Nonlinear free and forced vibration analysis of embedded functionally graded sandwich micro beam with moving mass

Abstract

In this study, nonlinear free and forced vibration analysis of an embedded functionally graded sandwich micro-beam with a moving mass is investigated. The velocity of moving mass is assumed constant. The structure is resting on nonlinear Pasternak foundation. The governing equation of motion is obtained using Hamilton's principle based on the Euler–Bernouli model with considering nonlinear terms in strain–displacement relation. Strain gradient elasticity theory is used to model the small scale effects. The micro-beam contains a homogenous core and two integrated functionally graded face-sheets. Mechanical properties except Poisson ratio are assumed to be variable based on the power-law distribution along the thickness direction. Galerkin's decomposition technique is implemented to convert nonlinear partial differential equation to a nonlinear ordinary differential equation. Multiple times scale method is applied to derive closed form approximate solution for free and forced vibration and nonlinear natural frequencies of the micro-beams. Accuracy of the obtained results using current issue may be justified by comparing with those obtained by existing results of the literature. The effect of some important parameters such as length scale parameter, power gradient index, nonlinear elastic foundation, aspect ratio, position, and velocity of moving mass and boundary conditions is studied on the various responses of the micro-beam such as nonlinear natural frequency, frequency response, and force–response curves.