IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties

IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties IUTAM BOOKSERIES Volume 27 Series Editors ...

Author: Alexander K. Belyaev | Robin S. Langley

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IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties

IUTAM BOOKSERIES Volume 27 Series Editors G.M.L. Gladwell, University of Waterloo, Waterloo, Ontario, Canada R. Moreau, INPG, Grenoble, France

Editorial Board J. Engelbrecht, Institute of Cybernetics, Tallinn, Estonia L.B. Freund, Brown University, Providence, USA A. Kluwick, Technische Universitt, Vienna, Austria H.K. Moffatt, University of Cambridge, Cambridge, UK N. Olhoff, Aalborg University, Aalborg, Denmark K. Tsutomu, IIDS, Tokyo, Japan D. van Campen, Technical University Eindhoven, Eindhoven, The Netherlands Z. Zheng, Chinese Academy of Sciences, Beijing, China

Aims and Scope of the Series The IUTAM Bookseries publishes the proceedings of IUTAM symposia under the auspices of the IUTAM Board.

For other titles published in this series, go to www.springer.com/series/7695

Alexander K. Belyaev r Robin S. Langley Editors

IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties Proceedings of the IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties held in St. Petersburg, Russia, July 5–9, 2009

Editors Alexander K. Belyaev Russian Academy of Sciences Inst. Problems of Mechanical Engineering Bolshoy Ave. V. O. 61 199178 St. Petersburg Russia [email protected]

Robin S. Langley University of Cambridge Dept. Engineering Trumpington Street CB2 1PZ Cambridge UK [email protected]

ISSN 1875-3507 e-ISSN 1875-3493 ISBN 978-94-007-0288-2 e-ISBN 978-94-007-0289-9 DOI 10.1007/978-94-007-0289-9 Springer Dordrecht Heidelberg London New York © 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: VTEX, Vilnius Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

The IUTAM Symposium on The Vibration Analysis of Structures with Uncertainties was held in Saint-Petersburg, Russia, on July 5 – July 9 2009. The members of the Scientific Committee were Alexander K. Belyaev (Chair) — Institute of Problems in Mechanical Engineering of the Russian Academy of Sciences, SaintPetersburg, Russia; Robin Langley (Co-Chair) — University of Cambridge, UK; Franz Ziegler (IUTAM Representative) — Vienna University of Technology, Austria; Antonio Carcaterra — University of Rome La Sapienza, Italy; Yakov BenHaim — Technion-Israel Institute of Technology, Israel; Christian Soize — Universite de Marne-la-Vallee, France; Dirk Vandepitte — Katholieke Universiteit Leuven, Belgium and Richard Weaver — University of Illinois at Urbana-Champaign, USA. The Symposium took place in Tsarskoe Selo in a palace designed by Prince Kochubei. Tsarkoe Selo (a suburb of St. Petersburg, also known as Pushkin) was a summer residence of the Tsar, and is well known for its many palaces, including the Katherine Palace which houses the famous Amber Room. The Symposium was aimed at the theoretical and numerical problems involved in modelling the dynamic response of structures which have uncertain properties due to variability in the manufacturing and assembly process, with automotive and aerospace structures forming prime examples. It is well known that the difficulty in predicting the response statistics of such structures is immense, due to the complexity of the structure, the large number of variables which might be uncertain, and the inevitable lack of data regarding the statistical distribution of these variables. The Symposium participants presented the latest thinking in this very active research area, and novel techniques were presented covering the full frequency spectrum of low, mid, and high frequency vibration problems. It was demonstrated that for high frequency vibrations the response statistics can saturate and become independent of the detailed distribution of the uncertain system parameters. A number of presentations exploited this physical behaviour by using and extending methods originally developed in both phenomenological thermodynamics and in the fields of quantum mechanics and random matrix theory. For low frequency vibrations a number of presentations focussed on parametric uncertainty modelling (for example, probabilistic models, interval analysis, and fuzzy descriptions) and on methods of propagating v

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this uncertainty through a large dynamic model in an efficient way. At mid frequencies the problem is mixed, and various hybrid schemes were proposed. It is clear that a comprehensive solution to the problem of predicting the vibration response of uncertain structures across the whole frequency range requires expertise across a wide range of areas (including probabilistic and non-probabilistic methods, interval and info-gap analysis, statistical energy analysis, statistical thermodynamics, random wave approaches, and large scale computations) and this IUTAM symposium presented a unique opportunity to bring together outstanding international experts in these fields. The lectures were arranged such that 12 of the presentations were keynote overviews and allocated 45 minutes, while the remaining 24 presentations were each allocated 20 minutes. In addition to this, there was much discussion and fruitful interaction, both during the technical sessions and over lunch and dinner. All of the presented papers are collected together in the Proceedings. The IUTAM grant and the financial support of the Russian Foundation for Basic Research are gratefully acknowledged. Also, we would like to express our sincere gratitude to Dr Dmitry Kiryan who took the trouble of preparing the camera ready manuscript of the Proceedings. Robin Langley’s participation was funded through the ITN Marie Curie project GA-214909 “MID-FREQUENCY – CAE Methodologies for Mid-Frequency Analysis in Vibration and Acoustics”. St. Petersburg, June 2010

Alexander K. Belyaev Robin S. Langley Symposium Co-Chairs

Contents

Part I Non-probabilistic and related approaches Linear Bounds on an Uncertain Non-Linear Oscillator: An Info-Gap Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yakov Ben-Haim and Scott Cogan 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Dynamics, Uncertainty and Robustness . . . . . . . . . . . . . . . . . . . . . . . 3 Example: Uncertain Cubic Non-Linearity . . . . . . . . . . . . . . . . . . . . . 4 Example: Multiple Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Robustness as a Proxy for Probability . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantification of uncertain and variable model parameters in non-deterministic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dirk Vandepitte, David Moens 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Numerical representation of parameter uncertainty and variability . 2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Discussion and extension of the definitions . . . . . . . . . . . . 3 Literature review on uncertain model and material data . . . . . . . . . . 3.1 Non-probabilistic models . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Probabilistic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Material data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Other model properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Alternative approaches: non-parametric model concept and info-gap theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Summary of observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 4 7 10 12 13 14 15 15 16 17 18 19 19 20 22 25 25 26 26 27 vii

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Vibrations of layered structures with fuzzy core stiffness/fuzzy interlayer slip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rudolf Heuer and Franz Ziegler 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Fuzzy sandwich beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Three-layer beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Modal analysis of the three-layer beam, hard-hinged support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Isosceles uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Constraints affected to the uncertain natural frequencies . 3.3 Some effects of non-symmetric uncertainty . . . . . . . . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vibration Analysis of Fluid-Filled Piping Systems with Epistemic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Hanss, J. Herrmann and T. Haag 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Classification, Representation and Propagation of Uncertainty . . . . 2.1 Uncertainty Classification and Representation . . . . . . . . . . 2.2 Uncertainty Propagation Based on the Transformation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Fluid-Filled Piping System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Modeling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Comprehensive Modeling and Simulation . . . . . . . . . . . . . . . . . . . . . 4.1 Modeling of Epistemic Uncertainties . . . . . . . . . . . . . . . . . 4.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Measures of Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fuzzy vibration analysis and optimization of engineering structures: Application to Demeter satellite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Massa, A. Leroux, B. Lallemand, T. Tison, F. Buffe, and S. Mary 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Aims of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Description of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Building of fuzzy optimization problem . . . . . . . . . . . . . . . 3 Fuzzy vibration analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 PAEM method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Numerical application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Fuzzy optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Design methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Improvement of the initial design . . . . . . . . . . . . . . . . . . . .

29 29 30 30 33 35 35 38 41 41 42 43 44 44 44 46 48 48 51 51 51 53 54 55 55 57 57 58 58 59 62 62 63 63 64 66

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5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Numerical dynamic analysis of uncertain mechanical structures based on interval fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . David Moens, Maarten De Munck, Wim Desmet, Dirk Vandepitte 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Interval finite element analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Interval fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 General concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Interval fields as uncertain input parameters . . . . . . . . . . . 3.3 Interval fields as uncertain analysis results . . . . . . . . . . . . . 4 Application of interval fields for vibro-acoustic analysis . . . . . . . . . 4.1 Vibro-acoustic analysis based on the ATV concept . . . . . . 4.2 Interval analysis based on structural FRF interval fields . . 4.3 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From Interval Computations to Constraint-Related Set Computations: Towards Faster Estimation of Statistics and ODEs under Interval and p-Box Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vladik Kreinovich 1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Interval Computations: Brief Reminder . . . . . . . . . . . . . . . . . . . . . . . 3 Constraint-Based Set Computations . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 73 74 74 75 77 79 79 80 81 82 83

85 85 88 89 98

Dynamic Steady-State Analysis of Structures under Uncertain Harmonic Loads via Semidefinite Program . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Yoshihiro Kanno and Izuru Takewaki 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2 Uncertain equations for steady state vibration . . . . . . . . . . . . . . . . . . 101 2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2.2 Uncertainty model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2.3 ULE in real variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3 Bounds for complex amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.1 Upper bound for modulus of displacement amplitude . . . . 103 3.2 Lower bound for modulus of displacement amplitude . . . 106 3.3 Bounds for phase angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4 Bounds for nodal oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

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Part II SEA related methods and wave propagation Universal eigenvalue statistics and vibration response prediction . . . . . . . . 115 R.S. Langley 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 2 Eigenvalue statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 2.1 The joint probability density function of the eigenvalues . 116 2.2 The modal density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 2.3 Universality of the “local” eigenvalue statistics . . . . . . . . . 119 2.4 Application to natural frequency statistics . . . . . . . . . . . . . 121 3 Application to response statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.1 Fundamental concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.2 Built-up systems: SEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.3 Built-up systems: the Hybrid method . . . . . . . . . . . . . . . . . 124 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Statistical Energy Analysis and the second principle of thermodynamics . 129 Alain Le Bot 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 2 First principle of thermodynamics in SEA . . . . . . . . . . . . . . . . . . . . . 130 3 Vibrational entropy, vibrational temperature . . . . . . . . . . . . . . . . . . . 133 4 Second principle of thermodynamics in SEA . . . . . . . . . . . . . . . . . . 134 5 Entropy balance in SEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Modeling noise and vibration transmission in complex systems . . . . . . . . . 141 Philip J. Shorter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 1.1 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 1.2 Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 1.3 How much information is needed for noise and vibration design? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 2 Modeling methods and frequency ranges . . . . . . . . . . . . . . . . . . . . . . 144 2.1 Low, mid and high frequency ranges . . . . . . . . . . . . . . . . . . 144 2.2 Low and High frequency modelling methods . . . . . . . . . . . 145 2.3 The Mid-Frequency problem . . . . . . . . . . . . . . . . . . . . . . . . 146 3 The Hybrid FE-SEA method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 3.1 Statistical subsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 3.2 The direct and reverberant fields of a statistical subsystem 148 3.3 Ensemble average reverberant loading . . . . . . . . . . . . . . . . 149 3.4 Coupling a deterministic and statistical subsystem . . . . . . 149 4 Application examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 4.1 Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

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4.2 Numerical applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.3 Industrial applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 A Power Absorbing Matrix for the Hybrid FEA-SEA Method . . . . . . . . . . 157 R.H. Lande and R.S. Langley 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 2 Cylindrical Waves and Energy Sinks . . . . . . . . . . . . . . . . . . . . . . . . . 158 2.1 The Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 2.2 The Cylindrical Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 3 Constructing the Power Absorbing Matrix . . . . . . . . . . . . . . . . . . . . . 162 3.1 Discretization of the Power Integral, and Matrix Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 3.2 Numerical Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.1 A Simple System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.2 System Randomization and Subsystem Response Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 The Energy Finite Element Method NoiseFEM . . . . . . . . . . . . . . . . . . . . . . . 171 Christian Cabos and Hermann G. Matthies 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 1.2 Literature Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 2 Components of NoiseFEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 3 Power Flow Between Structural Elements . . . . . . . . . . . . . . . . . . . . . 173 3.1 Transmission Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 174 3.2 The Coupling Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 4 Diffusive Energy Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4.1 Homogeneous Structural Elements . . . . . . . . . . . . . . . . . . . 178 4.2 Stiffened Subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5 Combining transport and coupling equations . . . . . . . . . . . . . . . . . . 180 6 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7 Validation of NoiseFEM with test structures . . . . . . . . . . . . . . . . . . . 182 8 Application of NoiseFEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

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Wave transport in complex vibro-acoustic structures in the high-frequency limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Gregor Tanner and Stefano Giani 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 2 Wave energy flow in terms of the Green function . . . . . . . . . . . . . . . 189 3 Linear phase space operators and DEA . . . . . . . . . . . . . . . . . . . . . . . 190 4 A numerical example: coupled two-domain systems . . . . . . . . . . . . 194 4.1 The hp-adaptive Discontinuous Galerkin Method . . . . . . . 194 4.2 FEM compared to DEA and SEA — results . . . . . . . . . . . 197 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Benchmark study of three approaches to propagation of harmonic waves in randomly heterogeneous elastic media . . . . . . . . . . . . . . . . . . . . . . 201 Alexander K. Belyaev 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 2 Method of integral spectral decomposition . . . . . . . . . . . . . . . . . . . . 202 3 The Fokker-Planck-Kolmogorov equation . . . . . . . . . . . . . . . . . . . . . 205 4 The Dyson integral equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Minimum-variance-response and irreversible energy confinement . . . . . . . 215 A. Carcaterra 1 Average Impulse Response and the Single Case . . . . . . . . . . . . . . . . 215 2 MIVAR: Minimum-Variance-Response . . . . . . . . . . . . . . . . . . . . . . . 217 3 Application of the theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 High-frequency vibrational power flows in randomly heterogeneous coupled structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 ´ Savin Eric 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 2 Transport model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 2.1 Radiative transfer in an open domain . . . . . . . . . . . . . . . . . 231 2.2 Radiative transfer in a bounded domain . . . . . . . . . . . . . . . 232 2.3 Radiative transfer with a sharp interface . . . . . . . . . . . . . . . 233 3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 3.1 Coupled beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 3.2 Coupled shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

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Uncertainty propagation in SEA using sensitivity analysis and Design of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Antonio Culla, Walter D’Ambrogio and Annalisa Fregolent 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 2 SEA equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 3 Uncertainty propagation in SEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 3.1 Approach using sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . 247 3.2 Approach using Design of Experiments . . . . . . . . . . . . . . . 248 4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 Phase reconstruction for time-domain analysis of uncertain structures . . . 255 L H Humphry 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 2 Explanation of minimum phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 2.1 Defining minimum phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 2.2 The Hilbert transform and analytic systems . . . . . . . . . . . . 256 2.3 The Hilbert Transform and minimum phase systems . . . . 257 2.4 Further interpretation of minimum phase . . . . . . . . . . . . . . 257 3 Using minimum phase reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 258 3.1 Approximating the Hilbert Transform . . . . . . . . . . . . . . . . . 258 3.2 Errors using MPR for non-minimum phase systems . . . . . 261 4 Application: peak shock prediction in uncertain structures . . . . . . . 263 4.1 Modelling an uncertain structure . . . . . . . . . . . . . . . . . . . . . 264 4.2 Ensemble average results . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 4.3 Changing the correlation of modal amplitudes . . . . . . . . . . 266 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Part III Probabilistic Methods Uncertain Linear Systems in Dynamics: Stochastic Approaches . . . . . . . . . 271 G.I. Schu¨eller and H.J. Pradlwarter 1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 2 Overview of Available Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 3 Response variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 3.1 Perturbation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 3.2 Spectral methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 3.3 Direct Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . 279 3.4 Random matrix approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 4 Computational Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

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Time domain analysis of structures with stochastic material properties . . 287 Giovanni Falsone and Dario Settineri 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 2 Preliminary concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 3 Application of the perturbation approach . . . . . . . . . . . . . . . . . . . . . . 289 4 Moments of the uncertain structure response . . . . . . . . . . . . . . . . . . . 290 5 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Vibration Analysis of an Ensemble of Structures using an Exact Theory of Stochastic Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 Christophe Lecomte 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 2 Description of the Stochastic System . . . . . . . . . . . . . . . . . . . . . . . . . 302 3 Expression of Mean, Variance, and Covariance . . . . . . . . . . . . . . . . . 304 3.1 Parameterized Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 3.2 Mean Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 3.3 Variance and Covariance of the Responses . . . . . . . . . . . . . 305 3.4 Multirank Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 3.5 Discussion of the Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 4 Stochastic Coefficients in the case of a Gaussian Probability Density Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 5 Application examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 5.1 Comparison to a Monte-Carlo Simulation . . . . . . . . . . . . . 310 5.2 Transition from low to high modal density . . . . . . . . . . . . . 311 5.3 Variance and covariance of responses at different frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 Structural Uncertainty Identification using Vibration Mode Shape Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 J. F. Dunne and S. Riefelyna 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 2 Maximum Likelihood Estimation of Uncertain Structural Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 2.1 Uncertainty Estimation via the Perturbation Method . . . . 319 3 ML estimates of uncertain point-mass position statistics using natural frequency information on a cantilever beam structure . . . . . 320 4 ML Estimation of uncertain point mass position on a plate structure using mode shape information . . . . . . . . . . . . . . . . . . . . . . . 323 5 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

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Extremely strong convergence of eigenvalue-density of linear stochastic dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 S. Adhikari and L. A. Pastur 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 2 Uncertainty quantification of dynamic response . . . . . . . . . . . . . . . . 333 3 Wishart random matrix model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 4 Density of eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 4.1 Linear eigenvalue statistic . . . . . . . . . . . . . . . . . . . . . . . . . . 336 4.2 Self averaging property and the Mar˘cenko-Pastur density 337 5 Numerical investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 5.1 Plate with randomly inhomogeneous material properties: parametric uncertainty problem . . . . . . . . . . . . 342 5.2 Plate with randomly attached spring-mass oscillators: nonparametric uncertainty problem . . . . . . . . . . . . . . . . . . . 343 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 Stochastic subspace projection schemes for dynamic analysis of uncertain systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 Prasanth B. Nair 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 3 Frequency domain analysis of linear stochastic structural systems . 349 3.1 Preconditioner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 3.2 Postprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 4 The algebraic random eigenvalue problem . . . . . . . . . . . . . . . . . . . . 354 4.1 Stochastic Basis Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 4.2 Bubnov-Galerkin Projection . . . . . . . . . . . . . . . . . . . . . . . . 356 4.3 Postprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 5 Numerical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 Part IV Probabilistic Methods, Applications Reliability Assessment of Uncertain Linear Systems in Structural Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 H.J. Pradlwarter and G.I. Schu¨eller 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 2 Methods of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 2.1 Representation of uncertain excitation . . . . . . . . . . . . . . . . 364 2.2 Uncertain structural systems . . . . . . . . . . . . . . . . . . . . . . . . 366 2.3 Stochastic conditional response . . . . . . . . . . . . . . . . . . . . . . 366 2.4 Conditional reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 2.5 Design point for stochastic structural systems . . . . . . . . . . 368 2.6 First excursion probability for stochastic systems . . . . . . . 370

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Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 3.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 3.2 Structural system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 3.3 Dynamic excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 3.4 Critical response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 3.5 Reliability of critical component . . . . . . . . . . . . . . . . . . . . . 375 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 On semi-statistical method of numerical solution of integral equations and its applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 D.G. Arsenjev, V.M. Ivanov, and N.A. Berkovskiy 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 2 Short scheme of semi-statistical method . . . . . . . . . . . . . . . . . . . . . . 380 3 Statement of the problem of blade cascade flow . . . . . . . . . . . . . . . . 381 4 Scheme of application of semi-statistical method to the problem of blade cascade flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 4.1 Main formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 4.2 Computation algorithm and optimization . . . . . . . . . . . . . . 384 5 Results of simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 6 Analysis of efficiency of the density adaptation . . . . . . . . . . . . . . . . 385 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 An efficient model of drill-string dynamics with localised non-linearities . 389 T. Butlin and R.S. Langley 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 2 Theoretical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 2.1 Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 2.2 Coupling to Non-Linearities . . . . . . . . . . . . . . . . . . . . . . . . . 393 2.3 Coupling to Subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 3 Example Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 3.1 Linear Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 3.2 Coupling to non-linear friction law . . . . . . . . . . . . . . . . . . . 397 3.3 Coupling to lumped inertia . . . . . . . . . . . . . . . . . . . . . . . . . . 398 3.4 Uncertainty Analysis of Stick-Slip Oscillation . . . . . . . . . 399 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 Equivalent thermo-mechanical parameters for perfect crystals . . . . . . . . . 403 V. A. Kuzkin and A. M. Krivtsov 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 2 Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 3 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 4 Equation of momentum balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 5 Equation of angular momentum balance . . . . . . . . . . . . . . . . . . . . . . 410

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6 Equation of energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 7 Constitutive relations for stress tensor and heat flux . . . . . . . . . . . . . 413 8 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 Analysis of offshore systems in random waves . . . . . . . . . . . . . . . . . . . . . . . . 417 Katrin Ellermann and Max Suell Dutra 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 2 Modeling aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 2.1 Modeling of environmental forces . . . . . . . . . . . . . . . . . . . . 419 2.2 Modeling of multibody systems . . . . . . . . . . . . . . . . . . . . . 420 3 Analysis of deterministic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 4 Analysis of random systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 4.1 Monte Carlo simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 4.2 Stochastic linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 5 Selected Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 Statistical Dynamics of the Rolling Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 Paul V. Krot 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 2 Cold Rolling Mills Chatter Vibrations . . . . . . . . . . . . . . . . . . . . . . . . 431 2.1 Rolling stand design and its modal analysis . . . . . . . . . . . . 431 2.2 Strip Elasto-Plastic Deformation . . . . . . . . . . . . . . . . . . . . . 433 2.3 Horizontal work rolls vibration . . . . . . . . . . . . . . . . . . . . . . 435 2.4 Contact friction force variation . . . . . . . . . . . . . . . . . . . . . . 436 2.5 Chatter detection and control . . . . . . . . . . . . . . . . . . . . . . . . 437 3 Hot Rolling Mills Torsional Vibrations . . . . . . . . . . . . . . . . . . . . . . . 438 3.1 Torsional vibration control and backlashes diagnostics . . 439 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 The application of robust design strategies on managing the uncertainty and variability issues of the blade mistuning vibration problem . . . . . . . . . 443 Y.-J. Chan and D. J. Ewins 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 2 Basic concepts of the blade mistuning problem . . . . . . . . . . . . . . . . 445 2.1 The Amplification Factor (its significance and range) . . . . 446 3 Casting blade mistuning as a robust design problem . . . . . . . . . . . . 447 3.1 The Taguchi method of robust design . . . . . . . . . . . . . . . . . 448 3.2 The robust optimisation method . . . . . . . . . . . . . . . . . . . . . 448 3.3 Application of robust design methods to the blade mistuning problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 4 Improving the robustness of bladed discs by parameter design . . . . 450 5 Improving the robustness of bladed discs by tolerance design . . . . . 452

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5.1 The Small Mistuning approach . . . . . . . . . . . . . . . . . . . . . . 453 5.2 The Intentional Mistuning approach . . . . . . . . . . . . . . . . . . 453 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 Localized modeling of uncertainty in the Arlequin framework . . . . . . . . . . 457 R. Cottereau, D. Clouteau, and H. Ben Dhia 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 2 The classical Arlequin method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 3 The continuous stochastic-deterministic Arlequin formulation . . . . 461 3.1 The stochastic monomodel . . . . . . . . . . . . . . . . . . . . . . . . . . 462 3.2 The Arlequin formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 4 The discretized stochastic-deterministic Arlequin formulation . . . . 463 5 Example of application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

Part I

Non-probabilistic and related approaches

Linear Bounds on an Uncertain Non-Linear Oscillator: An Info-Gap Approach Yakov Ben-Haim and Scott Cogan

Abstract We study a 1-dimensional cubic non-linear oscillator in the frequency domain, in which the non-linearity is roughly estimated but highly uncertain. The task is to choose a suite of linear computational models at different excitation frequencies whose responses are useful approximations to, or upper bounds of, the real non-linear system. These model predictions must be robust to uncertainty in the non-linearity. A worst case for the uncertain non-linearity is not known. The central question in this paper is: how to choose the linear computational models when the magnitude of error of the estimated non-linearity is unknown. A resolution is proposed, based on the robustness function of info-gap decision theory. We also prove that the non-probabilistic info-gap robustness is a proxy for the probability of success.

1 Introduction Structural reliability is an important concern for high consequence engineering systems and many qualitative and quantitative arguments come into play in certifying a design for an intended application. For example, component-level qualification tests as well as numerical simulations integrating component models into global structural analyses are an integral part of the certification process for aerospace structures and subassemblies. However, the dominant sources and degrees of lackof-knowledge in the studied system and its environment are often difficult to charYakov Ben-Haim Yitzhak Moda’i Chair in Technology and Economics, Faculty of Mechanical Engineering, Technion — Israel Institute of Technology, Haifa, Israel e-mail: [email protected] Scott Cogan FEMTO-ST Institute, Universit´e de Franche-Comt´e, Besanc¸on, France e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 1, © Springer Science+Business Media B.V. 2011

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acterize and it is clearly desirable that design decisions be as robust as possible to these uncertain quantities. In particular, non-linearities may be present in material behavior and joint properties, due to a wide variety of spatial and time-dependent phenomena (e.g. plastic strain, stick-slip, impact, aging, . . . ) and can modify the structural behavior in dramatic and often unsuspected ways. The importance of these effects is often observed during qualification tests at multiple excitation levels. While deterministic nonlinear dynamic analyses on complex global assemblies are not uncommon today, the computational burden of such computations is evidently much higher than for linear systems. In what follows, we propose a methodology that addresses both of these challenges — lack-of-knowledge and computational burden — by selecting a suite of linear computational models that approximates the upper bound of the real nonlinear system given uncertainty in non-linear stiffness properties. Moreover, we assume that a worst case for this uncertainty is not known. A method is proposed based on the info-gap robustness function, that allows the trade-offs between robustness and linear model design to be studied over a range of behaviors. The proposed methodology is illustrated on a single degree of freedom oscillator with an uncertain cubic stiffness term. A frequency domain approach is adopted here and the excitation is assumed to be periodic. The non-linear responses are approximated using a first-order harmonic balance technique which assumes that the system responds at the excitation frequency. Info-gap theory [1, 2] has been used in a wide range of engineering analysis problems. Matsuda and Kanno [7] study the info-gap robustness of structures based on plastic limit analysis with uncertain loads. Kanno and Takewaki [5, 6] study the robust design of structures under load uncertainty. Duncan et al. [4] develop an info-gap approach to robust decision making under severe uncertainty in life cycle design. Vinot et al. [9] develop a robust model-based test planning procedure. Pierce et al. [8] employ an info-gap technique to assess reliability of neural network-based damage detection.

2 Dynamics, Uncertainty and Robustness In this section we formulate the info-gap robustness function, which is a combination of 3 elements: system models, a performance requirement, and an uncertainty model. We consider two models in the time and frequency domains: the real non-linear system and an artificial computational model. Real system. The equation of motion in the time domain is expressed as: f = k1 x + jbx˙r + mx¨r + k3 xr3 where b and m are assumed known, but k1 , k3 and f may be uncertain.

(1)

Linear Bounds on an Uncertain Non-Linear Oscillator: An Info-Gap Approach

5

The response of this system will be calculated using the harmonic balance technique where the external excitation is assumed to be monoharmonic [10], f = F sin(ω t + φ ) with phase angle φ , and the amplitude of response is periodic, xr = Xr sin ω t. Balancing sine and cosine terms, and using the trigonometric equality sin3 a = 34 sin a − 14 sin 3a, yields: 3 − mω 2 Xr + k1 Xr + k3 Xr3 = F cos φ 4 bω Xr = −F sin φ

(2) (3)

where the term in sin(3ω t) has been neglected. Eliminating φ leads to the following polynomial equation in Xr as a function of ω : 2 3 2 2 2 + (bω ) Xr2 − F 2 = 0 (4) −mω + k1 + k3 Xr 4 The physically meaningful solutions of this equation correspond to the strictly real and positive roots. The solutions used in the computations that follow correspond to the upper portion of the non-linear frequency response function (between points A and B) as seen in fig. 1.

Fig. 1: Non-linear frequency response function with cubic stiffness Computational model. The equation of motion in the time domain is given by: F cos ω t = kc x + jbc x˙c + mx¨c + Fc cos ω t

(5)

and its transformation in the frequency domain by: F = (kc + j ω bc − ω 2 m)Xc + Fc

(6)

where m is the same as in eq. (4). We are free to choose the stiffness kc , damping √ bc , and the auxiliary force Fc . The dimensionless damping factor is ζc = bc /(2 kc m), which will typically be small, around 0.01.

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Yakov Ben-Haim and Scott Cogan

Performance requirement. At specified load, F, and driving frequency, ω , the amplitude of the response of the computational model, Xc , must be an upper bound of the amplitude of the response of the real system Xr . Specifically, eqs. (4) and (6) in general have complex roots. Let |Xc | denote the magnitude of the greatest real or complex root of eq. (6). Let |Xr | denote the greatest positive and purely real (not complex) root of eq. (4), evaluated with the method of harmonic balancing. The performance requirement is: |Xc | − |Xr | ≥ δ (7)

δ is the greatest allowed error in the bound. If δ ≥ 0 then |Xc | must bound |Xr | from above, with an excess no less than δ . For instance, when δ = 0, then |Xc | must be no less than |Xr |. If δ < 0 then |Xc | may be less than |Xr |, but by no more than δ . This performance requirement may at first appear insufficient to determine the computational model. One might be inclined to choose kc , bc and Fc in the computational model so that |Xc | will be very large. But how large? We don’t know how wrong our estimate of the non-linearity is, so we don’t know how large |Xr | could be; it might exceed even a very large value of |Xc |. On the other hand, if |Xc | is vastly greater than |Xr | then the computational bound is useless. The resolution of this indeterminacy is obtained from the robustness function, as we will see. Uncertainty model. We consider uncertainty in the linear and cubic stiffness coefficients of the real model, k1 and k3 . We also consider uncertainty in the load amplitude, F. Let k1 , k3 and F denote best estimates of these quantities (F may is a known but perhaps k3 and F, vary with frequency). Each of these quantities, k1 , unreliable estimate of the corresponding coefficient. We do not know by how much k1 , k3 and F. the estimate in fact errs. Let s1 , s3 and sF denote estimated errors of These error estimates do not constitute knowledge of a worst case. Rather, they reflect some information about the relative errors among the parameters. For instance, k1 may be known more reliably than k3 , in which case s1 < s3 . The absolute errors are unknown and unbounded. In some situations we may know the value one of the quantities with confidence, in which case its s-value is zero. We use a fractional-error info-gap model [2] to represent these uncertainties: k − k − F − F 1 k1 3 k3 U (h) = k1 , k3 , F : ≤ h, ≤ h, ≤ h , h ≥ 0 (8) s1 s3 sF Each of the 3 inequalities can be understood as a fractional error. For instance, considering the cubic coefficient, the inequality states that the fractional error of k3 −k3 the estimate, s3 , is bounded by the horizon of uncertainty, h. Each of the 3 fractional errors is dimensionless, and thus commensurable in terms of the horizon of uncertainty h. Since we do not know the magnitude of error — no realistic worst case is known — the horizon of uncertainty is unbounded. Thus h ≥ 0. When h = 0 then each estimate is correct: there is no uncertainty. The uncertainty set U (h) becomes more inclusive as h increases. The info-gap model is an unbounded family of nested sets of possible realizations of k1 , k3 and F.

Linear Bounds on an Uncertain Non-Linear Oscillator: An Info-Gap Approach

7

Robustness function. We now define the robustness of the linear computational model, eq. (6), to uncertainty in the load amplitude and in the linear and non-linear stiffnesses of the real oscillator, eq. (4). The robustness is the greatest horizon of uncertainty, h, up to which the performance requirement is satisfied for all realizations of the uncertain quantities:

(9) h = max h : min (|Xc | − |Xr |) ≥ δ k1 ,k3 ,F∈U (h)

This robustness function depends on the driving frequency and the nominal force, applied to the non-linear system. It also depends on the design parameters ω and F, of the linear computational model, Fc , kc and bc .

3 Example: Uncertain Cubic Non-Linearity In this section we illustrate the selection of a suite of linear computational models for robustly evaluating the upper bound of the response of an oscillator with an uncertain cubic non-linearity. We will consider multiple uncertainties in section 4. We assume that the other terms in the real model are known reliably. The known parameters are k1 = 0.3, m = 1, and F = 1. We consider various driving frequencies ω . The estimated cubic coefficient is k3 = 0.02 and its estimated error is s3 = 0.1. The dimensionless damping coefficient of the computational model is ζc = 0.01.

(a)

(b)

Fig. 2: (a) Robustness vs stiffness kc of linear computational model, for 3 values of Fc : 0.3, 0.5 and 0.7. δ = 0, ω = 0.8. (b) Robustness vs stiffness kc of linear computational model, for 3 values of δ : -1, 0 and 1. Fc = 0.5, ω = 0.8 Figs. 2a and 2b display the robustness,

h, vs. the stiffness, kc , of the linear computational model.

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Yakov Ben-Haim and Scott Cogan

Consider fig. 2a, which shows robustness curves for three different values of the h, the non-linear auxiliary load, Fc , of the linear model. At any level of robustness,

coefficient k3 can deviate from its estimated value, h around k3 , by as much as ±s3

k3 without violating the performance requirement, eq. (7). Larger variation entails k3 = 0.02. the possibility of violating the requirement. In this example s3 = 0.1 and Thus a robustness of 0.1 means that k3 can vary within ±0.01 around 0.02 without violating the bound requirement. In other words, a robustness of 0.1 means that the estimated non-linear coefficient, k3 , can err by as much as 50% and the linear computational model still provides an upper bound. The driving frequency in fig. 2a is ω = 0.8, so the linear system is at resonance if kc = 0.64. This explains the positive slope of the curves. As kc increases towards resonance, the value of Xc increases which allows the computational model to satisfy the upper bound requirement for a larger range of non-linearity. That is, the robustness increases as kc moves towards resonance from lower values. The curve in fig. 2a, if continued to higher kc values, would display reflection symmetry around kc = 0.64: the robustness would decline as kc moves past resonance to higher values. Fig. 2b shows robustness vs. kc for three choices of the margin-of-error parameter, δ . The robustness increases as δ increases: allowing larger margin of error induces greater robustness. The absolute value of the maximal response of the nonlinear system is in the range of 5 to 10 for robustness up to about 0.1. Thus a margin of error of 1 corresponds to about 10% or 20% of the non-linear response.

(a)

(b)

Fig. 3: (a) Robustness vs load Fc of linear computational model, for 3 values of kc : 0.4, 0.5 and 0.6. δ = 0, ω = 0.8. (b) Robustness vs load Fc of linear computational model, for 3 values of kc : 0.4, 0.5 and 0.6. δ = 0, ω = 0.9 Figs. 3a and 3b show the robustness vs. the auxiliary load of the linear model, for several values of the stiffness of the linear model. These figures, if extended to lower Fc values, would both be reflection-symmetric around Fc = 1, which is the

Linear Bounds on an Uncertain Non-Linear Oscillator: An Info-Gap Approach

9

value of f . This is because the amplitude of Xc , in eq. (6) is determined by the magnitude, but not the sign, of F − Fc . Next we note the monotonic increase of robustness with increase in Fc . Since the magnitude of F − Fc increases as Fc increases over this range of values, we see that |Xc | increases with increasing Fc . Thus the robustness increases monotonically over this range of Fc . The decreasing positive slope of the robustness curves in figs. 3a and 3b results from the non-linearity. The magnitude of the non-linear response, |Xr |, increases as the value of the non-linear coefficient, k3 , gets smaller at ω = 0.8. Thus the value of k3 which produces the maximum non-linear response gets smaller as the horizon of uncertainty increases. Furthermore, the magnitude of the non-linear response increases more, for the same small change in k3 , when k3 is small. Hence the increment of robustness, for each increment in Fc , decreases as Fc increases. Finally, comparing figs. 3a and 3b, we note that greater Fc is needed to achieve a given robustness at the larger driving frequency. This is because ω = 0.9 is further above the resonance of the linear system than ω = 0.8, so |Xc | is smaller in the former case at fixed Fc .

(a)

(b)

Fig. 4: (a) Robustness vs driving frequency ω for 3 values of kc : 0.5, 0.6 and 0.7. Fc = 0.5, δ = 0. (b) Robustness vs driving frequency ω for 3 values of Fc : 0.4, 0.5 and 0.6. kc = 0.6, δ = 0 Fig. 4a shows robustness,

h, vs. driving frequency, ω , for several values of stiffness of the linear computational model, kc . The figure shows that the robustness becomes small and vanishes (“detunes”) rather quickly as the driving frequency changes. However, we see that kc can be used to compensate for this by causing a shift in the ω -range with positive robustness. Fig. 4b shows robustness,

h, vs. driving frequency, ω , for several values of the auxiliary force, Fc . We again see the “detuning” of robustness. However, in contrast to fig. 4a, Fc does not compensate, at least not very efficiently in this example. Fig. 5a shows the robustness,

h, vs. the margin of error for the upper bound, δ , for several combinations of kc and Fc . The three solid curves show progressively

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Yakov Ben-Haim and Scott Cogan

(a)

(b)

Fig. 5: (a) Robustness vs error margin δ , for several combinations of kc and Fc . ω = 0.80. (b) Robustness vs error margin δ , for several combinations of kc and Fc . ω = 0.80 and ω = 0.82

greater robustness as Fc decreases, at fixed kc . This is consistent with fig. 3a if we recall that the extension of that figure to values of Fc below 1 is the mirror image of the figure which is shown. Now notice the dashed curve in fig. 4a, which very closely coincides with the middle solid curve. This is for a slightly lower value of kc and a substantially lower Fc . This demonstrates that kc and Fc can compensate each other, resulting in distinct (kc , Fc ) pairs with essentially the same robustness to uncertainty. Fig. 5b again shows robustness vs. error margin, at two different driving frequencies. The intermediate coinciding solid and dashed curves from fig. 4a are reproduced here, appearing in the upper part of fig. 5a. The pair of curves in the lower part of fig. 5a are at the same values of kc and Fc , but at a slightly larger driving frequency ω . We know from fig. 5b that the robustness can change dramatically with ω , which explains the strong separation of the curves at ω = 0.82 while the same parameters cause closely coinciding curves at ω = 0.80.

4 Example: Multiple Uncertainties In this section we consider uncertainty in all three elements of the non-linear model, as specified in the info-gap model, eq. (8): k1 , k3 and F. In all the calculations in this section we use k1 = 0.3, k3 = 0.02 and F = 1. The estimated error of k3 s1 is s3 = 0.1, which is 5 times the value of k3 . The estimated errors of k1 and F, and sF respectively, are either 0 or also 5 times the respective nominal values. The dimensional damping of the non-linear system is b = 0.02. The parameters of the

Linear Bounds on an Uncertain Non-Linear Oscillator: An Info-Gap Approach

11

linear computational model, when they are not varying on the horizontal axis, are kc = 0.5 or 0.6, Fc = 0.5 and ζc = 0.01, δ = 0, m = 1 and ω = 0.8 in this section. The main conclusion we will draw is that uncertainty in the cubic stiffness coefficient, k3 , dominates the uncertainty in the linear stiffness, k1 , and the load, F.

(a)

(b)

Fig. 6: (a) Robustness vs stiffness kc , for several combinations of s1 and sF . (b) Robustness vs auxiliary force Fc , for several combinations of s1 and sF . kc = 0.5

Fig. 6a shows robustness vs. the stiffness of the computational model, for different values of s1 and sF . These curves can be compared with the middle curve in fig. 2a, which is actually the same as the upper curve in fig. 6a. The most striking aspect of this figure is that introducing uncertainty in both k1 and F (bottom curve) reduces the robustness by no more than about 15%. We also note that the drop in robustness from the top to the middle curve is less than the subsequent drop to the bottom curve. That is, uncertainty in the linear stiffness influences the robustness more than uncertainty in the load. Fig. 6b shows robustness vs. auxiliary force, Fc , and should be compared with the middle curve in fig. 3a (which is the same as the upper curve in fig. 6b). We again see that uncertainty in the load, F, reduces the robustness less than uncertainty in the linear stiffness k1 , and that together they reduce the robustness by less than 15%. Fig. 7a shows robustness vs. driving frequency, ω , where the upper curve is the same as the middle curve in fig. 5a. We see the same effect as in figs. 6a and 6b. Fig. 7b shows robustness vs. the error margin, δ , displaying the same small impact of uncertainty in k1 and F, as compared to uncertainty in k3 .

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(a)

(b)

Fig. 7: (a) Robustness vs driving frequency ω , for several combinations of s1 and sF . s3 =0.1. kc =0.6. (b) Robustness vs error margin, δ , for several combinations of s1 and sF . s3 = 0.1. kc = 0.6

5 Robustness as a Proxy for Probability In this section we discuss a theorem which asserts that the non-probabilistic info-gap robustness is monotonically related to the probability that the non-linear system satisfies the performance requirement. This ‘proxy property’ is important since, when it holds, it implies that a computational model can be chosen which maximizes the probability of success, without knowing the probability distribution of the uncertain variables. The value of maximum probability will remain unknown. We begin with several definitions. Let q = (kc , bc , Fc ) denote the design variables of the computational model, eq. (5). Note that the real system, eq. (1), does not depend on q. Likewise, let c = (k1 , k3 , F) denote the uncertain parameters of the real system whose uncertainty is represented by an info-gap model such as eq. (8). Note that the computational model does not depend on c. For any design, q, of the computational model, let K(q) denote the set of all uncertain parameters c of the real system which satisfy the performance requirement in eq. (7): (10) K(q) = {c : |Xr (c)| ≤ |Xc (q)| − δ } We will suppose that a probability distribution exists for the parameters c of the real system, though this distribution is unknown. For any set, A, of coefficients c, let P(A) denote probability of this set. For any design, q, we define the probability of success as the probability that c takes a value which satisfies the performance requirement, eq. (7). Thus the probability of success of computational design q is the P-measure of the set K(q):

Linear Bounds on an Uncertain Non-Linear Oscillator: An Info-Gap Approach

Ps (q) = P[K(q)]

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(11)

All info-gap models, such as eq. (8), are nested, meaning that: h < h

=⇒ U (h) ⊆ U (h )

(12)

Given an info-gap model, U (h), let

h(q) denote the robustness to uncertainty in the real system as defined in eq. (9), for any design, q, of the computational model. We will say that robustness is a proxy for the probability of success if any change in the design of the computational model which enhances the robustness, does not reduce the probability of success. The following theorem asserts the proxy property for the system studied in this paper (see also lemma 1 in [3]). Theorem. Given the computational model, eq. (5), and the real system, eq. (1), whose uncertainty is represented by an info-gap model which is independent of q. If:

(13) h(q) >

h(q ) then:

Ps (q) ≥ Ps (q )

(14)

Proof. The proof derives from the following sequence of implications:

h(q) >

h(q ) ⇐⇒ |Xc (q)| > |Xc (q )| =⇒ K(q) ⊇ K(q ) =⇒ Ps (q) ≥ Ps (q ) Implication (14): Let us re-write the robustness in eq. (9) as:

h = max h : min (−|Xr (c)|) ≥ δ − |Xc (q)| c∈U (h)

(15) (16) (17)

(18)

The info-gap model is nested, meaning that the sets U (h) become more inclusive as h increases. Thus the inner minimum in eq. (18) cannot increase as h increases. Also, |Xr (c)| and U (h) are independent of q and |Xc (q)| is independent of c. Thus a change in q which enhances robustness can occur if and only if |Xc (q)| has increased. (15) implies (16): Any c in K(q ) satisfies |Xr (c)| ≤ |Xc (q )| − δ which is less than |Xc (q)| − δ . Hence that c also belongs to K(q). (16) implies (17): from the definition of Ps (·) in eq. (11) and since probability in non-decreasing on nested sets.

6 Conclusion We propose a methodology, based on info-gap theory, for designing a linear computational model to represent a non-linear model with uncertain linear and non-linear

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stiffnesses and uncertain load. It is assumed that a worst case for the uncertainties is not known. An info-gap robustness approach is used to study the tradeoffs between robustness and design. The proposed methodology is applied to a 1-dimensional non-linear oscillator which is analyzed in the frequency domain under periodic excitations. In practice, the analyst much decide on the level of robustness required in a given application. A suite of frequency dependent linear models is then generated to provide useful upper bounds on the uncertain non-linear responses. We show that the non-probabilistic info-gap robustness function can be used to choose a computational linear model for which the probability of bounding the non-linear model is maximized, without knowing the probability distribution of the parameters of the non-linear model. Acknowledgements The authors are pleased to acknowledge useful comments by Lior Davidovitch and Oded Gottlieb.

References 1. Yakov Ben-Haim, 2005, Info-gap Decision Theory For Engineering Design. Or: Why “Good” is Preferable to “Best”, appearing as chapter 11 in Engineering Design Reliability Handbook, Edited by Efstratios Nikolaidis, Dan M. Ghiocel and Surendra Singhal, CRC Press, Boca Raton. 2. Yakov Ben-Haim, 2006, Info-gap Decision Theory: Decisions Under Severe Uncertainty, 2nd edition, Academic Press, London. 3. Yakov Ben-Haim, 2007, Info-gap robust-satisficing and the probability of survival, De Nederlandsche Bank Working Papers, #138. 4. Duncan, S.J., Bras, B. and Paredis, C.J.J., 2008, An approach to robust decision making under severe uncertainty in life cycle design, Int. J. Sustainable Design, vol. 1, #1, pp.45–59. 5. Y. Kanno and I. Takewaki, 2006, Robustness analysis of trusses with separable load and structural uncertainties, International Journal of Solids and Structures, vol. 43, #9, pp.2646–2669. 6. Y. Kanno and I. Takewaki, 2006, Sequential semidefinite program for maximum robustness design of structures under load uncertainty, Journal of Optimization Theory and Applications, vol. 130, #2, pp.265–287. 7. Matsuda, Y. and Y. Kanno, 2008, Robustness analysis of structures based on plastic limit analysis with uncertain loads, Journal of Mechanics of Materials and Structures, vol. 3, pp.213– 242. 8. S.G. Pierce, K. Worden and G. Manson, 2006, A novel information-gap technique to assess reliability of neural network-based damage detection, Journal of Sound and Vibration, 293: #1–2, pp.96–111. 9. P. Vinot, S. Cogan and V. Cipolla, 2005, A robust model-based test planning procedure, Journal of Sound and Vibration, vol. 288, #3, pp.571–585. 10. K. Worden, and G.R. Tomlinson, Nonlinearity in structural dynamics: detection, identification and modeling, 2001, IOP, Bristol.

Quantification of uncertain and variable model parameters in non-deterministic analysis Dirk Vandepitte, David Moens

Abstract A multitude of models for non-deterministic structural analysis have been developed. They are all designed to predict how non-nominal input parameter values propagate through the different phases of the calculation procedure. A literature review on a number of publications that present practical examples shows that the relation between the numerical formalism that describes the uncertain or variable quantity and the physical reality is not so clear. In almost all cases the authors (have to) make assumptions on the non-deterministic nature of the physical quantity, especially for material properties. However, the sensitivity of the structural response to material parameter changes can be very significant. The authors recommend that the numerical formalism for model parameters should be well adapted to physically observed variations.

1 Introduction Numerical analysis is used throughout in technical analysis and scientific research. The paper takes structural finite element analysis as a reference. In many cases precise numerical data on one or more model parameters are not available, either because the parameter does not have a single value or because its value is not precisely known. Unless the analyst is satisfied with assumptions to assign certain values for each of these parameters, a non-deterministic analysis may be viable. However, some conditions must be met in order for a non-deterministic analysis run to yield Dirk Vandepitte K.U.Leuven, Department of Mechanical Engineering, PMA division PMA, Celestijnenlaan 300B, 3001 Heverlee, Belgium, e-mail: [email protected] David Moens K.U.Leuven Association, Lessius Hogeschool — Campus De Nayer Department of Applied Engineering, J. De Nayerlaan 5, 2860 Sint-Katelijne-Waver, Belgium, e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 2, © Springer Science+Business Media B.V. 2011

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practically relevant results. The paper discusses these conditions, the types of analysis that are feasible, the requirements on the input data, and the availability of useful data. In dynamic analysis the effect of uncertainty and variability depends to a very significant extent on frequency, with both effects strongly increasing with increasing frequency. Whereas natural frequencies of the first few modes usually do not change much for moderate changes of input parameter values, this effect is much more pronounced at medium and high frequencies. This paper considers only the low frequency regime, where individual local model parameters determine structural behaviour. They include three types: stiffness, mass and damping. Stiffness and mass parameters depend mainly on the geometrical lay-out and material selection of the structure and its components, and boundary conditions also play a role. Although local damping characteristics of specific materials and treatments may be important, damping is usually a rather global property of a built-up structure, and it is often modelled with one or just a few global model parameters. This paper focusses on local stiffness and mass parameters. There are two basic categories for non-deterministic analysis: probabilistic analysis is feasible in case of aleatory uncertainty, and non-probabilistic analysis can be used in case of epistemic uncertainty. Input data require a specific numerical formalism, with probability density functions for aleatory uncertainty and interval or fuzzy numbers for epistemic uncertainty. In addition to these distinct categories, there are intermediate categories. In a first section, this paper briefly presents a consistent structure for the representation of uncertain data in each of the cases of nondeterminism. The second section of the paper gives a wide selection of non-deterministic model data as they are reported in numerical analyses in journal articles and conference papers. Despite the apparent simplicity of data formats, the authors observe that in almost all cases the analyst makes assumptions on the nature of the nondeterminism of the problem and on the quantification of the uncertain or variable parameters. The unavailability of validated input data is a circumstance that is often encountered, but that does not justify inadvertent assumptions. It is observed that most authors assign the non-deterministic nature of the problem mostly to uncertainty and/or variability of the material characteristics. For this reason, this paper gives an overview of some sources of material data. It is shown that engineering material properties may be very sensitive to production-related aspects of structural components. This phenomenon is very pronounced for composite materials.

2 Numerical representation of parameter uncertainty and variability Engineering design is the activity of design and development of technical products. A technical product is built to fulfil a well specified function under more or less well prescribed conditions of utilisation. This process consists of a number of anal-

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ysis verifications on a virtual product. A common procedure for design verification is finite element analysis, a numerical method for the simulation of the effect of mechanical or thermal loads on a product. As most product parameters are undetermined in the initial phases of design, a range of non-deterministic properties have to be taken into account. This paper discusses the effects of non-determinism on engineering analysis using the finite element method (FE).

2.1 Definitions In literature, the use of the terminology error, uncertainty and variability is not unambiguous. Different researchers apply the same terminology but the meaning attached to these is rather inconsistent. This necessitates a profound clarification of the terminology for each publication which treats uncertainties. This work does not propose a new terminology, but applies the terminology proposed by O BERKAMPF [1]. Some additional nuances are, however, necessary in order to enable clear distinction between probabilistic and non-probabilistic quantities. The term variability covers the variation which is inherent to the modelled physical system or the environment under consideration. Generally, this is described by a distributed quantity defined over a range of possible values. The exact value is known to be within this range, but it will vary from unit to unit or from time to time. Ideally, objective information on both the range and the likelihood of the quantity within this range is available. Some literature refers to this variability as aleatory uncertainty or irreducible uncertainty, referring to the fact that even when all information on the particular property is available, the quantity cannot be deterministically determined. An uncertainty is a potential deficiency in any phase or activity of the modelling process that is due to lack of knowledge. The word potential stresses that the deficiency may or may not occur. This definition basically states that uncertainty is caused by incomplete information resulting from either vagueness, nonspecificity or dissonance [2]. Vagueness characterises information which is imprecisely defined, unclear or indistinct. It is typically the result of human opinion on unknown quantities (“the density of this material is around x”). Nonspecificity refers to the availability of a number of different models that describe the same phenomenon. The larger the number of alternatives, the larger the nonspecificity. Dissonance refers to the existence of conflicting evidence of the described phenomenon, for instance when there is evidence that a quantity belongs to disjoint sets. Possibly, limited objective information is available, for instance when a range of possible values is known. In most cases, however, information on uncertainties is subjective and based on some expert opinion. Others in literature refer to this uncertainty as reducible, epistemic or subjective uncertainty. An error is defined as a recognisable deficiency in any phase of modelling or simulation that is not due to lack of knowledge. The fact that the error is recognisable states that it should be identifiable through examination, and as such is not

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caused by lack of knowledge. A further distinction between acknowledged and unacknowledged errors is possible. Errors will not be considered further in this paper.

2.2 Discussion and extension of the definitions The above definitions of uncertainty and variability are fairly straightforward and comprehensible. However, they are not mutually exclusive, since a variability could be subject to lack of knowledge when information on its range or likelihood within the range is missing. This is for instance the case for every design dimension subject to tolerances, but without further specification of manufacturing process or supplier. The tolerances represent the bounds on the feasible domain, but there is no information on the likelihood of the possible values within these bounds. Consequently, because there is a lack of knowledge, such a variability is also an uncertainty. It is referred to here as an uncertain variability. Some vague knowledge may be available (“the mean value is approximately x”) but also nonspecificity may play an important role in the uncertainty, for instance in choosing an appropriate model to describe a random quantity. Opposed to the uncertain variability, a certain variability refers to a variability the range and likelihood of which are exactly known. On the other hand, it appears logical to state that every property in a numerical model corresponding to a physical quantity is a variability, since it will eventually have a range of possible values and a likelihood inside this range in the physical model. This argumentation implies that all uncertainties are also variabilities. In practice however, the majority of model properties are implemented as constant deterministic values in the numerical model. Though they are subject to variation, the influence of their variability on the analysis result is considered to be negligible. Often, uncertainties refer to a possible lack of knowledge in these deterministic properties. This type of uncertainty is referred to as invariable uncertainty. The word invariable in this case does not mean that the property cannot change over different analyses. According to the definition of uncertainty, it will change when additional information is gathered that decreases the amount of uncertainty. The invariable uncertainties typically occur in model properties for model parts that are difficult to describe numerically, but considered constant in the final physical product (connections, damping, . . . ). Other examples are design properties which have negligible variability but which are not defined exactly in an early design stage. Figure 1 gives a graphical illustration of the proposed subdivision of the definitions for uncertainty and variability. The group of variabilities may be further subdivided into two categories. Intersample variability is the property of a population of nominally identical realisations of a particular product, with each individual element of the population possibly exhibiting scatter. Intra-sample variability is a property of one particular realisation — of which other realisations possibly exist — that exhibits one or more properties that may change over time, due to temperature differences, ageing, . . .

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Fig. 1: Classification of variabilities and uncertainties in numerical modelling

3 Literature review on uncertain model and material data The authors have conducted a review of journal publications in the field of nondeterministic analysis in structural dynamics. Only the numerical examples are considered, with a focus on how the input data are described. Most of the papers that listed are published in the journal for Computer Methods in Applied Mechanics and Engineering. Uncertainty mostly applies on material constitutive data, and for this reason several materials related journals are listed as well.

3.1 Non-probabilistic models Non-probabilistic models are used in different conditions, usually when limited data are available, and when a probabilistic interpretation is not required. The following examples illustrate when these conditions are met: • concept models are used in early stages of engineering design, when only general and approximate information on a design case is available. M OENS et al. [3] have built a concept model of a truck-trailer combination. The model consists of discrete mass elements for the major components and systems in the truck, and individual springs. The truck manufacturer uses simple MatLab models to investigate vehicle dynamics in an early stage, and the objective of this study was to predict maximum response levels for different excitations. In a later stage of product design, M ASSA et al. [4] have used a fuzzy description to investigate different variants of a suspension triangle. The same authors have also used this approach on an impactor [5]. • investigation of the effect of production tolerance is done using interval analysis, possibly extended with subjective interpretation using fuzzy numbers. M OENS et al. [6] have investigated the effect of typical plate thickness on the natural frequencies of the stiffened cylinder of the C O ROT satellite. • uncertainty of specific model parameters occurs for specific quantities that are hard to quantify, such as the stiffness of a polymer layer in the windshield of a vehicle. The effect of uncertain thickness of the polymer layer on natural frequencies was also taken into account [7]. Another uncertain parameter that was investigated on the same structure was the curvature of the windshield [8].

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• model parameters with imprecise values may occur when incomplete data are available. N OOR et al. [9] have calculated the nonlinear response of a stiffened composite aircraft fuselage panel with a circular cut-out. Young’s modulus and Poisson ratio are sensitive to the alignment of fibre orientations, and they are modelled with triangular fuzzy numbers with a ±15% support. The flange width and the web height of the T-stiffener are also considered to be uncertain. • the imprecise effect of process steps occurs in several production processes. K HALED et al. [10] have used a fuzzy set approach in conjunction with FE analysis to predict the residual stress field in the heat affected zone of a welded structure that undergoes martensitic transformations during the cool-down part of the weld thermal cycle. In conclusion, non-probabilistic models are fed with input data that are only subjectively linked to realistic problem data. If the bounds of the interval are well defined, and if the non-probabilistic analysis procedure does not introduce artificial conservatism, the output is a realistic set of bounds on output quantities.

3.2 Probabilistic models Probabilistic models are used for several decades already, in a very wide range of applications. The list below is a selection of applications in structural dynamics and also in static structural analysis with random stiffness characteristics. 1. L IONNET and L ARDEUR [11] have developed a hierarchical model for the effect of variability on the booming noise in a passenger car. They identified three different sources of variability: engine vibrations, the dynamic stiffness of the engine mounts and the transfer function from the vibrating body to the noise level. Each of these sources is measured independently, and modelled with a normal distribution. Interior noise level at the driver’s ear shows measured intravariability level between 2 and 20 dB depending on the engine speed. The intervariability of booming noise was also measured, and it is found to be lower than the intra-variability one. 2. P ELLISSETTI et al. [12] conducted a reliability analysis on vibration levels of the I NTEGRAL satellite. Extensive material data were available. No less than 1319 independent random variables were defined with coefficients of variation ranging from 4% (for mass density) to 12% (mainly for composite material properties). Particular emphasis is given to the effect of the uncertainty in the damping on the reliability of the considered structure. Hence, various levels of the uncertainty in the damping have been investigated. In all the cases the damping ratios have been assumed to follow a log-normal distribution and to be mutually independent. 3. S CHU E¨ LLER [13] applied Monte Carlo simulation for the reliability analysis of a 12-storey building subjected to earthquake excitation. 244 random variables model the stiffness of confined reinforced concrete, and the covariance matrix is modelled with 80 Karhunen-Lo`eve terms.

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4. C HUNG et al. [14] developed a stochastic finite element model of Glare, a metal laminate with a glass fibre reinforced layer in between two aluminum layers. The properties of the metal layers and the Poisson ratio of the glass prepreg layers are considered to be deterministic. The elastic modulus of the fibre reinforced layer is assumed to be a random process, and it is modelled with a KarhunenLo`eve expansion. The covariance function for the prepreg layers is modelled with an exponential function with a correlation length which is longer in the fibre direction than in the transverse directions. The values for the correlation length are based on assumptions. 5. S ARKAR and G HANEM [15] have integrated a number of frequency-domain dynamic analysis procedures of randomly disordered structural systems in the medium frequency range into the stochastic finite element method with an application to the analysis of the dynamics of a coupled uncertain rod assembly subjected to an external excitation. Young’s modulus of each rod is assumed to be an independent and homogeneous Gaussian random field with a coefficient of variation equal to 5%. The autocovariance function of the process is chosen to be of the form R(x, y) = e−|x−y|/b , where b is the correlation length, assumed to be equal to half of the length of each rod. 6. AGARWAL and A LURU [16] propose a stochastic framework to handle uncertain coupled electromechanical interaction, arising from variations in material properties and geometrical parameters such as gap between the microstructures, applicable to the static analysis of electrostatic MicroElectroMechanical Systems. The stochastic mechanical analysis quantifies the uncertainty associated with the deformation of MEM structures due to the variations in material properties and/or applied traction, and the stochastic electrostatic analysis quantifies the uncertainty in the electrostatic pressure due to variations in geometrical parameters or uncertain deformation of the conductors. The Young’s modulus is assumed to be a uniformly distributed random variable with a mean value of 169MPa and a coefficient of variation of no less than 20%. 7. FALSONE and F ERRO [17] present a procedure that gives the exact relationship between the response and the random variables representing the structural uncertainties in structures that are built up of beam-like components, under the assumption that a point-discretisation method is used for the representation of the uncertain random field. An uncertain Young’s modulus is considered for each finite element, with “high” correlation (COV equal to 70%) for adjacent elements and “low” correlation (COV equal to 40%) for other cases 8. S TEFANOU and PAPADRAKAKIS [18] present a stochastic formulation of the triangular composite facet shell element for the case of combined uncertain material (Young’s modulus, Poisson’s ratio) and geometric (thickness) properties. These properties are assumed to be described by uncorrelated two-dimensional homogeneous stochastic fields. The spatial variability in Young’s modulus and thickness of the shells is described by two uncorrelated homogeneous stochastic fields with coefficient of variation equal to 10%. The same assumption is made for the stochastic fields describing the random variation of Young’s modulus and

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Poisson’s ratio. A prescribed form for the power spectral density function that characterises the two stochastic fields in both cases is assumed. Only the first two publications [11] and [12] give a reference to data that are based on measurements or on thorough analysis. The authors of the other papers content themselves with assumptions on the nature and the quantification of stochasticity. Quite different levels are assumed for the coefficients of variation (ranging from 4% to 30%). Sometimes the coefficients of variation are different for different properties, and sometimes they are not. Sometimes a correlation between different properties is assumed, and sometimes properties are independent. The models for spatial variation are very diverse, with different assumptions for correlation length. For uncertain variabilities, a representation by a single random quantity is generally not sufficient. Engineering scientist F REUDENTHAL [19] stated in 1961 that “. . . ignorance of the cause of variation does not make such variation random.”. By this, he means that when crucial information on a variability is missing, it is not good practice to model it as a probabilistic quantity represented by a single random PDF. On the contrary, in this case it is mandatory to apply a number of different probabilistic models to examine the effect of the chosen PDF on the result. For instance, when the range of the variability is known but the information on the likelihood is missing, all possible PDFs over the range should be taken into consideration in the analysis. The analyst will generally select only a few probabilistic models which he considers consistent with the limited available information or most appropriate to obtain as much knowledge as possible on the result. Another important criterion in the selection of the type of distribution is the nature of the distribution function itself and its relation to the phenomena that it represents. The risk function is a useful indicator in this respect. One conclusion is firm however, all authors apply variability on material characteristics, mostly on stiffness parameters. Material models are the most uncertain parameters in variability analysis (not taking into account damping).

3.3 Material data With the observation that material parameters are the main source for non-determinism in probabilistic models, the literature study is extended to materials data. The mechanical properties of most common structural materials, especially metals and unreinforced polymers, are relatively well known. However, the range of materials is very wide, and properties may differ with precise chemical composition, with thermal treatment and they may even be different with different manufacturers. In addition, material properties have some scatter. However, over all physical and mechanical properties that a material exhibits, mass and stiffness usually are fairly close to their nominal values, unlike strength, which depends strongly on chemical composition and heat treatment. Thickness of the unworked piece also plays a role in material strength, with strength decreasing with increasing thickness.

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Non-determinism in the properties of a specific material is a case of variability. Real materials are characterised in experimental measurements. The size of the set of measurements that are taken in identical conditions determines if a probability density function can be established with sufficient accuracy. If a sufficient number of individual measurements are available, the variability can be considered certain.

3.3.1 Metals and polymer materials databases The first source for metals and polymer materials data is a materials database. The MIL-handbook [20], which is now published on the web, and other web based databases such as matweb [21] and efunda [22] contain a large number of records for many material variants, even from different manufacturers. They usually specify nominal values, sometimes complemented with an indication of the probability distribution. These databases do not give any indication on the spatial scatter within a test coupon. The test procedure implies that stiffness values are averaged numbers of the length of the sensor that is used, whereas strength is based on a local value in the section of fracture. Experimental data on spatial scatter are not available.

3.3.2 Composites properties An important class of materials that will continue to gain importance is composites. The advantage of these materials is their excellent ratio of mechanical properties over mass density, which is a crucial asset especially in the transportation industry. These materials also offer wide opportunities for tailored solutions. The designer has many degrees of freedom, including the selection of raw materials for both the matrix and the fibre reinforcement, the architecture of the fibre reinforcement (unidirectional fibres, woven fabrics, knitted fabrics, braided fabrics, non-crimp fabrics, . . . , each with its own variants), the fibre volume fraction, the number of layers and the orientation of layers. For the analyst, this large set of design degrees of freedom translates into a wide range of model parameters, and inevitably also a wide range of uncertain or imprecise material data. Figure 2 illustrates some of these effects. The left hand side of the figure shows the variation of the elastic orthotropic stiffness constants for different orientations of a uniaxially reinforced glass fibre composite lamina with respect to the applied uniaxial tensile load. E11 is the modulus in the longitudinal direction along the fibre orientation, and Exx is the modulus in the loading direction that has an angle θ with respect to the fibre direction. The graph shows a significant decrease of stiffness with increasing misalignment of the fibre. The right hand side of the figure is valid for a cross-ply (0◦ -90◦ ) carbon-epoxy system. The graph shows the variation of the Young’s modulus for different alignments of the the fibre orientations with respect to the loading direction. The graphs show that the equivalent material stiffness depends strongly on the fibre placement. An imprecise placement of the fibre inevitably leads to a change of stiffness with respect to

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Fig. 2: Dependency of in-plane material parameters on the orientation θ of the major fibre axis to the loading direction; left: variation of the elastic constants of a continuous E-glass fibre lamina [23]; right: variation of the tensile Young’s modulus for a cross-ply carbon-epoxy composite [24]

the nominal values. The left hand side of the graph also shows that the orthotropic elastic constants are inter-related. Another geometrical parameter that determines the homogenised stiffness characteristics of a textile composite material is the so-called crimp factor. It is a measure of the waviness of the yarn through the thickness of the panel. A general tendency is that the equivalent modulus of a textile composite increases with decreasing crimp.

3.3.3 Multi-scale models for spatial variation of material properties Recent research has brought forward significant advances in models that describe different aspects of material non-homogeneity. Extensive research efforts are currently ongoing to develop a multi-scale modelling procedure at successive scales. Depending on the type of material, the micro-scale describes properties with a reference length in the order of 10−6 − 10−4 m, the meso-scale describes properties with a reference length in the order of 10−4 − 10−2 m, and entire component structural behaviour is described on the macro-scale, with reference lengths in the order of 10−2 − 100 m and above. The step from a lower level to a higher level is made using homogenisation procedures, that assign overall properties at a higher scale based on lower scale data. So far, these models are mainly deterministic. When these models will be well established, they present an excellent opportunity to introduce variability at the appropriate level, and to predict the propagation of their effect to a higher level, and ultimately to the entire component. C HARMPIS and S CHU E¨ LLER [25] have already made proposals to materials researchers to develop these models. Experiments will however always be required to validate these models. Multi-scale models also have the advantage that spatial variation of homogenised properties can be described based on lower scale characteristics. This presents op-

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portunities for realistic quantification of random fields, for which experimental data are currently missing. Some initial efforts to establish stochastic models for specific purposes have already been taken in other applications. As an example, the spatial distribution of crystal orientations affects plastic behaviour [26]. Other types of non-homogeneous materials, such as metal foam are also described using stochastic models [27].

3.4 Other model properties In addition to material properties and geometrical dimensions and shapes, other FE model characteristics exhibit some kind of uncertainty or variability as well. A delicate property is the boundary condition with which a structure is attached to the environment. Only one reference has been identified that addresses ncertainty on boundary conditions for buckling analysis of cylindrical shells with random boundary geometric imperfections [28]. FE models typically use either pinned of fixed conditions. In a pinned connection displacements are prescribed and rotations are free, and in a fixed connection both displacements and rotations are fixed. These conditions correspond to an infinitely stiff connection, which can never be realised in practice. The stiffness of the connection may be very small or very large, but it is always finite. The non-determinism has definitely a character of uncertainty, and an interval number or a fuzzy number seems to be the best representation. Damping is another unknown quantity. Physically realistic models for damping are not available, and it may even be hard to characterise damping from experiments. An interval number is again the most appropriate model.

3.5 Alternative approaches: non-parametric model concept and info-gap theory An alternative strategy is the non-parametric modelling concept, that was originally introduced by S OIZE in 2000 [29]. Rather than modelling the variability on each individual parameter, the generalised matrices of a mean reduced matrix model of the structure are replaced by random matrices whose probability model is constructed with the maximum entropy principle [30]. This is a promising unified approach that brings together uncertainty and probability. P ELLISSETTI et al. [12] have applied this concept in the reliability analysis of a satellite structure subjected to harmonic base excitation in the low frequency range with respect to the exceedance of critical frequency response thresholds. The results indicate that for low levels of uncertainty in the damping, the non-parametric model provides conservative predictions about the exceedance probabilities. For high levels of damping uncertainty the opposite is the case.

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The same research group has set up a procedure for the experimental identification and the validation of a non-parametric probabilistic approach allowing model uncertainties and data uncertainties to be taken into account in the numerical model developed to predict low- and medium-frequency dynamics of structures [31]. The analysis is performed for a composite sandwich panel representing a complex dynamical system which is sufficiently simple to be completely described and which exhibits not only data uncertainties, but above all model uncertainties. In a more recent paper [32] the same author has extended this approach to structural vibrations and vibro-acoustics. Another alternative approach is info-gap theory, introduced by B EN -H AIM. P IERCE et al. present a case study [33].

3.6 Summary of observations • Probabilistic methods provide more information than non-probabilistic methods; however, both families are highly complementary. • The number of publications on probabilistic methods exceeds the non-probabilistic ones. • Almost all publications refer to aleatory uncertainty in material parameters, but there are very few references to uncertainty on other important FE model parameters that are not precisely known, such as boundary conditions, although FE results are highly sensitive to them. • Very few publications refer to validated data, and most authors who publish in the leading scientific journal content themselves with assumptions on the nondeterministic nature of the model parameters. • Very different values are assumed for the coefficients of variation on material parameters such as Young’s modulus: from 4% to even 30% for isotropic materials. • Literature does not provide any evidence on values for spatial scatter; correlation length is based on assumptions, apparently related to the length of the component correlation between model parameters is not taken into account.

4 Conclusions Researchers follow diverse strategies when they introduce non-determinism in their engineering analysis, and the type of data that are available does not necessarily match with the objectives of the analysis. The availability of data determines the type of non-deterministic analysis that can be executed without unintentional misrepresentation of data and inadvertent introduction of unvalidated assumptions. Inversely, a specific type of analysis can only be executed when the model data are available in a suitable format. The appropriate data format depends on the phase of

Quantification of uncertain and variable model parameters in non-deterministic analysis

27

development of the structure that is considered and on the type of parameter that is modelled: material data, geometrical data, loads data, boundary conditions and spatial distribution of model parameters. The authors perceive a need for a coordinated effort by the scientific research community to collect reliable data on different types of model parameters in an appropriate format for non-deterministic analysis and to make available these data to their fellow researchers and to the engineering community. Acknowledgements This work was conducted with the financial support of the IWT Flanders through SBO-project 060043 Fuzzy Finite Elements, of FWO Flanders through project G.0476.04 and of the European Commission in the Marie Curie Research & Training Network MADUSE in the 6th framework programme.

References 1. Oberkampf, W., DeLand, S., Rutherford, B., Diegert, K., and Alvin, K., “A New Methodology for the Estimation of Total Uncertainty in Computational Simulation,” Proceedings of the 40th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, AIAA-99-1612, 1999, pp. 3061–3083 2. Klir, G. and Folger, T., Fuzzy Sets, Uncertainty and Information, Prentice Hall, Englewood Cliffs, 1988. 3. Moens, D., Vandepitte, D., “Interval sensitivity theory and its application to frequency response envelope analysis of uncertain structures,” Computer Methods in Applied Mechanics and Engineering, 2007, 196(21-24), pp. 2486–2496 4. Massa, F., Tison, T., Lallemand, B., “A fuzzy procedure for the static design of imprecise structures,” Computer Methods in Applied Mechanics and Engineering, 2006, 195(9-12), pp. 925–941 5. Massa, F., Rufin, K., Tison, T., Lallemand, B., “A complete method for efficient fuzzy modal analysis,” Journal of Sound and Vibration, 2008, 304(1-2), pp. 63–85 6. Moens, D., Vandepitte, D., “Recent advances in non-probabilistic approaches for nondeterministic dynamic finite element analysis,” Archives of Computational Methods in Engineering, 2006, 13(3), pp. 389–464 7. Moens, D., De Munck, M., Vandepitte, D., “Envelope frequency response function analysis of mechanical structures with uncertain modal damping characteristics,” Computer Methods in Engineering Science, 2006, 22(2), pp. 129–149 8. d’Ippolito, R., Tabak, U., De Munck, M., Donders, S., Moens, D., Vandepitte, D., “Modelling of a vehicle windshield with realistic uncertainty,” Proceedings of ISMA 2006, 2006, pp. 2023– 2032 9. Noor, A.K., Starnes, J.H.Jr., Peters, J.M., “Uncertainty analysis of composite structures,” Computer Methods in Applied Mechanics and Engineering, 2000, 185(2-4), pp. 413–432 10. Khaled, A.-T., Noor, A.K., “Uncertainty analysis of welding residual stress fields,” Computer Methods in Applied Mechanics and Engineering, 1999, 179(3-4), pp. 327–344 11. Lionnet, C., Lardeur, P., “A hierarchical approach to the assessment of the variability of interior noise levels measured in passenger cars,” Noise control engineering journal, 2007, 55(1), pp. 29–37 12. Pellissetti, M., Capiez-Lernout, E., Pradlwarter, H., Soize, C., Schu¨eller, G.I., “Reliability analysis of a satellite structure with a parametric and a non-parametric probabilistic model,” Computer Methods in Applied Mechanics and Engineering, 2008, 198(2), pp. 344–357

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13. Schu¨eller, G.I., “Efficient Monte Carlo simulation procedures in structural uncertainty and reliability analysis - recent advances,” Structural Engineering and Mechanics, 2009, 32(1), pp. 1–20 14. Chung, D.B., Gutierrez, M.A., de Borst, R., “Object-oriented stochastic finite element analysis of fibre metal laminates,” Computer Methods in Applied Mechanics and Engineering, 2005, 194(12-16), pp. 1427–1446 15. Sarkar, A., Ghanem, R., “Mid-frequency structural dynamics with parameter uncertainty,” Computer Methods in Applied Mechanics and Engineering, 2002, 191(47-48), pp. 5499–5513 16. Agarwal, N., Aluru, N.R., “Stochastic modeling of coupled electromechanical interaction for uncertainty quantification in electrostatically actuated MEMS,” Computer Methods in Applied Mechanics and Engineering, 2008, 197(43-44), pp. 3456–3471 17. Falsone, G., Ferro, G., “An exact solution for the static and dynamic analysis of FE discretized uncertain structures,” Computer Methods in Applied Mechanics and Engineering, 2007, 196(21-24), pp. 2390–2400 18. Stefanou, G., Papadrakakis, M., “Stochastic finite element analysis of shells with combined random material and geometric properties,” Computer Methods in Applied Mechanics and Engineering, 2004, 193(1-2), pp. 139–160 19. Freudenthal, A., “Fatigue Sensitivity and Reliability of Mechanical Systems, especially Aircraft Structures,” WADD Technical Report 61-53 1961 20. United States Department of Defense, “Military Handbook – metallic materials and elements for aerospace vehicle structures,” edition 5J, 2003, http://www.weibull.com/ knowledge/milhdbk.htm 21. http://www.matweb.com 22. http://www.efunda.com 23. Mallick, P.K., “Fiber-reinforced composites: Materials, Manufacturing and Design. Third edition,” 2008, CRC Press - Taylor & Francis Group 24. Kawai, M., Honda, N., “Off-axis fatigue behavior of a carbon/epoxy cross-ply laminate and predictions considering inelasticity and in situ strength of embedded plies,” International Journal of Fatigue, 2008, 30(10-11), pp. 1743–1755 25. Charmpis, D.C., Schu¨eller, G.I., Pellissetti, M.F., “The need for linking micromechanics of materials with stochastic finite elements: a challenge for materials science,” Computational Materials Science, 2007, 41(1), pp. 27–37 26. Sankaran, S., Zabaras, N., “Computing property variability of polycrystals induced by grain size and orientation uncertainties,” Acta Materialia, 2007, 55(7), pp. 2279–2290 27. Ramamurti, U., Paul, A., “Variability in mechanical properties of a metal foam,” Acta Materialia, 2004, 52(4), pp. 869–876 28. Schenck, C.A., Schu¨eller, G.I., “Buckling analysis of cylindrical shells with random boundary and geometric imperfections,” Proceedings of ICOSSAR01, June 2001, Swets & Zeitlinger, Lisse, CDrom 29. Soize, C., “A nonparametric model of random uncertainties on reduced matrix model in structural dynamics,” Probabilistic Engineering Mechanics, 2000, 15(3), pp. 277–294 30. Soize, C., “A comprehensive overview of a non-parametric probabilistic approach of model uncertainties for predictive models in structural dynamics,” Journal of Sound and Vibration, 2005, 288(3), pp. 623–652 31. Chen, C., Duhamel, D., Soize, C., “Probabilistic approach for model and data uncertainties and its experimental identification in structural dynamics: case of composite sandwich panels,” Journal of Sound and Vibration, 2005, 294(1-2), pp. 64–81 32. Soize, C., Capiez-Lernout, E., Durand, J.F., Fernandez, C., Gagliardini, L., “Probabilistic model identification of uncertainties in computational models for dynamical systems and experimental validation,” Computer Methods in Applied Mechanics and Engineering, 2008, 198(1), pp. 150–163 33. Pierce, S.G., Ben-Haim, Y., Worden, K., Manson, G., “Evaluation of neural network robust reliability using information-gap theory,” IEEE Transactions on Neural Networks, 2006, 17(6), pp. 1349–1361

Vibrations of layered structures with fuzzy core stiffness/fuzzy interlayer slip Rudolf Heuer and Franz Ziegler

Abstract Mainly the matrix in composite structures exhibits fuzzy randomness of the material parameters. When extending the work on two and symmetric, three layer beam-, plate- and shell structures based on the definition of an equivalent effective homogeneous model, to include either fuzzy interface slip or fuzzy core stiffness, we can avoid numerical analyses schemes and work out the effects on the dynamic properties of these fuzzy structures. Fully analyzed within the scope of this paper is a simply supported sandwich beam with fuzzy core material parameters. The analysis of this illustrative example is based on the interval representation (interval of confidence at a given level of presumption, i.e. α -cut) with a triangular fuzzy membership function of the core shear stiffness prescribed. Fuzzy membership functions of the natural frequencies are defined using fuzzy set theory, however, avoiding artificial uncertainties. Under time-harmonic excitation, the dynamic magnification factors and, with light modal structural damping taken into account, the fuzzy phase angles of the modal response are evaluated. Thus, modal superposition of forced vibrations becomes fuzzy in both, the time and the amplitude response. Where possible, envelope functions are defined.

1 Introduction Mainly the matrix in composite structures exhibits fuzzy randomness of the material parameters. Thus generally, analyses require application of the fuzzy finite element method, see e.g. [1] and [2]. Alternatively, for the formulation using stochastic finite Rudolf Heuer Civil Engineering Department, Vienna University of Technology Karlsplatz 13/E2063, A-1040 Vienna, e-mail: [email protected] Franz Ziegler Civil Engineering Department, Vienna University of Technology Karlsplatz 13/E2063, A-1040 Vienna, e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 3, © Springer Science+Business Media B.V. 2011

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Rudolf Heuer and Franz Ziegler

elements see e.g. [3]. When extending the work on two and symmetric three layer beam-, plate- and shell structures based on the definition of an equivalent effective homogeneous model, see [4]–[6], to include either fuzzy interface slip or fuzzy core stiffness, we can avoid numerical analyses schemes and analytically work out the effects on the dynamic properties of these fuzzy structures. Within the scope of this paper, a simply supported sandwich beam with fuzzy core material parameters is fully analyzed. It should be mentioned that the solution technique remains applicable to even polygonal shaped composite plates since a decomposition into two “membranes” has been explored, see again [5], and [7]. The analysis of the illustrative example, mentioned above, is based on the interval representation (interval of confidence at a given level of presumption, i.e. α -cut) with a triangular fuzzy membership function of the core shear stiffness prescribed. Fuzzy membership functions of the natural frequencies are defined using fuzzy set theory [8] and [9], however, avoiding artificial uncertainties. Under time-harmonic excitation, the dynamic magnification factors and, with light modal structural damping taken into account, the fuzzy phase angles of the modal response are evaluated. Where possible, envelope functions are defined. Since the action of imposed eigenstrains (e.g. piezo-electric strains in smart layers) is considered in the homogenized equivalent fuzzy beam, a fuzzy controller can be designed to annihilate the forced vibrations under the condition of no additional stress, see [10] for “impotent eigenstrains”, cf. Mura [11].

2 Fuzzy sandwich beams 2.1 Three-layer beams Sandwich structures are commonly defined as three-layer type constructions consisting of two thin face layers of high-strength material attached to a moderately thick core layer of low strength and density, Plantema [12], Stamm-Witte [13]. Dynamic response analyses require higher-order theories, a review of the equivalent single layer and the layer wise laminate theories is provided by Reddy [14], see also Irschik [15]. Effects of interlayer slip have been discussed for elastic bonding by Hoischen [16], Goodman and Popov [17], Chonan [18] and for more general interlayer slip laws by Murakami [19]. Heuer [20] presents complete analogies between various models of sandwich structures, even with or without interlayer slip, with homogenized single layer structures of effective parameters. For symmetrically three-layer beams with perfect bonds the following assumptions are made: (i)

The faces are rigid in shear; however, their individual bending stiffness is not neglected. (ii) The bending stiffness of the core is neglected with shear stiffness taken into account. Alternatively, for sandwich beams with elastic interlayer slip (for a more general linear visco-elastic-slip see again [4]), all three layers are considered to be rigid

Vibrations of layered structures with fuzzy core stiffness/fuzzy interlayer slip

31

in shear, with the linear elastic shear traction in the physical interfaces being proportional to the displacement jumps with an elastic interface stiffness understood. For both beam models mentioned above, the resulting equation in terms of the deflection w of the elastic centroid of the cross-section is derived by means of proper combinations of the equations of conservation of momentum, the kinematical relations and the constitutive equations; a detailed derivation is given in Heuer, Adam, Ziegler [21] with the crucial elimination of the cross-sectional rotation ψ of the homogenized beam performed, Fig. 1, w,xxxxxx − λ 2 w,xxxx +

κ ∗(0) = κ∗ =

1 B∞

μ μ λ2 1 ∗(0) ∗ w¨ ,xx − λ 2 w¨ = − p + p,xx + λ 2 κ,xx − κ,xxxx , B0 B∞ B∞ B0

1 3 12Bi ∑ h3 εx∗ zi dz , B0 i=1 i hi 2D1 d

1 h3

∗ εx dz − h1 εx∗ dz

h3

1

h1

(1)

i + ∑3i=1 12B εx∗ zi dz h3 i

hi

In the “homogenized” Eq. 1, p denotes the lateral load per unit of length. In smart layers, we may impose eigenstrains εx∗ , e.g. piezoelectric strain, for control purpose. Consequently, the eigen-curvature terms have been added to Eq. 1 for the sake of completeness. With the effective self-explaining parameters, inspection of Fig. 1 is understood, d = (h1 + h2 )/2, we have

μ = 2ρ1 h1 + ρ2 h2 ,

D1 = D3 = E1 bh1 ,

B0 = B1 + B3 , B∞ = B0 + 2D1 d 2 , −1 ≤ 1/4 B0 /B∞ = 1 + 3(1 + h2 /h1 )2

B1 = B3 = E1

bh31 , 12

(2)

Fig. 1: Geometry of a laterally loaded symmetric three-layer beam. Cross-sectional rotation of the homogenized beam shown. Fuzzy core shear stiffness considered

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Rudolf Heuer and Franz Ziegler

The shear coefficient in Eq. 1 is either proportional to the core’s shear modulus G2 in the case of perfectly bonded interfaces, 2b B∞ λ 2 = κ 2 G2 , h 2 D1 B 0

(3)

or, for the symmetric three-layer beam with elastic interlayer slip, it becomes proportional to the physical interface stiffness k,

λ2 = k

B∞ . D1 B 0

(4)

Thus, Eqs. 3 and 4, when substituted in Eq. 1, render qualitatively one and the same result with, e.g., hard-hinged supports of a single span beam understood. The gross bending moment and the shear force are related to the deflection w and its derivatives as follows,

B∞ μ ∗(0) ∗ (5) , Q = M,x M = −B∞ (w,xx + κ ) + 2 w,xxxx + w¨ + κ,xx λ B0 Fuzziness in Eq. 1 is simply introduced by inserting e.g., a triangular fuzzy membership function either for core’s shear modulus in Eq. 3 or for the physical interface stiffness in Eq. 4. The analysis of the illustrative example, mentioned above, consequently is based on the interval representation (interval of confidence at a given level of presumption, i.e. α -cut). However, the assumption of symmetry somewhat limits the application of Eq. 4 in this case. However, the important problem of a two-layer beam with interface slip, analyzed in detail in [4], yields a formula similar to Eq. 4, Fig. 2,

1 1 d2 , (6) λ 2 → λ22 = k + + D1 D2 B 0

Fig. 2: Geometry and stress resultants of a laterally loaded two-layer beam with linear elastic physical interface

Vibrations of layered structures with fuzzy core stiffness/fuzzy interlayer slip

33

to be substituted together with the alternated definition of B∞ = B0 +

D 1 D2 2 d D1 + D2

(7)

into Eq. 1. Subsequently, we consider fuzzy core stiffness of the three-layer beam in some detail.

2.2 Modal analysis of the three-layer beam, hard-hinged support For the single span beam with hard hinged support, the homogeneous Eq. 1 renders the simple ortho-normalized mode shapes,

ϕn (x) = An sin β1n x ,

β1n = nπ /l ,

An =

μl 2

2 λ 2 β1n + B∞ B0

− 12 (8)

with λ 2 substituted from Eq. 3. The corresponding (undamped) circular natural frequencies are,

2 (B0 /B∞ ) + γ2,n k2 (α ) ωn (α )/ωn∞ = 1 + γ2,n k2 (α ) when referred to

2 4 ωn∞ = β1n B∞ / μ

with the fuzzy nondimensional material constant of the core substituted, k2 (α ) = κ22 G2 (α ) / κ22 G2 0

(9)

(10)

(11)

The assigned constants are referred to the shear rigidity of the core; see again Fig. 1,

2 2b κ2 G2 0 = κ22 G2 (α = 1) , γ2,n = κ22 G2 0 2 . (12) β1n D1 h2 Hence, the normalizing factor in Eq. 8 becomes fuzzy too. The largest interval of confidence at α = 0 may include a core with vanishing stiffness (or a composite with no bond at the interface), Δ k2,0 = Δ κ22 G2 0 / κ22 G2 0 ≤ 1 . (13) The linear functions plotted in Fig. 3 result, both intersecting at α = 1, when, for the sake of simplicity an isosceles triangular distribution is assumed,

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Fig. 3: Uncertainty of the core shear stiffness: isosceles distribution assumed min k2 (α ) = 1 − Δ k2,0 (1 − α ) , max k2 (α ) = 1 + Δ k2,0 (1 − α ) .

(14)

Equation 14 is substituted into Eq. 9, when considering the α -cuts in Fig. 3, respectively, to render the interval of presumption of the natural frequencies. Since light modal damping is commonly considered in the steady state response of the modal oscillators to time-harmonic forcing, the dynamic magnification factor and the phase angle for every assigned forcing frequency (ω /ωn∞ ) become uncertain functions with parameter α . Constant modal damping coefficients are assumed, ξ = ξn 1. The dynamic magnification factor can be delineated in the proper form, see e.g., [22],

χn = 1 − 2 1 − 2ξ

2

ω ωn (α )

2

ω + ωn (α )

4 − 12 .

(15)

The phase angle is given for every assigned nondimensional forcing frequency (ω /ωn∞ ) as a function of alpha through, see again [22], tan φn = 2ξ

ω ωn∞

tan φn = 2ξ ωωn∞

ωn (α ) ωn∞

−

ω ωn∞

2

·

B0 /B∞ +γ2,n k2 (α ) 1+γ2,n k2 (α )

ωn∞ ω n (α )

−

−1

ω ωn∞

, 2 · B

1+γ2,n k2 (α ) 0 /B∞ +γ2,n k2 (α )

−1

(16)

When substituting Eq. 9 in Eqs. 15 and 16, they become explicitly dependent on the fuzzy variable, Eq. 11. The latter result is included in Eq. 16.

Vibrations of layered structures with fuzzy core stiffness/fuzzy interlayer slip

35

Again, considering the α -cut in Fig. 3, Eq. 14 is respectively substituted in Eqs. 9, 15 and 16, to render the interval of presumption of the dynamic magnification factor and of the phase angle as well.

3 Numerical results 3.1 Isosceles uncertainty The following two basic parameters are assigned for the three-layer beam of Section 2.2, with light modal damping ξn = ξ = 0.05 understood throughout, Eqs. 2 and 12: B0 /B∞ = 0.1 < 1/4 ,

γ2,n=1 = 0.25 .

(17)

In the course of numerical analyses, we found it most illustrative to refer the frequencies to the natural circular frequencies at α = 1, Ωn = ωn (α = 1), Eq. 9,

Ωn ωn∞

2 =

(B0 /B∞ ) + γ2,n 1 + γ2,n

(18)

Fig. 4: Uncertainty of the fundamental frequency ratio for various widths of fuzzy core stiffness Equation 9 is evaluated first to explore the influence of the width of uncertainty on the basic frequency, Fig. 4. Considering the largest uncertainty in the isosceles

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Rudolf Heuer and Franz Ziegler

distribution of Fig. 3, the intervals of uncertainty of the natural frequencies of higher order, n ≤ 10, are depicted in Fig. 5, however when referred to their values at α = 1, Eq. 18. These distributions become more informative when referred to the assigned fundamental frequency Ω1 , Fig. 6.

Fig. 5: Uncertainty of the natural frequencies for largest width of fuzzy core stiffness, however referred to the natural frequencies for α = 1

The envelope function of the dynamic magnification factor of the basic mode, i.e. putting n = 1 in Eq. 15, is plotted in Fig. 7 varying the width of uncertainty according to Fig. 3 in the α = 0.2-cut; the forcing frequency is referred to Ω1 . Considering maximum uncertainty but taking into account the whole range of parameter α render the envelope surfaces in Fig. 8. The α = 0.2-cut yields the envelope functions of the first three modes as plotted in Fig. 9. Complementary to Fig. 7, the variations of the range of uncertainty of the phase angle of the basic mode with the width of uncertainty according to Fig. 3 in the α = 0.2-cut are shown in Fig. 10; the forcing frequency is again referred to Ω1 . To complement Fig. 9, the ranges of uncertainty of the first three phase angles are plotted in Fig. 11.

Vibrations of layered structures with fuzzy core stiffness/fuzzy interlayer slip

37

Fig. 6: Uncertainty of the first five natural frequencies referred to the assigned fundamental frequency for α = 1

Fig. 7: Uncertainty of the first dynamic magnification factor for a single α = 0.2-cut, varying the maximum interval of the fuzzy core stiffness

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Rudolf Heuer and Franz Ziegler

Fig. 8: Uncertainty surfaces of the dynamic magnification factor of the basic mode

Fig. 9: Envelopes of the first three dynamic magnification factors for the single α = 0.2-cut

3.2 Constraints affected to the uncertain natural frequencies Constraints on the design uncertainty, say on the uncertainty of the core shear stiffness of the three-layered beam, are often based on limiting the maximum allowable variability of the natural frequencies in a given frequency window, see Tison et al [23]. By inspection of Fig. 6 it is easily recognized that such constraints can be expressed in assigning fractions of the frequency intervals that are maximum at α = 0. Thus, a 50% reduction is chosen and we refer to Ω1 ,

Vibrations of layered structures with fuzzy core stiffness/fuzzy interlayer slip

39

Fig. 10: Uncertainty of the first phase angle for a single α = 0.2-cut, varying the maximum interval of the fuzzy core stiffness: complement of Fig. 7

Fig. 11: Envelopes of the first three phase angles for the single α = 0.2-cut: complementing Fig. 9

ωn,max (α ) − ωn,min (α ) ωn,max (α = 0) − ωn,min (α = 0) ≤ , Ω1 2Ω 1

n = 1, 2, . . . , 5 (19)

The frequency window includes and is limited by the mode number 5 for some practical reasons. Hence, the α -cut is determined by solving the equation, Eq. 14 is substituted,

ω5,max (α ) − ω5,min (α ) ω5,max (α = 0) − ω5,min (α = 0) = Ω1 2Ω 1

(20)

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Rudolf Heuer and Franz Ziegler

Alternatively, it might fit the needs of precision design to limit the uncertainty relatively to the sure natural frequency of the same order of the mode, see Fig. 5, we refer thus to Ωn ,

ωn,max (α ) − ωn,min (α ) ωn,max (α = 0) − ωn,min (α = 0) ≤ , Ωn 2Ω n

n = 1, 2, 3, . . . (21)

In that case of relative uncertainty, the fundamental frequency interval renders the maximum tolerable uncertainty in the core shear stiffness by solving the equation for α , ω1,max (α ) − ω1,min (α ) ω1,max (α = 0) − ω1,min (α = 0) = (22) Ω1 2Ω 1 Since the largest width of uncertainty of the core shear stiffness is considered in Figs. 5 and 6, the constraints allow the definition of the α -cuts by solving either Eq. 20, to render α (n = 5) = 0.50, or alternatively, Eq. 22, to yield the less stringent condition at α (n = 1) = 0.44. The maximum bases of the allowable isosceles distributions of the uncertainty of the core shear stiffness in the more precise designs are plotted in Fig. 12.

Fig. 12: Reduction of the intervals of uncertainty under the conditions of 50% constraints of either uncertainty of the first five natural frequency intervals, n = 5; or of the relative natural frequency intervals, n = 1

Vibrations of layered structures with fuzzy core stiffness/fuzzy interlayer slip

41

3.3 Some effects of non-symmetric uncertainty We consider the extreme case of Δ k2,0 = 1, i.e. Eq. 14 when generalized reduces to min k2 (α ) = α , max k2 (α ) = 1 + Λ (1 − α ), where Λ > 1 renders the maximum core stiffness larger in Fig. 3. The extremes of natural frequencies thus become for the α = 0-cut, Eq. 9 is properly adapted,

max

ωn ωn,∞

2

ωn+1 min ωn+1,∞

= α =0 2 α =0

(B0 /B∞ ) + γ2,1 (1 + Λ )/n2 , 1 + γ2,1 (1 + Λ )/n2 = (B0 /B∞ ) ≤

1 , 4

2 4 ωn,∞ = n4 β1,1 (B∞ / μ ) ,

2 4 ωn+1,∞ = (n + 1)4 β1,1 (B∞ /μ )

(23)

A first effect of unsymmetric uncertainty in an ensemble of fuzzy beams is ob2 rendering the coefficient of non-symmetry, served by putting max ωn2 = min ωn+1 n+1 4 2 n − 1 n (B0 /B∞ ) Λ = Λ (n) = · (24) 4 − 1 > 0 γ2,1 1 − (B0 /B∞ ) n+1 n

The parameters assumed in section 3.1, (B0 /B∞ ) = 0.1, γ2,1 = 0.25 exclude a solution of Eq. 24 for n = 1 and yield the coefficients of non-symmetry: Λ (n=2) = 12.16, Λ (n=3) = 10.37, Λ (n=4) = 11.20, Λ (n=5) = 12.54, Λ (n=6) = 14.07, Λ (n=7) = 15.68, . . . Equation 24, when virtually considered for continuous n exhibits a singularity at n = 1, 28. It is moved to n = 1 for the smaller ratio of the flexural stiffness B0 /B∞ = 1/16 ≈ 0.063. Since max B0 /B∞ = 0.25, reported in Eq. 2, the singularity at n = 2 is still possible for the ratio B0 /B∞ = 0.198, but no effect on higher modes is observed for n ≥ 3. Consequently, for these two ratios of the flexural stiffness, a violation of the assumption of a triangular distribution of uncertainty is observed and consequently the solution for the coefficient of non-symmetry becomes invalid.

4 Conclusions For symmetric three layer structures and for two layer composites an exact homogenization is performed. Consequently, interval mathematics becomes applicable to define the intervals of confidence of the dynamic response for either fuzzy core shear stiffness and/or fuzzy stiffness of the physical interface between layers. A detailed study of a three layer simply hard supported beam with fuzzy core of an isosceles distribution is performed rendering deep insights into levels of confidence of the dynamic magnification factors and of the phase angles in time harmonic vibrations. Ensemble effects of such beams, e.g. overlapping of the intervals of confidence of their natural frequencies, however, under the assumption of a non-symmetric triangular distribution of the core stiffness uncertainty are shown to be limited to the first

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Rudolf Heuer and Franz Ziegler

and the second mode. Either fuzzy modal superposition or fuzzy control of vibrations can be based on the results of this short paper. Since the normalizing factor of the mode shapes is fuzzy too, the complexity of modal superposition is increased. Acknowledgements This research has been supported by the City-of-Vienna-research-grant “Hochschuljubil¨aumsfonds der Stadt Wien.”

References 1. M¨oller B. and Beer M. (2004) Fuzzy Randomness. Springer-Verlag, Berlin. 2. Hanss M. and Willner K. (2000) A fuzzy arithmetical approach to the solution of finite element problems with uncertain parameters. Mechanics Research Communications 27, 257–272. 3. Dasgupta G. (2008) Stochastic shape functions and stochastic strain-displacement matrix for a stochastic finite element stiffness matrix. Acta Mechanica 195, 379–395. 4. Adam C., Heuer R., Raue Annegret and Ziegler F. (2000) Thermally induced vibrations of composite beams with interlayer slip. J. Thermal Stresses 23, 747–772. 5. Heuer R. (2007) Equivalence in the analysis of thermally induced vibrations of sandwich structures. J. Thermal Stresses 30, 605–621. 6. Irschik H., Heuer R. and Ziegler F. (2000) Statics and dynamics of simply supported polygonal Reissner-Mindlin plates by analogy. Archive of Applied Mechanics 70, 231–244. 7. Heuer R., Irschik H. and Ziegler F. (1992) Thermally forced vibrations of moderately thick polygonal plates. J. Thermal Stresses 15, 203–210. 8. Zadeh L.A. (1965) Fuzzy sets. Information and Control 8, 338–353. 9. Viertl R. and Hareter D. (2006) Beschreibung und Analyse unscharfer Informationen — Statistische Methoden f¨ur unscharfe Daten. Springer-Verlag, Berlin (p. 32). 10. Ziegler, F. (2005) Computational aspects of structural shape control. Comp. Struct. 83/15–16, 1191–1204. 11. Mura T. (1991) Micromechanics of Defects in Solids. 2nd ed. Kluwer, Dordrecht. 12. Plantema F.J. (1966) Sandwich Construction. John Wiley, New York. 13. Stamm K. and Witte H. (1974) Sandwichkonstruktionen. Springer-Verlag, New York. 14. Reddy J.N. (1993) An evaluation of equivalent-single-layer and layerwise theories of composite laminates. Comp. Struct. 25, 21–35. 15. Irschik H. (1993) On vibrations of layered beams and plates. ZAMM 73, T34–T45. 16. Hoischen A. (1954) Verbundtr¨ager mit elastischer und unterbrochener Verd¨ubelung. Bauingenieur 29, 241–244. 17. Goodman J.R. and Popov E.P. (1968) Layered beam systems with interlayer slip. J. Struct. Div., ASCE 94, 2535–2547. 18. Chonan S. (1982) Vibration and stability of sandwich beams with elastic bonding. J. Sound and Vibration 85, 525–537. 19. Murakami H. (1984) A laminated beam theory with interlayer slip. J. Appl. Mech. 51, 551– 559. 20. Heuer A. (2004) Correspondence for the analysis of sandwich beams with or without interlayer slip. Mech. Advanced Materials Structures 11, 425–432. 21. Heuer R., Adam C. and Ziegler F. (2003) Sandwich panels with interlayer slip subjected to thermal loads. J. Thermal Stresses 26, 1185–1192. 22. Ziegler F. (1998) Mechanics of Solids and Fluids. 2nd corr. ed. Springer-Verlag, NewYork. 23. Massa F., Leroux A., Lallemand B., Tison T., Buffe F., Mary S. (2010) Fuzzy vibration analysis and optimization of engineering structures. In: Proc. IUTAM Symposium on The Vibration Analysis of Structures with Uncertainties (Eds. A.K. Belyaev and R.S. Langley), pp. 65–78 Springer Academic Publishers, Dordrecht, The Netherlands.

Vibration Analysis of Fluid-Filled Piping Systems with Epistemic Uncertainties M. Hanss, J. Herrmann and T. Haag

Abstract Non-determinism in numerical models of real-world systems may arise as a consequence of different sources: natural variability or scatter, which is often referred to as aleatory uncertainties, or so-called epistemic uncertainties, which arise from an absence of information, vagueness in parameter definition, subjectivity in numerical implementation, or simplification and idealization processes employed in the modeling procedure. Fuzzy arithmetic based on the transformation method can be applied to numerically represent epistemic uncertainties and to track the propagation of the uncertainties towards the output quantities of interest. In the current study, the fuzzy arithmetical approach is applied to the vibration analysis of a fluidfilled piping system with a structure attached. The investigation of this system is motivated by an automotive application, namely the brake pipes coupled to the floor panel of a car. The piping system is excited by a pressure pulsation in the fluid. Through fluid-structure interaction, this leads to a vibration of the pipes and thus of the structure attached. The uncertainties inherent to the system are of epistemic type and arise, among other things, from a lack of knowledge about the coupling elements between the pipes and the structure. Finite element simulations are performed to compute the vibration response of the system. These simulations are carried out multiple times in the framework of the fuzzy arithmetical algorithm to compute the uncertainty in the vibration response. Since a large number of simulations are needed, computational time is an important issue. In order to minimize the computational effort, substructuring in terms of the component mode synthesis (CMS) and model reduction techniques based on the Craig-Bampton method are used.

M. Hanss, J. Herrmann, T. Haag Institute of Applied and Experimental Mechanics, University of Stuttgart, Pfaffenwaldring 9, 70550 Stuttgart, Germany e-mail: [hanss, herrmann, haag]@iam.uni-stuttgart.de A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 4, © Springer Science+Business Media B.V. 2011

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1 Introduction Fluid-filled piping systems in industrial applications encounter the problem of acoustic excitation due to the operation of pumps and valves. The transition from fluid-borne sound to structure-borne sound induced on the pipe shell and attachment parts leads to undesired vibration and noise levels or even structural failure. A detailed understanding of vibration phenomena both in the fluid path and the structural path is needed in order to develop means to acoustically minimize the sound transmission via pipes and the excitation of attached panels such as the floor panel in brake-pipe systems. A finite element based substructuring technique including a model reduction according to the Craig-Bampton method [2] is used as an efficient simulation tool to investigate the dynamics of different pipe configurations [20, 17]. In order to validate the results of the numerical simulation, a hydraulic test bench with an innovative pulsation source is developed [15]. The test bench allows the reliable measurement of hydro- and vibro-acoustic transfer functions for different pipe configurations. The optimization of the mounting position of the piping assembly is a particularly efficient way to reduce structure-borne sound in a wide frequency range [14]. In addition to the mounting position, the design of the fastener elements (the socalled pipe clips) can be improved in order to reduce undesired sound transmission. Modern clip designs are characterized by flexible inserts or honeycomb-type structures to achieve an acoustical decoupling. However, an accurate modeling of these mechanical joints is extremely difficult due to the high level of uncertainty in material and design parameters. The vibration analysis of fluid-filled piping systems including such uncertainties is the scope of this research and the reason why in the following an introduction into the relevant concepts for the treatment of uncertainty is given.

2 Classification, Representation and Propagation of Uncertainty 2.1 Uncertainty Classification and Representation In general, non-determinism in numerical models may arise as a consequence of different sources, motivating some categorization of uncertainties. Although other classifications are possible in almost the same manner (e.g. [22]), the following categorization of uncertainties [23] proves to be well-suited in this context: aleatory uncertainties, such as natural variability or scatter caused by production, on the one side, and on the other side, epistemic uncertainties, which arise from an absence of information, rare data, vagueness in parameter definition, subjectivity in numerical implementation, or simplification and idealization processes employed in the modeling procedure. All these conditions manifest as uncertain model parameters and entail that the results that are obtained for simulations that only use one spe-

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cific set of values as the most likely ones for the model parameters cannot be considered as representative of the whole spectrum of possible model configurations. Furthermore, this fake exactness provided by the numerical simulation of models with uncertain but exact-valued parameters can significantly affect the comparison between numerical simulations and experimental testing. Namely, such a comparison may be rated as unsatisfactory if the crisp-valued simulation results do not well match the experimental ones, even though it might be absolutely satisfactory, if the uncertainties inherent to the models would have been appropriately taken into account in the simulation procedure. While aleatory uncertainties have successfully been taken into account by the use of probability theory and, in practice, by Monte Carlo simulation, the additional modeling of epistemic uncertainties still remains a challenging topic. As a practical approach to solve this limitation, a special interdisciplinary methodology to comprehensive modeling and analysis of systems is presented which allows for the inclusion of uncertainties — in particular of those of epistemic type — from the very beginning of the modeling procedure. This approach is based on fuzzy arithmetic, a special field of fuzzy set theory, which will be described in the following. The approach is applied to the simulation of the vibrational behavior of fluidfilled piping systems under consideration of the uncertainty that arises from simplifications in the modeling of the coupling elements. A special application of the theory of fuzzy sets, which is rather different from the well-established use of fuzzy set theory in fuzzy control, is the numerical implementation of uncertain model parameters as fuzzy numbers [18]. Fuzzy numbers are defined as convex fuzzy sets over the universal set R with their membership functions μ (x) ∈ [0, 1], where μ (x) = 1 is true only for one single value x = x ∈ R, the so-called center value or nominal value. For example, a fuzzy number p of triangular (linear) shape, expressed by the abbreviated notation [13] p = tfn(x, wl,i , wr,i ) ,

(1)

is defined by the membership function ∀ x ∈ R , (2) μ p(x) = min max 0, 1 − (x − x)/wl,i , max 0, 1 − (x − x)/wr,i or, more explicitly, by ⎧ 0 for x ≤ x − wl,i ⎪ ⎪ ⎨ 1 + (x − x)/wl,i for x − wl,i < x < x μ p(x) = . ⎪ 1 − (x − x)/wr,i for x ≤ x < x + wr,i ⎪ ⎩ 0 for x ≥ x + wr,i

(3)

However, any other shape of membership function may be selected if appropriate to quantify the uncertainty of a specific model parameter. The calculation with fuzzy numbers is referred to as fuzzy arithmetic and proves to be a non-trivial problem,

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especially with regard to the evaluation of large mathematical models with fuzzyvalued operands.

2.2 Uncertainty Propagation Based on the Transformation Method The problem of incorporating uncertainties into complex numerical models, such as finite element models, has already been addressed in a number of publications, of which the vast majority is based on stochastic descriptions of the uncertainties. In that context, the early papers of C ONTRERAS [1] and H ANDA AND A N DERSON [10], the monographs of G HANEM AND S PANOS [8] and K LEIBER AND H IEN [19], and the papers of E LISHAKOFF ET AL . (e.g., [4]) and S CHU E¨ LLER [25] are worthy of note. The alternative concept of using fuzzy descriptions of the uncertainties emerged more recently, and R AO AND S AWYER [24] presented an approach for its incorporation into the finite element method. However, since that approach uses the conventional concept of standard fuzzy arithmetic, based on interval computation, it suffers considerably from the overestimation effect [11, 13], also referred to as the dependency problem or conservatism. With the objective of reducing this effect while maintaining the computational effort to an acceptable level, M OENS AND VANDE PITTE [21] presented a fuzzy finite element approach which is based on the application of special optimization strategies of an approximative character. The achievements of this method are, for example, emphasized in [21] for the calculation of frequency response functions of undamped structures; however, its successful general applicability to arbitrary finite element problems and especially to the solution of complex real-world problems in both the frequency domain and the time domain still seems to pose a significant challenge. As a successful practical implementation of fuzzy arithmetic, which allows the evaluation of arbitrary systems with uncertain, fuzzy-valued model parameters, the transformation method [11] can be used. This method is available in a general, a reduced and an extended form, with the most appropriate form to be selected depending on the type of model to be evaluated [11, 13, 12]. Assuming the uncertain system to be characterized by n fuzzy-valued model parameters pi , i = 1, 2, . . . , n, the major steps of the method can briefly be described as follows: In the first step, each fuzzy number pi is discretized into a ( j) ( j) ( j) number of nested intervals Xi = [ai , bi ], assigned to the membership levels μ j = j/m, j = 0, 1, . . . , m, that result from subdividing the possible range of membership equally spaced by Δ μ = 1/m (Fig. 1). In a second step, the input intervals ( j) ( j) Xi , i = 1, 2, . . . , n, j = 0, 1, . . . , m, are transformed to arrays Xi that are obtained from the upper and lower interval bounds after the application of a well-defined combinatorial scheme [11, 13]. Each of these arrays represents a specific sample of possible parameter combinations and serves as an input parameter set to the problem to be evaluated. As a result of the evaluation of the model for the input arrays

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( j) Xi , output arrays Z( j) are obtained which are then retransformed to the output intervals Z ( j) = [a( j) , b( j) ] for each membership level μ j and finally recomposed to the fuzzy-valued output q of the system. In addition to the simulation part of the transformation method described above, the analysis part of the method can be used to quantify the influence of each fuzzyvalued input parameter pi on the overall fuzziness of the model output q. For these purposes, the standardized mean gain factors κi and the normalized degrees of influence ρi have been introduced [11, 13, 7], quantifying in an absolute and in a relative character, respectively, the effect of the uncertainty of the ith model parameter pi on the overall uncertainty of the model output q. In [11, 13], a standardization with respect to the nominal values is incorporated into the computation of the standardized mean gain factors κi and of the normalized degrees of influence ρi , whereas the approach that is proposed in [7] considers the influences of the overall input uncertainty on the overall output uncertainty.

