# Stability Analysis of Milling Process with Multiple Delays
## 4 AVSNIM with Spindle Speed-Dependent Discretization Algorithm
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Radial Immersion**} & **AVSNIM with Constant** & **AVSNIM with** \\  & **Discretization Parameter** & **Spindle-Speed-Dependent** \\  & **Discretization Parameters** \\ \hline
1.0 & 1857.21 & 857.57 \\
0.6 & 1784.73 & 1079.14 \\
0.1 & 1073.48 & 492.15 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calculation time required for the AVSNIM with constant discretization parameter and spindle-speed-dependent discretization parameters (unit: s). Figure 8: Numerical simulation of time step and discretization parameters, (**a**) time step of equal-step method; (**b**) time step of variable-step method; (**c**) discretization parameters obtained by the spindle-speed-dependent discretization algorithm. Another milling model to verify the proposed algorithm is from the literature [31]. In this model, a 12.75 mm diameter cutter with one flute is adopted, the helix angle of which is \(30^{\circ}\). The detailed values of the modal parameters and cutting force coefficients are as follows: the modal masses in \(x\) and \(y\) directions are \(m_{x}=m_{y}=0.23\,\mathrm{kg}\); the modal damping coefficients in \(x\) and \(y\) directions are \(c_{x}=c_{y}=15.73\,\mathrm{Ns}/\mathrm{m}\); the modal stiffness in \(x\) and \(y\) directions are \(k_{x}=k_{y}=8.4\times 10^{5}\,\mathrm{N}/\mathrm{m}\); the cutting force coefficients are \(K_{t}=5.36\times 10^{8}\,\mathrm{N}/\mathrm{m}^{2}\) and \(K_{n}=1.87\times 10^{8}\,\mathrm{N}/\mathrm{m}^{2}\). The simulation parameters are: down milling, \(\varepsilon=1\times e^{-3}\), \(\Delta m=2\), \(m_{p}=60\), the initial \(m_{d}=10\), the spindle speed \(\Omega\) ranges from 5000 to 30,000 rpm and the axial depth of cut \(w\) ranges from 0 to 25 mm. The simulation result is shown in Figure 10. The SLD in Figure 10 is constructed over a \(500\times 250\)-sized grid; the radial  
Figure 9: The SLD of the first milling model obtained by the proposed algorithm (**a**) radial immersion 1.0, \(m_{p}=200\); (**b**) radial immersion 0.6, \(m_{p}=200\); (**c**) radial immersion 0.1, \(m_{p}=160\). immersion is selected as 0.02. The reference stability limits demoed by the red line are obtained by the AVSNIM with a constant discretization parameter \(m=100\). As shown in Figure 10, the stability boundary is very complex, in which there are some islands. The stability limits obtained by the proposed algorithm are consistent with the reference stability limits and are the same as those obtained in the literature [31]. This shows that the algorithm proposed in this work is still effective when the stability boundary is very complex. It is worth noting that there is only a single time delay in this milling model, which is the same as the milling process with a uniform pitch milling cutter. This proves that the proposed algorithm is also suitable for the milling process with a uniform pitch milling cutter. Based on the above analysis, we can conclude that the proposed algorithm can not only maintain the calculation accuracy but also effectively reduce the calculation time, and hence can meet the needs in the engineering field well.