An analytical coupled technique for solving nonlinear large-amplitude oscillation of a conservative system with inertia and static non-linearity

Based on a new trial function, an analytical coupled technique (a combination of homotopy perturbation method and variational method) is presented to obtain the approximate frequencies and the corresponding periodic solutions of the free vibration of a conservative oscillator having inertia and static non-linearities. In some of the previous articles, the first and second-order approximations have been determined by the same method of such nonlinear oscillator, but the trial functions have not been satisfied the initial conditions. It seemed to be a big shortcoming of those articles. The new trial function of this paper overcomes aforementioned limitation. The first-order approximation is mainly considered in this paper. The main advantage of this present paper is, the first-order approximation gives better result than other existing second-order harmonic balance methods. The present method is valid for large amplitudes of oscillation. The absolute relative error measures (first-order approximate frequency) in this paper is 0.00 % for large amplitude A = 1000, while the relative error gives two different second-order harmonic balance methods: 10.33 and 3.72 %. Thus the present method is suitable for solving the above-mentioned nonlinear oscillator.