Nonlinear vibration of axially functionally graded non-uniform nanobeams

In this paper, the nonlinear vibration analysis of axially functionally graded (AFG) non-uniform nanobeams is performed based on Eringen's nonlocal theory and Euler–Bernoulli beam model. Hamilton's principle is used to derive the equations with consideration of Von–Kármán's geometric nonlinearity. The boundary conditions are taken as simply supported, clamped and clamped-simply supported ends. The solution procedure for the nonlinear frequencies is done using homotopy perturbation method (HPM) in conjunction with the generalized differential quadrature (GDQ) method for the first time. The small parameter in homotopy perturbation method is derived by the linear solution using GDQ method. The frequencies of the nanobeams are examined for various parameters such as the nonlinear amplitude, AFG power index, nonlocal value, different boundary conditions and rates of cross section change along the thickness and for pure ceramic, AFG and pure metal nanobeams.