# Stability Analysis of Milling Process with Multiple Delays
## 3 AVSNIM Considering the Helix Angle
### Algorithm Verification and Results Discussion  
For convenience of comparison and verification, the milling model used in [7; 13; 22] is employed for numerical simulation in this section. A 19.05 mm diameter variable pitch cutter with four flutes is adopted. The helix angle is \(30^{\circ}\) and the pitch angles are \(70^{\circ}\)-\(110^{\circ}\)--\(70^{\circ}\)-\(110^{\circ}\). The detailed values of the modal parameters and cutting force coefficients are as follows: the modal masses in \(x\) and \(y\) directions are \(m_{tx}=1.4986\) kg and \(m_{ty}=1.199\) kg, respectively; the natural angular frequencies in \(x\) and \(y\) directions are \(\omega_{nx}=563.55\times 2\pi\) rad/s and \(\omega_{ny}=516.27\times 2\pi\) rad/s, respectively; the damping ratios in \(x\) and \(y\) directions are \(\zeta_{x}=0.0558\) and \(\zeta_{y}=0.025\), respectively; the cutting force coefficients are \(K_{t}=6.79\times 10^{8}\) N/m\({}^{2}\) and \(K_{n}=2.56\times 10^{8}\) N/m\({}^{2}\). The SLDs constructed with the AVSNIM proposed in Section 3.1 are provided to compare with those obtained with the 1 st-SDM presented in [29] and equal-step numerical integration method (ESNIM) presented in [30]. The SLD is constructed over a \(200\times 100\)-sized grid, the machining condition is down-milling and the simulation parameters are set as follows: the spindle speed \(\Omega\) ranges from 2500 to 12,500 rpm and the axial depth of cut \(w\) ranges from 0 to 10 mm. The simulation is performed in Matlab(r) (The MathWorks Inc., Natick, MA, USA. Version: R2017b) on a laptop computer (Manufacturer: Hewlett-Packard, Made in China. Proessor: Intel(r) CoreTM i7-6700 HQ CPU, @2.60 GHz, Installed Memory (RAM): 8.00 GB, System type: Windows 10 home 64-bit Operating system.). The results are shown in Figures 4-6, in the SLD, the reference stability limits demoed by the red line are obtained by the 1 st-SDM with the discretization number \(m_{p}=600\). Figure 5: SLD of the two-DOF milling model with radial immersion 0.2 and \(m_{p}=60\), (**a**) SLD obtained by the 1 st-SDM; (**b**) SLD obtained by the ESNIM; (**c**) SLD obtained by the AVSNIM. The radial immersion in Figures 4-6 is selected as 0.3, 0.2 and 0.1, respectively. It is found that, compared with the other two methods, the differences between the simulation results of the AVSNIM and the reference values are smaller under the same computational parameters. Consequently, the computational accuracy of the proposed method is higher than those of the other two methods. For quantitative comparison, the mean relative error (MRE) of the 1 st-SDM, ESNIM and AVSNIM is provided. The MRE is defined as  
\[MRE=\frac{1}{n_{s}}\sum_{i=1}^{n_{s}}\left|\frac{w_{a,i}-w_{ref,i}}{w_{ref,i}}\right| \tag{24}\]  
where \(n_{s}\) is the total discrete points of the spindle speed in the SLD, \(|*|\) means the absolute value of \({}^{***}\), \(w_{a,i}\) is the approximate value of the stability limit obtained by the numerical algorithm at the \(i\)-th discrete spindle speed, \(w_{ref,i}\) is the reference value of the stability limit at the \(i\)-th discrete spindle speed. Figure 7 displays the MRE of stability limits obtained by 1 st-SDM, ESNIM and AVSNIM under the same simulation parameters that were used in Figures 4-6. It can be seen from Figure 7 that the  
Figure 6: SLD of the two-DOF milling model with radial immersion 0.1 and \(m_{p}=50\), (**a**) SLD obtained by the 1 st-SDM; (**b**) SLD obtained by the ESNIM; (**c**) SLD obtained by the AVSNIM. stability limit obtained by the AVSNIM has the minimum MRE under each radial immersion. At the same time, it can be seen that the smaller the radial immersion is, the more obvious the advantage of the variable step size algorithm is. This shows that the proposed algorithm is particularly suitable for the stability analysis of milling process under the small radial depth of cut.