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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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 << λd . Then ℘=℘hFP is implicitly defined by
∫
A
℘hFP f ( x)d λ d ( x) = ∫ h−1 ( A) f ( x)d λ d ( x), ( A ∈Π ) ,
(7)
which is referred to as the Frobenius–Perron–Operator [12], [15]. Note that via differentiation of this definition equation one can derive the operator ℘hFP in its common explicit form ℘hFP f ( y ) = ∫ δ ( y − h( x)) f ( x)d λ d ( x) , ( y ∈ E ) , with δ being Dirac’s delta distribution.
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C) More general, assume that all h ∈ supp µ are nonsingular. Then ℘f ( x) = ∫
℘hFP f ( x)d µ (h), ( x ∈ E ) .
H k (E)
(8)
From now on, we will postulate the examined measures ρ ∈ M 1 possess Lebesgue densities such that we can concentrate on operator ℘ instead of ˆ . analyzing operator ℘
3.2 Approximation of the Markov Operators In order to afford the mathematical toolbox developed so far being applied to numerical analysis we have to replace the operator ℘ by a finite dimensional approximation. For this, let (TN ) be a family of N – dimensional subspaces ( N < ∞) of ∞ L . We want to determine certain projections π N : L1 → TN such that lim π N f − f 1 = 0, ( f ∈ L1 ) . On the assumption that such projections exists, we can establish a N – dimensional approximation ℘N : TN → TN of ℘ fulfilling the condition ℘N π N = π N ℘ . Based on particular compactness suppositions with respect to ℘ , one strives to prove that lim PN π N − P 1 = 0 . For conservative deterministic systems, this is done by KIFER [9] and JUNGE [7], see also DELLNITZ and JUNGE [3]. A detailed study of dissipative deterministic systems can be found in GUDER AND KREUZER [4]. Galerkin projection: We want to specify the projections π N complying with ℘N π N = π N ℘ . For this purpose, we assume a λ d – partition Γ N := { I1 ,..., I N } of E to be given. The elements I i of the “cell space” Γ N are called “cells”. For each TN , we choose functions (φi )iN=1 such that ∑φi ≡1 . It is adequate here to define φi := χ Ii (i = 1,..., N ) , with χ Ii being the characteristic function on I i . We further set TN := span{ (φi ) }, such that TN is the space of step functions on Γ N . The projection π N = L1 → TN is established optimally by the Galerkin projection in terms of the inner product, which is chosen to be integration ( f , g ) ≡ ∫ f ⋅ g d λ d ( f , g ∈ L2 ) . That is to say, for all i = 1,.., N and f ∈ L1 , we set (π N f ,φi ) := ( f ,φi ) . Because of our special choice of the functions (φi )iN=1 , we can give an explicit declaration of π N f by
π N f = ∑αi χ I , i
with α i := [1/ λ d ( I i )] ⋅ ∫
Ii
f d λ d , ( f ∈ L1 ) .
(9)
Since ℘N is a linear operator, we can assign a matrix representation M N = M N (℘N ) with respect to the standard basis of ℜ N to it. Then each entry of M N =: (mijN ) , given by
Cell Mapping applied to Random Dynamical Systems mijN = [1/ λ d ( I j )] ⋅ ∫
Ij
p ( x, I i ) d λ d ( x ) ,
73 (10)
describes the probability for a transition from cell I j to cell I i (i, j = 1,.., N ) .
3.3 Approximation of Stationary Densities We are assuming E to be invariant under S , and, accordingly, M N is column stochastic 1 . Because of that, M N has always at least one right eigenvector to eigenvalue 1. These fixed points of M N and ℘N , respectively, yield approximations of stationary densities of ℘ , which are densities of stationary measures. Thus, the quintessence is to determine the transition probabilities mijN , in particular the integral term appearing in the definition equation (10) from above. For the sake of notation, we introduce the mapping S1 := S (1, ⋅, ⋅) : Ω× E → E . Let x ∈ E be arbitrary, then we denote for all A ⊂ Ω× E by [ A]x the set {ω ∈Ω|(ω , x)∈ A}. Therewith, for any x ∈ E and any i ∈{1,..., N } we have p ( x, I i ) = P ([ S1−1 ( I i )]x ) . In general, the sets S1−1 ( I i ) ⊂ Ω× E cannot be determined analytically so that we must rely on numerical approximations. There are well–known algorithms to find box coverings of the image of mapping S1 provided it fulfills a certain Lipschitz condition, see [2] and [7]. To be more precise, let Θ M := {O1 ,..., OM } be a finite partition of Ω . Then for all boxes Bki = Ok × I i with Ok ∈Θ M , I i ∈Γ N and for all (ω1 , x1 ) , (ω2 , x2 ) ∈ Bki mapping S1 is assumed to meet with S1 (ω1 , x1 ) − S1 (ω2 , x2 ) ≤ Lki (ω1 , x1 ) − (ω2 , x2 ) . Thereby, the appearing norms and the (local) Lipschitz constant Lki must be chosen suitably. The algorithms mentioned above can be exploited to calculate for each box Bki the image S1 ( Bki ) . With this information we are able to determine a box covering of S1−1 ( I i ) for each i ∈{1,..., N } , see Figure 1. That is, we can find sets RiM ⊂ {1,..., N } and RiN ⊂ {1,..., N } such that S1−1 ( I i ) ⊂ U k∈R M ,l∈R N Bkl =: Qi . i
(11)
i
We define Qi , j := [Qi ]x for x ∈ I j . It is straightforward to approximate for any x ∈ E and i ∈{1,..., N } the transition probability p ( x, I i ) by P ([Qi ]x ) , according to equation p ( x, I i ) = P([ S1−1 ( I i )]x ) from above. If we further define m% ijM , N := P (Qi , j ) , then probabilities mijN can be evaluated approximately via 1
If E is not invariant, we expand the theory by introducing sets I 0 := ℜd \E , I M ,N := I 0 ∪ I N , and switching from E to ℜd which is surely invariant but not compact, in opposition to our assumptions. To overcome this obstacle we set miM0 ,N := δ i 0 (the Kronecker delta), whereas the transition probabilities m0Mi ,N (i ≠ 0) can be calculated in an analogous manner as described in the following.
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N
∑ m% j =1
M ,N ij
.
(12)
All in all, if we define M M , N := (mijM , N ) then eigenvectors of M M , N to eigenvalue 1 yield approximations of stationary densities. The matrix M M , N is again column stochastic and therefore has always at least one eigenvector f M , N like this. Naturally, we are interested in convergence of the sequence f M , N for lim max {P (Ok )λ d ( I i )} = 0
M , N →∞ k ≤M ,i≤ N
(13)
to a stationary density f that has support on a set D ∈Π with positive Lebesgue measure: λ d ( D) > 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
79
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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X. L. Leng
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<∆K<0.55. That is to say, the influence of random disturbance on the response of the rotor system is very weak when its intensity, σ, is a little one; and such influence turns significant along with the increment of σ. From Figure 2(c) and 2(d), we can find that the orbit diagrams corresponding to ∆K =0.35 turn to be a family of unclosed curves. And the profile of those unclosed curves is similar to the corresponding periodical solution curves of the undisturbed system. In this research, such solution is named as the random disturbed periodical solution. Furthermore, the difference between Figure 2(c) and 2(d) shows that the higher the noise level is, the more violent the so-called disturbed periodical solution fluctuates around the corresponding periodical
A cracked rotor system under white noise disturbance
81
solution of the undisturbed system. From the Poincare diagram of the system, Figure 2(g) and 2(h) corresponding to ∆K =0.5, we can see that some similar phenomena occur. In this case, the solution of the disturbed rotor system is named as random disturbed quasi-periodical solution. In other words, here the Poincare diagram is a series of discrete points distributed around a closed curve, which represents quasi-periodic solutions of the undisturbed system, because of the effect of random disturbance. And along with the increment ofσ, the distribution of those discrete points turns more dispersed. From the power spectrum diagram of the system, Figure 2(e), 2(f), 2(i) and 2(j), we can see that each rank of the power spectrum of the system response is reduced in different range. It is also the effect of the random terms in Equation (1). Additionally, different samples of the random disturbance γ (t ) , simulated from Equation (2), are substituted into Equation (1). And the bifurcation diagrams, orbit diagrams, power spectrum diagrams and the Poincare diagrams of the disturbed system calculated with those different samples are approximately invariable. It is indicated that the samples of the white noise process simulated from Equation (2) can satisfy the ergodic condition numerically.
(a) bifurcation diagram (σ=0.001)
(b) bifurcation diagram (σ=0.01) 0.2
0.4
0.4
0.6
0.6
0.8
0.8
X
X
1 1
1.2 1.2
1.4
1.4
1.6
1.8 -0.8
1.6
-0.6
-0.4
-0.2
0 Y
0.2
0.4
0.6
(c) orbit (σ=0.001, △K=0.35)
0.8
1.8 -0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Y
(d) orbit (σ=0.01, △K=0.35)
0.8
1
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30
30
25
25
20
20
AM
35
AM
35
15
15
10
10
5
5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0
5
0
0.5
1
1.5
2
2.5
f/ω
(e) power spectrum (σ=0.001, △K=0.35)
4
4.5
5
0.5 0.4
0.4
0.3
0.3
0.2
0.2
0.1
X'
0.1
X'
3.5
(f) power spectrum (σ=0.01, △K=0.35)
0.5
0
0 -0.1
-0.1
-0.2
-0.2
-0.3
-0.3
-0.4
-0.4 -0.5
3
f/ω
-0.5 1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1
1.1
1.2
1.3
1.4
(g) Poincare diagram (σ=0.001, △K=0.5)
1.5
1.6
1.7
1.8
X
X
(h) Poincare diagram (σ=0.01, △K=0.5) 25
20
20
15
15
AM
AM
25
10
10
5
5
0 0
1
2
3
4
5
f/ω
(i) power spectrum (σ=0.001, △K=0.5)
0 0
1
2
3
4
5
f/ω
(j) power spectrum (σ=0.01, △K=0.5)
Figure 2. The influence of ∆K on system response (Ω=0.6, U=0.1, β=0.0)
Figure 3 shows the influence of Ω on the response of the rotor system subject to a white noise process, whenβ＝0，U＝0.1 and ∆K=0.62. Here, from the diagrams of the top Lyapunov exponent against the bifurcation parameter Ω , Figure 3(d), we can see that the interval space where the top Lyapunov exponent is little than zero turns narrow along with the increment
A cracked rotor system under white noise disturbance
83
of the noise level. And the drift of the bifurcation point can be identified clearly as the noise level being a large one.
(a) bifurcation diagram (σ=0.0)
( b) bifurcation diagram (σ=0.001)
σ=0.0 σ=0.001 σ=0.01
0.08
top Lyapunov exponent
0.06
0.04
0.02
0
-0.02
-0.04 0.525
(c) bifurcation diagram (σ=0.01)
(e)Poincare diagram (σ=0.001, Ω=0.531)
0.53 Ω
0.535
(d) top Lyapunov exponent against Ω
(f) Poincare diagram (σ=0.01, Ω=0.531)
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(g) power spectrum (σ=0.001, Ω=0.531)
(h) power spectrum (σ=0.001, Ω=0.531)
Figure 3. The influence of Ω on system response (U=0.1, β=0, ∆k=0.62)
4.
SURMARY AND CONCLUSION
In this paper, the Monte-Carlo method is applied 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. Our numerical simulation shows that random disturbance has a significant effect on the system response when the undisturbed response is a quasi-periodic one or a chaos one, and such effect is small when the undisturbed response is a periodical one. Along with the increment of the noise level, the effect mentioned above turns obvious. Additionally, disturbed responses calculated with different samples, which are simulated from Equation (2), have similar bifurcation diagrams, orbit diagrams, Poincare diagrams and power spectrum diagrams. We expect that the proposed method may become utilizable in the future fault diagnosing of rotating machinery．
ACKNOWLEDGEMENTS The supports of the National Natural Science Foundations of China (10325209, 50335030) and China Ph.D. Discipline Special Foundation are greatly acknowledged.
A cracked rotor system under white noise disturbance
85
APPENDIX A1. Nomenclature De k0 ∆K m U X,Y
β ξ,η T
ω ωc ωcr ωr Ωc Ω ϕ0 γ(t) σ
externaldampingratio stiffnessofuncrackedshaft stiffnesschangeratio massofrotor dimensionlessunbalanceparameter dimensionlessdeflection theanglebetweencrackandunbalance bodyfixedrotatingcoordinates,ξisthecrackdirection dimensionlesstime(=ωt) rotatingspeed pin-pincriticalspeedofuncrackedrotor pin-pincriticalspeedofcrackedrotor whirlspeed(nonsynchronousrotatingspeedrelative tosynchronousone) criticalspeedration(ωcr /ωc) rotatingspeedration(ω/ωc) initialphaseofunbalance astandardwhitenoiseprocess intensityofrandomdisturbance
A2. Some diagrams of the undisturbed case Some diagrams of the undisturbed case were added here, most of them coming from Zheng and Meng [8, 9].
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Appendix. Some diagrams of the undisturbed cases (1) Bifurcation diagram corresponding to Figure 2(a) and (b); (2) Diagram corresponding to Figure 2(c) and (d); (3) Poincare diagram corresponding to Figure 2(g) and (h)
REFERENCES 1. 2. 3. 4. 5.
6. 7. 8. 9.
Gasch R. “A survey of the dynamic behavior of a simple rotating shaft with a transverse crack”, Journal of Sound and Vibration, 162, pp. 313-332, 1993. Gao J, Zhu X. “Observation on the dynamic behaviour of cracked rotor”, Chinese Journal of Northwestern Polytechnical University, 10, pp. 434-439, 1992. Muszynska A. “Shaft crack detection”, Proc. 7th Machinery Dynamics Seminar, Edmonton, Canada, 1982. Meng G, Gasch R. “The nonlinear influences of whirl speed on the stability and response of a cracked rotor”, J. Mach. Vib., 4, pp. 216-230, 1992. Zhao Y, Lin JH. “Seismic analysis of rotor system under stationary non-stationary random earthquake excitations”, Chinese J. of Computational Mechanics, 19, pp. 7-11, 2002. Shinozuka M. “Simulation of Multivariate and Multidimensional Random Processes”, J. of Sound and Vibration, 19, pp. 357-367, 1971. Shinozuka M. “Digital Simulation of Random Processes and its Applications”, J. of Sound and Vibration, 25, pp. 111-128, 1972. Zheng JB, Meng G. “The nonlinear influence of whirl speed on bifurcation and chaos of a cracked rotor”, Chinese Journal of Vibration Engineering, 10, pp. 190-197, 1996. Zheng JB, Meng G. “Bifurcation and chaos response of a nonlinear cracked rotor”, International Journal of Bifurcation and Chaos, 8, pp. 597-607, 1998.
THE MAXIMAL LYAPUNOV EXPONENT FOR A STOCHASTIC SYSTEM X. B. Liu Institute of Vibration Engineering Research, College of Aerospace Engineering, Nangjing University of Aeronautics and Astronautics, 29 Yudao Street, Nangjing 210014, P.R. China
Abstract:
In this paper, the almost sure stability condition for a co-dimension twobifurcation system, that is on a three-dimensional central manifold and is driven parametrically by a real noise, is investigated. A model of enhanced generality is developed by assuming the real noise as an integrable function of an output of a linear filter system. The strong mixing condition, which is the essential theoretic basis for the stochastic averaging method, is removed in the present study. To solve the complicated problem encountered, the perturbation method introduced by L.Arnold etc. [1, 2] and the spectral analysis of an ndimensional linear filter system are employed in the construction of the asymptotic expansions of the invariant measures and the maximal Lyapunov exponents for the relevant system.
Key words:
Real noise, invariant measure, maximal Lyapunov exponent, asymptotic analysis.
1.
INTRODUCTION
The investigation on maximal Lyapunov exponents for dynamical systems that are excited by stochastic processes is the primary focus of the research interests in the field of random dynamical systems as well as stochastic bifurcation. This is mainly attributed to the fact that the sample or the almost-sure stability of the stationary solution of a random dynamical problem depends on the sign of the maximal Lyapunov exponent. The aim of this paper is to obtain the asymptotic expansion of the maximal Lyapunov exponent for a co-dimension two bifurcation system that is excited parametrically by a real noise, which, in the present paper, is 87 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 87–96. © 2007 Springer.
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assumed to be an integrable function of an n-dimensional OrnateinUhlenbeck process. It is well known that the asymptotic expression of the top Lyapunov exponent depends on the form of matrix B, which is included in the noise excitation term. Therefore in this paper, a general form of matrix B is employed and furthermore, for three special cases of the matrices B, for which the complexities of the singular points of an one-dimensional phase diffusion process arise, we will investigate the phenomena caused and discuss the findings thoroughly.
2.
THE SPECTRAL ANALYSIS FOR A LINEAR FILTER SYSTEM
In this section, we mainly recall the existing results for the spectral analysis of a n-dimensional linear filter system. As in Roy [3], consider a general linear filter system, which is governed by the following stochastic differential system
& (t ) u& (t ) = Au(t ) + W
(1)
& (t ) is an n-dimensional where A=(aij) n×n; aij are real or complex numbers. W & & & Gaussian white noise with E(W (t ))=0, E( W (t + τ ) W (t )) = Vδ (τ ) . V=(vij)n×n is a symmetric, non-negative defined constant matrix, and u=(u1, u2,…, un)T is a Ornatein-Uhlenbeck vector process. The matrix A is assumed to have a complete set of eigenvalues α1,…,αn along with the corresponding eigenvectors e1 ,…,en, which means that αi ≠αj if i≠j. Furthermore, the conditions given by Liberzon and Brockett [4] are assumed to be satisfied. For the diffusion process u(t), the differential generator Lu* and it’s adjoint, the Fokker Planck operator, Lu are respectively given by L∗u = aij u j
vij ∂ 2 ∂ vij ∂ 2 ∂ , Lu = − [aij u j ] + + 2 ∂ui ∂u j ∂ui 2 ∂ui ∂u j ∂ui
(2)
where the repeated indices indicate usual summation. Respectively, the Eigenvalue problems for the two operators are L∗uψ λ∗′ (u) = λ ′ψ λ∗′ (u), Luψ λ (u) = λψ λ (u)
(3)
Since the spectrum of Lu and Lu* are discrete and then both of them possess a same set of eigenvalues, which is expressed as λm = m1α1 + L + mnα n , where m=(m1,m2,…mn); m=m1+m2+…+mn, each mi is an non-negative integer. For a scalar stochastic function f(u), which is integrable function of u in 2 the sense that ∫ n [ f (u)] ψ 0 (u)du < +∞ and R
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89
E [ f (u) ] = ∫ n f (u)ψ 0 (u)du = 0
(4)
R
the spectral density functions for f(u) are obtained as S f (ω ) = −
2λm II f Ψ
∞
∑
m1 = 0,L, mn = 0
λ +ω 2 m
, Φ f (ω ) = − 2
2ω II f Ψ
∞
∑
m1 = 0,L, mn = 0
(5)
λm2 + ω 2
Where ψ m∗ (u), ψ m (u) are respectively the eigenfunctions of the operators of L∗u , Lu , and II f Ψ = f (u), ψ% m (u)
3.
E
f (u), ψ m∗ (u)
, ψ% m (u) = [ψ 0 (u) ] ψ m (u) −1
E
(6)
FORMULATION
Consider a typical deterministic co-dimension two bifurcation system which is on a three-dimensional central manifold and possesses one zeroeigenvalue and a pair of pure imaginary eigenvalues, i.e. 4
r& = µ1r + a1rz + (a2 r 3 + a3 r 2 z ) + O( r , z ) 4
z& = µ2 z + c1r 2 − z 2 + (c2 r 2 z + c3 z 3 ) + O( r , z ) & = ω + O( r , z 2 ) Θ
(7)
where µ1 and µ2 are the unfolding parameters, and a1, a2, a3, c1, c2, c3 and ω are real constants. In the vicinity of equilibrium point (r,z,Θ)=(0,0,ωt), via the transformation of r = ⎣⎡ x12 + x22 ⎦⎤
(1 2)
, Θ = arctan [ x2 x1 ] , z = x3 the model
of the linearization of the original system (7), which is subjected to a stochastic parametric perturbation, is obtained as x& = A 0 x − ε 2 A1x + ε f (u)Bx
(8)
Where ⎡ 0 A 0 = ⎢ −ω ⎢⎣ 0
ω 0 0
0⎤ ⎡δ 1 0 ⎥ , A1 = ⎢ 0 ⎢0 0 ⎥⎦ ⎣
0
δ1 0
0⎤ ⎡ b11 0 ⎥ , B = ⎢b21 ⎢b δ 2 ⎥⎦ ⎣ 31
b12 b22 b32
b13 ⎤ b23 ⎥ b33 ⎥⎦
(9)
µ1 and µ2 have been rescaled such that µ1 = −ε 2δ1 , µ2 = −ε 2δ 2 . f(u) is a scalar stochastic function of u(t), which has been defined in Equation (1). The following spherical polar transformation from (x1, x2, x3) to (ρ, θ, φ)
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X. B. Liu
x1 = R cosθ sin φ , x2 = R cosθ cos φ , x3 = R sin θ ρ = ln R, φ (t ) = ωt + ϕ (t ); θ ∈ [− π 2, π 2], φ , ϕ ∈ [0, 2π ]
(10)
yields a set of equations of the arguments of ρ, θ, φ and the noise process u, i.e. & (t ) ρ& = ρε , θ& = θε , φ& = φε , u& (t ) = Au(t ) + W
(11)
where
ρε = ε 2 ρ 2 + ε f (u) ρ1 , θε = ε 2θ 2 + ε f (u)θ1 , φε = ω + ε f (u)φ1 ρ 2 = −δ1 cos 2 θ − δ 2 sin 2 θ
1 2 1 θ1 = ( f z 2 − f r1 )sin 2θ + ( f z1 cos 2 θ − f r 2 sin 2 θ ) 2 1 θ 2 = (δ1 − δ 2 )sin 2θ , φ1 = fφ1 + tan θ fφ 2 2 1 f r1 = [ k1 + k2 cos 2φ + k3 sin 2φ ], f r 2 = b13 sin φ + b23 cos φ 2 1 fφ 1 = [k4 + k3 cos 2φ − k2 sin 2φ ], fφ 2 = b13 cos φ − b23 sin φ 2 f z1 = b31 sin φ + b32 cos φ , f z 2 = b33 k1 = b22 + b11 , k2 = b22 − b11 , k3 = b12 + b21 , k4 = b12 − b21
ρ1 = ( f r 2 + f z1 )sin 2θ + f r1 cos 2 θ + f z 2 sin 2 θ
(12)
The phase processes θ and φ together with the diffusion process, u(t), which is defined in Equation (1), form a dimension (n+2) vector diffusion process (θ(t),φ(t),u(t)) on [-π/2,π/2]×[0,2π]×Rn with the following generator and adjoint operator respectively L∗ε = L∗0 + ε L∗1 + ε 2 L∗2 ∂ ∗ ∂ ∂ ∂ L∗0 = L∗u + ω , L∗1 = f (u)φ1 , L2 = θ 2 + f (u)θ1 ∂θ ∂θ ∂φ ∂φ Lε = L0 + ε L1 + ε 2 L2 L0 = −ω
4.
∂ ∂ ∂ ∂ + Lu , L1 = − f (u) θ1 − f (u) φ1 , L2 = − θ 2 ∂φ ∂θ ∂φ ∂θ
(13)
(14)
ASYMPTOTICAL ANALYSIS
Corresponding to the Fokker Planck operator Lε , the invariant probability density function pε(θ, φ, u) satisfies the FPK equation, i.e.
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Lε pε = ( L0 + ε L1 + ε 2 L2 ) pε (θ ,φ , u) = 0
(15)
In the present paper, u(t) is assumed to be an ergodic Markov process on Rn, and then according to the multiplicative ergodic theorem of Oseledec, the maximal Lyapunov exponent for system (11) is 2π
π
λε = ρε , pε = ∫ dφ ∫ 2π dθ ∫ du [ ρε (θ ,φ , u) pε (θ ,φ , u)] −
0
Rn
2
(16)
In this work, the assumption ε<<1 holds and a formal expansion of pε (θ ,φ , u) = p0 (θ ,φ , u) + ε p1 (θ ,φ , u) + L + ε N pN (θ ,φ , u) + L
(17)
is sought such that the coefficients of p0 , p1 , p2 , L satisfy a set of the following recurrence equations, i.e. L0 p0 = 0, L0 p1 = − L1 p0 , L0 p2 = − L1 p1 − L2 p0 , L
(18)
and hence the maximal Lyapunov exponent for system (11) possess an asymptotic expansion as follows
ρε , pε = ρ 0 , p0 + ε ⎡⎣ ρ1 , p0 + ρ 0 , p1 ⎤⎦ +ε 2 ⎡⎣ ρ 2 , p0 + ρ1 , p1 + ρ 0 , p2 ⎤⎦ + L
(19)
In Equation (17), all the functions pε(θ,φ,u), p0(θ,φ,u), … are required to be 2π-periodic in variable φ, i.e. pε (θ ,φ , u) = pε (θ ,φ + 2π , u) p0 (θ ,φ , u) = p0 (θ ,φ + 2π , u) p1 (θ ,φ , u) = p1 (θ ,φ + 2π , u),L
(20)
The normalization condition of the probabilistic density function pε(θ,φ,u) then yields
∫
2π
∫
2π
0
0
π
dφ ∫ π2 dθ ∫ n dup0 = 1 -
2
R
π
2π
(21)
π
dφ ∫ π dθ ∫ n dup1 = ∫ dφ ∫ π dθ ∫ n dup2 = 0,L 2
-
2
R
0
2
-
2
R
Besides the conditions given in Equation (20) and Equation (21), each equation with the form L0p=q must satisfy the solvability condition, i.e.
∫
2π
0
dφ ∫ n duq (θ ,φ , u) = 0 R
(22)
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5.
EXPANSION OF INVARIANT MEASURE
To obtain the perturbation solution with the form in Equation (18) to the FPK equation (15), a study on the recurrence equations (18) will be conducted in the subsequent context. We first consider the equation of order ε0, since the set of the eigenfunctions ψ m (u) corresponding to Lu forms a complete set, then for the first equation in Equation (18), the solution p0(θ,φ,u) is sought in the form of
p 0 (θ , φ , u) =
∞
∑p
(0) m m1 = 0 ,L, mn = 0
(θ , φ )ψ m (u) (23)
Which leads to the fact that each of the coefficients pm(0) (θ ,φ ) is respectively the solution to [−ω ∂ ∂φ + λm ] pm(0) (θ ,φ ) = 0
(24)
By solving Equation (24), we know that there exists only one non-zero periodic solution, p0(0) (θ ,φ ) = p0(0) (θ ) , which corresponding to the eigenvalue λ0=0. Furthermore, via the normalization condition (21), we finally obtain p0 (θ ,φ , u) = F (θ )ψ 0 (u) 2π
(25)
where F(θ ) is a function of θ yet to be determined by the solvability condition of the third equation in Equation (18). We then consider the second equation in Equation (18), a calculation leads to it’s solution p1 (θ , φ , u) , which takes the expression as p1 (θ ,φ , u) =
∞ 1 (1) p0 (θ )ψ 0 (u) + ∑ 2π m1 = 0, m2 = 0,K, mn = 0
pm(1) (θ ,φ )ψ m (u)
(26)
m ≠1
where (1) m=0 ⎪⎧ p0 (θ ) 2π , (27) pm(1) (θ ,φ ) = ⎨ (0) (2) (11) (12) ⎪⎩ I f Ψ {Π m Λ 0 + Π m Λ 2 +Π m Λ11 + Π m Λ12 } 2π , m ≠ 0
and I f Ψ = f (u), ψ m∗ (u)
E
, Π m(0) = ( 2b33 − k1 ) 4α m
Π m(11) = {[b32α m + b31ω ] cos φ + [b31α m − b32ω ] sin φ } (ω 2 + α m2 )
Π m(12) = − {[b23α m + b13ω ] cos φ + [b13α m − b23ω ] sin φ } (ω 2 + α m2 )
Π m(2) = − {[ k2α m + 2k3ω ] cos 2φ + [ k3α m − 2k2ω ] sin 2φ } 4ω 2 + α m2
(28)
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where p0(1) (θ ) is an unknown function which contributes nothing to the expression of the top Lyapunov exponent. In addition, each pm(1) (θ ,φ ) contains the function F(θ ), which should be determined by the solvability condition of the third equation in Equation (18). To determine F(θ ) in Equations (25) and (28), the solvability condition of the third equation in Equation (18) is investigated, which leads to a FPK equation as 1 d2 d π π [σ 2 (θ ) F (θ )] − [ µ (θ ) F (θ )] = 0, θ ∈ [− , ] 2 2 dθ dθ 2 2
(29)
wherein the relevant diffusion coefficient and drift coefficient are respectively given as
σ 2 (θ ) = [ 4 β1 + β 2 2] sin 2 2θ + 2κ1 cos 2 2θ − 4κ 2 cos 2θ + 2κ 3 µ (θ ) = {[ 2β1 + β 2 4] − κ1} sin 4θ + {8 (δ1 − δ 2 ) + 2κ 4 − β 2 − 6κ 0 } sin 2θ − 4κ 5 tan θ
(30)
Then the solution to Equation (30) is F (θ ) =
1 1 c1 C − ⎡⎣(1 − F1 (θ ) ) (1 + F1 (θ ) ) ⎤⎦ 4 [ F2 (θ ) ] 4 2cos(θ )
(31)
where F1 (θ ) =
(κ1 − β ) cos 2θ − κ 2
β 2 − β (κ1 − κ 3 ) − 4κ 62 F2 (θ ) = (κ1 − β ) cos 2 2θ − 2κ 2 cos 2θ + β + κ 3 −8∆ + ( β + β 2 ) + (6κ 0 − κ1 + κ 3 − 4κ 5 )
(32)
c1 =
β 2 − β (κ1 − κ 3 ) − 4κ 62 β = 2β1 + β 2 4, κ 6 = [b13b32 − b23b31 ] S f (ω ), ∆ = δ1 − δ 2
C is a constant that is determined by the normalization condition π
∫ π F (θ )dθ = 1 . 2
−
2
Since the expressions of the maximal Lyapunov exponents depend on the forms of the matrix B, therefore in this paper, three special cases for B are investigated, i.e. Case I: b13=b23=0, which leads to κ 0 = κ 5 = κ 6 = 0, κ1 = κ 2 = κ 3 = −κ 4 Case II: b31=b13, b23=b32 and in addition, A1=A2=A.
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CaseIII: b31=b13, b23=b32, b21=-b12, b33=b22=b11 which leads to β1 = β 2 = 0 . For each case, we will respectively investigate the solution to Equation (29). For the first case, respectively the associated diffusion and drift coefficients are
σ 2 (θ ) = 8 ( β sin 2 θ + κ1 cos 2 θ ) cos 2 θ µ (θ ) = ( β − κ1 ) sin 4θ + [8∆ − 2κ1 − β 2 ] sin 2θ
(33)
Then via a direct integration, F(θ) is 1 ⎡ 2 β −8 ∆+ β 2 ⎤ 1 ⎡ −2 β −8 ∆+ β 2 ⎤ C ⎢− ⎥ 2 2 ⎥ β ⎦ ⎦ ⎡ β − β cos θ + κ cos θ ⎤ 4 ⎣ (34) F (θ ) = [ cos θ ] 2 ⎢⎣ β 1 ⎣ ⎦ 8 where C should be determined by the normalization condition. For the second case which is under the condition that σ2(θ)=A, B1=0 and 2B3=-A, we can obtain E (θ ) = ( 2 B2 A ) sin 2 θ + log cosθ , θ ∈ [− π 2, π 2] s (θ ) = secθ exp ⎡⎣ −α sin 2 θ ⎤⎦ , m(θ ) = (1 A ) cos θ exp ⎡⎣α sin 2 θ ⎤⎦ α = 1 − β 2 4κ 5 + 2∆ κ 5
(35)
To system (29) restricted on (0, π/2), the solution is ⎧ 2 α cos θ exp ⎡⎣α sin 2 θ ⎤⎦ , α > 0, ⎪ π ⎪ π Erfi[ α ] F (θ ) = ⎨ θ ∈ [0, ] 2 −α 2 ⎪ cos θ exp ⎡⎣α sin 2 θ ⎤⎦ , α ≤ 0, ⎪⎩ π Erf[ −α ]
(36)
On [-π/2, 0], the solution is of the same expression. For the third case, the assumption leads to
σ 2 (θ ) = 8κ 5 cos 2 2θ µ (θ ) = −4κ 5 sin 4θ + {8∆ + 4κ 5 } sin 2θ − 4κ 5 tan θ
(37)
and then the invariant measure is ⎧Cδ (θ + π 4 ) , θ ∈ [ − π 2, − π 4] 3 ⎪⎪ C F (θ ) = ⎨ [sec 2θ ]2 cosθ exp [ ∆ sec 2θ κ 5 ] , θ ∈ [ − π 4,π 4] (38) ⎪ 8κ 5 θ ∈ [π 4, π 2] ⎪⎩Cδ (θ − π 4 ) ,
MLE for a stochastic system
6.
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ASYMPTOTIC EXPANSION FOR TOP LYAPUNOV EXPONENT
Under the assumption that the Fokker Planck operator Lε defined by Equation (14) is an ergodic operator, therefore, on the domain [0,2π]× [-π/2,π/2]×Rn, the maximal Lyapunov exponent for the stochastic bifurcation system (11) is given as 2π
π
λε = ρε , pε = ∫ dφ ∫ 2π dθ ∫ du [ ρε pε ] 0
−
2
Rn
(39)
whose asymptotic expansion is
λε = ε 2 ⎡⎣ ρ 2 , p0 + f (u) ρ1 , p1 ⎤⎦ + ο (ε 2 )
(40)
For case one, the asymptotic expansion of the maximal Lyapunov exponent can be expressed as
β1 3 3 β2 β2 − − 16 8 2 + γ 1 γ 2 (2 + γ 1 ) ⎩ 3 γ 1 ⎡3 3 1 5 1 ⎤ + κ 5 [γ 2 ] 2 1 F ⎢ , + γ 1 ; + γ 1 ;1 − γ 2 ⎥ 4 6 + γ1 ⎣ 2 2 4 2 4 ⎦ γ 1γ 2 − 4γ 2 + 6 ⎡ 3 3 1 5 1 1 ⎤ F ⎢ , + γ 1 ; + γ 1 ;1 − γ 2 ⎥ − β2 γ 2 16 (6 + γ 1 )(2 + γ 1 ) ⎣ 2 2 4 2 4 ⎦ (γ 1γ 2 + 2) ⎡5 3 1 5 1 ⎤ F , + γ 1 ; + γ 1 ;1 − γ 2 ⎥ +3β1 γ 2 (6 + γ 1 )(2 + γ 1 ) ⎢⎣ 2 2 4 2 4 ⎦ ⎫ 2 γ2∆ ⎡3 1 1 3 1 ⎤⎪ F ⎢ , + γ 1 ; + γ 1 ;1 − γ 2 ⎥ ⎬ + ο (ε 2 ) − (2 + γ 1 ) ⎣ 2 2 4 2 4 ⎦ ⎪⎭ ⎧
λε = ε 2 ⎨−δ1 +
(41)
For the second case, the result is ⎧ 2⎡ κ β κ i exp[α ] α ⎤ 2 ⎪ε ⎢ −δ1 + 5 + 2 + 5 ⎥ + ο (ε ), α > 0 4 8 4 π Erfi[ α ] ⎦ ⎪ λε = ⎨ ⎣ ⎪ε 2 ⎡ −δ + κ 5 + β 2 + κ 5 exp[α ] −α ⎤ + ο (ε 2 ), α ≤ 0 ⎥ ⎪ ⎢ 1 4 8 4 π Erf[ −α ] ⎦ ⎩ ⎣
(42)
For the third case, we obtain
λε = ε 2 {−δ1 + (κ 5 4) + λ2(1) + λ2(2) } + ο (ε 2 )
λ2(1) = (1 2) {[ (δ 2 − δ1 ) − (κ 5 2) ] R1 + κ 5 R2 } λ2(2) = C {(δ 2 − δ1 )( R1 − 1)2κ 5 + (κ 5 2)( R1 − 2 R2 )}
(43)
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where R1 = π κ eκ ⎡⎣1 − erf ( κ ) ⎤⎦ 1 π κ eκ [1 − 2κ ] ⎡⎣1 − erf ( κ ) ⎤⎦ R2 = 2
7.
(44)
CONCLUSIONS
In this paper, the asymptotic expansion of the maximal Lyapunov exponent for a co-dimension two bifurcation system driven by a smallintensity real noise process is investigated. To consider a rather general model, the real noise is assumed to be an integrable function of the output of a linear filter system. The methods employed in the present study involve: (1) the asymptotic analysis introduced by Arnold et al. [1], (2) the spectrum analysis for Fokker Planck operator. In addition, in order to find the influences of the matrix B on the expressions of the maximal Lyapunov exponents, three special cases of the matrix B, for two of which the singularities of the diffusion coefficient processes arise, are considered and the relevant maximal Lyapunov exponents are evaluated.
ACKNOWLEDGEMENTS The support of the project offered by the National Natural Science Foundation of China under Key Grant No. 10672074 is gratefully acknowledged.
REFERENCES 1.
2. 3. 4.
Arnold L, Papanicolaou G, Wihstutz V. “Asymptotic analysis of the Lyapunov exponents and rotation numbers of the random oscillator and applications”, SIAM J. Appl. Math, 46, pp. 427-450, 1986. Arnold L. Random Dynamical Systems, Berlin, Springer, 1998. ROY RV. “Stochastic averaging of oscillators excited by colored Gaussian processes”, Int. J. Non-linear Mech, 29, pp. 461-475, 1994. Liberzon D, Brockett RW. “Spectral analysis of Fokker-Planck and related operators arising from linear stochastic differential equations”, SIAM J. Control Optim., 38, pp. 1453-1467, 2000.
STABILITY AND DENSITY ANALYSIS OF STOCHASTIC DUFFING OSCILLATORS W. V. Wedig Universität Karlsruhe, Institut für Technische Mechanik, D-76128 Karlsruhe, Germany, Kaiserstr. 12, E-mail:
[email protected]
Abstract:
The paper applies a generalized Hermite analysis with extended Fourier expansions to stochastic Duffing oscillators under additive white noise to investigate the stability of their stationary solutions by means of Lyapunov exponents. This analysis confirms that the top Lyapunov exponents remain negative for positively decreasing damping and mono-potential restoring. Known from purely numerical results, the top Lyapunov exponent becomes positive for a bi-potential restoring when the damping decreases with positive values. This was the first example of a dynamic system where additive white noise leads to instabilities.
Key words:
Lyapunov exponents, non-stationary densities, generalized Hermite analysis.
1.
DIMENSIONLESS SET-UP OF THE PROBLEM
A classical dynamic system of nonlinear mechanics is given by the Duffing oscillator under harmonic excitations [1] and [2]. In the stochastic case, the excitation is replaced by white noise and the equation of motion is
In this form, it possesses a linear damping term, the natural frequency, a cubic nonlinearity coefficient and an intensity parameter of delta-correlated white noise. The four parameters of the oscillator equation are reduced by
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In addition, the dimensionless time and noise, noted in (3), are introduced. The insertion of (2) and (3) into the second order oscillator equation (1) leads to the following two first order incremental state equations:
The stationary solutions of the Itô equations (4) and (5) are independent described by
Herein, the constant C stands for normalization. Note, that both densities are independent on the damping coefficient. The following analysis investigates the stability of the density distribution and higher order moments of the stationary displacement process. Both elements are shown in Figure 1 and 2.
Figure 1. Density distributions
Figure 2. Higher order moments
Following [3] and [4], a generalized Hermite analysis is applied to the Itô equations (4) and (5). The associated orthogonal polynomials are
They are defined by the orthogonal relation (8). In case of normal densities, The equation (11) follows from the orthogonal relation (8). For a unit nonthe equation (8) leads to Hermite polynomials satisfying the relations
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For non-normal densities with cubic restoring terms, as noted in (6), the polynomial coefficients in (7) are calculable by the equation system
The equation (11) follows from the orthogonal relation (8). For a unit non-linearity parameter e.g., the evaluation of (11) leads to the coefficients
The normalization constants of these polynomial coefficients are
Note that both matrices are of infinite dimension. The elements of the first six rows and columns only are calculated and explicitly given, above.
2.
NONSTATIONARY POLYNOMIAL MOMENTS
Applying Itô’s calculus to the product of both orthogonal polynomials (7) and (9) the associated incremental equations are derived to
Herein, differentiation with respect to the velocity and multiplication by the same coordinate are explained by the relation (10) and (9), respectively. The corresponding relations for the generalized polynomial (7) are
They follow from the orthogonality of the polynomial (7). For the special case of a unit cubic parameter value, the coefficient matrix (14) is calculated
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on the next page. Note, that the first elements under the main diagonal of the differentiation matrix, above, coincides with the differential rule of Hermite polynomials. All remaining elements are vanishing passing to the limit case of normal densities. Similar results and forms are derivable for the multiplication matrix (15). It possesses four diagonal lines with non-vanishing elements. The first lower diagonal elements coincide with the multiplication rule (9) of Hermite polynomials.
The second lower und upper diagonal elements are vanishing in the special case of normal densities, and the first upper diagonals degenerate to one in correspondence to (9). For unit cubic parameter value, the multiplication matrix (14) is calculated to
All left lower elements in (15) coincide with the corresponding ones in (14). Taking into account this identity, the insertion of the relations (14) and (15) into the polynomial equations (13) leads to the moments equation
Figures 3 and 4 show evaluations of the matrix differential equation (16). In the left picture, the time behavior of the first three polynomial moments is plotted for the initial conditions that the displacement moment is one and all other moments are vanishing at the beginning. In the right picture, one finds the associated phase portrait of the velocity moment versus the displacement moment for a vanishing nonlinearity parameter value in comparison with that of a unit parameter value. Densely neighbored graphs in both pictures indicate the convergence of the evaluation scheme that takes
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polynomials w sums are neglected, approximately. Obviously, this approach converges rapidly as it can be seen by the calculated graphs shown in Figure 3 and 4.
Figure 3. Polynomial moments versus time
Figure 4. Associated phase portrait
The matrix equation (16) is rewritten into the vector equation
For this purpose, only polynomials with odd index sums lead to the matrix
Herein, the polynomial equations with even index sums are decoupled. According to the structure, indicated by lines in the system matrix, above, the first diagonal matrix belongs to the two first order moments, the second diagonal matrix block belongs to the four third order polynomial moments and correspondingly for the higher order moments. Obviously, all diagonal matrix blocks are decoupled in the limiting case of a vanishing nonlinearity parameter value. Knowing all polynomial moments with odd and even index sums, the associated non-stationary density can be calculated to
Applying the normalization coefficients of both polynomials and multiplying by both polynomials, the integration over the state space lead to
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This justifies the expansion through the two orthogonal polynomials.
3.
STABILITY OF STATIONARY PROCESSES
Similar to the problems described in [5], the asymptotic stability of the stationary Duffing processes is investigated by the perturbation set-up
The insertion of the perturbations (20) into the stochastic Duffing equations and the linearization with respect to small perturbations leads to
Note, that the variation equation (22) possesses a quadratic fluctuation term determined by the stationary solutions of the stochastic Duffing equations (4) and (5). According to [5], the associated polar coordinate processes are
They are inserted into the variation equations (21) and (22). This leads to transformed perturbation equations of the following forms:
Herein, the first equation (24) describes the rotation of the phase process, which is periodic in the corresponding angle range and decoupled from the amplitude. The equation (25) determines the exponentially de- or increasing time behavior of the amplitude process. It can formally be integrated with respect to time, giving the top Lyapunov exponent
In this form, the integration is performed by infinitely increasing time. Provided there is a joint time invariant measure for the periodic angle process and the stationary Duffing process, the average (26) is replaced by
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This represents the space average in correspondence to the mean value (26). To evaluate the space average (27), one needs the stationary form of the three dimensional density distribution. It is expanded by a Fourier series of the two-periodic angle variable and the two polynomials, already used
Note, that the Fourier expansion of the angle density is only valid for the under-critical damping coefficient, noted in (29). For vanishing nonlinearity and overcritical damping, the rotation of the angle process is stopped, so that the angle density degenerates to a delta distribution which can not be represented by Fourier series. Applying Itô’s calculus, similar as shown in the two-dimensionless case, one obtains the moments equation
The stationary solutions of the equations (30) have to be calculated. For this purpose, only moments with even index sums are needed. The matrix equation (30) is written into the vector equation (31).
