Dynamical Systems: Discontinuity, Stochasticity and Time-Delay

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Dynamical Systems

Albert C.J. Luo Editor

Dynamical Systems Discontinuity, Stochasticity and Time-Delay

123

Editor Albert C.J. Luo Department of Mechanical and Industrial Engineering Southern Illinois University Edwardsville Edwardsville, IL 62026-1805 USA [email protected]

ISBN 978-1-4419-5753-5 e-ISBN 978-1-4419-5754-2 DOI 10.1007/978-1-4419-5754-2 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010924743 c Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Dynamics, vibrations, and control of dynamical systems with discontinuity, stochasticity, and time delay are presented in this book. Dynamical systems with discontinuity, stochasticity, and time delay exist extensively in practical systems. This book provides the reader with a better understanding of control of dynamical behaviors of complex dynamical systems. Recent developments in dynamical systems with discontinuity, stochasticity, and time delay are discussed along with topics normally associated with discontinuous systems including but not limited to impact systems, friction-induced systems, and impulsive systems. Also presented are classic vibration and control of dynamical systems. This content presented is based on the Second Conference on Dynamics, Vibration and Control, held in Chengdu-Jiuzhaigou, Sichuan, China, 2009 (DVC2009). The goal of this conference is to provide a place to exchange recent developments, discoveries, and progresses on dynamics, vibration, and control. This second conference is the continuation of the 2007 Arctic Summer Conference on Dynamics, Vibrations and Control. Papers and presentations relative to all areas pertaining to theoretical, symbolic, computational, and experimental aspects of dynamics, vibrations, and control were solicited. There were 57 papers initially submitted for presentation and publications. After peer review, only 34 papers were selected for publication in this book; these contributions are divided into four groups: Group 1 discusses nonlinear and discontinuous dynamical systems and includes

11 contributions that cover fractional dynamics, chaos and bifurcations in nonlinear dynamical systems, discontinuous dynamical systems, and applications in manufacturing and rotor dynamics. Group 2 discusses time-delay systems and includes four contributions that cover the method of time-continuous approximation and time-delay systems, the timedelay control for nonlinear dynamical systems, bifurcation and stability for neuronal systems with time delay. Group 3 discusses switching and stochastic systems and includes six contributions that cover nonlinear dynamics of switching and impulsive dynamical systems, neuron synchronization under noise excitation, nonequilibrium transition, and stochastic resonance.

v

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Preface

Group 4 discusses classic vibration and control with 13 contributions that cover

structural dynamics wave propagation in soil and foundations, fluid-induced vibration and control systems, and system identification. The conference organizers would like to take this opportunity to thank all volunteers for conference preparation and hotel arrangement. We would also like to express appreciation to Ms. Yi Sun who made all the necessary arrangements for the conference and special events in China. Edwardsville, IL

Albert C.J. Luo

Contents

Part I Nonlinear and Discontinuous Dynamical Systems 1

General Solution of a Vibration System with Damping Force of Fractional-Order Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . Z.H. Wang and X. Wang

3

2

An Analytic Proof for the Sensitivity of Chaos to Initial Condition and Perturbations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 13 J.H. Peng and J.S. Tang

3

Study on the Multifractal Spectrum of Local Area Networks Traffic and Their Correlations .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 23 Yan Liu and Jia-Zhong Zhang

4

A Boundary Crisis in High Dimensional Chaotic Systems . . . .. . . . . . . . . . . 31 Ling Hong, Yingwu Zhang, and Jun Jiang

5

Complete Bifurcation Behaviors of a Henon Map . . . . . . . . . . . . .. . . . . . . . . . . 37 Albert C.J. Luo and Yu Guo

6

Study on the Performance of a Two-Degree-of-Freedom Chaotic Vibration Isolation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 49 Jing-Jun Lou, Ying-Chun Wang, and Shi-Jian Zhu

7

Simulation and Nonlinear Analysis of Panel Flutter with Thermal Effects in Supersonic Flow. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 61 Kai-Lun Li, Jia-Zhong Zhang, and Peng-Fei Lei

8

A Parameter Study of a Machine Tool with Multiple Boundaries . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 77 Brandon C. Gegg, Steve C.S. Suh, and Albert C.J. Luo

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Contents

A New Friction Model for Evaluating Energy Dissipation in Carbon Nanotube-Based Composites . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 95 Yaping Huang and X.W. Tangpong

10 Nonlinear Response in a Rotor System With a Coulomb Spline. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .105 C. Nataraj and Karthik Kappaganthu 11 The Influence of the Cross-Coupling Effects on the Dynamics of Rotor/Stator Rubbing. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .121 Zhiyong Shang, Jun Jiang, and Ling Hong Part II Time-delay Systems 12 Some Control Studies of Dynamical Systems with Time Delay . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .135 Bo Song and Jian-Qiao Sun 13 Stability and Hopf Bifurcation Analysis in Synaptically Coupled FHN Neurons with Two Time Delays .. . . . . . . . . . . . . . . .. . . . . . . . . . .157 Dejun Fan and Ling Hong 14 On the Feedback Controlling of the Neuronal System with Time Delay .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .169 Hao Liu, Wuyin Jin, Chi Zhang, Ruicheng Feng, and Aihua Zhang 15 Control of Erosion of Safe Basins in a Single Degree of Freedom Yaw System of a Ship with a Delayed Position Feedback . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .177 Huilin Shang Part III

Switching and Stochastic Dynamical Systems

16 On Periodic Flows of a 3-D Switching System with Many Subsystems . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .189 Albert C.J. Luo and Yang Wang 17 Impulsive Control Induced Effects on Dynamics of Complex Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .203 Xiuping Han and Xilin Fu 18 Study on Synchronization of Two Identical Uncoupled Neurons Induced by Noise .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .217 Ying Wu, Ling Hong, Jun Jiang, and Wuyin Jin

Contents

ix

19 Non-equilibrium Phase Transitions in a Single-Mode Laser Model Driven by Non-Gaussian Noise . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .223 Yanfei Jin 20 Dynamical Properties of Intensity Fluctuation of Saturation Laser Model Driven by Cross-Correlated Additive and Multiplicative Noises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .233 Ping Zhu 21 Empirical Mode Decomposition Based on Bistable Stochastic Resonance Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .251 Y.-J. Zhao, Y. Xu, H. Zhang, S.-B. Fan, and Y.-G. Leng Part IV

Classic Vibrations and Control

22 Order Reduction of a Two-Span Rotor-Bearing System Via the Predictor-Corrector Galerkin Method .. . . . . . . . . . . . . . . .. . . . . . . . . . .263 Deng-Qing Cao, Jin-Lin Wang, and Wen-Hu Huang 23 Stiffness Nonlinearity Classification Using Morlet Wavelets . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .275 Rajkumar Porwal and Nalinaksh S. Vyas 24 Dynamics of Wire-Driven Machine Mechanisms: Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .285 Timo Karvinen and Erno Keskinen 25 Dynamics of Wire-Driven Machine Mechanisms, Part II: Theory and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .299 Erno Keskinen, Timo Karvinen, and Jori Montonen 26 On Analytical Methods for Vibrations of Soils and Foundations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .319 H.R. Hamidzadeh 27 Inversely Found Elastic and Dimensional Properties.. . . . . . . . .. . . . . . . . . . .341 Darryl K. Stoyko, Neil Popplewell, and Arvind H. Shah 28 Nonlinear Self-Defined Truss Element Based on the Plane Truss Structure with Flexible Connector . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .357 Yajun Luo, Xinong Zhang, and Minglong Xu

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Contents

29 Complex Frequency Analysis of an Axially Moving String with Multiple Attached Oscillators by Using Green’s Function Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .371 Le-Feng L¨u, Yue-Fang Wang, and Ying-Xi Liu 30 Model Reduction on Inertial Manifolds of Navier–Stokes Equations Through Multi-scale Finite Element . . . . . . . . . . . . . . . .. . . . . . . . . . .383 Jia-Zhong Zhang, Sheng Ren, and Guan-Hua Mei 31 Diesel Engine Condition Classification Based on Mechanical Dynamics and Time-Frequency Image Processing . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .397 Hongkun Li and Zhixin Zhang 32 Input Design for Systems Under Identification as Applied to Ultrasonic Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .409 Nishant Unnikrishnan, Yicheng Pan, Marco Schoen, Ajay Mahajan, Jarlen Don, and Tsuchin Chu 33 Development of a Control System for Automating of Spiral Concentrators in Coal Preparation Plants .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .421 Josh Hoelscher, Yicheng Pan, Manoj Mohanty, Jarlen Don, Tsuchin Chu, and Ajay Mahajan 34 On the Rough Number Computation and the Ada Language . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .437 Trong Wu Author Index. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .449 Subject Index . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .461

Contributors

Deng-Qing Cao School of Astronautics, Harbin Institute of Technology, P.O. Box 137, Harbin 150001, People’s Republic of China, [email protected] Tsuchin Chu Department of Mechanical Engineering and Energy Processes, Southern Illinois University Carbondale, 1230 Lincoln Drive, Carbondale, IL 62901-6603, USA, [email protected] Jarlen Don Department of Mechanical Engineering and Energy Processes, Southern Illinois University Carbondale, 1230 Lincoln Drive, Carbondale, IL 62901-6603, USA Dejun Fan MOE Key Laboratory for Strength and Vibration, School of Aerospace, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, People’s Republic of China and Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai, Shandong 264209, People’s Republic of China, [email protected] S.-B. Fan CSR Qingdao Sifang Locomotive and Rolling Stock Company, Chengyang, Qingdao 266111, People’s Republic of China Ruicheng Feng School of Mechano-Electronic Engineering, Lanzhou University of Technology, Lanzhou 730050, People’s Republic of China Xilin Fu School of Mathematical Sciences, Shandong Normal University, Jinan 250014, People’s Republic of China, [email protected] Brandon C. Gegg Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA, [email protected] Yu Guo Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805, USA, [email protected] H.R. Hamidzadeh Department of Mechanical and Manufacturing Engineering, Tennessee State University, Nashville, TN, USA, [email protected] Xiuping Han School of Mathematical Sciences, Shandong Normal University, Jinan 250014, People’s Republic of China, [email protected]

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Contributors

Josh Hoelscher Department of Mechanical Engineering and Energy Processes, Southern Illinois University Carbondale, 1230 Lincoln Drive, Carbondale, IL 62901-6603, USA Ling Hong MOE Key Lab for Strength and Vibration, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China, hongling@mail. xjtu.edu.cn Wen-Hu Huang School of Astronautics, Harbin Institute of Technology, P.O. Box 137, Harbin 150001, People’s Republic of China Yaping Huang Department of Coatings and Polymeric Materials, North Dakota State University, Fargo, ND 58108, USA, [email protected] Jun Jiang MOE Key Laboratory for Strength and Vibration, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China, [email protected] Wuyin Jin School of Mechano-Electronic Engineering, Lanzhou University of Technology, Lanzhou 730050, People’s Republic of China and Key Laboratory of Digital Manufacturing Technology and Application, The Ministry of Education, Lanzhou University of Technology, Lanzhou 730050, People’s Republic of China, [email protected] Yanfei Jin Department of Mechanics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China, [email protected] K. Kappaganthu Department of Mechanical Engineering, Villanova University, Villanova, PA, USA, [email protected] Timo Karvinen Department of Mechanics and Design, Tampere University of Technology, P.O. Box 589, 33101 Tampere, Finland, [email protected] Erno Keskinen Department of Mechanical Engineering, Tampere University of Technology, P.O. Box 589, 33101 Tampere, Finland, [email protected] Peng-Fei Lei School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, People’s Republic of China, [email protected] Yonggang Leng School of Mechanical Engineering, Tianjin University, Tianjin 300072, People’s Republic of China, Hleng [email protected] Hongkun Li School of Mechanical Engineering, Dalian University of Technology, Dalian 116023, People’s Republic of China and State Key Laboratory of Structural Analysis of Industrial Equipment, Dalian 116023, People’s Republic of China, [email protected] Kai-Lun Li School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shanxi 710049, People’s Republic of China, [email protected] Hao Liu School of Mechano-Electronic Engineering, Lanzhou University of Technology, Lanzhou 730050, People’s Republic of China

Contributors

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Yan Liu School of Mechatronics, Northwestern Polytechnical University, Xi’an 710072, People’s Republic of China, [email protected] Ying-Xi Liu Department of Engineering Mechanics, Dalian University of Technology, 2 Linggong Road, Dalian 116024, People’s Republic of China, [email protected] Jing-jun Lou College of Naval Architecture and Power, Naval University of Engineering, Wuhan 430033, People’s Republic of China, jingjun [email protected] Le-Feng Lu¨ Department of Engineering Mechanics, Dalian University of Technology, 2 Linggong Road, Dalian 116024, People’s Republic of China, [email protected] Albert C.J. Luo Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL62026-1805, USA, [email protected] Yajun Luo School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China, [email protected] Ajay Mahajan Department of Mechanical Engineering, University of Akron, Akron, OH, USA Guan-Hua Mei School of Energy and Power Engineering, Xi’an Jiaotong University, No. 28, Xianning West Road, Xi’an, Shaanxi 710049, People’s Republic of China, [email protected] Manoj Mohanty Department of Mining and Mineral Resources Engineering, Southern Illinois University Carbondale, 1230 Lincoln Drive, Carbondale, IL 62901-6603, USA Jori Montonen Department of Mechanics and Design, Tampere University of Technology, P.O. Box 589, FIN-33101 Tampere, Finland C. Nataraj Department of Mechanical Engineering, Villanova University, Villanova, PA, USA, [email protected] Yicheng Pan Department of Mechanical Engineering and Energy Processes, Southern Illinois University Carbondale, 1230 Lincoln Drive, Carbondale, IL 62901-6603, USA J.H. Peng Department of Mechanical Engineering, Shaoyang Polytechnic, Hunan, People’s Republic of China and Department of Physics, Shaoyang University, Shaoyang, Hunan, People’s Republic of China, [email protected] Neil Popplewell Mechanical and Manufacturing Engineering, University of Manitoba, 15 Gillson Street, Winnipeg, MB, Canada R3T 5V6, [email protected]

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Contributors

Rajkumar Porwal Department of Mechanical Engineering, Shri G. S. Institute of Technology and Science, 23 Park Road, Indore 452003, India, [email protected], [email protected] Sheng Ren School of Energy and Power Engineering, Xi’an Jiaotong University, No. 28, Xianning West Road, Xi’an, Shaanxi 710049, People’s Republic of China, [email protected] Marco Schoen Department of Mechanical Engineering, Idaho State University, Pocatello, ID, USA Arvind H. Shah Civil Engineering, University of Manitoba, 15 Gillson Street, Winnipeg, MB, Canada R3T 5V6, [email protected] Huilin Shang School of Mechanical and Automation Engineering, Shanghai Institute of Technology, Shanghai 20035, People’s Republic of China, [email protected] Zhiyong Shang MOE Key Laboratory for Strength and Vibration, Xi’an Jiaotong University, Xi’an, People’s Republic of China, [email protected] Bo Song School of Engineering, University of California, Merced CA 95344, USA Darryl K. Stoyko Mechanical and Manufacturing Engineering, University of Manitoba, 15 Gillson Street, Winnipeg, MB, Canada R3T 5V6, D [email protected] Jian-Qiao Sun School of Engineering, University of California, Merced, CA 95344, USA, [email protected] Steve C.S. Suh Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA J.S. Tang College of Mechanics and Aerospace, Hunan University, Changsha, Hunan, People’s Republic of China, [email protected] X.W. Tangpong Department of Mechanical Engineering, North Dakota State University, Fargo, ND 58108, USA, [email protected] Nishant Unnikrishnan Department of Mechanical Engineering and Energy Processes, Southern Illinois University Carbondale, 1230 Lincoln Drive, Carbondale, IL 62901-6603, USA Nalinaksh S. Vyas Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India, [email protected] Jin-Lin Wang School of Astronautics, Harbin Institute of Technology, P.O. Box 137, Harbin 150001, China X. Wang Institute of Science, PLA University of Science and Technology, 211101 Nanjing, People’s Republic of China

Contributors

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Yang Wang Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805, USA, [email protected] Ying-chun Wang Administrative Office of Training, Naval University of Engineering, Wuhan 430033, People’s Republic of China Yue-Fang Wang Department of Engineering Mechanics, Dalian University of Technology, 2 Linggong Road, Dalian 116024, People’s Republic of China and State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian 116024, People’s Republic of China, [email protected] Z.H. Wang Institute of Science, PLA University of Science and Technology, 211101 Nanjing, People’s Republic of China and Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, People’s Republic of China, [email protected] Trong Wu Department of Computer Science, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1656, USA, [email protected] Ying Wu School of Science, Xi’an University of Technology, Xi’an, Shaanxi 710054, People’s Republic of China, [email protected] Minglong Xu School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China, [email protected] Y. Xu CSR Qingdao Sifang Locomotive and Rolling Stock Company, Chengyang, Qingdao 266111, People’s Republic of China Aihua Zhang College of Electrical and Information Engineering, Lanzhou University of Technology, Lanzhou 730050, People’s Republic of China Chi Zhang School of Mechano-Electronic Engineering, Lanzhou University of Technology, Lanzhou 730050, People’s Republic of China H. Zhang CSR Qingdao Sifang Locomotive and Rolling Stock Company, Chengyang, Qingdao 266111, People’s Republic of China Jia-Zhong Zhang School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, People’s Republic of China, [email protected] Xinong Zhang School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China, [email protected] Y.-W. Zhang MOE Key Lab for Strength and Vibration, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China, [email protected] Zhixin Zhang School of Mechanical Engineering, Dalian University of Technology, Dalian 116023, People’s Republic of China, [email protected]

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Contributors

Y.-J. Zhao CSR Qingdao Sifang Locomotive and Rolling Stock Company, Chengyang, Qingdao 266111, People’s Republic of China, [email protected] P. Zhu Department of Physics, Simao Teacher’s College, Puer 665000, People’s Republic of China, [email protected] Shi-jian Zhu College of Naval Architecture and Power, Naval University of Engineering, Wuhan 430033, People’s Republic of China

Part I

Nonlinear and Discontinuous Dynamical Systems

Chapter 1

General Solution of a Vibration System with Damping Force of Fractional-Order Derivative Z.H. Wang and X. Wang

Abstract The generalized Bagley–Torvik system is a linear oscillator whose damping force is described by fractional-order derivative with order between 0 and 2. This paper shows that as a sequential fractional-order differential equation with constant coefficients whose general solution depends on more than two free (independent) constants, the generalized Bagley–Torvik equation actually admits a general solution that involves two free constants only and can be determined fully by the initial displacement and initial velocity.

1.1 Introduction Dynamical systems with fractional-order derivatives have found many applications in various problems in science and engineering such as viscoelasticity, heat conduction, electrode–electrolyte polarization, electromagnetic waves, diffusion wave, control theory and so on [1–5]. In order to model the forced motion of a rigid plate immersed in a kind of Newtonian fluid, for example, Bagley and Torvik proposed the following differential equation [5] Ax.t/ R C B0 D 3=2 x.t/ C C x.t/ D f .t/:

(1.1)

A peculiarity of this equation is the fractional-order derivative B0 D 3=2 x.t/ that is used to describe the damping force. Recently, the first author proved in [6] that the fractional-order derivative B0 D ˛ x.t/ appeared in

Z.H. Wang () Institute of Science, PLA University of Science and Technology, 211101 Nanjing, People’s Republic of China and Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, People’s Republic of China e-mail: [email protected]

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 1,

3

4

Z.H. Wang and X. Wang

Ax.t/ R C B0 D ˛ x.t/ C C x.t/ D 0 .A; B; C > 0I 0 < ˛ < 2/

(1.2)

acts always as a damping force, so that the motion governed by (1.2) is a result of external force, elastic restoring force and fractional-order damping force. A fundamental problem is how to formulate the initial conditions so that the evolution of the system is completely determined from the initial conditions [8]. Because the order of highest derivative in (1.2) is 2, it is expected that as in the case of classical vibration systems, the initial position and initial velocity determine completely the evolution of the vibration systems. A straightforward approach for solving this linear equation (1.2) is Laplacian transformation [1,7]. In this way, a solution in terms of the initial position and initial velocity can be easily obtained by using this method, if the Caputo’s fractional-order derivative is employed. However, in [8], the solution of the vibration system has to use the value of a fractional-order derivative at the initial point, except for the initial position and initial velocity, if the Riemann–Liouville’s fractional-order derivative is applied. Recently, Bonilla et al. established a theory for finding the general solution of sequential fractional-order differential equations (SFDEs) with constant coefficients [9]. As in solving linear ordinary differential equations with constant coefficients, the general solution of a SFDE can be obtained directly by the roots of a polynomial (be called characteristic roots for simplicity). This paper studies the general solution of the generalized Bagley–Torvik equation in dimensionless form x.t/ R C 0 D ˛ x.t/ C x.t/ D 0

. > 0; 0 < ˛ < 2; ˛ ¤ 1/:

(1.3)

Regarding the fractional vibration equation as a SFDE, its “order” is greater than 2, and it requires more than two free constants to describe its general solution. The main objective of this paper is to show that with Riemann–Liouville’s derivative or Caputo’s derivative, the general solution of (1.3) involves two free constants only, and it can be determined fully by the initial displacement and initial velocity.

1.2 Preliminaries for Sequential Fractional-Order Differential Equations The following homogeneous linear fractional differential equation # " n1 X n˛ k˛ x.t/ D 0 C ak .t/ a D aD

(1.4)

kD0

is called SFDE, where a; ak 2 R, and aD

k˛

D a D ˛ .a D .k1/˛ /

.k D 2; 3; : : : ; n/:

The fractional-order derivative a D ˛ can be defined in different ways [1,3], including Riemann–Liouville’s definition.

1

General Solution of a Fractional-Order Vibration System

5

Let ˛ 2 R; m 1 < ˛ m; m 2 N; a 2 R and f be a continuous function, then Riemann–Liouville fractional-order derivative is defined by RL ˛ a D f .t/

dm 1 D .m ˛/ dt m

Z

t a

f ./ d; .t /1C˛m

(1.5)

where .z/ is the Gamma function defined by Z

1

.z/ D

et t z1 dt

.<.z/ > 0/

0 ˛ satisfying .z C 1/ D z .z/. The derivative RL a D f .t/ requires that f .t/ is continuous only, so it is widely used in analysis.

1.2.1 Linear Dependence and Linear Independence of Functions Assume that x1 .t/; x2 .t/; : : : ; xn .t/ are n functions defined on Œa; b; they are called linearly dependent in Œa; b, if there exist constants c1 ; c2 ; : : : ; cn that are not zero simultaneously, such that c1 x1 .t/ C c2 x2 .t/ C C cn xn .t/ 0 .a t b/:

(1.6)

Otherwise, x1 .t/; x2 .t/; : : : ; xn .t/ are called linearly independent in Œa; b. To check the linear dependence in fractional calculus, it is convenient to use the generalized Wronsky determinant W˛ .x1 ; x2 ; : : : ; xn / of x1 .t/; x2 .t/; : : : ; xn .t/, defined by [9] ˇ ˇ ˇ ˇ x1 .t/ x2 .t/ xn .t/ ˇ ˇ ˛ ˛ ˇ ˇ a D ˛ x1 .t/ D x .t/ D x .t/ a 2 a n ˇ ˇ W˛ .x1 ; x2 ; : : : ; xn / D ˇ ˇ: : : : : :: :: :: :: ˇ ˇ ˇ ˇ ˇ D .n1/˛ x .t/ D .n1/˛ x .t/ D .n1/˛ x .t/ ˇ a

1

a

2

a

n

(1.7) The solutions x1 .t/, x2 .t/; : : : ; xn .t/ of (1.4) are linearly dependent in Œa; b if and only if there is a t0 2 Œa; b such that W˛ .t0 / D 0.

1.2.2 Characteristic Polynomial In classical calculus, the function et plays an important role in solving ordinary differential equations with constant coefficients, and it satisfies d t e D et : dt

6

Z.H. Wang and X. Wang

In fractional calculus, the ˛-exponential function [9] a/ e.t ˛

D .t a/

˛1

1 X k .t a/˛k ..k C 1/˛/

.t a/

(1.8)

kD0

satisfies the following fractional-order differential equation aD

˛

x.t/ D x.t/

.t > a/

(1.9)

in the sense of Riemann–Liouville fractional derivative. For the SFDE with constant coefficients, one has " RL n˛ a D

C

n1 X

# a/ a/ D p./e.t ; e.t ˛ ˛

k˛ ak RL a D

(1.10)

kD0

where p./ D n C

n1 X

ak k

(1.11)

kD1

is called the characteristic polynomial of (1.4). .t a/ Assume that (1.4) has a solution of the form c e˛ , then must be a root of .t a/ p./. Conversely, if p./ D 0, then (1.4) has a solution c e˛ . Thus, (1.4) has a .t a/ solution c e˛ if and only if is a root of p./. The ˛-exponential function can also be defined as following [9] a/ eO .t ˛

1 X k .t a/˛k D .k˛ C 1/

.t a/;

kD0

which satisfies the fractional-order differential equation (1.9) in the sense of Caputo fractional-order derivative. Caputo’s definition is given by C ˛ a D f .t/

1 D .m ˛/

Z a

t

f .m/ ./ d: .t /1C˛m

(1.12)

It requires that f .t/ has m-order continuous derivatives. For function f .t/, with m-order continuous derivative and starting from standstill, however, the Caputo fractional derivative gives the same value of Riemann–Liouville fractional derivative.

1

General Solution of a Fractional-Order Vibration System

7

1.2.3 General Solution of SFDE In what follows, we shall discuss the general solution of (1.4) in two cases as follows. Case 1. The polynomial p./ has simple roots only. Assume that 1 ; 2 ; : : : ; n are n different roots of p./, then (1.4) has the corresponding particular solutions x1 .t/ D e˛1 .t a/ ; x2 .t/ D e˛2 .t a/ ; : : : ; xn .t/ D e˛n .t a/ :

(1.13)

The functions in (1.13) are linearly independent and are the fundamental solutions of (1.4). In this case, the general solution of (1.4) is [9] x.t/ D c1 e˛1 .t a/ C c2 e˛2 .t a/ C C cn e˛n .t a/ ;

(1.14)

where c1 ; c2 ; : : : ; cn are arbitrary constants. Case 2. The polynomial p./ has repeated roots. Suppose that is a root of p./, with multiplicity l; .1 < l n/, then according to [9], @ .t a/ @2 .t a/ @l1 .t a/ a/ e e.t ; ; e ; : : : ; e (1.15) ˛ @ ˛ @2 ˛ @l1 ˛ are l linearly independent particular solutions of (1.4). Let 1 ; 2 ; : : : ; k be the different roots with multiplicities l1 ; l2 ; : : : ; lk , respectively, and l1 C l2 C C lk D n, then the general solution of (1.4) are the linear combination of the following fundamental solutions e˛1 .t a/ ;

@ 1 .t a/ @2 1 .t a/ @l1 1 1 .t a/ e˛ ; e ; : : : ; e ˛ @ @2 @l1 1 ˛

e˛2 .t a/ ;

@ 2 .t a/ @2 2 .t a/ @l2 1 2 .t a/ e˛ ; e˛ ; :::; e 2 @ @ @l2 1 ˛

:: : @ k .t a/ @2 k .t a/ @lk 1 k .t a/ e˛ ; e ; : : : ; e ˛ @ @2 @lk 1 ˛ iˇ h j .t a/ ˇ @ D @ e . ˇ j ˛

e˛k .t a/ ; where

@j k .t a/ e @j ˛

Dk

1.3 Analysis of the Characteristic Roots The order ˛ is assumed to be rational: k ˛ D D kˇ n

1 ˇD : n

8

Z.H. Wang and X. Wang

Then the generalized Bagley–Torvik equation (1.3) can be recast into a SFDE 0D

2nˇ

x.t/ C 0 D kˇ x.t/ C x.t/ D 0

(1.16)

and the characteristic equation reads p./ WD 2n C k C 1 D 0:

(1.17)

Obviously, with s D 1=, one has p./ D 0 if and only if q.s/ D 0, where q.s/ D 1 C s 2nk C s 2n . Thus, it is sufficient to consider the case 1 k < n only. Based on the theory introduced above, the general solution of a sequential fractional differential equation can be given directly if the characteristic roots are in hand. The case with repeated roots needs to be studied carefully, which can be carried out in four steps as follows. Step 1. p./ has no repeated roots with multiplicity greater than 2. In fact, if p./ has repeated root , satisfying 8 < p./ D 0 p 0 ./ D 0 : 00 p ./ D 0

, , ,

2n C k C 1 D 0 ; 2n2n1 C kk1 D 0 2n.2n 1/2n2 C k.k 1/k2 D 0

(1.18)

then, the latter two equations yield k 2n D ; 2n.2n 1/ k.k 1/ which is possible only if k D 2n. Step 2. p./ has a repeated real root with multiplicity 2 if k is odd, and it has no repeated real roots if k is even. In fact, the first two conditions in (1.18) implies that 2n D

k 2n > 0 and k D < 0: 2n k .2n k/

(1.19)

If the repeated root is real, then k must be odd. In this case, D O WD

2n 2n k

2n k k

k 2n

:

(1.20)

Step 3. p./ has no repeated roots of pure imaginary numbers if k is odd, and it has at most two pairs of conjugate pure imaginary roots with multiplicity 2 if k is even. In fact, if k is odd, then p.i !/ D 0 implies that ˙ ! 2n ˙ ! k i C 1 D 0 ) ! D 0:

1

General Solution of a Fractional-Order Vibration System

9

Obviously, D 0 is not a root of p./ D 0. When k is even, let k D 2m, 1 m < n=2, and denote 2 D s, then p./ D q.s/ WD s n C s m C 1. If q.s/ has a repeated negative root, then p./ has two pairs of repeated conjugate pure imaginary roots. As known in Step 2, q.s/ has a repeated negative root only if n is even and m is odd. It follows that p./ has a repeated complex roots with non-zero real part only, if both n and m are even. Step 4. p./ has two pairs of repeated conjugate complex roots with non-zero real part if n D 2i a, k D 2j b, where i > j and b is odd. Actually, let n D 2a, m D 2b, (1 b < n=4), then p./ D q.s/ D r.t/ WD t a C t b C 1 with t D 4 . If r.t/ has a repeated negative root, then q.s/ has two pairs of repeated conjugate pure imaginary roots ˙i with > 0. As known in Step 2, r.t/ has a repeated negative root only if a is even and b is odd. In this case, p./ has repeated roots as following p p 2 2 p .1 ˙ i/; p .1 ˙ i/: 2 2 Thus, the claim is confirmed if the same procedure is carried out repeatedly. As for ¤ , O the number of real roots of p./ keeps unchanged in .0; / O and .; O C1/, respectively, since the roots depend continuously on > 0. When D 0, p./ D 2n C1 has no real roots; thus, p./ has n pairs of conjugate simple complex roots for all 2 .0; /. O If p./ has a repeated real root at D , O then p./ has exactly two simple negative roots and .n 1/ pairs of conjugate simple complex roots for > . O Otherwise, p./ has n pairs of conjugate simple complex roots for > . O

1.4 The General Solution Based on the above analysis, we can express the general solution of the SFDE directly by using the roots of p./, in terms of ˛-exponential function et O t ˛ or e ˛ , and then prove that some .2n 2/ free constants depend on the rest two free constants. For simplicity, the general solution, in the terms of et ˛ , of the following equation x.t/ R C 0 D ˛ x.t/ C x.t/ D 0

> 0; ˛ D

2 3

(1.21)

is only addressed. With ˇ D 1=3, (1.21) can be changed to 0 D 6ˇ x.t/ C 0 D 2ˇ x.t/ C x.t/ D 0. For each ¤ O and > 0, the characteristic function p./ D 6 C 2 C 1 has no repeated roots but three pairs of conjugate simple roots 1 ; N 1 ; 2 ; N 2 ; 3 ; N 3 , and the general solution of the SFDE can be written as x.t/ D c1 x1 .t/ C cN1 xN 1 .t/ C c2 x2 .t/ C cN2 xN 2 .t/ C c3 x3 .t/ C cN3 xN 3 .t/;

(1.22)

10

Z.H. Wang and X. Wang t

i where xi .t/ D e1=3

.i D 1; 2; 3/. For t ! C0, one has

x.t/ D .c1 C cN1 C c2 C cN2 C c3 C cN3 /

t 2=3 .1=3/

t 1=3 .2=3/ 1 C c1 21 C cN1 N 21 C c2 22 C cN2 N 22 C c3 23 C cN3 N 23 .1/ t 1=3 C c1 31 C cN1 N 31 C c2 32 C cN2 N 32 C c3 33 C cN3 N 33 .4=3/ t 2=3 C O.t 2 /: C c1 41 C cN1 N 41 C c2 42 C cN2 N 42 C c3 43 C cN3 N 43 .5=3/ (1.23) C .c1 1 C cN1 N 1 C c2 2 C cN2 N 2 C c3 3 C cN3 N 3 /

In order that jx.0/j < 1, jx.0/j P < 1, it is required to have 8 c1 C cN1 C c2 C cN2 C c3 C cN3 D 0 ˆ ˆ ˆ ˆ < c C cN N C c C cN N C c C cN N D 0 1 1 1 1 2 2 2 2 3 3 3 3 : 3 3 3 3 3 N N ˆ c1 1 C cN1 1 C c2 2 C cN2 2 C c3 3 C cN3 N 33 D 0 ˆ ˆ ˆ : c1 41 C cN1 N 41 C c2 42 C cN2 N 42 C c3 43 C cN3 N 43 D 0

(1.24)

In this case, it holds jx.0/j R < 1 as well. Straightforward computation gives the coefficient determinant of the above linear system with respect to <.c2 /; =.c2 /; <.c3 / and =.c3 / as following D D 2.2 3 /.N 2 N 3 /.N 2 3 /.2 N 3 / .N 2 N 2 C N 3 N 3 C 2 3 C N 2 N 3 C 2 N 3 C N 2 3 /:

(1.25)

The assumptions i ¤ j and i ¤ N j imply that D ¤ 0. Thus, <.c2 /; =.c2 /; <.c3 / and =.c3 / can be determined uniquely from (1.24), in terms of <.c1 / and =.c1 /. As a result, the general solution of (1.21) depends on <.c1 / and =.c1 / only and can be determined fully by the initial displacement and initial velocity. The above procedure can be carried out to obtain the general solution of (1.3) in a similar way for D , O as well for any given rational number ˛ 2 .0; 2/. The computational complexity increases sharply as n increases. Acknowledgement This work was supported by NSF of China under Grant 10825207 and in part by FANEDD of China under Grant 200430.

1

General Solution of a Fractional-Order Vibration System

11

References 1. Podlubny I (1999) Fractional differential equations. Academic, San Diego, CA 2. Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, Amsterdam 3. Das S (2008) Functional fractional calculus for system identification and controls. Springer, Berlin 4. Bagley RL, Torvik PJ (1984) On the appearance of the fractional derivative in the behavior of real materials. ASME J Appl Mech 51:294–298 5. Bagley RL, Torvik PJ (1983) Fractional calculus – a different approach to the analysis of viscoe1astically damped structures. AIAA J 21(5):741–748 6. Wang ZH, Hu HY (2010) Stability of a linear oscillator with damping force of fractional-order derivative. Sci China: Phys, Mech Astron 53(2):1–8, doi:10.1007/s11433-009-0291-y 7. Suarez LE, Shokooh A (1997) An eigenvector expansion method for the solution of motion containing fractional derivatives. ASME J Appl Mech 64:629–635 8. Lorenzo CF, Hartley TT (2008) Initialization of fractional-order operators and fractional differential equations. ASME J Comput Nonlinear Dyn 3:021101 9. Bonilla B, Rivero M, Trujillo JJ (2007) Linear differential equations of fractional orders. In: Sabatier J, Agrawal OP, Tenreiro Machado JA (eds) Advances in fractional calculus. Springer, Dordrecht, pp 77–91

Chapter 2

An Analytic Proof for the Sensitivity of Chaos to Initial Condition and Perturbations J.H. Peng and J.S. Tang

Abstract An analytic method to prove the sensitivity of chaotic motion to initial states and perturbations is proposed in this paper. With the fundamental perturbation method, a second order nonlinear differential equation is expanded into a series of perturbation equations, and by means of variation of constants, the general solutions of the perturbation equations are obtained. Based on these general solutions, we prove that the chaotic motion is sensitively dependent on the initial conditions and perturbations.

2.1 Introduction Since Lorentz discovered the chaotic motion of air flow in his famous article “Deterministic nonperiodic flow” [1] and Li and Yorke first introduced the term chaos [2], considerable efforts, both theoretical and experimental, have been devoted to the study of chaotic motion of nonlinear dynamical system. Many important properties and applications of chaos have been obtained [3–12]. The property that chaotic motion is sensitively dependent on initial conditions is the most important nature of chaos [13]. With numeric computation technique the property of various chaotic systems has been checked [1, 2] and by means with the Lyapunov characteristic exponents, it can be verified quantitatively [14, 15]. But all these methods are numerical ones and we have not analytically proved the property of chaotic motion being sensitively dependent on the initial conditions still. In this paper, we propose an analytical proof for the property. Firstly, we expand the second nonlinear differentiate equation into a series perturbation equations, then, with the trial and error procedure and variation of constants method, we derive the general J.H. Peng () Department of Mechanical Engineering, Shaoyang Polytechnic, Hunan, People’s Republic of China and Department of Physics, Shaoyang University, Shaoyang, Hunan, People’s Republic of China e-mail: [email protected]

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 2,

13

14

J.H. Peng and J.S. Tang

solutions of each perturbation equation, and, finally based on the above solutions, we prove the sensitivity of chaos to initial conditions and perturbations analytically.

2.2 Perturbation Equations The governing equation of the second order nonlinear differential dynamical system being studied can be expressed as ( xR D f .x/ C "g.t; x/ P ; f; g 2 C r ; (2.1) x.0/ D a; x.0/ P Db where ".>0/ is a small arbitrary parameter, f .x/ is a nonlinear function of x, g.t; x/ P is the perturbation acting on the system. The fundamental perturbation method is employed to solve (2.1). Setting x.t/ D x0 .t/ C "x1 .t/ C "2 x2 .t/ C :

(2.2)

Substituting (2.2) into (2.1) and expanding f .x/ as well as g.t; x/ P into the Taylor’s series of x and xP yields @f .x0 / ."x1 C "2 x2 C / @x 1 @2 f .x0 / 2 C ."x1 C "2 x2 C / C 2Š @x 2 @g.t; xP 0 / C " g.t; xP 0 / C ."xP 1 C "2 xP 2 C / @xP 1 @2 g.t; xP 0 / 2 2 ."xP 1 C " xP 2 C / C ; C 2Š @xP 2 (2.3)

xR 0 .t/ C "xR 1 .t/ C "2 xR 2 .t/ C D f .x0 / C

(

x0 D x0 .0/ C "x1 .0/ C "2 x2 .0/ C xP 0 D xP 0 .0/ C "xP 1 .0/ C "2 xP 2 .0/ C

;

(2.4)

equating the sum of i th-order terms of " in (2.3) to zero, we obtain the zeroth-order perturbation equation and i th-order perturbation equations as follows "0 W xR 0 D f .x0 /; 8 <xR D @f .x0 / x C h .t; x ; xP / 1 1 1 0 0 ; "1 W @x : h1 .t; x0 ; xP 0 / D g.t; xP 0 /

(2.5)

(2.6)

2

Sensitivity of Chaos to Initial Condition and Perturbation

8 @f .x0 / ˆ <xR 2 D x2 C h2 .t; x0 ; x1 I xP 0 ; xP 1 / 2 @x " W ; 2 ˆ :h2 .t; x0 ; xP 0 ; x1 / D @g.t; xP0 / C 1 @ f .x0 / x 2 1 @xP 2 @x 2

15

(2.7)

8 @f .x0 / ˆ ˆ xR i D xi C hi .t; x0 ; x1 ; : : : ; xi 1 I xP 0 ; xP 1 I : : : I xP i 1 / ˆ ˆ ˆ @x ˆ ˆ ˆ ˆ hi .t; x0 ; x1 ; : : : ; xi 1 I xP 0 ; xP 1 I : : : I xP i 1 / ˆ ˆ ˆ ˆ ˆ < @g.t; xP 0 / @2 g.t; xP 0 / i x P C .xP 1 xP i 2 C xP 2 xP i 3 C / C D i 1 " W (2.8) : @xP @xP 2 ˆ ˆ i 1 2 ˆ @ g.t; x P / f .x / d ˆ 0 0 1 ˆ ˆ xP 1i 1 C .x1 xi 1 C x2 xi 2 C / C C .i 1/Š ˆ i 1 2 ˆ @ x P dx ˆ ˆ ˆ ˆ di 1 f .x0 / i 2 1 di f .x0 / i 1 ˆ ˆ x x C x1 :C 2 1 .i 2/Š dx i 1 i Š dx i

2.3 General Solutions of Perturbation Equations Equation (2.5) is the unperturbed equation of original system (2.1). If we can get the solutions, xi .t/ .i D 0; 1; 2; : : :/, of various equations given earlier, then the solution of (2.1) is obtained by inserting xi .t/ into (2.2). Noting that hi .t; x0 ; xP 0 I x1 ; xP 1 I : : : I xi 1 ; xP i 1 / .i D 1; 2; : : :/ is the non-homogeneous term of the i th-order perturbation equation .i D 1; 2; : : :/, hence the homogeneous parts of the i th-order perturbation equations .i D 1; 2; : : :/ are linear differential equations. The general solution of the i th-order perturbation equation .i D 1; 2; : : :/ can be derived from the fundamental solutions of the homogeneous parts of the i th-order equation with the method of variation of constants that is well known to us. For the sake of determining the general solution of the i th-order perturbation equation .i D 1; 2; : : :/, assume the formal solution of zeroth-order equation (2.5) is x0 .t/. According to the characteristic of the homogeneous parts of i th-order perturbation equation @f .x0 / xi ; i D 1; 2; : : : @x we suppose the trial solutions of them being Z xi1 .t/ D xP 0 .t/; xi 2 .t/ D xP 0 .t/ ŒxP 0 .t/2 dt; i D 1; 2; : : : xR i D

(2.9)

(2.10)

We verify these solutions satisfying (2.9) first. On the one hand, it is easy to get the second derivative as follows «0 .t/: (a) xR i1 .t/ D x Inserting it into (2.9), we have x «0 .t/ D

@f .x0 / xP 0 .t/: @x

(b)

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J.H. Peng and J.S. Tang

On the other hand, differentiating (2.5) one more time yields x «0 .t/ D

@f .x0 / xP 0 .t/: @x

(c)

The forms of (b) are the same with (c), hence the trial solution xi1 .t/ D xP 0 .t/ is a solution of (2.9). Now, we prove that the trial solution xi 2 .t/ is the real solution of (2.9). The first derivative and second derivative can be easily obtained as follows Z xP i 2 .t/ D xR 0 .t/ ŒxP 0 .t/2 dt C ŒxP 0 .t/1 ; Z «0 .t/ ŒxP 0 .t/2 dt C xR 0 .t/ŒxP 0 .t/2 xR 0 .t/ŒxP 0 .t/2 xR i 2 .t/ D x Z Dx «0 .t/ ŒxP 0 .t/2 dt Substituting representation (b) into this equation yields Z xR i 2 .t/ D x «0 .t/

ŒxP 0 .t/2 dt D

@f .x0 / xP 0 .t/ @x

Z

ŒxP 0 .t/2 dt:

(d)

Meanwhile, we substitute xi 2 .t/ into (2.9) xR i 2 .t/ D

@f .x0 / xP 0 .t/ @x

Z

ŒxP 0 .t/2 dt:

It is exactly the same with representation (d), and it shows that the trial solution xi 2 .t/ is the solution of (2.9). So far, the fundamental solutions of homogeneous equation (2.9) have been determined. It is obvious that xi1 .t/ and xi 2 .t/ are linearly independent, thereby the general solution of (2.9) is the linear combination of xi1 .t/ and xi 2 .t/, that is xi .t/ D c1 xi1 .t/ C c2 xi 2 .t/: According to the method of variation of constants, the general solution of the i th-order perturbation equation xR i D

@f .x0 / xi C hi .t; x0 ; x1 ; : : : ; xi 1 I xP 0 ; xP 1 ; : : : ; xP i 1 /; @x

i D 1; 2; : : : (2.11)

can be expressed as xi .t/ D ci1 .t/xi1 .t/ C ci 2 .t/xi 2 .t/:

(2.12)

Differentiating it with respect to t yields xP i .t/ D cPi1 .t/xi1 .t/ C cPi 2 .t/xi 2 .t/ C ci1 .t/xP i1 .t/ C ci 2 .t/xP i 2 .t/:

(2.13)

2

Sensitivity of Chaos to Initial Condition and Perturbation

17

Because there is only one equation (2.11) that the two unknown functions ci1 .t/ and ci 2 .t/ must fulfill, we can set another expression that the two unknown functions should hold. Representation (2.13) implying that the second derivatives of ci1 .t/ and ci 2 .t/ is undesired in xR i .t/ we can set cPi1 .t/xi1 .t/ C cPi 2 .t/xi 2 .t/ D 0;

(2.14)

xP i .t/ D ci1 .t/xP i1 .t/ C ci 2 .t/xP i 2 .t/:

(2.15)

so that differentiating (2.15) with respect to t one more time results in xR i .t/ D cPi1 .t/xP i1 .t/ C cPi 2 .t/xP i 2 .t/ C ci1 .t/xR i1 .t/ C ci 2 .t/xR i 2 .t/:

(2.16)

Substituting (2.12) and (2.16) into (2.11), one has @f .x0 / xi1 .t/ cPi1 .t/xP i1 .t/ C cPi 2 .t/xP i 2 .t/ C ci1 .t/ xR i1 .t/ @x @f .x0 / xi1 .t/ D hi : C ci 2 .t/ xR i 2 .t/ @x

(2.17)

Employing representations (2.9) and (2.10), this equation becomes cPi1 .t/xP i1 .t/ C cPi 2 .t/xP i 2 .t/ D hi :

(2.18)

Combining equations of (2.14) and (2.18), the condition is that the solutions of unknown variables ci1 .t/ and ci 2 .t/ are nontrivial is W D

xi1 xi 2 xP i1 xP i 2

D xi1 xP i 2 xi 2 xP i 2 ¤ 0:

(2.19)

That is

Z

2

1

W D xi1 xP i 2 xi 2 xP i 2 D xP 0 .t/ xR 0 .t/ ŒxP 0 .t/ dt C ŒxP 0 .t/ Z 2 xP 0 .t/ ŒxP 0 .t/ dt xR 0 .t/ D xP 0 .t/ŒxP 0 .t/1 D 1:

(2.20)

Condition (2.19) is fulfilled, and then the nontrivial solutions of ci1 .t/ and ci 2 .t/ are (

cPi1 .t/ D xi 2 .t/hi cPi 2 .t/ D xi1 .t/hi

:

(2.21)

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J.H. Peng and J.S. Tang

Integrating these equations yields (

ci1 .t/ D Ci1 C ci 2 .t/ D Ci 2 C

Rt

xi 2 .t/hi dt

Rt0t

xi1 .t/hi dt

t0

;

(2.22)

where Ci1 ; Ci 2 are the integral constants that are related to the initial conditions. With the initial conditions (2.4) and expressions (2.12), (2.15), we obtain Ci1 and Ci 2 as follows ( P xP 2 .0/x.0/ Ci1 D x2 .0/x.0/ : (2.23) Ci1 D xP 1 .0/x.0/ x1 .0/x.0/ P Substituting them into representations (2.22) yields (

P xP 2 .0/x.0/ C ci1 .t/ D x2 .0/x.0/ ci 2 .t/ D xP 1 .0/x.0/ x1 .0/x.0/ P C

Rt Rt0t t0

xi 2 .t/hi dt xi1 .t/hi dt

:

(2.24)

To simplify (2.24), we choose a special time ti1 D Ai1 and it fulfills Z

Ai1

P xP 2 .0/x.0/ x2 .0/x.0/

xi 2 .t/hi dt D 0:

(2.25)

t0

With the same method, we select ti 2 D Ai 2 and it holds Z P xP 1 .0/x.0/ x1 .0/x.0/

Ai1

xi1 .t/hi dt D 0:

(2.26)

t0

Therefore, representations (2.24) reduce to (

Rt ci1 .t/ D Ai1 xi 2 .t/hi dt ; Rt ci 2 .t/ D Ai 2 xi1 .t/hi dt

(2.27)

where Ai1 and Ai 2 are the constants determined by the initial conditions. Substituting them into (2.12), we get the general solution of i th-order perturbation equation Z t Z t xi .t/ D xi 2 .t/ xi1 .t/hi dt xi1 .t/ xi 2 .t/hi dt; i D 1; 2; : : : (2.28) Ai 2

Ai1

Differentiating it with respect to t results in Z

Z

t

xP i .t/ D xP i 2 .t/

t

xi1 .t/hi dt xP i1 .t/ Ai 2

xi 2 .t/hi dt; i D 1; 2; : : : Ai1

(2.29)

2

Sensitivity of Chaos to Initial Condition and Perturbation

19

2.4 Sensitivity to the Initial Conditions According to Ref. [16], the homoclinic point and heteroclinic point both are the core of the chaotic structure. Therefore, what the chaotic motion is sensitivity to the initial conditions is transferred to that of the homoclinic orbit or the heteroclinic orbit is sensitivity to the initial conditions. Setting that the solution fx0 .t/; xP 0 .t/g of unperturbed equation (2.5) of nonlinear second differentiating dynamical system (2.1) is a homoclinic orbit. In the light of the definition of homoclinic orbit, the orbit will tent to the saddle as t ! 1 [17]. It implies xP 0 .t/ ! 0; xR 0 .t/ ! 0 as t ! 1. Therefore, the fundamental solution xi 2 .t/ (see (2.10)) is a divergent function. That is, xi 2 .t/ ! 0 as t ! ˙1. Setting that there is a small change of the initial condition in (2.1) and it gives rise to the integral lower limits of (2.28) transforming from Ai1 and Ai 2 into A0i1 and A0i 2 .i D 1; 2; : : :/, Consequently, the general solution of i th order equation and its derivative transit from xi .t/, xP i .t/ to xi0 .t/, xP i0 .t/ .i D 1; 2; : : :/. The differences, which is caused by the small change of initial condition, between xi .t/ and xi0 .t/ are xi0 .t/ xi .t/ D xi 2 .t/

Z

Z

t A0i 2

xi1 .t/hi dt xi1 .t/

t A0i1

xi 2 .t/hi dt

Z t Z t xi 2 .t/ xi1 .t/hi dt xi1 .t/ xi 2 .t/hi dt A Ai1 "Z i 2 # Z t t D xi 2 .t/ xi1 .t/hi dt xi1 .t/hi dt A0i 2

Ai 2

D Axi 2 .t/ Bxi1 .t/;

i D 1; 2; : : :

(2.30)

and the differences between xP i .t/ and xP i0 .t/ are xP i0 .t/ xP i .t/ D xP i 2 .t/

Z

Z

t A0i 2

Z xP i 2 .t/

xi1 .t/hi dt xP i1 .t/

t A0i1

Z

t

xi 2 .t/hi dt

Ai 2

D AxP i 2 .t/ B xP i1 .t/; i D 1; 2; : : :

t

xi1 .t/hi dt xP i1 .t/

xi 2 .t/hi dt Ai1

(2.31)

The two equations given above show that only if Ai1 D A0i1 and Ai 2 D A0i 2 the function in all square brackets equal zero, so that xi0 .t/ D xi .t/, xP i0 .t/ D xP i .t/. Otherwise, xi0 .t/ ¤ xi .t/, xP i0 .t/ ¤ xP i .t/. With (2.25) and (2.26) we know that any small perturbation of initial condition will give rise to Ai1 ¤ A0i1 , Ai 2 ¤ A0i 2 in (2.30) and (2.31). Therefore, we have xi0 .t/ ¤ xi .t/, xP i0 .t/ ¤ xP i .t/. Inserting them in (2.2), we obtain x.t/ ¤ x 0 .t/, x.t/ P ¤ xP 0 .t/, that is to say the two orbits corresponding to two sets of little different initial conditions respectively are different. Equations (2.30) and (2.31) imply that xi0 .t/ xi .t/ ! 1 and xP i0 .t/ xP i .t/ ! 1

20

J.H. Peng and J.S. Tang

as t ! 1 because of the infinity of xi 2 .1/ and xP i 2 .1/, namely, the difference of the two orbits tends infinity. The above mentioned analysis clearly shows that the chaotic motion is sensitively dependent on the initial conditions.

2.5 Sensitivity to Perturbations In this section, we prove the sensitivity of chaotic motion to the perturbations with the same approach we used in the last section. To this end, we suppose that the perturbation acting on the system (2.1) changes from g.t; x/ P into P It makes hi .t; x0 ; x1 ; : : : ; xi 1 I xP 0 ; xP 1 ; : : : ; xP i 1 / of i th order perturbag 0 .t; x/. tion equation change into h0i .t; x0 ; x1 ; : : : ; xi 1 I xP 0 ; xP 1 ; : : : ; xP i 1 /xP 0 ; xP 1 ; : : : ; xP i 1 /. Hence, the solution of i th order perturbation equation .i D 1; 2; : : :/ transits from xi .t/ to xi0 .t/, xP i .t/ to xP i0 .t/ .i D 1; 2; : : :/. The chaotic orbit, fx0 .t/; x1 .t/; : : : ; xi .t/; : : : I xP 0 .t/; xP 1 .t/; : : : ; xP i .t/; : : :g of the system turns into the new orbit of fx0 .t/; x10 .t/; : : : ; xi0 .t/; : : : I xP 0 .t/; xP 10 .t/; : : : ; xP i0 .t/; : : :g. The difference between the two orbits can be determined by calculating the differences xi .t/ xi0 .t/ and xP i .t/ xP i0 .t/. Because the initial conditions of the two orbits are identical so that Ai1 and Ai 2 remain the same, and the unperturbed solution is independent of perturbation as well as, we can easily obtain the following results Z t xi1 .t/.h0i hi /dt xi0 .t/ xi .t/ D xi 2 .t/ Ai 2 t

Z xi1 .t/ xP i0 .t/ xP i .t/ D xP i 2 .t/

Ai1 t

Z

Ai 2 t

xi 2 .t/.h0i

hi /dt ;

xi1 .t/.h0i hi /dt

Z xP i1 .t/

Ai1

(2.32)

xi 2 .t/.h0i hi /dt ;

(2.33)

where i D 1; 2; : : :. These expressions state that as long as h0i hi does not equal to zero, xi0 .t/ xi .t/ and xP i0 .t/ xP i .t/ will not be vanished. Since h0i ¤ hi and lim xi 2 .t/ ! 1, lim xP i 2 .t/ ! 1, above two equations become

t !1

lim Œx 0 .t/ t !1 i

t !1

Z xi .t/ D lim xi 2 .t/ t !1

t Ai 2

Z

xi1 .t/.h0i hi /dt t

xi1 .t/ Z lim xi 2 .t/ t !1

Ai1 t Ai 2

xi 2 .t/.h0i hi /dt

xi1 .t/.h0i

hi /dt ! 1;

2

Sensitivity of Chaos to Initial Condition and Perturbation

Z 0 lim ŒxP i .t/ xP i .t/ D lim xP i 2 .t/

t !1

t !1

t Ai 2

Z

xi1 .t/.h0i hi /dt t

xP i1 .t/ Z lim xP i 2 .t/ t !1

Ai1 t Ai 2

21

xi 2 .t/.h0i hi /dt

xi1 .t/.h0i hi /dt ! 1;

i D 1; 2; : : :

These results clearly shows that any small change of the perturbation acting on the system will cause the orbit of the system to go far away from the original one. That is to say that the chaos possesses the property of sensitivity to the perturbation.

References 1. Lorentz EN (1963) Deterministic non-periodic flow. J Atmos Sci 20:130–141 2. Li T, Yorke JA (1975) Period three implies chaos. Am Math Monthly 82:985–992 3. Peng JH, Tang JS, YU DJ, et al (2002) Solutions bifurcations and chaos of the nonlinear Schrodinger equation with weak damping. Chin Phys 11:213–217 4. Lee CH, Hai WH, Lei S, et al (2001) Chaotic and frequency-locked atomic population oscillations between two coupled Bose–Einstein condensates. Phys Rev A64:053604 5. Hai WH, Zhang XL, Huang WL, et al (2001) Chaotic solution of the rf-driven Josephson system with quadratic damping. . Int J BifurcatChaos 11:2263–2269 6. Hai WH, Duan YW, Zhu XW, et al (1998) Stable orbits embedded in a chaotic attractor for a trapped ion interacting with a laser field. J Phys A Math Gen 31:2991–2996 7. Peng JH, Tang JS, Yu DJ, et al (2003) Suppressing Chaos by parametric perturbation at doubled frequency of periodic perturbation. Chin phys 12:17–21 8. Pecora IM, Carrol TL (1998) Synchronization in chaotic systems. Phys Rev Lett 64:821 9. Badola P, Tambe SS, Kulkami BD (1992) Driving systems with chaotic signals. Phys Rev A46:6735 10. Van Wiggeren GD, Roy R (1998) Communication with chaotic lasers. Science 279:1198–1200 11. Gauthier DJ (1998) Chaos has come again. Science 279:1156–1157 12. Lin WW, Wang YQ (2002) Chaotic synchronization in coupled map lattices with periodic boundary conditions. SIAN J Appl Dyn Syst 1:175–189 13. Eckman JP, Ruelle D (1985) Ergodic theory of chaos and strange attractors. Rev Mod Phys 57:617–656 14. Ruelle D (1980) Measures describing a turbulent flow. Ann N Y Acad Sci 357:1–9 15. He D-H, Xu J-X, Chen Y-H (2000) Study on Lyapunov characteristic exponents of a nonlinear differential equation system. Acta Phys Sin 49:833–837 16. Liu SD, Liu SS (1994) Solitons and turbulences. Shanghai Scientific and Technological Education Publishing House, Shanghai, p 102 17. Liu Z (1994) Perturbation criteria for chaos. Shanghai Scientific and Technological Education Publishing House, Shanghai, p 12

Chapter 3

Study on the Multifractal Spectrum of Local Area Networks Traffic and Their Correlations Yan Liu and Jia-Zhong Zhang

Abstract Due to the singularity in the Local Area Network (LAN) traffic, the multifractal spectrums are used to study the characteristics of network traffic from the viewpoint of nonlinear dynamic system. First, the multifractal spectrums of the LAN traffic are introduced and established to investigate the complex features of the systems. Then, the distributions of spectrum parameter vs. the network traffic are studied in detail, and some important phenomena, which are related with the complicated networks traffic, are captured. Furthermore, the correlations between multifractal spectrum and logarithm of mean traffic are presented, and they can be feasibly applied to the prediction for the networks traffic. Some conclusions can be drawn that the variation of width of multifractal spectrum is similar to that of network traffic in a sense. To some degree, the difference between maximum and minimum probability of multifractal spectrum is ahead to the oscillation of network traffic, and it is a fundamental route for the network traffic prediction.

3.1 Introduction There exists a rich variety of nonlinear phenomena in Network traffic. As the study of the characteristics of network traffic is in-depth, especially on the study of TCP data streams as well as the behaviors in scale of the global area network, it is found that the network traffic in larger time scales shows single fractal characteristics, while in smaller time scales shows the multifractal characteristics of network traffic obviously, that is, the network traffic behaves as a complex unsteady flow, and in particular, there exists a local breaking and bursting in the traffic [1–4]. Traditional methods of network traffic analysis and the models are focused on the self-similarity of the network traffic, in the relatively large time scale [5–8]. However, these methods and models could not be used to describe and investigate the J.-Z. Zhang () School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, People’s Republic of China e-mail: [email protected]

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 3,

23

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local characteristic of the time sequence of network traffic in small or fine scale in detail, and the breaking and bursting behaviors could not be captured. Hence, the multifractal is introduced to the analysis of the time-sequence of networks traffic in this study, especially for the rich variety of nonlinear information. Further, it is found that there exists the complex singularity in the time-sequence of network traffic in the global area network traffic. As for the local network, there are a few studies on such topics. With such background, the current work focuses on the essential features of the fractal structure in the time-sequence of local area networks traffic, especially the multilevel feature. First, the essential features of the local area network traffic are studied by the multifractal spectrum, and the relationship between the multifractal spectrum and the flux of the system is analyzed further. Some important results are obtained, and can be applied to the prediction of the network traffic.

3.2 Multifractal and the Spectral Parameters for Networks Traffic In a sense, multifractal is the decomposition of the fractal object, and the component will have its own fractal dimension. Assume that there exists a unit interval with a unit mass, and it is further divided into several subintervals with the length ı of each interval for. Then, a nonnegative mass is allocated in the kth sub-interval, and the sequence fk ; k 1g denotes a stochastic process. Hence, the exponent for singularity of k at time t0 can be defined as follows: ln.k / ; ı!0 ln ı

˛.t0 / D lim

(3.1)

where k denotes the mass of the interval containing t0 in time. Normally, singularity exponent ˛.t/ is also called as H˘older exponent, which describes the fractal dimension in the fractal geometric theory, namely, local fractal dimension, and can describe the probability of growing of the small interval. If there exists no limit in (3.1), then the singularity exponent could not be defined at time t0 ; If ˛.t/ is a constant, then the singularity of the sequence at the overall scale can be described by only one global exponent, which is the feature of single fractal; If ˛.t/ is the function of time t, that is, the feature of its scale is related to time, then the sequence has a multifractal characteristics. In comparison with the single fractal, the concept of multifractal extends the understanding of the scale, and the scale relevant to time could describe the nonregular behaviors in the local time interval. The multifractal is used to describe fractal dimension of the domain with a large number of small area, and ˛.t/ would extend the single fractal exponent (Hurst parameter) to the one with multivalues. To this end, it needs to know the probability of ˛.t/ with different values, in order to analyze the characteristics of

3

25

Multifractal Spectrum of LAN Traffic

network traffic. As the results, a spectrum f .˛/ composed by infinite sequence containing various ˛.t/, and f .˛/ fractal dimension of subsets with same ˛, namely, the probability of the occurrence of ˛ in the system, and it can be called multifractal spectrum. The value of f .˛/ values should be in the interval [0,1], and usually be the convex shape, which can be considered the probability density for the exponent ˛.t/. Multifractal spectrum f .˛/ could represent the nonuniform property of network traffic in the fractal structure, so that much more information on the structure can be gained, when compared with the simple single fractal dimension. If the multifractal spectrum ˛.t/ < 1 for a sequence, then the sequence in the small interval around a certain point will behave as bursting and breaking in all of the scales. If ˛.t/ > 1, it when demonstrate that the network traffic varies slightly without obvious bursting and breaking. Hence, it can use the range of ˛.t/ to determine bursting and breaking properties of network traffic. Multifractal spectrum width ˛ D ˛max ˛min could describe the property of nonuniform of the probability for the entire fractal structure, so that the features, related with the different area, different level, and different local area, can be described in detail. In addition, the multifractal spectrum parameters ˛max and f .˛max / represent the property of the minimum subset of the probability, and ˛min and f .˛min / represent the property of the maximum subset of the probability. Hence, multifractal spectrum is a measure for the complexity, non-regularity and nonuniform property of the fractal structure of the sequence.

3.3 Relationship Between Variables of Multifractal Spectrum A function q ."/, namely, distribution function, can be defined by the follows, q ."/ D

X

Pi ."/q D ".q/ ;

(3.2)

where .q/ is the mass function or structure function. If (3.2) can be satisfied, that is, there exists a power-type relationship between the distribution function and ©, then the mass function can be defined as .q/ D

ln q ."/ : ln "

(3.3)

According to the definition of multifractal, the generalized multifractal dimension by .q/ can be expressed as Dq D

ln q ."/ .q/ D : q1 .q 1/ ln "

(3.4)

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Following Legendre transform, there exist some relationships between ˛; f .˛/; .q/ and Dq , they are Dq D

1 Œq˛ f .˛/; q1

f .˛/ D q˛ .q/; ˛D

d.q/ : dq

(3.5) (3.6) (3.7)

By (3.5), Dq can be obtained as ˛ and f .˛/ are given, and ˛ can be obtained from (3.7), in combination with the derivation of .q/. Multifractal spectrum can be further obtained by (3.6) and (3.7), f .˛/ D q

d.q/ .q/: dq

(3.8)

Furthermore, following (3.4) and (3.7), yields, ˛D

d.q/ d D Œ.q 1/Dq : dq dq

(3.9)

It implies that ˛ can be also obtained if Dq is prescribed. In the above mentioned equation, q is the weighting factor, and the multifractal spectrum can be used to analyze the different level of fractal structure decomposed by q.

3.4 Numerical Simulation and Analysis LAN traffic sequence used in this study comes from the internal network of a central group of network monitoring system during March 16, 2005 to March 28, 2005. This type of LAN is the Ethernet host whose components come from hundreds of medium-sized local area networks, and the network traffic data is obtained by sniffing tools, recording the number of network packets and the amount of data with a time interval of 1 s, and resulting in a nonnegative time series. The transmission of information includes web browsing, file transfer, network operating systems, etc.

3.4.1 Multifractal Spectrum and Parameters Figure 3.1 shows the analysis of the multifractal spectrum of the time sequence obtained during March 16, 2005, and the multifractal spectrum parameters are shown in Table 3.1. It can be seen from the numerical simulation results that there exists a significant difference in the width of the multifractal spectrum and its shape.

3

27

Multifractal Spectrum of LAN Traffic

Fig. 3.1 Multifractal spectrum for real network traffic

Table 3.1 Date 8–9 9–10 10–11 11–12 14–15 15–16 16–17 17–18

Multifractal spectrum parameters for real sequence ˛min f .˛min / ˛max f .˛max / ˛ 0.4731 0.0576 2.7137 0.04 2.2406 0.3743 0.0381 1.9223 0.5211 1.548 0.6201 0.4739 2.6183 0.0415 1.9982 0.4121 0.0659 2.4348 0 2.0227 0.6153 0.2371 1.7835 0.2675 1.1682 0.1481 0 1.7947 0.1957 1.6466 0.3031 0 1.6556 0.077 1.3525 0.5515 0.1595 1.5614 0.2913 1.01

f 0.0176 0.483 0.4324 0.0659 0.0304 0.1957 0.077 0.1318

f D f .˛min / f .˛max /

In particular, the multifractal spectrum width during 17:00 to 18:00 is the narrowest, and the width during 8:00 to 9:00 is the widest, that means the latter shows the most non-uniform property of its network traffic. During 15:00 to 16:00, both ˛min and the f .˛min / are minimum, that implies the probability of occurrence of maximum traffic is lowest, and however the traffic is sensitive to the breaking. On the other hand, during 10:00 to 11:00, both ˛min and f .˛min / are maximum, then the probability of occurrence of maximum traffic is highest, and the traffic is difficult to become bursting and breaking. During 9:00 to 10:00, the f .˛max / is the maximum one, that means the probability of occurrence of minimum is the highest one. On the contrary, during 11:00 to 12:00, the f .˛max / is minimum, that is, the probability of occurrence of minimum is lowest. Furthermore, it is clear that the shape of the multifractal spectrum depends on f significantly, namely, its positive and negative value. For example, there exists a hooked-curve on the left of the multifractal spectrum during 10:00 to 11:00, and a hooked-curve on the right of the multifractal spectrum during 9:00 to 10:00. This is the results from the emergence of the maximum and minimum flow ratio of the probability, that is, when f > 0, probability of the maximum traffic is greater than the probability of the minimum traffic, and vice versa.

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3.4.2 Relationship Between Multifractal Spectrum Parameters and Traffic Variation From the simulation analysis given above, the multifractal spectrum parameters ˛ and f can describe the fluctuation of the network traffic to some extent. For analyzing the relationship between the multifractal spectrum and variation of the traffic further, Zi is introduced to measure the variation of average flux I.ti / in a certain time interval t, I.ti / : (3.10) Zi D ln I.ti 1 / Time scale t is set to 1 h, then I.ti / denotes the average flux of the i -instant traffic, I.ti 1/ means the i 1-instant average flux, so that there are a set of parameters, Zi ; ˛i and fi , at each instant. The time sequence collected during 15:00 pm on March 15, 2006 to 10:00 am on March 28, 2006, a total of 307 h, and the variation of ˛i ; fi vs. Zi are shown in Fig. 3.2. It is clear that ˛i will become greater as Zi is farther away from the origin, that is, greater the changes in traffic is, greater the ˛i becomes. At the same time, as jZi j is small, the probability of large jfi j larger becomes higher, that indicates there exists a relationship between the multifractal spectrum parameters and the average flow. In addition, Fig. 3.3 is the time history of the variance ln.VI/; ˛, the average flux rate ln(I ) and f , for the time sequence collected on March 16, 2005. It is clear that the variation of ˛ is very similar to that of variance ln.VI/, and there are five lowest points in the time history of ln(I ); they are 3:00 to 4:00, 7:00 to 8:00, 13:00 to 14:00, 16:00 to 17:00, and 19:00 to 20:00. And there are also seven lowest points in the time history of f , they are 2:00 to 3:00, 6:00 to 7:00, 9:00 to 10:00, 12:00 to 13:00, 15:00 to 16:00, 18:00 to 19:00, and 21:00 to 22:00. It can be seen, except two time intervals, the occurrence of the lowest points in the other time intervals of f lags that of ln(I ) for an hour. Once such phenomena is available for all of the network traffic, f can be chosen as a measure to predict the system, and the relationship will be investigated in depth, that is the next work.

Fig. 3.2 The distribution of Zi vs. ˛i and fi

3

Multifractal Spectrum of LAN Traffic

29

Fig. 3.3 The time history of variance ln.VI/; ˛, the average flux rate ln(I ) and f

3.5 Concluding Remarks The results show that the multifractal spectrum could describe the complex bursting and breaking behaviors for the local area network traffic, and can be considered a method for studying the complex nonlinear phenomena for the network system. Moreover, it is proved that there exists local singularity in the network traffic after the analysis of the real local area network traffic in detail. In particular, the important relationship between the multifractal spectrum and the variation of the traffic is founded and established. As a conclusion, all of the results presented can be applied to the prediction of traffic, deign, and performance evaluation, and play an important role for the network traffic.

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References 1. Levy VJ, Sikdar B (2001) A multiplicative multifractal model for TCP traffic [J]. In: IEEE symposium on computers and communications-proceedings, pp 714–719 2. Feldmann A, Gilbert AC, Willinger W (1998) Data networks as cascades: investigating the multifractal nature of internet WAN traffic [J]. Comput Commun Rev 28(4):42–55 3. Lacovoni G, Mance V, Verqni D (2000) Single source TCP behaviour: a multifractal analysis, conference record [J]. In: IEEE global telecommunications conference, vol 1, pp 323–328 4. Feldmann A, Gilbert AC, Willinger W, et al (1998) The changing nature of network traffic: scaling phenomena [J]. ACM SIGCOMM Comput Commun Rev 28(2):5–29 5. Jackson JK (1957) Network of waiting lines [J]. Oper Res 5:518–521 6. Beran J, Sherman R, Taqqu MS, et al (1995) Long-range dependence in variable-bit-rate video traffic [J]. IEEE Trans Commun 43(2):1566–1579 7. Rao Y, Xu Z, Liu Z (2004) Length requirement of self-similar network traffic [J]. Chin J Electron 13(1):175–178 8. Riedi R (1995) An improved multifractal formalism and self-similar measures [J]. J Math Anal Appl 189:462–490

Chapter 4

A Boundary Crisis in High Dimensional Chaotic Systems Ling Hong, Yingwu Zhang, and Jun Jiang

Abstract A crisis is investigated in high dimensional chaotic systems by means of generalized cell mapping digraph (GCMD) method. The crisis happens when a hyperchaotic attractor collides with a chaotic saddle in its fractal boundary, and is called a hyperchaotic boundary crisis. In such a case, the hyperchaotic attractor together with its basin of attraction is suddenly destroyed as a control parameter passes through a critical value, leaving behind a hyperchaotic saddle in the place of the original hyperchaotic attractor in phase space after the crisis, namely, the hyperchaotic attractor is converted into an incremental portion of the hyperchaotic saddle after the collision. This hyperchaotic saddle is an invariant and nonattracting hyperchaotic set. In the hyperchaotic boundary crisis, the chaotic saddle in the boundary has a complicated pattern and plays an extremely important role. We also investigate the formation and evolution of the chaotic saddle in the fractal boundary, particularly concentrating on its discontinuous bifurcations (metamorphoses). We demonstrate that the saddle in the boundary undergoes an abrupt enlargement in its size by a collision between two saddles in basin interior and boundary.

4.1 Introduction A hyperchaotic attractor is typically defined as chaotic behavior with at least two positive Lyapunov exponents. Such an attractor is common in high dimensional chaotic systems, namely at least two-dimensional maps or four-dimensional flows [1, 2]. The first example of a hyperchaotic attractor was presented in the folded-towel map and the 4-D R¨ossler system [3]. The experimental realization of hyperchaotic behavior was first observed in an electronic circuit [4]. There have been reports of chaos–hyperchaos transition in a driven system [5] and in a 9D model for a Rayleigh–B´enard convection [6]. The transition from chaos to hyperchaos is a blowout bifurcation [7] which occurs when a chaotic attractor lying L. Hong () MOE Key Lab for Strength and Vibration, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China e-mail: [email protected] A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 4,

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in some symmetric subspace becomes transversely unstable. Blowout bifurcations are related to some interesting complex phenomena such as chaotic synchronizations [8], riddle basins, on-off intermittency, [9] and metamorphoses of chaotic saddle for high dimensional chaotic systems [10]. Crises of chaotic attractors are extremely common and have been observed in an experiment [11]. Sudden changes in chaotic attractors with parameter variation have been called crises [12, 13]. Crises are due to a collision of chaotic attractors with an unstable periodic orbit, in which chaotic attractors undergo sudden discontinuous changes. Three types of crisis can be distinguished according to the nature of the discontinuous change that the crisis induces in the chaotic attractor: in the first type, a chaotic attractor is suddenly destroyed as the parameter passes through its critical value; in the second type, the size of the attractor in phase space suddenly increases; in the third type, two or more chaotic attractors merge to form one chaotic attractor. The crises of hyperchaotic attractors reported here are a high-dimensional chaotic phenomenon, whereas to our knowledge, few works dealing with this problem have been published to date. A generalized cell-mapping digraph (GCMD) method for global analysis of nonlinear systems was first introduced by Hsu [14]. Later, the GCMD method was developed [15] and has been used to deal with crisis bifurcations for deterministic and noisy nonlinear systems [16–18]. The current chapter studies a sudden change in a hyperchaotic attractor with two positive Lyapunov exponents. Such a change is called a hyperchaotic crisis following Grebogi’s definition of crisis in low dimensional chaotic systems [12, 13]. We shall study a hyperchaotic crisis involving the collision of a hyperchaotic attractor with a chaotic saddle in its fractal boundary. The origin and evolution of the chaotic saddle in the boundary are also investigated, particularly concentrating on its discontinuous bifurcations (metamorphoses). We illustrate this hyperchaotic crisis event by Kawakami map. The reminder of the chapter is outlined as follows. In Sect. 4.2, we study a hyperchaotic boundary crisis in Kawakami map. The chapter concludes in Sect. 4.3.

4.2 A Hyperchaotic Boundary Crisis in a Kawakami Map The model presented here is a map given by x.n C 1/ D axn C yn y.n C 1/ D xn2 b:

(4.1)

The equation is well known as Kawakami map. It was first introduced by Kawakami and Kobayashi in 1979 for an endomorphism study [19]. The strange hyperchaotic dynamics of the Kawakami map was thoroughly studied for a parameter region of 0:1 a 0:15, b D 1:6 [20, 21] including a hyperchaotic attractor with two Lyapunov exponents and unstable invariant sets in its basin boundary. In the present chapter, we confirm the previously reported results and further find

4

A Boundary Crisis in High Dimensional Chaotic Systems

33

a hyperchaotic boundary crisis involving the collision of a hyperchaotic attractor with a chaotic saddle in its fractal boundary. And we also investigate the origin and evolution of the chaotic saddle in the fractal boundary, particularly focusing on its discontinuous bifurcations (metamorphoses). To our knowledge, no attempt has been made regarding this problem. The domain D D f2:5 x 2:5; 2:0 y 3:5g is discretized into 300 300 cells when applying the FGCM method, 10 10 sampling points are used within each cell. We fix the parameter b D 1:6 and allow the parameter a to vary from 0:1 to 0:22. When a D 0:1, there exist a hyperchaotic attractor and three unstable solutions including a chaotic saddle in the basin interior as well as a saddle with two narrow disjointed strips and a unstable node embedded in the boundary. The global phase portrait is shown in Fig. 4.1. We reassure the above results by refining the 300 300 cell structure to 800 800 through 500 500. Grey denotes the hyperchaotic attractor. Light grey denotes the chaotic saddle in the basin interior. Black denotes the saddle in the boundary. The circle symbol ı denotes the unstable node in the boundary. The boundary is shown in (b) of figures. The color coding and symbol hold throughout the chapter. As a increases, the chaotic saddle in the basin interior becomes bigger and closer to the boundary. A discontinuous bifurcation (metamorphosis) happens in the interval a 2 .0:1648; 0:1649/. In the case, the chaotic saddle is touching the saddle in the boundary when a D 0:1648, creating a chaotic saddle in a fractal boundary when a D 0:1649. The chaotic saddle in the fractal boundary has a complicated structure and plays an extremely important role in an upcoming hyperchaotic crisis. Namely, it will collide with the hyperchaotic attractor when a increases further to 0:2164, leading to a hyperchaotic boundary crisis. The global phase portrait is shown in Figs. 4.2 and 4.3 when a D 0:1648 and a D 0:1649. b

a 3.5

3.5

3

3

2.5

2.5

2

2

1.5

1.5 1 y

y

1 0.5

0.5

0

0

−0.5

−0.5

−1

−1

−1.5

−1.5

−2 −2.5 −2 −1.5 −1 −0.5 0 x

0.5

1 1.5 2

2.5

−2 −2.5 −2 −1.5 −1 –0.5 0 0.5 1 1.5 x

2 2.5

Fig. 4.1 Global phase portrait of the Kawakami map with a D 0:1, b D 1:6. Grey denotes the hyperchaotic attractor. Light grey the chaotic saddle in the basin interior. Black the saddle in the boundary. The circle symbol ı the unstable node in the boundary. The boundary is shown in (b)

34

L. Hong et al.

a

b 3.5

3.5 3

3

2.5

2.5

2

2

1.5

1.5 1

y

y

1

0.5

0.5 0

0

−0.5

−0.5

−1

−1

−1.5

−1.5

−2 −2.5 −2 −1.5 −1 −0.5 0 x

0.5

1

1.5

2

−2 −2.5 −2 −1.5 −1 −0.5 0 x

2.5

0.5

1

1.5

2

2.5

Fig. 4.2 Just before the discontinuous bifurcation for the Kawakami map with a D 0:1648, b D 1:6 at a collision between two saddles, light grey one in the interior and black one in the boundary. The color coding is the same as that in Fig. 4.1

a 3.5

b 3.5

3

3

2.5

2.5

1

1 y

2 1.5

y

2 1.5

0.5

0.5 0

0

−0.5

−0.5

−1

−1

−1.5

−1.5

−2 −2.5 −2 −1.5 −1 −0.5 0 x

0.5

1

1.5

2

2.5

−2 −2.5 −2 −1.5 −1 −0.5 0 x

0.5

1

1.5

2

2.5

Fig. 4.3 Just after the discontinuous bifurcation for the Kawakami map with a D 0:1649, b D 1:6 creating a chaotic saddle in a fractal boundary. The color coding is the same as that in Fig. 4.1

A hyperchaotic crisis occurs in the interval a 2 .0:2164; 0:2165/ when the hyperchaotic attractor collides with a chaotic saddle in its fractal boundary. In the case, the hyperchaotic attractor together with its basin of attraction is suddenly destroyed as a control parameter passes through a critical value, leaving behind a hyperchaotic saddle in the place of the original hyperchaotic attractor in phase space after the crisis, namely, the hyperchaotic attractor is converted into an incremental portion of the hyperchaotic saddle after the collision. This hyperchaotic saddle is an invariant and nonattracting hyperchaotic set. The global phase portraits are shown in Figs. 4.4 and 4.5.

4

A Boundary Crisis in High Dimensional Chaotic Systems

a

35

b 3.5

3.5 3

3

2.5

2.5

2

2

1.5

1.5 1

y

y

1 0.5

0.5

0

0

−0.5

−0.5

−1

−1

−1.5

−1.5

−2 −2.5 −2 −1.5 −1 −0.5

0 x

0.5

1

1.5

2

−2 −2.5 −2 −1.5 −1 −0.5

2.5

0 x

0.5

1

1.5

2

2.5

Fig. 4.4 Just before the hyperchaotic crisis for the Kawakami map with a D 0:2164, b D 1:6 when the hyperchaotic attractor touches the chaotic saddle in the fractal boundary. The color coding is the same as that in Fig. 4.1 3.5 3 2.5 2 1.5 y

1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −2 −1.5 −1 −0.5

0 x

0.5

1

1.5

2

2.5

Fig. 4.5 Just after the hyperchaotic crisis for the Kawakami map with a D 0:2165, b D 1:6 leaving behind the hyperchaotic saddle

4.3 Concluding Remarks In this chapter, we have investigated hyperchaotic crises, where a hyperchaotic attractor collides with a chaotic saddle on its fractal boundary. In the case, the hyperchaotic attractor together with its basin of attraction is suddenly destroyed as a control parameter passes through a critical value, leaving behind a hyperchaotic saddle in the place of the original hyperchaotic attractor in phase space after the crisis.

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A collision with a chaotic saddle in a fractal boundary is the typical mechanism by which hyerchaotic attractors can be suddenly destroyed. In the hyperchaotic crises, the chaotic saddle in the boundary has a complicated pattern and plays an extremely important role. We also investigate the formation and evolution of the chaotic saddle in the fractal boundary, particularly concentrating on its discontinuous bifurcations (metamorphoses). We demonstrate that the saddle in the boundary undergoes an abrupt enlargement in its size by a collision between two saddles in basin interior and boundary. Acknowledgments This work is supported by the National Science Foundation of China under Grant Nos. 10772140 and 10872155 as well as the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

References 1. Baier G, Klein M (1990) Maximum hyperchaos in generalized Henon map. Phys Lett A 151:281–284 2. Baier G, Sahle S (1995) Design of hyperchaotic flows. Phys Rev E 51:R2712–R2714 3. Rossler OE (1979) An equation for hyperchaos. Phys Lett A 71:155–157 4. Matsumoto T, Chua LO, Kobayashi K (1986) Hyperchaos: laboratory experiment and numerical confirmation. IEEE Trans Circuits Syst CAS-33(11):1143–1147 5. Kapitaniak T, Thylwe KE, Cohen I, Wjewoda J (1995) Chaos–hyperchaos transition. Chaos Solitons Fractals 5(10):2003–2011 6. Reiterer P, Lainscsek C, Schurrer F (1998) A nine-dimensional lorenz system to study highdimensional chaos. J Phys A 31:7121–7139 7. Kapitaniak T, Maistrenko Y, Popovych S (2000) Chaos–hyperchaos transition. Phys Rev E 62(2):1972–1976 8. Kapitaniak T (2005) Chaos synchronization and hyperchaos. J Phys: Conf Ser 23:317–324 9. Ott E, Sommerer JC (1994) Blowout bifurcations: the occurrence of riddled basins and on–off intermittency. Phys Lett A 188:39–47 10. Kapitaniak T, Lai YC, Grebogi C (1999) Metamorphosis of chaotic saddle. Phys Lett A 259(6):445–450 11. Ditto WL, Rauseo S, Cawley R, Grebogi C (1989) Experimental observation of crisis-induced intermittency and its critical exponent. Phys Rev Lett 63:923–926 12. Grebogi C, Ott E, Yorke JA (1982) Chaotic attractors in crisis. Phys Rev Lett 48:1507–1510 13. Ott E (2002) Chaos in dynamical systems. Cambridge University Press, Cambridge 14. Hsu CS (1995) Global analysis of dynamical systems using posets and digraphs. Int J Bifurcat Chaos 5(4):1085–1118 15. Hong L, Xu JX (1999) Crises and chaotic transients studied by the generalized cell mapping digraph method. Phys Lett A 262:361–375 16. Hong L, Xu JX (2001) Discontinuous bifurcations of chaotic attractors in forced oscillators by generalized cell mapping digraph (GCMD) method. Int J Bifurcat Chaos 11:723–736 17. Hong L, Sun JQ (2006) Codimension two bifurcations of nonlinear systems driven by fuzzy noise. Physica D: Nonlinear Phenom 213(2):181–189 18. Xu W, He Q, Fang T, Rong H (2004) Stochastic bifurcation in Duffing system subject to harmonic excitation and in presence of random noise. Int J Non-Linear Mech 39:1473–1479 19. Kawakami H, Kobayashi O (1976) Computer experiment on chaotic solution. Bull Fac Eng Tokushima Univ 16:29–46 20. He DH, Xu JX, Chen YH (1999) A study on strange dynamics of a two dimensional map. Acta Physica Sin 48(9):1611–1617 21. He DH, Xu JX, Chen YH (2000) Study on strange hyper chaotic dynamics of kawakami map. Acta Mechanica Sin 32(6):750–754

Chapter 5

Complete Bifurcation Behaviors of a Henon Map Albert C.J. Luo and Yu Guo

Abstract In this paper, a methodology to analytically predict the stable and unstable periodic solutions for n-dimensional discrete dynamical systems is applied to investigate the Henon map. The positive and negative iterative mappings of discrete maps are introduced for the mapping structure of the periodic solutions. The complete bifurcation and stability of the stable and unstable periodic solutions with respect to the positive and negative mapping structures is given. The Poincare mapping sections of the Neimark bifurcation of periodic solutions is presented, and the chaotic layers for the discrete system with the Henon map are observed.

5.1 Introduction The Henon map in the discrete-time dynamic system was first introduced by Henon [1] in 1976 to simplify the three-dimensional Lorenz equations as a Poincare map; also, Henon observed chaos numerically in such a discrete system. These numerical results stimulated more attention on Henon map latter on. In 1979, Marotto [2] mathematically proved the existence of chaotic behavior of Henon map for certain parameters. Also, Curry [3] used Lyapunov characteristic exponent and frequency spectrum to measure the chaotic behavior of Henon map. In 1988, Cvitanovic et al. [4] investigated the topologic properties and multifractality of Henon map. In 1993, Gallas [5] numerically investigated the parameter maps for Henon map. In 2000, Zhusubaliyev et al. [6] did the bifurcation analysis of Henon map and further presented a more detailed parameter map. The aforementioned investigations were based on the numerical computation. In 2005, Gonchenko et al. [7] discussed the three-dimensional Henon map generated from a homoclinic bifurcation. In 2006, Hruska [8] developed a numerical algorithm to model the dynamics of a polynomial diffeomorphism of C 2 on its chain recurrent set and applied this algorithm to Henon map. In 2007, Gonchenko et al. [9] studied the bifurcation of periodic solution of A.C.J. Luo () Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805, USA e-mail: [email protected] A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 5,

37

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A.C.J. Luo and Y. Guo

the generalized Henon map, and further proved the existence of infinite cascades of periodic solutions in a generalized Henon map. In 2008, Lorenz [10] adopted a random searching procedure to determine the parameter maps of periodic windows embedded in chaotic solutions of Henon map. In 1992, Luo and Han [11] presented a geometric approach to the period doubling bifurcation and multifractality of a general one-dimensional iterative map. In 2005, Luo [12] investigated the mapping dynamics of periodic motions in a nonsmooth piecewise system. In this chapter, the method of mapping dynamics will be adopted to investigate the bifurcation behavior of the Henon map. The stable and unstable periodic solutions of the Henon map will then be investigated, and the eigenvalue analysis for each periodic solution of the Henon map will be carried out; the parameter maps of the Henon map can be obtained through analytical predictions which are beyond the existing results.

5.2 Analysis Consider the Henon map f1 .xk ; xkC1 ; p/ D xkC1 yk 1 C axk2 D 0 f2 .xk ; xkC1 ; p/ D ykC1 bxk D 0

(5.1)

where xk D .xk ; yk /T ; f D .f1 ; f2 /T and p D .a; b/T . Consider two positive and negative mapping structures as xkCN D PC.N / xk D PC ı PC ı PC xk „ ƒ‚ … N -terms xk D P.N / xkCN D P ı P ı P xkCN „ ƒ‚ … N -terms

(5.2)

Equations (5.1) and (5.2) give f.xk ; xkC1 ; p/ D 0 f.xkC1 ; xkC2 ; p/ D 0 :: : f.xkCN 1 ; xkCN ; p/ D 0 and

9 > > > = > > > ;

9 f.xkCN 1 ; xkCN ; p/ D 0 > > > f.xkCN 2 ; xkCN 1 ; p/ D 0 = :: > > : > ; f.xk ; xkC1 ; p/ D 0:

(5.3)

(5.4)

5

Complete Bifurcation Behaviors of a Henon Map

39

The switching of equation order in (5.2) shows (5.3) and (5.4) are identical. For periodic solutions for the positive and negative maps, the periodicity of the positive and negative mapping structures of the Henon map requires xkCN D xk

or

xk D xkCN

(5.5)

So the periodic solutions xkCj .j D 0; 1; : : : ; N / for the negative and positive mapping structures are the same, which are given by solving (5.3) and (5.5). However the stability and bifurcation are different because xkCj varies with xkCj 1 for the j th positive mapping and xkCj 1 varies with xkCj for the j th negative mapping. For a small perturbation, (5.1) for the positive mapping gives ˇ @xkCj ˇˇ @f @f C (5.6) D0 @xkCj 1 @xkCj @xkCj 1 ˇ x ; x kCj 1

where

2 @f

@xkCj 1

@f @xkCj

; x x kCj 1 kCj

;x x kCj 1 kCj

6@xkCj 1 @ykCj 17 7 6 7 6 4 @f2 @f2 5 @xkCj 1 @ykCj 1 x ; x kCj 1 kCj # " 1 2axkCj 1 (5.7) D b 0 3 2 @f1 @f1 6 @xkCj @ykCj 7 10 7 6 D6 D (5.8) 7 01 4 @f2 @f2 5

D

3

@f1

@f1

kCj

@xkCj @ykCj

; x x kCj 1 kCj

So one obtains

@x kCj DPC xkCj 1 D @xkCj 1 x " D

kCj 1

2axkCj 1 1 b 0

@f D @xkCj

1

@f @xkCj 1

#

x kCj 1

(5.9)

Similarly, for the negative mapping, ˇ @xkCj 1 ˇˇ @f @f C @xkCj @xkCj 1 @xkCj ˇ x

kCj 1 ; xkCj

D0

(5.10)

40

A.C.J. Luo and Y. Guo

With (5.7) and (5.8), the foregoing equation gives 1 @xkCj 1 @f @f D @xkCj x @xkCj 1 @xkCj x kCj kCj # " 1 0 1 D b b 2axkCj 1

DP .xkCj / D

(5.11)

Thus, the resultant perturbation of the mapping structure in (5.2) gives .N /

ıxkCN D DPC xk D DPC DPC DPC ıxk „ ƒ‚ … N -terms / ıxk D DP.N DP DP DP ıxkCN ıxkCN D „ ƒ‚ … N -terms where

9 = DPC xkCN j

Q ; D N j D1 DP xkCN j C1

/ DP.N C D / DP.N

(5.12)

QN

j D1

(5.13)

.N / / Consider the eigenvalues and C of DP.N .xkCN / and DPC .xk /, respectively. The following statements hold. ˇ ˇ ˇ ˇ .N / .N / ˇ ˇ ˇ 1. If ˇC 1;2 < 1 (or 1;2 < 1), the periodic solutions of PC .xk / (or P .xkCN /) are stable. ˇ ˇ ˇ ˇ .N / .N / ˇ ˇ ˇ 2. If ˇC 1 or 2 > 1 (or 1 or 2 < 1), the periodic solutions of PC .xk / (or P .xkCN /) are unstable. ˇ Cˇ ˇ ˇ ˇ ˇ < 1 (or D 1 and ˇ ˇ < 1), 3. If real eigenvalues C 1 D 1 and 2 1 2 the period-doubling (PD) bifurcation of the periodic solutions of PC.N / .xk / (or P.N / .xkCN /) occurs. ˇ ˇ ˇ ˇ C ˇ ˇ ˇ 4. If real eigenvalues ˇC 1 < 1 and 2 D 1 (or 1 < 1 and 2 D 1), then the saddle-node (SN) bifurcation of the periodic solutions relative to PC.N / .xk ) (or P.N / .xkCN /) occurs. ˇ ˇ ˇ ˇ ˇ ˇ ˇ 5. If two complex eigenvalues of ˇC 1;2 D 1 .or 1;2 D 1/, the Neimark bifurca-

tion (NB) of the periodic solutions of PC.N / .xk / (or P.N / .xkCN /) occurs.

5.3 Illustrations A numerical prediction of the periodic solutions of the Henon map is presented with varying parameter b for a D 0:85, as shown in Fig. 5.1. The dashed vertical lines give the bifurcation points. The acronyms “PD,” “SN,” and “NB” represent the

5

Complete Bifurcation Behaviors of a Henon Map

a

NB

1.5

NB SN PD

1.0 Iterative Points, xk

41

PD .5

SN P+(2) P+(4)

P+

0.0 −.5

PD

P+(2)

SN

−1.0 −1.5 −2

−1

0

1

2

Parameter, b

b

NB

1.5

NB

Iterative Points, xk

1.0 .5 0.0

P−(2)

P−

−.5

P−

SN

PD

−1.0 −1.5 −2

−1

0

1

2

Parameter b

c

NB

NB

1.5 SN Iterative Points, xk

1.0

PD PD

.5 0.0

P−

SN P+(2) P+(4)

P+

−.5

PD

P+(2) P−(2)

P−

SN

SN

PD

−1.0 −1.5 −2

−1

0

1

2

Parameter b

Fig. 5.1 Numerical predictions of periodic solutions of the Henon mapping: (a) positive mapping .PC /, (b) negative mapping .P / and (c) combination of the negative and positive mappings .a D 0:85/

42

A.C.J. Luo and Y. Guo

period-doubling bifurcation, saddle-node bifurcation and Neimark bifurcation, respectively. It is observed that the stable periodic solutions for positive mapping PC lie in b 2 .1:0; 1:0/. The stable period-1 solution of PC is in b 2 .1; 0:074/. At b D 1, the Neimark bifurcation (NB) of the period-1 solution occurs. At b 0:074, the period-doubling bifurcation (PD) of the period-1 solution occurs. This point is the saddle-node bifurcation (SN) for the period-2 solution of PC .2/ .2/ (i.e., PC ). The periodic solution of PC is in the range of b 2 .0:074; 0:3935/ and b 2 .0:82; 1:0/. Also, there is a periodic solution of PC.4/ existing in the range .2/ of b 2 .0:3935; 0:82/. At b D 1, the Neimark bifurcation (NB) of PC occurs. After the Neimark bifurcation, the stable periodic solutions for positive mapping PC do not exist any more. Such stable periodic solutions for positive mapping PC is shown in Fig. 5.1a. The stable solution for negative mapping P is in the ranges of b 2 .1; 1:0/ and b 2 .1:0; C1/. At b D 1, the Neimark bifurcation (NB) of the period-1 solution of P occurs. The period-1 solution of P is in b 2 .1; 1:0/ and b 2 .2:0735; C1/. The period-doubling bifurcation (PD) of the period-1 solution of P occurs at b 2:0735, and the bifurcation point is the saddle-node bifurcation (SN) for the period-2 solution of P (i.e., P.2/ ). The stable periodic solution of P.2/ is in b 2 .1:0; 2:0735/. At b D 1, the Neimark bifurcation (NB) of the periodic solution of P.2/ occurs. Such stable periodic solutions for positive mapping P are shown in Fig. 5.1b. The total bifurcation scenario for positive and negative mappings is plotted in Fig. 5.1c. The parameter ranges are in b 2 .1; C1/. From the numerical prediction, the stable periodic solutions of the Henon map are obtained. Herein, through the corresponding mapping structures, the stable and unstable periodic solutions for positive and negative mappings of the Henon maps are represented in Figs. 5.2 and 5.3. The acronyms “PD,” “SN,” and “NB” represent the period-doubling bifurcation, saddle-stable node bifurcation, and Neimark bifurcation, respectively. The acronyms “UPD,” “USN” represent the period-doubling bifurcation relative to unstable nodes and saddle-unstable node bifurcation, respectively. The analytical prediction of stable and unstable periodic solutions of positive mapping PC for a D 0:85 and b 2 .1; C1/ is presented in Fig. 5.2a– d. The periodic solution of the positive mapping is arranged in Fig. 5.2a. The real and imaginary parts and magnitude of eigenvalues for such periodic solutions are given in Fig. 5.2b–d, respectively. The stable periodic solutions for positive mapping PC lie in b 2 .1:0; 0:0745/, which is closer to numerical prediction. In other words, the stable period-1 solution of PC is in b 2 .1; 0:0745/. For b 2 .0:0745; 0:39555/, the unstable period-1 solution of PC is saddle. For b 2 .1; 1:0/, the unstable period-1 solution of PC is relative to the unstable focus. The corresponding bifurcations are Neimark bifurcation (NB) and perioddoubling bifurcation (PD). However, another period-1 solution of PC exists and which is unstable. For b 2 .2:07244; C1/, the periodic solution is of the unstable node. However, for b 2 .1; 2:07244/, the periodic solution is relative to saddle. Thus, the unstable period-doubling bifurcation (UPD) of the period-1 solution of PC occurs at b 2:07244. At this point, the unstable periodic solution is from an

Complete Bifurcation Behaviors of a Henon Map

a

b

2

NM

NM

Iteration Points, xk

PD

2

SN

1 PD SN

0 SN PD

−1

P+(2)

P+ −2

UPD

USN

−1

P+(4)

0

P+(2) 1

Real Part of Eigenvalue, Reλ1,2

5

NM

NM

NM

P+

d 2.5

1 PD PD

UPD

SN

USN

0 SN

SN

−1

−2

P+ −1

P+(2)

−1

PD

UPD

P+(2)

P+(4)

0

P+(2) 1

P+(4)

P+(2) 1

Parameter, b

2

NM

2.0

2

NM

P+(2)

P+

P+(4)

P+(2)

1.5 1.0

PD

PD

PD

SN

SN

SN

UPD USN

0.5 0.0

0

PD

Parameter, b

NM

PD

USN

PD

−1

Magnitude of Eigenvalue, |λ1,2|

Imaginary Part of Eigenvalue, Imλ1,2

2

SN

0

Parameter, b

c

SN

SN

1

−2

2

43

−1

0

1

2

Parameter, b

Fig. 5.2 Analytical predictions of stable and unstable periodic solutions for positive mapping .PC / of the Henon map: (a) periodic solutions, (b) real part of eigenvalues, (c) imaginary part of eigenvalues and (d) magnitude of eigenvalues (a D 0:85 and b 2 .1; C1/)

unstable node to saddle. Because of the unstable period-doubling bifurcation, the .2/ unstable periodic solution of PC for the unstable node is obtained for b 2 .1:0; 2:07244/. This unstable periodic solution is from unstable focus to unstable node during the parameter of b 2 .1:0; 2:07244/. At b 2:07244, the bifurcation of the unstable periodic solution of PC.2/ occurs between the saddle and unstable node. This bifurcation is called the unstable saddle-node bifurcation. The unstable periodic solution of PC.2/ relative to saddle exists for b 2 .0:3955; 0:8190/, while the stable period-4 solution of PC.4/ occur on the same interval. At b 0:3955, .2/ there is a period doubling bifurcation, where the PC periodic solution becomes .4/ unstable and PC periodic solution starts. At b 0:8190, there is a saddle node bi.4/ .2/ furcation where the PC periodic solution goes into the PC solution. At b D 1:0, the Neimark bifurcation (NB) between the periodic solutions of PC.2/ relative to the .2/ unstable and stable focuses occurs. The stable periodic solution of PC is existing for b 2 .0:0745; 0:3955/ and b 2 .0:8190; 1:0/.

