An integro-differential equation with variable delay arising in machine tool vibration

We rigorously derive an integro-differential equation as a model for the possible onset of regenerative chatter during a turning process using a lathe. The cut is made parallel to the axis of rotation of the spindle. The model allows the spindle speed to continuously vary with time, which results in the presence of a variable time delay determined from a threshold condition. We present a number of conditions sufficient for the elimination of chatter; these emphasize sufficiently low feed rate or sufficiently high spindle speed. Numerical simulations cast light on the effect of a sinusoidally varying spindle speed, a feature of some modern lathes. Spindle speed variation can cure chatter but does not necessarily do so and can fail at higher values of the tool feed rate.