On the separation of motions in systems with a large fast excitation of general form

Abstract

In this study dynamic systems are considered, in which motion can be described through a system of second-order ordinary differential equations with the right sides depending both on the slow time t and on the fast time τ= ωt ( ω≫1 is a big dimensionsless parameter). It is assumed that the right sides are large (they have the magnitude order ω) and depend both on generalised coordinates and on the generalised velocities of the system. A motion separating procedure is developed for the systems described in twi ways. The procedure enables separate systems to derive for fast (oscillating) and slow components of the solution. Each of these separated systems is simpler than the original one. The equivalence of both procedures is shown. The first of them is based on the multiple scales method, the second one generalises the averaging method of Krylov–Bogoliubov–Mitropolskii. Motion of a linear oscillator excited through the large fast oscillations of the damping coefficient is analysed as an example of the established method usage. It is shown, that the excitation significantly changes the effective mass and in consequence the natural frequency of the original system. The analytic results are compared with numerical ones.