44

A.C.J. Luo and Y. Guo

a

b NM

Iteration Points, xk

UPD

1 UPD

USN

0 USN

SN PD

UPD

−1

(2) −

P− −2

P −1

0

NM

2

USN

1

P−

Real Part of Eigenvalue, Reλ1,2

NM

2

P−(2)

1

USN

USN

USN

0 UPD UPD

−1

UPD

−1

d

NM

USN UPD USN

SN

0

PD

UPD

−5

UPD USN

−10

P−(2)

P− −15

−1

Parameter, b

2

1

NM

P−(2)

P−

P−

2.0

1.5 USN

1.0

UPD

SN

USN USN UPD UPD

PD

0.5

P− 0.0

0

NM

2.5

10 5

1 Parameter, b

Magnitude of Eigenvalue, |λ1,2|

Imaginary Part of Eigenvalue, Imλ1,2

NM

15

PD

0

Parameter, b

c

P− SN

−2

2

NM

P−

2

−1

0

1

2

Parameter, b

Fig. 5.3 Analytical predictions of stable and unstable periodic solutions for negative mapping .P / of the Henon map: (a) periodic solutions, (b) real part of eigenvalues, (c) imaginary part of eigenvalues and (d) magnitude of eigenvalues (a D 0:85 and b 2 .1; C1/)

Similarly, the analytical prediction of stable and unstable periodic solutions of negative mapping P for a D 0:85 and b 2 .1; C1/ is presented in Fig. 5.3a–d. The periodic solution of the negative mapping is plotted in Fig. 5.3a. The real part, and imaginary part and magnitude of the eigenvalues for such periodic solutions are presented in Fig. 5.3b–d, respectively. The stable periodic solutions for positive mapping P lie in b 2 .1; 1:0/ and b 2 .1:0; C1/, which is the same as in numerical prediction. The stable period-1 solution of P is stable focuses in b 2 .1; 1:0/ and stable nodes in b 2 .2:07244; C1/. For b 2 .1:0; 0:0745/, the unstable period-1 solution of P is from the unstable focus to unstable node. At b D 1, the bifurcation between the stable and unstable period-1 solution of P is the Neimark bifurcation (NB). For b 2 .0:0745; C1/, the unstable period-1 solution of P is of the saddle. Thus, the bifurcation between the period-1 solution of P between the unstable node and saddle occurs at b D 0:0745, which is called the unstable period-doubling bifurcation (UPD). For b 2 .0:0745; 0:3955/ and b 2 .0:8190; 1:0/, the unstable period-2 solution of P (i.e., P.2/ ) exists. For b 2 .1:0; 2:07244/, the stable period-2 solution of P (i.e., P.2/ ) is from the stable

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focus to the stable nodes. Thus, the point at b 0:3955 is the bifurcation of the unstable periodic solution of P.2/ , which is the unstable saddle-node bifurcation between the unstable node and saddle (i.e., USN). For the point at b D 1, the Neimark bifurcation between the periodic solutions of P.2/ relative to the unstable and stable focuses occurs. The point at b 2:07244 is the bifurcation of the stable periodic solution of P.2/ , which is the saddle-node bifurcation between the stable node and saddle (SN). For b 2 .1; 2:07244/, the unstable period-1 solution of P is saddle. At b 2:07244, the perioddoubling bifurcation (PD) of the period-1 solution of P takes place. Also for b 2 .0:3955; 0:8190/, there exists the unstable period-4 solution of P.4/ , which is again saddle. From the analytical prediction, the observations can be stated as follows. 1. The stable periodic solution of positive mapping PC is the unstable periodic solution of negative mapping P . 2. The stable periodic solution of negative mapping P is the unstable periodic solution of positive mapping PC . 3. The PD and SN bifurcations of the periodic solutions of positive mapping PC are the UPD and USN bifurcations of the periodic solutions of negative mapping P , vice versa. 4. The PD and SN bifurcations of the periodic solutions of negative mapping P are the UPD and USN bifurcations of the periodic solutions of positive mapping PC , vice versa. 5. If the unstable periodic solutions of positive mapping PC are saddle, the corresponding periodic solutions of negative mapping P are also saddle. In addition, the Neimark bifurcation between the periodic solution relative to the unstable and stable focuses is of great interest. The Poincare mapping relative to the Neimark bifurcation of the period-1 solution of positive mapping (or negative mapping) at a D 0:85 and b D 1 is presented in Fig. 5.4a. The most inside point xk ; yk .0:4237; 0:4237/ is the point for the period-1 solution of PC or P relative to the Neimark bifurcation. For the specified parameters, the initial values of .xk ; yk / used for simulation are given in Table 5.1. The most outside curve with the initial condition xk ; yk .1:0597; 0:4237/ is the biggest boundary for the strange attractors around the period-1 solutions with the Neimark bifurcation. The skew symmetry of the strange attractors in the Poincare mapping section is observed. The Poincare mapping relative to the Neimark bifurcation of the period-2 solution of positive mapping (or negative mapping) at a D 0:85 and b D 1 is presented in Fig. 5.4b. The two points xk ; yk .1:0846; 1:0846/ and .1:0846; 1:0846/ are the points for the period-2 solution of PC or P relative to the Neimark bifurcation. For the specified parameters, the input data for initial values are listed in Table 5.2. with the outer chaotic layer, the strange attractor near the periodic solutions of PC.2/ -1 (or P.2/ -1) disappears.

46

a 0.5 Iterative Coordinates yk

Fig. 5.4 Poincare mappings at the Neimark bifurcation of the Henon map: (a) period-1 (i.e., PC -1 or P -1) (a D 0:85 and b D 1) and (b) period-2 solution .2/ (i.e., PC -1 or .2/ P -1)(a D 0:85 and b D 1)

A.C.J. Luo and Y. Guo

0.0 −0.5 −1.0 −1.5

−0.5

0.0

0.5

1.0

1.5

Iterative Coordinates xk

b Iterative Coordinates yk

1.2

0.9 −0.9

−1.2 −1.2

−0.9

0.9

1.2

Iterative Coordinates xk

Table 5.1 Input data for Poincare mappings

Table 5.2 Input data for Poincare mappings

.xk ; yk /

.xk ; yk /

.0:4237; 0:4237/ .0:4737; 0:4237/ .0:5537; 0:4237/ .0:6237; 0:4237/

.0:7037; 0:4237/ .0:8037; 0:4237/ .0:9037; 0:4237/ .1:0597; 0:4237/

.xk ; yk /

.xk ; yk /

.1:1728; 1:0846/ .1:1541; 1:0846/ .1:1328; 1:0846/

.1:1128; 1:0846/ .1:0939; 1:0846/ .1:0846; 1:0846/

5.4 Conclusion In this chapter, a discrete dynamical system of the Henon map was investigated. The positive and negative iterative mappings of discrete maps were employed for the mapping structure of the periodic solutions. The complete bifurcation and stability

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of the stable and unstable periodic solutions with respect to the positive and negative mapping structures were analyzed. A complete picture of the periodic solutions of positive and negative mappings is given; the positive and negative mappings are in a pair. The Poincare mapping sections of the Neimark bifurcation of periodic solutions were illustrated, and the chaotic layers for the discrete system with the Henon map were observed.

References 1. Henon M (1976) A two-dimensional mapping with a strange attractor. Commun Math Phys 50:69–77 2. Marotto FR (1979) Chaotic behavior in the Henon mapping. Commun Math Phys 68:187–194 3. Curry JH (1979) On the Henon transformation. Commun Math Phys 68:129–140 4. Cvitanovic P, Gunaratne GH, Procaccia I (1988) Topological and metric properties of Henontype strange attractors. Phys Rev A 38(3):1503–1520 5. Gallas JAC (1993) Structure of the parameter space of the Henon map. Phys Rev Lett 70(18):2714–2717 6. Zhusubaliyev ZT, Rudakov VN, Soukhoterin EA, Mosekilde E (2000) Bifurcation analysis of the Henon map. Discrete Dyn Nat Soc 5:203–221 7. Gonchenko SV, Meiss JD, Ovsyannikov II (2006) Chaotic dynamics of three-dimensional Henon maps that originate from a homoclinic bifurcation. Regul Chaotic Dyn 11(2):191–212 8. Hruska SL (2006) Rigorous numerical models for the dynamics of complex Henon mappings on their chain recurrent sets. Discrete Continuous Dyn Syst 15(2):529–558 9. Gonchenko SV, Gonchenko VS, Tatjer JC (2007) Bifurcations of three-dimensional diffeomorphisms with non-simple quadratic homoclinic tangencies and generalized Henon maps. Regul Chaotic Dyn 12(3):233–266 10. Lorenz EN (2008) Compound windows of the Henon-map. Physica D 237:1689–1704 11. Luo ACJ, Han RPS (1992) Period doubling and multifractals in 1-D iterative maps. Chaos Solitons Fractals 2(3):335–348 12. Luo ACJ (2005) The mapping dynamics of periodic motions for a three-piecewise linear system under a periodic excitation. J Sound Vib 283:723–748

Chapter 6

Study on the Performance of a Two-Degree-of-Freedom Chaotic Vibration Isolation System Jing-Jun Lou, Ying-Chun Wang, and Shi-Jian Zhu

Abstract The nonlinear dynamics and the vibration isolation effectiveness of a two-degree-of-freedom nonlinear vibration isolation system are numerically studied. The complex nonlinear behavior in the force-frequency plane of the system is analyzed. Cascades of bifurcation of the system with different excitation amplitude are also obtained. The power flow transmissibility is analyzed to validate the performance of the system in vibration isolation. The numerical results show that the reduction of the line spectra when the system is chaotic is much greater than that when the system is nonchaotic, and that the overall effectiveness of vibration isolation at chaos is better than that at nonchaos.

6.1 Introduction More than three decades of intense studies of nonlinear dynamics have shown that chaos occurs widely in engineering and natural systems. Historically, it has usually been regarded as a nuisance and designed out if possible. It has been noted only as irregular or unpredictable behavior and often attributed to random external influences. Further studies showed that chaotic phenomena are completely deterministic and characteristic for typical nonlinear systems. These studies posed questions about the practical application of chaos. One of the possible answers is to control chaotic behavior in such a way as to make it predictable. Recently, there have been examples of the potential usefulness of chaotic behavior, and this has caused engineers and applied scientists to become more interested in chaos [1]. It was proposed by the authors the method using the chaotic regime of the nonlinear vibration isolation system for reduction of line spectrum of radiated noise

J.-J. Lou () College of Naval Architecture and Power, Naval University of Engineering, Wuhan 430033, People’s Republic of China e-mail: jingjun [email protected]

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 6,

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from a marine vessel in [2]. Experimental chaos in nonlinear vibration isolation system was observed, and the possible practical application of the chaos method in line spectra reduction was confirmed in [3]. In this work, the method of chaotic vibration isolation for reduction of line spectrum is numerically studied. As is well known, noise spectra of the radiated waterborne noise of surface and under-surface ships are generally in two categories. One is the broad-band noise having a continuous spectrum. The other is the line spectrum which contains lines at discrete frequencies. The nature of the radiated noise spectra changes as the navigation speed changes. At high speed, the signature is dominated by broad-band noise, while at low speed the signature is dominated by line spectrum, and the machinery is the leading noisemaker. Insertion of resilient isolators between the machinery and the base is one of the most common methods for controlling unwanted vibration. Isolators in service are usually assumed to be linear and the performance characteristics of isolators under the assumption of linearity have been widely reported [4, 5]. The linear vibration isolation system has vibration attenuation within a rather wide frequency range. But its ability in line spectrum reduction is limited. Thus it becomes important to include nonlinearity presented in practical isolators. Because of the limitations of the linear vibration isolators, nonlinear isolators were studied in some literatures. However, the investigation was constrained to the periodic vibration [6,7]. In spite of the achievement in the application of the chaotic vibration mechanics to chaotic vibratory rollers [8], it is neglected that vibration excitation and vibration isolation are two sides in the field of vibration engineering. And no efforts are made to make use of chaos in vibration isolation. The authors tried to utilize the characteristics of chaos to vibration isolation, and a method of chaotic vibration isolation was advanced for machinery vibration control and line spectra reduction [2]. When chaos takes place in a nonlinear vibration isolation system under a single frequency excitation, the line spectrum of response at the excitation frequency grows into a broad-band one. Therefore, the frequency configuration of the radiated noise is altered. What is more important, the concentrated energy spreads from the excitation frequency to a broad-band frequency range. To validate the effectiveness of the method of chaotic vibration isolation, the dynamics and the power flow transmissibility of the two-degree-of-freedom nonlinear vibration isolation system is studied in the present work.

6.2 Model of the Two-Degree-of-Freedom Nonlinear Vibration Isolation System A majority of stationary equipments are installed by means of vibration isolators attached to rigid supporting structures. There have been numerous publications that used a single-degree-of-freedom (SDOF) system to model both active and passive vibration isolation systems [9, 10]. However, some stationary equipments requiring vibration isolation are installed on nonrigid structures such as higher floors of

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Fig. 6.1 Two-degree-offreedom vibration isolation model

F0 cosWT

M1

C1

K1,K3

X1

M2 X2 K2

C2

multistory buildings. This problem also becomes critically important for objects installed in vehicles, such as car engines, machinery on surface and under-surface ships, etc. In these cases, the mass of the isolated object is substantially greater than the “effective mass” of the supporting structure [11]. As we all know, the equipment on a flexible base has six degrees of freedom. If the translational vibration in the vertical direction is considered only the isolation system with flexible base can be simplified as a two-degree-of-freedom system as shown in Fig. 6.1. In this work, the base is simplified as a single-degree-of-freedom lumped system, namely, a dynamical model of a combination of one equivalent mass, one equivalent stiffness, and one equivalent damper. In Fig. 6.1, M1 is the mass of the isolated equipment, M2 is the mass of the base, X1 is the displacement of the isolated equipment, and X2 is the displacement of the base. The elastic support of the base mass is assumed to be linear, and the stiffness and damping coefficient are K2 and C2 respectively. The isolator is of linear damping C1 and cubic hardening behavior with the first order term K1 and the third order term K3 . The excitation force F0 cos ˝T is a cosine function of the time T , and ˝ is a single, constant, and known input frequency. From Newtonian mechanics, the differential equations of motion of the system in Fig. 6.1 are ( M1 XR1 CC1 .XP1 XP2 / C K1 .X1 X2 /CK3 .X1 X2 /3 D F0 cos T CM1 g M2 XR 2 C1 .XP 1 XP2 / K1 .X1 X2 /K3 .X1 X2 /3 C C2 XP2 CK2 X2 D 0 (6.1) p For K1 ; Kp3 ¤ 0, by introducing p the dimensionless quantities x1 DX1 K3 =K1 , x2 D X2 K3 =K1 ; !1n D K1 =M1 , and t D T !1n , we put the differential equations of motion in a dimensionless form: p dX1 dx1 D !1n K1 =K3 ; dT dt

p dX2 dx2 D !1n K1 =K3 dT dt

(6.2)

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p d2 X1 d2 x1 2 D ! K =K ; 1 3 1n dT 2 dt2

p d2 x2 d2 X2 2 D ! K =K 1 3 1n dT 2 dt 2

(6.3)

Then, (6.1) grows into a dimensionless form 8 < xR 1 C 1 .xP 1 xP 2 / C .x1 x2 / C .x1 x2 /3 D f cos !t C G : xR 2 u1 .xP 1 xP 2 / u.x1 x2 / u.x1 x2 /3 C 2 xP 2 C u K2 x2 D 0 K1

(6.4)

where s

.p 1 D C1 M1 K1 ; uD

M1 ; M2

2 D u

f D

K3 F0 ; K13

!D ; !1n

s GD

C2 1 C1

K3 M1 g; K13 (6.5)

6.3 Dynamical Analysis In order to discuss the influence of the excitation amplitude f and excitation frequency ! on the dynamics of system (6.4), let 1 D 0:02;

2 D 0:2;

u D 2;

K2 =K1 D 100

As mentioned previously, for cases on high floors or on surface and under-surface ships, the mass of the isolated object is substantially greater that the “effective mass” of the supporting structure, and hence u D 2 is selected. In order to analyze the influence of the excitation frequency ! on the dynamics of system (6.4), the global bifurcation with regard to the change of the excitation frequency with the excitation amplitude fixed is analyzed, using the Poincar´e section method. The degree of periodicity of the response has been declared mainly on the basis of the Poincar´e map, but quantitative dynamic measures – such as the frequency response spectrum, the Lyapunov exponents, and the fractal dimensions – have been calculated in many specific situations, too. The global bifurcation diagram with f D 4:0 is shown in Fig. 6.2. Decreasing the excitation frequency from 5 to 0.8 with the step-size equal to 0.01, two cascades of period-doubling bifurcation occur which originates from a period 1 (P-1) motion. However, the excitation amplitude is so small that chaos does not appear. The global bifurcation diagram with f D 8:8 is shown in Fig. 6.3. Besides two cascades of period-doubling bifurcation, two sequences of chaotic motion occur. The global bifurcation process is as follows: (1) The system remains P-1 motion when ! 4:46. The phase plot of the P-1 motion when ! D 4:80 and f D 8:8 are shown in Fig. 6.4.

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Fig. 6.2 Global bifurcation diagram with regard to excitation frequency .f D 4:0/

Fig. 6.3 Global bifurcation diagram with regard to excitation frequency .f D 8:8/

(2) When ! D 4:45, the first period-doubling bifurcation occurs and a period 2 (P-2) motion is obtained. The phase plot of the P-2 motion when ! D 4:10 and f D 8:8 are shown in Fig. 6.5. (3) When ! is decreased to 3.96, a response of period 4 (P-4) is observed after twice period-doubling bifurcation. The phase plot of the P-4 motion when ! D 3:74 and f D 8:8 are shown in Fig. 6.6. (4) Further decrease of the excitation frequency leads to deeper bifurcation cascades, and responses of period 8 (P-8) and chaos at ! D 3:71 and ! D 3:70, respectively. The phase plot of the P-8 motion when ! D 3:71 and f D 8:8 are shown in Fig. 6.7. The phase plot and Poincar´e map of the chaotic motion when ! D 3:70 and f D 8:8 are shown in Fig. 6.8.

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Fig. 6.4 Phase plot of P-1 motion .f D 8:8; ! D 4:80/

Fig. 6.5 Phase plot of P-2 motion .f D 8:8; ! D 4:10/

(5) When 2:12 ! 3:60, the P-1 motion presents itself again. Then, another sequence of period-doubling bifurcation occurs, and the system remains P-2 motion when 1:96 ! 2:11. (6) However, this period-doubling bifurcation does not accelerate as the excitation frequency decreases, and instead it is broken by the P-1 motion when 1:18 ! 1:95.

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Fig. 6.6 Phase plot of P-4 motion .f D 8:8; ! D 3:74/

Fig. 6.7 Phase plot of P-8 motion .f D 8:8; ! D 3:71/

(7) When ! D 1:17, the third sequence of period-doubling bifurcation comes forth and the P-2 motion is obtained. (8) The P-4 and P-8 motions are observed when ! D 1:10 and ! D 1:09 respectively, and then chaos occurs. (9) Finally, the system rests on P-1 motion when ! 1:05.

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Fig. 6.8 Phase plot and Poincar´e map of chaotic motion .f D 8:8; ! D 3:70/

6.4 Analysis of the Power Flow In the third section, the dynamical evolution of the two-degree-of-freedom nonlinear vibration isolation system (6.4) in the control parameter space is analyzed. Two cascades of deep period-doubling bifurcation and two sequences of chaotic motion are observed when the excitation frequency is decreased from 5 to 0.8, while the excitation amplitude remains 8.8. However, the isolation characteristics of the system (6.4), rather than the dynamical evolution, is of much more concern. In the nonlinear theory of vibration isolation, one encounters the problem of defining a suitable performance index for the isolation. This is because a harmonic response with the same frequency as that of the excitation is not guaranteed. As mentioned previously, the response may contain subharmonics and superharmonics, and sometimes the response may even be nonperiodic, namely, chaotic. Usually, if other harmonics are present, a working index of isolation effectiveness may be defined as the ratio of the r.m.s. values of the response and the excitation. In the case of a chaotic response, the ratio of the power spectral densities of the response and the excitation may be used [7]. But the indices to be used for various types of excitation, unlike in a linear system, cannot be related through simple expressions. Furthermore, in the design of a vibration isolation system, what we care most is the attenuation of the vibratory energy. The power flow, hence, is a good candidate. As we all know, the force transmissibility is defined as the ratio of the force transferred to the base vs. the excitation force. Correspondingly, we can define the power flow transmissibility as [12] TP D

Pout Pin

(6.6)

where Pout is the power flow transferred to the base through the isolator, and Pin is the power flow into the whole system.

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Express the reciprocal of TP with decibel, and (6.6) becomes LP D 10 lg

1 Pin D 10 lg TP Pout

(6.7)

where LP is called the power flow attenuation rate (PFAR). However, this working index LP is devoid of any information about the response frequency content and cannot indicate the weakening of the line spectrum. One of the most advantages of the chaotic vibration isolation lies in isolating the line spectrum. Another index should be used. Following the definition of the vibration level difference, let DP D 10 lg

PLSin PLSout

(6.8)

as the power flow line spectrum drop (PFLSD), where PLSin is the input power flow at the frequency containing the line spectrum, and PLSout is output power flow at the same frequency. Thus, the isolation effectiveness can be analyzed throughout the bifurcation process from the point of view of energy. The evolution of the PFAR and the PFLSD at the excitation frequency of system (6.4) with f D 8:8 is shown in Fig. 6.9. The representative phase plots of P-1 and chaotic motions when ! D 4:80 and 3.70 throughout the two cascades of deep period-doubling bifurcation are shown in Figs. 6.4 and 6.8 respectively. The corresponding characteristics of the power flow when ! D 4:80 and 3.70 is shown in Figs. 6.10 and 6.11, and the corresponding PFLSD is 35.11 and 40.31 dB respectively. Namely, the reduction of the line spectra when the system is chaotic is 5 dB greater than that when the system is periodic. The values of the PFAR and the PFLSD at different excitation frequencies are listed in Table 6.1. As shown in Figs. 6.10 and 6.11 and Table 6.1, the nonlinear vibration isolation system is good for the reduction of the line spectra. For some parameters, the PFAR in chaotic state is 17 dB higher than that in P-1 state at best, and the PFLSD in chaotic state is 5–20 dB higher than that in P-1 state. Data in rows of No. 2–3 vs. that of No. 4–6 and data in rows of No. 10 vs. that of No. 11–12, furthermore, shows that the PFLSD in 1/2 (or 1/4, 1/8) subharmonic states is close to that in chaotic state. That is, the isolation effectiveness of the line spectrum is widely improved after a sequence of period-doubling bifurcations.

6.5 Conclusion The complex dynamic behaviour of the two-degree-of-freedom nonlinear vibration isolation system is studied numerically, and its ability in line spectrum reduction is also analyzed.

58 Fig. 6.9 Power flow attenuation rate and line spectra drop .f D 8:8, ! D .0:80 W 0:01 W 5//. (a) Bifurcation diagram, (b) power flow attenuation rate, (c) power flow line spectra drop

Fig. 6.10 Input and output power flow for P-1 motion .f D 8:8; ! D 4:80/

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Fig. 6.11 Input and output power flow for chaotic motion .f D 8:8; ! D 3:70/

Table 6.1 Values of the PFAR and PFLSD at different excitation frequencies PFAR Excitation Form PFLSD at excitation (dB) frequency of motion frequency (dB) No. 1 1.01 P1 21.92 36.69 2 1.06 Chaos 18.25 42.75 3 1.08 Chaos 18.62 41.45 4 1.09 P8 18.71 41.40 5 1.10 P4 18.73 41.40 6 1.12 P2 18.94 40.85 7 1.80 P1 18.06 28.16 8 2.07 P2 13.81 30.48 9 2.80 P1 11.35 20.20 10 3.70 Chaos 28.07 40.31 11 3.71 P8 27.98 40.45 12 3.74 P4 27.91 40.88 13 4.10 P2 28.07 36.82 14 4.80 P1 32.73 35.11

The dynamic behaviour distribution chart of the two-degree-of-freedom nonlinear vibration isolation system is obtained, which shows that there exists complex nonlinear behavior indeed in this system. Cascades of bifurcation in the two-degreeof-freedom nonlinear vibration isolation system with different excitation amplitudes are obtained. The isolation effectiveness is analyzed from the point of view of energy. For some parameters, the power flow attenuation rate in chaotic state is 17 dB higher than that in P-1 state at best, and the power flow line spectrum drop in chaotic state is 5–20 dB higher than that in P-1 state.

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It is also concluded that the isolation effectiveness of line spectrum is improved after once or twice period-doubling bifurcation, when the power flow attenuation rate and the power flow line spectrum drop are close to that in chaotic state. To validate the effectiveness of the method of chaotic vibration isolation, the vibration-isolation test rig with flexible foundation similar to the actual situation onboard of ships is designed, and meticulous experiment is also accomplished. Preliminary experimental results indicating better isolation in the chaotic regime will be published in near future. Acknowledgment This work was supported by National Natural Science Foundation of China under Grant 50675220.

References 1. Kapitaniak T (1997) Chaos for engineers: theory, application, and control. Springer, London 2. Lou JJ, Zhu SJ, He L, et al (2005) Application of chaos method to line spectra reduction. J Sound Vib 286:645–652 3. Lou JJ, Zhu SJ, He L, et al (2009) Experimental chaos in nonlinear vibration isolation system. Chaos Solitons Fractals 40:1367–1375 4. Crede CE (1951) Vibration and shock isolation. Wiley, New York 5. Snowdon JC (1979) Vibration isolation use and characterization. J Acoust Soc Am 66:1245–1279 6. Ravindra B, Mallik AK (1993) Hard Duffing-type vibration isolator with combined coulomb and viscous damping. Int J Non Linear Mech 28:427–440 7. Ravindra B, Mallik AK (1994) Performance of non-linear vibration isolation under harmonic excitation. J Sound Vib 170:325–337 8. Long YJ, Wang CL, Zhang P (1998) Road roller based on chaotic theory. J China Agric Univ 3:19–22 9. Harris CM (1988) Shock and vibration handbook, 3rd edn. McGraw-Hill, New York 10. Karnopp DC (1995) Active and semi-active vibration isolation. J Sound Vib 117:177–185 11. Rivin EI (2003) Passive vibration isolation. The American Society of Mechanical Engineers, New York 12. Lou JJ (2006) Application of chaos theory in line spectrum reduction [PHD dissertation]. Naval University of Engineering, Wuhan

Chapter 7

Simulation and Nonlinear Analysis of Panel Flutter with Thermal Effects in Supersonic Flow Kai-Lun Li, Jia-Zhong Zhang, and Peng-Fei Lei

Abstract With the consideration of thermal effect, an improved panel flutter model equation is established to study the dynamic behaviors of panel structures on supersonic aircrafts. The governing equation is approached by Galerkin Method, and then the resulting ordinary differential equations of the panel are obtained. By the numerical simulation, some essential nonlinear phenomena are discovered, and they play an important role in the stability of the panel in supersonic flow. Finally, Mach number and Steady temperature recovery factor are considered bifurcation parameters, Hopf bifurcation, and Pitchfork bifurcation, and other complex bifurcations at the equilibrium points are analyzed in detail, respectively, by seeking the eigenvalues of the Jacobian matrix of the dynamic system at bifurcation points. It can be concluded that there exist a rich variety of nonlinear dynamics, and they are essential for the stability of the panel in the supersonic flow.

7.1 Introduction The panel structures have been used frequently on supersonic aircrafts. As the aircrafts are flying at supersonic speed, the aerothermoelasticity has an enormous impact on the aircrafts. Under the combined effects of aerodynamics, thermodynamics, and structure dynamics, the panel structures on the aircrafts behave as periodic oscillation, quasi-periodic oscillation, chaotic motion, buckling, etc. These phenomena lead to a great deal of threat to the safety and life of the panel. From the viewpoint of nonlinear dynamics, the states of panel varying from static state to oscillation or dynamic buckling are the typical bifurcation behaviors, and such nonlinear phenomena could be utilized to improve the aerodynamic

J.-Z. Zhang () School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, People’s Republic of China e-mail: [email protected]

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 7,

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performance of the aircrafts. Therefore, using nonlinear theory to analyze the stability of the panel is essential for the modern aerodynamic design. Much of the early works have been devoted to panel flutter in supersonic speed [1, 2]. And the nonlinear behavior of panel flutter has been discovered and investigated [3, 4]. Also the chaos theory has been introduced by several scholars to investigate the panel flutter [5]. More recently, a panel flutter model with thermal effect has been established by Gee and Sipcic [6]. The introduction of thermal effect makes the problem more complicated. On the one hand, thermal effect reduces the stiffness of the panel because of aerodynamic heating. On the other hand, thermal stress is generated because of mismatch in the thermal expansion coefficients of panel and support structure. Also, it has been found that the change of temperature distribution in the panel does not synchronize with the change of temperature on the panel surface, that is, there is a time-lag between them because of heat transferring. An improved panel flutter model equation is established in this study, with the consideration of thermal effects. In order to simplify the model equation, the reduction of panel stiffness and the time-lag of heat transferring are neglected. Von Karman large deflection plate theory and Piston theory are used to obtain the strain in the panel and aerodynamic loads respectively. Aerodynamic heating is obtained from Busemann–Crocco’s solution of boundary layer equations. By Galerkin method, the vector form of governing equation is obtained. And then some nonlinear phenomena are discovered, and two main kinds of bifurcations of the dynamic system are analyzed following the nonlinear theory.

7.2 Governing Equation Figure 7.1 is the schematic of two-dimensional panel with hinged boundary condition in supersonic flow. It is assumed that the panel is infinite in spanwise direction, with a in length and h in thick.

Fig. 7.1 Two-dimensional panel with hinged boundary condition in supersonic flow

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7.2.1 Dynamic Loads and Heating The steady temperature in the panel caused by aerodynamic heating due to viscous flow is obtained from Busemann–Crocco’s solution as follows [7], 2 T1 : Tf D T1 C Rf Œ. 1/=2M1

(7.1)

According to Piston theory, the unsteady pressure on the outer surface of the panel gives pu D p1 C .2q1 =M1 /Dt w: (7.2) From isentropic relation between pressure and temperature, the unsteady temperature change can be obtained from (7.2) Tu D T1 C . 1/M1 T1 Dt w:

(7.3)

Actually, the temperature change on the outer surface is the result from viscous flow and the compression and expansion of air, so the actual temperature on the outer surface should be obtained by (7.1) and (7.3). Replacing T1 in (7.3) by Tf in (7.1), the actual unsteady temperature on the outer surface of the panel can be written in the form (7.4) Tu D Tf C Ru . 1/M1 T1 Dt w: In the equations given above, Dt w is defined by Dt w D V =U1 D w;x C w;t =U1 :

(7.5)

As the panel is under flow at constant Mach number, then the temperature in the inner side of panel maintains at the steady temperature Tf due to viscous flow and heat transferring. And the inner surface pressure on the panel is assumed to be equal to the pressure p1 , the pressure in the free stream.

7.2.2 Solution of Heat Transfer The heat conduction along the thickness direction of the panel is governed by the one-dimensional heat conduction equation. For this problem, Duhamel superposition integral is a closed-form solution [8], namely 1 x C Tu .t/ T .z; t/ D Tf C h 2 Z t 1 2 X .1/n 2 Tu .0/C Tu0 ./en d C nD1 n 0 1 x 2 t C ; en sin n h 2

(7.6)

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Tu D Tu Tf D Ru . 1/M1 T1 Dt w; c h2 D 2 ; k

(7.7) (7.8)

where cp ; p ; h, and kp are the specific heat capacity, density, thickness, and thermal conductivity of the panel, respectively. In fact, when aluminum alloy is adopted as the material of the panel, will be far less than 1 due to small thickness of the panel and good thermal conductivity of the material. So the distribution of temperature in the panel can be simplified as T .z; t/ D Tf C

x 1 C Tu .t/: h 2

(7.9)

That means the time-lag of heat transferring can be neglected to get the approximation of the temperature distribution in the panel.

7.2.3 Governing Equations The original governing equation of infinite two-dimensional panel flutter is @2 w @2 Mx C .Nx C NE / 2 C q.x; y/ D 0; 2 @x @x

(7.10)

where NE is an externally applied in-plane load and q.x; y/ is defined as q.x; t/ D p h

@2 w C .p1 pu /: @t 2

(7.11)

Introducing the thermal effect into Von Karman large deflection plate theory, the stress in the panel gives E x D 1 2

"

@u 1 C @x 2

@w @x

2

# @2 w z 2 .1 C /˛p Tp ; @x

Tp D T .x; t/ Tref ;

(7.12) (7.13)

where Tp denotes the difference between current temperature distribution and initial temperature distribution in the panel. In this study, it is assumed that Tref D T1 : And then the axial force and bending moment can be obtained as

(7.14)

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Analysis of Panel Flutter with Thermal Effects in Supersonic Flow

Nx D

Z a 2 @w Eh˛s Ts Eh C dx 2 2 1 2a.1 / 0 @x Z h=2 Z a ˛p E dx Tp dz; a.1 / 0 h=2 Z h=2 ˛p E @2 w Mx D D 2 Tp zdz; @x 1 h=2

65

(7.15) (7.16)

where Ts is the difference between current temperature and initial temperature in the support structure, it is defined as Ts D Tf Tref :

(7.17)

Substituting (7.11), (7.15), and (7.16) into the (7.10), and the nondimensional governing equation becomes 3 3 @4 wN O u @ wN C ‚RO u @ wN C ˆ R @xN 4 @xN 3 @tN@xN 2 ! Z O Rf @2 wN 1 @wN 2 @2 wN O C 12Rf RE C 12 6 2 dxN 1Cv @xN 2 @xN 0 @xN Z Z @2 wN 1 @wN @2 wN 1 @wN O O dxN C 6‚Ru 2 dxN C 6ˆRu 2 @xN 0 @xN @xN 0 @tN ‚ @wN @2 wN @wN Cƒ D 0: (7.18) C 2 Cƒ N @t @xN ˆ @tN

The coefficients in (7.18) are listed in Appendix 2. And the boundary condition of (7.18) gives w.0; N tN/ D w.1; N tN/ D 0;

@2 wN @2 wN N/ D .0; t .1; tN/ D 0; @xN 2 @xN 2

w. N x; N 0/ D g.x/; N

@wN .x; N 0/ D 0: @tN

(7.19) (7.20)

7.2.4 Galerkin Method Following Galerkin method, wN can be denoted as w. N x; N tN/ D

N X

wr .tN/ sin.r x/: N

(7.21)

rD1

Substituting (7.21) into (7.18) and integrating on the both side of the equation, the ordinary differential equations are obtained as follows

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. s/4 ws 2

ˆRO u

N X

. r/3 wr

rD1;r¤s

" RO f 2 O P s 12 ‚Ru . s/ w 1Cv 2

C 3. s/ ws

N X

. r/

2

w2r

s Œ1 .1/sCr s2 r 2 # O C 12Rf RE . s/2 ws

6‚RO u . s/2 ws

rD1 N X

C 2ƒ

wr

rD1;r¤s

N X

wP r

rD1

Œ1 .1/r C wR s r

rs ‚ Œ1 .1/sCr C ƒ wP s D 0; s2 r 2 ˆ

s D 1; : : : ; N: (7.22)

For the sake of simplicity, the resulting governing equations are rewritten in the first-order form wP s D wsCN ;

wP sCN D ba.s/ws bb.s/wsCN bc.s/wsCN

N X

bf .s; r/wr

rD1;r¤s

N X

N X

N X

bg.s; r/ws w2r

rD1;r¤s

bh.s; r/ws wrCN

rD1;r¤s

bi.s; r/wr bg.s; s/w3s bh.s; s/ws wsCN :

(7.23)

rD1;r¤s

The coefficients in (7.23) are shown in Appendix 2. And the Jacobian matrix of the dynamic system is @wP s @ws @wP s @wr @wP s @wsCN @wP s @wrCN

D 0; D 0; D 1; D 0;

@wP sCN D ba.s/ @ws

N X

bg.s; k/w2k

kD1;k¤s

N X

bh.s; k/wkCN

kD1;k¤s

3bg.s; s/w2s bh.s; s/wsCN ; @wP sCN D bf .s; r/ 2bg.s; r/ws wr bi.s; r/; @wr @wP sCN D bb.s/ bc.s/ bh.s; s/ws : @wsCN

(7.24)

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Analysis of Panel Flutter with Thermal Effects in Supersonic Flow

67

Also, the equilibrium positions of the dynamic system can be obtained by solving (7.25) as follows N X bg.s; s/ws 3 C ba.s/ws C bg.s; r/ws w2r rD1;r¤s

C

N X

Œbf .s; r/ C bi.s; r/wr D 0:

(7.25)

rD1;r¤s

Apparently, ws D 0; s D 1; : : : ; N are the equilibrium positions, and the stability of this position is analyzed in the following section.

7.3 Numerical Results In this section, it is assumed that Ru D 0, that means only the impact of Rf to the panel is considered. Runge–Kutta Method is used to investigate the dynamic system. When Rf D 0 and Rf D 0:056, the state of the panel changes from static state to periodic oscillation respectively as the Mach number M1 increases. And there exist two main kinds of bifurcations as the system parameters are varied. Letting M1 D 2, the state of the panel changes from static state in original position to buckling as the steady temperature recovery factor Rf increases, the bifurcation is also analyzed.

7.3.1 Rf D 0, Mach Number as the Bifurcation Parameter Rf D 0 means that the thermal effects are not taken into account. From Figs. 7.2 and 7.3, it is can be found that the state of the panel changes from static state to periodic oscillation as Mach number increases. As the parameters increase to the values as follows: Rf D 0;

M1 D 6:03177;

wi D 0;

i D 1; : : : ; N;

the real parts of a pair of conjugate complex eigenvalues of the Jacobian matrix equal to zero approximately and the real parts of the other eigenvalues are not equal to zero, namely, the system is at a critical state. So as the Mach number increases from 6.0317 to 6.0318, the Hopf bifurcation occurs in the dynamic system at about M1 D 6:03177. The bifurcation diagram is shown in Fig. 7.4.

68

Fig. 7.2 The time history of the dynamic system at Rf D 0; M1 D 6:0317

Fig. 7.3 The time history of the dynamic system at Rf D 0; M1 D 6:0318

Fig. 7.4 Bifurcation diagram of the dynamic system at Rf D 0

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Analysis of Panel Flutter with Thermal Effects in Supersonic Flow

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7.3.2 Rf D 0:056, Mach Number as the Bifurcation Parameter Taking thermal effect into account and assuming Rf D 0:056, it is can be found that as the Mach number increases from 1.73 to 1.74, the state of the panel varies from the static state in initial equilibrium position to buckling. The time histories are shown in Figs. 7.5 and 7.6. As the parameters increase to the values as follows, Rf D 0:056;

M1 D 1:73:75;

wi D 0;

i D 1; : : : ; N

one of the eigenvalues of the Jacobian matrix is real and equals to zero approximately, and the real parts of the other eigenvalues are not equal to zero. According to

Fig. 7.5 The time history of the dynamic system at Rf D 0:056; M1 D 1:73

Fig. 7.6 The time history of the dynamic system at Rf D 0:056; M1 D 1:74

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Fig. 7.7 Bifurcation diagram of the dynamic system at Rf D 0:056

Fig. 7.8 Time history of the dynamic system at Rf D 0:056; M1 D 2

the nonlinear theory, as the Mach number increases from 1.73 to 1.74, the Pitchfork bifurcation occurs in the dynamic system at about M1 D 1:73075. The bifurcation diagram is shown in Fig. 7.7. As Mach number increases from 2 to 2.8, several time histories and phase portraits are shown in Figs. 7.8–7.12. From the figures, it can be found that as Mach number increasing from 2 to 2.8, the panel changes its state from buckling to periodic oscillation. Some irregular oscillations and a state like chaos occur as the parameters are varied. Apparently, the bifurcation behavior at Rf D 0:056 is different from the bifurcation behavior at Rf D 0.

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Analysis of Panel Flutter with Thermal Effects in Supersonic Flow

Fig. 7.9 Phase portrait of the dynamic system at Rf D 0:056; M1 D 2:1

Fig. 7.10 Phase portrait of the dynamic system at Rf D 0:056; M1 D 2:5

Fig. 7.11 Phase portrait of the dynamic system at Rf D 0:056; M1 D 2:7

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Fig. 7.12 Phase portrait of the dynamic system at Rf D 0:056; M1 D 2:8

Fig. 7.13 Time history of the dynamic system at Rf D 0:05427; M1 D 2

7.3.3 M1 D 2, Rf as Bifurcation Parameter From Figs. 7.13 and 7.14, it can be found that the panel changes its state from static state to periodic oscillation as Rf increases. As the parameters are the values as follows, M1 D 2;

Rf D 0:0542714;

wi D 0;

i D 1; : : : ; N;

the real parts of a pair of conjugate complex eigenvalues of the Jacobian matrix equal to zero approximately and the real parts of the other eigenvalues are not equal to zero. From the nonlinear theory, as Rf increases from 0.05427 to 0.05428, Hopf bifurcation occurs in the dynamic system at about Rf D 0:0542714.

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Analysis of Panel Flutter with Thermal Effects in Supersonic Flow

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Fig. 7.14 Time history of the dynamic system at Rf D 0:05428; M1 D 2

Fig. 7.15 Time history of the dynamic system at Rf D 0:05596; M1 D 2

Fig. 7.16 Time history of the dynamic system at Rf D 0:05597; M1 D 2

As Rf continues to increase, the panel jumps suddenly from periodic oscillation to buckling at about Rf D 0:055965. The time histories are shown in Figs. 7.15 and 7.16, and the bifurcation diagram is demonstrated in Fig. 7.17.

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Fig. 7.17 Bifurcation diagram of the dynamic system at M1 D 2

7.4 Conclusions From the results presented above, some conclusions can be drawn as follows. As thermal effect is not taken into account, the state of the panel varies from static state to periodic oscillation at high Mach number (more than 6), which is verified as Hopf Bifurcation. Then, as thermal effect is taken into account, the state of the panel also varies from static state to periodic oscillation, however, at lower Mach number (less than 3). The pitchfork bifurcation is found as the Mach number is increasing. Further, as Mach number is assumed to be a constant, the state of the panel changes from static state to buckling as Rf increasing. So it is can be found that the thermal effect will make the panel unsteady in initial equilibrium position at lower Mach number (less than 3), the oscillation occurs easily than the situation without thermal effect.

Appendix 1 Nomenclature a c D E h k M p

Length of the panel Specific heat capacity Bending stiffness Young’s modulus Panel thickness Thermal conductivity Mach number Static pressure

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Analysis of Panel Flutter with Thermal Effects in Supersonic Flow

q Rf Ru T Tf U ˛

Dynamic pressure Steady temperature recovery factor Unsteady temperature recovery factor Temperature Temperature due to aerodynamic heating Velocity Coefficient of thermal expansion Poisson ration Density

Subscripts 1 p s u

Free stream Panel characteristic Support structure characteristic Condition at external surface

Appendix 2 Nondimensional Variables w h x xN D a p t tN D ; D p a4 h=D h ˆD a ˛s D ˛p 1 h ‚D U1 Eh3 NE a 2 RE D ; DD D 12.1 2 / 3 2a q1 ƒD DM 1 ˛p O Ru D Ru . 1/M1 T1 2 .1 C / ‚ ˛p 1 RO f D Rf . 1/M1 2 T1 2 .1 C / 2 ‚ wN D

75

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The Coefficients of the Governing Equation "

# Of R 12RO f C RE . s/2 ba.s/ D . s/4 C 12 1Cv bb.s/ D ‚RO u . s/2 ‚ bc.s/ D ƒ ˆ ˆRO u s Œ1 .1/sCr bf .s; r/ D 2 . r/3 2 s r2 bg.s; r/ D 3. s/2 . r/2 Œ1 .1/r bh.s; r/ D 6‚RO u . s/2 r s sCr Œ1 .1/ bi.s; r/ D 2ƒr 2 s r2 Acknowledgments This research is supported by Program for New Century Excellent Talents in University in China, No. NCET-07-0685.

References 1. Dugundji J (1966) Theoretical considerations of panel flutter at high super-sonic Mach number. AIAA J 4:1257–1266 2. Ashley H, Zartarian G (1956) Piston theory-a new aerodynamic tool for the aeroelastician. J Aeronaut Sci 23:1109–1118 3. Dowell EH (1966) Nonlinear oscillations of a fluttering plate. AIAA J 4:1267–1275 4. Dowell EH (1967) Nonlinear oscillations of a fluttering plate II. AIAA J 5:1856–1862 5. Sipcic SR (1990) Chaotic response of fluttering panel the influence of maneuvering. Nonlinear Dyn 1:243–264 6. Gee DJ, Sipcic SR (1999) Coupled thermal model for nonlinear panel flutter. AIAA J 37:642–650 7. Schlichting H (1979) Boundary-layer theory. McGraw-Hill, New York 8. Hildebrand FB (1976) Advanced calculus for applications. Prentice-Hall, Englewood Cliffs, NJ

Chapter 8

A Parameter Study of a Machine Tool with Multiple Boundaries Brandon C. Gegg, Steve C.S. Suh, and Albert C.J. Luo

Abstract The parameter study of a machine-tool with intermittent cutting is completed for eccentricity frequency and amplitude. The effects with respect to chip length are also incorporated, such that comparisons of the parameter maps can be accomplished. Specific areas within the parameter maps are studied, via switching components, to explain the complicated motions within. In such a case, the switching characteristics are shown in relation to the eccentricity frequency. The complexity of the periodic solution structure, with regard to the vector fields and mapping quantities, is discussed. Furthermore, the traditional definition of a stability boundary is extended beyond that in literature. The most useful data is the overlay of the number of mappings and minimum switching force product record. This aspect illustrates the extent and location of complexity in the machine-tool model studied herein.

8.1 Introduction The extent of parameter studies on machining systems are typically confined to descriptions in the frequency and depth of cut plane [1]. In such a case, the boundary for stable/unstable motions is defined. Other studies focus on the chip seizure (stick-slip effect) interaction, where a boundary can be defined in parameter space [2]. However, these motions are confined to limitations of a continuous system. Typically, a system with multiply interconnected domains is not studied for chip seizure or other phenomena associated with such a system. There are three parameters studied herein: eccentricity frequency and amplitude, and chip contact length. What makes this study further unique is the output dimension of the components. There are typically at least two switching components for a simple

B.C. Gegg () Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA e-mail: [email protected]

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 8,

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steady state motion. The eccentricity frequency and amplitude with initial conditions yield a steady state solution that can be characterized by the switching components characteristics; three measures are introduced. The typical characterization of a steady state response is noted by the magnitude of the displacement and velocity components at a zero phase measure of the system [3]. In a similar manner, one of the new parameters introduced herein is the magnitude of the displacement and velocity state in each principal direction. A new quantity, referred to as MAG., is multiplied by the minimum of the switching force product (MFP) components (MAG and MFP will be formally be defined herein). This unique measure in combination with the MAG. is a first application in literature. Additionally, the complexity is further quantified by the recording the total number of mappings (NOM), which is also a first application in literature. Due to varying dimensions of complexity in this system, combinations of the NOM, MFP, and MAG. are necessary to fully understand what phenomena may be inducing complex motion. Applications of these ideas and measures are not limited to machining. Rather, any system which contains boundaries in their continuous systems can be modeled by discontinuous systems theory. The ultimate implications of this study are the development of switching components and their use within a control scheme to produce a specific type of stability in a discontinuous system. If such switching components can be monitored in experiment, a control scheme can be adopted to manipulate these components to avoid such an interaction [4]. However, if the goal is to continually interact with a boundary, then avoidance of a sink boundary, or in this case a chip seizure can be completed. As far as this study is concerned, the modeling of a machine tool without control is adopted to observe the natural reaction of a system, which indeed will point out the requirement of such an approach to achieve robust operation.

8.2 Structured Motions by the Mapping Technique The mechanical model of 8.1 (A, B) is described by the chip adhesion dynamics, f0 .x; t; 0 / D 0: .CAD/

(8.1)

CAD denotes chip adhesion dynamics. The dynamics of the tool with no work-piece contact are (8.2) f1 .x; t; 1 / D 0: .TD/ TD denotes tool-piece dynamics. The dynamics of a reducing chip length process are f2 .x; t; 2 / D 0: .NC/ (8.3) NC denotes tool and work-piece dynamics, no cutting. The dynamics of an increasing chip length process are f3 .x; t; 3 / D 0: .CRC/

(8.4)

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A Parameter Study of a Machine Tool with Multiple Boundaries

79

CRC denotes tool and work-piece dynamics with cutting and reducing chip length. The dynamics are f4 .x; t; 4 / D 0: .CIC/ (8.5) CIC denotes tool and work-piece dynamics with cutting and increasing chip length. Parameters defining . i for i D 0; 1; 2; 3; 4/ the dynamics of (8.1–8.5) represent such characteristics such as mass, stiffness, damping, etc. In any case of the dynamics defined within these domains and on the boundaries, the interactions of these systems with the domain boundaries can be clearly understood by discontinuous systems theory of Luo [5]. The state of the tool is measured through the .x; N y/ N representing the tangential and normal directions with respect to contact of the work-piece (Figs. 8.1 and 8.2). Since the focus of the chip interactions are applied to the tool rake surface, the .x; N y/ N coordinate system is transformed to the .x; Q y/ Q coordinate system,

xN yN

D

cos ˛ sin ˛

a

sin ˛ cos ˛

xQ yQ

Dƒ

b η F2(t)

dy

ky

F2t

:

(8.6)

X1 x~

FP(t)

Xeq

A

y~

O

eX

a

eY

dx

F2n

Yeq Y1

m

kx

y x

xQ yQ

D1

m

d2 D2

F1(t)

β d1

B

Fig. 8.1 Cutting tool mechanical model: (a) external forces, (b) mechanical analogy

⋅ y~

( y~i ,V,ti ) Fig. 8.2 Periodic intermittent cutting motions P34 in the phase plane

P4

( yi+1,V,ti+1)

V

P3

y~

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The initial contact boundary of the tool and work-pieces is N sin ˇ C .Y1 Yeq y/ N cos ˇ D ı1 : .Xeq C x/

(8.7)

Such a boundary is related to the measure D1 .x; N y/ N D .Xeq C x/ N sin ˇ C .Y1 Yeq y/ N cos ˇ ı1 :

(8.8)

The onset of cutting boundary is N sin ˛ C .X1 Xeq x/ N cos ˛ D ı2 : .Yeq C y/

(8.9)

The final boundary is considered the chip disappearance boundary; where the chip begins to reduce in length until no effective force transmission is made through the chip-tool interface, Q y/; Lc yQ0 .x0 ; y0 / D y.x;

(8.10)

where Lc is the chip length, yQ0 is the initial tool position at the switching point on the chip-tool rake surface friction boundary, and yQ is the tool position at time t (see Fig. 8.3). x D .x; x/ P T and y D .y; y/ P T . The normalized governing equations characteristic of each domain or Eqs. (8.1–8.5) are of the form i h IrRQ .t/ C ƒ1 fD.i / gƒrPQ .t/ C ƒ1 fK.i /gƒQr.t/ D ƒ a.i / cos.t/ C b.i / t C c.i / ; (8.11) where rQ D .x; Q y/ Q T . The damping, stiffness, periodic amplitude and constants noted in Eq. (8.11) are defined in the appendix.

Fig. 8.3 Chip and tool-piece: (a) effective force contact and (b) route to loss of effective force contact

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A Parameter Study of a Machine Tool with Multiple Boundaries

81

8.3 Domains and Boundaries The four domains considered in this study are noted to overlap in several areas, and a formal comprehensive definition is necessary as in Gegg [6, 7]. Domain 1 is the vibration of the tool-piece without contacting the work-piece, †1 .x; y; x; P y/ P D f.x; y; x; P y/jD P 1 .x; y/ 2 .0; 1/gI domain 2 is the contact of the tool and work-piece without cutting, ( .x; y; x; P y/ P jD1 .x; y/ 2 .1; 0/g ; P y/ P D †2 .x; y; x; .x; y; x; P y/ P jD2 .x; y/ 2 .0; 1/g ; 9 .x; y; x; P y/jD P = 2 .x; y/ 2 .1; 1/ or and .x; y; x; .x; y/ 2 .1; 0/ P y/jD P 4 ; PQ x; if .x; y; x; P y/j P y. P y/ P 2 .1; V /I

(8.12)

(8.13)

(8.14)

domain 3 exists purely during reduced chip length, 9 8 .x; y; x; P y/jD P 1 .x; y/ 2 .1; 0/g; > ˆ > ˆ = < .x; y; x; P y/jD P 2 .x; y/ 2 .1; 0/g; †3 .x; y; x; P y/ P D ˆ .x; y; x; P y/jD P 4 .x; y/ 2 .0; Lc /g; > > ˆ ; : P .x; y; x; P y/j P y. Q x; P y/ P 2 .1; V /gI

(8.15)

and domain 4 is well defined by normal cutting, 8 9 P y/jD P < .x; y; x; 1 .x; y/ 2 .1; 0/; = †4 .x; y; x; P y/ P D .x; y; x; P y/jD P 2 .x; y/ 2 .1; 0/; : ; PQ x; .x; y; x; P y/j P y. P y/ P 2 .V; 1/:

(8.16)

The boundaries created by the domains noted in the above equations are P y/ P D f.x; y; x; P y/j' P 12 .x; y/ D '21 .x; y/ D D1 .x; y/ D 0g; (8.17) @†12 .x; y; x; ( ) P y/ P D @†24 .x; y; x;

.x; y; x; P y/j' P 24 .x; y/ D '42 .x; y/ D D2 .v/ D 0 if y. QP x; P y/ P > V; P y/ P D y. QP x; P y/ P V D0 .x; y; x; P y/j' P 24 .x;

(

@†32 D .x; y; x; P y/j' P 32 .x; y; x; P y/ P D

D4 .x; y/ D 0 D2 .x; y/ D 0

(8.18)

if D2 .x; y/ < 0;

if D2 .x; y/ < 0; (8.19) if y. QP x; P y/ P < V;

and PQ x; P y/j' P 34 .x; P y/ P D '43 .x; P y/ P D y. P y/ P V D 0 if D2 .v/ < 0; @†34 D .x; y; x; (8.20)

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as in Gegg [6, 7]; where V D VN = and is the eccentricity frequency applied to the work-piece. The discontinuous systems theory will now be applied to this machine-tool through the state and domain definitions.

8.4 Motion Switch Ability Conditions Development of the switching conditions is determined by application of discontinuous systems theory as in Gegg et al. [8, 9]. Accordingly, the only boundary which has the potential to produce a sink boundary is the chip-tool friction boundary as defined herein. The force conditions governing the passage of motion through the boundary of (8.3) are .3/

.4/

FyQ .Qx; yQ ; t/FyQ .Qx; yQ ; t/ > 0 on @†34

(passable motion);

(8.21)

FyQ.3/ .Qx; yQ ; t/FyQ.4/ .Qx; yQ ; t/

(non-passable motion):

(8.22)

0 on @†34

Appearance/disappearance of passable/non-passable motion, FyQ.3/ .Qx; yQ ; t/FyQ.4/ .Qx; yQ ; t/ 0 on @†34 :

(8.23)

The boundary @†34 is notation referring to the chip-tool friction boundary. The forces noted in (8.21) are derived from the state and the total forces acting on the tool-piece; hence, FyQ .Qx; yQ ; t/ D FD3 .Qx; yQ ; t/ D yRQ .i / .t/ D xR .i / sin ˛ C yR .i / cos ˛: .i /

.i /

(8.24)

The friction boundary exhibiting completely passable vector fields is defined by .3/

.4/

(8.25)

FyQ.3/ .Qx; yQ ; t/FyQ.4/ .Qx; yQ ; t/ > 0 on @†34 :

(8.26)

FyQ .Qx; yQ ; t/ > 0;

FyQ .Qx; yQ ; t/ > 0 on @†34 ;

which implies,

The non-passable motion through the friction boundary has switching components .3/

.4/

(8.27)

FyQ.3/ .Qx; yQ ; t/FyQ.4/ .Qx; yQ ; t/ < 0 on @†34 ;

(8.28)

FyQ .Qx; yQ ; t/ > 0;

FyQ .Qx; yQ ; t/ < 0 on @†34 ;

which implies,

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83

⋅ y~

Fig. 8.4 Vector fields for passable and non-passable with appearance and vanishing points a specific example

(4)

Fy~

Σ4

y~

(3)

Σ3

Fy~

Observe the simulation of Fig. 8.4. The boundary is initially non-passable, but becomes passable after motion along the boundary; where the vector field changes direction. Boundary four, the chip reduction boundary, is a permanently passable boundary as noted by Gegg et al. [10–12].

8.5 Parameter Study of (e, ) Consider the periodic motion, P34 D P3 ı P4 :

(8.29)

This motion structure implies that two switching points exist which defines a solution set. Since there are two switching points, there are two switching force products. Hence, ) .3;1/ .4;1/ FP.1/ D FyQ FyQ ; (8.30) FP.2/ D FyQ.3;2/ FyQ.4;2/ : The force components for this particular case are defined by domains three and four. The boundary of these two domains is the chip/tool friction boundary. As a result of limiting the output of the parameter study to one output for two input variables, only one of the force products can be shown on a contour or three-dimensional figure. A zero force product is known to be a predictive measure of the system encountering a change in the motion structure; hence, the minimum absolute value of the switching force products is recorded as the single output for the contour and three-dimensional figures. In general, the force components are ˇ ˇ

ˇ

ˇ ˇ ˇ ˇ ˇ FPmin D min ˇFP.k/ ˇ D min ˇFyQ.i;k/ FyQ.j;k/ ˇ ;

(8.31)

where i and j are the domains bordering the chip/tool friction boundary, and k is the kth switching force product for the steady state motion of the machine-tool system.

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Furthermore, the magnitude of the minimum absolute value force product and the orbit in the phase plane with respect to the switching points is a useful output. Hence, q Mag.e; / D min.jFP.k/ j/ .ı xQ mn /2 C .ı yQpq /2 C .ı xPQ rs /2 C .ı yPQuv /2 ; (8.32) where

9 ı xQ mn D max.xQ m / min.xQ n /; > > = ı yQpq D max.yQp / min.yQq /; ı xPQ rs D max.xPQ r / min.xPQ s /; > > ; ı yPQuv D max.yPQu / min.yPQv /I

(8.33)

for m; n; p; q; r; s; u; v 2 Œ1; w. The parameter w is the total number of switching points in the periodic orbit. Hence, max.xQ m / is the maximum value of xQ out of the w switching points and so on for the remaining measures of (8.33). Consider the steady state motion of a machine-tool where the above measures are recorded for a two parameter .e; / range allowing contouring and three-dimensional mesh plotting of the minimum force product component and the magnitude of the phase orbit. The parameters of most traditional reference in a parameter map are the frequency and amplitude. The related parameters in this study are the eccentricity frequency and amplitude e. The dynamical system parameters for the following results are me D 103 ; dx D 740 N s=mm; dy D 630 N s=mm; meq kx D ky D 560 kN=mm; k1 D 1 MN=mm; k2 D 100 kN=mm;

d1 D d2 D 0 N s=mm;

and the external force and geometry parameters are ı1 D ı2 D 103 m; D 0:7; Lc D 1:0 104 m; ˛ D rad; ˇ D 0:1 rad; D rad; 4 4 X1 D Y1 D 103 m; Xeq D Yeq D 5 103 m:

V D 20 mm=s;

The minimum force product noted in (8.31) are shown in the form of a contour plot in Fig. 8.5a, where the color variation is determined on a logarithmic scale. The three-dimensional view of a mesh plot is shown in Fig. 8.5b, where the minimum force product (MFP) is shown to vary with eccentricity amplitude and frequency, .e; /; respectively. The use of the logarithmic scale is necessary for both the MFP and the MAG (magnitude of (8.32)). Hence, the contour plot of Fig. 8.5a maintains the largest MFP in the neighborhood of the natural frequency groups of this machine-tool. The most useful components of Fig. 8.5a are the darkest areas, which imply the potential for chip seizure motion. There are apparent discontinuities in the contour plot which denotes a grazing of the chip/tool friction boundary. This phenomena causes the steady state of the machine-tool to jump to a new orbit with an MFP of zero or nearly zero. Although the MFP, Fig. 8.5a, b,

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Fig. 8.5 Minimum force product study for a machine-tool undergoing steady state motion with eccentricity amplitude e vs. eccentricity frequency : (a) ContourŒe; D min.FP.k/ /, (b) min.FP.k/ / vs: e vs: for Lc D 1 .mm/

uncovers the areas in the eccentricity amplitude and frequency .e; / range where chip seizure may occur, this says nothing about what motion actually occurs. Hence, the total number of mappings or domains traversed in steady state motion is recorded in a contour plot of Fig. 8.6a. The lighter colors denote the maximum number of mappings recorded (w D 30 for the current results). The three-dimensional mesh plot of the number of mappings is shown in Fig. 8.6b. The number of mappings in the steady state solution structure is very clearly noted in Fig. 8.6b. The level of mappings noted by the number one is the motion of the tool when there is no interaction with the chip/tool friction boundary. The noticeable jumps in the mesh are the areas that have an increased number of mappings which then denote increased

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Fig. 8.6 Number of mappings in steady state motion for a machine-tool undergoing steady state motion with eccentricity amplitude e vs. eccentricity frequency , for Lc D 1 .mm/

complexity in the system. A combination of the MFP and the NOM (number of mappings) will provide perhaps the most useful method of determining whether the chip seizure motion is occurring in the potential neighborhoods; see Fig. 8.7a. There are four notable areas that are bordered by the NOM’s outline. Region A is that near the second natural frequency group, where a chip seizure motion occurs in the steady state motion. The remaining regions (labeled by B) outlined by the NOM are grouped nearest the first natural frequency group. Such motions are expected to have chip seizure in the steady state structure with associated grazing motions. Furthermore, the use of the magnitude of the delta measure (MAG) and the MFP is useful to show the growth and reduction of the phase orbit in several planes, see Fig. 8.7b.

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Fig. 8.7 Number of mappings overlaid on (a) minimum force product and (b) magnitude for a machine-tool undergoing steady state motion with eccentricity amplitude e vs. eccentricity frequency , for Lc D 1 .mm/

8.6 Numerical Prediction of Eccentricity Frequency As a result of the parameter study of Figs. 8.5–8.7, interest of the phase and specific switching force components is developed. Hence, the numerical prediction of steady state motion for this machine-tool is shown via switching phase mod.ti ; 2 / and displacement y.Dy Q t /, Fig. 8.8a, b; respectively. The range of eccentricity frequency

2 Œ50:0; 600:0.rad=s/ is studied with the parameters, e D 0:275 mm and Lc D 1:0 mm:

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The P34 steady state motion dominates the largest frequency span, but is interrupted by the P234 steady state motion; see Figs. 8.8 and 8.9. As noted in Gegg [12], the route to an unstable state caused by the transient/steady state interference with chip seizure motion is observed with decreasing eccentricity frequency . The switching phase components are noted to nearly fill the spectrum, implying that the motion may be chaotic. Verification of the introduction of chip seizure motion is noted in Fig. 8.9a, b by the switching forces and force products. Table 8.1 summarizes the motions and changes in the steady state structure throughout the frequency range.

Fig. 8.8 Numerical and analytical predictions of (a) switching phase mod. ti ; 2 /, (b) switching displacement yQ for interrupted periodic motions over a range of eccentricity frequency for e D 0:275 mm and Lc D 1:0 mm

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Fig. 8.9 Numerical and analytical predictions of (a) switching forces .3/ .4/ FyQ and FyQ , and (b) switchingforce product

.3/

.4/

FyQ FyQ for interrupted periodic motions over a range of eccentricity frequency

for e D 0:275 mm and, Lc D 1:0 mm

Table 8.1 Summary of eccentricity frequencies for specific motions for e D 0:275; Lc D 1:0 Mapping Eccentricity Grazing bifurcation Chip seizure structure frequency

of boundary bifurcation P.034/n .0:1520k; 1656k .rad=s/ D3 W 0:1520k P034 .0:1656k; 0:1970k .rad=s/ D3 W 0:1656k .rad=s/ 0:1656k .rad=s/ P34 .0:1970k; 0:3869k .rad=s/ D3 W 0:1970k .rad=s/ 0:1970k .rad=s/ P234 .0:3869k; 0:4923k .rad=s/ D4 W 0:3869k .rad=s/ P34 .0:4923k; 0:6k .rad=s/ D4 W 0:4923k .rad=s/ 0:1970k .rad=s/

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8.7 Summary and Conclusions The steady state motion for a machine-tool has been studied over the three parameters: eccentricity frequency and amplitude, and chip length. The preliminary discussion of notable phenomena is developed through sketches and their governing equations. The steady state chip seizure with a near grazing motion is developed and observed in ensuing simulations. The parameters maps expressing the MFP, the NOM, and the MAG were presented alone and in two combinations. One combination shows the MFP and the NOM overlay; where the motions and complexity can be clearly defined. The second combination shows the MAG with the NOM overlaid to express the size of the orbit and extent the motion interacts with the chip/tool friction boundary. These parameter maps were completed for two chip lengths, where the motion structure could be observed for effects on the size and location of unstable/chaotic regions. The numerical prediction of two parameter ranges of the eccentricity frequency is completed; one for each chip length. Such is completed to explain the onset of the complex motion noted in Figs. 8.8 and 8.9. This study claims that the measure developed herein by observing not only a single quantity of the motion structure (which has been traditionally accepted protocol), but all the switching component in the motion structure and summarizing the motion with several output measures such as the NOM, the MFP and the MAG. are necessary and sufficient to characterize this machine-tool system and any interconnected dynamics of continuous systems.

Appendix The dynamical system damping parameters for this machine-tool system with free vibration of the tool-piece motion, domain †i , for i D 1, are 2 fD.i / g D 4

d11

.i /

d12

.i /

.i / d21

.i / d22

3 5;

(8.34)

where .i / D d11

1 dx ; m

.i / .i / d12 D d21 D 0;

.i / d22 D

1 dy : m

(8.35)

The stiffness parameters in domain †i for i D 1, are 2 .i / k11 fK.i / g D 4 .i / k21

.i / k12 .i /

k22

3 5;

(8.36)

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91

where .i / k11 D

1 kx ; m 2

.i / .i / k12 D k21 D 0;

.i / k22 D

1 ky : m 2

(8.37)

The external force parameters in domain †i , for i D 1, are h a.i / D ax.i /

ay.i /

iT ;

h b.i / D bx.i /

by.i /

iT ;

h c.i / D cx.i /

cy.i /

iT ;

(8.38)

where ax.i / D ay.i / D bx.i / D by.i / D cx.i / D cy.i / D 0;

(8.39)

respectively. The dynamical system damping parameters for this machine-tool system undergoing tool and work-piece in contact but no cutting, in domain †i , for i D 2, are 9 1 .i / 2 > Œdx C d1 sin ˇ; > d11 D > > m

> > > > 1 > .i / Œd1 cos ˇ sin ˇ; > d12 D = m

(8.40) > 1 .i / > d21 Œd1 sin ˇ cos ˇ; > D > > m

> > > > 1 > .i / 2 Œdy C d1 cos ˇ: ; d22 D m

The stiffness parameters in domain †i , for i = 2, are .i /

9 1 > 2 > Œk C k sin ˇ; x 1 > > m 2 > > > > 1 > > D Œk cos ˇ sin ˇ; = 1 2 m

> 1 > D Œk1 cos ˇ sin ˇ; > > > m 2 > > > > 1 > 2 ; D Œk C k cos ˇ: y 1 2 m

k11 D .i / k12 .i / k21 .i / k22

(8.41)

The external force parameters in domain †i , for i D 2 are ax.i / D e and .i /

me sin ; m

ay.i / D e

me cos ; m

bx.i / D by.i / D 0

9 1 > fk Œx sin ˇ y cos ˇ sin ˇ; = 1 1 1 m 2 1 > D fk1 Œx1 sin ˇ y1 cos ˇ cos ˇ; ; 2 m

(8.42)

cx D cy.i /

(8.43)

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respectively. The dynamical system damping parameters for this machine-tool system undergoing tool and work-piece in contact but with cutting, in domain †i , for i D 3, 4, are 9 1 Œdx C d1 sin2 ˇ C d2 cos ˛.cos ˛ .1/i sin ˛/; > = m

1 > Œd1 cos ˇ sin ˇ d2 sin ˛.cos ˛ .1/i sin ˛/; ; D m

(8.44)

9 1 Œd1 sin ˇ cos ˇ d2 cos ˛.sin ˛ C .1/i cos ˛/; > = m

1 > Œdy C d1 cos2 ˇ C d2 sin ˛.sin ˛ C .1/i cos ˛/: ; D m

(8.45)

.i / d11 D .i / d12

and .i /

d21 D .i / d22

The stiffness parameters in domain †i , for i D 3, 4, are .i /

9 1 Œkx C k1 sin2 ˇ C k2 cos ˛.cos ˛ .1/.i / sin ˛/; > = 2 m

1 > D Œk1 cos ˇ sin ˇ C k2 sin ˛.cos ˛ .1/.i / sin ˛/; ; m 2

(8.46)

9 1 .i / > Œk cos ˇ sin ˇ k cos ˛.sin ˛ C .1/ cos ˛/; = 1 2 m 2 1 > D Œky C k1 cos2 ˇ C k2 sin ˛.sin ˛ C .1/.i / cos ˛/: ; m 2

(8.47)

k11 D .i / k12

and .i / k21 D .i / k22

The external force parameters in domain †i , for i D 3, 4, are ax.i / D e

me sin ; m

ay.i / D e

me cos ; m

bx.i / D by.i / D 0

(8.48)

and cx.i / D

cy.i / D

1 ˚ k1 x1 sin ˇ y1 cos ˇ sin ˇ 2 m

C k2 x2 cos ˛ y2 sin ˛ cos ˛ .1/.i / sin ˛ ;

9 > > > > > > =

>

1 ˚ > > > k1 x1 sin ˇ y1 cos ˇ cos ˇ > 2 > m

; .i / C k2 x2 cos ˛ C y2 sin ˛ sin ˛ C .1/ cos ˛ ; (8.49)

respectively. The dynamical system damping parameters for this machine-tool system undergoing tool and work-piece in contact but with cutting, in domain †i , for i D 0, are

8

A Parameter Study of a Machine Tool with Multiple Boundaries .i / dQ11 D

93

1 d2 C d1 sin2 .˛ C ˇ/ C dx cos2 ˛ C dy sin2 ˛ ; 2m

(8.50)

.i / .i / .i / D dQ21 D dQ22 D 0. The stiffness parameters in domain †i , for i D 0, are and dQ12 .i / kQ11 D

1 k1 sin2 .˛ C ˇ/ C k2 C kx cos2 ˛ C ky sin2 ˛ ; 2 m

(8.51)

.i / .i / .i / and kQ12 D kQ21 D kQ22 D 0. The external force parameters in domain †i , for i D 0, are .i /

aQ x D e

me sin. ˛/; m

.i /

aQ y D 0;

(8.52)

V k1 cos.˛ C ˇ/ sin.˛ C ˇ/ C .kx ky / cos ˛ sin ˛ ; bQy.i / D 0 (8.53) bQx.i / D m

and cQx.i / D

1 fŒd1 VN k1 .VN t0 C yQ0 / cos.˛ C ˇ/ m 2 C k1 Œx1 sin ˇ y1 cos ˇg sin.˛ C ˇ/ C ŒVN .dx dy / ı C .VN t0 C yQ0 /.ky kx / cos ˛ sin ˛ C k2 xQ 2 ; (8.54)

respectively. The tilde noted parameters can be referred to in the .x; y/ coordinate system by iT iT h h ƒQa.i / D ƒ aQ x.i / aQ y.i / D a.i / D ax.i / ay.i / ; iT iT h h ƒbQ .i / D ƒ bQx.i / bQy.i / D b.i / D bx.i / by.i / ; and

h ƒQc.i / D ƒ cQx.i /

cQy.i /

iT

h D c.i / D cx.i /

cy.i /

(8.55) (8.56)

iT :

(8.57)

References 1. Traverso MG, Zapata R, Schmitz TL, Abbas AE (2009) Optimal experimentation for selecting stable milling parameters: a Bayesian approach. In: Proceedings of the ASME 2009 international manufacturing science and engineering conference MSEC2009-84032 2. Chandrasekaran H, Thoors H (1994) Tribology in interrupted machining: role of interruption cycle and work material. Wear 179:83–88 3. Wiercigroch M (1997) Chaotic vibration of a simple model of the machine tool-cutting process system. Trans ASME J Vib Acoust 119:468–475

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4. Navarro-Lopez EM (2009) An alternative characterization of bit-sticking phenomena in a multi-degree-of-freedom controlled drillstring. Nonlinear Anal Real World Appl 10(5):3162– 3174 5. Luo AC (2005) A theory for non-smooth dynamical systems on connectable domains. Commun Nonlinear Sci Numer Simul 10:1–55 6. Gegg BC, Suh CS, Luo ACJ (2008) Chip stick and slip periodic motions of a machine tool in the cutting process. In: ASME manufacturing science and engineering conference proceedings, MSEC ICMP2008/DYN-72052 7. Gegg BC, Suh CS, Luo ACJ (2008) Periodic motions of a machine tool with intermittent cutting. In: International mechanical engineering conference and exposition proceedings, IMECE ASME2008/VIB-67109 8. Gegg BC, Suh Steve, Luo ACJ (2008) Analytical prediction of interrupted cutting periodic motions in a machine tool. NSC2008-97, NSC, Porto, Portugal 9. Luo AC, Gegg BC (2004) Grazing phenomena in a periodically forced, linear oscillator with dry friction. Commun Nonlinear Sci Numer Simul 11(7):777–802 10. Gegg BC, Suh CS, Luo ACJ (2007) Periodic motions of the machine tools in cutting process. DETC2007/VIB-35166, Las Vegas, Nevada 11. Gegg BC, Suh S, Luo ACJ (2009) Interrupted cutting periodic motions in a machine tool with a friction boundary, Part I: modeling and theory. ASME J Manuf Sci Eng (in press 2010) 12. Gegg BC (2009) An investigation of the complex motions inherent to machining systems via a discontinuous systems theory approach. PhD dissertation, Texas A&M University, College Station, Texas

Chapter 9

A New Friction Model for Evaluating Energy Dissipation in Carbon Nanotube-Based Composites Yaping Huang and X.W. Tangpong

Abstract Being lighter and stiffer than traditional metallic materials, nanocomposites have great potential to be used as structural damping materials for a variety of applications. Studies of friction damping in the nanocomposites are largely experimental, and there has been a lack of understanding of the damping mechanism in nanocomposites. A new friction model is developed to study the energy dissipation at the interface between carbon nanotube (CNT) and polymer matrix under dynamic loading. Iwan’s distributed friction model is considered in order to capture the stick/slip phenomenon at the interface. The effects of several parameters on energy dissipation are investigated, including the excitation’s frequency and amplitude, and the interaction between CNT’s ends and matrix. A compliance number is introduced to evaluate the energy dissipation for different contact interfaces. Some of the results are compared well with experimental observations in the literature.

9.1 Introduction Friction damping refers to the conversion of kinetic energy associated with the motion of vibrating surfaces to thermal energy through friction between them [1]. Adding friction damping into a dynamic system can be a useful and practical means to control mechanical vibration passively, particularly in high temperature applications. There are various methods to introduce additional damping into a system, for example, through (1) the incorporation of a damper (such as a ring) in automotive and turbomachinery applications [2–5], (2) piezoelectric materials and shunted electrical circuits [6, 7], (3) electro-rheological/magneto-rheological fluids [8, 9], and (4) viscoelastic and elastomeric materials [10, 11]. While viscoelastic materials offer good damping performance, their thermal stability becomes an issue in high temperature environments and many of their mechanical X.W. Tangpong () Department of Mechanical Engineering, North Dakota State University, Fargo, ND 58108, USA e-mail: [email protected]

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 9,

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properties start to degrade as the temperature rises. Those conventional methods also pose challenges when it comes to integration into heterogeneous systems due to limitations on space, weight, thermal stability, and damper reliability [12]. One solution is to engineer the desirable amount of damping directly into composite materials to develop light-weight and durable structure damping composites that can be easily integrated into various systems. With the rapid development of nanotechnology in the last decade, an attractive opportunity rises as to engineer such high damping performance composite materials by adding nanoscale fillers such as carbon nanotubes (CNTs) into polymer composites (see Refs. 12, 13 and the references therein) for a variety of mechanical, civil, military, aerospace, and aeronautics applications. Due to CNT’s thermal stability, the CNT-reinforced nanocomposites can be used as structural damping materials for extreme temperature applications. The properties of nanocomposite, including the damping capacity, are highly dependent on the fabrication method and processing techniques used. The damping properties of nanocomposites have been studied experimentally [12–20]. Dynamic mechanical analysis (DMA) tests of nanocomposites with different weight fractions of CNTs showed that the reinforcement of CNTs could have significant influence to the material’s damping capacity, and both temperature and frequency affected the damping [14, 15]. Through mechanical cyclic test, the damping property of CNT-based composites has been found to be dependent on strain, temperature, CNTs’ weight fraction, and dispersion [16–18]. Friction damping can also be determined from frequency response of the material sample using an accelerometer and spectrum analyzer. Usually, a film of nanocomposite material is put in between piezoelectric, epoxy, or metal sheets to form a sandwich beam, and the vibration of the beam’s tip is then measured under cantilevered boundary condition. A critical weight fraction of the CNTs has been found to exist for maximum damping of the composite [12, 19, 20]. From these experimental studies, it was hypothesized that energy dissipation in nanocomposites was due to interfacial slippage between the CNTs and the matrix. In the limited modeling work on evaluating friction damping of nanocomposites, the stick–slip phenomenon between the CNTs and the polymer matrix has been well accepted [12, 13, 19–22]. Generally, CNT is modeled as a solid cylinder, and interfacial slippage takes place along the CNT–matrix interface when the interfacial shear stress reaches a critical value [19]. The molecular dynamics (MD) method was used to calculate the energy dissipation due to intertube friction in [21], where the boundary conditions of the nanotube cluster were considered to be periodic [21], though CNTs do not have continuous geometry. Shear lag analysis was also well accepted in modeling the interfacial slippage of CNT–matrix [13, 19]. Other modeling methods include (sandwiched) beam vibration analysis [12, 20, 23] and finite element analysis [19, 24]. These aforementioned models did not take into account the spatially distributed nature of the CNTs and did not consider varying interfacial stiffness across the CNT–matrix interface. A major property of the CNT is its high aspect ratio (length is much larger than diameter). When friction contact is across a spatially distributed interface, the interfacial stiffness is not constant across the interface and, therefore, should be treated in a statistical sense. The spatially

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varying interfacial stiffness, coupled with nonuniform pressure distribution as a result of material processing, could activate partial stick–slip motion, which is critical in determining energy dissipation. In this chapter, a friction model is developed considering the spatially distributed nature of CNT–matrix contact. Based on the developed model, dynamic analysis of energy dissipation at CNT–matrix interface considering multiple properties of the CNTs are performed. The results agree with some experimental results from the literature qualitatively.

9.2 Vibration Model To describe the spatially distributed nature of CNT–matrix contact, the distributedelement friction model of Iwan [25,26] is adopted in this work. This model is based on the concept of a large number of ideal elasto-plastic elements having different, and statistically distributed, yield levels. The model is capable of simulating hysteresis characteristics with nonlocal memory [27]. Figure 9.1 illustrates a parallel-series distributed-element model of Jenkin’s (or Maxwell-slip) elements, and each consists of a linear spring of stiffness K=N in series with a Coulomb damper of a slip force fi =N , where N is the number of elements. The force–deflection relation of one element is depicted in Fig. 9.2, and upon initial loading, the reaction force is F D

n X

fi =N C kx.N n/=N;

(9.1)

i D1

where n is the number of elements that have slipped, while the remainder stick. In the limit of very large N , the slip force is expressed in terms of the distribution function ', and (9.1) is written in the equivalent form: Z F D

kx

f '.f /df C kx

Z

0

'.f /df ;

xP > 0;

(9.2)

kx

k N

k N

*

f2 N

f1 N

Fig. 9.1 Distributed hysteresis model

1

k N

⋅⋅⋅

*

*

fi N

F

⋅⋅⋅

k N *

fN N

x

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Fig. 9.2 Force-deflection relation of a slip element

fi

a

*

fi /N k/N 1 −A

O

x A

b

Fig. 9.3 Distribution function '

yi Km

Km

Km

Km Km

•••

Ke

••• mi

Kc

Kc

Kc

Ke

xi Matrix

CNT

Fig. 9.4 Frictional contact model of a CNT and polymer matrix

where '.f /df is the fraction of elements satisfying f fi f C df . The force developed on other parts of the hysteresis loop is derived in a similar manner. The second term in (9.2) vanishes for large x, and the total slip force becomes Z

1

Fy D

f '.f /df :

(9.3)

0

The distribution function can be prescribed in analytical form [25, 28, 29], or it can be determined from experimental data. The distribution function is closely related to the topography of the contact interface, the contact compliance, and the coefficient of friction. In this chapter, the distribution function ' is expressed in an analytical form as illustrated in Fig. 9.3, similar to the one considered in [25]. The frictional interaction between a single CNT and the polymer matrix is described in Fig. 9.4. The CNT, matrix, and the friction interface between the two

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are each discretized into a large number of elements in the longitudinal direction. The frictional interface between each pair of CNT/matrix elements is modeled by Iwan’s distributed friction concept (depicted in Fig. 9.1). The equation of motion of the ith CNT element is mi xR i D Kc xi C1 Kc xi C Fi ;

(9.4)

where Kc is the axial stiffness of one CNT element, xi is the length of the ith element, and the friction force Fi is determined by (9.2). The total energy dissipation in one cycle of vibration is calculated by ED

N Z X i D1

T

Fi .yPi xP i /dt;

(9.5)

0

and the total number of elements .N / is determined by convergence study of the energy dissipation. The matrix’s motion is specified in a sinusoidal form as 2n y.x; t/ D y0 sin x cos.!t/; (9.6) L where y0 is the amplitude, ! is the excitation frequency, n is the spatial wave number and L is the CNT’s length. The friction force is, therefore, driving the CNT’s motion, but not the matrix’s. In this study, the matrix’s motion is specified, and therefore, the stiffness between matrix elements (parameter Km ) does not come into the equations of motion. The matrix’s response to dynamic loads will be included in a future work by the authors, and in that case, the friction force will also influence the matrix’s motion. It should be noted that the stiffness between the CNT’s two ends and the matrix is represented by a separate parameter Ke . This stiffness comes from the binding strength between the CNT’s ends and the matrix, and the binding mechanism is yet to be discovered [30, 31].

9.3 Results and Discussion The following nondimensionalized parameters are considered in this chapter. y D

y0 ; L

ˇD

! D q ; Kc m

f ; Fy

M D

tD

Kt ; Kc

Kc L EA D ; Fy Fy

eD

Ke ; Kc

E D

E ; Fy L (9.7)

where m is the elemental mass of the CNT and Kt is the tangential stiffness of the interface. Parameter e characterizes the ratio of the CNT’s end binding stiffness to its axial stiffness. Parameter ˇ is considered in the distribution function ' (see Fig. 9.3). Parameter M is introduced here to describe the relative stiffness of the CNT and the contact interface, and it is discussed in detail in Sect. 9.3.3.

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9.3.1 Excitation Frequency and Binding Strength The energy dissipation per cycle is calculated for a range of excitation frequency for different values of stiffness ratio e (Fig. 9.5). By varying the value of e, the axial stiffness of the CNT is fixed; therefore, larger values of e correspond to stronger interaction or stronger binding of the CNT’s ends with the matrix. As shown in Fig. 9.5, higher energy dissipation takes place around the natural frequencies of the CNT. Due to the symmetry of the model considered in this study, only the odd modes appear. It can also be observed that as the value of e increases, the CNT’s natural frequencies shift higher due to larger binding stiffness of the ends. At the same excitation frequency, the energy dissipation decreases as the CNT binds with the matrix stronger, or as the value of e increases (Fig. 9.6). As the interaction between the CNT’s ends and the matrix becomes stronger, the relative motion at the interface has smaller amplitude under the same excitation, and therefore, less energy dissipation takes place.

Fig. 9.5 Frequency response of energy dissipation per cycle; ˇ D 0:8; t D 0:2, M D 200; y D 0:05

Energy dissipation per cycle, E*

1 e = 0.1 0.5 0

0

1

2

3

1 e = 0.5 0.5 0 0

1

2

3

Frequency ratio, η

Fig. 9.6 Energy dissipation as a function of CNT’s binding stiffness with the matrix; ˇ D 0:8; t D 0:2, M D 200; y D 0:05

Energy dissipation per cycle, E*

0.038

η=0.7

0.036

η=0.3 0.034

0.032

0.03 0

η=1.2 0.1

0.2

0.3

Stiffness ratio, e

0.4

0.5

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A New Model for Evaluating Energy Dissipation

101

9.3.2 Excitation’s Amplitude Energy dissipation at the CNT–matrix interface is investigated as a function of excitation’s amplitude (Fig. 9.7). When the matrix’s displacement has small amplitude, the CNT–matrix interface experiences a combination of stick and slip motions since the matrix’s displacement is not large enough to activate full slip at the interface. The energy dissipation in the low excitation amplitude range, therefore, exhibits a nonlinear relationship as indicated in Fig. 9.7. As the matrix’s amplitude increases, the relative motion at the interface gradually progresses to full slip and energy dissipation changes linearly with respect to the excitation’s amplitude. Similar observation was made experimentally in [32], where energy dissipation in the CNT-based composite material was found to change with the strain amplitude of the composite nonlinearly in the low amplitude range, and then linearly as the strain amplitude increases (data with circle markers). The theoretical predictions presented in Fig. 9.8 [32], however, did not capture the nonlinear relationship in low strain amplitude range.

9.3.3 Compliance Number M In defining the parameter M in (9.7), Fy is the total slip force of the CNT–matrix contact interface, and Fy =L can be considered as the linear density of slip force along the interface. The slip force Fy is closely related to the surface topography of the contact interface, the contact pressure, and the coefficient of friction. For one CNT, its stiffness Kc is fixed, and the parameter M can be viewed as a compliance number that quantifies how easily full slip motion can be activated at the interface. The larger the compliance number is, the easier it is for full slip motion to take place at the interface. Figure 9.9 depicts how energy dissipation changes

Fig. 9.7 Energy dissipation as a function of excitation’s amplitude; t D 0:1; D 0:3, e D 0:2; ˇ D 0:8; M D 200

Energy dissipation per cycle, E*

0.5 0.4 0.3 0.2 0.1 0 0

0.05

0.1

0.15

0.2

Excitation's amplitude, y*

0.25

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Fig. 9.9 The effect of compliance number to energy dissipation; ˇ D 0:8; t D 0:2; D 0:5, e D 0:2; y D 0:05

Energy dissipation per cycle, E*

Fig. 9.8 Effect of shear strain amplitude on damping. Original figure used with permission of Nature Publishing Group [32]

1

x 10−4

0.8 0.6 0.4 0.2 0 0

500

1000

1500

2000

2500

Compliance number, M

with the compliance number. An optimal value of M exists for maximum energy dissipation. In the study shown in Fig. 9.9, the CNT’s stiffness Kc is fixed while the slip force Fy is varied. When the value of M is small, Fy is large, and it is not easy to activate slip at the interface; therefore, the energy dissipation is small. As the value of M increases, a combination of stick and slip motions begin to emerge, and the dissipation increases. At large values of M , or when Fy is small, the interface experiences full slip motion; however, since the slip force has small magnitude, the total dissipation is again not optimal. Similar study has been done to keep Fy fixed while changing the value of Kc , and the resulting energy dissipation shows

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A New Model for Evaluating Energy Dissipation

103

the same trend as depicted in Fig. 9.9 and an optimal value of M exists. In the light of these studies, damping of the nanocomposite can be varied by choosing different CNT fillers (single walled, multiwalled, different aspect ratios, etc.) for the polymer matrix. The value of Fy for various nanofillers will be investigated experimentally in a later study.

9.4 Summary A new vibration model has been developed to evaluate the energy dissipation at the CNT–polymer matrix interface. This model takes into account spatially distributed friction at the tube–matrix interface in a statistical sense, and it is capable of capturing the nonlinear stick/slip phenomenon. Energy dissipation has been found to have larger values around the CNT’s natural frequencies. At the same excitation frequency, energy dissipation decreases as the binding of CNT’s ends with the matrix becomes stronger. Strong end interaction of the CNT with the matrix constraints the motion of the CNT, limits the relative motion, and therefore lowers the energy dissipation. The model also predicts that the energy dissipation grows nonlinearly with the increase of excitation’s amplitude in small amplitude range, and the relationship becomes linear when the amplitude is large. Under low magnitude of excitation, the motion at the CNT–matrix interface is a combination of stick and slip motions, and under large excitation, full slip at the interface is activated. The model developed in this work is capable of capturing such transition of nonlinear to linear relationship of energy dissipation with excitation’s amplitude. Such results are consistent with some experimental observations in the literature. A compliance number (M ) is also introduced to quantify how easily full slip motion can be activated at the interface, and an optimal value of M exists for maximum energy dissipation.

References 1. Akay A (2002) Acoustics of friction. J Acoust Soc Am 111:1525–1548 2. Wickert JA, Akay A (2000) Damper for brake noise reduction brake drums. US Patent 6,112,865 3. Wickert JA, Akay A (1999) Damper for brake noise reduction. US Patent 5,855,257 4. Tangpong XW, Wickert JA, Akay A (2008) Finite element model for hysteretic friction damping of traveling wave vibration in axisymmetric structures. ASME J Vib Acoust 130:11005 5. Tangpong XW, Wickert JA, Akay A (2008) Distributed friction damping of traveling wave vibration in rods. Philos Trans R Soc A 366:811–827 6. Collinger JC, Wickert JA, Corr LR (2009) Adaptive piezoelectric vibration control with synchronized switching. J Dyn Syst-T ASME 131(4):041006 7. Tang J, Liu Y, Wang KW (2000) Semi-active and active-passive hybrid structural damping treatments via piezoelectric materials. Shock Vib Dig 32:189–200 8. Lindler JE, Wereley NM (1999) Analysis and testing of electrorheological bypass dampers. J Intell Mater Syst Struct 10:363–376

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9. Kamath GM, Wereley NM, Jolly MR (1999) Characterization of magnetorheological helicopter lag dampers. J Am Helicopter Soc 44:234–248 10. Liao WH, Wang KW (1997) On the analysis of viscoelastic materials for active constrained layer damping treatments. J Sound Vib 207:319–334 11. Brackbill CR, Lesieutre GA, Smith EC (2000) Characterization and modeling of the low strain amplitude and frequency dependent behavior of elastomeric damper materials. J Am Helicopter Soc 45:34–42 12. Koratkar N, Wei BQ, Ajayan PM (2002) Carbon nanotube films for damping applications. Adv Mater 14:997–1000 13. Suhr J, Koratkar N (2008) Energy dissipation in carbon nanotube composites: a review. J Mater Sci 43:4370–4382 14. Wang Z, Liang ZY, Wang B (2004) Processing and property investigation of single-walled carbon nanotubes (SWNT) buckypaper/epoxy resin matrix nanocomposites. Compos Part A: Appl Sci Manuf 35:1225–1232 15. Teo ETH, Yung WKP, Chua DHC (2007) A carbon nanomattress: A new nanosystem with intrinsic, tunable, damping properties. Adv Mater 19:2941–2945 16. Koratkar NA, Suhr J, Joshi A (2005) Characterizing energy dissipation in single-walled carbon nanotube polycarbonate composites. Appl Phys Lett 87:063102 17. Suhr J, Koratkar N (2006) Effect of pre-strain on interfacial friction damping in carbon nanotube polymer composites. J Nanosci Nanotechnol 6:483–486 18. Suhr J, Zhang W, Ajayan PM (2006) Temperature-activated interfacial friction damping in carbon nanotube polymer composites. Nano Lett 6:219–223 19. Zhou X, Shin E, Wang KW (2004) Interfacial damping characteristics of carbon nanotubebased composites. Compos Sci Technol 64:2425–2437 20. Koratkar NA, Wei BQ, Ajayan PM (2003) Multifunctional structural reinforcement featuring carbon nanotube films. Compos Sci Technol 63:1525–1531 21. Suhr J, Koratkar N, Ajayan PM (2004) Damping characterization of carbon nanotube thin films. Proc SPIE 5386:153–161 22. Liu A, Huang JH, Wang KW, Bakis CE (2006) Effects of interfacial friction on the damping characteristics of composites containing randomly oriented carbon nanotube ropes. J Intell Mater Syst Struct 17:217–229 23. Mahmoodi SN, Khadem SE, Jalili N (2006) Theoretical development and closed-form solution of nonlinear vibrations of directly excited nanotube-reinforced composite cantilevered beam. Arch Appl Mech 75:153–163 24. Kireitseu M, Hui D, Tomlinson G (2008) Advanced shock-resistant and vibration damping of nanoparticle-reinforced composite material. Compos Part B: Eng 39:128–138 25. Iwan WD (1966) A distributed-element model for hysteresis and its steady-state dynamic response. J Appl Mech 33:893–900 26. Iwan WD (1967) On a class of models for the yielding behavior of continuous and composite systems. J Appl Mech 89:612–617 27. Al-Bender F, Lampaert V, Swevers J (2004) Modeling of dry sliding friction dynamics: from heuristic models to physically motivated models and back. Chaos 14:446–460 28. Spanos P-TD (1979) Hysteretic structural vibrations under random load. J Acoust Soc Am 65:404–410 29. Segalman DJ (2001) An initial overview of Iwan modeling for mechanical joints, Technical Report, SAND2001–0811, Sandia National Laboratories 30. Mccarthy B, Coleman JN, Curran SA (2000) Observation of site selective binding in a polymer nanotube composite. J Mater Sc Lett 19:2239–2241 31. Qian D, Dickey EC, Andrews R, Rantell T (2000) Load transfer and deformation mechanisms in carbon nanotube-polystyrene composites. Appl Phys Lett 76:2868–2870 32. Suhr J, Koratkar NA, Keblinski P, Ajayan PM (2005) Viscoelasticity in carbon nanotube composites. Nat Mater 4:134–137

Chapter 10

Nonlinear Response in a Rotor System With a Coulomb Spline C. Nataraj and Karthik Kappaganthu

Abstract This chapter deals with the nonlinear analysis of a system with two rigid rotors connected by a spline coupling and mounted on isotropic bearings. The coupling has Coulomb friction. The system is found to have three unstable fixed points and a limit cycle above a certain critical speed. The response of the system at the limit cycle indicates transient chaos, which is usually an artifact of the crisis. The crisis route to chaos has been analyzed and a strange attractor is found.

10.1 Introduction Figure 10.1 illustrates a typical spline coupling used to drive a rotating shaft from a motor or other prime mover. Under insufficient lubrication conditions, such a coupling has been found to exhibit Coulomb friction. We will hence call this spline a “Coulomb spline.” Past analyses of such systems are very limited. Nataraj et al. [1] discovered that such a system can exhibit limit cycles under certain conditions. In fact, this was verified with a practical installation as well. In this chapter, the nonlinear dynamic behavior of the system is studied further. We will assume a rigid rotor model that simplifies some of the dynamics; such a model is valid for rotors operating below the first critical speed. First, the fixed points of the system and their stability are analyzed, then the possibility of creation or destruction of fixed points with changes in the spin speed is studied. A limit cycle is identified using analytical methods. This limit cycle exists only above a certain critical spin speed; this is the bifurcation point in the system. The numerical simulation of this limit cycle suggests the presence of transient chaos. Transient chaos and its cause is explained in [2]. In a paper published later by the same authors [3], they discuss the dependence of average lifetime of the system on the parameter. They suggest that transient chaos is one of the effects of crisis. They analyzed crisis and transient chaos in three different continuous systems C. Nataraj () Department of Mechanical Engineering, Villanova University, Villanova, PA, USA e-mail: [email protected] A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 10,

105

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Fig. 10.1 Spline coupling

including the Lorenz system. Transient chaos has also been observed in many other systems like Sine-Gordon system [4], Gear-Rattling model [5], Plasmas [6], and experimentally in [7]. In the next section, the mathematical model of the system is described, and the following section discusses the nonlinear phenomenon observed.

