Perturbation Analysis of Spindle Speed Variation in Machine Tool Chatter

Spindle speed variation has been shown to be an effective method for chatter control. In this paper, a single-degree-of-freedom regenerative type chatter equation is treated using perturbation methods. Rather than using the time coordinate, the angle of revolution is taken as the independent coordinate for maintaining a constant delay in the equations. The spindle speed is taken to be harmonically varying about a constant mean speed. Approximate analytical solutions are sought using the method of strained parameters, a perturbation technique. The amplitude of speed fluctuations (ϵ) is assumed to be small, and solutions are constructed using this parameter as the perturbation parameter. The stability lobes for constant spindle speeds are calculated exactly. By using the approximate perturbation analysis, the gain in stability is calculated for variable spindle speeds. The analysis is valid for (ϵ) values up to 0.02 (i.e., 2% of the constant mean speed). Solutions are verified using numerical simulations of the original equation.