Large-amplitude vibration of sigmoid functionally graded thin plates with porosities

Abstract

This research focuses on the large-amplitude vibration of sigmoid functionally graded material (S-FGM) thin plates with porosities. Porosities in S-FGM plates can happen due to technical issues during the preparation of S-FGMs. Two types of porosity distribution, i.e., even and uneven distribution, are taken into account. The material properties of S-FGM plates with porosities change smoothly along the thickness direction based on the sigmoid distribution law, which is described by modified piecewise functions. The geometrical nonlinearity is considered by applying the von Kármán non-linear plate theory. The nonlinear governing equation of S-FGM plates with porosities is derived using the D′Alembert's principle. By applying the Galerkin method with the first three modes, the governing equation is discretized to three ordinary differential equations. Then, the method of harmonic balance is used to solve these discretized equations. Analytical results are verified numerically with the adaptive step-size fourth-order Runge-Kutta method. The stability of the steady-state response is examined by means of the perturbation technique. Furthermore, the maximum amplitudes of each mode during the vibration period are obtained and shown in the neighborhood of the fundamental mode. Study demonstrates that the S-FGM plates with porosities possess hardening spring characteristics in nonlinear frequency response. Moreover, a complex multi-solution phenomenon occurs in the present dynamic system which is rooted from the nonlinear mode interaction. Finally, investigation is made on the effects of porosity along with other key parameters on large-amplitude vibration response of porous S-FGM plates.