( j)

Fig. 1: Decomposition of a fuzzy number pi into intervals Xi , j = 0, 1, . . . , m

Among other advantages of the transformation method, its characteristic property of reducing fuzzy arithmetic to multiple crisp-number operations entails that the transformation method can be implemented without major problems into an existing software environment for system simulation [13]. Expensive rewriting of the program code is not required. Instead, the steps of decomposition and transformation as well as of retransformation and recomposition can preferably be coupled to an existing, commercial software environment by a separate pre- and postprocessing tool.

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3 Fluid-Filled Piping System The investigated piping system consists of a fluid-filled elbow pipe with a uniform cross section and a steel plate which is attached to the pipe using two pipe clips. The hydraulic pipe is a curved steel brake pipe (lenghts: 0.7 m + 0.3 m, outer diameter: 4.7 mm) filled with water. The finite element based substructuring including fluidstructure coupling is summarized in the following section.

3.1 Modeling Approach 3.1.1 Finite Element Model The substructures of the piping system are modeled using the finite element method. The discretized fluid and the structural partitions are fully coupled by a fluidstructure interface [27] so that two coupling conditions hold, namely the Euler equation and the reaction force axiom. A finite element formulation based on the principle of virtual displacements leads to coupled discretized equations in terms of nodal structural displacements u and nodal acoustic excess pressures p

Ms 0 Ks −C u u¨ f(t) + = , p q(t) 0 Ka ρ0 CT Ma p¨ where ρ0 is the fluid density. Note that the index “s” denotes the structural partition, whereas the index “a” characterizes the acoustical fluid. The discretized equations include mass and stiffness matrices Ms,a and Ks,a , respectively, as well as the coupling matrix C. An alternative system representation uses the acoustic potential or particle velocities in the fluid domain as field variables [5]. However, from a practical engineering point of view, the pressure is preferably being considered as the field variable. Note that the mean flow velocity is neglected in this research since the superimposed mean flow is very small when compared with the hydroacoustic speed (Mach number M 1) [20]. Thus, the linear wave equation for a fluid at rest is used to describe the acoustic field.

3.1.2 Efficient Substructuring The described piping system is assembled using an efficient component mode synthesis which is essentially a combination between a model reduction and a subsequent substructuring technique. According to the Craig-Bampton method [2], a transformation with respect to the component interface degrees of freedom is fol-

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Fig. 2: Finite element model of the assembled piping system and boundary conditions; mode shape at f = 693 Hz

lowed by a model reduction in modal space. The reduction basis consists of constraint modes characterizing the static solution and fixed interface modes up to a specific frequency of interest. The Craig-Bampton method maintains the full number of DOFs at the interface. However, the associated model partition includes highfrequency dynamic content, which is not needed in a final reduced superelement. For this reason, an interface reduction method is applied to the substructure models received so far. Appropriate sets of Ritz vectors are determined to build the interface reduction bases in such a way that the relevant dynamic deflections and the acoustic pressure field on the interface is approximated well at low frequencies. The first set of Ritz vectors are eigenvectors from the local eigenvalue problem determined by the interface partition of mass and stiffness matrix as proposed by Craig and Chang [3]. It is important to augment these local interface eigenvectors by additional sets of Ritz vectors, namely the interface partition of rigid body and low-frequency modes computed from the eigenvalue problem of the free-floating substructure [17]. The overall reduction basis (denoted as T in the following) reduces the coupled system to nr nfull DOFs. The coupling between the reduced component models is defined by single point contraints at the component interface. This holds for both the solid and the fluid domain, and implicit scleronomic coupling conditions are established. They are used for the generation of an explicit transformation matrix Q from component coordinates in coordinates of the assembled piping system [20] which allows the subsequent summation of nsub substructure contributions

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M. Hanss, J. Herrmann and T. Haag nsub

∑

k=1

nsub

+∑

k=1

0 QTs,k Ms,k Qs,k q¨ s T q¨ a ρ0 QTa,k CT k Qs,k Qa,k Ma,k Qa,k Mg

nsub T

QTs,k Ks,k Qs,k −QTs,k Ck Qa,k Qs,k fs,k qs = ∑ QTa,k fa,k , qa 0 QTa,k Ka,k Qa,k k=1

(4)

Kg

where Ms = TTs Ms Ts , Ma = TTa Ma Ta , Ks = TTs Ks Ts , Ka = TTa Ka Ta , C = TTa CTs , fs = TTs f, and fa = TTa q. A detailed description of the substructuring technique and a way how to avoid locking problems for the applied interface reduction are presented in [17]. In the present work, a water-filled piping system with an elbow and attachment parts is assembled using nsub = 9 substructures. The reduced system has nr = 1532 DOFs as opposed to nfull = 56280 DOFs of the full finite element model. The piping system is clamped on the left end and an acoustic pressure excitation is applied as boundary condition. The finite element model of the assembled piping system with boundary conditions and a typical mode shape is depicted in Fig. 2. A harmonic analysis is performed to compute the vibro-acoustic transfer function H p1 →vz between the dynamic pressure p1 at the beginning of the pipe and the normal velocity vz at a representative measurement point on the plate. The threedimensional modeling approach of the assembled piping system comprises flexural vibrations of the pipe and the target structure, which are particularly important in automotive applications.

3.1.3 Modeling of Damping The influence of damping is integrated in the system of equations as a postprocessing step. The structural partition of the global damping matrix is given as Ds,k = αs,k QTs,k Ms,k Qs,k + βs,k QTs,k Ks,k Qs,k ,

(5)

assuming a Rayleigh damping model with the damping coefficients αs,k and βs,k for each substructure. The overall global viscous damping matrix is assembled by nsub

∑k=1 Ds,k 0 Dg = , 0 Da so that Eq. 4 can now be augmented accordingly. The frequency-dependent fluid damping model according to Theissen [26] is based on a complex wave number and includes wall friction effects, which is the dominant damping mechanism in thin pipes. Details of the fluid damping model and the generation of the fluid damping matrix Da are given in [16].

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3.2 Experimental Setup The experimental setup of the hydraulic test bench is illustrated in Fig. 3. The setup consists of a hydroacoustic pressure source and a hydraulic pipe with an attached target structure. The mounting of the target structure is realized with two clips as described in detail in Section 4.1. The pressure source consists of two piezo stack actuators which are arranged perpendicularly to the direction of wave propagation and on opposite sides of the hydraulic pipe. The piezostacks are driven by a power amplifier and a function generator and oscillate with opposite phase in order to excite pressure pulsations in the fluid column [15]. A sweep excitation is chosen in order to excite a wide frequency range. The repeated sweep signals have a cycle duration of 200 ms. The housing of the pressure source is designed in such a way that most of the energy goes into the brake pipe where the dynamic measurements are conducted. The supply pipe with the additional pump is required to fill and compress the fluid to ensure a stable fluid column without any air bubbles. The dynamic pressure pulsations are measured with piezoelectric pressure sensors, whereas the structure-borne sound on the target structure is measured with a tri-axial accelerometer. The used instrumentation captures both the fluid-borne sound and the resulting structural excitation of the target structure and, in particular, flexural vibrations. The estimation of transfer functions is performed as explained in [6].

Fig. 3: Experimental setup of the hydraulic test bench

4 Comprehensive Modeling and Simulation 4.1 Modeling of Epistemic Uncertainties Due to the pressure pulsations in the fluid and the strong fluid-structure coupling, the pipe is excited and the structural vibrations are transferred to the plate-like target

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structure by the clips. Thus, the clip design plays a dominant role in order to minimize the structure-borne sound induced on the target structure. An efficient way to acoustically decouple the pipe from the attachment part is an impedance change using different mass, stiffness and damping properties of the clip as compared to the pipe and the steel plate. An example is a built-in soft layer which leads to a considerable reduction of excitation. However, the accurate modeling of such clips is very challenging since the material, the damping and the design parameters are characterized by a high level of uncertainty as mentioned before. Three different clip designs are depicted in Fig. 4. Clip 1 is a relatively rigid plastic part, whereas

Fig. 4: Simplified finite element model of the mechanical joint and picture of Clip 1, Clip 2 and Clip 3 (courtesy of Tucker GmbH) Clip 2 is characterized by a honeycomb-type design in order to achieve the desired decoupling. A more advanced clip design with a soft layer both between pipe and clip, and between clip and plate is realized in Clip 3. All clips are mounted to the plate using steel bolts. The finite element model of the clips is also shown in Fig. 4. Instead of modeling every detail of the clip design, a simplified clip structure with uncertain material and design properties is assumed. Besides the mass, the stiffness and the damping properties of the clip, the coupling length between clip and plate is another important, but uncertain parameter for the accurate modeling of the piping system. Depending on the mounting of the clips on the plate, the effective coupling area might vary considerably and is therefore not exactly known. In the finite element model, the coupling length can be modified by using a different number of coupling nodes between clip and plate as depicted in Fig. 4. In total, four different uncertain parameters are identified: clip stiffness, clip density, structural damping coefficient and coupling length. In Table 1, the values of these uncertain parameters and their assumed uncertainty respectively, are summarized. The structural damping coefficient is a scaling factor of the Rayleigh damping parameters αs,k and βs,k . It ranges from 0 to 1 and thereby covers the worst-case intervals for αs,k and βs,k that are specified in Table 1. Thus, αs,k and βs,k are not independent but governed by the structural damping coefficient, which reduces the number of uncertain input parameters and thereby the computational effort. All parameters are chosen as trian-

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gular fuzzy numbers with nominal values xi and left-hand and right-hand deviations wl,i and wr,i . Even though the manufacturer provided the material data for the clips, it Table 1: Uncertain model parameters with nominal values xi and deviations wl,i and wr,i Parameter

1 Clip stiffness

2 Clip density

3 Coupling length

4a Struct. damp. αs,k

4b Struct. damp. βs,k

xi wl,i , wr,i Unit

300 200 , 200 MPa

1200 200 , 200 kg/m3

9 3, 3 mm

53.3 23.3 , 23.3 1/s

2.15 1.15 , 2.5 10−6 s

is a difficult task to decide what are the possible values for the previously mentioned input parameters to be accounted for by means of fuzzy numbers. The complicated geometry of the clips is the most obvious reason for this fact, but also anisotropy and nonlinearity of the material properties, as well as uncertain coupling conditions are to be mentioned in this context. For this reason, a manual optimization procedure, which involves a best fit to the mean value of the three measurement signals, has been conducted in order to find reasonable parameter combinations that are well capable of providing an adequate inclusion of the measured data, but also a preferably tight uncertainty band. A more systematic way to estimate the fuzzy-valued input parameters based on measurement data through a special inverse methodology is presented in [9]. However, due to certain requirements that are not met by the problem at hand, this identification algorithm cannot easily be applied here.

4.2 Simulation Results By the use of the transformation method, the parametric uncertainties that have been quantified in the previous section, are now propagated to the output of interest, namely the frequency response function Hp1 →vz from the input pressure p1 to the out-of-plane velocity vz of a characteristic node or the corresponding measurement point on the plate. For the simulation, a decomposition number of m = 3 is used to discretize the μ axis. For the complicated system that is investigated in this study, it is not obvious whether the input parameters pi have a monotonic influence on the output or not. For this reason, the general transformation method is used which is not only evaluating the extreme values of the input intervals but also some intermediate values. As a consequence, 354 evaluations of the finite-element model are needed theoretically which can practically be reduced to 337 evaluations through the consideration of recurring parameter combinations.

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Fig. 5 shows the results of the comprehensive simulation, together with the measurement signals for the three different clips. On the one hand, Fig. 5 shows that the large variation amongst the measured clips cannot be covered with a model with one specific set of crisp parameters. On the other hand, it becomes clear from the simulation results of the comprehensive model that the methodology that is persued in this paper, namely the inclusion of epistemic uncertainties by the use of fuzzy numbers, is definitely appropriate with respect to the achieved results. All three measurement curves are included within the bounds of the fuzzy-valued results for large ranges of frequencies. Thus, they offer a conservative and comprehensive prediction for the piping system under consideration. It is worthy of note that the clip design with the flexible inserts (Clip 3) leads to the lowest structure-borne sound level in almost the entire frequency range.

Fig. 5: Uncertain FRF |H p1 →vz | with measurement values for Clips 1 to 3

4.3 Measures of Influence In a further post-processing step, the transformation method of fuzzy arithmetic allows for the computation of normalized degrees of influence ρi , which express the relative influence of the uncertainty of the ith input parameter on the uncertainty of the output. Fig. 6 shows these degrees of influence for the present problem. For the frequency range under consideration, the uncertainties of both the clip density and the structural damping coefficient of the clip have a rather negligible influence on the uncertainty of the amplitude of the FRF H p1 →vz . The contributions of the coupling length and the stiffness of the clip make up approximately 50%. Especially the negligible influence of the structural damping coefficient is quite surprising, as it is the only mechanism of energy dissipation in the clip model. For future computations considering uncertainties, it would suffice to account for the uncertainty in the clip

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stiffness and the coupling length, while crisp values can be used for the other two input parameters.

Fig. 6: Normalized degrees of influence ρi of the input uncertainties on the uncertainty of the FRF |H p1 →vz |

5 Conclusions In the current study, a conservative inclusion of three measurement signals by a finite element model is achieved where uncertain parameters are used for a substructure that proves to be too complicated to be modeled in detail. An analysis of the contribution of the different input uncertainties reveals that only the stiffness of the clip and its coupling length have a significant influence on the uncertainty of the output. Thus, for future computations, the number of uncertain parameters can be reduced from four to two, which leads to a considerable reduction of the computational cost while practically achieving the same results.

References 1. Contreras H (1980) The stochastic finite-element method. Comp.&Struc. 12:341–348 2. Craig RR, Bampton MCC (1968) Coupling of substructures for dynamic analysis. AIAA Journal 6:1313–1319 3. Craig RR, Chang CJ (1977) Substructure coupling for dynamic analysis and testing. NASA CR 2781 4. Elishakoff I, Ren YJ (1998) The bird’s eye view on finite element method for structures with large stochastic variations. Comp. Meth. in Appl. Mech. and Eng. 168:51–61 5. Everstine GC (1981) A symmetric potential formulation for fluid-structure interaction. Journal of Sound and Vibration 79:157–160

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6. Ewins DJ (2003) Modal Testing: Theory, Practice, and Application. Research Studies Press 7. Gauger U, Turrin S, Hanss M, Gaul L (2007) A new uncertainty analysis for the transformation method. Fuzzy Sets and Systems 159:1273–1291 8. Ghanem RG, Spanos PD (1991) Stochastic Finite Elements: A Spectral Approach. Springer, New York 9. Haag T, Reuß P, Turrin S, Hanss M (2009) An inverse model updating procedure for systems with epistemic uncertainties. In: Proc. of the 2nd International Conference on Uncertainty in Structural Dynamics, Sheffield, UK 10. Handa K, Anderson K (1981) Application of finite element methods in the statistical analysis of structures. In: Moan T, Shinozuka M (eds) Proc. of the 3rd International Conference on Structural Safety and Reliability, Elsevier, Amsterdam, pp 409–417 11. Hanss M (2002) The transformation method for the simulation and analysis of systems with uncertain parameters. Fuzzy Sets and Systems 130(3):277–289 12. Hanss M (2003) The extended transformation method for the simulation and analysis of fuzzyparameterized models. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 11(6):711–727 13. Hanss M (2005) Applied Fuzzy Arithmetic – An Introduction with Engineering Applications. Springer, Berlin 14. Herrmann J, Haag T, Engelke S, Gaul L (2008a) Experimental and numerical investigation of the dynamics in spatial fluid-filled piping systems. In: Proc. of Acoustics, Paris 15. Herrmann J, Haag T, Gaul L, Bendel K, Horst HG (2008b) Experimentelle Untersuchung der Hydroakustik in Kfz-Leitungssystemen. In: Proc. of DAGA, Dresden 16. Herrmann J, Spitznagel M, Gaul L (2009) Fast FE-analysis and measurement of the hydraulic transfer function of pipes with non-uniform cross section. In: Proc. of NAG/DAGA, Netherlands 17. Herrmann J, Maess M, Gaul L (accepted 2009) Substructuring including interface reduction for the efficient vibro-acoustic simulation of fluid-filled piping systems. Mechanical Systems and Signal Processing 18. Kaufmann A, Gupta MM (1991) Introduction to Fuzzy Arithmetic. Van Nostrand Reinhold, New York 19. Kleiber M, Hien TD (1993) Stochastic Finite Element Method. John Wiley & Sons, New York 20. Maess M (2006) Methods for efficient acoustic-structure simulation of piping systems. Ph.D. thesis, Institute of Applied and Experimental Mechanics, University of Stuttgart 21. Moens D, Vandepitte D (2002) Fuzzy finite element method for frequency response function analysis of uncertain structures. AIAA Journal 40:126–136 22. M¨oller B, Beer M (2004) Fuzzy Randomness Uncertainty in Civil Engineering and Computational Mechanics. Springer, Berlin 23. Oberkampf WL (2007) Model validation under both aleatory and epistemic uncertainty. In: Proc. of NATO AVT-147 Symposium on Computational Uncertainty in Military Vehicle Design, Athens, Greece 24. Rao SS, Sawyer JP (1995) Fuzzy finite element approach for the analysis of imprecisely defined systems. AIAA Journal 33:2364–2370 25. Schu¨eller GI (2007) On the treatment of uncertainties in structural mechanics and analysis. Computers and Structures 85(5–6):235–243 26. Theissen H (1983) Die Ber¨ucksichtigung instation¨arer Rohrstr¨omung bei der Simulation hydraulischer Anlagen. Ph.D. thesis, RWTH Aachen 27. Zienkiewicz O, Taylor R (2000) The Finite Element Method. Butterworth-Heinemann, Oxford

Fuzzy vibration analysis and optimization of engineering structures: Application to Demeter satellite F. Massa, A. Leroux, B. Lallemand, T. Tison, F. Buffe, and S. Mary

Abstract The objective of this paper is to highlight that the fuzzy approaches can be employed in a design phase in order to build a robust engineering structure. An optimization problem, in which design variables, constraints and objectives functions are considered as fuzzy, is defined. To determine the optimized feasible design space, a specific methodology, based on genetic algorithms, is proposed.

1 Introduction Although the deterministic numerical simulations are more and more efficient, many sources of uncertainties can always affect structures behaviour prediction. Uncertainties are principally due to a lack of knowledge of the endogenous or exogenous parameters. Different approaches have then been developed to take these imperfections into account. Amongst these ones, the fuzzy subsets theory [1] allows to model both imprecise data and subjective data. This formalism has already been employed successfully by several researchers to study different problems [2]–[4], such as static, modal or dynamic, defined with uncertainties. In this paper, a nondeterministic approach is performed in an optimization process in order to improve an existing design of a structure by integrating the identified uncertainties. This study relies on an efficient propagation method to calculate fuzzy frequencies, which are used as constraints functions in the optimization problem, and on the use of geF. Massa, A. Leroux, B. Lallemand, T. Tison Laboratoire d’Automatique, de M´ecanique et d’Informatique industrielles et Humaines - LAMIH UMR CNRS 8530 - Universit´e de Valenciennes et du Hainaut Cambr´esis - Le Mont Houy - 59313 Valenciennes Cedex 9 - France - Tel: +33 (0)3.27.51.14.59 e-mail: [email protected] F. Buffe, S. Mary Centre National d’Etudes Spatiales - 18 Avenue Edouard Belin - 31401 Toulouse Cedex 9 e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 5, © Springer Science+Business Media B.V. 2011

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netic algorithms to explore the design space. The originality of the proposed method is that each generated individual are defined by fuzzy numbers. To note the selected designs and find the best one, some defuzzification criterions are used to evaluate the fuzzy objective functions.

2 Aims of the study 2.1 Description of the study The study is focussed on the lower wall of DEMETER micro satellite developed by the CNES French government space agency. DEMETER means Detection of Electro-Magnetic Emissions Transmitted from Earthquake Regions. The satellite measures ionospheric disturbances like ion density and extremely low frequency changes in the Earth’s magnetic field.

Fig. 1: Numerical model of DEMETER satellite

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The lower wall is modelled with shell elements and some of its parameter will be considered as fuzzy. The rest of the structure is deterministic and is represented by equivalent beam finite elements to keep a reasonable computational time. Only the lower wall globally influences the first modes shapes of the complete structure. The actual design is built in aluminium (Young modulus E = 72 GPa, density ρ = 2700 Kg·m−3 ) and the different thicknesses of plates are resumed in Fig. 2.

Fig. 2: Finite element model of lower wall To design the structure, the specifications, given by the engineers, are the following: 1. Take into account the production variability on the geometric characteristics (±0.1 mm on the thicknesses of plates of lower wall) 2. Verify the stiffness of the structure by studying the first three frequencies (the two first frequencies must be superior to 24 Hz and the third one superior to 65 Hz, a variability of 1 Hz on the thresholds is accepted) 3. Define the design space with the thicknesses e1 , e2 , e3 , e4 , e5 (±20% of variation of nominal value is authorized for each thickness) 4. Propose a robust design according to uncertainties 5. Minimize the mass of the lower wall, which represents 5 Kg of 100.78 Kg of the full structure. The next section traduces these specifications in terms of fuzzy numbers and details the different fuzzy data useful in the optimization process.

2.2 Building of fuzzy optimization problem By using the fuzzy set theory for modelling uncertainties, the general multiobjective optimization problem [5] can be written as follows:

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˜ p˜1 , . . . , p˜n p , q˜1 , . . . , q˜nq )) Minimize defuz ( F( Subject to g˜i ( p˜1 , . . . , p˜n p , q˜1 , . . . , q˜nq ) ⊆ R˜ i , i = 1 . . . I with p˜1 , . . . , p˜n p

(1)

where p˜1 , . . . , p˜n p and q˜1 , . . . , q˜nq are, respectively, the fuzzy design variables, which defines the initial design space, and the fuzzy imprecise parameters, which repre˜ p˜1 , . . . , p˜n p , q˜1 , . . . , q˜nq ) is the fuzzy sents the poorly defined data of the problem. F( global objective functions defined as a linear combination of each individual objective function f˜1 (p1 , . . . , pn p ), . . . , f˜no (p1 , . . . , pn p ), R˜ i are fuzzy restriction rules (or fuzzy constraint functions) applied to the fuzzy mechanical solutions described by the p˜1 , . . . , p˜n p . The operator “defuz” refers to a defuzzification step, presented in the chapter 4. The goal is to determine the feasible design space for which the fuzzy design vari˜ p˜1 , . . . , p˜n p , q˜1 , . . . , q˜nq ) and reables optimize the defuzzied objective functions F( spect the inclusion of the fuzzy constraint functions g˜i ( p˜1 , . . . , p˜n p , q˜1 , . . . , q˜nq ) in the fuzzy restriction rules R˜ i . The feasible design space is described by the membership functions of fuzzy design variables p˜1 , . . . , p˜n p . The fuzzy data used in the optimization problem are the following: The fuzzy design variables p˜ 1 , . . . , p˜n p represent the design space of the optimization problem. The crisp values of membership functions correspond to a previous or initial design whereas the supports represent the authorized range of variation. At this design stage, only three data (nominal value and bounds of each design variables) are generally available, triangular membership functions are so the simplest and common choice. The membership functions associated to thicknesses e˜i with i = 1 . . . 5 are defined Fig. 3.

Fig. 3: Fuzzy design variables

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The fuzzy imprecise parameters q˜1 , . . . , q˜nq represent poorly defined data, like material properties or geometric characteristics. In the present study, the production uncertainties are evaluated at ±0.1 mm and are taken into account on thicknesses e˜ j with j = 6 . . . 8. The membership functions associated to these thicknesses are presented Fig. 4.

Fig. 4: Fuzzy imprecise parameters

The fuzzy restriction rules R˜ i are constraints applied to fuzzy solutions g˜i ( p˜1 , . . . , p˜n p , q˜1 , . . . , q˜nq ) and are defined with the specifications. In this study, the fuzzy solutions f˜k with k = 1, . . . , 3 are calculated with the propagation method described in Section 3.1. The fuzzy restriction rules R˜ fk with k = 1, . . . , 3 , described Fig. 5, are composed by three areas: the good area for α = 1, the acceptable area for a included between 0 and 1, the poor area for α = 0.

Fig. 5: Fuzzy restriction rules

Finally, the fuzzy optimisation problem for the lower wall is defined as follows:

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Minimize defuz ( m˜ (e˜1 , e˜2 , . . . , e˜8 ) ) & Maximize rsp ( f˜1 , f˜2 , f˜3 ) Subject to f˜1 (e˜1 , e˜2 , . . . , e˜8 ) ⊆ R˜ f1 f˜2 (e˜1 , e˜2 , . . . , e˜8 ) ⊆ R˜ f2 f˜3 (e˜1 , e˜2 , . . . , e˜8 ) ⊆ R˜ f3 With e˜1 , e˜2 , . . . , e˜5

(2)

where m˜ and rsp are, respectively, the fuzzy mass of the structure and robustness criterion of the fuzzy solutions.

3 Fuzzy vibration analysis In a first time, a fuzzy modal analysis is performed by taking into account production uncertainties identified on the thicknesses. The different steps of the propagation method, called PAEM (Pad´e Approximants with Extrema Management), are recalled. The fuzzy frequencies and the non deterministic sensitivity of frequencies are presented.

3.1 PAEM method The PAEM method [2]–[3] requires that the membership function must be discretized according to the degree of confidence. The problem is then transformed into an interval problem for each α -cut level. This method has two steps: 1. Search of parameter combinations for each α -cut level, which lead to extreme solution variations. The sensitivity of the eigensolutions is evaluated between each level to determine how the response function is evolving. The modal quantities and their first sensitivities for each fuzzy parameter are determined for the crisp values (α = 1). The signs of the first-order sensitivities indicate the functional dependence of the response function and define the combinations of discrete fuzzy parameter values for the following α -cut level, which could supply the minimum and maximum variations. For each α -cut level, the first derivatives of the modal quantities are evaluated for the combinations of discrete fuzzy parameter values determined at the previous α -cut level. The signs of the derivatives are compared with those obtained at the previous α -cut level. If the sensitivities have the same signs, the response function is considered to be locally monotonic, and the determined combinations provide the minimum and maximum variations of the modal quantities for the current α -cut level. If the sensitivities have different signs, the response function cannot be considered as monotonic, giving rise to an extremum between these two α -cut levels. The combination nearest the extremum is chosen and the search is stopped for this variation. 2. Calculation of extreme solutions for these specific combinations in order to build the interval solution for each α -cut level. A high-order approximation using Pad´e

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rational functions is required in order to decrease the calculation time and maintain a good level of accuracy.

3.2 Numerical application In order to verify that the initial design is robust via uncertainties, a fuzzy modal analysis is performed with an uncertainty of ±0.1 mm considered on the eight thicknesses. The membership functions of the first three frequencies are presented in Fig. 6. It can be seen that all the fuzzy frequencies verify the restriction rules by considering the production uncertainties.

Fig. 6: Fuzzy frequencies and restriction rules

The PAEM method, which relies on a sensitivity analysis, allows to define the influence of each fuzzy parameter on the solutions (Fig. 7) for the complete variation space. This non deterministic sensitivity allows to detect the evolution of the functional dependence. The most influent parameters, in this study, are the thicknesses e6 , e2 and e8 , which represent 95% of the solution. The membership function of the fuzzy mass is described in Fig. 8. The defuzzification by “center of gravity” [6] leads to an equivalent mass of 100.78 Kg for the satellite. The aim of fuzzy optimization process will be to minimize this equivalent mass and to verify the other specifications too.

4 Fuzzy optimization This chapter explains briefly the main steps to solve a fuzzy optimization problem and focuses on the exploration of the design space of fuzzy design variables using a genetic algorithm and on the selection of the best updated fuzzy design variables

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Fig. 7: Influence of fuzzy parameters for the first frequency

Fig. 8: Membership function of mass

from the pool of these design variables. The last section presents results obtained after the fuzzy optimization of the lower wall of satellite.

4.1 Design methodology The fuzzy optimization process [5], resumed in Fig. 9, relies on 5 steps, namely the generation of different nominal values of design variables, the creation of fuzzy design variables, the determination of fuzzy solutions, the updating fuzzy design

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variables according to fuzzy solutions and fuzzy restriction rules and the notation of the fuzzy individuals to determine the best optimized design. It was shown in Massa et al. 2009 that to explore completely the design space, it is necessary to generate several fuzzy individuals, for which the nominal values of fuzzy design variables are randomly chosen. This step is performed by using genetic algorithms (GOAT Toolbox in Matlab). Several fuzzy populations will be created during the process. For each fuzzy individual, the fuzzy solutions are calculated with the PAEM method. The membership functions of fuzzy solutions and design variables will be updated. First of all, the restriction rules are applied to the fuzzy solutions. The intersection between the law and the output variables defines two α cuts levels LGA (Good-Acceptable limit) and LAP (Acceptable-Poor limit) for each output variable.

Fig. 9: Different steps of fuzzy optimization process These α -cuts levels represent the limits between the different levels of acceptability. If several restriction laws are used, the α -cuts level corresponding to the limit LGA is obtained by taking the maximum of LGAi and the limit LAP thanks to the

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maximum of LAPi . Secondly, these two limits are applied on the membership function of design variables. If no imprecise parameters are defined, the results being looked for on the input design variable are directly obtained. Otherwise, the process becomes iterative and in this case, the aim is to adjust the membership function of the design variables so that the LGA and LAP limits are as close as possible to 1 and 0 respectively. The membership functions of the fuzzy design variable’s membership functions represent the feasible design space that respects the inclusion of fuzzy constraint functions in the fuzzy restriction rules. Amongst the updated fuzzy design variables, it is important to select the best individuals which optimize the fuzzy objective functions. A notation step is then realized. In this study, the fuzzy sets are optimized to produce a minimum mass and a maximum robustness for the structure. Firstly, a mass criterion m˜ is evaluated by defuzzifying the corresponding fuzzy number using the “center of gravity” method [6]. Secondly, a fuzzy solution is considered to be robust with respect to a parameter if a large parameter variation implied a small solution variation. The dimensionless criterion rsp is evaluated. The smaller the criterion is, the more robust the solution is. A defuzzification by area is employed in order to take into account the dispersion of the considered parameter. defuz( f˜1 ), . . . , defuz( f˜3 ) (3) rsp = max min (defuz( e˜1 ), . . . , defuz( e˜5 ))

4.2 Improvement of the initial design The proposed methodologies are implemented with MATLAB and Structural Dynamics Toolbox. The CPU time for this fuzzy optimization is less to 3 hours (Pentium 4 2000 MHz). The fuzzy solutions obtained after the optimization are presented Fig. 10. It can be seen that the most restrictive constraint is those applied on the third frequency. The membership functions of fuzzy optimized design variables, which lead to feasible fuzzy solutions, are described in Fig. 11. These fuzzy sets supply to the designer some latitude to define the thicknesses of each plate. By considering a minimum mass and production uncertainties, the new selected thicknesses are presented Fig. 12. A comparison, between these last results and the initial ones presented Fig. 6, is performed Fig. 13. The objective to decrease the mass of the lower wall is reached. The defuzzification by “center of gravity” leads to an equivalent mass of 99.9 Kg. The gain is to 750 g, namely 15% of mass of the lower wall of the satellite.

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Fig. 10: Membership functions of fuzzy solutions and mass after optimization

Fig. 11: Membership functions of fuzzy design variables after optimization

5 Conclusion This paper highlights how the uncertainties, observed on the geometric parameters, can be taken into account in the design phase. The proposed methodology, based

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Fig. 12: New thicknesses of the lower wall

Fig. 13: Comparison of solutions and objective (before and after the optimization process)

on the fuzzy formalism, makes it possible to treat simultaneously propagation of uncertainties and multiobjective optimization problem. The main goal is to supply a robust optimal mechanical behavior that satisfies the product specifications. Our future experiments will consist of integrating fuzzy dynamic constraints and designer judgment and/or experience in optimization process. Acknowledgements The present research work has been supported by CNES French government space agency. The authors gratefully acknowledge the support of this institution.

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References 1. Zadeh L.A. (1965) Fuzzy sets, Information and Control 8:338–353. 2. Massa F., Ruffin K., Tison T., Lallemand B. (2008) A complete method for efficient fuzzy modal analysis, Journal of Sound and Vibration 309:63–85. 3. Massa F., Tison T., Lallemand B. (2009) Fuzzy modal analysis: Prediction of experimental behaviours, Journal of Sound and Vibration 322:135–154 4. Hanss M. (2002) The Transformation Method for the simulation and analysis of systems with uncertain parameters, Fuzzy Sets and Systems 130:277–289. 5. Massa F., Lallemand B., Tison T. (2009) Fuzzy multiobjective optimization of mechanical structures, Computer Methods in Applied Mechanics and Engineering 198:631–643. 6. Dubois D., Prade H. (1980) Fuzzy sets and systems: Theory and Applications, Bellman, R. Ed, Mathematics in science and engineering, Academic Press, London.

Numerical dynamic analysis of uncertain mechanical structures based on interval fields David Moens, Maarten De Munck, Wim Desmet, Dirk Vandepitte

Abstract This paper introduces the concept of interval fields for the dynamic analysis of uncertain mechanical structures in the context of finite element analysis. The theoretic background of the concept is explained, and it is shown how it can be applied to represent dependent uncertainty in the model definition phase and in the post-processing phase. Further, the paper concentrates on the calculation of interval fields resulting from a dynamic analysis. A procedure is introduced that enables the calculation of a joint representation of multiple output quantities of a single interval finite element problem while preserving the mutual dependence between the components of the output vector. The application of the approach is illustrated using a vibro-acoustic finite element analysis.

1 Introduction Intervals have been used extensively for the representation of parametric uncertainty in finite element models when the available information is insufficient to build representative probabilistic models [7]. Although the complexity of the interval concept is rather limited, the numerical problem arising at the core of the interval finite element method was proven to be non-trivial [5]. Many research activities have focussed on this problem over the last decade (see e.g., [4, 8, 9, 6]). Still, one of the principal shortcomings of the current interval finite element procedures is that they are intrinsically not capable of representing possible dependencies among uncertain input David Moens K.U.Leuven Association, Lessius Hogeschool – Campus De Nayer Department of Applied Engineering, J. De Nayerlaan 5, 2860 Sint-Katelijne-Waver, Belgium e-mail: [email protected] Maarten De Munck, Wim Desmet, Dirk Vandepitte K.U.Leuven, Department of Mechanical Engineering, division PMA, Celestijnenlaan 300B, 3001 Heverlee, Belgium, e-mail: maarten.demunck/wim.desmet/[email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 6, © Springer Science+Business Media B.V. 2011

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or output quantities. This poses important limits on the applicability of the interval concept in non-deterministic finite element analysis. These limits are surfacing in the model definition phase as well as in the post-processing phase. When an uncertain interval model is defined, global uncertain parameters as e.g. material and geometric properties are often modelled as a single interval parameter. In this case, the uncertain property, although confined within an interval, is presumed to have one constant value over the entire model. The interval scalar is a perfect concept for this type of uncertainty. However, if the uncertainty of the property is considered as a variation within a single realisation of the design, the intrinsic limitations of interval scalars pose strong limits on the applicability of the interval concept. A possible approach is to introduce a large set of interval scalars, ranging over all elements, specifying an interval for the considered uncertain parameter individually for each element. It is clear that this is not a realistic approach, as through the interval concept, all these parameters will be intrinsically independent. This does certainly not agree with reality, where spatial variation of a physical quantity always exhibits spatial dependence. As a result, interval numerical analysis based on classical interval scalars is limited to the analysis of uncertain parameters that are not expected to vary within one realisation of a design. The same problem is present at the output side of a numerical procedure if the analysis is aimed at multiple outputs. Classical interval finite element analysis based on interval scalars will aim at the interval solution for each individual output quantity, as such constructing a hypercubic approximation of the actual solution set [5]. However, the information on the mutual dependence between the components of an interval output vector is essential, especially when these outputs need to be postprocessed simultaneously. Similar as for the interval scalars for representation of spatial model uncertainty, the lack of dependence will enable the unrealistic combination of extreme limits for all output quantities, resulting in extremely unrealistic situations. This will finally cause very large conservatism in the analysis outcome. In the context of probabilistic numerical analysis, the topic of spatially or timevarying uncertain parameters has been addressed by the introduction of random fields. The objective of a random field is to represent a spatial variation of a specific model property by a stochastic variable defined over the region on which the variation occurs. A comprehensive overview of random fields is given in [10]. The principle of the random field approach is to express the non-deterministic model property as a field variable v(x, θ ) in which x represents the spatial variation and θ refers to its probabilistic behaviour. The specification of a random field generally comes down to the specification of the spatial evolution of the first two statistical moments of the field variable and a corresponding covariance function, expressing the spatial dependence of the field variable. The application of the concept of random fields in a numerical modelling framework requires some sort of discretisation of the spatially varying stochastic field over the defined geometry. The discretisation technique based on the K ARHUNEN L OEVE expansion [3] has gained particular attention in literature. It consists of a decomposition of the initial field into a superposition of a finite number of orthogonal random variables which are weighted with deterministic spatial functions and

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the eigenvalues of the covariance kernel. As such, the covariance kernel not only determines the spatial aspect of the random field, it also strongly influences the numerical approximation incorporated in most discretisation techniques. Therefore, in order for the method to have a predictive value in a design process, a well-founded assessment of the spatial correlations intrinsically incorporated in the assumed covariance kernel is necessary. However, the question on how to derive or approximate actual covariance functions in practical and realistic engineering applications seems far from answered. Therefore, the objective of this paper is to introduce the concept of interval fields as an alternative non-probabilistic approach for the representation of spatial uncertainty in the context of finite element analysis. After a short introduction on interval finite element analysis in section 2, section 3 describes the theoretic background of the interval field concept. It is shown how it can be applied to represent dependent uncertainty in the model definition phase, as well as in the post-processing phase. Section 4 finally concentrates on the calculation of interval fields based on a response surface methodology for a vibro-acoustic analysis case.

2 Interval finite element analysis In the interval finite element methodology, uncertain parameters are generally represented by interval scalars. An interval scalar is a convex subset of the domain of real numbers R. By definition, the range of the interval xI is bounded by its lower bound x and its upper bound x: xI = [x, x] = {x ∈ R | x ≤ x ≤ x} A set of interval scalars can be combined in an interval vector: ⎧ I⎫ z1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨zI ⎪

2 zI = . = z ∈ Rn | zi ∈ zIi . ⎪ ⎪ .⎪ ⎪ ⎪ ⎭ ⎩ I⎪ zn

(1)

(2)

This interval vector represents a multi-dimensional hypercube. This means that in this form, it cannot be used to incorporate mutual dependence between the vector components. Now consider a deterministic numerical analysis y = f (x), subject to multiple inputs and resulting in multiple outputs. Using the interval concept, the deterministic input vector x is replaced by an interval input vector xI . In this case, the exact solution set equals:

y = y | y = f (x) ∧ x ∈ xI (3) That is, the set y which contains all vectors y which result from applying the deterministic numerical analysis to all vectors x contained in the interval input vec-

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tor xI . In general, the elements of the output vector y are interdependent, as they are related through the deterministic numerical model and analysis and the nondeterministic inputs. Therefore, the solution set can adopt any shape in the multidimensional output space. In practice, the calculation of this exact solution set is extremely difficult. Therefore, most research focuses on the calculation of a conservative hypercubic approximation yI of the exact solution set, thus ignoring the interdependence between the output quantities. This conservative hypercubic approximation contains solutions that are not part of the exact solution. Several implementation strategies for interval numerical analysis have been proposed. Because global optimisation based strategies yield physically correct results, they are more and more acknowledged as the standard approach for non-intrusive interval FE analysis. The global optimisation based solution strategies actively search in the non-deterministic input interval space for the combination that results in the minimum or maximum value of an output quantity. All outputs of the numerical analysis are considered independently. For each output quantity of interest in the output vector yI , a separate global minimisation and maximisation is performed: yi = minx∈xI fi (x)

(4)

yi = maxx∈xI fi (x)

(5)

where fi (x) is the ith output of the deterministic numerical analysis. In theory, the global optimisation approach results in the exact interval vector. However, despite the smooth behaviour of typical objective functions, the computational cost of the global optimisation based approach remains high. Hence, most research on this method focuses on fast approximate optimisation techniques. In this domain, two different approaches can be distinguished: direct optimisation and response surface based optimisation. This paper in section 4 extends a recent algorithm based on Kriging response surfaces, developed by the authors [2], towards interval fields.

3 Interval fields 3.1 General concept One of the main shortcomings of the interval concept in the context of uncertain numerical analysis, is that it does not allow for the quantification of any sort of dependence between two uncertain quantities. This problem surfaces whenever a multi-dimensional analysis is performed, as already discussed in section 1. In order to cope with this problem, a new concept is introduced that enables the inclusion of mutual dependence in an interval vector. The explicit interval field concept is based on a superposition of nb deterministic base vectors ψ i using interval factors αiI . The interval field xF is defined as:

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nb

I xF = ∑ αx,i ψ x,i

(6)

i=1

The base vectors represent a limited set of reference patterns, each of which is scaled by an independent interval factor. The components of the interval field as such become coupled through the reference patterns. Once the reference patterns are chosen, the definition of the interval field comes down to the specification of the interval factors that define the field on x, which can be assembled in a classical (hypercubic) interval vector α Ix . In matrix notation, the interval field is denoted as: xF = [ψ x ] α Ix

(7)

The main question that needs to be addressed is how to quantify the base vectors and their respective interval factors. This discussion now distinguishes between interval fields used at the input side of a problem, and interval fields as a representation of uncertain analysis results.

3.2 Interval fields as uncertain input parameters The interval field concept as introduced in the previous section is now applied to represent uncertainty on a spatially varying model quantity. The spatial variation is captured in the base vectors, while the uncertainty is superimposed by the interval factors. For example, let’s consider a simple FE model of a simply supported beam, modelled by 101 beam elements. If we consider the height of the beam as an uncertainty, we introduce an interval hI to describe the range of the corresponding element property. If we consider all elements as independent, we can introduce a 101-dimensional hypercube hI each component of which is equal to hI . In the interval field notation, this comes down to: ⎧ I⎫ ⎪ ⎨h ⎪ ⎬ .. F I I h = [ψ h ] α h with [ψ h ] = [I]101×101 and α h = (8) ⎪ . ⎭ ⎪ ⎩ I h In this case, the interval field is identical to the hypercube hI . In the other extreme case, where all elements are fully dependent, we can introduce a single unity base vector to represent the variation pattern. The corresponding interval factor αhI than represents the actual range of the thickness: ⎧ ⎫ 1⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨1⎪ hF = . αhI with αhI = hI (9) .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 1

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The interval field concept now enables the introduction of additional patterns, such that deviations from this nominal pattern can be included with spatial coupling between the height of all beam elements: ⎡ ⎤ 1 ⎢1 ⎥ ⎢ ⎥ (10) hF = ⎢ . ψ h,2 ψ h,3 . . . ψ h,nb ⎥ α Ih ⎣ .. ⎦ 1 By defining the base vectors, the general shape of the spatial fields incorporated in the analysis is determined. The interval factors will only influence the relative contribution of each base vector to the actual field. In the beam example, based on this concept, one could limit the variation patterns to all possible linear functions between known intervals hI1 and hI2 at the supported ends in the following way: ⎡ ⎤ 1 0 ⎢ 0.99 0.01 ⎥ ⎢ ⎥ I ⎢ .. ⎥ hI1 = ψ F (11) h = ⎢ ... ⎥ h,1 ψ h,2 α h I . ⎢ ⎥ h2 ⎣ 0.01 0.99 ⎦ 0 1 If the interval FE problem with this interval field is now solved using the global optimisation approach as in eq. (4) and (5), the optimisation problem is restated as: yi = minx∈xF fi (x) yi = maxx∈xF fi (x)

(12) (13)

Converting the problem to the reduced domain of interval factors, the equivalent problem is formulated as: yi = minα x ∈α Ix fi ([ψ x ] α x )

yi = maxα x ∈α Ix fi ([ψ x ] α x )

(14) (15)

This clearly points out that the dimension of the search domain for the optimisation is drastically reduced by introducing the interval field concept in the analysis. In the beam example, the dimension of the search domain is reduced from 101 to only 2 for the case described above. Physically, this means that a very small but realistic subset of the original hypercubic search domain is selected as physical representation of the variation of the beam height. It is clear that a good and realistic estimation of the base vectors and their corresponding interval factors is mandatory for a realistic uncertainty assessment.

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3.3 Interval fields as uncertain analysis results Whenever FE analysis has multiple outputs, keeping track of the dependence between the output components is of high relevance whenever the output is used as a basis for post-processing. For example, if linear static analysis is performed, the output displacement vector u is generally post-processed to a stress distribution over the full FE model. If the outcome of the interval analysis is expressed in terms of a hypercubic interval displacement vector uI , this step would enable unrealistic displacement combinations, as all output components are intrinsically decoupled. Consequently, the post-processing will introduce a high amount of conservatism. In this case, a realistic assessment of the stress ranges resulting from the interval analysis can only be achieved if the output components are dependent. This is exactly the objective of the interval field. In general, two forms of the interval output field are distinguished: the explicit and the implicit form.

3.3.1 Explicit interval field In general, the same principle as described in section 3.2 can be applied to incorporate the dependence between the components of the output vector y. In this case, the outcome of the interval analysis is formulated as an interval field yF such that a close approximation of the actual solution set y given in eq. (3) is achieved: yF = [ψ y ] α Iy ≈ y

(16)

The final interval results after the post-processing phase should be derived from this formulation of the output vector interval field. If the function Π () represents the post-processing procedure in a deterministic case, the final interval results after the post-processing phase pI are calculated using an optimisation strategy, in which the interval factor hypercube α Iy defines the search domain: pi = miny∈yF Πi (y) = min Πi ([ψ y ] α y )

(17)

pi = maxy∈yF Πi (y) = max Πi ([ψ y ] α y )

(18)

α y ∈α Iy α y ∈α Iy

It is clear that these post-processing results confine a much narrower solution domain as would be the case if the optimisation was performed in the search domain defined by the hypercube yI . We can apply this explicit form of the output interval field in the case of linear static analysis. If the displacement output interval field uF is further processed, stresses should be derived from this interval field rather than the complete hypercube uI . In this case, the post-processing only involves the application of spatial operands ∇ on the output vector, which can be applied directly on the base vectors ψ u . The final interval solution is than easily assembled by performing the superposition of the post-processed displacement vectors, similar to eq. (16):

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∇ uF = [∇ψ u ] α Iu

(19)

The main challenge in the application of explicit interval fields at the output side of an FE analysis is in the determination of an appropriate set of base vectors for the solution, and their corresponding interval factors. The main shortcoming of the explicit interval field is that its definition only allows for a linear relation between the base vectors and the parameters representing the interval factors. Furthermore, in order to achieve an explicit interval field that produces no conservatism in its postprocessed results based on eq. (17) and (18), these parameters should be completely independent. In general, it is not an easy task to find this set of base vectors that will result in interval factors with these properties. For this reason, the next section introduces the implicit interval fields.

3.3.2 Implicit interval field The implicit interval field is an alternative mathematical formalism to express dependence between vector components at the output side of an analysis. It is based on implicit functions φy,i () of a set of predefined interval factors α y , describing the range of each individual component of the output vector: ⎫ ⎧⎧ ⎫ ⎪ ⎪ ⎬ ⎨ φy,1 (α y ) ⎪ ⎨⎪ ⎬

.. I , (20) α ∈ α yF = = φ y (α y ) , α y ∈ α Iy y y . ⎪ ⎪ ⎪ ⎭ ⎩ ⎩⎪ ⎭ φy,n (α y ) By introducing appropriate functions φi () for all components of the solution vector, the coupling between the components of the result vector is incorporated implicitly through the interval factors. The main advantage of this approach is that it no longer assumes a linear relationship between the interval factors and the output field. Furthermore, the set of interval factors is first chosen, after which the implicit relationship with the output vector components is determined through the numerical analysis. As such, a set of independent interval factors can always be guaranteed. In many cases, the actual set of interval uncertainties present at the input side of the problem is a very good candidate set, as they often are known (or considered) to be independent. Using the implicit interval field approach, post-processing of an output interval field comes down to an optimisation problem similar to the one derived for the explicit interval fields in section 3.3.1: pi = miny∈yF (Πi (y)) = min (Πi (φ y (α y )))

(21)

pi = maxy∈yF (Πi (y)) = max (Πi (φ y (α y )))

(22)

α y ∈α Iy α y ∈α Iy

An additional advantage of the interval field representation above is that even if the post-processing involves multiple output vectors y and z, still the exact solution can

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be achieved, as long as the required fields are represented using the same set of interval factors α yz . This can be seen from: pi = min Πi (y,, z) = min Πi φ y (α yz ) , φ z (α yz ) (23) α yz ∈α Iyz y ∈ yF z ∈ zF (24) pi = max Πi (y,, z) = max Πi φ y (α yz ) , φ z (α yz ) α yz ∈α Iyz y ∈ yF z ∈ zF The main challenge now is to determine the implicit function of each output vector component. In the application presented in the next section, the implicit interval field is calculated based on a response surface methodology.

4 Application of interval fields for vibro-acoustic analysis The interval field concept is now applied to a coupled vibro-acoustic analysis. The objective is to determine the interval sound pressure field response in an exterior point of a radiating structure, when interval uncertainty is present in the structural part of the analysis.

4.1 Vibro-acoustic analysis based on the ATV concept The vibro-acoustic analysis in this work focuses on the frequency response function between the sound pressure at an exterior point of a structure and a structural excitation force applied at an interior structural point. For this purpose, the concept of Acoustic Transfer Vectors (ATV) is applied. These ATV’s define the frequency dependent relation between the normal structural velocity on the radiating surface vns (ω ), and the resulting sound field at a specific exterior point p(ω ) through a set of linear equations: p(ω ) = ATV(ω )T vns (ω )

(25)

This procedure is clarified in figure 1. The harmonic response is obtained by dividing both sides of equation (25) by the interior harmonic excitation force: P(ω ) =

vns (ω ) p(ω ) = ATV(ω )T = ATV(ω )T ms (ω ) F(ω ) F(ω )

(26)

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Fig. 1: Visualisation of the ATV concept for the calculation of exterior radiated noise [1]

Based on this expression, the acoustic ATV analysis can be regarded as a postprocessing following a structural frequency response analysis resulting in the mobility ms (ω ). The next section uses this equivalence to translate the problem to an interval field analysis.

4.2 Interval analysis based on structural FRF interval fields The implicit interval field as described in section 3.3.2 is applied to describe the structural response FRF’s at all radiating points of the structure. The interval factors are chosen to be identical to the input uncertainty intervals xI . The structural output interval field ms ( ω ) F , representing the harmonic normal structural velocity response field at all radiating points, is expressed directly in relation to the input interval uncertainties xI through the implicit interval field formalism:

F (27) ms ( ω ) = φ ms (x) , x ∈ xI The implicit interval field is obtained by building response surfaces of each output component of ms (ω ) using the model uncertain properties x as parameters. An efficient way of building accurate response surfaces for numerical analysis with multiple outputs is provided by the Kriging response surface approach, developed by the authors [2]. This approach enables a very large number of outputs to be processed simultaneously, using a limited number of optimally located evaluation points in the approximation domain. This technique enables the quantification of the individual implicit interval field component functions φms ,i (x). Once the response surfaces are determined, the calculation of the radiating sound field is performed by applying the ATV-multiplication as a post-processing step on the structural results. Since the structural results are composed of a real and imaginary part that should not be decoupled in the analysis, the procedure as expressed in eq. (23) and (24) is applied, finally yielding:

Numerical dynamic analysis of uncertain mechanical structures based on interval fields

P(ω ) = minx∈xI Π φ ℜ(ms ) (x) , φ ℑ(ms ) (x) P(ω ) = maxx∈xI Π φ ℜ(ms ) (x) , φ ℑ(ms ) (x)

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(28) (29)

with Π representing the complex vector pre-multiplication with ATV(ω ) as expressed in eq. (26).

4.3 Numerical example The numerical example is shown in figure 2. The radiating box is fixed at the lower side, and contains an interior point where a local harmonic excitation is introduced. This excitation point is linked with structural beam elements to the outer surface of the box, consisting each of plate elements. Uncertainty is introduced as a 10% interval on the nominal plate thickness and rib heights.

Fig. 2: Numerical example consisting of a sound radiating box with internal structural excitation The interval field analysis as described above is applied, yielding the harmonic sound field response range for a frequency range up to 500 Hz. The procedure is further repeated for a set of different points in the exterior domain. Figure 3 illustrates the lower and upper value on the sound pressure response at 475 Hz, obtained on a cilindric surface around the radiating box. From this analysis, it becomes clear that the presented approach is very flexible. It allows for fast re-analysis of the same problem in different exterior points, as this only influences the acoustic transfer vectors. In fact, any re-analysis in different conditions that only affect the ATV’s can be performed easily, as e.g., the analysis in different acoustic environments.

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Fig. 3: Lower and upper value on the sound pressure response at 475 Hz, obtained on a cilindric surface around the radiating box

5 Conclusions This paper introduces the concept of interval fields for the representation of dependent uncertainty in numerical analysis. On the input side, this concept enables the definition of uncertain field variables ranging over an FE model through a very limited number of independent intervals, resulting in a strong reduction in dimension of the search domain for optimisation based interval analysis. As a result, the accuracy in the interval outcome is improved significantly. On the output side, the interval field concept enables the calculation of a joint representation of multiple output quantities of a single interval finite element problem while preserving the mutual dependence between the components of the output vector. The main advantage of this approach is in the substantial reduction of conservatism in post-processing steps. Finding the implicit interval field representation of a multiple output IFE analysis result comes down to expressing the coupling between the output vector components intrinsically incorporated in the numerical algorithm that solves the corresponding deterministic FE problem. This can be achieved by the application of fast and optimised response surface based approaches. The Kriging approach proved to be of great value in this context. The application of interval fields in a finite element context is illustrated for an ATV based vibro-acoustic analysis, where the uncertainty on the numerical model is

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first processed to structural interval FRF’s at the radiating surfaces of the structure. These structural FRF’s are calculated as a frequency dependent implicit interval field. In the next step, these interval fields are processed simultaneously to interval FRF’s representing the acoustic pressure range at specific points. Acknowledgements This work was conducted with the financial support of the IWT-Flanders through SBO-project 060043 Fuzzy Finite Elements and the Research Fund of the K.U.Leuven through CREA-project 07/015.

References 1. LMS Sysnoise manual, version 5.6., Leuven 2. De Munck M, Moens D, Desmet W, Vandepitte D (2008) An adaptive kriging based optimisation algorithm for interval and fuzzy frf analysis. In: Proceedings of the International Conference on Noise and Vibration Engineering, ISMA 2008, Leuven, pp 3767–3776 3. Ghanem R, Spanos P (1991) Stochastic finite elements: a spectral approach. Springer-Verlag, New-York 4. Hanss M (2005) Applied Fuzzy Arithmetic — An Introduction with Engineering Applications. Springer, Berlin 5. Moens D, Vandepitte D (2005) A survey of non-probabilistic uncertainty treatment in finite element analysis. Computer Methods in Applied Mechanics and Engineering 194(14–16):1527– 1555 6. Moens D, De Munck M, Vandepitte D (2007) Envelope frequency response function analysis of mechanical structures with uncertain modal damping characteristics. Computer Modelling in Engineering&Sciences 22(2):129–149 7. Moens D, De Munck M, Farkas L, De Gersem H, Desmet W, Vandepitte D (2008) Recent advances in interval finite element analysis in applied mechanics. In: Proceedings of the first Leuven Symposium on Applied Mechanics in Engineering LSAME.08, Leuven, pp 553–568 8. Muhanna R, Mullen R, Zhang H (2005) Penalty-based solution for the interval finite-element methods. Journal of Engineering Mechanics 131(10):1102–1111 9. Neumaier A, Pownuk A (2007) Linear systems with large uncertainties with applications to truss structures. Reliable Computing 13(1):149–172 10. Vanmarcke E (1993) Random fields: analysis and synthesis. MIT Press, Cambridge

From Interval Computations to Constraint-Related Set Computations: Towards Faster Estimation of Statistics and ODEs under Interval and p-Box Uncertainty Vladik Kreinovich

Abstract Interval computations estimate the uncertainty of the result of data processing in situations in which we only know the upper bounds Δ on the measurement errors. In this case, based on the measurement result x, we can only conclude that the actual (unknown) value x of the desired quantity is in the interval [ x − Δ , x+ Δ ]. In interval computations, at each intermediate stage of the computation, we have intervals of possible values of the corresponding quantities. As a result, we often have bounds with excess width. To remedy this problem, in our previous papers, we proposed an extension of interval technique to set computations, where on each stage, in addition to intervals of possible values of the quantities, we also keep sets of possible values of pairs (triples, etc.). In this paper, we show that in several practical problems, such as estimating statistics (variance, correlation, etc.) and solutions to ordinary differential equations (ODEs) with given accuracy, this new formalism enables us to find estimates in feasible (polynomial) time.

1 Formulation of the Problem Need for data processing. In many real-life situations, we are interested in the value of a physical quantity y that is difficult or impossible to measure directly. Examples of such quantities are the distance to a star and the amount of oil in a given well. Since we cannot measure y directly, a natural idea is to measure y indirectly. Specifically, we find some easier-to-measure quantities x1 , . . . , xn which are related to y by a known relation y = f (x1 , . . . , xn ); this relation may be a simple functional transformation, or complex algorithm (e.g., for the amount of oil, numerical solution to a partial differential equation). Then, to estimate y, we first measure or estimate the values of the quantities x1 , . . . , xn , and then we use the results x1 , . . . , xn of these measurements (estimations) to compute an estimate y for y as y = f ( x1 , . . . , xn ) University of Texas at El Paso, El Paso, TX 79968, USA, e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 7, © Springer Science+Business Media B.V. 2011

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x1 x2 -

f

y = f ( x1 , . . . , xn )

···

-

xn -

Computing an estimate for y based on the results of direct measurements is called data processing; data processing is the main reason why computers were invented in the first place, and data processing is still one of the main uses of computers as number crunching devices. Measurement uncertainty: from probabilities to intervals. Measurement are never 100% accurate, so in reality, the actual value xi of i-th measured quantity can differ from the measurement result xi . Because of these measurement errors def Δ xi = xi − xi , the result y = f ( x1 , . . . , xn ) of data processing is, in general, different from the actual value y = f (x1 , . . . , xn ) of the desired quantity y. def

It is desirable to describe the error Δ y = y− y of the result of data processing. To do that, we must have some information about the errors of direct measurements.

Δ x1 Δ x2 ...

f

Δy

-

Δ xn What do we know about the errors Δ xi of direct measurements? First, the manufacturer of the measuring instrument must supply us with an upper bound Δ i on the measurement error. If no such upper bound is supplied, this means that no accuracy is guaranteed, and the corresponding “measuring instrument” is practically useless. In this case, once we performed a measurement and got a measurement result xi , we know that the actual (unknown) value xi of the measured quantity belongs to the interval xi = [xi , xi ], where xi = xi − Δi and xi = xi + Δ i . In many practical situations, we not only know the interval [−Δi , Δ i ] of possible values of the measurement error; we also know the probability of different values Δ xi within this interval. This knowledge underlies the traditional engineering approach to estimating the error of indirect measurement, in which we assume that we know the probability distributions for measurement errors Δ xi . In practice, we can determine the desired probabilities of different values of Δ xi by comparing the results of measuring with this instrument with the results of measuring the same quantity by a standard (much more accurate) measuring instrument.

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Since the standard measuring instrument is much more accurate than the one use, the difference between these two measurement results is practically equal to the measurement error; thus, the empirical distribution of this difference is close to the desired probability distribution for measurement error. There are two cases, however, when this determination is not done: • First is the case of cutting-edge measurements, e.g., measurements in fundamental science. When we use the largest particle accelerator to measure the properties of elementary particles, there is no “standard” (much more accurate) located nearby that we can use for calibration: our accelerator is the best we have. • The second case is the case of measurements in manufacturing. In principle, every sensor can be thoroughly calibrated, but sensor calibration is so costly — usually costing ten times more than the sensor itself — that manufacturers rarely do it. In both cases, we have no information about the probabilities of Δ xi ; the only information we have is the upper bound on the measurement error. In this case, after we performed a measurement and got a measurement result xi , the only information that we have about the actual value xi of the measured quantity is that it belongs to the interval xi = [ xi − Δi , xi + Δ i ]. In such situations, the only information that we have about the (unknown) actual value of y = f (x1 , . . . , xn ) is that y belongs to the range y = [y, y] of the function f over the box x1 × . . . × xn : def

y = [y, y] = f (x1 , . . . , xn ) = { f (x1 , . . . , xn ) | x1 ∈ x1 , . . . , xn ∈ xn }. x1 x2 ···

f

y = f (x1 , . . . , xn )

-

xn -

The process of computing this interval range based on the input intervals xi is called interval computations; see, e.g., [4]. Outline. We start by recalling the basic techniques of interval computations and their drawbacks, then we will describe the new set computation techniques and describe a class of problems for which these techniques are efficient. Finally, we talk about how we can extend these techniques to other types of uncertainty (e.g., classes of probability distributions).

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2 Interval Computations: Brief Reminder Interval computations: main idea. Historically the first method for computing the enclosure for the range is the method which is sometimes called “straightforward” interval computations. This method is based on the fact that inside the computer, every algorithm consists of elementary operations (arithmetic operations, min, max, etc.). For each elementary operation f (a, b), if we know the intervals a and b for a and b, we can compute the exact range f (a, b). The corresponding formulas form the so-called interval arithmetic: [a, a] + [b, b] = [a + b, a + b]; [a, a] − [b, b] = [a − b, a − b]; [a, a] · [b, b] = [min(a · b, a · b, a · b, a · b), max(a · b, a · b, a · b, a · b)]; 1/[a, a] = [1/a, 1/a] if 0 ∈ [a, a]; [a, a]/[b, b] = [a, a] · (1/[b, b]). In straightforward interval computations, we repeat the computations forming the program f step-by-step, replacing each operation with real numbers by the corresponding operation of interval arithmetic. It is known that, as a result, we get an enclosure Y ⊇ y for the desired range. From main idea to actual computer implementation. Not every real number can be exactly implemented in a computer; thus, e.g., after implementing an operation of interval arithmetic, we must enclose the result [r− , r+ ] in a computer-representable interval: namely, we must round-off r− to a smaller computer-representable value r, and round-off r + to a larger computer-representable value r. Sometimes, we get excess width. In some cases, the resulting enclosure is exact; in other cases, the enclosure has excess width. The excess width is inevitable since straightforward interval computations increase the computation time by at most a factor of 4, while computing the exact range is, in general, NP-hard (see, e.g., [5]), 1 n 1 n even for computing the population variance V = · ∑ (xi − x)2 , where x = · ∑ xi n i=1 n i=1 (see [3]). If we get excess width, then we can use more sophisticated techniques to get a better estimate, such as centered form, bisection, etc.; see, e.g., [4]. Reason for excess width. The main reason for excess width is that intermediate results are dependent on each other, and straightforward interval computations ignore this dependence. For example, the actual range of f (x1 ) = x1 − x12 over x1 = [0, 1] is y = [0, 0.25]. Computing this f means that we first compute x2 := x12 and then subtract x2 from x1 . According to straightforward interval computations, we compute r = [0, 1]2 = [0, 1] and then x1 − x2 = [0, 1] − [0, 1] = [−1, 1]. This excess width comes from the fact that the formula for interval subtraction implicitly assumes that both a and b can take arbitrary values within the corresponding intervals a and b, while in this case, the values of x1 and x2 are clearly not independent: x2 is uniquely determined by x1 , as x2 = x12 . Why not use uniform distributions? Since we have no information about which values within a given interval are more probable and which are less probable, why

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not assume that these values are equally probable, i.e., that the distribution is uniform? Similarly, for several variables, why not assume a uniform distribution on the corresponding box x1 × . . . × xn — which is mathematically equivalent to assuming that we have n independent random variables xi uniformly distributed in the corresponding intervals xi . This is indeed one of the main ways how interval uncertainty is treated in engineering practice. To explain the limitations of this engineering approach, let us consider the simplest possible algorithm y = f (x1 , . . . , xi , . . . , xn ) = x1 + . . . + xi + . . . + xn . For simplicity, let us assume that the measured values of all n quantities are 0s x1 = . . . = xi = . . . = xn = 0, and that all n measurements have the same error bound Δ x ; Δ1 = . . . = Δ xi = . . . = Δn = Δx . In this case, Δ y = Δ x1 +. . .+ Δ xi +. . .+ Δ xn . Each of n component measurement errors can take any value from −Δ x to Δ x , so the largest possible value of Δ y is attained when all of the component errors attain the largest possible value Δ xi = Δx . In this case, the largest possible value Δ of Δ y is equal to Δ = n · Δ x . Let us see what the maximum entropy approach will predict in this case. According to this approach, we assume that Δ xi are independent random variables, each of which is uniformly distributed on the interval [−Δ , Δ ]. According to the Central Limit theorem, when n → ∞, the distribution of the sum of n independent identically distributed bounded random variables tends to Gaussian. This means that for large values n, the distribution of Δ y is approximately normal. A normal distribution is uniquely determined by its mean and variance. When we add several independent variables, their means and variances add up. For each uniform distribution Δ xi on the interval [−Δ x , Δ x ] of width 2Δx , the mean is 0 and 1 the variance is V = · Δx2 . Thus, for the sum Δ y of n such variables, the mean is 3 0, and the variance is equal to (n/3) · Δx2 . Hence, the standard deviation is equal to √ √ n σ = V = Δx · √ . 3 It is known that in a normal distribution, with probability close to 1, all the values are located within the k · σ vicinity of the mean: for k = 3, it is true with probability 99.9%, for k = 6, it is true with probability 1 − 10−6 %, etc. So, practically with √ certainty, Δ y is located within an√interval k · σ which grows with n as n. n For large n, we have k · Δ x · √ Δx · n, so we get a serious underestimation 3 of the resulting measurement error. This example shows that estimates obtained by selecting a uniform distribution can be very misleading.

3 Constraint-Based Set Computations Main idea. The main idea behind constraint-based set computations (see, e.g., [1]) is to remedy the above reason why interval computations lead to excess width.

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Specifically, at every stage of the computations, in addition to keeping the intervals xi of possible values of all intermediate quantities xi , we also keep several sets: • sets xi j of possible values of pairs (xi , x j ); • if needed, sets xi jk of possible values of triples (xi , x j , xk ); etc. In the above example, instead of just keeping two intervals x1 = x2 = [0, 1], we would then also generate and keep the set x12 = {(x1 , x12 ) | x1 ∈ [0, 1]}. Then, the desired range is computed as the range of x1 − x2 over this set — which is exactly [0, 0.25]. To the best of our knowledge, in interval computations context, the idea of representing dependence in terms of sets of possible values of tuples was first described by Shary; see, e.g., [6, 7] and references therein. How can we propagate this set uncertainty via arithmetic operations? Let us describe this on the example of addition, when, in the computation of f , we use two previously computed values xi and x j to compute a new value xk := xi + x j . In this case, we set xik = {(xi , xi + x j ) | (xi , x j ) ∈ xi j }, x jk = {(x j , xi + x j ) | (xi , x j ) ∈ xi j }, and for every l = i, j, we take xkl = {(xi + x j , xl ) | (xi , x j ) ∈ xi j , (xi , xl ) ∈ xil , (x j , xl ) ∈ x jl }. From main idea to actual computer implementation. In interval computations, we cannot represent an arbitrary interval inside the computer, we need an enclosure. Similarly, we cannot represent an arbitrary set inside a computer, we need an enclosure. To describe such enclosures, we fix the number C of granules (e.g., C = 10). We divide each interval xi into C equal parts Xi ; thus each box xi × x j is divided into C2 subboxes Xi × X j . We then describe each set xi j by listing all subboxes Xi × X j which have common elements with xi j ; the union of such subboxes is an enclosure for the desired set xi j . This implementation enables us to implement all above arithmetic operations. For example, to implement xik = {(xi , xi + x j ) | (xi , x j ) ∈ xi j }, we take all the subboxes Xi × X j that form the set xi j ; for each of these subboxes, we enclosure the corresponding set of pairs {(xi , xi + x j ) | (xi , x j ) ∈ Xi × X j } into a set Xi × (Xi + X j ). This set may have non-empty intersection with several subboxes Xi × Xk ; all these subboxes are added to the computed enclosure for xik . Once can easily see if we start with the exact range xi j , then the resulting enclosure for xik is an (1/C)approximation to the actual set — and so when C increases, we get more and more accurate representations of the desired set. Similarly, to find an enclosure for xkl = {(xi + x j , xl ) | (xi , x j ) ∈ xi j , (xi , xl ) ∈ xil , (x j , xl ) ∈ x jl }, we consider all the triples of subintervals (Xi , X j , Xl ) for which Xi × X j ⊆ xi j , Xi × Xl ⊆ xil , and X j ×Xl ⊆ x jl ; for each such triple, we compute the box (Xi +X j )×Xl ; then, we add subboxes Xk × Xl which intersect with this box to the enclosure for xkl .

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First example: computing the range of x − x. For f (x) = x − x on [0, 1], the actual range is [0, 0], but straightforward interval computations lead to an enclosure [0, 1] − [0, 1] = [−1, 1]. In straightforward interval computations, we have r1 = x with the exact interval range r1 = [0, 1], and we have r2 = x with the exact interval range x2 = [0, 1]. The variables r1 and r2 are dependent, but we ignore this dependence. In the new approach: we have r1 = r2 = [0, 1], and we also have r12 : r2 × × × × × r1 For each small box, we have [−0.2, 0.2], so the union is [−0.2, 0.2]. If we divide into more pieces, we get an interval closer to 0. Second example: computing the range of x − x2 . In straightforward interval computations, we have r1 = x with the exact interval range interval r1 = [0, 1], and we have r2 = x2 with the exact interval range x2 = [0, 1]. The variables r1 and r2 are dependent, but we ignore this dependence and estimate r3 as [0, 1] − [0, 1] = [−1, 1]. In the new approach: we have r1 = r2 = [0, 1], and we also have r12 . First, we divide the range [0, 1] into 5 equal subintervals R1 . The union of the ranges R21 corresponding to these 5 subintervals R1 is [0, 1], so r2 = [0, 1]. We divide this interval r2 into 5 equal sub-intervals [0, 0.2], [0.2, 0.4], etc. We now compute the set r12 as follows: • for R1 = [0, 0.2], we have R21 = [0, 0.04], so only sub-interval [0, 0.2] of the interval r2 is affected; • for R1 = [0.2, 0.4], we have R21 = [0.04, 0.16], so also only sub-interval [0, 0.2] is affected; • for R1 = [0.4, 0.6], we have R21 = [0.16, 0.36], so two sub-intervals [0, 0.2] and [0.2, 0.4] are affected, etc. r2 × ×

×

× × ×

×

×

× r1

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For each possible pair of small boxes R1 × R2 , we have R1 − R2 = [−0.2, 0.2], [0, 0.4], or [0.2, 0.6], so the union of R1 − R2 is r3 = [−0.2, 0.6]. If we divide into more and more pieces, we get the enclosure which is closer and closer to the exact range [0, 0.25]. How to compute rik . The above example is a good case to illustrate how we compute the range r13 for r3 = r1 − r2 . Indeed, since r3 = [−0.2, 0.6], we divide this range into 5 subintervals [−0.2, −0.04], [−0.04, 0.12], [0.12, 0.28], [0.28, 0.44], [0.44, 0.6]. • For R1 = [0, 0.2], the only possible R2 is [0, 0.2], so R1 − R2 = [−0.2, 0.2]. This covers [−0.2, −0.04], [−0.04, 0.12], and [0.12, 0.28]. • For R1 = [0.2, 0.4], the only possible R2 is [0, 0.2], so R1 − R2 = [0, 0.4]. This interval covers [−0.04, 0.12], [0.12, 0.28], and [0.28, 0.44]. • For R1 = [0.4, 0.6], we have two possible R2 : – for R2 = [0, 0.2], we have R1 − R2 = [0.2, 0.6]; this covers [0.12, 0.28], [0.28, 0.44], and [0.44, 0.6]; – for R2 = [0.2, 0.4], we have R1 − R2 = [0, 0.4]; this covers [−0.04, 0.12], [0.12, 0.28], and [0.28, 0.44]. • For R1 = [0.6, 0.8], we have R21 = [0.36, 0.64], so three possible R2 : [0.2, 0.4], [0.4, 0.6], and [0.6, 0.8], to the total of [0.2, 0.8]. Here, [0.6, 0.8] − [0.2, 0.8] = [−0.2, 0.6], so all 5 subintervals are affected. • Finally, for R1 = [0.8, 1.0], we have R21 = [0.64, 1.0], so two possible R2 : [0.6, 0.8] and [0.8, 1.0], to the total of [0.6, 1.0]. Here, [0.8, 1.0] − [0.6, 1.0] = [−0.2, 0.4], so the first 4 subintervals are affected. r3

×

× ×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

× r1

Limitations of this approach. The main limitation of this approach is that when we need an accuracy ε , we must use ∼ 1/ε granules; so, if we want to compute the result with k digits of accuracy, i.e., with accuracy ε = 10−k , we must consider exponentially many boxes (∼ 10k ). In plain words, this method is only applicable when we want to know the desired quantity with a given accuracy (e.g., 10%). Cases when this approach is applicable. In practice, there are many problems when it is sufficient to compute a quantity with a given accuracy: e.g., when we detect an outlier, we usually do not need to know the variance with a high accuracy, an accuracy of 10% is more than enough.

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Let us describe the case when interval computations do not lead to the exact range, but set computations do — of course, the range is “exact” modulo accuracy of the actual computer implementations of these sets. Example: estimating variance under interval uncertainty. Suppose that we know the intervals x1 , . . . , xn of possible values of x1 , . . . , xn , and we need to compute the 1 1 def n def n range of the variance V = · M − 2 · E 2 , where M = ∑ xi2 and E = ∑ xi . n n i=1 i=1 This problem is important, e.g., in detecting outliers. Outliers are useful in many application areas. For example, in medicine, to detect possible illnesses, we analyze the healthy population, compute the averages E[x] and the standard deviations σ [x] of different characteristics x, and if for some person, the value of a blood pressure, weight, body temperature, etc., is outside the corresponding 2- or 3-sigma interval [E[x]−k0 · σ [x], E[x]+k0 · σ [x]], then we perform additional tests to see if there is any hidden problem with this person’s health. Similarly, in geophysics, when we look for rare minerals, we know the typical values for a given area, and if at some location, the values of the geophysical characteristics are outliers (i.e., they are outside the corresponding interval), then these area are probably the most promising. Traditional algorithms for detecting outliers assume that we know the exact values xi of the corresponding characteristics but in practice, these values often come from estimates or crude measurements. For example, most routine blood pressure measurements performed at health fairs, in drugstores, at the dentist office, etc., are very approximate, with accuracy 10 or more; their objective is not to find the exact values of the corresponding characteristics but to make sure that we do not miss a dangerous anomaly. When we estimate the mean and the standard deviations based on these approximate measurements, we need to take into account that these values are very approximate, i.e., that, in effect, instead of the exact value xi (such as 110), we only know that the actual (unknown) value of the blood pressure is somewhere within the interval [ xi − Δ i , xi + Δ i ] = [110 − 10, 110 + 10] = [100, 120]. In all these situations, we need to compute the range on the variance V under the interval uncertainty on xi . def k

A natural way to to compute V is to compute the intermediate sums Mk = ∑ xi2 i=1

def k

and Ek = ∑ xi . We start with M0 = E0 = 0; once we know the pair (Mk , Ek ), we i=1

2 ,E +x compute (Mk+1 , Ek+1 ) = (Mk + xk+1 k k+1 ). Since the values of Mk and Ek only depend on x1 , . . . , xk and do not depend on xk+1 , we can conclude that if (Mk , Ek ) is a possible value of the pair and xk+1 is a possible value of this variable, then 2 ,E +x (Mk + xk+1 k k+1 ) is a possible value of (Mk+1 , Ek+1 ). So, the set p0 of possible values of (M0 , E0 ) is the single point (0, 0); once we know the set pk of possible values of (Mk , Ek ), we can compute pk+1 as

{(Mk + x2 , Ek + x) | (Mk , Ek ) ∈ pk , x ∈ xk+1 }.

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For k = n, we will get the set pn of possible values of (M, E); based on this set, we 1 1 can then find the exact range of the variance V = · M − 2 · E 2 . n n What C should we choose to get the results with an accuracy ε ·V ? On each step, we add the uncertainty of 1/C; to, after n steps, we add the inaccuracy of n/C. Thus, to get the accuracy n/C ≈ ε , we must choose C = n/ε . What is the running time of the resulting algorithm? We have n steps; on each step, we need to analyze C3 combinations of subintervals for Ek , Mk , and xk+1 . Thus, overall, we need n ·C3 steps, i.e., n4 /ε 3 steps. For fixed accuracy C ∼ n, so we need O(n4 ) steps — a polynomial time, and for ε = 1/10, the coefficient at n4 is still 103 — quite feasible. For example, for n = 10 values and for the desired accuracy ε = 0.1, we need 103 · n4 ≈ 107 computational steps — “nothing” for a Gigaherz (109 operations per second) processor on a usual PC. For n = 100 values and the same desired accuracy, we need 104 · n4 ≈ 1012 computational steps, i.e., 103 seconds (15 minutes) on a Gigaherz processor. For n = 1000, we need 1015 steps, i.e., 106 computational steps — 12 days on a single processor or a few hours on a multi-processor machine. In comparison, the exponential time 2n needed in the worst case for the exact computation of the variance under interval uncertainty, is doable (210 ≈ 103 step) for n = 10, but becomes unrealistically astronomical (2100 ≈ 1030 steps) already for n = 100. Comment. When the accuracy increases ε = 10−k , we get an exponential increase in running time — but this is OK since, as we have mentioned, the problem of computing variance under interval uncertainty is, in general, NP-hard. Other statistical characteristics. Similar algorithms can be presented for computing many other statistical characteristics. For example, for every integer d > 2, 1 n the corresponding higher-order central moment Cd = · ∑ (xi − x)d is a linear n i=1 def n

j

combination of d moments M ( j) = ∑ xi for j = 1, . . . , d; thus, to find the exact i=1

range for Cd , we can keep, for each k, the set of possible values of d-dimensional (1)

( j) def

(d)

tuples (Mk , . . . , Mk ), where Mk

k

= ∑ xij . For these computations, we need i=1

n ·Cd+1 ∼ nd+2 steps — still a polynomial time. n 1 n 1 n Another example is covariance Cov = · ∑ xi · yi − 2 · ∑ xi · ∑ yi . To compute n i=1 n i=1 i=1 def

covariance, we need to keep the values of the triples (Covk , Xk ,Yk ), where Covk = k

def k

def k

∑ xi · yi , Xk = ∑ xi , and Yk = ∑ yi . At each step, to compute the range of

i=1

i=1

i=1

(Covk+1 , Xk+1 ,Yk+1 ) = (Covk + xk+1 · yk+1 , Xk + xk+1 ,Yk + yk+1 ), we must consider all possible combinations of subintervals for Covk , Xk , Yk , xk+1 , and yk+1 — to the total of C5 . Thus, we can compute covariance in time n ·C5 ∼ n6 .

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95

Similarly, to compute correlation ρ = Cov/ Vx ·Vy , we can update, for each k, (2)

(2)

(2)

k

(2)

the values of (Ck , Xk ,Yk , Xk ,Yk ), where Xk = ∑ xi2 and Yk i=1

k

= ∑ y2i are needed i=1

to compute the variances Vx and Vy . These computations require time n ·C7 ∼ n8 . Systems of ordinary differential equations (ODEs) under interval uncertainty. A general system of ODEs has the form x˙i = f i (x1 , . . . , xm ,t), 1 ≤ i ≤ m. Interval uncertainty usually means that the exact functions fi are unknown, we only know the expressions of f i in terms of parameters, and we have interval bounds on these parameters. There are two types of interval uncertainty: we may have global parameters whose values are the same for all moments t, and we may have noise-like parameters whose values may different at different moments of time — but always within given intervals. In general, we have a system of the type x˙i = fi (x1 , . . . , xm ,t, a1 , . . . , ak , b1 (t), . . . , bl (t)), where fi is a known function, and we know the intervals a j and b j (t) of possible values of ai and b j (t). Example. For example, the case of a differential inequality when we only know the bounds f i (x1 , . . . , xn ,t) and f i (x1 , . . . , xn ,t) on fi (x1 , . . . , xn ,t) can be described as fi (x1 , . . . , xn ,t) + b1 (t) · Δ (x1 , . . . , xn ,t), def def where fi (x1 , . . . , xn ,t) = ( f i (x1 , . . . , xn ,t) + f i (x1 , . . . , xn ,t))/2, Δ (x1 , . . . , xn ,t) = ( f i (x1 , . . . , xn ,t) − f i (x1 , . . . , xn ,t))/2, and b1 (t) = [−1, 1].

Solving systems of ordinary differential equations (ODEs) under interval uncertainty. For the general system of ODEs, Euler’s equations take the form xi (t + Δ t) = xi (t) + Δ t · fi (x1 (t), . . . , xm (t),t, a1 , . . . , ak , b1 (t), . . . , bl (t)). Thus, if for every t, we keep the set of all possible values of a tuple (x1 (t), . . . , xm (t), a1 , . . . , ak ), then we can use the Euler’s equations to get the exact set of possible values of this tuple at the next moment of time. The reason for exactness is that the values xi (t) depend only on the previous values b j (t − Δ t), b j (t − 2Δ t), etc., and not on the current values b j (t). To predict the values xi (T ) at a moment T , we need n = T /Δ t iterations. To update the values, we need to consider all possible combinations of m + k + l variables x1 (t), . . . , xm (t), a1 , . . . , ak , b1 (t), . . . , bl (t); so, to predict the values at moment T = n · Δ t in the future for a given accuracy ε > 0, we need the running time n ·Cm+k+l ∼ nk+l+m+1 . This is is still polynomial in n.

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Other possible cases when our approach is efficient. Similar computations can be performed in other cases when we have an iterative process where a fixed finite number of variables is constantly updated. In such problems, there is an additional factor which speeds up computations. Indeed, in the modern computers, fetching a value from the memory, in general, takes much longer than performing an arithmetic operation. To decrease this time, computers have a hierarchy of memories — from registers from which the access is the fastest, to cash memory (second fastest), etc. Thus, to take full use of the speed of modern processors, we must try our best to keep all the intermediate results in the registers. In the problems in which, at each moment of time, we can only keep (and update) a small current values of the values, we can store all these values in the registers — and thus, get very fast computations (only the input values x1 , . . . , xn need to be fetched from slower-to-access memory locations). Additional advantage of our technique: possibility to take constraints into account. Traditional formulations of the interval computation problems assume that we can have arbitrary tuples (x1 , . . . , xn ) as long as xi ∈ xi for all i. In practice, we may have additional constraints on xi . For example, we may know that xi are observations of a smoothly changing signal at consequent moments of time; in this case, we know that |xi − xi+1 | ≤ ε for some small known ε > 0. Such constraints are easy to take into account in our approach. For example, if know that xi = [−1, 1] for all i and we want to estimate the value of a high-frequency Fourier coefficient f = x1 − x2 + x3 − x4 + . . . − x2n , then usual interval computations lead to an enclosure [−2n, 2n], while, for small ε , the actual range for the sum (x1 − x2 ) + (x3 − x4 ) + . . . where each of n differences is bounded by ε , is much narrower: [−n · ε , n · ε ] (and for xi = i · ε , these bounds are actually attained). Computation of f means computing the values fk = x1 − x2 + . . . + (−1)k+1 · xk for k = 1, . . . At each stage, we keep the set sk of possible values of ( fk , xk ), and use this set to find sk+1 = {( f k + (−1)k · xk+1 , xk+1 ) | ( fk , xk ) ∈ sk & |xk − xk+1 | ≤ ε }. In this approach, when computing f2k , we take into account that the value x2k must be ε -close to the value xk and thus, that we only add ≤ ε . Thus, our approach leads to almost exact bounds — modulo implementation accuracy 1/C. In this simplified example, the problem is linear, so we could use linear programming to get the exact range, but set computations work for similar non-linear problems as well. Toy example with a constraint. The problem is to find the range of r1 − r2 when r1 = [0, 1], r2 = [0, 1], and |r1 − r2 | ≤ 0.1. Here, the actual range is [−0.1, 0.1], but straightforward interval computations return [0, 1] − [0, 1] = [−1, 1]. In the new approach, first, we describe the constraint in terms of subboxes:

From Interval Computations to Set Computations

r2

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×

×

×

×

×

×

×

×

×

×

×

×

× r1

Next, we compute R1 − R2 for all possible pairs and take the union. The result is [−0.6, 0.6]. If we divide into more pieces, we get the enclosure closer to [−0.1, 0.1]. Towards possible extension to p-boxes and classes of probability distributions. Often, in addition to the interval xi of possible values of the inputs xi , we also have partial information about the probabilities of different values xi ∈ xi . An exact probability distribution can be described, e.g., by its cumulative distribution function Fi (z) = Prob(xi ≤ z). In these terms, a partial information means that instead of a single cdf, we have a class F of possible cdfs. A practically important particular case of this partial information is when, for each z, instead of the exact value F(z), we know an interval F(z) = [F(z), F(z)] of possible values of F(z); such an “interval-valued” cdf is called a probability box, or a p-box, for short; see, e.g., [2]. Propagating p-box uncertainty via computations: a problem. Once we know the classes Fi of possible distributions for xi , and a data processing algorithms f (x1 , . . . , xn ), we would like to know the class F of possible resulting distributions for y = f (x1 , . . . , xn ). Idea. For problems like systems of ODES, it is sufficient to keep, and update, for all t, the set of possible joint distributions for the tuple (x1 (t), . . . , a1 , . . .). From idea to computer implementation. We would like to estimate the values with some accuracy ε ∼ 1/C and the probabilities with the similar accuracy 1/C. To describe a distribution with this uncertainty, we divide both the x-range and the probability (p-) range into C granules, and then describe, for each x-granule, which p-granules are covered. Thus, we enclose this set into a finite union of p-boxes which assign, to each of x-granules, a finite union of p-granule intervals. A general class of distributions can be enclosed in the union of such p-boxes. There are finitely many such assignments, so, for a fixed C, we get a finite number of possible elements in the enclosure. We know how to propagate uncertainty via simple operations with a finite amount of p-boxes (see, e.g., [2]), so for ODEs we get a polynomial-time algorithm for computing the resulting p-box for y. For p-boxes, we need further improvements to make this method practical. Formally, the above method is polynomial-time. However, it is not yet practical

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beyond very small values of C. Indeed, in the case of interval uncertainty, we needed C2 or C3 subboxes. This amount is quite feasible even for C = 10. To describe a p-subbox, we need to attach one of C probability granules to each of C x-granules; these are ∼ CC such attachments, so we need ∼ CC subboxes. For C = 10, we already get an unrealistic 1010 increase in computation time. Acknowledgements This work was supported in part by the National Science Foundation grants HRD-0734825 and DUE-0926721, by Grant 1 T36 GM078000-01 from the National Institutes of ˇ Health, by Grant MSM 6198898701 from MSMT of Czech Republic, and by Grant 5015 from the Science and Technology Centre in Ukraine (STCU), funded by European Union. The authors are thankful to all the participants of the International Union for Theoretical and Applied Mechanics (IUTAM) Symposium on The Vibration Analysis of Structures with Uncertainties, Saint-Petersburg, Russia, July 6–10, 2009.

References 1. M. Ceberio, S. Ferson, V. Kreinovich, S. Chopra, G. Xiang, A. Murguia, and J. Santillan, “How To Take Into Account Dependence Between the Inputs: From Interval Computations to Constraint-Related Set Computations”, Proc. 2nd Int’l Workshop on Reliable Engineering Computing, Savannah, Georgia, February 22–24, 2006, pp. 127–154; final version in Journal of Uncertain Systems, 2007, Vol. 1, No. 1, pp. 11–34. 2. S. Ferson, RAMAS Risk Calc 4.0. CRC Press, Boca Raton, Florida, 2002. 3. S. Ferson, L. Ginzburg, V. Kreinovich, L. Longpr´e, and M. Aviles, “Computing Variance for Interval Data is NP-Hard”, ACM SIGACT News, 2002, Vol. 33, No. 2, pp. 108–118. 4. L.Jaulin, M. Kieffer, O. Didrit, and E. Walter, Applied Interval Analysis, Springer, London, 2001. 5. V. Kreinovich, A. Lakeyev, J. Rohn, and P. Kahl, Computational complexity and feasibility of data processing and interval computations, Kluwer, Dordrecht, 1997. 6. S. P. Shary, “Parameter partitioning scheme for interval linear systems with constraints”, Proceedings of the International Workshop on Interval Mathematics and Constraint Propagation Methods (ICMP’03), July 8–9, 2003, Novosibirsk, Akademgorodok, Russia, pp. 1–12 (in Russian). 7. S. P. Shary, “Solving tied interval linear systems”, Siberian Journal of Numerical Mathematics, 2004, Vol. 7, No. 4, pp. 363–376 (in Russian).

Dynamic Steady-State Analysis of Structures under Uncertain Harmonic Loads via Semidefinite Program Yoshihiro Kanno and Izuru Takewaki

Abstract In this paper we present an optimization-based method for finding confidential bounds for the dynamic steady-state response of a damped structure subjected to uncertain driving loads. The amplitude of harmonic driving loads is supposed to obey a non-probabilistic uncertainty model. We formulate a semidefinite programming problem, whose optimal value corresponds to a confidential bound for the characteristic amount of dynamic steady-state response, e.g. the modulus and phase angle of the complex amplitude of the displacement. Numerical examples demonstrate that sufficiently tight bounds can be obtained by solving the presented semidefinite programming problems.

1 Introduction Motivated by recent interests in non-stochastic uncertainty analysis of structures, we propose an optimization-based algorithm for estimating the dynamic structural response under uncertain harmonic loads. For stochastic uncertainty models, various methods have been developed for uncertainty analysis of structures; see, e.g. [23, 6], and the references therein. However, since it is often difficult to accurately estimate stochastic distributions of parameters of an uncertain structural system, we aim at investigating a non-stochastic uncertainty model. One of well-known non-stochastic methods is the so-called convex model approach [4]. The energy-bounded convex model was used to study dynamic problems subjected to uncertain impulsive loads [22] and uncertain seismic excitations [18]. Yoshihiro Kanno Department of Mathematical Informatics, University of Tokyo, Bunkyo, Tokyo 113-8656, Japan, e-mail: [email protected] Izuru Takewaki Department of Urban and Environmental Engineering, Kyoto University, Nishikyo, Kyoto 6158540, Japan, e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 8, © Springer Science+Business Media B.V. 2011

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Note that when we try to estimate a nonlinear function of uncertain parameters, the conventional convex model approach is valid only if the magnitude of uncertainty is small enough, because it is essentially based on the first-order approximation of the structural response with respect to the uncertain parameters. As a non-stochastic approach applicable to the uncertainty with an arbitrary large magnitude, Ben-Haim [3] proposed the info-gap decision theory, applications of which have been found in various fields. For computing the exact bounds of responses of an uncertain truss, mixed integer programming approaches have been developed for the member stress [11] and for the plastic limit-load factor [14]. The interval linear algebra [1] provides a conservative bound for the solution set of the uncertain linear equations (ULE). The interval algebra has been applied to uncertainty analysis in structural dynamics, in particular to uncertain eigenvalue problems for finding bounds for eigenvalues [8, 20, 10] and eigenmodes [9] of structures. The fuzzy theory is regarded as another non-stochastic uncertainty model, and has been applied to uncertain dynamic problems [17]. For the mathematical programs with uncertain data, Ben-Tal and Nemirovski [5] presented a unified methodology of robust optimization for convex optimization problems, in which given data in optimization problems are supposed to include non-stochastic uncertainties. Calafiore and El Ghaoui [7] proposed a method for finding an ellipsoidal bound for the solution set of ULE by using a semidefinite programming (SDP) relaxation. The authors formulated SDPs for various structures in order to obtain a confidential ellipsoidal bound for the static response [13, 15]. In this paper we consider the dynamic response of a damped structure under the uncertain driving load. We are particularly interested in the steady state of a forced oscillation induced by a harmonic driving load, where the amplitude of driving load is supposed to has non-stochastic uncertainties. We attempt to find conservative bounds for various characteristic parameters of the steady state, e.g. lower and upper bounds for the modulus and phase angle of the complex amplitude of nodal displacements. We show a close relation between the notion of robust optimization problem (in the sense of [5]) and the bound detection problem (or extremal-case detection problem) for an uncertain structural system. More specifically, we first formulate the extremal-case detection problem as a nonlinear programming problem. Then we show that the nonlinear programming problem can be rewritten as a robust optimization problem, which includes infinitely many constraint conditions. By using the S -lemma [19], we present a numerically tractable sufficient condition for the constraint conditions of the robust optimization problem. Finally, from the sufficient condition we construct a convex optimization problem which is guaranteed to provide a conservative approximation of the exact bound. The obtained goal formulation is an SDP problem, which can be solved efficiently by using the primal-dual interior-point method [12]. Hence, we can compute conservative bounds for dynamic steady-state structural response effectively by solving the SDPs.

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2 Uncertain equations for steady state vibration We here formulate the steady-state oscillation problem including the uncertainties as a system of uncertain linear equations (ULE) in the real variables.

2.1 Governing equations Consider a finite-dimensional linear elastic structure in the small deformations. We denote by d the number of degrees of freedom of displacements. Consider a harmonic driving load defined by feiω t , where f ∈ Rd is the load amplitude vector and ω ∈ R is the circular frequency. We denote by K , M , C ∈ Rd×d , respectively, the stiffness, mass, and damping matrices. The d’Alembert principle yields C u˙ˆ + K uˆ = feiω t , M u¨ˆ +C

(1)

where uˆ ∈ Cd is the displacement vector. Substitution of uˆ = ueiω t into (1) leads to (−ω 2 M + iω C + K )u = f,

(2)

where u ∈ Cd is the displacement amplitude vector driven by feiω t .

2.2 Uncertainty model Suppose that the amplitude of the load, f, in (2) is uncertain, while K , M , C , and ω are known precisely. Let ˜f ∈ Rd denote the nominal value, or the best estimate, of f. We describe the uncertainty of f by using an unknown vector ζ ∈ Rk . Assume that f depends on ζ affinely as f = ˜f + F 0 ζ ,

(3)

where F 0 ∈ Rd×k is a constant matrix. Note that the matrix F 0 represents the relative magnitude of the uncertainty of f j ( j = 1, . . . , d) and the relationship of the uncertainties among f1 , . . . , fd . See Example 1 below for an illustrative example. Suppose that ζ satisfies T j ζ , α ≥ T

j = 1, . . . , ,

(4)

where T j ∈ Rm j ×k is a constant matrix. We choose T 1 . . . , T so that the set of ζ satisfying (4) is bounded for any given α ≥ 0. We call ζ the vector of uncertain parameters, or unknown-but-bounded parameters. It follows (3) and (4) that f is supposed to be included in the set

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Fig. 1: A 29-bar truss subjected to dynamic loading

T j ζ ( j = 1, . . . , ) . F (α ) = f˜ + F 0 ζ | α ≥ T

(5)

We call F (α ) the uncertainty set of f. In (5) we see that the parameter α represents the level of uncertainty, i.e. the greater the value of α , the greater the range of possible variation of f. Hence, α is referred to as the uncertainty parameter [3]. It is also mentioned that F (0) = {˜f} is satisfied, which implies that the estimate ˜f is correct at α = 0. Example 1. Consider a plane truss shown in Fig. 1 with d = 20. Suppose that uncertain loads may possibly exist at all free nodes, and are independently running through the circles depicted with the dotted lines in Fig. 1. Relatively large forces are applied at the nodes (a) and (b) as the nominal load ˜f. Suppose that all circles in Fig. 1 have common radii. Then we put F 0 = f¯0 I 20 with k = 20. Since we consider 10 independent circles in Fig. 1, put = 10 and define T j ∈ R2×20 ( j = 1, . . . , ) by T 1 = I2 O · · · O , T 2 = O I2 · · · O , . . . , T 10 = O O · · · I2 , where O is the 2 × 2 zero matrix. Thus F (α ) in (5) represents the uncertainty model in which the nodal force at each node perturbs in the circle with the radius α f¯0 .

2.3 ULE in real variables Observe that (2) is rewritten as −ω 2 M + K Reu −ω C f = , 0 ωC −ω 2 M + K Imu

(6)

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103

d where Reu and Imu denote the real and imaginary parts of u ∈ C , respectively. Let Reu v= ∈ R2d . Then (6) is simply written as Imu

S 1 v = f, where S 1 , S 2 ∈ Rd×2d are defined by S 1 = K − ω 2 M −ω C ,

S 2 v = 0,

(7)

S2 = ω C K − ω 2 M .

Define V ⊆ R2d by V = {v | S 1 v = f, S 2 v = 0, f ∈ F (α )} ,

(8)

which is the set of all possible solutions to (7).

3 Bounds for complex amplitude Consider the displacement amplitude u = (uq ) ∈ Cd satisfying (2). We denote by |uq | and arg uq the modulus and the argument of uq , respectively. In this section we attempt to find upper and lower bounds for |uq | and arg uq .

3.1 Upper bound for modulus of displacement amplitude The maximum value rmax of |uq | is defined by rmax =

max

u∈Cd ,

f∈Rd

|uq | : (−ω 2 M + iω C + K )u = f, f ∈ F (α ) .

(9)

In this section, we formulate an SDP problem providing an upper bound of rmax . Define a constant matrix G ∈ R2×2d by Re uq = Gv , Im uq Gv. It should be clear that G Gv denotes the standard Euwhich yields |uq | = G clidean norm of the real vector G v ∈ R2 . Consequently, (9) is rewritten as Gv : v ∈ V } . rmax = max {G v∈R2d

(10)

Thus the optimization problem (9) containing the complex variables is reduced to the problem (10) in the real variables.

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Fig. 2: A schematic interpretation of the relation between (10) and (11). The radius r of a circle including VG corresponds to an upper bound of rmax

We next reformulate the optimization problem (10) into the form of the robust optimization in the sense of [5]. A key idea is represented in Fig. 2, where G V ⊆ R2 Gv | v ∈ V }. Observe that (10) is the maximization problem of is defined by G V = {G the Euclidean distance between the origin and the point G v in the two-dimensional space. In other words, among points in G V , we attempt to find the one which is the farthest from the origin. This is equivalent to find the minimum circle, which includes G V and is centered at the origin. The radius of the minimum circle coincides with rmax . Consequently, the problem (10) is equivalent to the following optimization problem: Gv (∀v ∈ V )} . rmax = min {r : r ≥ G r∈R

(11)

A rigorous proof of (11) can be found in [16]. Note that (11) is regarded as a robust optimization problem, because the conGv is required to be satisfied for all the possible v ∈ V . This straint condition r ≥ G condition is equivalently rewritten as v∈V

⇒

Gv. r ≥ G

(12)

Since it is difficult to deal with (12), we next attempt to replace (12) with a more tractable one which is guaranteed to be conservative. The basic idea with which we can address our tractable and conservative reformulation is stated in the following proposition. Proposition 1 ([19]). Let Q i ∈ Rn×n (i = 1, . . . , m) and P j ∈ Rn×n ( j = 1, . . . , k) be symmetric matrices. Consider the following two conditions: (a) : (b) :

x x ≥ 0 (i = 1, . . . , m), Qi 1 1 x x ≥ 0. Q0 1 1

x x = 0 ( j = 1, . . . , k); Pj 1 1

The implication “(a) ⇒ (b)” holds if there exist τ1 , . . . , τm and σ1 , . . . , σk satisfying

Dynamic Steady-State Analysis of Uncertain Structures via SDP m

k

i=1

j=1

Q 0 ∑ τi Q i + ∑ σ j P j ,

105

τ1 , . . . , τm ≥ 0.

Here, for symmetric real matrices A , B ∈ Rn×n , we write A B if the matrix A − B is positive semidefinite. Observe that the conditions (a) and (b) in Proposition 1 are represented by some quadratic inequalities. Similarly, we can rewrite the conditions appearing in (12) as finitely many quadratic inequalities by eliminating the uncertain Gv on the rightparameters ζ as follows. Firstly observe that the inequality r ≥ G hand side of (12) is rewritten as G G 0 v v −G ≥ 0, 1 0 r2 1

r ≥ 0.

We next consider the left-hand side of (11). Recall that V has been defined by (8). Define Ψˆ j ( j = 1, . . . , ) and Θˆ by Ψˆ j = T j F −1 S 1 −˜f , 0

Θˆ = S 2 O .

j = 1, . . . , ;

Then we see that v ∈ V if and only if ˆ v α − Ψ j 1 ≥ 0, j = 1, . . . , ;

v ˆ Θ = 0. 1

(13)

We can reduce (13) into quadratic inequities as v v ≥ 0, Ψj 1 1

j = 1, . . . , ;

v v = 0, Θ 1 1

where

Ψ j = diag(0, α 2 ) − Ψˆ j Ψˆ j ,

j = 1, . . . , ;

Θ = Θˆ Θˆ .

As a consequence, it follows from Proposition 1 that we obtain a sufficient condition for (11) as follows. Proposition 2. r ≥ 0 satisfies the condition (11) if there exists a vector (w, s) ∈ R × R satisfying G G 0 −G 0 r2

∑ w jΨ j + sΘ ,

w1 , . . . , w ≥ 0.

(14)

j=1

Proposition 2 provides a sufficient condition for the constraint condition of the problem (11). By replacing the constraint condition in (11) with (14), we obtain the following optimization problem:

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G G 0 −G − ∑ w jΨ j − sΘ O, w ≥ 0 . ρ ∗ = min ρ : ρ ,w,s 0 ρ j=1

(15)

2 , i.e. √ρ ∗ corresponds to an upper bound From Proposition 2 we obtain ρ ∗ ≥ rmax of |uq |. See, for a rigorous proof, [16]. Since (15) is an SDP problem [12] in the variables ρ ∈ R, w ∈ R , and s ∈ R, we can compute ρ ∗ efficiently by using the primal-dual interior-point method.

3.2 Lower bound for modulus of displacement amplitude The minimum value rmin of |uq | is defined by |uq | : (−ω 2 M + iω C + K )u = f, f ∈ F (α ) . rmin = min u∈Cd , f∈Rd

(16)

We reformulate the problem (16) in the complex variable to a problem in the real variables by using v and G . Then (16) is reduced to ˜f + F 0 ζ S Gv, 1 v = T j ζ ( j = 1, . . . , ) . rmin = min r : r ≥ G , α ≥ T S2 0 r,v,ζ (17) Note that (17) is an SOCP problem [2] in the variables r ∈ R, v ∈ R2d , and ζ ∈ Rk . Hence, we can compute the global optimal solution of (17) easily by using the primal-dual interior-point method. Thus, it is easy to compute the exact value of rmin , which is quite a contrast to computing rmax .

3.3 Bounds for phase angle In this section we consider the distribution of the phase angle of uq . The maximal value of arg uq is defined as arg uq : (−ω 2 M + iω C + K )u = f, f ∈ F (α ) . θmax := max (18) u∈Cd , f∈Rd

Throughout this section we assume that −π < θmax < π . We here restrict ourselves to the case in which θmax satisfies 0 < θmax < π . The other cases, in which this condition does not hold, can be dealt with similarly; see [16]. In a manner similar to Sect. 3.1, we reduce (18) to the form of robustoptiRe uq mization. A key observation is illustrated in Fig. 3. Recall that G v = and Im uq Gv | v ∈ V }. In the problem (18), we find the point which has the maximum G V = {G

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Fig. 3: A schematic interpretation of the relation between (18) and (19). For the normal vector a = (a1 , a2 ) of the line, the ratio −a1 /a2 corresponds to an upper bound of tan θmax

phase angle among points satisfying G v ∈ G V . On the other hand, we see that the line a G v = 0 depicted in Fig. 3 satisfies −a1 /a2 ≥ tan θmax , where a = (a1 , a2 ) is the normal vector of the line. Thus, finding θmax is equivalent to minimize −a1 /a2 as far as possible, among a such that a G v ≥ 0 holds for any G v ∈ GV . From this observation we obtain

−1/ tan θmax = min a2 : a1 = 1, a Gv ≥ 0 (∀v ∈ V ) ; (19) a∈R2

see [16] for a rigorous proof of (19). The problem (19) is a robust optimization reformulation of the problem (18). We next investigate the constraint condition of (19). In a manner similar to Proposition 2, we can show that a satisfies a G v ≥ 0 (∀v ∈ V ) if there exists a vector (w, s) ∈ R × R satisfying O Ga a G 0

∑ w jΨ j + sΘ ,

w1 , . . . , w ≥ 0.

(20)

j=1

By using (20), a conservatively approximated problem of (19) is obtained as

O G a ∗ − ∑ w jΨ j − sΘ O , w ≥ 0, a1 = 1 . (21) a2 = min a2 : a,w,s a G 0 j=1 For computing an upper bound for θmax , define ϕ ∗ by (if a∗2 ≤ 0), Arctan(−1/a∗2 ) ∗ ϕ = ∗ Arctan(−1/a2 ) + π (if a∗2 > 0).

(22)

Then we can show that ϕ ∗ ≥ θmax holds, i.e. ϕ ∗ corresponds to an upper bound for arg uq . See [16] for more details. Note that (21) is an SDP problem, and hence it can be solved easily. In a similar way, we can construct an SDP problem providing a lower bound for θmin , which is the minimum value of arg uq .

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4 Bounds for nodal oscillation Consider a structure in the two-dimensional space. For the p-th node we denote by u px , u py ∈ C the nodal displacements in the x- and y-th directions, respectively. In this section we consider to find the bounds for the amplitude of nodal oscillation, i.e. the bounds for ν p and ν py /ν px , where ν p ∈ R2 is defined by

ν p = (ν px , ν py ) = (|u px |, |u py |) . ˆ ∈ R4×2d by For finding an upper bound for ν p , define a constant matrix G ˆ v = (Reu px , Imu px , Reu py , Imu py ) , G ˆ v = ν p . Hence, the maximum value of ν p is obtained as which yields G

ν ˆ v : v ∈ V . (23) rmax = max G v

Observe that the problem (23) has the same form as (10) which has been investigated ν in Sect. 3.1. Hence, an SDP problem for computing an upper bound of rmax can be ˆ obtained by replacing G in the problem (15) with G . Similarly, we obtain the SOCP problem for computing the minimum value of ˆ . Conservative bounds for ν py /ν px can be obtained in a ν p by replacing G with G manner similar to Sect. 3.3.

5 Numerical experiments Consider a plane truss illustrated in Fig. 1, where W = 50 cm and H = 100 cm. As the nominal load ˜f, we consider the external forces 12 kN and 8 kN applied at the nodes (a) and (b), respectively, in the negative direction of y-axis. The uncertainty model of f is defined as discussed in Example 1, where f¯0 = 1.0 kN and α = 1.0. We assume the complex damping, i.e. the damping matrix C is given as ω C = 2β K , where β = 0.02. The first and second undamped fundamental natural circular frequencies are ω10 = 1.248 rad/s and ω20 = 1.872 rad/s, respectively. We denote by (ux , uy ) ∈ C2 the displacement amplitude of the node (a). We compute the upper and lower bounds for each of |ux |, arg ux , |uy |, and arg uy by solving the optimization problems in Sect. 3 using SeDuMi Ver. 1.05 [21], which implements the primal-dual interior-point method. The obtained results are depicted in Fig. 4, which also shows a number of displacement amplitudes computed from randomly generated samples of f. We can see that all generated amplitudes are included in the bounds, which confirms that the obtained bounds correspond to confidential bounds. Fig. 5 shows the variations of |ux | and |uy | with respect to ω (ω10 ≤ ω ≤ ω20 ). For ux , Fig. 6 illustrates the variation of bounds for |ux | and arg ux . Note that the conservative bounds for uncertain ω correspond to a union of infinitely many sectoral

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Fig. 4: Bounds and randomly generated samples for the complex amplitudes of the displacements of the 29-bar truss with ω = ω10

Fig. 5: Variations of bounds for the complex amplitudes of the displacements with respect to the frequency ω of the driving force. ‘· · · ’: nominal value; ‘—’: lower bounds; ‘– –’: upper bounds

bounds, while it is observed in Fig. 6 that a rough tendency of the distribution of the response may be captured intuitively from finitely many bounds. We next find the bounds for the distribution of the vector (|ux |, |uy |) by using the method presented in Sect. 4. Figs. 7 (a)–(c) depict the obtained bounds, each of which is the intersection of the two sets shown with the solid and dashed lines. It is observed in Fig. 7 (a) that the driving force with ω = ω10 yields the resonance, and that the oscillation is in the direction of the first eigenmode. The variations of upper and lower bounds for |ux |2 + |uy |2 with respect to ω (ω10 ≤ ω ≤ ω20 ) are shown in Fig. 7 (d).

