Higher-Order Approximate Solutions to a Strongly Nonlinear Duffing Oscillator

Abstract

This paper deals with the Duffing oscillator, which is generally a strong nonlinear oscillation of a conservative and single-degree-of-freedom system with odd nonlinearity. By coupling linearization of governing equation with the method of harmonic balancing method, the paper presents new, accurate, higher-order approximate solutions for the general strong nonlinear systems. Unlike the classical harmonic balance method, analytical approximate solutions are possible because linearization of governing differential equation is conducted prior to harmonic balancing. Simple linear algebraic equations are obtained instead of nonlinear algebraic equations without analytical solution. Using the approach, general approximate analytical expressions for the exact period and periodic solution are established. These approximate solutions are valid for small as well as large amplitudes of oscillation. In addition, it is not restricted to the presence of a small parameter such as in the classical perturbation method.