It is valid for every Fourier series term containing in a first block the zero moment, in a second block the three second order moments, in a third block five forth order moments and so on. The associated system matrix is
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It is calculated by the differentiation rule (14) and the restoring term expansion (15). To the quadratic variation term belongs the second matrix
For a unit non-linearity parameter value, these expansion elements are
They are calculated by the following expansion formula
Finally, the moment vector (32) is separated into real and imaginary parts by
The insertion of (32) into the expansion equation (31) leads to two equations, which are evaluated for every Fourier term, as follows.
Note that the diagonal matrices in (34) are abbreviations for the sum of the unit matrix I and the expansion matrix C of the quadratic variation term.
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4.
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EVALUATION OF TOP LYAPUNOV EXPONENTS
The integration of the joint density (28) with respect to displacement and velocity gives the marginal density of the stationary angle process by
Herein, the Fourier coefficients are calculated by the equation system (34).
Figure 5. Analysis of phase densities
Figure 6. Phase histograms by MC simulations
Figure 5 shows evaluations of the phase density (35) for a unit nonlinearity parameter value and three different under-critical damping coefficients. As indicated in the top space of the figure, the applied number of polynomials was eight and that one of Fourier terms was four, eight up to ten, so that the total number of polynomial coefficients was 25 leading to 500 unknowns in (35) in case of ten Fourier expansion terms. The associated results are sketched by solid lines. Thin lines belong to the expansion of four Fourier terms. The phase density with eight Fourier terms is covered by the solid lines of the ten term expansion that shows a rapid convergence. In Figure 6, Monte-Carlo simulation results are presented. They are obtained by means of an Euler Maruyama scheme applied to the stochastic equations (4), (5), (24) for the same parameter values as already used in Figure 5. The comparison of the histograms in Figure 6 with the smooth phase densities plotted in Figure 5 shows good agreements. The Monte-Carlo simulations are continued to evaluate the time mean value (26) of the top Lyapunov exponent in dependent on the damping coefficient for three values of the nonlinearity parameter applying the scan rate and the sample rate, noted in Figure 7. All values of the top Lyapunov
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exponents obtained for positive damping coefficients are negative indicating that the stationary Duffing processes are asymptotically stable. For overcritical damping coefficients greater than one, the Lyapunov exponents decrease with increasing nonlinearity parameter values so that the stationary Duffing processes are stabilized in this damping range. In the under-critical case, that the positive damping values are smaller than one, the stationary Duffing processes are slightly destabilized for increasing nonlinearities. For vanishing and negative damping values the mean value (26) is not applicable since the Duffing processes are increasing, infinitely.
Figure 7. Numerical Lyapunov exponents
Figure 8. Analytical Lyapunov exponents
The latter results are confirmed by analytical investigations shown in Figure 8 for the under-critical case that the absolute values of the damping coefficients are smaller than one. For increasing nonlinearity parameter values, the stationary solutions of the Duffing oscillator are destabilized in the positive damping range and stabilized in the negative one. These are analytical results of the top Lyapunov exponent taken from the space average (27) applying eight orthogonal polynomials for the displacement and velocity density variables and ten Fourier series terms. Integrating (27) for the analytical expansion (28) leads to the top Lyapunov exponent
Note, that the sum of the top Lyapunov exponent and the second one is equal to the damping term of the variation equation (22). Therewith, both Lyapunov exponents are known. They have the same sign and change their signs, simultaneously. In case of negative damping values, the evaluation of
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(36) leads to the smallest Lyapuov exponent, which is the top Lyapunov exponent for the negative time scale. This situation is completely different for the so-called Kramers oscillator [6] which is described by the incremental state equations
Its stationary displacement density possesses the form of bi-potentials, shown in Figure 9. The associated variation equations lead to the following phase equation and Lyapunov exponent:
Applying an Euler Maruyama scheme to the oscillator equations (37) and to the phase equation (38), the time average of the Lyapunov exponent (39) can be calculated in [6] and [7]. Plotted versus the damping coefficient, Figure 10 shows clearly that the top Lyapunov exponent becomes positive in the positive damping range, meanwhile the second one remains negative.
Figure 9. Bi-potential density
5.
Figure 10. Top Lyapunov exponents
SUMMARY AND CONCLUSION
The paper investigates the asymptotic stability of the stationary solutions of the stochastic Duffing and Kramers oscillator by calculating the top Lyapunov exponent of the linear perturbation equations. Replacing the perturbed state processes by polar coordinates, the amplitude process is separated und formally integrated leading to the time average of Oseledec’s multiplicative ergodic theorem. Evaluations of this form of the top Lyapunov
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exponent in the time domain are restricted to the case of positive damping values when the system solutions are stationary and asymptotically stable. To avoid this disadvantage the paper replaces the time average of Oseledec’s multiplicative ergodic theorem by the corresponding space average of Khasminskii-Fürstenberg provided that there is a joint invariant measure of the stationary system and phase processes. For the Duffing oscillator of interest the three-dimensional density of this invariant measure is calculated by means of Fourier and polynomial expansions. The integration with respect to the three density variables of the variation phase, the system velocity and the displacement leads to the top Lyapunov exponent for positive damping values of the dynamic system indicating that the stationary oscillator processes are asymptotically stable and can not be destabilized through increasing noise intensities.
REFERENCES 1. 2. 3.
4.
5.
6. 7.
Duffing G. Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung, Braunschweig, Vieweg und Sohn, 1918. Guckenheimer J, Holmes P. Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields, Heidelberg, Berlin, New York, Springer-Verlag, 1983. Wedig W. “Generalized Hermite Analysis of Nonlinear Stochastic Systems”, Proc. of the Conference on Stochastic Structural Dynamics, Namachchivaya, Hilton & Wen eds., University of Illinois, Urbana-Campaign, pp. 381-389, 1989. Wedig W. “Spectral Analysis of Non-Smooth Dynamical Systems”, Proc. of the Conference on Stochastic Structural Dynamics, Spencer & Johnson eds., Balkema Publishers, pp. 249-256, 1999. Wedig W. “Vertical Dynamics of Riding Cars under Stochastic and Harmonic Base Excitations”, IUTAM Symposium on Chaotic Dynamics and Control of Systems and Processes in Mechanics, Rega & Vestroni eds., Springer, Dordrecht, pp. 371-381, 2005. Arnold L, Imkeller P. “The Kramers Oscillator Revisited”, Lecture Notes in Physics, pp. 280-291, 2000. Simon M. Zur Stabilität stochastischer Systeme mit stochastischer Anregung, Dissertation Universität Karlsruhe, pp. 62-72, 2004.
UNCERTAINTIES IN DETERMINISTIC DYNAMICAL SYSTEMS AND THE COHERENCE OF STOCHASTIC DYNAMICAL SYSTEMS J. X. Xu, H. L. Zou School of Aerospace, Xi’an Jiaotong University, Xi’an, 710049, E-mail:
[email protected]
Abstract:
The uncertainties arising with the boundary crisis which destroys the Wada basin boundaries in the deterministic Hénon map are investigated in this work. It is shown that the region of disappearing chaotic attractor and its basin become a part of the new basin boundary, and then a complex deterministic evolution follows. Meanwhile, the effect of noise to the qualitative structure of the system is studied. The small noise can hold the Wada property, but noise larger than can destroy one attractor and therefore the Wada property is disappeared. Secondly, we study the Duffing equation with uncertainty of a parameter. A kind of linear feedback is used to make this system resist large degree of uncertainty.
Key words:
Uncertainty, indeterminate boundary crisis, generalized cell mapping.
1.
INTRODUCTION
One important property of chaos is its sensitive dependence on initial conditions. It makes people unable to predict long-term behavior of deterministic system. This unpredictable behavior could be regarded as a kind of uncertainty in the deterministic systems. Other uncertainties, for examples, may involve the domiciles of points on the fractal basin boundaries and riddled basins [1]. Comparing with local situations, these uncertainties mentioned above are global, namely these uncertainties concern the global dynamics behaviors of the deterministic systems. The global uncertainties are meaningful in many scientific fields such as neural 109 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 109–116. © 2007 Springer.
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dynamics [2], attitude control of satellite, et al. Uncertainties in this paper are mainly examined globally. Random excitations, like noises, can highly affect deterministic behaviors of deterministic systems under some conditions. Noise can make responses of the two coupled systems synchronous [3] and it also can turn chaos into regular motion. For some specified noisy systems, the regular stochastic resonance [4] can be reached which is characterized by the extremum of signal-to-noise ratio and average transition time as noise strength varies. Hence, the coherence and “order” in stochastic system may be established. Wada basin boundaries are usually considered more complicated than fractal basin boundaries. A point x on the basin boundary is a Wada point if every open neighborhood of x has a nonempty intersection with at least three different basins. A basin boundary is called a Wada basin boundary if all points of this basin boundary are Wada points [5]. Grebogi, Ott and Yorke [1] called the sudden disappearance of chaos and its basin that happens in noninvertible map boundary crisis. It is caused by the collision of saddle sets in the basin boundary with chaotic attractor. In ordinary differential equations, Thompson, Stewart and Ueda [7] called the abrupt disappearance of a chaotic attractor chaotic catastrophe. Its mechanism is the collision of chaotic attractor with stable manifold of a saddle set. Hong and Xu [8] found that the mechanism of boundary crisis in ordinary differential equations is also the collision of saddle sets with chaotic attractor. In the section 2, we first locate Wada basin boundaries with a chaotic attractor, and then investigate uncertainties arising with boundary crisis. The Hénon map is employed to analyze for illustrating these phenomena. In order to study effects of noise, we add noise to the Hénon map, and then the system turns to a stochastic system. When the stochastic perturbation is small, the property of Wada can be qualitatively held. On the contrary, the large stochastic perturbation can destroy one of the attractors and so the Wada basin boundaries will disappear. In the section 3, we study the Duffing equation with uncertainty of a parameter. A kind of linear feedback is used to make this system resist large degree of uncertainty. The global results in this section are obtained by improved generalized cell mapping which is not based on the theory of digraph and theory of partial set.
Uncertainties in deterministic systems and control
2.
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UNCERTAINTIES INDUCED BY BOUNDARY CRISIS AND EFFECT OF NOISE TO WADA BASIN BOUNDARIES Hénon map can be given by F ( x, y ) = (a − x 2 + by , x)
(1)
In the previous studies, Wada basin boundaries can arise in this map [1]. In this paper, we consider the situation of Wada basin boundaries in which one of the attractors is chaos. When the boundary crisis occurs, the chaotic attractor and its basin will suddenly disappear. With this disappearance, uncertainties can arise. The results in this section are obtained by direct point mapping method. Namely, a large number of initial points are used and their domiciles are obtained by direct method. We first fix parameters a = 0.71397 and b = 0.85. There are three attractors involving a six pieces of chaotic attractor and all the basin boundaries are Wada as shown in Figure 1. Then with the variation of parameter b, a boundary crisis caused by the collision of the chaotic attractor and an accessible periodic saddle on the basin boundary can happen which destroy the chaotic attractor and its basin. After bifurcation, two attractors are left. Figure 2 shows the p-2 attractor and its basin of attraction. Then an interesting question is that where the disappeared chaotic attractor and its basin go. We show that just after boundary crisis, most points of the disappeared chaotic attractor and its basin settle on the basin boundaries of the remaining two attractors. We use the term “most” here to describe the situation that the gab of basins of the left two attractors in the disappeared region gradually grows from zero. These phenomena are shown in Figure 3.
Figure 1. Wada basin boundaries of Hénon map with a = 0.71397 and b = 0.85. A chaotic attractor is shown in red. A p-2 attractor in illustrated by “+”. Basin of chaotic attractor is shown in green. Basin of p-2 attractor is shown in black. The infinity is the third attractor. Points that go to infinity are shown in white. (b) is the enlargement of rectangular in (a).
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Figure 2. Basins of attraction of Hénon map after bifurcation with a = 0.71397 and b = 0.85. A p-2 attractor in illustrated by “+”. Basin of p-2 attractor is shown in black. Points that go to infinity are shown in white.
Figure 3. The gap in the region {0.7 ≤ x ≤ 0.9, −0.6 ≤ y ≤ −0.25} gradually grows from zero after boundary crisis with b = 0.85. (a) a = 0.71398. (b) a = 0.72 . (c) a = 0.73 and (d) a = 0.74. (e) a = 0.75.
As shown in Figure 3 most points of the disappeared chaotic attractor and its basin will settle on the boundaries of the remaining two attractors just after the boundary crisis, there will be a suddenly increase in the fractal dimension of basin boundaries. So the degree of uncertainty suddenly increases [1].
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It is also find that the gap in the region of disappeared attractor and its basin after boundary crisis gradually grows from zero. “Zero” means that the domiciles of the points of the chaotic attractor and its basin are of uncertainty. However, this process shown in the Figure 3 is determinate. For definite value of the bifurcation parameter a, the global dynamical behavior of the system, the attractors and their basin of attraction including their “gaps” is determinate. One can see that for different values of a, the gap going to an attractor is variant both in size and region, and therefore exhibits another kind of uncertainty. Namely, the domiciles of the points near the gap are sensitive to the parameter. In the real system, there usually exists noise. So in the following we study whether Wada property can still exist under perturbation of small noise. We add noise to Hénon map (1).
F ( x, y ) = (a − x 2 + by, x + ξ (t )) ,
(2)
where ξ (t ) is Gaussian white noise. We first divide the region {-2.0<x<2.0,-2.5
Figure 4. Attractors, basins and basin boundaries for Hénon map (2) with noise intensity S = 0.00013 . A chaotic attractor is shown in red. A p-2 attractor in illustrated by “+”. Basin of chaotic attractor is shown in black. Basin of p-2 attractor is shown in green. Points that go to infinity are shown in white. The basin boundaries of p-2 attractor, the chaotic attractor and the attractor at infinity are shown in yellow.
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Figure 5. Attractors, basins and basin boundaries for Hénon map (2) with noise intensity S = 0.00014. A p-2 attractor is illustrated by “+”. Basin of p-2 attractor is shown in black. Points that go to infinity are shown in white. The basin boundaries of p-2 attractor, and the attractor at infinity are shown in yellow.
We fix parameters a = 0.7135, and b = 0.85. In this case, if noise intensity S < 0.00014 , the system (2) still has three attractors and the Wada basin boundaries like what shown in Figure 1. Figure 4 shows these results with noise intensity S = 0.00013. For the basin boundaries of all the three attractors surround each of the basins of the three attractors, we say that the Wada property still exists when perturbation is small. When noise intensity S = 0.00014, there are only two attractors. Figure 5 shows the attractors and their basins. So the Wada basin boundaries are destroyed by a relatively large noise.
3.
CONTROL UNCERTAINTY IN DUFFING EQUATION
When parameters of a system have some uncertainties, the dynamical behaviors will be interesting. In this section we study uncertainties of parameters in the forced Duffing equation. This model can be given by [9] u& = v, v& = βu − u 3 − δ v + r cos ωt
(3)
In this paper, we fix δ = 0.25, r = 0.15, and ω = 1.0. We study the system by Poincaré map with time T = 2π . When β = 1.0 , the system (3) has two attractors. Figure 6 shows the attractors, basins and basin boundaries by the improved generalized cell mapping method. If the parameter β of system (3) has uncertainties, the global behaviors (shown in Figure 6 and Figure 7) will probably undergo a sudden change. Let β be a random variable with variance σ and expectation 1.0. The variance σ can be used to measure the degree of uncertainty. The bigger it is, the more uncertainty the parameter β has. In this paper, we choose 10 random numbers to simulate β . Using the improved generalized cell
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mapping, we can find that when σ ≥ 0.13 one of the two attractors will be destroyed. Figure 7 shows that only one attractor exists when σ = 0.13. This phenomenon is dangerous in engineering. So it is of important significance to make system resist large degree of uncertainty. If σ reaches the critical value, one of the attractors will be destroyed by colliding with the saddle set on the basin boundaries. If we want to control the system to resist large degree of uncertainty, one way is to make the attractors farther away from the basin boundaries. We can add linear feedback ku with k > 0 to the second equation of system (3) because the two attractors of system (4) when r = 0.0 can be farther away than the case of k = 0, so we can get the controlled system: u& = v, v& = βu − u 3 − δ v + r cos ωt + ku
(4)
As an example, we choose k = 0.25 . Using the improved generalized cell mapping, now the controlled system (4) can resist uncertainty of β with σ = 0.25 . If we want to make system resist much larger degree of uncertainty, we can choose a much larger value of k .
Figure 6. Global behaviors of Duffing equation with δ = 0.25 , r = 0.15, ω = 1.0, and β = 1.0 . A p-1 attractor is illustrated by “+”. Its basin is shown in white. Another p-1 attractor is illustrated by “×”. Its basin is shown in black. The basin boundaries are shown in yellow. A p-1 saddle point is illustrated by red “□”.
Figure 7. Only one attractor exists in Duffing equation with with δ = 0.25 , r = 0.15, and ω = 1.0 when β ’s variance σ = 0.13 . The attractor is shown in black. Its basin is shown in white. An unstable solution is shown in light gray.
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CONCLUSIONS AND DISCUSSIONS
(1) The uncertainties arise with disappearance of the Wada basin boundaries in the deterministic Hénon map because the region of disappeared chaotic attractor and its basin become a part of the new basin boundary, and then a complex deterministic evolution follows. (2) The effect of noise to the qualitative structure of Hénon map is investigated. We find that the small noise can hold the Wada property, but large noise can not hold the Wada property. (3) A kind of linear feedback is used to make the Duffing equation system with parametric uncertainty to resist large degree of uncertainty.
ACKNOWLEDGEMENTS This work was supported by the National Science Foundation of China (NSFC) under grants no.10432010.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
9.
Ott E. Chaos in Dynamical systems in chapter 5, Cambridge university press, 1993. Xu JX, Gong YF, Ren W, Hu SJ, Wang FZ. “Propagation of periodic and chaotic action potential trains along nervous fibers”, Physica D, 100, pp. 212-224, 1997. He DH, Shi PL, Stone L. “Noise-induced synchronization in realistic models”. Physical Review E, 67, 027201, 2003. Benzi R, Sutera A, Vulpiani A. “The mechanism of stochastic resonance”, J physics A, 14, L453-L457, 1981. Nusse HE, Yorke JA. “Wada basin boundaries and basin cells”, Physica D, 90, pp. 242261, 1996. Grebogi C, Ott E, Yorke JA. “Basin boundary metamorphoses changes in accessible boundary orbits”, Physica 24D, pp. 243-262, 1987. Thompson JMT, Stewart HB, Ueda Y. “Safe, explosive, and dangerous bifurcations in dissipative dynamical systems”, Physical Review E, 49, pp. 1019-1027, 1994. Hong L, Xu JX. “Discontinuous bifurcations of chaotic attractors in forced oscillators by generalized cell mapping digraph (GCMD) method”, Int. J. of Bifurcation and Chaos, 11, pp. 723-736, 2001. Guckenheimer J, Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fieds, Springer, Chapter 2, 1983.
THE CELL MAPPING METHOD FOR APPROXIMATING THE INVARIANT MANIFOLDS W. Xu, Q. He, S. Li Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, China, E-mail:
[email protected]
Abstract:
The digraph cell mapping method (DCMM) is a powerful technique for global analysis of nonlinear systems. In this paper, we introduce two new notions to modify DCMM. After the modification, DCMM can successfully approximate the invariant manifolds of nonlinear systems. Furthermore, it is not only applicable to deterministic systems but also to stochastic systems. As an illustrative example, the safe basin erosion of a Duffing oscillator under deterministic or stochastic excitation is studied in detail. Numerical results show that the modified digraph cell mapping method (MDCMM) is an efficient tool to approximate the invariant manifolds of dynamical systems.
Key words:
The invariant manifolds, the modified digraph cell mapping method.
1.
INTRODUCTION
In recent years the phenomena that the qualitative and quantitative properties of nonlinear dynamic systems change drastically with a small change of system parameters have attracted many people’s attention [1, 2]. These discontinuous changes belong to the category of global bifurcations and are typically triggered by “collisions” (tangencies) of stable and unstable manifolds associated with the existing unstable solutions. The stable (unstable) manifolds are defined as the invariant sets of all initial conditions whose trajectories approach the saddle forward (backward) in time. It is well known that the stable and unstable manifolds can intersect at a point different from the saddle. If there exists one such intersection point, then 117 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 117–126. © 2007 Springer.
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there must be infinitely many, which indicate the presence of Smale horseshoe and chaotic sets. Hence, knowing these invariant manifolds is crucial as they govern the dynamical behavior of nonlinear systems. However, at present, the methods for approximating the invariant manifolds are very few [3, 4]. In this paper, aiming at the above question, we suggest a modified digraph cell mapping method to approximate the invariant manifolds. Comparing with the original method, two important notions are introduced to further divide the strongly connected sub-digraphs and transient cells in the modified method. Then the stable and unstable manifolds correspond with some particular sets of transient cells in a digraph. Numerical results show that the modified digraph cell mapping method (MDCMM) is a powerful tool for approximating the invariant manifolds.
2.
HOW TO APPROXIMATE THE INVARIANT MANIFOLDS
The cell mapping method (CMM) is a numerical technique for global analysis of nonlinear systems. Since first suggested by Hsu [5], a series of important developments have been made for it [5-8]. In 1995, Hsu developed a generalized cell mapping method using posets and digraphs [9]. For topological analysis, the global generalized cell mapping can be identical with a digraph, with a cell identical to a vertex. Then, the topological analysis of a cell mapping can be transformed to that of a digraph. We know that the stable (unstable) manifolds are defined as the invariant sets of all initial conditions whose trajectories approach the saddle forward (backward) in time. Then, a question is what characteristic sets correspond with these invariant manifolds in the analysis of digraph. In order to answer this question, two useful notions are introduced as follows. If a transient cell can reach a strongly connected sub-digraph (whether close or open), then we call the latter a domicile of the transient cell. If a transient cell locates in the path that an open strongly connected sub-digraph leads to a close strongly connected sub-digraph, then we call the open strongly connected subdigraph a route of the transient cell. Note that the notion of domicile is different from that used in the original digraph cell mapping method. In the original method, the domicile only concerns the close strongly connected sub-digraph, but not the open. The domicile characterizes the dependent relation between the transient cell and its corresponding strongly connected sub-digraph. The route denotes the development course of the strongly connected sub-digraph. For an open strongly connected sub-digraph, the transient cells having it as their common domicile are classified into one
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group. It forms the stable manifold of the corresponding saddle in the original system. Meanwhile, the transient cells having this strongly connected sub-digraph as their common route are classified into another group. This group forms the unstable manifold of this saddle. By the above classification, the original cell mapping dynamical system can be condensed, and the condensed digraph can reflect the topological structure of original dynamical system.
3.
BASIN EROSION OF DUFFING OSCILLATOR UNDER DETERMINISTIC EXCITATION
The problem of safe basin erosion has been discussed in previous study [10]. Here, as an illustrative example, we consider the application of MDCMM to basin erosion of Duffing oscillator. The equation of Duffing oscillator is given as
&& x + kx& − x + x3 = A sin(ωt + φ )
(1)
where the parameters k = 2 × 0.1, ω = 2 × 0.85, φ = π , and A = a × AE, in which AE = 0.2151. Now MDCMM is applied to the system (1). A cell structure of 800 × 800 cells is partitioned for interested domain Ω = {−1.8 ≤ x ≤ 0.2, −1.05≤ x& ≤ 1.05}
(2)
and 50 × 50 representative points are used within each cell. Numerical simulation results about the Poincaré cell mappings at different levels of coefficient a are shown in a series of pictures, Figures 1-8. In case a = 0.2 , Figure 1 shows six groups of cells, which correspond to the attractor denoted by S 1, the saddle denoted by D, the basin of attractor S 1 denoted by B1, the basin of sink cell denoted by B2, the stable manifold of saddle denoted by W S, the unstable manifold of saddle denoted by W U. From Figure 1 one can see that the safe basin B1 is intact. When a changes from 0.2 to 0.5546, refer to Figure 2, the stable manifold of saddle touches its unstable manifold. A homoclinic tangency occurs. After this critical point, refer to Figure 3, the homoclinic tangency turns into a homoclinic transverse intersection. When a changes to 0.82, refer to Figure 4, more transverse intersections between the stable and unstable manifolds are observed. At the same time, the safe basin becomes smaller. When a changes from 0.901 to 0.902, refer to Figures 5 and 6, it can be found that the P-1 attractor bifurcates into a P-2 attractor, that is, the period doubling bifurcation happens. As a is further
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raised, refer to Figures 7 and 8, the P-2 attractor bifurcates into a P-4 attractor, and the safe basin has disappeared.
Figure 1. Safe basin, attractor and invariant manifolds of system (1) with a = 0.2 .
Figure 2. Safe basin, attractor and invariant manifolds of system (1) with a = 0.5546.
Figure 3. Safe basin, attractor and invariant manifolds of system (1) with a = 0.6.
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Figure 4. Safe basin, attractor and invariant manifolds of system (1) with a = 0.82.
Figure 5. Safe basin, attractor and invariant manifolds of system (1) with a = 0.901.
Figure 6. Safe basin, attractor and invariant manifolds of system (1) with a = 0.902.
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Figure 7. Safe basin, attractor and invariant manifolds of system (1) with a = 0.977.
Figure 8. (a) Attractor and invariant manifolds of system (1) with a = 0.978 ; (b) Enlargement of the small rectangle region.
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Several numerical methods for calculating stable and unstable manifolds are only applicable to deterministic systems, therefore, in previous studies there is not the report on an efficient method for calculating the invariant manifolds of stochastic systems. However, in what follows it is shown that MDCMM is also a powerful tool to approximate the invariant manifolds of stochastic systems. We consider the basin erosion of Duffing oscillator under stochastic noise. The equation of stochastic Duffing oscillator is given as
&& x + kx& − x + x3 = ( A0 + σξ (t ))sin(ωt + φ )
(3)
where the parameters k , ω ,φ are the same as those used in (1), A0 = 0.1193 which corresponds to the homoclinic tangent point of Equation (2) (see Figure 2), ξ (t ) is a standard Gaussian process with intensity parameter σ . Now MDCMM is applied to study the changes of safe basin as the noise intensity σ increases. A cell structure of 400 × 420 cells is used for the same domain Ω (denoted by (2)), 50 × 50 representative points are selected within each cell, and 10 random sample trajectories are generated at each representative point. Numerical simulation results are presented in Figures 9-13. When σ is raised from 0.0 to 0.000021, refer to Figures 2 and 9, one can see the attractor S 1 becomes bigger, and the stable and unstable manifolds change from homoclinic tangency to homoclinic transverse intersection. Because of the homoclinic transverse intersection, a chaotic saddle is formed. As the value of σ is further raised, refer to Figures 10 and 11, more and more transverse intersections between the stable and unstable manifolds take place, which implies the chaotic saddle is growing. Moreover, one can find the growing chaotic saddle is close to the bigger attractor. When σ is raised from 0.000073 to 0.000074, refer to Figures 12 and 13, the attractor collides with the chaotic saddle, then the attractor and the associated basin suddenly disappear, at the same time, a huge chaotic saddle is created. That is, the boundary crisis happens. Because having no place to reside down, the trajectories starting from the former safe basin will escape from the domain Ω along the direction of the unstable manifold of chaotic saddle. The safe basin is completely eroded.
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Figure 9. Safe basin, attractor and invariant manifolds of system (3) with σ = 0.000021.
Figure 10. Safe basin, attractor and invariant manifolds of system (3) with σ = 0.000039.
Figure 11. Safe basin, attractor and invariant manifolds of system (3) with σ = 0.000070.
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Figure 12. Safe basin, attractor and invariant manifolds of system (3) with σ = 0.000073.
Figure13. Invariant manifolds of system (3) with σ = 0.000074 .
5.
CONCLUSIONS
Stable and unstable manifolds play an important role in the analysis of dynamical behavior of nonlinear systems. The structure and its variation of these manifolds are often responsible for chaotic crises, chaotic transients, fractal basin boundaries, etc. Therefore, approximating these invariant manifolds is crucial to study complicated nonlinear phenomena. In this paper, we introduce two new notions to modify DCMM. MDCMM can successfully approximate the invariant manifolds of nonlinear systems. Moreover, it has the same good performance in stochastic systems as in deterministic systems. As an application of MDCMM, the erosion of safe basins of a Duffing oscillator under deterministic or stochastic excitation is studied in detail. Numerical results show MDCMM is a powerful tool to approximate the invariant manifolds of nonlinear systems.
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ACKNOWLEDGEMENTS The authors are grateful for the support of the National Natural Science Foundation of China (Grant Nos. 10472091, 10332030).
REFERENCES 1.
Hong L, Xu JX. “Crises and chaotic transients studied by the generalized cell mapping digraph method”, Physics Letter A, 262, pp. 361-375, 1999. 2. Xu W, He Q, Fang T, Rong HW. “Stochastic bifurcation in Duffing system subject to harmonic excitation and in presence of random noise”, International Journal of Nonlinear Mechanics, 39, pp. 1473-1479, 2004. 3. Nusse HE, Yorke JA. Dynamics, Numerical Explorations, New York, Springer, 1998. 4. Krauskopf B, Osinga HM, et al. “A survey of methods for computing (un)stable manifolds of vector fields”, International Journal of Bifurcation and Chaos, 15, pp. 763791, 2005. 5. Hsu CS. “A theory of cell to cell mapping dynamical systems”, J. Applied Mechanics, 47, pp. 931-939, 1980. 6. Hsu CS. “A generalized theory of cell to cell mapping for nonlinear dynamical systems”, J. Applied Mechanics, 48, pp. 634-842, 1981. 7. Tongue BH, Gu K. “A higher order method of interpolated cell mapping”, J. Sound Vib., 125, pp. 169-179, 1988. 8. Guder R, Kreuzer E. “Control of an adaptive refinement technique of generalized cell mapping by system dynamics”, Nonlinear Dynamics, 20, pp. 21-32, 1999. 9. Hsu CS. “Global analysis of dynamical systems using posets and digraphs”, International Journal of Bifurcation and Chaos, 5, pp. 1085-1118, 1995. 10. Lansbury AN, Thompson JMT, Stewart H. B. “Basin erosion in twin-well duffing oscillator, two distinct bifurcation scenarios”, International Journal of Bifurcation and Chaos, 2, pp. 505-532, 1992.
PART 3
NONLINEAR DYNAMICS
DYNAMICAL FRACTAL DIMENSION: DIRECT AND INVERSE PROBLEMS L. Bevilacqua, M. M. Barros National Laboratory of Scientific Computation (LNCC) Av. Getúlio Vargas 333, Petropolis, E-mail:
[email protected],
[email protected]
Abstract:
This paper presents a new method to determine the fractal dimension of plane curves. Two main problems are described, namely the direct and the inverse problem. It is shown that the periods of harmonic oscillators obtained by folding wire like structures with the same geometry as the terms of a fractal sequence provide the necessary information to determine the fractal characteristics of that sequence. More interesting is the determination of the fractal properties of a given curve from a sequence of samples cut off the original curve. Building up harmonic oscillators with these samples it is shown that fractal characteristics can be identified. The dynamical response depends on the initial conditions providing complementary information about the fractal characteristics of the object. The last section deals with a simple random fractal and confirms the use of different initial conditions to characterize completely the fractal sequence.
Key words:
Fractal curves, dynamical dimension, random fractals.
1.
INTRODUCTION
One of the most common questions in applications of fractional geometry to actual problems, ranging from patterns formation in nature to time depending signals or material structures, is the determination of the corresponding fractal dimension if it really exists. Several methods have been used to solve this problem as, the box counting method or mass distribution method, to give two important examples, [1-2]. This paper proposes a new method that can be useful to characterize the fractal geometry using the dynamical properties of wire like structures 127 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 127–136. © 2007 Springer.
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folded with the same geometry as the fractal object. Some of these aspects will be presented in this paper but much more, we believe, is still hidden behind the dynamic properties of fractal structures. Let us take a classical example, the so called Koch triadic, defined by the sequence of curves as shown below: n=1
n=2
n= k
λ1
λ2
L0 λ1= L0/3 N1 = 4
λ2=
L0/32
N2 = 42
λk= L0/3k Nk = 4k
Figure 1. Formation of the Koch triadic.
The first element is the generator, all others are obtained by properly scaling down the generator and attaching it to the previous term as a support. In this way the elements of the sequence are self-similar, meaning that in each of them the generator is recuperated in a scale corresponding to the order of formation of that term. We will consider in this paper the generator term composed by p elements with length equal to L0/q where p and q are integers. The total number of elements in the term of order k is given by, N k = p k and the corresponding length of the elements are λk = L0 /q k. Clearly the Koch triadic belongs to this family of curves. The Euclidean lengths of these curves tend to infinity with k. The first successful attempt to introduce a new measure that would overcome this singularity was done by Hausdorff [3]. Applying the Hausdorff criterion to the family of curves defined here we obtain: D = log p / log q
(1)
D is the Hausdorff dimension and it can be shown that lim ( p / q D ) → finite k →∞ and non null. The dynamic dimension is obtained through the dynamic properties of simple harmonic oscillators built up after the geometry of the terms composing the fractal curve sequence. To visualize the procedure, consider wire like structures folded according to the pattern corresponding to the geometry of a particular term of the fractal series. Suppose, for instance, that the classical Koch curve is selected as the reference geometry. It is possible then to fold several pieces of wire with appropriate length Ln to obtain a sequence as shown in Figure 1. Now, a sequence of harmonic oscillators can be built up using the folded wires working as elastic springs as shown in Figure 2. Since we are dealing with plane curves each oscillator has three degrees of freedom. As is well known the general dynamic equations can be written in the following equivalent forms: k
Dynamical fractal dimension: direct and inverse problems && + Kw = 0 Mw
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Subjected to the complementary conditions: Rw = 0, or &&& − f = 0 Mw
(2-b)
Subjected to the complementary conditions: Sf = 0. Where M is the mass matrix, K is the rigidity matrix and w = {u,v,θ}T the displacement vector, f = {H,V,M}T the generalized force vector. The matrices R and S represent the constraints imposed on the generalized displacements and on the generalized forces as depicted in the Figure 2 and Figure 3. The constraint matrix S can always be chosen such that the displacement vector contains only one component of the generalized forces. Let us choose S such that only the horizontal component H is acting. The stored elastic energy Wk for the kth order term is given by: 2 EI 0Wk = ∑1 k ∫ N
λk
0
(M ( ) ) k i −1,i
2
(3)
ds
where E is the Young Modulus of the wire material and I0 the moment of inertia of the wire cross section. Nk is the total number of elements in the kth order term. Now all terms fit into a box L0 x h0 as can be seen in Figure 4. The bending moment along a segment (i-1, i) is (Figure 4):
(
)
M i(−k1,) i = H ⎡ yi(−k1) + yi( k ) − yi(−k1) s ⎤ ⎣ ⎦
(4)
Now introducing Equation (4) into (3), integrating over all segments λk and summing up we get: Tk2 = ( m0 h02 L0 EI 0 ) ( λk L0 ) N k Ω k
(5)
m0 is the mass attached at the end of the oscillator, Tk is the period, Ω k =
(∑
Nk
1
)
α i ( k ) N k with α i (k ) = ⎡⎣ zi2−1 + zi −1 zi + zi2 ⎤⎦ 3 and z j = y j h0 . v
θ
h0
L0
u
Figure 2. Harmonic oscillator corresponding to the second term of the Koch triadic with the generalized displacements.
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V M h0
L0
Figure 3. Harmonic oscillator corresponding to the second term of the Koch triadic with the generalized forces.
y h0
M i(−k1) i-1
M i(k )
λk
yi(−k1)
i
yi(k )
x
Figure 4. Bending moment along the segment (i-1,i) for a general term k.
Note that from the definition of h0 clearly zj ≤ 1 for all j, and consequently αi(k) ≤ 1 leading to Ωk ≤ 1as well. Now recalling the definition of Nk and λk and with Equation (1) we may write: log N k = − log ( λk L0 ) ( log p log q )
(6)
After some straightforward calculations we obtain: 2log (Tk T0 ) = log Ω k + (1 − D ) log ( λk L0 )
(7)
where D = (log p)/(log q) is the classical box dimension, that coincides with the Hausdorff dimension for this case, and T0 is a period of reference: T02 = m0 h02 L0 EI 0
The parameter D is the dynamical fractal dimension. It coincides with the box and the Hausdorff fractal dimensions provided that the mass is constant and the wire diameter as well. Now, if the sequence of oscillators corresponding to the terms of order, k = 1,2,….n, has a fractal characterization it is necessary that Equation (7) plotted on the plane log (Tk/T0) x log (λk/L0) will approach a straight line whose angular coefficient is equal to (1– D)/2.
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Proposition I. For all curves belonging to the family Nk = pk and λk/L0=(1/q)k the logarithm of the bilinear term Ωk tends to zero for increasing values of k. The proof is not given here. The interested reader can find it in [4].
2.
DIRECT PROBLEM
Let us examine now some examples. Consider the first 9 terms in sequence of the Koch triadic. The normalized periods (Tk/T0) of the corresponding harmonic oscillators for a couple, a horizontal force and a vertical force as function of the ratio (λk/L0) are depicted in Figure 5-a. The wire diameter is constant and the mass is also constant for all oscillators, We expect the normalized periods versus the relative length to be a straight line with slope (1-D)/2 for the case of the initial displacements corresponding to a couple and to be a curve whose slope tends to (1-D)/2 as λk/L0 tends do zero for initial displacements generated by a horizontal or vertical force. The fractal dimension of the Koch triadic is 1.26186 up to the fifth decimal term. The values obtained from the dynamic approach are given in the table below. Note that the slope used to evaluate the fractal dimension corresponds to the last segment connecting the last two points of the curve. The results are correct up to the fifth digit after the comma confirming the good results. It is also clear from Figure 5-a that the curve corresponding to the initial displacements generated by the horizontal force approaches asymptotically the straight line with slope -0.13093. Figure 5-b shows the ratio Dk/Dexact where Dk is the value of the fractal dimension obtained for the term of order k and Dexact is the exact value.
Slope of last segment Fractal dimension: D
Moment
Horizontal Force
Vertical Force
-0.13092975 1.26186
-0.13093495 1.26187
-0.13093002 1.26186
Figure 5. (a) Variation of the the normalized period with the length of the elementary segment. (b) Relative error Ddyn/Dexact for the three cases couple, vertical and horizontal forces.
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3.
INVERSE PROBLEM
More interesting is the inverse problem. The quest is to find out the topological characteristics of a given curve. As previously stated it can be considered as a folded wire as well. Suppose therefore that a piece of wire with length LT is folded reproducing exactly the curve whose horizontal projection has the length Ln. The question now is to find out whether some fractal structure, if any, can be associated to that geometry. Following a similar procedure as that introduced to obtain the dynamic fractal dimension for the direct problem, obtain first the period Tn of the harmonic oscillator corresponding to the original folded wire working as an elastic spring attached to some arbitrary mass M0. The second step is to cut off a subset from the original crumpled wire to obtain a new harmonic oscillator with natural period T1 (Figure 6). Then proceed successively, cutting off some subset from the previous configuration in order to obtain a series of periods Tn, Tn-1, …Tk, Tm corresponding to the lengths Ln, Ln-1, …Lk, Lm. If the original object belongs to a particular fractal sequence then it is possible to show that the corresponding dynamic fractal dimension can be obtained from the sequence of the periods referred above. Moreover there exists also a self-similar structure subjacent to the sequence Ln, Ln-1, …Lk, Lm. Consider the mth order term. For the case where the initial displacements are generated by a concentrated coupled, the period of free vibration is given by: Tm2 = ( m0 h02 Lm EI 0 ) ( λn Lm ) N m
(8)
Note that for this case the length of the elementary segment is λn corresponding to the term of order n. Equation (8) can be rewritten: Tm2 = ( m0 h02 Ln EI 0 ) ( N n λn Ln )( N m N n ) = Tn2 ( N m N n )
(9)
Now Lm is a subset of the original set Ln. The subset Lm is scaled relatively to Ln such that Lm = bmLn. We introduce now the following proposition: ….
Ln , Tn
….
Lk, Tk
Lm, Tm
Figure 6. Self-similar sequence of elastic springs representing pieces Ln, Lk, Lm of a fractal curve obtained by cutting off a segment out of the previous one.
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Proposition II. If Lm is the horizontal projection of a sample removed from a fractal curve whose horizontal projection is Ln , and if Lm = bmLn then Nm = (bm)D Nn provided that the curve family belongs to the class Nk = pk and λk = L0 /qk. Introducing this relation in (9) we obtain after some simple operations: log (Tm Tn ) = ( D 2 ) log ( bm )
(10)
With a similar procedure it is possible to show that for initial displacements generated by a horizontal or a vertical force we have: log (Tm Tn ) = (1 + D 2) log bm + (1 2 ) log ( Rm , n )
(11)
Where Rm ,n = ( Ω m L2n ) ( Ω n L2m ) . Now if Rm,n is close to unit the curve given by Equation (11) approaches a straight line with slope (1+D/2) from which it is possible to obtain D. From inspection of the parameter Rm,n it is possible to say that for short cuts, Lm<
Ln
Lj
Li
Lk
Figure 7. Example of a sequence of cuts used to infer the fractal dimension of the original curve Ln.
The results corresponding to the three initial conditions induced by a couple a horizontal force and vertical force are depicted in Figure 8-a and Figure 8-b. The reference term corresponds to the 9th term in the sequence and the cuts are such that two consecutive subsets scale always with the same ratio, that is Lm+1 = bLm. The representative line of the dynamical dimension was obtained by the classical interpolation of a straight line that minimizes the square root deviation from the points obtained in the numerical experiment.
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The slope of such lines that we designate by Saver are shown in the tables below. Clearly it is seen that the nonlinear term Rm,n introduces a large perturbation for the case of initial conditions generated by a horizontal force. In general we may extract several samples with the same size from different locations of the reference curve. It can be shown that the dispersion of the normalized periods is very small for samples larger than 0.5Ln. The critical point here is not the location of the sample but its size. Anyway the first attempts to identify the fractal characteristics are encouraging.
j
Cuts L j = 1/4(1.3) , j=0,1,2,3,... Couple Horiz.force Vert.force
Saver
D
0.62707315 1.62220225 1.63113080
1.2541 1.2444 1.2622
Cuts L j =1/ 4(2.0)j , j=0,1,2,3,... Couple Horiz.force Vert. force
Saver 0.63263124 1.63515805 1.63107795
D 1.2652 1.2703 1.2621
Figure 8. Inverse problem. Koch triadic used as test sequence. (a) cuts down scaled with b=1/1.3. (b) cuts down scaled with b=1/2.
4.
RANDOM FRACTALS
In this section we will examine a simple case of random fractals using again the Koch triadic as the fundamental matrix [5]. The random character is very simple, nevertheless very illustrative of the power of the dynamical dimension. Recall that the Koch triadic sequence can be built up using each preceding term to derive the next one. Let us call Gk the generator of the Koch triadic. Now the randomness introduced here is simply to allow the orientation of the GK’s to be taken by chance when assembling each the term in the sequence. In contrast with the well organized deterministic Koch triadic the terms obtained are not strictly self-similar as shown in the Figure 9 below. The HausdorfF dimension for this fractal sequence is the same as that for the deterministic, well organized, self-similar, Koch triadic sequence, that
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is D ≈1,26186. The reason is that the outer measure of the cover for this sequence is undistinguishable from that used for the deterministic Koch triadic. Therefore the Hausdorff dimension doesn’t detect the randomness of this sequence. Turning now to the dynamical dimensional we have at least two possibilities. Consider firs the case of the initial conditions generated by a couple. In that case the strain energy is the same for all elementary segments of a given term k. Therefore the strain energy corresponding to any two different terms, k and k+n scale exactly as determined by the Hausdorff dimension. Consequently the result obtained with the initial displacements induced by the couple should lead to the Hausdorff dimension. This is clearly apparent from the Figure 10. Six random sequences were generated independently. The straight lines representing the logarithm of the normalized periods versus the logarithm of the elementary segment lengths, all of them coincide. The straight line slope leads to the value of the Hausdorff dimension up to the fifth decimal digit (Figure 10-a). Now if we plot the normalized periods corresponding to the initial displacements induced by the horizontal force, the points are not aligned (Figure 10-b). The reason is that the strain energy doesn’t scale as required by the Hausdorff theory. In this case even small deviation from the classical deterministic triadic is detected by the perturbation in the energy distribution.
