Abstract
This paper investigates the multipulse global bifurcations and chaotic dynamics for the nonlinear oscillations of the laminated composite piezoelectric rectangular plate by using an energy phase method in the resonant case. Using the von Karman type equations, Reddy’s third-order shear deformation plate theory, and Hamilton’s principle, the equations of motion are derived for the laminated composite piezoelectric rectangular plate with combined parametric excitations and transverse excitation. 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. The energy phase method is used for the first time to investigate the Shilnikov type multipulse heteroclinic bifurcations and chaotic dynamics of the laminated composite piezoelectric rectangular plate. The paper demonstrates how to employ the energy phase method to analyze the Shilnikov type multipulse heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications. Numerical simulations show that for the nonlinear oscillations of the laminated composite piezoelectric rectangular plate, the Shilnikov type multipulse chaotic motions can occur. Overall, both theoretical and numerical studies suggest that chaos for the Smale horseshoe sense in motion exists.
Highlights
A piezoelectric material subjected to the mechanical force produces an electrical charge which is called the direct piezoelectric effect
The analysis shows that the existence of heteroclinic orbits depends only on an energy-phase criterion which is obtained from a reduced, one-degree-of-freedom Hamiltonian system
The nonlinear vibrations of the laminated composite piezoelectric rectangular plate are studied by applying the theories of the global bifurcations and chaotic dynamics for high-dimensional nonlinear systems
Summary
A piezoelectric material subjected to the mechanical force produces an electrical charge which is called the direct piezoelectric effect. Few researchers have made use of the energy phase method to study the Shilnikov type multipulse homoclinic and heteroclinic orbits and chaotic dynamics of high-dimensional nonlinear systems in engineering applications. Yao and Zhang [34, 35] utilized the energy-phase method to analyze the Shilnikov type multipulse heteroclinic or homoclinic orbits and chaotic dynamics in a parametrically and externally excited rectangular thin plate and a laminated composite piezoelectric rectangular plate. Zhang and Yao [42, 43] introduced the extended Melnikov method to the engineering field They came up with a simplification of the extended Melnikov method in the resonant case and utilized it to analyze the Shilnikov type multipulse homoclinic bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of the cantilever beam subjected to a harmonic axial excitation and two transverse excitations at the free end. Both theoretical and numerical studies demonstrate that chaos for the Smale horseshoe sense in the motion exists
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