Static/Dynamic Instability and Nonlinear Vibrations of FG Plates Resting on Elastic Foundation under Parametric Forcing Excitation

In this study, static/dynamic instability and nonlinear vibrations of FG plates resting on elastic foundation under parametric forcing excitation, are investigated. Based on CPT, applying the von-Karman nonlinear strain–displacement relation and the Hamilton’s principle, the governing nonlinear coupled partial differential equations are derived. By considering six vibration modes, the Galerkin’s procedure is used to reduce the equations of motion to nonlinear Mathieu equations. In the absence of elastic foundation, the validity of the formulation for analyzing the static buckling, dynamic instability and nonlinear deflection is accomplished by comparing the results with those of the literature. Then in the presence of the foundation and by deriving the regions of dynamic instability, it is shown that as the parameters of the foundation increases, the natural frequency and the critical buckling load increase and the dynamic instability occurs at higher excitation frequencies. The frequency response equations in the steady-state condition are derived by applying the multiple scales method, and the parametric resonance is analyzed. Then the conditions of existence and stability of nontrivial solutions are discussed. Moreover, the effects of the system parameters, including excitation frequency, amplitude of excitation, foundation parameters and damping, on the nonlinear dynamics of the FG plate are investigated. Also it is shown that the presence of the foundation has a considerable influence on the resonance characteristic curves.