On a Numerical Method for Simultaneous Prediction of Stability and Surface Location Error in Low Radial Immersion Milling

Abstract

This brief proposes a numerical approach for simultaneous prediction of stability lobe diagrams and surface location error in low radial immersion milling based on the direct integration scheme and the precise time-integration method. First, the mathematical model of the milling dynamics considering the regenerative effect is presented in a state space form. With the cutter tooth passing period being divided equally into a finite number of elements, the response of the system is formulated on the basis of the direct integration scheme. Then, the four involved time-variant items, i.e., the time-periodic coefficient item, system state item, time delay item, and static force item in the integration terms of the response, are discretized via linear approximations, respectively. The corresponding matrix exponential related functions are all calculated by using the precise time-integration method. After the state transition expression on one small time interval being constructed, an explicit form for the discrete dynamic map of the system on one tooth passing period is established. Thereafter, the milling stability is predicted via Floquet theory and the surface location error is calculated from the fixed point of the dynamic map. The proposed method is verified by the benchmark theoretical and experimental results in published literature. The high efficiency of the algorithm is also demonstrated.