IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty
IUTAM BOOKSERIES Volume 2
Aims and Scope of the Series The IUTAM Bookseries publishes the proceedings of IUTAM symposia under the auspices of the IUTAM Board.
For a list of related mechanics titles, see final pages.
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.
ISBN 978-1-4020-6331-2 (HB) ISBN 978-1-4020-6332-9 (e-book)
Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com
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CONTENTS
Preface ················································································································
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Opening Address ······························································································· xvii Welcome Address ·····························································································
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Contributed Papers PART 1
System Modeling with Uncertainty
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 Foam ·········································································································
1
M. Hernandez-Garcia, S. F. Masri, R. Ghanem and F. Arrate Data-Based Stochastic Models of Uncertain Nonlinear Systems ···········
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C. Proppe and C. Wetzel Overturning Probability of Railway Vehicles under Wind Gust Loads ·······································································································
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W. Schiehlen and R. Seifried Impact Systems with Uncertainty ····························································
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PART 2
System Dynamics with Uncertainty
S. K. Au and D. P. Thunnissen Uncertainty Propagation in Complex Engineering Systems by Advanced Monte Carlo Methods ·······················································
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T. F. Filippova Trajectory Tubes in Control and Estimation Problems under Uncertainty ····················································································
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A. Gaull and E. Kreuzer Cell Mapping Applied to Random Dynamical Systems ·························
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X. L. Leng Numerical Analysis of Bifurcation and Chaos Response in a Cracked Rotor System under White Noise Disturbance ··················
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X. B. Liu The Maximal Lyapunov Exponent for a Stochastic System ···················
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W. V. Wedig Stability and Density Analysis of Stochastic Duffing Oscillators ··········
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J. X. Xu and H. L. Zou Uncertainties in Deterministic Dynamical Systems and the Coherence of Stochastic Dynamical Systems ····························· 109 W. Xu, Q. He and S. Li The Cell Mapping Method for Approximating the Invariant Manifolds·································································································· 117
PART 3
Nonlinear Dynamics
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 ··································································································· ··
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N. D. Anh, N. Q. Hai and W. Schiehlen Application of Extended Averaged Equations to Nonlinear Vibration Analysis ···················································································
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Z. Q. Wu and Y. S. Chen Singularity Analysis on Constrained Bifurcations ··································
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H. Yabuno, Y. Kunitho, T. Inoue and Y. Ishida Nonlinear Analysis of Rotor Dynamics by Using the Method of Multiple Scales ·················································································· ··
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S. Yang and Y. Shen Nonlinear Dynamics of a Spur Gear Pair with Slight Wear Fault ···········
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K. Yunt and C. Glocker A Combined Continuation and Penalty Method for the Determination of Optimal Hybrid Mechanical Trajectories ···········································
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PART 4
Dynamics of High-Dimensional Systems
J. Awrejcewicz, G. Kudra and G. Wasilewski Numerical Prediction and Experimental Observation of Triple Pendulum Dynamics ················································································
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A. J. Dick, B. Balachandran, and C. D. Mote, Jr. Nonlinear Vibration Modes and Energy Localization in Micro-Resonator Arrays·······································································
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L. Q. Chen and X. D. Yang Parametric Resonance of an Axially Accelerating Viscoelastic Beam with Non-Typical Boundary Conditions ·······································
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F. L. Chernousko Dynamics of a Body Controlled by Internal Motions ·····························
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W. Lacarbonara, A. Paolone and F. Vestroni Linear and Nonlinear Elastodynamics of Nonshallow Cables ················
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S. Lenci and G. Rega Nonlinear Normal Modes of Homoclinic Orbits and their Use for Dimension Reduction in Chaos Control ············································
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A. Teufel, A. Steindl and H. Troger Rotating Slip Stick Separation Waves ·····················································
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M. H. Yao and W. Zhang Many Pulses Homoclinic Orbits and Chaotic Dynamics for Nonlinear Nonplanar Motion of a Cantilever Beam ··························
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PART 5
Control of Nonlinear Dynamic Systems
I. Ananievski Synthesis of Bounded Control for Nonlinear Uncertain Mechanical Systems ·················································································
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P. Barthels and J. Wauer Controlled Vibration Suppression of Structural Telescopic Systems ····································································································
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P. B. Gonçalves and D. Orlando Influence of a Pendulum Absorber on the Nonlinear Behavior and Instabilities of a Tall Tower ·····························································
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H. Y. Hu and M. L. Yu Robust Flutter Control of an Airfoil Section through an Ultrasonic Motor ·······································································································
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K. Czołczyński, A. Stefański, P. Perlikowski and T. Kapitaniak Periodization and Synchronization of Duffing Oscillators Suspended on Elastic Beam ·····································································
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Q. Y. Wang, Q. S. Lu, X. Shi and H. X. Wang Effects of Noise on Synchronization and Spatial Patterns in Coupled Neuronal Systems ······························································· ··
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PART 6
Dynamics of Time-Delay Systems
Y. F. Jin and H. Y. Hu Stability and Response of Stochastic Delayed Systems with Delayed Feedback Control ···························································· ··
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G. Stépán and T. Insperger Robust Time-Periodic Control of Time-Delayed Systems ·····················
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P. Wahi, G. Stépán and A. Chatterjee Self-Interrupted Regenerative Turning ···················································
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Z. H. Wang and H. Y. Hu Robust Stability of Time-Delay Systems with Uncertain Parameters ·······························································································
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J. Xu, M. S. Huang and Y. Y. Zhang Dynamics due to Non-Resonant Double Hopf Bifurcationin in Van Del Pol-Duffing System with Delayed Position Feedback ··········
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W. Q. Zhu and Z. H. Liu Stability and Response of Quasi Integrable Hamiltonian Systems with Time-Delayed Feedback Control ·················································· ··
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Author Index ······································································································
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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 dynamics and control of nonlinear systems with uncertainty, therefore, is a vital interdisciplinary topic related to both stochastic systems and 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 ix
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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 from 17 countries accepted invitation, and 40 of them made oral presentations at the symposium. The presentations cover the following topics: 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 deterministic 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 applications, S. K. Au et al. introduced an advanced Monte Carlo method, named Subset Simulation, to solve the problem of uncertainty propagation in
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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 more powerful computational techniques as well. For example, K. Yunt et al. represented the dynamics of a robotic manipulator with blockable degrees of freedom as a measure-differential inclusion and proposed a unified 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 functions, and applied his results to the rotor rub-impact prediction, etc. N. Q. Hai et al. applied the extended averaged equations to a nonlinear 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 estimation of the maximal possible averaged speed of motion of the robot and verified his results in a number of interesting experiments. B. Balachandran et al. made an analysis of nonlinear vibration modes and 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 vehicles with a drum brake, by means of the centre manifold reduction.
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S. Lenci et al. applied the method of nonlinear normal modes to analyzing the homoclinic orbits of a given hilltop saddle and the chaos control problem. 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 nontypical 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. 6. Dynamics of Time-Delay Systems. Time-delay systems have received more and more attention from the circle of mechanics over the past 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
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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 parametric 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 Symposium and the editorial work of the volumn. We gratefully acknowledge the financial supports from IUTAM and The National Natural Science Foundation of China. Finally, we greatly appreciate the successful cooperation with publisher Springer.
H. Y. Hu, Nanjing E. Kreuzer, Hamburg
1. Y. F. Jin 2. L. Q. Chen 3. W. Xu 4. J. Xu 5. Q. S. Lu 6. E. Kreuzer 7. W. Schiehlen 8. W. Lacarbonara 9. H. Y. Hu
10. S. F. Masri 11. G. Stépán 12. J. Awrejcewicz 13. S. Lenci 14. Z. H. Wang 15. H. L. Wang 16. L. Du 17. S. P. Yang 18. W. Zhang 19. P. B. Gonçalves 20. L. Bevilacqua 21. H. Troger 22. W. Q. Zhu 23. S. J. Hogan 24. T. F. Filippova 25. S. K. Au 26. N. Q. Hai 27. G. Rega
28. T. Kapitaniak 29. Q. Y. Jin 30. X. B. Liu 31. S. Li 32. X. L. Leng 33. A. K. Bajaj 34. C. Proppe 35. P. Barthels 36. I. Ananievski
37. A. Czanaky 38. J. X. Xu 39. K. Yunt 40. W. V. Wedig 41. B. Balachandran 42. H.Yabuno 43. F. L. Chernousko 44. T. Bodai
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 Chairman 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, especially 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
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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 schools offer 44 undergraduate programs, 127 master programs and 55 Ph.D. 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 symposium undoubtedly provides NUAA faculty members with an opportunity to demonstrate their recent achievements and to exchange their 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.
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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
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 hydroand 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
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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 Applied Mechanics, is represented by a number of scientists in IUTAM’s General Assembly. In particular, the Chinese delegates with IUTAM are
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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 International 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 distributed 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.
xxiv
Welcome address
IUTAM found that the proposal of Professor Haiyan Hu for such a symposium was not only very timely, but also well justified in the outstanding 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 CoChairman, 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
PART 1
SYSTEM MODELING WITH UNCERTAINTY
NONLINEAR MULTI-BODY DYNAMICS OF SEAT-OCCUPANT SYSTEMS USING EXPERIMENTALLY IDENTIFIED VISCOELASTIC MODELS OF POLYURETHNE FOAM A. K. Bajaj, P. Davies, R. Ippili, T. Puri Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907-2031, USA , E-mail:
[email protected] Abstract:
Most modern car seats are full-foam and thus the H-point location depends on quasi-static behavior of foam. In this work, planar multi-degree-of-freedom models of seat-occupant systems that incorporate nonlinear viscoelastic behavior of polyurethane foam are developed. The foam force is modeled as an additive sum of nonlinear elastic and linear viscoelastic effects and the model parameters are identified using a parameter identification technique. A possible model for seat-occupant interface is introduced. The resulting nonlinear integro-differential-algebraic model is used to determine the H-point for the system.
Key words:
Seat-occupant modeling, nonlinear viscoelastic foam models.
1.
INTRODUCTION
Rider comfort is of paramount importance to automotive manufacturers. The overall comfort is determined by a combination of factors like thermal comfort, static comfort, and dynamic comfort [1-3]. Dynamic comfort is related to vibration levels experienced by the occupant. Static comfort is related to posture and orientation of the occupant, and location of the occupant relative to certain critical points (the H-point) which is determined by the static equilibrium position of the occupant in the seat. 1 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 1–10. © 2007 Springer.
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Polyurethane foam is an important engineering material. Over the past few years the usage of polyurethane foam in automotive seating applications has increased because of its excellent vibration isolation properties, comfort features and durability. Many modern car seats are full foam, and the quasistatic and dynamic properties of foam significantly affect the static and dynamic comfort of the occupant [4]. Many researchers, e.g. Muksian and Nash [5] (also see [1]), have attempted to model the seat-occupant system response with lumped parameter models consisting of several masses, springs, and dampers. Most of these models are one-dimensional, do not represent the actual geometry of the seat-occupant system, and are unable to predict the substantial fore-aft motions of the occupant [6]. Nishiyama [6-7] developed models of vehicle-seat-occupant system for the study of dynamic response in vertical as well as fore-aft directions. In here, the occupant was assumed to undergo motions in both the vertical and fore-aft directions, and was modeled as a rigid body with six degrees of freedom. The seat back and seat bottom were each approximated by two linear springs and two linear dampers, thus modeling small vibrations around an equilibrium position of the occupant in the seat. This model was modified by Kim et al. [8] to incorporate the flexibility of the seat-back frame. The local elastic and damping properties of foam, a highly nonlinear material, depend on mean compression. Thus, various linear and nonlinear viscoelastic models have been developed to represent foam properties around a compression level and experimental techniques have been developed to estimate the model parameters [9-11]. More comprehensive seat-occupant models, capable of predicting both static settling of the occupant in the seat and the dynamic response of the occupant, must take into account these nonlinear viscoelastic properties. This work develops dynamic models of the seat-occupant system with a view to predicting the full planar dynamics of the system. It builds on methodology of Kim et al., and incorporates rigid body motions of the occupant, large-deformation models for foam in compression as well as a model of friction at the seat-occupant interface. The polyurethane foam in seat-back and seat-bottom is represented by a series of unidirectional springs characterized by nonlinear viscoelastic constitutive relations. The force generated in foam is in the form of an additive sum of nonlinear elastic and linear viscoelastic components. The nonlinear foam model parameters are estimated using an identification technique developed to extract these parameters from quasi-static compression tests data. The resulting seatoccupant model is used to predict the effects on static equilibrium position, of the changes in foam type, initial posture of the occupant, and the pressure distributions on occupant’s back and bottom.
Nonlinear seat-occupant dynamics
2.
3
MODELING OF SEAT-OCCUPANT SYSTEM
A general schematic of the two-dimensional multi-body models for the seat-occupant system is shown in Figure 1 and the definitions of the geometric parameters are shown in Figure 2. The occupant is modeled by three rigid bodies: AB, the torso, BC, the femur, and CD, the shin. Their centers of mass are at G1 , G2 and G3 , respectively. Point B is the hip- joint location (H-Point). The rigid components are connected by pin joints while the shin end D remains in contact with the foot rest. Five generalized coordinates describe the motion of the system: ξ , the absolute horizontal displacement of hip joint; ς , the absolute vertical displacement of hip joint; θ1 , the absolute angular orientation of the torso; θ 2 , the absolute angular orientation of the femur; and θ3 , the absolute angular orientation of the shin. The following assumptions were made while developing the dynamic model of the system: (a) foam in seat-back and seat-bottom is modeled by a
Figure 1. The mannequin-seat schematic with component characteristics. In this position, the seat-back as well as the seat-bottom foam is uncompressed.
Figure 2. The mannequin-seat model with geometry definitions.
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finite set of nonlinear viscoelastic springs; (b) the springs are attached to the occupant and are constrained to remain perpendicular to the body while sliding along the seat frame; (c) the weight of the occupant is taken into consideration; and (d) interfacial forces between the occupant and the seat are modeled like dry friction.
2.1 Quasi-static foam behavior Foam is a complex material exhibiting nonlinear viscoelastic behavior. It is viscoelastic that is, it does not recover immediately after the loading is removed and exhibits creep and stress relaxation. Much work on modeling and system identification of the dynamic behavior of foam has been conducted in our group [9-11]. Compression tests were performed on 3in cube foam blocks to measure the quasi-static behavior utilizing a MTS servo-hydraulic testing machine. The foam block was placed between two metallic plates. The upper plate was attached to the actuator which can be programmed to move with prespecified motion. The actuator displacement was varied linearly as a function of time to go from an unstressed state to a compression level of 67% and back to the unstressed state. Results of a 42 min cycle test are shown in Figure 3. The load cell measured the force in the foam block while the displacement was measured using an LVDT. Experiments were conducted on two different foams, designated as type `A' (soft) and type `D' (hard). The MTS system has a built-in data acquisition system though this data seemed to be very noisy. Thus, 48 dB/octave roll off, low-pass, anti-aliasing filters with the cut-off frequency set at 1 Hz , were used to filter the analog displacement and force signals from the MTS machine, prior to sampling. The sampling rate was set at 10 Hz. Figure 3 clearly shows that there is substantial difference between the force in foam during compression and relaxation phases. Furthermore, there is significant difference between the results for the first and subsequent cycles. The difference between the second and third cycle results is relatively small. The difference becomes smaller as the tests are repeated. For system identification purpose, third cycle data was used as not much variation was seen between the third and subsequent cycles. Foam Model: The quasi-static unidirectional compression behavior of foam was assumed to be governed by the following relation: M
t N
∑k x + ∫ ∑a e j =1
j
j
0
i =1
i
−α i ( t −τ )
x(τ )dτ = F (t )
(1)
Here, the first term on the left is a nonlinear elastic force Fe while the second (integral) term is the linear viscoelastic force Fv. The coefficients k j , j = 1, 2,..., M are the stiffness coefficients, ai ,α i , i = 1, 2,..., N are the
Nonlinear seat-occupant dynamics
5
Figure 3. The stress-strain response of a type ‘D’ foam in quasi-static tests with period 42 mins. Here, a 3in cube foam block is first compressed at a constant rate and then uncompressed at the same rate. The foam was subjected to three consecutive cycles.
viscoelastic parameters, F is the measured force, and x is the input displacement. The reciprocals of the real parts of the exponents α i give time-constants of the material. Model Parameter Identification: System identification techniques were utilized to estimate the material parameters of k j , j = 1, 2,..., M and ai ,α i , i = 1, 2,..., N from the measured experimental data (e.g., cycle 3 data in Figure 3). The steps followed were: (1) Average the upper and lower parts of the force response curve. This gives a curve which is assumed to be mostly influenced by the nonlinear elastic force; (2) Least-squares fit a polynomial to the averaged data to determine the stiffness coefficients; (3) Find the difference between the experimental data and the polynomial fit. This difference gives an approximation to the viscoelastic contribution; (4) Model the relationship between the sampled displacement x(t) and the output of step (3) as an (N, N-1) order auto-regressive moving average (ARMA) digital filter [12]. The coefficients of the digital model are directly related to the constants ai ,α i , i = 1, 2,..., N . The coefficients are estimated by using a modified least-squares fit to the experimental data as programmed in MATLAB function pem; (5) Use the filter model to regenerate the viscoelastic force and subtract it from the original experimental data to obtain a more accurate estimate of the nonlinear elastic term; (6) Repeat steps (2)-(5) until convergence is achieved; (7) Reconstruct the force with the estimated stiffness coefficients and the viscoelastic parameters. It was observed that typically ten iterations were sufficient to reach a steady-state in estimates. A typical result of system identification with this procedure is shown in Figure 4. Note that there is a very small difference between the experimental data and the estimated force in the 10th iteration. For this result, seven stiffness terms (M=7) and two viscoelastic terms (N=2) (a complex-conjugate pair) were required. Although higher-order models were tried, either the results didn’t improve or they gave unrealistic parameters.
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We should note that there were data sets for which pem did not converge. The following approach was adopted to solve this problem. The experimental data was perturbed by adding some uniformly distributed random noise whose amplitude is small (nearly one percent) compared to the maximum force in the experiment. The added noise was modeled by using the noise model structure in pem. It was seen that the identification process now achieved convergence for some realizations of noise though not in every case. Tests on simulated data showed that the procedure gives accurate results although a little bias is introduced in the parameter estimates. The mean values and standard deviations of the parameters estimates for the two foams ‘A’ and ‘D’ are shown in Table 1. These values will be used in the simulation of the dynamics of the seat-occupant system. The above foam model was incorporated in the seat-occupant model. For this, x(t) was expressed as the difference between the initial (δ0) and the present foam thickness (δ). Also, the force identified for a 3in cube foam block was scaled to take into account the actual contact area between the mannequin and seat cushions. Two more factors needed to be considered: (1) foam cannot apply a tension force, i.e., the foam force is zero for δ > δ0; and (2) the spring force in compression should grow very rapidly for large compression (> 90%-95%). These effects were incorporated in the foam force model by using sigmoid functions: sigmoid1,2 = 1 (1 + e −a (25δ −c) )
(2)
Here ‘a’ and ‘c’ are properly chosen values.
Figure 4. System identification result for foam type ‘A’ with M=7 and N=2.
2.2 Modeling of interfacial forces The interfacial forces between the occupant and the seat, including the frictional forces and the tangential shear forces, play an essential role in determining the dynamics of the system. With the exception of [5], where Coulomb friction was used to model the interaction between the torso and the seat back, very few researchers have reported on characterization of the
Nonlinear seat-occupant dynamics
7
Table 1. The values of parameters identified using the proposed parameter identification technique on experimental data in 42 min compression tests. M=7, N=2. Parameters Adjusted R2 k1, N/m k2, N/m2 k3, N/m3 k4, N/m4 k5, N/m5 K6, N/m6 K7, N/m7 |a1|, N/(m·s) ∠a1,o Re(α ), Hz Im(α ), Hz τ, s 1
1
Foam A Mean 0.9975 3.180×103 -2.837×105 1.339×107 -3.306×108 0.454×1010 -0.365×1011 0.201×1012 68.82 49.94 2.606×10-2 2.182×10-2 38.38
Standard Dev 3×10-4 16 0.058×105 0.066×107 0.373×108 0.109×1010 0.161×1011 0.0934×1012 10.91 3.17 0.221×10-2 0.095×10-2 3.17
Foam D Mean 0.9972 4.429×103 -5.036×105 3.669×107 -16.643×108 4.605×1010 -6.976×1011 4.439×1012 98.34 65.72 2.502×10-2 1.143×10-2 39.97
Standard Dev 5×10-4 109 0.361×105 0.414×107 0.236×108 0.711×1010 1.082×1011 0.653×1012 83.93 29.64 0.4289×10-2 0.4277×10-2 10.40
mannequin-seat interface. It was assumed here that these forces act on the occupant tangentially at the points where the viscoelastic spring forces act normally. Thus, the force at a given point is Fint= - µ F . This model represents the maximum friction force, and the orientation of each interface force is then always parallel to occupant's body. Two different coefficients of friction were used in the model, µ1 for seat back and µ2 for the seat bottom.
2.3 Equations of motion A constrained Lagrangian formulation was used to derive the equations of motion of the seat-occupant system. This accounted for the geometric constraint on motion since the foot has to always move along the foot rest. Thus, for a system with n generalized coordinates and m constraints, the equations of motion are given by d ⎛ ∂T ⎞ − ∂T + ∂U + f c = Q , r = 1, 2,...n r r dt ⎜⎝ ∂q′r ⎟⎠ ∂qr ∂qr
(3)
where T (qr , q&′r ) is the kinetic energy, U (qr ) is the potential energy associated with gravity as well as the elastic component of the forces in the viscoelastic springs, Qr (qr , q&′r , t ) represent the generalized forces that cannot be derived via a scalar potential function (including Fv, the viscoelastic components of foam forces), and qr is a generalized coordinate. Furthermore, f rc represents the constraint forces introduced due to the holonomic constraints on motion. Let the m constraints be of the form
φ j (q , q ,...qn ) = 0, j = 1, 2,....., m 1
2
(4)
A. K. Bajaj et al.
8 Then, the corresponding constraint forces are given by m
f rc = ∑ λ j j =1
∂φ j m ∂φ = ∑ λ j BrjT , where BrjT = j ∂qr j=1 ∂qr
(5)
and λj’s are the Lagrange Multipliers. The resulting equations (3) and (4) represent a system of (n + m) integro-differential-algebraic equations. This system can be transformed to a differential-algebraic system with the introduction of new state variables. In fact, 2N first-order differential equations are introduced to replace the integral terms in the model for each of the springs. Thus, the DAE for the seat-occupant system involving only one constraint has the form:
[ M ]{q}′′ + {B∗}
T
3.
λ = {Q}
(6)
SOLUTIONS FOR STATIC EQUILIBRIUM
There are two approaches to solving equations (6) and (4) for static equilibrium. In the first approach, one can set all the time derivatives terms to zero and then solve the resulting system of algebraic/transcendental equations for equilibrium points. In the second approach, the equations of motion can be integrated in time. The first approach requires good initial estimates of the states for the nonlinear solver, while the second may take a long time to reach steady state. The system of nonlinear algebraic/transcendental equations that determine the static equilibrium position is too complex to be solved symbolically. MATLAB’s function fsolve was used but the iterative scheme did not converge. Although it was possible to give initial guess for the generalized coordinates consistent with the constraints, it was not possible to provide a good initial guess for the constraint force λ. Thus, the equations of motion were solved numerically and the motion of the occupant observed as it settles into the seat. The transient response depends both on initial conditions and on damping properties, though the static equilibrium position (or positions) should not be influenced by these parameters. If there is more than one static equilibrium position, they can be achieved by starting integration with different initial conditions. The standard integration algorithms, such as the Runge-Kutta methods of different orders, cannot be used as the equations of motion are a differentialalgebraic system. The MATLAB function ode15s for DAEs also could not be used as the present system (Equations (4) and (6)) is of index higher than one. For a multibody system with n generalized coordinates and m holonomic constraints, there are (n-m) degrees of freedom. Since it is difficult to identify the independent coordinates of the system, a coordinate reduction
Nonlinear seat-occupant dynamics
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during the numerical integration process can help obtain independent equations [13]. Let, [S] be the (n-m)×n orthogonal complement of matrix {B*} determining the null space of {B*}. Then (6) can be transformed to
{q}′′ = [ A]−1 {D}
⎡[S ][M ]⎤ ⎧ [S ]{Q} ⎫ where [ A] = ⎢ , {D} = ⎨ ⎬ ⎥ ∗ ∗ ⎣ {B } ⎦ ⎩−{B }′ {q}′⎭
(7)
Equations (7) represent a set of n second-order differential equations that can be easily numerically integrated given appropriate initial conditions.
4.
NUMERICAL RESULTS AND DISCUSSION
Equations (7) were integrated to determine the static equilibrium position of the seat-occupant system. The geometric and inertia properties of mannequin are defined in [8]. The viscoelastic properties of foam were those identified through the identification process discussed above. Also recall that the foam force obtained from system identification has to be modified to take account of the contact area. Other important unknowns in the model are the coefficients of friction at the interfaces. Their values depend on both the friction forces and the tangential shear forces. For integration of equations of motion, initial conditions on the generalized coordinates also need to be specified. Since the foot is constrained, the initial conditions for the generalized coordinates ξ, ς , θ1 , and θ 2 were used with the constraint equation to solve for a consistent initial value for the coordinate θ 3 . The initial conditions were chosen such that all springs were uncompressed initially and the MATLAB tool ode15s was used for integration of equations (7). The equilibrium position was determined for different combinations of initial conditions and for different parameter sets corresponding to foam types ‘A’ and ‘D’. The friction coefficients were assumed to be 0.25. The results were verified by using WorkingModel 2D, a multibody dynamic simulation package which served as a reference. The results were specifically obtained for Chrysler LH seat (see [14] for details). The equilibrium position attained by the occupant in a specific case where the seat back consisted of 42 springs and the seat bottom had 48 springs, is shown in Figure 5. This position was reached in 10 secs of time integration. The equilibrium position of the occupant allowed for the computation of compression in each spring, and hence the distribution of forces acted on the occupant by the foam. These pressure distributions were also measured in an experimental set-up with the same mannequin, and the results were found to be similar qualitatively. This corroboration of results has provided confidence in our modeling approach for studying the nonlinear dynamics of seat-occupant systems.
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Figure 5. The equilibrium position of the seat-occupant system with 48 springs at the seat bottom and 42 springs at the seat back. Final configuration is reached in T=10 secs.
REFERENCES 1. 2. 3. 4.
5. 6.
7.
8. 9.
10. 11.
12. 13. 14.
Griffin MJ. Handbook of Human Vibration, London, Academic Press, 1990. Ebe K, Griffin MJ. “Factors affecting static seat cushion comfort”, Ergonomics, 44, pp. 901-921, 2001. Ebe K, Griffin MJ. “Qualitative models of seat discomfort including static and dynamic factors”, Ergonomics, 43, pp. 771-790, 2000. Inagaki H, Taguchi T, Yasuda E, Iizuka Y. “Evaluation of riding comfort: From the viewpoint of interaction of human body and seat for static, dynamic, long time driving”, SAE Paper 2000-01-0643, 2000. Muksian R, Nash CD. “A model for the response of seated humans to sinusoidal displacements of the seat”, Journal of Biomechanics, 7, pp. 209-215, 1974. Nishiyama S. “Vertical and lateral vibration analysis of vehicle-occupant dynamic interaction with simulation system”, Transactions of the Japan Society of Mechanical Engineers, 59, pp. 3239-3246, 1993. Nishiyama S. “Development of simulation system on vehicle-occupant dynamic interaction. “First report: Theoretical analysis and system verification”, Transactions of the Japan Society of Mechanical Engineers, 59, pp. 3613-3621, 1993. Kim SK, White SW, Bajaj AK, Davies P. “Simplified models of the vibration of mannequins in car seats”, Journal of Sound and Vibration, 264, pp. 49-90, 2003. White SW, Kim SK, Bajaj AK, Davies P, Showers DK, Liedtke PE. “Experimental techniques and identification of nonlinear and viscoelastic properties of flexible polyurethane foam”, Nonlinear Dynamics, 22, pp. 281-313, 2000. Singh R, Davies P, Bajaj AK. “Identification of nonlinear and viscoelastic properties of flexible polyurethane foam”, Nonlinear Dynamics, 34, pp. 319-346, 2003. Deng R, Davies P, Bajaj AK. “Flexible polyurethane foam modeling and viscoelastic parameters identification for automotive seating applications”, Journal of Sound and Vibration, 262, pp. 391-417, 2003. Ljung L. System Identification: Theory for the User, Upper Saddle River, Prentice Hall PTR, 1999. Amirouche FML. Computational Methods in Multibody Dynamics, Upper Saddle River, Prentice Hall, 1992. Puri T. Integration of Polyurethane Foam and Seat-Occupant Models to Predict the Settling Point of a Seat Occupant. West Lafayette, IN, Purdue University, MS Thesis, 2004.
DATA-BASED STOCHASTIC MODELS OF UNCERTAIN NONLINEAR SYSTEMS M. Hernandez-Garcia, S. F. Masri, R. Ghanem, F. Arrate Department of Civil and Environmental Engineering, Viterbi School of Engineering, University of Southern California, Los Angeles, California 90089, USA
Abstract:
A general methodology is presented for representing and propagating the effects of uncertainties in complex nonlinear systems through the use of a model-free representation that allows the estimation through analytical procedures of the uncertain system’s response bounds when it is excited by a different dynamic load than the one used to identify it. A nonparametric identification approach based on the use of the Restoring Force Method is employed to obtain a stochastic model of the nonlinear system of interest. Subsequently, the reduced-order stochastic model is used in conjunction with polynomial chaos representations to predict the uncertainty bounds on the nonlinear system response under transient dynamic loads. The proposed approach is applied to the damped hardening Duffing oscillator under sweptsine excitation.
Key words:
Uncertain nonlinear systems, stochastic restoring force method, polynomial chaos expansion.
1.
INTRODUCTION
This paper presents a study of a general methodology for representing and propagating the effects of uncertainties in complex nonlinear systems through the use of a reduced-order, reduced-complexity, model-free representation, that allows the estimation through analytical procedures of the uncertain system’s response bounds when it is excited by a different dynamic load than the one used to identify it. A nonparametric identification approach based on the use of the Restoring Force Method is employed to obtain a stochastic model of the nonlinear system of interest. Subsequently, 11 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 11–22. © 2007 Springer.
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the reduced-order stochastic model is used in conjunction with polynomial chaos representations to predict the uncertainty bounds on the nonlinear system response under transient dynamic loads.
2.
MODELING UNCERTAIN DYNAMIC SYSTEMS
The Stochastic Restoring Force Method (SRFM), as the name suggests, can be viewed as an extension to the classical deterministic Restoring Force Method (RFM), on which extensive literature is available [1-2]. The difference between these two methods is that in SRFM the parameters are random variables, while in the RFM the parameters are deterministic variables. The SRFM was implemented with the objective of estimating the uncertainty in nonlinear dynamical systems. This is accomplished by approximating both inputs (i.e. identified power series coefficients) and outputs (i.e. displacements) of the uncertain dynamic system through polynomial chaos expansions of random variables and random processes.
2.1 Polynomial Chaos Expansion The Cameron-Martin theorem [3] states that a general second-order stochastic process (i.e., finite variance) can be approximated by a spectral expansion in terms of a set of orthogonal basis function whose coefficients can be used to quantify and characterize the uncertainty. This approximation can be represented as: ∞
X (t ,θ ) = xˆ0 (t )Φ 0 + ∑ xˆi1 (t )Φ1 (ξi1 (θ )) + i1 =1 144 42444 3 ∞
i1
1st order terms
xˆi1 i2 (t )Φ 2 (ξi1 (θ ), ξi 2 (θ )) + … ∑∑ i1 =1 i2 =1 144444244444 3
(1)
2nd order terms
where Φ n (ξi1 (θ ), ξi2 (θ ),K , ξin (θ ) is a multidimensional Hermite polynomial of order n in terms of the random vector ξ = (ξi1 (θ ),K, ξin (θ )) of independent normal Gaussian distributed variables. Although in the Polynomial Chaos Expansion (PCE) the set of orthogonal basis are not just restricted to Hermite polynomials and Gaussian random variables [4], the Hermite Chaos expansion works theoretically for any random process (i.e., Gaussian or nonGaussian) [3; 5–7]. For convenience, the previous equation can be rewritten as
Data-based stochastic models of uncertain nonlinear systems ∞
X (t ,θ ) = ∑ x p (t )Ψ p (ξ (θ ))
13 (2)
p =0
where a one-to-one relationship exists between the Hermite polynomials Φ n (ξi1 , ξi2 ,L, ξin ) and Ψ p (ξ ) as well as between their coefficients xˆi1 i2 Lin and x p . In other words, for the multidimensional case, the subscript p indicates the place in the expansion of the term xˆi1 i2 Lin Φ n (ξi1 , ξi2 ,K , ξin ) .
2.2 Restoring Force Identification Considering that the motion of a general deterministic nonlinear SDOF system subjected to external excitation force can be mathematically represented by mx&&(t ) + r ( x(t ), x& (t )) = f (t )
(3)
where x(t ) is the displacement, m is the mass of the system, the term r ( x(t ), x& (t )) is the restoring force, and f (t ) is the applied force. In general, the restoring force depends on the displacement and velocity of the system. The main idea in the Restoring Force Method is that the restoring force r ( x(t ), x& (t )) can be expressed in terms of a truncated power series expansion [1-2] ∞
imax jmax
∞
r ( x, x& ) = ∑∑ aij x i x& j ≈ ∑∑ aij x i x& j i =0 j =0
(4)
i =0 j =0
max where the aij ’s are undetermined constants and ∑ imax xi x& j represents a =0 ∑ j =0 set of suitable basis functions in terms of displacement x(t) and velocity x& (t ) .With this representation in mind, the equation of motion for a nonlinear dynamic SDOF system can be expressed as
i
imax jmax
∑∑ x x& i =0 j =0
i
j
= f (t ) − mx&&
j
(5)
The power series expansion coefficients, which will be used for defining the original nonlinear element, can be obtained by performing a nonlinear regression in the time domain assuming that all time histories (acceleration, state variables and excitation) are available.
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2.3 Polynomial Chaos Representation of Stochastic Inputs The first step in the implementation of the SRFM is the representation of all the model inputs (i.e., the random coefficients aij ) in terms of a set of normal random variables. Adopting a one-dimensional Polynomial Chaos Expansion (PCE) in terms of standard Gaussian random variables γ ij (θ ) to characterize the uncertainty in the power-series coefficients, the random aij coefficients can be approximated by a truncated expansion at the finite term P [6]: P
aij (θ ) ≈ ∑ aij p Ψ p (γ ij (θ ))
(6)
p =0
Re-indexing and ordering the two-dimensional array of ai j coefficients into a vector of coefficients a* = {a1* ,K , ai* ,K an* } , the one-dimensional PCE is reduced to: P
ai* (θ ) = ∑ Aip*Ψ p (γ i (θ ))
(7)
p =0
The generalized Fourier coefficients Aip* should be chosen to characterize the cumulative distribution function (cdf ) of the identified power-series coefficients. This can be done by projecting the ai coefficients onto each polynomial basis Ψ p (γ i ) , Aip = *
ai* ,Ψ p (γ i ) Ψ p 2 (γ i )
=
1 1 −1 Fi (θ )Ψ p ( E −1 (θ ))dθ ∫ 0 p!
(8)
where Fi −1 (θ ) is the approximated inverse cumulative distribution function of every ai coefficient , E −1 (θ ) is the inverse Gaussian cdf and θ is a uniform variate ( θ ∈ [0,1] ). Due to the fact that one dimensional PCE does not characterize the existing correlation between all the power-series coefficients a i , it is necessary to induce those correlations into the PC expansion of the random a i [6]. Using Karhunen-Lòeve (K-L) expansion, it is possible to express the Gaussian random vector γ , with a covariance matrix Kγ matching the correlation matrix ρ a obtained for all a i coefficients, as a linear combination of a set of uncorrelated Gaussian random variables {ξi } [6-7]:
γ i = ∑ Bijξ j = γ i (ξ ) j
(9)
Data-based stochastic models of uncertain nonlinear systems
15
where matrix B = E Λ1/ 2 is obtained by decomposing the covariance matrix onto Kγ = E ΛE T . Based on the last expression, it is possible to generate a set of coefficients { a i } as a function of a set of uncorrelated standard Gaussian random variables {ξi } P
ai (ξ ) = ∑ Aip*Ψ p (γ i (ξ ))
(10)
p =0
This representation of a i coefficients makes them suitable for a multidimensional PC expansion in a set of independent standard Gaussian random variables {ξi } : R
ai (ξ (θ )) = ∑ Air Ψ r (ξ (θ ))
(11)
r =0
where the coefficients of this multidimensional polynomial chaos expansion are computed by approximating the multidimensional integral enclosed under the projection of a i (ξ ) onto Hermite polynomial basis using a GaussHermite quadrature: Air =
ai (ξ ),Ψ r (ξ ) Ψ r (ξ ) 2
=
1 Ψ r (ξ ) 2
∫
Ω
ai (ξ )Ψ r (ξ )d ξ
(12)
As a final step, after recasting the sub-index of each a i coefficient for matching the polynomial representation of restoring force, Equation (11) is given by: R
aij (ξ ) = ∑ Aijr Ψ r (ξ )
(13)
r =0
2.4 Polynomial Chaos Representation of Stochastic Outputs Following this basic idea, the uncertainty in the response of a stochastic dynamical system can be represented in the form of a multidimensional truncated PC expansion [4-6]: P
P
p =0
p =0
x(t ,θ ) = ∑ x p (t )Ψ p (ξ (θ )) , x& (t ,θ ) ≈ ∑ x& p (t )Ψ p (ξ (θ ))
(14)
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M. Hernandez-Garcia et al.
where the summation limit depends on the highest order of the Hermite polynomials (o), the dimension n = dim( ξ ) of the standard Gaussian vector ξ , and it is given by: P +1 =
(n + o)! n !o !
(15)
Using Equations (13) and (14), the restoring force equation of motion, described in Equation (5), for a stochastic SDOF system can be expressed as: i
⎞⎛ P ⎞ ⎛ R ⎞⎛ P ∑∑ ⎜ ∑ A i jr Ψ r (ξ ) ⎟ ⎜ ∑ x p (t )Ψ p (ξ ) ⎟ ⎜ ∑ x& p (t )Ψ p (ξ ) ⎟ i =0 j =0 ⎝ r =0 ⎠ ⎝ p =0 ⎠ ⎝ p =0 ⎠ imax jmax
j
(16)
Re-projecting the higher order terms for x(t ) and x& (t ) onto Hermite polynomial basis, it is possible to have a simplified expression for the general term xi x& j Q
xi (t ,θ ) x& j (t ,θ ) ≈ ∑ ωq Ψ q (ξ )
(17)
q =0
With Equations (16) and (17) in mind, the stochastic differential equation of motion for a SDOF is given mathematically by: imax jmax R
P
Q
m∑ && x(t )Ψ p (ξ ) + ∑∑∑∑ Aijr ωq Ψ r (ξ )Ψ q (ξ ) = F (t ) p =0
(18)
i =0 j =0 r =0 q =0
Finally, the system of deterministic differential equations in the stochastic modes xs (t ) is obtained after projecting Equation (18) onto each Hermite basis Ψ s (ξ ) imax jmax R
Q
mx&&s (t ) + ∑∑∑∑ Aijr ωq i =0 j =0 r =0 q =0
3.
Ψ r (ξ ), Ψ q (ξ ), Ψ s (ξ ) Ψ 2s
= F (t )
(19)
APPLICATION TO SDOF DUFFING OSCILLATOR
As an application of the Stochastic Restoring Force Method proposed in this work, the stochastic response of an uncertain SDOF damped Duffing oscillator subjected to a deterministic swept sine excitation will be determined. The mathematical model of this system is
Data-based stochastic models of uncertain nonlinear systems
&& x(t ,θ ) + c(θ ) x& (t ,θ ) + k (θ ) x(t ,θ ) + ε (θ ) x3 (t ,θ ) = F (t )
17
(20)
In this reference nonlinear system, the parameters corresponding to damping, linear stiffness and cubic stiffness were considered as a uniform, Gaussian and Gamma random variables respectively, with statistical characteristics shown in Table 1. The random parameters c(θ ) , k (θ ) , ε (θ ) where synthetically sampled Nobs = 5000 times. For each realization of the system, the non-parametric system identification procedure was carried out using third-order polynomial basis functions for obtaining an ensemble of aij coefficients characterizing the restoring force in terms of a power series expansion. Table 1. Uncertain Duffing oscillator parameters. Parameter
c(θ )
k (θ )
ε (θ )
pdf
Uniform
Gaussian
Gamma
Mean
1.20
24.00
2.40
Std. Deviation
0.30
6.00
0.60
Coef. Variation
0.25
0.25
0.25
Table 2. Identified dominant power-series coefficients. Coefficient
a01 (θ )
a10 (θ )
a30 (θ )
Mean
1.20
24.40
2.39
Std. Deviation
0.30
5.85
0.59
Coef. Variation
0.25
0.24
0.25
From the identified coefficients, it was observed that the dominant terms in the restoring force expansion are associated with coefficients a01 (equivalent viscous damping term), a10 (equivalent linear stiffness term) and a30 (associated with cubic stiffness term). The statistical properties of these random variables are summarized in Table 2. The ensemble of each dominant coefficients a01, a10 and a30 are then expanded in one-dimensional polynomial chaos basis using 3rd order Hermite polynomials to characterize their marginal cumulative distribution functions (cdf ). This is done by finding the generalized Fourier coefficients associated with each polynomial Ψ p in the PC expansion. Due to the fact that one-dimensional PC expansions are not able to characterize the existing correlation matrix ρ a among identified coefficients, it is necessary to induce those correlations into the PC expansions. As it was stated before, the correlations are induced into the covariance matrix Kγ among Gaussian random variables γ 1 , γ 2 and γ 3 by expanding each one in terms of multidimensional standard Gaussian random vector ξ = {ξ1 , ξ 2 , ξ3 } using a Karhunen-Loève expansion.
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M. Hernandez-Garcia et al.
Using the multidimensional expansion for γ i (ξ ) and the PC coefficients previously identified, it is possible to reconstruct the aij coefficients in a multidimensional fashion keeping their statistical characteristics (Table 3) as mean, variance and correlation coefficients. In Figure 1 the correlation matrix of the dominant identified coefficients a01, a10 and a30 is compared with the reconstructed coefficients in terms of multidimensional Gaussian random vector. Table 2. Reconstructed dominant power-series coefficients. Coefficient
* a01 (θ )
a10* (θ )
* a30 (θ )
Mean
1.20
24.40
2.39
Std. Deviation
0.30
5.85
0.59
Coef. Variation
0.25
0.24
0.25
Figure 1. Comparison between the correlation matrix among identified aij coefficients and reconstructed a*ij coefficients.
By comparing the statistical properties of reference parameters (Table 1) with the identified coefficients (Table 2), it is seen that the power-series expansion obtained by using the Restoring Force Method was able to identify the variability in the parameters of the reference nonlinear element. Similarly, by comparing the statistics of the identified coefficients with the reconstructed coefficients (Table 3) and the correlation matrices in Figure 1, it can be seen that multidimensional reconstructed coefficients conserved the stochastic properties of the one-dimensional identified power series coefficients. A more clear evaluation of the robustness of the non-parametric identification of the properties in the reference nonlinear element and efficiency of how the identified coefficients are reconstructed in terms of a
Data-based stochastic models of uncertain nonlinear systems
19
set of random Gaussian variables can be seen in Figure 2. In this figure, the top row of plots corresponds to histograms for 5000 realizations of uncertain parameters c(θ ), k (θ ) and ε (θ ) in the reference nonlinear element. In the middle row, the distributions of the dominant coefficients a01 (θ ) , a10 (θ ) and a30 (θ ) obtained after applying the non-parametric system identification procedure for each realization of the reference system are compared with the corresponding parent pdf. Since the identified coefficients clearly have the same distribution that the corresponding reference parameters, the Restoring Force Method was able to identify robustly the uncertainty associated to damping, linear stiffness and cubic stiffness in the reference Duffing oscillator.
Figure 2. Comparison of damping, linear stiffness and cubic stiffness associated coefficients distributions in the reference, identified and reconstructed cases. In each graph are shown the corresponding histograms and superposed reference pdf.
In the bottom row of Figure 2, the distributions of 5000 realizations of reconstructed power-series coefficients using multidimensional PC expansion are compared with the reference probability density functions. The * * reconstructed coercions a01 and a30 follow distributions significantly different from the reference and identified coefficient’s distributions. This occurs because the PC expansion of the aij coefficients approximated their marginal cumulative distribution function (cdf) by multidimensional Hermite polynomials in terms of standard Gaussian random variables. For the
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M. Hernandez-Garcia et al.
coefficient a10 , with a reference Gaussian distribution, the reconstructed a10* coefficient has the same Gaussian distribution as the corresponding reference and identified parameters.
Figure 3. Comparison of cumulative distribution functions of the damping, linear stiffness and cubic stiffness associated coefficients in the reference, identified and reconstructed cases.
As mentioned above, the idea behind a multidimensional PC expansion of random variables is to characterize their associated uncertainty by approximating their corresponding cdf using Gaussian random variables. In Figure 3, the cumulative distribution functions of dominant identified coefficients as well as the reconstructed coefficients using PC expansion are compared with the corresponding reference cdf. The set of deterministic differential equations (Equation (19)), corresponding to stochastic modes xs (t ) , obtained after performing a PC decomposition of the stochastic responses x(t ,θ ) and x (t ,θ ) is solved by using a standard explicit 4th-order Runge-Kutta integration scheme. Figure 4 shows the time evolution of the stochastic modes solution for the uncertain Duffing oscillator subjected to a deterministic swept sine excitation. A fourth-order Hermite polynomial is used to solve this problem. The top plot shows the mean ( x0 mode) and the first mode of the solution. The bottom plot represents the higher stochastic modes which decrease in amplitude as the order of the mode increases. These higher order modes are important in the time evolution of the second order moment in the solution of the stochastic differential equation of motion.
Data-based stochastic models of uncertain nonlinear systems
21
Figure 4. Time evolution of the stochastic modes solution for uncertain Duffing oscillator under deterministic swept sine excitation.
In Figure 5, the stochastic displacement and velocity time history are presented as the PC approximation to the mean solutions given by the random modes x0 (t ) and x&0 (t ) with a shaded region representing the two standard deviation bounds σ x and σ x& around the corresponding means µ x and µ x& . In the bottom row of plots, the phase diagram of the mean displacement versus mean velocity as well as the phase plot of the mean restoring force µr ( x , x& ) versus mean displacement µ x are shown.
4.
SUMMARY AND CONCLUSIONS
This study shows that uncertain nonlinear dynamic systems can be analyzed by implementing the Stochastic Restoring Force Method in conjunction with Polynomial Chaos approaches. This straightforward implementation permits a robust characterization of the model uncertainties in terms of stochastic power-series coefficients. Using this representation of the uncertain system and PCE approach for solving stochastic differential equations, it is possible to predict accurately the time evolution of dynamical systems in the presence of stochastic uncertainty.
