Nonlocal and surface effects on nonlinear vibration response of a graded Timoshenko nanobeam

The free and forced vibration of a graded geometrically nonlinear Timoshenko nanobeam supported by on a nonlinear foundation is considered in this paper. The main contribution of this study is to propose a new formulation for the dynamic response of this beam by combining nonlocal and surface elasticity in addition to employing the physical neutral axis method which eliminates the quadratic nonlinearity from the equation of motion. Using the principle of virtual work, a fourth-order nonlinear partial differential equation is formulated and Galerkin technique is employed to yield a fourth-order ordinary differential equation with cubic nonlinearity in the temporal domain. The method of multiple scales is employed to obtain the analytical expression of the nonlinear frequency of the beam and its frequency response curve from a primary resonance analysis. To assess the accuracy of this analytical solution, it is compared with a numerical solution obtained using the differential quadrature method. The obtained analytical results are successfully validated for particular cases of the considered problem with results published by other authors. The effects of surface elasticity, nonlocality, the physical neutral axis, the beam aspect ratio, the power-law index and the elastic foundation coefficients on the free and forced vibration response of the graded Timoshenko nanobeam are thoroughly investigated for different types of boundary conditions .