend to the exponential law (Leadbetter et al. 1986):

New Approach to Wave Weather Scenarios Modeling 605

 x F x   1  exp    x

(5)

Figs. 5 and 6 depict quantiles of distributions F*() and F*() as the q-q biplots. It is seen that the hypothesis that F*(x) belongs to a class of exponential distributions can be confirmed. Hence m and  should be nearly equal.

Fig. 2 Classes of climatic wave spectra. 1 – Hindcasting, 2 - Approximation

606 A.B. Degtyarev

Fig. 3 Frequency-directed climatic spectrum of complex sea. North-Eastern part of the Black Sea

h(t)

1-





i+1



tp tB

te

i

i+

h i h i1

h i 1

2

3

4

Fig. 4 Parameters describing storms and weather windows

5

t, days

New Approach to Wave Weather Scenarios Modeling 607

Fig. 5 Empirical distribution of storm duration F() /quantile bi-plot of exponential distribution (5)/. The Baltic Sea

Fig. 6 Empirical distribution of weather window duration F() /quantile bi-plot of exponential distribution (5)/. The Baltic Sea

Table. 2 gives correlation coefficients between different random functions in system {h+, h–, , } Table 2. Correlation coefficients  between impulse parameters Values

(h+,h–)

(h+,

(h–,)

(,)

(h+,)

(h–,)



-0.10.15

-0.150.05

-0.10.1

-0.10.1

0.50.8

-0.55-0.7

Hence, as the first approximation it is possible to consider parameters (h+, h), (h ,), (h,), (,) independent while parameters (h, ), (h+,) are dependent because their correlation coefficient is in range of 0.50.8. Hence, the fourdimensional distribution F (h+, h,  ,) can be expressed as a product of two twodimensional distributions F(h+, ) and F(h,), each of them being equal to +

F(x, y)  F(x) F(y | x)

(6)

i.e. to multiplication of marginal distribution F(x) and conditional distribution F(y|x) where x ={,} and y ={h+, h}. Different approximations for distribution F are given in report (Lopatoukhin et al. 2001). Examples of storms classification are given by author and other scientists in many papers (Boukhanovsky et al. 2000, Lopatoukhin et al. 2001, etc.). One of them (five storm classes specified for Black Sea (Lopatoukhin et al. 2001)) is shown in table 3. A matrix of probabilities that a certain storm category in 3 (for h=Z) will transform into another category is shown in Table 4. It follows from the table that there is some weak correlation between categories of consecutive storms.

608 A.B. Degtyarev Table 3. A classification of storm shapes Non-dimensional shape

Threshold Z=1.0 h t  , h t 

Threshold Z=2.0 h t  , h t 

Category

being mean seasonal wave height being mean seasonal wave height abscise is (t-tb)/

P, % N

ordinate is h/h+

Wave height h (cm)

Duration S P, % N Wave (hour) height h (cm)

h95%=207 S95%=45.5

I

mS=11.0

h=57

S=14.2

h=59

S=8.0

h5%=21

S5%=1.0

h5%=44

S5%=0.7

15% 33

mh=84

mS=28.7

6% 13

19% 41

S=22.4

h=63

S=10.3

S5%=5.0

h5%=43

S5%=1.8

mh=138

mS=44.9

10% 22

h95%=207 13% 20 mh=137

S95%=36.0 mS=19.6

h=75

S=25.4

h=61

S=11.0

h5%=33

S5%=8.5

h5%=66

S5%=5.0

h95%=277

S95%=110.5

13% 20 mh=134

mh=108

mS=40.9

h=63

S=23.3

h=60

S=25.1

h5%=44

S5%=12.2

h5%=42

S5%=3.5

h95%=181

S95%=184.5

h95%=197 S95%=135.8

V

S95%=38.3 mS=14.8

h=54

h95%=273 S95%=82.5

IV

h95%=267 24% 38 mh=121

mS=6.9

h5%=22

h95%=273 S95%=95.5

III

49% 78 mh=105

S95%=25.8

50% 110 mh=61

h95%=203 S95%=71.7

II

h95%=241

Duration S (hour)

1% 2 mh=181

mS=34.0

mh=104

mS=70.0

h=64

S=41.5

h=1

S=65.7

mS=118.8

h5%=31

S5%=9.5

h5%=180

S5%=53.1

According to (Goldman 1977), it is possible to reconstruct conditions of a hypothetical (artificial) storm that would lead to the highest practically possible waves at a location of interest. The idea is to look at a situation that did not happen as yet but can, in principle, happen in future.

New Approach to Wave Weather Scenarios Modeling 609 Table 4. Probability matrix of transformation of one storm category into another Storm category

I

II

III

IV

V

I

0.5

0.1

0.1

0.2

0.1

II

0.3

0.1

0.2

0.2

0.2

III

0.6

0.2

0.1

0.1

—

IV

0.3

0.2

0.2

0.3

—

V

0.2

0.3

0.2

0.4

—

5 Probabilistic Models for Long-Term Spatio-Temporal Wave Fields Reproducing At the quasi-stationary and synoptic intervals of variability the wave process is best described by the stationary auto-regression model AR(p) of order p (Boukhanovsky et al. 1998, Boukhanovsky and Degtyarev 1996), namely p

 t   k  t k   t , t = (t)

(7)

k 1

where k are coefficients to be computed using correlation function K(), t is white noise with a given distribution function, which has to be compatible with the nonlinear functional transformation f() of function t into, respectively, the Rayleigh or log-normal distribution of t. In (Lopatoukhin et al. 2001) it is shown that a stationary pulse-like random process is a good model for sequence of storms and fair weather intervals. A sample can be generated as follows: n



k 1



k 1



j 1



 (t )   wk  Z , t    j   j 

(8)

where j and j are, correspondingly, duration of storm and weather window (with threshold value Z),

 

 

Z  h   Z u t 0  t  ,  w( Z , t )  Z  h   Z u t      t      0 t  0  ∪ t       h+, h are the highest wave height in storm and the minimal wave height during the weather window. Function u(t) prescribes shape of the non-dimensional impulse. Triangular shape of this function looking as

610 A.B. Degtyarev

t  0  t   ,  u (t )  1 1     t 1     t  1  0 t  0  ∪ t  1  serves as a good first approximation. Parameter , as seen from fig. 4, sets asymmetry of function u(t). If =0.5, the function is symmetric. Actual generation of a series of random storms and weather windows is based on the Monte Carlo approach. It makes possible to reproduce the whole variety of values of {h+, h–, , }:

     |  ,h  F 

k  F1  1( k ) ,  k  F1  2( k )  k

1 h  |

h F

(k ) 3

k

 k

1 h  |

(k ) 4

| k



(9)

 

Here  i(k ) denotes a system of four pseudo random numbers. Stochastic model for extra-annual rhythms could be written as follows: .t)=m(t)+(t)t

(10)

Here m(t) and (t) are periodic functions, and t is a non-stationary process AP(p) so that p

 t   k (t ) t k   t

(11)

k 1

Coefficients k(t)=k(t+T) are periodic functions of time. A model that is capable to describe year-to-year variability of monthly mean wave heights will therefore require twelve values of m(t) and 78 values of K(t,). It is possible to reduce the number of dimensions by considering the following representation of PCSP:

 (t ) 



 (t ) expi t  k

k  

k

(12)

Here k(t) are stationary random processes (components) with mathematical expectation mk and co-variation function Kk() that can be obtained by expressing functions m(t) and K(t,) as the Fourier expansion series. A simpler model for PCSP can be obtained by expanding function (t) for each annual interval, as follows:

New Approach to Wave Weather Scenarios Modeling 611 q

 (t )  a0   ak cos  k t  bk sin  k t 

(13)

k 1

where ak and bk are random values, and q is the order of the model. For a stationary process it is possible to suppose that values ak and bk are independent, while for a non-stationary process they will be dependent. Table 5 gives average values of means(mak,mbk), variances(Dak,Dbk), co-variation Kak,bk, and correlation ak, bk for coefficients of the model of annual rhythms. Hence, instead of model (10-11) with 90 parameters, a simpler model (13) with 20 parameters may be used. Monthly mean values of wave heights in the Black Sea were used for the computations. Corresponding values of m(t) and D(t) are shown in Fig. 7. Table 5. Statistical parameters of coefficients ak, bk of monthly means wave height rhythms model (13). The Black Sea. Parameter

m,cm

D, cm2

Kak,bk (cm2) and ak, bk a0

a1

b1

a2

b2

a0

80

22

1

0.66

0.54

0.22

0.60

a1

19

42

20

1

0.21

0.65

0.47

b1

12

17

11

6

1

-0.26

0.52

a2

2

39

6

26

-7

1

0.15

b2

4

19

12

13

9

4

1

Note: co-variation Kak,bk is given below diagonal and correlation coefficient ak, bk is given above the diagonal.

3

D(t),cm

m(t),cm 80

( a)

(b)

1 2

2

1 2

60

1 40 0 20

1

2

2

4

6

8

10

12 month

0

2

4

6

8

10

12 month

Fig. 7 Mathematical mean (a) and variance (b) of monthly mean wave heights. All values are centered with respect of the mean annual wave height:(1)–The Baltic Sea, (2)–The Black Sea.

612 A.B. Degtyarev

6 Scenarios of Wave Weather as Initial Data in Seaworthiness Problems Such approach to wave weather evolution presentation in considered water space permits to develop mathematical model of sea waves for long-term periods. The main characteristic of this model is multiscale, i.e. taking into account different time intervals: quasistationarity, synoptic variability, season and year-to-year variability. So it is possible to obtain ensemble of wind wave fields realizations of any time length. Thus introduction of “climatic wave spectrum” and “wave climate” definitions result in expansion of design conditions base of external action for ship’s dynamic problems solution. Now instead of traditional poor set of some integral parameters (significant wave height, spectrum approximation, etc.) we can introduce new concept – wave weather scenario. It can take into account subject to considered problem the following items:

 peculiarities of wave formation conditions – wind waves, swell, complex sea;  geographical features of considered region;  variability of hydrometeorological conditions – features and characteristics of storms evaluation and good weather permanence;  scenarios of synoptic variability – alternation of storms and good weather states;  characteristics of season variability – features of summer, winter time and offseason for considered navigation regions, special missions carried out by ship, etc.;  long term presence of ship in given region or in known exploitation conditions comparable with life time of considered object. In this case it is possible to propose for using in problems of research design, seaworthiness safety estimation, risk assessment for ships and offshore structures the following scenarios of wave weather:

 short-term scenario – modelling of spatio-temporal wave fields realizations taking into account all counted before peculiarities;  scenario “storm” – wave actions modelling for typical storms in given region and season;  scenario “mission” – variation of wind and wave conditions and external actions on ship during specific mission carrying out: voyage, rescue operation, ship raising, survey operations, combat mission, etc.;  scenario “navigation” – consequence of ordinary scenarios “mission” covering long period including some seasons such as fishery, long navigation;  scenario “life time”- taking into account year-to-year and climatic variability of given region where ships and offshore structures operate; first of all it devotes to risk estimation in complicated expansive open-sea objects insurance.

New Approach to Wave Weather Scenarios Modeling 613

7 Wave Weather Sceneries Modeling Initial statistical information about wind and wave regime in given region is required for sceneries creation with the help of the methods described before. Obviously it is impossible to obtain such information by the way of measurement data only. The most fundamental starting point for derivation of equations governing the wave spectrum evolution is the equation for the conservation of the wave action density N (see, e.g., (Komen et al. 1994, Lavrenov 1998)):

N N N N N N    k   G    t  k

(14)

The action N is a function of latitude , longitude , wavenumber k, angle  between the direction of wave propagation and the parallel, angular frequency , and time t. G is net source function. It is represented as the sum of the input Gin by the wind, the nonlinear transfer Gnl by resonant wave-wave interaction, and the dissipation Gds. There are some other terms (interaction with slowly variable currents, etc.) which are normally small. They are not included in the propagation operator. Equation (14) describes functional relation between fields of atmosphere pressure, wind and waves. There are many calculation models based on (14) meant for obtaining time-spatial wave field. All they are differ from one another by sources function presentation and computational layout. Present spectral wind wave models based on equation (14) are rather well developed. They incorporate a representation of all significant mechanisms affecting the wave spectrum evolution and are quite sophisticated numerically. Being forced by wind data (or atmospheric pressure), and data on boundary layer stability, the models compute two dimensional (with respect of frequency and direction) spectrum S(,) at nodes ri of numerical grid at times tj. The first wave model which was realized as world famous software is WAMmodel (Ocean 1985). The theory and methods of numerical simulation are continuously improved. Now we have new results and models (WAVEWATCH (Tolman 1991), PHIDIAS (Van Vledder et al. 1994), TOMAWAC (Benoit et al. 1996), INTERPOL (Lavrenov 1998)) for deep and (SWAN (Ris 1997)) for shallow water. Specific character of computer presentation of hydrometeorological fields information is large volume of used data and long time for calculation. Hence, application of high performance computers is necessary. Any hydrodynamic model for wave fields calculation requires in initial wind data in meshes of net domain with assigned discretization in time domain. Available information till recently was heterogeneous, fragmentary and discrepant. A significant advance in numerical wave hindcasts resulted from the NCEP/NCAR meteorological reanalysis project (Kalnay et al. 1996), which produced global data series of great interest to wave modeling. The use of the reanalysis products to

614 A.B. Degtyarev

drive the wave model removed many of the inhomogeneities present in earlier data sets. For example results have shown in fig. 2,3 were obtained with the help of simulation using a 0.50  1.50 grid, covering the North Atlantic, Greenland, and the Norwegian and Barents seas. The full directional spectrum S(,θwas calculated at each grid points with 24 values in direction  and 25 values in frequency. The time step was 6 hours so that in any point r, a collection of more than 21000 spectra with more than 13 millions numerical values had to be considered. At the same time it is necessary to note that such technology is rather rough as long as reanalysis information is presented with rough space resolution. It takes possibility to obtain general representation about atmosphere processes evolution. With the object of improvement of atmosphere parameters it is possible to use special interpolation procedures (Reference data 2003) or to use regional models of atmosphere circulation. There are well-known such models as American model MM5 and WRF and European model HIRLAM. These models permit to calculate parameters of atmosphere boundary layer with high spatio-temporal resolution. These parameters include wind speed and direction, pressure, temperature, etc. Codes of all mentioned model are open source. It is possible also to obtain detailed manuals of these software. All these models permit to obtain initial information for statistical generalization, spectra parameterization, storms classification and scenario wave calculations. Then general algorithm for data preparation and realization of different scenarios looks like by the following: 1) Initial data of pressure fields preparation using reanalysis data for considered region or with the help of regional models of atmosphere circulation. Input of bathymetrical map, coastline and variation edge of the ice. 2) Verification of prepared data on the basis of comparison with natural observations. If occurrence of interpolation is bad than changing of model parameters for recalculation and go to i.1. 3) Wave fields calculation on the basis of model (14). Computational grid has to cover on the safe side considered region. It is necessary for taking into account influence of distant storms and incoming swell. Character of expansion of computational grid is defined by the expert evaluation taking into account geographical conditions of considered region. Time period of calculation depends on aims. 20-25 years are necessary for reliable statistical data. For purposes of extreme statistics this period has to be prolonged up to 30-40 years. 4) Verification of wave fields with the help of waves measurements in considered region (if we have long buoy records). Correction of model parameters and recalculation i.3 if big error of statistical characteristics exists (e.g. see (Reference data 2003)). 5) Assimilation of calculated data and measurements 6) Statistical treatment of obtained spatio-temporal wave fields and measurement data a) calculation of trivial statistics;

New Approach to Wave Weather Scenarios Modeling 615

b) determination of statistical parameters characterized storms and good weather periods (weather window); c) storms and weather windows classification; d) parameters of storms and weather windows interchange; e) climatic wave spectra classification; f) extreme statistics calculation; g) calculation parameters related with extreme waves. 7) Data assimilation for models of wave scenarios operation. 8) Using of year-to-year rhythmic model for climatic variation of wave weather reproduction in given region. 9) Superposition of climatic variations and results of probabilistic modeling of annual rhythmic. 10) Superposition of obtained results and results of stochastic modeling of storms and weather windows interchange. 11) Stochastic modeling of climatic spectra consecution corresponding to classes of storms and weather windows. 12) Time variation of frequency-directional spectra reproduction on the basis of obtained realization of average wave height and climatic spectra consecution. 13) Spatio-temporal wave fields generation for each wave spectrum. 14) Subject to solving problem and considered time scale reiteration of i.i.8-13 or collecting wave scenario ensembles.

8 Conclusion Considered models are in framework of united probabilistic-hydrodynamic approach. Such full set of the models changes general approach to problem of external forces acting on ship. Introducing of “climatic spectrum” definition permits to create ensembles of wave weather scenarios and to use them for marine object behavior simulation. Thus weather scenario is generalization and new level in assignment of external conditions in seaworthiness problems.

References Benoit M, Marcos F, Becq F (1996) Development of third-generation shallow water wave model with unstructured spatial meshing. Proc 25th Int Conf Coast Eng ASCE Bogdanov A, Degtyarev A, Stankova E, Shoshmina I (2006) Wave weather scenarios modeling using GRID technology. Proc. of the 9th Int conf STAB'2006 Rio de Janeiro 1: 279-286 Boroday IK, Netsvetaev YA (1982) Seakeeping of ships. Sudostroenie. Leningrad, Russian Boukhanovsky AV, Degtyarev AB (1996) The instrumental tool of wave generation modelling in ship-borne intelligence systems. Trans. of the 3d Int Conf CRF-96. St.Petersburg 1: 464-469

616 A.B. Degtyarev Boukhanovsky A, Degtyarev A, Lopatoukhin L, Rozhkov V (1997) Climatic spectra of wind waves. Shipbuild 4:14-18. Russian Boukhanovsky A, Degtyarev A, Lopatoukhin L, Rozhkov V (1998) Probabilistic modelling of wave climate. Izv. of RAS ”Physics of atmosphere and ocean”. 34 N2: 261-266. Russian Boukhanovsky AV, Lavrenov IV, Lopatoukhin LJ, Rozhkov VA, Divinsky BV, Kosy’an RD, Ozhan E, Abdalla S (1999) Persistence wave statistics for Black and Baltic seas. Proc. Int. MEDCOAST Conf. «Wind and wave climate of the Mediterranean and the Black Sea». Antalya, Turkey 199-210. Boukhanovsky A, Degtyarev A, Lopatoukhin L, Rozhkov V (2000) Stable states of wave climate: applications for risk estimation. Proc of the 7th Int conf STAB'2000. Launceston, Tasmania, Australia 2:831-846 Boukhanovsky A, Rozhkov V, Degtyarev A (2001) Peculiarities of computer simulation and statistical representation of time-spatial metocean fields. In: Comput Science - ICCS 2001, LNCS 2073, Springer, part I 463-472 Buckley WH (1988) Extreme and climatic wave spectra for use in structural design of ships. Nav Eng J Sept 36-57 Buckley WH (1993) Design wave climates for the world wide operations of ships. Part 1 establishments of design wave climate. Int. Mar Organisation (IMO), Selected Publications Comstock EN, Keane RG (1980) seakeeping by design. Nav Eng J April 157-178 Degtyarev AB, Boukhanovsky AV (1996) Probabilistic modelling of stormy wave fields. Proc of the Int Conf “Navy and Shipbuild Nowadays” St Petersburg February 26-29, 2:A2-29. Russian Goldman IL (1977) An approach to the maximum storm. Proc. 9 Ann. Offshore Conf. Houston, 2: 309-314 Kalnay E, Kanamitsu M, Kistler R, Collins W, Deaven D, Gandin L, Iredell M, Saha S, White G, Woollen J, Zhu Y, Leetmaa A, Reynolds R, Chelliah M, Ebisuzaki W, Higgins W, Janowiak J, Mo KC, Ropelewski C, Wang J, Roy Jenne, Dennis Joseph (1996) The NCEP/NCAR 40year reanalysis project. Bull of the Am Meteorol Soc. March 77, N3: 437-471 Komen GL, Cavaleri L, Donelan M, Hasselmann K, Hasselmann S, Janssen P (1994) Dynamics and modelling of ocean waves. Cambridge University Press Lavrenov IV (1998) Mathematical modelling of wind waves in a spatially-non-uniform ocean. Gidrometeoizdat Publ. St. Petersburg Russian Leadbetter M, Lindgren G, Rootzen H (1986) Extremes and related properties of random sequences and processes. Springer-Verlag, N.Y. Lopatoukhin L, Boukhanovsky A, Rozhkov V, Degtyarev A (1998) Climatic spectra of wind waves. Proc of the II Int conf on shipbuild – ISC”98, St.Petersburg, B: 375-382 Russian Lopatoukhin L, Rozhkov V, Ryabinin V, Swail V, Boukhanovsky A, Degtyarev A (2001) Estimation of extreme wind wave heights. WMO/TD-N 1041 Lopatoukhin L, Rozhkov V, Boukhanovsky A, Degtyarev A, Sas’kov K, Athanassoulis G, Stefanakos Ch, Krogstad H (2002) The spectral wave climate in the Barents Sea. Proc of the conf OMAE'2002, OMAE2002-28397. Oslo, Norway McCreight KK, Stahl RG (1985) Recent advances in the seakeeping assessment of ships. Nav Eng J May 224-233 Ocean wave modeling (1985) Plenum Press. NewYork Ochi MK (1978) Wave statistics for the design of ships and ocean structures. Trans Soc Nav Archit and Mar Eng 86: 47-76 Reference data. Wind and wave conditions for Barents Sea, Okhotsk Sea and Caspian Sea (2003) Russian Registry of Shipping Publ. St.Petersburg Ris RC (1997) Spectral modeling of wind waves in coastal areas. Commun on Hydraul and Geotech Eng, June, TUDelft N97-4 Sevastianov NB (1970) Stability of fishing vessels. Sudostr. Leningrad, Russian Sevastianov NB (1994) An algorithm of probabilistic stability assessment and standards. Proc. of the 5th Int conf STAB’94. V 5. Melbourne, Florida

New Approach to Wave Weather Scenarios Modeling 617 Tolman HL (1991) A third-generation model for wind waves on slowly varying, unsteady and inhomogeneous depths and current. J Phys Ocean 21, N6 :782-797 Van Vledder GPh, de Ronde JG, Stive MJF (1994) Performance of a stectral wind-wave model in shallow water. Proc 24th Int Conf Coast Eng ASCE: 753-762 Wilson BW (1965) Numerical prediction of ocean waves in the North Atlantic for December 1959. Deutsch. Hydrograph. Zeitschrift, 18, N 3

9 Damaged Ship Stability

Effect of Decks on Survivability of Ro–Ro Vessels Maciej Pawłowski School of Ocean Engineering and Ship Technology, TU of Gdansk, 80-952, Poland,

Abstract

The paper describes a methodology for accounting the effect of decks above the car deck on ro–pax vessels on their survivability in the damaged condition. The methodology can be regarded as extension of the SEM for multi-deck ro–pax vessels. The effect of decks appears to be ambiguous and depends on detailed subdivision of the ship, which proves the robustness of the original SEM. For trim cases, however, the decks can be very detrimental.

1 Introduction The static equivalent method (SEM) was developed in 1995 for ro–ro vessels with the large open main deck (vehicle deck). The method evolved from research carried out at the University of Strathclyde (Vassalos 1996, Vassalos 1997) based on a framework presented earlier by Pawłowski (1995). The SEM defines the critical significant wave height Hs (= Hs 50%) in terms of the median value, given damage stability. It is assumed that the damage opening is unrestricted in the vertical direction and the flow of water on the deck induced by waves is undisturbed by the presence of decks above the main deck, typical on ro–pax vessels. Now, the SEM will be extended to account for the effect of multi-decks for an unrestricted height of the damage opening. The presence of decks is believed to be detrimental for survivability of ro–pax vessels due to a greater heeling moment exerted by water accumulated on higher decks and the multi free surface effect. In particular, this is pertinent for damage cases with trim, despite some protection against flooding of the main deck provided by the higher decks.

2 The Heeling Moment Consider a damaged ro–ro vessel at the point of no return (PNR), as shown in Fig. 1, called also as the critical heel. The PNR occurs at a heel angle equal to max, where the GZ-curve has a maximum. This angle, for typical ferries is less than 10 M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_35, © Springer Science+Business Media B.V. 2011

621

622 M. Pawłowski

degrees. Reference is made here to the GZ-curve calculated traditionally, using the constant displacement method, allowing for free flooding of the vehicle deck when the deck edge is submerged, and corresponding to minimum stability. For compartments other than amidships, the GZ-curve should be obtained for a freely floating ship, exposed to a trimming moment of value equal to that produced by the water elevated on the decks at the PNR. This problem can be solved iteratively. In practice, however, a single iteration is normally sufficient. It is worth mentioning here that the widely used NAPA package is unable to calculate the GZ-curve of minimum stability. The amount of water on decks at the critical heel can be predicted from stability calculations considering a flooding scenario, in which the ship is damaged only below the vehicle deck. However, there is some amount of water on the (undamaged) deck inside the upper (intact) part of the ship and on the deck above, as shown in Fig. 1. The undamaged deck and the undamaged ship sides above the deck enclose the space above the vehicle deck, with the damage extending from below the deck downwards. For the ship with a side casing or wing tanks, the space is enclosed by the undamaged double hull beyond the flooded part of the double hull, and by the inner shell—in way of the flooded part of the wing spaces. Consequently, the freeboard can be interpreted as the depth of immersion of the deck edge (taken with a negative sign) measured at the inner shell of the wing tanks in the middle of damage.

Deck 1

d v1 D

v

h

WL

f

Fig. 1 A damaged ro–ro vessel at the PNR with water accumulated on decks

For given height D of Deck 1 above the sea level (the same as positive freeboard for this deck) and given sea state, the water head d on this deck can be found (as discussed later) with the help of the theory regarding accumulation of water on a deck with positive freeboard. The knowledge of d defines volume v1, accumulated on Deck 1 – the first deck above the vehicle deck (see Fig. 1). This is

Effect of Decks on Survivability of Ro–Ro Vessels 623

the volume, which in general combines both the elevated water that can drain off from the deck, when the damage is opened, and the dead water that stays on deck, when the damage is opened. The latter takes place particularly for the damage cases with trim. The dead water is a result of progressive flooding of the deck induced by waves that cannot drain off from the deck. It contributes to the heeling moment produced by water on deck but has nothing to do with the dynamic elevation of water induced by waves. The sea state that defines volume v1 is the critical sea state obtained by the original SEM, without taking into account the effect of decks above the vehicle deck. The heeling moment produced by water on Deck 1 equals v1l1, where l1 is the heeling arm with respect the axis of floatation of the damaged waterplane, passing through its center of floatation. The critical amount of water on the car deck, which in turns defines the critical sea state is such that the resultant moment vanishes at the PNR, that is VGZmax = vl + v1l1,

(1)

where V is the volume displacement of the intact ship (before damage), v is the elevated volume of water on the vehicle deck above the sea level, as shown in Fig. 1, and l is the heeling lever produced by this water with respect the axis of floatation of the damaged waterline, passing through the center of floatation (not shown in Fig. 1). Both the heeling arms l and l1 are measured with respect to the same axis of floatation of the damaged waterline, without the parts occupied by the floodwater. The critical amount of water on the vehicle deck can be found also with the omission of the GZ-curve, which is particularly useful for the flooding cases with trim. This characteristic value is such for which heeling moment produced by the elevated water reaches maximum. By solving equation (1), one gets the sought volume v of elevated water on the vehicle deck, its (static) elevation h above sea level, and the freeboard f. The said elevation of water has to be produced by the dynamic action of waves and ship motions. Knowing the two quantities we would like to estimate now the critical sea state without resorting to model experiments. This can be achieved only by utilizing the available theoretical knowledge on accumulation of water on the car deck (Pawłowski 2003, and 2004). From equation (1) it follows immediately that the critical rise of water on the vehicle deck with the effect of a higher deck is smaller. However, the rate of flooding such a deck is smaller as well due to the protection provided by the deck above while the outflow rate is unchanged. Thus, on the whole it is difficult to predict beforehand how this affects the critical sea state the ship can withstand, particularly for damage cases with no trim. The outcome depends on particulars of the design.

624 M. Pawłowski

3 Accumulation of Water on a Protected Deck The asymptotic dynamic elevation of water on the vehicle is defined by the equation: qin – qout = const = q1.5 (t1 = t2 ),

(2)

where qin is the nondimensional flow rate, ignoring the effect of the higher decks, qout is the nondimensional outflow rate, q1.5 (t1 = t2) is the inflow moment q1.5 (t1 ) calculated taking t1 to be t2 = D/, and  = Hsr/4 is the standard deviation of the relative wave elevation at the damage opening (Pawłowski 2003 and 2004). The two flow rates are given by the equations: qin = 1.5 q0.5 (t 1 ) + q1.5 (t 1 ), qout = 

3/2

F(t 0 ) + 1.5 q0.5 (t0 , t1 ) – 0.5 q1.5 (t0 , t1 ),

(3) (4)

where t = f/ is the nondimensional freeboard at orifice (the damage opening), t1 = h/ is the nondimensional height of the free surface on deck above sea level,  = d/ is the nondimensional depth of water on deck at orifice;  = t1t, and F(t) is the CDF of the standard density function f(t), calculated at t = t. The quantities qm (with m = 0.5 and 1.5) are in general moments of the order m of the standard density function f(t), defined as follows:

qm  t1   



t1

t  t1 m f t d t ,

qm  t0 , t1   

t1

t0

 t1  t  m f t dt

(5) (6)

As results from analytical studies for water on deck accumulation (Pawłowski 2003 and 2004), if the nondimensional height t2 of the edge of Deck 1 above sea level at the PNR is greater than 2.4, such a deck has no effect on the intensity of flooding the vehicle deck. Then, its effect on ship survivability can be ignored. A smaller height D <  2.4  = 0.6 Hsr has obviously a positive effect on ship's survivability, increasing the critical sea state Hs. The nondimensional height of the deck above sea level t2 = D/ plays the role of a parameter in equation (2). Due to the physics, the variable t1 is equal to or less than t2. Numerical solutions of this equation are shown in Fig. 2 to Fig. 4, equivalent one to another. Note that all the curves in these figures are terminated just at points corresponding to t1 = t2. It is clear from them that a restricted clearance above the vehicle deck reduces the water head, which is positive for ship survivability – the same elevation on a protected deck has to be induced by a higher sea state.

Effect of Decks on Survivability of Ro–Ro Vessels 625

Fig. 2 Nondimensional depth of water on deck  = d/ at opening versus nondimensional height of free surface above sea level t1 = h/, depending on nondimensional clearance t2 = D/

Fig. 3 Nondimensional depth of water on deck   = d/ at opening versus nondimensional freeboard at opening t0 = f /, depending on parameter nondimensional clearance t2 = D/ 

626 M. Pawłowski Non-dimensional Water Head

0,6

0,5

t1 = h/σ

τ = d/σ

0,4

0,3

0,5

0,2

2/3 1,0

0,1

t2 = ?

1,5

t0 = f/σ 0

-3

-2,5

-2

-1,5

-1

-0,5

0

0,5

1

1,5

2

2,5

3

Fig. 4 Nondimensional water head versus nondimensional freeboard at opening t = f /, depending on nondimensional clearance t2 = D/ 

We would like now to describe quantitatively the effect of clearance on the elevation of water on deck. The curves on the left-hand side in Fig. 4 (for negative freeboard) decay asymptotically as the inverse of the nondimensional freeboard t. To overcome this inconvenience, a transformation can be used in which the abscissa axis x = f /(3h – f), where the variable x   –1, 0while the ordinate axis presents y = t1/(1 + x)1/2. Such curves are shown in Fig. 5.

Fig. 5 Nondimensional water elevation versus the ratio f /(3h – f ), depending on nondimensional clearance t2 = D/ and its parabolic approximations

They are regular and can be accurately approximated by parabolic curves in the form: y = a0 + a1 x + a2 x2, where the coefficients a0, a1 and a2 are found with the

Effect of Decks on Survivability of Ro–Ro Vessels 627

help of the least squares method. They are linear functions of the inverse of t2, as shown in Fig. 6, which provides also their equations. Using graphs shown in Fig. 5, the critical sea state can be easily found with the help of the so-called reduction coefficient . Knowing the two particulars for water on the vehicle deck h and f from static calculations, the abscissa x = f /(3h – f) can be calculated. For given x-value, the following straightforward relationships hold:



 y x, t 2  t1 t     2  y x,   t1 t 2

(7)

which defines the reduction coefficient , where y = y(x, t2) is shown in Fig. 5 in graphical form or provided by approximations discussed earlier. As can be seen, the reduction coefficient  is the same as the ratio of the standard deviations of relative ship motion at opening for unrestricted opening with no protection and for the opening with a protection. As  ≤ 1, the critical sea state for the ship with a protected vehicle deck is larger than without a protection, which is obviously beneficial. 0,7 0,6 s0

0,5

y=-0,1195x + 0,6712

0,4 0,3 s1

y=-0,1647x + 0,448

0,2 0,1 y=-0,0959x + 0,2077 0

0

0,5

1

s2

1/tz 1,5

2

2,5

Fig. 6 Coefficients of the fitting polynomials as functions of nondimensional clearance t2 = D/

Because the quantity t2∞ = D/∞ is known, t2 can be found from equation (7) by iterations. Doing so, t2 = t2 (x, t2∞) can be obtained as function of x and t2∞. Hence, the ratio t2 /t2∞, equal to ∞/, defines the reduction coefficient  = (x, t2∞). With its help, finding the critical sea state is straightforward, as:  = ∞/, which follows from equation (7). The reduction coefficient  is shown in Fig. 7.

(8)

628 M. Pawłowski 1,5 1,0 S0 y=-0,0383x2 - 0,0836x + 1,0025

0,5 (σ7/D)2

0,0 0

1

2

3

4

5

-0,5 S2

y=-0,0716x2 - 0,0259x+0,0046

-1,0 -1,5

y=-0,1085x2 - 0,0276x + 0,0012

S1

-2,0

Fig. 7 Reduction coefficient  for negative freeboard along with parabolic fit (top), and coefficients of the fitting parabolas as functions of nondimensional clearance t2 ∞ = D/∞ (bottom)

The coefficient  approximates well by parabolic curves, whose coefficients are functions of (1/t2 ∞)2 = (∞/D)2, shown in Fig. 7. Because  = ¼ Hsr and Hsr = 0.76 Hs1.36, this leads to the critical sea state given by the equation Hs = (4/0.76)1/1.36 = 3.39  1/1.36.

(9)

Regarding positive freeboard, curves on the right-hand side in Fig. 4 have a different nature – the curve for unrestricted height of the opening decays exponentially with the nondimensional freeboard t, whilst the remaining curves, particularly for a smaller clearance look almost as segments of straight lines. To overcome the inconvenience of handling curves of various patterns, a similar transformation as before could be used for the abscissa axis x = f /(3d + f ), where this time x   0, 1, while the ordinate axis is unchanged. However, in the case of positive freeboard such a transformation, although mathematically efficient, is not very handy, as the quantity d is unknown. For positive freeboard, in order to find elevation of water on a deck above the vehicle deck, as can be seen in Fig. 4, the nondimensional water head  = d/, corresponding to unrestricted height of opening with t2 = ∞, is known function of the nondimensional freeboard t0 = f /, where f = D. Hence, the following results: d =  (t0).

(10)

It may be assumed, for the sake of simplicity, that  = ∞, where ∞ is the standard deviation of relative wave motion when ignoring decks above the vehicle deck. We err in such a case on the side of safety. All the quantities on the righthand side of equation (10) are then known; therefore this equation is a

Effect of Decks on Survivability of Ro–Ro Vessels 629

formulation. The function (t0) up to t0 ≈ 2.8 is given by a polynomial of the third degree:  = 0.6207 – 0.6205 t0 + 0.215 t02 – 0.0256 t03.

(11)

4 Two Decks It is easy to extend the method for two or more decks above the vehicle deck, if that proves necessary. For given height D1 of Deck 1 and D2 of Deck 2 above sea level (see Fig. 8) and given sea state, the water heads d1 and d2 on these decks can be found using graphs regarding accumulation of water on a deck with positive freeboard, as shown in Fig. 4. Water head d2 is defined by equation (10), in which freeboard f = D2. Regarding Deck 1, as this deck is protected by Deck 2, water head d1 is defined by a similar equation: d =  (t0, t2),

(12)

Deck 2

d2 v2 Deck 1

d1 v1

v

D1

h

D2

WL

f

Fig. 8 A damaged ro–ro vessel at the PNR with water accumulated on decks

where in general t2 = D/ . In this case, f = D1, whereas D = D2. And this should be taken as a rule: freeboard f is understood as height of the given deck above sea level, whereas D is a height of the deck above the deck under consideration relative to sea level, all measured at the PNR. The actual deck flooded and the next deck above, providing protection, form a kind of independent window – what happens

630 M. Pawłowski

below and above this window is of no interest for flooding the deck under consideration. Assuming again that  = ∞, all the quantities on the right-hand side of equation (12) are known; therefore this equation becomes a formulation that defines d. What is needed only is the knowledge of the nondimensional water head  = (t0, t2) as function of two variables t0 and t2. Graphs shown in Fig. 4, except for t2 = ∞, can be neatly approximated by parabolas, whose coefficients depend on the inverse of t2, as shown in Fig. 9. 0,8

y=-0,0825x2 - 0,1001x + 0,6165

-S1

0,7 0,6 0,5

y=-0,0258x2 - 0,0457x + 0,6249

S0

0,4 0,3 0,2

y=-0,1438x + 0,2158

S2 0,1 0

1/tz 0

0,5

1

1,5

2

2,5

Fig. 9 Coefficients of the fitting parabolas as functions of t2 = D/ 

The knowledge of water heads d1 and d2 on Deck 1 and Deck 2 defines volumes v1 and v2, accumulated on these decks (see Fig. 8). The critical amount of water on the vehicle deck, which defines the critical sea state is such that the resultant moment vanishes at the PNR, i.e. VGZmax = vl + v1l1 + v2l2,

(13)

where V is volume displacement of the intact ship (before damage), v is the elevated volume of water on the vehicle deck above the sea level, as shown in Fig. 8, and l is the heeling lever produced by this water with respect the axis of floatation of the damaged waterline, passing through the center of floatation (not shown in Fig. 8). All the heeling arms l, l1 and l2 are measured with respect to the same axis of floatation of the damaged waterline, without the parts occupied by the floodwater. By solving equation (13), one gets the sought volume v of elevated water on the vehicle deck and its (static) elevation h above sea level that has to be produced by the dynamic action of waves and ship motion, along with the freeboard f. Knowing the two quantities, the critical sea state can be found by exactly the same way, as described earlier for the case of one deck above the vehicle deck, with D = D1.

Effect of Decks on Survivability of Ro–Ro Vessels 631

Having found the critical standard deviation of the relative wave elevation at the damage opening , the whole methodology can be repeated, if necessary, starting with  = .

5 Conclusions From equation (13) it follows immediately that the critical elevation of water on the vehicle deck is smaller in comparison to one deck. However, due to the protection provided by Deck 1 it is difficult to predict beforehand how this per capita affects the critical sea state the ship can withstand, unless the ship has a trim. The outcome depends on particulars of the design. Trim is dangerous as it opens space for “dead” water, which stays on deck, once it entered there. Draining off the decks is therefore very beneficial for ship safety. The knowledge of water elevation on the vehicle deck and the immersion of the deck edge at the PNR along with the clearance to Deck 1 are sufficient for finding the critical sea state Hs (in terms of the median value) the ship is capable of withstanding at given flooding. The methodology is embedded at the SEM.

Acknowledgements Main body of the paper was developed during the author’s visit at The Ship Stability Research Centre of the University of Strathclyde, Glasgow from July to September 2005. Prof. D. Vassalos, the Director of the Centre, supported the visit.

References Pawłowski M (1995) A closed-form assessment of the capsizal probability – the si factor, Proceedings of WEGEMT Workshop on Damage Stability of Ships, Dan Technical Univ Cph, 11. Pawłowski M (2003) Accumulation of water on the vehicle deck, Proc of the Inst of Mech Eng, Part M: J of Eng for the Marit Environ (JEME), 217 M4: 201-211. Pawłowski M (2004) Subdivision and damage stability of ships, Euro-MTEC book series, ISBN 83-919488-6-2, Politechnika Gdańska, 2004, 311. Vassalos D, Pawłowski M and Turan O (1996) A theoretical investigation on the capsizal resistance of passenger ro–ro vessels and proposal of survival criteria, Final Report, Task 5, The North West European R&D Project. Vassalos D, Pawłowski M and Turan O (1997) Dynamic stability assessment of damaged passenger ro–ro ships and proposal of rational survival criteria, Mari Technol, 34 4: 241–266.

Experimental and Numerical Studies on Roll Motion of a Damaged Large Passenger Ship in Intermediate Stages of Flooding Yoshiho Ikeda*, Shigesuke Ishida**, Toru Katayama*, Yuji Takeuchi*** *Osaka Prefecture University;**National Maritime Research Institute; ***IHI Marine United Inc.

Abstracts

Measurements of roll motion of a two-dimensional scale model of a damaged large passenger ship are carried out during flooding process. The experimental results demonstrate that large and slow roll motion sometimes appears in the intermediate stages of flooding. The appearance of the large roll motion significantly depends on location and size of damage opening. It is also confirmed that the simulated results are in fairly good agreement with experimental ones.

1 Introduction In the previous paper, the authors pointed out on the basis of model tests that large and slow roll motion can appear in the intermediate stages of flooding for a damaged large passenger ship and the maximum roll angle depends on size and location of a damage opening (Ikeda 2003). The large roll is caused by shallow water accumulation on each deck of multiple decks in damaged compartments. In the present study, measurements of ship motions from start of flooding to final condition after flooding are carried out for a two dimensional model. The model is modeled the mid-ship parallel body of the ship designed for studies on large passenger ship safety by US coast guard, MARIN and Fincantieri. Numerical simulations are also carried out to validate the accuracy of the calculation method. The experimental results demonstrate that large and slow roll motion some times appears in the intermediate stages of flooding and a part of the bulkhead deck sink. In some cases the model used in the study can be capsized in the intermediate stages of flooding, even although she is safe in the final condition.

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_36, © Springer Science+Business Media B.V. 2011

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2 Model The model used in the experiment is a two-dimensional model of the parallel part of the mid-ship body of an un-building large passenger ship of 110,000GT designed by Fincantieri. The length, beam, draft and displacement of the model are 1.618m, 0.728m, 0.168m and 190.1N, respectively. The scale is 1/50. The model have a float to adjust the trim in flooded condition to that of the three dimensional ship. The model with a float is shown in Fig. 1. The GM value is adjusted to the corresponding one of the original threedimensional large passenger ship. The model has a watertight super-structure in above the bulkhead deck. In each compartment there are three decks as shown in Fig. 1, which are used for crew accommodations or void space. Arrangements for crew cabins and void space on the decks are shown in Fig. 3. All crew cabins are made with and without cabin doors, which are not watertight. A damage opening, whose size is 210mm length and 20mm or 60mm heights, is located on the side-hull below the bulkhead deck. The size and location of the opening is systematically changed in the experiments.

W.L.