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Figure 5. Stochastic response for uncertain the uncertain Doffing oscillator subjected to a deterministic swept sine excitation.
REFERENCES 1. 2. 3.
4. 5. 6.
7.
Masri SF and Caughey TK. “A nonparametric identification technique for nonlinear dynamic problems”, Journal of Applied Mechanics, 46, pp. 433–447, 1979. Worden K and Tomlinson GR. Nonlinearity in Structural Dynamics: Detection, Identification and Modelling, Institute of Physics, London, 2001. Cameron RH and Martin WT. “The orthogonal development of non-linear functionals in series of fourier-hermite functionals”, The Annals of Mathematics, 48, pp. 385–392, 1946. Xiu D and Karniadakis G. “The wiener-askey polynomial chaos for stochastic differential equations”, SIAM Journal on Scientific Computing, 24, pp. 619–644, 2002. Ghanem RG and Spanos PD. Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, 1991. Ghanem R, Masri SF, Pellissetti M and Wolfe R. “Identification and prediction of thocastic dynamical systems in a polynomial chaos basis”, Computer methods in applied mechanics and engineering, 194, pp. 1641–1654, 2005. Sakamoto S and Ghanem R. “Simulation of multi-dimensional non-gaussian nonstationary random fields”, Probabilistic Engineering Mechanics, 17, pp. 167–176, 2002.
OVERTURNING PROBABILITY OF RAILWAY VEHICLES UNDER WIND GUST LOADS C. Proppe, C. Wetzel Institut für Technische Mechanik, Universität Karlsruhe, Kaiserstr, 12, 76128 Karlsruhe, Germany, E-mail:
[email protected] Abstract:
Sufficient crosswind stability is an important criterion in the approval process of railway vehicles. However, crosswind stability is in conflict with demands for light-weight constructions (especially cabin cars) and higher driving velocities. In many countries, the approval process foresees stability predictions based on worst case scenarios, where uncertainties are taken into account by means of safety factors and comparison with reference vehicles. This procedure is a burden for innovations and hinders the interoperability of railway vehicles. Therefore, models have been proposed that take some of the uncertainties associated with the wind gusts and the aerodynamic coefficients of the carbody into account. In this paper, a consistent stochastic wind gust model is proposed, and probabilistic characteristic wind curves are computed by means of a reliability analysis of the train-environment system.
Key words:
Crosswind stability, wind gusts, reliability analysis.
1.
INTRODUCTION
Recent developments in railway engineering have been showing a trend to faster, more energy efficient and more comfortable trains with a higher capacity of passenger transportation. These efforts are directly leading to light-weight cars with distributed traction. Unfortunately, these developments significantly alter the crosswind stability in a negative manner. Therefore, crosswind stability has become a crucial issue of modern railway vehicle design that cannot be solved easily as all counter-measures 23 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 23–32. © 2007 Springer.
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C. Proppe, C. Wetzel
are very expensive. If a railway vehicle fails to be certified, ballasting is often the only possible solution. During the last 140 years about thirty wind-induced accidents have been reported. Most of these accidents happened in Japan and Switzerland on narrow gauges at highly endangered points (e.g. bridges or embankments) in nearly hurricane conditions [8-9]. But also on standard gauge track, incidents were reported. Due to the desired interoperability in Europe the leading operating companies of trains are working on Technical Specifications for Interoperability (TSI) to get a common rule for the certification of railway vehicles [19]. At the moment, most of the leading operating companies of trains in Europe are using approval processes which are based on worst case scenarios where uncertainties are taken into account by means of safety factors [14-15]. They are based on characteristic wind curves obtained from deterministic wind scenarios. The first formulation of this approach is generally attributed to Cooper [6]. Uncertainties enter only during the subsequent risk assessment process, where the risk of an incident is quantified based on a specific vehicle and connection [1]. Taking uncertainties underlying the computation of the characteristic wind curve into account, Carrarini [4], for the first time, proposes a probabilistic characteristic wind curve. In this paper the vehicle-environment system is analysed and a model for the crosswind analysis is proposed that includes the most significant uncertainties. They are represented as random variables, whose distributions are determined based on available data in the literature. Finally, crosswind stability is expressed as probability of failure, which can be computed by means of analytical or numerical approaches. The paper is organized as follows: the following section discusses the adopted model for the vehicle-environment system in detail. After that, the simulation procedure is introduced. In section 4, a representative cabin car is studied and the principle results are briefly stated, while section 5 contains the major conclusions.
2.
MODELING OF THE SYSTEM The system under study consists of two parts: the multibody vehicle model; the environmental model.
Overturning probability of railway vehicles under wind gust loads
25
The environmental model itself has two distinct components: the track (interaction with the vehicle model by means of the wheel/rail contact) and the aerodynamic model (interaction with the vehicle model by means of aerodynamic forces). The subsequent sections are dedicated to an in depth discussion of crucial modeling assumptions.
2.1 Vehicle model Commercial multi-body system software has been employed in order to accurately represent the vehicle, a cabin car. The elasticity of the carbody and the bogie frames has been neglected. On the other hand, nonlinearities of the spring and damper characteristics and the bump stops have been carefully taken into account. The latter are responsible for the orientation of the bogies during application of the wind loads. The train is assumed to move with constant velocity on the track.
2.2 Environmental model 2.2.1 Track model Sections of straight and curved track with constant cant deficiency have been investigated. The track is fitted with UIC 60 rails at standard track gauge of 1435 mm. Excitation by means of measured track irregularities of an intermediate quality German railway has been considered, in order to take effects of long wave track irregularities into account. The sleepers were modeled as rigid bodies. An elastic contact model has been adopted; the tangential forces have been computed by means of Kalkers Fastsim algorithm [12]. Vehicle overturn is described as critical wheel unloading, i.e. by the condition
Q ≤ δQ, Q0
(1)
where Q0 is the static wheel load, Q(t) the actual wheel load and δQ a safety margin usually taken as 10% or 5%. Other criteria, such as flange climbing (ratio of lateral to vertical wheel force) and track shift (sum of lateral forces on each axle) are less critical [14], when large aerodynamic loads are acting on the train.
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C. Proppe, C. Wetzel
2.2.2 Aerodynamic model The model for the crosswind consists of a superposition of the mean wind, the gust characteristic and the turbulent fluctuations. As the train speed is much higher than the velocity of the crosswind, the spatial correlation of the wind is neglected. Thus, the wind excitation is modeled, as if the train were running through a frozen wind field. Hence, the actual wind speed is a function of the track variable s. This function is transformed into the time domain by means of a reference velocity, which is the train speed v0. Two wind scenarios are investigated: a train coming out of a tunnel immediately being hit by a gust; a train traveling on an embankment under constant mean wind load being hit by a gust. Various shapes of the wind gust have been proposed in the literature. They have been reviewed and critically discussed in [5]. While former standards preferred ‘1-cos’ gust shapes [10-11], there are strong theoretical arguments [2] in favor of an exponential shape (cf. Figure 1) of the gust. However, as Carrarini [5] pointed out, direct computation of the aerodynamic forces and moments under assumption of stationary aerodynamics would lead to artificial results (and moreover pose severe problems to a correct numerical integration of the equations of motion). Taking unsteady aerodynamics via the aerodynamic admittance into account amounts to the application of a low pass filter (or a moving average) to the gust velocity time series. The gust shape introduces two parameters: gust amplitude and gust duration. They are represented as positive random as discussed in [7].
Figure 1. Representative mean gust shape.
Overturning probability of railway vehicles under wind gust loads
27
Turbulent fluctuations of the wind velocity are computed by a spectral decomposition of the von-Kármán spectral density function (cf. Figure 2), which is often used to describe air turbulence [20]. The turbulent fluctuations are assumed to be normally distributed. Von Karman Power Spectral Density
3
10
2
10
uo = 10[m/s] u = 20[m/s] o
1
S(f)
10
u = 30[m/s] o
0
10
−1
10
−2
10
−3
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
10
1
10
f [1/s]
Figure 2. von Kármán spectral density.
The resultant wind velocity v(t) is thus the sum of three different parts: mean wind, gust characteristic, and turbulent fluctuations. Similar approaches are common in the design of wind turbines [2]. The wind loads exerted on the vehicle are represented as concentrated loads. The coordinate system and the composition of the wind velocity vector acting on the train are shown in Figure 3.
Figure 3. Coordinate system and wind velocity vector.
The aerodynamic forces and moments are computed from the acting wind velocity vs(t) by means of experimentally determined aerodynamic coefficients:
28
C. Proppe, C. Wetzel 1 Fy / z (t ) = Cside / lift ( β ) ρ Avs2 (t ), 2 1 M x / y / z (t ) = Croll / pitch / yaw ( β ) ρ Alvs2 (t ). 2
(2)
The factors A and l are the related area and length dimension of the vehicle, resp. As the influence of the drag force on the crosswind stability is negligible the drag coefficient is not considered. The aerodynamic coefficients Cside/lift/roll/pitch/yaw depend nonlinearly on the angle
β = tan −1 (
v(t ) ) v0
(3)
For the calculation of vs(t), oscillations of the carbody are neglected and only the reference velocity is taken into account, as they are much smaller than the wind velocity. The aerodynamic coefficients are assumed to be random variables. Very little is known about their distribution [5], and their correlation is completely unknown. From the available experimental data, a normal distribution with a coefficient of variation of 10% seems to be reasonable.
3.
SIMULATION CONCEPT
For the calculation of the probability of failure Pf, it is necessary to evaluate integral
Pf =
∫p
Z*
( z*)dz *
(4)
Ωf
over the failure domain Ωf, where z* is the array of all stochastic variables of the system and pZ*(z*) the joint probability density function. The failure domain Ωf is the set of all arrangements of z* which forces the wheelunloading δQ to fall below the safety margin. For such a complex multibody system, the failure domain is not known explicitly but can only be evaluated pointwise. The integral in (4) can be simplified by using the law of conditional probability. The probability of failure is then obtained as u0 ,t
Pf =
∫ P( z | u
u0 , d
0
) p (u 0 )du 0 ,
(5)
Overturning probability of railway vehicles under wind gust loads
29
where P(z|u0) is the failure probability conditioned to the mean wind speed u0, p(u0) the pdf of u0 and z the array z* without u0. In order to evaluate (5), P(z|u0) and p(u0) have to be known. The latter can be obtained from meteorological measurements, while the former has to be computed. For simplification, the calculation of P(z|u0) is only done at certain predefined mean wind velocities which reduces the evaluation of (5) to the computation of the finite sum N
Pf = ∑ P( z | u0,i ) p (u0,i )∆u0,i ,
(6)
i =1
The conditional failure probability P(z|u0) can be evaluated either by analytical methods, such as FORM or SORM [18] or by numerical methods employing Monte Carlo simulation with variance reduction [17], eventually under application of a response surface [3]. Here, all distributions are mapped to a standard Gaussian space, in which the shortest distance to the failure domain, the so called design point, is computed. After that, importance sampling around the design point and line sampling [16] has been employed in order to obtain reliable estimates of the conditional failure probability.
4.
RESULTS
Preliminary studies indicated that the influence of the turbulence on the probability of failure can be neglected. Therefore, the following results refer to a wind model without atmospheric turbulence. Figure 4 shows the conditional failure probability versus the mean wind speed for the tunnel exit wind scenario for a typical cabin car traveling with 160 km/h on straight track. It can be clearly seen that the differences between the results obtained by FORM and the sampling based results are large (about 30%). The results show an exponential increase of the failure probability with increasing wind speed over a range of several orders of magnitude. Figure 5 compares the conditional failure probability for the tunnel exit and the embankment scenario. As can be expected, failure probabilities for the embankment scenario are lower than for the tunnel exit. Stated in another way, the cabin car can sustain mean wind speeds that are approximately 2 m/s higher at the same failure level.
30
C. Proppe, C. Wetzel
Figure 4. Conditional failure probability vs. mean wind speed for tunnel exit scenario.
Figure 5. Comparison of failure probability for tunnel exit and embankment scenario (FORM results).
Overturning probability of railway vehicles under wind gust loads
5.
31
CONCLUSIONS
In recent years due to the modern light weight constructions and due to the increasing interoperability in Europe, the crosswind stability of highspeed trains has come to the fore of the leading operating companies of rolling stock. To prove the crosswind stability of a railway vehicle, the state of the art is to calculate the deterministic characteristic wind curve and then to compare this characteristic wind curve with a reference model of an already existing vehicle. In contrast to this standard procedure, a consistent stochastic approach is proposed in which a probabilistic characteristic wind curve has to be computed. Two wind scenarios, referring to a tunnel exit and a situation on an embankment, are defined. In this way, the most important uncertainties of the vehicle-environment system are accounted for. By prescribing an acceptance level for the probability of failure conditioned to the mean wind speed, a critical mean wind speed can be inferred. For the first time, a probabilistic characteristic wind curve based on sampling techniques has been computed. Due to the use of very efficient variance reducing sampling algorithms, the effort is not much higher than for the computation of the design point. The results indicate that deviations from the FORM results are large. It is noted that for the cases under investigation, the failure probability increases exponentially with increasing mean wind speed. Further efforts are necessary in order to clarify the uncertainty modeling of the aerodynamic coefficients and the influence of non-stationary aerodynamics. For the former, due to the lack of data, resort to nonparametric models might by an interesting alternative. Furthermore, it is noted, that the applied procedures are still to complicate in order to enter design codes. However, sensitivity analyses of the conditional probability of failure may lead to considerable simplifications. Finally, the numerical techniques applied in this study can be generalized in order to couple reliability analysis and multi body system. In this way, a general framework for the uncertainty analysis of multi body systems can be obtained.
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REFERENCES 1. 2. 3. 4.
5. 6. 7.
8.
9. 10. 11. 12. 13. 14. 15. 16.
17.
18. 19. 20.
Andersson E, Häggström J, Sima M, Stichel S. “Assessment of train-overturning risk due to strong cross-winds”, J. of Rail and Rapid Transit, 218, pp. 213-223, 2004. Bierbooms W, Cheng PW. “Stochastic gust model for design calculations of wind turbines”, Wind Engineering and Industrial Aerodynamics, 90, pp. 1237-1251, 2002. Bucher CG, Burgound U. “A fast and efficient response surface approach for structural reliability problems”, Structural Safety, 7, pp. 57-66, 1990. Carrarini A. “A probabilistic approach to the effects of cross-winds on rolling stock”, Proc. European Congress on Computational Methods in Applied Sciences and Engineering, Jyväskylä, Finland, 24-28 July, 2004. Carrarini A. Reliability based analysis of the crosswind stability of railway vehicle, Dissertation Thesis, TU Berlin, 2006. Cooper RK. “The probability of trains overturning in high winds”, Proc. 5th Int. Conf. on Wind Engineering, Fort Collins, Colorado, USA, July, pp. 1185-1194, 1979. Delaunay D, Locatelly JP. “A gust model for the design of large horizontal axis wind turbines: completion and validation”, Proc. European Community Wind Energy Conference, Madrid, Spain, Sep. 10-14, pp. 176-180, 1990. Fujii T, Maeda T, Ishida H, Imai T, Tanemoto K, Suzuki, M. “Wind-Induced Accidents of Train/Vehicles and Their Measures in Japan”, Quarterly Report of Railway Technical Research Institute, 40, 1999. Gawthorpe RG. “Wind effects on ground transportation.” J. Wind Engineering Industrial Aerodynamics, 52, pp. 73-92, 1994. Hoblit FM. Gust Loads on Aircraft. AIAA, Washington DC, USA, 1988. IEC 61400-1 Wind Turbine Safety and Design, 1993. Kalker JJ. “A fast algorithm for the simplified theory of rolling contact”, Vehicle System Dynamics, 102, pp. 1-13, 1982. Lippert S. On side wind stability of trains. Report, Royal Institute of Technology – Railway Technology, Stockholm, 1999. Lippert S, Tengstrand H, Andersson E., Stichel S. “The effect of strong cross winds on rail vehicles”, VDI Berichte, 1568, pp. 221-241, 2000. Matschke G, Grab M, Bergander B. “Nachweis der Sicherheit im Schienenverkehr bei extremem Seitenwind”, Betrieb und Verkehr, 51, pp. 200-206, 2002. Pradlwarter HJ, Pellissetti MF, Schenk CA, Schuëller GI, Kreis A, Fransen S, Calvi A, Klein M. “Realistic and efficient reliability estimation for aerospace structures”, Computer Methods in Applied Mechanics and Engineering, 194, pp. 1597-1617, 2005. Proppe C, Pradlwarter HJ, Schuëller GI. “Equivalent linearization and Monte Carlo simulation in stochastic dynamics”, Probabilistic Engineering Mechanics, 18, pp. 1-15, 2003. Rackwitz R. “Reliability analysis – a review and some perspectives”, Structural Safety, 23, pp. 365-395, 2001. Schulte-Werning B, Gregoire R, Malfatti A. TRANSAERO - A European Initiative on Transient Aerodynamics for Railway System Optimization, Springer, Berlin, 2002 Simiu E, Scanlan RH. Wind Effects on Structures, Wiley, 1996.
IMPACT SYSTEMS WITH UNCERTAINTY W. Schiehlen, R. Seifried Institute of Engineering and Computational Mechanics, University of Stuttgart, 70569 Stuttgart, Germany
Abstract:
The coefficient of restitution is mostly required for impact analysis in multibody dynamics. Using a multiscale simulation approach the coefficient can be computed on a fast time scale. Thereby modal models with local contact models proof to be efficient and accurate models for the simulations on the fast time scale. For many impact systems the coefficient of restitution is assumed to be deterministic, depending on essential parameters such as material, shape and initial collision velocity. In this paper impacts on beams are investigated numerically and experimentally. The investigated beam impacts feature multiple impacts, resulting in an uncertainty for the coefficient of restitution.
Key words:
Multibody systems, multiscale simulation, multiple impacts, beam, experiments, coefficient of restitution, uncertainty.
1.
INTRODUCTION
Impacts occur in passive mechanical systems constraint by bearing with clearance, and in actively controlled mechanical systems like robots with colliding links. Such mechanical systems are often modeled as multibody systems to describe large nonlinear motions, and the impacts are treated by the coefficient of restitution, see e.g. Pfeiffer and Glocker [7] and Stronge [19]. The coefficient of restitution is considered as deterministic number depending on the material, the shape and the velocity of the colliding bodies see e.g. Goldsmith [3]. However, in experiments and simulations it was observed that for a sphere striking a beam the coefficient of restitution is uncertain due to multiple impacts resulting in random behavior. 33 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 33–44. © 2007 Springer.
34
2.
W. Schiehlen, R. Seifried
IMPACTS IN MULTIBODY SYSTEMS
The method of multibody systems allows the dynamical analysis of machines and structures, see References [8-10]. More recently contact and impact problems featuring unilateral constraints were considered too, see Pfeiffer and Glocker [7]. A multibody system is represented by its equations of motion as
M (y )&& y + k (y , y& ) = q(y , y& ),
(1)
where y(t) is the global position vector featuring f generalized coordinates, M the inertia matrix, k the vector of Coriolis and gyroscopic forces and q the vector of the applied forces. The continuous motion of the multibody system might be interrupted by collision. Collisions with non-zero relative velocity result in impacts and impact modeling is required. Using the instantaneous impact modeling the motion of the multibody system is divided into two periods with different initial conditions, see e.g. Glocker [2], Pfeiffer and Glocker [7] or Eberhard [1]. During impact the equations of motion (1) have to be extended by the impact force F which is assumed to act in normal direction to the impact points, M (y )&& y + k ( y , y& ) = q(y , y& ) + w N F.
(2)
The vector wN projects the impact force from the normal direction of the impact on the direction of the generalized coordinates. Due to the assumption of infinitesimal impact duration, the velocity changes in a jump, whereas the position remains unchanged. The equation of motion during impact is then formulated on velocity level, te
&& + k − q − w N F ) dt = M ( y& e − y& s ) − w N ∆P = 0, lim ∫ ( My
te → ts
(3)
ts
where the indices s and e mark the start and end of the impact, respectively. In the limit case t e → t s the quantities M and wN are constant and all but the impact forces vanish due to their limited amplitudes. However, the infinitely large impact force F yields a finite force impulse ∆P which results in the jump of the generalized velocities and the non-smooth behavior. The impact force F and, therefore, the impulse ∆P are still unknown. The coefficient of restitution e provides additional information for the assessment of the impulse. Using the kinetic coefficient of restitution due to Poisson, the impact duration is divided into a compression and a restitution phase. The
35
Impact systems with uncertainty
compression phase starts at time ts and ends with time tc , which is marked by the vanishing relative normal velocity. The restitution phase starts at time tc and ends at te . The kinetic coefficient of restitution is defined as the ratio of the impulses ∆Pc and ∆Pr during the compression and restitution of the impact, respectively. An impact with e = 1 is called elastic and indicates no energy loss, whereas an impact with e = 0 is called plastic or inelastic and indicates maximal energy loss, resulting in a permanent contact. However, it should be noted, that the terms ‘elastic’ and ‘plastic’ describe here only the impact behavior and have little to do with the material behavior. As shown in Reference [13, 17] the impulse during the compression phase reads as ∆Pc =
−g& Ns w M −1w N T N
(4)
where g& Ns is the relative normal velocity of the contact points before impact. The total impulse during impact follows as ∆P=∆Pc +∆Pr = (1+e ) ∆Pc
(5)
and using Equation (3) the generalized velocities after impact y& e can be computed. In the case of more than one impact occurring simultaneously or a permanent contact opening due to impact, respectively, the corresponding equations have to be solved simultaneously resulting in linear complementarity problems (LCPs), see Pfeiffer and Glocker [7]. The impact modeling using Poisson’s coefficient of restitution is a very efficient method for treating impacts in multibody systems if the coefficient of restitution is known. The coefficient of restitution is usually found by experiments or it is known from experience. However, the coefficient of restitution may be evaluated numerically by additional simulations on a fast time scale, too, see References [11-13]. This results in a multiscale simulation approach. The simulation on the slow time scale is interrupted by an impact. Then, for the impact, a detailed simulation with deformable bodies is performed on a fast time scale including elastodynamic wave propagation and elastic-plastic material phenomena. The generalized coordinates and velocities before impact are used as initial conditions for the simulations on the fast time scale. These simulations are limited to the impact duration and from the time-continuous impact force F the resulting impulse ∆P is computed and the kinetic coefficient of restitution follows as e=
w T M −1w N ∆Pr ∆P − ∆Pc = =− N ∆P − 1, ∆Pc ∆Pc g& Ns
(6)
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W. Schiehlen, R. Seifried
see References [13, 17] for more details. The coefficient of restitution is now fed back to the slow time scale. Then, the generalized velocities y& e after impact are computed using Equation (3-5).
3.
NUMERICAL MODELS
The computation on the fast time scale requires numerical models which include wave propagation within the bodies, and elastic or elastic-plastic deformation of the contact region. First of all, a complete Finite Element (FE) model of the impacting bodies is used. A small overall element length is required to comprise the wave propagation in the bodies and an additional refinement is necessary for the modeling of the contact region, see Reference [14]. Thus, FE-models for impact analysis are excessively time consuming and not suitable for larger impact systems as found in engineering. Therefore, in a more time efficient numerical approach, impact processes are divided into two parts, a small contact region and the remaining body featuring wave propagation, see Reference [13, 17]. This procedure is also called boundary approach. The contact is a nonlinear problem which is limited to a small region, while the wave propagation is a linear problem encompassing the entire body. Thus, combined models are developed in which the elastodynamic behavior of the impacting bodies is represented by a modally reduced model and the deformation of the contact region is presented by a local contact model based on FE-models of the contact region. The local contact model is than either concurrently computed or precomputed and then coupled with the reduced elastodynamic model of the impacting bodies, see References [12, 13, 15-17]. The efficiency and consistency of the combined models is demonstrated for the impact of a steel sphere (radius=15mm) on aluminum rods (radius=10mm, length=1000mm) with initial velocity of 0.3 m/s. The rods have elastic and elastic-plastic material behavior, respectively. The computed coefficients of restitution and computation times are summarized in Table 1. It turns out clearly that the simulation results obtained from the different models agree very well. It is also obvious that the completely nonlinear FE model is very time consuming, especially when including elastic-plastic material behavior. Using a modal model with concurrently computed FE-contact the computation time is reduced by 40-60%. A tremendous decrease in the computation time is achieved using the modal model with pre-computed FE contact. However, it should be noted that the pre-computation of the force-deformation diagram is time consuming, too, especially for elasto-plastic material behavior. The computation time corresponds to about 15 impact simulations with the nonlinear FE model.
37
Impact systems with uncertainty Table 1. Comparison of numerical models for sphere to rod impact model A. complete nonlinear FE-model B. modal model+concurrently computed FE-contact C. modal model+pre-computed FE-contact
coeff. of restitution elastic plastic 0.633 0.481 0.631 0.477
computation time [s] elastic plastic 462 937 285 354
0.632
0.04
0.477
0.05
Therefore, the benefit of the modal model with pre-computed FE-contact takes place especially when many impacts are investigated.
4.
ESSENTIAL PARAMETERS FOR THE COEFFICIENT OF RESTITUTION
The coefficient of restitution depends not only on the material parameters but also strongly on the contact geometry, the body geometry and the initial velocity. Early experimental results for the evaluation of the coefficient of restitution are summarized in Goldsmith [3], more recent numerical and experimental results are presented in Minamoto [6], Sondergard [18], Wu et al. [20], Zhang et al. [21] and References [11-17]. In Figure 1 the influence of the material properties and the initial velocity on the coefficient of restitution is presented for the impact of a steel sphere (radius 15mm) on two different aluminum rods (radius=10mm, length=1000mm). Rod 1 has a low yield stress of 205Mpa and rod 2 has a high yield stress of 575Mpa. The sphere has an initial velocity in the range of 0.05-0.5m/s, the rods are initially in rest. For the experimental evaluation a test bench with two Laser-Doppler-Vibrometer is used, see Hu et al. [4, 5].
Figure 1. Impact of a hard steel sphere on two aluminum rods (left: low yield stress 205 MPa, right: high yield stress 575 MPa).
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It is clearly seen from simulations and experiments that for both impact systems the coefficient of restitution decreases with increasing initial velocity. For rod 1 the measured coefficients and the ones obtained from simulations with elastic-plastic material behavior agree very well. However they are significantly lower than coefficients obtained from simulations with purely elastic material behavior. For rod 2, which has a high yield stress, simulations with elastic and elastic-plastic material behavior show for the investigated velocity range nearly identical behavior and agree well with experimental results. In References [13, 15] the influence of plastification on the coefficient of restitution for repeated impacts is investigated for both rods. The influence of the shape of the bodies on the coefficient of restitution is investigated in Reference [11] for the impact of a steel sphere on four elastic aluminum bodies with equal mass but different shape. These are a compact cylinder, a half-circular plate, a long rod and a slender beam. Figure 2 shows the computed coefficients of restitution of these impact systems for the velocity range 0.025-0.5m/s. The computed coefficient of restitution for the cylinder is close to e=1 for the investigated velocity range. For the impact on the cylinder the transformation of initial kinetic energy into waves and vibrations can be neglected. From the simulations for the rod and half-circular plate it is seen that the coefficient of restitution decreases steadily with increasing initial velocity. This indicates an increase of energy transformation from the initial rigid body motion into waves and vibrations with increasing velocity. The transverse impact on the beam excited very strong vibration phenomena in the beam resulting in multiple successive impacts within a very short time period. In sharp contrast to the previous impact systems the beam impact shows no clear pattern but a strong uncertainty, see also Reference [17].
Figure 2. Impact of a hard steel sphere on differently shaped aluminum bodies (□ compact cylinder, ◊ half circular plate, ○ rod, + beam).
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Impact systems with uncertainty
5.
UNCERTAINTY OF THE COEFFEICIENT OF RESTITUTION
The impact on a beam features multiple impacts which are caused by the strong bending vibrations of the beam, resulting from the first impact. The multiple impacts are the source of the uncertainty of the coefficient of restitution. Since more than one successive impact occur within a short time period efficient numerical methods for impact simulation on the fast time are even more important than for single impacts.
5.1 Comparison of Numerical Models A comparison of the simulation results using the different numerical models is presented in Figure 3 for the impact of a steel sphere (radius=15mm) with exactly the same initial velocity 0.2m/s on an elastic aluminum beam (radius=10mm, length=1000m). After the first impact the sphere still moves forward in its initial direction until a successive second impact occurs. This overall behavior is consistently observed in all simulations using the three different numerical models and shows the good overall agreement of the models. Moreover, it proves that the uncertainty is not a numerical problem. 2. impact beam 1. impact
sphere 1. impact
2. impact
Figure 3. Comparison of numerical models for beam impact (A: complete FEM, B: modal+concurrently computed FE-contact, C: modal+pre-computed FE-contact.
Table 2 summarizes the coefficients of restitution and computation times of the simulations. This shows again the good agreement of the modal models with FE-contact and the complete FE-model. It turns out that the complete FE-model is very time consuming. By using modal models the computation times can be reduced significantly. Using the modal model with concurrently computed FE-contact the computation time can be reduced by 97%.
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Using the modal model with pre-computed FE-contact the computation time can be reduced further, however the computation time for the forcedisplacement diagram has to be considered, which takes in this case about 1000s. This shows clearly, that for a larger and complex impact system, such as the transverse impact on a beam, the modal model with pre-computed FEcontact is the most efficient approach. Table 2. Comparison of numerical models for sphere to beam impact model A. complete nonlinear FE-model B. modal model+concurrently computed FE-contact C. modal model+pre-computed FE-contact
coeff. of restitution 0.707 0.700 0.717
computation time [s] 80564 2422 16
5.2 Experimental validation For the experimental validation of the simulation results an experimental setup, originally developed by Hu et al. [4, 5], was adapted to beam impacts, see Figure 4. The sphere and beam are suspended with thin Kevlar wires in a frame as pendula. The sphere is released by a magnet from a predefine height and it impacts on the beam along its symmetry line. Two LaserDoppler-Vibrometers are used for displacement and velocity measurement of sphere and beam in the central line of impact.
Figure 4. Experimental setup for sphere to beam impact.
Figure 5 shows for the three initial velocities v = 0.276 m/s, v = 0.287m/s and v = 0.303m/s the measured and simulated displacement of sphere and beam, as well as the velocity of the sphere. The measurement and simulation show for all three initial velocities, that within a few milliseconds several impacts occur. Although the initial velocities chosen are close together, the impact response is quite different which is due to the multiple impacts.
Impact systems with uncertainty
41
Figure 5. Impact on beam with initial velocity v = 0.276m/s (top), v = 0.287m/s (middle) and v = 0.303 m/s (bottom).
Figure 5 shows for all three velocities a very good agreement for the first impact as well as consistently a second impact after 4 ms. However, for the successive impacts significant differences occur resulting in an overall uncertainty. For the impact with an initial velocity v = 0.276 m/s the second impact yield only to a small velocity change. Therefore, after 5.2 ms a third impact
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W. Schiehlen, R. Seifried
occurs, which results in a large velocity change of the sphere. In this case experiment and simulation agree very well. This is also reflected by the good agreement of the measured and simulated coefficients of restitution which are em = 0.664 and es = 0.687, respectively. The impact with the initial velocity v = 0.287 m/s shows in the simulation a much stronger second impact than in the experiment. This results in a very different behavior of the following motion. Consequently the coefficient of restitution computed from measurement and simulations differ strongly and are em = 0.620 and es = 0.334. For the impact with initial velocity v = 0.303 m/s the experiment proves that sphere is in rest after the second impact and a third impact occurs after 5.7 ms. In the simulation the second impact is stronger as the one in the experiment. Thereby the sphere rebounds and no further impact occurs in the simulation. Measurement and simulation yield hereby nearly identical coefficients of restitution of em = 0.230 and es = 0.243.
5.3 Analysis of the coefficient of restitution In Figure 6 simulated and measured coefficients of restitution are presented for 53 different initial velocities of the sphere. Due to the multiple impacts the coefficient of restitution depends strongly on the initial velocity, however, without showing a clear pattern but strong uncertainty, see Reference [17]. The coefficients of restitution are in the range e=0.07-0.73. Small differences of the simulated and measured motion of beam and sphere after the first impact result in very different behavior of the successive impacts. As a result, the investigated impacts show significant differences of the measured and simulated coefficients of restitution, for different initial velocities. For the simulated and measured impacts presented in Figure 6 the mean value of the initial velocity of the sphere is v = 0.25 and the standard deviation is σ v = 0.0929 . The mean value of the simulated coefficients is es = 0.3981 and the standard deviation is σs = 0.2275 . This is in good accordance with the measured coefficients of restitution which have a mean value of e m = 0.3800 and a standard deviation of σ m = 0.2125 . This statistical analysis shows that although large deviations between measured and simulated impacts may occur, the overall behavior is represented accurately by the numerical models. In Figure 6 the mean value of the measurements and deviation intervals are added. Thereby the areas A-D corresponds to the intervals represented by the mean values and the deviations 0.5σ, σ, 1.5σ, 2σ , respectively. However, it turns out that using this statistical approach the interval D, defined by e = e m ± 2σ m , includes nonphysical negative values for the coefficient of restitution. This shows that
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Impact systems with uncertainty
mechanical aspects and the simple statistical evaluation of the coefficient of restitution are contradicting for this uncertain mechanical system. In the right plot of Figure 6 the numbers of multiple impacts are indicated for simulation and measurements. It turns out that only for very low velocities one impact occur. For higher velocities 2, 3 or 4 successive impacts occur, however no relationship between the coefficient of restitution and the number of multiple impacts is obvious.
D C B
2σ m σm
em
A
σm
2σ m
σv 2σ v
σv
v
2σ v
velocity [m/s]
Figure 6. Multiple impacts on an elastic aluminum beam.
6.
CONCLUSION
Measurements and simulations for the transverse impact of a steel sphere on an aluminum beam show multiple successive impacts within a very short time period, resulting in an uncertain behavior of the coefficient of restitution. For the evaluation of the numerical and experimental data a probabilistic approach using mean value and variance of the coefficient of restitution shows good overall agreement of simulation and measurement. However a simple statistical approach for describing the coefficient of restitution has its limitations in overcoming its uncertainty.
REFERENCES 1.
Eberhard P. Kontaktuntersuchungen durch hybride Mehrkörpersystem / Finite Elemente Simulation (in German), Shaker, Aachen, 2000.
44 2. 3. 4.
5.
6.
7. 8. 9. 10. 11. 12.
13.
14. 15.
16.
17.
18. 19. 20. 21.
W. Schiehlen, R. Seifried Glocker C. “On frictionless impact models in rigid-body systems”, Philosophical Transactions of the Royal Society of London, A359, pp. 2385-2404, 2001. Goldsmith W. Impact: The Theory and Physical Behaviour of Colliding Solids, London: Edward Arnold Ltd, 1960. Hu B, Eberhard P, Schiehlen W. “Comparison of analytical and experimental results for longitudinal impacts on elastic rods”, Journal of Vibration and Control, 9, pp. 157-174, 2003. Hu B, Eberhard P. Experimental and theoretical investigation of a rigid body striking an elastic rod, Institutsbericht IB-32, Stuttgart, Institute of Engineering and Computational Mechanics, 1999. Minamoto H. “Elasto / Visco-plastic impact of two equivalent spheres made of SUJ2”, Transactions of The Japan Society of Mechanical Engineers, Series C, 71, pp. 51-57, 2005 (in Japanese). Pfeiffer F, Glocker C. Multibody Dynamics with Unilateral Contacts, New York: John Wiley & Sons, 1996. Schiehlen W. “Multibody system dynamics: Roots and perspectives”, Multibody System Dynamics, 1, pp. 149-188, 1997. Schiehlen W. “Unilateral contacts in machine dynamics”, Unilateral Multibody Contacts, Pfeiffer F, Glocker Ch. (Eds.), Kluwer, Dordrecht, pp. 287-298. 1999. Schiehlen W, Eberhard P. Technische Dynamik, (in German) Teubner, Wiesbaden, 2004. Schiehlen W, Seifried R. “Three approaches for elastodynamic contact in multibody systems”, Multibody System Dynamics, 12, pp. 1-16, 2004. Schiehlen W, Seifried R. “Impact Mechanics in Mechanical Engineering”. Proceedings of the International Conference on Mechanical Engineering and Mechanics 2005 (ICMEM), Nanjing, China, October 26-28, pp. 2-10, 2005. Schiehlen W, Seifried R, Eberhard P. “Elastoplastic phenomena in multibody impact dynamics”, Computer Methods in Applied Mechanics and Engineering, in press, [doi:10.1016/j.cma.2005.08.011]. Seifried R, Hu B, Eberhard, P. “Numerical and experimental investigation of radial impacts on a half-circular plate”, Multibody Systems Dynamics, 9, pp. 265-281, 2003. Seifried R, Schiehlen W, Eberhard P. “Numerical and experimental evaluation of the coefficient of restitution for repeated impacts”, International Journal of Impact Engineering, 32, pp. 508-524, 2005. Seifried R, Eberhard P. “Comparison of Numerical and Experimental Results for Impacts” Proceedings of the ENOC-2005 Fifth EUROMECH Nonlinear Dynamics Conference, Eindhoven, The Netherlands, August 7-12, 2005, van Campen DH, Lazurko MD, van der Oever W. (Eds), pp. 399-408, 2005. Seifried R. Numerische und experimentelle Stoßanalyse für Mehrkörpersysteme (in German), Schriften aus dem Institut für Technische und Numerische Mechanik der Universität Stuttgart, Nr. 2, Aachen: Shaker Verlag, 2005. Sondergaard R, Chaney K, Brennen C. “Measurment of solid spheres bouncing off flat plates”, Journal of Applied Mechanics, 57, pp. 694-699, 1990. Stronge WJ. Impact Mechanics, Cambridge: Cambridge University Press, 2000. Wu C, Li L, Thornton C. “Rebound behavior of spheres for plastic impacts”, International Journal of Impact Engineering, 28, pp. 929-946, 2003. Zhang X, Vu-Quoc, L. “Modeling of the dependence of the coefficient of restitution on impact velocity in elasto-plastic collisions”, International Journal of Impact Engineering, 27, pp. 317-341, 2002.
PART 2
SYSTEM DYNAMICS WITH UNCERTAINTY
UNCERTAINTY PROPAGATION IN COMPLEX ENGINEERING SYSTEMS BY ADVANCED MONTE CARLO METHODS S. K. Au1, D. P. Thunnissen2 1
City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, E-mail:
[email protected] 2 School of Mechanical and Aerospace Engineering Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, E-mail:
[email protected] Abstract:
This paper presents a recently developed advanced Monte Carlo method called ‘Subset Simulation’ for efficient stochastic analysis of complex engineering systems. The method investigates rare failure scenarios by efficiently generating ‘conditional samples’ that populate progressively towards the rare failure region. In addition to reliability analysis and performance margin estimation, the conditional samples also provide information for sensitivity and data-mining purposes. Subset Simulation is based on a simple but important observation that a small failure probability can be expressed as a product of large conditional failure probabilities of some intermediate failure events. This perspective makes use of the defining properties of conditional probabilities, and so is valid for all applications. The method is illustrated with applications in structural and aerospace engineering. Recent and future development of the method will also be discussed.
Key words:
Monte Carlo method, structural reliability, subset simulation, uncertainty.
1.
INTRODUCTION
When designing new engineering systems subjected to uncertainties, it is desirable to quantify the predicted performance of a proposed design in terms of the reliability or performance margins with respect to specified design objectives. Let Θ ∈Rn denote the vector of random variables for which a probability model is available, say, in terms of the joint probability 45 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 45–54. © 2007 Springer.
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density function (PDF) p(θ). Many failure events in engineering risk analysis can be formulated as the exceedance of a critical response variable Y(Θ)≥0 over some specified threshold level y, i.e., P ( F ) = P (Y > y ) = ∫ p (θ )d θ F
(1)
Complementary to the failure probability is the performance margin that corresponds to the percentile of a given risk rolerance, say, p, through which the risk tolerance of a decision maker manifests. For example, the 90, 99, and 99.9 percentiles might provide a decision maker with low-, medium-, and high-confidence estimates in the upper-bound value of response, corresponding to a risk-tolerant, risk-neutral, or risk-averse decision maker, respectively. Monte Carlo simulation (MCS) [1] is the most established sampling technique and the benchmark for comparison by other techniques. In MCS, N random samples of the random variables are generated according to their specified probability distributions. The corresponding values of the response Y are then evaluated and analyzed statistically. A simple estimator for the upper q-percentile value can be obtained as the pN -th largest value of Y among the N samples, i.e., the pN -th order statistics. This estimator is asymptotically unbiased, although the highest percentile value that can be legitimately estimated can be significantly biased for finite N. Complementarily, the failure probability P(F)=P(Y >y) for a given y can be estimated simply as the fraction of samples with Y > y among the N MCS samples. The coefficient of variation (c.o.v.) of this failure probability estimate is given by
δ = (1 − P( F )) / P( F ) N ~ 1/ P( F ) N
for small P( F )
(1)
While MCS is applicable to all types of reliability problems, its computational efficiency is a practical concern when estimating small failure probabilities because information must be gained from samples that correspond to failure but these are rarely simulated. A rule of thumb is that one must generate at least 10 failure samples to get a reasonably accurate estimate of P(F), so if P(F)=0.001, at least 10,000 system analyses must be performed. This has motivated recent research to develop more efficient reliability algorithms. Over the past few decades, a number of reliability methods have been developed that are effective when the number of variables n is not too large or when the failure boundary has limited complexity. A common feature of most stochastic simulation methods is that they estimate the integral for P(F) by gaining information about the system behavior and then using such information to account for the failure probability. Excellent reviews can be found at different stages of
Uncertainty propagation in complex engineering systems
47
development, e.g., [1-5]. In recent years, attention has been focused on problems with complex system characteristics and with high dimensions (i.e., for large n) [5]. High-dimensional problems are frequently encountered in system reliability problems or those involving stochastic processes or random fields, whose discretized representation requires a large number of i.i.d. (independent and identically distributed) variables. Ideally, the dimension of a reliability problem should be determined based on modeling reasons rather than be limited by the capability of reliability methods. Stochastic simulation methods provide an attractive means for solving highdimensional problems, especially for complex systems where analytical results or knowledge about the dependence of the response on the excitation and modeling parameters are rarely available.
2.
SUBSET SIMULATION METHOD
Subset Simulation is an adaptive stochastic simulation procedure for efficiently computing small tail probabilities [7, 8]. Originally developed for reliability analysis of civil engineering structures, it stems from the idea that a small failure probability can be expressed as a product of larger conditional failure probabilities for some intermediate failure events, thereby converting a rare event simulation problem into a sequence of more frequent ones. During simulation, conditional samples are generated from specially-designed Markov chains so that they populate gradually each intermediate failure region until they reach the final target (rare) failure region. For a given y for which P(Y >y) is of interest, let 0 < y1 < y2 < … < ym = y be an increasing sequence of intermediate threshold values. By sequentially conditioning on the event {Y > yi}, the failure probability can be written as m
P (Y > y ) = P(Y > y1 )∏ P(Y > yi | Y > yi −1 )
(2)
i =2
The original idea is to estimate P(Y >y1) and {P(Y >yi | Y >yi-1): i = 2,…,m} by generating samples of Θ conditional on {Y(Θ)>yi): i = 1,…,m}. In implementations, y1, …, ym are generated adaptively using information from simulated samples so that the sample estimate of P(Y >y1) and {P(Y >yi | Y > yi-1): i = 2,…,m} always correspond to a common specified value of the conditional probability p0 (p0=0.1 is found to be a good choice). The efficient generation of conditional samples is highly-nontrivial but pivotal in the success of Subset Simulation, and it is made possible through the machinery of Markov Chain Monte Carlo (MCMC) simulation [9]. Markov Chain Monte Carlo is a class of powerful algorithms for generating
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samples according to any given probability distribution. It originates from the Metropolis algorithm developed by Metropolis and co-workers for applications in statistical physics [10]. A major generalization of the Metropolis algorithm was due to Hastings for applications in Bayesian statistics [11]. In MCMC, successive samples are generated from a specially designed Markov chain whose limiting stationary distribution tends to the target PDF as the length of the Markov chain increases. An essential aspect of the implementation of MCMC is the choice of ‘proposal distribution’ that governs the generation of the next sample from the current one. The efficiency of Subset Simulation is robust to the choice of the proposal distribution, but tailoring it for a particular class of problem can certainly improve efficiency. For example, some improvement in efficiency has been achieved when tailoring the method with respect to the nature of random variables [8] or to causal dynamical systems [12, 13].
2.1 Procedure The Subset Simulation procedure for adaptively generating samples of Θ conditional on {Y(Θ )> yi : i = 1,…,m} corresponding to specified target probabilities {P(Y(Θ)>yi) = p0i, i = 1,…,m} is summarized as follows. First, N samples {Θ0,k: k = 1,…,N} are simulated by direct MCS, i.e., they are i.i.d. as the original PDF. The subscript ‘0’ here denotes that the samples correspond to ‘conditional level 0’ (i.e., unconditional). The corresponding values of the tradable variable {Y0,k: k = 1,…,N} are then computed. The value of y1 is chosen as the (1-p0)⋅N-th value in the ascending list of {Y0,k: k = 1,…,N}, so that the sample estimate for P(F1) = P(Y>y1) is always equal to p0. Here, we have assumed that p0 and N are chosen such that p0⋅N is an integer. Due to the choice of y1, there are p0⋅N samples among {Θ0,k: k = 1,…,N} whose response Y lies in F1 = {Y>y1}. These are samples at ‘conditional level 1’ and are conditional on F1. Starting from each of these samples, MCMC is used to simulate an additional (1-p0)⋅N conditional samples so that there is a total of N conditional samples at conditional level 1. The value of y2 is then chosen as the (1-p0)⋅N-th value in the ascending list of {Y1,k: k = 1,…,N}, and it defines F2 = {Y >y2}. Note that the sample estimate for P(F2|F1) = P(Y >y2 | Y>y1) is automatically equal to p0. Again, there are p0⋅N samples lying in F2. They are samples conditional on F2 and provide ‘seeds’ for applying MCMC to simulate an additional (1-p0)⋅N conditional samples so that there is a total of N conditional samples at ‘conditional level 2.’ This procedure is repeated for higher conditional levels until the samples at ‘conditional level (m-1)’ have been generated to yield ym as the (1-p0)⋅N-th value in the ascending list of {Ym-1,k: k = 1,…,N} and that ym > y so that there
Uncertainty propagation in complex engineering systems
49
are enough samples for estimating P(Y >y). Note that the total number of samples is equal to N + (m-1)⋅(1-p0)⋅N. The whole procedure is illustrated in Figure 1. Approximate formulas have been derived for assessing the statistical error (in terms of c.o.v.) that can be estimated using samples generated in a single run [7]. response
response Monte Carlo Simulation
Monte Carlo Simulation
F1 b1
uncertain parameter space
failure probability estimate
a) Level 0 (initial phase): Monte Carlo simulation response
p0 failure probability estimate
uncertain parameter space
b) Level 0: adaptive selection of first intermediate threshold level response
Markov Chain Monte Carlo
F2 F1 b1
uncertain parameter space
b2
F1 b1
p0 failure probability estimate
c) Level 1: Markov Chain Monte Carlo simulation
uncertain parameter space
p0 2 p0 failure probability estimate
d) Level 1: adaptive selection of second intermediate threshold level
Figure 1. Schematic diagram of Subset Simulation Procedure
2.2 Data-mining using Markov chain samples The Markov chain samples generated during Subset Simulation can also be used to infer the probable scenarios that will occur in the case of failure. The conditional PDF p(θ |F) gives an idea of the probable cause of failure should it occur. By Bayes’ Theorem, P ( F | θ ) = P ( F ) p (θ | F ) / p (θ )
(3)
and hence P(F|θ ) will be insensitive to θ when the conditional PDF p(θ |F) is similar in shape to the unconditional PDF p(θ ). The distribution of some response quantity of interest evaluated at the Markov chain samples also gives information about the system performance when failure occurs. The conditional expectation can be estimated by averaging over the Markov chain samples [8, 22].
