Solutions to a general forced non-linear oscillations problem

Abstract

In this paper we present a mixed “multiple time scale – harmonic balance” (MHB) method for a quite general non-linear oscillations problem subject to a periodic force. The MHB differs from the more traditional multiple time scale (MS) method in that it uses two separate procedures for the solutions; (1) obtaining the form of solution using an almost trivial procedure and (2) balancing of the harmonic terms with a simple iteration. These procedures are demonstrated. The advantage of this alternative approach to MS is that it is easier to use. In fact, all the formulas for the transient (in terms of various harmonic components as well as the recovered solution to the given differential equation), and steady state solutions to the superharmonic, subharmonic and primary resonance cases are presented together. Finally, numerical results are presented to demonstrate the validity of the method. Examples for superharmonic, subharmonic and primary resonances are provided. These results indicate that MHB provide excellent approximations for sufficiently small values of ϵ in most cases. For several cases for parameters with moderately magnitudes, MHB still yields quite good approximations. Some observations on the so-called “steady state” solutions are included.