150

540

350

< DK3< < DK2< < DK1<

Fig. 1 Schematic view of 2-D model. 1618 2

3

4

5 áDK3ñ

77 34.5 4

áDK1ñ

404

1 53 53

áDK2ñ

6

230

341

341

341

Fig. 2 Deck position and size.

341

6

Experimental and Numerical Studies on Roll Motion R2

R2

R1

R1 R1

R2

R2

R1

R1

R1

R1 S

R1

R1

635

R1

R1

R1

R1

R1

R1

R1

R1

R1

R1

R1

R1

R1

R1

R1

R1

R1

R1

R1

R1

R1

R1

R1

R1

R1

R1 R1

R2

R2

R1

R2

R2

R2

R2

R1 : 66mm × 43mm S : 300mm × 43mm

(a) Crew cabin

R2 : 77mm × 48mm shaded : 32mm × 43mm

(b) Void space

Fig. 3 Crew cabin and void space.

3 Behavior of the Ship In calm water, the opening located on the side hull of the model is released, and the behavior of the model (roll and heave motions) is measured until end of flooding. An example of time histories of measured roll of the model for two-compartmentdamage is shown in Fig. 4. The result shows that the model rolls gradually, reaches a maximum angle, and returns to upright condition. In most cases with large roll motion in the intermediate stages of flooding, the maximum roll angles are around 17 degree. When a damage opening is located in lower position or large, such large roll motion does not appear.

636 Y. Ikeda et al.

20

Φ [deg]

model scale

10

0 0

200

400

time[sec] 600

Fig. 4 Time history of measured roll of model in flooding experiments.

The results of all experimental cases are tabulated in Table 1. The results demonstrate that large roll motion in the intermediate stages of flooding does not appear only when height of damage opening is low or middle. The initial heel angle also affects the roll motion, and large roll motion does not appear when the ship inclines to opposite direction (minus initial heel angle in the table) of damaged side. Table 1. Experimental conditions and appearance of large roll in the intermediate stages of flooding.

Case 1

maximum large roll roll angle motion [deg]

the opening

internal subdivision

size/ height

on DK2

small/ low

c.c.(without doors) – v 0 deg.

×

―

―

initial heel

time to maximum roll angle [sec.]

2

small/ middle c.c.(without doors) – v 0 deg.

×

―

―

3

small/ middle v – c.c.(without doors) 0 deg.

×

―

―

4

small/ middle v – v

0 deg.

×

―

―

5

small/ high

c.c.(without doors) – v 0 deg.

×

―

―

6

small/ high

c.c.(with doors) – v

0 deg.

○

15.5, 16.2, 16.9

130.4, 150.8,146.5

7

large/ high

c.c.(with doors) – v

0 deg.

○

16.5

126.9

8

large/ high

c.c.(with doors) – v

2 deg.

○

9

large/ high

c.c.(with doors) – v

-2 deg. ×

17.5

108.0

―

―

10

large/ high

v–v

0 deg.

×

―

―

11

large/ high

v–v

2 deg.

○

17.1, 16.9

35.6, 23.2

12

large/ high

v–v

-2 deg. ×

―

―

13

very large

c.c.(with doors) – v

0 deg.

―

―

×

Opening size: small : 210mm × 20mm, large: 210mm × 60mm, very large: 210mm × 105mm location: low: 100mm, middle: 158mm, high(small): 172.5mm, high(large): 162.5mm, very large: 142.5mm (Center of opening from baseline)

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Experimental and Numerical Studies on Roll Motion

4 Maximum Roll Angle In most cases with large roll angle in the intermediate stages of flooding, the maximum roll angles are around 17 degrees as shown in Fig. 5. Fig. 6 shows the comparison between the calculated roll angle by static calculation and measured maximum roll angle. It should be noted that in the calculation the super-structure above the bulkhead deck is assumed to be intact. The results demonstrate that the angle of 17 degrees almost coincides with the calculated one, which is up to about 20 degrees as shown by the solid line in Fig. 6 at the worst case of accumulation of water on each deck of three decks in the compartments. This means that shallow water accumulated on each deck create the almost largest roll moment, respectively. 20

Φ max[deg]

model scale

10 exp.

0

4200

12600 2 damage opening size [mm ]

Fig. 5 The maximum roll angle in the intermediate.

Φ [deg]

model scale

20

exp. cal.

10

0 0

0.05

3

0.1

flooded water volume[m ] Fig. 6 The comparison between the calculated roll angle by static calculation and measured maximum roll angle.

The time to the maximum roll angle from rest is shown in Fig. 7. The results show that the roll motion reaches the maximum angle in 100-150seconds in model

638 Y. Ikeda et al.

scale, or 13-18 minutes in full scale. The results also show that the time to maximum roll angle decreases with increasing size of damage opening. 200

Time to maximum roll[sec]

model scale

100 exp.

0

4200

12600 2 damage opening size [mm ]

Fig.7 Time to maximum roll angle in intermediate stages of flooding.

5 Flooding Simulation Simulations of flooding into the model are carried out. The simulation program for a PC was developed in National Maritime Research Institute. Flooding flow velocity through damage openings and holes on a wall to an adjacent compartment or to upper/lower decks is calculated by the Bernoulli equation with a modification factor. In the present simulation, the modification factor is assumed to be 0.8. Simulated results for Case 11, 6 and 2 are shown in Figs. 8-10 with experimental results. The simulated results are in fairly good agreement with the experimental ones. roll [deg.]

model scale

20

Cal.(moment:plus) Exp.

10

0 0

50

Time [sec]

100

Fig. 8 Simulated and experimental results of roll motion for Case 11.

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Experimental and Numerical Studies on Roll Motion roll[deg.]

20

model scale Cal.(moment:plus) Exp.

10

0 0

200

400 Time [sec] 600

Fig. 9 Simulated and experimental results of roll motion for Case 6. roll [deg.]

model scale

10

Cal.(moment:minus) Exp.

0

–10 0

50

Time [sec]

100

Fig. 10 Simulated and experimental results of roll motion for Case 2.

6 Capsizing in the Intermediate Stages A capsizing was observed for the model for which the damage opening extended to sidewall of the superstructure above the bulkhead deck. In the case, there is no transverse watertight wall to restrain flooded water over the bulkhead deck. Figs. 11 and 12 show calculated GZ curves and obtained roll angle for various volumes of water on each deck. The results suggest the ship can capsize in the intermediate stages of flooding if damage opening extending to above the unrestrained bulkhead deck. Since the designed ship has some partial watertight walls on the bulkhead deck, such a capsize could not occur in reality. It should be noted, however, that it may be important to guarantee not to spread flooded water widely on the bulkhead deck.

640 Y. Ikeda et al.

Φ [deg]

model scale

20

10 exp. the opening under BHD (cal.) the opening above and under BHD (cal.)

0 0

0.05 0.1 3 flooded water volume[m ]

Fig. 11 Calculated roll angles for the opening under BHD and above BHD.

GZ [m]

model scale

0.04

intact 3 ∇ w=0.010428m 3 ∇ w=0.0176m

0

–0.04 0

10

20 Φ [deg]

Fig. 12 GZ curve of model for water on each decks.

7 Conclusions Following conclusions are obtained. 1. Large and slow roll motion sometimes occurs for a damaged large passenger ship in the intermediate stages of flooding. 2. The large roll motion appears only when not so large damage opening is located in high. 3. The roll amplitude can be approximately predicted by a static calculation. 4. The simulation method used in the present study can predict the roll angle in the intermediate stages of flooding with enough accuracy.

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Acknowledgements The present study was carried out in the fiscal years of 2003 and 2004 as a part of the RR-S2 research panel of the Shipbuilding Research Association of Japan funded by the Nippon Foundation, to whom the authors express their gratitude.

References Umeda N, Kamo T, Ikeda Y (2004) Some remarks on theoretical modeling of damaged stability. Mar Technol 41:45-49. Ikeda Y, Shimoda S, Takeuchi Y (2003) Experimental studies on transient motion and time to sink of a damaged large passenger ship. Proc. of 8th Int Conf on the Stab of Ships and Ocean Veh 243-252. Kamo T, Shimoda S, Ikeda Y (2002) Effects of transient motion in intermediate stages of flooding on the final condition of a damaged PCC. Proc. of 1st Asea Pac Workshop on Mar Hydrodyn 26-31. Ikeda Y, Ma Y (2000) An experimental study on large roll motion in intermediate stage of flooding due to sudden ingress water. Proc. of 7th Int Conf on Stab of Ships and Ocean Veh 270-285.

Exploring the Influence of Different Arrangements of Semi-Watertight Spaces on Survivability of a Damaged Large Passenger Ship Riaan van’t Veer*, William Peters **, Anna-Lea Rimpela***, Jan de Kat**** * SBM Offshore, The Netherlands **US Coast Guard, USA *** STX Yards, Finland **** A.P. Moller-Maersk, Copenhagen,

Abstract The transient and progressive flooding of a complex labyrinth of compartments as found on large passenger ships is studied with a numerical non-linear time domain code. Earlier papers showed that such a tool can increase the awareness of how water floods through the ship and which parameters are important for ship survivability with respect to flooding. (Van’t Veer et al., 2002, 2003) This paper presents results from a study carried out to investigate the influence of different arrangements of semi-watertight space on the survivability of a damaged (unbuilt) large passenger ship. It is shown that the location of a single down-flooding point and its kind of protection against flooding can significantly influence the intermediate flooding conditions. It is shown that survivability decreases with increasing wave height and wave steepness. Different aspects in numerical transient and progressive flooding simulations are reported.

1 Introduction When a ship loses its watertight integrity, whether by collision, grounding or an explosion, it is subject to the risk of sinking or capsizing. The most effective way to protect the ship against progressive flooding is by internal subdivision using watertight transverse and/or longitudinal bulkheads. The consequence for the ship when losing its watertight integrity depends not only on the subdivision itself, but also on factors such as: the actual damage extent, its location, initial GM and the sea-state. Based on statistics of past accidents one defined the probabilistic approach for damage stability. This concept was adopted by IMO resolution A.265 (VIII) in 1973. It was introduced into SOLAS regulations for passenger ships in 1978. The probabilistic method includes a set of regulations which consists of a standard and a method of M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_37, © Springer Science+Business Media B.V. 2011

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calculating that standard. Recent study in the EU HARDER project has lead to a number of proposals (Ref. SLF 46 and 47 sessions at IMO) to modify the existing regulations considering up-to date collision databases and new insights in behaviour of damaged ships. In general, the subdivision of a ship is considered sufficient if the subdivision index A is not less than the required subdivision index R. The attained subdivision index A is derived by a summation of partial indices from all damage cases using the formula: A=∑pi*si, where pi accounts for the probability that only the compartment or group of compartments under consideration may be flooded, disregarding any horizontal subdivision, and si accounts for the probability of survival after flooding including the effect of horizontal subdivision. Within a given subdivision a watertight area can be further divided in non-watertight compartments, such as cabins. These compartment boundaries are not considered effective for the survivability of the ship and are as such not considered in traditional damage stability calculations. This paper focuses on the probability of survival (s-factor) using time-domain flooding simulations. The proposals from SLF 47 will be used to define survivability. Example simulations are performed using an unbuilt large passenger ship with a typical deck layout and using a three-compartment damage (beyond SOLAS). The damage reflects an imaginary striking bow-damage above the waterline and a bulbous bow penetration below the waterline. The damage length is 24 m or 0.1*Lpp, see Fig. 1. The location and size of the damage were not varied in this study.

Fig. 1 Imaginary three-compartment damage due to a striking bow above the waterline and a penetrating bulbous bow damage below the waterline.

1.1 Survivability A definition of survivability is difficult, because many aspects affect the survivability of the ship after an accident that leads to flooding. Considering the many threats for a ship with respect to flood water ingress the following aspects can be considered as limiting factors on the habitability:

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 a significant final equilibrium heel;  large intermediate heel angles, which for example prevent the use of life saving equipment, or lead to breakdown of crucial systems;  blockage or hindering of escape routes;  required closure of a subdivision leading to possible human life entrapment;  breakdown of active damage control possibilities. It remains difficult to pinpoint specific criteria for each threat but the following are related to current regulations or proposals concerning passenger ships:  final heel should not be larger than 7 degrees;  intermediate heel should not exceed 15 degrees;  people should be able to move to the mustering stations and remain safely on board until evacuation for at least 3 hours after initial damage. A drawback of the current regulations and general applied static stability assessments is that none of them can indicate survivability in terms of time. Nor is it possible to derive realistic intermediate flooding conditions based on progressive flooding in time, or to include the wave actions imposed on the ship in a proper realistic manner. In MSC 78/WP.14 paper performance based criteria are proposed for LPS, such that after a fire or collision: 1. The ship can return to port safely; or 2. The ship remains habitable for at least 3 hours for evacuation. In those 3 hours the ship condition must be such that all survival craft can be launched with their full complement of persons. In addition to the above, the following goal based criteria should be added in view of the recent revision of the SOLAS regulations, in particular Ch. II-1, Reg. 7-2; 1. 1. Horizontal evacuation route on bulkhead deck is not allowed to immerse in final stage of flooding; 2. 2. Any vertical escape hatch in the bulkhead deck is not allowed to immerse in intermediate of final stage of flooding.

1.2 Direct Simulation Approach This paper present results of time domain simulations from which direct assessment of the time varying survivability is possible. The ship motions are solved at each time instant accounting for the forces from flood water and waves. It is than possible to judge the results against the dynamic criteria as mentioned above. The benefit of this approach is that, within the assumptions of the approach, realistic intermediate flooding conditions are obtained, since the behaviour of the damaged ship is simulated in time starting with an intact ship.

646 R. van’t Veer et al.

The flood water progress through the ship is subject to the definition of compartments and openings between them. To assess the design, it is therefore essential to capture all relevant openings and compartment boundaries which will hinder the flow in its ingress and progression. The final flooded ship condition is now affected by the intermediate flooding stages. With the traditional static naval architectural approach one cannot assess the intermediate flooding stages since the distribution of water unknown, and one can only assume that the final equilibrium condition is not influenced by it. Clearly, there will be flooding scenarios where this is very true, so that the traditional methods used provide adequate assessment of the final flooding stage for many flooding conditions. The time domain simulations can be performed in calm water or in any seastate to study the effect of wind and waves. Again, the traditional flooding calculations are limited to the calm water condition, neglecting possible additional flooding due to wave ‘pumping’. Where calculations for the probabilistic method only require accurate modelling of the damage zone, the direct flooding simulations, as applied in this paper, require a detailed set-up of the internal space of the ship with all relevant openings such as doors and down-flooding openings between decks. This means that full knowledge of the ship layout in all its details is needed, which is a challenging task since the large passenger ships in particular are extremely complex. It also leads to discussion which details must at least be included since in reality one lacks computing power to model the ship as it will be built with all construction details. This paper discusses the latest findings in time domain flooding simulations and is a sequel to the initial study reported to IMO (SLF 46/INF.3). A review by experts of these first results was presented to IMO in SLF 46/8. The recommendations were followed and the results are reported in this paper.

2 Background of the Approach 2.1 Flooding of Compartments The time domain simulation program (FREDYN) has been described in previous publications, e.g. by (De Kat et al. 2000). A hydraulic flow model is used to calculate the progress of water flow between compartments. This model assumes that the flow velocity inside a compartment is zero. An empirical discharge coefficient accounts for the vena contracta of the flow through the opening between compartments, where the fluid is considered inviscid. Air that is displaced by flood water is assumed to vent freely. A previous study (Palazzi and De Kat, 2002) has shown that air flow within flooding compartments may reduce the rate of flood water ingress. However, the incorporation of this factor requires further detailed modelling that was beyond the scope of this project.

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2.2 Practical Assessment of Compartment Boundaries The first study report on time domain flooding (SLF 46/INF.3) showed that an accurate assessment of the flooding process requires an accurate model of internal compartments and the openings between them. Recently an assessment was published (SLF 47/INF.6) that considered the characteristics of three main categories of boundaries: 1) doors, 2) piping and 3) ventilation and windows. To characterise an opening the following parameters are identified:  hL = a static pressure head at which leaking starts,  AL = a portion of opening area through which leaking occurs,  hC = a static pressure head at which the obstruction collapses, and  AC = the opening area after collapse of the obstruction. The objective of the practical assessment was to define these parameters for different classes of openings. In some cases the parameters will be different depending upon the direction of the pressure on the opening obstruction. An obvious example of this would be hinged fire and joiner doors. The simulation program FREDYN has been modified so as to include the above effects in the internal geometry modelling. Watertight sliding doors used under the bulkhead deck are assumed not to leak or to collapse, and as such, the watertight sliding doors below the bulkhead deck are not included in the list of openings for numerical simulations. According to existing rules (MSC/Circ. 541) watertight subdivision should be taken above the bulkhead deck, if the deck will be submerged during any stage of flooding. If the area is not submerged in any stage the restricting structure (door) may be of weather tight type. 2.2.1 Weathertight Doors Weathertight doors do not remain watertight when a certain pressure difference over the door opening is present. Due to the lack of tested weather tight doors, the true collapse and leakage pressure are not known. But weathertight is assumed as high collapse pressure (hC = 4 m) and low leakage when the pressure exceeds 0.3 m. A Class Fire Doors are assumed to have zero leakage pressure, referring to the allowable gap beneath the fire door or 6 mm (resolution A.754 (18) and SOLAS regulation II-2/8.4.4.2). A moderate to high collapse pressure is assumed of 2 m. B Class Joiner Doors are typical cabin doors. A ventilation opening is permitted in the lower portion of such doors so that the leakage pressure is again zero. The collapse pressure is assumed to be low to moderate, 1.5 m. Table 1 summarizes the definition of openings considered in this study.

648 R. van’t Veer et al. Table 1. Definition of door types in simulations

Door type Weathertight doors (hinged splashtight doors) A Class fire doors B Class joiner doors

hL [m] 0.3

hC [m] 4.0

0.0 0.0

2.0 1.5

AL/AC 0.05 0.1 0.2

2.2.2 Piping It is assumed that all penetrations carried through subdivision watertight bulkheads below the bulkhead deck are constructed and arranged such that they remain intact. Open AC-canals, electrical cableways and grey/black water piping are assumed to not allow progressive flooding. The validity of this assumption, especially in the later flooding stages, requires further investigation and confirmation. In general, regulation prescribes that the connections between adjacent (partial) watertight compartments shall be located on centre line side from the watertight/ weather tight area. The ‘immersion limit line’ determines which area of the bulkhead deck is to be made watertight 2.2.3 Port-Lights and Windows It is assumed that all port-lights and windows remain intact during the simulations. Based on the required maximum allowable pressure head, and the requirement that the hull for intact stability is assumed to reach at least up to deck 6, this assumptions seems justified (see SLF 47/INF.6) It is noted that in the final JAIC report of the ‘MV Estonia’ accident the impact pressures from the waves pounding on the ship was indicated as a plausible cause leading to the collapse of the windows that contributed largely to the tragically sinkage and capsize. As such, SLF 47/INF.6 recommends to perform tests to establish the leakage and collapse pressure threshold for windows and port-lights, especially those that are located in the ‘intact stability hull’.

2.3 Probability of Survival The objective is to calculate time series of the ship motions, and in particular of the roll motions since criteria are based on heel. The time trace is evaluated and the probability of survival is zero (s=0) when the heel angle exceeds 15 degrees within 3 hours or when the final equilibrium is larger than 7 degrees. The latter

Exploring the Influence of Different Arrangements of Semi-Watertight Spaces

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criterion does not specifically mention that the final flooding condition should be reached within 3 hours. To limit the calculation time, simulations of 6 hours duration are performed. The time domain simulations are performed with the FREDYN code. The motion equations are solved at each time instant using pressure integration under the actual wetted surface (non-linear force component), but with linear, 3D diffraction and radiation forces. For simulations in irregular waves, long-crested Jonswap wave spectra were generated. Varying the random phases of the individual waves in the spectra leads to varying wave realisations with the same spectral density. All simulations in waves were carried out in beam seas with zero forward speed. In case of damaged ship simulations in waves we are concerned with transient random data. There is a clear defined beginning (the intact situation) and end (the final equilibrium for example) to the data. This means that one must repeat the experiment or simulation over and over again under similar (sea-state) conditions to obtain a collection of suitable records to derive the probability of survival. In Figure 2 a schematic view is given on the probability of survival for a particular damage condition. Below a certain sea-state the ship will not reach critical conditions and is considered survivable with s=1. Clearly, this includes the calm water result. With increasing sea-state it can be expected that survival time decreases and that at some point a criterion is exceeded and that for that particular run s=0. However, another simulation in the same sea-state but in a different seastate realisation could have s=1. This means that the survivability of the ship in that particular sea-state is a weighted average of all runs. Since long-duration time domain flooding simulations are still quite time consuming for complex ships such as a large passenger ship, calculation speed is about 2.5 times faster than real time, it is essential to limit the number of runs.

sign. wave height Hs [m] UNSAFE sea-states P(survival) = 0

shortest survival time in simulations

longest survival time in simulations

distribution of survival times

probability of survival P(survival)

Critical sea-states P(survival) = <0,1>

SAFE sea-states P(survival) = 1

probability function of survival P=1

P=0 lowest sea-state where capsize will occur

required rescue time

survival time [hours]

Fig. 2 Probability of survival can be obtained by given rescue time and estimated distribution functions of survival time (from numerical simulations) in the critical sea state area.

650 R. van’t Veer et al.

Therefore, the scenarios under investigation were limited to sea states of 3.5 m significant wave height and less. This is in agreement with statistical data that most flooding accidents occur in these sea states. The test conditions are based on the wave scatter diagram from the North Atlantic Annual Bales data, given in Fig. 3. To limit the number of simulations further, first a set of calculations was performed using the steepest waves with Hs ≤ 3.5 m. From this the most critical condition was selected and 10 simulations were executed for this scatter diagram entry. Based on the statistics of these runs the probability of survival could be calculated in that particular sea state. BALES Wave Scatterdiagram, North Atlantic 1

13.5

1

12.5 11.5

1

10.5

1

2

1

9.5

2

3

2

2

5

3

2

1

2

6

6

3

2

1

8.5

Hs [m]

1

7.5 6.5

1

11

9

7

4

3

1

5.5

5

23

10

8

4

3

1

24

26

10

8

5

3

1

35 30

26

11

9

5

3

1

17 43 44 30

23

13 10

6

4

1

3

30 51 42 41 28

27

13 13

8

5

2

3

42 43 30 38 21

20

9

10

5

2

1

18

20

5

4.5 7

3.5 1

2.5 1.5 0.5

4

3.2 4.5

6.3 7.5 8.5 9.7 10.9 12.4 13.9 15 16.4

Tpeak [s]

Fig. 3 Bales Wave Scatter diagram for the North-Atlantic.

Fig. 4 3D hull lines of the LPS. The knuckle point in the side is at deck level 7 (20.0 m). Lines extend up to deck 15 (40.0 m). Significantly more stations around the damage location.

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Table 2. Large passenger ship main particulars and intact loading condition

SHIP PARTICULARS Length overall

VALUE 289.605 m

Length between perpendiculars

242.280 m

Breadth moulded (deck 9) Breadth moulded (deck 8 and below) Bulkhead deck (deck 4) Summer load draft (moulded) Intact LOADING condition Draft Displacement Trim GM transverse

40.20 m 36.00 m 11.40 m 8.45 m 8.40 m 53010 tons 0.0 m 2.10 m

3 Large Passenger Ship Details A noted shipbuilder of large passenger cruise ships provided the general arrangement drawings of a large unbuilt passenger ship useful for studying dynamic flooding. Support was given to understand the complexity of the ship in all its relevant details. The main particulars can be found in Table 2 and a small body plan is given in Figure 4.

3.1 Loading Condition for Three Compartment Damage For the three-compartment damage, an intact GM condition of 2.10 m was defined by the ship designer. According to the proposed regulation from the document SLF 47/3/1 for SLF 47, Ch II-1 Part B Reg. 7.2 (3.1), the survivability factor for final flooding is: 1/ 4

Range   GZ s final  K   max  12 16  

where Range is not to be taken as more than 16 degrees and GZmax is not taken as more than 0.12 m. The factor K equals 1.0 when the equilibrium heel angle is

652 R. van’t Veer et al.

less than 7 degrees as zero when greater than 15 degrees, and varies between zero and unity for intermediate heel angles. For the three-compartment damage, the designer provided the following damage stability parameters:

 e  2.658

[deg]  Range  13.031 [deg]  s final  0.95 GZ max  0.134 [deg]

When during intermediate flooding the heel angle exceeds 15 degrees the si factor from this damage will reduce to zero, otherwise the maximum is to be taken from the sfinal and sintermediate, where the intermediate survivability factor is obtained from the proposed formula in regulation 7.2 (2): 1/ 4

Range   GZ sintermediate   max  7   0.05

The above approach requires calculating the GZ curve for the damage ship for all intermediate flooding stages, and as such, is based on a quasi-static approach which does not allow easy comparison with dynamic simulation results.

3.2 Compartment Details All compartments are modelled with permeability according to SOLAS regulation:  Permeability = 0.95 for spaces occupied by accommodation  Permeability = 0.85 for spaces occupied by machinery The watertight bulkheads define the subdivision. They extend up to deck 4, which is located 3.0 m above the calm water plane. The watertight doors in these bulkheads are assumed closed and intact during the simulations. 3.2.1 Downflooding Points

There are no connections on deck level 1, 2 or 3 between the subdivisions 9, 10 and 11, apart from watertight doors which remain intact by flooding. To escape from deck 1 in case of emergency, one or two vertical escape openings to deck 2 exist in each subdivision 9, 10 and 11, and similar number of escape openings connect deck 2 and 3. Such openings act as downflooding points between the decks, and as such, the location is very important for progressive flooding. Other

Exploring the Influence of Different Arrangements of Semi-Watertight Spaces

653

than these defined openings between decks, the decks are assumed to be watertight. The escape points are all protected by a small compartment and it will be shown that it is important to model these compartments and their protection (firedoors) properly. As an example the escape point in subdivision 9 between deck 2 and 1 is shown, see Fig. 5. An opening (stairs) to deck 3 is seen as well, which is located near the escape point to deck 1, but watertight boundaries restrict direct flooding between the openings in deck 2 and 3. All such boundaries need to be taken into account throughout the ship. The direction of the hinged doors might influence the flow behaviour as well, but this level of modelling has not been considered yet.

Fig. 5 Deck 2 detail near the centreline in subdivision 9. A protected downflooding point (ESC) to deck 1 is seen as well as a protected upflooding (UP, stairs) to deck 3. The small compartments and their protection need to be modelled.

3.2.2 Longitudinal Bulkheads

In several large compartments longitudinal bulkheads are present as seen in the drawings. Such boundaries limit the flood water flow between port and starboard extreme ends of the ship and it is important to include them in the set-up. It is important to recognize that many flow obstruction items (such as machinery equipment) will reduce the flow velocity in a large compartment. The hydraulic model assumes a zero flow velocity in a compartment and an instantaneous flood water distribution with always horizontal surface. Typical cabin areas consist of many bulkheads with openings between them, defining a large number of non-watertight cabins. The philosophy in this study was to group a number of cabins into a larger area and to create an opening reflecting the total area of all connections. An example is shown in Fig. 6.

654 R. van’t Veer et al.

Fig. 6 Typical example of watertight boundaries in a cabin area (deck 3). The openings between the area are not given. The circles represent down flooding points.

4 Simulation Results 4.1 Effect of a Single Downflooding Point in Calm Water and in Waves The effect of the type of protection of a single down flooding point can make a large difference in intermediate flooding condition. An example is given in Fig. 8. The protection of the down flooding point (escape area) was modelled as open, closed by a fire door, closed by a cabin door, and watertight. The drawing shows fire door protections near all escape areas, and this modelling was used in all other simulations. The down flooding point under consideration connects deck 3 and 2 and is located about 6 m from the damage in the shell at portside. It is the first

Exploring the Influence of Different Arrangements of Semi-Watertight Spaces

655

down flooding point the water reaches on this deck in this compartment. When the flooding continues in time one other down flooding point, located around centreline, is reached, see Fig..

Fig.7 Two downflooding points near frame 148 in compartment 9 (between frame 124-148) are presented by the circles. The solid lines are compartment boundaries. The door openings are schematically given. The protection of the one near the damage side (marked escape down) has been varied.

The simulation results suggest a strong effect on the intermediate flooding conditions in calm water, but not so much on the final flooding stage. See Figure 8. When the opening remains closed, downflooding to the lower deck can only take place after a considerable amount of flood water has entered ship and the downflooding through centreline openings occurs. The lower the collapse pressure of the door, the sooner deck 2 will be flooded which decreases the roll angle since the compartment on deck 2 extends from port to starboard. The difference between a fire door protection and the open door modelling was marginal. An interesting observation is that the maximum roll angle towards the damage is not so much changed (from 10 to 12 degrees) but that the time event of this maximum heel is very different (from half an hour to one and a half hours). All simulations show a first roll peak only a few minutes after damage, and a decreasing roll towards 2 degrees within 15 minutes. As soon as downflooding starts the list angle increases which can be explained by the fact that the water will flood to the lower corner of the compartment. Only when the filling grade of the lower compartment increases the flood water mass will bring the ship back to a smaller list angle.

656 R. van’t Veer et al.

Three compartment damage 0.00

OPEN

-2.00

FIRE

HEEL [deg]

-4.00

SPLASHTIGHT

-6.00 -8.00 -10.00

CLOSED -12.00 -14.00 0

3600

7200

10800

TIME [s]

Fig. 8 Effect of the protection of a single down flooding point on intermediate flooding conditions in calm water. Protection with Fire door 0

HEEL [deg]

CALM WATER -5

-10 Hs = 3.5 m, Tp =9.7 s

-15 0

3600

7200

TIME [s] Protection with Watertight door 0

HEEL [deg]

Hs = 3.5 m, Tp =9.7 s -5

-10

CALM WATER

-15 0

3600

7200

TIME [s]

Fig. 9 Response of the ship in calm water and in waves with two different protections of a single down flooding point (fire door and watertight door).

Exploring the Influence of Different Arrangements of Semi-Watertight Spaces

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The effect of the type of protection of this single downflooding point in waves is presented in Figure 9. It is clearly seen that in this sea state the effect of the protection of the downflooding point on the roll behaviour is very different than in calm water. The response of the ship is in fact very comparable in both simulations. This indicates that in (more extreme) waves the simulations in calm water are not a good measure for the response of the ship. The progressive flood water spills through the other downflooding point in compartment area 9 in a much earlier stage. This is an effect of the water being pumped into the ship by the higher waves.

4.2 Effect of Longitudinal Boundaries in Lower Compartments The drawings of the unbuilt design show a number of longitudinal semi-watertight boundaries. These have been modelled to represent the ship, but simplifications were made to obtain a useful numerical model. The methodology has been explained in chapter 3. The physical models used to validate the hydraulic flow model over the past years all had a simplified internal compartmentation with a limited number of compartments since a complex model as studied here is extremely difficult to realize for model testing purposes. A complex physical model such as the one made numerically for the present study has not been constructed or tested so far. The modelling of (non-watertight) longitudinal and transverse bulkheads and other major objects is important in damaged areas open to the sea. When neglecting such obstructions, the water will flood much faster into the ship leading to a different roll response. This is especially true for symmetrical, large compartments extending from port to starboard. The hydraulic flow model instantaneously ‘distributes’ the water between port and starboard shell and hence no roll moment will be present. The results in Figure 10 can be compared with those in Fig. 9 and the difference in response is found to be very significant. The survivability in waves is decreasing while the ship in calm water lacks the large intermediate heel angles. The results suggest that a correct modelling of all boundaries and the openings in those bulkheads is very important. Large machinery spaces which are open to the sea need to be modelled correctly.

4.3 Effect of Wave Steepness The shorter the wave period, the steeper the waves will be. In FREDYN longcrested waves are used and the spectrum is created by linear superposition. Thus, non-linear waves are not considered in this study. This would increase the local wave steepness in the wave crests.

658 R. van’t Veer et al. 'Open' compartment in deck 2 defintion 0

HEEL [deg]

CALM WATER -5 Hs = 3.5 m, Tp =9.7 s -10

-15 0

3600

7200

TIME [s]

Fig. 10 Effect of removing a single longitudinal boundary in the machine engine room. The results should be compared with those in Figure 9, which include the longitudinal boundary.

HEEL [deg]

Figure 11 shows results of simulations in different sea states of 2.5 m significant wave height. In Figure the results are shown for a number of sea states with a significant wave height of 3.5 m. Both figures suggest that the roll response is affected and that the steeper waves lead to, in general, a larger list after some time. The roll response of the damaged ship is detuned from the peak period of the waves so that vertical relative motions are the driving phenomenon for pumping water into the ship. The steeper the waves the more critical wave crests are experienced in the same amount of time and the greater tendency that the roll period will be detuned from the wave period.

4.00 2.00 0.00 -2.00 -4.00 -6.00 -8.00 -10.00 -12.00 -14.00 -16.00

Three compartment damage

Hs 2.5 Tp 6.3 Hs 2.5 Tp 7.5 Hs 2.5 Tp 8.5 Hs 2.5 Tp 9.7 Hs 2.5 Tp 10.9

0

3600

7200

10800

14400

18000

21600

TIME [s]

Fig. 11 Effect of wave steepness on the roll response in Hs = 2.5 m..

4.4 Effect of Sea State Realisation A sea state is constructed from superposition of a number of wave components with different wave amplitude and random phase, such that the spectral density defines the spectrum. Variation in the phases introduce different sea state

Exploring the Influence of Different Arrangements of Semi-Watertight Spaces

659

realisation and in critical conditions the behaviour of the ship in time will be different leading to variations in time to flood. Three compartment damage 10.00 Hs 3.5 Tp 7.5 Hs 3.5 Tp 8.5 Hs 3.5 Tp 9.7

HEEL [deg]

5.00 0.00 -5.00 -10.00 -15.00 -20.00

0

3600

7200

10800

14400

18000

21600

TIME [s]

Fig. 12 Effect of wave steepness on the roll response in Hs = 3.5 m. Hs = 3.5 m, Tp = 7.5 s, beam seas drifting 2.00 0.00

HEEL [deg]

-2.00 -4.00 -6.00 -8.00 -10.00 -12.00 -14.00 -16.00 0

3600

7200

10800

14400

18000

21600

TIME [s]

Fig. 13 Effect of seastate on the heel of the ship during flooding in beam seas condition.

In Figure 13 a number of time domain simulation results are given in different sea state realisations. The roll response is filtered and only the mean roll response is shown. Two criteria lines are given; 7 degrees heel and 15 degrees heel. As can be seen, in four realisations the mean heel remains below 7 degrees, while in two other realisations the mean heel is significant larger than 7 degrees after 6 hours of progressive flooding. The difference in the first heel towards the damage is marginal. The maximum heel remains below 15 degrees in all simulations within 6 hour time span, but as can be seen there is not always an equilibrium roll angle reached.

4.5 Survivability A large number of simulations should be carried out to investigate properly the survivability of the unbuilt large passenger ship in waves. The framework has been outlined in this paper.

660 R. van’t Veer et al.

The results so far suggest that the first roll response of the ship after damage is significant. But it should be kept in mind that many uncertainties as for example the effect of the striking ship have not been included. Depending on initial conditions and in terms of survivability this first transient heel may be neglected. In all simulations the ship heel reduced to a few degrees (around 4 deg) within half an hour. Depending on the significant wave height the ship remains at this list (Hs = 2.5 m) or increases slowly in time (Hs = 3.5 m). The variation in roll response between different sea state realisations increases with increasing wave height, as was expected. The time-to-flood or time-to-criteria can be summarised as follows, using: s = number of runs that comply / number of runs  Mean list below 7 degrees: Hs = 2.5 m Hs = 3.5 m

all simulations comply, s=1 s = 4 / 6 = 0.67

 Time-to-criteria of 7 degrees list: Hs = 2.5 m Hs = 3.5 m

T > 6 hours T = 4 hours (minimum)

 Maximum heel below 15 degrees Hs = 2.5 m Hs = 3.5 m

s=1 s=1

5 Conclusions The results in this paper suggest that the unbuilt large passenger ship design considered complies with damage stability criteria when subjected to a large threecompartment damage. The simulation results imply that accurate modelling of longitudinal and transverse bulkheads is important, as well as the proper modelling of the protection of downflooding points. However, the results in waves suggest that the ship, in more extreme conditions, is less sensitive to the modelling of the door type of a single downflooding point than in calm water. The scatter in the time-to-criteria increases with increasing wave height. More simulations are required to perform full insight in the survivability boundary with respect to the defined criteria. A time domain simulation model is a useful tool to investigate the behaviour of a damaged passenger ship in waves and to investigate critical points in the design for damage control options.

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Acknowledgement The authors would like to express their sincere gratitude to Mr. A. Serra from Fincantieri, who provided all data for the Large Passenger Ship. The research was sponsored by the US Coast Guard (DTCGG8-04-C-MSE080).

References De Kat JO, Kanerva M, Van’t Veer R.and Mikkonen I (2000) Damage survivability of a new RoRo Ferry, Proc 7th Int. Conf on Stab for Ships and Ocean Veh, STAB 2000, Launceston, Tasmania, Feb 2000. Palazzi L, De Kat JO (2002) Model experiments and simulations of a damaged ship with air flow taken into account, Proc 6th Int Stab Workshop, Oct. 2002, N.Y. MSC 78/WP.14, May 2004, IMO, London. SLF 45/3/5, (April 2002) Investigations and proposed formulations for the factors “p”, “r”, and “v”: the probability of damage to a particular compartment or compartments. SLF 45/INF.2, March 2002, Large passenger ship safety, with Annex: Damage stability “Time to Sink” Feasibility Study by Herbert Engineering Corp., USA. SLF 45/10/1, April 2002, Revision of the model test method specified in the 1995 SOLAS conference resolution 14. SLF 46/INF.3, June 2003, Time-to-flood simulations for a large passenger ship – initial study. SLF 47/INF.6, June 2004, Survivability investigation of large passenger ships. SOLAS edition 2001, Chapter II-Part B, Subdivision and stability; Chapter III-Part B, Requirements for ships and life-saving appliances. Van’t Veer R, De Kat JO, Cojeen P (2002) Large passenger ship safety: Time to sink, Proc 6th Int Ship Stab Workshop, Webb Inst. Van’t Veer R, Serra A (2003): Large Passenger ship safety: Time to sink simulations, RINA Conference on Passenger Ship Safety, March 2003.

Time-Based Survival Criteria for Passenger Ro-Ro Vessels Andrzej Jasionowski* Dracos Vassalos** Luis Guarin* *Safety At Sea Ltd ** The Ship Stability Research Centre (SSRC), University of Strathclyde

Abstract This paper outlines the study undertaken within the framework of research aiming to address the compound problem of the absolute time available for passenger evacuation on a damaged passenger/Ro-Ro vessel undergoing large scale flooding of car deck spaces. Deriving from extensive experimental information and utilising SEM principles, a methodology for predicting ship survival time is proposed that accounts for wave characteristics, water ingress/egress and vessel survivability. The progress achieved to date is discussed and aspects needing further investigation are highlighted.

Nomenclature Hs Tz Tp

   f

 a Q K dA l

Significant wave height Zero crossing period Modal period Significant wave steepness Wave length Spectral peakness parameter Residual or instantaneous freeboard Wave elevation Wave amplitude Flooding rate Flooding coefficient Flooding opening area Flooding opening length



Mean wave frequency Wave frequency mj jth spectral moment S() Wave energy spectrum n Random phase angle



M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_38, © Springer Science+Business Media B.V. 2011

663

664 A. Jasionowski et al.

(t) Envelope process Ha Ga



Number of waves in a high-run Number of waves in a group Spectral bandwidth

1 Introduction As history has repeatedly shown, maritime disasters bring about highly emotional societal response, in particular when large number of casualties is involved, leading to grief and often anger that overwhelms the ensuing political processes. These in turn often provoke imprudent deployment of efforts to improve safety, ultimately leading to new legislation that disregards many evolutionary aspects of ship design and operation primarily due to inadequate understanding of the complex processes involved in ship foundering. Serious deficiencies in safety standards, regarding in particular ship and passenger survival, are exemplified by recent well-publicised accidents of Ro-Ro passenger vessels, notably that of Herald of Free Enterprise and the Estonia, where due to very rapid deterioration in their stability no adequate time to orderly evacuate passengers and crew was available, resulting in large number of casualties in both accidents. Although strict new regulations have since been adopted internationally, arguably leading to safety improvements, the question of how long it takes a vessel to capsize from a breach in her hull has yet to be answered satisfactorily. Deriving from the above, it is the aim of this paper to call for a pro-active philosophy towards the problem of passenger survival and to encourage, through stimulating discussion, more research effort to better understand the processes involved in the loss of stability of Ro-Ro passenger vessels in case of large scale flooding in order to ensure that the minimum required time for such vessels to maintain their function as safe passenger-carrying platforms is provided. As mentioned above, to date no direct guidelines or regulations have been proposed as regards minimum ship survival time, which seems to defy logic in face of the new SOLAS Regulation 28-1-3, (SOLAS 2001), in force since 1 July 1999, which together with the IMO MSC/Circ.1033 set maximum evacuation time of passengers and crew at a level of no more than 60 minutes for newly built RoRo vessels. A series of full scale evacuation trials, e.g. (Wood 1996) as well as simulations by means of state of the art numerical techniques, (Christiansen 2000), have confirmed that this level seems to be? appropriate. The subject of survival time has recently been investigated in (Jasionowski et al. 1999), whereby limited experimental data, supplemented with extensive timedomain numerical simulations, (Turan 1992), have allowed for better understanding of concepts associated with survival time, as well as for proposal of quantitative survival time criteria. The developed formulae were based on the Static Equivalent Method (SEM), (Vassalos et al. 1996), with the relationship between the SEM

Time-Based Survival Criteria for Passenger Ro-Ro Vessels 665

floodwater elevation, h, and the relevant critical sea conditions, Hs, adjusted according to the available statistics on survival time. These statistics, in turn, were derived based on the concept of a band of critical sea states, see Figure 1, in which a ship with breach in her hull would sustain an acceptable attitude (heel less than 20deg) for time varying from above 60 min (“Safe Region”), down to a number of seconds (“Unsafe Region”), with associated probability of capsize in a given sea state varying from nearly zero to nearly 100%, respectively.

Capsize Boundary ‘Unsafe Region’

Hsig [m]

‘Uncertain Region’

Capsize Band ‘Safe Region’

GM [m]

Fig. 1 Definition of a capsize band and survival boundaries

Further studies (Jasionowski et al. 2001), however, led to better understanding the inadequacy of directly applying SEM to estimate ship survival time. This derives mainly from lack of highly accurate and consistent physical-model experiments allowing for precise quantification of the survival band. In the absence of such data a simplified numerical model was used instead leading to somewhat hasty generalisations of ship behaviour in near-capsize sea states. Although the approach pursued in (Jasionowski et al. 1999), associating survival time with survival bands is still valid, the availability now of the aforementioned experimental data, coupled with improvements in the modelling of damaged ship dynamics, (Letizia and Vassalos 1995) to (Specialist Committee for the Prediction of Extreme Motions and Capsizing 2002), allow for new line of thinking as regards dynamic behaviour of damaged ships in waves and consequently of the concept of ship survival time and factors determining it.