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3.
ILLUSTRATIVE APPLICATIONS
10
0
10
-1
-2 10 FO
10
-3
10
-4
0
Cumulative Distribution Function Value
P(Y>y)
The application of Subset Simulation is illustrated through the reliability analysis of a structure subjected to stochastic ground motions [8, 14] and the performance margin estimation of the critical temperature of spacecraft component [15, 16].
OP LS
1
2
3
4 y (%)
Figure 2. Reliability results for frame
1 0.9998 0.9996
0.9994 SS (NSS = 370)
0.9992
0.999 35
SS (NSS = 1850) MCS (NT = 10000) 36
37 38 39 40 SSPA Temperature, °C
41
42
Figure 3. Tail CDF of SSPA temperature
Figure 2 shows the complementary cumulative distribution function (complementary CDF = 1-CDF) of the peak interstory drift ratio of a sixstory moment resisting frame with elasto-plastic joints subjected to earthquake excitations. The failure probability P(F)=P(Y >y) for a given value of y can be readily read from the figure. The number of random variables involved is 6002, which comprises the moment magnitude, epicentral distance and the white noise sequence (6000 random variables) involved in generating the stochastic ground motion time history. The high dimensionality of the problem and its nonlinearity renders traditional variance reduction techniques such as importance sampling method inefficient [17]. In applying Subset Simulation, p0 was set to 10%. Three simulation levels were performed, each with 500 samples. The total number of samples used is therefore 500+450+450=1400. As an illustration of application to reliability-based design, the target design failure probabilities at three performance levels are set [18]: ‘Fully Operational (FO)’ in frequent events, ‘Operational (OP)’ in occasional events and ‘Life-Safe (LS)’ in rare earthquakes. For a reference return period of 50 years, assuming a Poisson arrival process for the occurrence of earthquakes, the performance criteria translate into design failure probabilities in a seismic event of 2.3×10-2, 1.4×10-2 and 2.1×10-3 for FO, OP and LS, respectively. These values are plotted in Figure 2 as squares. It
Uncertainty propagation in complex engineering systems
51
can be seen that the structure does not satisfy any one of the three criteria, although the situation is marginal at the LS level. As another illustration, Figure 3 shows the tail of the CDF of the critical temperature of a Solid-State-Power-Amplifier (SSPA) that is an important component of a space exploration rover [16]. The temperature was computed using a heat transfer model via SINDA/FLUINT [19]. Here, it is desired to estimate the percentiles of the SSPA temperature subjected to uncertainties in thermal properties, design (robustness) requirement and modeling error. Up to seven random variables have been studied so far, and more can be incorporated. In Figure 3, NSS and NT are the total number of samples used by Subset Simulation and direct MCS, respectively. The results for NSS=370 and NSS=1850 correspond to 100 and 500 samples used for each simulation level, with 3 levels to reach the tail region shown. Figure 3 indicates qualitatively that Subset Simulation requires much less computational effort than direct MCS. Direct MCS requires 10000 samples to yield on average 10 samples populating the tail probability regime in the figure. In contrast, Subset Simulation requires only 370 samples to produce the same number of samples (i.e., 10) populating the figure. There are 50 populating samples for NSS=1850 but it requires more computational effort whose justification depends on the required accuracy. Of course, a quantitative comparison of efficiency cannot be made here because the results have different statistical estimation error. 120
Prob.1 Prob.2.1 Prob.2.2 Prob.2.3 Prob.3
80 60
S MC
Unit c.o.v. ∆
100
40 20 0
-6
10
-4
10
P(F)
-2
10
Figure 4. Efficiency of Subset Simulation in Benchmark problems
Figure 4 gives a quantitative comparison of the efficiency of Subset Simulation with direct MCS at different failure probability levels for the benchmark study recently organized by Schueller and co-workers [20, 21]. The problems comprise high-dimensional static and dynamic problems with
52
S. K. Au, D. P. Thunnissen
uncertainties in system and loading properties. For stochastic algorithms, the c.o.v. for estimating P ( F ) is typically of the form δ = ∆ NT , the ‘unit c.o.v.’ ∆ plotted in Figure 4 therefore gives a measure of efficiency that is characteristic of the algorithm. The unit c.o.v. for Subset Simulation varies roughly in a logarithmic manner, i.e. ∆ is approximately proportional to log[1/ P ( F )] , while for direct MCS it grows drastically as ∆ ~ 1/ P ( F ) for small P ( F ) . The logarithmic character is a direct result of solving a rareevent simulation problem by a series of frequent conditional failure events. The logarithmic character holds for all the problems studied, suggesting that it is robust with respect to the type of applications; this is also supported by a theoretical study of efficiency [8].
4.
RECENT DEVELOPMENTS
The idea of Subset Simulation and the capability of efficiently generating conditional samples provided by MCMC have found to be extremely useful for reliability analysis and performance margin estimation. A method has been developed that takes advantage of the low computational effort required in meta-models but still maintaining unbiasedness in the reliability estimates [22]. MCMC has also been combined with importance sampling for efficient system analysis conditional on failure [23]. Along another direction, Subset Simulation has been extended to solving reliability design sensitivity problems [14], allowing the dependence of failure probability with design parameters to be estimated in a single simulation run. Essentially, if φ is the set of design parameters, the failure probability for a given design choice can be interpreted as a conditional failure probability, P(F|φ ), which can be expressed via Bayes’ Theorem as P ( F | φ ) = P ( F ) p (φ | F ) / p(φ )
(4)
The above equation indicates that the dependence of P(F|φ) on φ can be determined through the ratio of p(φ |F) and p(φ). The former can be obtained from MCMC samples generated during Subset Simulation, while the latter is specified by the user. There is a crucial conceptual breakthrough in interpreting (4). In the original design problem, φ is a deterministic parameter, but it has been ‘augmented’ as a random variable in the algorithm with an artificial PDF p(φ) whose choice is left to the user in relation to the regime of interest. In fact, (4) is identical to (3) except that the role of the conditioning variable in the problem are totally different; θ is represents a source of uncertainty while φ represents a controlled design choice. Some algorithms are being developed based on this idea, focusing on technical
Uncertainty propagation in complex engineering systems
53
issues in more efficiently estimating the conditional PDF [24] or in reliability optimization problems [25].
5.
CONCLUSIONS
Central to Subset Simulation is the generation of conditional samples that adaptively populate from the frequent (central) regime into the rare (tail) probability regime, thereby providing information for efficient estimation of failure probabilities and performance margins at the tails. In addition to reliability analysis, Subset Simulation can also be used for probabilistic datamining. It can also been extended to reliability design sensitivity and optimization problems through the concept of augmented design variables. With the increasing trend of using complex models for analysis, it is important that further stochastic analysis algorithms be developed with a balance in efficiency and robustness to applications.
ACKNOWLEDGEMENTS The research work presented in this paper is funded by the Singapore Ministry of Education through research grant ARC8/05. The support is gratefully acknowledged.
REFERENCES 1. 2. 3. 4. 5. 6.
7. 8. 9.
Rubinstein RY. Simulation and the Monte-Carlo Method. Wiley, 1981. Schueller GI, Stix R. “A critical appraisal of methods to determine failure probabilities”. Structural Safety, 4, pp. 293-309, 1987. Engelund S, Rackwitz R. “A benchmark study on importance sampling techniques in structural reliability”. Structural Safety, 12, pp. 255-76, 1993. Schueller GI. “Computational stochastic mechanics: recent advances”, Computers & Structures, 79, pp. 2225-2234, 2001. Rackwitz R. “Reliability analysis – a review and some perspectives”, Structural Safety, 23, pp. 365-395, 2001. Schueller GI, Pradlwarter HJ, Koutsourelakis PS. “A critical appraisal of reliability estimation procedures for high dimensions”, Probabilistic Engineering Mechanics, 19, pp. 463-474, 2004. Au SK, Beck JL. “Estimation of small failure probabilities in high dimensions by Subset Simulation”, Probabilistic Engineering Mechanics, 16, pp. 263-277, 2001. Au SK, Beck JL. “Subset Simulation and its application to probabilistic seismic performance assessment”, Journal of Engineering Mechanics, 129, pp. 1-17, 2003. Roberts C, Casella G, Monte Carlo Statistical Methods, Springer, 1999.
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10. Metropolis N, Rosenbluth A, Rosenbluth M, Teller A, “Equations of state calculations by fast computing machines”, Journal of Chemical Physics, 21(6), pp. 1087-1092, 1953. 11. Hastings W, “Monte Carlo sampling methods using Markov chains and their applications”, Biometrika, 57(1), pp. 97-109, 1970. 12. Ching J, Au SK, Beck JL. “Reliability estimation for dynamical systems subject to stochastic excitation using Subset Simulation with splitting”, Computational Methods in Applied Mechanics and Engineering, 194, pp. 1557-1579, 2005. 13. Ching J, Beck JL, Au SK. “Hybrid Subset Simulation method for reliability estimation of dynamical systems subjected to stochastic excitation”, Probabilistic Engineering Mechanics, 20(3), pp. 199-214, 2005. 14. Au SK. “Reliability-based design sensitivity by efficient simulation”. Computers & Structures, 83(14), pp. 1048-1061, 2005. 15. Thunnissen D. Propagating and mitigating uncertainty in the design of complex multidisciplinary systems, PhD thesis, California Institute of Technology, 2005. 16. Thunnissen DP, Au SK, Tsuyuki GT. “Uncertainty Quantification in Estimating Critical Spacecraft Component Temperatures”, AIAA Journal of Thermal Physics and Heat Transfer. To appear, 2005. 17. Au SK, Beck JL. “Importance sampling in high dimensions”. Structural Safety, 25(2), pp. 139-163, 2005. 18. Vision 2000: Performance based seismic engineering of buildings. Structural Engineers Association of California, Sacramento, California, 2000. 19. Cullimore B, Ring S, Johnson D. SINDA/FLUINT User’s Manual, C&R Technologies, Inc., Revision 17, Littleton, CO, 2003. 20. Au SK, Ching J, Beck JL. “Application of Subset Simulation Methods to Reliability Benchmark Problems”, Structural Safety. In print, 2005. 21. Schueller GI, Pradlwarter HJ. “Benchmark study on reliability estimation in higher dimensions of structural systems - an overview”, Structural Safety. In print, 2006. 22. Au SK. “Augmenting approximate solutions for consistent reliability analysis”, Probabilistic Engineering Mechanics. In print, 2006. 23. Au SK. “Probabilistic failure analysis by importance sampling Markov chain simulation”. Journal of Engineering Mechanics, 130(3), pp. 303-311, 2004. 24. Ching J, Hsieh YH. “Local estimation of failure probability function and its confidence interval with maximum entropy principle”. Probabilistic Engineering Mechanics. In print, 2006. 25. Taflanidis AA, Beck JL. “Reliability-based optimal design by efficient stochastic simulation”. Proc. of 5th Computational Stochastic Mechanics Conference, 21-23 June 2006, Rhodes, Greece.
TRAJECTORY TUBES IN CONTROL AND ESTIMATION PROBLEMS UNDER UNCERTAINTY T. F. Filippova Institute of Mathematics and Mechanics, Russian Academy of Sciences, 620219, Ekaterinburg, Russia, E-mail:
[email protected] Abstract:
The paper deals with the problems of control and state estimation for dynamical control systems described by differential equations with measure (or impulsive control) components. The problem is studied under uncertainty conditions with set-membership description of uncertain variables which are taken to be unknown but bounded with given bounds.
Key words:
Control, estimation, uncertainty, trajectory tubes, impulsive control, reachable set description.
1.
INTRODUCTION
The paper deals with the problems of control and state estimation for dynamical systems described by differential equations with measure or impulsive components. The problem is studied under uncertainty conditions with set-membership description of uncertain variables which are taken to be unknown but bounded with given bounds (e.g., the model may contain unpredictable errors without their statistical description). Models of this kind arise in a wide variety of applications ranging from space navigation to investment problems as well as ecological management. The solution to the impulsive differential system is introduced and studied here in the framework of the theory of uncertain dynamical systems through the techniques of trajectory tubes [1-3] (or set-valued state vectors) of the impulsive system: 55 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 55–64. © 2007 Springer.
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T. F. Filippova
dx(t ) = f (t, x(t ), u(t ))dt + B(t , x(t ), u(t ))dv(t ), x ∈ R n , t0 ≤ t ≤ T ,
(1)
with unknown but bounded initial condition x(t0 - 0) = x 0 , x 0 ∈ X 0.
(2)
Here u(t ) is a usual (measurable) control with constraint u(t ) ∈ U ⊂ R m, and v(t ) is an impulsive control function which is continuous from the right, with bounded variation on [t0 , T]. The current value of the state vector x(t ) is assumed to be restricted by a given map Y (t ) (the restriction of this kind may be generated e.g. by state observations with unknown but bounded disturbances). The trajectory tube to the system (1)-(2) is the set X [⋅] = { x[⋅] | x[t ] = x(t , t0 , x 0 ) | x 0 ∈ X 0}
(3)
of solutions to (1)-(2) with its t - cross-section X [t ] being the reachable set (the informational set) of the system (1)-(2) at instant t which is found under above given assumptions on uncertainty data and system constraints. The mathematical background for investigations of trajectory tubes ranging from theoretical schemes to numerical techniques may be found in [1]. In such problems the trajectories x[⋅] are discontinuous and belong to a space of functions with bounded variation. Among many problems related to treatment of dynamic systems of this kind let us mention the results devoted to a precise concept of a solution to (1) [4-5]. Based on the techniques of approximation of the discontinuous generalized trajectory tubes by the solutions of usual differential systems without measure terms [2-3] we study the properties of trajectory tubes and reachable sets of the impulsive control system under uncertainty. Numerical results using examples related to procedures of set-valued approximations of trajectory tubes and reachable sets are also presented. Another class of problems related to the addressed systems concerns the state estimation under state (viability) constraints. The redesign of the approach for impulsive control systems in the framework of the new solution concept is presented through the characterization of the reachable set as a level set of the value function regarded as a solution of related Hamilton – Jacobi – Bellman equations. The numerical simulation schemes developed for such problems require techniques of set-valued analysis, particularly its constructive methods – ellipsoidal or box-valued calculus. Based on the techniques of well-known ellipsoidal calculus we present a modified state estimation approach that uses the special structure of the studied impulsive control problem and is based on external ellipsoidal approximation of a convex union of ellipsoids. The examples of construction of such ellipsoidal
Trajectory tubes in control and estimation problems
57
external estimates of reachable sets of impulsive control systems are given also.
2.
NONLINEAR SYSTEMS
2.1 Trajectory Tubes to Impulsive Systems Based on the techniques of approximation of discontinuous generalized trajectory tubes to (1)-(2) by the solutions of usual differential systems without measures (impulsive controls) [2-3] it is possible to study the properties of generalized trajectory tubes and their cross-sections (reachable sets). Following the idea of [6] the information sets are treated here as level sets of the generalized solutions V(t,x) to the H-J-B (Hamilton – Jacobi – Bellman) equation, where V(t,x) is the value function of type V (t , x) = inf {φ (t0 , x[t0 ]) | x[i] = x(i, t0 , x 0 ), x[ i ] x[i] is a solution to (1) s.t. x[t ] = x }
(4)
with φ being a given function. For example, we may take φ (t,x) = φ (x) = d2(x,X0) with the initial set X0 defined in (2) and with d(x,M) being the distance function from x to M ⊂ R n . In this section for the simplicity we consider the case when B(t,x,u)=B(x,u) and f(t,x,u)=f(x,u). We assume also that the Lipschitz condition f ( x1, u) − f ( x2 , u) + B( x1, u) − B( x2 , u) ≤ L x1 − x2 , ∀u ∈U
is true and f ( x, u) + B( x, u) ≤ K (1+ x )
with some constants L,K>0. Assume also that the sets f ( x,U ) = U{ f ( x, u) | u ∈U }, B( x,U ) = U{B( x, u)l | u ∈U ,|| l ||≤ 1}
are convex. Let us introduce an auxiliary control system of type ⎧ x& (t ) = f ( x(t ), u(t )) + B( x(t ), u(t ))w(t ), ⎨ ⎩v&1(t ) =|| w(t ) ||,
(5)
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T. F. Filippova T
u(t ) ∈U , ∫ || w(t ) || dt ≤ µ ,
(6)
t0
with state variables x and v1 and control functions u(t), w(t).
Definition. A function x(i) with bounded variation and continuous from the right is called a generalized trajectory to (5)-(6) if there exist a function v1 also continuous from the right, with bounded variation, and a sequence of controls un(t),wn(t) for the system (5)-(6) such that the sequence of respective solutions {xn(t),v1n(t)} of (5)-(6) tends to {x(t),v1(t)} at every point to its continuity. The set of all such pairs {x(t),v1(t)} is a *-weak closure of the set of all classical solutions to (5)-(6). For all s ∈ [t0 , T+µ], y ∈ Rn , z, η ∈ R1 let us introduce the value function ∼
V (s, y, z,η ) = min{d 2 ( y(t0 ), X 0 ) + z 2 (t0 ) + η 2 (t0 )},
(7)
where the minimum is taken over all solutions {y( ⋅ ),z( ⋅ ), η ( ⋅ )} to the auxiliary control system: ⎧ y& (s) = α (s) f ( y(s), n(s)) + (1- α (s)) B( y(s), n(s))e(s), ⎪ ⎨ z&(s) = (1- α (s)) || e(s) ||, ⎪η& (s) = α (s), ⎩
(8)
with terminal conditions y(s)=y, z(s)=z, η(s)=η, and with ordinary (measurable) control functions α, n, e such that α ∈ [0,1], n ∈ U, ||e|| ≤ 1. The proof of the next theorem follows from the results of [4].
Theorem 1. The cross - section X [T] of the trajectory tube X [ ⋅ ] to the system (1)-(2) is a subset of the following set X [T ] ⊆ π y
U Lε (V% ),
(9)
0≤ε ≤µ
Lε (V% ) = {{ y, z,η}| V% (T + ε , y, z,η ) ≤ 0} .
(10)
Here πy M denotes the projection of a set M at the y-subspace,
π y M = { y | ∃z,η s.t. { y, z,η}∈ M } .
(11)
Trajectory tubes in control and estimation problems
59
It should be mentioned here the value function V% in optimization problem (7) may be found through the techniques of viscosity or minimax solutions to the corresponding H-J-B equation [3-6] ∂V% + max{α ∂V% f ( y, n) + (1 − α ) ∂V% B( y, n)e + (1 − α ) ∂V% || e || +α ∂V% ∂t ∂y ∂y ∂z ∂η , (12) {α ∈[0,1], n ∈U ,|| e ||≤ 1} = 0
with boundary condition ∼
V (t0 , y, z,η ) = d 2 ( y, X 0 ) + z 2 + η 2.
(13)
Theorem 1 gives a possibility to produce other upper estimates for the information sets X[t] through the comparison principle that allows connecting the given approach to the techniques of ellipsoidal or box-valued calculus developed for systems with linear structure [6-9].
2.2 Systems with State Constraints One of the principal points of interest of the theory of control under uncertainty conditions is to study the set of all solutions to (1)-(2) that satisfy a restriction on the state vector (the “viability” constraint) x[s]∈Y (s) , t0 ≤ s ≤ t ,
(14)
where Y( ⋅ ) is a convex compact valued multifunction. The viability constraint (14) may be induced by state constraints defined for a given plant model or by the so-called measurement equation y(t ) = G(t ) x + w,
where y is the measurement vector, G(t) - a matrix function, w - the unknown but bounded “noise”, w ∈ Q (t) ⊂ Rp. The problem consists in describing the set X [ ⋅ ] of solutions to the system (1)-(2), (14) which will be called as the viable trajectory tube or viability tube [1]. The Equation (14) may be expressed also as the constraint 0 ∈ G(s, x(s)) , t0 ≤ s ≤ t ,
(15)
with G being a given set-valued map. In this case we have to modify the value function (4) in the former problem as follows
60
T. F. Filippova t
V (t , x) = inf {φ (t0 , x[t0 ]) + ∫ d 2 (0, G(s, x(s))ds | x[i] = x(i, t0 , x 0 ), x[⋅] t0
(16)
x[i] is a solution to (1) s.t. x[t ] = x }
and also we have to introduce a modified function V% of type (7) (see details in [2-4]). It should be mentioned that the optimization problem (16) may be solved on the base of H-J-B approach and the following theorem gives the estimate of the viability tube.
Theorem 2. The cross - section X [T] of the viability tube X [ ⋅ ] to the system (1)-(2), (15) is a subset of the following set X [T ] ⊆ π y
3.
U Lε (V% ).
(17)
0≤ε ≤µ
LINEAR IMPULSIVE SYSTEMS WITH ELLIPSOIDAL CONSTRAINTS Let us consider the estimation problem for a linear control system dx(t ) = A(t )dt + B(t )du(t ), x(t0 - 0) = x 0 ∈ X 0 , t0 ≤ t ≤ T ,
(18)
with impulsive control u(t) restricted by a set U that will be defined further, X0 is convex and compact in Rn (in particular, X0 may be an ellipsoid in Rn ). Let E0 = { l ∈ R m | l ' Q0l ≤ 1 } be an ellipsoid in Rm with zero center defined by a symmetric positive definite matrix Q0 . Denote by Cm [t0 , T] the space of all continuous m-vector functions defined on [t0 , T] and denote also E = { y(⋅) ∈ С m[t0 , T ]| y(t ) ∈ E0 , ∀t ∈[t0 , T ]} . Let BVm[t0 , T] be the space of m-vector functions with bounded variation. Let us take U = E* with E* being the conjugate set to E (U=E* ⊂ BVm[t0 ,T] = (Cm [t0 , T])*) that is E * = {u(⋅) ∈ BV m[t0 , T ]|
∫
[ t0 , T ]
y′(t )du(t ) ≤ 1, ∀y(t ) ∈ E} .
Trajectory tubes in control and estimation problems
61
We assume in this section that admissible controls in (18) satisfy the restriction u(⋅) ∈ U=E*. Under this constraint from the structure of the set U it follows that jumps ∆u(ti ) = u(ti+1) − u (ti ) of any admissible control u(⋅) have to belong to the finite dimensional ellipsoid E 0* [9]. The following theorem describes the structure of the cross-section X [t] of trajectory tubes X [ ⋅ ] to the system (18) [9].
Theorem 3. The set X [t] is convex and compact for all t ∈ [t0 , T]. Any point x ∈ X [t] may be generated by a solution x[ ] = x( , t0 , x 0 ) with x[t] =x and with a corresponding admissible control u(⋅) which is piecewise constant and has only (n+1) jumps ∆u(ti ) belonging to E 0* .
.
.
The method of constructing the external (with respect to inclusion) estimates of trajectory tubes of a differential system with uncertainty is based on the ellipsoidal calculus [6-7] and on the new procedures of external approximation of a convex hull of the union of a variety of some ellipsoids [9]. Each of these ellipsoids coincides with the reachable set of the system (18) when only unique impulsive jump (at a prescribed instant t) of the admissible control function is allowed. The convex hull operation which we need to take additionally over the union of all these auxiliary ellipsoids in order to get the final reachable set X [t] is motivated by the above theorem 3. The following example shows how to find the reachable sets X [t] and their external ellipsoidal estimates. Example. Consider the following control system ⎧dx1(t ) = x2 (t )dt + du1(t ), ⎨ ⎩dx2 (t ) = du2 (t ).
(19)
Here we take X0 = {0}, t0 = 0 and assume that the set U is generated by the ellipsoid E 0 , ⎛ a2 0 ⎞ E0 = { l ∈ R 2 | l ' Q0l ≤ 1 } , Q0 = ⎜⎜ 2 ⎟⎟ , a, b > 0 ⎝0 b ⎠
Applying the results of theorem 1-3 we find the reachable set X [T] given at Figure 1.
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T. F. Filippova
Figure 1. The reachable set X [T] which coincides here with the convex hull of the union of two ellipsoids E1, E2.
Here we use the notation
x1* =
a2 a 2 + 0.25T 2b 2
< a, x2* =
E1 = { l ∈ R 2 |
a 2 + 0.5T 2b 2 a 2 + 0.25T 2b 2
> Tb,
(l1 - l2T ) 2 l2 2 + ≤ 1 }, a2 b2
E2 = { l ∈ R 2 | l ' Q0 −1l ≤ 1 }.
The tube of external ellipsoidal estimates of X [t] for t0 ≤ t ≤ T is shown at Figure 2.
Trajectory tubes in control and estimation problems
63
Figure 2. The external ellipsoidal estimates of the reachable set X [t] for t0 ≤ t ≤ T.
4.
CONCLUSIONS
We considered here the problems of control and state estimation for dynamical systems described by differential equations with measure or impulsive components. The problems were studied under uncertainty conditions with set-membership description of uncertain system disturbances. Based on the approximation approach of the discontinuous generalized trajectories by the solutions of usual differential systems without control measures we studied the properties of trajectory tubes and reachable sets of the impulsive control system under uncertainty. Numerical results related to procedures of set-valued approximations of trajectory tubes and reachable sets through the techniques of the ellipsoidal calculus were given.
ACKNOWLEDGEMENTS The research was supported by the Russian Foundation for Basic Researches (RFBR) under Projects 06-01-00483, 06-01-10807 and by the Program No. 15-F3 of the Presidium of the Russian Academy of Sciences.
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REFERENCES 1.
2.
3. 4. 5. 6. 7. 8. 9.
Kurzhanski AB, Filippova TF. “On the Theory of Trajectory Tubes - a Mathematical Formalism for Uncertain Dynamics, Viability and Control”, In: Kurzhanski AB eds. Advances in Nonlinear Dynamics and Control: a Report from Russia, Progress in Systems and Control Theory, Boston, Birkhauser, 1993, pp. 22-188. Filippova TF. “On the Generalized Solutions for Uncertain Systems with Applications to Optimal Control and Estimation Problems”, WSEAS Transactions on Systems, 4, pp. 481-486, 2005. Filippova TF. “Set-valued Solutions to Impulsive Differential Inclusions”, Mathematical and Computer Modelling of Dynamical Systems (MCMDS), 11, pp. 149-158, 2005. Pereira FL, Filippova TF. “On a Solution Concept to Impulsive Differential Systems”. Proc. of the 4th Math Tools Conference, S.- Petersburg, Russia, 2003. Vinter RB, Pereira FMFL. “A Maximum Principle for Optimal Processes with Discontinuous Trajectories”. SIAM J. Contr. And Optimization, 26, pp. 155-167, 1988. Kurzhanski AB, Valyi I. Ellipsoidal Calculus for Estimation and Control. Birkhauser, Boston, 1997. Chernousko FL. State Estimation for Dynamic Systems. Nauka, Moscow, 1988. Filippov AF. Differential Equations with Discontinuous Right-hand Side. Nauka, Moscow, 1985. Vzdornova OG., Filippova TF. “Estimates of Trajectory Tubes of Differential Systems of Impulsive Type”. Proc. of the 3th Conference on Mathematics, Informatics and Control. Irkutsk, Russia, pp. 1-12, 2004.
CELL MAPPING APPLIED TO RANDOM DYNAMICAL SYSTEMS A. Gaull, E. Kreuzer TUHH - Hamburg University of Technology, Institute of Mechanics and Ocean Engineering, 21071 Hamburg, Germany, URL: http://www.mum.tu-harburg.de/
Abstract:
The method of Cell Mapping is a numerical tool to analyze the long-term behavior of dynamical systems. For deterministic systems, the notion was first introduced by HSU and it is shown by GUDER AND KREUZER that Cell Mapping in this context represents an approximation of the Frobenius-Perron operator by a Galerkin method. Our purpose is to extend the concept to randomly perturbed dynamical systems or, more general, to Random Dynamical Systems. We show that time evolution of absolutely continuous measures and the corresponding densities can be described by Markov operators whose fixed points refer to stationary measures and densities, respectively. A projection of the Markov operator on densities onto a discrete basis set of characteristic functions leads directly to the reformulation of Cell Mapping against the background of stochastic dynamics.
Key words:
Cell Mapping, perturbations, Random Dynamical System, Markov operator, stationary measure, Galerkin projection.
1.
INTRODUCTION
In mathematical modeling we always have to deal with necessary simplifications: for technical dynamical systems neither there are equations that describe the systems completely nor accurate statements referring to the involved parameters can be given. If, in addition, the system’s properties are determined by strong nonlinearities, fluctuations according to parameters or states affect the dynamical system to a great extent. Analytical approaches to predict the behavior of nonlinear dynamical systems, which are perturbed in this manner, are in general doomed to failure. On the other hand, achieving 65 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 65–76. © 2007 Springer.
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success in computer experiments depends on the availability of appropriate numerical tools. Conventional solution methods for ordinary differential equations, which calculate trajectories via numerical integration starting from an arbitrary initial state, only allow, if at all, local propositions to the system's behavior. In the presence of strong nonlinearities, already small changes in the initial conditions may lead to considerable effects on the time evolution. Consequently, common integration techniques are not appropriate for predictions concerning the long-term behavior of dynamical systems represented by attractors. In view of this difficulty, much effort was in the past devoted to the search for methods which make it possible to overcome this deficiency and to end up in a characterization of asymptotic properties. Cell mapping is such a method, making allowance for the concept of dynamical flows in numerics. It is based on mapping (uncountable) subsets instead of mapping single points of the phase space and enables the identification of both the attractors’ geometric structure as well as the associated basins of attractions. In this contribution, we expand the concept of Cell Mapping to Random Dynamical Systems (RDS). We are able to consider perturbation of the system’s states as well as perturbations affecting the parameters. Because the technique of Cell Mapping inherently provides a stochastic approach to analyzing dynamical systems, it is applicable to problems of that kind very well. Long-term behavior of dynamical systems can be described by invariant subsets of the phase space, i.e. subsets which remain constant as a whole under the system’s mapping. A prominent class of invariant sets is given by attractors showing the special property of absorbing certain states of the phase space. Invariant subsets in turn are characterized by invariant measures, which reflect the probability of a set getting hit by a typical trajectory. Moreover, there is the concept of stationary measures, a notion which originally stems from stochastic analysis. As we will see, there is a close connection between invariant and stationary measures, tracing back to OHNO [11]. We will confine ourselves to investigating these stationary measures. It has proven fruitful for special RDS to regard them as Markov processes, whose associated semigroups can be applied to investigate time evolution of probability measures. We will characterize stationary measures as those measures which are fixed points of certain Markov operators. However, the latter are in general not accessible to the usual fixed point theorems, a fact which obviously complicates the problem. In either case, treating operators on measure spaces is rather challenging. We therefore simplify the task by only regarding those measures which are absolutely continuous and introduce Markov operators on densities, whose fixed points now represent densities of stationary measures. It must be mentioned that
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dynamical systems in general have an infinite number of invariant or stationary measures, but only those measures that exhibit support on a set with non-vanishing Lebesgue measure are physically relevant. In this context, we have to resort to numerical tools since analytic approaches are not feasible. Because of this, the above-mentioned Markov operators must be discretized. The phase space is subdivided into a finite number of subsets (“cells”) and time evolution of dynamical systems is described in terms of the partition obtained by this. All points within each cell are considered to be dynamically equivalent. We then gain a projection of the Markov operator on densities onto a discrete basis set of characteristic functions, each having support on a single cell. Based on this discretization approach we can approximately determine stationary densities and conjecture that these approximations converge to fixed points of the original Markov operator if the volume of the cells tends to zero. To prove this conjecture in general is, however, still an open problem today. The paper is organized as follows. In Section 2 we present the mathematical toolbox to treat RDS within the framework required henceforth. Then, we concentrate on the approximation of stationary measures in Section 3. In Section 4, we direct our attention to a concrete example of a dynamical system subjected to stochastic influences and analyze its stationary densities. We terminate this paper with conclusions in Section 5.
2.
MATHEMATICAL SET-UP
In this section we present the mathematical set-up to deal with dynamical systems under consideration in this paper. We confine ourselves mainly to giving the basic facts and results needed to develop the theory. Among others, we rely on the exposition of LIU [10]. For a comprehensive study we refer to the treatise of ARNOLD [1], which contains to a great extent a discussion of RDS in the context below.
2.1 Random Dynamical Systems In the following, let λ d be the Lebesgue measure on the d-dimensional Borel measure space (ℜd , B (ℜd )) and let E ⊂ R d be a connected compact subset with λ d ( E ) > 0 . We define Π := E ∩ B(ℜd ) and denote by H k ( E ) the space of C k transformations on ( E , Π ) , 0 ≤ k ≤ ∞ , endowed with the usual C k topology. Let M 1 be the set of probability measures on ( E , Π ) . The RDS studied in this contribution evolve in one-sided discrete time T = Z + starting at zero. Given a probability space (Ω, Ξ, Ρ) with Ω being a
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separable topological space whose topology is metrisable by a complete metric and Ξ := B (Ω) the Borel- σ -Algebra associated with it, we model the noise influencing our inherent deterministic system by the metric dynamical system (Ω, Ξ, Ρ,υ ) . That is to say, υ shall be a measurable transformation on (Ω, Ξ, Ρ) preserving Ρ . For our further considerations, we assume a measurable mapping ψ : Ω → H k ( E ) to be given. Therewith, we introduce a RDS over (Ω, Ξ, Ρ,υ ) by a measurable mapping S : T × Ω× E → E while putting S (0, ω , ⋅) := id E , where id E is the identity map on E , and further S (n, ω , ⋅) := ψ (υ n−1 (ω )) oKoψ (ω ) for n ≥ 1 . To simplify the notation, we set ψ 0 := id E and ψ n := ψ oυ n−1 (n ≥ 1) . Because of the invariance condition on υ , random elements ψ n are identically distributed. Taking this into consideration, we assume the probability law to be given by µ . With the concepts from above we can look at RDS from another point of view, utilizing the skew product of υ and S , viz, the measurable mapping Φ : T × Ω× E → Ω× E defined by Φ (n, ω , x) := Φ n (ω , x) := (υ n (ω ), S (n, ω , x)) . At this juncture, we are investigating motions in the extended phase space Ω× E instead of merely regarding motions in the phase space E. Henceforth we will denote a RDS depending on the particular situation by one of the pairs (υ , S ) and (υ , Φ ) .
Remarks: A) According to the definitions from above, randomness arises in our underlying deterministic dynamical system by “picking” at each time step n ≥ 1 a certain transformation h = ψ n (ω ) ∈ H k ( E ) with probability µ ({h}) and applying it to the space E . Clearly, the deterministic situation emerges from this, if µ is supported by a single h ∈ H k ( E ) , i.e. is given by the Dirac measure δ h . B) If the sequence of random elements (ψ n ) is independent then the related RDS yields a time homogeneous Markov process with state space E and a transition kernel p given for all points x ∈ E and subsets A ∈Π by p ( x, A) = µ{h | h( x) ∈ A} = Ρ{ω |ψ (ω ) ∈ A} . To be more precise, for any fixed x ∈ E let us define by ξ nx (ω ) := S (n, ω , x ) random elements ξ nx : Ω → E . Then the family (ξ nx ) represents a time homogeneous Markov process determined by the transition kernel above-quoted. Even more general, one can assume ξ 0x : Ω → E being any random element equipped with probability law ρ0 and being independent of the family (ψ n ) and define random elements ξ n : Ω → E by ξ n := S (n, ω , ξ 0 (ω )) , (n ≥ 1, ω ∈Ω) . The processes (ξ nx ) and (ξ n ) , respectively, are known as the one-point motions. From now on, RDS as introduced here are referred to as independent and identically distributed (i.i.d.) RDS.
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2.2 Generation of RDS from Differential Equations Apart from directly presetting random elements ψ and therewith deducing RDS as products of random mappings in the way described above, RDS can also be derived from Random Differential Equations (RDE) as well as from Stochastic Differential Equations (SDE). By this, we are able to bridge the gap to time continuous flows.
Random Differential Equations: Let θ : ℜ0+ × Ω → Ω be a measurable mapping such that all transformations θ t (⋅) := θ (t , ⋅) on (Ω, Ξ) are invariant with respect to Ρ and possess the semiflow property, that is, θO = id Ω and θt +s = θ s o θt ( s, t ∈ℜ0+ ) . Furthermore, we will assume the mapping θ to be absolutely continuous with respect to t. Our starting point is given by RDE x& (t ) = f (θt (ω ), x(t )) together with an initial condition x(0) = x . In addition to (deterministic) nonautonomous Ordinary Differential Equations (ODE), the right hand side of this RDE depends on the “parameter” ω which varies depending on θ , and for that reason, we say the RDE is driven by θ . For each fixed ω ∈Ω , accordingly, RDE can be understood as nonautonomous ODE and therefore solutions of them can be represented in terms of a cocycle ϕ : ℜ0+ × Ω → H k ( E ) given implicitly by equation t
ϕ (t , ω ) x = x + ∫ 0 f (θ s (ω ),ϕ ( s, ω ) x)ds ,
(1)
such that in particular we have ϕ (0, ω ) = id E .
Stochastic Differential Equations: Let (Ω B , Ξ B , PB ,( Bt )t∈R+ ) with 0 Bt := ( Bt1 ,..., Btd ) for all t ∈ℜ0+ be the canonical d–dimensional Brownian motion, that is, we take Ω B := {ω ∈ C 0 (ℜ0+ , E )|ω (0) = 0} equipped with the compact open topology. Moreover, Ξ B is the Borel- σ -algebra on Ω B and PB the so-called Wiener measure providing (Ω B , Ξ B , PB ,( Bt )t∈R+ ) to be a 0 Brownian motion with values in E. By (θt (ω )) s := ωt +s − ωt ( s, t ∈ℜ0+ ) random elements θt : Ω B → Ω B are defined in such way that all θ t are PB – invariant. Consequently, (Ω B , Ξ B , PB ,θ ) is the canonical measure-preserving dynamical system modeling a d – dimensional Brownian motion. Given a C k vector field X := ( X i )id=1 : E → E , where k has to be sufficiently large, we consider Stratonovich’s initial value problem dYt = X 0 (Yt )dt + ∑ i=1 X i (Yt ) • dBti , Y0 = x. d
(2)
Solutions of this problem are conceived as a stochastic semiflow
ϕ : ℜ0+ × Ω B → H k ( E ) that complies with the Itô formula
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t
f (ϕ (t , ω )) = f (ω ) + ∫ ( X 0 f )(ϕ ( s, ω )) ds + ∑ i=1 ∫ ( X i f )(ϕ ( s, ω )) • dBsi (3) d
0
0
for any C k map f : H k ( E ) → ℜ and t ∈ℜ0+ . Again, ϕ (0, ω ) = id E . Completely analogue to the i.i.d. case of a RDS, defining random elements ξt : Ω → E by ξt (ω ) := S (t , ω , ξ0 (ω )) (t ∈ℜ+ , ω ∈Ω) , where ξ0 : Ω → E is an appropriate measurable function, leads to a time homogeneous Markov process (ξt )t∈ℜ+ with state space E and transition kernel p given by 0
p (t , x, A) = P{ω ∈Ω | S (t , ω , x) ∈ A}, (t ∈ℜ0+ , x ∈Ε, A ∈ E ) .
(4)
Time discretization: Once we have derived the mapping ϕ in terms of solutions to a RDE or a SDE, we are able to deduce a RDS from it. By means of setting υ := θT for any fixed T ∈ℜ+ and choosing ψ := ϕ (T , ⋅) afterwards, we gain a RDS (υ ,ψ ) . This procedure apparently amounts to a time discretization of ϕ . Of course, this approach is reasonable solely if conclusions from the behavior of our RDS can be drawn back to solutions of the original RDE and SDE, respectively. Henceforth we will always act under this assumption.
2.3 Invariant and Stationary Measures We are now able to establish the notion of invariant measures. Given a RDS (υ ,ψ ) , we say a probability measure µ on (Ω× E , Ξ ⊗ Π ) is (υ ,ψ ) – invariant, if it shows the following properties: µ o Φ1−1 = µ , and µ o π Ω−1 = P with π Ω : Ω× E → Ω, π Ω (ω , x) := ω , being the canonical projection from Ω× E onto Ω. In case of RDS which can be described via stochastic kernels, there is also the term of stationary measures. A measure ρ ∈ M 1 is called stationary in case
ρ ( A) = ∫ p ( x, A) d ρ ( x), ( A ∈Π ) .
(5)
However, there is a close connection between invariant and stationary measures first recognized by OHNO [11]. Namely, a measure ρ ∈ M 1 is stationary if and only if the product measure µ = P ⊗ ρ on Ξ ⊗ Π is (υ ,ψ ) – invariant. The meaning of equation is then obvious: a measure ρ ∈ M 1 is stationary if it is (υ ,ψ ) – invariant on the average [1].
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APPROXIMATION OF STATIONARY MEASURES
3.1 Markov Operators In the following, we will restrict ourselves to those RDS (υ ,ψ ) which are derived from a Markov semigroup ( pn ) n∈T of stochastic kernels pn : E × Π → [0, 1] , that are defined by p1 ( x, A) := p ( x, A) and for all n ≥ 2 , pn ( x, A) := ∫ pn−1 ( y, A) p( x, λ d (dy )) . The Markov process corresponding to this semigroup is henceforth denoted by (ξ n ). Let ρ n := ξ n P for all n ∈T be the law of the random element ξ n such that ρ n ( A) reflects the probability of the dynamical system hitting a set ˆ : M 1 → M 1 given by A∈Π at time n . We introduce a Markov operator ℘ ˆ ρ ( A) := ∫ p( x, A)d ρ ( x), ( A∈Π ) . ℘
(6)
ˆ ρ n so that ℘ ˆ describes the time evolution As a result, we obtain ρ n+1 =℘ of measures ρ ∈ M 1 . Stationary measures correspond to fixed points of the ˆ ρ = ρ , or ρ ( A) = ˆ , i.e. measures ρ that satisfy ℘ Markov operator ℘ −1 ∫ ρ (ψ (ω ) ( A))dP(ω ) , ( A∈Π) . If we restrict ourselves to the special case of all ρ n being continuous with respect to the Lebesgue measure λ d on B(ℜd ) , then the well-known theorem of RADON–NIKODYM guarantees the existence of a density f n ∈ L1 with ρ n = f n λ d . We can establish a Markov operator ℘: L1 → L1 acting on densities given by ℘f n := f n+1 for all n ∈T . As a result, stationary measures ρ = f λ d correspond to fixed points of ℘ in terms of the relation ˆ ( f λ d ) = f λ d ⇔℘f = f . ℘ We consider the following special cases: A) If we assume p to have a bounded density q ∈ L∞ ( E × E , R0+ ) , i.e. p ( x, ⋅) shall be absolutely continuous λ d –a.e., then ℘ is explicitly declared via ℘f ( y ) = ∫ q ( x, y ) f ( x) d λ d ( x), ( y ∈ E ) .
B) The deterministic situation: Let µ = δ h with h ∈ H k being nonsingular with respect to λ d , i.e. λd h −1 0 . We conjecture the existence of such a density, although the proof of this is still an open problem so far.
Figure 1. Box covering of the preimage of I i under S1
4.
NUMERICAL EXAMPLE
In this section we illustrate the theory developed so far by a numerical example. The deterministic Duffing–van der Pol oscillator has become a exemplar for mathematicians as well as for engineers. We will briefly study this oscillator perturbed by white noise. We consider the equation
&& y (t ) = (α + σ 1ξ1 (t )) y (t ) + ( β + σ 2ξ 2 (t )) y& (t ) − y 3 (t ) − y 2 (t ) y& (t ) + σ 3ξ3 (t ) (14)
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where σ1, σ2 and σ3 are weighting parameters for the white noise processes ξ1, ξ 2 and ξ 3, respectively. The parameters assumed to be constant are given by: α = −1.0 , σ 1 = 0.0 , σ 3 = 0.0 . We will analyze qualitatively the shape of stationary densities of the noisy Duffing–van der Pol oscillator when parameters β and σ 2 are varied and thereby concentrate on the so-called phenomenological approach. Of course, β is the expectation of the process and if σ 2 = 0 , then we obtain the deterministic Duffing–van der Pol oscillator. In Figure 2, stationary densities are presented for three values of β leading to distinct regimes. The first row of the figure shows the deterministic situation whereas in the second row densities associated with the case of a multiplicatively perturbed oscillator are displayed. What we see here is, indeed, a survey of the characteristic states in the stochastic Hopf bifurcation regime.
Figure 2. Stationary densities of the Duffing-van der Pol oscillator for σ 2 = 0.0, 0.25 (rows) and β = -0.2 , −0.1 , 1.0 (columns)
5.
CONCLUSIONS
The Cell Mapping method is well suited for analyzing the global behavior of perturbed systems. Instead of studying the nonlinear system’s mapping, we can concentrate on linear operators such that results and techniques from mathematical analysis can be applied. Admittedly, convergence of the sequence f M , N to a stationary density f which is useful for practical
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purpose is still the tender point. It is not ensured that the sequence converges to a density which is “physically relevant”. The effort of numerical simulation rises significantly, but not exponentially with the complexity of the extended phase space. This important characteristic is ensured by an adaptive algorithm concerning the refinement of the cell partitions.
ACKNOWLEDGEMENTS The authors are indebted to the DFG (Deutsche Forschungsgesellschaft/ German Research Foundation) for funding the project under contract Kr 752/22-2.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
Arnold L. Random Dynamical Systems, Berlin, Springer, 1998. Dellnitz J, Hohmann A. “A Subdivision Algorithm for the Computation of Unstable Manifolds and Global Attractors”, Numerische Mathematik, 75, pp. 293-317, 1997. Dellnitz J, Junge O. “On the Approximation of Complicated Dynamical Behavior”, SIAM Journ. on Numer. Anal., 36, pp. 491-515, 1999. Guder R, Kreuzer E. “Generalized Cell Mapping to Approximate Invariant Measures on Compact Manifolds”, Intern. Journ. of Bifurc. and Chaos, 7, pp. 2787-2499, 1997. Hsu CS. “A theory of cell-to-cell mapping dynamical systems”, ASME Trans. J. Dyna. Syst. Meas. Control, 47, pp. 931-939, 1980. Hsu CS. “Generalized theory of cell-to-cell mapping for nonlinear dynamical systems”, Joumal of Applied. Mechanics, 48, pp. 834-842, 1981. Junge O. Mengenorientierte Methoden zur numerischen Analyse dynamischer Systeme, Paderborn: Universität, Fachbereich Mathematik/Informatik, Diss., 1999. Kifer Y. Ergodic Theory of Random Transformations, Boston, Birkhäuser, 1986. Kifer Y. Random Perturbations of Dynamical Systems, Boston, Birkhäuser, 1988. Liu PD. “A Dynamics of Random Transformations: Smooth Ergodic Theory”, Ergod. Th. & Dynam. Sys., 21, pp. 1279-1319, 2001. Ohno T. “Asymptotic Behaviors of Dynamical Systems with Random Parameters”, Publ. RIMS Kyoto Univ., 19, pp. 83-98, 1983. Rechard OW. “Invariant measures for many-one transformations”, Duke Math. J., 23, pp. 477-488, 1956. Schenk-Hoppé KR. “Bifurcation Scenarios of the Noisy Duffing-van der Pol Oscillator”, Nonl. Dynamics, 11, pp. 255-274, 1996. Schenk-Hoppé KR. The stochastic Duffing-van der Pol equation, Bremen, Universität, Fachbereich Mathematik/Informatik, Dissertation, 1996. Ulam S. A Collection of Mathematical Problems, New York, Inters. Publ., 1960.
