An updated Simpson-Based Method for Chatter Stability Prediction in Milling

Abstract

An updated Simpson-based method (USBM) is 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 method and the semi-discretization method, the state transition matrix of the milling system is constructed. The comparison of convergence rate is conducted to validate the accuracy of the proposed method. The results show that the proposed method converges faster than the benchmark methods. The stability lobe diagrams 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 evaluation. Meanwhile, the computation time analysis is also carried out. It is revealed that the proposed USBM has advantages in both accuracy and efficiency. Besides, the proposed method can accurately and efficiently predict the stability of milling with both large and low immersion conditions.