Stability of interrupted cutting by temporal finite element analysis

Chatter in milling and other interrupted cutting operations occurs at different combinations of speed and depth of cut from chatter in continuous cutting. Prediction of stability in interrupted cutting is complicated by two facts: (1) the equation of motion when cutting is not the same as the equation when the tool is free; (2) no exact analytical solution is known when the tool is in the cut. These problems are overcome by matching the free response with an approximate solution that is valid while the tool is cutting. An approximate solution, not restricted to small times in the cut, is obtained by the application of finite elements in time. The complete, combined solution is cast in the form of a discrete map that relates position and velocity at the beginning and end of each element to the corresponding values one period earlier. The eigenvalues of the linearized map are used to determine stability. This method can be used to predict stability for arbitrary times in the cut; the current method is applicable only to a single degree of freedom. Predictions of stability for a 1-degree of freedom case are confirmed by experiment.