NUMERICAL ANALYSIS OF BIFURCATION AND CHAOS RESPONSE IN A CRACKED ROTOR SYSTEM UNDER WHITE NOISE DISTURBANCE X. L. Leng Institute of Vibration Engineering Research, College of Aerospace Engineering, Nanjing Univ. of Aeronautics and Astronautics, Nanjing, 210016, P.R. China, E-mail:
[email protected] Abstract:
The Monte-Carlo method is used to investigate the bifurcation and chaos characteristics of a cracked rotor with a white noise process as its random disturbance. Special attention is paid to the influence of the stiffness change ratio and the rotating speed ratio on the bifurcation and chaos response of the system. Numerical simulations show that the effect of the random disturbance is significant as the undisturbed response of the cracked rotor system is a quasi-periodic or chaos one, and such effect is smaller as the undisturbed response is a periodic one.
Key words:
Random vibration, cracked rotor, bifurcation, chaos.
1.
INTRODUCTION
The dynamic behavior of a cracked rotor is of great importance to rotor crack detection. Up till now, many characteristics of a cracked rotor have been revealed, for example, unstable speed range near rotating speedω=2ωc /N, where ωc is the first pin-pin critical and N=1,2,3,4 [1]; large influence of crack position on system’s dynamic response [2]; and variation of response amplitude and phase angle with crack [3]. But all of these conclusions were based on the linear crack model, and weight dominance was assumed in 77 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 77–86. © 2007 Springer.
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almost all the analyses of horizontal cracked rotor and the influence of whirl speed on the closing and opening of crack was omitted．Meng and Gasch [4] analyzed the nonlinear influence of whirl speed. The angle between the crack center line and the line connecting the bearing and shaft center was used for determining the closing and opening of the crack, so the crack model is nonlinear and includes the cases with and without weight dominance, synchronous and non-synchronous responses. This nonlinear crack model is used in this paper, and the system’s equations of motion are nonlinear ones with time-varying coefficients. Random disturbance is often encountered in some rotor machines, such as the electric generators service in seismic zone, and the power-generating machine of an oceangoing ship. Thus, there is a need for analyzing the response problem of a rotor system subject to random disturbances. Zhao and Lin [5] used direct integration scheme in the analysis of the rotor bearing systems subjected to random earthquake excitation. But in their research, the rotor model is assumed to be a linear one, and the crack’s influence was not taken into account. In this paper, the rotor system mentioned above with a white noise process as its random disturbance is investigated numerically. And our particular focus is on the effect of the random disturbance on bifurcation and chaos character of the system, which will likely be utilized in the future fault diagnosing of rotating machinery．
2.
DYNAMICAL MODEL OF THE SYSTEM
For the simple Jeffcott rotor with a transverse crack subject to a random disturbance, taking whirl speed ωr into account, supposing the cross stiffness change ratio caused by rotor crack to be zero (Figure 1), and using the nonlinear crack model derived by Meng and Gasch [4], the non-dimensional equations of motion can be written as follow sin 2Φ ⎤ ⎧ X ⎫ ⎧ X&& ⎫ 2 De ⎧ X& ⎫ 1 ⎧ X ⎫ ∆K ⋅ f (Ψ ) ⎡1 + cos 2Φ ⎨ && ⎬ + ⎨ & ⎬+ 2 ⎨ ⎬− ⎨ ⎬ ⎢ 2 1 − cos 2Φ ⎥⎦ ⎩ Y ⎭ 2Ω ⎣ sin 2Φ ⎩Y ⎭ Ω ⎩Y ⎭ Ω ⎩Y ⎭ ⎧ 1 ⎫ ⎧cosτ ⎫ ⎧1 ⎫ ⎪ ⎪ = ⎨ Ω2 ⎬ + U ⎨ (1) ⎬ + σγ (t ) ⎨ ⎬ ⎩ sin τ ⎭ ⎩0 ⎭ ⎪⎩ 0 ⎪⎭
A cracked rotor system under white noise disturbance
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Ob heavy disk m
Y
η ks 2
ω rt + φr
c crack
ξ unbalance
mg
X
ψ (t ) β
σγ (t ) ωt + φ0
Figure 1. Schematic diagram of a nonlinear cracked rotor
where Φ = τ + β + ϕ0 Y Ψ = Φ − arctan( ) X 1 2 2 2 f (Ψ ) = + cos Ψ − cos3Ψ + cos5Ψ 2 π 3π 5π
Suppose γ (t ) acts as the random disturbance of the system, which is assumed to be a standard Gaussian white noise; and the positive constant σ stands for the noise level. Equation (1) is a nonlinear one with time varying coefficients, disturbed with a white noise process. In this paper, we take Ω and ∆K as varying parameters, and use a four-step Runge-Kutta method to integrate Equation (1). To illustrate the numerical result, tools such as orbit diagram, Poincare map and bifurcation diagram are used. Monte Carlo method can be used in numerical analysis of dynamics response of non-linear structure subject to random excitations. In its approach, a random process can be simulated as a series of cosine function with weighted amplitudes and random phase angles. This approach was presented by Shinozuka in 1970’s firstly [6, 7]. With a few simple modifications, similar approaches can be used in the numerical analysis of (a) wind-induced ocean wave elevation, (b) spatial random variation of material properties, (c) random surface roughness of highways and airport runways. In this paper, we suppose that γ (t ) is a standard white noise process, which can be simulated as
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γ (t ) = 2
S ⋅ ω0 N ω cos[ k 0 t + θ k ] ∑ N k =1 N
(2)
Here, N is a large integer number, ω0 is the truncation frequency in the simulation of γ (t ) , θ k (k = 1, 2,L , N ) are mutually independent random variables distributed between 0 and 2π , and s = 1.0 presents spectral density of a standard white noise process, γ (t ) . If θ k take a series of deterministic values, namely θ%k , where θ%k ∈ [0, 2π ] , the corresponding γ% (t ) obtained from Equation (2) will be a sample process of the white noise process γ (t ) . The process simulated in this method will be an ergodic one as N → ∞ , and the simulated spectral density converges as 1/ N in the mean square sense to the target spectral density [6,7]. In the following simulation, N and ω0 are taken as 500 and 25 respectively.
3.
RESULT AND DISCUSSION
As the rotor crack will decrease the rotor stiffness, the pin-pin critical speed of cracked rotor is smaller than that of the uncracked rotor. Therefore, the critical speed ratio Ωc < 1 for cracked rotor，and Ωc = 1 for uncracked system. Most of the response of the rotor system without random disturbance is omitted here to economize the paper, while some related results are presented [8, 9]. Figure 2 shows the influence of ∆K on the response of the rotor system subject to a white noise process, when β＝ 0, U ＝ 0.1 and Ω is near to 2/3 Ωc ( Ω ＝0.6). From the bifurcation diagram, Figure 2(a), we can see that the bifurcation curves just turn a little thicker; and the bifurcation value of parameter ∆K is almost invariant, compared to the curves while the rotor system is not disturbed. But as we can see in Figure 2(b), the bifurcation curves are thicker than those in Figure 2(a). Moreover, the bifurcation process cannot be distinguished clearly when 0.534 + < x1 f 4 > = 0, for Ψ = x2 z&1 : < x&2 z&1 > + < x2 f 3 > = 0, for Ψ = x1 z&1 : < x&1 z&1 > + < x1 f 3 > = 0.
(18) (19) (20)
After some calculations, from (18-20), one obtains 3 equations 96b23c14 r14 + 384b23c12 c22 r12 r22 + 96b23c24 r24 + 240b23c16 r16α1 + 2160b23c14 c22 r14 r22α1 +2160b23c12 c24 r12 r24α1 + 240b23c26 r26α1 + 210b23c18 r18α12 + 3360b23c16 c22 r16 r22α12 + 75 60b23c14 c24 r14 r24α12 + 3360b23c12 c26 r12 r26α12 + 210b23c28 r28α12 + 63b23c110 r110α13 + 1575 b23c18 c22 r18 r22α13 + 6300b23c16 c24 r16 r24α13 + 6300b23c14 c26 r14 r26α13 + 1575b23c12 c28 r12 r28α13 10 3 4 4 2 2 2 2 2 2 2 2 +63b23c10 2 r2 α1 + 32b21 (3c1 r1 α1 + 4c1 r1 (1 + 3c2 r2 α1 ) + c2 r2 (4 + 3c2 r2 α1 )) +32b22 (3c14 r1α 2 + c22 r2 (4 + 3c22α 2 ) + c12 (6c22 r2α 2 + r1 (4 + 6c22α 2 ))) + 128c12 r1ν 12 +96c14 r1α 2ν 12 + 192c12 c22 r1α 2ν 12 + 128c22 r2ν 22 + 192c12 c22 r2α 2ν 22 + 96c24 r2α 2ν 22 = 0 (21) 4 3 2 2 2 2 2 2 4 3 6 5 96b13c1 r1 + 192b13c1 c2 r1 r2 + 192b13c1 c2 r1r2 + 96b13c2 r2 + 240b13c1 r1 α1 + 720b13c16c22 r14 r2α1 + 1440b13c14c22 r13r22α1 + 1440b13c12c24 r12 r23α1 + 720b13c12c24 r1r24α1 + 240b13c26 r25α1 +210b13c18 r17α12 + 840b13c16c22 r16 r2α12 + 2520b13c16 c22 r15 r22α12 + 3780b13c14c24 r14 r23α12 + 37 80b13c14c24 r13r24α12 + 2520b13c12c26 r12 r25α12 + 840b13c12c26 r1r26α12 + 210b13c28 r27α12 + 63b13c110 r19α13 + 315b13c18c22 r18 r2α13 + 1260b13c18c22 r17 r22α13 + 2520b13c16c24 r16 r23α13 + 3780b13c16c24 r15 r24α13 + 3780b13c14c26 r14 r25α13 + 2520b13c14c26 r13r26α13 + 1260b13c12c28 r12 r27α13 + 315b13c12c28 r1 9 3 4 3 2 2 2 2 2 r28α13 + 63b13c10 2 r2 α1 + 32b11 (3c1 r1 α1 + c2 r2 (4 + 3c2 r2 α1 ) + 2c1 r1 (2 + 3c2 r2 (r1 + r2 ) 4 2 2 2 2 2 2 α1 )) + 32b12 (3c1 α 2 + 4c1 (1 + 3c2α 2 ) + c2 (4 + 3c2α 2 )) + 128c1 r1ν1 + 96c14 r13α1ν 12 +192c12c24 r1r22α1ν12 + 128c22 r2ν 22 + 192c12c22 r12 r2α1ν 22 + 96c24 r23α1ν 22 = 0 (22)
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96b13c14 r14 + 384b13c12 c22 r12 r22 + 96b13c24 r24 + 240b13c16 r16α1 + 2160b13c14 c22 r14 r22α1 +2160b13c12 c24 r12 r24α1 + 240b13c26 r26α1 + 210b13c18 r18α12 + 3360b13c16 c22 r16 r22α12 + 75 60b13c14 c24 r14 r24α12 + 3360b13c12 c26 r12 r26α12 + 210b13c28 r28α12 + 63b13c110 r110α13 + 1575 b13c18 c22 r18 r22α13 + 6300b13c16 c24 r16 r24α13 + 6300b13c14 c26 r14 r26α13 + 1575b13c12 c28 r12 r28α13 10 3 4 4 2 2 2 2 2 2 2 2 +63b13c10 2 r2 α1 + 32b11 (3c1 r1 α1 + 4c1 r1 (1 + 3c2 r2 α1 ) + c2 r2 (4 + 3c2 r2 α1 )) +32b12 (3c14 r1α 2 + c22 r2 (4 + 3c22α 2 ) + c12 (6c22 r2α 2 + r1 (4 + 6c22α 2 ))) + 128c12 r12ν 12 +96c14 r14α1ν 12 + 192c12 c22 r12α1ν 12 + 128c22 r22ν 22 + 192c12 c22 r12 r22α1ν 22 + 96c24 r24α1ν 22 = 0 (23)
for the three unknowns ε , α1 ,α 2 where c1 (ε ), c2 (ε ),ν 1 (ε ),ν 2 (ε ). The system (21-23) can be solved numerically as follows: 1) Step 1: let β = 0 (or k3 = β m1 = 0 ). Then, the system is linear, one obtains ε = α1 = α 2 = 0 , in other words, z1 (t ) ≡ x1 (t ); z2 (t ) ≡ x2 (t ) . 2) Step 2: let β = 0.1 (or k3 = 8), the system is weak nonlinear, ε , α1 ,α 2 are found in neighborhood of an initial condition (0,0,0); 3) Step 3: let β = 1 (or k3 = 80), ε , α1 ,α 2 are found in neighborhood of the initial condition ( ε 0 ,α10 ,α 20 ), which are determined in the step 2. 4) To do the same, step 3 will be applied for any other greater value up to the given one of β (or k3). 2a
DISPLACEMENT OF THE FIRST BODY
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Figure 2. Graphs of the free oscillation of the suspension system, with k3 = 8000N/m3 DISPLACEMENT OF THE FIRST BODY
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Figure 3. Graphs of the free oscillation of the suspension system, with k3 = 20000N/m3
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This procedure is established into a loop for software like Mathematica with “FindRoot” statement, or Matlab or Maple with “fsolve” statement, respectively. Once ε is found, the linear system (9) is determined, the calculations from (12-14) allow us to find out, first ν 1 ,ν 2 , next r1 , r2 then c1 , c2 ,θ1 ,θ 2 and then the responses of the linear system x1(t), x2(t). As a result, the solution Z P (t ) = [ z1 (t ), z2 (t ) ] T with the proposed method can be
obtained from (15). This response Z P (t ) is compared with a numerical simulation Z S (t ) in the Table 1 for m1 = 80kg , m2 = 1200kg , k1 = 30000 N/ m , k2 = 320000 N/m and different values of k3. The graphs obtained by the proposed method and by the numerical simulation technique are presented in the Figures 2. (a – b), Figures 3. (a–b). Table 1. Free oscillation amplitude
N0 1 2 3 4 5
k3 80 4000 8000 16000 20000
z1S max 1.9039 0.7798 0.7273 0.6719 0.6692
z1Pmax 1.9170 0.7945 0.7256 0.6465 0.6201
error 0.68% 1.88% -0.24% -3.77% -7.34%
z2S max 1.0294 0.5369 0.5513 0.5755 0.5825
z2Pmax 1.0217 0.5609 0.5728 0.5907 0.5959
error -0.75% 4.46% 3.90% 2.65% 2.30%
It can be seen from Table 1 that the proposed method can give the oscillation amplitudes with good accuracy in comparison with the simulation. Figures 2–3 show that the frequencies obtained by the two methods coincide in the free vibration for the system with arbitrary values of the nonlinearity.
3.2 Forced oscillation – nonlinear suspension system Consider the forced periodic vibration of the suspension system shown in Figure 1, which is governed by a differential equation system as follows && z1 = −ω12 (1 + µ ) z1 + ω22 z2 − β (1 + µ ) z13 − ζ (1 + µ ) z&1 − p cosν t , && z2 = ω12 µ z1 − ω22 z2 + βµ z13 + ζµ z&1 + p cosν t ,
(24)
ζ = d1 / m1 , p = k2 (m1 + m2 ) / 2m2 .
(25)
where
Now, for system (24), the corresponding linear one is considered && x1 = −ω12 (1 + µ ) x1 + ω22 x2 − ζ (1 + µ ) x&1 − p cosν t , && x2 = ω12 µ x1 − ω22 x2 + ζµ x&1 + p cosν t.
(26)
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Firstly, the forced vibration of system (26) can be found in a form x1 = c1 cosν t + s1 sinν t , x2 = c2 cosν t + s2 sinν t.
(27)
Substitution of (27) into (26), yields the following algebraic equations
ω12 s1 + µω12 s1 − ζν c1 − µζν c1 − ν 2 s1 − ω22 s2 = 0, ω22 s2 − µζν c1 − ν 2 s2 − µω12 s1 = 0, ζν s1 + µζν s1 − ν 2 c1 + ω12 c1 + µω12 c1 − ω22 s2 + p = 0, ω22 c2 − µζν s1 − ν 2 c2 − µω12 c1 − p = 0.
(28)
Four unknowns c1 , c2 , s1 , s2 can be found from (28). Then, according to (27), the solutions of the linear system x1(t) and x2(t) are determined. Now, one establishes the response of the nonlinear system (24) in the form z1 (t ) = x1 (t ) + α1 x13 (t ) = ( c1 cosν t + s1 sinν t ) + α1 ( c1 cosν t + s1 sinν t ) , 3 (29) z2 (t ) = x2 (t ) + α 2 x23 (t ) = ( c2 cosν t + s2 sinν t ) + α 2 ( c2 cosν t + s2 sinν t ) , 3
where, α1 ,α 2 are to be found from the GEA equation (16), with f1 = −ω12 (1 + µ ) x1 + ω22 x2 − ζ (1 + µ ) x&1 − p cosν t , f 2 = ω12 µ x1 − ω22 x2 + ζµ x&1 + p cosν t , f 3 = −ω12 (1 + µ ) z1 + ω22 z2 − β (1 + µ ) z13 − ζ (1 + µ ) z&1 − p cosν t , f 4 = ω12 µ z1 − ω22 z2 + βµ z13 + ζµ z&1 + p cosν t.
(30)
In order to get the lowest unknown moments we take the lowest polynomial functions Ψ ( X , Z ) and from (16), one gets: for Ψ = x2 z&1 < x&2 z&1 > + < x2 f 3 > = 0, for Ψ = x1 z&2 < x&1 z&2 > + < x1 f 4 > = 0.
(31) (32)
Substitution of (27), (29), (30) into (31), (32), after some calculations, one obtains two equations for two unknowns α1 ,α 2 , as follows: 3 p 4 (21 p 6α13 β (1 + µ )ν 18 (ν 2 − ω12 ) + 70 p 4α12 β (1 + µ )ν 14 (ν 2 − ω12 )(ζ 2 ((1 + µ )ν 3 − νω22 ) 2 + (ν 4 + ω12ω22 − ν 2 ((1 + µ )ω12 + ω22 )) 2 ) + 80 p 2α1β (1 + µ )ν 10 (ν 2 − ω12 )(ζ 2 ((1 + µ )ν 3 − νω22 ) 2 + (ν 4 + ω12ω22 − ν 2 ((1 + µ ) (33) ω12 + ω22 )) 2 ) 2 + 32( β (1 + µ )ν 6 (ν 2 − ω12 ) − α1ν 6 (ζ 2 (1 + µ )ν 2 + (ν 2 −ω12 )(ν 2 − (1 + µ )ω12 )) + α 2 (ζ 2ν 2 + (ν 2 − ω12 ) 2 ) 2 ω22 )(ζ 2 ((1 + µ )ν 3 −νω22 ) 2 + (ν 4 + ω12ω22 − ν 2 ((1 + µ )ω12 + ω22 )) 2 )3 ) = 0
3 p 4ν 2 (21 p 6α13 βµν 18 + 70 p 4α12 βµν 14 (ζ 2 ((1 + µ )ν 3 −νω22 ) 2 + (ν 4 +ω12ω22 − ν 2 ((1 + µ )ω12 + ω22 )) 2 ) + 80 p 2α1βµν 10 (ζ 2 ((1 + µ )ν 3 − νω22 ) 2 (34) + (ν 4 + ω12ω22 − ν 2 ((1 + µ )ω12 + ω22 )) 2 ) 2 − 32(− µν 6 ( β + α1ω12 ) + α 2 2 2 2 2 2 2 2 2 2 2 3 2 2 4 (ν − ω1 )(ζ ν + (ν − ω1 ) )(ν − ω2 ))(ζ ((1 + µ )ν − νω2 ) + (ν +ω12ω22 − ν 2 ((1 + µ )ω12 + ω22 )) 2 )3 ) = 0
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The amplitude of Z P (t ) and the maximum of a numerical simulation response Z S (t ) in stability stage are compared in Table 2 for m1 = 80kg , m2 = 1200kg , k1 = 30000 N/ m , k2 = 320000 N/m , d 1= 4800Ns/m, ν = 2π 1/s and different values of k3. Table 2. Forced oscillation amplitude
N0 1 2 3 4 5
k3 80 800 4000 8000 16000
z1S max 5.5717 4.8015 3.3624 3.0027 2.5791
z1Pmax 5.6994 4.5172 3.2369 2.9031 2.6512
error 2.24% -5.92% -3.73% -3.32% 2.79%
z2S max 75.6614 75.9515 76.3699 74.5263 73.3357
z2Pmax 76.0461 76.0369 76.0249 76.0192 76.0135
error 0.51% 0.11% -0.45% 2.01% 3.65%
It is seen from Table 2 that the proposed method can give results with very high accuracy for arbitrary values of nonlinearity.
4.
CONCLUSIONS
The method introduces the so-called GEA equation involving the variables of the original nonlinear and of the corresponding linear systems. Furthermore, the method presents a periodic solution of nonlinear systems by a polynomial of harmonic solution of its corresponding linear systems. Thus, a reliable way to determine the solution polynomial coefficients and the linear system can be derived. The technique is quite simple since it can use properties of harmonic functions and yields to a system of algebraic equations. The calculations, however, are more complicated than by the standard averaging method since the number of unknowns is more. The proposed method can be applied to both stochastic oscillations and deterministic ones. The GEA equation is established not using the assumption of small nonlinearity, thus, it can be applied to arbitrarily nonlinear systems. The corresponding linear system is essential for this methodology and should have expected properties of the nonlinear system, for example, if we are interested in periodic solutions of the nonlinear system then the corresponding linear system should be taken in the form that can possess them too. The term “corresponding” means the nonlinear and linear systems have the same dimension. The numerical results give good approximate solutions for the system within the quite large range of values of the nonlinearity. However, the technique should be tested for other nonlinear systems, for example, for systems with unsymmetrical properties. Some related questions on the set of GEA equations and the form of polynomials to be chosen to get a better approximate solution have to be discussed.
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ACKNOWLEDGEMENTS This research has been sponsored in part by the Vietnam Council for National Sciences.
REFERENCES 1. 2. 3. 4. 5. 6.
7. 8. 9.
Schmidt G, Tondl A. Nonlinear Vibration, Cambridge University Press, 1986. Roberts JB, and Spanos PD. Random Vibration and Stochastic Linearization, John Wiley, 1990. Anh ND, Schiehlen W. “An Approach to the Problem of Closure in the Nonlinear Stochastic Mechanics”, Int. J. of Mechanics, 29, pp. 109-123, 1994. Bogoliubov NN, Mitropolskii YA. Asymptotic methods in the theory of nonlinear oscillations, 4th ed., Moscow, 1974. Anh ND, Hai NQ. “A Technique of Closure Using a Polynomial Function of Gaussian Process”, Probabilistic Engineering Mechanics, 15, pp. 191-197, 2000. Anh ND, Hai NQ. “A Technique for Solving Nonlinear Systems Subject to Random Excitation”, IUTAM Symposium on Recent Developments in Nonlinear Oscillations of Mechanical Systems, Kluwer Academic Publishers, pp. 217-226, 2000. Anh ND, Hai NQ, Schiehlen W. “Nonlinear Vibration Analysis by an Extended Averaged Equation Approach”, Nonlinear Dynamics, accepted for publication (2006). Mueller PC, Popp K, Schiehlen WO. “Berechnungsverfahren stochastischer Fahrzengschwingungen”, Ingenieur –Archiv, 49, pp. 235-254, 1980. Roseau M. Vibrations in Mechanical Systems, Berlin Springer Verlag, 1989.
SINGULARITY ANALYSIS ON CONSTRAINED BIFURCATIONS Z. Q. Wu, Y. S. Chen Department of Mechanics, Tianjin University, Tianjin 300072, P.R. China, E-mail:
[email protected] Abstract:
The paper presents our recent work on the constrained bifurcation and its applications in engineering. For the bifurcation of systems with parameterized constraint either in single-sided or double-sided form, its classification is studied and the transition sets are obtained firstly. Three of the six sets are induced by the constraint boundary. For the bifurcation defined by piecewise, continuous functions, its transition sets can not be found by computing the transition sets of unconstrained bifurcation for each piece. Bases on these results, some related work on the rotor rub-impact prediction problem, two mode interactions of systems without internal resonance, and periodic solutions of non-smooth systems with bilinear hysteresis, are briefly described. It is found that, for self-excited vibration of the van der Pol system, remarkable vibration reduction can be achieved either by coupling a Duffing oscillator or by adding a hysteretic force.
Key words:
Constrained bifurcation, singularity theory, mode interaction.
1.
INTRODUCTION
Singularity theory applied to bifurcation with no constraint had been well developed by the end of 1980s. It was thoroughly summarized by Golubitsky and Schaeffer in their book [1]. The theory was widely used in determining the possible bifurcation types in nonlinear systems. In research on bifurcation of nonlinear systems of practical background, one often meets the question that the variation of the state variable of the bifurcation equation is restricted. For example, the value of the variable for 157 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 157–165. © 2007 Springer.
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the bifurcation of the periodic solutions is non-negative because it represents the amplitude, and it is also non-negative when it represents the density of the reacting material in equilibrium bifurcation problems of nonlinear chemical reaction systems. For these constrained bifurcation problems popular in engineering, there are few results and methods in the existing singularity theory which can be used in a straight forward way. The paper presents our recent work on constrained bifurcation and its applications in engineering. Section 2 and section 3 discuss the classification of bifurcation with single/double-sided parameterized constraints, respectively. Section 4 analyzes the bifurcation of periodic solutions in a hysteretic system. Section 5 summarizes the results of the paper.
2.
BIFURCATION OF SINGLE-SIDED CONSTRAINT For the bifurcation equation
g ( x, λ ; α ) = 0
(1)
with the simplest single-sided constraint x > 0, its transition sets are firstly given in [2] by casting it into a new unconstrained one through a nonlinear transformation. The result was used in the classification of ten elementary bifurcations [3] and of the unsymmetric bifurcations of the periodic solution of a rotating shaft with unsymmetrical stiffness [4]. Here we consider the effects of the more general constraint
δ ( x − β (λ ) ) > 0
(δ = ±1)
(2)
on the bifurcation classification. β (λ) denotes the parameterized constraint boundary. By introducing a nonlinear transformation
x = δ u2 + β ,
(3)
one can convert the constrained bifurcation (1) and (2) into the unconstrained one
G (u , λ ; α ) = g (δ u 2 + β , λ ; α ) = 0 .
(4)
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Its transition sets are deduced by the method given in [1] and listed below.
BI
⎧ g (β , λ ;α ) = 0 ⎪ ⎨ g x (β , λ ;α )β λ ⎪ + g (β , λ ;α ) = 0 λ ⎩
⎧ g (β , λ ;α ) = 0 ⎩ g x (β , λ ;α ) = 0
HI ⎨
⎧ g ( x, λ ; α ) = 0 ⎪ ⎪ g x ( x, λ ; α ) = 0 B ⎨ ⎪ g λ ( x, λ ; α ) = 0 ⎪δ ( x − β ) ≥ 0 ⎩
⎧ g ( β , λ ;α ) = 0 ⎪ g ( x, λ ; α ) = 0 ⎪ DLI ⎨ ⎪ g x ( x, λ ; α ) = 0 ⎪⎩ x ≠ β
H
⎧ g ( x, λ ; α ) = 0 ⎪ ⎪ g x ( x, λ ; α ) = 0 ⎨ ⎪ g xx ( x, λ ; α ) = 0 ⎪δ ( x − β ) ≥ 0 ⎩
and
⎧ g ( xi , λ ; α ) = 0 ⎪ ⎪ g x ( xi , λ ; α ) = 0 ⎪ DL ⎨δ ( xi − β ) ≥ 0 ⎪i = 1, 2 ⎪ ⎪( x1 ≠ − x2 ) ⎩
It is easy to shown that the six sets are also the transition sets of the constrained bifurcation (1) and (2). The capital B, H, and DL denote the bifurcation set, the hysteretic set and the double limit point set respectively. It is obvious that the first three, BI, HI and DLI, are induced by the constraint boundary, while the remaining three are subsets of those for the unconstrained bifurcation (1) on which the bifurcation points of the corresponding bifurcation diagrams are located in the region satisfying the constraint (2). Among these six parts, only set BI is influenced by the dependency of the constraint boundary on the bifurcation parameter, λ.
3.
BIFURCATION OF DOUBLE-SIDED CONSTRAINT In a similar way as used in section 2, the constrained bifurcation
⎧⎪ g ( x, λ ; α ) = 0 ⎨ ⎪⎩ β1 ( λ ) ≤ x ≤ β 2 ( λ )
(5)
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can be transformed into the unconstrained one
G (u , λ ; α ) = g (U ( x), λ ; α ) = 0
(6)
by the transformation u=U(x) that is solved from the relation
(β − β ) ( x − β1 )( x − β 2 ) = − 1 2 4
2
u2 . 1+ u2
Then by calculating the transition sets of bifurcation (6), those sets for bifurcation (5) are obtained as follows:
⎧ g ( βi , λ ;α ) = 0 ⎪ ⎨ g x ( β i , λ ; α ) β iλ BI ⎪ ⎩ + gλ ( βi , λ ;α ) = 0
⎧ g ( βi , λ ;α ) = 0 ⎨ HI ⎩ g x ( β i , λ ; α ) = 0 , (i = 1, 2)
(i = 1, 2) ⎧ g ( βi , λ ;α ) = 0 ⎪ g ( x, λ ; α ) = 0 ⎪ ⎨ DLI ⎪ g x ( x, λ ; α ) = 0 ⎪⎩ x ≠ β i (i = 1, 2)
H
⎧ g ( x, λ ; α ) = 0 ⎪ ⎪ g x ( x, λ ; α ) = 0 ⎨ ⎪ g xx ( x, λ ; α ) = 0 ⎪β ≤ x ≤ β 2 ⎩ 1
⎧ g ( x, λ ; α ) = 0 ⎪ ⎪ g x ( x, λ ; α ) = 0 B ⎨ , ⎪ g λ ( x, λ ; α ) = 0 ⎪β ≤ x ≤ β ⎩ 1 2
and
⎧ g ( xi , λ ; α ) = 0 ⎪ ⎪ g x ( xi , λ ; α ) = 0 ⎪ DL ⎨ β1 ≤ x ≤ β 2 ⎪i = 1, 2 ⎪ ⎪( x1 ≠ − x2 ) ⎩
According to the above results, one-mode motions of a rotating shaft were analyzed in [5]. It was shown that there are much more types of
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bifurcation patterns when the clearance constraint between the shaft and the bearing was taken into account. The resonant periodic solutions of the system consist of 11 different types of bifurcation patterns, among of which the following four types are more likely to appear, (1) patterns without impact and jump, (2) jump patterns without impact, (3) impact patterns without jump and (4) patterns with impact and jump. Based on the obtained transition sets and the bifurcation diagrams, parameter conditions either for rub-impact phenomena or non-impact-rub phenomena can be easily derived. The method proposed in [5] can be used to predict rub-impact phenomena in more complicated rotor systems. The paper [6] considered the two mode interaction problems in a system composed by van der Pol oscillator (VDP) and resonantly forced Duffing oscillator. It was shown that the quasiperiodic solution bifurcation is the double-sided constrained one in some cases. From the obtained bifurcation diagrams, it was observed that the oscillation of VDP can be greatly reduced, or even quenched. Because the system is not internally resonant, the result can be used in the design of a wide band self-vibration suppressor.
4.
PERIODIC BIFURCATIONS OF NON-SMOOTH SYSTEMS
The non-smooth system consists of a mass, a linear elastic spring, a negative damping (van del Pol type) and a hysteretic damper F(x) as shown in Figure 1. Its dimensionless governing equation of motion can be written
Figure 1. Mechanical model and the hysteretic force
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as
⎡⎛ λ ⎤ ⎞ && x + x = ε ⎢⎜ − x 2 ⎟ − F ( x ) ⎥ ⎠ ⎣⎝ 4 ⎦
(7)
where x is the displacement of the mass in the vertical direction, λ/4 is the coefficient of the van der Pol damping, F(x) is the bilinear hysteretic force and ε is a small positive parameter. For its periodic solution in the form
x = y cos ( t + γ ) ,
(8)
the averaged Equation (9) was deduced in [7] by using the averaging method.
y 1 2π ⎧ 2 ⎪⎪ y& = 8 ( λ − y ) + 2π ∫0 F ( y sinψ )dψ ⎨ ⎪γ& = 1 2π F ( y cosψ )dψ ⎪⎩ 2π ∫0
(9)
Setting y& = 0 in Equation (9) yields the so-called governing bifurcation equation for the amplitude of the self-excited periodic solution of system (7):
⎧Y ( y, λ ; α% ) = 0 ⎨ ⎩y ≥ 0
(10)
where α% is a three-dimensional parameter vector, and,
⎧ 2 ⎪Y1 ( y, λ ; α% ) = y ( y − λ ) ⎪ 16acy 16a 2 c ⎪ − Y ( y, λ ; α% ) = ⎨Y2 ( y, λ ; α% ) = y 4 − λ y 2 + π π ⎪ 16abc ⎪ 4 2 ⎪⎩Y3 ( y, λ ; α% ) = y − λ y + π
0≤ y≤a a ≤ y ≤ a+b y ≥ a+b (11)
For the bifurcation problem defined as above by the three piecewise, continuous functions, its transition sets were obtained in [8] by considering the bifurcation as three sub-bifurcations
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0 ≤ y ≤ a,
(II) Y2 ( y, λ ; α% ) = 0,
a ≤ y ≤ a + b, and
(III) Y3 ( y, λ ; α% ) = 0,
y ≥ a + b.
Figure 2 presents the transition sets of the bifurcation problem (11) in the same ranges of the parameters as the paper [7]. It is found in region II (see [7]) that instead of a single bifurcation diagram, there exist other bifurcation diagrams, IIb, IIc, IId, IIe, IIf (see Figure 3 for detail), which are qualitatively different from the only bifurcation diagram IIa obtained in [7]. Therefore the classification of the bifurcation problem (11) can not be solved just by computing the transition sets of three unconstrained bifurcations
Yi ( y, λ ;α ) = 0 (i = 1, 2,3) . To verify the validity of the above theoretical results, a lot of computation work is done. The numerical results indeed confirm the analytical predictions. The constrained bifurcation diagrams show that, by adding the hysteretic force, the vibration of the von der Pol system can be greatly reduced. It implies that the materials or devices that have hysteretic behavior, like shape memory alloy, can be used in passive control on self-excited vibration.
Figure 2. Transition sets of bifurcation (11)
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Figure 3. The constrained bifurcation diagrams
5.
CONCLUSIONS
The classification of the bifurcation of parameterized constraints is studied. The transition sets are derived for bifurcation of the two kinds of constraint, the single-sided one and the double-sided one. The bifurcation defined by piecewise, continuous function, its transition sets can be found out from a series of sub-bifurcations of single/double-sided constraints. The constraint may be physical, mechanical or resulted from the process of solving equations. As long as the constraint exists, one has to use the results here in analyzing the related bifurcation in order to get all possible bifurcation patterns.
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165
ACKNOWLEDGEMENTS The work is supported by the National Natural Science Foundation of China (No. 10472078) and the New Century Excellent Young Researcher Funding of the Education Ministry of China.
REFERENCES 1. 2. 3. 4. 5.
6. 7. 8.
Golubitsky M, Schaeffer DG. Singularities and Groups in Bifurcation Theory (I) New,York, Springer-Verlag, 1985. Wu ZQ, Chen YS. “New bifurcation patterns in elementary bifurcation problems with single side constraint”, Applied Mathematics and Mechanics, 22, pp. 1260-1267, 2001. Wu ZQ, Chen YS. “Classification of bifurcations for nonlinear dynamical problems with constrains ”, Applied Mathematics and Mechanics, 23, pp. 535-541, 2002. Chen FQ, Wu ZQ, Chen, YS. “Bifurcation and universal unfolding for a rotating shaft with unsymmetrical stiffness”, Acta Mechanica. Sinica, 18, pp. 181-187, 2002. Wu ZQ, Chen YS. “Prediction for the rub-impact phenomena in rotor systems”, 2001 ASME Design Engineering Technical Conferences, Pittsburgh, Pennsylvania, USA, Sep. 9-12, 2001. Wu ZQ, Zhang JW. “Mode interactions in nonlinear self-excited nonlinear systems”, Journal of Tianjin University (accepted). Ding Q, etc., “Dynamic analysis of a self-excited hysteretic system”, Journal of Sound and Vibration, 245, pp. 151-164, 2001. Wu ZQ, etc., “Bifurcation analysis on a self-excited hysteretic system”, International Journal of Bifurcation and Chaos, 14, pp. 2825-2842, 2004.
NONLINEAR ANALYSIS OF ROTOR DYNAMICS BY USING THE METHOD OF MULTIPLE SCALES H. Yabuno1, Y. Kunitho1, T. Inoue2, Y. Ishida2 1
Graduate School of Systems and Information Engineering, University of Tsukuba, Tsukuba, 305-8573, Japan 2 Graduate School of Engineering, Nagoya University Nagoya, 464-8603, Japan, E-mail:
[email protected] Abstract:
The method of multiple scales is modified to nonlinear analysis in rotor systems. Amplitude equations for forward and backward whirling modes are directly derived and the method makes it easier to understand resonance mechanism. As an example, we analyze near the major critical speed the nonlinear dynamics of a horizontally supported Jeffcott rotor and show that nonlinear and gravity effects cause the backward whirling mode in addition to the forward one. Some experiments are performed and the validity of the theoretical results is confirmed.
Key words:
Rotor dynamics, nonlinear, whirling motion, the method of multiple scales.
1.
INTRODUCTION
As operating speed of rotating machinery is increased, nonlinear oscillation are often encountered. Since about 50 years ago, rotor dynamics under various type of nonlinear force have analyzed. For example, Yamamoto reported that nonlinear oscillations are produced due to radial clearance of the ball bearings in the supports in 1954. Furthermore, it has been shown that other nonlinear phenomena can be produced by oil film in a journal bearing, clearance of squeeze film damper, elongation of shaft, and so on: comprehensive survey is found in [1-2]. The dynamics of rotor systems consist of the forward and backward whirling modes [3] and also 167 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 167–176. © 2007 Springer.
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the nonlinear characteristics are very different from those of rectilinear systems. Widely used analytical methods like averaging method, the method of multiple scales and so on [4], are intended to analysis of nonlinear phenomena in rectilinear systems and there are few methods which are specialized to rotor dynamics. In the first half of this paper, we propose an analytical method for nonlinear rotor dynamics by modifying the method of multiple scales. In the latter half, we apply the method to the equations of motion of a horizontally supported Jeffcott rotor and examine the nonlinear characteristics in the vicinity of major critical speed. Furthermore, we conduct experiments using a simple apparatus and discuss the validity of the theoretical results.
2.
THE METHOD OF MULTIPLE SCALES FOR ROTOR DYNAMICS
2.1 Typical equations of motion of rotor system To modify the method of multiple scales for its application to nonlinear analysis of rotor dynamics, we first characterize the equation of motion of a vertically supported Jeffcott rotor as shown in Figure 1a. The center of gravity, geometrical center, and eccentricity of the disk are G, M, and e. We consider the case of when the rotational speed Ω is positive (counterclockwise). The z-axis of the coordinate system o − x − y − z coincides with the bearing center line. The equations of motion with respect to the displacements of the point G in x and y directions can be expressed as d 2 xG (1) = Fx dt 2 d 2 yG (2) m = Fy dt 2 where Fx and Fy are the x and y components of restoring force F by stiffness of the shaft and the bearings of the supporting points. Furthermore, xG and yG are expressed by x and y as m
d2 ( x + e cos Ωt ) dt 2 d2 yG = m 2 ( y + e sin Ωt ) dt
xG = m
(3) (4)
Nonlinear analysis of rotor dynamics
169
The restoring force F is a function of the distance between z axis and the geometric center M, i.e., r = x 2 + y 2 and can be assumed as
Figure 1. Analytical model
F (r ) = −k1r − k3 r 3 − k5 r 5 ;"
(5)
Then, the x and y components of the restoring force are
Fx = F (r ) cos θ = − k1 x − k3 ( x 2 + y 2 ) x − k5 ( x 2 + y 2 ) 2 x −"
(6)
Fy = F (r )sin θ = − k1 y − k3 ( x 2 + y 2 ) y − k5 ( x 2 + y 2 )2 y −"
(7)
As a result, the governing equations of motion can be expressed as m
d2x + k1 x + k3 ( x 2 + y 2 ) x + " = emΩ 2 cos Ωt 2 dt
(8)
d2y (9) + k1 y + k3 ( x 2 + y 2 ) y + " = emΩ 2 sin Ωt dt 2 where the phase difference of external force due to the unbalance is always π /2 in the case of rotor systems. All lengths are nondimensionalized using the shaft span l and the time is nondimensionalized using T = k1 / m . We denote the resulting dimensionless quantities corresponding to x and t by x∗ and t ∗, respectively. We introduce the following dimensionless parameters: m
β3 =
k3 l 2 ∗ e Ω , e = ,ω = k1 l T
(10)
In this way, we can obtain the following dimensionless equations of motion.
x + x + β 3 ( x 2 + y 2 ) x = eω 2 cos ωt
(11)
y + y + β3 ( x 2 + y 2 ) y = eω 2 cos ωt
(12)
where (˙) represents the derivative with respect to t ∗ .
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2.2 Introduction of complex coordinate and separation of forward and backward whirling modes First, we describe the above equations on complex coordinate by the relationship: (13) z = x + iy Equations (6) and (7) are transformed into the complex form: 2
z + z + β 3 z z + " = eω 2 eiωt
(14)
The transformation is employed to separate the forward and backward modes. In nonlinear analysis, the procedure is much more essential than in linear analysis. To examine the characteristics of modes, we consider the following linear homogeneous problem by neglecting external and nonlinear terms: (15) z+z=0 The solution is expressed as z = A+ eit + A− e − it
(16)
The complex amplitudes, A+ and A− , are rewritten in the from
A+ = a+ eiϕ+ , A− = a− eiϕ−
(17)
By the way, linear vibration problem in the case of one-degree of rectilinear system is governed with (18) x+x=0 and the solution is x = A+ eit + A− e − it
(19)
The complex amplitudes, A+ and A− , of Equation (17) are complex conjugate each other. On the other hand, in rotor systems, the first and second terms of Equation (14) are independent and do not generally have the relationship of complex conjugate each other. The first term exhibits the forward whirling mode. The direction of locus of the geometrical center is positive, which is the same as the rotation direction of the shaft (ω > 0). The second term is the backward whirling mode whose direction is negative. The locus for each mode is circler and the existence of both modes results in elliptic locus. Next, we take into account the effect of unbalance and discuss resonance mechanism near the major critical speed, i.e., ω ≈ 1 .The governing equation is z + z = eω 2 eiωt
(20)
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Obviously, only the secular term proportional to exp(it ) is produced in the solution, but the secular term proportional to exp(−it ) is not produced. Therefore, only the forward whirling mode is excited in the resonance. It is shown in the next section that backward whirling mode is excited in addition to the forward whirling mode in the case of horizontal support of the shaft.
2.3 Application of the method of multiple scales to rotor dynamics We propose the application of the method of multiple scales to nonlinear analysis of rotor dynamics through an example. We consider a horizontally supported Jeffcott rotor as Figure 1b. The gravity effect has been taken into account and the governing equations of motion are follows: d 2x dx +c + x + β 3 ( x 2 + y 2 ) x + " = eω 2 cos ω t dt 2 dt d2y dy +c + y + β 3 ( x 2 + y 2 ) y + " = eω 2 sin ω t − g 2 dt dt
(21) (22)
The equations are rewritten in dimensionless complex form: 2
z + cz + z + β 3 z z + " = eω 2 eiωt − ig
(23)
where g is dimensionless gravity effect ( g ≡ mg /(kl )) . We analyze the case when the rotational speed is in the neighborhood of the critical speed: ω ≈ 1 . The parameter values are c = 1.2 × 10−2, β = 2.52 × 103 , g = 3.46 × 10−3 which correspond to those of experimental setup mentioned later. We introduce the detuning parameter defined by
ω = 1 + σ = 1 + ε 2σˆ
(24) The third-order uniform expansion of the solution of the equation of motion is determined letting the complex displacement: z (t ) = ε z1 (t0 , t2 ) + ε 3 z3 (t0 , t2 )
(25)
where t0 = t is the fast scale associated with variations occurring at the natural frequency, and t2 = ε 2t is the stretched time scale governing the nonlinear slow variations. Also, we perform the scaling of some parameters according to
c = ε 2 cˆ, g = ε gˆ , e = ε 3eˆ
(26)
where ˆ denotes “of order O(1) ” and ε is a bookkeeping device. Equating coefficients of like powers of ε yield the following equations of the orders O(ε ) and O(ε 3 ) .
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O(ε ) : D02 z1 + z1 = −igˆ
(27) 2
ˆ 0 z1 − β 3 z1 z1 + ee ˆ O(ε ) : D z + z3 = −2 D0 D2 z1 − cD 3
2 0 3
iω t0
(28)
where D0 = ∂ / ∂t0 , D2 = ∂ / ∂t2 . The solution of Equation (25) is
z1 = A+ (t2 )eit0 + A− (t2 )e − it0 − igˆ
(29)
The complex amplitudes of the forward and backward modes, A+ and A− , are governed with amplitude equations which are independently from the solvability conditions of Equation (26). This is completely different from the procedure of the method of multiple scales for rectilinear systems as summarized in Figure 2. As mentioned in the preceding section, the first and second terms in Equation (f-1) are complex conjugate each other, but in Equation (f-3) are not. Furthermore, the solvability conditions of Equation (f-2), which are f ( A+ , A− ) = 0 and f ( A+ , A− ) = 0 , are equivalent. On the other hand, the solvability conditions of Equation (f-4), which are f + ( A+ , A− ) = 0 and f − ( A+ , A− ) = 0 , are not equivalent and lead to amplitude equations for the forward and backward whirling modes, respectively. We substitute Equation (27) into Equation (26) and obtain 2
2
ˆ + + β ( A+ A+ + 2 A− A+ D03 z3 + z3 = −{2iD2 A+ + icA 2 2 ˆ ˆ ˆ iσˆ t2 }eit0 +2 g A+ − g A− ) − ee 2 2 ˆ − − β (2 A+ A− + A− A− +{2iD2 A− + icA − gˆ 2 A+ + 2 gˆ 2 A− )}e − it0 + NST One degree-of –freedom rectilinear system 2 x1 = A+ (Å )eit0 + AÄ (Å )eÄit0 O(è) : D0 x1 + x1 = 0
(30)
(f-1)
Complex conjugate 3 O(è) : D0 x2 + x2 = f (A+ ; AÄ )eit0 + fñ(A+ ; AÄ )eÄit0 + nst 2
0
Rotor system D02 z1 + z1 = Äi^ g
Same eq.