10.2 Mathematical Modeling The system being considered consists of two rigid rotors coupled by a Coulomb spline. A schematic of the system is shown in Fig. 10.2. The equations of motion of the system as derived in [8] are M qR C .D ˝G/qP C Kq D F;

(10.1)

where M , D, G, and K are the mass, damping, gyroscopic, and stiffness matrices respectively, given by: M D DD GD

m 0 c 0 0 Ip

0 m 0 c Ip 0

10

Nonlinear Response in a Rotor System With a Coulomb Spline l1

107

l2

y

V Ω x

W

Fig. 10.2 Schematic diagram of the system

k KD y 0

0 kz

q D ŒV W T is the vector of displacements of the point of interconnection along y and z axes respectively. The parameters of the system are:

m: Equivalent mass Ip : Equivalent polar moment of inertia c: External viscous damping coefficient ky and kz : Equivalent stiffness in y and z directions respectively ˝: Spin speed of the rotor The force acting due to the spline coupling is F D

Ci c .1 C l1 = l2 /

1=2 .WP ˝V /2 C .VP ˝W /2 T P sin k Ci c D ns

VP C ˝W WP ˝V

(10.2) (10.3)

where is the coefficient of friction l1 ; l2 are the lengths of the shaft, and k is the angular position of kth spline. The equations should be nondimensionalized to identify key parameters. The nondimensional form of the equation is

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0

Mn qn C .Dn ˝n Gn /qn C Kn qn D Fn ; where Mn D Dn D Gn D Kn D

1 0 2 0 0 ˛ 1 0

0 1

(10.4)

0 2

˛ 0 0 2

qn D ŒV = l1 W= l1 T

(10.5)

This nondimensional q form has been q derived using the substitutions D !y t; I ky kz !z ˝ pc ˝n D !y , !y D ; ˛ D mlp2 . is the m ; !z D m ; D !y ; D 2 km

orthotropic parameter. The nondimensional force is given by [8] Fn D

.w0 ˝n v/2 C .v0 C ˝n w/2 D 0:64T

1=2

0 v C ˝n w 0 w ˝n v

1 C l1 = l2 m!y2 l12

1

(10.6)

(10.7)

10.3 Analysis In this chapter, the nonlinear behavior of the isotropic system is analyzed, i.e, with is equal to one. The nondimensional form of the equation, (10.4) is used for the analysis. The only nonlinearity in the system is from the force exerted by the spline. The coupling between the two differential equations in (10.4) is primarily due to the gyroscopic matrix. The system is studied for variation in response with changes in the nondimensional spin speed, ˝n . The system has fixed points at v1 D 0; v2 D p ; 1 C 2

w1 D 0; w2 D

p 1 C 2

(10.8) (10.9)

10

Nonlinear Response in a Rotor System With a Coulomb Spline

v3 D p ; 1 C 2

w3 D p 1 C 2

109

(10.10)

No fixed points are created or destroyed with changes in ˝n ; further, the eigenvalues of the system linearized about each of the fixed points suggest that each of these points is unstable for all values of ˝n . Owing to the isotropy, the limit cycle, if it exists, must be a circular orbit in y–z plane. For such a solution, v D Ao cos !n and w D Ao sin !n . Substituting these into (10.4) and solving ˝n < !n No limit cycle exists ˝n > !n Limit cycle exists; with s ˝n ˛ !n D C 2 Ao D

1C

˝n ˛ 2

(10.11)

2

: 2!n

(10.12) (10.13)

The critical value of the spin speed from (10.11), (10.12) is given by 1 ˝n; critical D p 1˛

(10.14)

The limit cycle and the transients obtained at ˝n D 1:3 are shown in Figs. 10.3–10.7. The parameters used for the simulation are given in Table 10.1. For these chosen values, ˝n; critical D 1:1952. The bifurcation diagram is shown in Fig. 10.8. It has also been observed that, but for certain initial conditions close to the limit cycle, the system settles into a stable limit cycle only after very large chaotic transients. These transients are shown in Figs. 10.9–10.13. The presence of such transient chaos suggests the presence of strange attractors and crisis. The presence of crisis as a possible route of chaos is explained in [2] and [3]. The phenomenon of crisis occurs when there is a “collision between a chaotic attractor and a coexisting unstable fixed point or periodic orbit” [2]. In system,

this w the crisis occurs because of the presence of an unstable fixed points at v 2 2 and

v3 w3 . Numerical simulations about v1 w1 show that it is a strange attractor. Different ODE solvers in MATLAB give similar results. The system response at subcritical speeds is shown in Figs. 10.14–10.17. A zoomed in view of the response in v and w is shown in Figs. 10.18 and 10.19 respectively. The response seems to divide itself, however it is difficult to establish the presence of fractals in the four dimensional space. In order to rule out the possibility of a quasi periodic orbit, the volume of the state space is analyzed. Proceeding as in [9], the state space form of (10.4) is xP D f.x/;

(10.15)

110

C. Nataraj and K. Kappaganthu

0.08 0.06 0.04

w

0.02 0 −0.02 −0.04 −0.06 −0.05

0

0.05

0.1

v Fig. 10.3 Orbit in a limit cycle

0.1 0.08 0.06 0.04

vdot

0.02 0 −0.02 −0.04 −0.06 −0.08 −0.1

−0.05

0

0.05 v

Fig. 10.4 Phase plane v vs. vP

0.1

10

Nonlinear Response in a Rotor System With a Coulomb Spline

111

0.15

0.1

wdot

0.05

0

−0.05

−0.1 −0.08 −0.06 −0.04 −0.02

0

0.02

0.04

0.06

0.08

0.1

w

Fig. 10.5 Phase plane w vs. w P

0.3

v

0.2 0.1 0 −0.1

0

50

100

150

200

250

150

200

250

τ 0.1

w

0.05 0 −0.05 −0.1 0

50

100 τ

Fig. 10.6 Variation of v and w

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C. Nataraj and K. Kappaganthu 0.1

vdot

0.05 0 −0.05 −0.1 0

50

100

150

200

250

150

200

250

τ 0.3

wdot

0.2 0.1 0 −0.1 0

50

100

τ Fig. 10.7 Variation of vP and w P Table 10.1 Parameter values

Parameter

Value

˛

0.3 0.1 0.01867 1

where i h 0 0 T xD vwv w 2

0

(10.16) 3

v 0 6 7 w 6 7

0 6 7 6 7 v C ˝ w n 0 0 6 7 6 2v ˝n ˛w v h 7 i 1=2 7 6 2 2 f D6 7: w0 ˝n v C v0 C ˝n w 6 7

6 7 0 6 7 w ˝ v n 0 0 6 7 2 42w ˝n ˛v w h i 2 0 2 1=2 5 0 w ˝n v C v C ˝n w (10.17) Using divergence theorem, the volume rate of the state space is given by Z 0 V D gradf dV : (10.18)

10

Nonlinear Response in a Rotor System With a Coulomb Spline

113

0.08 0.075 0.07

Amplitude

0.065 0.06 0.055 0.05 0.045 0.04

0

0.5

1

1.5

2

2.5

Ωn

Fig. 10.8 Bifurcation diagram

0.06 0.04

w

0.02 0 −0.02 −0.04 −0.06

−0.08 −0.06 −0.04 −0.02

0

0.02

0.04

0.06

0.08

v Fig. 10.9 Transient chaos followed by a limit cycle

For the system to exhibit a chaotic motion, the volume in the state space should shrink exponentially fast [9], and there should be no stable fixed points or limit cycles. The second part of the condition has already been shown. To look at the

volumes, the system is linearized about v1 w1 . The linearization matrix of the system is as shown in (10.19).

114

C. Nataraj and K. Kappaganthu 0.1 0.08 0.06 0.04

vdot

0.02 0 −0.02 −0.04 −0.06 −0.08 −0.1 −0.08 −0.06 −0.04 −0.02

0

0.02

0.04

0.06

0.08

v Fig. 10.10 Phase plane in transient chaos (v vs. vP /

0.1 0.08 0.06 0.04 wdot

0.02 0 −0.02 −0.04 −0.06 −0.08 −0.1 −0.08 −0.06 −0.04 −0.02

0 w

0.02

0.04

0.06

0.08

Fig. 10.11 Phase plane in transient chaos (w vs. w/ P

2

3 0 0 1 0 6 7 0 0 0 1 7; AD6 41 C L1 L2 2 C L3 ˝n ˛ C L4 5 L5 2 C L6 ˝n ˛ C L7 2 C L8

(10.19)

10

Nonlinear Response in a Rotor System With a Coulomb Spline

115

0.1 0.05 v

0 −0.05 −0.1

0

200

400

600

800

1000

600

800

1000

τ 0.1 0.05 w

0 − 0.05 −0.1

0

200

400 τ

Fig. 10.12 Variation of v and w in transient chaos

0.1

vdot

0.05 0

−0.05 −0.1

0

200

400

0

200

400

τ

600

800

1000

600

800

1000

0.1

wdot

0.05 0

− 0.05 −0.1

τ Fig. 10.13 Variation of vP and w P in transient chaos

116

C. Nataraj and K. Kappaganthu x 10−7 2 1.5 1 0.5

w

0 −0.5 −1 −1.5 −2 −2

−1

0

1

2 x 10−7

v Fig. 10.14 Chaotic response

3

x 10−5

2

vdot

1

0

−1

−2

−3 −2.5

−2

−1.5

−1

−0.5

0 v

0.5

1

1.5

2

2.5 x 10−7

Fig. 10.15 Phase plane view of chaos (v vs. vP /

where L1 D

˝n .wd ˝n v/ .vd C ˝n w/ 3=2 ; .wd ˝n v/2 C .vd C ˝n w/2

(10.20)

10

Nonlinear Response in a Rotor System With a Coulomb Spline 3

117

x 10−5

2

wdot

1 0 −1 −2 −3 −2.5 −2 −1.5 −1 −0.5

0 w

0.5

1

1.5

2

2.5 x 10−7

Fig. 10.16 Phase plane view of chaos (w vs. w/ P

4

x 10−7

v

2 0 −2 −4 0

100

200

300

400

500

300

400

500

τ 4

x 10−5

vdot

2 0 −2 −4 0

100

200 τ

Fig. 10.17 Variation of v and vP in chaos

L2 D

˝n .wd ˝n v/2 .wd ˝n v/2 C .vd C ˝n w/2

3=2 ;

(10.21)

118

C. Nataraj and K. Kappaganthu 1.5

x 10−7

1

w

0.5 0 −0.5 −1 −1.5 −1.5

−1

−0.5

0 v

0.5

1

1.5 x 10−7

Fig. 10.18 Zoomed view of the chaotic orbit

x 10−7

1

v

0.5

0

−0.5

−1 270

271

272

τ

273

274

275

Fig. 10.19 Zoomed view of the variation of v

L3 D

L4 D

.wd ˝n v/2 .wd ˝n v/2 C .vd C ˝n w/2 .wd ˝n v/ .vd C ˝n w/ 2

.wd ˝n v/ C .vd C ˝n w/

L5 D

2

3=2 ;

˝n .wd ˝n v/2 2

3=2 ;

.wd ˝n v/ C .vd C ˝n w/

2

3=2 ;

(10.22)

(10.23)

(10.24)

10

Nonlinear Response in a Rotor System With a Coulomb Spline

L6 D

L7 D

119

˝n .wd ˝n v/ .vd C ˝n w/ 3=2 ; .wd ˝n v/2 C .vd C ˝n w/2 .wd ˝n v/ .vd C ˝n w/ .wd ˝n v/2 C .vd C ˝n w/2

L8 D

3=2 ;

.vd C ˝n w/2 .wd ˝n v/2 C .vd C ˝n w/2

3=2 :

(10.25)

(10.26)

(10.27)

The trace of the A matrix is given by (10.28). Since the second part of the equation is less than zero, according to the comparison lemma [10], the volume relation is given by (10.29). Hence, the volume implodes with time and the response is chaotic. (10.28) tr.A/ D 4

1=2 .wd ˝n v/2 C .vd C ˝n w/2 V ./ < V .0/e4

(10.29)

Finally, Fig. 10.20 shows the convergence of the largest Lyapunov exponent for the parameter values listed in Table 10.1; the converged value is 0.1483.

2

Largest Lyapunov Exponent

0 −2 −4 −6 −8 −10 −12

0

200

400

600

800

1000 1200 1400 1600

Iteration Number

Fig. 10.20 Largest Lyapunov exponent

120

C. Nataraj and K. Kappaganthu

10.4 Conclusion This chapter analyzed the nonlinear behavior of a system with two rigid rotors connected by a Coulomb spline and mounted on isotropic bearings. The fixed points and limit cycles were identified analytically. The transient chaos exhibited near the limit cycle indicated crisis and chaos. Owing to the higher dimensionality of the system numerical analysis alone was performed. The strange attractor at the origin was analyzed and chaos has been established satisfactorily within the limitations of numerical simulations.

References 1. Nataraj C, Nelson HD, Arakere N (1985) Effect of coulomb spline on rotor dynamic response. Washington, DC, pp 225–233 2. Grebogi C, Ott E, Yorke JA (1983) Crisis, sudden changes in chaotic attractors and transient chaos. Physica D 7:181–200 3. Grebogi C, Ott E, Yotke JA (1986) Critical exponent of chaotic transients in nonlinear dynamical systems. Phys Rev Lett 57(11):1284–1287 4. Bartuccelli M, Christiansen PL (1986) Horseshoe chaos in the space-independent double sinegordon system. Wave Motion 8:581–594 5. de Souza SLT, Caldas IL (2001) Basins of attraction and transient chaos in a gear-rattling model. J Vib Control 7(6):849–860 6. He K (2003) Critical phenomenon, crisis and transition to spatiotemporal chaos in plasmas. Space Sci Rev 107:475–494 7. Stouboulos IN, Miliou, AN, Valaristos AP, Kypriandis IM, Anagnostopoulos AN (2007) Crisis induced intermittency in a fourth-order autonomous electric circuit. Chaos Solitons Fractals 33(4):1256–1262 8. Nataraj C (1987) Periodic oscillations in mechanical systems with nonlinear components. PhD thesis, Arizona State University 9. Strogatz SH (2000) Nonlinear Dynamics and Chaos. Westview Press, Boulder County, CO 10. Khalil HK (2002) Nonlinear systems. Prentice Hall, Upper Saddle River, NJ

Chapter 11

The Influence of the Cross-Coupling Effects on the Dynamics of Rotor/Stator Rubbing Zhiyong Shang, Jun Jiang, and Ling Hong

Abstract In this chapter, the influence of cross-coupling effects on the rubbingrelated dynamics of rotor/stator systems is investigated. The model considered in this chapter is a four-dof rotor/stator system, which takes into account the dynamics of the stator and the deformation on the contact surface as well as the cross-coupling effects. The stability of the synchronous full annular rub solution of the model is first analyzed. Then, the cross-coupling effects on the stability of the system at different system parameter planes are studied. It is found that the cross-coupling damping of the stator benefits the synchronous full annular rubs and that of the rotor has a little influence on the response. While the cross-coupling stiffness of the stator always reduces the stability domain of the response, the cross-coupling stiffness of the rotor may either increase or decrease the stability domain depending upon its value.

11.1 Introduction In order to improve the efficiency, the gap between the rotor and the stator of a rotating machine is setting smaller and smaller, while the rotor/stator rubbing is becoming more and more common. Rotor/stator rub is a serious malfunction in the operation of a rotating machine, which can seriously degrade the machine performance and can even lead to disastrous consequences of the machine. There are a large amount of works on the rub-related phenomena in the rotor/stator systems in order to get deep insights into the dynamical behaviors and their relationship with system parameters. It has now become well known that rotor-to-stator rubbing can induce periodic synchronous full annular rubs [1–4], sub and superharmonic motions [5, 6], quasiperiodic partial rubs [7, 8], chaotic responses [9, 10] as well as the destructive self-excited dry friction backward whirl [2, 11, 12]. Since rotor-to-stator

J. Jiang () MOE Key Laboratory for Strength and Vibration, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China e-mail: [email protected]

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rubbing is usually caused by other malfunctions that induce large rotor deflections covering the clearance between the rotor and the stator, an effective way to prevent from the rotor-to-stator rubbing is to suppress the vibration amplitude of the rotor, i.e., through active magnetic bearing or active controlled journal bearings [13–15]. When the rotor/stator rubbing is unavoidable, i.e., without the vibration suppression measures or due to the failure of the suppression measures in the extreme running condition, it is still highly expected that the rotating machine can adapt to the changing environments and optimize its performance to reduce the rubbing induced degradation and, especially, to avoid the occurrence of the destructive instability. In this chapter, the influences of the cross-coupling effects on the rotorto-stator rubbing will be investigated. The aims are twofolds: at one hand, to take into account more realistic effects in the rotor/stator model and study their influence on the rub-related behaviors of the system; on the other hand, to examine the possibility to use these cross-coupling effects to develop new control approaches in order to reduce the rubbing severity. The chapter is arranged as follows. In Sect. 11.2, the model of the rotor/stator system is introduced. The stability of the synchronous full annular rub solution is carried out in Sect. 11.3. The influence of cross-coupling effects on the stability of the synchronous full annular rub solution is studied in Sect. 11.4. Finally in Sect. 11.5, the conclusions of this work are drawn.

11.2 The Rotor/Stator Model with Cross-Coupling Effects The schematic of the rotor/stator model studied in this chapter is shown in Fig. 11.1. A weightless shaft supported by two ideal bearings has effective transverse stiffness kr and rotates at an angular speed !. A rigid disk of mass mr is mounted at the midpoint of the shaft. Concentric with the disk is an annular stator (an auxiliary bearing) of mass ms . The stator is elastically supported by a symmetrical set of springs with isotropic radial stiffness ks . A clearance ı exists between the rotor and the stator. To consider the deformation at the contact surface between the rotor and stator, a symmetrical set of fictive springs with isotropic radial stiffness kc is

Fig. 11.1 Left: the schematic plot of the rotor-to-stator system; Right: the section view on the plane of the rotor and the stator ring

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assumed being laid in the inner ring of the stator to model the contact stiffness. The rotor also possesses a mass eccentricity of e. The equations that govern the motion the rotor/stator system in the complex form can be written as: mr rRr C .cr jr /Prr C .kr jQr /rr C F D mr e! 2 ej!t ms rRs C .cs js /Prs C .ks jQs /rs F D 0

s F D kc .1 C j/ rr rs ı jrrrr r rs j ;

(11.1)

where rr D yr C jzr and rs D ys C jzs are the complex deflections and cr and cs the damping of the rotor and the stator, respectively. F represents the resultant contact force on the contact surface. is the friction coefficient. Besides, the cross-coupling effects represented by the cross-coupling damping terms, with r and s for the rotor and the stator, respectively, and the cross-coupling stiffness terms, with Qr and Qs for the rotor and the stator, respectively, are also included in the model. Additionally, we define D 1, if jrr rs j ı and D 0, if jrr rs j < ı. Equation (11.1) can be formulated into the nondimensional form as rOr00 C .2r jr /Orr0 C .1 jr /Or C FO D ˝ 2 ej ; p Msr rOs00 C .2s Msr ˇsr js /Ors0 C .ˇsr js /Ors FO D 0; rOr rOs FO D ˇcr .1 C j/ rOr rOs ; jOrr rOs j

(11.2)

where0 represents thepdifferentiation with respect to the nondimensional time D !0 t with !0 D kr =mr being the natural frequency of the rotor. The other nondimensional variables are defined as following: F ! rr rs ms ks kc ı ; rOs D ; FO D ; Msr D ; ˇsr D ; ˇcr D ; D ; ˝ D ; e e ekr mr kr k e !0 p r s Msr ˇsr cr cs r Qr Qs r D p ; s D p ; r D p ; s D p ; r D ; s D kr kr 2 kr mr 2 ks ms kr mr ks ms rOr D

11.3 The Solution and the Stability Analysis It is well known that for a linear rotor system without rubbing, the cross-coupling stiffness may induce the instability of the rotor when the cross-coupling stiffness goes above a critical value. It has been shown that for a rotor system with rubbing, the dry friction at the contact surface may induce the instability of the rotor/stator system when the dry friction exceeds a critical value [12]. It is, therefore, of great interest to investigate the interaction effect between the cross-coupling effects and the dry friction on the rubbing behaviors of the rotor/stator system.

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When D 1 in (11.1) or (11.2), the governing equation has a steady-state periodic solution with constant amplitude and frequency that equals to the rotating speed of the rotor. This solution is called a synchronous full annular rub solution. As known that only a stable solution corresponds to a physical observable response, so the stability of the solution is of great interest. It is known that the destructive dry friction backward whirl of a rotor/stator system always occurs after the synchronous full annular rub loses stability. The stability analysis will provide useful information for the design of the rotor/stator system.

11.3.1 Synchronous Full Annular Rub Solution To carry out the stability analysis, the explicit form of the solution must be first available. By doing so, an equivalent form of the resultant contact force is adopted [16] 1 0 O 1 j F A ˇ ˇ : (11.3) FO D ˇcr .1 C j/ @rOr rOs p 1 C 2 ˇˇFO ˇˇ Since the synchronous full annular rub solution has a frequency equal to the rotating speed, the stability analysis of the solution will become easier when (11.2) is transformed into a rotating coordinate system with the frequency equal to the rotating speed of the rotor. To do so, we introduce the transformations rOr D r e jt ; rOs D s e jt ;

FO D ˚e jt :

(11.4)

After substituting (11.4) into (11.2), we get the equations in the rotating coordinates as

r00 CŒ2r C j.2˝ r / r0 CŒ1 ˝ 2 C˝r C j.2˝r r / r D ˝ 2 ˚; Msr s00 CŒ2s Cj.2˝Msr s / s0 CŒˇsr ˝ 2 Msr C˝s Cj.2˝s ks / s D ˚; ˚ ˚ D ˇcr .1 C j/ r s p1j 2 j˚j ; (11.5) 1C

p where s D s Msr ˇsr . Using Cr und Cs to denote the dynamical stiffness of the rotor and the stator, Cr D 1 ˝ 2 C ˝r C j.2˝r r /; Cs D ˇsr ˝ 2 Msr C ˝s C j.2˝s s /. Since the amplitude of the solution we seek is constant, the terms containing differentiation with respect to will be cancelled. In this way the algebraic equations on the complex amplitudes of the rotor and the stator as well as the corresponding resultant contact force are obtained as

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Cr r D ˝ 2 ˚ Cs s D ˚ ˚ D ˚: .1 C j/ˇcr r s p1j 2 j˚j 1C

(11.6)

It is seen that the equations governing the motion of the rotor and the stator are linear. However, the system with rubbing becomes nonlinear due to the presence of the resultant contact force. In this case, the complex amplitudes of the rotor and the stator are the functions of the resultant contact force. From the first two equations of (11.6), we derive

r D .˝ 2 ˚/=Cr : (11.7)

s D ˚=Cs After substituting (11.7) back into the third equation of (11.6) and doing some manipulation, the magnitude of the resultant contact force can be written as q .1 C R1 /K ˙ Œ.1 C R1 /2 C I12 .R22 C I22 / K 2 I12 ; (11.8) j˚j D .1 C R1 /2 C I12 p where K D ˇcr 1 C 2 is a real number and Cr C Cs ˝2 R1 D < .1 C j/ˇcr ; R2 D < .1 C j/ˇcr ; Cr Cs Cr Cr C Cs ˝2 ; I2 D = .1 C j/ˇcr I1 D = .1 C j/ˇcr Cr Cs Cr with <./ and =./ standing for the real and the imaginary part of a complex number. The resultant contact force is then given by .1 C j/ˇcr ˚D 1C

˝2 Cr

Cr C Cs K C .1 C j/ˇcr Cr Cs j˚j

:

(11.9)

Substituting (11.9) into (11.7), the complex amplitudes of the rotor, r , and the stator, s , can be determined. The synchronous full annular rub solution is then obtained by substituting r and s into (11.4), which should also meet the condition j r s j .

11.3.2 Stability Analysis To carry out the stability analysis on the synchronous full annular rub solution, the complex amplitude of the rotor and the stator will be first written in their component form as:

r0 D Yr0 C jZr0 D Br exp.j˛/; s0 D Ys0 C jZs0 D Bs exp.jˇ/:

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The synchronous full annular rub solutions are in the form rOr0 D r0 ej D Br ej.C˛/ D yr0 ./ C jzr0 ./; rOs0 D s0 ej D Bs ej.Cˇ / D ys0 ./ C jzs0 ./ Defining the state vector in the inertial coordinate system oT n x./ D yr zr ys zs yr0 z0r ys0 z0s : Equation (11.2) can be written as first-order differential equations in the state space xP D Ax C g.x; / D G.x; /;

(11.10)

where A and g(x; ) are the coefficient matrix and the nonlinear vector as 2

3 0 0 0 0 1 0 0 0 6 0 0 0 0 0 1 0 0 7 6 7 6 0 0 0 0 0 0 1 0 7 6 7 6 7 0 0 0 0 0 0 0 1 7 6 AD6 7 6 .1 C ˇcr / .r ˇcr / ˇcr ˇcr 2r r 0 0 7 6 7 6 r ˇcr .1 C ˇcr / ˇcr ˇcr r 2r 0 0 7 6 7 4 Bcr Bcr .Bsr C Bcr / .Ks Bcr / 0 0 2„s Gs 5 Bcr Bcr Ks Bcr .Bsr C Bcr / 0 0 Gs 2„s

T ˚ and g.x; /D 0 0 0 0 ˇ cr Hy C˝ 2 cos ˝ ˇ cr Hz C˝ 2 sin ˝ Bcr Hy Bcr Hz . The variables above are defined as followings Bsr D ˇsr=Msr ; Bcr D ˇcr=Msr ; „r D r=Msr ; „s D s=Msr ; Gs D

s s ; Ks D Msr Msr

Hy D =RŒ.yr ys / .zr zs /; Hz D =RŒ.yr ys / C .zr zs / p with R D .yr ys /2 C .zr zs /2 : After linearizing (11.10) about the synchronous full annular rub solution, which is denoted by ˚ x0 D Br cos '˛ Br sin '˛ Bs cos 'ˇ Bs sin 'ˇ ˝ Br sin '˛ T ˝Br cos '˛ ˝Bs sin 'ˇ ˝Bs cos 'ˇ ; with '˛ D ˝ C˛; 'ˇ D ˝ Cˇ, a set of time-variant linear differential equations is obtained (11.11) ı xP D ŒJ.x0 ; /ıx;

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where J.x0 ; / 88

ˇ ˇ D @G @x ˇ

xDx0

ˇ ˇ D A C @g @x ˇ

xDx0

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is the so-called Jacobian matrix.

ıx D x x is the perturbation to the synchronous full annular rub solution. Since the Jacobian matrix is periodic time-dependent, it cannot be directly used to derive the information of stability of the analyzed solution. We thus make the following transformation: (11.12) ıx D ŒT ıu; 0

where 2

Œt1 0 0 6 0 Œt2 0 ŒT D 6 4 0 0 Œt1 0 0 0

3 0 0 7 7 ; Œt1 D cos '˛ sin '˛ ; Œt2 D sin 'ˇ cos 'ˇ : 0 5 sin '˛ cos '˛ cos 'ˇ sin 'ˇ Œt2

Substituting (11.12) into (11.11) and after some simple manipulation, we get ıu0 D ŒJn ıu;

(11.13)

where [Jn ] D ŒT1 .ŒJ.x 0 ; £/ŒT ŒT0 /. The stability of the synchronous full annular rub solution can now be determined through the examination of the sign of the real parts of the characteristic roots of the Jacobian matrix, , which is solved from ŒŒJn I88 D 0:

(11.14)

The solution is stable if all the real parts of its characteristic roots are less than zero. Otherwise it is unstable.

11.4 The Cross-Coupling Effects on the Stability To demonstrate the influence of the cross-coupling effects on the stability of the synchronous full annular rub solution, the stability domains of the solution are drawn on different parameter planes at different cross-coupling coefficients. In the following analysis, some system parameters are fixed: r D 0:05; s D 0:05; ˇcr D 200:0; D 2:0. In the following figures, two solid curves SNl and SNr define the lower and the upper existence boundaries of the synchronous full annular rub solution. Outside the region enclosed by SNl and SNr , the synchronous no-rub response always exists. Inside the region enclosed by SNl and SNr , the dotted curves with different symbols show the stability boundaries (Hopf bifurcation) of the synchronous full annular rub solution at different cross-coupling coefficients. Outside each stability domain of the synchronous full annular rub solution, the quasiperiodic partial rub response or the dry friction backward whirl response with heavy or destructive rubbing severity may appear.

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11.4.1 Stability Domains in the Plane of ˝ In this case, the mass ratio and the stiffness ratio are fixed to Msr D 0:2; ˇsr D 2:0. The stability domains of the synchronous full annular rub solutions in the parameter plane of ˝ at different cross-coupling coefficients are studied. Figure 11.2 demonstrates the influence of the cross-coupling stiffness on the stability domains. In Fig. 11.2 (left), it is found that with the increase of the crosscoupling stiffness of the rotor, r , from 0.0 to 0.4, both the lower and the upper existence boundaries of the solution move rightward. So the existence region of the solution has almost no change with the increase of r while the stable domain shrinks. In fact, the increase of the stable domain may be observed when r is tuned between 0.0 and 0.2. As pointed out in [3], the dry friction and the cross-coupling stiffness of the rotor play contrary roles in stabilizing the synchronous full annular rub solution and both can destabilize the solution. So when the two parameters are well balanced for the given system parameters, the synchronous full annular solution is stable. As either of the two parameters is too large to be compensated by the other one, the solution will become unstable. In Fig. 11.2 (left), the lower branches of the stability boundary at r D 0:2 and r D 0:4 are due to the too large crosscoupling stiffness of the rotor. In Fig. 11.2 (right), it is easily seen that the increase of the cross-coupling stiffness of the stator has no influence on the lower boundary of the existence region of the synchronous full annular solution, but will shift the upper boundary to the right. So the increase of s will enlarge the existence region of the solution. Meanwhile it will also monotonically reduce the stability domain of the solution, indicating the disadvantage for the appearance of the cross-coupling stiffness of the stator in the rotor-to-stator contact systems. The influence of the cross-coupling damping on the stability of the synchronous full annular rub solution is demonstrated in Fig. 11.3. It is found from Fig. 11.3 (left)

Fig. 11.2 Stability chart of the synchronous full annular solution in the parameter plane of ˝ at different cross-coupling stiffness coefficients. Left: in the case that s D r D s D 0, where (inverted open triangle) stands for the stability boundary at r D 0, (open circle) at r D 0:2, (cross) at r D 0:4. Right: in the case that r D r D s D 0, where (inverted open triangle) stands for the stability boundary at s D 0, (open circle) at s D 0:2, (cross) at s D 0:4

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Fig. 11.3 Stability chart of the synchronous full annular solution in the parameter plane of ˝ at different cross-coupling damping coefficients. Left: in the case that r D s D s D 0, where (inverted open triangle) stands for the stability boundary at r D 0, (open circle) at r D 0:2, (cross) at r D 0:4. Right: in the case that r D s D r D 0, where (inverted open triangle) stands for the stability boundary ats D 0, O at s D 0:2, (cross) at s D 0:4

that the cross-coupling damping of the rotor, r , has a little influence on the size of the existence region of the synchronous full annular rub solution when r changes from 0.0 to 0.2 then to 0.4. But it has almost no effect on the stability domain of the solution. The effect of the cross-coupling damping of the stator is significant in comparison. First, it enlarges the existence region of the full annular rub solution by keeping the lower boundary unchanged and moving the upper boundary rightward. Secondly, it increases the stability domain of the solution remarkably. So the cross-coupling damping of the stator will benefit the synchronous full annular rub response by avoiding the contact severity between the rotor and the stator through a synchronous full annular rub.

11.4.2 Stability Domains in the Plane of ˝ˇsr Below the mass ratio and the friction coefficient are fixed to Msr D 0:2; D 0:10. The cross-coupling damping of the rotor, r , on the stability domains of the synchronous full annular rub solution in the parameter plane of ˝ ˇsr are explored. It is noticed from Fig. 11.4 that the speed range by which the synchronous full annular rub solution exists increases significantly with the increase of the stiffness ratio between the stator and the rotor. Meanwhile, the speed range of the stable synchronous full annular rub solution becomes large when the stiffness ratio ˇsr is reduced, indicating that small stiffness ratio between the stator and the rotor may benefit the system to have a relatively large range of mild rubbing in the case that rubbing between the rotor and the stator is unavoidable. The influence of the cross-coupling damping of the rotor on both the existence region and the stable domain of the full annular rub solution is seen to be very little.

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Fig. 11.4 Stability chart of the synchronous full annular solution in the parameter plane of ˝ ˇsr at different cross-coupling damping of the rotor in the case that r D s D s D 0, where (inverted open triangle) stands for the stability boundary at r D 0:0, (open circle) at r D 0:4, (cross) at r D 0:8

With the increase of r , the lower and the upper boundaries of the solution move simultaneously to the right, while the stability boundaries of the solution shift in the similar manner to produce seemingly unaltered stable domains.

11.4.3 Stability Domains in the Plane of ˝Msr To study the stability domains of the synchronous full annular rub solution in the parameter plane of ˝Msr , the stiffness ratio and the friction coefficient are fixed to ˇsr D 2:0; D 0:10. From Fig. 11.5 it is seen that the influence of the mass ratio on the existence region of the synchronous full annular rub solution is small. It is, however, observable that the speed range of the stable synchronous full annular rub solution will take its maximal value around Msr D 0:2. In order to get an overall view on the influence of the cross-coupling effects on the stability of the synchronous full annular rub solution, the stability domains of the solution under the influence of the cross-coupling effects of the rotor and of the stator are drawn in Fig. 11.5 (left) and (right), respectively. It is found that the cross-coupling stiffness of the rotor slightly enlarges the stability domain in this case and the cross-coupling damping of the rotor reduces it as mentioned above (see Fig. 11.5 (left)). On the other hand, the increase of the cross-coupling damping of the stator significantly enlarges the stable domain of the synchronous full annular rub solution, while the increase of the cross-coupling stiffness of the stator may largely reduce the stable domain of the solution, as shown in Fig. 11.5 (right).

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Fig. 11.5 Stability chart of the synchronous full annular solution in the parameter plane of ˝Msr . Left: under the cross-coupling effects of the rotor with s D s D 0, where (inverted open triangle) stands for the stability boundary at r D r D 0:0; (open circle) at r D 0:0; r D 0:3; (cross) at r D 0:3; r D 0:0. Right: under the cross-coupling effects of the stator with r D r D 0, where (inverted open triangle) stands for the stability boundary at s D s D 0:0; (open circle) at s D 0:3 s D 0:0; (cross) at s D 0:0; s D 0:3

11.5 Conclusions With the goal to understand the influence of cross-coupling effects on the rubbingrelated dynamics of rotor/stator systems, a model for a rotor/stator system including both the dynamics of the stator and the deformation on the contact surface as well as the cross-coupling damping and stiffness is set up in this chapter. After solving the synchronous full annular rub solution of the model, the stability analysis of the solution is then carried out. Our results show that except the cross-coupling damping of the rotor, the other three cross-coupling terms all have influence on the stability domains of the synchronous full annular rub solution. While the cross-coupling stiffness of the stator can monotonically reduce the stability domain, the cross-coupling stiffness of the rotor has some optimal value that may achieve a maximal stability domain for the given system parameters. The cross-coupling damping of the stator will benefit the rotor-to-stator contact system through significantly enlarging the stable domain of the synchronous full annular rub solution. Since the synchronous full annular rub is a mild rub in comparing with the heavy rub, such as the quasiperiodic partial rubs and the destructive self-excited dry friction backward whirls. So an appropriate adaptation of the cross-coupling effects that can stabilize the synchronous full annular rubs in the rotor/stator system may prevent system from quick damages under rubbing. Acknowledgments The authors are grateful for the financial support by the National Natural Science Foundation of China (NSFC) under the grant No. 10872155, 10472086 and 10772140.

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References 1. Muszynska A (1989) Rotor-to-stationary element rub-related vibration phenomena in rotating machinery – literature survey. Sound Vib Dig 21:3–11 2. Black HF (1968) Interaction of a whirling rotor with a vibrating stator across a clearance annulus. Int J Mech Eng Sci 10:1–12 3. Jiang J, Ulbrich H (2001) Stability analysis of sliding whirl in a nonlinear jeffcott rotor with cross-coupling stiffness coefficients. Nonlinear Dyn 24:269–283 4. Yu JJ, Goldman P, Bently DE, Muszynska A (2002) Rotor/seal experimental and analytical study on full annular rub. ASME J Gas Turbine Power, 124:340–350 5. Childs DW (1979) Rub-induced parametric excitation in rotors. J Mech Design 101:640–644 6. Ehrich FF (1988) High order sub-harmonic response of high speed rotors in bearing clearance. ASME J Vib Acoust Stress Reliab 110:9–16 7. Day WB (1987) Asymptotic expansions in nonlinear rotor-dynamics. Quart Appl Math 44:779–792 8. YB Kim, Noah ST (1996) Quasi-periodic response and stability analysis for a nonlinear jeffcott rotor. J Sound Vib 190:239–253 9. Choi SK, Noah ST (1994) Mode-locking and chaos in a Jeffcott rotor with bearing clearances. ASME J Appl Mech 61:131–138 10. Abdul Azeez MF, Vakakis AF (1999) Numerical and experimental analysis of a continuous overhung rotor undergoing vibro-impacts. Int J Nonlinear Mech 34:415–435 11. Choi YS (2002) Investigation on the whirling motion of full annular rotor rub. J Sound Vib 258:191–198 12. Jiang J, Ulbrich H (2005) The physical reason and the analytical condition for the onset of dry whip in rotor-to-stator contact systems. ASME J Acoust Vib 127:594–603 13. Sun L, Krodkiewski JM (1998) Self-tuning adaptive control of forced vibration in rotor systems using an active journal bearing. J Sound Vib 213:1–14 14. Knospe C, Hope R, Tamer S, Fedigan S (1996) Robustness of adaptive unba-lance control of rotors with magnetic bearings. J Vib Control 2:33–52 15. Qiu JH, Tani J, Kwon T (2003) Control of self-excited vibration of a rotor system with active gas bearings. ASME J Vib Acoust 125:328–334 16. Jiang J, Ulbrich H, Chavz A (2006) Improvement of rotor performance under rubbing conditions through active auxiliary bearings. Int J Non-Linear Mech 41/8:949–957

Part II

Time-delay Systems

Chapter 12

Some Control Studies of Dynamical Systems with Time Delay Bo Song and Jian-Qiao Sun

Abstract This chapter presents a summary of recent studies of controlling dynamical systems with time delay. The time delay can be uncertain and timevarying with known lower and upper bounds. Two methods for approximate solutions of the system with time delay are discussed, namely the method of semi discretization and the method of continuous time approximation. The spectral properties of mapping based methods are discussed. We also demonstrate the supervisory control to handle uncertainties in time delay. Several control examples are presented in the chapter.

12.1 Introduction Time-delayed systems have been studied using discretization techniques with an extended state vector. Pinto and Goncalves [1] fully discretized a nonlinear SDOF system to study control problems with time delay. Klein and Ramirez [2] studied MDOF delayed optimal regulator controllers with a hybrid discretization technique, where the state equation was partitioned into discrete and continuous portions. Yang and Wu [3] and Stepan [4] have studied structural systems with time delay. A study on stability and performance of feedback controls with multiple time delays is reported in [5] by considering the roots of the closed-loop characteristic equation. A method using Chebyshev polynomials to approximate general nonlinear functions of time has been developed to handle linear and nonlinear time-delayed dynamical systems with periodic coefficients [6–9]. The method has also been applied to study optimal control problems. A temporal finite element method has been proposed in [10] to study the stability of time-delayed systems with parametric excitations. The work reported in [11] makes use of the piece-wise exact solution of linear differential equations with a single time delay to create a map in order to study the stability

J.-Q. Sun () School of Engineering, University of California, Merced CA 95344, USA e-mail: [email protected]

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of the system. A series of papers in [12–14] have studied optimal feedback gain designs based on the mapping of an extended state vector. For deterministic delayed linear systems, a survey of methods for stability analysis is presented in [15]. An excellent survey of stability and control of time-delayed systems can be found in [16]. There have also been many studies of control systems with unknown and timevarying time delays. Chen et al. derived sufficient conditions for the existence of the guaranteed cost output-feedback controller in terms of matrix inequalities for uncertain dynamical systems with time delay [17]. The Lyapunov method is used in [18] for the stability analysis of systems with time-varying delay with known lower and upper bounds. The Lyapunov function dependent on the known upper bound of uncertain state-delays is derived in the study of model predictive controls (MPC) for a constrained linear digital systems with uncertain state-delays [19]. A class of iterative learning control systems with uncertain state delay and control delay is studied in [20]. Robust stability of uncertain linear systems with interval time-varying delay is studied in [21]. Stability of systems with bounded uncertain time-varying bounded delays in the feedback loop is studied in [22]. The stability problem is treated in the integral quadratic constraint (IQC) framework. Kwon, Park and Lee [23] investigated delay-dependent robust stability for neutral systems with the help of the Lyapunov method. The system has time-varying structured uncertainties and interval time-varying delays. A compensation scheme that consists of a fuzzy-PID controller and a neural network compensator is proposed for realtime control over the network is studied in [24]. This scheme reduces the influence of time delays on stability while maintaining the system performance. According to [25], given a finite-dimensional linear time invariant (LTI) plant and an upper bound on the admissible time delay, there is no general theory for designing a controller to handle an arbitrarily large uncertain delay. The authors show that given a finite-dimensional LTI plant and an upper bound on the admissible time delay, there exists a linear periodic controller which robustly stabilizes the plant. Robust stability for systems with random time-varying delay with a known probability distribution is studied in [26]. The resulting system model has stochastic parameters. Sufficient conditions for the exponential mean square stability of the system are derived by using the Lyapunov functional method and the linear matrix inequality (LMI) technique. When the uncertain time delay is bounded with known lower and upper bounds, we can consider the supervisory control [27–30]. The supervisory control proposes to use several estimates of uncertain parameters for the system model. For each estimate of the parameter, a control is designed to achieve the desired performance. A supervisor monitors the real-time response of the system, selects a plant model according to a switching criterion, and implements the corresponding control. In this chapter, we review some recent control studies of dynamical systems with time delay. We first review two approximate methods for computing the response of time-delayed systems, discuss their properties and applications to feedback controls. We also introduce the supervisory control of systems with unknown time delay. The influence of the range of the unknown time delays on the supervisory control is discussed. A number of examples are included to demonstrate the theoretical discussions.

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12.2 Methods of Solution 12.2.1 Semi-Discretization Consider a linear periodic system with time delay xP .t/ D A.t/x.t/CAd .t/x.t / C B.t/u.t/;

(12.1)

where x 2 Rn and u 2 Rm . A.t/ 2 Rnn , Ad .t/ 2 Rnn and B.t/ 2 Rnm are periodic matrices with period T . We shall consider a feedback control with or without time delay in the following forms u.t/ D Kx.t/

or

u.t/ D Kx.t /;

(12.2)

where K 2 Rmn is the gain matrix. In the closed loop system, the control simply modifies the matrix A.t/ or Ad .t/. When we introduce the method of semi-discretization, we can focus on the following system without loss of generality, xP .t/ D A.t/x.t/CAd .t/x.t /:

(12.3)

Because of the time delay, the state vector of the system is no longer just x.t/, but .xT .t/; xT .t 1 //T for all 0 < 1 , which has an infinite dimension. The time delay significantly complicates the solution process of the system. Let us discretize the period T into an integer k intervals of length t such that T D kt. For the sake of simplicity, we assume that the time delay D Nt where N is an integer. When an integer N cannot be found, discretization of the time delay will be approximate [31], or a continuous time approximation can be adopted as discussed in Sect. 12.2.2. Consider (12.3) in a time interval t 2 Œti ; ti C1 , where ti D i t, i D 0; 1; 2; : : : ; k. In each small time interval Œti ; ti C1 , the delayed responses x.t / and the time dependent coefficients are assumed to be constant. We denote x.ti / D x..i N /t/ D xi N ; A.ti / D Ai ; Ad .ti / D Ad i :

(12.4)

Equation (12.3) becomes xP .t/ Ai x.t/ D Ad .t/x.t / t 2 Œti ; ti C1 ; i D 0; 1; 2; : : : ; k:

(12.5)

The general solution of the equation is x.t/ D eAi .t ti / xi C

Z

t

O

eAi .t ti t / Ad .tO/x.tO /dtO;

ti

t 2 Œti ; ti C1 ; i D 0; 1; 2; : : : ; k:

(12.6)

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The integration on the RHS of the above equation can be computed by assuming that Ad .t/x.t / is a constant or a linear function of time over the small interval Œti ; ti C1 . The work in [32] has studied the accuracy of these approximation schemes. As an example, we show the case when Ad .t/x.t / is assumed to be constant over Œti ; ti C1 : The response xi C1 D x.ti C1 / at time ti C1 can then be expressed in the following mapping (12.7) xi C1 D Qi xi C Pi xi N : where

Z Pi D

ti C1

eAi .t / Ad i d; Qi DeAi t ;

(12.8)

ti

Define an .N C 1/n dimensional state vector as

T yi D xTi xTi1 xTi2 : : : xTiN :

(12.9)

A mapping of the state vector over the interval Œti ; ti C1 can be found as yi C1 D Hi yi ;

(12.10)

where the transition matrix from time ti to ti C1 is 3 0n.N 1/n Pi Qi 5: Hi D 4 Inn 0n.N 1/n 0nn 0.N 1/nn I.N 1/n.N 1/n 0.N 1/nn 2

(12.11)

The mapping of the state vector over one period T D kt is therefore yj C1 D ˆy j ;

(12.12)

where the mapping matrix ˆ is given by ˆ D Hk1 Hk2 H1 H0 :

(12.13)

Note that the index j (j D 0; 1; :::) refers to the number of periods, i.e. yj is the state vector at the beginning of the j th period. The stability of the control system is determined by the eigenvalues of ˆ. Let jjmax denote the largest absolute value of eigenvalues of the matrix ˆ. Then, jyj C1 j jjmax jyj j:

(12.14)

When jjmax < 1, ˆ is a contraction, and the control system is asymptotically stable. The stability boundary is given by jjmax D 1. Equation (12.14) indicates that the smaller jjmax is, the faster the system converges to zero. jjmax therefore also provides a measure of the control performance.

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12.2.2 Continuous Time Approximation Consider a nonlinear system with one time delay given by, xP D f .x .t/ ; x .t / ; t/ CBu .t/ ;

(12.15)

where x 2 Rn ; u 2 Rm , f is a nonlinear function of its arguments, and B D fBij g is the control influence matrix. Following the idea of semi-discretization, we discretize the delayed part of the state vector .x .t t1 / ; 0 < t1 /. Let N be an integer such that D =N . i D i .i D 1; 2; : : : ; N /. We introduce a finite forward difference approximation of the derivatives of .x .t i / ; 1 i N / as xP .t i / D

1 Œx .t i 1 / x .t i / :

(12.16)

Note that other approximation schemes including, for example, the central difference and Gear integration method for ordinary differential equations, can be used, and that the discretization of the time delay interval can be nonuniform. Define M D n.N C 1/ dimensional extended state vector as y .t/ D Œx .t/ ; x .t 1 / ; x .t 2 / ; : : : ; x .t N /T Œy1 .t/ ; y2 .t/ ; y3 .t/ ; : : : ; yN C1 .t/T : We obtain an equation for the vector y .t/ 2 RM . 2 3 f y1 .t/ ; yN C1 .t/ ; t 2 3 B 6 1 7 6 7 Œy .t/ y .t/ 6 1 2 6 7 607 7C6 : 7 u .t/ yP .t/ D 6 : 6 7 4:7 :: 6 7 :5 4 1 5 0 ŒyN .t/ yN C1 .t/ O .t/ : Of.y; t/ C Bu

(12.17)

(12.18)

For a linear system xP D Ax .t/ CA x .t / CBu .t/ ;

(12.19)

where A is the Rnn state matrix and A is the Rnn state matrix related to the delayed response, we have an equation for y .t/ 2 RM as 2

3 A 0 0 A 6 1 7 2 3 1 6 B 0 7 6 I I 0 7 6 7 607 : 6 7 6 7 :: yP .t/ D 6 7 y .t/ C 6 : 7 u .t/ 6 7 4 :: 5 :: 6 7 : 6 7 0 4 1 1 5 0 0 I I O .t/ C Bu O .t/ ; Ay

(12.20)

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O 2 RM M and BO 2 RM m . Recall that i need not to be spaced uniformly where A in the time interval Œ0; . Nonuniform sampling allows the method to handle more than one independent time delays [14].

12.3 Spectral Properties of the Methods Both the methods of semi-discretization and CTA focus on approximation of temporal responses of the system over a short time interval. As N ! 1, the approximate solution approaches the exact one in time domain at the rate depending on the order of the local approximation. This has been verified by means of extensive numerical simulations [12–14, 31]. These methods are not specifically developed to meet frequency dominant requirements such as accurate representation of the open-loop or closed-loop poles and zeros of the original system. The methods have been mostly validated with time domain numerical solutions. Their properties in frequency dominant are studied next with the help of numerical examples of a linear time-delayed system. Since the CTA formulation can be made completely equivalent to semi-discretization for linear time-invariant systems with a single time delay, the spectral properties of the CTA method are the same as that of semi-discretization. Consider a linear spring–mass–dashpot oscillator subject to a delayed PD control. The closed-loop characteristic equation is given by s 2 C cs C k C kp es C kd ses D 0;

(12.21)

where we take c D 0:2, k D 4, and D =2. kp and kad are the feedback gains. The state matrix with the CTA method reads 3 2 0 1 0 0 6 k c kp kd 7 7 6 7 6 1 1 7 6 I I 0 0 7 6 7 6 O D6 :: (12.22) A 7: : 7 6 7 6 7 6 :: 7 6 : 5 4 1 1 0 0 I I From the stability chart in [13], we know that the system is stable when .kp ; kd / D .0:5; 0:5/. There are two pairs of the dominant poles with real parts approximately equal to 1 and 0:4. Figures 12.1 and 12.2 show the roots of the characteristic equation (12.21) and the eigenvalues of the state matrix (12.22) constructed with the forward central difference and fourth order Gear’s integration algorithm [33]. Extensive simulations show that the CTA method is able to capture the dominant poles of the system only, and completely misses the fast and high frequency poles.

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a

b 50

20 15 10

Im(s)

Im(s)

5 0

0 −5

−10 −15 −50

−40

−20 Re(s)

−20 −20

0

−10 Re(s)

0

Fig. 12.1 Closed-loop poles of the linear oscillator under a delayed PD control. “” denotes the roots of the characteristic equation (12.21). “C” denotes the eigen values of the state matrix (12.22) constructed with forward finite difference approximation. .kP ; kD / D .0:5; 0:5/. k D 4: c D 0:2. D =4. N D 20. (b) is the zoomed view of (a) in the indicated range

a

b 50

20 15 10

Im(s)

Im(s)

5 0

0 −5 −10 −15

−50 −100

−50 Re(s)

0

−20 −20

−10 Re(s)

0

Fig. 12.2 Closed-loop poles of the linear oscillator under a delayed PD control. denotes the roots of the characteristic equation (12.21). “C” denotes the eigen values of the state matrix (12.22) constructed with the fourth order Gear’s integration scheme. .kP ; kD / D .0:5; 0:5/. k D 4. c D 0:2. D =4. N D 20. (b) is the zoomed view of (a) in the indicated range

The central finite difference for CTA yields more accurate solutions in time domain. But it introduces a set of lightly damped poles as shown in Fig. 12.3. This causes difficulties when the method is used in control design. On the other hand, the backward finite difference for CTA is unstable, even though it also accurately

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Im(s)

20

0

−20

−40 −3

−2.5

−2

−1.5 Re(s)

−1

−0.5

0

Fig. 12.3 Closed-loop poles of the linear system with central finite difference approximation of the delayed portion of the response under a delayed PD control. “” denotes the roots of the characteristic equation (12.21). “C” denotes the eigen values of the state matrix (12.22). .kP ; kD / D .0:5; 0:5/. k D 4. c D 0:2. D =2. N D 40 40

Im(s)

20

0

−20

−40 −30

−20

−10

0

10 Re(s)

20

30

40

50

Fig. 12.4 Closed-loop poles of the linear system with backward finite difference approximation of the delayed portion of the response under a delayed PD control. “” denotes the roots of the characteristic equation (12.21). “C” denotes the eigen values of the state matrix (12.22). .kP ; kD / D .0:5; 0:5/. k D 4. c D 0:2. D =2. N D 40

captures the dominant poles as shown in Fig. 12.4. Furthermore, all the methods can capture the right most dominant poles when the system is unstable. The results are not presented here for the sake of space. Why can the CTA method accurately predict temporal response x .t/ of the timedelayed system even when it misses all the fast and high frequency poles? Recall that the response of time-delayed systems lives in an infinite dimensional state space. x .t/ is a projection of the infinite dimensional response .x .t/ ; x .t t1 / ; 0 < t1 / on to the finite dimensional space Rn . However, different objects in a higher dimensional space can have the same projection in a lower dimensional space. The CTA method aims at accurate time domain solutions

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of x .t/ for all t > 0, much like the numerical algorithms for integrating ordinary differential equations, such as Runge–Kutta methods whose frequency domain properties are also rarely discussed. These methods in time domain provide one projection of the infinite dimensional response, while the methods in frequency domain such as Pad´e approximations of the transfer function provide a different projection [34, 35]. In principle, the solutions x .t/ obtained by both the time and frequency domain methods with “equivalent accuracy” should be very close to each other, while the solutions obtained by the frequency domain methods may have an advantage of containing more accurate information about the poles and zeros. The question is then, can we construct a time domain method that accurately predicts both the temporal responses and the poles of the system?

12.4 Control Formulations 12.4.1 Full-State Feedback Optimal Control Within the framework of continuous time approximation, we can formulate a fullstate feedback optimal control problem. Define a performance index as 1 J D 2

Z1

yT Qy C uT Ru dt;

(12.23)

0

where Q D QT 0 and R D RT > 0. When the linear system (12.20) is considered, the full state feedback control u D Ky is the LQR control determined by the O B; O Q; R/ [36]. When the nonlinear system (12.18) is considered, we matrices .A; have a nonlinear optimal control problem on hand [37]. Note that the extended state vector y contains the current and past system response x .t/. The full state feedback control does not consider possible transport delays since the current state x .t/ is included in the control.

12.4.2 Output Feedback Optimal Control Assume that there is a transport delay p . We consider a control of the form u D Kx.t p / for the linear system. First, we select a discretization scheme such that p is one of the points i of the time discretization. Assume that p D k . Define an output equation as v D Cy D ykC1 D x.t p /;

(12.24)

where ykC1 is the .k C 1/th elemental vector defined in (12.17). According to [36], if a control gain K for the linear system in (12.20) can be O BKC O found such that the closed-loop system characterized by the matrix A

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is stable, the system is output stabilizable. When the system is output stabilizable, an optimal control gain can be found in the following optimization problem: Find a control gain K such that the performance index i 1 h 1 (12.25) J D yT0 Py0 D tr Py0 yT0 ; 2 2 is minimized where y0 is an initial condition of the extended state vector y.t/, subject to the constraint of the Lyapunov equation T O BKC/ O BKC/ O O .A P C P.A C CT KT RKC C Q D 0:

(12.26)

This is a nonlinear matrix algebraic optimization problem. The Matlab function fminsearch can be used to find the optimal control. The optimal gain is in general a function of the initial condition y0 . This is not a desirable feature of the output feedback control. A common approach to select initial conditions is to replace the term y0 yT0 by its statistical average EŒy0 yT0 , i.e., the autocorrelation function of y0 . For more discussions, the reader is referred to [36]. It should be noted that for a given initial value of the control gain to start searching for the optimal one, even the best searching algorithm only gives a local minimum of the performance index J . There are many research issues with the output feedback design that need further studies. For example, how to help the searching algorithm land on a much deeper local minimum? How to select the design matrices Q and R to improve the control performance under certain constraints? In the current formulation, when is the system output stabilizable? These turn out to be tough technical questions to answer.

12.4.3 Optimal Feedback Gains via Mapping Another way to obtain optimal gains for output feedback controls is via mapping. This approach has been studied extensively in [12, 13] with semi-discretization. For the linear system (12.20), a mapping of the response can be constructed as y.k C 1/ D ˆ.K/y.k/; k D 0; 1; : : :

(12.27)

where u D Kx.t p / has been substituted. The mapping ˆ is therefore a function of the control gain K. This mapping can be found either by the method of semidiscretization or directly from the solution of Equation (12.20) given by O At

Zt

y .t/ D e y0 C

O

O .t1 / dt1 : e A.t t1 / Bu

(12.28)

0

We have found that the mapping constructed via CTA is completely equivalent to the mapping via semi-discretization. When the system (12.20) is periodic with multiple

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independent time delays, we can combine the method of semi-discretization with CTA to construct the mapping for the response of the system [31, 32]. Consider a bounded and compact region Rmn such that K 2 . We can find the domains of stability and optimal control gains in the region to minimize the largest magnitude of the eigenvalues of ˆ. This leads to the following optimization problem (12.29) min Œmax j.ˆ/j subject to jjmax < 1: K2

This formulation offers a different approach to the design of output feedback controls for linear periodic systems with or without time delay. The control performance criterion is the decay rate of the mapping ˆ over one mapping step. In the frequency domain, we have found that the optimal feedback gains designed by the mapping method maximize the damping of the dominant closed-loop poles of the system that are closest to the imaginary axis of the s-plane [13].

12.5 Supervisory Control Recall the system in (12.30). The time delay is assumed to be slowly time-varying, and lie in an interval Œmin ; max , where the minimum and maximum time delays are assumed to be known. Assume that we have obtained a set of optimal feedback gains for the set of time delays sampled in the interval Œmin ; max . We present the switching algorithm for selecting a gain to implement in real time. The actual time delay is such that min max . We discretize Œmin ; max into M 1 intervals, so that min D 1 < 2 < M D max . Consider M models of the time-delayed system as xP i D f .xi .t/ ; xi .t i / ; t/ CBui .t/ ; 1 i M :

(12.30)

Consider the feedback control ui D Ki xi .t i / where Ki 2 . Each Ki is found by imposing (12.29) subject to an additional constraint: Ki must be stable for all j (1 j M ). Let Ki Opt 2 be the optimal gain for i and the associate eigenvalue with the smallest magnitude ji .ˆ/jmin < 1. Check if Ki Opt stabilizes the system in (12.30) for all other time delays j (1 j M ). Following the concept of the supervisory control [27–30], we define an estimation error as (12.31) ei D xi .t/ x .t/ ; 1 i M ; where x .t/ is the output of the system with unknown time delay. In the experiment, x .t/ would be obtained from measurements. Consider a positive function of the estimation error Fi .ei / > 0. An example is Fi .ei / D jjei jj2 . Define a switching index i .t/ such that P i .t/ C i i .t/ D Fi .ei /; .i > 0/ i .0/ D 0;

(12.32)

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where the parameter i defines the bandwidth of the low pass filter. The general solution for i .t/ can be obtained as i .t/ D ei t i .0/ C

Zt

ei .t / Fi .ei .//d:

(12.33)

0

The hysteretic switching algorithm in [29, 30] is stated as follows. Assume that the system is sampled at time interval t. At the kth time step, the system is under control with the gain Kj and the associated switching signal is j .k/. At the .k C 1/th step, if there is an index i such that i .k/ < .1 / j .k/ where > 0 is a small number, we switch to the gain Ki . Otherwise, we continue with the gain Kj . is known as the hysteretic parameter and prevents the system from switching too frequently.

12.6 Numerical Examples 12.6.1 Linear Time Invariant System Consider a second-order autonomous system under a delayed PI control. 2 3 3 2 0 1 0 0 0 0 xP .t/ D 40 0 1 5 x.t/ 4 0 0 05 x.t /; 0 k c ki kp 0

(12.34)

where x D .x; x; P x/ R T . The feedback control u D Œki ; kp ; 0 x.t / has an uncertain transport delay Take k D 4, c D 0:2. The discretization number of the time delayed response is set to be N D 20 for all sampled time delays [12–14]. min D 0:0419 and max D 0:2094. We pick five different time delays to design optimal feedback gains according to the method outlined in the previous section. The optimal gains associated with the five time delays are listed in Table 12.1. The associated stability domains in the gain space are shown in Fig. 12.5. It should

Table 12.1 Optimal PI feedback gains and jjmax for the four sampled time delays of the linear time invariant system Time delay ki kp jjmax 1 2 3 4 5

D 0:0419 D 0:0733 D 0:1047 D 0:1571 D 0:2094

0.2000 0.2000 0.2000 0.2000 0.2000

1:8400 2:3500 2:8600 2:8600 3:3700

0.9992 0.9982 0.9966 0.9938 0.9907

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1.5

ki

1

0.5

0

−0.5 −4

−2

0

2

4

6

kp

Fig. 12.5 Stability domains (lines) in the gain space and the optimal feedback gains (o) for the autonomous systems with four different time delays i (i D 1; 2; 3; 4; 5). The stability boundaries become taller and narrower, and move upward along ki axis as time delay increases

be pointed out that when the optimal gains of all the controls with different time delays fall in the intersection of the stability domains, it is possible to use the hysteretic algorithm to switch among the predesigned controls and to keep the system stable all the time. When an optimal gain is out of the intersection, the control with that gain can destabilize the system with some time delay in the range Œmin ; max . This property limits the size of the unknown time delay range Œmin ; max because the stability domains change significantly with the time delay, particularly for periodic systems [13]. Figure 12.6 shows the closed loop response of the system under the feedback control with all four different time delays when the system true time delay is taken to be 4 and is assumed to be unknown. As it can be seen from the figure, when the control designed for the time delay that is close to 4 is implemented, the performance is acceptable. Otherwise, the performance can deteriorate as seen in the left-upper sub-figure. Next, we examine how well the hysteretic switch algorithm works. Assume that we start with a control gain K1 designed for 1 , while the system delay is 4 . Figure 12.7 shows that the hysteretic algorithm is able to switch the control to K3 and K4 since both controls have a similar performance. Figure 12.8 shows the switch signal .t/ and the control index.

12.6.2 Periodic System Consider the Mathieu equation with a delayed PID feedback control 2

0 xP .t/ D 4 0 4" sin 2t

1 0 .ı C 2" cos 2t/

3 2 0 0 1 5 x.t/ 4 0 0 ki

0 0 kp

3 0 0 5 x.t /; kd (12.35)

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5

τ=τ1, K=K2

x1

τ=τ1, K=K1 0

−5

0

0

20

40

60

5

−5

0

20

5

x1

−5

0

60

τ=τ1, K=K5

τ=τ1, K=K3, K3 0

40

0

20 40 Time (s)

60

−5

0

20 40 Time (s)

60

Fig. 12.6 Response of the autonomous system under feedback controls designed for a specific gain when the system true time delay is 1 and is assumed to be unknown. When the feedback gains .K2 ; K3 ; K5 ) are designed for the time delay close to the actual one, the control performance is quite good. K4 and K5 are the same. When the mismatch gap is large, i.e. when K5 designed for 5 is implemented for the system with time delay 1 , the performance deteriorates

5

x1

τ=τ1, K=K5 0

−5 0

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30 Time (s)

40

50

60

5

x1

τ=τ1, Switch from K5 0

−5 0

10

20

30 Time (s)

40

50

60

Fig. 12.7 The closed loop response of the system under the switched control when the initial gain of the control is K5 designed for 5 while the system true time delay is 1 (bottom), as compared to the case when the gain is fixed at K5 (top)

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Control Index

4 3 2 1 0

Switching Signal

2

10

20

30

40

50

60

70

80

10

20

30

40 Time (s)

50

60

70

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x 10−3

1

0 0

Fig. 12.8 Switch signal (lower figure) and the control index (upper figure) of the hysteretic switching algorithm for the closed loop response in Fig. 12.7 Table 12.2 Optimal PD feedback gains and jjmax for the five sampled time delays of the periodic system. note that the mapping step for the periodic system is one period, while the mapping step for the LTI system is only one time delay Time delay kp kd jjmax 1 2 3 4 5

D0.5498 D0.6676 D0.7854 D0.9032 D1.0210

3:6634 4:0000 2:8000 2:6000 1:8000

0:0990 0:8000 0:6000 0:6000 0:6000

0.0130 0.0083 0.0141 0.0213 0.0347

where x D .x; x; P x/ R T . The period of the system is T D . We select the parameters to be " D 1; ı D 4, and N D 20 for all sampled time delays. The uncontrolled system is parametrically unstable. Next, we show the closed-loop response of the system under a switching PD control with time delay in the range [0.5498, 1.0210]. Five time delays are sampled from the interval, and their optimal gains are listed in Table 12.2. The stability domains in the gain space are shown in Fig. 12.9. Note that the optimal gains of all the controls with different time delays fall in the intersection of the stability domains. Hence, it is possible to use the hysteretic algorithm to switch among the predesigned controls and to keep the system stable all the time. Another interesting phenomenon as shown in Fig. 12.9 and also in [13] is that the stability domain in kp kd gain space grows along the kd direction as time delay increases. Figure 12.10 shows the closed loop response of the system under the feedback control with the first four different time delays when the system true time delay is

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kd

0 −1 −2 −3 −4 −5 −6

−4

−2 kp

0

2

Fig. 12.9 Stability domains (lines) in the gain space and the optimal feedback gains (o) for the periodic system with five different time delays i (i D 1; 2; 3; 4; 5). The stability boundaries move down along kd axis as time delay increases

2

x1

1

2

τ=τ1, K=K1

1

0

0

−1

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−2 0

10

20

2

x1

1

−2 0

10

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τ=τ1, K=K3

1

0

0

−1

−1

−2 0

τ=τ1, K=K2

10 Time (s)

20

−2 0

τ=τ1, K=K4

10 Time (s)

20

Fig. 12.10 Response of the periodic system under PD feedback controls designed for a specific gain when the system true time delay is 1 and is assumed to be unknown

taken to be 2 and is assumed to be unknown. As it can be seen from the figure, when the control designed for the time delay that is close to 2 is implemented, the performance is better.

12

Control of Systems with Time Delay

151

2 τ=τ1, K=K4

x1

1 0 −1 −2 0

5

10 Time (s)

15

20

2 τ=τ1, Switch from K4

x1

1 0 −1 −2 0

5

10 Time (s)

15

20

Fig. 12.11 The closed loop response of the periodic system under the switched PD control when the initial gain of the control is K4 designed for 4 while the system true time delay is 1 (bottom), as compared to the case when the gain is fixed at K4 (top)

Control Index

4 3 2 1 0

5

10

15

20

5

10 Time (s)

15

20

Switching Signal

0.16 0.14 0.12 0.1 0

Fig. 12.12 Switch signal (lower figure) and the control index (upper figure) of the hysteretic switching algorithm for the closed loop response in Fig. 12.11

Next, we start with a control gain K4 designed for 4 . Figure 12.11 shows the closed loop response. The hysteretic algorithm switches the gain to reduce the switch signal .t/ as shown in Fig. 12.12.

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12.6.3 An Experimental Example We experimentally study time-delayed feedback control of a rotary flexible joint made by Quanser. The test apparatus is shown in Fig. 12.13, which consists of a rotary flexible joint mounted on top of a rigid rotary platform. Two encoders are used in the system. One measures the angular position of the platform, the other measures the angular displacement of the flexible joint relative to the platform. The state equation of the system is of fourth order given by xP D Ax C bu;

(12.36)

where 2

P ˛ x D Œ; ˛; ; P T;

0 60 AD6 40

3

0 1 0 0 0 17 7; 689:86 57:658 0 5 0 1359:2 57:658 0

b D Œ0; 0; 107:39; 107:39T :

(12.37)

(12.38)

(12.39)

is the angular position of the platform, and ˛ is the angular position of the flexible joint relative to the platform. By examining the measured step response of the open-loop system, we have found that the system has a time delay of 0:002 s. An additional transport delay 0:008 s between the input and the output of the system is digitally introduced, leading to a total time delay D 0:01 s. We consider the time-delayed control u D k x.t /. The closed-loop system reads xP D Ax bk x.t /: (12.40)

Fig. 12.13 Experimental setup of the flexible rotary joint by Quanser

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153

The mapping over one delay time step is constructed based on the solution given in (12.28). Consider a region in the gain space 0 < k1 < 100; 50 < k2 < 50;

(12.41)

0 < k3 < 5; 0 < k4 < 5: We look for optimal gains in this region. We initially discretize the domain by dividing each gain range into ten partitions. Successive refinement of the grid leads to the global minimum of jjmax D 0:65128 in the domain (12.41) and the associated optimal feedback gains kopt D Œ16:256; 14:428; 1:089; 0:656:

(12.42)

The optimal feedback gains reported in [13] under the same condition by means of the mapping approach are kopt D Œ12:321; 15:432; 0:864; 0:452 with jjmax D 0:6511. The difference between these two sets of gains is due to the different approximation schemes used to compute the delayed response and the mapping. Figure 12.14 compares the closed-loop response of the rotary flexible joint system with the optimal feedback gains kopt and with the LQR controller designed for the system without time-delay. The control performance of the feedback controller designed for the system with time-delay is obviously much better than that of the LQR control. For the completeness, we list the parameters for the LQR control design here: Q D diag.Œ2; 500; 4; 500; 1; 1/, R D 2 and k D Œ35:355; 39:574; 1:974; 0:873.

Joint Angle (θ+α)

1

0.5

0

−0.5

−1 0

1

2

3 Time (s)

4

5

6

Fig. 12.14 Comparison of the rotary flexible joint response under the feedback control with the optimal gains and the LQR controller designed for the system without time-delay. Solid line: the feedback control with optimal gains. Dashed line: the LQR control for the system without time-delay

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12.7 Concluding Remarks We have presented a review of recent studies of analysis and control of dynamical systems with time delay. Methods of solution including semi-discretization and continuous time approximation are reviewed, and their spectral properties are discussed. Examples of supervisory control of systems with unknown time delay and experimental validation are presented also.

References 1. Pinto OC, Goncalves PB (2002) Control of structures with cubic and quadratic non-linearities with time delay consideration. Journal of the Brazilian Society of Mechanical Sciences 24(2), 99–104 2. Klein EJ, Ramirez WF (2001) State controllability and optimal regulator control of timedelayed systems. International Journal of Control 74(3), 281–289 3. Yang B, Wu X (1998) Modal expansion of structural systems with time delays. AIAA Journal 36(12), 2218–2224 4. Stepan G (1998) Delay-differential equation models for machine tool chatter. In: Moon FC (ed) Dynamics and Chaos in Manufacturing Processes, Wiley, New York, pp 165–192 5. Ali MS, Hou ZK, Noori MN (1998) Stability and performance of feedback control systems with time delays. Computers and Structures 66(2-3), 241–248 6. Deshmukh V, Butcher EA, Bueler E (2008) Dimensional reduction of nonlinear delay differential equations with periodic coefficients using chebyshev spectral collocation. Nonlinear Dynamics 52(1–2) 7. Deshmukh V, Ma H, Butchern EA (2006) Optimal control of parametrically excited linear delay differential systems via Chebyshev polynomials. Optimal Control Applications and Methods 27, 123–136 8. Ma H, Deshmukh V, Butcher EA, Averina V (2005) Delayed state feedback and chaos control for time periodic systems via a symbolic approach. Communications in Nonlinear Science and Numerical Simulation 10(5), 479–497 9. Ma H, Butcher EA, Bueler E (2003) Chebyshev expansion of linear and piecewise linear dynamic systems with time delay and periodic coefficients under control excitations. Journal of Dynamic Systems, Measurement, and Control 125, 236–243 10. Garg NK, Mann BP, Kim NH, Kurdi MH (2007) Stability of a time-delayed system with parametric excitation. Journal of Dynamic Systems, Measurement, and Control 129(2), 125–135 11. Kalmar-Nagy T (2005) A novel method for efficient numerical stability analysis of delaydifferential equations. In: Proceedings of American Control Conference, Portland, Oregon, pp 2823–2826 12. Sheng J, Elbeyli O, Sun JQ (2004) Stability and optimal feedback controls for time-delayed linear periodic systems. AIAA Journal 42(5), 908–911 13. Sheng J, Sun JQ (2005) Feedback controls and optimal gain design of delayed periodic linear systems. Journal of Vibration and Control 11(2), 277–294 14. Sun JQ (2009) A method of continuous time approximation of delayed dynamical systems. Communications in Nonlinear Science and Numerical Simulation 14(4), 998–1007 15. Niculescu SI, Verriest EI, Dugard L, Dion JM (1998) Stability of linear systems with delayed state: A guided tour. In: Proceedings of the IFAC Workshop: Linear Time Delay Systems, Grenoble, France, pp 31–38 16. Gu K, Niculescu SI (2003) Survey on recent results in the stability and control of time-delay systems. Journal of Dynamic Systems, Measurement, and Control 125, 158–165

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17. Chen WH, Guan ZH, Lu XM (2004) Delay-dependent output feedback guaranteed cost control for uncertain time-delay systems. Automatica 40(7), 1263–1268 18. He Y, Wang QG, Lin C, Wu M (2007) Delay-range-dependent stability for systems with timevarying delay. Automatica 43(2), 371–376 19. Hu XB, Chen WH (2004) Model predictive control for constrained systems with uncertain state-delays. International Journal of Robust and Nonlinear Control 14(17), 1421–1432 20. Ji G, Luo Q (2006) Iterative learning control for uncertain time-delay systems. Dynamics of Continuous Discrete and Impulsive Systems-Series a-Mathematical Analysis 13, 1300–1306 21. Jiang XF, Han QL (2008) New stability criteria for linear systems with interval time-varying delay. Automatica 44(10), 2680–2685 22. Kao CY, Rantzer A (2007) Stability analysis of systems with uncertain time-varying delays. Automatica 43(6), 959–970 23. Kwon OM, Park JH, Lee SM (2008) On delay-dependent robust stability of uncertain neutral systems with interval time-varying delays. Applied Mathematics and Computation 203(2), 843–853 24. Lin CL, Chen CH, Huang HC (2008) Stabilizing control of networks with uncertain time varying communication delays. Control Engineering Practice 16(1), 56–66 25. Miller DE, Davison DE (2005) Stabilization in the presence of an uncertain arbitrarily large delay. Ieee Transactions on Automatic Control 50(8), 1074–1089 26. Yue D, Tian E, Zhang Y, Peng C (2009) Delay-distribution-dependent robust stability of uncertain systems with time-varying delay. International Journal of Robust and Nonlinear Control 19(4), 377–393 27. Morse AS (1996) Supervisory control of families of linear set-point controllers - Part 1: Exact matching. IEEE Transactions on Automatic Control 41(10), 1413–1431 28. Morse AS (1997) Supervisory control of families of linear set-point controllers - Part 2: Robustness. IEEE Transactions on Automatic Control 42(11), 1500–1515 29. Hespanha JP, Liberzon D, Morse AS (1999) Logic-based switching control of a nonholomic system with parametric modeling uncertainty. Systems & Control Letters 38(3), 167–177 30. Hespanha JP, Liberzon D, Morse AS (2003) Hysteresis-based switching algorithms for supervisory control of uncertain systems. Automatica 39(2), 263–272 31. Insperger T, Stepan G (2001) Semi-discretization of delayed dynamical systems. In: Proceedings of ASME 2001 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Pittsburgh, Pennsylvania 32. Elbeyli O, Sun JQ (2004) On the semi-discretization method for feedback control design of linear systems with time delay. Journal of Sound and Vibration 273, 429–440 33. Carnahan B, Luther HA, Wilkes JO (1969) Applied Numerical Methods. John Wiley and Sons, New York 34. Franklin GF, Powell JD, Emami-Naeini A (1986) Feedback Control of Dynamic Systems. Addison-Wesley, Reading, Massachusetts 35. Vijta M (2000) Some remarks on the Pad´e-approximations. In: Proceedings of the 3rd TEMPUS-INTCOM Symposium, Veszpr´e,. Hungary, pp 1–6 36. Lewis FL, Syrmos VL (1995) Optimal Control. John Wiley and Sons, New York 37. Slotine JJE, Li W (1991) Applied Nonlinear Control. Prentice Hall, Englewood Cliffs, New Jersey

Chapter 13

Stability and Hopf Bifurcation Analysis in Synaptically Coupled FHN Neurons with Two Time Delays Dejun Fan and Ling Hong

Abstract This chapter presents an investigation of stability and Hopf bifurcation of the synaptically coupled nonidentical FHN neurons with two time delays. By regarding the sum of the two delays as a parameter, it is shown that under certain assumptions, the steady state of the model is absolutely stable; Under another set of conditions, there is a critical value of the parameter, the steady state is stable when the parameter is less than the critical value and unstable when the parameter is greater than the critical value. Thus, oscillations via Hopf bifurcation occur at the steady state when the parameter passes through the critical values. Then, explicit formulas are derived by using the normal form method and center manifold theory to determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions.

13.1 Introduction The FHN equation has been derived as a simplied model of Hodgkin–Huxley (HH) equation by FitzHugh and Nagumo. They reduce a four-dimensional HH equation [1] to a two-dimensional system called the FitzHugh–Nagumo (FHN) model [2, 3] by extracting excitability of the dynamics of the behavior in the HH equation. A complete topological and qualitative investigation of the FHN equation with a cubic nonlinearity has been done by Bautin [4], and a rich variety of nonlinear phenomena is observed including a hard oscillation, separatrix loops, bifurcations for equilibria and limit cycles. To understand information processing in the brain, complex dynamics and bifurcations of oscillatory phenomena in D. Fan () MOE Key Laboratory for Strength and Vibration, School of Aerospace, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, People’s Republic of China and Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai, Shandong 264209, People’s Republic of China e-mail: [email protected]

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 13,

157

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coupled FHN neurons have been much investigated [5–7]. It is known that time delays always occur in the signal transmission for real neurons. The observations of a finite time delay in synaptic communication between neurons [8] have stimulated some theoretical studies on coupled time-delay oscillators. Bifurcations and synchronization have been investigated in coupled identical neurons with delayed coupling [9, 10]. However, to our knowledge, there is little work in the literature to deal with delay-coupled nonidentical neurons. For experimental relevance, the questions of prime importance are about the effects of time delays on dynamical behaviors. Recently, Wang et al. [11] has studied the following model: dV1 .t/ dt dW1 .t/ dt dV2 .t/ dt dW2 .t/ dt

D V1 .t/3 C aV1 .t/ W1 .t/ C C1 tanh.V2 .t //; D V1 .t/ b1 W1 .t/;

(13.1)

D V2 .t/3 C aV2 .t/ W2 .t/ C C2 tanh.V1 .t //; D V2 .t/ b2 W2 .t/;

where V1 .t/ and V2 .t/ represent the transmembrane voltage, W1 .t/ and W2 .t/ should model the time dependence of several physical quantities related to electrical variables. Constants a; b1 ; b2 ; C1 ; C2 are positive, represents time delay. In [11], this nonidentical FHN neurons with time-delay coupling has been numerically investigated involving the effects of time delays and coupling strength on bifurcations and synchronization. In order to describe the model more reasonable, we introduce second time delay in model (13.2), and for convenience, we rewrite this system as the following form: xP 1 .t/ D x13 .t/ C ax1 .t/ x2 .t/ C C1 tanh.x3 .t 1 //; xP 2 .t/ D x1 .t/ b1 x2 .t/;

(13.2)

xP 3 .t/ D x33 .t/ C ax3 .t/ x4 .t/ C C2 tanh.x1 .t 2 //; xP 4 .t/ D x3 .t/ b2 x4 .t/: In this chapter, we regard the delay D 1 C 2 as a parameter to investigate the stability and bifurcation to the model (13.3). It is shown that under certain assumptions the steady state of the model ia absolutely stable; Under another set of conditions, there is a critical value of the parameter, the steady state is stable when the parameter is less than the critical value and unstable when the parameter is greater than the critical value. Specifically, there exists a sequence of values of , 0 < 0 < 1 < < j <

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Bifurcation Analysis in FHN Neurons with Delays

159

such that the zero equilibrium loses its stability when passes through 0 , and a Hopf bifurcation occurs when passes through each critical value j . Thus, oscillations via Hopf bifurcation occur at the steady state when the parameter passes through the critical value. Meanwhile, using the center manifold theory and normal form method due to Hassard et al. [12], we derive the algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions on the center manifold. The rest of this chapter is organized as follows. In Sect. 13.2, we shall consider the stability of the zero equilibrium and the existence of the local Hopf bifurcation. In Sect. 13.3, the stability and direction of periodic solutions bifurcating from Hopf bifurcations are investigated by using the normal form theory and the center manifold theorem. We would like to mention that there are several articles on the bifurcation for neural network models with delays, we refer the readers to [13–23] and references therein.

13.2 Stability Analysis Obviously, the origin (0,0,0,0) is an equilibrium of system (13.3), linearizing it gives xP 1 .t/ D ax1 .t/ x2 .t/ C C1 .x3 .t 1 //; xP 2 .t/ D x1 .t/ b1 x2 .t/;

(13.3)

xP 3 .t/ D ax3 .t/ x4 .t/ C C2 .x1 .t 2 //; xP 4 .t/ D x3 .t/ b2 x4 .t/: The characteristic equation associated with system (13.4) is given by 4 C A3 C B2 C C C D E. C b1 /. C b2 /e D 0;

(13.4)

where A D b1 C b2 2a; B D b1 b2 2a.b1 C b2 / C a2 C 2; C D a2 .b1 C b2 / 2ab1 b2 C b1 C b2 2a; D D a2 b1 b2 C 1 ab1 ab2 ;

(13.5)

E D C1 C2 ; D 1 C 2 : In this section, we first study the distribution of roots of (13.4). Clearly, i! .! > 0/ is a root of (13.4) if and only if ! satisfies ! 4 B! 2 C D D EŒ.b1 b2 ! 2 / cos ! C !.b1 C b2 / sin !; A! 3 C C ! D EŒ!.b1 C b2 / cos ! .b1 b2 ! 2 / sin !:

(13.6)

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Taking square on the both sides of (13.6) and summing them up, we obtain .! 4 B! 2 CD/2 C! 2 .A! 2 CC /2 D E 2 Œ! 2 .b1 Cb2 /2 C.b1 b2 ! 2 /2 : (13.7) Letting p D A2 2B; q D B 2 C 2D 2AC E 2 ; u D C 2 2BD E 2 b12 C b22 ; 2

vDD z D !2:

(13.8)

E 2 b12 b22 ;

Then (13.7) becomes z4 C pz3 C qz2 C uz C v D 0:

(13.9)

Since the form of (13.9) is identical to that of (13.6) in [24], thus, we can get Lemmas 1 and 2 analogously. The proofs are omitted. Lemma 1. ([24]) If v < 0, then (13.9) has at least one positive root. Denote h.z/ D z4 C pz3 C qz2 C uz C v: Then we have

(13.10)

h0 .z/ D 4z3 C 3pz2 C 2qz C u:

(13.11)

4z3 C 3pz2 C 2qz C u D 0:

(13.12)

Set Let y D z C

p 4,

then (13.12) becomes y 3 C p1 y C q1 D 0;

where p1 D Define

q 2

3 2 p ; q1 16

D

pq 8

C 4u :

p 3 1 C ; 2 p 3 1 C i 3 ; "D r 2 r q1 p q1 p 3 y1 D C C 3 ; 2 2 r r q1 p q1 p y2 D 3 C " C 3 "2 ; 2 2 D

q 2

p3 32

1

(13.13)

13

Bifurcation Analysis in FHN Neurons with Delays

r y3 D

3

q1 p C "2 C 2

Let zi D yi

p ; 4

161

r 3

q1 p ": 2

.i D 1; 2; 3/:

(13.14)

(13.15)

Lemma 2. ([24]) Suppose that v 0, then we have the following: (i) If 0, then (13.9) has positive roots if and only if z1 > 0 and h.z1 / < 0 (ii) If < 0, then (13.9) has positive roots if and only if there exists at least one z 2 fz1 ; z2 ; z3 g such that z > 0 and h.z / 0 Suppose that (13.9) has positive roots. Without loss of generality, we assume that it has four positive roots, denoted by zk .k D 1; 2; 3; 4/. Then (13.7) has four positive roots, q !k D zk ; .k D 1; 2; 3; 4/ (13.16) By (13.16), we have !k4 B!k2 C D !k .b1 C b2 / A!k3 C C !k b1 b2 !k2 h sin !k D ; 2 i E .b1 C b2 /2 !k2 C b1 b2 !k2

4 !k B!k2 C D b1 b2 !k2 C A!k3 C C !k !k .b1 C b2 / i h cos !k D : 2 E .b1 C b2 /2 !k2 C b1 b2 !k2 (13.17) Thus, denoting 4 !k B!k2 C D !k .b1 C b2 / A!k3 C C !k b1 b2 !k2 i h ; a D 2 E .b1 C b2 /2 !k2 C b1 b2 !k2

(13.18) !k4 B!k2 C D b1 b2 !k2 C A!k3 C C !k !k .b1 C b2 / h b D ; 2 i E .b1 C b2 /2 !k2 C b1 b2 !k2

k.j /

8 1 ˆ .arccosb C 2j /; a 0; < ! k D ˆ : 1 .2 arccosb C 2j /; a < 0; !k

(13.19)

where k D 1; 2; 3; 4 and j D 0; 1; 2; : : : ; then ˙i!k is a pair of purely imaginary oC1 n roots of (13.4) with D k.j / . Clearly, the sequence k.j / is increasing, and .j /

limj !C1 k

j D0

D C1 .k D 1; 2; 3; 4/.

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For convenience, we let

where

S4

.j / C1 kD1 fk gj D0

D fi gC1 i D0 , such that

0 < 1 < 2 < < i < ;

(13.20)

o n 0 D min 1.0/ ; 2.0/ ; 3.0/ ; 4.0/ :

(13.21)

Applying Lemmas 1, 2, and Corollary 2.4 of Ruan and Wei [16], we have the following results. Lemma 3. ([24]) Assume that (H) A > 0; A.B E/ > C E.b1 C b2 /; D > Eb1 b2 ; ŒC E.b1 C b2 /ŒA.B E/ C C E.b1 C b2 / > A2 .D Eb1 b2 /. (i) If one of the followings holds (a) v < 0; (b) v 0; D 0; z1 > 0 and h.z1 / 0; (c) v 0, D < 0, and there exists a z 2 fz1 ; z2 ; z3 g such that z > 0 and h.z / 0, then all roots of (13.4) have negative real parts when 2 Œ0; 0 /. (ii) If the conditions .a/ .c/ of (i) are not satisfied, then all roots of (13.4) have negative real parts for all 0. Proof. When D 0, (13.4) becomes 4 C A3 C .B E/2 C ŒC E.b1 C b2 / C D Eb1 b2 D 0:

(13.22)

By the Routh–Hurwite criterion, all roots of (13.22) have negative real parts if and only if A > 0, A.B E/ > C E.b1 C b2 /, D > Eb1 b2 , ŒC E.b1 C b2 /ŒA.B E/ C C E.b1 C b2 / > A2 ŒD Eb1 b2 : From Lemmas 1 and 2, we know that if .a/–.c/ of (i) are not satisfied, then (13.4) has no roots with zero real part for all 0; If one of the .a/; .b/ and .c/ holds, when ¤ k.j / (k D 1; 2; 3; 4; j D 0; 1; 2; : : :), (13.4) has no roots with zero real part and 0 is the minimum value of so that (13.4) has purely imaginary roots. This completes the proof. t u Let ./ D ˛./ C i!./

(13.23)

be the root of (13.4) near D k.j / satisfying

˛ k.j / D 0;

! k.j / D !k ;

(13.24)

then, from Lamma 2.5 of Hu and Huang [25] the following conclusion holds. Lemma 4. ([25]) Suppose h0 .zk / ¤ 0, where h.z/ is defined by (13.10). If D k.j / , then ˙i!k is a pair of simple purely imaginary roots of (13.4). Moreover, ˇ d.Re.// ˇˇ ˇ .j / ¤ 0; d D k

(13.25)

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Bifurcation Analysis in FHN Neurons with Delays

163

ˇ d.Re.// ˇˇ ˇ .j / d D

and the sign of

is consistent with that of h0 .zk /.

k

Applying Lemmas 3 and 4, we obtain Theorem 1 immediately. .j /

Theorem 1. Let k Suppose (H) hold,

and 0 be defined by (13.19) and (13.21), respectively.

(i) If the conditions (a) v < 0; (b) v 0; D 0; z1 > 0 and h.z1 / 0; (c) v 0, D < 0, and there exists a z 2 fz1 ; z2 ; z3 g such that z > 0 and h.z / 0 are not satisfied, then the zero solution of system (13.3) is asymptotically stable for all 0. (ii) If one of the conditions (a), (b) and (c) of (i) is satisfied, then the zero solution of system (13.3) is asymptotically stable when 2 Œ0; 0 /. (iii) If one of the conditions (a), (b) and (c) of (i) is satisfied, and h0 .zk / ¤ 0, then the system (13.3) undergoes a Hopf bifurcation at (0,0,0,0) when D i (i D 0; 1; 2; : : :).

13.3 The Direction and Stability of Hopf Bifurcation In the previous section, we have obtained some conditions to ensure that the system (13.3) undergoes a single Hopf bifurcation at the origin when passes through certain critical values. In this section, we study the direction, stability, and the period of the bifurcating periodic solutions. The method we used is based on the normal form method and the center manifold theory introduced by Hassard et al. [12]. Without loss of generality, we denote the critical value j .j D 0; 1; 2; : : :/ by Q D Q1 CQ2 at which system (13.3) undergoes a Hopf bifurcation, where Q1 < Q2 and D Q C D .Q1 C / C Q2 , then D 0 is Hopf bifurcation value of system (13.3). We choose the phase space as C D C.ŒQ2 ; 0; C 4 /, where for convenience in computation we use C 4 instead of R4 . Letting X.t/ D .x1 .t/; x2 .t/; x3 .t/; x4 .t//T and Xt ./ D X.t C / 2 C .Q2 0/, we can transform system (13.3) into an operator of the form: XP .t/ D L .Xt / C G.; Xt /;

(13.26)

L ./ D B.0/ C B1 .Q1 / C B2 .Q2 /;

(13.27)

with and

0

1 C1 3 3 .Q1 / C B C 3 B C 0 B C G.; / D B C; B 3 .0/ C2 3 .Q / C C 2 @ 3 A 1 3 0 13 .0/

(13.28)

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where ./ D .1 ./; 2 ./; 3 ./; 4 .//T 2 C , and 2

a 1 6 1 b1 B D6 40 0 0 0

2

3 0 0 0 0 7 7; a 1 5 1 b2

0 60 B1 D6 40 0

0 0 0 0

C1 0 0 0

2

3 0 07 7; 05 0

0 6 0 B2 D6 4 C2 0

0 0 0 0

0 0 0 0

3 0 07 7: 05 0

By the Riesz representation theorem, there exists a 4 4 matrix function .; /, whose elements are of bounded variation, such that Z L ./ D

0

Œd.; / ./;

for 2 C:

(13.29)

Q 2

For 2 C , define the operator A. / as 8 < d./ ; A. /./ D R d : 0 Œd.; /./; Q 2

2 ŒQ2 ; 0/; D 0:

(13.30)

We further define the operator R. / as R. /./ D

2 ŒQ2 ; 0/; D 0:

0; G.; /;

(13.31)

Then, the system (13.26) is equivalent to the following operator equation XP t D A. /Xt C R. /Xt : Letting C D C.Œ0; Q2 ; C 4 /, for A

(13.32)

2 C , A is defined by

8 < d'.s/ ; .s/ D R ds : 0 './d.; 0/; Q 2

s 2 .0; Q2 ; s D 0;

(13.33)

and a bilinear form Z

0

Z

'. N /d././d;

h .s/ ; ./i D '.0/.0/ N Q 2

(13.34)

D0

where ./ D .; 0/. Then A.0/ and A are adjoint operators. From the discussion in Sect. 13.2, we know that ˙i!k are eigenvalues of A.0/ and therefore they are also eigenvalues of A , without loss of generality, we write ˙i!k as ˙i!0 , that is, A.0/q./ D i!0 q./;

A q .s/ D i!0 q .s/:

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Bifurcation Analysis in FHN Neurons with Delays

165

It is not difficult to verify that the vectors q./ D .q1 ; q2 ; q3 ; 1/T ei!0 1 i!0 s . 2 ŒQ2 ; 0/ and q .s/ D D .s 2 Œ0; Q2 / are the eigenN .q1 ; q2 ; q3 ; 1/e vectors of A.0/ and A corresponding to the eigenvalue i!0 and i!0 , respectively. By direct computation, we obtain 1 i!0 Q2 e Œ1 C .i!0 a/.i!0 C b2 /; C2 1 ei!0 Q2 Œ1 C .i!0 a/.i!0 C b2 /; q2 D C2 .i!0 C b1 / q3 D i!0 C b2 :

q1 D

(13.35)

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i 2 h N 2 g21 D q1 3q1 qN1 C C1 q32 qN3 ei!0 Q1 C qN3 3q32 qN 3 C C2 q12 qN1 ei!0 Q2 : D (13.38) Because each gij in (13.38) is expressed by the parameters and delay in system (13.3), we can compute the following quantities: g21 g21 i jg02 j2 g11 g20 2jg11 j2 C D ; 2!0 Q 3 2 2 Re.c1 .0// 2 D ; Re.0 .// Q ˇ2 D 2Re.c1 .0//; Im.c1 .0// C 2 Im.0 .Q // T2 D : !0

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From the discussion in Sects. 13.2 and 13.3, we have the following result immediately. Theorem 2. The direction of the Hopf bifurcation of the system (13.3) at the origin when D i .i D 0; 1; 2; : : :/ is supercritical (subcritical) if 2 > 0 .2 < 0/, that is, there exist the bifurcating periodic solutions for > i . < i /; The bifurcating periodic solutions on the center manifold are stable (unstable) if ˇ2 < 0 .ˇ2 > 0/; The period of the bifurcating periodic solutions increases (decreases) if T2 > 0 .T2 < 0/. Acknowledgment This research is supported by the National Science Foundation of China under Grant No. 10772140 and Harbin Institute Technology (Weihai) Science Foundation with No. HIT(WH)ZB200812.

References 1. Hodgkin AL, Huxley AF (1952) A quantitative description of membrane and its application to conduction and excitation in nerve. J Physiol 117:500–544 2. FitzHugh R (1961) Impulses and physiological state in theoretical models of nerve membrane Biophys J 1:445–466 3. Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc IRE 50:2061–2070 4. Bautin AN (1975) Qualitative investigation of a particular nonlinear system. J Appl Math Mech 39:606–615 5. Tetsushi U, Hisayo M, Takuji K, Hiroshi K (2004) Bifurcation and chaos in coupled BVP oscillators. Int J Bifurcat Chaos 4(14):1305–1324 6. Tetsushi U, Hiroshi K (2003) Bifurcation in asymmetrically coupled BVP oscillators. Int J Bifurcat Chaos 5(13):1319–1327 7. Kunichika T, Kazuyuki A, Hiroshi K (2001) Bifurcations in synaptically coupled BVP neurons. Int J Bifurcat Chaos 4(11):1053–1064 8. Dhamala M, Jirsa VK, Ding M (2004) Enhancement of neural synchronization by time delay. Phys Rev Lett 92:028101 9. Nikola B, Dragana T (2003) Dynamics of FitzHugh–Nagumo excitable system with delayed coupling. Phys Rev E 67:066222 10. Nikola B, Inse G, Nebojsa V (2005) Type I vs. type II excitable system with delayed coupling. Chaos Solitons Fractals 23(2):1221–1233 11. Wang Q, Lu Q, Chen G, Feng Z, Duan L (2009) Bifurcation and synchronization of synaptically coupled FHN models with time delay. Chaos Solitons Fractals 39:918–925 12. Hassard BD, Kazarinoff ND, Wan YH (1981) Theory and applications of Hopf bifurcation. Cambridge University Press, Cambridge 13. Yuan Y, Campbell SA (2004) Stability and synchronization of a ring of identical cells with delayed coupling. J Dyn Differ Equ 16(3):709–744 14. Guo S, Huang L (2003) Hopf bifurcating periodic orbits in a ring of neurons with delays. Pyhsica D 183:19–44 15. Wei J, Ruan S (1999) Stability and bifurcation in a neural network model with two delays. Physica D 130:255–272 16. Ruan S, Wei J (2003) On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dyn Continuous Discrete Impulsive Syst A: Math Anal 10:863–874

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17. Wei J, Velarde M (2004) Bifurcation analysis and existence of periodic solutions in a simple neural network with delays. Chaos 14(3):940–952 18. Wei J, Yuan Y (2005) Synchronized Hopf bifurcation analysis in a neural network model with delays. J Math Anal Appl 312:205–229 19. Wang L, Zou X (2004) Hopf bifurcation in bidirectional associative memory neural networks with delays: analysis and computation. J Comp Appl Math 167:73–90 20. Fan D, Wei J (2008) Hopf bifurcation analysis in a tri-neuron network with time delay. Nonlinear Anal: Real World Appl 9:9–25 21. Wu J, Faria T, Huang YS (1999) Synchronization and stable phase-locking in a network of neurons with memory. Math Comput Model 30:117–138 22. Wu J (1998) Symmetric functional differential equations and neural networks with memory. Trans Am Math Soc 350:4799–4838 23. Wu J (2001) Introduction to neural dynamics and single transmission delay. Walter de Gruyter, Berlin 24. Li X, Wei J (2005) On the zeros of a fourth degree exponential polynomial with applications to a neural network model with delays. Chaos Solitons Fractals 26:519–526 25. Hu H, Huang L (2009) Stability and Hopf bifurcation analysis on a ring of four neurons with delays. Appl Math Comput 213:587–599

Chapter 14

On the Feedback Controlling of the Neuronal System with Time Delay Hao Liu, Wuyin Jin, Chi Zhang, Ruicheng Feng, and Aihua Zhang

Abstract For an individual Hindmarsh–Rose model neuron with time delay and dynamic threshold, the dynamic considerations of firing depending on time delay and synaptic intensity are studied in this chapter. It is found out that with the scaling of the delay time and synaptic intensity ", neural firing patterns transform among tonic spiking, busting, and resting firing states each other; as an example, with two groups of the time delays and synaptic intensity " together, the neuronal chaotic firing behavior could be controlled to period-1 or period-3 activities, respectively.

14.1 Introduction As we all know, there is a growing evidence that the hysteresis phenomenon exists in the dynamical system inevitably, i.e., the systemical development direction depends not only on the current state but also on their passed state; specifically, there is always a time-lag response to the input of system. Many scientists, including neuroscientists and biologists, have done a lot of research on dynamical system with time delay, acquiring a great deal of significant achievements, moreover, controlling the dynamical system by time delay skillfully [1–3]. For example, it is important to control chaos with feedback of time delay. Due to the time delay, the system appears to have abundant dynamical behaviors and its characteristics have been altered, such as a delay of cells, the propagation delay, and synaptic delay in the biological systems [4]. Then, it is necessary and important to study the time delays in transmition of neuro-information among the neurons. Initially the neuron system with time delay is studied, which consists of W. Jin () School of Mechano-Electronic Engineering, Lanzhou University of Technology, Lanzhou 730050, People’s Republic of China and Key Laboratory of Digital Manufacturing Technology and Application, The Ministry of Education, Lanzhou University of Technology, Lanzhou 730050, People’s Republic of China e-mail: [email protected]

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an individual neuron or a pair of neurons with delayed coupling, obtaining a sense of results especially in research of stability of system; Yu and Peng Jian-hua have controlled the chaotic activities to one 5 spikes/burst orbit embedded in the chaotic attractor by one nonlinear time-continuous feedback perturbation stimulus of membrane potential [5]. Adding the time delay to the feedback signal, the dynamical system could be controlled to some target orbits. Besides the method of OGY, the adaptive control, OPF control, transfer and displace control, periodic disturbances with parameters, and periodic vibration are widely used in dynamical system [6–8]. Pyragas proposes two kinds of control strategy to the continuous system based on the idea of OGY method for discrete data, i.e., by using the feedback controlling, add one bit disturbance with continuous time on the system. The disturbance cannot change the unstable period orbits (UPOs), but these UPOs could stabilize under some conditions [9, 10]. The feedback controlling with time delay has been widely adapted to control dynamical system and has been realized in experimental study. For neural system, it is useful and also important to use explicitly the time delays in the description of the transfer of information between the neurons, owing to a single neuron that might influence a recurrent loop through an autosynapse and/or through synaptic connections involving other neurons, and synaptic communication between neurons depends on propagation of action potentials along the axons. Diez-Martinez and Segundo studied experimentally the pacemaker neuron in the crayfish stretch receptor organ and showed that as the transmission delay time was increased the discharge patterns went from periodic spikes to trains of spikes separated by silent intervals [4, 11].