2 Hypothesis Capsizing of a damaged ship is an extremely complex phenomenon, arising from interactions of highly non-linear ship-floodwater-waves system, thus displaying characteristics of an apparent chaotic behaviour. However, contrary to this notion,

666 A. Jasionowski et al.

research results presented in (Vassalos et al. 1996) created a breakthrough by identifying that damaged ship capsizing was of quasi-static nature with the dynamic processes involved of secondary importance. Thus, ship survival was defined by correlation of static stability related features (the SEM floodwater elevation h) with the mean significant wave height in which the ship capsized. With the question of survival time posed, a hypothesis is put forward in this paper that this method can be improved if the wave environment is considered on the basis of individual waves or groups of waves as an integral element of the capsizing process. Incidence of these groups can then be used to more accurately identify when a capsize event is likely to occur whilst encountering a random sea and hence to statistically determine the survival time. In other words, it is suggested here that the vessel capsize is governed predominantly by two interacting phenomena, namely (a) ship slow attitude variation (quasi-static) due to water accumulation and (b) the ability of waves to pump water onto the Ro-Ro deck in rates higher than the water egress, such ability being primarily a function of the statistical characteristics of wave groups. This concept, put in the context of a risk-based approach, is illustrated in Table 1 and Figure 2, and discussed thereafter in the following sections. Table 1. Procedure of determining survival time based on wave group statistics Input

Hs, Tz, JONSWAP, mj, Average group length Ga and average high run Ha for different height levels, Vcritical from SEM

Step 1

Water ingress per wave group?

(C) vs (B) Water egress per wave group? Net inflow per wave group? Step 2

How many wave groups (Ncritical) are necessary to exceed (A) vs (D) Vcritical ? Step 3 (E)

What is the probability that Ncritical will occur within Tcritical? in a given sea state?

3 The Risk-Based Approach to Capsize In order to systematically explain the methodology adopted in realising the concept presented in Figure 2, an outline is given of the experimental data used, followed by a brief discussion on the assumptions and adopted simplified techniques used for quantification of flooding by waves as well as wave groups, followed finally by discussion of the results and trends obtained.

Time-Based Survival Criteria for Passenger Ro-Ro Vessels 667

Fig. 2 Risk-based concept of ship capsizing

3.1 Experimental Survivability Data The model tests were performed at Denny Tank in Dumbarton, the model testing facility of the University of Strathclyde. For the tests a 1:40 scale GRP model of Passenger Ro-Ro (PRR1) was used, see Table 2 and Figure 3 for the details. The model was equipped with 14 wave probes on the car deck and another two wave probes in front of the opening to measure the amount of floodwater on the car deck. Only the bilge keels were mounted as external appendages. The sea conditions were modelled according to JONSWAP wave energy spectrum, generated on the basis of linear theory of random processes. The parameters of the spectrum were determined according to the following relations:

668 A. Jasionowski et al.

 Tz 

Hs



, Tp 

2    , Tp  C  Hs , C  g

2  , g 

Tp 1.49  0.102    0.0142   2  0.00079   3

Where wave steepness was chosen as 1/25 and 1/20. The spectral peakness parameter was chosen as 3.3. A range of sea states of Hs=1.0 – 6.25 [m], each of which was represented by at least 5 different time realisations, were pre-tested to ensure modelling of the environment with high accuracy (±1mm model scale in Hs). The model was removed from the tank and the wave measured by a fixed wave probe. Table 2. Particulars of PRR1 vessel Length between perpendiculars

170.00 m

Subdivision Length Breadth Depth to subdivision deck (G-Deck) Depth to E-Deck Draught Displacement intact KMT KG

178.75 m 27.80 m 9.00 m 14.85 m 6.25 m 17301.7 t 15.522 m 12.892 m

Fig. 3 D901 damage case of PRR1

Time-Based Survival Criteria for Passenger Ro-Ro Vessels 669

The model was placed in the tank, free to drift, beam-on to the waves and the survivability was tested for approximately five successive sea states, again, each one repeated at least five times, so that a clear distinction between capsize and survival cases could be derived. In total, 627 experiments were performed. The vessel conditions tested are given in Table 3. The reported survivability is expressed in terms of Hs measured at the fixed probe with no model in the basin. Table 3. Initial conditions of the study cases Case No. Permeability

Initial Initial Draught Trim (m) (deg)

KG (m) Final Final Draught Trim (m) (deg)

Final Heel (deg)

Residual Freeboard (m)

1

0.95

6.250

0.000

12.200

6.922

0.784

2.500

1.472

2

0.95

6.250

0.000

12.892

6.916

0.788

3.200

1.308

3

0.95

6.250

0.000

13.456

6.904

0.794

4.100

1.102

4

0.95

6.250

0.000

14.114

6.867

0.807

6.000

0.680

5

0.95

6.250

-1.000

12.200

6.942

-1.947

2.400

1.476

6

0.95

6.250

-1.000

12.892

6.937

-1.960

3.000

1.336

7

0.95

6.250

-1.000

13.456

6.927

-1.956

3.900

1.128

8

0.95

6.250

-1.000

14.114

6.893

-1.957

6.100

0.630

9

0.95

6.250

1.000

12.200

6.917

3.353

2.500

1.477

10

0.95

6.250

1.000

12.892

6.910

3.370

3.200

1.314

11

0.95

6.250

1.000

13.456

6.896

3.365

4.100

1.110

12

0.95

6.250

1.000

14.114

6.847

3.375

6.700

0.531

13

0.95

5.750

0.000

12.200

6.391

0.746

2.400

2.027

14

0.95

5.750

0.000

12.892

6.387

0.750

3.000

1.886

15

0.95

5.750

0.000

13.456

6.377

0.758

3.700

1.726

16

0.95

5.750

0.000

14.114

6.352

0.774

5.300

1.364

17

0.95

6.750

0.000

12.200

7.451

0.793

2.600

0.918

18

0.95

6.750

0.000

12.892

7.443

0.794

3.300

0.757

19

0.95

6.750

0.000

13.456

7.429

0.796

4.400

0.505

20

0.95

6.750

0.000

14.114

-

-

-

-

21

0.95

6.250

-0.600

12.892

6.929

-0.842

3.100

1.319

22

0.7

6.250

0.000

12.200

6.735

0.573

1.800

1.828

23

0.7

6.250

0.000

12.892

6.732

0.576

2.200

1.734

24

0.7

6.250

0.000

13.456

6.727

0.580

2.900

1.570

25

0.7

6.250

0.000

14.114

6.708

0.591

4.300

1.250

26

0.7

6.250

-1.000

12.200

6.748

-2.223

1.600

1.864

27

0.7

6.250

-1.000

12.892

6.745

-2.239

2.100

1.746

28

0.7

6.250

-1.000

13.456

6.741

-2.234

2.600

1.628

29

0.7

6.250

-1.000

14.114

6.728

-2.228

3.900

1.327

670 A. Jasionowski et al.

Only fifteen experiments, 101-116, of Case 2 were analysed with regard to the concept discussed in this paper. The overview of the experimentally derived survivability in terms of critical Hs is given in Table 4. Samples of time series for non-capsize and capsize cases are given in Figure 4 and Figure 5, respectively.

Fig. 4 Time series for survivability tests of PRR1 vessel, “survive” case, Hs=4.0m Table 4. Ship survivability Run No.

Target Wave Height Hs(m)

Comments

Run 101

4.0

Passed

Run 102

4.0

Passed

Run 103

4.0

Passed

Run 104

4.0

Passed

Run 105

4.0

Passed

Run 106

4.25

Passed

Run 107

4.25

Capsized

Run 108

4.25

Capsized

Run 109

4.25

Capsized

Run 110

4.25

Capsized

Run 111

4.25

Passed

Run 112

4.5

Capsized

Run 113

4.5

Capsized

Run 114

4.5

Capsized

Run 115

4.5

Capsized

Run 116

4.5

Capsized

Time-Based Survival Criteria for Passenger Ro-Ro Vessels 671

Fig. 5 Time series for survivability tests of PRR1 vessel, “capsize” case, Hs=4.5m

3.2 Quasi-Static Nature of Capsize Details of the SEM are given in (Vassalos et al. 1996). The main assumption considers the floodwater on the car deck to be sustained as a result of wave action at a level higher than the average free surface level. Figure 6 demonstrates this concept by comparing the water amounts at the instant of capsize, derived numerically (SEM) or experimentally, with the equivalent volume estimated for standard static conditions. As can be seen, the amount of floodwater on the car deck prior to capsize at any given vessel attitude exceeds the amount derived from standard static stability calculations.

Fig. 6 Correlation between critical heel angle and critical amount of floodwater on the car deck at the instant of capsize

Righting lever [m]

672 A. Jasionowski et al. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 0 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1

KG=12.892

5

10

15

20

25

30

35

40

45

50

Intact

55

60

Damaged

Heel angle [deg]

Fig. 7 Righting levers of PRR1 vessel in intact and damaged conditions

This quasi-static accumulation of floodwater will lead eventually to capsize, once some critical value is exceeded. For instance it can be seen in Figure 5 that after about 2,000 seconds, about 1,000 tonnes of water is accumulated on the car deck, which however, did not lead to capsize. Once this value is exceeded two minutes later, the vessel capsized. The 1,000 tonnes correspond to approximately 15 degrees of heel, Figure 6, which is somewhere in the middle between the angle of maximum restoring and vanishing stability, Figure 7. Considering strictly static conditions, the floodwater corresponding to maximum restoring at 12deg (700tonnes), should be capable of capsizing the vessel, provided the water does not flow out as the ship heels. However, since the floodwater will flow out, the actual amount should be described by some form of probability distribution function, PDF, see curve (A) in Figure 2, spanning the range between maximum restoring (12 deg; 700t) and vanishing stability (18deg; 1,200t). The form of this distribution should be derived based on observations of ship behaviour during capsize events recorded either experimentally or derived by numerical simulations. Furthermore, the above distribution will influence the water egress rate distribution, curve (B) in Figure 2, which additionally must take into account the geometry of the opening as well as internal distribution of volume on the car deck at a given ship attitude. Clearly, conditions involving trim, for instance, will lead to reduced floodwater egress than when the trim is zero. Both of these aspects are part of research work at SSRC towards the development of time-based survival criteria.

3.3 Dynamic Behaviour To further elucidate the viability of an approach based on hydrostatic properties of the vessel, spectral analyses of the sample signals presented in Figure 4 and Figure 5 are shown below, Figure 8 to Figure 11, with some statistics given in Table 5.

Time-Based Survival Criteria for Passenger Ro-Ro Vessels 673 6 wave-101 wave-114

5

S [m2s/rad]

4 3 2 1 0 0.2

0.3

0.4

0.5

0.6

0.7

0.8

ω [rad/s]

0.9

1

1.1

1.2

1.3

1.4

Fig. 8 Spectral analysis of wave elevation (Run 101 and Run 114) 6 heave-101 heave-114

S [m2s/rad]

5 4 3 2 1 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ω [rad/s]

1

1.1 1.2 1.3 1.4

Fig. 9 Spectral analysis of heave motion (Run 101 and Run 114) 6 roll-101 roll-114

S [deg2s/rad]

5 4 3 2 1 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ω [rad/s]

1

1.1 1.2 1.3 1.4

Fig. 10 Spectral analysis of roll motion (Run 101 and Run 114)

674 A. Jasionowski et al. 6 relative-101 relative-114

S [m2s/rad]

5 4 3 2 1 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ω [rad/s]

1

1.1 1.2 1.3 1.4

Fig. 11 Spectral analysis of relative motion between car deck and wave elevation (Run 101 and Run 114) Table 5. Significant values -

Wave_fx (m) Wave_tr (m) Heave (m) Roll (deg) Relative (m)

Run 101 4.23

3.91

3.57

3.85

4.42

Run 114 4.65

4.35

3.83

4.61

4.51

The two examples, Run101 and 114, correspond to the sea states where the ship survives five successive runs at Hs=4m, and sea states where the vessel systematically capsizes at Hs=4.5m, respectively. The heave motion in both examples does not display any peculiarities. Since the waves are much longer than the vessel beam the ship simply follows the wave elevation. Hence, the spectra of the heave response correspond closely to the wave spectra. The roll, on the other hand, shows a considerable relative increase in the response between the two cases. This is a result of considerably more wave energy in the sea state with Hs=4.5m exciting the ship at its natural roll frequency (about 0.5 [rad/s]). However, since the roll motion of a damage ship is greatly damped, there is only about 1deg difference in the significant roll, which ultimately does not contribute noticeably to the signal of relative motion between the wave and the edge of the car deck in both cases, Figure 11. Therefore, since the motion responsible primarily for the amount of flooding into the ship does not demonstrate a meaningful change, it can be argued that any increase in the ingress rates derive primarily from the wave action.

3.4 Prediction of Flooding As mentioned above, water ingress/egress is the reason that a vessel capsizes. Therefore, any time-based survival criterion must take time-dependence of flooding

Time-Based Survival Criteria for Passenger Ro-Ro Vessels 675

into account. As is shown in Figure 12, the cumulative probability distribution function of flooding rates (ingress and egress) confirm that for the sea states where the ship does not capsize (Run101-105, Hs=4.0m), the flooding rates are lower than for the case when the vessel capsizes for every considered sea state (Run 112116, Hs=4.50m). How is this conditioned on the sea state is the key question to be established in this study. The corresponding probability distribution functions, Figure 13, reveal a known peculiarity, that water egress is higher than water ingress, thus the PDF of flooding rates take normal-type shape, with non-zero mean.

Fig. 12 Cumulative probability distribution function of flooding rates (Runs 101-116)

Fig. 13 Probability distribution function of flooding rates (Runs 101 and Run 114)

676 A. Jasionowski et al. x(t) a

f

t1

t2

Tz

Fig. 14 Simplified modelling of water ingress on a Ro-Ro car deck

In addressing the process of floodwater ingress caused by the action of waves, only a simplified model was adopted in this research in order to identify any conforming trends in properties of groups of waves, which lead to capsize, Figure 14. More specifically, the following idealistic model was adopted in the first instance. t2

Q  K   2  g  h  dA  dt

(1)

t1

Where:

t1 

Tz d 4

(2)

t2 

Tz d 4

(3)

d

a Tz  sin 1   4 f

(4)

 (t )  a  sin   t 

(5)

h   (t )  f

(6)

dA  h 1

(7)

The level of validity of this simplified approach is demonstrated in Figure 15.

Time-Based Survival Criteria for Passenger Ro-Ro Vessels 677 Run 114, Flloding rates per individual waves 800

SUM( dQ/dt ) [m3/s]

700

Experimental measurment Theoretical

600 500 400 300 200 100 0 0

500

1000 1500 Time [s]

2000

2500

Fig. 15 Comparison between experimental measurements with predictions of cumulative water ingress by model ( 1 ), K = 0.13, f = f(t) – instantaneous freeboard from experiments

This model was used in establishing relevant the flooding rates caused by a group of waves for the different cases discussed in Sect. 3.5.3.

3.5 Analysis of Wave Groups An interesting phenomenon often observed in wind generated seas is a sequence of high waves having nearly equal periods, commonly known as wave groups, (Ochi 1998). It has been known that such wave groups often cause serious problems for the safety of marine systems when the period of the individual waves in the group are close to the marine system’s natural motion period. This is not because the wave heights are exceptionally large but because of motion augmentation due to resonance, which may induce failure (capsizing) of the marine system. The physical explanation of the wave group phenomenon has yet to be clarified, however there are suggestions that the wave field does not consist of independently propagating Fourier components but instead consists wholly or in part of wave groups of a permanent type. As evidence, results are presented of field and laboratory observations indicating that harmonic components of waves propagate at higher phase velocities than those predicted by linear theory. Many studies on stochastic analysis of group waves in random seas have been carried out, primarily concerning the frequency of occurrence of the phenomenon. These studies may be categorised into two approaches: one treats a sequence of large wave heights as a Markov chain problem, the other considers the phenomenon as a level-crossing problem associated with the envelope of a random process. Only the latter approach is considered in this research.

678 A. Jasionowski et al.

3.5.1 Envelope Process The probabilistic analysis of random phenomena based on the envelope process was first introduced by (Rice 1945) in communication engineering. It is based on the mathematically rigorous spectral analysis approach in the frequency domain. The wave spectrum is a source of information from which the probabilistic prediction of various wave properties can be achieved in the probability domain. Assumptions most commonly introduced at this stage are that waves are considered to be a steady-state Gaussian ergodic random process, and the wave spectral density function is narrow-banded. Under these conditions, the probability functions applicable to various wave properties such as the frequency of occurrence of group waves, etc, in a given sea can be analytically derived. The envelope process represents a measure of change of wave amplitudes in the time domain and is defined as a pair of symmetric curves that pass through the wave crests and troughs. For a mathematical presentation of the envelope process, the wave profile can be written as, (Ochi 1998):

 t   Re  an  ei t i n

n

n

(8)

Where an is a normal random variable with zero mean and a variance mo. Let



 be the mean frequency defined as:

m1 m0

(9)

Where mj is jth moment of wave energy spectrum ( 10 ).

m j    j  S    d

(10)

The wave profile can be expressed as:

 t   Re  an  ei  t i n

n it

n

 t   Re  t   ei t   eit Where:

 t   eit

is a slow amplitude

(11) (12)

Time-Based Survival Criteria for Passenger Ro-Ro Vessels 679

– –

 t  e

is the envelope process

i .t

is a carrier wave

Fig. 16 Analysis of wave group statistics. A “group” of waves above a certain level is constituted by at least two successive level crossings (Run 114)

The envelope process for a given wave record can be evaluated from (14) after applying the concept of the Hilbert transform (13):

~

 t  

1







  

 t    d

(13)



~

 t    2   2

(14)

An example of the envelope process derived is shown in Figure 16. 3.5.2 Wave Groups Statistics Drawing on the concept of envelope process, a wave group can be defined as the up-crossing of the envelope above a certain level, a principle credited to (Rice 1945) and (Longuet-Higgins 1984). Following this approach various authors have developed methods to evaluate mean values of the length of time a wave group persists, the number of wave crests in the group, etc. The group length is defined as the time interval between two successive up-crossings of a given level by the wave envelope function, see Figure 16, and a run of high waves is formed by a number of successive high waves, which exceed a specified level. The number of waves in a high-run, Ha, and in a group, Ga, can be found as follows, (Kyllikki 1989):

680 A. Jasionowski et al.

m0 1 1  2 H a        2  Ga     H a     e

(15)

2 2m0

(16)

Where:



m2  m0 1 m12

(17)

The corresponding length of a group and the high run can be found by multiplying ( 15 ) and ( 16 ) by the zero crossing period Tz, ( 18 ).

Tz  2   

m0 m2

(18)

It is important to underline here that the above expressions having been derived as a level crossing problem, imply indirectly that a wave group can be composed of only one wave.

Fig. 17 Duration of group for different wave amplitude levels (Runs 101); comparison between theoretical prediction and experimental data, (assumed minimum one wave per group or two waves per group)

The deficiency of this assumption, pointed for instance by (Ochi 1998), was determined by simple analysis of time domain signals of the envelope process derived from experimental wave records, Figure 17. The theoretical predictions of

Time-Based Survival Criteria for Passenger Ro-Ro Vessels 681

group lengths, agrees reasonably well only for higher values of the envelope level, Hs

Fig. 18 Duration of group for different wave amplitude levels (Runs 101-116)

Demonstrable improvement in theoretical predictions of the statistical properties of group waves can be achieved if (a) exceedance of a specified level and (b) at least two wave crests during the exceedance are considered. (Ochi 1998), provides such formulation. However it has not been considered in this study as yet. Instead, the group lengths as well as high-run length for different envelope function levels have been derived based on time series analyses and under the assumption that a group must consist of at least two waves, as shown in Figure 18 and Figure 19.

Fig. 19 Duration of high-run for different wave amplitude levels (Runs 101-116)

A somewhat disappointing conclusion derived from the above test is that the duration of either the group length or the high-run length, are indistinguishable between the sea states considered. This implies that no inference can be made as to the characteristics of average statistics of wave groups that lead to capsize. This will be further verified with other data during future work.

682 A. Jasionowski et al.

3.5.3 Other Analyses In the meantime more rigorous examination was undertaken of the composition of the wave groups chosen ad-hock based on visual observations from the sixteen runs under examination, see samples shown in Figure 20 and Figure 21. In this case the groups of waves were identified simply by trough-to-crest excursions of the envelope process.

Fig. 20 Sample of a wave group leading to near-capsize, Run 101, Hs=4.0m

Fig. 21 Sample of a wave group leading to capsize, Run 114, Hs=4.5m

The individual characteristic of each of the groups examined were as follows:    

Maximum, Mean and Total stored wave energy Maximum, Mean and Total water ingress Maximum and Mean wave height Total number of waves

Probability distributions of the above properties in the individual wave groups were also examined. Figure 22 shows that the wave composition of groups for either near or actual capsizes does not show any distinguishable patterns. In fact none of the examined characteristics for the sixteen runs allowed as yet identification of prevailing

Time-Based Survival Criteria for Passenger Ro-Ro Vessels 683

trends, e.g. Figure 23 or Figure 24. Reasons for this outcome include the very small data sample size processed to date, the possibility that each of the groups examined could lead to a different event (potentially a capsize) if the conditions prior to encountering the group, e.g. water egress, were different, see Figure 28. Interestingly, the latter implies that the curve (B) shown in Figure 2 should be dependent on the wave properties as well. 6 5

[m]

4 3 2 1

G_max_wave G_mean_wave

0 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 Run No

Fig. 22 Maximum and mean wave heights in the selected groups 50

G_max_flooding

45

G_mean_flooding

40 35

[m]

30 25 20 15 10 5 0 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 Run No

Fig. 23 Maximum and mean floodwater ingress rates in the selected groups

Fig. 24 Cumulative probability distributions of the floodwater ingress rates in the selected groups

684 A. Jasionowski et al.

Notwithstanding the lack of any clear breakthrough in developing the concept discussed in this paper so far, some more general observations indicate that it is a promising approach. For example, Figure 25 shows time series of average wave group height and the flooding rate estimated per group of waves. 8 AvrWaveInGroup

6

Dvolf_dt

4 2 0

0

500

1000

1500

2000

2500

-2 -4 -6

Fig. 25 Time series of average wave group height (wave group based on trough-to-crest excursion in wave envelope function) and the flooding rate estimated per group of waves, Run 101

As can be seen, the flooding rates show a distinguishable relationship between inflow/outflow and the average wave height in the group, see Figure 26.

dQ/dt [m3/s] (per group)

8 6 4 2 0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

-2 -4 -6

Average wave height in a group [m]

Fig. 26 Relationship between the measured water ingress/egress and the average wave height in a wave group Run 101

Time-Based Survival Criteria for Passenger Ro-Ro Vessels 685

Fig. 27 Floodwater ingress measured experimentally [Experiments with “capsize” and “survive” outcomes]

Fig. 28 Floodwater ingress and egress (tonne) measured experimentally

Furthermore, as can be seen in Figure 27, the cumulative water ingress, when inspected for each encountered wave group, shows clear differences in the character between the survive and capsize cases. This again underlines the importance of considering both floodwater ingress as well as egress. For instance as is shown in Figure 28, the egress could not deplete the floodwater accumulation on the car deck, and as a result the vessel capsized.

4 Conclusions This paper presents a new approach for predicting ship survival time together with the progress achieved to date in its development. Many aspects need further investigation to either validate or disprove different elements of the method.

686 A. Jasionowski et al.

The main aspect of the new approach, derived from the SEM principles, is a direct association of the characteristics of the waves with the process of water ingress and egress, and thus with vessel survivability. Although vessel capsize is a case of limit state behaviour, a physical process that is highly sensitive to variation of many governing factors, it is anticipated that observed regularities in capsizing will lead to identification of a functional representation of this sensitivity in the near future.

Acknowledgements This research has been funded partly internally by the Ship Stability Research Centre of the University of Strathclyde, and partly by the Marie Curie Industry Host Fellowship, Contract No MCFH/ 2000/00240-SI2.327090/SaS. The experimental data have been derived during Phase II of the research into the time-based survival criteria for Ro-Ro vessels funded by the UK MCA, with partial support of the European Commission DG Research project HARDER, Contract No GRD1-199910721. All this support is gratefully acknowledged.

References Christiansen, G (2000) A Report on the State of the Art of Evacuation and Crowd Simulation. The Ship Stab Res Centre, SSRC-05-00-GC-01-IR De Jan K, Van’t Veer R (2001) Mechanisms And Physics Leading To The Capsize Of Damaged Ships. 5th Int Workshop, Univ of Trieste Jasionowski A (2001) An Integrated Approach to Damage Ship Survivability Assessment. PhD thesis, Univ of Strathclyde, Glasgow Jasionowski A, Vassalos D (2001) Numerical Modelling Of Damage Ship Stability In Waves. 5th Int Workshop On Stability And Oper Safety Of Ships, Trieste Jasionowski A, Dodworth K, Vassalos D (1999) Proposal of Passenger Survival-Based Criteria for Ro-Ro Vessels. Int Shipbuilding Prog 46 (448) Jasionowski A, Vassalos D, Chai, S H, Samalekos P (2001) Time-based survival criteria for RoRo vessels, Phase II. Draft Final Report for UK MCA Kyllikki TK (1989) Prediction of Critical Conditions for Extreme Vessel Response in Random Seas. PhD dissertation, Univ of California Letizia L, Vassalos D (1995) Formulation of a Non-Linear Mathematical Model for a Damaged Ship With Progressive Flooding. Int Symposium on Ship Safety in a Seaway, Kaliningrad, Russia Longuet-Higgins M S (1984) Statistical Properties of Wave Groups in a Random Sea State. Phil Trans R Soc London, A 312, pp 219-250 Ochi M (1998) Ocean Waves, The Stochastic Approach. Series 6, Camb Univ Press, ISBN 0 521 56378 Rice S O (1945) Mathematical analysis of random noise. Bell System Tech J 23 and 24. Reprinted in: Selected Papers in Noise and Stochastic Processes. Wax N (ed), Dover Publ, NY, 1954, pp 1-162 SOLAS (2001) Consolidated Edition

Time-Based Survival Criteria for Passenger Ro-Ro Vessels 687 Specialist Committee for the Prediction of Extreme Motions and Capsizing (2002)Final Report and Recommendations to the 23rd ITTC Tucker M J, Challenor P G, Carter D J T (1984) Numerical simulation of a random sea: a common error and its effect upon wave group statistics. Appl Ocean Res 6 (2): 118-122 Turan O (1992) Dynamic Stability Assessment of damaged Ships by Time-Domain Simulation. PhD Thesis, Univ of Strathclyde Vassalos D, Pawłowski M, Turan O (1996) A theoretical investigation on the capsizal resistance of passenger Ro/Ro vessels and proposal of survival criteria. Final Report, Task 5, The North West European R&D Project Wood AG (1996) Validating Ferry Evacuation Standards. RINA Int Conf in association with The Nautical Inst on: “Escape, Evacuation & Rescue Design for the Future”

Pressure-Correction Method and Its Applications for Time-Domain Flooding Simulation Pekka Ruponen Napa Ltd Abstract The principle idea of using pressure-correction method for time-domain flooding simulation is presented. Special attention is paid on the handling of large openings and relevant modifications to the equations. Practical examples of application are also discussed. First, progressive flooding in a complex system of rooms and openings of a passenger ship is presented. The study concentrates on the effect of input parameters for non-watertight structures. The second example demonstrates cross-flooding calculation with the air compressibility taken into account.

1 Introduction Napa Ltd and Helsinki University of Technology have in close co-operation developed a novel time-domain simulation method for progressive flooding. The principle idea of applying pressure-correction technique for flooding simulation was first presented in Ruponen (2006) based on the assumption that all openings are relatively small. Some examples of the validation of this method were presented in Ruponen et al. (2007). In this paper, an extension of this simulation method is presented, along with a brief revisit to the theoretical background. A more comprehensive description is given in Ruponen (2007). The pressure-correction technique is very suitable for the flooding cases, where also air compression and the resulting airflows are significant. A detailed presentation of the theory for calculation of air pressures and airflows is given in Ruponen (2006) and Ruponen (2007). In this paper just a practical example of applying the flooding simulation for calculation of cross-flooding time is presented. The main emphasis is on the effect of air compression inside the equalizing tank.

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_39, © Springer Science+Business Media B.V. 2011

689

690 P. Ruponen

2 Pressure-Correction Method 2.1 Background Usually flooding simulation methods are based on the volumes of floodwater that are integrated explicitly from the flow velocities that are calculated from Bernoulli’s equation. The water height differences are then calculated from the volumes of water with the heel and trim angles taken into account. However, the applied simulation method is based on a completely different approach, where the volumes are calculated on the basis of water heights and the heel and trim angles. This is reasonable as the water height is physically more meaningful than the volume of water since it represents the hydrostatic pressure. Consequently, the progress of the floodwater can be solved implicitly on the basis of the pressures in the rooms and the velocities in the openings. The ship model for flooding simulation can be considered as an unstructured and staggered grid (Fig. 1). Each modelled room is used as a single computational cell. However, the flux through a cell face is possible only if there is an opening that connects the rooms (cells). 11111 00000 000001111 11111 0000 00000 11111 0000 1111 0000 00000 11111 00001111 0000 1111 00000 11111 000001111 11111 0000 1111 0000 1111 00000 11111 00000 11111 00000 11111 0000 1111 0000 1111 00000 11111 00000 11111 00000 11111 0000 1111 0000 1111 00000 11111 00000 11111 00000 11111 0000 1111 0000 1111 00000 11111 000001111 11111 00000 11111 0000 0000 1111 00000 11111 00000 11111 00000 11111 0000 1111 0000 1111 00000 11111 00000 11111 00000 11111

Fig. 1 Staggered grid for flooding simulation

2.2 Governing Equations At each time step the conservation of mass must be satisfied in each flooded room. The equation of continuity for water is:



 t d   



w

v  dS

(1)

S

where w is density, v is the velocity vector and S is the surface that bounds the control volume . The normal vector of the surface points outwards from the control volume, hence the minus sign on the right hand side of the equation. The mass balance for water, i.e. the residual of the equation of continuity, in the room i can be expressed as:

Pressure-Correction Method and Its Applications 691

m w,i   w S fs ,i

dH w,i dt

  w  Qw, k

(2)

k

where Sfs is the area of free surface in the compartment (assumed to be constant during the time step), Hw is the water height and Qw is the volumetric water flow through an opening in the compartment. The index k refers to an opening in the room i. The velocities in the openings are calculated by applying Bernoulli’s equation for a streamline from point A that is in the middle of a flooded room to point B in the opening: B

dp

 A







1 2 u B  u A2  g hB  hA   0 2

(3)

where p is air pressure, u is flow velocity and h is height from the reference level. It is assumed that the flow velocity is negligible in the center of the room (uA = 0). The equation (3) applies for inviscid and irrotational flow. The pressure losses in the openings and pipes are taken into account by applying semi-empirical discharge coefficients. Consequently, the mass flow through an opening k is:

m w,k   w Qw,k   w C d ,k Ak u k

(4)

where Qw,k is the volumetric flow through the opening, Cd,k is the discharge coefficient, Ak is the area of the opening and uk is velocity. Basically equation (4) applies only to very small openings. In the next chapter, the handling of tall openings is considered in detail. Bernoulli’s equation for water flow through the opening k that connects the compartments i and j (positive flow from i to j) can be written in a form of a pressure loss:

1 K k m w,k m w,k  Pi  Pj k 2

(5)

where the absolute value is used to define the direction of the flow. The dimensional pressure loss coefficient is defined as:

K k 

1

 w C d ,k 2 Ak2

(6)

When constant air pressure is assumed, the total pressure difference for an opening k that connects the compartments i and j is:

692 P. Ruponen

Pi  Pj k

  w g   f i, k   f  j , k 

(7)

where the following auxiliary function is used:

f i, k   maxH w,i  H o ,k ,0 

(8)

where Hw is the height of the water level and Ho is the height of the opening, measured from the same horizontal reference level. It is also possible to deal with openings that can be formed when structures (e.g. closed doors or down-flooding hatches) collapse under the pressure of the floodwater.

2.3 Pressure-Correction Equation The initial values are denoted with an asterisk and the corrections with an apostrophe. The linearization of equation (5) results in (Ruponen, 2006 and 2007):

K w ,k m w* ,k m w,k  Pi  Pj

(9)

Consequently, by using equations (2) and (9), the following pressure-correction equation can be derived (Ruponen, 2006 and 2007):

F i, k   H w ,i  F  j , k   H w , j 3  w S fs ,i  H w ,i  m w* ,i k 2 t K w ,k  w Qw* ,k

(10)

where the following auxiliary function is used:

F i, k   maxsign H w,i  H o ,k ,0

(11)

and t is time step (constant) and the mass balance is:

 m w* ,i   w  Q w* ,k   w S fs ,i H w* ,i

(12)

k

The underlined terms in equations (10) and (12) are zero if the room is filled up with water. The water heights are updated by adding the solved corrections and the results are used as initial values for the next iteration round. Some underrelaxation is usually needed. The iteration is continued until all mass balances are smaller than the required convergence criterion.

Pressure-Correction Method and Its Applications 693

2.4 Equation of Motion The calculation of progressive flooding with the pressure-correction method needs to be combined with the simulation of ship motions. In this paper it is assumed that the sea is calm and that both the trim and heave motions are quasi-stationary. The dynamic roll motion (t) is solved from the following simplified equation:

Axx ,tot    B xx ,tot    M st    M ext Vw,i 

(13)

where Axx,tot is the sum of inertia and added mass and Bxx,tot represents linear damping. Mst is the righting moment and Mext is the heeling moment due to floodwater. The natural roll period and critical damping ratio for the intact ship can be obtained from the roll decaying test or seakeeping calculations.

3 Extended Handling of Tall Openings 3.1 Calculation of Volumetric Flow In the previous chapter, a pressure-correction equation was derived by assuming that the openings are so small that they can be considered as points. In practice, this simplification is not always valid. Therefore, it is essential that the simulation method is extended to deal with tall openings as well. A more realistic, yet simple, representation for an opening with significant vertical height, such as an open door, is a straight line with a given area. A similar approach was used in Dillingham (1981) for calculation of two-dimensional flow over a bulwark and in Pawlowski (2003) for an opening with a constant width. However, in practice this method requires an additional assumption that the flow velocity is always perpendicular to the opening. The opening line can be considered as three separate openings since the sections are treated individually. This is illustrated in Fig 2. The section AB corresponds to an opening point with the same area since the flow through this section depends on both water heights but not on the vertical location of the opening. Consequently, no separate handling is needed. The section CD corresponds to an opening point for airflow. Consequently, the shape of the opening has no effect on the computation of airflow. In the following, it is assumed that all rooms are fully vented, and consequently air pressure is constant.

694 P. Ruponen

D 11 00 00 11

11 00 C 11 00 11 00

H w,max

0 B1 1 0 1 0

Ho,max

Ho,min

11 00 C 11 00 11 00

Lo

Hw,min

A 11 00 00 11

1 0 0 1

l 1 0 B 1 0 1 0

1 0 0 1

∆Ho β

reference level Fig. 2 Opening line and co-ordinate system definitions

The section BC needs to be treated with a different way since the volumetric flow through this section must be integrated. This affects also the pressure-correction equation for water heights. A detailed description of the applied methods is given in the following. The volumetric water flow through the section BC is obtained by integration of the flow velocity u over the corresponding part of the opening line (see Fig. 2): BC

Qw. BC  b 

 u  dl

(14)

0

where b is the width of the opening. Similarly to the case of a one-dimensional opening point, let us consider a streamline from point E that is in the middle of the flooded room, to point F that is in the opening between the points B and C. The following heights for the points along the streamline are used:

hE  H C hF  H B  l  sin 

(15)

where l is the distance from the point B along the opening line and  is the angle between the reference level and the opening line (see Fig. 2). The flow velocity is obtained from equation (3) and the pressure losses are taken into account in the form of a discharge coefficient. Consequently, the following equation for the volumetric water flow through the section BC can be obtained:

Pressure-Correction Method and Its Applications 695 BC

Q w.BC  C d b  2 g H w,max  H B  l  sin  dl

(16)

0

where Cd is the discharge coefficient for the opening when the jet discharges into air. It is practical to assume that the changes of the water height are rather small, so that Hw,max  HC and Hw,min  HB. Consequently, equation (16) can be written as: BC

Q w. BC  C d b 



2 g H C  H B  l  sin  dl

(17)

0

This can be evaluated analytically:

Qw. BC  C d b

2  2g 3  sin 

3 3   2  H  H  BC sin  2   H H  B C B  C   

(18)

The inclination angle of the opening line, i.e. the angle between the water levels and the opening (Fig. 2), can be evaluated on the basis of the vertical distance between the end points of the opening line, Ho. Consequently:

sin  

H o Lo

(19)

Equation (18) can be presented in a simpler form since the vertical distance between the points B and C is:

H C  H B  BC  sin 

(20)

Hence, the volumetric flow, equation (18) can be written as:

Qw. BC  C d  b 

3 2  2g H C  H B  2 3  sin 

(21)

The area of the opening between the points B and C is:

ABC  b  BC  b 

H C  H B  sin 

(22)

696 P. Ruponen

Therefore, equation (21) can be simplified to:

2 2g H C  H B  H C  H B 3 sin   C d ABC 2 2 g  H C  H B  3

Q w. BC  C d b

(23)

The basic form of the rearranged equation for the volumetric flow (23) is the same as in the equation for the flow through a one-dimensional opening, multiplied by the coefficient 2/3. This relation is used in the following when the pressure-correction equation is derived.

3.2 Pressure-Correction Equation The equation (23) can be presented in the form of a pressure loss, similarly to equation (5). The mass flow of water through the section BC is:

m w, BC   w  C d  ABC 

2 2 g H C  H B  3

(24)

and the square of the mass flow, divided by two is:

1 4 2 m w, BC m w, BC   w2  C d2  ABC   g H C  H B  2 9

(25)

This can be rearranged to:

 1  1  2 4 2 2    w  C d  ABC 9

    m w, BC m w, BC   w g  H C  H B    

(26)

Moreover, this can be simplified by applying the dimensional pressure loss coefficient, defined in (6). Therefore, equation (26) can be written as:

1 9   K w m w, BC m w, BC   w g H C  H B  2 4

(27)

Pressure-Correction Method and Its Applications 697

The basic form of this equation is similar to the pressure loss equation for onedimensional opening (5), but the dimensional pressure loss coefficient is multiplied by a constant factor of 9/4. Therefore, the same form of the pressure-correction equation can be used when this additional coefficient is taken into account. The total volumetric water flow through the opening k consists of two parts:

Qw* ,k  Qw* , AB ,k  Qw* , BC ,k

(28)

Consequently, the pressure-correction equation (10) for water heights in the case of two-dimensional openings can be rewritten as:



  K 

H w ,i  H w , j



4 G i, j   H w ,i  G  j , i   H w , j    9 K w , BC ,k  w Qw* , BC ,k 

*  w, AB ,k  w Qw, AB ,k 3  w S fs ,i  H w ,i  m w* ,i 2 t k

(29)

where the following auxiliary function is used:

G i, j   maxsign H w,i  H w, j ,0

(30)

The first term on the left hand applies for the water flow discharging into water since the mass flow depends on both water heights. The second term on the left hand side is for the water flow discharging into air since this part of the mass flow only depends on the water height on the maximum pressure side. The underlined term is taken into account only if the room is not filled with water. In principle, the basic form of the pressure-correction equation is exactly the same as in the case of one-dimensional opening points, equation (10). Discontinuities should be avoided during the iteration process in order to ensure convergence,. Therefore, the points B and C are kept constant during the time step. This simplification should not cause significant error if the applied time step is sufficiently short. The applicability of this implementation has been tested by performing comparative simulations, where the tall openings were modelled with several individual points. An example of the comparisons is presented in Fig. 3. In this case, 20 evenly distributed opening points give practically the same result as one vertical opening line. The difference is increased as the number of opening points is decreased. When the whole opening is submerged (volume of floodwater is larger than 10 m3) the modelling of the opening does not affect anymore.

698 P. Ruponen 12

3

volume [m ]

10 8 6 4 20 points 4 points 2 points line

2 0 0

10

20

30

40

50

time [s]

Fig. 3 Comparison of different modelling techniques for a tall opening

4 Simulation of Progressive Flooding 4.1 Case Study The studied case is a two-compartment damage in a medium sized passenger ship of 40 000 GT. The flooded compartments on two deck levels contain crew cabins, laundry and store areas. All doors are considered to be initially closed. The modelled rooms and openings (i.e. the computational grid) are shown in Fig. 4.

Fig. 4 Modelled rooms and openings, deck 2 (left) and deck 1 (right)

Pressure-Correction Method and Its Applications 699

Simulations were performed with a constant time step of 1.0 s. The applied convergence criterion corresponds to a water height difference of 0.05 mm. It was checked that a shorter time step or a stricter criterion did not affect the results. The applied permeabilities were according to the SOLAS regulation. The values for leaking and collapsing pressure heads and leaking ratios, presented in IMO SLF47/INF.6 (2004) were used as a starting point. The initial values for Aratio (i.e. the effective leaking area divided by the full area) are quite high. Therefore, these values were reduced by 50 % for the other simulations. In addition, it was tested how an increment of 10 % in the critical pressure head for collapsing affects the results. The applied values are listed in Table 1. In all cases, Hleak = 0.0 m and Cd = 0.6 were used for all openings. Table 1. Applied parameters for leaking and collapsing of non-watertight doors.

A-class fire doors: Aratio Hcoll (m) 2.00 0.10 2.00 0.05

Case SLF47/INF.6 Aratio–50% Aratio–50% Hcoll +20%

2.40

B-class joiner doors: Hcoll (m) Aratio 1.50 0.20 1.50 0.10

0.05

1.80

0.10

4.2 Results Time histories for heeling and total volume of floodwater are presented in Fig. 5. In the studied case the heeling is very minimal due to the large initial stability and symmetry in the flooding process. The applied Aratio values seem to have a remarkable effect on the time-to-flood. On the other hand, a 20% increase in the critical pressure head for collapsing has much smaller effects on the flooding process. 1400

0.4

volume of floodwater [m3]

0.3

heel [deg]

0.2 0.1 0.0 SLF47/INF.6 Aratio −50% Hcoll +20% & Aratio −50%

-0.1 -0.2

0

5

10

15

20 25 time [min]

30

35

40

1200 1000 800 600 400 SLF47/INF.6 Aratio −50% Hcoll +20% & Aratio −50%

200 0

0

5

10

15

20 25 time [min]

30

35

Fig. 5 Results of the sensitivity analysis for heel and total volume of floodwater

40

700 P. Ruponen

5 Simulation of Cross-Flooding 5.1 Introduction The pressure-correction method for flooding simulation can easily be extended to include also air pressures in the flooded rooms and air flows through the openings (Ruponen, 2007). The IMO Resolution MSC.245(83) for assessment of the crossflooding pays special attention to the effect of counter air pressure in rooms, where the ventilation level is restricted. In the resolution it is stated that full ventilation can be assumed if the area of the air pipes is at least 10% of the area of the cross-flooding openings. Thus, for example the effect of the air pipe length is not considered at all for large pipes. An efficient and practical method for assessing the cross-flooding time with all kinds of arrangements is dedicated time-domain flooding simulation, where the compression of air and the resulting airflows are properly taken into account. This kind of simulation is accepted as an alternative method for the assessment of cross-flooding time in IMO Resolution MSC.245(83). Peters et al. (2003) have applied a time-domain flooding simulation and seakeeping tool for studying the effectiveness of various cross-flooding arrangements in naval ships. Vredeveldt and Journée (1991) and Xia et al. (1999) have considered cross-flooding and the counter air pressures with a simplified model. In this paper, a combination of these two approaches is presented. U-shaped voids are used in passenger ship designs since there are several benefits. Firstly, the voids give good protection against minor side damages, for example when steering the vessel into quay. Furthermore, they prevent oil leakage into sea as the voids form a kind of double hull. Due to these benefits, also the Aindex is slightly increased. If the breadth of the side spaces is reasonably small, the U-shaped void does not significantly reduce the volume of the machinery space, and consequently, the design is feasible.

