Abstract
This paper investigates the multi-pulse global heteroclinic bifurcations and chaotic dynamics for the nonlinear vibrations of a simply supported rectangular thin plate by using an extended Melnikov method in the resonant case. The rectangular thin plate is subjected to spatially and temporally varying transversal and in-plane excitations, simultaneously. The equations of motion for the rectangular thin plate are derived from the von Kármán equation. Applying the method of multiple scales and Galerkin’s approach to the partial differential governing equation, the four-dimensional averaged equation is obtained for the case of 1:2 internal resonance and primary parametric resonance. From the averaged equations obtained, the theory of normal form is used to derive the explicit expressions of normal form with a double zero and a pair of pure imaginary Eigenvalues. Based on the explicit expressions of normal form, the extended Melnikov method is used to analyze the multi-pulse heteroclinic bifurcations and chaotic dynamics of the rectangular thin plate. The contribution of the paper is the simplification of the extended Melnikov method. The extended Melnikov function can be simplified in the resonant case and does not depend on the perturbation parameter. The necessary conditions of the existence for the Shilnikov type multi-pulse chaotic dynamics of the rectangular thin plate are analytically obtained. Numerical simulations also display that the Shilnikov type multi-pulse chaotic motions can occur in the rectangular thin plate. Overall, both theoretical and numerical studies demonstrate that the chaos for the Smale horseshoe sense exists in the rectangular thin plate.
Highlights
Due to the high-speed, lightweight and energy-saving requirements in the aerospace and aviation industry, many structures become thinner
This paper focuses on the Shilnikov type multi-pulse orbits and chaotic dynamics for a supported at four-edge, rectangular thin plate subjected to spatially and temporally varying transversal and in-plane excitations, simultaneously
The nonlinear vibrations of the supported rectangular thin plate are studied by applying the theories of the global bifurcations and chaotic dynamics for highdimensional nonlinear systems
Summary
Due to the high-speed, lightweight and energy-saving requirements in the aerospace and aviation industry, many structures become thinner. The study on the second theory of the Shilnikov type multi-pulse chaotic dynamics were stated by Haller and Wiggins [30] They presented the energy phase method to investigate the existence of the multi-pulse jumping homoclinic and heteroclinic orbits in perturbed Hamiltonian systems. Few researchers have made use of the energy phase method to study the Shilnikov type multi-pulse homoclinic and heteroclinic bifurcations and chaotic dynamics of highdimensional nonlinear systems in engineering applications. Yao and Zhang [32] utilized the energy-phase method to analyze the Shilnikov type multi-pulse heteroclinic orbits and chaotic dynamics in a parametrically and externally excited rectangular thin plate. Both theoretical and numerical studies demonstrate that chaos for the Smale horseshoe sense in the motion exists
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