## 3 Results and Discussion
### Numerical Simulation Results  
Another modern approach is related to simulating the cutting process using different analysis software (e.g., ANSYS 19, Hob3D v.2022, AdvantEdge v.7.8), with results quite close to the experimental methods [17; 18; 19; 20; 23; 30]. In this paper, the SolidWorks 2019 software for geometric modeling and the ANSYS program for process simulation are used for the gear hobbing process. A hob was used with the module m = 3.5 mm; the number of origins was z\({}_{1}\) = 1; cutting direction--R; material HHS; and other geometric parameters according to Table 1, namely: II--DIN 3972; I--STAS 11482. The input gear data: the module m = 3.5 mm; the outside diameter d\({}_{\text{g}}\) = 112 mm; the number of teeth z\({}_{2}\) = 30; and the pressure angle \(\alpha\) = 20\({}^{\circ}\). In order to obtain this gear teeth number after processing, a correlation is required between the rotation speeds of the two elements: gear and hob (Equation (1)):  
\[\frac{\text{n}_{\text{g}}}{\text{n}_{\text{h}}}=\frac{\text{z}_{1}}{\text{z}_ {2}} \tag{1}\]  
where: n\({}_{\text{g}}\)--rotational speed of the gear (rpm); n\({}_{\text{h}}\)--rotational speed of the hob (rpm); z\({}_{1}\)--number of origins of the hob; and z\({}_{2}\)--number of the gear teeth. To analyze the milling process, the cutting speed v\({}_{\text{c}}\) is determined with the Equation (2) [37]:  
\[\text{v}_{\text{C}}=\frac{312}{\text{t}^{0.33}\cdot\text{s}^{0.5}}\ (\text{m}/ \text{min}), \tag{2}\]  
where s = axial feed (mm), and t = hob edge endurance (min)  
There are many studies for the calculation of the wear and edge endurance of the hob, depending on the process parameters [2; 6; 13; 14; 15]. For this study, the s and t values were adopted (s = 4 mm and t = 240 min [37]), and the speed v\({}_{\text{c}}\) = 25.56 m/min is obtained. The cutting speed will determine the rotation speed of the hob n\({}_{\text{h}}\) = 95 [rpm] (Equation (3)):  
\[\text{n}_{\text{h}}=\frac{1000\cdot\text{v}_{\text{C}}}{\pi\cdot\text{d}_{ \text{h}}}\ (\text{rpm}), \tag{3}\]  
where d\({}_{\text{h}}\) (m), v\({}_{\text{c}}\) (m/min). From Equation (1), results in the rotation speed of the gear n\({}_{\text{g}}\) = 3.16 (rpm). In the gear hobbing process, the depth of cut a\({}_{\text{p}}\) = 0.15 mm and the feed rate v\({}_{\text{f}}\) = 4 mm/min were used. Referring to the stiffness behavior, the hob was considered rigid, and the gear was flexible. The kinematic rotation joints, gear-base, and hob-base were considered rigid. The geometric model of the hob, made with SolidWorks, was imported into ANSYS software. The global reference coordinate system has been attached to the center of mass of the workpiece (Figure 5). Figure 5: Geometric model for the assembly. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Time** & **Minimum** & **Maximum** & **Average** \\
**t (s)** & \(\varepsilon_{\mathbf{1\ min}}\) (mm/mm) & \(\varepsilon_{\mathbf{1\ max}}\) (mm/mm) & \(\widetilde{\varepsilon}_{\mathbf{1}}\) (mm/mm) \\ \hline \(1.1755\times 10^{-38}\) & - & - & - \\ \(9.9221\times 10^{-4}\) & \(-0.12521\) & \(4.2826\times 10^{-8}\) & \(-1.7264\times 10^{-5}\) \\ \(4.2896\times 10^{-3}\) & \(-0.96972\) & \(1.1543\times 10^{-4}\) & \(-2.2078\times 10^{-4}\) \\ \(5.0076\times 10^{-3}\) & \(-1.2395\) & \(3.3848\times 10^{-5}\) & \(-2.6823\times 10^{-4}\) \\ \(5.0749\times 10^{-3}\) & \(-1.2643\) & \(2.5838\times 10^{-5}\) & \(-2.7514\times 10^{-4}\) \\ \(5.6075\times 10^{-3}\) & \(-1.4533\) & \(1.5112\times 10^{-4}\) & \(-3.0484\times 10^{-4}\) \\ \(6.1808\times 10^{-3}\) & \(-1.5139\) & \(6.9237\times 10^{-5}\) & \(-3.3336\times 10^{-4}\) \\ \(6.2767\times 10^{-3}\) & \(-1.5149\) & \(1.1148\times 10^{-4}\) & \(-3.3702\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Minimum principal elastic strain.