Generator G1 : Koch Triadic
Figure 9. Example of three first terms of a random fractal generated by the Koch triadic.
Figure 10. Numerial experiment with six random Koch triadi.
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5.
CONCLUSIONS
As far as we know, the concept of dynamical fractal dimension has not been explored before. This first attempt has shown to be encouraging. The method is very rich in information because of the sensitiveness to the initial conditions among others still hidden characteristics. If this property introduces some perturbations as for the case of the identification problem on the other hand it provides meaningful information for random fractals. The investigation is only beginning and we think that this approach opens a very promising research field. The response of structures with several degrees of freedom, varying geometric and mechanical properties of the oscillators, exploring space curves and multi-fractals, renormalization, uniqueness of the dynamical fractal dimension, wave propagation, are only a few questions to be investigated. Applications can be found in several fields from biological membranes and fibers [6] till composite materials.
AKNOWLEDGEMENTS We would like to thanks the National Research Council – CNPq – whose support has been decisive to carry on this research line.
REFERENCES 1. 2. 3. 4.
5. 6.
Feder J. Fractals, Plenum Press, New York and London, 1988. Gouyet JF. Physics and Fractal Structures, Masson, Springer, NY 1996. Falconer K. Fractal Geometry: Mathematical Foundations and Applications, John Wiley&Sons, Chichester and New York, 1990. Bevilac1qua L, Barros MM. “Dynamical Properties of Fractal strucrure: a new approach to the fractal dimension”, to appear Journal of the Brazilian Society for Mechanical Sciences. Stuttgart University, Scientific/Educational MATLAB Database, University of Stuttgart, Germany, 2005. Bassingthwaighte L, Liebovitch LS, Bruce JW. Fractal Physiology, American Physiological Society, 1994.
RAY STABILITY FOR RANGE-DEPENDENT BACKGROUND SOUND SPEED PROFILES T. Bódai, A. J. Fenwick, M. Wiercigroch Centre for Applied Dynamics Research, Department of Engineering, Kings College, University of Aberdeen, Aberdeen, AB24 3UE, UK, E-mail:
[email protected]
Abstract:
We list several applications of underwater acoustics and discuss underwater sound propagation in the framework of ray theory. The concept of ray chaos is introduced. It is shown that Poincaré sections and Lyapunov exponents provide equivalent descriptions of simple environmental models. We consider a class of environments not previously studied – those where there are transitions between single- and double duct sound speed versus depth profiles. To examine the behaviour of propagation in transition regions the concept of launching basins is introduced, and its use in the design of system performance is indicated.
Key words:
Underwater acoustics, ray chaos, transitions, launching basins.
1.
INTRODUCTION
Many applications of underwater acoustics involve transmitting a signal in order to collect data to solve an inverse problem. These include: propagation measurements, to find out properties of the media, based on signals transmitted at one point and received at other locations; backscatter sensing, to detect, identify and locate underwater objects of interest, with transmitters and receivers at the same point. Examples of each type of application we cite are: - thermometry by means of acoustic tomography [7]; sonar for detecting underwater vehicles [1]. It is also worth mentioning that passive acoustics may be used to monitor and classify underwater sound sources. An example of this is monitoring of underwater geological activities to predict possible natural disasters [4]. 137 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 137–146. © 2007 Springer.
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The different applications are associated with models appropriate to different configurations of environment, measurement equipment, etc. Therefore, a characteristic of a component might affect the outcome differently in different applications. An example is the ocean acoustic medium and the characteristic of interest is the perturbation of the sound speed structure. To examine the effect of environmental conditions on sound propagation, this study uses ray theory, an approach which has proven its worth. It has been established that internal waves and similar oceanographic noise phenomena, which constitute a perturbation of the sound speed structure, are responsible for the instability of acoustic rays [2]. It is believed that in several applications, instability of rays affects the efficiency of signal transmission. Knowledge of the mechanism complimented with the present environmental study can give guidance for the designing of underwater signal transmission systems. In the next section, the ray theory equations are introduced together with the particular sound speed profiles used to model propagation. In the third section several ways of checking stability are considered and applied. The fourth section describes the problem of transition, after which the idea of launching basins is developed followed by results. In the last section we summarize the paper.
2.
RAY THEORY
Sound propagation in the limit of small pressure perturbations is governed by the hyperbolic wave equation, whose equivalent in the frequency domain is the elliptic Helmholtz equation:
∆p + k 2 p = 0,
(1)
where k = ω/c is the wave number, ω is the frequency of the wave field, and c, the speed of sound, is a function of the spatial variables. In the case of duct propagation this equation may be replaced by the parabolic wave equation [10]. We consider only cylindrically symmetric ocean models [5]:
2ik 0 ∂ rψ + ∂ zzψ + k 02 ( n 2 − 1)ψ = 0
(2)
The parabolic equation is written for ψ, the envelope of pressure in the farfield (k0r >> 1): p =ψ
2 ei (k0r −π / 4) , π k0 r
(3)
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where k 0 = ω /c 0 is the reference wave number, c0 is an arbitrary constant reference sound speed, n = c0 /c is the index of refraction. The solution ∞ may be expressed in the form of a ray series, ψ = exp(iωt) j =0 A j /(iω ) j. Taking the terms of the leading order in ω yields the eikonal equation:
∑
∂ rτ +
c0 1− n2 (∂ zτ ) 2 + =0 2 2c 0
(4)
Reference [6] shows that there is a corresponding Hamiltonian system of equations for the rays given by the canonical form,
z′ = ∂ p H ,
p ′ = −∂ z H ,
(5)
where the Hamiltonian is
c0 2 1 − n 2 H= p + . 2 2c0
(6)
According to the mechanical analogy the first- and second terms are related to the kinetic and potential energy, respectively. The pair of canonical variables are depth z, and the vertical slowness p = ∂zτ. The independent variable is range, which plays a similar role to time in dynamical systems. The connection of the vertical slowness with the ray angle is given by the formula c0p = tan(φ) .
Figure 1. Examples of ray paths and ray trajectories with both Munk- and Double duct profiles, shown on the left. Initial conditions were: (z0,φ0) = (1 km,–5° ) – Munk, (z0,φ0 ) = (1.3 km,–5° ) – Double duct. Perturbation parameters were: A = 0.01, R = 10 km – unchanged for all computations
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The parabolic approximation is valid for small ray angles and small deviation of c from c0. In that case, the potential function can be linearised about c0 with respect to the normalised sound speed C = n–1: V = [1 – C –2] /2. The solutions of the ray equations are governed by the sound speed structure. A stratified ocean can be described by a single sound speed versus depth profile. One which is taken to represent the conditions in much of the world’s oceans is the Munk profile [8], given by
C M = 1 + ε (e −η + η − 1),
(7)
where η = 2(z – za)/B. As shown in Figure 1 this describes a single duct wave guide created by a minimum in the sound speed profile at z = za. Constants that we take here are c0 = 1.49 km/s, ε = 0.0057, za = 1 km and B = 1 km. The double duct profile we use was taken in the eastern North Atlantic [11]: CDd
4 ⎧ i z < 1.5 ⎪ ∑ ai z = ⎨ i =0 ⎪b + c( z − 1.5) z > 1.5 ⎩
(8)
Coefficients are a0 = 1.49323, a1 = 0.0471063, a2 = 0.147473, a3 = 0.145517, a4 = 0.045226, b = 1.49222, c = 0.023624. With the model of a stratified ocean, the ray system is separable, which in the above cases allows for analytic solution by integration. If the sound speed is range dependent, this is not the case. Range dependency is the true character of the ocean, and leads to substantial differences in propagation behaviour. Range dependency converts the autonomous 2D system (5) into a driven system and is a necessary condition for ray chaos [9]. The simplest way to take variability in range into account is given by the following sound speed perturbation [11]:
2z − ⎛ 2πr ⎞ δC = A e B sin ⎜ ⎟ B ⎝ R ⎠ 2z
(9)
The perturbation is superimposed onto the background, e.g. C = CM + δC. Equation (9) is a single mode approximation of an internal wave field which is periodic in range and decays exponentially with depth. The perturbation strength is controlled by A; R is the internal wave length. In Figure 1 examples of ray paths and ray trajectories are shown, with each of the profiles introduced above, shown on the left.
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STABILITY
One benefit of having a model with a simple harmonic perturbation is that we can construct the stroboscopic Poincaré section of the ray trajectories. The trajectory is sampled at ranges which are multiples of the wavelength of the harmonic perturbation [3]. The associated Poincaré return map, which converts one stroboscopic point to the next, is unique and area preserving. As is known for Hamiltonian systems, trajectories of the Poincaré return map that form an invariant curve or fill an area are respectively associated with regular or irregular motion. On a more basic level, regularity is associated with the number of integrals of the motion, which also has an implication for stability of trajectories. Namely, regular trajectories are stable; irregular trajectories are unstable or chaotic. By this, the stability of trajectories can be explored graphically by constructing their Poincaré sections [6]. While the graphical technique is generally applicable, it is central in studying Hamiltonian systems. The reason is that in such systems there are no attractors, and hence there is much more to explore, the myriad trajectories with distinct Poincaré sections. Figure 2 illustrates this point showing portraits of Poincaré sections for each profile. With a series of initial conditions the Poincaré sections can cover the phase space to a degree so that they map out the stable and unstable regions. Stability can be analysed numerically as well. First we present the fundamental ideas, and then cite a simplified calculation. Let us consider the dynamical system in its canonical form,
x ′( x0 , r ) = f ( x, r ).
(10)
The solution x(x0,r) ∈ R n is now the function of range as well as the initial values. The prime denotes partial differentiation with respect to range. We introduce the deformation gradient as
X = ∂ x0 x.
(11)
To analyse the divergence of nearby trajectories two different versions of the variational equations can be derived for the deformation gradient X and the perturbation δx:
∂ r X = ∂ x f ( x, r ) ⋅ X , ∂ r δx = ∂ x f ( x, r ) ⋅ δx
(12,13)
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Figure 2. Portraits of Poincaré sections with Munk- (left) and Double duct profiles (right). Initial conditions were: z0 = 1 km, –φ0 = 1,…,12° – Munk, z0 = 1.3 km, –φ0 = 1,…,15°, z0 = 0.35 km, φ0 = –0.5, 0°, z0 = 0.325 km, φ0 = –7, –7.5, –8° and z0 = 0.2 km, φ0 = 1,2° – Double duct
Equation (12) is to define the Lyapunov exponents, Equation (13) will be applied at a simplified calculation. The connection between the initial and actual values of the perturbation,
δx = X ⋅ δx0 ,
(14)
is a linear map. Long range instability is analysed via the singular values of the deformation gradient si, in terms of the Lyapunov exponents
ln(si ) . r →∞ r
ν i = lim
(15)
The singular values si are the square roots of the real eigenvalues of the matrix X ⋅ X T, which are always nonnegative. For one degree of freedom Hamiltonian systems the product of singular values is always unity, therefore the Lyapunov exponents come in a plus-minus pair. That is, the greater is either zero or positive, denoting stable or chaotic motion. In practice, it is not feasible to calculate the Lyapunov exponents by the above formula, but different numerical algorithms may be employed. One algorithm [12] that has been used in case of ray systems will now be discussed. Equation (13) is integrated to find the perturbation, by which the distance between reference and perturbed trajectories are approximated by the following measure: d = |δz| + |δp|. Initial conditions are chosen arbitrarily,
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but so that d(0) = 1. The greater Lyapunov exponent is approximated by taking the stability exponent
νL =
ln(d ) r
(16)
at some sufficiently large r. The calculation might need to be reinitialized to avoid overflow problems. Nondimensionalization of depth and range was achieved dividing by the internal wavelength. Based on this measure we can develop a method to explore the space of initial conditions with respect to stability by numerical means. This is done by estimating the larger Lyapunov exponents associated with a grid of initial conditions. Figure 3 presents these results. The equations were integrated over a range of 5000 km. Reflection of rays from a flat sea surface and sea bottom was taken into account by reversing the ray angle; the ranges of the points of reflection were determined exactly employing a trick devised by Hénon [13]. The integration was carried out using fourth order Runge-Kutta integrator with a fixed step size of 0.1 km.
4.
TRANSITIONS
As previously noted the sound speed profile in much of the world’s oceans is considered to be well approximated by a Munk profile. There are locations, however, where the sound speed profile has two minima instead of just one. Between these profiles there is a smooth transition. Transition between Munk- and Double duct background profiles is modeled in this paper by the following scheme:
1 C bg = [C M + C Dd − tanh(a (r − rt ))(C M − C Dd )] 2
(17)
The parameters a and rt of the hyperbolic tangent function are interpreted as the rate and the midrange of the transition, respectively. Clearly, for values of it close to ±1, which occur at ranges much greater or smaller than rt, the background profile is either the Double duct or the Munk profile. That is, a certain ray passing through a transition will have two segments of significantly different propagation dynamics. Having seen the different stability character of the two profiles above (Figure 3), it is clear that some rays will retain the same stability, i.e. remain stable or unstable, but there are others which switch stability. Figure 4 demonstrates in two ways what happens when an initially stable ray becomes unstable after transition. On the left it is indicated by the divergence of initially closely spaced rays, and
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on the right – by the area filling Poincaré section of the post-transition ray trajectory segment. The larger Lyapunov exponent is not calculated since there is no meaning for a range average taken over the transition region. When there is no transition, either the Poincaré sections or the map of the larger Lyapunov exponents can be used to define those depths and ray angles for which rays launched with those parameters are stable. These regions in the case of either the Munk- or the Double duct profile are enclosed by Poincaré sections of chaotic rays. The space of initial conditions is thus divided into two regions. This provides guidance as to desirable depths and beam-widths for transmitting and receiving systems. When there is a transition, we are interested in the same question. Where are the points in the plane of initial conditions from where rays launched are stable even after transition, i.e. points associated with stable-to-stable matches of ray segments? The regions associated with various types of behaviour, e.g. stable-to-stable matches of ray segments etc., can be referred to as launching basins, and they define a partition of the space of initial conditions. The launching basins can again be used to guide the design of long range sonar systems. It is not straightforward to infer the launching basins graphically. Instead, we calculate a table of the larger Lyapunov exponents for the post-transition segments of rays, as was done in the end of the previous section for single profiles, and project these values to the plane of initial conditions before transition by backward ray tracing. Figure 5.a shows results of these calculations for the transition scenario from Figure 4. Stability of pretransition ray segments, on the other hand, is already assessed and results are
Figure 3. Maps of the larger Lyapunov exponent for the two profiles, with resolution given by a grid of 200×200 initial conditions in a regime: 0 < z0 < 4 km and –15° < φ0 < 15°
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presented on the left in Figure 3. Concerning the four types of behaviour in case of a single transition, information on stability of pre- and post-transition ray segments is retained in a qualitative sense. This is done by setting up a threshold when any ray with a greater Lyapunov exponent below or above this threshold is regarded stable or unstable. The threshold here set is 0.05 1/10km. Hence, two shades are retained in each of the two latter mentioned figures, which combination in a single figure draws out the launching basins. Figure 5.b shows the launching basins for the discussed transition scenario.
Figure 4. Initially stable ray gets instable after transition. Initial conditions are the same as in Figure 1; perturbation is achieved by 0.001° and 0.002° differences in take-off angle. The parameters of the transition scenario were chosen to be a = 0.05 and rt = 204 km
Figure 5. Map of larger Lyapunov exponents for Double duct profile projected to the plane of initial conditions (left), and launching basins (right)
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CONCLUSIONS
Through the connection with nonlinear dynamics ray theory has been shown to provide a powerful means of investigating underwater acoustic propagation phenomena.
ACKNOWLEDGEMENT The second author would like to thank the Royal Academy of Engineering for supporting his secondment to Aberdeen from QinetiQ within the framework of the Industry to academia Fellowship Scheme. The PhD studentship awarded to the first author by the College of Physical Sciences, the University of Aberdeen is gratefully acknowledged.
REFERENCES 1. 2.
3. 4. 5. 6. 7. 8. 9. 10. 11.
12. 13.
Balanov A, Janson N, Wang C, Wiercigroch M. “Multiple delay differential systems in a sensing problem”, Technical report, 2002. Brown MG, Colosi JA, Tomsovic S, Virovlyansky AL, Wolfson MA, Zaslavsky GM. “Ray dynamics in long-range deep ocean sound propagation”, J. Acoust. Soc. Am., 113, pp. 2533–2547, May 2003. Guckenheimer J, Holmes P. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. Howe BM. Acoustics, ocean observatories, and the future. Jensen FB, Kuperman WA, Porter MB, Schmidt H. Computational Ocean Acoustics, Springer-Verlag, New York, 2000. Lichtenberg AJ, Lieberman MA. Regular and stochastic motion, Springer-Verlag, New York, 1982. Munk W, Worcester P, Wunsch C. Ocean Acoustic Tomography, Cambridge University Press, 1995. Munk W. “Sound channel in an exponentially stratified ocean, with application to sofar”, J. Acoust. Soc. Am., 55, pp. 220–226, February 1974. Palmer DR, Brown MG, Tappert FD, Bezdek HF. “Classical chaos in nonseparable wave propagation problems”, Geophysical Research Letters, 15, pp. 569–572, 1988. Tappert FD. The parabolic approximation method, volume 70 of Wave Propagation and Underwater Acoustics, chapter 5, pp. 224–287. Springer Berlin, Heidelberg, 1977. Wiercigroch M, Cheng AHD, Simmen J, Badiey M. “Nonlinear behavior of acoustic rays in underwater sound channels”, Chaos, Solitons, & Fractals, 9, pp. 193–207, 1998/1/2. Smith KB, Brown MG, Tappert FD. “Ray chaos in underwater acoustics”, J. Acoust. Soc. Am., 91, pp. 1939–1949, 1992. Hénon M. “On the numerical computation of Poincaré maps”, Physica D: Nonlinear Phenomena, 5, pp. 412-414, 1982/9.
APPLICATION OF EXTENDED AVERAGED EQUATIONS TO NONLINEAR VIBRATION ANALYSIS N. D. Anh1, N. Q. Hai2, W. Schiehlen3 1
Institute of Mechanics, Vietnamese Academy of Science and Technology, Hanoi, Vietnam Hanoi Architectural University, Hanoi, Vietnam 3 Institute of Eng. Num. Mechanics, University of Stuttgart, 70550 Stuttgart, Germany 2
Abstract:
The paper presents an extended averaged equation approach to the investigation of nonlinear vibration problems. The proposed method is applied to a nonlinear suspension system with two-degree-of-freedom. The results in analyzing the oscillations with arbitrary values of the nonlinearity show the efficiency of the method.
Key words:
Extended averaged equation, suspension system, cubic nonlinearity.
1.
INTRODUCTION
Research on vibration phenomena in nonlinear systems with an aim to reduce undesired oscillations has a long tradition, see e.g., Schmidt and Tondl [1]. Recently, great interest of researchers is devoted to new methods for investigating nonlinear vibrations preferably applicable to wider classes of nonlinear systems including weak and strong nonlinearity, subject to deterministic and/or random excitations, see e.g., Roberts and Spanos [2], Anh and Schiehlen [3]. The method of moment equations is well known for analysis of random nonlinear vibration phenomena and gives also good approximate solutions for systems with strong nonlinearity, Anh and Hai [5-6]. The question is whether the method can be extended to deterministic systems. This method related to the classical averaging method attributed to Bogoliubov and Mitropolsky [4] was recently extended to deterministic vibration systems by Anh, Hai and Schiehlen [7]. The main idea of the 147 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 147–156. © 2007 Springer.
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proposed method is that such solutions of the original nonlinear system are searched in the form of a function of the corresponding linear system solution which has the expected properties of the solution of the nonlinear system. In this paper, engineering applications of the general extended averaged (GEA) equation are presented. For illustration, a nonlinear suspension system with two-degree-of-freedom is investigated showing the efficiency of the proposed approach.
2.
GEA EQUATION FOR DETERMINISTIC SYSTEMS Consider a nonlinear system with harmonic excitation
&Z = F ( Z ) + P cosν t, Z = ( z , z ,..., z ) T 1 2 n
(1)
where F ( Z ) = ( F1 ( Z ), F2 ( Z ),..., Fn ( Z ) ) T, and P = [ p1 , p2 ,..., pn ] T is a constant vector. Furthermore, a corresponding linear system is introduced X& = AX + P cosν t , X = ( x1 , x2 ,..., xn )T, A = { aij } (i, j = 1,2,...,n),
(2)
where A is a constant matrix. In general, the matrix A is fully unknown. The term “corresponding” means the nonlinear and linear systems have the same dimension, i.e, the numbers of components of the vectors Z and X are equal to each other. Furthermore, the linear system 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 [7]. If the nonlinear response is based on the linear one, it can be expressed in a form of polynomial up to an order N as follows: N
N
N
zi = xi + ∑ ∑ ... ∑ α k1k2 ...kn x1k1 x2k2 ...xnkn , k1 + k2 + ... + kn ≥ 2. k1 = 0 k2 = 0
(3)
kn = 0
The problem is reduced to the problem of determining X(t) or the matrix A, respectively, and the parameters α k1k2 ...kn . One possibility is to apply GEA equations. For an arbitrary differentiable function Ψ(t, Z, X) one gets n ⎞ d Ψ ∂Ψ n ∂Ψ ∂Ψ ⎛ n = +∑ ( Fi ( Z ) + pi cosν t ) + ∑ ⎜ ∑ aij x j + pi cosν t ⎟ (4) dt ∂t i =1 ∂zi i =1 ∂xi ⎝ j =1 ⎠
Denoting the averaging operator as T
1 (.)dt. T →+∞ T ∫ 0
< . > = lim
(5)
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and supposing the averaged value of Ψ doesn’t depend on the time, one gets, see Anh, Hai and Schiehlen [7] n
∑ i =1
n ⎞ ∂Ψ ∂Ψ ⎛ n ( Fi ( Z ) + pi cosν t ) + ∑ ⎜ ∑ a ij x j + p i cosν t ⎟ = 0 x ∂zi ∂ i =1 i ⎝ j =1 ⎠
(6)
In principal the function Ψ can be arbitrary, however, in order to get in (6) the possible lowest unknown moments we consider a form Ψ = g ksrq (t ) zkm zsd xrp xql , where m, d, p, l = 0, 1, 2, ... and g ksrq (t ) are functions of t, respectively. Equation (6) could be referred to as a GEA equation, which is similar to the moment equations in the theory of random vibrations, where the averaging operator is taken in the probabilistic sense, see e.g., Roberts and Spanos [2], Anh and Schiehlen [3], Anh and Hai [5-6]. For the exact solution, all extended averaged equations have to be satisfied. One might expect that the accuracy of the technique may be better if more averaged equations are satisfied. However, to obtain a closed equation system, the number of equations will be taken equal to the number of unknowns. The proposed method has been applied to some benchmarks with convincing accuracy checked by numerical simulation, Anh, Hai and Schiehlen [7]. In the following section, for illustration, some vibrations of an engineering system are investigated showing the efficiency of the approach.
3.
ANALYZING NONLINEAR OSCILLATIONS
3.1 Free oscillation of a nonlinear suspension system Consider the free oscillation of a suspension system, Mueller, Popp and Schiehlen [8], Roseau [9] which is represented schematically in Figure 1 by two bodies of mass m1 and m2 linked with each other by a nonlinear spring (k3), a linear one (k1) and a shock damper with viscous damping (d1). Mass m2 is contacting with the ground through a linear spring (k2). The free vibration without damping d1=0 is governed by nonlinear equations as follows && z1 = b11 z1 + b12 z2 + b13 z13 , && z2 = b21 z1 + b22 z2 + b23 z13 ,
(7)
k k1 k , ω 22 = 2 , β = 3 , b11 = −ω12 (1 + µ ), m1 m2 m1 b12 = ω 22 , b13 = − β (1 + µ ), b21 = ω12 µ , b22 = − ω 22 , b23 = βµ , µ = m1 / m 2 .
(8)
where it is denoted z1 = y1 − y 2 , z 2 = y 2 , ω12 =
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y1 (t) d1
k3 , k1
y2(t) m2
k2
ξ(t) = pcosνt
Figure 1. Nonlinear suspension system
The linear system corresponding to the nonlinear one (7) has the form && x1 + a11 x1 + a12 x2 = 0, && x2 + a21 x1 + a22 x2 = 0,
(9)
a11 = ω12 (1 + µ )(1 + ε ), a12 = − ω22 , a21 = −ω12 µ (1 + ε ), a22 = ω22 ,
(10)
here ε is the frequency correction due to nonlinear terms. The general solution of the system (9) has the form x1 = c1r1 sin(ν 1t + θ1 ) + c2 r2 sin(ν 2 t + θ 2 ), x2 = c1 sin(ν 1t + θ1 ) + c2 sin(ν 2 t + θ 2 ),
(11)
here (12) r1 = a12 /(ν 1 − a11 ), r2 = a12 /(ν 2 − a11 ). The frequencies ν 1 and ν 2 are determined from the characteristic equation
(a
11
−ν 2 )
a21
a12 =0 a ( 22 −ν 2 )
(13)
The constants c1, c2 and phase values θ1 ,θ 2 are determined by initial conditions of the free oscillation, e.g. x1 (0) =1, x&1 (0) =1, x2 (0) = 1, x&2 (0) = 1.
(14)
The initial conditions (14) allow to represent c1, c2, ν 1 ,ν 2 as functions of ε. Now, one takes the response of Equation (7) in the form z1 (t ) = x1 (t ) + α1 x13 (t ), z2 (t ) = x2 (t ) + α 2 x23 (t )
(15)
Consequently, one has got 3 unknowns ε , α1 ,α 2 which can be found from the GEA equation which for an arbitrary differentiable function Ψ ( X , Z ) = Ψ ( x1 , x&1 , x2 , x&2 , z1 , z&1 , z2 , z&2 ) takes the form
Application of extended averaged equations to nonlinear analysis ∂Ψ ∂Ψ ∂Ψ ∂Ψ x&1 + f1 + x&2 + f2 + ∂x1 ∂x&1 ∂x2 ∂x&2 ∂Ψ ∂Ψ ∂Ψ ∂Ψ + z&1 + f3 + z&2 + f 4 = 0, ∂z1 ∂z&1 ∂z2 ∂z&2
151
(16)
where f1 = − a11 x1 − a12 x2 , f 2 = − a21 x1 − a22 x2 , f3 = b11 z1 + b12 z2 + b13 z13 , f 4 = b21 z1 + b22 z2 + b23 z13 .
(17)
In order to get the lowest unknown moments we take the lowest polynomial functions Ψ ( X , Z ) and from (16), one gets: for Ψ = x1 z&2 : < x&1 z&2 > + < 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 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
Constrained bifurcation analysis (I) Y1 ( y, λ ; α% ) = 0,
163
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.
Constrained bifurcation analysis
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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K. Yunt, C. Glocker
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]
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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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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 ) =
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n =0
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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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5.
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)<0.001). Consider only the bending vibration described by the transverse displacement U(X, T), where T is the time and X is the axial coordinate. The axial speed is assumed to be a small simple harmonic variation about the constant mean speed, v (T ) = v0 + ε v1 sin ΩT
(1)
where ε is a small dimensionless parameter. Under the quasi-static stretch assumption, the governing equation takes the dimensionless form [6] 1
1 u ,tt +2γ 0u ,t + ( γ 0 2 − 1) u , xx + vf 2u , xxxx = ε k12u , xx ∫ u , x 2 d x − 2εγ 1 sin ωtu , xt (2) 2 0 − 2εγ 0γ 1 sin ωtu , xx −εωγ 1 cos ωtu , x −εα u , xxxxt +O ( ε 2 )
where the dimensionless variables and parameters are respectively defined as P0 X ρA ρA ,t =T , γ 0 = v0 , γ 1 = v0 2 L P0 P0 ρ AL εL EI I EA ρ AL2 η Ω, kf 2 = ,α = 3 , k1 = ω= P0 P0 L2 P0 ε L ρ AP0 u=
U
,x=
(3)
For two ends with the same torsion spring stiffness, setting the transverse displacements to zero and balancing the bending moment at both ends lead to the boundary conditions in the dimensionless form u ( 0, t ) = u (1, t ) = 0, u , xx ( 0, t ) − ku , x ( 0, t ) = u , xx (1, t ) + ku , x (1, t ) = 0
(4)
Parametric resonance of an axially accelerating viscoelastic beam
3.
219
MULTI-SCALE ANALYSIS
The method of multiple scales will be employed to solve Equation (1). A first order uniform approximation is sought in the form u ( x, t ; ε ) = u0 ( x, T0 , T1 ) + ε u1 ( x, T0 , T1 ) + O ( ε 2 )
(5)
Substitution of Equation (5) into Equation (1) and then equalization of coefficients of ε0 and ε in the resulting equation lead to u0 ,tt +2γ 0u0 ,t + ( γ 0 2 − 1) u0 , xx + vf 2u0 , xxxx = 0
(6)
1
1 u1 ,tt +2γ 0u1 ,t + ( γ 0 − 1) u1 , xx + vf u1 , xxxx = k12u0 , xx ∫ u0 , x 2 d x − 2u0 ,T0T1 (7) 2 0 − 2γ 0u0 , xT1 −2γ 1 sin ωT0 u0 , xT0 +γ 0 u0 , xx − γ 1ω cos ωT0 u0 , x −α u0 , xxxxT0 2
2
(
)
The solution to Equation (6) has been given by Wickert and Mote [7] as ∞
u0 ( x, T0 , T1 ) = ∑ ⎡⎣φn ( x) An (T1 ) ei ωnT0 + φn ( x) An (T1 ) e − i ωnT0 ⎤⎦
(8)
n =1
where ωn and φn are respectively the n-th natural frequency and complex mode function. Under conditions (4), the mode function is [8]
φn ( x) = C1n {ei β x − i k ( ei β + ei β ) ( β1n − β3n ) + ( ei β − ei β ) ⎡⎣k2 + ( β1n + β4n )( β3n + β4n ) ⎤⎦ ( β1n − β4n ) iβ x e − i k ( ei β + ei β ) ( β2n − β3n ) + ( ei β − ei β ) ⎡⎣k2 + ( β2n + β4n )( β3n + β4n ) ⎤⎦ ( β2n − β4n ) i k ( ei β + ei β ) ( β1n − β2n ) + ( ei β − ei β ) ⎡⎣k2 + ( β1n + β4n )( β2n + β4n ) ⎤⎦ ( β1n − β4n ) i β x (9) e + i k ( ei β + ei β ) ( β2n − β3n ) + ( ei β − ei β ) ⎡⎣k2 + ( β3n + β4n )( β2n + β4n ) ⎤⎦ ( β3n − β4n ) ⎡ i k ( ei β + ei β ) ( β1n − β3n ) + ( ei β − ei β ) ⎡k2 + ( β1n + β4n )( β3n + β4n )⎤ ( β − β ) ⎣ ⎦ 1n 4n 1n
1n
3n
1n
3n
2n
3n
2n
3n
1n
2n
1n
2n
2n
3n
2n
3n
2n
3n
1n
3n
1n
3n
⎢ −1+ ⎢⎣i k ei β2n + ei β3n ( β2n − β3n ) + ei β2n − ei β3n ⎡⎣k2 + ( β2n + β4n )( β3n + β4n ) ⎤⎦ ( β2n − β4n ) i k ei β1n + ei β2n ( β1n − β2n ) + ei β1n − ei β2n ⎣⎡k2 + ( β1n + β4n )( β2n + β4n )⎦⎤ ( β1n − β4n ) ⎤ i β x ⎪⎫ ⎥ e 4n ⎬ i k ei β2n + ei β3n ( β2n − β3n ) + ei β2n − ei β3n ⎡⎣k2 + ( β3n + β4n )( β2n + β4n ) ⎤⎦ ( β3n − β4n ) ⎥⎦ ⎭⎪
( ( (
) )
)
( (
(
) )
)
where βjn ( j=1,2,3,4) are four roots of the equation
kf4 β jn4 + (1 − γ 2 ) β jn2 − 2ωn β jn − ωn2 = 0
(10)
that can be solved with the frequency equation e ( 1n 2 n ) ( β1n − β2n )( − i k + β1n + β2n )( β3n − β4n )( i k + β3n + β4n ) + e ( 1n 3n ) ( β3n i β +β −β1n )( − i k + β1n + β3n )( β2n − β4n )( i k + β2n + β4n ) + e ( 1n 4 n ) ( β2n − β3n )( i k + β2n i( β2 n + β3 n ) +β3n )( β1n − β4n )( − i k + β1n + β4n ) + e ( β2n − β3n )( − i k + β2n + β3n )( β1n (11) i ( β 2 n + β4 n ) −β4n )( i k + β1n + β4n ) + e ( β3n − β1n ) ( i k + β3n + β1n )( β2n − β4n )( − i k + β2n i( β3 n + β4 n ) + β4 n ) + e ( β1n − β2n )( i k + β1n + β2n )( β3n − β4n )( − i k + β3n + β4n ) = 0 i β +β
i β +β
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If the variation frequency ω approaches two times of any natural frequency of the system (6), principal parametric resonance may occur. To explore the n-th principal resonance, it does not lose generality for u0 to include only the n-th mode vibration u0 ( x, T0 , T1 ) = φn ( x) An (T1 ) ei ωnT0 + cc
(12)
where cc stands for the complex conjugate of all preceding terms on the right hand of an equation. A detuning parameter σ is introduced to quantify the deviation of ω from 2ωn and ω is described by ω = 2ωn + εσ (13) Substitution of Equations (12) and (13) into Equation (14) yields u1 ,tt +2γ 0u1 ,t + ( γ 0 2 − 1) u1 , xx +vf 2u1 , xxxx = ⎡⎣ −2 An ( i ωnφn + γ 0φn′ ) + i γ 0γ 1φn′′φn ei σ T1 1 (14) ⎤ ⎛1 1 ⎞ − i αωn Anφn′′′′+ k12 ⎜ φn′′∫ φn′2 d x + φn′′∫ φn′φn′ d x ⎟ An2 An ⎥ ei ωnT0 + cc + NST 0 ⎝2 0 ⎠ ⎦⎥
where the dot and the prime denote derivation with respect to the slow time variable T1 and the dimensionless spatial variable x respectively, and NST stands for the terms that will not bring secular terms into the solution. Equation (14) has a bounded solution only if a solvability condition holds. The solvability condition demands that the possible secular term coefficient at right hand of the Equation (14) be orthogonal to every solution of the homogeneous problem. That is − 2 An ( i ω nφ n + γ 0φ n′ ) + i γ 0 γ 1φ n′′φ n e i σ T1 − i αω n Anφ n′′′′+ 1 ⎛1 1 ⎞ k12 ⎜ φ n′′∫ φ n′ 2 d x + φ n′′∫ φ n′φ n′ d x ⎟ An2 An , φ n = 0 ⎜2 ⎟ 0 0 n ⎠ ⎝
(15)
where the inner product is defined for complex functions f and g on [0,1] as 1
f , g = ∫ f ( x) g ( x)d x
(16)
0
Application of the distributive law of the inner product to Equation (15) leads to
An + αµn An + γ 1 χ n Am ei σ T1 + k12κ n An2 An = 0
(17)
where 1
µn =
1
−
1
i ω n ∫ φ nφ n d x + γ 0 ∫ φ n′φ n d x 1
κn =
1
1 i ω n ∫ φ n′′′′φ n d x 2 0 0
1
0
1
, χn =
1 i γ 0 ∫ φ n′′φ n d x 2 0
1
1
0
0
i ω n ∫ φ nφ n d x + γ 0 ∫ φ n′φ n d x
1
1 1 φ nφ n′′ d x ∫ φ n′ 2 d x + ∫ φ n′φ n′ d x ∫ φ nφ n′′ d x ∫ 40 20 0 0 1
1
0
0
i ω n ∫ φ nφ n d x + γ 0 ∫ φ nφ n′ d x
(18)
Parametric resonance of an axially accelerating viscoelastic beam
4.
221
STEADY-STATE RESPONSE Express the solution to Equation (17) in polar form An (T1 ) = an (T1 ) ei ϕn (T1 )
(19)
Here an(T1) and ϕn(T1) are respectively the amplitude and the phase angle of the response in the n-th principal parametric resonance. For mode functions given by Equation (9), it can be numerically verified that Re ( µn ) > 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.
225
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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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.
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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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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
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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
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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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(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
14
16
18
λ/π
20
22
24
26
28
30
∞ ∞ (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=λ1<0,
α4= d12,1 / 2 >0,
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 λ1<0 and λn>0, n≥2, [8] and thus proves that α6<0. The sign of α i, i ≥8, is not known in advance. The coefficients αi are at the base of reduced order (single d.o.f., indeed) models (ROMs), and αi, i ≥6, summarize the effects of non-flatness of the manifold where the homoclinic loops lie. In fact, they vanish for the first family of b.c., but not for the second [8]. The equation of motion associated with (8) is
x +α2x+2α4kx3+3α6k2x5+4α8k3x7+…=0,
(9)
while the homoclinic orbit of the hilltop saddle, whose determination is the main target of this section, can be obtained by solving
x = − α 2 x 2 − α 4 kx 4 − α 6 k 2 x 6 − α 8 k 2 x8 − ... .
(10)
The solution of (10) is simplified by the change of variables τ =2t√(–α2) and y=(–α4k/α2)x2, which is used in the following.
3.1 A hierarchy of ROMs Equations (8) and (9) provide a hierarchy of approximate ROMs, obtained by considering an increasing number l of higher order nonlinearities ai,l in the ith slave amplitude, as accounted for by the αi, i≥6:
x +α2x+2α4kx3=0, (unrefined ROM) 3 2 5 x +α2x+2α4kx +3α6k x =0, (first refined ROM) 3 2 5 3 7 x +α2x+2α4kx +3α6k x +4α8k x =0. (second refined ROM)
(11) (12) (13)
The first model is the well-known Duffing equation, which can also be obtained by the classical (linear) Galerkin projection of the dynamics onto the first linear mode. It provides an exact result for the first family of b.c., for which αi =0, i ≥6, whereas an unrefined minimal approximation for the second family, where minimal means the simplest ROM approximating the homoclinic orbit. The homoclinic loop of Equation (11) is y3,h(τ)=2/[1+cosh(τ)].
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For the second family of b.c. it is possible to improve the approximation -first refined ROM - by keeping α6 in Equation (9), thus obtaining the homoclinic solution y5,h(τ)=2/[1+βcosh(τ)], where β=(α42–4α2α6)1/2/α4. As α6 is negative, this model is softening, which appears inconsistent with the original system. However, for α6 sufficiently small (which occurs just above the buckling threshold), the softening “error” manifests itself only for large values of x, and far from the region occupied by the homoclinic loop. When |α6| increases, the escape region approaches the homoclinic orbit, and y5,h(τ) first is no longer a reliable approximation of the true homoclinic orbit and then disappears. Above this threshold, the ROM (12) is meaningless, and one must retain higher order terms. Yet, below the threshold, the present approximation can be practically used, although with a Γ-dependent degree of reliability. The possibility of having an analytical expression of the homoclinic orbit makes model (12) interesting in any case. To visualize the effects of the non-planarity, we have drawn in a 3D section of the phase space (Figure 1) the invariant manifold Σ with the homoclinic loop y5,h(τ), its projection on the plane (y, y ), and its approximation y3,h(τ).
Figure 1. The invariant manifold Σ, y5,h(τ) (very thick line), its projection on the plane (y, y ) (thick line) and its approximation y3,h(τ) (thin line) in the space (y, y , a2).
A second refined model is obtained by considering terms up to α8 in (9). It reproduces the hardening behaviour of the original system if α8>0, 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
-300
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)
Nonlinear normal modes of homoclinic orbits and chaos control
5.
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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
A. Teufel et al.
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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.
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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
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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
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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 ):
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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.
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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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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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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 << 1) on the unperturbed system (7).
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We analyze the dynamics of the perturbed system and the influence of small perturbations on M . Based on the analysis in references [6-9], we know that M along with its stable and unstable manifolds are invariant under small, sufficiently differentiable perturbations. It is noticed that q0 may persist under small perturbations, in particular, M → M ε . So, we obtain M = M ε = {(u , I , γ ) u = q0 , I1 ≤ I ≤ I 2 , 0 ≤ γ ≤ 2π } .
(15)
Considering the later two equations of system (5) yields
I = − µ I − f 2 sin γ , γ = σ 2 − η2 I 2 + β1u12 − f 2 I cos γ .
(16)
It is known from the aforementioned analysis that the last two equations of system (5) are of a pair of pure imaginary eigenvalues. Therefore, the resonance can occur in system (16). Also introduce the scale transformations T (17) µ → εµ , I = I r + ε h , f 2 → ε f 2 , T1 → 1 .
ε Substituting the above transformations into Equation (16) yields
h = − µ I r − f 2 sin γ − ε hµ , γ = −2η 2 I r h − ε ( f 2 I cos 2γ + η 2 h 2 ) . (18)
When ε = 0 , Equation (18) becomes h = − µ I r − f 2 sin γ , γ = −2η 2 I r h .
(19)
The unperturbed system (19) is a Hamiltonian system with the Hamiltonian function (20) Ηˆ D (h, γ ) = − µ I r γ + f 2 cos γ + η 2 I r h 2 . The singular points of system (19) are given as ⎛ ⎛ µI ⎞ µI ⎞ p0 = (0, γ c ) = ⎜ 0, − arcsin r ⎟ and q0 = (0, γ s ) = ⎜ 0, π + arcsin r ⎟ .(21) f2 ⎠ f2 ⎠ ⎝ ⎝ It is known that the singular point p0 is a center. The singular point q0 is a saddle which is connected to itself by a homoclinic orbit.
5.
THE k-PULSE MELNIKOV FUNCTION
We use the generalized Melnikov method to investigate the homoclinic orbits with several consecutive fast pulses rather than just one. Since the motion along the hyperbolic manifold is slow in this problem, this theory simplifies considerably due to the facts that the k -pulse Melnikov function does not depend on the small parameter ε , and that the non-folding condition is automatically satisfied and thus not needed.
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In order to show the existence of multi-pulse homoclinic orbits, it is important to obtain the expression of the k -pulse Melnikov function. Firstly, we compute 1-pulse Melnikov function at the resonance I = I r . The Melnikov function M ( I ,θ 0 , µ ) on both homoclinic manifolds W s ( M ) and W u ( M ) is equal to ∞ ∂H ∂H u2 ∂H I ∂H γ M ( I r , γ 0 , µ , β1 , δ1 , ε1 ) = ∫ ( g u1 + g + g + g )dT1 −∞ ∂u ∂u2 ∂I ∂γ 1 1
⎡ ε ε ⎤ 4µ 3 ε2 = − 2 ε12 + 4 β1µ I r2 1 − f 2 I r ⎢cos(γ 0 + 2 β1 1 ) − cos(γ 0 − 2 β1 1 ) ⎥ . (22) δ1 δ1 ⎥⎦ δ1 3δ1 ⎢⎣ Then, the k -pulse Melnikov function is obtained as ∆γ M k ( I r , γ 0 , µ , β1 , δ1 , ε1 ) = M k ( I r , γ k −1 − (k − 1) , µ , β1 , δ1 , ε1 ) 2 (23) 2µ ε 1 1 1 = 2 f 2 I r sin γ k −1 sin( k ∆γ ) − 2 1 ( k ∆γ ) + 2 µ I r2 ( k ∆γ ) 2 3β1 2 2
where ∆γ = 4 β1
ε1 ∆γ , γ k −1 = γ 0 + (k − 1) . 2 δ1
The non-folding condition can be written Dγ 0 M k ( I r , γ 0 , µ , β1 , δ1 , ε1 ) ≠ 0 .
(24)
We obtain two expressions as follows 1 k ∆γ ( µ2ε1 − 3β1µ I r2 ) 2 (25) <1, 1 3 β f I r 1 2 sin( k ∆γ ) 2 1 k ∆γ ≠ nπ , n = 0, ± 1, ± 2, . (26) 2 The main aim of the following research focuses on identifying simple zeroes of the k -pulse Melnikov function. Define a set that contains all such simple zeroes
{
}
Z −n = ( I r , γ k −1 , µ , β1 , δ1 , ε1 ) M k = 0, Dγ 0 M k ≠ 0 .