0
z1 = A+ (t2 )eit0 + AÄ (t2 )eÄ it0 Ä i^ g (f-3) Independent
D02 z3 + z3 = f+ (A+ ; AÄ )eit 0 + fÄ (A+ ; AÄ )eÄ it0
0
(f-2)
Not need
Not same eq.
(f-4)
0
Figure 2. Comparison of methods of multiple scales for rectilinear and rotor systems
Nonlinear analysis of rotor dynamics
173
where NST denotes terms not to proportional to eit0 or e − it0 The condition not to produce the secular term proportional to eit0 in the solution of z3 is + β ( A+
2
ˆ + − ee ˆ iσˆ t2 2iD2 A+ + icA 2 A+ + 2 A− A+ + 2 gˆ 2 A+ − gˆ 2 A− ) = 0
(31)
Also, the condition not to produce the secular term proportional to e − it0 is 2
2
ˆ − − β (2 A+ A− + A− A− − gˆ 2 A+ + 2 gˆ 2 A− ) = 0 2iD2 A− + icA
(32)
In the case of rectilinear systems, the complex conjugate of the first condition corresponding to Equation (29) is generally equivalent to the second condition corresponding to Equation (30). Therefore, one of them leads to the modulation equation of the rectilinear systems. However, in rotor systems, as seen from Equations (29) and (30), these conditions are generally not equivalent and the first and second conditions lead to amplitude equations of the forward and backward whirling modes, respectively. As a result, the approximate solution is expressed as follows:
z = a+ (t )ei (ωt +φ+ ( t )) + a− (t )ei ( −ωt +φ− ( t )) − ig + O(ε 3 )
(33)
where the time variations of a+ , a− , φ+ , and φ− are governed with the following equations which are obtained from Equations (29) and (30): da+ 1 1 1 (34) = − ca+ − β 3 g 2 a− sin(ϕ+ + ϕ− ) − e sin ϕ+ dt 2 2 2 dϕ 1 a+ + = −σ a+ + β a+3 + β 3 a+ a−2 + β 3 g 2 a+ dt 2 (35) 1 1 2 − β 3 g a− cos(ϕ+ + ϕ − ) − e cos ϕ+ 2 2 da− 1 1 (36) = − ca− + β 3 g 2 a+ sin(ϕ+ + ϕ− ) dt 2 2 dϕ 1 a− − = σ a− − β3 a−3 − β a+2 a− − β 3 g 2 a− dt 2 (37) 1 2 + β g a+ cos(ϕ+ + ϕ − ) 2
2.4 Production of backward whirling mode due to nonlinearity and gravity effect From Equations (32), (33), (34), and (35), we have frequency response curves for forward and backward modes and for x and y directions, as Figures 3 and 4, respectively. The transformation from a+ and a− to ax and a y is performed by using relationship
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-3
1.0
1.0
a-
1.5 x10
a+
1.5 x10
0.5
0.5
0.0 -0.05
0.00
0.05
0.10
σ (a) Forward whirling mode
0.0 -0.05
0.15
0.00
0.05
0.10
0.15
σ (b) Backward whirling mode
Figure 3. Frequency response curve (—— : stable, - - - - : unstable)
-3
-3
2
2
ay
3 x10
ax
3 x10
1
1
0 -0.05
0.00
0.05
0.10
0 -0.05
0.15
σ (a) x direction
0.00
0.05
0.10
σ (b) y direction
0.15
Figure 4. Frequency response curve (—— : stable, - - - - : unstable)
ax = (a+ cos ϕ + + a− cos ϕ− ) 2 + (a+ sin ϕ+ − a− sin ϕ− ) 2 a y = (a+ cos ϕ + − a− cos ϕ − ) 2 + ( a+ sin ϕ+ + a− sin ϕ− ) 2 Because the amplitude of x and y directions are different, the locus is elliptic orbit. The first resonance peak has large horizontal component and the second one has large vertical component. By the way, Equation (30) is rewritten as 2
2
2iD2 A− + 2icˆ A− − β3 gˆ 2 ( A+ A− − A+ A− ) = 0
(38)
In the case when both nonlinear effect β3 and gravity effect gˆ does not exist, Equation (36) is 2
2
D2 A− + cˆ A− = 0
(39) Therefore, the backward whirling mode decays and can not be excited. In the linear system, the backward whirling mode occurs in the case when the linear stiffness of supporting points depends on the directions [1]. However, co-existence of nonlinear and gravity effects causes backward whirling mode even in the case without dependency of the linear stiffness on the direction at the supporting points.
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3.
EXPERIMENT
We show experimental setup in Figure 5. The span and diameter of the shaft are 12mm and 700mm, respectively. The mass and diameter of the disk are 8.21kg and 0.3mm, respectively. Displacement of the disk in x and y directions are measured by laser sensors. Figure 6 shows experimentally obtained frequency response curves. As theoretically predicted, the backward whirling mode in addition to the forward one is produced and the frequency response curves are different between x and y directions. Also, the first and second resonance has large horizontal component and large vertical component, respectively. In those cases, the loci are experimentally depicted as Figures 7a and 7b, respectively.
Figure 5. Experimental setup
2
ay [mm]
ax [mm]
2
1
0
1
0 9.0
9.5
10.0
10.5
Ω/2π [Hz] (a) x direction
11.0 11.5
9.0
9.5
10.0
10.5
Ω/2π [Hz] (b) y direction
Figure 6. Experimental frequency response curve
11.0 11.5
176 3
3
2
2
1
1
y [mm]
y [mm]
H. Yabuno et al.
0
0
-1
-1
-2
-2
-3
-3 -3
-2
-1
0
1
2
x [mm] (a) Ω/2π = 9.9 [Hz]
3
-3
-2
-1
0
1
2
x [mm] (b) Ω/2π = 10.6 [Hz]
3
Figure 7. Experimentally obtained loci in the first and second resonance
4.
SUMMERY
We proposed a method for nonlinear analysis of rotor systems, which is based on the modification of the method of multiple scales. The method can directly derive amplitude equations for forward and backward whirling modes which generally characterize dynamics of rotor systems. By using the method, we analyze nonlinear dynamics of horizontally supported Jeffcott rotor. It is clarified from the result that the coexistence of gravity and cubic nonlinearity causes the backward whirling mode in addition to the forward one. Furthermore, experiments are conducted and theoretically predicted phenomena are quantitatively confirmed.
ACKNOWLEDGEMENTS The authors wish to thank Mr. T. Kashimura, graduate student at the University of Tsukuba, for his assistance. This work is supported by TEPCO Research foundation.
REFERENCES 1. 2. 3. 4.
Yamamoto T, and Ishida Y. Linear and Nonlinear Rotor dynamics, Wiley, 2001. Lalanne M, and Ferraris G. Rotor dynamics Prediction in Engineering, Wiley, New York, 1990. Ehrich FF. Handbook of Rotor dynamics, Krieger, Malabar, 1999. Nayfeh AH, and Mook DT. Nonlinear Oscillations, Wiley, 1979.
NONLINEAR DYNAMICS OF A SPUR GEAR PAIR WITH SLIGHT WEAR FAULT S. Yang, Y. Shen Department of Mechanical E-mail:
[email protected] Engineering,
Shijiazhuang
Railway
Institute,
050043,
Abstract:
This paper is focused on the nonlinear dynamics of a spur gear pair with slight wear fault, where the backlash, time-varying stiffness and wear fault are all included in the model. The Incremental Harmonic Balance Method (IHBM) is applied to research the periodic solution of this system. Based on the Kronecker’s notation, step function and sign function, the general forms of the periodic solutions are founded, which is useful to obtain the periodic solutions precisely. At last the typical frequency-response diagrams are obtained to illustrate the different properties of fault gear system compared with the faultless one.
Key words:
Nonlinear dynamics, incremental harmonic balance method, fault gear system.
1.
INTRODUCTION
Gear system is an important part in mechanical engineering and other engineering fields, so that many papers on nonlinear dynamics of gear system and many important results have been obtained [1 – 5]. Some typical methods, such as the analytical, numerical and experimental methods, have been adopted to analyze the nonlinear dynamics of gear system. The effective analytical methods include Harmonic Balance Method [2], MultiScale Method jointed with piecewise technique [3] and Incremental Harmonic Balance Method (IHBM) [4 – 5]. But the existing investigations are mainly focused on the dynamics of faultless gear system, and only a little works have been done on the nonlinear dynamics of fault gear system by numerical technique due to the complexity of gear system [6 – 8]. In order to 177 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 177–185. © 2007 Springer.
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control the dynamical behavior of gear system effectively, it is necessary to understand the detail dynamical behaviors of fault gear system. In this paper the authors investigate a dynamical model of gear pair with slight wear fault, which is considered as the high-frequency parametric excitation in the model. By IHBM and some other means, such as Kronecker’s notation, step function and sign function, we obtain the general forms of the periodic solution of gear system with slight wear fault. Then some typical frequency-response diagrams are obtained and the differences with the faultless gear system are illustrated.
2.
DYNAMICAL MODEL OF SPUR GEAR PAIR
The researched model is shown in Figure 1, where the shaft and bearing is assumed to be rigid. According to the Newtonian law of motion, the equations of the torsion are d 2θ a dθ dθ de ⎛ + c ⎜ Ra ⋅ a − Rb ⋅ b − 2 dt dt dt dt ⎝ Ra K ( t ) f ( Raθ a − Rbθb − e ( t ) ) = Ta ,
Ia
⎞ ⎟ ⋅ Ra + ⎠
d 2θb dθ dθ de ⎞ ⎛ − c ⎜ Ra ⋅ a − Rb ⋅ b − ⋅ Rb − Ib 2 dt dt dt dt ⎟⎠ ⎝ Rb K ( t ) f ( Raθ a − Rbθb − e ( t ) ) = −Tb
(1)
where I a and I b are the mass moment of inertia, θ a and θb are the angular displacement, Ra and Rb are the radius of the base circle of active and passive gears respectively, Ta and Tb are the torque on active and passive gears, e ( t ) is the static transmission error. Obviously this system is semi-definite, and it could be transformed into (2) by letting x = Raθ a − Rbθb − e ( t ) m
d2x dx +c + k[1 + 2ε1 cos ω1 t + 2ε 2 cos ω2 t ] f ( x ) = F 2 dt dt
(2)
where the K ( t ) is time-varying stiffness, m is the equivalent mass representing the total inertia of the gear pair, F is the average force transmitted through the gear pair and 2ε1 cos ω1 t , 2ε 2 cos ω2 t are corresponding to the periodic contact ratio and effect of wear fault on meshing stiffness.
Nonlinear dynamics of a spur gear pair with slight wear fault θb
θa
Ta
c K( t )
Tb
Ra
Ia
179
Rb
c
a
Ib
e (t )
Figure 1. A spur gear pair model.
After some variable transforms, one could get the dimensionless form of (2) shown as (3) d2x dx + 2ε1µ + [1 + 2ε1 cos Ω1t + 2ε 2 cos(l Ω1t )] f ( x) = f 0 2 dt dt
where l =
ω2 and the nonlinear displacement function due to backlash is ω1
⎧⎪ x − 1 f ( x ) = ⎨0 ⎪⎩ x + 1
3.
(3)
x >1 −1 ≤ x ≤ 1 . x < −1
COMPUTATION SCHEME FOR THE PERIODIC SOLUTIONS BY IHBM For convenience, (3) could be re-written as
Ω 2 x + 2ε1µ Ωx + [1 + 2ε1 cosτ + 2ε 2 cos lτ ] f ( x) = f 0
(4)
where τ = Ωt , and the symbol • denotes derivative with respect to τ . The periodic solution taking N harmonic terms and the increment could be expressed as N
x0 = a0 + ∑ [an cos(nτ ) + bn sin( nτ )] n =1
N
∆x = ∆a0 + ∑ [∆an cos(nτ ) + ∆bn sin(nτ )]
(5)
n =1
Substituting x = x0 + ∆x into (4) and expanding all the terms into Taylor series one could get
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S. Yang, Y. Shen d 2 ∆x d ∆x + 2εµ Ω + [1 + 2ε1 cosτ + 2ε 2 cos(lτ )] f ′( x0 )∆x = f 0 2 dτ dτ ⎡ d 2 x0 ⎤ dx − ⎢Ω 2 + 2εµ Ω 0 + [1 + 2ε1 cosτ + 2ε 2 cos(lτ )] f ( x0 ) ⎥ 2 dτ dτ ⎣ ⎦
Ω2
(6)
Here the higher order terms of the small increment ∆x are neglected. T T Letting a = [ a0 , a1 ," , aN , b1 ," , bN ] , ∆a = [ ∆a0 , ∆a1 ," , ∆aN , ∆b1 , " , ∆bN ] . and thenapplying Galerkin’s procedure one could obtain the linearized equations C ∆a = R
(7)
where ⎡ [C ] C = ⎢ 11 ⎣ [C21 ]
[C12 ] ⎤ , R = ⎡ R1 ⎤ ⎢⎣ R2 ⎥⎦ [C22 ] ⎥⎦
(8)
The explicit forms for the elements of Jacobi matrix C and corrective vector R are
[C11 ]ij = −δ ij j 2Ω2π + [C11 ]ij
NL
, i = 0,1," , N ; j = 0,1," , N
[C12 ]ij = 2εµπδ ij jΩ + [C12 ]ij
NL
, i = 0,1," , N ; j = 1,", N
[C21 ]ij = −2εµπδ ij jΩ + [C21 ]ij
NL
[C22 ]ij = −δ ij j 2 Ω2π + [C22 ]ij
NL
, i = 1," , N ; j = 0,1," , N , i = 1," , N ; j = 1," , N
R10 = 2π f 0 + R10NL , R1i = π [i 2 Ω 2 ai − 2εµ iΩbi ] + R1NL i , i = 1," , N
R2i = π [i 2 Ω 2bi + 2εµ iΩai ] + R2NL i , i = 1," , N
(9)
In the above equations the superscript NL denotes the nonlinear parts, and δ ij is Kronecker’s notation. And then the nonlinear parts of the elements of C and R could be calculated explicitly as
[C11 ]ij
NL
M
{
= ∑ H (um ) ⎡⎣ Aij (θ m +1 ) − Aij (θ m ) ⎤⎦ m =0
}
+ε1 ⎡⎣ Aij* (θ m +1 ) − Aij* (θ m ) ⎤⎦ + ε 2 ⎡⎣ Aij** (θ m +1 ) − Aij** (θ m ) ⎤⎦ ,
Nonlinear dynamics of a spur gear pair with slight wear fault
[C12 ]ij
NL
M
181
{
= ∑ H (um ) ⎡⎣ Bij (θ m +1 ) − Bij (θ m ) ⎤⎦ m=0
}
+ε1 ⎣⎡ Bij* (θ m +1 ) − Bij* (θ m ) ⎦⎤ + ε 2 ⎣⎡ Bij** (θ m +1 ) − Bij** (θ m ) ⎦⎤ ,
[C21 ]ij
NL
M
{
= ∑ H (um ) ⎡⎣Cij (θ m +1 ) − Cij (θ m ) ⎤⎦ m=0
}
+ε1 ⎡⎣Cij* (θ m +1 ) − Cij* (θ m ) ⎤⎦ + ε 2 ⎡⎣Cij** (θ m +1 ) − Cij** (θ m ) ⎤⎦ ,
[C22 ]ij
NL
M
{
= ∑ H (um ) ⎡⎣ Dij (θ m +1 ) − Dij (θ m ) ⎤⎦ m=0
}
+ε1 ⎡⎣ Dij* (θ m +1 ) − Dij* (θ m ) ⎤⎦ + ε 2 ⎡⎣ Dij** (θ m +1 ) − Dij** (θ m ) ⎤⎦ , M
N
m=0
j =0
{
⎡ ⎤ ⎡ ⎤ R1NL i = − ∑ H (um ) ∑ a j ⎣ Aij (θ m +1 ) − Aij (θ m ) ⎦ + b j ⎣ Bij (θ m +1 ) − Bij (θ m ) ⎦
+ε1a j ⎡⎣ Aij* (θ m +1 ) − Aij* (θ m ) ⎤⎦ + ε1b j ⎡⎣ Bij* (θ m +1 ) − Bij* (θ m ) ⎤⎦
}
+ε 2 a j ⎡⎣ Aij** (θ m +1 ) − Aij** (θ m ) ⎤⎦ + ε 2 b j ⎡⎣ Bij** (θ m +1 ) − Bij** (θ m ) ⎤⎦ M
+ sgn( x) ∑ H (um ) {[ Ei (θ m +1 ) − Ei (θ m ) ] m=0
}
+ε1 ⎡⎣ E (θ m +1 ) − Ei* (θ m ) ⎤⎦ + ε 2 ⎡⎣ Ei** (θ m +1 ) − Ei** (θ m ) ⎤⎦ , * i
M
N
m=0
j =0
{
⎡ ⎤ ⎡ ⎤ R2NL i = − ∑ H (um )∑ a j ⎣Cij (θ m +1 ) − Cij (θ m ) ⎦ + b j ⎣ Dij (θ m +1 ) − Dij (θ m ) ⎦ +ε1a j ⎡⎣Cij* (θ m +1 ) − Cij* (θ m ) ⎤⎦ + ε1b j ⎡⎣ Dij* (θ m +1 ) − Dij* (θ m ) ⎤⎦
}
+ε 2 a j ⎡⎣Cij** (θ m +1 ) − Cij** (θ m ) ⎤⎦ + ε 2b j ⎡⎣ Dij** (θ m +1 ) − Dij** (θ m ) ⎤⎦ M
+ sgn( x) ∑ H (um ) {[ Fi (θ m +1 ) − Fi (θ m ) ] } m =0
}
+ε1 ⎡⎣ Fi (θ m +1 ) − Fi * (θ m ) ⎤⎦ + ε 2 ⎡⎣ Fi ** (θ m +1 ) − Fi ** (θ m ) ⎤⎦ *
(10)
where M represents the number of roots for the equation x(τ ) = 1 in the interval ( 0, 2π ) . Letting θ 0 = 0 , θ M +1 = 2π and assuming these roots are θl ( l = 1," , M ), one could get the step function H (um ) , where u0 , u1 ," , uM
S. Yang, Y. Shen
182
is the sign of x(τ ) −1 re spectively in the intervals [θ 0 θ1 ] , [θ1 θ 2 ] , " ,
[θ M θ M +1 ] . And
then the expressions to calculate other terms in (10) are
given as Aij** (θ ) =
θ ⎡ sin(l + i + j )θ sin(l + i − j )θ sin(l − i + j )θ sin(l − i − j )θ ⎤ , + + + 2 ⎢⎣ (l + i + j )θ (l + i − j )θ (l − i + j )θ (l − i − j )θ ⎥⎦ Bij** (θ ) = −
θ ⎡ cos(l + i − j )θ cos(l − i − j )θ + 2 ⎢⎣ (l + i − j )θ (l − i − j )θ
cos(l − i + j )θ cos(l + i + j )θ ⎤ , − (l − i + j )θ (l + i + j )θ ⎥⎦ Cij** (θ ) = Bij** (θ ) ,
Dij** (θ ) =
θ ⎡ sin(l − i + j )θ sin(l + i − j )θ sin(l − i − j )θ sin(l + i + j )θ ⎤ , + − − 2 ⎢⎣ (l − i + j )θ (l + i − j )θ (l − i − j )θ (l + i + j )θ ⎥⎦ ⎡ sin(l + i )θ sin(l − i )θ ⎤ , + Ei** (θ ) = θ ⎢ (l − i )θ ⎥⎦ ⎣ (l + i )θ ⎡ cos(l − i )θ cos(l + i )θ ⎤ − Fi ** (θ ) = θ ⎢ (l + i )θ ⎥⎦ ⎣ (l − i )θ
(11)
where the other expressions could be found similar to the forms in [4 – 5]. Base on the above equations, one could get the periodic solutions of gear system with high precision. That is to say, according to an initial guess of a the increment ∆a could be solved and then the next initial value a would be obtained. After some iterations the steady a are obtained if the increment ∆a is smaller than the given error estimate.
4.
RESULTS AND ANALYSIS Here the basic parameters for the gear system are selected as
ε1 = ε 2 = 0.05 , ε 2 = 0.005 , f 0 = 0.9 , µ = 0.5 . The l is chosen as 8 so that the order for periodic solution must be selected much larger. Here we selected it as N = 51 . The convergence of the periodic solution is examined by calculating the error estimate
Nonlinear dynamics of a spur gear pair with slight wear fault
183
max( ∆a ) < 10−9
(12)
Based on the IHBM and the above equations, the typical frequencyresponse diagrams are shown as Figure 2, where the solid lines and symbol ‘o’ denote the analytical and numerical solutions respectively. And here only the stable solutions are illustrated whose stability could be confirmed by the Floquet theory. Compared with the faultless gear system [4 – 5], it could be found that more super- or sub-harmonic solutions occur commonly, that maybe makes the dynamical behaviors more complicated. If we research the dynamical behavior in the range of primary resonance, namely the frequency around Ω = 1 , the Figure 3 would be obtained, where the solid and dot lines denote the dynamical behavior of gear pair with and without fault. It could be found that although the differences of amplitude between them are not very notable, the difference of resonance ranges is very significant. 4.0
2.0
3.5
1.0 xmin
xmax
3.0 2.5
-1.0
2.0 1.5
0
0.5
1.0
Ω
1.5
2.0
2.5
-2.0
0.5
1.0
Ω
1.5
2.0
2.5
Figure 2. The frequency-response: xmax-Ω and xmin-Ω, where the solid lines and symbol ‘o’ denote the analytical and numerical solutions respectively. 2.0
3.5
1.0 xmin
4.0
0
xmax
3.0
-1.0
2.5
0.6
0.8
Ω
1.0
1.2
0.6
0.8
Ω
1.0
1.2
Figure 3. Detail comparison of frequency-response between gear pair: xmax-Ω and xmin-Ω, where the solid and dot lines denote the behavior of gear pair with and without fault.
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S. Yang, Y. Shen
184
When the two important system parameters, µ and f 0, are changed, the dynamical behavior may be changed abruptly. If the damping ratio µ is changed and all the other parameters keep fixed, the frequency-response curves are shown in Figure 4, where the solid line, symbol ‘o’ and dot denote µ = 0.5, µ = 0.8 and µ = 1.2 respectively. One could find that with the increase of the damping ratio the minimum amplitude of response increases accordingly, and the single-side and double-side impact would vanish gradually. The parameter f 0 has the similar effect on dynamical behavior, shown in Figure 5, where the solid, dot and dash-dot line represents f 0 = 0.9 , f 0 = 3.0 and f 0 = 6.0 respectively. It could be concluded that with the increase of the amplitude of excitation the minimum response amplitude would all increase accordingly, and the single-side and double-side impact would vanish gradually. 2.0
xmin
1.0 0
-1.0 -2.0 0.5
Ω
1.0
1.5
Figure 4. Detail comparison of frequency-response between gear pair: xmin-Ω, where the solid line, symbol ‘o’ and dot denote µ = 0.5, µ = 0.8, and µ = 1.2 respectively.
7.0
xmin
5.0 3.0 1.0 -1.0 0.5
Ω
1.0
1.5
Figure 5. Detail comparison of frequency-response between gear pair: xmin-Ω, where the solid, dot and dash-dot line represents f0 = 0.9, f0 = 3.0 and f0 = 6.0 respectively.
Nonlinear dynamics of a spur gear pair with slight wear fault
5.
185
CONCLUSIONS
In this paper the nonlinear dynamics of a spur gear pair with wear fault is researched by IHBM, and the general forms of periodic solutions with high precision are established. It could be concluded that there are more sup- and sub-harmonic response seen from the amplitude-frequency diagrams and the resonance range for fault gear pair is larger the one of faultless gear pair. These Two results would be benefit to understand the dynamical behavior of gear system with fault.
ACKNOWLEDGEMENTS The authors are great thankful to the support by the National Natural Science Foundation of China (No. 50625518, No. 10472073), the Natural Science Foundation of Hebei Province (E2006000383) and the fund for scientific project of Education Office of Hebei Province (No. 2005125).
REFERENCES 1. 2. 3.
4.
5.
6. 7. 8.
Wang JJ, Li RF, Peng XH. “Survey of nonlinear vibration of gear transmission systems”, ASME Journal of Applied Mechanics Review, 56, pp. 309-329, 2003. Kahraman A, Singh R. “Non-linear dynamics of a spur gear pair”, Journal of Sound and Vibration, 142, pp. 49-75, 1990. Theodossiades S, Natsiavas S, Goudas I. “Dynamic analysis of piecewise linear oscillators with time periodic coefficients”, International Journal of Non-linear Mechanics, 35, pp. 53-68, 2000. Shen YJ, Yang SP, Pan CZ. “Nonlinear dynamics of a spur gear pair with time-varying stiffness and backlash”, Journal of Low Frequency Noise, Vibration and Active Control, 23, pp. 178-187, 2004. Shen YJ, Yang SP, Liu XD. “Nonlinear dynamics of a spur gear pair with time-varying stiffness and backlash based on incremental harmonic balance method”, International Journal of Mechanical Science, 48, pp. 1256-1263, 2006. Parey A, Tandon N. “Spur gear dynamic model including defects: a review”, The Shock and Vibration Digest, 35, pp. 465-478, 2003. Wojnarowski J, Onishchenko V. “Tooth wear effects on spur gear dynamics”, Mechanism and Machine Theory, 38, pp. 161-178, 2003. Kuang J, Lin A. “The effect of tooth wear on the vibration spectrum of a spur gear pair”, ASME Journal of Vibration and Acoustics, 123, pp. 311-317, 2001.
A COMBINED CONTINUATION AND PENALTY METHOD FOR THE DETERMINATION OF OPTIMAL HYBRID MECHANICAL TRAJECTORIES K. Yunt, C. Glocker IMES - Center of Mechanics, ETH Zentrum, Tannenstrasse 3, CH-8092 Zurich, Switzerland, E-mail:
[email protected],
[email protected] Abstract:
The aim of this report is to propose a unified framework for the determination of non-smooth trajectories for robotic manipulators with blockable DOF along with a computational scheme. The benefits to represent the dynamics as a measure-differential inclusion will be presented. The optimal control problem will be transcribed into a Nonlinear Programming Problem (NLP) and transformed from the infinite dimensional representation into a finite dimensional representation. The relation to mathematical programs with equilibrium constraints (MPEC) will be established. A numerical scheme will be proposed for the determination of the state and costate trajectories, which can bear discontinuities.
Key words:
Non-smooth analysis, hybrid, optimal control, nonconvex optimization.
1.
INTRODUCTION
There has been much interest in the research of modeling discontinuities and nonlinearities in multibody systems. The discontinuities arising from impacts and stick-slip transitions are primarily contact phenomena, which concur temporally and spatially. The spatial concurrence of discontinuity is due to the fact that discontinuities on velocity level (e.g. collisions) can occur along with discontinuities on acceleration level (e.g. stick-slip transitions). Recent research showed that such rigid-body systems can best 187 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 187–196. © 2007 Springer.
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be described by variational inequalities which lead to nonlinear and linear complementarity type of systems to be solved in order to obtain the accelerations/velocities and forces. The main concern in the research so far has been to determine the forces and accelerations/velocities for autonomous/uncontrolled mechanical systems. In the modeling considered in this work, impulsive forces can arise autonomously, due to effects such as collisions or controlled/nonautonomously, due to actions as blocking of manipulator degrees of freedom suddenly. The introduced framework will have the ability to model control of hybrid mechanical systems with discontinuous transitions among different system modes. The existence of force and impulsive/discrete type of controls will through the solution of the complementarity problem take effect on the course of system trajectories. The physical realization of impactive blocking can be achieved in several ways, however its modeling can be seen as a Coulomb like friction with controlled/adjustable height of the set-valued signum relation. Further, the height of the set-valued signum relation must be high enough to reduce the relative velocity to zero immediately and must be zero if unblocked. The relevance of unilateral and friction modeling in mechanics is therefore closely related to the set-valued impulsive control of systems with blockable DOF. In [1] it has been shown that the determination of the accelerations of a mechanical system subject to unilateral constraints without friction can be represented as a primal and dual quadratic programming problem. Further, it is shown that a generalization of the Gauss’ variational principle is valid in the case of unilateral constraints without friction. In [2], it is shown that a quadratic programming problem can be obtained if Tresca type friction, for which the normal force is decoupled from the tangential force, exists and that the equations of motion along with the linear-complementarity conditions constitute necessary Karush-Kuhn-Tucker conditions of optimality for the quadratic programming problem. In [3] a MPEC is defined as an optimization problem in which the essential constraints are defined by parametric variational inequality or complementarity systems. One of the many representations of a MPEC can be stated as follows: min f ( x, z ), z ∈ S ( x), x ∈ U ad , z ∈ Z . x,z
(1)
The problem stated in (1) includes a subclass of so-called bilevel programs, where S assigns each x ∈ Uad the solution of a “lower-level” optimization problem. In the case where the complementarity system arises from mechanical systems without Coulomb friction, a so-called subclass of MPEC, namely, bilevel programs apply. In references [4], [5], detailed treatment of complementarities and optimization can be found. References [3], [6], treat MPEC and bilevel programs extensively. The control action is represented by x ∈ Uad. The differential measures of control can be
A method for determinating optimal hybrid mechanical trajectories
189
considered as the variables of the “higher-level” optimization problem whereas the contact forces and states are the variables of the “lower-level” problem. By analogy, the measure-differential inclusion, that describes the dynamics as a balance of measures, can be considered as the necessary conditions of a “lower-level” optimization problem represented by the saddle-region restraining set S. If Coulomb type friction exists at the contacts, then the optimal control problem is subjected to variational inequalities and the mechanical quadratic programming problem does not exist any more. In the special case of robotic manipulators with blockable degrees of freedom, let IB denote the index set of blockable DOF of a scleronomic mechanical system. The dynamics of a mechanical system Sv with blockable degrees of freedom (DOF) can be formulated on the measure-differential level by considering the dynamic balance as an equality of measures: M (q ) du − h(q, u + ) dt − W (q ) dT − B (q ) τ dt = 0,
(2)
dTi ∈ − dN i Sgn( γ i+ ),
∀i ∈ I B , dN i ∈ R0+ ,
(3)
γ i+ dN i = 0,
∀i ∈ I B ,
(4)
Here dN and dT represent differential measures of controls, respectively and du denotes the differential measure of the generalized velocity. M(q) is the symmetric positive definite mass matrix and h(q, u+) represents the vector with gyroscopic and coriolis accelerations as well as smooth potential forces such as gravity, B(q) is the generalized direction of single-valued bounded controls, W(q) is the generalized direction of set-valued impulsive unbounded differential measures of control. The entity γ +j denotes the linear or rotational (RCBV) relative velocity at link j. The system has controls of mixed measure and ordinary type. For review of set-valued force laws and MDI in multibody dynamics the reader is referred to [2]. The concept of measure differential inclusion and its applications to mechanics stems from J. J. Moreau, and related works of him are given in [7, 8]. The non-smooth optimal control problem subject to a mechanical dynamical system described as a measure-differential inclusion can be stated as follows: min J = Φ (q ( t f ), u + (t f ), t f ) + ∫
(τ , d Γ , t f )
tf t0
g (u + , q, τ ) dt ,
du = f ( u + , q, τ , t ) dt + PΛ ( q, t ) d Λ + PΓ ( q, t ) d Γ, (d Λ, d Γ) ∈ ϒ( d Λ, d Γ, q, u + ), Π ( u + , q, τ ) ≤ 0, Ψ ( q (t0 ), u − (t0 ), q(t f ), u + (t f )) = 0, t0 fixed, t f free, t ∈ [ t0 , t f ],
(5)
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with absolutely continuous positions q ∈ Rn, right continuous bounded variation (RCBV) generalized velocities u ∈ Rn, control variables τ ∈ Rm, unilateral force differential measures dΛ ∈ Rp, impulsive and set-valued control differential measures dΛ ∈ Rr. Further, the Lebesgue measurable system dynamics is given by f : R n × R n × R m × R → R n . The set-valued variational constraints on the measure variables ϒ : R p × R r × R n × R n → R k, influence matrix of contact force differential measures PΛ ∈ R n× p, influence matrix of control differential measures PΓ ∈ R n×r , state and control constraints Π: R n × R n × R m → R l and boundary constraints Ψ: R n × R n × R n × R n → R q are incorporated in the optimal control problem. The end state cost Φ : R n × R n × R → R , integrand of the cost functional g : R n × R n × R m → R constitute the goal function to be minimized. In the special case of robotic manipulators with blockable degrees of freedom the set-valued variational constraints arise from Equations (3) and (4).
2.
THE TRANSCRIPTION OF THE OPTIMAL CONTROL PROBLEM INTO A NLP
In this subsection the equalities and inequalities that represent the finite dimensional optimization of a scleronomic rigidbody mechanical system with f DOF, m blockable joints and s controls is presented. Consider the problem: min f ( y ) h1 ( y ) = 0,..., hm ( y ) = 0, g1 ( y ) ≤ 0,..., g r ( y ) ≤ 0,
The corresponding augmented Lagrangian function that is being successively minimized can be obtained as: La ( y k , λ k , µ k ) = f ( y k ) + ( λ k ) h k + T
ck k T k 1 h ) h + k ( 2 2c
r
∑{(µ j =1
) − ( µ kj ) 2 }
k +1 2 j
Here y = proxC(x) denotes the nearest point y ∈ C to x. The vectors
λk , µ k denote the Lagrange multipliers of the equality and inequality constraints, which are updated as follows:
λik +1 = λik + c k hi ( y k ),
i = 1,..., m,
µ kj +1 = proxR ( µ kj + c k g j ( y k )),
j = 1,..., r.
+ 0
In order to formulate the equality and inequality conditions of the optimization problem the impulsive and set-valued blocking control will be decomposed into four complementarities making use of the relations:
A method for determinating optimal hybrid mechanical trajectories
γ R+ dTR = 0, γ L+ dTL = 0, γ R+ dN = 0, γ L+ dN = 0, γ R+ ≥ 0, γ L+ ≥ 0, dTR ≥ 0, dTL ≥ 0, dN ≥ 0.
191 (6)
after introduction of the slack variables that are related by following equations: dTR = dT + dN , dTL = − dT + dN , γ + = γ R+ − γ L+ .
(7)
The four linear complementarities, each consisting of one equality and 2 inequality constraints will be replaced by the Fischer-Burmeister function in the following manner: 0 ≤ x ⊥ y ≥ 0 ⇔ Φ ( x, y ) = x 2 + y 2 − x − y = 0,
(8)
which has first been introduced in [12] in the framework of nonlinear programming. The subdifferential (in the sense of convex analysis [10]) of this function at the origin is a closed convex set, given by the unit circle with its center at point (-1, -1) and at every other point the subdifferential of it consists of a single value. The general set of equalities and inequalities for the finite dimensional optimization can be stated as follows: i = 1,..., N
min f ( y ), y
n
∑m k =1
jk
r
tf
k =1
N
(qi , ω ) (u( i +1) k − uik ) − ∑W jk (qi , ω ) Tik − h j (qi , ui , ω )
s
−∑ B jk (qi , ω )τ k
tf
= 0, N Φ (γ ( i +1) j , Tij + N ij ) = 0,
j = 1,..., n,
k =1
n
j = 1,..., r ,
Φ (γ ( i +1) j − ∑Wkj (qi , ω ) u( i +1) k , −Tij + N ij ) = 0,
j = 1,..., r ,
Φ (γ (i+1) j , N ij ) = 0,
j = 1,..., r ,
k =1
n
Φ (γ ( i +1) j − ∑Wkj (qi , ω ) u( i +1) k , N ij ) = 0,
j = 1,..., r ,
τ min ≤ τ ij ≤ τ max , j j
j = 1,..., s,
qN +1, j − q dj = 0,
j = 1,..., n,
u N +1, j − u = 0,
j = 1,..., n,
k =1
d j
tmin ≤ t f ≤ tmax .
Here mjk(qi,ω), hj(qi, ui, ω) and Bjk(qi,ω) denote the respective entries of the PD symmetric mass matrix, the vector of smooth and single-valued forces and the generalized forced directions of the controls, respectively.
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K. Yunt, C. Glocker
Wjk(qi,ω) denote the entries of the matrix of generalized directions of the impulsive control differential measures dT. It is assumed that for all DOF desired end positions qd and end velocities ud are specified. The temporal discretization of the generalized positions are based on the sweeping process that stems from J. J. Moreau.
2.1
Description of The Algorithm
The minimization of nonconvex non-smooth optimization problems is a relatively new and intensively researched field. The most promising methods in minimizing such problems are classified in two main groups, namely, bundle methods and subgradient methods. Though bundle methods are Table 1. Optimization parameters, initial and final states of the optimization #of disc. points #of dual variables #of complementarities α f [rad] β0 [rad / sec]
200 2810 800 3π / 4 0
penalty c Increment in ω α 0 [rad] α f [rad/sec] β f [rad]
1e5 0.1 0 0
π
#of primal variables kstage α 0 [rad/sec] β 0 [rad] β f [rad/sec]
1201 40 0 0 0
shown to be more efficient, large scale problems (number of variables > 500) are not easily handled because of the data storage requirements. In the used application, the number of variables easily exceeds 1000 so an augmented Lagrangian based subgradient method is developed. The augmented Lagrangian function is Lipschitz as a function of the primal variables y and is tangentially regular in the sense of Clarke [9], which is constructed by the discretized equations in the previous subsection. The algorithm consists of three iterations which are embedded in each other. The most outer iteration is adjusting the continuation parameter. The continuation parameter ω is increased so that when ω becomes one the system of equalities and inequalities fully represents the discretized non-smooth mechanical system. The intermediate iteration performs for each given ω a number of successive minimizations of the resulting augmented Lagrangian function Lak and updates after every iteration the dual multiplier vectors λk . The inner iteration performs the minimization of the augmented Lagrangian Lak for a given ω and Lagrange multiplier vector λk by a nonlinear conjugate gradients method, which requires the gradient of Lak with respect to y explicitly. The continuation parameter ω has been increased gradually to one and the intermediate systems of equations are partially minimized in order to approximate the optimal state and Lagrange multiplier trajectories iteratively. The penalty parameter c is increased as the value of ω is gradually increased to one. At the start of each intermediate iteration the
A method for determinating optimal hybrid mechanical trajectories
193
vector λk is initialized to zero. In [11] properties of augmented Lagrangians and subgradient methods can be found in their general guidelines.
3.
NUMERICAL EXAMPLE
The example mechanical system can be seen in Figure (1). It has two rotational degrees of freedom denoted by a and b. The DOF a is controlled continuously in a single valued manner, whereas DOF b can be blocked. In Table 2 the numerical values of the mechanical properties along with the torque limitations on the continuous actuator t are given. The setting of the time optimal control problem is summarized in Table 1. The DOFβ is blocked at times t1=0 sec, t3=0.546 sec, t5=2.678 sec and t7=3.395 sec, and is released at times t2=0.4435 sec, t4=1.5 sec, t6 =2.678 sec, whereas the total maneuver takes 3.4 sec. When the link is blocked, the whole system possesses one mechanical DOF, whereas when released it has two DOF. The transitions at times t3, t5 and t7 are impactive as can be seen in Figure (3). Table 2. System parameters of the manipulator m1
m2
m3
m4
[ kg ]
[ kg ]
[ kg ]
[ kg ]
1
1
1
1
Θ1
Θ2
Θ3
Θ4
l1
[kg m 2 ] [kg m 2 ] [kg m 2 ] [kg m 2 ] [m] 0.05
0.05
0.05
0.05
1
l2
τ
[ m]
[ Nm]
1
< 2
Figure 1. The 2-DOF planar manipulator with 1 blockable DOF
The control history has a time-optimal bang-bang character. At t2 the system switches from 1-DOF mode to 2-DOF mode. At times t3, t7 the system switches to the 1-DOF system. The speciality of blocking action at t5 is that the system switches from the 2-DOF mode to itself, and the blocking is
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K. Yunt, C. Glocker
triggered at a time instant when the relative velocity is zero as can be seen in Figure (3).
4.
DISCUSSIONS AND CONCLUSIONS
A numerical method is presented for the determination of optimal trajectories for mechanical structure-variant systems with blockable DOF. The method benefits from a sound modeling approach for structure-variant systems based on the measure-differential inclusion approach (MDI), which has first been mentioned in the works of J. J. Moreau such as [7] and [8]. As a consequence, the location and time of phase transitions where the system changes DOF is not prespecified but is determined as an outcome of the optimization. The method minimizes over modes as well, and chooses a sequence of modes and transitions which possess a certificate of optimality. The proofs of convergence and the certificate of optimality are detailed in [13]. Though the underlying system might undergo structure-variant phase changes such as impactive phase transitions a mixed integer approach is not necessary. Normal Braking Moment
Frictional Moment 30
T in [Nm sec]
N in [Nm sec]
30 20 10 0
20 10 0 −10 −20 −30
0
0.5
1
1.5 2 time in [sec]
2.5
3
0
0.5
1
Moment of the actuated Link
1.5 2 time in [sec]
2.5
3
2.5
3
Slack Velocity
2 3
[rad/sec]
τ in [Nm]
1 0
2 1
−1 −2 0
0.5
1
1.5 2 2.5 time in [sec]
3
3.5
0
0
0.5
1
1.5 2 time in [sec]
Figure 2. The optimal history of blocking control and smooth control
A method for determinating optimal hybrid mechanical trajectories v
v
α
β
3
2.5
2
[rad/sec]
2
[rad/sec]
195
1.5 1
1 0
0.5 −1
0 −0.5
−2 0
0.5
1
1.5 2 time in [sec]
2.5
3
0
0.5
vα−vβ
1.5 2 time in [sec]
2.5
3
Generalized Positions α
3 2
β
3
1
[rad]
[rad/sec]
1
0
2
−1 1
−2 −3 0
0.5
1
1.5 2 time in [sec]
2.5
0
3
0
0.5
1
1.5 2 time in [sec]
2.5
3
Figure 3. The optimal position and velocity trajectories
Λα
Λβ
400
600
200
400
0
200
−200 0 −400 −200 −600 −400 −800 −600
−1000
−800
−1200 −1400
−1000 0
0.5
1
1.5 2 time in [sec]
2.5
3
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Figure 4. The optimal costate trajectories with discontinuities
Knowledge regarding adjoint variables is not necessary and the dual vector converges to the adjoint state as the optimization proceeds as can be seen in Figure (4). The obtained costate trajectories are discontinuous and
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non-smooth, which are themselves described in terms of two discontinuous second-order differential equations. The sufficiency condition for a convergence to an at-least locally optimal solution is the existence of a nonempty feasible set, which is known as the slater condition. The time stepping based discretization scheme provides a means to circumvent the difficulties associated with event-driven algorithms that lead to mixed integer programming problems for mechanical systems. The global convergence of the combined continuation scheme and the augmented Lagrangian for solving the underlying problem is enhanced by the convexification induced utilizing augmented Lagrangian based minimization. To the best knowledge of the authors, the approach presented here, is the first NLP scheme proposed for large-scale non-smooth optimization with complementary constraints by making use of Lipschitz continuous augmented Lagrangian based subgradient method.
REFERENCES 1. 2. 3.
4. 5. 6. 7. 8.
9. 10. 11. 12. 13.
Moreau JJ. “Quadratic programming in mechanics: Dynamics of one-sided constraints”, SIAM Journal of Control, 4, pp. 153-158, 1966. Glocker Ch. “Set-Valued Force Laws”, Dynamics of Non-Smooth Systems, Lecture Notes in Applied Mechanics, 1, Springer-Verlag, Berlin, 2001. Outrata J, Kocvara M, Zowe J. “Non-smooth Approach to Optimization Problems with Equilibrium Constraints”, Nonconvex Optimization and its Applications, Kluwer Academic Publishers, Dordrecht, 28, 1998. Cottle RW, Pang JS, Stone RE. “The Linear Complementarity Problem”, Academic Press, Boston, 1992. Murty KG. “Linear Complementarity, Linear and Nonlinear Programming”, Helderman-Verlag, 1988. Luo ZQ, Pang JS, Ralph D. “Mathematical Programs with Equilibrium Constraints”, Cambridge University Press, Cambridge, 1996. Moreau JJ. “Bounded Variations in time”, In: Topics in Non-smooth Mechanics, Edts: J. J. Moreau, P. D. Panagiotopoulos, G. Strang, Birkhauser, Basel, pp. 1-74, 1988. Moreau JJ. “Unilateral Contact and Dry Friction in Finite Freedom Dynamics”, Nonsmooth Mechanics and Applications, CISM Courses and Lectures, 302, Springer Verlag, Wien, 1988. Clarke FH. “Optimization and Non-smooth Analysis”, SIAM Classics in Applied Mathematics, Wiley, New York, 1983. Rockafellar RT. “Convex Analysis”, Princeton Landmarks in Mathematics, Princeton University Press, 1970. Bertsekas DP. “Nonlinear Programming”, 2nd Ed., Convex Analysis and Optimization, Optimization and Computation Series, Athena Scientific, Massachusetts, 1999. Fischer A. “Solution of monotone complementarity problems with locally Lipschitzian functions”, Mathematical Programming, 76, pp. 513-532, 1997. Yunt K, Glocker Ch. “Trajectory Optimization of Hybrid Mechanical Systems using SUMT”, IEEE Proc. of Advanced Motion Control, Istanbul, 2006, pp. 665-671.
PART 4
DYNAMICS OF HIGH-DIMENSIONAL SYSTEMS
NUMERICAL PREDICTION AND EXPERIMENTAL OBSERVATION OF TRIPLE PENDULUM DYNAMICS J. Awrejcewicz, G. Kudra, G. Wasilewski Department of Automatics and Biomechanics, Technical University of Łódź, 1/15 Stefanowskiego St., 90-924 Łódź, Poland, E-mail:
[email protected] Abstract:
Numerical and experimental studies of a triple physical pendulum are performed. The experimental setup of the triple pendulum with the first body externally excited by the square-shape function of time is build and described. The mathematical model of the real system is presented and the real rig parameters are estimated. Then, numerical analysis of the model is performed obtaining a good agreement with the results obtained from the experiment. It is shown that the presented model can be used as a tool for fast prediction of the real system behavior.
Key words:
Triple pendulum, experiment, identification, chaos, Lyapunov exponents.
1.
