Modified Lindstedt–Poincare methods for some strongly non-linear oscillations: Part I: expansion of a constant

Abstract

In this paper, a modified Lindstedt–Poincare method is proposed. In this technique, a constant, rather than the non-linear frequency, is expanded in powers of the expanding parameter to avoid the occurrence of secular terms in the perturbation series solution. Some examples are given here to illustrate its effectiveness and convenience. The results show that the obtained approximate solutions are uniformly valid on the whole solution domain, and they are suitable not only for weakly non-linear systems, but also for strongly non-linear systems.