## 3 Stability Analysis and Discussion
### Single DOF Model of VHCVSS
#### 3.1.2 Simulation and Analysis  
Based on the 1-DOF model in Section 3.1.1 and the method in Section 2, the computer programs are written in MATLAB(R) 2009a (MathWorks, Natick, USA) and implemented on a personal computer (Lenovo, Beijing, China). The multi-parameters calculation corresponding to milling stability is carried out for above defined processes. Considering that the difference of each process is mainly reflected in the change of the variable speed parameters and the geometric parameters of variable helix cutter, therefore except for the core parameters, i.e., RVF and RVA parameters associated with VSS and the pitch at the  
Figure 3: Schematic drawing of the relationship of the segmentation of the axial depth of cut and time delay. end of the tool (\(\psi\)) and helix angle (\(\beta\)) parameters associated with VHC, other parameters are the same as that in [7] and remain unchanged in the subsequent calculation. For the convenience of analysis, two related variations are defined to analyze the effect of the cutting parameters on chatter quantitatively firstly. One is the stable area ratio [49], which can be expressed by  
\[\chi\ =\ \frac{A_{0}}{A} \tag{26}\]  
and is applied to characterize the ratio between the stable area \(A_{0}\) and the total area \(A\) for a certain SLD. Another is the contribution factor \(\kappa\) that is to measure the contribution of a certain process to Process IV. For Processes II and III, their contributions to process IV can be respectively expressed as \(\kappa_{II}\)and \(\kappa_{III}\), and be defined by  
\[\kappa_{II}\ =\ \frac{\Delta\chi_{III}}{\Delta\chi_{II}+\Delta\chi_{III}} \tag{27}\] \[\kappa_{III}\ =\ \frac{\Delta\chi_{III}}{\Delta\chi_{II}+\Delta\chi_{III}}\]  
with \(\Delta\chi_{II}\ =\ \chi_{IV}-\chi_{II}\) and \(\Delta\chi_{III}\ =\ \chi_{IV}-\chi_{III}\). Figure 4 illustrates a number of SLDs corresponding to the 1-DOF systems for Processes I, II, III, and IV. Here, two different helical endmills with the number of teeth \(N=4\) and the diameter \(D\) = 20 mm are considered, i.e., Cutter-1: a conventional one with the pitches at the end of the tool \(\psi\) = [90, 90, 90, 90]\({}^{\circ}\) and the helix angles \(\beta\) = [30, 30, 30, 30]\({}^{\circ}\), Cutter-2: a VHC with \(\psi\) = [75, 85, 95, 105]\({}^{\circ}\) and \(\beta\) = [35, 30, 35, 30]\({}^{\circ}\). The material of workpieces is Al7050. The other calculation parameters are: up milling, the radial immersion ratio \(\alpha_{\rm D}\) = 0.5, the feed per tooth \(f_{z}\) = 0.1 mm, the cutting-force coefficients \(K_{t}\ =\ 1.07\times 10^{8}\)N/m\({}^{1+q}\) and \(K_{r}\ =\ 0.4\times 10^{8}\)N/m\({}^{1+q}\), the nonlinear parameter in cutting force \(q\) = 0.75, the modal mass \(m_{x}\)= 3.1663 kg, the natural frequency \(\omega_{x}\) = 400 Hz and the damping ratio \(\xi_{x}\ =\ 0.02\). The black thin lines in Figure 4 are associated with Process I, they are the reference stability boundaries of the traditional milling with Cutter-1; the red dotted lines correspond to process III with Cutter-2; the cyan thin lines in the subgraphs (a)-(d) of Figure 4 respectively are the stability boundary of Process II with different RVF (RVF = 0.05, 0.1, 0.2, and 0.5) but the same RVA (RVA = 0.1) when Cutter-1 is used; the SLDs shown by the green thick lines are related to Process IV, who uses Cutter-2 on the basis of above Processes II. Obviously, it can be found from Figure 4 that compared to Process I, Processes II and III do lead to better milling stability gains, especially for low-speed region. Also, their corresponding values of \(\chi\) are always larger (see Figure 5). The phenomena above imply the active influence of VHC or VSS on milling stability by the disruption of regeneration chatter, and they are also coincident with that in [35, 36, 37, 38, 40, 41, 42]. Then, Process IV is focused on. As seen from the green lines in Figure 4, Process IV almost always leads to the highest axial depth of cut for every adopted RVF.