Discontinuity Mapping for Near-Grazing Dynamics in Vibro-Impact Oscillators

Abstract

Vibro-impact oscillators may undergo zero-velocity impacts, also known as grazing contacts. Near-grazing dynamics refer to low-velocity impacts. The conventional technique of local stability analysis suffers singularity when applied to grazing dynamics. Discontinuity mapping conceived by Nordmark provides a powerful tool to analyze and predict the plethora of complex phenomena due to grazing. This article intends to help beginners in vibro-impact dynamics better understand the concept of discontinuity mapping through a lucid derivation of the discontinuity mapping. The fundamental approach consists of three steps. First, a Poincaré map is introduced for the oscillatory dynamics without impacts. Second, impact dynamics near a grazing contact point are approximated using a series expansion to generate the so-called discontinuity mapping. Finally, the overall oscillations involving low-velocity impacts are analyzed using a combination of the Poincaré map and the discontinuity mapping derived in the previous two steps. We first present a transparent derivation of the discontinuity mapping for a generic one-degree-of-freedom vibro-impact oscillator. Then, the approach is applied to a linear mass spring oscillator whose vibrations are restricted by a rigid wall.