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

Shuang-bao LI;Wei ZHANG

doi:10.1007/s10483-012-1609-9

Applied Mathematics and Mechanics >

2012 , Vol. 33 >Issue 9: 1115 - 1128

DOI: https://doi.org/10.1007/s10483-012-1609-9

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

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1. College of Science, Civil Aviation University of China, Tianjin 300300, P. R. China;
2. College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, P. R. China

Received date: 2011-07-20

Revised date: 2012-05-06

Online published: 2012-09-10

Supported by

Project supported the National Natural Science Foundation of China (Nos. 10732020, 11072008, and 11102226), the Scientific Research Foundation of Civil Aviation University of China (No. 2010QD 04X), and the Fundamental Research Funds for the Central Universities of China (Nos. ZXH2011D006 and ZXH2012K004)

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Abstract

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.

Key words： ordered Banach space; separated property; quasi-weak convergence; fixed points

Cite this article

Shuang-bao LI;Wei ZHANG . Global bifurcations and multi-pulse chaotic dynamics of rectangular thin plate with one-to-one internal resonance[J]. Applied Mathematics and Mechanics, 2012 , 33(9) : 1115 -1128 . DOI: 10.1007/s10483-012-1609-9

References

[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 amodel 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 ofthe American Mathematical Society, 348(10), 3835-3887 (1996)
[4] Camassa, R., Kovacic, G., and Tin, S. K. A Melnikov method for homoclinic orbits with manypulses. 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 truncationof the damped-forced nonlinear Schrödinger 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 vibrationsnearly 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 thinplate. Nonliner Dynamics, 4(4), 389-408 (1993)
[11] Chang, S. I., Bajaj, A. K., and Krousgrill, C. M. Nonlinear vibrations and chaos in harmonicallyexcited 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, rectangularlaminated 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 excitedthin plate. Nonlinear Dynamics, 24(3), 245-268 (2001)
[14] Zhang, W. Global and chaotic dynamics for a parametrically excited thin plate. Journal of Soundand Vibration, 239(5), 1013-1036 (2001)
[15] Anlas, G. and Elbeyli, O. Nonlinear vibrations of a simply supported rectangular metallic platesubjected 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 asymmetric cross-ply laminated thin plate under parametric excitation. International Journal ofBifurcation and Chaos, 16(2), 325-347 (2006)
[17] Zhang, W., Yang, J., and Hao, Y. X. Chaotic vibrations of an orthotropic FGM rectangular platebased 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 platesubjected 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 gradedmaterial rectangular plate with one-to-one internal resonance. International Journal of NonlinearSciences and Numerical Simulation, 11(5), 351-362 (2010)
[20] Zhang, W. and Li, S. B. Resonant chaotic motions of a buckled rectangular thin plate withparametrically 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)

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