Analytical expressions for chatter analysis in milling operations with one dominant mode

In milling, an accurate prediction of chatter is still one of the most complex problems in the field. The presence of these self-excited vibrations can spoil the surface of the part and can also cause a large reduction in tool life. The stability diagrams provide a practical selection of the optimum cutting conditions determined either by time domain or frequency domain based methods. Applying these methods parametric or parameter traced representations of the linear stability limits can be achieved by solving the corresponding eigenvalue problems. In this work, new analytical formulae are proposed related to the parameter domains of both Hopf and period doubling type stability boundaries emerging in the regenerative mechanical model of time periodical milling processes. These formulae are useful to enrich and speed up the currently used numerical methods. Also, the destabilization mechanism of double period chatter is explained, creating an analogy with the chatter related to the Hopf bifurcation, considering one dominant mode and using concepts established by the Pioneers of chatter research.