Stability of non-sticking periodic oscillations obtained via the averaging method in discontinuous systems. I. Smooth outside of the discontinuity surfaces systems

TL;DR

This paper extends Bogolyubov's theorem to discontinuous systems, demonstrating the stability of certain periodic oscillations in systems that are smooth outside discontinuity surfaces, relevant for mechanical systems with dry friction.

Contribution

It justifies the second Bogolyubov's theorem for discontinuous systems with transversal intersections, expanding the theorem's applicability to more realistic mechanical models.

Findings

Proves stability of velocity in vibration-induced displacements.

Validates periodic solutions in discontinuous systems with small parameters.

Extends classical averaging methods to systems with discontinuities.

Abstract

In this paper the statement of the second Bogolyubov's theorem on periodic solutions of smooth systems with small parameter is justified for discountinuous systems. It is assumed that the generating solution intersects the discontinuity hyperplanes transversally and that the system is smooth outside of these hyperplanes. Such a situation is natural for mechanical systems with dry friction and without constraints and sticking of oscillations. To illustrate the result we prove stability of the velocity of vibration-induced displacement.