## 6 Comparison Results and Discussions  
### Bending Stress and Contact Stress  
The maximum bending and contact stresses are taken from the middle position along the tooth width without considering stress concentration on the edge of the end surface. The maximum principal stress during a mesh cycle is used to evaluate the bending stress. Meanwhile, the maximum CPRESS is extracted to reveal the contact stress. As presented in Figures 19 and 20, the number of gear teeth engaged in a mesh cycle of the HCR internal gear drive is significantly more than that of the involute drive. However, the contact bandwidth on the HCR pinion flank is smaller than that of the involute one. Furthermore, the maximum bending stress position is not consistent with the attachment position of the strain gauge in the experiment, which is closer to the tooth root under the strain gauge.  
Figure 18: Schematic of the FEMs: (**a**) HCR internal spur gear pair, and (**b**) involute internal spur gear pair.  
Figure 21 shows the comparison of maximum bending stress and contact stress of the FEA results between the involute internal spur gear pair and the HCR internal spur gear pair, as well as the comparison between the maximum average experimental values of bending stress and FEA results. It can be seen that the maximum bending stress of the FEA results is higher than that of FEA.  
Figure 19: Bending and contact stress results of HCR internal spur gear pair at 540 r/min and 600 Nm.  
Figure 20: Bending and contact stress results of involute internal spur gear pair at 540 r/min and 600 Nm.  
the HCR pinion is reduced by about 35.99% (as shown in Table 4) because of the high contact ratio compared with the involute pinion. The maximum contact stress of the tooth flanks is almost equal. The main reason for this is that the contact stress is decided by both the contact ratio and induced curvature. The HCR internal gear drive designed in Section 2 has a higher induced curvature than the involute one. Meanwhile, the contact is convex-to-convex for the HCR internal gear pair and convex-to-concave for the involute internal gear pair. Therefore, the combination of the above factors results in the small contact area and higher contact stress of the HCR internal spur gear pair.  
The differences between the experimental results and FEA results are within the acceptable limits. The experimental bending stress and FEA results on the measurement point of the involute internal spur gear pair have very good consistency. However, the FEA result (249.8 Mpa) is higher compared to the experimental bending stress (208 Mpa) on the measurement point of the HCR internal spur gear pair, with a maximum percentage difference of 16.7%, which may be due to the error in the attachment position of strain gauges and the frictional effects of the gear drive in real experimental conditions. Extracting the tooth surface contact stress in the tooth-width direction of the gear, as shown in Figure 22, shows that the contact stress distribution is not completely uniform.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Comparison Items** & **Involute** & **HCR** & **\% Decrease** \\ \hline maximum bending stress & 405.4 & 259.5 & 35.99\% \\ (FEM) & 453.9 & 452 & 0.42\% \\ bending stress of measurement point & 352 & 249.8 & 29.03\% \\ (FEM) & 353.6 & 208 & 41.18\% \\ bending stress of measurement point & 353.6 & 208 & 41.18\% \\ (TEST) & \(-\)0.45\% & 16.73\% & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Percentage decrease of stress between involute and HCR internal gear pair.  
Figure 21: Comparison of bending and contact stress between involute and HCR internal gear pairs.