5.3 Air Flow Calculation The pressure losses in the openings and pipes are taken into account by applying semi-experimental discharge coefficients. Consequently, the mass flow through an opening is:

m   Q   Cd Au

(31)

where Q is the volumetric flow through the opening, Cd is the discharge coefficient and A is the area of the opening. For pipes and ducts, the discharge

Pressure-Correction Method and Its Applications 701

coefficient is usually calculated from the sum of the pressure loss coefficients ki, so that:

Cd 

1

1   ki

(32)

i

The flooding process is assumed to be isothermal. Therefore, Boyle’s law can be applied and the density of air is assumed to be linearly dependent on the pressure:

a 

0 p0

p

(33)

where 0 is the density of air at the atmospheric pressure p0.

5.4 Case Study The studied case is a U-shaped void, surrounding machinery spaces in a modern large passenger ship design. The tested damage case includes several compartments that are flooded immediately, and thus considered to be open to sea. Consequently, the initial heeling angle after the damage is large (10 degrees). The sides of the U-shaped void are connected by a cross-duct that consists of four parts due to the longitudinal girders. In order to study the effect of the detail level in the modelling, three different models of the cross-duct were tested (Fig. 6):  Level A: each girder defines a room partition  Level B: cross-duct is considered as a single room  Level C: cross-duct is considered as a single opening in the centerline

Fig.6 Schematic representation of the different detail levels for modelling of the cross-duct

It is somewhat questionable what values should be used for the discharge coefficients of the openings. In general, Cd  0.6 has usually been used in flooding

702 P. Ruponen

simulations. However, this can be far too conservative when flooding through a cross-duct with several girders is considered. Most of the pressure losses are likely to take place in the inlet, where the discharging jet is initially formed. The subsequent openings will then have a smaller effect on the jet, and therefore, also the pressure losses are likely smaller. In Level A, Cd = 0.65 is used for all openings in the duct. In Levels B and C, the applied discharge coefficients were obtained by using the formula in the IMO Resolution MSC.245(83). This is based on the RANSE computations, presented in Pittaluga and Giannini (2006). In Level B, Cd = 0.65 was used for the opening on the damaged side and 0.45 for the other one and in Level C, Cd = 0.40 was used to represent the whole duct. Air compression inside the cross-duct was ignored (Levels A and B) since the volume of the duct is minimal when compared to the volume of the whole U-void.

5.4 Effect of the Cross-Duct Modelling Comparisons of the heeling angle and over pressure in the equalizing side of the void are presented in Fig. 7. The equalization process is estimated very similarly with all tested modelling levels of the cross-duct. However, Level C gives much larger maximum over pressure in the void during the flooding. This results from the fact that in Level C the initial volume of air is larger than in Levels A and B. 10

D = 250 mm

heel [deg]

7 6 5 4 3

Level A Level B Level C D = 250 mm

30 over pressures [kPa]

8

25 20 15 10 5

2 1

35

Level A Level B Level C

9

0

10 20 30 40 50 60 70 80 90 100 110 120 time [s]

0

10 20 30 40 50 60 70 80 90 100 110 120 time [s]

Fig. 7 Comparison of the time histories for the heeling angle and over pressure in the equalizing side with different modelling levels of the cross-duct

5.5 Air Pipe Dimensioning The simulations for the dimensioning of the air pipes were performed with the cross-duct modelling Level C. The pipe diameter was increased until there was no difference, when compared to the results from the simulation with full ventilation

Pressure-Correction Method and Its Applications 703

in the void. The results for the heeling angle, over pressure and volume of floodwater in the equalizing side are presented in Fig. 8. The area of the largest pipe (D = 400 mm) is 9.9% of the area of the crossflooding openings. The results for the heeling angle and volume of water are very close to the results from the simulation with fully vented void. 10

8

6 5

8 7 6 5

4

4

3

3

2 1

D = 250 mm D = 300 mm D = 350 mm D = 400 mm fully vented

9

heel [deg]

7 heel [deg]

10

D = 250 mm D = 300 mm D = 350 mm D = 400 mm fully vented

9

2 0

10 20 30 40 50 60 70 80 90 100 110 120 time [s]

1

0

10 20 30 40 50 60 70 80 90 100 110 120 time [s]

Fig. 8 Heeling angle and over pressure in the equalizing side with different sizes of air pipes

The effect of air pipe diameter is not notable in the beginning of the equalization process. Thereafter, near the equilibrium condition, the effect is more significant. On the basis of these results, it seems that in this particular case, the criterion of 10% area for the air pipes in the IMO Resolution MSC.245(83) seems to be reasonable. However, no general conclusions can be made without systematic simulations with many different ship designs.

5.6 Realistic Damage Case The presented simulations were based on the statutory approach, where the damaged rooms are considered to be immediately flooded. For comparison, a separate case, where only the U-void is damaged, was calculated both with the statutory approach and with a modelled damage opening of 5.0 m2. Simulations were performed with two different air pipe diameters. The results for the time histories of heeling are presented in Fig. 9. The formula of the IMO Resolution MSC.245(83) gives an equalization time of 131 s, which is slightly conservative when compared to the simulations. When also the damage opening is modelled, the compression of air in the damage side of the void slows down the flooding from the sea. Consequently, the maximum heeling angle is decreased with smaller air pipes, even though the equalization time is longer. It might be reasonable to take this phenomenon into account with damages to rooms that are located far below the water line.

704 P. Ruponen 2.5

stat. & D = 250 mm stat. & D = 400 mm 2 5 m dam. & D = 250 mm 2 5 m dam. & D = 400 mm

2.0 heel [deg]

Level C 1.5 1.0 0.5 0.0

0

10 20 30 40 50 60 70 80 90 100 110 120 time [s]

Fig. 9 Comparison of heeling angle with the statutory case and with 5.0 m2 damage opening, simulated with two different air pipe diameters

6 Conclusions Application of a pressure-correction technique for time-domain flooding simulation is an effective method for calculation of complex flooding cases. The iterative approach allows feasible time steps without compromising the accuracy. Even the best simulation methods cannot provide realistic results if the input parameters for potential openings, such as closed doors, are not known, accurately enough. Therefore, systematic studies are necessary in order to increase the reliability of flooding simulations. Time-domain flooding simulation, with the compression of air taken into account, is a very efficient and practical tool for assessing the cross-flooding times more realistically than the simplified formula in the Resolution MSC.245(83). Further systematic CFD calculations and large scale model tests can provide the necessary input data for flooding simulations.

Acknowledgements Anna-Lea Routi from STX Finland has provided valuable help and comments, especially concerning the presented case study for cross-flooding simulation, which is gratefully acknowledged.

References Dillingham J (1981) Motion Studies of a Vessel with Water on Deck, Marine Technol, 18:38-50. IMO Resolution MSC.245(83) (2007) Recommendation on a Standard Method for Evaluating Cross-Flooding Arrangements, Int. Marit Organ, adopted on 12 Oct 2007.

Pressure-Correction Method and Its Applications 705 IMO SLF47/INF.6 (2004) Large Passenger Ship Safety: Survivability Investigation of Large Passenger Ships, submitted by Finland, 11. June 2004. Pawlowski M (2003) Accumulation of Water on the Veh Deck, Proc. Inst Mech. Eng. 217 Part M: J. Eng for the Marit Environment, pp. 201-211. Peters AJ, Galloway M, Minnick PV (2003) Cross-Flooding Design Using Simulations, Proc of the 8th Int Conf on Stab of Ships and Ocean Veh, Madrid, Spain, 2003, 743-755. Pittaluga C, Giannini M(2006) Pressure Losses Estimation for Structural Double Bottom by CFD Technique, CETENA Technical Report. Ruponen P (2006) Pressure-Correction Method for Simulation of Progressive Flooding and Internal Air Flows, Schiffstechnik – Ship Technol Res, 53:63-73. Ruponen P (2007) Progressive Flooding of a Damaged Passenger Ship, Doctoral Dissertation, Helsinki Univ of Technol, TKK Dissertations 94. Ruponen P, Sundell T, Larmela M (2007) Validation of a Simulation Method for Progressive Flooding, Int. Shipbuild. Prog., 54, pp. 305-321. Solda GS (1961) Equalisation of Unsymmetrical Flooding, Transactions of Royal Inst of Naval Archit, RINA, Vol. 103: 219-225. Vredeveldt AW, Journée JMJ (1991) Roll Motion of Ships due to Sudden Water Ingress, Calculations and Experiments, RINA’91, Int Conf on Ro-Ro Safety and Vulnerability the Way Ahead, London, United Kingdom, 17-19. April 1991, Vol. I. Xia J, Jensen JJ, Pedersen PT (1999) A Dynamic Model for Roll Motion of Ships Due to Flooding, Schiffstechnik – Ship Technol Res, 46: 208-216.

10 CFD Applications to Ship Stability

Applications of 3D Parallel SPH for Sloshing and Flooding Liang Shen, Dracos Vassalos University of Strathclyde, Glasgow, UK

Abstract Smoothed particles dynamics scheme is applied in sloshing and flooding problems in this paper. New solid boundary condition is used to simulate complex geometry. Parallelization of SPH scheme is carried out using MPI standard which makes 3D simulation acceptable. The numerical solutions obtained have been compared with both experimental results and other numerical solutions.

1 Introduction The smoothed particles hydrodynamics (SPH) is a meshless scheme, developed by (Lucy 1977) and (Gingold and Monaghan 1977) for astrophysical applications. It has been extended to hydrodynamics problems with free surface flow (Monaghan 1994). The advantages of SPH can be summarized as follow: SPH is conceptually both simple and easy for coding. The Lagrangian nature of SPH means that changes in density and flow morphology are automatically accounted for without the need for mesh refinement or other complicated procedures.

1.1 Basic Algorithm 1.1.1 Governing Equations The governing equations for fluid flow are the mass and momentum conservation. In Lagrangian form, these governing equations can be written as

1 D   u  0  Dt Du 1   p  g  v 2u Dt 

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_40, © Springer Science+Business Media B.V. 2011

(1)

709

710 L. Shen and D. Vassalos

where  is the fluid particle density; t is time; u is the particle velocity; p is pressure at the particles; g is gravitational acceleration; and v is the kinematic viscosity.

1.2 SPH Formulation The mass density at an arbitrary position is determined by a weighted average of the neighbouring particles: N

 ( x )   mbWb ( x )

(2)

b 1

And, the velocities are N

v ( x )   Vb vbWb ( x)

Vb 

b 1

mb

b

(3)

Thus, we obtain the divergence of the velocity N

va  Vv (vb  va )  Wb ( xa )

(4)

b 1

Where,

Wb ( xa ) 

dW 1 ( xa  xb ) dr rab

rab  xa  xb

(5)

1.3 Kernel Function The most commonly used kernel functions, Wab, belong to a family of spline

curves, and Wab is the gradient of interpolating kernel function. In our applications we refer to cubic B-spline, defined as follows:

 3 2 3 3 1  r  r if 0  r  1 4  2  1 3 W r , h    2  r  if 1  r  2 h 4  0 ifr  2

(6)

Applications of 3D Parallel SPH for Sloshing and Flooding

Here,

W

ab

711

r  xa  xb / h and h is called smoothed length. is chosen to satisfy

dV  1

1.4 Equation of State The constitutive model is determined by the equation of state. To close the system of equations, the most common used EOS is on derived from a relation proposed by Batchelor [4].

    p      1   0   ,

(7)

with y=7 The density variation in fluid flow is proportional to the square of Mach number, M. An estimation of the upper bound of the velocity of the particles is

v  2 gH

(8)

(Monaghan 1992) shows 0.1 is a good estimation of Mach number for this problem and using Eq. (8) as an estimation for sound speed. The bulk modulus, k, is re-evaluated as



200 gH



(9)

1.5 Viscosity Basic SPH formulism suffers the absence of dissipation of energy. In order to increase the stability of SPH, many forms of artificial viscosity have been proposed, but the most commonly used artificial viscosity is obtained by writing the momentum equation as

P P dva   mb ( b2  a2   ab ) aWab b  a dt b

(10)

Where ab is given by 2  α cabμ ab  βμ ab vab  rab  0; Π ab  { ρab 0 v ab  rab  0;

And

(11)

712 L. Shen and D. Vassalos

 ab 

hvab  rab rab2   2

(12)

The expression for ab contains a term that is linear in the velocity differences, which produces a shear and bulk viscosity.

2 Boundary Conditions In SPH method, in order to apply the correct boundary conditions for the equations in the SPH formulation, the detection of boundary particles is needed to impose correct boundary conditions on these particles. Concerning bounded domain problems, there are several strategies and mathematical artifacts that allow for modeling the presence of boundaries with different degrees of accuracy. In the first approach, the boundaries are replaced by interacting particles which exert a repellent force on fluid particles. In general, Lennard-Jones force is applied (Monaghan 1994). The repellent forces on the boundaries produce oscillations and this force only depends on the distance between particles and boundaries. Quasi-ghost particles (figure 1) are defined to satisfy the wall boundary no penetration condition (Dalrymple and Knio 2000). Several layers of ghost particles are lying within some specified distance (smoothing length) from the boundary. They carry the attributes with time history like fluid particles but the position is fixed and velocity is zero at every time step. The effects of the fictitious ‘ghost’ particles are explicitly included in the summation for the fields and for their gradients.

Fig. 1 Quasi-ghost particles

Dab means the internal force on particles due to ghost particles which depends on pressure gradient.

Applications of 3D Parallel SPH for Sloshing and Flooding

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Fig. 2 Dynamic ghost particles

Quasi-ghost particles can give some accurate results in our tests but they disturb the fluid domain too much. Alternatively dynamic ghost particles boundary conditions (figure 2) are available for the tests which give some more accurate results particular in the pressure evaluation on the boundary. Different from quasi-ghost particles, dynamic ghost particles are mirroring particles from fluid domain near the wall boundary and they carry the same attributes as the fluid particles but the opposite velocity in normal direction of wall boundary. Dynamic ghost particles boundary conditions suffer penetration problem because of tensile instability. To avoid this kind unphysical phenomenon we restrain the particle motion and velocity when they are too close to solid boundary.

3 Time Evolution and Parallel SPH 3.1 Time Stepping and Evolution The numerical integration of the ordinary differential equations for the physical variables at each particle can be carried out by standard methods with a time-step control that involves the courant condition, the force terms, and the viscous diffusion term. The time step should be defined following Courant Friedrichs Levy (CFL) condition based on the local smoothing length and local sound speed. Then the resulting ordinary differential equation system can be integrated in time by schemes such as Runge-Kutta, Leap-Frog or any Predictor-Corrector, to ensure at least second order convergence in time.

714 L. Shen and D. Vassalos

3.2 Parallel SPH In general, SPH calculation is very computationally demanding, both in memory and in CPU time. Firstly, SPH method usually involves a large number of particles to be geometrically enough to model the deformation of fluid body. In three dimensional cases, the SPH model may involve several millions of particles. Secondly, the evolution of the particle information should be very time consuming. Besides the governing equations themselves, neighbour search, boundary treatment, and interactions between particles manifold the complexity of the problem.

Fig. 3 Parallel SPH

For these reasons, standard MPI technique is adopted for parallel SPH code. The computational domain is decomposed into several parts for each process working.

4 Benchmark Results 4.1 Case 1 - Validation Case for Sloshing Tests have been carried out in 2001 and 2002 by the GIS-Hydro in the wave basin of La Seyne/mer – France (BGO First facilities). The experimental set up consists of a rectangular barge model supporting two rectangular tanks partly filled with water. Regular and irregular wave tests results are available for beam seas. Barge motions and internal wave elevations have been measured. Results are free to use which can be provided by Principia.The barge has the following features:

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Table 1. Barge characteristics Length

3m

Width

1m

Height

0.267m

Mass without tank

127 kg

Draft

0.108m

Moulded volume

0.285m3

Center of gravity above keel 0.237m Giration radius (in roll)

0.414m

The model is installed in the basin in the middle of the testing section; with its longitudinal axis parallel to the wave-maker line (only beam waves were considered). The mooring system is ensured by 4 cables equipped by springs. The mooring lines are attached at the 4 corners of the barge at 0.262m height according to the keel. The natural sway period of the mooring system (direction of the waves) is 20s. The reference point for the measurements of the translation of the barge motion is at the deck level (0.267 m above the keel level).

Fig. 4 Sketch of the model

The two tanks are set on the deck of the barge, at mid-ship, with their length in the transverse direction. The elevation of the inner bottom of the tanks, with respect to the keel, is about 0.3m. The characteristic of the two rectangular tanks are the following:

716 L. Shen and D. Vassalos Table 2. Tank characteristics Length 0.8m Width 0.25m Height 0.6m Mass

37kg

The water motion inside the tank is measured by 5 probes:  3 in the tank 2 (higher filling level) at 24mm, 180mm and 350mm from the wall closest to the wave-maker.  2 in the tank 1 at 25mm and 180mm from the corresponding wall. The sloshing motion in the tanks is recorded with a video camera. Waves elevation is measured with 5 probes. All are set on the longitudinal axis of the basin at different spacing.

Fig. 5 View of the model test set-up in BGO First

4.1.1 Definition of the Test Cases Sloshing in tank without considering the barge: the discretized domain is a rectangular tank animated with a motion which is deduced from the motion of the barge. The input are the height of the free surface, the 6 DOF (degree of freedom) displacement of the tank (motions of the barge) and the position of the tank according to the barge centre of gravity.

Applications of 3D Parallel SPH for Sloshing and Flooding

717

Table 3. Test cases Test cases irregular waves Motions of the barge Filling level Irreg 1

6 DOF

39cm

Irreg 2

6 DOF

19cm

The following figures show an example of the 6 DOF motions of the barge (and of the tank) in irregular waves.

Fig. 6 Irregular wave motion in time series

Fig. 7 Results comparison of wave evaluation

718 L. Shen and D. Vassalos

The numerical results in Figure 7 were carried out by SPH code and Fluent which both give good results of wave height on probes.

4.2 Case 2 – 2D Rectangular Box To validate flooding case with SPH concerning coupling free motion tank we proposed numerical models and simulate the cases with both SPH code and Fluent.

0.02 0.02 0.01

0.06

0.02

0.04

0.02

0.1 Case A

Case B

Case C

Fig. 8 Validation case for 2d transient flooding under free motion. Table 4. Tank characteristics Weight

7.5kg

Moment of inertia

0.075kg.m2

Length

0.1m

Height

0.1m

Gravity center above bottom 0.025m Horizontal gravity center

0.05m

Thickness

0.01m

Drought

0.075m

Fig. 9 Case A From left to right: 0.1s, 0.2s, 0.3s, 0.4s. From top to bottom: VOF, SPH

Applications of 3D Parallel SPH for Sloshing and Flooding

Fig. 10 Case B From left to right: 0.1s, 0.2s, 0.3s, 0.35s. From top to bottom: VOF, SPH

Fig. 11 Case C From left to right: 0.1s, 0.2s, 0.3s, 0.4s. From top to bottom: VOF, SPH

Fig. 12 Surge, heave and roll motion of damaged box

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Figures 9-12 give the results of water surface and motion in time series which prove SPH’s ability of handling transient flooding coupling the response of tank.

Fig. 13 From top to bottom: Flow 3D,SPH,Fluent

Applications of 3D Parallel SPH for Sloshing and Flooding

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4.3 Case 3 – 3D Rectangular Box Flooding test of Roro ship is proposed in 24th ITTC to investigation the stability under damaged flooding situation. Simplified model was tested by (Cho 2005). In figure 13 flooded water surfaces in the damaged tank was given at 1.5s, 3s, 4.5s, 6s, 7.5s, 9s. More than one million of particles are used in SPH. Because of efficiency parallel scheme, SPH could simulate 3D flooding correctly and quickly.

5 Summary and Conclusions In this paper we presented the applications of SPH on the hydrodynamics problem: slamming, sloshing and flooding. SPH scheme shows the ability to predict correct answer to the violent free surface flow. After parallelization of SPH scheme and variable particles smooth length, large 3D simulation became acceptable for SPH. Further application of this method will concern real ship flooding and coupling SPH with other CFD method.

References Lucy LB (1977) A numerical approach to the testing of fission hypothesis. J Astron 82:1013-24. Gingold RA, Monaghan JJ (1977) Smooth particle hydrodynamics: theory and application to non-spherical stars. Month Notices Roy Astron Soc, 181:375-89. Monaghan JJ (1994) Simulating free surface flows with SPH. J Comput Phys 110:399-406. Oger G, Doring M, Alessandrini B, Ferrant P (2006) Two-dimensional SPH simulations of wedge water entries. J Comput Phys 213:803-22. Dalrymple RA, Knio O (2000) SPH modelling of water waves. Proc, Coastal Dynamics. Cho (2005) Investigation of dynamic characteristics of the flooding water of the damaged compartment of an ITTC RORO Passenger. Maritime and Ocean Eng Res Inst, KORDI, Korea.

Simulation of Wave Effect on Ship Hydrodynamics by RANSE Qiuxin Gao, Dracos Vassalos University of Strathclyde, UK

Abstract The application of advanced numerical methods based on the solution of RANSE and VOF equations on the prediction of ship hydrodynamics is presented. The test cases selected are restrained and free oblique motions of a container ship with the effect of incoming wave. The computed and measured results are compared. The general agreements between calculations and experiments are satisfying.

1 Introduction The simulation of wave influence on ship hydrodynamics by using state-of-art CFD is gaining increasing attention in recent years. The advantage of the approach over traditional strip theory or linear/non-linear potential theory is improved accuracy and capability of simulation of ship motions in heavy sea (Ducrozet et al. 2005). The disadvantage of demanding computer resources and effort is becoming less bottlenecked after high performance parallel computing technology was put into use. Nevertheless, the simulation of ship manoeuvring in sea is a troublesome task numerically. The difficulty lies on wave generation and dynamic motion. The wave generation using RANSE method is a paramount task for the study of ship manoeuvring in waves. Numerical wave generation and diffusion may disturb physical wave propagation and affect the quality of numerical manoeuvring wave tank (diffraction problem in manoeuvring scope). Meanwhile, the free motions of the ship sailing in the seaway make numerical simulation more complicated (radiation problem in RANSE scope). So far, there is little relevant work available in the literature. There were only a few publications addressing added wave resistance using RANSE approach. One research work was presented by Weynouth etc using CFDSHIP-IOWA0 (Weymouth 2005). They studied head sea effects on diffraction problem, forced heaving and pitching motions. The uncertainty studies were carried out and the systematic calculations of parametric effects were conducted. The comparisons of M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_41, © Springer Science+Business Media B.V. 2011

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724 Q. Gao and D. Vassalos

results between RANSE code, strip theory, potential code with measurement show that RANSE code performed much better than other codes. Cura Hochbaum and Vogt studied combined steady drift and turning motions in calm water, and straight ahead motion in head wave by HSVA code NEPTUN (Hochbaum and Vogt 2003). The force and moment results agree in general well with experiment measurements. However, ship was restrained without attitude change, and the calculations of manoeuvring motions were for calm water condition. (Ducrozet et al. 2005) adopted a HOS scheme in their SWENSE approach for the non linear water wave simulation. The comparison shows that the wave quality is pretty satisfying for the practical purpose. However, no result of application on ship hydrodynamics was presented. In this paper, the application of computational fluid dynamics on the study of ship manoeuvrability by solving RANSE and VOF equations is presented. Firstly, numerical formulations will be described with the focus on numerical wave maker. Secondly, the method will be used to the simulation of steady yaw motion of a container model in head wave. Then, the attempt was made for free running model calculations. The accuracy of numerical results will be evaluated by model test data. Finally, future work for improving numerical quality and application on simulation of critical ship motion in severe sea will be proposed.

2 Numerical Formulation The Reynolds averaged Navier-Stokes equations (RANSE) with SST -ω turbulence model for closure were solved. The VOF method was adopted to capture free surface interface. The governing equations can be written as follows.

2.1 Continuity Equation The continuity equation is written as:

u  0

(1)

2.2 RANSE Equation u 1  u  u   p  g   2 u t 

(2)

Simulation of Wave Effect on Ship Hydrodynamics by RANSE 725

2.3 Turbulence Model   K     uK   P          T t K 

  K    

      u        G  Y  D  S t

(3)

(4)

2.4 VOF Equation  rw     rw u   0 t Where: u g

  

rw K



P

(5)

Velocity vector Gravity vector Stress tensor Mixture density Mixture viscosity Volume fraction of water Turbulence energy Specific dissipation rate Pressure

2.5 Boundary Conditions The inlet is located at one ship length in front of bow where velocity components and volume fraction were imposed. The wave was generated by digital wave maker at inlet using following values (the same as in model tests): – Wave length /L=1.0 – Wave amplitude a/L=0.008 – Wave number k=2/ –

Wave frequency for deep water =

– –

Model speed V=1.89m/s Wave encounter frequency

gk

e    kV

726 Q. Gao and D. Vassalos

– –

Period T=2/e Wave elevation at inlet

–

X velocity u   a e

kz

–

Z velocity w   a e

kz

   a sine t  sin  e t 

cos e t 

From above relationship, we can obtain the wave number is 0.98 and corresponding angular frequency of incoming wave is 3.1/s. Wave encounter angular frequency is 4.95/s. The period is 1.27 seconds. Wave amplitude is 0.0512m. The incoming wave was generated at the start of the calculation by analytic solution of deep water wave through user defined function (UDF). It takes a few periods to eliminate initial disturbance. After about 5 periods, the force record becomes periodic with period 1.27s. The outlet is at two ship length behind the stern. The hydrostatic pressure is specified. Velocity components and free surface elevation were given on side boundary, which is located at one ship length from centreplane. Wall function was used on hull boundary to save computer time. Using these parameters, the simulation of wave effect on steady oblique motion was performed. Yaw angles are 0 and 10 degrees.

2.6 Numerics Second order upwinding interpolation was used for convection flux. SIMPLE method was applied to obtain pressure. Geometric reconstruction of volume fraction was used to calculate wave elevation.

3 Test Case A container model (Hamburg Test Case) adopted in EU FP6 IP project VIRTUE was selected for numerical analysis and validation. The body plan and profile of bow and stern are shown in Figure 1. The main particular of the vessel is given in the table 1. The calculations were carried out on a 16 processors cluster. The mesh was medium sized with 1.3m cells. The grid sensitivity studies were carried out on a relevant study. The grid effects are small for unsteady calculations (less than 3% in general). Two test cases were made. One is captive steady yaw motion with incoming wave. The other is free sailing motion with wave effects. The results are given below.

Simulation of Wave Effect on Ship Hydrodynamics by RANSE 727

Fig. 1 HTC body plan Table 1. Main particulars of HTC Scale 1:24 Length between perpendiculars Length waterline Breadth max. moulded Draught fore moulded Draught aft moulded Displacement bare hull moulded Wetted surface area Centre of buoyancy from AP Block coefficient Length – breadth ratio Breadth – draught ratio

Ship 153.70 153.10 27.50 10.30 10.30 28332 5577 75.97

Model 6.404 6.380 1.146 0.429 0.429 2.049 9.682 3.165 06.51 5.589 2.670

Unit M M M M M m3 m2 M -

4 Captive Yaw Motion in Wave Manoeuvring in sea waves was traditionally studied by simplified numerical approach utilizing model test based semi-empirical formula. The accuracy of simulation was case dependent. More reliable numerical tools are in demanding. Recent CFD development provides an alternative to tackle the problem. In EU project VIRTUE, manoeuvring in calm water and wave were studied intensively. The results of calculations of manoeuvring in calm water covering steady yaw, steady turning, oscillatory sway and oscillatory yaw were in good agreement with model test measurement. The simulation of ship manoeuvring in wave is latest development and representative results will be presented below.

728 Q. Gao and D. Vassalos

4.1 Captive Yaw Angle 10 Degrees The first case is captive yaw motion (10 degrees) in wave as shown in snapshot below.

Fig. 2 Captive yaw motion

The harmonic wave was generated using digital wave maker. The model was in oblique motion at Froude number 0.238 and yaw angle 10 degrees. The X forces are given in Figure 3, where the results from SSRC, MARIN and measurements were compared. The results from MARIN were obtained by a potential solver FreDyn with viscous correction by PANASSOS. The HSVA experiment data includes those with and without wave. As we can see, the surge force is constant without wave. The surge force oscillates at the period given above when there is wave. Both computed and measured surge forces of steady yaw in waves oscillate around the value from steady measurement in calm water. However, as can be seen, the amplitudes of oscillations were different. The computed amplitude by FLUENT is close to that from measurement. The predicted amplitude by MARIN is lower than data. It seems that viscous-wave interactions were not captured accurately by potential solver. 0.1

SSRC HSVA_wave HSVA_nowave MARIN

X

0.05

0

-0.05

-0.1

4

6

t

Fig.3 Surge force

8

10

Simulation of Wave Effect on Ship Hydrodynamics by RANSE 729

The sway force in Figure 4 shows similar trend as surge force. The computed sway force from both MARIN and SSRC oscillates periodically after about 5 periods when there is incoming wave. The time record of sway forces from calculations displays sinusoidal feature. However, the measured sway force exhibits strong high frequency oscillation. It seems that wave generated in model test suffer from short wave disturbance. The amplitudes of oscillations between calculation and measurement were generally consistent. The averaged sway forces in one period from calculation and measurement are close to that from steady measurement. The agreements between calculations and measurement are acceptable. 0.15

SSRC HSVA_wave HSVA_nowave MARIN

0.1

Y

0.05

0

-0.05

4

6

t

8

10

Fig. 4 Sway force

The yaw moment was shown in Figure 5. The agreements between calculations and measurement seem pretty satisfying. The computed amplitudes and phase angles are close to the measured one. The unsteady time record of forces oscillates around the value from steady measurement. Averaged yaw moment by FLUENT is slightly larger than other results. Similar results were obtained in the calculations of steady yaw and turning where predicted yaw moments are slightly larger than measurement. The similar conclusion could be drawn for yaw moment as for surge and sway forces. Additional to the calculations of drift angle 10 degrees, the calculations of wave effect at straight forward condition (without yaw) were carried out as well.

4.2 Captive Straight Ahead Motion in Head Wave As there was no drift angle in the calculation, the sway force, roll moment and yaw moment are all zero. Both computed and measured surge forces shown in Figure 6 oscillate around the value from steady measurement. However, as can be

730 Q. Gao and D. Vassalos

seen, the amplitudes of oscillations were depending on the solvers. The computed amplitude by FLUENT is close to the measured one. However, it is much smaller by MARIN’s solver. The averaged surge forces from calculations and measurement in one period were close to that from steady measurement without wave. The conclusion for surge force in yaw angle 10 degrees applies to straight ahead condition. 0.06

SSRC HSVA_wave HSVA_nowave MARIN

0.05

0.04

N

0.03

0.02

0.01

0

4

6

t

8

10

Fig. 5 Yaw moment 0.1 SSRC HSVA_wave HSVA_nowave MARIN

X

0.05

0

-0.05

-0.1

4

6

t

8

10

Fig. 6 Surge force in head wave

5 Free Running in Wave Ship sails in seaway in free running condition. Ship attitude is not restrained as in captive condition. The dynamic change of attitude in sea wave was simulated in the calculation. Ship speed is constant at 18 knots. Only surge force can be

Simulation of Wave Effect on Ship Hydrodynamics by RANSE 731

measured in model test. In the section below, the comparison of calculation with measurement is made. Only calculation without drift was made. The surge force was given in Figure 7. The computed and measured surge forces oscillate with same period 1.27s as in captive condition. The computed amplitude of oscillation in free condition was close to that in captive condition while measured amplitude of oscillation in free condition was slightly larger than that in captive condition. The computed surge force exhibits numerical wave disturbance while measured surge force oscillates periodically. The magnitude of averaged surge forces in one period in calculation and measurement were slightly larger than that in captive condition. The sinkage and trim in free condition tends to increase surge force a few percentage. In general, the agreements between computed and measured surge force were good. 0.1

SSRC HSVA

X

0.05

0

-0.05

-0.1 2

4

t

6

8

Fig. 7 Surge force in free motion

The heave motion was given in Figure 8 (no data is available from model tests). 0.1

SSRC

Heave

0.05

0

-0.05

2

4

t

Fig. 8 Heave

6

8

732 Q. Gao and D. Vassalos

Model moves vertically with wave induced heave force. The period of oscillation was the same as wave period 1.27s. The amplitude of heave motion is determined by wave induced heave force. The averaged heave motion is related to steady hydrodynamic heave force. The pitch motion was given in Figure 9.

0.04 SSRC

Pitch

0.02

0

-0.02

2

4

t

6

8

Fig. 9 Pitch

Model moves with free pitching of period 1.27s. The oscillation of pitch motion in the calculation was not strictly periodic. There was strong effect from numerical wave. The amplitude of pitch motion is determined by wave induced pitch moment. The averaged pitch in one period is related to steady hydrodynamic pitch moment. From maneuvering calculations in sea wave and comparison with model tests, the accuracy of CFD tools was generally encouraging. Further implementation of the tool on naval hydrodynamics will be strengthened.

6 Future Work CFD tools can be effectively used to study the wave effect on ship hydrodynamics in variety of motion modes. For the purpose of practical application, the following work needs to be established in the future:  Model test data base with wave and motion signals as well as force components are needed for validation of CFD results  Although bow wave can be generated reliably, following sea as well as oblique sea need to be generated by RANSE approach

Simulation of Wave Effect on Ship Hydrodynamics by RANSE 733

 For the calculation of large amplitude ship motion, the quality of deforming mesh needs to be improved

7 Conclusion Based on the computational results, the following conclusions are drawn:  The effects of wave on time-averaged forces from calculation in captive condition were small. Averaged X, Y, and N were close to those from measurement without wave.  The forces from measurement in waves suffer from high frequency disturbance. The reason for this whether it is due to wall reflection or wave quality needs to be clarified.  The computed and measured surge forces in free condition averaged in one period were slightly larger than those in captive condition.

References Ducrozet G,.Bonnefoy F, Le Touze D, Ferrant P (2005) Development of a Fully Nonlinear Water Wave Simulator based on Higher Order Spectral Theory. 20th Workshop on Water Waves and Float Bodies, Norway. Hochbaum C, Vog M (2003) Towards the Simulation of Seakeeping and Manoeuvring based on the Computation of the Free Surface Viscous Ship Flow. 24th Symposium on Naval Hydrodyn. Weymouth GD, Wilson RV, Stern F (2005) RANS Computational Fluid Dynamics Predictions of Pitch and Heave Motions in Head Seas. J of Ship Res 49(2): 80-97.

A Combined Experimental and SPH Approach to Sloshing and Ship Roll Motions Luis Pérez-Rojas, Elkin Botía-Vera, José Luis Cercos-Pita, Antonio Souto-Iglesias* Gabriele Bulian**, Louis Delorme*** * ETSIN, ** DINMA, ***Eurocopter

Abstract Passive anti-roll tanks have been used for a long time in ships to damp their roll motion. The coupled roll motion response of a single degree of freedom (SDOF) system to which a passive anti-roll tank has been attached is considered in the present paper. The performance of the anti-roll tank has been studied both experimentally and numerically, with weakly compressible SPH. The sloshing flows inside the tank comprise the onset of breaking waves. In order to characterise the wave breaking effects on the response curves, tests have been performed with liquids of different viscosity, the increasing viscosity preventing the onset of breaking waves. The capabilities of SPH to treat this coupling problem are assessed and the results show that SPH is able to capture a part of the physics involved in the addressed phenomena but further work remains still to be done.

1 Introduction Roll motion is one of the most important responses of a ship in waves. Antiroll tanks have been used for a long time in ships to damp their roll motion. A free surface roll-damping tank was first introduced by Philip Watts at the end of the 19th century (Watts, 1883). Frahm in 1911 presented a U-tube form roll damper that became popular at the end of the second world war. The mechanical equivalent to an antiroll tank is a damped vibration absorber, (Anderson et al., 2002, Ikeda and Nakagawa, 1997). These absorbers, in the field of civil engineering, are usually rectangular tanks filled with water and are named tuned liquid dampers (TLD). The TLD systems rely on the sloshing waves that appear at the free surface of the fluid to produce a counter force and/or torque thus dampening the original forcing motion if the correct frequency ratio is met. In the existing literature two approaches can be found to characterise the behaviour of a TLD to external excitations. The

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_42, © Springer Science+Business Media B.V. 2011

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first one consists of imposing a periodic motion on the TLD by using a shaking table or a forced roll motion device and measuring the response in terms of lateral force or moment (Tait et al., 2005; Souto-Iglesias et al. 2006). The other approach, more complex, and the one the present paper deals with, is to consider the motion response of the coupled system tank-structure, subjected to external excitation in terms of force, moment or even induced motion to the tank interfaced with an elastic structure. With this second approach the damping characteristics, inertia and restoring terms are also relevant in the dynamic analysis. Real motions of the structure are the outcome of this process, that can be compared with design limit states (Delorme et al, 2006; Attari and Rofooei, 2008). In contrast to the civil engineering case, where rectilinear motions are usually the main concern, the dampening effect on roll given by an anti-roll tank is obtained through the moment generated by the tank with angular motion. Van den Bosch and Vugts (1966) coupled the roll motion equation with experimentally determined fluid moments for an oscillating free surface tank and Francescutto and Contento (1999) exploited a mechanical equivalence to provide a 2-DOF analytical model which is nonlinear in respect to roll, and linear in respect to the DOF associated to the fluid sloshing. SDOF models for roll are often used to simulate the roll behaviour in beam seas (Bulian and Francescutto, 2004) and an example of analytical nonlinear SDOF descriptions of roll motion coupled with direct CFD calculations for the free surface tank can be found in Armenio et al, (1996). Regarding experimental work on the coupling problem, Rognebakke and Faltinsen (2003) analyzed the coupled problem in the sway case of a box excited by waves, in comparison to experimental data. Also, a few studies have been reported using the shaking table for horizontal excitations (Sun and Fujino, 1994), but not much in relation to angular motions (Pirner and Urushadze, 2007). In this paper, a SDOF (single degree of freedom) structure with a partially filled tank is considered, with roll motion modelled by means of an "exact" (from the dynamics point of view) 1-DOF approach. This means that the damper acts as an angular damper whilst most of the previously described works correspond to horizontal excitation. The moment created by the fluid with respect to the rolling axis is simulated and results for the roll angle are compared. The motion is excited by the moment created by a transversally (in a tank fixed reference system) moving mass with imposed motion. Experiments have been performed using fluids of different viscosity in order to assess the influence of breaking. A similar approach was taken by Pirner and Urushadze (2007), because water was not a suitable liquid to be used in footbridges which was the problem they were interested in. The numerical simulations have been performed using SPH. SPH has been applied with success to shallow depth sloshing problems with periodic oscillation in sway (Landrini et al. 2003) and roll (Souto et.al., 2006) motions. It had also been applied to a coupled motion problem (Delorme et al, 2006), showing promising results that had been compared with those obtained with a multimodal approach. Nevertheless, in Delorme et al (2006), comparisons with experimental

A Combined Experimental and SPH Approach to Sloshing and Ship Roll Motions 737

results were not possible because they were not available at that time. In the present paper, an experimental SDOF model has been modeled and it will serve as a benchmark data supplier for the comparisons with the SPH computations. The problem represents a significant challenge specially for SPH due to the extent in time of the real phenomena to be simulated. It is also important to underline that the present experimental/numerical approach allows to remove the difficulties usually encountered in a correct modelling of the actual ship roll motion. Indeed, when numerical simulations are compared with experimental tests carried out on ship models excited by waves, it is almost never completely clear which is the real source of discrepancy between experimental results and numerical prediction, i.e. whether the reason is to be sought in the modelling of sea-ship interaction or in the modelling of sea-tank interaction. In the present tests, being the dynamics of the mechanical system practically known “exactly” (at least at a reasonable level of accuracy, with some question mark on damping at small rolling angles), any significant discrepancy is likely to be sought in the SPH computational method.

2 Experiments. Test Cases The experiments were conducted with the tank testing device of the CEHINAV group (see Souto et al 2006 for a detailed description). The standard forced motion configuration of the device, used regularly in the design of anti-roll tanks, was modified by disconnecting the driving electrical engine from the tank holding structure, in order to allow the free motion of the tank.

508mm

y

x



470mm

900mm

m m 62

Fig. 1 Tank dimensions

The tank chosen for the simulations is depicted with its dimensions in figure 1. It is narrow along the z-direction, i.e. the direction perpendicular to the paper, in

738 L. Pérez-Rojas et al.

comparison with the horizontal and vertical dimensions. This is intended to have predominantly a two dimensional flow, since the code used for the simulations will be a 2D one, faster than the 3D ones, and which can be more easily modified and validated than a 3D one. The rotating shaft is placed 470mm above the base line of the tank. Having the tank below the rotating shaft makes it possible in this case to have a still stable configuration. Moreover, having the base of the tank at a considerable distance from the rotation axis makes the flow more interesting in terms of propagation of generated waves. There is a horizontal linear guide 600mm long, placed at the rotation centre to simplify some terms in the motion equation (fig. 2). This linear guide consists of a controllable electrical engine that laterally moves a weight with a specified motion. This weight is intended to generate the heeling moment and roughly mimic the wave action on the roll motion.

Fig. 2 Sliding mass device

The natural frequency of the rigid system with the empty tank ω0 , as will be later discussed, is 3.26 rad/s. The water depth (H) whose first sloshing frequency matches ω0 has been chosen for the experiments (92mm). The test matrix is defined by choosing 3 moving weight frequencies 0.9 ω0, ω0 and 1.1 ω0 and 4 moving weight motion amplitudes (50, 100, 150 and 200mm). Three different liquids have been used, namely: water, sunflower oil and glycerin. For all cases, the moving mass is the same m=4.978 Kg. A summary of the test cases is reported in table 1. It is important to define which are the Reynolds numbers of our experimental flows. The characteristic velocity will be related to the bore front propagation velocity of the equal height dam-break ( gH ); the characteristic length will be taken as the water depth H. The physical constants of the three liquids and the corresponding Reynolds numbers are documented in table 2.

