Application of AFF and HPM to the systems of strongly nonlinear oscillation

This study is concerned with two well-known systems of nonlinear oscillators. One of them is a system consists of a mass grounded two springs which one of those springs is linear and the other is nonlinear, and the other one is dealt with nonlinear large amplitude free vibrations of a slender cantilever beam with a rotationally flexible root and carrying a lumped mass at an intermediate position along its span. The main objective was to obtain highly accurate analytical solutions for free vibration of conservative oscillators with inertia and static type cubic nonlinearities. Two methods are studied to analyze the dynamic system behavior. One method makes use of a set of Amplitude–Frequency Formulation (AFF) and the other method applies Homotopy Perturbation Method (HPM). Taking advantage of some simple mathematical operations on these methods, we can obtain their natural frequencies. The computed results agree with those analytical and numerical results given in the literatures. The results indicate that the present analysis is accurate, and provide us a unified and systematic procedure which is simple and more straightforward than the other similar methods.