Higher Approximate Solutions of the Duffing Equation : Primary Resonance in the Soft Spring System

The Duffing system with soft spring is analysed by using a new algorithm which computes nonlinear steady-state oscillations and their stability with high speed and accuracy, based on the lines of the harmonic balance method. Detailed results of the frequency response of the primary resonance, up to the 9th harmonics, are presented. The following are demonstrated : a new unstable region, an isolated branch, the bifurcation of the non-odd order solutions, subharmonic oscillations (order 1/2 and 1/4), random almost periodic oscillations and superharmonic resonances (order 2 to 7). Some of these are confirmed by numerical simulation.