Harmonic and Subharmonic Response for the Duffing Equation ẍ+αx+βx3 = F cosωt (α>0)

Abstract

The Duffing equation ẍ+αx+βx3 = F cosωt (α>0) is known to possess periodic solutions with frequency ω/n for all integral n, provided that β is sufficiently small. For n=1 and n≠1, these solutions are designated in the literature as harmonic and subharmonic solutions respectively. In this paper, a classification of the various types of periodic solutions is given, in which it is shown that there are two types of harmonic solutions and two types of subharmonic solutions for sufficiently small β. The perturbation method is employed to find the approximate response curves for each of the four types of periodic solutions. A comparison is made between the response curves obtained in the non-linear case (β≠0) and linear case (β=0) and some properties of the solutions in the non-linear case are discussed. A comparison is then made between the perturbation method and the Rauscher method, which is an iteration method that assumes F small instead of β. This comparison is not made in all generality, but only for one particular type of subharmonic solution. The main result obtained is that for F small, the two methods yield similar results for larger values of β than might have been anticipated.