An Iterative Approach to Obtaining the Nonlinear Frequency of a Conservative Oscillator with Strong Nonlinearities

An iterative procedure is suggested for obtaining the higher-order approximate solutions of a conservative system comprising an oscillator with cubic and quintic restoring force function. The proposed method is similar to the traditional harmonic balance methods but, unlike them, the errors obtained from the previous step are considered at the current step to increase the accuracy of the solution. A comparison of results with those obtained by exact solution and other approximate analytical techniques confirms the accuracy of the method. It is shown that the achieved approximate solutions are valid for both small and large amplitudes of oscillation and can meet the exact solutions with a high level of accuracy in the lower-order of approximations. Furthermore, using the obtained analytical solutions, the effect of cubic and quintic terms on the frequency is discussed.