# Stability Analysis of Milling Process with Multiple Delays
## 1 Introduction  
Chatter is a self-excited vibration induced in the machining process which reduces the machining efficiency and surface quality, accelerates the tool wear and shortens the tool durability. There have been a lot of research on the mechanism of chatter [1; 2; 3; 4; 5; 6], among which regenerative effect was usually considered as a main inducement. Since the regenerative chatter is caused by the phase differences between wavy surfaces left by the adjacent teeth, it can be suppressed by adjusting the pitch angle between the adjacent teeth. The use of variable pitch cutters to improve the stability of the milling process is built on this idea. Different from the uniform pitch cutter, when the variable pitch cutter is used, the dynamics model of cutting vibration changes from DDEs with single delay to DDEs with multiple delays. The stability analysis of the milling process with multiple delays has been studied through different methods. [PERSON] et al. [7] adopted the frequency domain method to analyze the milling stability of the variable pitch cutter and presented a method to select the optimal pitch angles. [PERSON] [8; 9] proposed an analytical design method for nonconstant pitch milling cutters to improve the stability of the milling process and verified it through cutting experiments. The results showed that chatter stability can be improved significantly even at slow cutting speeds by properly designing the pitch angles. The modified semi-discretization method (SDM) was used by [PERSON] et al. [10] to investigate the stability of variable pitch and variable helix milling tools. Considering the tool runout and thepitch angle, [PERSON] et al. [11] proposed a unified stability prediction method for a milling process with multiple delays based on the SDM. [PERSON] et al. proposed an improved full-discretization method (FDM) [12] and the variable-step numerical integration method [13] for milling chatter stability prediction with multiple delays. [PERSON] and [PERSON] [14] studied the stability boundary in a milling process with variable-pitch cutters based on the cluster treatment of characteristic roots method and presented an optimization procedure for the geometry design of variable pitch cutters. The enhanced multistage homotopy perturbation method was presented by [PERSON] et al. [15] to analyze the milling stability influenced by the variable cutting edge pitch, the variable helix, and the variable rake angle. The spectral element approach was first introduced by [PERSON] et al. to study the stability of delay systems with single delay [16]. Later, it was generalized to time-periodic delay equations with multiple delays [17]. [PERSON] et al. [18] generalized the spectral element method for time-periodic DDEs with multiple delays and distributed delay and given an explicit formula for the construction of the matrix approximation of the monodromy operator. An improved SDM was presented by [PERSON] et al. [19; 20] to predict the stability of milling process with multiple delays. On the basis of the multifrequency solution, [PERSON] et al. [21] presented a new method for the identification of the chatter stability lobes in milling with non-uniform pitch and variable helix tools. [PERSON] et al. [22] proposed an extended Adams-Moulton-based method for the stability prediction of milling processes with multiple delays. After the stability analysis is completed, the stability limits under different processing parameters are usually displayed in one diagram, which is known as the SLD. The SLD is a powerful tool, which can not only help to select the optimal process parameters to realize the chatter-free machining, but also assist the geometric design of the cutting tools. A variety of methods have been proposed to improve the accuracy or calculation efficiency of the SLD. The improved FDM was used in [12] to reduce the calculation time. However, similar to the SDM, it was also based on the equal-step discretization technique.