Dynamical Systems: Discontinuity, Stochasticity and Time-Delay 1441957537, 9781441957535

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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.

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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

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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

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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

. 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 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

5

Complete Bifurcation Behaviors of a Henon Map

45

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

5

Complete Bifurcation Behaviors of a Henon Map

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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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63

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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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

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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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A Parameter Study of a Machine Tool with Multiple Boundaries

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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A Parameter Study of a Machine Tool with Multiple Boundaries

85

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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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

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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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99

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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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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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

106

C. Nataraj and K. Kappaganthu

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

108

C. Nataraj and K. Kappaganthu 00

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

112

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]

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 11, 

121

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Z. Shang et al.

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

11

Cross-Coupling Effects on the Dynamics of Rotor/Stator Rubbing

123

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 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

10

20

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

80

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

−1

−2 0

10

20

2

x1

1

−2 0

10

20

2

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.

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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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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.

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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, 

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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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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)

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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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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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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)

and 1 i!0 Q1 e Œ1 C .i!0 C a/.i!0  b2 /; C1 1 ei!0 Q1 Œ1 C .i!0 C a/.i!0  b2 /; q2 D  C1 .i!0  b1 / q3 D i!0  b2 : q1 D 

(13.36)

Then, we can choose D D .qN1 q1 C qN2 q2 C qN3 q3 C 1/ 

Z

0

Q 2

Z



qN .  /d./q./d:

(13.37)

D0

Then hq  .s/; q./i D 1: Following the algorithms given in Hassard et al. [12] and using a computation process similar to that of Wei and Ruan [15], we can obtain the coefficients which will be used in determining the important qualities: g20 D g11 D g02 D 0; 

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

c1 .0/ D

(13.39)

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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]

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 14, 

169

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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 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, 

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

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

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5.0 time, t

7.5

−50.0 0.0

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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

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50.0

−8.5

P1

−6.5

d

P1

25.0

−5.0

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−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

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P1

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2.5

5.0 time, t

7.5

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−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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A.C.J. Luo and Y. Wang

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
t0 and impulse function Ik .xi / W D!Rn ; kD0; 1; 2; : : :. !1; t1 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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211

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

xi1(t)(i=1,2,3,4)

20 10 0 −10 −20 0

5

10

15

20

25

30

35

40

45

50

5

10

15

20

25

30

35

40

45

50

5

10

15

20

25 t/s

30

35

40

45

50

xi3(t)(i=1,2,3,4)

xi2(t)(i=1,2,3,4)

40 20 0 −20 −40 0 60 40 20 0 0

xi2(t)(i=1,2,3,4) xi1(t)(i=1,2,3,4)

10

xi3(t)(i=1,2,3,4)

Fig. 17.1 Variable state of the ring networks with four Lorenz coupled without impulsive control

10

5 0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1 t/s

1.2

1.4

1.6

1.8

2

15 10 5 0

5 0

Fig. 17.2 Variable state of the ring networks with four Lorenz coupled for  D 0:5

xi3(t)(i=1,2,3,4) xi2(t)(i=1,2,3,4) xi1(t)(i=1,2,3,4)

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Impulsive Control Induced Effects on Dynamics of Complex Networks

213

10 5 0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0

0.05

0.1

0.15

0.2

0.25 t/s

0.3

0.35

0.4

0.45

0.5

15 10 5 0 10 5 0

xi3(t)(i=1,2,3,4)

xi2(t)(i=1,2,3,4)

xi1(t)(i=1,2,3,4)

Fig. 17.3 Variable state of the ring networks with four Lorenz coupled  D 1

100 50 0 −50 −100 0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

−200 0

1

2

3

4

5 t /s

6

7

8

9

10

200 100 0 −100 −200 200 100 0 −100

Fig. 17.4 Synchronization of the variable states of the networks with four Lorenz coupled for  D 3:5

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

xi3(t)(i=1,2,3,4)

xi2(t)(i=1,2,3,4)

xi1(t)(i=1,2,3,4)

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X. Han and X. Fu 100 50 0 −50

−100

8

8.2

8.4

8.6

8.8

9

9.2

9.4

9.6

9.8

10

8

8.2

8.4

8.6

8.8

9

9.2

9.4

9.6

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Fig. 17.4 (continued)

x11−x21

20 0 −20 0

x12−x22

50 0 −50 0 x13−x23

50 0 −50

−100 0

Fig. 17.5 The synchronization errors between the first and second node of the four Lorenz coupled networks for  D 3:5

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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100 50 0 −50 −100 −150 200

150

100

50 x12

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−50 −100 −20 −40 −150 −200 −80 −60

0 x11

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Fig. 17.6 The dynamical behavior of the first node of networks for  D 3:5

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]

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 18, 

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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.

a

b 2 I=3.3

5

1.5

4

1

3 2 1

0

x

x

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0

−0.5

−1

−1

−2

−1.5 −2 3.25

I=3.3

−3 3.3

3.35 z

3.4

3.45

−4 2.4

2.5

2.6

2.7

2.8

2.9

z

Fig. 18.1 The projections of attractors of HR neurons in z  x plane for I D 3:3, the projections of noiseless attractors (a); the projections of attractors synchronized by noise, k D 2:4 (b)

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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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a

b

η=4.7

40

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40

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−40

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h

h

Fig. 18.2 The projections of attractors of HH neurons in V h plane for  D 4:7, the projection of noiseless attractors (a) the projection of attractors synchronized by noise (b)

a

b 0.15

b=0.2404

0.15 b=0.2404

0.1

0.1

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0.05

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w 0

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−0.1 −0.2

0

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0.4 v

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−0.1 −0.5

0

0.5

1

v

Fig. 18.3 The projections of attractors of FHN neurons in v  w plane for b D 0:2404, the projection of noiseless attractors (a) the projection of attractors synchronized by noise (b)

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]

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 19, 

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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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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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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

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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

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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

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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

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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, 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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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)

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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
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

242

P. Zhu 0.8

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

20

Dynamical Properties of Intensity Fluctuation

243

a 1.5 1.25 l= 0.95,D>0

Pst(I)

1

l= 0.4545,D=0

0.75 0.5 0.25

l=0.2,D 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

246

P. Zhu

a

1

C(s)

0.8 0.6 0.4

—————————— 0.9 —————————— 0 —————————— 0.9

0.2 0

b

0

0.05

0.1 s

0.15

0.2

1

C(s)

0.8 0.6 —————————— 1.2 —————————— 0.1 —————————— 0

0.4 0.2 0

0

0.02 0.04 0.06 0.08 0.1 0.12 0.14 s

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.

20

Dynamical Properties of Intensity Fluctuation

247

a 0.14

Tc

0.12 0.1 0.08 -0.75 -0.5 -0.25

b

0

0.25 0.5 0.75

1

C(s)

0.8 ————————— 0.9 0.5 —————————— —————————— 0 —————————— 0.5 —————————— 0.9

0.6 0.4 0.2 0

0.2

0.4

s

0.6

0.8

1

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]

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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

Part IV

Classic Vibrations and Control

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

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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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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]

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 23, 

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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. 23.2 (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]

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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.

306

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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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E. Keskinen et al.

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 ΠXtip 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

314

E. Keskinen et al. MOT

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, 

319

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H.R. Hamidzadeh

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

26

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321

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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323

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

324

H.R. Hamidzadeh

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

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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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331

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-

334

H.R. Hamidzadeh

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.

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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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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