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

doi:10.1007/s10483-012-1609-9

Applied Mathematics and Mechanics (English Edition) ›› 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

Shuang-bao LI¹, Wei ZHANG ²

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:2011-07-20 Revised:2012-05-06 Online:2012-09-10 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)

Abstract

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

2010 MSC Number:

O322
O302

Shuang-bao LI;Wei ZHANG . Global bifurcations and multi-pulse chaotic dynamics of rectangular thin plate with one-to-one internal resonance. Applied Mathematics and Mechanics (English Edition), 2012, 33(9): 1115-1128.

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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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