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

Several perturbation methods are commonly used to predict the free and forced response of weakly non-linear oscillators. The Krylov-Bogoliub ov-Mitropolsky (KBM) and multiple-time-scales (MTS) methods use expans ions of dependent variables, ordinary time derivatives, and some syste m parameters to convert the equations of motion into a set of first or der differential equations. Each of these equations represents the slo w time modulations of the amplitude and phase of the zeroth order solu tion(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 th e sense that to any order of approximation these two methods will prov ide identical results when they use the same parameter expansions and identical additional constraints. The KBM method attempts to reduce un needed algebraic calculations by tailoring the derivative expansions t o the simplest applicable form. This, however, requires some experienc e or a trial and error approach to establish the intermediate expansio n variables and the implicit and explicit dependence of perturbation s olutions on the different time scales. This relation depends not only on the problem at hand but also on the parameter expansions used in th e 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 usin g a compact derivative operator type notation. However, these operator s do not commute in general. (C) 1997 Academic Press Limited.