Mode Shapes of Directional Matrix and Chatter in Milling

The directional matrix transforms the dynamic feed vector to the dynamic milling forces, and plays a central role in the milling chatter analysis. Through modal analysis of the zero-order directional matrix (DM), this paper presents algebraic expressions for the mode shapes of DM in up, down and symmetric milling configurations. Analysis on the eigenvectors of DM reveals physical insights into the geometrical features of its characteristic mode shapes. The two mode shapes are demonstrated to be linear for a lower radial immersion with real eigenvalues, and are elliptical for a higher immersion with complex eigenvalues, and become a circle at full immersion. The up and down-milling processes at the same immersion have their mode shapes pointing to the right and left side of the feed direction, respectively, with their modal angles differing by the radial immersion angle. The dominant mode shape of DM is also shown to be the mode shape of regenerative chatter for milling with a symmetrical structural dynamics. For all types of elliptical chatter mode shapes, their vibration trajectories are shown to move in a counterclockwise direction. These mode shape predictions are verified by experiments.