Necessary discrete condition for error control of time-domain methods in milling stability prediction

Stability prediction is an efficient way to avoid milling chatter which is one of the major limitations in increasing efficiency of milling operations. Prediction accuracy is an important topic since it affects the feasibility of process parameter optimization, especially when gradient calculation is needed. However, milling operation is an interrupted cutting process because of the cut in and out phenomenon of tool teeth, which means the accelerations of tool motion at the moments where cutter tooth enters and exits of the cut are not continuous. This characteristic was always neglected in the general discrete methods which leads to incorrect conclusions about the order of truncation errors. As a result, the order of error cannot reach the nominal one, especially in milling with multiple delays such as variable pitch milling, and the related algorithms are always non-uniformly convergent. In this paper, the necessary discrete condition for reaching the nominal error order is identified. Also, a novel discrete scheme is developed, which can keep the non-uniformly convergent error within the nominal order for stability prediction in both uniform and variable pitch milling processes.