# Stability Analysis of Milling Process with Multiple Delays
## 3 AVSNIM Considering the Helix Angle
### Algorithm Description  
For the milling process with multiple delays, the cutters may experience several free and forced vibrations in one spindle period. When the hysteresis effect caused by the helix angle is considered, the free vibration and forced vibration intervals will vary with the axial depth of the cut. In order to achieve accurate discretization, the first thing is to determine the free vibration and forced vibration interval of the cutter in one spindle period \(T\). Let \(\phi_{j}\) be the pitch angle between the \((j-1)\)-th and the \(j\)-th tooth, \(w\) be the axial depth of cut, \(\varphi_{st}\) and \(\varphi_{ex}\) be the tool start angle and exit angle, respectively. The angle swept by each tooth from cutting into the workpiece to cutting out of the workpiece can be expressed as  
\[\phi_{s}(w)=\varphi_{ex}-\varphi_{st}+\phi_{lag}(w) \tag{7}\]  
Figure 1: Schematic of two-degree of freedom (DOF) milling model with variable pitch milling cutter, (**a**) schematic of milling model; (**b**) z-direction view; (**c**) distribution of the cutter teeth; (**d**) the lag angle and tooth sweep angle. Then, the free vibration angle interval before the \(j\)-th tooth cutting into the workpiece \(\phi_{fr,j}\) can be expressed as  
\[\phi_{fr,j}(w)=\phi_{j}-\phi_{s}(w) \tag{8}\]  
In general, the distribution of free vibration and forced vibration in one spindle period can be divided into two categories: one where, at most, one tooth is in cutting during one spindle period; the other where more than one tooth is in cutting simultaneously at a certain time in one spindle period. If \(\phi_{fr,j}(w)>0\) holds for all \(j\) (\(j=1,2,\cdots,N\)), it corresponds to the first case. There are \(N\) free vibration intervals and \(N\) forced vibration intervals distributed alternately in one spindle period and each forced vibration angle interval \(\phi_{fo,j}(w)\) equals to \(\phi_{s}(w)\), as shown in Figure 2a. Else if \(\phi_{fr,j}(w)\leq 0\), it means that the \(j\)-th tooth has already cut into the workpiece before the \((j-1)\)-th tooth cuts out of the workpiece, which belongs to the second case. Under these circumstances, the continuous forced vibration angle intervals are combined as one forced vibration angle interval, as shown in Figure 2b,c. The length of this continuous forced vibration angle interval \(\phi_{foc}(w)\) can be calculated as  
\[\phi_{foc}(w)=\phi_{fo,j-1}(w)+\phi_{j} \tag{9}\]  
According to the above analysis, one spindle period is finally divided into \(N_{d}\) (\(N_{d}\leq N\)) free vibration and \(N_{d}\) forced vibration intervals that are alternately distributed. For the convenience of description, we mark one pair of adjacent free vibration and forced vibration intervals as one segment, hence one whole spindle period is divided into \(N_{d}\) segments, as described in Figure 2c. Once the distribution of free vibration and forced vibration intervals of the cutter is obtained, the spindle period can be discretized according to the following rules. In the free vibration interval, the vibration displacement of the cutter can be solved analytically, so there is no need to discretize. For the forced vibration interval, the following strategies are adopted for discretization. Firstly, the sum of all forced vibration angle intervals of the cutter is calculated as  
\[\phi_{fo}(w)=2\pi-\sum_{i=1}^{N_{d}}\phi_{fr,i}(w) \tag{10}\]  
Figure 2: The distribution of free vibration and forced vibration angle interval in one spindle period, (**a**) at most one tooth is in cutting; (**b**) more than one tooth is in cutting simultaneously; (**c**) combine the continuous forced vibration angle interval. Then, \(\phi_{fo}(w)\) is discretized into \(m_{p}\) intervals; each interval has a size of \(\Delta\phi_{p}(w)\), and  
\[\Delta\phi_{p}(w)=\frac{\phi_{fo}(w)}{m_{p}} \tag{11}\]  
where \(m_{p}\) is the pre-set discretization parameter.