IUTAM Symposium on Dynamics and Controlof Nonlinear Systems with Uncertainty

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Aims and Scope of the Series

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For a list of related mechanics titles, see final pages.

Volume 2

IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty Proceedings of the IUTAM Symposium held in Nanjing, China, September 18-22, 2006

Edited by

H. Y. Hu Nanjing University of Aeronautics and Astronautics, Nanjing, China

and

E. Kreuzer Hamburg University of Technology, Hamburg, Germany

A C.I.P. Catalogue record for this book is available from the Library of Congress.

Published by Springer,P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

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Printed on acid-free paper

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ISBN 978-1-4020-6331-2 (HB)ISBN 978-1-4020-6332-9 (e-book)

CONTENTS

Opening Address ·······························································································

Welcome Address ·····························································································

Contributed Papers

PART 1

A. K. Bajaj, P. Davies, R. Ippili and T. Puri Nonlinear Multi-Body Dynamics of Seat-Occupant Systems Using Experimentally Identified Viscoelastic Models of Polyurethne

1

M. Hernandez-Garcia, S. F. Masri, R. Ghanem and F. Arrate Data-Based Stochastic Models of Uncertain Nonlinear Systems ··········· 11

C. Proppe and C. Wetzel Overturning Probability of Railway Vehicles under Wind Gust Loads ······································································································· 23

W. Schiehlen and R. Seifried Impact Systems with Uncertainty ···························································· 33

System Modeling with Uncertainty

Foam ·········································································································

Preface ················································································································

xvii

xxi

v

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PART 2

S. K. Au and D. P. Thunnissen

T. F. Filippova

A. Gaull and E. Kreuzer Cell Mapping Applied to Random Dynamical Systems ························· 65

X. L. Leng

X. B. Liu The Maximal Lyapunov Exponent for a Stochastic System ··················· 87

W. V. Wedig Stability and Density Analysis of Stochastic Duffing Oscillators ·········· 97

J. X. Xu and H. L. Zou

W. Xu, Q. He and S. Li The Cell Mapping Method for Approximating the Invariant Manifolds······························································································· 117

PART 3

L. Bevilacqua and M. M. Barros Dynamical Fractal Dimension: Direct and Inverse Problems··············· 127

T. Bódai, A. J. Fenwick and M. Wiercigroch Ray Stability for Range-Dependent Background Sound Speed Profiles ··································································································· 137

N. D. Anh, N. Q. Hai and W. Schiehlen Application of Extended Averaged Equations to Nonlinear Vibration Analysis ················································································· 147

Z. Q. Wu and Y. S. Chen Singularity Analysis on Constrained Bifurcations ································ 157

System Dynamics with Uncertainty

Nonlinear Dynamics

by Advanced Monte Carlo Methods ······················································· 45

Trajectory Tubes in Control and Estimation Problems under Uncertainty ···················································································· 55

Numerical Analysis of Bifurcation and Chaos Response 77

and the Coherence of Stochastic Dynamical Systems ·························· 109 Uncertainties in Deterministic Dynamical Systems

Uncertainty Propagation in Complex Engineering Systems

in a Cracked Rotor System under White Noise Disturbance ··················

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Contents vi

H. Yabuno, Y. Kunitho, T. Inoue and Y. Ishida

S. Yang and Y. Shen Nonlinear Dynamics of a Spur Gear Pair with Slight Wear Fault ········ 177

K. Yunt and C. Glocker

of Optimal Hybrid Mechanical Trajectories ········································· 187

PART 4

J. Awrejcewicz, G. Kudra and G. Wasilewski Numerical Prediction and Experimental Observation of Triple Pendulum Dynamics ·············································································· 197

A. J. Dick, B. Balachandran, and C. D. Mote, Jr. Nonlinear Vibration Modes and Energy Localization in Micro-Resonator Arrays···································································· 207

L. Q. Chen and X. D. Yang

Beam with Non-Typical Boundary Conditions ····································· 217

F. L. Chernousko Dynamics of a Body Controlled by Internal Motions ··························· 227

W. Lacarbonara, A. Paolone and F. Vestroni Linear and Nonlinear Elastodynamics of Nonshallow Cables ·············· 237

S. Lenci and G. Rega Nonlinear Normal Modes of Homoclinic Orbits and their Use for Dimension Reduction in Chaos Control ·········································· 247

