Size-dependent nonlinear free and forced vibration analyses of a functionally graded microplate subjected to transverse patch loading

Abstract

This paper presents a semianalytical methodology for the nonlinear vibration of a functionally graded microplate under transverse patch loadings. The higher-order shear deformation theory (HSDT) is combined with the modified strain gradient theory (MSGT) to model the microplate. The power-law function is used to model the functionally graded material. Hamilton’s principle is used to obtain the governing partial differential equations of motion, which are then solved using Galerkin’s method. The nonlinear free and forced vibration responses are obtained using the incremental harmonic balance (IHB) method, where the incremental part is performed using the arc-length continuation methods. The dependence of the steady-state amplitude on the amplitude of initial perturbation is studied using time history plots. These are plotted using the Newmark-𝛽 method. The effects of various parameters such as the power-law index, thickness of plate, thickness to material length scale parameter ratio, damping coefficient, different boundary conditions, and different positions of patch loading and its concentrations on the nonlinear free and forced vibration characteristics are examined in detail.