Singularity in the discrete-time model of impacting mechanical systems

TL;DR

This paper investigates the mathematical nature of bifurcations in impacting mechanical systems, revealing that a square-root singularity appears only in the trace of the Jacobian matrix near grazing incidences.

Contribution

It demonstrates that the square-root singularity in the Jacobian matrix is confined to its trace, not the determinant, refining the understanding of bifurcations in impact oscillators.

Findings

Square-root singularity appears only in the trace of the Jacobian.

Determinant of the Jacobian remains invariant across grazing.

Provides insight into the bifurcation structure of impacting systems.

Abstract

It is known that many peculiar nonlinear vibration problems in impacting systems are caused by grazing incidences. Such bifurcation phenomena are normally investigated through the Poincare map. The discrete-time map of a simple impact oscillator was derived by Nordmark, which showed that there should be a square-root singularity in the Jacobian matrix close to the grazing condition. In this paper we show that the square root singularity will be expressed only in the trace of the Jacobian matrix, while the determinant remains invariant across the grazing condition.