The influence of tangential and torsional vibrations on the stability lobes in metal cutting

Chatter vibrations in machining are studied. Special attention is paid to the effect of structural vibrations in the cutting speed direction. These so-called tangential vibrations may dominate the dynamics in drilling, sawing or grinding, where the torsional dynamic stiffness is much lower than the lateral, axial or radial dynamic stiffness. The inclusion of tangential vibrations in the chip thickness model leads to a delay differential equation (DDE) with state-dependent delay in the time domain. It is shown that the system is equivalent to a DDE with constant delay if it is described in terms of the spindle rotation angle. The analysis of metal cutting vibrations in the angular domain is very useful because in this case the delay is a constant parameter. Numerical examples for turning and drilling processes are presented. An extension of the universal oriented transfer function method for the stability analysis in the frequency domain is presented, which includes the tangential regenerative effect. An additional term appears in the oriented transfer function, which can stabilize or destabilize critical eigenmodes and may also lead to new instabilities in comparison with the conventional analysis.