A Structured Approach to Analytical Multi-Degree-of-Freedom Time-Invariant Stability Analysis for Machining

A general, fully analytical stability solution for multi-degree-of-freedom machining is derived based on the formulation of an eigenvalue problem. The structure and process are assumed to be linear and time-invariant. A specific-energy-based cutting process model, which has been adopted by many researchers and practitioners, is used here to represent the process. This model is attractive for industrial application since it requires minimal experimentation for its application over a wide range of cutting conditions and tool geometry. The analysis permits arbitrary orientation of the structure’s orthogonal modes relative to the coordinates in which the cutting process is modeled. The final solution differs from others in that the analytical methods are carried through to the final result without need for the numerical methods used by other investigators when applying Nyquist-based analysis for more degrees of freedom than one. This final result for limiting width of cut and chatter phase, being in analytical form, clearly shows the effects of each individual mode on the stability behavior. En route to the multi-degree-of-freedom solution, one- and two-degree-of-freedom cases are addressed to demonstrate the approach.