A hybrid multi-step method based on 1/3 and 3/8 Simpson formulas for milling stability prediction

Chatter is an unfavorable phenomenon frequently experienced in the milling process, which severely reduces machining productivity and surface quality. The prediction of milling stability is essential for realizing chatter-free machining. This paper presents a hybrid multi-step method to predict the milling stability. Firstly, the milling dynamic system including regenerative effect is described as time-periodic delay differential equations (DDEs) that are then re-expressed as state-space equations. Secondly, the time period is divided into free vibration duration and forced vibration duration. With the forced vibration duration being divided into small time intervals, the integral response is approximated by hybrid multi-step Simpson formulas, where the fourth state item is approximated by the 3/8 Simpson formula and the remaining state items are approximated by the 1/3 Simpson formula. Finally, the Floquet transition matrix over the time period is constructed by a linear discrete map, and the milling stability is predicted via the Floquet theory. By using the benchmark examples from literatures, the convergence rates and stability boundaries of the proposed method are compared with typical methods, such as the updated numerical integration method (UNIM) and the fourth-order full-discretization method (4th FDM). Numerical results show that the proposed method can achieve high stability prediction accuracy with high efficiency. Furthermore, a series of milling experiments are carried out to verify the reliability and feasibility of the proposed method. According to the waveform and spectrum analysis of lots of synchronously measured sound and displacement signals, it can be found that the predicted stability boundaries are in good agreement with the experimental results.