## 3 Stability Analysis and Discussion
### 2-DOF Model of VHCVSS
#### 3.2.2 Simulation and Analysis  
Here, multi-parameters calculation for the milling stability of Processes I, II, III, and IV, is carried out based on the two DOF model in Section 3.2.1 and the method in Section 2. Similar to the situation in Section 3.1.2, the other parameters in the subsequent calculation remain unchanged except for the core parameters RVF, RVA, \(\boldsymbol{\psi}\), and \(\boldsymbol{\beta}\). The mainly parameters are selected from the representative papers [54], and are: down milling, half immersion, \(q\ =\ 1\), \(K_{r}\) = \(2.56\times 10^{8}\)N/m, \(K_{t}\) = \(6.79\times 10^{8}\)N/m, \(N\) = \(4\), \(f_{z}\) = \(0.1\) mm, \(\omega_{x}\) = \(563.6\) Hz, \(\omega_{y}\) = \(516.21\) Hz, \(\zeta_{x}\) = \(0.0558\), \(\zeta_{y}\) = \(0.025\), \(m_{x}\) = \(1.4986\) kg, and \(m_{y}\) = \(1.199\) kg. Figure 8 shows some SLDs corresponding to the 2-DOF systems for Processes I, II, III, and IV. The black thin lines are the stability boundaries corresponding to Process I for reference, whose core parameters are: RVF = 1, RVA = 0, \(\boldsymbol{\psi}\) = [90, 90, 90]\({}^{\circ}\) and \(\boldsymbol{\beta}\) = [30, 30, 30, 30]\({}^{\circ}\). The cyan thin lines are related to the stability boundaries of Process II with RVF = 0.5, RVA = 0.1 and the same \(\boldsymbol{\psi}\) and \(\boldsymbol{\beta}\) in Process I. The boundaries plotted by red dotted lines in Figure 8a-d are associated with Process III, where four VHCs with the different \(\boldsymbol{\beta}\) (\(\boldsymbol{\beta}\) = [30, 30, 30, 30]\({}^{\circ}\), \(\boldsymbol{\beta}\) = [32, 30, 32, 30]\({}^{\circ}\), \(\boldsymbol{\beta}\) = [34, 30, 34, 30]\({}^{\circ}\), and \(\boldsymbol{\beta}\) = [36, 30, 36, 30]\({}^{\circ}\)) but the same \(\boldsymbol{\psi}\) (\(\boldsymbol{\psi}\) = [75, 85, 95, 105]\({}^{\circ}\)) are utilized. The green thick lines exhibit the SLDs related to Process IV for the relevant parameters combinations of Process II and Process III. Comparing with the curves in the Figure 8, it can be found that the combined  
process (i.e., Process IV) still shows a better cutting stability suppression advantage than that of Processes I, II, and III. For different combinations of variable helix angles \(\beta\), the values of \(\chi\) associated with Process IV are the biggest (see Figure 9). For Process IV, the values of \(\chi\) will increase for a bigger helix angle increment \(\Delta\beta\) (defined by \(\beta_{1}-\beta_{2}\)). This shows that the increasing of \(\Delta\beta\) plays a positive role in enhancing the stability area for VHCVSS. Thus, a bigger \(\Delta\beta\) should be suggested in practice. Figure 8: Comparisons of the SLDs associated with Processes I, II, III, and IV for the 2-DOF system. Note that the SLDs for Processes III and IV are plotted under the conditions of the same pitch angles (\(\psi\) = [70, 110, 70, 110]\({}^{\circ}\)) but different helix angles: (**a**) \(\beta\) = [30, 30, 30, 30]\({}^{\circ}\); (**b**) \(\beta\) = [32, 30, 32, 30]\({}^{\circ}\); (**c**) \(\beta\) = [34, 30, 34, 30]\({}^{\circ}\); and (**d**) \(\beta\) = [36, 30, 36, 30]\({}^{\circ}\). Figure 9: Comparison of the values of \(\chi\) corresponding to the SLDs in Figure 8.  
Figure 10 shows the contribution factor diagram of Processes II and III to Process IV under different \(\beta\). Obviously, the corresponding values of \(\kappa_{III}\) are all greater than 61%, Process III still has a significant advantage in the contribution of the combination effect, compared to Process II. However, it should be noted that the contribution factor changed most significantly when the variable helix angles change from \(\beta\) = [30; 30; 30; 30]\({}^{\circ}\) to \(\beta\) = [32; 30; 32; 30]\({}^{\circ}\). This is mainly due to the fact that the former actually constructs a VPC milling process that is consistent with the model described in [49], while the latter is a VHC one.