## 5 Numerical Examples
### Influence of Design Parameters on Curvature Limit Lines  
Previous studies indicate that the meshing performance in face worm gear drives can be controlled by adjusting the grinding wheel deflection angle, semi-cone angle, and worm axial installation distance. Therefore, this section investigates the influence of these parameters on curvature interference limit lines through three numerical cases. The selected cases are listed in Table 5, and their corresponding curvature interference limit lines are illustrated in Figures 12-14.  
Figure 11: Curvature interference limit lines on cross-sections of \(\Sigma_{1}^{(S)}\) and \(\Sigma_{2}^{(S)}\): (**a**) \(i\)-flank; (**b**) \(e\)-flank.  
Figure 10: Curvature interference limit lines on the face worm gear tooth surface \(\Sigma_{2}^{(S)}\).  
Figure 9: Curvature interference limit lines on the worm helical surface \(\Sigma_{1}^{(S)}\).  
The influence of the grinding wheel semi-cone angle on the curvature interference limit line is illustrated in Figure 12. The effect of the grinding wheel secondary inclination angle on the curvature interference limit line is shown in Figure 13. The analysis of Figures 12 and 13 reveals that curvature interference limit lines exhibit minimal sensitivity to the grinding wheel deflection angle and semi-cone angle. Alterations to the grinding wheel parameters displace the curvature interference limit lines toward the tooth surfaces, yet these limit lines remain sufficiently distant from the tooth surface. This prevents the occurrence of curvature interference.  
However, the curvature interference limit lines exhibit significant sensitivity to the worm axial installation distance, as shown in Figure 14. Reducing the worm axial installation distance can cause the limit lines to progressively migrate toward the tooth  
Figure 12: Influence of grinding wheel half taper angle \(\delta_{\text{g}}^{(S)}\), (\(S=1,2\)) on curvature interference limit lines: (**a**) \(i\)-flank; (**b**) \(e\)-flank.  
Figure 13: Influence of grinding wheel secondary tilt angle \(\varepsilon_{S}\), (\(S=1,2\)) on curvature interference limit lines: (**a**) \(i\)-flank; (**b**) \(e\)-flank.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Nomenclature (Unit)** & **Symbols/Formulas** & **Results** \\ \hline Radius of grinding wheel (\({}^{\circ}\)) & \(r_{\text{g}}\) & 150 \\ Half taper angle of grinding wheel for \(i\)-flank (\({}^{\circ}\)) & \(\delta_{\text{g}}^{(1)}\) & 65 \\ Tilt angle of grinding wheel for \(i\)-flank (\({}^{\circ}\)) & \(\varepsilon_{1}\) & 5 \\ Half taper angle of grinding wheel for \(e\)-flank (\({}^{\circ}\)) & \(\delta_{\text{g}}^{(2)}\) & 60 \\ Tilt angle of grinding wheel for \(e\)-flank (\({}^{\circ}\)) & \(\varepsilon_{2}\) & \(-5\) \\ Operating centre distance (mm) & \(a_{\text{d}}=r_{\text{f1}}+r_{\text{g}}/\cos\varepsilon_{\text{s}}(S=1,2)\) & 165.57 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Operating parameters for machining worm.  
surfaces. An excessive reduction in the axial installation distance may allow the limit lines to penetrate the tooth surfaces. Consequently, this may lead to curvature interference.  
In conclusion, this study demonstrates that the geometric parameters of the grinding wheel utilize negligible influence on the curvature interference limit lines within appropriate process parameter ranges. However, the axial installation distance requires stringent control to prevent interference.