Complex dynamics of a vibration machine caused by a relay feedback control

Abstract

In this paper we describe some nonlinear phenomena observed in a vibration machine with an unbalanced rotor driven by a DC motor controlled by a relay system. We demonstrate that this type of control leads to the appearance of several complex kinds of behavior caused by interactions between slow and fast dynamics. Our numerical results are verified through comparison with an experimental vibration system. In order to understand the mechanism leading to these phenomena, we apply a modeling approach based on the reduction of a 5D autonomous non-smooth vector field to a 1D discontinuous map for the phase of the switching events. We demonstrate that there are two different reasons causing this map to be discontinuous, so that depending on parameters it may be either a gap map or a circle diffeomorphism. In both parameter domains the map exhibits a period adding structure. We explain also which features of the behavior of the modeled vibration machine are captured by the 1D map and which are not.