Approximate Solution of the Quadratic Nonlinear Oscillator by Iteration Procedure

Abstract

An analytical technique has been developed based on an iteration method to determine higher order approximate periodic solutions for a nonlinear oscillator with discontinuity for which the elastic force term anti-symmetric and quadratic. 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 large oscillations. A modified approximate analytic solution of the quadratic nonlinear oscillator "ẍ + x2 sgn(x) = 0" has been obtained based on an iteration procedure. Here we have used the truncated Fourier series in each iterative step. The approximate frequencies obtained by the technique shows a good agreement with the exact frequency and periodic solutions with the exact ones. The percentage of error between exact frequency and fourth approximate frequency of the method is as low as 0.00003%.The method is mainly illustrated by the quadratic nonlinear oscillator but it is also useful for many other nonlinear problems.