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Nonlinear vibrations of a composite laminated cantilever rectangular plate with one-to-one internal resonance

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Abstract

The nonlinear vibrations of a composite laminated cantilever rectangular plate subjected to the in-plane and transversal excitations are investigated in this paper. Based on the Reddy’s third-order plate theory and the von Karman type equations for the geometric nonlinearity, the nonlinear partial differential governing equations of motion for the composite laminated cantilever rectangular plate are established by using the Hamilton’s principle. The Galerkin approach is used to transform the nonlinear partial differential governing equations of motion into a two degree-of-freedom nonlinear system under combined parametric and forcing excitations. The case of foundational parametric resonance and 1:1 internal resonance is taken into account. The method of multiple scales is utilized to obtain the four-dimensional averaged equation. The numerical method is used to find the periodic and chaotic motions of the composite laminated cantilever rectangular plate. It is found that the chaotic responses are sensitive to the changing of the forcing excitations and the damping coefficient. The influence of the forcing excitation and the damping coefficient on the bifurcations and chaotic behaviors of the composite laminated cantilever rectangular plate is investigated numerically. The frequency-response curves of the first-order and the second-order modes show that there exists the soft-spring type characteristic for the first-order and the second-order modes.

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Acknowledgements

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through grant Nos. 11072008 and 10732020, the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB).

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Authors and Affiliations

  1. College of Mechanical Engineering, Beijing University of Technology, Beijing, 100124, P.R. China

    W. Zhang & M. H. Zhao

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  1. W. Zhang
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  2. M. H. Zhao
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Correspondence to W. Zhang.

Appendix

Appendix

All coefficients given in Eq. (17) are presented as follows:

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Zhang, W., Zhao, M.H. Nonlinear vibrations of a composite laminated cantilever rectangular plate with one-to-one internal resonance. Nonlinear Dyn 70, 295–313 (2012). https://doi.org/10.1007/s11071-012-0455-6

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  • DOI: https://doi.org/10.1007/s11071-012-0455-6

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