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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.

This research deals with the nonlinear vibration of the functionally graded nano-beams based on the nonlocal elasticity theory considering surface and flexoelectric effects. The flexoelectric functionally graded nano-beam is resting on nonlinear Pasternak foundation. Cubic nonlinearity is assumed for foundation. It is assumed that the material properties of the nano-beam change continuously along the thickness direction according to different patterns of material distribution. In order to include coupling of strain gradients and electrical polarizations in equation of motion, the nonlocal, nonclassical nano-beam model containing flexoelectric effect is employed. In addition, the effects of surface elasticity, di-electricity, and piezoelectricity as well as bulk flexoelectricity are accounted in constitutive relations. The governing equations of motion are derived using Hamilton principle based on first shear deformation beam theory and the nonlocal strain gradient elasticity theory considering residual surface stresses. The differential quadrature method is used to calculate nonlinear natural frequency of flexoelectric functionally graded nano-beam as well as nonlinear vibrational mode shape. After validation of the present numerical results with those results available in literature, full numerical results are presented to investigate the influence of important parameters such as flexoelectric coefficients of the surface and bulk, residual surface stresses, nonlocal parameter, length scale effects (strain gradient parameter), cubic nonlinear Winkler and shear coefficients, power gradient index of functionally graded material, and geometric dimensions on the nonlinear vibration behaviors of flexoelectric functionally graded nano-beam. The numerical results indicate that, considering the flexoelectricity leads to the decrease of the bending stiffness of the flexoelectric functionally graded nano-beams.