Accurate approximate analytical solutions to an anti-symmetric quadratic nonlinear oscillator

Abstract

In this paper, an analytical technique has been developed based on a modified harmonic balance method to determine higher-order approximate periodic solutions for a nonlinear oscillator with discontinuity for which the elastic force term, is an anti-symmetric and quadratic term. Usually, a set of nonlinear algebraic equations is solved with this method. However, analytical solutions of these algebraic equations are not always possible, especially in the case of a large oscillation. In this article, different parameters of the same nonlinear problems are found, for which the power series produces desired results even for the large oscillation. We found out that a modified harmonic balance method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones have been demonstrated and discussed. The method is mainly illustrated by the anti-symmetric quadratic nonlinear oscillator but it is also useful for many other nonlinear problems. Key words: Anti-symmetric oscillator, approximate solutions, harmonic balance method, perturbation.