THE KBM DERIVATIVE EXPANSION METHOD IS EQUIVALENT TO THE MULTIPLE-TIME-SCALES METHOD

Abstract

Several perturbation methods are commonly used to predict the free and forced response of weakly non-linear oscillators. The Krylov-Bogoliubov-Mitropolsky (KBM) and multiple-time-scales (MTS) methods use expansions of dependent variables, ordinary time derivatives, and some system parameters to convert the equations of motion into a set of first order differential equations. Each of these equations represents the slow time modulations of the amplitude and phase of the zeroth order solution(s).In this paper, a simple correspondence between the expansions of ordinary time derivatives employed in these two methods is used to show that, except for notation, these two methods are identical in the sense that to any order of approximation these two methods will provide identical results when they use the same parameter expansions and identical additional constraints. The KBM method attempts to reduce unneeded algebraic calculations by tailoring the derivative expansions to the simplest applicable form. This, however, requires some experience or a trial and error approach to establish the intermediate expansion variables and the implicit and explicit dependence of perturbation solutions on the different time scales. This relation depends not only on the problem at hand but also on the parameter expansions used in the solution procedure. By using the most general expansion for the time derivatives, the MTS method establishes this dependence as a part of the analysis. In this method, the algebraic details are hidden by using a compact derivative operator type notation. However, these operators do not commute in general.