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(a) ω ∈ [(0.75)1/2 ω10 , (0.85)1/2 ω10 ]

(b) ω ∈ [(0.85)1/2 ω10 , (0.95)1/2 ω10 ]

(c) ω ∈ [(0.95)1/2 ω10 , (1.05)1/2 ω10 ]

(d) ω ∈ [(1.05)1/2 ω10 , (1.15)1/2 ω10 ]

Fig. 6: Variations of bounds for |ux | and arg ux with respect to the frequency ω of the driving force

6 Conclusions We have proposed tractable formulations based on the semidefinite program (SDP) in order to find confidential bounds for the dynamic steady-state behaviors of a structure subjected to uncertain driving loads, where the amplitude of the driving load is assumed to obey a non-probabilistic uncertainty model. In the numerical examples, it has been illustrated that the bounds provided by the proposed method are sufficiently tight even for a moderately large magnitude of uncertainty.

Dynamic Steady-State Analysis of Uncertain Structures via SDP

(a) ω = ω10

(c) ω = ω20

111

(b) ω = 1.2ω10

(d) variations of bounds |ux |2 + |uy |2 with respect to ω

for

Fig. 7: Bounds and randomly generated samples for the nodal oscillation

References 1. Alefeld, G., Mayer, G.: Interval analysis: theory and applications. J. Comput. Appl. Math., 121, 421–464 (2000). 2. Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program., B95, 3–51 (2003). 3. Ben-Haim, Y.: Information-gap Decision Theory: Decisions under Severe Uncertainty (2nd ed.). Academic Press, London (2006). 4. Ben-Haim, Y., Elishakoff, I.: Convex Models of Uncertainty in Applied Mechanics. Elsevier, New York (1990). 5. Ben-Tal, A., Nemirovski, A.: Robust optimization — methodology and applications. Math. Program., B92, 453–480 (2002). 6. Beyer, H.-G., Sendhoff, B.: Robust optimization — A comprehensive survey. Comput. Methods Appl. Mech. Eng., 196, 3190–3218 (2007). 7. Calafiore, G., El Ghaoui, L.: Ellipsoidal bounds for uncertain linear equations and dynamical systems. Automatica, 40, 773–787 (2004). 8. Chen, S., Lian, H., Yang, X.: Interval eigenvalue analysis for structures with interval parameters. Finite Elem. Anal. Des., 39, 419–431 (2003).

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9. Gao, W.: Interval natural frequency and mode shape analysis for truss structures with interval parameters. Finite Elem. Anal. Des., 42, 471–477 (2006). 10. El-Gebeily, M.A., Abu-Baker, Y., Elginde, M.B.: The generalized eigenvalue problem for tridiagonal symmetric interval matrices. Internat. J. Control, 72, 531–535 (1999). 11. Guo, X., Bai, W., Zhang, W.: Extreme structural response analysis of truss structures under material uncertainty via linear mixed 0-1 programming. Int. J. Numer. Methods Engrg., 76, 253–277 (2008). 12. Helmberg, C.: Semidefinite programming. Europ. J. Operational Research, 137, 461–482 (2002). 13. Kanno, Y., Takewaki, I.: Confidence ellipsoids for static response of trusses with load and structural uncertainties. Comput. Methods Appl. Mech. Eng., 196, 393–403 (2006). 14. Kanno, Y., Takewaki, I.: Worst-case plastic limit analysis of trusses under uncertain loads via mixed 0-1 programming. J. Mech. Materials Struct., 2, 245–273 (2007). 15. Kanno, Y., Takewaki, I.: Semidefinite programming for uncertain linear equations in static analysis of structures. Comput. Methods Appl. Mech. Eng., 198, 102–115 (2008). 16. Kanno, Y., Takewaki, I.: Semidefinite programming for dynamic steady-state analysis of structures under uncertain harmonic loads. Comput. Methods Appl. Mech. Eng., doi:10.1016/j.cma.2009.06.005. 17. Moens, D., Vandepitte, D.: A fuzzy finite element procedure for the calculation of uncertain frequency-response functions of damped structures: Part 1 — Procedure. J. Sound Vibrat., 288, 431–462 (2005). 18. Pantelides, C.P., Tzan, S.-R.: Convex model for seismic design of structures — I: Analysis. Earthquake Engrg. Struct. Dyn., 25, 927–944 (1996). 19. P´olik, I., Terlaky, T.: A survey of S -lemma. SIAM Review, 49, 371–418 (2007). 20. Qiu, Z., Chen, S.H., Elishakoff, I.: Natural frequencies of structures with uncertain but nonrandom parameters. J. Optim. Theory Appl., 86, 669–683 (1995). 21. Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimiz. Methods Software, 11/12, 625–653 (1999). 22. Tzan, S.-R., Pantelides, C.P.: Convex models for impulsive response of structures. J. Eng. Mech. (ASCE), 122, 521–529 (1996). 23. Zang, C., Friswell, M.I., Mottershead J.E.: A review of robust optimal design and its application in dynamics. Comput. Struct., 83, 315–326 (2005).

Part II

SEA related methods and wave propagation

Universal eigenvalue statistics and vibration response prediction R.S. Langley

Abstract It is well known that, under broad conditions, the local statistics of the eigenvalues of a random matrix tend towards a universal distribution if the matrix is sufficiently random, regardless of the statistics of the matrix entries. This distribution is the same as that associated with a class of random matrix known as the Gaussian Orthogonal Ensemble (GOE). The underlying reason for this behaviour is explored here, and it is concluded that the source of the universal statistics lies in the Vandermonde determinant, which appears in the Jacobian of the transformation between the entries of a matrix and its eigenvalues and eigenvectors. Attention is then turned to the application of this result to natural frequency statistics, and to the prediction of the response statistics of a random built-up system. Recent work is reviewed in which it is shown that, away from low frequencies, the mean and variance of the system response can be predicted in an efficient way without knowledge of the statistics of the underlying system uncertainties.

1 Introduction There is interest in predicting the statistical properties of the eigenvalues of a random matrix across a wide range of applications in science and engineering. At one extreme of scale the eigenvalues might represent the energy levels of a complex nucleus, while at another they might represent the natural frequencies of a large engineering structure with manufacturing uncertainties. There is little obvious physical similarity between these two examples, and very different eigenvalue statistics might therefore be expected. However, it has been found that many of the statistical results derived initially in nuclear physics appear to be more generally valid. For example, the Gaussian Orthogonal Ensemble (GOE) is a special type of ranR.S. Langley Cambridge University Engineering Department, Cambridge, CB2 1PZ, UK e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 9, © Springer Science+Business Media B.V. 2011

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dom matrix first introduced in nuclear physics to represent a random Hamiltonian (Wigner 1955, Dyson 1962). The matrix is symmetric, with zero mean, and the entries are uncorrelated Gaussian random variables, with the diagonals having twice the variance of the off-diagonals. This matrix bears little resemblance to those that arise in the mathematical description of an engineering structure, and yet key features of the eigenvalue statistics associated with the GOE have been found to apply to the natural frequencies of metal blocks (Weaver 1989), elastic plates (Bertelsen et al 2000, Langley and Brown 2004), and membranes (Bohigas 1984). This suggests some degree of “universality” in the GOE eigenvalue statistics. As stated by Forrester et al (2003): “It is hard not to be fascinated by random matrix theory. Everyone who works in this field has experienced the amazement of obtaining truly universal behaviour by diagonalising a large random matrix”. The underlying cause of this universal behaviour is the subject of the first part of the present work (Section 2). One obvious limitation is that universality can be claimed only in a local sense, i.e. among a group of neighbouring eigenvalues with nearly constant mean spacing, since, for example, long range variations in the mean spacing will clearly be system dependent, and differs for a plate, a metal block, and the GOE. The second part of the present work (Section 3) is focussed on employing universal eigenvalue statistics in the prediction of the response statistics of complex built-up systems. Initially the use of an extended version of statistical energy analysis (SEA) to predict the mean and variance of the system response is described (Lyon and DeJong, 1995, Langley and Cotoni, 2004). This approach requires that each component in the system has universal eigenvalue statistics, and this can be excessively restrictive at lower frequencies, where stiff beams and frames are relatively deterministic. A method which overcomes this restriction is then reviewed: the hybrid method, in which the finite element method is coupled to SEA (Shorter and Langley, 2005, Langley and Cotoni, 2007). It is shown that very efficient response predictions can be obtained using this approach. Possible extensions to the method are then discussed in the concluding remarks.

2 Eigenvalue statistics This section considers the statistics of the eigenvalues of a random matrix. A standard linear eigen-problem is considered initially, and the results obtained are then related to structural dynamics problems in which the eigenvalues are the squares of the system natural frequencies.

2.1 The joint probability density function of the eigenvalues Consider an M × M random symmetric matrix A having eigenvalues λ j and normalised eigenvectors u j ( j = 1, 2, . . . , M) so that

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117

Au j = λ j u j ,

(1)

A = UΛ U , UUT = I ,

(2) (3)

T

where Λ is a diagonal matrix containing the eigenvalues, and the j-th column of the matrix U consists of the j-th eigenvector. The fact that the matrix A is symmetric implies that it is fully defined by M(M +1)/2 entries, which can be taken to be those in the lower triangle, i.e. A jk with j ≥ k. Similarly, the matrices Λ and U are together fully defined by M(M + 1)/2 entries, and these can be taken to be (necessarily) the M eigenvalues and (non-uniquely) the lower triangle of U, excluding the diagonal terms, i.e. U jk with j > k. It is convenient to express the M(M − 1)/2 selected values of U as the entries of a vector u, so that the complete set of M(M + 1)/2 variables describing the eigenvalues and eigenvectors can be written as (λ , u), where λ is a vector containing the eigenvalues. The concern here is to derive an expression for the joint probability density function (jpdf) of the eigenvalues given the jpdf of the matrix entries, pA (A) say. The first step towards this is to write the jpdf of the variables (λ , u) in the form pλ ,u (λ , u) = |J|pA (A),

(4)

where J is the Jacobian of the transform between the M(M + 1)/2 variables (λ , u) and the M(M + 1)/2 variables in the lower triangle of A. It is well known (see for example Mehta 1991, and Muirhead 1982) that the Jacobian has the form |J| = V (λ ) f (u) , V (λ ) = ∏ λi − λ j ,

(5) (6)

i> j

where V (λ ) is known as the Vandermonde determinant, and f (u) is an algebraic function of the entries of u. It follows from equations (4) to (6) that the jpdf of the eigenvalues, p(λ ) say, is given by T p(λ ) = V (λ ) . . . f (u)pA (UΛ U )du . (7) s

It is convenient to express this result in the form p(λ ) = V (λ ) exp[−g(λ )] , . . . f (u)pA (UΛ UT )du , g(λ ) = − ln

(8) (9)

s

where the detailed form of the function g(λ ) will clearly depend upon the jpdf of the matrix entries, pA (A). The subscript s appearing in equations (7) and (9) is to indicate that the result is made symmetric if necessary, by averaging over permutations of λ . This is because the eigenvalues are considered to be “unordered” with no special significance to labelling.

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Equation (8) has been derived here for the special case of a symmetric matrix, but the result can readily be generalised to include Hermitian matrices to yield p(λ ) = [V (λ )]β exp[−g(λ )],

(10)

where β = 1 for the symmetric case and β = 2 for the Hermitian case. Furthermore, the case β = 4 corresponds to a quaternion-real, self-dual matrix A (Mehta, 1991). For β = 2 and β = 4 the definition of g(λ ) is modified slightly by a change in the properties of the eigenvector matrix (U is respectively unitary and symplectic for β = 2 and β = 4, as opposed to orthogonal for β = 1) but in each case the resulting expression is very similar in form to equation (9). It was noted above that the eigenvalues are considered to be “unordered”. The jpdf of the eigenvalues when ordered by magnitude, with λ j+1 ≥ λ j , is also of interest. If pord (λ ) represents the jpdf of the ordered eigenvalues, then pord (λ ) = M!p(λ ), λ j+1 ≥ λ j , = 0, otherwise, p(λ ) =

∑

pord (λ1 , λ2 , . . . , λM ),

(11) (12)

perm

where the summation in equation (12) is over all possible permutations in the listed sequence of the eigenvalues. In effect pord (λ ) is non-zero only over a fraction 1/M! of the M-dimensional domain. Equation (12) is equivalent to mapping this fraction over the whole domain by a series of reflections.

2.2 The modal density An insight into the physical consequences of the jpdf given by equation (10) can be obtained by considering the most probable values of the eigenvalues, i.e. those values for which the jpdf is a maximum. The jpdf can be rewritten in the form p(λ ) = exp ∑ β ln λi − λ j − g(λ ) , (13) i> j

from which it follows that for each λr

∂ ln p = ∂ λr

β

∂g

∑ λr − λ j − ∂ λr = 0

at λ = λ sp .

(14)

j=r

Here the superscript “sp” is used to represent the most probable values, which are associated with the stationary point of the jpdf. Clearly the function g(λ ) plays a central role in determining the values λ sp that satisfy equation (14). For example, in the special case of a random matrix having statistical properties defined by the

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Gaussian Unitary Ensemble (GUE) then β = 2 and ∂ g/∂ λr = 2λr , and it is found that the solutions to equation (14) obey a “semi-circle” law (Mehta, 1991). This means that the spacing between two consecutive (ordered) eigenvalues is given approximately by √ √ sp sp sp sp λ j+1 − λ j = π [2M − (λ j )2 ]−1/2 , − 2M ≤ λ j ≤ 2M. (15) In contrast, Leff (1964) has shown that for a system with β = 2 and g = 0, the corresponding result (when the eigenvalues are scaled to the same range as the GUE) is √ √ sp sp sp sp λ j+1 − λ j = (π /M)[2M − (λ j )2 ]1/2 , − 2M ≤ λ j ≤ 2M. (16) Clearly these results are very different, demonstrating (as is well known) that the most probable values of the eigenvalues are strongly influenced by the function g(λ ), and hence by the detailed statistics of the matrix A, via equation (9). If the sp sp size of the system M is large, then it is reasonable to assume that λ j+1 − λ j will be approximately constant across a set of N M neighbouring eigenvalues. In this case a key question is whether the statistics of these N eigenvalues might be independent of the statistics of A, other than through the specification of the mean spacing μ = sp sp λ j+1 − λ j . The inverse of the mean spacing is referred to as the modal density ν , and it is well known in structural dynamics and acoustics that the form of the modal density varies significantly between different types of system. Rather than conform to a circle law such as equation (15), the modal density of a beam, plate, membrane, and acoustic volume are respectively proportional to frequency raised to the power of −1/2, 0, 1 and 2 (Cremer and Heckl, 1988). Given these different behaviours, the question raised above is of considerable practical importance: is it possible that the statistics of a set of N neighbouring natural frequencies are independent of both the type of system under consideration and the details of the way in which the system is randomised? This question is addressed in the following section.

2.3 Universality of the “local” eigenvalue statistics Following the above discussion, consider a set of N M neighbouring eigenvalues sp − λ jsp is approximately constant. The eigenvalfor which the mean spacing μ = λ j+1 ues that fall within this set will be labelled λn j ( j = 1, 2, . . . , N) while the remaining eigenvalues will be labelled λ f j ( j = 1, 2, . . . , M − N). From equations (13) and (11), the probability density function of the complete set of ordered eigenvalues can be written in the form pord (λn , λ f ) = exp{ ∑ β ln λni − λn j + ∑ β ln λni − λ f j − g(λn , λ f ) + r(λ f )}, i> j

i, j

(17) where the function r(λ f ) is defined accordingly. Equation (17) can also be written in the form

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pord (λn , λ f ) = exp{ ∑ β ln λni − λn j } exp{s(λn , λ f )} = exp{h(λn , λ f )},

(18)

i> j

where the function s(λn , λ f ) contains the final three terms in the exponent on the right hand side of equation (17), and the function h(λn , λ f ) is defined accordingly. It is now of interest to explore the functional dependency of pord (λn , λ f ) on the eigenvalues λn j , and one way of doing this is to consider a Taylor series expansion of the function h(λn , λ f ) in the vicinity of the modal values of the eigenvalues. The first derivative which appears in this expansion is zero, by virtue of equation (14). The m-th derivative has the form ∂ m h (−1)m−1 (m − 1)!β (−1)m−1 (m − 1)!β ∂ m g = +∑ − . ∂ λnim λ =λ sp i∑ ∂ λnim λ =λ sp (λnisp − λnspj )m (λnisp − λ fspj )m j =j (19) Now if N is large and λni lies towards the centre of λn , then λni will be distant from each λ f j and the second term of the right hand side of equation (19) will be negligible relative to the first term. Furthermore, the first term can be expected to dominate the equation if for m > 1 m μm ∂ g 1, (20) (m − 1)!β ∂ λnim λ =λ sp where μ is the spacing between the modal values of the eigenvalues. Under this condition, the function s(λn , λ f ) that appears in equation (18) is relatively insensitive to changes in the values of λn ; the functional dependency of the probability density function pord (λn , λ f ) on λn is then dominated by the first term on the right hand side of equation (18). In summary, the function g(λn , λ f ) depends on the probability density function of the random matrix A via equation (9). If the matrix statistics are such that the function g(λn , λ f ) satisfies equation (20), then the functional dependency of pord (λn , λ f ) on λn is approximately independent of g(λn , λ f ), and we can write β pord (λn ) = C ∑ λni − λn j ,

(21)

i> j

where C is a normalisation constant. In this case the probability density function of a “local” group of eigenvalues is independent of the statistics of the matrix A and a “universal” statistical distribution is obtained. The resulting distribution is just the Vandermonde determinant; this expression arises universally in the Jacobian of the transformation between the entries of a matrix and its eigenvalues and eigenvectors, as described by equations (4–6). To take the Gaussian ensembles as an example, the matrix statistics are such that (Mehta 1991) g(λ ) = B + (β /2) ∑ λ j2 , j

(22)

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where B is a normalisation constant. The modal √ spacing of the eigenvalues towards the centre of the distribution is of the order 1/ M (see equation (15) for example), and thus equation (20) is clearly satisfied for large M. The “universality” of eigenvalue statistics is often interpreted as meaning that many systems of sufficient complexity tend to yield eigenvalue statistics that conform to the Gaussian ensembles. This then poses a mystery as to why a matrix A, which possibly has very different statistics to the Gaussian ensembles, should yield such eigenvalue statistics. The conjecture here is that the Gaussian ensembles are not the key underlying cause or explanation of universality; rather the Vandermonde determinant is the common factor to all problems in eigenvalue statistics, and if equation (20) is met then this determinant completely dominates the local statistics of the eigenvalues, regardless of the detailed statistics of the matrix A. The Gaussian ensembles meet the condition expressed by equation (20), and are therefore one special example of this effect. In this regard Leff (1964) has shown directly that equation (21), the Vandermonde determinant, yields the normal universal statistical properties.

2.4 Application to natural frequency statistics The previous sections concerned the statistics of the eigenvalues of a random matrix. In vibration analysis however, the eigenvalues represent the square of the natural frequencies, and the concern is with the statistics of the natural frequencies ω j . Writing λ j = ω 2j and applying the Jacobian of the transformation to equation (10), the probability density function of the natural frequencies can be written as p(ω ) = [V (ω )]β exp{−g[λ (ω )]} ∏(ωi + ω j )β ∏(2ωk ). i> j

(23)

k

If g(λ ) satisfies the condition in equation (20), then the equivalent function g(ω ), which contains additional terms arising from the two product terms in the above equation, will also satisfy this condition providing ω j μ . Hence away from the origin the natural frequencies can be expected to display universal statistics, providing this is true of the eigenvalues. This result agrees with numerical simulations (Langley and Brown, 2004) and experimental measurements (Weaver 1989, Bertletsen et al 2000).

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3 Application to response statistics 3.1 Fundamental concepts The statistics of the natural frequencies of a random system can be described directly by the joint probability density function p(ω1 , ω2 , . . . , ωn ) or alternatively by a series of functions introduced independently in random matrix theory (for example Mehta, 1991) and in random point process theory (Stratonovich, 1967). The n-point correlation function Rn of random matrix theory is defined as Rn (ω1 , ω2 , . . . , ωn ) =

N! (N − n)!

. . . p(ω1 , ω2 , . . . , ωn )dωn+1 dωn+2 . . . dωn .

(24) This function is referred to in random point process theory as the n-th distribution function fn (ω1 , ω2 , . . . , ωn ), and physically the quantity dP = fn (ω1 , ω2 , . . . , ωn )dω1 dω2 . . . dωn ,

(25)

is the probability that at least one eigenvalue occurs in each of the small nonoverlapping regions ωr ≤ ω ≤ ωr + dωr , (r = 1, 2, . . . , n), without reference to the occurrence of eigenvalues outside these regions. The first distribution function is just the modal density of the system, f1 = 1/μ = ν . Now the vibration response of a random system at a prescribed frequency ω can usually be written as a modal summation in the form (26) T = ∑ cn H(ωn ), n

where the random coefficients cn depend on the system mode shapes, and the function H is related to the modal frequency response function. For example, the vibrational energy of a system driven by a unit point load at location x0 is given by cn , 2 2 2 n (ωn − ω ) + (ηωn ω )

T =∑

(27)

where η is the loss factor, and cn = φn2 (x0 ) where φn is the n-th mode shape. The first two statistical moments of the response can be expressed in terms of the distribution functions f n (ω1 , ω2 , . . . , ωn ) as follows (Stratonovich, 1967) ∞

E[T ] = E[cn ]

Var[T ] =

E[c2n ]

∞

E[cn ]2

−∞

H(ωn ) f1 (ωn )dωn ,

(28)

H 2 (ωn ) f1 (ωn )dωn +

−∞ ∞

∞

−∞ −∞

H(ωn )H(ωm )g2 (ωn , ωm )dωn dωm ,

(29)

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g2 (ωn , ωm ) = f2 (ωn , ωm ) − f1 (ωn ) f1 (ωm ).

(30)

The function g2 is known in random point process theory as the second cumulant function, and in random matrix theory as the 2-level cluster function. It can be seen from equations (28)-(30) that the mean and variance of the system response depend only on the system modal density and on the function g2 ; moreover, if universal eigenvalue statistics are taken to apply, then the function g2 is independent of the detailed statistics of the system randomness and depends only on the modal density (Mehta, 1991, Langley and Brown, 2004). As an example, Langley and Brown (2004) have shown that equations (27) to (30) lead to the results E[cn ]πν , 2ηω 3

(31)

Relvar[T ] = r2 (α , m) ,

(32)

E[T ] =

α=

E[c2n ] E[cn ]2

(33)

,

m = ωην .

(34)

Here “Relvar” represents the relative variance, and the parameter m is termed the modal overlap factor. The function r is given by

1 1

1 α −1+ r2 = 1 − e−2π m + E1 (π m) cosh (π m) − sinh (π m) , πm 2π m πm (35) where E1 is the exponential integral (Abromowitz and Stegun, 1964).

3.2 Built-up systems: SEA The theory presented in the previous section is applicable to single structural components, such as beams and plates, in which the damping is uniformly distributed and the mode shapes are not localized. Langley and Cotoni (2004) have extended the theory to built-up systems to yield an augmented version of statistical energy analysis (SEA). With this approach the system is divided into a number of “subsystems” and the aim is to derive the mean and variance of the vibrational energy of each of the subsystems. The resulting equations are

ωη j E j + ∑ ωη jk n j (E j /n j − Ek /nk ) = Pin, j

or

ˆ = Pin , CE

(36)

k

2 −1 −1 ˆ 2 Var(E j ) = ∑(C−1 jk ) Var(Pran,k ) + ∑ ∑ [(C jk −C js )Es ] Var(Cran,ks ), k

k s=k

(37)

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R.S. Langley 2 2 Var(Pran,k ) = Pin, k r ( αk , m k ) ,

Var(Cran,ks ) =

2 2 Cks r (αks , mk ),

mk = ωηk nk , −1 ηk = 1/(ω nkCkk ).

(38) (39) (40) (41)

Equation (36) represents the standard set of SEA equations, in which E j is the vibrational energy of subsystem j, n j is the modal density of the subsystem, η j is the loss factor, and Pin, j is the power input from external sources. The coefficients η jk are known as the coupling loss factors, and they are normally calculated by considering wave transmission between adjoining subsystems (see, for example, Lyon and DeJong, 1995). The energy E j strictly represents the ensemble average of the subsystem energy; although not explicitly stated, SEA invokes the universality hypothesis by default, since the equations contain no information regarding the detailed nature of the system uncertainties. In the variance equations, equations (39)–(41): Eˆ s = Es /ns , C−1 jk is the jk-th entry of the inverse of the matrix C that appears in equation (36), and the function r 2 is given by equation (35). The only “non-SEA” parameters that appear in equations (39) to (41) are the terms αk and αks ; αk has been defined in section 3.1, while αks depends on the nature of the coupling between subsystems k and s; in the majority of cases this has the value 2. An example of the application of equations (36)–(41) is shown in Figure 1 (Langley and Cotoni, 2004). The results concern two plates that are coupled along a common edge. Plate 1 is subject to a point load, and Monte Carlo simulations have been performed on an ensemble of systems generated by adding small masses in random locations. Results are shown for the mean and relative variance of the bending energy in each plate, and good agreement with theory is demonstrated. It can be noted that while the driven plate has the largest mean energy, the relative variance is greater for the second plate. This is because the response of the second plate is affected by the randomness in both plates, while the first plate response is little affected by the second plate. This type of behaviour is further illustrated in Figure 2 for a three plate system (Langley and Cotoni, 2004): the relative variance increases with increasing distance from the excited plate. Further examples of the application of universal statistics to response prediction have been given by Cotini et al (2005) and Langley and Cotoni (2005). In addition Weaver and his co-workers have presented fundamental studies of the applicability of GOE statistics to dynamic systems, for example: Weaver (1989), Lobkis et al (2000).

3.3 Built-up systems: the Hybrid method The method described above has been extended to systems which are modeled by using a combination of the finite element (FE) method and SEA. This approach

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Fig. 1: Mean and relative variance for the bending energies in a two plate system

Fig. 2: Mean and relative variance for the bending energies in a three plate system

is described by Shorter and Langley (2005) and Langley and Cotoni (2007): the system is represented as an assembly of statistical subsystems and deterministic FE components. An example of the type of results yielded by the method is shown in Figure 3 for the case of a beam/panel system (Langley and Cotoni, 2007). In this case the beam response and the in-plane motion of the panels were represented using FE, while the bending motion of the panels was modeled using SEA.

4 Conclusions It has been conjectured here that the occurrence of universal eigenvalue statistics is driven by the appearance of the Vandermonde determinant in the Jacobian of the transformation between the entries of a matrix and the associated eigenvalues and eigenvectors. The main point is that the jpdf of the eigenvalues can be written in the form of equation (10), in which the function g(λ ) depends on the statistics of the matrix entries: although this function affects the long range modal density via equation (14), the function does not affect the local eigenvalue statistics if equation (20)

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Fig. 3: The response of a beam framework and plate assembly to a force applied to the framework, as calculated by Monte Carlo simulations (irregular curves) and the hybrid method (smooth curves). Upper figures: the response of a panel. Lower figures: the response at a point on the framework. Left figures: the mean response, together with the ensemble of Monte Carlo simulations. Right figures: the relative variance

is satisfied. It is argued that systems which display universal statistics satisfy equation (20). A more rigorous and thorough version of the current analysis could be used to explore in more detail the relation between equation (20) and conditions that apply directly to the matrix probability density function. The second part of this work has reviewed a number of recent developments in predicting the response of random built-up systems. At low frequencies it is known that eigenvalue statistics are not universal (see for example Soize, 2005), and the methods discussed could potentially be combined with parametric methods to extend their range of applicability. Acknowledgements The work reported in the second part of this paper is the result of long term collaboration with Drs Phil Shorter and Vincent Cotoni of ESI US R&D Inc., San Diego, USA.

References 1. Ambromowitz, M. & Stegun, I.A. (1964). Handbook of Mathematical Functions. Dover, New York. 2. Bertelsen, P., Ellegaard, C. & Hugues, E. (2001). Acoustic chaos. Physica Scripta 90, 223– 230.

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3. Bohigas, O., Giannoni, M.J. & Schmit, C. (1984). Spectral properties of the Laplacian and random matrix theory. J. Physique Lett. 45, L1015–L1022. 4. Cotoni, V., Langley, R.S. & Kidner, M.R.F. (2005). Numerical and experimental validation of variance prediction in the statistical energy analysis of built-up systems. Journal of Sound and Vibration 288, 701–728. 5. Dyson, F.J. (1962). Statistical theory of energy levels in complex systems, Parts I, II, and III. J. Maths. Phys. 3, 140–156, 157–165, 166–175. 6. Forrester, P.J., Snaith, N.C. & Verbaarschot, J.J.M. (2003). Developments in random matrix theory. Journal of Physics A: Mathematical and General 36, R1–R10. 7. Langley, R.S. & Brown, A.W.M. (2004). The ensemble statistics of the energy of a random system subjected to harmonic excitation. Journal of Sound and Vibration 275, 823–846. 8. Langley, R.S. & Cotoni, V. (2004). Response variance prediction in the statistical energy analysis of built-up systems. Journal of the Acoustical Society of America 115, 706–718. 9. Langley, R.S. & Cotoni, V. (2005). The ensemble statistics of the vibrational energy density of a random system subjected to single point harmonic excitation. Journal of the Acoustical Society of America 118, 3064–3076. 10. Langley, R.S. & Cotoni, V. (2007). Response variance prediction for uncertain vibro-acoustic systems using a hybrid deterministic-statistical method. Journal of the Acoustical Society of America 122, 3445–3463. 11. Leff, H.S. (1964). Class of ensembles in the statistical theory of energy-level spectra. Journal of Mathematical Physics 5, 763–768. 12. Lobkis, O.I., Weaver, R.L. & Rozhkov, I. (2000). Power variances and decay curvature in a reverberant system. Journal of Sound and Vibration 237, 281–302. 13. Lyon, R.H. & DeJong, R.G. (1995). Theory and Application of Statistical Energy Analysis, Second Edition. Butterworth-Heinemann, Boston. 14. Mehta, M.L. (1991). Random Matrices, Second Edition. Academic Press, San Diego. 15. Muirhead, R.J. (1982). Aspects of multivariate statistical theory. John Wiley & Sons, Inc. New York. 16. Shorter, P.J. & Langley, R.S. (2005). Vibro-acoustic analysis of complex systems. Journal of Sound and Vibration 288, 669–700. 17. Soize, C. (2005). A comprehensive overview of a non-parametric probabilistic approach of model uncertainties for predictive models in structural dynamics. Journal of Sound and Vibration 288, 623–652. 18. Stratonovich, R.L. (1967). Topics in the Theory of Random Noise, Vol. 2. McGraw-Hill, London. 19. Weaver, R.L. (1989). On the ensemble variance of reverberation room transmission functions, the efect of spectral rigidity. Journal of the Acoustical Society of America 130, 487–491. 20. Wigner, E.P. (1955). Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. 62, 548.

Statistical Energy Analysis and the second principle of thermodynamics Alain Le Bot

Abstract Statistical Energy Analysis is a statistical method in vibroacoustics entirely based on the application of energy balance that is the first principle of thermodynamics. In this study, the definition of vibrational entropy is introduced for sub-systems containing energy and modes. The rate of entropy production at interfaces between sub-systems is also derived. Finally, in steady-state condition, an entropy equilibrium is reached. The meaning of entropy and some implications of this entropy balance are also discussed.

1 Introduction Statistical Energy Analysis [1, 2] is born from the application of statistical physics concepts to vibroacoustical systems. The idea is quite simple. When the number of modes of vibroacoustical systems is so large that the solving of governing equations becomes unpracticable, it is preferable to give up a deterministic description of the system and to adopt statistical methods. This standpoint is reasonable. It has been followed in several other fields in physics, kinetic theory of gases instead of point mechanics, statistical theory of turbulence instead of Navier-Stoke’s equation, statistical behaviour of granular material and some others. In all these cases, the number of entities is very large so that a complete description of all of them would lead to a set of equations untractable in practice and whose solution would be very sensitive to errors of modelling. In addition, the amount of produced information would be huge, the major part being useless. The application of concepts and methods of statistical physics to the audio frequency range instead of thermal vibration raises some difficulties. First of all, the number of entities in vibroacoustics (the modes) is not so large than in thermics Alain Le Bot Laboratoire de tribologie et dynamique des syst`emes CNRS, Ecole centrale de Lyon, 36, av. Guy de Collongue 69134 Ecully, France, e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 10, © Springer Science+Business Media B.V. 2011

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(atoms or molecules). Statistical methods applied to poor populations are fragile. This explains why results of SEA are sometimes disappointing. SEA is too oftently used outside its validity domain. Secondly, small populations lead to large fluctuations. While it is almost impossible (or at least very difficult) to observe fluctuations in thermodynamical systems, large fluctuations are common in SEA and it is sometimes necessary to deal with large population of similar systems, the so-called Gibb’s ensemble, to increase the quality of SEA results. Finally, the equipartition of energy among modes is sometimes violated. Poor population and low modal overlap are conditions favourable to a deviation from the equilibrium state. This is why it is so urgent to extend SE to non-equilibrium states [3, 4, 5] by relaxing the diffuse field assumption following the example of non-equilibrium thermodynamics. Up to now, SEA is entirely based on the application of energy balance that is the first principle of thermodynamics. But nothing is told on entropy in SEA [6, 7]. Conceptually, both quantities energy and entropy are necessary for a complete derivation of statistical physics and therefore SEA. Energy is indeed necessary to describe the state of the system. Entropy is necessary to measure the loss of information induced by the renouncement to a full description of the system. In this regard, different levels of approximations would lead to different values of entropy. In particular, starting from a complete theory (null entropy), a sequence of theories each of them being an approximation of the preceeding one, leads to an increasing sequence of entropy values. In SEA, the level of approximation has been stated, but the entropy has not yet been defined. This is an important shortcoming. Entropy is also necessary to define the temperature. The classical definition of temperature T with [8], ∂S 1 = (1) T ∂E where S is the entropy and E the internal energy, shows that the temperature measures the rate of increasing of entropy (and therefore the loss of information on the system) with energy. A positive temperature means that entropy is a non-decreasing function of internal energy. The loss of information during the approximation process is therefore more important for energetic systems. It is rather common in SEA literature to define the vibrational temperature as being the modal energy. This result is true but a formal proof requires to introduce the entropy [9]. This is the purpose of this paper.

2 First principle of thermodynamics in SEA SEA is a simple method to assess the vibrational energy of complex systems. SEA is entirely based on some statistical considerations and the application of energy balance (first principle) and, as we shall see, the application of entropy balance (second principle).

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In SEA, a complex system is sub-divided into n components called sub-systems. Sub-systems are considered as tanks containing vibroacoustical energy Ei with i = 1, . . . n whatever their nature, structural component or acoustical cavity (Fig. 1). Within sub-system i, the vibrational energy Ei is repartited among Ni modes. Modes are therefore the sites which carry vibrational energy, exactly in the same way that molecules carry kinetic energy in kinetic theory of gases and that atoms carry vibrational energy in solids. In the two latter cases, this energy (kinetic or vibrational) is called heat. In SEA, the equivalent of heat is therefore the vibrational energy itself. The only difference between a true heat and the vibrational heat introduced in SEA is the frequency of the underlying vibration, thermal range for the former and audio frequency range for the latter. The heat in SEA is thus defined as the vibrational energy Ei in broadband. The analysis is confined to a frequency band Δ ω about the central frequency ω (rad/s). No strict definition is given for the width of the frequency band, but it is commonly admitted that octave bands are well suited. The first principle of thermodynamics can now be introduced. All sub-systems can receive energy from sources, driving forces in the structural case or noise inj sources in acoustics. The power being injected into sub-system i is noted Pi . But they also dissipate vibrational energy by natural mechanisms such as damping of vibration, absorption of sound by walls, attenuation of sound... The power being dissipated is noted Pidiss . Finally, the vibrational energy can be exchanged with adjacent sub-systems. The net exchanged power between sub-sytem i and j is noted Pi j . In steady-state condition, the energy balance for sub-system i reads, Pidiss + ∑ Pi j = Pi . inj

(2)

j=i

inj

The power being injected Pi is assumed to be known, but the powers being dissipated Pidiss and being exchanged Pi j must be expressed in terms of vibrational energies (Fig. 2). The power being dissipated by internal losses is, Pidiss = ωηi Ei , Fig. 1: Decomposition of complex structure in SEA. The vibroacoustical problem is solved by splitting the complex structure into n elements. In all these sub-systems, the vibration field is diffuse i.e. homogeneous and isotropic. The SEA approach consists in writing the exchange of energy between these subsystems

(3)

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Fig. 2: Energy balance in SEA. Sources supply energy to sub-systems. Part of this energy is dissipated by natural processes. Energy is finally exchanged between adjacent sub-systems. Vibrational energy Ei of sub-systems results from the balance of sources, dissipation and exchanges

where ηi is the damping loss factor usually determined by a direct measurement. The power supplied by the sub-system i to the sub-system j is, Pi→ j = ωηi j Ei ,

(4)

where ηi j is the coupling loss factor. The net exchanged power between sub-systems i and j is Pi j = Pi→ j − Pj→i and therefore [10], Pi j = ω (ηi j Ei − η ji E j ) .

(5)

Coupling loss factors are phenomenological constants attached to the junction between two sub-systems. Their values must be determined in each case either by a direct measurement or by using some predictive relationships. These theoretical relationships have been derived for a large number of cases, coupling between adjacent plates, acoustical cavities, connection between plate and beam, sound radiation, vibrational response and so on. We do not enter into the discussion of the validity of a particular method to derive such relationships. A large literature is devoted to this problem and many discussions on the efficiency of these relationships can be found. For the purpose of the present discussion, it is enough to admit that for every junctions, two coupling loss factors ηi j and η ji exist. Coupling loss factors verify the reciprocity relationship, Ni ηi j = N j η ji .

(6)

The reciprocity relationship highlights the importance of the modal energies Ei /Ni . Substituting Eq. (6) into Eq. (5) leads to, Ei Ei − Pi j = ωηi j Ni , (7) Ni Ni showing that the net exchanged power is proportional to the difference of modal energies. The SEA equation is simply obtained by substituting Eqs. (3,5) in Eq. (2),

Statistical Energy Analysis and the second principle of thermodynamics

⎛ ⎜ ω⎝

N1 ∑ j η1 j

−Nl ηlk ..

−Nk ηkl

. Nn ∑ j ηn j

133

⎞⎛

⎞ ⎛ inj ⎞ P1 E1 /N1 ⎟ ⎜ .. ⎟ ⎜ .. ⎟ ⎠⎝ . ⎠ = ⎝ . ⎠ En /Nn

(8)

inj Pn

This is a linear system on vibrational temperatures whose matrix is symmetric.

3 Vibrational entropy, vibrational temperature In order to rigorously define the entropy of vibrational systems within the context of SEA, it is important to well understand the approximation process applied to equations of motion to derive SEA equations. The equations of motion are the reference theory which gives an access to every physical quantities. There is no loss of information, the entropy is null. SEA is a simplified theory which does not give access to the detail of all physical quantities. There is a loss of information. In SEA, the vibrational energy is repartited among modes and the exact repartition of energy is not known. This exact repartition is refered as microstate of the system. A microstate of the system is a list of modes with a specification of their energy level such as: Mode 1 Mode 2 ... Mode N

Energy e1 Energy e2 Energy eN

The sum of all modal energies is the energy of the overall system E = ∑Ni=1 ei . Indeed, the solving of the equations of motion provides this list and therefore the knowledge of the microstate. Conversely, one can admit that the knowledge of this list gives of the total information on the system. Actually, a complete description of the state of system in steady condition is given by a list of modal amplitudes (complex-valued numbers) rather than modal energies (real-valued numbers). But in this text, we will not discuss the question of entropy induced by the loss of information when one neglects phase shifts between modal amplitudes. The problem is therefore to quantify the loss of information between the knowledge of a microstate, that is the repartition ei with i = 1 . . . N and the knowledge of a macrostate, that is the total energy E and the number N of modes. The question is to count the number of microstates that correspond to a given macrostate. With this idea in mind, a SEA system can be viewed as a set of N linear oscilators whose natural frequencies is an increasing sequence ω1 , ω2 . . . ωN (Fig. 3). And the number W of microstates attached to a macrostate E, N is given by the structure function [9], N N−1 2π E Ω (E) = (9) ω N − 1! and, W=

Ω (E) δE hN

(10)

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Fig. 3: Principle of a SEA system. The vibrational energy E is repartited among N linear oscillators whose natural frequencies is an increasing sequence ω1 , ω2 . . . ωN . The detailed repartition of the energy e1 , e2 . . . eN is a microstate whereas the knowledge of the total energy E and the number N of oscillators is a macrostate

where δ E is the uncertainty on E and h is a constant introduced during the discretization of phase space into equiprobable cells. The final step is to apply Boltzmann’s definition of entropy, S = k logW (11) with the result,

2π E S(E, N) = kN 1 + log hω N

(12)

This is the microcanonical entropy of a SEA sub-system. Eq. (12) gives the complete expression of the “vibrational entropy” of any SEA sub-system. The temperature is obtained as for any thermodynamic system with, 1 ∂S = (13) T ∂E N with the result,

E (14) kN As it was expected from Eq. (7) giving the power flow between two SEA subsystems, the “vibrational temperature” is well defined by the modal energy. In the literature, this result is generally obtained from an analogy between SEA and thermodynamics. This is now a logical consequence of the expression of the entropy obtained from Boltzmann’s definition. T=

4 Second principle of thermodynamics in SEA The second principle of thermodynamics states that entropy of isolated system cannot decrease. This is also true within the context of SEA. The problem that must be considered is the mixing of vibrational energy of two adjacent sub-systems. Let us consider two sub-systems with energies E1 , E2 and mode counts N1 and N2 . If these sub-systems are isolated, their entropies are respectively S(E1 , N1 )

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and S(E2 , N2 ) given by Eq. (12). Now, if these two sub-systems are connected together, the final state has energy E1 + E1 and the total number of modes is N1 + N2 . When equilibrium is reached, the final entropy of the overall system is S(E1 + E2 , N1 + N2 ), once again given by Eq. (12). The difference between the final entropy and the sum of initial entropy is therefore the entropy created during the mixing process.

Δ S = S(E, N) − [S(E1 , N1 ) + S(E2 , N2 )]

(15)

By substituting Eq. (12), it yields,

Δ S = k(N1 + N2 ) log

E1 + E2 E1 E2 − kN1 log − kN2 log N1 + N2 N2 N2

(16)

This is the entropy created by mixing the energy of the two sub-systems. This entropy production is non-negative Δ S > 0. This result stems from the convexity of the function f (x, y) = −y log(x/y). The fact that Δ S is non-negative can be interpreted in terms of loss of information. The loss of information for sub-system i is the log of the number of microstates corresponding to Ei , Ni . And the loss of information for the overall system is the log of the number of microstates corresponding to E = E1 + E1 , N = N1 + N1 . But there is many more possibilities to share the energy E = E1 + E1 over N = N1 + N1 modes than the sum of possibilities to share on the one hand, E1 over N1 modes and, on the second hand, E2 over N2 modes. The difference between the two is exactly the loss of information during the mixing process.

5 Entropy balance in SEA Coming back to the initial situation of n sub-systems in interaction, we are now in position to state an entropy balance of the overall system in SEA. Three questions must be examined: sources, dissipation and mixing. The variation of entropy is driven by the variation of vibrational energy around a vibrational temperature. Since the number of modes N is always fixed, Eq. (13) gives the infinitesimal variation of entropy dS for an infinitesimal variation δ E, dS =

δE T

(17)

For sources, the injected power is Pi . The vibrational energy δ E supplied to the sub-system during time dt is δ E = Piinj dt. It follows that the increase rate of entropy by sources is, inj inj inj P P Ni dSi = i =k i (18) dt Ti Ei inj

The last equality stems from Eq. (14).

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Similarly, dissipation processes of vibration within sub-systems leads to a modification of vibrational level. This energy is dissipated that is transformed into heat (in the sense of thermics). In classical thermodynamics, dissipation processes induce an increase of entropy by creating heat inside the system. But, in SEA, that we have called vibrational heat is the vibrational energy itself. And dissipation leads to a decreasing of vibrational energy. In other words, dissipation tends to cool the system and therefore induces a decreasing of vibrational entropy. dSidiss Pdiss = − i = −kωηi Ni , dt Ti

(19)

where the last equality is deduced from Eq. (3). The mixing of energy at interfaces between sub-systems also induces a variation of entropy. As we have seen in previous section, this variation is always non-negative. Let us develop this point. The power being exchanged between subsystems i and j is Pi j . Therefore, the vibrational entropy introduced in or extracted from sub-system i is −Pi j /Ti . Following the same reasoning, the vibrational entropy introduced in sub-system j is −Pji /T j . Since Pi j = −Pji , the net vibrational entropy introduced in the entire system by the mixing process is, N j Ni dSi j 1 1 = Pi j − − = kω (ηi j Ei − η ji E j ) (20) dt T j Ti E j Ei The last equality stems from Eq. (5). For the entire system, the vibrational entropy introduced in the system is, inj

n dS dSi j dSdiss dS = ∑ i + i +∑ dt dt dt i> j dt i=1

(21)

It is easy to deduce from Eq. (2) that this entropy is null, dS =0 dt

(22)

It means that there is no production of entropy for the entire SEA system. But there is an exchange of entropy with exterior which exactly balances the production of entropy by mixing processes (Fig. 4).

6 Conclusion In this paper, it has been shown that the second principle of thermodynamics can be stated in the context of SEA. Explicit relationships have been derived for the vibrational entropy of sub-systems and the production of entropy at interfaces of

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Fig. 4: Entropy balance in SEA. Sources supply entropy to sub-systems. Dissipation processes extract entropy since they are responsible of a cooling of the sub-system. Entropy is also produced by mixing process at interfaces between adjacent sub-systems. Vibrational entropy Si of sub-systems results from the balance of sources, dissipation and mixing

sub-systems. SEA is up to now only based on the application of the energy balance. An entropy balance also applies.

7 Discussion G. Tanner: What is the meaning of the number of modes N if one considers the continuum limit which is natural in acoustics? A. Le Bot: This problem is also encoutered in classical mechanical statistics. The usual response of physicists is to say that basically nothing is continous, everything is quantum that is discrete. I think that this response is not satisfactory for vibroacoustics. In physical statistics, we start from systems which have a huge but finite number of states and we derive continuum mechanics. But in statistical vibroacoustics we start from a continous equation and our problem is not to know if this equation is an approximation of a more fundamental equation but rather to degrade it with further statistical assumptions. What was discussed in this talk is how to quantify the amount of information that has been lost during this second level statistical process. Y. Ben-Haim: Could you extend this formalism to open systems? For instance, when vibroacoustic systems radiate and they have a lost of energy and they also exchange entropy because energy is leaving the system and going into vacuum or air. A. Le Bot: I just said that this approach is limited to the case where the surface of constant energy is closed but not that the system itself is closed. I don’t know if this a possibility to apply this formalism to open systems, but I feel that this formalism is not simply well-suited for non-finite systems which, by nature, cannot be in thermal equibrium.

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P. Shorter: Let’s me make a comment on the number of configurations. It’s not only a story of number of modes. If you consider a plate with only two modes but in a very large ensemble of similar plates, they may have varying damping, material properties or boundary conditions. So the variation of entropy is taken over this space of configurations. But, if you look at a single system, a single system has a very smooth response, while it has a huge number of modes. But that doesn’t definetely suppose it has only one value, the one given by a SEA model. It’s value can be different from the one you’ve got from an ensemble average. A. Le Bot: In the expression of the vibrational entropy I propose to use, it is just included the missing information relative to fact that we don’t know the exact repartition of energy over modes. But if we want to measure the effect of particular statistical properties such as the boundary conditions are not known, we must modify this entropy expression. And we can even imagine to add all these individual entropies for boundary conditions, size of system, properties of material and so on, to get a more accurate expression of vibrational entropy. F. Ziegler: You mentioned that when you define the “vibrational temperature” as modal energy divided by Boltzmann constant and when you calculate the vibrational temperature of a plate for usual acoustical levels, you get very a high temperature about ten power eleven Kelvins! So, what are the physical consequences of this fact? Can we will not use this theory of thermodynamics only because of the improper scale of these numbers? A. Le Bot: I think that there is absolutely no physical consequences of this fact. In astronomy, for instance, measurements are so accurate and signals so tiny that they must aware of noise level induced by mechanical vibration. They have very large structures which are in thermal equilibrium with surrounding air and therefore, the “vibrational temperature” is equal to the true thermal temperature of hundred Kelvins. But in our case, the sources of vibration have a mechanical origin and therefore we deals with very large vibration, which are not in thermal equilibrium with usual thermal frequencies. This is why we get so high vibrational temperatures. But this very hot vibration is confined into a relatively low and narrow frequency band which is disconnected from thermal frequencies. The value of Boltzmann constant is not well-suited in SEA for usual vibrational systems and therefore, the question which raises is the definition of an appropriate scale for vibrational temperature. A. Belayev: Antonio was the first to introduce the vibrational entropy in SEA. A. Carcaterra: I just worked on that subject some years ago. And it is a very interesting theoretical point of view. But my concern was that it is not really obvious how we can use this additional concept of vibrational entropy in terms of improving the SEA

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models. How is it possible to use an entropy balance equation in order to get more information on SEA models or in order to get an advantage in the description of vibroacoustical systems? A. Le Bot: This is indeed an important question and may be the only question of interest in fact. Is it useful to use the entropy concept in vibroacoustics? I now work on this problem for a couple of years and I sincerely hope that it is useful! But let me read some lines from Ingo Muller who recently wrote an interesting paper in Entropy journal on historical aspects of thermodynamics: “In the nineteenth century — after the formulation of the seconf law — there was a noisy controversy between energetics, represented by Oswald, and thermodynamics favoured by Boltzmann. Energeticists maintained that the entropy was not needed. There were wrong, but they did have a point, albeit only at small temperatures. Planck was involved — in a minor role — in the discussion as an unaprreciated supporter of Boltzmann’s thermodynamic view. It was this controversy which prompted Planck to issue his oft-quoted dictum: The only way to get revolutionary advances in science accepted is to wait for all scientists to die.” Acknowledgements The author wishes to acknowledge the ANR (National Research Agency, France) for its financial support with the project ANR-BLANC DyVIn.

References 1. Lyon R.H. and DeJong R. (1995) Theory and Application of Statistical Energy Analysis. Butterworth-Heinemann, Boston 2. Lyon R.H. (2003) Fluctuation theory and (very) early statistical energy analysis. J. Acoust. Soc. Am. 113:2401–2403 3. Le Bot A. (2006) Energy exchange in uncorrelated ray fields of vibroacoustics. J. Acoust. Soc. Am. 120:1194–1208 4. Le Bot A. (2007) Derivation of statistical energy analysis from radiative exchanges. J. Sound Vib. 300:763–779 5. Maxit L. (2003) Extension of sea model to subsystems with non-uniform modal energy distribution. J. Sound Vib. 265:337–358 6. Carcaterra A. (1998) An entropy approach to statistical energy analysis. In: Proc. of Internoise 98 Christchurch, New-Zealand 7. Carcaterra A. (2002) An entropy formulation for the analysis of energy flow between mechanical resonators. Mech. Syst. Sig. Proc. 16:905–920 8. Pauli W. (1973) Statistical Mechanics. Dover Publications Inc., New-York 9. Le Bot A. (2009) Entropy in Statistical Energy Analysis. J. Acoust. Soc. Am. 125:1473–1478 10. Lyon R.H., Maidanik G. (1962) Power flow between linearly coupled oscillators. J. Acoust. Soc. Am. 34:623–639

Modeling noise and vibration transmission in complex systems Philip J. Shorter

Abstract This paper is concerned with modeling the transmission of noise and vibration in complex vibro-acoustic systems. For many systems of practical interest, the justification for statistical analysis methods arises not because of a need to “model uncertainty” but rather because of a need to “reduce model complexity”. This distinction is important; analysis methods which increase model complexity (by requiring large amounts of input information) are seldom of practical use. This paper discusses physical and dynamic complexity and reviews existing analysis methods based on Finite Elements (FE), Boundary Elements (BEM) and Statistical Energy Analysis (SEA). A “Hybrid FE-SEA” method is then presented for coupling low and high wavenumber descriptions of various parts of a system. The method can be viewed as a rigorous way to add detail to an SEA model or to remove detail from an FE/BEM model. The theory is discussed and references are given to recent numerical and industrial applications.

1 Introduction This paper is concerned with modeling the transmission of noise and vibration in complex vibro-acoustic systems. Before discussing modelling methods in detail it is useful to first review the systems of interest and to discuss the information that we wish to extract from a model. The systems of interest in this paper are engineering structures for which noise and vibration either: (i) impact the perceived quality of a product or (ii) adversely affect the performance of a system. In the former case we may, for example, be interested in designing aircraft, automobiles, trains or ships in such a way that interior occupants are shielded from exterior noise. In the latter case, we may, for example, be interested in ensuring that sensitive electronic equipment Philip J. Shorter ESI Group 12555 High Bluff Drive, Suite 250, San Diego, CA 92130, USA e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 11, © Springer Science+Business Media B.V. 2011

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in a satellite survives the vibro-acoustic environments encountered during launch. Alternatively, we may be interested in how machinery noise is transmitted to the hull of a submarine and radiated to the surrounding water.

1.1 Complexity One characteristic of the systems that we are interested in modelling is that they are usually “complex”. This complexity can arise because the system itself consists of a large number of physical components. For example, Figure 1a shows a single module of the International Space Station that could be viewed as being “physically complex”.

Fig. 1: Example of a “physically complex” system: (a) Node 3 of International Space Station in the Shuttle Cargo Bay, (b) SEA model, (c) random loads used in SEA model. Courtesy of Thales-Alenia [1] A detailed model of this system that included every last pipe and bolt would typically be unmanageable due to the vast amount of input information needed to define the model. Some form of simplification is therefore typically required in order to ensure that an analysis remains tractable. Complexity can also arise because the system can respond in a large number of different ways. For example, the system may have a large number of modes across the frequency range of interest and is therefore “dynamically complex”. For example, a typical automotive sedan contains around 3×106 structural modes and 1×106 acoustic modes below 10 kHz; a 2 m section of a typical commercial aircraft fuselage can contain around 4×105 structural modes and 8×106 acoustic modes below 10 kHz; a typical launch vehicle can contain around 1×108 acoustic modes below 10 kHz. A detailed model of the response of such systems across the audible frequency range becomes unmanageable because of the vast amount of information that is required to describe the response.

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1.2 Uncertainty One characteristic of a complex system is that a large amount of “information” is required in order to describe the response of the system. At a given frequency, the system can respond in any one of many possible different states. To determine exactly which state the system will respond in requires that an equally large amount of input information be specified. One can view this as conservation of information (to get very detailed response information for a system with a large number of response degrees of freedom it is necessary to specify very detailed input information about the properties of the system). In practice, there is limited information about the properties of a complex system. A lack of information may arise, for example, because of the precision to which material properties and boundary conditions of the system are known, or because the system is still at the design stage (the detailed properties of a system have not yet been finalized). In some instances it may be possible to try to gain information about a system, perhaps by performing detailed measurements or model updating. However, in practice there is usually a cost (and time delay) associated with obtaining information. This then places a limit on the amount of additional information that can be obtained (particularly in a design situation). In summary, we do not have sufficient information about the properties of a complex system to be able to precisely define its response. We could view this lack of information as representing “uncertainties” in the properties of the system. However, it is important to note that the primary problem is “a lack of information” rather than a requirement to “model uncertainties”. Reference [3] shows the measured responses of a population of beer cans. Clearly, something about this system is uncertain but we do not know precisely what is uncertain. Attempting to find out precisely what is uncertain would require us to obtain and specify additional information. The cost of obtaining this information experimentally may be quite considerable. Even for a system as simple as a beer can “we don’t know what we don’t know and it’s usually too expensive to find out exactly what we don’t know”. Since the main problem is a lack of information, a modelling method that requires us to provide more information (ie., an explicit description of the uncertainties in a system) often exacerbates the problem.

1.3 How much information is needed for noise and vibration design? The previous sections have highlighted that the systems of interest are “complex” and there is a limited amount of “information” about their properties. However, there is still a need to model the transmission of noise and vibration through such systems. It is therefore natural to ask “how much detail is needed in order to model the response of a complex system?”

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Typically, the motivation for creating a model is to: (i) quickly identify the main transmission paths by which noise and vibration are transmitted from external sources to various receiving locations of interest and (ii) to help guide design changes or counter measures that can modify the transmission of noise and vibration. Such changes may, for example, involve the addition of isolation and/or damping treatments along the main transmission paths (or the removal of treatments on paths that are not dominant in order to reduce overall weight or cost). In many instances a model is also used to support the design of a product. Most design questions arise early in the design cycle when a product is not fully specified. The sooner that a model can be created, the more likely it is that it will be used to answer design questions (and therefore the more likely it is that it will have an impact on the product). In such instances, a simpler model that can answer design questions is often preferable to a more complex model [5]. Detail is also often only needed along the dominant transmission paths in a system (for example, creating an extremely complex structural model of a system may be of little benefit if the dominant path in the system is airborne transmission through a simple acoustic leakage path; a better approach in this example would be to create a simple model initially that can be used to estimate and rank all transmission paths, followed by a more detailed model of the airborne leakage path). In summary, in order to address design questions, a vibro-acoustic model should contain enough detail to be able to diagnose the dominant transmission paths in a system and should include the main design parameters of interest (for example, isolation and damping treatments). However, the complexity of a model should match the minimum amount of information that needs to be extracted from the model. The complexity of a model should not exceed one’s knowledge of the properties of a system.

2 Modeling methods and frequency ranges 2.1 Low, mid and high frequency ranges In wave propagation problems it is common practice to define different frequency ranges depending on the relative size of the components of a system compared with a wavelength. When a component is small compared with a wavelength its response typically involves a small number of local modes. When a component is large compared with a wavelength (and supports wave propagation in several directions) its response typically involves a large number of local modes. The frequency range in which all components in a system are small compared to a wavelength is often referred to as “the low frequency” range. The frequency range in which all components in a system are large compared with a wavelength is often referred to as “the high frequency” range. The frequency range for which some components are small compared with a wavelength and some components are large compared with a wave-

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length is referred to as “the mid-frequency range”. It should be noted that while the term “frequency” is commonly used, the classification is perhaps more closely related to the “dynamic complexity” of the components in a system.

2.2 Low and High frequency modelling methods The choice of modelling method depends on the frequency range of interest. In the low frequency range, the response can be characterized by a small number of structural and acoustic modes. Finite elements, Boundary Elements and Infinite Elements are well suited for modelling the response (see for example [6], [7], [8]). The development of techniques such as Automated Multi-Level Substructuring [9] and Fast Multipole Boundary Elements [10], [11] have significantly improved the computational efficiency of such methods in recent years and expanded the types of problems that can be addressed. However, it should be noted that these methods are still “low frequency” methods. The amount of input information that is required or assumed in such methods increases rapidly with the number of degrees of freedom in the model (low frequency methods increase rather than reduce the amount of required input information as the dynamic complexity of a system increases). At high frequencies a system has (by definition) a large amount of dynamic complexity. Rather than increasing the complexity of a model in order to match the dynamic complexity of a system, it is better to take advantage of the dynamic complexity of the system and to use statistics to reduce model complexity. This approach was initially adopted more than a century ago in the development of statistical mechanics [12] and room acoustics [13], [14]. In both approaches a statistical description of the degrees of freedom in a system is adopted and a state of maximum disorder (or maximum “entropy”) is assumed in order to reduce the complexity of an analysis. The extension of this approach1 to vibro-acoustic problems was pioneered by Lyon and Smith in the early 1960’s and led to the development of Statistical Energy Analysis (SEA) [15]–[17]. Broadly speaking, SEA represents a field of study in which statistical descriptions of a system are employed in order to simplify the analysis of complicated vibro-acoustic problems [18]. While such a definition encompasses many different methods and analysis techniques, SEA is often viewed (perhaps for historical reasons) as being a method for describing the storage and transfer of vibrational and acoustic energy between subsystems of “weakly” coupled modes. In particular, the energy storage capacity of a given subsystem is described by a parameter termed the modal density; the coupling between subsystems is described by parameters termed coupling loss factors (CLFs). Applying conservation of energy to each subsystem 1 It should be noted when comparing SEA to thermodynamic analogies that the application of SEA does not require that an individual system have a very large number of modes (ie., it is not required that the response of a single system match the SEA ensemble average in order for an ensemble average prediction to be of value). Such misperceptions lead to erroneous conclusions about the range of applicability of SEA.

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then results in a set of simple linear simultaneous equations for the subsystem energies. In the wave approach to SEA, a complex subsystem is represented as a collection of propagating wavetypes. The energy storage capacity of a given subsystem is related to the expected group velocity and dimension of the subsystem. The CLFs at a given junction are found by calculating the local transmission of energy into the ‘direct fields’ of a number of receiving subsystems, due to the presence of a ‘diffuse reverberant field’ in an excited subsystem. A typical modern SEA code consists of a large library of different subsystems and junctions that can be used to model the transmission paths in a complex system [20]. Figure 1b shows an example of an SEA model of a node of the International Space Station, while Figure 1c shows the structural and acoustic loads that are used to represent the random environment in the payload bay of the Shuttle. The SEA model represents a simplification of the actual flight hardware in Figure 1a, however the model contains a sufficient amount of detail to represent the main structural and acoustic transmission paths and provides good predictions of the overall levels on key components when compared with test data [1]. It should be noted that the model is not necessarily “simple” (the model still contains approximately 1000 subsystems, each with several wavefields), however the use of SEA provides a “rational” way to reduce the complexity of an overall model. This underlying philosophy is equally valid today as it was in the early 1960’s (reference [21] provides an excellent summary of the philosophy that led to the initial derivation of SEA). However, it is natural to question what are the limitations of SEA?

2.3 The Mid-Frequency problem The two main assumptions in the wave approach to SEA are: (i) each subsystem is large compared with a wavelength so that there is a significant amount of uncertainty in the dynamic properties of the subsystem (ie., the reverberant field of each subsystem is “diffuse” when viewed across a large enough ensemble) and (ii) the modal density and CLFs of a subsystem (whose dynamic properties are sufficiently uncertain) can be approximated by the subsystem and junction formulations available in a given SEA code. The second assumption can perhaps be viewed as an implementation issue. The subsystem and CLF formulations in a modern SEA code are significantly more generic than the codes of 50 years ago and can therefore be used to model a much broader class of systems. Research in this area is also still ongoing (for example, the use of FE periodic structure theory to derive propagating wavetypes and connection impedances enables significantly more detail to be included in the definition of an SEA subsystem [22], [23]). However, there are still instances in which a component may be difficult to characterize using the subsystems available in a given SEA code. This does not necessarily mean that SEA is not applicable to the component (or that SEA itself is only applicable for flat plates, singly curved shells and simple cylinders – a misperception that is often encountered) but rather that it is

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not always possible to predict the SEA parameters using the analytical formulations available in a given SEA code. In such instances, one must therefore look for alternative ways to update the unknown parameters in an SEA model. This may be achieved using: (i) measured data [24], (ii) updating assumed impedances in existing SEA formulations [25], [26] or (iii) if the component doesn’t contain too many modes, by post-processing local finite element models [17], [27], [28]. All of these approaches have been used in SEA applications during the past 50 years. The assumption of a significant amount of uncertainty in the dynamic properties of each component is, however, more problematic. In the high frequency range the assumption is valid (by definition). However, in the mid-frequency range not all components are “dynamically complex” and an assumption of significant uncertainty in the subsystem properties is not necessarily appropriate for all subsystems. It should be noted that SEA does not “break down” at mid and low frequencies (it is still possible to make an ensemble average SEA prediction and this prediction still has meaning). However, the ensemble assumed in the SEA model is “too uncertain” at mid and low frequencies and does not make use of all available information about a system. It is then perhaps natural to ask whether it is possible to add “detail” to an SEA model. Alternatively, one might ask whether it is possible to remove “detail” from an FE/BEM model. The two questions are related and the following sections provide an approach to this problem using a “Hybrid” (or “Coupled”) FE-SEA method. The presentation in the following sections provides a qualitative overview of the method (the reader is referred to references [32], [33], [34] for a more detailed derivation).

3 The Hybrid FE-SEA method Consider the problem of predicting the ensemble average response of a system of coupled subsystems, some of which are known precisely (and therefore modelled “deterministically”), some of which are uncertain (and therefore modelled “statistically”). The “statistical subsystems” may support several wavetypes (with different directions of propagation) and are typically large compared with a wavelength. One of the key steps in coupling the different subsystems is to choose an appropriate description of the response in the statistical subsystems. The following section discusses this in more detail.

3.1 Statistical subsystem Consider the subsystem illustrated in Figure 2a. The subsystem represents a structural or acoustic component in a vibro-acoustic system and is assumed to be large compared with a wavelength. The subsystem could, for example, represent a large reverberant acoustic space or perhaps a region of the skin of an aircraft fuselage

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that contains a large number of local modes. Various “connection regions” can be defined on the boundary of the subsystem. These connection regions describe areas of the boundary that are used to connect the subsystem to adjacent subsystems or to external loads. Broadly speaking, they represent regions through which energy can flow into or out of the subsystem. The connection regions can be of arbitrary dimension (although it is common to refer to connections as being point, line or area connections depending on the dimension of the connection and subsystem). Additional regions of the subsystem boundary may also be known precisely. These may, for example, be regions of the boundary that lie within a few wavelengths of the connections. These regions and the connection regions are collectively referred

Fig. 2: A statistical subsystem (a) can be represented in terms of the superposition of a “direct field” (b) and a “reverberant field” (c)

to as the “deterministic boundaries” of the subsystem. The remaining boundaries are referred to as the “random boundaries” of the subsystem. In most instances, the deterministic boundary represents a relatively small portion of the overall subsystem boundary and can therefore be represented by a small number of “degrees of freedom”. In contrast, the random boundary typically involves a large number of degrees of freedom.

3.2 The direct and reverberant fields of a statistical subsystem Suppose now, that a given connection region is given a prescribed displacement. Interest lies in determining the force on the connection due to the prescribed displacement. The response of the subsystem can be described in terms of two separate wavefields. The first is termed the “direct field” and describes the response of the subsystem in the absence of the random boundary (as shown in Figure 2b). The direct field is deterministic and only depends on the properties of the deterministic boundary and the way in which waves propagate within the subsystem. The force that the direct field applies to the connection can be represented by an impedance matrix (termed the “direct field impedance”). For example, for an acoustic subsystem, this impedance is given by the “radiation impedance” of the connection. The second field is termed the “reverberant field” and describes the field that results from the scattering of the direct field at the random boundaries of the subsystem

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(as shown in Figure 2c). The reverberant field also applies a force to the connection. This force is termed the “reverberant loading”. The total response within the subsystem is found by superimposing the direct and reverberant fields. It should be noted that no approximations have been made at this stage (the use of these two displacement fields is a mathematically exact way to describe the response of a given component).

3.3 Ensemble average reverberant loading A detailed representation of the random boundary would require a large number of response degrees of freedom. One could therefore view the random boundary as being “dynamically complex” (a large amount of information would be required to provide an exact description of the random boundary). Due to this complexity, the reverberant field is very sensitive to small changes in the properties of the random boundary. Rather than attempting to describe the reverberant field in detail it becomes simpler to instead assume that the random boundary is uncertain and to adopt a statistical description of the reverberant field (across an assumed ensemble). The force that the reverberant field applies to the connection degrees of freedom therefore also become statistical and can be represented by a random load. Clearly the statistics of this reverberant loading will depend on the ensemble of “random boundaries” that is assumed in an analysis. However, it transpires that as the uncertainty in the random boundary increases, the statistics of the reverberant field tend to simple limits. In the state of “maximum entropy”, the reverberant field attains a special set of statistics that are termed “diffuse” (a general definition of the term diffuse is given in [33]). It can be shown that the (ensemble average) reverberant loading that a “diffuse reverberant field” applies to a connection is proportional to the radiation impedance of the connection [33]. The constant of proportionality depends on the incident power in the reverberant field (and can be related to the energy and modal density of the reverberant field). This remarkably simple result forms the basis of the “diffuse field reciprocity relationship” and considerably simplifies the analysis of complex systems that are coupled at arbitrary connections.

3.4 Coupling a deterministic and statistical subsystem Consider now a single statistical (or SEA) subsystem connected to a single deterministic subsystem as illustrated in Figure 3. From a qualitative point of view the (ensemble average) response of the SEA subsystem can be represented by the superposition of a “direct field radiation impedance” and a random “diffuse reverberant loading” applied to the connection degrees of freedom. The magnitude of the reverberant loading depends on the energy in the reverberant field. This is perhaps intuitive for room acoustics problems. Consider, for example, a plate (with a small

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number of structural modes) that is placed in contact with a large reverberant acoustic cavity. The acoustic cavity adds reactive and resistive impedance to the modes of the plate (which can be represented by a complex “modal radiation impedance” matrix). The reverberant field also applies a random loading to the modes of the plate (which can be represented by a “modal cross-spectral force” matrix). If the cavity is heavily damped then the reverberant loading becomes negligible and the cavity appears as an energy sink. If the cavity is lightly damped then the reverberant loading is significant and a significant amount of energy is injected back into the structure. In the limit of light damping the plate and cavity attain a state of “equipartition of energy”. It should be noted that the cavity cannot be represented by only applying impedance to the plate (an assumption adopted in certain versions of “fuzzy structure” theory [35]). Nor can the cavity be represented by just a random reverberant

Fig. 3: A statistical (“SEA”) subsystem that is coupled to a deterministic (“FE/BEM”) subsystem can be represented by a “direct field radiation impedance” and a “diffuse reverberant load” loading (an assumption that was adopted prior to the initial development of SEA). Finally, it should be noted that the representation above is only true when looking at the response across an ensemble (it does not mean, for example, that the reverberant field of an individual realization of the ensemble will necessarily be “diffuse”). The previous results are completely general and can be used to model the response of an arbitrary number of FE, BEM and SEA subsystems coupled by various point, line and area type connections. For a more detailed quantitative derivation, the reader is referred to [32], [33], [34].

4 Application examples This section provides a brief overview of a number of applications of the Hybrid FE-SEA method. The generation of numerical reference results using placeMonte Carlo simulations is discussed initially. This is then followed by a list of references to different application examples.

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4.1 Monte Carlo simulations The Hybrid FE-SEA method predicts the ensemble average response of a set of coupled subsystems. In order to validate the method it is therefore necessary, in principle, to compare the Hybrid predictions with a response that has been averaged over a sufficiently large ensemble. One approach is to use an FE Monte Carlo simulation in which small masses are placed at random locations across the structure. For structural subsystems that are large compared with a wavelength, this typically introduces a sufficient amount of uncertainty to approximate the maximum entropy ensemble assumed in a Hybrid FE-SEA analysis. Figure 4a shows an example in which two SEA plates are point connected to an FE frame [32]. The response in the receiving plate is plotted in Figure 4b and good agreement is observed between the ensemble average response from the FE Monte Carlo simulation and from the Hybrid model (the example is discussed in more detail in [32]).

Fig. 4: (a) Example of transmission of energy between two SEA plates via an FE frame, (b) response in receiving plate predicted by Hybrid model (dark line), FE monte Carlo simulation average (white line), individual realizations from FE simulation (gray lines)

4.2 Numerical applications It is beyond the scope of the current paper to list all numerical validations of the Hybrid FE-SEA method. However, the following is a brief list of selected references sorted by connection and subsystem. The reader is referred to these references for additional numerical validation examples (reference [38] provides a brief overview). – Point connections ◦ SEA beams point connected to FE beam elements [41] ([42]) ◦ SEA plates point connected to FE beam elements [38]

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◦ SEA plates point connected to FE plate elements [32] ◦ SEA plates point connected to SEA plates [43] – Line connections ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦

SEA plates line connected to FE shell (local bracket) [47][48] SEA plates line connected to FE shell (frame) [44][47][48] SEA plates line connected to FE solid (welding example) [36] SEA plates line connected to FE shell (vibration isolator) [45] SEA plates line connected to FE shell (frame-plate) [50] SEA plates line connected to FE shell (frame- cavity) [47][48] SEA plate line connected to FE shell (local input power) [41] SEA cylinder line connected to FE shell (cylinder-plate) [51] SEA plates line connected to FE shell (strip plate) [49]

– Area connections ◦ ◦ ◦ ◦ ◦ ◦ ◦

FE structure area connected to SEA fluid (piston radiation) [41] FE structure area connected to SEA fluid (panel radiation) [41] FE structure area connected to SEA fluid (panel TL) [41][46] FE structure area connected to SEA fluid (unbaffled panel) [39] FE structure area connected to SEA fluid (trim+panel TL) [40] FE cavity area connected to SEA infinite fluid (leak TL) [41] FE cavity area connected to SEA cavity (slit/seal TL) [52]

4.3 Industrial applications The Hybrid FE-SEA method has been applied to a large number of industrial applications in recent years. Many of the models in these applications are company proprietary and are therefore not available in the public domain. However a number of references are available and are listed in this section (the following list is by no means exhaustive but instead provides specific examples in different industries). Broadly speaking the applications can be separated into three categories: (i) adding “quick” SEA acoustics and trim to existing FE structural/acoustic models, (ii) adding local “detail” to existing SEA models (particularly at junctions and excitation locations) and (iii) creating efficient “system level” models. A list of references in each category is given below: – Adding “quick” acoustics to existing FE models ◦ ◦ ◦ ◦ ◦

Modelling acoustic transmission loss of an automotive dash [53] Modelling acoustic radiation from damped train panels [54] Modelling diffuse field excitation of a satellite antenna [55] Modelling acoustic radiation from an engine casing [56] Modelling acoustic insertion loss of large HVAC ducts [57]

– Adding “local junction details” to existing SEA models

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Modelling transmission loss of automotive door seals [52] Modelling structure-borne transmission in trains [58][59] Modelling local component response in a spacecraft [60] Modelling local equipment response in a launch vehicle [61]

– Creating efficient models of system level response ◦ Modelling automotive structure-borne noise [28][62][63] ◦ Modelling structure-borne noise in an aircraft fuselage [64] Reference [61] highlights how the Hybrid FE-SEA method can be used to add local FE “detail” to a system level SEA model (in this case, an SEA model of a launch vehicle). The SEA subsystems add an impedance boundary condition to the FE structural components and also apply a random distributed loading across the connection regions (that depends on the system level response of the launch vehicle). Recovering this type of local response is typically not feasible using a pure FE or SEA model. Reference [63] shows how the Hybrid FE-SEA method can be used to create efficient system level models of noise and vibration transmission (using a combination of Hybrid FE-SEA and EFM/EIC methods). The model is primarily intended for diagnosing structure-borne transmission paths in a trimmed vehicle and investigation the impact of various treatments on these transmission paths. The model solution takes approximately 2 min/frequency point and predictions of interior sound pressure levels for the trimmed vehicle are typically within approximately 5 dB of measured (1/3rd octave band) results.

5 Concluding remarks This paper has discussed methods for modeling noise and vibration in complex systems. A Hybrid FE-SEA method was presented for modeling the response at “midfrequencies”. The advantage of the method is that it provides a generic way to add “necessary detail” to an SEA model (or to remove “unnecessary detail” from an FE/BEM model). The method can be viewed as a generalization of the wave approach to SEA and therefore also clarifies some of the assumptions made in the traditional wave approach to SEA. In order to apply the Hybrid FE-SEA method it is necessary to decompose a system into regions that will be modeled with FE/BEM and regions that will be modeled with SEA. Typically, this classification can be made based on the expected size of the various components in a system compared with a wavelength. It should be noted that the SEA “regions” do not need to be restricted to regions that look like simple flat plates or cylinders. SEA is ideally suited for modeling complex components that are geometrically irregular and which contain a large number of modes. Finally, it should be noted that improved Hybrid FE-SEA modeling methods are still being actively researched and developed; the

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recent introduction of generic SEA subsystems based on wave propagation in FE periodic theory is one such example.

References 1. P.C. Marucchi-Chierro, S. Destefanis, “ISS Node 3 – Cupola launch coupled Vibro–Acoustic analysis”, Proceedings of ESI European Vibro-Acoustic Users Conference, Cologne 2007. 2. C. Shannon, The Mathematical Theory of Communication. Univ. of Illinois Press, 1963. 3. F. Fahy, Foundations of Engineering Acoustics. Academic Press, 2001. p275 (see also [4]). 4. M. Kompella, B. Bernhard, “Measurement of the statistical variation of structural-acoustic characteristics of automotive vehicles”, Proc. SAE Noise and Vibration Conf., Warrendale, USA: Soc. Auto. Eng., 1993. 5. L. Chwif et al., “On simulation model complexity”, Proceedings of the 2000 Winter Simulation Conference, J. A. Joines, R. R. Barton, K. Kang, and P. A. Fishwick, eds. 6. O.C. Zienkiewicz et al., The Finite Element Method, Butterworth-Heinemann; 6th Edition 2005. 7. C. Brebbia et al., Boundary Elements, McGraw-Hill, 2nd edition 1992. 8. R. Astley, “Infinite elements for wave problems: a review of current formulations and an assessment of accuracy”. International Journal for Numerical Methods in Engineering, 49, (7), 951–976. 2000. 9. J. Bennighof et al., “An automated multi-level substructuring method for the eigenspace computation in linear elastodynamics” SIAM J. Sci. Comput. 25:2084–2106, 2004. 10. N. Gumerov, R. Duraiswami, “A broadband fast multipole accelerated boundary element method for the three dimensional Helmholtz equation”, J. Acoust. Soc. Am. 125 191 (2009). 11. N. Gumerov, R. Duraiswami, Fast Multipole Methods for the Helmholtz equation in three dimensions, Elsevier Science, 2005. 12. J. Maxwell, “Molecules”, Nature, Sept. 1873, pp. 437–441. 13. W. Sabine, Collected Papers on Acoustics, Penisula Publishing (in particular, see the article “Reverberation” initially published in 1900). 14. H. Kuttruff, Room Acoustics, Taylor & Francis; 4 edition, 2000. 15. P.W. Smith, “Response and radiation of structural modes excited by sound” J. Acoust. Soc. Am. 34(5) 640–647, 1962. 16. R. Lyon, G. Maidanik, “Power flow between linearly coupled oscillators”. J. Acoust. Soc. Am. 34(5) 623–639, 1962. 17. R. Lyon, Statistical Energy Analysis of Dynamical Systems, MIT Press, 1975. 18. R. Lyon, “Statistical Energy Analysis and structural fuzzy”, J. Acoust. Soc. Am. 97(5), 2878– 2881, 1995. 19. F. Fahy, “Statistical Energy Analysis: a critical overview”, Phil. Trans. R. Soc. Lond. A (1994) 346, 431–447. 20. VA One 2008.0, ESI Group. http://www.esi-group.com. 21. R. Lyon, P. Smith, “Sound and structural vibration” NASA-CR-160, March 1965. http://ntrs.nasa.gov/search.jsp?R=982436&id=1&qs=N%3D42949662837. 22. V. Cotoni, R. Langley, P. Shorter, “A statistical energy analysis subsystem formulation using finite element and periodic structure theory” Journal of Sound and Vibration, 318 (4–5). pp. 1077–1108, 2008. 23. V. Cotoni et al., “A statistical energy analysis subsystem formulation using finite element and periodic structure theory” Proc. Novem 2009. 24. B. Cimerman, T. Bharj, G. Borello, “Overview of the experimental approach to statistical energy analysis” Proc. SAE N&V conference, paper 97NV169, 1997. 25. J. Manning, “Formulation of SEA parameters using mobility functions”, Proc. Royal Soc. 346 (1681), 1994.

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26. J. Manning, “Hybrid SEA for mid-frequencies”, Proc. SAE Noise and Vibration Conference, Chicago 2007. 27. T. Onsay, A. Akanda, G. Goetchius, “Vibro-Acoustic Behavior of Bead-Stiffened Flat Panels: FEA, SEA, and Experimental Analysis”, Proceedings of SAE, Traverse City, 1999. 28. B. Mace, P. Shorter, “Energy Flow Models from Finite Elements”, Journal of Sound and Vibration, 233(3), 369–389, 2000. 29. A. Charpentier, S. Prasanth, K. Fukui, “Using the Hybrid FE-SEA model of a trimmed full vehicle to reduce structure borne noise from 200Hz to 1kHz” Proc. Internoise 2008. Shanghai. 30. V. Cotoni et al., “Numerical and experimental validation of variance prediction in the statistical energy analysis of built-up systems” Journal of Sound and Vibration, 288(3), 701–728, 2005. 31. R. Langley, V. Cotoni, “Response variance prediction for uncertain vibro-acoustic systems using a hybrid deterministic-statistical method”, J. Acoust. Soc. Am. 122(6), pp. 3445–3463, 2007. 32. P. Shorter, R. Langley, “Vibro-Acoustic analysis of complex systems”. Journal of Sound and Vibration, 288(3), 669–699, 2005. 33. P. Shorter, R. Langley, “On the reciprocity relationship between direct field radiation and diffuse reverberant loading”, Journal of the Acoustical Society of America 117, 85–95. 2004. 34. P. Shorter, “Modeling noise and vibration in complex systems”, Proc. Novem 2009, Oxford, UK. 2009. 35. C. Soize, “Modeling and numerical method in the medium frequency range for vibroacoustic predictions using theory of structural fuzzy”, Journal of the Acoustical Society of America, 94(2), 1993. (see also [18]). 36. P. Shorter, Y. Gooroochurn, B. Rodewald, “Advanced vibro-acoustic models of welded junctions”, Proceedings of InterNoise2005, Brazil, 2005. 37. R. Langley, P. Shorter, “The wave transmission coefficients and coupling loss factors of point connected structures” J. Acoust. Soc. Am., 113(4), 1947–1964, 2003. 38. V. Cotoni, P. Shorter, R. Langley, “Numerical and experimental validation of a hybrid finite element-statistical energy analysis method” J. Acoust. Soc. Am. 122(1), 2007. 39. R. Langley, “Numerical evaluation of the acoustic radiation from planar structures with general baffle conditions using wavelets” J. Acoust. Soc. Am. 121(2) 766–777, 2007. 40. P. Shorter, S. Mueller, “Modeling the mid-frequency response of poroelastic materials in Vibro-Acoustics applications”, Proc. SAPEM 2008, Bradford 2008. 41. VA One 2008 Validation and QA Manual. ESI Group. 2008. 42. K. De Langhe, “High frequency vibrations: contributions to experimental and computational SEA parameter identification techniques”, Ph.D Thesis, KU Leuven, 1996. 43. R. Langley et al., “Predicting the response statistics of uncertain structures using extended versions of SEA”, Keynote address. Proc. Internoise 2005. 44. P. Shorter et al., “Modeling structure-borne noise with the Hybrid FE-SEA method”, Proc. Eurodyn 2005. 45. B. Gardner et al., “Modeling vibration isolators at mid and high frequency using Hybrid FESEA analysis”, Proceedings of InterNoise 2005, Brazil, 2005. 46. V. Cotoni et al., “Efficient models of the acoustic radiation and transmission properties of complex trimmed structures”, Proceedings of InterNoise2005, Brazil, 2005. 47. P. Shorter et al., “Numerical and experimental validation of the Hybrid FE-SEA method”, Proceedings of InterNoise 2005, Brazil, 2005. 48. P. Shorter et al., “Numerical and experimental validation of the Hybrid FE-SEA method”, Proceedings of Novem 2005. 49. K. Heron, “Hybrid methods for predictions at mid-frequencies”, Proceedings of Novem2005 (nb. reference uses a modified form of the direct field impedance at a line connection from the one presented in the current work). 50. R. Langley et al., “A Hybrid FE-SEA method for the analysis of complex vibro-acoustic systems”, Proceedings of Novem2005. 51. T. Kuba et al., “Improving SEA predictions of structure-borne noise by describing junction details with FE” Proceedings of ICSV13, 2006.

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52. J. Cordioli et al., “Numerical investigation of the transmission loss of seals and slits for airborne SEA predictions”. SAE N&V Conference, Chicago 2009. 53. P. Shorter, Q. Zhang, A. Parrett, “Using the Hybrid FE-SEA method to predict and diagnose component Transmission Loss”, Proc. SAE N&V Conf, Chicago, 2007. 54. U. Orrenius, M. Almgren, Y. Jiang, L. Feng, “Damping treatments on train interior panels: vibration and radiation effects”, Joint Baltic-Nordic Acoustics Meeting 2006, 8-10 November 2006, Gothenburg, Sweden. 55. J. Larko et al., “Prediction of the dynamic response of the NASA ACTS antenna to wide spectrum acoustic loading” Proc. Spacecraft and Launch Vehicle Dynamic Environments Workshop, El Segundo 2006. 56. Z. Cui et al., “VA One modelling of engine cover radiation” University of.Kentucky VibroAcoustics Consortium Meeting, Nov 2006. 57. D. Herrin et al., “Predicting insertion loss of large duct systems utilizing the diffuse reciprocity relationship”, Proc. Noisecon 2008. 58. L´eo Baur`es, “Coupled FE-SEA vibroacoustic analysis of floating floors for trains”, Proceedings of German Acoustical Society (DAGA) 2007. 59. J. Sapena et al., “Interior noise structureborne path prediction in a high speed train using “FE/SEA” hybrid modeling methodologies”, Proc. Novem 2009. 60. M. Yang, N. Tengler, “Derivation of preliminary ascent vibro-acoustic environments for the Crew Exploration Vehicle”, Proc. Spacecraft and Launch Vehicle Dynamic Environments Workshop, El Segundo 2006. 61. B. Prock, “Vibro-Acoustic Hybrid modeling and analysis of the ARES IX roll control system”, Proc. Spacecraft and Launch Vehicle Dynamic Environments Workshop, El Segundo 2008. 62. A. Charpentier, K. Fukui, “Using the Hybrid FE-SEA method to predict structure-borne noise transmission in a trimmed automotive vehicle”, Proc. SAE N&V Conference 2007. 63. A. Charpentier, P. Sreedhar, and K. Fukui, “Efficient Model of Structure-Borne Noise in a Fully Trimmed Vehicle from 200Hz to 1kHz”, Proceedings of INTER-NOISE 08, Shanghai, China, paper 0701, 2008. 64. V. Cotoni et al., “Modeling methods for commercial aircraft”, Proc. ICSV 2007. 65. R. Langley, J. Cordioli, “Hybrid deterministic-statistical analysis of vibro-acoustic systems with domain couplings on statistical components” Journal of Sound and Vibration. In Press.

A Power Absorbing Matrix for the Hybrid FEA-SEA Method R.H. Lande and R.S. Langley

Abstract A key ingredient in the Hybrid deterministic-statistical approach to the vibroacoustic analysis of complex uncertain structures (Shorter and Langley 2005) is the power absorbing dynamic stiffness matrix. The derivation of such a matrix associated with the flexural motion of a thin, flat, homogeneous plate, based on the cylindrical wave representation of the displacement field over a single convex domain (Lande 2005), is presented here. Key theoretical concepts are discussed, and the numerical results from a simple application are compared against Monte Carlo simulation results.

1 Introduction During the last decade, a novel and computationally efficient approach to the vibroacoustic analysis of complex uncertain structures has been developed and commercialized (Shorter and Langley 2005, and Cotoni et al 2007): it involves the combination of Finite Element Analysis (FEA) and Statistical Energy Analysis (SEA) into a Hybrid deterministic-statistical formulation, which aims to predict the ensemble average response of a complex uncertain built-up structure. In short, the Hybrid analysis provides a set of equations which is similar to the classical SEA equations (Lyon 1975) although the derivation is very different. The equations provide estimates for the ensemble average of the kinetic energy of modally dense structural subsystems with random properties, such as plates or acoustic cavities, as well as the power input to these subsystems due to excitation transmitted via the deterministic master R.H. Lande Det Norske Veritas (DNV), Veritasveien 1, NO-1322 Hovik, Norway e-mail: [email protected] R.S. Langley Cambridge University Engineering Department, Cambridge, CB2 1PZ, UK e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 12, © Springer Science+Business Media B.V. 2011

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structure, to which the subsystems are attached. The Hybrid analysis also provides a set of equations for the ensemble mean square response of the master structure. The first set of equations represents the SEA part of the problem, and the second set represents the FEA part; two fundamentally different approaches combined into a hybrid method of analysis. The key link between the two sets of equations is the socalled power absorbing dynamic stiffness matrix associated with a modally dense and uncertain subsystem. The work presented herein is concerned with the derivation of a power absorbing dynamic stiffness matrix associated with the flexural motion of a thin, flat, homogeneous plate, based on the cylindrical wave representation of the displacement field (Lande 2005): the power absorbing dynamic stiffness matrix is obtained by including only the cylindrical waves propagating from the outer boundary towards the interior of the plate.

2 Cylindrical Waves and Energy Sinks 2.1 The Governing Equations The governing equation of motion for the flexural response is taken from the classical linear thin plate theory (Rayleigh 1945), where the out-of-plane displacement w is uncoupled from the in-plane displacements. The cylindrical wavefield to be developed requires a local polar coordinate system (r, ϕ ) located somewhere in the interior of the plate. Adopting a harmonic time dependence exp(iω t), where t is time, ω is the circular frequency of vibration and i is the imaginary unit with the property i2 = −1, the equation of free flexural vibration is given by the differential equation (Rayleigh 1945) {∇2 + (ikB )2 }{∇2 + (kB )2 }w = 0, where the Laplacian differential operator ∇2 is required in polar form, and kB is the wavenumber associated with plane bending waves in a plate. In the present case there are no forces or constraints acting in the interior of the plate, and therefore the general solution to the differential equation cannot involve those admissible functions that are singular at the origin (Rayleigh 1945), and so the general solution follows as w(kB , r, ϕ ) = 2a0 J0 (kB r) +

n=∞

∑ {an exp(inϕ ) + bn exp(−inϕ )} 2Jn (kB r) + . . .

n=1

n=∞

c0 I0 (kB r) +

∑ {cn exp(inϕ ) + dn exp(−inϕ )} In (kB r) ,

(1)

n=1

where an , bn , cn and dn are complex amplitudes whose values depend on the boundary conditions; Jn (kB r) is the n-th order Bessel function of the first kind, and In (kB r) = (−i)n Jn (ikB r) is the n-th order modified Bessel function of the first kind (Abramowitz and Stegun 1972). Further progress requires the solution (1) to be expressed in terms of cylindrical waves, as discussed in the next section.

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The power absorbing dynamic stiffness matrix derived herein is obtained from the expression for the net power flow through the boundary of the plate, so the externally applied distributed forces and moments acting on the boundary need to be expressed in terms of a local curvilinear coordinate system (ν , σ , z) located on the plate edge: the z-axis of this local system is in the direction of the out-of-plane motion; the σ -axis is directed along the curvilinear boundary in the counter-clockwise sense; and the ν -axis is normal to the boundary and directed away from the plate. The total time averaged flexural vibrational power Pflex transmitted through a closed boundary may then be written as (Bobrovnitskii 1996), ⎫ ⎧ ∗ ⎬ ∂ T w 1 ⎨ ∂ ν (iω w)∗ Sz − + −iω mσ ds , (2) Pflex = Re ⎭ 2 ⎩ ∂σ ∂ν boundary

where: Sz is the distributed transverse shear force, acting along the z-axis; mσ is the distributed bending moment, acting about the σ -axis, where counter-clockwise rotation about this axis defines the positive direction for mσ ; Tν is the distributed twisting moment, acting about the ν -axis, where counter-clockwise rotation about this axis defines the positive direction for Tν , and where Sz − ∂ Tν /∂ s = f z is the distributed Kirchhoff shear force (Love 1944). Further progress requires the terms in the power integral (2) to be expressed in polar form. The displacement w is given by Eq (1), and the externally applied loads acting on the boundary are considered next. Expressed in terms of the local curvilinear system, the Kirchhoff shear force fz and the flexural moment mσ acting at a point on the plate boundary are given as (Rayleigh 1945, and Love 1944), 2 ∂ ∂ 2w 1 ∂ w ∂ 2w ∂ ∂ w fz − = + + + (1 − μ ) , (3) D ∂ ν ∂ ν2 R ∂ ν ∂ σ 2 ∂ σ ∂ ν∂ σ and −

∂ 2w mσ +μ = D ∂ ν2

∂ 2w 1 ∂ w + ∂σ2 R ∂ν

,

(4)

where D is the flexural rigidity of the plate, μ is Poisson’s ratio, and R is the local radius of curvature (and which is positive for a locally convex boundary segment). The transformation from the local curvilinear system to the polar coordinate system is given by ∂

∂ cos ψ − sin ψ ∂r ∂ν , ψ = ϕ −θ , = (5) 1 ∂ ∂ sin ψ cos ψ r ∂ϕ ∂σ where θ is the angle of the local ν -axis, measured with respect to the same reference axis from which the polar angle ϕ of the boundary point is measured. In the case of a straight boundary (R → ∞), Eq (5) may be substituted into Eqs (3) and (4) to obtain

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2 2 − mDσ = 1 − (1 − μ ) sin2 ψ ∂∂ rw2 − (1 − μ ) sin 2ψ 1r ∂∂r∂wϕ + . . . 2 1 − (1 − μ ) cos2 ψ r12 ∂∂ ϕw2 + 1 − (1 − μ ) cos2 ψ 1r ∂∂wr + (1 − μ ) sin 2ψ

1 ∂w , r2 ∂ ϕ

(6) and 3 3 − Dfz = cos ψ 1 + (1 − μ ) sin2 ψ ∂∂ rw3 − sin ψ 1 + (1 − μ ) cos2 ψ r13 ∂∂ ϕw3 + . . . 3 − sin ψ 1 + (1 − μ ) 1 − 3 cos2 ψ 1r ∂∂r2 ∂wϕ + . . . 3 cos ψ 1 + (1 − μ ) 1 − 3 sin2 ψ r12 ∂∂r∂ wϕ 2 + . . . 2 cos ψ 1 + (1 − μ ) 1 − 3 sin2 ψ 1r ∂∂ rw2 + . . . 2 −2 cos ψ 1 + (1 − μ ) 1 − 3 sin2 ψ r13 ∂∂ ϕw2 + . . . 2 − sin ψ 1 − (1 − μ ) 2 − 9 cos2 ψ r12 ∂∂r∂wϕ + . . . − cos ψ 1 + (1 − μ ) 1 − 3 sin2 ψ r12 ∂∂wr + 2 (1 − μ ) sin ψ 4 cos2 ψ − 1 r13 ∂∂ ϕw . (7) In the special case where ψ = 0, which yields the identities ∂ /∂ ν = ∂ /∂ r, ∂ 2 /∂ ν 2 = ∂ 2 /∂ r2 , ∂ /∂ σ = 1/r ∂ /∂ ϕ and ∂ 2 /∂ σ 2 = 1/r2 ∂ 2 /∂ ϕ 2 + 1/r ∂ /∂ r, Eqs (6) and (7) reduce to Eqs (3) and (4) with R → ∞, as expected. The general equations valid for any curvilinear boundary have been given by Lande (2005).

2.2 The Cylindrical Waves The cylindrical propagating wavefield may be obtained from Eq (1) by noting the (1) (2) (1) well-known relation 2Jn (ξ ) = Hn (ξ ) + Hn (ξ ), where ξ = kB r, and where Hn (2) and Hn are the Hankel functions of the first and second kind, respectively, both of order n. These two functions are generally identified as propagating cylindrical waves (Cremer and Heckl 1973), as may be appreciated by noting the large argument form of the Hankel functions (Abramowitz and Stegun 1972), indicating that √ (1) (2) to exp(iξ )/ ξ for ξ /n 1 while Hn (ξ ) is proportional Hn (ξ ) is proportional √ to exp(−iξ )/ ξ . As the time dependency exp(iω t) has been assumed, it is clear (1) that Hn represents an incoming cylindrical wave propagating towards the origin (2) (ξ = kB r = 0) located somewhere in the interior of the domain, while Hn represents an outgoing cylindrical wave propagating towards the boundary. The terms inside the curly brackets of Eq (1) represent the angular dependence of the displacement field, and physically these terms give rise to two spiralling wavefields for each order n > 0: for example, given the adopted time dependency, the waves of amplitude an are spiralling in the clockwise direction, while the waves of amplitude bn are spiralling in the counter-clockwise direction. The order n of the (1) wave Hn × exp(inϕ ) represents the number of spirals that connect to the origin. The sum of terms involving In (ξ ) = In (kB r) in Eq (1) represents a nearfield, or an evanescent field (Cremer and Heckl 1973), where the field variable decreases expo-

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nentially in magnitude in the radial direction away from the boundary and towards the origin. This may be appreciated by noting the large argument form of√In (ξ ) (Abramowitz and Stegun 1972), indicating that it is proportional to exp(ξ )/ ξ for ξ /n 1 and thus only important near the boundary. In wave terminology, the function In (ξ )×exp(inϕ ) may be considered to represent a cylindrical evanescent wave, although Fahy (1987) points out that exponential functions with a real argument should not be called waves, since they do not allow transportation of energy. However, Bobrovnitskii (1992) has shown in the case of an Euler-Bernoulli beam that two evanescent functions associated with different ends of the beam can interact in such a way that net energy transportation does occur. Moreover, Lande and Langley (2005) have shown in the case of a thin isotropic plate that two evanescent cylindrical functions associated with the regions near the boundary and near the origin may also interact and allow energy transportation. Further, given the adopted time dependency, the term In (ξ )×exp(inϕ ) represents an edge wave that circulates reactive power around the boundary in the clockwise sense. By itself the wave will not carry energy in the radial direction, unless it interacts with other evanescent cylindrical waves (Lande and Langley 2005). The imaginary part of the Hankel functions, that is, the Bessel function of the (1) (2) second kind, Yn = Im{Hn } = −Im{Hn } (Abramowitz and Stegun 1972), is singular at the origin. This singularity allows, and is required for, energy transportation in each propagating wave: in the case of the incoming waves, the singularity represents an energy sink for the power absorbed at the outer boundary of the plate; in the case of the outgoing waves, the singularity represents an energy source for the power transmitted outwards and across the boundary. In the case where there are no forces or constraints acting in the interior of the plate, the net power flow must be zero, and thus there may be no net singularity anywhere within the domain. The implication of this is that for a given order n, the two corresponding wave (1) (2) functions Hn × exp(inϕ ) and Hn × exp(inϕ ) must have equal amplitudes so that the two singularities cancel each other. The power absorbing wavefield is then obtained simply by removing the singularity associated with the outgoing waves, i.e. (2) by removing all terms involving Hn . As opposed to the case of plane waves propagating in a waveguide, there is in the case of cylindrical waves no distinct value of the order n which would identify the transformation of a wave from the propagating state to the evanescent state. In other words, there is no distinct cut-on frequency associated with cylindrical waves. However, as the order n of the Hankel functions increases, the propagating cylindrical waves eventually transform into non-oscillating functions over the domain of interest. The large order form of the n-th order Bessel function Jn (ξ ) (Abramowitz and Stegun 1972) indicates that, for a given n such that ξ /n 1, Jn (ξ ) is proportional to (ξ /n)n . Hence, as ξ /n decreases below unity, Jn (ξ ) tends to zero. The large order form of Yn (ξ ) (Abramowitz and Stegun 1972) indicates that, for a given n such that ξ /n 1, Yn (ξ ) is inversely proportional to −(ξ /n)n . Hence, as ξ /n decreases below unity, Yn (ξ ) tends to negative infinity, and the extent of the domain affected by the singularity also increases for increasing n. For all practical purposes, the Han-

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kel function Hn = Jn + iYn can be considered to no longer represent a propagating wave over the domain of interest when the order n increases in value much beyond the maximum value of the argument ξ . The inclusion of larger order terms introduces non-oscillating components into the displacement field, which, in addition to the evanescent components already included, results in numerical ill-conditioning. As a rule of thumb, the significant contributions to the approximate representation of the wavefield can be obtained from the lower order functions n ∈ [0, 2ξmax ]. The inclusion of higher order functions is not required — in fact, their inclusion does more harm than good by introducing numerical ill-conditioning into the final equations (Lande 2005). The displacement field of a power absorbing plate may now be obtained from Eq (1) by: (i) approximately representing the field as a superposition of incoming and outgoing cylindrical propagating waves; and (ii) retaining only the incoming propagating wavefield. Hence, w(kB , r, ϕ ) ≈ (1) 2(k r) (1) a0 H0 (kB r) + ∑n=1B max {an exp(inϕ ) + bn exp(−inϕ )} Hn (kB r) + . . . , 2(k r) c0 I0 (kB r) + ∑n=1B max {cn exp(inϕ ) + dn exp(−inϕ )} In (kB r),

(8)

where an , bn , cn and dn are the independent complex amplitudes to be determined approximately from the boundary conditions. It is to be expected that a power absorbing plate may be modelled with sufficient accuracy using cylindrical waves of orders n up to 2(kB r)max , a value above which the inclusion of higher order terms will only add relatively negligible resistive dynamic stiffness in addition to local reactive dynamic stiffness at the plate boundary. (2) Although Hn (iξ ) is found to be proportional to In (ξ ) in the far field, i.e. for (2) large argument ξ (Abramowitz and Stegun 1972), Hn (iξ ) is singular at the origin and is therefore associated with direct power flow (Lande and Langley 2005); thus, it may not replace In (ξ ) in Eq (8).

3 Constructing the Power Absorbing Matrix 3.1 Discretization of the Power Integral, and Matrix Assembly The power integral (2) may be discretized as Pflex ≈

ω ω ∗T ω ∗T Im qb Γ b fb = Im q∗T Im qb Db qb , b Fb ≡ 2 2 2

(9)

where the integrals along each boundary edge have been approximated via quadrature: the quadrature weights are all collected in the discretization matrix Γ b , where the subscript b refers to the cylindrical wavefield element boundary coordinate system; the distributed forces and moments evaluated at each node (or collocation

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point) are collected in the vector fb and the transverse displacements and rotations in the vector qb ; the vector Fb = Γ b fb contains the nodal forces and moments and it also defines the power absorbing dynamic stiffness matrix Db , via Fb ≡ Db qb . The construction of Db follows by expressing the local degrees of freedom qb and associated distributed generalized forces fb in terms of the polar coordinate system and thus in terms of the cylindrical waves. In terms of the local polar coordinate system the required degrees of freedom are the quantities w, ∂ w/∂ r and ∂ w/∂ ϕ , all evaluated at the collocation points around the boundary. These quantities are all expressed in terms of Hankel functions with argument ξ = kB r, so that ∂ w/∂ r = kB ∂ w/∂ ξ . The quantities w, ∂ w/∂ ξ and ∂ w/∂ ϕ may be collected in a local polar coordinate degree of freedom vector pq , where the subscript q refers to the generalized displacement degrees of freedom, ⎧ ⎫ w(ξ ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂w ⎬ (ξ ) pq = ∂ ξ = Hc , ⎪ ⎪ ⎪ ⎪ ∂ w ⎪ ⎪ ⎩ (ξ ) ⎭ ∂ϕ where

⎡

Hw ⎢ ∂ Hw ⎢ H = ⎢ ∂ξ ⎣ ∂H w ∂ϕ

Iw ∂ Iw ∂ξ ∂ Iw ∂ϕ

⎤ ⎥ ⎥ ⎥, ⎦

(1) (1) Hw = H(1) , 0 (ξ ) Hn (ξ ) ◦ cos(ϕ n) Hn (ξ ) ◦ sin(ϕ n)

(10)

(11)

(1) and where Iw is obtained simply by replacing Hn (ξ ) by In (ξ ) in Eq (11); the other matrix blocks of H may be written in similar but more complicated forms (Lande 2005). Here, the column vectors ξ and ϕ contain the variables ξ j = kB r j and ϕ j evaluated at the collocation points; thus, w(ξ ) denotes the column vector obtained by evaluating the cylindrical wave displacement field at all collocation points for a given wavenumber kB . Likewise, ∂ w(ξ )/∂ ξ denotes the column vector obtained by evaluating 1/kB ∂ w/∂ r at all collocation points for the same wavenumber. The matrix H thus represents the collection of all the included cylindrical wave functions, and the column vector c contains the associated wave amplitudes. In the above, n is a row vector from 1 to nmax , where nmax is the maximum wave function order; the (1) (1) notation Hn (ξ ) thus refers to a matrix, the rs-th element of which is Hn(s) [ξ (r)], where n(s) and ξ (r) denote the s-th and r-th element of n and ξ , respectively. The notation (. . . ) ◦ (. . . ) refers to the Hadamard product of two matrices (Horn and Johnson 1985), involving element-wise multiplication. From Eqs (6) and (7), the required derivatives of w(kB , r, ϕ ) with respect to ξ = kB r and ϕ are of orders up to and including the 3rd order. In terms of the local polar coordinate system the required quantities relating to the distributed loads are all the nine partial derivatives ∂ w/∂ ξ , ∂ w/∂ ϕ , ∂ 2 w/∂ ξ 2 , ∂ 2 w/∂ ξ ∂ ϕ , ∂ 2 w/∂ ϕ 2 , ∂ 3 w/∂ ξ 3 , ∂ 3 w/∂ ξ 2 ∂ ϕ , ∂ 3 w/∂ ξ ∂ ϕ 2 and ∂ 3 w/∂ ϕ 3 , all evaluated at the collocation points around the boundary. These quantities may be collected in the local polar coordinate distributed force vector p f , where the subscript f refers to generalized forces, (12) p f = Gc,

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where G is similar to H in form but contains the block matrices ∂ Hw /∂ ξ , ∂ Hw /∂ ϕ , ∂ 2 Hw /∂ ξ 2 , ∂ 2 Hw /∂ ξ ∂ ϕ , etc (and similarly for Iw ), corresponding to the aforementioned nine partial derivatives. The displacement vectors qb and pq are related through (Lande 2005) (0)

qb = Kq Tq pq ,

(13) (0)

where Kq is a diagonal matrix involving simply the wavenumber kB , and Tq is a transformation matrix which involves the cos(ψ ) and 1/ξ sin(ψ ) terms required for the evaluation of the edge rotations, −∂ w/∂ ν ; see Eq (5). Similarly, the distributed load vectors fb and p f are related through fb = T f p f ,

(14)

where the transformation matrix T f involves all the terms associated with the coefficients of the partial derivatives in Eqs (6) and (7). Given Eqs (10) and (13), the displacement vector qb may be written as (0)

qb = Kq (Tq H)c = Kq Aq c .

(15)

Similarly, from Eqs (9), (12) and (14), the generalized force vector Fb follows as Fb = Γ b fb = Γ b (T f G)c = Γ b A f c .

(16)

Since Fb = Db qb , the power absorbing dynamic stiffness matrix Db now follows from Eqs (15) and (16) by collocation as Db = Γ b (A f C)(Aq C)+ K−1 q ,

(17)

where the plus sign in superscript denotes the pseudo-inverse (Strang 1976), obtained by Singular Value Decomposition (SVD), and where a diagonal scaling matrix C has been included in order to reduce the level of numerical ill-conditioning in Aq . Hence, the SVD is applied to (Aq C) rather than to Aq .

3.2 Numerical Issues The major obstacle associated with the development of the cylindrical wavefield element dynamic stiffness matrix is the ill-conditioning of the matrix H and thus of Aq . As may be appreciated by the large order form of the Hankel functions and of the modified Bessel functions, there are columns of both H and G which will contain numbers of very large and of very small magnitude. Even with the use of column scaling, the resulting matrices will be ill-conditioned, and even more so if too many cylindrical wave functions are included. For this reason, independently of the number of collocation points, there should be a limit to the number of cylin-

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drical wave functions included, and it turns out that the rule of thumb referred to previously, which was based on physical considerations, does provide a satisfactory requirement. Further, to ensure convergence, there needs to be a sufficient number of collocation points along the boundary. As another rule of thumb, it has been found (Lande 2005) that satisfactory power convergence is obtained by using at least eight collocation points per unit half wavelength of a plane bending wave propagating in the interior of the plate. From these considerations, it is clear that the matrix Aq will be highly rectangular with a larger number of rows than of columns; hence the need for the pseudoinverse. However, provided the number of cylindrical wave functions is limited according to the aforementioned rule of thumb, there is no need to consider such details as the optimal choice of a singular value tolerance level for the SVD; the computations are rather straightforward.

