Developing homotopy perturbation method to investigate the nonlinear vibration of a Porous FG-Beam subjected to the external excitation

In this paper large amplitude vibration analysis of a porous FG-beam rested on a Winkler foundation and subjected to a harmonic loading is analytically investigated. The material properties of the porous FG beam are assumed to vary continuously according to a simple power law. Employing Von Karman’s geometric nonlinearity, implementing the Galerkin’s method and assuming doubly clamped immovable end boundary conditions, the governing nonlinear partial differential equation is reduced to a nonlinear ODE. Because of the large coefficient of the nonlinear term and due to existence of the external harmonic loading effect, none of the traditional perturbation methods leads to a valid solution. So, in order to solve this nonlinear nonhomogenous equation, the modified homotopy perturbation method is developed and it is called developed homotopy perturbation method (DHPM). For validating, the time response results obtained by DHPM and numerical methods are compared for various values of excitation amplitudes and frequencies. The increasing of nonlinear frequency obtained by DHPM with those of existence literature revealed a good agreement with a desired accuracy. Finally, the frequency response curves are presented for different values of volume fraction exponent together with porous volume fraction and the effects of nonlinearities on the frequency response curves are discussed in detail.