Analysis of Stability in High Speed Milling Machining by Means of Spectral Decomposition

A technique to study the critical chatter conditions arising in the high speed milling machining process is proposed, by means of evaluating system eigenvalue stability and eigenvalue sensitivity with respect to system parameters. Starting from the equations of motion of a general machine tool system, a set of linear, ordinary, time dependent parameter, Periodic Delay Differential Equations (PDDEs) may be the result. Three approaches (multi-step, full-discretization and semi-discretization) based on the extended Floquet theory and time discretization techniques are analyzed. The semi-Discretization (SD) method is considered in order to derive an eigenvalue sensitivity formula and to show the limits of this approach. A different approach not based on the Floquet theory is proposed. A set of linear, ordinary, constant parameter, PDDEs is obtained by applying a spectral decomposition and a generalized harmonic balance technique. The stability of the solution is evaluated by solving a non standard eigenvalue problem, making it possible to predict the occurrence of chatter vibration during milling. The proposed approach makes it possible to obtain a straightforward formula for the sensitivity of eigenvalues, and of the critical chatter condition, with respect to the variation of a system parameter. A numerical example is presented in order to test the proposed approach. Finally, strengths and limits of the proposed technique are critically discussed and a comparison with the SD method is carried out