4 Numerical Results 4.1 A Simple System A simple application of the power absorbing dynamic stiffness matrix obtained herein involves a single aluminium plate surrounded by a stiff aluminium beam frame, both with Young’s modulus E = 70 GPa, density ρ = 2700 kg/m3 , Poisson’s ratio μ = 0.3, and a constant loss factor of η = 0.02, associated with a complex Young’s modulus, E(1 + iη ). The plate is 1 mm thick and has dimensions 1×1.234 m2 . The four beams of the frame are attached to the plate along their neutral axes and have a rectangular cross-section, with a depth of 20 mm and a width of 10 mm. The system is vertically unconstrained, although all in-plane motion of the plate and thus of the beams has been fully constrained; hence, the plate experiences flexural motion only, and the beams experience flexural motion as well as torsional motion. A transverse unit load was applied to one of the 1 m long beams at the driving point located 0.3 m from one end. A second response measurement point was located on the same beam but 0.38 m from the other end. The driving point mobility modulus and the cross mobility modulus associated with the measurement point were then calculated.

4.2 System Randomization and Subsystem Response Prediction The spatial mass distribution of the nominally homogenous plate was randomized, and the Hybrid equations (Shorter and Langley 2005) were used to predict the square root of the ensemble mean of the modulus squared of the driving point mobility

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and cross mobility, as well as the ensemble average plate energy, here represented by the average plate kinetic energy multiplied by two, over a frequency range of 10–300 Hz. For comparison, the same results were obtained from a Monte Carlo simulation, in which the spatial mass distribution of the plate was randomized by adding a total of 20% of the nominal plate mass, distributed over the interior of the plate by 10 individual point masses randomly located on the plate. For every individual ensemble realization, the location of each of the 10 point masses was taken from an ensemble of coordinates with a statistically uniform distribution over the interior of the plate. The ensemble size was 300 samples. Regarding the estimation of the plate energy, this was done in two ways: (i) via the total time averaged kinetic energy of the plate, 1/4ω 2 q∗T p M p q p (where M p is the mass matrix of the plate and q p the corresponding degrees of freedom); and (ii) by calculating the displacement modulus squared at some randomly pre-defined location in the interior of the plate and thus estimating the total kinetic energy from the kinetic energy density evaluated at that point, assuming a perfectly diffuse field in the plate and therefore a spatially homogeneous kinetic energy density. In the cases considered herein, the collocation point resolution was set to a fixed value so that it corresponded to 8 points per half plane wavelength in the plate at the highest frequency, in this case 300 Hz.

4.3 Results Figures 1 and 2 show the cloud of results from the 300 samples of the Monte Carlo computation of the driving point mobility (Fig. 1) and cross mobility (Fig. 2) for the coupled system (loss factor η = 0.02), compared with the Hybrid prediction using the power absorbing dynamic stiffness matrix obtained by the cylindrical wavefield approach, indicated by the continuous curve. The ensemble average result is shown by the dashed-dotted curve. The dashed curve is the result for the heavily damped and uncoupled frame only (loss factor η = 0.2). From fuzzy structure theory (Soize 1986, 1993), it is to be expected that the ensemble average response of the frame, when coupled to the plate, should be similar to the result of the uncoupled but heavily damped frame. The results here confirm, as expected from fuzzy structure theory, that the average effect of the uncertain plate (the adjunct substructure) on the response of the frame (the master structure) is to add significant damping but also a little modal inertia. The Hybrid Method, with the power absorbing matrices obtained herein, is able to estimate the ‘right’ amount of added damping and inertia as sensed by the frame (although the estimate is not perfect). Discrepancies are present at frequencies below 50 Hz, however, the response in the plate at these low frequencies is not expected to be sufficiently diffuse across the ensemble, and so the hybrid equations are not expected to be valid at these frequencies. Figure 3 shows the results for the plate energy, which confirm that the power absorbing dynamic stiffness matrix obtained herein does provide a satisfactory estimate of the power injected into the plate subsystem via the excited frame. It is

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encouraging that the prediction includes the “boosts” of additional energy response around the frequencies of some of the uncoupled frame modes (not shown here but which appear at the frequencies 144 Hz, 216 Hz and 256 Hz; Lande 2005).

5 Conclusions A key ingredient in the Hybrid deterministic-statistical approach to the vibroacoustic analysis of complex uncertain structures (Shorter and Langley 2005) is the power absorbing dynamic stiffness matrix, required as a link between two sets of equations which couple FEA with SEA. The derivation of such a matrix associated with the flexural motion of a thin, flat, homogeneous plate has been presented here. The derivation is based on the use of cylindrical waves, and the displacement field of a power absorbing system within a closed boundary is obtained by excluding the cylindrical waves propagating outwards towards the boundary. Numerical results have indicated that the use of the cylindrical wave formulation to obtain the power absorbing dynamic stiffness matrix provides satisfactory predictions for simple built-up systems involving isotropic flat plates. Further study is required to investigate the performance of the method for more complicated systems, especially those involving irregular plate geometries and curved boundaries. The potential strength of the method is that there should in principle be no difficulties in dealing with curved boundaries of flat plates, though this remains to be demonstrated in future work.

Fig. 1: Monte Carlo (MC) simulation results for the driving point mobility, compared with the Hybrid prediction using the power absorbing dynamic stiffness matrix obtained by the cylindrical wavefield approach. Scatter: results from 300 MC samples. Dashed-dotted line: MC ensemble average. Dashed line: result for heavily damped uncoupled frame only, with loss factor η = 0.2. Continuous line: Hybrid prediction (of the root-mean-square), with loss factor η = 0.02 for both the plate and the frame

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Fig. 2: Monte Carlo (MC) simulation results for the cross mobility, compared with the Hybrid prediction (of the root-mean-square) using the power absorbing dynamic stiffness matrix obtained by the cylindrical wavefield approach. Key as in Fig. 1

Fig. 3: The plate energy. Scatter: 300 MC samples; each individual curve is twice the total kinetic energy of the plate. Dashed-dotted line: MC ensemble average. Dashed line: estimate of the MC ensemble average based on the kinetic energy density, obtained by evaluating the displacement modulus squared at a pre-defined random point in the interior of the plate during the simulation, assuming a diffuse reverberant field and therefore homogenous kinetic energy density over the plate. Continuous line: Hybrid prediction using the cylindrical wavefield approach

Acknowledgements Financial support received by the first author from Universities UK for an Overseas Research Studentship Award during 2002–2004, and from the Research Council of Norway for a Doctoral Scholarship during 2002–2005, is gratefully acknowledged. Both authors would also like to acknowledge ESI US R&D for continuous interest and financial support during the course of the work.

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References 1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover Publications, New York (1972) 2. Bobrovnitskii, Yu.I.: On the energy flow in evanescent waves. J. Sound Vib. 152, 175–176 (1992) 3. Bobrovnitskii, Yu.I.: Calculation of the power flow in flexural waves on thin plates. J. Sound Vib. 194, 103–106 (1996) 4. Cotoni, V., Shorter, P., Langley, R.: Numerical and experimental validation of a hybrid finite element-statistical energy analysis method. J. Acoust. Soc. Am. 122, 259–270 (2007) 5. Cremer, L., Heckl, M.: Structure-Borne Sound. Springer-Verlag, Berlin (1973) 6. Fahy, F.J.: Sound and Structural Vibration. Academic Press, London (1987) 7. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985) 8. Lande, R.H.: The Development of Power Absorbing Matrices for the Vibration Analysis of Complex Uncertain Structures. PhD-thesis, Department of Engineering, University of Cambridge (2005) 9. Lande, R.H., Langley, R.S.: The energetics of cylindrical bending waves in a thin plate. J. Sound Vib. 279, 513–518 (2005) 10. Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover Publications, New York (1944) 11. Lyon, R.H.: Statistical Energy Analysis of Dynamical Systems: Theory and Applications. MIT Press, Massachusetts (1975) 12. Rayleigh (J.W.S., Lord): The Theory of Sound. 2nd edn (two volumes). Dover Publications, New York (1945) 13. Shorter, P.J., Langley, R.S.: Vibro-acoustic analysis of complex systems. J. Sound Vib. 288, 669–699 (2005) 14. Soize, C.: Probabilistic structural modelling in linear dynamic analysis of complex mechanical systems I: theoretical elements. Rech. Aerosp. 5, 23–48 (1986) 15. Soize, C.: A model and numerical method in the medium frequency range for vibroacoustic predictions using the theory of structural fuzzy. J. Acoust. Soc. Am. 94, 849–865 (1993) 16. Strang, G.: Linear Algebra and its Applications. Academic Press, New York (1976)

The Energy Finite Element Method NoiseFEM Christian Cabos and Hermann G. Matthies

Abstract The most costly part in the prediction of structure borne sound for a complex vehicle such as a ship lies in the build-up of the analysis model. Since often a finite element model is available from other structural mechanical analyses, NoiseFEM, the approach described in this paper, uses such a model as a starting point for an energy flow analysis. The methodology focuses on stiffened shell structures and combines ideas from diffusive energy propagation methods and statistical energy analysis. Modifications are introduced to simplify the direct re-use of finite element models. Since 1997, NoiseFEM has been applied and validated in numerous shipbuilding projects.

1 Introduction We describe the propagation of structure borne noise in the form of power-flow equation, a process in some ways similar to statistical energy analysis (SEA), see [2] for an overview of power flow methods and their relationsship to SEA. Our point of departure is the formulation of elliptic differential equations which describe the steady state conditions under structure borne noise excitation. As in other similar approaches this is arrived at with a kind of averaging procedure. We pay particular attention at the question of how to couple different subsystems in such a way that the resulting equations become independent of arbitrarily chosen subdivisons of a system. In this way the coupling coefficient becomes consistent, agreeing with the well-known SEA coupling coefficient in the weak coupling limit.

Christian Cabos Germanischer Lloyd, Hamburg, e-mail: [email protected] Hermann G. Matthies Technical University Braunschweig, e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 13, © Springer Science+Business Media B.V. 2011

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1.1 Motivation The main motivation for the development of the prediction method GL NoiseFEM lies in the search for an effective procedure to analyse the structure borne sound in a ship structure. Design times in ship-building are significantly shorter than in the car or aircraft industries. Typically, designs are only used for one ship or a small series in the order of up to ten ships. Within a few months, the construction has to be verified and optimised with respect to several disciplines such as strength, seakeeping, vibration and noise. For strength and vibration analysis, the finite element method (FEM) is the standard methodology. Build-up of the model for this purpose is laborious and — due to the short design times — time critical. Modelling work can amount to up to two person months, often interfered with frequent design changes. The analysis results are needed in time by the yard to order steel plates and stiffeners. Both from a cost perspective and because of the tight time schedule, the analysis model should be built up such that it can be used for several analysis disciplines, see [4]. The size of a ship structure and its relation to the length of relevant waves in the structure is such that it is at present not feasible to compute the propagation of sound througout the whole ship structure using a dynamic FEM analysis like for low frequency vibrations. The FE model would have to be very fine in relation to the structure dimensions in order to resolve the wavelength of the sound waves, and thus would have a prohibitive number of degrees of freedom. Incidentally, it would also not allow the re-use of the finite element model used for strength and vibration analysis. Therefore a more approximate procedure is needed, and here we look at the way the sound energy spreads in the ship structure. When using FE Models for an SEA analysis, the standard approach would be to regard every finite element as an SEA subsystem. This directly leads to several practical problems and possible violations of SEA model requirements because • finite element models do not typically contain sufficient model information (e.g. stiffener dimensions) to correctly compute SEA quantities such as the coupling loss factor, • finite elements typically are not large enough to yield an SEA subsystem with sufficient modal density, and • subsystems derived from finite elements do not fulfil the constraints on coupling strength between subsystems. An alternative would be to reconstruct proper, in general coarser, SEA subsystems from several finite elements. Doing so is again laborious since an automatic procedure would require a high frequency vibration analysis. Therefore, this approach is not followed here. Rather, energy flow equations and SEA coupling relations are combined in a way to remove or reduce the above problems.

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1.2 Literature Overview Whereas in the low-frequency range (i.e. vibrations) the formulation and practical solution of the dynamic equations poses no problem — at least in principle — for higher frequencies and subsequently shorter wave-lengths one runs into the problem alluded to above of very small element subdivisions in relation to the structure, which make the solution — at least presently — of the discretised wave equation infeasible. One approach in the mid-frequency range is the variational theory of complex rays (VTCR), see [26, 18, 27]. Here homogeneous parts of the structure are considered with a known wave-like solution “ansatz”. Then conditions have to be formulated — in a variational way — to allow those parts to be joined together. It is therefore akin to a two-scale approach. Methods which extend to higher frequencies are based on power-flow considerations. The use of SEA on ship structures has some history, see [25, 31, 16, 15], and [13]. Energy finite element methods have e.g. been appplied to ship structures in [11, 5] and [29].

2 Components of NoiseFEM In NoiseFEM, different methods modelling diffusive energy transport are combined. Namely, Statistical Energy Analysis is used to compute the energy flow between subsystems. Here, finite elements from a global FE model of a ship are used as subsystems. To take care of problems with strong coupling, a correction for the coupling matrix is introduced. Plate strip theory is employed to find the transmission coefficients at line couplings with an attached stiffener. SEA subsystems are characterized by a constant value of modal energy. This does not allow finding the direction of energy flow within a subsystem. For that reason, within subsystems the Power Flow Finite Element Method (PFFEM, [23]) is applied. To take care of stiffened subsystem, the sound conductivity used in PFFEM is generalized to the case of coupled wavetypes. The named approaches are described in the following sections.

3 Power Flow Between Structural Elements Statistical Energy Analysis can predict the flow of energy through complex structures in case they can be modelled through weakly coupled subsystems with a sufficiently high modal overlap. In SEA, the coupling loss factor ηi j relates the difference in total subsystem energies Ei and E j to the net power flow between these subsystems

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Pi j,net = ω (ηi j Ei − η ji E j ).

(1)

Here, ω denotes a center frequency in the band of interest. Together with the internal power loss Pid = ωηi,d Ei within a subsystem i, a global system of equations can be assembled. Given the input power Pi into each subsystem this system is then used to solve for the unknown Ei . E.g. for a 2-subsystem model the SEA equations read d c E P P P (Ad + Ac ) 1 = 1d + 1c = 1 , (2) E2 P2 P2 P2 where Pic denotes the power leaving subsystem i due to coupling. The diagonal matrix Ad describes the effect of internal damping (and energy loss due to radiation, if acoustic cavities are not contained in the model) and the coupling loss factors ηi j are used to determine the coupling matrix Ac : c η12 −η21 E1 P1 Ac = with Ac = ω . (3) E2 P2c −η12 η21 A correction to this coupling relation is sought in the following sections to account for strong coupling effects. The sound temperature Ti = Ei /Ni is defined as the energy per mode of the subsystem. Using the sound temperatures as primary variables in (2) instead of the subsystem energies leads to a symmetric system matrix, see [21].

3.1 Transmission Coefficients Transmission coefficients, denoted here by the symbol τ , describe the ratio of transmitted to incident power at a junction. For weakly coupled subsystems, they are typically related easily to the coupling loss factors η . E.g. for the line junction between two 2D subsystems the coupling loss factor is ([21])

η12 =

cg,1 b1 τ12 , πω A1

(4)

where A1 is the area of subsystem one, b1 the coupling length, and, following the wave approach to SEA, τ12 is the transmission coefficient averaged over the incidence angle of the planar waves directed towards the junction. The wave approach has the advantage that transmission coefficients can be computed with moderate effort for many junction types. The prediction quality of an SEA analysis depends largely on the precision of the coupling loss factors and thereby also on the transmission coefficients. In NoiseFEM, the wave approach is used to compute the transmission coefficients. Therefore, the following assumptions common to SEA are made:

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• The subsystem response can be decomposed into plane wave components propagating in different directions, and the power and energies of all waves can be summed up incoherently. • Proportionality holds between the total energy of plane waves and their total power. Based on these assumptions the response of each subsystem is now split into • the waves incident to the junction with total energy E + and power flow P+ orthogonal to the junction and • the waves leaving the junction with total energy E − and power flow P− orthogonal to the junction. Fig. 1 shows the notation for the case of two subsystems. We consider the case of n subsystems coupled at a common junction. Upper case letters without subscripts denote vectors of the subscripted quantities. In matrix form, the definition of τi j yields (5) Pj− = ∑ τi j Pi+ or P− = τ P+ with τ = (τi j ) . i

Fig. 1: Notation for deriving the coupling matrix In shipbuilding, stiffeners and girders have a major influence on the attenuation of structure borne sound. Their effect has therefore to be considered accurately, when computing the transmission coefficient of a junction. Commonly, commercially available SEA codes use the wave approach applied to semi-infinite systems for the computation of the transmission coefficients. Major inaccuracies can arise, if stiffeners or girders attached to the junction are modelled as beam elements, i.e. by only considering their bending and torsional stiffness and mass properties. If instead, these elements are modelled as plate strips (see [20, 10]), waves propagating on the web and the flange can be taken account of. Figure 2 shows the difference in bending wave transmission coefficient (averaged over angle of incidence) for the stiffened junction between two semi-infinite plates. Because of the notable discrepencies, plate strip theory is used to compute transmission coefficients in NoiseFEM.

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Fig. 2: Differing bending wave transmission coefficients when modelling the shown T-girder as a beam or as plate strips (on the right). A cross section of the girder attached to the plates is shown on the left

3.2 The Coupling Matrix Once the transmission coefficients between the subsystems coupled at a junction are known, the standard SEA way of forming the coupling matrix Ac is given by (3) in conjunction with e.g. (4) for line junctions. In general terms, the coupling matrix Ac describes the relation between the vector of subsystem energies E = (E1 , . . . , En ) and the net power Pc = (P1c , . . . , Pnc ) leaving the subsystems. In this section we are looking for alternative ways to compute the matrix Ac from the transmission coefficients. The objective is to correct Ac for the case of strong coupling (i.e. large τ ) instead of correcting the relationship between the individual coupling loss factors ηi j and transmission coefficients τi j . The reason is that equation (2), not (1), is used to compute the SEA response to external power sources. In particular for junctions with more than two subsystems a correction of the single matrix entries appears artificial as compared to a consistent correction of the matrix. As in (3) we are looking for a relation between E and Pc . Ac is derived on the basis of the transmission coefficient τ , see also [7]. In NoiseFEM, each wave type is regarded as a separate subsystem. Therefore, two plates joint at a common edge would e.g. lead to six coupled subsystems: bending, longitudinal and shear waves on each plate.

3.2.1 Strong Coupling Regime The matrix of transmission coefficients τ has the reflection coefficients on its diagonal. Therefore the sought net power loss of the subsystems is

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Pc = P+ − P− = (I − τ )P+ ,

(6)

where I denotes the identity matrix. If C is the diagonal matrix containing the proportionality factors between energy and power flow orthogonal to the junction, then e.g. CE − = P− and with (5) we have CE = P+ + P− = (I + τ )P+ .

(7)

Combining (7) with (6) results in Pc = Ac E

with

Ac = (I − τ )(I + τ )−1C.

(8)

This form of the coupling matrix is used in NoiseFEM, it has also been proposed in [22]. The SEA coupling relation can be altered accordingly.

3.2.2 Special Cases A relation between ηi j and τi j can be derived from (8) in special cases. E.g. in the case of two subsystems with ε := τi j = τ ji we have 1 ε 1−ε ε 1 −1 τ= , so Ac = C. (9) ε 1−ε 2 1 − ε −1 1 τ

Comparing this result with (3) yields ωηi j = 12 1−iτj i j Ci . This formula is only a consequence of (8) for this case and is not used directly in implementation. For weak coupling, i.e. small ε , the denominator in (9) is near one and Ac is the standard SEA coupling matrix. Now regard the special case P2+ = 0, i.e. incident waves coming from only one direction. Then from (8) and (7) P1c =

1 ε ε (2 − 2ε )C1 E1+ = C1 E1 . 2 1−ε 2−ε

This is in agreement with [21] and [28]. It is not used in NoiseFEM, because P2+ = 0 appears to be in conflict with the assumption of a reverberant sound field. Both examples show, that (8) leads to simple correction formulas for the transmission coefficients for strong coupling in particular cases. Relation (8) in turn covers the more general case of an arbitrary transmission matrix τ .

3.2.3 Mesh Independence When modelling a ship with finite elements, plates subdivided into several elements typically occur in many places. This is in particular due to the compatibility requirements for the common edge of adjacent finite elements. In this case, the coupling

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between the neighbouring elements is perfect, i.e. the transmission coefficient τi j is one. The case of two subsystems is then covered by (9) with ε = 1, i.e. the entries of the coupling matrix become infinitely large. This corresponds to the fact that even for nonzero net power flow between two perfectly coupled two subsystems, the difference in subsystem energies should be zero, see (1). As one single structural element can also be thought — in a “Gedankenexperiment” — of as consisting of say two parts cut apart arbitrarily, the assemblage procedure — this is what happens once we discretise with finite elements — has to produce the same answer no matter how the subdivision is. For a consistent derivation of the sound conductivity, the transmission coefficient has to be defined in a certain unique way in order to make this a consistent assumption in the strong coupling limit — which is what we have in the just described situation. With this kind of definition, the derived equations are free of subjective judgements of what is a subsystem. The final equations always come out the same.

4 Diffusive Energy Transport 4.1 Homogeneous Structural Elements Under certain conditions the flow of sound energy in homogenous damped structural systems such as beams and plates can be described through a partial differential equation similar to the equation for heat propagation. This has also been termed vibrational conductivity approach ([19]), for a marine application see e.g. [29]. For the conditions under which this approach is valid see e.g. [9] or [19] and [14]. In particular, it has been found that the approach is justified in the case of a reverberant sound field, when many wave reflections occur. The following analysis (see also [6]) focuses on plates as the dominant structural system type in ships. It can be generalized to beams. In the practical cases of stiffened shell structures the above assumptions can be regarded to be valid. Although in this case the frequency averaged sound energy density within one subsystem is rather homogenous, it is not constant but rather the gradient of the energy density bears valuable information, namely the direction of energy flow, c2g ∇e, (10) I=− ηd ω see [19]. Here, cg is the group velocity and e is the frequency averaged energy density for the wave type under consideration, ηd is the damping loss factor and I is the intensity vector. Since the coupling relations between subsystems become symmetric if the sound temperature instead of the energy densities are used as primary variables, equation (10) is divided by n = N/A, the number of modes N per area A of the plate

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in the frequency band under consideration. Now, t = e/n denotes the local sound temperature, derived from the energy density. This leads to I = −Λ ∇t

with Λ =

Δω , aπηd

(11)

where Δ ω is the width of the frequency band and a = 1 for bending waves and a = 2 for inplane waves in plates. From the power balance pin = pd + ∇ · I where pin and pd denote the input power per area and the power loss per area due to damping, resp., we can now derive the elliptic equation for the distribution of energy in a lightly damped plate. Assuming that the time averaged kinetic and potential energy densities agree, the dissipated power is given by pd = ηd ω e, and one arrives at pin = − ∇ ·

c2g ∇e + ηd ω e. ηd ω

(12)

Changing variables again to sound temperatures, we have pin = − ∇ · Λ ∇t + cηd ω t,

(13)

where c is the number of modes per area of the subsystem in the frequency band. The solution of the propagation equation for built-up structures can be achieved with finite element procedures leading to energy finite element methods, see [3, 29].

4.2 Stiffened Subsystems For the derivation of the vibrational conductivity equation for a stiffened subsystem assume that the stiffeners are positioned in parallel at positions x1 to xn−1 on the plate, see Fig. 3. We are now looking for a generalization of equation (13) to the case of three coupled temperature variables, i.e. ∇Λ0 ∇t + γ0 t = 0

with t = (tb ,t ,ts ) ,

(14)

so t is a three-dimensional vector with components for bending, longitudinal and shear waves. In this case Λ0 is a (3 × 3)-matrix. For a homogeneous system, Λ0 is diagonal, where as in the case of a plate with unsymmetric stiffeners, the wave types couple and Λ0 should become a full matrix. The technique for deriving (14) is described in [17]. For the case of evenly distributed disconituities and one wave type, see also [1] and [12]. In case of the configuration shown in Fig. 3, equation (13) is valid in each of the homogenuous plate strips, whereas the coupling condition at the discontinuities can be written as

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Fig. 3: Nomenclature for the derivation of the homogenized equations

Pxi− ,xi+

= −Bi t(xi+ ) − t(xi− )

i = 1 . . . n − 1,

with the coupling matrix Bi . When homogenizing a second order differential equation, the zero order term can be averaged (see [17]), i.e. γ0 = γ , where γ = cηd ω . The elements of the homogenized matrix Λ0 are then found by solving the auxiliary problem

ξ Λ0 ξ = inf Π (u, ξ ), u

where ξ is a constant vector chosen to find the appropriate element of Λ0 . The functional Π is in this case given by

Π=

n

∑

xi

i=1 xi−1

n−1

(ξ + ∇u) Λ (ξ + ∇u)dx + ∑

i=1

ui− ui+

u L B i− ui+

and ui− := u(xi− ), ui+ := u(xi+ ). The infimum is taken over all admissible sound temperature distributions u which are periodic on the interval [x0 , xn ].

5 Combining transport and coupling equations The SEA equations are now combined with the energy transport equations within subsystems. In the case of constant energy density e = E/A in a subsystem, the e agree. In NoiseFEM, the more general case of a quantities T = E/N and t = N/A non-constant energy distribution in a subsystem is assumed. To combine with (13), the coupling relation (2) is enforced between the values of t at the edges of the connected subsystems (15) pc = (I − τ )(I + τ )−1CNt.

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In (2), the total energy loss due to damping in a subsystem is described via the matrix Ad . This energy loss is on the other hand already included in (13) for the localized quantities. For that reason, the term Ad is omitted from the coupling relations. In fact, by integrating cηd ω t over the area of the subsystem, one can see that it is equivalent to the total loss ωηd E for the case of constant energy distribution. To summarize, in NoiseFEM, the equations (15) and pin = − ∇ · Λ ∇t + cηd ω t,

(16)

are solved for the unknown sound temperatures t.

6 Discretization To solve (16), the spatial distribution of t is approximated by bilinear functions on the (four node or three node) surface elements. This leads to three unknown sound temperature values (bending, longitudinal, shear) at each vertex of each element. The coupling relation (15) relates the linear temperature distribution at the two adjoining edges, see Figure 4.

Fig. 4: Calculation of the global conductivity matrix from the FE model. The matrix (1) (1) Kh couples the dofs of element 1 (nodes 1,2,3,4). Kh is calculated from (16). The matrix KSEA couples all dofs at the structural joint (node 3,4,5,8). This matrix is calculated from (15)

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7 Validation of NoiseFEM with test structures The first step of validation of NoiseFEM was the comparison with detailed FE vibration analysis of test structures. Examples are a longitudinal box structure used in [24] and a stiffened double bottom test structure, see [7] and Figure 5. The advantage of a validation against vibration computations is that damping and source levels are known. The double bottom consists of 8 mm steel plates with 200 mm×16 mm stiffeners. The internal loss factor has been chosen as η = 0.03. The finite element mesh used for the NoiseFEM computations is shown on the left hand side of Figure 5. The same model geometry was used to also conduct an SEA analysis of the structure. Results of the fine mesh FE vibration computation were averaged over frequency of excitation and over the excitation point within the excited subplate (rain-on-theroof excitation in the subplate marked by thick black edges). The FE model and the results for the 200 Hz third octave band are shown on the right hand side in Figure 5. This model consists of ca. 167000 degrees of freedom and has 1850 eigenmodes below 230 Hz.

Fig. 5: Test structure for comparing NoiseFEM, SEA and vibration analysis. On the right hand side, the result of an averaged vibration calculation in the 200 Hz third octave band is shown. Rain-on-the-roof excitation is applied at the plate marked with an arrow

The comparison of the SEA, NoiseFEM and vibration results are shown in Figure 6. For the vibration analysis, the structure was simply supported at the outer edges of upper and lower deck. The graph shows that SEA results tend to predict too high attenuation away from the excitation point. This has also been observed in the other validation examples. A deckhouse mockup structure has been assembled for the validation with a more complex built-up structure. This structure has been used to compare both against

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Fig. 6: Frequency averaged out-of-plane velocity level of the double bottom test structure along the diagonal cut plane shown in Figure 5, 200 Hz third octave band. Black lines show the cut structure, one tick mark refers to 1 m. Colored lines show computation results above the corresponding cut structural part, SEA (blue), NoiseFEM (red), frequency averaged vibration analysis (green), frequency and spaced averaged vibration analysis (yellow), one tick mark corresponds to 5 dB

detailed vibration analysis and mesurements. For a more detailed account of this validation see [30].

8 Application of NoiseFEM Germanischer Lloyd has conducted numerous consultation projects focussing on the prediction of noise propagation in ships using NoiseFEM since 1997, see e.g. [11, 5, 8]. The projects have been conducted mainly on behalf of shipyards with the purpose of optimizing the structural design or planning of sound isolation measures during the design stage of the ship. The handled ship types include mega yachts, cruise liners, ferries, container ships and naval vessels. The typical procedure involves the modelling of the ship structure according to the drawings submitted by the yard and the application of sound sources derived from input given by the manufacturers of the main and auxiliary engine, propeller, high velocity air condition devices, seawater pumps, exhaust gas system, etc.. The predicition of structure borne noise velocity levels is followed by computing sound intensities mainly for identifying the major transfer paths of sound energy. Sound pressure levels in receiving rooms are derived from the structure borne noise levels with a semi-empirical approach representing the ship isolation. A major practical

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benefit arises from the possibility to assess the relative participation of the different noise sources. In most projects, measurements have been conducted after the ship had been built. NoiseFEM has then been validated both concerning sound velocity level predictions and applied source levels.

9 Conclusions A method for the prediction of structure borne sound in complex stiffened shell structure has been presented. It combines different approaches from the field of sound energy propagation. The chosen strong coupling correction makes the method robust with regard to the subdivision of subsystems appropriate for SEA. It is therefore suitable for energy flow analysis with the help of global ship FE models. The method has proven useful in many commercial consultation projects focussing on the prediction of structure borne sound propagation in ships in the design stage.

References 1. Belov, V.D., Rybak, S.A., Tartakovskii, B.D.: Propagation of vibrational energy in absorbing structures. Sov. Phys. Acoust. 23(2), 115–119 (1977) 2. Bernhard, R.J.: The family of EFA equations and their relationship to SEA. In: Proceedings of NOVEM. Lyon (2000) 3. Bouthier, O.M., Bernhard, R.J.: Simple Models of the Energetics of Transversely Vibrating Plates. Journal of Sound and Vibration 182(1), 149–164 (1995) 4. Cabos, C., Asmussen, I.: Integrating global strength, vibration and noise analyses for ships using one computational model. In: H. Mang, F. Rammerstorfer, J. Eberhardsteiner (eds.) Fifth World Congress on Computational Mechanics. Vienna (2002) 5. Cabos, C., Jokat, J.: Computation of structure-borne noise propagation in ship structures using Noise-FEM. In: O. Tan (ed.) Practical Design of Ships and Mobile Units, pp. 927–934. Elsevier (1998) 6. Cabos, C., Matthies, H.G.: A method for the prediction of structure-borne noise propagation in ships. In: Proceedings of the 6th international congress on sound and vibration, pp. 2349– 2356. Technical University of Denmark (1999) 7. Cabos, C., Matthies, H.G.: Prediction of Sound Propagation in Stiffened Shell Structures with an Energy Finite Element Method. In: Proceedings of the 17th International Congress on Acoustics. Rome (2001) 8. Cabos, C., Worms, C., Jokat, J.: Application of an Energy Finite Element Method to the Prediction of Structure Borne Sound Propagation in Ships. In: Proceedings of Internoise. The Hague (2001) 9. Carcaterra, A., Sestieri, A.: Energy Density Equations and Power Flow in Structures. Journal of Sound and Vibration 188(2), 269–282 (1995) 10. Craik, R.J.M., Bosmans, I., Cabos, C., Heron, K.H., Sarradj, E., Steel, J.A., Vermeir, G.: Structural transmission at line junctions: a benchmarking exercise. Journal of Sound and Vibration 272, 1086–1096 (2004)

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11. Flehmke, A., Jesse, A., Jokat, J., Schellin, T.E.: Hydrodynamics, structural optimization and noise control of a fast monohull ferry design. In: FAST. Sydney (1997) 12. Heckl, M.: Structure-borne Sound Propagation on Beams with Many Discontinuities. Acustica 81(5), 439–449 (1995) 13. Hynn¨a, P., Klinge, P., Vuoksinen, J.: Prediction of structure-borne sound transmission in large welded ship structures using SEA. Journal of sound and vibration 180(4), 583–607 (1995) 14. Ichchou, M.N., Jezequel, L.: Comments on simple models of the energy flow in vibrating membranes and on simple models of the energetics of transversely vibrating plates. Journal of Sound and Vibration 195(4), 679–685 (1996) 15. Iino, K., Honda, I.: Total Noise Prediction System for a Passenger Cruise Ship. In: 5th International Symposium on Practical Design of Ships and Mobile Units, pp. 1648–1661. Newcastle (1992) 16. Irie, Y., Nakamura, T.: Prediction of Structure Borne Sound Transmission Using Statistical Energy Analysis. Bulletin of the M.E.S.J. 13(2), 60–72 (1984) 17. Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer-Verlag, Berlin (1994) 18. Ladev`eze, P., Chevreuil, M.: A new computational method for transient dynamics including the low- and the medium-frequency ranges. Int J. for Numerical Methods in Engineering 64(4), 503–527 (2005) 19. Langley, R.S.: On the Vibrational Conductivity Approach to High Frequency Dynamics for Two-Dimensional Structural Components. Journal of Sound and Vibration 182(4), 637–657 (1995) 20. Langley, R.S., Heron, K.H.: Elastic wave transmission through plate/beam junctions. Journal of sound and vibration 143(2), 241–253 (1990) 21. Lyon, R.H., DeJong, R.G.: Theory and Applications of Statistical Energy Analysis. Butterworth Heinemann, Boston, Massachusetts (1995) 22. Moens, I., Vandepitte, D., Sas, P.: Vibro-acoustic Energy Flow Models Implemented by Finite Elements. In: Proceedings of ISMA 23 (1998) 23. Nefske, D.J., Sung, S.H.: Power Flow Finite Element Analysis of Dynamic Systems: Basic Theory and Application to Beams. Transactions of the ASME Journal of Vibration Acoustics Stress, and Reliability in Design 111, 94–100 (1989) 24. Nishino, H., Ohlrich, M.: Application of Wave Intensity Analysis for Predicting MidFrequency Vibration Transmission in Extended Plate Structures. In: Proceedings of Internoise. Den Haag (2001) 25. Plunt, J.: Methods for predicting noise levels in ships. Parts 1 and 2. Ph.D. thesis, Chalmers University of Technology, G¨oteborg (1980) 26. Riou, H., Ladev`eze, P., Rouch, P.: Extension of the variational theory of complex rays to shells for medium-frequency vibrations. J. of Sound and Vibration 272, 341–360 (2004) 27. Riou, H., Ladev`eze, P., Sourcis, B.: The multiscale VTCR approach applied to acoustics problems. J. of Computational Acoustics 16(4), 487–505 (2008) 28. Sarradj, E.: The uncertain relationship between transmission coefficient and coupling loss factor. In: Proceedings of NOVEM. Lyon (2000) 29. Vlahopoulos, N., Garza-Rios, L.O., Mollo, C.: Numerical Implementation, Validation, and Marine Applications of an Energy Finite Element Formulation. Journal of Ship Research 43(3), 143–156 (1999) 30. Wilken, M., Cabos, C., Semrau, S., Worms, C., Jokat, J.: Prediction and Measurement of Structure-borne Sound Propagation in a Full Scale Deckhouse-Mock-up. In: 9th International Symposium of Practical Design of Ships and other Floating Structures, pp. 653–659. L¨ubeck (2004) 31. Yoshikai, T., Hattori, K., Sato, T., Tashiro, S., Takahashi, K., Koshino, T.: Noise Prediction Program on Board Ships. J. S. N. A. Japan 150, 158–173 (1981)

Wave transport in complex vibro-acoustic structures in the high-frequency limit Gregor Tanner and Stefano Giani

Abstract Recently, a new approach towards determining the distribution of mechanical and acoustic wave energy in complex built-up structures has been proposed in [1]. The technique interpolates between Statistical Energy Analysis (SEA) and ray tracing containing both these methods as limiting cases. The method has its origin in studying solutions of wave equation with an underlying chaotic raydynamics — often referred to as wave chaos. Within the new theory — Dynamical Energy Analysis (DEA) — SEA is identified as a low resolution ray tracing algorithm and typical SEA assumptions can be quantified in terms of the properties of the ray dynamics. DEA calculations are made for a range of two-component model systems. The results are compared with both SEA results and FEM calculations using most advanced hp-adaptive discontinuous Galerkin methods.

1 Introduction Wave energy distributions in complex mechanical systems can often be modelled well by using a thermodynamical approach. The flow of wave energy is assumed to follow the gradient of the energy density just like heat energy flows along the temperature gradient [2]. To simplify the treatment, it is often suggested to partition the full system into subsystems and to assume that each subsystem is internally in ‘thermal’ equilibrium. Interactions between directly coupled subsystems can then be described in terms of coupling constants determined by the properties of the wave dynamics at subsystem boundaries alone. These ideas form the basis of Statistical Energy Analysis (SEA) [3, 4]. Gregor Tanner School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK, e-mail: [email protected] Stefano Gianim same address, e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 14, © Springer Science+Business Media B.V. 2011

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A method similar in spirit but very different in applications is the so-called ray tracing technique. The wave intensity distribution at a specific point r is determined here by summing over contributions from all ray paths starting at a source point r0 and reaching the receiver point r. It thus takes into account the full flow of ray trajectories. The method has found widespread applications in room acoustics [5] and seismology [6] as well as in determining radio wave field distributions in wireless communication [7] and in computer imagining software [8]. A discussion of ray tracing algorithms used for analysing the energy distribution in vibrating plates can be found in [9, 10]. Both methods — that is, SEA and ray tracing — are in fact complementary in many ways. Ray tracing can handle wave problems well, in which the effective number of reflections at walls or interfaces is relatively small. It gives estimates for the wave energy density with detailed spatial resolution and works for all types of geometries and interfaces. SEA can deal with complex structures carrying wave energy over many sub-elements including potentially a large number of reflections and scattering events, albeit at the cost of reduced resolution. In addition, the quality of SEA predictions may depend on the geometry of the subsystems, and error bounds are often hard to obtain. Ray tracing and SEA have in common that they predict mean values of the energy distribution and do not contain information about wave effects such as interference, diffraction or tunnelling. Both methods are thus expected to hold in the high frequency or small wave length limit and when the small scale fluctuations in the wave solutions are averaged out, for example, due to a finite resolution at the receiver. It will be shown here that SEA can be derived from a ray picture and is indeed a low resolution version of a ray tracing method. Ray tracing is thus superior to SEA, however, at a large computational overhead [10]. We introduce a new technique here, Dynamical Energy Analysis (DEA), which interpolates between SEA and a full ray tracing analysis. DEA enhances the range of applicability of standard SEA considerably and makes it possible to give quantitative estimates for the applicability of an SEA treatment. Methods similar in spirit have been discussed before in the context of wave chaos, see [11], and structural dynamics [12]. In particular Langley’s Wave Intensity Analysis (WIA) [13, 14] and Le Bot’s thermodynamical approach [15, 16, 17, 18] make use of the details of the underlying ray-flow dynamics. Our approach differs in as far as we consider multi-reflection in terms of linear operators directly and use a basis function representation of these operators leading to SEA-type equations. The work presented here is closely related to the IUTAM conference contributions by Cabos [19], Langley [20], Lande and Langley [21], Le Bot [22], Savin [23] and Shorter [24]. In particular the theoretical discussions in [20, 21, 22, 23] focusing on universality aspects building the foundations of SEA as well as the limits of the SEA approach are directly tied in with the results presented in this paper. The wealth of problems which can be treated with SEA, modified SEA or hybrid SEA/FEM methods have been reported in [24, 19]. The paper is structured as follows: after briefly discussing the approximations necessary to reduce wave transport equations to flow equations in Sec. 2, we intro-

Wave transport in complex vibro-acoustic structures in the high-frequency limit

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duce classical, linear flow operators in Sec. 3 and derive the DEA equations. We treat some simple two-domain problems in Sec. 4 both numerically using an advanced hp-adaptive discontinuous Galerkin method and using DEA.

2 Wave energy flow in terms of the Green function We assume that the system as a whole is characterised by a linear wave equation describing the overall wave dynamics including damping and radiation. We restrict the treatment here to stationary problems with continuous, monochromatic energy sources - generalising the results to the time domain with impulsive sources is straightforward. To simplify the notation, we will in the following assume scalar wave fields only. The general problem of determining the response of a system to external forcing with frequency ω at a source point r0 can after appropriate rescaling be reduced to solving the equation 2 ω − Hˆ G(r, r0 , ω ) = δ (r − r0 ) (1) where Hˆ is a scalar wave operator and G(r, r0 , ω ) represents the Green function. The wave energy density at a point r induced by a source at r0 is then given as

εr0 (r, ω ) ∝ ω 2 |G(r, r0 , ω )|2 .

(2)

The linear wave operator Hˆ can in a natural way be associated with a ray dynamics via the Eikonal approximation, see for example [1]. Using small wave length asymptotics, one can write the Green function G(r, r0 , ω ) in Eq. (1) as sum over all classical rays going from r0 to r for fixed “kinetic energy” of the hypothetical ray-particle. One obtains [27] G(r, r0 , ω ) ∝

∑

A j eiS j (ω ) .

(3)

j:r→r0

The action S j (ω ) is usually the dominant ω — dependent term and is for homogeneous media proportional to the length of the ray, that is, S = kL = ω L/c where L is the distance between source and receiver point and k is the wave number. The amplitudes A j contain contributions due to damping, conversion or transmission/reflection coefficients and geometrical factors, see [29, 28, 1, 30, 31, 32, 33, 5, 11] for details. The representation (3) has been considered in detail in quantum mechanics [27, 34, 35]. It is valid also for general wave equations in elasticity such as the biharmonic [33] and the Navier-Cauchy equation [28]; in the latter case, G becomes matrix valued. Note that the summation in Eq. (3) is typically over infinitely many terms where the number of contributing rays increase (in general) exponentially with the length of the trajectories included.

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The wave energy density, Eq. (2), can now be expressed as a double sum over classical trajectories, that is,

εr0 (r, ω ) ∝

∑

π

A j A j ei(S j −S j −(ν j −ν j ) 2 )

(4)

j, j :r0 →r

= ρ (r, r0 , ω ) + off-diagonal terms where the dominant terms (often denoted the diagonal terms) originate from pairs j = j , that is, ρ (r, r0 , ω ) = ∑ |A j |2 , (5) j:r0 →r

which gives rise to a smooth background signal. The off-diagonal contribution is responsible for the fluctuating part of the signal due to (overlapping) resonances. It has been shown in [1] that the background part, Eq. (5), is equivalent to a ray tracing treatment. That is, the smooth part of the energy density can be described in terms the flow of ‘fictitious’ non-interacting particles emerging from the source point r0 uniformly in all directions and propagating along ray trajectories. This makes it possible to relate wave energy transport with classical flow equations and thus thermodynamical concepts which are at the heart of an SEA treatment. This connection has also be highlighted in [17, 18]. In DEA [1], the classical flow is determined in terms of linear phase space operators. We adopt a boundary mapping approach here, that is, we describe the flow operators in terms of boundary operators which leads in a natural way to substructuring and SEA type equations as will be shown in the next section.

3 Linear phase space operators and DEA We give here a brief sketch of the derivation of the DEA flow equations; for details see [1]. The time dependence of a density of ray trajectories can be described in terms of a linear phase space operator L τ (X, X ), also called a Perron-Frobenius operator in dynamical system theory, that is

ρ (X, τ ) =

dX L τ (X, X )ρ0 (X );

with L τ (X, X ) = δ (X − ϕ τ (X )),

(6)

where X = (r, p) denotes the phase space coordinate with r position and p the momentum (or velocity) coordinate and ϕ τ (X) is the phase space flow giving the position of a particle after time τ starting at τ = 0 in r with velocity p. Furthermore, ρ0 denotes the initial ray density at time τ = 0. We consider in the following the situation of a source localised at a point r0 emitting waves continuously at a fixed frequency ω . Standard ray tracing techniques estimate the wave energy at a receiver point r by determining the density of rays starting in r0 and reaching r after some unspecified time. This can be written in the form

Wave transport in complex vibro-acoustic structures in the high-frequency limit

ρ (r, r0 , ω ) =

∞ 0

dτ

dp

dX w(X , τ ) L τ (X, X )ρ0 (X ; ω )

191

(7)

with initial density

ρ0 (X ; ω ) = δ (r − r0 )δ (ω 2 − H(X )),

(8)

ˆ It can be and H(X) is the Hamilton function related to the wave operator H. shown [1] that Eq. (7) is equivalent to the diagonal approximation, Eq. (5). We included here a weight function w(X, τ ), which may contain damping and reflection/transmission coefficients and we assume that w is multiplicative, that is, w(ϕ τ1 (X), τ2 )w(X, τ1 ) = w(X, τ1 + τ2 ),

(9)

which is fulfilled for (standard) absorption mechanism and reflection processes. In order to solve the stationary flow problem (7) we adopt a boundary mapping technique. That is, we map the ray density emanating continuously from the source (0) onto the boundary; the resulting boundary layer density ρB is equivalent to a source density on the boundary producing the same ray field in the interior as the original source field after one reflection, see Fig. 1 a). Densities on the boundary are mapped back onto the boundary by a boundary operator T (Xs , Xs ; ω ) = w(Xs )δ (Xs − φω (Xs )), where Xs = (s, ps ) represents coordinates on the boundary (where s parametrises the boundary and ps denotes the momentum components tangential to the boundary at s) and φω (Xs ) is the boundary map, see Fig. 1 b).

Fig. 1: a) Mapping from source distribution to boundary distribution; b) mapping under boundary operator T ; c) mapping from boundary distribution into interior The stationary density on the boundary induced by the initial boundary distribu(0) tion ρB (Xs , ω ) is then obtained using

ρB (ω ) =

∞

∑ T n (ω ) ρB

(0)

(ω ) = (1 − T (ω ))−1 ρB (ω ) , (0)

(10)

n=0

where T n contains trajectories undergoing n reflections at boundaries. The resulting density distribution on the boundary, ρ (Xs , ω )) forming the solution of the linear

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problem (10), can now be mapped back into the interior, Fig. 1c). One obtains the density (7) after projecting down onto coordinate space. To evaluate (1 − T )−1 it is convenient to express the operator T in a suitable set of basis functions defined on the boundary. Depending on the topology of the boundary, complete function sets such as a Fourier basis for two dimensional plates or spherical harmonics for bodies in three dimensions may be chosen. Denoting the orthonormal basis {. . . , Ψ0 , Ψ1 , Ψ2 , . . .}, we obtain

Tnm =

=

dXs dXs Ψn∗ (Xs ) T (Xs , Xs ; ω ) Ψm (Xs )

(11)

dXs Ψn∗ (φω (Xs )) w(Xs ) Ψm (Xs ) .

The treatment is reminiscent to the Fourier-mode approximation in the wave intensity analysis (WIA) [13, 14]; note, however, that the basis functions cover both momentum and position space here and can thus resolve spacial density inhomogeneities unlike WIA. For more details see again [1]. So far, we have sketched the method treating the system as one entity. In many applications, it may be useful to split the full system into subsystems and to consider the dynamics within each subsystem separately. Coupling between sub-elements can then be treated as losses in one subsystem and source terms in the other. Typical subsystem boundaries are surfaces of reflection/transmission due to sudden changes in the material parameters or local boundary conditions due to for example bends in plates. We denote the subsystems {P1 , . . . PN } and describe the full dynamics in terms of the subsystem boundary operators T i j ; power flowing from Pj to Pi is possible only if the two subsystems are connected and one obtains T i j (Xsi , Xsj ) = wi j (Xsj ) δ (Xsi − φω (Xsj )) ij

(12)

where φωi j is the boundary map in subsystem j mapped onto the boundary of the adjacent subsystem i and Xsi are the coordinates of subsystem i. The weight wi j contains, among other factors, reflection and transmission coefficients characterising the coupling at the interface between Pj and Pi . A basis function representation of the full operator T as suggested in Eq. (11) is now written in terms of subsystem boundary basis functions Ψni with ij Tnm =

∗

dXsi dXsj Ψ i n (Xsi ) T i j (Xsi , Xsj ) Ψmj (Xsj ).

(13)

The equilibrium distribution on the boundaries of the subsystems is then obtained by solving the systems of equations (10), that is, (1 − T )ρB = ρB0 .

(14)

Here, T is the full operator including all subsystems and the equation is solved for the unknown energy densities ρ = (ρB1 , . . . , ρBN ) where ρBi denotes the (Fourier)

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coefficients of the density on the boundary of subsystem i. Equations similar to (14) have been considered by Craik [4] in the context of SEA. Up to now, the various representations given are all equivalent and correspond to a description of the wave dynamics in terms of the ray tracing ansatz (7). Traditional ray tracing based on sampling ray solutions over the available phase space is rather inefficient, however. Convergence tends to be fairly slow, especially if absorption is low and an exponentially increasing number of long paths including multiple reflections need to be taken into account. An SEA treatment emerges when approximating the individual operators T i j in terms of the lowest order basis function (or Fourier mode), that is, the constant function Ψ0j = (ABj )−1/2 with ABj , the area of the boundary of Pj . The matrix T i j is then one-dimensional and gives the mean transmission rate from subsystem Pj to Pi . It is thus equivalent to the coupling loss factor ηi j used in standard SEA equations. The resulting full N-dimensional T matrix (with N, the number of subsystems) yields a set of SEA equations using the relation (14). An SEA approximation is justified if the ray dynamics within each subsystem is sufficiently chaotic such that a trajectory entering subsystem j ‘forgets’ everything about its past history before exciting Pj again. In other words, correlations within the dynamics must decay fast on the time scales of the staying time, that is, the time scale it takes for a typical ray to leave the cavity Pj . This makes it possible to quantify the applicability of an SEA approach in terms of two simple inequalities. SEA is expected to work well if k 1/L i τesc

i τcorr

(15) (16)

i and τ i where k is the wave number, L is a typical dimension of the system and τesc corr is the escape and correlation time in subsystem i. The time scales can be obtained directly from the subsystem operator T ii as defined in (13). The condition (15) represents the usual high-frequency assumption. The second condition, (16), will be fulfilled if the subsystems’ boundaries are sufficiently irregular (small τcorr ), the subsystems are dynamically well separated (large τesc ) and absorption and dissipation is small (large τesc ) — conditions typically cited in an SEA context. Under these circumstances, SEA is an extremely efficient method compared to standard ray tracing techniques. However, for subsystems with regular features, such as rectangular cavities or corridor-like elements, incoming rays are directly channelled into outgoing rays thus violating the equilibration hypothesis and introducing memory effects. Likewise, strong damping may lead to significant decay of the signal before reaching the exit channel introducing geometric (system dependent) effects. These features can be incorporated in DEA by including higher order basis functions for each subsystem boundary operator T i j . This leads to a smooth interpolation from SEA to a full ray-tracing treatment. The maximal number of basis functions needed to reach convergence are expected to be relatively small thus making the

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new method more efficient than a full ray tracing treatment — in particular in the small damping regime.

4 A numerical example: coupled two-domain systems The method has been implemented numerically for a coupled two-domain system; the wave energy distribution has been calculated using DEA and compared to the numerically exact solutions obtained from an advanced hp-adaptive FEM method. In particular, we consider the following acoustic two domain model problems in 2D with point source located at r0 in the interior of one of the two domains. The corresponding wave equation has the form ⎧ ⎨ −∇ · (μ −1 (r)∇G(r, r0 )) − (ω + id)2 G(r, r0 ) = δ (r − r0 ) in Ω , (17) ⎩ on ∂ Ω , G(r, r0 ) = 0 where μ is the mass-density, ω is the frequency of the wave emitted at the source and d is a damping parameter. Furthermore, δ (r − r0 ) is the delta function centred in r0 . Here we take the domain Ω ⊂ R2 to be the union of two polygons, see Figs. 2 and 3, which represent the two domains. We impose homogeneous Dirichlet boundary conditions on the outer boundary ∂ Ω . In the following we will consider the situation, where the wave velocity is constant within each sub-domain, but may jump discontinuously at the interface between the two domains giving rise to reflection/transmission phenomena. In the following, we will briefly describe the FEM method used. We will then present results both for FEM solutions and using the DEA approximations.

4.1 The hp-adaptive Discontinuous Galerkin Method In recent years the discontinuous Galerkin (DG) method for elliptic problems have become increasingly popular; a unified presentation and analysis for DG methods is contained in [36]. The main reason for this increase of interest in DG methods is that allowing for discontinuities across elements gives extraordinary flexibility in terms of mesh design and choice of shape functions. hp-adaptive DG methods, which are based on locally refined meshes and variable approximation orders, are very popular for many different types of PDE problems, since they can achieve exponential rates of convergence even in the presence of singularities [40, 37, 38, 39]. We consider shape-regular meshes Th that partition Ω ⊂ R2 into open triangles {K}K∈Th . Each element K ∈ Th can then be affinely mapped onto the generic ref The diameter of an element K ∈ Th is denoted by hK . Due to our erence element K. assumptions on the meshes, these diameters are of bounded variation, that is, there

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is a constant b1 ≥ 1 such that b−1 1 ≤ hK /hK ≤ b1 ,

(18)

whenever K and K share a common edge. We store the elemental diameters in the mesh size vector h = {hK : K ∈ Th }. Similarly, we associate with each element K ∈ Th a polynomial degree pK ≥ 1 and define the degree vector p = {pK : K ∈ Th }. We assume that p is of bounded variation as well, that is, there is a constant b2 ≥ 1 such that (19) b−1 2 ≤ pK /pK ≤ b2 , whenever K and K share a common edge. For a partition Th of Ω and a degree vector p, we define the hp-version discontinuous Galerkin finite element space Vh by Vh = { v ∈ L2 (Ω ) : v|K ∈ P pK (K), K ∈ Th }.

(20)

Here, P pK (K) is the space of polynomials of total degree ≤ pK . Next, we define some trace operators that are required for the DG methods. To this end, we denote by EI (Th ) the set of all interior edges of the partition Th of Ω , and by EB (Th ) the set of all boundary edges of Th . Furthermore, we define E (Th ) = EI (Th ) ∪ EB (Th ). The boundary ∂ K of an element K and the sets ∂ K \ ∂ Ω and ∂ K ∩ ∂ Ω will be identified in a natural way with the corresponding subsets of E (Th ). Let K + and K − be two adjacent elements of Th , and r an arbitrary point on the interior edge κ ∈ EI (Th ) given by κ = ∂ K + ∩ ∂ K − . Furthermore, let v and q be scalar and vector-valued functions, respectively, that are smooth inside each element K ± . By (v± , q± ), we denote the traces of (v, q) on κ taken from within the interior of K ± , respectively. Then, the averages of v and q at r ∈ κ are given by 1 {{v}} = (v+ + v− ), 2

1 {{q}} = (q+ + q− ), 2

respectively. Similarly, the jumps of v and q at r ∈ κ are given by [[v]] = v+ nK + + v− nK − ,

[[q]] = q+ · nK + + q− · nK − ,

respectively, where we denote by nK ± the unit outward normal vector of ∂ K ± , respectively. On a boundary edge κ ∈ EB (Th ), we set {{v}} = v, {{q}} = q, and [[v]] = vn, with n denoting the unit outward normal vector on the boundary ∂ Ω . For a mesh Th on Ω and a polynomial degree vector p, let Vh be the hp-version finite element space defined in (20). We consider the (symmetric) interior penalty discretization of (17): find Gh ∈ Vh such that Ah (Gh , v) = Fh (v) for all v ∈ Vh , where

(21)

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Gregor Tanner and Stefano Giani

Ah (u, v) :=

∑

K∈Th

− +

∑

K

μ −1 ∇h u · ∇h v − (ω + id)2 uv dr−

κ ∈E (Th ) κ

∑

κ ∈E (Th ) κ

Fh (v) :=

Ω

{{μ −1 ∇h v}} · [[u]] + {{μ −1 ∇h u}} · [[v]] ds

c [[u]] · [[v]]ds,

δ (r − r0 )v dr .

Here, ∇h denotes the element wise gradient operator. Furthermore, the function c ∈ L∞ (E (Th )) is the discontinuity stabilisation function that is chosen as follows: we define the functions h ∈ L∞ (E (Th )) and p ∈ L∞ (E (Th )) by r ∈ κ ∈ EI (Th ), κ = ∂ K ∩ ∂ K , min(hK , hK ), h(r) := hK , r ∈ κ ∈ EB (Th ), κ ∈ ∂ K ∩ ∂ Ω , r ∈ κ ∈ EI (Th ), κ = ∂ K ∩ ∂ K , max(pK , pK ), p(r) := r ∈ κ ∈ EB (Th ), κ ∈ ∂ K ∩ ∂ Ω , pK , and set

c = γ p2 h−1 ,

(22)

with a parameter γ > 0 that is independent of h and p. All the simulations has been done using AptoFEM (www.aptofem.com) on a parallel machine; results for the considered model problems are displayed in Fig. 2.

Fig. 2: Solutions for the Green function in configuration A and B: 1st row: |G|2 without damping; 2nd row: ReG with damping d = 0.6. In both cases: wavenumber on left hand side (Area 1) — k1 = 100; wavenumber on right hand side (Area 2) — k2 = 200

Wave transport in complex vibro-acoustic structures in the high-frequency limit

197

4.2 FEM compared to DEA and SEA — results Two different configurations are considered as shown in Fig. 2 (together with a wave solution) as well as in the insets of Fig. 3. Estimates for the acoustic energy induced by a point source in subsystem 1 is obtained by using DEA and will be compared to the FEM results. For the DEA analysis a Fourier basis both in position and momentum space is used. For details of implementing the boundary operator see [1]. The two set-ups can be characterised as follows: • configuration A comprises two subsystems of irregular shape with a line of intersection relatively small compared to the total length of the boundaries; the two subsystems are thus well separated and SEA is expected to work well. • configuration B consists of two domains where the line of intersection is of the order of the size of the system; the only dynamical barrier is posed by the BC at the interface itself. The standard SEA assumption of weak coupling and a quasi-stationary distributions in each subsystem may thus be violated. (This configuration has also been studied in [41, 26]). Note, that SEA results are in general insensitive to the position of the source, whereas actual trajectory calculations may well depend on the exact position especially for strong damping and for sources placed close to or far away from points of contact between subsections. The DEA calculations have been done for fixed wave number in both domains, that is, k1 = k2 . We have used finite basis sets up to n1 , n2 = −N, . . . N with N ≤ 8. This gives rise to matrices of the sizes dimT = 2(2N + 1)2 with basis functions covering position and momentum coordinates uniformly in both subsystems. Energy distributions have been studied as a function of the damping rate d. Note, that in the limit d → 0, the matrix T has an eigenvalue one with eigenvector corresponding to an equidistributed energy density in both domains, see [1]. In the case of no damping, the ray dynamics explores the full phase space uniformly on the manifold H(X) = ω 2 in the long time limit. Eq. (14) is singular for d = 0 and the solutions become independent of the source distribution ρ0 for d → 0. One obtains

ρ¯ 1 ε1 = lim = 1, ¯ 2 d→0 ε2 d→0 ρ lim

where ρ¯ i denotes the mean ray density in domain i averaged over the area of the domain and εi is the corresponding mean energy density obtained from Eq. (2). Results for the relative energy density distribution for the two-domain systems are shown in Fig. 3. Increasing the basis size - indicated here by the index N — leads to fast convergence as is evident from the figures. An SEA-like treatment corresponds to N = 0, here. SEA works remarkably well for configuration A, for which the main SEA assumptions, that is, irregular shape, well separated subsystems and relative small damping, are fulfilled. The deviations between SEA and the result for N = 4 are of

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the order of a few percent. Given that SEA describes the energy densities here in terms of a system of only two coupled equations, this clearly shows the power of SEA compared to, for example, ray tracing methods. In configuration B, the division into two subsystems is less clear-cut and deviations from SEA due to the strong coupling between the cavities may be expected. Indeed, one finds a higher energy density in area 2 than expected from SEA — energy dissipates into area 2 before an equilibrium distribution is attained in area 1. Increasing the basis size leads to fast convergence of the DEA results to the true energy ratios. DEA makes it possible to resolve the wave intensity distribution within each of the subsystems. The boundary distribution obtained from Eqs. (10) and (14) can be mapped back into the interior. The spatial resolution of the wave energy density contains important information about, for example, the (acoustic) radiation characteristics of sub-elements in the high frequency limit.

Fig. 3: Energy in domain 1 relative to energy in area 2 for depending on the overall damping parameter d: SEA results (full line) and DEA results (dashed lines) are compared with FEM solutions in the frequency range k = 100 − 102

5 Conclusions We have shown that ray tracing methods and SEA are closely related and that the latter is indeed an approximation of the former by smoothing out the details of ray dynamics within individual subsystems. We propose a numerical technique which interpolates between SEA and full ray tracing by resolving the ray dynamics on a finer and finer scale. This is achieved by expressing the dynamics in terms of linear boundary operators and representing those in terms of a set of basis functions on the boundary. The resolution of the dynamics is now determined by the number of boundary functions taken into account. We have tested the new method for two two-coupled domain models and compared the results with FEM calculations. We found that DEA accounts correctly for correlations in the underlying ray dynamics and applies in situations where stan-

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dard SEA shows systematic deviations from the FEM solutions. DEA is thus a more universal tool to predict average energy distributions at a small computational overhead. DEA and SEA are high-frequency methods, that is, they will give estimates for the energy density distribution in a regime not accessible to ‘numerically exact’ PDE solvers such as finite element methods (FEM). SEA has thus become a valued alternative widely used in the engineering community for estimating energy distributions in wave problems with high frequency noise sources, see for example [19, 24]. DEA is expected to give good results in the high frequency limit; it will offer more robust and reliable estimates in this regime compared to SEA and may thus form the basis of a high-frequency black-box tool. Acknowledgements The author would like to thank Oscar Bandtlow, Brian Mace, David Chappel, Cathleen Seidel and Stewart McWilliam for stimulating discussions. Support from the EPSRC and the EU (through the FP7 IAPP grant MIDEA) is gratefully acknowledged.

References 1. Dynamical energy analysis – Determining wave energy distributions in vibro-acoustical structures in the high-frequency regime, Tanner G 2009 Journal of Sound and Vibration 320 1023. 2. Statistical analysis of power injection and response in structures and rooms, Lyon R H 1969 Journal of the Acoustical Society of America 45 545 3. Lyon R H and DeJong R G 1995 Theory and Application of statistical energy analysis (2nd edn. Boston, MA: Butterworth-Heinemann) 4. Craik R J M 1996 Sound transmission through buildings: using statistical energy analysis (Gower, Hampshire) 5. Kuttruff H 2000 Room Acoustics, 4th edition, Spon Press, London 6. Cerven´y V 2001 Seismic Ray Theory (Cambridge University Press) 7. Mc Kown J W and Hamilton Jr R L 1991 Ray Tracing as a Design Tool for Radio Networks, IEEE Network Magazine 27 8. Glasser A S (Ed) 1989 An Introduction to Ray Tracing (Academic Press) 9. Prediction of vibrational energy distribution in the thin plate at high-frequency bands by using the ray tracing method, Chae K-S and Ih J-G 2001 Journal of Sound and Vibration 240 263 10. Vibrations in several interconnected regions: a comparison of SEA, ray theory and numerical results, Kulkarni S, Leppington F G and Broadbent E G 2001 Wave Motion 33 79 11. Wave chaos in acoustics and elasticity, Tanner G and Sondergaard N 2007 Journal of Physics A 40 R443 12. Advanced Statistical Energy Analysis, Heron K H 1994 Philosophical Transactions of the Royal Society London A 346 501. 13. A wave intensity technique for the analysis of high frequency vibrations, Langley R S 1992 Journal of Sound and Vibration 159 483 14. Wave Intensity Analysis of High Frequency Vibrations, Langley R S and Bercin A N 1994 Philosophical Transactions of the Royal Society London A 346 489. 15. A vibroacoustic model for high frequency analysis, Le Bot A 1998 Journal of Sound and Vibration 211 537 16. Energy transfer for high frequencies in Built-up structures, Le Bot A 2002 Journal of Sound and Vibration 250 247 17. Derivation of statistical energy analysis from radiative exchanges, Le Bot A 2007 Journal of Sound and Vibration 300 763

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18. Entropy in statistical energy analysis, Le Bot A 2009 J Acoust Soc Am 125 1473 19. NoiseFEM — a validated energy finite element method for structure borne sound prediction, Cabos C and Matthies H G 2009 IUTAM proceedings 20. Universal eigenvalue statistics and vibration response prediction, Langley R S 2009 IUTAM proceedings 21. A power absorbing matrix for the hybrid FEA-SEA approach to vibroacoustic analysis of complex uncertain structures, Lande R H and Langley R S 2009 IUTAM proceedings 22. Statistical energy analysis and the second princible of thermodynamics, Le Bot A 2009 IUTAM proceedings 23. High-frequency vibrational power flows in randomly heterogeneous coupled structures, Savin E 2009 IUTAM proceedings 24. Modeling noise and vibration transmission in complex systems, Schorter P J 2009 IUTAM proceedings 25. Energy flow models from finite element analysis, Mace B R and Shorter P J 2000 Journal of Sound and Vibration 233 369 26. Statistical energy analysis, energy distribution models and system modes, Mace B R 2003 Journal of Sound and Vibration 264 391; Statistical energy analysis: coupling loss factors, indirect coupling and system modes, Mace B R 2005 Journal of Sound Vibration 279 141 27. Gutzwiller M C 1990 Chaos in Classical and Quantum Mechanics (Springer, New York) 28. Short wave length approximation of a boundary integral operator for homogeneous and isotropic elastic bodies, Tanner G and Søndergaard N 2007 Physical Review E 75 036607 29. Wave chaos in the elastic disk, Søndergaard N and Tanner G 2002 Physical Review E 66 066211 30. Cremer L, Heckl M and Ungar E E 1988 Structure-borne Sound (Springer Verlag, Berlin 2nd edition) 31. Elastic wave transmission through plate/beam junctions, Langley R S and Heron K H 1990 Journal of Sound and Vibration 143 241 32. Ray Splitting and Quantum Chaos, Bl¨umel R, Antonsen Jr. T M, Georgeot B, Ott E and Prange R E 1996 Physical Review Letter 76 2476 33. Semiclassical theory of flexural vibrations of plates, Bogomolny E and Hugues E 1998 Physical Review E 57 5404 34. St¨ockmann H-J 1999 Quantum Chaos – an introduction (CUP, Cambridge) 35. Haake F 2001 Quantum Signatures of Chaos, (2nd edn., Springer, Berlin) 36. Unified analysis of discontinuous Galerkin methods for elliptic problems, Arnold D N, Brezzi F, Cockburn B and Marini L D 2001 SIAM J. Numer. Anal. 39(5) 1749 37. Adaptive Discontinuous Galerkin Finite Element Methods for Nonlinear Hyperbolic Conservation Laws, Hartmann R and Houston P 2002 SISC 24(3) 979 38. An Optimal Order Interior Penalty Discontinuous Galerkin Discretization of the Compressible Navier–Stokes Equations, Hartmann R and Houston P 2008 J. Comp. Phys. 227 9670 39. Energy Norm A Posteriori Error Estimation of hp–Adaptive Discontinuous Galerkin Methods for Elliptic Problems, Houston P, Sch¨otzau D and Wihler T P 2007 M3AS 17(1) 33 40. Discontinuous hp-finite element methods for advection-diffusion-reaction problems, Houston P, Schwab C and S¨uli E 2002 SIAM J. Numer. Anal. 39(6) 2133 41. The SEA of two coupled plates: an investigation into the effects of subsystem irregularity, Mace B R and Rosenberger J 1998 Journal of Sound and Vibration 212 395

Benchmark study of three approaches to propagation of harmonic waves in randomly heterogeneous elastic media Alexander K. Belyaev

Abstract A benchmark study of the one-dimensional stochastic elastic wave propagation is carried out. Three approaches which are the method of integral spectral decomposition, the Fokker-Planck-Kolmogorov (FPK) equation and the Dyson integral equation are applied to solve several boundary value problems. The merits and the shortcomings of each approach became apparent, that is, one can choose an appropriate approach based on the above conclusions on the strong and weak sides of each approach. The discussed approaches cover all problems of the harmonic wave propagation in heterogeneous or stochastic media, therefore, by means of a preliminary analysis of the problem, one can choose an appropriate approach.

1 Introduction A number of approaches have been applied to analysis of stochastic wave propagation, see e.g. [1], nevertheless they deal with the wave propagation in medium with a single random function. A more realistic approach requires modelling random elastic media in terms of two random fields which are random elastic modulus and random mass density. The equation governing propagation of harmonic one-dimensional wave of frequency ω and ampitude u is as follows E u + ρω 2 u = 0 , (1) where E is Young’s modulus, ρ is the mass density and a prime indicates the differentiation with respect to x. Random elastic modulus and random mass density assume the following form Alexander K. Belyaev Institute of Problems in Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia, e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 15, © Springer Science+Business Media B.V. 2011

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Alexander K. Belyaev

E (x) = E [1 + ε (x)] , ρ (x) = ρ (1 + r (x)) .

(2)

Here denotes the mean value of a random function whereas non-dimensional centered random functions ε and r describe the randomness of the mass density and Young’s modulus, respectively. Since the medium is assumed to be statistically homogeneous, E and ρ do not depend upon x. Two boundary conditions will be discussed in what follows: the Dirichlet boundary condition which implies a prescribed displacement H at x = 0 u(0) = H

(3)

and the Neumann boundary condition with a prescribed axial stress at x = 0 E u (0) = Σ .

(4)

Despite the randomness in medium’s parameters, H and Σ are deterministic. The intent of the paper is a benchmark study of three approaches to wave propagation in random elastic media. They are (i) the method of integral spectral decomposition, (ii) the Fokker-Plank-Kolmogorov equation and (iii) the Dyson integral equation. The merits and the deficiencies of each approach are discussed in each Section, and they are briefly outlined in Conclusions.

2 Method of integral spectral decomposition Application of this method to the wave problem can be found e.g. in [2] and [3]. In the framework of this approach the medium parameters are represented in the form of the Fourier-Stiltjes integrals E (x) = E 1 + Ξ (k) exp (ikx) dk , ρ (x) = ρ 1 + R (k) exp (ikx) dk , (5) where Ξ (k), and R (k) are the random Fourier spectra, and Ξ (k) = R (k) = 0. Subsequently, u(x) can be represented by its spectral representation u(x) = u (x) +

U (k) μ (x, k) exp (ikx) dk ,

(6)

where a deterministic function μ (x.ω ), produces the frequency-amplitude modulation of the homogeneous stochastic process U (k), U (k) = 0. Substituting Eqs. (5) and (6) into Eq. (1) and taking expectations yields d 2 u + λ02 ω 2 u + dx2

∞

−∞

∞ d2 μ dμ 2 SEu (k) + ik dk + λ0 Sρ u (k) μ dk = 0 , dx2 dx −∞

(7)

Benchmark study of three approaches to propagation. . .

203

where λ0 = ω / E / ρ denotes an “averaged” wave number. While deriving the latter equation we use the property of stochastic orthogonality of random spectra Ξ (k)U ∗ (k1 ) = SEu (k) δ (k − k1 ) ; R (k)U ∗ (k1 ) = Sρ u (k) δ (k − k1 ) where SEu and Sρ u denote the spectral densities of the random processes. Substituting (5) and (6) into Eq. (1), multiplying the result with R∗ (k1 ) and Ξ ∗ (k1 ), respectively, and taking mathematic expectations one obtains another two equations

2 d2μ dμ 2 + SEu (k) + + 2ik μ λ − k 0 dx2 dx 2 d u d u + ik u SE ρ (k) = 0 , (8) + SE (k) + λ02 dx2 dx

2 d2 μ dμ 2 + μ λ0 − k Sρ u (k) + + 2ik dx2 dx 2 d u d u + ik u Sρ (k) = 0 , (9) + SE ρ (k) + λ02 dx2 dx

where the Wiener-Khinchin condition of statistical orthogonality of the spectra has been used. During the estimations, the third moments have been omitted, i.e. we introduce a closure approximation otherwise the moments are governed by an infinite hierarchy of the coupled ordinary differential equations. Therefore, the analysis is restricted by the mean field and the spectral density matrix, which is typical for the correlation theories, cf. [4]. Since Eqs. (7)-(9) are linear in u and μ , the solution is sought in the form u = B exp (λ x) ,

μ = M exp (λ x) ,

(10)

where λ is the eigenvalue. Inserting Eq. (10) into Eqs. (7)–(9) yields the characteristic equation for λ

∞ λ 2 (λ + ik)2 SE + 2λ 2 λ (λ + ik) SE ρ + λ 4 Sρ dk 0 0 = 0. (11) λ 2 + λ02 − 2 2 (λ + ik) + λ0 −∞

Let us study some typical cases. First, assume Young’s modulus and the mass density to be statistically independent random functions of diffuse type with the following spectral densities SE (k) =

σE2 αE , π k2 + αE2

Sρ (k) =

σρ2 αρ , π k2 + αρ2

SE ρ (k) = 0 ,

(12)

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where σE and σρ are the standard deviations while αE−1 and αρ−1 are the correlation radii. Such modelling is typical for complex engineering structures, cf. [5]. The integral in Eq. (11) is evaluated by the method of residue calculus to yield

λ 2 + λ02 −

σE2 λ 2 (λ + αE )2 (λ + αE )2 + λ02

−

σρ2 λ04 = 0. 2 λ + αρ + λ02

(13)

Analysis of this characteristic equation which is a polynomial of sixth order is rather laborious, see to this aim [3] where an asymptotic analysis has been briefly outlined for weakly heterogeneous media (σE 1, σρ 1). We will perform the complete analysis of the case in which the randomness in Young’s modulus E and mass density ρ is fully correlated, that is σ2 α E (x) = E [1 + bE q (x)] , ρ (x) = ρ 1 + bρ q (x) , Sq (k) = π k2 + α 2

(14)

where q(x) is a centred exponentially correlated random function. The representation (14) models the materials with imperfections and is rather flexible. Choosing coefficients bE and bρ in a proper way one can study a broad class of the problems, for instance, in order to study wave propagation in a medium with random elastic properties and deterministic mass density one should assume bρ = 0. By means of the representation (14) we will see the restrictions, shortcomings and merits of the approach. Equations (7)-(9) lead to the characteristic equation

2 P (λ ) = λ 2 + λ02 (λ + a)2 + λ02 − σ 2 bE λ (λ + α ) + bρ λ02 = 0 (15) This polynomial of fourth order is reduced to a quadratic equation in λ (λ + α ) and has four roots

√ α 2 1 − σ 2 bE bρ ± Δ α 2 λ1,2,3,4 = − ± − λ0 (16) 2 2 1 − σ 2 b2E 2 with the discriminant Δ = σ 2 bE − bρ + α 2 λ0−2 σ 2 b2E − 1 . In order to study the limitations of the approach, one should recall that the imaginary part of λ ensures the wave character of the propagating disturbance. In other words, if the characteristic equation (15) has only real eigenvalues, the wave motion is impossible. Provided that σ bE > 1 a polynomial P (λ ) with four real roots should have two positive maxima and one negative minimum. Not complicated, but rather cumbersome analysis of four inequalities yields the set of conditions for absence of any wave motion

σ bE > 1; λ02 <

α 2 σ bE − 1 . 4 σ bρ + 1

(17)

The case σ bE < 1 is analysed by analogy and results in the set of conditions

Benchmark study of three approaches to propagation. . .

σ bE < 1; σ bE + bρ > 2,

205

1 − σ 2 bE bρ 4λ02 1 − σ 2 b2E

<α

2

σ < λ02

2 bE + bρ 1 − σ 2 b2E

2

(18)

Inequalities (17) and (18) may be roughly viewed as follows: if the standard variation in the Young modulus is larger than its mean value (σ bE > 1, Eq. (17)) or the standard variation in the mass density is larger than its mean value (σ bE < 1, Eq.(18)), then the wave motion does not exist since these inherently positive parameters become “too often negative”. This means that essentially heterogeneous media cannot be handled by means of this approach. The conditions (17) and (18) reduce to known conditions for medium with a single random field (random mass density or random Young’s modulus) obtained in [2] and [3]. Only two eigenvalues, say λ1 and λ2 , satisfy the Sommerfeld radiation condition, i.e. u =

2

∑ Bn exp(λnx) ,

n=1

S μ (k, x) = − Suqq

2

bE λn (λn + ik) + bρ λ02

n=1

(λn + ik)2 + λ02

∑ Bn

(19) exp(λn x).

In view to obtain Bn we consider e.g. the Dirichlet boundary condition (3). As follows from Eq. (6) the boundary condition (3) is fulfilled and the second moments become trivial at x = 0 if u(0, k) = H, μ (0, k) = 0 which yields the following two equations for Bn B1 + B2 = H, B1

bE λ1 (λ1 + ik) + bρ λ02 (λ1 + ik) + λ02 2

+

bE λ2 (λ2 + ik) + bρ λ02 (λ2 + ik)2 + λ02

=0

(20)

These equations are resolved for B1 and B2 that completes the solution of the problem. The merits of the approach are (i) the approach is valid for arbitrary spectral densities and (ii) it can be easily generalised to 2D and 3D problems. The shortcomings are (i) only mean field and the second moments are obtained, (ii) difficult to apply to nonlinear problems and (iii) applicable only for weakly heterogeneous media.

3 The Fokker-Planck-Kolmogorov equation The second approach to the stochastic wave propagation is based on the theory of continuous Markov processes. The state transition probability function for such processes is governed by a linear partial differential equation, known as the FokkerPlanck-Kolmogorov (FPK) equation, cf. [6] and [7]. To begin with, we consider the case of wave propagation in medium with deterministic Young’s modulus (ε = 0) and random mass density. Assume that in Eq. (2) the random function r = ξ (x) is a spatial white noise of intensity s. To apply FPK

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equation we introduce the new variables ν1 = u, ν2 = u and recast Eq. (1) as two first-order differential equations

ν1 = ν2 ,

ν2 = −λ02 ν1 − λ02 ξ ν1 .

(21)

Provided that ξ is a Gaussian random function, the state transition probability function p is governed by the FPK equation N N N ∂p ∂ ∂2 =−∑ [χn p] + ∑ ∑ [γmn p] . ∂x n=1 ∂ νn n=1 m=1 ∂ νn ∂ νm

(22)

In this case N = 2 and the drift χn and diffusion γmn coefficients are given by

χ1 = ν2 ,

χ2 = −λ02 ν1 ,

γ11 = γ12 = γ21 = 0 ,

s γ22 = λ04 ν12 , 2

(23)

cf. [2]. Substituting Eq. (23) into FPK equation, multiplying it consequently with ν1 and ν2 and integrating over the space of ν1 and ν2 results in the following equations for the mean values of the phase variables ν1 − ν2 = 0, ν2 + λ02 ν1 = 0 .

(24)

This system of equations is solved by means of the substitution νk = Vk exp (λ x), k = 1, 2 which results in the characteristic equation λ 2 + λ02 = 0 with the roots λ1,2 = ±iλ0 . The eigenvalue λ2 = −iλ0 satisfies the Sommerfeld radiation condition since it describes harmonic waves propagating in the positive x direction. The Dirichlet boundary condition, Eq. (3), yields ν1 = H exp (−iλ0 x) i.e. the mean field in stochastic medium with spatial white noise random mass density coincides with the deterministic field in the homogeneous medium. To obtain equations for the second moments we multiply the FPK equation consequently by ν12 , ν1 ν2 and ν22 and integrate the result over the space of ν1 and ν2 , 2 ν1 − 2 ν1 ν2 = 0, ν1 ν2 + λ02 ν12 − ν22 = 0, 2 ν2 − sλ04 ν12 + 2λ02 ν1 ν2 = 0 .

(25)

To solve the latter equations we substitute νk vn = Wkn exp (Λ x) , k, n = 1, 2, and arrive at the following characteristic equation

Λ 3 + 4λ02Λ − 2sλ04 = 0

(26)

which has one real positive root and two complex conjugated roots. Only one root of Eq. (26) satisfies the Sommerfeld radiation condition, namely 1 −1 √ 13 1 −1 , h= Λ = λ0 −h + h − i 3 h + h sλ0 + s2 λ02 + p2 . (27) 3 3 2

Benchmark study of three approaches to propagation. . .

207

Boundary condition (3) which is ν12 (0) = H 2 leads to the following result

ν12 = H 2 exp (λ x) , ν1 ν2 = −H

2Λ

2

exp (Λ x) ,

ν22

1 2 2 = H λ0 − Λ exp (Λ x) . (28) 2 2

The latter equation indicates an exponential decay of the second moments of the stochastic wave that contradicts to the equation for the mean field which states that the mean value of the stochastic waves propagates without decay, cf. [2]. For realistic modelling of the elastic and mass parameters, the spatial white noise should be pre-filtered. The further analysis is restricted to the fully correlated randomness in the Young modulus and the mass density, Eq. (14), which allows us to compare the result with that of the integral spectral decomposition and draw some conclusions. Introducing the variables ν1 = u, ν2 = u , ν3 = q leads to three firstorder equations α bE ν2 ν3 − λ02 1 + bρ ν3 ν1 bE ν2 ν1 = u, ν2 = − ξ , ν3 = −αν3 + ξ (29) 1 + bE ν3 1 + b E ν3 where the latter equation is a filter equation and ξ is the spatial white noise of intensity s = 2ασ 2 . The FPK equation (22) has now a higher order (N = 3) α bE ν2 ν3 − λ02 1 + bρ ν3 ν1 ∂p ∂ ∂ 2 ∂ + αρ =− [ν2 p]+ λ0 [ν3 p] + ∂x ∂ ν1 ∂ ν2 1 + b E ν3 ∂ νn 2 ασ 2 b2E ∂ 2 2 ∂2p ν2 p 2 2∂ p + p − 2 ν σ α b ασ . (30) + E 2 1 + bE ν3 ∂ ν22 ∂ ν2 ∂ ν3 1 + bE ν3 ∂ ν32 The equation obtained is much more complicated than the previous one, cf. Eqs. (22) and (23), since its coefficients are no longer polynomials but rational functions. One obtains the equations for ν1 , ν2 , ν1 ν3 and ν2 ν3 by multiplying Eq. (30) consequently with ν1 , ν2 , ν2 (1 + bE ν3 )2 and ν2 ν3 (1 + bE ν3 )2 and integrating over the space of the phase variables. The result is ν1 − ν2 = 0 , ν1 ν3 − ν2 ν3 + α ν1 ν3 = 0 , λ02 1 + bE bρ σ 2 ν1 + λ02 bE + bρ ν1 ν3 + 1 + b2E σ 2 ν2 + + 3α bE σ 2 ν2 + 2bE ν2 ν3 + α ν2 ν3 = 0 , λ02 σ 2 bE + bρ ν1 + λ02 1 + 3bE bρ σ 2 ν1 ν3 + 2bE σ 2 ν2 + + α bE ν2 + 1 + 3b2E σ 2 ν2 ν3 + α 1 + 6b2E σ 2 ν2 ν3 = 0. (31) The latter equations are obtained by means of the Gaussian closure. The FPK equation is known to deliver an infinite hierarchy of coupled partial differential equation [6], [7]. Thus to obtain a solution it is necessary to introduce a closure. The simplest closure is a Gaussian closure, i.e. one assumes that ν1 and ν2 are quasi-

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Alexander K. Belyaev

Gaussian. Since ν3 and ξ are Gaussian, we can use the properties of the Gaussian distribution and to express the unknown higher order moments in terms of lower order moments, e.g. in terms of moments up to order 2 by means of expressions ν1 ν32 = σ 2 v1 etc. The expression for the vanishing determinant of the system with closure at the second order is the characteristic equation. This expression is very bulky and is not shown here. It can be proved that this equation does not coincide with the characteristic equation of the integral spectral decomposition (15). The explanation is that we multiplied of the the FPKequation with the polynomials third and forth order, namely, v2 1 + 3b2E σ 2 and v2 v3 1 + 3b2E σ 2 . This means that the moments of third and forth orders are always involved to this technique and a closure is unavoidable. The higher moments obtained were expressed in terms of the moments up to the second order which automatically means the reduction in accuracy of the approach. The characteristic equation of the integral spectral decomposition approach, Eq. (15), is obtained if we introduce another set of variables

ν1 = u ,

ν2 = u (1 + 3bE q) ,

ν3 = q .

The results is three first-order stochastic differential equations ν1 (1 + bE ν3 ) − ν2 = 0, ν2 + λ02 1 + 3bρ ν3 ν1 = 0, ν3 + αν3 = ξ . Stochastic averaging of the first and the second equation yields ν1 + bE (λ + α ) ν1 ν3 − ν2 = 0, ν2 + λ02 ν1 + bρ ν1 ν3 = 0

(32)

(33)

Another two equations are obtained by multiplying the first two equations (32) with ν3 and taking expectations ν1 ν3 + bE σ 2 ν1 + α ν1 ν3 − ν2 ν3 = 0, ν2 ν3 + λ02 ν1 ν3 + bρ σ 2 ν1 + α ν2 ν3 = 0,

(34)

While deriving Eqs. (33) and (34) two assumptions were done: firstly, a standard assumption of the FPK-equation ξ ν1 = ξ ν2 = 0 and secondly v3 and ξ and ν1 and ν2 are quasi-Gaussian which permits a Gaussian closure are Gaussian ν1 ν32 = σ 2 v1 etc. Since Eqs. (33) and (34) are linear in the moments, the substitution νk vn = Wkn exp (Λ x) , k, n = 1, 2 yields the characteristic equation (17). This means that the new variables suit better than the previous ones due to the fact that the newly introduced variable ν2 = u E/ E is non-dimensional axial stress and has clear physical sense. Let λ1 and λ2 denote the eigenvalues which satisfy the Sommerfeld radiation condition then the general solution is ν1 = A1 exp (λ1 x) + A2 exp (λ21 x). The Dirichlet boundary condition, Eq. (3) yields the following equations x=0,

ν1 = H ,

ν1 ν3 = 0 ,

(35)

Benchmark study of three approaches to propagation. . .

209

which allows one to determine the integration constants −1 λ 2 + λ12 bρ λ02 + bE λ1 (λ1 + α ) , A2 = H − A1 . A1 = H 1 − 02 λ0 + λ22 bρ λ02 + bE λ2 (λ2 + α )

(36)

Provided that the axial stress is prescribed at x = 0, i.e. E u (0) = Σ , the Neumann boundary condition (4) reduces to ν2 = Σ / E ,

x=0,

ν2 ν3 = 0 ,

(37)

which leads to a system of two equations for two integration constants. Multiplying Eq. (32) by ν1 and ν2 and averaging yields the system of three firstorder ordinary differential equations for the second moments (ν12 , ν1 ν2 , ν22 ) 2 ν1 − 2 ν1 ν2 = α ν1 ν1 ν3 − ν1 ν1 ν3 − ν1 ν1 ν3 , ν1 ν2 + λ02 ν12 − ν22 = −2bρ λ02 ν1 ν1 ν3 − −bE ν1 ν2 ν3 − ν2 ν1 ν3 − α ν2 ν1 ν3 , 2 ν2 + λ02 ν1 ν2 = −2bρ λ02 [ν1 ν2 ν3 + ν2 ν1 ν3 ] . (38) The main advantage of the FPK equation is that the moments of any order can be obtained. There exists even a certain class of problems which allows for an exact solution for the state transition probability function, cf. [8]. The deficiencies of the approach are as follows (i) the approach is applicable only for the random processes with rational spectral densities which are obtained by means of the white noise filtering, (ii) each filter increases the order of the system of equation, (iii) difficulties increase rapidly with the order of the system, (iii) the approach is applicable only for one-dimensional waves and (iiii) the approach appears to be accurate only for the systems with the polynomial coefficients. As the characteristic equation of this approach coincides with that of the integral spectral representation, only weakly heterogeneous random media can be analysed by means of the FPK equation.

4 The Dyson integral equation In contrast to the previous approaches which deal with the random mass density and random Young’s modulus an alternative modelling of 1D random media is proposed. We define a new independent variable y and a new dependent variable U y = a

x 0

dξ , a (ξ )

U (y) =

√ z u (x, ω ) ,

(39)

√ where a = E/ρ is the velocity of sound and z = E ρ is the acoustic impedance. The original problem is thus reduced to the wave propagation in elastic medium

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Alexander K. Belyaev

in terms of impedance and velocity of sound that seems to be much more suitable for the problems of wave propagation. In what follows a random elastic medium is modelled by a medium with the following statistically independent and statistically homogeneous random parameters: the sound speed a and the impedance z. Substitution of Eq. (39) into Eq. (1) yields √ ω2 d 2U 1 d2 z + + e (y) U = 0 , e (y) = − √ . (40) dy2 z dy2 a2 Introducing a new centred random function which will be referred to as heterogeneity enables Eq. (40) to be written as follows d 2U + [κ + ε (y)]U = 0 , dy2

κ=

ω2 + e . a2

(41)

For the Neumann boundary condition dU/dx = 1 at y = y0 the boundary value problem is equivalent to the single differential equation d 2U 2 + λ + ε (y) U = δ (y − y0 ) 2 dy

(42)

provided that the solution satisfies the Sommerfeld radiation condition. Equation (42) has a delta-function in the right hand side. Hence, its solution U(y) = G(y, y0 ) is Green’s function of Eq. (42) and the mean field U (y) = G (y, y0 ) is the averaged Green’s function. The latter is known to be the solution of the Dyson integral equation, cf. [1], [9] G (y, y0 ) = G0 (y − y0 ) +

G0 (y − r1 )M(r1 , r2 ) G (r2 , y0 ) dr1 dr2 .

(43)

Here Green’s function for the homogeneous material is given by G0 (y − y0 ) = −

1 e−iκ |y−y0 | = 2κ i 2π

∞ iκ (y−y0 ) e −∞

κ 2 − k2

dk ,

κ >0

(44)

and M denotes the kernel of an integral operator referred to as the mass operator in quantum field theory. This kernel contains an infinite number of terms. Even a single term corresponds to summation of an infinite subsequence of the series in perturbation theory. In what follows, we truncate M at its first term (so-called Bourett’s approximation, also called the first-order smoothing approximation) M (r1 , r2 ) = ε K0 ε = Bε (r1 , r2 ) G0 (r1 − r2 )

(45)

where K0 is an integral operator with kernel G0 and Bε (r1 , r2 ) is the correlation function of the random field ε .

Benchmark study of three approaches to propagation. . .

211

Since ε is assumed to be statistically homogeneous, its correlation function depends only on the difference of the arguments, i.e. Bε (r1 , r2 ) = Bε (r1 − r2 ). The Dyson equation (43) takes the form of an integral equation with a difference kernel which is solved by means of the spatial Fourier transform, see for details [5] −1 2 ˆ ˆ ε (k) Gˆ (k) = Gˆ −1 (k) − 4 π (k) B . G 0 0

(46)

The Fourier transform is inverted to recover the averaged Green’s function 1 G(r) = 2π

∞ −∞

eiκ r

κ2 −

∞ −∞

Bε (ρ ) G0 (ρ ) e−iκρ d ρ − k2

dk

(47)

where r = y − y0 is substituted for brevity. As before, we study an exponential correlation function for the heterogeneity Bε (r) = σ 2 exp (α |r|). Evaluating the integral in the denominator yields 1 G(r) = 2π

∞ −∞

eiκ r dk

κ2 +

σ2

iκ +α iκ (iκ +α )2 +k2

− k2

.

(48)

Let k1 and k2 denote the poles of the integrand that satisfy the Sommerfeld radiation condition and lay in the upper half-plane

2 α + iκ 2 1 2 2 √ k1,2 = σ , Im k1,2 > 0 . κ − (α + iκ ) ± κ 2 + (α + iκ )2 + 4 iκ 2 (49) One is often interested in the problem of propagation of long waves in a medium √ with a small-scale heterogeneity (α k). Under the assumptions α α k σ and α k expression (49) for poles k1 and k2 reduces to σ2 k1 = −κ 1 − i 3 , k2 = −κ + iα (50) κ α The contribution of the pole k2 becomes small even within the correlation radius R = α −1 , i.e. k2 models the near field which can be neglected in case of the long travelling distances. Hence, the influence of heterogeneity is taken into account by k1 which may be transformed to σ 2 α2 (51) k1 = −κ 1 − i 3 α κ κ2 Inspection of this equation indicates that the imaginary part √ of k1 can be considerable even for waves in weakly heterogeneous media (σ α k) with short range perturbations (α k). Considerable attenuation is explained by the accumulation

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Alexander K. Belyaev

of the dispersion effects. Application of the residue calculus in Eq. (48) renders an averaged Green’s function eik1 |r| σ 2R G(r) = , k1 = −κ 1 − i 3 , Im k1 > 0 . (52) 2ik1 κ The latter equation for the mean field may be obtained in a simpler way by setting k = 0 in the integral over ρ in the denominator in Eq. (47), i.e. 1 G (r) = 2π

∞ −∞

eiκ r

κ2 −

∞ −∞

dk .

(53)

Bε (ρ ) G0 (ρ ) d ρ − k2

Evaluation of the integral in the denominator yields 1 G (r) = 2π

∞

eiκ r dk

κ 2 + iκ (iσκ +α ) − k2 2

−∞

(54)

If the random medium with short range heterogeneity (α k) is considered, the only pole in the upper half-plane is k1 , Eq. (50). In this case the residue calculus delivers Eq. (52). In other words, the second pole which represents a near field does not appear if one sets k = 0 in the integral over ρ in Eq. (47). The substantiation is that the correlation radius R is small and one can consider only the leading term in the expansion of exp (−iκρ ) in terms of κρ , i.e. to set exp (−iκρ ) = 1. Other correlation functions Bε (r) = σ 2 exp (−α |r|) cos η r , α Bε (r) = σ 2 exp (−α |r|) cos η r + sin η |r| η

(55)

model the wave propagation in random media with a hidden periodicity. The first one describes a non-differentiable in a mean square sense stochastic process whereas the second one models a mean-square differentiable stochastic process. Only one pole of the integrand in the spectral spatial decomposition of the mean field has been observed to determine the far field while the others represent some near fields vanishing within the correlation radius, see [5] for detail. Despite the broad variety of properties of correlation functions the equation for the mean field in the random media takes the form of Eq. (52) for any of the above correlation functions. It allows one to study the general case of random medium with an arbitrary short range heterogeneity. To this end, we model the correlation function of the medium as follows Bε (r) = σ 2

δR (r) ; δR (0)

∞ −∞

δR (r)dr = 1 ;

R=

1 2Bε (0)

∞

Bε (r)dr = −∞

1 , (56) 2δR (0)

Benchmark study of three approaches to propagation. . .

213

where δR (r) is a window-function which has support only for small r, |r| < R. The previous analysis shows that when studying a random medium with the short range heterogeneity (α k) one must rather take into account the shift of the pole k1 (at k1 = −κ if the heterogeneity was supposed to vanish) to the upper half-plane due to the heterogeneity rather than the appearance of the new poles. Substituting the correlation function (56) into Eq. (47) yields 1 G(r) = 2π

∞ −∞

eiκ r

κ 2 + 2iκδσR (0) 2

+R −R

dk .

(57)

δR (ρ )e−iκ |ρ | d ρ − k2

The case of the heterogeneity of small extend is considered, i.e. κ R 1, that allows one to take only the leading term in the expansion of exp (−iκ |ρ |), in a series in terms of κ |ρ |, i.e. 1 G(r) = 2π

∞ −∞

eiκ r dk . κ 2 − iσ 2 Rκ −1 − k2

(58)

The residue calculus yields Eq. (52) for an arbitrary heterogeneous elastic medium with short range fluctuations. Since the velocity of sound a is random, G(r) is a random field which should be averaged over a. For a given probability density function for the velocity of sound, the mean field can be estimated. Averaged decay in statistically homogeneous medium d allows for a closed form expression σ 2R 1 d ξ Im κ 1 − i 3 = a(ξ ) κ x0 σ 2R 1 Im κ 1 − i 3 > 0 . a =− a κ

a d = − x − x0

x

(59)

Assuming that the decay is small leads to the following simple equation for the averaged decay d = a

∞ 2 1 σ R 1 −2 3 1 a = ω Bε (r)dr . a 2κ 2 4 a

(60)

−∞

One of the main merits of the Dyson equation is that the solution method does not require the random fields to be Gaussian. This allows one to overcome the known shortcoming of the normal distribution, such as the impedance or the velocity of sound can be negative. Any probability distribution function which satisfies these physical constraints with probability 1 can be taken in the framework of the present approach. Another merits are (i) the approach is applicable to essentially heterogeneous media and (ii) a closed form expression for the average attenuation is obtained. The approach shortcomings are (i) it is difficult to satisfy the boundary condi-

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Alexander K. Belyaev

tions and (ii) in order to find the standard deviation of the random fields, one should apply the Bethe-Salpeter equation, e.g. [1], [9], however its solution is extremely laborious.

5 Concluding remarks A benchmark study of the one-dimensional stochastic elastic wave propagation is carried out. Three approaches which are the method of integral spectral decomposition, the Fokker-Planck-Kolmogorov equation and the Dyson integral equation were applied to solve several boundary value problems. The merits and the shortcomings of each approach became apparent after solving some problems and were outlined in each Section. One can choose an appropriate approach based on the above conclusions on the strong and weak sides of each approach. For example, it was shown that only weakly heterogeneous elastic media can be analysed in the framework of the method of integral spectral decomposition and the FPK equation. Though Eqs. (17) and (18) give the conditions of the methods’ inapplicability only for random parameters of diffuse type, one should expect no drastic difference for another typical correlation functions. Dealing with essentially heterogeneous media, one is recommended to apply the Dyson integral equation. Summarising, we can say that the approaches discussed cover all problems of the harmonic wave propagation in heterogeneous or stochastic media, therefore, by means of a preliminary analysis of the problem, one can find an appropriate approach. Acknowledgements The present study was financially supported by the Scientific Program of the Russian Academy of Sciences (coordinator Academician Nikita F. Morozov).

References 1. Sobczyk, K.: Stochastic wave propagation. Amsterdam: Elsevier 1984. 2. Makarov, B.P.: Nonlinear problems of statistical dynamics of machines and devices (in Russian). Moscow: Mashinostroenie 1983. 3. Naprstek, J.: Propagation of longitudinal stochastic waves in bars with random parameters. In: Structural Dynamics (Augusti, G., Borri, C., Spinelli, P., eds) pp. 51–60. Rotterdam: A.A.Balkema 1996. 4. Yaglom, A.M.: Correlation theory of stationary and related random functions, vol. 1. New York: Springer-Verlag 1987. 5. Belyaev, A.K., Ziegler, F.: Uniaxial waves in randomly heterogeneous elastic media. Int. J. Probabilistic Engineering Mechanics, 13, No. 1, pp. 27–38, 1998. 6. Bolotin, V.V.: Random vibrations in elastic systems. The Hague: Nijhoff 1984. 7. Roberts, J.B., Spanos, P.D.: Random vibration and statistical linearization. Chichester: Wiley and Sons 1990. 8. Dimentberg, M.F.: Statistical dynamics of nonlinear and time-varying systems. New York: Wiley and Sons 1988. 9. Tatarsky, V.I.: Wave propagation in turbulent atmosphere (in Russian). Moscow Nauka 1969.

Minimum-variance-response and irreversible energy confinement A. Carcaterra

Abstract This paper discusses the question of the energy confinement in mechanical structures in the light of the uncertainties affecting the natural frequencies of the system. More precisely, recent studies have shown that energy can be introduced to a linear system with near irreversibility, or energy within a system can migrate to a subsystem nearly irreversibly, even in the absence of dissipation, provided that the system has a particular natural frequency distribution. In this paper the case of uncertainty in the system’s natural frequency is discussed and a remarkable statistical property of the natural frequency is derived for a permanent energy confinement within a part of the system. The results demonstrate the existence of a special class of linear non-dissipative dynamic systems that exhibit nearly-irreversible energy confinement-IEC if they satisfy a minimum-variance-response-MIVAR property. In this case, if the probability density function of the natural frequencies has a special shape, the conservative system shows an unexpected decaying impulse response.

1 Average Impulse Response and the Single Case In complex built-up structures, comprised of many individual structural components and modes, the spatial redistribution of vibratory energy throughout the structure, in many ways, seem similar to reduction of vibration due to classical dissipation mechanisms and “appear” as damping, but distinct from dissipation of vibratory energy as heat. In fact, the phenomenon consists of a fast energy transfer from the part of the structure directly excited to a second subsystem that receives and stores permanently this energy. The question has been considered in some papers reported in the A. Carcaterra Department of Mechanics and Aeronautics, University of Rome, ‘La Sapienza’ Via Eudossiana, 18, 00184, Rome, Italy Department of Mechanical Engineering, Carnegie Mellon University Pittsburgh, Pennsylvania 15213 A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 16, © Springer Science+Business Media B.V. 2011

215

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A. Carcaterra

references [1]–[8], that illustrate and is some cases provide theoretical fundaments for this phenomenon. The present work shows, for a system with random natural frequencies, the properties the pdf of the resonant frequencies must satisfy to observe such an unusual effect. In particular it is shown that an energy confinement effect is related to a requirement for a minimum variance response. The main statements of the theory are formulated in this and in the next section, while in section 3 an application of the theory is discussed. In many problems of mechanics, a combination of harmonic functions represents the impulse response h(t) of a conservative linear dynamical system, which commonly yields a discrete eigenfrequency spectrum that can be expressed as: N

h(t) = ∑ αi G(ωi ) sin ωit

(1)

i=1

where αi G(ωi ), ωi , N represent mode shapes dependent factors, system natural frequencies and the number of the modes involved in the system response. Except the special and simple case where frequencies ωi are integer multiples of a fundamental frequency, the properties of a harmonic series such as h(t) can be, in general, quite complicated. However, for conservative systems, expression (1) does not vanish asymptotically but behaves as an almost periodic function. The theory outlined in this paper investigates special occurrences of random natural frequencies ωi in equation (1). In this case, the function σ = G(ω ) sin ω t, appearing in each term of equation (1), is also random. Assume that each natural frequency ωi has the same probability density function pω . This physically means we are considering systems with a cluster of modes with the same average natural frequency that spread randomly over a given frequency range. Such a class of systems reveal a surprising theoretical and technical interest, as shown in the examples ahead. Expected value E{σ } of σ is: E{σ } =

+∞ −∞

pω (ω ) G (ω ) sin ω t d ω

(2)

Equivalently, in terms of the probability density function of σ , pσ , the same expectation has the form: E {σ} =

+∞ −∞

pσ ( σ ) σ d σ

(3)

where:

dσ = pω ( ω ) dω The expected value of the impulse response h(t) is therefore: pσ (σ )

N

+∞

i=1

−∞

E{h} = ∑ αi

pω (ω )G(ω ) sin ω t d ω ,

N

α = ∑ αi , i=1

(4)

E{h} = α E{σ } (5)

Minimum-variance-response and irreversible energy confinement

217

Asymptotic expansion of the integral (2) shows E{h} obeys the time asymptotic property (6) lim E{h} = 0 t→∞

A comparison between h and E{h}, from equation (1) and (2), respectively, reveals that the two functions do not share the fundamental asymptotic property (6): the single sample h, being the system conservative, is almost periodic and does not exhibit a vanishing impulse response, that is indeed the case for E{h}. Property (6) has a deep implication in the energy behavior of the considered system. The vanishing impulse response in the absence of any dissipation effect, implies the energy injected by the impulse is irreversibly stored in some part of the system far away from the point of energy injection. Therefore, we can conclude that property (6) implies an irreversible energy confinement-IEC far away from the point at which h is computed. This property, besides its theoretical importance, could also have a potential attractiveness in the design of new shock absorbers, so that it can be regarded, in some circumstances, as a desirable property. In this light, we can ask if there are particular populations of systems, i.e. some probability density functions pω , that can make the expected value E{h} as close as possible to the samples h(t). This requirement has its strict statistical translation: find the function pω making minimum the variance of the impulse response h. If this property holds, called here minimum-variance-response-MIVAR, we expect that each sample h(t) of the population will exhibit a property close to the one expressed by equation (6). Therefore this paper investigates, for linear and conservative systems, the link between the irreversible energy confinement and the minimum-variance-response. Note that the resonant frequencies of a single sample of a vibrating structure with N modes, all with the same pω , can be itself viewed as a collection of N samples of natural frequencies ωi , i = 1, 2, . . . N, of the random variable ω . Thus, its N eigenfrequencies distribuite within the uncertain system with a modal density n(ω ) directly related to the pdf pω , namely n(ω ) = N pω . This note is of practical use. In fact, this shows how the result of the statistical analysis can be directly applied even to the single sample translating the result forpω as a result for the natural frequency distribution n(ω ) within a single structural sample.

2 MIVAR: Minimum-Variance-Response Although our aim is to find pω leading to the minimum-variance-impulse-responseMIVAR, the problem is more conveniently formulated in terms of pσ ; once this is known, pω is obtained with some cares by equation (3). The analysis of MIVAR, needs some preliminary definitions and results. Def 1) The set of functions σi = G(ωi ) sin ωi t, i = 1, 2, . . . , N, defines a random point (or a random vector) σ = [σ1 , σ2 , . . . , σN ]T in a space Σ ; σ exists within the hypercube C ≡ { ExE x....x E}, with:

218

A. Carcaterra

E ≡ [ −Gmax , Gmax ] and Gmax = max {G1 , G2 , . . . , GN } with this definition the impulse response is a linear function of the random vector σ : h(σ ) = α T · σ = ∑Ni=1 αi σi . Def 2) The joint probability of σ is: P(σ ) = ∏Nk=1 pσ (σk ) , with: C

P(σ ) dC = 1 ,

Def 3) The variance D2 of h is: D2 =

E{h} = C

C

h(σ )P(σ ) dC

(7)

[h (σ ) − E {h(σ )} ]2 P(σ ) dC

Lemma 1) With E{∗} = ∗¯ , h¯ = α σ¯ , and from equation (7) (first): ∂ 1 ∂ σ ) dC = 0 → σ ) P(σ ) dC P( P( ¯ P(σ ) ∂ h¯ C ∂h C ∂ = ¯ [log P(σ )] P(σ ) dC = 0 C ∂h i.e.:

∂ log P = 0 ∂ h¯ The last equation is equivalent also to: ∂ h¯ ¯ log P = 0 ∂h

(8)

Lemma 2) From equation (7) (second): C

∂ h ¯ P(σ ) dC = 1 ∂h

→

h C

1 ∂ σ ) P(s, I) dC = 1 P( P(σ ) ∂ h¯

i.e.:

∂ log P = 1 (9) ∂ h¯ Discuss now the way to minimize the distance D between h and h¯ in the space Σ . The difference between equations (8) and (9) produces: h

¯ · (h − h)

∂ log P = 1 ∂ h¯

¯ and a condition on the scalar product between the functions (h − h) The Schwartz inequality applied to equation (10) leads to: ¯ · (h − h)

2 2 ∂ ∂ 2 ¯ log P ≤ (h − h) log P ∂ h¯ ∂ h¯

that, using equation (10) and the Def 3 for D, becomes:

(10)

∂ log P ∂ h¯

.

(11)

Minimum-variance-response and irreversible energy confinement

D2 ≥

1

∂ ∂ h¯

2 log P

219

(12)

The right-hand side of (12) presents a lower bound for D, depending on the probability function P or, equivalently, depending on pσ and through equation (3) on pω . Among the possible choices for P, those providing a minimum for the variance D match the lower bound exactly: D2 =

1

2

(13)

∂ log P ∂ h¯

As the minimum for D is reached through equation (13), the Schwartz inequality (11) becomes: 2 ∂ ¯ (h − h) · log P = 1 ∂ h¯ ¯ and ∂¯ log P becomes unitary; this imi.e. the scalar product between (h − h) ∂h plies the two functions are proportional (i.e. they are parallel functions, since their scalar product is 1): ∂ ¯ (h − h) ¯ log P = a(h) (14) ∂ h¯ ¯ is a proportionality constant. where a(h) Condition (14) represents a differential equation in terms of P and its solution leads to a family of exponential functions P(σ ) = ∏Nk=1 pσ (σk ). The solution to Eq. (14), originally obtained by Pitman and Koopman in the context of the theory of estimators, is given as:

¯ ε (σ ) + ζ (σ ) + β (h) ¯ pσ (σ ) = exp α (h) ¯ β (h) ¯ , ε (σ ), and ζ (σ ) are arbitrary functions of their respective arguwhere α (h), ments. Gauss function is part of this family of solutions:

1 1 (σ − σ )2 (15) pσ (σ ) = √ exp − 2 r2 r 2π pσ (σ ) has a shape that depends on the functions σ (G, ω ), σ and on the parameter r. Together with equation (15), equation (3) provides the probability density function pω (ω ) of the natural frequencies that minimizes D, i.e. that provides a MIVAR: dσ 1 1 ( σ − σ )2 (16) pω (ω ) = √ exp − 2 2 r d ω r 2π Equation (16) shows that time appears as a parameter in the pdf that minimizes D. The attention is here limited to time-invariant pdfs, thus equation (16) is taken here

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A. Carcaterra

just using a particular time t0 . The choice for t0 , selection of the frequency interval [ωmin , ωmax ] for pω (ω ), which also depends the choice of t0 , deserve some cares. In fact, in equation (3) both pω (ω ) and pσ (σ ) must be positive implying the derivative d σ /d ω must be positive within [ωmin , ωmax ]. Moreover, pω (ω ) through equation (3), must satisfy conditions (7), used to derive lemmas 1 and 2. Since integration domain of our integrals is finite E ≡ [ −Gmax , Gmax ], r and σ (t0 ) must satisfy the constraints: (17) r Gmax , σ (t0 ) ∈ E These guarantee that pσ (σ ) has its main peak within the interval E and therefore (at least approximately) satisfies equations (7) . Therefore, the explicit expression for pω (ω ) 1 pω (ω ) = √ exp r 2π

1 (G (ω ) sin ω t0 − σ (t0 ))2 − 2 r2 d G(ω ) (18) sin ω t0 + t0 G(ω ) cos ω t0 dω

where r, σ (t0 ), t0 can be regarded as arbitrary parameters but verifying restraints (17). This is the fundamental result of this paper: the pdf from equation (18) leads to a MIVAR (minimum-variance-response) and therefore produces an IEC (irreversible energy confinement). The previous theory applies to the impulse response of a random system, and, more in general, to any physical quantity h(t) that is a suitable linear combination of harmonic terms with random frequencies as in equation (1).