(27)
There are two simple zeroes of the k -pulse Melnikov function in the interval γ k −1 ∈ [ 0, π ] , that is 1 ( k ∆γ ) ( µ2ε1 − 3β1µ I r2 ) γ k −1,1 = arcsin 2 , (28) 1 (3β1 f 2 I r ) sin( k ∆γ ) 2 γ k −1, 2 = π − γ k −1,1 . (29)
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Then, the k -pulse Melnikov function (23) has simple zeroes in γ k −1 at some γ k −1 = γ k −1,1 and γ k −1 = γ k −1, 2 = π − γ k −1,1 .
6.
NUMERICAL RESULTS OF CHAOTIC MOTIONS
We choose averaged Equation (4) to do numerical simulations. Numerical approach is utilized to explore the existence of the multi-pulse chaotic motions for the nonlinear nonplanar oscillations of a cantilever beam. In Figure 2, the chaotic response of the cantilever beam is discovered when we choose the parametric excitation, transverse excitation in the z direction, parameters and initial conditions as F1 = 48.4 , f 2 = 198.4 , c = 0.2 , σ 1 = 11.4 , σ 2 = 6.8 , α1 = 1.0 , α 2 = −4.0 , α 3 = 1.1 , β1 = 4.9 , β 2 = −3.9 , x10 = 3.1385 , x20 = 4.45 , x30 = 4.35 , x40 = 5.16 .
x3
x2
x2
x1 (a)
x1 (b )
Figure 2. The multi-pulse chaotic responses for the nonlinear nonplanar oscillations of the cantilever beam.
7.
CONCLUSIONS
The multi-pulse homoclinic orbits and chaotic dynamics for the nonlinear nonplanar oscillations of the cantilever beam subjected to a harmonic axial excitation and two transverse excitations at the free end are investigated for the first time by using the analytical and numerical approaches when the averaged equations have one non-semisimple double zero and a pair of pure imaginary eigenvalues. The generalized Melnikov method is employed to explore the existence of complex motions by identifying the existence of multi-pulse homoclinic orbits in the perturbed phase space. Based on the aforementioned analytical and numerical studies, it is found that the multi-pulse homoclinic orbits depend on dissipative perturbations and periodic excitations. It can be conjectured that the transfer of energy
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between the in-plane and out-of-plane modes occurs through the multi-pulse homoclinic orbits. Numerical simulations finished in this paper indicate that there exist chaotic responses for the nonlinear nonplanar oscillations of the cantilever beam under certain parametric excitation. We find that the parametric excitation F1 , transverse excitation in the z direction f 2 and damping coefficient c have important influence on the chaotic motions.
ACKNOWLEDGEMENTS The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through grants Nos. 10372008 and 10328204, the National Science Foundation for Distinguished Young Scholars of China (NSFDYSC) through grant No.10425209 and the Natural Science Foundation of Beijing (NSFB) through grant No. 3032006.
REFERENCES 1.
Crespo da Silva, M MR, Glynn CC. “Nonlinear flexural-flexural torsional dynamics of inextensional beams, I. Equation of motion,” Journal of Structural Mechanics, 6, pp. 437-448, 1978. 2. Zaretzky CL, Crespo da Silva, M MR. “Experimental investigation of non-linear modal coupling in the response of cantilever beams,” Journal of Sound and Vibration, 174, pp. 145-167, 1994. 3. Nayfeh AH, Pai PF. “Non-linear non-planar parametric responses of an inextensional beam,” International Journal of Non-linear Mechanics, 24, pp. 139-158, 1989. 4. Arafat HN, Nayfeh AH, Chin CM. “Nonlinear nonplanar dynamics of parametrically excited cantilever beams,” Nonlinear Dynamics, 15, pp. 31-61, 1998. 5. Yao MH, Zhang W. “Multi-Pulse Shilnikov Orbits and Chaotic Dynamics in Nonlinear Nonplanar Motion of a Cantilever Beam”, International Journal of Bifurcation and Chaos, 15, pp. 3923-3952, 2005. 6. Wiggins S, Global Bifurcations and Chaos-Analytical Methods, New York, Berlin, Springer-Verlag, 1988. 7. Kovacic G, Wiggins S. “Orbits homoclinic to resonance with an application to chaos in a model of the forced and damped sine-Gordon equation”, Physica D, 57, pp. 185-225, 1992. 8. Kaper TJ, Kovacic G. “Multi-bump orbits homoclinic to resonance bands”, Transactions of the American Mathematical Society, 348, pp. 3835-3887, 1996. 9. Camassa R, Kovacic G, Tin SK. “A Melnikov method for homoclinic orbits with many pulse”, Archive for Rational Mechanics and Analysis, 143, pp. 105-193, 1998. 10. Haller G, Wiggins S. “Orbits homoclinic to resonance: the Hamiltonian”, Physica D, 66, pp. 298-346, 1993.
PART 5
CONTROL OF NONLINEAR DYNAMIC SYSTEMS
SYNTHESIS OF BOUNDED CONTROL FOR NONLINEAR UNCERTAIN MECHANICAL SYSTEMS I. Ananievski Institute for Problems in Mechanics of the Russian Academy of Sciences, 119526, 101-1 prosp, Vernadskogo, Moscow, Russia
Abstract:
A problem of designing a control for Lagrange mechanical systems of a general form under uncertainty is investigated. A rheonomic system is considered under the assumption that the coefficients of a quadratic polynomial which represents the kinetic energy are unknown but lie within given bounds and the system undergoes uncontrolled bounded disturbances. Stability theory based control is constructed which enables one to drive the system to a prescribed terminal state in finite time. The proposed algorithm can be treated as a linear feedback control with the gains which are differentiable functions of the phase variables. The gains increase and tend to infinity as the phase variables tend to zero; nevertheless, the control forces are bounded and meet the imposed constraint.
Key words:
Lagrange mechanical system, constraint control, Lyapunov method, finite time.
1.
INTRODUCTION
The aim of the paper is creating effective control methods for Lagrangian systems under uncertainty. All mechanical systems, as well as many electromechanical ones, are governed by Lagrange’s equations. For this reason, Lagrangian systems are extremely important in mechanical engineering. In practice, it is also important to take into account uncontrolled external disturbances, as well as errors in the identification of the parameters of the system. 277 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 277–286. © 2007 Springer.
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We consider a problem of designing feedback controls for nonlinear mechanical systems with many degrees of freedom under following assumptions: - the number of degrees of freedom exceeds the dimension of the control force vector; - some of the parameters of the system are unknown; - the system is subjected to unknown bounded disturbances. A lot of approaches to designing control algorithms for dynamical systems under uncertainty are based on the stability theory methods and consist in constructing feedback controls which ensure the asymptotic stability of the desired motion (in particular, the terminal state) of the system. Apart from these approaches, we are searching for the control which drives the system to a prescribed terminal state in finite (unfixed) time. Besides, the desired control must satisfy the following requirements: - the control must be presented in closed-loop form; - the control must be bounded and meet the imposed constraints; - the control must cope with uncertain bounded disturbances; - the control must be robust with respect to the perturbation of the parameters of the system. Constructing controls possessing the above properties is a challenging scientific problem, rather topical for various applications, including robotics, aerospace, etc. In recent years, new approaches to constructing constrained controls for steering perturbed mechanical systems with many degrees of freedom into a prescribed terminal state in finite time have been developed [1 – 3]. These and some others methods, not mentioned here, lead to control laws which are, generally speaking, discontinuous functions of time and, therefore, difficult for applying in practice. In [4 – 5] an approach has been proposed which provides a control law depending continuously on the phase variables everywhere except for the terminal state (in the terminal state, the control is not assigned). The proposed algorithm can be treated as a linear feedback control with the gains which are differentiable functions of the phase variables. The gains increase and tend to infinity as the phase variables tend to zero; nevertheless, the control forces are bounded and meet the imposed constraint. In [4] the approach was applied to a fully actuated Lagrangian scleronomic mechanical system and ensures steering the system to the terminal state in a finite time under the assumption that the matrix of the kinetic energy is known and the system undergoes uncertain bounded disturbances. In [5] this approach was extended to scleronomic systems with uncertain matrix of inertia as well as to rheonomic systems whose matrix of inertia is known. In the present paper the approach is applied to a rheonomic system under the assumption that the matrix on inertia is not known exactly.
Synthesis of bounded control for uncertain mechanical systems
2.
279
STATEMENT OF THE PROBLEM We consider a mechanical system governed by Lagrange's equations d ∂T ∂T − = S +u dt ∂q ∂q
(1)
where q, q ∈ R n are the vectors of the generalized coordinates and velocities, u is the vector of control forces, and S is the vector of unknown forces (disturbances). Both scleronomic and rheonomic systems are of our concern. In the most general rheonomic case the kinetic energy has the form of a full quadratic polynomial whose coefficients depend on time explicitly 1 T (t , q, q ) = q Τ A(t , q )q + a Τ (t , q )q + a0 (t , q ) 2
We assume that the positive-definite continuously differentiable symmetric matrix of kinetic energy A(t , q ) is represented in the form A(t , q ) = A0 (t , q ) + A1 (t , q )
where A0 (t , q ), A1 (t , q ) are also symmetric continuously differentiable matrices and A0 (t , q ) is known and positive-definite, but A1 (t , q ) is unknown. The eigenvalues of matrices A(t , q ) and A0 (t , q ) belong to the interval [m, M ], 0 < m < M , for any t , q , i.e., 2
m z ≤ z Τ A(t , q ) z ≤ M z 2
2
2
m z ≤ z Τ A0 (t , q ) z ≤ M z , ∀t ∀z , q ∈ R n
And the matrix A1 and the partial derivatives of the matrix A0 , A1 , and A are bounded uniformly in t , q ,i.e., A1 ≤ M 1 ,
∂A1 ∂A ≤ C1 , ≤ C , i = 1,…, n ∂qi ∂qi
∂A0 ∂A1 ≤ D0 , ≤ D1 , M 1 , C , C1 , D0 , D1 > 0 ∂t ∂t
Here and everywhere below, we denote by ⋅ the Euclidean norm of a vector or a matrix. The vector-valued function a(t , q ) and the function a0 (t , q ) are assumed to be unknown and satisfy the following conditions
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⎛ ∂a ⎞ ∂a ∂a0 ∂a − ≤ D2 , ⎜ ⎟ − ≤ D3 , D2 D3 > 0 ∂q ∂t ⎝ ∂q ⎠ ∂q
The unknown perturbation vector S may be an arbitrary vector-valued function, including a discontinuous one, satisfying some existence conditions for the solution of system (1.1) and meeting the constraint
S (t , q, q ) ≤ S0 , S0 > 0 The vector of control forces is also bounded
u (t , q, q ) ≤ U , U > 0
(2)
The problem is to construct a continuous feedback control law meeting constraint (2) and specify a domain of admissible initial states from which system (1) governed by this control comes to the terminal state q = q , q=0 in finite time, whatever the vector a(t , q ) , the function a0 (t , q ) , and the disturbance S , satisfying the above conditions, be. Without loss of generality we assume that the terminal state coincides Otherwise we can make the with the phase space origin q = 0, q=0. appropriate coordinate transformation.
3.
SYNTHESIS OF CONTROL
To illustrate the ideas behind the method let us consider an unperturbed scleronomic mechanical system controlled with linear feedback (PDcontroller) d ∂T ∂T − = −α q − β q dt ∂q ∂q
(3)
where 1 T = q Τ A(q)q 2
and feedback factors α and β are some positive real constants. Researchers and engineers often use such PD-controller for stabilizing a system. It is not difficult to prove that the phase space origin of system (3) is globally asymptotically stable. However, such linear feedback control has some disadvantages. First, the control forces generated by the above PD-controller are not bounded and do not satisfy constraint (2). On the other hand the PD-
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controller does not use the control possibilities to full extent. The closer the system is to the terminal state the smaller are the control forces. The control forces tend to zero as the trajectory tends to the phase space origin. This implies infinite time of motion. To make the control bounded and more effective we choose the feedback factors α and β as functions of time t and the phase variables q, q . The feedback factors increase and tend to infinity as the phase variables tend to zero; nevertheless, the control forces are bounded and meet the imposed constraint. The control brings the system to the terminal state in finite time and, moreover, it copes with some uncertain bounded disturbances. We define the control as follows u (t , q, q ) = −α (t , q, q ) A0 (t , q) q − β (t , q, q ) q
(4)
where 3U 2 3U , α (t , q, q ) = 8V (t , q, q ) 2 2 MV (t , q, q ) 1 1 1 = q Τ A0 (t , q) q + β (t,q,q)q 2 + α (t,q,q) q Τ A0 (t , q) q V (t,q,q) 2 2 2
β (t , q, q ) =
(5) (6)
Relations (4) and (5) define the functions α (t , q, q ), β (t , q, q ), and V (t , q, q ) implicitly. The control u is defined through feedback factors α and β , the feedback factors α and β are defined by means of function V . The function V , in turn, depends on the feedback factors α and β . Apart from control algorithm used in [4] for the scleronomic system and in [5] for the rheonomic one, the matrix which occurs in (4) and (6) does not coincide with the matrix of inertia A of the system. It represents ``the known part'' of A only. The justification of the proposed control law is based on Lyapunov’s direct method. The function V plays a principal role in the present investigation. Given this function one can find the feedback factors α and β through the above relations, and, consequently, the control u according to this formula. In addition, the function V has the dimension of energy and serves as a Lyapunov function for the system under consideration. It tends to zero as the trajectory approaches the terminal state. Since the function V appears in the denominators in these relations, the feedback factors tend to infinity as the trajectory approaches the origin. Nevertheless, the proposed control does not go beyond the admissible boundaries. The peculiarity of the investigation is that we do not express the function V , as well as the functions α and β , in explicit form. We establish their properties and carry out all other necessary reasoning for the functions defined implicitly.
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The justification of the control law is based on the following statements [4 – 5]. Theorem 1. In the domain q 2 + q 2 > 0 there exist continuously differentiable positive functions α (t , q, q ), β (t , q, q ), and V (t , q, q ) satisfying (5) and (6). Theorem 2. The function V satisfies the inequalities 1 1 1⎛ 2 3⎛ 2 4 2 2 2 2 4 2 2 2 ⎞ 2⎞ ⎜ mq + ( m q + 6U q ) ⎟ ≤ V ≤ ⎜ Mq + ( M q + 2U q ) ⎟ 8⎝ 8⎝ ⎠ ⎠
Let (t0 , q0 , q0 ) be the initial state. Denote
δ1 =
5M 1 3 ⎛ M ⎞5 , δ 2 = ⎜1 + 1 ⎟ m ⎠m 2 2M m m ⎝
δ3 =
CM 1 ⎞ 2 nC 6 5n ⎛ + ⎜ C1 + ⎟ m ⎠ m M m m⎝
δ4 =
2 D3 mM
+
2 D0 2 5M 1 D3 + m m2 1
δ = δ1U − δ 2 ( D2 + S0 ) − δ 3V0 − δ 4V0 2 Theorem 3. If δ > 0 then
δ 1 V (t ) ≤ − V 2 (t ) 3 along the trajectory of system (1) and the time of motion τ (t0 , q0 , q0 ) from the state (t0 , q0 , q0 ) to the phase space origin satisfies the inequality 6
1
τ (t0 , q0 , q0 ) ≤ V 2 (t0 , q0 , q0 ) δ Theorems 1 and 2 have been proved in [4] for the scleronomic case. The fact that the functions α , β , and V depend on time in the present paper virtually does not influence the proof. One can find in [4 – 5] also the proof of Theorem 3 for scleronomic mechanical systems. For the rheonomic systems the proof employs similar ideas. The inequality δ > 0 gives us the following sufficient condition for driving system (1) to the terminal state by control (6)
Synthesis of bounded control for uncertain mechanical systems U>
δ δ2 δ 1 ( D2 + S0 ) + 3 V0 + 4 V0 2 δ1 δ1 δ1
283 (7)
The above sufficient condition relates the maximum admissible values of the control and disturbances, as well as the value of the Lyapunov function V at the initial point (t0 , q0 , q0 ) . It characterizes the superiority of the control force over the disturbances which one needs to apply the above control, and describes the domain of admissible initial states, from which the proposed control brings the system to the terminal state. It is important to note that the sufficient condition stated is not necessary for driving the system to the terminal state by the proposed control. One can see that the control does not depend on S0 , C , C1 , M 1 , and Di , i = 0,… ,3. To utilize the control it is sufficient to know the matrix A0 , the upper bound of eigenvalues M , and the phase variables q and q. Hence, the proposed control law can be formally applied when inequality (7) does not hold. Computer simulation shows the algorithm’s efficiency far beyond this sufficient condition. To explain this phenomena let us note that the sufficient condition stated guarantees monotonic decrease of the Lyapunov function V along the trajectory. However, it may happen that the function V is not monotone and tends to zero, as the trajectories tend to the terminal state. The results of the computer simulation show such behavior of various systems.
4.
THE COMPUTER SIMULATION RESULTS
To illustrate the efficiency of the proposed algorithm let us consider a controlled rotation of a rod (Figure 1). The rod rotates in a horizontal plane about one of the ends driven by a control torque u. We assume that a particle of an unknown mass moves uncertainly along the rod, therefore, the moment of inertia of the rod is unknown and depends on time.
Figure 1. A controlled rotation of a rod and with a particle moving along the rod.
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Denote the angular coordinate and velocity by q, q , and the distance from the rotation axis to the particle by L(t ). With the above notation the terms in the kinetic energy expression have the form A(t ) = mL2 (t ), a ≡ 0, a0 (t ) =
m 2 L (t ) 2
The system dynamic is governed by the equation mL2 (t )q + 2mL(t ) L (t )q = u
The parameters of the system and the law of the motion of the particle along the rod L(t ) were chosen as follows 1 m = 1, U = 10, L(t ) = 1 + sin ω t 2
The proposed approach provides a continuous bounded feedback control which steers the rod to the terminal state q = 0, q = 0 in a finite time. Figures 2 and 3 illustrate the behavior of the system subject to the control described above. Figure 2 shows the phase portrait of the system. The thick solid line depicts the trajectory in case ω = 0, and the dashed line the trajectory in case ω = 2. The thin solid lines depict the level sets of the Lyapunov function V .
Figure 2. The phase portrait.
The graphs in Figure 3 show the time history of the function V (t ) (the dashed line) and the value of the control torque vector u (t ) (the solid line) along the trajectory under consideration.
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Figure 3. The control torque u (t ) and the function V (t ) .
5.
COMPARISON WITH TIME-OPTIMAL CONTROL
To compare the proposed control with time-optimal one let us consider a simple linear system with one degree of freedom q = u
(8)
The time-optimal control for steering such a system to the phase space origin has the following form ⎧ − q 2 − > ≥ q q 1, if 0 and ⎪ 2 ⎪⎪ 2 uopt (q, q ) = ⎨−1, if q < 0 and q > q 2 ⎪ 1, otherwise ⎪ ⎪⎩
Figure 4 depicts the graphs of the function uopt (q, q ) (step function) and the function u (q, q ) defined by relations (4 - 6).
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Figure 4. The time-optimal control and the proposed control.
6.
CONCLUSIONS
The proposed approach enables one to construct a feedback control law which is a smooth function of phase variables, meets the imposed constraints, and brings a mechanical system of a general form to a prescribed terminal state in finite time. Figure 4 shows that for system (8) the proposed control is very similar to the time-optimal one.
ACKNOWLEDGEMENTS This work was carried out under the Grant for Support of Leading Scientific Schools (SS-9831.2006.1), was supported by the Russian Foundation for Basic Research (Grants 05-01-00647 and 05-01-00563).
REFERENCES 1. 2. 3. 4. 5.
Piatnitski YS. “Decomposition principle in control of mechanical systems” Dokl. Akad. Nauk SSSR, 300, pp. 300-303, 1988. Chernousko FL. “Synthesis of the control of a non-linear dynamical systems”, Prikl. Mat. Mekh, 56, pp. 179-191, 1992. Ananievski IM. “Control of mechanical system with unknown parameters by a bounded force”, J. Appl. Maths Mechs, 61, pp. 52-62. Ananievski IM. “Continuous feedback control of perturbed mechanical systems”, J. Appl. Maths Mechs, 67, pp. 143-156. Ananievski IM. “Control of nonlinear mechanical systems under uncertainty”, Proceedings of the Fifth EUROMECH Nonlinear Dynamics Conference (ENOC-2005), Eindhoven, pp. 790-799.
CONTROLLED VIBRATION SUPPRESSION OF STRUCTURAL TELESCOPIC SYSTEMS P. Barthels, J. Wauer Institut für Technische Mechanik, Universität Karlsruhe, 76128, Karlsruhe, Germany. E-mail:
[email protected]
Abstract:
Bending vibrations of telescopic systems of structural components are analyzed during extending and retracting motion. For a physical model, consisting of geometrically non-linear Timoshenko beams which are connected with some clearance in their contact areas, the governing boundary value problem is derived by applying Hamilton’s principle. Galerkin’s method based on admissible shape functions is used as a discretization procedure to generate a system of coupled, non-linear, time-varying, ordinary differential equations. Linearization about the static equilibrium position and model reduction by modal truncation for different telescopic lengths leads to a multiplicity of simple linear reduced models. On the basis of these models, an adaptive state regulator and an adaptive full state observer (Luenberger observer) are designed for vibration suppression using the Optimal Linear Quadratic Regulator (LQR). The adaptive controller and observer are applied to the significantly more complicated geometrically non-linear system with clearance so that the robustness of the controlled system can be studied during telescopic motions.
Key words:
Vibration suppression, non-linear and time-varying systems, non-linear Timoshenko beams, contact, clearance, model reduction, adaptive state regulator, adaptive Luenberger observer, LQR.
1.
INTRODUCTION
Telescopic systems of structural components are found, e.g., in mobile cranes, stacker cranes, rack feeders, fork lifters (see Figure 1a). Since the main duty of these machines is loading and unloading goods to and from racks, trucks etc., a great number of acceleration and deceleration operations occur. Due to these operations combined with the extending and retracting 287 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 287–296. © 2007 Springer.
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motion of the sections and the eccentric mass of the transported good, bending vibrations of the system appear which are perpendicular to the telescopic axis. These vibrations lead to a significant reduction of the performance due to the required waiting time for attenuation and to safety problems so that controlled vibration suppression seems to be beneficial.
Figure 1.
a) Fork lifter,
b) System model.
For practical use, two different approaches have to be studied for two different cases of reaching the target position. For the case of an automated target positioning, the initial and final positions are known. In this case, the rigid body movement as well as the vibration suppression is achieved by a drive control, which is done in [1]. The present contribution considers the case when the target positioning is performed by an operator. In this case, the use of a drive control is no longer possible as the final position is unknown for the system. Hence an additional actuator becomes necessary for vibration suppression. Before developing a controller concept for preventing harmful vibrations in detail, an appropriate modeling of such systems has to be introduced.
2.
PHYSICAL MODEL
From the viewpoint of mechanics, a non-linear field problem with variable geometry has to be considered. The clearance produces non-linear effects and the different segments are pre-stressed due to their self-weight and the weight of the transported good. In [2] the modeling is shown for systems with a non-eccentric load. The different segments of the telescopic system
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289
are supposed to be slender and modeled as Bernoulli/Euler beams. In the present contribution the load is supposed to act eccentrically on the system (see Figure 1b). The segments are modeled as Timoshenko beams and for large deformations, the beams are modeled geometrically non-linear. The discrete contact forces, realized via spring-damper systems, are introduced in the form of distributed line loads (by using Dirac impulse functions δ (.) ), so that elementary boundary conditions remain. The procedure is illustrated in Figure 1b for a two-section telescopic beam system. Beam 1 is fixed at a rigid vehicle unit with the prescribed displacement u0 ( t ) . Beam 2 carries an eccentric point load. The deformation of the beams is represented by the horizontal displacements u1 ( z1 , t ) and u2 ( z2 , t ) , the vertical displacements w1 ( z1 , t ) and w2 ( z2 , t ) and the angles α1 ( z1 , t ) and α 2 ( z2 , t ) . Assuming that the longitudinal axes of the beams are inextensible, one obtains the kinematic relation
wi , zi = −1 + 1 − ui2, zi
(1)
between the horizontal displacements ui ( zi , t ) and the vertical displacements wi ( zi , t ) ( i = 1, 2 ). The model is specified by the following parameters: length (to be equal in most practical applications) of the beams l , constant cross-sectional areas A1,2 , constant cross-sectional moments of inertia I1,2 , density ρ , Young’s modulus E and shear modulus G of the two flexible components, mass mL and eccentricity aL of the load, mass of the vehicle mT and given telescopic lengths l A ( t ) and lL ( t ) . The contact is defined by the given number n of contact points, the clearance lS , spring stiffness c , and damping coefficient d . In the axial direction it is assumed that there is no friction and that the entire force flow leads from the upper part into the lower part through the lowest contact point.
3.
FORMULATION
3.1 Boundary value problem Applying Hamilton’s principle
δ∫
t1 t0
t1
(T − U ) dt + ∫ t
0
Wvirt dt = 0,
(2)
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the governing boundary value problem can be derived. T is the kinetic energy, U the potential energy and Wvirt the virtual work of the forces without potential of the considered system. For a better understanding, the contact formulation takes place by simple linear springs during the derivation of the field equations and will be replaced by one-sided spring-damper elements in the final equations of motion. With Equation (1) and for the special case in which the beam segments contact each other at the two points z1 = l and z2 = 0 only, the potential energy reads in a third order approximation l ⎛1 1 1 ⎞ U = ∫ ⎜ GA1u1,2 z1 − GA1α1u1, z1 + GA1α12 + EI1α1,2 z1 ⎟ dz1 0 2 2 ⎝2 ⎠ l ⎛1 1 1 ⎞ + ∫ ⎜ GA2 u2,2 z2 − GA2α 2u2, z2 + GA2α 22 + EI 2α 2,2 z2 ⎟ dz2 0 2 2 2 ⎝ ⎠ l l 1 1 − ρ A1 g ∫ ( l − z1 ) u1,2 z1 dz1 − ρ A2 g ∫ ( l − z2 ) u2,2 z2 dz2 0 0 2 2 lA 1 − ρ lA2 g ∫ u1,2 z1 dz1 + 2 ⎡⎣u2 ( 0 ) − u1 ( l A ) ⎤⎦ α1 ( l A ) 0 2 l lL 1 A − mL g ∫ u1,2 z1 dz1 + ∫ u2,2 z2 dz2 + 2 ⎡⎣u2 ( 0 ) − u1 ( l A ) ⎤⎦ α1 ( l A ) 0 0 2 2 1 ⎡ ⎤ 1 − mL gaL ⎢α 2 ( lL ) − α 23 ( lL ) ⎥ + c ⎡⎣u2 ( 0 ) − u1 ( l A ) ⎤⎦ 6 ⎣ ⎦ 2 2 1 + c ⎡⎣u2 ( l − l A ) − u1 ( l ) ⎤⎦ . 2
(
)
(
) (3)
Assuming that the vertical velocities w1,t and w2,t are negligible for the kinetic energy of the system, one obtains in a third order approximation T= +
(I α 2 (∫
ρ
l
0
(
2 1 1,t
+ A1 ( u1,t + u0,t )
2
mL 2 2 aLα 2,t ( lL ) + ( u2,t ( lL ) + u0,t 2
) dz + ∫ ( I α + A (u + u ) ) dz ) ) ) − m (u (l ) + u ) α (l ) a α (l ). l
1
0
2
2 2,t
2,t
L
2
2
2,t
0,t
2
2
L
0,t
2,t
L
L
2
L
(4) With a force FR applied by an actuator on the lower segment of the system at z1 = lR , the virtual work reads Wvirt = FRδ u1 ( lR ) .
(5)
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Evaluating Hamilton’s principle (2) introducing T , U and Wvirt according to Equations (3), (4) and (5), respectively, yields the governing field equations
ρ A1 ( u0,tt + u1,tt ) + GA1 (α1, z − u1, z z ) + ⎡⎣1 − σ ( z1 − l A ) ⎤⎦ g ( mL + ρ lA2 ) u1, z z 1
1 1
(
+ ρ A1 g ⎡⎣( l − z1 ) u1, z1 ⎤⎦ + δ ( z1 − l A ) g ( mL + ρ lA2 ) α1 − u1, z1 z1
1 1
)
(6)
= δ ( z1 − lR ) FR + FK1 ,
ρ A2 ( u0,tt + u2,tt ) + GA2 (α 2, z − u2, z z ) + ⎡⎣1 − σ ( z2 − lL ) ⎤⎦ gmL u2, z z 2
2 2
(
+ ρ A2 g ⎡⎣( l − z2 ) u2, z2 ⎤⎦ − δ ( z2 ) g ( mL + ρ lA2 ) α1 ( l A ) − u2, z2 z2
)
2 2
(7)
−δ ( z2 − lL ) mL ⎡⎣ gu2, z2 − ( u0,tt + u2,tt ) + α 2,tt aLα 2 + α 2,2 t aL ⎤⎦ = − FK 2 ,
ρ I1α1,tt + GA1 (α1 − u1, z ) − EI1α1, z z 1
1 1
−δ ( z1 − l A ) g ( mL + ρ lA2 ) ⎡⎣u2 ( 0 ) − u1 ( l A ) ⎤⎦ = 0,
(8)
ρ I 2α 2,tt + GA2 (α 2 − u2, z ) − EI 2α 2, z z 2
2 2
⎡ ⎛ 1 ⎞⎤ +δ ( z2 − lL ) aL ⎢ mL aLα 2,tt − ( u0,tt + u2,tt ) α 2 − mL g ⎜1 − α 22 ⎟ ⎥ = 0 ⎝ 2 ⎠⎦ ⎣
(9)
for the two bodies with the contact forces FK1 = δ ( z1 − l A ) ( CK (ξ1 ) + ξ1,t DK (ξ1 ) ) + δ ( z1 − l ) ( CK (ξ 2 ) + ξ 2,t DK (ξ 2 ) ) , FK2 = δ ( z2 ) ( CK (ξ1 ) + ξ1,t DK (ξ1 ) ) + δ ( z2 − l + l A ) ( CK (ξ 2 ) + ξ 2,t DK (ξ 2 ) )
(10)
where ⎡ l ⎞ l ⎞ 1⎛ l ⎞ l ⎞⎤ 1⎛ ⎛ ⎛ CK (ξ ) = c ⎢ξ − ⎜ ξ + S ⎟ sign ⎜ ξ + S ⎟ + ⎜ ξ − S ⎟ sign ⎜ ξ − S ⎟ ⎥ , 2⎝ 2⎠ 2 ⎠ 2⎝ 2⎠ 2 ⎠⎦ ⎝ ⎝ ⎣ ⎡ 1 l ⎞ 1 l ⎞⎤ ⎛ ⎛ DK (ξ ) = d ⎢1- sign ⎜ ξ + S ⎟ + sign ⎜ ξ − S ⎟ ⎥ , (11) 2⎠ 2 2 ⎠⎦ ⎝ ⎝ ⎣ 2 ξ1 ( t ) = u2 ( 0, t ) − u1 ( l A , t ) , ξ 2 ( t ) = u2 ( l − l A , t ) − u1 ( l , t ) .
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The corresponding boundary conditions read u1 ( 0, t ) = 0, u1, z1 ( l , t ) = α1 ( l , t ) , α1 ( 0, t ) = 0, α1, z1 ( l , t ) = 0, u2, z2 ( 0, t ) = α 2 ( 0, t ) , u2, z2 ( l , t ) = α 2 ( l , t ) , α 2, z2 ( 0, t ) = 0, α 2, z2 ( l , t ) = 0.
(12)
3.2 Discretization The discretization of the non-linear, time-varying, coupled partial differential equations (6), (7), (8) and (9) together with the corresponding boundary conditions (12) is based on Galerkin’s method. For that, the approximate solutions u1 ( z1 , t ) , u2 ( z2 , t ) , α1 ( z1 , t ) and α 2 ( z2 , t ) are represented by a series expansion N
N
i =1
i =1
u1 ( z1 , t ) = ∑ qi ( t ) U1i ( z1 ) , α1 ( z1 , t ) = ∑ qi ( t ) Φ1i ( z1 ) , N
N
i =1
i =1
u2 ( z 2 , t ) = ∑ q N + i ( t ) U 2 i ( z 2 ) , α 2 ( z 2 , t ) = ∑ q N + i ( t ) Φ 2 i ( z 2 ) .
(13)
U1i ( z1 ) , U 2i ( z2 ) , Φ1i ( z1 ) and Φ 2i ( z2 ) are modes of ordinary Timoshenko beams fulfilling the boundary conditions (12). The discretization applying Galerkin’s procedure leads to a system of coupled, non-linear, time-varying, ordinary differential equations of the type
= f ( q, q , l A , lL , FR , u0,tt ) . M (q ) q
4.
(14)
VIBRATION SUPPRESSION CONCEPT
To suppress vibrations, a state space control concept is introduced with a collocated actuator-sensor pair at z1 = lR . For fixed telescopic lengths l A = const. and lL = const. and with FR = 0 and u0,tt = 0 , the time-invariant system (14) is linearized about its static equilibrium position q 0 and finally reformulated with the coordinate transformation ql = q − q 0 as l + Cql = b∗R FR + b∗0u0,tt , Mq u1 ( lR , t ) = ⎡⎣U11 ( lR ) ,...,U1N ( lR ) ,0,...,0 ⎤⎦ ( ql + q 0 ) = c*RT ql + u R 0 .
(15)
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4.1 Model reduction Equation (15) represents a 2 N -degree-of-freedom system. The objective of an order reduction is to find, for a given model of high order, a model of significantly lower order whose dynamic behavior approximates the original behavior as well as possible in a specified frequency bandwidth. For this purpose, the system equations (15) are transformed to modal coordinates according to ql = Θy
(16)
where y is the vector of modal amplitudes and Θ is the matrix of the normalized mode shapes Θ = ( Θ1 , Θ 2 ,..., Θ 2 N ) ordered by increasing natural frequencies (ω1 ≤ ω2 ≤ ω3 ≤ ...) . Substituting Equation (16) in Equation (15) and left multiplying by ΘT one obtains y + diag (ωi2 ) y = ΘT b*R FR + ΘT b*0 u0,tt , N N
** b b** Ω R 0 *T **T u1 ( lR , t ) = c R Θy + uR 0 = c R y + u R 0 .
(17)
If the modes are divided in low frequency modes which respond dynamically, and high frequency modes which respond statically, Equation (17) reads y1 ⎞ ⎛ Ω1 0 ⎞⎛ y1 ⎞ ⎛ b**R 1 ⎞ ⎛ b** ⎞ ⎛ I 0 ⎞ ⎛ 01 + = F + ⎟⎜ ⎟ ⎜ ** ⎟ R ⎜ ** ⎟ u0,tt , ⎜ ⎟ ⎜ ⎟ ⎜ ⎝ 0 0 ⎠⎝ y 2 ⎠ ⎝ 0 Ω 2 ⎠⎝ y 2 ⎠ ⎝ b R 2 ⎠ ⎝ b0 2 ⎠ **T **T u1 ( lR , t ) = c R 1 y1 + c R 2 y 2 + u R 0 .
(18)
The reduced state space model finally reads I ⎞⎛ y1 ⎞ ⎛ 0 ⎞ ⎛ 0 ⎞ ⎛ y 1 ⎞ ⎛ 0 ⎜ ⎟ = ⎜ ⎟⎜ ⎟ + ⎜ ** ⎟ FR + ⎜ ** ⎟ u0,tt , b0 1 ⎠ y1 ⎠ ⎝ −Ω1 0 ⎠⎝ y1 ⎠ ⎝ b R 1 ⎠ ⎝N ⎝N N N b0 z z bR A
(19)
u1 ( lR , t ) = ⎡⎣c ,0,...,0 ⎤⎦ z + c Ω b FR + c Ω b u0,tt + u R 0 .
T d d0 R cR ** T R1
** T R2
−1 ** R2 2
** T R2
−1 ** 2 0 2
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4.2 Linear quadratic regulator The idea within the state space approach using the Optimal Linear Quadratic Regulator (LQR) is to seek a linear state feedback with constant gain r FR = −r T z
(20)
such that the following quadratic cost functional is minimized: ∞⎛ 1 ⎞ J = ∫ ⎜ z T Qz + FR2 ⎟ dt. 0 κ ⎝ ⎠
(21)
Q is semi-positive definite and κ > 0 . The solution of this problem is
rT = κ bTR P
(22)
where P is the symmetric positive definite solution of the algebraic Riccati equation PA + AT P + Q − κ Pb R bTR P = 0.
(23)
The existence and the uniqueness of the solution of the algebraic Riccati equation are guaranteed if ( A, b R ) is a controllable pair and ( A, H ) is observable where HT H = Q [3]. Techniques for solving the algebraic Riccati equation are automated in most control design software applications and will not be discussed here. In this contribution, Q is chosen in such a way that z T Qz represents the total energy (potential and kinetic energy) of the system. The value of κ is selected to achieve reasonably fast closed-loop poles without excessive values of the control effort [3].
4.3 Full state observer The state feedback assumes that the states are known at all times. For the present problem, the state vector z cannot be measured directly, but the displacement u1 ( lR , t ) of the lower segment and the acceleration of the rigid vehicle unit u0,tt ( t ) are measurable. The objective of a Luenberger Observer is to find from this information an approximate value zˆ of the state vector z . The Luenberger Observer is written as
Controlled vibration suppression of structural telescopic systems
(
)
zˆ = Azˆ + b R FR + b 0u0,tt + k u1 ( lR , t ) − uˆ1 ( lR , t ) , uˆ1 ( lR , t ) = c zˆ + d R FR + d 0 u0,tt + u R 0 . T R
295 (24)
Combining Equations (19) and (24), one obtains
( z − zˆ ) = ( A − kc ) ( z − zˆ ) . T R
(25)
Equation (25) shows that the error of the observer goes to zero if the eigenvalues of A − kcTR (the observer poles) have negative real parts. Due to the collocation of the actuator and the sensor ( b*R = c*R ), the observer poles correspond to the closed-loop poles (the eigenvalues of A − b R rT ) for ⎛0 I ⎞ k =⎜ ⎟ r. ⎝ I 0⎠
(26)
To ensure that the observer is faster than the closed-loop, the observer poles are placed on the left of the closed-loop poles. Due to Equation (26) this can be achieved by repeating the procedure illustrated in paragraph 4.2 using a new value κ * > κ . Hence the regulator and the observer can both be designed from a single root locus.
4.4 Telescopic operations with clearance For real telescopic operations, the parameters of the controller and of the observer are determined for different telescopic lengths lT = lL + l A (with l A = 0 for 0 ≤ lL < l ) and interpolation leads to an adaptive controller and an adaptive observer. The position z1 = lR of the actuator-sensor pair has to be selected in such a way that the system is observable and controllable for any length lT . Due to the Luenberger Observer, straightforward measurements of the acceleration of the rigid vehicle unit u0,tt ( t ) , of the telescopic length lT and of the displacement u1 ( lR , t ) of the lower segment are sufficient to operate the controller. This makes it possible to apply the adaptive controller and observer, developed for the reduced linear system model, to the significantly more complicated, geometrically non-linear and time-varying system with clearance (14). Due to this approach, the influence of clearance, the robustness of the controlled system and the danger of spillover instability can be studied during telescopic motions.
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5.
SIMULATION RESULTS AND CONCLUSIONS
Results are presented here for a two-sectional system with the following parameters: l = 1.35 m , A1,2 = 0.001 m 2 , mL = 17.897 kg , aL = 0.5 m , lS = 0.01 m , I1,2 = 0.83 ⋅ 10−8 m 4 , ρ = 7850 kg/m3 , E = 2.1 ⋅ 1011 N/m 2 , G = 0.8 ⋅ 1011 N/m 2 , c = 108 N/m , d = 103 Ns/m , N = 4 , n = 3 . The controller influences the first two modes with κ = 1500 and κ * = 3000 . The system starts from an initial point without any initial velocity, is accelerated during 3 seconds ( u0,tt = 5 m/s 2 ) and moves with constant velocity during 4 seconds before it is decelerated during 3 seconds ( u0,tt = −5 m/s 2 ). During the simulation the telescopic length lT increases as shown in Figure 2a. Figure 2b shows the position u2 ( l , t ) of the telescope tip relative to its base during the motion and illustrates the vibration suppression by state control. 1.4
1
1
relative position [m]
telescopic length [m]
1.2
0.8 0.6 0.4
0.5
0
l
0.2
L
controlled without control
l 0 0
Figure 2.
A
5
10 time [s]
15
a) Telescopic length,
20
-0.5 0
5
10 time [s]
15
20
b) Relative position.
The present contribution shows that controlled vibration suppression of structural telescopic systems with clearance is possible. For practical use, the dynamics of the actuator and of the sensor which have been assumed to be perfect in this contribution must be taken into account due to their possible destabilizing influence on the closed-loop. The influence of structural damping, which is neglected in this contribution, will improve the stability of the closed-loop and reduce the danger of spillover instability.
REFERENCES 1.
2. 3.
Barthels P, Wauer J, Mittwollen M, Arnold D. “Vibration Suppression for Telescopic Systems of Structural Members with Clearance”, Proc. 9th Int. Conf. on Energy and Environment 2005, “Vibration & Control-3”, Cairo/Sharm El-Sheikh, Egypt, 2005. Barthels P, Wauer J. “Modeling and Dynamic Analysis of Telescopic Systems of Structural Members with Clearance”, Nonlinear Dynamics, 42, pp. 371-382, 2005. Preumont A. Vibration Control of Active Structures, Dordrecht/Boston/London, Kluwer Academic Publishers, 2002.
INFLUENCE OF A PENDULUM ABSORBER ON THE NONLINEAR BEHAVIOR AND INSTABILITIES OF A TALL TOWER P. B. Gonçalves, D. Orlando Department of Civil Engineering, Pontifical Catholic University, PUC-Rio, 22453-900 Rio de Janeiro, Brazil, E-mail:
[email protected]
Abstract:
In this paper, passive and hybrid control devices are used to improve the dynamic response of a tall tower. A pendulum attached to the top of the tower is used as a vibration absorber. The tower is modeled as a bar with variable cross-section with concentrated masses. First, the vibration modes and frequencies of the tower are obtained analytically. The primary structure and absorber together constitute a coupled system which is discretized as a two degrees of freedom nonlinear system, using the normalized eigenfunctions and the Rayleigh-Ritz method. In order to improve the effectiveness of the control during the transient response, a hybrid control system is suggested. The added control force is implemented as a non-linear variable stiffness device based on position and velocity feedback. The obtained results show that this strategy of nonlinear control is attractive, has a good potential and can be used to minimize the response of slender structures under various types of excitation.
Key words:
Tower, vibration control, hybrid control, nonlinear vibrations, time delay.
1.
INTRODUCTION
Towers, due to its height and slenderness, are vulnerable to the occurrence of extreme vibrations caused by dynamic loads, such as wind and earthquakes. The high vibration levels induced by these loading conditions can cause problems with signals, discomfort and, worse, compromise the structure’s integrity. The action of wind is of utmost importance in towers, since it generates flexural vibrations, causing large displacements and rotations at the top of the tower. These vibrations in towers usually are 297 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 297–306. © 2007 Springer.
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caused by the vortex shedding. This type of vibration occurs around the wind speed where the periodic vortex-shedding frequency coincides with one of the natural frequencies of the structure. In towers, the worst case occurs when the vortex-shedding frequency coincides with the frequency of the first vibration mode of the tower [1]. An alternative to minimize these vibrations is the application of control strategies. There are basically three types of structural control: active, passive and hybrid. Basic concepts, experiments and practical applications of these devices are found in Korenev & Reznikov [1] and Soong & Dargush [2]. In the passive control, the magnitude of the control forces depends only on the physical properties of the auxiliary system. The main advantage of the passive absorber is that it does not need any external energy source and the control is always stable. However there are some limitations in the use of this technology, since the passive devices are designed to work efficiently within a small frequency range. One passive control device proposed in literature is the pendulum absorber. Pendulum absorbers were studied by Mustafa & Ertas [3] and Yaman & Sen [4], among others. The hybrid control approaches combine active controllers with passive devices. The active portion of a hybrid system requires much less power than a similar active system, while providing better structural response than the passive system alone [5]. Here a hybrid control approach is proposed based on the simultaneous use of a pendulum absorber with an external force applied at the pendulum-tower connection. The active force is based on the basic ideas of the switched stiffness approach [6]. Here, the tower is modeled as a bar with variable cross-section [7] and concentrated masses. First, the vibration modes and frequencies of the tower are obtained analytically. Using these vibration modes as interpolating functions, the natural frequencies and modes of the column-pendulum system are obtained by the Rayleigh-Ritz method. For a pendulum tuned to the lowest frequency of the tower, only the two first vibration modes and frequencies of the tower-pendulum system are important, since the subsequent frequencies of the cantilevered tower are much higher than the first ones. So, using these two vibration modes, a two degrees of freedom nonlinear system is obtained. These equations are either solved numerically, using the Runge-Kutta method, or analytically by the use of the GalerkinUrabe method. Floquet theory is used in the stability analysis of the responses.
