Asymptotic stability of oscillations of two-bodies vibrating screen with one-sided obstacle without clearances

TL;DR

This paper proves the asymptotic stability of resonant periodic oscillations in a two-body vibrating screen model with a one-sided obstacle, using a nonsmooth version of Bogolyubov's theorem, revealing two-frequency oscillations.

Contribution

It introduces a rigorous proof of the stability of resonant oscillations in a nonlinear vibrating system with obstacles, extending classical methods to nonsmooth dynamics.

Findings

Oscillations are asymptotically stable under resonance conditions.

Periodic oscillations have two frequencies due to the obstacle.

The method applies a nonsmooth analog of Bogolyubov's theorem.

Abstract

We prove asymptotic stability of periodic oscillations in two-bodies vibrating screen model under assumption that the frequencies w1 and w2 of the generating system (without obstacle and periodic driving) satisfy the assumption w1:w2=1:2. We also assume that the frequency of the external periodic driving equals to w1. These settings correspond to nonlinear resonance which is a well known phenomenon in industrial implementation of the vibrating screen. The justification is performed over nonsmooth analog of the second Bogolyubov's theorem proposed by the author in his previous papers. It is rigorously proven that the periodic oscillations obtained have two frequencies, in contrast with the case when the obtacle is absent.