## 6 Comparison Results and Discussions
### Contact Ratio  
Considering all the effective strain samples under the test conditions, the average contact ratios of the gear pairs in one mesh cycle for every working state are solved, and the average contact ratios under various working conditions of the HCR internal spur gear drive and involute internal spur gear drive are obtained through surface fitting, as shown in Figure 23a,b.  
It is clear from the experimental results that while the rotational speed has little effect on contact ratio, the input torque has a decisive influence on it. The percentage increases in the average contact ratio of the HCR gear pair compared to the involute gear pair under rotational speeds of 360 r/min and continuously varied increasing input torque are 97.14%, 108.33%, 128.17%, 151.13%, 159.57%, 166.95%, 162.4%, 174.71%, and 186.3%, as shown in Table 5. It can be seen that the contact ratios of both the involute internal spur gear pair and the HCR internal spur gear pair increase with input torque due to the large elastic deformation of the gear teeth. Moreover, the increase in contact ratio of the HCR gear pair is significantly faster than that of the involute pair. However, the actual measured contact ratio for the HCR gear pair is much smaller than the design value under light loading conditions, which may be caused by geometrical errors such as profile and pitch error. In the case of heavy loading, the deformation of gear teeth exceeds the error amount and the design tooth clearance, which leads to a significant improvement in the contact ratio. The comparison has shown that the contact ratio of the HCR gear pair exceeds 7 at input torques above 1000 Nm, but this reaches no more than 3 for the involute gear pair under the same conditions. This gap is essentially caused by differences in the tooth profile, including root bending stress under different rotational speeds and input torque conditions, as shown in the Figure 24. The tooth root stress of the involute standard gear increases with the increase in torque, as shown in Figure 24b. When the input torque is lower than 800 nm, the tooth root stress of the HCR gear increases with the increase in torque. In addition, when the input torque is higher than 800 Nm, the contact ratio increases with the increase of torque, and more teeth participate in meshing, which reduces the tooth root stress of a single tooth, as shown in Figure 24a.  
Figure 22: Contact stress curve of tooth surface in the direction of tooth width.  
Figure 23: Average contact ratio under different rotational speeds and input torque conditions: (a) HCR internal spur gear drive; (b) involute internal spur gear drive.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Input Torque (Nm)**} & \multicolumn{2}{c}{**Contact Ratio**} & \multirow{2}{*}{**\% Increase**} \\ \cline{2-3} \cline{5-5}  & **Involute** & **HCR** & \\ \hline
100 & 1.75 & 3.45 & 97.14 \\
200 & 2.04 & 4.25 & 108.33 \\
300 & 2.13 & 4.86 & 128.17 \\
400 & 2.21 & 5.55 & 151.13 \\
500 & 2.3 & 5.97 & 159.57 \\
600 & 2.36 & 6.3 & 166.95 \\
800 & 2.5 & 6.56 & 162.4 \\
1000 & 2.61 & 7.17 & 174.71 \\
1200 & 2.7 & 7.73 & 186.3 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Percentage increase in contact ratio.