Influence of a pendulum absorber on the behavior of a tower
2.
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PROBLEM FORMULATION
The tower is modeled as a clamped-free column with variable crosssection [7]. Platforms, antennas and equipments are modeled as discrete masses along the tower, as shown in Figure 1. The pendulum is considered as a discrete element along the tower. l m
w(L)
Kp
L2
v
θ
Mc P (x, t)
l
vx
h1
L Mx
EIx
L1
m
qx
vy
h
x
h2
Figure 1. Variable cross-section tower with a pendulum absorber: main parameters and reference system.
The behavior of the column-pendulum system shown in Figure 1 is described by the following Lagrange function [8]: L
2
1 1 n ⎛ ∂w ⎞ ⎛ ∂w( L1 ) ⎞ Lg = ∫ M o (1 + η x ) ⎜ ⎟ dx + M c ⎜ ⎟ 2 2 ⎝ ∂t ⎠ ⎝ ∂t ⎠ 0
2
(1)
L
1 n+2 ⎡1 ⎤ − ∫ ⎢ EI o (1 + η x ) w2 , xx − w2 , x ( N o (1 + η x) n +1 ) ⎥ dx 2 2 ⎦ 0⎣ 2 1 ⎡⎛ ∂w( L) ⎞ ⎛ ∂w( L) ⎞⎛ ∂θ + m ⎢⎜ ⎟ + 2l ⎜ ⎟⎜ 2 ⎣⎢⎝ ∂t ⎠ ⎝ ∂t ⎠⎝ ∂t
2 ⎤ ⎞ 2 ⎛ ∂θ ⎞ + l θ cos( ) ⎟ ⎜ ⎟ ⎥ ⎠ ⎝ ∂t ⎠ ⎦⎥
1 − mgl (1 − cos(θ )) − K pθ 2 2
where EI o is the flexural stiffness at the base of the column, N 0 , the normal force at the base, M o , the mass per unit length at the base, M c , a concentrated mass at a distance L1 from the base, L , the length of the column, m , the mass of the pendulum, K p , the stiffness of the pendulum and l , the pendulum length. Finally η and n are parameters that define the variation of the column cross-section and g is the acceleration of the gravity. First the free vibration modes and frequencies of the column (with and without concentrated masses) are obtained analytically using symbolic algebra. Then, these modes are used together with the Rayleigh-Ritz method
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to obtain the free vibration modes and frequencies of the column-pendulum system. Finally, a two degrees of freedom model, capable of describing with precision the behavior of the system in the neighborhood of the basic frequency of the column, is derived, from which the following set of nonlinear equations of motion, in the non-dimensional form, is obtained 2 ⎧ ⎛ ωc ⎞ ωc ⎪(1 + µ )ζ ,ττ +2ξ c ζ ,τ + ⎜ ⎟ ζ + µθ ,ττ cos(θ ) − µθ ,τ 2 sin(θ ) ωe ⎪ ⎝ ωe ⎠ ⎪ 2 ⎛ω ⎞ ⎪ = ζ s ⎜ c ⎟ sin(τ ) ⎨ ⎝ ωe ⎠ ⎪ 2 ⎪ ⎪ µθ , +2µξ ω p θ , + µζ , cos(θ ) + µ ⎛ ω p ⎞ sin(θ ) = 0 ⎜ ⎟ ττ p ⎪ ττ ωe τ ⎝ ωe ⎠ ⎩
(2)
where ζ = x / L , τ = ωe t , ωc is the natural frequency of the column; ξ c is the damping ratio of the column; ω p is the pendulum frequency, ξ p is damping ratio of the pendulum absorber; µ is the mass ratio; ζ s is the amplitude of the excitation force and ωe is the excitation frequency. The external control force is applied directly to the main structure, in the opposite direction of the excitation force. It is given, in its non-dimensional form, as: Fc = f tanh( βζζ ,τ )ζ
(3)
being a function of the displacement and velocity of the column. The control force depends on two parameters: f and β . Considering the external control force, the state equations are ⎧ y1 = y2 ⎪ 2 ⎡ ⎛ ω ⎞2 ⎛ ωc ⎞ ⎪ ωc c y −⎜ ⎟ y ⎪ y 2 = ⎢ζ s ⎜ ⎟ sin(τ ) − f tanh( β y1 y2 ) y1 − 2ξ c ω ω e 2 ⎝ ωe ⎠ 1 ⎪ ⎣⎢ ⎝ e ⎠ ⎪ − µ y 4 cos( y3 ) + µ y4 2 sin( y3 ) ⎤⎦ /(1 + µ ) ⎨ ⎪ ⎪ y3 = y4 2 ⎪ ⎪ y = −2ξ ω p y − y cos( y ) − ⎛ ω p ⎞ sin( y ) ⎜ ⎟ 3 3 p ⎪ 4 ωe 4 2 ⎝ ωe ⎠ ⎩
(4)
where y1 is the displacement, y2 , the velocity and y 2 , the acceleration of the column, and y3 is the displacement, y4 , the velocity and y 4 , the acceleration of the pendulum absorber.
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2.1 Algebraic non-linear equations The nonlinear equations of motion can not be solved analytically. So, the solution can be either obtained by numerical integration or approximately by the use of perturbation techniques or harmonic expansions. Here an approximate solution is obtained by the use of the Galerkin-Urabe method, which transforms the system (2) into a system of nonlinear algebraic equations [8]. First consider that the solution of the system, subjected to a harmonic excitation of frequency ωe , is of the form: w = w cos(ωe t ) ,
θ = θ cos(ωet + ϕ )
(5)
where ϕ is the phase between the response of the column and that of the pendulum Consider now that there is a phase angle ψ of the force relative to the response of the tower. Thus the force can be written in the form: Fo sin(ωe t + ψ ) = Fc cos(ωe t ) + Fs sin(ωe t)
(6)
Substituting expressions (5) and (6) into the system of equations of motion (2), multiplying these equations by the weight functions φ1 = cos(ωe t ) and φ2 = sin(ωe t ) , respectively, and integrating each one of the four resultant equations from 0 to 2π / ω , the following nonlinear system of nonlinear algebraic equations is obtained 2 ⎧ ⎛ ωc ⎞ ⎛ ⎞ θ ⎪ζ (−1 − µ + ⎜ ⎟ ) − µθ cos(ϕ ) ⎜ J 0 (θ ) − J 2 (θ ) + J1 (θ ) ⎟ 2 ⎪ ⎝ ⎠ ⎝ ωe ⎠ ⎪ 2 ⎛ω ⎞ ⎪ = ζ s ⎜ c ⎟ cos(ψ ) ⎪ ⎝ ωe ⎠ ⎪ 2 ⎪ ⎪−2ζξ ωc + µθ sin(ϕ ) ⎛ J (θ ) − J (θ ) + θ J (θ ) ⎞ = ζ ⎛⎜ ωc ⎞⎟ sin(ψ ) ⎜ 0 ⎟ c s 2 1 ⎪⎪ ωe 2 ⎝ ⎠ ⎝ ωe ⎠ (7) ⎨ ⎪− µ θ cos(ϕ ) − ζ cos(2ϕ ) J (θ ) − J (θ ) − 2µξ ω p θ sin(ϕ ) ( ) p 2 0 ⎪ ωe ⎪ 2 ⎪ ⎛ ωp ⎞ + 2µ ⎜ ⎪ ⎟ cos(ϕ ) J1 (θ ) = 0 ⎪ ⎝ ωe ⎠ ⎪ ⎛ ⎛ ω p ⎞2 ⎞ ωp ⎪ ⎜ 2⎜ sin( ϕ ) µ J1 (θ ) − θ ⎟ + µζ J 2 (θ )sin(2ϕ ) + 2µ θ cos(ϕ ) = 0 ⎟ ⎪ ⎜ ⎝ ωe ⎠ ⎟ ωe ⎝ ⎠ ⎩⎪
(
)
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where J 0 (θ ) , J1 (θ ) and J 2 (θ ) are Bessel functions. This system of nonlinear equations is already in its non-dimensional form. The nonlinear algebraic equations possess as variables the frequency of excitation ωe , the amplitudes ζ and θ , and the phase angles ϕ and ψ .
3.
RESULTS
First the behavior of the column-absorber system with the passive control is investigated. Then the behavior of the system with the proposed hybrid control is analyzed. The main parameters of the system used in this investigation are: ωc = 1.255438 rad/s, ξ c = 0.7% , ξ p = 0.0% , µ = 0.04 (4.0% of the modal mass of the first mode) and ζ s = 0.007 . Figure 2 shows a comparison of the response of the column with and without a pendulum absorber using (a) a linearized formulation for the pendulum and (b) a nonlinear formulation for the pendulum, including both geometric and inertial nonlinearities. The nonlinearity of the pendulum has a considerable and beneficial influence on the results. For slender towers under harmonic excitation the absorber is most efficient when ωc ≈ ω p ≈ ωe , allowing some variations in the value of the frequency of excitation in the neighborhood of this point. Also, the absorber is less efficient under loads with short duration and during the initial transient response.
(a) Linearized pendulum
(b) Nonlinear pendulum
Figure 2. Comparison of the steady-state response of the column with (continuous line) and without (dotted line) the pendulum absorber for ω p / ωc = 1.00 .
To improve the effectiveness of the control during the transient response, a hybrid control system is suggested. The added control force is a non-linear
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variable stiffness device based on position and velocity feedback. Figure 3 clarifies the influence of the present control strategy on the behavior of the tower. Figures 3.a and 3.b show, respectively, the uncontrolled and controlled response of the tower (maximum normalized displacement vs. time). A marked decrease in vibration amplitudes is obtained. The same is observed for velocities and accelerations. Figure 3.c shows the evolution of the control force, while Figure 3.d shows the response of the pendulum. First, while the response of the pendulum increases steadily from rest, the added control force acts to control the response of the tower but soon tends to zero as the pendulum reaches its full potential, absorbing most of the vibration energy. In this analysis it was adopted ωc = ω p = ωe . In this example, the control force is computed considering f = 1.00 and β = 6000 . To evaluate the efficiency of the hybrid control, results were obtained for ωe / ωc = 0.8991 resulting in ωe = 1.13 rad/s. This point coincides with the point where the absorber-tower system reaches the maximum amplitude (first resonance – see Figure 2). Figure 4 shows a comparison of the tower and pendulum steady-state response with and without the active force. The results show that the hybrid control system practically eliminates the vibrations in the main resonance region of the coupled system.
(a) Column without control
(b) Column with hybrid control
(c) Control force Figure 3. Behavior of the system with hybrid control.
(d) Pendulum absorber
f = 1.00 and β = 6000 .
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Figure 4. Comparison of the amplitude of displacement with and without the control force.
In active, hybrid or adaptive control systems where feedback strategies are used, a certain amount of time is necessary to obtain and process the signal and, after that, evaluate and apply the control force. This time delay may cause a deterioration of the control system and can even cause instability. So, the influence of time delay is an essential step in the design of a given control system. Here the time delay, Td, is given as a percentage of the period of the tower response, T. Based on these observations, a parametric analysis was conducted to evaluate the critical values of f and β as a function of the time delay. The results are presented in Figures 5 and 6, where the variations of the critical values are shown as a function of the time delay. The worst case occurs when Td/T =0.5. From the results, one can conclude that reasonable values of f and β can be used without instability problems due to time delay. However if higher values of f and β are required, several compensation methods including modifications of phase shift of the measured state variables in the modal domain and methods of updating the measured quantities can be found in literature [9].
Figure 5. Variation of the critical value of β as a function of time delay.
Influence of a pendulum absorber on the behavior of a tower
Figure 6. Variation of the critical value of
4.
305
f as a function of time delay.
CONCLUSIONS
In the present paper, an analysis of a tower-pendulum system was carried out. Results show that the pendulum is effective in reducing the vibration amplitudes of the tower and that the non-linearity of the pendulum cannot be neglected is this class of problem. To improve the efficiency of the control, a hybrid control mechanism is proposed. Results demonstrate that the control force acts when the pendulum absorber starts to move. After the absorber reaches the amplitude necessary to control the oscillations of the column, the amplitude of the control force diminishes significantly. It is also observed that this control can practically eliminate the oscillations of the system in the main resonance region of the coupled pendulum-tower system. Results show that time delay has a major influence on the stability of the controlled response and that a proper choice of the active force parameters to avoid instability is an essential step in the control design. A detailed analysis of the results can be found in Orlando [8], where it is shown that the control can be effective even when the tower is under other types of load, including short pulses. However additional studies are necessary so that the efficiency of the hybrid control can be properly evaluated.
ACKNOWLEDGEMENTS The authors acknowledge the financial support of the Brazilian research agencies CAPES and CNPq.
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REFERENCES 1. 2. 3. 4.
5.
6.
7. 8. 9.
Korenev BG, Reznikov LM. Dynamic Vibration Absorbers: Theory and Technical Applications, Chichester, UK, John Wiley & Sons Ltd, 1993. Soong TT, Dargush, GF. Passive Energy Dissipation Systems in Structural Engineering, Chichester: John Wiley & Sons, 1997. Mustafa, G, Ertas A. “Dynamics and Bifurcations of the Coupled Column-Pendulum Oscillator ”, Journal of Sound and Vibration, 182, pp. 393-413, 1995. Yaman M, Sen S. “The Analysis of the Orientation Effect of Non-Linear Flexible Systems on Performance of the Pendulum Absorber ”, International Journal of NonLinear Mechanics, 39, pp. 741-752, 2004. Oueini SS, Nayfeh H, Pratt JR. “The Review of Development and Implementation of Active Non-Linear Vibration Absorber”, Archive of Applied Mechanics, 69. pp. 585-620, 1999. Wu B, Liu F, Wei D. “Approximate analysis method for interstory shear forces in structures with active variable stiffness systems”. Journal of Sound and Vibration, 286, pp. 963-980, 2005. Qiusheng L, Hong C, Guiqing L. “Static and Dynamic analysis of Straight Bars with Variable Cross-Section”. Computers & Structures, 59, pp. 1185-1191, 1994. Orlando D. “The Use of Pendulum Absorbers for Vibration Control of Slender Towers”, M.Sc. Dissertation, Catholic University, PUC-Rio, Rio de Janeiro, Brazil, 2006. Soong YT. “State-of-the–art review: Active control in civil engineering”. Eng. Structures, 10, pp. 74-83, 1988.
ROBUST FLUTTER CONTROL OF AN AIRFOIL SECTION THROUGH AN ULTRASONIC MOTOR H. Y. Hu, M. L. Yu Institute of Vibration Engineering Research, School of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China E-mail:
[email protected]
Abstract:
The paper presents how to realize the robust flutter suppression of an airfoil section through the control surface driven by an ultrasonic motor. The study starts with the combined theoretical and experimental modeling for the aeroservoelastic system with the damping uncertainties in plunge, and the modeling error of the servo, taken into consideration. Then, it turns to the design of the H ∞ controller and the µ-synthesis controller at a given flow speed. The numerical simulations and wind tunnel tests show that both controllers are able to suppress the airfoil flutter effectively, and increase the flutter speed by 23.4%. In addition, the µ-synthesis controller works better than the H ∞ controller not only in the suppression rate, but also in the robustness with respect to the time delay in the control loop.
Key words:
Flutter suppression, robust control, ultrasonic motor, delay.
1.
INTRODUCTION
The past decade has witnessed an increasing interest in the active flutter suppression with rapid development in both control theory and actuator technique [1]. Now the active flutter suppression is facing to several challenges, such as the new concept and development of actuators of high efficiency and reliability, the design of robust controllers, the integration of sensors, controllers and actuators. Among them, the challenge due to flap actuators is essential. The earlier flap actuators were mainly hydraulic, featuring large output torque, but excessively slow response. With recent progresses in functional materials, many kinds of novel flap actuators, especially those made of piezoelectric materials, have become available 307 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 307–316. © 2007 Springer.
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[2,3]. For example, Ardelean et al. invented a type of piezoelectric V-stack actuators [4], and Hall et al. proposed a piezoelectric actuator of double Xframe [5]. A recent study of authors indicates that the ultrasonic motor is a kind of promising flap actuators for flutter suppression of light wings [6], featuring not only small size and light weight, but also large output torque, quick response, and direct mechanical transmission. For example, an ultrasonic motor is able to output the torque of 10-100 times as large as a conventional electromagnetic motor of the same size or the same weight at low speed. The control design for active flutter suppression is always subject to uncertainties, such as the errors in modeling both aeroelastic systems and servos. For instance, the ultrasonic motor has to be greatly simplified in modeling since it is a complicated mechatronic system. Hence, the robust flutter suppression has drawn much attention. For example, Waszak designed an H ∞ controller and a µ-synthesis controller for BACT system so as to take the uncertainties in flow speed and dynamic pressure into account [7]. Borglund and Nilsson studied the µ-synthesis controller for a model of high-aspect-ratio composite wing to improve the robustness of controller with respect to the uncertain dynamic flow [8]. Motivated by the flutter suppression of light flights, this study aims at the robust flutter suppression of a two dimensional airfoil section through the control surface driven by an ultrasonic motor.
2.
MODEL FOR AEROSERVOELASTIC SYSTEM
The aeroservoelastic system of concern is a two-dimensional airfoil section of NACA0012 with a flap servo shown in Figure 1 and is subject to a steady flow at speed U as shown in Figure 2. The system involves three degrees of freedom, i.e., the plunge h along a guide rail, the pitch α around a hinge, and the flap β of control surface. A linear spring is attached to the airfoil section at its elastic axis to provide the plunge stiffness kh , and a pair of linear springs is installed at the same axis to offer the pitch stiffness k1 . The elastic axis of the airfoil section is located at a distance ab from the mid-chord, and the hinge of the control surface is located at a distance cb to the elastic axis, where b is the semi-chord of the airfoil. The mass center of
U kh +α
kα ab b
Figure 1. Airfoil section with a flap servo.
ultrasonic motor kβ
xa +h cb
xβ
+β
b
Figure 2. Simplified model of mechanics.
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the airfoil section is located at a distance xα from the elastic axis. The distance from the mass center of the control surface to the hinge is xβ . To avoid the excessively large pitch of the airfoil section in any possible flutter, a pair of linear springs is installed to provide an elastic stop of stiffness k2 via a lever when the pitch angle exceeds a specific value. Hence, the total pitch stiffness of the airfoil section is piecewise linear. The linear vibration of the airfoil section near the equilibrium yields ⎡ m Shα Shβ ⎤ ⎡ h(t ) ⎤ ⎡ch 0 0 ⎤ ⎡ h(t ) ⎤ ⎡ kh 0 0 ⎤ ⎡ h(t ) ⎤ ⎡ − L(t ) ⎤ ⎡ 0 ⎤ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎢S I S ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ hα α αβ ⎥ ⎢α(t ) ⎥ + ⎢ 0 cα 0 ⎥ ⎢α (t ) ⎥ + ⎢ 0 kα 0 ⎥ ⎢α (t ) ⎥ = ⎢ Ma (t ) ⎥ + ⎢ 0 ⎥ βc (t ) ⎣⎢ Shβ Sαβ Iβ ⎦⎥ ⎣⎢ β (t ) ⎦⎥ ⎣⎢ 0 0 cβ ⎦⎥ ⎣⎢ β (t ) ⎦⎥ ⎣⎢ 0 0 kβ ⎦⎥ ⎢⎣ β (t ) ⎥⎦ ⎣⎢ M β (t ) ⎦⎥ ⎢⎣ k0 ⎥⎦ (1)
where m is the mass of the airfoil section, Iα = mrα2 is the inertial moment of the airfoil section about the elastic axis, I β is the inertial moment of the control surface and the rotor of ultrasonic motor, S hα = mxα is the static moment of the airfoil section about the elastic axis, S hβ is the static moment of the control surface and the rotor of ultrasonic motor about the elastic axis, Sαβ is the static moment of the control surface and the rotor of ultrasonic motor about their axis, ch and cα are the damping coefficients of the airfoil section corresponding to the plunge and the pitch, kh and kα are the stiffness coefficients of the airfoil section corresponding to the plunge and the pitch, kβ and cβ are the stiffness coefficient and the damping coefficient of ultrasonic motor, − L(t ) , M α (t ) and M β (t ) are Theodorsen’s aerodynamic force and torque, β c is the input command of flap angle to the ultrasonic motor, and k0 is constant. To describe Theodoren’s aerodynamic load briefly, Equation (1) can be recast as the following matrix form s (t ) + Cs q s (t ) + K s q s (t ) = f ae (t ) + bβ c (t ) . M sq
(2)
where Theodorsen’s aerodynamic load f ae (t ) = f a (t ) + fc (t ). Here, f a (t ) is the non-circular part including the structural variables only, i.e.,
s (t ) , f a (t ) = −U 2 K a q s (t ) − UCa q s (t ) − M a q
(3)
where − πab −bT1 ⎤ ⎡ π ⎢ 2 2 M a = ρ b ⎢ − πab πb (1/ 8 + a ) 2b 2T13 ⎥⎥ , ⎢⎣ −bT1 2b 2T13 −b 2T3 / π ⎥⎦ 2
π −T4 ⎡0 ⎤ ⎢ ⎥, Ca = ρ b ⎢0 πb(0.5 − a ) bT16 ⎥ ⎢⎣0 bT17 −0.5bT4T11 / π ⎥⎦ 2
0 ⎤ ⎡0 0 ⎢ K a = ρ b ⎢0 0 T15 ⎥⎥ . (4) ⎢⎣0 0 T18 / π ⎥⎦ 2
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fc (t ) is the circular part yielding
fc (t ) = −U 2 Hq a (t ) − UCc q s (t ) − U 2 K c q s (t ) ,
(5)
where H = ρ R [ 0.0075 0.10055] , R = [2π −2πb(0.5 + a) bT12 ]T ,
Cc =
T bT11 1 1 1 ] , S2 = [0 1 10 ] , (6) ρ bRS1 , K c = ρ bRS2 , S1 = [1 b( − a) 2 2π π 2 2
q a ∈ R 2 is a set of Theodorsen’s aerodynamic variables related to Jones’ approximation and yields q a (t ) =
U Pq a (t ) + S1q s (t ) + US 2q s (t ) , P = diag[−0.0045 −0.3] . b
(7)
If a set of state variables q w = [q Ts q Ts q Ta ]T ∈ R8×1 is introduced for the entire system, the substitution of Equations (3), (5) and (7) into Equation (2) leads to the following state equation q w (t ) = A wq w (t ) + B w β c (t ) .
(8)
It should be pointed out that the inertial, stiffness and damping parameters in the above model need to be determined from the combined computations and measurements. Section 3 will focus on the identification of actuator parameters since it is a routine work to determine the parameters of an airfoil section.
3.
MODEL OF SERVO
When the input command β c (t ) of flap angle is applied to the ultrasonic motor, as shown in Equation (1), the flap angle β (t ) of the control surface is just the output angle of the ultrasonic motor provided that the output torque of the ultrasonic motor is much larger than the aerodynamic torque M β (t ) on the control surface. In this case, the control surface driven by the ultrasonic motor looks like a system of single degree of freedom, and yields I β β(t ) + cβ β (t ) + k β β (t ) = k0 β c (t ) .
(9)
Given the input command β c (t ) of flap angle, substituting β (t ) determined from Equation (9) into the first two equations in Equation (1) can greatly simplify the analysis of the entire system. Especially when β c (t ) ≡ 0 , the self-lock property of the ultrasonic motor leads to β (t ) ≡ 0 such that the entire system becomes a system of two degrees of freedom. In this study, the parameters in Equation (9) are identified from the measured frequency response function as following
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k0 , ωβ − ω + 2iζωβ ω 2
(10)
2
where ωβ = kβ / I β , ζ = cβ /(2 I β kβ ) and k0 = k0 / I β . Given the measured frequency response function G (ω ) , ωβ , ζ and k0 can be determined by minimizing the following error function n
n
i =1
i =1
J (k , ωβ , ζ ) = γ ∑ ∆µi2 + ∑ ∆θ i2 ,
(11)
where ∆µ = µexp − µpred and ∆θ = θ exp − θ pred are respectively the amplitude error and the phase error, γ = 500 is a weighted factor. In the frequency range 1~9Hz, the fitting results read ωβ = 357.07 rad/s, ζ = 0.598 and k0 = 0.9715 . 0 -5
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a. Amplitude response b. Phase response Figure 3. Frequency response of the servo, where the solid line is the theoretical prediction, D is the measured response without air flow, ▲ is the measured data in a wind tunnel test.
The fitted frequency response function shown in Figure 3 indicates that the deviations of amplitude and phase are less than 0.03 and 10D in the frequency range 1~ 9Hz. That is, the ultrasonic motor features quick response. Of course, the deviation increases when the frequency is higher than 9Hz. To check the feasibility of Equation (9) when the control surface and the ultrasonic motor are put into use, their dynamic property is measured in a wind tunnel test. In Figure 3, ▲ gives the measured frequency response at U = 22m/s , a little bit lower than the flutter speed of the aeroservoelastic system. It is easy to see that the deviations of both amplitude and phase are small in the frequency range 5 ~ 9Hz, which just covers the frequency range 5 ~ 7Hz of flutter. As a result, it is possible to neglect the effect of aerodynamic load on the frequency response of the servo and to simplify the design of control strategy.
4.
DESIGN OF ROBUST CONTROLLERS
Uncertainties always exist in modeling a real system. In the model established in Section 3 for the aeroservoelastic system, there are mainly two kinds of uncertainties. One is the friction along the guide rail and the other is the simplified model of single degree of freedom for the servo including both control surface and ultrasonic motor. The equivalent damping
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coefficient corresponding to the friction is not a constant, but a function in the vibration amplitude and vibration frequency, and the simplified model for the servo works only around the flutter frequency. To describe the above uncertainties, rewrite Equation (8) as q (t ) = [ A w + Ea Σ(θ )Fa ]q(t ) + [B w + Eb Σ(θ )Fb ]β c (t ) ,
(12)
where A w and B w are the matrices of nominal system, Ei , Fi , i = a, b are the uncertainties in the model, ∑(θ ) is the uncertainties of system parameters. The above matrices for uncertainties can be normalized in scaling. More specifically, let Wing in Figure 4 be the nominal model of the experimental system, Wh , Wa and Wb be the multiplication perturbations along the plunge direction, the pitch direction and the flap rotation, Wu be the weight function for adjusting input, Wd 1 be used to adjust the response velocity of the system, Wd 2 be the dynamic property of disturbance, wh , zh , wα , zα , wb , zb be the evaluation signals to ensure the multiplication robustness, wd , zd be the evaluation signal for the disturbance response. The numerical tests suggest the following weighted functions [9] Wh =
0.5s + 1.87 0.5s + 0.186 s + 0.866 200s + 10000 . (13) , Wα = , Wb = , Wu = s + 187 s + 186 s + 173 s + 10000
The design for an H ∞ controller is quite popular in publications. Hence, this subsection focuses on the µ-synthesis controller as following. Dolyle proposed the concept of µ-synthesis [10] based on the fact that any distributed uncertainties can be collected into a block-diagonal matrix ∆∈∆ set , where ∆ set is a set of block-diagonal matrices. To present his idea, let w and z be the input and output signals in the loop of uncertainties, d be the disturbance input and e be the evaluation signal for control performance. Then, the following transfer relation holds in the Laplace domain ⎡ z ( s ) ⎤ ⎡ M11 ( s ) M12 ( s ) ⎤ ⎡ w ( s ) ⎤ ⎢d ( s ) ⎥ = ⎢ M ( s ) M ( s ) ⎥ ⎢ e( s ) ⎥ . ⎣ ⎦ ⎣ 21 ⎦ 22 ⎦⎣
(14)
The analysis of M11 ( s ) determines the robust stability of controlled system, and the analysis of M 22 ( s ) gives the robustness of the controlled system. Given a matrix M11 ∈ ^ n×n , the µ value of M11 is defined as
µ∆ (M11 ) =
1 min{σ max (∆) | ∆ ∈ ∆ set ,det(I − M11∆) = 0}
,
(15)
where σ max (∆) is the maximal singular value of the block-diagnal matrix ∆ . That is, µ∆ (M11 ) measures the smallest uncertainty ∆ that causes system instability. The weaker µ∆ (M11 ) is, the more robust the system. Dolyle proved that the necessary and sufficient condition of the robust stability of the above system with uncertainty ∆∈∆ set is µ∆ [M11 ( s )] < γ if ∆ < 1/ γ holds. The µ-synthesis of a system with the uncertainty ∆
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yielding ∆ < 1/ γ is to design the controller such that µ∆ [M11 ( s )] < γ holds. The resolution of this problem can be conducted though successive iterations. Given min{ h , α } as the control target, it is quite straightforward to determine both H ∞ controller and µ-synthesis controller with help of MATLAB Toolbox. For each fixed flow speed, the design procedure gives a controller corresponding to the system in Figure 4, as well as the system equation as following q con (t ) = A con q con (t ) + B con y w (t ) , β c (t ) = Econ q con (t ) + Dcon y w (t ) ,
(16)
where y w (t ) is the measured signal. The controller design here results in a continuous system, but the data acquisition process and the digital controller are not. In this study, hence, the continuous controller is digitalized with sampling frequency of 1000Hz. Normally, the controllers designed above are of high orders and should be simplified. In this study, the Bode chart and the Hankel singular values indicate that the order of both controllers can be reduced from 12 to 9. 20
zh zα
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5.
Critical flutter speed Uf=22.3m/s
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zu
Figure 5. Variations of real parts of eigenvalues with an increase of flow speed.
NUMERICAL SIMULATIONS Let the output equation for Equation (8) be y w (t ) = E wq w (t ) .
(17)
The combination of Equations (8), (16) and (17) leads to a set of dynamic equations for the entire aeroservoelastic system. The digital controllers used in wind tunnel tests suggest that the numerical simulations should turn to the corresponding difference equations as follows q total (n + 1) = A total q total (n) , y total (n) = Etotal qtotal ( n) ,
(18)
where ⎡ Aw A total = ⎢ ⎣B con E w
B w Econ ⎤ ⎡E , Etotal = ⎢ w ⎥ A con ⎦ ⎣0
0 ⎤ ⎡q ⎤ , q total = ⎢ w ⎥ . (19) ⎥ Econ ⎦ ⎣q con ⎦
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The system parameters are set as L=0.3m, b=0.1m, a=-0.5, c=0.5, ρ=1.225 Kg/m3, m=1.85Kg, mcon=0.053Kg, Iα=3.142×10-3Kg·m, Shα=0.0309Kg·m, Shβ=8.608×10-4Kg·m, Sαβ=1.215×10-4Kg·m, ωh≡ kh / m /(2π) =5.9Hz, ωα≡ kα / Iα /(2π) =4.5Hz, ωβ≡ kβ / I β /(2π) =61.34 Hz. For a given flow at speed U, the computation of eigenvalues gives Figure 5, where the critical flutter speed of flow is 22.3m/s when the controller does not work. Given such a critical flutter speed, solving Equation (18) gives the time history of dynamic response of system.
6.
WIND TUNNEL TESTS
Figures 6 and 7 show the test rig, where a two-dimensional airfoil section is installed in an open wind tunnel and the control surface is directly driven by an ultrasonic motor. Two angular sensors measure the pitch angle of the airfoil section and the relative flap angle with respect to the airfoil section, respectively. A Polytec/PDV-100 laser vibrometer senses the plunge velocity and outputs the plunge displacement after on-line integration. Those measurements in three channels go into an A/D converter and then into the control computer, which establishes the control strategy and sends them to a D/A converter and the amplifier of the ultrasonic motor. As a result, the motor drives the control surface to suppress the possible flutter.
Figure 6. Test rig installed in an open wind tunnel; 1. outlet, 2. inlet, 3. airfoil section, 4. seating, 5. vibrometer, 6. power amplifier, 7. charge amplifier, 8. controller.
Figure 7. Ultrasonic motor and pitch mechanism; 1. ultrasonic motor, 2. angular sensor, 3. seating, 4. pitch springs, 5. elastic stop along pitch spring.
The wind tunnel tests indicates that the critical flutter speed of the uncontrolled airfoil section is 22.4m/s, which well coincides with the prediction, and that the controller can successfully suppress the flutter as soon as it starts as shown in Figure 8. However, the controller fails to work when the flow speed reaches 27.7m/s. Figure 9 presents a comparison of both H ∞ controller and µ-synthesis controller in suppressing the flutter caused by a small pitch disturbance, and indicates that the µ-synthesis controller is better than the H ∞ controller in such a test of flutter suppression.
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Figure 8. H ∞ suppression of a limit cycle flutter at flow speed 22.6m/s.
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Figure 9. Flutter suppressions subject to a small disturbance at flow speed 25m/s.
In order to remove both DC component and noise in measurements, a digital filter is introduced into the control loop. For a better performance, the digital filter should be of two orders or even higher. As such, the group delay in the digital filter is so large that both controllers exhibit poor performance. When the group delay of the digital filter reaches 0.02s or so, for example, the flutter decays when the H ∞ controller starts to work, but a new flutter of lower frequency appears soon as shown in Figure 10a. Normally, the µsynthesis controller works better than the H ∞ controller as shown in Figure 10b, regarding to the robustness with respect to the time delay. In the design of both robust controllers, it is certainly possible to take the digital filter with a group delay as a linear sub-system and include it into the mathematical model of entire aeroservoelastic system. In such a case, the design guarantees the stability of controlled system with the digital filter of given group delay, but fails to do so when the group delay becomes either longer or shorter. 18
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CONCLUDING REMARKS
The ultrasonic motor, as a kind of light flap actuator of good frequency response and large output torque, is applicable to the active flutter suppression of a two-dimensional airfoil section. To improve the robustness of active flutter suppressions, it is possible to apply either H ∞ controller or µ-synthesis controller to the flutter suppression after the disturbance is introduced to describe the uncertainties in system damping and servo modeling as the multiplication disturbances to the nominal system. The numerical simulations and wind tunnel tests show that both robust controllers work effectively and increase the critical flutter speed from 22.4m/s to 27.5m/s, i.e., by 23.4%. Both robust controllers are able to suppress the limit cycle flutter of uncontrolled system when the flow speed is beyond 22.4m/s. Relatively, the µ-synthesis controller works better than the H ∞ controller, especially when there is a time delay caused by the digital filter in the control loop.
REFERENCES 1. 2. 3. 4.
5.
6.
7.
8. 9. 10.
Librescu L, Marzocca P. “Advances in the Linear/Nonlinear Control of Aeroelastic Structural Systems”, Acta Mechanica, 178, pp. 147-186, 2005. Giurgiutiu V. “Active-materials Induced-strain Actuation for Aeroelastic Vibration Control”, Shock and Vibration Digest, 32, pp. 355-368, 2000. Niezrecki Ch, Brei D, Balakrishnan S, et al. “Piezoelectric Actuation: State of the Art”, Shock and Vibration Digest, 33, pp. 269-280, 2001. Ardelean EV, et al. “Flutter Suppression Using V-Stack Piezoelectric Actuator”, Proceedings of the 11th AIAA/ASME/AHS Conference on Adaptive Structures, pp. 7-10, 2003. Hall SR, Tzianetopoulou T, et al. “Design and Testing of a Double X-frame Piezoelectric Actuator”, Proceedings of SPIE – The International Society for Optical Engineering, 3985, pp. 26-37, 2000. Yu ML, Hu HY. “Flutter Suppression of Two-dimensional Aerofoil Section Using Ultrasonic Motor as Actuator”, Journal of Vibration Engineering, 18, pp. 418-425, 2005. Waszak RM. “Robust Multivariable Flutter Suppression for Benchmark Active Control Technology Wind-Tunnel Model”, Journal of Guidance, Control and Dynamics, 24, pp. 147-153, 2001. Borglund D, Nilsson U. “Robust Wing Flutter Suppression Considering Aerodynamic Uncertainty”, Journal of Aircraft, 41, pp. 331-334, 2004. Waszak RM, Andrisani D. “Uncertainty Modeling via Frequency Domain Model Validation”, AIAA 99-3959, 1999. Zhou K, Doyle J. Essential of Robust Control, New York, Prentice-Hall, 1998.
PERIODIZATION AND SYNCHRONIZATION OF DUFFING OSCILLATORS SUSPENDED ON ELASTIC BEAM K. Czołczyński, A. Stefański, P. Perlikowski, T. Kapitaniak Division of Dynamics, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Lodz, Poland
Abstract:
We consider the dynamics of chaotic oscillators suspended on the elastic beam. We show that for the given conditions of the beam oscillations, initially uncorrelated chaotic oscillators become periodic and synchronous. In the case of the mismatch of the excitation frequency in each oscillator we observe beating-like behavior. We argue that the observed phenomena are generic in the parameter space and independent of the number of oscillators and their location on the elastic structure.
Key words:
Chaotic behavior, synchronization, beating of oscillations.
1.
INTRODUCTION
The phenomenon of synchronization in dynamical and, in particular, mechanical systems has been known for a long time. In the last decade of the XX century the idea of synchronization has been adopted for chaotic systems [1-4]. In the current studies we consider the synchronization of nonlinear chaotic oscillators located on (coupled through) elastic structure. We present a numerical study of a realistic model of two double well-potential Duffing oscillators suspended on the elastic beam. Both oscillators are externally excited by periodic force with a frequency ω. We show that for the given conditions of the elastic structure, initially uncorrelated chaotic oscillators can synchronize in periodic regime. One can observe the phenomena of the periodization of oscillators, i.e., the behavior 317 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 317–322. © 2007 Springer.
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of nonlinear oscillators becomes periodic as a result of interaction with the elastic structure. We argue that the observed phenomena are generic in the parameter space and independent of the number of oscillators and their location on the elastic structure.
2.
THE MODEL We consider two double well-potential Duffing oscillators described by: yi + d y yi + k y yi + kd yi3 = F cos (ωi t )
(1)
where dy, ky, kd, F, ωi are constant and i = 1,2. Oscillators (1) have three equilibria: unstable for y0=(0,0) and two stable for y-1=(-1,0) and y1=(1,0). In our numerical analysis we assumed dy=0.168, ky=-0.5, kd=0.5 and ω=1, i.e.; oscillators (1) show chaotic behavior [5]. We assume that two identical Duffing oscillators are suspended on the elastic beam in the points x=l1, x=l2 as shown in Figure 1. The evolution of the system is described by 4 ∂2 z ∂z 2 ∂ z + d + c = p ( x, t ) ∂t 2 ∂t ∂x 4
(2)
Figure 1. Two Duffing oscillators suspended on the elastic beam. The beam was discretized by 5 mass elements u1-5 located on elastic string.
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where c2=EJl/M and d=αM describes internal damping of the beam and p(x,t) describes the signal transmitted by the ith oscillator to the beam, p ( x ,t ) = (d y ( yi − z ) + k y ( yi − z ) + kd ( yi3 − z 3 )) / M for x=li and p(x,t)=0 for x≠li. We assumed that the beam is simply supported at both ends so we have the following boundary conditions; y (0, t ) = 0 , y (l , t ) = 0 , d 2 y (0, t ) / dx 2 = 0 and d 2 y (l , t ) / dx 2 = 0 . In our studies Equation (2) has been discretized in such a way that the i =5 continuous beam of mass M = ∑ i =1 ui was replaced by the massless beam on which k discrete identical masses u are located (ku=M). The number of discrete masses k was selected in such a way as to have the first two eigenfrequencies of continuous and discrete beam approximately equal. As the result of discretization one obtains the following set of ODEs m1 y1 + d y1 ( y1 − z2 ) + k y1 ( y1 − z2 ) + kd 1 ( y1 − z2 ) = F1 sin (ω1t ) 3
m2 y2 + d y 2 ( y 2 − z4 ) + k y 2 ( y2 − z4 ) + kd 2 ( y2 − z4 ) = F2 sin (ω2 t ) 3
u1 z1 + ϑ u1 z1 + k11 z1 + k12 z2 + k13 z3 + k14 z4 + k15 z5 = 0 u2 z2 + ϑ u2 z2 + k21 z1 + k22 z2 + k23 z3 + k24 z4 + k25 z5 = = d y1 ( y1 − z2 ) + k y1 ( y1 − z2 ) + kd 1 ( y1 − z2 )
3
(3)
u3 z3 + ϑ u3 z3 + k31 z1 + k32 z2 + k33 z3 + k34 z4 + k35 z5 = 0 u4 z4 + ϑ u4 z4 + k41 z1 + k42 z2 + k43 z3 + k44 z4 + k45 z5 = = d y 2 ( y 2 − z4 ) + k y 2 ( y2 − z4 ) + kd 2 ( y2 − z4 )
3
u5 z5 + ϑ u5 z5 + k51 z1 + k52 z2 + k53 z3 + k54 z4 + k55 z5 = 0
where kij coefficient of stiffness matrix of the beam, ϑ coefficient of damping matrix of the beam. In our numerical simulations we have assumed that all Duffings oscillators evolve on chaotic attractors when they start to interact with a beam.
3.
SYNCHRONIZATION, PERIODIZATION AND BEATING OF THE OSCILLATIONS
In Figure 2(a,b) we show bifurcation diagrams yi versus c-2 describing behavior of two oscillators suspended on the beam. We assumed that the oscillators are located symmetrically on the beam. In the case of increasing c-2 (Figure 2(a)) for low values of c-2 oscillators behave periodically and are synchronized. Synchronization is lost at c-2=250. But the oscillators are still periodic. For larger values of control parameter the behavior of the
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oscillators becomes uncorrelated and chaotic. When c-2 decreases (Figure 2(b)) oscillators become periodic at c-2=360 but synchronization is not observed. The comparison of Figure 2(a) and Figure 2(b) shows dynamical hysteresis in the neighborhood of c-2=250 and multistability (coexistence of different attractors for smaller values of c-2. In the considered system in the case of periodic behavior of both oscillators four different modes of oscillations are possible; (i) both oscillators evolve around upper stable fixed point, (ii) both oscillators evolve around lower stable fixed point, (iii) left oscillator evolve around lower stable fixed point while the right one around upper stable fixed point, (iv) opposite to the case (iii). Modes (i) and (ii) are symmetrical in which both oscillator are synchronized.
Figure 2. Bifurcation diagrams y1,2 versus c-2; (a) c-2 increases, (b) c-2 decreases.
Figure 3. Initial conditions leading to different modes of oscillations; (a) 4 modes exist, (b) 2 modes exist.
All modes together with the initial conditions which guarantee their appearance are shown in Figure 3(a). The ranges of initial conditions leading
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to each mode are indicated respectively in light grey, dark grey, white and black. In our calculations we took z10 −50 = z10 −50 = y10 = y 20 = 0 and allowed y10 and y20 to vary in the interval (y10,y20) [-1,1]×[-1,1]. When at c-2=250 symmetrical modes (i) and (ii) disappear. The initial conditions leading to the two survived modes ((iii) and (iv)) are shown in Figure 3(b). Structure of Figure 3(b) shows that there exist regions in the phase space where small uncertainty of the initial conditions can lead the system behavior to different attractors. Now let us assume that the frequencies of excitation in both oscillators are slightly different, i.e., ω1–ω2=ε, where ε<<1. In this case one observes a beating behavior shown in Figure 4(a-d). It should be noted here that the period of the beating indicated in Figure 4(a) is equal to Tb=2π/ε, i.e., as known in the linear theory of oscillations.
Figure 4. Time series showing beating of oscillators; (a,b) periodic beating, (c-d) chaotic beating. (a) 1/c-2=100; (b) 1/c-2=117.5; (c) 1/c-2=118; (d) 1/c-2=250.
Periodization of the oscillators behavior can also be observed in the case when oscillators are not located symmetrically on the beam [6]. In the case of n oscillators all oscillators synchronize in phase and symmetrical
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oscillators perform complete synchronization for low values of c-2, but these results will be presented elsewhere [7].
4.
CONCLUSIONS
To sum up, we have investigated the possibility of the synchronization of nonlinear chaotic oscillators located on (coupled through) elastic structure. In the numerical study we have considered a realistic model of two double well-potential chaotic Duffing oscillators suspended on the elastic beam. We identified the phenomenon of the periodization of oscillators in which their behavior becomes periodic and synchronous as a result of the interaction with the elastic structure. In the case of the mismatch of the excitation frequency ω in each oscillator we have observed chaotic beating-like behavior. We have shown the analogy of the observed behavior with the phenomena known from the linear theory of oscillations.
