Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach

In this study, a novel size-dependent beam model is proposed for nonlinear free vibration of a functionally graded (FG) nanobeam with immovable ends based on the nonlocal strain gradient theory (NLSGT) and Euler-Bernoulli beam theory in conjunction with the von-Kármán's geometric nonlinearity. It is assumed the material properties of the nanobeam changes continuously in the thickness direction according to simple power-law form. To remove the stretching and bending coupling due to the unsymmetrical material variation along the thickness, the formulation of the problem is developed based on a new reference surface. The Hamilton's principle is utilized to derive the equations of the motion and the corresponding boundary conditions. The partial nonlinear differential equation describes the nonlinear vibration of FG nanobeam is reduced to an ordinary nonlinear differential equation with cubic nonlinearity via Galerkin's approach under the assumption that the axial inertia is negligible. A closed-form solution is obtained for nonlinear frequency by the novel Hamiltonian approach, and some illustrative numerical examples are given in order to study the effects of the strain gradient length scale, the nonlocal parameters, vibration amplitude and various material compositions on the ratio of nonlinear frequency to linear frequency (the nonlinear frequency ratio).