A Combined Experimental and SPH Approach to Sloshing and Ship Roll Motions 739 Table 1. Test matrix (repeated for each liquid: water, sunflower oil and glycerin)

 / 0

50mm 100mm 150mm 200mm

A

0.9

0.9/50 0.9/100 0.9/150 0.9/200

1.0

1.0/50 1.0/100 1.0/150 1.0/200

1.1

1.1/50 1.1/100 1.1/150 1.1/200

Table 2. Physical properties (units SI) of the liquids: ρ for density, μ , ν for the dynamic and kinematic viscosity. Re for Reynolds number -

ρ

Water

998 8.94•10-4 8.96•10-7 97546

Oil

900 0.045

μ

Glycerine 1261 0.934

ν

Re

5•10-5

1748

7.4•10-4

118

According to the obtained Reynolds number, the water cases will be fully turbulent and the glycerin ones will be completely laminar. In the oil cases, both regimes will coexist. In order to simulate the water cases, due to the small thickness of the boundary layers and the high resolution it would imply, a free slip condition will be used. We will mainly focus on the water cases in the present paper, using the other liquids’ cases as reference regarding the onset of splashing and breaking waves and the influence of these phenomena on the damping effect of the TLD.

3 Model 3.1 Analytical Model of the SDOF Structural System An analytical model of the SDOF structural system used in the experiments is needed in order to have it incorporated into the structure part of the SPH code. It was obtained by rigorously analyzing the dynamics of the system and by obtaining the coefficients after carefully analyzing a set of tests with the empty tank and thereafter finding a data-consistent damping term model. The analytical model used to describe the behaviour of the system is, in general, as follows:

I

0



 m m2 (t )    2m m (t )m (t )    g  S G sin( )

 m  g   m (t )  cos( )  Qdamp (t )  Q fluid (t )

(1)

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Qdamp (t )   K df  sign()  B  

(2)

where: 

 [rad] is the roll angle.



g [m / s 2 ] is the gravitational acceleration.

 I o [ kg  m 2 ] is the polar moment of inertia of the rigid system.  m [kg ] is the mass of the moving weight.   m  t  [m] is the instantaneous (imposed) position of the excitation weight

along the linear guide (tank-fixed reference system).  m  t  [m/s] and m  t  [m/s 2 ] are the first and second time derivatives of

 m  t  [m] .  SG =M R G [kg  m] is the static moment of the rigid system with respect to the rotation axis.  M R [kg] is the total mass of the rigid system.



G [m] is the(signed) distance of the centre of gravity of the rigid system

with respect to the rotation axis (tank-fixed reference system).  Qdamp  t    K df  sign   B   [N  m] is the assumed form of roll damping



moment with a:



 A dry friction term  K df  sign 

with K df [N  m] being the dry friction

coefficient.  A linear damping term  B   with

 B [N  m/  rad/s ] being the linear damping coefficient.

 Q fluid  t  [N  m] is the fluid moment. 150 100 50 0 -50 -150 400 300 200 100 0 -100 -200 -300 -400 1500 1000 500 0 -500 -1000 -1500 0 -2000

2.5

5

7.5

10

12.5

15

17.5

Fig. 3  m  t  top, m  t  middle, m  t  bottom

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A Combined Experimental and SPH Approach to Sloshing and Ship Roll Motions 741

3.2 Analytical Model Assessment 3.2.1 General

By using a set of inclining as well as decay tests, the unknown parameters can be determined, including the natural frequency of the rigid system ω0 . The values of these parameters can be found in table 3. In the case of decay tests m=0 , whereas in the case of forced rolling tests the motion  m (t ) of the shifting mass is imposed during each experiment. During each experiment  m (t ) is directly measured whilst its derivatives are obtained from numerical derivation after fitting a least square cubic spline to the moving weight motion signal in order to mitigate the noise influence in the derivatives. See figure 3 for an example of the sliding mass motion curves. Table 3. Mechanical parameters of the rigid system Quantity Units

Value

SG

kg  m

-29.2

Io

kg  m 2

26.9

K df

N m

0.540

B

N  m /  rad / s  0.326

0

rad / s

3.26

3.2.2 Free Decays Without Fluid: Comparison Between Simulations and Experiments

Figure 4 shows a comparison between an experimental decay and a simulated decay using the parameters reported in Table 3. The agreement is excellent in the range of roll angles above about 2-3deg. For smaller oscillations, the assumed damping model underestimates the actual damping of the system and therefore the simulated time histories will be slightly under-damped in the tail region. A better modeling of the friction damping or a modification of the damping characteristics of the system would be therefore necessary in the case of interest in the region of small amplitude roll motions.

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Fig. 4 Example of comparison between experimental and simulated free decay angles

3.2.3 Forced Roll Without Fluid: Comparison Between Simulations and Experiments

A series of tests with the empty tank and the moving mass have been performed in order to check the capability of the model to reproduce the experimentally measured rolling motion of the system. The test matrix was described in section 2. Figure 5 shows an example of comparison between experiments and simulations. The match for the decay and forced roll tests is good enough to ensure that the model of the rigid system is adequate to proceed with the liquid coupling simulations. 20

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Fig. 5 Comparison between roll angle of the empty tank in experiments and simulations. Empty tank, nominal maximum shift 200mm, nominal oscillation period 1.764s (1.1 0 )

A Combined Experimental and SPH Approach to Sloshing and Ship Roll Motions 743

3.3 SPH Formulation 3.3.1 General

The SPH formulation used for this paper is the weakly compressible one (Monaghan, 1994), in which the closure of the system formed by the compressible mass and momentum conservation equations is achieved by means of an equation of state (EOS) that correlates pressure with density. This EOS is defined in such a way that the density variations are very small, thus modelling a quasi-incompressible fluid. Another important aspect of the computation is the use or not of normalized and complete moving least squares (MLS) kernels (Dilts, 1999) for performing the interpolations. They are crucial if periodic reinizialization of the density field with the aim of resetting the compatibility between the volume, the mass and the density field is implemented(Colagrossi and Landrini, 2003). MLS kernels are important when fluid fields are evaluated close to the boundaries and density reinitialization is important to remove some oscillations in the pressure field. In the present case there is no need to evaluate any specific field (pressure for instance) because we are interested only in the motion of the tank which acts as an integrator of all the fluid loads. Therefore, we think using those techniques would not contribute much in this particular case. Other important point is the viscosity and the eventual need for an artificial viscosity term. In the first SPH approaches to free surface incompressible flows (Monaghan, 1994), the main aim was to solve inertia and gravity driven flows. An artificial viscosity term was included. This term prevented particles crossing their trajectories as well as providing some additional diffusion that increased the stability of the time integration. The approach was later seen to model accurately viscous laminar flows (Monaghan 2005) and works as a very dissipative (often too dissipative) term in high Reynolds number flows which are simulated with a much lower numerical Re. A very promising approach, not yet implemented by the authors, relies on Riemann solvers to resolve the interaction between particles; this makes a big difference in the viscous interactions (Le Touzé et al., 2008). 3.3.2 Boundary Conditions.

The implementation of the boundary conditions will be achieved by using Ghost Particles (Colagrossi and Landrini, 2003), which work very well for a rectangular domain like the present one. For more details on the SPH formulation with application to the assessment of localised values like the wave impact pressures, see Delorme et al, (2009).

744 L. Pérez-Rojas et al.

4 Results of the Simulations 4.1 General The natural period of the system is approximately 1.927s. The liquid height has been chosen so that the linear theory first sloshing period will also be 1.927s. Simulations regarding off-resonance conditions have been performed but the most interesting cases correspond to the resonance ones, and in particular those for which the differences between the three fluids in the dampening effects are worth noting, which correspond to the smallest moving mass amplitudes: 50mm and 100mm. For 50mm the roll angles are large enough to obtain results of reasonable accuracy. However, as discussed in section 3.2.2, for these cases inaccuracies have been observed concerning the modelling of damping, and such inaccuracies could have influenced also the results from SPH simulations. Therefore, the 100mm amplitude cases will be in principle the ones that merit further study. The resonance cases of higher amplitudes show in turn other important features, like for instance that some fluids are able to limit the structure motions whilst others, due to their viscosity (and consequently due to their different dissipation mechanism), do not provide under the same conditions enough damping to put an upper limit the roll angle in resonance conditions.

4.2 Convergence Analysis A convergence analysis in terms of the particle number has been performed using 800, 3200 and 12800 particles. The convergence to a specific solution is very good during the simulation range. A comparison for the glycerine case simulation for resonance condition with 100mm weight displacement is shown in figure 6. 7.5

800 particles 3200 particles 12800 particles

5

2.5

0

-5 -7.5 0

time (s)

5

10

15

20

Fig. 6 Convergence analysis

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A Combined Experimental and SPH Approach to Sloshing and Ship Roll Motions 745

4.3 Resonance Cases 4.3.1 General

The magnitude whose value will be discussed is the reduction ratio in amplitude between the partially filled tank and the empty tank roll angles (percentage) in a specific period of time. A ratio close to 100% will mean that the liquid has no dampening effect. A ratio close to 0%, will mean that the resulting amplitude is very small. The ratio will be established taking the maximum values of the partially filled tank roll angle and the empty tank roll angle. The values of this ratio are reported in table 4, with a row corresponding to the SPH simulations. Table 4. Amplitude reduction ratio ( amplitude partially filled tank / amplitude empty tank) A

50mm 100mm 150mm 200mm

Glycerin 28.4

43.0

58.7

66.2

Oil

13.1

31.4

51.4

61.9

Water

6.4

23.6

46.7

59.3

SPH

10.4

32.2

45.5

58.5

The differences in this ratio for the three liquids are most substantial for the small amplitude cases, as expected. We will describe with detail the 100mm amplitude case, which is the most significant. 4.3.2 Amplitude = 100mm

In this case, the reduction ratio ranges from 23.6 for water to 43.0 for glycerin (table 4, analysis performed from 0 to 35s, end time in the empty tank experiment). The time evolution of the roll angle for the three liquids can be appreciated in figure 7 and in more detail in figure 8. In figure 7 it can be also appreciated that the tank roll angle increase is stopped by the effect of the three liquids, contrary to what happens with the empty tank. There is a substantial difference between water and glycerin and the dynamics are quite different as well, as can be seen in figure 9 (bottom and top). On the other hand, there is not such a big difference between the global dynamics of the oil and water cases, but the differences in the dampening characteristics are very substantial. As can be appreciated in figure 9, in the water case there is the formation of a bore that will develop into a a plunging breaking wave, whereas in the oil case, a mild spilling wave is formed with presumably significantly less dissipation. A specific dye was used for the sunflower oil instead of the fluorescine used for water and glycerin and that cannot be dissolved in oil. This is the reason for the different shade of the oil picture in figure 9.

746 L. Pérez-Rojas et al. 40

PHI_EMPTY_EXP PHI_EXP_WATER PHI_EXP_OIL PHI_EXP_GLYCERIN SPH

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Fig. 7 Roll angle. Nominal maximum shift 100mm, nominal oscillation period 1.927s 20

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Fig. 8 Roll angle (zoom of figure 7)

The SPH results show that the water amplitude reduction ratio error is around 9% in this case. The simulations were run with 800, 3200, and 12800 particles and the convergence in terms of the roll angle curve was good as was commented in section 4.2. It is important to note that the numerical Reynolds number is two orders of magnitude less than the physical one, when comparing with water. It seems that this could justify the results being more similar to those for oil but it is not yet clear at this stage. When analyzing individual frames, the dynamics as simulated by SPH is more similar to that of water, as can be seen in figures 10 and 11 which can be easily assimilated to the water case amongst the three of figure 9. Velocity field samples are plotted in figures 10 and 11. It can be seen that the gradients in the breaking area are significant. This is an indication of the violence of the impact of the jet with the free surface, in which a substantial amount of energy is dissipated.

A Combined Experimental and SPH Approach to Sloshing and Ship Roll Motions 747

Fig. 9 Maximum shift 100mm, oscillation period 1.927s. t/T0=8.66 (left) and t/T0=8.85 (right). Glycerin (top), oil (middle), water (bottom). The orange colour arrow signals the moving mass position

Fig. 10 SPH velocity field, t/T0=8.66, (see fig. 9)

Fig. 11 SPH velocity field. t/T0=8.85 (see fig. 9)

748 L. Pérez-Rojas et al.

5 Conclusions and Future Work The roll motion response of a single degree of freedom (SDOF) structural system to which a rigid rectangular partially filled liquid tank has been attached has been considered. The SDOF system has been described analytically and this description tested experimentally by applying to the system periodic excitations, finding that the response is accurately reproduced by the considered analytical model. The antiroll tank performance has been assessed both experimentally and numerically with weakly compressible SPH. In order to characterise the wave breaking effects on the response curves, tests have been performed with liquids of different viscosity, the increasing viscosity preventing the onset of breaking waves. It is not completely clear at this stage how to exactly quantify the influence of wave breaking on the damping characteristics of the three liquids in the cases studied in this paper. Nevertheless, from the analysis of the experiments described in the paper, it seems substantial. The capabilities of SPH to treat this coupling problem have been assessed. From the comparisons with the experiments, it seems SPH is able to capture part of the dissipation effects due to wave breaking which is reflected in reasonably accurate damping reduction ratios. Nevertheless, it is not at this stage possible to separate those effects from the shear ones in the boundary layers and in the bulk of the fluid, for which the SPH approach is not yet completely consistent with the Reynolds numbers of the experiments. Further work has to be done in these regards. It would be very interesting to compare these results with those obtained with other methods. To achieve this, application to angular motions of the techniques used for linear TLDs will be assessed as well as the possibility of treating this problem with a commercial CFD. The work presented in this paper has dealt with a very controlled situation, with a simple dynamical system that can be analytically modelled with a level of accuracy such that the underlying analytical model for the dynamic system can be considered practically “exact” in engineering terms. According to this, the majority of discrepancies between experiments and simulations with the partially filled tank could be attributed to the SPH simulation (apart from the case of small amplitude oscillations where the inaccurate modelling of damping could have been a not negligible source of differences). Taking into account the fact that the agreement between experiments and simulation was, in general, good, we could conclude that the SPH approach, although still needing improvements on some aspects, could serve as a practical tool for the assessment of tanks’ performances. The next steps that could be followed along this path are:

 Introduction and consequent experimentation/simulation of cases with the tank equipped with baffles as dissipation means.  Implementation of a mathematical model for ship motions. In such case SPH could be used as a means to study, e.g., the effect of water on deck, in addition to the study of anti-rolling devices. Moreover, effects like parametric excitation induced by vertical ship motions on the fluid could also be studied.

A Combined Experimental and SPH Approach to Sloshing and Ship Roll Motions 749

References Anderson JG, Semercigil SE and Turan OF (2000) A Standing-Wave-Type Sloshing Absorber to Control Transient Oscillations J Sound Vib 232 5 Armenio V, Francescutto A, La Rocca M (1996) On the Roll Motion of a Ship with Partially Filled Unbaffled and Baffled Tanks - Part 1: Mathematical Model and Experimental Setup. Int J of Offshore and Polar Eng 6: 4 278-282 Armenio V, Francescutto A, La Rocca M (1996b) On the Roll Motion of a Ship with Partially Filled Unbaffled and Baffled Tanks - Part 2: Numerical and Experimental Analysis. Int J of Offshore and Polar Eng 6: 4 283-290 Attari NKA, Rofooei FR (2008) On lateral response of structures containing a cylindrical liquid tank under the effect of fluid/structure resonances. J of Sound and Vib 318: 4-5 1154-1179 Bulian G and Francescutto A (2004) A simplified modular approach for the prediction of the roll motion due to the combined action of wind and waves. Proc. Instn Mech Engrs, 218 Part M: J of Eng for the Marit Env 2004 189-212 Colagrossi A, Landrini M (2003) Numerical simulation of interfacial flows by smoothed particle hydrodyns. J of Comput Phys 191: 448–475 Delorme L, Bulian G, Mc Cue L, Souto-Iglesias A (2006) Coupling between sloshing and ship roll motion: Comparison between first order potential theory and SPH. 26th Symposium on Naval Hydrodyns. Italian Ship Model Basin (INSEAN). U.S. Off of Nav Res (ONR). 17-22 septiembre 2006, Rome (Italy). Delorme L, Colagrossi A, Souto-Iglesias A, Zamora-Rodriguez R, Botia-Vera E (2009) A set of canonical problems in sloshing, Part I: Pressure field in forced roll-comparison between experimental results and SPH. Ocean Eng 36: 2 168-178 Dilts G (1999) Moving least squares particle hydrodynamics: consistency and stability. Int J of Numer Methods in Eng 44 8: 1115–1155 Francescutto A and Contento G (1999) An Investigation of the Applicability of Simplified Mathematical Models to the Roll-Sloshing Problem. Int J of Offshore and Polar Eng 9: 2 97-104 Ikeda T, Nakagawa N (1997) Non-linear vibrations of a structure caused by water sloshing in a rectangular tank. J of Sound and Vib 201: 1 23-41 Le Touzé D, Oger G and Alessandrini B (2008) Smoothed particle hydrodynamics simulation of fast ship flows. In Proc. 27th Symp on Nav Hydrodyn Monaghan JJ (1994) Simulating free surface flows with SPH. J of Comput Phys 110: 2 390–406. Monaghan JJ (2005) Smoothed particle hydrodynamic simulations of shear flow. Monthly Notices of the Royal Astronomical Soc. 365, 199-213. Pirner M, Urushadze S (2007) "Liquid damper for suppressing horizontal and vertical motions - parametric study", J of Wind Eng and Ind Aerodynamics 95, 1329–1349 Rognebakke OF and Faltinsen OM (2003) “Coupling of Sloshing and Ship Motions”, J of Ship Res 47, 3, 208–221. Souto-Iglesias A, Delorme L, Rojas PL and Abril S (2006) “Liquid moment amplitude assessment in sloshing type problems with SPH”. Ocean Eng 33 , 11-12. Sun LM, Fujino Y (1994) "A Semi-Analytical Model for Tuned Liquid Damper (TLD) with Wave Breaking", J of Fluids and Structures, 8, 5, 471-488. Tait MJ, AA el Damatty and N Isyumov (2005) “An Investigation of Tuned Liquid Dampers Equipped With Damping Screens Under 2D Excitation”. Earthquake Eng & Structural Dyn 34 (7): 719-735. Van Den Bosch JJ and Vugts JH (1966) “On roll damping by free-surface tanks”, Trans RINA Watts P (1883) “On the Method of Reducing the Rolling of Ships at Sea”. Trans. I.N.A., p. 165.

11 Design for Safety

Design for Safety with Minimum Life-Cycle Cost Romanas Puisa* Dracos Vassalos* Luis Guarin** George Mermiris* *University of Strathclyde, Glasgow, UK ** Safety At Sea Ltd, Glasgow, UK

Abstract The paper outlines an approach to multidisciplinary ship design via a software platform maintaining a holistic view on the overall ship quality. The platform integrates design and first-principles design evaluation tools that estimate performance indices of risk, costs, earnings and ship functionality. The platform has built-in mechanisms that determine dominant design parameters, derive parametric models and perform gradual optimisation of constantly updated response surfaces, thus guiding designers towards cost-effective design solutions. The applications aspects and results of the platform are also presented here.

1 Background Traditionally, ship design has not purely been engineering endeavour but it has also involved comprehensive business, economical and social studies, among others prompting in the process, intense consultations between all stake holders of a future vessel. The latter sets design requirements and constraints representing the input to the technical side of the ship design iterations also referred to as the design “spiral” (Gale 2003). In the ship design “spiral” the ship designers move through the design process in a number of steps, each dealing with a particular synthesis or analysis task. After all the steps have been completed, the design is unlikely to be balanced (or even feasible). Thus, a second cycle begins and all the steps are repeated. Typically, a number of cycles (design iterations) is required to arrive at a satisfactory solution. This process is not sequential, unless the design is entirely developed by one person. Although modern design methods are capable of producing very good designs, these designs are unlikely to be optimal. This is because the actual process requires a great deal of design time and thus designers are unable to explore the complete design solution space. Moreover, without being able to recognise the effects of slight modifications on the design all at once, designers may adversely M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_43, © Springer Science+Business Media B.V. 2011

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affect other design requirements while concentrating on a particular design aspect. Lastly and more importantly, with safety being treated as a constraint rather than a design objective, meeting safety requirements cost-effectively is left to chance. The implications of this are twofold:  The safety level of the vessel might not be as expected; the level of safety associated with most of prescriptive regulations is largely unknown (explicitly)  The safety of the vessel might have costed more than it should, have the right trade-offs taken place, this in turn might have led to reduced earnings and to a less competitive design These are particularly true for new design concepts or design concepts incorporating innovative features for which operational experience is very limited or does not exist at all. Attaining safety objectives cost effectively and the potential for increased profitability have led to the introduction and implementation of risk-based design. In simple terms, risk-based design (RBD) is a goal-based design process incorporating multi-disciplinary, multi-objective performance verification in which explicit safety criteria are among the design goals. Hence, the intuitively appealing approach to RBD has been a software platform that integrates design and design evaluation tools under one umbrella, allowing to instantly reflect design changes in all design objectives considered. The current market offers a number of such platforms of which most distinctive are VIP (Wu, Duffy et al. 2007), ANSYS EKM1, NOESIS OPTIMUS2 and ENGINEOUS iSIGHT3. These integration platforms have primarily focused on tool integration, which is the only prerequisite for formal design optimisation. Hence these platforms are rather optimisation platforms of which structure of the integrated process (e.g., no. of design parameters) is predetermined. However in practice this structure is dynamic because design describing information grows with iterations, Figure 1. Therefore, the design process must be regarded as a dynamic optimisation problem that should be readdressed every time when new design data becomes available. This should not be confused with dynamic programming but seen as a greedy algorithm (Black 2005) that follows the methodology (see Subsection 2.2) of making the locally optimal choice at each design stage in order to gradually converge to some satisfactory solution. It is worth noting that design by experience is efficient and effective only until a point when the design complexity is still manageable by a group of designers, see Figure 1. In this context the term manageable refers to a situation when design spiral iterations lead to trade-off solutions. With increasing number of design parameters the degree of freedom of a designed object raises exponentially, according to SN law. Here S corresponds to the number of states a design www.ansys.com/Products/ekm.asp www.noesissolutions.com 3 www.engineous.com/products.cfm 1 2

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parameter has (for continuous parameters S is theoretically infinity) while N stands for the actual number of design parameters. Thus for example, an introduction of two additional discrete parameters (e.g., size of two circular openings) with five options coming from a manufacture, will multiply the total degree of freedom by 52 and rise design complexity by an order of magnitude. In this context the term degree of freedom refers to the number of design variants that can be produced by altering states of design variables.

Fig. 1 Design complexity vs. ability to deliver trade-off solutions.

Bearing in mind the above, SAFEDOR has developed an integrated design platform for risk-based design, setting a primary focus on a design environment as such and making sure that tool integration and design facilitation are addressed. The following subsections give an overview of key features of the SAFEDOR platform (SP), skipping those pertaining to tools integration that are common to other platforms.

2 Concept 2.1 Achieving Holism In simple terms, the holism is achieved by providing an interface for monitoring the reflection of any design changes in resultant variations of all design objectives. Figure 2 illustrates the principle of holism, indicating that a design change made in design tools (e.g., NAPA) is propagated down to design evaluation tools that produce safety, costs, earnings and functionality measures. Additionally, links amongst design objectives (state parameters) are also maintained, thus helping to understand and make use of the correlation between them (see Subsection 3.2).

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In summary, the SAFEDOR platform implements holism by integrating tools, maintaining data transfer and process control, publishing design and state parameters and visualising design iterations and providing access to associated data.

2.1.1 Tool Integration The implementation of tool integration, data management and process control derives from the earlier work in VIP (Wu, Duffy et al. 2007) and brings in new features like management of dynamic input/output. A mechanism for preparation of simple scripts takes care of it. This allows maintaining links amongst tools of which input/output is changing over time. CFD codes of which output content is based on convergence characteristics of internal algorithms is a case in point. Such CFD codes are used for functionality or safety (e.g., fire simulation) analysis.

Fig. 2 Interpretation of the holistic view

2.1.2 Parameter Publishing Tools integration allows performing design spiral activities and hence generating design data. The publishing of this data for further manipulation in the SAFEDOR platform is called parameter publishing. In simple terms, the platform automates the access to design data, bringing them in one place whereat they can be easy analysed and manipulated. The automated and scripting-based implementations of parameter publishing are available.

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2.1.3 Visualisation Data, usually quantitative, generated over design iterations and then published for further analysis are typically strongly scattered and difficult to use for comparison of different design variants or any other qualitative analysis. Therefore, they must be mapped into another level and presented in a more lucid way. Figure 3 exemplifies the way of comparing design variants in terms of tree design objectives. These design variants are all Pareto optimal solutions, hence equal in terms of these three design criteria. Additionally, a user can instantly access data (e.g., first-principles analysis files) associated with any design version. Other graph types are also available with the platform.

Fig. 3 Example trade-off design variants. The highlighted design is compared to the initial design.

2.2 Design Towards Trade-Offs The features outlined in the preceding sub-section are prerequisite for monitoring the design process and, by comparing various design variants, ensuring that the actual design is a best compromise. The question of how to achieve that compromise is answered in this subsection. As shown in Figure 1, the exponentially increasing design complexity indicates that the design by experience become ineffective. Instead, data analysis and decision making support mechanisms are necessary. To this end, the platform implements the following

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– – – –

suggestion of a right direction towards improvement in all design objectives, sensitivity analysis towards determination of dominant/critical design parameters, derivation of response surfaces and gradual multi-objective optimisation of response surfaces

2.2.1 Estimation of the Right Direction The technique analyses previous design iterations and estimates the next design change that would potentially lead in the direction of desired design objective values. This technique is practical when there is no much data about the design sensitivity to design variables yet. This contrasts with statistical methods discussed in the following paragraphs as they are effective only when there is enough data to derive correlation trends from. The right design direction (a vector) is expressed as needed changes in design parameters. Specifically, the design direction x is expressed as

 x1i  x1 j  x  x   2i 2j x  x i  x j        xni  xnj 

(1)

where n is the total number of design variables published, i, j=1,2,3,…K for i≠j and K as the number of design iterations/versions made so far. The problem here is to find such i and j that make x lead towards pre-set design goals and not diverge from them. This is done by analysing all the combinations of i and j and choose the one that maximises the following dot product Risk

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Fig. 4 Illustration of the search for the right change in design parameters.

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

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where =(f1*,f2*,f3*, ,fm*)T is a point/vector of specific target values for all for all design objectives, c=(f1(xc),f2(xc),…,fm(xc))T is a current design solution to be advanced towards  in terms of objective values fi, for i = 1,2,…,m.. The right side of the dot product in (2) yields the difference between xi and xj in terms of design objectives. Graphically the presented technique is shown in Figure 4. Figure 4 shows a case of five design variants of which the third one is considered as a starting point for next iterations. The two design objectives, risk and cost, are minimised towards pre-set target values, which can be expressed as percentages of initial values. The fifth and first design variants are the solution for (2) and hence if we apply x (1) to the third variant we may move towards the target in the direction shown as the grey arrow (potential move). The grey arrows connecting design variants with each other indicate the search algorithm goes through all point pairs until the one maximising (2) is found4. 2.2.2 Sensitivity Analysis. The purpose of sensitivity analysis (SA) is to determine dominant or critical design variables with respect to resultant variations in design objectives considered. It is rather obvious that this kind of analysis is crucial because its outcome is often not intuitive, provided the number of parameters describing a ship concept. In the SAFEDOR platform implements the sampling-based SA (Helton et al., 2006) where the sampling per se is performed (1) manually by designers while altering design parameters or (2) automatically by running design of experiments. Then each design variant corresponds to a sample (an observation) containing unique design variables and related design objective values. This allows applying statistical techniques measuring the strength and significance of correlation amongst parameters. Currently, partial Spearman correlation coefficients are calculated, following by tests on statistical significance against the null hypothesis of no correlation. The significance level is chosen arbitrary, although  has been used. Dominant design parameters are determined automatically, reflecting new changes in the design. The actual procedure of statistical analysis has been implemented as an external script written in language R. Language R is a scripting approach to R Project for Statistical Computing5

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 2K  i  - the number of combinations.

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2.2.3 Response Surface Analysis The determination of dominant design parameters sets a basis for further statistical analysis of the design problem at hand. In particular, the multivariate regression analysis or response surface analysis is of great interest to expressing relationships as closed form equations. As with determination of dominant parameters, response surfaces for selected state variables are derived on the fly. Users of the platform can also choose when the response surfaces are to be updated. The procedure of response surface derivation has also been implemented as an external script written in language R. 2.2.4 Gradual Optimisation Once response surfaces have been updated they are composed into a multiobjective optimisation problem. An optimisation result is a set of Pareto designs that are candidate solutions and should be further analysed against their feasibility. Figure 5 demonstrates the concept of gradual optimisation that delivers solutions of which closeness to the real optimum is stipulated by amount of design data available. As derivation of response surfaces follows some saturation points (Pi in Figure 5) in terms of new design data, then the response surface optimisation delivers corresponding local optima, x1*,x2*,…xk*, to be considered by designers. It is intuitive that as k grows, the closer becomes xk* to some real optimal solution xk*. Formally, *

lim xk  x *

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Fig. 5 Solutions of gradual local optimisation.

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An illustration of the gradual optimisation is shown in Figure 6. The figures illustrates a case of two design objectives, flooding risk and building cost, varied over the number of bulkheads. The optimal solution for these two objectives problem is denoted as x* and it is close to the intersection of the two curves. Note, the both risk and cost curves are not real curves although exhibit typical shapes when estimating the cost and flooding risk for variable number of transversal bulkheads. Initially there is a situation when only two initial points, x1 and x2, (two variants of the number of bulkheads) are known. Then linear response surface, f(x1,x2), is derived based on these points to represent the variation of flooding risk; a response surface for the cost has the same shape as the linear cost curve and hence not shown here. Optimisation of response surface f(x1,x2) yields first optimum x1*, which corresponds to x3’—a local optimal number of bulkheads. Then the second optimal solution x2* for the number of bulkheads is found, based on the response surface f(x1, x2, x3) (linear form, again) enriched with new data. The new local optimum corresponds to x4’ and has additionally advanced towards global optimum x*. The gradual optimisation procedure is so repeated until a found local optimum is satisfactory enough.

f (x1, x2) cost f (x1, x2, X3' ) X* x1

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Fig. 6 Cost vs. flooding risk optimisation with two local solutions.

The technique of gradual optimisation of response surfaces has analogy with stochastic approximation methods (Robbins and Monro 1951) that attempt to find zeroes or extrema of functions which cannot be computed directly due to lack of data. The comparison of the both methods is a subject of future work.

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2.3 Design Methodology The features implemented in the SAFEDOR imply a design methodology to be followed when working with the platform. Figure 7 outlines such a methodology, indicating the decision support options that have been introduced. The presented decision support options have their own user graphical interfaces that ease their application. The next section presents a case study of platform application, outlining results achieved for subdivision design. Process setup, tools Publishing design and state parameters of interest

Design iterations / design spiral

Decision support options - Experience / past designs - Direction towards tradeoffs - Dominant parameters - Response surface models - Gradual optimisation

Feasibility check passed

Concept completed Fig. 7 Summary of decision support options available with the SAFEDOR platform.

3 Case Study The case study outlined below concerns about subdivision design of a cruise liner. The study case is not exhaustive, aiming at demonstrating the sensibility of the gradual optimisation technique (see Subsection 2.2) in the first place.

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3.1 Problem Statement Design of watertight subdivision is driven by three design objectives: – building cost, – total risk (safety level) and – failure performance of the fire main system. Building cost is estimated using an approximate cost model developed at SSRC. The cost model is a function of main particulars, number of bulkheads ( N BT ) and openings, spaces areas etc., as shown in (4). In principle, the cost is a linear function of its parameters.

C  f LBP , D , B , C B ,  , N BT , N O , Ai  

(4)

The total risk is estimated according to the methodology by (Jasionowski and Vassalos 2006) and stands for the expected number of fatalities from flooding and fire accidents. The risk from flooding is estimated by combining simulation of passenger evacuation by HELIOS (Majumder et al. 2005) and first-principles flooding simulation by Proteus3 (Jasionowski 2001). The expected number of fatalities from a fire accident is estimated by zone model-based simulations of fire, followed by evacuation simulation in HELIOS. The methodology is comprehensively presented by (Guarin et al. 2004). The failure performance of the fire main system, Figure 8, is implicitly estimated by post-accident systems availability analysis software HELIOSSAVANT (Vassalos et al. 2009), developed by SSRC under SAFEDOR.

Fig.8 Schematic representation of the fire main system.

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The failure performance is derived from calculation results shown in Figure 9 and expressed as

FFM  2  max PMZ , i   min PMZ , i 

(5)

where PMZ,i is failure probability within main vertical zone i, for i = 1,2,…,6.. By seeking a minimum of (5) the independence of fire main components installed in different zones is increased, while simultaneously reducing the failure probability of the whole system. Quantitative Analysis Fire Main Unavailable Given Fire Casualty on Any Deck in Any FZ.

0.14 0.12

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_MZ2

_MZ3

_MZ4

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_MZ6

Fig. 9 Fire main failure probabilities give a fire accident within any A-class compartment (Vassalos et al. 2009). The abscise axis stands for main vertical zones, the ordinate axis shows the corresponding failure probabilities.

100,000 GT 3,500 pax Lbp 266.23 m Loa 298.816 m Beam 32.2 m Draft 8 m Fig. 10 Used cruise model and its main particulars.

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The naval architecture tool NAPA6 has been used to develop a ship model that included a subdivision and a superstructure with deck layouts, Figure 10. The ship model is a copy of an operating cruise liner of which details are not further disclosed, due to confidentiality reasons. The design process began by arbitrary placing transversal bulkheads and further moving them forward and back to arrive at an initial subdivision index “A” of 0.733. Each variation of bulkhead positions has been stored as a separate design variant within a database linked to the SAFEDOR platform. The variation of design parameters was automatically propagated down to integrated design evaluation tools that delivered total risk, building cost (4) and fire main failure performance (5) values for each variation. This formed an initial data set that has further been used for sensitivity analysis, derivation of response surfaces and gradual local optimisation. The next subsection discusses the outcome of this design exercise.

3.2 Application Results The application results are presented graphically as variation curves for all three design objectives, preceding by corresponding variation of index “A” in Figure 11. In total, eleven design iterations have been performed of which one, V-1.2.4.1, is based on optimisation results of response surfaces. The response surfaces of linear form were derived based on seven points generate arbitrary, they correspond to designs V-1 to V-1.2.4. 0.74 0.739 0.738 0.737 0.736 0.735 0.734 0.733 0.732 0.731 0.73

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Fig. 11 Variation of attained subdivision index “A” over design iterations.

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The trends in Figure 12 and Figure 13 show that risk and cost are creeping down while the failure performance of the fire main system is only slightly affected. The solution of optimisation, V-1.2.4.1, causes a peak in index “A” as well as descents in all three design objectives. It is worth noticing the strong correlation between index “A” and the total risk, as shown Figure 15. Hence by optimising either of them independently, the other is also improved. This is at least true locally, provided that the total risk includes the flooding risk component. 4.000E-05 3.950E-05 3.900E-05 3.850E-05 3.800E-05 3.750E-05 3.700E-05 3.650E-05

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Fig. 13 Variation of building cost (€) over design iterations.

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Fig. 14 Variation of failure performance of the fire main system over design iterations.

4.000E-05 3.950E-05 3.900E-05

y = -0.0003x + 0.0003 R2 = 0.6163

3.850E-05 3.800E-05 3.750E-05 3.700E-05 3.650E-05 0.7305

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Fig. 15 Correlation between index “A” and the total risk (per ship year).

All performed design iterations, except V-1.2, are Pareto optimal solutions and it is up to a designer to decide which one is to be a starting point for next iterations. In the actual case we have chosen the design obtained through optimisation of response surfaces, that is V-1.2.4.1 as it has the highest index “A” value as well as favourable rates for the design objectives, in view of initial design V-1. Figure 16 compares design variables, positions of transversal bulkheads, of initial and new designs.

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Fig. 16 Changes in transversal bulkhead positions needed to migrate from version V-1 to version V-1.2.4.1.

4 Concluding Remarks The presented SAFEDOR platform is a multi-disciplinary design tool on its own. It possesses features of optimisation platforms (e.g., modeFRONTIER7, OPTISLANG8) and brings in innovative functionally that captures the dynamics of a design process. As a result, incremental improvements through design optimisation becomes a secondary purpose of the platform, while the primary one is design from scratch towards trade-offs or cost-effective concepts. The platform provides design decision support, which is based on statistical data analysis and gradual local optimisation of response surfaces. This allows making sure that majority of design iterations lead to improvement in all design objectives. Gradual optimisation of response surface proves effective for design of subdivision, deck layouts and functional elements. However, this approach is resultant only at later design stages, when there is enough data to derive parameter correlations from. Therefore, another technique that suggests a design direction potentially improving all design objectives should be used initially. 7 8

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The model of Risk Control Option (RCO) has not been implemented explicitly. Instead, due to the possibility to compare design variants in terms of cost/risk ratio, design changes that lower the ratio can be considered as RCOs therefore. The platform has a flexible architecture enabling new data mining and design decision support modules to be accommodated in the near future.

Acknowledgements The work reported in this paper has been carried out under the SAFEDOR Project, IP-516278, with partial funding from the European Commission. Contribution of the CAD Centre of Strathclyde University is also gratefully acknowledged.

References Black PE (2005) Greedy algorithm. In: Dict of Algorithms and Data Structures, US Natl Inst of Standards and Technol Gale P (2003) Ship Design and Construction. In: Lamb T (ed) SNAME Guarin L, Majumder J, Shigunov V (2004) Fire and Flooding Risk Assessment in Ship Design for Easy of Evacuation. Design for Safety, Japan Helton JC, Johnson JD, Salaberry CJ, Storlie CB (2006) Survey of sampling based methods for uncertainty and sensitivity analysis. Reliability Eng and System Safety 91:1175–1209 Jasionowski A (2001) An Integrated Approach to Damage Ship Survivability Assessment. PhD thesis, Univ of Strathclyde, Glasgow Jasionowski A, Vassalos D (2006) Conceptualising Risk. In: Proc of 9th Int Conf on Stab of Ships and Ocean Veh, Rio J. Majumder J, Vassalos D, Guarin L, Vassalos GC (2005) Automated Generation of Shipboard Environment Model for Simulation and Design. Int Conf on Comput Sci, Atlanta, USA Robbins H, Monro S (1951) A Stochastic Approximation Method. Ann of Math Stat 22 (3): 400– 407 Vassalos D, Jasionowski A, Cichowicz J (2009) Probabilistic Assessment of Post-Casualty Availability of Ship Systems. STAB, St Petersburg Wu Z, Duffy AHB, Whitfield RI (2007) Virtue Integrated Platform: Holistic Support for Distributed Ship Hydrodynamic Design. 16th Int Conf on Eng Des (ICED `07), Paris, France

12 Stability of Naval Vessels

Stability Criteria Evaluation and Performance Based Criteria Development for Damaged Naval Vessels Andrew J. Peters & David Wing QinetiQ, Haslar Marine Technology Park, (UK)

Abstract The current quasi-static damage stability criteria used by the UK MoD are largely based on the Sarchin and Goldberg work published in 1962. These criteria ensure a level of stability performance after damage. Like the intact stability criteria, the inherent level of safety in these criteria and the link to the dynamic performance of the vessel in waves is little known. A methodology has been developed to evaluate the dynamic stability performance of naval vessels after damage. The evaluation of the current criteria has been performed using a large number of time-domain ship motion simulations using a computer program capable of simulating a damaged vessel with subsequent water ingress and flooding. This gives an insight into the level of safety inherent in the current damage stability standards. A selection of damage cases were conducted using a frigate hullform with geometric variations made to the internal subdivision. A range of loading conditions from those passing the current criteria through to those failing in each of the geometric damage case variations were systematically assessed in a range of wave conditions representative of post-damage sea states. The results from the dynamic study were then compared to the current damage stability criteria terms to identify how the current criteria relate to the dynamic damage performance in waves.

1 Introduction In 1990 the Cooperative Research Navies (CRNAV) Dynamic Stability group was established with the aim of deriving dynamic stability criteria for naval vessels. To derive such criteria, the group needed to evaluate in-service and new ship designs, in moderate to extreme seas in terms of their relative safety and probability of capsize. This would ensure that new vessels continued to be safe, while avoiding high build and life-cycle costs associated with over-engineering. To achieve these objectives the numerical simulation program FREDYN was developed, and continues to be applied extensively, both to intact and damaged M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_44, © Copyright QinetiQ Limited 2010, by permission of QinetiQ Limited.

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ships. This time-domain program is able to take account of nonlinearities associated with drag forces, wave excitation forces, large-angle rigid-body dynamics and motion control devices. The current CRNAV group comprises representatives from UK MoD, Naval Sea Systems Command (NAVSEA), the Australian, Canadian, French and the Netherlands navies, as well as the U.S. Coast Guard, Defence Research & Development Canada, (DRDC), Maritime Research Institute in the Netherlands (MARIN), Naval Surface Warfare Center, Carderock Division (NSWCCD) and QinetiQ. The objective of this paper is to discuss the work that is currently being conducted to assess dynamic performance whilst damaged of naval type vessels in waves and wind. The first findings from this work are presented here. The Royal Navy currently uses static and quasi-static stability criteria to ensure a level of stability performance after damage. Like the intact stability criteria, the inherent level of safety in these criteria and the link to the dynamic performance of the vessel in waves is little known. A methodology has been developed to evaluate the dynamic stability performance of naval vessels after damage. In a similar manner to the probabilistic intact calculations previously conducted by McTaggart 2000, probabilistic calculations have been conducted to evaluate the level of safety inherent in the current damage stability standards. A selection of damage cases based on a modern frigate hullform with geometric variations made to the internal subdivision was assessed. A range of loading conditions, from those passing the current criteria through to those failing in each of the geometric damage case variations was created.

2 Damage Stability Criteria As with many static-based stability criteria adopted around the world, the origins date back to data and information gathered over many years. This applies especially to the great Pacific Typhoon of December 1944, which struck vessels of USN Pacific Fleet causing the loss of 790 men and three destroyers (see Calhoun, 1981.). Following this incident a review of stability assessment was undertaken, which resulted in new stability criteria for U.S. Navy ships (Sarchin and Goldberg, 1962). This covers the intact and damaged stability criteria, which has been adopted by many Navies around the world including the UK MoD. The current stability criteria and damage extents that UK naval vessels have to be able to survive are defined in DEFSTAN 02-109 (The Defence Standard). This document also states the minimum acceptable intact and damaged stability standards for the vessels for which the UK MoD is responsible. The current damage lengths are defined as follows:  Vessels of waterline length less than 30m; any single main compartment.  Vessels of waterline length between 30m and 92m; any two adjacent main compartments. A “main compartment” is to have a minimum length of 6m.

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 Vessels of waterline length greater than 92m; damage anywhere along its length, extending 15% of the waterline length, or 21m whichever is greater. Significant subdivision is common practice in naval ship design. These internal arrangements introduce the potential for both symmetric and asymmetric flooding when damaged. The current Royal Navy stability criteria are based largely upon the criteria originally suggested by Sarchin and Goldberg in 1962. This traditional damage stability analysis using quasi-static approximations cannot account for the behaviour in a seaway or for example, the head of water on a bulkhead bounding a damaged region. For this example of the V-line requirements for the Royal Navy, a dynamic allowance over and above the static damage waterline is included in order to account for vessel motions in a seaway. It has until recently not been possible to asses the suitability of these damage criteria for modern vessels.

