## 5 Application of the Elastic TE Compensation Method for the RV Speed Reducer
### Modified RV Speed Reducer Prototype Obtained by TE Compensation  
An RV speed reducer prototype model is used for numerical analysis of the error sensitivity index. According to the flowchart in Figure 6 and parameters of the initial RV speed reducer prototype in Table 3, initial clearances \(\Delta(\varphi)_{i}\) and normal deformation \(\delta_{i}\) of the RV speed reducer prototype before modification are shown in Figure 7. The \(m\)-th pinwheel is the first pinwheel in the meshing state and the \(n\)-th pinwheel is the last pinwheel in the meshing state. There are nine pinwheels from the \(m\)-th to the \(n\)-th pinwheel, which are in meshing states with the cycloidal gear before modification of the cycloidal gear. The maximum contact force \(F_{\max}\) and normal deformation \(\delta_{i}\) of the initial RV speed reducer prototype before modification can be obtained after 22 iterations, as shown in Tables 4 and 5. To transfer the torque \(T_{c}\), the maximum contact force \(F_{\max}\) between a cycloidal gear and a pinwheel of the RV speed reducer prototype before modification is 1910.3 N. The maximum normal deformation \(\delta_{\max}\) of a cycloidal gear before modification is 0.0116 mm.  
In order to verify the feasibility and accuracy of the proposed elastic TE compensation method, the TE of the RV speed reducer is simulated by using Adams, as shown in Figure 8. The input angular velocity of the RV speed reducer is set as a sine function \(V=A\sin(Bt)+C\), where \(A\) is 4615, \(B\) is 30, and \(C\) is 2005. The function completes one cycle at a time, the input shaft completes one rotation with a change of the rotation direction, and the TE completes one accumulation. The mean of the TE is smoother, and the change of the rotation direction of the cycloidal gear  
cycloidal gears with normal deformation are set as flexible bodies, and the contact penetration value between a cycloidal gear and pinwheels is set as \(10^{-3}\).  
The simulated input shaft rotation angle after running 0.4 s is 826.2965\({}^{\circ}\) and the simulated output shaft rotation angle after running 0.4 s is 8.1977\({}^{\circ}\). The total TE of the initial prototype model has accumulated twice after running 0.4 s and its value for a period is 1.4559\({}^{\prime}\). The total TE is divided into TEs that are caused by the pinwheel radius error, the shifting tooth profile modification error, and the cumulative error of the circumference of the cycloidal gear in inverse proportion. By substituting the TEs into Equation (26), the error compensation amount of each TE factor is obtained by Equation (27). The 3D model of the modified RV speed reducer prototype is modified based on modifications for TE compensation of prototype components. The parameters of the modified RV speed reducer prototype are shown in Table 6.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Pinwheel center circle radius \(r_{p}\)/mm & 76 \\ Pinwheel radius \(r_{p}\)/mm & 2.9721 \\ Number of teeth of cycloidal gear \(z_{c}\) & 39 \\ Number of pinwheels \(z_{p}\) & 40 \\ Number of center teeth \(z_{1}\) & 22 \\ Number of teeth of planetary gear \(z_{2}\) & 60 \\ Eccentricity \(a\)/mm & 1.5 \\ Isometric tooth profile modification \(\Delta r_{p}\)/mm & \(-\)0.022 \\ Shifting tooth profile modification \(\Delta r_{p}\)/mm & \(-\)0.0266 \\ Short-formed coefficient \(K^{\prime}_{1}\) & 0.7898 \\ Modifications of cumulative error of circumference \(\Delta\delta_{8}\) & \(-\)0.011 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Parameters of the modified RV speed reducer prototype.  
Figure 8: RV speed reducer prototype model with constraint settings in Adams.