3 Application of the theory The examples given in this section illustrate an application for the theory described above. The system under consideration consists of set of resonators with random natural frequencies ωi (i = 1, 2, . . . , N), connected in parallel to a common principal structure of natural frequency ωM , see Figure 1. The system does not possess any means of energy dissipation. The coupled equation of master-cluster system are: ⎧ ⎨ M x¨M (t) + KM xM (t) + ∑Ni=1 ki [xM (t) − xi (t)] = 0 (19) ⎩ m x¨i (t) − ki [xM (t) − xi (t)] = 0 where M, KM , xM are the mass, stiffness and displacement of the master structure, respectively; m, ki , xi represent the same quantities for the oscillators in the attached set. The solution of the second equation provides:

Minimum-variance-response and irreversible energy confinement

221

Fig. 1

xi (t) =

ki sin ωi t ∗ xM (t) = ωi sin ωi t ∗ xM (t) m ωi

that introduced into the first produces: N

M x¨M (t) + KM + ∑ ki

xM (t) + m h(t) ∗ xM (t) = 0

(20)

i=1

where the coupling effect of the cluster on the master is collapsed in the last term, characterized by the kernel h(t) = ∑Ni=1 ωi3 sin ωi t , again of the form (1) with G (ω ) ≡ ω 3 , αi = 1. Let us consider together with equation (20) a companion equation obtained re¯ placing h by h: N ¯ ∗ xM (t) = 0 M x¨M (t) + KM + ∑ ki xM (t) + m h(t) (21) i=1

¯ =N h(t)

ωmax 0

pω (ω ) ω 3 sin ω t d ω

(22)

How the cluster affects the master motion in equation (21), can be investigated ¯ ¯ Ω ) of h(t). After some mathematics: examining the Fourier transform H( H¯ (Ω ) = − j Ω

π 4

+∞ Ω 2 pω (Ω ) + −∞

H¯ (Ω ) = 0

π pω (ζ ) ζ 3 2 (ζ − Ω )

d ζ , for Ω ∈ [0, ωmax ] (23)

elsewhere

Equation (23) shows the coupling term between the master and the cluster in ¯ Ω ). Namely an equation (21) acts as a damper, because of the imaginary part of H(

222

A. Carcaterra

equivalent viscous damping Ceq = π4 Ω 2 pω (Ω ) appears as a function of the natural frequencies pdf. This phenomenon amounts to an irreversible energy confinementIEC within the cluster of resonators. It is important to remark that a similar effect for the companion equation (20) is not shown. Therefore, the IEC effect appears ¯ Therefore the conclusion: if only if in equation (20) h is replaced by its average h. pω (ω ) is of the form given by equation (18) with G(ω ) ≡ ω 3 , then the population of h exhibits a minimum deviation from its average h¯ and the IEC effect is close to be satisfied also for equation (20). As already mentioned, this conclusion has a practical implication even for the single case: if the modal density within the cluster is n(ω ) = N pω (ω ), then it approaches, and in the closest way, a IEC. As an example, consider a master structure, with an uncoupled natural frequency ωM = 1, with N = 100 attached oscillators. Assuming t0 = π4 it follows that, within the frequency interval ω ∈ [0, 2], σ ∈ [0, 8], so that the derivative d σ /d ω is positive. The values of r and σ¯ (t0 ) (r = 0.4 and σ¯ (t0 ) = 0.6) are selected to be consistent with inequalities (17), and to assure that the function represented by equation (18) has its peak around ω = ωM = 1. Figure 2 shows the modal density of the attached oscillators from equation (18).

Fig. 2

Figure 3 shows the master response following an impulse applied at t=0, which illustrates how a significant part of its energy is transferred to the set of oscillators and remains there without returning back to the master, producing an irreversible energy confinement. We conclude the paper with an example of a build-up aerospace structure equipped by a special patented vibration absorber [9] developed on the base of the present theory and consisting of a cluster of beam shaped resonators all clamped to

Minimum-variance-response and irreversible energy confinement

223

Fig. 3

the same structural base. The lengths of the beams, i.e. the natural frequencies of the set, obey the rules of selection discussed in the previous sections. On these ideas, a device has been designed for use on board of UNISAT (UNIversity SATellite), see Figs. 4-6, which is a permanent space project developed at the placePlaceTypeUniversity of PlaceNameRome La Sapienza by the Gauss Group. UNISAT is a small scientific satellite (14-20) kg depending on the payload), launched periodically starting in 2000. The latest version of UNISAT will be equipped with the vibration suppressor described in this manuscript. Severe vibrations affect the electronic instrumentation of the satellite during lift-off of (by a DNEPR rocket launched from place country-region Russia) and the present device designed to reduce the shock and vibration the structure that carries the electronic package. Moreover, due to the safety requirements imposed, all electronic equipment in the satellites on board the launcher must be switched-off during the lift-off, making it unfeasible to use any active vibration suppressor. The material used for the absorber is steel (ρ = 7780 kg/m3 , E = 187.5 GPa, ) and the cluster of resonators is actually made of beams manufactured by milling it from a stack of steel sheets, each with a thickness of h=0.6mm. The maximum allowed space for the device is 90mm x 90mm x 40mm, with a maximum allowed mass of 0.15 kg. Measurements to validate the estimated equivalent damping given by equation (23) and effectiveness of the absorber are performed following the procedure below. As the first step, the best location for attachment point P is identified considering both the response and the geometric constraints (Figs. 5 and 6). An electro-dynamic shaker was used to excite the structure with a spectrum similar to that under operating conditions while monitoring several sensitive positions of interest. The point where the maximum amplitude response develops is identified as P (Fig. 6). The

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Fig. 4: UNISAT (version n. 3 – launched in 2004). View of the satellites room in the rocket before the launch

Fig. 5: Left: electrodynamic exciter acting on the bottom panel of the satellite, Right: inner view of the satellite structure

Minimum-variance-response and irreversible energy confinement

225

Fig. 6: Selection of the “point P” at which the device is applied

Fig. 7: Experimental drive-point FRF

peak frequency is identified from the drive point frequency response function (FRF) at P (Fig. 7) for determining the tuning frequency and the bandwidth of the damper. To verify the effectiveness of the built-up device, shown in figure 8, it was installed on the satellite plate (see Fig. 9). Figure 10 presents a comparison of the attenuated drive point FRF with attached absorber as well as the one without the absorber. The results, given for the frequency range covered by the set of first modes of the beams in the cluster, show significant reduction in the FRF amplitude, and expected to provide sufficient protection.

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Fig. 8: View of the build-up device — tuning frequency 440 Hz, total weigth 130 g

Fig. 9: View of the final installation of the device on board of UNISAT

Finally, figure 11 shows the same comparison but for the frequency bandwidth 2200-3000 Hz. This bandwidth covers the set of natural frequencies of the second modes of the beams producing an additional attenuation of the amplitude of vibration.

Minimum-variance-response and irreversible energy confinement

227

Fig. 10: Comparison of the FRFs at “point P” with (red curve) and without (blu curve) the vibration suppression device

Fig. 11: Experimental evidence of the second modes effect

References 1. A.D. Pierce, V.W. Sparrow, D.A. Russel, ‘Foundamental structural-acoustic idealization for structure with fuzzy internals’, Journal of Vibration and Acoustics, vol. 117, 339–348 (1995). 2. M. Strasberg, D. Feit, ‘Vibration damping of large structures induced by attached small resonant structures’, Journal of Acoustical Society of America, vol. 99, 335–344 (1996). 3. R.J. Nagem, I. Veljkovic, G. Sandri, ‘Vibration damping by a continuous distribution of undamped oscillators’, Journal of Sound and Vibration, vol. 207, 429–434 (1997).

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4. R.L. Weaver, ‘The effect of an undamped finite degree of freedom ‘fuzzy’ substructure: numerical solution and theoretical discussion’, Journal of Acoustical Society of America, vol. 101, 3159–3164 (1996). 5. A. Carcaterra, A. Akay, ‘Transient energy exchange between a primary structure and a set of oscillators: return time and apparent damping’, Journal of Acoustical Society of America, vol. 115, 683–696 (2004). 6. I. Murat Koc¸, A. Carcaterra, Zhaoshun Xu, Adnan Akay, ‘Energy sinks: vibration absorption by an optimal set of undamped oscillators’, Journal of Acoustical Society of America, vol. 118(5), 3031–3042 (2005). 7. A. Carcaterra, A. Akay, ‘Theoretical foundation of apparent damping and energy irreversible energy exchange in linear conservative dynamical systems’, Journal of Acoustical Society of America, vol. 121, 1971–1982 (2007). 8. A. Akay, Z. Xu, A. Carcaterra, and I.M. Koc, ‘Experiments on vibration absorption using energy sinks’, J. Acoust. Soc. Amer., vol. 118, 3043–3049 (2005) 9. A. Carcaterra and A. Akay, “Damping device”, 2006, International Patent Number: WO 2006/103291 A1.

High-frequency vibrational power flows in randomly heterogeneous coupled structures ´ Savin Eric

Abstract The purpose of this paper is to expound some recent developments for the modeling and numerical simulation of high-frequency (HF) vibrations of randomly heterogeneous structures, and outline some perspectives of future research. The mathematical-mechanical model is based on a microlocal analysis of quantum or classical linear wave systems. The theory shows that the energy density associated to their mean-zero solutions—including the strongly oscillating (HF) ones— satisfies a Liouville-type transport equation, or a radiative transfer equation in a random medium at length scales comparable to the small wavelength. Its main limitation to date lies in the consideration of energetic boundary and interface conditions consistent with the boundary and interface conditions imposed to the solutions of the wave system. The corresponding power flow reflection/transmission operators are derived formally and rigorously for elastic media, including slender structures, near the doubly-hyperbolic and hyperbolic-elliptic sets, ignoring however the glancing set. Yet a radiative transfer model in bounded media with general transverse or diffuse boundary conditions for the power flows is detailed in the paper. Some direct numerical simulations are presented to illustrate the theory.

1 Introduction Engineering structures exhibit typical transport and diffusive behaviors in the higher frequency range of vibration. These regimes are described by linear transport equations for the energy density associated to the oscillating solutions of the NavierCauchy equation for elastic wave propagation. More generally this result holds for all symmetric hyperbolic systems, including quantum waves and classical acoustic or electromagnetic waves [1, 2, 3, 4, 5, 6, 7]. It has been specialized to slender elastic ´ Savin Eric Aeroelasticity and Structural Dynamics Dpt., ONERA, 29 avenue de la Division Leclerc, F-92322 Chˆatillon cedex, France, e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 17, © Springer Science+Business Media B.V. 2011

229

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´ Savin Eric

structures, typically beams, plates and shells, and fluid-saturated poro-visco-elastic media in [8, 9, 10, 11] for applications to aerospace structures. The main limitation to date of the theory lies in the consideration of boundary and interface conditions for quadratic quantities (e.g., the energy and power flow densities) consistent with the boundary and interface conditions imposed to the displacement and stress fields. This issue is dealt with in [12] starting from an approximate high-frequency solution of the Navier-Cauchy equation for which an explicit formula of the associated energy density can be derived, the latter being also the energy density associated to the true solution, in the L2 -norm sense. This approximate solution is chosen for example as the superposition of Gaussian beams weighted by the FBI transform of the initial conditions away from the boundaries. Yet it is always possible to construct Gaussian beams which satisfy Dirichlet or Neumann boundary conditions, and their energy densities are transported along the bicharacteristic lines of the underlying wave equation. Thus the reflections of energy flows are described in phase space by those of the Gaussian beams. Notably the physical space subset yielding transverse rays is transported along the bicharacteristic lines reflected on the boundaries according to Descartes law (specular reflections). Based on these results, a formal derivation of high-frequency power flow reflection/transmission coefficients at interfaces between slender sub-structures such as beams or plates has been proposed in [11]. More recently, a systematic derivation of the elastic energy density boundary conditions for Dirichlet or Neumann boundary conditions imposed on the displacement or stress fields near the doubly-hyperbolic set (where both types of elastic waves are incident on the boundary) and the hyperbolic-elliptic set (where transverse elastic waves are incident on the boundary with an angle greater than their critical angle) has been proposed in [13]. The analysis however does not account for the glancing set and the transport of energy along an interface by so-called gliding rays. The proofs follow the original ideas of Miller [14], who in return also considered the glancing set for scalar waves. The analysis of surface effects (glancing) in the elastic case for polarized waves is the subject of ongoing research. The purpose of this paper is to expound a transport model for the high-frequency energy density evolution in bounded elastic media with general transverse or diffuse boundary conditions for the power flows. Specular-like transverse reflections and transmissions at interfaces can be treated as a particular case of the proposed model, provided that the reflection/transmission kernels have been formally identified—e.g. by the approaches cited above. Direct numerical simulations (DNS) for assemblies of beams or shells illustrate the transport regime in such systems and the onset of a diffusive regime at late times. The latter is characterized by energy equilibration rules which can be exhibited by the different schemes implemented for DNS, namely the discontinuous “Galerkin” finite element method [15] and the MonteCarlo method [16]. The examples presented in this article underline the crucial role played by boundary and interface conditions in the emergence of a diffusive regime in piecewise homogeneous media, similarly to the effects of random heterogeneities of their densities or elastic modulii [3, 17]. These results are in basic contradiction with the vibrational conductivity analogy of the structural-acoustics litterature [18, 19, 20, 21], where a diffusive regime is assumed improperly for ho-

High-frequency vibrational power flows in randomly heterogeneous coupled structures

231

mogeneous, unbounded media. The paper is organised as follows. First, the transport model for the evolution of the energy density in a bounded heterogeneous medium is outlined. Then two numerical examples are introduced in Sect. 3, which resumes and extends the numerical simulation of transient transport in an assembly of random thick beams [11] and presents some new simulations of multi-group transport in an assembly of thick cylindrical shells. At last Sect. 4 offers some conclusions and directions of future works.

2 Transport model In this section one summarizes the basic results obtained in [2, 3] for the transport properties of high-frequency waves in random elastic media, and in [5] for viscoelastic materials with memory effect. The boundary conditions to be added to the evolution model outlined in Sect. 2.1 below are detailed in Sect. 2.2 for specular-like or diffusive-like reflections at outer boundaries and in Sect. 2.3 for reflections and transmissions at interfaces.

2.1 Radiative transfer in an open domain For an heterogeneous, visco-elastic medium with statistically isotropic random perturbations of its mechanical parameters, such as the density and the Young or shear modulus, and embedded in the open domain O ⊆ Rd (R is the set of all real numbers and d = 1, 2 or 3), the evolution of the high-frequency energy density follows the multigroup radiative transfer equations [2, 3, 5]:

∂t wα + Hα wα =

M

∑

d−1 β =1 S

σαβ (x, |k|, kˆ · kˆ )(wβ (x, k ,t) − wα (x, k,t))dμ (kˆ ) , (1)

which couple M different energy propagation modes, or rays, for 1 ≤ α ≤ M. Here Hα is the Hamiltonian associated to the eigenfrequency λα (x, k) = cα (x)|k| of the mode α of which energy celerity is cα , Hα f ≡ {λα , f } = ∇k λα · ∇ x f − ∇x λα · ∇k f (the usual Poisson’s bracket), Sd−1 is the unit sphere of Rd with the uniform probability measure μ , and σαβ ≥ 0 is the scattering cross-section which gives the rate of conversion of an energy ray β in the direction kˆ to another ray α in the ˆ at position x and wavenumber |k|; the standard notation k = |k|kˆ is direction k, used throughout this paper. In a deterministic medium all scattering cross-sections are zero and right-hand-sides vanish in Eq. (1), which reduces to Liouville transport equations. The total scattering cross-section Σ α ≥ 0 for a mode α is defined by:

Σα (x, k) =

M

∑

d−1 β =1 S

σαβ (x, |k|, kˆ · kˆ ) dμ (kˆ ) .

(2)

´ Savin Eric

232

Here the σαβ ’s are assumed to depend on the directions kˆ and kˆ through their scalar product solely, yielding isotropic scattering, but more general scattering processes can be considered by which they depend on k and k . Note that the scalar Wigner measures {wα }1≤α ≤M , hereafter called phase-space energy densities or specific intensities, are non-negative if the initial data are non-negative. Then the space-time energy density E (x,t) ∈ R+ and power flow density Π (x,t) ∈ Rd of this medium are recovered by: E (x,t) =

1 M 2 α∑ =1

Rd

wα (x, k,t) dk , Π (x,t) =

1 M cα (x) 2 α∑ =1

Rd

wα (x, k,t)kˆ dk . (3)

These results are adapted to elastic waves in slender structures in [8, 10, 11], but they also apply to acoustic waves, electromagnetic waves, or the Schr¨odinger equation in random media [2, 3, 4, 6, 7].

2.2 Radiative transfer in a bounded domain The above system (1) is considered in the phase space Ω = D × Sd−1 where D is a bounded subset of O, and for t ∈ [0, T ], T < +∞ but arbitrary. Wavenumber |k| > 0 is a fixed parameter, and we shall assume as usual that the celerities cα ∈ W 1,∞ (D) are positive functions with possible isolated discontinuities if D contains interfaces between different materials; see Sect. 2.3 below for further analyses. As for the scattering cross-sections, the right-hand-side collision operator in (1) is continuous on L2 (Ω ) for σαβ ∈ L2 (Ω × Sd−1 ) and Σα ∈ L∞ (Ω ), 1 ≤ α , β ≤ M. The initial conditions are known square integrable functions, wα (x, k, 0) = w0α (x, k). Well-posedness of the problem is ensured by setting proper boundary conditions [22]. For example, denoting by nˆ the outward unit normal to the (smooth) boundary ∂ D and introducˆ ∈ ∂ D × Sd−1 ; ±kˆ · n(x) ˆ ing the boundaries Γ± = {(x, k) > 0}, an incoming flux can ˆ be imposed as cα (x)wα (x, k,t)|kˆ · n(x)| = gα (x, k,t) on Γ− for t > 0. Generalized boundary reflection laws can also be considered in the form: ˆ = gα (x, k,t) − cα (x)wα (x, k,t)kˆ · n(x) M

+

∑

ˆ ˆ β =1 k ·n(x)>0

ˆ kˆ )cβ (x)wβ (x, k ,t)kˆ · n(x) ˆ d μ (kˆ ) on Γ− , (4) ραβ (x, k,

ˆ kˆ ), 1 ≤ α , β ≤ M, are boundary reflectance coefficients for t > 0, where ραβ (x, k, ˆ at mode conversion from a ray β in the direction kˆ to a ray α in the direction k, x ∈ ∂ D. Note that these kernels depend only on the local geometry of the boundary (they may also depend on |k|). They shall satisfy: c kˆ · nˆ M β ˆ kˆ ) dμ (k) ˆ ≤ 1 a.e. x ∈ ∂ D , 0≤ ∑ ρ (x, k, ˆ · nˆ β α ˆ n(x)<0 ˆ c k k· α β =1

High-frequency vibrational power flows in randomly heterogeneous coupled structures

233

for the total flux reflected by the boundary for an incident ray α ; the second equality holds for a conservative boundary (total reflection). For example, specular reflection without mode conversion corresponds to gα ≡ 0 and: ˆ kˆ )dμ (kˆ ) = 2η (x)δαβ ⊗ δ kˆ − kˆ + 2(nˆ ⊗ n) ˆ kˆ ραβ (x, k, (5)

ˆ = d−1 dμ (k) ˆ = 1. η ≤ 1 is a bounddμ (k) with the normalization 2 ±k· ˆ n(x)>0 S ˆ ary loss factor modeling absorption: η = 1 holds for total reflection, and η < 1 holds for partial absorption at the boundary. A common example in radiative heat transfer with M = 1 is the gray diffuse boundary condition for which η (x) = 1 − ε (x), where ε > 0 is the emissivity of the boundary; in that case g(x, k,t) in Eq. (4) is the blackbody radiative flux, which is proportional to the emissivity. Another example is diffuse, or isotropic reflection (Lambertian reflectance) ˆ kˆ ) = 2η (x) independently of the observer’s angle of which corresponds to ρ (x, k, ˆ view k.

2.3 Radiative transfer with a sharp interface Assume that the physical domain D is partitioned into non-overlapping sub-domains and their interface γD ⊂ D is a smooth, bounded manifold of codimension 1 oriented by its unit normal nˆ D (ignoring edges and corners at a first glance). More specifically, γD is any interface between these sub-domains or between a sub-domain and the exterior of D and consequently, it may also be a part or the whole of ∂ D. If the celerities cα are continuous across γD , the continuity of the power flow + ˆ ˆ = c− w − k ˆ ˆ D if kˆ · nˆ D > 0 (foron γD is ensured by the conditions c+ D α wα k · n α α ·n − − + + ˆ ˆ ˆ ward flux) and cα wα k · nˆ D = cα wα k · nˆ D if k · nˆ D < 0 (backward flux), where ˆ D , k,t) for a.e. x ∈ γD , and w+ w± α (x, k,t) = limh→0+ wα (x ± hn α (x, k,t) = 0 when± ever x ∈ ∂ D; likewise, cα (x) = limh→0+ cα (x ± hnˆ D ) a.e. x ∈ γD . If however the − celerities are discontinuous across γD , such that c+ α = cα , these expressions may be generalized to the form: + ˆ ˆ D (x) = c+ α (x)wα (x, k,t)k · n M − ˆ kˆ )c− (x)w− (x, k ,t)kˆ · nˆ D (x) dμ (kˆ ) ταβ (x, k, ∑ β β

β =1

kˆ ·nˆ D (x)>0

−

if kˆ · nˆ D (x) > 0, and:

+ ˆ kˆ )c+ (x)w+ (x, k ,t)kˆ · nˆ D (x) dμ (kˆ ) ραβ (x, k, β β kˆ ·nˆ D (x)<0

(6)

´ Savin Eric

234 − ˆ ˆ D (x) = c− α (x)wα (x, k,t)k · n M + ˆ kˆ )c+ (x)w+ (x, k ,t)kˆ · nˆ D (x) dμ (kˆ ) ταβ (x, k, ∑ β β

β =1

kˆ ·nˆ D (x)<0

−

− ˆ kˆ )c− (x)w− (x, k ,t)kˆ · nˆ D (x) dμ (kˆ ) ραβ (x, k, β β kˆ ·nˆ D (x)>0

(7)

± ± and ταβ are the left (−) and right if kˆ · nˆ D (x) < 0. In the above equations ραβ (+) reflectance/transmittance coefficients, respectively, of the interface γD where the cα ’s may be discontinuous. They satisfy the conservation rules: M

∑

ˆ kˆ )v(kˆ )(kˆ · nˆ D (x)) dμ (kˆ )dμ (k) ˆ τβ±α (x, k,

ˆ nˆ D (x)<0 ±kˆ ·nˆ D (x)<0 ±k·

β =1

−

ˆ kˆ )v(kˆ )(kˆ · nˆ D (x)) dμ (kˆ )dμ (k) ˆ ρβ±α (x, k,

ˆ nˆ D (x)>0 ±kˆ ·nˆ D (x)<0 ±k·

≤

ˆ kˆ · nˆ D (x)) dμ (k) ˆ a.e. x ∈ γD v(k)(

ˆ nˆ D (x)<0 ±k·

ˆ n|d ˆ μ ), stating that the total flux impinging the boundary for, say, any v ∈ L1 (Sd−1 , |k· is either reflected or transmitted with or without losses; a strict inequality holds in the former case, while a strict equality holds in the latter case. Note that these ± ± conditions relate transmittances ταβ and reflectances ραβ by the requirement that the normal energy flow Π · nˆ D be conserved across the interface; it turns out that the energy density itself is not necessarily continuous across that interface. ± ± ± ˆ ˆ Example 1. Simply choosing ραβ ≡ 0 and ταβ ≡ δ (c∓ α k − cβ k ) (no scattering on γD ) yields a continuous upwind flux, as above for continuous celerities.

− = 0 and no incoming flux, Example 2. If γD is a part of the boundary ∂ D with ταβ + wβ = 0 for 1 ≤ β ≤ M, the boundary flow of Eq. (4) with gα ≡ 0 is directly recovered.

Example 3. For d ≥ 2 and considering specular-like transverse reflection and transmission of polarized elastic waves on a sharp interface, the reflectance and transmit± ± tance coefficients ραβ and ταβ have the form: ± ˆ kˆ )dμ (k) ˆ = R ± (x, kˆ )δ (kˆ − R ˆ ± (kˆ )) , ραβ (x, k, αβ αβ ± ˆ kˆ )dμ (k) ˆ = T ± (x, kˆ )δ (kˆ − Tˆ ± (kˆ )) , ταβ (x, k, αβ αβ

(8)

ˆ ± are one-to-one mappings from ω ∓ (x) into ω ± (x) and ω ∓ (x), ˆ ± and T where R D D D αβ αβ respectively, with ωD± (x) = {kˆ ∈ Sd−1 ; ±kˆ · nˆ D (x) > 0}. More precisely, they are defined on the open hyperbolic subsets C ± (x, Θ ) of ωD± (x), which are the intersections with Sd−1 of the cones of vertex x, axis ±nˆ D (x), and solid angles Ωd sin2 (Θ /2), where Θ are critical angles arising in conjunction with Descartes law

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stating that the tangential wave vector remains constant in the reflection and trans−− (x) mission process at a point x on the interface γD . They are such that Θ ≡ θαβ − +− − ++ + −+ ˆ ˆ ˆ for Rαβ , Θ ≡ θαβ (x) for Tαβ , Θ ≡ θαβ (x) for Rαβ , and Θ ≡ θαβ (x) for Tˆ + αβ , ±± ± ). Let T γ be the tangent space to γ at x and let where θαβ (x) = arcsin(c± /c x D D α β Px = Id − nˆ D (x) ⊗ nˆ D (x) be the orthogonal projection from Rd into that space, where Id stands for the identity matrix in Rd ; then Descartes law writes: c± β

c± β ± ˆ ˆ ˆ± ˆ ˆ P ( k) = R x αβ ∓ Px Tαβ (k) = Px k , c± c α α

∀ kˆ ∈ Sd−1 ,

∀ α , β = 1, 2 . . . M .

ˆ ± and T ˆ ± have the properties (reciprocity): In addition, the mappings R αβ αβ ˆ ± (k)) ˆ = kˆ , kˆ ∈ C ∓ (x, θ ±± ) , −Tˆ ∓ (−T ˆ ± (k)) ˆ = kˆ , kˆ ∈ C ∓ (x, θ ∓± ) , ˆ ± (−R −R βα αβ αβ βα αβ αβ and (symmetric reflection without mode conversion — see also Eq. (5)): ˆ ˆ± ˆ D ⊗ nˆ D )kˆ , R αα (k) = (Px − n

±± kˆ ∈ C ∓ (x, θαα ) ≡ ωD∓ (x) .

± ± = Tαβ = 0. The On the elliptic sets ωD± (x) \ C ± (x, Θ ), α = β , one simply has Rαβ ± ± ˆ · nˆ D = Tˆ (k) ˆ · nˆ D = 0 but they are ignored in ˆ (k) glancing sets are such that R αβ αβ our formulation because a theoretical model lacks at present to describe them, espe± ± cially the corresponding kernels Rαβ and Tαβ in the elastic case. The latter are otherwise positive functions and, at least, smooth on C ∓ (x, Θ ). A rigorous derivation of the reflection/transmission operators (8) for elastic media with Dirichlet or Neumann boundary conditions away from the glancing set is proposed in [13], while a formal derivation for slender elastic structures is proposed in [11]. Both are adapted from the ideas developed by Miller for scalar waves in [14] (this work also analyzes rigorously the glancing set). This case has also been studied numerically by Jin & Yin [23].

3 Numerical examples In Sect. 3.1 the example of [11] of scalar transport in coupled Timoshenko beams is resumed and extended to random heterogeneities. Sect. 3.2 deals with two coupled thick cylindrical shells. In these examples the radiative transfer equations (1) with specular-like boundary and interface (at the junctions) conditions are solved numerically by two distinct schemes: a Runge-Kutta “Galerkin” discontinuous finite element method is used for the first example, while a fully vectorized direct Monte-Carlo method is used for the second example.

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3.1 Coupled beams We first consider an assembly of two random Timoshenko beams ]−L, 0[∪]0, L[, L > 0, with a junction at x = 0 such that beam #2 forms an angle φ with beam #1. Three types of energy rays propagate in such a system: a pure shear transverse mode at the celerities cT j ( j = 1 or 2 for either beam #1 or beam #2) denoted by subscript T, a pure bending mode at the celerities cP j denoted by subscript Pb, and a longitudinal mode at the celerities cP j denoted by subscript Pn; see [10, 11] for some details on the derivation. Therefore the energy density evolution in this system is depicted by three coupled, one-dimensional radiative transfer equations (M = 3 and d = 1 such that kˆ ≡ sign(k)) written as: 1 ∂t wα + kˆ ∂x wα + η wα = Σα (|k|)(wα (−k) − wα (k)) , cα

α = T, Pb, Pn ,

(9)

where the coupling is ensured by the left (−) and right (+) power reflection and ± ± (φ ), Tαβ (φ ) of the junction. Indeed, for thick beams transmission kernels Rαβ no coupling of modes occurs through the collision operator on the right-hand± ± and Tαβ as functions of φ and side in Eq. (1) [10, 11]. The expressions of Rαβ beams mechanical and geometrical characteristics are derived in [11], using a highfrequency model compatible with the propagation characteristics obtained by the transport theory. This study has also shown that the power reflection coefficients at a beam end for either Dirichlet (clamped) or Neumann (free) boundary condition were all equal to 1, without mode conversion of any type; therefore specular reflections are considered at the boundaries x = ±L. The total scattering cross-sections are √ 2 (Gaussian model) Σ α (k0 ) = 2π rα k02 e−k0 , k0 = k, where is a correlation length of the parameters random fluctuations and rα is a scaling factor such that |rα | ≤ 1. All computations are performed for the initial conditions: − ln 2 w0T (x, k) = e

x−x0 r0

2

⊗ δ (kˆ − ε0 ) ,

w0Pb = w0Pn ≡ 0 ,

with x0 = −0.1 × L, r√0 = 0.4 × L, and ε0 = +1. Beam celerities are such that cT2 /cT1 = cP2 /cP1 = 2, their Poisson’s ratio being ν1 = ν2 = 0.3, and the junction angle is φ = π /3. The coupled radiative transfer equations (9) are solved numerically by the Runge-Kutta “Galerkin” discontinuous finite element method introduced in [25] (see also [15], for example, for a general reference). In the proposed scheme, the reflectance/transmittance coefficients are directly taken into account by the numerical fluxes. In this example, the assembly is partitioned into 100 uniform spatial elements. Legendre polynomials up to the fourth order are used as local basis functions on them, and Eqs. (9) are solved in time by a second-order SSP (Strong-StabilityPreserving) Runge-Kutta scheme [24]. The Courant number for the simulations is fixed at υ ≡ cT2 ΔΔ xt = 2.5 10−2 . Fig. 1 and Fig. 2 display our results for either rα = 0 or rα = 0.3 and k0 = 1.5, respectively, for the scattering cross-sections. The first case corresponds to pure transport without scattering in a deterministic, piecewise

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homogeneous assembly, and the second case corresponds to full radiative transfer in a randomly heterogeneous assembly. The time scale is T = L/cT2 , the time needed by the wave front to reach the extremity x = L of the homogeneous assembly starting from the junction. The thin vertical line indicates the location of the junction. The wavefronts of shear (α = T) and bending/axial (α = P) energy densities propagating at the speeds cT j and cP j , respectively, are marked by small capital letters. A small amount of damping η = 0.02 m−1 has been added such that the total energy within the system at a fixed wavenumber |k| evolves as (see Eq. (3)): E (t) =

∑

α =T,Pb,Pn

e−η cα t Eα0 ,

Eα0 =

L −L

w0α (x, ±|k|)dx .

(10)

This exact result is recovered numerically by our scheme, as shown by Fig. 3 and Fig. 4. They display E jh (t) computed within each 0 theh overall mechanical L energies h h h beam by E1 (t) = −L E (x,t)dx, E2 (t) = 0 E (x,t)dx, and their sum, normalized by the total energy introduced into the system E (0) = ∑α Eα0 = ET0 . The computed and exact total energies perfectly match on these plots: one concludes that the proposed scheme in dimension one is numerically conservative. These results exhibit the following features: one observes a beating phenomenon of the total energy within each subsystem, by which it is transferred back and forth from the first beam to the second one and the other way round. At late time an equipartition regime holds between both beams. It is described by diffusion equations as explained in [17] for example. It should be noted that this diffusive regime also arises in the case of a piecewise homogeneous assembly (see Fig. 3), the onset of diffusion being prompted by the mixing of wave fronts and modes at the junction, and to a lesser extent at the boundaries. This mixing process is accelerated by the presence of random heterogeneities, as seen on Fig. 4. Fig. 1: Evolution of the energy density in an assembly of two homogeneous Timo= π3 , shenko beams with φ √ cT2 /cT1 = cP2 /cP1 = 2, ν1 = ν2 = 0.3, η = 0.02 m−1 , and T = L/cT2

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Fig. 2: Evolution of the energy density in an assembly of randomly heterogeneous φ= Timoshenko beams with √ π 3 , cT2 /cT1 = cP2 /cP1 = 2, ν1 = ν2 = 0.3, η = 0.02 m−1 , T = L/cT2 , and rα = 0.3, k0 = 1.5 for the scattering cross-sections

Fig. 3: Evolution of the total vibrational energies in an assembly of two homogeneous Timo= π3 , shenko beams with φ √ cT2 /cT1 = cP2 /cP1 = 2, ν1 = ν2 = 0.3, η = 0.02 m−1 , and T = L/cT2 . Thick solid line: beam #1, thick dashed line: beam #2, thin solid line: computed total energy, thin dashed line: exact total energy

Fig. 4: Evolution of the total vibrational energies in an assembly of two randomly heterogeneous Timoshenko beams with√φ = π3 , cT2 /cT1 = cP2 /cP1 = 2, ν1 = ν2 = 0.3, η = 0.02 m−1 , T = L/cT2 , and rα = 0.3, k0 = 1.5 for the scattering cross-sections. Thick solid line: beam #1, thick dashed line: beam #2, thin solid line: computed total energy, thin dashed line: exact total energy

3.2 Coupled shells Next we consider an assembly of two thick cylindrical shells, the first one with a constant radius and the second one with a variable radius. Both cylinders have the same length L which is also twice the constant radius of the first shell. The angle

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between the shells is again denoted by φ , and their mean surfaces are denoted by D1 and D2 , respectively. Five types of energy rays propagate in such a system: a pure shear transverse mode at the celerities cT j ( j = 1 or 2 for either shell #1 or shell #2) denoted by subscript T, and coupled longitudinal and transverse modes P and S each of multiplicity 2 corresponding to either in-plane (subscript n) or bending energies (subscript b), at the celerities cP j and cS j respectively; see [8] for some details on the derivation. Then the energy density evolution in this system is depicted by five coupled radiative transfer equations (1) with M = 5—since α = T, Pb, Pn, Sb or Sn—and d = 2, where the coupling is ensured by the left (−) ± ± and right (+) power reflection and transmission kernels Rαβ (φ , kt ), Tαβ (φ , kt ) of the shell junction. Their expressions as functions of φ , the incident tangent wave vector kt = Pxt k on the junction line γD xt , and shells mechanical and geometrical characteristics are derived in [11], using a high-frequency model compatible with the propagation characteristics obtained by the transport theory within plates. As the curvature tensor has no influence on the high-frequency regime in this model, the same reflection/transmission kernels as for flat plates are used in this example. Neumann boundary conditions are considered at the extremities of the shells, and the corresponding power reflection coefficients have been derived in [11] as well. Here it has been shown that the shear mode is uncoupled to the in-plane and bending modes by boundary reflections, however the longitudinal P and transverse S modes are coupled. Otherwise all these modes are fully coupled by the junction. The initial condition is an isotropic shear impact w0T (x, k) = δ (x − x0 ), independently of k, all other modes being initially unloaded. Here x0 = (−L/10, 0) and the junction line is the circle γD = {x = (0, y), y ∈ [0, π L]} in curvilinear coordinates. The Young’s modulii, densities, and thicknesses of both shells are equal, their Poisson’s ratios are ν1 = 0.1 and ν2 = 0.4, and the junction angle is φ = π /12. The coupled radiative transfer equations (1) are solved in this case by a direct Monte-Carlo method [16]. The computer code developed for this application has been fully vectorized, in such a way that numerical simulations of O(106 ) ray paths with O(102 ) time steps can be performed in less than one hour of CPU time on a personal computer. Fig. 5 through Fig. 7 below display our results for pure transport without scattering in a deterministic, piecewise homogeneous assembly for illustration purposes. Propagation in random shells was considered in [8]. The time scale is T = L/cT2 , and a small amount of damping η = 0.02 m−1 has been added as for the beam assembly of the first example. Fig. 5 displays the evolution of the energy density within the shell assembly, while Fig. 6 displays the evolution of the overall mechanical energies within each shell and their sum. In addition Fig. 7 displays the ratios EPhj (t)/EShj (t) of the total longitudinal and transverse energies within each shells, respectively, for the same data as above except η = 0, where: EPhj (t) =

∑

1 α =Pn,Pb D j ×S

ˆ , whα (x, k,t)dx dμ (k)

j = 1, 2 ,

and a similar expression for EShj (t). For an isotropic collision operator, their diffusive limit at large times is proved to be the ratio of the celerities cS j /cP j to the power of

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the physical dimension, d = 2 in the present case, independently of the source type and scatterers [3, 17]. Here this universal equilibration rule is apparently recovered for anisotropic scatterers constituted by the junction and the boundaries. Again, it can be concluded that the numerical scheme is conservative, and that it exhibits the equilibration rule of energies between both shells in the late-time diffusive regime. The emergence of the latter is prompted here by the sole contribution of the energy rays numerous reflections and transmissions at the boundaries and the junction, together with mode conversions. A quite unexpected observation is the promptness of the stabilization of the energy ratios with time. This result, however, ignores the contribution of the glancing rays possibly trapped by the boundaries and the junction line, because a theoretical model lacks at present to describe them. Indeed it is known from experiments that the vibrational energy is very much likely to be guided by the interfaces, stiffeners or junctions in a real-life complex structure, so these effects should not be disregarded. Further developments are needed on this issue. Fig. 5: Evolution of the energy density in an assembly of two homogeneous cylindrical π , ν1 = 0.1, shells with φ = 12 ν2 = 0.4, η = 0.02 m−1 , and T = L/cT2

Fig. 6: Evolution of the total vibrational energies within each shell of the assemπ bly with φ = 12 , ν1 = 0.1, ν2 = 0.4, η = 0.02 m−1 , and T = L/cT2 . Thick solid line: shell #1, thick dashed line: shell #2, thin solid line: computed total energy

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Fig. 7: Evolution of the ratio c2P j EPhj (t)/c2S j EShj (t) of the total longitudinal and transverse energies within each shell of the assembly π , ν1 = 0.1, ν2 = with φ = 12 0.4, η = 0, and T = L/cT2 . Solid line: shell #1 ( j = 1), dashed line: shell #2 ( j = 2)

4 Conclusions A transport model for the elastic energy propagation in bounded heterogeneous media has been proposed. It accounts for general transverse reflection/transmission effects at boundaries and interfaces, including specular-like reflections according to Descartes law as a particular case. This model is amenable to be solved numerically by the discontinuous finite element method or the Monte-Carlo method, among other possible schemes. Both methods have been implemented and tested on examples of assemblies of thick beams or cylindrical shells. DNS illustrate the transport regime for these systems, and the onset of a diffusive regime at late times can also be exhibited. It is characterized by energy equilibration rules very much comparable to the assumption of modal equipartition invoked in the statistical energy analysis (SEA) of structural-acoustics systems [26]. However a rigorous mathematical model of the diffusive regime for bounded media lacks at present. It should be able to predict the equilibration ratios as function of the sub-systems mechanical parameters, as well as their unexpectedly rapid stabilization with time in two dimensions. Another (still) open issue is the consideration of the glancing set of energy rays — the diffractive and gliding ones — for the part of the power flows possibly trapped by the interfaces and junctions. Both aspects are the subject of ongoing research. Acknowledgements The author acknowledges the support of the French National Research Agency (contract n. ANR-06-CIS6-003) for some parts of this work.

References 1. G´erard P., Markowich P.A., Mauser N.J., Poupaud F.: Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. L (1997) 323–379 2. Guo, M., Wang, X.-P.: Transport equations for a general class of evolution equations with random perturbations. J. Math. Phys. 40 (1999) 4828–4858

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3. Papanicolaou G.C., Ryzhik L.V.: Waves and transport. In Caffarelli, L., E, W., eds.: Hyperbolic Equations and Frequency Interactions. Volume 5 of IAS/Park City Mathematics Series, American Mathematical Society, Providence, RI (1999) 305–382 4. Erd¨os L., Yau H.-T.: Linear Boltzmann equation as the weak coupling limit of a random Schr¨odinger equation. Comm. Pure Appl. Math. LIII (2000) 667–735 5. Akian, J.-L.: Wigner measures for high-frequency energy propagation in visco-elastic media. Technical Report RT 2/07950 DDSS, ONERA, Chˆatillon (2003) 6. Powell J.M., Vanneste J.: Transport equations for waves in randomly perturbed Hamiltonian systems, with application to Rossby waves. Wave Motion 42 (2005) 289–308 7. Lukkarinen J., Spohn H.: Kinetic limit for wave propagation in a random medium. Arch. Rat. Mech. Anal. 183 (2007) 93–162 ´ Transient transport equations for high-frequency power flow in heterogeneous cylin8. Savin E.: drical shells. Waves Random Media 14 (2004) 303–325 ´ Radiative transfer theory for high-frequency power flows in fluid-saturated, poro9. Savin, E.: visco-elastic media. J. Acoust. Soc. Am. 117 (2005) 1020–1031 ´ High-frequency vibrational power flows in randomly heterogeneous structures. In 10. Savin E.: Augusti, G., Schu¨eller, G.I., Ciampoli, M., eds.: Proceedings of the 9th International Conference on Structural Safety and Reliability ICOSSAR 2005, Rome, 19–23 june 2005. Millpress Science Publishers, Rotterdam (2005) 2467–2474 ´ A transport model for high-frequency vibrational power flows in coupled heteroge11. Savin E.: neous structures. Interaction Multiscale Mech. 1 (2007) 53–81 12. Bougacha S., Akian J.-L., Alexandre R.: Gaussian beams summation for the wave equation in a convex domain. Comm. Math. Sci. 7 (2009) 973–1008 13. Akian J.-L.: Semiclassical measures for tridimensional elastodynamics: boundary conditions for the hyperbolic set. Asym. Anal. (2009) submitted 14. Miller L.: Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary. J. Math. Pure Appl. 79 (2000) 227–269 15. Hesthaven J.S., Warburton T.: Nodal Discontinuous Galerkin Methods. Springer, New York, NY (2008) ´ Sentis R.: M´ethodes de Monte-Carlo pour les Equations ´ 16. Lapeyre B., Pardoux E., de Transport et de Diffusion. Springer-Verlag, Berlin (1998) ´ Diffusion regime for high-frequency vibrations of randomly heterogeneous struc17. Savin, E.: tures. J. Acoust. Soc. Am. 124 (2008) 3507–3520 18. Nefske, D.J., Sung, S.H.: Power flow finite element analysis of dynamic systems: basic theory and application to beams. ASME J. Vib. Acoust. Stress. Reliab. Des. 111 (1989) 94–100 19. Bouthier, O.M., Bernhard, R.J.: Models of space-averaged energetics of plates. AIAA J. 30 (1992) 616–623 20. Langley, R.S.: On the vibrational conductivity approach to high frequency dynamics for twodimensional structural components. J. Sound Vib. 182 (1995) 637–657 21. Le Bot, A., Ichchou, M.N., J´ez´equel, L.: Energy flow analysis for curved beams. J. Acoust. Soc. Am. 102 (1997) 943–954 22. Bensoussan, A., Lions, J.-L., Papanicolaou, G.C.: Boundary layers and homogenization of transport processes. Publ. RIMS, Kyoto Univ. 15 (1979) 53–157 23. Jin, S., Yin, D.: Computational high frequency waves through curved interfaces via the Liouville equation and geometric theory of diffraction. J. Comp. Phys. 227 (2008) 6106–6139 24. Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43 (2001) 89–112 ´ Discontinuous finite element solution of radiative transfer 25. Berton, B., Candapin, D., Savin, E.: equations for high-frequency power flows in slender structures. Preprint (2009) 26. Lyon, R.H., DeJong, R.G.: Theory and Application of Statistical Energy Analysis, 2nd ed. Butterworth-Heinemann, Boston, MA (1995)

Uncertainty propagation in SEA using sensitivity analysis and Design of Experiments Antonio Culla, Walter D’Ambrogio and Annalisa Fregolent

Abstract A limit of Statistical Energy Analysis (SEA) is that of providing only the mean values of the mechanical energy of a vibrating system. In the proposed paper, the variability of SEA solution under uncertain SEA parameters (coupling loss factors and internal loss factors) is investigated by comparing a sensitivity approach and a Design of Experiment (DoE) approach. Uncertainties of the SEA parameters depend on uncertainties in the physical properties of the considered mechanical system (Young modulus, material density, geometry, . . . ). Numerical results are derived using a benchmark structure made by three aluminum plates with a common junction.

1 Introduction In Statistical Energy Analysis, the studied systems belong to a random population of similar systems [1]. Systems are considered similar if their physical parameters are slightly different. SEA considers a structure as the union of several subsystems. Each of them is a modal group, i.e. a set of similar modes. For instance, considering two plates welded together, six modal groups can be identified, one set of flexural modes and two sets of in plane modes for each plate.

Antonio Culla Dipartimento di Meccanica e Aeronautica, Universit`a di Roma “La Sapienza”, Via Eudossiana 18, 00184 Roma, Italy, e-mail: [email protected] Walter D’Ambrogio Dipartimento di Ingegneria Meccanica, Energetica e Gestionale, Universit`a dell’Aquila, Piazza E. Pontieri 2, 67040 Roio Poggio (AQ), Italy, e-mail: [email protected] Annalisa Fregolent Dipartimento di Meccanica e Aeronautica, Universit`a di Roma “La Sapienza”, Via Eudossiana 18, 00184 Roma, Italy, e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 18, © Springer Science+Business Media B.V. 2011

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SEA estimates the mean value of the energy stored in the modal groups constituting the studied system. The mean value provided by SEA equations is in principle the average response of a set of similar systems. However, SEA equations are represented by a linear system of equations for each frequency, or better for each frequency band. The solution of each linear system gives the energy of each subsystem in a given frequency band. No average operation is explicitly performed, but all the statistics is not visible to the user. In general, this is not a problem because many simple relationships used by physicists and engineers are the result of more complicated mathematical procedures. Unfortunately, in this case this simple model holds only under many strong hypotheses, listed in Section 2. The linear system results from some mathematical manipulations, that include also averages on frequency bands, on the classical equations of motion of multi degrees of freedom systems, and the observance of the strong hypotheses mentioned before. The coefficients of the linear system, named coupling loss factor (CLF) and internal loss factor (ILF), are the result of these average processes and account for the parameters of the native physical system. Therefore, SEA gives the energy of each modal group belonging to the studied system. This energy is the most representative sample of a statistical population of similar systems and on a frequency band. No information is given about the dispersion of the data around the result. In order to provide a true statistic solution it is necessary, at least, to know the variance of the result. A correct procedure should calculate the variance of the energy by starting from the equation of motion following a similar procedure like that performed for the mean calculation. Lyon [1, 2] estimates the variance by using a particular probability distribution of natural frequencies. Therefore this procedure neglects a direct dependence of the modal parameters on the randomness of the physical properties. However, he concludes that “There is a considerable area of interesting research work that needs to be done in analysing variance of interacting systems.” Radcliffe and Huang [3] study the problem by introducing a stochastic perturbation in SEA equations, by using a first order approximation of them in terms of this random perturbation and by calculating the variance of the new linear equations. They state that the lack of information of SEA solution may be filled by calculating the solution variance due to the random perturbation of SEA parameters (coupling loss factor, injected power, . . . ). Langley and Cotoni [4] follow the Lyon’s approach and study the problem by considering the Gaussian orthogonal ensemble, detailed in Weaver [5]. Some other authors [6, 7] assume that uncertainties lie on the system parameters and they achieve results not exhibiting the same tendency of the Lyon’s prediction. Bussow et al. [8] propose an analytical approach for the investigation of the problem of uncertain system parameters. The analysis is performed by partial derivatives of the energies. It shows the effect of uncertainties of a given parameter. All these methods to approach the uncertainty problems use either parametric or nonparametric models [9]. In the first case the physical properties of the system are considered to be uncertain. The uncertainty of the response is calculated by considering the propagation of the physical uncertainties through the model. If the system is very complex and random, its natural frequencies statistics can be assumed, that

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is the statistics of the response can be considered independent of the statistics of the physical quantities. This is the nonparametric case. In this paper a parametric approach is adopted. It proceeds from the belief of the authors that SEA is a methodology able to analyse well both the actual nonparametric models (very complex and random systems) and the models (not so complex and random) which could be solved also by a FE approach (simple structure, not much uncertain, forced by random loads). Moreover, aim of this paper is not only to evaluate the variability of SEA solution, but also to calculate the sensitivity of the energies stored into the subsystems by considering uncertainties on ILF’s and CLF’s. To be more precise, the nominal values of CLF’s and their range of variability are those resulting from a previous analysis. The effect of uncertainties is modeled by using both a sensitivity approach and a DoE approach [10, 11]. DoE provides a regression model of energies: the coefficients of this model show the influence of the uncertain parameters on the energy stored in each subsystem. Finally, a comparison between sensitivity and DoE results is presented.

2 SEA equations Under some particular hypotheses, it is possible to assume that the transmitted power between two subsystems is proportional to the difference of the energy stored in each subsystem. A list of these hypotheses is presented below: • all the modes of a subsystem must be similar (i.e. they must have almost the same energy, damping, coupling with the other subsystems and they must be almost excited by the same input power), • the coupling between the subsystems must be conservative, • the eigenfrequencies must be uniformly probable in the frequency range, • the force exciting the subsystems must be random and not-correlated, • the interactions between the subsystems must be weak. Thus, the SEA equations of Nsub coupled subsystems can be written as follows: Pi,in j = ω ηi Ei + ω

Nsub

∑

(ηi j Ei − η ji E j )

(1)

j=1, j=i

where i and j are indexes of the subsystems, ηi and ηi j are the internal loss factors (ILF) and the coupling loss factors (CLF), respectively, Pi,in j is the power injected into the subsystem i, E is the energy in a given subsystem and ω is the central frequency of the considered band. Equations (1) represents the energy balance of the subsystems. The power dissipated in the subsystem i is: Pi,d = ω ηi Ei The power transmitted from subsystem i to the subsystem j is:

(2)

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Pi j = ω (ηi j Ei − η ji E j )

(3)

The solution of the linear system (1) provides the energy stored in each subsystem. The set of equations (1) can be rewritten in a more convenient way as follows: p = ω Ce

(4)

where the coefficients of matrix C are combinations of ILF’s and CLF’s as shown in the following equations: ⎧ C = −η ji ⎪ ⎨ ij ⎪ ⎩C j j = η j +

i, j = 1 . . . Nsub , i = j Nsub

∑

η ji

(5)

i=1, i= j

If ni and n j are the modal densities of subsystems i and j, the following reciprocity relationship holds: ηi j ni = η ji n j (6) By enforcing reciprocity, under the assumption that only the η ji with j > i are known, it is: ⎧ ⎧ ⎨−η ji ⎪ if j > i ⎪ ⎪ ⎪ n = C j ⎪ i j ⎪ if j < i ⎩−η ji ⎨ ni (7) ⎪ ⎪ Nsub ⎪ ⎪ ⎪ ⎪ ⎩ C j j = η j − ∑ Ci j i=1, i= j

3 Uncertainty propagation in SEA SEA gives only the mean value of the energy of a set of similar systems. This is not a complete statistical information, because at least the dispersion of the data around the mean is lacking. The correct variance of the solution could be obtained by working directly on the equation of motion as it was done to provide the SEA equations. Here a study on the variability of SEA results is performed. Therefore, the goal of this research is not a way to achieve the variance matching the classical SEA solution, but to understand how much the energies (SEA solution) depend on uncertainties on CLF’s and ILF’s. SEA equations are deterministic, and CLF’s are deterministic functions of the physical parameters as well. The solution of this deterministic set of equations, the energies of the modal groups, depends on the ILF’s, the CLF’s and the injected powers. In order to study the variability of SEA solution, many techniques can be followed (Monte Carlo, sensitivity, Design of Experiments, etc.). Let us consider a given mechanical system made of Nsub subsystems. CLF’s depend on the material properties and the geometric parameters of the coupled sub-

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systems. Therefore, a given ηi j depends, for instance, on the Young modulus of the systems i and j, Yi and Y j , and on the thickness of these subsystems, ti and t j . The energy of each subsystem is calculated by solving equation (4) with the obvious implication that energies depend on the CLF’s and the ILF’s of the considered system. By defining a range of variability of the physical parameters, a procedure can be developed in order to obtain the range of variability of the CLF’s. At this point, both a sensitivity approach and a DoE procedure are developed to account for the dependence of the energy on the variability of SEA coefficients.

3.1 Approach using sensitivity Sensitivity to loss factors is evaluated in correspondence to nominal values ηˆ of the CLF’s and ILF’s. To compare different sensitivity factors, it is assumed that changes Δ ηkl in the coupling loss factors are not arbitrarily chosen, but are those corresponding to variations of the physical parameters. ∂ e Δ ekl = Δ ηkl (8) ∂ ηkl η =ηˆ and similarly for ILF’s. To find ∂ e/∂ ηkl , it is necessary to differentiate the solution of Eq. (4): e=

1 −1 C p ω

⇒

1 ∂ C−1 ∂e = p ∂ ηkl ω ∂ ηkl

(9)

and similarly if internal loss factor ηk are considered instead of ηkl . Here it is assumed that the injected power is not affected by changes in CLF’s and ILF’s. The derivative of C−1 can be easily obtained from the identity C C−1 = I

∂ C−1 ∂ C −1 = −C−1 C ∂ ηkl ∂ ηkl

(10)

where ∂ C/∂ ηkl can be computed from Eq. (7): ⎧ 1 ⎪ ⎪ ⎪ nk ⎪ ⎪ ⎪ ⎪ ∂ Ci j ⎨ nl = −1 ∂ ηkl ⎪ n ⎪ ⎪ ⎪− k ⎪ ⎪ ⎪ ⎩ nl 0 and similarly for ∂ C/∂ ηk :

if i = j and i = k if i = l and j = l if i = l and j = k if i = k and j = l else

(11)

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∂ Ci j = ∂ ηk

1 if i = j and i = k 0 else

(12)

The variance of the energy stored in a given subsystem i can be approximately evaluated from the variance of CLF’s and ILF’s as follows: Nsub k−1 Nsub ∂ Ei 2 2 ∂ Ei 2 2 σE2i = ∑ ∑ σηkl + ∑ σηk (13) k=1 l=1 ∂ ηkl k=1 ∂ ηk

3.2 Approach using Design of Experiments In Design of Experiments (DoE), the values of the variables that affect an output response are appropriately modified by a series of tests, to identify the reasons for changes in the response. This does not prevent from performing numerical tests whenever this may be convenient for a better understanding of the numerical problem under investigation. Since many experiments involve the study of the effects of two or more variables or factors, it is necessary to investigate all possible combinations of the levels of the factors. This is performed by factorial designs which are very efficient for this task. Specifically, if p factors at two levels are considered, a complete series of experiments requires 2 p observations and is called a two-level 2 p full factorial design. Usually, each series of experiments should be replicated several times using the same value of the factors to average out the effects of noise. Of course, this is unnecessary if experiments are numerical. A feature of two-level factorial design is the assumption of linearity in the effect of each single factor and of multi-linearity in interactions among factors. To account for possible non linear effects, quadratic terms can be introduced, as in the following regression model for two factors: f = α0 + α1 x1 + α2 x2 + α12 x1 x2 + α11 x12 + α22 x22 + ε

(14)

where the α ’s are parameters whose values are to be determined, the variables x1 and x2 are defined on a coded scale from −1 to +1 (the low and high levels of the two factors) and ε is an error term. Of course, a three level (low level −1, intermediate level 0, high level +1) factorial design, involving 3 p observations, is a possible option if quadratic terms are important. However, a more efficient alternative is the Central Composite Design (CCD) that starts from the 2 p design augmented with the center point i.e. a single observation with all factors at intermediate level, and axial runs where each factor is considered at two levels (the low level −1 and the high level +1) while the remaining factors are at the intermediate level, for a total of 2p observations.

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Overall, a central composite design for p factors requires n = 2 p + 2p + 1 observations instead of 3 p observations required by the three level factorial design, with advantages for p ≥ 3. For p control factors, the experimental response can be expressed as a regression model representation of a 2 p full factorial experiment (involving 2 p terms), augmented with p quadratic terms: p

p i−1

i=1 p i−1 m−1

i=1 j=1

f = α0 + ∑ αi xi + ∑ ∑ α ji x j xi + . . . + + ∑ ∑ ··· i=1 j=1

∑

n=1

p

αnm··· ji xn xm · · · x j xi + ∑ αii xi2 + ε

(15)

i=1

The expression contains 2 p + p parameters α , each one providing an estimate of the effect of a single factor (linear or quadratic) or of a combination of them. Note that Eq. (15) is linear in the parameters α , and it can be rewritten as: ⎧ ⎫ α0 ⎪ ⎪ ⎪ ⎪

⎨ α1 ⎬ f = 1 x1 · · · x2p (16) +ε .. ⎪ ⎪ . ⎪ ⎪ ⎩ α pp ⎭ having arranged the parameters in a vector α . A different equation can be written for each observation by varying the factors (x1 , . . . , x p ) as indicated by CCD. By arranging the experimental responses in a vector f, a linear relationship between f and α can be expressed in matrix notation as: f = Xα + ε

(17)

where X is a (2 p + 2p + 1) × (2 p + p) matrix. The least square estimate of α is:

αˆ = (XT X)−1 XT f

⇒

ˆf = Xαˆ

(18)

where ˆf is the fitted regression model. The difference between the actual observations vector f and the corresponding fitted model ˆf is the vector of residuals e = f − ˆf. The residuals account both for the modelling error ε and for the fitting error due to the least square estimation. The total sum of squares SST is the sum of the squared deviations of each response fi from its average value f¯ = ∑ni=1 fi /n: n

SST = ∑ ( fi − f¯)2 = ∑ fi2 − n f¯2 = fT f − ¯fT ¯f

(19)

i=1

SST can be partitioned into a sum of squares due to the model SSR and one due to residual SSE : (20) SST = SSR + SSE

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It can be shown that the sum of squares of the residuals can be computed as: SSE = fT f − αˆ T XT f

(21)

A low value of the ratio SSE /SST between the error sum of squares and the total sum of squares indicates that the chosen regression variables provide a good fit.

4 Results The studied structure is a system of three Aluminum plates with the same thickness of 3 mm and different sizes: plate 1 (600 mm × 400 mm) along x axis, plate 2 (300 mm × 400 mm) and plate 3 (400 mm × 400 mm). These plates are welded along the 400 mm side (Figure 1). Power is input to plate 1 only. Fig. 1: Three plates system: power is input to plate 1 (along x axis)

The considered problem concerns the variability of the SEA solution, the energy of each subsystem, represented by the flexural modes of each plate, when the variability of CLF’s and ILF’s (6 parameters) is taken into account, while input power is assumed to be constant at all frequencies with a value P1 of about 30 mW. A range of variability of the Young modulus and the thickness is considered by varying these parameters of ±10% around the nominal values: 7 × 1010 Pa and 3 mm, respectively. The correspondence between the variability of the physical parameters and the variability of the CLF’s is preserved, because the nominal value of the CLF’s, shown in Table 1, corresponds to the nominal value of the physical parameters. Further-

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more, the variation of the CLF’s, shown in Table 2, is the maximum difference from the nominal value obtained, by varying the physical parameters of ±10%. It should be noted that the Δ ηi j are about 50% of the corresponding ηi j as a consequence of 10% variations of physical parameters. The reciprocity relationship (6) is used to get the ηi j and the Δ ηi j with i < j. The modal densities are n1 = 12.46 · 10−3 , n2 = 6.233 · 10−3 and n3 = 8.311 · 10−3 . The ILF’s are varied of ±10% around 0.01. Table 1: Nominal CLF’s ηˆ i j Frequency [Hz]

ηˆ 21

100 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000

15.440 13.808 12.203 10.913 9.759 8.691 7.710 6.894 6.138 5.443 4.865 4.347 3.838 3.427

ηˆ 31

Table 2: Variations Δ ηi j of the CLF’s ηˆ 32

values to be multiplied by 10−3

11.580 10.356 9.152 8.185 7.319 6.518 5.783 5.170 4.603 4.083 3.649 3.260 2.878 2.570

11.288 10.015 8.739 7.688 6.731 5.959 6.042 6.252 5.533 4.762 4.167 3.668 3.198 2.831

Frequency [Hz]

Δ η21

100 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000

7.992 7.250 6.405 5.725 5.117 4.555 4.037 3.607 3.209 2.842 2.537 2.264 1.995 1.778

Δ η31

Δ η32

values to be multiplied by 10−3

5.993 5.437 4.803 4.294 3.838 3.416 3.028 2.705 2.406 2.132 1.903 1.698 1.496 1.333

6.307 5.237 4.587 4.067 3.600 3.076 2.991 3.536 3.127 2.642 2.285 1.999 1.739 1.541

Sensitivities of energies in the three subsystems, with respect to coupling loss factors and internal loss factors, are evaluated according to the procedure outlined in section 3.1. In practise, each value represents the first order approximation of variation of the energy stored in a given subsystem (1, 2, or 3) due to a change Δ ηi j of a given CLF or Δ ηi of a given ILF. Results are shown in Table 3 with reference to the 500 Hz third octave band, and to both CLF’s and ILF’s and in Figures 2, 3 and 4 for CLF’s only but for all third octave bands. A DoE procedure, with Central Composite Design requiring 26 + 2 · 6 + 1 = 77 experiments for each of 14 third octave bands from 100 Hz to 2000 Hz, is used. Table 4 shows the regression coefficients α at 500 Hz. Only the linear, quadratic and bilinear terms of the regression model are shown. The fit is very good because low values of SSE /SST (not shown) are found. The results show that the α of the linear terms are always the most relevant. Furthermore, all energies decrease as the internal loss factors η1 , η2 and η3 increase, as expected. By comparing the sensitivities and the linear terms of the regression model, it can be noticed that they are quite similar: not only they indicate an energy variation in the same direction, but the values provided by the two models are very close to each other as well.

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Table 3: Sensitivitiesa Δ e at 500 Hz index loss factor 1 2 3 4 5 6 a

η21 η31 η32 η1 η2 η3

Table 4: α valuesb at 500 Hz

E1

E2

E3

index

loss factor

E1

E2

E3

−0.0839 −0.0984 −0.0008 −0.2412 −0.0894 −0.1161

0.0641 0.0123 −0.0057 −0.1039 −0.0557 −0.0614

0.0198 0.0861 0.0065 −0.1366 −0.0622 −0.0954

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

η21 η31 η32 η1 η2 η3 η21 η21 η21 η31 η21 η32 η21 η1 η21 η2 η21 η3 η31 η31 η31 η32 η31 η1 η31 η2 η31 η3 η32 η32 η32 η1 η32 η2 η32 η3 η1 η1 η1 η2 η1 η3 η2 η2 η2 η3 η3 η3

−0.1101 −0.1288 −0.0081 −0.2504 −0.0882 −0.1137 0.0316 0.0408 −0.0037 0.0113 −0.0071 −0.0021 0.0363 0.0039 0.0132 −0.0013 −0.0094 0.0023 0.0008 0.0004 −0.0010 0.0141 0.0089 0.0115 0.0044 0.0052 0.0060

0.0828 0.0158 −0.0046 −0.1046 −0.0572 −0.0603 −0.0233 −0.0181 −0.0128 −0.0019 0.0005 0.0001 −0.0050 0.0149 0.0019 −0.0002 −0.0020 0.0019 0.0004 0.0026 −0.0029 0.0045 0.0048 0.0055 0.0008 0.0031 0.0015

0.0262 0.1117 0.0121 −0.1372 −0.0614 −0.0969 −0.0081 −0.0224 0.0165 0.0016 −0.0018 −0.0005 −0.0311 −0.0187 −0.0022 −0.0001 0.0002 −0.0040 −0.0004 −0.0026 0.0027 0.0058 0.0056 0.0081 0.0006 0.0040 0.0023

All values are multiplied by 103

b

All values are multiplied by 103

Figures 2, 3 and 4 show the regression coefficients α of the energies of the three subsystems. For the sake of comparison with sensitivities, only the linear terms of the regression model are drawn. In both cases, the trends are very similar. Also, it can be noticed that the magnitude of the sensitivities is always smaller than the magnitude of the corresponding regression coefficients. This is not unexpected because the regression model includes quadratic and multi-linear terms that can compensate for the noted difference.

Uncertainty propagation in SEA using sensitivity analysis and Design of Experiments Fig. 2: Energy of subsystem 1: regression coefficients α (solid) vs sensitivities (dash–dotted) with respect to CLF’s η21 (+), η31 () and η32 (o)

Fig. 3: Energy of subsystem 2: regression coefficients α (solid) vs sensitivities (dash–dotted) with respect to CLF’s η21 (+), η31 () and η32 (o)

Fig. 4: Energy of subsystem 3: regression coefficients α (solid) vs sensitivities (dash–dotted) with respect to CLF’s η21 (+), η31 () and η32 (o)

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5 Conclusions In this paper the effect of uncertainties of the loss factors is modeled by using both a sensitivity and a DoE approach. The present technique to model uncertainties is parametric, i.e. a model of the uncertainties of the system parameters is necessary to evaluate the variability of the solution. The results obtained by DoE and sensitivity are compared and they show a good agreement. However, the DoE approach gives more information than sensitivity, because it allows to calculate the dependence of SEA solution both on the parameters (CLF’s and ILF’s) and on their combinations. In addition to the results on the uncertainty propagation, an important information is obtained for the design of a system with controlled energy levels. In fact, by assuming that the energy of one subsystem must not exceed a given level, the presented analysis allows to evaluate the sensitivity of such energy to the uncertain parameters and, consequently, where to act in order to reduce vibration and noise. Next activities will consider the study of more complicated systems to investigate the dependency of the SEA solution on the parameters of not directly connected subsystems and on the injected power. Acknowledgements This research is supported by MIUR grants.

References 1. Lyon, R., De Jong, R.: Theory and Applications of Statistical Energy Analysis. The MIT Press, Cambridge (U.S.A.) (1995) 2. Lyon, R.: Statistical analysis of power injection and response in structures and rooms. Journal of the Acoustical Society of America 45(3), 545–565 (1969) 3. Radcliffe, C.J., Huang, X.: Putting statistics into the statistical energy analysis of automotive vehicles. Journal of Vibration and Acoustics 119(4), 629–634 (1997) 4. Langley, R., Cotoni, V.: Response variance prediction in the statistical energy analysis of builtup systems. Journal of the Acoustical Society of America 115(2), 706–718 (2004) 5. Weaver, R.: Spectral statistics in elastodynamics. Journal of the Acoustical Society of America 85, 1005–1013 (1989) 6. Culla, A., Carcaterra, A., Sestieri, A.: Energy flow uncertainties in vibrating systems: Definition of a statistical confidence factor. Mechanical Systems and Signal Processing 17(3), 635–663 (2003) 7. de Langhe, R.: Statistical analysis of the power injection method. Journal of the Acoustical Society of America 100(1), 294–304 (1996) 8. Bussow, R., Petersson, B.: Path sensitivity and uncertainty propagation in SEA. Journal of Sound and Vibration 300(3–5), 479–489 (2007) 9. Langley, R., Cotoni, V.: Response variance prediction for uncertain vibro-acoustic systems using a hybrid deterministic-statistical method. Journal of the Acoustical Society of America 122(6), 3445–3463 (2007) 10. Montgomery, D.: Design and Analysis of Experiments, 6th edn. Wiley, New York (2005) 11. D’Ambrogio, W., Fregolent, A.: Reducing variability of a set of structures assembled from uncertain substructures. In: Proceeding of 26th IMAC. Orlando (U.S.A.) (2008)

Phase reconstruction for time-domain analysis of uncertain structures L H Humphry

Abstract Using the Hybrid method (FE+SEA) it is possible to estimate the frequency response of an uncertain structure. For many applications (shock, auralisation) a time domain response is sought. Problems to be overcome when taking Hybrid method results into the time domain are a) the Hybrid method frequency response has no phase information, and b) the Hybrid method frequency response is smoothed in frequency and shows no modal peaks. In this paper the first problem of phase reconstruction is tackled, using minimum phase reconstruction. Explanation of minimum phase reconstruction and its limitations are given, and application to shock problems described.

1 Introduction Structural vibration problems are conventionally divided into two sets: low frequency problems, where the wavelengths of vibration are long compared with the structural dimensions; and high frequency problems, where the wavelengths are short. For these problems there are well developed tools for simulating and predicting vibrations: Finite Element Analysis (FEA) [9] for low frequencies, and Statistical Energy Analysis (SEA) [6] for high frequencies. Another class of problem exists, when a structure contains different elements that show both low and high frequency behaviour simultaneously. This is a “midfrequency” problem, and can be tackled using the Hybrid method [5]. The Hybrid method couples together both FEA and SEA modelled subsystems into one structural vibration model. The Hybrid method gives a frequency response result which is averaged over an ensemble of statistically similar structures, so can cope with uncertainty about the nature of a structure.

L H Humphry Department of Engineering, University of Cambridge, CB2 1PZ, UK, e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 19, © Springer Science+Business Media B.V. 2011

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For many applications a time domain result is sought, for example shock and auralisation problems. For both applications structural excitation can be expected across a large range of frequencies (both low and high) and so for structures with significant resonances extending over this range neither FEA nor SEA is suitable. The Hybrid method is a more suitable tool for such problems. Hybrid method results are frequency domain based, but can be transformed using the inverse Fourier transform for time domain solutions. Two problems are encountered when transforming to the time domain, a) Hybrid method results lack phase information, and b) ensemble averaging has the effect of smoothing the frequency response in frequency and so removing modal peaks in the spectrum. Both of these frequency domain features have a large effect on the time domain response. The work contained in this paper seeks solve the problem of phase reconstruction, to allow time domain modelling using the Hybrid method, and is an extension of work described in a previous paper by the same author [4]. The meaning of minimum phase is introduced in Section 2, then a discussion of how minimum phase reconstruction can be carried out in practise is given in Section 3. Initial experiences of using minimum phase reconstruction for the shock response of uncertain structures are given in Section 4, and concluding remarks given in Section 5.

2 Explanation of minimum phase For a system F(u) the phase of the system is defined as θ (u), where F(u) = |F(u)|e jθ (u) . In the case of Hybrid method produced frequency responses, the magnitude |F(ω )| is known but the phase is not. Phase information is needed for accurate time domain realisation. In some cases phase retrieval is possible, but this is usually ambiguous. It has been shown that, except for certain special cases, there exists an infinity of functions f i (t) whose Fourier transforms Fi (ω ) have the same modulus |F(ω )| on the real frequency axis [2].

2.1 Defining minimum phase One special case where a unique relationship between the magnitude and phase of a system exists is when is a system is known to be minimum phase [7]. A system is defined as minimum phase if the system and its inverse are causal and stable.

2.2 The Hilbert transform and analytic systems For a general system G(x), the Kramers-Kronig relations state that if G(x) is analytic in the negative imaginary half of the complex plane (i.e. lower half plane, LHP), the

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real and imaginary parts of G(x) = R(x) + jI(x) will be related: R(x) = H {I(x)}

(1)

I(x) = −H {R(x)}

(2)

where H {F(x)} is known as the Hilbert Transform and is defined as H {F(x)} =

1 π

∞ F(τ ) −∞

x−τ

dτ .

(3)

Any real physical system must be causal, i.e. no response is seen before an input is applied. It can be shown that causality and stability in the time domain implies that in the frequency domain a system is analytic in the LHP [1]. Thus, for any stable real physical system H(ω ) = HR (ω ) + jHI (ω ) it is possible to reconstruct HR (ω ) from HI (ω ) and vice versa.

2.3 The Hilbert Transform and minimum phase systems With a unique mapping between real and imaginary parts of a real system’s frequency response, it might be thought that a similar relationship exists between the phase θ (ω ) and magnitude |H(ω )|. By taking the complex logarithm of H(ω ),

it can be seen that

L(ω ) = ln(H(ω )) = ln |H(ω )| + jθ (ω )

(4)

ln |H(ω )| = H {θ (ω )}

(5)

θ (ω ) = −H {ln |H(ω )|} .

(6)

However, Eqs. 5 and 6 hold only when the function L(ω ) is analytic in the LHP, so must have no poles in this region. L(ω ) will have poles whenever H(ω ) has a pole or a zero. Therefore a unique mapping between the phase and magnitude of a response H(ω ) exists only when H(ω ) has no pole or zeros in the LHP. If this is true for H(ω ) it must also be true for H −1 (ω ) where poles and zeros reverse positions. Thus, the response and its inverse are both causal and stable, and so the system is minimum phase by the definition given in Section 2.1.

2.4 Further interpretation of minimum phase A minimum phase system can be also be thought of as the “fastest decaying” system [8]. Formally, for all causal signals fi (n) that have a given magnitude spectrum |F(ω )|, the minimum phase signal fmp (n) is the one that maximises initial energy:

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L H Humphry K

K

n=0

n=0

∑ | fmp (n)|2 ≥ ∑ | fi (n)|2 ,

for

K = 0, 1, 2, · · · , N − 1 .

(7)

Minimum phase systems occur frequently when studying structural vibration frequency responses. The driving point response of any physical system must be minimum phase, with one minimum phase zero between every pole. To see this, the driving point response is described by mobility — the velocity at a point resulting from a prescribed force input at the same point. This is causal and stable for all real systems. The inverse of the driving point mobility is the driving point impedance — the force that results from applying a prescribed velocity input. This is also causal and stable for all real systems. Thus, the drive point response function and its inverse are causal and stable and so both must be minimum phase functions. For non-drive point responses, the inverse function is not a simple physical function like drive point impedance, and so it cannot be assumed that the inverse is stable. Thus, the non-drive point response cannot be assumed to be minimum phase.

3 Using minimum phase reconstruction For a general system with magnitude frequency response |F(ω )| the phase cannot be retrieved uniquely. By assuming that the system is minimum phase the phase can be found using the Hilbert transform, this is called minimum phase reconstruction (MPR). This section details artefacts encountered when implementing an MPR algorithm, and the errors that result in the time domain when taking a minimum phase assumption for a non-minimum phase system.

3.1 Approximating the Hilbert Transform This unique mapping given by the Hilbert transform, from phase to magnitude in minimum phase systems, is for continuous functions only. Such a function is not realisable on a digital computer, and it is to be expected that artefacts will occur as a result. For discrete systems there exists the discrete Hilbert transform (DHT) [3]. For a signal f (k) with discrete Fourier transform F(k), the DHT g(k) is given by the inverse discrete Fourier transform of G(k), where G(k) = H(k) · F(k) , and

⎧ ⎨ − j, H(k) = 0, ⎩ + j,

k = 1, 2, · · · , N2 − 1 k = 0, N2 k = N2 + 1, N2 + 2, · · · , N − 1.

(8)

(9)

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For large N the DHT approximates the continuous time Hilbert transform well. Errors in the DHT approximation of the Hilbert transform occur due to the using the DFT approximation of the Fourier transform. By using an FFT length that is long enough to give a good approximation of the Fourier transform, errors in the DHT can be reduced. An MPR algorithm has been written in MatLab that reconstructs the minimum phase realisation of a magnitude spectrum |F(ω )|, using Eq. (6) and the DHT in the frequency domain. To test the accuracy of the MPR algorithm, the MPR was validated using the magnitude spectrum of a driving point impulse response — a minimum phase signal. The spectrum HD (ω ) for the driving point response of a random system was calculated using a modal summation, HD (ω ) =

N

φ2

∑ ωn2 + j2β ωn nω − ω 2 ,

(10)

n=1

where modal amplitudes φn had a Gaussian distribution, resonant frequencies ωn had a Rayleigh distribution, and damping ratio β was assumed constant. The DHT algorithm was validated by reconstructing the phase of HD (ω ) to give Hˆ D (ω ). Figure 1 shows the Bode plot for a driving point response and the MPR using the DHT. It can be seen that the MPR does not reproduce the phase of original signal exactly. The magnitudes of HD (ω ) and Hˆ D (ω ) are equal, but the phases are not quite. The MPR, Hˆ D (ω ), shows an approximately linear phase term that increases the phase by 2π over the frequency range calculated. This error in calculated phase is due to the FFT approximation of a Fourier transform used in the calculation of the DHT. Though the phase error at high frequencies (i.e. above the Nyquist frequency) look to be large, these are all actually 2π − θe , where θe is the small phase error at the symmetric point below the Nyquist frequency. Phase errors are worst at the Nyquist frequency, there being a π rads difference between the phase of HD (ω ) and Hˆ D (ω ) here. As long as the magnitude of the response is low here, these errors around the Nyquist frequency do not give a significant error in the time domain. The corresponding time domain signals hD (t) and hˆ D (t) show only a very slight error, and only in low magnitude high frequency components which decay very rapidly. Quantitatively, the normalised L2-norm error of hD (t) and hˆ D (t) is given by ∑ (hD (nT ) − hˆ D (nT ))2 εL2 = n , (∑n hD (nT )2 )(∑n hˆ D (nT )2 )

n = 0, 1, . . . , N − 1 ,

(11)

and has a very low value of εL2 = 2.77 × 10−4 . This error can be reduced still further by increasing the frequency window of the spectrum. Errors can also occur for MPR when a system of low damping is used. The main effect of these errors is an increase in the decay rate of the MPR signal hˆ D (t). These errors are not found for spectra produced from systems with higher damping. It has been found that this error comes from the discrete approximation of HD (ω ) used in computer calculations. By increasing the frequency resolution of the discrete

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Fig. 1: Bode Plot of Driving Point Response Spectrum

approximation of HD (ω ) the error seen in hˆ D (t) is reduced. A rule of thumb has been found for satisfactory reproduction of hˆ D (t): by ensuring that there are 5 frequency data points covering the half power bandwidth of every resonance of HD (ω ), no noticeable error in the time domain is observed. Though the Hilbert transform provides a unique mapping between phase and magnitude for minimum phase signals, for work on a computer the DHT must be used, and this has been shown to produce errors. The solution has been to increase the frequency range and resolution of the spectrum being analysed. Put simply, one way to reduce the errors that result from using the discrete Hilbert transform is to increase the size and resolution of the sampled space to better approximate the continuous space. This is a limitation of the algorithm, and currently cannot be avoided. It should be noted that the use of the continuous variable ω to describe spectra here is not strictly correct for discrete spectra modelled using computers. However, effort has been made to make data lengths long enough to reduce the artefacts that occur from approximating a continuous function by a finite sampled function. By reducing these artefacts, the discrete and continuous functions become equivalent, and so the term ω (and t for time) is used where formally a discrete variable would be more appropriate.

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3.2 Errors using MPR for non-minimum phase systems Though the driving point response of a system is minimum phase, non-driving point responses typically are not. As explained in Section 2, the phase of a non-minimum phase response HNMP (ω ) cannot be determined uniquely from its magnitude. By assuming that the system is minimum phase, the MPR algorithm can be used to (incorrectly) reconstruct of the phase of HNMP (ω ). In this section two non-minimum phase systems are presented — first a stretched string with an analytic solution, then a more complicated dispersive drill-string model — and errors resulting from MPR are noted.

3.2.1 Analytic Stretched String One simple non-minimum phase system is that of a stretched flexible string given an impulsive force input at some point along the string. The string has length L, tension P, mass per unit length m and wave speed of c = P/m. Figure 2 shows the response of a point some distance away from the driving point, and the MPR response found using the incorrect assumption of minimum phase and the MPR algorithm.

Fig. 2: Stretched string impulse response Some obvious features of the MPR response are seen: • • • •

Echoes are preserved, and have the same spacing as the original signal, The magnitude of the response is preserved, Decay characteristics are preserved, Initial pulse delay is lost.

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Apart from the lack of initial delay, the MPR looks remarkably similar to the original signal. If this delay is added to the MPR, a perfect reconstruction of the original is found. This delay term is found to be tdelay = |x0 − x| c , i.e. the length of time taken for a wavefront to travel from the drivepoint x0 to the listening point x. It is found that by introducing the delay tdelay back into the MPR at any point along the string, the modified MPR predicts the original signal extremely well. Thus, for the impulse response of a string the only error introduced by using a minimum phase assumption is the loss of the initial delay. For this string system the admittance transfer function for a point x < x0 can be written analytically as

u(x) 1 sin(k(L − x0 )) sin(kx) = HS (ω , x) = , (12) Fin Pk sin(kL) where the wavenumber k = ω /c. As k → − j∞, it can be shown that the factor e jk(L−x0 ) e jkx e jk(x−x0 ) sin(k(L − x0 )) sin(kx) → →0 → sin(kL) 2j 2 je jkL

(13)

for x < x0 . Thus, there is a zero at negative infinity, and so ln(HS (ω , x)) → −∞. As explained in Section 2.3, minimum phase systems have no poles or zeros in the LHP. This non-minimum phase zero can be removed by multiplying HS (ω , x) by e− jk(x−x0 ) , to give the MPR: HMP (ω , x) = HS (ω , x)e− jk(x−x0 ) .

(14)

It can be seen that this extra phase term is a pure time delay, and has the effect of removing the delay tdelay = (x0 − x)/c seen in Fig. 2.

3.2.2 Dispersive Drill-string Figure 3 shows the sonogram of an impulse response of a more complicated nonminimum phase system. This system is an idealised model of a periodic drill string, and the response shown is the end to end impulse response. Fig. 3(a) shows the dispersive nature of the system, and shows echoes occurring after the initial impulse. The MPR shown in Fig. 3(b) shows three main features of interest: • • • •

Initial delays are lost at all frequencies The magnitude of the response is preserved, Decay characteristics are preserved, Echoes are preserved.

For dispersive systems it is found that different frequency components experience a different amount of time delay removal, such that all frequency components have a

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Fig. 3: Sonogram of end to end impulse response: (a) original and (b) MPR

large response at t = 0. The removal of delay terms means that waves are changed in phase with one another and so response pulse shapes are changed in the time domain. Unlike the previous string model, this system has no analytic solution and so it is not obvious exactly what changes the MPR is making to the signal. However, the lack of initial delay in the MPR response can be explained by remembering the ‘fastest decay’ definition of minimum phase systems given in Section 2.4. Clearly, any initial delay must be removed if Eq. (7) is to be satisfied, and this is true for all systems. The other features of MPR seen in Figs. 2 and 3 (magnitude, decay and echoes preserved) have been observed for a range of non-minimum phase systems, including strings, beams, acoustics ducts, and random spectra. In summary, for all non-minimum phase systems using MPR will introduce errors. When the nature of a system is known, it may be possible to reconstruct the original signal perfectly by post processing the MPR, though this will only be the case for a few simple systems. The errors introduced by MPR are often limited to the removal of initial delays in the signal.

4 Application: peak shock prediction in uncertain structures An uncertain structure was simulated by using a random spectrum. The MPR was found of the spectrum, and a half cosine pulse applied to both the original and MPR spectra. The time domain shock responses were obtained using the IFFT, and the magnitude of the peak response found. In this section a comparison of results given by the original non-minimum phase system and its MPR are presented.

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4.1 Modelling an uncertain structure The inertance of the structure is of interest for shock analysis, and the inertance transfer function was modelled using the modal summation HD (ω ) =

N ω 2 φn (x1 )φn (x2 ) a(x2 ) =∑ 2 , F(x1 ) n=1 ωn + j2β ωn ω − ω 2

(15)

where x1 defines the position of the input force, x2 defines the listening position for the acceleration output, and damping ratio β is assumed constant. Modal amplitudes φn are randomly assigned with Gaussian distribution, and resonant frequencies ωn have a Rayleigh distribution. The transfer function described in Eq. (15) does not decay at high frequencies, but instead has a constant trend line. For a true impulse response an infinite frequency range would be needed. To avoid this, the response for a non-idealised impulse was used. The impulse was assumed to be a half cosine pulse, centred on t = 0 and with area = 1. By widening an idealised impulse in this way the high frequency components of the shock are lost. Also, this is a more realistic model of a shock input that may occur in practise. By taking the Fourier transform of the cosine pulse the decay at high frequencies is seen:

π π 2 cos ω2b π

=F Hshock (ω ) = 2 cos t , (16) b π2 − ω 2 2b b b2 where b defines the pulse width. The power spectrum Hshock has a “sinc” type structure, with rapidly decaying lobes that decrease with increasing frequency. After the second side-lobe the amplitude of the spectrum is at most 2% of the DC amplitude, and so it is assumed that after this point the effect of higher frequencies is negligible to the overall response. Thus, the rectangular windowing that must be used to create a finite length spectrum will not cause large artefacts. Multiplying the spectra of the cosine pulse and the random structure gives the shock response of the structure. Figure 4 shows the shock response for the original system, its MPR, and for comparison the zero-phase system with the same magnitude as the original. It can be seen that the MPR response is much closer to the original than the zero-phase system. The MPR response over-predicts the value of the peak shock response, and this is expected due to the “fastest decay” property of minimum phase systems.

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Fig. 4: Impulse responses for random structure

4.2 Ensemble average results Due to the random nature of the structure modelled above, the exact nature of the shock response varies depending on the random spectrum used. Using an ensemble of statistically similar structures, the peak shock response was found for original, MPR, and zero-phase spectra. Table 1 shows the results found from an ensemble of 100 random spectra. The sampling frequency was set to 44100 Hz, and an impulse duration of 160μ s. Table 1: Peak shock response results for an ensemble of 100 random filters

Mean Max Relative Variance Max

Original

Minimum Phase

Zero Phase

3.07 · 10−4

3.90 · 10−4

0.24

0.25

8.39 · 10−4 0.18

Frequency resolution was varied in accordance with the rule of thumb, at least five data points covering every spectral peak. There were 100 modes in the frequency range 0–22050 Hz (i.e. the Nyquist frequency), and a damping ratio of β = 0.005 was used. “Mean Max” refers to the ensemble mean value of the peak, “Relative Variance Max” to the relative variance of peak response results across the ensemble. These results show that using the MPR gives a prediction of peak shock response that is comparable with the true result, and much closer in value than results predicted using a zero phase spectrum. However, the minimum phase filter over-predicts the response, by 27% in this case.

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The over-prediction in the MPR peak shock response value is found for a wide range of ensembles, with different damping and modal density values. This overprediction is typically within 50% of the original response; an acceptable error when compared with safety factors of 300–400% typically used to design against shock failure. Due to being an over prediction, the MPR result provides a conservative estimate for the mean peak shock response. The variance of results does not change noticeably for MPR predicted responses, and so can be used to give likelihood estimates for shock responses of uncertain structures.

4.3 Changing the correlation of modal amplitudes The random spectra used in Sections 4.1 and 4.2 have modal amplitudes φn (x1 ) and φn (x2 ) that are uncorrelated, produced using the “randn” command in MatLab. Modal amplitudes are only truly uncorrelated in a structure when the listening point x2 is in the far field, i.e. r = |x1 − x2 | ∞. For random structures the level of correlation between modal amplitudes at different points depends on the distance between these points, r. When r = 0 (driving point response), the correlation R(r) = 1 for all systems. For a 2-D system the diffuse field correlation function R(r) is given by: (17)

R(r) = J0 (kr) ,

where J0 is a zeroth order Bessel function of the first kind, and k is the wave number of the system. To allow shock responses to be calculated at a variety of listen positions, modal amplitudes φn (x1 ) and φn (x2 ) were chosen such that they were correlated according to Eq. (17). Again an ensemble of similar spectra was used to find the peak response values for original and MPR shock responses. Results are given in Table 2 for a system where the distance r was low, meaning that the correlation between modal amplitudes was high. Table 2: Peak shock response results for an ensemble of 100 correlated filters Original Mean Max Relative Variance Max

Minimum Phase

5.13 · 10−4 5.53 · 10−4 0.34 0.30

Zero Phase 9.16 · 10−4 0.32

When the distance between input and listening points is low it can be seen that the MPR still over-predicts the mean shock response of the ensemble, but is closer to the true result than for the uncorrelated far field case. When r = 0, MPR gives a very accurate prediction of the mean, and this is to be expected as at r = 0 a driving point response is seen.

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5 Conclusions The Hybrid method is currently not suited to time domain applications: frequency domain results produced by the Hybrid method lack phase information and are smoothed in frequency. Using an assumption of minimum phase, the phase can be reconstructed using the Hilbert transform. A minimum phase reconstruction algorithm has been implemented, and it has been found that for accurate MPR the frequency range and resolution of the spectra being studied must both be high. Applying MPR to non-minimum phase systems produces errors in the time domain, but these are limited to a removal of any delay in waveforms. Using MPR to predict the shock response of uncertain (non-minimum phase) structures has proved very acceptable — MPR gives a mean ensemble shock response that over-predicts by less than 50% and provides a conservative estimate. MPR gives much more accurate results than using a zero phase assumption. The problem of Hybrid method frequency smoothing has not been tackled, future work must concentrate on this area to allow Hybrid method models to be used for time domain applications. Acknowledgements Financial support from the EPSRC gratefully acknowledged. This paper is part of ongoing research under the supervision of Prof. R S Langley, University of Cambridge.

References 1. Aki K, Richards PG (2002) Quantitative seismology, 2nd edn, University Science Books, New York, p 167 2. Burge R, Fiddy M, Greenway A, Ross G (1976) Phase Problem. Proc of the Royal Society of London Series A — Mathematical Physical and Engineering Sciences 350(1661):191–212 3. Cizek V (1970) Discrete hilbert transform. IEEE Transactions on Audio and Electroacoustics 18(4):340–343 4. Humphry LH, Langley RS Predicting shock response in uncertain structures using the Hybrid method. In: Proceedings of the 7th International Conference on Modern Practise in Stress and Vibration Analysis, Institute of Physics, Cambridge 5. Langley R, Bremner P (1999) A hybrid method for the vibration analysis of complex structuralacoustic systems. Journal of the Acoustical Society of America 105(3):1657–1671 6. Lyon RH (1975) Statistical Energy Analysis of Dynamical Systems. M.I.T. Press 7. Oppenheim AV, Schafer RW (1975) Digital Signal Processing, Prentice-Hall International, London, p 346 8. Smith JO (2007) Introduction to Digital Filters with Audio Applications, W3K Publishing, http://www.w3k.org/books/. Accessed 18th May 2009 9. Turner MJ, Clough RW, Martin HC, Topp LC (1956) Stiffness and deflection analysis of complex structures. Journal of the Aeronautical Sciences 23(9):805–823

Part III

Probabilistic Methods

Uncertain Linear Systems in Dynamics: Stochastic Approaches G.I. Schu¨eller and H.J. Pradlwarter

Abstract This paper provides an overview along with a critical appraisal of available methods for uncertainty propagation of linear systems subjected to dynamic loading. All uncertain structural properties are treated as random quantities by employing a stochastic approach. The loading can be either of deterministic or stochastic nature, described by withe noise, filtered white noise, and more generally, by a Gaussian stochastic process. The assessment of the variability of the uncertain response in terms of the mean and variance is described by reviewing the random eigenvalue problem and procedures to evaluate the first two moments of the stochastic (uncertain) response. Computational procedures which are efficiently applicable for general FE-models are in the focus of this work.

1 Introductory Remarks Structures are generally exposed to loading conditions which are associated with uncertainties. In some cases, like environmental loading, such as wind, earthquakes, water waves, etc., structures are also dynamically excited. For certain classes of problems, see e.g. aerospace structures, the uncertainties in the material and structural properties are of importance. It is also to be noted, that structures of interest for the engineering practice are generally large, usually including tens of thousands of degrees of freedom. Their modeling is carried out almost exclusively by Finite Elements.

G.I. Schu¨eller Institute of Engineering Mechanics, University of Innsbruck, A-6020 Innsbruck, Austria, EU, email: [email protected] H.J. Pradlwarter Institute of Engineering Mechanics, University of Innsbruck, A-6020 Innsbruck, Austria, EU, email: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 20, © Springer Science+Business Media B.V. 2011

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If the analysis is to be done realistically, i.e. by accounting quantitatively for the uncertainties in the loads and in the structural and material properties, respectively, one is faced with a multi-fold challenge, i.e. with 1. 2. 3. 4. 5.

The probabilistic description of the loading (probabilistic modeling) Structural modeling (by mechanical modeling including Finite Elements) Material modeling Structural analysis in terms of computational mechanics (FEM) Probabilistic description of material and structural properties.

Naturally, the analysis tools should reflect most recent developments in all these fields. Simplifications in even one of these fields will adversely affect the credibility and consequently the acceptability of the entire analysis by the engineering practice. Yet, for the time being, time variant effects caused by dynamic excitation, such as deterioration effects, etc. will not be included. This work focuses on the assessment of the current status of the procedures for stochastic linear structural analysis in dynamics which consider also uncertainties of the structural properties besides their ability to determine the stochastic response due to random excitation. Not only the number of DOFs — which affect mainly the computational resources required for a single response computation — play an important role, but also the number of uncertain parameters involved. In general, large complex structures involve also a large number of uncertain structural parameters. Moreover, dynamic excitations are usually modeled by a large number of independent random variables for a suitable approximation of a stochastic process. A similar situation arises in case random fields are modeled. In this context, attempts are made to quantitatively assess the potential of some of the methods for high dimensional problems, by referring to recent benchmark studies [53, 54].

2 Overview of Available Methods Not all available procedures are generally applicable. Even the most versatile method, i.e. direct Monte Carlo Simulation, is — for reasons of computational efforts — limited to structures with a somewhat smaller number of DOFs. Although the structure is still considered to react linearly, the structural response is slightly or moderately non-linear with respect to the uncertainty expressed by the uncertain quantity θ . This can be recognized immediately by considering the displacement of a linear spring with uncertain properties k(θ ) = k0 (1 + θ ) loaded by a constant force f . The displacement is then a nonlinear function of the uncertain variable θ u(θ ) =

f f f = = (1 − θ + θ 2 − θ 3 + . . . ) k(θ ) k0 (1 + θ ) k0

(1)

Many restrictions are imposed by this basic non-linearity with respect to uncertainties of linear systems with uncertain structural properties. Obviously, a linear approximation of the variability of the response is accurate only for cases where |θ |

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273

is small, say a few percents. Hence, the domain of uncertainties plays an essential role regarding the applicability, accuracy and efficiency of various procedures for uncertainty propagation towards the structural response. The non-linearity also implies non-Gaussian properties of the response even for linear systems and assuming θ to be Gaussian distributed. Characteristics of the loading in terms of stationarity or non-stationarity, etc. are also of importance w.r.t. the applicability of various methods. In the following uncertainty propagation procedures are reviewed in terms of the first two statistical moments of the response, i.e. the mean response and the covariance matrix.

3 Response variability 3.1 Perturbation Method Perturbation procedures have been used from the early developments of stochastic finite elements [8]. These follow all the steps of a deterministic analysis and are therefore applicable for arbitrarily large FE-models. This method can be used advantageously in case the random fluctuations are small compared to the unperturbed system, such that terms of order two, three and higher are negligible. The method is based on a Taylor series expansion in terms of a set of zero mean random variables θ = {θi }ki=1 . Further it will be assumed that these random quantities are uncorrelated, which in fact can always be assured by a suitable coordinate transformation. Before discussing linear dynamic applications of the perturbation approach, the principle is first exemplified by a simple static consideration in which the stiffness and also the loading deviates randomly from the nominal value. Assuming that the variations of the stiffness matrix are small, a first order Taylor expansion suffices k

K (θ ) = K 0 + ∑ θi K Ii ;

K 0 = K (00) ;

i=1

K Ii =

∂ K , ∂ θi θ =00

(2)

where the terms K Ii denote the partial derivatives of the stiffness matrix w.r.t. the random variables. Similarly, the loading f (θ ) might also be described by a linear relation f (θ ) = f 0 +

k+m

∑

θi f Ii .

(3)

i=k+1

Since it is known that the response is a nonlinear function of the random quantities {θi }ki=1 , higher order terms are generally needed for an accurate representation.

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u (θ ) = u 0 +

k+m

1 k+m k+m

i=1

i=1 j=1

∑ θi u Ii + 2 ∑ ∑ θi θ j uIIij + · · ·

(4)

The above assumptions allow to establish a set of recursive equations for the solution of the static problem K (θ )uu(θ ) = f (θ ) , which read:

K 0 ]−1 f 0 ; u 0 = [K

(5)

K 0 ]−1 ( f Ii − K Ii u 0 ) u Ii = [K

(6)

These recursive relations can be easily verified by differentiating the static global equilibrium equation (5). It should be noted that a single factorization suffices. Including higher order terms, however, will dramatically increase the computational efforts, especially in case the number k + m of random variables will be large. The perturbation approach can be used to approximate the first two moments of the stochastic response, i.e. the mean vector and the covariance matrix. EII {uu} ≈ u 0 +

1 k 2 II ∑ σi u ii 2 i=1

k C ov {uu} = E (uu − E{uu})(uu − E{uu})T ≈ ∑ σi2 u Ii (uuIi )T ,

(7)

I

i=1

where σi denotes the standard deviation of the uncertain variable θi and E{·} denotes the mean or mathematical expectation. The achieved accuracy will only be satisfactory for small levels of uncertainty, e.g. where the coefficient of variation of the linear response does not exceed a few percents. Regarding the series expansion of the static response for stochastic systems, more theoretical considerations to perturbation in general are provided in [26]. Recent work in Refs. [12, 11] provide an alternative accurate approach, not based on perturbation, for the problem as outlined above. The perturbation procedure, applied to linear static problems, can also be used to analyze the variability of the dynamic response of uncertain linear structures. The uncertain mass matrix M (θ ) and damping matrix C (θ ) are represented analogously as a linear first order Taylor expansion: k

M (θ ) = M 0 + ∑ θi M Ii ; i=1

k

C (θ ) = C 0 + ∑ θiC Ii

(8)

i=1

Since for large FE-models, modal analysis can be used efficiently, this leads immediately to the random eigenvalue problem: K (θ )φ i (θ ) = M (θ )φ i (θ )λi (θ )

(9)

A first order Taylor expansion of the eigenvalues λi (θ ) and eigenvectors φ i (θ ) around θ = 0 reads,

Uncertain Linear Systems in Dynamics: Stochastic Approaches k

λi (θ ) = λi (00) + ∑ θ j j=1

∂ λi ; ∂θj

275 k

φ i (θ ) = φ i (00) + ∑ θ j j=1

∂φi , ∂θj

(10)

allowing the evaluation of the first order approximation of the first two moments of the modal properties in a straight forward manner: k

∂λj ∂ θl

E{λi } = λi (00) ;

Cov(λi , λ j ) = ∑ σl2 ∂∂ λθi l

E{φ i } = φ i (00) ;

Cov(φ i , φ j ) = ∑ σl2 ∂ θ i l

l=1 k

l=1

T ∂φ ∂φ j ∂ θl

(11)

The procedure to determine the partial derivative of the eigenvalues and eigenvectors dates back to the work by [15]. The derivatives for the eigenvalues can be computed in a straight forward manner:

∂ λi ∂K ∂ M K Ij − λi (00)M M Ij ]φ i (00) = φ Ti [ − λi ]φ = φ Ti (00)[K ∂θj ∂θj ∂ θ j i θ =00

(12)

The computation of the derivatives of eigenvectors, however, are computationally more demanding. The original solution by [15] required a complete set of modes to determine the derivatives of eigenvectors which is inefficient for larger systems. This strong limitation has been removed, thus allowing applicability in presence of a truncated set of modes [42, 34, 32]. In the following the simplified eigensolution [42] is shown, in which the derivative is represented as linear combination of the available modes:

∂φi = Φ · c (i, j) ∂θj (i, j)

cl

=

φ Tk f (i, j) λl − λi

for λl = λi ;

f (i, j) =

∂ λi 0 K Ij − λi M Ij ]φ i M φ i − [K ∂θj

(13)

(14)

(i, j)

The missing component ci is determined from the derivative of the norm of the eigenvectors Φ T M Φ = I , leading to the solution: (i, j)

ci

1 (i, j) = φ Ti M Ij φ i + φ Ti M 0 ∑ cl φ l 2 l=i

(15)

The above solution is only applicable for distinct eigenvalues. The reader might consult e.g. [44, 30, 7, 63] for procedures which are capable to treat multiple natural frequencies. The above presented first order perturbation method for dynamic analysis give reasonable quality results only when the coefficients of variation of the uncertain system parameters are small. An approach which does not rely on perturbation is proposed in [49] for capturing statistical moments of the eigenvalues by a dimensional decomposition. This work has been extended in [50] to complex valued

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stochastic eigensolutions deriving also approximation for the variability of eigenvectors. A wealth of works exist to improve either the computational efficiency of the perturbation approach in dynamics or to increase the accuracy [37]. A modal approach combined with sensitivity analysis and perturbation is proposed in [43] for assessing the modal response variability and associated frequency response functions. The work in [3] introduces an unconditional stable step-by-step procedure, capable of evaluating the linear response for a modified system. An improved accuracy can be obtained by applying the procedure in [41] which uses the solution of the first order perturbation as a starting point. Two basis vectors lead to a sequence of (2×2) reduced order eigenvalue problems for each mode φ of interest: ∂φ φˆ (θ ) = ζ1 (θ )φ (0) + ζ2 (θ ) ∑ θi (16) ∂ θi i Explicit expressions for the modal solution are given in terms of parameters arising from the discretization of the random field or uncertain structural parameters. A first order Taylor expansion in the random parameters θ about their mean values E{θ } as (θ − E{θ }) (17) u (t, θ ) = u (t, E{θ }) + ∇ θ u (t, θ ) θ =E{θ }

is employed in [2] and denoted as First Order Second Moment (FOSM) approximation, where ∇ θ u (t, θ ) denotes the gradient of the response the random quantities θ . For a given mean vector E{θ } and covariance matrix C θ , the variance of the displacement response is given by: C θ · ∇θ u(t, θ ) C u = ∇Tθ u(t, θ ) ·C (18) θ =E{θ }

θ =E{θ }

A spectral stochastic approach for the dynamic analysis in the frequency domain based on perturbation has been proposed recently [56]. The solution is based on the explicit dependence of the impulse response function on the uncertain structural parameters. The variability of the dynamic response has also been studied for special type of linear structures, mostly truss structures, by approaches closely related to the perturbation approach [16, 9].

3.2 Spectral methods 3.2.1 Karhunen-Lo`eve expansion The simplest representation of a stochastic process by a spectral representation is the Karhunen-Lo`eve expansion, capable to describe the first two moments of a stochas-

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tic process [28, 35, 18] m

h (t) = h (0) (t) + ∑ θ j h ( j) (t) ,

(19)

j=1

where {hh( j) (t)}mj=0 are deterministic functions of t, which might denote time or spatial coordinates. The set {θ j }mj=1 are uncorrelated zero mean random quantities, which can be normalized to unit variance by proper scaling the functions {hh ( j) (t)}mj=1 E[θ j ] = 0 , E[θi θk ] = δik , (20) where δik denotes the Kronecker’s delta symbol. The above Karhunen-Lo`eve (KL) expansion allows for a straight forward representation of the mean vector μ h and of the covariance matrix C hh :

μ h = E[hh(t)] = h (0) (t) C hh = E[(hh(t) − μ h )(hh(t) − μ h )T ] =

(21)

m

∑ h( j) (t)hh( j)T (t)

(22)

j=1

Theoretically the expansion has an infinite number of terms for a continuous stochastic process, and n terms for a discrete scalar process with n components. Given a n-dimensional covariance matrix C , the KL terms {hh( j) (t)}nj=0 are uniquely specified by the eigenvectors φ j and eigenvalues λ j of the symmetric covariance matrix (23) h ( j) (t) = φ j (t) λi , after solving the standard eigenproblem C hh Φ = Φ Λ , where the columns of matrix Φ contain the eigenvectors. The actual number m necessary to describe the second moment properties of the stochastic process with sufficient accuracy depend on the eigenvalues and also on the frequency content of the response of interest. Assuming the eigenvalues decreasingly ordered, and accepting a tolerance ε , the number m < n can be estimated accordingly. Summarizing, the KL-expansion allows for a simple description of the first two moment properties of a stochastic process [18, 22, 1, 29, 52]. It might also be used directly to specify the first two moments of the process. In such case, the KL-terms {hh ( j) (t)} do not need to be orthogonal. It should be stressed, that the KL-expansion is applicable for Gaussian and non-Gaussian stochastic processes. However, for a non-Gaussian process, the first two moments are insufficient for a complete probabilistic description of the process.

3.2.2 Polynomial chaos expansion When expressing the structural variability or uncertainty by a Gaussian random field, or by interpreting some structural quantities by random variables with Gaus-

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sian properties, all structural matrices can be expressed in the form of the KLexpansion, as e.g. the stiffness matrix K (θ ) m

K (θ ) = K (0) + ∑ θ j K ( j) ,

(24)

j=1

and analogously the mass matrix M (θ ) and damping matrix D(θ ), where θ j are standard Gaussian variables, i.e have zero mean and unit standard deviation. The response, however, is non-Gaussian and these non-Gaussian response properties can be approximated by the so called “Polynomial chaos expansion” [18, 17, 47]. Polynomial chaos expansion is a Galerkin projection scheme based on the Wiener integral representation. Before the dynamic case is addressed, as a basic introduction the simple static displacement response is considered K (θ )]−1 f u (θ ) = [K

(25)

for a deterministic load vector f . The polynomial chaos expansion uses a finite set of P terms {uu(i) }Pi=0 to approximate the response P

u (θ ) = ∑ Ψi (θ )uu(i) ,

(26)

i=0

where the functions Ψi (θ ) are scalar Hermite polynomials with the advantageous properties

Ψ0 (θ ) ≡ 1;

E[Ψi (θ )] = 0 for i > 0;

E[Ψi (θ )Ψj (θ )] = δi j · E[Ψi2 (θ )] .

(27)

Multiplying both sides of eqn. (26) by the scalar function Ψk (θ ) and taking the expectation on both sides leads to P

E[Ψk (θ )uu(θ )] = ∑ E[Ψk (θ )Ψi (θ )]uu(i) .

(28)

i=0

Taking into account the orthogonality of the Hermite polynomials on the r.h.s. of the above relation, the vector u (k) must satisfy the relation u (k) =

E[Ψk (θ )uu(θ )] . E[Ψk2 (θ )]

(29)

The expectation E[Ψk2 (θ )] is known exactly in closed form, the nominator E[Ψk (θ )uu(θ )], however, needs to be evaluated numerically. Two different approaches are available to compute the coefficient vector u (k) . The first establishes an extended linear equation system [17] which is solved either directly or in an iterative manner [47]. Alternatively, Monte Carlo sampling is applied — as discussed in the next section — to obtain an estimate for the nominator E[Ψk (θ )uu(θ )].

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It can be shown that any non-Gaussian distribution can be represented by the polynomial chaos (PC) representation for any variability with finite variance. However, the polynomial chaos expansion suffers from the so called “curse of dimensionality”, which means an exponential growth of computational efforts and required number of terms P as a function of the number m of independent random quantities. Since in practical application the number P is necessarily finite, and also sufficiently small to be competitive with simple “direct” Monte Carlo simulation, the rate of convergence plays an important role [13]. This serious drawback can be mitigated by a recently suggested reduction scheme [10], i.e. by introducing two well separated coarse and fine scales respectively, which allow to expand the range of applicability for higher dimensional problems. These developments, however, address mainly static stochastic Finite Element problems. Only recently, the polynomial chaos expansion has been applied in [19] to represent the joint statistics of the random eigenvalue problem. For expanding the range of applicability of the PCE, condensation techniques are proposed in [31] to compute the non-stationary random vibration response of an uncertain linear system. In a recent work [36], the PCE is used to approximate the variability of the linear dynamic response in context with robust structural optimization.

