Nonlinear Vibrations and Chaos in Rectangular Functionally Graded Plates with Thermo-mechanical Coupling

Abstract

We analyze the nonlinear dynamics of a simply supported, rectangular, and functionally graded plate in terms of a newly derived coupled system of thermo-elasticity and energy equations, which is then expanded here in derivations and explored for chaotic responses through a parameter study in the state space.1 The plate properties vary linearly in thickness. Three-dimensional stress-strain relations are considered in general case and nonlinear strain-displacement relations are deployed to account for the plate’s large deflection. A lateral harmonic force is applied on the plate, and there is a heat generation source within it and the surfaces are exposed to free convection. By integrating over the thickness, four new thermal parameters are introduced, which together with the midplane displacements constitute a system of seven partial differential equations. These equations are changed into ordinary differential equations in time using Galerkin’s approximation and solved by using the 4th order RungeKutta method. Finally, a parameter study is performed and the appropriate conditions resulting in chaotic solutions are determined by using numerical features such as the Lyapunov exponent and power spectrum.