Improving accurate vibration periods and responses of a rigid rod model

Abstract

In this paper, we construct a residue harmonic balance solution procedure to improve the accuracy of approximate vibration period and steady response for a strongly nonlinear oscillator from the motion of rigid rod rocking back and forth on a circular surface without slipping. In this new solution procedure, all the unbalanced residuals due to Fourier truncation are considered into next order approximation by iteration to improve the accuracy. The presented approximate analytical results are compared with the other existing solutions such as Newton harmonic balance method, variational approach method, amplitude-frequency formulation, He’s energy balance method and the exact ones which are shown in graphs. It is evident that the method considered here is quickly convergent and only the second order approximation leads to high accuracy of the steady state solutions. In addition, four numerical examples are presented to highlight the effects of system parameters on the nonlinear vibration period, and excellent agreement between approximate periods and exact ones in the steady state. It is predicted that the solution method can be found wide application in the other nonlinear oscillation problems.