Multi-pulse chaotic motions of high-dimension nonlinear system for a laminated composite piezoelectric rectangular plate

Abstract

This paper investigates the multi-pulse global bifurcations and chaotic dynamics of the high-dimension nonlinear system for a laminated composite piezoelectric rectangular plate by using an extended Melnikov 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. 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 for the first time to investigate the Shilnikov type multi-pulse homoclinic bifurcations and chaotic dynamics of the laminated composite piezoelectric rectangular plate. The necessary conditions of the existence for the Shilnikov type multi-pulse chaotic dynamics of the laminated composite piezoelectric rectangular plate are analytically obtained. Numerical simulations also illustrate that the Shilnikov type multi-pulse chaotic motions can also occur in the laminated composite piezoelectric rectangular plate. Overall, both theoretical and numerical studies demonstrate that the chaos in the Smale horseshoe sense exists for the laminated composite piezoelectric rectangular plate.