Analysis vibrations of two-direction functionally graded beam with integrated piezoelectric layers on resting elastic foundation using differential quadrature

In this paper a two dimensional elasticity for free vibrations and the effect of elastic foundantion on a two-direction functionally graded beams with integrated surface piezoelectric layers with combination of differential quadrature method and space-state method is presented here. Differential quadrature method in axial direction and space-state method in transverse direction is used. It’s considered that two parameters model or winkler-pasternak for elastic foundation which has been considered two kinds of boundary conditions include simply support and clamped-clamped. Also، It is assumed that beam properties in thickness and axial direction varying exponentially and poison factor is constant which has been considered the effects of materials properties gradient index and number waves on free vibrations beams. The obtained results show that this method has good accuracy and high speed of convergence.