# Stability Analysis of Milling Process with Multiple Delays
## 4 AVSNIM with Spindle Speed-Dependent Discretization Algorithm
On the contrary, if \(\Delta m\) is too large, a discretization parameter larger than the required value may be obtained, thereby increasing the final calculation load. The values of these three parameters can be selected according to the actual engineering requirements. Numerical simulation is performed to verify the above algorithm; two milling models are studied in this section. The first milling model is the same as used in Section 3.2. The simulation parameters are: down milling, \(\varepsilon=1\times e^{-3}\), \(\Delta m=2\), \(m_{p}=200\), the initial \(m_{d}=40\), the spindle speed \(\Omega\) ranges from 2500 to 12,500 rpm and the axial depth of cut \(w\) ranges from 0 to 15 mm. Firstly, the time step under constant discretization parameter and the discretization parameters based on the spindle-speed-dependent algorithm are simulated. The results are shown in Figure 8. Figure 8a,b describes the time step under the equal-step method and the variable-step method, respectively. It can be found that the time step in the equal-step algorithm is only related to the spindle speed, while in the variable-step method, due to the effect of the helix angle, the step size depends on both the spindle speed and the axial depth of cut. Figure 8c shows the discretization parameters obtained through the spindle-speed-dependent algorithm. It can be seen that under the given relative error \(\varepsilon\), the discretization parameters obtained by the spindle-speed-dependent algorithm are smaller than the reference discretization parameter, which means that the time required to obtain the SLD will be effectively reduced. At the same time, we can see that the value of the discretization parameter is higher in the low-spindle-speed region than that in the high-spindle-speed region in general, which proves the previous conclusion that, in the low-speed region, in order to maintain the accuracy of the numerical calculation, larger discretization parameters are required. However, the discretization parameters do not monotonically decrease with the increase in spindle speed, but there are some fluctuations. There may be two incentives for these fluctuations. One is that they may be caused by numerical calculation, because the bisection method used in this article may randomly converge to the upper or lower boundary of the relative error limit, which usually causes some numerical fluctuations. The second is that they may be caused by the different sensitivity of the eigenvalues of Floquet transition matrix to the processing parameters (mainly the spindle speed here). Subsequently, the SLD is provided to verify the effectiveness of the proposed spindle-speed-dependent discretization algorithm. Figure 9 shows the SLD obtained through the algorithm proposed above; the SLD is constructed over a 200 \(\times\) 150-sized grid; the radial immersion in Figure 9a-c is selected as 1.0, 0.6 and 0.1, respectively. The reference stability limits demoed by the red line are obtained by the AVSNIM with a large reference discretization parameter. From Figure 9a-c, we can see that the stability limits obtained by the proposed algorithm are very close to the reference stability limits. This shows that the proposed algorithm is reliable in terms of calculation accuracy. Meanwhile, since the radial immersion in Figure 9 ranges from 1.0 to 0.1, this indicates that the proposed algorithm is applicable to all kinds of radial immersion. Finally, the calculation time required for the AVSNIM with spindle speed-dependent discretization parameters is listed in Table 2 to compare with the AVSNIM with constant discretization parameter. As outlined in Table 2, the AVSNIM with spindle-speed-dependent discretization parameters can significantly reduce the calculation time.