Third-order updated full-discretization method for milling stability prediction

Based on third-order Newton interpolation polynomial and direct integration scheme (DIS), this paper proposes a method to generate stability lobe diagram in milling process. The dynamic model of milling process with consideration of regeneration effect is described by time periodic delay-differential equation (DDE). Then, the DDE is rewritten as state space equation by a transformation. After equally discretizing the time delay into a series of small time intervals, the state space equation of milling system is integrated on the small time interval. Both the state term and delayed term are interpolated by third-order Newton interpolation polynomial, and the periodic-coefficient matrix is interpolated by first-order Newton interpolation polynomial. The state transition matrix which reflects the discrete mapping relation of dynamic responses for current tooth pass period and immediate previous tooth pass period is obtained directly. The accuracy of the proposed method is evaluated by comparing with benchmark methods in terms of the rate of convergence. The efficiency of the proposed method is verified through the comparison of computational time with existing methods. The proposed method is proved to be an accurate and efficient method by the comparison results. The distinction between up-milling and down-milling operations is also analyzed by comparing the stability lobe diagrams for these two operations. Besides, according to the analysis of rate of convergence, the number of substitutions, which are used to convert the variables located out of the required range into the required range, may affect the results of stability lobe diagrams. Moreover, the stability lobe diagram cannot be generated by using fourth-order updated full-discretization method.