Nonlinear Free Vibration Analysis of Bi-directional Functionally Graded Rectangular Plates

In the present study, nonlinear free vibration analysis of bi-directional functionally graded simply supported rectangular plates is investigated analytically for the first time. For this purpose, with the aid of Hamilton’s principle and von Karman nonlinear strain-displacement relations, the partial differential equations of motion are developed. Afterward, the nonlinear partial differential equations are transformed into the time-dependent nonlinear ordinary differential equations by applying the Galerkin method. The nonlinear equation of motion is then solved analytically by the Modified Lindstedt-Poincare method to determine the the plate nonlinear frequencies. The volume fraction distribution is assumed to be continuously graded in both the length and width directions of the plate. Finally, the effects of some system parameters such as the vibration amplitude, FG indexes and aspect ratio on the nonlinear frequency are discussed in detail. To validate the analysis, the results of this paper are compared with both the published data and numerical method, and good agreements are found.