Application of He's homotopy perturbation method to nonlinear shock damper dynamics

In order to obtain the equations of motion of vibratory systems, we will need a mathematical description of the forces and moments involved, as function of displacement or velocity, solution of vibration models to predict system behavior requires solution of differential equations, the differential equations based on linear model of the forces and moments are much easier to solve than the ones based on nonlinear models, but sometimes a nonlinear model is unavoidable, this is the case when a system is designed with nonlinear spring and nonlinear damping. Homotopy perturbation method is an effective method to find a solution of a nonlinear differential equation. In this method, a nonlinear complex differential equation is transformed to a series of linear and nonlinear parts, almost simpler differential equations. These sets of equations are then solved iteratively. Finally, a linear series of the solutions completes the answer if the convergence is maintained; homotopy perturbation method (HPM) is enhanced by a preliminary assumption. The idea is to keep the inherent stability of nonlinear dynamic; the enhanced HPM is used to solve the nonlinear shock absorber and spring equations.