3.3 Direct Monte Carlo Simulation Monte Carlo methods have been first developed by physicists [39, 25] and are nowadays used in many fields ranging from finance, social science, chemistry, medicine, mathematics, engineering [23, 51, 14] etc. In this work, only Monte Carlo methods related with structural uncertainties and reliability will be discussed. One main application area is the generation of samples from given probability distributions and their propagation to the response, providing information on the likely spread of the response by scatter plots or moment estimates like mean response, standard deviation, correlations, etc. In other words for simulating and understanding the effects of uncertainty and variability in structural design. Several aspects favor procedures based on MCS in engineering applications: 1. Considerably smaller growth rate of the computational efforts with increasing dimensionality than analytical procedures. 2. Generally applicable, well suited for parallel processing and computationally straightforward. 3. Non-linear complex structural behavior does not complicate the basic procedure. 4. Manageable efforts for complex systems. Contrary to numerical solutions of analytical procedures, the employed structural model and the type of stochastic loading do for MCS not play a decisive role. For this reason, MCS procedures might be structured according to their purpose i.e. where sample functions are generated either for the estimation of the overall distribution or for generating rare adverse events for reliability assessment.

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The most basic application of Monte Carlo procedures is the generation of independent samples of the structural properties and of the loading by making use of an appropriate random number generator. Since each call to the random number generator is supposed to return an independent random number, the stochastic process needs to be represented as a function of independent random quantities. For non-standard distributions, realizations are computed by the Inverse transformation method, the composition method and the Acceptance-Rejection method [14]. The consideration of correlations in most cases is not simple, in fact actually quite difficult in general. Correlations are only easy to handle for Gaussian distributed variables, where they can always be transformed into independent quantities by using a linear transformation. Non-Gaussian correlated quantities are therefore often transformed to correlated Gaussian variables which can be established as linear combination of independent quantities, and then transformed back to the original non-Gaussian distributed variables (see e.g. [33]). Similarly as for non-Gaussian variables, are non-Gaussian distributed stochastic processes transformed to Gaussian processes, which are actually used for the generation of samples as a function of independent random quantities, and then again transformed back to non-Gaussian properties [21, 22]. For cases where the complexity of the problem does not allow for an explicit formulation of a joint probability function, but only to define ratios of the density for various random quantities, the Metropolis-Hastings algorithm [40, 24, 6] can be applied. This algorithm allows to simulate the posterior density in Bayesian updating problems [45]. Another alternative to generate random samples of correlated random variables is Gibbs-sampling [5]. The ability to generate independent samples of the structural properties and of the loading provides the basis for studying the associated variability of the response. Suppose all structural properties are uniquely specified as function of a set of k independent standard normal variates θ = {θl }kl=1 and the loading also by a function of m independent standard normal variates ξ = {ξl }m l=1 . These functions are in general non-linear whenever non-Gaussian properties are described. As a consequence, the equation of motion for linear structural analysis reads then ¨ θ , ξ ) + D (θ )u(t; ˙ θ , ξ ) + K (θ )uu(t; θ , ξ ) = f (t; ξ ) , M (θ )u(t;

(30)

where M , D and K denotes the mass, damping and stiffness matrix, respectively. In context with Monte Carlo Simulation, it is essential to note, that any point ζ = (θ , ξ ) in a (k + m)-dimensional space, specifies uniquely the equation of motion. Let {ζ (i) }Ni=1 be a set of independent random realizations, drawn from the (i) distribution of (θ , ξ ). Each realization ζ leads then to a deterministically defined structural problem, for which a variety of commercial and open-source FE analysis tools are available to determine the static/dynamic, linear/non-linear response. Suppose the responses of interest, e.g. extreme stresses, displacements or acceleration, (i) (i) rl , associated with the realization ζ , are represented by the components of the (i) vector r . Statistical estimates for the mean value μˆ l , for the standard deviation σˆ L and for the correlation coefficients ρˆ l j between different response quantities, are then obtained by

Uncertain Linear Systems in Dynamics: Stochastic Approaches

μˆ l =

1 N (i) ∑ rl ; N i=1

ρˆ l j =

281

1 N (i) ∑ (rl − μˆ l )2 N − 1 i=1

(31)

N 1 (i) (i) (r − μˆ l )(r j − μˆ j ) , ∑ N σˆ l σˆ j i=1 l

(32)

σˆ l2 =

which provide then information on the variability of the response. Characteristic for all estimates obtained by Monte √ Carlo Simulation is the the slow convergence rate, which is proportional to 1/ N. To increase the efficiency, two basic routes can be followed. One route of procedures aims at a reduced computer time by exploiting the fact that the same structure with similar properties has to be analyzed many times. The Neumann expansion [64] is a typical example for such a procedure. Typically, these procedures exploit the fact that the set of solutions do not deviate by far from a reference solution. For linear dynamic systems with uncertain structural properties, accurate and fast solutions for the eigensolution have been suggested in [62, 48] by using vector iteration and component mode synthesis. Other methods to increase the number of independent samples apply parallel processing [46, 62, 27, 55]. Parallel processing is especially well suited to generate the response of independent realizations since all tasks are completely independent and require little communication. The second main application of Monte Carlo sampling is the evaluation of higher dimensional integrals [23, 51] needed for reliability analysis and for the identification of the failure domains by random search algorithms [61]. In order to compute the failure probability, a high-dimensional integral needs to be evaluated, p f = E[g(θ , ξ ) ≤ 0] =

···

I f (g(θ , ξ ))q(θ )q(ξ ) d θ1 · · · d θk · d ξ1 · · · d ξm , (33)

where I f (g) is an indicator function, assuming the value of 1 for g ≤ 0 and zero otherwise. In the above representation, it is implied that the performance function g = g(θ , ξ ) is formulated such that the function is positive whenever, all response quantities rl (θ , ξ ) of interest are in the safe domain, i.e. when they fulfill all response or performance criteria, otherwise g must become zero or negative. Two basic difficulties must be overcome. First, the dimension d = k + m is usually high, precluding any available deterministic integration scheme. Second, the performance function is usually not known explicitly in terms of the random variables (θ , ξ ), but must be computed point-wise by applying suitable deterministic FE-analysis tools. Both difficulties favor the application of Monte Carlo simulation as fundamental approach to cope with the above integral. Applying Direct MCS, the above integral is estimated by using the following approximation: pˆ f =

1 N (i) ∑ I f (g(ζ )), N i=1

(i) (i) ζ = (θ (i) , ξ )

(34)

Its convergence rate, which in the mean square sense is most appropriately measured by the coefficient of variation CoV( pˆ f ) = (1 − p f )/(N p f ) of pˆ f , is inde-

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pendent of the dimensionality of the random vector ζ . Its main disadvantage is its inefficiency in estimating small failure probabilities p f due to the large number (proportional to 1/p f ) of samples, or equivalently system analyses, needed to achieve an acceptable level of accuracy. So called variance reduction procedure are applied for reliability estimation. These methods, dealing with the efficient reliability evaluation, are described within the section discussing the reliability evaluation.

3.4 Random matrix approach The random matrix approach has its origin in nuclear physics to describe eigenvalues and eigenfunctions of the Hamiltonian operator approximated by finite high dimensional matrices [38]. These matrices must satisfy certain constraints, like symmetry and (semi) positive definiteness in order to belong to the set of structural matrices. A random matrix solution for uncertainty propagation has been proposed in [57, 58, 59, 60] where a so called non-parametric approach is suggested. In this approach, it is argued that uncertainties do not only arise from uncertainties of the structural parameters and and the excitation, but also from uncertainties induced by its mathematical or mechanical modeling, which always involves some abstractions, insufficient understanding of the actual true mechanics and simplifications. Clearly, both model uncertainties and parametric uncertainties lead to uncertainties of the structural matrices which are modeled by random variables using the probabilistic approach. In the proposed approach it is assumed that the mean of the structural matrices (stiffness, damping and mass) is known and that the inverse of these random matrices possesses finite higher moments. A reduced modal space is used which allows to consider large complex structural linear systems. To describe the variability of these matrices, the maximum entropy principle is applied, leading for each structural matrix to a single parameter — the so called dispersion parameter — which specifies uniquely the random properties of these structural matrices. These dispersion parameters need to be calibrated requiring measurement data of the variability of the response. In [4], e.g. the variability of the frequency response function of an aerospace structure is compared using a parametric and the non-parametric approach, respectively. The variability induced by uncertainties of the modal damping, however, could not be represented by the non-parametric approach for reasons which are yet not fully clarified [20].

4 Computational Efficiency The quantification and propagation of uncertainty in structural dynamics demand multiple computational efforts when compared with deterministic analysis. Naturally, to be useful, the computational efficiency of a particular method of analysis plays an important role, particularly w.r.t. practical applications. In this context two

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aspects are essential, namely the development of the hardware and software, respectively. While from the user’s point of view there is no possibility to influence the developments in the hardware, it is certainly possible to improve the efficiency on the software side. For this, three aspects appear to be important, i.e. (1) Model reduction (Subspace methods, including those applying Karhunen-Lo`eve expansion [28, 35]). (2) Improvement of algorithms (Advanced Monte Carlo simulation procedures). (3) Parallel Processing (very well suited for Monte Carlo Simulation.) A combination of all three aspects leads to a considerable reduction in the computational efforts, and consequently an increase of the computational efficiency. Since the tremendous computational efforts are often used as an argument for not taking into account uncertainties in the analysis, the application and further development of methods for increasing the efficiency is of paramount importance.

5 Summary The critical appraisal of the various method in terms of applicability, efficiency, accuracy and limitations is summarized as follows: Perturbation is straightforward to apply for general FE-models. Its accuracy is only satisfactory in case the coefficients of variation of the structural parameters are small, say less than 5%. Only first order perturbation is recommended to investigate the variability of the response. For such cases, the computational efforts increase approximately linearly with the number of uncertain quantities. Spectral methods are well suited to represent statistical moment properties of a stochastic process or of random fields. The Karhunen-Lo`eve representation describes the mean and Covariance matrix, while the polynomial chaos expansion (PCE) is — in principle — capable to describe any distribution with finite variance. Unfortunately, the computational efforts increase exponentially with the order and the number of involved uncertain quantities, which put some practical restrictions to this method. Direct Monte Carlo is the most generally applicable procedure known to investigate the variability — as well the reliability — of any type of structure (linear/nonlinear). It is well suited to study the variability of the response for cases where few hundred samples might suffice to obtain a reliable estimate for the variably. The strength of this approach is that its accuracy is completely independent of the number of uncertain quantities and of the complexity of the investigated structure. The Random Matrix approach is designed to capture the variability of complex large structures. Using the entropy principle, the variability of structural matrices are controlled by a single parameter, the so called dispersion parameter. This dispersion parameter needs to be calibrated from data. Since measured data include also deviations due to modeling errors, the variability due to modeling uncertainties are included by this approach.

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Acknowledgements This work was partially supported by the Austrian Research Fund (FWF) under the contract number P19781-N13, which is gratefully acknowledged by the authors.

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Time domain analysis of structures with stochastic material properties Giovanni Falsone and Dario Settineri

Abstract Aim of the present work is the presentation of a time domain approach for the evaluation of the dynamical response of uncertain structures excited by deterministic excitations. In the case of linear systems, the time domain analysis is necessary when the non-stationary response is required. This occurs when the input is non-stationary or when the transient analysis is required. The proposed approach is based on the use of the perturbation technique. The formulation here presented is able to overcome one of the main drawbacks related to the use of the perturbation techniques, that is the difficult extension to perturbation degrees higher than one. In the present work this extension is made feasible both from an analytical point of view and from a computational point of view.

1 Introduction The study of uncertain structures has become more and more important in these last years. The uncertain structures are characterized by the fact that one or more of their mechanical and/or geometrical properties cannot be defined deterministically. The importance of this kind of study is above all related to some structural problems, as the structural reliability, for which neglecting the effective uncertain nature of the structural parameters is not possible. It is obvious that for these systems the traditional deterministic analyses cannot be applied, but alternative approaches have to be taken into account. If it is possible to characterize the uncertain parameters stochastically, then the probabilistic approaches can be used. Among the probabilistic approaches, the statistical ones, based on the Monte Carlo simulations, are the Giovanni Falsone Dipartimento di Ingegneria Civile, Universita di Messina, e-mail: [email protected] Dario Settineri Dipartimento di Ingegneria Civile, Universita di Messina, e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 21, © Springer Science+Business Media B.V. 2011

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simplest from a theoretical point of view. In fact, they need the realizations of a sufficiently high number of samples of the uncertain parameters and the solution of the corresponding deterministic problems [1]. However, increasing the structural degrees of freedom and the number of uncertain parameters, the computational effort attained by the statistical methods becomes very high, above all for nonlinear structures. For this reason, some alternative non-statistical methods have been proposed in the literature [2]–[8]. In particular, the perturbation approaches have had the greatest diffusion [9]–[12]. The fundamental drawback related to the use of the perturbation approaches lies on the consistent loss of accuracy when the level of uncertainty of the structural parameters increases, if a suitable degree of perturbation is not used. Unfortunately, increasing the degree of perturbation over one-two usually implies too complicated formulations and heavy computations. With reference to the dynamic analyses, while in the frequency domain it is possible to find in the literature some interesting approaches giving very accurate results [19], and an approach even the exact solution [20], in the time domain the applications are not numerous and almost all based on the application of the first order perturbation approach. In this paper a perturbation approach is used for the dynamic analysis of uncertain structures subjected to deterministic inputs. The uncertainty, for simplicity considered only on the structural stiffness is treated in such a way that it is defined by a set of random variables affecting the stiffness matrix in a linear form. The formulation presented is very simple and easily extendible to high degree of perturbation, allowing an acceptable level of accuracy even in the case of not extremely low level of uncertainties.

2 Preliminary concepts The numerical analysis of an uncertain structure requires, first, its FE discretization and, if the uncertain structural parameters are modeled as continuous random fields, these have to be approximated by discrete random variables. In the literature many methods allow the discretization of the random fields in random variables [13]. They may be roughly classified into two groups: the point-discretization methods, in which, for example, the midpoint [14], the local averaging [15] and the shape function method [16] are included; and the series expansion methods, among which the Karhunen-Loeve series expansion method [17] is perhaps the most used one. In the following we will make reference to the first class of methods. As a consequence, each uncertain parameter is assumed to be constant in each FE and it is characterized by a single random variable. For simplicity we make reference to those structures in which the uncertainties are only in the stiffness matrix. As a consequence, the n (n =structural DOF) differential equations governing the structural motion can be written as: ˙ α ) + K(α )u(t, α ) = f (t) M u(t, ¨ α ) +Cu(t,

(1)

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where α = [α1 α2 · · · αN ]T is the vector collecting all the random variables characterizing the uncertainties in the various FEs; u is the nodal displacement vector; K is the stochastic stiffness matrix; M and C are the mass and damping deterministic matrices; at last, f is the deterministic nodal force vector. It is important to note that the relationship between the stiffness matrix and the vector α is often linear. This surely happens, for example, when only the material Young moduli of a structure are uncertain. In other cases, for example when the Poisson moduli are uncertain, the relationship is nonlinear. Nevertheless, by applying a suitable random variable transformation, it is always possible to make this relationship linear with respect to the new variables. For example, if the stiffness matrix is proportional to 1/β , β being a random variable, it is always possible to apply the variable transformation α = 1/β , in order to make linear the relationship between K and α [18]. This means that it is always possible to write: n

K(α ) = K0 + ∑ Ki αi

(2)

i=1

If the state variable vector zT (t, α ) = uT (t, α ) u˙T (t, α ) is introduced, the motion equation, once that eq. (2) has been taken into account, can be rewritten as follows: z˙(t, α ) =

n

D0 + ∑ Di αi z(t, α ) +V f (t)

(3)

i=1

where:

D0 =

0

I

M −1 K M −1C

;

Di =

0 ; M −1 Ki 0 0

0 V= I

(4)

3 Application of the perturbation approach The application of the perturbation approach requires the assumption about the dependence of the structural response z(t, α ) on the random variables α having the following form: n

n

n

n

n

z(t, α ) = z0 (t) + ∑ zi (t)αi + ∑ ∑ zi j (t)αi α j + ∑ ∑ i=1

i=1 j=1

n

∑ zi jk (t)αi α j αk + · · ·

(5)

i=1 j=1 k=1

where z0 (t), zi (t), zi j (t), etc., are deterministic vectors representing the solutions of the differential equations obtained by replacing the expression of z(t, α ) given in eq. (5) into eq. (3) and by applying the polynomial identity principle, that is:

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z˙0 (t) = D0 z0 (t) +V f (t) z˙i (t) = D0 zi (t) + Di z0 (t) z˙i j (t) = D0 zi j (t) + Di z j (t))

(6)

z˙i jk (t) = D0 zi jk (t) + Di z jk (t) .. . They are first order differential equations all characterized by the same dynamical matrix D0 . They differ each other for the input: in the first one the input is the original one, while for the j-th equation the input depends on the responses of the ( j − 1)-th equations. This allows us to solve all the system equations by a cascade approach. For example, making use of the transition matrix, Θ0 (t), related to the dynamical matrix, D0 , that is Θ0 (t) = exp(D0t), the cascade solutions of eqs. (6), under the assumption of zero initial conditions (z(t)|t=0 = 0) can be written as follows: t

z0 (t) =

0

t

zi (t) =

Θ0 (t − τ )V f (τ )dτ

Θ0 (t − τ )Di z0 (τ )dτ

0

t

zi j (t) =

0

Θ0 (t − τ )Di z j (τ )dτ

t

zi jk (t) =

0

(7)

Θ0 (t − τ )Di z jk (τ )dτ

.. . Depending on the form of the input f (t), the previous cascade integrals may be solved in closed form. In the most cases a step-by-step procedure is necessary.

4 Moments of the uncertain structure response Once that the cascade deterministic solutions are obtained by solving the integrals appearing in eqs. (7), the statistical moments of the global structural response z(t, α ) can be obtained starting from eq. (5) and applying properly the mean operator. For example, the first two response moments, omitting for simplicity the time dependence on the symbols and by applying a third order perturbation, have the following expression:

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291

E[z] = z0 n

+ ∑ zi E[αi ] i=1 n n

+ ∑ ∑ zi j E[αi α j ] i=1 j=i n n n

+ ∑ ∑ ∑ zi jk E[αi α j αk ] i=1 j=i k= j

[2]

E[z[2] ] = z0

(8)

n

+ ∑ zˆi E[αi ] i=1 n n

+ ∑ ∑ zˆi j E[αi α j ] i=1 j=i n n n

+∑ ∑

∑

+∑ ∑

∑∑

+∑ ∑

∑∑∑

+∑ ∑

∑ ∑ ∑ ∑ zˆi jkl pq E[αi α j αk αl α p αq]

zˆi jk E[αi α j αk ] i=1 j=1 k=1 n n n n

zˆi jkl E[αi α j αk αl ] i=1 j=1 k=1 l=1 n n n n n

zˆi jkl p E[αi α j αk αl α p ] i=1 j=1 k=1 l=1 p=1 n n n n n n i=1 j=1 k=1 l=1 p=1 q=1

where:

zˆi = zi ⊗ z0 + z0 ⊗ zi zˆi j = zi j ⊗ z0 + z0 ⊗ zi j + zi ⊗ z j zˆi jk = zi jk ⊗ z0 + z0 ⊗ zi jk + zi j ⊗ zk + zi ⊗ z jk zˆi jkl = zi jk ⊗ zl + zi ⊗ z jkl + zi j ⊗ zkl

(9)

zˆi jkl p = zi jk ⊗ zl p + zi j ⊗ zkl p zˆi jkl pq = zi jk ⊗ zl pq In eqs. (8) and (9) the apex into square brackets indicates the power made by the Kronecker product, that is: z[2] = z ⊗ z, where the symbol ⊗ indicates the Kronecker product between two matrices; the symbol E[(•)] indicates the mean of (•). As a consequence E[z] is a vector collecting all the response means and E[z[2] ] is a vector collecting all the second order statistical moments of the response. If the second order approach is used, the terms in zi jk must be zero in the eqs. (5), (6) and (7) and, as a consequence, the eqs. (8) and (9) are reduced as follows:

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n

n

E[z] = z0 + ∑ zi E[αi ] + ∑ ∑ zi j E[αi α j ] i=1 j=i n n

i=1

n

n

n

E[z[2] ] = z0 + ∑ zˆi E[αi ] + ∑ ∑ zˆi j E[αi α j ] + ∑ ∑ [2]

n

+∑

n

i=1 j=i

i=1 n n

n

∑ z˜i jk E[αi α j αk ]

i=1 j=1 k=1

∑ ∑ ∑ z˜i jkl E[αi α j αk αl ]

(10)

i=1 j=1 k=1 l=1

and z˜i jk = zi j ⊗ zk + zi ⊗ z jk z˜i jkl = zi j ⊗ zkl

(11)

At last, if a first order perturbation approach is used, then the terms zi j must be imposed zero and the response mean and second order response moments are approximated as follows: n

E[z] = z0 + ∑ zi E [αi ] i=1

n

n

n

(12)

E[z[2] ] = z0 + ∑ zˆi E [αi ] + ∑ ∑ z˜i j E[αi α j ] [2]

i=1 j=i

i=1

where: z˜i j = zi ⊗ z j

(13)

In many cases of practical interest the random variables αi are assumed to have zero mean and symmetric probability density function; this implies that all the moments of odd order are zero. In these cases the relationships above reported are notably simplified. For example, for the third order perturbation approach, eqs. (8) are simplified as follows: n

n

E [z] = z0 + ∑ ∑ zi j E [αi α j ] E

i=1 j=i n n n n n n [2] zˆi j E [αi α j ] + zˆi jkl E[αi α j αk αl ] z[2] = z0 + i=1 j=i i=1 j=1 k=1 l=1 n n n n n n

∑∑

+∑ ∑

∑∑∑∑

∑ ∑ ∑ ∑ zˆi jkl pq E[αi α j αk αl α pαq ]

(14)

i=1 j=1 k=1 l=1 p=1 q=1

Finally, under the very common assumption of Gaussian zero-mean variables, the last equation is reduced further on, all the even moments of order greater than two being expressed in terms of the second order ones, that fully characterize the variables. In this way, the first of eqs. (14) remains unchanged, while the second one is rewritten in the form:

Time domain analysis of structures with stochastic material properties

n n n n [2] E z[2] = z0 + ∑ ∑ zˆi j E [αi α j ] + ∑ ∑ i=1 j=i

n

n

+∑ ∑

n

n

n

n

293 3

∑ ∑ zˆi jkl ∑ E[αi α j ]E[αk αl ]

i=1 j=1 k=1 l=1

n

n

15

∑ ∑ ∑ ∑ zˆi jkl pq ∑ E[αi α j ]E[αk αl ]E[α p αq]

(15)

i=1 j=1 k=1 l=1 p=1 q=1

where the number over the summation symbols refers to the number of possible distinct products obtained by the index permutation, that is, for example: 3

∑ E[αi α j ]E[αk αl ] = E[αi α j ]E[αk αl ] + E[αi αk ]E[α j αl ] + E[αi αl ]E[αk α j ]

(16)

An ultimate reduction in the computational effort related to the solution of the previous expressions is related to the assumption of independence among the random variables; this implies that all the moments E[αi α j ] are zero when i = j. Hence, the first two response moments are given as: n E [z] = z0 + ∑ zii E αi2

E z[2]

i=1

[2] = z0 + n

n

∑ zˆii E

i=1 n n

+ 15 ∑ ∑

n n 2 αi + 3 ∑ ∑ zˆii j j E αi2 E α 2j

∑ zˆii j jkk E

(17)

i=1 j=1

2 2 2 αi E α j E αk

i=1 j=1 k=1

The above reported relationships, depending on the assumptions made on the random variables and depending on the degree of the perturbation chosen for the analyses, are able to evaluate the approximated solution in terms of the first two order moments of the response state variables. Analogous relationships can be obtained for higher order response moments. The level of the accuracy obviously depends on the degree of the perturbation used in the analyses. We expect that, in the case of zero-mean Gaussian random variables describing the structural uncertainties, if their variances is not relatively high, a third degree perturbation approach is able to give acceptable results.

5 Application As example, the beam represented in Fig 1 is taken into account. It is characterized by the following geometrical and physical parameters: L = 10 m, the cross section area is A = 0.04 m2 , the mass density factor is μ = 257850 kg/m3 , the damping factor is ξ = 0.05.

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Fig. 1

The beam is discretized by N = 6 finite elements, with length equal to l = 2.50 m. The only uncertain parameter is chosen to be the Young modulus in each finite element, having the form: Ei = E0 (1 + αi ) , i = 1, 2, 3, 4, where E0 = 2 × 1011 N/m2 is the mean value of the Young modulus, assumed equal for all the elements. The random variables αi are assumed to be zero-mean Gaussian and uncorrelated, whit standard deviation σ = 0.2. Assuming the external action constant, f (t) = F = 10000 N, the results represented in Figs. 2–5, in terms of mean and second order moment of displacement and velocity of the central node deflection, have been obtained. In Figs. 6-9 analogous results have been obtained for the case f (t) = F sin(ω f t), whit F = 10000N and ω f = 16rad/ sec. In all the results represented in these figures the improving accuracy augmenting the perturbation order is evident. Moreover, the not very high value of the standard deviation of the random variables guarantees a very good accuracy when the third order perturbation is used.

6 Conclusions A very simple perturbation approach for the time domain dynamic analysis of uncertain structures has been presented. It lies on the assumption that the structural uncertainty is represented by a set of random variables appearing linearly on the differential motion equations governing the problem. The simplicity of the formulation allows to easily consider perturbations of order greater than one or two, that are the orders usually considered in the literature works. The results of the application have evidenced that, for not too high levels of uncertainties, a good accuracy can be reached even for a third order perturbation approach.

Time domain analysis of structures with stochastic material properties

Fig. 2: Mean response of the central node displacement

Fig. 3: Mean response of the central node velocity

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Fig. 4: Second moment of the central node displacement

Fig. 5: Second moment of the central node velocity

Time domain analysis of structures with stochastic material properties

Fig. 6: Mean response of the central node displacement

Fig. 7: Mean response of the central node velocity

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Fig. 8: Second moment of the central node displacement

Fig. 9: Second moment of the central node velocity

Time domain analysis of structures with stochastic material properties

299

References 1. Papadrakakis, M. and Kotsopoulos, A. (1999), Parallel solution method for stochastic finite element analysis using Monte-Carlo simulation, Computer Methods in Applied Mechanics and Engineering, Vol. 168, 305–320. 2. Liu, K.L., Mani, A. and Belytschko, T. (1987), Finite element methods in probabilistic mechanics, Probabilistic Engineering Mechanics, Vol. 2 (4), 201–213. 3. Ghanem, R.G. and Spanos, P.D. (1991), Stochastic Finite Elements: a Spectral Approach; Springer- Verlag, N.Y. 4. Matthies, H.G., Brenner, C.E., Bucher, C.G. and Soares, C.G. (1997), Uncertainties in probabilistic numerical analysis of structures and solids — Stochastic finite elements, Structural Safety, Vol. 19 (3), 283–336. 5. Sudret, B. and DerKiureghian, A. (2000), Stochastic finite element methods and reliability: a state-of-art report, Technical Report UCB/SEMM-2000/08, Department of Civil and Environmental, University of California. 6. Schueller, G.I. (2001), Computational stochastic mechanics — recent advances, Computer & Structures, Vol. 79, 2225–2234. 7. Noh, H.-C. (2004), A formulation for stochastic finite element analysis of plate structures with uncertain Poisson’s ratio, Computer Methods in Applied Mechanics and Engineering, Vol. 193, 4847–4873. 8. Stefanou, G. and Papadrakakis, M. (2004), Stochastic finite element analysis of shells with combined random material and geometric properties, Computer Methods in Applied Mechanics and Engineering, Vol. 193, 139–160. 9. Nakagiri, S. and Hisada, T. (1982), Stochastic Finite Element Method Applied to Structural Analysis with Uncertain Parameters, Proceedings of the International Conference on Finite Element Methods, Aechen, Germany, 206–211. 10. Elishakoff, I., Ren, Y.J. and Shinozuka, M. (1995), Improved Finite Element Method for Stochastic Structures, Chaos, Solitons & Fractals, Vol. 5 (5), 833–846. 11. Impollonia, N. and Muscolino, G. (2002), Static and Dynamic Analysis of Non-Linear Uncertain Structures, Meccanica, Vol. 37, 179–192. 12. Van den Nieuwenhof, B. and Coyette, J.-P. (2003), Modal approaches for the stochastic finite element analysis of structures with material and geometric uncertainties, Computer Methods in Applied Mechanics and Engineering, Vol. 192, 3705–3729. 13. Li, C.C. and Der Kiureghian, A. (1993), Optimal discretization of random fields, Journal of Engineering Mechanics, Vol. 119 (6), 1136–1154. 14. Der Kiureghian, A. and Ke, J.-B. (1988), The stochastic finite element method in structural reliability, Probabilistic Engineering Mechanics, Vol. 3 (2), 83–91. 15. Vanmarcke, E.H. and Grigoriu, M. (1983), Stochastic finite element analysis of simple beams, Journal of Engineering Mechanics, Vol. 109 (5), 1203–1214. 16. Liu, K.L., Belytschko, T. and Mani, A. (1986), Random field finite elements, International Journal of Numerical Methods in Engineering, Vol. 24 (10), 1831–1845. 17. Spanos, P.D. and Ghanem, R. (1989), “Stochastic finite element expansion for random media”, Journal of Engineering Mechanics, Vol. 115(5). 18. Falsone, G. and Impollonia, N. (2002), A New Approach for the Stochastic Analysis of Finite Element Modelled Structures with Uncertain Parameters, Computer Methods in Applied Mechanics and Engineering, Vol. 191, 5067–5085. 19. Falsone, G. and Ferro, G. (2005), A Method for the Dynamical Analysis of FE Discretized Uncertain Structures in the Frequency Domain, Computer Methods in Applied Mechanics and Engineering, Vol. 194, 4544–4564. 20. Falsone, G. and Ferro, G. (2007), An exact solution for the static and dynamic analysis of FE discretized uncertain structures, Computer Methods in Applied Mechanics and Engineering, Vol. 196 (21–24), 2390–2400.

Vibration Analysis of an Ensemble of Structures using an Exact Theory of Stochastic Linear Systems Christophe Lecomte

Abstract An exact theory of stochastic linear systems is presented. These are made of a nominal deterministic system disturbed by a random component (itself being a deterministic disturbance weighted by a random variable whose probability density function is known). Both rank-one and multi-rank disturbances are covered, so that the theory is applicable to a wide range of situations where some parameters of a dynamic system (such as mass, stiffness, Young modulus, damping coefficient, etc. of some components) are random. Exact expressions of the statistics of the response of the stochastic system are given for any inputs and outputs, and at any frequency. An exact closed-form expression of the statistics in terms of special (error) functions is available in the case of normal variables (having a Gaussian probability density function). All expressions can therefore be evaluated precisely and efficiently. The theory is applied to a few structural dynamic systems and shown to be applicable from low to high frequencies without particular restriction in the mid-frequency range. The theory is also shown to be more precise and several orders of magnitude faster than a Monte-Carlo approach applied to the same stochastic linear systems.

1 Introduction In structural vibration analysis, the treatment of uncertainties becomes necessary in several situations. In one situation, components of a structure may be imperfectly known, say because they are not perfectly measurable. This would be the case, for example, of a vehicle suspension component whose distribution of mass is unknown. In another situation, as is the case here, the interest may be in the vibrational behavior of an ensemble of structures that vary from each other rather Christophe Lecomte University of Cambridge, Department of Engineering, Trumpington Street, Cambridge, CB21PZ, United Kingdom. Now at University of Southampton, ISVR and S3RI, University Road, Highfield, Southampton S017 1BJ, United Kingdom, e-mail: [email protected], [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 22, © Springer Science+Business Media B.V. 2011

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than in the behavior of its individual members. This would typically be the case of the design or refinement of an assembly line. In both situations, the variability of the imperfectly known or varying properties is important only if the behavior of the structure or the ensemble is sensitive to it. This sensitivity is usually higher in the frequency ranges with higher modal density, which generally occurs at higher frequencies. Therefore, the higher the modal density, the less it makes sense to consider a particular occurrence of a system because a tiny variation in the properties of the system may change the local behavior of the system completely. But, at the same time, the higher the modal density, the simpler the statistics of the system or the ensemble become. Several statistical methods making use of this property may therefore usually be used at high modal density. The region of medium modal density might be somewhat more challenging, as a system may at the same time be quite sensitive to variation of parameters while not exhibiting the simpler statistical properties. The subject has been under extensive investigation for several decades and is still actively discussed in the literature, including in other chapters of this book. The reader is referred to [9] for a recent review of existing work. Herein, the paper is concerned with the ensemble statistics of discrete stochastic systems whose properties are perfectly known [7, 8]. They are made of a main deterministic structure to which random subsystems are coupled. Each of those random disturbances is represented as the product of a deterministic matrix by a random variable with known probability density function. It is shown, for simplicity, in the particular case of a single disturbance, that exact and computable expressions of the ensemble statistics may be derived when the disturbance is either rank-one [7] or multi-rank [8]. A significant advantage of the resulting exact theory discussed here is that it is applicable in any frequency range and at any modal density. In the next section, the stochastic system is formally described, introducing some assumptions and notations. In Sect. 3, exact expressions of the ensemble statistics are derived, namely mean, variance, and covariance, of the stochastic system for a rank-one disturbance. The exact expressions for multi-rank disturbances are then presented, referring to [8] for a proof and derivation, and it is explained that the theory is very general, computationally relatively inexpensive, and robust for any frequency range or modal density. The randomness is isolated into a discrete number of scalar integrals which have analytic expressions and may be efficiently evaluated for particular probability density functions. Such exact analytic expressions are presented, in Sect. 4, for the case of Gaussian probability density functions, i.e. in the case of normal variables. Finally the application of the theory is illustrated on several problems.

2 Description of the Stochastic System The stochastic system is now introduced with some assumptions and notations. In the work presented in this chapter, a stochastic system is defined as an ensemble of linear dynamic systems made of a main deterministic subsystem to which

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303

a random subsystem is coupled, as illustrated in Fig. 1. The random subsystem is a known disturbance weighted by a random variable, s, whose probability density function, p(s), is known.

Fig. 1: The stochastic system is made of a deterministic system to which a random component is added. It is an ensemble of systems, parameterized by the random variable, s, with probability density function, p(s) All the systems have a finite dimension N and the linear relations between all degrees of freedom, are written in matrices A and D, respectively for the deterministic system and the disturbance. Expressed in the frequency domain, this results in the following discrete linear system whose parameters are the random variable, s, and the circular frequency ω = 2π f , [A(ω ) − sD(ω )] x(ω , s) = f

(1)

where f is the frequency, f ∈ C N×1 is a force vector, x(ω , s) ∈ C N×1 is the response vector. The matrix A(ω ) ∈ C N×N is the dynamic stiffness matrix corresponding to the undisturbed or deterministic system. As noted, it can be complex as well as real, and there is no restriction on its properties (i.e. it does not have to be symmetric, hermitian, or anything else). However, one can assume that it is full-rank (invertible) everywhere in the frequency range of interest except, possibly, at a finite set of discrete values of ω corresponding to the eigenvalues of the deterministic system. For an undamped deterministic system, one could have A(ω ) = K − ω 2 M where K and M are the stiffness and mass matrices respectively. For a damped system, one could have A(ω ) = K − iω C − ω 2 M

or

A(ω ) = K(1 + iζ ) − ω 2 M

(2)

where C is a damping matrix or where ζ is a damping coefficient (in the case of proportional damping). The disturbance matrix D(ω ) ∈ C N×N may be entirely general. It can be presented in the form D(ω ) = Dl (ω )Dr (ω )T where both matrices Dl (ω ) and Dr (ω ) ∈ C N×n are full-rank and n is the rank of D(ω ) and could in principle vary with ω . For simplicity of derivation in the next section, it will be assumed that the disturbance matrix D is rank-one (n=1) and its left and right matrices will be denoted as the (lower case) vectors they are, D(ω ) = dl (ω )dr (ω )T with .T denoting a transpose matrix or vector. The statistics in the case of a general, multi-rank, disturbance have been derived in [8]. The probability density function p(s) describing the random variable s may be anything. For some particular probability density func-

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tions, the statistics of the system may be expressed and evaluated analytically. This is for example the case for Gaussian probability function if s is a normal variable, as will be shown in Sect. 4. For any given member of the stochastic system (i.e. for a given value of s) and at any frequency, the whole response vector or a transfer function, g(ω , s), in the direction of any output vector c, may be expressed as x(ω , s) = [A(ω ) − sD(ω )]−1 f and g(ω , s) = cT x(ω , s). Derivation of the statistics of these functions is given in the next section.

3 Expression of Mean, Variance, and Covariance The statistics are derived from an exact rank-one update expression of the response.

3.1 Parameterized Response An exact expression of the parameterized response (as a function of the random variable) may easily be obtained as rank-one update of the deterministic response, x(ω , 0) = A(ω )−1 f, at s = 0. Introducing the update Δ x(ω , s) = x(ω , s) − x(ω , 0) in Eq. 1, one has [A(ω ) − sD(ω )] [x(ω , 0) + Δ x(ω , s)] = f.

(3)

This, in turn, considering that D(ω ) = dl (ω )dr (ω )T is rank-one, gives the exact rank-one update expressions of the response and any transfer function, s1 (ω ) x(ω , s) = x(ω , 0) + B1 (ω )f −1 (4) s1 (ω ) − s s1 ( ω ) g(ω , s) = g(ω , 0) + b1 (ω ) −1 . (5) s1 (ω ) − s These expressions are proved in [7] by two different ways: a direct, transfer function, approach that leads to probably the easiest and shortest proof, and a modal approach that gives a physical insight into the equations, notably that s1 (ω ) is the only finite eigenvalue of the matrix pencil (A(ω ), D(ω )). The update can be seen as an application of the Sherman-Morrison formula [10] commonly used in linear algebra and it has also been used, in another modal form, in the context of stochastic systems [5, 4] (The current work has however been developed independently since the author was unaware of this recent work before the symposium). An advantage of these rank-one update expressions is that the random variable s is isolated in the fraction s1 (ω )/(s1 (ω ) − s) while the following rank-one matrix, B1 , and scalar functions b1 and s1 are all independent of s,

Vibration analysis using an exact theory of stochastic linear systems

B1 (ω ) = s1 (ω )A(ω )−1 dl (ω )dr (ω )T A(ω )−1 −1 and b1 (ω ) = cT B1 (ω )f. s1 (ω ) = dr (ω )T A(ω )−1 dl (ω )

305

(6) (7)

This permits the statistics of the ensemble of structures to be evaluated efficiently.

3.2 Mean Response The exact expression of the mean response, x(ω ) = E x(ω , s) , and of the mean transfer function, g(ω ) = E g(ω , s) , among the ensemble of structures are obtained by integration of Eqs. 4 and 5 by the probability density function, p(s). Assuming ∞ p(s)ds = 1, one has that s is real and considering that, by definition, −∞ x(ω ) = x(ω , 0) + B1 (ω )f (e1 (ω ) − 1)

(8)

g(ω ) = g(ω , 0) + b1 (ω ) (e1 (ω ) − 1) .

(9)

The scalar function e1 (ω ) is the first stochastic coefficient which is actually the only term dependent on the probability density function, e1 (ω ) = s1 (ω )

∞

−∞

1 p(s)ds. s1 (ω ) − s

(10)

It may be evaluated analytically for several particular probability density functions. Such an explicit expression is presented in Sect. 4 when s is a normal variable (i.e. when p(s) is a Gaussian probability density function).

3.3 Variance and Covariance of the Responses Similarly, the variance of the response or the transfer function may be easily evaluated from Eqs. 4 and 5. In this section, expressions for these are derived in a more general form: i.e. considering covariance between responses to two forces and at two frequencies that may be different from each other. Subindices .A and .B are used to distinguish between the two responses, xA (ωA , s) and xB (ωB , s). Besides the need to carry these indices, the derivation is identical to the derivation of the variance of a single response. The covariance, var xA (ωA , s), xB (ωB , s) is defined as var xA (ωA , s), xB (ωB , s) =

H E xA (ωA , s) − xA (ωA ) xB (ωB , s) − xB (ωB ) (11)

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where .H denotes the complex conjugate transpose. The parameterized and mean responses are available from Eq. 4 and Eq. 8, respectively. Direct substitution of these expressions in Eq. 11 gives var xA (ωA , s), xB (ωB , s) =

H (12) B1 (ωA )fA fH B B1 (ωB ) v11 (ωA , ωB ) − e1 (ωA )e1 (ωB )

where a bar, ., denotes a complex conjugate and where the variance of the two fractions is introduced, s1 (ωA ) s1 (ωB ) . (13) v11 (ωA , ωB ) = E (s1 (ωA ) − s) (s1 (ωB ) − s) Two cases have to be considered: • If s1 (ωA ) = s1 (ωB ), then v11 (ωA , ωB ) = q11 (ωA ) where, by integration by parts, q11 (ω ) = −s1 (ω )2

∞

−∞

1 ∂ p(s) ds. s1 (ω ) − s ∂ s

(14)

• Otherwise (if s1 (ωA ) = s¯1 (ωB )), by considering the partial fraction decomposition, v11 (ωA , ωB ) =

s1 (ωB )e1 (ωA ) − s1 (ωA )e1 (ωB ) . s1 (ωB ) − s1 (ωA )

(15)

In the latter case, v11 is a simple function of the scalar function s1 and the first stochastic coefficient, e1 , while in the former case, the integral q11 (ω ) may be evaluated analytically for several particular probability density functions. The term q11 (ω ) is known as the second stochastic coefficient. The resulting explicit expression is also presented in Sect. 4 when s is a normal variable. The covariance of any pair of transfer functions follows trivially from the covariance of the responses [7].

3.4 Multirank Disturbance Similar expressions of mean and covariance exist when the disturbance matrix is multi-rank. The expression of the mean transfer function is presented below and reference [8] provides a proof and discussion of a more general form of the theorem.

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Theorem 1 (Mean response for a multi-rank stochastic linear system). From [8]. Let’s consider a linear operator, A(ω ) ∈ C N×N , invertible everywhere except, possibly, at a finite number of values of ω ; a multi-rank disturbance, D(ω ) = Dl (ω )Dr (ω )T , where Dl (ω ) and Dr (ω ) ∈ C N×n have full rank n ≤ N; and a random variable, s, with probability density function p(s). Let’s gather the right and left eigenvectors of T −1 Dr (ω ) A(ω ) Dl (ω ), respectively rj (ω ) and lj (ω), into matrices Rs (ω ) = r1 (ω ) . . . rn (ω ) and Ls (ω ) = l1 (ω ) . . . ln (ω ) , normalize them so that I = Ls (ω )T Dr (ω )T A(ω )−1 Dl (ω ) Rs (ω ), and denote the corresponding eigenvalues δj (ω ) = sj (ω )−1 for j = 1, . . . , n, Dr (ω )T A(ω )−1 Dl (ω ) rj (ω ) = δj (ω )rj (ω ) Dl (ω )T A(ω )−T Dr (ω ) lj (ω ) = δj (ω )lj (ω ).

(16) (17)

For any input and output vectors, f and c, the mean, g(ω ) of the transfer function, g(ω , s) = cT [A(ω ) − sD(ω )]−1 f, is equal to n

g(ω ) = cT A(ω )−1 f + ∑ bj (ω )(ej (ω ) − 1)

(18)

j=1

where, for j = 1, . . . , n, ej (ω ) =

∞ −∞

p(s)

sj (ω ) ds = sj (ω ) − s

∞ −∞

p(s)

δj (ω )−1 ds δj (ω )−1 − s

bj (ω ) = cT A(ω )−1 Dl (ω )rj (ω ) lj (ω )T Dr (ω )T A(ω )−1 f .

(19) (20)

The structure of Eq. 18 is very similar to the particular case of Eq. 9. The main difference is the consideration of a set of scalar functions, bj (ω ), and first stochastic coefficients, ej (ω ). Similarly, as detailed in [8], the expression of the variance in the general case is in terms of scalar functions vjk (ω ), j, k = 1, . . . , n which themselves are function of the scalar functions sj (ω ) and ej (ω ), or equal to the following second stochastic coefficients, qjj (ω ) = −sj (ω )2

∞

−∞

1 ∂ p(s) ds, sj (ω ) − s ∂ s

j = 1, . . . , n.

(21)

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3.5 Discussion of the Theory Now briefly discussed are some properties of the theory. First, since no really limiting assumption was made on the system and the disturbance, the theory is applicable and valid in a wide range of situations. Second, the compact form of the expressions allows easy interpretation. For example, one can see that, for a given system with rank-one disturbance, the variance and covariance of the transfer functions are proportional to how much v11 differs from |e1 |2 . One may also identify stable points (with zero variance) of a transfer function that are not affected by the value of the random variable. Similarly, one may see that the mean g is equal to the nominal transfer function, cT A−1 f, when the product b1 (e1 − 1) is equal to zero. Third, the entire evaluation of the stochastic properties has a limited numerical complexity: At most, the only solutions to evaluate could be x = A−1 f and xl = A−1 dl . One would then only need to evaluate the inner products of the solution vectors by the vectors c and dr . This cost decreases even more in particular cases. For example, if f equals dl , only one solution is required. The remaining integrals defining the stochastic coefficients may be expressed as special functions for particular cases of probability density functions as shown in the next section. Fourth, among the three sets of constants, bj , ej , and vjk , only the bj ’s are functions of the input and output vectors. One may therefore formally isolate the input and output vectors by working with a matrix like the B1 matrix of Eq. 6. Finally, the formulation for a rank-one disturbance does not necessitate the explicit solution of any eigenproblem as all results may be obtained from evaluation of transfer functions. In the case of a multi-rank disturbance, the only eigenvalue to solve will only be of dimension equal to the rank of the disturbance. All these properties are further discussed in [7, 8].

4 Stochastic Coefficients in the case of a Gaussian Probability Density Function Now consider the analytic expression of the two sets of stochastic coefficients in the case of a normal random variable. It has already been shown that once these coefficients are available, analytic expressions of all the mean and covariance statistics of the ensemble of structures are also available. By definition, the probability density function of a normal random variable, s, with mean value, s, ˆ and standard deviation, σ , is the Gaussian function, 1 − (s−s)ˆ2 2 ˆ σ) = e 2σ . (22) pg (s; s, 2πσ 2 In this case, the stochastic coefficients are denoted

Vibration analysis using an exact theory of stochastic linear systems (g)

ej (ω ; s, ˆ σ ) = sj (ω ) (g)

∞

1

−∞ sj (ω ) − s ∞ 1 2

qjj (ω ; s, ˆ σ ) = −sj (ω )

309

ˆ σ )ds pg (s; s,

−∞ sj (ω ) − s

(23)

∂ pg (s; s, ˆ σ) ds ∂s

(24)

for j = 1, . . . , n. On one hand, the first stochastic coefficients are easily evaluated by √ 2σ ). This gives using the change of variable, t = (s − s)/( ˆ

sj (ω ) − sˆ 1 (g) √ , j = 1, . . . , n (25) ˆ σ ) = s j (ω ) I ej (ω ; s, −1 2πσ 2 2σ ∞

e−t I−1 (b) = dt . −∞ (b − t) 2

(26)

The expression of the integral I−1 (b) depends on the sign of the imaginary part, ℑ(b), of its argument. In the case of a real argument notably, the integral does not converge except if it is evaluated in a principal value sense. From [1, 6], one indeed has that

π w(b) i I−1 (b) = iπ w(b)

I−1 (b) =

if ℑ(b) > 0,

(27)

if ℑ(b) < 0 ,

(28)

√ and I−1 (b) = 2 π F(b) in a principal value sense if b is real. The Faddeeva and Dawson’s functions in these expressions are, respectively, ⎛ ⎞ b b 2i 2 2 2 −b ⎝ t −b2 ⎠ 1+ √ w(b) = e e dt et dt . (29) and F(b) = e π 0

0

On the other hand, the second stochastic coefficients may be evaluated by explicit substitution of the pg (.) function and simple algebra operations. This gives (g)

qjj (ω ; s, ˆ σ) =

(g) 1 2 s , j = 1, . . . , n (30) ( ω ) s ( ω ) − s ˆ e ( ω ; s, ˆ σ ) − s ( ω ) j j j j σ2

each of these being a simple function of the scalar function sj and the corresponding (g)

first stochastic coefficient, ej .

5 Application examples The application of the theory is now illustrated. Specifically, the mean, variance, and covariance for some benchmark systems are evaluated. The matrices of the first benchmark were extracted from the Harwell-Boeing collection [3]. They can be retrieved from the Matrix Market, a free service of the National Institute of Standards

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and Technology [2]. The term benchmark BCSST 11 refers to the system made of the matrices K = BCSST K11 and M = BCSST M11 respectively. Another system is a chain of alternating unit masses and springs, free at one extremity and attached to a fixed point at the other, so that the stiffness matrix is a tridiagonal matrix with diagonal coefficients {1, 2, . . . , 2} and non-zero off-diagonal coefficients equal to −1 while the mass matrix is the identity matrix, M = I. This system is referred to as the Spring-Mass benchmark. Its dimension, N, is the number of masses. In all the examples, a slight proportional damping is introduced so that the parameterized matrix has the form (31) A(ω ) = K(1 + iζ ) − ω 2 M. The input and output vectors as well as the disturbances and their standard deviation are described below. All random variables are normal with zero mean.

5.1 Comparison to a Monte-Carlo Simulation Comparison is given of the exact mean obtained from the theory to the approximate mean obtained from a Monte-Carlo simulation for the BCSST 11 benchmark. Consider the frequency range 0 ≤ ω 2 ≤ 2000 where the system has 26 eigenvalues. The input and force vectors are both equal to a vector whose components are all ones, T c = f = 1 . . . 1 while the disturbance is an added random stiffness at n = 10 equidistant positions on the diagonal of K, Dl = Dr = e147 e294 . . . e1470 , with σ = 106 and where ek denotes the k-th unit vector. A small proportional damping, ζ = 10−6 , is used. The deterministic and (exact) mean responses are presented in the left plot of Fig. 2, together with the response of 20 sample structures randomly selected from the stochastic ensemble (a small Monte-Carlo simulation). Looking at the mean (also shown in the right plot of the same figure), one may see that resonances have been smoothed out, especially at higher frequency, and that some resonances and zeros have entirely disappeared in the averaging process. The smooth mean is in contrast of the average of the 20 sampled responses shown in the second, central, plot of the figure. If the number of samples were increased, the Monte-Carlo average would eventually converge to the exact mean. The total number of samples might however need to be huge in order to give a precise approximation of the mean. This is particularly true if some resonances are lightly damped. In the present case, the relative error on the absolute value of the approximation of the mean at ω 2 = 1000 would still be well in the few percent with 100.000 samples, while the error on the approximation of its imaginary part would be almost of the same order as its exact counterpart. The exact mean, on the other hand, is precisely evaluated at relatively low cost.

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Fig. 2: Illustration of Monte-Carlo simulation on the BCSST 11 benchmark. Left: Twenty sampled random transfer functions are plotted in light gray together with the (deterministic) nominal and (exact) mean transfer functions in thin and thick black lines respectively. Center: The (Monte-carlo) average from the 20 sampled transfer functions. Right: The exact mean transfer function again

5.2 Transition from low to high modal density In order to study the application of the theory to varying modal density, consider the Spring-Mass benchmark. Since its eigenvalues are concentrated in the range ω 2 ∈ [0, 4] (for ζ = 0), one may increase the modal density in this region by simply increasing its dimension, N. Consider the displacement at the free mass from a force applied at the same position, f = c = e1 and use a small value of proportional damping, ζ = 10−7 . The disturbance is a change of the stiffness applied to the first two hundred-ths of degrees of freedom rounded up to next integer number(the displace- ment of the first 5 masses if N = 250). This corresponds to Dl = Dr = e1 e2 . . . en for n = N/50, as illustrated in Fig. 3.

Fig. 3: Illustration of the Spring-Mass benchmark. The interest is in the transfer function, x1 / f1 , at the first mass when a disturbance is added to each of the first N/50 masses The deterministic response and the ensemble statistics are presented in Fig. 4 for an increasing dimension of the system. The larger the dimension of the system, the higher the modal density of the deterministic response is in the range 0 ≤ ω 2 ≤ 4. Despite this apparent complexity, the mean response becomes simpler when the dimension is large enough. At N = 1000, the mean response is so smooth that it is impossible to distinguish the influence of the many resonances. This can be understood intuitively by inspection of a smaller frequency range. In Fig. 5, twenty samples of the random responses selected consistently with the Gaussian probability density

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pg (s; 0, 0.3) are superposed on the deterministic and average response. For N =100, the sampled responses are relatively tightly gathered around the deterministic nominal response. The resulting (exact) ensemble average exhibits smoothed out resonances while average zeros appear to be at the same location as the deterministic nominal zeros. The increased modal density at N = 250 is such that the samples are distributed more regularly in the whole frequency range. This results in averaged resonances and zeros that are much smoother versions of their deterministic equivalents. As the modal density still increases, the sensitivity of the responses is such that the ranges of disturbed resonances and zeros overlap each others, which results in the smooth mean shown in Fig. 4.

Fig. 4: Statistics of the Spring-Mass benchmark. Each column corresponds to a varying dimension, N = 100, 250, 1000. The rows correspond successively to the g(ω )), absolute value of the deterministic and mean responses, abs(g(ω , 0)) and abs( and to the variance var[g(ω , s)] It is worth noting that although the deterministic nominal response and each individual sampled random response are mostly real, the mean response has an imaginary part of the same order as its real part. A similar smoothing process occurs in Fig. 4 for the variance of the response, although the variance does not become entirely smooth at higher dimension. A remarkable aspect of the theory, that appears in this example, is that even though it explicitly uses the possibly highly irregular deterministic function in its expressions, it is able to predict smooth means or variances without any apparent numerical problem. This is a sign of robustness of the theory and its resulting expressions.

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Fig. 5: Zoomed transfer functions of Fig. 4. Twenty sampled random responses are plotted in light gray while the (deterministic) nominal and (exact) mean responses are plotted with thin and thick black lines respectively

5.3 Variance and covariance of responses at different frequencies The final illustration and discussion concern the evaluation of the variance and covariance. The variance of the benchmark problem of Sect. 5.1 is presented in the right plot of Fig. 6 together with the 20 samples. The variance has local maxima in

Fig. 6: Variance and covariance of the transfer function of the BCSST 11 benchmark problem of Sect. 5.1 at different frequencies. Left: Covariance var[g(ωA , s), g(ωB , s)]. Right: Variance var[g(ω , s)] = var[g(ω , s), g(ω , s)] and 20 (Monte-Carlo) sampled responses pockets where the resonances are likely to be while lower variance corresponds to regions where the response is unlikely to vary much from one sample to another. As indicated above, a zero variance would indicate stable or interpolation points where all the samples have exactly the same value. The covariance of the same transfer function at different frequencies is presented in the left map of Fig. 6. The diagonal coefficients mostly dominate so that the norm of the covariance matrix without its diagonal is just above 2% of its norm with diagonal. While the diagonal of the covariance matrix is the variance of the transfer function, the off-diagonal coefficients indicate how the variations of the transfer function from its mean correlate at any two different frequencies.

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6 Conclusion A very general exact theory of stochastic systems has been presented that allows the study of the vibrations of ensembles of structures. The stochastic systems comprise a deterministic structure on which a disturbance is added. This disturbance is described by a deterministic matrix and a random variable whose probability density function is known. Presented and briefly discussed are the general expressions of mean, variance, and covariance of all the responses and transfer functions of the ensemble of structures. It has been shown that all the expressions are in terms of deterministic terms and some stochastic coefficients. Analytic expressions for these stochastic coefficients, that are functions of special functions, are given in the case of a normal random variable. All the expressions are analytic, without any approximation, and they may be evaluated at relatively low cost. The explicitly steps of the derivation are given for the expressions for the case of a rank-one disturbance following [7] and the multi-rank expressions that were derived in [8] presented. The evaluation of the statistics for a couple of benchmarks were shown and the apparent robustness, flexibility, and efficiency of the theory is observed. It is significant that the theory is applicable from low to high modal density or frequency, and that the mean and variance transition without problem from a region of local smoothing of poles and zeros at low modal density to a region of global smoothness at high modal density. The expression for the covariance between the values of the same transfer function at different frequencies was also presented and evaluated on one of the benchmarks. In the context of stochastic linear systems presented here, the exact theory appears to be potentially several orders of magnitude more efficient than a Monte Carlo approach. Acknowledgements The author thanks Robin Langley for having inspired him to look at this problem, for his thoughtful feedback, and for the kind invitation to participate at the symposium. He also appreciates the encouragement and support received from many of his colleagues, including Will Graham, Bill Fitzgerald, Ann Dowling, Paul Barbone, and Jim Woodhouse. He thanks Neil Ferguson for having proofread this paper. The author finally gratefully acknowledges the financial support to travel to the symposium provided by The Royal Society under an International Travel Grants Scheme, as well as the financial support to participate to the conference from IUTAM via the conference organisers.

References 1. Abramowitz M, Stegun I (1972) Handbook of Mathematical Functions. National Bureau of Standards, New York 2. Boisvert RF, Pozo R, Remington K, Barrett R, Dongarra JJ (1997) The Matrix Market: A web resource for test matrix collections. In: Boisvert RF(ed) Quality of Numerical Software, Assessment and Enhancement, Chapman & Hall, London, pp 125–137 3. Duff IS, Grimes RG, Lewis JG (1992) Users’ guide for the Harwell-Boeing sparse matrix collection (Release I). Tech. Rep. RAL 92-086, Chilton, Oxon, England

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4. Falsone G, Ferro G (2007) An exact solution for the static and dynamic analysis of FE discretized uncertain structures. Computer Methods in Applied Mechanics and Engineering 196(21–24):2390–2400, 10.1016/j.cma.2006.12.003 5. Falsone G, Impollonia N (2002) A new approach for the stochastic analysis of finite element modelled structures with uncertain parameters. Computer Methods in Applied Mechanics and Engineering 191(44):5067–5085, 10.1016/S0045-7825(02)00437-1 6. Gautschi W (1970) Efficient computation of the complex error function. SIAM Journal on Numerical Analysis pp 187–198, 10.1137/0707012 7. Lecomte C (2010a) An analytical theory of rank-one stochastic dynamic systems. In preparation 8. Lecomte C (2010b) A theory of multi-rank stochastic dynamic systems. In preparation 9. Schu¨eller GI, Pradlwarter HJ (2010) Uncertain linear systems in dynamics: Stochastic approaches. In: IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, A.K. Belyaev and R.S. Langley, eds. Springer-Verlag. 10. Sherman J, Morrison WJ (1950) Adjustment of an inverse matrix corresponding to a change in one element of a given matrix. The Annals of Mathematical Statistics 21(1):124–127, 10.1214/aoms/1177729893

Structural Uncertainty Identification using Vibration Mode Shape Information J. F. Dunne and S. Riefelyna

Abstract A maximum likelihood estimation (MLE) approach is applied via the perturbation method, to vibrating beam and plate structures with uncertainty. In both cases the focus is on estimating the statistical properties of the position of an attached point-mass. For the beam structures, frequency information with a closed form Jacobian is used in two specific cases and compared with a numerically determined Jacobian and a Finite Element implementation. Application of MLE to a simply-supported-plate with an attached point-mass of unknown position uses mode shape information in the form of the coefficients obtained from the Rayleigh-Ritz method. The paper shows that appropriate combinations of this mode shape information allows variance reduction of parameter estimates when using relatively small numbers of random samples.

1 Introduction Uncertainty arises naturally in many different types of engineering structure as a result of variability in geometry, material properties, manufacture, and assembly. Obvious examples include automotive vehicle-bodies, aircraft, and ship structures. Response prediction of uncertain structures can be approached in a probabilistic manner if complete knowledge of the probabilistic distributions of key system parameters is known. When this is not the case, non-probabilistic approaches such as possibilistic methods, interval analysis, convex methods, fuzzy modelling, and evidence theory should be used [1]. These methods can be practically useful in the absence of alternatives but they can also lead to grossly conservative predictions. J. F. Dunne School of Engineering and Design, The University of Sussex Falmer, Brighton, BN1 9QT, UK e-mail: [email protected] S. Riefelyna e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 23, © Springer Science+Business Media B.V. 2011

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Attempts to address this problem have been made to solve the inverse problem by obtaining uncertain parameter distributions from a limited number of response measurements. This is known as uncertainty identification and is of profound practical importance since in most cases it is either not practical, or even impossible, to directly measure the unknown properties from an ensemble of assembled structures. At its most basic level the problem implicitly requires solution of three subproblems: First, from a finite set of measurements, missing information in the model needs to be filled in. The second problem is to invert the model making the uncertain parameter values explicit. And the third problem is then to develop an uncertainty propagation method to allow the statistical properties of the response variables to be related back to the uncertain system parameters. More generally, in addition to physical uncertainties, wider issues of uncertainty associated with measurement and estimation needs to be included. In most cases involving physical parameter uncertainty, the first problem is necessarily bi-passed assuming the model structure is known with certainty and only explicit parameter properties are needed. Furthermore because the second step is often extremely difficult or even impossible, alternative approaches have been explored which actually avoid explicit model inversion. One of the most general uncertainty identification methods uses maximum likelihood estimation (MLE) to obtain the statistical properties of unknown parameters for relatively simple structures. Fonseca and Friswell et al (2005) [2] developed an MLE approach which allowed, within the same framework, approximate approaches such as the perturbation method and a Monte Carlo simulation method to be used to construct the most likely estimator for uncertain parameter statistics. This was successfully tested on a number of very simple structures using natural frequency measurements. By contrast, Shiryayev and Page et al (2007) [3] constructed a pseudo-inverse model using a neural network to obtain mean values of uncertain parameters of a bolted-joint. And elsewhere, the estimation of model parameter distributions has been approached using evidence theory by McGill and Ayyub (2008) [4]. The wider questions of measurement and estimation uncertainty in model-based predictions has also been addressed by Fraccone and Ruzzene et al (2008) [5] in connection with uncertainty of bladed-discs in gas-turbine engines. One particular difficulty with model-based uncertainty estimation is, for computational reasons, the necessary restriction to small amounts of response data. Any technique that can exploit small amounts of data will be useful in practice. In this paper, attention focuses on the use of MLE and the perturbation method with both natural frequency and mode shape information. The possibility of using mode shape information opens-up the scope to reduce scatter in estimated parameter statistics, in particular, in standard deviation estimates. The objective of the paper is to examine the MLE approach on beam and plate structures using as much closed form information as possible and then to apply the MLE to estimate the position statistics of a point mass on plate structure. First, the MLE approach is briefly outlined.

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2 Maximum Likelihood Estimation of Uncertain Structural Parameters Estimation of the randomly uncertain parameters from the response of a structure can be expressed as an inverse problem associated with the uncertainty propagation problem [2]: y = f (x) (1) where the statistical properties of a set of uncertain parameters x = [x1 x2 . . . xn ]T are to be obtained from y = [y1 y2 . . . ym ]T . This is particularly relevant when construction of the inverse operation, namely: x = f −1 (y)

(2)

is either ill-conditioned, irreducible, or impossible. The Maximum likelihood approach proceeds by assuming the distribution of random parameters is known such that X ∼ D(θx ) in which the vector of parameters θx is to be determined. In the MLE approach, the log-likelihood function l(θx ) is sought in the form: N

l(θx ) = ∑ log f (y(i) |θx )

(3)

i=1

where f (y(i) |θx ) is the conditional density associated with a set of N response measurements given the parameter vector. By finding the response vector that maximises l(θx ), this provides a basis for generating the most likely parameter estimates.

2.1 Uncertainty Estimation via the Perturbation Method Construction of the set of conditional densities f (y(i) |θx ) needed in equation (3) can be approximated by various means but the simplest approach is to linearise equation (1) and to assume that the uncertain variables belong to a normal joint-density function X∼Nn (μx , Σ x ). Linearisation about an assumed point x0 (near to the mean) thus allows the response vector y to be correspondingly normal Y ∼Nm (μy , Σ y ). The perturbation solution becomes a means of constructing the log-likelihood function using uncertainty propagation methods. With these assumptions, the approximate log-likelihood function [2] is obtained as: N 1 l(μx , Σx ) = − (Nm log 2π + N log |Σ y | + ∑ (y(i) −μy )T Σy−1 (y(i) −μy ) . . . 2 i=1

(4)

where approximately: y = f 0 + J 0 (x − x0 ) and the response mean and covariance matrix are respectively approximated as:

(5)

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μy = f 0 + J 0 (μx − x0 ) and

(6)

T

Σ y = J 0 Σx J 0 (7) and where J 0 = [∂ f ∂ x1 ∂ f ∂ x2 . . . ∂ f ∂ xn ] is the Jacobian evaluated at the expansion point x0 . The choice of independent response variable y depends on a number of factors including whether a mathematical model can easily be constructed to provide a computationally efficient means of generating response samples as a function of the uncertain parameters. Frequency information has been used on beam structures in [2] and therefore, before development of a computationally efficient means of generating natural frequencies and mode shapes for use of perturbation-based ML estimates of uncertainty in plate problems, some frequency-based results for beam structures are discussed. This switch in problem complexity is also justified because equation (7) always generates a singular covariance matrix Σy if an attempt is made to use a value m > n, namely an attempt to use a response vector y with more components than those in the uncertain parameter vector x.

3 ML estimates of uncertain point-mass position statistics using natural frequency information on a cantilever beam structure In most practical applications of the perturbation method only numerical evaluation of the Jacobian J 0 in equation (5) is possible. This often applies (owing to resulting complexity) even when a closed form frequency equation can be constructed of the form F(λ , x) = 0, where λ 2 is a nondimensional frequency parameter. To create an uncertain structure with a closed form J 0 , and with simpler form of likelihood function, two example-cases are useful, both involving a simple beam with a pointmass M attached at an uncertain position x. Case I is a mass-less cantilever beam of length L and elastic stiffness EI; Case II is a cantilever beam of mass per-unitlength m, of elastic stiffness EI, and also point-mass M attached, but where the exact frequency equation is constructed using the Receptance Method and where an exact Jacobian is constructed in closed form using symbolic computation (Maple). This offers a good comparison with the use of a Finite Element solution.

Fig. 1: Flexible Beam with stiffness EI and a point mass attached

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Case I: the natural frequency for the mass-less version of the beam shown in Figure 1 is ω 2 = 3EI/Mx3 . Similarly the frequency equation becomes F(λ , x) = λ 2 −

L3 = 0 and the Jacobian is x3

3 3 5 dλ = − L 2 x− 2 . dx 2

A random sample of 100 normally-distributed random positions of x has been generated with a target mean μx = 0.75 m, and σx = 0.05 m, and the log-likelihood functions for the mean and standard deviation are shown in Figure 2 where both numerical and closed form Jacobians are used.

Fig. 2: Log-Likelihood functions for point-mass position mean and standard deviation for a mass-less cantilever with elastic stiffness EI: – numerically evaluated Jacobian; • closed form Jacobian

Case II: Natural frequencies for a version of the beam shown in Figure 1 with distributed mass per-unit-length m, can be obtained using the Receptance Method [6] from the frequency equation: F(λ , x) = F8 (λ L) F8 (λ x)F7 [λ (L − x)] + F9 (λ L) F10 (λ x)F7 [λ (L − x)] + ... F8 (λ x) F9 (λ L) F9 [λ (L − x)] − F7 (λ L) F10 (λ x) F9 [λ (L − L)] − 4mFλ4M(λ L) = 0 (8) 4 where ω 2 = λ mEI , and where the functions in equation (8) are defined as: ⎫ F4 (λ x) = cos(λ x) cosh(λ x) + 1 ⎪ ⎪ ⎪ F7 (λ x) = sin(λ x) + sinh(λ x) ⎪ ⎬ F8 (λ x) = sin(λ x) − sinh(λ x) (9) ⎪ F9 (λ x) = cos(λ x) + cosh(λ x) ⎪ ⎪ ⎪ ⎭ F10 (λ x) = cos(λ x) − cosh(λ x) Using symbolic computation (Maple), the Jacobian associated with equation (1) can be constructed in closed functional form and then used subsequently to construct the log-likelihood function to estimate μx and σx . The natural frequencies can

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be generated by solving for the roots of equation (8). As an intermediate comparison of the benefits of using the Receptance Method, an alternative solution for this problem was obtained using a commercial Finite Element code assuming the following parameter values: E = 210 GPa, I = 8.33 · 10−9 m4 , L = 1 m, M = 0.1 kg, m = 7.8 kg/m. Figure 3 shows the comparison of the first four frequencies as a function of x.

Fig. 3: Natural frequency as a function of point-mass position for first four modes of a cantilever beam with finite mass distribution and elastic stiffness EI: — FEM, and - Receptance method

The FE solution involved use of 100 beam elements, where it was only practically possible to generate response data with the point-mass positions at the nearest nodal point. The Receptance Method by contrast allows arbitrary (continuously) random positioning of the point-mass. Again with a target μx = 0.75 m and σx = 0.05 m, the FEM estimates are: μˆ x = 0.77 m (error: 2.6%) and σˆ x = 0.0503 m (error: 0.6%) whereas the corresponding Receptance Method estimates are: μˆ x = 0.7475 m (error: 0.34%) and σˆ x = 0.0508 m (error: 1.6%). The log-likelihood function for the Receptance Method is shown in Figure 4.

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Fig. 4: Log-Likelihood functions via Receptance Method for the point-mass position mean and standard deviation for a cantilever beam with finite mass distribution and elastic stiffness EI

4 ML Estimation of uncertain point mass position on a plate structure using mode shape information Plate problems provide a good vehicle for testing the use of mode shape information because in general, frequencies and mode shapes are relatively easy to generate in functional form and at the same time are not limited by the m > n constraint discussed at the end of Section 2. The equation of motion for small displacements u(x1 , x2 ,t) of a thin homogeneous linear plate is given as: 4 ∂ u ∂ 4u ∂ 4u ∂ 2u =0 (10) γ 2 +D + 2 + ∂t ∂ x14 ∂ x12 ∂ x12 ∂ x24 with D=

Et 3 12(1 − ν 2 )

(11)

where ν is Poisson’s ratio, E is Young’s modulus, t is the thickness, and γ is the mass per-unit-area. Well-known solutions are available for the linear plate of regular geometry and various boundary conditions [7]. With the addition of a point mass M at an arbitrary position (k, h) on the plate, only an approximate solution can be expected such as the FEM, the Rayleigh Ritz solution [8]-[9], or Discrete Singular Convolution [10]. Here an energy form Rayleigh-Ritz solution [9] is adopted for a rectangular plate for use with the MLE approach of Section 2. This is obtained by assuming a solution in the form: u(x1 , x2 ,t) = w(x1 , x2 )e jω t

(12)

By adopting expedient basis functions X1m (x1 ) and X2n (x2 ), the mode shape can be expanded in series-form as follows:

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w(x1 , x2 ) = ∑ ∑ Amn X1m (x1 )X2n (x2 )

(13)

m n

The Rayleigh-Ritz solution [9] is obtained by solving the eigenvalue problem: D[K]{A} = ω 2 [S]{A}

(14)

where the reduced stiffness matrix can be obtained from: )(X X + X X )+ (X1m X2n + X1m X2n 1p 2q 1p 2q Kmnpq = X X − X X X X ) dx1 dx2 2(1 − ν )(X1m X1p 2n 2q 1m 1p 2n 2q (15)

... and the reduced mass matrix from: Smnpq = γ = where X1m

(X1m X1p X2n X2q ) dx1 dx2 + MX1m (k)X1p (k)X2n (h)X2q (h)

∂ X1m ∂ x1 ,

= X2n

∂ X2n ∂ x2 ,

= X1m

∂ 2 X1m ∂ x12

= and X2n

∂ 2 X2n . ∂ x22

(16)

The square matrices

[K] and [S] in equation (14) are obtained by converting the 4-dimensional matrices obtained from equations (15) and (16) by appropriate re-assignment of the indices. Here only a plate, simply-supported on all sides (or the so called SSSS problem) with an attached mass M, will be considered, for which expedient basis functions [9] are: (17) X1m (x1 ) = sin(mπ x1 /a) and

X1n (x2 ) = sin(nπ x2 /b)

(18)

Note for M = 0, the free-vibration SSSS rectangular plate characteristics [7] are:

π x1 π x2 2 w j (x1 , x2 ) = ) sin( j2 ) sin( j1 a b γ ab

and

ωj =

D γ

j1 π a

2

+

j2 π b

(19)

2 (20)

where j1 and j2 are positive integers. To verify the convergence of the Rayleigh-Ritz solution, application is now shown for a 1 m square SSSS plate with, and without the point mass attached. In this case, Poisson’s ratio ν = 0.3, Young’s modulus E = 210GN/m2 , the plate-thickness t = 2 mm, and the mass per-unit-area γ = 16kg/m2 . Figure 5 shows the solution for the plate involving MR = 2 (or rather involving four Rayleigh-Ritz coefficients). The eigenvectors and eigenvalues are in complete agreement with equations (19) and (20) respectively. Interestingly however, great care has to be taken with roundoff error in extracting the eigenvectors associated with modes 2 and 3. Although the

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natural frequencies (eigenvalues) are insensitive to round-off and small truncation errors (particularly associated with the integrations in equations (15) and (16)), the eigenvectors are extremely sensitive, even when the reduced stiffness matrix has terms of the order 10−11 which should otherwise be zero. A norm-based convergence criterion does however show that when care is exercised, the level of agreement in the mode shapes is also excellent. Now with a point mass of M = 0.3 Kg attached to the plate at the position; x1 = 0.3 m and x2 = 0.25 m, the convergence of the Rayleigh-Ritz solution with an increasing number of coefficients is also shown in Figures 5 and 6 corresponding to the use of MR = 6, 7, and 8, in equation (13) (or rather the use 36, 49, and 64 coefficients). It is evident from the frequencies shown on Figures 5 and 6, that for this particular location of the mass, full convergence has (for practical purposes) occurred by MR = 6. But Low et al [9] found that in some cases MR = 10 was needed for convergence (or rather 100 coefficients were needed). It is evident also for this particular position of the mass, that the mode-2 frequency (but not the mode2 shape) is virtually identical to the case when no point-mass is present.

Fig. 5: Simply-supported plate modes 1–4 via Rayleigh-Ritz method MR = 2: PointMass = 0.0 Kg; and MR = 6 for Point-Mass = 0.3 kg at x1 = 0.3, x2 = 0.25 Now with an ability to generate mode shape information via the Rayleigh-Ritz method, the response information can be used in the MLE-perturbation approach of section 2. Here in fact, subsets of the Rayleigh-Ritz eigenvector coefficients A(i) , obtained from equation (14), will be used as the appropriate response variables y = [y1 y2 . . . ym ]T in the MLE approach, where the Jacobian is obtained numerically. As an example of this approach Figure 7 shows the Log-likelihood function obtained for the standard deviation estimate of the horizontal position of the point mass. In this case just the first two components in the mode-1 eigenvector are used with MR = 6, and with a random response sample size=50. The target σx1 = 0.05m and the estimate obtained was σˆ x1 = 0.047. This result verifies that mode shape information can indeed be used. But this was obtained at quite considerable relative computational cost. And yet, most of the mode shape information was not used (and

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Fig. 6: Simply-supported plate modes 1–4 via Rayleigh-Ritz method for MR = 7 and MR = 8. Point-Mass = 0.3 kg at x1 = 0.3, x2 = 0.25

indeed could not be used with the perturbation method for the reason explained at the end of Section 2). An attempt is now made to use more of the eigenvector information. By noting that (in principle) any combination of pairs of eigenvector components (for each mode) could be used for example when there are two uncertain random variables in the problem, namely n = 2 in x = [x1 x2 . . . xn ]T . The question is precisely how should this information be used. The strategy adopted is as follows: For the case n = 2, each combination of eigenvector components is used to generate statistical estimates for the uncertain parameters. If these response variables are genuinely independent then the multiplicity of estimates of the same variables can be averaged to achieve substantial variance reduction. Thus instead of using 50 or 100 random samples, if these estimates are indeed independent, the number of samples could be dramatically reduced. This strategy is now tested in estimating the standard deviation of the horizontal position of a point-mass, again with a target σx1 = 0.05m. (Both the horizontal and vertical positions of the point-mass are however varied randomly but only the horizontal position statistics are estimated). A random sample size=10 was adopted to test this strategy which enabled all of the Rayleigh-Ritz eigenvector data to be generated in a fraction of the time needed for 50 samples. Using for convenience MR = 5, the raw estimates σˆ x1 are shown in Figure 8 against their Rayleigh-Ritz Coefficient Group for mode-1, mode-2, and mode-3. Coefficient Group 1 corresponds to use of eigenvector components 1 & 2; Group 2 corresponds to use of eigenvector components 3 & 4 and so on. Figure 8 shows however that some estimates (outliers) are in error by an order of magnitude or more (indeed they are hard up against their upper bounds). The majority of estimates are however scattered around the target value. By setting a threshold to deem any estimate more than three times the target as an outlier, these are then excluded. The resulting raw estimates and their corresponding (outlier-excluded) averages are shown in Table 1.

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Fig. 7: Log-Likelihood function obtained using mode 1 shape information for the estimate of the standard deviation of one dimension of a point mass position attached to a simply a supported plate. Mode shape obtained via Rayleigh-Ritz method, likelihood function via Perturbation method

5 Discussion of Results The single estimate σˆ x1 using the first two Rayleigh-Ritz coefficients, whose loglikelihood function is shown in Figure 7, does indeed confirm that mode shape information can be used for uncertainty estimation. But the estimates obtained using combinations of Rayleigh-Ritz coefficients as shown in Figure 8 suggests that not all of this information is being seen as independent. The reason is not clear why some combinations appear to be independent, and others not. Clearly in these cases, maximum likelihood estimates are not being found. But the averages obtained at the bottom of Table 1 (by excluding outlier estimates) suggests that some estimates are indeed statistically independent. A check on the magnitude of determinant of each Jacobian J 0 associated with the outliers estimates often indicates very near-singular behaviour but there are instances where the outlier Jacobian is actually not singular suggesting some other mechanism at work justifying further investigation.

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Fig. 8: Individual estimates (including outliers) of the standard deviation of an uncertain horizontal point-mass position on a simply supported square plate using a 25-term set of Rayleigh-Ritz coefficients in the MLE-perturbation approach with a random sample size=10, and a target σx1 = 0.05m

Table 1: Individual non-outlier estimates of the standard deviation of an uncertain horizontal point-mass position on a simply supported square plate using a 25-term set of Rayleigh-Ritz coefficients in the MLE-perturbation approach with a random sample size=10, and a target σx1 = 0.05m

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6 Conclusions Application of an MLE approach to beam problems with a randomly positioned point-mass attached has been shown to be accurate using measured natural frequency information despite being obtained using relatively non-robust log-likelihood functions. The use of mode shape information in the form of Rayleigh-Ritz coefficients applied to a plate problem shows that the corresponding log-likelihood functions are also not particularly robust but accurate estimates of parameter statistics can still be obtained. By appropriate combinations of the Rayleigh-Ritz coefficients the paper shows that it is possible to achieve variance reduction in the estimates using small random response sample sizes, particularly in estimating the standard deviation of an uncertain point mass position.

References 1. Mace B, Worden K, Manson G (2005) Preface — Uncertainty in Dynamics. Journal of Sound and Vibration 288:423–429. 2. Fonseca JR, Friswell MI, Mottershead JE, Lees AW (2005) Uncertainty identification by the maximum likelihood method. Journal of Sound and Vibration 288:587–599. 3. Shiryayev OV, Page SM, Pettit CL, Slater JC (2007) Parameter estimation and investigation of a bolted joint model. Journal of Sound and Vibration 307:680–697. 4. McGill WL, Ayyub BM (2008) Estimating parameter distributions in structural reliability assessment using Transferable Belief Model. Computers and Structures 86: 1052–1060. 5. Fraccone GC, Ruzzene M, Volovoi V, Cento P, Vining C (2008) Assessment of Uncertainty in response estimation for turbine engine bladed disks. Journal of Sound and Vibration 317:625– 645. 6. Bishop RED, Johnson DC (1960) The Mechanics of Vibration. Cambridge University Press. 7. Newland DE (1994) Mechanical Vibrations Analysis and Computation. Longman Scientific & Technical. 8. Tongue BH (2002) Principles of Vibration. Oxford University Press. 9. Low KH, Chai GB, and Tan GS (1997) A comparative study of vibrating loaded plates between the Rayleigh-Ritz and Experimental methods. Journal of Sound and Vibration 1992: 285–297. 10. Sec¸gin A, Sarig¨ul AS (2008) Free vibration analysis of symmetrically laminated thin composite plates by using discrete singular convolution (DSC) approach: Algorithm and verification. Journal of Sound and Vibration 315: 197–211.

Extremely strong convergence of eigenvalue-density of linear stochastic dynamical systems S. Adhikari and L. A. Pastur

Abstract Eigenvalue problems play a crucial role in the stability and dynamics of engineering systems modeled using the linear mechanical theory. When uncertainties, either in the parameters or in the modelling, are considered, the eigenvalue problem becomes a random eigenvalue problem. Over the past half a century, random eigenvalue problems have received extensive attentions from the physicists, applied mathematicians and engineers. Within the context of civil, mechanical and aerospace engineering, significant work has been done on perturbation method based approaches in conjunction with the stochastic finite element method. The perturbation based methods work very well in the low frequency region which is often sufficient for many engineering applications. In the high frequency region however, which is necessary for some practical applications, these methods often fail to capture crucial physics, such as the veering and modal overlap. In this region one needs to consider the complete spectrum of the eigenvalues as opposed to the individual eigenvalues often considered in the low frequency applications. In this paper we consider the density of the eigenvalues of a discrete or discretised continuous system with uncertainty. It has been rigorously proved that the density of eigenvalues of random dynamical systems reaches a non-random limit for large systems. This fact has been demonstrated by numerical examples. The implications of this result for the response calculation of large stochastic structural dynamical systems have been highlighted.

S. Adhikari School of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, UK, e-mail: [email protected] L. A. Pastur Department of Theoretical Physics, B.I.Verkin Institute for Low Temperature Physics & Engineering, 47 Lenin Ave., Kharkov, Ukraine, e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 24, © Springer Science+Business Media B.V. 2011

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1 Introduction The predictions from high-resolution numerical models may sometimes exhibit significant differences with the results from physical experiments due to uncertainty. Such uncertainties include, but are not limited to (a) parameter uncertainty (e.g, uncertainty in geometric parameters, friction coefficient, strength of the materials involved); (b) model uncertainty (arising from the lack of scientific knowledge about the model which is a priori unknown); (c) experimental error (uncertain and unknown errors percolate into the model when they are calibrated against experimental results). When substantial statistical information exists, the theory of probability and stochastic processes offer a rich mathematical framework to represent such uncertainties. In a probabilistic setting, the data (parameter) uncertainty associated with the system parameters, such as the geometric properties and constitutive relations (i.e. Young’s modulus, mass density, Poisson’s ratio, damping coefficients), can be modeled as random variables or stochastic processes using the so-called parametric approach. These uncertainties can be quantified and propagated, for example, using the stochastic finite element method [8, 12]. Recently, the uncertainty due to modelling error has received attention as this is crucial for model validation [11]. The model uncertainty problem poses serious challenges as the parameters contributing to the modelling errors are not available a priori and therefore precludes the application of a parametric approach to address such issues. Model uncertainties do not explicitly depend on the system parameters. For example, there can be unquantified errors associated with the equation of motion (linear or non-linear), in the damping model (viscous or non-viscous), in the model of structural joints. The model uncertainty may be tackled by the so-called non-parametric method such as the statistical energy analysis [13, 14] random matrix approach [20, 21, 22, 1, 2, 3, 5]. The equation of motion of a damped n-degree-of-freedom linear dynamic system can be expressed as ¨ + Cq(t) ˙ + Kq(t) = f(t) Mq(t) (1) where f(t) ∈ Rn is the forcing vector, q(t) ∈ Rn is the response vector and M ∈ Rn×n , C ∈ Rn×n and K ∈ Rn×n are the mass, damping and stiffness matrices respectively. In order to completely quantify the uncertainties associated with system 1 we need to obtain the probability density functions of the random matrices M, C and K. Using the parametric approach, such as the stochastic finite element method, one usually obtains a problem specific covariance structure for the elements of system matrices. The nonparametric approach [20, 21, 22, 2, 1, 3, 5] on the other hand results in a central Wishart distribution for the system matrices. In a recent paper [4] it was shown that a single Wishart matrix with properly selected parameters can be used for systems with both parametric uncertainty and nonparametric uncertainty. The calculation of the statistics of dynamic response can be expressed in terms of the eigenvalues and eigenvectors of a random matrix. This paper is focused on the density of eigenvalues of such random systems. The outline of the paper is as follows. In 2 dynamic response of linear stochastic systems is discussed. A brief overview of random matrix models in probabilistic

Extremely strong convergence of eigenvalue-density of linear stochastic dynamical systems

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structural dynamics is given in 3. The density of the eigenvalues are discussed 4. In 5 the accuracy of the proposed results regarding the density of the eigenvalues are numerically verified. Based on the study taken in the paper, a set of conclusions are drawn in 6.

2 Uncertainty quantification of dynamic response Assuming all the initial conditions are zero and taking the Laplace transform of the equation of motion 1 we have 2 ¯ = ¯f(s) s M + sC + K q(s) (2) ¯ denotes the Laplace transform of the respective quantities. The aim here where (•) ¯ is to obtain the statistical properties of q(s) ∈ Cn when the system matrices are random matrices. The undamped eigenvalue problem is given by Kφ j = ω 2j Mφ j ,

j = 1, 2, . . . , n

(3)

where ω 2j and φ j are respectively the eigenvalues and mass-normalized eigenvectors of the system. We define the matrices

Ω = diag[ω1 , ω2 , . . . , ωn ] ∈ Rn×n so that

Φ T Ke Φ = Ω 2

and and

Φ = [φ 1 , φ 2 , . . . , φ n ] ∈ Rn×n Φ T MΦ = In

(4) (5)

where In is an n-dimensional identity matrix. Using these, Eq. 2 can be transformed into the modal coordinates as 2 (6) s In + sC + Ω 2 q¯ = ¯f where and (•) denotes the quantities in the modal coordinates: C = Φ T CΦ ,

q¯ = Φ q¯

and

¯f = Φ T ¯f

(7)

For simplicity let us assume that the system is proportionally damped with deterministic modal damping factors ζ1 , ζ2 , . . . , ζn . Therefore, when we consider random systems, the matrix of eigenvalues Ω 2 in equation 6 will be a random matrix of dimension n. Suppose this random matrix is denoted by Ξ ∈ Rn×n :

Ω2 ∼ Ξ

(8)

Since Ξ is a symmetric and positive definite matrix, it can be diagonalized by a orthogonal matrix Ψ r such that

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Ψ Tr ΞΨ r = Ω 2r

(9)

Here the subscript r denotes the random nature of the eigenvalues and eigenvectors of the random matrix Ξ . Recalling that Ψ Tr Ψ r = In , from equation 6 we obtain

where

−1 ¯f q¯ = s2 In + sC + Ω 2 2 −1 T Ψ r ¯f = Ψ r s In + 2sζ Ω r + Ω 2r

(10)

ζ = diag[ζ1 , ζ2 , . . . , ζn ] ∈ Rn×n

(12)

(11)

The response in the original coordinate can be obtained as −1 ¯ = Φ q¯ (s) = ΦΨ r s2 In + 2sζ Ω r + Ω 2r q(s) (ΦΨ r )T ¯f(s) n xTrj ¯f(s) =∑ 2 x . 2 rj j=1 s + 2sζ j ωr j + ωr j

(13)

Here

Ω r = diag[ωr1 , ωr2 , . . . , ωrn ] and Xr = ΦΨ r = [xr1 , xr2 , . . . , xrn ]

(14) (15)

are respectively the matrices containing random eigenvalues and eigenvectors of the system. The Frequency Response Function (FRF) of the system can be obtained by substituting s = iω in Eq. 13. In the next section we discuss the derivation of the random matrix Ξ .

3 Wishart random matrix model We start with the fact that the baseline model of the system under consideration is known. Since proportional damping model is assumed, the baseline model consist of the mass and stiffness matrices given by M0 ∈ Rn×n and K0 ∈ Rn×n . These matrices are in general large banded matrices and can be obtained using the conventional finite element method [25, 7]. In addition to this, it is assumed that the dispersion parameters associated with these matrices are known. The dispersion parameter [20, 21] is a measure of uncertainty in the system and it is similar to normalized variance of a matrix. For example, the dispersion parameter associated with the mass matrix is defined as E M − M0 2F δM = (16) M0 2F where •F denotes the Frobenius norm of a matrix. The dispersion parameter associated with the stiffness matrix can be defined in a similar way. The dispersion pa-

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rameters δM and δK can be obtained using stochastic finite element method [1, 2, 3] or experimental measurements [5]. From the mathematical analysis in the previous section it can be seen that the statistics of dynamic response depends only on the distribution of the eigenvalues and eigenvectors of the matrix Ξ . Adhikari has shown that [4] the matrix Ξ can be modelled as a Wishart random matrix so that Ξ ∼ Wn (p, Σ ). We refer to the books by Muirhead [17], Gupta and Nagar [10] and Tulino and Verd´u [23] for discussions on Wishart random matrices and and related mathematical details. The parameters p and Σ can be obtained from the available data regarding the system, namely M0 , K0 , δM and δK . It was shown that

Ξ ∼ Wn (p, Σ )

(17)

where

Σ = Ω 20 /θ ,

p = n+1+θ

and

θ=

(1 + βH ) − (n + 1) δH2

(18)

The constant βH and the dispersion parameter δH can be obtained as

βH =

n

∑ ω02j

j=1

2

n

/ ∑ ω04j

(19)

j=1

and

δH =

β n2 + 2β n + n2 − 1 + β δK − nβK − β nβK − β n − n − β βK + 1 − β + βK δM 2 (1 + βK ) (−1 − βM + nδM ) (−1 − βM + nδM + 3δM ) + +

(−2n − 2nβM − 2β − 2β β1 − 2β n − 2β nβM ) δK δM (1 + βK ) (−1 − βM + nδM ) (−1 − βM + nδM + 3δM )

(β βK βM + βK βM + β βM + βK + β βK + β + 1 + βM ) δM (1 + βK ) (−1 − βM + nδM ) (−1 − βM + nδM + 3δM )

β βM 2 + 2β βM + 2βM + β + 1 + βM 2 δK + (1 + βK ) (−1 − βM + nδM ) (−1 − βM + nδM + 3δM )

(20)

where βM = {Trace (M0 )}2 /Trace M0 2 and βK = {Trace (K0 )}2 /Trace K0 2 (21) These relationships completely defines all the parameters of the Wishart random matric necessary for uncertainty quantification of structural dynamic systems.

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4 Density of eigenvalues From equations 10 and 11 it is clear that the spectral properties of the Wishart random matrix Ξ play a key role in uncertainty quantification of stochastic dynamical systems. In this section we specifically look into the density of the eigenvalues. Our main result is that the density of the eigenvalues have the ‘self averaging’ property. This implies that the density of the eigenvalues of nominally identical systems are almost identical. In the next two subsections we aim to present a rigorous proof.

4.1 Linear eigenvalue statistic Let Ξ be an n × n random matrix and {λl }nl=1 its eigenvalues. Then the (empirical) eigenvalue density is n

ρn (λ ) = n−1 ∑ δ (λ − λl ),

(22)

l=1

where δ is the Dirac delta-function. Denote ρ n (λ ) the expectation of ρn , i.e.,

ρ n (λ ) = E{ρn (λ )}.

(23)

The symbol E{. . .} denotes the mathematical expectation, i.e., the averaging with respect to the corresponding probability law. To prove the selfaveraging of ρn one has to show that, say, its the variance Var{ρn (λ )} := E{ρn2 (λ )} − E2 {ρn (λ )}

(24)

tends to zero as n → ∞, and if possible, to find (or to estimate) the rate of decay of the variance. However, while ρ n (λ ) = E{ρn (λ )}, i.e., the second term above, is well defined, this is not the case for the first term. Indeed, we have by definition

ρn2 (λ ) = n−2

n

∑

δ (λ − λl1 )δ (λ − λl2 )

l1 ,l2 =1 n −2 2

=n

∑δ

(λ − λl1 ) + n−2

l1 =1

∑

l1 =l2

δ (λ − λl1 )δ (λ − λl2 ).

(25) (26)

and we see that summand δ 2 (λ − λl1 ) of the first sum on the r.h.s is not well defined (it is often said that the square of delta-function is infinity). To avoid this we have to ‘smooth’ the delta-function, i.e., to replacing it with a smooth function having a well pronounced peak. If we denote this function u, then we have n

n−1 ∑ u(λ − λl ) = l=1

u(λ − μ )ρn (μ )d μ

(27)

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337

instead of ρn (λ ). This happens, in particular, when one computes ρn (λ ) numerically. This is because one first finds the eigenvalues and then draws a continuous envelope curve which corresponds to smoothing ρn with a function u whose peak has a width bigger that the distance (of the order O(n−1 )) between the eigenvalues. This is why we will not deal with ρn (λ ) itself but rather with so called linear eigenvalue statistics, defined for any sufficiently smooth test function ϕ as n

Nn [ϕ ] = n−1 ∑ ϕ (λl ) =

ϕ (μ )ρn (μ )d μ

(28)

l=1

Note that ρn in equation 22 correspond formally to ϕ (μ ) = δ (λ − μ ) for a given λ . In the next subsection, we consider the density of eigenvalues within these general frameworks of the Random Matrix Theory.

4.2 Self averaging property and the Mar˘cenko-Pastur density Let Ξ be n × n real symmetric or Hermitian Wishart matrix with p degrees of freedom and an n × n covariance matrix Σ , i.e., Ξ ∼ Wn (p, Σ ). Denote {λl }nl=1 its eigenvalues and consider the linear eigenvalue statistic (see 28) n

Nn [ϕ ] = n−1 ∑ ϕ (λl ),

(29)

l=1

corresponding to a real or complex valued test function ϕ . It can be shown (see [15, 19, 9]) that lim

n→∞→, p→∞, p/n→c∈[1,∞)

E{Nn [ϕ ]} =

ϕ (λ )ρ (λ )d λ ,

(30)

where ρ can be found by solving a certain functional equation for its Stieltjes transform ρ (λ )d λ f (z) = , ℑz = 0. (31) λ −z In the case, where Σ = In and c > 1 we have

1 (a+ − λ )(λ − a− ), λ ∈ [a− , a+ ], (32) ρ (λ ) = λ∈ / [a− , a+ ], 0, 2πλ √ where a± = (1 ± c)2 . If c = 1 then

1 (4 − λ )/λ , λ ∈ (0, 4], ρ (λ ) = λ∈ / (0, 4]. 0, 2π

(33)

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The density of eigenvalues given by equations 32 or 33 is now known as Mar˘cenkoPastur (MP) density. This density will be considered in the next section in the numerical examples. Let us show that the fluctuations of Nn [ϕ ] around its expectation E{Nn [ϕ ]} vanish sufficiently fast in the limit n → ∞ →, p → ∞, p/n → c ∈ (0, ∞)

(34)

To this end we obtain a bound for the variance Var{Nn [ϕ ]} = E{|Nn [ϕ ]|2 } − |E{Nn [ϕ ]}2 of Nn [ϕ ]. The bound is √ 4 3 Var{Nn [ϕ ]} ≤ 2 Tr Σ 2 (max |ϕ (λ )|)2 . n p λ ∈R

(35)

It is valid for real symmetric as well as for hermitian Wishart matrices. We give below its proof for real symmetric matrices. The proof for hermitian matrices is practically the same. It can be shown that if we want to keep the spectrum of Ξ bounded for all n, p of (34) rather than escaping to infinity, we have to assume that in the limit (34): max n−1 TrΣ 2 ≤ C < ∞. p,n

(36)

(the same, in fact stronger, condition is necessary to prove (30)). Assuming this and max |ϕ (λ )| < ∞,

(37)

Var{Nn [ϕ ]} = O(n−2 )

(38)

λ ∈R

we obtain from (35) that

under condition (34) and (36)–(37). Note that if {λl }nl=1 were independent identically distributed random variables, then the variance of their linear statistics is equal to n−1 Var{ϕ (λ1 )}, i.e., is O(n−1 ) for any ϕ such that Var{ϕ (λ1 )} < ∞. This is a manifestation of strong statistical dependence between eigenvalues of Wishart (and many other) random matrices, known also as the repulsion of eigenvalues and/or the rigidity of spectrum (see e.g. [16]). Proof of (35). The proof can be given by following these steps: (i). Given the standard Gaussian random variables {ξl }ql=1 E{ξl } = 0, E{ξl ξm } = δlm

(39)

and a differentiable function Φ : Rq → C of q variables, consider the random variable

Extremely strong convergence of eigenvalue-density of linear stochastic dynamical systems

Ψ = Φ (ξ1 , . . . , ξq ).

339

(40)

Then its variance admits the bound

∂ Φ 2 , Var{Ψ } ≤ ∑ E ∂ ξl l=1 q

(41)

known as the Poincar´e inequality (see e.g. [6]). (ii). Given an n × n real symmetric or hermitian matrix A(t) depending on a parameter t and a function ϕ : R → C, consider the matrix function ϕ (A(t)). Then we have d Trϕ (A(t)) = Trϕ (A(t))A (t)). (42) dt Note now that we can write the Wn (p, Σ ) real symmetric Wishart matrix as

Ξ = p−1 RXXT R,

(43)

where R is a positive definite n × n matrix such that R2 = Σ and X = {Xα j }αp,n, j=1 is a p × n random matrix whose entries are the standard Gaussian random variables E{Xα j } = 0, E{Xα j X∗β k } = δαβ δ jk .

(44)

By using this one can check easily that the entries {Ξ jk }nj,k=1 of Ξ are

Ξ jk = p−1 thus

p

n

∑ ∑ R jl Xα l Xα m Rmk

(45)

j,k=1 α =1

E{Ξ jk } = Σ jk

(46)

as it should be. Note now that it follows from the spectral theorem for real symmetric matrices and (29) that (47) Nn [ϕ ] = Trϕ (Ξ ). Take in (41) n−1 Trϕ (Ξ ) as Ψ and {Xα j }αp,n, j=1 as {ξl }ql=1 , hence q = np. This yields Var{Nn [ϕ ]} ≤ n−2

p

∑

∂ Trϕ (Ξ ) 2 ∑ E ∂ Xα j . n

(48)

α =1 j=1

Take now in (42) Xα j as t, p−1 RXX T R as A and use the formula (see (45))

∂ (p−1 RXXT R)lm = Rl j (XT R)α m + (RX)l α R jm . ∂ Xα j This yields after a simple algebra

(49)

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Var{Nn [ϕ ]} ≤

4 E TrΞ ϕ (Ξ )Σ ϕ (Ξ ) . (np)2

(50)

Now we use (iii). The Schwarz inequality for traces |TrAB|2 ≤ TrAA∗ TrBB∗ with A = Ξ ϕ and B = Σ ϕ . We obtain that |TrΞ ϕ (Ξ )Σ ϕ (Ξ )| ≤ (TrΞ 2 ϕ (Ξ )ϕ (Ξ ))1/2 (TrΣ 2 ϕ (Ξ )ϕ (Ξ ))1/2 . Two more inequality to use are (iv) |TrAB| ≤ ||A||TrB, valid for any matrix A and a positive definite B, where ||A|| is the Euclidian norm of A, and (v) ||ψ (Ξ )|| ≤ maxx∈R |ψ (x)|, valid a real symmetric (hermitian) Ξ . We obtain from the above Var{Nn [ϕ ]} ≤

4 (max |ϕ (x)|)2 (TrΣ 2 )1/2 ({ETrΞ 2 })1/2 (np)2 x∈R

(51)

It follows from (44) and (45) that if n ≤ p, then {ETrΞ 2 } ≤ 3p2 TrΣ 2 . Plugging this in (51), we obtain (35). In the next section, the validity of equations and 38 are examined using numerical examples.

5 Numerical investigations In the previous section it was proved that for large random dynamical systems, the density of eigenvalues reaches a non-random limit. In this section we examine the validity of this result using numerical examples. We also verify if one of most widely used asymptotic density, namely the Mar˘cenko-Pastur density, is valid for structuraldynamic systems. A rectangular cantilever steel plate is considered to illustrate the convergence of the eigenvalue-density. The deterministic properties are assumed to be E¯ = 200 × 109 N/m2 , μ¯ = 0.3, ρ¯ = 7860kg/m3 , t¯ = 3.0mm, Lx = 0.998m, Ly = 0.59m. The plate is divided into 50 elements along the longer length and 30 elements along the shorter length for the numerical calculations. The resulting system has 4650 degrees-of-freedom. In 1 the density of first 1200 eigenvalues of the deterministic system is compared with the Mar˘cenko-Pastur density. Except in the very low

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frequency region, the Mar˘cenko-Pastur density agree well with the Finite Element results.

Fig. 1: The density of first 1200 eigenvalues of the baseline model

The (non-normalized) density of the natural frequencies (square root of the eigenvalues) of a plate can be obtained using the analytical expression derived by Xie et al. [24] as Lx ky ρ th 1 ρ th 1/4 Lx + ky ν= ω −1/2 (52) + π D 2 D π where D =

Eth3 . 12(1− μ 2 )

The quantity ν is also known as the modal density. The initial

decay in the density observed in 1 can be explained by the ω −1/2 term in Eq. 52. From Eq. 52 it is clear that the constant modal density of a plate, often used in many approximate analyses, is only applicable in the high frequency. The result in 1 shows that Wishart random matrix model can be used even in the lower frequency range where the modal density is not constant. Two different cases of uncertainties are considered. In the first case it is assumed that the material properties are randomly inhomogeneous. In the second case we consider that the plate is ‘perturbed’ by attaching spring-mass oscillators at random locations. The first case corresponds to a parametric uncertainty problem while the second case corresponds to a non-parametric uncertainty problem.

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5.1 Plate with randomly inhomogeneous material properties: parametric uncertainty problem It is assumed that the Young’s modulus, Poissons ratio, mass density and thickness are random fields of the form E(x) = E¯ (1 + εE f1 (x)) , μ (x) = μ¯ 1 + εμ f2 (x) (53) ¯ ¯ ρ (x) = ρ 1 + ερ f3 (x) and t(x) = t (1 + εt f4 (x)) (54) The two dimensional vector x denotes the spatial coordinates. The strength parameters are assumed to be εE = 0.10, εμ = 0.10, ερ = 0.08 and εt = 0.12. The random fields f i (x), i = 1, · · · , 4 are assumed to be correlated homogenous Gaussian random fields. An exponential correlation function with correlation length 0.2 times the lengths in each direction has been considered. The random fields are simulated by expanding them using the Karhunen-Lo`eve expansion [8, 18] involving uncorrelated standard normal variables. A 5000-sample Monte Carlo simulation is performed to obtain the eigenvalues of the system. In 2 100 samples of the density of first 1200 eigenvalues are shown, alongside the fitted Mar˘cenko-Pastur density and density obtained from the baseline model. The density of the eigenvalues of the random realization are quite close.

Fig. 2: The density of eigenvalues of the plate with randomly inhomogeneous material properties

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5.2 Plate with randomly attached spring-mass oscillators: nonparametric uncertainty problem In this example we consider the same plate but with non-parametric uncertainty. The baseline model is perturbed by attaching 10 spring mass oscillators with random natural frequencies at random nodal points in the plate. The natural frequencies of the attached oscillators follow a uniform distribution between 0.2 kHz to 4.0 kHz. The nature of uncertainty in this case is different from the previous case because here the sparsity structure of the system matrices change with different realizations of the system. Again a 5000-sample Monte Carlo simulation is performed to obtain the eigenvalues. In 3, 100 samples of the density of first 1200 eigenvalues are shown, alongside the fitted Mar˘cenko-Pastur density and density obtained from the baseline model. The density of the eigenvalues of the random realization are quite close.

Fig. 3: The density of eigenvalues of the plate with randomly attached oscillators

6 Conclusions The density of eigenvalues of structural dynamical systems with uncertainty is considered in this paper. Due to the positive definiteness nature of a real system, it can be modeled using a Wishart random matrix with suitable parameters. These parameters in turn can be explicitly obtained form the baseline model and dispersion

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parameters corresponding to the mass and stiffness matrices of the system using the closed-form expressions given the paper. It was shown that for large random systems, the density of eigenvalues reaches a non-random limit. It particular, it was rigorously proved that for an n-dimensional system, the variance associated with a suitable linear statistic of the eigenvalues is in the order O(n−2 ). This result shows that if a system is large, then the detailed nature of random perturbation do not effect the eigenvalue-density. Under certain simplified assumptions, this asymptotic density can be suitably represented by the so called Mar˘cenko-Pastur density. Two numerical examples involving a cantilever plate with parametric and non-parametric uncertainty have been used to investigate the validity of analytical results. Using direct Monte Carlo simulations, it was indeed observed that eigenvalue-densities of nominally identical systems do not differ from each other and are very close to the Mar˘cenko-Pastur density. It is important to note that the original matrices are not Wishart matrices, but the eigenvalue-density is close to that of Wishart matrices. The strong convergence of the eigenvalue-density perhaps explains why many random matrix based methods (e.g. statistical energy analysis) are so useful for highfrequency vibration problems where the sizes of the underlying matrices are very large. This convergence property may also opens up the possibility of calculating other useful quantities such as the response statistics. Acknowledgements The author gratefully acknowledges the support of UK Engineering and Physical Sciences Research Council (EPSRC) through the award of an Advanced Research Fellowship and The Leverhulme Trust for the award of the Philip Leverhulme Prize.

References 1. Adhikari S (2007a) Matrix variate distributions for probabilistic structural mechanics. AIAA Journal 45(7):1748–1762 2. Adhikari S (2007b) On the quantification of damping model uncertainty. Journal of Sound and Vibration 305(1–2):153–171 3. Adhikari S (2008) Wishart random matrices in probabilistic structural mechanics. ASCE Journal of Engineering Mechanics 134(12):1029–1044 4. Adhikari S (2009) On the validity of random matrix models in probabilistic structural dynamics. In: 2nd International Conference on Uncertainty in Structural Dynamics (USD), Sheffield, UK 5. Adhikari S, Sarkar A (2009) Uncertainty in structural dynamics: experimental validation of wishart random matrix model. Journal of Sound and Vibration 323(3–5):802–825 6. Bogachev V (1991) Gaussian Measures. American Mathematical Society, Providence, USA 7. Dawe D (1984) Matrix and Finite Element Displacement Analysis of Structures. Oxford University Press, Oxford, UK 8. Ghanem R, Spanos P (1991) Stochastic Finite Elements: A Spectral Approach. SpringerVerlag, New York, USA 9. Girko V (2001) Theory of Stochastic Canonical Equations, vols. I, II. Kluwer, Dordrecht 10. Gupta A, Nagar D (2000) Matrix Variate Distributions. Monographs & Surveys in Pure & Applied Mathematics, Chapman & Hall/CRC, London 11. Hemez FM (2004) Uncertainty quantification and the verification and validation of computational models. In: Inman DJ, Farrar CR, Lopes Jr V, Steffen Jr V (eds) Damage Prognosis for Aerospace, Civil and Mechanical Systems, John Wiley & Sons Ltd., London, United Kingdom

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12. Kleiber M, Hien TD (1992) The Stochastic Finite Element Method. John Wiley, Chichester 13. Langley RS, Brown AWM (2004) The ensemble statistics of the energy of a random system subjected to harmonic excitation. Journal of Sound and Vibration 275:823–846 14. Langley RS, Cotoni V (2007) Response variance prediction for uncertain vibro-acoustic systems using a hybrid deterministic-statistical method. Journal of the Acoustical Society of America 122(6):3445–3463 15. Mar˘cenko V, Pastur L (1967) The eigenvalue distribution in some ensembles of random matrices. Math USSR Sbornik 1:457–483 16. Mehta ML (1991) Random Matrices, 2nd edn. Academic Press, San Diego, CA 17. Muirhead RJ (1982) Aspects of Multivariate Statistical Theory. John Wiely and Sons, New York, USA 18. Papoulis A, Pillai SU (2002) Probability, Random Variables and Stochastic Processes, 4th edn. McGraw-Hill, Boston, USA 19. Silverstein JW, Bai ZD (1995) On the empirical distribution of eigenvalues of a class of largedimensional random matrices. Journal of Multivariate Analysis 54:175–192 20. Soize C (2000) A nonparametric model of random uncertainties for reduced matrix models in structural dynamics. Probabilistic Engineering Mechanics 15(3):277–294 21. Soize C (2001) Maximum entropy approach for modeling random uncertainties in transient elastodynamics. Journal of the Acoustical Society of America 109(5):1979–1996, part 1 22. Soize C (2005) Random matrix theory for modeling uncertainties in computational mechanics. Computer Methods in Applied Mechanics and Engineering 194(12–16):1333–1366 23. Tulino AM, Verd´u S (2004) Random Matrix Theory and Wireless Communications. Now Publishers Inc., Hanover, MA, USA 24. Xie G, Thompson DJ, Jones CJC (2004) Mode count and modal density of structural systems: relationships with boundary conditions. Journal of Sound and Vibration 274:621–651 25. Zienkiewicz OC, Taylor RL (1991) The Finite Element Method, 4th edn. McGraw-Hill, London

Stochastic subspace projection schemes for dynamic analysis of uncertain systems Prasanth B. Nair

Abstract We present stochastic subspace projection schemes for dynamic response analysis of linear stochastic structural systems. The underlying idea of the numerical methods presented here is to approximate the response process using a set of stochastic basis vectors with undetermined coefficients that are estimated via orthogonal/oblique stochastic projection. We present a preconditioned stochastic conjugate gradient method based on the conjugate orthogonality condition for approximating the frequency response statistics of stochastic structural systems. We also outline a new stochastic projection scheme for solving the generalized algebraic random eigenvalue problem. Some preliminary results are presented for a model problem to illustrate the performance of the proposed methods.