A. Teufel, A. Steindl and H. Troger Rotating Slip Stick Separation Waves ··················································· 257

M. H. Yao and W. Zhang Many Pulses Homoclinic Orbits and Chaotic Dynamics for Nonlinear Nonplanar Motion of a Cantilever Beam ························ 267

PART 5

I. Ananievski Synthesis of Bounded Control for Nonlinear Uncertain Mechanical Systems ·············································································· 277

Dynamics of High-Dimensional Systems

Control of Nonlinear Dynamic Systems

of Multiple Scales ·················································································· 167 Nonlinear Analysis of Rotor Dynamics by Using the Method

A Combined Continuation and Penalty Method for the Determination

Parametric Resonance of an Axially Accelerating Viscoelastic

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Contents vii

P. Barthels and J. Wauer Controlled Vibration Suppression of Structural Telescopic Systems ·································································································· 287

P. B. Gonçalves and D. Orlando

H. Y. Hu and M. L. Yu Robust Flutter Control of an Airfoil Section through an Ultrasonic Motor ····································································································· 307

K. Czołczyński, A. Stefański, P. Perlikowski and T. Kapitaniak

Suspended on Elastic Beam ··································································· 317

Q. Y. Wang, Q. S. Lu, X. Shi and H. X. Wang

PART 6

Y. F. Jin and H. Y. Hu

G. Stépán and T. Insperger Robust Time-Periodic Control of Time-Delayed Systems ··················· 343

P. Wahi, G. Stépán and A. Chatterjee Self-Interrupted Regenerative Turning ················································· 353

Z. H. Wang and H. Y. Hu

Parameters ····························································································· 363

J. Xu, M. S. Huang and Y. Y. Zhang

W. Q. Zhu and Z. H. Liu Stability and Response of Quasi Integrable Hamiltonian Systems with Time-Delayed Feedback Control ·················································· 383

Author Index ···································································································· 393

Dynamics of Time-Delay Systems

Influence of a Pendulum Absorber on the Nonlinear Behavior and Instabilities of a Tall Tower ··························································· 297

Periodization and Synchronization of Duffing Oscillators

in Coupled Neuronal Systems ······························································· 323 Effects of Noise on Synchronization and Spatial Patterns

with Delayed Feedback Control ···························································· 333

Robust Stability of Time-Delay Systems with Uncertain

Stability and Response of Stochastic Delayed Systems

Dynamics due to Non-Resonant Double Hopf Bifurcationin in Van Del Pol-Duffing System with Delayed Position Feedback ······· 373

Contents

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viii

PREFACE

The last decade has witnessed an increasing interest towards the

dynamics and control of nonlinear engineering systems from the scientists engaged in nonlinear dynamics and the control engineers. Both groups of people have recognized the importance of interaction between nonlinear dynamics and robust control during their efforts to improve the dynamic performance of engineering systems with uncertainty, which comes from either the random excitations, such as wind and earthquake, or the modelling errors of real systems including their sensors, controllers and actuators. The

deterministic systems. This volume contains the papers presented at the IUTAM Symposium

on Dynamics and Control of Nonlinear Systems with Uncertainty, which was sponsored by the International Union of Theoretical and Applied Mechanics (IUTAM) and held at Nanjing University of Aeronautics and Astronautics, China, 18-22 September, 2006. The aim of the symposium was to bringing together the scientists to discuss the advances in dynamics and control of nonlinear systems, especially those with uncertainties in either system modeling or excitation.

