Chaotic Wave Motions and Chaotic Dynamic Responses of Piezoelectric Laminated Composite Rectangular Thin Plate Under Combined Transverse and In-Plane Excitations

摘要

In the present work, the chaotic wave motions and the chaotic dynamic responses are investigated for a four-edge simply supported piezoelectric composite laminated rectangular thin plate subjected to the transverse and the in-plane excitations. Based on the reductive perturbation method, the complicated partial differential nonlinear governing equation of motion for the piezoelectric composite laminated rectangular thin plate subjected to the transverse and the in-plane excitations is transformed into an equivalent and soluble nonlinear wave equation. The heteroclinic orbit and resonant torus are obtained for the unperturbed nonlinear wave equation. The topological structures of the unperturbed and the perturbed nonlinear wave equations are investigated on the fast and the slow manifolds. The persistence of the heteroclinic orbit is studied for the perturbed nonlinear wave equation through the Melnikov method. The geometric analysis is utilized to prove that the heteroclinic orbit goes back to the stable manifold of the saddle point on the slow manifold under the perturbations. The existence of the homoclinic orbit is conformed for the perturbed nonlinear wave equation by the first and the second measures. When the homoclinic orbit is broken, the chaotic motions occur in the Smale horseshoe sense for the piezoelectric composite laminated rectangular thin plate subjected to the transverse and the in-plane excitations. Numerical simulations are finished to study the influence of the damping coefficient on the propagation properties of the piezoelectric composite laminated rectangular thin plate subjected to the transverse and the in-plane excitations. Both theoretical study and numerical simulation results indicate the existence of the chaotic wave motions and the chaotic dynamic responses of the piezoelectric composite laminated rectangular thin plate subjected to the transverse and the in-plane excitations.