A simple expansion based HPM for damped nonlinear oscillator

A new amplitude expansion-based homotopy perturbation method (AE-HPM) is used to analyze the nonlinear behavior of a damped oscillator. The proposed method employs a simple amplitude expansion concept in addition to the traditional homotopy perturbation method to determine the solution and amplitude frequency relationship for the damped system. Three examples with linear damping are considered to demonstrate the simplicity, efficiency, and effectiveness of the proposed approach. The study considers a quintic oscillator, a strongly nonlinear oscillator characterized by cubic nonlinearity combined with a harmonic restoring force and a microelectromechanical system (MEMS). Analytical results obtained by the AE-HPM reveal that the amplitude of oscillation decays exponentially with the damping parameter, while the nonlinear stiffness terms strongly affect the frequency response. Comparative analysis is carried out for the amplitude with the corresponding numerical results and shows that the proposed method achieves superior accuracy, faster convergence and broader applicability with the absolute error remaining below 0.1 over the entire time interval considered for weak to strong damping effect while the traditional HPM method fails. Therefore, the proposed method provides a simple, powerful and reliable analytical tool for investigating nonlinear damped oscillatory systems.