Added Stability Lobes in Machining Processes That Exhibit Periodic Time Variation, Part 1: An Analytical Solution

An added family of stability lobes, which exists in addition to the traditional stability lobes, has been identified for the case of periodically time varying systems. An analytical solution of arbitrary order is presented that identifies and locates multiple added lobes. The stability limit solution is first derived for zero damping where a final closed-form symbolic result can be realized up to second order. The un-damped solution provides a mathematical description of the added lobes’ locations along the speed axis, an added-lobe numbering convention, and the asymptotes for the damped case. The derivation for the damped case permits a final closed-form symbolic result for first-order only; the second-order solution requires numerical evaluation. The easily computed analytical solution is shown to agree well with the results of the computationally intensive numerical simulation approach. An increase in solution order improves the agreement with numerical simulation; but, more importantly, it allows equivalently more added lobes to be predicted, including the second added lobe that cuts into the speed regime of the traditional high-speed stability peak.