Two updated methods based on Simpson formula for chatter stability prediction in milling

Two updated methods based on the Simpson formula, i.e., the updated Simpson-based method (USBM) and the updated Adams-Simpson-based method (UASM), are presented for milling stability analysis. Firstly, the delay differential equation (DDE) is employed to describe the milling process mathematically. Then, the tooth passing period is divided into two subintervals, i.e., the free and forced vibration intervals. Only the forced vibration interval is divided into many equal small-time intervals. Subsequently, the DDE in the state space is solved based on direct integration. By combining the two-step Simpson formula and the semi-discretization method, the state transition matrix of the milling system is constructed using the USBM and UASM. The comparisons of convergence rates are conducted to validate the accuracy of the proposed method. The results show that the proposed USBM and UASM converge faster than the benchmark methods. The stability lobe diagrams and the arithmetic mean of relative error (AMRE) for the one degree of freedom (one-DOF) and two degrees of freedom (two-DOF) milling systems are also obtained by different methods for further accuracy evaluation. Meanwhile, the computational time analysis is also carried out. It is revealed that the proposed USBM and UASM are accurate and efficient methods for milling stability analysis. Besides, the proposed methods can accurately and efficiently predict the stability of milling with both large and low radial immersion conditions. Finally, the effectiveness of the proposed methods is verified by experiments.