The scientific committee, appointed by the Bureau of IUTAM, includes the following members:

F. L. Chernousko, Moscow, Russia E. Kreuzer, Hamburg, Germany (Co-Chairman) H. Y. Hu, Nanjing, China (Chairman) A. H. Nayfeh, Blacksburg, USA G. Rega, Rome, Italy W. Schiehlen, Stuttgart, Germany (IUTAM representative) K. Sobczyk, Warsaw, Poland G. Stepan, Budapest, Hungary H. Troger, Wien, Austria

is a vital interdisciplinary topic related to both stochastic systems and dynamics and control of nonlinear systems with uncertainty, therefore,

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The committee selected the participants to be invited and the

presentations to be given at the symposium. As a result, 53 active scientists

1. System Modeling with Uncertainty. Uncertainties arising in system modelling may have a great influence on system dynamics and control. In modelling of mechanical systems, the descriptions of backlash and friction, as well as hysteresis, most likely introduce uncertainties and have drawn increasing attention over the past years. For example, W. Schiehlen et al. observed in both experiments and computations that for a sphere striking a beam, the coefficient of restitution was uncertain due to multiple impacts resulting in chaotic behaviour. Based on this observation, they proposed an efficient numerical approach for modelling, and verified the numerical models by experiments. S. F. Masri et al. investigated some significant issues in modelling uncertain parameters of hysteretic nonlinear systems subject to deterministic excitation. He found that the parameters, such as the yielding parameter, in the model were uncertain, and discussed also the uncertainties in the identified coefficients and statistic features. In studying practical engineering systems, A. K. Bajaj et al. developed a modelling technique to predict the static equilibrium position of an occupant seated in a car seat of polyurethane foam, and then used the model to identify the system parameters. C. Proppe et al. proposed a consistent stochastic model for wind gust and computed the probabilistic characteristic wind curves by using a reliability analysis of the train-environment system.

2. System Dynamics with Uncertainty. The studies on the dynamic systems with uncertainties, including both stochastic systems and deter-ministic systems, are mainly a combination of theoretical and numerical analysis. For instance, E. Kreuzer et al. extended the concept of Cell Mapping to the global dynamics of randomly perturbed dynamical systems. J. X. Xu et al. investigated the noise effect at different scales on the boundary of Wada basin, including the crisis coming from a strong perturbation. W. Xu et al. discussed the approximation of the invariant manifolds of nonlinear systems with uncertainty by means of the method of digraph cell mapping. W. V. Wedig and X. B. Liu studied the estimation and computation of the maximal Lyapunov exponent for stochastic bifurcation systems, respectively. T. Filippova studied the control and state estimation of dynamical systems with uncertainty, described by differential equations with measure (or impulsive control) component. For engineering appli-cations, S. K. Au et al. introduced an advanced Monte Carlo method, named Subset Simulation, to solve the problem of uncertainty propagation in

from 17 countries accepted invitation, and 40 of them made oral presen- tations at the symposium. The presentations cover the following topics:

Prefacex

Preface complex engineering systems. X. L. Leng analyzed the bifurcation and chaotic response of a cracked rotor system under white noise disturbance.

3. Nonlinear Dynamics. Recent studies on nonlinear dynamics have coped with either subtle academic problems or practical engineering problems, with help of new mathematical and physical tools, and more and

framework for the determination of non-smooth trajectories. T. Bodai et al. introduced the concept of ray chaos to the study on underwater sound propagation and found that the combination of nonlinear dynamics and ray theory provides a powerful tool in analyzing underwater sound problems. L. Bevilacqua et al. explored the concept of dynamic fractal dimension, proposed a new method to determine the fractal dimension of plane curves and discussed its possible applications to dynamic problems. Z. Q. Wu et al. studied the problem of constrained bifurcations, including the bifurcation of a dynamic system with a parameterized constraint in either single-sided or double-sided form and the bifurcation defined by piecewise, continuous

suspension system of two degrees of freedom. H. Yabuno et al. implemented the method of multiple scales to analyze the nonlinear rotor dynamics of two degrees of freedom. S. P. Yang et al. applied the Incremental Harmonic Balance Method to study the nonlinear dynamics of a spur gear pair with slight wear fault, where the backlash, time-varying stiffness and wear fault were all included in the model.

