## 4 Numerical Example and Verification
### Verification of the Proposed Model  
The calculated results including contact stress, LTE, and GR are presented and compared with those obtained by the current model based on the nonlinear Hertz contact stiffness method [5; 15], which was developed specifically for loaded contact analysis of cycloid drives assumed as a two-dimensional plane problem. In the absence of experimental validation, this comparison serves to verify the proposed model. Without loss of generality, three different cases are considered here. One is the standard profile without tooth profile modification. The other two cases are NNM and PPM with the same radial clearances. The corresponding TPM parameters for the cases are listed in Table 3. The applied torque was set to 350 Nm.  
Figure 9 compares the contact stress values obtained by the current model and the proposed model for the cases of no modification, NNM, and PPM at 120\({}^{\circ}\) and 300\({}^{\circ}\) crankshaft angles, which represent different mesh phases. Here, the contact stress calculated by the proposed model comes from the central line of the contact width along the longitudinal direction. The maximum and minimum values occurring on the edges and the middle of the face-width of the cycloid gear, as well as their mean values, are plotted as error bars with red lines in Figure 9. It can be seen that the proposed model agrees well with the current model in terms of the roller numbers which the load is distributed on and magnitudes of contact stress for the three cases, although edge contact leads to increasingly concentrated stresses, which could further accord with the practical situation. Aside from this, it was also observed that, in the presence of TPM, the contact stresses increase significantly as the number of contacting tooth pairs decreases. This can be explained by the fact that the tooth profile modification causes more contacting tooth pairs to separate.  
Figure 10 shows the LTE and the GR of the example cycloid gear pair calculated by the current model and the proposed model for the three cases. These time histories illustrate that the same shape and period of the LTE and the GR were obtained by both models, varying periodically as the crankshaft rotates, as expected, while the average absolute value of LTE and the peak-to-peak value of the GR obtained by the proposed model are a little smaller than those obtained by the current model.  
The results shown in Figures 9 and 10 by the proposed model are in good agreement with the predictions of the current model, which verifies the effectiveness of the proposed model. From Figures 9 and 10, we can also see the effects of different types of TPM on the load distribution and contact stress, LTE, and GR.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Value** \\ \hline \(n_{2}\) & Cycloid gear tooth & 15 \\  & number & 16 \\ \(n_{1}\) & Roller number & 16 \\ \(a\) & Roller position & 90 mm \\ \(\rho\) & Roller radius & 7.2 mm \\ \(e\) & Eccentricity & 4.5 mm \\ \(B\) & Width of cycloid gear & 17 mm \\ \hline \hline \end{tabular}
\end{table}
Table 2: Basic design parameters of the example cycloid gear pair.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Standard** & **NNM** & **PPM** \\ \hline \(\Delta a=0\) μm & \(\Delta a=-16\) μm & \(\Delta a=8\) μm \\ \(\Delta\rho=0\) μm & \(\Delta\rho=-8\) μm & \(\Delta\rho=16\) μm \\ \hline \hline \end{tabular}
\end{table}
Table 3: Tooth profile modification (TPM) parameters of three cases for the example cycloid gear pair.  
Figure 9: Comparison of contact stress on tooth pairs at 120\({}^{\circ}\) and 300\({}^{\circ}\) crankshaft angles obtained by the current model and the proposed model for the cases of (**a**) standard, (**b**) NNM, and (**c**) PPM profiles.