The two-degree of freedom oscillator is presented to model the dynamics of a machine tool system in the cutting process. The chip dynamics are presented through a discontinuous system with a velocity boundary (frictional force). The closed form solutions are presented for the normalized linear set of ordinary differential equations. The basic mappings are introduced to investigate the stick-slip motion in such a mechanical model. The mapping structures for the periodic motions will be developed. The numerical prediction of the phase trajectory, over a range of the excitation frequency is presented through the discontinuity. The predictions are verified by numerical illustrations of the phase trajectory, velocity and forces time histories. The main contributions are the forces and force product distribution at the switching points for this machine-tool.

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