ACKNOWLEDGEMENTS This work is supported by the Ministry of Science and Higher Education (Poland) under the project no. 2490/T02/2006/31.
REFERENCES 1. 2. 3. 4. 5. 6.
7.
Pecora L, Carroll TS. Physical Review Letters, 64, pp. 821, 1990. Boccaletti S, Kurths J, Osipov G, Valladares DL, Zhou CS. “The synchronization of chaotic systems”, Prysics Reports, 366, pp. 1, 2002. Chen G. (ed.) Controlling Chaos and Bifurcations, CRC Press, Boca Raton, 1999. Kapitaniak T. Controlling Chaos, Academic Press, London 1996. Dowell E, Pezeshki C. “On the understanding of chaos in Duffing’s equation including a comparison with experiment”, Journal of Applied Mechanics, 53, pp. 229, 1986. Czołczynski K, Kapitaniak T, Perlikowski P, Stefański A. “Periodization of Duffing oscillators suspended on elastic structure: mechanical explanation”, Chaos Solitons and Fractals, in press. Czołczynski K, Perlikowski P, Stefański A, Kapitaniak T. “Synchronization of selfexcited oscillators suspended on elastic structure”, Chaos Solitons and Fractals, in press.
EFFECTS OF NOISE ON SYNCHRONIZATION AND SPATIAL PATTERNS IN COUPLED NEURONAL SYSTEMS Q. Y. Wang1,2, Q. S. Lu1, X. Shi3, H. X. Wang4 1
Beijing University of Aeronautics and Astronautics, Beijing 100083, China, E-mail:
[email protected] 2 Inner Mongolia Finance and Economics College, 010051, Huhhot, China 3 Beijing University of Posts and Telecommunications, Beijing 100876, China 4 Nanjing University of Science and Technology, Nanjing 210094, China
Abstract:
The effects on synchronization and spatiotemporal patterns in coupled neurons are investigated. Firstly, noise-induced and noise-enhanced synchronization can be observed in coupled neuronal systems as the noise level attains suitable values. Secondly, the collective behaviour of a square lattice Hodgkin– Huxley neuronal network model with white noise is studied. It is shown that when the noise level reaches an intermediate value, the Hodgkin–Huxley neurons in this square lattice will exhibit an ordered circular structure. However, as the noise level increases, the ordered circular structure is distorted, and eventually totally destroyed when the noise level is high enough. This manifests the existence of spatial coherence resonance in the network and may be instructive to understand information processing of neural systems in the presence of noise.
Key words:
Neuronal systems, noise, synchronization, spatial patterns.
1.
INTRUDUCTION
It is commonly accepted that a single neuron in the vertebrate cortex connects to more than 10,000 postsynaptic neurons via a synapses–forming complex network. Synchronization is of great importance for signal encoding and transduction in information processing of neurons in two respects: (1) a single neuron may faithfully encode temporal information in 323 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 323–332. © 2007 Springer.
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the timing of successive spikes, (2) a group of neurons can respond collectively to a common synaptic current through synchronization. Moreover, coupled neurons can display various synchronous rhythmic activities due to their interactions. Physiological experiments have indicated the existence of synchronous motion of neurons in different areas of the brain of some animals [1–5]. Complete synchronization of coupled Hindmarsh–Rose neurons with ring structure was investigated by means of Lyapunov stability theory [6]. It was noted in three types of regular networks that different coupling styles had different critical values, which depended on specific coupling styles when the neurons achieved complete synchronization [7]. Phase synchronization was studied in a small world chaotic neural network [8]. They found that the phase synchronization are absent in the regular network can be greatly enhanced by random shortcuts between the neurons. However, in neural systems, noise arises from many different sources, such as the quasi-random release of neurotransmitter by the synapse, the random switching of ion channels, and most important random synaptic inputs from other neurons. Noise was seen traditionally as a disturbance which limits the accuracy of information transfer, while it is true that intrinsic noise presented in excitable cells can play a relevant role in the detection and processing of signals. In recent years, noise sources are viewed as important dynamical components, and it may influence synchronization and spatial patterns in neural systems in different ways. Perc studied effects of spatiotemporal additive noise on the spatial dynamics of excitable neuronal media. It was shown that there existed an optimal noise intensity at which spatial coherence resonance was maximal [9]. Effects of small-world connectivity on noise-induced temporal and spatial order in neural media have also been investigated in [10]. Although there have been many results for the effects of noise on neural systems, they are still worthy of being studied because of complexity of neural systems. In this paper, we mainly discuss noise effects on synchronization of coupled neurons, including noise-induced and noise-enhanced synchronization. At the same time, spatial patterns in a square lattice neuronal network are also investigated.
2.
NOISE-INDUCED SYNCHRONIZATION
2.1 Noise-induced complete synchronization In neural systems, different neurons connected to another group of neurons may receive a common input signal, which often approaches a
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Gaussian distribution as a result of integration of many independent synaptic currents. In general, when two identical neurons start from different initial conditions, the spike timings of them may not coincide. However, when a randomly fluctuating injection current is applied to two neurons, the spike timings can be driven into synchronization. Consider two uncoupled identical neurons without coupling driven randomly by a common additive Gaussian white noise (GWN), and the equations are given as follows: 1 = f (w 1 ) + ξ (t ) w 2 = f (w 2 ) + ξ (t ) w
(1)
where w i = ( xi , yi , zi )(i = 1, 2) denotes the state variables of the i -th neuron governed by Hindmarsh-Rose (HR) model, and ξ (t ) = (ξ (t ),0,0) with ξ (t ) being GWN. Consider the synchronization error e = w 2 − w1 , which is assumed to be small, and linearize the synchronization error system, then we obtain e = D f (w )e
(2)
where D f (w ) stands for the Jacobian matrix on the trajectory of a neuron. This linearized equation has the same form as the noise-free system, while the trajectory of w here is different from that of the noise-free system. Based on the linearized dynamics, Lyapunov exponents are well defined similar to that in the deterministic case. According to Equation (2), two neurons will reach CS when the largest Lyapunov exponent becomes negative. We calculate the largest Lyapunov exponent λ1 as a function of the noise intensity D for I = 1.0,1.31,1.4,3.0, 4.0 , which correspond respectively to the quiescent state, subthreshold oscillation, periodic bursting, 0.14
4
(a)
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I= I= I= I= I=
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Figure 1. (a) The largest Lyapunov exponents λ1 vs. the noise intensity; (b) the mean synchronization error the mean synchronization error e vs. the noise intensity for I = 1.0 (○), I = 1.31 (△), I = 1.4 (◇), I = 3.0 (☆) and I = 4.0 (＊).
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chaotic and periodic spiking of neurons without noise. In parallel, the average synchronization error between two neurons defined by
e =
( x2 − x1 ) 2 + ( y2 − y1 ) 2 + ( z2 − z1 ) 2
, in which the angle bracket
means time averaging, is also computed. It is seen from Figure 1(a) that λ1 undergoes a transition from positive to negative at a critical noise intensity. Beyond this critical point, two identical HR neurons with the same input converge to an identical spike sequence after a short transient time, and the synchronization error vanishes as shown in Figure 1(b).
2.2 Noise-induced phase synchronization Nonidentity is common in neurons, and it is important to study the synchronization phenomena of coupled nonidentical neurons. Due to the nonidentity of neurons, coupled neurons will not achieve complete synchronization but a weaker one, that is, phase synchronization (or frequency synchronization). For a neuron stimulated by noise, it will preserve spiking but the fluctuation causes stochastic firing and the spike sequences are influenced significantly. Under this situation, spikes are counted when the upstroke of the voltage variable reaching a given threshold peak. Then the residence time between two successive spiking events is Tk = τ k +1 − τ k , which is usually called interspike interval. Correspondingly, the mean firing period is 1 n defined by T = lim ∑ Tk and the mean firing frequency by n →∞ n k =1 ω = 2π / T . With this definition, we investigate the frequency synchronization of two stochastic neurons. It is found that a common noise can induce complete synchronization in two identical neurons, while a common noise cannot induce frequency synchronization in two neurons with different external currents. Then we explore the noise effects on frequency synchronization. Fix D2 = 0.05 (the noise intensity added to the second neuron) while let D1 (the noise intensity added to the first neuron) changeable. We consider two uncoupled HR neurons with different external current. Specifically, choose I = 1.25, 1.3 for excitable states, I = 1.45, 1.5 for periodic bursting states, I = 3.0, 3.05 for chaotic firing states and I = 3.6, 3.65 for periodic spiking state, respectively. The dependence on the noise intensity of the relative ω1 − ω2 mean frequency difference ∆ω = is plotted in Figure 2. ω1 + ω2
Effects of noise on synchronization and spatial patterns (a)
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Figure 2. The diagrams of the relative mean frequency difference ∆ω versus the noise intensity D1 for uncoupled neurons with different firing patterns, where (a) excitable states (○) and periodic bursting (△); (b) chaotic firing pattern (□) and periodic spiking (◇).
It is shown that mean frequency locking occurs with different values of the noise intensity D1 and D2. It is also clear that there is a greater range of the noise intensity for the spiking neurons than that for the bursting neurons, with which the frequency synchronization is achieved.
3.
NOISE-ENHANCED SYNCHRONIZATION
In fact, as in neuroscience, systems are often coupled besides the random forcing. In this section, we study the interplay between noise and the coupling of the systems to the effect of synchronization.
3.1 Noise-enhanced complete synchronization Previously, we have considered complete synchronization of two coupled HR neurons without noise and derived the critical values of coupling strength for different firing patterns of neuron with varying external current. For weak coupling strength such as C = 0.1, two coupled identical HR neurons will not reach complete synchronization when they are out of a quiescent state. Under this situation, we consider the effect of the common noise added to them. Considering two electrically coupled HR neurons driven by a common GWN with the network model 1 = f (w1 ) + C ( x2 − x1 ) + ξ (t ) w w 2 = f (w 2 ) + C ( x1 − x2 ) + ξ (t )
Similarly, consider synchronization error, we know that two neurons will reach complete synchronization when the largest conditional Lyapunov
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exponent becomes negative. The largest conditional Lyapunov exponent λ1c and the average synchronization error e is calculated as functions of the noise intensity D as shown in Figure 3. It can be seen that λ1c undergoes a transition from positive to negative at a critical noise intensity, and the synchronization error vanishes beyond this critical value. 0.1
3 I = 1.0 I = 1.31 I = 1.4 λ=0 I = 3.0 I = 4.0
(a) 0.08
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noise intensity D
Figure 3. (a) The largest conditional Lyapunov exponents λ1c vs. the noise intensity; (b) the mean synchronization error e vs. the noise intensity when I = 1.0 (○), I = 1.31 ( ), I = 1.4 ( ), I = 3.0 ( ) and I = 4.0 (＊).
Finally, we compute the critical values of the noise intensity with respect to the external direct current, which correspond to various firing patterns of a single neuron under uncoupled and coupled situations. It is obvious from Figure 4 that the critical value of the noise intensity increases with the external direct current for two uncoupled or weakly coupled HR neurons. Recalling that with increasing current I , the firing pattern of a single HR neuron transits from bursting to spiking, which means that the bursting firing pattern is easier to induce synchronization by noise than the spiking firing pattern. With coupling, the critical values of noise intensity are less than those without coupling, which implies that weak coupling enhances the sensitivity of synchronization of two neurons to the noise stimulus. 3
2.5
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cr
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C = 0.1 1.5
C = 0.2
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Figure 4. The critical values of the noise intensity for complete synchronization of two uncoupled HR neurons (○) and two weakly coupled HR neurons (△ and ◇) with a common GWN vs. the external direct current added to each neuron.
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Noise-enhanced phase synchronization
Furthermore, we explore noise-enhanced phase synchronization. Firstly, we consider the situation when two neurons lie in the excitable regimes with I1 = 1.25, I 2 = 1.3. With different values of the coupling strength, we also calculate the relative mean frequency difference as the function of the noise intensity D1 with D2 = 0.05 . From Figure 5(a), it is clearly seen that a larger coupling strength corresponds to a greater range of the noise intensities where both neurons become entrained. That is, coupling enhances stochastic synchronization of two excitable HR neurons. This is because that with the coupling strength increasing, coupling plays more important role on stochastic synchronization than the noise. 0.18
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Figure 5. The diagrams of the relative mean frequency difference ∆ω vs. the noise intensity D1 with different coupling strength for different firing patterns with (a) excitable, (b) periodic bursting.
Secondly, we consider the situation when two neurons exhibit periodic bursting with I1 = 1.25, I 2 = 1.3. The relative mean frequency difference is calculated as a function of the noise intensity D1 with D2 = 0.05 for this firing patterns when the coupling strength is fixed at C = 0.1, 0.2, 0.3. Figure 5(b) presents the numerical results. It is evident that with the coupling strength increasing, the noise intensity with which the mean frequency locking occurs increases first and then decreases and its range becomes larger. In all above simulation results, the frequency locking is considered occurring when the relative mean frequency difference is smaller than 0.01.
4.
SPATIAL PATTERNS IN A SQUARE-LATTICE HH NEURONAL SYSTEMS
In this section, we focus on spatial patterns in a square–lattice HH neuronal network in the presence of noise. Noise-induced ordered circular
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structure can be observed at an intermediate noise level and it can be destroyed when the noise level becomes high enough. Hence, there exists an optimal noise level, at which circular structure is well-preserved and exhibits maximal order. This phenomenon is called spatially coherence resonance. Spatial patterns of an HH neuronal network of size 128 × 128 with the nearest diffusive coupling are studied as the noise level σ changes. It is observed that for lower noise levels, neurons can not excite large amplitude spikes to depict any particularly outstanding spatial structures. While, for an intermediate noise level, noise-induced patterns emerge in space, which orderly propagate through the neurons with circular waves as shown in Figures 6(a) and (b). It is obvious that spatial patterns of the circular structure are characterized by layers with one smaller cycle being surrounded by other larger one. Furthermore, the number of ordered cycles increases with the increment of the noise level. It is well-understood that larger noise can evoke more spikes in a given time interval, and thus local excitations can propagate more closely through space, which accounts for the increase of the number of ordered cycles.
Figure 6. Spatial pattern formation in the square lattice HH neuronal network for different noise levels. All figures depict values of Vi , j on a 128 × 128 square grid at a given time t. The noise level σ is: (a) 5, (b) 15, (c) 25, (d) 60 , respectively. Here, the diffusive coefficient is D = 0.5.
However, Figure 6(c) shows that with the noise level increasing further, ordered spatial patterns are distorted by strong noisy perturbations. It is a fact that the stronger the noise level is, the more are the neurons fired
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simultaneously and seemingly randomly. Thus, a neuron be can frequently and randomly affected by others, which results in deformation of the
originally ordered circular structures. When the noise level is high enough, the circular structure can not persist and is replaced by disordered random portraits as observed in Figure 6(d). Hence, it is shown that there exists an optimal level of noise at which the spatial resonance in the square lattice HH neurons is resonantly pronounced.
5.
CONCLUSIONS AND DISCUSSIONS
Synchronization and spatial patterns were investigated in coupled neural systems when neurons are subjected to noise. It was shown that noise could not only induce synchronization of neurons but also enhance synchronization of neurons with weak coupling. Furthermore, we studied the effects of noise on different firing regimes. At the same time, spatial patterns in coupled neuronal system were also studied. Results showed that there existed ordered circular waves with a layered structure in this network at an intermediate noise level. As the noise level increases, the ordered circular waves could be deformed and finally dominated by some random patterns.
ACKNOWLEDGEMENTS This work was supported by the National Natural Science Foundation of China (No. 10432010, 10572011 and 60504018).
REFERENCES 1.
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Whittington MA, Traub RD and Jefferys JGR. “Synchronized oscillations in interneuron networks driven by metabotrophic glutamate receptor activiation”, Nature, 373, pp. 612615, 1995. Hansel D and Sompolinsky H. “Synchronization and computation in a chaotic neural network”, Phys. Rev. Lett., 68, pp. 718-721, 1992. Fujii H, Ito H, Aidhhara KI and Tsukada M. “Dynamical cell assembly hypothesis– Theoretical possibity of spatio–temporal coding in the cortex”, Neural Networks, 9, pp. 1303-1350, 1996. Gray CM, König P and Singer AK. “Oscillatory response in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties”, Nature, 338, pp. 334-337, 1989. Gray CM and McCormick DA. “Chattering cells: Superficial pyramidal neurons contributing to the generation of synchronous osciallators in the visual cortex”, Science, 274, pp. 109-113, 1996.
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Shi X and Lu QS. “Complete synchronziation of coupled Hindmarsh-Rose neurons with ring structure”, Chin. Phys. Lett., 21, pp. 1695-1698, 2004. 7. Wang QY, Lu QS, Chen GR and Guo DH. “Chaos synchronization of coupled neurons with gap junctions’, Phys. Lett. A, 356, pp. 17-25, 2006. 8. Wang QY and Lu QS. “Phase synchronization in small world chaotic neural networks”, Chin. Phys. Lett., 6, pp. 1329-1332, 2005. 9. Perc M. “Spatial coherence resonance in excitable media”, Phys. Rev. E. 72, 016207, 2005. 10. Perc M. “Effects of small-world connectivity on noise-induced temporal and spatial order in neural media”. Chaos, Solitons and Fractals (in press).
PART 6
DYNAMICS OF TIME-DELAY SYSTEMS
STABILITY AND RESPONSE OF STOCHASTIC DELAYED SYSTEMS WITH DELAYED FEEDBACK CONTROL Y. F. Jin, H. Y. Hu Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, China. E-mail:
[email protected] Abstract:
The stability and response for delayed dynamical systems with delayed feedback control under additive or multiplicative Gaussian white noise excitations are studied by using the stochastic averaging method. The stochastic differential equation with time delay is transformed into Itô stochastic differential equation without time delay first. Then, the averaged Itô stochastic differential equations for the system are established and the stationary solutions of the averaged Fokker-Planck equations are derived. Finally, the analytical expressions of the response and stability conditions are derived for both cases through two examples. The boundedness conditions of the mean square of the amplitude for additive Gaussian white noise are obtained. Meanwhile, the moment stability condition for the case of multiplicative Gaussian white noise depends on the noise intensity, the time delays and the feedback gains. The numerical simulation results demonstrate the effectiveness of the proposed method
Key words:
Time delay, stochastic averaging method, moment stability.
1.
INTRODUCTION
Unavoidable time delays always exist in the feedback of controlled systems so that the effect of time delays on the system performance has drawn much attention from scientists in many fields, such as biology, optics, engineering and neural networks [1-3]. Especially, the stability of the dynamic systems with delayed feedback has been intensively studied over the past decades [4-6]. Hsu [4] proposed several numerical methods to study the relationship of a single time delay and a system parameter when the 333 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 333–342. © 2007 Springer.
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system undergoes instability. Stepan [5] investigated the stability conditions for un-damped systems. Wang and Hu [6] studied the robust stabilization to nonlinear delayed systems via delayed state feedback by the averaging method. However, it is not possible for deterministic delay differential equations to capture the essence of the real systems in any randomly fluctuating environment. Hence, the frame models must be replaced with stochastic differential delay equations, which can take into account the random perturbations in the real world. The analytical result for the delayed dynamic systems under any random excitation is very limited because of the complexity of probabilistic models and the infinite dimensions of delayed dynamic systems [7-11]. Mohammed [7] gave the existence and uniqueness of solutions to stochastic functional differential equations. Mackey et al. [8] presented the solution moment stability for linear stochastic differential delay equations. Grigoriu et al. [9] analyzed the stability of a linear control system with deterministic and random time delays subject to Gaussian white noise. Guillouzic et al. [10] developed the approximate Fokker-Planck equation for the stochastic differential delay equation. But the studies in this field are still in their infancy due to the difficulties caused by both time delays and random excitations. In this paper, the stochastic averaging method [11-13] is proposed to investigate the response and stability for the generalized delay system with time-delayed feedback control and additive or multiplicative Gaussian white noise. In Section 2, the stochastic averaging method is used to determine the averaged Itô stochastic differential equations of the system for the general case; In Section 3, two examples are discussed in detail to verify the theoretical results. Finally, some conclusions are drawn in Section 4.
2.
AVERAGED EQUATIONS OF THE GENERALIZED SYSTEM
Consider a stochastic delayed SDOF system with delayed feedback control force described by the following general form x(t ) + ω 2 x(t ) + ε g ( x(t ), x (t ), x(t − τ 1 ), x (t − τ 1 )) = ε f ( x(t ), x (t ), x(t − τ 2 ), x (t − τ 2 )) + ε 1 2 h( x(t ), x (t ))ξ (t ).
(1)
where 0 < ε 1 is the small parameter, f (0,0,0,0) = g (0,0,0,0) = 0 , x ∈ R , ε f is the control force, τ 1 ≥ 0 and τ 2 ≥ 0 are the delays in the control plant
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and the controller, respectively. ξ (t ) is a stationary Gaussian white noise process, whose mean and covariance function are given by
ξ (t ) = 0 , ξ (t )ξ (t ′) = 2 Dδ (t − t ′) .
(2)
In order to apply the stochastic averaging procedure, one can introduce the following transformation x(t ) = A(t ) cos θ , θ = ωt + ϕ (t ) , x (t ) = −ω A(t )sin θ .
(3)
Substituting Equation (3) into Equation (1), one obtains ⎧ A cos θ − Aϕ sin θ = 0, ⎨ 12 ⎩ω A sin θ + ω Aϕ cos θ = ε ( g − f ) − ε hξ (t ).
(4)
where f := f ( A(t ) cos(ωt + ϕ (t )), −ω A(t )sin(ωt + ϕ (t )), A(t − τ 2 ) cos(ω (t − τ 2 ) + ϕ (t − τ 2 )), −ω A(t − τ 2 )sin(ω (t − τ 2 ) + ϕ (t − τ 2 ))), g := g ( A(t ) cos(ωt + ϕ (t)), −ω A(t)sin(ωt + ϕ (t)), A(t − τ 1 ) cos(ω (t − τ 1 ) + ϕ (t − τ 1 )), −ω A(t − τ 1 )sin(ω (t − τ 1 ) + ϕ (t − τ 1 ))). h := h( A(t ) cos(ωt + ϕ (t )), −ω A(t )sin(ωt + ϕ (t )))
Solving Equation (4), one obtains the following equations for A(t ) and
ϕ (t ) ⎧ ε ( g − f )sin θ (t ) ε 1 2 h sin θ (t )ξ (t ) A ( t ) , = − ⎪ ω ω ⎪ ⎨ 12 ⎪ϕ (t ) = ε ( g − f ) cos θ (t ) − ε h cos θ (t )ξ (t ) . ⎪⎩ A(t )ω A(t )ω
(5)
Denote by a probability space and let Ω := (Ω, Σ, P ) 2 C ([− max{τ 1 ,τ 2 },0], R ) . A(t ) and ϕ (t ) are slowly varying processes while the average value of the instantaneous phase is a fast varying process. Thus, A(t − τ i ) and ϕ (t − τ i ) (i = 1, 2) can be replaced by A(t ) and ϕ (t ) over one period T = 2π ω , respectively. ( A,θ ) is an approximate twodimensional diffusion process. Using the stochastic averaging method, the corresponding Itô stochastic differential equation of ( A,θ ) can be obtained
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(6)
where B1 (t ) and B2 (t ) are independent standard Wiener processes. The corresponding drift and diffusion terms can be derived as follows 1 2π 1 2π a1 ( A) = a1 ( A,θ )dθ , a2 ( A) = a2 ( A,θ )dθ , ∫ 2π 0 2π ∫0 b11 ( A) =
εD πω 2
∫
2π
0
h 2 sin 2 θ dθ , b22 ( A) =
εD π A2ω 2
∫
2π
0
h 2 cos 2 θ dθ .
(7)
where a1 ( A,θ ) =
⎡ h sin θ ∂h h 2 cos 2 θ ⎤ ε ( g1 − f1 )sin θ + ε D ⎢− ⋅ + ⎥, Aω 2 ⎦ ω ω ∂x ⎣
a2 ( A,θ ) =
ε ( g1 − f1 ) cosθ ε Dh cosθ ⎡ sin 2θ ∂h ∂h ⎤ − ⋅ + cos 2θ ⋅ ⎥ , ⎢ Aω Aω ∂x ∂x ⎦ ⎣ ω
f1 := f ( A cosθ , −ω A sin θ , A cos(θ − ωτ 2 ), −ω A sin(θ − ωτ 2 )) , g1 := g ( A cos θ , −ω A sin θ , A cos(θ − ωτ 1 ), −ω A sin(θ − ωτ 1 )) .
From Equation (6), the amplitude A(t ) is a single variable Markov process with a FPK equation in one-dimensional space ∂p ∂ 1 ∂2 =− [a1 (α ) p] + [b11 (α ) p ] , ∂t ∂α 2 ∂α 2
(8)
where α is the state variable for A(t ) , and the initial condition of Equation (8) is p (α , t α 0 , t0 ) .
3.
STABILITY AND RESPONSE ANALYSIS FOR THE STEADY-STATE MOMENTS
From Equation (6), the response and stability conditions of the first-order and the second-order moments are derived and the numerical simulations are presented for the following two cases through two examples.
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3.1 Case I: additive Gaussian white noise Consider the delayed Van der Pol oscillator with linear delay feedback control and additive Gaussian white noise x(t ) + ω 2 x(t ) − ε [1 − x 2 (t − τ 1 )]x (t ) = ε [ux(t − τ 2 ) + vx (t − τ 2 )] + ε 1 2ξ (t ) ,
(9) According to Equation (7), one has a1 ( A) =
ε εD ⎡ − A4ω 2 w1 + 4 A2ω w2 + 4 D ⎤⎦ , b11 ( A) = 2 . 2 ⎣ ω 8 Aω
(10)
where w1 = 2 − cos 2ωτ 1 , w2 = ω − u sin ωτ 2 + ω v cos ωτ 2 . From the first equation of Equations (6), (8) and (10), A(t ) is a time homogeneous diffusion process. Thus, the stationary probability density, if it exists, can be obtained as follows ⎡ U (α ) ⎤ , ps (α ) = Cα exp ⎢ − D ⎥⎦ ⎣
(11)
where the normalization constant C and the potential function U (α ) are determined by the following equations
(
)
−1
C = ω 2 w1 π D ⎡1 + erf w2 w1 D ⎤ exp ( − w22 w1 D ) , ⎣ ⎦ 2 2 U (α ) = α ω (α ω w1 − 8w2 ) 16 .
Using the definition of the nth steady-state moment and Equation (11), one can derive the expressions of the first-order and the second-order moments as follows 14
E[ A] = −
E[ A2 ] =
⎛ w22 ⎞ 2 w2 C ⎛ w22 ⎞ exp ⎜ ⎟ ⎜ ⎟ ω w1 ⎝ ω w12 ⎠ ⎝ 2 Dw1 ⎠ ⎡ ⎛ 3 w2 ⎞ ⎛ 1 w2 ⎞ ⎤ ⋅ ⎢ BesselK ⎜ , 2 ⎟ − BesselK ⎜ , 2 ⎟ ⎥ , ⎝ 4 2 Dw1 ⎠ ⎝ 4 2 Dw1 ⎠ ⎦ ⎣
⎛ w2 ⎛ w22 ⎞ ⎛ 4 DC ⎡ ⎢ π w D w exp ⋅ + ⎜ ⎟ ⋅ ⎜⎜ 1 + erf ⎜⎜ 1 2 2 32 ω w1 ⎢ ⎝ Dw1 ⎠ ⎝ ⎝ Dw1 ⎣
⎞ ⎞⎤ ⎟ ⎟ ⎥ . (12) ⎟ ⎟⎥ ⎠ ⎠⎦
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where BesselK(n, x) is the modified Bessel functions of the second kind, erf( x) is the error function. From Equation (12), the boundedness conditions of the first-order and the second-order moments require that
ω − u sin ωτ 2 + ω v cos ωτ 2 < 0 , ⎛ w2 ⎛ w2 ⎞ ⎛ w1 D + π w2 exp ⎜ 2 ⎟ ⋅ ⎜1 + erf ⎜ ⎜ ⎝ Dw1 ⎠ ⎝⎜ ⎝ Dw1
⎞⎞ ⎟⎟ > 0 . ⎟⎟ ⎠⎠
(13)
To verify the efficacy of the above theoretical result, Figure 1 shows the first-order and the second-order steady-state moments obtained by the stochastic averaging method and the numerical simulation results of Equation (9). Obviously, the approximate analytical results coincide with the exact results very well for the small time delay. Moreover, the steady-state moments are decreased with an increase of v , which is consistent with the physical instinct. 0.70
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v
-1.2
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Figure 1. The first-order and the second-order steady-state moments with ω = 1 , ε = 0.1 , u = 0 , τ1 = 0.1 , τ2 = 0.2 and D = 0.05 .
3.2 Case II: multiplicative Gaussian white noise Consider a delayed SDOF system with linear delay feedback control and parametric Gaussian white noise, which is given by the following equation x(t ) + 2εζ x (t ) + ω 2 x(t ) = ε ux(t − τ ) + ε vx (t − τ ) + ε 1 2 x(t )ξ (t ) ,
(14)
where ζ ≥ 0 is the damping ratio and the other system parameters are defined as above.
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The time-averaged version of the stochastic averaging procedure can be applied to obtain the averaged amplitude process of A(t ) as follows dA = ε [−
B1 D 3D + ] Adt + [ε 2 ]1 2 AdB (t ) . 2 2 16ω 8ω
(15)
where B1 = 2ζ + u sin ω0τ ω0 − v cos ω0τ . Using the Itô differential rule and taking the ensemble average, one can derive the equation of the nth moment as follows ⎡ B (n + 2) D ⎤ dE[ An ] n = ε n ⎢− 1 + ⎥ E[ A ] . 2 dt ω 2 16 ⎣ ⎦
(16)
According to the asymptotic moment stability definition, the nth moment is asymptotic stable if 1 ⎡ u sin ωτ (n + 2) D ⎤ + v cos ωτ + ⎥. 2⎣ 8ω 2 ⎦ ω
ζ > ⎢−
(17)
It is seen that the moment stability conditions depend on the orders of the moments. With the order n of the moment increasing, the stability condition is increasingly more stringent. Meanwhile, the moment stability conditions depend on the noise intensity for the case of multiplicative Gaussian white noise. When Equation (14) degenerates to a SDOF system without delayed state feedback, the moment stability condition (17) is consistent with that obtained in [11]. To understand the above theoretical results, one can turn to a case study, where the orders n of the moments are chosen as n = 1 and 2. Then, the stability conditions for the first-order and the second-order moments can be obtained from Equation (17). That is, 1 ⎡ u sin ωτ D ⎤ + v cos ωτ + ⎥. ω 2⎣ 2ω 2 ⎦
ζ > ⎢−
(18)
In Figure 2, the first-order and the second-order moments stability boundaries are obtained by the proposed method and the numerical results of Equation (14) in the (ζ ,τ ) plane. It is obvious that the approximate analytical results coincide with the exact results very well. When the point (ζ ,τ ) falls into the unstable region, the first-order and the second-order steady-state moments are unstable. Otherwise, the corresponding moments are stable.
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Figure 2. Regions of stability for the moments in the (ζ, τ) plane with ω = 1 , ε = 0.1 , u = 0.5 , v = 0.5 and D = 0.05 , (a) the first-order moment; (b) the second-order moment.
To gain insight in the stability of the nth steady-state moment, one can find all points of intersection with the axes of u and v . From Equation (17), one can obtain the points of intersection S1 and S 2 as shown in Figure 3: S1 = (− 2ζω sin(ωτ ) + (n + 2) D 8ω sin(ωτ ),0),
S2 = (0, 2ζ cos(ωτ ) − (n + 2) D 8ω 2 cos(ωτ )) .
(19)
Obviously, the stable region of the nth steady-state moment increases with an increase of ζ and decreases with an increase of noise intensity D . Moreover, when the orders n of the moments increase, one may choose the positive displacement feedback u and negative velocity feedback v to stabilize the nth moment.
Figure 3. Stable region of the nth moment in the
(u , v) plane.
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4.
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CONCLUSIONS
In this paper, the stochastic averaging method is proposed to investigate the response and stability for the dynamic system with delayed feedback control and additive or multiplicative Gaussian white noise. The delayed feedback force can be expressed in terms of the system states without time delay and harmonic functions of delay first. Then, the averaged Itô stochastic differential equations for the generalized system response are derived. Two examples are used to illustrate the general results. This study makes use of the proposed method to reduce the delay differential equation of infinite dimensions to an ordinary differential equation, which simplifies the complexity of stochastic differential delay equations. Moreover, the results in this study may serve as a basis for the design of an optimal feedback control for such type of nonlinear stochastic systems with time delayed.
ACKNOWLEDGEMENTS This work was supported in part by the National Natural Science Foundation of China under Grant No. 10532050, in part by the China Postdoctoral Science Foundation and Jiangsu Postdoctoral Science Foundation. The first author would like to thank Prof. W. Q. Zhu for his valuable suggestions.
REFERENCES 1. 2. 3. 4. 5. 6.
7. 8. 9.
Hale J. Theory of functional differential equations, New York, Springer-Verlag, 1977. Gopalsmy K. Stability and oscillations in delay differential equations of population dynamics, Dordrecht, Kluwer, 1992. Hu HY, Wang ZH. Dynamics of controlled mechanical systems with delayed feedback, Heidelberg, Springer, 2002. Hsu CS. “Application of the τ -decomposition method to dynamical systems subject to the retarded follower forces”, Journal of Applied Mechanics, 37, pp. 258-266, 1970. Stepan G. Retarded dynamical systems: stability and characteristic functions. England: Longman Scientific and Technical, 1989. Wang ZH, Hu HY. “Robust stabilization to nonlinear delayed systems via delayed state feedback: the averaging method”, Journal of Sound and Vibration, 279, pp. 937-953, 2005. Mohammed SEA. Stochastic functional differential equations. Boston, Pitman, 1984. Mackey MC, Nechaeva I.G. “Solution moment stability in stochastic differential delay equations”, Physical Review E, 52, pp. 3366-3376, 1995. Grigoriu M. “Control of Time Delay Linear System with Gaussian White Noise”, Journal of Engineering Mechanics, 12, pp. 89-96, 1997.
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10. Guillouzic S, L’Heureux L, Longtin A. “Small delay approximation of stochastic delay differential equations”, Physical Review E, 59, pp. 3970-3982, 1999. 11. Lin YK, Cai GQ. Probabilistic Structural Dynamics, Advanced Theory and Applications, New York, McGraw-Hill, 1995. 12. Zhu WQ. Random Vibration, Science Press, Beijing, 1992. 13. Elbeyli O, Sun JQ. “Feedback Control Optimization of Nonlinear Systems under Random Excitations”, Communications in Nonlinear Science and Numerical Simulation, 11, pp. 125-136, 2006.
ROBUST TIME-PERIODIC CONTROL OF TIME-DELAYED SYSTEMS G. Stépán, T. Insperger Department of Applied Mechanincs, Budapest University of Technology and Economics, 1521, Budapest, Hungary, E-mail:
[email protected],
[email protected]
Abstract:
Position control of a single body is considered with delayed discrete feedback. The act-and-wait control concept is applied: the feedback gains are constant for a sampling period (act), then they are zero for a certain number of periods (wait), then they are constant again, etc. Stability of the system is investigated for different act-and-wait periods. It is shown that if the period of gain variation is larger than the feedback delay, then the system performance changes radically. Stability, rate of decay and robustness issues are discussed.
Key words:
PD control, periodic control, time delay, stability, dead-beat control.
1.
INTRODUCTION
Time delay often arise in feedback control systems due to acquisition of response and excitation data, information transmission, on-line data processing, computation and application of control forces. In spite of the efforts to minimize time delays, they can not be eliminated totally even with today’s advanced technology due to physical limits. The information delay is often negligible, but for some cases, it still may be crucial, for example, in space applications [1], in systems controlled through the internet [2] or in robotic applications with time-consuming control force computation [3]. Due to the delay of the control feedback, the governing equation is a delay-differential equation (DDE). DDE’s usually have infinite dimensional phase spaces [4-5], therefore the linear stability conditions for the system parameters are complicated and often do not have an analytical form. However, there exist several methods to analyze control systems with delayed feedback [6-8]. 343 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 343–352. © 2007 Springer.
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Digital effects in the feedback loop endow the systems with a semidiscrete nature. These systems can be transformed into finite dimensional discrete maps via the state augmentation method even if the feedback loop contains time delay (in addition to the digital effects). The order of these feedback systems is equal to the dimension of the resulted discrete map that increases with the time delay. Gain parameters and sampling time do not always provide stable motion and fast settling time, since these parameters are often hedged by other technical conditions. For these cases, application of periodic controllers (e.g., time varying feedback gains) may stabilize or speed up the control. The idea of stabilizing by parametric excitation comes from the classical example of the pendulum: the upper position of a pendulum can be stabilized by vertically vibrating its pivot point [9]. The problem of stabilization by means of time-periodic feedback gains has been presented by Brockett as one of the challenging open problems in control theory [10]. With the exception of some papers on discrete-time systems [11-12], the problem has received little attention and only partial results for special classes of systems have been derived concerning sinusoidal and piecewise constant gain variations [13-15]. The adequate problem for time-delayed systems, i.e. stabilizing by time periodic feedback gains, was composed by Stépán and Insperger [16]. They developed the so-called act-and-wait control concept for both continuoustime [16-17] and discrete-time [18] systems. In this paper, the act-and-wait concept will be applied to the position control of a single block with discrete delayed feedback. The efficiency of the act-and-wait control concept is demonstrated via stability charts and simulations. Optimal control parameters are determined, and robustness issues are discussed.
2.
OVERVIEW OF ACT-AND-WAIT CONTROL FOR DISCRETE-TIME SYSTEMS Consider the linear discrete-time system with input delay in the form (1) x( j + 1) = Ax( j ) + Bu( j − n)
where x ∈ R r is the state, u ∈ R q is the input, A ∈ R r ×r, B ∈ R r ×q are constant and j ∈ Z . We assume that the feedback delay n ∈ Z+ is a given parameter of the system and cannot be tuned during the control design. There might be several reasons for such time delay, e.g., acquisition of response and excitation data, information transmission, on-line data processing, computation and application of control forces. Note that the general form of Equation (1) includes the semi-discrete system (t ) + Bu ( j − n), x (t ) = Ax t ∈ [ j , j + 1) , (2)
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too. This system describes a continuous-time system under digital control like computer controlled machines [4-19]. The solution of system (2) over the interval [j, j+1) results in the discrete system (1) with
A = eA ,
1
B = ∫ e A (1− s ) d s B .
0
(3)
2.1 Time-independent controller Consider the autonomous (time-independent) controller u( j ) = Dx( j ) ,
(4)
q×r
with D ∈ R . State augmentation of Equation (1) with Equation (4) results in the discrete map z j +1 = Ψz j , (5) with z j = ( xT ( j ), uT ( j − 1),… , uT ( j − n) ) ∈ R r + qn . The coefficient matrix T
⎛ A 0 " 0 B⎞ ⎜D 0 " 0 0⎟ (6) Ψ=⎜ 0 I " 0 0⎟ ⎜# % #⎟ ⎜ 0 0 " I 0⎟ ⎝ ⎠ is actually the (r + qn) × (r + qn) monodromy matrix of the system [20]. The identity submatrices I below the diagonal in Ψ represent the delay effect in the feedback. Stability properties are determined by the eigenvalues of matrix Ψ, which are also called characteristic multipliers or poles. The system is asymptotically stable if all the (r+qn) poles are in the open unit disc of the complex plane. It can easily be seen that in general cases, the poles cannot be controlled by the control parameters, i.e. by the elements of matrix D (see [18]).
2.2 Act-and-wait controller Consider now the time-dependent act-and-wait controller u( j ) = g ( j )Dx( j ) .
(7)
where g ( j ) is the k-periodic act-and-wait switching function defined as
{
g ( j ) = 1 if 0
j = hk , h ∈ Z . otherwise
(8)
From now on, integer n will be called delay parameter and integer k will be called period parameter. While n is a given system parameter, k can be chosen during the control design. If k=1 then g ( j ) ≡ 1. This case corresponds to the autonomous controller (4).
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If k ≥ 2 then g ( j ) alternates between one and zero. In the first discrete step, g ( j ) = 1 and the control is active (act), while in the following (k-1) number of steps, g ( j ) = 0 and the control term is switched off (wait), then in the (k+1)st step, the control is active again, etc. In this case, the connection between two subsequent sampling instants can be given by the map z j +1 = G j z j , (9) where z j is the same vector as in Equation (5) and the coefficient matrix reads 0 " 0 B⎞ ⎛ A ⎜ g ( j )D 0 " 0 0 ⎟ (10) Gj =⎜ 0 I " 0 0⎟. ⎜ # % #⎟ ⎜ 0 0 " I 0 ⎟⎠ ⎝ Due to Equation (8), G j is k-periodic: G j = G j + k with G 0 ≠ G1 = … = G k −1 . Coupling the solution over k subsequent steps (i.e., over a full act-andwait period) with initial state z0 results in the discrete map z k = G k −1G k − 2 … G 0 z 0 . (11)
Φ Stability properties of the control system are determined by the eigenvalues of the monodromy matrix Φ. In general case, this system has still (r+qn) number of poles, and stability cannot be assessed in all cases. However, if the period parameter k is chosen larger than the delay parameter n, then the monodromy matrix reads ⎛ M A k − n B A k − n +1B " A n −1B ⎞ ⎜ 0 0 " 0 ⎟ Φ k ≥ n +1 = ⎜ 0 # # # # ⎟ ⎜0 " 0 0 0 ⎟⎠ ⎝
(12)
with M = A k + A k − n −1BD. In this case, the system has qn poles at zero and the remaining r poles are the eigenvalues of the r × r matrix M. Consequently, if the pair ( A k , A k − n −1B) is controllable, then the poles of the system can arbitrarily be placed and deadbeat control can be attained. For further details on the general system (1) with controller (7), see [18].
3.
CASE STUDY: 1 DOF PD POSITION CONTROL
The 1 DOF mechanical model of PD position control is shown in Figure 1. The position of the block of mass m is sensed and control force Q is applied to push the block into the desired zero position. The system is governed by the differential equation (13) m x(t ) = Q .
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Figure1. Mechanical model.
Figure 2. Sampling effect and control force in act-and-wait control.
Let ∆t be the sampling time, and let τ = n∆t denote the feedback delay in the control, where the integer n is the delay parameter. The control force Q is updated at each sampling instant tj =j∆t ( j ∈ Z ) by means of the delayed discrete values x(t j − n ) and x (t j − n ) : Q = − g (t )( P x(t j − n ) + D x (t j − n )),
t ∈ [t j , t j +1 ) .
(14)
where g(t) is the k∆t-periodic act-and-wait switching function: g (t ) =
{10
if if
t ∈ [t0 + hk ∆t , t1 + hk ∆t ), h ∈ Z . t ∈ [t1 + hk ∆t , tk + hk ∆t ), h ∈ Z
(15)
If k = 1 , then g (t ) ≡ 1 This corresponds to the traditional control with constant gains P and D. If k ≥ 2 , then g (t ) alternates between zero and one according to the actand-wait concept (see Figure 2).
3.1 Stability analysis Using Equation (14), the governing equation reads mx(t ) = − g (t )( P x(t j − n ) + D x (t j − n )), t ∈ [t j , t j +1 ) .
(16)
Introduce the dimensionless time t = t / ∆t , and by abuse of notation, drop the tilde immediately. Then, Equation (16) becomes x(t ) = − g ( j )( p x( j − n) + d x ( j − n)), t ∈ [ j , j + 1) . (17)
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where ∆t 2 ∆t (18) P, d = D. m m are the dimensionless proportional and differential gain parameters, and g ( j ) is given by Equation (8). Equation (17) can be written in the compact form of the semi-discrete system (2) subjected to controller (7) with r=2, q=1 and p=
( )
( )
()
( ).
