Comparison of higher order versions of the method of multiple scales for an odd non-linearity problem

Abstract

The method of multiple scales is one of the important perturbation techniques widely used. The method yields transient solutions as well as steady state solutions in contrast to some other techniques which yield only the steady state solution. When employing higher order expansions, slightly different versions of the method appear in the literature. The oldest version, called reconstitution method, is due to Nayfeh (see, for example, Refs. [1–3]). Generally speaking, in this method, for primary resonances, the damping and forcing terms are re-ordered such that they balance the effect of non-linearities. The nearness of the external excitation frequency to one of the natural frequencies is represented by using only one correction term. The time derivatives for each time scale do not vanish separately, but their sum vanishes for finding the steady state solutions. In contrast, Rahman and Burton [4] showed that the reconstitution method (which will be called MMS I) cannot capture well the steady state Lindstedt–Poincare solutions. MMS I yielded extra solutions which are not physical for the simple duffing oscillator. Rahman and Burton [4] then suggested an alternative version (MMS II) to handle the problem. The excitation frequency and the damping should be expanded in a series and require that each time-scale derivative vanish independently. This method was presented for finding the steady state solutions. However, the unsteady solutions cannot be retrieved using the method. Boyac y and Pakdemirli [5] applied this new version as well as MMS I to partial differential equations with arbitrary quadratic and cubic non-linearities and found similar results to Ref. [4]. Hassan [6] applied MMS II to the case of superharmonic resonances and compared his results with the harmonic balance method. Later, Lee and Lee [7] improved MMS II by showing how to calculate unsteady solutions as well as the steady state solutions (MMS II modified). Similar to MMS I, the suggested modified version make series expansions unnecessary for the frequency, damping and excitation amplitude. In this version, damping and excitation are scaled to appear in the first non-linear order. In MMS II, only the steady state solutions can be retrieved