A Time Marching Integration for Semianalytical Solutions of Nonlinear Oscillators Based on Synchronization

Abstract

An efficient method is proposed in this study for solving the semi-analytical solutions of periodic responses of nonlinear oscillators. The basic ideas come from the fact that any periodic response can be described by Fourier series. By transforming the Fourier series into a system of harmonic oscillators, we thus establish an efficient numerical scheme for tracking the periodic responses, as long as a synchronized motion can be achieved between the system of harmonic oscillators and the nonlinear oscillators considered. The presented method can be implemented by conducting time marching integration only, but it is capable of providing semi-analytical solutions straightforwardly. Different from some widely used methods such as harmonic balance method and its improved forms, this method can solve solutions involving high order harmonics without incorporating any tedious derivations as it is totally a numerical scheme. Several typical oscillators with smooth as well as nonsmooth nonlinearities are taken as numerical examples to test the validity and efficiency.