INTRODUCTION
A great role played by a pendulum in the history of mechanics and nonlinear dynamics is observed. It is caused by simplicity of that system on the one hand, and due to many fundamental and spectacular phenomena exhibited by a single pendulum on the other hand. In mechanics and physics investigations of single and coupled pendulums are widely applied [1, 2]. The subject of a study can be either a mathematical model or a real physical system. Usually these two objects are investigated simultaneously, and a problem of mathematical model and experimental rig matching arises. Although a single or a double pendulum (in their different forms) are quite often studied experimentally [3-5], a triple physical pendulum is rather rarely presented in literature from a point of view of real experimental 197 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 197–205. © 2007 Springer.
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object. For example, in the work [6] the triple pendulum excited by horizontal harmonic motion of the pendulum frame is presented and a few examples of chaotic attractors are reported. There are two aspects of the interest in the pendulum dynamics. The first one is that the single and coupled pendulums are the very rich sources of many fundamental phenomena of non-linear dynamics. The second one is the possibility of modeling of many natural and technical objects by the use of system of pendulums. The example is the piston – connecting rod – crankshaft system modeled as a triple physical pendulum with rigid limiters of motion [7]. This work is a continuation of earlier studies of authors [7, 8] on a triple physical pendulum. In those studies a numerical model of triple physical pendulum with rigid limiters of motion was formulated. Such a system can exhibit impacts as well as a sliding solution, with permanent contact with the obstacle on some time intervals. Special numerical tools for non-linear dynamics analysis of that system exhibiting discontinuities was developed and tested. Also the possible application of the numerical model was presented: the piston – connecting rod – crank-shaft system of a combustion engine [8]. On the present stage of investigations the experimental rig of triple pendulum without obstacles is build. The mathematical model presented here is a special case of the models from being studied by us earlier [7, 8].
2.
EXPERIMENTAL RIG
The experimental rig (see Figure 1) of the triple physical pendulum consists of the following subsystems: pendulum, driving subsystem and the measurement subsystem. The pendulum possesses a stand and three links with adjustable lengths and masses suspended on the tripod and joined by the use of radial and axial needle bearings. The periodic square-shape in time external forcing acting on the first body is implemented by the use of the direct-current motor of our own construction with optical commutation. The voltage conveyed to the engine inductors is controlled by the use of special digital system again of our construction in order to obtain desired amplitude and frequency of the square-shape in time forcing. The measurement of the angular position of the three links is realized by the use of the precise rotational potentiometers. Then the Lab View measure-programming system is used for experimental data acquisition and presentation in the computer.
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Figure 1. Experimental triple pendulum.
3.
MATHEMATICAL MODELLING
Figure 2 presents idealized physical concept of the real pendulum presented in Figure 1. Its mathematical description follows our earlier works [7, 8], where the governing equations of the triple physical pendulum in nondimensional form have been reported. The system is idealized assuming that it is an ideally plane system of coupled links, moving in the vacuum with linear damping in joints. Each of the pendulums has a mass center lying in the lines including the joints, and one of the principal central inertia axes (zci) of each link is perpendicular to the movement plane.
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Figure 2. Model of the triple pendulum.
The system is governed by the following set of differential equations + N ( ψ ) ψ 2 + Cψ + p ( ψ ) = fe ( t ) , M (ψ) ψ
(1)
where: ⎡ B1 M ( ψ ) = ⎢ N12 c12 ⎢⎣ N13c13 ⎡ c1 + c2 C = ⎢ −c2 ⎢⎣ 0
N12 c12 B2 N 23c23
−c2 c2 + c3 −c3
N13c13 ⎤ ⎡ 0 N 23c23 ⎥ , N ( ψ ) = ⎢− N12 s12 ⎢⎣ − N13 s13 B3 ⎥⎦
N12 s12 0 − N 23 s23
N13 s13 ⎤ N 23 s23 ⎥ , 0 ⎥⎦
⎧ψ 2 ⎫ ⎧ f e1 ( t ) ⎫ 0 ⎤ ⎧⎪ M 1s1 ⎫⎪ ⎪ ⎪ 2 ⎪ 12 ⎪ −c3 ⎥ , p ( ψ ) = ⎨ M 2 s2 ⎬ , fe ( t ) = ⎨ 0 ⎬ , ψ = ⎨ψ 2 ⎬ , c3 ⎥⎦ ⎪⎩ψ 32 ⎪⎭ ⎩⎪ M 3 s3 ⎭⎪ ⎩⎪ 0 ⎭⎪ (2)
cij = cos(ψ i - ψj), si j = sin(ψi - ψj) and ψi ( i = 1,2,3) is the rotation angle of the i-th link. The vector of parameters follows µ = [ B1 , B2 , B3 , N12 , N13 , N 23 , M 1 , M 2 , M 3 , c1 , c2 , c3 ] .
(3)
Note the parameters of external forcing fe1(t) (amplitude q and frequency f ) are treated separately.
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More details regarding mathematical model and relations between both model and physical parameters shown in Figure 2 are given in references [7-9]. The model parameters are then estimated by the global minimum searching of the criterion-function of the output signals ψi(t) from model integrated numerically and real pendulum matching assuming the same inputs (external forcing fe1). Together with the model parameters also initial conditions of the numerical simulation are estimated. The sum of squares of deviations between corresponding samples of signals from model and experiment, for few different solutions serves as a criterion function. A minimum is searched applying simplex method. For the parameters estimation two periodic experimental series are used for different forcing frequencies f = 0.65 Hz and f = 0.85 Hz and for the same forcing amplitude q = 2 Nm. The following set of parameters is obtained:
B1 = 0.19574 kg m 2 ,
B2 = 0.16368 kg m 2,
B3 = 0.02022 kg m 2 ,
N12 = 0.12972 kg m 2 ,
N13 = 0.02540 kg m 2,
N 23 = 0.03164 kg m 2 ,
M 1 = 10.1009 kg m 2 s 2 ,
M 2 = 7.5154 kg m 2 s 2,
M 3 = 1.3926 kg m 2 s 2
c1 = 0.06015 kg m 2 s , c2 = 0.00261 kg m 2 s , c3 = 0.00171 kg m 2 s , (4) Note that the model parameters are optimal (if the global minimum is found) in the sense of the best matching of output signals from both model and real pendulum rather than in the sense of the best real physical values approximation. From this point of view even somehow artificial values of damping coefficients are not very important, since the model serves for a prediction of behavior of the real pendulum, as it will be shown in the next section.
4.
NUMERICAL PREDICTION AND EXPERIMENTAL OBSERVATION
In this section we present results of investigations of experimental triple pendulum and the corresponding mathematical model with parameters (4) and forcing amplitude q = 2 Nm. Figure 3 shows periodic solutions observed experimentally and numerically for the forcing frequencies f = 0.65 Hz and f = 0.85 Hz. A good agreement between experimental and numerical series is obtained, and hence these solutions have been used in the model parameters identification. Figure 4 contains a bifurcation diagram for the mathematical model with the forcing frequency f as a bifurcation parameter. Chaotic window for f ∈
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(0.698, 0.771) is observed being confirmed well by the experimental observations (chaotic zone for f ∈ (0.695, 0.774)). 0.65 Hz
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Figure 5. Chaotic solution for forcing frequency f = 0.73 Hz observed experimentally and numerically.
Figure 5 presents comparison of chaotic solutions for forcing frequency f = 0.73 Hz obtained experimentally and numerically. Both solutions start from the same initial conditions and their divergence can be observed. The characteristic feature of this attractor is manifested through full rotations of each of the links from time to time. Figure 6 contains two projections of the Poincaré section of the attractor of the mathematical model for f = 0.73 Hz. The chaotic character of the presented attractor is confirmed by the Lyapunov exponents (4.4, 1.9, 0.4, -0.7, -2.3, -4.7) shown in Figure 7, where three positive exponents are exhibited. Lyapunov exponents are estimated from the differential equations and the method of Wolf [10] has been used. Transient motion of length 104 s is ignored and computation time reaches 104 s. The period of the applied Gramm-Schmidt reorthonormalization is equal to 0.5 s. 25
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5.
CONCLUDING REMARKS
Because of the method of parameter estimation used, the model parameter values are not optimal in the sense of the best real physical values approximation, but rather in the sense of the best matching of output signals from the model and the real pendulum. Good agreement between both numerical simulation results and experimental measurements presented in the paper lead to conclusion that used mathematical model of triple pendulum with its parameters estimated can be also applied as a tool for quick searching for various phenomena of nonlinear dynamics exhibited by a real pendulum as well as for their explanation. There are two sources of differences between results of numerical simulation and experimental observations. Firstly, the mathematical model may be not sufficiently complex for describing some real physical phenomena in the triple pendulum. It especially concerns the damping in the joints of the real pendulum, where a more complicated phenomena then linear damping may exist. Secondly, the method of global minimum finding, for the criterion-function, in the case of multi-dimensional problem (the simplex method) does not belong to perfect ones. In other words sometimes it is not clear, that we have found a global minimum and not just a local one.
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ACKNOWLEDGEMENTS This work has been supported by the Ministry of Science and Information (grant No 4 T07A 031 28).
REFERENCES 1.
Skeledon AC and Mullin T. “Mode interaction in a double pendulum”, Phys. Lett., A166, pp. 224-229, 1992. 2. Skeledon AC. “Dynamics of a parametrically excited double pendulum”, Physica, D75, pp. 541-558, 1994. 3. Blackburn JA, Zhou-Jing Y, Vik S, Smith HJT, and Nerenberg MAH. “Experimental study of chaos in a driven pendulum”, Physica, D26, pp. 385-395, 1987. 4. Heng H, Doerner R, Hubinger B, and Martienssen W. “Approaching nonlinear dynamics by studying the motion of a pendulum. I. Observing trajectories in state space”, Int. J. Bifurcation and Chaos, 4, pp. 751-760, 1994. 5. Bishop SR and Clifford MJ. “Zones of chaotic behaviour in the parametrically excited pendulum”, J. Sound and Vib, 189, pp. 142-147, 1996. 6. Zhu Q and Ishitobi M. “Experimental study of chaos in a driven triple pendulum”, J. Sound and Vib, 227, pp. 230-238, 1999. 7. Awrejcewicz J and Kudra G. “The piston-connecting rod-crankshaft system as a triple physical pendulum with impacts”, Int. J. Bifurcation and Chaos, 15, pp. 2207-2226, 2005. 8. Awrejcewicz J, Kudra G, and Lamarque CH. “Investigation of triple physical pendulum with impacts using fundamental solution matrices”, Int. J. Bifurcation and Chaos, 14, pp. 4191-4213, 2004. 9. Awrejcewicz J, Supeł B, Kudra G, Wasilewski G, and Olejnik P. “Numerical and experimental study of regular and chaotic behaviour of triple physical pendulum”, Fifth EUROMECH Nonlinear Dynamics Conference, ENOC2005, Eindhoven, The Netherlands, pp. 1817-1824, 2005 (CD Rom). 10. Wolf A, Swift JB, Swinney HL, and Vastano JA. “Determining Lyapunov Exponents from a time series”, Physica, D16, pp. 285-317, 1985.
NONLINEAR VIBRATION MODES AND ENERGY LOCALIZATION IN MICRO-RESONATOR ARRAYS A. J. Dick, B. Balachandran, C. D. Mote, Jr. Department of Mechanical Engineering, University of Maryland, College Park, MD 207423035, USA. E-mail:
[email protected] Abstract:
In recent decades, a particular localization phenomenon called intrinsic localized modes has been studied with great interest in physics and other fields. Here, this localization is studied in the context of micro-electromechanical systems to determine if they can be realized by using nonlinear normal modes or nonlinear vibration modes. Various analytical methods are explored, and a modified version of the invariant manifold approach is shown to be successful in predicting the amplitude profile of an intrinsic localized mode and initiating localizations within the considered micro-array.
Key words:
Nonlinear vibration, intrinsic localized modes, micro-resonators.
1.
INTRODUCTION
Localizations caused by a system’s intrinsic properties have been studied by physicists and other scientists for a number of decades. One of the earliest publications on this phenomenon is Anderson’s 1958 paper, in which the occurrences of localization within random lattices are documented [1]. Recently, the focus of these studies has shifted to the presence of localizations in homogeneous anharmonic lattices [2]; that is, systems that consist of a perfectly periodic array of coupled components that behave in a nonlinear fashion. Although these localizations display the same characteristics of localizations as that due to a defect within a harmonic lattice, these types of localizations are solely due to the combination of a system’s nonlinearity and its discreteness. 207 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 207–216. © 2007 Springer.
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1.1 Localization in anharmonic lattices These types of localizations were uncovered through two separate paths of study and as such, they are referred to as both “discrete breathers” (DBs) and “intrinsic localized modes” (ILMs). These localizations have been studied in many different systems and while their main characteristics remain similar, a variety of terms are used to convey more specific information about the event. Localized vibration modes (LVM) have been studied in solid-state materials such as the charge transfer solid PtCl studied by Swanson et al. [3] with the aid of resonance Raman spectra. A high power microwave source has been used to produce intrinsic localized spinwave modes (ILSM) in quasi-one-dimensional antiferromagnetic chains [4]. Solitons have been optically induced in two-dimensional photonic structures and studied [5]. Josephson junctions, which consist of two superconductive materials separated by an ultra-thin insulating layer, have been coupled together to form arrays where localized rotation modes called rotobreathers have been studied [6]. The phase difference across the junction is monitored and this difference can be modeled in the same manner as the rotation of a forced, damped pendulum. A recent addition to the different systems in which localization has been studied is micro-electro-mechanical systems (MEMS) [7-8].
1.2 Micro-scale cantilever arrays The micro-electro-mechanical system considered in studies [7-8] is an array of coupled micro-scale cantilever beams. Since the system is uniformly excited, it is necessary that the array consist of repeating cantilever pairs of two different sizes. This is done so that the dispersion curves are folded over and the highest frequency vibration mode is at the zone center. The following form of the Klein-Gordon equations are used to model the behavior of the two cantilevers that make up a unit cell [9], which is illustrated in Figure 1a. m ma xa,i + a xa,i + k2a xa,i + k4a xa3,i + k I 2 xa,i − xb,i − xb,i −1 = ma α τ (1) mb mb xb,i + xb,i + k2b xb,i + k4b xb3,i + k I 2 xb,i − xa,i +1 − xa,i = mb α
(
τ
(
) )
Representative system parameter values are provided in Table 1. To obtain an ILM, the array is initially excited in a sinusoidal fashion with a chirp that ends above the array’s highest frequency, then this excitation frequency is held constant at this value for a period of time, and finally,
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Figure 1. a) unit-cell of micro-cantilever array and b) example of localization within an array of coupled nonlinear oscillators.
the excitation is removed. The array is initialized with small, random displacements. Due to the ease of observing the behavior of this system, different types of ILMs can be observed and additional studies have been conducted. The first type of ILM is called an “unlocked ILM”. These localizations have relatively low energy levels and as a result, are able to move freely from site to site within the array. The second type of localization is characterized by a higher level of energy. These “locked ILMs” become spatially fixed within the array. Occasionally, this type of ILM can become synchronized with the constant frequency excitation and this subcategory is referred to as “pinned ILMs”. Experiments and numerical simulations have provided insights into how these different types of ILM behave within a micro-cantilever array. Work has also been conducted on the realization of pinned ILMs to develop a method for manipulating the position of the localization while a constant frequency excitation is applied [10]. It has been found that the addition of an artificial impurity into the system could be used to manipulate the ILM. In experimental work, lasers have been used to soften cantilevers within an array in order to manipulate a pinned ILM. Recent work on the nonlinear behavior of microscale resonators suggests that when joined together to form an array, favorable conditions do exist for the occurrence of intrinsic localized modes [11-13]. It is conceivable that the unique behavior of ILMs can be exploited to improve the designs of micro-scale systems and perhaps even contribute to the development of a new technology designed around the ability of a pinned ILM to focus energy within an array. It is for this reason that in this work, analyses has been conducted to determine if these ILMs can be realized as nonlinear vibration modes or nonlinear normal modes; this can help to eliminate some of the uncertainty about the choice of initial conditions associated with this unique nonlinear phenomenon. The rest of this article is organized as follows. In the next section, the analyses conducted to calculate an amplitude profile of a pinned ILM are
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Table 1. Micro-cantilever array parameters [7]. Parameter Value Mass of large cantilever, ma 5.46*10-13 kg Mass of small cantilever, mb 4.96*10-13 kg Time constant, τ 8.75*10-3 sec Linear stiffness of large cantilever, k2a 0.303 N/m 0.353 N/m Linear stiffness of small cantilever, k2b Nonlinear stiffness, k4 = k4a = k4b 5.0*108 N/m³ Interconnect stiffness, kI 0.2041 N/m Acceleration magnitude, α0 1.0*104 m/s²
presented. The third section contains the results of these analyses, and a discussion of them.
2.
ANALYSES
Previous studies have focused on the behavior of the ILMs within the array and how they interact with each other. Within this study, the details of the ILM and the form of the associated amplitude profile are investigated. To conduct the analyses, a reference amplitude profile is first obtained by examining the displacement data from the numerical simulations.
2.1 Simulation profile In order to obtain a profile of the amplitude that accurately represents the ILMs observed in the simulations, multiple profiles are averaged. To account for the oscillation of the cantilevers, each profile is normalized by the displacement at the center of the localization. To focus on the ILM, the range of the profiles to be averaged is limited to only the cantilevers in the immediate vicinity of its center. Data that includes many periods of oscillation is selected from the second excitation phase of the simulation, following the decay of any transient localization. The normalized profiles are averaged to produce a single profile to describe the oscillation amplitudes of the cantilevers while a pinned ILM is present, as shown in Figure 2a. Additionally, the standard deviation is calculated for the normalized profiles in order to determine how accurately the average represented the ILM profile of the form:
x±1 = r1 x0 ,
x±2 = r2 x±1
(2)
In (2), the symmetry of the ILM is recognized with the plus/minus symbol. The amplitude ratio, r1, is the ratio of the amplitude of the cantilevers x±1 one
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(b)
Figure 2. a) average normalized profile of cantilever amplitudes from simulation with average plus standard deviation and average minus standard deviation and b) array model simplification for studying ILMs.
site away from the center of the ILM to the amplitude of the cantilever at the center x0. The second amplitude ratio, r2, is the ratio of the amplitude of the cantilevers x±2 two sites away from the center of the ILM to the amplitude of x±1. Only two amplitude ratios are presented because additional calculations suggest that the two ratios repeat with an increasing amount of deviation while moving away from the center of the ILM. This pattern agrees with the repeating cantilever pairs and decreasing displacement levels. From the simulations conducted, the amplitude ratios are found to have the values of r1 = -0.21 and r2 = -0.70. The negative sign in the amplitude ratio indicates that the two cantilevers are oscillating 180° out of phase. x±1 = c1 x0 + c2 x03 ,
x±2 = c3 x±1 + c4 x±31
(3)
In an effort to accommodate the nonlinearity of the cantilevers, a cubic term is added to each relationship, as shown in (3). Instead of normalizing the profiles, a least-squares minimization is used to identify the values of c1 and c2. This is done by comparing the expected and true displacement values for x±1. A second minimization is used for c3 and c4 with the expected displacement value of x±2 calculated from the expected displacement value of x±1. The identified coefficient values are c1 = -0.2060, c2 = 2.4657×108, c3 = -0.6679, and c4 = 4.3477×1010. The significantly larger magnitude of the value for c4 is due to the larger influence of transient localizations attributed to smaller displacement levels in x±2 as a result of the localization.
2.2 Array model With a reference profile from the simulations, a number of analytical methods are investigated to study this localization phenomenon. To perform these analyses, the damping and excitation are disregarded resulting in (4).
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The coupling terms and nonlinear terms are grouped together and represented by the function G given by (5). x j ( t ) + ω 2j x j ( t ) + G j = 0
(4)
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(5)
k
Here the coefficients g1,j,k and g2,j,k are the coefficients of the nonlinear terms and the linear coupling terms, respectively. The index j corresponds to the influenced oscillator and the index k corresponds to the influencing oscillator. The index j in (4) determines whether a large, lower frequency cantilever (j, even) is considered or a small, higher frequency cantilever (j, odd) is considered. In order to simplify the model, symmetry about the center of the localization is considered. To this end, the influence of the cantilever at x-1 on x0 is doubled so that the influence of the cantilever at x+1 can be ignored. The second step in to neglect the influence of x-2 on x-1. As the amplitude of x-2 in the simulation profile is only about fourteen percent of the amplitude of x0, its affect on x-1 through the coupling is significantly less and it is neglected. Through these two simplifications, the profile of the ILM can be modeled as shown in (6) with two coupled nonlinear oscillators. m ma xa + a xa + k2a xa + k4a xa3 + k I ( 2 xa − xb ) = 0 τ mb mb xb + xb + k2b xb + k4b xb3 + 2 k I ( xb − xa ) = 0
(6)
τ
2.3 Analytical profile Given the nonlinear nature of the array, it is investigated whether an intrinsic localized mode can be realized by using nonlinear normal modes or nonlinear vibration modes as a basis. The methods for calculating nonlinear normal modes that have been examined include the Method of Multiple Scales Approach, the Restricted Normal Mode Approach, and multiple variations of the Invariant-Manifold Approach [14, 15]. In using the method of multiple scales, multiple time scales are used, as shown in (7), to produce an approximate solution of a nonlinear differential equation [e.g., 16]. Here, ε is used as the non-dimensional scaling parameter and it is such that ε 1 . The coupling terms and nonlinear terms are assumed to be of a scale smaller than the other terms.
Nonlinear vibration modes and localization in micro-resonator arrays Tn = ε nt ,
xi ( t ) =
N
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213 (7)
n =0
x0, k (T0 , T1 ) = a (T1 ) cos (ωk T0 + β (T1 ) ) ,
x0, j (T0 , T1 ) = 0
(8)
By using this approach, the analytical approximation (8) is obtained as a solution of (6); the response component of the influencing oscillator k is harmonic while the influenced oscillator j has no response component. The response component of oscillator j for the slower time scale is then calculated and as a result of the coupling between the two oscillators, the response of the jth oscillator is determined as a function of the response of the kth oscillator [17]. By selecting the necessary values for j and k, the desired equations are derived to relate the oscillation amplitudes of the cantilevers when an ILM occurs and produce the corresponding profile. In order to employ the Restricted Normal Mode Approach, the simplified model is necessary. As shown in (9), the two cantilevers are assumed to be oscillating at the same frequency but with different amplitudes A and B. The amplitude values are defined as the product of a variable R and a sinusoidal function of a variable θ. An additional variable p is defined as the amplitude ratio A B . X a (t ) = A sin (ω t ) = R sin (θ ) sin (ω t ) X b (t ) = B sin (ω t ) = R cos (θ ) sin (ω t )
(9)
On substituting (9) into (6) and eliminating the frequency variable ω, a fourth-order polynomial in p is obtained. The four roots of the polynomial are functions of the system parameters and the variable R. These functions are used to calculate the values of the amplitudes A and B. By choosing the appropriate values of R, the resulting A and B values are calculated with the appropriate root pn to obtain the amplitude ratios. The amplitude ratios are used to produce an amplitude profile for the ILM. The third method examined is the Invariant-Manifold Approach. In this approach, a manifold is constructed to describe the behavior of one cantilever as a function of another cantilever. The form of the manifold equations is selected based on the form of the terms in G. A two-dimensional real-variable manifold, as shown by (10), is chosen and here, Xjk and Vjk are the associated displacement equation and velocity equation, respectively. x j ( t ) = X jk ( xk , vk ) ,
v j ( t ) = V jk ( xk , vk )
(10)
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∂ X jk
⎡ −ω 2 x − G ⎤ = V k k ⎥⎦ jk ∂vk ⎢⎣ k ∂ V jk ⎡ −ω 2 x − G ⎤ = −ω 2 X − G vk + j jk j k k ⎥⎦ ∂xk ∂vk ⎣⎢ k
∂xk ∂ V jk
vk +
(11)
After making use of Equations (6), (10) and (11) are obtained and the unknown coefficients are determined. The manifold equations are used to relate the oscillation amplitudes of the cantilevers when an ILM is present. An amplitude profile is calculated with these equations. After the initial implementation of the Invariant-Manifold Approach, it is determined that the three methods examined are unable to accurately predict the profile of an ILM in this system. This appears to be due to the dominant linear behavior of the system and the presence of only linear coupling. To accommodate these characteristics, a modified version of the Real-Variable Invariant-Manifold Approach is used. While this method conforms to most of the requirements of the Invariant-Manifold Approach, it differs because of the inclusion of linear terms in the manifold equations, as shown in (12). By following the standard procedure, the three coefficients, Γ1,j,k, Γ2,j,k, and Γ3,j,k are identified. X jk = Γ1, j , k xk + Γ 2, j , k xk3 ,
V jk = Γ3, j , k vk
(12)
Once the coefficients of the manifold equations are identified, the displacement equation is used by substituting in different values of j and k to obtain the amplitude values necessary to produce the amplitude profile of an ILM. The stability of the analytical approximations obtained from each of the three methods is determined by using Floquet theory.
3.
RESULTS AND DISCUSSION
The results obtained by using the different methods are collected and presented in Table 2 and Figure 3. As can be seen, the results obtained by using the modified Invariant-Manifold Approach are favorable. A comparison of the amplitude profile calculated with the Modified Invariant-Manifold method is found to agree the best with the simulation amplitude profile. By using a Floquet analysis and conducting simulations, the resulting periodic solutions are determined to be stable. Numerical simulations confirm that a pinned ILM can be placed within the array, as shown in Figure 4. The model used for the cantilever array is equivalent to models used for piezoelectric micro-resonators. Hence, it is expected that it will be possible for this type of localization phenomena to occur within micro-resonator
Nonlinear vibration modes and localization in micro-resonator arrays Table 2. Amplitude ratios form simulation and analyses. %Diff % Diff Ration Ratio r1 % Diff value Linear Nonlinear r2 value Linear Simulation, Linear -0.21 -------0.70 ---Simulation, Nonlinear -0.2060 -------0.6679 ---Method of Multiple -0.2816 25% 27% -0.6200 13% Scales Approach Restricted Normal -0.1910 10% 7.9% -0.5190 35% Modes Approach Real-/ComplexNo ----No ---Variable Invariant Soln. Soln. Manifold Approaches Modified Invariant-0.1940 8.2% 6.2% -0.7118 1.7% Manifold Approach
Method
215
% Diff. Nonlinear ------7.7% 29% ---
6.2
Figure 3. Amplitude profile comparison: a) method of multiple scales approach, b) restricted normal mode approach, and c) modified invariant-manifold approach.
Figure 4. Pinned ILM placed into micro-cantilever array.
arrays. In a micro-resonator array, effects due to artificial impurities can be introduced by applying a DC bias in order to manipulate or stop the ILM after it has been created. A better understanding of this type of localization can enable enhanced performance of resonator arrays and perhaps even contribute to the development of new types of technologies.
ACKNOWLEDGEMENTS Support received for this work through AFOSR grant no. F49620-0310181 and ARO grant no. W911NF0510076 is gratefully acknowledged.
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REFERENCES 1. 2. 3.
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6. 7.
8.
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14. 15. 16. 17.
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Anderson PW. “Absence of diffusion in certain random lattices”, Physical Review, 109, pp. 1492-1505, 1958. Campbell DK, Flach S, Kivshar VS. “Localizing energy through nonlinearity and discreteness”, Physics Today, 57, pp. 43-49, 2004. Swanson BI, Brozak JA, Love SP, Strouse GF, Shreve AP, Bishop AR, Wang WZ, Salkola MI. “Observation of intrinsically localized modes in a discrete low-dimensional material”, Physical Review Letters, 82, pp. 3288-3291, 1999. Schwarz UT, English LQ, Sievers AJ. “Experimental generation and observation of intrinsic localized spin wave modes in an antiferromagnet”, Physical Review Letters, 83, pp. 223-226, 1999. Fleischer JW, Segev M, Efremidis NK, Christodoulides DN. “Observation of twodimensional discrete solitons in optically induced nonlinear photonic lattices”, Nature, 422, pp. 147-150, 2003. Ustinov AV. “Imaging of discrete breathers”, Chaos, 13, pp. 716-724, 2003. Sato M, Hubbard BE, Sievers AJ, Ilic B, Czaplewski DA, Craighead HG. “Observation of locked intrinsic localized vibration modes in a micromechanical oscillator array”, Physical Review Letters, 90, 044102, pp. 1-4, 2003. Sato M, Hubbard BE, English LQ, Sievers AJ, Ilic B, Czaplewski DA, Craighead HG. “Study of intrinsic localized modes in micromechanical oscillator arrays”, Chaos, 13, pp. 702-715, 2003. Dauxois T, Peyrard M, Willis CR. “Discrete effects on the formation and propagation of breathers in nonlinear Klein-Gordon equations”, Physical Review E, 48, pp. 4768-4778, 1993. Sato M, Hubbard BE, Sievers AJ, Ilic B, Craighead HG. “Optical manipulation of intrinsic localized vibrational energy in cantilever arrays”, Europhysics Letters, 66, pp. 318-323, 2004. Balachandran B, Li H. “Nonlinear phenomena in microelectromechanical resonators”, Proc. of the IUTAM Symposium on Chaotic Dynamics and Control of Systems and Process in Mechanics, Rome, Italy, Jun. 8-13, 2003, pp. 97-108. Dick AJ, Balachandran B, DeVoe DL, Mote CD Jr. “Parametric identification of piezoelectric microscale resonators”, Proc. of the 5th Euromech Nonlinear Dynamic Conference, Eindhoven, Netherlands, Aug. 7-12, 2005, 19-239. Dick AJ, Balachandran B, DeVoe DL, Mote CD Jr. “Parametric identification of piezoelectric microscale resonators”, J. of Micromech. And Microeng., 16, pp. 15931601, 2005. Nayfeh AH. Nonlinear Interactions: Analytical, Computational, and Experimental Methods, New York, Wiley, 2000. Pak CH. Nonlinear Normal Modes Dynamics: for Two Degree-of-Freedom Systems, Seoul, Inha University Press, 1999. Nayfeh AH. Perturbation Methods, New York, Wiley, 2003. Dick AJ, Balachandran B, Mote CD Jr. “Intrinsic localized modes and nonlinear normal modes in micro-resonator arrays”, Proc. of the ASME IMECE, Orlando, Florida, USA, Nov. 5-11, 2005, pp. 80255. Nayfeh AH, Balachandran B. Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods, New York, Wiley, 1995.
PARAMETRIC RESONANCE OF AN AXIALLY ACCELERATING VISCOELASTIC BEAM WITH NON-TYPICAL BOUNDARY CONDITIONS L. Q. Chen, X. D. Yang Department of Mechanics, Shanghai University, Shanghai 200444, China, E-mail:
[email protected] Abstract:
Principal parametric resonance in transverse vibration is investigated for axially accelerating viscoelastic beams constrained by rotating sleeves with torsion springs. The method of multiple scales is applied to calculate the steady-state response. Expression of the amplitude of the steady-state response is derived from the solvability condition of eliminating secular terms. The stability of straight equilibrium and nontrivial steady-state response are analyzed by use of the Lyapunov linearized stability theory. Numerical examples are presented to highlight the effects of speed pulsation, viscoelasticity, and nonlinearity and the stiffness of the torsion spring.
Key words:
Principal, parametric resonance, axially accelerating beam, method of multiple scales, viscoelasticity, stability
1.
INTRODUCTION
Axially moving beams can represent many engineering devices. Understanding transverse vibrations of axially moving beams is important for the design of the devices. A major problem is the occurrence of large transverse vibrations due to tension or axial speed variation. In many axially moving systems, the axial transport speed is a constant mean velocity with small periodic fluctuations. In addition, modeling of dissipative mechanisms is an important topic of axially moving material vibrations, and viscoelasticity is an effective approach to model the damping mechanism because some beamlike engineering devices are composed of some viscoelastic materials. Therefore, it is significant to analyze transverse vibration of axially accelerating viscoelastic beams. 217 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 217–226. © 2007 Springer.
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Boundary conditions have important influence on vibration of distributed parameter systems. In all available studies on axially accelerating nonlinear beams [1-6] boundary conditions under consideration were definitely cast into two types, pinned or clamped ends. However, in some engineering circumstances, such clear distinction cannot be easily made. For example, the boundary conditions of a belt on two pulleys may be neither pinned nor clamped. Hence, certain non-typical boundary conditions should be investigated to account some engineering systems more exactly. For the purpose, this paper study transverse vibration of an axially accelerating viscoelastic beam constrained by rotating sleeves with torsion springs.
2.
PROBLEM FORMULATIONS
A uniform axially moving viscoelastic beam, with density ρ, stiffness constant E, viscosity coefficient η, cross-sectional area A, moment of inertial I and initial tension P0, travels at the time-dependent axial transport speed v(T) between two sleeves with torsion springs. The sleeves rotate about fixed points separated by distance L (I/(AL2) 0, Im ( µn ) = 0; Re (κ n ) = 0, Im (κ n ) > 0
(20)
Substituting Equations (19) and (20) into Equation (17) and separating the resulting equation into real and imaginary parts give a n = ⎡⎣α Re ( µ n ) + γ 1 Im ( χ n ) sin θ n − γ 1 Re ( χ n ) cos θ n ⎤⎦ a n (21) 1 a nθ n = a nσ + 2γ 1 ⎡⎣ Re ( χ n ) sin θ n + Im ( χ n ) cos θ n ⎤⎦ a n − v12 Im (κ n ) a n3 2
where
θ n = σ T1 − 2ϕn
(22)
For the steady-state response, the amplitude an and the new phase angle
θn in Equation (21) are constant. It is obvious that Equation (21) possesses a
singular point at the origin (trivial zero solution), which represents the straight equilibrium configuration of the beam. In addition, there may exist nontrivial periodic solution with amplitudes given by an1,2 =
1 k1 Im (κ n )
2σ ± 4 γ 12 χ n − α 2 ⎡⎣ Re ( µ n ) ⎤⎦ 2
2
(23)
Equation (23) is the closed form solution of the amplitude of nontrivial steady-state response. From Equation (23), it can be concluded that the nontrivial steady-state solutions exist only if the following conditions hold,
α≤
γ1 χn Re ( µn )
σ ≥ σ 1,2 = ∓2 γ 12 χ n − α 2 ⎡⎣ Re ( µ n ) ⎤⎦ 2
(24) 2
(25)
Numerical examples are presented to highlight the effects of parameters. Consider an axially moving beam with vf=0.8, γ0=2.0, γ1=0.25, k1=0.2, α=0.0002 and k=1.0, if no other indications. In this case, ω1=7.0217, β11=4.3442, β21=-1.2454+2.8314i, β31=-1.2454-2.8314i, β41=-1.8534 and ω2=31.6548, β12=7.6026, β22=-1.2498+6.2285i, β32=-1.2498-6.2285i, β42=5.1030. Equation (18) gives µ1=63.8961, χ1=-2.2638+0.0088i and µ2=853.6940, χ2=-0.7068+0.8697i, κ1=67.2032i and κ2=155.1480i. Figure 1 illustrates the effect of the axial speed variation amplitude in the first two principal parametric resonances, in which the solid lines denote γ1=0.25 and
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the dot lines denote γ1=0.2. Figure 3 shows the effect of the viscosity coefficient, in which the solid lines denote α=0.0002, 0.0001 and the dot lines denote α=0.008, 0.0002. Figure 4 displays the effect of nonlinearity, in which the solid and dot lines respectively stand for k1=0.2 and k1=0.25. The amplitudes of steady-stage response increase with the decrease of the nonlinear term coefficient, while the existence interval is independent of the coefficient. Figure 4 demonstrates the effect of the spring stiffness, in which the solid lines denote k=1.0 and the dot lines denote k=3.0.
Figure 1. Effect of the axial speed variation amplitude in the first two principal parametric resonances.
Figure 2. Effect of the viscosity in the first two principal parametric resonances.
Figure 3. Effect of the nonlinearity in the first two principal parametric resonances.
Parametric resonance of an axially accelerating viscoelastic beam a1
223
a2
σ
σ
Figure 4. The effect of the spring stiffness in the first two principal parametric resonances.
5.
STABILITY
To determine the stability of the trivial solution, suppose that the perturbed solutions of Equation (17) take the form σT
1 i 1 (26) ⎡⎣ pn (T1 ) + i qn (T1 ) ⎤⎦ e 2 2 where pn and qn are real functions. Substituting Equations (26) and (20) into Equation (17) and separating the resulting equation into real and imaginary parts yield 1 ⎡σ ⎤ pn = −⎡⎣α Re ( µn ) + γ 1 Re ( χn ) ⎤⎦ pn + ⎢ − γ 1 Im ( χn ) ⎥ qn − k12 Im (κ n ) ( pn2 + qn2 ) qn 2 4 ⎣ ⎦ (27) 1 2 ⎡σ ⎤ qn = − ⎢ + γ 1 Im ( χn ) ⎥ pn − ⎡⎣α Re ( µn ) − γ 1 Re ( χn )⎤⎦ qn + k1 Im (κ n ) ( pn2 + qn2 ) pn
An (T1 ) =
⎣2
4
⎦
The Jacobian matrix of the right hand function of Equation (27), calculated at (0, 0), is σ ⎛ ⎞ − γ 1 Im ( χ n ) ⎜ −α Re ( µ n ) − γ 1 Re ( χ n ) ⎟ 2 (28) ⎜ ⎟ σ − − − + Im Re Re γ χ α µ γ χ ⎜ ( n) ( n ) 1 ( n )⎟ 1 ⎝
⎠
2
with its characteristic equation 2
2 2 ⎛σ ⎞ λ 2 + 2α Re ( µn ) λ − γ 12 χ n + α 2 ⎡⎣ Re ( µn ) ⎤⎦ + ⎜ ⎟ = 0 ⎝2⎠
(29)
The Routh-Hurwitz criterion yields the stability conditions σ < σ1 = −2 γ 12 χn − α 2 ⎡⎣Re ( µn ) ⎤⎦ or σ > σ 2 = 2 γ 12 χn − α 2 ⎡⎣Re ( µn ) ⎤⎦ (30) 2
2
2
2
Otherwise, the trivial solution is unstable. The Lyapunov linearized stability theory indicates that the stability of a nonlinear system coincides with that of
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the corresponding linear system. Hence there exists an instability interval [σ1,σ2] of trivial solution. The instability condition of the trivial solution coincides with the existence condition of the first steady-state response, and the stability condition of the trivial solution coincides with the existence condition of the second steady-state response. The stability of the nontrivial steady-state response can be determined by the following equation derived from Equation (21) on the condition that an≠0. an = ⎡⎣α Re ( µn ) + γ 1 Im ( χ n ) sin θ n − γ 1 Re ( χ n ) cosθ n ⎤⎦ an (31) 1 θ n = σ + 2γ 1 ⎡⎣ Re ( χ n ) sin θ n + Im ( χ n ) cosθ n ⎤⎦ − v12 Im (κ n ) an2 2 The Jacobian matrix of right hand function of Equation (31), calculated at (an1,2, θn1,2), is 2 2 ⎡ ±γ 1 γ 12 χ n − α 2 ⎡⎣ Re ( µ n ) ⎤⎦ an1,2 ⎤⎥ 0 ⎢ 2 −2α Re ( µ n ) ⎢⎣ −v1 Im (κ n ) an1,2 ⎥⎦ Here the definition of the amplitude of steady-state response
(32)
α Re ( µn ) + γ 1 Im ( χ n ) sin θ n1,2 − γ 1 Re ( χ n ) cosθ n1,2 = 0
1 2 is used. The characteristic equation of matrix (32) is
σ + 2γ 1 ⎡⎣ Re ( χ n ) sin θ n1,2 + Im ( χ n ) cosθ n1,2 ⎤⎦ − v12 Im (κ n ) an21,2 = 0
(33)
λ 2 + 2α Re ( µ n ) λ ± γ 1v12 Im (κ n ) an21,2 γ 12 χ n − α 2 ⎡⎣ Re ( µ n ) ⎤⎦ = 0 (34) 2
2
According to the Routh-Hurwitz criterion, the first (large) steady-state response is always stable, and the second (small) steady-state response is always unstable. For an axially moving beam with vf=0.8, γ0=2.0, γ1=0.25, k1=0.2, α=0.0002 and k=1.0, the equilibrium and the nontrivial steady-state response are depicted in Figure 5 for the first two principal parametric resonances, in which the solid or dot lines stand for stable or unstable solutions respectively.
Figure 5. The stable and unstable responses in the first two principal parametric resonance.
Parametric resonance of an axially accelerating viscoelastic beam
6.
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ON ANOTHER NONLINEAR BEAM MODEL
If the quasi-static stretch assumption is not used, the dimensionless form of the governing equation is [6] 3 u ,tt +2γ 0u ,t + ( γ 0 2 − 1) u , xx + vf 2u , xxxx = ε k12u , xx u , x 2 −2εγ 1 sin ωtu , xt (35) 2 − 2εγ 0γ 1 sin ωtu , xx −εωγ 1 cos ωtu , x −εα u , xxxxt +O ( ε 2 ) The preceding analysis procedure can still be employed to derive the amplitude of steady-state response. In fact, all formulas are the same, with only one exception. In this case, a coefficient in Equation (18) are defined by 1
κn =
1
3 3 φnφn′′φn′2 d x + ∫ φnφn′′φn′φn′ d x ∫ 40 20 1
1
0
0
(36)
i ωn ∫ φnφn d x + γ 0 ∫ φnφn′ d x
A numerical example is presented to demonstrate the differences between Equations (1) and (35). For an axially moving beam with vf=0.8, γ0=2.0, γ1=0.25, k1=0.2, α=0.0002 and k=1.0, Figure 6 shows the relationship between the amplitude and the detuning parameter for first two principal parametric resonances, in which the dot and solid lines represent the results based on Equations (1) and (35). The nontrivial solution amplitude derived from equation (35) is smaller, and the instability intervals are the same. Besides, numerical calculations indicate that the two models have the same tendencies to change with related parameters.
Figure 6. The stable and unstable responses in the first two principal parametric resonance.
7.
CONCLUSIONS
This paper applies the method of multiple scales to treat an axially accelerating viscoelastic beams constrained by rotating sleeves with torsion
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springs. The Lyapunov linearized stability theory is applied to prove that here exists an instability interval of the detuning parameters on which the straight equilibrium is unstable, and the first (second) nontrivial steady-state response is always stable (unstable). Numerical calculations show that the lower order principal resonance has the larger instability interval and the larger steady-stage response amplitude. The instability interval increases with the increasing axial speed variation amplitude, the decreasing viscosity coefficient and constraint spring stiffness. The amplitude of steady-stage response increases with the decreasing nonlinear term coefficient. The amplitude of stable response increases with the increasing axial speed variation amplitude and the decreasing viscosity coefficient. Two nonlinear beam models predict the same changing tendencies. The quasi-static stretch assumption leads to larger amplitudes, while the existence intervals are same.
ACKNOWLEDGEMENTS This work was supported by the National Natural Science Foundation of China (Project No. 10472060), Natural Science Foundation of Shanghai Municipality (Project No. 04ZR14058), Shanghai Municipal Education Commission Scientific Research Project (No. 07ZZ07), and Shanghai Leading Academic Discipline Project (Project No.Y0103).
REFERENCES 1. 2. 3.
4. 5.
6.
7. 8.
Chakraborty G, Mallik AK. “Parametrically excited nonlinear traveling beams with and without external forcing”, Nonlinear Dynamics, 17, pp. 301-324, 1998. Ravindra B, Zhu WD. “Low dimensional chaotic response of axially accelerating continuum in the supercritical regime”, Achieve of Applied Mechanics, 68, pp. 195-205, 1998. Öz HR, Pakdemirli M, Boyaci H. “Non-linear vibrations and stability of an axially moving beam with time-dependent velocity”, International Journal Non-Linear Mechanics, 36, pp. 107-115, 2001. Parker RG, Lin Y. “Parametric instability of axially moving media subjected to multifrequency tension and speed fluctuations”, Journal of Applied Mechanics, 68, pp. 49-57, 2001. Marynowski K, Kapitaniak T. “Kelvin-Voigt versus Bügers internal damping in modeling of axially moving viscoelastic web”, International Journal of Non-Linear Mechanics, 37, pp. 1147-1161, 2002. Chen LQ, Yang XD. “Steady-state response of axially moving viscoelastic beams with pulsating speed: comparison of two nonlinear models”, International Journal of Solids and Structures, 42, pp. 123-126, 2005. Wickert JA, Mote CDJr. “Classical vibration analysis of axially moving continua”, Journal of Applied Mechanics, 57, pp. 738-744, 1990. . Chen LQ, Yang XD. “Vibration and stability of an axially moving viscoelastic beam with hybrid supports”, European Journal of Mechanics A/Solid, 25, pp. 996-1008, 2006.
DYNAMICS OF A BODY CONTROLLED BY INTERNAL MOTIONS F.L. Chernousko Institute for Problems in Mechanics, Russian Academy of Sciences, pr. Vernadskogo, 101-1, 119526, Moscow, Russia, E-mail:
[email protected] Abstract:
Progressive motions of a body containing internal moving masses are analyzed in the presence of resistance forces acting between the body and the environment. The cases of Coulomb’s dry friction and nonlinear resistance forces are considered. The internal masses perform periodic motions subject to constraints imposed on the displacements, velocities, and accelerations. Optimal relative periodic motions of internal masses are determined that correspond to the maximal average speed of the system as a whole. Experimental data confirm the obtained theoretical results.
Key words:
Dynamics, multibody system, internal mass, friction, mobile robots.
1.
INTRODUCTION
It is well-known that a rigid body containing internal masses can move progressively in a resistive medium, if the internal masses perform special periodic motions inside the body. This effect is utilized in certain projects of mobile robots and underwater vehicles. In the paper, simple models of this phenomenon are analyzed. A mechanical system is considered that consists of a rigid body of mass M and an internal mass of mass m that can move inside the body. By contrast to [1], the internal mass is supposed not to interact with the medium outside the body. Rectilinear progressive motions of the system along a horizontal line are studied. Certain periodic motions of the internal mass m relative to body M are analyzed, namely, two-phase and three-phase motions. In the two-phase/ three-phase motion, the relative velocity/acceleration of mass m is piecewise 227 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 227–236. © 2007 Springer.
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constant, and the period includes two/three intervals of constant velocity/ acceleration. Various kinds of resistance forces acting upon body M are considered: Coulomb’s dry friction, linear and nonlinear resistance depending on the velocity of the body. These forces can be anisotropic, i.e., different for onward and backward motions of the body. Optimal parameters of periodic motions are determined that result in the maximal average speed V of body M under the constraints imposed on the displacement, velocity, and acceleration of mass m. The obtained results (see also [2, 3]) allow to evaluate the maximal possible speed of mobile systems whose motion is based on the displacement of internal masses. Experimental results confirm the practical realizability of the principle of motion considered in the paper. This principle is of interest for mobile robots, especially, for mini-robots moving inside tubes and in aggressive media.
2.
STATEMENT OF THE PROBLEM
Figure 1. Mechanical systems.
Consider a rigid body of mass M that can move horizontally in a resistive medium. Another body of mass m moves horizontally inside the body (Figure 1). For brevity, these bodies will be called “body M ” and “mass m”, respectively. Denote by x and v the absolute coordinate and velocity of body M, respectively, and by ξ , u, w, the displacement of mass m, its velocity and acceleration relative to body M, respectively. The kinematic equations of mass m relative to body M are
ξ = u, u = w .