14.2 HR Model Neuron with Time Delay Many biological systems operate under the influence of time-delayed feedback mechanisms, and excitable cells can exhibit dissimilar firing patterns to various time delay . The influence of time delay on chaotic system has attracted particular attention extensively, and an individual neuron can exhibit different intrinsic oscillatory activities introduced by external currents [12, 13]. However, it is also interesting to analyze discharge activities induced by the time-delayed synaptic interaction between neurons without external current. In this case, the properties of discharge activities of each neuron depend on the synaptic intensity " and the time delay . Here, the one time-delay HR model neuron with external stimulus I is introduced, and the change of discharge patterns of the neuron are studied on synaptic intensity " and the time delay , as well as their dynamical behavior, and the model is described as follows: 8 3 2 ˆ ˆ <xP D y ax C bx z C I C ".x.t // ; (14.1) yP D c dx 2 y ˆ ˆ :zP D rŒS.x / z

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where x is the membrane potential of neuron, y is a recovery variable associated with the fast current for a sodium current or a potassium current, and z is a slowly changing adaptation current for a calcium current [14]. The model used the following parameters: a D 1:0; b D 3:0; c D 1:0; d D 5:0; D 1:6; r D 0:013; s D 4:0; I D 3:1. The first equation of model (14.1) describes the effect of feedback connection. x.t / is the membrane potential at the earlier time t , here " is the synaptic intensity and is the time delay.

14.3 Influence of Synaptic Intensity " on the Neuronal Discharge The dynamic variation of neuronal discharge depending on the synaptic intensity " are first studied in the absence of external stimulus with a fixed time-delay selffeedback input, here D 14 ms. It is seen from Fig. 14.1 that there is no firing for a smaller " and the membrane potential exhibits only subthreshold oscillations. When synaptic intensity " a

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increases greater than the threshold "th , for different synaptic strength, the neuron displays different discharge patterns (shown in the left of Fig. 14.1) and phase space (shown in the right of Fig. 14.1), transforming among resting, tonic, and bursting state. For fixed D 14 ms, with synaptic intensity " changing from 1.5 to 4:6 ms1 , the neuronal spike changes from period-1 to period-5 as shown in Fig. 14.1a–e, respectively; sometimes, the period-2, period-3, period-4, etc. are generally called bursting spike. We also found out that the neuron firing behavior disappears and the membrane potential keeps at a constant value for too bigger synaptic intensity of ", (e.g., " 15 ms1 ).

14.4 Influence of Time Delay £ on the Neuronal Discharge Next, the time delay is the other one important factor in neuroinformation transferring and disposal for neural system, so the influence depending on time delay of feedback control can also be investigated in this section, the synaptic intensity is fixed at 3:18 ms1 . The neuron displays rest state when the delay time smaller than 5 ms, as shown in Fig. 14.2a. When rises up to 5 ms gradually, the neuron exhibits likewise different discharge patterns (shown in the left of Fig. 14.2) and phase space (shown in the right of Fig. 14.2), as well as realizing the transition from testing, spiking, bursting state. For fixed " D 3:18 ms1 , with time delay changing from 5 to 14 ms, the neuronal firing changes from period-1 to period-4 as shown in Fig. 14.2b–e, respectively. With positive feedback with time delay, both " and could affect the firing patterns of the HR neuron in the absence of the stimulus current, along with increasing of the synaptic intensity or time delay, the neuron will change from periodic spike to burst spike actives.

14.5 Controlling Chaotic Discharge by Feedback Control with Time Delay The controlling of chaotic discharges of the HR model neuron is investigated in this work. One external stimulus current is introduced to the first equation of neuron model [shown in (14.1)] to generate chaotic discharges in the absence of time-delay part; here, the external stimulus is set as 3:12 A. The neuronal discharge represents chaotic firing, as shown in Fig. 14.3a, and its chaotic characteristics, chaotic saddle, could be easily found in the return map of ISI (interspike interval) as shown in Fig. 14.3b. Then the feedback control with time delay is added to the model too; the results suggested that the neuronal chaotic discharges could be controlled to tonic firing, as two examples, when the synaptic strength " D 1:8 ms1 and time

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delay D 6 ms, the chaotic discharges are replaced by period-1 discharges (see Fig. 14.3c, d), and when " D 1:76 ms1 and D 12 ms the neuron displays period3 activities (see Fig. 14.3e, f).

14.6 Conclusions The firing responses of an individual HR model neuron with time delay by synapse have performed numerical investigations in this work; for the different feedback inputs, the neuron shows rich and complex dynamical firing activities. The scaling of synaptic intensity " and time delay could trigger many kinds of periodic

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oscillation in the absence of the stimulus current, and the neuronal chaotic discharge could also be controlled to period-1 or period-3 discharge by feedback control with time delay. Acknowledgments We are grateful for the support of the National Natural Science Foundation of China under Grant Nos. 30670529 and 10572056.

References 1. Just W, Bernard T, Ostheimer M, Reibold E, Benner H (1997) Mechanism of time-delayed feedback control. Phys Rev Lett 78(2):203–206 2. Hu HY, Wang ZH (1998) Stability analysis of damped SDOF systems with two time delays in state feedback. J Sound Vib 214:213–225

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3. Ji CJ, Leung AYT (2002) Resonances of a non-linear sdof system with two time-delays in linear feedback control. J Sound Vib 253(5):985–1000 4. Sainz-Trap´aga M Masoller C, Braun HA Huber MT (2004) Influence of time-delayed feedback in the firing pattern of thermally sensitive neurons. Phys Rev E 70:1–11 5. Yu HJ, Peng JH (2005) Chaotic control of the Hindmarsh–Rose neuron model. Acta Biophysica Sin 21(4):295–300 6. Ott E, Grebogi C Yorke JA (1990) Controlling chaos. Phys Rev Lett 64(11):1196–1199 7. Sinha H (1991) An efficient control algorithm for nonlinear systems. Phys Lett A 156:475–478 8. Huberman BA Lumer EL (1990) Dynamics of adaptive system. IEEE Trans on CAS 37(4):547–550 9. Pyragas K (1993) Experimental control of chaos by delayed self controlling feedback. Phys Lett A 180:99–102 10. Pyragas K (1992) Continuous control of chaos by self-controlling feedback. Phys Lett A 170:421 11. Diez-Martines O, Segundo JP (1983) Behavior of a single neuron in a recurrent excitatory loop. Biol Cybern 47:33–41 12. Gong YF, Xu JX, Ren W, Hu SJ, Wang FZ (1996) Bifurcation, chaos and control in nervous system, Acta Biophys Sin 12:663 13. Zheng YH, Lu QS (2008) Synchronization in ring coupled chaotic neurons with time delay. J Dyn Control 06(3):208–212 14. Hindmarsh JL, Rose RM (1984) A model of the nerve impulse using two first-order differential equations Nature 296:162

Chapter 15

Control of Erosion of Safe Basins in a Single Degree of Freedom Yaw System of a Ship with a Delayed Position Feedback Huilin Shang

Abstract A single degree of freedom nonlinear yaw system of a ship with autopilot force and wave exciting force is investigated in this chapter. The delayed position feedbacks are applied to control erosion of safe basins in the dynamical system for increasing the ship’s safe possibility. Considering time delay as a variable parameter and employing the fourth Runge–Kutta and Monte Carlo methods, the evolutions of boundary and area of safe basins with time delay are presented. For a short delay, the mechanism of the evolution of safe basin is studied analytically. It is found that the delayed position feedback can be used as a good strategy to control the erosion of safe basins of the rolling motion system.

15.1 Introduction It is well known that the unacceptably large motions in engineering dynamical system under consideration may lead to the failure of the system [1–5]. For example, ship at sea may capsize under wave excitation when the roll-motion angle of the ship is too large [2–4]. The safe basin is induced to study the phenomena which is the union of basins of attraction of all bounded solutions for a system [2,3]. The erosion of safe basins due to the penetration of tongues from the “dangerous” attractors is an unwanted and dangerous phenomenon from a practical point of view. Therefore, it is necessary to investigate the basin erosion and its possible control. The control methods, which are used to enlighten the loss of safety of the structure, have been devoted much attention during decades [6–10]. Various integrity measures, such as GIM [6], LIM [6], and IF [7], are applied to study the evolution of the system. Variation of a system parameter may induce or control the basin erosion globally [8–10]. It is found that the erosion of safe basins can be aggravated by the Gaussian white

H. Shang () School of Mechanical and Automation Engineering, Shanghai Institute of Technology, Shanghai 20035, People’s Republic of China e-mail: [email protected]

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or bounded noise excitation [8] but be controlled by the increasing of the damping [9, 10] or the decreasing of the excitation amplitude [11]. In this chapter, we aim to control the erosion of safe basins in nonlinear systems by the delayed position feedbacks. Many obtained results [12–14] shows that applying a delayed feedback in a dynamical system also may be a good approach to control dynamical motions of the system. We study a nonlinear yaw system of a ship, where the autopilot force acted on and the wave exciting force are taken into account (15.1) xR C c xP C x x 3 D G cos t; which is a softening Duffing oscillator and x in (15.1) represents the roll-motion angle of the ship. By adding the delayed position feedback to the system, one may obtain that xP D y; yP D cy x C x 3 C G cos t C A.x.t / x/;

(15.2)

where is time delay and A the feedback gain. When D 0, the system (15.2) can be reduced to the system (15.1). For a system modeled by an ordinary differential equation (ODE), the safe basin is defined [2]. However, it is difficult to describe and study the safe basin for DDE, since initial conditions are completely different from those for ODE. Solutions of DDE are determined by initial conditions z.t/ D z0 for t D 0 and z.t/ D .t/ for t < 0. Thus, regions of attraction of various solutions are located within a polyhedron composed of the initial phase space with the thickness . It is unintuitively to observe dynamical behaviors on such regions of attraction. However, the regions of attraction can be projected to the initial phase space given by z.0/ D fz1 .0/; z2 .0/; : : : ; zn .0/gT for controlled systems with delayed feedbacks, since there is not any signal to be returned into systems before t D 0. Therefore, the safe basin in (15.2) can be studied on the plane where the x-axis is x.0/ and the y-axis y.0/. In this chapter, the effect of the delay or the feedback gain on the erosion of safe basins is discussed in some detail in system (15.2). Section 15.2 presents variations of boundaries of safe basins numerically when one changes the delay in the delayed position feedback. In Sect. 15.3, the variation of the basin area with the delay is displayed and the mechanism of the variation is explained when the delay is short. In Sect. 15.4, results are summarized and discussed.

15.2 Effects of the Time Delay on the Basin Boundary To determine boundaries of the solutions of the system (15.2), we generate figures of safe basins when ˝ D 1:0; c D 0:01; A D 0:2, and increases from 0 to 7 =5 as shown in Figs. 15.1 and 15.2 for G D 0:1 and G D 0:3, respectively. The region of attraction is drawn in the sufficiently large space region defined as

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2 x.0/ 2; 2 y.0/ 2 by generating a 640 240 array of starting conditions for each of those starting points. The escaping set for infinite time is approximated with good accuracy by a study with 10,000 excited circles. The time step is taken as 1/4,000 of the period of excitation. The white and black regions are numerical approximations to the safe basin and the basins of attraction of unbounded solutions, respectively. When G ¤ 0, the system (15.2) is a nonautonomous system. It is well known that the boundary of the safe basin in a nonautonomous system may be fractal and have a noninteger dimension without time delay [10, 11]. Figure 15.1 presents a sequence of safe basins for (15.2) when G D 0:1 and the delay varies. For D 0, the safe basin is fractal as shown in Fig. 15.1a. With increasing from 0 to 4 =5 (see Figs. 15.1a–e), the safe basin is enlarged, and its boundary becomes smoother and smoother. When grows to 2 =25 (see Fig. 15.1e), the fractal dimension is near 2.0. As the delay increases from 4 =5 to (see Fig. 15.1f, g), the safe basin becomes small but its boundary is still completely smooth. Then a small increase for the value of the delay makes the fingers of the basin return back to the boundary (see Fig. 15.1h). Figure 15.1h–l shows the evolution of the basin by fractal with increasing. When D 27 =25, fingers from fractal suddenly begin to erode the inner region of the safe basin, which leads to the strong erosion as shown in Fig. 15.1i. As the delay continues to grow, the fractal structure also becomes more and more visible. It follows from Fig. 15.1h–l that such evolution of the erosion in the inner

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region of the basin is performed continuously until there is only a trivial point (see Fig. 15.1l). Accordingly, the change of the topological structure of the safe basin in Fig. 15.1 confirms that a short delay can reduce but a long delay may aggravate the fractal erosion of the safe basin. Figure 15.2 shows the change of the boundary and range of the safe basin of (15.2) when G D 0:3 and the delay varies. It is drawn in a similar way to Fig. 15.1. For D 0, the safe basin is not smooth and the range is small as shown in Fig. 15.2a. With varying from 0 to 13 =25 (see Fig. 15.2a–d), the area of the safe basin increases with delay though the boundary of the safe basin is still not smooth. With the delay increasing further, the safe basin begins to shrink and the fingers occur in the boundary (see Fig. 15.2e–h), which suggests the fractal behavior. And such erosion in the inner region of the basin is performed continuously until there is only a trivial point as the safe basin (see Fig. 15.2h). Similar to the case for G D 0:1, it follows from Fig. 15.2 that a small value of the time delay is helpful to enlarge the safe basin, but a large value provides the opposite action or aggravates the erosion.

15.3 Effects of the Delay on the Area of the Safe Basin Figures 15.1 and 15.2 indicate that the delay can reduce the erosion or enlarge the extent of safe basins of (15.2). However, the evolution of safe basins with the delay is not simple. When the delay is long enough, it can lead to the sudden erosion of the safe basin. To quantify those phenomena displayed in Figs. 15.1 and 15.2, we define that the area of the “initial basin” is 100% for c D 0:01; G D 0, and D 0 in (15.2) (see Fig. 15.3), since the basin boundary is smooth. Then the change of the basin area with the amplitude G for the different levels of the delay can be observed in Fig. 15.4 where c D 0:01; ˝ D 1:0, and G varies from 0 to 0.8. For A D 0:2 (the dashing-dot line in Fig. 15.4), the area of the safe basin with delayed position feedback control is smaller than that without control. For A D 0:2 (solid lines in Fig. 15.4), the area of the safe basin under delayed position feedback for a small value of the amplitude is bigger than the uneroded “initial basin,” which show the erosion of the safe basin is successfully reduced. The area of the safe basin of the system (15.2) decreases with the increasing of the amplitude under

Fig. 15.3 The initial safe basin

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τ =π 50

τ =0

τ=

π 25 0.2

0.4 G

0.6

0.8

Fig. 15.4 Variation of the basin area with the increasing the amplitude of the excitation for the system (15.2) under different values of time delay where A D 0:2 for solid lines and A D 0:2 for the dashing-dot line

each different value of the delay. As time delay increases from 0, the basin area is first enlarged to maximum value and then shrink (see the solid lines D 4 =5 and D ). Besides, the comparison of the solid lines D 0 and D =5 shows that a small delay can control the basin erosion under different values of G when A is positive. When the delay is short, it is worthwhile to expand the delay variable x.t / in a Taylor series despite occasional warnings [15]. One can write x.t / D x.t/ x.t/ P C

(15.3)

Substituting (15.3) in (15.2), one can obtain that

xP D y; yP D .c C A/y x C x 3 C G cos t:

(15.4)

The unperturbed system can be expressed as

xP D y; yP D x C x 3 ;

(15.5)

which is a Hamiltonian system. Then the system has a hetero-clinic orbit to twosaddle points .1; 0/ and (1, 0), which can be given by p 2 x.t/ D tanh t; 2

p

p 2 2 y.t/ D ˙ sech t: 2 2

(15.6)

184

H. Shang

Obviously, when the feedback gain A is positive, the system (15.4) can be considered as the system (15.1) with a larger damping. As we know, the increasing of the damping can reduce the erosion of safe basins [9, 10]. Therefore, one can explain why the safe basin in (15.2) is enlarged when the delay increases from 0. Besides, one can also conclude that the safe basin in (15.2) will be eroded when the feedback gain is negative and the delay increases from 0, which is verified by the dashing-dot line in Fig. 15.4.

15.4 Conclusions Some investigation has been made on the evolution of the broaching attracting basin of a nonlinear yaw equation model of a ship with delayed position feedbacks. The fourth Runge-Kutta and Monte Carlo methods are employed to observe effects on safe basins when time delay is considered as the control parameter. For a short delay, the mechanism of the evolution of safe basin is analyzed. Some important results are obtained as below: (a) Similar to the case in ordinary differential system, the increasing of excitation amplitude can lead to the erosion of the safe basin under delayed position feedback control. (b) The basin area is not a monotonic function of the delay for the delayed position feedback control. (c) For negative feedbacks, the increasing of the delay aggravates the erosion of safe basins. For positive feedbacks, the delay can be indeed used to reduce the erosion of safe basins. The erosion can be reduced when the delay is short, but a long time delay makes the erosion more sudden and severe. (d) The sudden erosion of the safe basin caused by the increasing of the delay in delayed position feedback controlled system can be ascribed to the hetero-clinic tangency of the manifolds. It is well known that the hetero-clinic tangency of the stable and unstable manifolds may yield chaos. The results provide the possibility to control chaos that occurs in the system. Acknowledgments This work is supported by Shanghai Municipal Education Commission under Grant No. YYY08004, Shanghai Leading Academic Discipline Project under Grant No. J51501, and National Natural Science Foundation of China under Grant No. 10902071.

References 1. Freitas M, Viana R, Grebogi C (2003) Erosion of the safe basin for the transversal oscillations of a suspension bridge. Chaos Solitons Fractals 18(4):829–841 2. Thompson J, Rainey F, Soliman MS (1995) Ship stability criteria based on chaotic transients from incursive fractals. Philos Trans R Soc Lond A 332(1):149–167 3. Thompson J, McRobie F (1993) Indeterminate bifurcation and the global dynamics of driven oscillators. In: Proceedings of 1st European nonlinear oscillation’s conference. Hamburg, Germany

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Control of Safe Basin by Delayed Feedbacks

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4. Senjanovic I, Parunov J, Cipric G (1997) Safety analysis of ship rolling in rough sea. Chaos Solitons Fractals 8(4):659–680 5. Soliman M (1995) Fractal erosion of basins of attraction in coupled nonlinear systems. J Sound Vib 182(5):729–740 6. Soliman M, Thompson J (1989) Intergrity measures quantifying the erosion of smooth and fractal basins of attraction. J Sound Vib 35(3):453–475 7. Rega G, Lenci S (2005) Identifying, evaluating, and controlling dynamical integrity measures in non-linear mechanical oscillators. Nonlinear Anal 63(5–7):902–914 8. Gan C (2005) Noise-induced chaos and safe basin in softening Duffing oscillator. Chaos Solitons Fractals 25(5):1069–1081 9. Bishop S, Galvanetto U (1993) The influence of ramped forcing on safe basins in a mechanical oscillator. Dyn Stab Syst 8(2):73–80 10. Moon F, Li G (1985) Fractal basin boundaries and homoclinic orbits for periodic motion in a two-well potential. Physica Rev Lett 55(14):1439–1442 11. Xu J, Lu Q, Huang K (1996) Controlling erosion of safe basin in nonlinear parametrically excited systems. Acta Mechanica Sin 12(3):281–288 12. Shang H, Xu J (2008) Multiple periodic solutions in Lienard oscillator with delayed position feedbacks. J Tongji Univ (Nat Sci) 36(7):962–966 13. Xu J, Chung K (2003) Effects of time delayed position feedback on a van der Pol–Duffing oscillator. Physica D 180(1–2):17–39 14. Maccari A (2006) Vibration control for parametrically excited Li´enard systems. Nonlinear Mech 41(1):146–155 15. Driver D (1977) Ordinary and delay differential equations. Springer, New York

Part III

Switching and Stochastic Dynamical Systems

Chapter 16

On Periodic Flows of a 3-D Switching System with Many Subsystems Albert C.J. Luo and Yang Wang

Abstract In this chapter, the stability and bifurcation of periodic flows in a switching system of multiple subsystems with transport laws at switching points is presented. The periodic flows and stability for linear switching systems are discussed as an example. Analytical prediction of the periodic flow in such linear switching systems is carried out and parameter maps of stability are given. The methodology presented in this chapter can be applied to nonlinear switching systems. The further results on chaos, stability, and bifurcation of periodic flows in nonlinear switching systems will be presented in sequel.

16.1 Introduction Consider a C ri -continuous system (ri > 1) on an open domain Di Rn , in the time interval t 2 Œtk1 ; tk

xP .i / D F.i / x.i / ; t; p.i / 2 Rn I

T

/ .i / .i / x.i / D x.i ; x ; : : : ; x 2 Di : n 1 2

(16.1)

T

.i / p1.i / ; p2.i / ; : : : ; pm 2 Rmi is a i parameter vector. On the domain Di Rn , the vector field F.i / xP .i / ; t; p.i / with the parameter vector p.i / is C ri -continuous in x.i / for time interval t 2 Œtk1 ; tk . .i / With an initial condition x.i /.tk1 / D xk1 , the dynamical system in (16.1) possesses a continuous flow as The time is t and xP .i / D dx.i /=dt. p.i / D

/ .i / I ; t; p x.i / .t/ D ˆ .i / x.i k1

/ .i / .i / .i / x.i x : D ˆ ; t ; p k1 k1 k1

(16.2)

A.C.J. Luo () Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL62026-1805, USA e-mail: [email protected]

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 16,

189

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To investigate the switching system consisting of many subsystems, the following assumptions of the i th subsystem should be held. (A1)

/ F.i / x.i / ; t; p.i / 2 C ri I ˆ .i / x.i ; t; p.i / 2 C ri C1 I on Di for t 2 Œtk1 ; tk ; k (16.3) (A2) jjF.i / jjK1.i /.const/I jjˆ .i / jj K2.i /.const/ on Di for t 2 Œtk1 ; tk ;

(16.4)

(A3) x.i / D ˆ .i / .t/ … @Di for t 2 .tk1 ; tk /:

(16.5)

(A4) The switching of any two subsystems possesses the time continuity. To investigate the switching system, a set of dynamical systems in finite time intervals will be introduced first. From such a set of dynamical systems, the dynamical subsystems in a resultant switching system can be selected. Definition 1. From dynamical systems in (16.1), a set of dynamical systems on the open domain Di in the time interval t 2 Œtk1 ; tk for i D 1; 2; : : : ; m is defined as SD f Si j i D 1; 2; : : : ; mg ; (16.6) where ( Si xP

.i /

DF

.i /

.i /

.i /

x ; t; p

ˇ ) ˇ .i / 2 D Rn ; p.i / 2 Rmi I i 2 R ˇ .i / : / ˇ x .tk1 / D x.i k1 I t 2 Œtk1 ; tk I k 2 N n ˇx

(16.7) From Assumptions (A1)–(A3), the subsystem possesses a finite solution in the finite time interval and such a solution will not reach the corresponding domain boundary. From the set of subsystems, the corresponding set of solutions for such subsystems can be defined as follows. Definition 2. For the i th dynamical subsystems in (16.1), with an initial condition

/ .i / .i / .i / .i / 2 D for k 2 N and, there is a unique solution x .t/ D ˆ ; t; p x . x.i i k1 k1 For all i D 1; 2; : : : ; m, a set of solutions for the i th subsystem in (16.1) on the open domain Di in the time interval t 2 Œtk1 ; tk is defined as n o S D ‚.i / ji D 1; 2; : : : ; m ;

(16.8)

where ˇ

n ˇ .i / ‚.i / x.i / .t/ ˇx.i / .t/ D ˆ .i / xk1 ; t; tk1 ; p.i / I

o t 2 Œtk1 ; tk ; k 2 N : (16.9)

16

On Periodic Flows of a 3-D Switching System with Many Subsystems

191

This class of switching systems exists in network systems and control systems [1] and widely applied to electronic power systems [2]. The survey of such switching systems can be referred to [3]. To obtain the complex responses of switching systems, numerical schemes can be developed. For instance, Danca [4] used an explicit Euler method to give a numerical integration scheme for switching dynamical systems to obtain approximate numerical solutions. Grune and Kloeden [5] used the Taylor series scheme to develop a higher order numerical integration scheme to carry out numerical simulations of switching systems. Gokcek [6] used the Floquet theory to discuss the stability of switching systems. Indeed, approximate numerical methods can help one understand the behaviors of switching systems. However, for the traditional analytical and numerical methods, it is very difficult to handle the discontinuity for the switching of two systems, and it is impossible to provide an accurate solution for switching systems. In 2005, Luo [7] developed a local theory of singularity for such switching systems in order to exactly switch from a subsystem to another subsystem (also, see Luo [8]). In 2008, Luo and Wang [9] presented a general concept of switching systems to investigate switching dynamics of multiple linear oscillators. With periodic and random piecewise forcing, the solutions of dynamical oscillation were presented. Periodic motions of switching systems with vector field switching were investigated. This chapter will apply the methodology for switching system in Luo and Wang [9] to 3-D switching dynamical systems. The periodic flow and corresponding stability of the 3-D switching system will be investigated. The parameter map for the 3-D linear switching systems will be presented and numerical illustrations for periodic flow will be carried out.

16.2 Methodology for Periodic Flows To describe the switching of subsystems, consider a switching set for the i th subsystem to be ˇ n o / ˇ .i / x D x.i / .tk /; k D 0; 1; 2; : : : : (16.10) †.i / D x.i k ˇ k From the solution of the i th subsystem, a mapping Pi for a time interval Œtk1 ; tk is defined as / / Pi W †.i / ! †.i / or Pi W x.i ! x.i k1 k

for i D 1; 2; : : : ; m:

(16.11)

Define a time difference parameter for the i th subsystem for time interval Œtk1 ; tk ˛k.i / D tk tk1

(16.12)

which is set arbitrarily. For simplicity, introduce two vectors herein T

.i / .i / f .i / D f1 ; f2 ; fm.i / I

T

.i / .i / .i / x.i / D x1 ; x2 ; : : : ; xm :

(16.13)

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A.C.J. Luo and Y. Wang

From the solution in (16.9) for the i th subsystem, the foregoing equation gives for (i D 1; 2; : : : ; m)

/ .i / .i / .i / .i / .i / D x x D 0: f .i / x.i ; x ˆ ; t ; t ; p k k1 k k1 k k1

(16.14)

Suppose the two trajectories of the i th and j th subsystems in phase space at the switching time tk is continuous, i.e., for i; j 2 f1; 2; : : : ; mg at time tk , .j /

xk

.i / .i / .i / .j / .j / D xk or x1;k ; : : : ; xm;k D x1;k ; : : : ; xm;k

(16.15)

If the two solutions of the i th and j th subsystems at the switching time tk are discontinuous, for instance, an impulsive system needs a transport law. From Luo [2], a vector for the transport law is introduced as T

.ij/ .ij/ .ij/ (16.16) g.ij/ D g1 ; g2 ; : : : ; gm so the transport law between the i th and j th subsystems can be written as

/ .j / g.ij/ x.i D0 ; x k k

at time tk :

(16.17)

In other words, one obtains

9 .ij/ .ij/ .j / .i / .i / .j / .i / .i / ; x2.k/ ;> D g1 x1.k/ ; : : : ; xm.k/ D g2 x1.k/ ; : : : ; xm.k/ x1.k/ > =

.j / xm.k/

D

.ij/ gm

.i / .i / x1.k/ ; ; : : : ; xm.k/

for i; j 2 f1; 2; : : : ; mg:

> > ;

:

(16.18)

From the transport law, a transport mapping is introduced as for i; j 2 f1; 2; : : : ; mg .ij/

P0

W †.i / ! †.j / ;

(16.19)

i.e.,

.ij/ / .j / .ij / .i / .i / .j / .j / ! x :(16.20) !x or P W x ; : : : ; x ; : : : ; x P0 Wx.i 0 k k 1.k/ m.k/ 1.k/ m.k/ .l lnC1 /

P D P0 n

.l l2 /

ı Pln ı ı Pl2 ı P0 1

ı Pl1 Pln : : :l2 l1

(16.21)

The algebraic equations for the transport mapping are given in (16.18). The mapping Pi for the i th subsystem for time t 2 Œtk1 ; tk and the transport mapping at time t D tk are sketched in Figs. 16.1 and 16.2. The initial and final points of mapping Pi / / / are x.i and x.i . Similarly, the initial and final points of mapping Pj are x.j and k1 k k .j / xkC1 . The mappings relative to the subsystem are sketched by a solid curve. The two mappings are connected by a transport mapping at t D tk , which is depicted

16

On Periodic Flows of a 3-D Switching System with Many Subsystems

a

b

( j)

xk+1

Pi (i )

Pi

(i )

xk−1

193

xk−1

Pj

(ij)

P0 (i )

xn2

xk

xn2

xk xn1

( j)

xk+1

xn1 .ij/

Fig. 16.1 (a) Mapping Pi and (b) transport mapping P0

( j)

b

a

in phase plane (n1 C n2 D n)

Pi

(i)

xk−1

(i)

xk−1

xk

Pi (i)

(i)

xk

xk

( j)

xk+1

x

x tk−1

Pj

(ij)

P0

tk

tk − 1

tk

tk + 1

t

t

.ij/

Fig. 16.2 (a) Mapping Pi and (b) transport mapping P0

in the time history

by the dashed line. In phase plane, there is a nonnegative distance governed by (16.17). However, the time-history of flows for the switching system experiences a jump at time t D tk . If the transport law gives a special case to satisfy (16.14), the solutions of two subsystems are C 0 -continuous at the switching time t D tk . The jump phenomenon will disappear. Consider a flow of the resultant switching system to have a mapping structure for t 2 [njD1 tkCj 1 ; tkCj as where Plk 2 f Pi j i D 1; 2; : : : ; ng :

(16.22)

T

.l1 / .l1 / ; : : : ; xm.k/ at t D tk and final state Consider an initial state xk.l1 / D x1.k/

T .lnC1 / .lnC1 / .lnC1 / xkCn D x1.kCn/ ; : : : ; xm.kCn/ at t D tkCn , respectively. .l

/

.l /

.l lnC1 /

nC1 xkCn D P xk 1 D P0 n

.l l2 /

ı Pln ı ı Pl2 ı P0 1

.l /

ı Pl1 xk 1 :

(16.23)

with each time difference, one has the total time difference tkCn tk D

Xn mD1

m/ ˛.l kCm :

(16.24)

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A.C.J. Luo and Y. Wang

In addition, (16.25) gives the following mapping relations 9 > > > > > > > > > > > > > > =

.l1 / .l1 / xkC1 D Pl1 xk.l1 / ) Pl1 W xk.l1 / ! xkC1 .l /

.l l2 / .l1 / xkC1 .l2 / Pl2 xkC1 ) .l2 / P0.l2 l3 / xkC2

.l l2 /

.l /

.l /

2 xkC1 D P0 1

) P0 1

.l2 / xkC2 D

.l2 / .l2 / Pl2 W xkC1 ! xkC2

.l3 / xkC1 D :: : .l /

1 2 W xkC1 ! xkC1

.l2 / .l3 / ) P0.l1 l3 / W xkC2 ! xkC2

.l /

.l /

.l /

n n n n xkCn D Pln xkCn1 ) Pln W xkCn1 ! xkCn

.l

/

.l lnC1 / .ln / xkCn

nC1 xkCn D P0 n

.l lnC1 /

) P0 n

> > > > > > > > > > > > > ; .lnC1 / >

:

(16.25)

.ln / W xkCn ! xkCn

Mapping relations in (16.20) yields a set of algebraic equations as 9

.l1 / .l1 / .l2 / > f .l1 / xkC1 D 0; ; xk.l1 / ; ˛k.l1 / D 0; g.l1 l2 / xkC1 ; xkC1 > > = :: : >

> > .lnC1 / .ln / .ln / .ln / .ln / D 0; g.ln lnC1 / xkCn D 0: ; f .ln / xkCn ; xkCn1 ; ˛kCn ; xkCn

(16.26)

If there is a periodic motion, the periodicity for tkCn D T C tk is .l

/

.l

/

.l

/

.l1 / .l1 / nC1 nC1 nC1 xkCn D xk.l1 / for lnC1 D l1 or x1;kCn D x1;k ; : : : ; xm;kCn D xP m;k ;

(16.27)

where T is time period. The resultant periodic solution of the switching system is for i D 1; 2; ; n ˇ ˚ ) x.li / .t/ D x.li / .t/ˇ t 2 ŒtkCnsCi 1 ; tkCnsCi for s D 0; 1; 2; : : : ; .l mod .i;n/C1 / .li / D xkCnsCi for s D 0; 1; 2; : : : xkCnsCi

(16.28)

From (16.23) and (16.24), the corresponding switching points for the periodic motion can be determined. From the time difference parameter, the time interval parameter is defined as .lm / D qkCm

m/ ˛.l kCm

T

and

Xn mD1

.lm / qkCm D 1:

(16.29)

If a set of the time interval parameters for switching subsystems during the next period is the same as during the current period, the periodic flow is called the equi-time-interval periodic flow. The pattern of the resultant flow for the switching system during the next period will repeat the pattern of the flow during the current period. If a set of the time interval parameters for the second period is different from the first period, the periodic motion is called the non-equi-time-interval periodic motion. For this flow, the switching pattern during the next period is different

16

On Periodic Flows of a 3-D Switching System with Many Subsystems

195

from the current one. For a general case, during two periods, only one pattern to make (16.28) can be satisfied. Hence, this switching pattern can be treated as a periodic flow with two periods. To determine the stability of such a periodic motion, the Jacobian matrix can be computed, i.e., .l l /

DP D DP0 n 1

.l l2 /

DPln DPl2 DP0 1

DPl1 ;

(16.30)

where for j D 1; 2; : : : ; n, 2 DPlj D 4

D

3

.l /

j @xkCs

2

.lj / .lj / .lj / ; x2.kCs/ ; : : : ; xm.kCs/ @ x1.kCs/

3

5 5 D 4 .l / .lj / .lj / .lj / j @xkCs1 @ x1.kCs1/ ; x2.kCs1/ ; : : : ; xm.kCs1/ mm 3mm 3 2 1 2 .lj / .lj / 5 4 @f 5 4 @f .l /

j @xkCs

.l /

mm

j @xkCs1

mm

because from (16.15) one obtains 2 3 3 2 .lj / .lj / @f @f 5 4 C 4 .l / 5 .lj / j @xkCs1 @xkCs1 mm

(16.31) 2 4

mm

.l /

j @xkCs

.l /

j @xkCs1

3 5

mm

(16.32)

mm

Similarly, from the transport law, one obtains 2 3 3 2 3 2 .lj C1 / .lj lj C1 / .lj lj C1 / @x @g 4 @g 5 4 kCs 5 C 4 .l / 5 .lj / .lj / j C1 @xkCs @xkCs @xkCs mm

D 0:

D 0;

(16.33)

mm

3 3 2 2 .l / .l / .lj C1 / j C1 j C1 .lj C1 / ; x ; : : : ; x @ x @x 1.kCs/ 2.kCs/ m.kCs/ .l l / 5 5 D 4 .l / DP0 j j C1 D 4 kCs .lj / .lj / .lj / j @xkCs @ x1.kCs/ ; x2.kCs/ ; : : : ; xm.kCs/ mm mm 31 2 3 2 .lj lj C1 / .lj lj C1 / @g 4 @g 5 D 4 .l / 5 : .lj / j C1 @xkCs @xkCs (16.34) mm mm If the magnitudes of two eigenvalues of the total Jacobian matrix DP are less than 1 (i.e., j˛ j < 1, ˛ D 1; 2; : : : ; m:/, the periodic motion is stable. If the magnitude of one of two eigenvalues is greater than 1 (j˛ j > 1,˛ 2 f1; 2; : : : ; mg/, the periodic motion is unstable. If one of eigenvalues is positive one (C1) and the rest of eigenvalues are in the unit cycle, the periodic flow experiences a saddle-node bifurcation. If one of eigenvalues is negative one (1) and the rest of eigenvalues are in the unit cycle, the periodic flow possesses a period-doubling bifurcation. If a pair of complex eigenvalues is on the unit cycle and the rest of eigenvalues are in the unit cycle, a Neimark bifurcation of the periodic flow occurs.

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16.3 Analytical Predictions For switching system, periodic motions can be predicted by constructing certain mapping structure and the corresponding stability can be determined by evaluating eigenvalues of Jacobian matrix. To illustrate the stability of the 3-D switching system, consider a linear switching system as an example that includes two 3-D subsystems with two matrices for j D 0; 1; 2; : : : 3 a11 a12 a13 A.1/ D 4a21 a22 a23 5 for t 2 ŒtkC2j ; tkC2j C1 ; a31 a32 a33 3 2 b11 b12 b13 A.2/ D 4b21 b22 b23 5 for t 2 ŒtkC2j C1 ; tkC2j C2 ; b31 b32 b33 T

/ t t0 .i / .i / e ; A .t t /; A for i D 1; 2: Q.i / D A.i 0 1 2 3 2

(16.35)

The linear switching system is expressed by P .i / D A.i / X.i / C Q.i / X

(16.36)

The two subsystems are continuously connected at the switching points. Without losing generality, the parameters for a dynamical system are fixed and the parameters for another subsystem are varied. For instance, select the parameters as a11 D a13 D a23 D b11 D b31 D 1I a22 D a31 D 1I a12 D a21 D 2I a32 D 3I a33 D 3I b21 D 1:5:

(16.37)

To understand the time intervals of the two subsystems for periodic motion, introduce a new parameter as t2kCi t2kCi 1 .i / D q2kCi 1 T X2 Xn .i / 1D q2kCi 1 i D1

.i /

kD1

for i D 1; 2 and for k D 1; 2; : : : ; n:

(16.38)

If q2kCi 1 D q .i / , the i th subsystem possesses the equi-time interval. Otherwise, the i th subsystem possesses the non-equi-time interval. Consider a simple periodic motion with a mapping structure as P D P2 P1 D P21 , and the corresponding time interval parameter can be set as q .1/ and q .2/ . For q .1/ D 0 (q .2/ D 1/, the switching system is formed by the second subsystem only. For q .1/ D 1 (q .2/ D 0/, the switching system is formed by the first subsystem only. Switching points are recorded in Fig. 16.3 for a periodic flow with P D P21 , where P1 is given as a .1/ .1/ mapping from an initial state xk to the final state xkC1 in the first subsystem of

16

On Periodic Flows of a 3-D Switching System with Many Subsystems

Switching, x1(k)

a

100

50 x1(k+1)

0 x1(k)

−50 −100

b

197

0

200

800

1000

800

1000

400 600 Parameter, b32

100

Switching, x2(k)

50 x2(k)

0 x2(k+1)

−50 −100

c

0

200

400 600 Parameter, b32

400

Switching, x3(k)

200 x3(k+1)

0

x3(k)

−200 −400

0

200

400 600 Parameter, b32

800

1000

Fig. 16.3 Periodic motion scenario with P21 for switching 3-D systems (a) switching x1.k/ , (b) switching x2.k/ and (c) switching x3.k/ . q .1/ D 0:25I a11 D a13 D a23 D b11 D b31 D b23 D 1I a22 D a31 D b22 D 1I a12 D a21 D 2I a32 D 3I a33 D 3I b21 D 1:5I b12 D .1/ .1/ .1/ .2/ .2/ .2/ 0:5I b13 D 2I b33 D 2:0I A1 D A2 D A3 D A1 D A2 D 1I A3 D 1I T D 4

the switching system, and P2 represents the mapping between an initial state x.2/ kC1 and the final state x.2/ for the second subsystem. The solid line stands for the kC2 stable periodic response in shaded area and the dashed line represents the unstable response in white area with the parameter (b33 D 2:0; q .1/ D 0:25), where q .1/ is the time interval of first subsystem as defined in (16.38). The similarity of

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the switching points and periodic motion can be observed as parameter b32 varying from 0 to 1,000. The corresponding eigenvalues are given in Fig. 16.4. Once there is amplitude of eigenvalues of Jocabian matrix greater than one, the switching system is unstable. Only if all amplitudes of eigenvalues are less than one, switching system is stable. It is consistent with the results of switching points and eigenvalues. When one real eigenvalue approaches to positive one (C1), switching points of periodic flow goes to infinite and turns to unstable infinite. Meanwhile, if one real eigenvalue is negative one (1), switching points are finite numbers shown as solid points. As long as parameter b32 is increasing, the stable region vanished to zero and values of switching points are expanding. Parameter map of stability region is given in Fig. 16.5 with respect to b32 and b33 . For the boundary of stable and unstable region, b33 and b32 goes to negative infinity in the left-hand plane, and cusps can be found along the boundary in right-hand plane.

b 6.0

5.0

4.0 4.0 Eigenvalues

Magnitude of Eigenvalues

a

3.0 2.0 1.0

2.0 0.0 −2.0 −4.0

0.0 0

200

400 600 Parameter, b32

800

−6.0

1000

0

200

400 600 Parameter, b32

800

1000

Fig. 16.4 Eigenvalue analysis for periodic motions with P21 for switching 3-D systems (a) magnitudes and (b) real and imaginary parts of eigenvalues. q .1/ D 0:25I a12 D a21 D 2I a32 D 3I a33 D 3I a11 D a13 D a23 D b11 D b31 D b23 D 1I a22 D a31 D b22 D 1I b21 D .1/ .1/ .1/ .2/ .2/ .2/ 1:5I b12 D 0:5I b13 D 2I b33 D 2:0I A1 D A2 D A3 D A1 D A2 D 1I A3 D 1I T D 4

a

b 8.0

8.0 Unstable Parameter, b33

Parameter, b33

Unstable 4.0 0.0

−4.0

4.0 0.0 −4.0 Stable

Stable −8.0

0

200

400 600 Parameter, b32

800

1000

−8.0 −20

0

20 40 60 Parameter, b32

80

100

Fig. 16.5 Parameter map for switching 3-D systems, (a) stable and unstable motion regions (b) zoomed view. q .1/ D 0:25; a11 D a13 D a23 D b11 D b31 D b23 D 1; a22 D a31 D b22 D 1; a32 D 3; T D 4; a12 D a21 D 2; a33 D 3; b21 D 1:5; b12 D 0:5; b13 D 2; .1/ .1/ .1/ .2/ .2/ .2/ A1 D A2 D A3 D A1 D A2 D 1; A3 D 1; b12 D 0:5; b13 D 2; b23 D 1

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199

e

a

10.0

T

T

P1

P2

P1

P2

P1

0.0

−5.0

2.5

2.0

5.0 time, t

7.5

−100.0 0.0

10.0

f

T

P2

P1

P2

State, x2

State, x2

P1

5.0 time, t

7.5

P1

T P1 P2

P2 0.0

10.0

P1

P1

−25.0

−12.0 0.0

4.0

2.5

5.0 time, t

7.5

−50.0 0.0

10.0

2.5

400.0 P2

5.0 time, t

g

T

5.0

7.5

10.0

T

P2 200.0

−3.0 P1

P1

P2

State, x3

State, x3

2.5

50.0

−8.5

P1

−6.5

d

P1

25.0

−5.0

−10.0 0.0

P1

P1

−1.5

c

P2

P2 0.0

−50.0

−10.0 0.0

b

P2

50.0 State, x1

State, x1

5.0

P1

P2

0.0 P1

P1

−200.0

2.5

5.0 time, t

7.5

10.0

−400.0 0.0

h

2.0

2.5

5.0 time, t

7.5

10.0

22.0 P1 7.0

x3

x3

−2.0

P2

−6.0

−10.0

−8.0

P1 P2

P2

P1

−23.0 10.0 5.0 0.0 1 −5.0 x

−2.0 x2

−10.0

9.0 5.0 1.0 −3.0 x 1

−38.0 4.0

2.0

0.0 x2

−2.0

−4.0

Fig. 16.6 Periodic flows of P21 for switching 3-D systems. q .1/ D 0:25; a12 D a21 D 2; b13 D 2; b12 D 0:5; a11 D a13 D a23 D b11 D b31 D b23 D 1; a22 D .2/ a31 D b22 D 1; a32 D 3; a33 D 3; b33 D 2:0; b21 D 1:5; A3 D 1; T D .1/ .1/ .1/ .2/ .2/ 4; A1 D A2 D A3 D A1 D A2 D 1: t0 D 0; (a–d) stable periodic flow x1 .t0 / 6:6304; x2 .t0 / 4:3082; x3 .t0 / 2:6031; b32 D 0; (e–h) unstable flow x1 .t0 / 0:7232; x2 .t0 / 3:0431; x3 .t0 / 10:1835; b32 D 100

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16.4 Numerical Illustrations From the analytical prediction, numerical illustrations of periodic flows can provide a comprehensive understanding of the switching systems. Consider the switching system as defined in (16.37) and parameters in (16.36), a stable periodic flow is given in Fig. 16.6(a)–(d) with b32 D 0; q .1/ D 0:25; A.2/ D 1; T D 4; and 3 .1/ .1/ .2/ .2/ A.1/ DA DA DA DA D1. From analytical prediction, the initial condition 1 2 3 1 2 for this periodic flow is x1 .0/ D 6:6304, x2 .0/ D 4:3082, x3 .0/ D 2:6031. The time histories for three state variables (xi , i D 1; 2; 3/ in the periodic flow of the 3-D linear switching system are presented in Fig. 16.6(a)–(c). The solid point represents the initial points and hollow points are switching points when subsystem is switching from one to another. It is observed that all the switching points are continuous but nonsmooth. An unstable flow of P21 is given in Fig. 16.6(e)–(h) with b32 D 100. Even the initial condition is chosen as x1 .0/ D 0:7232, x2 .0/ D 3:0431, x3 .0/ D 10:1835 from analytical prediction, the periodic flow is easily destroyed by a disturbance and the values of each state (xi ,i D 1; 2; 3) will goes to infinity as shown in Fig. 16.6(e)–(g) in time history. The numerical simulations are consistent with analytical predictions.

16.5 Conclusions In this chapter, a switching system of multiple subsystems with transport laws at switching points is discussed. A frame work for periodic flows of such a switching system is presented. To show applications, periodic flows and stability for linear switching systems are discussed as an example. Analytical prediction of periodic flows in such linear switching systems is carried out, and parameter maps for periodic motion stability are developed. Numerical simulations are demonstrated for illustration of stable and unstable motions. For linear switching systems, bifurcations of the periodic flows cannot be observed. This framework can be applied to the nonlinear switching systems. The further results on stability and bifurcation of periodic flows in nonlinear switching systems will be presented in sequel.

References 1. Morse AS (1997) Control using logic-based switching. Lecture notes in control and information sciences, vol 222. Springer, London 2. Sachdev MS, Hakal PD, Sidhu TS (1997) Automated design of substation switching systems. Developments in power system protection, 6th international conference, pp 369–372 3. Liberzon D, Morse AS (1999) Basic problems in stability and design of switched systems. IEEE Control Syst 19(5):59–70 4. Danca MF (2008) Numerical approximations of a class of switch dynamical systems. Chaos Solitons Fractals 38:184–191

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5. Grune L, Kloeden PE (2006) High order numerical approximation of switching system. Syst Control Lett 55:746–754 6. Gokcek C (2004) Stability analysis of periodically switched linear system using Floquet theory. Math Probl Eng 2004(1):1–10 7. Luo ACJ (2005) A theory for non-smooth dynamic systems on the connectable domains. Commun Nonlinear Sci Numer Simul 10:1–55 8. Luo ACJ (2006) Singularity and dynamics on discontinuous vector fields. Elsevier, Amsterdam 9. Luo ACJ, Wang Y (2009) Switching dynamics of multiple linear oscillators. Commun Nonlinear Sci Numer Simul 14:3472–3485

Chapter 17

Impulsive Control Induced Effects on Dynamics of Complex Networks Xiuping Han and Xilin Fu

Abstract Control and synchronization of complex networks have been extensively investigated in many research and application fields. Previous works focused upon realizing synchronization by varied methods. There has been little research on the dynamics of synchronization manifold of complex networks by now. It was known that dynamics of the single system can be changed very obviously after inputting particular impulse signals. For the first time, above impulsive control of complex networks is considered in this chapter. Complex networks can realize to synchronize with such impulsive control. And dynamics of the synchronous state of complex networks can be induced to different orbit. The orbit may be an equilibrium point, a periodic orbit,or a chaotic orbit, which is determined by a parameter in the outer impulse signal. Strict theories are given.

17.1 Introduction Recently, complex networks have received rapidly increasing attentions from different fields. Such as from internet to world wide web, from communication networks to social organizations, from food webs to ecological communities, etc. They widely exist in our life and are presently prominent candidates to describe the sophisticated collaborative dynamics in many sciences [1, 3, 5, 8, 19, 21]. So far, the dynamics of complex networks has been extensively investigated. Control and synchronization are typical topics that have attracted lots of interests [11, 13, 17, 19–21]. Synchronization is a fundamental phenomenon that enables coherent behavior in networks as a result of interactions. And several different approaches including adaptive synchronization [26], robust synchronization [16], and impulsive control [9] have been introduced to solve the above problem. Among these approaches,

X. Fu () School of Mathematical Sciences, Shandong Normal University, Jinan 250014, People’s Republic of China e-mail: [email protected]

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 17,

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studies show that impulsive control strategy [2, 10, 15, 22–24] is very effective and robust while with low cost. In past decades, it has been widely applied in many fields, such as space techniques, information science, control system, dynamical nerve cell networks, and communication security etc. It allows stability of a complex network only by small impulses being sent to the receiving systems at the discrete impulsive instances, which can reduce the information redundancy in the transmitted signal and increase robustness against the disturbances. In this sense, impulsive control schemes have been applied to numerous chaos-based communication systems for cryptographically secure purposes and detailed experiments have been carried out [6, 7, 12]. Previous work on impulsive control and synchronization of complex networks are focused on normal impulsive effects. The impulsive input only contains state variable. Little work has been done for the synchronization of networks with special impulsive control, which contains outer signals. Control of chaotic systems to periodic motions has been discussed using proper impulsive input [25]. It is presented in [18] that chaos exists in a class of impulsive differential equation. Two chaotic models for single impulsive differential systems have been discussed in [14]. Impulsive control induced effects on single and coupled systems with two systems have been discussed in [4]. In this chapter, impulsive control induced effects on dynamics of complex networks and the stability of complex networks with different dynamical nodes by such impulsive control is discussed, where the impulse effects have outer input signals. It can be seen that the complex networks realize synchronization with such special impulsive control. The dynamics of complex networks are affected by changes of outer input signals. Complex networks can realize synchronization and its synchronization manifold can be changed with such impulsive control signals.

17.2 Main Results Consider a network consisting of N nodes, in which each node is an n-dimensional dynamical system. The state equations are xPi D fi .xi / C

N X

bij xj ;

i D 1; 2; : : : ; N;

(17.1)

j D1;:::;N

where xi 2 Rn , B D .bij /N N denote the coupling configuration matrix. Then bij D bj i D 1 if there is a connection between node i and j (i ¤ j ); otherwise, bij D bj i D 0. In this model, it is required that the coupling coefficients satisfy N P bi i D bij . And 2 Rnn is the inner connecting matrix in each node. It j D1;j ¤i

is called complex networks with identical dynamical nodes if f1 D f2 D D fN , otherwise it is with different dynamical nodes.The impulsive control with special

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205

functions of above complex networks will be discussed. The first part is the stability analysis of the presented dynamical networks with different dynamical nodes with impulsive control.

17.2.1 Stability Analysis of the Presented Dynamical Networks with Different Dynamical Nodes with Impulsive Control Consider the impulsive control of complex networks (17.1) with different dynamics of each node. That is, fi ¤ fj ; .i ¤ j; i; j D 1; 2; : : : ; N /. It can be described by 8 ˆ ˆ ˆ xPi D fi .xi / C <

N P j D1;:::;N

t ¤ tk ;

bij xj ;

ˆ xi .t/ D Ik .xi .t//; t D tk ; ˆ ˆ : C xi t0 D xi 0 ; k D 0; 1; : : : ;

(17.2) i D 1; 2; : : : ; N;

where fi .xi /WD!Rn ; t1 >t0 and impulse function Ik .xi / W D!Rn ; kD0; 1; 2; : : :. !1; t1
Few studies were focused on complex networks with different dynamical nodes by impulsive control for existing results. And in this chapter, the impulsive function is different from existing results. Choose Ik .xi / D zk Bk C Ck xi xi , Ck is an n n constant matrix, Bk is a column vector, and zk is a discrete model outer input state variable, such as from the Logistic model etc. Denote xi D T T .xi1 ; xi 2 ; : : : ; xi n /T ; i D 1; 2; : : : ; N; X D x1T ; x2T ; ; xN ; fi .xi /DAi xi C T T T T nn 'i .xi /; Ai 2 R ; '1 .x1 / ; '2 .x2 /; : : : ; 'N .xN / DF .X /; 2

A1 : : : 6 :: : : AD4 : : 0 :::

3 0 :: 7 : : 5 AN

System (13.2) can be rewritten as 8 < XP D AX C F .X / C .B ˝ /X; t ¤ tk ; X D zk .IN ˝ Bk / C .IN ˝ Ck /X .IN ˝ In /X; t D tk ; : T .t0 //T D X0 ; X.t0C / D X.t0 / D .x1T .t0 /; x2T .t0 /; ; xN

(17.3)

where In ; IN are n n and N N unit matrix respectively. We have the following theorem from results [25, 26]. Theorem 1. Suppose the series fzk g of impulsive differential equation (17.3) as zkC1 D zk .1 zk / and the impulsive interval is , then the solution of system (17.3) is changed with different value of in Logistic model. Especially, the trivial

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solution of system (17.3) is global asymptotical stable as 0 < 1 for impulsive Bk ; Ck ; k D 0; 1; 2; : : : That is the impulsive controlled networks in (17.2) is global asymptotical stable at origin.

17.2.2 Impulsive Control Induced Effects on Dynamics of Complex Networks with Identical Dynamical Nodes Suppose that f1 D D fN D f , complex networks (17.1) can be described by xPi D f .xi / C

N X

bij xj D Axi C '.xi / C

N X

bij xj ;

i D 1; 2; : : : ; N:

j D1

j D1;:::;N

(17.4) And suppose the following assumption holding. Assumption 1. There exists a constant L > 0 such that all the nonlinear functions k'.xi / '.xj /k Lkxi xj k for all i ¤ j.i; j D 1; 2; : : : ; N /. Definition 1. Complex networks (17.4) is called realize synchronization if lim kxi .t/ xj .t/k D 0 for all i; j D 1; 2; : : : ; N .

t !1

Consider the impulsive control of the presented model: 8 N P ˆ ˆ ˆ x P D Ax C '.x / C bij xj ; i i i ˆ < j D1

t ¤ tk ; (17.5)

xi .t/ D Ik .xi .t//; t D tk ; ˆ ˆ ˆ ˆ : x.t0C / D x0 ; k D 0; 1; : : : ;

where Ik .zk ; xi / D zk Bk C Ck xi xi , Ck is an n n constant matrix, Bk is a column vector and zk is a discrete model outer input state variable, such as from the N P xi is the synchronization state of the Logistic model etc. Assume that s D N1 complex networks. Then, we have

i D1

8 N N P P ˆ ˆ sP D N1 xP i D As C N1 '.xi / D '; N ˆ ˆ ˆ i D1 i D1 ˆ < s D zk Bk C Ck s s; t D tk ; ˆ ˆ ˆ N ˆ P ˆ ˆ : s t0C D N1 xi .t0 /; t0 > 0: i D1

t ¤ tk ; (17.6)

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Impulsive Control Induced Effects on Dynamics of Complex Networks

Let ei D .ei1 ; ei 2 ; : : : ; ein /T D xi s; equation is ePi D Aei C

N X

bij xj C

j D1

ei D Ck ei ei ;

207

i D 1; 2; : : : ; N , the dynamical error

N 1 X .'.xi / '.xj //; N

t ¤ tk ;

i D1

t D tk ;

i D 1; 2; : : : ; N:

(17.7)

Definition 2. Complex network (17.4) is called realize synchronize if lim kxi .t/ t !1

s.t/k D 0 for all i D 1; 2; : : : ; N . And the synchronization state is the orbit of s. Denote max .X / as the maximum eigenvalues of a square matrix X . Then, we have Theorem 2. Let ˇk Dmax CkT Ck ; 1 D max .max ..ACbi i /T C.AC bi i ///; 2 D max . /; b D max bii :

i D1;2;:::;N

i D1;:::;N

(i) If D 1 C 2b2 C 4L < 0 ( is a constant), and there exists a constant ˛.0 ˛ < /, such that lnˇk ˛.tk tk1 / 0; k D 1; 2; : : : , then the impulsive control coupled system (17.5) realize synchronization. (ii) If D 1 C 2b2 C 4L 0 ( is a constant) and there exists a constant ˛ 1, such that ln.˛ˇk / C .tkC1 tk / 0; k D 1; 2; : : : ; then ˛ D 1 implies that the trivial solution of the system (17.7) is stable and ˛ > 1 implies the trivial solution is globally asymptotically stable. That is, the impulsive control coupled system (17.5) realize synchronize. ! N 1 P e T ei . For t 2 .tk1 ; tk Proof. Construct a Lyapunov function V .e/ D 2 1 i .k D 1; 2; : : :/, the time derivative of along the trajectory of system (17.7) is VP .e1 ; : : : ; eN / 00 1T N N N N X X X 1 1 1 X T @@Aei C ePi ei C eiT ePi D bij ej C .'.xi / '.xj //A ei D 2 2 N i D1

0

C eiT @Aei C

i D1

N X j D1

bij ej C

j D1

j D1

11

N 1 X .'.xi / '.xj //AA N j D1

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0

D

N N X 1X T @ei ..A C bi i /T C .A C bi i //ei C 2 eiT bij ej 2 i D1 j D1;j ¤i 1 N 1 X C 2eiT .'.xi / '.xj //A N j D1

X 1X max ..A C bi i /T C .A C bi i //eiT ei C 2 N

N

i D1

C

eiT

1 N

N X

.'.xi / '.xj //

j D1

X 1X max ..A C bi i /T C .A C bi i //eiT ei C 2 N

N

i D1

C

eiT bij ej

i D1 j D1;j ¤i

N X i D1

N X

N X

bij 2 jjeiT jjjjej jj

i D1 j D1;j ¤i

N N 1 XX jjei jjjj.'.xi / '.xj //jj N i D1 j D1

X 1X max ..A C bi i /T C .A C bi i //eiT ei C 2 N

N

i D1

C

L N

i D1 j D1;j ¤i

N N X X

jjei jjjjxi xj jj

i D1 j D1

N 1 X T ei ei 1 2

! C

i D1

C

L N

N N X X

N X

2 bij

i D1 j D1;j ¤i

1 T ei ei C ejT ej 2

jjei jj.jjxi sjj C jjs xj jj/

i D1

L N

N X

i D1 j D1

N 1 X T ei ei 1 2

C

N X

N N X X

! C

N X

N X

2 bij

i D1 j D1;j ¤i

.jjei jj2 C jjei jjjjej jj/

i D1 j D1

1 T e ei CejT ej 2 i

bij 2 jjeiT jjjjej jj

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Impulsive Control Induced Effects on Dynamics of Complex Networks

0

209

1

! N N N X 2 X 1 X T @bi i eiT ei C 1 ei ei C bij ejT ej A 2 2 i D1 i D1 j D1;j ¤i 00 1 1 N N N X 1 L X @@ X eiT ei C ejT ej A jjei jj2 A C C N 2 i D1 j D1 j D1 0 0 11 ! N N N N X X 1 X T 2 @X @ 1 bi i eiT ei C ei ei C bij ejT ej AA 2 2 i D1 i D1 i D1 j D1;j ¤i 0 0 11 N N N X X L @X 1 @ C eiT ei C ejT ej AA NeiT ei C N 2 i D1 i D1 j D1 0 0 11 ! N N N N X X 1 X T 2 @X @ 1 bi i eiT ei C ei ei C bij ejT ej AA 2 2 i D1 i D1 i D1 j D1;j ¤i ! N L X 2NeiT ei C N i D1 ! ! N N N N X X X 2 X 1 T T T 1 ei ei C eiT ei bi i ei ei C bi i ei ei C 2L 2 2 i D1 i D1 i D1 i D1 ! ! N N N X X X 1 eiT ei C 2L 1 eiT ei C b2 eiT ei 2 i D1 i D1 i D1 ! N 1 X T .1 C 2b2 C 4L/ ei ei : (17.8) 2 i D1

Let D 1 C 2b2 C 4L, then VP .e/ V .e/ and C C ; : : : ; eN tk1 exp..t tk1 //; V .e1 .t/; : : : ; eN .t// V e1 tk1 (17.9) t 2 .tk1 ; tk ; k D 1; 2; : : : On the other hand, it follows C 1 T C C T tk eN tkC e1 tk e1 tk C C eN 2 1 T T .Ck e1 .tk // Ck e1 .tk / C C .Ck eN .tk // Ck eN .tk / D 2 max CkT Ck V .e.tk //

V .e.tkC // D

ˇk V .e.tk //:

(17.10)

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The following results come from (17.9) and (17.10). For t 2 .t0 ; t1 , V .e1 .t/; : : : ; eN .t// V .e1 .t0C /; : : : ; e= n.t0C // exp..t t0 //; which leads to V .e1 .t1 /; : : : ; eN .t1 // V .e1 .t0C /; : : : ; eN .t0C // exp..t1 t0 // and V .e1 .t1C /; : : : ; eN .t1C // ˇ1 V .e1 .t1 /; : : : ; eN .t1 // ˇ1 V .e1 .t0C /; : : : ; eN .t0C // exp..t1 t0 //. In general, for t 2 .tk ; tkC1 ; .k D 0; 1; 2; /; V .e1 .t/; : : : ; eN .t// V .e1 .t0C /; : : : ; eN .t0C //ˇ1 : : : ˇk exp..t t0 //: Therefore, we have the following results. (i) If < 0 and there exists a constant ˛.0 ˛ < / such that ln ˇk ˛.tk tk1 / 0; k D 1; 2; : : : : Then we obtain that for t 2 .tk ; tkC1 , V .e1 .t/; : : : ; eN .t// V .e1 .t0C /; : : : ; eN .t0C //ˇ1 ˇk exp..t t0 //

D V .e1 .t0C /; : : : ; eN .t0C //ˇ1 ˇk exp.˛.t t0 // exp.. C ˛/.t t0 // V .e1 .t0C /; : : : ; eN .t0C //ˇ1 ˇk exp.˛.tk t0 // exp.. C ˛/.t t0 //

D V .e1 .t0C /; : : : ; eN .t0C //ˇ1 exp.˛.t1 t0 //ˇ2 exp.˛.t2 t1 // ˇk exp.˛.tk tk1 // exp.. C ˛/.t t0 // V .e1 .t0C /; : : : ; eN .t0C // exp.. C ˛/.t t0 //:

Namely, V .e1 .t/; : : : ; eN .t// V .e1 .t0C /; : : : ; eN .t0C // exp.. C ˛/ .t t0 //; t t0 . We can conclude that the trivial solution of system (17.7) is globally exponentially stable from the theories in [2, 10, 22]. That is ei1 ; : : : ; ei n ! 0.i D 1; 2; N / as t ! 1. Then, complex networks (17.5) synchronize up with each node quickly with above impulsive control inputs. (ii) If 0 and there exists a constant ˛ 1, such that ln.˛ˇk / C .tkC1 tk / 0; k D 1; 2; : : : . For t 2 .tk ; tkC1 , then V .e1 .t/; : : : ; eN .t// V .e1 .t0C /; : : : ; eN .t0C //ˇ1 ˇ2 ˇk exp..t t0 // V .e1 .t0C /; : : : ; eN .t0C //ˇ1 ˇ2 ˇk exp..tkC1 t0 // V .e1 .t0C /; : : : ; eN .t0C //ˇ1 exp..t2 t1 //ˇ2 exp..t3 t2 // ˇk exp..tkC1 tk // exp...t1 t0 // V .e1 .t0C /; : : : ; eN .t0C //

1 exp...t1 t0 //; ˛k

which implies that the conclusion (ii) of Theorem 2 holds. That is complex networks achieve synchronize. t u Remark. The solution of system (17.6) exists if the conditions in Theorem 2 hold for system (17.5). That is, the impulsive system (17.6) is soluble.

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The following theorem can be derived for system (17.5): Theorem 3. Suppose that conditions in Theorem 2 are satisfied by impulsive differential system (17.5). And impulsive input functions are taken as Ik .zk ; xi / D H.zk C "xi / xi D zk Bk C Ck xi xi ; i D 1; 2; : : : ; N; where zkC1 D g.zk /; k D 0; 1; 2; : : : , then we have (a) (1) H.0/ D g.0/ D f .0/ D 0; the map H is a topological transmission from D ! D, which is defined in C 2 ; (2) The map g W Y ! Y D is a chaotic map in C 2 in the Devaney sense, and Y is compact. Then, the system (17.6) is also chaotic in the Devaney sense as D 0 if conditions .1/ and .2/ hold. (b) If the series fzk g.k D 0; 1; 2; : : :/ is . 1/-period, that is z D g .z/ (for all z 2 fzk g) and g./ is continuous, complex networks (17.5) synchronize, and the synchronization manifold s.t/ is T -period, that is s.t/ D s.t C T /, for any t 2 Œ0; C1/. It can be seen from numerical simulations that the synchronization orbit of complex networks may be an equilibrium point as 0 < 1, periodic orbits as 1 < < 3:5, or a chaotic orbit as > 3:57.

17.3 Numerical Simulations To demonstrate the above-derived theoretical results, some typical examples of chaotic systems are used as the dynamical node of the impulsively coupled system. Such as the typical Lorenz system etc. A single Lorenz system is described by 8 < xP i1 D c1 .xi 2 xi1 /; xP D c3 xi1 xi1 xi 3 xi 2 ; : i2 xP i 3 D xi1 xi 2 c2 xi 3 : When c1 D 10; c2 D Then we have 2

8 3 ; c3

(17.11)

D 28; Lorenz system has a chaotic attractor.

3 2 3 0:1 0 0 10 10 0 A D 4 28 1 0 5 ; D 4 0 0 0 5 ; 0 00 0 0 83

and 1 D max ..A C bii /T C .A C bii // D 27:2938: We can obtain that the bound of above chaotic system is 39:2462. The nonlinear part of the system is '.xi1 ; xi 2 ; xi 3 / D .0; xi1 xi 3 ; xi1 xi 2 /T . It follows that L in theorem is 39:2462.

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Consider the coupled systems with four Lorenz systems as the ring chain, 2 3 1 0 0 1 6 0 1 1 0 7 7 where B D 6 4 0 0 1 1 5 : For the case, choose the matrix Bk D .0; 1; 0/; 1 0 0 1 Ck D .0:1; 0:1; 0:1/ and ˛ D 1:01; ˇ D 0:01; 2 D 1; D 186:9592 the same value as before. The impulsive interval for synchronization should be less than 0:0246. Then, Figs. 17.1–17.3 show the stable of the origin solution of the four Lorenz

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systems coupled. Figures 17.4–17.6 display the synchronization and the node dynamics of the coupled networks with four Lorenz systems as D 3:5. It can be seen from the numerical simulations that ring networks with four Lorenz system coupled cannot achieve synchronization for certain given values of coupling strength matrix . It is shown by Fig. 17.1. However, the networks (17.2) will be stable to origin quickly with the presented impulsive control, where the impulsive response is corresponding with the outer input variables. Figures 17.2

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and 17.3 display the stable of the origin of networks. And the synchronization manifold can be changed with the outer input impulsive functions. The same conclusions can be obtained for the synchronizing coupling strengths, that is, the synchronization manifold is changed with impulsive inputs.

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17.4 Conclusion Control and synchronization of complex networks have been extensively investigated in many research and application fields. Previous works focused upon realizing synchronization by vary methods. There has been little research on the dynamics of synchronization manifold of complex networks by now. In this chapter, impulsive control induced effects on dynamics of complex networks are investigated. And the stability of complex networks with different dynamical nodes by such impulsive control is discussed too,where the impulse effects have outer input signals. Complex networks realize synchronization with such special impulsive control. The dynamics of complex networks are affected by changes of outer input signals. Complex networks can realize synchronization and its synchronization manifold can be changed with such impulsive control signals. The synchronization orbit may be an equilibrium point, a periodic orbit,or a chaotic orbit, which is determined by a parameter in outer impulse single. Strict theories are given, and some numerical simulations confirm the correctness of theoretical results. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 10871120), the China Tianyuan Youth Founding of Mathematics (Grant No. 10826029), the Science & Technology Development Funds of Shandong Education Committee (Grant No. J08LI10).

References 1. BarabKai AL, Albert R, Jeong, H (1999) Mean-field theory for scale-free random networks. Physica A 272:173–187 2. Fu XL, Yan BQ, Liu YS (2005) Introduction of impulsive differential system. Academic Press of China, Beijing 3. Goto S, Nishioka T, Kanehisa M (1998) LIGAND: Chemical database for enzyme reactions. Bioinformatics 14:591–599

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4. Han XP, Lu JA (2010) Impulsive control induced effects on dynamics of single and coupled ODE systems. Nonlinear Dyn 59:101–111 5. Horne AB, Hodgman TC, Spence HD, Dalby AR (2004) Constructing an enzyme-centric view of metabolism. Bioinformatics 20:2050–2055 6. Itoh M, Yamamoto N, Yang T, Chua LO (1999) Performance analysis of impulsive synchronization. In: Proceedings of the 1999 European conference on circuit theory and design, Stresa, Italy, pp 353–356 7. Itoh M, Yang T, Chua LO (2001) Experimental study of impulsive synchronization of chaotic and hyperchaotic circuits. Int J Bifurcat Chaos 11:1393–1424 8. Jeong H, Tombor B, Albert R, Oltvai ZN, BarabKai AL (2000) The large-scale organization of metabolic networks. Nat Lond 407:651–654 9. Khadra A, Liu XZ, Shen X (2005) Impulsively synchronizing chaotic systems with delay and application to secure communication. Automatica 41:1491–1502 10. Lakshmikantham V, Bainov D, Simeonov P (1989) Theory of impulsive differential equations. World Scientific, Singapore 11. Li C, Chen G (2004) Synchronization in general complex dynamical networks with coupling delays. Phys A 343:263–278 12. Li ZG, Wen CY, Soh YC (2001) Analysis and design of impulsive control systems. IEEE Trans Automat Control 46:894–897 13. Li C, Xu H, Liao X, Yu J (2004) Synchronization in small-world oscillator networks with coupling delays. Phys A 335:359–364 14. Lin W (2002) Some problems in chaotic systems and their applications. Fudan University’s doctoral dissertation 15. Liu XZ, Willms AR (1996) Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft. Math Prob Eng 2:277–299 16. Liu B, Liu XZ, Chen GR, Wang HY (2005) Robust impulsive synchronization of uncertain dynamical networks. IEEE Trans Circuits Syst I 52:1431–1440 17. LRu, J, Yu X, Chen G (2004) Chaos synchronization of general complex dynamical networks. Phys A 334:281–302 18. Ruan J, Lin W (1999) Chaos in a class of impulsive differential equation. Commun Nonlinear Sci Numer Simul 4:166–169 19. Strogatz SH (2001) Exploring complex networks. Nature 410:268–276 20. Wang X, Chen G (2002) Synchronization in scale-free dynamical networks: Robustness and fragility. IEEE Trans Circuits Syst I 49:54–62 21. Watts DJ, Strogatz SH (1998) Collective dynamics of small-world. Nature 393:440–442 22. Yang T (2001) Impulsive control theory. Springer, Berlin 23. Yang T, Chua LO (1997) Impulsive control and synchronization of nonlinear dynamical systems and application to secure communication. Int J Bifurcat Chaos 7:645–664 24. Yang T, Chua LO (1997) Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication. IEEE Trans Circuits Syst I 44:976–988 25. Yang T, Yang CM, Yang LB (1997) Control of Rossler systems to periodic motions using control method. Phys Lett A 232:356–361 26. Zhou J, Lu J, LRu, J (2006) Adaptive synchronization of an uncertain complex dynamical network. IEEE Trans Automat Control 51:652–656

Chapter 18

Study on Synchronization of Two Identical Uncoupled Neurons Induced by Noise Ying Wu, Ling Hong, Jun Jiang, and Wuyin Jin

Abstract In this paper, noise-induced synchronization between two identical uncoupled neurons is investigated by using Hodgkin–Huxley .H H/ and FHN models with sinusoidal stimulations. The numerical results show that the value of the critical noise intensity for synchronizing two neurons is much less than the magnitude of mean size of the attractor of the original system, the deterministic dynamics structure of the original attractor is not swamped under noise, and the deterministic feature of the attractor of the original system is affected by noise slightly for FHN neurons. This finding is significantly different from the previous work (Phys Rev E 67:027201, 2003).

18.1 Introduction Noise-induced synchronization is widely studied [1–9]. Theoretical results have inspired experimental work since the noise-induced synchronization was observed in a pair of uncoupled sensory neurons of biological system [10]. It was confirmed that phase synchronization could be achieved between two coupled non-identical HR neurons when the intensity of common noise exceeds a critical value [9]. The same, two uncoupled identical HR or HH neurons are able to achieve complete synchronization [6, 8], and two uncoupled non-identical HR neurons or HH neurons are able to achieve generalized synchronization or phase synchronization respectively [7,8]. Specially, the results of Ref. [6] show that the value of the critical noise intensity for synchronizing two identical uncoupled neurons is roughly equal to the magnitude of mean size of the attractor of the original system, and the randomicity of noise swamps the deterministic dynamics structure of the original attractor.

Y. Wu () School of Science, Xi’an University of Technology, Xi’an, Shaanxi 710048, People’s Republic of China e-mail: [email protected]

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In this paper, noise-induced synchronization between two identical uncoupled HH neurons and FHN neurons is numerically investigated. The results show that the value of the critical noise intensity for synchronizing two neurons is much less than the mean size of the attractor of the original system, and the deterministic dynamics structure of the original attractor is not swamped under noise, but the deterministic dynamics structure of the original attractor in FHN models is changed slightly by noise. This result is completely different from the conclusion of Ref. [6], which means that the value of critical noise intensity reaching or exceeding the magnitude of size of the attractor of the original system for synchronizing two identical uncoupled neurons would not be necessary.

18.2 Numerical Results and Discussion In Ref. [6], the equations of two identical uncoupled HR neurons are given as follows: 8 3 2 < xP 1;2 D y1;2 ax1;2 C bx1;2 z1;2 C I C k.t/ 2 (18.1) yP D c dx1;2 y1;2 : 1;2 zP1;2 D rŒS.x1;2 / z1;2 the parameter I is external input current, .t/ is white noise with zero mean, and h.t/.t /i D ı./; k is noise intensity. Other parameters can be obtained in [6]. The result is given in Fig. 18.1, Fig. 18.1a corresponds to attractors without noise, and Fig. 18.1b to synchronized attractors with noise intensity k D 2:4. Obviously, in the noiseless conditions, neurons is chaotic firing for I D3:3. After noise-induced synchronization, the deterministic dynamics structure of the original attractors is swamped by noise.

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The differential equations of single H H model are given as follows: 8 dV ˆ ˆ C D gNa m3 h.V VNa / gk n4 .V VK / gL .V VL / C Iext ˆ ˆ dt ˆ ˆ ˆ ˆ ˆ dm ˆ ˆ ˆ < dt D ˛m .V /.1 m/ ˇm .V /m ˆ dn ˆ ˆ D ˛n .V /.1 n/ ˇn .V /n ˆ ˆ ˆ dt ˆ ˆ ˆ ˆ ˆ dh ˆ : D ˛h .V /.1 h/ ˇh .V /h dt

(18.2)

where V is the membrane potential, the definitions of the other variables could be found in [11], and, the parameters C D 1 F=cm2 ; gNa D 120 mS=cm2 ; gK D 36 mS=cm2 ; gL D 0:3 mS=cm2 ; VNa D 50 mV; VK D 77 mV; VL D 54:4 mV; Iext is an external periodic signal current, Iext D Ishift C sin.2 ft/, where Ishift D 10 A =cm2 , is the amplitude of current shift, and f D f0 is the stimulus frequency, f0 D 20 Hz is basic stimulus frequency. In this work, the double precision fourthorder Runge Kutta method is used with integration time step 0.05, and the threshold Vth D 40. As parameter varies, there are a lot of firing types, such as quasi-periodic, bursting, chaotic, and periodic motions. Adding white noise .t/ with zero mean to external input current, the differential equations of two uncoupled H H models are given as follows: 8 dV1;2 ˆ ˆ C D gNa m1;2 3 h1;2 .V1;2 VNa / gk n1;2 4 .V1;2 VK / gL .V1;2 VL / ˆ ˆ dt ˆ ˆ ˆ C Iext C k.t/ ˆ ˆ ˆ ˆ ˆ < dm1;2 D ˛m .V1;2 /.1 m1;2 / ˇm .V1;2 /m1;2 (18.3) dt ˆ ˆ ˆ dn ˆ 1;2 ˆ ˆ D ˛n .V1;2 /.1 n1;2 / ˇn .V1;2 /n1;2 ˆ ˆ dt ˆ ˆ ˆ ˆ : dh1;2 D ˛h .V1;2 /.1 h1;2 / ˇh .V1;2 /h1;2 dt where k is noise intensity. In numerical simulation, the initial conditions are .V1 ; m1 ; n1 ; h1 /0 D.60; 0:2; 0:4; 0:45/ and .V2 ; m2 ; n2 ; h2 /0 D.65; 0:1; 0:4; 0:45/, the data of n 104 are ignored to avoid transients. According to the definition in [6], the mean size of attractor of H H model without noise corresponding to variable V is SV 106:8, and the mean size is not changed as stimulus frequency varies. In Fig. 18.2, we give V h projections of the attractors for f D 4:7f0 . The critical noise intensity for synchronizing two H H neurons is k D 4:3. The critical noise intensity are much less than the magnitude of the mean size of the original attractor .SV 106:8/. Figure 18.2a corresponds to the attractors without noise which is chaotic firing, Fig. 18.2b to synchronized attractors. Obviously, after synchronizing the original attractors is not swamped by noise, which is completely different from the conclusion of Ref. [6].

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The differential equations of two uncoupled FHN models with noise are given as follows: 8 dv ˆ < " i D vi .vi a/.1 vi / wi C k.t/ dt (18.4) ˆ : dwi D vi dw b C r sin.ˇt/ dt in which, i D 1, 2 and other parameters are " D 0:02; d D 0:78; a D 0:5; r D 0:27, and ˇ D 14:0 respectively, single FHN model can show a lot of firing types with parameter b changing. Figure 18.3 gives the projections of attractors of FHN neurons in v w plane for b D 0:2404. Figure 18.3a corresponds to the attractors without noise which is chaotic firing, Fig. 18.3b to synchronized attractors. Obviously, after synchronizing the original attractor is not swamped by noise, but are changed obviously, which

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are disturbed by noise slightly. The result is different from the conclusion of HH neurons. The values of critical noise intensity for synchronizing two FHN neurons is 0.06, which is much less than the magnitude of the mean size of the original attractor .Sw D 0:43/. That is same as the result of HH neurons.

18.3 Conclusions By numerically investigating noise-induced synchronization between two uncoupled FHN and HH models, we find that the value of the critical noise intensity reaching or exceeding the magnitude of the mean size of the attractor of the original system for synchronizing two models is not necessary. This result is completely different from conclusion of Ref. [6]. It is interesting that the deterministic dynamics structure of the original attractor of neurons is swamped by noise for HR model, but that of the original attractor of neuron is remained for HHmodel, and the original attractor of neuron is slightly changed by noise for FHN models. These differences mean that the mechanism of noise-induced synchronization is worthy to further detected. Acknowledgments The project is grateful for the financial support by the National Natural Science Foundation of China under Grant Nos. 10772140, 10872155, and the first author is grateful for the financial support by the Foundation of Xi’an University of Technology under Grant Nos. 108-210808, 108-210809.

References 1. Maritan A, Banavar JR (1994) Chaos, noise, and synchronization [J]. Phys Rev Lett 72:1451 2. Pikovsky AS (1994) Chaos, noise, and synchronization [J]. Phys Rev Lett 73:2931 3. Lai CH, Zhou CS (1998) Synchronization of chaotic maps by symmetric common noise [J]. Europhys Lett 43:376 4. Zhou CS, Kurths J (2002) Noise-enhanced phase synchronization of chaotic oscillators [J]. Phys Rev Lett 88:230602 5. Toral R, et al (2001) Analytical and numerical studies of noise-induced synchronization of chaotic systems [J]. Chaos 11:665 6. He DH, Shi PL, Stone L (2003) Noiseinduced synchronization in realistic models [J]. Phys Rev E 67:027201 7. Wu Y, Xu JX, He DH, et al (2005) Generalized synchronization induced by noise and parameter mismatching in Hindmarsh–Rose neurons [J]. Chaos Solitons Fractals 23:1605 8. Zhou CS, Kurths J (2003) Noise-induced synchronization and coherence resonance of a Hodgkin–Huxley model of thermally sensitive neurons [J]. Chaos 13:401 9. Shuai JW, Durand DM (1999) Phase synchronization in two coupled chaotic neurons [J]. Phys Lett A 264:289 10. Neiman AB, Russell DF (2002) Synchronization of noise-induced bursts in noncoupled sensory neurons [J]. Phys Rev Lett 88:138103 11. Jin WY, Xu JX, Wu Y, et al (2004) An alternating periodic-chaotic ISI sequence of HH neuron under external sinusoidal stimulus. Chinese Phys 13:335

Chapter 19

Non-equilibrium Phase Transitions in a Single-Mode Laser Model Driven by Non-Gaussian Noise Yanfei Jin

Abstract The non-equilibrium phase transition of a single-mode laser model driven by non-Gaussian noise is studied in this paper. The stationary probability distribution (SPD) and its extremal equation are derived by using the path integral approach and the unified colored noise approximation. It is found that there is a critical relation between the noise intensity and the correlation time so that there is a transition line separating the mono-stable region and the bi-stable region. Given the noise intensity and the correlation time, the single-mode laser system undergoes a successive phase transition by varying the departure of the non-Gaussian noise from the Gaussian noise. Meanwhile, as indicated in the phase diagram, when the noise intensity and the correlation time are varied, the system undergoes a reentrance phenomenon.

19.1 Introduction Noise may play a constructive role and induce new ordering phenomena in some non-equilibrium systems [1–4] even though it is usually considered as a source of disorder and chaos. This phenomenon is the so-called noise induced phase transition. The noise induced phase transitions in systems far away from thermal equilibrium have been widely investigated in many fields [5–11], such as physics, chemistry, and biology. Gudyma [7] examined the mechanisms of action of colored multiplicative noise in a positionally disordered semiconductor with Moss–Burstein shift. It is found that the action of multiplicative noise causes non-equilibrium first-order phase transition of the disorder-order-type in electron subsystem of semiconductor. Van den Broeck et al. [8] found that the lattice model with multiplicative noise could undergo a non-equilibrium phase transition to a symmetry-breaking state. Zaikin et al. [9] showed the non-equilibrium first-order Y. Jin () Department of Mechanics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China e-mail: [email protected]

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phase transition in a nonlinear lattice of over-damped oscillators with both additive and multiplicative noise terms, which is induced by additive noise. Castro et al. [10] reported the reentrance phenomena induced by colored multiplicative noise and correlated additive and multiplicative white noises. Jia et al. [11] found that the cross-correlation between noises could induce the reentrance phenomena in a bi-stable kinetic model. The above studies proved that the noise acts an important role in the study of non-equilibrium phase transitions. The single-mode laser model with random fluctuation has received much attention from both theorists and experimentalists [12–18] and can be regarded as a particular prototype of a nonlinear problem in non-equilibrium statistical mechanics. Zhu [12] investigated the transient properties of two different singlemode laser models with loss noise and gain noise. Cao and Liang et al. [13–15] studied the effects of white and colored cross-correlation between additive and multiplicative noises on the phase transition in the dye laser system respectively. Luo et al. [16] developed a two-dimensional decoupling theory, which applied to singlemode dye laser system with colored noise. Xie and Mei [17] studied a single-mode laser model with cross-correlated additive and multiplicative noises term and found that nearby threshold the presence of negatively correlated noises slows down the decay of fluctuation. Jin et al. [18] investigated the relaxation time of a single-mode dye laser system driven by cross-correlated additive and multiplicative noises and found that the correlation intensity speeds up the intensity fluctuation decay of laser slightly above threshold. Most of existing theoretical works assume that the noise source yields a Gaussian distribution (either white or colored). Recently, some experiments in sensory, biological, and physical systems [19, 20] have indicated that the noise sources may yield non-Gaussian distributions. Fuentes and Horacio et al. [21–23] investigated the effects of non-Gaussian noises on the mean first-passage time and stochastic resonance in a bi-stable system. Meanwhile, the effect of non-Gaussian noise sources in the noise-induced transition for a genetic model is studied by Horacio and Toral [24]. Goswami et al. [25] studied the barrier crossing dynamics in presence of additive and multiplicative non-Gaussian noise. This included the multiplicative colored non-Gaussian noise, which can induce resonant activation. To the best knowledge of the author, no attention has been paid to the noise-induced phenomena in the laser model subjected to non-Gaussian noises because of mathematical difficulties. It is desirable, hence, to gain an insight into the non-equilibrium phase transitions of a single-mode laser model subjected to the non-Gaussian noise. The paper is organized as follows: In Sect. 19.2, the stationary probability distribution (SPD) and its extremal equation are derived by using the path integral approach and the unified colored noise approximation. It is found that there is a critical relation between the noise intensity and the correlation time so that there is a transition line separating the mono-stable region and the bi-stable region. Analyzing the roots of the extreme equation of SPD, the effects of non-Gaussian noise on nonequilibrium phase transition is discussed in Sect. 19.3. Finally, several concluding remarks are drawn in Sect. 19.4.

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19.2 Approximate Stationary Probability Distribution Consider a single-mode laser loss model with the non-Gaussian noise described by the following Langevin equation IP D 2. I /I C 2I .t/;

(19.1)

where I is the laser intensity, is the pump parameter, .t/ is the noise term with the non-Gaussian distribution, and can be modeled as a Markov process by the following Langevin equation [21] P D

1 1 d Vq ./ C .t/; d

(19.2)

where .t/ is the Gaussian white noise and can be characterized by the following mean and variance h.t/i D 0; and

˝

˛ .t/.t 0 / D 2Dı.t t 0 /

2 D Vq ./ D ; ln 1 C .q 1/ .q 1/ D 2

(19.3)

(19.4)

where parameter q denotes the departure of the non-Gaussian noise .t/ from the Gaussian distribution, the parameters and D represent the noise correlation time and the noise intensity respectively. When q ! 1; .t/ degenerates to Gaussian colored noise. From [21–23], the stationary probability distribution of .t/ is given by Pqst ./ D

1=.q1/ 2 1 1 C .q 1/ ; Zq D 2

(19.5)

here Zq is the normalization factor. And the mean and variance of .t/ are derived as follows h.t/i D 0; ( ˝ 2 ˛ .t/ D

2D ; .53q/

1;

q 2 .1; 5=3/ : q 2 Œ5=3; 3/

(19.6)

The multiplicative noise .t/ in (19.1) turns to the additive one by setting x D ln I xP D 2. e x / C 2.t/:

(19.7)

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From (19.2)–(19.6), one can see that the introduction of the non-Gaussian noise involves many mathematical difficulties. Thus, the following Gaussian noise approximation in the region jq 1j 1 (q > 1 and q < 1) is adopted by applying the path integral approach [21–22] " ˝ 2 ˛ #1 1 1 dVq ./ 2 1 D 1 C .q 1/ 1 C .q 1/ D ; d D 2 D 2 eff (19.8) where eff D .2.2 q/=.5 3q//. It is easy to see that (19.2) can be rewritten as a re-normalized Ornstein– Uhlenbeck process with the effective noise correlation time eff and the effective multiplicative noise intensity Deff D .2.2 q/=.5 3q//2 D. Especially, when q ! 1, there are eff ! and Deff ! D. Based on (19.8), (19.2) has been transformed into a Markov process. And using the unified colored noise approximation, (19.7) can be rewritten as xP D

2 2. e x / .t/; C 1 x 1 C 2e eff eff Œ1 C 2e x eff

(19.9)

where eff is defined as (19.8). The corresponding Fokker–Plank equation determined by (19.9) can be derived as @ @2 @P .x; t/ D ŒA.x/P .x; t/ C 2 ŒB.x/P .x; t/; @t @x @x

(19.10)

where A.x/ D

2. e x / 8e x eff Deff ; 1 C 2eff .1 C 2e x eff /3

B.x/ D

4Deff : .1 C 2e x eff /2

Substituting the transformation x D ln I , the stationary probability distribution (SPD) of the laser intensity can be obtained from (19.10) ˆ.I / N

st .I / D ; exp B.I / D

(19.11)

where ˆ.I / D B.I / D

5 3q 2q

2

. ln I C I / .2 q/I.2 I / ; 8 4.5 3q/

2.2 q/I .5 3q/ C 4.2 q/I

(19.12)

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Non-equilibrium Phase Transitions in a Single-Mode Laser Model

227

and N is a normalization constant. When q D 1, (19.11) is consistent with the result obtained in [4]. From (19.11)–(19.12), the extremal equation of the SPD st .I / are determined by the following equation of third order I 3 C a2 I 2 C a1 I C a0 D 0;

(19.13)

where 1 8D.2 q/2 .5 3q/2 ; 16 2 .2 q/2 1 .5 3q/Œ.5 3q/ 8.2 q/ ; a1 D 16 2 .2 q/2 1 .5 3q/ 2.2 q/ : a2 D 2 .2 q/

a0 D

When the form of st .I / changes between the bimodal structure and the unimodal structure, the critical parameter Dc satisfies the following equation Dc D

.5 3q/ 3 2 4 C : 216 .2 q/

(19.14)

That is, when the correlation time is fixed, the SPD is bimodal for D > Dc and it is unimodal for D < Dc . Equation (19.14) can be rewritten according to the effective noise correlation time eff and the effective multiplicative noise intensity Deff as 2 1 3 Deff D eff .2 C eff / : (19.15) 27

19.3 General Analysis From (19.11) and (19.15), the effects of the non-Gaussian noise on the nonequilibrium phase transition of the single-mode laser are discussed in this section. In Fig. 19.1, the transition lines for different values of q are plotted in the .; D/ plane. It is seen that the transition line decreases with the increase of q and a minimum value of noise intensity D is needed to induce the phase transition. When .; D/ falls into the region above the transition line, the extreme equation (19.13) has three extreme points. Otherwise, (19.13) only has one extreme point. The corresponding SPD st .I / is plotted with different values of q shown in Fig. 19.2. In Fig. 19.2, the noise correlation time and the noise intensity D are fixed as D 1 and D D 0:5, which are chosen above the boundary of q D 1:5 and below the boundaries of q D 1 and q D 0:5. The SPD st .I / corresponding to q D 1 and q D 0:5 has a single peak when the laser intensity I in interval [0,1]. When q D 1:5,

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Y. Jin

Fig. 19.1 Phase diagram Dc vs. for different values of q with D 1

1 q=1 q=1.5 q=0.5

0.9 0.8 0.7

ρst(I)

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.5

1

1.5 I

2

2.5

3

Fig. 19.2 The SPD st .I / for different values of q with D 1, D D 0:5; D 1

the curve of st .I / exists an infinite maximum at I D 0 and a finite maximum at 0 < I < 1. The SPD st .I / appears a bimodal structure and the corresponding extreme equation (19.13) has two maximum points and one minimum point, which is consistent with the results shown in Fig. 19.1. Thus, the SPD st .I / can switch from unimodal to bimodal structure by increasing the parameter q for fixed and D. Therefore, the parameter q plays an important role in the noise induced transitions. Figure 19.3 gives the transition line for the case of q D 1:5. Points A and C are chosen from the region below the transition line, while point B is laid above the transition line. The SPD st .I / for the points A, B, C indicated in Fig. 19.3 are presented in Fig. 19.4. The SPD st .I / at point A appears a bimodal structure. Then fixed the value of the correlation time and increased the noise intensity D beyond some threshold phase, the SPD st .I / corresponding to B has a unimodal structure. If further fixed the noise intensity D and increased the correlation time beyond

19

Non-equilibrium Phase Transitions in a Single-Mode Laser Model

229

Fig. 19.3 Phase diagram Dc vs. for q D 1:5 and D 1 1 corresponding to ponit A corresponding to point B corresponding to point C

0.9 0.8 0.7 ρst(I)

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.5

1

1.5 I

2

2.5

3

Fig. 19.4 The SPD st .I / for different combinations of .; D/ with D 1 and q D 1:5

some threshold value, the SPD st .I / corresponding to C goes back to the bimodal structure. This type of non-equilibrium transition phenomenon is called reentrance phenomenon. Thus, the reentrance phenomenon is found in this laser model when the noise source is the non-Gaussian noise.

19.4 Conclusion Remarks The non-equilibrium phase transition of a single-mode laser model driven by nonGaussian noise is studied in this paper. The stationary probability distribution (SPD) and its extremal equation are derived by using the path integral approach and the

230

Y. Jin

unified colored noise approximation. It is found that there is a critical relation between the noise intensity and the correlation time so that there is a transition line separating the mono-stable region and the bi-stable region. Given the noise intensity and the correlation time, the single-mode laser system undergoes a successive phase transition by varying the departure of the non-Gaussian noise from the Gaussian noise. Meanwhile, as indicated in the phase diagram, for some regions of values of noise intensity and correlation time, the system turns to a bi-stable phase. Then, for fixed value of correlation time and increased noise intensity beyond some threshold value, the system undergoes a transition to a mono-stable phase. If the noise intensity is further fixed and the correlation time is increased beyond some threshold value, the system goes back to a bi-stable phase. This type of non-equilibrium transition phenomenon is called reentrance phenomenon. The single-mode laser model with random fluctuation is a particular prototype in describing the effects of noises and may be subject to various kinds of noise sources. This study, therefore, extends the application of the single-mode laser model by introducing the non-Gaussian noise. The phenomena found in this paper provide a basis for experimental research and technological applications of the laser system. Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grant Nos. 10702025, 10972032, and 70771005, in part by the Excellent Young Scholars Research Fund of Beijing Institute of Technology under Grant No. 2008Y0175, Beijing Municipal Commission of Education Project under Grant No. 20080739027, and the Ministry of Education Foundation of China under Grant No. 20070004045.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Nicolis G, Prigogine I (1976) Selforganization in nonequilibrium system. Wiley, New York Haken H (1977) Synergetics. Springer, Berlin Horsthemke W (1984) Noise-induced transitions. Springer, New York Hu G (1994) Stochastic force and nonlinear systems. Shanghai Science and Technological Education Publishing House, Shanghai Fulinski A (1995) Relaxation, noise-induced transitions, and stochastic resonance driven by non-Markovian dichotomic noise. Phys Rev E 52:4523 Zhu SQ (1989) Multiplicative colored noise in a dye laser at steady state. Phys Rev A 40:3441 Gudyma YV (2004) Nonequilibrium first-order phase transition in semiconductor system driven by colored noise. Physica A 331:61 Van den Broeck C, Parrondo JMR, Toral R (1994) Noise-induced nonequilibrium phase transition. Phys Rev Lett 73:3395 Zaikin AA, Garcia-Ojalvo J, Schimansky-Geier L (1999) Nonequilibrium first-order phase transition induced by additive noise. Phys Rev E 60:R6275 Castro F, Sanchez AD, Wio HS (1995) Reentrance phenomena in noise induced transitions. Phys Rev Lett 75:1691 Jia Y, Li JL (1997) Reentrance phenomena in a bistable kinetic model driven by correlated noise. Phys Rev Lett 78:994 Zhu SQ (1990) White noise in dye-laser transients. Phys Rev A 42:5758 Cao L, Wu DJ, Lin L (1994) First-order-like transition for colored saturation models of dye lasers: effects of quantum noise. Phys Rev A 49:506

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14. Cao L, Wu DJ (1999) Cross-correlation of multiplicative and additive noises in a single-mode laser white-gain-noise model and correlated noises induced transitions. Phys Lett A 260:126 15. Liang GY, Cao L, Wu DJ (2002) Moments of intensity of single-mode laser driven by additive and multiplicative colored noises with colored cross-correlation. Phys Lett A 294:190 16. Luo XQ, Zhu SQ, Chen XF (2001) Effects of colored noise on the intensity and phase in a laser system. Phys Lett A 287:111 17. Xie CW, Mei DC (2004) Effects of correlated noises on the intensity fluctuation of a singlemode laser system. Phys Lett A 323:421 18. Jin YF, Xu W, Xie WX, Xu M (2005) The relaxation time of a single-mode dye laser system driven by cross-correlated additive and multiplicative noises. Physica A 354:143 19. Bezrukov SM, Vodyanoy I (1997) Stochastic resonance in non-dynamical systems without response thresholds. Nature 385:319 20. Goychuk I, H¨anggi P (2000) Stochastic resonance in ion channels characterized by information theory. Phys Rev E 61:4272 21. Fuentes MA, Toral R, Wio HS (2001) Enhancement of stochastic resonance: the role of nonGaussian noises. Physica A 295:114 22. Fuentes MA, Wio HS, Toral R (2002) Effective Markovian approximation for non-Gaussian noises: a path integral approach. Physica A 303:91 23. Revelli JA, Sanchez AD, Wio HS (2002) Effect of non-Gaussian noises on the stochastic resonance-like phenomenon in gated traps. Physica D 168:165 24. Wio HS, Toral R (2004) Effect of non-Gaussian noise sources in a noise-induced transition. Physica D 193:161 25. Goswami G, Majee P, Ghosh PK, Bag BC (2007) Colored multiplicative and additive nonGaussian noise-driven dynamical system: mean first passage time. Physica A 374:549

Chapter 20

Dynamical Properties of Intensity Fluctuation of Saturation Laser Model Driven by Cross-Correlated Additive and Multiplicative Noises Ping Zhu

Abstract Dynamical properties of the intensity fluctuation of a saturation laser model driven by cross-correlated additive and multiplicative noises are investigated. Using the Novikov theorem and the projection operator method, we obtain the analytic expressions of the stationary probability distribution Pst .I /, the relaxation time Tc , and the normalized correlated function C.s/ of the system. By numerical computation, we discussed the effects of the cross-correlated strength and the cross-correlated time , the quantum noise intensity D, and the pump noise intensity Q for the fluctuation of the laser intensity. Above the threshold, weakens the stationary probability distribution, speeds up the startup velocity of the laser system from start status to steady work, and enhances the stability of laser intensity output; however, strengthens the stationary probability distribution and decreases the stability of laser intensity output; when < 0, speeds up the startup; on the contrary, when > 0, slows down the startup. D and Q make the associated time exhibit extremum structure, that is, the startup time possesses the least values. At the threshold, cannot generate the effects for the saturation laser system, expedites the startup velocity and enhances the stability of the startup. Below threshold, the effects of and not only relate to and , but also relate to other parameters of the system.