4. Dynamics of High-Dimensional Systems. A deep insight into the nonlinear dynamics, such as internal resonance, bifurcation and chaos, of high dimensional systems plays an important role in creating new control methods and strategies. Some scientists focused on the high dimensional systems with good background of real engineering applications, with help of analysis, computation and experiments, and made important progresses. For example, F. L. Chernousko studied the dynamics and control of a simple mobile robot, which consists of a rigid body and an internal lumped mass swinging inside the robot. The swing of the internal lumped mass and the external friction of the rigid body jointly drive the robot. He gave an

energy localization for the micro-resonator arrays in MEMS. H. Troger et al. analyzed the wave propagation of a brake squeal occurring in high speed

more powerful computational techniques as well. For example, K. Yunt

N. Q. Hai et al. applied the extended averaged equations to a nonlinear functions, and applied his results to the rotor rub-impact prediction, etc.

grees of freedom as a measure-differential inclusion and proposed a unified et al. represented the dynamics of a robotic manipulator with blockable de-

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vehicles with a drum brake, by means of the centre manifold reduction.

B. Balachandran et al. made an analysis of nonlinear vibration modes and robot and verified his results in a number of interesting experiments.estimation of the maximal possible averaged speed of motion of the

W. Lacarbonara et al. investigated the linear and nonlinear elastodynamics of nonshallow cables by using the method of multiple scales. Meanwhile, other scientists tried to understand deeply the nonlinear dynamics of the classic high-dimensional systems. For instance, J. Awrejcewicz et al. analyzed the global complicated dynamics of a triple-pendulum and partly verified the observed results in experiments. W. Zhang et al. analyzed the multiple pulses homoclinic orbits and chaotic motion of a cantilever beam subject to a harmonic axial excitation and two transverse excitations at the free end, on the basis of the generalized Melnikov method. L. Q. Chen et al. studied the parametric resonance of an axially accelerating viscoelastic beam with non-typical boundary conditions by using the method of multiple scales.

5. Control of Nonlinear Dynamic Systems. Addressed under this topic are two kinds of problems. One is about the design of control or robust control strategies, new actuators and their integrations for specific engineering applications, and the other is about chaos control and synchronization. For example, H. Y. Hu et al. studied the robust flutter suppression of the airfoil section with the control surface driven by an ultrasonic motor, and discussed the effect of a time delay arising from digital filter on the stability of the controlled system. P. Barthels et al. dealt with the controlled vibration suppression of structural telescopic systems and discussed the design of robust controller. P. B. Goncalves et al. used a pendulum absorber of large amplitude movement to improve the dynamic response of a tall tower, and pointed out that this strategy of nonlinear control was attractive and had a great potential in engineering. I. Ananievski investigated the synthesis of bounded control for nonlinear uncertain mechanical systems and illustrated the results through the numerical simulations of a controlled plane rotation of a bar attached to a movable base. T. Kapitaniak et al. investigated the synchronization of chaotic oscillators suspended on the elastic structure and found that the behaviour of the oscillators became periodic under certain conditions. Q. S. Lu et al. studied the noise effect on the synchronization and transition of firing patterns in coupled neurons.

years since almost all controlled systems involve unavoidable time delays. For example, G. Stepan et al. studied the position control of a single body with delayed discrete feedback by using the so-called “act and wait” scheme, and analyzed the stability and robustness of the controlled system. W. Q. Zhu et al. investigated the delay effect on the stability and bifurcation of a

6. Dynamics of Time-Delay Systems. Time-delay systems have re- ceived more and more attention from the circle of mechanics over the past

Prefacexii

the homoclinic orbits of a given hilltop saddle and the chaos control problem. S. Lenci et al. applied the method of nonlinear normal modes to analyzing

Preface kind of quasi-integrable Hamiltonian systems with delayed feedback control. Y. F. Jin et al. studied the moment stability of stochastic delayed systems with delayed feedback control and additive/multiplicative Gaussian white noise, by means of the method of stochastic averaging. Z. H. Wang et al. analyzed the robust stability of time-delay systems with respect to para-metric uncertainties. J. Xu et al. and P. Wahi et al. studied the double-Hopf bifurcation of time-delay systems, respectively.

The papers in each part of the volume are arranged in alphabetical order with respect to the surname of the lecturer.