(1)
The dynamic equations of body M can be presented as follows
x = v,
v = − µ w − r ( v ),
µ = m /( M + m ),
(2)
where r ( v ) is the resistance force acting upon body M divided by its mass.
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229
For the case of anisotropic Coulomb’s dry friction (Figure 1a), we have r ( v ) = f + g if v > 0,
r ( v ) = − f − g if v < 0,
(3)
where g is the gravity acceleration and f + , f − are the coefficients of friction. If the inequalities − f + g ≤ µ w ≤ f − g hold and body M is at rest ( v = 0) at some time instant, then the body will stay at rest. The function r ( v ) is given by r ( v ) = k + v, if v ≥ 0 ;
r ( v ) = k − v, if v ≤ 0
(4)
for the anisotropic linear resistance and by r ( v ) = κ + v v , if v ≥ 0;
r ( v ) = κ − v v , if v ≤ 0
(5)
for the anisotropic quadratic resistance (Figure 1b). Here, k + , k − , κ + , and κ − are positive coefficients. In the isotropic case, we have k + = k − and κ + = κ − .
3.
RELATIVE MOTION
We consider periodic motions of mass m relative to body M within a fixed interval: 0 ≤ ξ (t ) ≤ L , where L > 0 is given. Also, we impose conditions ξ (0) = ξ (T ) = 0 and u(0) = u (T ) = 0 , where T is a period of motion, and require that the maximal displacement ξ (θ ) = L is attained at some θ ∈ (0, T ) . Let is restrict ourselves with two classes of relative periodic motions of mass m : two-phase and three-phase motions.
Figure 2. Two-phase motion.
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Figure 3. Three-phase motion.
In the two-phase motion (Figure 2), the period [0, T] consists of two intervals, where u(t ) is constant. In the three-phase motion (Figure 3), the period [0, T ] consists of three intervals, where w(t ) is constant. It can be shown that these two-phase and three-phase motions have, under the imposed periodicity conditions, the least possible number of intervals, where the velocity u(t ) and acceleration w(t ) , respectively, are constant. Denote by τ i the durations of the intervals introduced above. For the two-phase motion, we have u(t ) = u1 for t ∈ (0,τ 1 ), u(t ) = −u2 for t ∈ (τ 1 , T ), T = τ 1 + τ 2 ,
(6)
where u1 and u2 are positive constants. The two-phase motion is determined by two parameters, u1 and u2 , and other parameters are expressed in terms of u1 and u2 as follows:
τ 1 = θ = L / u1 , τ 2 = L / u2 , T = L(u1−1 + u2 −1 ) .
(7)
For the three-phase motion, we have w(t ) = w1 for t ∈ (0,τ 1 ), w(t ) = − w2 w(t ) = w3
for t ∈ (τ 1 ,τ 1 + τ 2 )
for t ∈ (τ 1 + τ 2 , T ), T = τ 1 + τ 2 + τ 3
(8)
where w1 , w2 and w3 are positive constants. All other parameters can be expressed in terms of w1 , w2 , and w3 [2 , 3]. The parameters introduced above should satisfy the constraints 0 < ui ≤ U , i = 1,2 ,
(9)
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for the two-phase motion and 0 < wi ≤ W , i = 1,2,3 ,
(10)
for the three-phase motion, respectively. Here, U and W are the maximal admissible velocity and acceleration of the relative motion. By substituting Equations (6) and (8) into (2) and using the periodicity conditions, we analyze possible motions of body M . We determine such motions that: 1) the velocity v (t ) of body M is T-periodic; 2) the conditions v (0) = v (T ) hold; 3) the average speed V = [ x (T ) − x (0)]/ T of body M is maximal with respect to the parameters u1 , u2 / w1 , w2 , w3 for the two-phase / three-phase motions, under the respective constraints (9) / (10). Some results of the analysis are presented below; the complete proof and more details can be found in [3].
4.
DRY FRICTION: TWO-PHASE MOTION
Figure 4. Modes of two-phase motion.
For the case of dry friction (3), we assume that v (0) = v (T ) = 0 . It occurs that two modes, a and b , are possible in the two-phase motion (Figure 4). In mode a , body M is never in the state of rest (always v ≠ 0 ), whereas in mode b there is an interval of rest where v = 0 . Let us introduce non-dimensional variables and functions: ui = u0 xi , i = 1,2, u0 = ( Lf − g / µ )1/ 2 , U = u0 X , V = 0.5µ u0Φ , x* ( c ) = {( c / 2)( c − 1) −1[1 − 3c + (9c 2 + 2c − 7)1/ 2 ]}1/ 2 , Φ 0 ( x, c ) = x[2c + x 2 (1 − c )]( c + x 2 ) −1 , c = f + / f − .
(11)
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Consider first the case where there is no upper bound in (9) so that U → ∞ . Then the maximal average speed is attained in mode a , if c ≤ 1 , and mode b , if c > 1 . The optimal motion is defined by formulas: x1 = 1, x2 = c, Φ = 1, if
c ≤ 1;
x1 = x* ( c ), x2 = c / x * ( c ), Φ = Φ 0 ( x * ( c ), c ), if
c > 1.
(12)
In the friction is isotropic ( f + = f − , c = 1) we have, according to (11) and (12): x1 = x2 = 1, u1 = u2 = u0 = ( Lfg / µ )1/ 2 , Φ = 1, τ 1 = τ 2 = ( µ L / fg )1/ 2 , T = 2τ 1 , V = 0.5( µ Lfg )1/ 2 , see Figure 4c. For the general case of finite U in (9), the two-phase motions are realizable, if X ≥ max( c1/ 2 , c ) . The optimal motion is defined by formulas:
x1 = X , x2 = c / X , Φ = Φ 0 ( X , c ), if
c1/ 2 ≤ X < 1;
x1 = 1, x2 = c, Φ = 1, if
X ≥ 1;
c ≤ 1 and
x1 = c / X , x2 = X , Φ = Φ ( c / X , c ), if 0
1 < c ≤ X < c / x * ( c );
x1 = x * ( c ), x2 = c / x * ( c ), Φ = Φ 0 ( x* ( c ), c ), if
c / x* (c) ≤ X .
Using formulas (11), one can return to the original dimensional parameters.
5.
DRY FRICTION: THREE-PHASE MOTION
Figure 5. Modes of three-phase motion.
In the three-phase motion, modes a and b can occur (Figure 5). It is shown [3] that mode b which contains an interval of rest but does not include backward motion of body M corresponds to a higher maximal average speed V than mode a.
Dynamics of a body controlled by internal motions
233
For the sake of brevity, let us restrict ourselves with the isotropic friction ( f + = f − = f ) and introduce the non-dimensional variables: wi = ( fg / µ ) yi , i = 1,2,3, W = ( fg / µ )Y , V = ( µ Lfg / 2)1/ 2 F . (13)
The three-phase motions are possible, if Y > 1 . The optimal solution is given by the relationships: y1 = 1, y2 = Y , y3 = 1, F =
Y
1/ 2
Y −1 , if 1 < Y ≤ 2 + 5; (Y + 1)1/ 2
Y +1 Y y1 = 1, y2 = Y , y3 = ,F = , if Y −3 Y +1
(14)
Y > 2 + 5.
For the particular case where the upper constraint (10) is absent and Y → ∞ , we obtain from (14) and (13): w1 = w3 = fg / µ , w2 → ∞, V = ( µ Lfg / 2)1/ 2 , v (t ) = 0, if
t ∈ (0, T / 2), v (t ) = 2 fg (T − t ), if
T = 2(2 µ L / fg ) , τ 1 = τ 3 = T / 2, τ 2 = 0, 1/ 2
t ∈ (T / 2, T ),
v = 2(2 µ Lfg )1/ 2 .
The velocity diagram of body M for this case (Figure 5c) contains a jump v at the instant t = T / 2 .
6.
GENERALIZATIONS AND EXPERIMENTS
The problem of optimal control for a body containing a moving internal mass in the presence of the isotropic dry friction is considered in [4], where the acceleration of the internal mass is subject to the constraint w(t ) ≤ W . The obtained optimal acceleration occurs to be piecewise constant with three intervals of constancy but, by contrast to the three-phase motion, the instants when ξ (t ) = 0, u(t ) = 0, and v (t ) = 0 , do not coincide. The case of one or more internal masses moving in two directions in the vertical plane inside body M is considered in [5, 6]. Due to the vertical motion of the internal mass, the pressure of body M exerted upon the horizontal plane changes and, therefore, the friction force also changes. Thus, the additional increment of the average speed of the system is attained.
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Figure 6. Experimental models.
The principle of motion presented above is implemented in experimental models shown in Figure 6. In Figure 6a, the internal motion is performed by an inverted pendulum [7]. In Figure 6b, the cart carries eccentric rotating wheels. The experiments have shown the realizability of motions induced by internal masses. The obtained experimental data confirm the theoretical results.
Figure 7. Vibro-robot in a tube.
Mini-robots that utilize the same principle and can move inside tubes have been designed [8]. Such vibro-robots consist of two parts which vibrate with respect to each other with the frequency 20 ÷ 40 Hz and can move inside tubes (both straight and curved) of the diameter 4 ÷ 70 mm with a speed 10 ÷ 30 mm/s (Figure 7).
7.
NONLINEAR RESISTANCE
Let us consider briefly the cases (4) and (5) of nonlinear resistance depending on the velocity of body M. We restrict ourselves to the two-phase motions defined by (6) and assume that the periodicity conditions
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ξ (0) = ξ (T ) = 0, u(0) = u(T ) = 0, and v (0) = v (T ) = v0 are imposed, where v0 is to be chosen so that v (t ) is T-periodic. It follows from Equations (2) that, for the piecewise linear resistance (4), the periodic solution v (t ) is given by v (t ) = − µ (u1 + u2 )(1 − e2 )(1 − e1e2 ) −1 exp( − k− t ),
t ∈ (0,τ 1 ),
v (t ) = µ (u1 + u2 )e2 (1 − e1 )(1 − e1e2 ) exp[− k + (T − t )], t ∈ (τ 1 , T ), −1
v0 = µ (1 − e1e2 ) −1[u1e2 (1 − e1 ) − u2 (1 − e2 )],
(15)
e1 = exp( − k−τ 1 ), e2 = exp( − k+τ 2 ),
where the parameters u1 , u2 , τ 1 , τ 2 , and T satisfy Equation (7). Let us calculate the total displacement x of body M by integrating Equations (15) over the period T and evaluate the average speed V = x / T . Using also Equations (7), we obtain V = µ L(1 − e1 )(1 − e2 )(1 − e1e2 ) −1 (τ 1τ 2 ) −1 ( k+ −1 − k − −1 ) .
(16)
It follows from (16) that V > 0 , only if k+ < k− , what is physically quite natural. Moreover, it can be shown easily that for the linear resistance, i.e., if k+ = k − , the progressive motion of body M is impossible for arbitrary periodic motion of the internal mass. For given µ , L, k+ , and k− , the average speed V from (16) depends on two parameters τ 1 and τ 2 , or u1 and u2 , see (7). Maximizing V with respect to these parameters subject to the constraint (9), we obtain Vmax = µU 2 L−1 (1 − e1 )(1 − e2 )(1 − e1e2 ) −1 ( k + −1 − k− −1 ), e1 = exp( − k− L / U ), e2 = exp( − k + L / U ).
(17)
If k− L / U 1, Equation (17) reduces to Vmax = µU ( k − − k + )( k+ + k− ) . Note that Vmax → ∞ as U → ∞ . Similar result ( Vmax → ∞ as U → ∞ ) is true also for the case of the quadratic resistance (5). However, for the case (5), by contrast to the case (4), the average speed is positive even for the isotropic resistance. If κ + = κ − = κ in (5), we have V = −U (2 Lκ ) −1 (1 − µ Lκ )log(1 − µ 2 L2κ 2 ) > 0 .
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CONCLUSIONS
Periodic progressive motions of a rigid body controlled by internal masses and moving in resistive media are analyzed. For certain classes of periodic motions, optimal controls are found that correspond to the maximal average velocity of the body under various constraints imposed on relative displacements, velocity, and acceleration of internal motions. The maximal velocity of the body in the case of Coulomb's friction is V = λ ( µ Lfg )1/ 2 , where λ ∼ 1 depends on the constraints imposed. For the piecewise linear and quadratic resistance, the maximal speed of the body V → ∞ as the velocity of internal motions increases indefinitely. Experimental data confirm the obtained theoretical results. The principle of motion considered above can be used for robots moving inside tubes and in aggressive media.
ACKNOWLEDGEMENTS This work was supported by the RFBR (Project 05-01-00647) and the Program for Support of Russian Scientific Schools (Grant 9831. 2006. 1).
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
Chernousko FL. “The Optimum Rectilinear Motion of a Two-mass System”, Journal of Applied Mathematics and Mechanics, 66, pp. 1-7, 2002. Chernousko FL. “On the Motion of a Body Containing a Movable Internal Mass”, Doklady Physics, 50, pp. 593-597, 2005. Chernousko FL. “Analysis and Optimization of the Motion of a Body Controlled by a Movable Internal Mass”, Journal of Applied Mathematics and Mechanics, 70, pp. 915-941, 2006. Figurina TY. “Optimal Control of Motion of Two Bodies Along a Straight Line”, Journal of Computer and Systems Science International, 47, 2007, to be published. Bolotnik NN, Zeidis IM, Zimmermann K, Yatsun SF. “Dynamics of Controlled Motion of Vibration-driven System”, Journal of Computer and Systems Science International , 45, pp. 831-840, 2005. Chernousko FL, Zimmermann K, Bolotnik NN, Yatsun SF, Zeidis I. “Vibration-driven Robots”, Proc. of the Workshop on Adaptive and Intelligent Robots: Present and Future, Moscow, Russia, Nov. 24-26, pp. 26-31, 2005. Li H, Furuta K, Chernousko FL. “A Pendulum-driven Cart via Internal Force and Static Friction”, Proc. of the International Conference “Physics and Control”, St.-Petersburg, Russia, Aug. 24-26, pp. 15-17, 2005. Gradetsky V, Solovtsov V, Kniazkov M, Rizzotto GG, Amato P. “Modular Design of Electro-magnetic Mechatronic Microrobots”, Proc. of the 6th International Conference on Climbing and Walking Robots CLAWAR, Catania, Italy, Sept. 17-19, pp. 651-658, 2003.
LINEAR AND NONLINEAR ELASTODYNAMICS OF NONSHALLOW CABLES W. Lacarbonara, A. Paolone, F. Vestroni Department of Structural and Geotechnical Engineering, University of Rome La Sapienza via Eudosssiana 18, 00184 Rome Italy, E-mail:
[email protected] Abstract:
A geometrically exact mechanical model describing large motions of nonshallow elastic cables is employed to investigate the linear and nonlinear properties of the cable planar modes. Considering the potential and kinetic linear modal energy content, it is shown that the elasto-static and elastodynamic modes are located around the various crossover lines and they become closer for deeply sagged cables. The nonlinear characteristics of the modes of different type are documented and general properties are unfolded.
Key words:
Nonshallow cables, nonlinear normal modes, crossovers, veering, reducedorder models, two-to-one internal resonances.
1.
INTRODUCTION
The linear and nonlinear dynamics of suspended shallow elastic cables [1-8] have received considerable attention due to their use in various engineering applications. On the contrary, little attention has been addressed to the linear and nonlinear vibration properties of nonshallow cables. Following previous works by the authors [11, 12], a nonlinear mechanical model of nonshallow cables, describing the fully coupled longitudinal and transverse dynamics, is employed first, via its linearization, to systematically explore the linear modal properties; then, its third-order perturbation is treated with the asymptotic method of multiple scales to unfold the nonlinear properties of the individual planar modes away from internal resonances. Higher-order expansions are also carried out in the neighbourhood of 2:1 internal resonances between the considered mode and a high-frequency 237 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 237–246. © 2007 Springer.
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mode to study the effects of the auto-parametric energy transfer on the nonlinearity of the low-frequency modes. Interesting phenomena are documented highlighting the importance of these interactions. Reducedorder descriptions of the nonlinear modal properties have been investigated although they are briefly summarized for conciseness.
2.
MECHANICAL MODEL
Denoting, in Figure 1, with (O, i, j) the orthonormal basis of a fixed inertial reference frame, the reference configuration C0 of the cable axis is described by the vector p*0 ( x* ) := x*i + y* j where y ( x) is the catenary solution [1] which, in nondimensional form, reads y ( x) = 1 γ ⎡⎣cosh γ 2 − cosh γ (1 2 − x ) ⎤⎦ where x := x∗ A , y := y ∗ A , A is the distance between the suspension points placed at the same level, γ := mg A/H 0∗ is solution of the geometric compatibility condition
i
O j
p0*
y,* v *
p* P0 u* P
x,* u*
C C0
Figure 1. The geometry of the cable model with the inertial reference frame.
sinh(γ / 2) = ( γ / 2)η0 . In the previous equation, η0 := L0 /A, L0 is the initial total length of the cable, H 0∗ is the horizontal projection of the axial force N 0 ( x) := N 0∗ H 0∗ = cosh γ (1 2 − x ) . Here, the star denotes dimensional variables. The sag-to-span ratio is d := 1 γ ( cosh γ 2 − 1) . Under a finite planar motion, the cable material point P0 is displaced to the current place P described by p* ( x* , t * ) := p*0 ( x* ) + u* ( x* , t * ) where u * := u *i + v* j is the displacement vector. The cable elongation is e := cos θ 0 | p*′ | −1 where the prime indicates differentiation with respect to x* , cos θ 0 ( x* ) := a0 ( x* ) ⋅ i with a 0 denoting the unit vector along the tangential direction in C0 , the dot stands for the standard inner product in Euclidean space, | ⋅ | represents the magnitude of
Linear and nonlinear elastodynamics of nonshallow cables
239
the vectorial argument. Due to the deformation process, an incremental axial force is generated; thus, the current force is N 0∗ + N ∗ and must be directed, for the balance of angular momentum, along the current tangential direction a := x*′ / | x*′ | . Therefore, the balance of linear momentum, after incorporating the equilibrium equation in C0 , ( N0*a 0 )′ + fg* = 0 - where f g* := (m0 g )secθ 0 j is the gravity force density - can be written as * − ⎡⎣ N 0* (a − a 0 ) ⎤⎦′ − ( N * a )′ = 0 m0 secθ 0 u
(2)
where m0 represents the cable mass per unit reference length in C0 , the overdot indicates differentiation with respect to time. Exploiting the fact that the axial strains are small for typical engineering materials and loading conditions, assuming that the cable is made of a hyperelastic material, a linear constitutive elastic law relating the incremental axial load to the elongation is assumed in the form Nˆ * (e) = E A e where E is Young’s modulus of elasticity and A is the area of the undeformed cable cross section. The following nondimensional variables and parameters are introduced: u := u ∗ A , v := v∗ A , t := ωct ∗, ωc 2 := H 0∗ m0 A 2 , Nˆ := k e, k := E A H 0∗ . It can be shown [11] that typical values of k for engineering cables are within the range [102, 10 4 ]. The ensuing nondimensional equations of motion, in componential form, become [11,12] ⎧ cos θ 0 ⎫′ (secθ 0 )u − ⎨ [ N 0 (u ′ − e) + Nˆ (1 + u ′)]⎬ = 0 ⎩ 1+ e ⎭ ⎧ cos θ 0 ⎫′ (secθ 0 )v − ⎨ [ N 0 (v′ − tan θ 0 e) + Nˆ (v′ + tan θ 0 )]⎬ = 0 ⎩ 1+ e ⎭
(3)
The field equations are supplemented with the boundary conditions u(0, t ) = 0 and u(1, t ) = 0.
3.
LINEAR MODAL CHARACTERISTICS
The linear as well as nonlinear free motions [11,12] of linearly elastic nonshallow cables depend on two parameters, namely, γ (geometric flexibility parameter) and k (elastic stiffness relative to the geometric stiffness), contrary to shallow cables [1-2] whose linear motions depend solely on Irvine’s elasto-geometric parameter λ . Irvine’s parameter is related to the two characteristic parameters for nonshallow cables according 1 to λ 2 := γ 2 k /η with η = ∫ cos3 θ 0 dx. 0
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(a)
60
(b)
55
4A 3S
25
20
3A
15
2S 2A
ωn
45 40 35 30 25
1S 1A
10
50
20 15 10
5
5 0
0 1
2
3
4
5
λ/π
6
7
8
9
10
0
4
8
12
λ/π
16
20
24
28
30
Figure 2. Variation of the frequencies of the in-plane modes with λ π when (a) γ = 0.75 and (b) γ = 1.5.
Three regions of static regimes were accordingly identified [12]: shallow profiles, γ ∈ [0,0.5] , nonshallow profiles, γ > 1 , and transition profiles, γ ∈ [0.5, 1] . In Figures. 2a and 2b, the frequencies of the transition cables with γ = 0.75 and those of the nonshallow cables with γ = 1.5 are shown, respectively. The shaded areas denote regions of non physically admissible cable parameters. The frequencies and mode shapes were obtained applying the Ritz-Galerkin approach to the linearized equations of motion using the sine functions as admissible functions. The modes are then classified on the basis of their potential and kinetic energy contents. In particular, considering the ratio of the modal elastic strain energy to the modal geometric energy and the ratio of the modal longitudinal kinetic energy component to the total kinetic energy, it turns out that at the principal crossovers there is a peak of strain energy of a quasistatic nature since the associated longitudinal kinetic energy component is negligible. These modes are referred to as elasto-static modes, previously highlighted by Irvine and Caughey [2] as modes with quasi-static stretching. On the contrary, corresponding to the secondary crossovers, the maximum strain energy is accompanied by a kinetic energy which is mostly of the longitudinal type. Hence, these modes are elasto-dynamic modes also referred to by Triantafyllou [3] as elastic modes. The elasto-static modes lie within a relatively extended stiffness region around the principal lowestorder crossovers whereas the elasto-dynamic modes are mostly localized around their crossovers. Away from the crossovers, the modes become geometric. These properties are general and relate to different static regimes of cables from shallow to deeply nonshallow configurations where the elasto-static and elasto-dynamic modes become closer contrary to shallow taut cables where a wide separation occurs. Further, calculating the modal elongations associated with the elastostatic modes and those exhibited by the elasto-dynamic modes, it turns out
Linear and nonlinear elastodynamics of nonshallow cables
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that (i) at the elasto-static crossovers, the strain mode is nearly constant, (ii) at the first elasto-dynamic crossover, the lowest skew-symmetric strain mode occurs with one node; (iii) at the higher elasto-dynamic crossovers, the strain modes form a complete sequence of symmetric and skew-symmetric modes.
4.
NONLINEAR MODAL CHARACTERISTICS
The nonlinear properties of the individual planar modes away from internal resonances are unfolded and discussed. To this end, the method of multiple scales is applied directly to the third-order perturbation of the equations of motion and boundary conditions, Equation (3), thus yielding the individual nonlinear normal mode in the form [10] u m ( x, t ) ≈ a cos(ωm∞ t + ψ m ) φm ( x) +
a2 ⎡cos2(ωm∞ t + ψ m ) Φ( x) + Ψ ( x) ⎤⎦ (4) 2 ⎣
where a is the amplitude of the motion at leading order, φm is the mth linear mode shape, and the second-order functions are expressed in the eigenbase as ∞ ⎡ ∞ ⎡Λ ⎤ Λ jmm ⎤ 1 jmm Φ = ∑⎢ 2 φ , Ψ = ⎥ ⎢ 2 ⎥ φ j , Λ jkh := ∫0φ j ⋅ N 2 (φk , φh ) dx (5) ∑ j 2 j =1 ⎢ ω j − 4ωm ⎥ j =1 ⎢ ω j ⎣ ⎦ ⎣ ⎦⎥
where N 2 ( v, w ) indicates the vectorial operator of the quadratic forces. The cable oscillates with the nonlinear frequency ωm∞ := ωm − Γ ∞mm am2 around the displaced configuration given by p( x) = p 0 ( x) + Ψ ( x) a 2 2 . The effective nonlinearity coefficient Γ ∞mm thus regulates the bending of the backbone of the system oscillating in the mth mode. It can be regarded as a nonlinear modal constitutive parameter embodying the combined modal effects of the nonlinear quadratic and cubic forces according to Γ ∞mm := Π ∞mm + ∆ mm where Π ∞mm := ∆ mm :=
1 8ωm 3 8ωm
1
∫φ
0 m
1
∫φ
0 m
⋅ ⎡⎣ N 2 (Φ, φm ) + N 2 (φm , Φ) + 2 ( N 2 ( Ψ , φm ) + N 2 (φm , Ψ ) ) ⎤⎦ dx
(6) ⋅ N 3 (φm , φm , φm ) dx
In Equation (6), Π ∞mm denotes the softening contribution of the quadratic forces which, in principle, depends, through Φ( x) and Ψ ( x) , from all of the cable eigenfunctions as emphasized by the superscript ∞, whereas ∆ mm is the hardening contribution of the cubic forces depending only on the
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considered active mode φm . Using the full-basis Galerkin discretization, the softening contribution is [10]
Π ∞mm =
1 8ωm
∞
∑ Λ (Λ j =1
jmm
mmj
⎡ 3ω 2j − 4ωm2 ⎤ ⎥ + Λ mjm ) ⎢ 2 2 2 ⎢⎣ ω j (ω j − 4ωm ) ⎥⎦
(7)
The coefficient in Equation (7) diverges when ω j = 2ωm due to a 2:1 internal resonance between the jth and mth mode. To account for the 2:1 interaction, the generating solution must include the two interacting modes and a third-order expansion has to be pursued. The equations governing the slow modulations of the amplitudes and phases of the interacting modes are i 1 Am = ( Λ mmn + Λ mnm ) An Am ei δ t + Γˆ ∞mm Am2 Am + Γˆ ∞mn Am An An 4 8ωm i 1 An = Λ nmm Am2 e − i δ t + Γˆ ∞nn An2 An + Γˆ ∞mn An Am Am 4 8ωn
(8)
where Am , An are the complex-valued amplitudes of the interacting modes at first order, the bar indicates the complex conjugate, δ is a small parameter expressing the detuning of the internal resonance, ωn = 2ωm + δ . The part of the nonlinearity coefficient of the mth mode due to the quadratic forces in the presence of the 2:1 resonance, Equation (7) is modified as follows: the summation does not include the nth term which, on the contrary, is 9 (4ωn2 ) Λ nmm ( Λ mmn + Λ mnm ) . For the definition of the other coefficients, see [9]. The nonlinear characteristics of the modes are investigated considering variations of Γˆ ∞mm with λ . Previous results relating to shallow cables [7-8] have shown that the lowest mode is initially hardening, then it becomes softening around its crossover and then hardening before diverging due to a 2:1 resonance with the third symmetric mode. In Figure 3a, the effective nonlinearity coefficient of the lowest mode of the transition cables is shown. The mode is a skew-symmetric geometric mode with two half-waves. The thick lines denote the coefficients Γ ∞mm and Γˆ ∞mm obtained with the individual mode assumption and with the 2:1 internal m) resonance (thicker line); the dashed line indicates the coefficient Γ (mm obtained with the one-mode discretization, retaining in Equation (8) the active linear mode only. The coefficient Γ ∞mm indicates a hardening mode almost everywhere except for a region around λ ≈ 3.9π where a 2:1 internal resonance between the third mode and the considered first mode is activated and the coefficient of the individual mode consequently diverges. However, the coefficient Γˆ ∞ mm
243
Linear and nonlinear elastodynamics of nonshallow cables 6.E+4
ω3 = 2 ω1
(a)
n=3
4.E+4
2.E+4
Γ
(∞)
0.E+0
Γ ^
-2.E+4
m=1 (∞)
-4.E+4
Γ
-6.E+4
(m)
-8.E+4 2
4
6
1.E+6
8
λ/π
10
12
14
m=4
5.E+5
ω7 = 2 ω3
16
(b)
Γ
ω9 = 2 ω3
(∞)
λ = 10 π
0.E+0
ω10 = 2 ω4
m=3 -5.E+5
(∞) Γ ^
n=9 -1.E+6
λ=4π -1.5E+6
Γ
-2.E+6 1
2
3
4
5
6
7
8
9
10
λ/π
11
λ = 10 π
(m)
12
13
14
15
16
17
m) Figure 3. (a) Variation of Γ ∞mm , Γˆ ∞mm , Γ (mm with λ π of (a) the lowest mode (m=1) and (b) the third mode (m=3) of the transition cables with γ = 0.75.
indicates that the mode preserves its hardening nature. The one-mode discretization captures the qualitative character of the mode although it greatly overestimates the nonlinear modal stiffness for higher λ . In Figure 3b, the nonlinearity of the second symmetric mode of the transition cables is investigated. This mode undergoes a crossover with the second skew-symmetric mode when λ is slightly below 4π where the mode
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becomes elasto-static. For low values of λ , the mode is slightly hardening, thereafter around the crossover it becomes softening. There are minor divergences for 2:1 internal resonances between the considered mode and the seventh or ninth modes, respectively. Further increasing λ , the mode (m = 4) becomes hardening again with a strong divergence in the region around the 2:1 internal resonance with the ninth mode. The coefficient Γˆ ∞mm accounting for the interaction is always negative indicating a hardening behaviour. Right above the crossover between the ninth (elasto-static mode) and the tenth mode (geometric mode), the nonlinearity decreases significantly due to the softening contribution delivered by the coupled elasto-static mode. The nonshallow regime, γ = 1.5, is investigated next for the lowest symmetric modes; namely, the third symmetric mode, m=5 (Figure 4a), the fourth symmetric mode, m=7 (Figure 4b), and the ninth symmetric mode, m=17 (Figure 4c). The third mode is hardening in the whole range except for the region where it becomes elasto-static while undergoing a crossover with the third skew-symmetric mode. In Figure 4(b), the mode predicted with Γ ∞mm follows the same pattern as the third mode of the transition cable; it is hardening, then around the crossover it becomes softening, then hardening again and diverges due a 2:1 resonance with the ninth symmetric mode (n=17). In this case, Γˆ ∞mm indicates that, in the interaction region the mode is hardening, then softening with a change of curvature occurring where the interacting mode undergoes a crossover. Away from the resonance region, the mode regains its hardening feature. As shown in Figure 4c, the interacting mode is hardening except for the region where it undergoes the crossover and its significant softening contribution to the low-frequency coupled mode explains why this mode becomes softening in the interaction.
5.
CONCLUDING REMARKS
A mechanical model describing finite motions of nonshallow cables around their initial catenary configurations has been employed to investigate the linear and nonlinear vibration characteristics of individual in-plane modes. The partial-differential equations of motion and boundary conditions have been asymptotically treated with the method of multiple scales, overcoming the drawbacks of a discretization process. The investigations into nonshallow cables indicate that the geometric modes are hardening. Conversely, in the neighborhood of the localized regions where the frequencies undergo crossovers, the modes turn into elasto-static or elastodynamic modes and exhibit a softening-type nonlinearity.
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Linear and nonlinear elastodynamics of nonshallow cables 5.E+5
(a) 0.E+0
-5.E+5
Γ
m=5
(∞)
Γ
-1.E+6
(m)
λ = 16 π
λ = 5.7 π
-1.5E+6
-2.E+6 2
4
6
8
10
12
14
16
18
20
1.E+7
(b) m=8
Γ
(∞)
5.E+6
λ = 16.5 π 0.E+0 (∞) Γ ^
m=7
ω18 = 2 ω10
-5.E+6
λ = 7.8 π
Γ
-1.E+7 2
4
6
8
10
12
14
16
18
20
(m)
22
24
26
28
5.E+7
30
(c)
0.E+0
m = 17
-5.E+7
λ = 19 π λ = 16.5 π
-1.E+8
-1.5E+8
-2.E+8
2
4
6
8
10
12
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18
λ/π
20
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∞ ∞ (m) Figure 4. (a) Variation of Γ mm , Γˆ mm , Γ mm with λ π of (a) the fifth, (b) the seventh, and (c) the seventeenth mode of the transition cables with γ = 1.5.
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The phenomenon inherent in the change of the nonlinearity may be explained considering that the relevant mode, around the crossovers, exhibits a shape with an appreciable transverse displacement inducing stretching which is quite sensitive to the upward or downward displacement directions. A significant drift is caused by the quadratic geometric forces towards the upper configurations where the tension in the cable can vanish thus leading to unsymmetrical softening behavior. Further, in the region of 2:1 interaction with a symmetric high-frequency mode, a higher-order analysis has shown that the geometric mode can preserve its hardening nature (although with reduced stiffness) or may become softening in the case of strong and largely detuned interaction with a softening elasto-static high-frequency mode. Moreover, from the convergence analysis carried out on reduced-order models, the contribution of the elasto-static modes turns out to be important; they must be included in the reduced-order description although they are far from the considered individual modes.
REFERENCES 1. 2.
Irvine HM. Cable Structures, Dover Publications Inc., New York (1984). Irvine HM, Caughey TK. “The linear theory of free vibrations of a suspended cable”, Proceedings of the Royal Society of London, Series A, pp. 299–315, 1974. 3. Burgess JJ, Triantafyllou MS. “The elastic frequencies of cables”, Journal of Sound and Vibration, pp. 153–165, 1988. 4. Triantafyllou MS. “The dynamics of taut inclined cables”, Quarterly Journal of Mechanics and Applied Mathematics, pp. 421–440, 1984. 5. Luongo A, Rega G, Vestroni F. “Planar non-linear free vibrations of an elastic cable”, International Journal of Non-Linear Mechanics, pp. 39–52, 1984. 6. Nayfeh AH, Arafat HN, Chin CM, Lacarbonara W. “Multimode interactions in suspended cables”, Journal of Vibration and Control, pp. 337–387, 2002. 7. Rega G, Lacarbonara W, Nayfeh AH. “Reduction methods for nonlinear vibrations of spatially continuous systems with initial curvature”, Solid Mechanics and Its Applications, pp. 235–246, 2000. 8. Arafat HN, Nayfeh AH. “Non-linear responses of suspended cables to primary resonance excitations”, Journal of Sound and Vibration, pp. 325–354, 2003. 9. Lacarbonara W, Rega G, Nayfeh AH. “Resonant non-linear normal modes. Part I: analytical treatment for structural one-dimensional systems”, International Journal of Non-Linear Mechanics, 38, pp. 851–872, 2003. 10. Lacarbonara W. “Direct treatment and discretizations of nonlinear spatially continuous systems”, Journal of Sound and Vibration, pp. 849–866, 1999. 11. Lacarbonara W, Paolone A, Vestroni F. “Shallow versus nonshallow cables: linear and nonlinear vibration performance”, Proc. Fifth EUROMECH Nonlinear Dynamics Conference, Eindhoven, The Netherlands, August 7–12, 2005. 12. Lacarbonara W, Paolone A, Vestroni F. “Elastodynamics of nonshallow suspended cables: Linear modal properties”, Journal of Vibration and Acoustics, in press, 2005.
NONLINEAR NORMAL MODES OF HOMOCLINIC ORBITS AND THEIR USE FOR DIMENSION REDUCTION IN CHAOS CONTROL S. Lenci1, G. Rega2 1
Department of Architecture, Buildings and Structures, Polytechnic University of Marche, via Brecce Bianche, 60131, Ancona, Italy, E-mail:
[email protected] 2 Department Structural and Geotechnical Engineering, University of Rome “La Sapienza”, via A. Gramsci 57, 00197, Rome, Italy, E-mail:
[email protected] Abstract:
A method for controlling nonlinear dynamics and chaos is applied to the infinite dimensional dynamics of a buckled beam subjected to a generic spacevarying time-periodic transversal excitation. The homoclinic bifurcation of the (unstable) rest position is identified as the undesired dynamical event, and is analytically detected by the Holmes and Marsden theorem [1]. The homoclinic orbits of the unperturbed systems, which are required to apply to theorem, are detected by the nonlinear normal modes technique. A comparison of the outcomes of infinite- vs finite-dimensional analyses is also performed.
Key words:
Buckled beam, infinite-dimensional system, homoclinic orbit, nonlinear normal modes, dimension reduction, homoclinic bifurcation, optimal control of chaos.
1.
INTRODUCTION
Nonlinear dynamics of many structures are governed by partial differential equations (PDEs), i.e., they are infinite dimensional. These equations are often over-complicated as the actual system dynamics activates only few spatial modes, so that reduced order, finite and lowdimensional, models governed by few ordinary differential equations (ODEs) are usually introduced. Then, the question arises of evaluating how reliable the approximations are in providing accurate descriptions of the true dynamics [2, 3]. Of course, the answer depends on the kind of investigated 247 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 247–256. © 2007 Springer.
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phenomena, since, depending on several circumstances, reduced order models (ROMs) can be able to exactly, accurately, or poorly describe various nonlinear dynamical features. This question is addressed in this work with respect (i) to the homoclinic behaviour, and (ii) to the application of a control of chaos technique. Nonlinear normal modes (NNMs) [4-6] is a technique suitable to provide accurate ROMs of continuous systems. Following Rosenberg’s ideas [4], the method was mainly applied to regular nonlinear oscillations, although the ensuing ROMs also exhibit complex dynamics ending up to chaos. In this work the NNM technique is instead used to detect a homoclinic orbit of a given hilltop saddle, playing a meaningful role in system dynamics. When compared with the classical cases, the main technical difference is that the dominant mode is no longer resonant (as no resonance frequency occurs) but rather unstable, i.e., structurally buckled. The analysis is conducted with reference to the buckled beam, but the treatment can be easily extended to other mechanical cases of interest. Furthermore, the Hamiltonian (conservative) case is considered to guarantee the existence of the homoclinic orbits, and to simplify the computations. When damping and/or excitations are added, depending on the kind and size of the perturbations the homoclinic orbits may or may not survive, their disappearance being associated with the occurrence of an homoclinic bifurcation. Indeed, in this work, detecting the homoclinic orbits is a necessary prerequisite for studying homoclinic bifurcation in infinite dimension, which can be done by means of the Holmes and Marsden theorem [1] which generalizes the classical Melnikov method. The homoclinic bifurcation triggers the transversal intersection of stable and unstable manifolds, and is at the base of such complex phenomena as fractal basin boundaries, sensitivity to initial conditions, transient or steady chaos. An objective of this work is to compare the outcomes of infinite- vs finite-dimensional analyses in terms of homoclinic bifurcation thresholds. In this respect, ROMs obtained by the classical (linear) Galerkin method, which projects the dynamics onto a planar subspace, may give incorrect results even from a qualitative viewpoint, so that more refined analytical techniques are needed to overcome this, and other, drawbacks [2-3]. A phenomenon of this type occurs herein. More precisely, it will be shown how for a class of boundary conditions (b.c.) the linear Galerkin method correctly captures the homoclinic bifurcation, while for others it does not. In the latter case, where the homoclinic orbit lies on a non-flat manifold, the NNMs become crucial, while in the former they are useless. As a matter of fact, the first class of b.c. is very restricted and somehow special, so that NNMs are indeed necessary to study the homoclinic bifurcation in infinite dimension with a sufficient degree of generality. This is the main motivation of this work.
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A second objective is to check/extend a method previously developed by the authors to control nonlinear dynamics and chaos in low-dimensional systems [7], based on shifting the homoclinic bifurcations in parameters space by optimally modifying the temporal shape of the excitation. Besides validating the temporal shape obtained in the low-dimensional framework – therein, the solely to be possibly exploited –, the present extension to infinite dimension also allows us to change the spatial shape of the excitation, thus enlarging the capabilities of control.
2.
MECHANICAL MODEL AND MODAL EXPANSION
The dimensionless partial differential equation governing the planar nonlinear dynamics of an initially straight buckled beam is 1
+w′′′′+Γw′′–kw′′ ∫ ( w' ) 2 dz =ε[F(z,t)–δ w ], w
(1)
0
where w(z,t) is the time dependent transversal displacement of a point at z∈[0,1], dot (prime) represents time (space) derivative, Γ>Γcr is the axial governing parameter (positive=compression, Γcr=buckling threshold), k is the stiffness due to membrane effects (the unique source of nonlinearity considered in this work), δ is the coefficient of viscous damping, and F(z,t)=
∑
N
n =1
f n ( z ) sin(nωt + ψ n )
(2)
is the external T=2π/ω time periodic, spatially distributed, excitation. ε is a small parameter introduced to stress the smallness of damping and excitations. ε=0 is referred to as “unperturbed” or “conservative” problem. The solutions of (1), and in particular the homoclinic solutions we are interested in, are sought in the modal expansion form w(z,t)=Σnan(t)wn(z),
(3)
where wn(z) are the eigenfunctions of the linearized equation and are called linear normal modes, and an(t) are the modal amplitudes. The functions wn(z) span planar invariant (with respect to the linearized equations) manifolds. By using (3) we obtain the Hamiltonian H=(1/2)Σn an2 +V, where (being λn the eigenvalues associated to wn(z)) the potential V is given by V=(1/2) Σ ∞n =1 λn(an)2+(k/4)( Σ ∞n =1 Σ ∞m =1 anamdn,m)2, dm,n=dn,m=
1
∫ w 'w 0
n
m
' dz . (4)
dm,n are nonlinear coupling coefficients, which depend only upon Γ. There are two families of boundary conditions (b.c.) [8]. The first is
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characterized by the property d n,m = –κnδnm . The equations of motion strongly simplify and, in particular, if we start from an initial condition such that an≠0, a n ≠0, for a given n, and ai= ai =0, i≠n, then ai and ai , i≠n, remain null for all time, while the time evolution of an is governed by the Duffing n +λnan+k κ n2 an3 =0. This means that the eigenspaces spanned by equation a the linear modes remain planar also in the nonlinear case. The hinged-hinged and guided-hinged b.c. belong to the first family [8]. All of the others belong to the second family, for which we have a full nonlinear coupling among modes, and the invariant manifolds no longer remain planar when nonlinearities are considered.
3.
HOMOCLINIC ORBITS (UNPERTURBED SYSTEM)
The two symmetric unperturbed homoclinic orbits of the hilltop saddle (representing the unstable vertical position) are detected by the NNM technique. Following [9] (see also [3] for an overall framework), we consider the first modal amplitude (spanning the unique unstable manifold) as the main (master) variable x and the others as secondary (slave) variables. The key idea of the method consists in assuming the slave variables as time-independent functions of the master one, ai=ai(x), i≥2, thus determining the associated modal (or slave) equations as in [8]:
⎛
2Vai′′+[1+ Σ ∞j =1 (a j ' ) 2 ] ⎜⎜ ai '
⎝
∂V ∂V ⎞ ⎟ =0, i≥2, − ∂x ∂ai ⎟⎠
(5)
where use is made of the fact that the Hamiltonian vanishes on the homoclinic orbits. The solution of (5) is sought in polynomial form ai(x)=ai,3kx3+ai,5k2x5+…, i ≥2.
(6)
To determine the unknowns coefficients ai,l, we insert (6) in (5) and expand in Taylor series. We obtain a recursive set of equations which yield
− 6d12,1 ai ,3 + Σ ∞n =1 a n ,3 (d1,1 d i ,n + 2d1,i d1,n ) d1,1d1,i ai,3= , ai,5= , i≥2. (7) 9λ1 − λi 25λ1 − λi For the first family of b.c. we have d1,i=0, i ≥2, so that, from (7), ai,3=ai,5=…=0. This means that ai(x)=0, i.e., the nonlinear invariant manifold coincides with the linear one, namely, it is planar. This case corresponds to what Rosenberg [4] called similar normal modes. For the second family of b.c. the coefficients ai,3 and ai,5 are not trivial and, on average, ai,5 is one order of magnitude smaller than ai,3 [8], showing how the higher order nonlinearities (Taylor coefficients ai,l) in the slave
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modal amplitudes are important only for some isolated cases. Furthermore, still on average, both ai,3 and ai,5, in absolute value, are decreasing functions of i. This means that high order slave amplitudes are practically unessential. By inserting the functions (6) in the expression (4) of V we get V(x)=(1/2){α2x2+α4kx4+α6k2x6+α8k3x8+…}, α6= d12,1 Σ ∞n =1 d12,n
α2=λ10,
18λ1 − λ n , α8= Σ ∞n =1 [(43λ1–λn)an,3an,5+6 d12,1 (an,3)2], … (8) (9λ1 − λ n ) 2
In what follows we assume that Γ∈]Γcr,1;Γcr,2[, which is the region of main practical interest. This implies that λ10, n≥2, [8] and thus proves that α60, which always occurs apart from special cases [8]. The homoclinic solution cannot be computed in closed form, but can be easily determined in a pseudoanalytical form. Apart from guaranteeing the hardening behaviour at infinity and providing a better approximation of the true homoclinic orbits, this model has no new qualitative features. A comparison among the three ROMs is made in terms of the relevant potentials V4(x), V6(x) and V8(x) (Figure 2a) and phase portraits (Figure 2b) for the fixed-fixed case, by looking at the corrections due to the higher order nonlinearities. Apart from the quantitative differences, which are of the order of 10÷15%, we see that there is a non monotonic convergence toward the conjectured limit potential V∞ giving the exact homoclinic solutions.
Nonlinear normal modes of homoclinic orbits and chaos control 600
20
V∞ (conjectured) B V4 V8
x 7,hom (t) x 3,hom (t)
V6
x√k
kV 0
253
x3
x 5,hom (t)
x7
x5
A
x ∞,hom(t) (conjectured)
-20
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a)
0
x√k
3.5
b)
0
x√k
2.5
Figure 2. a) The potentials kV4(x√k), kV6(x√k) and kV8(x√k) and b) the phase portrait of the homoclinic loops x3,h(t), x5,h(t) and x7,h(t). Fixed-fixed b.c., (Γ–Γcr,1)/(Γcr,2–Γcr,1)≅0.64.
An overall comparison can also be made in terms of the analytical expressions of the homoclinic solution provided by the three models: wh(z,t)=x3,h(t)w1(z), wh(z,t)=x5,h(t)w1(z)+ Σ ∞n =1 an,3k[x5,h(t)]3wn(z), wh(z,t)=x7,h(t)w1(z)+ Σ ∞n =1 {an,3k[x7,h(t)]3wn(z)+an,5k2[x7,h(t)]5wn(z)}.
(14)
They are seen to be more and more refined as the approximation of the master modal amplitude is improved by accounting for (a theoretically infinite number of) slave modal amplitudes an,l at different l-orders.
4.
