Explanation for Low-Speed Stability Increases in Machining: Application of a Continuous Delay Model

This paper investigates the analysis of delay integro-differential equations to explain the increased stability behavior commonly observed at low cutting speeds in machining processes. In the past, this improved stability has been attributed to the energy dissipation from the interference between the workpiece and the tool relief face. In this study, an alternative physical explanation is described. In contrast to the conventional approach, which uses a point force acting at the tool tip, the cutting forces are distributed over the tool-chip interface. This approximation results in a second order delayed integro-differential equation for the system that involves a short and a discrete delay. A method for determining the stability of the system for an exponential shape function is described, and temporal finite element analysis is used to chart the stability regions. Comparisons are then made between the stability charts that use the conventional point force and those that use the distributed force model for continuous and interrupted turning.