Higher accuracy approximate solution for oscillations of a mass attached to a stretched elastic wire by rational harmonic balance method

Abstract

A second-order modified rational harmonic balance method is used for approximately solve the nonlinear differential equation that governs the oscillations of a system typified as a mass attached to a stretched elastic wire for which the restoring force for this oscillator has an irrational term with a parameter lambda that characterizes the system. A frequency-amplitude relation is constructed and this frequency is valid for the complete range of oscillation amplitudes A and parameter lambda, and excellent agreement of the approximate frequencies with the exact one is demonstrated and discussed. The discrepancy between the approximate frequency and the exact one never exceed 0.12%. This error corresponds to lambda = 1. while for lambda < 1 the relative error is much lower. For example, its value is lower than 0.017% for lambda = 0.5.