An asymptotic analysis of the response of non-linear damped systems subjected to step function excitation

Abstract

Two forms of a unified type of Krylov-Bogoliubov method are considered for the purpose of deriving approximate solutions of non-linear ordinary differential equations that describe the underdamped and overdamped motion of systems subjected to step function excitation. The method of analysis, based upon power series expansions of the phase and amplitude in terms of the non-linearity parameter e and, alternatively, the parameter μ=e/(e+1), explicitly accounts for the effect of linear damping, which is proportional to velocity, in such a way that the approximate solutions of the non-linear equations reduce in the limit to the exact solutions of the corresponding linear problems as the non-linearity parameter tends to zero. To assess the relative accuracies of the two variations of the fundamental method, numerical calculations were performed using (a) approximate formulas for the amplitude and phase and (b) the fourth order Runge-Kutta method of numerical integration. These numerical results are compared graphically, and the merits of the two approaches are discussed.