Abstract
The dimensionless partial differential equations governing thedynamics of a thin flexible isotropic plate with an external load arederived and investigated. The period doubling bifurcations, as well asthe chaotic dynamics, are detected and analyzed. The algorithms leadingto the reduction of the original equations to those of a difference setof ordinary differential and algebraic equations are proposed, comparedto other known methods, and then applied to the problem.
Among others, it is shown that, in spite of the system complexity, theFeigenbaum scenario exhibited by one-dimensional maps also governs theroute to chaos in the continuous system under consideration.
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References
Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley-Interscience, New York, 1979.
Chang, S. I., Bajaj, A. K., and Davies, P., ‘Bifurcations and chaotic motions in resonantly excited structures’, in Bifurcation and Chaos-Theory and Applications, J. Awrejcewicz (ed.), Springer-Verlag, Berlin, 1995, pp. 217–249.
Nayfeh, A. H. and Balachandran, B., ‘Modal interactions in dynamical and structural systems’, Applied Mechanics Review 42, 1989, 5175–5201.
Johnson, J. M. and Bajaj, A. K., ‘Amplitude modulated and chaotic dynamics in resonant motion of strings’, Journal of Sound and Vibration 128, 1989, 87–107.
Awrejcewicz, J., ‘Strange nonlinear behavior governed by a set of four averaged amplitude equations’, Meccanica 31, 1996, 347–36.
Thompson, J. M. T. and Stewart, H. B., Nonlinear Dynamics and Chaos, Wiley, Chichester, 1986.
Sathyamoorthy, M., ‘Nonlinear vibration analysis of plates: A review and survey of current developments’, Applied Mechanics Review 40, 1987, 1553–1561.
Landa, P. S., Nonlinear Oscillations and Waves in Dynamical Systems, Kluwer, Dordrecht, 1996.
Landa, P. S., ‘Turbulence in nonclosed fluid flows as a noise-induced phase transition’, Europhysics Letters 36(b), 1996, 401–406.
Landa, P. S., ‘What is the turbulence?’ (afterward in the paper of Yu. L. Klimontovitch, Izv. VUZov), Prikladnaya Nelineynaya Dinamika 2, 1995, 37–41 [in Russian].
Guckenheimer, J. and Holmes, P. J., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
Schuster, H. G., Deterministics Chaos, Physik-Verlag, Weinheim, 1984.
Kuklin, P. M. (ed.), Selfwaves Processes in the Diffusion Systems, IPF, Gorkij, ANSSSR, 1981 [in Russian].
Stoliarov, N. N. and Riabov, A. A., ‘Stability and postcritical behavior of rectangular plates with variable thickness’, in Investigation of Plates Theories, Kazan, 1982, pp. 195–145 [in Russian].
Arnold, V. I., Ordinary Differential Equations, Mir, Moscow, 1977 [in Rusian].
Arnold, V. I., Theoretical Problems of Ordinary Differential Equations, Mir, Moscow, 1978 [in Russian].
Babin, A. V. and Vishik, M. N., ‘Attractors of the evolutionary partial differential equations and estimation of their dimension’, Uspekhi Matematiceskich Nauk 38(4), 1983, 133–187.
Krysko, V. A., ‘Dynamical stability loss of rectangular shells with finite deflections’, Prikladnaja Mekhanika 15(11), 1979, 58–62 [in Russian].
Krysko, V. A. and Fedorov, P. B., ‘Stability loss of thin comical shells subjected to a heat impact’, Prikladnaja Mekhanika 16(5), 1980, 126–129 [in Russian].
Krysko, V. A. and Dediukin, I. Ju., ‘On the criterions of dynamical stability loss’, Prikladnaja Mekhanika 30(10), 1994, 67–71 [in Russian].
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Awrejcewicz, J., Krysko, V.A. Feigenbaum Scenario Exhibited by Thin Plate Dynamics. Nonlinear Dynamics 24, 373–398 (2001). https://doi.org/10.1023/A:1011133223520
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DOI: https://doi.org/10.1023/A:1011133223520