= 0 1 , B = 0 , D = − p x(t ) = x (t ) , A x(t ) −d 0 0 1
T
(19)
If k = 1 (no act-and-wait), then the monodromy matrix of the system can be given based on Equations (3) and (6): 1 ⎛ 1 ⎜ 0 1 ⎜ Ψ = ⎜ − p −d 0 0 ⎜ # # ⎜ 0 0 ⎝
0 1/ 2 ⎞ 0 1 ⎟ 0 0 ⎟. (20) 0 0 ⎟ # ⎟ 1 0 ⎟⎠ This system has n+2 number of poles that can not arbitrarily be placed using the two control parameters p and d. If k ≥ n + 1 (act-and-wait case), then the monodromy matrix of the system can be given according to Equation (12), and the stability is determined by the eigenvalues of the 2 × 2 matrix " " " " % 0 "
0 0 0 1
(
)
M = 1 − (k − n − 1/ 2) p k − (k − n − 1/ 2)d . −p 1− d
(21)
In this case the system has only two non-zero poles. According to [18], it can be shown that the control parameters p and d can always be tuned so that both eigenvalues of M are zero, that is, the control is deadbeat.
3.2 Stability charts, optimal control gains In this subsection, series of stability charts are determined for Equation (17) with different delay and period parameters. The convergence of the system is analyzed using the decay ratio ρ =| µ1 |1/ k , where µ1 is the critical (largest in modulus) eigenvalue of the monodromy matrix. Decay ratio ρ characterizes the average error decay over a single sampling period as opposed to the critical eigenvalue µ1 that characterizes the average error decay over the principal period k: | z j +1 | ≤ ρ | z j | and | z j + k | ≤ | µ1 || z j | . (22) Decay ratio is a measure that will be used to compare systems with different periods k of the act-and-wait controller for a given delay parameter n of the system.
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For the case k=1, stability charts are presented in Figure 3 for different delay parameters n. The charts were determined via point-by-point evaluation of the monodromy matrix Ψ over a 200 × 200 sized grid of dimensionless proportional and differential gains p and d. Contour plot was used to obtain the transition curves associated to different decay ratios ρ = 1,0.95,0.9,… . Obviously, the stability boundaries are the transition curves where ρ = 1 . The optimal gain parameters p* and d*, which result in the fastest convergence of the system, are denoted by black dots and the corresponding decay ratio ρ* is given. As it can be seen, the larger the delay parameter n is, the worse the system performance is. (Note the different scales for cases n=1, 2 and n=5, 10 in Figure 3.)
Figure 3. Stability charts for (17) with k=1 (constant gains) and different n.
For reference, in the case n=1, analytical computation of the optimal control parameters gives p∗ = 1/ 27 , d ∗ = 17 / 54 with the corresponding minimal decay ratio ρ ∗ = 2 / 3. (see [19]). In Figure 4, stability charts are presented for delay parameter n=5 and for different period parameters k. The optimal gains are denoted by black dots and the corresponding decay ratios are given. As it can be seen, some improvement can be observed in the system’s performance for increasing k when k ≤ 5 . For the case k=6, when the control waits for 5 sampling periods that is just equal to the time delay, the control performance changes radically (see the different scales for k=5 and 6). The optimal decay index is zero that corresponds to deadbeat control: the system converges to zero within finite time. For the cases k=7 and 8 the stable regions get slightly smaller and smaller. Note that for k ≥ n + 1 , the stability boundaries are the straight lines k 1⎞ 1⎞ ⎛ ⎛ (23) d = 2 + ⎜ n − + ⎟ p, d = ⎜n + ⎟ p 2 2⎠ 2⎠ ⎝ ⎝ (see cases k=6,7,8 in Figure 4), and the optimal control parameters are 1 2k + 2n + 1 p∗ = , d∗ = , (24) k 2k as they can be derived in closed form by eigenvalue analysis of matrix M. p = 0,
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Figure 4. Stability charts for Equation (17) with n=5 and k>1 (act-and-wait control).
In Figure 5, numerical simulations are shown for delay parameter n=5 and for different period parameters k with the associated optimal control gains. The best control performance is obtained for the case k = 6. For k > 6, the control performance is still equivalent to deadbeat, but the period of convergence slightly increases with k.
Figure 5. Numerical simulations for (17) with n=5 for the optimal control gains.
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3.3 Robustness Although, dead beat control can clearly be achieved, it should be mentioned that it is very sensitive to parameter variations. For example, if n=5, k=6, then the optimal gains are p*=0.1667 and d*=1.9167, and the associated decay index is ρ*=0. If the gains are perturbed by 10% around the optimal point, i.e., p = p ∗ + 0.1 p∗ cos ϕ and d = d ∗ + 0.1d ∗ sin ϕ with ϕ ∈ [0, 2π ) as running parameter, then the worst performance (largest decay ratio) is obtained at ϕ = 1.8544 [rad] with the corresponding gains p=0.1620, d=2.1007 and with the corresponding decay index ρ=0.9073. This is a significant change in the decay. However, this is still better than the autonomous case with k=1 where the optimal decay index is ρ*=0.8987 and the same 10% perturbation of the optimal gains results in ϕ=2.1897 [rad], p=0.0025, d=0.0908 and ρ=0.9570. This shows that the act-and-wait control system is sensitive to small perturbations but its robustness is still better than that of the corresponding autonomous control system (k=1).
4.
CONCLUSIONS
Position control of a single block was considered with discrete delayed feedback. The act-and-wait control concept was applied in order to improve control performance: the feedback gains are constant (act) for the first sampling period, then they are zero for a certain number of sampling periods (wait). It was shown analytically that if the period of gain modulation is larger than the time delay itself, then number of relevant eigenvalues decreases, and in the optimal case, deadbeat control can be attained. The act-and-wait concept provides an alternative for control systems with significant feedback delays. The traditional way is the continuous use of small control gains p and d according to the k=1 case in Figures 4 and 5, when a cautious, slow feedback is applied with small gains resulting slow convergence. The other, alternative way is the act-and-wait control concept, when large control gains are used for a short time (for a sampling period) and zero gains for long time (for a period equal to the time delay). Although it might seem unnatural not to actuate during the wait period, act-and-wait concept is still a natural control logic for time-delayed systems. This is the way, for example, that one would adjust the shower temperature considering the delay between the controller (tap) and the sensed output (skin).
ACKNOWLEDGEMENTS This research was supported in part by the Hungarian National Science Foundation under grant no. OTKA F047318 and OTKA T043368 and by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.
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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.
Kim WS, Bejczy AK. “Demonstration of a high-fidelity predictive preview display technique for telerobotic servicing in space”, IEEE Transactions on Robotics and Automation, 9, pp. 698-704, 1993. Munir S, Book WJ. “Control techniques and programming issues for time delayed internet based teleoperation”, Journal of Dynamic Systems, Measurements and Control, 125, pp. 205-214, 2003. Kovács LL, Insperger T, Stépán G. “Teaching-in force control of industrial robots used in medical applications”, Proceedings of 15th CISM-IFToMM Symposium on Robot Design, Dynamics and Control, Montreal, Canada, June 14-18, 2004, CD-ROM Rom04-46. Hale JK, Lunel SMV. Introduction to Functional Differential Equations, New York, Springer-Verlag, 1993. Stépán G. Retarded Dynamical Systems, Harlow, Longman, 1989. Insperger T, Stépán G. “Semi-discretization method for delayed systems”, International Journal for Numerical Methods in Engineering, 55, pp. 503-518, 2002. Butcher EA, Ma H, Bueler E, Averina V, Szabó Zs. “Stability of time-periodic delaydifferential equations via Chebyshev polynomials”, International Journal for Numerical Methods in Engineering, 59, pp. 895-922, 2004. Olgac N, Sipahi R. “An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems”, IEEE Transactions on Automatic Control, 47, pp. 793977, 2002. Stephenson A. “On a new type of dynamical stability”, Memoirs and Proceedings of the Manchester Literary and Philosophical Society, 52, pp. 1-10, 1908. Brockett R. “A stabilization problem”, in Open Problems in Mathematical Systems and Control Engineering, 16, pp. 75-78. Springer-Verlag. Aeyels D, Willems JL. “Pole assignment for linear time-invariant systems by periodic memoryless output feedback”, Automatica, 28, pp. 1159-1168, 1992. Leonov GA. “The Brockett problem for linear discrete control systems”, Automation and Remote Control, 63, pp. 777-781, 2002. Moreau L, Aeyels D. “Stabilization by means of periodic output feedback”, Proceedings of the 37th IEEE Conference on Decision & Control, Phoenix, Arizona, USA, pp. 108109, 1999. Leonov GA. “Brockett’s problem in the theory of stability of linear differential equations”, St. Petersburg Math. Journal, 13, pp. 1-14, 2002. Allwright JC, Astolfi A, Wong HP. “A note on asymptotic stabilization of linear systems by periodic, piece-wise constant output feedback”, Automatica, 41, pp. 339-344, 2005. Stépán, G., Insperger,T., “Stability of time-periodic and delayed systems - a route to actand-wait control”, Annual Reviews in Control, 30, pp. 159-168, 2006. Insperger T. “Act-and-wait concept for time-continuous control systems with feedback delay”, IEEE Transactions on Control Systems Technology, 14, pp. 974-977, 2006. Insperger T, Stépán G. “Act-and-wait control concept for discrete-time systems with feedback delay”, IEE Proceedings - Control Theory & Applications, in press, 2006. Stépán G, Steven A, Maunder L. “Design principles of digitally controlled robots”, Mechanism and Machine Theory, 25, pp. 515-527, 1990. Åström KJ, Wittenmark B. Computer Controlled Systems: Theory and Design, Englewood Cliffs, NJ, Prentice-Hall, 1984.
SELF-INTERRUPTED REGENERATIVE TURNING P. Wahi 1, G. Stépán 2, A. Chatterjee 1 1
Mechanical Engineering, Indian Institute of Science, Bangalore, 560012, India. E-mail:
[email protected],
[email protected] 2 Applied Mechanics, Budapest University of Technology and Economics, Budapest, H-1521, Hungary. E-mail:
[email protected]
Abstract:
A preliminary study of self-interrupted regenerative turning is performed in this paper. To facilitate the analysis, a new approach is proposed to model the regenerative effect in metal cutting. This model automatically incorporates the multiple-regenerative effects accompanying self-interrupted cutting. Some lower dimensional ODE approximations are obtained for this model using Galerkin projections. Using these ODE approximations, a bifurcation diagram of the regenerative turning process is obtained. It is found that the unstable branch resulting from the subcritical Hopf bifurcation meets the stable branch resulting from the self-interrupted dynamics in a turning point bifurcation. Using a rough analytical estimate of the turning point tool displacement, we can identify regions in the cutting parameter space where loss of stability leads to much greater amplitude self-interrupted motions than in some other regions.
Key words:
Regenerative turning, multiple regenerative effect, global bifurcation.
1.
INTRODUCTION
In this paper, we study the self-interrupted regenerative turning arising after the loss of linear stability of steady cutting. Self-interrupted machining occurs due to a loss of contact between the tool and the workpiece when the relative vibrations between the tool and the workpiece grow large. Selfinterruption can give rise to globally stable periodic solutions under broad parameter ranges where the primary bifurcation to instability is subcritical [1 – 4]. 353 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 353–362. © 2007 Springer.
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We develop a new approach to model the regenerative effect in turning operations. The dynamics of the cutting process is modeled using a direct approach which leads to a partial differential equation (PDE) along with an ordinary differential equation (ODE). The PDE describes the cut surface while the ODE describes the motions of the cutting tool. This model automatically incorporates self-interruption as well as the multipleregenerative effects accompanying it. The regenerative effect has been established as one of the most important sources of relative vibrations between the tool and the workpiece [1 – 14]. It arises from the variation in the chip thickness due to a wavy workpiece surface generated by a tool vibrating under an external perturbation. Due to this wavy surface the instantaneous chip thickness depends on the instanttaneous tool displacement as well as the tool displacement in the previous revolutions. Accordingly, the mathematical models of the regenerative vibrations are traditionally delay differential equations (DDEs). These DDEs are nonlinear in nature primarily due to the nonlinear dependence of the cutting force on the chip thickness. Linear stability analysis of these DDE models gives the operating parameters for stable stationary cutting [1, 2, 4, 12, 13]. However, using various analytical and numerical techniques from nonlinear dynamics theory, it has been established that these nonlinear delayed models exhibit subcritical instability which means that unstable periodic motions exist around the otherwise stationary cutting [2, 13]. This makes the results of linear stability analysis less reliable from engineering viewpoint. The phenomenon has been reported in the experiments of Hanna and Tobias [6] and Shi and Tobias [9]. It has also been shown that these models do not have any globally stable periodic solutions except under special operating conditions and/or incorporating a specific strength of structural nonlinearity [2, 13]. The dynamics of such large amplitude solutions can be studied by incorporating the loss of contact between the tool and the workpiece giving rise to self-interrupted cutting [3]. This loss of contact also results in multiple-regenerative effects, wherein displacements from more than one previous revolution have to be considered for determining the chip thickness [8 – 12] (see section 2 for details). These possibilities require tracking of the delay to be used in the DDE model during phases of contact. As will be seen below, our approach requires no such delay tracking while retaining the ability to model self-interruption, at the cost of introducing an extra PDE.
2.
REGENERATIVE TURNING DYNAMICS In this paper, we consider the following SDOF model for tool vibration
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x(t ) + 2ζ x(t ) + x(t ) = PC (t )3/ 4
(1)
where the parameter P is proportional to the chip width and C (t ) is the instantaneous chip thickness. Given the governing equation for the toolworkpiece dynamics, i.e., Equation (1), it remains to determine the instantaneous chip thickness C (t ) . In cutting with sustained contact, the instantaneous chip thickness C (t ) is given by C (t ) = C0 + x(t − τ ) − x(t )
(2)
where C0 is the nominal chip thickness and τ = 2π / Ω is the time period of one revolution of the workpiece. However, when the relative displacement becomes so large that the tool leaves the workpiece, Equation (2) no longer represents the instantaneous chip thickness. Delayed displacements from more than one immediate pass have to be considered (hence, the possible delays of 2τ , 3τ , etc.). Our approach to model chip formation is somewhat similar in spirit to that of Batzer et al. [11], in that we both define the cut surface by essentially the same function (defined below). However, we derive a PDE governing the evolution of this function; we introduce a new solution method and we also present new results specific to turning operations. Reference plane discontinuity at φ=2π gives chip thickness
angular coordinate L(φ,t)
φ=2π
φ Ω
x
actual tool position = L(0,0) - vt + x nominal tool position = L(0,0) - vt
φ=0 (tool position)
Figure 1. A schematic of turning process.
Lab Fixed
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A schematic of the turning process is shown in Figure 1. The tool travels along the negative-direction with a velocity and the workpiece rotates about the positive-axis at an angular rate Ω . The tool velocity v = (C0 Ω) /(2π ) . We define a function L(φ , t ) , φ ∈ [0, 2π ] , which represents the perpendicular distance of points on the cut surface from a reference plane as shown in Figure 1. The angle φ is measured in lab-fixed coordinates, φ = 0 ≡ 2π representing the position of the tool. Even though φ = 0 and φ = 2π are the same points in the lab-fixed coordinate system, they imply different things for the workpiece. φ = 0 represents the freshly exposed surface just past the tool, while φ = 2π represents the one-revolution old surface just reaching the tool. This function L(φ , t ) can have a discontinuity at φ = 0 ≡ 2π which determines the instantaneous chip thickness C (t ) by
C (t ) = L(2π , t ) − L(0, t )
(3)
till it is positive, i.e., whenever the tool is in contact with the workpiece. When the tool leaves the workpiece, the chip thickness becomes zero and the discontinuity in the function L(φ , t ) vanishes, i.e., L(2π , t ) = L(0,t). At any instant t , the nominal position of the tool is L(0,0) − vt . Due to vibrations, the actual position of the tool is L(0,0) − vt + x(t ) . When there is contact between the tool and the workpiece, it follows that L(0, t ) = L(0,0) − vt + x(t ) . Equation (3) then gives (setting the constant L(0,0) to zero) C (t ) = max{L(2π , t ) + vt − x(t ),0}
(4)
valid whether the tool is in contact with the workpiece or not. The above value of C (t ) is to be used in Equation (1). Now, consider a material point on the cut surface at some angle φ ∈ (0, 2π ) . Over a small time duration Δt , the workpiece rotates by an angle Δφ = ΩΔt . Therefore, L(φ + ΩΔt , t + Δt ) = L(φ , t ) . It follows that ∂L ∂L = −Ω ∂t ∂φ
(5)
The above PDE governs the evolution of L(φ , t ) for φ ∈ (0, 2π ) . The boundary condition for this PDE is obtained by rewriting Equation (3) as L(0, t ) = L(2π , t ) − C (t )
(6)
where C (t ) is given by Equation (4). Equation (6) determines L(0, t ) based on L(2π , t ) and C (t ) . This automatically and implicitly incorporates all regenerative effects. We now define L (φ , t ) = L(φ , t ) + vt . This leads to C (t ) = max{L (2π , t ) − x(t ),0}
(7)
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The PDE is modified to ∂L ∂L =v−Ω ∂t ∂φ
(8)
and the boundary condition after using Equation (7) becomes L (0, t ) = L (2π , t ) − C (t ) = min{L (2π , t ), x(t )}
(9)
Equations (7), (8) and (9) together define the kinematics of the chip formation. These in combination with Equation (1) determine the complete dynamics of the regenerative turning process.
3.
FINITE-DIMENSIONAL REDUCTION USING GALERKIN PROJECTION
Equation (8) is very similar to the PDE obtained in [14] where a Galerkin projection was used to obtain a finite set of ODEs. On similar lines, to obtain a finite set of ODEs, we approximate the function L = (φ , t ) as L (φ , t ) = a0 (t )(1 −
φ φ N −1 kφ ) + a1 (t ) + ∑ ak +1 (t )sin( ) 2π 2π k =1 2
(10)
where N is a finite number that we choose. We emphasize that our choice of the shape functions represents a smooth function over the domain φ ∈ [0, 2π ] . However, the function L (φ , t ) potentially has slope discontinuities due to loss and reestablishment of contact. Our choice of shape functions leads to an artificial smoothening of these discontinuities in the Galerkin approximation. However, it is shown in [15] using comparison with a fine mesh finite-difference solutions that this does not introduce significant errors in the approximation and a good match in physical quantities of interest is obtained even with a modest number of shape functions. From Equation (10), we note that L (0, t ) = a0 (t ) and L (2π , t ) = a1 (t ) . Hence, the chip thickness (Equation (7)) and the boundary condition (Equation (9)) are given as C (t ) = max{a1 (t ) − x(t ),0}
(11)
a0 (t ) = min(a1 (t ), x(t ))
(12)
Hence, a0 (t ) acts as a dummy variable. In particular, we take
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x(t ), if x(t ) < a1 (t ), a (t ), otherwise.
(13)
Substituting Equation (10) in Equation (8) and rearranging, we get an equation which cannot hold identically, and gives us the framework for the Galerkin projection. We call the left hand side (LHS) of the equation as the residual and make this residual orthogonal to the shape functions corresponding to theeal r variables, i.e., ak (t ) , k = 1, 2, , N . This gives us N equations
φ φ N −1 kφ ) + ( ) + ∑ ak +1 (t )sin( ) − v a t 1 ∫0 2π 2π k =1 2 N −1 Ωa0 (t ) Ωa1 (t ) ka (t ) kφ φ − + + Ω∑ k +1 cos( )}i dφ = 0 2π 2π 2 2 2π k =1
(14)
φ φ N −1 kφ ) + a1 (t ) + ∑ ak +1 (t )sin( ) − v 0 2π 2π k =1 2 N −1 Ωa0 (t ) Ωa1 (t ) ka (t ) kφ mφ − + + Ω∑ k +1 cos( )}isin( )dφ = 0 2π 2π 2 2 2π k =1
(15)
2π
{a0 (t )(1 −
and
∫
2π
{a0 (t )(1 −
for m = 1, 2, , N − 1 . Substituting for a0 (t ) from Equation (12) and a0 (t ) from Equation (13) gives us the ODEs in terms of ai (t ) , ai (t ) , i = 1, 2, , N , x(t ) and x(t ) . These equations, i.e., Equations (11), (14) and (15) along with Equation (1) define our finite-dimensional ODE model for the global turning dynamics.
4.
GLOBAL BIFURCATION STRUCTURE OF TURNING
We use Equations (1), (11), (14) and (15) to obtain the global bifurcation structure of the turning process using a fixed arc-length based continuation scheme in conjunction with a numerical scheme for obtaining periodic solutions of ODEs. The bifurcation diagram is presented in Figure 2. It can be seen in a zoomed view of the boxed portion that the unstable branch from the subcritical Hopf bifurcation (see [2]) and the stable branch from the self-interrupted motion meet in a turning point bifurcation (at least within our finite dimensional approximation). The point of impending loss of contact occurs on the unstable branch, close to the turning point. At our
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level of numerical precision and finite dimensional (Galerkin) modeling, it may be unreliable to try to pinpoint the separation between these two points and for all practical purposes, the two points may be treated as coincident. Numerically, we have observed for various values of Ω , that the value of P/Pcritical at the turning point lie in the interval [0.92, 0.95] for ζ ∈ [0,0.2] . This compares well with the analytical estimate of 0.92 obtained by Stépán and Kalmár-Nagy [1]. This observation may have practical relevance in that steady cutting is stable under large perturbations when P is below 0.92 Pcritical. 1.6 1.4
x
extremum
1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 0
0.1
0.2
0.3
0.4
0.5
0.6
P
1
0.8
0.6
0.4 Sustained contact
0.2 Turning point
0 Contact loss
-0.2 0.425 0.43 0.435
0.44 0.445
0.45 0.455
0.46 0.465
Figure 2. Bifurcation diagram for ζ = 0.1 , Ω = 2 , v = 1/ π and N = 25 . Thick lines represent stable solutions while thin lines represent unstable solutions.
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4.1 Amplitudes at turning points The turning point roughly corresponds to the minimum value of the amplitude of the solution for which the tool can leave the workpiece. We use the usual τ -delayed model before loss of contact, i.e. Equation (2) for chip thickness and as a rough approximation, assume a pure sinusoidal solution x(t ) = A cos(ωt ) about the static deflection xstatic. In addition, we assume that ω = ωcritical to get the condition for loss of contact as A cos(ωcritical t − ωcriticalτ ) − A cos(ωcritical t ) = C0
Replacing ωcritical t by a yet to be determined phase θ gives us
−2 A sin(
2θ − ωcriticalτ ω τ )sin( critical ) = C0 2 2
(16)
In addition, for impending loss of contact, we must have 2θ − ωcriticalτ ω τ ⎫ ⎧ max ⎨−2 A sin( )sin( critical ) ⎬ = C0 θ 2 2 ⎩ ⎭
which gives the amplitude for which the tool leaves the workpiece as A=
C0
2sin(
ωcriticalτ 2
(17)
)
In the limiting case of zero damping, i.e., ζ = 0 , we can obtain analytical estimates for sin(ωcriticalτ / 2) [13], and hence for A as well. Each lobe of the stability boundary for ζ = 0 consists of three parts [13]: a curved part corresponding to ωcriticalτ = (2 M + 1)π , a horizontal part corresponding to P = 0 (with ωcritical = 1 and a vertical part corresponding to ωcriticalτ = 2 M π . For ζ > 0 , the stability lobe is a continuous curve which approaches the ζ = 0 curves as ζ → 0 . Accordingly, for small ζ , and for the first stability lobe we have sin(
ωcriticalτ 2
) ≈ 1, for τ ∈ (0, π ) and sin(
ωcriticalτ 2
τ
) ≈ sin( ), for τ ∈ (π , 2π ). 2
A numerically obtained plot of sin(ωcriticalτ / 2) versus τ (for values corresponding to the first stability lobe for different nonzero values of ζ is presented in Figure 3. It can be seen that the ζ = 0 approximation is reasonably good for light damping. Figure 3 shows that sin(ωcriticalτ / 2)
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monotonically decreases with increasing τ ∈ (0, 2π ) . Equation (17) therefore implies that the amplitude of the stable periodic solutions will monotonically increase with increasing τ (or equivalently decreasing Ω ) within the stability lobe. Thus, the position within a given stability lobe range can have an important effect, should stability be lost.
1 0.9
sin ( ω critical τ /2)
0.8 0.7 ζ=0.1 ζ=0.05 ζ=0.01 ζ=0 approximation
0.6 0.5 0.4 0.3 0.2
Ω increases
0.1 1
2
3
τ
4
5
6
Figure 3. sin(ωcriticalτ / 2) versus τ for the first stability lobe for different values of ζ .
5.
CONCLUDING REMARKS
A new approach to model the regenerative effect in turning operations has been developed. The dynamics of the cutting process has been modeled using a direct approach which leads to a partial differential equation (PDE) along with an ordinary differential equation (ODE). The PDE describes the cut surface while the ODE describes the motions of the cutting tool. This model automatically incorporates self-interruption due to the loss of contact between the tool and the workpiece, and the multiple-regenerative effects accompanying it. Using some lower dimensional ODE approximations, a bifurcation diagram of the regenerative turning process has been obtained. It has been found that the unstable branch of periodic solutions resulting from the subcritical Hopf bifurcation and the stable periodic branch resulting from the self-interrupted dynamics meet in a turning point bifurication. A rough analytical estimate of the turning point tool displacement under light damping has also been obtained. This estimate helps to identify regions in the space of cutting parameters where loss of stability leads to much greater amplitudes of self-interrupted motions than in some other regions.
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ACKNOWLEDGEMENTS Anindya Chatterjee thanks the Department of Science and Technology (DST) for financial support.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
Stépán G, and Kalmár-Nagy T. “Nonlinear regenerative machine tool vibrations”, Proc. of the DETC’97, Sacramento, California, USA, 1997. Kalmár-Nagy T, Stépán G, and Moon FC. “Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations”, Nonlinear Dynamics, 26, pp. 121-142, 2001. Stépán G. “Modeling nonlinear regenerative effects in metal cutting”, Proceedings of the Royal Society of London A,359, pp. 739-757, 2001. Stépán G. Retarded Dynamical Systems, London, UK, Longman Group, 1989. Tobias SA. Machine-Tool Vibration, London, UK, Blackie and Sons Ltd., 1965. Hanna NH, and Tobias SA. “A theory of nonlinear regenerative chatter”, ASME Journal of Engineering for Industry, 96, pp. 247-255, 1974. Tlusty J, and Ismail F. “Basic non-linearity in machining chatter”, Annals of the CIRP, 30, pp. 299-304, 1981. Kondo Y, Kawano O, and Sato H. “Behavior of chatter due to multiple regenerative effect”, ASME Journal of Engineering for Industry, 103, pp. 324-329, 1981. Shi M, and Tobias SA. “Theory of finite amplitude machine tool instability”, International Journal of Machine Tool Design and Research, 24, pp. 45-69, 1984. Balachandran B. “Nonlinear dynamics of milling process”, Proceedings of the Royal Society of London A, 359, pp. 793-819, 2001. Batzer SA, Gouskov AM, and Voronov SA. “Modeling vibratory drilling dynamics”, ASME Journal of Vibration and Acoustics, 123, pp. 435-443, 2001. Pratt JR, and Nayfeh AH. “Chatter control and stability analysis of a cantilever boring bar under regenerative cutting conditions”, Proceedings of the Royal Society of London A, 359, pp. 759-792, 2001. Wahi P, and Chatterjee A. “Regenerative tool chatter near a codimension 2 Hopf point using multiple scales”, Nonlinear Dynamics, 40, pp. 323-338, 2005. Wahi P, and Chatterjee A. “Galerkin projections for delay differential equations”, ASME Journal of Dynamic Systems, Measurement, and Control, 127, pp. 80-87, 2005. Wahi P. “A study of delay differential equations with applications to machine tool chatter”, PhD Dissertation, Indian Institute of Science, Bangalore, India, 2005.
ROBUST STABILITY OF TIME-DELAY SYSTEMS WITH UNCERTAIN PARAMETERS Z. H. Wang, H. Y. Hu Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, China Abstract:
This paper presents a brief survey on the robust stability analysis of time-delay systems against parametric uncertainties. It focuses on the case when the systems involve uncertain parameters falling into some intervals respectively, and their characteristic quasi-polynomials depend linearly on the uncertain parameters. The “Edge Theorem”, the method of stability switch and a graphic test are applied for the robust stability analysis, and are demonstrated through an example of time-delay system of single degree of freedom.
Key words:
Time delay, robust stability, uncertain parameters.
1.
INTRODUCTION
Uncertainties, including parametric uncertainty and structural uncertainty, always exist in both deterministic and stochastic dynamical systems. They may come from the simplification in system modeling, the measurement errors of system parameters, environment changes, etc. In many applications, the system dynamics must be preserved against the uncertainties from parametric disturbance and structural disturbance [1]. In addition, in the design phase of a controlled time-delay system, the feedback gains have to be determined. Hence, the determination of the admissible gains for the stability falls into the category of robust stability analysis [2]. This article addresses the robust stability problem of dynamical systems with time delays against parametric uncertainties. The system of concern is governed by a linear delay differential equation as following
x(t ) = A(p)x(t ) + B(p)x(t − τ ) ,
( x ∈ R n , A, B ∈ R n×n ),
(1)
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where p = [ p1 , p2 , , ps ] ∈ R s is the parameter vector. The trivial equilibrium x = 0 of Equation (1) is asymptotically stable if and only if all the roots of the following characteristic quasi-polynomial m n −1
D (λ , p,τ ) := λ I − A − B exp(−λτ ) = λ n + ∑∑ cij (p)e − jλτ λ n −1−i
(2)
j =0 i =0
have negative real parts, or in short, D(λ , p,τ ) is Hurwitz. A typical problem of robust stability is the so-called interval stability [4-5]. That is, the parameters are uncertain, but fall into some given intervals, say, p i ≤ pi ≤ pi ,
τ ≤τ ≤τ
The interval stability requires that the equilibrium is asymptotically stable for all parametric combinations [1, 3-4]. When τ = 0 , the robust stability of Equation (1) has been extensively studied. An astonishing result is Kharitonov’s theorem [1], which asserts that a polynomial λ n + a1λ n −1 + + an −1λ + an , with independent coefficients ai ’s falling into given intervals respectively, is Hurwitz if and only if four special polynomials are Hurwitz. An extension of Kharitonov’s theorem is the “Edge Theorem” that deals with the D-stability [1]. When τ > 0 is fixed, the “Edge Theorem” works too, if the coefficients of D(λ , p,τ ) depend linearly on the uncertain parameters p i ’s [4]. However, the robust stability analysis is usually a tough problem if the coefficients of D(λ , p,τ ) depend nonlinearly on pi ’ s, or on τ < τ , or on both [5-6]. This paper focuses on the time-delay systems for which the coefficients of D(λ , p,τ ) depend linearly on pi ’s. In Section 3, three cases are considered: (i). the delay τ > 0 is fixed; (ii). all the pi ’s are fixed; and (iii). both the pi ’s and the delay τ are uncertain. For the first case, the robust stability can be tested by means of the “Edge Theorem” and Nyquist plots [4]. For the second case, the robust stability analysis can be carried out on the basis of stability switches [3, 7-9]. For the third case, the stability of the whole family with such interval parameters can be confirmed by determining whether the vertex quasi-polynomials with one parameter are asymptotically stable or not, and whether the edge quasi-polynomials with two parameters have pure imaginary roots or not [10]. For demonstration, the robust stability analysis of a delayed oscillator arising from machine tool dynamics [11] is given in Section 4. Finally, Section 5 presents some concluding remarks.
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2.
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ROBUST STABILITY AGAINST PARAMETRIC UNCERTAINTY
The study begins with an assumption that the coefficients cij (p) depend linearly on the uncertain parametrers p , then for each given τ , the family Ωτ = {D(λ , p,τ ) : p ∈ Q, τ ≤ τ ≤ τ }
is a polytope spanned as the convex hull of the vertex quasi-polynomials Ωτ = conv{D1 (λ ,τ ), D2 (λ ,τ ),
where Q := {p = ( p1 , p2 ,
, ps ) ∈ R s :
, Dr (λ ,τ )} ,
p i ≤ pi ≤ pi , i = 1, 2,
(3) , s}. If τ = τ ,
then Ω = Ωτ is a polytope. Otherwise, the whole family Ω = ∪ Ωτ τ ≤τ ≤τ
(4)
is not a polytope. Let the set of all edges of Ωτ be denoted by E[Ωτ ] , where an edge of Ωτ is in the form of conv{ pi (λ ) , p j (λ ) } for some i and j [4]: eij (λ ,τ ) = (1 − µ ) Di (λ ,τ ) + µ D j (λ ,τ ) , µ ∈ [0, 1] .
(5)
2.1 The case of τ = τ In this case, Ω = Ωτ is a polytope, so that the “Edge Theorem” holds true. [4] shows that a polytope Ω of quasi-polynomials of order n given by Equation (3) is Hurwitz if and only if E[Ω] , the set of all edges of Ω , is Hurwitz. To check the Hurwitz stability of an edge quasi-polynomial eij (λ ,τ ) , generated by pi (λ ) and p j (λ ) , one requires to check the stability of a vertex quasi-polynomial and to confirm that eij (λ ,τ ) has no roots on the imaginary axis for all µ ∈ [0, 1] , because the roots of eij (λ ,τ ) depends continuously on µ . Moreover, [4] presented a graphical test. Let E1 , E2 , …, Et be the edges of the polytope defined in Equation (3), pk 0 (λ ) and pk1 (λ ) be the vertex quasi-polynomials of Ek . Then, Ω is Hurwitz if and only if for each edge Ek , one has (a) the Nyquist plot of pk 0 (iω ) /(iω + 1) n does not encircle the origin of the complex plane; and (b) the Nyquist diagram of pk1 (iω ) / pk 0 (iω ) does not cross the non-negative part of the real axis.
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For example, if D(λ , p,τ ) depends linearly on two parameters falling into p i ≤ pi ≤ pi , ( i = 1, 2 ), let D0 (λ ) = D (λ , p1 , p 2 ,τ ) , D1 (λ ) = D(λ , p1 , p2 ,τ ) , D2 (λ ) = D(λ , p1 , p2 ,τ ) , D3 (λ ) = D(λ , p1 , p 2 ,τ ) .
Then, the family of quasi-polynomials is Hurwitz if the Nyquist plot of D0 (iω ) /(iω + 1) n does not encircle the origin of the complex plane, and the Nyquist plots of D1 (iω ) / D0 (iω ) , D3 (iω ) / D0 (iω ) , D2 (iω ) / D1 (iω ) and D3 (iω ) / D2 (iω ) do not intersect (−∞,0] , respectively [4].
2.2 The case of τ < τ and p i = pi , (i = 1, 2, , s) In such a case, the robust stability for all τ ≤ τ ≤ τ can be analyzed by using the concept of stability switches [3,7-9]. In fact, as the delay τ varies, the stability of a quasi-polynomial D(λ , p,τ ) changes only when τ passes through a critical delay for which the quasi-polynomial has a root on the imaginary axis, because the roots of D(λ , p,τ ) depends continuously on τ . The critical values of τ can be determined from Re[ D(iω ,τ )] = 0 ,
Im[ D(iω ,τ )] = 0 .
(6)
The oscillation frequency ω must be a root of a function F (ω ) with even order terms only, and can be usually found numerically at first. Once a pair of roots λ = ±iω and the corresponding critical values of τ in Equation (6) are in hand, the changing direction of λ (τ ) can be determined from S ≡ sgn[d(Reλ ) / dτ
λ = iω
].
(7)
If S > 0 , then D(λ , p,τ ) admits one new pair of conjugate complex roots with positive real part as τ passes through the critical delay from the left to the right. And if S < 0 , then D(λ , p,τ ) loses one new pair of conjugate complex roots with positive real part as τ passes through the critical delay from the left to the right. For simplicity, let D(λ , p,τ ) = α (λ ) + β (λ )e − λτ , deg α > deg β , one has 2
2
F (ω ) = α (iω ) − β (iω ) ,
S = sgn F ′(ω ) .
If F (ω ) has exactly one positive simple root ω0 , then F (ω ) > F (ω0 ) = 0 for all ω > ω0 , and F (ω ) < F (ω0 ) = 0 for all ω ∈ [0, ω0 ) . It follows F ′(ω0 ) > 0 . This fact indicates that each crossing of the real part of characteristic roots at
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a critical value τ k of time delay corresponding to ±iω0 must be from the left to the right. Hence, if D(λ , p,τ ) with τ = 0 is Hurwitz, the numbers of characteristic roots with positive real part are 0, 2, 4, , 2i, on the intervals [0,τ 0 ) , (τ 0 ,τ 1 ) , (τ 1 ,τ 2 ), ,(τ i −1 ,τ i ), , respectively. That is, D(λ , p,τ ) is Hurwitz for τ ∈ [0,τ 0 ) , and is unstable for τ ∈ [τ 0 , +∞) . If D(λ , p,τ ) with τ = 0 is unstable, then there exists at least one pair of conjugate characteristic roots with positive real part for τ ∈ [0,τ 0 ) , and consequently, D(λ , p,τ ) is unstable for any given delay. If F (ω ) has two or more positive simple roots ω1 > ω2 > > ω p > 0 , then the difference between two critical values of time delay corresponding to a given pair of roots ±iω j satisfies
τ j , k +1 − τ j , k =
2π
ωj
<
2π
ω j +1
= τ j +1, k +1 − τ j +1,k
(8)
for k = 1, 2, , j = 1, 2, , p − 1 . It is easy to see that both sgn[ F ′(ω2 j −1 )] > 0 and sgn[ F ′(ω2 j )] < 0 , j ≥ 1 are true, since F (ω ) > F (ω1 ) = 0 holds for all ω ∈ (ω1 , + ∞) and all possible ω ∈ (ω2 k +1 , ω2 k ) , and F (ω ) < F (ω1 ) = 0 holds for all possible ω ∈ (ω2 k , ω2 k −1 ). Hence, the crossing real parts of characteristic roots at τ 2 j −1,k (corresponding to ±iω2 j −1 ) must be from the left to the right, and the crossing at τ 2 j , k (corresponding to ±iω2 j ) must be from the right to the left. Then, as τ increases, D(λ , p,τ ) always admits a new pair of conjugate roots with positive real parts for each crossing at τ 2 j −1,k , and reduces such a pair for each crossing at τ 2 j ,k . Given a long time delay τˆ , Equation (8) indicates that the interval [0, τˆ] includes more τ 1, k corresponding to ±iω1 than τ 2,l to ±iω2 , more τ 3,m corresponding to ±iω3 than τ 4, n to ±iω4 , and so forth. Hence, D(λ , p,τ ) must have eventually some roots with positive real parts when τ is long enough. As a result, D(λ , p,τ ) becomes unstable after a finite number of stability switches. In general, more elaborate tools [3, 7, 9-10] are required for deriving the polynomial F (ω ) , say, the Sylvester resultant as used in [10].
2.3 The case of τ < τ and p i ≤ pi , (i = 1, 2, , s) Now, Ω is not a polytope, but Ωτ is a polytope for each τ ≤ τ ≤ τ . Thus, the “Edge Theorem” applies for each fixed τ ≤ τ ≤ τ . More precisely, the non-polytope Ω is Hurwitz if and only if for each edge quasi-polynomial eij (λ ,τ ) = (1 − µ ) Di (λ ,τ ) + µ D j (λ ,τ ) , (i) the two vertex quasi-polynomials
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are Hurwitz for all τ ≤ τ ≤ τ ; (ii) eij (iω ,τ ) ≠ 0 holds for any ω ∈ R , µ ∈ [0,1] and all τ ≤ τ ≤ τ . Here the vertex quasi-polynomials have a parameter τ , and their stability can be analyzed by following the idea of stability switch discussed above. As for the edge quasi-polynomial, the inequality eij (iω ,τ ) ≠ 0 is equivalent to z 2 Di (iω ,τ ) + D j (iω ,τ ) ≠ 0 ,
(9)
where z 2 = (1 − µ ) / µ ≥ 0 for all µ ∈ [0, 1] . Let a = (Re[ Di (iω ,τ )]) 2 + (Im[ Di (iω ,τ )]) 2 , b = Re[ Di (iω ,τ )] ⋅ Re[ D j (iω ,τ )] + Im[ Di (iω ,τ )] ⋅ Im[ D j (iω ,τ )] , c = (Re[ D j (iω ,τ )]) 2 + (Im[ D j (iω ,τ )]) 2 .
Then, condition (9) is equivalent to a z 4 + 2b z 2 + c ≠ 0 .
(10)
If the vertex quasi-polynomials are assumed to be Hurwitz, then a > 0 and c > 0 hold. Thus, condition (10) holds if and only if b≥0,
or,
b < 0 and b 2 − ac < 0 .
According to the definition of b = b(ω ,τ ) , the coefficients of b are polynomials with respect to ω , cos ωτ and sin ωτ , and the leading coefficient with respect to ω is positive and independent of cos ωτ and sin ωτ . It is easy to get a polynomial b0 (ω ) (≤ b(ω ,τ )) with positive leading coefficient, by replacing cos ωτ and sin ωτ in b(ω ,τ ) with 1 or −1 . Then, it is straightforward to find out the maximal root ω0 for polynomial b0 (ω ) numerically. If ω > ω0 , then b = b(ω ,τ ) ≥ b0 (ω ) > 0 . Thus, one only needs to check the conditions on the rectangle [0, ω0 ] × [τ ,τ ] . If min b(ω ,τ ) ≥ 0 or
[0,ω0 ]×[τ ,τ ]
max (b 2 − 4ac) < 0 ,
[0,ω0 ]×[τ ,τ ]
the edge family eij (λ ,τ , µ ) is Hurwitz for any τ ∈ [τ ,τ ] and µ ∈ [0,1] . Let b(ωˆ ,τˆ) ≡ min[0,ω0 ]×[τ ,τ ] b(ω ,τ ) . Then, one has ∂ b(ω ,τ ) ∂ω
(ωˆ ,τˆ )
= 0 and
∂ b(ω ,τ ) ∂τ
(ωˆ ,τˆ )
= 0,
(11)
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where ∂b / ∂ω and ∂b / ∂τ are polynomials with respect to ω , cos ωτ and sin ωτ . Thus, the resultant elimination can be used to find out the extreme points (ωˆ ,τˆ) and the corresponding extreme values. The same procedure can be performed to b 2 − 4ac . In general, it is not an easy task to check condition (11) analytically. Maple/Matlab commands (say, plots [implicitplot] in Maple, or ezplot in Matlab), however, enable one to check condition (11) easily. It is reasonable to have such a graphical test. In fact, as pointed out in [6], the problem of robust stability is an NP-hard problem, which is usually interpreted as an indication of inherent intractability, due to the uncertainty of time delays.
3.
AN ILLUSTRATIVE EXAMPLE
To show the main results in Section 2, this section checks the robust stability of an oscillator with delayed position feedback governed by x(t ) + 2ξ x(t ) + (1 + p ) x(t ) − px(t − τ ) = 0 with the characteristic function D(λ , ξ , p,τ ) = λ 2 + 2ξλ + (1 + p) − pe − λτ ,
(12)
where ξ > 0 and p > 0 are two parameters [11]. When τ = 0 , the polynomial is Hurwitz. Hence, the stability may change only if τ passes through a critical delay for which D(iω , ξ , p,τ ) =0 holds. At the critical delays, ω must be a root of the following polynomial F (ω ) = ω 4 + (−2 − 2 p + 4ξ 2 )ω 2 + 1 + 2 p := ω 4 + c2ω 2 + c0 .
