Applicability of homotopy perturbation method for analysis the nonlinear vibration of thin circular functionally graded plate on nonlinear elastic foundation

In this paper the nonlinear forced vibration of thin circular functionally graded plate resting on Winkler-Pasternak and nonlinear Winkler foundation is investigated. The governing equations of the system are constructed using the von-Karman geometric nonlinear theory. Plates with edges elastically restrained against rotation and inplane displacement are analyzed. In this study, it is assumed that the plate can be subjected to any periodic distributed lateral loading with respect to time. The Galerkin method is used to reduce the nonlinear partial differential equations to a nonlinearly second-order ordinary differential equation for the central deflection. The Homotopy Perturbation Method is employed to analysis the vibration of the circular plate which is subjected to the vibration amplitude of the same or bigger order of the plate thickness. The effects of various parameters including large vibration amplitude, elastic foundation parameters and volume fraction on frequency ratio and variation of radial and circumferential stresses are considered. Based on the results, with increasing the vibration amplitudes the frequency ratio is increased, consequently. On the other hand, the variation of frequency ratio versus volume fraction is reversely. According to the results, the first approximation of HPM leads to highly accurate solutions for geometrically nonlinear vibration of circular thin FGM plates. Additionally, volume fraction and vibration amplitude affect the frequency ratio significantly.