Global bifurcations and multi-pulse chaotic dynamics of rectangular thin plate with one-to-one internal resonance

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• 作者：Shuang-bao Li (1) shuangbaoli@yeah.net
  Wei Zhang (2)
• 关键词：rectangular thin plate &#8211 ; global bifurcation &#8211 ; multi ; pulse chaotic dynamics &#8211 ; extended Melnikov method
• 刊名：Applied Mathematics and Mechanics
• 出版年：2012
• 出版时间：September 2012
• 年：2012
• 卷：33
• 期：9
• 页码：1115-1128
• 全文大小：423.4 KB
• 参考文献：1. Wiggins, S. Global Bifurcations and Chaos: Analytical Methods, Springer-Verlag, New York (1988)
  2. Kovacic, G. and Wiggins, S. Orbits homoclinic to resonances with an application to chaos in a model of the forced and damped sine-Gordon equation. Physica D, 57(1–2), 185–225 (1992)
  3. Kaper, T. J. and Kovacic, G. Multi-bump orbits homoclinic to resonance bands. Transactions of the American Mathematical Society, 348(10), 3835–3887 (1996)
  4. Camassa, R., Kovacic, G., and Tin, S. K. A Melnikov method for homoclinic orbits with many pulses. Archive for Rational Mechanics and Analysis, 143(2), 105–193 (1998)
  5. Haller, G. and Wiggins, S. Multi-pulse jumping orbits and homoclinic trees in a modal truncation of the damped-forced nonlinear Schr枚dinger equation. Physica D, 85(3), 311–347 (1995)
  6. Haller, G. Chaos Near Resonance, Springer-Verlag, New York (1999)
  7. Hadian, J. and Nayfeh, A. H. Modal interaction in circular plates. Journal of Sound and Vibration, 142(2), 279–292 (1990)
  8. Yang, X. L. and Sethna, P. R. Local and global bifurcations in parametrically excited vibrations nearly square plates. International Journal of Non-Linear Mechanics, 26(2), 199–220 (1991)
  9. Yang, X. L. and Sethna, P. R. Non-linear phenomena in forced vibrations of a nearly square plate: antisymmetric case. Journal of Sound and Vibration, 155(3), 413–441 (1992)
  10. Feng, Z. C. and Sethna, P. R. Global bifurcations in the motion of parametrically excited thin plate. Nonliner Dynamics, 4(4), 389–408 (1993)
  11. Chang, S. I., Bajaj, A. K., and Krousgrill, C. M. Nonlinear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance. Nonlinear Dynamics, 4(5), 433–460 (1993)
  12. Abe, A., Kobayashi, Y., and Yamada, G. Two-mode response of simply supported, rectangular laminated plates. International Journal of Non-Linear Mechanics, 33(4), 675–690 (1998)
  13. Zhang, W., Liu, Z. M., and Yu, P. Global dynamics of a parametrically and externally excited thin plate. Nonlinear Dynamics, 24(3), 245–268 (2001)
  14. Zhang, W. Global and chaotic dynamics for a parametrically excited thin plate. Journal of Sound and Vibration, 239(5), 1013–1036 (2001)
  15. Anlas, G. and Elbeyli, O. Nonlinear vibrations of a simply supported rectangular metallic plate subjected to transverse harmonic excitation in the presence of a one-to-one internal resonance. Nonlinear Dynamics, 30(1), 1–28 (2002)
  16. Zhang, W., Song, C. Z., and Ye, M. Further studies on nonlinear oscillations and chaos of a symmetric cross-ply laminated thin plate under parametric excitation. International Journal of Bifurcation and Chaos, 16(2), 325–347 (2006)
  17. Zhang, W., Yang, J., and Hao, Y. X. Chaotic vibrations of an orthotropic FGM rectangular plate based on third-order shear deformation theory. Nonlinear Dynamics, 59(4), 619–660 (2010)
  18. Yu, W. Q. and Chen, F. Q. Global bifurcations of a simply supported rectangular metallic plate subjected to a transverse harmonic excitation. Nonlinear Dynamics, 59(1–2), 129–141 (2010)
  19. Li, S. B., Zhang, W., and Hao, Y. X. Multi-pulse chaotic dynamics of a functionally graded material rectangular plate with one-to-one internal resonance. International Journal of Nonlinear Sciences and Numerical Simulation, 11(5), 351–362 (2010)
  20. Zhang, W. and Li, S. B. Resonant chaotic motions of a buckled rectangular thin plate with parametrically and externally excitations. Nonlinear Dynamics, 62(3), 673–686 (2010)
  21. Chia, C. Y. Nonlinear Analysis of Plate, McMraw-Hill, California (1980)
  22. Timoshenko, S. and Woinowsky-Krieger, S. Theory of Plates and Shells, McGraw-Hill, New York (1959)
  23. Nayfeh, A. H. and Mook, D. T. Nonlinear Oscillations, Wiley, New York (1979)
• 作者单位：1. College of Science, Civil Aviation University of China, Tianjin, 300300 P. R. China2. College of Mechanical Engineering, Beijing University of Technology, Beijing, 100124 P. R. China
• ISSN：1573-2754

文摘

Global bifurcations and multi-pulse chaotic dynamics for a simply supported rectangular thin plate are studied by the extended Melnikov method. The rectangular thin plate is subject to transversal and in-plane excitation. A two-degree-of-freedom nonlinear nonautonomous system governing equations of motion for the rectangular thin plate is derived by the von Karman type equation and the Galerkin approach. A one-toone internal resonance is considered. An averaged equation is obtained with a multi-scale method. After transforming the averaged equation into a standard form, the extended Melnikov method is used to show the existence of multi-pulse chaotic dynamics, which can be used to explain the mechanism of modal interactions of thin plates. A method for calculating the Melnikov function is given without an explicit analytical expression of homoclinic orbits. Furthermore, restrictions on the damping, excitation, and detuning parameters are obtained, under which the multi-pulse chaotic dynamics is expected. The results of numerical simulations are also given to indicate the existence of small amplitude multi-pulse chaotic responses for the rectangular thin plate.