3 Damage Criteria Assessment Criteria To investigate the inherent level of safety in the current damage stability criteria (DEFSTAN 02-109) a new methodology was required to systematically compare the dynamic performance of a damaged vessel to the current static damage criteria based on GZ parameters. There are several degrees of increased complexity involved in damage dynamic stability investigations over conducting intact dynamic stability studies. In the intact studies, the variables which directly influence the intact stability criteria values, for a particular vessel, can be largely restricted to KG and displacement. With a damage stability investigation, there is the added complexity of the load condition (tank states) and the size and shape of the damage, all which significantly influence the damage stability criteria values. This results in a complex matrix of simulations to isolate parameters and investigate the influence on dynamic damage stability performance. The high level methodology that has been used to assess the current damage stability criteria is similar to that used previously by the CRNAV group to investigate the intact stability criteria, based on the work by McTaggart (2000). Dynamic stability performance was systematically assessed in a range of wave conditions, with the vessel in carefully selected damage and loading conditions. Results from the dynamic study were then compared to the current damage stability criteria terms to identify trends and which criteria are most closely linked to the dynamic damage performance in waves.

4 Numerical Modelling The FREDYN program was designed to enable the simulation of motion of an intact steered ship in wind and waves. Unlike currently available frequency-domain programs, FREDYN is able to take account of non-linearities associated with the

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drag forces, excitation forces and rigid-body dynamics. The approach is a physical one, where all factors are considered. Non-linearities have to be considered as they arise from    

Effect of large angles on excitation forces, Rigid-body dynamics with large angles, Drag forces associated with hull motions, wave orbital velocities and wind, or Integration of wave induced pressure up to free surface.

The theory for predicting the large amplitude motions with FREDYN has been described by McTaggart and De Kat (2000) and by Van’t Veer and De Kat (2000). The derivation of the equations of motions for a ship subjected to flooding through one or more damage openings is based on the conservation of linear and angular momentum for six coupled degrees of freedom and has been described by De Kat and Peters (2002). The latest version of this software (9.8) was used for this study and can model the influence of damaged compartments and cross-flooding ducts on the vessel’s behaviour in waves.

5 A Relative Damage Loss Index FREDYN allows multiple load and environmental conditions to be tested and the dynamic performance of the vessel to be evaluated. The output from FREDYN consists of motions and relative water heights and so a new measure was required which could provide an overall measure of the damage stability performance in waves. To measure the performance of the vessel after damage the initial focus was on the safety of the vessel and crew following a damage incident rather than on residual mission capability. A number of measures were developed to establish a relative probability of effective loss of the vessel in each damage simulation. These measures of damage performance relate directly to the fundamental modes of loss of the vessel after sustaining damage. These measures provide an indication of the inherent safety of the vessel when damaged, and are as follows:  Roll angle, related to the likelihood of capsize;  Pitch angle, related to plunging and the ease of movement (for evacuation) longitudinally along the vessel;  Reserve of buoyancy, related to sinking (vertical stability); and  Gunwale submergence, related to loss of weather deck area and the ability to evacuate effectively. For each damage simulation the time spent and the number of excursions over a range of roll and pitch angles was calculated. The reserve of buoyancy in the hull was also evaluated throughout the simulation. For the evaluation of the gunwale submergence, six water height points were positioned at 0.25 length between perpendiculars (LBP), 0.5 LBP and 0.75LBP on each side of the vessel on the

Stability Criteria Evaluation and Performance Based Criteria Development 777

gunwale. During each simulation, the amount of time each water height point was submerged was calculated. Post analysis of the output was conducted to identify when adjacent pairs of water height points were submerged at the same time indicating significant submergence of a large portion of the weather deck. Together with each of these four performance measures, an acceptable limit was required to define the point where the vessel was deemed to have been “effectively lost”, i.e. no longer safe for the crew to be onboard and the complete catastrophic loss of the vessel imminent. The selection of these limits has an element of subjectivity to it. In this study the limits chosen were identical in all of the test cases, so the relative performance difference between the test cases could be evaluated. The roll angle limit was set at 85% of the range of positive stability of the GZ curve in the damage condition for each case. This provides a 15% margin for exceeding the range of positive stability and a complete capsize occurring. The time spent over this roll angle limit is assumed analogous to the probability of loss due to roll for the damage case, load and wave condition tested. If the vessel actually capsized during one of the runs the probability of capsize of the run was taken as 100% in the analysis. The pitch angle limit was set to 15 degrees. This was selected based on discussions with operators at the Royal Navy damage control school (DRIU) and naval officers, which suggested that at 15 degrees pitch angle, moving along the decks becomes very difficult and moving damage control equipment becomes very restricted. The 15 degree pitch limit indicates where evacuation becomes difficult and therefore was selected as a suitable limit to use for the relative performance measure. The time spent over this pitch angle therefore is analogous to the probability of loss due to pitch for the damage, load and wave condition. The reserve of buoyancy output from FREDYN is defined by the buoyancy remaining in the hull up to the weather deck. A limit was selected as 2000 Tonnes of equivalent buoyancy, as this relates to approximately half of the original displacement of the vessel in the intact condition. This reserve of buoyancy was considered necessary to keep the crew inside the vessel and allow for the possibility of escape. It was not envisaged that the reserve of buoyancy would be one of the limiting factors due to the relatively short floodable damage length used in this study. The deck edge water height points were used to capture excessive immersion of the gunwale and weather deck. Regularly immersing pairs of points on the gunwales indicates that there is significantly reduced area of the weather deck and that safe crew evacuation would become very difficult with the gunwales deeply submerged and the weather deck awash. Following discussion within CRNAV it was concluded that if two of the adjacent water height points were simultaneously submerged by more than 0.2m, the vessel was considered to be effectively lost. Therefore the percentage of time spent with two adjacent water height points submerged above 0.2m indicates the probability of effective loss of the vessel. In order to produce a measure for the probability of the effective loss of the vessel in the damage case, load and wave condition, each one hour simulation was conducted five times with different wave realisations at the same wave height and

778 A.J. Peters and D. Wing

modal period. The five hours of simulation in each wave and heading combination is believed to provide sufficient time and wave encounters to evaluate the performance of the vessel. By analysing these four measures for ‘effective loss’ of the vessel, the limiting measure in each case was identified and used as the relative probability of loss for that damage, heading and sea condition. Using global wave statistics (Bales 1982) in combination with the probability of Royal Naval vessels being in the waves (Haywood 2006), a relative probability for the vessel being in a particular wave condition was calculated. Multiplying the relative probability of loss for each wave condition by the probability of the ship encountering the wave conditions, produces a relative risk of loss of the vessel after damage. These relative risks of loss for each wave and heading can then be summed together for all wave conditions to provide an overall relative risk of loss measure. The Relative Damage Loss Index (RDLI) term provides a measure of the dynamic damage stability performance of the vessel in a particular damage scenario and load condition for a 99.9% probability [Haywood 2006] of the waves the vessel is likely to experience after becoming damaged. Using these RDLI values calculated from a range of loading and transverse damage extent cases allowed the performance to be compared to the static damage stability criteria. This allows relationships between the static criteria and the dynamic damage performance to be derived.

6 Vessel Damage Cases and Load Condition Selection Computer models of a modern frigate were required to perform simulations; the basic static stability model and the FREDYN dynamic stability model. A static stability model was required to provide the basic hydrostatic inputs for FREDYN; it also serves as a benchmark test to validate the FREDYN model. PARAMARINE was chosen as the software for which the static stability model would be produced. Graphics Research Corporation (GRC) develops PARAMARINE with specific funding from the UK MoD. QinetiQ has rigorously tested and validated PARAMARINE against pure mathematical models on the behalf of the UK MoD.

6.1 Damage Length It was decided that the damage length that would be used for the vessel would remain fixed in all cases in the study. The damage length and shape that was used related to a ‘significant damage’ event previously defined [Peters 2007] as the 95th percentile of the damage data for this size of vessel, based on commercial damage statistics using the data from the HARDER project (HARDER 2003,

Stability Criteria Evaluation and Performance Based Criteria Development 779

Lutzen 2001). This damage length is similar to that required in the current DEFSTAN 02-109 criteria. The location of the damage opening was fixed, giving flooding into three compartments. With the damage length fixed for the study, the variables that could influence the stability criteria parameters were restricted to those that define the transverse extent of damage flooding, such as the position of the longitudinal subdivision. The other major parameters that influence the damage stability criteria are the vessel’s intact displacement and load condition. To set the boundaries for the study a modern frigate hull hullform was used; taking a typical DEFSTAN 02-109 asymmetric damage case as the baseline. A PARAMARINE model of a modern frigate design was used to generate the test cases for the investigation. A typical DEFSTAN 02-109 three compartment damage case for the vessel was selected as the start point for the investigation. This arises from the current DEFSTAN 02-109 damage length of 15% LBP which equates to the ‘significant’ damage length for this vessel in the study. This damage case incorporates flooding into zones including large machinery spaces as well as two pairs of Dieso wing tanks. A typical light loading condition was used to set the displacement and fluid levels in the tanks to be the same for all the cases tested.

6.2 Loading Condition At the light loading displacement, the limiting KG value was calculated for the current DEFSTAN 02-109 stability criteria in this initial damage case. The values of all the damage stability parameters were calculated at this limiting KG condition to identify which was the limiting criterion. Varying the vessel’s KG at the fixed displacement allowed three loading conditions to be set which provided a pass, marginal and fail condition against the current DEFSTAN 02-109 damage stability criteria. Each criteria term was calculated at the three load conditions. KG values at the pass and fail conditions were set equal to the limiting KG values for the more asymmetric and less asymmetric damage cases discussed below to give common load conditions between the damage cases. The second variable investigated in the study was the transverse damage extent i.e. the longitudinal subdivision. The transverse damage extent and transverse subdivision were systematically varied to change the amount of damage and flooding of the vessel. These changes to the transverse damage extent affect the damage stability criteria and so allow for variation of the criterion for a fixed displacement and KG load condition. The three damage cases are presented in Figure 1. The transverse damage extent was increased and decreased to create two further damage cases with differing transverse flooding extents. At each of these new damage cases the limiting KG value was calculated. Each of the criteria terms was calculated at the limiting KG condition to identify the limiting criterion. The KG was then varied to create a pass and fail load case against the current stability

780 A.J. Peters and D. Wing

criteria for the three damage cases. This is shown pictorially in Figure 2 where the limiting KG cases for each of the three damage scenarios are indicated by the markers with black centres.

Most Asymmetric (Case 1)

Mid Asymmetric (Case 2)

Least Asymmetric (Case 3)

Fig. 1 Damage cases GHJ Damage Case and KG Load Condition Matrix 7.2

7.15

7.1

KG [m]

7.05 Case 3 KG Conditions Case 2 KG Conditions Case 1 KG Conditions

7

6.95

6.9

6.85

6.8 Relative Transverse Damage Extent

Fig. 2 KG load matrix for the 3 damage cases

For each the limiting KG condition in each of the three damage cases, the limiting criteria term is different. This allowed the effect of each criteria parameter and the current DEFSTAN02-109 criteria limit to be investigated in comparison to the dynamic stability performance.

6.3 FREDYN Simulations Each of the load cases and damage scenarios were set up in FREDYN V9.8 using QinetiQ’s current standard practice for creating FREDYN models, by exporting the geometry and compartment definitions from the PARAMARINE model in the format for use with FREDYN.

Stability Criteria Evaluation and Performance Based Criteria Development 781

Each case was set up to examine the vessel at two beam seas headings. At the 90 degree heading the vessel is in beam seas with the damage opening towards the waves. At the 270 degree heading the vessel is in beam seas with the damage opening away from the waves. In each case a total of 32 different wave height and period combinations were used to cover the range of waves that the vessel would likely encounter if damaged (Bales 1982). The wave height was set to a maximum of 6m significant wave height, as it has been shown that the probability of a Royal Navy vessel being in a sea state 6 or less is 99.9% based on data from the past 40 years (Haywood 2006). The simulations were set up using QinetiQ’s computer clusters to allow multiple simulations to be conducted simultaneously. Following each simulation the roll, pitch, reserve of buoyancy and water height data were automatically analysed and the statistics of the motions collected. The motion and water height data were averaged over the five different realisations for each run. The relative probability of loss for roll, pitch, reserve of buoyancy and the water height combinations were then calculated, based upon the pre-defined limits. The largest and hence limiting relative probability of loss was selected as the probability of loss for that damage scenario, heading and wave condition.

7 Comparison Between DEFSTAN 02-019 Damage Criteria and the Dynamic Stability Performance The RDLI calculated for each of the eleven combinations of damage extent and load condition were plotted in a number of ways to identify how the dynamic damage performance compares to the current stability criteria. The load condition, the transverse damage extent and wave height were also examined to identify the effect on the dynamic stability performance. The current damage stability criteria values were plotted for the eleven cases on a linear and logarithmic scale to identify relationships between the criteria and the dynamic performance of the vessel. Linear and logarithmic scales were used for the plots as they were previously found to highlight trends during studies on intact stability criteria performance [Peters 2007]. Linear, log and power fit trend lines were then used to fit to the data in order to rank the criteria. Figure 3 shows the relationship between the damage list angle and the RDLI. It is clear that the damage list angle stability criterion gives a poor relationship (based on the best R2 fit of 0.71) with the RDLI. This is of particular interest as this damage stability criterion is often a dominating factor in the certification of naval ships due to their inherent asymmetry in the internal subdivision. This figure shows that in isolation, it is not a particularly strong measure of the damage performance of this vessel. The current 20 degree limit from DEFSTAN 02-109 relates to a 28% RDLI value for this vessel when using the derived trend line, but the results show a variation in RDLI between 12% and 58%.

782 A.J. Peters and D. Wing

Figure 4 shows the Area A1 criteria versus the RDLI. This criterion shows a slightly improved relationship compared to the static list angle, with an exponential curve R2 fit of 0.85. The current DEFSTAN 02-109 stability criteria again relates to an RDLI of close to 28%, which is very close to that given by the current damage list angle criteria. Damage List Angle Vs RDLI - All Headings - All Waves 100.000 y = 0.0012x 3.3694 R2 = 0.7197

90.000 80.000

RDLI (%)

70.000 60.000

Damage data

50.000

DEFSTAN 02-109

40.000

Power (Damage data)

30.000 20.000 10.000 0.000 0.000

5.000

10.000

15.000

20.000

25.000

30.000

Damage List Angle (Degs)

Fig. 3 Damage list angle criteria vs. RDLI all headings and waves

Area A1 Vs RDLI - All Headings - All Waves Area A1 (m.rads) 0.000

RDLI (%)

100.00

0.025

0.050

0.075 y = 70.449e-65.275x R 2 = 0.851

Damage data 10.00

DEFSTAN 02-109 Expon. (Damage data)

1.00

Fig. 4 GZ Area A1 criteria vs. RDLI all headings and waves

Figure 5 shows the results of the GZc/GZmax criteria in comparison with the RDLI data. This criterion shows improvement over the two previous criteria (list angle and A1 area criteria), with a R2 fit of 0.93 using a linear data fit. It is interesting to note that the data is predominately below the current DEFSTAN 02-109 stability criteria (all but one case), giving an RDLI of 62% for this criterion at its present value. This suggests that this criterion is not particularly good in these cases if used in isolation, even with the good data fit (R2=0.93).

Stability Criteria Evaluation and Performance Based Criteria Development 783 GZc/GZmax Vs RDLI - All Headings - All Waves 100.000 y = 148.92x - 27.699 R2 = 0.931

90.000 80.000

RDLI (%)

70.000 60.000

Damage data

50.000

DEFSTAN 02-109

40.000

Linear (Damage data)

30.000 20.000 10.000 0.000 0.000

0.200

0.400

0.600

0.800

1.000

GZc/GZmax

Fig. 5 GZc/GZmax damage criteria vs. RDLI all headings and waves

Figure 6 shows the A1/A2 damage criteria versus the RDLI. An exponential fit to the data from the eleven test cases produced an R2 fit of 0.992 and is the best fit of all the current damage criteria. This criterion appears to show an excellent (R2 = 0.992) relationship to the dynamic performance of the vessel in waves after damage. The A1/A2 criterion provides a relationship between the restoring and disturbing energy on the vessel after damage. This is based on the wind heeling curve and a fixed 15 degree roll back angle. No wind effects were included in the FREDYN calculations and so the only disturbance was from the waves in the simulations, which has been shown to be the dominating disturbing effect. Area A1 / A2 Vs RDLI - All Headings - All Waves Area A1 / A2 (m.rads) 0.000 100.00

0.500

1.000

1.500

2.000

2.500

3.000

3.500 y = 93.052e-1.3616x R2 = 0.991

RDLI (%)

10.00 Damage Data DEFSTAN 02-109 Expon. (Damage Data) 1.00

0.10

Fig. 6 A1/A2 damage criteria vs. RDLI all headings and waves

The range of values from the eleven test cases provided an even spread of data for the A1/A2 criteria and did not show clustering of data on the chart like the GZc/GZmax criteria. The criteria provides a good fit to the data and a good spread of results, meaning this criterion is potentially a very good indicator of the dynamic damage stability performance after a damage incident.

784 A.J. Peters and D. Wing

The current DEFSTAN 02-109 A1/A2 criteria limit is shown on the plot, Figure 6. The current criterion limit value relates to an RDLI of 12% which is significantly lower than the 28% of the next two closest fitting criteria. Like the list angle criteria, this criterion is often the limiting damage criteria for Frigates.

8 Comparison Between Other Potential Damage Stability Criteria and the Dynamic Stability Performance Together with the current static damage stability criteria analysis, additional potential alternative criteria measures were calculated for each of the eleven test cases. These potential measures were plotted against the RDLI in a number of ways to identify if an alternative measure could be used to relate to the dynamic performance of the vessel after damage. The damaged GM was the first alternative measure investigated, as this is a measure that is often used to give an indication of residual damage stability performance both by naval architects and naval staff. Damage GM versus the RDLI for all waves and headings is shown in Figure 7. Damage GM Vs RDLI - All Headings - All Waves GM [m] 0.000

0.500

1.000

1.500

2.000

2.500

3.000

3.500

RDLI (%)

100.00

10.00

Damage Data

1.00

Fig. 7 Damage GM alternative criteria vs. RDLI all headings and waves

From Figure 7, it can be seen that the damage GM does not show a close relationship with the RDLI in the cases tested. This suggests that the damaged GM is not a good indicator of the dynamic damage stability performance. The area under the GZ curve has been shown to be a good measure of the intact dynamic stability performance [Peters 2007]. It was therefore hypothesised that the area under the GZ curve from the angle of list to the range of positive stability (RPS) would provide good indication of the dynamic damage performance in waves.

Stability Criteria Evaluation and Performance Based Criteria Development 785

In each of the eleven cases tested, the area under the GZ curve was calculated and plotted against the RDLI using a log scale. Figure 8 shows the relationship between the area under the damaged GZ curve and the RDLI. Area list to RPS Vs RDLI - All Headings - All Waves Area list to RPS (m.rads) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 100.00 y = 98.745e-50.542x R2 = 0.8221

RDLI (%)

10.00 Damage Data Expon. (Damage Data) 1.00

0.10

Fig. 8 Area under GZ from list angle to RPS alternative criteria vs. RDLI all headings and waves

As can be seen in Figure 8, the damaged GZ curve area provides an indication of the dynamic stability performance with a reasonable spread of the data points, but the R2 data fit is 0.822, which is not as good as the top three ranked current damage stability criteria. In order to improve on this criterion another alternative measure was considered. This investigated the relationship between the areas under the GZ curve prior to damage divided by the area under the damaged GZ curve from the angle of list to the range of positive stability. This is shown in Figure 9 below plotted on a log scale. Intact GZ Area/Damaged GZ Area Vs RDLI - All Headings - All Waves Intact GZ Area / Damaged GZ Area 0.00 90.10 80.10

5.00

10.00

15.00

20.00

25.00 y = 4.27x - 11.454 R2 = 0.9138

RDLI (%)

70.10 60.10 50.10 40.10

Damage Data Linear (Damage Data)

30.10 20.10 10.10 0.10

Fig. 9 Intact area under GZ / Area under GZ from list angle to RPS alternative criteria vs. RDLI all headings and waves

Figure 9 shows the data fit with the inclusion of the intact GZ area term. With an R2 value of 0.913, it shows an improvement in comparison to the damaged GZ area in isolation. Unfortunately this criterion does not provide a reasonable spread of data as the data are predominantly clustered at the left hand side of the chart.

786 A.J. Peters and D. Wing

Using this hypothesis the area from the angle of list to 30 degrees and then to 40 degrees was calculated and plotted against the RDLI using a log scale. Figure 10 shows the area from the angle of list to 30 degrees, which shows an improvement (R2 = 0.894) compared to the full GZ area from the angle of list to the range of positive stability. There is also a greater spread of data. Area list to 30 degs Vs RDLI - All Headings - All Waves Area list to 30 degs (m.rads) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 100.00

y = 87.871e-108.27x R2 = 0.8946

RDLI (%)

10.00 Damage Data Expon. (Damage Data) 1.00

0.10

Fig. 10 Area from list angle to 30 degrees alternative criteria vs. RDLI all headings and waves

Figure 11 shows the area from the angle of list to 40 degrees compared to the RDLI. This shows a much improved R2 data fit of 0.97 compared to that from Figure 8 and 9. This criterion shows the greatest potential as an alternative criterion that could be used in conjunction with the higher ranked current criteria such as GZc/GZmax and the A1/A2 criteria.

Area list to 40 degs Vs RDLI - All Headings - All Waves Area list to 40 degs (m.rads) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 100.00

y = 144.49e-68.388x R2 = 0.9708

RDLI (% )

10.00 Damage Data Expon. (Damage Data) 1.00

0.10

Fig. 11 Area from list angle to 40 degrees alternative criteria vs. RDLI all headings and waves

Stability Criteria Evaluation and Performance Based Criteria Development 787

A suitable criteria limit for the area from the angle of list to 40 degrees would be 0.04 mrads, which equates to a 10% RDLI. This 10% RDLI is in close alignment with the level associated with the current A1/A2 damage criteria for the cases tested. Further investigation is required to extend the data set to cover other geometries and conditions.

9 GM and Wave Height Effect on Relative Damage Loss Index Figures A1-A6 (Appendix A) shows the plots of the RDLI against significant wave height to examine how the RDLI changes as the sea state increases. The values presented are calculated by averaging the results from each of the wave periods at the specific significant wave height. Figures A1-A6 show the RDLI versus the significant wave height for each of the three transverse damage extents, showing curves for each of the GM conditions tested at 90 degree (damage opening towards the waves) and 270 degree (damage opening away from the waves) headings respectively. Figures A1 and A7 show the data for the most asymmetric of the three damage extents tested, at 90 and 270 degree headings. On comparing the largest GM load condition data it can be seen there are very similar results for the RDLI at both the 90 and 270 degree headings. Figures A2 and A5 show the mid asymmetric damage case at the 90 and 270 degree headings respectively. For the largest GM case, the RDLI value increases steadily to 20% at 90 degree heading and 11% at 270 degree heading at the 4m significant wave height. The RDLI then increases rapidly reaching 85% and 66% at the 6m significant wave height at the 90 and 270 degree headings respectively. This shows that the 270 degree heading (damage opening away) provides a lower RDLI across the wave height range. Examining the limiting GM curve for this damage case (GM=0.73m at A1/A2 criteria limit), there is a similar pattern with a steady increase in RDLI to the 3m wave height at both headings. Above the 3m significant wave height, the RDLI at the 90 degree heading increases rapidly to 100% in the 5m waves. At the 270 degree heading the rate of increase in RDLI is much lower with the RDLI reaching 40% at the 5m significant wave height. The GM case which is outside the current criteria limit starts with an RDLI of over 40% at both headings and both rapidly increase with wave height. The RDLI value at the 270 degree heading then levels off at around the 85% level. The 90 degree heading reaches an RDLI of 100% level at the 4m significant wave height and remains at that level. This shows that the 270 degree (opening away) heading has a better survival probability than the 90 degree (opening towards) in this low GM condition.

788 A.J. Peters and D. Wing

10 Results and Conclusion Performance based measure of the damage performance of a vessel for assessment of the strengths of static and quasi-static based damage stability criteria has been conducted. The RDLI is a measure of the probability of the vessel no longer being viable for the crew to remain onboard safely for an hour after damage, considering all waves up to sea state 6. This is used to compare the current static damage stability criteria with a measure of the dynamic stability performance in waves. When the current DEFSTAN 02-109 damage criteria and current levels of criteria are examined in comparison with the RDLI, the GZc/GZmax and A1/A2 are the two current measures that show a good logarithmic relationship with the dynamic performance of the vessel after damage. However, it is only the A1/A2 criteria (R2 = 0.99) which shows a suitable current level for the criteria, with the current DEFSTAN 02-109 criteria level equivalent to an RDLI of 12%. This indicates that this criteria level provides an 88% possibility of survival for 1 hour after damage, considering both beam sea headings and all waves up to sea state 6. The alternative criteria measures that were examined highlighted two key points. The first shows that the damaged GM value, which is often used as rule of thumb for damage stability performance was shown to have a poor relationship to the dynamic performance in all of the cases tested. The second point was that the area under the GZ curve from the angle of list to 40 degrees was shown to be a good alternative measure for the dynamic performance of the vessel after damage. A value of 0.04 mrads for this criterion would provide an RDLI value of 10%, which is in line with that from the current A1/A2 criteria. The cases tested during the study have shown interesting relationships between the static stability criteria and dynamic damage performance. The addition of results from further damage cases for this and other vessel types are required to identify if the conclusions regarding the criteria with the strong relationships to dynamic performance are still valid. It is clear from the study that the significant wave height has a great influence on the survivability of the vessel after damage. Using a 10% RDLI value as an acceptable level of risk of loss of the vessel in a seaway, it is clear from the current criteria limiting cases that the current stability criteria reach the 10% RDLI at or just above the significant wave height of 2m, which equates to a sea state 4. When the significant wave height reaches 3m there is often a considerable increase in the probability of the effective loss of the vessel. This indicates that if the vessel was damaged with a significant damage length and the significant wave height was above 2m, then the general guidance would be to consider evacuation of the crew or to prepare for rapid evacuation. During the study the 270 degree (damage opening away from waves) heading was shown to be the safer of the two headings with a lower RDLI, in all of the waves. This was particularly evident in waves above 2m where the difference in performance was larger. This suggests that if the vessel suffers asymmetric damage and if it is

Stability Criteria Evaluation and Performance Based Criteria Development 789

possible to influence the vessels heading, then the damage opening should be positioned away from the waves.

Acknowledgements The authors would like to gratefully acknowledge the permission granted by the UK MoD for publishing the results of the investigation.

References Bales SL, Lee WT and Voelker JM (1982) Standardised Wave and Wind Environments for NATO Operational Areas, DTNSRDC Calhoun, CAPT C. Raymond Calhoun, USN (ret.), “Typhoon: The Other Enemy - The Third Fleet and the Pacific Storm of Dec, 1944”, Naval Inst Press, Annapolis, Maryland, 1981. De Kat JO, Peters AJ (2002) “Damage Stability of Frigate Sized Ships”, Int Marit Assoc of the Mediterr Conf, Crete. Haywood M, Sea State following damage, UK UNCLASSIFIED, BMT Customer report, March 2006. IMO SLF paper 47/INF.4, Development of Revised SOLAS Chapter II-1 Parts A, b and B-1, Bottom damage statistics for draft regulation 9, June 2004 Lutzen M, Ship Collision Damage, PhD Thesis, Technical Univ of Denmark, 2001 McTaggart K and De Kat JO, “Capsize Risk of Intact Frigates in Irregular Seas”, Transactions SNAME, 2000 McTaggart KA, Improved Modelling of Capsize Risk in Random Seas, Defence Res Establishment Atlantic, July 2000. MoD Defence Standard 02-109 (NES 109), UK Ministry of Defence, Stability Standards for Surface Ships, Part 1, Conventional Ships, 2000 Peters AJ “Cross-flooding of Frigate Sized Vessels” March 2001 – Commercial in confidence USCG and the UK MoD Peters AJ, A Comparison of Ship Capsize Risk with Static Stability Criteria – Working Paper, QINETIQ/D&T/SEA/WP0702785/1.0, March 2007. Peters AJ, FREDYN Support 2006/2007, QINETIQ/D&TS/SEA/CR0704929, April 2007. Peters AJ, Development of Accidental Structural Loss Templates for Naval Vessels, QINETIQ/D&TS/SEA/TWP0701501, July 2007. Sarchin TH. and Goldberg, L.L., “Stability and Buoyancy Criteria for US Naval Surface Ships”, Transactions SNAME, 1962 SOLAS Consolidated Edition, 2001, Int Marit Organ, London, 2001 U.S. Navy, Naval Ship Engineering Center, Design Data Sheet – Stability and Buoyancy of U.S. Naval Surface Ships, DDS 079-1, U.S. Navy, currently Naval Sea Systems Command, Washington, DC, 1 August 1975. Van’t Veer, R. and De Kat, J.O., “Experimental and Numerical Investigation on Progressive Flooding and Sloshing in Complex Compartment Geometries”, Proc of the 7th Int Conf on Stab for Ships and Ocean Veh, STAB 2000, Vol. A, Launceston, Tasmania, Feb. 2000, pp. 305-321 HARDER Project 1999-2003 “Harmonization of Rules and Design Rationale”, Project funded by the European Community, Contract No G3RD-CT-1999-00028

790 A.J. Peters and D. Wing

Appendix A S ignifica nt W a ve He ight V s RDLI - He a ding 90 - Da m a ge Ca se 1 (M ost Asym m e tric) 100 90 80

RDLI (%)

70 GM = 0.87m

60

GM = 0.79m

50

GM = 0.73m

40

GM = 0.61m

30 20 10 0 1

2

3

4

5

6

Sig W ave Height (m )

Fig. A1. S ignifica nt W a ve He ight V s RDLI - He a ding 90 - Da m a ge Ca se 2 (M id Asym m e tric) 100 90 80

RDLI (% )

70 60

GM = 0.79m

50

GM = 0.73m

40

GM = 0.61m

30 20 10 0 1

2

3

4

5

6

Sig Wave He ig h t (m )

Fig. A2. S ignifica nt W a ve He ight V s RDLI - He a ding 90 - Da m a ge Ca se 3 (Le a st Asym m e tric) 100 90 80

RDLI (% )

70 GM = 0.79m

60

GM = 0.73m

50

GM = 0.61m

40

GM = 0.57m

30 20 10 0 1

2

3

4

Sig Wave He ig h t (m )

Fig. A3.

5

6

Stability Criteria Evaluation and Performance Based Criteria Development 791 S ignifica nt W a ve He ight V s RDLI - He a ding 270 - Da m a ge Ca se 1 (Most Asym m e tric) 1 00 90 80

RDLI (%)

70 GM = 0 .87 m

60

GM = 0 .79 m

50

GM = 0 .73 m

40

GM = 0 .61 m

30 20 10 0 1

2

3

4

5

6

Sig W ave Height (m )

Fig. A4. S ignifica nt W a ve He ight V s RDLI - He a ding 270 - Da m a ge Ca se 2 (M id Asym m e tric) 100 90 80

RDLI (% )

70 60

GM = 0.79m

50

GM = 0.73m

40

GM = 0.61m

30 20 10 0 1

2

3

4

5

6

Sig Wave He ig h t (m )

Fig. A5. S ignifica nt W a ve He ight V s RDLI - He a ding 270 - Da m a ge Ca se 3 (Le a st Asym m e tric) 100 90 80

RDLI (% )

70 GM = 0.79m

60

GM = 0.73m

50

GM = 0.61m

40

GM = 0.57m

30 20 10 0 1

2

3

4

Sig Wave He ig h t (m )

Fig. A6.

5

6

A Naval Perspective on Ship Stability Arthur M. Reed Carderock Division, Naval Surface Warfare Center

Abstract From a naval perspective, three areas have been identified as critical for examining the performance of vessels in extreme seas: the physics of large-amplitude motions; verification, validation and accreditation (VV&A) of tools for these conditions; and performance-based criteria. In the physics of large-amplitude motions, three topics are most important: hydrodynamic forces, maneuvering in waves, and largeamplitude roll damping. In the VV&A arena, the challenge remains for performing this function for extreme seas conditions, where linear concepts such as response amplitude operators are not applicable. The challenge of performance-based criteria results from the fact that it is on the leading edge of our knowledge base.

1 Introduction A navy has the same concerns relative to stability failures that all ship owners, designers and operators have. The significant differences arise from the fact that a navy is not governed by IMO regulations; that the naval vessel is often much more costly than a commercial vessel; and that the naval vessel may not have the luxury of avoiding dangerous weather conditions when performing its missions, while a commercial vessel may be able to choose an alternate route. In addition to these differences, a navy often has access to more research and development funds to investigate these issues than the commercial builder and operator. As a consequence of the above, the US Navy has been conducting extensive research on the physics of stability failures; investigating the processes by which computational tools for predicting stability failures can be verified, validated, and accredited (VV&A’d) for use in certifying the dynamic behavior of vessels; and developing performance-based stability criteria. In the physics of large-amplitude motions, three topics have been identified as most important: hydrodynamic forces, maneuvering in waves, and large-amplitude roll damping. In the VV&A arena, performing this function for extreme seas conditions, where linear concepts such as response amplitude operators are not applicable, continues to be a challenge. The challenge of performance-based criteria results from the fact that it is on the leading edge of our knowledge base and therefore, treading paths that have previously been unexplored. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_45, © Springer Science+Business Media B.V. 2011

793

794 A.M. Reed

This paper is divided into three major sections, each of which provides an introduction to, and hopefully some insight into, the three areas mentioned above from a naval perspective: physics of large-amplitude motions, VV&A of tools for predicting stability failures, and performance-based criteria.

2 Physics of Large-Amplitude Motions In order to obtain insight into the behavior of naval vessels in extreme seas, the US Navy has performed many experiments in extreme seas and has also performed computations corresponding to the experimental conditions. Comparisons of the experiments and computations have shown a number of discrepancies, particularly in following and stern-quartering seas, where the frequency of encounter is low. Examination of the differences between the experiments and predictions has identified several possible deficiencies in our understanding of the physics of large-amplitude motions. These include the hydrodynamic forces, the modeling of maneuvering in waves, and time-domain roll damping for large roll angles. The three following subsections will discuss what we have learned, and are currently doing, in these three areas.

2.1 Hydrodynamic Forces Traditionally, the modeling of large-amplitude motions in extreme seas has relied on the use of what we call blended methods—methods that employ a combination of linear and nonlinear computations to compute the fluid forces on the vessel (Beck & Reed, 2001). Typically, this means that the hydrostatic restoring forces and the Froude-Krylov (incident wave) exciting forces are calculated fully nonlinearly; and that the radiation (corresponding to linear added-mass and damping) and diffraction forces are calculated linearly. To determine if the blended method assumptions are correct and to develop an understanding of the forces on a vessel undergoing large-amplitude motions, a numerical experiment was performed using a variety of computational tools. These computational tools ranged from linear to blended to fully nonlinear. The complete experiment is documented in a massive report, Telste & Belknap (2008); Belknap & Telste (2008) contains a more complete summary than is included herein. 2.1.1 Vessels Selected For this investigation, two hulls representative of contemporary naval combatant hull forms were chosen. The first was NSWCCD Models 5415 & 5514, the preliminary design hull form for the DDG-51 class (Hayden, et al., 2006). The

A Naval Perspective on Ship Stability 795

other was the tumble-home version of the hull forms in the ONR Topside Series. These were used to investigate the effect of topside shape on motions, as the topside varied from flared to wall-sided to tumblehome with the same underwater form across the 3 variants (Bishop, et al, 2005). The body plans of the two hulls investigated are shown in Figure 1, and the full-scale particulars, for which the force predictions were made, are shown in Table 1.

a)

b) Fig. 1 Body plans of hull forms used in the Force Study, a) ONRTH hull form b) Model 5514

Simulated Conditions As several of the computational tools employed in the numerical experiment do not incorporate full motion solvers, the investigation was designed to employ only prescribed motions, during which the forces and moments on a ship’s hull were predicted throughout the motion cycle. The forces and moments were computed in a ship-fixed coordinate system with the origin at the center of gravity, with x positive toward the bow, y positive to port, and z positive upward. The moments were computed about the center of gravity.

796 A.M. Reed Table 1 Full-scale particulars of the ONRTH and Model 5514 hull forms

ONRTH

Model 5514

Length (m)

154

142

Beam (m)

18.8

18.84

5.5

6.51

Volume (m )

8540

9150

LCB (m aft of FP)

79.6

72.1

KG (m)

5.5

6.51

GM (m)

4.25

3.02

Draft (m) 3

The numerical experiment was divided into three tasks, each of which corresponds to one of the classical linear force decompositions, although at nonlinear amplitudes. Task 1 corresponded to the classic radiation problem, where the hull was forced to oscillate in calm water. Task 2 corresponded to the classic wave exciting force problem, where the hull was held fixed while it encountered waves. Task 3 was simulated motion in waves, where the vessel underwent prescribed motions in waves. For Task 1, the hulls were forced in heave, pitch, and roll, while at zero speed and at a Froude number (FN) of 0.3. For each of these modes of motion, the hulls were oscillated at frequencies of 0.2070 rad/sec, 0.3831 rad/sec, and 1.1 rad/sec. The hulls undergoing heave were forced at five amplitudes, varying from 5 to 80 percent of the calm water draft. For pitch, the hulls were forced at 5 pitch angles, varying from 1° to 5°. Finally, roll was forced at five amplitudes, varying from 5° to 65°. The pitch and roll rotations were all prescribed about the center of gravity. In all modes, the maximum values were chosen to be well outside of the traditional linear regime, and well into the region where geometric nonlinearities could be expected to be significant. Task 2 required simulation of the flow about the hulls, fixed at the calm waterline, in monochromatic waves of a single wave length; equal to the ship length (/L = 1). The calculations were performed at five equally spaced headings, from head to following seas, and at four wave steepnesses: H/ = 1/60, 1/20, 1/15, 1/10. The flow was simulated at both zero speed and at FN = 0.3. Again, the extreme wave steepnesses were selected to push the calculations into the region where the geometric nonlinearities would be significant. Task 3 simulated large-amplitude motions in beam and following seas. In both conditions, the vessel was forced to move such that the heave amplitude equaled the wave amplitude, and at the center of gravity, the vessel’s waterplane was tangent to the wave’s surface. In the following-seas’ condition, the vessel simulates heave and pitch motion; while in the beam-seas’ condition, the vessel simulates heave and roll. For the following-seas’ case, the wave length was chosen

A Naval Perspective on Ship Stability 797

as twice the vessel length, /L = 2, and the wave steepness was chosen as H/ = 1/20. The maximum pitch corresponded to the maximum wave slope and the pitch frequency was equal to the frequency of encounter. In the beam-seas’ condition, the wave length was chosen as equal to the vessel length, /L = 1, and wave steepness was chosen as H/ = 1/10. The maximum roll corresponded to the maximum wave slope, and the roll frequency corresponded to the wave frequency. In both cases, the amplitude of the vessel’s motions was large, but the motions relative to the wave were small. Simulation Tools Exercised Eight simulation tools were chosen for use in the Force Study. These are listed in Table 2, which provides the names of the codes, the abbreviations used to denote them on the plots of the results, a brief characterization of the theory incorporated in each code, and a bibliographic reference to each code. The codes were selected to provide a range of state-of-theart capabilities. With the exception of FREDYN, all of the codes were run by their developers; FREDYN was run at NSWCCD. Table 2 Simulation tools included in the Force Study (2-D—strip theory, 3-D—fully 3-dimensional, L—linear theory, B—blended theory, NL—nonlinear theory)

Program Name

Abbreviation

Type of Theory

AEGIR-1 AEGIR-2

A1 A2

3-D, L 3-D, B

FREDYN

FD

2-D, B

LAMP-1 LAMP-3 LAMP-4

L1 L3 L4

3-D, L 3-D, B 3-D, NL

NF

3-D, NL

NS

2-D, NL

NFA

1

NSHIPM O

References Kring, et al. (2004) De Kat & Paulling (1989), de Kat 1994), de Kat, et al. (1994) Liut, et al. (2002)

Dommermuth, et al (2006), Dommermuth, et al. (2007) Telste & Belknap (2008)2

Due to the fact that NFA required large amounts of computer time, it did not predict all of the cases discussed. Additionally, because the predicted forces were not decomposed into components, none of its results appear in the figures presented. 2 NSHIPMO has not been documented or reported. The description in Telste & Belknap (2008), based on a private communication from Prof. Robert F. Beck of the University of Michigan, provides the best description available. 1

798 A.M. Reed

The codes were selected to include fully linear theory, blended codes—as defined earlier in the paper, and nonlinear codes—although the extent of the nonlinearity varied. Most of the codes were three-dimensional codes, although two of them were strip theory codes: one linear (FREDYN) and one nonlinear (NSHIPMO). Results The size limitation on this paper precludes the presentation of all but a few of the results from the Force Study, which constitutes literally many 1000’s of plots. From Task 1, time-histories of the vertical component of the radiation force on the ONRTH model during forced pitch of at FN = 0 and  = 1.1 rad/sec is presented in Figures 2 and 3, for pitch amplitudes of 1° and 5°, respectively. In Figure 2, the force time-histories for all of the computational tools, except LAMP-4 (L4) and NSHIPMO (NS), the two nonlinear codes, appear as similar sinusoidal curves.

Fig. 2 Time-history of ship-fixed vertical force from Task 1 predictions for ONRTH hull undergoing forced pitch at FN = 0 and  = 1.1 rad/sec with a pitch amplitude of 1°

This shows that all of the linear and blended codes, which have linear radiation forces, are producing quite similar results. For pitch amplitudes as small as 1°, the two nonlinear codes, LAMP-4 and NSHIPMO, show significant deviations from the linear radiation forces over a significant fraction of a cycle. The deviations from linearity are even more dramatic at 5° amplitude, where there is a significant deviation from the linear radiation force over the entire cycle. It is quite remarkable how similar the predictions are from the two nonlinear codes, considering that one is a strip theory and the other a 3-dimensional code. The

A Naval Perspective on Ship Stability 799

noise in the predictions by the two nonlinear codes may be related to discontinuous jumps in geometry from one time step to another.

Fig. 3 Time-history of ship-fixed vertical force from Task 1 predictions for ONRTH hull undergoing forced pitch at FN = 0 and  = 1.1 rad/sec with a pitch amplitude of 5°

Figure 4 presents the vertical-force time history for a zero speed followingseas’ case of the ONRTH hull in Task 2. The hull is at FN = 0, and the wave length corresponds to /L = 1, with a steepness H/ = 1/15. The force has been decomposed into three components: the diffraction force, the Froude-Krylov force, and the hydrostatic force. There is more variation, compared to Task 1, amongst the force predicted by the various codes for this case. This is in part due to known inconsistencies between how the various codes represent the incident waves and how the pressures on the hull are calculated. For instance, it is known that NSHIPMO uses Wheeler stretching (Wheeler, 1969) to correct the pressure within an incident wave to produce zero pressure on the incident wave surface. The LAMP codes use the fully nonlinear Bernoulli equation to compute the pressure for a linear wave. Some indications of the impact of this inconsistency for the incident wave appear in Fig. 3. The hydrostatic force calculations by the two linear codes present constant values, as anticipated, and are in agreement. The hydrostatic forces predicted by the blended and nonlinear codes are consistent with the exception of NSHIPMO, which varies, probably due to the use of Wheeler stretching.