1 Introduction Traditional numerical methods for tackling deterministic problems in structural dynamics use either a set of basis functions or basis vectors to approximate the response quantities; for example, the finite element method, Rayleigh-Ritz method, classical modal analysis, subspace iteration schemes and Krylov methods. More recently, rational extensions of such numerical schemes to solve randomly parametrized algebraic and operator equations have been the focus of much interest in the numerical analysis and stochastic finite element analysis communities; see, for example, references [1]–[6]. The fundamental idea used in such schemes is to employ a set of stochastic trial functions or basis vectors to represent the solution process. The undetermined coefficients in the subspace approximation are then estimated by

Prasanth B. Nair Computational Engineering and Design Group, School of Engineering Sciences, University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 25, © Springer Science+Business Media B.V. 2011

347

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employing stochastic projection schemes which ensure that the original governing equations are satisfied in a probabilistic sense. In this paper, we present numerical methods based on stochastic subspace projection theory for dynamic response analysis of linear stochastic structural systems. We propose different types of stochastic basis for solving randomly parametrized complex linear algebraic equations (which arise in frequency response analysis of uncertain systems) and the algebraic random eigenvalue problem. Orthogonal and oblique stochastic subspace projection schemes are then introduced for solving this class of problems in linear stochastic structural dynamics. We show that these formulations allow for the possibility of efficiently approximating the statistics of the natural frequencies, mode shapes, and the forced response in the low-frequency region without resorting to computationally expensive Monte Carlo simulation. The remainder of this paper is organized as follows: Section 2 outlines some preliminary background material and describes the problems studied in this paper. Section 3 presents a conjugate gradient method based on the conjugate orthogonality condition for frequency domain analysis of linear stochastic structural systems. Section 4 presents a stochastic reduced basis projection scheme for the generalized linear algebraic random eigenvalue problem. Section 5 presents some numerical studies on a model problem. Section 6 concludes the paper and highlights further work that is currently underway.

2 Preliminaries We focus on stochastic projection schemes for the following matrix system of second-order stochastic ordinary differential equations encountered in linear structural dynamics M(ξ )¨x(ξ ,t) + C(ξ )˙x(ξ ,t) + K(ξ )x(ξ ,t) = f(t) ,

(1)

where M(ξ ), C(ξ ), and K(ξ ) ∈ Rn×n are random coefficient matrices, x(ξ ,t) ∈ Rn denotes the system response to the random excitation vector f(t) ∈ Rn , and t ∈ R+ denotes time. Note that the overdots represent derivatives with respect to time. Solution of (1) entails computing the statistics of the response process x(ξ ,t) given the joint pdf P(ξ ) of the random parameter vector ξ = {ξ1 , ξ2 , . . . , ξ p } ∈ R p . In addition, we shall make the following assumptions: A1 The system under consideration is proportionally damped, i.e., C(ξ ) = γ1 K(ξ )+ γ2 M(ξ ), where γ1 and γ2 are deterministic scalars. A2 The coefficient matrices K and M are symmetric and can be represented in a polynomial chaos (PC) basis,1 i.e., 1

This assumption is not limiting since the Cameron-Martin theorem guarantees that such a representation exists provided the variance of K and M is finite. PC representations of the coefficient matrices can be arrived at for uncertainty in the material properties, boundary conditions, geometry, etc; see, for example [5] and the references therein.

Stochastic subspace projection schemes for dynamic analysis of uncertain systems P

P

i=0

i=0

K(ξ ) = ∑ Ki ϕi (ξ ), and M(ξ ) = ∑ Mi ϕi (ξ ),

349

(2)

where Ki , Mi ∈ Rn×n are deterministic symmetric matrices and {ϕi (ξ )} is a set of PC basis functions with the following properties2 :

ϕ0 = 1 , ϕi = 0 , ∀i > 0 and ϕi ϕ j = δi j , Kronecker delta function and the expectation operator · is dewhere δi j is the fined as · = ·P(ξ )dξ . A3 All the coefficient matrices are non-singular — this assumption can be easily relaxed later to allow for analysis of systems with zero eigenvalues. A4 For notational simplicity, we shall assume that the excitation vector f is deterministic. Our objective is to develop numerical methods for the following problems outlined below: P1:Approximate the dynamic response statistics in the frequency domain by solving the following system of complex linear random algebraic equations (3) K(ξ ) − ω 2 M(ξ ) + jω C(ξ ) x(ξ , ω ) = f(ω ) , √ where ω is the excitation frequency of interest and j = −1. P2:Approximate the statistics of the natural frequency and mode shapes by solving the following generalized linear algebraic random eigenvalue problem K(ξ )φ (ξ ) = λ (ξ )M(ξ )φ (ξ ) ,

(4)

where λ and φ denote the eigenvalue and eigenvector of a particular mode.

3 Frequency domain analysis of linear stochastic structural systems In this section, we focus on problem P1, which involves approximating the dynamic response statistics in the frequency domain. The governing equations in the frequency domain can be compactly written as a matrix system of complex linear random algebraic equations in the following form D(ξ , ω )x(ξ , ω ) = f(ω ) ,

(5)

where D(ξ , ω ) = K(ξ )− ω 2 M(ξ )+jω C(ξ ) ∈ Cn×n is the stochastic dynamic stiffness matrix. It follows from Assumption A2 that D admits a PC expansion of the 2

Note that we have orthonormalized the PC basis functions in order to simplify the expressions that follow while formulating the stochastic conjugate gradient algorithm.

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form

PD

D(ξ , ω ) = ∑ Di (ω )ϕi (ξ ) ,

(6)

i=0

where Di (ω ) = Ki − ω 2 Mi + jω Ci ∈ Cn×n are deterministic matrices and PD is the total number of terms in the PC expansion of D. For a given frequency point ω , the following theorem [4] tells us which subspace in which the frequency response x lives. Theorem 1. If D(ξ , ω ) is a non-singular matrix and the degree of its minimal random polynomial3 is m, then the solution of (5) lies in the following stochastic Krylov subspace (7) Km D(ξ , ω ), r0 = span r0 , D(ξ , ω )r0 , . . . , [D(ξ , ω )]m−1 r0 , where r0 = f(ω ) − D(ξ , ω )x0 and x0 is the initial guess4 for the response process. The above theorem suggests that we can use the stochastic (or parametrized) basis vectors spanning the Krylov subspace to approximate the solution of (5), i.e., m

x(ξ , ω ) ≈ ∑ ci (ω )ψ i (ξ , ω ) ,

(8)

i=1

where ψ i (ξ , ω ) are frequency dependant stochastic basis vectors spanning the stochastic Krylov subspace. The undetermined coefficients ci can be computed by Bubnov-Galerkin projection, which involves enforcing the orthogonality condition m

D(ξ , ω ) ∑ ci (ω )ψ i (ξ , ω ) − f(ω ) ⊥ ψ j (ξ , ω ), j = 1, 2, . . . , m.

(9)

i=1

In this orthogonal projection scheme, the following definition of Hermitian orthogonality is used to enforce the above condition. Definition 1 (Hermitian Orthogonality): Two random vectors x(ξ ), y(ξ ) ∈ Cn are orthogonal if

(x(ξ ), y(ξ )) = x(ξ )H y(ξ ) = x(ξ )H y(ξ )P(ξ )dξ = 0,

(10)

where the superscript H denotes the complex conjugate transpose. Application of the orthogonality condition (9) leads to a reduced-order deterministic system of equations for the undetermined coefficients. The statistics of the response at the frequency point ω can be subsequently approximated using (8). 3 For a diagonalizable random matrix, D(ξ ), the degree of the minimal random polynomial can be defined as the maximum number of distinct eigenvalues over all possible values of ξ . 4 For example, the initial guess for the response at the frequency point ω can be chosen to be the nominal response D(ξ , ω )−1 f(ω ).

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However, directly computing the basis vectors spanning the Krylov subspace and carrying out Bubnov-Galerkin projection (particularly for a large number of frequency points) is computationally inefficient. In addition, numerical instabilities may arise when a large number of basis vectors is employed in the approximation. Numerical studies based on this formulation can be found in the literature; see, for example, [4], [7]. In recent work [8], it has been found that by developing stochastic variants of classical Krylov iterative methods, better computational efficiency and numerical stability can be achieved. It was shown that for linear random algebraic equations with a Hermitian positive definite coefficient matrix (say D(ξ )), the solution process can be efficiently approximated by applying a stochastic conjugate gradient algorithm to minimize the following error function

1 H x (ξ )D(ξ )x(ξ ) − xH (ξ )f 2 over the parametrized affine space x0 + Km . This idea has been successfully applied by the authors’ group to solve a class of steady-state stochastic partial differential equations. Compared to the original stochastic reduced basis formulations in [4], conjugate gradient schemes are significantly faster and require less computer memory for large-scale problems with many random variables. For a detailed analysis of the connections between this approach and the standard approach of applying deterministic Krylov methods to the high-dimensional deterministic equations arising from the Ghanem-Spanos projection scheme; see [8]. The stochastic dynamic stiffness matrix D is not Hermitian positive definite since DH = D and hence we cannot use a stochastic version of the conjugate gradient method to solve (5). This is unfortunate, since the conjugate gradient method enables the efficient application of a three-term recursion relation to extract an orthogonal basis for the Krylov subspace Km . It is possible in theory to derive alternative Krylov methods based on the Petrov-Galerkin condition (also known as an oblique projection scheme) to deal with non-Hermitian random matrices, i.e., m

D(ξ , ω ) ∑ ci (ω )ψ i (ξ , ω ) − f(ω ) ⊥ D(ξ , ω )ψ j (ξ , ω ) , j = 1, 2, . . . , m . (11) i=1

Unfortunately, it is difficult to construct numerically stable implementations of projection schemes based on the Petrov-Galerkin condition and in addition, this approach would require significantly more computational effort and memory for stochastic coefficient matrices expressed in a PC basis. It can be noted that, due to assumption A2, the stochastic dynamic stiffness matrix D is a complex symmetric random matrix i.e., DT = D, where the superscript T denotes the standard transpose operator. This enables us to exploit an idea originally proposed by Van der Vorst and Melissen [9] in the context of developing Krylov methods for deterministic complex symmetric matrix equations encountered in lin-

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ear eddy current problems5 . The key idea is to replace the Hermitian orthogonality condition defined in (10) with the following conjugate orthogonality condition. Definition 2 (Conjugate Orthogonality Condition:) Two random vectors x(ξ ), y(ξ ) ∈ Cn are said to be conjugate orthogonal if

(12) (x(ξ ), y(ξ )) = x(ξ )T y(ξ ) = 0 , where x denotes the complex conjugate of x, the superscript T denotes the standard transpose operator and (·, ·) denotes the standard Hermitian inner product defined earlier in (10). It is to be noted here that the coefficient matrix D is symmetric with respect to the conjugate inner product; this can be shown as follows: T

uT Dv = uH Dv = D uH v = (Du)H v = (Du)T v . When orthogonality between the residual and the basis vectors are enforced in the sense of Definition 2, we are essentially computing the undetermined coefficients by enforcing the following stochastic Petrov-Galerkin condition m

D(ξ , ω ) ∑ ci (ω )ψ i (ξ , ω ) − f(ω ) ⊥ ψ j (ξ , ω ), j = 1, 2, . . . , m. i=1

In other words, the residual is enforced to be orthogonal to the complex conjugate of the basis vectors. An outline of the preconditioned conjugate gradient algorithm based on the conjugate orthogonality condition (12) is given next. This algorithm solves (5) via minimization of the modified error function

1 T x (ξ )D(ξ )x(ξ ) − xT (ξ )f 2 over the parametrized affine space x0 + Km . We refer to this algorithm as the stochastic preconditioned conjugate orthogonal conjugate gradient (sPCOCG) algorithm. Note that in the description of the sPCOCG algorithm, we have not explicitly shown the dependence of the coefficient matrix and other quantities on the frequency point of interest, ω . sPCOCG(x(ξ ), f, D(ξ ), M, ε , kmax , P) D 1. Compute ri = ∑Pj=0 ∑Pl=0 (fi − ϕi ϕ j ϕl D j xl ), for i = 0, 1, 2, . . . , P ρ 0 = ∑Pi=0 rH ri , k = 1. i 2. Do While ρ k−1 > ε ∑Pi=0 fH i fi and k < kmax

a. zi = Mri , for i = 0, 1, 2, . . . , P b. τ k−1 = ∑Pi=0 zTi ri

5

Complex symmetric matrices also arise in a number of other areas, including the time-dependant Schr¨odinger equation, inverse scattering problems and underwater acoustics.

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c. if k = 1 then β = 0 and pi = zi for i = 0, 1, 2, . . . , P else β = τ k−1 /τ k−2 and pi = zi + β pi , for i = 0, 1, 2, . . . , P D d. wi = ∑Pj=0 ∑Pl=0 ϕi ϕ j ϕl D j pl , for i = 0, 1, 2, . . . , P e. α = τ k−1 /(∑Pi=0 pTi wi ) f. xi = xi + α pi , for i = 0, 1, 2, . . . , P g. ri = ri − α wi , for i = 0, 1, 2, . . . , P h. ρ k = ∑Pi=0 rH i ri i. k = k + 1 The inputs to the sPCOCG algorithm are: (1) PC expansions of D and the initial guess, (2) convergence tolerance ε , (3) maximum number of iterations or basis vectors kmax , (4) preconditioner matrix M and (5) the maximum number of terms in the PC expansion of the solution process (which is related to the truncation order of the solution), P. In practice, we monitor the L2 norm of the true residual to ensure that unnecessary iterations are avoided [8]. Note that in Step 1 of the algorithm f0 = f and fi = 0 ∀i > 0 due to assumption A4. Also note that the subscripts on the vectors x, r, z, p denotes the PC expansion coefficient number. The main computational effort in this scheme involves multiplying the stochastic coefficient matrix with a random vector in a PC basis (see Step 2d) and multiplying the preconditioner with a vector in Step 2a. The memory requirements of the sPCOCG algorithm are very modest since we only need to store 4 vectors (x, r, z, p) in a PC basis apart from the coefficient matrix D and the preconditioner M. In theory, the sPCOCG algorithm can breakdown before convergence if we encounter a quasi-null vector, i.e., a non-zero vector v(ξ ) whose conjugate norm vT (ξ )v(ξ ) = 0. Even though, this possibility cannot be ruled out, we haven’t encountered quasi-null vectors in our numerical studies. It is desirable in practical implementations of the sPCOCG algorithm to check for quasi-null vectors. If a quasi-null vector is encountered, then the algorithm can be restarted or recourse can be made to computationally more expensive residual error minimization schemes.

3.1 Preconditioner A key ingredient of the sPCOCG algorithm is the preconditioning matrix M since the choice of this matrix plays a crucial role in determining the convergence rate. Ideally, we would like to choose the matrix M such that the distributions of the eigenvalues of the random matrix MD(ξ , ω ) has a high degree of statistical overlap; in other words, this random matrix numerically behaves like a matrix with a few number of distinct eigenvalues. For simplicity of implementation, a deterministic preconditioner is desirable and one possible choice is the inverse of the nominal dynamic stiffness matrix at the frequency point ω , i.e., M = D(ξ , ω )−1 . In practice, we don’t need the explicit inverse of the nominal dynamic stiffness matrix since in Step 2a, we only need to solve the deterministic system of equations

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D(ξ , ω ) z = r .

(13)

The above system of equations can be efficiently solved for any value of ω if the eigenvalues and eigenvectors of the nominal system are available. Because of the proportional damping assumption A1, the eigenvectors of the nominal system can be used as basis vectors to diagonalize the preconditioner D(ξ , ω ). Hence, the solution of (13) can be efficiently computed as follows: (14) Λ 0 − ω 2 In + jω (γ1 Λ 0 + γ2 In ) z = (Φ 0 )T r, z = Φ 0z ,

where In ∈ Rn ×n denotes the identity matrix. Λ 0 ∈ Rn ×n and Φ 0 ∈ Rn×n are matrices containing the first n eigenvalues and corresponding eigenvectors of the matrix pair (K0 , M0 ), respectively.

3.2 Postprocessing The final output of the sPCOCG algorithm is a reduced basis representation for the frequency response in the following form P

x(ξ , ω ) = ∑ xi (ω )ϕi (ξ ) .

(15)

i=0

Using the standard properties of the orthonormalized PC basis functions, it can be shown that the mean of the frequency response is given by x0 (ω ) while its covariance matrix is given by ∑Pi=1 xi xH i .

4 The algebraic random eigenvalue problem In this section, we develop stochastic projection schemes for symmetric generalized algebraic random eigenvalue problems of the form (problem P2). K(ξ )φ i (ξ ) = λi (ξ )M(ξ )φ i (ξ ) ,

(16)

where φ i (ξ ) ∈ Rn and λi (ξ ) ∈ R denote the eigenvector and eigenvalue of mode i, respectively. A stochastic reduced basis method for randomly parametrized eigenvalue problems was proposed in [10]. In this method, the eigenparameter statistics are approximated by solving a sequence of reduced-order random eigenproblem for each mode of interest. However, the accuracy of this approach may deteriorate when the structural system has high statistical modal overlap factor. This observation suggests that formulations which use a global set of stochastic basis vectors may lead to bet-

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ter approximations for the eigenvalue statistics. The fundamental idea used here is to approximate the first m eigenvectors of (16) in a subspace spanned by a set of stochastic basis vectors, i.e., (ξ ) = Ψ (ξ )C , Φ

(17)

where Ψ (ξ ) = [ψ 1 (ξ ), ψ 2 (ξ ), . . . , ψ m (ξ )] ∈ Rn×m and C = [c1 , c2 , . . . , cm ] ∈ Rm×m denote matrices of basis vectors and undetermined coefficients, respectively. The matrix of undetermined coefficients can be computed by substituting (17) into (16) and imposing the Bubnov-Galerkin condition (K(ξ ) − λ (ξ )M(ξ ))Ψ (ξ )C ⊥ Ψ (ξ ).

(18)

This leads to a reduced-order deterministic generalized eigenvalue problem of the following form KR C = MR CΩ , (19) where KR = Ψ T (ξ )K(ξ )Ψ (ξ ), MR = Ψ T (ξ )M(ξ )Ψ (ξ ) ∈ Rm×m are reducedorder matrices, and Ω ∈ Rm×m denotes the diagonal matrix of eigenvalues of (19). In the sections that follow we present a stochastic subspace projection scheme based on this idea for solving the algebraic random eigenvalue problem when the underlying random matrices are expressed in a PC basis. We discuss the choice of basis vectors and how the solution of the reduced-order eigenvalue problem (19) can be used to approximate the eigenvalue statistics. Note that in contrast to the approach of Ghanem and Ghosh [6], the present method does not involve the solution of a nonlinear system of equations.

4.1 Stochastic Basis Vectors In the present work, we employ the solution of the following linear random algebraic system of equations as stochastic basis vectors K(ξ )Ψ (ξ ) = M(ξ )Φ 0 ,

(20)

where Φ o ∈ Rn×m is a matrix containing the first m eigenvectors of the deterministic eigenvalue problem K0 φ = λ M0 φ . The above set of stochastic basis is motivated by the classical subspace iteration method for solving deterministic eigenvalue problems [11]. Equation (20) can be readily solved using a stochastic version of the conjugate gradient method6 (with the matrix K0 as preconditioner). Subsequently, the matrix of stochastic basis vectors can be directly written in a PC expansion as Ψ (ξ ) = ∑Pj=0 Ψ j ϕ j (ξ ), where P is the number of terms used in PC expansion of the solution of (20).

6

This is essentially the sPCOCG algorithm with complex variables replaced by real variables.

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4.2 Bubnov-Galerkin Projection Note that we have assumed for simplicity of notation that the number of terms in the PC representation of K(ξ ) and M(ξ ) is P. Using the PC representation of the basis vectors, the elements of the reduced-order matrices in (19) can be efficiently computed as follows: P

P

KR = ∑ ∑

P

P

P

∑ ϕi ϕ j ϕk Ψ Ti K jΨ k and MR = ∑ ∑

i=0 j=0 k=0

P

∑ ϕi ϕ j ϕk Ψ Ti M jΨ k .

(21)

i=0 j=0 k=0

For the special case of zero-eigenvalues, the matrix K(ξ ) is a semi-definite matrix and hence we rewrite the eigenvalue as κ (ξ ) = λ (1ξ ) . Now the problem statement involves approximating the m largest values of κ (ξ ). The stochastic subspace for approximating the eigenvectors can hence be obtained by solving the linear random algebraic system of equations M(ξ )Ψ (ξ ) = K(ξ )Φ 0 . The expressions for the reduced-order matrices arising from Bubnov-Galerkin projection using these new basis vectors can be subsequently derived following the steps outlined earlier.

4.3 Postprocessing If the basis vectors are rich (i.e., the first m eigenvectors of (16) has significant components along it), then the solution of (19) leads to a good approximation for (ξ ) = Ψ (ξ )C. We can hence write the stochastic invariant subspace of (16), i.e., Φ approximations for the first m random eigenvalues of (16) as cT Ψ T (ξ )K(ξ )Ψ (ξ )ci λi (ξ ) = Ti T , i = 1, 2, . . . m, ci Ψ (ξ )M(ξ )Ψ (ξ )ci where ci ∈ Rm denotes the eigenvector corresponding to its i-th eigenvalue of (20). Since both the numerator and denominator of the preceding equation can be written as a PC expansion, λi (ξ ) can also be easily expressed in a PC basis by solving a small system of deterministic linear algebraic equations. The resultλi (ξ ) can be postprocessed for the eigenvalue statistics. ing PC expansions for (ξ ) = The eigenvector statistics can be approximated using the relationship Φ Ψ (ξ )C. Note that it is also possible to employ the eigenvalue and eigenvector approximations to calculate the forced response statistics in the time/frequency domain.

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5 Numerical Studies In this section, we present some numerical studies on a model problem involving a plate structure clamped at one end in a state of plane stress. The Youngs modulus of the structure is modeled by a stationary Gaussian random field. The pdfs of the natural frequencies and the scatter in the frequency response function for this model problem are shown in Figures 1 and 2, respectively. There are around 100 modes in the frequency range [0, 600] Hz. The results obtained using the proposed methods are compared against those calculated using Monte Carlo simulation (MCS) with a sample size of 100, 000.

Fig. 1: Probability density functions of the natural frequencies (Hz) for the model problem

Fig. 2: Scatter in the frequency response function as a result of parameter uncertainty in the model problem (2000 realizations) The absolute value of the mean and standard deviation of the frequency response function computed using the sPCOCG algorithm are compared against results ob-

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tained using MCS in Figures 3 and 4. Note that the results computed using the sPCOCG algorithm are obtained by setting the PC expansion order of the solution px = 1, 2. It can be seen that good accuracy can be obtained, particularly for the frequency range [0, 400], using a second-order approximation.

Fig. 3: Comparison of absolute value of mean frequency response function computed using the sPCOCG algorithm (with order px = 1, 2) with results obtained from Monte Carlo simulation We have also obtained preliminary results using a first-order implementation of the proposed method for solving the algebraic random eigenvalue problem. In other words, we first solve (21) using a stochastic conjugate gradient algorithm with the solution truncated to first-order. The resulting stochastic basis vectors are then used to approximate the eigenvalue statistics.

Fig. 4: Comparison of absolute value of standard deviation of frequency response function computed using the sPCOCG algorithm (with order px = 1, 2) with results obtained from Monte Carlo simulation

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Fig. 5: Comparison of eigenvalue pdf approximations for the first 6 modes obtained using a first-order implementation of the proposed method with results obtained using Monte Carlo simulation The eigenvalue pdfs of the first 6 modes obtained using a first-order approximation are compared against MCS results in Figure 5. It can be noted that the approximations for the first 3 modes are very encouraging. The relative lack of accuracy for the higher modes can be attributed to the fact that we are using a first-order approximation and in addition no shift parameters have been incorporated to accelerate convergence for the higher modes. It is expected that better accuracy can be achieved by increasing the PC expansion order of the stochastic basis vectors along with a shifting strategy in an iterative fashion as outlined in the next section.

6 Concluding Remarks In this paper, we outlined new numerical schemes based on stochastic reduced basis approximations for: (1) dynamic analysis of linear stochastic structural systems in the frequency domain and (2) solution of the generalized algebraic random eigenvalue problem. Numerical studies on a model problem suggest that the proposed preconditioned conjugate gradient algorithm based on the conjugate orthogonality condition (sPCOCG algorithm) performs well and provides good accuracy. The results obtained using a first-order implementation of the proposed method for solving the algebraic random eigenvalue problem are encouraging. Ongoing work is focusing on the development of an iterative stochastic subspace iteration method, wherein the following linear random algebraic system of equations are solved to update the stochastic basis vectors

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(K(ξ ) − μ M(ξ )) Ψ k (ξ ) = M(ξ )Ψ k−1 (ξ ) ,

(22)

where the superscript k denotes the iteration counter and μ is a shift parameter which ensures that the algorithm converges to eigenvalues clustered around this value. After updating the basis, the reduced-order eigenvalue problem (20) is solved to update the approximation for the eigenvalues and eigenvectors. It is to be noted here that when a non-zero shift parameter is chosen, the stochastic coefficient matrix K − μ M in the preceding equations will be symmetric indefinite and as a result we can no longer use the stochastic conjugate gradient method. We are hence developing a stochastic residual error minimization scheme on the lines of the MINRES algorithm [12] to solve (22). This approach is expected to lead to improvements in accuracy for the higher modes. Acknowledgements This work is supported by EPSRC Grant EP/F006802/1.

References 1. Ghanem, R. and Spanos, P., Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, 1991. 2. Deb, M.K., Babuska, I.M. and Oden, J.T., “Solution of stochastic partial differential equations using Galerkin finite element techniques,” Computer Methods in Applied Mechanics and Engineering, 190, 2001, pp. 6359–6372. 3. Sarkar, A. and Ghanem, R., “Mid-frequency structural dynamics with parameter uncertainty,” Computer Methods in Applied Mechanics and Engineering, 191, 2002, pp. 5499–5513. 4. Nair, P.B. and Keane, A.J., “Stochastic reduced basis methods,” AIAA Journal, 40, 2002, pp. 1653–1664. 5. Nair, P.B., “Projection schemes in stochastic finite element analysis,” CRC Engineering Design Reliability Handbook, Chapter 21, editors: E. Nikolaidis, D.M.E. Ghiocel and S.E. Singhal, CRC Press, Boca Raton, Florida, 2004. 6. Ghanem, R. and Ghosh, D., “Efficient characterization of the random eigenvalue problem in a polynomial chaos decomposition,” International Journal for Numerical Methods in Engineering, 72, 2007, pp. 486–504. 7. Bah, M.T., Nair, P.B., Bhaskar, A. and Keane, A.J., “Forced response analysis of mistuned bladed disks: A stochastic reduced basis approach,” Journal of Sound and Vibration, 263, 2003, pp. 377–397. 8. Hakansson, P. and Nair, P.B., “Conjugate gradient methods for randomly parameterized linear random algebraic equations,” submitted for review. 9. Vand der Vorst, H.A. and Melissen, J.B.M., “A Petrov-Galerkin type method for solving Ax = b, where A is a symmetric complex matrix,” IEEE Transactions on Magnetics, 26, 1990, pp. 706–708. 10. Nair, P.B. and Keane, A.J., “An approximate solution scheme for the algebraic random eigenvalue problem,” Journal of Sound and Vibration, 260, 2003, pp. 45–65. 11. Bathe, K.J. and Wilson, E.L., “Large eigenvalue problems in dynamic analysis,” ASCE Journal of Engineering Mechanics, 98, 1972, pp. 1471–1485. 12. Paige, C.C. and Saunders, M.A., “Solution of sparse indefinite systems of linear equations,” SIAM Journal of Numerical Analysis, 12, 1975, pp. 617–629.

Part IV

Probabilistic Methods, Applications

Reliability Assessment of Uncertain Linear Systems in Structural Dynamics H.J. Pradlwarter and G.I. Schu¨eller

Abstract A numerical procedure for the reliability assessment of uncertain linear structures subjected to general Gaussian loading is presented. In this work, restricted to linear FE systems and Gaussian excitation, the loading is described quite generally by the Karhunen-Lo`eve expansion, which allows to model general types of non-stationarities with respect to intensity and frequency content. The structural uncertainties are represented by a stochastic approach where all uncertain quantities are described by probability distributions. First, the critical domain within the parameter space of the uncertain structural quantities is identified, which is defined as the region which contributes most to the excursion probability. Each point in the space of uncertain structural parameters is associated with a certain excursion probability caused by the Gaussian excitation. In order to determine the first excursion probability of uncertain linear structures, an integration over the space of uncertain structural parameters is required. Special attention is devoted to the efficiency of the proposed approach when dealing with realistic FE models, characterized by a large number of degrees of freedom and also a large number of uncertain parameters.

1 Introduction The potentials and abilities of modeling the behavior of fluids, solids and complex materials subjected to various forces and boundary conditions by Finite Element analyses (FEA) provides a key technology for further developments in the industrialized world. Supported by the available relatively inexpensive and continuously H.J. Pradlwarter Institute of Engineering Mechanics, University of Innsbruck, A-6020 Innsbruck, Austria, EU, e-mail: [email protected] G.I. Schu¨eller Institute of Engineering Mechanics, University of Innsbruck, A-6020 Innsbruck, Austria, EU, e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 26, © Springer Science+Business Media B.V. 2011

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growing computer power, the expectations on FEA are shifting towards reliable and robust computational simulations and predictions of physical events. For achieving this goal, answers to many open issues in computational mechanics need to be provided as discussed e.g. in [9]. The need to design, given uncertain material properties, production processes, operating conditions and fidelity of mathematical computational (FE) models to represent reality, leads to the concern of reliability and robustness of computer-generated predictions. Without some confidence in the validity of simulations, their value is obviously diminished. Since uncertainty is always present in non-trivial realistic applications, uncertainty propagation through the FEA is one of the important issues which must be addressed for further developments. In the this paper, a computational efficient reliability estimation procedure for uncertain linear system subjected to dynamic stochastic loading is presented. The approach is designed to cope with large FE-models in terms of degrees of freedom, a large number of uncertain input quantities and high variabilities in terms of coefficient of variation (e.g. ≥ 10%). Uncertainties with respect to dynamic loading and of the parameters describing the mechanical properties of the FE model are propagated by a stochastic approach: Inherent irreducible (aleatory) uncertainties as well as reducible (epistemic) uncertainties due to insufficient knowledge or modeling capabilities are translated into a probability distribution defining the input of the stochastic analysis. The reliability will be assessed in terms of the first excursion probability of critical responses. Several procedures to solve the first excursion problem for deterministic structural systems subjected to Gaussian excitation have been developed within the last decades [2, 12]. In a further step, the the domain of uncertain structural parameters which contributes most to the failure probability is identified. On this basis, the unconditional total failure probability is estimated by Importance sampling or by Line sampling within the space of uncertain parameters To demonstrate the applicability of the proposed approach for general FE-models, a realistic FE model for a twelve story building with more than 24,000 DOF’s and 200 uncertain quantities is analyzed.

2 Methods of Analysis 2.1 Representation of uncertain excitation Dynamic excitations acting on the structure are in many cases uncertain. However, although it might be impossible to describe these excitations in a deterministic sense, information on the expected range and its variability is usually available. Such information can be described by the mean value, the standard deviation of the fluctuation, and also the correlation in space and time. It is supposed herein that a Gaussian distributed stochastic process can be used as a suitable mathematical model to describe the uncertain dyanmic excitation.

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Any Gaussian distributed process is most conveniently described by the so called Karhunen-Lo`eve presentation (see e.g. [8, 6, 5, 13]) n

f(t; ξ ) = f(0) (t) + ∑ ξi f(i) (t).

(1)

i=1

In the above, all vectors on the right hand side do have deterministic properties and the independent random variables assume the following relations, E[ξi ] = 0 ,

E[ξi2 ] = 1 ,

E[ξi ξ j ] = 0

for i = j ,

(2)

where E[·] denotes the the mean or expectation. The representation (1) specifies uniquely the mean μk (t) and the variance σk2 (t) or standard deviation σk (t) for each degree of freedom k, and the correlation of the uncertain excitation:

μ f (t) = f(0) (t) n

(3)

σ 2fk (t) = ∑ [ fk (t)]2 i=1 n

(i)

(4)

E[ f j (t1 ) fk (t2 )] = ∑ f j (t1 ) fk (t2 ) (i)

(i)

(5)

i=0

However, the deterministic vector valued functions f(i) (t) are usually not quantified a priori. They need to be determined from the symmetric covariance matrix Cf (t j ,tk ). In practical applications, each of the deterministic Karhunen-Lo`eve excitation terms f(i) (t) can be described by a constant vector F(i) defined over the range of degrees of freedom and a scalar function h(i) (t) as function of time t, f(i) (t) = F(i) · h(i) (t)

(6)

A typical example for such a separation could be an earthquake excitation in a specified direction, f(t) = −MI · a(t) ,

(7)

in which M is the mass matrix and I is vector with 1’s for all degrees of freedom in the considered direction and zeros elsewhere. Hence the nodal forces f(t) are for any time t fully correlated, and only the correlations at different times t j = tk need to be described: n

f(tk ) = −MI · a(0) (tk ) + ∑ ξi a(i) (tk ) i=1

(8)

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2.2 Uncertain structural systems Linear structural systems are described by a constant symmetric mass matrix M, damping matrix D and stiffness matrix K. These matrices are realistically considered to be associated with uncertainties. In the proposed stochastic parametric approach [1], the uncertainties of these matrices are also conveniently modeled as functions of independent random variables

Θ = {Θ1 , Θ2 , . . . , ΘM } ,

(9)

such that for any fixed set Θ = θ , these structural matrices are uniquely specified. Hence, for any fixed set (vector) θ , the structural matrices M(θ ), D(θ ) and K(θ ) can be treated deterministically. It is common practice, to assume the random variables Θi to be standard normally distributed, i.e. with zero mean E[Θi ] = 0 and unit variance E[Θi2 ] = 1. This assumption, however, does not imply that the structural components follow a Gaussian distribution or that correlations among structural parts do not exist.

2.3 Stochastic conditional response Since the structural system is assumed to be linear, the law of superposition is valid. This law implies, that the response u(t; θ ) for fixed structural properties θ and dynamic excitation f(t) has, analogous to (1), also a Karhunen-Lo`eve representation, n

u(t; θ , ξ ) = u(θ )(0) (t) + ∑ ξi u(i) (t; θ ) ,

(10)

i=1

where each deterministic term u(i) (t; θ ), i = 0, 1, . . . , n, is the solution of a deterministic dynamic analysis, involving the constant symmetric mass matrix M(θ ), damping matrix D(θ ) and stiffness matrix K(θ ) in the equation of motion: M(θ )u¨ (i) (t, θ ) + D(θ )u˙ (i) (t, θ ) + K(θ )u(i) (t, θ ) = f(i) (t),

∀0 ≤ i ≤ n

(11)

Hence, to specify the variance of the displacement response, n + 1 deterministic analyses are required. Suppose, the critical responses of interest, e.g. stresses, strains, accelerations, displacements, etc., are comprised in the vector y(t, θ ). Each component yi (t, θ ) of the vector must fulfill certain conditions in order to be regarded as reliable. It will be assumed that these critical types of structural response can be represented by a ¨ θ ), linear combination of the displacement or acceleration response u(t, θ ) or u(t, y(t, θ ) = Q(θ )u(t, θ ) ,

(12)

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where Q is a constant matrix, independent of time t, and a function of the structural parameters which might depend on θ . Similarly, as the displacement response, the variability due to the random excitation can be cast into a Karhunen-Lo`eve representation: n

y(t; θ , ξ ) = y(0) (t; θ ) + ∑ ξi y(i) (t; θ )

(13)

i=1

For the efficiency of the presented approach, it should be stressed that a single modal FE analysis is sufficient to compute the Gaussian distributed response. The associated Karhunen-Lo`eve representation (13) provides the basis for the reliability assessment conditional on the structural parameters θ .

2.4 Conditional reliability 2.4.1 First excursion probability In this section, the conditional first excursion problem p f (ξ , θ ) is discussed for random Gaussian excitation and for the case where a random realization of Θ assumes a certain set of deterministic parameters θ . The first excursion probability is then defined as the probability that the critical response exceeds at least once within the considered time period [0, T ] the threshold bi of the i-th component of the critical response y(t; θ ) [7]. The threshold bi might consist of a lower limit bi and an upper limit b¯ i for a single critical response. pi, f (θ ) = P{max[yi (t; θ ) : 0 ≤ t ≤ T ] ≥ b¯ i ∪ min[yi (t; θ ) : 0 ≤ t ≤ T ] ≤ bi } pi, f (θ ) =

gi (θ ,ξ )≤0

(14)

q(ξ ) d ξ

gi (θ , ξ ) = min[b¯ i − max[yi (t; θ , ξ ) : 0 ≤ t ≤ T ], min[yi (t; θ , ξ ) : 0 ≤ t ≤ T ] − bi ]

(15)

This conditional first excursion probability corresponds to the case of stochastic excitation, specified by the random vector ξ and its probability density function q(ξ ), subjected to a deterministic structural system. In the next section, the integration p f = p f (θ )q(θ ) d θ over the whole domain of the probability density function q(θ ) will be discussed in detail, leading to the unconditional first excursion problem p f . To simplify the notations and the reliability problem, each critical response will be considered separately. Hence, the subscript “i” is dropped in the following developments. For this case, where the critical response y(t) can be represented by a KarhunenLo`eve representation, as given in (10), several efficient numerical procedures have

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been developed recently to calculate the first excursion probability (e.g. [2, 12]) which are not described here any further.

2.5 Design point for stochastic structural systems In this section, the domain which contributes most to the unconditional total failure probability p f will be determined. Theoretically, the failure probability p f |θ , conditioned on specific realizations θ , need to be integrated over the whole domain of pΘ (θ ):

pf =

p f (θ )pΘ (θ ) d θ

(16)

Since the number of uncertain structural parameters, characterized by the random quantities {Θ1 , Θ2 , . . . , ΘM }, is in general large, numerical integration is not feasible. Basically, the above integral can be estimated by Monte Carlo sampling pˆ f =

1 N

N

∑ p f (θ ( j) ) ,

(17)

j=1

where N independent realizations θ ( j) are drawn from the the distribution pΘ (θ ) and the associated conditional failure probability p f (θ ( j) ) is determined as described in the previous section. Such an approach is only efficient in case the variability of p f (θ ( j) ) is small, i.e. might be within one order of magnitude. However, in case the structural variability is high, the conditional first excursion probabilities will vary over many orders of magnitude and direct Monte Carlo approaches will be very inefficient. Considering equation (16), it is not difficult to recognize that sampling in the neighborhood of the structural design point θ ∗ , characterized by p f (θ ∗ )pΘ (θ ∗ ) ≥ p f (θ )pΘ (θ ),

(18)

contributes most to the total (unconditional) failure probability. In order to identify this domain, composed of the standard normal variables {ξ , θ }, the point {ξ ∗ , θ ∗ } with the minimal distance β to the failure surface is determined at the most critical time t1∗ . Since the uncertainty in the excitation and of the structural parameters, specified by ξ ∗ and θ ∗ , respectively, are independent and therefore orthogonal in the standard normal space, this distance is specified according to Pythagoras by

β 2 (ξ , θ ) = ||ξ ||2 + ||θ ||2 .

(19)

Figure 1 shows a sketch of the failure domain for this case, where β (ξ , θ ) can only be determined in the complete space (ξ , θ ). The threshold b is reached for

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Fig. 1: Structural Design Point θ ∗

||ξ ||2 =

b2

(20)

γ 2 (t ∗ ; θ ) n

γ 2 (t ∗ , θ ) ≥ γ 2 (t, θ ) = ∑ [y(i) (t)]2 ,

t ∈ [0, T ]

(21)

b = min[b¯ − y (t ; θ ), y(0) (t ∗ ; θ ) − b]

(22)

i=1 (0) ∗

The failure point (θ ∗ , ξ ∗ ), with the highest probability density and defined as the nearest failure point to the origin in standard normal space, is derived by imposing the necessary condition

∂ β 2 (ξ , θ ) = 0, ∂ θk

k = 1, 2, . . . , K,

(23)

∂ γ (t ∗ , θ ∗ ) . ∂ θk

(24)

which leads to the solution

θk∗ =

b2

γ 3 (t ∗ , θ ∗ )

·

An accurate solution of the above relation can only be obtained in an iterative manner, where convergence is reached for s > S with ε (S) smaller than the tolerance, ∗(s+1)

θk

∗(s)

= (1 − w)θk

+w

b2

γ 3 (t ∗ , θ ∗(s) )

ε (s+1) = ||θ ∗(s+1) − θ ∗(s) ||/||θ ∗(s+1) ||,

·

∂ γ (t ∗ , θ ∗(s) ) ∂ θk

(25) (26)

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where w = 0.5 leads to a stable and fast convergence. However, there is not much gain in evaluating θ ∗ very accurately. A first or second step (s = 0, 1) is usually sufficiently accurate for the integration procedures as developed in the next section. The above iterative evaluation of the structural design point θ ∗ requires a gradient computation which might hamper the efficiency of the approach in case the number of uncertain structural parameters is large. For such cases, a gradient estimation procedure [11, 10] might be used to improve the computational efficiency.

2.6 First excursion probability for stochastic systems In this section, the procedure to estimate the total unconditional first excursion probability by integrating over the complete space of uncertain structural parameters is shown (see eqn. (16)). The proposed approach is to limit the application of LS to the subspace of the structural parameters θ . This in fact leads to robust results, even for quite large uncertainties of the structural parameters, since the important directions α [i] (θ ) are then always computed in the optimal directions. The procedure is outlined as follows: 1. Determine the design point θ ∗ with acceptable accuracy and compute

βS = ||θ ∗ || θ∗ αS = , βS

(27) (28)

where the index “S” denotes the subspace of the structural uncertainties. 2. Generate samples {θ ⊥( j) }Nj=1 using direct MCS in the subspace of θ , which are perpendicular to the vector θ ∗ . 3. Compute for each parallel line for, say, five discrete points θ ( j) (ck ), k = 1, . . . , 5, as shown in Figure 2

θ ( j) (ck ) = θ ⊥( j) + ck · α S , ck = βS + (k − 3)Δ , k = 1, 2, . . . , 5 Δ ≈ 0.6

(29) (30) (31)

( j)

the associated conditional failure probability p f (ck ). ( j)

4. Estimate the conditional failure probability p f along each line, 1 ( j) pf = √ 2π

∞ −∞

e−c

2 /2

( j)

p f (c) dc , ( j)

(32)

with quadratic interpolation of the function ln(p f (c)) by using the available discrete values. This estimation will be discussed subsequently in more detail.

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Fig. 2: Line Sampling in the uncertain structural paramameter space 5. Estimate the mean and variance of the failure probability p f (t) applying basic statistics. (i)

It should be noted that p f (c) > 0 for −∞ < c < ∞, since the position along each line specifies uniquely the properties of a structure subjected to random excitation. (i) (i) Since p f (c) is always a positive quantity, it is proposed to represent p f (c) as the exponential of a linear or quadratic polynomial, (i)

p f (c) = exp[a0 + a1 c + a2 c2 /2] ,

(33)

where the coefficients are obtained by solving a least square problem (i)

a0 + a1 cl + a2 c2l /2 = ln[p f (cl )] ,

l = 1, 2, . . . , 5

(34)

The advantage of this approximation is that the infinite integral in eq. (32) can be represented in closed form for a2 < 1 a21 1 (i) . (35) pf = √ exp a0 + 2(1 − a2 ) 1 − a2 (i)

A linear approximation of ln[p f (c)] in the least square solution should be applied (a2 = 0), whenever a2 > 0.4, which might be due to random fluctuation of the esti(i) mates p f (cl ). Experience showed that the results using either a linear or quadratic

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approximation are quite similar. However, a linear approximation is less sensitive (i) to random fluctuations of the estimate p f (cl ). Hence the integration over all points along the line is efficiently approximated, requiring only three to five Finite Element analyses per line.

3 Numerical example 3.1 General remarks The method, as developed in section 2, is now exemplified within this section. Since reliability evaluations in structural dynamics are computationally very demanding, the computational efficiency of the proposed method is naturally in the focus of interest. However, to be of practical value, the efficiency of procedures should be demonstrated by applying them to models which reflect the complexity of relevant engineering structures. By choosing a realistic FE model, modeling a 12-story building made of reinforced concrete, the requirement of practical applicability is shown.

3.2 Structural system 3.2.1 Geometry A twelve story building with an additional cellar floor made of reinforced concrete is considered. The FE-model consists of 4046 nodes and 5972 elements using shell and 3-D beam elements for modeling the girders, resulting in 24,276 degrees of freedom. Figure 3 provides a view of the building showing the displacement field according to the first three mode shapes. The building is axis-symmetric and consists of 13 floors of 24.0×24.0 m. The foundation plate is 0.5 m thick and rests on soil modeled by elastic springs. Figure 4 shows a plan view for all floors, with the exception that the cellar floor is surrounded by cellar walls of 0.3 m thickness. The height of each story is 4.0 m. The weight of the structure is carried by four groups of concrete walls forming a cross. The four groups of supporting concrete walls are connected by four girders of type g−1 the height of which is 1.2 m and a width identical to the walls.

3.2.2 Uncertain structural properties Concrete and reinforced concrete are widely used in building constructions. The stiffness of reinforced concrete construction parts depends on many factors which

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Fig. 3: First three mode shapes of twelve story building

Fig. 4: Floor plan of twelve story building supported by croncrete walls (w) and girders of type g−1 and type g−2. Critical strains are considered at positions p−1 and p−2 are still difficult to predict and to control. Due to inhomogeneities and possible cracking, the stiffness under dynamic loading reveals uncertainties. Moreover, it might depend also on the loading history (see e.g. [3]). At the present state-of-theart, quantitative models for reinforced concrete structures, which are both feasible and realistic (including bond-slip, cracking, etc.) are not used yet for ordinary structural analysis as they are still topics of active research. A practical way to cover

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all these uncertainties is to take a single quantity, such as the Young’s modulus, to represent the uncertainties in the stiffness, with the understanding that this quantity should cover the uncertainty of the stiffness and not only of the physical elastic constant. All structural uncertainties are assumed to be represented as a function of 244 independent standard normal variables. The mean value for the Young’s modulus for the foundation plate and cellar walls are assumed to be 2.8 · 1010 N/m2 and its variability is modeled by a coefficient of variation of 0.215 and correlation coefficients in the order of 0.86. The Young’s moduli of the remaining primary reinforced concrete parts are assumed somewhat higher to be 3.2 · 1010 N/m2 with a coefficient of variation of 0.144. It implies also a large correlation of the stiffness, which is reflected by a correlation coefficient in the order of 0.7. The live loads (masses) are represented by a function of five random variables per floor, which are assumed to be independent between the floors and correlated within the floors. The mean value is assumed to be 100 kg/m2 and the coefficient of variation of 0.224 with correlation coefficients of 0.8.

3.3 Dynamic excitation In this numerical example, the dynamic excitation is due to earthquake ground motion in terms of ground accelerations. Future ground motions are highly uncertain, regarding its amplitudes, frequency content and durations, respectively. In this work, an approach based on filtered white noise is used. To cover the unknown frequency content and random amplitudes of the acceleration, the model proposed in [4] is applied, where the acceleration is represented as the output of the response of a linear filter excited by white noise. For illustration, the variance of the horizontal acceleration and some KarhunenLo`eve functions a( j) (t) are shown in Figure 5.

Fig. 5: Variance of horizontal acceleration (left) and Karhunen-Lo`eve functions a( j) (t), j = 10, 30, 60, 80 of the horizontal acceleration (right) over time

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3.4 Critical response Under dynamic earthquake loading, two reinforced structural components are likely to exceed the critical strain. The strain in vertical direction of the walls in the basement, forming a cross, might be critical. Since the structure is symmetric, and the horizontal accelerations are in both directions independent and identically distributed, it suffices to consider a single position, indicated by p−1 in figure 4. Girders form the remaining critical parts, denoted as g−1 in figure 4, which connect separated walls. The critical part is the curvature of the girder at the connection to the walls. The position is indicated by p−2 in figure 4. Accepting as limit state function a vertical strain of 0.0016 for the walls without having to expect serious damage and a maximum acceptable curvature of 0.004 in the girders g−1, the curvature in the girder are more likely to be exceeded, where floor five showed the largest standard deviation of the curvature. Hence, the reliability analysis will focus on the reliability of the girders for not exceeding the limit state function the curvature of 0.004. Taking into account the curvature due to dead loads of −0.00041, the limit state function has the lower and upper bounds are b = −0.004 + 000041 = −0.003959 and b¯ = 0.004 + 000041 = 0.004041, respectively.

3.5 Reliability of critical component 3.5.1 Estimation by Direct Monte Carlo Simulation Before evaluating the first excursion probability, the Karhunen-Lo`eve terms of the critical response, i.e. the curvature at girder g − 1, are determined. These terms y( j) (t, θ ) depend only on the set of uncertain structural parameters θ and their evaluation requires a single FE analysis for each distinct set θ ( j) . Hence, the investigation of the variability of the response due to uncertain structural properties is the computationally expensive part. The associated first excursion problem is determined by line sampling. The failure probability assumes for the nominal system (θ = 0) the value pˆ f = 4.67×10−7 and a standard deviation of this estimate of σ pˆ f = 1.46×10−7 . However, it is well known that the reliability is quite sensitive with respect to the variations of structural properties. This sensitivity is shown by the results of Direct Monte Carlo simulation (see eqn. (17)) by randomly sampling over the uncertain structural parameters {θ ( j) }1600 j=1 . −4 Direct Monte Carlo sampling leads to the estimate pˆ f = 2.32 × 10 and the standard deviation of σ p f = 3.43 × 10−5 . Figure 6 shows the histogram of all independent estimates. It is observed, that the estimate for the failure probability varies over many orders of magnitude. Hence, the procedure of Direct Monte Carlo Simulation can not be used in an efficient manner.

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Fig. 6: 1600 estimates by direct Monte Carlo of the stochastic structure

3.5.2 Critical domain of uncertain structural parameters The domain of the structural parameters which contributes most to the total failure probability has been derived in section 2.6. Eqn. (24) provides a quantitative guidance to determine this domain. Table 1: The first three most important components (k) of the structural design point θ ∗ computed by three iterations s = 1, 2, 3 s 0 1 2 3 γ ∗(s) 0.00070 0.00190 0.00110 0.00111 k 26 244 114

∗(0)

θk

0 0 0

∗(1)

θk −1.92 −1.52 −0.69

∗(2)

θk −2.07 −1.18 −0.76

∗(3)

θk −2.05 −1.17 −0.75

Its solution, shown in Table 1, requires the gradient of the standard deviation due to the dynamic excitation with respect to the uncertain structural parameters. To avoid direct differentiation for all 244 uncertain structural parameters, the gradient estimation procedure is employed [11, 10] to compute these components which significantly influence the standard deviation of the critical response. It was observed that the variability of γ (t ∗ , θ ) is governed by only three random variables, namely the numbers 26, 244 and 114. Number 26 controls the stiffness of the girder of all floors, number 244 controls the log-normally distributed damping ratio, and the number 114 the local stiffness deviation at the considered girder at floor 5. These three random variables cause 97% of the total variability of γ (t ∗ , θ ).

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3.5.3 Line sampling over uncertain structural parameter space The procedure outlined in section 2.7 is applied to compute efficiently the total first excursion probability of the uncertain structural system.

Fig. 7: Left: Five discrete conditional failure probabilities along twenty lines in the direction of the structural design point using βS = 2.48 and Δ = 0.6. Right: Twenty independent estimates of the total first excursion probability ( j)

Figure 7 shows at the left five discrete values p f (ck ) of the conditional failure probability at five points along 20 random lines by using βS = 2.48 and Δ = 0.6. The significant increase of the conditional failure probabilities in the direction of the design point demonstrates the high sensitivity of the failure probability with respect to the particular uncertain structural parameters. Figure 7 at the right shows twenty independent estimates of the total first excursion probability. To obtain these results 100 FE analyses (20×5) have been carried out. The estimated mean is pˆ f = 0.00046 with a standard deviation of σ pˆ f = 0.000040. When compared with results of Direct Monte Carlo sampling in Figure 6 in the uncertain parameter space, one observes that these results are approximately by a factor of two larger. LS, however, explores the important domain systematically and is therefore more reliable. This can be also seen by the substantial smaller coefficient of variation of approximately 8% compared with 16% and and a 16 times larger number of FE analyses when using Direct MCS.

4 Conclusions The developments as shown here allow to draw the following conclusions: 1. The reliability evaluation of linear structural systems (general FE models) with uncertain structural parameters subjected to general Gaussian dynamic excitation is feasible. To the authors knowledge, the feasibility of a reliability evaluation in

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4. 5.

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dynamics for a large FE model and a large number of structural uncertainties is demonstrated the first time in a numerical example. A novel procedure to identify the critical uncertain parameters has been introduced. Efficiency is gained by performing modal analysis, impulse response functions combined with Fast Fourier Transforms (FFT) and a new Line Sampling approach which focuses on the important failure domain within the uncertain parameter space. The presented approach is accurate and robust, also for cases where the uncertainties of the structural parameters are large. The reliability of linear systems is quite sensitive to structural uncertainties — as demonstrated in the numerical example — and therefore must not be ignored.

Acknowledgements This work has been supported by the Austrian Science Foundation (FWF) under contract number P19781–N13 (Simulation Strategies for FE Systems under Uncertainties), which is gratefully acknowledged by the authors.

References 1. Adomian G (1983) Stochastic Systems. Academic Press, New York 2. Au S, Beck J (2001) First excursion probabilities for linear systems by very efficient importance sampling. Probabilistic Engineering Mechanics 16:193–207 3. Chryssanthopoulos M, Dymiotis C, Kappos A (2000) Probabilistic evaluation of behaviour factors in ec8-designed r/c frames. Engineering Structures 22:1028–1041 4. Clough R, Penzien J (1975) Dynamics of structures. McGraw-Hill, Intern. Student ed., Auckland 5. Ghanem R, Spanos P (1991) Stochastic Finite Elements: A Spectral Approach. SpringerVerlag ¨ 6. Karhunen K (1947) Uber lineare Methoden in der Wahrscheinlichkeitsrechnung. Amer Acad Sci, Fennicade, Ser A 37:3–79 7. Lin Y (1967) Probabilistic Theory of Structural Dynamics. McGraw-Hill Inc., McGraw-Hill Company, New York 8. Lo`eve M (1948) Fonctions aleatoires du second ordre, supplemeent to P. Levy. In: Processus Stochastic et Mouvement Brownien, Gauthier Villars, Paris 9. Oberkampf W, DeLand S, Rutherford B, Diegert K, Alvin K (2000) Estimation of total uncertainty in computational simulation. Tech. rep., Sandia National Laboratories, Albuquerque, NM SAND2000-0824, USA 10. Pellissetti MF, Pradlwarter HJ, Schu¨eller GI (2007) Relative importance of uncertain structural parameters, part II: Applications. Computational Mechanics 40(4):637–649, 10.1007/s00466-006-0128-8 11. Pradlwarter HJ (2007) Relative importance of uncertain structural parameters, part I: Algorithm. Computational Mechanics 40(4):627–635, 10.1007/s00466-006-0127-9 12. Pradlwarter HJ, Schu¨eller GI (2004) Excursion probability of non-linear systems. International Journal of Non-Linear Mechanics 39(9):1447–1452, 10.1016/j.ijnonlinmec.2004.02.006 13. Schenk C, Schu¨eller G (2005) Uncertainty Assessment of Large Finite Elements Systems. Springer-Verlag, Berlin/Heidelberg/New York

On semi-statistical method of numerical solution of integral equations and its applications D.G. Arsenjev, V.M. Ivanov, and N.A. Berkovskiy

Abstract This paper is devoted to the semi-statistical method of numerical solution of integral equations. The main advantage of this method is the possibility of optimizing the nodes of the integration domain and control of the computation accuracy by means of the sample variance. In addition to this, in order to improve the accuracy one obtains an average solution by statistically independent estimations at a small number of the integration nodes. In this paper the application of semi-statistical method to the problem of blade cascade flow is considered. Rather accurate results have been obtained for the turbine blades grills. These results are compared with the solutions from other computational methods.

1 Introduction The aim is analysis of the effectiveness of application of the semi-statistical (adaptive-stochastic) method to the problems of computational and engineering practice. The main advantage of this method is the possibility of optimizing the nodes of the integration domain and control of the computation accuracy by means of the sample variance. In addition to this, in order to improve the accuracy one obtains an average solution by statistically independent estimations at a small number of the in-

D.G. Arsenjev SPbSTU, Polytechnicheskaya, 29, St. Petersburg, Russia, 195220 e-mail: [email protected] V.M. Ivanov SPbSTU, Polytechnicheskaya, 29, St. Petersburg, Russia, 195220 e-mail: [email protected] N.A. Berkovskiy SPbSTU, Polytechnicheskaya, 29, St. Petersburg, Russia, 195220 e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 27, © Springer Science+Business Media B.V. 2011

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tegration nodes. A less attractive feature of this method is a low rate of convergence which is typical for all statistic methods. The substantiation for the present research was a quite successful application of semi-statistical method to the test problems [1, 2, 3]. The problem of plane lattice cascade flow with ideal incompressible fluid was chosen for simulation. With the help of semi-statistical method quite accurate results were obtained for the grills whose parameters were taken from engineering practice. These results were compared with the solutions from other computational methods. Attempts to accelerate the convergence rate lead to modernization of the method (removing overshoots in the average sum). As a result, for all considered problems we obtained the solutions with the satisfactory accuracy, the adaptive algorithm of “placed” the nodes in the integration domain in accordance with the theoretical considerations. However in some cases the solution by the semi-statistical method turned out to be more expensive than that from the deterministic methods. It was caused by imperfection of code implementation and the necessity of further search for the new ways of acceleration the convergence of the semi-statistical method and, in particular, the optimization mechanism.

2 Short scheme of semi-statistical method The object of the semi-statistical method is solving the following integral equations

ϕ (x) − λ

K(x, y)ϕ (y)dy = f (x),

(1)

S

where S is a smooth (m − 1)-dimensional surface in Rm ; x ∈ S, y ∈ S, λ ∈ R, K denotes kernel of the equation, f is a given function and ϕ is sought. This algorithm is described in detail in [1]. The scheme of its application in general case is as follows. a) Using a random number generator one produces N independent points x1 , x2 , . . . , xN (vectors) on surface S with an arbitrary probability density p(x) (random integration grid). b) These points are substituted consequently in eq. (1), then we obtain N equations of the sort ϕ (xi ) − λ K(xi , y)ϕ (y)dy = f (xi ) (i = 1, N) (2) S

c) Integrals in (2) are replaced by the sums with the help of the Monte-Carlo method [1, 2]. The result is the system of linear algebraic equations

On semi-statistical method of numerical solution of integral equations

ϕi −

λ N K(xi , x j ) ϕ j = f (xi ) N −1 ∑ p(x j ) j=1

381

(3)

j=i

Here ϕi (i = 1, N) denotes the vector of unknown variables of system (3). Having solved eq. (3), one obtains ϕi which is understood as the approximated value ϕ (xi ) of the solution of integral equation (1). Approximated value ϕ (x)∀x ∈ S is defined by “retracing” by means of the following algorithm:

ϕ (x) ≈ f (x) +

λ N K(x, xi ) ∑ p(xi ) ϕi N i=1

(4)

The accuracy of approximation of the integral in (2) increases with growth of N. Hence, one can expect that the error of approximation of ϕi and ϕ (x) from eqs. (3) and (4) can be made arbitrary small with increase in N. As the number of points is sometimes not sufficient to achieve the predefined accuracy (this number can not be made arbitrary great since it is not possible to solve to large system of equations), it is recommended to carry out calculation m times with N points and average the results. This technique yields nearly the same result if we would throw N × m points, because the random points are statistically independent in each iteration. d) Following [1] one obtains the following estimation for the optimal density of integration nodes in terms of the approximated solution ϕ (x) (K(x j , y)ϕ (y)) 2 p(y) j=1 N

∑

popt (y) = C(N − 1) . N K(x j , y)ϕ (y) N ∑ K(x j , xi )ϕ (x j ) ∑ p(x j ) i=1 j=1

(5)

i= j

Having generated the points with the density popt (y) constructed by eq. (5) in terms of the approximated values ϕi one obtains a more accurate solution of eq. (1). Using eq. (5) we calculate the value of optimal density more accurately. The process can be repeated until the density becomes constant.

3 Statement of the problem of blade cascade flow We consider a plane grill of profiles with the distance t Fig. (1) in the potential flow of ideal fluid. The input and the output angle of the flow are β1 and β2 , respectively. The absolute value of normed speed of the flow on the profiles is sought.

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D.G. Arsenjev, V.M. Ivanov, and N.A. Berkovskiy

This problem reduces to solution of the following integral equation, cf. [5], 1 · w (l) dl = b(s), (6) w(s) + K (s, l) − L C

where w(s) denotes the normed speed of flow and 1 K (s, l) = · t

∂x ∂s

· sin 2tπ · (y (s) − y (l)) − ∂∂ ys sinh 2tπ · (x (s) − x (l))

; cos 2tπ · (y (s) − y (l)) − cosh 2tπ · (x (s) − x (l))

b(s) = −2 ·

t ∂x ∂y − · cot(β1 ) + cot(β2 ) + . ∂s ∂s L

Fig. 1: Profile grill. Here w is the flow velocity, t is the grill pitch, β1 is the input angle of the flow, β2 is the output angle of the flow, C is contour of the blade profile Here s and l denote the arch length at different points of the profile contour, the arches being counted from the middle of the exit edge in the positive direction (counterclockwise); x(l), y(l) are the coordinates of the profile’s point with the arch length l; C is a contour of the blade profile; L is the length of the contour of the blade profile. The direction of the unit tangent vector {∂ x/∂ s, ∂ y/∂ s} corresponds to counterclockwise tracking of the contour. The front of the grill lattice has direction of the ordinate axis rather than the abscise axis, as in [5]. The speed in [5] is corresponds to unit output fluid flow rate whereas in the present study the flow speed on the grill output is equal to unity. This is achieved by multiplication of the speed from eq. (1) by sin(β2 ). This norm is used in the code by Ural Polytechnic Institute (UPI), the computations being carried out by method of rectangles with the optimal setting of the integration nodes [4, 6]. These solutions are taken for the benchmark study here.

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4 Scheme of application of semi-statistical method to the problem of blade cascade flow 4.1 Main formulas In this problem contour C plays the part of surface S and we look for the solution w(s) of integral equation (6). Let Wk (s) denote the solution of the integral equation (1) with N generated points at k-th iteration then the averaged solution after k iterations wk (s) 1 k wk (s) = · ∑ Wk (s) (7) N m=1 The standard deviation of k-th iteration is as follows: 1 k δk (s) = 2 · ∑ Dl (s), k l=1

(8)

where Dl denotes the sample dispersion of l-th iteration N 1 Dl (s) = ∑ N (N − 1) m=1

2 K (s, lm ) − L1 ·Wk (lm ) + b (s) −Wk (s) . p (lm )

Here l1 , l2 , . . . , lN are random points on the segment [0, 2π ] at k-th iteration, N is the number of points at each iteration (is the same for all iterations in this particular simulation) and s is the point of observation. The values of w(lm ) are the result of the approximate solution of integral equation (1) at the k-th interation. The simulation experience has shown that the deviation (calculation error) does not exceed triple standard deviation and, as a rule, is under the standard confidential (95%) interval.

Fig. 2: The equidistant points of simulation, 50 points on both upper and lower parts of the contour

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Analytic definition of the blade contour Integral equation (6) on smooth contour C is of Fredholm type and has a unique solution [5]. For a twice differentiated contour the kernel of equation (6) can be considered as being continuous since it has a removable singularity at s = l, cf. [5]. However the contour under consideration is described by a spline curve whose first derivative has jumps at a finite set of points. This obstacle leads to discontinuities in the kernel at the points of discontinuities of the first derivative however this does not affect the computation accuracy. Additionally, the spline can always be approximated by a finite Fourier series and the problem can be solved on the infinitely differentiated contour, as shown in [4]. Both approaches were tested, and the difference in the solutions obtained by spline and finite Fourier series was not considerable. The semi-statistical method was applied to eq. (1) in terms of the general scheme [1] without any additional preparation of regularization type. Values of the speed were calculated at 100 points the blade contour, 50 equidistant points being taken on both upper and lower parts of the contour, see 2. These values were multiplied by sin(β2 ), the result was compared to the solution obtained by the method of rectangles on the same contour. Both results were compared to the solution given by the UPI code.

4.2 Computation algorithm and optimization The simulation was performed iteratively. At each iteration a number of random points on the segment [0, 2π ] were generated with the density calculated at the previous iterations (adaptive algorithm). At the first iterations the points were generated with the uniform density on the segment [0, 2π ] which corresponds to a nearly uniform distribution of the points on the contour of the blade. The results were determined more accurately by iterations and the approximate solution after iteration number i was considered as the arithmetic average of the solutions obtained at the previous iterations. Based upon this approximate solution the optimal density was calculated by the method described in [1]. The present algorithm is more efficient than that in [1] since it utilises a more accurate approximation of the exact solution. The simulation experience has shown that in the case of extended contours very strong overshoots are possible at some iterations and these overshoots can not be smoothed even for a great number of iterations. However, if the solution is very inaccurate then the sample dispersions are also considerable at the control points, the latter being calculated during the computation. We can introduce a constraint which finds the terms with a great dispersion. The present code ensures that averaging takes place only at the iterations with the relative dispersion calculated in terms of the sample iteration is under 100%. The solutions obtained at other iterations (usually not more than one percent from the total number except for the points near the edge), are considered as overshoots and excluded from averaging sum. In the case extended

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blade this improvement leads to the undoubted advantage in the quality of computations. Using the semi-statistical method we calculated the speed at 150 points distributed over the blade contour with an equal increment in parameter u (i.e. with nearly equal increment in the arch length), and the values of speed in the checkpoints (not uniformly distributed over the contour) were calculated with the help of interpolation. The sample dispersion is used as an index of accuracy of the approximation. The computing costs were proved to need for calculating the values of kernel in the generated points, and for this reason the aspect of decreasing number of generated points and ensuring the accuracy of computation is of crucial importance. It can be achieved by optimization of the integration grid.

5 Results of simulations Let us first introduce some denotations. The quantity m in Figures 3–5 stands for the number of observation points, wm is the speed at the point number m calculated by the semi-statistical method, w1m is the speed at point number m calculated by means of the method of rectangles, w2m is the speed at point number m calculated by means of the UPI code and |wm − w1m | denotes the absolute deviation (calculation error) of calculation of the speed at point number m by means of semi-statistical method in comparison to the method of rectangles. Under the words “speed calculated by means of semi-statistical method (4 ∗ 400)” we understand 4 iterations of semi-statistical method with 400 points generated in every iteration. Figures 3–5 display the results of simulations. From given above examples (3) it is evident, that semi-statistical method commutated the speed with a good precision in all the points of contour, except for some points in the edgings, which are not important for practical issues.

6 Analysis of efficiency of the density adaptation It is interesting to observe as to how the adaptive algorithm works while choosing the optimal density. It appears that the points become denser on the edges and back, that is exactly those same places of profile, where the simulation accuracy is very low during the first iterations. It can be seen from 4. Figure 5 shows the results of simulations obtained after five iterations by using the adaptive algorithm for choice of the optimal density and the results obtained after five iterations with the uniform distribution of generated points. The results of adaptive algorithm are seen to be more accurate for the same number of generated points.

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Fig. 3: Results of simulations: a) Speed calculated by the method of rectangles and the UPI code; b) Speed calculated by the semi-statistical method (150 ∗ 400) and the method of rectangles; c) The absolute discrepancy of the results of the semistatistical method (150 ∗ 400) and the method of rectangles

It is clear from 5 that when one uses the adaptive algorithm the standard deviation decreases faster than that with uniform distribution. It allows one to reduce the number of points necessary to achieve the predefined accuracy.

7 Conclusion The following conclusions can be drawn: a) Rather accurate results can be obtained by solving the problem of potential flow of profile cascade when the semi-statistical method is used. b) In accordance with the theoretical computations the adaptive algorithm works for optimization of nodes in the integration domain. It improves the convergence by reducing the sample dispersion. c) The convergence rate is rather low for extended domains. There still remains an important problem of the convergence acceleration which is needed to make the semi-statistical method successful for extended domains and competitive to deterministic methods with respect to the simulation rate. Clearly, one of the ways of solving the problem is improvement of the adaptive algorithm of optimization.

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Fig. 4: a) Histogram of the optimal density after two iterations; b) Speed calculated by means of the semi-statistical method (two iterations with 400 points each) and the method of rectangles. c) The symbol “×” marks the borders of intervals from the histogram in 3a), the numbers 1, 2, . . . 10 denoting numbers of these intervals. The bold points are the checkpoints (the numbers 1, 11, 21, . . . , 91)

Fig. 5: The results of simulations: a) the speed calculated by the semi-statistical method (5 ∗ 400) with the adaptive algorithm and the method of rectangles; b) the speed calculated by semi-statistical method (5∗400) without adaptive algorithm and the method of rectangles

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References 1. Arsenjev D.G., Ivanov V.M., Kul’chitsky O.Y. Adaptive methods of computing mathematics and mechanics. Stochastic variant. World Scientific Publishing Co., Singapore (1999). 2. Arsenjev D.G., Ivanov V.M., Korenevsky M.L. Semi-statistical and projection-statistical methods of numerical solving integral equations. World Scientific and Engineering Academy and Society (WSEAS) Transactions on Circuits and Systems, Issue 9, Volume 3: 1745–1749 (2004). 3. Arsenjev D.G., Ivanov V.M., Berkovskiy N.A. Application of semi-statistical method to inside Dirichlet problem in three-dimensional space (in Russian). Nauchno-Technicheskie Vedomosti St. Petersburg State Polytechnical University, No. 4(38), pp. 52–59 (2004). 4. Isakov S.N., Tugushev N.U., Pirogova I.N. Report on scientific research. Development of computational software for calculating field of velocities and profile losses in grills in compressors and turbine blades (in Russian). Ural Polytechnical Institute, Svedrdlovsk, 1984. 5. Jhukovskiy M.I. Aerodynamic computing of flow in axial turbo-machines. Mashgiz, Leningrad (1967). 6. Vochmyanin S.M., Roost E.G., Bogov I.A. Computing cooling systems for gas turbine blades. Bundled software GOLD. International Academy of Sciences of Higher School, St. Petersburg Division (1997).

An efficient model of drill-string dynamics with localised non-linearities T. Butlin and R.S. Langley

Abstract High amplitude vibration of drill-strings can lead to damage or loss of down-hole equipment costing the industry hundreds of millions of dollars per year. Theoretical models of drill-string dynamics are often limited to lumped parameter models of subsections of the drilling assembly that are difficult to correlate with real systems, or high resolution finite elements models that are more realistic but computationally demanding and do not provide clear insight into the fundamental mechanisms at play. In addition, drill-strings are subject to many uncertainties: to model these using Monte-Carlo simulations can be prohibitively slow. This paper presents a digital filter implementation of the transfer matrix method that allows time-domain coupling of the full length of a periodic drill-string to other systems or arbitrary, localised non-linearities. In this way the dynamics of the complete drillstring are included and the computational effort is focussed on those parts of the system that are harder to model while retaining the efficiency of the linear model where it applies. The efficiency makes the use of Monte-Carlo simulations to explore uncertainties a feasible approach.

1 Introduction The oil-well drilling process is in principle straightforward: a motor on the surface drives a steel pipe, providing a cutting torque for the bit at the far end. But as oil fields become more remote the geometry and operating environment become increasingly extreme leading to a wide range of technical challenges, including path control, pressure balancing, communications with downhole equipment and vibration problems.

T. Butlin and R.S. Langley Cambridge University Engineering Department, Trumpington Street, Cambridge, CB2 1PZ, England, e-mail: [email protected], [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 28, © Springer Science+Business Media B.V. 2011

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High amplitude vibration costs the industry hundreds of millions of dollars each year. Torsional vibration in particular can cause fatigue of the drill-string, unthreading of a section of the pipe under extreme amplitudes, or damage to the drill bit if it reverses direction. Theoretical modelling of drill-string vibration presents significant challenges. Ideally a given model should realistically capture all the known physical effects that influence the system, be computationally efficient, provide clear insight into observed phenomena and account for the effects of uncertainties. Clearly these are in conflict: a high degree of realism is often at the expense of efficiency and clarity of insight. This conflict is evident within the literature. Lumped parameter models are difficult to correlate with real systems but are computationally efficient (e.g. Aarrestad and Kyllingstad [1], Leine et al. [2] and Richard et al. [3, 4]). Detailed finite element models of the bottom-hole assembly or other subsection of the drill-string may be more realistic but are computationally demanding and do not necessarily provide clear insight into the fundamental mechanisms at play (e.g. Khulief et al. [5] and Sampaio et al. [6]). A further difficulty associated with models of subsections such as the bottom-hole assembly, rather than the complete drill-string, is the modelling of the boundary conditions where it joins to the rest of the drill-string. Several authors consider the transmission-line analogy for torsional waves, for example Halsey et al. [7]. The model captures the dynamics of the full length of the drill-string in an efficient and clear manner but does not easily allow for the possibility of internal reflections for varying cross-sections and does not consider non-linear interactions. Tucker and Wang [8, 9] extend the principle to a control strategy based on torsional wave rectification for a drill-string of arbitrary geometry. This is applied to a relatively simple example but it is not clear how the method might efficiently generalise to more complicated cases. In addition, very few authors tackle the issue of uncertainties within drill-string dynamics. Many mention a need for a stochastic approach, but models that do generally assume a stochastic input to a deterministic system (e.g. Spanos et al. [10]) rather than considering system uncertainties. There remains a need for an efficient model of the full drill-string that can include non-linearities and uncertainties. This article describes a possible approach. Each stage of the strategy that is described could be implemented in many ways: the implementation summarised here illustrates the principle. The chosen priorities of the current model are to: • • • •

include the dynamics of the whole drill-string; enable integration with other models; allow for the inclusion of strong non-linearities; be computationally efficient.

The approach taken is to model in the frequency domain those parts of the system that can be effectively approximated as linear, then couple this to the non-linear parts of the system in the time-domain.

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2 Theoretical Framework The transfer matrix method allows efficient computation of the frequency response functions from one part of a pipe to another. This means that the full dynamics of a continuous pipe with a periodically varying cross-section are included, but are described by the states at just two parts of the drill-string. The inherent limitation of this model is that it is based in the frequency-domain and is therefore linear. If non-linearities occur locally (such as at a point of contact with the borehole wall) this can be overcome by obtaining a finite impulse response filter from the frequency-domain model and solving the problem in the time-domain. The gain in efficiency remains as the filter coefficients are pre-calculated. The same method that allows for inclusion of non-linearities can also be used to couple this model to a more detailed analysis of a subsection such as the bottom-hole assembly. This combines the benefits of the detailed analysis of a region of interest with realistic and efficient inclusion of the rest of the drill-string. The efficiency of the model makes the use of Monte-Carlo simulations a feasible approach, so that the effects of uncertainties can be explored. We begin by applying the transfer matrix method to those subsections of the drill-string that could reasonably be represented as linear systems.

2.1 Linear Model Consider the uniform pipe illustrated in Figure 1 with properties given by the shear modulus, G, the density, ρ and the second moment of area of the cross-section, J.

Fig. 1: Definition of variables for a uniform pipe The equation of motion for the torsional dynamics of a pipe of uniform crosssection can be written: GJ θ − cθ˙ − ρ J θ¨ = 0. (1) Letting θ = θ¯ eiω t gives: GJ θ¯ − iω cθ¯ + ω 2 ρ J θ¯ = 0. This can be written as a first-order matrix differential equation:

(2)

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1 θ¯ 0 GJ = GJ θ¯ iω c − ω 2 ρ J 0

θ¯ GJ θ¯

or in terms of angular velocity rather than displacement, this becomes: ¯ ¯ iω 0 θ˙ θ˙ GJ = ¯ c + i ωρ J 0 GJ θ GJ θ¯

(3)

(4)

The angular velocity θ˙ has been chosen for the first state to avoid zero-frequency singularities which occur if angular displacement is chosen, using common boundary conditions. T With y = θ¯˙ GJ θ¯ , this can be written: y = Ay

(5)

The dynamics of two points along the length of the pipe a distance of L apart are described by the transfer matrix: yb = eAL ya .

(6)

Consider joining two pipes end-to-end (representing pipe and joint), illustrated in Figure 2. Each can be described by the above equations of motion but with different material and geometric properties. The states of each end of this combined section are related by a new transfer matrix equation: yn+1 = eA2 L2 eA1 L1 yn = Tyn

(7)

where T = eA2 L2 eA1 L1 is the transfer matrix of the combined section. If N pipe-joint sections are connected the dynamics of the ends are given by: yN = TN y0 = Hy0

Fig. 2: Definition of variables for a pipe and joint section The matrix A can be decomposed into its right and left eigenvectors:

(8)

An efficient model of drill-string dynamics with localised non-linearities

A = UΛ VT

393

(9)

where U and V are the matrix of right and left eigenvectors respectively and Λ is a diagonal matrix of the corresponding eigenvalues. If the dynamic stiffness matrix is symmetric, then the right and left eigenvector matrices satisfy U = JV. The eigenvalues form a pair with opposite signs whose corresponding eigenvectors represent waves travelling in opposite directions. The transfer matrix can be written: T = eAL = UeΛ L VT

(10)

The transfer matrix for a pipe-joint segment then becomes: N = UeΛ NL VT TN = eA2 L2 eA1 L1

(11)

The transfer matrix for the whole length of drill-pipe relates the surface and downhole states: ¯ ¯ H11 H12 θ˙N θ˙0 = (12) ¯ H H TN T¯0 21 22 where T¯ = GJ θ¯ . The downhole angular velocity can be expressed in terms of the downhole and surface torques, eliminating the surface torque: H12 ¯ H11 H22 − H12 H21 ¯˙ TN + θ¯˙N = θ0 H22 H22

(13)

A simple boundary condition is to assume zero torque downhole and specify the surface velocity giving the transfer function from the surface velocity to downhole velocity: θ¯˙N H11 H22 − H12 H21 = . (14) ¯ ˙ H22 θ0 In order to obtain a time-domain impulse response the inverse fourier transform of the frequency response can be estimated using the inverse discrete fourier transform. The discrete-time impulse response directly gives an approximation to the filter coefficients for a FIR digital filter, allowing the response to arbitrary inputs to be computed efficiently. It also allows coupling to other subsystems (such as a finite element model of a bottom hole assembly) and localised non-linearities.

2.2 Coupling to Non-Linearities The decoupling of the linear and non-linear parts of the model retain the efficiency of linear models and focus the computational effort on localised non-linear sections. This opens the possibility of an efficient model of a complete drill-string that contains several subsystems coupled by regions of strong non-linearity.

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With the drillstring itself represented as an efficient linear subsystem, computation can be focussed on the non-linear aspects of the system. As a starting point, consider a periodic drill-string in frictional contact with the bottom where the torsional force and angular velocity are related non-linearly. The drill-string dynamics can be written:

θ¯˙N = GN T¯N + G0 θ¯˙0

(15)

where GN and G0 are the transfer functions given in Equation 13. In the time-domain this becomes: θ˙N = gN ∗ TN + g0 ∗ θ˙0 (16) which can be written explicitly as a convolution summation:

θ˙N (k) =

k

k

n=0

n=0

∑ gN (k − n)TN (n) + ∑ g0 (k − n)θ˙0 (n).

(17)

The torque and angular velocity are related by some friction law: TN = f (θ˙N )

(18)

which may be non-linear. At a given time-step, Equation 17 and 18 must be solved simultaneously to find the current angular velocity and torque. This is made relatively straightforward if Equation 17 is expressed: k ˙ θ˙N (k) = gN (0) TN (k) + ∑k−1 n=0 gN (k − n)TN (n) + ∑n=0 g0 (k − n)θ0 (n) ˙ = gN (0) TN (k) + θhistory

(19)

which is simply a straight line. The intersection of this with the non-linear frictionvelocity curve gives the current angular velocity and torque without iteration in a method directly analogous to the digital waveguide method used to model the bowstring interaction of stringed instruments (McIntyre and Woodhouse [11]). Combining these principles allows localised non-linearities to be modelled at any point along the drill-string. If a non-linearity exists after n segments, then the relevant transfer functions can be written:

θ¯˙na = Gna T¯na + G0a θ¯˙nb = Gnb T¯nb + GNb

θ¯˙0a T¯Nb

(20)

where the subscripts a and b identify the two subsystems. As convolution summations these become: θ˙na = gna ∗ Tna + g0a ∗ θ˙0a (21) θ˙nb = gnb ∗ Tnb + gNb ∗ TNb . Assuming an external torque Text is applied at this interface then the boundary conditions are:

An efficient model of drill-string dynamics with localised non-linearities

θ˙na = θ˙nb Tna + Text = Tnb .

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(22)

These can be written in the same way as Equation 17 separating the current states from the past states. Eliminating either Tna or Tnb returns the problem to the intersection of a straight line with the non-linear friction law, which allows the current states to be evaluated without iteration. This enables efficient implementation of a model that could take into account a drill-pipe and bottom hole assembly together with localised non-linearities at arbitrary points along the length of the drill-string.

2.3 Coupling to Subsystems When the non-linearity cannot be described as an instantaneous state relationship then the above method does not apply so directly. Nevertheless the digital filter can be coupled to an arbitrary differential equation. Let yN and FN represent the vectors of displacement and force states respectively at the end of the drill-pipe and ynew and Fnew represent the end states of a new subsystem that couple to the drill-pipe. If the equations of motion of the new subsystem can be written: d ynew = Fnew (23) dt then the coupled equations of motion could be written: d ynew = Fnew + FN . dt

(24)

Using a digital filter expression for the drill-pipe, yN can be written: yN = gN ∗ FN + g0 ∗ y0 .

(25)

Separating instantaneous from past states and rearranging in the same way as before leaves a system of linear equations that can be solved in conjunction with the differential equation at each time step. For the case of one displacement state and one force state, and by compatability, the differential equation becomes: yN − yhistory d yN = Fnew + dt gN (0)

(26)

where gN (0) is the first coefficient of the relevant impulse response. In principle this allows coupling to any system that can be described by differential equations.

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3 Example Simulations Here we present a proof-of-concept implementation for some specific cases that illustrate the different ways in which the above framework can be applied. The purely linear system is considered first in order to confirm initial expectations of the system behaviour. Two types of coupling are then considered: to a non-linear friction law and to a lumped inertia. Finally the effects of drilling speed, contact torque and uncertainty in damping upon the occurrence of stick-slip oscillation is explored using a Monte-Carlo simulation.

3.1 Linear Behaviour The material properties chosen for these initial tests are based on those of steel and the geometry is nominally based on possible drill-pipe parameters. The distributed damping is arbitrarily taken to be c = 10 Nmsrad−1 per unit length. The number of sections N has been chosen to be 100, representing a 1 km drill-string. First a uniform pipe is considered, where the geometry of the joint is taken to be identical to the pipe. Figure 3 (a) shows the transfer function of this uniform free-free pipe from surface to downhole angular velocity for 0–10 Hz. A wider frequency range shows these peaks continue at regular intervals of the same amplitude. Resonant frequencies are consistent with those expected for a uniform pipe of these dimensions. The impulse response (Figure 3b) is also consistent with expectations: there is an initial delay of 0.31 seconds until the impulse arrives, consistent with a group velocity of 3.2 kms−1 . The subsequent delays of 0.62 seconds are also consistent with the impulse returning to the top of the drill-string and back again. The amplitude of each reflection decays due to the damping at a rate consistent with the expected time constant of 2ρ J/c = 0.58 seconds. Small artifacts surrounding each impulse are apparent, which result from band-limiting the frequency response. Figures 3 (c) and (d) show the equivalent plots for the drill-string with periodically varying cross-section. The effect of the periodicity on the frequency response is to introduce stop-bands. As expected these occur at frequencies near the resonance of an individual section: the first section-resonance occurs at 160 Hz. Between each of these stop bands the resonances of the whole pipe still occur (not shown) and as expected there are 100 resonant peaks between each stop-band. The reflections that occur at the boundaries between radius changes cause dispersion, seen in Figure 3 (d) by the change of shape of the impulses. The first arrival times are still consistent with the same group velocity.

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Fig. 3: Transfer functions and impulse responses of (a)–(b): uniform pipe, (c)–(d): periodic pipe

3.2 Coupling to non-linear friction law To illustrate a non-linear coupling, the same drill-pipe as above is coupled to a discontinuous Coulomb friction law at the bottom end, with a static friction torque 20% higher than the dynamic friction torque. Figure 4a (solid line) shows the response to an input angular velocity profile (dashed line) for a frictional torque of 1×104 Nm and with a rotation speed of ω = 0.6 rpm. The values may not be representative, but provide a useful illustration. There is a delay before the angular velocity becomes non-zero. This corresponds to the delay before the first arrival of the torsional wave in addition to a build up of torque to the static friction limit. Dispersion due to the periodicity of the pipe is also visible. Subsequent reflections are apparent that have a period consistent with the expected group velocity and settle to the input velocity after a few seconds. The computation time for this 18 second simulation was 15 seconds. To explore the effect of an alternative friction model, the same simulation was run with additive white noise applied to the dynamic frictional torque, ±5% of the magnitude of the nominal dynamic torque. The results are shown in Figure 4b.

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Fig. 4: Example simulations using a non-linear friction law (a) without and (b) with white noise

It can be seen that the response is very different: the amplitude of the reflection after slip first occurs is sufficiently large to cause the bit to stick, causing the drill to settle into a stick-slip limit cycle. This will be explored further in Section 3.4.

3.3 Coupling to lumped inertia The above coupling is a special case: the non-linearity is an instantaneous state relationship that does not depend on past values or derivatives. To demonstrate coupling to a differential equation, we consider coupling to a lumped inertia. The bottom-hole torque must balance the acceleration torque of the inertia such that: θ˙h − θ˙N = Jinertia θ¨N (27) gN (0) This can simply be formulated as a finite difference equation which can readily be solved in the time domain. Figure 5 shows the step response as the inertia is increased (in equal steps from zero to 1×104 kgm2 ). The thin solid line indicates the zero inertia case, which is simply the step response of the drill-string alone and clear steps at each reflection are evident. The thick solid line indicates the highest inertia response which closely resembles the expected response for a simple lumped inertia on a torsional spring. The theoretical step response for the equivalent lumped parameter system is shown as a dot-dashed line. The intermediate cases are shown by dotted lines. The convergence towards the lumped inertia and spring model shows how the drill-pipe dynamics gradually become less important. This provides a proof-of-concept implementation that demonstrates the possibility of coupling the digital filter models to more general systems that can be described by differential equations.

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Fig. 5: Step response for a lumped inertia on the end of a drill-string. Thin solid line: zero inertia, thick solid line: inertia=1 × 104 kgm2 , dotted lines: intermediate cases, dot-dashed line: theoretical mass-on-spring response for largest inertia

3.4 Uncertainty Analysis of Stick-Slip Oscillation Short computation times enable the possibility of running multiple simulations. There are a number of wide-ranging questions of interest that could be explored, for example: • • • • •

Over what parameter space does stick-slip occur? How sensitive are predictions to damping values / models, or other parameters? What is the effect of different friction models? What is the effect of different transient profiles of the input velocity? How well does a given surface control strategy minimise torsional reflections?

Here we consider some aspects of the first three questions. We choose to vary the frictional torque and input velocity and identify whether or not stick-slip oscillation occurs for each simulation (1×104 < Tlimit < 1×106 Nm and 6 < ω < 60 rpm, each over 10 equal steps). The algorithm used here to determine whether or not sustained stick-slip occurs is to simply identify if zero-sliding velocity occurs for more than 10% of the last 2 out of 10 reflections after the first slip of a transient simulation (thus the simulation times adapt depending on the input angular velocity and frictional torque). In addition, damping is varied from 1–10 Nmsrad−1 per unit length of drill-pipe (in five equal steps). Simulations were carried out using both of the frictional models mentioned in Section 3.2, giving a total of 1000 simulations. The total computation time was approximately six hours. Figure 6a shows the occurrence of stick-slip oscillations for each simulation using the simple Coulomb friction law. Dots represent simulation runs for which stickslip never occurred. Circles and squares increasing in size represent simulations for which stick-slip occurred for each damping value used. For each value of damping there is a clear boundary separating cases that did or did not settle to stick-slip oscil-

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lations: the boundaries are approximately shown as grey dividing lines (dark shades correspond to high damping). The results are consistent with the intuition that slow rotational speeds and high frictional torques (which could be thought of as a high normal force on the cutting bit) are more likely to lead to stick-slip. There is also a clear trend with damping: as damping increases the region of the parameter space for which stick-slip occurs in general becomes smaller. A closer look reveals some exceptions, the reasons for which could be explored using a higher resolution parameter set in the regions of interest. The effect of an alternative friction law can easily be considered. A simple extension is to add white noise to the constant frictional torque. It has already been seen to have an effect on the occurrence of stick-slip oscillations in Section 3.2 for one case: here we explore the effect of additive white noise (5% of the nominal torque) on the stick-slip boundaries. Figure 6b shows that the region of the parameter space for which stick-slip occurs becomes significantly larger: the boundary lies at approximately three times the rotational speed or half the torque compared to the constant friction simulations.

Fig. 6: Exploring the effect of rotational speed, frictional torque, damping and friction model on the occurrence of stick-slip oscillations. Dots indicate tests for which stick-slip never occurred. Legend shows symbols used for tests where stick-slip did occur and their corresponding damping values in Nmsrad−1 per unit length. Grey lines indicate approximate boundary between cases that did or did not stick-slip for each damping value tested (dark shades correspond to high damping) The tests demonstrate how the efficiency of the model strategy makes the use of Monte-Carlo simulations for exploring uncertainties a feasible approach, without neglecting the dynamics of the complete length of the drill-string or the effects of the periodically varying cross-section. In addition, there is scope for generalising the specific model considered here to more complicated cases without losing the efficiencies gained.

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4 Conclusions High amplitude vibration of oil-well drill-strings cost the industry hundreds of millions of dollars per year. Deriving effective theoretical models has proved challenging: lumped parameter models, though efficient, are too simplistic to effectively correlate with real systems, while high resolution finite element models may be more realistic but are computationally demanding and may not provide clear insight into the fundamental mechanisms at play. In addition, drill-strings are highly uncertain: efficient models are needed if these are to be considered using Monte-Carlo techniques. This paper presented a strategy for modelling the entire length of a drill-string, by efficiently describing the linear dynamics and focussing the computation on localised non-linearities. The linear dynamics are represented by digital filters obtained using periodic structure theory. Only the states coupled to non-linearities and inputs need to be retained allowing efficient time-domain simulation. Some example simulations were described that show the effectiveness of this approach. In particular a sequence of 1000 simulations was used to explore the occurrence of stick-slip as a function of three parameters using two friction models. The results showed that stick-slip is most likely for high frictional torque and at low rotational speeds, consistent with field observations. The boundary between sustained stick-slip and decaying oscillations was shown to be sensitive to damping and frictional noise. While these conclusions may have been expected, the emphasis here is on the effectiveness of the analysis tools presented in tackling questions of this kind. Acknowledgements The authors are grateful to Schlumberger Cambridge Research for providing financial and technical support for this project and for giving permission to publish this work.

References 1. T. V. Aarrestad and A. Kyllingstad. Rig suspension measurements and theoretical models and the effect on drillstring vibrations. Society of Petroleum Engineers, 1993. 2. R. I. Leine, D. H. van Campen, and W. J. G. Keultjes. Stick-slip whirl interaction in drillstring dynamics. Journal Of Vibration And Acoustics-Transactions Of The Asme, 124(2):209–220, April 2002. 3. T. Richard, C. Germay, and E. Detournay. Self-excited stick-slip oscillations of drill bits. Comptes Rendus Mecanique, 332(8):619–626, August 2004. 4. T. Richard, C. Germay, and E. Detournay. A simplified model to explore the root cause of stick-slip vibrations in drilling systems with drag bits. Journal Of Sound And Vibration, 305(3):432–456, August 2007. 5. Y. A. Khulief, F. A. Al-Sulaiman, and S. Bashmal. Vibration analysis of drillstrings with self-excited stick-slip oscillations. Journal Of Sound And Vibration, 299(3):540–558, January 2007. 6. R. Sampaio, M. T. Piovan, and G. Venero Lozano. Coupled axial/torsional vibrations of drillstrings by means of non-linear model. Mechanics Research Communications, 34(5–6):497– 502, 2007.

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7. G. W. Halsey, A. Kyllingstad, and A. Kylling. Torque feedback used to cure slip-stick motion. Society of Petroleum Engineers, 1988. 8. W. R. Tucker and C. Wang. On the effective control of torsional vibrations in drilling systems. Journal of Sound and Vibration, 224(1):101–122, July 1999. 9. R. W. Tucker and C. Wang. Torsional vibration control and cosserat dynamics of a drill-rig assembly. Meccanica, 38(1):143–159, 2003. 10. P. D. Spanos, A. M. Chevallier, and N. P. Politis. Nonlinear stochastic drill-string vibrations. Journal Of Vibration And Acoustics-Transactions Of The Asme, 124(4):512–518, October 2002. 11. M. E. McIntyre and J. Woodhouse. On the fundamentals of bowed-string dynamics. Acustica, 43:93–108, 1979.

Equivalent thermo-mechanical parameters for perfect crystals V. A. Kuzkin and A. M. Krivtsov

Abstract Thermo-elastic behavior of perfect single crystal is considered. The crystal is represented as a set of interacting particles (atoms). The approach for determination of equivalent continuum values for the discrete system is proposed. Averaging of equations of particles’ motion and long wave approximation are used in order to make link between the discrete system and equivalent continuum. Basic balance equations for equivalent continuum are derived from microscopic equations. Macroscopic values such as Piola and Cauchy stress tensors and heat flux are represented via microscopic parameters. Connection between the heat flux and temperature is discussed. Equation of state in Mie-Gruneisen form connecting Cauchy stress tensor with deformation gradient and thermal energy is obtained from microscopic considerations.

1 Introduction Determination of the relation between parameters of discrete and continuum systems is one of the challenging problems for modern physics. There were intensive investigations in this area for the last several decades. However the problem is far from its final solution. At the beginning the problem was only of a fundamental interest. However, practical interest is increasing now. The increase is caused by fast development of discrete [1] and discrete-continuum [17, 16] methods for simulation of mechanical behavior of bodies under mechanical and thermal loadings. Various methods for transition from discrete system to equivalent continuum are considered V. A. Kuzkin Institute for Problems in Mechanical Engineering RAS, V.O., Bolshoj pr. 61, St. Petersburg, 199178, Russia, e-mail: [email protected] A. M. Krivtsov Institute for Problems in Mechanical Engineering RAS, V.O., Bolshoj pr. 61, St. Petersburg, 199178, Russia, e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 29, © Springer Science+Business Media B.V. 2011

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in literature. Long wave approximation is used in [2]. The concept of quasicontinuum is proposed in [10]. Localization functions are used in [4, 5, 18]. These approaches give the opportunity to spread mechanical parameters determined in lattice nodes on all volume of the body. Decomposition of motions on slow macroscopic and fast thermal ones is used for description of thermal properties. There are different approaches for decomposition. In papers [4, 5, 18, 21] the decomposition of particle velocities is carried out using localization functions. As a result, dependencies of stress tensor and heat flux on parameters of the discrete system were obtained and analyzed. Another approach was proposed in [20]. Fourier transformation was used for decomposition of displacements and particle velocities. Different methods for decomposition were discussed. It was noted that the result of the decomposition is not unique. It should depend on characteristic time and spatial scales of the problem. The approach based on averaging of equations of motions and application of the long wave approximation [2] was proposed in papers [7, 8]. The derivation of expressions for stress tensors for ideal crystals was carried out in book [7]. Only pair potentials were considered. Thermal motion was neglected. The influence of thermal oscillations on mechanical properties was considered in [7, 8] for one-dimensional case. The proposed approach gives an opportunity to carry out analytical derivations. In particular, the equation of state in Mie-Gruneisen form was obtained in papers [7, 8, 9]. In the present paper a generalization of approaches proposed in [7, 8] for two and three-dimensional cases is carried out. The expressions connecting Piola and Cauchy stress tensors and heat flux with parameters of the discrete system are derived. The approach for derivation of constitutive relations for Cauchy stress tensor and heat flux is discussed.

2 Hypotheses Let us consider discrete system consisted of particles, which form the infinite ideal crystal lattice in d-dimensional space (d = 1, 2 or 3). Crystals with simple structure are investigated only (i.e. crystals that are invariant to translation on any vector connecting lattice nodes). For the sake of simplicity let the particles interact via pairwise potential of Lennard-Jones type. Generalization of the approach discussed in the present paper for the case of multibody interatomic potentials is considered in papers [11, 12]. Two main principles are used for transition from discrete system to equivalent continuum: the long wave approximation [2] and decomposition of particles’ motions into slow continuum and fast thermal one [8, 20]. First let us focus on decomposition. In literature it is carried out using different types of averaging such as spatial averaging, time averaging, averaging over phase space or over frequency spectrum, etc. It was noted in paper [20] that unique decomposition is impossible because rules for a choice of averaging parameters like averaging time, represen-

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tative volume, etc. do not exist. The only possible rule for these parameters is that they should depend on time and spatial scales of the problem being solved. Let us denote average and oscillating (thermal) components of physical value f as f and f respectively. Obviously, def f = f − f .

f = f + f,

(1)

Different expressions for the averaging operator are proposed in literature. The following operator was used in paper [8] for one dimensional case t+T /2 n+Λ /2 1 fn = ∑ fk (t)dt , T Λ t−T /2 k=n− Λ /2

(2)

where fk is magnitude of physical value f for particle number k. Parameters T and Λ should satisfy the following relations 1 Λ N, Tmin T Tmax , where Tmin and Tmax are minimal and maximal periods of oscillations in the system, N is the total number of particles. Obviously these limitations are too weak. Direct and inverse Fourier transformations were used for decomposition in paper [20]. The direct transformation gives 1 F(ν ) = √ 2π where F (ν ) =

F(ν ), ν < νcuto f f 0, ν ≥ νcuto f f

∞ −∞

f eiν t dt ,

(3)

ν) = F(

0, ν < νcuto f f F(ν ), ν ≥ νcuto f f

(4)

Here F is Fourier transform of value f ; i is imaginary unit; νcuto f f is cut-off frequency, which should be taken in the range 0.5–50 THz [2]. Inverse Fourier transformation was used in order to obtain f and f ∞ 1 √ F e−iν t d ν f = 2π −∞

1 f = √ 2π

∞ −∞

−iν t d ν . Fe

(5)

In the framework of the given approach the choice of cut-off frequency is almost arbitrary. In papers [4, 5, 21] the following relations were used for decomposition of particle velocities fk = f (x,t) + fk (x,t),

∑M mk fk ψ (x − xk ) . f (x,t) = k=1 ∑M k=1 mk ψ (x − xk )

(6)

Here fk , mk , xk are velocity, mass and radius-vector of particle number k; x is coordinate of the spatial point where velocity is calculated; ψ is localization function; M is

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the total number of particles in the system. Decomposition (6) can be considered as spatial averaging with weight determined by function ψ . Note that according to this approach thermal component of the velocity fk (x,t) is continuum value (it is deter mined at points between particles). Several thermal velocities fk1 (x0 ,t), fk2 (x0 ,t), ... and one continuum velocity f (x0 ,t) are simultaneously determined in one spatial point x0 . However, formally it does not lead to any contradictions. This type of decomposition as well as all mentioned above is not unique. It strongly depends on the choice of localization function. In particular, if localization area surrounds single atom, then thermal component of the velocity is equal to zero. Since the unique decomposition does not exist, the theory should not be based on particular method of decomposition. In addition, the results of the theory should not qualitatively change with the change of the method. Properties of particular methods for averaging are not used in the present paper, unless otherwise stated. Let us speak about averaged f and thermal f components of physical value f , which are connected by formula (1). The second important statement used in the present paper is the long wave approximation [2]. The idea of the approximation is the following: an average component of any physical value is assumed to be slowly changing in space on the distances of an order of the interatomic distance. Then the average component can be considered as a continuum function of a space variable and can be expanded into power series with respect to interatomic distance. The resulting series should converge rapidly. Exactly this assumption allows to make transition from a discrete system to an equivalent continuum.

3 Kinematics Let us use material description of the equivalent continuum. Two configurations of continuum and discrete system are considered: reference and actual. For the sake of simplicity let us take undeformed configuration of the crystal lattice as the reference one. Radius-vectors of equivalent continuum in the reference and actual configurations are denoted as r and R respectively. Two ways of particles’ identification are used. On the one hand, the position of the particle is determined by its radius-vector. On the other hand, let us use local numbering [7]. Starting with one reference particle let us mark all its neighbors by index α . Let us denote a vector connecting the reference particle with its neighbor number α as aα . By the definition vectors aα have the following property aα = −a−α . (7) The same vectors in an actual configuration are represented as a sum of averaged ˜ α . They can be expressed in terms of veccomponent Aα and thermal component A tors aα and displacements of particles as Aα = aα + uα − u,

˜α = A uα − u.

Equivalent thermo-mechanical parameters for perfect crystals

407

α are average and thermal components of displacements respecHere u, uα and u, u tively. The introduced way of particle identification has several properties. Let us use the following definition. A physical value determined by the state of one particle is called a single-particle value. For example, particle’s mass, radius-vector, velocity, displacement etc. Let f (r0 ) be one-particle value that corresponds to the reference particle with radius-vector r0 in the reference configuration. Denote value f , which corresponds to particle number α , as fα (r0 ). Then the following two designations are equivalent (8) fα (r0 ) ≡ f (r0 + aα ). In the framework of this approach the magnitude of physical value f at the point r0 can be represented in the following equivalent forms f (r0 ) = fα (r0 − aα ) = f−α (r0 + aα ) .

(9)

One can show that for multiplication of two one-particle values f and g the following identities are satisfied [ fα g] (r0 ) = [ f g−α ] (r0 + aα )

[ f−α g] (r0 ) = [ f gα ] (r0 − aα ) .

(10)

Hereinafter square brackets mean that all values in them are calculated at the same point. Let us also consider values that depend on the state the reference particle and its neighbor number α , notably vector connecting two particles and force acting between particles. These values have the following property hα (r0 ) = −h−α (r0 + aα ) .

(11)

If h is a force acting between particles, then equation (11) is a specific form of Newton’s third law. The following identities are satisfied for one-particle value f and the value h [ fα hα ] (r0 ) = − [ f h−α ] (r0 + aα )

[ f h−α ] (r0 ) = − [ fα hα ] (r0 − aα ).

(12)

Let us consider kinematics of the discrete system in the long wave approximation. It is assumed that average values of particle radius-vectors are identical to positions of corresponding points of continuum media. Thus if some particle has radius-vector r in the reference configuration, then the average value of the radiusvector in actual configuration is equal to R(r). The average position of its neighbor number α is determined by vector R(r + aα ). Then one can show that vectors Aα and aα , connecting the particles, are related by the following formula ◦

Aα = R(r + aα ) − R(r) ≈ aα · ∇ R , ◦

(13)

where ∇ is nabla-operator in the reference configuration. Here the long wave approximation was used, which allows to leave the first order terms only. One can

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see that expression (13) is similar to the formula used in continuum mechanics that connects vectors dr and dR. Using equation (13) one can derive relations between vectors Aα , aα and measures of deformation used in nonlinear theory of elasticity [14]. For example, the following identity fulfills for Cauchy-Green measure G A2α = aα aα · ·G,

def

◦

◦

G = (∇ R) · (R ∇) .

(14)

4 Equation of momentum balance Let us obtain the equation of motion for the equivalent continuum. Thereto let us write down the equation of motion for the reference particle and use the decomposition of motions α (Aα + A ˜ α) , ˜ α) , u¨ = ∑ F m (15) mu¨ = ∑ Fα (Aα + A α

α

where Fα is a force acting on the reference particle from its neighbor α ; m is particle’s mass. The first equation from (15) describes slow motion of the system. The motion can be considered as motion of continuum media. The second equation describes thermal oscillations. One can see that both equations are coupled via the argument of the force Fα . However, if the dependence of the force on the distance between particles is linear, then equations become independent. It reflects the wellknown fact that harmonic models can not describe coupled thermo-mechanical effects such as thermal expansion [13]. Let us conduct the following transformations in the first equation from (15). 1 mu¨ = ∑ Fα = ∑ F−α = ∑ Fα + F−α . 2 α α α

(16)

Force Fα satisfies Newton’s third law, i.e. Fα (r−aα ) = −F−α (r) (see formula (11)). Averaging this expression and using long wave approximation one obtains ◦ F−α (r) ≈ − Fα (r) + aα · ∇ Fα (r) (17) Substituting formula (17) into equation (16) and dividing both parts by volume of elementary cell in the reference configuration V0 one obtains ◦ m 1 (18) u¨ =∇ · aα Fα V0 2V0 ∑ α Let us compare formula (18) with equation of motion for continuum in Piola’s form [14]. ◦ ρ0 u¨ =∇ ·P, (19)

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409

where P is Piola stress tensor, ρ0 is a density in the reference configuration. Comparing equations (18) and (19) one can deduce that P=

1 aα F α , ∑ 2V0 α

ρ0 =

m . V0

(20)

Strictly speaking the first formula from (20) is satisfied only with accuracy of tensor with zero divergency. This tensor corresponds to some equilibrium stress field in the crystal. Let us conduct the same derivations in an actual configuration. Equation of motion for the particle has form (16). Let us rewrite formula (17) in an actual configuration. F−α (R) ≈ − Fα (R) + Aα · ∇ Fα (R). Fα (R − Aα ) = −F−α (R) ⇒ (21) Here it was used that in the long wave approximation A−α ≈ −Aα . Substituting expression (21) into equation (16) and dividing both parts by the volume of elementary cell in the actual configuration V one obtains m 1 u¨ = V 2V

A · ∇ Fα . α ∑ α

(22)

Let us conduct the following transformations in the right side of formula (22) 1 1 1 A · ∇ F A ∇ · Fα = ∇ · F (23) − A α α α α α ∑ 2V ∑ 2V ∑ 2V α α α The second term in the right side of the given equation can be written down in the following form using equation (13) ◦

1 1 V0 R F A = Fα = 0 , (24) · a ∇ · ∇ · ∇ α α α ∑ 2V 2V0 ∑ V α α ◦

where Piola’s identity ∇ · VV0 R ∇ ≡ 0 was used (see, for example, [14]). Then equation of motion (22) has the following form 1 m u¨ = ∇ · Fα . (25) A α V 2V ∑ α The requirement of equivalence of discrete and continuum systems leads to the following expressions for Cauchy stress tensor and density in the actual configuration 1 m τ= Aα Fα , ρ= . (26) ∑ 2V α V If thermal motion is not taken into account, then expression (26) coincides with expressions derived in work [7].

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It is known that Cauchy stress tensor is symmetrical in systems without moment interactions. Let us consider tensor τ determined by formula (26). Force Fα can be represented as def

˜ α )2 )(Aα + A ˜ α ), Fα = −Φα ((Aα + A

Φ (A2 ) = −

Π (A) A

(27)

Substituting the given expression into formula (26) one obtains

τ =−

1 2V

∑ α

1 Φα Aα Aα − 2V

∑ Aα α

˜α . α A Φ

(28)

The first tensor in the right side of formula (28) is symmetrical indeed. However the symmetry of the second tensor is not evident. Further it will be shown that antisymmetrical part of this tensor is small with respect to symmetrical part.

