Difference Discretization Method for Milling Stability Prediction

Abstract

This paper proposes a difference discretization method (DDM) under a difference frame for the prediction of milling stability. In this method, the dynamic milling process is described by a delay differential equation (DDE) with two degrees of freedom rather than the traditional state-space form with a single discrete time delay. After discretization, only the velocity and acceleration in the DDE are approximated by the first- and second-order central difference for each smaller time interval, while the other items are kept unchanged. Then, the criterion for the optimal discretization interval number is put forward and derived based on the largest effective time interval (also called the critical time interval). The use of the critical time interval cannot only obtain sufficient accuracy, but also promotes as much efficiency as possible. Subsequently, a new DDM (NDDM) with varied discretization interval numbers as the milling rotating speeds is developed. Finally, the effectiveness of the proposed algorithm is demonstrated by using a benchmark example for a two-degrees-of- freedom milling model compared to the full discretization method (FDM) and the Hermite-interpolation full discretization method (HFDM). The results show that the proposed method has satisfactory stability charts and is able to increase the efficiency by 100% or more.