Higher-order Approximations of Nonlinear Oscillator with Coordinate-dependent Mass

Abstract

This paper analyses a nonlinear oscillator with coordinate-dependent mass based on the presented methods of multi-term harmonic balance (MHB) and iterative residue harmonic balance (IRHB). The proposed methods calculate higher-order approximations. After using the MHB, a group of complicated nonlinear algebraic equations are obtained which are cumbersome to calculate analytically. This limitation is overcome in the presented other method by using the IRHB. In the solution procedure of IRHB method, the higher-order approximations to angular frequencies and periodic responses can be determined due to linear residue equations. Results show that the presented solutions give high accuracy and better results than those obtained by other existing ones from the homotopy perturbation method and the frequency-amplitude formulation. The advantage of the IRHB method is that it balances the all residues step by step and the present second-order approximations almost coincide with the corresponding exact solutions. Thus, the presented IRHB method could be applied to other strongly nonlinear oscillator systems.