5 Equation of angular momentum balance It is known from continuum mechanics [15] that the symmetry of Cauchy stress tensor follows from equation of angular momentum balance for elementary volume. In the discrete case elementary cell plays the role of elementary volume. Let us write down the averaged equation of angular momentum balance for elementary cell (moments are calculated with respect to the center of the cell determined by vector R). α = − ∑ A ˜ α ×F α + ∑ α . ×∑F (29) uα × F m u˙˙ = u u× α

α

α

Transforming the second term in the right side of the given equation using the long wave approximation one obtains −α (R + Aα ) ≈ − α (R) = − u −α − Aα · ∇ −α . ×F u×F u×F uα × F (30) Let us substitute the result into equation (29) and resolve it with respect to ˜ Aα × Fα . 1 ˜ α = 1 ∑ Aα × Φ −α − m α A ˜ α = − 1 ∑ Aα · ∇ u ×F u˙˙ . u× Aα × F ∑ 2 α 2 α 2 α (31) Using expression (28) for the stress tensor one can transform formula (31) to the following form E · ×τ A =

1 2V

∑ Aα · ∇ α

−α + ρ u × ×F u¨ . u

(32)

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411

Here A denotes antisymmetrical part of the tensor. One can see that if there is no thermal motion, then τ A ≡ 0. Let us show that in general case τ A is small in comparison with τ S . The first term in formula (32) is small in the long wave approximation. Let us show that if operator contains spatial averaging, then the second term is also small. Consider the following identity (33) u˙˙ . u˙ ˙= ρ (R + u) × u× ρ The right side of formula (33) is a derivative of angular momentum, which corresponds to thermal motion. Angular momentum is calculated with respect to the origin of coordinates. Let the averaging operator include spatial averaging over significantly large volume and let us assume that thermal motion does not lead to macroscopic rotation of the volume. Then expressions (33) are equal to zero. As a result τ A has the same order as terms that were neglected in long wave assumption. Consequently, tensor (26) can be considered as approximately symmetrical. Thus averaging operator proposed above can not be arbitrary. It should include spatial averaging. Otherwise one can not prove the symmetry of tensor τ determined by formula (26).

6 Equation of energy balance For the sake of simplicity let us assume that volumetrical forces and volumetrical heat sources are equal to zero. Derivations are carried out in the reference configuration. In this case averaged specific total energy per volume V0 has the following form 1 1 ˜ α) , u˙ )2 + ρ0 E = ρ0 (u˙ + Π (Aα + A (34) ∑ 2 2V0 α where E is particle’s total energy divided by the mass, i.e. discrete analog for mass density of the energy. Let us introduce the following designations

ρ0 E = ρ0 (K + U ) , 1 ρ0 K = ρ0 u˙ 2 , 2

1 1 ˙ 2 ˜ u + ρ0 U = ρ0 Π (A + A ) α α . 2 2V0 ∑ α

(35)

Values K and U correspond to mass densities of macroscopic kinetic and internal energies. Calculating derivatives of kinetic and potential energies taking into account formulas (15), (20) one can obtain

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◦

2 1 d 1 ˙ 2 ˙ · = ρ0 u˙ · u¨ + u u · Fα , u¨ = ∇ ·P · u˙ + ∑ u˙ + ρ0 u˙ 2 dt V0 α (36) 1 d ˙ α ) ˜ α ) = 1 ∑ Fα · (A ˙ α +A Π (Aα + A ∑ 2V0 dt α 2V0 α

dΠ was used. Let us conduct the following transformations dAα ◦ ˙ + aα ) − u(r)) ˙ (37) ≈ ∑ aα Fα · ·u˙ ∇ ∑ Fα · A˙ α = ∑ Fα · (u(r

where formula Fα =

α

α

α

Summarizing expressions (36) and taking into account formulas (20), (37) one obtains the expression for derivative of the total energy with respect to time ◦

˙ + ρ0 E˙ =∇ · (P · u)

1 α · u˙ . u˙ α + F ∑ 2V0 α

(38)

◦

˙ SubUsing equation of momentum balance one can show that ρ0 K˙ = ∇ ·P · u. stituting this expression into equation (38) one obtains ◦

1 ˙ ˙ F u u ) . ρ0 U˙ = P · · u˙ ∇ + · ( + α α 2V0 ∑ α

(39)

Comparing the last expression with energy balance equation for a continuum media [15] one can conclude that expression for divergency of heat flux in the reference configuration h has form ◦

∇ ·h = −

1 1 1 α · ( α · −α · F F F u˙ α + u˙ ) = − u˙ α − u˙ . (40) ∑ ∑ ∑ 2V0 α 2V0 α 2V0 α

Let us represent the right side of this expression in the form of divergency. Using the first identity form (12) in the right side of formula (40) one obtains −α · α · −α · α · u˙ α (r) = − F u˙ (r + aα ), F u˙ (r) = − F u˙ α (r − aα ). F (41) Substituting formulas (41) into formula (40) and applying the long wave approximation one can obtain ◦ 1 ◦ ◦ 1 ˙α ˙ F = F · − u u a · a · . (42) ∇ ·h =∇ · − ∇ α α α α 2V0 ∑ 2V0 ∑ α α Using this expression one can write down three representations for heat flux in the reference configuration1

1

Note that heat flux is determined with the accuracy of vector with zero divergency.

Equivalent thermo-mechanical parameters for perfect crystals

413

1 1 1 α · ( α · α · u˙ α + u˙ ) = − u˙ α = − u˙ . aα F aα F aα F ∑ ∑ ∑ 4V0 α 2V0 α 2V0 α (43) Analogous formulas can be obtained for heat flux in the actual configuration H h=−

H=−

1 4V

∑ Aα α

1 α · ( F u˙ α + u˙ ) = − 2V

∑ Aα α

1 α · F u˙ α = − 2V

∑ Aα α

α · F u˙

(44) ◦

V0 Here formula H = V R ∇ · h relating heat fluxes in different configurations was used [6]. Note that different expressions for h and H in formulas (43), (44) are equal with accuracy of terms, which were neglected in the long wave approximation.

7 Constitutive relations for stress tensor and heat flux Expressions (26), (35) connecting micro- and macro-parameters allow to derive nonlinear constitutive relations (equations of state) for thermo-elastic behavior of the crystal. This problem is considered in detail in works [7, 9]. Only main ideas and results are shown below. The relation between “cold” component of Cauchy stress 2 τ0 and Cauchy-Green measure of deformation can be obtained substituting formulas (13), (27) into formula (26) (see [7] for details). ◦ ◦ 1 (R ∇) · ∑ Φ (aα aα · ·G)aα aα · (∇ R) , (45) τ0 = − 2V0 |G| α where |G| is a determinant of tensor G. Equations of state connecting thermal comde f

ponent of Cauchy stress τT = τ − τ0 with thermal energy were obtained in pa˜α per [9]. The expansion of Cauchy stress and internal energy with respect to A was conducted. In particular, in the first approximation the following system was obtained

1 ˜ αA ˜α , 2Φα Aα EA + Φ α Aα Aα E + 2Φ α Aα Aα Aα Aα · · A τT = − ∑ 2V α (46) nΦ

1 d def (n) ˜ αA ˜ α , Φα = UT = − ∑ Φα E + 2Φα Aα Aα · · A 2 α d(A2α )n ˜ αA ˜α = Here UT is a thermal energy per unit volume. It was assumed that A def 1 2 ˜ 2α in order to close system (46). In the framework of the asκ E, κ 2 = A d

sumption system (46) takes form

2

Stress in the crystal in the absence of thermal motion.

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(d + 2)Φ α + 2Φ α A2α Aα Aα ∑ 1 def τ T = Γ UT , Γ = α . V ∑ d Φα + 2Φ α A2α

(47)

α

Expression (47) is generalized Mie-Gruneisen equation, where Γ is tensor Gruneisen coefficient. Note that more accurate equations of state can be obtained leaving higher order terms in expansions (46). For further details about the approach for derivation of equations of state see paper [9]. Let us consider propagation of small thermal disturbances. Assume that the amplitude of thermal oscillations is small in comparison with interatomic distance. ˜ α and leaving terms of Expanding expression for heat flux (44) with respect to A ˜ 2α only one obtains order of A H=

1 2V

∑ α

def d Φ ˜ α Φ (A2α )Aα E + 2Φ (A2α )Aα Aα Aα · · A u˙ α , Φ = 2 . dAα

(48)

Expression (48) is satisfied for arbitrary nonlinear elastic deformations. Let us consider the case when discrete system is free from internal mechanical loads and constraints. In this case deformations are caused by thermal expansion only. Then the reference and actual configurations approximately coincide. Linearizing expression (48) assuming that Aα ≈ aα one obtains ˜ αu ˜ αu ˙ α = ∑ 3 Cα · · A ˙ , (49) H = ∑ 3 Cα · · A α

α

def where 3 Cα = 2V10 Φ (a2α )aα E + 2Φ (a2α )aα aα aα . Let us represent the expression for heat flux in the form of divergency. S 1 3 ˜ α ˙ − ∑ 3 Cα · · u u˙ α ≈ ∇ · H = ∑ 3 Cα · · A a C · · u˙ α . (50) uu α α ∑ 2 α α α Here the following identity was used ∑α 3 Cα = 0. From formula (50) it follows that, in contrast to classical Fourier law, the heat flux depends on the set of symmetrical S , u tensors uu u˙ α . Let us try to connect heat flux with temperature. Classical ideal gas definition of temperature is used 2 dkT = m u˙ ,

(51)

where k is Boltsman constant. Equation (51) can be transformed taking into account equation of motion (15) α · ˜ α . (52) · uA dkT = m u· u˙ ˙− ∑ F u ≈m u u˙ ˙+ ∑ Φ E + 2Φ aα aα · · α

α

Equivalent thermo-mechanical parameters for perfect crystals

415

˜ α is carried out. The second order Here the expansion into series with respect to A terms are leaved only. Using the definition of tensor 3 Cα let us write down the resulting system for connection between heat flux and temperature. S 1 3 ˙ α H = ∇· aα Cα · · u u ˙ − ∑ 3 Cα · · uu ∑ 2 α α (53) S 2V0 m 3 . u u ¨+ ∑ 2 aα · Cα · · u dkT = E · · uu uα − 2 α aα According to system (53) the thermal state at the given point is determined by sym S u , u uα . In general, these tensors are independent. Theremetrical tensors3 u fore system (53) is not closed.

8 Concluding remarks The generalization of the approach proposed in [7] that allows to carry out the transformation from discrete system to equivalent continuum was presented. Two main principles were used for the transformation: the decomposition of particles’ motions into continuum and thermal parts, and the long wave assumption [2]. The review of different methods for decomposition was given. It was shown that all of them contain uncertain parameters. Therefore, the result of decomposition is principally nonunique. Thus, one can conclude that derivations should not be based on any specific decomposition type. The connection between kinematics of discrete system and kinematics equivalent continuum was analyzed. Equivalent Cauchy-Green measure of deformation for discrete system was introduced. The transition form single particle’s equation of motion to equation of motion for equivalent continuum was carried out. Expressions connecting Cauchy and Piola stress tensors with parameters of the discrete system were derived. It was shown that discrete analog of Cauchy stress tensor can be non-symmetrical. Spatial averaging is necessary for the symmetry of this tensor. Thus, averaging operator cannot be arbitrary and should contains spatial averaging. The energy balance equation for discrete system was considered. The equation was transformed to the form similar to energy balance equation for a continuum system. As a result, the expression connecting heat flux with parameters of discrete system was obtained. Propagation of small thermal disturbances in undeformed crystal was analyzed. It was shown that thermal state at the S point is determined by the set of independent symmetrical tensors u u , u uα . This fact does not allow to connect heat flux with temperature. Equation of state in generalized Mie-Gruneisen form connecting Cauchy stress tensor with deformation gradient and thermal energy is obtained from microscopic considerations. S S α using long wave u˙ α can be represented via time derivatives of tensors uu Tensors u approximation. 3

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Acknowledgements This work was supported by Russian Foundation for Basic Research (grants No. 08-01-00865a, 09-05-12071-ofi-m).

References 1. M.P. Allen, D.J. Tildesley. Computer Simulation of Liquids. Clarendon Press, Oxford. (1987). 385 p. 2. M. Born, K. Huang. Dynamical theory of crystal lattices. Oxford: Clarendon Press, (1988). 3. B.L. Glushak, V.F. Kuropatenko, S.A. Novikov. Issledovanie prochnosti materialov pri dinamicheskix nagruzkax. Nauka, (1992), p. 295. 4. R.J. Hardy. Formulae for determining local properties in molecular-dynamics simulations: Shock waves. Journal of Chemical Physics 76, pp. 622–628, (1982). 5. W.G. Hoover. Smooth particle applied mechnics — The state of the art. Advanced series in nonlinear dynamics, Vol. 25, World Scientific, (2006). 300 p. 6. V.I. Kondaurov, V.E. Fortov. Foundations of thermo-mechanics of condensed matters. M.: Izd. MFTI, (2002). 336 p. (in Russian). 7. A.M. Krivtsov. Deformation and fracture of bodies with microstructure. M.: Fizmatlit, (2007). 302 p. 8. A.M. Krivtsov. From nonlinear oscillations to equation of state in simple discrete systems. Chaos, Solitons & Fractals 17, 79, (2003). 9. A.M. Krivtsov, V.A. Kuzkin. Derivation of equations of state for perfect crystals with simple structure. Mechanics of Solids, (2010) (paper in press). 10. I.A. Kunin. Theory of elastic media with microstrucrutes. Springer-Verlag, (1982). 11. V.A. Kuzkin. Equivalent thermo-mechanical parameters for perfect crystals with arbitrary multibody potential. Proc. of XXXVII Summer School-Conference “Advanced Problems in Mechanic”. St. Petersburg. pp. 421–431, (2009). 12. V.A. Kuzkin, A.M. Krivtsov. Thermo-mechanical effects in perfect crystals with arbitrary multibody potential. Proc. of Joint U.S.-Russia conference on Advances in Material Science. Prague. (2009). pp. 30–34. 13. G. Leibfrid. Microscopic Theory of Mechanical and Thermal Properties of Crystals, Moscow, GIFML, (1962), 312 p. 14. A.I. Lurie. Nonlinear theory of elasticity. North-Holland. Amsterdam. (1990). 617 p. 15. V.A. Palmov. Vibrations of elasto-palstic bodies. Springer-Verlag, Berlin (1998). 16. R.E. Rudd, J.Q. Broughton. Coarse-grained molecular dynamics: Nonlinear finite elements and finite temperature. Phys. Rev. B 72, 144104, (2005). 17. G.J. Wagner, W.K. Liu. Coupling of atomistic and continuum simulations using a bridging scale decomposition. J. Comput. Phys. 190, pp. 249—274, (2003). 18. E.B. Webb, J.A. Zimmerman, S.C. Seel. Reconsideration of Continuum Thermomechanical Quantities in Atomic Scale Simulations. Mathematics and Mechanics of Solids 13, (2008), pp. 221–266. 19. Y.B. Zeldovich, J.P. Raiser. Physics of shock waves and high temperature hydrodynamic events. Academic Press, New York, (1967), p. 785. 20. M. Zhou. Thermomechanical contimuum representation of atomistic deformation at arbitrary size scales. Proc. R. Soc. A 461 (2005) pp. 3437–3472. 21. J.A. Zimmerman, E.B. Webb, J.J. Hoyt, R.E. Jones, P.A. Klein, D.J. Bammann. Calculation of stress in atomistic simulation. Modelling Simul. Mater. Sci. Eng. 12 (2004) pp. 319–332.

Analysis of offshore systems in random waves Katrin Ellermann and Max Suell Dutra

Abstract This paper addresses the analysis of the motion of floating offshore systems in waves. The focus lies on different approaches for the determination of probability density functions. The analysis of the dynamical behavior of systems in ocean waves is an important part in offshore engineering. These structures involve components with very distinct nonlinear characteristics which affect the dynamics. The systematic analysis of the nonlinear dynamics of floating structures is generally facilitated by additional assumptions, such as waves modeled by periodic functions or linearizations of parts of the equations of motion. These assumptions make conclusions about the first passage time or upcrossing probabilities difficult or even impossible. This paper addresses different approaches for the description of random forces acting on floating structures and for the analysis of the resulting motion.

1 Introduction The dynamics of an offshore system is influenced by various effects which are inherently random or nonlinear in nature: These effects can result from different sources such as hydrodynamic forces, coupling of different vessels or nonlinear restoring forces of mooring systems as described in [3]. Therefore, results from a linear analysis such as frequency response calculations may be comparatively easy to obtain, but their validity is usually limited to small amplitude motions. In an environment characterized by random wave and wind force accidents resulting from the dynamical response of floating vessels are a potential danger [8]. Accessing the probabilities of large amplitude motions or collisions mathematically Katrin Ellermann University of Technology, e-mail: [email protected] Max Suell Dutra Universidade Federal do Rio de Janeiro, e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 30, © Springer Science+Business Media B.V. 2011

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is a difficult task [4]. It not only requires a detailed description of the mechanical problem but also results in a high computational effort. The first step in the investigation is the development of a suitable model. Mathematical descriptions of offshore systems show a wide variety — from relatively simple one or two degree-of-freedom models to large-scale multibody systems and discretized descriptions of the fluid-structure interaction or flexible components. The modeling process is usually a trade-off between complex and simple formulations: While the former give a more precise description of the mechanical interrelation of different components, the later are significantly easier to evaluate for multiple sets of parameter values or initial conditions at practicable computational costs. A systematic evaluation of the equations of motion requires both, a precise model which is yet simple enough to evaluate numerically. It is therefore important to treat the modeling process as an integral part of the investigation: Different techniques for the analysis require specific formulations of the equations of motion and a large number of different approaches have been presented in literature [5] and the references therein. In special cases, analytical solutions can be obtained, e.g. from the Fokker-Planck equation [2]. Once an analytical solution has been found for a specific model, it is easily evaluated for any operating condition. On the other hand, they are limited to simple models and are seldom applied to complex systems. Numerical techniques on the other hand can be used for larger sets of equations. Unfortunately, the results usually do not allow for any conclusion about different operating conditions. Each set of parameter values and initial conditions has to be treated separately, which results in high computational costs. Nevertheless, the Monte Carlo simulations are the most commonly used technique for the investigation of random systems, even for the investigation of rare events such as accidents or high-amplitude motion. Monte Carlo Simulation can be applied to approximate probability density functions, gain information about spectral components of the response or estimate exceedance probabilities. In order to apply this method different considerations of the modeling process are presented in this paper: Methods which allow for a gradual refinement of a mechanical model are addressed as well as the combination of linear and nonlinear descriptions for different components of a floating system.

2 Modeling aspects Offshore systems often exhibit distinctly nonlinear phenomena. Even when the excitation by waves is assumed to be periodic, they show responses ranging from harmonic or subharmonic to chaotic motion. Depending on the system’s parameters, these different types of responses can be coexisting, which makes the initial conditions crucial for the steady state response.

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2.1 Modeling of environmental forces In this section, we assume different cases where the excitation is only nearly harmonic. There are different ways to model an excitation fe (t) which is in some sense close to a periodic function: The first model would be the sum of a harmonic function and a random disturbance: fe (t) = sin(ω t) + ξ .

(1)

While this is a formulation which is mathematically simple and easily implemented in a numerical code, it is not capable of describing the wave excitation very well. Depending on the location and weather, wave excitation forces are better approximated by a narrow-banded process. As a second model for the excitation we take (2) fe (t) = y y¨ + 2DF ωF y˙ + ωF2 y = ξ ,

(3)

which constitutes a simple filter with the parameters DF and ωF . When the input of this filter ξ is random white noise, this filter gives a narrow-banded spectrum as shown in Fig. 1.

Fig. 1: Sample trajectory (left) and spectrum (right) of a narrow-banded filter, corresponding to (3) Considering that seaway spectra have often been analyzed experimentally and are commonly described by well-known spectral forms such as the JONSWAP spectrum αg 5ωm4 (4) SJ (ω ) = 5 exp − 4 γ r , ω 4ω filters can also be designed to give the required shape. One way to achieve this is the use of auto regressive moving average (ARMA) filters, given by m n −k −k 1 + ∑ dk z . (5) H(z) = ∑ ck z k=0

k=0

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The coefficients ck and dk of the filter are chosen such that the difference between the filter’s output signal and the required seaway spectrum is minimal. Fig. 2 shows a sample trajectory and the spectrum obtained from an ARMA-filter with orders m = 5 and n = 5.

Fig. 2: Sample trajectory (left) and spectrum (right) of an ARMA-filter (m=5, n=4)

2.2 Modeling of multibody systems The dynamical behavior of ocean systems can come from different sources such as the fluid structure interaction or the coupling of different connected bodies. For a moored floating body such as a buoy or a barge, the catenary system contributes a progressive stiffness term in the equations of motion. So as an approximation we obtain the classical Duffing equation x¨ + γ x˙ + (1 + ε x2 )x = fe (t)

(6)

with the coefficients γ , ε . As the Duffing oscillator has been studied previously in various different contexts and is therefore used to test the numerical techniques described below. A second model is given by ˙ x| ˙ + bx x˙ + c1 x + c2 x|x| + c3 x3 = fe (t), (m p + ax )x¨ + 0.5BT cd ρ x|

(7)

which constitutes a simple one degree-of-freedom model of a moored barge considering the surge motion x. Herein, m p is the mass of the barge, ax is the added mass, bx added damping, B and T are width and draft, cd the drag coefficient, and ci are the coefficients describing the mooring system. As a next step of a gradually refined model, the model of a barge is extended by adding a crane with a swinging load as indicated in Fig. 3. This gives a simple multibody system M(y)¨y + k(y, y˙ ) − q(y, y˙ ) = fe (t), (8) where M is the total mass matrix

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Fig. 3: Model of a floating crane ⎡

M =⎢⎣

m p + ml + ax lml cos α

lml cos α −l 2 m

⎤ ⎥ ⎦,

(9)

l

the vector of Coriolis and gyroscopic forces k is ⎡ ⎤ −lml α˙ 2 sin α ⎦, k=⎣ 0 and the vector of external forces q is ⎤ ⎡ ˙ x| ˙ − bx x˙ − c1 x − c2 x|x| − c3 x3 −0.5BT cd ρ x| ⎦. q=⎣ −glml sin α

(10)

(11)

Herein the mass of the load is denoted by ml , the length of the hoisting rope is l, the swing angle is α and g is the acceleration due to gravity.

3 Analysis of deterministic systems Before addressing the analysis for the cases of a random excitation, the results for harmonically forced oscillators are reviewed. There are various techniques which allow for the approximation of bifurcation diagrams such as numerical path following [1] or different perturbation techniques [6]. For the periodically forced Duffing oscillator (6) the amplitude of the motion depends on the excitation frequency: It is well-known that for a certain range of the parameter ω there are two different stable solutions. The initial condition uniquely defines which of these two attractors will be obtained as time t → ∞. This co-existence of different steady-state solutions is similar for other nonlinear dynamical equations. Figs. 4 and 5 shows the different attractors for the moored barge (7) and their basins of attraction. As the attractors shown in Fig. 4 describe the dynamical behavior under the assumption that the excitation is purely harmonic, the question arises as of how much it changes when a random perturbation is included in the model of the forcing as described in section 2.

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Fig. 4: Different attractors in phase space for the model of a moored barge

Fig. 5: Basins of attraction (in black) for the attractors given in Fig. 4

4 Analysis of random systems In the case of a narrow-banded random forcing function, the response of each of the random systems described above may still be near the unperturbed solution but occasional large deviations may occur. This also includes the possibility of jumps between the areas around the attractors of the dynamically forced system. In order to describe this effect, this section addresses the numerical determination of probability density functions.

4.1 Monte Carlo simulation For the analysis of randomly forced dynamical systems, the Monte Carlo simulation is a versatile and well-known technique. It allows for the numerical approximation of probability density functions or first passage times. Monte Carlo simulation is

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discussed extensively in the literature. Here, only some results obtained from Monte Carlo Simulation will be used for comparison.

4.2 Stochastic linearization An alternative approach for the investigation of nonlinear random systems is the local statistical linearization which has recently been proposed [7]: While nonlinearities might be significant for the behavior of a specific system in the global sense, it might still be useful to apply linearizations locally. At the core of the technique, the difference between the nonlinear and an equivalent linearized system is to be minimized in some appropriate sense. Linearization techniques were used in very different areas of engineering and mathematics and provide efficient tools to treat even highly complicated systems, but usually do not address any specific nonlinear effects. Here, the linearization is applied locally in the vicinities of a number of linearization points, thus the overall structure of the nonlinear system can be approximated closely. The process here describes the change in shape of a probability density as time evolves. Considering a single Gaussian distribution as the initial condition, this distribution is shifted and stretched under a linear transformation but it keeps its Gaussian shape. The linearization gives a good approximation as long as the standard deviation is sufficiently small. If the initial distribution is described as the sum of components each with a Gaussian shape, the mapping is applied to each component individually. This procedure results in a time-stepping scheme in which each distribution {pi , μi ,Ci }t with the probability component pi , the mean value μi and the standard deviation Ci is mapped onto a new distribution in the next time step t + Δ t {pi , μi ,Ci }t → {pi , μi ,Ci }t+Δ t

∀i.

(12)

Distributions for which a prescribed maximum of the standard deviation is exceeded, are split into a sum of distributions with a smaller standard deviations before applying (12). On the other hand, distributions that contract in a part of the state space can efficiently be combined. For details on the technique we refer to the literature [7].

5 Selected Results The parameters used for the integration can be found in Table 1. Under the assumption of periodic forcing, this system has shown sub- and superharmonic responses over a certain range of operating conditions. Here, we present what happens in slightly less idealized and randomly perturbed conditions.

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Table 1: Parameter for the barge Aw B Iyy T cd

= = = = =

1.4 · 103 m2 25 m 936 · 103 kg·m2 1.69 m 0.2

c1 c2 c3 g hm

= = = = =

21.2 · 103 N /m 9.44 · 103 N /m2 13.8N /m3 9.81m /s2 163 m

mp xs zs ρ

= = = =

1920 · 103 kg −4.1 m −3.9 m 1 · 103 kg /m3

Fig. 6 depicts the influence of a small perturbation while the main component of the excitation is nearly harmonic corresponding to the excitation given in (1).

Fig. 6: Influence of an added disturbance, time histories and probability density functions It can be seen that for small perturbations the system’s response remains in certain bounds for a long time while sudden jumps to a completely different range of amplitudes may occur. With an increased amplitude of the perturbation — relative to the amplitude of the harmonic component of the excitation — the jumps become more frequent until the different ranges can no longer be distinguished. This behavior also becomes obvious in the probability density plots given in Fig. 6. For small disturbances, different areas with higher probabilities can clearly be

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distinguished from areas where the probability density is practically nil. For higher disturbances the areas become more blurred. An example generated by a narrow band filter is given in Fig. 7 and the corresponding probability density function.

Fig. 7: Excitation by narrow band filter, time histories and probability densities The results show a similar behavior as with the added disturbance. Here, the sharp shape of the pdf becomes blurred as the frequency spectrum widens. Fig. 8 shows an example trajectory in which a JONSWAP spectrum has been used for the excitation in connection with an ARMA-filter and the corresponding PDF. As a final example Figs. 9 and 10 refer to the model of a floating crane. Again, both Figures refer to the same numerical example only at different time steps. It should be noted that for the parameters used here, the crane would show a period-3 motion for harmonic forcing, very much like attractor 4 in Fig. 4. Fig. 9 shows the probability density averaged over the first three periods. In the surface plot, the peak of the initial condition at [0, 0]T is very pronounced, which means that the different components of the distribution quickly spread out in different directions. On the other hand, the contour plot shows that the maximum in each time-step (dotted line) still follows the trajectory of the deterministic system (dark solid line) closely. Only a few periods later, the trajectory corresponding to the deterministic system shows an almost closed loop corresponding to the period-3 motion, Fig. 10.

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Fig. 8: Excitation by JONSWAP spectrum (filter), time histories and probability densities

Fig. 9: Probability density for moored crane as surface and contour plots, averaged over [t = 0;t = 3T ] — The dark solid line in the contour plot refers to the trajectory of the corresponding deterministic system, the dots to the maxima of the probability density in each time-step

The probability density for this case shows two peaks which no longer correspond to the initial conditions but rather to the loops in trajectory, i.e. places in phase space, where the system passes by relatively several times over three periods of the forcing.

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Fig. 10: Probability density for moored crane as surface and contour plots, averaged over [t = 4T ;t = 7T ] — The dark solid line in the contour plot refers to the trajectory of the corresponding deterministic system

6 Conclusions Different models for simple offshore structures and the excitation due to ocean waves are described and analyzed numerically. Two techniques, Monte Carlo simulation and statistical linearization, are used for the determination of probability densities. The examples are chosen such that they give a complex dynamical behavior for the given parameters. The results imply that there is no single technique which is applicable to all models the environmental forces equally well in all circumstances: They differ in the amount of numerical effort and also the precision by which the characteristics of environmental effects are met in the mathematical model. Acknowledgements The support of the German Research Foundation within the Emmy Noether Program is greatfully acknowledged.

References 1. Allgower EL, Georg K (1993) Continuation and path following. Acta Numerica pp. 1–64 2. Cai GQ, Lin YK (1996) Exact and approximate solutions for randomly excited mdof nonlinear systems. International Journal of Non-Linear Mechanics 31(5):647–655 3. Ellermann K (2003) Verzweigungsuntersuchungen meerestechnischer Systeme. Fortschritt-Berichte VDI, Reihe 11, Nr. 318, VDI Verlag GmbH, D¨usseldorf 4. Ellermann K (2005) Dynamics of a moored barge under periodic and randomly disturbed excitation. Ocean Engineering 32:1420–1430, http://dx.doi.org/doi:10.1016/j.oceaneng.2004.11.004 5. Jensen JJ (2001) Load and global response of ships. Elsevier Ocean Engineering Book Series, Elsevier, Amsterdam/. . . 6. Nayfeh AH (2000) Nonlinear Interactions — Analytical, Computationals, and Experimental Methods. John Wiley & Sons, Inc., New York 7. Pradlwarter HJ (2001) Non-linear stochastic response distributions by local statistical linearizations. Non-linear Mechanics 36:1135–1151 8. Tucker MJ, Pitt EG (2001) Waves in Ocean Engineering. Ocean Engineering Books Series, Elsevier Science, Amsterdam

Statistical Dynamics of the Rolling Mills Paul V. Krot

Abstract Uncertainty of technology and equipment parameters is particular for the rolling mills dynamics. Contact friction produces a wide band stochastic impacts which cause chatter vibration in the middle and higher natural frequencies range (100–1000 Hz) of the mill. Strip elasto-plastic deformation as a nonlinear spring in the stand depends on random technology parameters (rolls bending, strip tensions, mill speed). The transfer functions were used for principal mode distribution and chatter vibrations control in the cold rolling mills. The second cause of uncertainty is wear (backlashes) which makes drive train an essentially nonlinear system. The backlashes gaps and rolling loads are always uncertain at the beginning of transient process in the hot rolling mills causing dynamic torques scattering in the multibody drive trains. The low frequency (10–20 Hz) torsional vibrations are investigated. The output dynamic load distribution parameters were obtained. Statistical aspects of the mill control and diagnostics are discussed.

1 Introduction The plants subjected to random loadings and having uncertainty of parameters cover a wide area of scientific research and applications which number includes rolling mills and their drive trains. The rolling mills equipment is operated at a wide range of vibration impacts and the torsional torques. It causes excitation of the difficultly predicted chatter vibrations, rolls and costly gears failures since designers and maintenance staff not always know statistical parameters of loads and current wear of the elements as angular and radial backlashes. Numerous books are commonly known in dynamics of randomly loaded structures [1] also with elasto-plastic deformation [2]. These problems are characteristic for many kinds of heavy industry equipPaul V. Krot Iron and Steel Institute of Ukraine National Academy of Science, ac. Starodubov sq. 1, 49050, Dnepropetrovsk, Ukraine, e-mail: paul [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 31, © Springer Science+Business Media B.V. 2011

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ment, e.g., mining or drilling machines with the unpredictable material parameters and inclusions [3]. Different methods have been developed for random systems analysis: stochastic and fuzzy finite element techniques [4]; statistical energy analysis; random matrix theory and transfer matrixes [5] with the frequency response functions (FRF) analysis. Modal density parameter and statistical overlap factor are used for eigenvalues distribution [6]. Statistical analysis of the dynamical systems has a close relation to a control theory. Modern strategies of the rolling mills control are based on adaptive models including fuzzy logic algorithms and adaptive neural nets for parameters prediction to meet very high demands on steel strips flatness, roughness and thickness tolerance (±5 microns). Nevertheless two basic problems remain not solved to the end in the rolling mills dynamics. The first problem is chatter vibration in the high speed thin strip cold rolling mills where standard Hydraulic Automatic Gauge Control (HAGC) systems are not able to control high frequency friction oscillations (100–200 Hz) because the cut frequency of the control loops is much less (about 10–15 Hz). The tuned mass dampers are not quite reliable in the rolling mills because of stochastic parameters and natural modes variation. Therefore active vibration control is required for chatter suppression. An improved HAGC with the increased flow rate, fast switching valves and novel control method has been developed for chatter problem solving [7], [8]. Another approach is developing based on vibration monitoring of chatter for its early diagnostics and rolling parameters control [9]. Some recent works are known in the field of chaotic vibrations in a cold rolling [10], [11] and it is a subject for further developing. Chaotic vibrations in metals cutting were already studied [12] and methods of friction instability early detection (diagnostics) are developed for their active control [13]. The essential complexities arise with the analysis of adjacent stands interaction in the rolling mills due to regenerative mechanism of chatter. It was also considered but by the deterministic models with a constant time delay [14]-[16]. Strip elasto-plastic properties and its material damping ability were investigated in [17]-[22]. The second problem of the rolling mills dynamics is the transient torsional vibrations in the drive trains during strip capture by the rolls. State of art in the deterministic models of rolling mills had been outlined in [23]-[26]. Parametrical vibrations are considered in [27] due to periodic gear coupling stiffness variation. The fewer studies are known on statistical dynamics of the rolling mills because of difficulties of analytical description of the random loads and model parameters for the multibody systems with the essential nonlinearities. Basically it was carried out a durability calculation and loads spectrums identification [28], torque amplification factors (TAF) and damping [29], strips geometry spectral and correlation analysis [30]. Excitation of chaotic vibrations in torsional system with the backlashes under periodic loading is investigated in [31], [32]. Some methods of backlashes identification (diagnostics) in the transmissions and gearboxes were proposed [33]. Methods of model based control and intelligent observers were investigated to prevent friction instability in the hot rolling mills [34].

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2 Cold Rolling Mills Chatter Vibrations 2.1 Rolling stand design and its modal analysis Rolling stand design is described below (see fig. 1a) so that to explain where stochastic and controllable stiffness change can appear during work. A 4-high cold rolling stand (quarto) consists of massive cast housing fixed by four feet, the top and bottom set of work rolls (WR) and backup rolls (BUR), placed with its necks into chocks with the roller bearings and hydrostatic sliding bearings accordingly. The chocks are placed into the housing where side backlashes appear (designated as δ1 and δ2 in fig. 1a) because of removable flat liners deterioration. Therefore stands are designed so that WR chocks are shifted forward (last stands) or backward (first stands) from BUR axis by a = 10 − 15 mm. There are four hydraulic blocks fitted to each rack of housing which contains 4 big cylinders for upper BUR balancing during stand maintenance (WR changing) and 8 or 16 (depending on stand design) smaller cylinders which are involved in strip flatness regulation by the positive or negative WR bending. Bending cylinders are regulated automatically in the last stand or manually in the previous stands depending on the WR wear and strip flatness. In general WR bending causes random contact stiffness deviations between the WR and BUR.

Fig. 1: Design of stand quarto (a), spring-mass model (b) and vibration modes (c) The two cylinders for rolling load creation and strip thickness regulation are placed between the upper cross-beam and BUR chocks. Some mills have HAGC cylinders at the bottom of stands. The small changes of stiffness occur due to HAGC oil column and rolls diameters. Deformation of every element has influence on the stand summary module (4–6 MN/mm). In order to obtain vibration response spe-

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cific for different measuring points a detailed spring-mass model has been designed (see fig. 1b) where masses are ×103 kg: m0 (80.0) — upper cross-beam of housing with the HAGC units; m1 (26.7) — upper BUR chocks; m2 (45.0) — upper BUR; m3 (8.7) — upper WR with chocks; m4 (9.1) — bottom WR with chocks; m5 (45.0) — bottom BUR; m6 (24.3) — bottom BUR chocks. The stiffness parameters are (MN/mm): C07 (29.01) — housing racks and top cross-beam.; C01 (22.02) — HAGC cylinders and upper BUR chocks; C12 (87.99) — oil film of sliding bearings and upper BUR bending; C23 (19.89) — upper WR and BUR contact; C34 (20.00) — WR bending with the contact indentation and strip deformation; C45 (18.89) — lower WR and BUR contact; C56 (87.99) — oil film of the sliding bearings and lower BUR bending; C67 (38.95) — lower BUR chock and rolling line tuning jack; C17 (25.00) — upper BUR weight balancing cylinders; C37 = C47 (15.00) — WR negative bending cylinders; C13 = C46 (15.00) — WR positive bending cylinders. Unlike all known models the WR bending and BUR balancing forces are included as stiffness instead of external forces. Such approach allows fulfilling a statistical analysis of the stand vibration under the variable operating schedules and helps to find methods for chatter active control. Stand vibration is assumed symmetrical for both operator and drive side (without skewness) because that is proved by measurements. Positive movement is assumed upward. The natural frequencies spectrum for the average values of stand parameters covers a band of 50–600 Hz. It was shown in work [35] by the detailed FEM analysis that 4-high rolling stand cumulative mass fraction is about 0.80–0.85 for the first 4–9 modes. Also experimental modal analysis of the rolling stand was fulfilled in [36]. The modes of vibrations are represented in fig. 1c. It is commonly admitted that feedback loop which created in the rolling stands by the rolling load, strip thickness, tensions and speed interdependence is a main cause of instability [37]-[40]. Tandem mill chatter excitation by this mechanism can occurs only due to symmetrical modes near the strip plane with the opposite phases of the upper and lower pairs of WR and BUR motion. Chatter excitation through the strip thickness variation (δ h = 20 − 50 microns) can only be initiated by the modes with the several masses movement near the strip plane. The WR out of phase movement cause only strip or BUR periodic defects (Ra = 2 − 3 microns). Vibration measurements on the four rolls chocks have shown that for the thick strips (h1 > 0.8 mm) chatter is not exciting because vibration node is note in the WR gap [16]. The results of the statistical calculations show that newly introduced parameters of stiffness (C13 , C46 , C37 , C47 ) change natural frequencies and mode shapes. Some of frequencies change more: N3, N6 or less (see fig. 2a). In practice first mode does not appear. Node displacement of the principal vibration mode (∼ 118 Hz) can be achieved due to the HAGC stiffness decreasing by 25% from the nominal value or increasing by 50% the rolling level tuning unit stiffness at the bottom of stand. Both actions can be done only before rolling process during mill set-up procedure. Bending system of the WR influences on stand principal mode frequency not significantly (N1 in fig. 2a). But bending stiffness variation can move node from the masses 2 and 3 interval into the masses 3 and 4 gap (see fig. 2b). Therefore it can be used for chatter active control not only by the high frequency periodical impacts

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but also due to comparatively slow stiffness control in the stands. It does not require oil pumping station with a high flow rate and fast switching valves to produce quick control impacts of large amplitude.

Fig. 2: Natural frequencies (Hz) scattering (a) and principal mode deviation (b)

2.2 Strip Elasto-Plastic Deformation The rolled strip elasto-plastic properties play a main role in the mill vibration. By the various estimations, the rigid-plastic theory can serve until the fraction of external energy to the whole energy of elastic deformation of a material, at least, not less than three. It becomes important in cold rolling of thin strips where elastic restoring of strip thickness (h1 − he ) is comparable with an absolute reduction (h1 − h0 ) especially in the last stands of the tandem mills. Strip deformation scheme is represented in fig. 3a. In a modern rolling theory the strip properties are described by the nonlinear formulae including an initial yield stress, tensions, strain and strain rate with other rolling parameters (see [41]). Statistical deviations of the strip properties and rolling parameters as a rule are not taken into account. It does not allow reliable predicting the dynamic phenomena of chatter vibrations in the rolling mills. The strip stiffness Ks and its viscous damping Cs are described with the fraction derivatives of a rolling load deviation by the strip thickness reduction and its deformation rate. Also dynamics sensitivity factor λ is introduced to estimate strip deformation rate: KS = −

∂P , ∂ h1

CS = −

∂P , ∂ h˙ 1

λ=

∂σ ∂ lg(d ε /dt)

Stand and strip stiffness graphs are represented in fig. 3b. An initial nonlinear stand deformation b due to backlashes is eliminated by the stand preloading P0 . There is a variety of operation points around a mill set-up load value P∗ and thickness h1 . Curve (1) corresponds to back and/or front tensions increasing; curve (2) — to a less input thickness; curve (3) — to the working point of mill set-up; curve (4) — to an increment in the output thickness and curve (5) — to a possible incre-

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ment in the friction coefficient and in the average yield stress or a decrement in the back and/or in the front tensions. It can be seen from fig. 3b that random rolling parameters can change strip stiffness significantly (tangent of inclination angle at a working point) and hence natural frequencies and modes. It is known that a quick

Fig. 3: Strip and WR contact scheme (a); stand and strip deformation diagram (b) jump up of λ factor appears when d ε /dt > 103 s−1 . It is particular for the material hardening properties. Strip deformation conditions in the 5-stand tandem mill 2030 (length of WR) are given in Table 1. Parameters combination gives minimal

Table 1: Strip rolling parameters in the 5-stand tandem cold rolling mill 2030

strip damping value in stands N3 and N4 which are most susceptible to chatter vibrations in practice. Deformation velocity is rising up to stand 4 and is close to critical value 103 s−1 . The mill automatic monitoring systems or mill operators for chatter canceling commonly use rolling speed decreasing which is addressed to the

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strip deformation rate and its damping ability respectively. Beside it strip reduction schedule is proposed for chatter control due to more damping in the strip material in the last stands. It was tested in the stands and approximately every step down by 1% of ε gave 20 m/min additional mill speed. Also strip specific tension reducing by 5 N/mm2 gives maximal speed increasing by 40–50 m/min [16].

2.3 Horizontal work rolls vibration The chocks and stand housing wear give to WR an additional degree of freedom in the backlashes gap and makes it possible vertical and horizontal vibration modes interaction. WR and its chocks may start vibrate within stand housing gap under the random variation of the strip tensions difference Δ T (tension rollers misbalance), rolling load P (rolls eccentricities, strip yield stress) and M (drive torque fluctuations and torsional vibrations). The measurements of horizontal forces were carried out in the tandem mill stands [42] and housing backlashes size was restricted in the maintenance rules. Some companies (SMS Demag, Asko and Dofasco) have patented the flat hydraulic cylinders for chocks pressing to the housing sides during rolling. But it was shown in [43] by experiments that critical rolling speed increases only by 10%. Also restriction of the WR chocks vertical motion worsens HAGC operating conditions. Possibly first probabilistic approach to chatter vibrations based on WR chocks horizontal instability in the tandem cold rolling mills has been done in [44]. Simplified empirical criterion was derived for chatter probability estimation: n 1 E ·UCi · eΣ i Πi (1) ∑ Πi ≥ n , Πi = hi · σ0i · fi · Fi , i=1 where Πi — dimensionless chatter probability factor; i — stand number; E — strip modulus (MPa); Uci = V1 α /h0 — mean deformation rate (s−1 ); eΣ i — summary WR and BUR eccentricity (mm); σ0i = σinit +m(100(H −hi−1 )/H)K — yield stress at ith stand entry (MPa); σinit , m and K — initial yield stress before rolling and material hardening constants; H, h0 and hi — initial, entry and exit strip thickness (mm); fi — contact friction factor; Fi — chatter vibration main frequency (Hz, s−1 ); n — number of stands. Condition (1) means chatter increased probability in ith stand if it more than 1/n value (equal for all stands probability). But only rolling load component and tensions differences are used in [44] for horizontal forces balance composition. Therefore new formula is proposed here for steady state condition: 0<

ΔT M ·i·η a·P − + F · Kf − , RBUR + RW R 2 RW R

(2)

where P — rolling load; a — WR chocks displacement; RBUR and RW R — BUR and WR radiuses; F and K f — summary bending force and friction factor in the WR chocks and piston contacts; Δ T = T0 − T1 — strip entry and exit tensions difference

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(T0 > T1 is assumed); M — drive torque; i and η — drive train gears ratio and efficiency (0.8–0.9). It was taken into account WR bending forces and reactions from drive torque at the WR chocks. Every component in equation (2) may be used for comparing with others. In practice a half difference of the strip tensions is used as a most scattered parameter.

2.4 Contact friction force variation When any of the surface defects pass contact zone a stochastic wide band input disturbance appears which causes stand vertical vibration in the middle and higher frequency range (100–1000 Hz). The maximal bound of frequency is determined by the rolling speed and defect size along the strip in the rolling direction. Intensity of disturbances is dependable on the defect size across the strip including imperfection dots of the cooling liquid film. Contact friction stability is estimated in the rolling theory by the neutral angle γ (see fig. 3a) and strip slip forward which is determined by the strip and rolls velocities: S = (V1 − VW R )/VW R . The both parameters S and γ are related by a simple formula: S = γ 2 RW R /h1 . If to admit constant angle α of contact, WR radius R and mean specific contact pressure p the condition of steady rolling process is as following: T1 (t) − T0 (t) α α , 1− > 2 2 · f (t) 4 · f (t) · p · b · R where α — contact arc angle; f (t) — variable contact friction factor; T0 (t), T1 (t) — entry and exit strip tensions (T0 > T1 ); p — specific contact pressure; b — strip width; R — indented WR radius. It was noted in [16] that despite friction factor f (t) dependence on many rolling parameters the most critical for mill stability parameter is a WR temperature because it causes minimal point displacement in the friction and speed relation f = F(V ) to the higher speeds while all other parameters shift lowest point of critical friction only up and down. But another mechanism exists of friction control by the vibration which is commonly used in ultrasound materials processing techniques. For the description of vibration influence on friction without contact loosing the “effective friction factors” were introduced in [45]: Φ0 Φ0 2 Φ0 , fN = f · 1 − fS = f · 1 − , fs = f · 1 − , f ·N f ·N N where f — initial friction factor without vibration; N — normal load; Φ0 — amplitude of periodical load Φ = Φ0 sin ω t; f S — factor for tangent horizontal vibrations parallel to strip displacement (tensions vibration); fs — factor for tangent horizontal vibrations normal to strip displacement (axial WR vibration); fN — factor for vertical vibrations normal to strip displacement (vertical WR vibration). If to con-

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sider WR or BUR eccentricities then periodical force is: Φ0 = m0 eω 2 , where m0 e — static misbalance, ω — frequency of rolls rotation. Under the stand rolling load vibration by ±5% and tensions by ±15% a friction factor nonlinearly (proportionally to ω 2 ) is decreasing (maximal drop is up to −30%) with a WR speed increasing. Beyond a certain limit of speed or rolling parameters deviations rolling process stability may be violated and then WR slipping with chatter appears. It is worth to note that term “effective friction factor” is not identified with a physical friction factor but it only formalizes friction dependence on vibration. Alternative approach to chatter excitation is developed in [46] based on distributed friction model.

2.5 Chatter detection and control The different methods are known for chatter detection and control in metal processing [47]. Some techniques may be suitable for rolling mills but are not yet applied anywhere. Taking into account the cold rolling process particularities the new methods of chatter control were proposed based on adjacent stands vibration analysis [48]. Strip material transition from elastic to plastic deformation is essential to rolling process and nonlinear strip stiffness will always influence on vibration spectrum. That is a physical basis for chatter early detection [16]. Chatter diagnostics is provided due to use of the commonly known physical phenomenon — synchronization of the mechanical oscillatory systems due to elastic links between them. The rolling stands in tandem mill are synchronizing through the elastic strip. Synchronization is accompanied by the “grabbing” of close natural frequencies of adjacent stands and drift of their phases of vibration to certain current value, identical in all stands. It results in frequencies deviation decreasing from average value and the correlation coefficients increasing in each pair of stands. Vibration frequencies and phases begin to change even for insignificant exchange of energy between stands. The two main conditions for chatter early detection were derived as: σn ≤ σmin , Kmean ≥ Kmax , where σn — current root mean square (RMS) deviation of the principal modes frequencies in the adjacent stands within a chatter frequency range (Hz); σmin — minimal difference equal to spectrum frequency step (Hz). In the second condition Kmean — mean correlation coefficient of vibration in the n − 1 pairs of adjacent stands; n — number of stands; Kmax — maximal correlation coefficient for steady rolling without chattering (Kmax < 0.2 − 0.3). Chatter is detecting by the 5–8 s earlier than by spectrum maximum amplitude alarming separately in every stand as it implemented in the known vibration monitoring systems.

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3 Hot Rolling Mills Torsional Vibrations The hot rolling mills typical geared drive train includes (see fig. 4a): 1 — strip; 2 — WR; 3 — spindles with the sliding bushes or cardan universal joints; 4 — pinion stand; 5 — root coupling; 6 — one or two stages gearbox; 7 — motor shaft coupling; 8 — single or twin electric drives (5–12 MW). Quick wear causes angular and radial backlashes which characteristic with a dead zone see in fig. 4b. Backlash is opened before transient for uncertain part δ1 . Then coupling stiffness is tan(β ).

Fig. 4: Geared drive train of the hot rolling mill (a) and backlashes characteristic (b) The angular and radial backlashes make a drive train essentially nonlinear multibody system. Drive train peak torque Mmax produced by input static load Mst and corresponding torque amplification factor (TAF = Mmax /Mst ) can be described by a polynomial function: Mmax = a0 + a1 · Mst + a2 · Mst2

(3)

where ai — constants (vary in time of mill operation) describing drive train design and mill current technical condition; Mst — rolling torque applied to WR. Constant a0 describes drive train losses, a1 summarize linear system design and a2 takes into account nonlinear properties. Coefficient a1 corresponds to TAF which for the linear systems is constant and only depends on the spring-mass parameters and input load rate (front edge profile and temperature of slabs). Damping principally influences on transient duration. The nonlinear functions Mmax = F(Mst ) and TAF = F(Mst ) obtained from the hot rolling mill torques measurements are given in fig. 5 where discrete values correspond to certain Mst levels during mill operation. Measurements have shown that input load Mst has normal distribution. Mean value Mst∗ and standard deviation σMst of the static load were determined by electrical ∗ motor current. A mean value of the output dynamic load Mmax can be described as: ∗ Mmax

∞

= −∞

Mmax · P(Mst ) · dMst ,

(4)

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Fig. 5: Nonlinear statistical relations of Mmax (a) and TAF (b) by the static load Mst

where P(Mst ) — normal density distribution function of Mst (for simplicity variable Mst is taken with a zero mean, that does not influence on generality of result). Then substituting Mmax from (3) into (4) and integrating by Mst gives: ∗ Mmax

=

∞ 0

a0 + a1 Mst + a2 Mst2

1 √

σMst 2π

−

e

2 Mst 2 2·σM st

2 dMst = a0 + a2 · σM st

(5)

2 Mean value of Mmax linearly depends on dispersion σM of input load Mst . It is st similarly possible to show that a dispersion σMmax also depends on a dispersion of the input load σMst 2 4 σMmax = a1 · σM + 2 · a22 · σM (6) st st

Based on the above derived relations (5) and (6) the output loads statistical parameters can be estimated for every coupling along the drive trains. Obviously every coupling will have its own ai coefficients varying in time of mill operation.

3.1 Torsional vibration control and backlashes diagnostics Statistical analysis of the input loads allows improving continuous hot rolling mills control from the dynamics viewpoint. Spectrums of input loads in the five consecutive stands of a continuous hot rolling mill are represented in fig. 6 both for Mst and Mmax . The fields of scattering look like the discrete lines due to different rolled strip thickness. Input loads frequencies are calculated as a double time for strip biting in WR. Because of rolling speed is increasing due to strip elongation the input load frequencies shift to a higher band with a stand number. Also frequency response function (FRF) of the motor shaft is shown in fig. 6. Mill stands have almost equal lowest natural frequencies of the drive trains. So as it could be seen from fig. 6 the input load frequencies lay in the resonance bands of the stands N4 and N5. Therefore drive trains dynamics can be reduced by the strip reductions control in the stands in order to avoid loads scattering fields coinciding with the resonance bands in every stand. But output strip thickness should re-

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Fig. 6: Input loads spectrums for the five stands and the FRF of motor shaft

main the same. Implementation of such control strategy in the continuous hot rolling mills requires backlash values which can be identified based on measurements data nonlinear regression because curves Mmax (Mst ) and TAF(Mst ) inclination is proportional to backlash gap. The kurtosis and skewness of dynamic loads distribution may be used as the additional parameters for diagnostics.

4 Conclusions Uncertainty accounting of the cold and hot tandem rolling mills technology and equipment parameters is essential for effective active control of chatter vibrations in the stands and torsional dynamics monitoring in the geared drive trains. Acknowledgements Author is thankful to colleagues from the Iron and Steel Institute and people from the industrial plants for assistance with the experimental measurements. Also author is appreciated to the IUTAM Symposium organizing committee for financial support.

References 1. Bolotin VV (1984) Random Vibration of Elastic Systems. Martinus Nijhoff Publ., p 484 2. Palmov VA (1998) Vibrations of elasto-plastic bodies. Springer-Verlag, p 311 3. Dokukin AV, et al. (1978) Statistical dynamics of the mining machines. Mashinostroenie, Moscow, p 239 (in Russian) 4. De Gersem H, et al. (2004) Interval and fuzzy element analysis of mechanical structures with uncertain parameters. In: Proc of the Int. Conf. on Noise and Vibration Engineering ISMA2004. Leuven, Belgium 5. Langley R (1996) A transfer matrix analysis of the energetics of structural wave motion and harmonic vibration. Proceedings of the Royal Society of London Series A 452:1631–1648 6. Langley R (1999) A non-Poisson model for the vibration analysis of uncertain dynamic systems. Proceedings of the Royal Society of London Series A 455:3325–3349 7. Schlacher K, Fuchshumer S, et al. (2005) Active vibration rejection in steel rolling mills. In: Proceedings of the 16th World IFAC Congress. Prague, Czech Republic

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8. Keintzel G, et al. (2008) Active chatter damping in cold rolling mills — Siemens VAI’s revolutionary product. Rolling and Processing Conference. Linz, Austria 9. Prykhodko IY, Krot PV, et al. (2006) Vibration monitoring system and the new methods of chatter early diagnostics for tandem mill control. In: Proc. of the Int. Conf. “Vibration in rolling mills”. Inst. of Materials, Minerals and Mining, London 10. Luo Z, Wang X, et al. (2005) Modeling and prediction of violent abnormal vibration of large rolling mills based on chaos and wavelet neural networks. In: Proc. of the 2nd International symposium on neural networks, Chongqing, China 11. Bar A, Bar O, Swiatoniowski A (2006) Vibration of mill rolls during the cold rolling of a steel strip by the “chaotic approach”. In: Proc. of the “Steel Rolling 2006”, the 9th International and 4th European Conference, France 12. Wiercigroch M, Cheng AH-D (1997) Chaotic and stochastic dynamics of orthogonal metal cutting. In: Proc. of the 1st Int. Symposium “Multi-Body Dynamics: Monitoring and Simulation Techniques — COMADEM’97”, University of Bradford 13. Al-Regib E (2000) Machining systems stability analysis for chatter suppression and detection. Ph.D. dissertation, University of Michigan 14. Hu PH, Ehmann KF (2001) Regenerative effect in rolling chatter. Journal of Manufacturing Process 3(2):82–93 15. Chen Y, Liu S, et al. (2002) Stability analysis of the rolling process and regenerative chatter on 2030 tandem mill. Proc. Inst. Mech. Eng., Part C: J. of Mechanical Engineering Science 216(12):1225–1235 16. Krot PV, et al. (2008) Regenerative chatter vibrations control in the tandem cold rolling mills. In: 4th European Conference on Structural Control, ECSC 2008. Eds: A.K. Belyaev, D.A. Indeitsev and H. Irschik, IPME RAS, St. Petersburg, Russia 17. Dukmasov VG (1987) Dynamic Features of Rolling Stand-Strip System. Steel in Translation 17:177–180 18. Pawelski O, et al. (1988) Calculation of the vibrational behavior of high-speed cold rolling tandem mills. Stahl und Eisen 108(7):49–54 19. Johnson RE (1994) The effect of friction and inelastic deformation on chatter in sheet rolling. Proceedings of the Royal Society of London Series A 445:479–499 20. Guo RM (1994) Material damping effect in cold rolling process. Iron and Steel Engineer 71(1):29–39 21. Forouzan MR, et al. (2008) Analysis of chatter vibration in cold strip rolling. Part I: system equivalent damping. In: Proc. of the 12th Int. Conf. “METALFORMING-2008”, Krakow 22. Yarita I, Furukawa K, et al. (1978) An analysis of chattering in cold rolling for ultrathin gauge steel strip. Trans. of the Iron and Steel Inst. of Japan 18(l):1–11 23. Kozhevnikov SN (1975) International symposium on the dynamics of heavy machinery of the mining and metallurgy industries. Int. Applied Mechanics 11(4):456–460 24. Bol’shakov VI (2001) Loads and Strength of Metallurgical Equipment. Materials Science 37(2):311–318 25. Larin VB, et al. (1997) Problems of the dynamics of rolling-mills (review). Int. Applied Mechanics 33(3):175–193 26. Adamia R (1978) Optimization of rolling mills dynamic loadings. Metallurgy, Moscow, p 232 (in Russian) 27. Krot PV (2002) Parametrical vibrations in the rolling mills. Trans. of the Dnepropetrovsk National Mining Academy 3(13):15–21 (in Russian) 28. Griese F, Schweer W, et al. (1963) Investigations on the statistical laws for spindle torques and housing loads in a blooming mill train. Stahl und Eisen 83(12):715–723 29. Kotsar SL, Polyakov BN, et al. (1974) Statistical analysis and mathematical modeling of blooming. Metallurgy, Moscow, p 280 (in Russian) 30. Zheleznov YD, et al. (1974) Statistical research of accuracy of the thin strips rolling. Metallurgy, Moscow, p 274 (in Russian) 31. Li H, Wen B, Zhang J (2001) Asymptotic method and numerical analysis for self-excited vibration in rolling mill with clearance. Shock and Vibration 8:9–14

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32. Tjahjowidodo T, et al. (2007) Quantifying chaotic responses of mechanical systems with backlash component. Mech. Systems and Signal Processing 21:973–993 33. Krot PV (2008) Nonlinear vibrations and backlashes diagnostics in the rolling mills drive trains. In: Proc. of the 6th EUROMECH Nonlinear Dynamics Conference — ENOC2008, IPME RAS, St. Petersburg, Russia 34. Belli P (2004) On the origin of torsional vibrations in hot rolling mills and a possible remedy. Trans. of the ASME: Journal of Dynamic Systems, Measurement, and Control 126:811–823 35. Guo R-M, Urso AC, Schunk JH (1993) Analysis of chatter vibration phenomena of rolling mills using finite element methods. Iron and Steel Engineer 70(7):29–39 36. Moore JM (1991) Development of modal analysis for rolling mill. ECSC No.7210-EA/821 37. Nieb JR, Nicolas VT (1991) Automated monitoring and control of vibrations and chatter in rolling processes. Iron and Steel Engineer 68(7):33–42 38. Kimura Y, et al. (2003) Analysis of chatter in tandem cold rolling. ISIJ Int. 43:77–84. 39. Boulton G, Domanti T (2005) Vibration identification and suppression in hot and cold rolling mills. In: Proc of the AISTech 2005, Charlotte NC 40. Farley TWD, et al. (2006) Understanding mill vibration phenomena. In: Proc. of the Int. Conf. “Vibration in rolling mills”. Inst. of Materials, Minerals and Mining, London, UK 41. Roberts WL (1988) Flat Processing of Steel. Marcel Dekker Inc. 42. Petit B, Decrequy A, et al. (2005) Global approach of the 3rd octave chatter vibrations at Arcelor Mardyck cold rolling mill and analysis of technological interactions. La Revue de Metallurgie — CIT 7-8:535–541 43. Weber J, et al. (2000) Reduction of chatter in cold rolling. ECSC No.7210-PR/030 44. Prikhodko IY, et al. (2004) Method for cold rolling of strips in multistand mill. Patent RU2225272. Russia, B21B1/28 45. Blekhman II, Dzhanelidze GY (1964) Vibrational transportation. Nauka, Moscow (in Russian) 46. Marjuta AN, Krot PV (1997) High frequency rolling mills chatter — mathematical identification and simulation. In: Proc of the 1st Int. Symp. “Multi-Body Dynamics: Monitoring and Simulation Techniques — COMADEM’97”. University of Bradford 47. Faassen R, et al. (2003) Prediction of regenerative chatter by modelling and analysis of highspeed milling. Int. J. of Machine Tools and Manufacture 43(14):1437–1446 48. Krot PV, Prikhodko IY, et al. (2008) Method of diagnosing of sympathetic vibration and control of multistand cold rolling mill and device for implementation of diagnosing. Patent RU2338609. Russia. B21 B 37/00

The application of robust design strategies on managing the uncertainty and variability issues of the blade mistuning vibration problem Y.-J. Chan and D. J. Ewins

Abstract The blade mistuning vibration problem refers to the often-dramatic scatter - and thus increase - in vibration response levels when the nominally-identical blades on a bladed disc are slightly different to each other. For example, when a set of blades have eigenfrequencies varying randomly with a 0.5% scatter about the design value, the vibration response of the individual blades can vary dramatically, with individual blades displaying increases of as much as 100-500% as compared with the level which would be observed on every blade in a perfect (i.e. “tuned”) bladed disc. As both the vibration response levels and variations of blades are physical in nature, the variation is called aleatory uncertainty or simply variability. Although the problem became a research topic over 40 years ago with more than 400 papers already published, industry still faces shortened fatigue lives due to these extreme vibration responses, with high levels of scatter and uncertainty as to the likely incidence of extreme response levels. In the paper, the blade mistuning problem is viewed as a robust design problem, where the maximum blade response on the bladed disc is the robustness. The dependence of robustness on selected design parameters is discussed, and a procedure for managing the consequences of blade mistuning on manufactured bladed discs, rather than trying to eliminate the problem, is presented.

Y.-J. Chan Imperial College London, Exhibition Road, South Kensington, London SW7 2AZ, UK e-mail: [email protected] D. J. Ewins University of Bristol, University Walk, Bristol BS8 1TR, UK e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 32, © Springer Science+Business Media B.V. 2011

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1 Introduction In designing bladed discs in turbomachines, all blades on each stage are assumed to be identical, and this is called a tuned bladed disc. However, due to manufacturing tolerances, variations in material properties and wear in service, each blade on a disc is slightly different to the others. The potential consequences of these small differences, called mistuning, are a huge variation of peak responses across blades. For example, a 5% variation in the blade cantilever frequencies on a 92-bladed turbine disc can lead to one blade suffering responses which are over 500% of that observed on every blade on a tuned bladed disc [11], and a less severe example on a 64-bladed disc is shown in Fig. 1.

Fig. 1: The responses of blade in a 64-bladed mistuned bladed disc, as compared with the response observed on a tuned bladed disc The existence of extreme responses leads to much shortened fatigue lives. The phenomenon is described as the blade mistuning problem. Research into the blade mistuning problem began more than 40 years ago and more than 400 papers discussing various aspects of mistuned bladed discs have been published. Extensive knowledge has been gained, but industry still faces the occurrance of shortened fatigue lives due to extreme vibration responses, probably due to mistuning of bladed discs. In this paper, resolution of the blade mistuning problem is sought by management, instead of elimination. In the first section of the paper, the basic concepts of the blade mistuning problem are introduced. The second section is focused on casting the blade mistuning problem as a robustness problem and finding a suitable strategy to manage this problem under the robust design framework. The two methods eventually adopted are discussed in the third and fourth sections.

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The uncertainty in physical parameters can come from (i) a lack of knowledge (e.g. imprecise observations) or from (ii) variation of the parameter itself. There are two methods to distinguish the two types of uncertainties. In one method, the uncertainties from the two sources are called epistemic and aleatory uncertainty, respectively [2]. Alternatively, by using a risk analysis concept [1], the two sources of randomness are called uncertainty and variability respectively. Because the latter type of terminology is clearer, the random nature in both mistuning and the vibration responses is called “variability” in this paper.

2 Basic concepts of the blade mistuning problem Due to the existence of upstream vanes (and other factors), rotating bladed discs experience forced excitation because the distribution of air or gas pressure is uneven around the annulus. As the number of blades on a bladed disc is usually not equal to the number of vanes, the blades on a bladed disc do not vibrate in phase. Instead, the n cycles of variation of pressure around the blades lead to an n-th engine order (nEO) excitation on the bladed disc. The mode shapes of a bladed disc assembly are grouped into families according to the mode shape of the individual blade and there are N modes in each family for an N-bladed disc, and every mode shape varies sinusoidally around the annulus. There are less than N distinct natural frequencies in a mode family on a tuned bladed disc because of the existence of double modes: two independent modes with the same radial shape pattern, but circumferentially separated by a 90 degrees phase shift to each other, share the same natural frequency. When a tuned bladed disc is exposed to nEO excitation, only the pair of double modes with n nodal diameters are excited in each family. With the existence of mistuning, the double natural frequencies are no longer repeated, and occur as distinct, but close, natural frequencies which is known as mode splitting. In addition to mode splitting, the mode shapes are no longer sinusoidal in a mistuned bladed disc due to mode distortion. As a result, all modes in a family are excited to some degrees under any engine order excitation pattern. In research related to the blade mistuning problem: 1. the pattern of blade to blade variation around a mistuned bladed disc is called a mistuning pattern, comprising mistuning parameters; and 2. the dimensions and material properties are generalised into design parameters. While the individual blade properties are usually normalised in analysis, there are two main groups of design parameters of interest: interblade coupling which shows the degree of influence of the motion of one blade to another, and damping.

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2.1 The Amplification Factor (its significance and range) As shown in Fig. 1, the responses of blades on a mistuned bladed disc can be normalised to the peak response observed in any and every blade on a tuned bladed disc. The maximum normalised response in a mistuned bladed disc in a given resonant region, which shows the potential of the early failure of blades, is called the amplification factor, A. In high-cycle fatigue, the fatigue life becomes longer if the dynamic stresses are reduced. In a particular superalloy (PWA 1484), the fatigue life is inversely proportional to the 16th power of the dynamic stress [13]. This means the fatigue life would double if the peak response of a blade is reduced by as little as 4%. From a structural dynamics point of view, the governing equation of any mistuned bladed disc is shown in Eq. (1): (K + Δ K) − ω 2 (M + Δ M) x = f (1) For the sake of clear presentation, the amplification factor is described as a function of the design parameters, the mistuning parameters and the excitation pattern in this paper. This is called the amplification factor function: A = f (Design parameters, Mistuning pattern, Excitation order)

(2)

Because mistuning parameters are random variables in practice, the amplification factor is also random. Some of the previous research in the blade mistuning problem was dedicated to understand the extremity of amplification factors in bladed disc designs, and can be divided into two groups. The first group of research studies sought the maximum amplification factor possible on any bladed disc design because the extreme amplification factors indicate how short the fatigue life can be. In 1966, Whitehead derived an upper bound for the amplification factor called the Whitehead factor [17, 18] using aeroelastic principles. The same factor is proven to be the upper bound using structural dynamics principles by Lim et al in 2004 [9]. The Whitehead factor depends only on the number of blades on a bladed disc, N: √ 1 (3) 1+ N Amax = 2 Efforts have been made to improve the bound to incorporate other design parameters [14] but the Whitehead factor remains the most practical version. Besides a theoretical approach, the maximum amplification factor was found by optimising the amplification factor function. Different optimisation strategies have been applied to find the maximum amplification factor [12, 15, 4]. Although the results sought from optimisation are confined to a particular bladed disc design, the advantage of an optimisation-based method is that the results of optimisation always refer to particular mistuning patterns.

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In the second group, the probabilities of all higher amplification factors are obtained by a stochastic analysis, with Monte Carlo simulation being the most popular method. To carry out Monte Carlo simulations on finite element models of bladed discs, which usually have hundreds of thousands of degrees of freedom, reduced order models such as the Fundamental Mistuning Model (FMM) [5] have been developed to express the properties of bladed discs using relatively small matrices (in the case of the FMM, N×N matrices). Monte Carlo simulations do not perform well at predicting the low but significant probabilities of extreme amplification factors (Fig. 2). Curve-fitting [3] and importance sampling [4] have been developed to suit the needs of designers.

Fig. 2: The probability density function (pdf) of the amplification factor on a 24bladed disc under mistuning with standard deviation of 3% and the maximum mistuning of 10%

3 Casting blade mistuning as a robust design problem The problem with mistuned bladed discs is that a small variability at the input of the system, represented by the small scatter between nominally-identical across blades, leads to a high variability at the output. In other words, bladed discs are non-robust systems. Robustness refers to the variability of the output (as a function of the variability of the input) and is different to sensitivity in a strict sense: sensivity refers to the change of the output for a incremental change of the input, and it can be described by derivatives.

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The actions taken to improve the robustness of systems are known as robust design methods. There are two robust design methods available, namely the Taguchi method and the robust optimisation method. The two methods are described briefly before the approach adopted in this paper is outlined.

3.1 The Taguchi method of robust design The first robust design method is called the Taguchi method, which consists of three steps [16]: 1. System design: robustness of a system is improved by replacing the original system by a potentially more robust counterpart. For example, a simple voltage source-and-ammeter circuit is replaced by a Wheatstone bridge to measure the resistance of a resistor. 2. Parameter design: robustness of the system is improved by changing the design parameters. Parameter design involves (i) reducing the variability of the function output without considering the mean of the function output and (ii) moving the mean of the function output to the target value [10]. In parameter design, every parameter, both controllable or uncontrollable, is discretised into 2 or 3 levels. A small number (usually 30-40) of experiments or simulations are carried out on combinations guided by an orthogonal array to find the dependence of robustness on individual controllable parameters [8]. The system can be adjusted to improve the robustness accordingly. 3. Tolerance design: if the desired level of robustness is not achieved in parameter design due to the input variability, tolerances of the input are specified by balancing the life-cycle cost of a product and the cost of imposing those tolerances. The robustness measure in the Taguchi method is called signal-to-noise ratio (SNR), expressed in decibels (dB). In problems where the desired output has a finite value and the goal being having output as close as to the target value (socalled “Nominal the best” in the Taguchi method), the SNR is defined as the square of the ratio of the target value, μ , to the standard deviation, σ , of the the output of 2 the function. In other words, SNR = 10 log σμ 2 [10].

3.2 The robust optimisation method In the robust optimisation method, the input variability and robustness of the function representing the system in question, called the goal function, becomes the input and output of a new function called the robustness function. The robustness function is then optimised either by a deterministic approach or a randomised approach. In problems where the robustness function can be expressed as simple equations, a deterministic approach, such as sensitivity analysis, can be applied. In contrast, the

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randomised approach has to be used in problems where the values of the robustness function is known only at specific points. There are three measures available to quantify robustness in the robust optimisation method [2]: 1. In the robust counterpart measure (also known as the “worst case”), the robustness of the system is quantified by the highest output of the goal function due to all possible input combinations. In this case, the robust optimisation method improves the “worst case” by minimising the highest output of the goal function. 2. By using the expectancy measures, the mean or the variance of the output of the goal function, or a weighted combination of the two parameters, is used as the measure of robustness. 3. The probabilistic threshold measure of robustness is the probability of the output of the goal function being higher than a given threshold. This measure of robustness can be transformed into expectancy measures by rewriting the goal function in a different format. In robust optimisation, the robustness measure is minimised if either the robust counterpart or probabilistic threshold measure is adopted.

3.3 Application of robust design methods to the blade mistuning problem The goal in the blade mistuning problem is to achieve a bladed disc design with “acceptable” robustness. Potential improvements can be gained either by changing the bladed disc design slightly or by imposing tighter tolerances, which are similar to parameter design and tolerance design in the Taguchi method, respectively. These two approaches are discussed in the following two sections in this paper. However, the terminology and methods of the robust optimisation method are used. The scheme of experiments and the signal-to-noise ratio defined in Taguchi’s method do not suit the needs of the blade mistuning problem. Although the amplification factor and the mistuning pattern of a mistuned bladed disc are random variables, the relationship between the distribution parameters (e.g. mean and standard deviation) of the mistuning parameters and those of the amplification factor is deterministic in a bladed disc design. According to the robust optimisation approach, the amplification factor function (Eq. (2)) is transformed into robustness functions according to principles based on either interval analysis or probabilistics: • If an approach based on interval analysis is adopted, the ampliciation factor function is transformed into a maximum amplification factor function, dependent on the maximum allowable mistuning: Amax = f1 (Design parameters, Tolerances, Excitation order)

(4)

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and the input variability and robustness are defined by the tolerances (the maximum allowable mistuning) on a bladed disc and the maximum amplification factor (Amax ) respectively. • Otherwise, the robustness function based on probabilistics relates the 99.9thpercentile of the amplification factor distribution (A99.9 ) as robustness, and the mistuning distribution as the input variability: A99.9 = f2 (Design parameters, Mistuning distribution, Excitation order)

(5)

The standard deviation of the mistuning parameters is usually adopted as the single parameter to quantify the mistuning distribution. However, good approximations of either form of the robustness function have not been found in the past 40 years. Direct search methods, for example the Monte Carlo simulation and optimisation analysis, have to be used to evaluate of the robustness of bladed disc designs.

4 Improving the robustness of bladed discs by parameter design The aim of the present investigation is to produce a “robustness map” showing the general trend of how design parameters affect the maximum amplification factor of bladed discs. The dependence of the amplification factor of a pair of coupled oscillators on design parameters was first shown in a contour plot by Yoo et al [19], but practical systems are involved in the present investigation. Three basic numerical models are used to investigate the robustness of bladed discs in this paper: • A lumped parameter model of a 6-bladed disc (Fig. 3) • An FMM-reduced order model of a flat 6-bladed disc • An FMM-reduced order model of a flat 24-bladed disc (a modified finite element model is shown in Fig. 5) Details involved in reducing the order of a model by the FMM algorithm are shown in Reference [5]. The flat 6-bladed disc is created by removing blades from the flat 24-bladed disc. To simulate the wide variety of bladed discs present in gas turbines, a range of interblade coupling parameters are assigned to the lumped parameter model and stiffening rings are attached to the finite element models.

Fig. 3: The layout of a typical lumped parameter model of a bladed disc. (In simulations, k = 1Nm−1 and m = 1 kg, and only kc and the damping loss factor η are adjusted)

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Fig. 4: The layout of the finite element model of the flat 24-bladed disc, with a stiffening ring attached

Figs. 5 and 6 show robustness maps of the 6-bladed lumped parameter model and the FMM reduced-order model, respectively. By comparing the behaviour of the results from the reduced order model with those from the lumped parameter model, bladed disc having stiff discs have lower intercoupling stiffness than those with flexible discs.

Fig. 5: The “robustness map” showing the maximum amplification factors of 6-bladed lumped parameter models

It is found that small changes of design parameters do not change the maximum amplification factor significantly, except where the intercoupling stiffness in a bladed disc design is low. The maximum amplification factor of bladed discs with small interblade coupling decreases slightly with additional damping. However, the decrease in the amplification factor is a secondary effect compared with the decrease in the tuned responses of blades as shown in Fig. 7.

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Fig. 6: The “robustness map” showing the maximum amplification factors of 6-bladed FMM reduced order models

Fig. 7: The impact of the decrease in the maximum amplification factor under increasing damping loss factor on a 6-bladed lumped parameter model

5 Improving the robustness of bladed discs by tolerance design As shown in the investigation in the last section, the robustness of flexible bladed disc designs is not improved by changing the design parameters slightly. Alternative methods have to be sought to improve the robustness of those designs. In this section, the mistuning parameters existing on bladed discs are controlled, either by (i) limiting the maximum mistuning parameter to exist on a bladed disc or by (ii) specifying mistuning patterns to be occured on bladed discs. The approaches are called Small Mistuning and Intentional Mistuning, respectively.

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5.1 The Small Mistuning approach Although extreme amplification factors usually occur in bladed discs with relatively small amounts of mistuning, the change of the maximum amplification factor on the tolerances is gradual, i.e. the line of the maximum (or the 99.9th-percentile) amplification factor on mistuning parameters is smooth, with the y-intercept located at 1 (which is the tuned case). Two opportunities arise from this fact, namely (i) controlling the maximum amplification factor by limiting the tolerances and (ii) finding the sensitivity of the maximum amplification factor on maximum mistuning. The models used in the previous section are reused here. The dependence of the extreme amplification factors on mistuning strength in selected 6-bladed lumped parameter and 24-bladed FMM models is shown in Figs. 8 and 9. It shows that the maximum (or the 99th-percentile) amplification factor can be controlled by limiting the maximum or the standard deviation on mistuning. This is a realistic option because the standard deviation of mistuning parameters on a new bladed disc, using up-to-date manufacturing techniques, is typically 0.5% [7].

Fig. 8: The maximum amplification factors on two 6-bladed disc lumped parameter models

5.2 The Intentional Mistuning approach Extreme amplification factors only occur in bladed discs with particular mistuning patterns, even though those patterns cannot be predicted analytically. As a result, it is suggested that those patterns should be avoided on mistuned bladed discs by specifying some other mistuning patterns intentionally at the design stage. A “linear” mistuning pattern (Fig. 10) is put on a 24-bladed FMM reduced order model to show the potential of the intentional mistuning approach. It was first proposed as a candidate by Jones [6], because he found that the amplification factor associated with the pattern can be lower than 1.

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Fig. 9: The 99.9th-percentile amplification factors on two 24-bladed disc designs. Damping loss factor η = 0.002

Fig. 10: The “linear” and “ladder” mistuning patterns as applied to a 24-bladed disc

Before implementing the approach, there are two issues to be resolved: (i) because the strength of intentional mistuning is as important as its pattern, it has to be determined efficiently and (ii) the intentional mistuning pattern may itself be perturbed by manufacturing tolerances and wear and so the robustness of the amplification factor of a bladed disc under such circumstances, called “additional mistuning”, has to be studied. The straightforward approach to examine the issues is to run separate Monte Carlo simulations on several “shortlisted” intentional mistuning patterns of different strengths. However, this is computationally expensive, and the samples used in one simulation cannot be reused to improve the results of other simulations. As a result, an importance sampling-based method is developed to generate a “master pool” of Monte Carlo samples, and the pdf of amplification factors under a particular intentional pattern under further mistuning is studied by re-weighting the samples in the pool. By using this approach, the effect of slightly different intentional mistuning patterns (e.g. a “ladder” pattern shown in Fig. 10) can be studied at a realistic cost.

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Provided that a significant amount of intentional mistuning is provided (e.g. 6%, as shown in Fig. 11, or more), the peak of the pdf of the amplification factor can be shifted to a lower level even if the intentional mistuning pattern itself lead to an amplification factor above 1.

Fig. 11: The effect of intentional mistuning on the distribution of the amplification factor on a 24-bladed FMM reduced order model It is believed that the intentional mistuning approach can improve robustness in a variety of bladed discs given a suitable strength of intentional mistuning is found. However, some patterns, such as the linear mistuning pattern, are difficult to implement because many types of blade are involved, and the blades are only slightly different to each other. This issue is more acute if the bladed disc is manufactured as an assembly of separate rotor and blades.

6 Conclusions Bladed discs are highly non-robust structures because small, unavoidable, variations in blades can lead to extremely high responses and shortened fatigue lives. Parameter design and tolerance design approaches have been successfully applied to the blade mistuning problem to improve the robustness of bladed discs, i.e. by reducing either the maximum amplification factor or the chance of an extreme amplification factor occuring. It is found that the robustness of relatively rigid bladed discs (which have low interblade coupling) can be improved by adding damping (e.g. the maximum amplification factor is reduced by more than 15% in a 6-bladed disc design), and that of relatively flexible bladed discs can be improved by putting tighter tolerances on mistuning parameters (e.g. putting a 1% tolerance can reduce the 99.9-th percentile amplification factor from 1.8 to 1.5 in a 24-bladed disc de-

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sign). Intentional mistuning has also been shown to be a viable method to improve the robustness of bladed discs, but practical issues related to implementing such a scheme exist.

References 1. Anderson EL, Hattis D (1999) Uncertainty and variability. Risk Analysis 19(1):47–49 2. Beyer HG, Sendhoff B (2007) Robust optimization — a comprehensive survey. Computing methods in applied mechanics and engineering 196:3190–3218 3. Bladh R, Pierre C, Castanier MP, Kruse MJ (2002) Dynamic response predictions for a mistuned industrial turbomachinery rotor using reduced-order modeling. Journal of Engineering for Gas Turbines and Power 124(2):311–324 4. Chan YJ, Ewins DJ (2009) A comprehensive set of procedures to estimate the probability of extreme vibration levels due to mistuning. In: Proceedings of ASME Turbo Expo 2009: Power for Land, Sea and Air, Orlando, Florida, USA, GT2009-59088 5. Feiner DM, Griffin JH (2004) Mistuning identification of bladed disks using a fundamental mistuning model — part I: Theory. Transactions of the ASME, Journal of Turbomachinery 126(1):150–158 6. Jones K (2008) Minimizing maximum modal force in mistuned bladed disk forced response. Transactions of the ASME, Journal of Turbomachinery 130:011011 7. Klauke T (2007) Schaufelschwingungen realer integraler Verdichterr¨ader im Hinblick auf Verstimmung und Lokalisierung. PhD thesis, Brandenburgischen Technischen Universit¨at Cottbus 8. Lee KH, Eom IS, Park GJ, Lee WI (1996) Robust design for unconstrained optimization problems using the Taguchi method. AIAA Journal 37(5):1059–1063 9. Lim SH, Castanier MP, Pierre C (2004) Predicting mistuned blade amplitude bounds and stress increases from energy formulations. In: 9th National Turbine Engine HCF Conference 10. Park GJ, Lee TH, Lee KH, Hwang KH (2006) Robust design: an overview. AIAA Journal 44(1):181–191 11. Petrov EP, Ewins DJ (2003) Analysis of the worst mistuning patterns in bladed disk assemblies. Transactions of the ASME, Journal of Turbomachinery 125(4):623–631 12. Petrov EP, Iglin SP (1999) Search of the worst and best mistuning patterns for vibration amplitudes of bladed disks by the optimization methods using sensitivity coefficients. In: Proceedings of the 1st ASSMO UK Conference. Engineering Design Optimization, Ilkley, UK, pp 303–310 13. Reed RC (2006) The Superalloys: Fundamentals and Applications. Cambridge University Press, New York, USA, www.cambridge.org/9780521859042 14. Rivas-Guerra AJ, Mignolet MP (2003) Maximum amplification of blade response due to mistuning: localization and mode shape aspects of the worst disks. Transactions of the ASME, Journal of Turbomachinery 125(3):442–454 15. Scarselli G, Leece L (2005) Non deterministic approaches for the evaluation of the mistune effects on the rotor dynamics. In: AIAA 2004 Conference, Palm Springs, California 16. Taguchi G, Chowdbury S, Wu Y (2004) Taguchi’s quality engineering handbook. John Wiley & Sons 17. Whitehead DS (1966) Effect of mistuning on vibration of turbomachine blades induced by wakes. Journal of Mechanical Engineering Science 8(1):15–21 18. Whitehead DS (1998) The maximum factor by which forced vibration of blades can increase due to mistuning. Journal of Engineering for Gas turbines and Power 120:115–119 19. Yoo HH, Kim JY, Inman DJ (2003) Vibration localization of simplified mistuned cyclic structures undertaking external harmonic force. Journal of Sound and Vibration 261:859–870

Localized modeling of uncertainty in the Arlequin framework R. Cottereau, D. Clouteau, and H. Ben Dhia

Abstract This paper discusses the coupling and interaction of a classical continuum model with another continuum model with random parameters. The former model, deterministic, aims at representing a part of the domain where the local fluctuations of the parameters, such as Young’s modulus, do not influence the output of interest in a significant manner, and where a homogenized model is sufficient to predict this output. The latter model, stochastic, stands for the part of the domain where the local behavior is of interest and the fluctuations of the parameters cannot be considered only in a homogenized way. The coupling of these models is performed in the Arlequin framework. This paper focuses on the technical definitions of the spaces and operators introduced in the Arlequin framework for that particular problem, and on the definition of the corresponding discretized formulations. A simple example is shown, emphasizing the gain in computational power to compute the mean and confidence intervals in the region of interest.

1 Introduction Classical deterministic models provide global predictions that are satisfactory for many industrial applications. However, when one is interested in a very localized behavior or quantity, or when multiscale phenomena come into play, these models R. Cottereau ´ Laboratoire MSSMat, Ecole Centrale Paris, CNRS UMR 8579, Grande voie des vignes, 92295 Chˆatenay-Malabry cedex, France, e-mail: [email protected] D. Clouteau ´ Laboratoire MSSMat, Ecole Centrale Paris, CNRS UMR 8579, Grande voie des vignes, 92295 Chˆatenay-Malabry cedex, France, e-mail: [email protected] H. Ben Dhia ´ Laboratoire MSSMat, Ecole Centrale Paris, CNRS UMR 8579, Grande voie des vignes, 92295 Chˆatenay-Malabry cedex, France, e-mail: [email protected] A.K. Belyaev, R.S. Langley (eds.), IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, IUTAM Bookseries 27, DOI 10.1007/978-94-007-0289-9 33, © Springer Science+Business Media B.V. 2011

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may not be sufficient. For instance, the limited heterogeneity of a material modeled as a continuum might have no influence on its behavior on a large scale, while the study of a local stress intensity factor would strongly depend on the local heterogeneity of the mechanical parameters. Likewise, the prediction of the outbreak of a fracture in a structure might be performed with homogeneous models, while the incorporation of atomic modeling would be necessary to follow the exact path of that fracture. Unfortunately, for these problems, the information necessary to parameterize the relevant, very complex, models is usually not available. Stochastic methods have therefore been proposed and now appear unavoidable in multiscale modeling. Although the use of stochastic models and methods has expanded rapidly in the last decades, the related numerical costs are still often prohibitive. Hence, the application of these methods in a complex or industrial context remains limited. An important field of research is therefore concerned with the reduction of the costs associated with the use of stochastic methods, for example by using iterative methods specially adapted to the structure of the matrices arising in the Stochastic Finite Element (FE) method [11, 14], using reduced bases for the representation of random fields [9], or using special domain decomposition techniques for parallel resolution on clusters of computers [16]. The present paper proposes an alternative to these purely mathematical/numerical approaches through the coupling of two models: one deterministic and one stochastic. The general goal is that of modeling a global problem in a mean or homogeneous way where it yields sufficient accuracy, while retaining a stochastic model where needed. Hence, additional complexity is added in the model only where required, and the general approach is both more elegant and numerically cheaper than a global all-over stochastic model would be. Further, the cuts on computational costs mean that industrial applications come within reach. More specifically, we discuss here the interaction and coupling of a classical continuum model with another continuum model with random parameters. The former model, deterministic, aims at representing a part of the domain where the local fluctuations of the parameters, such as Young’s modulus, do not influence the output of interest in a significant manner, and where a homogenized model is sufficient to predict this output. The latter model, stochastic, stands for the part of the domain where the local behavior is of interest and the fluctuations of the parameters cannot be considered only in a homogenized way. The coupling of these models is performed in the Arlequin framework [2, 4, 5, 6, 3]. Note that the choice of two continuous models is by no means a restriction of the contents of this paper, and that the Arlequin method can accomodate different models [8]. However, considering similar models allows us to concentrate more particularly on the specific aspects of the coupling of a deterministic model with a stochastic one. The framework of this paper is very different from that of classical micromechanics [24, 7] and homogenization [13]. In these, the objective is to find a mean, or homogenized, behavior for a material, that will allow its study on a higher scale. In our case, we wish to study the local behavior of a small subdomain of that material, while the influence of the rest of the domain is taken into account in some homogenized way. Even when homogenization techniques are embedded within a stochastic

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FE framework, with both scales actually represented, the coupling does not go both ways, and only the low scale influences the high one (see for instance [21]). This type of one-way coupling approach is also very classical in climate modeling [22], where the influence of the small, unresolved, details at the global scale are taken into account through stochastic models more or less tuned on a lower scale. Nevertheless, we will re-use some notions explored extensively in homogenization techniques. In particular the notion of size of a Representative Volume Element, with respect to the correlation length of the parameters of the medium and/or the number of realizations of that medium (see in particular [12, 13, 20, 23]) will be discussed in relation with the size of the coupling zone between our two models. This work is closely related to previous works in the literature [18, 8]. However, in [18], the theoretical basis, different from the Arlequin formulation, is less general. In particular, it is only aimed at coupling a deterministic Boundary Element method with a stochastic FE method. In the recent work [8], the authors aim at coupling two stochastic models, one continuous, and one atomistic. However, many theoretical questions are left out. In particular, the coupling is performed between realizations of the stochastic operators, while we try to describe here the coupling at the level of the stochastic operators. The definition of the coupling operators is explicit, and the question of stability of the mixed problem can therefore be discussed (this will be done in a forthcoming paper). Also, it opens up the possibility of choosing among different numerical schemes, while the approach in [8] is limited to the Monte Carlo technique.

2 The classical Arlequin method We first recall the Arlequin framework in the classical case of the coupling of two deterministic continuum models, with different meshes [2, 4, 5, 3]. This will allow to introduce more gently the relevant material for the coupling of a deterministic and a stochastic continuum models, which is of interest in this paper. Further, this will emphasize the novelty of the work presented here, and will make it easier for the reader already at ease with the Arlequin method. Let us consider a domain Ω of Rd , with smooth boundary ∂ Ω separated into Dirichlet and Neumann boundaries ΓD and ΓN , such that ΓD ∪ ΓN = ∂ Ω , ΓD ∩ ΓN = 0, / and ΓD = 0/ (figure 1, left). We consider Poisson’s equation, with a parameter K0 , considered here constant, a bulk loading field f (x), defined on Ω , and a surface loading field g(x), defined on ΓN . Assuming for simplicity that the Dirichlet boundary condition is homogeneous, the weak formulation for this problem reads: find u0 ∈ V0 such that a0 (u0 , v) = 0 (v), ∀v ∈ V0 , (1) where a0 : V0 × V0 → R and 0 : V0 → R are defined, respectively, by a0 (u, v) = Ω K0 ∇u · ∇v d Ω , and 0 (v) = Ω f v d Ω + ΓN g · nv d Γ , and

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Fig. 1: Description of the model problem (left), and zoom on the zone where the two models are superposed (right), with the definition of the different domains: the homogenized domain Ω and the stochastic patch Ωs , itself partitioned into a free zone Ω f and a coupling zone Ωc = Ωs \Ω f V0 = {v ∈ H 1 (Ω ), v|ΓD = 0}. This problem can be shown to have a unique solution u0 , which may, for example, be approximated by the Finite Element method. Depending on the problem at hand, for example when there is a localized defect, it may be interesting to consider two very different meshes. A patch Ωs ⊂ Ω is therefore selected around the defect, and further partitioned into a free zone Ω f and a coupled zone Ωc , with Ω f ∩ Ωc = 0, / and Ω f ∪ Ωc = Ωs . A coarse finite element basis (supported by a coarse mesh) will be used on Ω to account for large scale deformations and stresses, and a refined one (supported on a fine mesh) will be used on Ωs to reproduce more accurately the local effects around the defect. The Arlequin method allows to couple these two problems through the resolution of the following mixed problem: find (u0 , us , Φ ) ∈ V0 × Vs × Vc such that ⎧ ⎪ ⎨a0 (u0 , v) +C(Φ , v) = 0 (v), ∀v ∈ V0 (2) as (us , v) −C(Φ , v) = 0, ∀v ∈ Vs , ⎪ ⎩ C(Ψ , u0 − us ) = 0, ∀Ψ ∈ Vc where the bilinear functions a0 : V0 × V0 → R and as : Vs × Vs → R are weighted restrictions of a0 , defined, respectively, by a0 (u, v) = Ω α0 K0 ∇u · ∇v d Ω and as (u, v) = Ωs αs K0 ∇u · ∇v d Ω , the coupling functional C : Vc × Vc → R is defined by

C(u, v) =

Ωc

(κ0 uv + κ1 ∇u · ∇v) d Ω ,

(3)

with κ0 and κ1 two constants (see for example [5] for details), the weights are chosen such that

Localized modeling of uncertainty in the Arlequin framework

⎧ α0 , αs ≥ 0 ⎪ ⎪ ⎪ ⎨α + α = 1 0 s ⎪α0 = 1 ⎪ ⎪ ⎩ α0 , αs constant

in Ω in Ω , in Ω /Ωs in Ω f

461

(4)

and the functional spaces are Vs = {v ∈ H 1 (Ωs )} and Vc = {v ∈ H 1 (Ωc )}. Note that, for simplicity, this has been derived in the case when the patch Ωs is totally included inside the domain Ω . In particular, the patch does not intersect the Dirichlet boundary condition, i.e. ∂ Ωs ∩ ΓD = 0, / and the loads are outside the patch, i.e. that is f (x ∈ Ωs ) = 0. However, more general results can be obtained without any further theoretical difficulty [3]. Several important propositions have been derived in different papers and summed up and completed in [3]. We recall two such propositions below, in the simplified case considered, and without the corresponding proofs. Proposition 1 (Stability). Under classical regularity hypotheses on the domains Ω0 , Ωs , Ωc and Ω f , with Ωc ∪ Ω f = Ωs , Ωc ∩ Ω f = 0, / and meas(Ωs ) = 0, assuming the hypotheses (4) on the weights, and supposing K0 > 0, the Arlequin problem (2) admits a unique solution (u0 , us , Φ ) in V0 × Vs × Vc . Proposition 2 (Consistency). Under the hypotheses (4) on the weights, the solution uarl of the Arlequin problem (2), defined by in Ω0 \Ωs u0 , uarl = α0 u0 + αs us in Ωs verifies uarl = u0 , where u0 is the unique solution of the monomodel reference problem (1). Further, if the restriction to Ω f of the displacement field solution is regular, we also have u0 = us = u0 in Ω f .

3 The continuous stochastic-deterministic Arlequin formulation In this section, we introduce the main result of this paper, i.e. the mixed formulation for the coupling of a stochastic continuous model with a deterministic continuous model. This is an extension of the results described in the previous section, and the main novelties are in the choice of the coupling and lift operators. We start by introducing the stochastic monomodel, where the physical parameter is modeled as stochastic on the entire domain Ω , and then move on to the coupled formulation.

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3.1 The stochastic monomodel Let us therefore consider that the physical parameter is modeled by a random field K(x) ∈ L 2 (Θ , R), where (Θ , F , P) is a complete probability space, with Θ a set of outcomes, F a σ -algebra of events, and P : F → [0, 1] a probability measure. We suppose (as in [1] for example) that this field is bounded and uniformly coercive, that is to say ∃ Kmin , Kmax ∈ (0, +∞), such that P(K(x) ∈ [Kmin , Kmax ], ∀x ∈ Ω ) = 1. The weak formulation of the corresponding stochastic boundary value problem now reads: find u0 ∈ W0 such that A 0 (u0 , v) = L0 (v), ∀v ∈ W0

(5)

where respectively, by A 0 (u, v) = A 0 : W0 ×W0 → R and L0 : W 0 → R are defined, E [ Ω K∇u · ∇v d Ω ], and L0 (v) = Ω f E[v] d Ω + ΓN g · nE[v] dΓ , E[·] denotes the mathematical expectation, and W0 = L 2 (Θ , V0 ). The above hypothesis on the parameter field K(x) ensures that the stochastic bilinear form A 0 is continuous and coercive on W0 × W0 . The Lax-Milgram theorem can therefore be used [1] to prove the existence and uniqueness of the solution u0 . An approximation of this solution can be obtained, for example, by using a Stochastic FE method [10, 17] or a Monte Carlo method [15]. Note that the requirement of boundedness of the mechanical parameter field is a quite stringent one. Another set of conditions has been derived in [19] that relaxes the upper bound property, while still preserving the solvability of the associated stochastic boundary value problem.

3.2 The Arlequin formulation We now wish to superpose, in the Arlequin framework, two models: one deterministic, in Ω ; and one stochastic, in Ωs . We will therefore consider two models of the parameter field: a deterministic one, K0 , supposed constant on the domain Ω ; and a stochastic one, K(x), modeled as a random field on Ωs . We further suppose that K(x) verifies the conditions described above, and that E[K(x)] = K0 . The stochasticdeterministic Arlequin problem reads: find (u0 , us , Φ ) ∈ V0 × Ws × Wc such that ⎧ ⎪ ⎨a0 (u0 , v) +C(Φ , v) = 0 (v), ∀v ∈ V0 (6) As (us , v) −C(Φ , v) = 0, ∀v ∈ Ws , ⎪ ⎩ C(Ψ , u0 − us ) = 0, ∀Ψ ∈ Wc where the bilinear forms a0 : V0 × V0 → R, As : Ws × Ws → R, and C : Wc × Wc → R are defined by

Localized modeling of uncertainty in the Arlequin framework

a0 (u, v) = As (u, v) = E

C(u, v) = E

α0 K0 ∇u · ∇v d Ω ,

and

Ω

Ωc

463

Ωs

αs K ∇u · ∇v d Ω ,

(κ0 uv + κ1 ∇u · ∇v) d Ω ,

the fields α0 (x) and αs (x) verify the conditions (4), and the functional spaces Ws and Wc are given by Wc = v(x) + θ Ic (x)|v ∈ H 1 (Ωc ), θ ∈ L 2 (Θ , R), E [θ ] = 0 , and

Ws = L 2 (Θ , H 1 (Ωs )),

/ Ωc ) = 0. and where the indicator function I(x) is such that Ic (x ∈ Ωc ) = 1 and Ic (x ∈ Note that the space Wc can be seen as composed of functions with a spatially varying mean and perfectly spatially correlated randomness. Note that, thanks to the specific structure of the space Wc , the last line of the system (6) can be written equivalently,

(κ0 (Ψ + θ Ic )(u0 − us ) + κ1 ∇Ψ · ∇(u0 − us )) d Ω C(Ψ , u0 − us ) = E Ωc

= C(E [Ψ ] , u0 − E [us ]) + E θ (u0 − us )d Ω = 0, ∀Ψ ∈ Wc .

Ωc

Therefore, this condition imposes that the mean of the field u should be equal s to the field u0 , in all points of Ωc , and that the variability of Ωc (E [us ] − us )d Ω should cancel. In other words, this means that some degree of homogenization takes place within the coupling zone. In particular, if that zone is not big enough with respect to the correlation lengths of the fields K(x) or us (x), the Arlequin scheme is expected to yield results that would be different from those obtained with the stochastic monomodel. It would mean that there is not enough localization of the variability and stochasticity for the Arlequin scheme to make sense. The next section addresses the issue of the discretization of the mixed formulation to obtain computable estimates of the different solution fields.

4 The discretized stochastic-deterministic Arlequin formulation The general idea here is to use a classical FE approach to discretize the first equation of the Arlequin system (6) and a stochastic FE method for the second and third ones. It should be noted that a Monte Carlo resolution of this formulation is not

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straightforward because the coupling equation (the third one) works on the mean of us , which is not available when one considers only one realization of that random field. A modified scheme to solve this problem using of Monte Carlo simulations will be described in a forthcoming paper. We therefore associate to the domain Ω a mesh T , composed of elements E, to the domain Ωs a mesh Ts , composed of elements Es , and to the domain Ωc a mesh Tc , composed of elements Ec . We look for approximate functions of the elements of V0 , Vs and Vc in the functional spaces V0H = {v ∈ P1 (E), v|ΓD = 0}, VsH = {v ∈ P1 (Es )}, and VcH = {v ∈ P1 (Ec )}, composed of linear functions on each of the elements of the meshes. We then choose the bases {v0 (x)}1≤≤m0 , {vs (x)}1≤≤ms , and {vc (x)}1≤≤mc for the functions in V0H , VsH , and VcH , respectively. We introduce the matrices A0 , C0 , and Cs , with elements α0 K0 ∇v0i · ∇v0j d Ω , A0,i j = Ω0

C0,i j =

Ωc

and Cs,i j =

0 c κ0 vi v j + κ1 ∇v0i · ∇vcj d Ω ,

Ωc

κ0 vsi vcj + κ1 ∇vsi · ∇vcj d Ω .

For the space Ws , we choose an approximating space as the span of the polynomial chaos basis [10], of order n and degree p, in conjunction with the previous basis H,n,p and the elements of its bafor the spatial dimension. We denote this space Ws s s sis {wk (x) = vk (x)Γˆ }1≤k≤ms ,0≤≤N−1 , where N is the number of elements in the polynomial chaos basis, which depends both on n and p. We approximate both the parameter field K(x), and the solution us (x) in that basis, us (x) ≈

ms N−1

∑ ∑ uks vsk (x)Γˆ,

k=1 =0

and finally obtain the matrix A for the stochastic part of the Arlequin system as N

A j,JL = ∑ ci jJ i=1

Ωs

αs (x)ki (x)∇vs (x) · ∇vsL (x) d Ω ,

where ci jJ = E Γˆi [ξ ]Γˆj [ξ ]ΓˆJ [ξ ] and ki (x) = E[K(x)Γˆi (ξ )]. Note that the double indices ( j, ) and (J, L) each correspond to only one index in the matrix form of the system. We denote by M the ms × (N − 1) matrix with general term

Localized modeling of uncertainty in the Arlequin framework

Mk = δ

Ωc

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vsk (x)d Ω ,

where the δ is related to the vector ordering of the coefficients usk . We further denote A = A0,0L =

Ω0

αs K0 ∇vsi · ∇vsj d Ω ,

and Ac = A0,JL the sub-matrix that corresponds to the coupling of the mean part of the unknown field with the fluctuating part, and As = A j=0 ,J=0L that corresponding to the fluctuating part. We finally get the form of the matrix system for the Arlequin problem (6): ⎡ ⎤⎡ ⎤ ⎡ ⎤ 0 C0 0 A0 0 U0 F ⎢ 0 A Ac −Cs 0 ⎥ ⎢ U ⎥ ⎢0⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ 0 ATc As 0 M⎥ ⎢ Us ⎥ = ⎢0⎥ , (7) ⎢ T ⎥⎢ ⎥ ⎢ ⎥ ⎣C0 −CTs 0 0 0 ⎦ ⎣ Φ ⎦ ⎣0⎦ Λ 0 0 0 MT 0 0 where the coordinates of the vector F are defined by Fi = 0 (vi ), U0 , U, Us , and Φ are the vectors of coordinates of u0 , E[us ], us − E[us ], and Φ , in the bases of V0H , VsH , WsH,n,p \VsH , and VcH , respectively, and Λ is the vector of the Lagrange multipliers enforcing the homogenization in the coupling zone. Note that the controlling parameters for the size of that matrix are n and p, and that in most cases, As will be a very large matrix, much larger than the other ones appearing in the system (7). However, it is much smaller than the matrix that would be obtained by applying directly a stochastic FE approach to the entire model.

5 Example of application For illustrative purposes, we consider the indented domain of Fig. 2, with −3 < x < 3 and −1 < y < 1, with a neck in the zone around x = 0, loading f (x, y) = 1 in the zone 2.5 < x < 3 (right side of the plate), and homogeneous Dirichlet boundary conditions on the left side of the plate. The boundary conditions read: u = 0, x = −3 ∇u = 0, on ∂ Ω \{x = −3} The resolution of this problem with a homogeneous parameter K0 = 1, using a FE scheme based on Eq. (1) leads to the intensity field |K0 ∇u| represented in Fig. 3. In this figure, we also plot the point-wise mean and the point-wise 90%-confidence interval of the intensity field |K∇u| of the corresponding stochastic problem given in Eq. (5). For that problem, the parameter field is modeled as a random field K(x)

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R. Cottereau, D. Clouteau, and H. Ben Dhia

Fig. 2: Scheme of the model problem: indented domain Ω , homogeneous Dirichlet boundary conditions on the left side, and bulk forces in an area on the right side

as in [19], with a mean E[K(x)] = K0 = 1, a dispersion parameter δ = 0.2 and a correlation length c = 0.15 m.

Fig. 3: Intensity field |K∇u| corresponding to the solution of the deterministic monomodel (Eq. (1) (solid line), and of the stochastic monomodel, Eq. (5) (the dashed line indicates the point-wise mean and the grey shade indicates the pointwise 90%-confidence interval) Finally, we solve the corresponding coupled Arlequin problem with a region of interest Ωs contained in the zone −1.7 < x < 1.7, and with a coupling region Ωc of length 0.6 on each side (i.e. contained within −1.7 < x < −1.1 and 1.1 < x < 1.7). We obtain the results plotted in Fig. 4, where it can be seen that the coupled Arlequin problem gives the same solution as the full heterogeneous problem in the region of interest. Note also that, at the limit of the coupling region, the approximation obtained from the Arlequin method seems to deteriorate compared to the full scale simulation. As the ensemble mean and the space mean are intricately woven in our approach, this was only to be expected. However, if the coupling zone is large enough, this effect does not contaminate the solution in the free zone Ω f . Further numerical studies on the convergence of this coupled approach, depending on the size of the coupling zone and the correlation length, will be performed in future work.

Localized modeling of uncertainty in the Arlequin framework

467

Fig. 4: Intensity field |K∇u| corresponding to the solution of the stochastic monomodel, Eq. (5) (the solid line indicates the point-wise mean and the grey shade indicates the point-wise 90%-confidence interval), and corresponding values obtained through the resolution of the coupled Arlequin problem of Eq. (6) (dashed lines for the mean and upper bound of the 90%-confidence interval)

6 Conclusion We have shown here a method for coupling a probabilistic model of continuum mechanics with a deterministic one. The numerical costs associated with the resolution of a probabilistic model are heavily lowered, which renders its use in an industrial setting reasonable. The framework that was described here can very easily be extended to other problems, be it with different physics (continuum mechanics, molecular dynamics, nonlinear constitutive relation), using the available literature on the Arlequin method, or involving two probabilistic models. For the latter, a discussion will be necessary to choose the adequate coupling operator. In an upcoming paper, we will discuss further the particularities of the method, and in particular: the stability of the mixed formulation, the size of the coupling zone with respect to the definition of representative volume elements in homogenization techniques [12, 13, 20, 23], and the possibility to solve the coupled system using the Monte Carlo simulation technique.

References 1. Babuˇska I, Tempone R, Zouraris GE (2004) Galerkin Finite Element aproximations of stochastic elliptic partial differential equations. SIAM J Numer Anal 42(2):800–825 2. Ben Dhia H (1998) Multiscale mechanical problems: the Arlequin method. Comptes-Rendus de l’Acad´emie des Sciences - series IIB 326(12):899–904 3. Ben Dhia H (2008) Further insights by theoretical investigations of the multiscale Arlequin method. Int J Multiscale Comp Engr 6(3):215–232 4. Ben Dhia H, Rateau G (2001) Mathematical analysis of the mixed Arlequin method. ComptesRendus Acad Sci (Series I - Math) 332(7):649–654 5. Ben Dhia H, Rateau G (2005) The Arlequin method as a flexible engineering design tool. Int J Numer Meths Engr 62(11):1442–1462 6. Ben Dhia H, Zammali C (2007) Level-Sets fields, placement and velocity based formulations of contact-impact problems. Int J Numer Meths Engr 69(13):2711–2735

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7. Cailletaud G, Forest S, Jeulin D, Feyel F, Galliet I, Mounoury V, Quilici S (2003) Some elements of microstructural mechanics. Comp Mat Sci 27:351–374 8. Chamoin L, Oden JT, Prudhomme S (2008) A stochastic coupling method for atomic-tocontinuum Monte Carlo simulations. Comp Meths Appl Mech Engr 197(43-44):3530–3546 9. Doostan A, Ghanem RG, Red-Horse J (2007) Stochastic model reduction for chaos representation. Comp Meths Appl Mech Engr 196(37-40):3951–3966 10. Ghanem RG, Spanos PD (1991) Stochastic finite elements: a spectral approach, SpringerVerlag 11. Ghanem RG, Kruger R (1996) Numerical solution of spectral stochastic finite element systems. Comp Meths Appl Mech Engr 129(3):289–303 12. Kanit T, Forest S, Galliet I, Mounoury V, Jeulin D (2003) Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int J Solids Struct 40:3647–3679 13. Ostoja-Starzewski M (2006) Material spatial randomness: from statistical to representative volume element. Prob Engr Mech 21(2):112–132 14. Pellissetti MF, Ghanem RG (2000) Iterative solution of systems of linear equations arising in the context of stochastic finite elements. Adv Engr Sftw 31(8-9):607–616 15. Robert CP, Casella G (2004) Monte Carlo statistical methods, Springer 16. Sarkar A, Benabbou N, Ghanem RG (2009) Domain decomposition of stochastic PDEs: theoretical formulations. Int J Numer Meths Engr 77(5):689–701 17. Stefanou G (2009) The stochastic finite element method: past, present and future. Comp Meths Appl Mech Engr 198(9-12):1031–1051 18. Savin E, Clouteau D (2002) Elastic wave propagation in a 3-D unbounded random heterogeneous medium coupled with a bounded medium. Application to seismic soil-structure interaction (SSSI). Int J Numer Meths Engr 54(4):607–630 19. Soize C (2006) Non-gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators. Comp Meths Appl Mech Engr 195:26–64 20. Soize C (2008) Tensor-valued random field for meso-scale stochastic model of anisotropic elastic microstructure and probabilistic analysis of representative volume element size. Prob Engr Mech 23:307–323 21. Spanos PD, Kontsos A (2008) Multiscale Monte Carlo finite element method for determining mechanical properties of polymer nanocomposites. Prob Engr Mech 23:456–470 22. Williams PD (2005) Modelling climate change: the role of unresolved processes. Phil Trans R Soc A 363:2931–2946 23. Xu XF, Chen X (2009) Stochastic homogenization of random elastic multi-phase composites and size quantification of representative volume element. Mech Mat 41(2):174–186 24. Zaoui A (2002) Continuum micromechanics: a survey. J Engr Mech 128(8): 808–816

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