Eigenvalue Analysis of Mode-Coupling Chatter for Machine-Tool Stabilization

The mode-coupling chatter phenomenon, due to self-excited vibrations occurring during many machining operations, is studied in this paper. Although mode-coupling chatter has been widely treated in the specialized literature, the authors propose here an approach that is rather different from the classical one, namely, an eigenvalue analysis of chatter based on system theory. A mathematical model of the workpiece- tool system is established first, so as to define the system equations. A closed-form equation, expressing the stability condition of the system as a function of the system parameters, and particularly of the angle between the feed direction and the piece surface, is then obtained from the eigenvalues of the system. Moreover, a study of the eigenvectors of the system enables one to analytically express the tool trajectories, both in the stable and in the unstable case. The model thus formulated and the related considerations about the system stability are then employed in a real case. Namely, a wood-cutting machine subject to mode-coupling chatter is considered, and a technique for stabilization, based on the considerations made on the model, is presented. The authors believe that the eigenvalue analysis of mode-coupling chatter presented in this paper provides a more detailed and complete analysis of the phenomenon than what is currently present in the literature.