We wish to thank all participants of this IUTAM Symposium, and all organizers, especially Prof. Z. H. Wang, Scretary-General of Local Organizing Committee, for their enthusiastic and valuable contributions to the Sympo-sium and the editorial work of the volumn. We gratefully acknowledge the financial supports from IUTAM and The National Natural Science Founda-tion of China. Finally, we greatly appreciate the successful cooperation with publisher Springer. H. Y. Hu, Nanjing E. Kreuzer, Hamburg

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i

OPENING ADDRESS

Dear colleagues, Ladies and Gentlemen,

It is my great pleasure to announce the opening of the IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty. As Chair-man of both Scientific Committee and Local Organizing Committee of the symposium, I wish to extend my warm welcome to all the participants. As President of Nanjing University of Aeronautics and Astronautics, I would like to extend the warm welcome of my university to all the guests, especi-ally to those who have traveled all the way to Nanjing for this important event. Welcome to the symposium, welcome to Nanjing!

Nanjing is an ancient city with its history of more than 2,400 years. It used to be one of four important ancient Chinese capitals, as well as the capital of Republic of China from 1912 to 1949. The long history of the capital laid a solid foundation of culture for Nanjing. Now, we are standing at the ruins of the Ming Palace, which was founded in 1360s but destroyed in a series of wars later. For example, the ruins of the Imperial Temple of Ming Dynasty, including an ancient well in the present central garden, was found during the construction of this library. From the library, you can see the Purple Mountain, which is famous not only for the relics, but also for the first astronomical observatory in China.

Now, I would like to give you a brief introduction to Nanjing University of Aeronautics and Astronautics, which is abbreviated as NUAA. The university was founded in 1952. In the early stage, NUAA was mainly an educational institution of aeronautical technology, following the educational system of former Soviet Union. With rapid developments since 1980s, NUAA has been among the top universities in China. At present, there are about 1,400 faculty members and over 1,300 administrative and technical staff with the university. More than 22,000 full time students, including 7,000 graduate students pursuing Ph.D. and Master degrees, are studying on

xvii

its two campuses. One is the old campus where we are holding the symposium, and the other stands beside the highway from Lukou Airport to downtown. NUAA has 12 schools, covering most fields of engineering, natural science, management, economics, law, arts and humanities. These

programs. Among them, NUAA features the excellent education and research of aerospace engineering and civil aviation, with great contributions made to the China’s aerospace industry and civil aviation.

The research of engineering dynamics in NUAA stemmed from the early study of Professor Azhou Zhang, my Ph.D. supervisor, on structural dynamics in 1960s. Professor Zhang and his colleagues successfully established the Institute of Vibration Engineering Research. I was greatly honored to serve as the fourth director of the institute from 1994 to 1996. The institute has made many important achievements in theoretical and experimental modeling, analysis and simulation, design, control and fault diagnosis of a great variety of dynamic systems in engineering, including many airplanes, helicopters, rockets, ground vehicles, bridges and tall buildings designed in China. It has become an important national research center of vibration engineering, and has served as the host of the Chinese Society for Vibration Engineering since its inception in 1986. The

ideas with all the participants. I believe, the symposium will not only promote the further research in nonlinear dynamics and control at NUAA, but also initiate and enhance the cooperation of NUAA with other research institutions around the world.

The last decade has witnessed numerous advances in the dynamics and control of nonlinear engineering systems, reported in part at a series of successful IUTAM symposia such as those in Stuttgart, 1990; London, 1993; Eindhoven, 1996; Cornell, 1997; Rome, 2003. On one hand, the scientists in nonlinear dynamics have developed new control strategies, such as the OGY control and Pyragas’ delayed feedback control, for nonlinear engineering systems from their good understanding of nonlinear dynamics. On the other hand, the control engineers have paid considerable attention to various intelligent and robust controls for an increasing number of complicated dynamic systems with uncertainty since most information used to support decisions is approximate by nature. Both groups of people have recognized the importance of interaction between nonlinear dynamics and robust control in their efforts to improve the dynamic performance of engineering systems with uncertainty. Therefore, the dynamics and control of nonlinear systems with uncertainty has become a vital interdisciplinary topic.

schools offer 44 undergraduate programs, 127 master programs and 55 Ph.D.

symposium undoubtedly provides NUAA faculty members with an oppor- tunity to demonstrate their recent achievements and to exchange their

xviii Opening address

Opening address

Nowadays, the concept of dynamics and control implies the combination of dynamic analysis and control synthesis. It is essential to gain an insight into the dynamics of a nonlinear system with uncertainty if any new control strategy is designed to utilize nonlinearity. However, the new control strategy to be proposed must be robust enough so that any small disturbances do not alter the desired target of control. Such a concept is calling for more attention to the modelling and simplification of dynamic systems subject to uncertain environment, the fine analysis and robust design of controlled dynamic systems resulting in new control strategies due to understanding of nonlinear phenomena and artificial intelligence, the combination of passive control, active control and semi-active control, as well as the interaction among sensors, controllers and actuators.