20.1 Introduction Recently, people have been more and more interested in statistical fluctuations of laser radiation which determine the limits on the use of lasers in almost every application. The statistical properties of a sing-mode laser that contains both additive and multiplicative noises are discussed by the experimental measurements and theoretical analysis [1–7]. Meanwhile, importance of the saturation effects for

P. Zhu () Department of Physics, Simao Teacher’s College, Puer 665000, Peoples’s Republic of China e-mail: [email protected] A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 20,

233

234

P. Zhu

the behavior of the laser is shown [8–12]. Zhu [13] discussed the saturation effects in a laser with additive and multiplicative white noises. Cao et al. [14–16] analyzed the effects of saturation in the transient process of a dye laser with additive and multiplicative noises. Recently, the effects of correlations between additive and multiplicative noises on the statistical fluctuation of a single-mode laser model have attracted the close attention [7, 17–19]. In [7], Zhu investigated the steadystate properties of the cubic model of single-mode laser with correlations between additive noise and a multiplicative noise. Long et al. [18] studied the phase lock induced by correlations between an additive noise and a multiplicative noise in the cubic model of a single-mode laser. Liang at al. discussed moments of intensity of single-model laser driven by additive and multiplicative colored noises with crosscorrelation [20]. The saturation laser model possessing the theoretical and applied values is as typical as the cubic model of a single-mode laser model. In 1992, Zhu [21] investigated the saturation effects of uncorrelated additive and multiplicative noises for a saturation laser model. Thereafter, Zhu et al. [22,23] presented the correlation function and the relaxation time of a saturation laser model with correlated additive and multiplicative white noises and discussed the effects of correlated white noises. The associated relaxation time and the correlation function are important physical quantity to characterize the dynamic behavior of a stochastic process, and hence are usually used to describe the fluctuation behavior of a nonlinear system. Researches [24–27] on the problem have shown the important physical feature of the associated relaxation time and the correlation function of the probability fluctuation in a nonlinear stochastic system. Applying the means of the projection operator method, Xie and Mei investigated the dynamical properties of a bistable kinetic model with correlated white noises [28], and Mei et al. [29, 30] investigated the effects of cross-correlation for the relaxation time and the correlated function of a bistable system and showed the dynamical properties of a bistable system with cross-correlated white noise. Zhu discussed the effects of cross-correlated additive and multiplicative colored noise sources for the associated relaxation time and the intensity correlation function of a bistable system [31, 32]. From these researches, we found that the correlation strength between additive and multiplicative noises play an important role in the processes of a nonlinear stochastic system. As the ongoing studying works further deepens, people have been more and more interested in the stochastic system with cross-correlated additive and multiplicative colored noises. Jin et al. [33] discussed the relaxation time of a single-mode dye laser system driven by cross-correlated additive and multiplicative noises.The case of a full account of the saturation with cross-correlated additive and multiplicative noises are not further investigated. This chapter is organized as follows: in Sect. 20.2, making use of the approximatic Fokker–Plank equation (AFPE) for a saturation laser model with crosscorrelated additive and multiplicative noises, we solve the AFPE for stationary probability distribution (SPD) of the laser system. Employing the means of the projection operator method, in which the effects of the memory kernels are taken into account, the analytic expressions of the associated relaxation time and the normalized correlation function on the saturation laser model with cross-correlated

20

Dynamical Properties of Intensity Fluctuation

235

noises were derived. In Sect. 20.3, based on the numerical results, we discuss the effects of the coupling strength , the cross-correlated tine , the quantum noise intensity D, the pump noise intensity Q for the stationary probability distribution, the associated relaxation time and the correlation function, so that we show further dynamical properties of intensity fluctuation of saturation laser model driven by cross-correlated additive and multiplicative noises. The discussion and conclusion of the results conclude the paper.

20.2 Stationary Probability Distribution and Relaxation Time and Correlated Function The complex laser field E of a laser model with a full account of the saturation effects follows the Langevin equation [8]

e

e

F1 E dE D kE C C p.t/E C q .t/; dt 1 C A j E j2 =F1

(20.1)

where K is the cavity decay rate for the electric field and F1 D a0 C K is the gain parameter; a0 and A are real and stand for net gain and self-saturation coefficients. The random variables q .t/ and p .t/ are complex and present the quantum and pump noise. Performing the polar coordinate transform E D rei on (20.1), one can obtain two equation of the field amplitude r and phase . Then, the Langevin equation of the field amplitude r can be written as follows [34]:

e

rP D Kr C

e

F1 r 1C

Ar 2 F1

C

D C rp.t/ C q.t/: r

(20.2)

Defining the laser intensity is as I , and then we have I D r 2 . Thus, (20.2) can be rewritten as the Langevin equation of I IP D 2KI C

F1 I 1C

AI F1

1

C 2D C 2I 2 q.t/ C 2Ip.t/:

(20.3)

The multiplicative noise p.t/ and the additive noise q.t/ are considered to be Gaussian-type noise, with zero mean, and hq.t/q.t 0 /i D 2Dı.t t 0 /;

(20.4)

hp.t/p.t 0 /i D 2Qı.t t 0 /;

(20.5)

p jt t 0 j DQ exp hq.t/p.t 0 /i D hp.t/q.t 0 /i D p 0 ! 2 DQı.t t / as ! 0;

(20.6)

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P. Zhu

where D and Q stand for the strength of additive and multiplicative noises, respectively. The parameter measures the strength of correlations between q.t/ and p.t/. is the correlation time of the correlation between q.t/ and p.t/. When the limit ! 0, the colored correlation becomes the white one. Employing the Novikov theorem [35], Fox’s approach [36] and ansatz of Hanggi et al. [37], the approximate Fokker–Planck equation (FPE) to (20.3) is given by [38–40] @P .I; t/ D LFP P .I; t/; @t LFP D

(20.7)

@2 @ f .I / C 2 G.I /; @I @I

(20.8)

where f .I / D 2KI C

2F1 I 1 C 2D C 2 D C 3b0 I 2 C 2QI ; 1 C AI =F1

3 G(I) D 4 DI C 2b0 I 2 C QI 2 ; p DQ

b0 D 1 C 2K 1

and

K F1

(20.9)

(20.10)

:

(20.11)

Note that this approximate Fokker–Plank equation holds under the condition 1C2K.1 FK1 / > 0. Thus, when the system is operated above the threshold (a0 >0), there is no restriction on 0; when the system is operated at the p threshold (a0 D 0), b0 D DQ, and (20.7) reduces to the case of the correlated additive and multiplicative white noises; when the system is operated below the threshold (a0 < 0), must satisfy 0 < .K C a0 /=2Ka0. In the case of a stationary state, the probability density function Pst .I / of (20.7) can be obtained below: 1. When a0 > 0, the probability density function is given by ˇ2

ˇ1 A 1 2 I C1 Pst .I / D N QI C 2b0 I C D F1 2 1 2

QI C b0 1 6 exp 4ˇ3 arctan q C ˇ4 arctan I 2 DQ b02

s

!

3

A 7 5; F1 (20.12)

for jj 1, where ˇ1 D

K F12 .AD F1 Q/ 1; 2Q 2Œ.AD F1 Q/2 C 4b02 F1 A

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Dynamical Properties of Intensity Fluctuation

237

F12 .AD F1 Q/ ; 2Œ.AD F1 Q/2 C 4b02 F1 A F12 .AD C F1 Q/ K b0 ˇ3 D q 1 ; 2 2 DQ b02 Q .AD F1 Q/ C 4b0 F1 A

ˇ2 D

and

5p 2b0 F12 A : ˇ4 D .AD F1 Q/2 C 4b02 F1 A

2. When a0 D 0, the probability density function is given by ˇ2 p A 1 ˇ1 2 I C1 Pst .I / D N.QI C 2 DQI C D/ F1 s !# " p 1 1 A QI 2 C DQ C ˇ4 arctan I 2 exp ˇ3 arctan p ; 2 F 1 DQ.1 / (20.13) for 0 jj < 1 and p 1 Pst .I / D N.QI C 2 DQI 2 C D/ˇ1 " exp

1

QI 2

A I C1 F1

ˇ2

˛1 1 C ˇ4 arctan I 2 p C DQ

s A F1

!# : (20.14)

for jj D 1, where p F12 .AD C F1 Q/ K ˛1 D DQ 1 : Q .AD F1 Q/2 C 42 DQF1 A ˚ 3. When a0<0, discriminant D4.b02 DQ/D4DQ 2 =Œ1 C 2K.1K=a0 C K/2 1 , plus-minus of which is determined by taking values of , K, a0 and . (a) > 0, we have 1

Pst .I / D N.QI C 2b0 I 2 C D/ˇ1

A I C1 F1

ˇ2

1˛2 0 p q s !# " 2 I C b .b DQ/ Q 0 A 1 0 C B q @ p ; A exp ˇ4 arctan I 2 F 1 Q I C b0 C .b02 DQ/ (20.15)

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P. Zhu

where F12 .AD C F1 Q/ K 1 : ˛2 D q 2 2 2 b02 DQ Q .AD F1 Q/ C 4b0 F1 A

b0

(b) D 0, we have Pst .I / D N.QI C 2b0 I

1 2

C D/

ˇ1

"

A I C1 F1

ˇ2

1 ˛3 exp C ˇ4 arctan I 2 QI C b0

where

˛3 D b0

s A F1

!# ;

(20.16)

F12 .AD C F1 Q/ K 1 : Q .AD F1 Q/2 C 4b02 F1 A

(c) < 0, we have

1

Pst .I / D N.QI C 2b0 I 2 C D/ˇ1 2

A I C1 F1

ˇ2

1

QI 2 C b0 1 6 exp 4ˇ3 arctan q C ˇ4 arctan I 2 2 DQ b0

s

3 ! A 7 5: F1 (20.17)

In (20.12)–(20.17), N is the responding normalization constant. The normalization constant N is given by the equation Z

1

Pst .I /dI D 1:

0

Then, expectation values of the nth power of the laser intensity I are given by Z

1

hI n i D

I n Pst .I /dI:

(20.18)

0

For a general laser model where a stationary state exists, the stationary correlation function is defined by C.s/ D hıI.t C s/ıI.t/ist D lim hıI.t C s/ıI.t/i; t !1

where ıI.t/ D I.t/ hI.t/i:

(20.19)

20

Dynamical Properties of Intensity Fluctuation

239

A normalized correlation function is C.s/ D

hıI.t C s/ıI.t/ist : h.ıI /2 ist

(20.20)

The associated relaxation time which describes the fluctuation decay of the laser intensity variable I is defined by Z

1

Tc D

C.t/dt;

(20.21)

0

By using the projection operator method [29], the zeroth approximation for the relaxation time is given by Tc D 01 D

h.ıI /2 ist : hG.I /ist

(20.22)

Similarly, the first-order approximation for the relaxation time is given by 1 1 ; (20.23) Tc D 0 C 1 where 1 D and 1 D

hG.I /f 0 .I /ist C 02 ; h.ıI /2 ist

(20.24)

hG.I /Œf 0 .I /2 ist 03 C 20 : 1 h.ıI /2 ist 1

(20.25)

Employing (20.9), (20.10), and (20.18), we have 0 D

1 D

4k1 ; 2 st hI ist

(20.26)

h.I /2 i

8Œ.K C 2Q/k1 C F1 k2 C

3b0 2 k12

h.I /2 ist hI i2st

C 02 ;

(20.27)

9 16.K C 2Q/2 > > = 3 1 k1 C 16F12 k4 C 36b02 k0 C 0 20 ; 1 D > C32.K C 2Q/F1 k2 1 1 .h.I /2 ist hI i2st / ˆ > ˆ ; : C48b0 .K C 2Q/k12 C 48b0 F1 k5 (20.28) 8 ˆ ˆ <

where 1

k0 D D C 2b0 hI 2 ist C QhI ist;

(20.29)

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P. Zhu 3

k1 D DhI ist C 2b0 hI 2 ist C QhI 2 ist ; 1

(20.30)

3

k12 D DhI 2 ist C 2b0 hI ist C QhI 2 ist ; + * + * * k2 D D

1

* k4 D D

C 2b0

I

2 1C AI F

+

st

* C 2b0

I

4 1C AI F 1

I

3 2

1C AI F1 I

3 2

1C AI F

CQ

2

+

I2 1C AI F

CQ

2

1

*

4

1

st

st

(20.31) +

I2 1C AI F

(20.32)

;

(20.33)

+ 4

1

st

; st

st

and *

+

1

I2

k5 D D

1C

AI F1

2 st

*

I

C 2b0

1C

AI F1

+

* CQ

2 st

+

3

I2 1C

AI F1

2

(20.34) : st

Here, we see that the zeroth approximation of the relaxation time Tc D 01 is good agreement with that by virtue of the Stratonovich-like ansatz [41]. When D 0 and Q D 0, the above results fall back to (2.29)–(2.31) presented in [24]. In other words, the Stratonovich-like ansatz completely neglects the memory kernel. Performing converse the Laplace transformation, we easily get the correlation function of the laser intensity fluctuation C.s/ D ˇ exp. s/ C .1 ˇ/ exp. C s/; where ˇD

1 C

(20.35)

(20.36)

and

1p 0 C 1 ˙ .0 1 /2 41 : (20.37) 2 2 To see the effects of cross-correlation noises on the laser intensity fluctuation, by virtue of the expression of the associated relaxation time (20.23) and the correlation function (20.35) and by applying the method of numerical calculation, we have plotted the curves of C.s/ and Tc (a0 > 0) in Figs. 20.4–20.8, respectively, so that we can discuss properties of the intensity fluctuation of a saturation laser model with cross-correlation noises. ˙ D

20.3 Discussion and Conclusion In order to illustrate the effects of the coupling strength and the cross-correlation time between additive and multiplicative white noises, by the means of numerical calculations, we plot the curves of the stationery probability distribution (SPD) of

20

Dynamical Properties of Intensity Fluctuation

a

241

0.14

=-1 =-0.6 =-0.3 =0 =0.3 =0.6 =1

0.12 Pst(I)

0.1 0.08 0.06 0.04 0.02 0

0

5

10

I

15

20

b 0.25

t=0 t=0.1

Pst(I)

0.2 0.15 0.1

————————— t=0.5 ————————— t=1

0.05 0

5

10

I

15

20

Fig. 20.1 The steady-state laser intensity distribution function Pst .I / vs the variable I above the threshold a0 > 0. The parameters chosen are a0 D 5, A D 1, D D 1, Q D 0:5 and K D 30. (a) is fixed to be 0.3 and takes different values. (b) is fixed to be 0.2 and takes different values

the saturation laser model vs. the laser intensity variable I for above the threshold .a0 > 0/, at the threshold .a0 D 0/, and below the threshold .a0 < 0/ under the condition of 1 C 2K.1 K=.a0 C K// > 0 in Figs. 20.1–20.3. When the laser is operated above the threshold .a0 > 0/, the SPD of the system vs. the variable I is plotted in Fig. 20.1. In Fig. 20.1a, is fixed to be 0:3 and takes different values. We see that the SPD exhibits one-maximum structure, the height of the peak of Pst .I / decreases as the coupling strength increases from 1 to 1. In Fig. 20.1b, is fixed to be 0:2 and takes different values, we see that for D 0 and the smaller value of , Pst .I / always gives a higher value as I ! 0 and decreases monotonously as I increases; however, when is increased continuously, Pst .I / starts to exhibit one-maximum structure, and the height of the peak of the SPD decreases as increases. When the laser is operated at the threshold (a0 D 0), the effects of the crosscorrelation time cannot occur, and the SPD of the system vs. the variable I is plotted in Fig. 20.2. From Fig. 20.2, we see that the values of the SPD decreases as increases for a fixed I , that is, the coupling strength attenuates the SPD; under the case of the parameters chosen, when takes larger values ( > 0), Pst .I / possesses no-maximum structure, and Pst .I / decreases monotonously as I

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0.9

Pst(I)

0.6

0.5 0

0.4

0.5 0.9

0.2

0

2

4 I

6

8

Fig. 20.2 The steady-state laser intensity distribution function Pst .I / vs the variable I for different values of the coupling strength at threshold a0 D 0. The parameters chosen are A D 1, D D 1, Q D 2 and K D 30

increases; however, when takes smaller values( < 0), Pst .I / possesses onemaximum structure and the height of the speak of the SPD increases as decreases. When ( D 0), the Pst .I / I curve is agreeable with the result given by [13]. When the laser is operated ˚ below the threshold (a0 < 0), discriminant D 4 b02 DQ D 4DQ 2 =Œ1 C 2K.1 K=a0 C K/2 1 , plus–minus of which is determined by taking values of , K, a0 , and , and the approximate Fokker–Plank equation [20.8] must satisfy the condition 1 C 2K.1 FK1 / > 0. For example, if K D 60 and a0 D 5, 0 < 0:092. In Fig. 20.3a, the crosscorrelation time takes D 0:05, When jj > 0:4545, > 0; When jj D 0:4545, D 0; When jj < 0:4545; the SPD vs. the variable I for different values of the coupling strength is plotted. Pst .I / possesses no-maximum structure and Pst .I / decreases monotonously as I increases. The coupling strength enhances the SPD. In Fig. 20.3b, is fixed to be 0.5 and takes different values, the parameters chosen make the discriminant < 0, and corresponding the SPD is plotted. Pst .I / possesses no-maximum structure and Pst .I / decreases monotonously as I increases. The correlation time weakens the SPD. When the laser starts working, it is always from below the threshold to at the threshold, and again to above threshold. So we discuss mainly the character of the associated time and the correlation function above the threshold and at the threshold. The associated relaxation time Tc gives dynamical information about the time scale of the evolution of a spontaneous fluctuation of the system in the steady state, which means that it reflects the evolution velocity of the system from an arbitrary initial state to the steady state. The associated relaxation time distribution diagrams of the saturation laser model vs. different parameter variables are given in Figs. 20.4–20.6. In Fig. 20.4a, the associated time cures of Tc are symmetrical on the axes D 0. When < 0, Tc decreases as increases. On the contrary, when > 0, Tc increases as increases. When D 0, Tc is unchange as increases. In Fig. 20.4b, the associated time Tc monotonously decreases as increases. For

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Fig. 20.3 The steady-state laser intensity distribution function Pst .I / vs. the variable I for different values of the coupling strength below the threshold a0 < 0. The parameters chosen are a0 D 5, K D 60, A D 1, D D 1, and Q D 2. (a) is fixed to be 0.05 and takes different values. (b) is fixed to be 0.5 and takes different values

a 0.065 =-1 =-0.5

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Fig. 20.4 (a) The relaxation time Tc as a function of the cross-correlated time for different values of the coupling strength . Parameters chosen are a0 D 5, K D 60, A D 1, D D 1 and Q D 2. (b) The relaxation time Tc as a function of the coupling strength for different values of the cross-correlated time . The parameters chosen are a0 D 5, K D 60, A D 1, D D 1 and Q D 2

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Fig. 20.4 (continued)

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Fig. 20.5 (a) The relaxation time Tc as a function of the quantum noise intensity D for different values of the coupling strength . Parameters chosen are a0 D 5, D 0:9, K D 120, A D 1, and Q D 2. (b) The relaxation time Tc as a function of the quantum noise intensity D for different values of the cross-correlated time . The parameters chosen are a0 D 5, D 0:5, K D 120, A D 1, and Q D 2

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40 Q

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Fig. 20.6 (a) The relaxation time Tc as a function of the pump noise intensity Q for different values of the coupling strength . Parameters chosen are a0 D 5, D 0:9, K D 120, A D 1, and D D 1. (b) The relaxation time Tc as a function of the pump noise intensity Q for different values of the cross-correlated time . The parameters chosen are a0 D 5, D 0:5, K D 120, A D 1, and DD1

< 0, Tc decreases as takes the larger values; for > 0, the effects of are entirely opposite. Figure 20.5 shows the the effects of the quantum noise intensity D, the coupling strength , and the correlation time for the associated relaxation time. Tc –D curves exhibit one-minimum structure. When D takes smaller values, Tc decreases as D increases, and the effects of attenuate the associated relaxation time. When D takes larger values, Tc increases as D increases, enhances the associated relaxation time. In contrast, the effects of are entirely opposite for the associated relaxation time. Figure 20.6 displays the the effects of the pump noise intensity Q, the coupling strength , and the correlation time for the associated relaxation time. Tc –Q curves exhibit one-minimum structure. When Q takes smaller values, Tc decreases as Q increases and the effects of and do not almost occur. When Q takes larger

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a

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Fig. 20.7 The normalized correlation function C.s/ as a function of decay time s for a0 D 5, A D 1, K D 60, D D 1 and Q D 1:5. (a) is fixed to be 0.2 and takes different values. (b) is fixed to be 0.2 and takes different values

values, Tc increases as Q increases, attenuates the associated relaxation time, however, the effects of are entirely opposite. The correlated function C.s/ describes the fluctuation decay of the variable ıI with time in the stationary state. To see that the effect of the coupling strength for the intensity fluctuation, C.s/ as functions of the decay time s are plotted in Fig. 20.7. It is clear that the correlation function C.s/ is an exponential function of the variable I . In Fig. 20.7a, the cross-correlated time is fixed to be 0:2 and takes different values. C.s/ increases as increases from 0:9 to 0:9. In Fig. 20.7b, the correlated strength is fixed to be 0:8 and takes different values. C.s/ increases as increases from zero to 1. When the laser system is operated at the threshold, the associated relaxation time and the correlation function distribution diagrams are given in Fig. 20.8. In Fig. 20.8a, the relaxation time decreases as increases. In Fig. 20.8b, C.s/ decreases as increases from 0:9 to 0:9 for a fixed value of the decay time s. From the foregoing, we can further understand that dynamical properties of the saturation laser intensity, and see that the cross-correlated additive and multiplicative noises play important roles in a laser model with a full account of the saturation effects.

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Fig. 20.8 The laser system is operated at the threshold (a0 D 0). (a) The relaxation time Tc as a function of the coupling strength . Parameters chosen are K D 120, A D 1, and Q D 2. (b) The normalized correlation function C.s/ as a function of decay time s for different values of the coupling strength . Parameters chosen are A D 1, K D 60, D D 1 and Q D 2

1. When the laser system is operated at the threshold and below the threshold, Pst .I / mainly concentrated on the region of I ! 0, so it cannot almost possess the laser intensity output. When the laser system is operated above the threshold, Pst .I / possesses one-maximum structure, and the laser system possesses the output of the steady laser intensity. The coupling strength weakens the peak of the SPD, and the position of the peak shifts to the smaller intensity variable I as increases; in contrast, the cross-correlation time generates entirely opposite effects. 2. The relaxation time Tc reflects the evolution velocity of the system from arbitrary initial state to the steady state, which can characterize the startup time of the laser system from start to steady work. When the laser system is operated above the threshold, the coupling strength speeds up the startup; When < 0, the crosscorrelation time speeds up the startup; When > 0, the cross-correlation time blows down the startup; The quantum noise intensity D and the pump noise

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intensity Q makes the relaxation time exhibit the extremum structures, that is, the startup time possesses the least values at D D D0 and Q D Q0 , which are as functions of other parameters, respectively. 3. The correlation function reflects the stability of laser intensity of laser output. When the laser system is operated above the threshold, which the laser system possesses the steady output of laser intensity, enhances the stability of the intensity output of the saturation laser system, however, weakens the stability of the intensity output. 4. When the laser system is operated at threshold, the coupling strength expedites the startup and enhances the stability of the startup.

References 1. Kaminish K, Roy R, Short R, Mandel L (1981) Investigation of photon statistics and correlations of a dye laser. Phys Rev A 24:370–378 2. Dixit SN, Sahni PS (1983) Nonlinear stochastic processes driven by colored noise: application to dye-laser statistics. Phys Rev Lett 50:1273–1276 3. Jung P, Leiber T, Risken H (1987) Dye laser model with pump and quantum fluctuations; white noise. Z Phys B 66:397–407 4. Roy R, Yu AW, Zhu S (1985) Quantum fluctuations, pump noise, and the growth of laser radiation. Phys Rev Lett 55:2794–2797 5. Zhu SQ (1989) Multiplicative colored noise in a dye laser at steady state. Phys Rev A 40:3441–3443 6. Young MR, Singh S (1988) Statistical properties of a laser with multiplicative noise. Opt Lett 13:21–23 7. Zhu SQ (1993) Steady-state analysis of a single-mode laser with correlations between additive and multiplicative noise. Phys Rev A 47:2405–2408 8. Zhu SQ, Yu AW, Roy R (1986) Statistical fluctuations in laser transients. Phys Rev A 34:4333–4347 9. Zhu SQ (1990) White noise in dye-laser transients. Phys Rev A 42:5758–5761 10. Aguado M, Hernandez-Garcia E, San Miguel M (1988) Dye-laser fluctuations: Comparison of colored loss-noise and white gain-noise models. Phys Rev A 38:5670–5677 11. Schenzle A, Boyd RW, Raymer MG, Narducci LM (eds) (1986) In optical instabilities. Cambridge University Press, Cambridge 12. Hernandez-Garcia E, Toral R, San Miguel M (1990) Intensity correlation functions for the colored gain-noise model of dye lasers. Phys Rev A 42:6823–6830 13. Zhu S (1992) Saturation effects in a laser with multiplicative white noise. Phys Rev A 45:3210–3215 14. Cao L, Wu DJ, Luo XL (1992) Effects of saturation in the transient process of a dye laser. I. White-noise case. Phys Rev A 45(9):6838–6847 15. Cao L, Wu DJ, Luo XL (1992) Effects of saturation in the transient process of a dye laser. II. Colored-noise case. Phys Rev A 45(9):6848–6856 16. Cao L, Wu DJ, Luo XL (1993) Effects of saturation in the transient process of a dye laser. III. The case of colored noise with large and small correlation time. Phys Rev A 47(1):57–70 17. Cao L, Wu DJ (1999) Cross-correlation of multiplicative and additive noises in a singlemode laser white-gain-noise model and correlated noises induced transitions. Phys Lett A 260:126–131 18. Long Q, Cao L, Wu DJ, Li ZG (1997) Phase lock and stationary fluctuations induced by correlation between additive and multiplicative noise terms in a single-mode laser. Phys Lett A 231:339–342

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19. Xie CW, Mei DC (2004) Effects of correlated noises on the intensity fluctuation of a singlemodle laser system. Phys Lett A 323:421–426 20. Liang GY, Cao L, Wu DJ (2002) Moments of intensity of single-mode laser driven by additive and multiplicative colored noises with colored cross-correlation. Phys Lett A 294:190–198 21. Zhu SQ, Yin JP (1992) Saturation effect in a laser at steady state. Phys Rev A 45:4969–4973 22. Zhu P, Chen SB, Mei DC (2006) Intensity correlation function and associated relaxation time of a saturation laser model with correlated noises. Chin Phys Lett 23(1):29–30 23. Zhu P, Chen SB, Mei DC (2006) Effects of correlated noises in a saturation laser model. Modern Physics Letters B 20(23):1481–1488 24. Hernandez-Machado A, San Migud M, Sancho JM (1984) Relaxation time of processes driven by multiplicative noise. Phys Rev A 29:3388–3396 25. Casadememunt J, Mannell R, McClintock PVE, Moss FE, Sancho JM (1987) Relaxation times of non-Markovian processes. Phys Rev A 35:5183–5190 26. Sancho JM, Mannell R, McClintock PVE, Frank M (1985) Relaxation times in a bistable system with parametric, white noise: theory and experiment. Phys Rev A 32:3639–3646 27. Hernandez-Machado A, Casademunt J, Rodeigaez MA, Pesqueraet L, Noriega JM (1991) Theory for correlation functions of processes driven by external colored noise. Phys Rev A 43:1744–1753 28. Xie CW, Mei DC (2004) Dynamical properties of a bistable kinetic model with correlated noises. Chin Phys 42:192–199 29. Mei DC, Xie CW, Zhang L (2003) Effects of cross correlation on the relaxation time of a bistable system driven by cross-correlated noise. Phys Rev E 68:051102–051108 30. Mei DC, Xie CW, Xiang YL (2003) The state variable correlation function of the bistable system subject to the cross-correlated noise. Physica A 343:167–174 31. Zhu P (2006) Effects of self-Correlation time and cross-correlation time of additive and multiplicative colored noises for dynamical properties of a bistable system. J Stat Phys 124(6):1511–1525 32. Zhu P (2007) Associated relaxation time and intensity correlation function of a bistable system driven by cross-correlation additive and multiplicative coloured noise sources. Eur Phys J B 55:447–452 33. Jin YF, Xu W, Xie, WX, Xu M (2005) The relaxation time of a single-mode dye laser system driven by cross-correlated additive and multiplicative noises. Physica A 353:143–152 34. Fox RF, Roy R (1987) Steady-state analysis of strongly colored multiplicative noise in a dye laser. Phys Rev A 35:1838–1842 35. Novikov EA (1964) Functionals and the random force method in turbulence theory. Zh Exp Teor Fiz 47:1919–1926 (English transl.: Sov Phys JETP 20 (1964), 1290–1295) 36. Fox RF (1986) Uniform convergence to an effective Fokker-Planck equation for weakly colored noise. Phys Rev A 34:4525–4527 37. Hanggi P, Mroczkowski TT, Moss F (1985) McClintocket PVE: bistability driven by colored noise: theory and experiment. Phys Rev A 32:695–698 38. Wu DJ, Cao L, Ke SZ (1994) Bistable kinetic model driven by correlated noises: steady-state analysis. Phys Rev E 50:2496–2502 39. Jia Y, Li JR (1996) Steady-state analysis of a bistable system with additive and multiplicative noises. Phys Rev E 53:5786–5792 40. Fujisaka H, Grossmann S (1981) External noise effects on the fluctuation line width. Z Phys B 43:69–75 41. Sreaonovich RL (1967) Topics in the theory of random noises, vol II. Chap. 7. Gordon and Breach, New York

Chapter 21

Empirical Mode Decomposition Based on Bistable Stochastic Resonance Denoising Y.-J. Zhao, Y. Xu, H. Zhang, S.-B. Fan, and Y.-G. Leng

Abstract The empirical mode decomposition (EMD) of weak signals submerged in a heavy noise was conducted and a method of stochastic resonance (SR) used for noisy EMD was presented. This method used SR as pre-treatment of EMD to remove noise and detect weak signals. The experiment result proves that this method, compared with that using EMD directly, not only improve SNR, enhance weak signals, but also improve the decomposition performance and reduce the decomposition layers.

21.1 Introduction In the quest of accurate time and frequency localization, Huang et al. [1, 2] proposed the empirical mode decomposition (EMD) scheme which offers a different approach in time-series processing. This method can decompose the signal into a set of oscillatory modes by taking advantage of the characteristic time scales embedded in the data. So there is no need for a basis function and no need for transformation [3]. But EMD method can not eliminate boundary problem because it uses cubic splines method to obtain signal’s instantaneous average. If EMD is used in strong noise condition it will affect the decomposition performance, especially increase decomposition layers and lower efficiency of arithmetic, even may lead EMD lose definitude significance, so denoising process must be performed before EMD decomposition. References [4] and [5] use wavelet and SVD to denoise and get good effect. But these methods are not good at weak signals submerged in heavy noise. At the meantime, stochastic resonance (SR), which is put forward by Benzi [6] wherein he addresses the problem of the primary cycle of recurrent ice ages, has a strong development for its advantages in weak signal’s enhancement and detection. With the cooperation effect of signal and noise in non-liner system, SR can put Y.-J. Zhao () CSR Qingdao Sifang Locomotive and Rolling Stock Company, Chengyang, Qingdao 266111, People’s Republic of China e-mail: [email protected]

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 21,

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noise power into lower-frequency signal which can enhance weak signal and reduce noise at the same time. In a word, this paper proposes a new method of weak signals detection based on SR theory and EMD decomposition. Firstly, do denoising procession with SR and then do EMD decomposition. Experiments proved that this method is better than direct EMD decomposition method.

21.2 Empirical Mode Decomposition Empirical Mode Decomposition (EMD) is a novel method for adaptive of non-linear and non-stationary signals. It can decompose any non-linear signal into several Intrinsic Mode Functions (IMFs), and a residue.

21.2.1 IMFs The components resulting from EMD, called Intrinsic Mode Functions (IMFs), each admit an unambiguous definition of instantaneous frequency. By definition, an Intrinsic Mode Function (IMF) satisfies two conditions 1. The number of extreme and the number of zero crossing may differ by no more than one 2. The local average is zero where the local average is defined by the average of the maximum and minimum envelopes discussed in the following section. These properties of IMFs allow for instantaneous frequency and amplitude to be defined unambiguously.

21.2.2 The Sifting Processing In order to obtain the separate components called IMFs, we perform a sifting process. The goal of sifting is to subtract away the large-scale features of the signal repeatedly until only the fine-scale features remain. A signal x.t/ is then divided into the fine-scale detail, h.t/ and a residual, m.t/ so x.t/ D m.t/ C h.t/. This detail becomes the first IMF and the sifting process is repeated on the residual m.t/ D x.t/ d.t/. The sifting process requires that a local average of the function be defined. If we knew the components before, we would naturally define the local average to be the lowest frequency component. Since the goal of EMD is to discover these components, we must approximate the local average of the signal. Huang’s solution to find a local average creates maximum and minimum envelopes around the signal by

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using natural cubic splines through the respective local extreme. The local average is approximated as the mean of the two envelopes. The first IMF, c1 .t/ of a signal, x.t/, is found by iterating through the following loop. 1. Find the local extreme of x.t/. 2. Find the maximum envelope eC .t/ of x.t/ by passing a natural cubic spline through the local maxima. Similarly find the minimum envelope e .t/ with the local minima. 3. Compute an approximation to the local average, m.t/ D .eC .t/ C e .t//=2. 4. Find the proto-mode function h1 .t/ D x.t/ m.t/. 5. Check whether h1 .t/ is an IMF. If h1 .t/ is not an IMF, repeat the loop on h1 .t/. If h1 .t/ is an IMF then set c1 .t/ D h1 .t/. The sifting indicates the process of removing the lowest frequency information until only the highest frequency remains. The sifting procedure performed x.t/ on can then be performed on the residual r1 .t/ D x.t/ c1 .t/ to obtain r2 .t/ and c2 .t/, repeat the process as described above for n time, then n IMFs of signal x.t/ could be got. Then 8 r2 .t/ D r1 .t/ c2 .t/ ˆ ˆ ˆ < r3 .t/ D r2 .t/ c3 .t/ (21.1) :: ˆ ˆ : ˆ : rn .t/ D rn1 .t/ cn .t/ At last, the signal x.t/ is decomposed into several Intrinsic Mode Functions (IMFs), and a residue. n X x.t/ D ci .t/ C rn .t/ (21.2) i D1

Residue rn .t/ is the mean trend of x.t/. The IMFs c1 ; c2 ; : : : ; cn include different frequency bands ranging from high to low. The frequency components contained in each frequency band are different and they change with the combination signal x.t/, while rn .t/ represents the central tendency of signal x.t/.

21.2.3 Examples In order to illustrate the performance of EMD, we use a combination of two pure sine waves, which are added together as x.t/ D a1 sin.2 f1 t/ C a2 sin.2 f2 t/

(21.3)

The values of f1 and f2 are respective 10 and 30 Hz, amplitudes are both 1.5.

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Fig. 21.1 The EMD decomposed result of pure mixed signal

As shown in Fig. 21.1, mixed signal x.t/ is decomposed into two IMFs: c1 ; c2 and the residue r2 c1 is corresponding with sine wave of 30 Hz, c2 is the other sine wave of 10 Hz. It can be seen from this example that, with the EMD method, signal can be decomposed into some different time-scales IMFs by which the characteristics of the signal can be presented in different resolution ratio. However, signals are not always pure in practice. They usually mix much noise which can affect EMD performance. In Fig. 21.2, the mixed signal is still composed by two sine waves 30 and 10 Hz, but mixed noise of intensity D D 0:2. While the amplitude of 30 Hz is still 1.5, but the one of 10 Hz changes into 0.1, belongs to weak signals. As shown in Fig. 21.2, c1 c7 are IMFs decomposed form EMD, and r7 is the residue. It can be seen from this figure that the first IMF c1 is mainly composed of high frequency noise, and for the noise existence, the sine wave of 30 Hz is decomposed into c2 and c3 , the wave is serious distortion especially in time domain. c4 c6 are supposed to depict sine wave of 10 Hz, but due to small amplitude, they cannot be distinguished in despite of in frequency spectrum. Therefore, when signal is very weak and mixed with noise, direct EMD performance is bad. It will result distorted IMFs and cannot detect the weak signal. So before EMD operation denoising process is necessary.

21.3 Stochastive Resonance Stochastic resonance (SR) is a phenomenon in which a nonlinear system heightens the sensitivity to a weak signal input and when noise with an optimal intensity is presented simultaneously. So SR doesn’t remove but make use of noise to gain the optimal and desirable signal-to-noise ration (SNR) of output signals.

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Fig. 21.2 The EMD decomposed result of noisy and weak signal

21.3.1 Bistable System SR is most often considered by the example of the motion equation of a light particle in a bistable potential field. It is disturbed by a small periodic signal and additive white noise: xP C f .t/ D A sin.!t/ C n.t/; (21.4) where x is the particle displacement, A sin !t is a weak periodic signal at frequency !; f .x/ D dU.x/=dx; U.x/ is a symmetric double-well potential We will consider the simplest case of such a potential, namely, U.x/ D ax 2 =2Cbx 4 =4n.t/ is white noise of intensity D, i.e. n.t/n.t C / D Dı./. As it follows from [7–9], under the condition of adiabatic elimination and small parameters (frequency, amplitude and intensity of noise are less than 1), the power spectrum of (21.4) is described that high frequency noise is weakened, and spectra energy is center around in the low frequency area, especially, there is a peak at the frequency of !. However, the adiabatic elimination SR theory in small parameters cannot meet larger signals in practice.

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So the method of twice sampling stochastic resonance is recommended. According to this method, we first compress the measurement signals to low frequency ones in a linear mode to meet the requirement of small parameters. Secondly, analyze the compressed data spectrum to acquire characteristics of signals. At last we reconstruct the compressed data to origin according to the linear mode mentioned earlier.

21.3.2 Examples Figure 21.3 is a given example to illustrate the implementation of stochastic resonance under larger parameters signal condition. As shown in Fig. 21.3, the f0 value is much more than 1, belongs to large parameter signals, so the twice sampling

Fig. 21.3 Implement of SR of large parameters single, where f0 D 10 Hz; fs D 2;000 Hz, and data length is 2,048. Parameter values are a D 0:1; b D 1; A D 0:5, and D D 0:6

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stochastic resonance method is adopted to compress it to small parameter signal. The twice sampling frequency is selected 8 Hz, then the compressed frequency is calculated 0.04 Hz. In the figure, (c) and (d) graphs display waveform and spectrum of compressed signal, (c) and (d) is the SR output of compressed signal. Compared with graph (c) and (e), because high-frequency noise in original signal is weakened, the waveform of SR output becomes smooth, at the same time the amplitude of signal is heightened, as shown in graph there is a peak at the frequency f0 .

21.4 Test of EMD Based on SR Denoising We still use the 10 and 30 Hz sine wave signals. Sampling frequency is 2 kHz and data length is 2,048. Amplitude of 30 and 10 Hz signal is 1.5 and 0.1. The noise of intensity is 0.2. Parameters of SR are set as: a D 0:1; b D 1, twice sampling frequency is 8 Hz. Figure 21.4 shows signal’s waveform and spectrum before and after SR process. The 10 Hz weak signal nearly submergence in noise is seen in graph (b), but after SR system the weak signal is enhanced as shown in graph (d). Figure 21.5 gives the EMD decomposition results of signal which is processed by SR. It is clear that the high frequency noise is filtered by SR system and 10-Hz weak signal is enhanced. It can be seen from frequency domain, the first IMF c1 is 30 Hz frequency component, it is smooth and obvious. IMF c2 is correspondence with 10 Hz signal. IMF c3 c6 , for their small amplitudes, consider as residue.

Fig. 21.4 Waveform and spectrum of original signal before and after SR

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Fig. 21.5 EMD decomposition result of SR output

Figures 21.2 and 21.5 use the same original signal. In Fig. 21.2, we decompose signal with EMD directly, for the reason of noise the first IMF c1 is mainly highfrequency noise component. 30 Hz frequency component is decomposed into IMF c2 ; c3 and is distorted. Ten hertz correspondence with c4 c6 , for the low amplitude it is hard to detect. Compared with Figs. 21.2 and 21.5, we can conclude that SR C EMD method is better than only EMD method. For large signal the former performance is better than the latter and for weak signal only the former can be detected and decomposed. Otherwise, SR C EMD method can reduce EMD decomposition’s layer, as shown in figures, there are eight layers in Fig. 21.5 but only six in Fig. 21.2.

21.5 Conclusion In order to use EMD decomposition in weak signals under noisy background, this paper puts forward a new method: using SR as pre-treatment and then performing EMD decomposition. The experiment result proved that this method,

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compared with EMD directly, not only improve SNR, enhance weak signals, but also improve the decomposition performance and reduce the decomposition layers of signals. Acknowledgments This work is supported by national 863 project fund Grant # 2007AA04Z414 and National Natural Science Foundation of China Grant # 50675153.

References 1. Huang NE, Shen Z, Long SR, et al (1998) The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis [J]. Proc R Soc Lond A 454:903–995 2. Huang NE (1996) Computer implicated empirical mode decomposition method, apparatus, and article of manufacture [P]. U.S. Patent Pending 3. Deng Y (2001) Comment and modification on EMD and Hilbert transform method [J]. Chinese Sci Bull 46(3):257–263 4. Dai G-P, Liu B (2007) Instantaneous parameters extraction based on wavelet denoising and EMD [J]. Acta Meteor Sin 28(2):158–162 5. Wang T, Wang Z, Xu Y (2005) Empirical mode decomposition and its engineering applications based on SVD denoising [J]. J Vib Shock 24(4):96–98 6. Benzi R, Sutera A, Vulpiana A (1981) The mechanism of stochastic resonance. J Phys A l4(11):L453–L4572 7. Leng Y-G, Wang T-Y, et al (2004) Power spectrum research of twice sampling stochastic resonance response in a bistable system [J]. Acta Phys Sin 53:(3):717–723 8. Leng Y-G, Wang T-Y (2003) Numerical research of twice sampling stochastic resonance for the detection of a weak signal submerged in a heavy noise [J]. Acta Phys Sin 52(10):2432–2437 9. Leng Y-G (2004) Mechanism analysis of the large signal scale-transformation stochastic resonance and its engineering application study. Papers of PHD, Tianjin University, Tianjin

Chapter 22

Order Reduction of a Two-Span Rotor-Bearing System Via the Predictor-Corrector Galerkin Method Deng-Qing Cao, Jin-Lin Wang, and Wen-Hu Huang

Abstract The predictor-corrector Galerkin method (PCGM) is employed to obtain a lower order system for a two-span rotor-bearing system. First of all, a 32-DOF nonlinear dynamic model is established for the system. Then, the predictor-corrector algorithm based on the Galerkin method is employed to deal with the problem of order reduction for such a complicated system. The first six modes are chosen to be the master subsystem and the following six modes are taken to be the slave subsystem. Finally, the dynamical responses are numerically worked out for the master and slave subsystems using the PCGM, and the results obtained are used to compare with those obtained by using the SGM. It is shown that the PCGM provides a considerable increase in accuracy for a little computational cost in comparison with the SGM in which the first six modes are reserved.

22.1 Introduction Increasing demands for high performance rotating machinery have made the rotor dynamic problems more and more complex, and more and more attention has been drawn to the nonlinear dynamics of large-scale rotor-bearing systems. Usually, a large rotating machine consists of two or more shafts which are rigidly or flexibly coupled together to form a continuous rotor supported on three or more hydrodynamic journal bearings. Such a rotating machine is referred to as a multi-span rotor-bearing system. Up to now, research efforts on dynamics of multi-span rotor-bearing systems are much fewer than those on single-shaft rotor systems, e.g., the rigid or flexible Jeffcott rotor. Ding and Krodkiewski [1] and Krodkiewski et al. [2] proposed a general mathematical model of multi-bearing rotor systems for the straightforward formulation of an approach for on-site identification

D.-Q. Cao () School of Astronautics, Harbin Institute of Technology, P.O. Box 137, Harbin 150001, People’s Republic of China e-mail: [email protected]

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of bearing alignment changes during system operation. Based on the approach proposed in [1, 2], procedures for balancing large multi-bearing rotors have been established in [3]. Hu et al. [4] designed a test rotor supported on four bearing to validate that the vibration behaviors of statically indeterminate rotor-bearing systems with hydrodynamic journal bearings are significantly dependent on the relative lateral alignment of bearing housing. The study of Ding and Leung [5] indicated that the non-synchronous whirls of two flexibly coupled shafts may affect each other. Finite element method (FEM) and computer simulation technology have been widely used in designing and analyzing rotor-bearing systems. And in the machine structure analysis, a refine discretization is usually necessary to obtain a reliable dynamic model. In this process, a large set of second order differential equations of motion for the multi-bearing system is established. Additionally, when one or more nonlinear elements such as fluid-bearing, fluid seal, etc., are included in the system, which is often case, the large order nonlinear systems are usually costly to solve in terms of computer time and storage, especially over long-time intervals. Hence, it is essential to reduce the order of the large-scale nonlinear dynamic system, and subsequently to get a lower order dynamic system which is an approximate representative of the original one. The traditional order reduction methods, such as the Guyan order reduction method [6] and standard Galerkin method (SGM) may also be applied to nonlinear dynamic systems. Although the traditional order reduction methods proved to be efficient in constructing approximate solutions for nonlinear dynamic systems, it has itself own limitation that accurate results may only be achieved through the inclusion of many modes in the reduced order system. If only a few nonlinear elements exist, the large order system can be reduced using a fixed-interface component mode synthesis procedure (CMS) [7] in which the degrees of freedom (DOF) associated with nonlinear elements are retained in the physical coordinates while the linear subsystem, the DOF of which far exceeds the DOF of the nonlinear subsystem, are truncated to a few dominant modes. However, if there are complex nonlinear terms in the system, i.e., the DOF of the nonlinear subsystem is not small enough, the practicability of fixed-interface CMS is worth of further study. An ideally order reduction method is sought to provide a reduced order model that only contains a few modes. In pursuit of the goal, a large dynamical system is transformed to modal coordinates and split into a master subsystem and a slave subsystem. Then the nonlinear Galerkin method (NLGM) is developed, in which a lower order subsystem is constructed by estimating and approximating the slave subsystem as function of the master subsystem. The approximate relation between the master subsystem and slave subsystem was given the name of approximate inertial manifolds (AIM) [8–11]. One simple method for constructing the AIM has been proposed in [11] by ignoring the time derivative term of the slave subsystem. The AIM can be obtained by iteration. In particular, during numerically integrating the reduced order system obtained via NLGM, constructing the AIM at each time step is tedious and very costly. In order to avoid the disadvantage of the NLGM and at the meantime to improve SGM, Garcia-Archilla et al. [12] proposed a so-called postprocessed Galerkin method (PPGM). The PPGM, which is as simple as the SGM

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and has some advantage of NLGM, has been used to solve the nonlinear dynamics of shells in [13] and the dissipative equation in fluid dynamics under periodic boundary conditions [14]. In the application of PPGM, the SGM is used and the AIM only needs to be calculated when output is required. For the merit and fault of NLGM and PPGM, we refer to the comments in [15, 16]. For the rotor-bearing system, since the oil film forces are nonlinear functions of the displacements, the velocities at bearings and the rotating speed of the rotor, the NLGM may be not available due to the limitation on the displacements at bearings. In fact, by ignoring the time derivative term of the slave subsystem to get the AIM may overtop the limitation on the displacements. In order to get an order reduction of a large-scale rotor-bearing system, the predictor-corrector Galerkin method (PCGM) has been proposed in [17] based on the ideals of NLGM and PPGM. In [17], a large nonlinear dynamical system is split into a master subsystem, a slave subsystem, and a negligible subsystem. In a general way, the lower modes whose frequencies are close to the frequencies of external periodic excitations dominate the dynamic behaviors of the large dynamical system, and the corresponding modes can be chosen to be the so-called master subsystem. In order to save the calculating time, the order of the slave subsystem may be chosen to be the same as the master subsystem, while the order of the negligible subsystem may far exceed the order of master subsystem. A 32-DOF nonlinear dynamic model is established for a two-span rotor-bearing system in this paper. The predictor-corrector algorithm based on the Galerkin method is employed to deal with the problem of order reduction for such a complicated system. The first six modes are chosen to be the master subsystem and the following six modes are taken to be the slave subsystem. The dynamical responses are numerically worked out for the master and slave subsystems using the PCGM, and the results obtained are used to compare with those obtained by using the SGM. It is shown that the PCGM provides a considerable increase in accuracy for a little computational cost in comparison with the SGM.

22.2 Modeling of the Two-Span Rotor-Bearing System Consider a two-span rotor-bearing system as shown in Fig. 22.1. It consists of four disks and two shafts, which are rigidly coupled together via a coupling. The shaft is treated as a free-free body and is modeled by FEM based on Euler beam theory. Each node has four degrees of freedom. On the free-free rotor, all external forces can be applied, no matter whether they are linear or nonlinear, static or time-dependent. The dynamic responses of the two-span rotor-bearing system are governed by the following differential equations of motion. M zR C DPz C Kz D f .z; zP; t/ D fn .z; zP/ C fe .t/ C g;

(22.1)

where M; C; K 2 R3232 are the mass, damping, and stiffness matrices, and fn .z; zP/; fe .t/; g 2 R32 are the nonlinear fluid film force, unbalance force, and

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Fig. 22.1 The sketch of the two-span rotor-bearing system

gravitational force vectors, respectively. The short bearing model [18] is employed to describe the fluid film forces at four bearings. The displacement vector is z D Œx1 ; y1 ; ˇ1 ; 1 ; x2 ; y2 ; ˇ2 ; 2 ; : : : ; xr ; yr ; ˇr ; r T 2 R32 ; where r is the number of nodes, xi ; yi , and ˇi ; i are the lateral displacements and rotation angles of the i th nodal point along the horizontal and vertical direction, respectively. The ms .s D 1; 2; 3; 4/ is the mass of the sth disk, es is the eccentricity of the sth disk, and Js is the transverse moment of inertia of the sth disk as shown in Fig. 22.1.

22.3 The Predictor-Corrector Galerkin Method The original system described by (22.1) can be split into three parts by the modes of the linearized part of (22.1), i.e., a master subsystem, a slave subsystem, and a negligible subsystem. To do this, the solution z is assumed to be in the form z D „ D ˆm C ‰s

C Zp ;

(22.2)

where DŒ; ; T ; „DŒˆm ; ‰s ; Zp ;ˆm DŒ'1 ; : : : ; 'm ;‰s D Œ'mC1 ; : : : ; 'mCs ; Zp D Œ'mCsC1 ; : : : ; 'mCsCp , and 'i0 s are the eigenmodes of the linearized part of (22.1). Substituting (22.2) into (22.1) and multiplying with „T from the left, (22.1) can be written as P t/; M R C C P C K D f .„; „; (22.3) where M ; C and K are the diagonal matrices in which the diagonal elements respectively are mi D 'iT M 'i ;

c i D 'iT C 'i

and k i D 'iT K'i ;

P t/ for i D 1; 2; : : : n. and the generalized force is f i D 'iT f .„; „;

(22.4)

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The matrices M ; C ; K and the force vector f are rearranged in the form. 2 2 2 3 3 3 Mm 0 0 Cm 0 0 Km 0 0 M D 4 0 M s 0 5 ; C D 4 0 C s 0 5 ; K D 4 0 Ks 0 5 ; 0 0 Mp 0 0 Cp 0 0 Kp 2 3 fm f D 4f s 5: fp Then, (22.3) can be rewritten as P R P P M m CC m CK m D f m .ˆm C‰s CZp ; ˆm C‰ s P CZp ; t/; (22.5a) M s R C C s P C Ks

P P D f s .ˆm C‰s CZp ; ˆm C‰ s P CZp ; t/; (22.5b)

P P M p R C C p P C K p D f p .ˆm C‰s CZp ; ˆm C‰ s P CZp ; t/: (22.5c)

The excitation frequency of the rotor is far from the higher modes, the contribution of higher modes is considered to be insignificant. Thus, by neglecting (22.5c) or setting D 0, the entire system is approximated by M m R C C m P C K m D f m .ˆm C ‰p ; ˆm P C ‰p P ; t/; M p R C C p P C K p D f p .ˆm C ‰p ; ˆm P C ‰p P ; t/:

(22.6a) (22.6b)

In the process of numerical integration of the system, the refined calculation of the master subsystem is necessary. Equation (22.6b) corresponds to the slave subsystem, for which the solution could be approximately carried out. Therefore, the time step T for (22.6b) may be chosen as several times as the time step t for (22.6a). The solving procedures of the PCGM proposed in [17] can be described as P and .t/; P .t/ are assumed 1. Set T D k t (k > 1 is an integer). .t/; .t/ to be known at a given time t.D mT D mkt for an integer m). 2. To calculate the force vector at time t, ( P C ‰p P .t/; t/ f m .ˆm .t/ C ‰p .t/; ˆm .t/ : P C ‰p P .t/; t/ f p .ˆm .t/ C ‰p .t/; ˆm .t/ 3. The dynamic responses .t C T / and P .t C T / of the slave subsystem are predicted by (22.6b) at time t C T . Then, the dynamic response of the slave subsystem at time t C j t are given by the interpolation method as 8 j Œ .t C T / .t/ ˆ < .t C j t/ D .t/ C ; k

for j D 1; : : : ; k 1: P P ˆ : P .t C j t/ D P .t/ C j .t C T / .t/ ; k

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4. The dynamic responses .t C j t/ and .t C j t/ of the master subsystem are solved by integration of (22.6a) at time t C j t. When j D 1, the force vector f m has been obtained from the second step; when j > 1, the force vector f m is f m .ˆm .t C .j 1/ t/ C ‰p .t C .j 1/ t/; P C .j 1/ t/ C ‰p P .t C .j 1/ t/; t C .j 1/ t/; ˆm .t where .t C .j 1/ t/ and P .t C .j 1/ t/ are given from the third step. P C k t/ of the Up to t C T , the dynamic responses .t C k t/ and .t P C T /. master subsystem are obtained, i.e., .t C T / and .t 5. To repeat the processes from Step 2 to Step 5, we can get the solution of the system for next time step T .

22.4 Numerical Results and Discussion A two-span rotor-bearing model with specific material constants and structural parameters listed in Table 22.1 is now presented to explore the complicated nonlinear dynamic behavior of the system. Numerical calculations based on the procedures of PCGM are carried out. The first six modes are kept as the master subsystem; the

Table 22.1 The geometrical and physical parameters for the rotor-bearing system Physical properties Value L1 D L2 =2 D L3 =3 L4 L5 D L6 =2 D L7 =3 m1 D m2 m3 m4 J1 D J2 J3 J4 e1 D e3 D e4 e2 Journal and shaft radius Bearing length Radial clearance Dynamic viscosity Mass density Young’s modulus Number of elements Number of node (r)

0.38267 m 0.25 m 0.414 m 50.31 kg 45.32 kg 31.63 kg 0:514 kg m2 0:416 kg m2 0:203 kg m2 3 105 m 4 105 m 0.057 m 0.03 m 0:2 103 m 0.018 Pa s 7; 850 kg=m3 2:06 1011 N=m2 7 8

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slave subsystem consists of the following six modes and the remained modes are neglected during numerical integration. At the meantime, the SGM with the first six modes is also selected to reduce the number of DOF. We should note that the contribution of the slave subsystem is neglected in the integration of SGM. The frequency-response curves of displacements at the first bearing and the first disk from the left are shown in Fig. 22.2a, b, respectively. It can be observed from Fig. 22.2 that the dynamical responses of the original system and the PCGM-based reduced system are nearly indistinguishable, whereas the SGMbased reduced system exhibits an appreciable error after 4,740 rpm (after first order critical speed), especially the frequency-response curves at the disk as shown in

a

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Amplitude, x2

1.2 1 0.8 0.6 0.4 0.2 0 0

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Fig. 22.2 Comparison of frequency–response curves using different methods: (a) left bearing; (b) the first disk from the left

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Fig. 22.3 Comparison of bifurcation diagrams for different order reduction methods: (a) the original system; (b) the PCGM; (c) the SGM

Fig. 22.2b. The first jump phenomenon appears at the rotating speed 4,440 rpm on the frequency-response curves of both original system and reduced order system obtained through PCGM. The jump phenomenon in the reduced order system

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a Re sponse

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Fig. 22.4 Comparison of three-dimensional spectrum plots for different order reduction methods: (a) the original system; (b) the PCGM; (c) the SGM

obtained by the SGM is delayed near 300 rpm. Observing Figs. 22.2 and 22.3 we find that a period-doubling bifurcation (Fig. 22.3a–c) results in a transition from synchronous whirl to non-synchronous whirl with whirling frequency being equal to half of rotating speed (Fig. 22.4a–c), when the fist jump phenomenon

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take place. In the original system, with the increases of the rotating speed, the synchronous motion again occurs in the speed range 5,460–6,240 rpm, after rotating speed 6,240 rpm more complex nonlinear dynamic behaviors begin evolvement as shown in Figs. 22.3a and 22.4a. It is delightful that the dynamic phenomena of reduced order system obtained via the PCGM are coincident with those of the original system (Figs. 22.3b and 22.4b). However, the reduced order system obtained via the SGM can not reproduce the actual nonlinear dynamic phenomena of original system (Figs. 22.3c and 22.4c) after the fist bifurcation point in the original system. In order to achieve the accurate results, more modes should be included for the SGM.

22.5 Conclusions A 32-DOF nonlinear dynamic model has been established for the two-span rotorbearing system. A lower order subsystem with 6-DOF has been taken as the master subsystem. The master subsystem associated with the slave subsystem which also has 6-DOF has been numerically solved using the predictor-corrector Galerkin method proposed in [17]. It has been shown that the PCGM provided a considerable increase in accuracy for a little computational cost in comparison with the SGM in which the first six modes were reserved. The numerical results indicated that the influence of some higher modes should be taken into account. The PCGM can achieve the request accuracy and reduce the order of the large-scale nonlinear dynamical system without losing essential dynamical behaviors of the original system. Acknowledgments This research was supported by National Natural Science Foundation of China (10772056, 10632040) and the Natural Science Foundation of Hei-Long-Jiang Province, China (ZJG0704).

References 1. Ding J, Krodkiewski JM (1993) Inclusion of static indetermination in the mathematical model for non-linear dynamic analyses of multi-bearing rotor systems. J Sound Vib 164(2):267–280 2. Krodkiewski JM, Ding J, Zhang N (1994) Identification of unbalance change using a non-linear mathematical model for multi-bearing rotor systems. J Sound Vib 169(5):685–698 3. Ding J (1997) Computation of multi-plane imbalance for a multi-bearing rotor system. J Sound Vib 205(3):364–371 4. Hu W, Miah H, Feng NS, et al (2000) A rig for testing lateral misalignment effects in a flexible rotor supported on three or more hydrodynamic journal bearings. Tribol Int 33:197–204 5. Ding Q, Leung AYT (2005) Numerical and experimental investigations on flexible multibearing rotor dynamics. J Vib Acoust 127:408–415 6. Guyan RJ (1965) Reduction of stiffness and mass matrices. AAIA J 3(2):380 7. Sundararajan P, Noah ST (1998) An algorithm for response and stability of large order nonlinear systems-application to rotor systems. J Sound Vib 214(4):695–723 8. Devulder C, Marion M (1992) A class of numerical algorithms for large time integration: the nonlinear Galerkin method. SIAM J Numer Anal 29(2):462–483

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9. Foias C, Manley O, Temam R (1993) Iterated approximate inertial manifolds for Navier–Stokes equations in 2-D. J Math Anal Appl 178:567–583 10. Sthindl A, Troger H (2001) Methods for dimension reduction and their application in nonlinear dynamics. Int J Solids Struct 38:2131–2147 11. Titi ES (1990) On approximate inertial manifolds to the Navier–Stokes equations. J Math Anal Appl 149:540–570 12. Garcia-Archilla B, Novo J, Titi ES (1998) Postprocessing the Galerkin method: a novel approach, to approximate inertial manifolds. SIAM J Numer Anal 35:941–972 13. Sansour C, Wriggers P, Sansour J (2003) A finite element post-processed Galerkin method for dimensional reduction in the non-linear dynamics of solids: applications to shells. Comput Mech 32:104–114 14. Garcia-Archilla B, Novo J, Titi ES (1999) An approximate inertial manifolds approach to postprocessing the Galerkin method for the Navier–Stokes equations. Math Comput 68:893–911 15. Rega G, Troger H (2005) Dimension reduction of dynamical systems: methods, models, applications. Nonlinear Dyn 41:1–15 16. Matthies HG, Meyer M (2003) Nonlinear Galerkin methods for the model reduction of nonlinear dynamical systems. Comput Struct 81:1277–1286 17. Cao DQ, Wang JL, Huang WH (2010) The predictor-corrector Galerkin method and its application in a large-scale rotor-bearing system. J Vib Shock 29(2):100–105 18. Adiletta G, Guido AR, Rossi C (1996) Chaotic motions of a rigid rotor in short journal bearings. Nonlinear Dyn 10:251–269

Chapter 23

Stiffness Nonlinearity Classification Using Morlet Wavelets Rajkumar Porwal and Nalinaksh S. Vyas

Abstract A methodology based on wavelet transform using standard and low-oscillation Morlet wavelets is presented to distinguish between symmetric and asymmetric polynomial form of stiffness nonlinearities. Free vibration response of the system is wavelet transformed and ridges are estimated. Characteristics of ridges in conjunction with analytical solutions from Krylov–Bogoliubov method are used to classify the nonlinearities. Numerical simulations are performed on the quadratic and mixed parity nonlinear oscillator to illustrate the procedure.

23.1 Introduction Presence of many forms of the nonlinearities in the restoring force and dissipative forces makes identification and parameter estimation of nonlinear systems quite involved. Classification and identification of nonlinearities is one of the important aspects of system parameter estimations. A brief description of various identification schemes can be found in Kerschen et al. [4] and Worden and Tomlinson [14]. Wavelet transform technique for nonlinear system identification is being used more frequently [4, 10]. The capability of wavelet transform to capture instantaneous frequency and amplitude envelope makes it suitable for the identification of nonlinear systems. Transient vibration of a weakly nonlinear system can be modeled as a signal whose amplitude envelope and instantaneous frequency change with time. Closed form solutions obtained from Krylov–Bogoliubov technique establish the relationships for amplitude envelope and instantaneous frequency for a weakly nonlinear system [9]. Instantaneous frequency and amplitude envelope of the given signal can also be determined numerically using the wavelet transform.

R. Porwal () Department of Mechanical Engineering Shri G. S. Institute of Technology and Science, 23 Park Road, Indore 452003, India e-mail: [email protected]; [email protected]

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The numerical results obtained from wavelet transform and closed form solutions from Krylov–Bogoliubov technique are used in conjunction to determine the system parameters. Staszewski [12] applied the analytic wavelet transform techniques to nonlinear systems of the Duffing type. Lardies and Ta [5] analyzed the system containing nonlinear damping. In another paper, Ta and Lardies [13] addressed the systems having polynomial type of nonlinearity on damping and stiffness. They demonstrated the application of the procedure on the systems with symmetric polynomial type of stiffness nonlinearities. Most parameter estimation procedures are based on an inherent assumption about the form of nonlinearities. However, for complex engineering systems, it is difficult to recognize the actual form of nonlinearities. A third degree polynomial form of stiffness is considered in the present study and a methodology based on wavelet transform using standard and low-oscillation Morlet wavelets is suggested to distinguish between the symmetric and asymmetric polynomial forms of stiffness nonlinearity.

23.2 Response of the Nonlinear System For weakly nonlinear systems given by yR C y D "F .y; y/ P I

(23.1)

the response through the Krylov–Bogoliubov method is represented as y D A./ cos Œ C ./ ;

(23.2)

where A./ and ./ are calculated by [8].

" Z 2 F .A cos ; A sin / sin d ; 2 0

" Z 2 P D F .A cos ; A sin / cos d : 2 A 0

AP D

(23.3)

(23.4)

The instantaneous frequency Œ!./ of the response signal is time derivative of the P phase i.e., Œ1 C ./. A damped single degree of freedom system with nonlinear stiffness modeled by a polynomial of degree three, executing free vibration, initiated by providing some initial displacement, can be modeled as yR C 2 yP C y C "2 y 2 C "3 y 3 D 0;

y.0/ D 1 ;

y.0/ P D 0:

(23.5)

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In absence of quadratic term, the response of the system is given by [8] y./ D A0 e

Z cos C 0

3 2 "3 A ./d C 0 : 8

(23.6)

Here constants A0 and 0 depend on the initial conditions. Amplitude envelope A./ and instantaneous frequency !./ are given by the following expressions A./ D A0 e ; !./ D 1 C

(23.7)

3 2 8 "3 A ./:

(23.8)

The above relationship involves dependence of the frequency of oscillations on the amplitude of oscillations. Presence of quadratic term makes the problem little involved. The Krylov– Bogoliubov method, which is first order averaging technique is not able to incorporate the effect of quadratic nonlinearity [8]. In order to use Krylov–Bogoliubov method and to have a better representation of response for quadratic and mixedparity nonlinear oscillators, Porwal and Vyas [11] analyzed the positive displacement and negative displacement motion separately. Motion during positive half cycle is governed by yR C 2 yP C y C "2 y jyj C "3 y 3 D 0 ;

for y > 0

(23.9)

Response during the ith positive half cycle is given by

ypi ./ D Api e cos !pi C pi ;

(23.10)

where !pi

2 3"3 Ypi 4"2 Ypi C D 1C 3 8

! ;

(23.11)

Api and pi are constants which depend on the conditions at the beginning of the half cycle. Ypi is the amplitude of oscillation and !pi is the average frequency of oscillation. Governing equation of motion during negative half cycle is yR C 2 yP C y "2 y jyj C "3 y 3 D 0 ;

for y < 0:

(23.12)

Response during the jth negative half cycle is given by

ynj ./ D Anj e cos !nj C nj ;

(23.13)

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where !nj

2 3"3 Ynj 4"2 Ynj C D 1 3 8

! ;

(23.14)

Constants Anj and nj depend on the conditions at the beginning of half cycle. Ynj is the amplitude of oscillation and !nj is the average frequency of oscillation. A semi analytical approach to determine all these quantities for the complete motion is described by Porwal and Vyas [11]. The frequency changes from half cycle to half cycle as per the above equations, (23.11) and (23.14). Wavelet transform is used to detect and characterize the system using this fact.

23.3 Wavelet Transform The continuous wavelet transform (CWT) for a finite energy signal y./ using admissible wavelet function g./ is defined by the integral [3] Z 1 1 b dt ; where y./ 2 L2 .R/ y./g Wy .a; b/D hy; g.; a; b/i D p a a 1 (23.15)

g b is complex conjugate of g b . Variables a and b are scale and a a translational parameters, respectively. The above mentioned definition of wavelet transform i.e., (23.15) can be converted to the frequency domain using Parseval identity [3] p Z 1 a Y .!/G.a!/ej!b d!: (23.16) Wy .a; b/ D 2 1 Here Y .!/ is the Fourier transform of y./ and G.!/ is the Fourier transN form of g./. The bar over G.a!/ i.e., G.a!/ denotes complex conjugate of G.a!/. p This result can be interpreted as an inverse Fourier transform of the function aY .!/ G .a!/. Equation (23.16) is used here for computation in order to take advantage of an efficient Fast Fourier Transform (FFT) algorithm.

23.3.1 Wavelet Transform of the Response Signal The system response represented by (23.6), or (23.10) and (23.13) can be written in general as y./ D A./ cosŒ./ ;

P where ./ D !./:

(23.17)

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Under the assumption of small damping, the response signal can be considered as an asymptotic signal if oscillation due to the phase term ./ is more significant than that coming from the amplitude term A./ [2]. A reasonable approximation of wavelet transform of an asymptotic signal (23.17) turns out to be [2, 6] p a A.b/ej.b/ GN .a!/ : (23.18) Wy .a; b/ D 2 23.3.1.1 Standard Morlet Wavelet Standard Morlet wavelet function with central frequency !0 is given by g./ D

1 2 e =2 ej!0 : . /1=4

(23.19)

Its Fourier transform is G.!/ D .4 /1=4 e.!!0 /

2 =2

:

(23.20)

!0 D 5 is used here to meet the admissibility criterion [2]. Therefore, a reasonable approximation of wavelet transform of an asymptotic signal (23.17) is Wy .a; b/ D

p a 2 A.b/ej.b/ .4 /1=4 e.a!!0 / =2 : 2

(23.21)

The wavelet transform is presented as a 2D plot of normalized scalogram

ˇ result of ˇ ˇWy .a; b/ˇ2 =a . The magnitude of normalized scalogram is represented by proportional intensity on the plot as a function of .a; b/. Ridge of the wavelet transform is defined to extract useful information out of the spread normalized scalogram. Wavelet ridge is closely related to instantaneous frequency of the signal. A ridge is defined as the locus in time–frequency plane along which normalized scalogram attains the maxima; mathematically @ @a

ˇ ˇ ! ˇWy .a; b/ˇ2 a

D 0:

(23.22)

Substituting the transform (23.21) into (23.22) and simplification gives the locus of the ridge as ar .b/ D

!0 ; !.b/

(23.23)

where ar .b/ is the scale along the ridge. In order to determine the scale corresponding to ridge ar .b/ the maxima of normalized scalogram is noted for each

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time location b in the time–frequency representation calculated by (23.16) and the corresponding scale is obtained. Subsequently, the instantaneous frequency !.b/ at a particular time can be determined.

23.3.1.2 Complete Morlet Wavelet Time resolution of standard Morlet wavelet depends on !0 (Table 23.1) whose value is kept greater than 5 for the mean of wavelet function to approach zero, a condition to fulfill admissibility requirement. It is therefore, unable to capture any phenomenon which is more temporal in nature. In order to increase time resolution and capture the temporal phenomenon, use of complete Morlet wavelet has been suggested. This allows us to use lower value of central frequency !0 . Since the central frequency is lower than the standard Morlet wavelet, it is known as low-oscillation Morlet wavelet. !0 D 1 is used in the present work to have better time resolution . Initially low oscillation Morlet wavelet was used as a pattern matching tool to detect and classify P waves in ECG signal by Michaelis [7]. Later Addison et al. [1] lucidly explained the underlying mathematics and applied it to sonic echo NDT signal used for the analysis of structure elements. The complete Morlet wavelet is defined as follows 1 g.t/ D . /1=4

2 !0 2 j!0 e e =2 : e 2

(23.24)

Its Fourier transform is G.!/ D .4 /1=4 e.!

2 C! 2 0

/=2 .e!!0 1/ :

(23.25)

With complete Morlet wavelet, the wavelet transform of an asymptotic signal (23.17) can be written approximately as [6] p a 2 2 A.b/ej.b/ .4 /1=4 e.a! C!0 /=2 .ea!!0 1/ : Wy .a; b/ D 2

(23.26)

The wavelet ridge is obtained by employing the expression (23.22), thus ar ! 1 log ar D : !0 ! ar ! !0

Table 23.1 Properties of standard Morlet wavelet functions Time center Frequency center Time spread Frequency spread !o !0 1 ! 1 p p b a ! 2 !0 2

(23.27)

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Equation (23.27) is solved numerically for instantaneous frequency (!) after determining locus of ridge (ar ) from the maxima of normalized scalogram at each time instant.

23.4 Illustration Following three representative cases are numerically simulated: 1. "2 and "3 are present i.e., asymmetric mixed parity oscillator. 2. "2 is present and "3 is 0 i.e., asymmetric quadratic oscillator. 3. "2 is 0 and "3 is present i.e., symmetric cubic oscillator. The steps involved in classifying the system using ridges of wavelet transform are shown schematically in Fig. 23.1 for a mixed parity nonlinear oscillator in which both "2 and "3 are eqal to 0:1 i.e., case (1). Damping factor D 0:01 is assumed in all the simulations. Figure 23.1a is a simulated response of the system obtained through Runge–Kutta method. Results in Fig. 23.1b–d are obtained using ˙ standard Morlet wavelet while the results in Fig.23.1e–g are obtained using low oscillation Morlet wavelet. Wavelet transform of the response is shown in Fig. 23.1b. Ridge obtained over the relevant scale is shown in Fig. 23.1c. Locus of ridge is converted to variation of instantaneous frequency with time in Fig. 23.1d. The trend of the graph shows that the frequency decreases continuously as the amplitude of oscillation decreases. The corresponding results in Figs. 23.1e–g indicate that the low oscillation Morlet wavelet is able to capture the change in instantaneous frequency form half cycle to half cycle. The instantaneous frequency shows oscillating behavior due to the different governing law during positive and negative motion and this fact is also evident from (23.11) and (23.14). The results obtained for the other two cases are shown in Fig. 23.2. Figure 23.2a–c belongs to case (2). Figure 23.2a shows the response of the system for "2 D 0:1 and "3 D 0. Instantaneous frequencies obtained using standard Morlet wavelet and low oscillation Morlet wavelet are shown in Fig. 23.2b, c, respectively. Here the two wavelets yield different loci for the instantaneous frequency. In absence of "3 , the frequency locus obtained by the standard Morlet wavelet remains in the close vicinity of unity. The low-oscillation Morlet wavelet yields an oscillatory frequency plot. Figure 23.2d–f belongs to case (3) and shows the results for "2 D0 and "3 D0:1. Results obtained are same for both wavelets since the instantaneous frequency changes smoothly with the amplitude of oscillation as given by (23.8).

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Fig. 23.1 (a) Response of a mixed-parity nonlinear oscillator "2 D 0:1 and "3 D 0:1 (b) Wavelet transform of the response using standard Morlet wavelet (c) Ridge of the wavelet transform (d) Instantaneous frequency of the response (e) Wavelet transform of the response using low-oscillation Morlet wavelet (f) Ridge of the wavelet transform (g) Instantaneous frequency of the response

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Fig. 23.2 (a) Simulated response for case 2 (b) Instantaneous frequency using standard Morlet wavelet for case 2 (c) Instantaneous frequency using low-oscillation Morlet wavelet for case 2 (d) Simulated response for case 3 (e) Instantaneous frequency using standard Morlet wavelet for case 3 (f) Instantaneous frequency using low-oscillation Morlet wavelet for case 3

23.5 Conclusions The frequency loci obtained through standard Morlet wavelet are non-oscillatory. Locus of frequency in this case is better explained by considering corresponding symmetric case i.e., the (23.8). Standard Morlet wavelet averages out the effect of asymmetric nonlinearity over a number of response cycles due to its higher time spread. This averaged out results are used to find the presence of "3 . Lowoscillation Morlet wavelet, due to its ability to capture the temporal phenomenon, is able to distinguish between the symmetric and asymmetric forms easily. It gives non-oscillatory frequency locus for symmetric nonlinearity while for asymmetric nonlinearity the frequency locus is oscillatory. The oscillatory locus of frequency is better explained by the (23.11) and (23.14).

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References 1. Addison PS, Watson JN, Feng T (2002) Low-oscillation complex wavelets. J Sound Vib 254(4):733–762 2. Carmona R, Hwang W, Torresani B (1998) Wavelet analysis and its application: practical time frequency analysis, vol 9. Academic, San Diego, CA 3. Chui CK (1992) An introduction to wavelets. Academic, San Diego, CA 4. Kerschen G, Worden K, Vakakis AF, Golinval JC (2006) Past, present and future of nonlinear system identification in structural dynamics. Mech Syst Signal Process 20:505–592 5. Lardies J, Ta MN (2005) A wavelet based approach for the identification of damping in nonlinear oscillator. Int J Mech Sci 47:1262–1281 6. Mallat S (1998) A wavelet tour of signal processing. Academic, San Diego, CA 7. Michaelis M, Penz S, Black C, Sommer G (1993) Detection and classification of p waves using gabor wavelets. In: Computers in cardiology 1993. Proceedings of IEEE, pp 531–534 8. Mickens RE (1995) Oscillations in planar dynamic systems. World Scientific Publishing, Singapore 9. Nayfeh AH (1973) Perturbation methods. Wiley, New York 10. Peng Z, Chu F (2004) Application of the wavelet transform in machine condition monitoring and fault diagnostics: a review with bibliography. Mecha Syst Signal Process 18:199–221 11. Porwal R, Vyas NS (2008) Damped quadratic and mixed-parity oscillator response using Krylov–bogoliubov method and energy balance. J Sound Vib 309:877–886 12. Staszewski W (1998) Identification of non-linear systems using multi-scale ridges and skeletons of the wavelet transform. J Sound Vib 214(4), 639–658 13. Ta MN, Lardies J (2006) Identification of weak nonlinearities on damping and stiffness by the continuous wavelet transform. J Sound Vib 293:16–37 14. Worden K, Tomlinson G (2001) Nonlinearity in structural dynamics: Detection, identification and modelling. Institute of Physics Publishing, Bristol, Avov and Philadelphia, PA

Chapter 24

Dynamics of Wire-Driven Machine Mechanisms: Literature Review Timo Karvinen and Erno Keskinen

Abstract The state-of-the art of the mechanical properties of wire ropes and the dynamics simulation of wire rope mechanisms is reviewed in this paper. A special emphasis is put on the tension dependent Young’s modulus and the damping of the wire rope in the part dealing with the mechanical properties. In the part discussing the dynamics simulation, the simplification of the complex system and the connection between the rope and the pulley are accentuated. There is plenty of literature on modeling the material properties and they can be predicted accurately. There is still room for new developments in the dynamics simulation of wire rope systems.

24.1 Introduction During the past two decades or so, considerable interest has been shown in the mechanical characteristics of helically wound steel cables for use in both onshore and offshore applications such as bridges, oil drilling platforms, cranes, and elevators. Most wire ropes are constructed from either a single strand or from several strands that are wound around a core. This core may be a strand in itself or it may be a fibrous or deformable element. The strand is constructed of wires that are wound around a central wire. The design of wire rope cross-sections dates back to the late 1800s and has been continuously improved ever since. A typical wire rope is shown in Fig. 24.1. Several modeling approaches are based on the nonlinear equations of equilibrium of a thin helical rod [2] and consider the torsion and bending stiffness of the wires. The theory of wire ropes is presented very extensively in the book of [3]. The parameters affecting the wire rope properties are: Outer diameter Wire diameter

T. Karvinen () Department of Mechanics and Design, Tampere University of Technology, P.O. Box 589, 33101 Tampere, Finland e-mail: [email protected]

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 24,

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Fig. 24.1 Typical six-stranded wire-rope with (a) independent wire rope core (IWRC), (b) fibre core, [1] reprinted with permission from Professional Engineering Publishing

Number of wires Number of strands Strands lay angle Lay type: ordinary lay, Lang’s lay Core type Diameter of core

Material properties of wire ropes are needed in order to perform the dynamic analysis accurately. Most papers dealing with the dynamic analyses related to ropes are not actually covering the ropes but belt–pulley systems. The same analysis can be extended to wire rope – pulley systems using different material properties. A lot of emphasis is put on the detailed modeling of the belt–pulley contact in the literature but we try to find ways to simplify a complex system consisting of the rope system, mechanisms, hydraulics, and control systems to more manageable one often needed in the dynamics analysis.

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24.2 Mechanical Properties of Wire Ropes The wires or strands can be modeled as thin helical rods using [2] curved rod theory. Global relationships between deformation of the cable and the resultant axial force and bending/torsional moments are established accordingly [3]. There also exist semicontinuous models in which each layer of helical wires is replaced with an equivalent orthotropic medium. Interwire and/or interstrand frictional forces are considered by using contact mechanics to account for stick and slip friction transition [7]. The individual layers of wires in an axially preloaded multilayered spiral strand are treated as a series of partly self-stressed cylindrical orthotropic sheets whose nonlinear elastic properties are averaged to form an equivalent continuum. The theory is based on the main assumption that with zero axial load on the strand, the wires in each layer are just touching each other. Under dynamic or cyclic loading, detailed local models overestimate the cable damping for cables with large radii of curvature. There are analytical approaches and FEM methods [4, 9] developed to calculate cables’ overall mechanical properties which are: axial, bending, torsional stiffness, and hysteresis characteristics. Depending on the accuracy of the model, leading to complex mathematical problem, the solution may be unsuitable for largescale engineering applications. The most detailed wire cable models have primarily been developed for static monotonic loading. The mechanical properties exhibit the following features:

Young’s modulus is tension dependent Stretch consists of two parts: constructional and elastic stretch Lay-angle dependent properties Interwire friction factor used in simulations D 0:12

24.2.1 Young’s Modulus Important concepts in studying the Young’s Modulus are the no-slip and full-slip limits. The no-slip Young’s modulus is the upper bound to the effective Young’s modulus. Small disturbances do not induce interwire slippage over the line-contact patches between adjacent wires in various layers and the wires stick together behaving as a solid bar. The full-slip Young’s modulus is the lower bound limit whereby wires in linecontact throughout the strand undergo gross sliding with the interwire frictional forces becoming insignificant compared with the axial force changes within individual helical wires. The Young’s modulus depends on the loading; therefore one should measure it at conditions similar to expected working conditions. Analytical expressions are found in the literature for the effective Young’s modulus taking into account the geometry, lay angle, number of strands etc. Even though the expressions are analytic, the calculation of numerical values is quite involving and it may take some time to write the equations in the computer. The equations of these models are not presented in this review, apart from some simple expressions. All the equations

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Fig. 24.2 Rope axial stiffness theory vs. experiments, [7] reprinted with permission from ASCE

needed in the calculations can be found in the corresponding papers in the reference list. There are also models for calculating the stiffness matrix relating the axial force, moment, and extension and torsion responses [5, 6]. Figure 24.2 illustrates the nondimensional effective Young’s modulus divided by the steel’s Young’s modulus as a function of the load range obtained both theoretically and experimentally. The figure shows a descending trend for the Young’s modulus with the increasing load. A similar curve is presented in Fig. 24.3 for other type of wire rope. Similar curves can be drawn for any wire rope by applying the calculation procedure and the equations found in, e.g., [7, 8]. Many models found in the literature are computer based and involve certain iterative procedures. This potential drawback for practical applications is overcome by some simple methods. The full-slip values of the Young’s modulus can be calculated from a simple formula (Hruska’s approach) proposed by [33] M P

H D

n P

ALi cos3 ˛i cos3 ˇj

Erope j D1 i D1 D n , M Esteel P P cos ˇj ALi = cos ˛i j D1

i D1

(24.1)

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Fig. 24.3 Variation of Young’s modulus as function of loading, [8] reprinted with permission from Elsevier

in which AL ; ˛, and ˇ are the total cross-sectional area of wires, the lay angle of wires in a strand, and the lay angle of a strand. Raoof [1] developed simplified but still accurate procedures for predicting the no-slip and full-slip axial stiffnesses of wire ropes, with the proposed formulations being amenable to simple hand calculations using a pocket calculator. The method is based on fitting a curve through the data calculated using a more complicated model. The comparison between Hruska’s and Raoof’s simple models for the fullslip values is shown in Fig. 24.4. Zhu [9] performed experiments in which the cable was repeatedly loaded–unloaded, and loaded–unloaded after being bent repeatedly without tension. It was found out that the elastic modulus changes depending on the loading: the repeated load applications will increase the elastic modulus of the new wire cable, and the increased elastic modulus can be lost after bending of the cable at zero tension.

24.2.2 Damping Just like in the case of Young’s modulus, analytical models exist for cable damping [10–12]. The theory for prediction of damping is based on the orthotropic sheet theory with wire compliances derived from contact stress theories. Later on, [31] developed simplified, hand-based procedures for obtaining the maximum values of axial and/or torsional frictional specific loss by fitting curves through the data calculated using a complicated model. A general observation is that the damping

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Fig. 24.4 Comparison of different models for calculating Young’s modulus of rope, [1] reprinted with permission from Professional Engineering Publishing

increases by increasing the number of wires and/or decreasing the helix angle. The results for damping are presented in a nondimensional form, the x-axis being the load range divided by the mean load, which should be checked in the paper, and the y-axis the axial energy loss per cycle divided by the maximum elastic energy, U=U . This ratio provides the most direct method of quantifying damping irrespective of the mechanism involved. For low damping it can be shown that the equivalent logarithmic decrement is ıeq D U=2U . The traditional Coulomb friction model tends to grossly overestimate cable damping for large radii of curvature in connection with vortex shedding dynamic instabilities, overhead transmission lines, and underwater cables where the maximum amplitude of vibration is of the order of one cable diameter [10]. Figures 24.5 and 24.6 show characteristic damping curves for strands of different diameters. The curves have a narrow peak at a certain value of loading indicating maximum damping. There is no easy way to predict at which value of loading the maximum damping occurs. Damping curves can also be presented as a function of the Young’s modulus [10]. Figure 24.7 illustrates the damping as a function of radius of curvature. Similar curves as those found in Fig. 24.7 can be obtained as a function of core-wire radius ratio, helix angle, and number of wires [12]. It has also been found that the service time (or number of cycles) increases damping as shown in Fig. 24.8. Besides analytical models, there exist numerical approaches such as the Rayleigh damping used for FEM calculations [9]) ŒC D ˛ŒM C ˇŒK; ˛D2

˛ C !i2 ˇ D 2i !i

!1 !2 .!2 1 !1 2 / ; !22 !12

ˇD

!2 2 !1 1 !22 !12

(24.2)

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Fig. 24.5 Axial energy dissipation in strands, [11] reprinted with permission from Professional Engineering Publishing

in which ˛ and ˇ are the damping coefficients multiplying the mass and stiffness matrices and !1 and !2 the lower and upper limits of the frequency range of interest. The Kelvin–Voigt model for damping is employed in simulation of vibrations and rope stability in one reference [13]. The wear of the rope in bending is discussed, e.g., in the papers of [14, 15].

24.3 Dynamics Simulation of Wire Rope Systems There are not many papers in the literature dealing with the system-level analysis of wire rope systems. Usually the system consists of a belt and two pulleys. Some very large system level models including the wire rope system, mechanisms, hydraulics, and control can be found in [16] and [17]. Also if there is a more complicated mechanism, the system does not fully describe the mechanics and omits, e.g., the wire stiffness and/or damping.

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Fig. 24.6 Theoretical axial energy dissipation in strands, [12] reprinted with permission from Professional Engineering Publishing

24.3.1 Belt–Pulley Systems There exists no generally accepted friction model between the rope and the pulley. Two different theories, both physically motivated but based on different belt–pulley friction assumptions, are developed in the literature. The creep model assumes that the existence of friction depends completely on the relative motion (creep) between the belt and the pulley surfaces. Only kinetic friction is considered in the belt–pulley interaction, the contact arc is divided into two zones: an adhesion zone in which the belt moves at the same speed as the pulley and the sliding zone in which the belt slides (creeps) on the pulley surface. Only the sliding zone contributes to moment transmission through friction forces, whereas the friction remains zero in the adhesion zone. Important concepts in the creep model are the slip angle and the contact region, which are shown in Fig. 24.9. Another approach in the friction modeling is the shear model, which addresses the shearing deformation of the belt assuming it inextensible. The creep model is much more used than the shear model. The shear model is mathematically more complicated than the creep model. The solutions presented in the literature are in most cases for the steady-state and usually for a belt-two-pulley system. Important quantities in the analysis of belt, wire rope, and pulley systems are the normal force, the friction force, the maximum moment that can be trans-

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Fig. 24.7 Comparison of damping of single-layer strands as function of radius of curvature between two methods, [12] reprinted with permission from ASCE

mitted and the slip angle. [18] compared different models which were: the full model taking into account all the effects of inertia including the acceleration due to stretching, the engineering and alternate solutions which take into account the centrifugal acceleration but not the tangential acceleration, and the Capstan solution neglecting the inertia in the momentum equations. It was found that the three first approaches gave very similar results for the normal and frictional forces of a belt and two pulleys systems, but the Capstan solution neglecting the inertia gave significantly different results, which is illustrated in Fig. 24.10. Also the slip angles had notable differences. Different approaches can be found in the literature such as those using the curved Euler–Bernoulli beam including the bending stiffness [19], some models are solved analytically and numerical values are possible to obtain rather easily [18], and some models are solved numerically [19–21].

24.3.2 Complex Wire Rope Systems Despite the establishment of the analysis capabilities, very few actual wire rope system designs are accomplished through their use and there is a lack of unified design methodology for wire rope systems based on the analysis. [22] discusses the

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Fig. 24.8 Damping against service time, [12] reprinted with permission from ASCE

Fig. 24.9 Normal force in different zones of belt calculated using different models, [18] reprinted with permission from ASME

design of wire rope systems, but dynamics is not considered and the emphasis is on the minimization of the rope weight. The models found in the literature are rather detailed and there have not been attempts to simplify the systems, probably due to the fact that the system has been a belt-two-pulley system. [23] presented an efficient

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Fig. 24.10 Schematic showing slip angle and contact zone, [18] reprinted with permission from ASME

method for calculating the eigensolutions and the dynamic response of a serpentine belt drive consisting of several pulleys. The efficiency in the solution is due to the discretization of the belt spans, which can easily be generalized to a system of arbitrary number of pulleys. Hong [24] presented a method for representing complex cable–pulley mechanism configurations and motions. In this study only the motion of pulleys, ropes, and blocks is considered and the cables must be oriented in x or y direction and the orientation change is not allowed. Verho [25] analyzed a wire rope

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Fig. 24.11 Reducing wire rope mass for moment of inertia of pulley, [25] reprinted with permission from author

Fig. 24.12 Comparison of full (solid line) and simplified (dashed line) wire rope models, [25] reprinted with permission from author

mechanism in which the moment of inertia of the ropes was reduced to the pulley and a single value tension approximation along the entire belt was presented as 1 Ii D Iip C si C .Li 1 C Li / m0t Ri2 (24.3) 2 for which the notation shown in Fig. 24.11. The simplified and the full model including different tension between different pulleys and without the wire rope mass reduction are compared in Fig. 24.12 in which the quantity on the y-axis is the acceleration of a mass to be accelerated with the mechanism.

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There also exist FEM approaches for dynamic analysis and other numerical models in which cable and pulley elements are developed [26]. Kumaniecka [13] developed a model for vibrations and stability of ropes with the Kelvin–Voigt model for rope damping. Most system level simulations involve control of elevators and cranes [27–29]; Characteristic to these studies is the simplified mechanical model in which the mass and stiffness of rope are neglected, taken as a constant and damping constant or zero and the emphasis is on the control model. The rope is often carrying a point mass. Such studies are, e.g., swinging prevention of the load [30] and improvement of riding comfort in the elevator [32].

24.4 Conclusions The state-of-the-art of modeling the mechanical properties of wire ropes and the dynamics analysis of wire rope systems has been presented in the paper. Plenty of literature is found on predicting the Young’s modulus and damping of the wire rope taking into account all the geometrical parameters such as the wire and strand diameter, the lay angle and lay type, respectively. Several modeling approaches exist and they seem to produce accurate results, even though coding the equations in the computer and obtaining the numerical results is not very simple. The literature is scarce discussing the system level dynamic analysis of wire rope systems. Most papers are dealing with the very detailed analysis of the belt pulley contact in the steady-state case. Several papers are found on the control of wire rope systems in which the mechanical model is often very simplified. The emphasis in these papers is usually in the control of the actuator to minimize the vibration of the load, for instance. We conclude that there is room for significant developments in the system level dynamic analysis of wire rope systems.

References 1. Raoof M, Davies TJ (2003) Simple determination of the axial stiffness for large-diameter independent wire rope core or fibre core wire ropes. J Strain Anal 38(6):577–586 2. Love AEH (1944) A treatise on the mathematical theory of elasticity. Dover, New York, p 643 3. Costello GA (1990) Theory of wire rope. Springer, New York 4. Ma J, Ge S, Zhang D (2008) Distribution of wire deformation within strands of wire ropes. J China Univ Min Technol 18:475–478 5. Elata D, Eshkenazy R, Weiss MP (2003) The mechanical behavior of a wire rope with an independent wire rope core. Int J Solids Struct 41:1157–1172 6. Raoof M, Kraincanic I (1995) Characteristics of fibre-core wire rope. J Strain Anal 30(3):217–226 7. Raoof M, Kraincanic I (1995) Analysis of large diameter steel ropes. J Eng Mech 121(6):667–675 8. Raoof M, Kraincanic I (1995) Simple derivation of the stiffness matrix for axial/torsional coupling of spiral strands. Comput Struct 55(4):589–600

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9. Zhu ZH, Meguid SA (2007) Nonlinear FE-based investigation of flexural damping of slacking wire cables. Int J Solids Struct 44:5122–5132 10. Raoof M, Huang YP (1991) Upper-bound prediction of cable damping under cyclic bending. J Eng Mech 117(12):2729–2747 11. Raoof M (1991). The prediction of axial damping in spiral strands. J Strain Anal 26(4):221–229 12. Raoof M (1991) Methods for analyzing large spiral strands. J Strain Anal 26(3):165–174 13. Kumaniecka A, Niziol J (1994) Dynamic stability of a rope with slow variability of the parameters. J Sound Vib 178(2):211–226 14. Ridge IML, Zheng J, Chaplin CR (2000) Measurement of cyclic bending strains in steel wire rope. J Strin Anal 35(6):545–558 15. Urchegui MA, Tato W, G´omez X (2008) Wear evolution in a stranded rope subjected to cyclic bending. J Mat Eng Perform 17(4):550–560 16. Keskinen E, Montonen J, Launis S, Cotsaftis M (1999) Simulation of wire and chain mechanism in hydraulic driven booms. Proceedings of the IASTED international conference applied modelling and simulation, Cairns, QLD, 1–3 September 1999 17. Sun G, Kleeberger M, Liu J (2004) Complete dynamic calculation of lattice mobile crane during hoisting motion. Mech Mach Theory 40:447–466 18. Bechtel SE, Vohra S, Jacob KI, Carlson CD (2000) The stretching and slipping of belts and fibers on pulleys. J Appl Mech 67:197–206 19. Kong L, Parker RG (2005) Steady mechanics of belt–pulley systems. J Appl Mech 72:25–34 20. Kong L, Parker RG (2005) Microslip friction in flat belt drives. Proc Inst Mech Eng C: J Mech Eng Sci 219:1097–1106 21. Rubin MB (2000) An exact solution for steady motion of an extensible belt in multipulley belt drive systems. J Mech Des 122:311–316 22. Velinsky SA (1993) A stress based methodology for the design of wire rope systems. J Mech Des 68:69–73 23. Parker RG (2004) Efficient eigensolution, dynamic response, and eigensensitivity of serpentine belt drives. J Sound Vib 270:15–38 24. Hong DW, Cipra RJ (2003) A method for representing the configuration and analyzing the motion of complex cable–pulley systems. ASME J Mech Des 125:332–341 25. Verho A (2003) Katapulttimekanismin mallinnus ja simulointi (in Finnish). MSc Thesis, Tampere University of Technology 26. Hashemi SM, Roach A (2006) A dynamic finite element for vibration analysis of cables and wire ropes. Asian J Civil Eng (Build Hous), 7(5):487–500 27. Lee H (2003) A new approach for the anti-swing control of overhead cranes with high speed load hoisting. Int J Control 76(15):1493–1499 28. Otsuki M, Ushijima Y, Yoshida K, Kimura H, Nakagawa T (2006) Application of nonstationary sliding mode control to suppression of transverse vibration of elevator rope using input device using gaps. JSME Int J C 49(2):385–394 29. Otsuki M, Yoshida K, Nagakagi S, Nakagawa T, Fujimoto S, Kimura H (2004) Experimental examination of non-stationary robust vibration control for an elevator rope. Proc Inst Mech Eng I: J Syst Control Eng 218:531–544 30. Yanai N, Yamamoto M, Mohri A (2001) Feed-back control of crane based on inverse dynamics calculation. Proceeding of the 2001 IEEE/RSJ, international conference on intelligent robots and systems, Maui, Hi, October 29–November 3, 2001 31. Raoof M, Davies TJ (2005) Simple determination of the maximum axial and torsional energy dissipation in large diameter spiral strands. Comput Struct 84:676–689 32. Kang J-K, Sul S-K (2000) Vertical vibration control of elevator using estimated car acceleration feedback compensation. IEEE Transactions on Industrial Electronics 47(1):91–99 33. Strzemiecki J, Hobbs RE (1988) Properties of wire rope under various fatigue loadings. CESLIC Report SC6, Civil Engineering Department, Imperial College, London

Chapter 25

Dynamics of Wire-Driven Machine Mechanisms, Part II: Theory and Applications Erno Keskinen, Timo Karvinen, and Jori Montonen

Abstract A systematic approach, where the mass conservation theorem is applied to a wire continuum, leads to system equations, in which the dynamics of a complete wire (or chain) and pulley assembly are coupled with the motion of a boom mechanism. This methodology makes it possible to develop a simulation model of complete crane and lift systems where the wire and chain wheels communicating with the wire element can transmit the effect of boom displacements into the dynamic variations of wire tension.

25.1 Introduction Wires and chains are widely used machine elements in devices carrying out work processes where large amplitude displacements and high capacity tools subject to tensional loads are needed. In boom systems, the wires are typically used in lifting winches whereas the chains can provide a compact solution in telescopic boom configurations. The modeling problem of wires and chains rests upon the basic question of how to represent the elastic property of their internal structure into a simple enough small degree of freedom model. Along the line of very fast technology development aiming at delivering more precise and more task-efficient tools, compact library models are preferable in system level simulations of large technical systems since, today, low cost computational capability brings fast models closer to realtime applications needed in man-in-the-loop simulations. A typical example is the use of virtual product models as training simulators to give operators good practice and familiarity with the boom behavior in critical lifting maneuvers. The better real dynamics is coded into the computational model behind the screen animation, the better touch and feeling of real operation may be reached in the maneuvers.

