Nonlinear free vibration analysis of the functionally graded beams

Nonlinear natural oscillations of beams made from functionally graded material (FGM) are studied in this paper. The equation of motion is derived according to the Euler-Bernoulli beam theory and von Karman geometric nonlinearity. Subsequently, Galerkin's solution technique is applied to obtain the corresponding ordinary differential equation (ODE) for the FGM beam. This equation represents a kind of a nonlinear ODE containing quadratic and cubic nonlinear terms. This nonlinear equation is then solved by means of three efficient approaches. Homotopy perturbation method is applied at the first stage and the corresponding frequency-amplitude relationship is obtained. Frequency-amplitude formulation and Harmonic balance method are then employed and the consequent frequency responses are determined. In addition, Parameter Expansion Method is utilized for evaluating the nonlinear vibration of the system. A parametric study is then conducted to evaluate the influence of the geometrical and mechanical properties of the FGM beam on its frequency responses. Different types of material properties and boundary conditions are taken into account and frequency responses of the system are evaluated for different gradient indexes. The frequency ratio (nonlinear to linear natural frequency) is obtained in terms of the initial amplitude and compared for different materials and end conditions.