800 A.M. Reed

a)

b)

c) Fig.4 Time-history of the ship-fixed vertical force from Task 2 predictions for the ONRTH hull. The hull is at FN = 0, in steep following seas, /L = 1, H/ = 1/15, a) diffraction force, b) FroudeKrylov force, c) hydrostatic force. (Belknap & Telste, 2008)

A Naval Perspective on Ship Stability 801

a)

b)

c) Fig. 5 Time-history of ship-fixed vertical force from Task 3 predictions for Model 5514 hull, while contouring following seas at FN = 0, /L = 2, H/ = 1/20, a) hydrodynamic force, b) Froude-Krylov force, c) hydrostatic force. (Belknap & Telste, 2008)

802 A.M. Reed

Task 3 predictions of the vertical force time-history from a following-seas’ case for Model 5514 are presented in Figure 5. The hull is at FN = 0, and the wave length corresponds to /L = 2, with a steepness H/ = 1/20. As presented, the force has been decomposed into three components: the hydrodynamic (radiation and diffraction) force, the Froude-Krylov force, and the hydrostatic force. The hydrostatic and Froude-Krylov forces shown in Figure 5 are an order of magnitude greater than the hydrodynamic forces. However, the hydrostatic and Froude-Krylov forces are 180° degrees out of phase with each other, so they largely cancel each other. Thus the difference between the hydrostatic and FroudeKrylov forces is the same order of magnitude as the hydrodynamic force, rendering an accurate calculation of the hydrodynamic force very important. Both the hydrostatic and Froude-Krylov forces calculated by all of the codes are in remarkable agreement—there is no difference in the hydrostatic force, and the differences in the Froude-Krylov force predictions are small. The hydrodynamic forces presented in Figure 5 show significant variation between the codes. As it was impossible to distinguish between the radiation and diffraction components of the hydrodynamic force, one cannot identify the sources of the difference. The AEGIR codes and FREDYN produced similar force levels, as did all of the LAMP codes, NSHIPMO predicted forces that vary between the AEGIR/FREDYN force levels and the LAMP force levels. It is not obvious why the three LAMP codes produced such similar forces. Although LAMP-1 and LAMP-3 produced the same force, as they should, this difference may be due to their use of the fully nonlinear form of Bernoulli’s equation. Conclusions from the Force Study. Thousands of calculations were made and compared of the forces and moments on two hulls: oscillating in various modes of motion in calm water, fixed in waves, and contouring waves. The results are presented in the form of time-history plots showing simulated ship motions at two speeds, for a variety of headings and wave/motion amplitudes. (These results have also been compiled into a report containing 15,240 pages.) It was not the purpose of the study to evaluate any one code relative to another, but rather to evaluate the differences between various complexities of theory. In general, codes with a consistent level of theory produced quite consistent results. In many of the forces and moment predictions, the results from Task 1 demonstrated the importance of nonlinearity in the radiation forces. An obvious indicator of nonlinearity is the departure of the components of force and moment from a simple sinusoidal form. Spikes in the forces and moments may originate from the consideration of geometric nonlinearities. The results from the Task 2 predictions indicated that a nonlinear wave model, as well as an appropriate evaluation of the pressure on the hull, is required. This subject area needs further investigation. Task 3 showed that cancellation of forces and moments arising from the Froude-Krylov forces and hydrostatics indicated the importance of determining the wave radiation and diffraction force and moment components accurately, even when they are small.

A Naval Perspective on Ship Stability 803

A surprising finding was that the body-exact strip theory is capable of capturing important nonlinearities—comparable to the fully nonlinear 3-dimensional codes. This result provides hope for the development of fast codes to predict dynamic stability failures, on the order of real time.

2.2 Maneuvering in Waves The state-of-the-art in the theory of maneuvering in waves is to superimpose a maneuvering model on a seakeeping code. Experience with several different maneuvering models has shown that a Abkowitz-type maneuvering model (Abkowitz, 1969), which does an excellent job of predicting calm water maneuvering, produces nonsensical results in waves. To gain insight into this issue, an experiment was performed where the traditional maneuvering tests—turning circles and zig-zags, were run in both calm water and in regular waves, of varying wave lengths and steepnesses using a model of a combatant-type hull form. Figures 6, 7 and 8 show the time-histories of a series of turning circles performed under identical conditions except for increased wave steepness. In each figure, the model was at FN = 0.3, the rudder angle, is 20°, and the wave length to model length, /L, was 1. Between the three figures, the wave steepness, H/increased from 1/90 to 1/60 to 1/30, respectively. Each figure shows the corresponding calm water turning circle as a dashed line and the tuning circle in waves as a solid line. Each run was composed of 4-1/2 complete circles. In Figure 6, for H/ = 1/90, there was virtually no difference between the turning circles in calm water and in waves—the diameter is unchanged and there is no “drift” of the turning circle. The turning circles in waves in Figure 7, for H/ = 1/60, shows no increase in the turning diameter, but there is a slight amount of drift in the turning circle, roughly 10 percent of the diameter over four and a half circles. The plot shows quite different results for the H/ = 1/30 case, Figure 8. In this case, there is an increase in the turning diameter of about 10 percent, with significant drift of the circles—roughly 50 percent of the turning diameter over 4-1/2 turns. In shorter waves, the increase in turning diameter and drift occur in less steep waves than was examined for this case. Although there is no consensus as to the source of this drift and the increase in turning diameter, one plausible explanation is the second-order drift force, which increases as the square of the wave amplitude. Although it is also likely that there are other issues involving the interaction of maneuvering with ship motions in waves.

804 A.M. Reed

Fig. 6 Time-history turning circle in waves, FN = 0.3,  = 20°, /L = 1, H/ = 1/90; red dashed line in calm water, solid blue line in waves

2.3 Large-amplitude Roll Damping The state-of-the-art for predicting frequency-domain roll damping is the component-based model of Ikeda (Ikeda, et al., 1978; Himeno, 1981). This model is semi-empirical, and is based on small amplitude experiments with models of merchant hull forms. Time-domain roll damping is generally based on the Ikeda frequency-domain model, and uses averaging of the frequency-domain rolldamping components to produce the time-domain roll damping used in simulations. Thus, there is a need to develop a physics-based, time-domain rolldamping model, which is applicable to large-amplitude motions and naval vessels, as well as to commercial vessels. As discussed in Himeno (1981), frequency-domain roll damping is typically divided into seven components attributed to the hull, bilge keels, and rudder. Four of the components: friction, eddy making, lift, and wave making are attributed to the hull; and normal force, hull pressure, and wave-making roll damping are attributed to the bilge keels and rudder. The seven terms are generally treated as linear, in the sense that in the equation of motion for roll, the roll-damping contribution to the equation of motion is only a linear function of the roll velocity. However, the coefficients multiplying the roll velocity may be functions of the roll frequency and the amplitude of the roll. In

A Naval Perspective on Ship Stability 805

the frequency domain, this necessitates iteration for the values of roll damping, which correspond to the resulting amplitude of roll at each frequency.

Fig. 7 Time-history turning circle in waves, FN = 0.3,  = 20°, /L = 1, H/ = 1/60; red dashed line in calm water, solid blue line in waves

Fig. 8 Time-history turning circle in waves, FN = 0.3,  = 20°, /L = 1, H/ = 1/30; red dashed line in calm water, solid blue line in waves

806 A.M. Reed

This type formulation is not amenable for use in the time domain, except when reduced to some average representation, such as an equivalent linear damping. This may be adequate for small-amplitude roll, but it is not adequate when the amplitude becomes large. For instance, for many ships, when the roll angle reaches an angle of around 30°, one of the bilge keels emerges (Figure 9) and the bilge-keel roll damping is approximately halved—ignoring the potentially large transient associated with the bilge keel exiting and entering the water. Bilge-keel component predictions are also dependent on ship geometry, including bilge keel size, as well as roll amplitude. Although neglected, consideration of the bilge-keel wave-making damping and added mass effects at large roll should also be included (Bassler & Reed, 2009). Modifications as elementary as adding a simple jump in the linear roll-damping coefficient at the angle for which the bilge keels emerge renders the total solution for the simple roll equation highly nonlinear, even in the frequency domain. The US Navy has performed forced oscillation experiments (Bassler, et al., 2007; Fullerton, et al., 2008) to study these phenomena and determine rolldamping characteristics at large roll angles. Roll decay experiments have also been carried out with instrumented appendages—including bilge keels, rudders, and skegs, to assess their contribution to roll damping (Grant, et al., 2007; Etebari, et al., 2008). Initial comparisons between experimental results and unsteady RANS simulations have shown some promising results (Miller, et al., 2008).

Fig. 9 RANS simulation of the ONRTH, FN = 0.30, ω = 4.83 rad/s, φ = 30 deg, where the bilge keel partially emerged from the water (Miller, et al., 2008).

In the time domain, it has been suggested that treating the ship with its bilge keels as a slender fish-like body (Lighthill, 1960; Newman & Wu, 1974; Newman, 1975) might provide a reasonable analytical basis. Another path the US Navy is currently exploring is the application of low-aspect lifting-surface theory, as studied by Bollay (1936, 1939), Gersten (1961, 1964), and van Zwol (2004) for steady flows, and by Brown & Michael (1954, 1955) and Howe (1996) for unsteady vortex shedding.

A Naval Perspective on Ship Stability 807

3 Verification, Validation and Accreditation3 If decisions regarding the design and construction of ships, each costing hundreds of millions of dollars, if not a few billion dollars, are going to be made based on the stability predictions of a simulation tool, there must be a reasonable assurance that the tool provides acceptably accurate results. The process by which a tool may be determined to be sufficiently accurate is known as verification, validation and accreditation (VV&A).

3.1 An Overview of VV&A The definition of VV&A follows. Quoting from a US Navy VV&A presentation, “Verification, Validation, and Accreditation are three interrelated but distinct processes that gather and evaluate evidence to determine, based on the simulation’s intended use, the simulation’s capabilities, limitations, and performance relative to the real-world objects it simulates.” Beck, et al. (1996); AIAA (1998); and DoD (1998, 2003, 2007) provide different, although consistent, definitions of the three components of VV&A. The DoD definitions for these three terms are provided below, each followed by a practical commentary relevant to computational tools for predicting dynamic stability. 1.) Verification—the process of determining that a model or simulation implementation accurately represents the developer’s conceptual description and specification, i.e., does the code accurately implement the theory that is proposed to model the problem at hand? 2.) Validation—the process of determining the degree to which a model or simulation is an accurate representation of the real world from the perspective of the intended uses of the model or simulation, i.e., does the theory and the code that implements the theory accurately model the relevant physical problem of interest? 3.) Accreditation—the official determination that a model or simulation, . . . is acceptable for use for a specific purpose, i.e., is the theory and the code that implements it adequate for modeling the physics relevant to a specific platform? In other words, are the theory and code relevant to the type of vessel for which it is being accredited? US Navy experience with attempting to verify ship-dynamics’ software has been that the documentation for many hydrodynamic codes, particularly the theoretical basis, is neither complete nor rigorous enough for the verification process to be separated from the validation process. Under these circumstances, when one finds that the computations do not adequately model the physical reality, one is left to ponder whether the code is not accurately modeling the intended physics or whether the intended physics are not adequate for the problem. In this case, the dilemma becomes: should one attempt to debug the code or should one abandon 3

An expansion of Reed (2008)

808 A.M. Reed

use of the code because its underlying physics model is not adequate? Attempting to resolve this dilemma can be expensive, in terms of both time and money. Another issue related to verification of software is the actual quality of the code and the documentation of the code itself. Often the coding does not follow any consistent standard and there is often insufficient guidance to link the actual code back to its theoretical basis. As for the actual verification of the code, this is best done by means of unit tests, where each module and block of modules is exercised against known or expected solutions. When properly constructed, these unit tests will not only test the module against normal execution, but also against unexpected or unanticipated inputs, to determine if the code handles error exceptions correctly via error traps or error returns. Many codes are not designed robustly enough so as to deal with anomalous inputs—they expect that the input will always be correct and that all modules that produce input for other modules provide correct input. Rationally, this is a rather naïve assumption. A second observation with regard to VV&A relates to the question of how one performs validation for a code used for predicting total (as opposed to partial) dynamic stability failures, events that are essentially binary. Either there was a failure or there was not. One can certainly contemplate comparing the failure predictions from a simulation to a model test or full-scale vessel. However, the failure is hopefully a rare occurrence and is fraught with many unknowns: What were the actual local environmental characteristics at the instant of failure and in the few minutes leading up to failure?; What were the actual mass properties of the vessel at the time of failure?; Was the vessel actually intact at the time of failure, or had it in fact taken on water, leading to a failure in what was actually a damaged state?; Was the vessel on autopilot or under manual steering, etc.? In the case of model-scale tests, some of the full-scale issues can be resolved, but for a free-running model it is still difficult to characterize the waves that the model is encountering, particularly if they are irregular seas. The question of the autopilot steering algorithms is particularly challenging: Can a simulation accurately model the actions of a helmsman?; What is the range of human performance or the actual autopilot on the vessel?; Particularly in these days of smart autopilots that “learn,” can the actual autopilot algorithm in the time leading up to the instant of failure be known? Thus, there are a number of issues that must be resolved before one can conclude that any computational tool is ready to use for establishing performancebased stability criteria, or certifying a ship design as meeting the criteria. In order to accommodate the validation of simulations for predicting stability failures, situations must be examined that are not easily characterized using techniques that are routinely used for seakeeping validation. Nonlinear dynamics methods appear to show significant promise. There are two aspects of nonlinear dynamics that appear to apply to validation. The first is nonlinear time-series analysis and the second is bifurcation analysis.

A Naval Perspective on Ship Stability 809

3.2 Nonlinear Time-Series Analysis In nonlinear time-series analysis (cf. Kantz & Schreiber, 2004), the same timeseries analysis is applied to motions measured on a physical model (or ship) and to simulations of the same vessel, in the same environment, as observed during the measurements. The results of the two sets of analysis are compared to each other, often graphically, to determine whether they have produced similar results. McCue, et al. (2008) provide examples of nonlinear time-series analysis, applied as it might be for validation of simulations. Both qualitative and quantitative metrics that may apply were examined. Some qualitative measures include: reconstructed attractors, correlation integrals, recurrence plots, and Poincaré sampling; possible quantitative measures are: correlation dimension, Lyapunov exponent comparison, system entropy, and approximations to the equations of motion (EOM).

a)

b) Fig. 10 Recurrence plot for DTMB Model 5514 Run 312,  = 0.15 below the diagonal,  = 0.1 above the diagonal, a) measured data, b) numerical simulation. (McCue, et al., 2008)

810 A.M. Reed

Figure 10 presents two figures from McCue, et al. (2008) that compare recurrence analysis of measured data from DTMB Model 5514 (Hayden et al., 2006) to numerical simulations. The recurrence plot presents graphically how often a motion trajectory in state space returns to a trajectory near the initial one. State variables include quantities such as roll angle, roll velocity, pitch angle, and pitch angular velocity. In generating a recurrence plot, all quantities are nondimensionalized by their standard deviation. The quantity  is the distance allowance between the two states, so the smaller  is, the closer two states must be to correlate. As can be observed, there is a significant difference in the density of

Correlation Integral

10-1

a)

10-2

10-3

10-4 10-1

ε

100

Correlation Integral

10-1

10-2

10-3

10-4 10-1 b)

ε

100

Fig. 11 Correlation integral versus  for DTMB Model 5514 Run 312, a) measured data, b) numerical simulation. (McCue, et al., 2008)

A Naval Perspective on Ship Stability 811

points between these two recurrence plots for the same conditions. Figure 11 provides the correlation integral, which represents the density of the data on the recurrence plot versus for both the measured model data and the numerical simulations. As is seen, there is a significant difference in the slopes between the two figures, indicting that the simulation is missing some aspect of the experimental results.

a)

b) Fig. 12 Phase space and Poincaré plot for DTMB Model 5514 Run 312 with Poincaré sampling every 36.5 seconds., a) measured data, b) numerical simulation. (McCue, et al., 2008)

Figure 12 shows phase space and Poincaré plots for the same run from Hayden, et al. (2006), as described for Figures 10 and 11. The phase plot is the standard plot in which displacement is plotted against velocity, in this case roll angle against roll-angular velocity. The Poincaré component of the plot, the plus signs, shows the displacement and velocity at a common time interval, in this case

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36.5 seconds, full scale. If the motions are purely periodic, then the points will converge, as in Figure 5b, while if the motion is not periodic at the selected period, they will not converge to a single location. This figure shows quite different shapes for the trajectories in phase space, and the Poincaré sampling shows convergence for the computed results and quite scattered results for the model experiments. While nonlinear time-series analysis techniques can easily illustrate differences between measurements and predictions, there is still much to be investigated. The range of time-series analysis techniques which may be applicable to dynamicstability failure prediction certainly has not been exhausted. However, these comparisons are at best qualitative; quantitative methods, particularly for physical understanding and for comparing experimental and computed results, are needed. Bifurcation analysis techniques may provide this necessary additional insight.

3.3 Bifurcation Analysis There are at least four bifurcations that have been observed in ship dynamics which could be used to analyze whether or not a dynamic- stability code is producing the correct dynamic behavior: Fold bifurcation, Flip bifurcation, Hopf bifurcation, and Homoclinic bifurcation. All of these bifurcations are discussed in Belenky & Sevastianov (2007). Bifurcation analysis would appear to be appropriate for application to the lateral-plane aspects of dynamic stability. A brief discussion of these bifurcations follows. Fold bifurcation (also known as tangent instability, jump to large-amplitude, or hysteresis) can be found in roll and yaw (Spyrpou, 1997; Belenky & Sevastianov, 2007: Sect. 4.5.2 for roll, Sect. 6.5.6 for yaw). It is responsible for direct broaching, and is observed as a dramatic increase in the response amplitude, with a small change of control parameter—usually excitation frequency for roll and commanded heading for yaw. Fold bifurcation can be detected when the eigenvalues of the variation equation are real and leave the unit circle in a positive direction. It has been observed in a model test reported by Francescutto, et al. (1994). Flip bifurcation, which is also known as period doubling instability, can be found in roll and yaw (Spyrpou, 1997; Belenky & Sevastianov, 2007: Sect. 4.5.3 for roll, Sect. 6.5.6 for yaw). It is observed first as a loss of symmetry of a phase trajectory (the cycle becomes egg shaped), followed by a series of period doublings, eventually leading to deterministic chaos. Flip bifurcation can be observed when the eigenvalues are real and leave the unit circle in a negative direction. A chaotic stage of the bifurcation can be detected when the Melnikov function crosses zero. In this case, branches of the invariant manifold have an infinite number of intersections, resulting in fractalization of the safe basin. Hopf bifurcation, also known as flutter, can be found for surge in sternquartering seas (Spyrpou, 1996; Belenky & Sevastianov, 2007: Sect. 6.5.2). Hopf bifurcation can be observed as relatively small-amplitude oscillations around the

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surf-riding equilibrium. It can be detected by a combination of the limit cycle with unstable focus equilibrium. It has been observed in a model test by Kan (1990). Homoclinic bifurcation can be found in surge (Belenky & Sevastianov, 2007: Sect. 6.3.5), and is responsible for surf-riding. It is observed as the appearance of an equilibrium co-existing with periodical surging, then as the only option. The control parameter for this bifurcation is the nominal Froude number. Homoclinic bifurcation can be detected in the phase plane as the connection of a saddle point to itself (or to a saddle point on the next wave). It is reported as having been observed in both full-scale and numerous model tests. Bifurcation analysis has not been formally reported as a code validation technique, but in conjunction with analytical models of dynamic instabilities and model- and full-scale observations, appears to show promise as a validation technique—does the code demonstrate the physical bifurcation behavior that would be expected, based on analytical models and experimental observations? Spyrou, et al. (2009) employed continuation analysis, in conjunction with a specialized version of LAMP (Liut, et al., 2002), to examine the occurrence of surf-riding and broaching. Figure 13 provides an example of the results that they have produced. While the work of Spyrou, et al. (2009) was not constructed as a validation study, what they have done indicates that some type of bifurcation analysis has promise as a validation technique. This is a line of investigation that should be pursued further.

3.4 The Problem of Rarity Another issue for the VV&A of simulations for dynamic stability is the “problem of rarity,” where the time between events is long compared to the wave period (Belenky, et al., 2008a,b). Large numbers of realizations may be required to observe dynamic stability failures, either in a simulation or experimentally. Even if these events are observed, direct comparison between realizations is difficult due to the stochastic nature of the failure event. One method that may help to resolve this problem is the use of deterministic critical-wave groups. This would enable direct comparison of realizations, while also capturing the worstcase conditions of the stochastic environment necessary to assess the ship’s stability performance. Themelis & Spyrou (2007, 2008) demonstrated the production of deterministic critical-wave groups using simulation tools, and Bassler, et al. (2009) has shown that it may also be possible to produce them experimentally.

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a)

b) Fig. 13 Output of LAMPCont for the sample ship in quartering seas, illustrating some of the stability variants with a fixed propeller speed in a wave,  = 200 m and H = 4 m, a) ship’s yaw angle relative to the wave direction vs. rudder deflection, b) position of the ship CG on the wave relative to the wave crest. (Spyrou, et al., 2009)

A Naval Perspective on Ship Stability 815

4 Performance-Based Criteria4 With few exceptions, the navies of the world are still employing hydrostatic-based stability criteria, which reflect outgrowths or extensions of the works of Rahola (1939) and Sarchin & Goldberg (1962). However, the need to develop risk-based stability criteria is recognized. There have been a number of recent papers and reports relating to the subject of dynamic stability assessment, cf. Alman, et al. (1999), McTaggart (2000), McTaggart & de Kat (2000), and Hughes & Perrault (2008). Many of these have been motivated by the work of the Naval Stability Standards Working Group (NSSWG) a collaborative effort between the Royal Australian Navy, the Canadian Navy, the Royal Netherlands Navy, the British Royal Navy, the US Coast Guard, and the US Navy. The following subsections summarize the current status of several naval stability standards: NATO, Naval Stability Standards Working Group (NSSWG), and the US Navy. The US Navy standard appears to be the first quantitative performance-based standard.

4.1 Existing Naval Stability Standards In 2003, NATO initiated an effort to develop a goal-based standard for naval vessels that could guide navies and classification societies in the development of rules for naval vessels. The intent was to develop regulations for naval vessels that paralleled the IMO regulations for commercial vessels (IMO regulations do not apply to naval vessels). In 2007, NATO issued several documents relating to standards for classing naval vessels, all under the umbrella of a Naval Ship Code (NATO, 2007a). The introduction to the Naval Ship Code states, “The overall aim of the Naval Ship Code is to provide a framework for a naval surface-ship safetymanagement system based on and benchmarked against IMO conventions and resolutions that embraces the majority of ships operated by Navies.” The code further goes on to state “. . . it therefore contains safety-related issues that correspond in scope to that which is covered by IMO publications but which reflect the fundamental nature of naval ships.” Rudgley, et al. (2005) provided an overview of the process and the overall philosophy for the development of the Naval Ship Code. The Naval Ship Code is composed of ten chapters: Chapter I General provisions Chapter II Structure Chapter III Buoyancy and Stability Chapter IV Machinery Installations Chapter V Electrical Installations 4

Based on the author’s section on naval standards in ITTC (2008)

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Chapter VI Fire Safety Chapter VII Escape, Evacuation and Rescue Chapter VIII Radiocommunications Chapter IX Safety of Navigation Chapter X Carriage of Dangerous Goods Chapter III, Bouyancy and Stability, includes dynamic stability and capsize. This chapter was developed in the second half of 2006 by a study group composed primarily of representatives from: Australia, Canada, Italy, Netherlands, Spain and the United Kingdom, with input from the Naval Stability Standards Working Group (NSSWG). In parallel to the Naval Ship Code, NATO produced another document, Guide to the Naval Ship Code (NATO 2007b). The discussion of Chapter III in the Guide states “Due to the variety of available Naval Standards on stability and on-going work in other bodies to understand the dynamics of the stability problem and the measure of safety provided by current standards, it was decided not to develop another detailed quasi-static stability standard.” Thus, the Naval Ship Code provides only the most generic guidance with regard to dynamic stability and capsize. The Buoyancy and Stability chapter is divided into eight “Regulations” numbered 0 through 7. Regulations 1–7 are subdivided into four sections: Functional Objectives, Performance Requirements, Verification Methods, and Definitions (optional). Four of these Regulations (0-Goal, 1-General, 4-Reserve of Stability, and 7-Provision of Operational Information), explicitly mention capsize or dynamic stability. The Regulation 0-Goal specifically states: 1 The buoyancy, freeboard, main sub-division compartment and stability characteristics of the ship shall be designed, constructed and maintained to: 2. Provide adequate stability to avoid capsizing in all foreseeable intact and damaged conditions, in the environment for which the ship is to operate, under the precepts of good seamanship. The “Performance Requirements” listed under Regulation 1-General further elaborate: 4

The ship shall: 1. Be capable of operating in the environment defined in the Concept of Operations Statement. 2. Have a level of inherent seaworthiness including motions tolerable by equipment and persons onboard, controllability and the ability to remain afloat and not capsize.” 3. Be designed to minimize the risk faced by hazards to naval shipping including but not limited to the impact of the environment causing dynamic capsize, broach or damage to crew & equipment . . . .” 4. Be provided with operator guidance, as required in Regulation 7- Operator Guidance, to facilitate safe handling of the ship. The “Verification Methods” section of Regulation 1-General states: 6 Verification that the ship complies with this chapter shall be by the Naval Administration.

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7

The burden of verification falls upon the Naval Administration. All decisions that affect compliance with the requirements of this chapter shall be recorded at all stages from Concept to Disposal and these records be maintained throughout the life of the ship. Thus the Naval Ship Code contains no specific dynamic-stability or capsize criteria, nor does it specify any procedures by which a vessel can be determined to be in compliance with the Code. Neither Regulation 4-Reserve of Stability nor Regulation 7-Provision of Operational Information provides any additional detail on how the requirements are to be met.

4.2 Developments Relating to Standards for Navies Among other topics, the NSSWG has been examining existing naval stability standards against a dynamic stability metric. Hughes & Perrault (2008) presents preliminary results of this study. In the study, a wide variety of static stability metrics (16 in total) for 12 naval vessels have been correlated against a dynamic stability assessment performed using an older version of FREDYN (De Kat & Paulling, 1989; de Kat, 1994; de Kat, et al., 1994). The correlation coefficients for many of these metrics (e.g. GM, GZmax, GZ30°, range, A0°-40°, A0°-range, etc.) were all in the high 0.8s or 0.9s (the majority were in the 0.9s, many were in the 0.99s). There was no static-stability parameter or metric that consistently had the highest correlation coefficient across all 12 ships. This indicates that the current static stability-based naval stability standards are not significantly deficient—at least for ships that fit the current hull-form mold. However, it should be noted that there is no means to determine how much stability margin any of these ships have. The NSSWG is continuing its assessment of existing naval stability standards. It is anticipated that the FREDYN correlation will be repeated with an updated version of the code. However, this will require substantial computational effort and will not be undertaken lightly.

4.3 US Navy Stability Criteria The one navy that has been identified as applying a performance-based dynamicstability criteria is the US Navy. Based on some model tests of tumble-home hull forms in the late 1990s, it was found that the criteria of DDS 079-1 (US Navy, 2003) did not provide the equivalent margin against capsize for tumble-home ship designs as it does for traditional wall-sided and flared designs. Therefore, an intensive effort was instituted in 2000 to develop a dynamic stability-based standard.

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Current US Navy dynamic-stability criteria is a relative criteria, whereby the new vessel design is assessed against an existing naval vessel designed for an equivalent mission—this ensures that a frigate is not judged against an aircraft carrier, or vice versa. The dynamic- stability criteria has five components, which are as follows: a. The annual probability of capsize without wind effects, for the appropriate range of sea states, when multiplied by a margin of 1.10, is less than or equal to that of the equivalent mission benchmark ship. b. For each sea state, the capsize probability shall be determined for the specified range of modal periods. The worst-case capsize probability for each sea state/modal period multiplied by a factor of 1.10 shall be less than or equal to that of the worst capsize risk for the benchmark ship taken in the same sea state over the same range of speeds, headings, and modal periods. c. In any given sea state, it is shown on the capsize- and broaching-risk polar diagrams, that regions of high capsize probability, 60 percent or higher, are not adjacent to regions of high broaching probability, 60 percent or higher. d. Regions of zero capsize probability, as shown on a capsize-risk polar diagram, do not transition to regions of 80 percent or higher capsize probability over a 5-knot range of speed or 15° heading change. e. There shall be no region of 100 percent capsize probability in the defined mission sea states. These criteria are applied over a range of sea states and modal periods. The sea states range from 5 to 8, with sea states 7 and 8 being subdivided into three significant wave heights, each of which has three modal periods. For each significant wave height and modal period, an assessment of capsize probability is performed over a range of speeds, 0 to 30 kt in 5-kt increments, and headings, 0° to 180°, in 15° increments. For each speed-heading combination, twenty-five 30-minute simulations are performed in a different realization of the sea state being investigated, resulting in up to 12-1/2 hours of simulated operation at each speed-heading combination. Component (a) of the US Navy dynamic- stability criteria is intended to determine that the overall capsize risk is acceptable. Component (b) ensures that the capsize risk in all of the sea states is not excessive, by limiting it to being no worse than the worst risk for the benchmark ship. Component (c) is intended to ensure that the ship has a region of the speed-polar plot where it can operate without having to choose between having a high risk of either a capsize or broach, while (d) ensures that there are no locations where the ship transitions too rapidly in speed or heading from safe operation to high risk of capsize, and finally, (e) ensures that there are no absolutely unsafe areas on the capsize-speed polar plot where the ship has a 100 percent probability of capsize. The implementation of the above standard has four components. The first component defines the code to be used for the assessment and the physical model against which the code will be validated. The second component defines the process for setting up the computational-model simulation of a ship for the dynamic stability. The third component defines the code validation process against

A Naval Perspective on Ship Stability 819

model tests, and the criteria for the validation process. Finally, the last component provides the details of the capsize-risk assessment. The motivation for the US Navy to employ a relative capsize-risk assessment approach was the recognition that simulation tools are not absolutely accurate, but it was assumed that the biases of the code would be independent of the details of the hull form. There are two major weaknesses of the US Navy’s relative capsize criteria. The first is that there is no way of knowing the level of safety or margin that the benchmark ship has against capsize. The second relates to the assumption that the computational tools will have a uniform bias against all hull forms—the reality is that it has been found not to be consistently true, although it is not clear why. To supplement the relative capsize-risk assessment methodology just described, Belknap, et al. (2005) investigated a methodology for assessing annual- and lifetime-capsize risk based solely on regular-wave capsize model tests. This methodology relies on mapping the model test-based capsize probabilities onto the joint-probability distribution of a given wave length and period in both a given sea state and in a modal period. This joint-probability distribution is based on the work of Longuet-Higgins (1957). The results of these calculations indicate that the lifetime capsize risk for a typical naval vessel is on the order of a fraction of a percent. Intuitively, this seems to be a reasonable absolute lifetime capsize risk. However, many issues relating to the linear superposition of nonlinear experimental results, using the decomposition of nonlinear seas by means of a joint-probability distribution, need to be resolved regarding this methodology.

5 Conclusions From a naval perspective, three areas critical to the performance of vessels in extreme seas have been discussed: physics of large-amplitude motions; verification, validation and accreditation (VV&A) of tools for these conditions; and performancebased criteria. Hydrodynamic forces, maneuvering in waves, and large-amplitude roll damping have been highlighted as being very important to the physics of large-amplitude motions. The Force Study that examined the impact of large-amplitude responses in calm water, large-wave amplitudes on wave excitation, and prescribed largeamplitude motions in beam and following seas on hull forces was introduced. For maneuvering in waves, the impact of waves of increasing amplitude on the traditional calm-water turning circles was described. Some concerns regarding the prediction of damping in the time domain for large-amplitude roll motions was also introduced. In VV&A, the challenge is performing this function for extreme-seas’ conditions where linear concepts such as response-amplitude operators are not applicable. The use of nonlinear time-series analysis and bifurcation analysis in validation were discussed as possible techniques, and some example were provided.

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Performance criteria have been discussed from a naval perspective. The current NATO criteria are presented, along with the current NSSWG efforts. The US Navy’s dynamic- stability criteria are introduced.

Acknowledgements The author would like to thank Vadim Belenky for his discussions regarding the paper, particularly those relating to the bifurcation analysis; John Telste and Tim Smith for the data plots they provided; Chris Bassler for his production support and editorial comments; and Suzanne Reed for her editorial work on this paper.

References Abkowitz MA (1969) Stability and Control of Marine Vehicles. MIT Press, Cambridge, MA, vi+352 p. AIAA (1998) Guide for the Verification and Validation of Computational Fluid Dynamics Simulations. AIAA G-077-1998 Guide, American Inst of Aeronautics and Astronautics. Alman PR, PV Minnick, R Sheinberg, WL Thomas, III (1999) Dynamic capsize vulnerability: Reducing the hidden operational risk. Trans. SNAME, 107:245–80, Jersey City, NJ. Bassler C, J Carneal, P Atsavapranee (2007) Experimental Investigation of Hydrodynamic Coefficients of a Wave-Piercing Tumblehome Hull Form, Proc. 26th Int’l Conf on Offshore Mech and Arctic Eng OMAE2007, San Diego, CA. Bassler CC, AM Reed (2009) An Analysis of the Bilge Keel Roll Damping Component Model. Proc. 10th Int’l Conf. Stability of Ships and Ocean Veh (STAB 09), St. Petersburg, Russia, 17 p. Bassler CC, MJ Dipper, GE Lang (2009) Formulation of Large-Amplitude Wave Groups in an Experimental Model Basin. Proc 10th Int’l Conf. Stab of Ships and Ocean Veh (STAB 09), St. Petersburg, Russia, 13 p. Beck RF, AM Reed (2001) Modern Computational Methods for Ships in a Seaway. Trans. SNAME, 109:1–51, Jersey City, NJ. Beck RF, AM Reed, EP Rood (1996) Application of modern numerical methods in marine hydrodynamics. Trans. SNAME, 104:519–37, Jersey City, NJ. Belenky VL, JO de Kat, N Umeda (2008a) Toward Performance-Based Criteria for Intact Stability. Marine Tech., 45(2):122–123. Belenky VL, NB Sevastianov (2007) Stability and safety of ship: Risk of Capsizing. Second Edition. SNAME, Jersey City, NJ, xx+435 p. Belenky V, KM Weems, W-M Lin (2008b) Numerical Procedure for Evaluation of Capsizing Probability with Split Time Method Proc. 27th Symp. Naval Hydro, Seoul, Korea, 25 p. Belknap WF, BL Campbell, MJ Dipper, WT Lee (2005) Method for Computing Relative Annual and Lifetime Capsize Risks, Carderock Division, Naval Surface Warfare Center Report NSWCCD-50-TR-2005/085, 26 p. Belknap W, J Telste (2008) Identification of Leading Order Nonlinearities from Numerical Forced Motion Experiment Results. Proc 27th Symp Naval Hydro, Seoul, Korea, 18 p. Bishop RC, WF Belknap, C Turner, B Simon, JH Kim (2005) Parametric Investigation on the Influence of GM, Roll Damping, and Above-Water Form on the Roll Response of Model 5613. Carderock Division, Naval Surface Warfare Center Report NSWCCD-50-TR2005/027, 185 p.

A Naval Perspective on Ship Stability 821 Bollay W (1936) A New Theory for Wings with Small Aspect Ratio. Ph.D. Thesis, Caltech, ii+86 p. Bollay W (1939) A Nonlinear Wing Theory and its Application to Rectangular Wings of Small Aspect Ratio. Zeit. angew. Math. Mech. (ZAMM), 19(1):21–35. Brown CE, WH Michael (1954) Effect of leading edge separation on the lift of a delta wing. J. Aeronautical Science, 21:690–706. Brown CE, WH Michael, Jr. (1955) On Slender Delta Wings with Leading-Edge Separation. Nat Advisory Committee for Aeronautics, Technical Note 8430, i+27 p. de Kat JO (1994) Irregular Waves and their Influence on Extreme Ship Motions, Proc 20th Symp Naval Hydro, Santa Barbara, CA, pp. 48–67. de Kat JO, R Brouwer, KA McTaggart, WL Thomas, III (1994) Intact Ship Survivability in Extreme Waves: New Criteria from a Res and Navy Perspective, Proc 5th Int’l Conf Stab of Ships and Ocean Veh (STAB 94), Florida Inst of Technol, Melbourne, FL, Vol. 1, 26 p. de Kat JO, JR Paulling (1989) The Simulation of Ship Motions and Capsizing in Severe Seas, Trans. SNAME, 97:139–168, Jersey City, NJ. DOD (1998) DoD Modeling and Simulation (M&S) Glossary. DoD 5000.59-M, U. S. Dep of Defense. DOD (2003) DoD Modeling and Simulation (M&S) Verification, Validation, and Accreditation (VV&A). DoD Instruction 5000.61, U. S. Dep of Defense. DOD (2007) DoD Modeling and Simulation (M&S) Management. DoD Directive 5000.59, U. S. Dep of Defense. Dommermuth DG, TT O’Shea, DC Wyatt, T Ratcliffe, GD Weymouth, KL Hendrikson, DK P Yue, M Sussman, P Adams, M Valenciano, (2007) An Application of Cartesian-Grid and Vol of-Fluid Methods to Numerical Ship Hydrodyn. Proc. 9th Int’l Conf Numerical Ship Hydro, Ann Arbor, MI. Dommermuth DG, TT O’Shea, DC Wyatt, M Sussman, GD Weymouth, DKP Yue, P Adams, R Hand, (2006) The Numerical Solution of Ship Waves Using Cartesian-Grid and Vol of-Fluid Methods. Proc 26th Symp Naval Hydro, Rome, Italy. Etebari A, P Atsavapranee, C Bassler, J Carneal (2008) Experimental Analysis of Rudder Contribution to Roll Damping, Proc 27th Int’l Conf on Offshore Mec and Arctic Eng. OMAE 2008, Estoril, Portugal. Francescutto A, G Contento, R Penna (1994) Experimental Evidence of Strong Nonlinear Effects in the Rolling Motion of a Destroyer in Beam Seas. Proc 5th Int’l Conf of Stabof Ships and Ocean Veh (STAB 94), Florida Inst of Technol Melbourne, FL, Vol 1, 13 p. Fullerton AM, TC Fu, AM Reed (2008) The Moments on a Tumblehome Hull Form Undergoing Forced Roll. Proc 27th Symp Naval Hydro, 12 p. Gersten K (1963) A Nonlinear Lifting-Surface Theory Especially for Low-Aspect-Ratio Wings. AIAA J 1:924–925. Gersten K (1961) Nichtlineare Tragflachentheorie insbesondere flir Tragflügel mit kleinem Seitenverhaltnis. Ingenieur-Archiv, 30:431–452. Grant DJ, A Etebari, P Atsavapranee, (2007) Experimental Investigation of Roll and Heave Excitation and Damping in Beam Wave Fields, Proc 26th Int’l Conf on Offshore Mech and Arctic Eng, OMAE2007, San Diego, CA. Hayden DD, RC Bishop, JT Park, SM Laverty (2006) Model 5514 Capsize Experiments Representing the Pre-Contract DDG 51 Hull Form at End of Service Life Conditions. Carderock Division, Naval Surface Warfare Center Report NSWCCD-50-TR-2006/020, 82 p. Himeno Y, 1981, Prediction of Ship Roll Damping-State of the Art, Dept. of Naval Architec and Marine Eng, Univ. of Michigan, Report 239. Howe MS (1996) Emendation of the Brown & Michael equation, with application to sound generation by vortex motion near a half-plane. J. Fluid Mechanics, 329:89101. Hughes T, D Perrault (2008) Critical Review of Naval Stability Standards. Defence R&D Canada Report DRDC Atlantic ECR 2008-174, 38 p. Ikeda Y, Y Himeno, and N Tanaka, 1978, A Prediction Method for Ship Roll Damping, Report of the Dep of Naval Archit, Univ of Osaka Prefecture, No. 00405.

822 A.M. Reed ITTC (2008) The Specialist Committee on Stability in Waves: Final Report and Recommendations to the 25th ITTC. Proc 25th ITTC, Fukuoko, Japan, 36 p. Kan M (1990) Surging of large-amplitude and surf-riding of ships in following seas. Naval Architec and Ocean Eng, Soc of Naval Archit of Japan, 28:49–62. Kantz H, T Schreiber (2004) Nonlinear time series analysis. Cambridge Univ Press, Cambridge, UK, xvi+369 Kring DC, WM Milewski, NE Fine (2004) Validation of a NURBS-Based BEM for Multihull Ship Seakeeping. Proc. 25th Symp. Naval Hydro., St. John’s, Newfoundland and Labrador, Canada. Lighthill MJ (1960) A Note on the Swimming of Slender Fish. J. Fluid Mech., 9:305–17 Liut DA, KW Weems, W-M Lin (2002) Nonlinear Green Water Effects On Ship Motions and Structural Loads. Proc 24th Symp. on Naval Hydro, Fukuoka, Japan. Longuet-Higgins, MS (1957) The statistical analysis of a random, moving surface. Phil. Trans. Roy. Soc., A 249:32187. McCue LS, WR Story, AM Reed (2008) Nonlinear Dynamics Applied to the Validation of Computational Methods. Proc 27th Symp Naval Hydro, Seoul, South Korea, 10 p. McTaggart KA, JO de Kat (2000) Capsize risk of intact frigates in irregular seas. Trans. SNAME, 108:147–77. McTaggart KA (2000) Ship capsize risk in a seaway using fitted distributions to roll maxima. J Offshore Mech and Arctic Eng, 122:141–146. Miller RW, CC Bassler, P Atsavapranee, JJ Gorski (2008) Viscous Roll Predictions for Naval Surface Ships Appended with Bilge Keels Using URANS. Proc 27th Symp Naval Hydro, Seoul, South Korea. NATO (2007a) Buoyancy, Stability and Controllability. Chapter III of Naval Ship Code, NATO Naval Armaments Group, Maritime Capability Group 6, Specialist Team on Naval Ship Safety and Classification, Allied Naval Eng Pub ANEP–77, vii+121 p. NATO (2007b) Guidance on NSC Chapter III Buoyancy and Stability, Part B: Application. Chapter 3, Guide to the Naval Ship Code, NATO Naval Armaments Group, Maritime Capability Group 6, Specialist Team on Naval Ship Safety and Classification, 91 p. Newman JN (1975) Swimming of slender fish in a non-uniform velocity field. J. Australian Mathematical Soc, Series B, Applied Mathematics, 19(1):95–111. Newman, JN, TY Wu (1974) Hydromechanical aspects of fish swimming. in Swimming and Flying in Nature, TY Wu, CJ Brokaw, CJ Brennan, Editors; Plenum Press, Vol. 2, pp. 615–634. Rahola J (1939) The judging of the stability of ships and the determination of the minimum amount of stability especially considering the vessel navigating Finnish waters. PhD Thesis, Technical Univ of Finland, Helsinki, viii+232 Reed AM (2008) Discussion of: Belenky, VL, JO de Kat, N Umeda (2008a). Marine Tech., 45(2):122–123. Rudgley, G, ECA. ter Bekke, P Boxall, R Humphrey (2005) Development of a NATO ‘Naval Ship Code’. RINA Conf on Safety Regulations and Naval Class II, RINA, London, United Kingdom, 8 p. Sarchin TH, LL Goldberg (1962) Stability and buoyancy criteria for U. S. Naval surface ships. Trans. SNAME, 72:418–58. Spyrou KJ (1996) Dynamic Instability in Quartering Seas: The Behavior of a Ship During Broaching. J. Ship Research, 40(1):4659. Spyrou KJ (1997) Dynamic Instability in Quartering Seas—Part III: Nonlinear Effectson Periodic Motions. J Ship Res, 41(3):210–223. Spyrou, KJ, KM Weems, V Belenky (2009) Patterns of Surf-Riding and Broaching-to Captured by Advanced Hydrodynamic Modelling. Proc 10th Int’l Conf Stab of Ships and Ocean Veh (STAB 09), St. Petersburg, Russia, 15 p. Telste JG, WF Belknap (2008) Potential Flow Forces and Moments from Selected Ship Flow Codes in a Set of Numerical Experiments. Carderock Division, Naval Surface Warfare Center Report NSWCCD-50-TR–2008/040, 15,240 p.