E. Keskinen () Department of Mechanical Engineering, Tampere University of Technology, P.O. Box 589, 33101 Tampere, Finland e-mail: [email protected]

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 25,

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Wires are consisting of fiber groups and chains are consisting of link-pin sections. Despite the complicated internal structure, their dominating mechanical property is axial elasticity, which may be taken as their main behavior in modeling. This has been shown by many authors in papers mentioned in a comprehensive reference list of an adjunct state-of-art review paper written by corresponding authors Karvinen and Keskinen [1]. A homogenization process produces an average axial stiffness property, which defines the effective elastic modulus of the material carrying the axial loads. It has been shown [2] that a similar lumped volume model governing the dynamic pressure variation within a fluid volume in fluid power circuits can be developed for the tensional state in the homogenized wire continuum also. Two worked case studies show the feasibility of proposed methodology in linking together the dynamical model of a loading crane [3] with a hydraulic winch for first one (Fig. 25.1a), and of an elevator boom with a chain-driven extension mechanism [4] for second one (Fig. 25.1b).

a

b

Fig. 25.1 Hydraulic boom systems. Winch equipped loading crane (a), chain-driven elevating platform (b)

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25.2 Wire Mechanics 25.2.1 Wire Topology Wire, chain, and wheel arrangements represent configurations where one-parametric elastic continuum is stretched to form spans between a group of wheels mounted on a steady or moving reference frame in Fig. 25.2. One end is winched into or from a drum whereas the other one is connected to the work object. In technical systems, the wire typically follows the motion of a boom mechanism, as shown on Fig. 25.3. Topologically, the winch systems represent open kinematic chains since the work object attached to wire end may be moved freely inside the working space covered by the boom.

Fig. 25.2 Topology of a rigidly mounted wire-wheel-winch-object arrangement

Fig. 25.3 Winch system mounted on a moving boom mechanism in an open kinematic chain

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Fig. 25.4 Chain synchronizing the motion of links in a multistage telescope boom

The situation is different if the wire or chain is synchronizing the motion of individual links in a multiredundant mechanism in Fig. 25.4. In such configurations, the wire is forming together with the link elements a closed kinematic loop. This property is useful since it reduces the number of actuators to be controlled during complicated boom maneuvers.

25.2.2 Elastic Properties of Wire and Chain Elements In order to model the wire response, a constitutive model of the wire material that links the strains and strain rates to the tensional stress, is needed One starts from a very general dependence D f ."; "/ P

(25.1)

in which the wire stress is a nonlinear function of strain and strain rate. The effective elastic modulus for small strains reads then @f O ."/ D E./: EO D @"

(25.2)

Correspondingly, the effective viscous modulus takes form @f ."/ D ./: O @P"

(25.3)

O P D E./P " C ./R O ":

(25.4)

O D The stress rate is then

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Based on its fiberous structure, wire behaves much more elastically than a steel rod with the same cross-sectional area [5]. This means that in wire dynamical considerations the actual elastic modulus should be replaced by EO D c./E

(25.5)

where the reduction factor c usually varies in range 0.35. . . 0.85 depending on the stress level. For a chain, the situation in Fig. 25.5 is more complicated since the chain consists of a finite number of sections having local elasticities in the pin-joints and distributed elasticity on side plates. As a model, each chain section will be replaced by an equivalent spring assembly where the sideplates have a linear elastic behavior with pitch stiffness coefficient k1 D

AE

(25.6)

see Fig. 25.6, but the pin-joint has a nonlinear load-displacement relationship of the form N D ab

(25.7)

leading to load-dependent joint stiffness coefficient k2 .N / D

Fig. 25.5 Chain section

@N D ab bN bC1 @

(25.8)

l

Dl

N

Fig. 25.6 Spring models in series representing one section of the chain

l

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Chain homogenization is a process where the whole chain is replaced by pitch and joint spring pairs in series leading to continuous material model with cross-sectional area A and equivalent elastic constant EO D

E 1C

(25.9)

k1 k2 .N /

25.2.3 Kinematics of Wire Motion Suppose the wire is stretched over guiding wheels located between the winch drum and work object. The total wire length L in a multispan arrangement consists of lengths `i along wheel-to-wheel free spans and overlapping lengths si along the contact zones on wheels in Fig. 25.7 X X LD `i C si : (25.10) i

i

The expression of span and overlapping lengths varies depending on the different cases shown in Fig. 25.8 and Table 25.1. If the position vectors of wheel centers and wheel radii are known, the distance di between wheel centers is in all cases di2 D .R i C1 R i /.R i C1 R i / i−1

si+1

(25.11)

i+1

i

si+2

si

Fig. 25.7 Definition of span and overlapping lengths in wire kinematics

B

B A

A I

II

B A

Fig. 25.8 Four basic cases in wire length computation

B II

I

A IV

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Table 25.1 The basic situations of span topology Case Contact on wheel A Contact on wheel B I Outside Outside II Inside Inside III Outside Inside IV Inside Outside

Fig. 25.9 Span length computation for case I

bi i+1

f1 bi

Ri

i

i f2

di

ai

i+1 Ri+1

i

The span length for cases I and II in Fig. 25.9 is `i D

q

di2 Ri2

(25.12)

where Ri D Ri C1 Ri , but for cases III and IV `i D

q

di2 †Ri2 ;

(25.13)

where ˙Ri D Ri C1 C Ri . The inclination angle ˛ of the center line is in all cases ˛i D arctan

Yi C1 Yi Xi C1 Xi

(25.14)

with the replacement ˛ < 0 ) ˛ ! ˛ C 2 for avoiding negative angles. The angular difference between wire span and center line is for cases I and II ˇi D arctan

Ri `i

(25.15)

ˇi D arctan

†Ri `i

(25.16)

whereas for cases III and IV

The direction angles of contact points on the wheels can be calculated from casedependent formulae below.

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Fig. 25.10 Overlapping length

f1i i e1

f2i

i

e2

Table 25.2 Direction angles of separation points for basic cases Case Contact on wheel A Contact on wheel B I II III IV

2i 1i 1i 2i

D ˛ C ˇ C =2 D ˛ ˇ C 3 =2 D ˛ ˇ C =2 D ˛ C ˇ C 3 =2

1iC1 1iC1 1iC1 1iC1

D 2i D 2i D 2i C D 2i

The overlapping lengths in Fig. 25.10 si are now easily computed from angular differences between the angles of in-wheel and out-wheel separation points in Table 25.2 ˇ ˇ (25.17) si D Ri ˇ'1i '2i ˇ For further need, the unit vectors for span lines for in-wheel and out-wheel directions are e i1 D sin '1i i C cos '1i j e i2 D sin '2i i cos '2i j

(25.18a) (25.18b)

25.2.4 Tension Dynamics Because a lifting wire or a chain drive is highly loaded during operation, the tensional state does not vary so much between the spans. A lumped parameter model may therefore be accurate enough to model dynamic variation of the average tensional state N along the whole wire length. The mass conservation theorem applied to the control volume filled of wire continuum with instantaneous mass M D V reads dM D V P C Q D 0 dt

(25.19)

in which Q is the total flow rate of the wire material flowing out from the control volume. For a purely axially deforming solid continuum, the density rate is related to axial strain rate by the expression

P D P"

(25.20)

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Combining this with the mass conservation equation then leads to kinematic equation of the wire continuum Q (25.21) "P D V Because tensional force is linked to stress by N D A, the state equation for the wire tension gets form O /Q C .N E.N O /QP NP D A V

(25.22)

By making use of relations V D AL P QP D AL

(25.23) (25.24)

for wire volume and flow rate, the state equation finally reads O /LP C .N E.N O /LR : NP D A L

(25.25)

The rate of wire length LP is contributed by the feeding speed vwinch of winch drum (the sink), by the speed of work object vobject and by the stretching effect of moving guide wheels LP D ˝Rwinch C vwinch e rope C vobject e rope

X

vi e i1 C e i2 :

(25.26)

i

The reaction forces F j acting on the wheel bearings moving by velocities vj can be computed from the dynamic wire tension by vector expressions F i D Ni e i1 C e i2 :

(25.27)

25.3 Boom Mechanics Hydraulic booms are multibody systems, which can be modeled using either relative coordinates between the links or using absolute coordinates for link positions. The latter approach needs also equations of kinematic constraints to couple the motion of separate links together or, alternatively, contact force equations to describe more physically the dynamics of joints [6]. In a boom mechanism, the links are connected to other links, actuators, and wire wheels as illustrated in Fig. 25.11. If absolute coordinates are used, the differential equation of motion for each link reads M yR C Ky D G C C C H C L; (25.28)

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Fig. 25.11 Link element connected to hydraulic actuator, neighboring link, and wire wheel

θ rj S

R

T where the first two elements (S, ) of vector y D S u v are the rigid body coordinates of link center of gravity. The remaining components are modal coordinates of the vibratory motion in axial and bending directions. The position of nodal points fixed to the moving link can be calculated using kinematic transformation R D R.r; y/ D ŒX Y T , where the link state variables y and local link positions r are related to the global Cartesian variables. This makes it possible to follow also the global positions of wheel centers with link motion. When the wheels are moving with boom link, their velocity is given for wheel i by vi D J .r i /yP

(25.29)

where J D @R=@y is the Jacobian between the Cartesian and state vector spaces evaluated at the wire wheel node. The right-hand-side terms in (25.28), respectively, represent gravitational loading, concentrated forces from motion constraints, actuators loading, see (25.37), and the last term is the wire wheel force L D J.ri /T Fi :

(25.30)

The wire reaction F i can be computed from (25.27) completing the equation system in wire–boom interaction.

25.4 Fluid Power Circuit The power source in lifting booms is very often oil hydraulics. Hydraulic volume element consists of subvolumes in actuator chamber (a), hose (h), and pipe (p). If lumped parameter approach is used, the dynamic equations for pressure variations in plus (i D C) and minus (i D ) volumes of actuator circuit are pPi D B

Qi ; Vi

(25.31)

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where the effective volume of fluid Vi is given by Vi D

X B 1C Vij ˇij

(25.32)

j Da;h;p

from addition of subvolumes having actual volume Vij and equivalent bulk modulus ˇij . The actuator volumes are related to link positions by formulae VCa D AC .z zmin /;

where zD

Va D A .zmax z/;

(25.33)

p .R C R /.R C R /

(25.34)

is the instantaneous cylinder length connecting link nodal points to cylinder eyes at plus and minus ends, see Fig. 25.12. Contribution to flow into the volume is coming from valve flow qi and from the displacement flow in actuator piston generated by the link movements Qi D qi C Ai xP i

(25.35)

with actuator chambers length rates xP C D zP, xP D Pz and cylinder speed zP D

1 P P /.R C R /: .R C R z

(25.36)

Once the dynamic pressures are integrated from (25.24), then the actuator load can be evaluated by H H D J T .R C R /; (25.37) z where H D pC AC p A is the resulting hydraulic force. The flows through the valve ports depend on the pressure differences between the pressures pC and p in z

V+a , p+

x+

A V+h R+

Fig. 25.12 Variables of an hydraulic actuator

V+p R−

x +

A

−

V−a , p−

−

V−h V−p

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Fig. 25.13 Generation of ramp functions for open loop driving mode

u

u

Q

Quick

N

Normal

C

Creep

t

actuator volumes and pressures ps and pt in the supply and tank lines. For a double action actuator driven by a 4/3 directional control valve, the flows into plus and minus volumes are given in pushing direction (positive control input u>0, switching code I D1) by p qC D sgn.ps pC /cu jps pC j; p q D sgn.p pt /cu jp pt j

(25.38a) (25.38b)

and in pulling direction (negative control input u < 0, switching code I D 1) by p qC D sgn.pC pt /cu jpC pt j; p q D sgn.ps p /cu jps p j

(25.39a) (25.39b)

1

=2 with c D qnom u1 max pnom obtained by measuring volumetric flow qnom for fixed pressure difference Pnom and full input voltage umax . The classical way to govern the valve in a servo-loop control is to apply PIDcontrol Z t e dt; (25.40) u D KD eP C KP e C KI 0

where e is the error between desired and measured values of controlled positions in the system. For open loop control, a systematic way is to use ramp functions from an electronic ramp generator, a family of which is shown on Fig. 25.13.

25.5 System Dynamics Examples As applications of the presented simplified wire dynamical model, two cases respectively corresponding both kinematically and from control point of view to open and closed loop structures are considered.

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25.5.1 Hydraulic Winch System The system is an articulated three link system with rotational joints and hydraulic actuators at the first two links and a third link that is an expandable boom supporting a winch moving a chain with a hook at its extremity to carry a load, see Figs. 25.1a and 25.14. Based on the component level models presented in previous chapters, a complete system model has been built up and tested on the following maneuver consisting of ten ramp inputs to the valves in an open loop control time-sequence. The above sequence consists of ten actions in four phases corresponding to lifting, boom shortening, tilting, and winching actions in Table 25.3.

MOT CYL3

CYL4

CYL2 CYL1

CYL1

CYL2

CYL3

CYL4

MOT

DCV1

DCV2

DCV3

DCV4

DCV5

Fig. 25.14 Three link open loop winching system Table 25.3 Sequence of ramp functions in lifting task of the hydraulic winch system Valve Actuator Switch Ramp Time DCV1 DCV1 DCV2 DCV2 DCV3 DCV3 DCV4 DCV4 DCV5 DCV5

CYL1 CYL1 CYL2 CYL2 CYL3 CYL3 CYL4 CYL4 MOT MOT

1 0 1 0 1 0 1 0 1 0

N N N N N N N N Q N

0.0 2.0 2.0 3.5 1.5 3.0 1.5 3.0 0.0 1.5

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The dynamic behavior of this system has been simulated using the model above and the results are shown on Figs. 25.15–25.17 giving respectively the time variation of wire tension N , boom contraction force H , boom stress and tip trajectory Rtip D Œ Xtip Ytip T in cartesian space.

12000

N H

N [N]

H [N]

0

Fig. 25.15 Wire tension N and actuator force H in boom shortening cylinder

−4000

12000

0 −4000 5 t [s]

0

s [N/m2] x 10 8 2.5

0

Fig. 25.16 Bending stress variation middle in the first stage of the telescopic link

−1

0

5

t [s]

Ytip [m] 4.5

Fig. 25.17 Trajectory R tip of the boom tip

2 3.4

5 Xtip [m]

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Large variation of wire tension is observed on Fig. 25.15 as a result of load motion and low frequency boom oscillation, which correlates with boom stress on Fig. 25.16. However, the high frequency generation in the interval between 2 and 3 s. of boom wire elasticity due to nonlinear friction effect during boom retraction is almost totally filtered and is only significant above threshold stress value. The wandering behavior of tip trajectory on Fig. 25.17 is due to the absence of control and provides an element on its need. It should be noted that from these plots important design properties can be directly fixed concerning system and actuator parameters.

25.5.2 Chain Driven Elevating Boom The second case under study is a hydraulic telescopic boom system consisting of two links, the first one being a chain-driven expandable boom with a rotary joint at origin to change the latitude angle. The second link is fixed by a rotary joint to first one, and is carrying at its tip an orientable platform, see Figs. 25.1b and 25.18. The problem here is to keep the orientation of the platform during operation cycle of 20 s by applying into platform actuator a PI control subject to orientation error with the vertical. A similar control has been earlier applied to drive line guiding in an Excavator-based sheet-piling system [7]. The system model has been built up and tested on the following maneuver consisting of eight ramp inputs governing the actuators of first two joints and the boom extension. The sequence of eight actions corresponds to three phases for boom extension, tilting, and latitude decrease actions in Table 25.4. The actuator of last joint works during the actions as a part of servo loop for controlling platform vertical orientation. Simulation of the system corresponding to parameter values of a real industrial device has been performed. Typical results corresponding to time evolution of chain tension N and cylinder force H are displayed in Fig. 25.19 while tip trajectory R tip D Œ Xtip Ytip T in cartesian space is given in Fig. 25.20. A large variation is observed as in previous case for wire tension. A reason may be that for the first phase between 0 and 10 s, nonlinear friction modes are generated during boom extension and are transmitted to chain tension through its elasticity. Mode amplitude is larger than in second phase as they are driven unstable by boom expansion. This follows from the large and increasing wandering of tip trajectory on Fig. 25.20 during first period with expansion corresponding to the vertical part. To analyze further the effect of PI-controller, a stationary situation where only the platform actuator is acting to maintain platform orientation along the vertical has been considered for two values of the gains. A step drop on time point t D 0 turns the platform out from the initial vertical orientation to a small negative inclination d D 0:015 rad. The step response is given in Fig. 25.21 and will be compared to the orientation history shown in Fig. 25.22 obtained from the previous maneuver. Both results show the superposition of two low frequency (f 2) and high frequency (f 20) oscillations coming respectively from boom and from arm natural

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DCV3

q

DCV4 DCV2

qd = 0

CYL3

DCV1 CYL1 CYL2

Fig. 25.18 Chain driven elevating platform

Table 25.4 Sequence of ramp functions in lifting task of the hydraulic elevating platform Valve Actuator Switch Ramp Time DCV1 DCV2 DCV3 DCV4 DCV5 DCV6 DCV7 DCV8

CYL1 CYL2 CYL3 CYL4 CYL5 CYL6 CYL7 CYL8

1 0 1 0 1 0 1 0

C N C N N N C N

14:0 18:0 14:0 18:0 0:0 10:0 10:0 14:0

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Dynamics of Wire-Driven Machine Mechanisms

Fig. 25.19 Chain tension N and actuator force H in boom extension

5000

315 5000

N H

N [N]

H [N]

0

20

0

0 t [s]

Fig. 25.20 Trajectory R tip of the boom tip

Ytip [m] 27.5

22.5

11.5

8

Xtip [m]

Fig. 25.21 Step responses of platform orientation ™ at stationary position

q [rad] x10 −3 2

−18

0

KP = 1, KI = 2 KP = 2, KI = 1

5

t [s]

316 Fig. 25.22 Platform orientation ™ during maneuver

E. Keskinen et al. θ [rad] 0.03

0

−0.03

0

20 t [s]

vibrations, in the evolution of platform orientation. The behaviors of the two frequencies are different for different values of the gains in the controller. When the gain for proportional part is small but integral part large, the response is, after slight overshooting, converging to a constant amplitude oscillation. For larger proportional gain value but smaller integral gain value, the overshooting peak is eliminated, but the motion is still converging towards the same vibration. This oscillation is still not steady-state nor limit-cycle vibration. The reason for this motion is simply the high flexibility of the long extension boom as compared with its low internal damping. The time required for this oscillation to die out is therefore relatively long. PI-controller applied to the platform orientation control only is not capable to compensate this vibration mode. This behavior clearly exhibits the limitation of a simple PI-controller to realize the correct functional transformation required to give the closed loop system asymptotically stable behavior, as it will be shown elsewhere.

25.6 Conclusion The problem of finding a simple enough PC workable simulation model of mechanical systems including chains and/or wires in transmission from hydraulic actuators, and allowing a study of vibration modes, has been addressed to. It was shown that by homogenization procedure, these continuous parts may in first approximation be represented by their stiffness characterizing their tension during time evolution. The resulting model description has been studied for two industrial applications kinematical corresponding to open and closed loop structures. In both cases, wire or chain tension has been directly obtained with the other variables and system vibrations can be analyzed. The present simulation model allows to directly compare the consequences of different choice of nominal parameters onto system dynamics. The effect of various controllers can as well be discussed. In this sense, the proposed model is providing an adequate tool for system design.

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References 1. Karvinen T, Keskinen E (2009) Dynamics of wire-driven machine mechanisms, Part I – literature review. Dynamical system with discontinuity, stochasticity and time-delay. (A.C.J. Luo, ed., Springer), 285–298 2. Keskinen E, Keskiniva E, Riitahuhta A (1995) Utilization of integrated simulation techniques for rapid prototyping of mechatronic machines. Invited Lecture Proceedings ICRAM’95, vol I. p 111 3. Keskinen E, Montonen J, Launis S, Cotsaftis M (1999) Simulation of wire and chain mechanisms in hydraulic driven booms. In: IASTED international conference on applied modelling and simulation, Cairns, QLD, September 1–3 1999 4. Keskinen E, Montonen E, Launis S, Cotsaftis M (1999) Cartesian trajectory control of hydraulic elevating platforms. In: IASTED International conference on robotics and applications, Santa Barbara, CA, 28–30 October 1999 5. Keskinen EK, Iltanen M, Salonen T, Launis S, Cotsaftis M, Pispala J (2000) Man-in-the-loop training simulator for hydraulic elevating platforms. In: Proceedings of 17th IAARC/IFAC/IEEE international symposium on automation and robotics in construction, Taipei, Taiwan, 18–20 September 2000 6. Keskinen EK, Iltanen M, Salonen T, Launis S, Cotsaftis M, Pispala J (2001) Dynamics of training simulator for mobile elevating platforms. In: Arai E, Arai T, Takano M (eds) Human friendly mechatronics. Elsevier Science B.V., Amsterdam 7. Keskinen E, Launis S, Cotsaftis M, Raunisto Y (2001) Performance analysis of drive line steering methods in excavator-mounted sheet-piling systems. Comput Aided Civil Infrastruc Eng 16(4):229–238

Chapter 26

On Analytical Methods for Vibrations of Soils and Foundations H.R. Hamidzadeh

Abstract Research on dynamics of soils and foundations has yielded several fundamental methods for formulation of interaction problems. This paper is intended to survey the development of the current state-of-practice for design and analysis of dynamically loaded foundations. Extensive studies in this field utilize various linear mathematical models for interaction between foundations and different soil media. The effective analytical, numerical, and experimental techniques and their methodologies, which are well established for treating problems in dynamic soil-foundation interaction are outlined. Described techniques are categorized based upon formulation procedures and their applications. Some areas are indicated where further research is needed.

26.1 Introduction The possible occurrence of extreme dynamic excitation, either natural or manmade, has a major influence on the design of buildings and machine foundations. A primary concern in designing foundations is the knowledge of how they are expected to respond when subjected to dynamic loadings. The validity of the mathematical analysis depends entirely on how well the mathematical model simulates the behavior of the real foundation. Over the past decades, our ability to analyze mathematical models for dynamics of foundations has been improved by the use of different analytical and numerical techniques. In most of these analyses, it is common to assume that the footing is rigid and the medium is a homogeneous elastic half-space. Extensive efforts have been confined in the development of procedures and computer simulations to tackle some practical problems that arise in this field, while other important problems have been neglected. It should be noted that interaction between foundations

H.R. Hamidzadeh () Department of Mechanical and Manufacturing Engineering, Tennessee State University, Nashville, TN, USA e-mail: [email protected]

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 26,

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for noncircular footings was not treated in a satisfactory manner and significant deficiencies remain in most of the previous analyses. This paper will discuss some of the issues of dynamics of soils and foundations from a practical point of view. Since this topic is quite broad, a brief description of methodology will be outlined, while details will be given for a few procedures that have proven to be effective and accurate. One of the main objectives of this review paper is to survey different available techniques for solving the dynamic response of foundations when subjected to harmonic loadings. Special attention is directed to the dynamic response of the surface of the medium due to concentrated dynamic loads, response of foundations, coupled vibrations of foundations, interaction between two foundations, experimental aspects of soils and foundations, and laboratory simulations.

26.2 Surface Response Due to Concentrated Forces In the field of propagation of disturbances on the surface of an elastic half-space, the first mathematical attempt was made by Lamb [1]. He gave integral representations for the vertical and radial displacements of the surface of an elastic half-space due to a concentrated vertical harmonic force. Evaluation of these integrals involves considerable mathematical difficulties, due to the evaluation of a Cauchy principal integral and certain infinite integrals with oscillatory integrands. Nakano [2] considered the same problem for a normal and tangential force distribution on the surface. Barkan [3] presented a series solution for the evaluation of integrals for the vertical displacement caused by a vertical force on the surface, which was given by Shekhter [4]. Pekeris [5, 6] gave a greatly improved solution to this problem when the surface motion is produced by a vertical point load varying with time, like the Heaviside function. Elorduy et al. [7] developed a solution by applying Duhamel’s integral to obtain the harmonic response of the surface of an elastic half-space due to a vertical harmonic point force. Heller and Weiss [8] studied the far field ground motion due to an energy source on the surface of the ground. Among the investigators who considered the three-dimensional problem for a tangential point force, Chao [9] presented an integral solution to this problem for an applied force varying with time like the Heaviside unit function. Papadopulus [10] and Aggarwal and Ablow [11] have presented solutions, in integral expressions, to a class of three-dimensional pulse propagation in an elastic half-space. Johnson [12] used Green’s functions for solving Lamb’s problem, and Apsel [13] employed Green’s functions to formulate the procedure for layered media. Kausel [14] reported an explicit solution for dynamic response of layered media. Davies and Banerjee [15] used Green’s functions to determine responses of the medium due to forces that were harmonic in time with a constant amplitudes. The solution was derived from the general analysis for impulsive sources. Kobayashi and Nishimura [16] utilized the Fourier transform to develop a solution for this problem, and expressed the results in terms of the full-space Green’s functions, which include

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infinite integrals of exponential and Bessel’s function products. Banerjee and Mamoon [17] provided a solution for a periodic point force in the interior of a three-dimensional, isotropic elastic half-space by employing the methods of synthesis and superposition. The solution was obtained in the Laplace transform as well as the frequency domain. Hamidzadeh [18] presented mathematical procedures for determination of the dynamic response of surface of an elastic half-space subjected to harmonic loadings and provided numerical results for displacement of any point on the surface in terms of properties of the medium and of the exciting force. The solution was analytically formulated by employing double Fourier transforms and was presented by integral expressions. Hamidzadeh [19] and Hamidzadeh and Chandler [20] provided dimensionless response for an elastic half-space and compared their results with other available approximate results. Considering a semi-infinite elastic solid subjected to a vertical concentrated harmonic force as illustrated in Fig. 26.1, the radial and vertical displacements on the surface of the medium due to applied load can be expressed as: Fo .u1 C iu2 / ei!t Gr Fo .v1 C i v2 / ei!t v.r/ D Gr

u.r/ D

(26.1) (26.2)

where a0 D r! =G is frequency factor, Fo and ! are amplitude and angular frequency of the applied force, respectively. G and are shear modulus and density of the medium, respectively. u and v are radial and vertical displacement at any point on the surface. u1 C iu is a complex non-dimensional radial displacement function. v1 C iv2 is a complex non-dimensional vertical displacement function. Figure 26.2 presents numerical results computed for the displacements on the surface of semi-infinite elastic medium. The displacements of a point on the surface

F0eiΤt

v u

Semi-infinite Solid

Fig. 26.1 Surface of a semi-infinite elastic solid subjected to a vertical concentrated harmonic force

322

H.R. Hamidzadeh 0.2 −u1,u2

0.1

−u1

0 −0.1

u2

−0.2 0

5

10

15

5 10 Frequency Factor - a0

15

0.4

−v1,v2

0.2 −v1 0 −0.2 −0.4 0

v2

Fig. 26.2 Complex non-dimensional radial and vertical functions vs. frequency factor a0 (Hamidzadeh’s – lines; and Rucker’s – symbols)

of the medium depend largely upon in-phase and quadrature components of the non-dimensionalized complex displacement functions. These components are functions of frequency factory. The range of frequency factor covered is more than sufficient for practical purposes of considering near-field displacements. Far-field displacements can be determined using Lamb’s equations. The values of u1 , u2 , v1 , and v2 are for Poisson’s ratio of 0.25.

26.3 Dynamic Response of Foundations Advances in the development of solutions for soil-foundation interaction problems are categorized in the following sections based on the formulation procedures.

26.3.1 Assumed Contact Stress Distributions The first attempt to solve the vertical vibration of a massive circular base on the surface of an elastic medium was made by Reissner [21]. He adopted Lamb’s [1] approach and developed a solution by assuming a uniform stress distribution on the surface of the medium. He established an estimated solution for determining the vertical steady state response of circular footings. He also calculated the displacement of the center of the base and introduced the amplitude of vibration in terms of

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non-dimensional parameters. It has been proven that his results overestimated the amplitude due to his consideration of the displacement at the center of the base. Reissner and Sagoci [22] presented a static solution for the torsional oscillation of a disc on the medium. Miller and Pursey [23, 24] considered the vertical response of a circular base due to a force uniformly distributed on the contact surface. Quinlan [25] and Sung [26] independently extended Riessner’s [21] approach to solve the problem of vertical vibration of circular and infinitely long rectangular footings. In their analyses, they considered three different harmonic stress distributions: uniform, parabolic, and stress produced by a rigid base under static conditions. They showed that the vibration characteristics of semi-infinite media effectively vary with the type of stress distributions and elastic properties of the medium. Arnold et al. [27] considered four vibrational modes (vertical, horizontal, torsional, and rocking) for a circular base on the surface of elastic media. By assuming harmonic static stress distributions for all modes, they evaluated the dynamic responses using an averaging technique. They also verified this work with experimental results using a finite model for the infinite medium. Bycroft [28] followed the same approach for four modes of vibration by determining complex functions to represent the in-phase and out-of-phase components of displacement of a rigid massless circular plate. Bycroft [29] later carried out some tests to verify his previous theoretical work. Thomson and Kobori [30] and Kobori et al. [31–35] considered the dynamic response of a rectangular base. They provided computational results for components of the complex displacement functions by assuming a uniform stress distribution on the contact area of the base and medium. Their analysis was for an elastic half-space, viscoelastic half-space, and layered viscoelastic media.

26.3.2 Mixed Boundary Value Problems Harding and Sneddon [36] and Sneddon [37] gave a static solution for a rigid circular punch pressed into an elastic half-space. By the use of the Hankel transform, the appropriate mixed boundary value problem was reduced to a pair of dual integral equations representing the stress distribution and uniform displacement under the rigid punch. Several investigations have been conducted to extend the static solution to the corresponding dynamic problems. Awojobi and Grootenhuis [38] and Awojobi [39–42] used the Hankel transform to present the complete dynamic problem by dual integral equations. They gave an analytical solution for the vertical and torsional oscillations of a circular body and the vertical and rocking modes of an infinitely long strip foundation. The evaluation of stress distributions and uniform displacements was based on the extension of the Titchmarsh’s [43] dual integral equation and a method of successive approximations. Robertson [44, 45] followed the same procedure and reduced the dual integral equations into Fredholm integral equations and gave series solutions to these equations for the vertical and torsional response of a rigid circular disc. Gladwell [46–48] developed a solution for the

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mixed boundary value problems for circular bases resting on an elastic half-space or elastic strata. He solved the integral equations and presented the displacements of four different modes of vibration in series forms. Karasudhi et al. [49] treated the vertical, horizontal, and rocking oscillations of a rigid strip footing, on an elastic half-space, by reducing the dual integral equations into the Fredholm integral equations. Housner and Castellani [50] conducted an analytical solution based on the work done by the total dynamic force and determined the weighted average vertical displacement for a cylindrical body. To determine the free field displacements of an elastic medium, for four different modes of vibration, for a cylindrical body, Richardson [51] followed Bycroft’s [28] method and provided a solution to this problem. Luco and Westmann [52] solved the mixed boundary value problems for four modes of vibration by considering a massless circular base. Their procedure reduced the resulting dual integral equations to the Fredholm integral equations. They calculated the complex displacement functions for a wide range of frequency factors. In a separate publication [53], they followed the same procedure for the determination of the response of a rigid strip footing for the three modes of vertical, horizontal, and rocking vibration. The vibration of a circular base was also treated by Veletsos and Wei [54] for horizontal and rocking vibrations. Bycroft [55] extended his earlier work to present approximate results for the complex displacement functions at higher frequency factors. Veletsos and Verbic [56] introduced the vibration of a viscoelastic foundation. Clemmet [57] included hysteretic damping in the Richardson [51] solution. Luco [58] provided a solution for a rigid circular foundation on a viscoelastic half-space medium. These investigations were based on circular or infinitely long strip foundations. Few investigators have paid attention to the dynamic responses of a rectangular foundation, due to the difficulty of the asymmetric problem. Elorduy et at. [7] introduced a numerical technique based on the uniform displacements for a number of points on the contact surface of the rectangular footings. In their analysis, they employed an approximate solution for the surface motion of a medium due to a vertical point force. They gave complex displacement functions for vertical and rocking modes with different ratios of length to width of rectangular footings. By extending the Bycroft [29] idea of an equivalent circular base for a rectangular foundation, Tabiowo [59] and Awojobi and Tabiowo [60] gave a solution to this problem. They also introduced another solution by superimposing the solution of two orthogonal infinitely long strips and gave the displacement at the intersection of these strips for different frequencies. Wong and Luco [61, 62] solved this problem for the three modes of vertical, horizontal, and rocking vibrations. They used the approach reported by Kobori et al. [31] to provide an approximate solution to a footing, which was divided into a number of square subregions. They assumed that the stress distribution for each subregion is uniform with unknown magnitude, while all the subregions experienced uniform displacements. Their solution considered the coupling effect for viscoelastic medium, and the complex stiffness coefficients were tabulated [62] for different loss factors. Hamidzadeh [18] and Hamidzadeh and Grootenhuis [63] presented an improved version of Elorduy’s method to obtain the dynamic responses for three modes of vertical, horizontal, and rocking vibration for rectangular foundations. In their analysis,

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a uniform displacement for each node of rectangular subregions were assumed. They utilized the impedance matching technique to formulate the dimensionless response for foundations. As reported by Hamidzadeh and Grootenhuis [63], vertical, horizontal, and rocking displacements of a rigid massless base resting on an elastic half-space can be expressed conveniently in terms of two non-dimensional displacement functions, one in-phase with the motion (F1 ) and the other in quadrature (F2 ) such that: for vertical displacements one may write: Uz D

F .FV1 C FV2 / Gc

(26.3)

P .FH1 C iF H2 / Gc

(26.4)

for horizontal displacement it yields Ux D

and finally for rocking motion it results x D

M .FR1 C iF R2 / : Gc3

(26.5)

As shown in Fig. 26.3. F , P , and M are the applied harmonic vertical force, horizontal force, and the moment. G is the shear modulus of the medium, and c is the half width of the square base. The numerical values of the displacement functions F1 and F2 , for the different uncoupled motions with subscripts V for vertical, H for horizontal, and R for rocking, have been computed for a square base. The abscissa is the frequency factor a D !c=c2

(26.6)

where ! is the excitation frequency in rad/s, d is half the width of one side and c2 is the velocity of shear waves in the medium. The displacement functions have

F P

M

Fig. 26.3 A rectangular rigid foundation on the surface of an elastic half-space subjected to harmonic vertical, horizontal, and rocking excitation forces and moment

326

H.R. Hamidzadeh

−V1,V2

0.2

−V1

Vertical

0.1

−H1,H2

V2 0 0 0.2 0.2 −H1

0.4

0.8

1

1.2

1.4

1.6

Horizontal

0.1 H2 0 0 0.2 0.2 −R1

−R1,R2

0.6

0.4

0.6

0.8

1

1.2

1.4

1.6

Rocking

0.1 R2 0 0

0.2

0.4

0.6 0.8 1 Frequency Factor - a

1.2

1.4

1.6

Fig. 26.4 Variation of non-dimensional displacements functions vs. frequency factor a for vertical, horizontal, and rocking motions

been calculated for a frequency factor up to 1.6 where necessary for the subsequent computation of the resonant response of a foundation block. The results presented in Fig. 26.4 are for a Poisson’s ratio of 0.25, which is a reasonable value to take for certain types of soil. Similar sets of curves have been computed for values of Poisson’s ratio of 0, 0.31, and 0.5 (given in [18]). The in-phase function, F1 , gives a measure of the stiffness of the elastic halfspace whereas the quadrature function, F2 , is dependent upon the amount of energy radiated into the half-space. The amplitude response at resonance of a massive foundation block, is therefore controlled solely by the quadrature function. To evaluate the dynamic response of a foundation block the displacement functions and the impedance matching technique will be used. The assembled systems of a mass, m, or of an inertia, I , and a half-space for the three motions can be split into components of mass or inertia and medium. The axis of rocking has been taken through the centre point, O, in the base of the block. The impedance of the rigid body is defined as the ratio of the applied force divided by the resultant velocity. Since the applied force and moment are harmonic, the displacements given by (26.5) can be transformed into velocities. By adding the component impedances, the non-dimensional amplitude of vibration for each mode can be expressed as: Vertical #1=2 " Fv 12 C Fv 22 jUz jGc D (26.7) F .1 C a2 bFv 1/2 C .a2 bFv 2/2 Horizontal jUx jGc P

" D

FH 12 C FH 22 .1 C a2 bF H 1/2 C .a2 bFH 2/2

#1=2 (26.8)

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Rocking jx jGc3 M

" D

FR 12 C FR 22

327

#1=2

.1 C a2 b 0 FR 1/2 C .a2 b 0 FR 2/2

;

(26.9)

where a is the frequency factor, b and b 0 are the mass and inertia ratios defined as: m

c 3 I b0 D 5

c bD

(26.10) (26.11)

and is the density of the medium. The amplitude factors have been plotted in Figs. 26.5–26.7 as a function of the frequency factor a for constant square base for a Poisson’s ration of 0.25. It can be seen from these frequency response curves that the lower the mass or inertia ratio the lower the maximum value of the amplitude factor and the higher the value of the frequency factor at which this maximum occurs. It may seem rather surprising at first that a foundation block with a high mass or inertia ratio will respond more vigorously at resonance than another block with lower values for the same ratio, when each is exposed to the same disturbing forces or moments. The energy radiated into the infinite half-space will be less in proportion to the kinetic energy of the vibrating system for the higher values of the mass or inertia ratio than for the lower values, and hence there will be less effective damping at resonance. For very low values of the mass ratio for vertical and for horizontal forcing, all the energy is radiated into the half-space and there is then no resonant response.

Non-dimensionalized amplitude - V

0.5

0.4

0.3

90 80 70 60 50 40 30

Poisson ratio = 0.25

20 0.2

b=10

0.1

0 0

0.2 0.4 0.6 0.8 1 1.2 Non-dimensionalized frequency - a

1.4

Fig. 26.5 Non-dimensional amplitude vs. frequency at different mass ratios for vertical harmonic vibration

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H.R. Hamidzadeh

Non-dimensionalized amplitude - U

1.6

90 80 70 60 50

1.4 1.2

Poisson ratio = 0.25

40 1

30

0.8

20

0.6 b=10 0.4 0.2 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Non-dimensionalized frequency - a

Fig. 26.6 Non-dimensional amplitude vs. frequency at different mass ratios for horizontal harmonic vibration

Non-dimensionalized rocking amplitude

7

60 Poisson ratio = 0.25

6

50

5 40

4

30

3 2

20 b=10

1 0 0.1

0.2

0.3

0.4 0.5 0.6 0.7 0.8 0.9 Non-dimensionalized frequency - a

1

1.1

Fig. 26.7 Non-dimensional amplitude vs. frequency at different Inertia ratios for rocking harmonic vibration

Another important parameter that has a significant effect on the dynamic response is the Poisson’s ratio. Previous numerical results [1, 12] have shown that the resonant amplitude factor will decrease when the Poisson’s ratio of the medium is increased. The effects of mass ratio and of Poisson’s ratio on the frequency factor at resonance and the amplitude factor for the three modes are shown in

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On Analytical Methods for Vibrations of Soils and Foundations

Fig. 26.8 Non-dimensional amplitude vs. non-dimensional resonant frequency for different mass and Poisson ratios for vertical harmonic vibration

329

0.9 ν = 0.0

Resonant Amplitude Factor

0.8

b=120

0.7

80 0.25

0.6

60

0.31

0.5

40

0.4

20 0.5

0.3

10 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Resonant Frequency Factor

Resonant Amplitude Factor

15

ν = 0.0 b=80

60

10

0.25 0.31

5

0.5

40

20 10 5

0

0

0.2

0.4 0.6 0.8 Resonant Frequency Factor

1

1.2

Fig. 26.9 Non-dimensional amplitude vs. non-dimensional resonant frequency for different mass and Poisson ratios for horizontal harmonic vibration

Figs. 26.8–26.10. These results play an important part in the design of a foundation block. The variation of the resonant frequency factor and the amplitude factor can be explained physically by the change in stiffness, which accompanies a variation in Poisson’s ratio. An approximate method for computation of compliance functions of rigid plates resting on an elastic or viscoelastic half-space excited in all directions was reported by Rucker [64]. The proposed method provides compliances for vertical, horizontal, rocking, and torsional motion for rectangular foundations. Another approximate solutions for harmonic response of an arbitrary shape foundation on an elastic half-space was reported by Chow [65]. In his analysis, the contact area was discretized into square subregions, and the influence of the square subregion was

330

H.R. Hamidzadeh 1.6

Resonant Amplitude Factor

1.4 1.2

b′ = 60 ν = 0.0 0.25 0.31 40

1 20

0.8

10

0.6

5 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

1.2

Resonant Frequency Factor

Fig. 26.10 Non-dimensional amplitude vs. non-dimensional resonant frequency for different inertia and Poisson ratios for rocking harmonic vibration

approximated by that of an equivalent circular base. He then compared his results with those of Wong and Luco [62] and Hamidzadeh and Grootenhuis [63]. The dynamic stiffness of a rigid rectangular foundation on the half-space was considered by Triantafyllidis [66] who provided solutions for the mixed-boundary value problem. The problem was formulated in terms of coupled Fredholm integral equations of the first kind. The displacement boundary value conditions were satisfied using the Bubnov–Galerkin method. The solution yielded the influence functions and the stiffness functions characterizing the dynamic interaction between the foundation and half-space. The presented analytical method considered the coupling between normal and shear stress distributions acting on the contact area between the footing and the half-space.

26.3.3 Lumped Parameter Models Based on the theoretical work for the response of a rigid massless circular plate, Hsieh [67] was able to present equivalent mass-spring-dashpot models for all four modes of vibration. The calculated dynamic parameters for each mode were frequency dependent. Following the Hsieh’s approach, Lysmer [68] provided frequency independent values for equivalent spring and damping constants. The idea of developing an equivalent mass-spring-dashpot model for a rigid mass on the surface of a half-space has attracted many investigators such as Lysmer and Richart [69], Weissmann [70], Whitman and Richart [71], Hall [72], Veletsos and Wei [54], Roesset et al. [73], Oner and Janbu [74], Hall et al. [75], and Veletsos [76]. Approximate expressions for the dynamic stiffness coefficients in the frequency domain are

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well established, and are summarized in a text by Wolf [77]. Lumped parameter models to represent the soil-structure interaction for embedded foundations were developed by Wolf [78]. Dobry and Gazetas [79, 80] presented a method to determine the effective dynamic stiffness and damping coefficient of a rigid footing by considering variations of foundation shape, soil type, and length ratio of the base. The analytical methods employed were those of Wong and Luco [60], and they verified some of their computations by several in-situ experiments. A different approach based on the subgrade reaction method is also reported by many investigators such as Terzaghi [81, 82], Barkan [3], and Girard [83].

26.3.4 Computational Methods The Finite Element Method (FEM) has been applied to discretize foundations, the most crucial problem for discretization of foundations by FEM is transmitting waves through artificial finite boundaries. This is considered to be a drawback, which is due to the difficulty encountered in proper modeling of an unbounded soil medium and satisfying the wave radiation condition. The problem of a rigid circular base on an elastic half-space has been considered numerically by many investigators such as Duns and Butterfield [84], Lysmer and Kuhlemeyer [85], and Seed et al. [86]. However, these solutions have not been generalized to cover all modes of vibration due to numerical limitations. Day and Frazier [87] suggested the use of artificial boundaries far away from the region of interest to avoid the undesirable wave reflections. Bettess and Zienkiewicz [88] recommended the use of infinite elements and Roesset and Ettouney [89], and Kausel and Tassoulas [90] proposed transmitting or non-reflecting boundaries to circumvent the problem. Chuhan and Chongbin [91] presented an approximate solution for a viscoelastic medium and strip foundation by considering two-dimensional wave equations in conjunction with the use of Galerkin weighted residual approximations. A frequency-dependent compatible infinite element was presented, then by coupling the infinite elements with ordinary finite elements the system was used to simulate the propagation of waves. The boundary element method (BEM) has been used as an effective numerical technique for solving elastodynamic problems. In this method, boundary integral representation provides an exact formulation, and the only approximations are those due to the numerical implementation of the integral equations. The technique is suitable for infinite and semi-infinite domain due to employment of Green’s functions, which satisfy the radiation condition at far-field. The first application of this technique on soil-structure interaction was performed by Dominguez [92], and Dominguez and Roesset [93] in frequency domain. Karabalis and Beskos [94] determined the frequency and time domain solutions for dynamic stiffness of a rectangular foundation resting on an elastic half-space medium. Spyrakos and Beskos [95, 96] used the time domain boundary element method to consider dynamic response of two-dimensional rigid and flexible foundations.

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Numerical solutions for a layered medium are reported by Luco and Apsel [97] and Chapel and Tsakalidis [98]. In these solutions, formulation for Green’s functions were made using the Hankel transform for each layer. Kausel [14] presented an explicit closed-form solution for the Green’s functions corresponding to dynamic loads acting on layered strata. These functions embody all the essential mechanical properties of the medium. The solution is based on a discretization of the medium in the direction of layering, which results in a formula yielding algebraic expressions for the displacement of all layers. Determination of response for a layered medium to dynamic loads, prescribed at some location in the soil, can be achieved by using the stiffness matrix method presented by Kausel and Roesset [99]. In their solution the external loads, applied at the layer interfaces, are related to the displacements at these locations through stiffness matrices, which are functions of both frequency and wave number. The proposed method can treat simultaneous solutions for multiple loadings. Israil and Ahmad [100] investigated the dynamic response of rigid strip foundations on different media of a viscoelastic half-space, viscoelastic strata on a half-space, and viscoelastic strata on a rigid bed. An advanced boundary element algorithm was developed by incorporating isoparametric quadratic elements. The effect of Poisson’s ratio and material damping, layer depth, embedment, and type of contact at the foundation-medium interface was studied.

26.4 Coupled Vibrations of Foundations In the analyses for the pure rocking or the horizontal mode, it is usual to assume that the medium is rigidly stiff for either shear or compression deformations. However, it is known from characteristics of the soil that it can resist elastically for both applied compression and shear. Therefore, the problem of horizontal or rocking motion for a massive footing on the surface of elastic half-space media should be considered as a coupled motion. Wong and Luco [61,62], Rucker [64], and Triantafyllidis [66] have presented solutions for coupled vibrations of rectangular foundations on an elastic and viscoelastic half-space. Another approximate solution is based on simultaneous horizontal and rocking motions. Hall [72] is among the early investigators who used the solution for pure sliding and rocking of circular bases. He followed the Hsieh [67] approach in deriving the equations for simultaneous motion and presented numerical results for certain cases. Richart and Whitman [101] compared the experimental results obtained by Fry [102] with a similar prediction that Hall introduced for a circular base. They found that their comparisons were satisfactory. Karasudhi et al. [49] and Luco and Westmann [53] provided an approximate solution to the problem of the coupled motion of an infinitely long rigid strip. Veletsos and Wei [54] introduced an approximate solution to evaluate the stiffness and damping coefficients for a massless rigid circular footing. They also compared these coefficients with their corresponding value for simultaneous motion. Ratay [103] considered simultaneous motions when the circular base is excited by a harmonic horizontal force. He studied the variation of the frequency response curve due to variation of involved parameters. Beredugo

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and Novak [104] studied the simultaneous motions of a circular base embedded in the surface of a homogeneous elastic medium. Using the finite lumped parameter model for circular embedded foundation, Krizek and Gupta [105] gave a solution to this problem. Clemmet [57] improved the horizontal translation model and considered the rocking displacement of the base due to a horizontal force. Wolf [106] and many others studied the practical problem of soil-structure interaction by allowing simultaneous motions for circular foundations. A number of publications on the simultaneous motions of rectangular foundations are available. These are based on the sub-grade reaction method discussed by Barkan [3] and Girard [83]. Wolf [77, 78] investigated the problem by allowing simultaneous motions for the circular foundation. It should be noted that for all modes except torsional, Wolf’s analysis was done by applying relaxed boundary conditions; this allowed one of the components of the surface tractions under the foundation to be zero. In his analysis, Wolf considered the coupled horizontal and rocking motions of a rectangular foundation based on the Wong and Luco [61] theoretical results. Hamidzadeh and Minor [107] utilized the procedure reported in reference [63] and developed a method for simultaneous horizontal and rocking motions of square foundations on an elastic half-space. They compared their results with those of coupled responses determined by Wong and Luco [60, 62]. The comparison indicates satisfactory agreement for low dimensionless frequencies.

26.5 Interactions Between Foundations To the authors’ knowledge, little attention has been directed to the dynamics of foundations on the surface of a homogeneous elastic half-space. The available solutions are limited to circular bases, infinitely long strips, or for foundations that are very far apart. Iljitchov [108] introduced this problem and gave a poor estimation of the effect of vertical vibration of one foundation on the other. This problem was considered in detail by Richardson [51], Richardson et al. [109], and Warburton et al. [110, 111]. They studied the dynamic responses of two circular bases and provided numerical results for active and passive bases. Their solution was based on averaging techniques for the displacements of both footings. Lee and Wesley [112] presented a solution to the dynamic responses of a group of flexible structures on the surface of an elastic half-space medium. MacCalden and Matthiesen [113] presented theoretical and experimental results for far-field bases. Utilizing the method reported by Richardson et al. [109], Clemmet [57] improved the solution for horizontal and rocking motion of a circular base and introduced hysteretic damping for the media. He verified his results for the vibration of passive and active bases with the experimental results of Tabiowo [59]. Snyder et al. [114] employed a two dimensional finite element method to study this problem for circular and infinitely long rigid strip bases. Hamidzadeh [18] investigated the interactions of two foundations resting on the surface of a homogeneous elastic half-space. In his analysis, the two rectangular foundations were separated by a certain distance. The mathemati-

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cal model was developed by considering displacement components for each base. The first component was due to the base vibration and the second one caused by the induced displacement due to the reaction force at the other base. The analysis allowed displacement and rotation in every direction. The results of analysis were provided for two different cases of active-active and active-passive foundations.

26.6 Experimental Studies Many different laboratory and in-situ experiments have been carried out in this field. These experimental works can be divided into two main groups: First, the measurement of dynamic properties for the medium and secondly, measurement of frequency response for foundations. For measurement of shear modulus, various techniques have been developed by investigators. Among researchers, Awojobi [115] employed refraction and reflection surveys. Jones [116, 117] measured the velocity of Rayleigh waves to evaluate the shear modulus. Maxwell and Fry [118] measured the velocity of shear and compression waves to determine the shear modulus and Poisson’s ratio. Stokoe and Richart [119] and Beeston and McEvilly [120] measured the velocity of shear waves using cross-hole tests. The calculated response characteristics of vertical vibration of a rigid circular footing were used by Jones [116], Dawance and Guillot [121], and Grootenhuis and Awojobi [122] to determine shear modulus and Poisson’s ratio. In addition to the above in-situ experiments, many laboratory tests have also been conducted. Hardin and Drnevich [123, 124], and Cunny and Fry [125] recommended resonant column tests. Lawrence [126] used a pulse technique to measure shear modulus. Theirs and Seed [127], and Kovacs et al. [128] used a cyclic simple shear test for low frequency cyclic loading. They measured the shear modulus and the damping ratio of the soil for very small strains by recording the free response of the vertical vibration of a circular footing. In development of experimental methods for determination of frequency response for foundations, few experiments have been performed to determine the frequency response of a footing on the surface of an elastic half-space. This is due to the difficulties involved in creating a suitable environment for the field test and establishing a finite model for an infinite elastic medium. Jones [116], Kanai and Yoshizawa [129], Bycroft [29], Dawance, Guillot [121], and Awojobi [115] presented some experimental frequency responses for a circular footing in the field. Eastwood [130], Arnold et al. [27], Chae [131], and Tabiowo [59] established a finite model for a half-space and did some tests on the dynamic response for circular bases. Eastwood [130] and Tabiowo [59] gave a number of experimental results for the response of a rectangular base, but Eastwood did not determine the dynamic elastic constants. Kanai and Yoshizawa [129] tested an actual building for the rocking mode. Hamidzadeh [18, 132] and Hamidzadeh and Grootenhuis [63] reported experimental results using a laboratory model that simulated an elastic half-space medium

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subjected to dynamic loading. The model was used in an attempt to experimentally determine the two important elastic properties of shear modulus and Poisson’s ratio for the medium. The finite size model was also used to conduct experiments to verify the validity of established theories. In the case of two circular footings on an elastic medium, MacCalden [113] gave results for the vibrations of active and passive circular bases using in-situ tests. Extensive surveys of experimental studies for the interaction between soil and structure are reported by Luco et al. [133, 134] and Wong et al. [135]. All the reported investigations were performed on the Millikan Library Building, which has been the subject of a large number of forced vibration tests. These experimental results showed that forced vibration tests can be used to obtain estimates of the foundation impedance functions

26.7 Conclusions A comprehensive literature review on the dynamic response of rigid foundations subjected to loadings is presented. Significant progress has been established in simulation of the embedded foundations and foundations on elastic half-space and layered media. Techniques described here can be utilized with some degree of confidence to estimate dynamic responses for a number of problems in the field of soil-foundation interaction. A considerable amount of the performed research has been restricted to simple and idealized soil-foundation configurations subjected to harmonic excitations, and the medium is treated linearly. Further advances in development of mathematical models are needed for flexible foundations, structures supported by piles, interaction between foundations, and the development of a lowfrequency measuring system for in-situ testing.

References 1. Lamb H (1904) On the propagation of tremors over the surface of an elastic solid. Phil Trans Roy Soc 203(A):1–42 2. Nakano H (1930) Some problems concerning the propagation of the disturbances in and on semi-infinite elastic solid. Geophys Mag Tokyo 2:189–348 3. Barkan DD (1962) Dynamics of bases and foundations. McGraw-Hill, New York 4. Shekhter OY (1948) Consideration of inertial properties of soil in the computations of vertical forced vibrations of massive foundation. NII Symposium, 12, Vibratasii, Osnovaniy i Fundementov, Moscow 5. Pekeris CL (1955) The seismic buried pulse. Proc Nat Acad Sci USA 41:629–639 6. Pekeris CL (1955) The seismic surface pulse. Proc Nat Acad Sci USA 41:469–480 7. Elorduy J, Nieto JA, Szekely EM (1967) Dynamic response of bases of arbitrary shape subjected to periodic vertical loading. Proceedings of the international symposium on wave propagation and dynamic properties of earth materials, Albuquerque, University of New Mexico, pp 105–123

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8. Heller LW, Weiss RA (1967) Ground motion transmission from surface sources. Proceedings of the international symposium on wave propagation and dynamic properties of earth materials, Albuquerque, University of New Mexico, pp 71–84 9. Chao CC (1960) Dynamical response of an elastic half-space to tangential surface loadings. J Appl Mech ASME 27:559–567 10. Papadopulus M (1963) The use of singular integrals in wave propagation problems with application to the point source in a semi-infinite elastic medium. Proc Roy Soc London A 276:204–237 11. Aggarwal HR, Ablow CM (1967) Solution to a class of three-dimensional pulse propagation problems in an elastic half-space. Int J Eng Sci 5:663–679 12. Johnson LR (1974) Green’s function for Lamb’s problem. Geophys J Roy Astr Soc 37:99–131 13. Apsel RJ (1979) Dynamic Green’s functions for layered media and applications to boundary value problems, Ph.D. Dissertation, University of California, San Diego, CA 14. Kausel E (1981) An explicit solution for the Green functions for dynamic loads in layered media, MIT Research Report R 81–13, Cambridge, MA 15. Davies TG, Banerjee PK (1983) Elastodynamic Green’s function for half-space, Report GT/1983/1, Department of Civil Engineering, State University of New York at Buffalo 16. Kobayashi S, Nishimura N (1980) Green’s tensors for elastic half-space – an application of boundary integral equation method. Memories-Fac Eng Kyoto Univ XLII:228–241 17. Banerjee PK, Mamoon SM (1990) A fundamental solution due to a periodic point force in the interior of an elastic half-space. Int J Earthquake Eng Struct Dyn 19:91–105 18. Hamidzadeh HR (1978) Dynamics of rigid foundations on the surface of an elastic half-space, Ph.D. Dissertation, University of London, England 19. Hamidzadeh HR (1986) Surface vibration of an elastic half-space. Proc SECTAM XIII 2:637–642 20. Hamidzadeh HR, Chandler DE (1991) Elastic waves on semi-infinite solid due to a harmonic vertical surface loading. Proc Canadian Cong Appl Mech 1:370–371 21. Reissner E (1936) Stationare, axialsymmetrische durch eine schuttelnde masseerregte schwingungen eines homogenen elastischen habraumes. Ingenieur Archiv 7(6):381–397 22. Reissner E, Sagoci HF (1944) Forced torsional oscillations of an elastic half-space. J Appl Phys 15:652–662 23. Miller GF, Pursey H (1954) The field and radiation impedance of mechanical radiators on the free surface of a semi-infinite isotropic solid. Proc R Soc Lond A 223:521–541 24. Miller GF, Pursey H (1955) On the partition of energy between elastic waves in a semi-infinite solid. Proc R Soc Lond A 233:55–69 25. Quinlan PM (1953) The elastic theory of soil dynamics. Symp Dyn Test Soils ASTM STP 156:3–34 26. Sung TY (1953) Vibration in semi-infinite solids due to periodic surface loadings. Symp Dyn Test Soils ASTM STP 156:35–64 27. Arnold RN, Bycroft GN, Warburton GB (1955) Forced vibrations of a body in an infinite elastic solid. J Appl Mech 22:391–400 28. Bycroft GN (1956) Forced vibrations of circular plate on a semi-infinite elastic space and on an elastic stratum. Phil Trans R Soc A, 248(948):327–368 29. Bycroft GN (1959) Machine foundation vibration. Proc Inst Mech Eng 173:18 30. Thomson WT, Kobori T (1963) Dynamic compliance of rectangular foundations on an elastic half-space. J Appl Mech Trans ASME 30:579–584 31. Kobori T, Minai R, Suzuki T, Kusakabe K (1966) Dynamical ground compliance of rectangular foundations. In: Proceedings of 16th Japan national congress on applied mechanical engineering, pp 301–315 32. Kobori T, Minai R, Suzuki T (1966) Dynamic ground compliance of rectangular foundation on an elastic stratum. In: Proceedings of 2nd Japan national symposium on earthquake engineering, pp 261–266 33. Kobori T, Minai R, Suzuki T, Kusakabe K (1968) Dynamic ground compliance of rectangular foundation on a semi-infinite viscoelastic medium. Annual Report, Disaster Prevention Research Institute of Kyoto University, vol 11A. 349–367

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34. Kobori T, Suzuki T (1970) Foundation vibrations on a viscoelastic multilayered medium. In: Proceedings of the 3rd Japan earthquake engineering symposium, Tokyo, pp 493–499 35. Kobori T, Minai R, Suzuki T (1971) The dynamical ground compliance of a rectangular foundation on a viscoelastic stratum. Bull Disas Prev Res Inst Kyoto Univ 20:289–329 36. Harding JW, Sneddon IN (1945) The elastic stresses produced by the indentation of the plane surface of a semi-infinite solid by a rigid punch. Proc Camb Phil Soc 41:16–26 37. Sneddon IN (1959) Fourier transform. McGraw Hill, New York 38. Awojobi AO, Grootenhuis P (1965) Vibration of rigid bodies on semi-infinite elastic media. Proc R Soc A 287:27–63 39. Awojobi AO (1966) Harmonic rocking of a rigid rectangular body on a semi-infinite elastic medium. J Appl Mech ASME 33:547–552 40. Awojobi AO (1969) Torsional vibration of a rigid circular body on an infinite elastic stratum. Int J Solids Struct 5:369–378 41. Awojobi AO (1972) Vertical vibration of a rigid circular body and harmonic rocking of a rigid rectangular body on an elastic stratum. Int J Solids Struct 8:759–774 42. Awojobi AO (1972) Vertical vibration of a rigid circular foundation on Gibson soil. Geotechnique 22(2):333–343 43. Titchmarsh EC (1937) Introduction to the theory of Fourier integrals. Oxford, New York 44. Robertson IA (1966) Forced vertical vibration of a rigid circular disc on a semi-infinite elastic solid. Proc Camb Philos Soc 62A:547–553 45. Robertson IA (1967) On a proposed determination of the shear modules of an isotropic, elastic half-space by the forced torsional oscillations of a circular disc. Appl Sci Res 17:305–312 46. Gladwell GML (1968) The calculation of mechanical impedances relating to an indenter vibrating on the surface of a semi-infinite elastic body. J Sound Vib 8:215–228 47. Gladwell GML (1968) Forced tangential and rotatory vibration of a rigid circular disc on a semi-infinite solid. Int J Eng Sci 6:591–607 48. Gladwell GML (1969) The forced torsional vibration of an elastic stratum. Int J Eng Sci 7:1011–1024 49. Karasudhi P, Keer LM, Lee SL (1968) Vibratory motion of a body on an elastic half-space. J Appl Mech 35:697–705 50. Housner GW, Castellani A (1969) Discussion of “Comparison of footing vibration tests with theory” by F.E. Richart, Jr. and R.V. Whitman. J SMFD ASCE 95:360–364 51. Richardson JD (1969) Forced vibrations of rigid bodies on a semi-infinite elastic medium. Ph.D. Thesis, University of Nottingham 52. Luco JE, Westmann RA (1971) Dynamic response of circular footings. J EMD ASCE 97:1381–1395 53. Luco JE, Westmann RA (1972) Dynamic response of a rigid footing bonded to an elastic half-space. J Appl Mech ASME 39:527–534 54. Veletsos AS, Wei YT (1971) Lateral and rocking vibration of footings. J SMFD ASCE 97:1227–1248 55. Bycroft GN (1977) Soil-structure interaction at higher frequency factors. Int J Earthquake Eng Struct Dyn 5:235–248 56. Veletsos AS, Verbic B (1974) Basic response functions for elastic foundation. J EMD ASCE 100:189–201 57. Clemmet JF (1974) Dynamic response of structures on elastic media. Ph.D. Thesis, Nottingham University 58. Luco JE (1976) Vibrations of a rigid disc on a layered visco-elastic medium. Nuclear Eng Des 36:325–340 59. Tabiowo PH (1973) Vertical vibration of rigid bodies with rectangular bases on elastic media. Ph.D. Thesis, University of Lagos, Nigeria 60. Awojobi AO, Tabiowo PH (1976) Vertical vibration of rigid bodies with rectangular bases on elastic media. Int J Earthquake Eng Struct Dyn 4:439–454 61. Wong HL, Luco JE (1976) Dynamic response of rigid foundations of arbitrary shape. Int J Earthquake Eng Struct Dyn 4:579–587

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62. Wong HL, Luco JE (1978) Table of impedance functions and input motions for rectangular foundations, Report CE 78–15, Department of Civil Engineering, USC, Los Angeles, CA 63. Hamidzadeh HR, Grootenhuis G (1981) The dynamics of a rigid foundation on the surface of an elastic half-space. Int J Earthquake Eng Struct Dyn 9:501–515 64. Rucker W (1982) Dynamic behavior of rigid foundations of arbitrary shape on a half-space. Int J Earthquake Eng Struct Dyn 10:675–690 65. Chow YK (1987) Vertical vibration of three-dimensional rigid foundations on layered media. Int J Earthquake Eng Struct Dyn 15:585–594 66. Triantafyllidis T (1986) Dynamic stiffness of rigid rectangular foundations on the half-space. Int J Earthquake Eng Struct Dyn 14:391–411 67. Hsieh TK (1962) Foundation vibration. Proc Inst Civ Eng 22:211–226 68. Lysmer J (1965) Vertical motion of rigid footings. Department of Civil Engineering, University of Michigan, Report to WES Contract Report, vol 3. p 115 69. Lysmer J, Richart FE Jr (1966) Dynamic response of footings to vertical loading. J SMFD Proc ASCE 92:65–91 70. Weissmann GF (1966) A mathematical model of a vibrating soil-foundation system. Bell Syst Tech J 45(1):177–228 71. Whitman RV, Richart FE (1967) Design procedures for dynamic loaded foundations. J SMFD ASCE 93:169–193 72. Hall JR Jr (1967) Coupled rocking and sliding oscillations of rigid circular footings. In: Proceedings of International Symposium on Wave Propagation and Dynamic Properties of Earth Materials, University of New Mexico, Albuquerque, NM, pp 139–149 73. Roesset JM, Whitman RV, Dobry R (1973) Model analysis for structures with foundation interaction. J STD ASCE 99:399–416 74. Oner M, Janbu N (1975) Dynamic soil-structure interaction in offshore storage tank. In: Proceedings of the international conference on soil mechanics and foundation engineering, Istanbul, March 1975 75. Hall JR Jr, Kissenpfenning JF, Rizzo PC (1975) Continuum and finite element analyses for soil–structure interaction analysis of deeply embedded foundations. In: 3rd international conference on structural mechanics in reactor technology, vol 4. Part K, Paper K 2/4 76. Veletsos AS (1975) Dynamics of structure-foundation systems. In: Proceedings of the symposium on structural and geotechnical mechanics, Honoring Newmark NM, University of Illinois, pp 333–361 77. Wolf JP (1985) Dynamic soil–structure interaction, Prentice-Hall, Englewood Cliffs, NJ 78. Wolf JP, Somaini DR (1986) Approximate dynamic model of embedded foundation in time domain. Int J Earthquake Eng Struct Dyn 14:683–703 79. Dobry R, Gazetas G (1986) Dynamic response of arbitrary shaped foundation. J Geotech Eng ASCE 112:109–135 80. Dobry R, Gazetas G, Strohoe KH (1986) Dynamic response of arbitrary shaped foundation II. J Geotech Eng ASCE 112:136–154 81. Terzaghi K (1943) Theoretical soil mechanics. Wiley, New York 82. Terzaghi K (1955) Evaluation of coefficients of subgrade reaction. Geotechnique 5:297–326 83. Girard J (1968) Vibrations des massifs sur supports elastiques. Ann Inst Tech Batiment et Trayaux Publics 23–24:407–425 84. Duns CS, Butterfield R (1967) The dynamic analysis of soil–structure system using the finite element method. In: Proceedings of the international symposium on wave propagation and dynamic properties of earth materials, University of New Mexico, Albuquerque, NM, pp 615–631 85. Lysmer J, Kuhlemeyer RL (1971) Closure to finite dynamic model for infinite media. J EMD ASCE 97:129–131 86. Seed HB, Lysmer J, Whitman RV (1975) Soil structure interaction effects on the design of nuclear power plants. In: Proceedings of the symposium on structural and geotechnical mechanics, Honoring N.M. Newmark, University of Illinois, 220–241 87. Day SM, Frazier GA (1979) Seismic response of hemispherical foundation. J Eng Mech Div ASCE 105:29–41

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88. Bettess P, Zienkiewicz OC (1977) Diffraction and refraction of surface waves using finite and infinite elements. Int J Numer Methods Eng 11:1271–1290 89. Roesset JM, Ettouney MM (1977) Transmitting boundaries: a comparison. Int J Numer Anal Methods Geomech 1:151–176 90. Kausel E, Tassoulas JL (1981) Transmitting boundaries: a closed form comparison. Bull Seism Soc Am 71:143–159 91. Chuhan Z, Chongbin Z (1987) Coupling method of finite and infinite elements for strip foundation wave problems. Int J Earthquake Eng Struct Dyn 15:839–851 92. Dominguez J (1978) Dynamic stiffness of rectangular foundation, Report R78–20, Department of Civil Engineering, MIT, Cambridge, MA 93. Dominguez J, Roesset JM (1978) Dynamic stiffness of rectangular foundations, Report R78–20, Department of Civil Engineering, MIT, Cambridge, MA 94. Karabalis DI, Beskos DE (1984) Dynamic response of 3-D rigid surface foundations by time domain boundary element, Int J Earthquake Eng Struct Dyn 12:73–93 95. Spyrakos CC, Beskos DE (1986) Dynamic response of rigid strip foundation by time domain boundary element method. Int J Numer Method Eng 23:1547–1565 96. Spyrakos CC, Beskos DE (1986) Dynamic response of flexible strip foundation by boundary and finite elements. Soil Dyn Earthquake Eng 5:84–96 97. Luco JE, Apsel RJ (1983) On the Green’s function for a layered half space parts I & II. Bull Seism Soc Am 73:909–929, 931–951 98. Chapel F, Tsakalidis C (1985) Computation of the Green’s functions of elastodynamics for a layered half-space through a Hankel transform – applications to foundation vibration and seismology. In: Proceedings of the numerical methods in geomechanics, Nagoya, Japan, pp 1311–1318 99. Kausel E, Roesset JM (1975) Dynamic stiffness of circular foundations. J Eng Mech Div ASCE 111:771–785 100. Israil ASM, Ahmad S (1989) Dynamic vertical compliance of strip foundations in layered soils. Int J Earthquake Eng Struct Dyn 18:933–950 101. Richart FE Jr, Whitman RV (1967) Comparison of footing vibration test with theory. J SMFD ASCE 93:143–168 102. Fry ZB (1963) Development and evaluation of soil bearing capacity, foundation of structures. WES Technical Report No. 3, 632 103. Ratay RT (1971) Sliding-rocking vibration of body on elastic medium. J SMFD ASCE 97:177–192 104. Beredugo JO, Novak M (1972) Coupled horizontal and rocking vibration of embedded footings. Can Geotech J 9(4):477–497 105. Krizek RO, Gupta DC, Parmelee RA (1972) Coupled sliding and rocking of embedded foundations. J SMFD ASCE 98:1347–1358 106. Wolf JP (1975) Approximate soil-structure interaction with separation of base mat from soil lifting-off. In: 3rd international conference on structural mechanics in reactor technology, vol 4. park K, pp. K 3/6 107. Hamidzadeh HR, Minor GR (1993) Horizontal and rocking vibration of foundation on an elastic half-space. Proc Canadian Cong Appl Mech 2:525–526 108. Iljitchov VA (1967) Towards the soil transmission of vibrations from one foundation to another. In: Proceedings of the international symposium on wave propagation and dynamic properties of earth materials, University of New Mexico, Albuquerque, NM, 641–654 109. Richardson JD, Webster JJ, Warburton GB (1971) The response on the surface of an elastic half-space near to a harmonically excited mass. J Sound Vib 14:307–316 110. Warburton GB, Richardson JD, Webster JJ (1971) Forced vibrations of two masses on an elastic half-space. J Appl Mech ASME 38:148–156 111. Warburton GB, Richardson JD, Webster JJ (1972) Harmonic response of masses on an elastic half-space. J EI ASME 94:193–200 112. Lee TH, Wesley DA (1973) Soil–structure interaction of nuclear reactor structure considering through-soil coupling between adjacent structures. Nucl Eng Des 24:374–387

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113. MacCalden PB, Matthiesen RB (1973) Coupled response of two foundations. In: 5th world conference on earthquake engineering, Rome, 1913–1922 114. Snyder MD, Shaw DE, Hall JR Jr (1975) Structure–soil–structure interaction of nuclear structures. In: 3rd international conference on structural mechanics in reactor technology, vol 4. Park K, K2/9 115. Awojobi AO (1964) Vibrations of rigid bodies on elastic media, Ph.D. Thesis, University of London 116. Jones R (1958) In-situ measurement of the dynamic properties of soil by vibration method. Geotechnique 8(1):1–21 117. Jones R (1959) Interpretation of surface vibrations measurements. In: Proceedings of the symposium on vibration testing of road and runways, Koninklijke/Shell-Laboratorium, Amsterdam 118. Maxwell AA, Fry ZB (1967) A procedure for determining elastic moduli of in-situ soils by dynamic technique. In: Proceedings of the international symposium on wave propagation and dynamic properties of earth materials, University of New Mexico, Albuquerque, NM, pp 913–920 119. Stokoe KH, Richart FE Jr (1974) Dynamic response of embedded machine foundations. J GTD ASCE 100(GT4):427–447, Proc. Paper 10499 120. Beeston HE, McEvilly TV (1977) Shear wave velocities from down hole measurements. Int J Earthquake Eng Struct Dyn 5:181–190 121. Dawance G, Guillot M (1963) Vibration des massifs de foundations de machines. Ann Inst Tech Batiment et Travaux Publics, 116(185):512–531 122. Grootenhuis P, Awojobi AO (1965) The in-situ measurement of the dynamic properties of soils. Proceedings of symposium vibration in civil engineering, Institute of Civil Engineering, pp 181–187 123. Hardin BO, Drnevich VP (1972) Shear modulus and damping in soils. I. Measurement and parameter effects. J SMFD ASCE 98:603–624 124. Hardin BO, Drnevich VP (1972) Shear modulus and damping in soils. II. Design equations and curves. J SMFD ASCE 98:667–692 125. Cunny RW, Fry ZB (1973) Vibratory in situ and laboratory soil moduli compared. J SMFD ASCE 99:1055–1076 126. Lawrence FV Jr (1965) Ultrasonic shear wave velocities in sand and clay. Report R65–05., WES, Department of Civil Engineering, MIT, Cambridge, MA 127. Theirs GR, Seed HB (1968) Cyclic stress–strain characteristics of clay. J SMFD ASCE 94:555–569 128. Kovacs WD, Seed HB, Chan CK (1971) Dynamic moduli and damping ratios for a soft clay. J SMFD ASCE 97:59–75 129. Kanai K, Yoshizawa S (1961) On the period and the damping of vibration in actual buildings. BERI 39:477 130. Eastwood W (1953) Vibrations in foundations. Struct Eng 82:82–98 131. Chae YS (1967) The material constants of soils as determined from dynamic testing. In: Proceedings of the international symposium on wave propagation and dynamic properties of earth materials, University of New Mexico, Albuquerque, NM, 759–771 132. Hamidzadeh HR (1987) Dynamics of foundation on a simulated elastic half-space. Proc Intl Symp Geotech Eng Soft Soils 1:339–345 133. Luco JE, Trifunac MD, Wong HL (1987) On the apparent changes in dynamic behavior of a nine-story reinforced concrete building. Bull Seism Soc Am 77:1961–1983 134. Luco JE, Trifunac MD, Wong HL (1988) Isolation of soil–structure interaction effects by full-scale forced vibration tests. Int J Earthquake Eng Struct Dyn 16:1–21 135. Wong HL, Luco JE, Trifunac MD (1977) Contact stresses and ground motion generated by soil–structure interaction. Int J Earthquake Eng Struct Dyn 5:67–79

Chapter 27

Inversely Found Elastic and Dimensional Properties Darryl K. Stoyko, Neil Popplewell, and Arvind H. Shah

Abstract The ability to simultaneously measure a homogeneous, isotropic pipe’s elastic properties and wall thickness from its known mass density, outer diameter and the cut-off frequencies of three ultrasonic guided wave modes is demonstrated for a typical steel pipe. This inverse procedure is based upon simulated results computed by using an efficient Semi-Analytical Finite Element (SAFE) forward solver. The Young’s modulus, shear modulus, and wall thickness agree very well with those found from conventional but destructive experiments. On the other hand, Poisson ratios agree within their assessed uncertainties.

27.1 Introduction Material and dimensional information constitute fundamental knowledge for assessing the current behaviour or “health” of a structure. From a practical perspective, in situ measurements should be used that are quick, reliable and non-destructive. An ultrasonic based approach is one plausible candidate. Indeed ultrasonic body waves are employed commonly to accurately measure fine dimensions [8]. Single or “focused guided waves,” on the other hand, can propagate over tens of metres so they have been used to remotely interrogate inaccessible locations [2–4, 7, 10, 13, 15]. Procedures which employ guided waves are attractive because their multi-modal and dispersive behaviour can simultaneously provide information over a range of frequencies [14]. The behaviour of a single, essentially non-dispersive mode is interpreted relatively easily [3, 10], but it is difficult to implement. Even if excited, a single mode is likely converted to additional modes at geometrical discontinuities [3, 10]. These modes are generally dispersive so that the nature of a propagating wave packet changes as it travels along a structure. The objective here is to develop a procedure involving several guided waves, that can be automated to N. Popplewell () Mechanical and Manufacturing Engineering, University of Manitoba, 15 Gillson Street, Winnipeg, MB, Canada R3T 5V6 e-mail: [email protected]

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 27,

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display material and dimensional data and can be extended, in the future, to indicate a structure’s condition. Although the procedure could be applied to any plate-like structure, it is illustrated by using a homogeneous, isotropic steel pipe. Such pipes are employed ubiquitously in industry [1]. Defining a pipe’s unknown character from its measured response to a specified excitation is an example of an inverse procedure. Even a computational inversion procedure, in which errors are generally less than those arising from experimental measurements, may not produce a unique solution [9]. Moreover, an inversion is often based upon a more computationally efficient forward solver [9]. (A forward solver determines the response when both the excitation and pipe’s character are known.) This indirect approach is adopted here as an uncertainty analysis can be also performed straightforwardly. The forward solver is based upon a standard SemiAnalytical Finite Element (SAFE) formulation [21]. Therefore, the novelty lies not in the numerical technique but in the insight gained from an original presentation and physical interpretation of the results. On the other hand, the inherent modal decomposition produced by SAFE is crucial to understanding why simple features of a pipe’s temporal and corresponding frequency behaviour can be exploited in the inverse problem. An overriding concern is that a pipe’s properties should be measured as simply as possible. Therefore, a short duration excitation is applied radially at an easily accessible external surface of the pipe. No effort is made to avoid dispersive wave modes in contrast to common practice. The modes are received at a single off-set transducer, which is linked to a computer processing capability. Both the transmitting and receiving transducers’ dimensions are assumed to be much smaller than the excited modes’ predominant wavelengths. Therefore, they are idealized as acting at points. Software incorporates a Discrete Fourier Transform (DFT) whose output is employed as input to a curve fitting scheme for the receiving transducer’s temporal signal. The aim of this contorted procedure is to refine the frequency values of only the predominant modal contributions in the received response [19]. These values correspond to the pipe’s cut-off frequencies, which are common to all non-nodal locations [6]. Therefore, the choice of measurement location is relatively unimportant. However, the measured cut-off frequencies still have to be reconciled with their SAFE counterparts. This task is accomplished by taking the “true” set of pipe properties as the one for which three measured and computed cut-off frequencies are closest. Likely uncertainties are estimated from a sensitivity assessment around the selected set of properties. Agreement is shown to be generally good with classically but more tediously performed destructive experiments.

27.2 Computational Overview The SAFE forward solver provides the computational foundation so that it is outlined first. Its use in finding an inverse solution is described later. An infinitely long pipe, half of which is illustrated in Fig. 27.1, is considered. The pipe is assumed

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Fig. 27.1 A pipe’s discretization

to be uniformly right circular, homogeneous, linearly elastic and isotropic. It has Lam´e constants and , density , a constant mean radius R, outside radius r0 and thickness H , in addition to traction free, inner and outer surfaces. Right hand cylindrical and Cartesian coordinate systems .r; ; z/ and .x; y; z/, respectively, are shown in Fig. 27.1. Their common origin is located at the geometric centre of a generic cross-section of the pipe with the z axis directed along the pipe’s longitudinal (axial) axis. The point excitation, Ft .; z; t/, is applied normally to the external surface at y D 0 in the plane z D 0 by the transmitting transducer. (In the cylindrical coordinate system, the excitation’s application coincides with D 0.) To circumvent convergence difficulties associated with a point application, the excitation is approximated by using a “narrow” pulse having a uniform amplitude over a circumferential distance 2r0 0 . This narrow pulse is represented by using a Fourier series of “ring-like” loads having separable spatial and time, t, variations. In particular, ( F .; z; t/ D F0 p.t/•./•.z/ t

D

1 X

p.t / •.z/F0 ; 2r0 0

0;

sinc.n0 / jn e •.z/F0 p.t/; 2r0 nD1

0 0 otherwise (27.1)

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where use has been made of the Fourier series for a rectangular pulse. In (27.1) F0 and p.t/ are a vector and a function that describe the radial and temporal variations of the excitation, respectively, n is the circumferential wave-number, ı is the Dirac p delta function, and j D 1. The F0 is a vector of zeros except for a single element corresponding to the excitation’s specified position and direction. Application of the Fourier transform integral to the series in (27.1) transforms the excitation vector from the axial, z, domain to the wave-number, k, domain. The result is: Ft .; k; t/

1 X sinc.n0 / jn e F0 p.t/; 2r0 nD1

(27.2)

in which the “sifting” property of the Dirac delta function has been applied. In (27.1) and (27.2) p.t/ is taken as the commonly used Gaussian modulated sine wave, which has a non-dimensional form (always indicated by a star superscript) of H p.t/ D p .t/ D

(

0; t <0 2 ea.ht / sin.h!0 t/; t 0:

(27.3)

The constant a, , !0 , and h are: a D 2:29595 1010 s2 ; D 1:4 10

5

s;

!0 D .5 10 / rad=s; and 5

h D 0:28

(27.4a) (27.4b) (27.4c) (27.4d)

here. Consequently the excitation has a 70 kHz centre frequency and over 99% of its energy is contained within a 35–107 kHz bandwidth. Therefore, the Fourier integral transform of p .t/, pN .!/ where ! is the circular frequency, may be reasonably assumed to be contained within this finite bandwidth. The adopted p .t/ and jpN .!/j, where an over bar indicates a Fourier transformed variable, are illustrated in Fig. 27.2. Having described the excitation, its effect on the pipe has to be considered next. The pipe is discretized into N layers through its thickness, where N is six in Fig. 27.1. The thickness of the kth layer is Hk , and it extends radially from rk to rkC1 . For simplicity, the Hk are considered to be identical. Each layer corresponds to a one-dimensional finite element in the pipe’s radial direction for which a quadratic interpolation function is assumed. A conventional finite element approach is applied, layer by layer, to approximate the elastic equations of motion [21] in which the displacements, ut .r; ; z; t/, take the form ut .r; ; z; t/ D N.r/Ut .; z; t/:

(27.5)

The N.r/ contains the set of interpolation functions assembled over the entire pipe. On the other hand, Ut .; z; t/ assimilates the corresponding array of nodal

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345

Fig. 27.2 Applied excitation in (a) time and (b) frequency

displacements in which the easily measured radial displacement at the pipe’s external surface is principally of interest. Like the similarly approximated excitation Ft .; z; t/, the Ut .; z; t/ is assumed to be circumferentially periodic, i.e., Ut .; z; t/ D

1 X

ejn Utn .z; t/:

(27.6)

nD1

Consider, on the other hand, a single temporally harmonic component of Ft .; z; t/, F.; z; t/, having (circular) frequency !. This excitation component produces the harmonic response component U.; z; t/. The Fourier series of these two variables take the form: U.; z; t/ D ej!t

1 P

ejn Un .z/

(27.7a)

ejn Fn .z/:

(27.7b)

nD1

and F.; z; t/ D ej!t

1 P nD1

Equations (27.7a) and (27.7b) are substituted into approximate equations of motion obtained from Hamilton’s principle [21]. The result is transformed into the

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wave-number domain by applying the Fourier integral transform and making use of (27.2). The result for the nth circumferential wave-number is:

N n C jnkn .K3 jnK5 / U Nn K1 C jnK2 C n2 K4 ! 2 M U 2 N n D FN n ; C kn K6 U

(27.8)

where the Ki are stiffness matrices and M is the mass matrix. Details are given in [21]. Proceeding in a classical modal analysis fashion, (27.8) takes the form of an eigensystem in the special case when FN n is the null vector. Integer values are always assigned to n. Either kn or ! is assigned when the wave-number or frequency is presumed. Then a linear or quadratic eigensystem is produced in ! 2 or kn , respectively. In the latter, more commonly encountered case, (27.8) may be rewritten in the linear form: Nn 0 U D N ; (27.9) ŒA.n; !/ kn B N Fn kn Un where: A.n; !/ D

0 I ; (27.10a) K1 C jnK2 C n2 K4 ! 2 M jn .K3 jnK5 / I 0 BD (27.10b) 0 K6

and 0 (I) is the null (identity) matrix. Normal modes are found for the nth circumferential wave-number by solving the homogeneous form of (27.9). This results in 12N C 6 eigenvalues or axial wave-numbers. A real (complex) valued wave-number corresponds to a propagating (evanescent) wave. Moreover, half the wave-numbers correspond to solutions for the positive z coordinates; the other half represent solutions for the negative z R coordinates. In addition to the wave-numbers, right and left eigenvectors, nm and L nm respectively, are associated with the mth eigenvalue. They are partitioned into the upper and lower halves. ˚ R R R T nm D nmu nml (27.11a) and T ˚ L L L nm D nmu nml

(27.11b)

that are represented by the subscripts u and l, respectively. The pipe’s response is obtained, for the nth circumferential wave-number and only those axial cross sections having positive z, by linearly superimposing the admissible 6N C3 right eigenvector solutions. Applying first the inverse Fourier transform to this sum, and then Cauchy’s residue theorem, gives the nth circumferential mode of the response. The result is [21]:

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Un .; z; t/ D e

j!t

where

6N C3 L T F0 R jknm z jsinc.n0 / X nml nmu e ; 2 r0 B nm mD1

L T R Bnm •mp D nm Bnp ;

347

(27.12)

(27.13)

in which bi-orthogonality relations [20] have been used. Moreover, •mp is the Kronecker delta. The linear response to a multi-frequency excitation can be found by merely superimposing the responses caused by each individual frequency component. Hence: j U .; z; t/ D 4 2 r0 t

Z

1

j!t p.!/e N # "6N C3 1 L T X X nml .!/F0 R nmu .!/ejknm .!/z ejn d! sinc.n0 / B .!/ nm nD1 mD1 1

(27.14) after summing all the circumferential harmonic components.

27.3 Simplifying Features It is demonstrated in [17,18], by using the modal decomposition capability of SAFE, that the “peak” magnitudes of a pipe’s radial Frequency Response Function (FRF) occur at its modal cut-off frequencies where the wave number is zero. A sharp increase there arises because the corresponding Bnm in (27.12) and (27.13) tends to zero as a cut-off frequency is approached, i.e., a singularity happens in the nm mode’s FRF. Although details are omitted here for brevity, this feature exists because there is a repeated root at each cut-off frequency. Hence a defective1 eigensystem exists. Consequently the corresponding left and right eigenvectors are orthogonal to the B matrix defined in (27.10) and Bnm also becomes zero [20]. It is interesting that a cut-off frequency can be interpreted as a juncture at which a travelling wave problem transitions to a vibration problem because the wavelength, 2 =knm , becomes infinitely long. Therefore, the response at a cut-off frequency may be termed “vibration”-like [6, 12]. Also note from (27.14) that a modal response at a cut-off frequency becomes advantageously independent of an observation point’s axial location. Moreover, cut-off frequencies can be calculated without knowing the corresponding eigenvectors of (27.8) with kn D 0 and FN n D 0. Cut-off frequencies depend, through the stiffness and mass matrices, on the pipe’s elastic properties, mass density, and geometrical dimensions. It is assumed

1 A defective eigensystem is one in which an eigenvalue is repeated, say integer r times, but fewer than r unique (right) eigenvectors exist for the repeated eigenvalue [20].

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here that the pipe’s outer diameter and mass density are readily available, which leaves the elastic properties and wall thickness to be determined. Two independent elastic constants are sufficient to characterize a homogeneous isotropic material. The Buckingham theorem [5] is used to reduce the dimensional space and make calculations more tractable. Then, for a given circumferential wave-number n and order m, the non-dimensional cut-off frequency ratio c D !F.n;m/

c !F.n;m/

!ref

(27.15)

p c is the cut-off frequency of is defined where !ref D 1=H .= / and !F.n;m/ c can be expressed in terms of the nonthe F.n; m/ flexural mode.2 The !F.n;m/ dimensional parameters .H=R/ and .=/. Note that the desired elastic properties and wall thickness can be calculated from .H=R/, .=/, and !ref , as well as the presumed mass density and outer diameter [18, 19]. The inequality 0 < .H=R/ < 2 arises physically because the lower and upper bounds relate to a pipe having no wall thickness and a solid pipe, respectively. On the other hand, .=/ can be bounded reasonably as 0 .=/ . 10 from the standard elasticity relation 0 0:5, where is Poisson’s ratio.

27.4 Graphical Relations Between the Cut-Off Frequencies and Cylinder Properties The key to making the inverse problem tractable is a succinct yet clear presentation showing the dependence of the forward solved cut-off frequencies upon the independent !ref , .H=R/ and .=/. Transformations to obtain practical engineering properties, which also require the use of f D !=.2 / where f is frequency, are performed later. The presentation’s construction may be envisaged by initially considering all the , H , , and !ref to be unity. Then and R are each varied uniformly within the physically viable ranges described earlier. Forward computations c c , !F.11;1/ and to determine three non-dimensional cut-off frequencies, say !F.10;1/ c !F.12;1/ , are performed within these ranges by SAFE. The corresponding values of .H=R/ and .=/, given !ref is simply 1 rad=s, are also noted. In general c D !F.i;1/

c !F.i;1/

!ref

for all i;

(27.16)

c c so that a particular !F.i;1/ is identical to !F.i;1/ at this juncture. Moreover c c c !F.10;1/ !F.11;1/ !F.12;1/

(27.17)

2 Modes are labelled by using the standard convention employed in [16]. Only flexural modes are considered here although the extension to torsional or longitudinal modes is obvious.

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c c regardless of the value of !ref . Then the largest of the three !F.i;1/ , !F.12;1/ , is c taken as an extreme but arbitrary 200;000 rad=s (i.e., fF.12;1/ is 100 kHz). The non-dimensional cut-off frequency for the selected values of .H=R/ and .=/ are available so that equation (27.16) is used to find the corresponding !ref . The asc c sociated values of !F.10;1/ and !F.11;1/ can be determined similarly. Two points

c c c !F.10;1/ ; !F.11;1/ ; !F.12;1/ , computed for the assumed (unity) and calculated !ref , are converted to frequencies, in kHz, and graphed. They are joined by a line along which .H=R/ and .=/ are each constant, but !ref varies. The line must also pass c and !ref are zero there. through the graph’s origin because, physically, the !F.i;1/ The effects of other variations in !ref are calculated straightforwardly. The same procedure produces a set of similar, closely spaced lines after perturbing .H=R/ and .=/. The resulting overall behaviour is presented in Fig. 27.3. c (or, alThe shaded solution space of Fig. 27.3a shows the dependence of !F.12;1/ c c c c c ternatively fF.12;1/ ) upon !F.10;1/ (fF.10;1/ ) and !F.11;1/ (fF.11;1/ ). Note that three cut-off frequencies are required to determine the three unknown .H=R/, .=/ and !ref . The solution space seems to be a narrow bounded plane whose width increases c progressively with deepening shades, i.e., higher fF.12;1/ . This is somewhat deceptive, however, because a computed inverse solution has been found empirically to

a

b

c

d

Fig. 27.3 Dependence of the three cut-off frequencies on (a) each other, (b) fref D !ref =2 , (c) .H=R/, and (d) .=/. A dashed line indicates a boundary

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exist only in a thin volume, not on a planar approximation. Previous research [18,19] supports this contention because it was determined that cut-off frequencies should be measured ideally within 0.01%. The shadings of Figs. 27.3b and 27.3c indicate that the cut-off frequencies depend strongly upon fref or !ref and .H=R/, particularly at their highest values. Interestingly, a quite uniform shading emerges when these two figures are superimposed. Therefore, similar changes in fref and .H=R/ counterbalance. On the other hand, the sizeable swathes of a given shading seen in Fig. 27.3d suggest that the c , i D 10; 11; 12, alter little with large .=/ modifications. A comparison fF.i;1/ of Fig. 27.3c, d also intimates that shadings across the shorter width of the solution space have similar tendencies for the .=/ and smaller .H=R/ variations. Therefore, the effect of .=/ may be concealed, to some extent, by a greater one from .H=R/. Figure 27.4 is a magnification of the computed solution space and nearby regions contained within the boxes shown in Fig. 27.3. The illustrated points, which are offset slightly from the surface, correspond to the cut-off frequencies extracted from simulated and experimentally measured time histories, to be presented later. The off-sets arise from uncertainties and errors, the overall magnitudes of which are

a

b

c

d

Fig. 27.4 Magnified version of Fig. 27.3 near the experimental and simulated data; dependence of the three cut-off frequencies on (a) each other, (b) fref D !ref =2 , (c) (H/R), and (d) (=)

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351

intimated by the off-set’s shortest distance to the computed surface. The errors arise principally from the temporal curve fitting procedure to find the cut-off frequencies. Numerical studies [19] suggest that all the discernible cut-off frequencies in the measurement bandwidth should be incorporated to reduce the error. Uncertainties, on the other hand, may originate from any questionable assumption of SAFE like complete uniformity, no out-of-roundness, homogeneity, etc.

27.5 Inversion Scheme The transmitter is pulsed using the previously described excitation and the response c , is measured by the nearby receiving transducer. Three cut-off frequencies, !O F.n i ;mi / are “extracted” from the measured transient response using the temporal curve fitting procedure described in [19]. If these cut-off frequencies are exact and the SAFE modelling is perfect, the following relations hold: c c !O F.n D0 !ref !F.n i ;mi / i ;mi /

for i D 1; 2; 3;

(27.18)

c where !F.n are the corresponding non-dimensional cut-off frequencies i ;mi / predicted by SAFE. Equation (27.18) provides three relations in three unknowns [!ref , .H=R/, and .=/]. The solution of the non-linear equations provides a characterization of the pipe. Unlike the idealized simulation, experimental noise and errors can cause a measured point to lie outside the space spanned by the SAFE computer solutions, as seen in Fig. 27.4. To overcome this discrepancy, the point in the solution space “nearest” the measurement is sought. This point is located by minimizing the objective function: 3

2 X c c ! O : !ref !F.n D F.n ;m / ;m / i i i i

(27.19)

i D1

It is found by using the robust direct search method described in [11]. The c c and the nearest !ref !F.n are projections between the extracted !O F.n i ;mi / i ;mi / shown in Fig. 27.4 for an essentially precise numerical simulation and imprecise experimental data. Differences between the measured and predicted cut-off frequencies are employed to estimate the uncertainty in the recovered .H=R/, .=/ and !ref . First, c and the nearthe vector norm of the difference between the measured !O F.n i ;mi / c 3 est found !ref !F.ni ;mi / is computed. Then the variation in each .H=R/, .=/, and !ref required to produce a change in an individual cut-off frequency equal to the previously defined vector norm is computed, holding all but one of these three

3 If any single component of the difference vector is less than 100 Hz, it is replaced by 100 Hz. This appears from Table 2 of [19] to be a reasonably conservative upper bound of the uncertainty on the extracted cut-off frequencies.

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parameters constant, for each individual cut-off frequency. The largest permissible variation in each of the three parameters is termed the uncertainty. These uncertainties are propagated by using standard uncertainty estimation techniques for each of the variables derived from the three properties calculated by the inversion scheme.

27.6 Illustrative Examples Two examples are presented next that illustrate the inversion technique. A numerical simulation using a priori known material properties and dimensions is given first. This is followed by a real experimental example for a similar pipe.

27.6.1 Numerical Simulation An idealized 80-mm Diameter Nominal (DN), Schedule 40, seamless, carbon steel pipe is considered first. Its dimensional and material properties are summarized in Table 27.1. This particular pipe is selected because it is commercially important. At the end of 1997, for example, there was approximately 64,900 km of such pipe in industrial use as energy-related pipeline in Alberta, Canada [1]. Consequently it has been studied extensively as in, for example, [2, 3, 10]. The radial displacement is calculated on the pipe’s outer surface at D 0 and z D z=H D 5:1, by using (27.14) as described in [19]. The resulting time history is presented in Fig. 3a of [19]. The corresponding DFT and temporal curve fit are given in Figs. 3d and 3c, respectively, of [19]. Table 2 of [19] compares the cut-off frequencies obtained from the computed FRF, DFT, and temporal curve fit. The inversion procedure and uncertainty estimation are applied for the cut-off frequencies from the temporal curve fit. The results are summarized in Table 27.1. This table shows that the assigned and recovered material and dimensional properties generally agree within their estimated uncertainties.

Table 27.1 Comparing assigned values with those computed from inversion Property Assigned value Computed value Young’s modulus, E (GPa) 216:9 216 ˙ 2 Lam´e constant [Shear modulus], [G] (GPa) 84:3 84:5 ˙ 0:5 Lam´e constant, (GPa) 113:2 109 ˙ 4 Ratio of Lam´e constants, .=/ 1:34 1:29 ˙ 0:04 Poisson’s ratio, 0:287 0:282 ˙ 0:004 Outer diameter, D0 (mm) 88:8 – Thickness, H (mm) 5:59 5:596 ˙ 0:007 Mean radius, R (mm) 41:6 41:60 ˙ 0:05 Thickness to mean radius ratio, .H=R/ 0:134 0:1345 ˙ 0:0002 Mass density, (kg=m3 ) 7,932 –

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353

Table 27.2 Comparing conventionally measured values with those recovered ultrasonically Property Conventional approach Ultrasonic approach Young’s modulus, E (GPa) 202 ˙ 6 200 ˙ 10 Lam´e constant [Shear modulus], [G] (GPa) 79 ˙ 2 82 ˙ 3 Lam´e constant, (GPa) 99 ˙ 54 68 ˙ 8 Ratio of Lam´e constants, .=/ 1:3 ˙ 0:7 0:84 ˙ 0:06 Poisson’s ratio, 0:28 ˙ 0:07 0:228 ˙ 0:009 Outer diameter, D0 (mm) 88:80 ˙ 0:09 – Thickness, H (mm) 5:6 ˙ 0:1 5:59 ˙ 0:01 Mean radius, R (mm) 41:6 ˙ 0:1 41:60 ˙ 0:05 Thickness to mean radius ratio, .H=R/ 0:134˙0:003 0:1343 ˙ 0:0004 Mass density, (kg=m3 ) 7;700 ˙ 200 –

27.6.2 Experimental Corroboration An actual 80 mm DN, Schedule 40, seamless, carbon steel pipe was examined experimentally next. The radial displacement was measured on the pipe’s outer surface at 0 and z D z=H 5. The time history is presented in Fig. 4a, while the corresponding DFT and temporal curve fit are given in Figs. 4c and 4b, respectively, of [19]. The inversion procedure and uncertainty estimation were applied to the cut-off frequencies obtained from the temporal curve fit. Results from the ultrasonic measurements are summarized in Table 27.2. An 18 cm or so length was cut from one end of the pipe. Plates and grips were welded onto this short sample after which the sample was heat treated to relieve residual stresses. Two nominally identical, three-element strain gauge rosettes were bonded to the specimen. Then the instrumented sample was mounted in a standard pseudo-static test frame and loaded in either tension-compression or torsion. The tension-compression results were used to estimate Young’s modulus, E; the torsional test gave the shear modulus, G. These values are also reported in Table 27.2, along with the independently determined mass density, outer diameter, and wall thickness. It can be seen that E, G, and dimensional information found from the ultrasonic measurements correlate very well with those obtained pseudo-statically. However, the conventional, unlike the ultrasonic approach, calculates , .=/, and from the E and G values and standard elasticity relationships whose uncertainties, therefore, are amplified.