Faced with the above trend, Prof. Edwin Kreuzer and I proposed that this IUTAM symposium focuses on both nonlinear dynamics with uncertainty and robust control. As a result, some renowned scientists of nonlinear stochastic dynamics joined us. They will definitely bring us some fresh ideas of studying uncertain dynamics.

Compared with previous IUTAM symposia on dynamics and control of nonlinear engineering systems, the Scientific Committee of this symposium has invited more active young scientists. I believe, the symposium will offer a forum for young participants to demonstrate their recent achievements, find and discuss various open problems in this field. Most young scientists believe that they are being faced relatively with more challenges in the field of mechanics than their supervisors. Hopefully, they feel to have more opportunities than their supervisors when they leave the symposium with open and interesting problems.

Finally, I wish to thank all members of Scientific Committee and Local Organizing Committee for their valuable work. I wish the symposium a tremendous success! And I also wish everybody a nice stay in Nanjing. September 18, 2006 H. Y. Hu Nanjing University of Aeronautics

and Astronautics, China

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WELCOME ADDRESS

Mr. President and Mr. Chairman, Dear Colleagues from all over the world, Ladies and Gentlemen,

It is my honour and pleasure to welcome all of you on behalf of the International Union of Theoretical and Applied Mechanics here in China.

As we have learnt, Nanjing University of Aeronautics and Astronautics was established in 1952, and already in 1996 it succeeded in becoming one of the hundred key universities of China. NUAA is devoted to teaching and research in science and engineering with special emphasis to aeronautics and astronautics. And there are key programmes in engineering mechanics, too. Thus, NUAA is a perfect place to hold an IUTAM Symposium.

Let me use this Opening Ceremony for a short look on the past and present activities of IUTAM.

Organized meetings between scientists in the field of mechanics were initiated 84 years ago, namely in 1922, when Prof. Theodore von Kármán and Prof. Tullio Levi-Civita organized the world’s first conference in hydro- and aero-mechanics. Two years later, in 1924, the First International Congress was held in Delft, The Netherlands, encompassing all fields of mechanics that means analytical, solid and fluid mechanics, including their applications. From then on, with exception of the year 1942, International Congresses in Mechanics have been held every four years. The 20th Congress took place in Chicago, USA, at the turn of the century highlighted by a poster featuring the history of mechanics.

In particular, when the mechanics community reassembled in Paris for the Sixth Congress in 1946, out of the congress series an international union was formed, and as a result IUTAM was created and statutes were adopted. After one year, in 1947, the Union was admitted to ICSU, the International Council for Science. This council coordinates activities among various other

xxi

scientific unions to form a tie between them and the United Nations Educational, Scientific and Cultural Organization, well known as UNESCO.

Today, IUTAM forms the international umbrella organization of about 50 national Adhering Organizations of mechanics from nations all over the world. Furthermore, a large number of international scientific organizations of general or more specialized branches of mechanics are connected with IUTAM as Affiliated Organizations. As a few examples, let me mention: the European Mechanics Society (EUROMECH), the International Association of Computational Mechanics (IACM), the International Association for Vehicle System Dynamics (IAVSD), and the International Commission of Acoustics (ICA).

Within IUTAM the only division used so far is related to solid and fluid mechanics as indicated by our two Symposia Panels. But more recently nine Working Parties with up to five members each have been established by the General Assembly of IUTAM devoted to specific areas of mechanics. These areas are:

Non-Newtonian Fluid Mechanics and Rheology, Dynamical Systems and Mechatronics, Mechanics of Materials, Materials Processing, Computational Fluid and Solid Mechanics, Biomechanics, Nano- and Micro-Scale Phenomena in Mechanics, Geophysical and Environmental Mechanics, Education in Mechanics and Capacity Building.

