On the multi-scale analysis of strongly non-linear forced oscillators

We consider the resonant response of strongly non-linear oscillators of the form ü + 2ϵηu + mu + ϵƒ(u) = 2ϵpcosΩt, where ƒ(u) is an odd non-linearity, ϵ need not be small, and m = −1, 0, or + 1. Approximate solutions are obtained using a multiple-scale approach with two procedural steps which differ from the usual ones: (1) the detuning is introduced in the square of the excitation frequency Ω and as a deviation from the so called backbone curve and (2) a new expansion parameter α = α(ϵ) is defined, enabling accurate low order solutions to be obtained for the strongly non-linear case.