Shilnikov Type Multi-Pulse Orbits of Functionally Graded Materials Rectangular Plate

Abstract

Multi-pulse chaotic dynamics of a simply supported functionally graded materials (FGMs) rectangular plate is investigated in this paper. The FGMs rectangular plate is subjected to the transversal and in-plane excitations. The properties of material are graded in the direction of thickness. Based on Reddy’s third-order shear deformation plate theory, the nonlinear governing equations of motion for the FGMs plate are derived by using the Hamilton’s principle. The four-dimensional averaged equation under the case of 1:2 internal resonance, primary parametric resonance and 1/2-subharmonic resonance is obtained by directly using the asymptotic perturbation method and Galerkin approach to the partial differential governing equation of motion for the FGMs rectangular plate. The system is transformed to the averaged equation. From the averaged equation, the theory of normal form is used to find the explicit formulas of normal form. Based on normal form obtained, the energy phase method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics for the FGMs rectangular plate. The analysis of global dynamics indicates that there exist the multi-pulse jumping orbits in the perturbed phase space of the averaged equation. From the averaged equations obtained, the chaotic motions and the Shilnikov type multi-pulse orbits of the FGMs rectangular plate are found by using numerical simulation. The results obtained above mean the existence of the chaos for the Smale horseshoe sense for the simply supported FGMs rectangular plate.