Nonlinear Vibrations of Sandwich FG Plates Resting on Nonlinear Pasternak Foundation under Primary Resonance Excitation Using Modified FSDT

In this study, the nonlinear vibration of a sandwich FG plate resting on a nonlinear Pasternak foundation which is simultaneously subjected to transverse harmonic forcing excitation and in-plane static force is investigated. Based on the Modified First-Order Shear Deformation Theory (FSDT), applying the von-Karman nonlinear strain–displacement relation and the Hamilton’s principle, the governing nonlinear coupled partial differential equations are derived. Then, the Galerkin’s procedure is used to reduce the equations of motion to nonlinear ordinary differential equations. In the absence of foundation, the validity of the formulation for analyzing the modified shear correction factors for shear stresses is accomplished by comparing the results with those reported in the literature. By applying the multiple scales method and considering the second order nonlinear approximation of solution, the primary resonance of the system under the transverse forcing excitation is analyzed. Under the steady-state condition, the frequency-response, the force-response and the damping-response equations are derived. Then the conditions of existence and stability of multiple coexisting non-trivial solutions for amplitude of the responses are discussed and the saddle node bifurcation points of the characteristic curves are derived. It is shown that, the variation of the system parameters in the resonance boundary may cause the jump phenomenon. Moreover, the effects of the system parameters including, excitation frequency, foundation parameters, damping, and amplitude of the harmonic and in-plane forces on the system nonlinear dynamics are investigated. Also it is shown that the presence of the foundation has a considerable influence on the resonance characteristic curves.