Singularity in the discrete-time model of impacting mechanical systems

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.