Bifurcation and chaos in nonsmooth mechanical systems

• 作者: Awrejcewicz, J.
• 其他作者: Lamarque, Claude-Henri.
• 出版: Singapore ;River Edge, NJ : World Scientific 2003.
• 稽核項: 1 online resource (xvii, 543 pages) :illustrations (some color).
• 叢書名: World Scientific series on nonlinear science. Series A, Monographs and treatises ;v. 45
• 標題: Differential EquationsGeneral. , Electronic books. , Differential equations, Nonlinear. , Chaotic behavior in systems. , MATHEMATICS , Bifurcation theory. , MATHEMATICS Differential Equations -- General.
• ISBN: 9812564802 , 9789812564801
• ISBN: 9812384596 , 9789812384591
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• 附註: Includes bibliographical references (pages 507-530) and index. 1. Introduction to discontinuous ODEs. 1.1. Introduction. 1.2. Filippov's theory. 1.3. Aizerman's theory. 1.4. Examples. 1.5. Boundary value problem -- 2. Mathematical background for multivalued formulations. 2.1. Origin of nonlinearities. 2.2. Smooth and nonsmooth nonlinearities. 2.3. Examples and dynamical equilibria. 2.4. Existence and uniqueness. 2.5. Stochastic frame -- 3. Numerical schemes and analytical methods. 3.1. Numerical schemes. 3.2. Analytical methods -- 4. Properties of numerical schemes. 4.1. Dynamics of systems with friction or elastoplastic terms. 4.2. Systems with impacts. 4.3. Conclusion -- 5. Bifurcations of a particular van der Pol-Duffing oscillator. 5.1. The analysed system and the averaged equations. 5.2. "0" type bifurcations. 5.3. Complex bifurcations. 5.4. Observations of strange attractors using numerical simulations -- 6. Stick-slip oscillator with two degrees of freedom. 6.1. Introduction. 6.2. Disc -- flexible arm oscillator. 6.3. Two horizontally situated masses -- 7. Piecewise linear approximations. 7.1. Introduction. 7.2. Exact and approximated models. 7.3. Approximation and global dynamic behavior. 7.4. Numerical results. 7.5. Conclusion -- 8. Chua's circuit with discontinuities. 8.1. Introduction. 8.2. Mechanical realizations of Chua's circuit. 8.3. Generalized double scroll Chua's circuit -- 9. Mechanical system with impacts and modal approaches. 9.1. Introduction. 9.2. Single degree of freedom system. 9.3. Two degrees of freedom systems. 9.4. Conclusion -- 10. One DOF mechanical system with friction. 10.1. Introduction. 10.2. Modelling the pendulum with friction. 10.3. Numerical results. 10.4. The Melnikov analysis. 10.5. Conclusion. 11. Modelling the dynamical behaviour of elasto-plastic systems. 11.1. Rheological systems with "friction" -- 12. A mechanical system with 7 DOF. 12.1. Mathematical model. 12.2. Numerical results and comments for finite k3 -- 13. Stability of singular periodic motions in single degree of freedom vibro-impact oscillators and grazing bifurcations. 13.1. Introduction. 13.2. Mechanical system and change of coordinates. 13.3. Local expansion of the Poincaré map. 13.4. Stability of the nondifferentiable fixed point. 13.5. Applications. 13.6. Conclusion -- 14. Triple pendulum with impacts. 14.1. Introduction. 14.2. Investigated pendulum and governing equations (without impacts). 14.3. Introduction of the obstacles. 14.4. Calculation of the fundamental solution matrices for dynamical systems with impacts. 14.5. Simplification of the system. 14.6. The method used for integration of the system and its accuracy. 14.7. Numerical examples. 14.8. Concluding remarks -- 15. Analytical prediction of stick-slip chaos. 15.1. Introduction. 15.2. The Melnikov's method. 15.3. Analyzed system. 15.4. Analytical results -- 16. Thermoelasticity, wear and stick-slip movements of a rotating shaft with a rigid bush. 16.1. Introduction -- 17. Control for discrete models of buildings including elastoplastic terms. 17.1. Introduction. 17.2. Reminder about Prandtl rheological model. 17.3. The studied models with n DOF. 17.4. Existence and uniqueness results. 17.5. Numerical scheme. 17.6. Control procedure. 17.7. Algorithm of control. 17.8 Numerical results for a system with 3 DOF. 17.9. Extension to nonlinear cases. 17.10. Conclusion.
• 摘要: This book presents the theoretical frame for studying lumped nonsmooth dynamical systems: the mathematical methods are recalled, and adapted numerical methods are introduced (differential inclusions, maximal monotone operators, Filippov theory, Aizerman theory, etc.). Tools available for the analysis of classical smooth nonlinear dynamics (stability analysis, the Melnikov method, bifurcation scenarios, numerical integrators, solvers, etc.) are extended to the nonsmooth frame. Many models and applications arising from mechanical engineering, electrical circuits, material behavior and civil engineering are investigated to illustrate theoretical and computational developments.
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