A Naval Perspective on Ship Stability 823 Themelis N, KJ Spyrou (2007) Probabilistic Assessment of Ship Stability, Trans. SNAME, 115:181–206. Themelis N, KJ Spyrou (2008) Probabilistic Assessment of Ship Stability Based on the Concept of Critical Wave Groups, Proc 10th Int’l Ship Stab Workshop, Daejeon, Korea. US Navy (2003) Stability and buoyancy of U.S. Naval surface ships. Design Data Sheet DDS 079-1, Version 1.21, 81 p. van Zwol, JA (2004) Design aspects of submerged vanes. M. Sc. Thesis, Delft Univ of Technol, Delft, The Netherlands, 126 p. Wheeler JD (1969) Method of calculating forces produced by irregular waves. Proc Offshore Technol Conf (OTC 1006), Vol. 1, pp. 71–82, Houston, TX.

13 Accident Investigations

New Insights on the Sinking of MV Estonia Andrzej Jasionowski* Dracos Vassalos** * Safety at Sea Ltd (SaS), Glasgow, UK; ** The Ship Stability Research Centre (SSRC), Dept. of Naval Architecture and Marine Engineering, The Universities of Glasgow and Strathclyde, Glasgow, UK,

Abstract The paper presents the latest results from the ongoing research study on the sinking of Estonia, aimed at establishing a verifiable loss scenario by using stateof-the-art numerical and experimental tools to address all pertinent issues: flooding mechanism, coupled flooding-ship-sea dynamics, deterioration of watertight integrity and the abandonment process. The strategy in approaching this problem and the new insights derived from the adopted process are presented leading to early conclusions on the likely loss scenario.

1 Introduction The foundering of MV Estonia on 28 September 1994, with reported loss of 852 lives, is one of the biggest peacetime catastrophes of Western Europe. However, it seems that efforts expended on explaining the circumstances of this loss have not been commensurate with the magnitude of the disaster. No comprehensible description of the chain of events leading to the loss of the vessel has been derived to date. This paper aims to briefly summarize some findings during the studies undertaken in contribution to these efforts and carried out by the authors within the partnership of the SSPA Consortium1. The paper presents a judiciously chosen set of key information, which has been deemed pertinent to the argument presented. It is stressed here that it is only a brief summary of an on-going investigation, presented here for the purpose of exchange of information and public discussion.

1 www.safety-at-sea.co.uk/mvestonia M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_46, © Springer Science+Business Media B.V. 2011

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1.1 Status of the Evidence Available 1.1.1 Heeling Most survivors state that the vessel heeled substantially, see Figure 1. Many persons onboard have not managed to escape, and hence, quite likely, considerable heeling developed rapidly, within a few minutes from an initiating event and thus prevented persons to abandon the ship. List Development 120

100

List [deg]

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Time Reference Testimonies

WP2.1 Evaluation

JAIC

Fig. 1 The process of heeling as reported by witnesses. Established by Rutgersson et al, (2006).

The explanation of the cause of the heeling offered to date is the water on deck, reaching the car deck through open bow doors, see Figure 2.

Fig. 2 The primary JAIC hypothesis of the loss of the vessel is the detachment of the bow visor, and subsequent ingress of floodwater onto the car deck through a partially opened ramp.

New Insights on the Sinking of MV Estonia 829

1.1.2 Capsizing A visual picture taken by one of the survivors clearly demonstrates the ship at over 90 deg angle of heel. Therefore, the vessel capsized, that is she heeled beyond some 40deg, at which attitude the restoring capacity of the ship’s watertight enclosure reaches its maximum, see Figure 3. Theoretically, once the vessel heels to this angle, capsizing becomes imminent.

Fig. 3 GZ curve for MV Estonia, when buoyancy is considered to either, top of the car deck (D4) or tip of the funnel (D7). The latter, obviously hypothetical, is shown to visualise the physics of the real capsizing process ignored in typical routine ship stability calculations. Loss of the buoyancy above D4 must take some time.

However, in practice, capsizing must take a finite amount of time, driven primarily by the process of flooding of all the spaces of the superstructure, Jasionowski and Vassalos (2002). 1.1.3 Sinking The vessel, see Figure 4, rests at the bottom of the Baltic Sea, and hence it is obvious that most of the buoyancy of the ship has been lost.

Fig. 4 Side profile of MV Estonia, centre plane view.

As is shown in Figure 5 below, for the MV Estonia to sink, flooding must amount to at least 10,792 m3 in the spaces below the car deck, in addition to complete flooding of all other spaces on the ship. In case of any air pockets remaining in

830 A. Jasionowski and D. Vassalos

any space from the car deck upwards, the flooding below would have to be higher, proportional to the volume of these air pockets. To date, no plausible explanation of the process, by means of which this flooding took place during the ship loss, has been offered.

Fig. 5 The whole “body” of the MV Estonia would displace 78,006 m3 of water if fully submerged. The volume that could flood internal spaces from the car deck upwards to the tip of the funnel is 55,284 m3. The ship’s weight at the time of her loss would displace 11,930 m3. Therefore, the minimum amount of floodwater required to ingress below the car deck spaces for the ship to sink is: 10,792 m3 = 78,006 m3 – 55,284 m3 – 11,930 m3, or 64% of all the floodable space below the car deck.

1.1.4 Loss Scenario Many different and more or less complete scenarios that offer to explain an appropriate sequence of occurrences of the above three elements of the loss have been proposed, as reviewed in Jasionowski and Vassalos (2002). However, none has put forward a consistent sequence of events which could be considered plausible. In particular, no clear explanation exists for: (a) The perceived prolonged capsizing process, that is heeling beyond 40deg until capsize/sinking. It seems from Figure 1 that some 20-30 minutes are assumed for this process to take place. (b) The sinking process, that is, an explanation of how and when 10,792 m3 of water reached below the car deck for the vessel to disappear from the radars at 01:52. The following is a hypothesis addressing these gaps.

New Insights on the Sinking of MV Estonia 831

1.2 Study on Possible Loss Mechanisms 1.2.1 Heeling It is conceivable that a considerable heeling angle could be induced by flooding spaces below the car deck. As is shown for a hypothetical flooding case into the forward spaces, such heeling angle can reach some 20deg, see Figure 6. A study has been undertaken to investigate the possibility of larger angles of heel occurring due to transient flooding effects. The study has comprised a series of numerical simulations of the vessel response when subjected to damages below the car deck occurring randomly according to historical data on collisions, see Figure 7. 3

GZ curves when Decks 4,5,6 (up to 22.2m) considered not floodable

Free surface effects due to flooding of aft mashinery spaces on Deck0 and Deck1

2.5 2

4000t

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40

50

60

70

80

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1000t GZ curves when Decks 4,5,6 flood

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Fig. 6 Free surface effects due to flooding of the forward spaces below the car deck.

Fig. 7 Sample MC simulations set-up, distribution of damage location, length and Hs. Damages assumed below the car deck.

832 A. Jasionowski and D. Vassalos

A model of MV Estonia, Figure 8, has been subjected to these damages, Figure 9, and a statistic derived of the maximum heel angles recorded during the initial stages of flooding simulations, as shown in Figure 10.

Fig. 8 Digital model of MV Estonia, aft and front views, PROTEUS3.

Fig. 9 A sample snapshot of the simulation of flooding below the car deck. 1

0.12

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Fig. 10 Probability distribution for heel angles recorded during the first 20 minutes from hull breach. Angles in excess of some 20deg result from up-flooding the car deck.

New Insights on the Sinking of MV Estonia 833

It is evident from this study, as performed to date, that angles of heel in excess of some 20deg could not result from flooding of the spaces below the car deck alone. The car deck (CD) must have also flooded. It could be argued that such flooding on the CD could indeed result from firstly flooding of the spaces below through breaching the hull below a height of 7.65m from the base plane, and then the car deck through either up-flooding or also a breach of the hull somewhere between 7.65m and 13.4 m height. Availability of the information on the likelihood of occurrence of different collision damages recorded historically as well as the likelihood of the expected time for capsizing for each of these damages allows for identifying which of the damages would be the most likely to conform to assumptions such as the period of about 30 minutes time to capsize. Use can be made of the Bayesian theory, which states that the conditional probability that a specific space d became flooded, given that a damage occurred in an “ordinary” collision and capsizing occurred subsequently within t = 30min time, can be expressed by the following equation ( 1 ):

p D T d t  

p D   p E D  pT D&E E

 p D

Where

D

(1)

 p E D  pT D&E

E

pE D is the probability mass function that a specific environmental

condition occurred during a collision event;

pT D & E is the conditional probability

that capsize occurs within specific time and for given damage and environmental condition; and pD is the prior probability that specific the damage extent d occurred. The result is shown in Figure 11.

Fig. 11 Distribution of conditional probability

pD T

that damage D = d occurred (given that a

capsize event occurred within time T = t), see the dimensionless color scale on the LHS.

It would appear that the most likely damage, given the above assumptions, would be a 2-compartment flooding in the aft, where the machinery is located.

834 A. Jasionowski and D. Vassalos

A sample simulation of one of the damages possible at this location is shown in Figure 12. It seems that at least qualitatively, the mode of the loss conforms to some rather established facts, such as sinking with the stern first.

Fig. 12 Capsize and sinking of MV Estonia after flooding through damage in the way of engine room and car deck.

Hence, a hypothesis that the initial heeling resulted from flooding of spaces below the car deck, in conjunction with an event of the car deck becoming subjected to water inflow (up-flooding, bow visor loss, etc) carries substantial credibility. However, according to statements of the three engine room crew, who were present in some 2/3 of the length of the vessel at the very onset of the accident, see Figure 13, no substantial flooding was reported. In total, 22 people from spaces below survived, and none reported any substantial flooding. Hence, any scenario initiating with a breach below the car deck is highly unlikely.

Fig. 13 Sketch by the crew (red) and passenger (blue) survivors marking their presence at the initial phase of the vessel foundering. None reports seeing any substantial of water in these spaces.

New Insights on the Sinking of MV Estonia 835

Therefore, it is concluded that the heeling of the ship was caused primarily by water flooding the car deck as the initiating event. Whether the water entered through bow doors or through any other means is left out of discussion at present. It is worth noting that through a reverse engineering argument, it can be established that an amount of some 2,500 m3 of water entered the car deck, leading the vessel to 40deg + heel, which accumulated within some 30 minutes, between 01:00 and 01:30 (last radio communication). It is hypothesized here that at this time the vessel entered the capsize phase, as discussed later on. 1.2.2 Capsizing The capsizing process is one of the more puzzling elements of the loss mechanisms. The interpretation of the survivors’ statements leads to the perception that the capsizing process (heeling beyond 40deg) has taken “considerable” time. From Figure 3 it can be inferred that for such prolonged capsizing to materialize, the process of filling the superstructure spaces by water must have delayed the capsizing process, and hence that it took rather longer time than intuitively expected. Therefore, considerable effort has been spent on verifying numerical and indeed common sense assumptions on how fast these spaces could flood.

Fig. 14 Comparison of the predictions of the process of flooding across Deck 4 in idealized conditions, performed by PROTEUS3 and FLUENT models, Strasser.

Figure 14 shows that a “decent” degree of detail in representing the internal geometry of the upper spaces is sufficient for representing the flooding process and thus for accurate modelling of the time it took the vessel to capsize. According to results from these simulations the capsizing has never taken more than 2-3 minutes with all the windows assumed broken.

836 A. Jasionowski and D. Vassalos

Although puzzling initially, it becomes more plausible that in fact capsize happened relatively fast. Compensating for some simplifications in the model, it is suggested that according to predictions it took some 3-4 minutes. This would imply that MV Estonia has de-facto floated up-side down. Considering the conditions prevailing at the time, it may in fact be argued that the survivors testimonies support this hypothesis. Namely, 30 survivors claim that MV Estonia sank by stern. However there are 9 survivors who saw MV Estonia sinking by the bow, since they saw the stern, e.g. propellers. It is suggested here that there is no contradiction in these statements and that all of them saw the vessel in an upside condition.

Fig. 15 MV Estonia in an up-side attitude; 30 survivors claim that MV Estonia sank by stern, and 9 survivors state that the vessel sank by bow, with one statement about visible propellers. Could all these survivors have seen MV Estonia floating bottom-up?

1.2.3 Sinking If the ship did float up-side down, then the centre casing becomes submerged some 2 to 8m below the free surface and hence is subject to considerable pressure. Since the design of the centre casing was only as a fire-resistant structure and was fitted with many non-Watertight doors, see Figure 18, it is highly likely that it would break and let water into the spaces “below” the car deck. In fact, at a water head pressure of 5m and an opening of 2m2 (1 door), the amount that could flood into these spaces in 15-18 minutes would be sufficient for the vessel to sink, see Figure 5.

New Insights on the Sinking of MV Estonia 837

Fig. 16 A likely loss sequence of the loss of MV Estonia. Flooding of the spaces below the car deck commenced once MV Estonia capsized. The multitude of doors in the centre casing collapsed due to excessive water head pressure of 2 to 8 m. Some 2 m2 of opening in the centre casing would be sufficient to allow for 10,792 m3 of water to enter the spaces below the car deck within 15-20 minutes.

1.2.4 Loss Scenario Therefore, a complete sinking sequence can now be proposed. As is shown in Figure 16, the sinking sequence can be broken into three phases. (1) Firstly, water accumulation on the car deck took place. It must have started relatively rapidly with the vessel heeling to high angles and thus preventing persons onboard from abandonment. At this stage of this investigation no firm suggestions on details of the initial water inflow are proposed, though from Figure 1, or Figure 17 repeated below, it would appear that the initial large heel of some 30deg developed within 5 minutes. On average, the water inflow between 01:00 and about 01:30 must have been in the order of some 83 m3/min. (2) Secondly, once the amount of some 2,500m3 accumulated on the car deck the vessel capsized, that is, it turned up-side down, within some 3-4 minutes. It is

838 A. Jasionowski and D. Vassalos

List [deg]

180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 01:00

Phase 2

suggested here that 3-4 minutes would be sufficient for many to rememebr the vessel to have been at 90deg+ attitude “for some time”. (3) Thirdly, the sinking would commence as described above, through the submerged centre casing. The hypothetical (yet to be verified) time sequence is shown in Figure 17 below.

Phase 1

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Fig. 17 Could it be that heeling, capsizing and sinking followed such trend? Some 83 m3/min on average have flooded into the car deck between 01:00 and 01:30. Capsizing would take place in some 3-4 minutes. As the vessel turned turtle, all spaces from a height of 7.65 m upwards fill up with water at a rate of some 15,000 m3/min through many broken windows. Finally, at 01:34 water starts flooding the spaces “below” the car deck (now up) at some 600 m3/min through the centre casing. In total, over 72,106 m3 of water enters MV Estonia between 01:00 and 01:52.

Fig. 18 The centre casing was fitted with many fire doors. In an up-side down attitude the centre casing is some 5m below the sea surface, hence it would buckle under pressure and let water reach spaces “below” the car deck.

New Insights on the Sinking of MV Estonia 839

2 Conclusions This article is a summary of the findings of the investigation carried out within the SSPA partnership to date. The preliminary conclusions are that:  MV Estonia heeled because of an inflow of some 2,500 m3 of water on the car deck between 01:00 and 01:30. The cause of the inflow is not addressed at present. Any substantial flooding below the car deck is unlikely to have been the initiating event because (a) many survivors come from the lower deck spaces forward and (b) the three engine room crew report no substantial amount of water in any of the spaces aft at the onset of the foundering.  Because of the water on deck, MV Estonia capsized within a course of some 34 minutes, during which all the spaces from 7.65 m upwards filled up with 55,284 m3 of floodwater. It would seem possible that 39 survivors report MV Estonia floating up side down.  The centre casing on the car deck becomes submerged to some 5 m water head pressure on average. Some doors collapsed and allowed the spaces below the car deck to fill up with water. An opening of 2 m2 is sufficient for the requisite 10,792 m3 to enter these spaces between 01:34-01:52, most likely with the aft spaces flooding faster.  MV Estonia sinks stern first. This is offered as a preliminary explanation of the mechanism of the loss of MV Estonia. The investigation is ongoing

Acknowledgements This research has been sponsored by the Swedish Agency for Research and Development VINNOVA, whose support is hereby gratefully acknowledged.

References Karpinen T (1999) “More thoughts on the Estonia accident”, The Naval Architect, July/Aug. Lawson D (2005) ”Engineering disasters – Lessons to be learned”, John Wiley, ISBN 1860584594. Rutgersson O, Schreuder M and Bergholtz J (2006) “Research study of Sinking Sequence of M/V Estonia, WP2.1 – Review of Evidence and Forming of Loss Hypothesis“, Dep of Shipping and Marine Technol, Chalmers, 10 Oct , available at safety-at-sea.co.uk/mvestonia. Jasionowski Andrzej and Vassalos Dracos (2002) “Shedding Light Into The Loss Of MV Estonia”, RINA conf “Learning From Marine Incidents II”, London, UK, 13-14 March. Strasser Clemens, PhD student

Experimental Investigation on Capsizing and Sinking of a Cruising Yacht in Wind Naoya Umeda*, Masatoshi Hori*, Kazunori Aoki*, Toru Katayama** and Yoshiho Ikeda** * Osaka University , 2-1 , Yamadaoka , Suita , Osaka , 565-0871 , JAPAN ** Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai , Osaka 599-8531, JAPAN

Abstract This paper describes the main finding of the investigation of the sinking accident of a 6.45 metre-long cruising yacht in September 2003 in Japan. Responding to the request from Japan’s Marine Accident Inquiry Agency, the authors executed a model experiment in beam wind for identifying the time-to-sink as well as stability calculations with and without water inside the yacht. The results indicate it could capsize when the wind velocity exceeds a threshold. Because of an opened hatch on the deck, water initially enters into the cabin. After capsizing, the water ingress process stops. However, if wind is strong enough to incline the capsized yacht, the yacht starts to return to upright condition but this transition stage provides an opportunity to further flooding. By systematically changing wind velocity in the experiment, the time to sink was recorded. In conclusion, the critical wind velocities for capsizing and sinking were estimated, and reasonably well explain the reason of this accident.

1 Introduction On 15 September 2003, a 6.45 metre-long cruising yacht capsized and then sunk in Lake Biwa, which is the largest lake in Japan, and seven of its twelve crew drowned. This yacht having a cabin and a fin keel has the stability range of more than 110 degrees and no significant waves existed in the period of the accident. Thus the reason why the yacht sank seemed to be puzzling. To identify the reason of this accident, the Marine Accident Inquiry Agency requested the first author to investigate this accident. It is well established that a cruising yacht could capsize in heavy weather, often due to breaking waves. Several model experiments of capsizing and re-righting of cruising yachts have been reported. (e.g. Hirayama et al. 1994, 1995, Nimura et al. 1994, Deakin 2000) These were executed with breaking waves generated in model M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_47, © Springer Science+Business Media B.V. 2011

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842 N. Umeda et al.

basins. In this accident, however, no significant waves existed on the lake. Thus, it was impossible for us to simply apply the established knowledge from the existing experiments. Actual accident reports often indicate that capsize of sailing yacht does not directly result in sinking. This is because air can be trapped inside upside-down hull. Then there is a chance for the yacht to re-right with the attack of succeeding large waves. (Renilson et al., 2000) Why the capsizing of the yacht in the accident leaded to sinking was also an unsolved problem for us. To provide possible solutions of these problems, we calculated restoring arm curves of the yacht and executed an experiment with a 1/5.7857 scaled model of the yacht, which had the cabin, mast, sail, keel and so on, in a towing tank with a blower. As a result, we realised the process from capsizing to sinking in model scale. By using these calculated and experimental results together with the witness reports, we presumed a possible scenario where the yacht capsizes and then sinks.

2 Accident The yacht that sank was built as one of a class produced for pleasure cruising and/or racing in 1980’s. It was rigged as sloop, and has a cabin, fin keel, rudder and an outboard motor. The hull was made of GRP. The cabin had two openings; a companionway and a hole for ventilator on deck at the bow. The position of this bow hole was slightly shifted to the port side and its area was 0.0064m2. Here the ventilator itself was not equipped at the accident. The principal dimensions and body plan are shown in Table 1 and Figure 1, respectively. It was allowed to carry ten adults by the Administration. Here two children can be regarded as equivalent to one adult by law. On 15 September 2003, the yacht departed from a harbour situated in the western shore in Lake Biwa by using the outboard motor at 16:30. Seven adults and five children were onboard. This means the number of crew can be below the maximum allowed one. The companionway and the bow hole were open. None of the crew except for two children wore life-vests. Table 1. Principal particulars of the yacht at the light ship condition.

Length over all Maximum breadth Maximum depth Draught at midship Ship displacement Initial trim angle by stern Metacentric height Area of main sail

6.45 m 2.48 m 1.17 m 1.30 m 1.045 ton 2.1 degrees 0.935 m 12.56 m2

Experimental Investigation on Capsizing and Sinking 843

Fig. 1 General arrangement of the yacht.

At 16:33, the skipper stopped the motor and the yacht started to sail with a main sail only. Then the yacht sailed closed-hauled on port tack with a leeward heel of about 15 degrees and estimated speed of about 1.3 knot. Here the main sheet was fixed by using the cam cleat. Four adults and four children were on the port side, one adult in the centre and two adults and one child on the starboard side. All of them were on deck. At 16:49, the skipper ordered to tack about and helmed up. Then the yacht was on starboard tack but the yawing motion continued beyond the expected closehauled course on the new tack. At the same time the roll towards the new lee side violently increases. It was probably because the skipper failed to helm amidships and to release the main sheet from the cleat. Because of significant roll towards port side, one adult and one child situated on the starboard-side deck dropped onto the main sail and then all other crew dropped into the water and the yacht completely capsized. Shortly after the yacht started to re-right but its stern was under water due to flooding. At 16:50, the yacht sunk from the stern. It was a fine day. According to the measured data from a Shiga University’s observation buoy situated with the distance of 4.5 km from the accident, the mean wind speed ranges from 7.4 m/s to 8.5 m/s and the maximum was from 11.5 m/s to 11.8 m/s. Because of the very limited fetch and duration, waves were very short. Three adults and two children survived but the others including the skipper were drowned or still missing.

844 N. Umeda et al.

3 Hydrostatic Calculation Hydrostatic calculation was carried out for the yacht with and without flooded water inside the cabin. Weights and positions of the crew and equipment were estimated with available data from the Shiga Prefecture Police. Weight and trim of the light ship were obtained from the ship yard data. GZ (light ship) GZ (C-cond.)

GZ (A-cond.) GZ (D-cond.)

GZ (B-cond.) GZ (E-cond.)

0.6

GZ (m)

0.4 0.2 0 -0.2 0

45

90

135

180

-0.4 -0.6 Heel angle (degrees)

Fig. 2 Restoring arm curves without flooded water.

As shown in Figure 2, in the light ship condition, the angle of vanishing stability is 116 degrees and the maximum restoring arm is 0.45 m. Recent research of the Japan Craft Inspection Organization (Takaishi, 2000) recommends the angle of vanishing stability to be 100 degrees or over. Thus, the yacht seems to have sufficient stability. This is because the weight of the ballast in the fin keel is about 31.6 % of the light ship displacement. In the loading condition at the accident (the condition A in the figure), however, the restoring arm is significantly reduced because of weight of the crew (593 kg) on the deck. The ship has a port-side heel of 12 degrees due to unsymmetrical loading, the angle of vanishing stability of 81 degrees and the maximum righting arm is 0.096 m. Since the IMO Intact stability code requires the maximum righting arm of 0.2 m, static stability in this loading condition can be regarded as insufficient. This is because the crew weight on deck corresponds to 60 % of the light ship displacement. The condition B indicates the case all crew are shifted to lee side with 300 mm, which results in the initial heel of 36 degrees. The condition C is the case one adult and one child moved to the centre of main sail. Here restoring arm is always negative. The condition E will result in the case all other crew dropping into the water from the condition C. Here the static stability increases. The condition D also results in the case all the crew dropping into the water, which is almost identical to the light ship condition. Because of opening of the cabin, a certain amount of flooded water could exist once capsizing occurs. Figure 3 indicates restoring arm curves with a certain amount of flooded water inside the cabin but here we ignore further water ingress and egress through the opening. Figure 4 shows change of trim angle due to heel.

Experimental Investigation on Capsizing and Sinking 845 with flooded water 0.5 0.0m3 1.0m3 2.0m3 3.0m3 4.0m3

0.4

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0.3 0.2 0.1 0 -0.1 0

45

90

135

180

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Fig. 3 Restoring arm curves with flooded water. with flooded water

Trim angle (degrees)

60 50

0.0m3

40

1.0m3 2.0m3

30

3.0m3

20

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10 0 -10 0

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Fig. 4 Trim angles with flooded water. Here the positive trim means the trim by the stern.

In case of the flooded water of 0 and 1 m3, slopes of the righting arm at the capsized condition (180 degrees) are very steep. Thus, the capsized yacht is very stable. In case of 2m3, however, the slope becomes zero. This means the capsized yacht could start to re-right if flooding exceeds this amount of water. This fact was confirmed for a similar yacht by a full-scale experiment and calculation of Nomoto (2000). If water flooding progresses further, the trim by the stern increases with heel and the righting arm curves cannot be properly calculated.

4 Model Experiment For investigating the dynamic process from capsizing to sinking, a model experiment was conducted at the towing tank of Osaka Prefecture University, which can realise wind on the water surface with a blower. The tank is 70 m in long, 3 m in wide and 1.5 m in deep.

846 N. Umeda et al.

The 1/5.7857 scaled model of the yacht was used in the experiment because the capsized model with the mast should not touch the bottom of the tank. As a result, the model length was 1.1 m. Because of flow separation due to the fin keel and sail, the scale effect due to viscosity can be regarded as negligibly small. And the scale effect of air compression is estimated as about 10 % or less in the draught of the capsized yacht. (Nomoto, 2000) The model was equipped with a mast, a boom, a set of spreaders, a main sail, a main sheet, a main sheet traveller, a kicking strap, a main halyard, a fore stay, a back stay, an upper side stay, a lower side stay, a rudder, a tiller, a fin keel, an outboard motor and so on. The internal layout of the cabin including bulkheads and longitudinal frames and thickness of the hull were also geometrically scaled. By adjusting the weight inside the fin keel, the vertical and longitudinal centres of gravity of the light ship were set to correspond to the full scale ship and were confirmed by an inclining test. The roll test in calm water indicated that the natural roll period at upright condition was 3.6 seconds in full scale and no periodic roll was observed at capsized condition because of large roll damping due to the sail. If the skipper failed to helm amidships and to release the main sheet, the yacht suddenly suffered beam wind loading after the tacking about. Based on this assumption, the model was placed to beam wind conditions in the tank with a constant wind velocity. The model was initially held upright but at the start of the experiment the restraint was released. As a result, sudden beam wind loading was simulated. In the experiment, the weight corresponding to one adult and one child was initially attached to the mast at the height of the centre of the main sail, and the weight was designed to mechanically leave the mast at the heel of 120 degrees. Once the restraint of the yacht model was suddenly removed, the model rolled significantly beyond the static equilibrium of heel and stopped at the equilibrium angle of energy. The static equilibrium can be estimated with the comparison between wind heeling moment and restoring moment as shown in Figure 5. Here the wind heeling moment, MW, is assumed to be calculated with the following formulae.

MW  12  AV 2 AS0 zs 0 CD cos2   12  A V 2 AH 0 zH 0 CD cos2 

(1)

 12  A V 2 AH 90 zH 90CD sin 2  for

0   /2

M W  12  AV 2 AH 180 z H 180C D cos 2   12  AV 2 AH 90 z H 90CD sin 2 

(2)

Experimental Investigation on Capsizing and Sinking 847

for

 /2  

where : heel angle, A: air density, V: wind speed, CD: drag coefficient, AS: lateral projected area of sail, AH: lateral projected area of hull, zS: height of centre of lateral projected area of sail above the water from the centre of underwater hull, zH: height of centre of lateral projected area of hull above water from the centre of underwater hull and sail. And the suffices such as “90” indicate the heel angle. The value of CD is assumed be 1.11, based on the IMO weather criterion (Japan, 2005). The restoring moment between 0 and 120 degrees in heel corresponds to that of the condition E and that beyond 120 degrees does to that of the condition D because of the loss of weight. As shown in Figure 5, static equilibria exist even in the case of the wind speed of 11 m/s.

Moment (N-m)

WGZ (exp) wind heel moment (V=9m/s) wind heel moment (V=11m/s) 8000 6000 4000 2000 0 -2000 0 -4000 -6000

45

wind heel moment (V=8m/s) wind heel moment (V=10m/s)

90

135

180

Heel angle (degrees)

Fig. 5 Comparison of restoring moment and wind heeling moment in case of model experiment. Restoring Energy (exp) heeling energy (V=9 m/s) heeling energy (V=11m/s)

heeling energy (V=8 m/s) heeling energy (V=10m/s)

energy (N-m)

8000 6000 4000 2000 0 -2000

0

45

90

135

180

Heel angle (degrees)

Fig. 6 Comparison of restoring energy and wind heeling energy in case of model experiment.

848 N. Umeda et al.

As shown in Figure 6, the equilibrium of energy does not exist even with the wind speed of 8m/s, In the experiment, however, the yacht model capsized only with 10 m/s or over in wind speed. (See Figure 7) It can be presumed that energy dispassion due to roll damping up to 90 degrees is about 1kN-m. Once capsizing occurs, the roll motion immediately stops because of large roll damping due to the main sail around the capsized condition. However, the static heel angle here is not 180 degrees because of wind heeling moment acting on the fin keel and capsized canoe body as shown in Figure 8. If this additional heel angle from 180 degrees is large enough, part of companionway and (/or) the ventilator hole can emerge and then water ingress and air egress can start. As indicated in the restoring arm curves with flooded water inside the hull, if flooded water amount exceeds 2 m3, capsized condition of the model becomes unstable and then the model starts to re-right.

Fig. 7 Photo of the yacht model just before capsizing under the wind velocity of 11 m/s.

Fig. 8 Photo of the capsized yacht model under the wind velocity of 11 m/s.

Experimental Investigation on Capsizing and Sinking 849

If the model starts to re-right, the trim by the stern also increases. This agrees with the results of righting arm calculation with flooded water. When the roll angle reaches about 270 degrees, the stern including companionway completely submerges. Thus, if the ventilator hole on the bow deck is closed, air inside the cabin is trapped and therefore the model cannot sink. In contrast, if the ventilator hole is open, air inside the cabin gradually escaped through the hole and finally the model sinks as shown in Figure 9.

Fig. 9 Photo of the sinking yacht with the full-sized bow hole under the wind velocity of 11 m/s.

Table 2. Summary of model experiment. Here O: fully open, o: 20% open, X: closed, b: bow hole and ch: companionway.

wind speed (m/s) 8 9 10 11 12 13.8 14.5 18

b

ch

b

ch

b

ch

b

ch

O

O

X

O

O

X

o

O

non-capsize non-capsize sink sink sink sink sink

capsize capsize re-right re-right

capsize capsize fully re-right

non-capsize Sink Sink Sink

850 N. Umeda et al.

The experiment was carried out for several combinations of open or close bow hole and companionway as well as different wind velocities. The results are summarised in Table 2 and the time to sink from upside down is plotted in Figure 10. The threshold for capsizing exists between 9 and 10 m/s. If either the bow hole or the companionway is closed, the yacht model did not sink. This is because air flow requires both the bow hole and the companionway are open. In particular, in the cases the companionway is closed and wind speed is strong enough, the yacht model completely re-rights and sails again. However, these cases require stronger wind than the cases of sink because flooding from capsized condition with small opening is more difficult. With the same reason, when the wind speed increases time-to-sink decreases. If the bow hole is smaller, time-to-sink can be longer because of reduction of air flow speed but sinking cannot be prevented in the long run. 300 bow hole small size

time (s)

250 200

bow hole full size

150 100 50 0

0

5

10 15 wind velocity (m/s)

20

Fig. 10 Time to sink of the yacht model from upside down in full scale. Here the companionway was open.

5 Scenario of Accident Based on the above investigation and witness reports, the authors presume the following scenario of the accident. When the tacking about was executed, the skipper failed to helm amidships and to release the main sheet. Then the yacht drastically rolled toward leeward side. Because of roll, the crews on deck were moved to leeward and then one adult and one child dropped into the main sail. Subsequently all others dropped into the water. Under this condition the static equilibrium existed as shown in Figure 11, but the dynamic equilibrium did not exist, as shown in Figure 12, even with energy dispassion due to roll damping taken into account. Then the roll angle exceeds the angle of vanishing stability and therefore capsizing cannot be avoided. This scenario could work even with the wind speed of 8 m/s. The measured data near the place of accident shows that the mean wind speed at the accident was about 8 m/s as described before. In addition, the authors also investigated the

Experimental Investigation on Capsizing and Sinking 851

effect of replacement of crew positions at the tacking and confirmed that this is not the primary cause of the accident. Restoring Moment (accident) wind heel moment (V=8m/s) wind heel moment (V=11m/s)

wind heel moment (V=10m/s) wind heel moment (V=9m/s)

8000 Moment (N-m)

6000 4000 2000

0 -2000 0

45

90

135

180

-4000 -6000 Heel angle (degrees)

Fig. 11 Comparison of restoring moment and wind heeling moment in case of accident.

6000

restoring energy wind heeling energy

energy (N-m)

4000 2000 0 -2000 0

45

90

135

180

-4000 -6000 Heel angle (degrees)

Fig. 12 Comparison of restoring energy and wind heeling energy with 8m/s of wind speed

Once capsized, air inside the cabin was completely trapped. However, the wind moment acting on the fin keel and canoe body could further incline the yacht from the fully capsized condition and the bow hole and part of the companionway could emerge. Then air can escape and water can enter into the cabin gradually. When the flooded water inside the cabin exceeds 2 m3, the capsized yacht becomes unstable and the yacht starts to re-right. Further flooding during this transient process induces large trim by the stern. When the yacht almost re-rights, the companionway is completely submerged due to this trim by the stern. If the bow hole is closed here, air inside the cabin can be trapped and the yacht can be safe. Unfortunately, the bow hole was open. As a result, air inside the cabin escaped through the bow hole, and therefore the yacht sunk. The critical wind speed of this

852 N. Umeda et al.

scenario is 10m/s while the measured maximum wind speed was about 11 m/s. This means the yacht could sink if she suffered a gusty wind at that time. After the accident, several countermeasures were proposed. Shiga Prefecture enforced a new local rule for mandating the use of life vests for people on all powered pleasure boats in 2004. The Japan Sailing Federation (2004) warned that a small cruising yacht could capsize due to uncontrolled tacking or jibing. And Masuyama (2004) proposed to add small but sufficient buoyancy-aid inside the cabin of a small cruising yacht for preventing its sinking.

6 Another Accident A sister yacht with the exactly same design also capsized on 28 October 2001 in Osaka Bay without the loss of human life. In this case the yacht capsized and then completely re-righted without sinking. The difference between the two yachts is discussed here. The yacht at the accident in Osaka Bay had six adults and six children as her crew and all of them were on deck. Thus, the righting arm of this yacht could be drastically reduced like the yacht in Lake Biwa. The companionway and the ventilator at the bow were open. The mean wind speed was about 8 m/s with some gust and the wave height was about 0.5 m. The yacht ran with its outboard motor without sail in head wind. When she met a gust, the helmsman could not keep her course despite his steering effort and eventually the yacht suffered beam wind and waves from her port side. Then the yacht capsized toward starboard side. Although the windage area is small, the yacht could capsize because of insufficient righting arm due to crew weight on deck. Two or three minutes passed with the turn turtle capsize and then the yacht rerighted up to an upright condition. When the yacht completely re-righted, she was almost even keel and flooded water was found in the cabin. The main difference between the two accidents is that the yacht in Osaka Bay capsized toward starboard side while the yacht in Lake Biwa did toward port side. Since the position of the bow hole is slightly shifted to the port side, the case of capsize toward port side the bow hole can be emerge with smaller additional wind-induced heel from turn turtle capsize. As a result, the yacht in Osaka Bay water ingress during the process from capsizing to re-righting could be smaller. The even keel at the re-righted moment also indicated small flooded water amount. It can be also pointed out that small roll damping at the capsized condition without sail could reduce time to re-right, which can limit water ingress.

Experimental Investigation on Capsizing and Sinking 853

7 Conclusions As a result of hydrostatic calculations and model experiments from capsizing to sink, following conclusions were obtained: 1. Because of so many crews on deck, the restoring arm of the yacht was drastically reduced. 2. If the skipper failed to helm amidships and to release the main sheet in case of tacking about with the wind speed of 8 m/s or over, the yacht could capsize. 3. If wind speed is more than 10 m/s, additional heel can allow air egress and water ingress. When the flooded water exceeds 2 m3, the yacht could start to re-right. 4. Because of the large amount of water inside, trim by the stern could occur and the companionway could submerge. If the bow hole is open, air egress through the bow hole and water ingress through the companionway could result in sinking of the yacht. 5. The measured wind speed at the accident exceeds the above critical wind speeds for capsizing and re-righting. 6. The outcomes could depend on the direction of capsizing because of offset of the position of the bow hole.

Acknowledgements This investigation was carried out under the contract between Japan’s Marine Accident Inquiry Agency and the first author in 2004. The authors thank Capt. A. Sawa, Director General of the Kobe Local Marine Accident Inquiry Commisioner’s Office for his support during this investigation. The authors are also grateful to Professor Y. Masuyama from Kanazawa Institute of Technology for his useful discussion. In addition, they would like to thank Mr. K. Ohnishi for his technical assistance during the experiment,

References Deakn B (2000) “Model Tests to Study Capsize and Stability of Sailing Multihull, Proceedings of the 7th Int Conf on Stab of Ships and Ocean Veh, Launceston, Vol. B, 619-639. Hirayama T, Miyagawa K, Takayama T (1994) “Capsizing and Restoring Characteristics of a Sailing Yacht in Oblique and Breaking Waves“, J of the Kansai Soc of Naval Archit, No. 221:117-122, in Japanese. Hirayama T, Miyagawa K, Takayama T and Satake H (1995) “Capsizing and Restoration of a Sailing Yacht in Breaking Isolated Triangular Transient Waves and Breaking Long Crested Transient Waves”, J of the Kansai Soc of Naval Archit, No. 223:59-66, in Japanese.

854 N. Umeda et al. Japan (2005) “Proposal on draft explanatory notes to the severe wind and rolling criterion”, SLF 48/4/5, Int Maritime Org (London). Japan Sailing Federation (Technical Committee), 2004, “For Preventing Capsizing and Sinking of Sailing Cruisers”, in Japanese. Masuyama Y (2004) “An Investigation of Sinking of Small Sailing Cruiser”, KAZI, May issue, 182-187, in Japanese. Nimura T, Ishida S and Watanabe I (1994) “On the Effect of Hull Forms and Other Factors on the Capsizing of Sailing Yachts”, J of the Soc of Naval Archit of Japan, Vol. 17:57-67, in Japanese. Nomoto K (2000) “Stability of a Sailing Yacht Floating Upside-down”, Proc of the 7th Int Conf on Stab of Ships and Ocean Veh, Launceston, Vol. B, 572-585. Renilson MR, Steel J and Tuite A (2000) “A Preliminary Investigation into the Effect of a Coach House on the Self-Righting of a Modern Racing Yacht”, Proc of the 7th Int Conf on Stab of Ships and Ocean Veh, Launceston, Vol. B, 562-571. Takaishi Y (2000) “On Stability Standard of Sailing Boats”, Proc of the 7th Int Conf on Stab of Ships and Ocean Veh, Launceston, Vol. B, 586-594.

Author Index Aoki, K. Bassler, C. C. Belenky, V. L. Blok, J. J. Boonstra, H. Botía-Vera, E. Bulian, G. Cercos-Pita, J. L. Daalen, E. F. Gvan Degtyarev, A.B. Delorme, L. Ferreiro, L. D. Francescutto, A. Fujiwara, T. Gao, Q. Guarin, L. Hashimoto, H. Hori, M. Ikeda, Y. Ishida, S. Jasionowski, A. Kat, J. Ode Katayama, T. Kawahara, Y. Kotaki, M. Levadou, M. Lin, Woei-Min Lorca, O. M. Maekawa, K. Matsuda, A. McCue, L. S. Mermiris, G. Minami, M. Munif, A. Nechaev, Y. Neves, M. A. S

841 3 3,295,531,555,589 501 501 735 3,735 735 501 599 735 141 3 331 723 663,753 217,267,379 217,841 331,465,487,633,841 65,277,633 47,103,663,827 643 331,487,633,841 465 487 307 555 231 465 267,379 181,415,433 753 277 331 79 231,449

M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3, © Springer Science+Business Media B.V. 2011

855

856 Author Index

Nowacki, H. Odabasi, A. Y. Ohkura, Y. Paulling, J. R. Pawłowski, M. Pérez, N. A. Pérez-Rojas, L. Peters, A. J. Peters, W. Puisa, R. Reed, A. M. Rimpela, A. L. Rodriguez, C. A. Ruponen, P. Santen, J van Sawada, H. Shen, L. Souto-Iglesias, A. Spyrou, K. J. Taguchi, H. Takeuchi, Y. Themelis, N. Troesch, A. W. Ucer, E. Umeda, N. Urano, S. Vassalos, D. Veer, Rvan´t Vivanco, J. E. M. Weems, K. M. Wing, D. Yu, Han-Chang

141 119 217 347,555 573,621 231 735 773 643 753 531,793 643 231,449 689 193 65,277 709 735 3,25,253,363,399,515 65,277 633 515 181,415 119 3,217,267,369,841 217 47,103,663,709,723,753,827 307,643 449 295,531,555 773 295