In this paper, a two-degree of freedom dynamical system with discontinuity is developed to describe the vibration in the cutting process. The analytical solutions for the switchability of motion on the discontinuous boundary are presented. the switching sets based on the discontinuous boundary is introduced and the basic mappings are introduced to investigate periodic motion in such a mechanical model. The mapping structures for the stick and non-stick motions are discussed. Numerical predictions of motions of the machine tool in the cutting process are presented through the two-degree of freedom system with discontinuity. The phase trajectory and velocity and force responses are presented and the switchability of motion on the discontinuous boundary is illustrated through force distribution and force product on the boundary.