A precise and efficient updated third-order full-discretization approach for chatter stability analysis of the milling process

Abstract

Abstract Stability prediction of milling is of great significance as the regenerative chatter can reduce the machining quality and limit the efficiency of productivity. The stability lobe diagrams (SLDs) are the most popular used prediction approach, which is determined by solving the delay-differential equations (DDEs) describing the milling dynamic system. In this study, a precise and efficient updated third-order full-discretization approach (PE3rdFDM) considering the analytical solution of the free vibration is proposed to determine the SLDs. In each time interval discretized, the state term is defined approximately by the third-order Hermite interpolation polynomial and the derivative values needed to conduct the Hermite interpolation are provided by the original DDEs. To handle the time-delay term, the original integral of the equation obtained by directly integrating the DDE is divided into two parts. For the part with the time delay term, the updated numerical integration formula derived in the past literature is used for approximation. Moreover, the precise integration (PI) algorithm is utilized to calculate the matrix exponentials efficiently and accurately. At last, the transition matrix is established to determine SLDs. To comprehensively appraise the performance of the proposed approach, comparisons between the proposed approach and other prediction approaches are carried out. It includes the analysis of the convergence rate, SLDs obtained by various prediction approaches in different milling conditions, analysis of time cost, the sum of absolute error (SAE), and the arithmetic mean of relative error (AMRE). The above indicators are introduced in the study to estimate the prediction accuracy of the various approaches quantitatively. The results show that the proposed approach not only has high calculation efficiency but also has high prediction accuracy. It is very suitable to carry out the stability prediction in all kinds of milling conditions.