Once a positive root of F (ω ) = 0 is in hand, the critical delays can be determined from sin(ωτ ) = −2ξω / p , cos(ωτ ) = (1 + p − ω 2 ) / p . Let us firstly consider the robust stability independent of the delay. If F (ω ) = 0 has no real roots, then D(λ , ξ , p,τ ) is Hurwitz for all τ ≥ 0 . Because c0 = 1 + 2 p > 0 , it is easy to see that F (ω ) = 0 has no real roots if and only if c22 − c0 < 0 , or, c22 − c0 ≥ 0 and c2 > 0 [3]. As shown in Figure 1, D(λ , ξ , p,τ ) is Hurwitz for all τ ≥ 0 if and only if (ξ , p) takes values in sub-region II+III+IV. Hence, D(λ , ξ , p,τ ) with (ξ , p ) ∈ [ξ , ξ ] × [ p, p ] is Hurwitz for all τ ≥ 0 if and only if the four vertex quasi-polynomials and the four exposed edge quasi-polynomials are Hurwitz for all τ ≥ 0 . For the robust stability with finite delay, the Nyquist plot, shown in Figure 2, of D(iω ,0.1,0.3, 2) /(1 + iω )2 indicates that D(λ ,0.1,0.3, 2) is
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Hurwitz. If the parameters ξ , p and τ admit ±20% variation, one needs to check the system stability for all ξ ∈ [0.08, 0.12] , p ∈ [0.24, 0.36] and τ ∈ [1.6, 2.4] . As discussed above, one requires to examine: z
the stability of four vertex families vi (λ ,τ ) for all τ ∈ [1.6, 2.4] , where v1 = D(λ ,0.08,0.24,τ ) , v2 = D(λ ,0.12,0.24,τ ) , v3 = D(λ ,0.12,0.36,τ ) , v4 = D(λ ,0.08,0.36,τ ) ,
z
zero-exclusion of the exposed edge families eij (λ ,τ , µ ) , where e12 = (1 − µ )v1 + µ v2 , e23 = (1 − µ )v2 + µ v3 , e34 = (1 − µ )v3 + µ v4 , e41 = (1 − µ )v4 + µ v1 . 10 9 2
p=2ξ −1
8 7
p=2ξ2+2ξ
6
p
II
I
5
IV
III
4 2
3
p=2ξ −2ξ
2 1 0 0
0.5
1
1.5
2
ξ
2.5
3
3.5
4
Figure 1. A division of the parameter plane (ξ , p ) . In sub-region I (the left to the curve of p = 2ξ 2 + 2ξ ), F (ω ) has real roots only so that stability switch occurs. If (ξ , p ) is taken from the sub-region II+III+IV (the right to the curve of p = 2ξ 2 + 2ξ ), F (ω ) has no real roots so that D (λ , ξ , p,τ ) is Hurwitz for all given τ ≥ 0 . 0.5 0.4 0.3 0.2
Im
0.1
ω→+∞ ω→−∞
0
−0.1 −0.2 −0.3 −0.4 −0.5 0
0.2
0.4
0.6
Re
0.8
1
Figure 2. The Nyquist plot of D(iω ,0.1,0.3, 2) /(1 + iω ) 2 shows that D(λ ,0.1,0.3, 2) is Hurwitz.
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3.1 The stability of the vertex quasi-polynomials At first, one checks the stability of v1 = D(λ ,0.08,0.24,τ ) . In this case, F (ω ) = ω 4 − 2.4544ω 2 + 1.48 has two positive roots 1.0324 and 1.1783, and the corresponding minimal critical delays are 5.3506 and 3.4329, respectively. Thus, the vertex quasi-polynomial v1 is Hurwitz for all τ ∈ [0,3.4329) . Next, for the vertex family v2 , F (ω ) = ω 4 − 2.4224ω 2 + 1.48 has no real roots. Hence, v2 is Hurwitz for all given τ ≥ 0 . Then, for the vertex family v3 , F (ω ) = ω 4 − 2.6624ω 2 + 1.72 has two positive roots 1.0502 and 1.2488, and the corresponding minimal critical delays are 3.7299 and 4.2438, respectively. Thus, v3 is Hurwitz for all τ ∈ [0,3.7299) . Finally, for the vertex family v4 , F (ω ) = ω 4 − 2.6944ω 2 + 1.72 has two positive roots 1.0193 and 1.2866, and the minimal critical delays are 5.7027 and 2.9149, respectively. Thus, v4 is Hurwitz for all τ ∈ [0, 2.9149) . As a result, the 4 vertex quasi-polynomials are Hurwitz if τ ∈ [0, 2.9149) .
3.2 The stability of the edge quasi-polynomials To check whether the edge family eij (λ ,τ , µ ) is Hurwitz or not, for any τ ∈ [τ ,τ ] and µ ∈ [0,1] , one needs to find out an ω0 > 0 such that b > 0 if ω > ω0 , and then to determine whether the condition (11) holds or not over [0, ω0 ] × [τ ,τ ] . For the family e12 , it is easy to see
b(ω ,τ ) ≥ ω 4 − 2.9216ω 2 − 0.096ω + 1 . Thus, b > 0 if ω > ω0 := 1.6112 . In the Maple platform, straightforward application of the commands plots [implicitplot] and plot3d show that b and d has a definite sign over [0, 1.6112] × [0, 2.9149] , namely, b > 0 and d < 0 . It means that e12 (iω ,τ ) ≠ 0 holds for any ω ∈ R , µ ∈ [0, 1] and all τ ∈ [0, 2.9149) . Next, one has b(ω ,τ ) ≥ ω 4 − 3.1424ω 2 − 0.144ω + 1 for e23 . Thus, b > 0 if ω > ω0 := 1.6971 . The graphical test shows that b > 0 holds over [0, 1.6971] × [0, 2.9149] . Hence, e23 (iω ,τ ) ≠ 0 holds for any ω ∈ R , µ ∈ [0,1] and all τ ∈ [0, 2.9149) . Finally, the same procedure can be used for the edge families e34 and e41 . For e34 , b > 0 if ω > ω0 := 1.7814 . The graphical test shows that over [0,1.7814] × [0, 2.9149] , b does not have a definite sign, but d < 0 definitely. Hence, e34 (iω ,τ ) ≠ 0 holds for any ω ∈ R , µ ∈ [0, 1] and all τ ∈ [0, 2.9149) . This is the same case for e41 . Thus, the four exposed edge families are all Hurwitz. As a result, D(λ , ξ , p,τ ) given in Equation (12) is Hurwitz for all ξ ∈ [0.08, 0.12] , p ∈ [0.24, 0.36] and τ ∈ [0, 2.9149) .
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4.
CONCLUSIONS
From the view point of computational complexity, the robust stability of a time-delay system with uncertain parameters including the time delays is a NP-hard problem. This paper presents some basic facts about the robust stability analysis for a special case when the characteristic quasipolynomials depend linearly on the uncertain parameters (not including the delay), and it shows how to combine the “Edge Theorem”, the analysis of stability switch and a graphic routine to check the robust stability. It is worthy to note that the robust stability analysis is still an open problem for the general time-delay systems with uncertain parameters.
ACKNOWLEDGEMENTS This work was supported by FANEDD of China under Grant 200430, and in part by the National Natural Science Foundation of China under Grants 10372116, 10532050.
REFERENCES 1. 2.
Huang L. Fundamentals of Stability and Robustness, Beijing: Science Press, 2003. Datta A, Ho MT, Bhattacharyya SP. Structure and Synthesis of PID Controllers, London: Springer-Verlag, 2000. 3. Hu HY, Wang ZH. Dynamics of Controlled Mechanical Systems with Delayed Feedback, Berlin: Springer-Verlag, 2002. 4. Fu MY, Olbrot AW, Polis MP. “Robust stability for time-delay systems: the edge theorem and graphical tests”, IEEE Transactions on Automatic Control, 34, pp. 813-820, 1989. 5. Niculescu SI, Delay Effects on Stability: A Robust Control Approach, London: SpringerVerlag, 2001. 6. Blondel VD, Tsitsiklis JN. “A survey of computational complexity results in systems and control”. Automatica, 36, pp. 1249-1274, 2000. 7. Marshall JE, Goreki H, Walton K, Korytowski W. Time-Delay Systems, Stability and Performance Criteria with Applications. New York: Ellis Horwood, 1992. 8. Hu HY. “Using delayed state feedback to stabilize periodic motions of an oscillator”. Journal of Sound and Vibration, 275, pp. 1009-1025, 2004. 9. Wang ZH, Hu HY. “Stabilization of vibration systems via delayed state difference feedback”. Journal of Sound and Vibration, 297, pp. 117-129, 2006. 10. Wang ZH, Hu HY, Küpper T. “Robust Hurwitz stability test for linear systems with uncertain commensurate time delays”, IEEE Transactions on Automatic Control, 49, pp. 1389-1393, 2004. 11. Kalmár-Nagy T, Stépán G, Moon FC. “Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations”, Nonlinear Dynamics, 26, pp. 121-142, 2001.
DYNAMICS DUE TO NON-RESONANT DOUBLE HOPF BIFURCATIONIN IN VAN DEL POL-DUFFING SYSTEM WITH DELAYED POSITION FEEDBACK J. Xu1, M. S. Huang2, Y. Y. Zhang3 1
School of Aerospace and Applied Mechanics, Tongji University, 200092, Shanghai, China Department of Communication Science and Engineering, Fudan University, 220437, Shanghai, China 3 Department of Mathematics, University of Houston, 77204, Houston, Texas, USA 2
Abstract:
In this paper, the van der Pol-Duffing system with delayed position feedback is investigated A critical point with 1: 2 frequency ratio is obtained. The center manifold reduction (CMR) is employed to classify various solutions bifurcating from the point. The approximate expressions provided by the CMR are valid for values of the time delay and feedback gain close to the double Hopf point but invalid for those of the bifurcation parameters far from the bifurcation point. A called perturbation-incremental scheme (PIS) is proposed to overcome such disadvantage. The consistency between numerical and PIS solutions shows the PIS is efficient. The analytical approximation derived from the PIS can reach any required accuracy.
Key words:
Delay differential equation, double Hopf bifurcation, feedback control, nonlinear dynamics, chaos.
1.
INTRODUCTION
Time delay is not taken into account modeling in the traditional conception, i.e. its effects on systems are neglected, although the delay is ubiquitous in the nature. In fact, time delays often occur due to the transformation lag in the input-output systems. And it has been well known 373 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 373–382. © 2007 Springer.
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that time delay may evoke notable changes of dynamics behavior of many systems. In studying effects of the time delayed position feedback [1] or the time delayed velocity feedback [2] on a van der Pol-Duffing oscillator, we find that it is possible for the double Hopf bifurcations (associated with two pairs of pure imaginary eigenvalues) to appear in the system with varying the feedback gain and the delay. The problem is remained there. Motivated by it, this paper deals with delay-induced non-resonant double Hopf bifurcations, chaos and quantitative computation of the bifurcating solutions derived from the bifurcation in a van der Pol-Duffing system with delayed feedback. The initial model that is considered is a van der Pol-Duffing system
x + ω02 x − (α − γ x 2 ) x + β x3 = 0.
(1)
Adding a time delay linear position feedback to system (1), one can get x + ω02 x − (α − γ x 2 ) x + β x3 = A( xτ − x),
(2)
where α , β , γ > 0, xτ = x (t − τ ) and τ is a time delay. In this paper only negative position feedback is discussed, i.e. A < 0 . And the positive case can be studied in the similar way. Rescaling x → ε x in Equation (2) and yields x + ω02 x − (α − εγ x 2 ) x + εβ x 3 = A( xτ − x),
(3)
since the stability of the trivial solution of Equation (3) is mainly investigated. As one of non-resonant cases, the paper is focus on discussing 1 : 2 double Hopf bifurcation occurring in the system (1) with varying A and τ .
2.
DELAY-INDUCED DOUBLE HOPF BIFURCATION WITH NON-RESONANCE The characteristic equation of Equation (3) is given by
λ 2 − αλ + ω02 = A ( e − λτ − 1)
(4)
Let λ = κ + iω , where κ and ω are real and ω1 , ω2 are positive since the roots of Equation (4) appear in pairs. For A < 0 and α > 0 , substituting λ and κ = 0 into Equation (4) yields
Dynamics due to delay-induced double Hopf bifurcation
α2
375
2
⎛ α2 ⎞ 2 2 ω± = A − +ω ± ⎜ A− ⎟ − α ω0 , 2 2 ⎠ ⎝ 2 0
(5)
and
τ ± [ j] =
ω02 − ω±2 ⎞ ⎤ 1 ⎡ −1 ⎛ ⎢ 2 jπ − cos ⎜1 + ⎟⎥ , A ⎠⎦ ω± ⎣ ⎝
(6)
where j = 1, 2, 3, …. It can be proved that dκ dτ
⎧> 0 on τ − , =⎨ ⎩< 0 on τ + , κ =0
(7)
which implies that the curves determined by τ − and τ + are boundaries of the Hopf bifurcation. As an example, we consider ω1 : ω2 = 1 : 2 , where ω1 = ω− , ω2 = ω+ and τ − = τ + . These yield the critical point to be τ c = 5.7171625, Ac = −0.3126549, ω1 = 0.657901, ω2 = 0.930413 when α = 0.276023.
3.
CENTER MANIFOLD REDUCTION
To obtain various dynamical behaviors in a neighborhood of the above double Hopf point, we reduce Equation (3) on its center manifold. Letting A = Ac + σ 1ε , τ = τ c + σ 2 ε , then Equation (3) can be written as u1 = u2 , u2 = − ⎛⎜⎝ ω02 + Ac ⎞⎟⎠ u1 + α u2 + Ac u1τ c + ε ⎜⎜⎜⎝ −σ 1u1 + σ 1u1τ c ⎟⎟⎟⎠ ⎛
⎞
(8)
+ Ac u1(τ c +εσ 2 ) − Ac u1τ c + ε ⎛⎜⎝ −γ u12u2 − β u13 ⎞⎟⎠ ,
where u1 = x, u1τ = u1 ( t − τ ) . Equation(8) can be expressed in a form of the functional differential equation, given by ut = D(0)ut + D(σ 1 , σ 2 )ut + ε Qut ,
where ut (θ ) = u (t + θ ), −τ ≤ θ ≤ 0, u = ( u1 , u2 ) , T
(9)
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⎧ dφ (θ ) ⎪ D(0)φ = ⎨ dθ ⎪⎩ L(0)φ
θ ∈ [−τ , 0), ⎧ 0 D(σ 1 , σ 2 )φ = ⎨ ⎩ L(σ 1 ,σ 2 )φ θ = 0,
θ ∈ [−τ , 0), θ = 0,
and ⎧ 0 Qφ = ⎨ ⎩ F (t , φ )
forθ ∈ [−τ , 0), forθ = 0,
0 ⎧ ⎫ F (t , φ ) = ⎨ ⎬, 2 3 ⎩−γφ1 (0)φ2 (0) − βφ1 (0) ⎭
with L(0)φ = ∫
0
⎡⎣ dη ( s ) ⎤⎦ φ ( s ),
−τ c
0
0
−τ
−τ
u1τ = u1 ( t − τ ) = ∫ δ (θ + τ ) u1 (t + θ )dθ = ∫ δ (θ + τ ) u1t dθ ,
F : R+ × C → R 2 , F ( t , ut ) = ( 0, −γ u12t ( 0 ) u2t (0) − β u13t (0) ) , T
0 δ (s) ⎤ ⎡ dη ( s ) = ⎢ ⎥ ds. 2 ⎣ −(ω0 + Ac )δ ( s ) + Acδ ( s + τ c ) αδ ( s ) ⎦
For ε = 0 , there are two pairs of eigenvalues with zero real parts in Equation (9), denoted by Λ = ±iω1 , ±iω2 . In this case, there exists a 4-dimensional invariant manifold in the state space C which can be split into two subspaces as C = PΛ ⊕ QΛ , where PΛ is a four-dimensional space spanned by the eigenvectors corresponding to the eigenvalues Λ , and QΛ is the complementary space of PΛ . Let Φ be a basis for the invariant subspace corresponding to PΛ and Ψ to QΛ and Ψ,Φ = I , where I is the 4 × 4 identity matrix and the bilinear form is given by
ψ ,φ = ψ T (0)φ (0) − ∫
0
−τ
∫
θ
0
ψ T (ξ − θ )[dη (θ )]φ (ξ )d ξ ,
(10)
where C ∗ is the dual space of C , φ ∈ C and ψ ∈ C ∗ . With aids of Equation (10), one can obtain Φ and Ψ . Now projecting ut to the four-dimensional center manifold by Ψ, ut yields Ψ,Φ v = Ψ, D(0)Φ v + Ψ, D(σ 1 , σ 2 )Φ v + ε Ψ, QΦv ,
and
(11)
Dynamics due to delay-induced double Hopf bifurcation ⎡ 0 ω1 ⎢ −ω 0 v = ⎢ 1 ⎢ 0 0 ⎢ 0 ⎣ 0
0 0 0 −ω2
0⎤ 0 ⎥⎥ v + Dε v + ε Nε (v), ω2 ⎥ ⎥ 0⎦
377
(12)
where Dε is the O(ε ) linear term and Nε (v) represents the nonlinear terms contributed from the original system to the center manifold. The relationship between u and v is expressed as ⎧ v1 ⎫ ⎪v ⎪ ⎧u ⎫ ⎪ ⎪ u = ⎨ 1 ⎬ = Φ ( 0) ⎨ 2 ⎬. ⎩u2 ⎭ ⎪v3 ⎪ ⎩⎪v4 ⎭⎪
(13)
It should be noted that some details are not represented here due to limit pages. Readers can see Ref. [1] or [2].
4.
CLASSIFICATION OF DYNAMICAL BEHAVIORS
Using method of normal forms given in Ref. [3], one can transform Equation (12) in a polar coordinate form K r1 = µ1r1 + r1 3 + br1 r22 , K r2 = µ2 r2 + cr1 2 r2 + d r23,
(14)
µ 1 = −0.00479719σ 1 − 0.00204471σ 2 , µ 2 = 0.0134786σ 1 + 0.00127686σ 2 .
(15)
where
It follows from Equation (13) that the approximation can be obtained, where v1 = Re ( r1eiθ1 ) , v2 = − Im ( r1eiθ1 ) , v3 = Re ( r2 eiθ2 ) , v4 = − Im ( r2 eiθ2 ) . According to the case of Figure 7.5.5 in [4], one can easily classify dynamics near the double Hopf point as shown in Figure 1. Figure 1 shows that double Hopf bifurcation with frequency ratio 1: 2 at ( Ac ,τ c ) = (−0.3126549, 5.7171625) leads to seven different bifurcation sets of the system with the time delay and feedback gain varying. The trivial
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solution is stable on Region I, which is called the death island (see Ref. [1]). A Hopf bifurcation leads to a stable periodic solution with the frequency in ω1 when ( A,τ ) crosses from Region I to II. A stable quasi-periodic solution occurs in the system for values of ( A,τ ) belonging to Region III by a saddle-node bifurcation. There no any stable harmonic solutions on other regions.
Figure 1. Classification of bifurcating solutions derived from double Hopf bifurcation in A − τ plane when ω1 : ω2 = 1: 2 , ω0 = 1 , β = γ = 1 .
(a) Region I
(b) Region II
(c) Region III (d) Region IV Figure 2. Numerical simulation corresponding to regions in Figure 1 for (a) A = −0.275,τ = 5.5 , (b) A = −0.3,τ = 5.5 , (c) A = −0.32,τ = 5.5 , (d) A = −0.35,τ = 5.6
To verify the validity the analytic results shown in Figure 1, the numerical simulation for all regions in Figure 1 is represented in Figure 2. The numerical simulation matches well with qualitative analysis represented in
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Figure 1. It should be noted that some complex phenomena such as chaos from torus-breaking are found in Region IV in numerical analysis, where no any stable harmonic solutions are found analytically (see Figure 2(d)). This suggests that the theoretical analysis can provide basic information to find the complex dynamics of the system. To make sure that the chaotic motions may occurs in Region IV further, we compute two distinct values of Region IV and find two chaotic attractors, as shown in Figure 3 where the Poincaré section is given by {( x, y ) xτ = 0} .
(a) A = −0.4,τ = 5.7 (b) A = −0.375,τ = 5.6 Figure 3. The chaotic attractors on the Poincaré section for values of ( A,τ ) located on Region IV.
5.
QUANTITATIVE COMPUTATION
It has been seen that the qualitatively analytical results derived from CMR are in good agreement with those from numerical simulation. Now, we observe the validity of the quantitative analysis by using CMR. It should be noted that the approximate solution given in Equation (13) is used here. First, the analytical and numerical results are compared at two values ( A,τ ) = (−0.311,5.70) and ( A,τ ) = (−0.3326,5.617) located on Region II and III, respectively, as shown in Figure 4. It can be seen from Figure 4 that the analytical results are in quite good with those from numerical simulation even for a quasi-periodic solution (see Figure 4(b)). However, it is found that the analytical solution can not expressed that from numerical simulation quantitatively with values of ( A,τ ) are varied far away from the critical point ( Ac ,τ c ) . This implies that when ( A,τ ) is close to the double Hopf bifurcation point, analytical approximations solved by applying CMR and method of normal forms have good accuracy not only for periodic solutions but also for quasi-periodic ones. However, when ( A,τ ) goes far away from the bifurcation point, analytical result becomes invalid, as shown in Figure 5. Aiming to solve this problem, we extend and develop a semi-analytical/numerical method, called the perturbation-incremental
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scheme (PIS) [5] to express the periodic solution for delayed systems. For this end, we introduce a new time variable ϕ given by
Figure 4. Comparison between analytical and numerical solutions, where the solid line represents analytical solutions and the dashed line numerical ones.
Figure 5. Comparison between the approximate analytical solution (D) , PIS solution (×) and numerical simulation (solid) when A and τ vary along σ 2 = −15σ 1 starting from ( Ac ,τ c )
dϕ = Φ (ϕ ), Φ (ϕ + 2π ) = Φ (ϕ ), dt
(16)
so that periodic solution of Equation (2) can be rewritten as x(ϕ ) = a cos ϕ + b , where a is amplitude, b is the bias and a > 0 . If t − τ corresponds to ϕ1 , then xτ = x(ϕ1 ) = a cos ϕ1 + b . Taking ( A0 ,τ 0 ) close to the double Hopf point as a starting point, then one can obtain the corresponding periodic solution by CMR as follow x = x0 = a0 cos ϕ + b0 , Φ = Φ 0 , ϕ1 = ϕ10 .
(17)
The solution governed by Equation (17) can be used as the initial solution for the iteration in the PIS since it has good accuracy as mentioned above. The Φ 0 is expanded in a Fourier’s series M
Φ 0 = ∑ ( Pj cos jϕ + Q j sin jϕ ). j =0
(18)
Dynamics due to delay-induced double Hopf bifurcation
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For any ( A,τ ) on the region where the periodic solution exists, incrementing ( A,τ ) from ( A0 ,τ 0 ) to ( A0 + ∆A,τ 0 + ∆τ ) yields incremental of all other variables. One substitutes these into Equation (2) and uses harmonic balance to obtain linear algebraic equations in terms of variables ∆a , ∆b , ∆Pj , ∆Q j , given by M
An ∆a + Bn ∆b + An ,0 ∆P0 + ∑ ( An , j ∆Pj + Bn , j ∆Q j ) = Rn ,
(19)
j =1
where n = 0,1, 2," , 2 M + 2 . Here, the right term Rn of Equation (19) is residue to prevent the incremental process from deviating required accuracy. Now, one can solve Equation (19) through an equation solver such as the Gaussian procedure to obtain ∆a , ∆b , ∆Pj , ∆Q j and ∆Φ , ∆ϕ1 , giving arise to new variables a , b , Φ , ϕ1 , replacing a0 , b0 , Φ 0 , ϕ10 with a , b , Φ , ϕ1 as the initial values of the next iteration, and then repeating the incremental process until Rn → 0 for any n . Thus, the stable periodic solution of Equation (2) is finally obtained when ( A,τ ) is changed from ( A0 ,τ 0 ) to ( A0 + ∆A,τ 0 + ∆τ ) . In this way, even if τ is changed to be a value far away from the double Hopf point, periodic solutions of Equation (2) can be gained, as shown in Figure 5 (see symbol crossing labels). It follows from Figure 5 that the results derived from PIS have required accuracy even for those values of ( A,τ ) far away from the double Hopf bifurcation point. Therefore, it may be considered as a tool of quantitative analysis.
6.
CONCLUSIONS
In this paper, the sufficient and necessary condition for existence of double Hopf bifurcation point is first given by a linear analysis for the system under consideration when the time delay and feedback gain are regarded as bifurcation parameters. Then, the CMR and method of normal forms are employed to reduce the delayed differential equation (DDE) to be a set of 4-dimensional ordinary differential equations. Subsequently, dynamics in the neighborhood of the 1: 2 double Hopf bifurcation point is classified qualitatively. The analytical approximations of the stable periodic and quasi-periodic solutions are expressed quantitatively. It is seen that the approximate solutions are in good agreement with those from the numerical simulations for values of the bifurcation parameters close to the double Hopf point. In addition, there are more related works that one can be considered further. For instance, one may consider the other two cases: strong and weak resonance. One can also add nonlinear position or velocity feedback to the
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original system. Although we only consider a two-order delay differential equation, it is a basic element in a large scale of network. Based on the results given in this paper, one may investigate some network models with coupling delays. Further possible applications can be extended to a machine tool system in which quasi-periodic oscillations are shown analytically and experimentally by Stepan [6]. In [7], a robotic application is described with experimental results for force control and with a CMR and normal form analysis for non-holonomic position control, 3rd order system with single delay, several closed-form non-resonant cases and a resonance problem is also outlined. For a neural network application with 3 neurons, quasiperiodic oscillations are shown close to double Hopf point parameters [8] . Finally, this paper provides a method combining the CMR with PIS for DDEs. Such method is efficient in not only qualitative but also quantitative analyses for those values far from the double Hopf bifurcation point.
ACKNOWLEDGEMENTS This work is supported by Key Project of National Nature and Science Foundation of China under grant No. 10532050 and Project of National Outstanding Young Science Foundation under grant No. 10625211.
REFERENCES 1. 2.
3. 4. 5. 6. 7. 8.
Xu J, Chung KW. “Effects of time delayed position feedback on a van der Pol-Duffing oscillator”, Physica D, 180, pp. 17-39, 2003. Xu J., Yu P. “Delay-reduced bifurcations in an non-autonomous system with delayed velocity feedbacks”, International Journal of Bifurcation and Chaos, 14, pp. 2777-2798, 2004. Nayfeh AH. Method of normal forms, New York, John Wiley & Sons, 1993. Guckenheimer J, Holmes P. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, New York, Springer-Verlag, 1983. Chan HSY, Chung KW, Xu Z. “A perturbation-incremental method for strongly nonlinear oscillators”, International Journal of Nonlinear Mechanics, 31, pp. 59-72, 1996. Stepan G. “Modelling non-linear regenerative effects in metal cutting”, Philosophical Transactions of the Royal Society, 359, pp. 739-757, 2001. Stepan G. Haller G. “Quasiperiodic oscillations in robot dynamics”, Nonlinear Dynamics, 8, pp. 513-528, 1995. Campbell SA., Edwards R. and van den Driessche P. “Delayed coupling between two neural network loops”, SIAM. Journal on Applied Mathematics, 65, pp. 316-335, 2004.
STABILITY AND RESPONSE OF QUASI INTEGRABLE HAMILTONIAN SYSTEMS WITH TIME-DELAYED FEEDBACK CONTROL W. Q. Zhu, Z. H. Liu Department of Mechanics, Zhejiang University, 310027, Hangzhou, China, E-mail:
[email protected]
Abstract:
The stochastic averaging method for quasi integrable Hamiltonian systems under time-delayed feedback control is firstly introduced. Then, the effect of time-delayed feedback control on the response is investigated by solving the associated averaged FPK equation. The asymptotic Lyapunov stability with probability one of quasi integrable Hamiltonian systems with time-delayed feedback control is studied by using the largest Lyapunov exponent of the averaged systems. Finally, three examples are given to illustrate the proposed method and the effects of time-delayed feedback control on the stability and response.
Key words:
Time delay, feedback control, stochastic averaging, stationary response, stochastic stability.
1.
INTRODUCTION
Time delay is one of the most important factors that should be considered cautiously in applying the feedback control technique, since feedback control needs time to measure and estimate the system states and to compute and implement the control forces, and application of unsynchronized control forces due to time delay may result in a degradation of control performance and may even destabilize the system. Systems with time delay under deterministic excitation have been studied extensively [1-3] while the study on those systems under random excitation is very limited [4, 5]. The stochastic averaging method for quasi integrable Hamiltonian systems has 383 H. Y. Hu and E. Kreuzer (Eds.), IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, 383–392. © 2007 Springer.
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been proposed by the present first author and his coworkers [6] and used for response prediction [6], stability analysis [7] and optimal control [8] of MDOF stochastically excited nonlinear systems. In this paper, the stochastic averaging method is extended to study the stability and response of quasi integrable Hamiltonian systems with time-delayed feedback control.
2.
THE STOCHASTIC AVERAGING METHOD FOR QUASI INTEGRABLE HAMILTONIAN SYSTEMS WITH TIME-DELAYED FEEDBACK CONTROL
Consider an n-degree-of-freedom quasi integrable Hamiltonian system with time-delayed feedback control governed by the following equations: ∂H ′ Q i = ∂Pi ∂H ′ ∂H ′ Pi = − − ε cij′ + ε Fi (Qτ , Pτ ) + ε 1/ 2 f ikWk (t ) ∂Qi ∂Pj i, j = 1, 2,..., n; k = 1, 2,..., m
(1)
where Qi and Pi are generalized displacements and momenta, respectively; H ′ = H ′(Q, P ) is twice differentiable Hamiltonian; ε is a small positive parameter; ε cij′ = ε cij′ (Q, P) represent the coefficients of quasi linear dampings; ε 1/ 2 f ik = ε 1/ 2 f ik (Q, P) represent the amplitudes of stochastic excitations; ε Fi (Qτ , Pτ ) with Qτ = Q(t − τ ) and Pτ = P (t − τ ) denote feedback control forces with time delay τ ; Wk (t ) are Gaussian white noises in the sense of Stratonovich with correlation functions E[Wk (t )Wl (t + T )] = 2 Dkl δ (T ) . System (1) can be converted into Itô stochastic differential equation by adding Wong-Zakai correction terms, i.e. ∂H ′ ∂Pi ⎡ ∂H ′ ∂f ⎤ ∂H ′ dPi = − ⎢ + ε cij′ − ε Fi (Qτ , Pτ ) − ε Dkl f jl ik ⎥ dt + ε 1/ 2σ ik dBk (t ) (2) ∂Pj ∂Pj ⎥⎦ ⎢⎣ ∂Qi i, j = 1, 2,..., n; k = 1, 2,..., m dQi =
where Bk (t ) are standard Wiener processes and σσT = 2fDf T . Assume that Hamiltonian H ′ is separable and of the form n
H ′ = ∑ H i′(qi , pi ) , H i′ = i =1
1 2 pi + G (qi ) 2
(3)
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where G (qi ) ≥ 0 is symmetric with respect to qi = 0 , and with minimum at qi = 0 . Then the associated Hamiltonian system has a family of periodic solutions around the origin and the solution to Equation (2) is of the form Qi (t ) = Ai cos Φ i (t ), Pi (t ) = − Ai
d Θi sin Φ i (t ), Φ i (t ) = Θi (t ) + Γ i (t ), dt
(4)
For system (2), Ai (t ) and Γi (t ) are slowly varying processes and the averaged value of the instantaneous frequency d Θi / dt is equal to ωi ( Ai ) . Thus, for small time delay, we have the following approximate expressions: Qi (t −τ ) = Ai (t − τ )cos Φi (t −τ ) ≈ Ai (t )cos[ωi × (t − τ ) + Γi (t )] = Qi (t )cosωτ i −
Pi
ωi
sin ωτ i
d Θ (t − τ ) Pi (t −τ ) = − Ai (t −τ ) i sin Φi (t − τ ) ≈ − Ai (t )ωi sin[ωi × (t − τ ) + Γi (t )] dt = Pi cosωτ i + Qi (t )ωi sin ωτ i
(5)
The time-delayed feedback control forces ε F (Qτ , Pτ ) can be converted into those without time delay by using Equation (5), i.e., ε Fi (Qτ , Pτ ) ≈ ε Fi (Q, P;τ ) . The term ε Fi (Q, P;τ ) + ε Dkl f jl ∂fik / ∂Pj in Equation (2) can be split into two parts: one has the effect of modifying the conservative forces and the other modifying the damping forces. The first part can be combined with −∂H ′ / ∂Qi to form an overall effective conservative forces −∂H / ∂Qi with a new Hamiltonian H = H (Q, P;τ ) and with ∂H / ∂Pi = ∂H ′ / ∂Pi . The second part may be combined with −ε cij′ ∂H ′ / ∂Pj to constitute to an effective damping forces −ε cij ∂H / ∂Pj with cij = cij (Q, P;τ ) . With these accomplished, Equation (2) can be rewritten as
the following Itô equations for quasi Hamiltonian systems with τ as a parameter: dQi =
∂H dt ∂Pi
∂H ∂H )dt + ε 1/ 2σ ik dBk (t ); i, j = 1, 2,..., n; k = 1, 2,..., m. dPi = −( + ε cij ∂Qi ∂Pj
(6)
Suppose that the Hamiltonian system with modified Hamiltonian H is still integrable. Then the stochastic averaging method for quasi integrable Hamiltonian systems[6] can be applied to the system governed by Equation (6). The dimension and form of the averaged Itô and FPK equations depend upon the resonance of the associated Hamiltonian system.
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In non-resonant case, the averaged Itô equations are of the form dI r = ε U r (I;τ )dt + ε 1/ 2Vrk (I;τ )dBk (t ), r = 1, 2,..., n, k = 1, 2,..., m
(7)
and the averaged FPK equation is of the form ⎧ ∂ ⎫ 1 ∂2 ∂p [ar (I;τ ) p ] + [brs (I;τ ) p ]⎬ = ε ⎨− 2 ∂I r ∂I s ∂t ⎩ ∂I r ⎭
(8)
The drift and diffusion coefficients in Equations (7) and (8) can be founded in [6]. The exact stationary solution p(I;τ ) of FPK Equation (8) with vanish probability potential flow at boundary can be obtained if some compatibility conditions are satisfied. The approximate stationary probability density p (q, p;τ ) of system (1) is then obtained from p(I;τ ) [6]. If the action-angle variables I , θ for Hamiltonian system with Hamiltonian H can not be obtained, then the averaged Itô equations for independent integrals H i (i = 1, 2,..., n) of motion may be derived and the associated FPK equation can be solved to yield the stationary probability density p (H;τ ) and then p (q, p;τ ) [6]. If the associated Hamiltonian system is resonant with α weak resonant relations of the form kruωr = 0(ε ), u = 1, 2,...,α ; r = 1, 2,..., n , then the averaged Itô equations are of the form dI r = ε mr (I, Ψ;τ )dt + ε 1/ 2σ rk (I, Ψ;τ )dBk (t ) d Ψ u = ε mn + u (I, Ψ;τ )dt + ε 1/ 2σ n + u , k (I, Ψ;τ )dBk (t )
(9)
r = 1, 2," , n; u = 1, 2," ,α ; k = 1, 2," , m
and the averaged FPK equation is of the form ⎡ ∂ 1 ∂2 ∂p ∂ (ar p) − (an +u p) + (brs p ) = ε ⎢− 2 ∂I r ∂I s ∂t ∂ψ u ⎣ ∂I r ⎤ 1 ∂2 ∂2 (br , n + u p ) + (bn + u , n + v p) ⎥ + 2 ∂ψ u ∂ψ v ∂I r ∂ψ u ⎦
(10)
where Ψ = [Ψ1 , Ψ 2 ," , Ψ α ]T , ψ u = kru Θ r . The drift and diffusion coefficients of Equations (9) and (10) can be found in [6]. The stationary probability densities, i.e., p (I, ψ;τ ) and p (q, p;τ ) can be obtained similarly [6].
Stability and response of quasi integrable Hamiltonian systems
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THE LARGEST LYAPUNOV EXPONENT AND ASYMPTOTIC LYAPUNOV STABILITY WITH PROBABILITY ONE
Now consider a quasi integrable Hamiltonian system with parametric excitations of Gaussian white noises and time-delayed feedback control and study the asymptotic Lyapunov stability with probability one of the trivial solution Q=P=0. In non-resonant case, suppose that the drift and diffusion coefficients in Equation (7) are linear or homogeneous of degree one. Otherwise, we linearize Equation (7) at I = 0 . The linearized (or homogenous of order one) equations of Equation (7) are of the form dI r = ε Fr (I;τ )dt + ε 1/ 2 Gru (I;τ )dBu (t ) r = 1, 2,..., n; u = 1, 2,..., m
(11)
Introduce the following new variables: 1 2
n
ρ = log I , I = ∑ I r , α r = I r I , r = 1, 2,..., n.
(12)
r =1
The Itô equations for ρ and α r are obtained from Equation (11) by using Itô differential rule as follows: d ρ = ε Q(a;τ )dt + ε 1/ 2 Σu (a;τ )dBu (t )
(13)
dα r = ε mr (a;τ )dt + ε 1/ 2σ ru (a;τ )dBu (t )
(14)
r = 1, 2,..., n; u = 1, 2,..., m
where a = [α1 ,α 2 ,...,α n ]T . The expressions for Q, Σu , mr and σ ru can be found in [7]. Let a ' = [α1 ,α 2 ,...,α n −1 ] be an (n − 1) -dimensional vector diffusing process with α n replaced by α n = 1 − Σ nr =−11α r . Define the Lyapunov exponent of system (11) as the asymptotic rate of exponential growth of the square-root of I. Suppose that (n-1)-dimensional vector diffusion process a ' is ergodic over the interval 0 ≤ α r ≤ 1, r = 1, 2,..., n − 1 , and the invariant measure p (a ';τ ) of this vector process can be obtained from solving the reduced FPK equation associated with the first n − 1 equations of Itô equations (14). The largest Lyapunov exponent of averaged system (11) is then given by
λ1 (τ ) = E[ε Q(a ';τ )] = ∫ ε Q(a ';τ ) p(a ';τ )da '
(15)
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The necessary and sufficient condition for asymptotic Lyapunov stability with probability one of the trivial solution of Equation (11) is λ1 < 0 . This is also the approximate condition for the asymptotic Lyapunov stability with probability one of the trivial solution of original system (1). The largest Lyapunov exponent for averaged Itô Equation (9) can be obtained similarly and the stability of system (1) in resonant case can also be determined approximately by using the largest Lyapunov exponent.
4.
EFFECTS OF TIME-DELAY IN FEEDBACK CONTROL ON STABILITY AND RESPONSE
The effects of the time-delay in feedback control on the stability and response of controlled quasi integrable Hamiltonian systems are studied using the following three examples. Example 1. Consider a Duffing-van der Pol oscillator with delayed linear feedback control subject to Gaussian white noise excitation. The equation of motion is
X + ω0′2 X + α X 3 = ε (b − X 2 ) X − ε (aX τ ) + ε 1/ 2W (t )
(16)
For this example, the time-delayed feedback control force
ε aPτ ≈ ε a( P cos ω ′τ + Qω ′ sin ω ′τ )
(17)
and the modified Hamiltonian is H (q, p;τ ) =
1 2 1 2 2 1 4 p + ω0 q + α q 2 2 4
(18)
where ω ′ is the average frequency and ω02 = ω0′2 + εω ′ sin ω ′τ . The approximate stationary probability density of system (16) is p(q, p;τ ) =
⎡ H 1 ⎛ db(H ;τ ) C ⎞ ⎤ exp ⎢∫ (19) − 2a(H ;τ ) ⎟ dH ⎥ ⎜ 0 2π ⎠ ⎦ H = ( 1 p2 + 1 ω 2 q 2 + 1 α q4 ) ⎣ b(H ;τ ) ⎝ dH 2
2
0
4
Some numerical results for stationary marginal probability density p (q;τ ) obtained by using the proposed stochastic averaging method and from digital simulation are shown in Figure 1. The parameters are a = 1.0 , b = 1.0 , ω0′ = 1.0 , α = 0.5 , ε = 0.01 , 2 D = 0.2 . It is seen that the analytical results obtained by using the proposed method agree well with those from digital simulation even for longer delay time and that the time delay in feedback control causes phenomenological bifurcation of response.
Stability and response of quasi integrable Hamiltonian systems (a)
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Figure 1. Stationary marginal probability density p (q;τ ) of system (16) with velocity feedback. (a) τ = 0 , (b) τ = 2 , (c) τ = 5 . — by using the proposed stochastic averaging method; ● from digital simulation.
Example 2. Consider two linear oscillators coupled by linear and polynomial type nonlinear dampings subject to external excitations of two uncorrelated Gaussian white noises and delayed bang-bang feedback control. The equations of motion of the system are of the form X1 + α11 X 1 + α12 X 2 + β1 X 1 ( X 12 + X 22 ) + ω12 X 1 = u1τ + W1 (t ) X2 + α 21 X 1 + α 22 X 2 + β 2 X 2 ( X 12 + X 22 ) + ω22 X 2 = u2τ + W2 (t )
(20)
where α ij , β i , ωi (i, j = 1, 2) are constants and α ij , β i , Dii and uiτ are of the same order of ε ; Wi (t )(i = 1, 2) are uncorrelated Gaussian white noises with intensities 2 Dii . uiτ = −bi sgn ( X i (t − τ ) ) (i = 1, 2) are the delayed feedback bang-bang control forces which can be approximately expressed as uiτ ≈ −bi cos ωiτ sgn ( Pi )
(21)
Nonresonant Case. In this case, the approximate stationary probability density of the displacements and velocities of original system (20) is β1 2 4b1 cos ω1τ 1 ⎛ ⎞ 2ω1 I1 ⎟ ⎜ α11ω1 I1 + I1 + D11 ⎝ 4 π ⎠ (22) β 2 2 4b2 cos ω2τ 1 ⎛ ⎞ β1ω1 − I2 + 2ω2 I 2 ⎟ − I1 I 2 ] |I = ( q2 +ω 2 p2 ) / 2ω ⎜ α 22ω2 I 2 + i i i i i D22 ⎝ 4 π ⎠ D11ω2
p(q1 , p1 , q2 , p2 ;τ ) = C ′ exp[−
where C ′ is a normalization constant. Primary Resonant Case. ω1 = ω2 = ω . Let θ1 − θ 2 = ϕ . The approximate stationary probability density of the displacements and velocities of original system (20) is then
W. Q. Zhu, Z. H. Liu
390 p ( q1 , p1 , q 2 , p 2 ;τ ) = C ′ exp[ − ( +
4b1 cos ω1τ π D11
+2
α 12ω D11
α 11ω
I1 +
α 22ω
β1
I2 +
I12 +
β2
I 22 D11 D22 4 D11 4 D22 4b cos ω 2τ β 1 2ω1 I1 + 2 2ω 2 I 2 + 1 I1 I 2 (1 − cos 2ϕ ) (23) D11 2 π D22
I1 I 2 cos ϕ )] I i = ( qi2 + ωi2 pi2 ) / 2 ωi ϕ =θ1 −θ 2
where C ′ is another normalized constant. Some numerical results obtained by using the proposed stochastic averaging method and from digital simulation in nonresonant case are shown in Figure 2. The parameters are α11 = 0.01 , α12 = 0.01 , β1 = 0.01 , ω1 = 1.0 , D11 = 0.01 , b1 = 0.01 , α 21 = 0.01 , α 22 = 0.01 , β 2 = 0.02 , ω2 = 0.707 , D22 = 0.01 , b2 = 0.01 . It is seen from these figures that the proposed method yields quite accurate results even for large delay time and time delay deteriorates the control effectiveness remarkably. 1.3
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Figure 2. Stationary response of the first oscillator of system (20) in nonresonant case. (a) Stationary marginal probability density p ( q1;τ ) . (b) Stationary mean-square value E[Q12 (τ )] . — by using the proposed stochastic averaging method; ● ▲▼ from digital simulation.
Example 3. Consider two linear oscillators coupled by nonlinear dampings and stochastic parametric excitations subject to time-delayed feedback control. The equations of motion of the system are of the form X1 + α11′ X 1 + α12′ X 2 + β1 X 1 ( X 12 + X 22 ) + ω1′2 X 1 = −η1 X 1τ + k11 X 1W1 (t ) + k12 X 2W2 (t ) ′ X 1 + α 22 ′ X 2 + β 2 X 2 ( X 12 + X 22 ) + ω2′2 X 2 = X2 + α 21 −η 2 X 2τ + k21 X 1W1 (t ) + k22 X 2W2 (t )
(24)
where α ij′ and βi are damping coefficients; kij are constants; ωi′ are the natural frequencies of the two linear oscillators; −ηi X iτ represent the timedelayed feedback control forces; W j (t ) are independent Gaussian white noises in the senses of Stratonovich with intensities 2 Dij . Assume that α ij′ , βi , D j and ηi are of the order ε .
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Following the procedure described in Section 3, we obtain the following expression for the largest Lyapunov exponent
λ1 (τ ) =
1 F (1)λ1 − F (0)λ2 2 F (1) − F (0)
(25)
where
⎡ 2(λ − λ2 ) 2aα1 + b − ∆ ⎤ F (α1 ) = exp ⎢ 1 ln ⎥ ,if ∆ > 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)
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Figure 3. The largest Lyapunov exponent λ1 (τ ) of system (24). —Analytical result. ● ▲▼ Results form digital simulation.
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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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