Grazing Orbits and Related Local ifurcations of an Oscillator with Continuous and Piecewise-Linear Restoring Force

One critical case for the motion of a periodically excited oscillator with continuous and piecewise-linear restoring force is that the motion happens to graze a switching plane between two linear regions of the restoring force. This article presents a numerical scheme for locating the periodic grazing orbit first. Then, through a brief analysis, the article shows that the grazing phenomenon turns the stability trend of the periodic orbit so abruptly that it may be impossible to predict an incident local bifurcation with the variation of a control parameter from the concept of smooth dynamic systems. The numerical simulation in the article well supports the scheme and the analysis, and shows an abundance of grazing phenomena in an engineering range of the excitation frequency.