HOMOCLINIC BIFURCATIONS (PERTURBED SYSTEM)
When excitation and damping are added to the unperturbed homoclinic orbits of the previous section, the stable and unstable manifolds split, and, depending on the relative magnitude of the perturbations, they keep disjoint or intersect. The intermediate critical case corresponds to manifolds tangency, and represents the homoclinic bifurcation threshold. The homoclinic bifurcation can be analytically detected by the Holmes and Marsden theorem [1], which is a generalization to infinite-dimensional systems of the classical, finite-dimensional, Melnikov’s theory. The theorem is quite technical, and requires determining the manifold Σ where the unperturbed homoclinic solutions lie (see Figure 1), which has been done in the previous sections. The application is then relatively straightforward, and, as in the classical case, relies on the Melnikov function, which is given by [10] Ml,r(m)=–(δ/k)β0± Σ nN=1 nωcos(nm+ψn)[γn1β1(nω)+γn2β2(nω)+γn3β3(nω)+…],
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∫
∞
−∞
β1(ω)=
∫
xh2 (t ){1 + ∑i∞=2 [ ai ' ( xh (t ))]2 }dt , γnj = ∞
−∞
xh (t ) cos(ωt ) dt , βi(ω)=
∫
∞
−∞
∫
1
0
f n (ζ ) w j (ζ ) dζ ,
ai ( xh (t )) cos(ωt ) dt , i≥2. (15)
The βi (ω), i ≥2, account for the non planarity of the manifold Σ. In fact, for the first family of b.c. we have ai(x)=0, and then βi(ω)=0, i≥2, so that the Melnikov function (15) gets simpler, becoming identical to the Melnikov function of the Duffing equation obtained with the (linear) Galerkin projection on the first mode [10]. This proves that, in the present case, even the unrefined ROM is able to exactly capture the homoclinic bifurcation. This is a consequence of the flatness of Σ, and no longer holds for the second family of b.c.. In any case, it must be emphasized that “the full power of the (Holmes and Marsden) theorem is necessary since in the infinite dimensional case, the perturbed manifolds … do not lie in Σ” [1]. For the following purposes, it is useful to rewrite (15) in the form Mr,l(m)=const.[1±γ11h(m)/ γ 11h ,cr (ω ) ],
(16)
where h(m)= Σ nN=1 hncos(nm+ψn) and the expressions of hn and γ 11h ,cr (ω ) can be obtained by comparison with (15). We use this representation because we assign γ11 the role of overall excitation amplitude, while the relative amplitudes γnj/γ11 and the phase ψn of the superharmonics determine the (temporal) shape of the excitation [10]. According to the theorem, we have homoclinic intersection of the right (left) manifolds (i.e., right (left) Melnikov chaos) if there exists m∈[0,2π] such that Mr,l(m) have simple zeros, respectively, which occurs if and only if γ11> γ 11h ,cr (ω ) /M r,l = γ 11r ,,lcr (ω ) ,
(17)
being M l=max m∈[0,2π]{h(m)}, M r=–min m∈[0,2π]{h(m)}. For a generic excitation, the curves γ 11r ,,lcr (ω ) separate, in the frequency/amplitude parameter space (ω, γ11), the zones – respectively below (above) the critical curves – where right/left homoclinic intersections do not (do) occur. In the case of harmonic excitation we have h(m)=cos(ωt+ψ1) and r M =M l=1. This shows that γ 11h ,cr (ω ) represents the coinciding right and left homoclinic bifurcation thresholds for the time-harmonic excitation, which is considered as a reference to measure the improvement obtained with control. The strip above γ 11h ,cr (ω ) and below γ 11r ,,lcr (ω ) is called saved (i.e., controlled) region, and represents the zone where the time-unharmonic excitation is theoretically effective. Its maximum enlargement constitutes the objective of the control method (Sect. 5). To quantitatively measure the increment of the critical thresholds we use the gains [7], which are the ratios G r,l= γ 11r ,,lcr (ω ) / γ 11h ,cr (ω ) =1/M r,l.
(18)
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OPTIMAL CONTROL OF CHAOS
In this section we apply to the buckled beam a method for controlling nonlinear dynamics and chaos developed by the authors [7] and previously applied only to single d.o.f. mechanical systems. The idea of the method is to increase the homoclinic bifurcation thresholds, or enlarge the saved region, or increase the gain. This result is pursued by optimally varying both the spatial (Sect. 5.1) and temporal (Sect. 5.2) shapes of the excitation.
5.1 “Optimizing” the spatial shape of the excitation We assume fn(z)=γn1w1(z) as practically “optimal” spatial shape of the excitation. Indeed, entailing γn2=γn3=…=0, it permits to eliminate the influence of the non-flatness of Σ in the oscillating part of the Melnikov functions (15), which become M l,r(m)= –(δ/k)β0± Σ nN=1 nωcos(nm+ψn)γn1β1(nω),
(19)
while still having it in the constant part β0, where it is unavoidable. In addition, this choice permits to get rid of the non-resonance conditions involved in the Holmes and Marsden theorem [1], thus contributing to enlarge the range of applicability of the method [10]. For the first family of b.c. (for which βi(ω)=0, i ≥ 2) the Melnikov function is already of the form (19). Thus the assumption on the excitation spatial shape is not ‘necessary’, but it is still appropriate because w1(z) is the only spatial component of fn(z) affecting the homoclinic bifurcation, so that it permits to minimize the control cost, because we only use what is effective.
5.2 Optimizing the temporal shape of the excitation The choice of the optimal temporal shape is based on the observation that the Melnikov function (19) is formally identical to that obtained for the reduced order models based on the Galerkin projection. The solely difference is in the expression of β0, which accounts for the non-flatness of the manifold Σ in the case of the second family. For the first family of b.c., on the other hand, the Melnikov functions are identical. According to this property we can conclude that the features of the control method highlighted in the previous applications to finite-dimensional systems [7] also hold for the possibly underlying infinite-dimensional systems. They are now summarized for the sake of completeness. It is clear from (18) that the best temporal shape of the excitation is the one providing the largest gain. However, we can control only the right (left) homoclinic bifurcation, irrespective of what happens to the other, or we can
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control simultaneously the right and left homoclinic bifurcations, each choice having its own advantages/disadvantages. i) “One-side” right (left) control. To increase as much as possible γ 11r ,cr ( γ 11l ,cr ), we have to maximize G r (G l ) by varying the temporal-shape coefficients hj and ψj, j=2,3,…. The solution of these problems is reported in [7], where it is shown that the optimal gains are G2=1.4142, G3=1.6180, G4=1.7321,…, G∞=2, the index being the number of superharmonics used for control. ii) “Global” control. To control both right and left homoclinic bifurcations, the relevant gains Gr and Gl must be increased at the same time. This entails maximizing G=min{Gr,Gl} by varying hj and ψj, j=2,3,…. The solution of this problem [7] provides G3=1.1547, G5=1.2071, G7=1.2310,…, G∞=1.2732, the even super harmonics being not involved in this case. The results of this section further confirm the generality of the proposed control of chaos method, which applies to a variety of mechanical systems/ models within a common dynamic framework, as discussed in detail in [7].
REFERENCES 1.
Holmes P, Marsden J. “A partial differential equation with infinitely many periodic orbits: Chaotic oscillations of a forced beam,” Arch. Rat. Mech. Anal., 76, pp. 135-165, 1981. 2. Steindl A, Troger H. “Methods for dimension reduction and their application in nonlinear dynamics,” Int. J. Solids Struct., 38, pp. 2131-2147, 2001. 3. Rega G, Troger H. “Dimension Reduction of Dynamical Systems: Methods, Models, Applications,” Nonlinear Dynamics, 41, pp. 1-15, 2005. 4. Rosenberg RM. “On nonlinear vibrations of systems with many degrees of freedom,” Adv. Appl. Mech., 9, pp. 155-242, 1966. 5. Shaw SW, Pierre C. “Normal modes for nonlinear vibratory systems,” J. Sound Vibr., 164, pp. 85-124, 1993. 6. Vakakis AF. “Non-linear normal modes (NNMs) and their applications in vibration theory: An overview,” Mech. Syst. Signal Proc., 11, pp. 3-22, 1997. 7. Lenci S, Rega G. “A unified control framework of the nonregular dynamics of mechanical oscillators,” J. Sound Vibr., 278, pp. 1051-1080, 2004. 8. Lenci S, Rega G. “Dimension reduction of homoclinic orbits of buckled beams via nonlinear normal modes technique,” in press on Int. J. Non-linear Mech., 2007. 9. Rand RH, Lecture Notes on Nonlinear Vibrations, Cornell University, available on line at www.tam.cornell.edu/randdocs/, 2003. 10. Lenci S, Rega G. “Optimal control of the homoclinic bifurcation in buckled beams: infinite-dimensional vs reduced order modeling,” submitted, 2007.
ROTATING SLIP STICK SEPARATION WAVES A. Teufel, A. Steindl, H. Troger Institute for Mechanics and Mechatronics, Vienna University of Technology, A-1040 Vienna, Austria
Abstract:
We consider a thick-walled elastic tube (cylinder) which is fixed in space at its outer surface. Inside the tube a rigid shaft rotates about the common axis of the tube-shaft system. The inner diameter of the tube is assumed to be smaller than the outer diameter of the shaft. Hence the tube is compressed. The tangential contact force is assumed to be given by Coulomb’s law of dry friction. For the infinite dimensional continuous tube shaft system solutions are found, which are rotating slip-stick and slip-stick-separation waves with different wave numbers. To understand some special features of these solutions we also consider a simple one degree of freedom oscillator, however, with a more complicated dry friction contact force which now depends on the relative velocity possessing a decreasing and increasing part.
Key words:
Finite and infinite dimensional oscillators with friction, bifurcation analysis, Coulomb friction, three-parameter friction model
1.
INTRODUCTION
The motivation to consider this problem follows from the task to understand brake squeal and to find means to avoid it. Brake squeal occurs, for example, for many modern high speed trains, where it is highly undesired. Brake squeal is a friction induced oscillation and must be avoided. Contrary to the usual approach, using low dimensional systems to investigate friction induced oscillations, in [1] a continuous linearly elastic system model with an infinite number of degrees of freedom is treated (Figure 1). The friction model is Coulomb’s law, that is, (i) gliding friction is constant and independent of the relative velocity, (ii) there is a jump between positive and 257 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 257–266. © 2007 Springer.
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Figure 1. Elastic tube, fixed in space at the outer radius Ro, surrounding a rigid shaft rotating with angular velocity Ω , the radius of which Ri is larger than the inner radius of the undeformed tube by the value δ .
Figure 2. Friction oscillator with one degree of freedom. The belt moves with velocity V .
negative relative velocities and (iii) static friction has the same value as gliding friction. Under these assumptions, it is shown in [1] numerically by means of a Finite Element analysis, that interesting new solutions exist, which cannot be found in low dimensional systems with Coulomb friction. For example, there exists rotating stick-slip or rotating stick-slip-separation waves where, for example, local domains exist, where there is no contact between the tube and the shaft. Another interesting phenomenon occurred for the continuous model, namely that the elastic continuum at contact, when it was moving in the same direction as the rigid shaft, under certain conditions was moving faster than the shaft. We call this phenomenon “overshooting” or “reverse slip”. Since this is a very interesting effect, we also asked the question whether this overshooting may occur for the simple friction oscillator shown in Figure 2, that is, that the mass moving in the direction of the moving belt moves faster than the belt. We show that this may happen but only if we use a more complicated friction model than Coulomb’s model. We also give an explanation for this fact.
2.
MECHANICAL MODEL WITH ONE DEGREE OF FREEDOM
We show that “overshooting”, as defined before, may occur for the elastically constrained mass on a moving belt (Figure 2) with one degree of freedom, if we use a more complicated friction characteristic than the one of Coulomb. We assume that the friction force in the contact area is described by the non-smooth, three-parameter function depicted in Figure 3 taking into
Rotating slip stick separation waves
259
account both a decreasing and increasing part of the friction force depending on the relative velocity ⎡ f (V − y ) = sgn(V − y ) ⎢α − ⎢⎣
(
β+ γ
⎛
⎞⎤
⎝
⎠⎦
) ⎜⎜1 − 1 + V1 − y ⎟⎟⎥⎥ + γ (V − y), 2
(1)
where y is the velocity of the mass and V is the velocity of the moving belt. The non-smooth function f (V − y ) of the dynamic friction force for constant normal load depending on the relative velocity V − y contains three parameters: α accounts for the static friction force, β measures the maximum
Figure 3. Three parameter (α , β , γ ) model of the dynamic friction force depending on the relative velocity.
Figure 4. Stratifications of the parameter plane spanned by friction drop β and belt velocity V for β = 0.4 , and γ = 0.08 . The loci of grazing, switching, crossing and Hopf bifurcations are marked by GR, SW, CC and Ho, respectively.
drop and γ defines the asymptotic slope of the friction characteristics, as illustrated in Figure 3. The equations of motion of the elastically constrained single mass on the moving belt in nondimensional form are (cf. [7]), x = y, y = − x + f (V − y ),
(2)
where x is the displacement of the mass. System (2) has a stationary solution y = 0, x = f (V ) , which is stable for belt velocities V > β / γ . At the critical value V = β / γ a subcritical Hopfbifurcation occurs ([8]). System (2) constitutes a Filippov system (cf. [2]), where the two dimensional phase space has a discontinuity at the line V − y = 0 .We call a
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segment, that stays on the discontinuity y = V for a finite time, a sticking motion. The maximum length of a sticking segment in the phase space is bounded by two tangent points, where the vector field (2) is tangent to the discontinuity at x = ±α , y = V , as it is explained in [2] (first frame in Figure 5). In the following we shall be concerned with the appearance of periodic steady-state motions with a sticking segment, mechanically referred to as stick-slip oscillations, depending on the shape of the friction characteristic (1). For the numerical calculations we use the software package SLIDECONT ([9]), SW
CR
CR2
Ho
SW2
GR
stable unstable
Figure 5. Sequence of phase plane plots related to Figure 4 for quasistatically increasing V at β / α = 0.75 .
which treats the non-smooth system (2) as an extended smooth system in a six-dimensional phase space, x0 = V y 0 = 0 x1 = y1 y1 = − x1 + α −
(
β+ γ
(
β+ γ
) ⎛⎜⎝1 − 1 + V1− y ⎞⎟⎠ + γ (V − y ),
(4)
) ⎛⎜⎝1 − 1 − V1+ y
(5)
2
1
1
x2 = y2 y 2 = − x2 − α +
(3)
2
⎞ ⎟ + γ (V − y2 ), 2 ⎠
where ( x0 , y0 ) describe the motion on the discontinuity y = V of (2) if −α ≤ x ≤ α .The two motions ( x1 , y1 ) and ( x2 , y2 ) account for the dynamics of (2) in the domains of the phase space where y < V and y > V , respectively.
Rotating slip stick separation waves
261
First we calculate the stability boundary in the parameter space (Figure 4), where the bifurcation parameter V is drawn versus the depth β / α of the friction characteristic. We note that physically only values β / α < 1 make sense. The third parameter γ = 0.008 . Let us consider three typical values of β / α , namely β / α = 0.20 , β / α = 0.50 and β / α = 0.75 and see from Figure 4 which behaviour we get, if we quasistatically increase the speed V of the moving belt. We take first β / α = 0.20 . After the belt starts from its rest position, immediately a stick slip oscillation sets in according to the first frame in Figure 5. Besides this stable limit cycle also an unstable equilibrium exists. A third solution appears at the curve “Ho”. Here due to a Hopf bifurcation the stationary state becomes stable and is separated by an unstable periodic solution from the still stable stick slip oscillation (frame 4 of Figure 5). If the speed of the belt is increased further, the curve “GR” is reached (frame 6 of Figure 5). Now the unstable limit cycle touches the discontinuity y = V and both solutions disappear with the consequence, that the mass attains the stable equilibrium position. For β / α = 0.50 we start with the stick slip oscillation again, but at the curve “SW” overshooting according to frame 2 of Figure 5 occurs. Again after crossing the curve “Ho” an additional unstable limit cycle appears (frame 4 of Figure 5). If the speed is increased further, at “SW” the overshooting disappears and finally at “GR” the cycles annihilate each other and only the equilibrium position remains. The sequence of phase plots for β / α = 0.75 , which is depicted in Figure 5, starts as before, but at the curve “CC” the overshooting has extended to the full length of the sticking domain, as can be seen from frame 3 of Figure 5. If the bifurcating unstable limit cycle, which bifurcated at “Ho”, touches the discontinuity y = V at the tangent point x = α , the stable stick-slip cycle disappears by a bifurcation that is termed grazing in [2].
3.
CONTINUOUS INFINITE DIMENSIONAL SYSTEM
We consider now the system of Figure 1, where an elastic thick-walled annular tube, which is fixed in space at its outer surface, surrounds a rigid shaft rotating with angular velocity Ω about the common axis. We assume that the radius of the shaft is larger than the undeformed inner radius of the tube by the mismatch δ (Figure 1). Friction between the rotating shaft and the surrounding tube is taken into account by Coulomb’s law as explained in the Introduction. The linear planar equations of motion for the elastic continuum together with the corresponding boundary and contact conditions are derived in polar coordinates ( r ,ϑ ) in ([3]) and read
262
A. Teufel et al. 1 ( ∇u + ∇uT ) , 2 ρ u = div σ ,
ε=
σ=
ν
(1 + ν )(1 − 2ν )
(6) tr ( ε ) I +
1 ε 1 +ν
where σ , ε and u = ( u , v ) are the stress tensor, the strain tensor and the displacement vector, respectively. The boundary conditions at the outer boundary of the elastic tube are: T
u (ξ ,ϕ , t ) ≡ v (ξ ,ϕ , t ) ≡ 0.
The conditions at the tube shaft contact are
σ rr (1,ϕ , t ) = − p (ϕ , t ) , σ rϕ (1,ϕ , t ) = − q (ϕ , t ) , u (1,ϕ , t ) ≥ δ ,
p (1, ϕ , t ) ≥ 0,
vrel (ϕ , t ) = 1 − v (1,ϕ , t ) ,
p ( u (1,ϕ , t ) − δ ) = 0,
q ≤ fp, vrel ( q − fp ) = 0,
where the ratio ξ = R0 / Ri and γ = ρ Ri2 Ω 2 / E is introduced. The physical quantities have been scaled by u ← u / Ri , δ ← δ / Ri , σ ← σ / E , r ← r / Ri , t ← Ωt ,
System (6) together with the boundary conditions is a nonlinear system of second order partial differential equations in radial and circumferential direction and in time. The Coulomb friction assumption and the nonpenetration condition at the interface between cylinder and shaft introduce non-smooth nonlinear terms. It is shown in ([3]), that the boundary value problem has the following (unstable) steady state sliding solution ⎞ 1 ⎛ξ2 ⎜ − r ⎟, 2 ξ −1⎝ r ⎠ 2 ⎞ ⎞⎛ 1 ⎛ξ 1 ve = δ f 2 ⎟⎟ , ⎜ − r ⎟ ⎜⎜1 + 2 ξ −1⎝ r ⎠ ⎝ ξ (1 − 2ν ) ⎠ 1 1 ⎛ 2 1 ⎞ pe = δ 2 ⎜ξ + ⎟ > 0. + − 1 ν 1 2ν ⎠ (ξ − 1) ( ) ⎝ qe = fpe
ue = δ
To simplify the further analysis we follow ([3]) and assume that the displacement field may be decomposed in the form
263
Rotating slip stick separation waves u ( r , ϕ , t ) = u (ϕ , t ) X ( r ) , v ( r , ϕ , t ) = v ( ϕ , t ) X ( r ) ,
(7)
where X (r ) =
1
r (ξ 2 − 1)
(ξ
2
− r 2 ).
(8)
Here X ( r ) is taken from the steady state sliding solution, c denotes the nondimensional wave speed and u , v are the radial and tangential displacements of the tube at the contact surface with the shaft ( r = 1) . Inserting (7) and (8) into system (6) we obtain a system of partial differential equations together with boundary conditions for the radial displacement u and tangential displacement v in the form u − bu'' − Dv' + gu = P, v − av'' + Du' + hv = Q, P ≥ 0, u ≥ δ , P ( u − δ ) = 0, Q ≤ fP, Q (1 − v ) − fP 1 − v = 0.
Here ( ⋅)′ and (⋅) denote the derivatives w.r.t. the circumferential angle ϕ and the dimensionless time t , respectively. P is the normal pressure between the cylinder and the shaft and Q is the friction force. The parameters a, b, d , f , g , h depend on the angular velocity Ω , the material properties and geometry of the cylinder. δ is the mismatch between the radii at the contact circle. Assuming a k − mode traveling wave (TW) solution in circumferential direction, u (ϕ , t ) = U (θ ) , v (ϕ , t ) = V (θ ) , with θ = ϕ − ct ,
the PDE can be reduced to a BVP of nonlinear ordinary differential equations
(c (c
− b )U'' − DV' + gU = P, − a )V'' + DU' + hV = Q, P ≥ 0, U ≥ δ , P (U − δ ) = 0, Q ≤ fP, Q (1 + cV' ) − fP 1 + cV' = 0, {U ,U' ,V ,V'}( Θ ) = {U ,U',V,V'}( 0 ) , 2 2
with Θ = 2π / k . Due to the nonlinear contact and friction forces several different types of solution segments are possible. For simplicity the following list gives the conditions for left traveling waves ( c < 0 ):
264
A. Teufel et al. Sticking solution: The cylinder rotates with shaft. U ≡ δ , P = gδ − dV ′ , V ′ ≡ −1/ c, Q = − hV , Q ≤ fP Slip: The cylinder slides along the shaft. U ≡ δ , P = gδ − dV', V' < −1/ c, Q = ( c 2 − a )V'' − hV
Counter-slip (overshooting): The cylinder moves faster than the shaft. U ≡ δ , P = gδ − dV' , V' > −1/ c, Q = − ( c 2 − a )V'' + hV . Separation: The contact between the bodies is lost. U > 0, Q = 0:
(c
2
− b )U'' − dV' + gU = 0,
(c
2
P = 0 and
− a )V'' + dU' + hV = 0.
At the end of the separation region U ′ jumps back to 0. At the borders between the different regions appropriate switching conditions must be fulfilled, e.g. P = 0 at the start of the separation and U = δ at the end. For the sticking region we have V ′ = −1/ c at the start and Q = fP at the end. The nonlinear BVP is solved with the program Boundsco ([5]), which is able to compute the switching points between different regions automatically. In order to vary parameters in the system, we use the continuation algorithm Hom ([6]).
5000
0.0008
4500
0.00075
4000 3500
δ=0.776e-3 δ=0.729e-3 δ=0.453e-3 τ1 τ2 τ3
0.0007
2500
U(θ)
P(θ)
3000 2000
0.00065 0.0006
1500 0.00055
1000 500
0.0005
δ=0.005 δ=0.743E-3
0 -500 0
0.2
0.4
0.6
0.8 θ
1
1.2
0.00045 1.4
1.6
Figure 6. Pressure for a slip-stick solution depending on the parameter δ for a mode-4 solution.
0
0.2
0.4
0.6
0.8 θ
1
1.2
1.4
1.6
Figure 7. Radial displacement u showing clear separation for small mismatch δ . For example for δ = .000729 we have from left to right: slip, separation, slip and stick. If δ decreases, the length of the second slip interval goes to zero.
We present now some numerical results. In Figure 6 the pressure is drawn as function of the circumferential angle ϑ for a mode-4 solution ( 0 < ϑ < π / 2 ) for various values of the mismatch δ as parameter. While for
Rotating slip stick separation waves
265
larger values of δ just a slip-stick solution occurs, the contact pressure P becomes zero, if δ decreases below a certain value. If we decrease the mismatch δ further, a separation interval occurs, as it can be seen in Figure 7, which shows the radial displacement for different values of δ and the loci of the switching points τ i . If the mismatch δ becomes very small, the switching points τ 2 and τ 3 coalesce and the slip region right of the separation zone vanishes. We conclude from Figure 6, that the nonlinear form of the pressure distribution most likely is responsible for the complicated behavior (overshooting), which is found for the simple oscillator only for more complicated friction characteristics than Coulomb’s one. In Figure 8 a phase plane plot for mode-8 traveling waves with reverse slip (overshooting) is depicted. Figure 9 shows the relative velocity and the friction force. The short segment between ϑ = 0.62 and ϑ = 0.73 , where the relative velocity is negative, is clearly visible. Mode 8, δ=0.0001, ξ=1.5, f=0.3
0.002
100
0.001
50
0 -0.001
1
0
0
-50
-0.002
-1
-100
-0.003 -0.004 -0.0004
2
150
Q
V’
Mode 8, δ=0.0001, ξ=1.5, f=0.3 200
Separation Slip-Stick
0.003
1+cV’
0.004
-150
q 1+cV’
-200 -0.0002
0
0.0002
0.0004
V
Figure 8. Phase plane plot of the tangential displacement of a travelling wave with reverse slip (overshooting).
0
0.1
-2 0.2
0.3
0.4
0.5
0.6
0.7
0.8
θ
Figure 9. Friction force and relative velocity for a mode-8 travelling wave with reverse slip.
In order to calculate the stability of the computed solutions, we have to investigate the linearized PDE. For simplicity we consider only the simple slip-stick solution. We used two methods to estimate the stability of the solution. First by replacing the spatial derivatives by finite differences, we obtain a large system of ODEs. Although one obtains many eigenvalues at once, these eigenvalues are usually quite inaccurate. In order to improve the accuracy of selected eigenvalues, we derive as alternative a BVP for the eigenfunctions of the system. For the considered parameter values no stable traveling wave solutions were found. However, numerical simulations indicated that there might be some stable oscillations about a traveling wave solution, which can be considered as motion on a torus.
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CONCLUSIONS
The existence of stick-slip, slip-separation and stick-slip-separation waves, either rotating in the same direction as the rigid shaft or rotating in the opposite direction, has been demonstrated. Phenomena like reverse slip or loss of contact have been discussed. By means of a simple one-degree of freedom oscillator we show that reverse slip, which occurs for the tube shaft problem for Coulomb friction, can also occur for the one degree of freedom oscillator, however, only if a more complicated friction function depending on the relative velocity is assumed. We conclude that this corresponds to the complicated pressure distribution for the continuous system.
REFERENCES 1. 2. 3.
4. 5.
6. 7. 8. 9. 10. 11. 12.
Moirot F, Nguyen QS, Oueslati A. “An example of stick-slip and stick-slip-separation waves”, European Journal of Mechanics A/Solids, 22, pp. 107-118, 2002. Kuznetsov Y, Rinaldi S, and Gragnani A. “One-parameter bifurcations in planar Filippov systems”, Int. J. Bifurcation and Chaos, 13, pp. 2157-2188, 2003. Nguyen QS, Oueslati A, Lorang A. “Brake Squeal: A Problem of Dynamic Instability and Stick-Slip-Separation Waves?”, Proceedings of the EUROMECH457, S. Bellizzi et al. (eds.) Press of the ENTPE, Lyon, pp. 99-102. Teufel A, Troger H. “Stick-slip-separation waves in rotating shaft bush system”, Submitted to PAMM 2006. Oberle H.J, Grimm W, Berger E. BNDSCO, Rechenprogramm zur Lösung beschränkter optimaler Steuerungsprobleme, Benutzeranleitung M 8509, Techn. Univ. München, 1985. Seydel R. “A continuation algorithm with step control”, Numerical methods for bifurcation problems, ISNM 70, Birkhäuser, pp. 480-494, 1984. Galvanetto U and Bishop S. “Dynamics of a simple damped oscillator undergoing stickslip vibrations”, Meccanica, 13, pp. 337-347, 1999. Elmer F. “Nonlinear dynamics of dry friction”, J. Phys. A: Math. Gen, 30, pp. 60576063, 1997. Dercole F and Kuznetsov Y, User guide to SlideCont2.0, 2004. Teufel A, Steindl A, Troger H. “On non-smooth bifurcations in a simple friction oscillator”, PAMM, 5, pp. 139-140, 2005. Teufel A. “Smooth and non-smooth bifurcation analysis in applied mechanical systems”, PHD-Thesis, Vienna University of Technology, 2006. Steindl A. “Bifurcations of stick-slip-separation waves in a brake-like system”, PAMM, 6, pp. 337-338, 2006.
MANY PULSE HOMOCLINIC ORBITS AND CHAOTIC DYNAMICS FOR NONLINEAR NONPLANAR MOTION OF A CANTILEVER BEAM M. H. Yao1, W. Zhang2 1
College of Mechanical Engineering, Beijing University of Technology, Beijing 100022, P. R. China, E-mail:
[email protected] 2 College of Mechanical Engineering, Beijing University of Technology, Beijing 100022, P. R. China, E-mail:
[email protected] Abstract:
The many pulses homoclinic orbits with a Melnikov method and chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam are investigated in this paper for the first time. The cantilever beam studied here is subjected to a harmonic axial excitation and two transverse excitations at the free end. A generalized Melnikov method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of the cantilever beam. The analysis of global dynamics indicates that there exist the multi-pulse jumping orbits in the perturbed phase space of the averaged equation. Numerical simulations are given to verify the analytical predictions. It is also found from the results of numerical simulation in three-dimensional phase space that the multi-pulse orbits exist for the nonlinear nonplanar oscillations of the cantilever beam.
Key words:
Cantilever beam, nonlinear nonplanar oscillations, parametric and external excitations, many pulses orbits, genaralized Melnikov method, chaotic dynamics
1.
INTRODUCTION
The nonlinear nonplanar dynamics of the cantilever beams are the subjects of interest because of their importance in many applications to spacecraft stations, satellite antennas, machine tools and flexible manipulators. 267 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 267–276. © 2007 Springer.
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Therefore, research on the nonlinear nonplanar dynamics of the cantilever beams has received considerable attention in the past two decade. Crespo da Silva and Glynn [1] formulated a set of integral-partial differential governing equations of motion describing the nonlinear nonplanar oscillations of an inextensional cantilever beam. Zaretzky and Crespo da Silva [2] gave an experimental investigation for the nonlinear nonplanar motion of the cantilever beams excited by a periodic transverse base excitation. Nayfeh and Pai [3] used the Galerkin procedure and the method of multiple scales to investigate the nonlinear planar and nonplanar responses of the inextensional cantilever beams and found that the nonlinear geometric terms produce a hardening effect and dominate the nonplanar responses for all modes. Arafat et al. [4] studied the nonlinear nonplanar response of the cantilever inextensional metallic beams to a principal parametric excitation and found that there exist the bifurcations and chaotic motion. Recently, Yao and Zhang [5] utilized the energy-phase method to analyze the Shilnikov type multi-pulse heteroclinic orbits and chaotic dynamics in the nonlinear nonplanar oscillations of the cantilever beam. Some new phenomena on the global bifurcations and chaotic dynamics are discovered in high-dimensional nonlinear systems, such as the multi-pulse Shilnikov orbits. However, due to lack of analytical tools and methods to investigate the global bifurcations and chaotic dynamics for high-dimensional nonlinear systems, it is extremely challenging to develop the theories of the global bifurcations and chaotic dynamics for highdimensional nonlinear systems. Despite the challenge, certain progress has been achieved in this field in the past two decades. Wiggins [6] divided four-dimensional perturbed Hamiltonian systems into three types and utilized the Melnikov method to study the global bifurcations and chaotic dynamics for these three basic systems. Based on research given by Wiggins [6], Kovacic and Wiggins [7] developed a new global perturbation technique which may be used to detect the Shilnikov type single-pulse homoclinic and heteroclinic orbits in four-dimensional autonomous ordinary differential equations. Kaper and Kovacic [8] investigated the existence of several types of multi-bump homoclinic orbits to resonance bands for completely integral Hamiltonian systems subjected to small amplitude Hamiltonian and damped perturbations. Camassa et al. [9] extended the Melnikov method to investigate multi-pulse jumping of the homoclinic and heteroclinic orbits in a class of perturbed Hamiltonian systems. In the meantime, the energy-phase method was first presented by Haller and Wiggins [10] where single-pulse homoclinic orbits to a resonance in the Hamiltonian case were studied.
Many pulse homoclinic orbits of a cantilever beam
269
This paper focuses on the multi-pulse homoclinic orbits and chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam subjected to a harmonic axial excitation and two transverse excitations at the free end. The generalized Melnikov method presented by Camassa and Kovacic et al. [9] is employed to analyze the multi-pulse homoclinic orbits and chaotic dynamics for the nonlinear nonplanar oscillations of the cantilever beam. The analysis indicates that there exist the multi-pulse jumping orbits for the averaged equation.
2.
EQUATIONS OF MOTION AND PERTURBATION ANALYSIS
We consider a cantilever beam with length L , mass m per unit length and subjected to a harmonic axial excitation and two transverse excitations at the free end, as shown in Figure 1a. Assume that the beam considered here is the Euler-Bernoulli beam. A Cartesian coordinate system, Oxyz , is adopted which is located in the symmetric plane of the cantilever beam. The s denotes the curve coordinate along the elastic axis before deformation. The ξ , η and ς are the principal axes of the cross section for the cantilever beam at position s , as shown in Figure 1b. The symbols v( s , t ) and w( s , t ) denote the displacements of a point in the middle line of the cantilever beam in the y and z directions, respectively. For later convenience, the harmonic axial excitation may be expressed in the form 2 F1 cos Ω1t . The transverse excitations in the y and z directions are represented in the forms 2 F2 ( s ) cos Ω 2t and 2 F3 ( s ) cos Ω 2t , respectively. The non-dimensional nonlinear governing equations of nonplanar motion for the cantilever beam under combined parametric and forcing excitations are of the following form ′ s s v + cv + β y v iv + F1 cos(Ω1t )v′′ = (1 − β y ) ⎡ w′′∫ v′′w′′ds − w′′′∫ v′′w′ds ⎤ 1 0 ⎣⎢ ⎦⎥ ″ s s 2 1 − (1 − β y ) ⎡ w′′∫ ∫ v′′w′′dsds ⎤ − β y [v′(v′v′′ + w′w′′)′]′ 0 1 ⎣⎢ ⎦⎥ βγ , (1a) ′ s ′ ⎤ 1 ⎡ s d2 − ⎢ v′∫ (v′2 + w′2 )ds ds ⎥ − F1 cos(Ω1t ) ⎡⎣v′(v′2 + w′2 ) ⎤⎦ 2 ∫0 1 2⎣ dt ⎦ + F2 ( s ) cos(Ω 2t )
{
}
270
M. H. Yao, W. Zhang s s w + cw + wiv + F1 cos(Ω1t ) w′′ = −(1 − β y ) ⎡v′′∫ v′′w′′ds − v′′′∫ w′′v′ds ⎤ 0 ⎣⎢ 1 ⎦⎥ ″ s s 2 1 − (1 − β y ) ⎡v′′∫ ∫ v′′w′′dsds ⎤ − [ w′(v′v′′ + w′w′′)′]′ ⎣⎢ 0 1 ⎦⎥ β
γ
{
′
(1b)
′
}
2 s d s ′ ⎤ 1⎡ − ⎢ w′∫ (v′2 + w′2 )ds ds ⎥ − F1 cos(Ω1t ) ⎣⎡ w′(v′2 + w′2 ) ⎦⎤ 2 ∫0 1 2⎣ dt ⎦ + F3 ( s ) cos(Ω 2 t )
η
Y
Y
η
ξ
ς
F2(t)
v(s, t ) F1(t)
ς F3(t) Z
(a )
ξ
X
s + u (s, t )
w(s, t )
X
(b)
Figure 1．The model of a cantilever beam with length L , mass m per unit length and subjected to a harmonic axial excitation and transverse excitations at the free end, (a) the model; (b) a segment.
The boundary conditions are v(0, t ) = w(0, t ) = v′(0, t ) = w′(0, t ) = 0 ,
(2a)
v′′(1, t ) = w′′(1, t ) = v′′′(1, t ) = w′′′(1, t ) = 0
(2b)
We apply the Galerkin procedure and the method of multiple scales to Equation (1) to obtain the averaged equations. We investigate the case of the ratio β y = ω12 ≈ 1/ 4 . In addition, principal parametric resonance-1/2 subharmonic resonance for the first mode and fundamental parametric resonance-primary resonance for the second mode are considered. The resonant relations are represented as 1 (3) Ω1 = Ω 2 , ω12 = β y = Ω12 + εσ 1 , 1 = ω22 = Ω12 + εσ 2 , 4 where σ 1 and σ 2 are two detuning parameters. For convenience of the following analysis, let Ω1 = 1 . The averaged equations in the Cartesian form are obtained as follows
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271
1 1 x1 = − cx1 − (σ 1 + α1 F1 ) x2 + ( 2α 2 − 3α 3 ) x2 ( x12 + x22 ) − β1 x2 ( x32 + x42 ) , (4a) 2 16 1 1 x2 = (σ 1 − α1 F1 ) x1 − cx2 − ( 2α 2 − 3α 3 ) x1 ( x12 + x22 ) + β1 x1 ( x32 + x42 ) , (4b) 2 16 1 1 1 x3 = − cx3 − σ 2 x4 − β 2 x4 ( x12 + x22 ) + ( 2α 2 − 3α 3 ) x4 ( x32 + x42 ) , (4c) 2 2 8 1 1 1 1 2 2 x4 = − f 2 + σ 2 x3 − cx4 + β 2 x3 ( x1 + x2 ) − ( 2α 2 − 3α 3 ) x3 ( x32 + x42 ) . (4d) 2 2 2 8 In order to conveniently analyze the multi-pulse homoclinic orbits with the generalized Melnikov method and chaotic dynamics, we need to reduce averaged Equation (4) to a simpler normal form. It is seen that there are Z 2 ⊕ Z 2 and D4 symmetries in averaged Equation (4) without the parameters. Therefore, these symmetries are also held in normal form. Normal form with parameters can be written as u1 = u2 , (5a) u2 = − µ1u1 − µ2u2 + η1u13 + β1u1 I 2 ,
(5b)
I = − µ I − f 2 sin γ ,
(5c)
I γ = σ 2 I − η 2 I + β1 Iu − f 2 cos γ .
(5d)
3
2 1
The scale transformations may be introduced as follows
µ2 → εµ2 , µ → εµ , f 2 → ε f 2 .
(6)
Then, normal form (5) can be rewritten as the form with the perturbations ∂H + ε g u1 = u2 , (7a) u1 = ∂u2 ∂H (7b) u2 = − + ε g u2 = − µ1u1 + η1u13 + β1u1 I 2 − εµ 2u2 , ∂u1 ∂H (7c) I= + ε g I − ε f 2 sin γ = −εµ I − ε f 2 sin γ , ∂γ ∂H Iγ = − (7d) + ε g γ − ε f 2 cos γ = σ 2 I − η 2 I 3 + β1 Iu12 − ε f 2 cos γ , ∂I where the Hamiltonian function H is of the form 1 1 1 1 1 1 H (u1 , u2 , I , γ ) = u22 + µ1u12 − η1u14 − β1 I 2u12 − σ 2 I 2 + η 2 I 4 , (8) 2 2 4 2 2 4 u1 u2 I γ and g , g , g and g are the perturbation terms induced by the dissipative effects g u1 = 0 , g u2 = − µ2u2 , g I = − µ I − f 2 sin γ , g γ = − f 2 cos γ . (9)
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M. H. Yao, W. Zhang
UNPERTURBED DYNAMICS
When ε = 0 , it is noted that system (7) is an uncoupled two-degreeof-freedom nonlinear system. The I variable appears in (u1 , u 2 ) compone nts of system (7) as a parameter since I = 0 . Consider the first two decoupled equations u1 = u2 , u2 = − µ1u1 + η1u13 + β1 I 2u1 .
(10)
Since η1 < 0 , system (10) can exhibit the homoclinic bifurcations. It is known that the singular point q0 = (0 , 0) is the saddle point and the singular points q± ( I ) = ( B , 0) are center pints. There exists homoclinic loop Γ 0 which consists of one hyperbolic saddle point q0 and a pair of homoclinic orbits u± (T1 ) . In order to calculate the phase shift and the energy difference function, we need to obtain the equations of a pair of homoclinic orbits which are given as u1 (T1 ) = ±
2ε1
δ1
sec h
(
)
ε1 T1 , u2 (T1 ) = ∓
2ε1
δ1
th
(
)
ε1 T1 sech
(
)
ε1T1 . (11)
Let us turn our attention to the computation of the phase shift. Substituting the first equation of Equation (11) into the fourth equation of the unperturbed system of Equation (5) yields 2β ε (12) γ = σ 2 − η2 I 2 + 1 1 sec h 2 ( ε1 T1 ) .
δ1
Integrating Equation (12) yields
γ (T1 ) = ωr T1 +
2β1 ε1
δ1
tanh
(
)
ε1T1 + γ 0 ,
(13)
where ωr = σ 2 − η 2 I 2 . At I = I r , there is ωr ≡ 0 . Therefore, the phase shift may be expressed as
⎡ 4β ε ⎤ 4β ∆γ = ⎢ 1 1 ⎥ = − 1 β1 I r2 − µ 2 + σ 1 (1 − σ 1 ) . η1 ⎢⎣ δ1 ⎥⎦ I = I r
4.
(14)
DISSIPATIVE PERTURBATIONS
After obtaining detailed information on the nonlinear dynamic characteristics of (u1 , u2 ) components for the unperturbed system (7), the next step is to examine the effects of small perturbation terms (0 < ε 0 ∆ 2aα1 + b + ∆ ⎥⎦ ⎢⎣ 2aα1 + b ⎤ ⎡ 4(λ − λ2 ) F (α1 ) = exp ⎢ 1 arctan ⎥ ,if ∆ < 0 −∆ −∆ ⎦ ⎣ ⎡ −4(λ1 − λ2 ) ⎤ F (α1 ) = exp ⎢ ⎥ ,if ∆ = 0 ⎣ 2aα1 + b ⎦ 1 1 (2) ∆ = b 2 − 4ac, λ1 = F11 − b11(1) , λ2 = F22 − b22 , 2 2 (2) (2) (1) (1) (2 ) (1) (2) a = b11 + b22 − b11 − b22 , b = b11 + b22 − 2b11 , c = b11(2)
(26)
Numerical results for the largest Lyapunov exponent have been obtained ′ = 0.01, for system (24) with parameter values α11′ = 0.02, α12′ = 0.01, α 21 ′ α 22 = 0.01, β1 = 0.01, β 2 = 0.01, k11 = 1, k12 = 1, k21 = −1, k22 = 1, ω1′ = 1, ω2′ = 1.414, η1 = 0.03, η2 = 0.03, D11 = 0.02, D22 = 0.02 and given in Figure 3. It is seen that the proposed method yields quite accurate results even for large delay time and time delay in feedback control deteriorates the stability remarkably. λ 1(τ)
0.05
τ=1.5 τ=1 τ=0.5
0.04
τ=0
0.03 0.02 0.01 0 -0.01 -0.02 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
D 0.015
λ 1(τ)
τ=1.5
0.01
τ=1
0.005 0 -0.005 -0.01 -0.015
τ=0 τ=0.5
-0.02 -0.025 0
0.01
0.02
0.03
0.04
0.05
η
Figure 3. The largest Lyapunov exponent λ1 (τ ) of system (24). —Analytical result. ● ▲▼ Results form digital simulation.
W. Q. Zhu, Z. H. Liu
392
5.
CONCLUSIONS
In this paper the stochastic averaging method for quasi integrable Hamiltonian systems with time-delayed feedback control has been proposed. The largest Lyapunov exponent for such systems has been evaluated approximately by using the averaged Itô equations. The effects of timedelayed feedback control on the stability and response of the systems have been examined through three examples. It has been shown that the proposed method is quite accurate even for large delay time and the time delay in feedback control may cause phenomenological bifurcation, significantly degrade the performance of the controller and even destabilize the system.
ACKNOWLEDGEMENTS The work reported in this paper is supported by the National Natural Science Foundation of China under a key grant No. 10332030 and the Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20060335125.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
Malek-Zavarei M, Jamshidi M. Time-Delay System Analysis, Optimization and Application, New York, North-Holland, 1987. Hu HY, Wang ZH. Dynamics of Controlled Mechanical Systems with Delayed Feedback, Berlin, Springer-Verlag, 2002. Pu JP. “Time Delay Compensation in Active Control of Structure”, ASCE Journal of Engineering Mechanics, 124, pp. 1018-1028, 1998. Grigoriu M. “Control of Time Delay Linear Systems with Gaussian White Noise”, Probabilistic Engineering Mechanics, 12, pp. 89-96, 1997. Di Paola M, Pirrotta A. “Time Delay Induced effects on Control of Linear Systems under Random Excitation”, Probabilistic Engineering Mechanics, 16, pp. 43-51, 2001. Zhu WQ, Huang ZL, Yang YQ. “Stochastic Averaging Method of Quasi Integrable Hamiltonian Systems”, ASME, Journal of Applied Mechanics, 64, pp. 975-984, 1997. Zhu WQ, Huang ZL. “Lyapunov Exponent and Stochastic Stability of Quasi Integrable Hamiltonian Systems”, ASME, Journal of Applied Mechanics, 66, pp. 211-217, 1999. Zhu WQ, Deng ML. “Optimal Bounded Control for Minimizing the response of Quasi Integrable Hamiltonian Systems”, International Journal of Non-Linear Mechanics, 39, pp. 1535-1546, 2004.
AUTHOR INDEX
A Ananievski, I., 277 Anh, N.D., 147 Arrate, F., 11 Au, S.K., 45 Awrejcewicz, J., 197
B Bajaj, A.K., 1 Balachandran, B., 207 Barros, M.M., 127 Barthels, P., 287 Bevilacqua, L., 127 Bódai, T., 137
C Chatterjee, A., 353 Chen, L.Q., 217 Chen, Y.S., 157 Chernousko, F.L., 227 Czołczyński, K., 317
D Davies, P., 1 Dick, A.J., 207
F Fenwick, A.J., 137 Filippova, T.F., 55
G Gaull, A., 65 Ghanem, R., 11
Glocker, C., 187 Gonçalves, P.B., 297
H Hai, N.Q., 147 He, Q., 117 Hernandez-Garcia, M., 11 Hu, H.Y., 307, 333, 363 Huang, M.S., 373
I Inoue, T., 167 Insperger, T., 343 Ippili, R., 1 Ishida, Y., 167
M Masri, S.F., 11 Mote Jr., C.D., 207
O Orlando, D., 297
P Paolone, A., 237 Perlikowski, P., 317 Proppe, C., 23 Puri, T., 1
R Rega, G., 247
J Jin, Y.F., 333
K Kapitaniak, T., 317 Kreuzer, E., 65 Kudra, G., 197 Kunitho, Y., 167
L Lacarbonara, W., 237 Lenci, S., 247 Leng, X.L., 77 Li, S., 117 Liu, X.B., 87 Liu, Z. H., 383 Lu, Q.S., 323
393
S Schiehlen, W., 33, 147 Seifried, R., 33 Shen, Y.J., 177 Shi, X., 323 Stefański, A., 317 Steindl, A., 257 Stépán, G., 343, 353
T Teufel, A., 257 Thunnissen, D.P., 45 Troger, H., 257
V Vestroni, F., 237
Author Index
394 W Wahi, P., 353 Wang, H.X., 323 Wang, Q.Y., 323 Wang, Z.H., 363 Wasilewski, G., 197 Wauer, J., 287 Wedig, W.V., 97 Wetzel, C., 23 Wiercigroch, M., 137 Wu, Z.Q., 157
X Xu, J., 373 Xu, J.X., 109 Xu, W., 117
Y Yabuno, H., 167 Yang, S.P., 177 Yang, X.D., 217 Yao, M., 267
Yu, M.L., 307 Yunt, K., 187
Z Zhang, W., 267 Zhang, Y.Y., 373 Zhu, W.Q., 383 Zou, H.L., 109
IUTAM BOOK SERIES 1.
Eberhard, Peter (Ed.): IUTAM Symposium on Multiscale Problems in Multibody System Contacts ISBN 978-1-4020-5980-3
2.
Hu, H. Y.; Kreuzer, E. (Eds.): IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty Proceedings of the IUTAM Symposium held in Nanjing, China, September 18-22, 2006 ISBN 978-1-4020-6331-2
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