The terms of reference of the Working Parties include recommendations to the General Assembly regarding timely subjects for IUTAM Symposia, to maintain contact with the relevant Affiliated Organizations and sister International Unions, to identify important growth areas of the field, and to assist the Bureau and the General Assembly in discussions on position statements. Professors Felix Chernousko and Hiroshi Yabuno whom I am greeting, too, are members of the Working Party on Dynamical Systems and Mechatronics.

IUTAM carries out an exceptionally important task of scientific cooperation in mechanics on the international scene. Each national Adhering Organization of IUTAM, like The Chinese Society of Theoretical and

General Assembly. In particular, the Chinese delegates with IUTAM are

Applied Mechanics, is represented by a number of scientists in IUTAM’s

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Welcome address

Professor Yilong Bai, Chinese Academy of Sciences, Beijing; Professor Erjie Cui, Beijing Institute of Aerodynamics; Professor Wei Yang, Tshinghua University, Beijing; Professor Zhemin Zheng, Chinese Academy of Sciences, Beijing. Professor Zheng is also serving as a member of the Bureau of

IUTAM.

Mechanics is a very well developed science in China represented at most universities and some national laboratories. Since 1949 more than 280 IUTAM symposia have been held worldwide, many of them in China. This decade has witnessed four IUTAM Symposia in China.

In 2002 the IUTAM Symposium on Complementary-Duality Variational Principles in Nonlinear Mechanics in Shanghai chaired by Wanxie Zhong.

In 2004 the IUTAM Symposium on Mechanics and Reliability of Actuating Materials in Beijing chaired by Wei Yang.

In 2005 the IUTAM Symposium on Mechanical Behaviour and Micro-mechanics of Nanostructured Materials in Beijing chaired by Yilong Bai.

And this year IUTAM holds a Symposium in Nanjing.

As I mentioned before, IUTAM organizes not only symposia but also international congresses all over the world. Two years ago the 21st Inter-national Congress of Theoretical and Applied Mechanics was held in Warsaw, Poland. With 1515 participants the Warsaw Congress was a major event in mechanics also described as the Olympics of Mechanics. The Twenty-second International Congress of Theoretical and Applied Mechanics will be held in Adelaide, Australia, from 24th to 30st August 2008, what means in two years from now. Announcements of this forthcoming congress will be widely dis-tributed and published in many scientific journals. The Chinese member elected to the standing Congress Committee of IUTAM is Professor Gengdong Cheng, Dalian University of Technology.

The present Symposium is exceptionally interesting because it deals with new developments in mechanics. The Symposium covers important approaches:

Modelling and identification of nonlinear systems, Stability and bifurcation of nonlinear systems with uncertainty, Nonlinear dynamics of controlled systems, Control of chaos and stochastic oscillations.

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IUTAM found that the proposal of Professor Haiyan Hu for such a

symposium was not only very timely, but also well justified in the outstand-ing research carried out in this field at the NUAA. Thus, the proposal for the Symposium was readily accepted and granted by the General Assembly of IUTAM. There is no doubt that IUTAM considers nonlinear systems as an important field of mechanics.

Finally, I would like to mention that to sponsor a scientific meeting is one thing, but to organize one is another. A heavy burden is placed on the shoulders of the Chairman and his associates who are in charge of the scientific program and the practical local arrangements. All who have tried this before know very well how much work has to be done in organizing such a meeting.

Thus, we are very thankful, not only to the International Scientific Committee, but also to the Chairman, Professor Haiyan Hu, to the Co-Chairman, Professor Edwin Kreuzer, to the Secretary-General, Dr. Zaihua Wang and to all associates who assisted them in carrying the heavy load and responsibility.

It is up to you now, Ladies and Gentlemen, to harvest the fruits of the Organizers’ work. Contribute your share to make this IUTAM Symposium a meeting that will be long remembered as a very successful one!

On behalf of IUTAM, I greet you all and wish you great success! September 18, 2006 W. Schiehlen Representative of IUTAM

University of Stuttgart, Germany

Welcome addressxxiv