A Generalized Runge-Kutta Method for Stability Prediction of Milling Operations With Variable Pitch Tools

Abstract

This paper extends the generalized Runge-Kutta method (GRKM) to predict the machining stability of milling systems with variable-pitch tools. Different from the uniform cutters with fixed pitch angles, the variation of tooth distribution angles of variable pitch cutters significantly affects the stability diagrams of the milling systems. From the viewpoint of the regenerative chatter, the milling system with non-uniform tools is governed by a delayed differential equation (DDE) with multiple delays. Afterwards, the GRKM, an approach verified with high computational accuracy and efficiency for DDEs with a single delay, is extended to tackle the milling systems with multiple delays based on Floquet theory. Besides the pitch angles, other geometry parameters of the cutter are also taken into consideration, such as the helix angle, which is proved with limited influence on the stability lobes. With the objective of maximizing productivity, the resultant stability charts provide valuable reference for the geometry design of variable-pitch cutters and for the choice of machining parameters, i.e. the spindle speed and the depth of cut.