A whole discretization method for milling stability prediction considering the discrete vibration velocities

The discretization methods have been widely used for chatter stability prediction and can be divided into two categories: semi-discretization methods (SDMs) by discretizing the time-delayed displacements, and full-discretization methods (FDMs) by discretizing both the present and time-delayed displacements. In this paper, a whole discretization method is proposed to achieve a faster and more accurate prediction for the milling stability. In addition to the discrete vibration displacements, the discrete vibration velocities are included in the solution by rebuilding the integrated matrix. The prediction error caused by the separate discretization of the cutting force coefficient matrix and vibration state variables can be avoided. The milling stability is determined with the discrete map of the state based on the Floquet theory. The proposed model is compared with the zeroth-order SDM and the first-order FDM which also approximate the solution with two neighboring discrete state variables. It is shown that the local discretization error based on the proposed method achieves o((Delta t)5), which is three orders higher compared to the zeroth-order SDM and the first-order FDM. In addition, when the same prediction errors are required, the computational time of the proposed method is 11-14% of the zeroth-order SDM, and 24-27% of the first-order FDM. The results prove that considering vibration velocity can increase the effi-ciency and accuracy of the chatter stability prediction compared to existing SDM and FDM al-gorithms.