27.7 Conclusions The ability to simultaneously measure a homogeneous, isotropic pipe’s elastic properties and wall thickness by using the cut-off frequencies of three ultrasonic guided wave modes is demonstrated. Young’s modulus, the shear modulus, and wall thickness agree very well with values found conventionally. Poisson’s ratios agree within

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assessed uncertainties. However, dimensional uncertainties cannot be reflected realistically by the ultrasonic approach’s implicit assumption that dimensions do not change along or around a pipe. On the other hand, conventionally found dimensional uncertainties are more indicative of actual pipe variations as they are based upon direct measurements. Acknowledgments All three authors acknowledge the financial support from the Natural Science and Engineering Research Council (NSERC) of Canada. The first author also wishes to acknowledge financial aid from the Society of Automotive Engineers (SAE) International, University of Manitoba, Province of Manitoba, Ms. A. Toporeck, and the University of Manitoba Students’ Union (UMSU). The assistance of Messrs. B. Forzley, I. Penner, D. Rossong, V. Stoyko and Dr. S. Balakrishnan with preparation of the experimental sample is much appreciated. Recognition is given to Dr. M. Singh and Mr. A. Komus for the use and help with the load frame.

References 1. Alberta Energy and Utilities Board (1998) Pipeline performance in Alberta 1980–1997. Alta Energy and Util Board, Calgary, AB 2. Alleyne D, Cawley P (1996) Excitation of Lamb waves in pipes using dry-coupled piezoelectric transducers. J Nondestruct Eval 15:11–20 3. Alleyne D, Lowe M, Cawley P (1998) Reflection of guided waves from circumferential notches in pipes. J Appl Mech, Trans ASME 65(3):635–641 4. Alleyne D, Pavlakovic B, Lowe M et al. (2001) Rapid, long range inspection of chemical plant pipework using guided waves. In: Thompson D, Chimenti D, Poore L (eds) Review of progress in quantitative nondestructive evaluation, vol 20. American Institute of Physics, Melville, NY, pp 180–187 5. Buckingham E (1914) On physically similar systems: illustrations of the use of dimensional equations. Phys Rev 4:345–376 6. Gazis D (1959) Three-dimensional investigation of propagation of waves in hollow circular cylinders. J Acoust Soc Am 31(5):568–578 7. Hay T, Rose J (2002) Flexible PVDF comb transducers for excitation of axisymmetric guided waves in pipe. Sens Actuators A (Phys) A100 (1):18–23 8. Krautkr¨amer J, Krautkr¨amer H (1977) Ultrasonic testing of materials, 2nd edn. Springer, Berlin 9. Liu G, Han, X (2003) Computational inverse techniques in nondestructive evaluation. CRC Press, Boca Raton, FL 10. Lowe M, Alleyne D, Cawley P (1998) Mode conversion of a guided wave by a partcircumferential notch in a pipe. J Appl Mech, Trans ASME 65(3):649–656 11. Mathworks, Incorporated, The (2008) Genetic algorithm and direct search toolboxTM 2 user’s guide. The Mathworks, Inc, Natick (http://www.mathworks.com/products/gads/ technicalliterature.html) 12. Moiseuev N, Friedland S (1980) Association of resonance states with the incomplete spectrum of finite complex-scaled Hamiltonian matrices. Phys Rev A 22:618–624 13. Mu J, Zhang L, Rose J (2007) Defect circumferential sizing by using long range ultrasonic guided wave focusing techniques in pipe. Nondestruct Test Eval 22(4):239–253 14. Rattanawangcharoen N (1993) Propagation and scattering of elastic waves in laminated circular cylinders. Doctoral Dissertation, University of Manitoba 15. Rose J, Quarry M (1999) Feasibility of ultrasonic guided waves for non-destructive evaluation of gas pipelines. Gas Research Institute, Chicago, IL 16. Silk M, Bainton K (1979) Propagation in metal tubing of ultrasonic wave modes equivalent to Lamb waves. Ultrasonics 17:11–19

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17. Stoyko D, Popplewell N, Shah A (2007) Modal analysis of transient ultrasonic guided waves in a cylinder. In: Yao Z, Yuan M (eds) Computational Mechanics Proceedings ISCM 2007: Abstract page 411, Full manuscript (pp 1215–1225) on accompanying CD-ROM, 211 FP-136PopplewellN.pdf Tsinghua University Press and Springer, Beijing 18. Stoyko D, Popplewell N, Shah A (2008) Feasibility of finding properties of homogeneous, isotropic pipes using nondestructive guided waves. In: Brennan M (ed) Proceedings of the 7th European conference structural dynamics: Abstract page 29, Full manuscript on accompanying CD-ROM, E152.pdf Institute of Sound Vibration Research, Southampton, Hampshire 19. Stoyko D, Popplewell N, Shah A (2009) Ultrasonic measurement of dimensional and material properties. In: Thompson D, Chimenti D (eds) Review of progress in quantitative nondestructive evaluation, AIP Conference Proceedings 1096. American Institute of Physics, Melville, NY, pp 1267–1274 20. Wilkinson J (1965) The algebraic eigenvalue problem. Oxford University Press, Oxford 21. Zhuang W, Shah A, Dong S (1999) Elastodynamic Green’s function for laminated anisotropic circular cylinders. J Appl Mech 66:665–674

Chapter 28

Nonlinear Self-Defined Truss Element Based on the Plane Truss Structure with Flexible Connector Yajun Luo, Xinong Zhang, and Minglong Xu

Abstract A finite-element method based on the self-defined truss element is developed and used to model the plane truss structure with the flexible connector, moreover, the dynamic characteristic of the corresponding model is analyzed in this paper. Firstly, a kind of new type truss structure is analyzed where the flexible connectors between trusses include clearance effects. A self-defined truss element is defined based on the mechanical analysis and then used to build the finite-element model. And the nonlinear elastic-damper model and the Coulomb friction model are adopted to analyze the nonlinear nodal forces from the clearance field. Secondly, a nonlinear numerical solution method is developed based on the Newmark implicit integrate method together with Newton–Raphson iterated method and then used to solve the nonlinear dynamic model. Finally a numerical example is performed by the method above and the effects of several key parameters (such as the contact stiffness and the clearance) on the dynamic characteristic are analyzed. The results validate the numerical solution method and show that the nonlinear finite-element model is effective.

28.1 Introduction Presently, various connectors are wildly adopted in the modern space structure such as the deployable antenna and the solar plate for the requirement of the launching, deformation, and attitude control. The dynamic analyses of the connectors are usually primary and key in the full structural analyses. Moreover, the connectors will result in variety and complexity of the dynamic analysis of the space structure. Common connectors, such as the spherical joint [1] and the sleeve connector [2] in the deployable structure and the pre-load connector in the truss structure [3], generally include clearance, friction, impact, and other nonlinear cases. Furthermore,

X. Zhang () School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China e-mail: [email protected]

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 28,

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the errors induced by manufacturing and assembling usually also result in the randomness of the structural parameters in the practical structure. Therefore, the modeling of the connectors is not easy for the complicated cases above. At present, main reports such as the 3-D spherical joint model [1], the sleeve-type connector model considering the Coulomb friction and clearance effect [2, 4], the bilinear hysteresis model [3, 5], are contributive to the development of this field. The other work for the connectors, for instance, Flores et al. [6] studied the nonlinear dynamical behavior of the translational joints with clearance, Shi and Atluri [7] used the Ramberg–Osgood function to describe the nonlinear elastic performance, and Gaul and Lenz [8] used the Valanis model to analyze the slip and adhesion characteristic of the contact field, make also contribution to the design and application of the connectors and other clearance mechanisms. Conclusively, the clearance effect is still the main research focus because it widely exists in the connectors and easily results in the impact and friction effect. The friction models include the Coulomb friction model, LuGre friction model, and so on. The impact models are mainly classified into two types, one is the linear impact model [9] and the other is the nonlinear elastic-damper model [10–12]. The two models are put forward based on Hertz contact theory, but the damping forces are considered based on the different hypothesizes. Generally, the latter is employed because it is closer to actual case than the former. However, with the development of the modern space structure technology, many new connectors which are with better performance and can satisfy more special requirements are being designed, so some new modeling methods need to be developed now. Truss structure is still used widely in the space structure. The connectors between the truss and the main body structure, or between the truss and the support structure, or among the trusses have various construction for different application forms. The truss structure with the flexible connectors which belong to the slideable connector is applied in a space structure for satisfying some special requirements in this study. Firstly, a self-defined truss element which has considered the clearance effect is defined based on the construction of the flexible connector. The nonlinear elastic-damper model and Column friction model are adopted here. Secondly, the nonlinear dynamic equations of the full system are obtained. Finally, the nonlinear dynamic equations are solved in the time domain using the Newmark method for time integration, in which the Newton–Raphson method is used for handling the non-linear behavior within each time step. Furthermore, the dynamic characteristic and response are analyzed by the numerical simulation. And the dynamic effects of some key parameters on the full system are also analyzed. In summary, the dynamic modeling based on the self-defined truss and the corresponding nonlinear numerical solution method are explored. Moreover, they are shown as effective approaches for the dynamic analysis of a new type truss structure with the flexible connector.

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28.2 Finite-Element Dynamic Modeling 28.2.1 Introduction of the Structure The flexible truss structure is shown in Fig. 28.1. The flexible connector consists of six parts: a hollow sphere, a sleeve, two linked springs, a linkage, an inserted nut, and a hollow rod. The hollow sphere is used as the joint to connect the adjacent trusses. The sleeve is used to connect the hollow sphere and the hollow rod, in which is embedded two linked spring and a linkage. One end of the sleeve is welded with the hollow sphere. The outer ends of the two linked springs are pressed on the right end and the left end of the sleeve by a pre-load, respectively. The truncated cone which locates in the middle of the linkage is used to separate the two adjacent springs. It is likely to contact with the sleeve during the motion. The inserted nuts in the ends of the hollow rod are designed because the hollow rod is a thin wall structure and is hard to be directly connected with other components. They can realize the threaded connection between the linkages and the hollow rods. Obviously, the flexibility of the connector and the full truss structure is depended on the stiffness of the linked spring based on the design of this flexible connector. That is, the truss structure with various flexibilities can be obtained by assembling the linked springs with different stiffness levels. It is just a highlight of this connector. Moreover, the clearance between the sleeve and linkage is considered in the model as shown in Fig. 28.1. Compared with the traditional truss structure, the flexible truss structure is with more complex nonlinearity and modeling hardness. The diagram of the self-defined truss element used to simulate the flexible truss in the finite-element model is shown in Fig. 28.2. One truss is modeled as one

linkage

inserted nut

hollow sphere hollow rod sleeve linked spring

Fig. 28.1 The construction of the flexible connector

i

E1

j

E2

Fig. 28.2 The model of the self-defined truss element

q

E3

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self-defined truss element with three sub-elements. The sub-element E1, E3, and E2 in the self-defined element are used to describe the left and right flexible connection and the middle hollow rod, respectively. The node i and p are located at the center of the hollow spheres, respectively. The node j and q are located approximately at the two ends of the hollow rods.

28.2.2 Motion Equations of the Self-Defined Truss Element There are two contact pairs: one is located between the linkage and the right end of the sleeve (the contact pair a); the other is located between the sleeve and the truncated cone of the linkage (the contact pair b). The point contact can be assumed because the area of the contact surface is very small. Both the contact forces and the friction forces at the two contact pairs are considered. The stiffness of the sleeve and the linkage is much larger than that of the linked spring so they are assumed as rigid body and only the inertia is considered. Through the force analysis, the equilibrium equations of the connector can be written as follows: ME1 fuR E1 g C CE1 fuP E1 g C KE1 fuE1 g D fFE1 g;

(28.1)

where ME1 , CE1 , KE1 , and fFE1g are the mass matrix, damping matrix, stiffness matrix, and load vector, respectively. The sub-element E3 and E1 are located symmetrically along the axial direction. The material, size, force analysis, and boundary conditions of the sub-element E3 are the same as those of the sub-element E1. So its motion equations can be directly obtained. For the sub-element E2, the traditional plane frame element is adopted here. Finally, the motion equations of the self-defined element can be obtained by the superposition of all the dynamic matrix and vectors along the direction of the corresponding coordinates according to the finite-element assembly method. The finite-element equations have been given as follows. Me fuR e g C Ce fuP e g C Ke fue g D fFe g

(28.2)

To expand the load vector fFe g, it can be written as below: fFe g D fFCe g C fFN e g;

(28.3)

where fFCe g and fFN e g are the load vectors about the clearance effect and the nodal load, respectively. In order to analyze the numerical solution method of the nonlinear motion equations conveniently, the summary nodal force is defined as, fRe g D Ke fue g fFCe g

(28.4)

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Nonlinear Self-Defined Truss Element Based on the Plane Truss Structure

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and then the motion equations of the self-defined element can be revised as Me fuR e g C Ce fuP e g C fRe g D fFe g;

(28.5)

where the external load vector fFe g is equal to fFN e g.

28.3 Nonlinear Nodal Forces Based on the nonlinear elastic-damper model [12], the contact force at every contact pair is the sum of the elastic restoring force and the damping restoring force. For example, at the contact pair a, FjC D Fje C Fjd ;

(28.6)

where the expression of the elastic restoring force Fje and the damping restoring force Fjd are written as follows, respectively. Fje D Ka "nja h."ja / Fjd

(28.7)

D Ca "ja "Pja h."ja /;

(28.8)

where n is a factor. h."mn / is a Heaviside function used to describe the direction of the contact force. "ja is the relative displacement of the middle of the contact pair a. Similarly, the contact force of the contact pair b can also be obtained. In the Coulomb friction model, the friction forces FjF at the contact pair a are expressed as: FjF D FjC sign.xP i xP j / D .Ka "nja h."ja /CCa "ja "Pja h."ja //sign.xP i xP j /; (28.9) where is the friction factor and the function sign(x) is used to describe the direction of the friction force. By substituting the concrete expressions of the contact force and friction force into the contact load vector fFCe g, the nonlinear nodal force vector of the local element can be obtained. Furthermore, the tangent stiffness matrix of the nodal force vector of the self-defined element in the local coordinate can be derived from (28.4). Kte D

@.Ke fue g fFCe g/ @fRe g @fFCe g D D Ke ; @fue g @fue g @fue g

(28.10)

where the expressions of every elements in the tangent stiffness matrix and the contact stiffness matrix can be written as Kte .i; j / D Ke .i; j / KCe .i; j /;

i; j D 1; 2; 3; 4; 5; 6

(28.11)

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KCe .i; j / D

@fFCe .i /g ; @fue .j /g

i; j D 1; 2; 3; 4; 5; 6;

(28.12)

where Ke and KCe are the elastic stiffness matrix and the contact stiffness matrix, respectively.

28.4 The Numerical Solution Method and Example 28.4.1 The Numerical Solution Method In this search, the Newmark implicit integral method is employed for time integration, and then the Newton–Raphson method is employed for equilibrium iteration in each time step. Therefore, the equation can be written as the following iterative form. uP g CnC1 Kte fk ue g D fnC1 Fe g fnC1 R g Me fnC1 k uR e g C Ce fnC1 k e k1 k1 e

(28.13)

where ue g fnC1 u g fk ue g D fnC1 k k1 e

(28.14)

Then, (28.4) and (28.10) can be modified as nC1 nC1 C nC1 fnC1 k1 Re g D fRe .fk1 ue g/g D Ke fk1 ue g fFe .fk1 ue g/g nC1 t K k1 e

D Kte .fnC1 u g/ D Ke KCe .fnC1 u g/ k1 e k1 e

(28.15) (28.16)

Equations (28.15) and (28.16) are substituted into (28.13), then nC1 Me fnC1k uR e g C Ce fnC1k uP e g C ŒKe KCe .fnC1 Fe g Ke fnC1 k1 ue g/fk ue g D f k1 ue g

CfFCe .fnC1 k1 ue g/g (28.17)

Equation (28.17) is just the iteration formula of the self-defined element at the time tnC1 .

28.4.2 Numerical Example A truss structure with only one flexible truss is considered in the numerical example. Its structural diagram is given as shown in Fig. 28.3. Both the left and right ends are fixed. A transverse force Fy and an axial force Fx are simultaneously applied at the node 2. The responses of both node 2 and node 3 along all DOF directions can be excited under these loading conditions. As shown in Fig. 28.4, the length of the

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Nonlinear Self-Defined Truss Element Based on the Plane Truss Structure

Fig. 28.3 The construction of the single flexible truss structure

y 1

b 2

0 −0.005 0

0.5 Time (s)

−0.005 −0.01

0

0.5 Time (s)

1

2

x

-3 x 10

0 −1 −2

0

0.5 Time (s)

−3

1

-4 x 10

f

1 0 −1 −2

4

1

−1 −2

Displacement (m)

Displacement (m)

0

2

0

e

0.005

c

-4

1

1

0.01

x 10

3

Fx

Angle (rad)

0.005

−0.01

d

Displacement (m)

Displacement (m)

0.01

Fy

Angle (rad)

a

2

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0

0.5 Time (s)

1

4

0

0.5 Time (s)

1

0.5 Time (s)

1

-3 x 10

2 0 −2 −4

0

Fig. 28.4 The displacement response curves. (Node 2: (a) X; (b) Y; (c) ™ Node 3: (d) X; (e) Y; (f) ™ Factor of random excitation: solid line 0.25; dashed line 0.5)

truss is 0.42 m, the length of the connector is 0.051 m, the stiffness of the linkage spring is 1,580 N/m, and all the parts are made of steel.

28.4.2.1 Random Excitation A group random signal with 0–100 Hz frequency range is adopted as the excited force by the gain amplifier. The different excited force levels are adopted because the stiffness level along the axis direction is significant difference compared with that of the transverse direction. In this example, the corresponding gain factors of Fx and Fy are 0.25 and 1:25, respectively. Simultaneously, the gain factors of Fx and Fy are increased twice and changed as 0.5 and 2:5 in order to compare the effects of the different levels of the excited force. The clearance value is 127 m in the connector.

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The displacement responses of the nodes 2 and 3 are given as shown in Fig. 28.4. The responses along the transverse and rotation directions are mostly constrained by the clearance. However, the responses along the axial direction are obviously different under two kinds of excitation. It is found that the excited force with greater amplitude generates the response with greater level. The nonlinearity along the axial direction is not strong though the friction effect has been considered. And the axial displacements at nodes 2 and 3 are almost synchronous as shown in Fig. 28.4a, d. It is the cause that the axial stiffness of the hollow rod between the two nodes is much larger than that of the connectors. The conclusions above are also explained from Fig. 28.5 wherein the corresponding displacement spectrum curves are given. It is indicated that the first frequency is 38.0 Hz in Fig. 28.5a, d. And it is found that the curves are much unorganized for the significant nonlinearity from the other four sub-figures. The position curves at the two nodes with clearance are given in Fig. 28.6 in order to observe the contact motion law along the transverse direction. The two boundaries just express the value of the clearance. If the positions are between the up and below boundaries, it expresses no contact, otherwise contact. The excess parts of the positions compared with two boundaries are just the penetration of the

b

15

0.3

10

38.0Hz

5

d

0.2 0.1 0 0

100 200 Frequency (Hz)

1 Spectrum (rad)

0.4

0 0

0 0

0.2 0.15 0.1 0.05

100 200 Frequency (Hz)

0.4 0.2 100 200 Frequency (Hz)

1.5 Spectrum (rad)

5

Spectrum (m)

10

38.0Hz

0.6

f 0.25

15

0.8

0 0

100 200 Frequency (Hz)

e 20

Spectrum (m)

c

20 Spectrum (m)

Spectrum (m)

a

0 0

100 200 Frequency (Hz)

1

0.5

0 0

100 200 Frequency (Hz)

Fig. 28.5 The displacement spectrum curves. (Node 2: (a) X; (b) Y; (c) ™ Node 3: (d) X; (e) Y; (f) ™ Factor of random excitation: solid line 0.25; dashed line 0.5)

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a

Position (m)

2

x 10−4

1 0 −1 −2

b 2 Position (m)

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0

0.1

0.2

0.3

0.4

0.5 Time(s)

0.6

0.7

0.8

0.9

1

0.2

0.3

0.4

0.5 Time(s)

0.6

0.7

0.8

0.9

1

x 10−4

1 0 −1 −2

0

0.1

Fig. 28.6 The position curves. (Node: (a) node 2; (b) node 3. Contact pair: solid line a; dashed line b)

contact pair. In spite of two excitations with the different amplitudes, the position curves at every node are almost synchronous, respectively. The other phenomenon is that the repeated impact curves at the node 2 between the two boundaries are clearer than those at the node 3. The possible explanation is that the node 2 is actively excited, but the node 3 is passively excited.

28.4.2.2 Contact Stiffness The dynamic responses are compared in the present example which the contact stiffness is considered as 1.5E7 and 1.5E8 N/m. A sine signal with the first mode frequency 38.0 Hz is adopted as the excited force. Moreover, the gain factors of Fx and Fy are 0.5 and 2.5 N, respectively. The curves of the displacement responses and the corresponding displacement spectrums are given as shown in Figs. 28.7 and 28.8, respectively. In Fig. 28.7, it can be seen that the contact stiffness has little effect on the axis responses of the system, but it has obvious effect on the other two responses at every nodes. From sub-figure (b, c, e, f), the smaller contact stiffness can result in the greater response level. As shown in Fig. 28.8, the spectrum curves have several peak values. And it is found that all the peak frequencies are several times of the excited frequency. It is obvious that the nonlinear dynamic system has generated

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Amplitude (m)

1 0 −1 −2

d

Amplitude (m)

2

2

2 0

0

0.5 Time (s)

−4

1

e

x 10−3

4

0

0

0.5 Time (s)

0

0.5 Time (s)

0

f

x 10−4

4

2 0

−4

1

1

−2

1

−2

−1

x 10−3

−1

−2

1

−2

c

x 10−4

Amplitude (rad)

4

Amplitude (m)

Amplitude (m)

2

b

x 10−3

Amplitude (rad)

a

0

0.5 Time (s)

1

0.5 Time (s)

1

x 10−3

2 0

−2

0

0.5 Time (s)

1

−4

0

Fig. 28.7 The displacement response curves. (Node 2: (a) X; (b) Y; (c) ™ Node 3: (d) X; (e) Y; (f) ™ Contact stiffness: dashed line 1.5E7 N/m; solid line 1.5E8 N/m)

Amplitude (m)

e

15

10

5

0 0

38.0Hz

76.0Hz

200 400 Frequency (Hz)

1

0 0

200 400 Frequency (Hz)

Amplitude (m)

76.0Hz

2

3 2 1

200 400 Frequency (Hz)

f 10

1.5

1

0.5

0 0

4

0 0

200 400 Frequency (Hz)

Amplitude (m)

5

38.0Hz

c

3 Amplitude (m)

10

0 0

d

b

15

Amplitude (m)

Amplitude (m)

a

200 400 Frequency (Hz)

8 6 4 2 0 0

200 400 Frequency (Hz)

Fig. 28.8 The displacement spectrum curves. (Node 2: (a) X; (b) Y; (c) ™ Node 3: (d) X; (e) Y; (f) ™ Contact stiffness: dashed line 1.5E7 N/m; solid line 1.5E8 N/m)

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a

Position (m)

4

x 10−4

2 0 −2 −4 0

b 4 Position (m)

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0.1

0.2

0.3

0.4

0.5 Time (s)

0.6

0.7

0.8

0.9

1

0.1

0.2

0.3

0.4

0.5 Time (s)

0.6

0.7

0.8

0.9

1

x 10−4

2 0 −2 −4

0

Fig. 28.9 The position curves. (Nodes: (a) node 2; (b) node 3. Contact stiffness: dashed line 1.5E7 N/m; solid line 1.5E8 N/m)

the primary resonance and the superharmonic resonance phenomenon. Moreover, both of the displacement spectrum curves in sub-figure (a, d) have generated only one superharmonic resonance at the frequency 76.0 Hz. But more superharmonic resonances are observed in the other four sub-figures. This result reflects that the nonlinear performance of the axis motion of the system is weaker than that of the other two directions. From Fig. 28.9, it can be observed that the position curves of the two nodes are vibrated around the up and below contact boundaries (as shown by dash-dot). Because the excited forces are only actuated at the node 2, the position curves in Fig. 28.9a have obvious periodicity compared with that in Fig. 28.9b. And an obvious phenomenon is that the amplitude of the penetration is inversely proportional to the contact stiffness.

28.4.2.3 Clearance In this example, the contact stiffness is 1.5E8 N/m, the excited force is same so that in the example of contact stiffness, and the clearance value is adopted two cases: 0 m (without clearance) and 127 m (with clearance). Figures 28.10 and 28.11 are given the displacement response curves and the corresponding spectrum curves. From Fig. 28.10, it can be seen that the clearance has

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b

x 10−3

2

1

Amplitude (m)

Amplitude (m)

2

0 −1 −2 0

0 −1 −2 0

0.5

e

1 0 −1 −2 0

0.5

0.5

2

1 0 −1

f

x 10−4

2

1 0 −1 −2 0

0.5

Time (s)

Amplitude (m)

x 10−3

−2 0

x 10−3

Time (s)

Amplitude (m)

Amplitude (m)

2

2

1

Time (s)

d

c

x 10−4

Amplitude (m)

a

Time (s)

x 10−3

1 0 −1 −2 0

0.5 Time (s)

0.5 Time (s)

Fig. 28.10 The displacement response curves. (Node 2: (a) X; (b) Y; (c) ™ Node 3: (d) X; (e) Y; (f) ™ Clearance: dashed line with; solid line without)

a

b

c

10−2

d

76.0Hz

10−1

0

Amplitude (m)

Amplitude (m)

38.0Hz 100

200 400 Frequency (Hz)

e

101

10−2

10−4

10−6 0

100

100

10−1

10−2 0

76.0Hz

Amplitude (m)

Amplitude (m)

38.0Hz

200 400 Frequency (Hz)

10−2

10−4

10−6 0

102 Amplitude (rad)

100

200 400 Frequency (Hz)

100

10−2

10−4 0

200 400 Frequency (Hz)

f 102 Amplitude (rad)

101

200 400 Frequency (Hz)

100

10−2

10−4 0

200 400 Frequency (Hz)

Fig. 28.11 The displacement spectrum curves. (Node 2: (a) X; (b) Y; (c) ™ Node 3: (d) X; (e) Y; (f) ™ Clearance: dashed line with; solid line without)

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Nonlinear Self-Defined Truss Element Based on the Plane Truss Structure

a Position (m)

2

0 −1

2

Position (m)

x 10− 4

1

−2

b

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0

0.05

0.1

0.15

0.2

0.25 0.3 Time (s)

0.35

0.4

0.45

0.5

0.1

0.15

0.2

0.25 0.3 Time (s)

0.35

0.4

0.45

0.5

x 10− 4

1 0

−1 −2

0

0.05

Fig. 28.12 The position curves. (Nodes: (a) node 2; (b) node 3. Clearance: dashed line with; solid line without)

little effect on the axis motion of the system by observing the sub-figure (a, d). However, the clearance has great effect on the other responses by the obvious different result in the sub-figure (b, c, e, f). The corresponding spectrum curves also expressed the results as shown in Fig. 28.11. Only one hyperhormonic resonance peak is observed in sub-figure (a, d), but a series of ones are observed in the other four sub-figures. For the case of without clearance, the response curves and spectrum curves become more regular and their amplitudes are also reduced significantly than that of with clearance as shown in Figs. 28.10 and 28.11b, c, e, f. From Fig. 28.12, the penetration is always generated and the curve can be regarded approximately as the sinusoidal curve when the clearance value is zero.

28.5 Conclusions In this paper, a nonlinear finite-element model is developed to analyze the dynamic problems of a new type plane truss structure with the flexible connector that included clearance, friction, and axial constraint based on the self-defined truss element. And then the model is solved by means of the nonlinear numerical solution method based on the Newmark implicit integrate method together with Newton–Raphson intern method. An example: single flexible truss structure is also analyzed using the nonlinear numerical solution method.

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There are two conclusions in the numerical example. On the one hand, the stiffness level along axial direction is lower than the other directions, so that the first mode can be clearly observed only in the axial response for the case of random excitation. On the other hand, both the contact stiffness and the clearance mainly affect the responses of the two non-axial DOFs. And both of them induce the superharmonic resonance phenomena. Obviously, the dynamic motion of the single flexible truss structure has strong nonlinearity. In summary, the developed nonlinear self-defined element represents the nonlinear cases and the dynamic properties of the system well. In the future, the method above will be thoroughly discussed in the complex plane truss structure. Acknowledgments This work was supported by the Civil Space Advanced Project of China under Grant No. C4120061309.

References 1. Tzou HS, Roog Y (1991) Contact dynamics of a spherical joint and a jointed truss-cell system. AIAA J 29(1):81–88 2. Ferri AA (1998) Modeling and analysis of nonlinear sleeve joints of large space structures. AIAA J Spacecr Rockets 25(5):354–360 3. Junjiro O, Tetsuni S, Kenji M (1995) Passive damping of truss vibration using preloaded joint backlash. AIAA J 33(7):1335–1341 4. Bindemann AC, Ferri AA (1995) Large amplitude vibration of a beam restrained by a nonlinear sleeve joint. J Sound Vib 184(1):19–34 5. Hsu ST, Griffin JH, Bielak J (1989) How gravity and joint scaling affect dynamic response. AIAA J 27(9):1280–1287 6. Flores P, Ambr´osio J, Claro JCP, Lankarani HM (2008) Translational joints with clearance in rigid multibody systems. J Comput Nonlinear Dyn 3(011007):1–10 7. Shi G, Atluri SN (1992) Nonlinear dynamic response of frame-type structures with hysteretic damping at the joints. AIAA J 30(1):234–240 8. Gaul L, Lenz J (1997) Nonlinear dynamics of structures assembled by bolted joints. Acta Mech 125:169–181 9. Dubowsky S, Deck JF, Costello H (1987) Dynamic modeling of flexible spatial machine systems with clearance connections. J Mech Transm Autom Des 109(1):87–94 10. Lankarani HM, Nikravesh PE (1990) A contact force model with hysteresis damping for impact analysis of multibody systems. ASME J Mech Des 112:369–376 11. Lankarani HM, Nikravesh PE (1994) Continuous contact force models for impact analysis in multibody systems. Nonlinear Dyn 5:193–207 12. Yigit AS, Ulsoy AG, Scott RA (1990) Spring-dashpot models for the dynamics of a radially rotating beam with impact. J Sound Vib 142(3):515–525

Chapter 29

Complex Frequency Analysis of an Axially Moving String with Multiple Attached Oscillators by Using Green’s Function Method Le-Feng Lu, ¨ Yue-Fang Wang, and Ying-Xi Liu

Abstract In the present paper, the eigenvalue problem of an axially moving string with multiple attached mass-spring oscillators is investigated. Closed form transcendental equations for the natural frequencies are obtained by means of the Green’s function method. The maximum variance rate of eigen-frequencies of the string is presented to indicate the coupling strength between the modes of subsystems. The Galerkin’s discretization method is analyzed so as to determine the approximate eigenvalues for large numbers of oscillators. The results of the traveling mass model are also presented as a limit case.

29.1 Introduction Axially moving continuum, rotating flexible machineries, and fluid-conveying pipes are examples of gyroscopic systems. A convective acceleration component in the governing equations renders complex, speed-dependent modes. In finding the eigensolution of this problem, Meirovitch [1, 2] proposed a method for the solution of the eigenvalue problem for linear gyroscopic systems, which permits a formulation in terms of real symmetric matrices. Parker [3] adopted a perturbation analysis to determine approximate eigenvalue loci for axially moving materials on elastic foundation. Wang et al. [4] investigated the eigenvalue problem for axially moving strings based on Hamilton system and obtained the modal functions and symplectic orthogonality conditions through a symplectic analysis. For those continuous system subjected to complex constraints the system characteristics can be significantly influenced by the coupling between subsystems. For example, for moving oscillator systems the presence of mass-spring oscillator Y.-F. Wang () Department of Engineering Mechanics, Dalian University of Technology, 2 Linggong Road, Dalian 116024, People’s Republic of China and State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian 116024, People’s Republic of China e-mail: [email protected] A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 29,

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varies the inertia of the system, which results in location-dependent eigenvalues that are different from conventional analytical eigenvalue solutions. Stannisie [5] determined exact eigen-solutions for mass moving on an elastic foundation by performing the integration in the Stieltjes sense. Kukla [6, 7] studied the eigenvalue problem of a beam with intermediate elastic supports and presented the exact solution of the coupled system by means of Green’s function method. The Green’s function, with its intuitive concept and definite physical meaning, is simple yet effective for problems with presence of Dirac-Delta function. From the point of view of the Green’s function theory, reference [5] utilizes the properties of Green’s function that was equivalent to the method of [6, 7]. The Green’s function method is used to analyze the eigenvalue problem for the axially moving string with attached mass-spring oscillator in this paper. The theorem of construction of Green’s function is applied to obtain the explicit Green’s function of the coupling system without complicated computation. In the numerical demonstration, the maximum variance rate of the eigen-frequencies of the moving string related to coupling strength between the string and the masses is calculated. The model of attached mass is also analyzed as the limit case of the coupled oscillator. Finally, the Galerkin’s method is compared with the Green’s function method for validity of the Galerkin’s discretization method.

29.2 Formulation of the Eigenvalue Problem Consider a simply supported, axially moving string travelling with a velocity c with N attached undamped linear oscillators, as shown in Fig. 29.1. Assume the i th oscillator arrives at the left end of the string at time ti ; i D 1; : : : ; N , of which the coefficient of acting time is „i D H.t ti / H.t .ti C 1=c//, where H./ is the Heaviside unit-step function. By neglecting the flexural, shear, and torsion rigidity, the nondimensional governing equations of transverse vibration of the system can be written in the form [8]

yi (t)

mi

ki

yi+1(t)

mi+1

k i+ 1

w (x,t) x

Fig. 29.1 Axially moving string with attached multiple oscillators

c

29

Complex Frequency Analysis of an Axially Moving String

wR C .wP C cw0 / C 2c wP 0 C .c 2 1/w00 D

N X

373

ki „i .yi w/ı.x xi / C fw ;

i D1

(29.1) ki yRi C .yi w.xi ; t// D fi ; ti t ti C 1=c; mi

i D 1; : : : ; N;

(29.2)

where w.x; t/ is the transverse displacement of the string, the modal damping of the string, yi ; ki and mi the deflection, the stiffness coefficient and mass of the i th oscillator, respectively. fw and fi are external forces on the string and mass, respectively. ı./ is the Dirac-Delta function. Let the general solutions of the foregoing be w.x; t/ D Ref.x/et g;

yi .t/ D Yi et ;

(29.3)

which leads to an eigenvalue problem .c 2 1/ 00 .x/ C .2c C c/ 0 .x/ C .2 C /.x/ D

N X

"i .xi /„i ı.x xi /;

i D1

(29.4) where "i D mi ki 2 =.mi 2 C ki /. By applying the Green’s function method, the solution of (29.4) can be obtained, Z

N 1X

.x/ D 0

"i „i .xi /ı.x xi /G.xI /d D

i D1

N X

"i „i .xi /G.xI xi /;

i D1

(29.5) where G.xI / is the Green’s function of the (29.4). A set of N homogeneous equations can be derived by substituting x D xi ; i D 1; : : : ; N into (29.5), yielding .xi / C

N X

"j „j .xj /G.xi I xj / D 0;

i D 1; : : : ; N:

(29.6)

j D1

The following equation must hold to require a non-trivial solution of (29.6): ˇ ˇ"1 „1 G.x1 I x1 / C 1 ˇ ˇ "1 „1 G.x2 I x1 / ˇ ˇ :: ˇ : ˇ ˇ " „ G.x I x / 1 1

N

1

"2 „2 G.x1 I x2 / "2 „2 G.x2 I x2 / C 1 :: :

:: :

"2 „2 G.xN I x2 /

ˇ ˇ ˇ ˇ ˇ ˇ D 0: ˇ ˇ "N „N G.xN I xN / C 1ˇ (29.7) "N „N G.x1 I xN / "N „N G.x2 I xN / :: :

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29.3 Construction of Green’s Function G.xI / should satisfy the equation according to the definition of the Green’s function, as .2 C /G.xI / C .2c C c/G 0 .xI / C .c 2 1/G 00 .xI / D ı.x /

(29.8)

and boundary condition G.0I / D G.1I / D 0:

(29.9)

Generally, (29.8) with (29.9) can be solved by taking advantage of properties of the Green’s function, e.g., the continuous condition and jump condition at x D . The tedious procedure can be eliminated by using the construction of Green’s theorem [9], which permits explicit expression of the Green’s function, as follows. For the second linear differential operator L D P .x/d2 =dx 2 C Q.x/d=dx C R.x/, if u1 .x/ and u2 .x/ satisfy Lu1 D 0;

au1 .0 / C a0 u01 .0 / D 0;

Lu2 D 0;

bu2 .1 / C b 0 u02 .1 / D 0

and W .u1 ; u2 / ¤ 0, then the Green’s function for L can be expressed as ( 1 u1 .x/u2 ./; x < ˇ G.xI / D ; P ./W .u1 ; u2 / ˇxD u ./u .x/; < x 1

(29.10)

(29.11)

2

where 0 ; 1 are two boundaries and W .u1 ; u2 / is the Wronskian determination deˇ ˇ fined as ˇ u .x/; u2 .x/ ˇ ˇ: (29.12) W .u1 ; u2 / , ˇˇ 10 u1 .x/; u02 .x/ ˇ With the following fundamental properties of the Green’s function, it is easy to verify that the expression given by (29.11) is the Green’s function. That is to say that G.xI / is continuous G.xI / D G.xI C /

(29.13)

and the derivative dG=dx satisfies the jump condition at x D ˇ ˇ dG ˇˇ 1 dG ˇˇ : D ˇ ˇ dx C dx P .x/

(29.14)

There are two independent solutions to the homogeneous differential (29.8) which satisfy each of the boundary conditions given by (29.9) (

1 .x/ D exp. x/ exp.C x/;

0x

2 .x/ D exp. .x 1// exp.C .x 1//;

; x1

(29.15)

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where

375

p .2c C c/ ˙ 2 c 2 C 42 C 4 : ˙ D 2.c 2 1/

By applying the above-mentioned method, one can get 1 ˇ G.xI / D 2 .c 1/W .1 ; 2 / ˇxD

(

1 .x/2 ./;

0x

1 ./2 .x/;

x1

;

(29.16)

where W .1 ; 2 / ¤ 0 holds. Substitution of (29.16) into (29.7) leads to the frequencies. In order to solve this equation, the eigenvalue of the unconstrained system is adopted as an initial solution p mi D I ki =mi ; nstr D ˙ I!n ; p 1 1p 2 D ; !n D .c 1/.2 C 4 2 n2 .c 2 1//; I D 1: (29.17) 2 2 Particularly, for a conservative system, the initial eigenvalues of the string alone are 2 str n D n .1 c /I;

n D ˙1; ˙ 2; : : :

(29.18)

It is worth pointing out that there exists a singularity in the Green’s function, i.e., c D 1, which is the critical speed of the system and corresponds to a non-oscillatory eigenvector .x/ D 0.

29.4 Results and Discussions The numerical examples presented hereafter aim to examine the effect of one or several attached mass-spring oscillators on the eigenvalues of an axially moving string, especially when the eigen-frequency of the oscillator is close to the first eigen-frequency of the translating string. For the problem of a single oscillator, the parameters c D 0:25; k D 3:4, and m D 0:375 are assigned. The eigenvalues of a conservative system (i.e., D 0) and a dissipative system with D 0:04 are shown in Figs. 29.2 and 29.3, respectively, both varying with the location of the oscillator x0 . The eigenvalues of the coupled system derived from the eigenvalues of unconstrained string and oscillator are marked by solid lines and dot-dashed lines, respectively. When c < 1, the conservative system is positive definite and gyroscopic, thus only imaginary eigenvalues exist [10]. It is noted that both the real and imaginary parts of the eigenvalues of the constrained system are symmetric with respect to x0 D 0:5 due to the symmetry of the Green’s function. The distributions of the eigen-frequencies (the imaginary part of the eigenvalues) of the constrained system are in accordance with the principle of eigenvalue inclusion [11], which applies only for conservative gyroscopic system.

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Fig. 29.2 The first four eigenvalues of the single moving oscillator model with D 0

Fig. 29.3 The first four eigenvalues of the single moving oscillator model with D 0:04

As shown in Fig. 29.3a, it is not the case that the distributions of the real part of the eigenvalues of the constrained system have the similar principle. With the above assigned parameters the eigen-frequency of the attached oscillator .!m D 3:0111/ is very close to the first natural frequency of the unconstrained

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Table 29.1 The maximum variance rate of the eigen-frequencies of the coupled system Im.1str / Im.2str / Im.3str / Im.m1 / Im.m2 / (%) (%) (%) (%) (%) Single-oscillator model 34.99 10.01 4.32 41.43 Double-oscillator model 39.16 12.27 6.73 63.61 20.35 Double-mass model 34.23 22.53 19.49

moving string. Therefore, both the real and the imaginary part of the first eigenvalue of the constrained string change significantly with the varying attached oscillator, which is related to the resonance-type of the eigenvalue. Other eigenvalues have also changed but not significantly. In order to determine the coupling strength ofˇthe subsystems variance rate defined by ˇı quantitatively, the maximum ; i D 1; 2; : : : is used, where ˛i D sup ˇIm ui ci ˇ min Im ui ; Im ci ui and ci represent the i th eigenvalues of the unconstrained system and the constrained system, respectively. The maximum variance rate is tabulated in Table 29.1. It can be seen that the variance rate of the eigen-frequencies of the moving string drops with the increasing order of the modes. Moreover, the effect of higher modes is relatively small on low-frequency response. This implies that the transient solutions of the coupled system can be approximated by using the expansion of first few modes of the system. For the eigenvalue problem of an axially moving string with two massspring oscillators, an equal stiffness of spring k1 D k2 D k and different masses m1 and m2 are used. In this case, the parameters are chosen to be m1 D 0:375; m2 D 0:3; t1 D 0; t D 1 and t2 D t1 C t. The whole process of movement can be divided into three parts I W t 2 Œ0; t ; „1 D 1; „2 D 0; x1 D ct; x2 D 0; II W t 2 Œt; 1=c ; „1 D 1; „2 D 1; x1 D ct; x2 D c.t t/; III W t 2 Œ1=c; t C 1=c ; „1 D 0; „2 D 1; x1 D 0; x2 D c.t t/: As shown in Fig. 29.4, the eigenvalue curves are no longer symmetric with respect to the central point .t D 2:5/ on the time axis due to different masses of the oscillators. Nonetheless, the eigenvalue curves will be mirrored about t D 2:5 as a result of the symmetry of the Green’s function if the two masses are interchanged. For Parts I and III, the eigenvalues are identical to those of the first and second oscillators individually placed on the string, whereas for Part II another additional eigenvalue appears when the second oscillator steps in, with less variance of frequency than that of the first one. The maximum variance rate of other eigenvalues resembles the one from the problem of single oscillator, though it becomes larger in magnitude. Strictly speaking, the limit case of moving oscillators will be the one with infinitely large spring stiffness, which is not equivalent to the problem of moving mass problem [12]. However, (29.7) is still valid for the eigenvalue problem of coupled systems with attached masses since there is no additional assumption but "i D mi 2 . The variance of the first three eigenvalues of attached two masses model

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Fig. 29.4 The first three eigenvalues of multiple two oscillators model

is depicted in Fig. 29.5 with c D 0:25; D 0:04; m1 D 0:375, and m2 D 0:3. The maximum variance rate of the first three eigenvalues is also listed in Table 29.1. Compared to the model of attached oscillator, the effect of moving mass on the first three eigenvalues of the string are significant. This means that the transient solutions of the coupled mass-string system should be approximated using much more modal expansions of the system than that of the attached oscillator system. It should be noted that the eigenvalue problem from (29.7) will be complicated for very large N . The Galerkin’s discretization method is used to approximately approach the eigenvalue solutions of (29.7), for which the eigen-functions of the stationary string n D sin.n x/ are chosen. The Jacobian coefficient matrix of discrete ODEs is shown in the Appendix. The first two eigen-frequencies of moving string under double oscillators with the parameters above are computed for different orders of expansion: M D 4; 8; 16, respectively, at various moments in Table 29.2. It can be seen that the Galerkin’s method can capture the dynamical characteristics of the string when locations of the oscillators vary. The results are more and more accurate with increasing orders of expansion in comparison with the calculations of determination (29.7). Although it has been demonstrated that Galerkin’s discretization method using Fourier series expansion can be effective to analyze the dynamics of the moving systems, the expansion is still questionable for insufficient

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379

Fig. 29.5 The first three eigenvalues of the moving masses model

Table 29.2 Comparison of the first two eigen-frequencies of moving string under double oscillators between Galerkin’s discretization method of different order of expansion with the present paper Galerkin’s discretization method

t 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

The present paper

M D4

Im.1str /

Im.2str /

Im.1str /

Im.2str /

M D8 Im.1str /

Im.2str /

M D 16

2.9452 2.3667 2.0876 1.9162 1.8152 1.7938 1.8541 2.0149 2.2528 2.5472 2.9452

5.8905 6.1510 6.4804 6.6021 6.5723 6.4340 6.5381 6.6127 6.5137 6.1662 5.8905

2.9461 2.4109 2.1211 1.9379 1.8343 1.8116 1.8727 2.0369 2.2822 2.5807 2.9461

5.9023 6.1777 6.5547 6.7000 6.6618 6.5172 6.6218 6.7144 6.5942 6.1951 5.9023

2.9453 2.3907 2.1042 1.9292 1.8259 1.8041 1.8648 2.0274 2.2672 2.5657 2.9453

5.8918 6.1632 6.5117 6.6448 6.6168 6.4596 6.5798 6.6569 6.5478 6.1797 5.8918

Im.1str / 2.9452 2.3789 2.0959 1.9226 1.8205 1.7988 1.8594 2.0210 2.2601 2.5566 2.9452

Im.2str / 5.8906 6.1568 6.4947 6.6223 6.5922 6.4465 6.5567 6.6337 6.5293 6.1727 5.8906

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theoretical foundation. From the eigenvalue point of view, with the Galerkin’s discretization method, the transient responses of the moving string with attached oscillators will be more and more reliable with increasing orders of expansion.

29.5 Conclusion The eigenvalue problem of an axially moving string coupled with multiple linear oscillators is investigated by Green’s function method. The Green’s function in an explicit form is obtained by the theorem of Green’s function construction, and the analytical transcendental equation is directly obtained. The numerical examples show that both the real and the imaginary parts of eigenvalues are varying. The maximum variance rate is defined to analyze the coupling strength of the subsystems. It is demonstrated that only the first eigenvalue changes significantly when the eigen-frequency of the oscillator is close to that of the string’s first eigenvalue, while all the eigen-frequencies of the string are significantly influenced by the moving mass model. Furthermore, the validity of the Galerkin’s method is presented for a number of oscillators, which can be used to approximately solve the eigenvalue problem without complicated computations of the determinant equation. Acknowledgement The authors are grateful to the Natural Science Foundation of China (Project 10721062) and to the National 863 (Project 2007AA04Z405) for their fundings.

Appendix The Jacobian matrix of the discrete ordinary differential equations of the moving string with multiple linear mass-spring oscillators 8 ˆ J.2i 1; 2i / D 1; 1 i M C 2 ˆ ˆ ˆ ˆ N P ˆ ˆ ˆ kn i2 .xn /; J.2i; 2j / D ; 1 i D j M J.2i; 2j 1/ D .i /2 .1 c 2 / 2 ˆ ˆ ˆ nD1 ˆ ˆ N ˆ P ˆ ˆ <J.2i; 2j 1/ D 2cd ij 2 kn i .xn /j .xn /; J.2i; 2j / D 4cd ij ; 1 i ¤ j M nD1

ˆ ˆ J.2i; 2.M C n/ C 1/ D 2kn i .xn /; 1 i M C 2 ˆ ˆ ˆ ˆ ˆ J.2.M C n/; 2j 1/ D kn =mn j .xn /; 1 n N; 1 j M ˆ ˆ ˆ ˆ ˆ ˆ J.2.M C n/; 2.M C n/ 1/ D kn =mn ; 1 n N ˆ ˆ ˆ :J.i; j / D 0; for other i; j

where dij D ij.1 .1/i Cj /=.i 2 j 2 /.

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381

References 1. Meirovitch L (1974) A new method of solution of the eigenvalue problem for gyroscopic systems. AIAA J 12(10):1337–1342 2. Meirovitch L (1975) A modal analysis for the response of linear gyroscopic systems. J Appl Mech 42:446–450 3. Parker RG (1998) On the eigenvalues and critical speed stability of gyroscopic continua. J Appl Mech 65:134–140 4. Wang YF, Huang LH, Liu XT (2005) Eigenvalue and stability analysis for transverse vibrations of axially moving strings based on Hamiltonian dynamics [J]. Acta Mech Sin 21(5):485–494 5. Stanisic MM (1985) On a new theory of the dynamic behavior of the structures carrying moving masses. Arch Appl Mech 55(3):176–185 6. Kukla S (1991) The Green function method in frequency analysis of a beam with intermediate elastic supports. J Sound Vib 149(1):154–159 7. Kukla S, Zamojska I (2007) Frequency analysis of axially loaded stepped beams by Green’s function method. J Sound Vib 300:1034–1041 8. Pesterev AV, Yang B, Bergman LA (2001) Response of elastic continuum carrying multiple moving oscillators. J Eng Mech 127(3):260–265 9. Gerlach UH (2007) Linear mathematics in infinite dimensions: signals, boundary value problems, and special functions. Lecture Notes of Ohio State University, http://www.math.ohiostate.edu/ gerlach/math/BVtypset/ 10. Lee KY, Renshaw AA (2000) Solution of the moving mass problem using complex eigenfunction expansions. J Appl Mech 67:823–827 11. Yang B (1992) Eigenvalue inclusion principles for distributed gyroscopic systems. J Appl Mech 59:650–656 12. Pesterev AV, Bergman LA, Tan CA, Tsao TC, Yang B (2003) On asymptotics of the solution of the moving oscillator problem. J Sound Vib 260:519–536

Chapter 30

Model Reduction on Inertial Manifolds of Navier–Stokes Equations Through Multi-scale Finite Element Jia-Zhong Zhang, Sheng Ren, and Guan-Hua Mei

Abstract Multilevel finite element method is used to approach the Approximate Inertial Manifolds (AIMs) from viewpoint of nonlinear dynamics, in the computational fluid dynamics. By this method, an unknown variable is divided into two components, namely, the large eddy and small eddy components. With the introduction of an AIMs, the interaction between large eddy and small eddy components, which is negligible if standard Galerkin algorithm is used to approach the original governing equations, is considered essentially, and consequently a coarse grid finite element space and a fine grid incremental finite element space are introduced to approach the two components. By this method, the flow field of incompressible flows around airfoil is simulated numerically as an example, and velocity and pressure distributions of the flow field are obtained accurately. The results show that there exists less degrees-of-freedom in the discretized system in comparison with the traditional methods, and large computing time can be saved by this efficient method. The small eddy component can be captured by AIMs, and an accurate result can also be obtained.

30.1 Introduction Most of dynamic systems encountered in engineering are nonlinear continuous dynamic systems, which have a rich variety of nonlinear dynamical phenomena, such as the large and complex fluid-structure interaction systems. Normally, the finite element method or other numerical methods are used to approach the solutions of the governing equations, due to the difficulty of obtaining a solution in analytical form. As the results, the resulting equations are generally second order in time dissipative evolution equations with many degrees-of-freedom in sense of

J.-Z. Zhang () School of Energy and Power Engineering, Xi’an Jiaotong University, No. 28, Xianning West Road, Xi’an, Shaanxi 710049, People’s Republic of China e-mail: [email protected]

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 30,

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dynamics. For such kind of equations, there are several classical numerical schemes to approximate them, such as Newmark, Wilson-, Houbolt, and the Runge–Kutta scheme with higher precision if the system is transferred into phase space. However, great difficulties will arise from analyzing the nonlinear dynamics both qualitatively and quantitatively in a finite dimensional phase space of higher dimension [1]. For example, the analysis of nonlinear dynamical systems, based on the numerical schemes mentioned above, requires considerable computing time due to the large number of degrees-of-freedom, and some numerical round-off errors will have a strong influence on the long-term behaviors of the systems or the bifurcation analysis if the systems have a cluster of bifurcation points [2–5]. In other words, model reduction is the key to such obstacle and currently urgent for the bifurcation analysis by large scale numerical computation. Therefore, it is natural to reduce the model from higher to lower dimensions and to achieve an acceptable approximation of the original dynamics before large-scale numerical analysis is applied to the original system. Indeed, this reduction technique can be reached for some certain dissipative systems, by neglecting inessential degrees-of-freedom of the system and keeping the topology of the solutions unchanged [1]. Under such background, a number of researchers have developed many practical numerical algorithms [2]. For the linear dynamic system, the component mode synthesis techniques can be used to analyze the dynamic behaviors of the system, and much computing time will be saved, and an approximate result can be acceptable. However, for the nonlinear dynamic system, there are a few methods for the model reduction. Most of the numerical algorithms are developed based on the component mode synthesis techniques, which can be used for linear dynamical systems with acceptable approximate results, while few rigorous theoretical studies or the error estimate has been carried out on the influence of such reduction on the long-term behaviors, though a lot of numerical experiments are given [6–10]. Strictly speaking, due to the strong nonlinearities of some dissipative autonomous dynamical systems, the reduction of the systems has a greater influence on the solution at a certain degree, in a mathematically precise way [11, 12]. Fluid dynamics, a kind of continuous dynamic system, is governed by a set of nonlinear dissipative evolutionary equations, and there are many nonlinear phenomena, such as separation in boundary layer, soliton and turbulence, and some other open problems in it. In particular, the connections between fluid mechanics, partial differential equations and nonlinear dynamical system, and the global attractors and turbulence, are the essential heart of understanding of many important problems of natural science. There are a number of numerical analysis of Navier–Stokes equations based on Finite Element Method, and most of them are the adaptations of traditional Galerkin procedure [1]. However, an important deficiency of the existing numerical methods in the computation fluid dynamics is the cost of computingtime, that is, there are a large number of degrees-of-freedom after the system is approached by the discretization, and the system is the one with higher dimension from viewpoint of nonlinear dynamics. Hence, in the nonlinear continuous dynamic systems, it is the current aim to reduce the original system to a system with less degrees-of-freedom.

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On the other hand, it is well known that the asymptotic behaviors of some higher dimensional or even infinite dimensional dissipative dynamic systems evolve to a compact set known as a global attractor, which is finite-dimensional [13]. That means such kind of dynamic systems can be described by the deterministic flow on a lower dimensional attractor. Hence, it opens the way for the reduction of the dynamics of infinite-dimensional dissipative equations to finite-dimensional systems, or higher dimensional dissipative equations to a lower dimensional system. Consequently, various schemes have been used to construct a finite system for reproducing the asymptotic dynamics of the original dynamic system [14–19]. One of the schemes is the traditional Galerkin method, which ignores the small spatial structure of a solution. However, an important and well-known aspect of nonlinear dynamics is the sensitive dependence of the solution on the perturbations. Such perturbations include small variations in initial and boundary values, as well as numerical errors if a numerical computation method is adopted. A slight perturbation to the system may produce very important effects and significant changes in the system’s configuration after a long time [20]. The Center Manifold Theory can be applied to the system with a small number of modes, whose eigenvalues are close to the imaginary axis, but a small parameter variation from the critical value or a large parameter variation for some cases will have the effect that additional modes will become unstable, and the originally low dimensional system will not be valid anymore [1]. The theory of Inertial Manifolds (IMs) has shown that the long time dynamic behaviors of dissipative partial differential equation (PDE) can be fully described by that of a set finite ordinary differential equation to which the PDE is reduced on IMs. Roughly speaking, the IMs can be considered as a reduction method. In fact, the methods used to construct the IMs are the adaptations of various theories in the studies of center manifolds and integral manifolds. Then, the stability and bifurcation can be investigated based on the ordinary differential equation with relatively low dimension on the IMs, and some nature of nonlinear phenomena can be explained availably and feasibly. For decades, the theory of Inertial Manifolds is a novel technique for model reduction [1, 21–23]. Unfortunately, the existence of IMs usually holds only under the very restrictive spectral gap condition. Hence in practical applications, the concept of Approximate Inertial Manifolds (AIMs) has been introduced [12, 23, 24]. However, there are few studies on the IMs and AIMs for the second order in time nonlinear dissipative autonomous dynamic systems. In [25], AIMs for second order in time partial differential equations with delay are constructed, and some important results have been given. In light of Approximate Inertial Manifolds developed in infinite dynamic systems and Mode Analysis in linear structural dynamics, a combined method is presented to reduce the second order in time nonlinear dissipative autonomous dynamic systems with many degrees-of-freedom, which are encountered frequently in engineering, and the influence of model reduction on the long-term behaviors has been studied in detail, and the error estimate is also given [26]. Additionally, the AIMs has been introduced to the dynamic snap-through buckling analysis of shallow arches under impact load [27].

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In this study, following Optimum finite element nonlinear Galerkin algorithm for the Navier–Stokes equations [28], the multilevel finite element method is adapted to approach the AIMs for Navier–Stokes equations, and the governing equations can be projected onto the space dominated by the Inertial Manifolds, which is finite dimension.

30.2 Multilevel Finite Element Method For the sake of simplicity, the Approximate Inertial Manifolds approached by multilevel finite element method is given for the incompressible flow. The governing equations with elementary variables for the incompressible flow are 8 @uj ˆ ˆ D0 ˆ ˆ @x < j ˆ ˆ @ui @ui 1 @p @2 ui ˆ ˆ : C uj D @x C i @t @xj

@xj @xj

.i D 1; 2/:

(30.1)

The boundary conditions: D 1 [ 2 ; where 1 is boundary for the velocity, 2 the essential boundary condition for the pressure. And on boundary 1 , ui D uQ i .i D 1; 2/, on boundary 2 pij nj D pQi .i D 1; 2/, nj is the outward normal unit vector component on 2 , and tensor p @ui @uj : C pij D ıij C

@xj @xi The initial conditions:

u.xi ; 0/ D u0 .xi /: In a sense, the nonlinear phenomena are the features of the interaction between the higher mode and the lower mode in the system. The nonlinear Galerkin method is the new numerical method for the computational fluid dynamics. Following the Inertial Manifolds and the nonlinear dynamics, and it decomposes the unknown into two components, that is, the large eddy and small eddy, the essential interaction between them is the key for the schemes. For spectral method, such two components are approached by the two subspaces spanned by the Eigen functions of the positive definite operator. For the finite element methods, such kind subspaces are spanned by a coarse grid finite element space and a fine grid incremental finite element space. For the coarse grid finite element space, the quadrangle element is adopted to mesh the domain, as shown in Fig. 30.1. A quadrangle element with four nodes, namely, initial element, is shown in Fig. 30.2 as the initial mesh, and then it is refined by adding other five nodes, resulting in a refined quadrangle element with nine nodes, as shown in Fig. 30.3. It is

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Reduction on Inertial Manifolds of Navier–Stokes Equations

Fig. 30.1 The mesh of the domain Fig. 30.2 Quadrangle element with four nodes

Fig. 30.3 Quadrangle element with nine nodes

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clear that node 1, 2, 3, 4 in Fig. 30.3 are the same as that in Fig. 30.2, and hence such element is referred to as multilevel element by this regular refinement for the mesh. Accordingly, the shape function is constructed in the initial element, and the basis functions span the fundamental finite element space. Further, the shape function is constructed with the other five nodes to provide the basis function for the incremental finite element space. For clarity, the shape functions for the quadrangle element with four nodes are listed as follows, 1 1 .1 /.1 / N2 D .1 C /.1 /; 4 4 1 1 N3 D .1 C /.1 C / N4 D .1 /.1 C /: 4 4

N1 D

As for the quadrangle element with nine nodes, the hierarchical functions for the variable u can be presented as the following, remaining the fundamental shape functions unchanged, H1 D N1 H2 D N2 H3 D N3 H4 D N4 1 3 1 . /.1 / HN 6 D .1 C /.3 / HN 5 D 12 12 1 3 1 . /.1 C / HN 8 D .1 /.3 / HN 7 D 12 12 1 3 . /.3 /: HN 9 D 12 Following the Inertial Manifolds, the velocity in the element, namely, u.e/ , can be decomposed into u.e/ D y .e/ C z.e/ ; where y .e/ denotes the large eddy component, z.e/ the small eddy component. More, the following expressions can be obtained y

.e/

D

4 X

.e/ Ni yi ;

i D1

.e/

z

D

9 X

.e/ HN i zi :

i D5

30.3 The Weak Form of the Navier–Stokes Equations Following the Galerkin procedure, the weak form of the governing equations for incompressible flow can be obtained as Z

@uj ıp d D 0 @xj

(30.2)

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Z

389

@ui @ui 1 @p @2 ui C uj ıui d D 0 C @t @xj

@xi @xj @xj

By the Green formulas, (30.2) can be rewritten as Z Z @ uj .ıp/d D uN n ıp d @xj

1

(30.3)

(30.4)

where uN n is the norm component on 1 , uN n D uN j nN j : Similarly, (30.3) can be rewritten as the following form, Z

@ui @ui C uj @t @xj

Z @uj @ui @ p C .ıui / d D pNi ıui d ıui C ıij C

@xj @xi @xj

2

(30.5) The terms in the left hand side in (30.5) denotes the inertial force, normal stress, visual power from viscosity, respectively. And the terms in the right hand side denotes the visual power both from outer forces on the boundary and mass force.

30.4 Numerical Methods The pressure-correction method is applied to the numerical analysis of the governing equations. The Euler scheme is used to approach the derivative 8 .nC1/ @uj ˆ ˆ ˆ D0 ˆ ˆ < @xj ˆ ˆ .nC1/ .n/ .n/ .n/ ˆ uO ui 1 @p .nC1/ @2 ui ˆ .n/ @ui ˆ : i D C C uj t @xj

@xi @xj @xj

.i D 1; 2I n is the time step/

(30.6) For such implicit scheme, an intermediate velocity uO is introduced availably, and the momentum equations can be rewritten as the following form, .n/

uO i ui t

.n/ .n/ @ui

C uj

@xj

.n/

D

@2 ui 1 @p .n/ C

@xi @xj @xj

.i D 1; 2I n is the time step/ (30.7)

However, the variable p is explicit in the scheme, and its value can be set as that at the last step. Subtracting (30.6) from (30.7) gives, u.nC1/ i

t uO i D

@p .nC1/ @p .n/ @xi @xi

! (30.8)

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A scalar variable is also introduced, and let .nC1/

ui

uO i D

@ @xi

(30.9)

The pressure p at the next step can be expressed p .nC1/ D

C p .n/ t

(30.10)

Following the divergence of (30.9) and the continuity equation, a Poisson equation relevant to can be given as @2 @2 @Ou1 @Ou2 C D C @x1 @x2 @x12 @x22

(30.11)

Then, can be obtained from (30.11), and substitute it into (30.9) and (30.10), both u and p can be solved together.

30.5 Large Eddy and Small Eddy Components As stated above, the velocity in the domain can be decomposed into large eddy and small eddy components, namely, u D y C z. In the element, the velocity can be expressed as X X HN i z.e/ u.e/ D y .e/ C z.e/ D Hi yi.e/ C i : Similarly, the pressure in the element can be expressed as p .e/ D

X

.e/ i Pi

C

X

N i pN .e/ : i

The weight function in (30.5) can be chosen as ıui , and set ıui D HN j (j D N k (k D 5; 6; : : :; 9). Equations (30.4) and 5; 6; : : :; 9). As for (30.4), set ıp D ‰ (30.5) can be rewritten consequently as .e/

.e/ .e/

.e/ .e/

.e/

.e/ .e/

.e/ .e/

.e/

A.e/ zP1 C B .e/ z1 z1 C C .e/ z2 z1 C D1 pN .e/ C F1 z1 C F2 z2 D E1

.e/ .e/ .e/ .e/ .e/ .e/ .e/ .e/ .e/ A.e/ zP.e/ z1 z2 C C .e/ z.e/ C F3.e/ z.e/ 2 CB 2 z2 C D2 pN 1 C F4 z2 D E2

M1.e/ z1 C M2.e/ z2 D G .e/

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Furthermore, the final governing equations can be expressed in the following form, APz1 C Bz1 z1 C C z2 z1 C D1 pN C F1 z1 C F2 z2 D E1

(30.12)

APz2 C Bz1 z2 C C z2 z2 C D2 pN C F3 z1 C F4 z2 D E2

(30.13)

M1 z1 C M2 z2 D G

(30.14)

It is clear that the relationship between yi and zi can be expressed in the implicit form by (30.12) and (30.13), that is, coefficients are the function of yi . As yi is known, zi can be obtained from this relationship. Also, the Euler scheme is used to approach the derivative in time, uP ˛r D

C1/ / u.i u.i ˛r ˛r t

.˛ D 1; 2I r D 1; 2; : : : ; Nu / :

At t D ti C1 , the implicit scheme is chosen as the following, .i C1/

Az1

.i C1/ .i C1/ .i C1/ .i C1/ .i C1/ .i C1/ C t Bz1 z1 C Cz2 z1 C D1 pN .i C1/ C F1 z1 C F2 z2

/ D tE1 C Az.i

1 C1/ C1/ .i C1/ C1/ .i C1/ C1/ C1/ Az.i C t Bz.i z2 C Cz.i z2 C D2 pN .i C1/ C F3 z.i C F4 z.i 2 1 2 1 2 / D tE2 C Az.i 2

As z.i / at i step is known, z.i C1/ can then be obtained from the iterative following Newton–Raphson method, J .i C1/;.k/ x .i C1/;.kC1/ D J .i C1/;.k/ x .i C1/;.k/ R.i C1/;.k/ where 2

3 z1 x D 4 z2 5 : pN The residual is 2

3 R1 R D 4 R2 5 : R3 Herein, J is the Jacobean Matrix.

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30.6 Numerical Examples The NACA0012 airfoil is analyzed as a numerical example. The parameters of the systems can be listed as Kinematic viscosity of air D 1:8 105 N s=m2 , density D 1:205 kg=m3 , chord length c D 0:42 m, freestream velocity u D 100 m=s. Figures 30.4–30.6 are the results of the flow field using coarse mesh, and it is clear that there is no vortex in the boundary layer. As the small eddy components are considered, that is, the Approximate Inertial Manifolds is used to approach the solution of the system, some detail features of the system can be captured. Figures 30.7–30.9 are the results from multilevel finite element method. Referring to Fig. 30.10, it can be seen that the Approximate Inertial Manifolds can capture the small eddies in the boundary layer, in comparison with the traditional Galerkin method. It is well-known that such small eddies in the boundary layer is very important for the flow control in the engineering.

Fig. 30.4 Velocity of the flow from the traditional method

Fig. 30.5 Streamline of the flow from the traditional method

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Fig. 30.6 Pressure distribution from traditional method

Fig. 30.7 Velocity of the flow from AIMs

Fig. 30.8 Streamline of the flow from AIMs

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Fig. 30.9 Pressure distribution from AIMs

Fig. 30.10 Streamline of the flow from [29]

30.7 Concluding Remarks In comparison with the traditional methods, the Approximate Inertial Manifolds approached by the multilevel finite element method is feasible for the numerical analysis of complex flow, and much computing time can be saved with an accurate result. In particular, following Approximate Inertial Manifolds, the solution space can be decomposed into two subspace, namely, large eddy and small eddy components, and the method can capture the small eddies and a more accurate solution can be given. In sense of dynamics, the Approximate Inertial Manifolds can be considered as efficient method for model reduction for the nonlinear continuous dynamic systems. As the further work, some improvements will be developed for the numerical results. Acknowledgments This research is supported by Program for New Century Excellent Talents in University in China, No. NCET-07-0685.

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References 1. Steindl A, Troger H (2001) Methods for dimension reduction and their application in nonlinear dynamics. Int J Solids Struct 38:2131–2147 2. Zhang JZ, Liu Y, Chen DM (2005) Error estimate for influence of model reduction of nonlinear dissipative autonomous dynamical system on long-term behaviours. Appl Math Mech 26: 938–943 3. Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical system, and bifurcations of vector fields. Springer, New York 4. Wiggins S (1990) Introduction to applied nonlinear dynamical systems and chaos. Springer, New York 5. Seydel R (1994) Practical bifurcation and stability analysis: from equilibrium to chaos. Springer, New York 6. Friswell MI, Penny JET, Garvey SD (1996) The application of the IRS and balanced realization methods to obtain reduced models of structures with local nonlinearities. J Sound Vib 196: 453–468 7. Fey RHB, van Campen DH, de Kraker A (1996) Long term structural dynamics of mechanical system with local nonlinearities. ASME J Vib Acoust 118:147–163 8. Kordt M, Lusebrink H (2001) Nonlinear order reduction of structural dynamic aircraft models. Aerosp Sci Technol 5:55–68 9. Slaats PMA, de Jongh J, Sauren AAHJ (1995) Model reduction tools for nonlinear structural dynamics. Comput Struct 54:1155–1171 10. Zhang JZ (2001) Calculation and bifurcation of fluid film with cavitation based on variational inequality. Int J Bifurcat Chaos 11:43–55 11. Murota K, Ikeda K (2002) Imperfect bifurcation in structures and materials. Springer, New York 12. Marion M, Temam R (1989) Nonlinear Galerkin methods. Siam J Numer Anal 26:1139–1157 13. Temam R (1997) Infinite-dimensional dynamical system in mechanics and physics. Springer, New York 14. Titi ES (1990) On approximate inertial manifolds to the Navier–Stokes equations. J Math Anal Appl 149:540–557 15. Jauberteau F, Rosier C, Temam R (1990) A nonlinear Galerkin method for the Navier–Stokes equations. Comput Methods Appl Mech Eng 80:245–260 16. Schmidtmann O (1996) Modelling of the interaction of lower and higher modes in twodimensional MHD-equations. Nonlinear Anal Theory Methods Appl 26:41–54 17. Chueshov ID (1996) On a construction of approximate inertial manifolds for second order in time evolution equations. Nonlinear Anal Theory Methods Appl 26:1007–1021 18. Rezounenko AV (2002) Inertial manifolds for retarded second order in time evolution equations. Nonlinear Anal 51:1045–1054 19. Laing CR, McRobie A, Thompson JMT (1999) The post-processed Galerkin method applied to non-linear shell vibrations. Dyn Stab Syst 14:163–181 20. Zhang JZ, van Campen DH, Zhang GQ, Bouwman V, ter Weeme JW (2001) Dynamic stability of doubly curved orthotropic shallow shells under impact. AIAA J 39:956–961 21. Foias C, Sell GR, Temam R (1985) Varietes Inertielles des Equations Differentielles Dissipatives. C R Acad Sci Paris Ser I Math 301:139–141 22. Chow SN, Lu K (2001) Invariant manifolds for flows in Banach space. J Differ Equ 74:285–317 23. Foias C, Manley O, Temam R (1987) On the interaction of small and large eddies in turbulent flows. C R Acad Sci Paris Ser I Math 305:497–500 24. Foias C, Sell GR, Titi ES (1989) Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations. J Dyn Differ Equ 1:199–244 25. Rezounenko AV (2004) Investigations of retarded PDEs of second order in time using the method of inertial manifolds with delay. Ann Inst Fourier (Grenoble) 54:1547–1564 26. Zhang JZ, Liu Y, Cheng DM (2005) Error estimate for the influence of model reduction of nonlinear dissipative autonomous dynamical system on the long-term behaviors. J Appl Math Mech 26(7):938–943

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27. Zhang JZ, Liu Y, Lei PF, Sun X (2007) Dynamic snap-through buckling analysis of shallow arches under impact load based on approximate inertial manifolds. Dyn Continuous Discrete Impulsive Syst Ser B (DCDIS-B) 14:287–291 28. He YN, Li KT (1999) Optimum finite element nonlinear Galerkin algorithm for the Navier– Stokes equations. Math Numer Sin 21(1):29–38 (in Chinese) 29. Ren X, Li BH, Yin XY, Gao G (2007) Calculation of airfoil flows using GAO-YONG turbulence equations. J Aerosp Power 22(1):73–78 (in Chinese)

Part IV

Classic Vibrations and Control

Chapter 31

Diesel Engine Condition Classification Based on Mechanical Dynamics and Time-Frequency Image Processing Hongkun Li and Zhixin Zhang

Abstract In this research, mechanical structure dynamics for diesel engine working process are investigated in detail for diesel engine vibration signal analysis and pattern recognition. Time domain vibration signal can be looked on as several impulse forces’ responses according to mechanical dynamics analysis. Different part vibration signal can be used for different components’ fault diagnoses. It is very useful to determine the best suitable vibration signal for analysis according to structure dynamics analysis. Hilbert spectrum is used to construct time-frequency distribution because of its performance for nonstationary signal analysis. Time-frequency image technology is investigated in this research for diesel engine fault diagnosis. Euclidean distance is used to distinguish engines’ different working conditions. A single cylinder 1135 direct injection diesel engine with different working conditions classification is as an example to testify the effectiveness of this method. It can be concluded that this new approach can improve the accuracy for diesel engine condition classification.

31.1 Introduction Due to the complexity of mechanical systems and variation of working conditions, diesel engines often operate in an off-design condition, which could leads to poor performance, heavy wearing and even a breakdown of diesel engine [1]. Accurate feature extraction and pattern recognition technology is urgently needed based on different information, such as vibration signal, thermal parameters, etc. Condition monitoring and fault diagnosis is very important to prevent break down of diesel engines. Vibration signal is often used to evaluate diesel engine working H. Li () School of Mechanical Engineering, Dalian University of Technology, Dalian 116023, People’s Republic of China and State Key Laboratory of Structural Analysis of Industrial Equipment, Dalian 116023, People’s Republic of China e-mail: [email protected]

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condition and predict the occurrence of faults without interrupting its operation, because vibration signal contains lots of information about diesel engine condition [2]. Diesel engine vibration is a very complicated phenomenon caused by impulsive forces of moving components and external forces, such as cylinder combustion pressure, valve vibration, impulsive injection pressure, initial force, which occur simultaneously with external load disturbances. The vibration signals from cylinder head and cylinder block are typical nonstationary and nonlinear characteristics. At the same time, there is much noise interference for practical monitored signal analysis. Effective feature extraction and pattern recognition are very helpful for diesel engine fault diagnosis and preventative maintenance. Nowadays, timefrequency technology has been broadly investigated in vibration signal analysis. It has been testified for effectiveness in nondestructive evaluation applications. Among them, Wigner–Ville distribution, Wavelet analysis, and a recently developed method, Hilbert spectrum (HS), have been broadly investigated [2, 3]. In this research, HS is used to construct time-frequency image for diesel engine pattern recognition. Experimental data of a DI135 diesel with different conditions is used to evaluate the improved methodology for system pattern recognition and fault diagnosis. According to the result analysis, it can be concluded that this method is very promising for diesel engine preventative maintenance.

31.2 Diesel Engine Structure Dynamics Diesel engine’s work in complicated conditions. There are many impulse forces during its working condition. Complex physical and chemical processes will happen during the combustion process. At the same time, the movement of valves and pistons make the signals fluctuate during a working cycle and the difference in the max amplitude for cylinder head vibration signal. Thus, it is very necessary to divide different impulse forces during its working condition for the monitored vibration signal analysis. Figure 31.1 shows the piston–crank working condition. During one working cycle, the crank will rotate twice for a four-stroke diesel engine. It contains intake stroke, compress stroke, combustion stroke, and exhaust stroke. In every working stroke, there are different impulses for diesel engine cylinder head and cylinder body. The surface vibration signal is the combination of response for the different impulses. It can be used to analyze engine working condition. Diesel engine vibration analysis has been broadly used for its pattern recognition. Diesel engine vibration signal belongs to typical nonstationary and nonlinear signal. It is the response from its internal impulse forces. Among them, Cylinder pressure is a very important parameter for diesel engine working conditions estimation. But it is also very difficult to determine in real condition because it cannot give quantitative description. Cylinder pressure is directly related with the combustion process in a working cycle. The heat released with cylinder pressure and crank angle can be expressed with (31.1). The pressure with crank angle relationship is shown in Fig. 31.2.

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Fig. 31.1 Sketch map about the structure

Fig. 31.2 Sketch map of cylinder pressure

The maximum cylinder pressure will be after top dead centre (TDC). Although cylinder pressure is very useful for pattern recognition, the difficulty for monitoring limits its application. dV 1 dp dQ D p C V d 1 d . 1/ d

(31.1)

At the same time, vibration signal is closely related with cylinder pressure, which can be used to reflect engine working condition. For a single-cylinder diesel

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Fig. 31.3 Cylinder head vibration signal of singe diesel

engine, the vibration signal monitored on the cylinder head is shown as Fig. 31.3. The TDC information can give time domain information for the vibration signal. It is much convenient for practical vibration signal analysis. Vibration signal time information can be changed from time domain to angle domain, which is more suitable to practical vibration signal analysis. During one working cycle, there are four vibration responses corresponding to impulse forces shown in Fig. 31.3b. They are combustion process impulse (CPI) responses, which contain fuel injection impulse and combustion cylinder pressure impulse, exhaust valve close (OVC) impulse response, throttle force impulse (TI) response, and inlet valve close (IVC) response. Different impulse is corresponding to vibration signal. Thus, it can be used to separate vibration signal for feature extraction and fault diagnosis [4]. Obviously, different impulse response vibration signal can be used to classify different fault. If there is a fault in the inlet valve, it is very important to analyze IVC vibration signal. But if there is a fault in the combustion process, it is very important to analyze the CPI vibration signal. The combustion process contains lots of information. It is directly corresponding to diesel engine cylinder pressure and performance. Thus, combustion process vibration signal is used to diesel engine classification in this research. Mathematically, the relationship between a vibration signal Y .t/ of a linear timeinvariant system and its signal vibration source X.t/ can be expressed by (31.2). Y .t/ D h.t/ X.t/

(31.2)

Where: h.t/ is the impulse response of X.t/ and symbol denotes convolution of two functions. It can be further expressed as (31.3). Y .t/ D hi .t/ I.t/ C hg .t/ P .t/ C he .t/ E.t/ C hd .t/ D.t/ C n.t/

(31.3)

The vibration of an engine cylinder head can be simplified as a linear system of multi-inputs and single output as shown in Fig. 31.4. In Fig. 31.4, hi .t/; hg .t/; he .t/;

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Fig. 31.4 Model of cylinder head vibration

hd .t/ stand for the response paths of vibrations caused by combustion pressure impulsive response function, exhaust valve closing impulsive response function impact, throttle impulsive response function, and inlet valve closing impulsive response function. At the same time, the vibration signals generated by impulse forces follow fixed time regularity in every engine working cycle. Thus, the impulse response functions in the time domain can be separated and their individual characteristics studied.

31.3 Classification by Time Frequency Image 31.3.1 Hilbert Time-Frequency Spectrum It is obvious that diesel engine vibration signal is full of nonstationary characteristics. Traditional method, such as Fourier transform, is not enough to satisfy diesel engine vibration signal analysis. Time frequency distribution method should be used for its analysis. As stated above, Hilbert spectrum is used in this research for diesel engine vibration signal analysis. To get a meaningful Hilbert spectrum, a new signal decomposition method was introduced by Huang named as empirical mode decomposition [3]. In 2000, Professor Ma put forward a new signal analysis theory, named as local wave method, which is the development of EMD and HS [5]. For an arbitrary time series, X.t/, original data is decomposed to n intrinsic mode components Ci .t/ and a residual component rn as (31.4) according to empirical mode decomposition (EMD). n X X.t/ D Ci .t/ C rn .t/ (31.4) i D1

After obtaining each IMF component, (31.5) can be deduced by using Hilbert transform. It gives both the amplitude and the frequency of the real part of each component as a function of time.

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X.t/ D RP

n X

aj .t/ei j .t / D RP

j D1

n X

aj .t/ei

R

!j .t /t

(31.5)

j D1

Both the amplitude and the instantaneous frequency can be represented in a threedimensional plot, in which the amplitude can be contoured on a time-frequency plane. The TFD of the amplitude is expressed by the Hilbert spectrum, H.!; t/, as shown in (31.6). n R X aj .t/ei !j .t /t (31.6) H.!; t/ D RP j D1

HS is developed from instantaneous frequency. It is very suitable for nonstationary signal analysis. As for diesel engine working process, its vibration signal is full of nonstationary characteristics. Thus, it can be used for vibration signal analysis and condition classification. At the same time, the Hilbert time-frequency spectrum can be also looked as a two-dimension image. Nowadays, image classification has broadly used on face classification. It has been investigated and applied in other areas, which can help solve many practical problems. It will be helpful by using image recognition on time-frequency distribution analysis [6]. Image classification technology can be used to recognize diesel engine working condition.

31.3.2 Euclidean Distance There are many methods for image analysis and classification. Among them, Euclidean distance (ED) is greatly investigated and used in image analysis because of its simplicity. In mathematics, the Euclidean distance or Euclidean metric is the “ordinary” distance between two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theory. By using this formula as distance, Euclidean space becomes a metric space. The ED between points P D .p1 ; p2 ; : : : ; pn / and Q D .q1 ; q2 ; : : : ; qn /, in Euclidean n-space, is defined as: p .p1 q1 /2 C .p2 q2 /2 C C .pn qn /2 v u n uX Dt .pi qi /2

ED D

(31.7)

i D1

For two time-frequency images XI.m; n/ and YI.m; n/, the ED can be calculated according to (31.8). v uX n um X ED D t .XI.i; j / YI.i; j //2 i D1 j D1

(31.8)

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In (31.8), XI and YI are the standard time-frequency image and undetermined image. ED is the distance between the two images. The smaller the ED, the similar the two images [7]. In practical question analysis, EDmin D min.EDm /. If ED is satisfied with some scope, it can be looked as the two images are identical. That means the engine is working with standard condition. Therefore, a threshold value EDTh is determined to define the relationship between test sample and standard image. If EDTh > EDmin , it means the test sample is same as the standard image. Otherwise, it means that they are in a different working condition.

31.3.3 Condition Classification As stated above, mechanical dynamics is used to determine the vibration signal for analysis in this research. Then, Hilbert spectrum is calculated to construct timefrequency image. Euclidean distance is used to classify time-frequency image for diesel engine pattern recognition. Thus, a quantitative description for machine condition can be determined by the above method.

31.4 Example Analysis To testify the effectiveness of this developed method, a single cylinder diesel engine is used to simulate different working condition for pattern recognition. The experiment was carried on in a lab of Institute of Internal combustion engine, Dalian University of Technology. The testing platform is a single cylinder, direct injection diesel engine. There are four simulated working conditions for pattern recognition.

31.4.1 Experiment The data monitoring system is shown in Fig. 31.5. The main technical parameters of the experimental engine are shown in Table 31.1.

DI 135 Diesel engine

Vibration sensor

TDC Signal

Cylinder Pressure CBB366

Fig. 31.5 Sketch map of data acquisition for experiment

PDM-2000

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135 140 2.0 17.5 14.7 1,500 ®10 5 0:32 150 22 15

The vibration signal transducer was located on the cylinder head to monitor the combustion process. At the same time, there is not any vibration signal influence from other cylinder effect just for single cylinder diesel engine. Thus, the vibration signal measured from a cylinder head can be considered a classification signal for the working condition, including combustion and fuel injection process, for a single cylinder. It is very useful to estimate the combustion process. It is usually used to classify different working conditions.

31.4.2 Vibration Signal Analysis The monitored vibration signal was obtained from four working conditions, normal condition (NC), nozzle fault condition (NF), inlet fault (IF), and reduced injection fuel angle (RI). The Vibration signals were monitored with speed 1,500 rpm with 25% load. The sampling frequency and analysis frequency are 25,600 and 10,000 Hz, respectively. The time domain signal corresponding to the combustion process for different working conditions is shown in Fig. 31.6 according to diesel engine dynamics analysis. TDC is used to determine the time domain information for vibration signal. For the combustion process, the total monitored signal length is 512. It is obvious that the vibration signals are very similar in time domain. It is unable to separate different working conditions just according to vibration signal.

31.4.3 Hilbert Spectrum Recognition Based on empirical mode decomposition and Hilbert–Huang transform, Hilbert spectrum can be obtained for different working conditions shown in Fig. 31.7. The HS is a broad band frequency signal as shown in Fig. 31.7. The energy is also concentrated around TDC. At the same time, there is not much difference for the HS in different working conditions although there is difference in the time-frequency image. But it is not clear to distinguish different conditions.

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Fig. 31.6 Time domain signal in different conditions; (a) NC, (b) NF, (c) IF, (d) RI

Fig. 31.7 Time-frequency image for different working condition: (a) NC, (b) NF, (c) IF, (d) RI

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Fig. 31.8 Recalculation time-frequency image for different working condition: (a) NC, (b) NF, (c) IF, (d) RI

As stated above, Hilbert spectrum can be looked as a two-dimension timefrequency image. Thus, image processing technology can be used for feature extraction and fault diagnosis. At the same time, it is better to give more concentration for the energy distribution. That small energy can be removed, which can improve the expression for Hilbert spectrum. HS is recalculated and a new time-frequency image is determined in this research. After energy distribution recalculation, new time-frequency image can be obtained shown in Fig. 31.8.

31.4.4 ED Discriminant It is much clearer for different diesel engine working condition classification after the time-frequency image recalculation. The energy is also as not dispersed as shown in Fig. 31.7. It is more suitable for condition recognition. For quantitative description, ED is used in this research for different condition classification. There is much difference for different working cycles although they are in the same working

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Fig. 31.9 Time domain vibration signal for diesel engine cylinder head

Table 31.2 Classification results for different working conditions ED1 ED2 ED3 ED4 Test 1 0.3146 5.6163 5.6614 5.6650 Test 2 5.5007 0.9064 5.3414 5.3844 Test 3 5.6607 5.5260 0.8699 5.3159 Test 4 5.6844 5.5224 5.3372 0.7945

Classification result NC NF IF RI

condition shown in Fig. 31.9. Cyclostationary is very common for diesel engine [8]. It is not accurate just according to one cycle vibration signal. To reduce cyclostationary effect for pattern classification, multicycle time-frequency image average is used. Based on five time-frequency image average, four standard images can be obtained. These images can be looked on as the typical image for different working conditions. Then, ED is used to classify different working conditions for test samples based on above calculation equation (31.8). It can give a quantitative description for pattern recognition. The distance with different standard image is shown in Table 31.2 for different working condition classification. Its corresponding condition is also given in Table 31.2 according to ED. According to Table 31.2, the smaller is ED, the more similar the two images. That means the test sample is similar with standard image. The time image reflects machine working condition. It also means the similar of two monitored conditions. It can be recognized that two monitored signals are in the same condition if ED is very small. Therefore, ED is used to estimate the similar of two images. In this research, a threshold EDTh is defined equal to 1. When ED between the standard image and test image is smaller or equal to EDTh , it means that the test sample is same as the standard image. They are in the same working condition. Otherwise, they are in different working conditions. ED can be used to estimate the similarity between test sample and standard sample. According to ED, sample “Test 1” can be accurately classified as NC where ED is smaller than 1. It is also the same as practical condition. For other working condition classification, it can have the identical result. It is obvious that different conditions can be recognized according to ED. This method is convenient for practical vibration signal analysis and pattern recognition with determined working conditions.

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31.5 Conclusions Mechanical dynamics analysis is very important for machine pattern recognition and condition monitoring. It is the basis for engine vibration signal analysis. Diesel engine structure dynamics and impulse response characteristics are investigated in detail in this research. It is very important for time domain vibration signal determination for analysis and pattern classification. It can effectively reduce the calculation and improve the accuracy of pattern recognition based on mechanical dynamics analysis. Hilbert spectrum is used to construct time-frequency distribution, which is very effective for nonstationary and nonlinear signal analysis. Time-frequency distribution can be looked as a two-dimension image. Thus, image recognition method is investigated in this research. Euclidean distance is used to classify engines according to different working conditions. Different working condition classification of diesel engine is as an example to testify the effectiveness of this method. According to the classification result analysis, it can be concluded that this method is suitable for diesel engine pattern recognition and fault diagnosis. This developed method is much more suitable for separated structure analysis and fault diagnosis of diesel engine. For multicylinder with single body diesel engine, its structure dynamics should be further investigated. Acknowledgments The support from Chinese National Science Foundation (Grant No. 50805014) for this research is gratefully acknowledged. The first author also wishes to acknowledge to financial aid from State key laboratory of mechanical system and vibration, Shanghai Jiao Tong University (Grant No. VSN-2008-04) and State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology (Grant No. GZ0817).

References 1. Logan K, Inozu B, Roy P, Hetet JF, Chesse P, Tauzia X (2002) Real-time marine diesel engine simulation for fault diagnosis. Mar Technol 39(1):21–28 2. Zhou PL, Li HK (2005) Pattern recognition on diesel engine working conditions by wavelet Kullback–Leibler distance method. Proc Inst Mech Eng Part C J Mech Eng Sci 219(9):879–888 3. Huang NE, Shen Z, Long SR, Wu MLC, Shih HH, Zheng QN, Yen NC, Tung CC, Liu HH (1998) The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc R Soc Lon Ser A Math Phys Eng Sci 454:903–995 4. Li HK, Zhou PL, Ma XJ (2006) Marine diesel engine fault diagnosis by using improved Hilbert spectrum. J Ship Res 4:378–387 5. Ma XJ, Yu B, Zhang ZX (2000) A new approach to time-frequency analysis – local wave method. J Vib Eng 25(s):219–224 6. Wang WJ, McFadden PD (1993) Early detection of gear failure by vibration analysis-II: interpretation of the timefrequency distribution using image processing techniques. Mech Syst Signal Process 7(3):205–215 7. Lv C, Wang GZ (2005) Noise fault diagnosis based on time-frequency domain model. J Vib Shock (in Chinese) 24(2):54–57 8. Antoni J, Daniere J, et al (2002) Effective vibration analysis of IC engines using cyclostationarity. Part I – a methodology for condition monitoring. J Sound Vib 257(5): 815–837

Chapter 32

Input Design for Systems Under Identification as Applied to Ultrasonic Transducers Nishant Unnikrishnan, Yicheng Pan, Marco Schoen, Ajay Mahajan, Jarlen Don, and Tsuchin Chu

Abstract An input design system identification (IDSI) method is outlined in this chapter based on input/output data gathered from random excitation of a system so as to excite all modes. The data set is then used to compose a new set of more focused input data, from which the system is excited again and identified. In this chapter, the input design method is used for the system identification of an ultrasonic sensor pair (transmitter–receiver) so as to obtain an accurate model for it. This model is essential for the analysis of a 3D position estimation system that uses ultrasonic transducers. A single transmitter is attached to the point of interest, and its position is triangulated based on signals received at multiple receivers fixed in a 3D environment. An accurate model for the response of the ultrasonic receivers is essential in the eventual optimization of the entire system. This chapter only presents results for a single pair (called the system in this chapter), but the results will be applicable for the entire 3D system. Results are given that show the comparison of the actual output signal of the system and the output of the model obtained from the IDSI method as well as the identifiability indicator and an identified state-space model. It is shown that the IDSI improves system identification even in the presence of excessive noise.

32.1 Introduction The field of system identification has been developed for a multitude of purposes. As each discipline gives its motivation, there is a common goal: one wants to infer accurately the characteristics of a particular system from its input/output data. There exists a great multitude of proposed and developed identification methods for linear, stochastic, and autonomous systems [1–4]. The main focus of this work

T. Chu () Department of Mechanical Engineering and Energy Processes, Southern Illinois University Carbondale, 1230 Lincoln Drive, Carbondale, IL 62901-6603, USA e-mail: [email protected]

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is the parametric system identification methods, in particular the Observer/Kalman filter Identification method (OKID) [2] and the indirect closed-loop system identification method [5, 6]. In general, one seeks to obtain a system representation that is as accurate and close to the original system as possible. The system identification methods used here employ autoregressive with exogenous input (ARX) model representations and least-squares estimation for the determination of its parameters. The objective of this work is to present a new input design which improves the accuracy of the identification results of ultrasonic sensors. Experimental results show that the new method is effective in identifying ultrasonic transducers with high process and/or measurement noises.

32.2 Input Design Formulation for ARX Models Consider a stochastic, discrete time, autonomous system given in state space form: xkC1 D Axk C Buk C wk yk D Cxk C Duk C vk

(32.1)

where x 2
(32.2)

Equation (32.2) is known as the steady state filter innovation model, where "k D yk C xO k contains the new information. The steady state Kalman filter gain K is guaranteed if the system is detectable and .A; Q1=2 / is stabilizable. Rewriting (32.2) in input/output description and using AN D A.In KC /, one can derive an ARX model representation yk D

p X

C ANi 1 AKyki C

i D1

p X

C ANi 1 Buki C "k

(32.3)

i D1

provided that A.In KC / is asymptotically stable, and the model order p is large enough such that ANp Š 0. Defining the matrix coefficient ai and bi ai D C ANi 1 AK

and bi D C ANi 1 B

(32.4)

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Then the ARX model for such a system can be given as yk D

p X

ai yki C

i D1

p X

bi uki C "k

(32.5)

i D1

Since the system model is unknown, the parameter matrices for the ARX model (ai and bi ) have to be estimated from the input/output data. Note that the finite ARX models given in (32.3) and (32.5) represent a truncated infinite transversal. If one considers each summation of the finite ARX model series as a contributor for the current estimate of the output, the ARX model of an open-loop system can be represented in matrix format as follows: 2

3 2 yNk1 a1 6 yN 7 6 0 6 k2 7 6 6 7 6 6 yNk3 7 D 6 0 6 : 7 6: 6 : 7 6: 4 : 5 4: yNkp 0

0 a2 0 :: : 0

0 0 a3 :: : 0

:: :

32 3 2 0 yk1 b1 7 6 6 07 7 6 yk2 7 6 0 76 7 6 0 7 6 yk3 7 C 6 0 7 6 6 :: 7 6 :: 7 7 6: : 5 4 : 5 4 :: ykp ap 0

with yQk D

p X

0 b2 0 :: : 0

0 0 b3 :: : 0

:: :

32 3 0 uk1 7 6 07 7 6 uk2 7 76 7 0 7 6 uk3 7 (32.6) 7 6 :: 7 6 :: 7 7 : 54 : 5 ukp bp

yNki

(32.7)

i D1

If one wants to design a new input sequence to yield a new ARX model with p2 < p model order, one can use (32.6) to write D ƒy C Qu

(32.8)

where 3 2 yNkp2 C1 ap2 C1 7 6 6 0 6 yNkp2 C2 7 6 7 6 7; ƒ D 6 y N D6 k3 6 0 7 6 6 : 6 :: 7 4 :: 4 : 5 0 yNkp 2

2 3 2 bp2 C1 ykp2 1 6 0 6ykp 2 7 2 6 7 6 6 7 6 y D 6ykp2 3 7 ; D 6 0 6 : 7 6 :: 4 :: 5 4 : ykp2

0

3

ap2 C2 0 :: :

ap2 C3 :: :

:: :

0

0

ap2 C1

0

0 0

0 0 0 :: : bp2 C1

0

0 0

bp2 C2 0 :: :

bp2 C3 :: :

:: :

0

0

0 0 0 :: :

7 7 7 7 7 5 3 7 7 7 7 7 5

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and

2

3 ukp2 1 6 ukp 2 7 2 6 7 6 7 uQ D 6 ukp2 3 7 6 7 : :: 4 5 ukp2

The coefficients ai and bi of the ARX model are matrices for multiple input and multiple output systems. Since each element of (32.8) represents a contribution to the current output, the influence of this summation can be minimized by setting D0

(32.9)

For a linear time-invariant system, the matrices ƒ and are constant. If ƒ and

are known and is invertible, one can solve for the input which satisfies (32.9) uQ D 1 ƒy or

a y ; uQ kp2 j D bp1 2 Cj p2 Cj kp2 j

j D 1; 2; : : : ; p

(32.10)

where p D p p2 . Using (32.4) and rewriting ƒ and in (32.8) yields 2 Np2 C A AK 6 0 6 6 0 ƒD6 6 :: 4 : 0 2 Np2 CA B 6 0 6 6

D6 0 6 :: 4 : 0

C ANp2 C1 AK 0 :: :

0 0

C ANp2 C2 AK :: :

:: :

0

0

0

C ANp2 C1 B 0 :: :

C ANp2 C2 B :: :

:: :

0

0

0

0 0

0 0 0 :: :

3 7 7 7 7 (32.11) 7 5

C ANp1 AK 0 0 0 :: :

3 7 7 7 7 7 5

(32.12)

C ANp1 B

If is invertible, and using (32.10), the revised input uQ can be written as ukp2 j D .C ANp2 Cj B/1 C ANp2 Cj AKykp2 j

for i D 1; 2; : : : ; p (32.13)

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If C 2 <non , B 2
(32.14)

32.3 System Identification Algorithm A number of different parametric system identification algorithm can be employed for the proposed input design. The one used in this work is the Observer Kalman Filter System Identification Algorithm

(OKID). Consider an output vector yM D yp ypC1 : : : yL1 and the information matrix u k D k yk where L is the data length. Forming the following overall information matrix from experimental input/output data: 2 3 up1 : : : uL1 up 6p1 p2 : : : L2 7 6 7 VN D 6 : (32.15) : :: 7 : : : : 4 : : : : 5 0

1

:::

Lp1

and the Markov Parameter sequence

YN D D CBN CANBN CAN2 BN : : : CANp1BN

where BN D B KD K . Then one can write yM D YN VN C " C C ANq xO

(32.16)

(32.17a)

where xO D xO 0 xO 1 xO 2 : : : xO Lp2 and " D "p "pC1 "pC2 : : : "L1 . If p is large enough, the term CANp becomes zero and one can write yM D YN VN C "

(32.17b)

Employing a least-square method, one can compute the sequence YN , which is the observer Markov parameters:

1 YN D yM VN T VN VN T

(32.18)

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where YN D YN0 YN1 YN2 : : : YNq and YN0 D D i h YNk D C ANk1 BN D YNk.1/ YNk.2/

(32.19a) (32.19b)

To recover the system Markov parameters, one can compute Y0 D YN0 D D Yk D YNk.1/

k X

YNi.2/ Yki

(32.20a) (32.20b)

i D1

To recover the system description in state-space form, one can employ an eigensystem realization method such as the following: Construct a block Hankel matrix from the system Markov parameters 3 2 Y1 Y1 : : : Y1 6Y1 Y1 : : : Y1 7 7 6 H0 D 6 : (32.21) 7 :: : : 5 4 :: : : Y1 Y1 : : : Y1 Using the singular value decomposition, the block Hankel matrix H0 is factorized as follows: (32.22) H0 D R˙S T nn are two orthogonal matrices and RT R D Im , where R 2 <mm and S 2 <

s0 , and s D diag 1 ; 2; : : : n . i are the sinS T S D In , ˙ D 00 gular values and are in the following order: 1 2 : : : r > 0, where r D rankfH0 g. The order of the identified system is determined by examining the magnitudes of the singular values. The singular values with relatively high magnitudes are associated with the number of states of the system, while the singular values with relatively low magnitude are assumed to be noise related. A construction of a minimum order system representation can be established as follows:

AO D ˙n1= 2 RnT H1 Sn ˙n1= 2

(32.23)

BO D ˙n1= 2 SnT Er

(32.24)

CO D

(32.25)

T Em Rn ˙n1= 2

T

D Im 0m : : : 0m and m are the number of where ErT D Ir 0r : : : 0r , Em outputs and r the number of inputs. To see what the input design does, consider the case for which p D 3, p2 D 2. Formulating the new output: yQ1 D CAK yQ1 C CBQu0 C "1 D "1

(32.26)

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yQ2 D CAK yQ0 C CBQu1 C "2 and using uQ 1 D B 1 AKy1 and yQ1 yields yQ2 D "2

(32.27)

Continuing in the same fashion yields: yQ3 D CAKCAK"1 CAKCBu1 C "3 yQ4 D CA2 KCAK"1 C CAK"3 C "4 CA2 KCBu1 C CBu3

(32.28) (32.29)

yQ5 D CA3 KCAK"1 C CA2 K"3 C CAK"4 C "5 CA3 KCBu1 C CABu3 C CBu4 (32.30) From the above equations, the output for the second identification experiment is composed out of a series of residuals " and the open-loop Kalman filter gain Markov parameter and a series of original inputs times the open-loop system Markov parameters [7]. It was shown that the residual has a minor contribution to the output of the system when compared to the truncated terms of the ARX model; therefore, the output of the system excited by the new input is dominated by the open-loop Markov parameters.

