Error analysis of time-domain methods for milling stability prediction

Abstract

Stability prediction is an efficient way to avoid milling chatter which is one of the major limitations in increasing efficiency of milling operations. Since Insperger et al. [1] presented the semi-discretization methods (SDM) and demonstrated that the local discretization error is o(h3), in order to increase prediction accuracy, full-discretization methods (FDMs) [2,3], Simpson method [4] and spectral methods [5,6] etc. are proposed. These methods use polynomial interpolation to approximate the original system. The prediction errors are always analyzed based on Taylor expansions. Mathematically, the error analysis based on Taylor’s theorem requires that the original function has continuous derivatives of equivalent orders, that is, if a k-order polynomial interpolation is adopted, the original function must have continuous k-order derivatives [7]. However, milling operation is a highly interrupted cutting process, and the accelerations at the moments where cutter tooth enters and exits of the cut are obviously not continuous. Thus, even the order of local error at somewhere is rather high, the global error is not improved yet, and it may cause nonuniform convergence. In this paper, problems are revealed through simulations with specific milling parameters. It is concluded that the accuracy of the representative time-domain methods can hardly reach as high as o(h3). Moreover, the discretization scheme to overcome the problems and improve applicability of the existing methods is also discussed.