32.4 Input Design for System Identification The input given in (32.14) is used in subsequent identification experiments as depicted in Fig. 32.1. In the first step, the system is excited by a random input and the input/output data is recorded. From that data set, the corrected input is computed by

Fig. 32.1 Flowchart of proposed IDSI algorithm

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Fig. 32.2 Input signal design with repeated

employing (32.14) and the estimated ARX model parameter matrices. The corrected input is used for the second identification experiment. The original input is windowed p data points at a time, and p2 data points are substituted by the newly computed values and used for the second identification experiment as shown in Fig. 32.2. The new ARX model order p2 is chosen to be less than p such that the information matrix for the parameter estimation remains of full rank.

32.5 Experimental Results The method proposed in this paper is used for the identification of the mathematical model of an ultrasonic transmitter–receiver pair shown in Fig. 32.3. The 3D system uses a single transmitter and multiple receivers fixed in a 3D reference frame [8]. The ultrasonic transducers being used are 75 KHz transducers and do not come with a good mathematical model. A good model would greatly help in the development of a simulation procedure that is being used to select the optimum electronic components for the signal conditioning circuit being developed. A white Gaussian, zero-mean, noise signal was fed in to the system (a single transmitter– receiver pair) and the output recorded as shown in Fig. 32.4a, b. The output was then used to revise the input, which was then again fed in to the system, and the revised output was recorded as shown in Fig. 32.5a, b.

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Fig. 32.3 Experimental set-up

Fig. 32.4 Original: (a) input and (b) output signal

Fig. 32.5 Revised: (a) input and (b) output signal

The IDSI algorithm was then used to identify the system model and develop the state space model [9] as shown below: xkC1 D Axk C Buk yk D C xk C Duk

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where 2

0:0041635977039 0:98480731601734 0:03653877050420 6 0:98480731601734 0:00488328785749 0:11368525167570 6 6 0:0365387705042 0:1136852516757 0:58870091911101 6 6 A14 D 60:06288097942857 0:01614990771161 0:73027542330511 6 6 0:01417428012868 0:01394298451211 0:16010304585432 6 4 0:01972613525657 0:03142712127999 0:04006585406940 0:00501883955118 0:00774233608568 0:03179518814198 3 0:06288097942857 0:01614990771161 7 7 0:73027542330511 7 7 7 0:62566436272868 7 7 0:10215569036939 7 7 0:13144819221398 5 0:01969274099346 3 2 0:01417428012868 0:01972613525657 0:00501883955118 60:01394298451211 0:03142712127999 0:007742336085687 7 6 6 0:16010304585433 0:04006585406941 0:031795188141987 7 6 7 6 A57 D 6 0:10215569036939 0:13144819221398 0:01969274099345 7 7 6 6 0:96043352146054 0:18334817689745 0:041597296513137 7 6 4 0:18334817689745 0:84572873359289 0:43739360310728 5 0:04159729651314 0:43739360310727 0:89607292609058 3T 2 3 2 0:01153858488056 0:01153858488056 6 0:00079517351442 7 6 0:00079517351442 7 7 6 7 6 6 0:01506526977112 7 6 0:01506526977112 7 7 6 7 6 7 6 7 6 ; C D 6 0:00202684478110 7 ; B D 6 0:00202684478110 7 7 6 7 6 6 0:00396350806899 7 6 0:00396350806899 7 7 6 7 6 4 0:00639750685067 5 4 0:00639750685067 5 0:00078894718034 0:00078894718034

D D 5:716827158140736e 006 and A D Œa14 A57 Note that this is a discrete time state-space model with a sampling time of 2 MHz. For the identification, the following parameters were selected: number of states n D 7, p D 75, p2 D 50. Figure 32.6 shows the response of the IDSI identified model of the ultrasonic transducer pair. Figure 32.7 shows the actual response of the system. Note that the model response is similar in terms of predicting the magnitude response as well as the time delay in triggering of the signal. It is not very good in predicting the duration of the signal response.

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Fig. 32.6 Response of the IDSI sensor system model

Fig. 32.7 Actual response of ultrasonic sensor system

32.6 Conclusions and Recommendations An IDSI method is outlined in this chapter based on input/output data gathered from random excitation of a system so as to excite all modes. The data set is then used to compose a new set of more focused input data, from which the system is excited again and identified. In this chapter, the input design method is used for the system identification of an ultrasonic sensor pair (transmitter–receiver) so as to

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obtain an accurate model for it. This model is essential for the analysis of a 3D position estimation system that uses ultrasonic transducers. A single transmitter is attached to the point of interest, and its position is triangulated based on signals received at multiple receivers fixed in a 3D environment. An accurate model for the response of the ultrasonic receivers is essential in the eventual optimization of the entire system. This chapter only presents results for a single pair (called the system in this paper), but the results will be applicable for the entire 3D system. Results are given that show the comparison of the actual output signal of the system and the output of the model obtained from the IDSI method. This work has applications in the identification of all complex linear and nonlinear systems. It is particularly focused toward those systems that can be easily excited by a random noise signal and the output response can be easily recorded.

References 1. Ljung L (1987) System identification – theory for the user. Prentice-Hall, Englewood Cliffs, NJ 2. Juang J-N (1994) Applied system identification. PRT Prentice-Hall, Englewood Cliffs, NJ 3. Gustavsson I, Ljung L, Soderstrom T (1977) Survey paper: identification of processes in closedloop-identifiability and accuracy aspects. Automatica 13:59–75 4. Chen CW, Juang J-N, Huang J-K (1993) Adaptive linear identification and state estimation. In: Leondes CT (ed) Control and dynamic systems: advances in theory and applications, vol 57, multidisciplinary engineering systems: design and optimization techniques and their application. Academic, New York, pp 331–368 5. Huang J-K, Hsiao M-H, Cox DE (1996) Indirect identification of linear stochastic systems with known feedback dynamics. J Guid Control Dyn 19(4):836–841 6. Huang J-K, Lee HC, Schoen MP, Hsiao M-H (1996) State-space system identification from closed-loop frequency response data. J Guid Control Dyn 19(6):1378–1380 7. Schoen MP (1997) Input design for system under identification. PhD dissertation, Old Dominion University 8. Mahajan A, Walworth M (2001) 3D Position sensing using the difference in time-of-flights to various receivers from a single transmitter of wave energy. IEEE Trans Robot Autom 17(1):91–94 9. Schoen MP, Kuo C-H, Chinvorarat S, Huang J-K (1997) Parameter identifiability for system under identification using ARX models. In: Proceedings of the 18th ASEM conference, Virginia Beach, VA, October 23–26, pp 175–181

Chapter 33

Development of a Control System for Automating of Spiral Concentrators in Coal Preparation Plants Josh Hoelscher, Yicheng Pan, Manoj Mohanty, Jarlen Don, Tsuchin Chu, and Ajay Mahajan

Abstract Spiral concentrators have been widely used in coal preparation plants in Illinois and elsewhere to clean 1 mm 150 m particle size coal fraction. The major factors which have made spiral concentrator so popular include its low capital and operating cost, no chemical reagent or dense medium requirement, and the ease of operation/maintenance. Spirals are capable of providing excellent clean coal recovery although at a relatively high ash content. Like any other water-only separation process, spirals are also susceptible to continuously fluctuating feed quality and solids content in the feed slurry, which are quite common in a plant. The main objective of the project is to develop an inexpensive control system for spiral to automatically adjust the splitter position with fluctuating feed characteristics to maintain the desired effective separation specific gravity (density cut-point).

33.1 Introduction This chapter addresses the improvement of product quality and quantity from spiral circuits, which typically clean 70% of the feed to the fine coal circuit of a preparation plant. Spiral concentrators are widely used in coal preparation plants in Illinois and elsewhere because of the ease of their operation, low cost, and the ability to achieve high clean coal yield. However, the product quality obtained from single stage spiral cleaning is relatively inferior. In addition, unlike many other density based cleaning circuits, spiral circuit has no means of being controlled from a plant control room through a PLC. In fact, spiral circuit is the only link in the chain of a modern day preparation plant, which has yet to be automated. Spiral concentrators are used in coal preparation plants to clean 1 mm 150 m particle size coal fraction, which is too fine to be effectively cleaned by a heavy

T. Chu () Department of Mechanical Engineering and Energy Processes, Southern Illinois University Carbondale, 1230 Lincoln Drive, Carbondale, IL 62901-6603, USA e-mail: [email protected]

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 33,

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medium cyclone and too coarse for froth flotation cells or flotation columns. Spiral is a flowing film separator, in which the lightest particles move to the outermost section of the spiral profile, whereas the heaviest particles remain in the inner most section. There are usually two splitters at the discharge end of a spiral to produce three product streams, i.e., clean coal product, middlings and tailings, respectively. The splitter position, which decides the clean coal yield and product quality, is typically set at one point during initial installation and rarely adjusted again. This phenomenon results in a significant loss of clean coal to the tailings stream with fluctuating feed characteristics and solid loading in the feed stream to the spiral. To explain this phenomenon, let us just consider one splitter (for simplicity) in the following spiral profile schematic (Fig. 33.1). A preliminary test conducted to prepare data indicated a 20% reduction (from 75.9 to 55.9%) in clean coal yield to the product stream resulting because of a change in the feed solids content from 20 to 10%. A higher product ash content of 12.8% in comparison to 10.6% was caused because of the above reduction in feed solids content. Understandably, the reduction in clean coal yield and clean coal ash content was caused by a reduction in specific gravity of separation (cut-point). It was possible to maintain clean coal yield at the original level of nearly 76% by a manual adjustment of the splitter position one step in-ward (from the original location between the sections D and E to a new location between C and D in Fig. 33.1). Similar adjustments of the splitter position are required to maintain the same density cut-point to deal with many other fluctuations, commonly encountered in the mine and plant operating environment, which affect the feed flow rate, feed washability, distribution of feed flows in the spiral bank, etc. Past studies [1, 8] indicate that it is essential to maintain the same density cut-point in each spiral in a spiral-bank to achieve the maximum yield from a spiral circuit. Common drawbacks of single-stage spiral operation, listed by many investigators [2, 4, 6], include high density cut-point and misplacement of rock to clean coal. Two-stage spiral operations were recommended to rectify the above mentioned draw-backs. Luttrell et al. [5] and [3] examined a variety of circuits for two-stage spiral operation and concluded that a rougher-cleaner circuit, where the middling

Fig. 33.1 Six sections (A–F) of the profile of the last turn of a test spiral; the solid arrow represents the original splitter position, whereas the dotted arrows represent various possible positions of the splitter

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and clean streams are remixed and retreated by a second spiral, produces the lowest density cut-point and rejects most misplaced rocks. However, the downside to this two-stage configuration is that it requires substantially more number of secondary spirals since both clean coal middling products must be rewashed [4]. By comparing the performance obtained from two-stage middling treatment spiral circuits and the recently commercialized, PrepTech’s SX7 compound (two stages of spiral cleaning in a single taller unit) spiral, Bethel and Arnold [2] recommended the use of the latter over the former both from technical and economic angles. In light of the above discussion, the main goal of this project was to improve upon the spiral cleaning performance, so that fine clean coal yield and quality can be significantly improved in coal preparation plants in Illinois and elsewhere. The specific objectives include the development of (1) a low cost microprocessor-based control system for automatic adjustment of the splitter position in each spiral (not each start) to maintain the desired density cut-points irrespective of the fluctuations in the plant feed; and (2) a technique to monitor the particle mass (function of both size and density) across the spiral profile at the discharge end, and then based on this knowledge adjust the splitter setting to obtain a desired clean coal yield. Three types of systems were evaluated: mechanical strain gage type, ultrasonic type, and a pressure sensor mat type. The criteria for selecting the right type of control system was based on several key factors, such as cost, reliability, robustness, ease of retrofitting to existing spiral banks, and minimal maintenance. Coal spirals are widely used in preparation plants to clean 1 mm 150 m particle size coal fraction, which is too fine to be effectively cleaned by a heavy medium cyclone and too coarse for froth flotation cells or flotation columns. Spiral cleaning is achieved by two types of flow of the feed coal slurry: primary downward flow and a secondary circular transverse flow on the spiral profile. The latter creates a decreasing density gradient across the spiral profile (rock has higher density than coal), whereas the former helps in carrying the density gradient up to the discharge end, where it is split into three (typically) product streams, namely clean coal, middling, and tailings stream. Understandably, the density gradient and thus the gradient of ash content for different particle size fractions will not be the same, since mass is a function of both particle size and density. Luttrell et al. [4] explain the misplacement of rock in the clean coal stream due to the opposite direction of flow pattern in the lower and upper sections of the spiral. The product splitters are typically so placed that the entire upper section and also the upper portion of the lower section report to the clean coal launder. Understandably, the ideal location for the product splitter position is not fixed. It is a function of the amount of solids on the spiral profile (solid loading), amount of total slurry (volumetric flow) on the profile and also the type of coal (washability characteristics) being treated at a given point of time. Since all of these three conditions fluctuate in a plant environment, the ideal location for the splitter position also shifts. A control system based on the current study will constantly sense the distribution of solid and liquid on the spiral profile and accordingly an actuator will alter the splitter position to the ideal location.

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33.2 Technical Approach The problem can be stated as follows: Develop a technique to monitor the particle size (through the specific gravity of the slurry), and then based on that knowledge control the splitter setting. This problem has the following three components: 1. Find a technique that can monitor the particle size in real time 2. Actuate the splitter 3. Develop a microprocessor based control system that uses the sensor data and actuates the splitter appropriately The whole system should be: 1. 2. 3. 4. 5.

Economical Reliable Robust Easily retrofitted to existing set-ups Zero or very little maintenance

This chapter describes the control system, actuator, and the GUI developed for the project. For demonstration purposes, the mechanical strain gage system, described in the chapter, is outlined below. The main idea is to estimate the specific gravity of the slurry at multiple lateral positions as it flows of the slide. Figure 33.2 shows a schematic of the concept which includes a sensing system, a microcontroller, and an actuated splitter. The slurry will flow in to cups whose volume is known. As these cups fill up to the brim, a strain will be recorded on the cantilevers which will be monitored by the microprocessor. We are only interested in the maximum strain, as that indicates the weight of the slurry when it completes filling up the cups. This is then used to estimate the specific gravity of the slurry on either side of the divider, and an appropriate change can be made to the actuator

Fig. 33.2 Control system concept

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driven divider. Once reliable readings are taken, then a mechanism could be used to empty out the cups, or one could use an “off-center center of gravity” that would topple the cup contents as soon as it is completely filled. Once the cups are emptied, the process can start all over again. The greatest advantage of this method is its simplicity, robustness, and economical nature. The draw back on it is that it relies on the correlation of specific gravity to particle size. We do intend to study this correlation, and if it exists, then this will be the “approach of choice” for us. The system is implemented using five subsystems: 1. Metal arms with mounted strain gages to monitor the stress because of weight in the channels. 2. Wheatstone bridges to create variable voltage outputs based on the weights on these arms. 3. Differential amplifiers which amplify the difference in the values obtained from the Wheatstone bridges. 4. A PIC microcontroller whose program can monitor the voltages provided by the amplifiers to compute the difference in weight applied to the arms and control a motor to direct the flow of liquid in attempt to equalize the weight applied to each arm. 5. A motor which will move an apparatus which will redirect the flow of the coalfilled liquid in order to equalize the weight in the channels.

33.2.1 Metal Arms and Wheatstone Bridge Figure 33.3 has shown the laboratory setup for the control system. The arms have had strain gages mounted on them, which offer approximately 120 Of resistance

Fig. 33.3 Laboratory set up for control system

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unstrained. When pressure is applied, this value can increase up to approximately 1 . The Wheatstone bridges, which are identical, are accomplished by three equal resistances in addition to the resistance that the gage offers. Each is equivalent to 120 , which is accomplished by 110 and 10 resistances in series.

33.2.2 Differential Amplifiers To amplify the voltages received from the wheatstone bridges, amplifier tlv2464, which includes four amplifiers, two of which are used for the two bridges. Figure 33.4 shows the schematic for the signal conditioning circuiting. The diagram is given below. R1 is set at 1 k with R2 at 820 k being used for the gain control. R3 is 1:5 M

and is used to offset the approximately 1.4 V at the output when no stress is on either arm due to the slightly unbalanced bridge and high gain. R2 will change as different gain requirements arise.

33.2.3 PIC Microcontroller and Motor The PIC microcontroller used in this system is the PIC18f4550. It is clocked at 12 MHz, and it continuously monitors the voltages output by both differential amplifiers. Based on the difference in these values, with attention paid to which is greater, it will instruct the motor to move to one of five degree positions: 30, 15, 0, 15, 30. The voltages read in are translated to hexadecimal numbers between 0 and 255, and a table is given below to illustrate the logic of the system. The hexadecimal values given above are subject to immediate change due to further system calibration. This also applies to the degree positions as they are more fine-tuned. The high hexadecimal values shown for the right arm being greater are due to the difference being less than 0, and the value being “wrapped around.” Avoiding inconclusive logic in the 50 to B0 value range is accomplished by noting which arm is experiencing the

Fig. 33.4 Circuiting schematic

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greater weight. The motor in place is controlled by digital pulses, spaced 20 ms apart. Based on the length of the pulse (1.25–1.75 ms), the motor will position itself from 0ı to 180ı , respectively. The motor’s 90ı position is taken as our neutral position, with our positions for the motor being effectively 60, 75, 90, 105, and 120ı from left to right. These values are accomplished by applying the proportion of degree movement to total pulse range to obtain the desired pulse length, with 1.5 ms being 90ı , or our neutral. The values for 60, 75, 90, 105, and 120ı are 1.417, 1.4583, 1.5, 1.5417, and 1.583 ms respectively. Of course, these values are also subject to further refinement. In addition to this, the current system also includes an RS232 chip which allows us to communicate to the PC through the HyperTerminal program. Currently, this program is used to monitor the voltage values in hex, as well as the current position the motor is at for debug and calibration purposes. A Graphical User Interface (GUI) has been designed that does the following:

Sets up communication with the PIC microcontroller Sets up the number of sensor channels Sets up communication with the motor and displays the position Collects data, stores it, and plots it

33.2.4 Demonstration This section shows a demonstration of the strain gage based mechanical system as hooked to the motor. One arm (with strain gage 1) was pushed down, and this causes the motor to move in one direction (eventually this will be controlling the splitter). An applied force on the other arm causes the motor to move in the opposite direction. Given below are screen captures of the GUI in the set-up phase as well as the operation phase (see Fig. 33.5). In the set-up phase, set the serial port to COM 1 or 2. Set the Baud rate to 9,600. Then set the number of channels, and pick on online or offline operation. In the operation phase, as one arm was pushed down, the strain gages show a strain, and the motor rotates a predetermined angle; as highlighted in Fig. 33.5.

Fig. 33.5 Computer based GUI

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Fig. 33.6 Actual system assembly

33.2.5 Assembly Figure 33.6 shows how the samples will be taken from the slurry flow. Since the spiral has a curved profile from the inside to the outside, the blue bars are adjustable in length and angle to accommodate for any misalignment. The blue bars are held in place by clamping blocks. Attached to the end of the blue bars will be the cups that will draw samples from the flow. On the top of the blue bars, strain gages will be placed as far away from the cups as possible without disrupting the function of the clamping blocks. Both of the cups will be attached to a central shaft that for preliminary experiments will be turned by hand and in the final stages driven by a motor.

33.3 Mechanical System Design A new system has been designed that takes advantage of the different specific gravity of the slurry at different points of the cross section of the spiral. Different specific gravity implies that the force per unit area of impact will be different at different points of the cross section. Hence, a mechanical fixture has been designed that has metal fingers in contact with the floor and are in the normal flow direction of the slurry. The different fingers form one side of the fixture to the other will experience different forces and will want to rotate the fixture. The fixture is constrained by a spring, but the rotation is monitored, and gives an indication of the change in the specific gravity of the slurry at the different locations. The change in the angle of the vertical rod is used as a sensor input to the control system. This mechanical system needs to be tuned but is expected to be robust, cheap, and yet effective (see Fig. 33.7). On the metal fingers are replaced by fins in

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Fig. 33.7 Alternate design for sensing slurry specific gravity Fig. 33.8 A more robust design

the shape of a vertical wing. The principles behind the two systems are the same. However, with the probes being shaped like a wing creates less disruption in the flow. The fins are able to rotate to adjust to the flow. The entire mechanism rotates because of a change in the flow (see Fig. 33.8). Before testing the idea on the spiral, a mathematical approach was taken to see if the mechanism would work as desired. A few assumptions were made and are as follows: no loss due to friction in the bearings, only drag force acts on the wings, motion is constrained to the horizontal plane. To further test mathematically, the left

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fin is placed in section C, while the right fin is placed in section D. Measurements have been taken for flow rate and pulp density for all six sections. (Measurements for four different situations are shown in Sect. 33.5.) If the force acting on the left wing is greater than that on the right wing, a moment will occur and the spring will deflect to counteract the moment. By summing moments about the center, the equation simplifies and becomes FDL

b b FDR D FS L; 2 2

(33.1)

where FDL is the drag force acting on the left wing, FDR is the drag force acting on the right wing, and FS is the force being applied by the spring. FDL and FDR can be written as FDL D CDL L AL VL2 ;

(33.2)

CDR R AR VR2 ;

(33.3)

FDR D

where CDL and CDR are the drag coefficients, L and R are the pulp densities for the left and right, AL =AR are the cross sectional areas of the fin, VL =VR are the flow velocities for left and right fins respectively. The cross sectional areas for both of the fins are equal and substitution and simplifying (33.1) becomes CDL L VL2 CDR R VR2 D 2

LFS : bAC

(33.4)

The coefficient of drag can be found by calculating the Reynolds number and using Figure 11.6 from Crowe et al. [7]. Figure 11.6 is a graph that gives the coefficient of drag with respect to the Reynolds number for a fin with the length being four times that of the width. The fin for the mathematical model has a length of 4 cm and a width of 1 cm. In calculating the Reynolds number, two more assumptions are made. The first being the height does not change between section C and D and since the slurry consists of roughly 20–30% solids, the dynamic viscosity of the slurry is taken to be that of water at 30ı C for both sections. The dynamic viscosity of the two sections of slurry may be higher or lower than that of water; however, the difference in the viscosity between the two sections should be rather small. Table 33.1 shows the calculations made.

Table 33.1 Calculations made Section Volumetric Pulp flowrate density width .kg=m3 / (m) Section LPM C 4.27 1.50 0.0508 D 8.8 1.34 0.0698

Flow height Area (m) .m2 / 0.003 0.0001524 0.003 0.0002094

Values taken from Run 1 from previous experiments Further results can be seen in Sect. 33.5

Assumed Velocity viscosity (m/s) .Ns=m2 / 0.467 7.97E04 0.700 7.97E04

Fin width Reynolds (m) number 0.01 8.79E C 00 0.01 1.18E C 01

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A problem occurs when using the figure relating the Reynolds number with the coefficient of drag. The problem occurs because the Reynolds number is around 9–10. The Reynolds number for the given application of the fin ranges from 200 to 107 . Instead of using the fin design, if a circular cylinder is used, then from Figure 11.6 the coefficient of drag for each of the sections is 2.9 and 2.8 respectively. This small change limits the ability to notice any small changes present in the flow. Even if the correct dynamic viscosity for the slurry at different solid percents and pulp densities are found, the coefficient of drag may be too small to have any affect. However, if the velocity or volumetric flow rate for a given section is needed to be measured, this mechanism may be useful.

33.4 The Need for a Model The need for a model arose out of noticing trends in the data collected on a previous washability analysis. In the washability analysis, eight runs were made, four on a rougher coal spiral and four on a rougher cleaner coal spiral. For each set of four, the volumetric feed rate and solid % were varied as seen in Table 33.2. For each of the spiral sections (A–F), the volumetric flow rate, solid %, pulp density, slurry mass flow rate, water mass flow rate, and the ash percentage of product/tailings per size fraction were measured or calculated. In looking over the data, several correlations were seen in the data sets for each of the coal spiral designs. In Table 33.3, one can see how the volumetric flowrate and solid % affects the pulp density. As the volumetric flowrate increases from 75 to 150 LPM and the solid % remains roughly constant at 33%, the pulp density of 1:34 kg=m3 shifts roughly one trough section from C to D. Likewise with a lower solid %, the pulp density of 1:34 kg=m3 shifts from B to C. (The highlighted box may be considered an error during the sampling.)

Table 33.2 Volumetric feed rate and solid %

Four sets of data of primary interest Volumetric feed rate Rougher coal spiral LPM Run 1 150 Run 2 75 Run 3 90 Run 4 150

Table 33.3 The volumetric flow rate and solid % affects the pulp density Pulp density per trough section Flow rate LPM Solid % A B C D 150 33.78 1.59 1.58 1.50 1.34 75 33.44 1.54 1.49 1.36 1.21 150 20.03 1.48 1.68 1.33 1.20 90 20.34 1.77 1.34 1.23 1.17

E 1.20 1.21 1.25 1.17

Solid % 33.78 33.44 20.34 20.03

F 1.19 1.17 1.10 1.09

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Fig. 33.9 Pulp density variation in a spiral cross section

Figure 33.9 is a graphical representation of the above table for the six locations on the last part of the spiral cross-section (1–6) going from the inside to the outside. A splitter is superimposed to illustrate how a comprehensive model would be beneficial for developing a way to control the splitter’s position. Some observations: 1. There is a considerable density change in the middle, but not as much at the inner or the outer sides. 2. For the same LPM, i.e., LPM 150, there is a shift of density from 1.3 to 1.5 at location 3 and 1.2 to 1.35 at location 4 as the solid % changes from 20 to 30%. This is the location where the splitter position can make a difference. 3. For the same solid %, i.e., 20%, there is a smaller change in density due to the LPM. The change is 1.24–1.32 at location 3 when the flow changes from 90 to 150, and 1.18 to 1.25 at location 4 for the same change in flow. 4. Overall, for different flows and solid %, the change in density could be anywhere from 1.2 to 1.5 at location 3 and 1.15 to 1.35 at location 4. This is significant and a strong proponent for a control system for the splitter. However, the pulp density is not an accurate measurement for the amount of ash in each section. In comparing the pulp density measurements of sections D and E from the 90 LPM 20.34% Solids and the product ash percentage for those same sections, a pulp density measurement of 1:17 kg=m3 correlates to a 8.75 or 7.09% ash content.

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Fig. 33.10 Product ash % vs. spiral trough section Table 33.4 Product ash % vs. spiral trough section Initial conditions Trough section LPM 150 75 90 150

Solid % 33.78 33.44 20.34 20.03

Product ash %

1 26.21 23.65 21.46 26.54 C 100 Mesh

2 21.44 16.96 13.92 16.25

3 18.56 13.79 11.17 13.61

4 13.89 11.38 8.75 11.20

5 12.03 9.08 7.09 9.08

In Fig. 33.10, the data presented is the product ash % vs. spiral trough section. In the chart, the 150 LPM 20.03% Solid line corresponds with the 75 LPM 33.44% solid line from section 3 to 5 (Table 33.4). Since two different flow rates at different solid % behave almost identically, it may be practical to measure these two variables plus the solid material density (if needed) in order to control the splitter position. The solid material density may need to be measured depending on the affect of higher ash content (higher solid density) on the spiral. The data collected thus far on the spiral is inconclusive. The first question raised is on how the spiral changes with time even though the slurry remains relatively unchanged. Since each sample measurement was taken only once, how does one distinguish the difference between an accurate measurement and a measurement taken in error? Currently, there are only four sets of data for a rougher coal spiral with only one sample measurement for each variable being taken per section. In the test spiral, the ash content remains relatively unchanged. One can speculate that if

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the ash content for all size fractions increases, then there will be a shift in the cut point toward the outer section of the spiral. However, it is not known how much of a shift will occur for a given flow rate and solid %. One way to answer these questions would be to develop a model for the coal spiral for analysis. Since the slurry is a two phase flow and current CFD techniques are unable to model the flow easily, a comprehensive empirical model may be best suited for the time being. In developing the model, the volumetric feed rate, solid %, and the percent ash content would need to be varied. The model would also need to have multiple measurements taken for each dependent variable in question. Further guidelines for the model are as follows: 1. At the top of the spiral, the volumetric feed rate, solid %, and ash content should be varied from high, medium, and low. 2. The ash content should be varied from high, medium, and low for all particle sizes and not just one specific particle size. 3. At the bottom, the volumetric flow rate, pulp density, solid %, and % ash should be measured or calculated for each section. 4. It is necessary to take multiple measurements to acquire a mean measurement for each variable measured. (Three or more measurements should be taken.) 5. If an extensive project is conducted, other variables need to be considered in order to be as thorough as possible to eliminate the need to go back and conduct more trials. (For instance, taking measurements necessary to calculate flow velocity along the spiral profile.) 6. Automation of the sampling and measuring may need to be implemented in order to expedite and simplify the collection process. After a good model for the coal spiral is developed, trends taking place within the spiral may be able to be seen and actions can be taken to automate the spiral. Further analysis may need to be done on-site at a coal preparation plant to determine local trends in the spiral. This may be a better option since taking measurements periodically from the bottom of the spiral and back calculating can give the same results as varying the feed conditions in an experimental setup. If on-site analysis is considered, measurements need only be taken from one spiral from one bank since one bank is designed to provide the same flowrate and slurry mix to each of the individual spirals.

33.5 Conclusions Spiral concentrators have been widely used in coal preparation plants in Illinois and elsewhere to clean 1 mm 150 m particle size coal fraction. The major factors which have made spiral concentrator so popular include its low capital and operating cost, no chemical reagent or dense medium requirement, and the ease of operation/maintenance. Spirals are capable of providing excellent clean coal recovery although at a relatively high ash content. Like any other water-only separation

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process, spirals are also susceptible to continuously fluctuating feed quality and solids content in the feed slurry, which are quite common in a plant. The main objective of the project is to develop an inexpensive control system for spiral to automatically adjust the splitter position with fluctuating feed characteristics to maintain the desired effective separation specific gravity (density cut-point). The splitter actuation and the control system worked perfectly, but the sensing system requires additional work. Two mechanical designs were investigated for the sensing system, and both did not have the required resolution for effectively estimating the specific gravity of the slurry across the profile of the spiral. Future work entails investigation of other techniques such as ultrasonic, inductive and capacitive based sensing. An intriguing concept of using a model was also investigated, whereby it may be possible to use simpler measurements, thereby circumventing the difficult problem of estimating the specific gravity of the slurry in real time. Further investigation of this approach also remains as future work. Acknowledgments The authors would like to acknowledge the grant provided by ICCI (# 06-1/2.1A.6) to support this work.

References 1. Abbot J (1982) The optimization of process parameters to maximize the profitability from a three-component blend. In: Proceedings – 1st Australian coal preparation conference, April 6–10, New Castle, Australia, pp 87–105 2. Bethell PJ, Arnold BJ (2003) Comparing a two-stage spiral to two stages of spirals for fine coal preparation. In: Honaker RQ, Forrest WR (eds) Symposium on recent advances in gravity separation, SME Annual Meeting, Littleton, CO, pp 107–114 3. Luttrell GH, Kohmuench JN, Stanley FL, Trump GD (1999) An evaluation of multi-stage spiral circuits. In: Proc. 16th International Coal Preparation Conference, pp 79–88. 4. Luttrell GH, Honaker RQ, Bethell PJ, Stanley FL (2003) Proceedings – 20th international coal preparation exhibition and conference, Lexington, Kentucky, April 29–May 1, pp 69–78 5. Luttrell GH, Kohmuench JN, Stanley FL, Trump GD (1998) Improving spiral performance using circuit analysis. Minerals and Metallurgical Processing Journal 15(4):16–21 6. Prinsolo TR, Abela RL (1998) Multiple stage fine coal spiral concentrators. In: Proceedings of the international coal preparation congress, Brisbane, Australia, October 4–10, pp 245–253 7. Crowe CT, Elger DF, Roberson JA (2005) Engineering fluid mechanics, 8th edn. Wiley, New York, p 441, Appendix A14 8. Yoon RH, Phillips DI, and Luttrell, GH., (2000) “Applications of Advanced Fine Coal Cleaning and Dewatering Technologies,” Proceedings, 7th Asia-Pacific Economic Cooperation (APEC) Clean Fossil Energy Technical Seminar, APEC Experts Group on Clean Fossil Energy, Chapter VI (3), Taipei, China, March 7-8, 2000, 10 pp

Chapter 34

On the Rough Number Computation and the Ada Language Trong Wu

Abstract This chapter reports a numerical computation problem from theoretical viewpoint. It shows computer systems are not capable for computation of real numbers correctly due to the differences of algebraic structures between real numbers and model numbers. These two classes of numbers are not isomorphic. This chapter defines several classes of real numbers such as model numbers, dyadic numbers, finite dyadic numbers, limited dyadic numbers, and rough numbers, and studies their structures. For numerical computation, this chapter proposes using the concept of computer model numbers to approximate rough numbers for computation and selecting the Ada language for computation. The Ada language allows a user to declare variables with appropriate number of digits or to specify a suitable size of “small” for computation. In this way, the user can preset the desired accuracy for all variables.

34.1 Introduction In 1982, Pawlak proposed a concept called rough sets and used in the theory of knowledge for the classification of features of objects [13]. He considered X to be a set and R to be a relation over X . The pair .X; R/ is called a rough space, and R is called the rough relation. If xRy, then one could say that x is too close to y, x and y are indiscernible, and x and y belong to the same elementary set. The concept of rough sets is usually used in knowledge representation. Later, Wu defined a new class of real numbers and called rough number [17, 19], which is one-dimensional rough set in rough set theory. Consider E be the set of all real numbers within the range of a given computer system M , and let R be a relation over E. The pair .E; R/ is called a rough space and R is called the rough relation. If x; y 2 E and .x; y/ 2 R, we say that x T. Wu () Department of Computer Science, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1656, USA e-mail: [email protected]

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-5754-2 34,

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and y are indistinctive in the rough space .E; R/ with respect to the given computer system M . Wu called x and y rough numbers, and they are approximated by the same model number [4,5], a number that the computer system can represent exactly. In fact, if E is the set of all real numbers within the range of a given computer system M , and let R be a relation on E. If x, y 2 E and .x; y/ 2 R in the rough space .E; R/ then R is an equivalence relation on E. Equivalence classes of the relation R are called basic model intervals [15], the smallest interval with model number endpoints. The set of all basic model intervals in .E; R/ is denoted by E=R. The definition of a rough number is a one-dimensional rough set that is a special case of rough sets given by Pawlak [13]. The difficulty of numerical computation is that one must work with two distinct number systems. Solving any numerical computation problem consists of the following three parts. (1) The problem is given in the real number system. (2) The computation is done in the model number system for the given machine, M . (3) The results must be converted from model numbers into real numbers. These two number systems have different algebraic structures, and they are not isomorphic. In general, moving from part (1) to part (2) can create errors and so can moving from part (2) to part (3). This chapter reports the theoretical foundation that a computer system is not capable of computing real numbers accurately within its constraints. In order to manage the computation, we will introduce some basic algebraic structures, the structures of dyadic numbers, and rough numbers. Then, we will consider computing rough numbers with the Ada programming language.

34.2 Some Basic Algebraic Structures For solving a numerical computation problem, there exists a fundamental crisis in the algebraic structure of numbers. Different classes of numbers have different algebraic structures. Therefore, we must study some basic algebraic structures of numbers. In fact, the algebraic structures of real numbers, dyadic numbers, and rough numbers have different algebraic structure. We will begin with the definition of abelian group, then extend it to include a ring and a field [7]: Definition 1. An abelian group (G; C) is a set G together with a binary operation namely, addition, “C”, such that the following axioms are satisfied: For all a; b 2 G, such that a C b 2 G. For all a; b; c 2 G, then .a C b/ C c D a C .b C c/. There is an identity element, e in G such that a C e D e C a, for all a 2 G. For all a 2 G, there exists an inverse, .a/, such that .a/ C a D a C .a/ D e. (5) For all a; b 2 G, such that a C b D b C a. (1) (2) (3) (4)

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Example 1. The set of all integers, I , with usual addition, C; .I; C/, is an abelian group. A ring structure is a special case of an abelian group. By adding conditions to an abelian group to form a ring: Definition 2. We say that .S I C; / is a ring, if .S I C/ is an abelian group, defining as a mapping from S S ! S , and satisfying the following axioms: (1) For all a; b 2 S such that a C b D b C a. (2) For all a; b; c 2 S , then .a b/ c D a .b c/. (3) Multiplication is distributed over addition: that is for all a; b; c 2 S , the left and right distributive laws hold: a .b C c/ D a b C a c; and .b C c/ a D b a C b c: Example 2. The set of all integers I with usual addition + and multiplication , .I; C; / is a ring. This ring has no zero-divisor. A field is a commutative ring accompanied by unity with respect to multiplication and every nonzero element has a multiplicative inverse. Definition 3. A Field .F I C; / is a ring with commutative law and multiplicative unity such that each nonzero element has a multiplicative inverse that satisfies the following axioms: (1) For all a; b 2 F such that a b D b a. (2) There exists 1 such that a 1 D 1 a D a. (3) For each nonzero element a in F , there is an inverse .1=a/ such that a.1=a/ D .1=a/ a D 1. Example 3. The set of all rational numbers,real numbers, and complex numbers with usual addition + and multiplication are fields. Let F be a field, the set of all polynomials in the indeterminate, x, with coefficients in F and written as F Œx, then the following theorem (without proof) determines a ring of polynomials [7]: Theorem 1. Let F be a field, the set of all polynomials a0 Ca1 xCa2 x 2 C Can x n , written as F Œx, where n can be any nonnegative integer and ai .i D 0; 1; : : : ; n/ are all in F . Then, F Œx is a ring under the operation induced by the operations in F . It is known that the set of all real numbers, denoted R, together with arithmetic addition and multiplication forms a field. Human arithmetic, C and , always returns exact results. On the other hand, most computer systems can store only certain real numbers exactly in the memory. Some of these real numbers are 100.5, 70.875, 12.375, 0.5, 87.125, etc. Some other real numbers such as 0.1, 0.2, 0.3, 0.4, 0.6, etc. cannot be stored exactly in the memory. This tells us that the computer system does not actually have these real numbers. Therefore, the set of all numbers represented by the computer for computation with the usual addition, C, and multiplication, operations are not the same as the set of all real numbers. Even if we allow a computer system to have as many bits as required to store a floating-point number; it is still not able to have their multiplicative inverse for all numbers in the computer

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system. Because of these, computer numbers do not form a field. The structures of the set of all real numbers and the set of all the numbers in the computer system are quite different. They are not isomorphic. This leads to a computation of C and over the set of all numbers represented by a computer system that can induce some errors. Therefore, we need to find out the actual structure of computer-represented numbers. What kind of arithmetic can computer systems do? So, we will now turn to study the set of dyadic numbers [10].

34.3 The Structure of Dyadic Numbers We will require that for each nonnegative real number x a b-adic expansion, where b is an integer greater than 1. Actually, we want to write a real number x as the sum of multiples of powers of b, where the multiples are nonnegative integers less than b. Clearly, the b-adic expansion of the number may fail to be unique in a decimal expansion, 0:9999 : : : (all nines) and 1.0000. . . (all zeros) are to be expansions of the same real number. When b D 2, the b-adic expansions are then called dyadic expansions and numbers written as dyadic expansions are called dyadic numbers [10]. Theorem 2. For each real number x, we have its dyadic expansion over a finite field (BI C; ): x D sgn

n X

where ai 2 B D f0; 1g and sgn 2 fC; g:

ai 2i ;

(34.1)

i Dn

A computer system is a finite state machine; therefore, it is capable of representing only a finite set of numbers internally. Thus, any attempt to use a digital computer to do arithmetic in the set of all real numbers is doomed to failure. The set of all real numbers is an infinite set, and most of the elements in the set cannot be represented by a computer system. Therefore, for theoretical reasons, we may assume that a computer system can have any finite number of bits to store its numbers, integers and floating-point numbers. For any nonnegative integer n, we consider the set of all numbers with the representation: n X ai 2i ; ai 2 B; sgn 2 fC; g: (34.2) y D sgn i Dn

This representation contains two parts, one is with index 0 i n and the other is with an index 1 i n. Actually, these two parts are the integer part of y and fraction part of y, respectively. In this, we define a structure for the set of all numbers with the representation, denoted D and call the set of all finite (term) dyadic numbers, and the term of finite means any finite value: ) ( n X ai 2i ; ai 2 B; sgn 2 fC; g; n 2 f0; ZC g : D D f yj y D sgn i Dn

(34.3)

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From Definitions 2, 3, and 2.4, we see that all finite (term) dyadic numbers D comprise neither a ring nor a field. More over, it is not a ring of polynomials of 2 over a finite field B D .f0; 1gI C; /. In fact, each computer system can have only a fixed number of bits of memory space to store a number of its type such as integer, float, double ; : : :; etc. An integer within the computer-predefined range often can be represented exactly. However, a real number within the range usually cannot be represented correctly. Today, most computer systems implement the IEEE floating-point number format [9] for storing real numbers. For example, a 32-bit floating-point number format is divided into three areas: a sign bit sign, an 8-bit exponent E, and a 23-bit mantissa M : sign .1 M / 2E 127 , where sign D ‘0’ indicates a positive value, sign D‘1’ indicates a negative value. The 1 in the .1 M / is the hidden bit, M is a 23-bit mantissa, and exponent E is in f0; 1; 2; : : :255g. It is clear that a floating-point number representation can be given in the set of D in (34.3). Again, computer systems can only use a limited number of bits to represent a floating-number. Therefore, ring computation, C and , can cause an overflow or underflow. Most computer systems use floatingpoint number arithmetic for numerical computation, and most computer users are not aware that floating-point number arithmetic often can create rounding-errors during the course of a computation. This error is unavoidable. In general, the output of a numerical computation program from one machine can be different from the output from another machine.

34.4 The Structure of Rough Numbers The Ada programming language [5, 6, 15] calls a real number a model number, if the real number can be stored exactly in a computer system with a radix of power of 2. Therefore, the set of all model numbers is a subset of dyadic numbers, and we call it the set of limited dyadic numbers with respect to a given computer system. The term limited reflects the given computer system’s word size for floating-point numbers. For those real numbers that cannot be stored exactly in a given computer system, we call them rough numbers with respect to the given computer system. In order to study this large class of numbers, it is necessary to define rough numbers precisely and mathematically. Definition 4. Let E be the set of all real numbers within the range of a given computer system M , and let R be a relation on E. The pair .E; R/ is called rough space and R is called the rough relation. If x; y 2 E and .x; y/ 2 R, we say that x and y are indistinctive in the rough space .E; R/ with respect to the given computer system M . We call x and y rough numbers, and they are approximated to the same model number. If a real number is a model number with respect to a given machine M , then it is a special case rough number. Most computer systems use a downward rounding policy to approximate a rough number with a model number. Thus, there are infinitely many rough numbers approximated by the same model number, such that all rough numbers z 2 Œx; y/ are approximated by the same value of the model

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number x for computation. Throughout this chapter, we use downward rounding policy to handle the approximation of rough numbers [15, 17, 19]. Theorem 3. Let E be the set of all real numbers within the range of a given computer system M , and let R be a relation on E. If x; y 2 E and .x; y/ 2 R are in the rough space .E; R/, then R is an equivalence relation on E (Readers may verify this theorem). Equivalence classes of the relation R are called basic model intervals [15], the smallest interval with model number endpoints. The set of all basic model intervals in .E; R/ is denoted by E=R. The definition of a rough number is a special case of the one-dimensional rough sets given by Pawlak [13]. In reality, for any rough number z whose value is within the range of a given computer system M , there exists the smallest closed-open interval, such that k k1 z 2 i C n ;i C n 2 2

for some positive integers n and k;

(34.4)

where the lower bound is the approximation of z; i is the integer part of z; k is an integer with 0 k 2n 1, and n D n.be ; bm /, a function of the number of bits in the field of exponential be , and the number of bits in the field of mantissa bm . The difference, d , between the rough number z, and its approximation given in (34.4) is given by k1 d Dz i C n : (34.5) 2 This is also called rounding-error. Usually, a programmer is either unaware of the existence of rounding-errors or unable to reduce or control them in the course of a computation. We will show that the set of all rough numbers over the set of all computer systems has the dyadic number structure. Most computer users are not aware that the computation of real numbers and rough numbers are not the same. Even more confusing, some programming languages such as FORTRAN, COBOL, Pascal, etc. allow programmers to declare variables with type real in their programs for computations. To make it clear to all computer users, we must study the algebraic structures for the set of all real numbers and the set of all rough numbers over computer systems. This not only provides the theoretical foundation for computer arithmetic, but also interprets rounding-errors from an algebraic theoretical viewpoint.

34.5 Hierarchical Classes and Computation In mathematics, we have learned properties of several classes of numbers such as integers, rational numbers, irrational numbers, and real numbers. However, a computer system can have only a limited word size; therefore, only some integers and some rational numbers can be computed correctly in a computer system.

34

On the Rough Number Computation and the Ada Language Rough numbers

Real Number

a,b

443

f f ( a), f ( b) f ( a + b)

a+ b

Dyadic Numbers

Finite Dyadic Numbers

Limited Dyadic Numbers (Model numbers)

Fig. 34.1 Hierarchical structures of dyadic numbers

Other numbers cannot even be represented exactly in a computer system. Computer systems handle real numbers in a binary fashion. For each real number x, we have its dyadic expansion over the finite field .BI C; /. Among all the dyadic numbers, only a limited number of dyadic numbers can be computed in a computer system. The relationship of the set of real numbers and the hierarchical structures of dyadic numbers is shown in Fig. 34.1. On the set of real numbers, we perform a field computation, which is an exact computation. However, when a real number within the given range is stored or read into a computer system, it is converted into a dyadic number. The computation over the set of limited dyadic numbers is the dyadic number computation. Let f be a mapping that takes each real number into its dyadic representation. Consider real numbers a and b within a given range mapped into their dyadic representations f .a/ and f .b/ respectively. In general, we should have f .a C b/ ¤ f .a/ ˚ f .b/:

(34.6)

The set of real numbers has a field structure, while the set of finite dyadic numbers is not a field. f does not preserve the structure, and f cannot even be a local homomorphism between the set of real numbers within a given range of machine M and the set of limited dyadic numbers. The addition ‘C’ (or multiplication ‘ ’) on the right hand side of (34.6) is a field addition, and the addition ‘˚’ on the left hand side is a ring addition. They are two different additions. The ‘˚’ is the dyadic number addition. To adjust the inequality (34.6), we will add in an error term, Err: f .a C b/ ¤ f .a/ ˚ f .b/ C Err:

(34.7)

This error term, Err, is called rounding-error. Unfortunately, most programming textbooks do not teach students the addition ‘C’ in program is not real number addition, but dyadic number addition instead. Many computer users are misled and disappointed that computer systems cannot provide accurate results for computations.

444

T. Wu

Among all commonly used programming languages, the Ada programming language is a unique language that allows the user to define a specific model number for their rough numbers for dealing with computation. In the next section, we will introduce the Ada language briefly. Then, we will study its special feature in the defining model numbers for the computation of rough numbers.

34.6 The Ada Language and Rough Number Computation In the 1970s, the US Department of Defense (DoD) realized that it was spending too much for its software, particularly in its real-time or embedded systems. The DoD found that they used several hundred special-purpose programming languages for its real-time systems. The DoD decided to design a new language to replace all of these languages, and this new language would be a general-purpose language as well. Following a number of definitions of requirements, developments, and evaluations, the ANSI standard for the Ada language issued in 1983 [5,6]. The document containing the final requirements for the Ada 83 language was called Steelman. The team that developed the language was CII Honeywell Bull in France, and the leader of the design team was Jean D. Ichbiah. The language was named Ada after Augusta Ada Byron, Countess of Lovelace (1815–1852), and the daughter of Lord Byron. She worked as an assistant for Charles Babbage’s mechanical analysis engine. It was very well believed that she was the world’s first programmer. After the Ada language had several years of real-world application and users learned object-oriented programming techniques, the DoD established the Ada 9X revision project in 1988. It was based on the original Ada 83, but included additions such as type extensions, hierarchical libraries, a greater ability to manipulate points or reference types, object-oriented programming without incurring overhead, and a tasking model. Ada 9X remained very strongly type checking. Later, this revised language was standardized and named the Ada 95 language [2]. The Ada language’s real number types are subdivided into float-point types and fixed-point types. In the float-point type, values are numbers with the format ˙:dd : : : d 10˙dd . For fixed-point types, values are numbers with the formats ˙dd:ddd; ˙dddd:0 or ˙ 0:00ddd [1, 3, 18]. From Definition 4, we have learned that all rough numbers z 2 Œx; y/ are approximated by the same value of the model number x for computation. For the float-point number types, a model number, other than zero that can be represented exactly by a given computer, is of the form: sign mantissa .radix/.exponent/ :

(34.8)

In this form, sign is either C1 or 1; mantissa is expressed in a number base given by radix and exponent is an integer. The Ada language allows the user to specify or to declare the number of significant decimal digits needed. This is downward rounding of a rough number to its approximated model number. A floating-point type declaration with or without the optional range constraint is shown: Type T is digit D Œrange L R:

(34.9)

34

On the Rough Number Computation and the Ada Language

445

Table 34.1 Ada floating-point types Type

Size (bize)

Precision (digits)

f–float d–float g–float h–float

32 64 64 128

6 9 15 33

In addition, most Ada compilers provide the types long float and long long float (used in package standard) and f float, d float, g float, and h float (used in package system) [8, 14]. The size and the precision of each of the Ada floating-point types are given in Table 34.1. The goal of computation is accuracy. Higher accuracy will provide more reliability in the real-time environment. Sometimes, a single precision or a double precision of floating-point numbers in FORTRAN 77 [9] is not enough for solving some critical problems. In the Ada language, one may use the floating point number type: long long float (h float) by declaring “digit 33” to use 128 bits for floating point numbers provided by Vax Ada [14] for example to achieve a precision of 33 decimal digits accuracy; the range of the exponent is about from 10134 to 10C134 or 448 to C448 of base 2 [1, 2, 4–6, 8, 14] for the range. The author employed this special accuracy feature in the computation of the hypergeometric distribution function [16]. Similarly, for fixed-point types, the model numbers are in the form: sign mantissa small:

(34.10)

The sign is either C1 or 1; mantissa is a positive integer; small is a certain positive real number. The model numbers are defined by a fixed-point constraint, the number small is chosen as the largest power of two that is not greater than the delta of a fixed accuracy definition. The Ada language permits the user to determine or declare a possible range and an error bound which is called delta for computational needs. Examples are the follows: Overhead has a delta of 0.01 and Overhead has a range from 1:0E5 to 1:0E5:

(34.11)

These indicate small is 0.0078125 which is 27 , and model numbers are in the following interval Œ12; 800; 000 small; C12; 800; 000 small:

(34.12)

The predetermined range provides a reliable programming environment. The user assigned error bound delta guarantees an accurate computation. These floatingpoint number and fixed-point number types not only provide good features for

446

T. Wu

real-time critical computations, but also give an extra reliability and accuracy for general-purpose numerical computations. This result can be provided by interval computation [11, 12] too. But using the primitive Ada programming method can eliminate the more complicated implementation of interval computation.

34.7 Conclusions In this chapter, we have reviewed some basic algebraic structures such as abelian group, ring, and field. We defined sets of rough numbers, dyadic numbers, finite dyadic numbers, limited dyadic numbers, and model numbers; from these definitions, we have learned that the difficulty of numerical computation is one must actually work with two distinct number systems. Solving any numerical computation problem consists of the following three parts: (1) the problem is given in the real number system, (2) the computation is done in the model number system for the given machine, and (3) the results must be converted from model numbers into real numbers. Real numbers and model numbers have two different algebraic structures, and they are not isomorphic. In general, starting from the problem to computation on a machine can create some errors and going from the machine computation to reporting results in real numbers can induce errors too. This chapter has reported that a computer system is not capable of computing real numbers accurately within its constraints from a theoretical viewpoint. Computation over the set of real numbers requires performing a field computation. However, when a real number within the given range is stored or read into a computer system, it is converted into a dyadic number. Computation over the set of limited dyadic numbers is dyadic number computation. Let f be a mapping that takes each real number into its dyadic number representation. If a and b are two real numbers within a given range and they map into their dyadic representations f .a/ and f .b/, respectively, then in general, we should have f .a C b/ ¤ f .a/ ˚ f .b/. The addition, C, on the right-hand side is the addition of real numbers. The mapping f does not preserve the algebraic structure, so the set of real numbers and the set of dyadic numbers are not isomorphic. To avoid overloading and possible confusion, we will introduce a new addition ‘˚’ for the dyadic number addition. To adjust the inequality, we will add in an error term, Err. Therefore, we have f .a C b/ D f .a/ ˚ f .b/ C Err. To do this, we use the special features of the Ada programming language either to declare variables with a specific value of “small” for fixed-point numbers or to designate a value of delta for float-point numbers. In this way, the error of the computation result is controlled within the acceptable level.

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447

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Aberth O (1988) Precise numerical analysis. Wm. C. Brown Publishers, Dubuque Ada 9x Mapping/Revision Team (1993) Ada 9X rationale. Intermetrics Inc., Cambridge, MA Alefeld G, Herzberger J (1983) Introduction to interval computations. Academic, New York Barnes JGP (1989) Programming in Ada. 3rd edn. Addison-Wesley, Reading, MA Barnes JGP (1992) Programming language in Ada. 4th edn. Addison-Wesley, Reading, MA Barnes J (1995) Programming in Ada 95. Addison-Wesley, Reading, MA Herstein IN (1971) Topics in algebra. University of Chicago, Chicago, IL IBM AIX (1992) Ada/6000 user’s guide. IBM Canada Ltd. Laboratory, North York, ON, M3C 1W3 IEEE Std 754-1985 (1985) IEEE standard for binary floating-point arithmetic IEEE Inc. New York, NY Kelley JL (1955) General topology. D. Van Nostrand Company, Inc., Princeton, NJ Moore RE (1966) Interval analysis. Prentice-Hall, Englewood Cliffs, NJ Moore RE (1979) Methods and applications of interval analysis. SIAM studies in applied mathematics, vol 2. Society for Industrial and Applied Mathematics, Philadelphia, PA Pawlak Z (1982) Rough sets. Int J Comput Inform Sci 11(5):341–356 Vax Ada language reference manual (1985) Digital Equipment Corporation, Maynard, MA Watt DA, Wichmann BA, Findlay W (1987) Ada language and methodology. Prentice-Hall, Englewood Cliffs, NJ Wu T (1993) An accurate computation of the hypergeometric distribution function. ACM Trans Math Softw 19(1):33–43 Wu T (1994) Rough number structure and computation. In: Proceedings of the 3rd international workshop on rough sets and soft computing, pp 360–367 Wu T (1995.) Ada programming language for numerical computation. In: Proceedings of IEEE 1995 national aerospace and electronics conference, pp 853–859 Wu T (1998) Rough numbers and computations. In: 1998 IEEE world congress on computational intelligence proceedings Fuzz-IEEE’98, pp 845–890

Author Index

A Abbas, A.E., 77 Abbot, J., 422 Abdul Azeez, M.F., 121 Abela, R.L., 422 Aberth, O., 444–445 Ablow, C.M., 320 Addison, P.S., 280 Adiletta, G., 266 Aggarwal, H.R., 320 Aguado, M., 234 Ahmad, S., 332 Ajayan, P.M., 96, 101–102 Akay, A., 95 Al-Bender, F., 97 Albert, R., 203 Alefeld, G., 444 Ali, M.S., 135 Alleyne, D., 341, 352 Ambr´osio, J., 358 Anagnostopoulos, A.N., 106 Andrews, R., 99 Antoni, J., 407 Apsel, R.J., 320, 332 Arakere, N., 105 Arimoto, S., 157 Arnold, B.J., 422–423 Arnold, R.N., 323, 334 Ashley, H., 62 Atluri, S.N., 358 Averina, V., 135 Awojobi, A.O., 323–324, 331, 333–334

B Bag, B.C., 224 Bagley, R.L., 3 Baier, G., 31 Bainov, D., 204, 210

Bainton, K., 348 Bakis, C.E., 96 Banavar, J.R., 217 Banerjee, P.K., 320–321 Barab´ai, A.L., 203 Barkan, D.D., 320, 331, 333 Barnes, J.G.P., 438, 441, 444–445 Bartuccelli, M., 106 Bautin, A.N., 157 Bechtel, S.E., 293–295 Beeston, H.E., 334 Benner, H., 169 Bently, D.E., 121 Benzi, R., 251 Beran, J., 23 Beredugo, J.O., 333 Bergman, L.A., 372, 377 Bernard, T., 169 Beskos, D.E., 331 Bethell, P.J., 422–423 Bettess, P., 331 Bezrukov, S.M., 224 Bielak, J., 358 Bindemann, A.C., 358 Bishop, S., 177–178, 184 Black, C., 280 Black, H.F., 121 Bonilla, B., 4–7 Bouwman, V., 385 Boyd, R.W., 234 Brackbill, C.R., 95 Braun, H.A., 169–170 Buckingham, E., 348 Bueler, E., 135 Butcher, E.A., 135 Butterfield, R., 331 Bycroft, G.N., 323–324, 334

449

450 C Caldas, I.L., 106 Campbell, S.A., 159 Cao, D.-Q., 263–272 Cao, L., 224, 234, 236 Carlson, C.D., 293–295 Carmona, R., 279 Carnahan, B., 140 Carrol, T.L., 13 Casadememunt, J., 234 Casademunt, J., 234 Castellani, A., 324 Castro, F., 223–224 Cawley, P., 341, 352 Cawley, R., 32 Chae, Y.S., 334 Chan, C.K., 334 Chandler, D.E., 321 Chandrasekaran, H., 77 Chao, C.C., 320 Chapel, F., 332 Chaplin, C.R., 291 Chavz, A., 124 Chen, C.H., 136 Chen, C.W., 409 Chen, D.M., 384 Chen, G.R., 158, 203 Chen, S.B., 234 Chen, W.H., 136 Chen, X.F., 224 Chen, Y.-H., 13, 33 Cheng, D.M., 385 Chesse, P., 397 Childs, D.W., 121 Chinvorarat, S., 417 Choi, S.K., 121 Choi, Y.S., 121 Chongbin, Z., 331 Chow, S.N., 385 Chow, Y.K., 329 Christiansen, P.L., 106 Chu, F., 275 Chu, T., 409–435 Chua, D.H.C., 96 Chua, L.O., 31, 204 Chueshov, I.D., 385 Chuhan, Z., 331 Chui, C.K., 278 Chung, K., 178 Cipra, R.J., 295 Cipric, G., 177 Claro, J.C.P., 358 Clemmet, J.F., 324, 333 Cohen, I., 31

Author Index Coleman, J.N., 99 Collinger, J.C., 95 Costello, G.A., 285, 287 Costello, H., 358 Cotsaftis, M., 291, 300, 313 Cox, D.E., 410 Crede, C.E., 50 Crowe, C.T., 430 Cunny, R.W., 334 Curran, S.A., 99 Curry, J.H., 37 Cvitanovic, P., 37

D Dai, G.-P., 251 Dalby, A.R., 203 Danca, M.F., 191 Daniere, J., 407 Das, S., 3, 5 Davies, T.G., 320 Davies, T.J., 286, 289–290 Davison, D.E., 136 Dawance, G., 334 Day, S.M., 331 Day, W.B., 121 de Jongh, J., 384 de Kraker, A., 384 de Souza, S.L.T., 106 Deck, J.F., 358 Deng, Y., 251 Deshmukh, V., 135 Devulder, C., 264 Dhamala, M., 158 Dickey, E.C., 99 Diez-Martines, O., 170 Ding, J., 263–264 Ding, M., 158 Ding, Q., 264 Dion, J.M., 136 Ditto, W.L., 32 Dixit, S.N., 233 Dobry, R., 330–331 Dominguez, J., 331 Don, J., 409–435 Dong, S., 342, 344–346 Dowell, E.H., 62 Dragana, T., 158 Driver, D., 183 Drnevich, V.P., 334 Duan, L., 158 Duan, Y.W., 13 Dubow Sky, S., 358 Dugard, L., 136

Author Index Dugundji, J., 62 Duns, C.S., 331 Durand, D.M., 217

E Eastwood, W., 334 Eckman, J.P., 13 Ehrich, F.F., 121 Elata, D., 288 Elbeyli, O., 136, 138, 140, 144–146 Elger, D.F., 430 Elorduy, J., 320, 324 Emami-Naeini, A., 143 Eshkenazy, R., 288 Ettouney, M.M., 331

F Fan, D., 157–166 Fan, S.-B., 251–259 Fang, T., 32 Faria, T., 159 Fedigan, S., 122 Feldmann, A., 23 Feng, N.S., 264 Feng, R., 169–174 Feng, T., 280 Feng, Z., 158 Ferri, A.A., 357–358 Fey, R.H.B., 384 Findlay, W., 438, 441–442 FitzHugh, R., 157 Flores, P., 358 Foias, C., 264, 385 Fox, R.F., 235–236 Frank, M., 234 Franklin, G.F., 143 Frazier, G.A., 331 Freitas, M., 177 Friedland, S., 347 Friswell, M.I., 384 Fry, Z.B., 332, 334 Fu, X.L., 203–215 Fuentes, M.A., 224–226 Fujimoto, S., 297 Fujisaka, H., 236 Fulinski, A., 223

G Gallas, J.A.C., 37 Galvanetto, U., 177–178, 184 Gan, C., 177–178

451 Gao, G., 394 Garcia-Archilla, B., 264–265 Garg, N.K., 135 Garvey, S.D., 384 Gaul, L., 358 Gazetas, G., 331 Gazis, D., 342, 347 Ge, S., 287 Gee, D.J., 62 Gegg, B.C., 77–93 Gerlach, U.H., 374 Ghosh, P.K., 224 Gilbert, A.C., 23 Girard, J., 331, 333 Gladwell, G.M.L., 323 Gokcek, C., 191 Goldman, P., 121 Golinval, J.C., 275 G´omez, X., 291 Goncalves, P.B., 135 Gonchenko, S.V., 37 Gonchenko, V.S., 37 Gong, Y.F., 170 Goswami, G., 224 Goto, S., 203 Goychuk, I., 224 Grebogi, C., 32, 105, 109, 170, 177 Grebogi, E.C., 32 Griffin, J.H., 358 Grootenhuis, G., 324–325, 330, 333–334 Grootenhuis, P., 323, 334 Grossmann, S., 236 Grune, L., 191 Gu, K., 136 Guan, Z.H., 136 Guckenheimer, J., 384 Gudyma, Y.V., 223 Guido, A.R., 266 Guillot, M., 334 Gunaratne, G.H., 37 Guo, S., 159 Guo, Y., 37–47 Gupta, D.C., 333 Gustavsson, I., 409 Guyan, R.J., 264

H Hai, W.H., 13 Hakal, P.D., 191–192 Haken, H., 223 Hall, J.R. Jr., 330, 332–333 Hamidzadeh, H.R., 319–335 Han, Q.L., 136

452 Han, R.P.S., 38 Han, X.P., 203–215, 342 Hanggi, P., 236 H¨anggi, P., 224 Hardin, B.O., 334 Harding, J.W., 323 Harris, C.M., 50 Hashemi, S.M., 297 Hassard, B.D., 159, 163, 165 Hay, T., 341 He, D.H., 13, 33, 217–219, 221 He, K., 106 He, L., 50 He, Q., 32 He, Y.N., 136, 386 Heller, L.W., 320 Henon, M., 37 Hernandez-Garcia, E., 234 Hernandez-Machado, A., 234, 240 Herstein, I.N., 438–439 Herzberger, J., 444 Hespanha, J.P., 136, 145–146 Hetet, J.F., 397 Hildebrand, F.B., 63 Hindmarsh, J.L., 171 Hiroshi, K., 158 Hisayo, M., 158 Hodgkin, A.L., 157 Hodgman, T.C., 203 Hoelscher, J., 421–435 Holmes, P., 384 Honaker, R.Q., 422–423 Hong, D.W., 295 Hong, L., 31–36, 121–131, 157–166, 217–221 Hope, R., 122 Horne, A.B., 203 Horsthemke, W., 223 Hou, Z.K., 135 Housner, G.W., 324 Hruska, S.L., 37 Hsiao, M.-H., 410 Hsieh, T.K., 330, 332 Hsu, C.S., 32 Hsu, S.T., 358 Hu, G., 223, 227 Hu, H.Y., 4, 162, 169 Hu, S.J., 170 Hu, W., 264 Hu, X.B., 136 Huang, H.C., 136 Huang, J.H., 96 Huang, J.-K., 409–410, 417 Huang, K., 178 Huang, L.H., 159, 162, 371

Author Index Huang, N.E., 251, 398, 401 Huang, W.H., 263–272 Huang, W.L., 13 Huang, Y., 95–103 Huang, Y.P., 289–290 Huang, Y.S., 159 Huber, M.T., 169–170 Huberman, B.A., 170 Hui, D., 96 Huxley, A.F., 157 Hwang, W., 279 I Ikeda, K., 384 Iljitchov, V.A., 333 Inozu, B., 397 Inse, G., 158 Insperger, T., 140, 145 Israil, A.S.M., 332 Itoh, M., 204 Iwan, W.D., 97–98

J Jackson, J.K., 23 Jacob, K.I., 293–295 Jalili, N., 96 Janbu, N., 330 Jauberteau, F., 385 Jeong, H., 203 Ji, C.J., 169 Ji, G., 136 Jia, Y., 223–224, 236 Jiang, J., 31–36, 121–131, 217–221 Jiang, X.F., 136 Jin, W.Y., 169–174, 217–221 Jin, Y.F., 223–230, 234 Jirsa, V.K., 158 Johnson, L.R., 320, 328 Jolly, M.R., 95 Jones, R., 334 Joshi, A., 96 Juang, J.-N., 409–410 Jung, P., 233 Junjiro, O., 357–358 Just, W., 169

K Kalmar-Nagy, T., 135 Kamath, G.M., 95 Kaminish, K., 233 Kanai, K., 334

Author Index Kanehisa, M., 203 Kang, J.-K., 297 Kao, C.Y., 136 Kapitaniak, T., 31–32, 49 Kappaganthu, K., 105–120 Karabalis, D.I., 331 Karasudhi, P., 324, 332 Karnopp, D.C., 50 Karvinen, T., 285–297, 299–316 Kausel, E., 320, 331–332 Kawakami, H., 32 Kazarinoff, N.D., 159, 163, 165 Kazuyuki, A., 158 Ke, S.Z., 236 Keblinski, P., 101–102 Keer, L.M., 324, 332 Kelley, J.L., 440 Kenji, M., 357–358 Kerschen, G., 275 Keskinen, E.K., 285–297, 299–316 Keskiniva, E., 300, 307 Khadem, S.E., 96 Khadra, A., 203 Khalil, H.K., 119 Kilbas, A.A., 3 Kim, N.H., 135 Kim, Y.B., 121 Kimura, H., 297 Kireitseu, M., 96 Kissenpfenning, J.F., 330 Kleeberger, M., 291 Klein, E.J., 135 Klein, M., 31 Kloeden, P.E., 191 Knospe, C., 122 Kobayashi, K., 31 Kobayashi, O., 32 Kobayashi, S., 320 Kobori, T., 323–324 Kohmuench, J.N., 422 Kong, L., 293 Koratkar, N.A., 96, 101–102 Kordt, M., 384 Kovacs, W.D., 334 Kraincanic, I., 287–289 Krautkr¨amer, H., 341 Krautkr¨amer, J., 341 Krizek, R.O., 333 Krodkiewski, J.M., 122, 263–264 Kuhlemeyer, R.L., 331 Kukla, S., 372 Kumaniecka, A., 291, 297 Kunichika, T., 158 Kuo, C.-H., 417

453 Kurdi, M.H., 135 Kurths, J., 217 Kusakabe, K., 323–324 Kwon, O.M., 136 Kwon, T., 122 Kypriandis, I.M., 106

L Lacovoni, G., 23 Lai, C.H., 217 Lai, Y.C., 32 Laing, C.R., 385 Lainscsek, C., 32 Lakshmikantham, V., 204, 210 Lamb, H., 320, 322, 328 Lampaert, V., 97 Lankarani, H.M., 358 Lardies, J., 276 Launis, S., 291, 300, 313 Lawrence, F.V. Jr., 334 Lee, C.H., 13 Lee, H.C., 297, 410 Lee, K.Y., 375 Lee, S.L., 324, 332 Lee, S.M., 136 Lee, T.H., 333 Lei, P.-F., 61–76, 385 Lei, S., 13 Leiber, T., 233 Lenci, S., 177 Leng, Y.-G., 251–259 Lenz, J., 358 Lesieutre, G.A., 95 Leung, A.Y.T., 169, 264 Levy, V.J., 23 Lewis, F.L., 143–144 Li, B.H., 394 Li, C., 203 Li, G., 177–178, 184 Li, H.K., 397–408 Li, J.L., 223–224 Li, J.R., 236 Li, K.-L., 61–76 Li, K.T., 386 Li, T., 13 Li, W., 143 Li, X., 160–162 Li, Z.G., 204, 234 Liang, G.Y., 224, 234 Liang, Z.Y., 96 Liao, W.H., 95 Liao, X., 203 Liberzon, D., 136, 145–146, 191

454 Lin, C.L., 136 Lin, L., 224 Lin, W.W., 13, 204 Lindler, J.E., 95 Liu, A., 96 Liu, B., 203, 251 Liu, G., 342 Liu, H.H., 169–174, 398, 401 Liu, J., 291 Liu, S.D., 19 Liu, S.S., 19 Liu, X.T., 371 Liu, X.Z., 203–204 Liu, Y., 23–29, 95, 384–385 Liu, Y.S., 204, 210 Liu, Y.-X., 371–380 Liu, Z., 19, 23 Ljung, L., 409 Logan, K., 397 Long, Q., 234 Long, S.R., 251, 398, 401 Long, Y.J., 50 Lorenz, E.N., 13, 38 Lou, J.J., 49–60 Love, A.E.H., 285, 287 Lowe, M., 341, 352 L¨u, J., 203, 205 Lu, J.A., 203–205 Lu, K., 385 L¨u, L.-F., 371–380 Lu, Q.S., 158, 170, 178 Lu, X.M., 136 Luco, J.E., 324, 330, 332–333, 335 Lumer, E.L., 170 Luo, A.C.J., 37–47, 77–93, 189–200 Luo, Q., 136 Luo, X.L., 234 Luo, X.Q., 224 Luo, Y., 357–370 Lusebrink, H., 384 Luther, H.A., 140 Luttrell, G.H., 422–423 Lv, C., 403 Lysmer, J., 330–331

M Ma, H., 135 Ma, J., 287 Ma, X.J., 400–401 MacCalden, P.B., 333, 335 Maccari, A., 178 Mahajan, A., 409–435 Mahmoodi, S.N., 96

Author Index Maistrenko, Y., 32 Majee, P., 224 Mallat, S., 279–280 Mallik, A.K., 50, 56 Mamoon, S.M., 321 Mance, V., 23 Mandel, L., 233 Manley, O., 264, 385 Mann, B.P., 135 Mannell, R., 234 Marion, M., 264, 384–385 Maritan, A., 217 Marotto, F.R., 37 Masoller, C, 169–170 Matsumoto, T., 31 Matthies, H.G., 265 Matthiesen, R.B., 333, 335 Maxwell, A.A., 334 Mccarthy, B., 99 McClintock, P.V.E., 234 McEvilly, T.V., 334 McFadden, P.D., 402 McRobie, A., 385 McRobie, F., 177 Meguid, S.A., 287, 289–290 Mei, D.C., 224, 234, 239 Mei, G.-H., 383–394 Meirovitch, L., 371 Meiss, J.D., 37 Meyer, M., 265 Miah, H., 264 Michaelis, M., 280 Mickens, R.E., 276–277 Miliou, A.N., 106 Miller, D.E., 136 Miller, G.F., 323 Minai, R., 323–324 Minor, G.R., 333 Mohanty, M., 421–435 Mohri, A., 297 Moiseuev, N., 347 Montonen, E., 300 Montonen, J., 291, 299–316 Moon F., 177–178, 184 Moore, R.E., 446 Morse, A.S., 136, 145–146, 191 Mosekilde, E., 37 Moss, F.E., 234, 236 Mroczkowski, T.T., 236 Mu, J., 341 Murota, K., 384 Muszynska, A., 121

Author Index N Nagakagi, S., 297 Nagumo, J., 157 Nakagawa, T., 297 Nakano, H., 320 Narducci, L.M., 234 Nataraj, C., 105–120 Navarro-Lopez, E.M., 78 Nayfeh, A.H., 275 Nebojsa, V., 158 Neiman, A.B., 217 Nelson, H.D., 105 Nicolis, G., 223 Niculescu, S.I., 136 Nieto, J.A., 320, 324 Nikola, B., 158 Nikravesh, P.E., 358 Nishimura, N., 320 Nishioka, T., 203 Niziol, J., 291, 297 Noah, S.T., 121, 264 Noori, M.N., 135 Noriega, J.M., 234 Novak, M., 333 Novikov, E.A., 236 Novo, J., 264–265

O Oltvai, Z.N., 203 Oner, M., 330 Ostheimer, M., 169 Otsuki, M., 297 Ott, E., 32, 105, 109, 170 Ott, Y.J.A., 32 Ovsyannikov, I.I., 37

P Pan, Y., 409–435 Papadopulus, M., 320 Park, J.H., 136 Parker, R.G., 293–294, 371 Parmelee, R.A., 333 Parrondo, J.M.R., 223 Parunov, J., 177 Pavlakovic, B., 341 Pawlak, Z., 437–438, 442 Pecora, I.M., 13 Pekeris, C.L., 320 Peng, C., 136 Peng, J.H., 13–21, 170 Peng, Z., 275

455 Penny, J.E.T., 384 Penz, S., 280 Pesqueraet, L., 234 Pesterev, A.V., 372, 377 Phillips, D.I., 422 Pikovsky, A.S., 217 Pinto, O.C., 135 Podlubny, I., 3, 5 Popovych, S., 32 Popplewell, N., 341–354 Porwal, R., 275–283 Powell, J.D., 143 Prigogine, I., 223 Prinsolo, T.R., 422 Procaccia, I., 37 Pursey, H., 323 Pyragas, K., 170

Q Qian D, 99 Qiu, J.H., 122 Quarry, M., 341 Quinlan, P.M., 323

R Rainey, F., 177–178 Ramirez, W.F., 135 Rantell, T., 99 Rantzer, A., 136 Rao, Y., 23 Raoof, M., 286–294 Ratay, R.T., 332 Rattanawangcharoen, N., 341 Raunisto, Y., 313 Rauseo, S., 32 Ravindra, B., 50, 56 Raymer, M.G., 234 Rega, G., 177, 265 Reibold, E., 169 Reissner, E., 322–323 Reiterer, P., 32 Ren, S., 383–394 Ren, W., 170 Ren, X., 394 Renshaw, A.A., 375 Revelli, J.A., 224–225 Rezounenko, A.V., 385 Richardson, J.D., 324, 333 Richart, F.E., 330, 332, 334 Ridge, I.M.L., 291 Riedi, R., 23

456 Riitahuhta, A., 300, 307 Risken, H., 233 Rivero, M., 4–7 Rivin, E.I., 51 Rizzo, P.C., 330 Roach, A., 297 Roberson, J.A., 430 Robertson, I.A., 323 Rodeigaez, M.A., 234 Roesset, J.M., 330–332 Rong, H., 32 Roog, Y., 357–358 Rose, J., 341 Rose, R.M., 171 Rosier, C., 385 Rossi, C., 266 Rossler, O.E., 31 Roy, P., 397 Roy, R., 13, 233–235 Ruan, J., 204 Ruan, S., 159, 162, 165 Rubin, M.B., 293 Rucker, W., 329, 332 Rudakov, V.N., 37 Ruelle, D., 13 Russell, D.F., 217

S Sachdev, M.S., 191–192 Sagoci, H.F., 323 Sahle, S., 31 Sahni, P.S., 233 Sainz-Trap´aga, M., 169 San Migud, M., 234, 240 San Miguel, M., 234 Sanchez, A.D., 223–225 Sancho, J.M., 234, 240 Sansour, C., 265 Sansour, J., 265 Sauren, A.A.H.J., 384 Schenzle, A., 234 Schlichting, H., 63 Schmidtmann, O., 385 Schmitz, T.L., 77 Schoen, M.P., 409–420 Schurrer, F., 32 Scott, R.A., 358 Seed, H.B., 331, 334 Segalman, D.J., 98 Segundo, J.P., 170 Sell, G.R., 385 Senjanovic, I., 177 Seydel, R., 384

Author Index Shah, A., 341–354 Shang, H., 177–184 Shang, Z., 121–131 Shaw, D.E., 333 Shekhter, O.Y., 320 Shen, X., 203 Shen, Z., 251, 398, 401 Sheng, J., 136, 140, 144–147, 149, 153 Sherman, R., 23 Shi, G., 358 Shi, P.L., 217–219, 221 Shih, H.H., 398, 401 Shin, E., 96 Shokooh, A., 4 Short, R., 233 Shuai, J.W., 217 Sidhu, T.S., 191–192 Sikdar, B., 23 Silk, M., 348 Simeonov, P., 204, 210 Singh, S., 233 Sinha, H., 170 Sipcic, S.R., 62 Slaats, P.M.A., 384 Slotine, J.J.E., 143 Smith, E.C., 95 Sneddon, I.N., 323 Snowdon, J.C., 50 Snyder, M.D., 333 Soderstrom, T., 409 Soh, Y.C., 204 Soliman, M.S., 177–178 Somaini, D.R., 331 Sommer, G., 280 Sommerer, J.C., 32 Song, B., 135–154 Soukhoterin, E.A., 37 Spanos, P.-T.D., 98 Spence, H.D., 203 Spyrakos, C.C., 331 Sreaonovich, R.L., 240 Srivastava, H.M., 3 Stanisic, M.M., 372 Stanley, F.L., 422–423 Staszewski, W., 276 Steindl, A., 384–385 Stepan, G., 135, 140, 145 Sthindl, A., 264 Stokoe, K.H., 334 Stone, L., 217–219, 221 Stouboulos, I.N., 106 Stoyko, D., 341–354 Strogatz, S.H., 109, 113, 203 Strohoe, K.H., 331

Author Index Strzemiecki, J., 288 Suarez, L.E., 4 Suh, C.S., 77–93 Suhr, J., 96, 101–102 Sul, S.-K., 297 Sun, G., 291 Sun, J.Q., 32, 135–154 Sun, L., 122 Sun, X., 385 Sundararajan, P., 264 Sung, T.Y., 323 Sutera, A., 251 Suzuki, T., 323–324 Swevers, J., 97 Syrmos, V.L., 143–144 Szekely, E.M., 320, 324

T Ta, M.N., 276 Tabiowo, P.H., 324, 331, 333–334 Takuji, K., 158 Tamer, S., 122 Tan, C.A., 377 Tang, J.S., 13–21, 95 Tangpong, X.W., 95–103 Tani, J., 122 Taqqu, M.S., 23 Tassoulas, J.L., 331 Tatjer, J.C., 37 Tato, W., 291 Tauzia, X., 397 Temam, R., 264, 384–385 Teo, E.T.H., 96 ter Weeme, J.W., 385 Terzaghi, K., 331 Tetsuni, S., 357–358 Tetsushi, U., 158 Theirs, G.R., 334 Thompson, J.M.T., 177–178, 395 Thomson, W.T., 323 Thoors, H., 77 Thylwe, K.E., 31 Tian, E., 136 Titchmarsh, E.C., 323 Titi, E.S., 264–265, 385 Tombor, B., 203 Tomlinson, G., 96, 275 Toral, R., 217, 223–226, 234 Torresani, B., 279 Torvik, P.J., 3 Traverso, M.G., 77 Triantafyllidis, T., 330, 332 Trifunac, M.D., 335

457 Troger, H., 264–265, 384–385 Trujillo, J.J., 3–7 Trump, G.D., 422 Tsakalidis, C., 332 Tsao, T.C., 377 Tung, C.C., 398, 401 Tzou, H.S., 357–358

U Ulbrich, H., 121, 123–124, 128 Ulsoy, A.G., 358 Unnikrishnan, N., 409–420 Urchegui, M.A., 291 Ushijima, Y., 297

V Vakakis, A.F., 121, 275 Valaristos, A.P., 106 van Campen, D.H., 384–385 Van den Broeck, C., 223 Van Wiggeren, G.D., 13 Velarde, M., 159 Veletsos, A.S., 324, 330, 332 Velinsky, S.A., 293 Verbic, B., 324 Verho, A., 295–296 Verqni, D., 23 Verriest, E.I., 136 Viana, R., 177 Vijta, M., 143 Vodyanoy, I., 224 Vohra, S., 293–295 Vulpiana, A., 251 Vyas, N.S., 275–283

W Walworth, M., 416 Wan, Y.H., 159, 163, 165 Wang, B., 96 Wang, C.L., 50 Wang, F.Z., 170 Wang, G.Z., 403 Wang, H.Y., 203 Wang, J.L., 263–272 Wang, K.W., 95–96 Wang, L., 159 Wang, Q.G., 136, 158 Wang, T.-Y., 251, 255 Wang, W.J., 402 Wang, X., 4–10, 203 Wang, Y., 189–200

458 Wang, Y.-C., 49–60 Wang, Y.F., 371–380 Wang, Y.Q., 13 Wang, Z.H., 4–10, 96, 169, 251 Warburton, G.B., 323, 333–334 Watson, J.N., 280 Watt, D.A., 438, 441–442 Watts, D.J., 203 Webster, J.J., 333 Wei, B.Q., 96 Wei, J., 159–162, 165 Wei, Y.T., 324, 330, 332 Weiss, M.P., 288 Weiss, R.A., 320 Weissmann, G.F., 330 Wen, C.Y., 204 Wereley, N.M., 95 Wesley, D.A., 333 Westmann, R.A., 324, 332 Whitman, R.V., 330–332 Wichmann, B.A., 438, 441–442 Wickert, J.A., 95 Wiercigroch, M., 78 Wiggins, S., 384 Wilkes, J.O., 140 Wilkinson, J., 347 Willinger, W., 23 Willms, A.R., 204 Wio, H.S., 223–226 Wjewoda, J., 31 Wolf, J.P., 331, 333 Wong, H.L., 324, 330, 332–333, 335 Worden, K., 275 Wriggers, P., 265 Wu, D.J., 224, 234, 236 Wu, J., 159 Wu, M.L.C., 136, 398, 401 Wu, T., 437–446 Wu, X., 135 Wu, Y., 217–221

X Xiang, Y.L., 234 Xie, C.W., 224, 234, 239 Xie, W.X., 224, 234 Xu, H., 203 Xu, J.-X., 13, 32–33, 170, 178, 217, 219 Xu, M., 224, 234, 357–370 Xu, W., 32, 224, 234 Xu, Y., 251–259 Xu, Z., 23

Author Index Y Yamamoto, M., 297 Yamamoto, N., 204 Yan, B.Q., 204, 210 Yanai, N., 297 Yang, B., 135, 372, 375, 377 Yang, C.M., 204–205 Yang, L.B., 204–205 Yang, T., 204–205, 210 Yen, N.C., 398, 401 Yigit, A.S., 358 Yin, J.P., 234 Yin, X.Y., 394 Yoon, R.H., 422 Yorke, J.A., 13, 105, 109, 170 Yoshida, K., 297 Yoshizawa, S., 157, 334 Young, M.R., 233 Yu, A.W., 233–235 Yu, B., 401 Yu, D.J., 13 Yu, H.J., 170 Yu, J.J., 121, 203 Yu, X., 203 Yuan, Y., 159 Yue, D., 136 Yung, W.K.P., 96

Z Zaikin, A.A., 223 Zamojska, I., 372 Zapata, R., 77 Zartarian, G., 62 Zhang, A., 169–174 Zhang, C., 169–174 Zhang, D., 287 Zhang, G.Q., 385 Zhang, H., 251–259 Zhang, J.-Z., 23–29, 61–76, 383–394 Zhang, L., 234, 239, 341 Zhang, N., 263–264 Zhang, P., 50 Zhang, W., 96 Zhang, X.L., 13, 357–370 Zhang, Y., 31–36, 136 Zhang, Z.X., 397–408 Zhao, Y.-J., 251–259 Zheng, J., 291 Zheng, Q.N., 398, 401 Zheng, Y.H., 170 Zhou, C.S., 217 Zhou, J., 203, 205

Author Index Zhou, P.L., 398, 400 Zhou, X., 96 Zhu, P., 233–248 Zhu, S., 233–234, 242 Zhu, S.J., 49–60 Zhu, S.Q., 223–224, 233–235

459 Zhu, X.W., 13 Zhu, Z.H., 287, 289–290 Zhuang, W., 342, 344–346 Zhusubaliyev, Z.T., 37 Zienkiewicz, O.C., 331 Zou, X., 159

Subject Index

A Ada programming language, 437–446 Aerodynamic heating, 62–63, 79 Aerothermoelasticity, 61 Algebraic structures, 438–440, 442, 446 Approximate inertial manifolds (AIM), 385–386, 392, 394 Attached masses, 377 Attached mass-spring oscillators, 371–372, 375, 377, 380 Automation, 341, 421–435 Axially moving string, 371–380

B Basin of attraction, 34, 35, 177, 180 Belt-pulley systems, 286, 292–294 Bifurcation, 31, 33, 34, 36–47, 52–62, 67–74, 105, 109, 113, 127, 157–166, 195, 200, 270–272, 384–385 Bifurcation analysis, 37, 157–166, 384 Buckling, 61, 67, 69–70, 73–74, 385 Bursting and breaking, 23–25, 27, 29

C Carbon nanotube (CNT), 95–103 Cell mapping method, 32 Chain elasticity, 313 Chain mechanisms, 300 Chaos, 13–21, 31, 37, 49–50, 52–53, 55, 62, 70, 105–106, 109, 113–117, 120, 169, 184, 204, 223 Chaotic attractor, 31, 109, 170, 211 Chaotic discharge, 172–174 Chaotic motion, 13, 19–20, 52–53, 56–57, 59, 61, 113 Chaotic solutions, 38 Chaotic vibration isolation, 49–60

Clean coal, 421–423, 434 Clearance, 122, 268, 357–358, 360, 363–364, 367–370 CNT-matrix interface, 96–97, 101, 103 Coal preparation plant, 421–435 Complex networks, 203–215 Complex wire systems, 293–297 Contact stress distribution, 322–323 Continuous time approximation, 137, 139–140, 143, 154 Control, 3, 34, 35, 49–50, 56, 78, 95, 122, 135–154, 169–170, 172–174, 177–184, 191, 203–215, 286, 291, 297, 306, 310–311, 313, 316, 357, 392, 421–435, 442 Coupled system, 204, 207, 212–213, 372, 375, 377 Coupled vibration of foundations, 320, 332–333 Coupling configuration matrix, 204 Coupling strength, 158, 214, 235, 240–248, 371, 377, 380 Crisis, 31–35, 105, 109, 120, 438 Cross-correlated noises, 224, 233–248 Cross-coupling effects, 121–131 Cut-off frequencies, 342, 347–353 Cutting dynamics, 78–81, 91–92

D Damping, 3–10, 51, 90–92, 95–96, 102–103, 106–107, 123, 128–131, 178, 184, 265, 276, 279, 281, 287, 289–291, 297, 316, 324, 327, 330–334, 360–361, 373 Damping force of fractional-order derivative, 3–10 Discontinuous boundaries, 78–79, 82 Distributed friction, 99, 103

461

462 Distribution function, 25, 97–99, 241–243, 445 Dynamical behavior, 121, 158, 169–170, 178, 215, 272, 358 Dynamical error equation, 207 Dynamical properties, 233–248 Dynamic buckling, 61 Dynamics, 32, 37–38, 49–50, 52, 61, 78–79, 90, 96, 105, 121–131, 157, 191, 203–215, 217–218, 221, 224, 263, 265, 285–296, 299–316, 319–320, 333, 378, 384–386, 394, 398–401, 403–404, 408

E Eigenvalue problem, 332, 371–373, 377–378 Elastic properties, 287, 299, 302–304, 323, 335, 347–348 Elevating platform, 300, 314 Empirical mode decomposition (EMD), 251–259, 401 Energy dissipation, 95–103, 291–292 Equilibrium point, 211, 215 Erosion, 177–184 Explicit form, 124, 380

F Feedback, 135–137, 140, 143–150, 152–153, 169–174, 177–184 Feedback control, 135, 137, 144–150, 152–153, 169–174, 182, 184 FHN neuron, 157–166, 217, 220–221 Flexible connector, 357–370 Flow field, 392 Flows around airfoil, 393 Fluid dynamics, 384 Foundations, 319–335 Fractal, 23–26, 32–36, 52, 109, 180, 182 Fractal dimension, 24–25, 52, 180 Fractional-order exponential function, 6, 9 Friction damping, 95–96

G Galerkin method, 62, 65–67, 263–272, 372, 378, 380, 385–386, 392 Galerkin’s discretization, 372, 378–380, 384 Generalized Bagley-Torvik equation, 4, 8 General solution, 3–10, 15–19, 137, 146, 373 Graphical user interface (GUI), 424, 427 Green’s function, 320, 331–332, 371–380 Gyroscopic system, 371, 375

Subject Index H Hankel matrix, 414 H˘older exponent, 24 Henon map, 37–47 Hetero-clinic orbit, 183 High dimensional chaotic system, 31–36 High order system, 264 Hindmarsh-Rose model, 170–172, 221 Homogeneous, 4, 15–16, 319, 333, 342–343, 346, 348, 353, 373–374 Hopf bifurcation, 67, 72, 74, 127, 157–166 Hydraulic boom systems, 300, 307–308, 313 Hydraulic winch, 300, 311–313 Hysteretic friction, 97–98

I Impulsive control, 203–215 Inertial manifolds with time delay, 383–394 Input design, 409–420 Intensity fluctuation, 233–248 Interaction between foundations, 319, 330, 333–335 Interfacial stiffness, 96–97 Inverse problem, 342, 348 Isotropic, 108, 120, 122, 321, 342–343, 348, 353

J Jacobian matrix, 66–67, 69, 72, 127, 195–196, 380

K Kalman filter, 410, 415 Krylov-Bogoliubov method, 275–277

L Least-squares estimation, 410 Limit cycles, 105, 109–110, 113, 120, 157, 316 Line spectrum reduction, 49–50, 57 Local area networks traffic, 23–29 Low-oscillation Morlet wavelets, 276, 280–283 Lumped parameter models, 306, 330–331, 333 Lyapunov exponent, 13, 31, 32, 37, 52, 119, 125 Lyapunov function, 136, 207

Subject Index M Machine tool vibration, 81, 90 Markov parameters, 413–415 Maximum variance rate, 372, 377–378, 380 Microcontroller, 424–427 Mixed boundary value problems, 323–330 Mixed parity oscillator, 277, 281–282 Model number, 438, 441–442, 444–446 Model reduction, 383–394 Multifractal spectrum, 23–29 Multilevel finite element method, 386–388 Multi-span rotor, 263, 304

N Nanocomposites, 96, 103 Navier-Stokes equations, 383–394 Negative mapping, 38–39, 41–42, 44–45, 47 Neurons, 157–166, 169–174, 217–221 Nodes, 42, 44–45, 204–211, 215, 266, 364–365, 367, 369, 386–388 Noise, 49–50, 178, 217–221, 223–230, 233–248, 251–252, 254–255, 257–258, 351, 398, 410, 414, 416, 420 Non-dimensionalization, 65, 75, 99, 107–108, 123, 288, 290, 372 Non-equilibrium phase transitions, 223–230 Non-Gaussian noise, 223–230 Nonlinear continuous dynamics, 383–384, 394 Nonlinear dynamic model, 265, 272 Nonlinear dynamics, 49, 61, 105, 263–265, 268, 272, 358, 365, 383–386 Nonlinear Galerkin method (NLGM), 264–265, 385–386 Nonlinearity classification, 275–283 Nonlinear partial differential equation, 384 Nonlinear system identification, 275 Nonlinear vibration isolation system, 49–52, 56–57, 59 Non-stick motion, 79 Non-uniform, 25, 27, 97, 139–140 Numerical simulation, 26–29, 90, 105, 109, 120, 140, 191, 200, 211–215, 219, 351–353, 358

O Observer/Kalman filter identification (OKID), 410, 413 Optimal gains, 144–147, 149, 153 Order reduction, 263–272 Oscillation, 61, 67, 70, 72–74, 82, 157, 159, 171, 174, 191, 277–279, 281, 313, 316, 323–324

463 P Panel flutter, 61–76 Period, 38, 43, 52–53, 79, 137–138, 149, 163, 166, 170, 180, 194, 313 Periodic flow, 189–200 Periodic oscillation, 61, 67, 70, 72–74 Periodic solutions, 37–47, 124, 159, 163, 166, 171–174, 194 Perturbation, 13–21, 39–40, 127, 170, 371, 385 Perturbation method, 14 Perturbed equations, 13–18, 20 Phase space, 32, 34, 35, 163, 171–173, 178, 192, 384 Pipe, 308, 342–348, 352–354, 371 Pitch-fork bifurcation, 70, 74 Positive mapping, 39, 41–45 Power flow attenuation rate (PFAR), 57–60 Power flow line spectrum, 57, 59–60 Power flow transmissibility, 50, 56 Predictor-corrector Galerkin method, 263–272 Q Quadratic oscillator, 277, 281 R Reynolds number, 430–431 Ridge, 279–282 Rolling motion, 183 Rotor-bearing system, 263–272 Rotor dynamics, 263 Rotor/stator rubbing, 121–131 Rough number, 437–446 Rough set, 437–438 S Safe basin, 177–184 Saturation laser model, 233–248 Second-order nonlinear dynamical system, 14 Self-defined truss element, 357–370 Self similarity, 23 Semi-Analytical Finite Element (SAFE), 342, 347–348, 351 Semi-discretization, 137–140, 144–145, 154 Sensitivity to initial condition, 13–21 Separation specific gravity, 422, 435 Servo control, 310, 313 Simulations, 26–29, 45, 61–76, 90, 105, 109, 120, 140, 191, 200, 211–215, 219, 264, 281, 287, 291–297, 299, 313, 316, 319–320, 335, 351–353, 358–359, 416

464 Single-mode laser model, 223–230, 234 Single-stage spiral operation, 422 Singularity exponent, 24 3-D Siwthing system, 189–200 Soil, 319–335 Spiral circuit, 421–423 Spiral concentrator, 421–435 Spline coupling, 105–107 Stability, 39, 46, 62, 65, 95–96, 105, 122–131, 135–136, 138, 140, 145–147, 149–150, 157–166, 191, 195–196, 198, 200, 204–206, 215, 248, 291, 297, 385 Stability analysis, 123–127, 131 Stable period-doubling bifurcation, 42–44 stable saddle bifurcation, 42 Standard Morlet wavelets, 279–283 Stick-motion, 97, 101–103 Stick/slip, 95–97, 101–103 Stochastic resonance, 251–259 Subsystems, 189–200, 223, 264–269, 272, 371, 377, 380, 425 Sub-threshold oscillation, 171 Surface responses, 320–322 Switch ability, 82–83 Switching system, 189–200 Symmetric oscillator, 281 Synaptic intensity, 170–173 Synchronization, 158, 203–204, 206–207, 211–215, 217–221 Synchronization manifold, 204, 211, 214–215 Synchronous full annular rub solution, 121–122, 124–131 System identification, 275, 409–420

Subject Index T Time-delay, 135–154, 157–166, 169–174, 178–184, 418 Time-delay coupling, 157–166 Time sequence, 24, 26, 28, 311 Transition, 31, 103, 138, 172, 223–230, 271, 287, 347 Two-stage spiral operations, 422

U Ultrasonic guided waves, 353 Ultrasonic receiver, 420 Ultrasonic transducer, 409–420 Ultrasonic transmitter-receiver pair, 409–420 Unstable period-doubling bifurcation (UPD), 42–45 Unstable saddle bifurcation, 43, 45 Unsteady flux, 23

V Vibration system, 3–10

W Wall thickness, 348, 353 Wavelet transforms, 275–276, 278–282 Weak signal, 251–252, 254–255, 257–259 Wheatstone bridge, 425–426 Wire-driven machine, 285–297, 299–316 Wire elasticity, 313 Wire mechanisms, 285–297, 299–316 Wire rope, 285–292, 296–297 Wire-rope mechanisms, 295–296 Wire-rope systems, 291–297 Wire tension, 287, 289, 296, 300, 302, 306–307, 312–313