# Transmission Efficiency of Cycloid-Pinion System Considering the Assembly Dimensional Chain
## 5 Comparison of Transmission Efficiency of Different Structures
### Tolerance Analysis of Different Structure
\(A_{0}\) is the clearance between the crankshaft and the inner bore of the cycloid gear. Assembly dimensional chain information is shown in Table 3. In the cycloid system, the cylindrical surface of the pin tooth is a clearance fit to the pin gear, as in Figure 9. B\({}_{0}\) is the solution target. In the case of cycloid pin gear systems with pinion sleeve construction, the pinion sleeve has a clearance fit with the pin gear as shown in Figure 10. Similarly, the maximum clearance between the cycloid and the input crank is 26 \(\upmu\)m. Taking this value as a reference base, the recommended maximum engagement clearance at the pin teeth is 24 \(\upmu\)m. With this as the design objective, a clearance fit was employed. The recommended fit was obtained as H7/h6. The upper tolerances of the hole and shaft are ES = 15 \(\upmu\)m and es = 0 \(\upmu\)m, respectively. Meanwhile, the corresponding lower tolerances are EI = 0 \(\upmu\)m and ei = \(-\)9 \(\upmu\)m. The machining precision required for direct contact between the housing and  
Figure 8: Number of teeth meshed depending on system clearance. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Link** & **Part** & **Tolerance (\(\upmu\)m)** \\ \hline Decreasing link A\({}_{1}\) & Crankshaft diameter & \(-\)18 \\ Increasing link A\({}_{2}\) & Cycloid bore diameter & \(+\)8 \\ Decreasing link B\({}_{1}\) & Pin tooth diameter & \(-\)13 \\ Increasing link B\({}_{2}\) & Pin gear bore diameter & \(+\)9 \\ Decreasing link C\({}_{1}\) & Pin tooth diameter & \(-\)9 \\ Increasing link C\({}_{2}\) & Sleeve diameter & \(+\)15 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Assembly dimensional chain. the pin tooth is very high. In comparison, the machining economy of the pin tooth and the sleeve is significantly improved. To summarize, the information of each dimensional chain is shown in Table 3.  
The clearance between the crankshaft and the cycloid is 26 \(\upmu\)m by the extreme value method, i.e., \(A_{0}=\Delta R_{b}=26\)\(\upmu\)m. Similarly, the clearance between the pin tooth and the pin gear, and the clearance between the pin tooth and the pinion sleeve are 22 \(\upmu\)m and 24 \(\upmu\)m, respectively, i.e., \(B_{0}=\Delta r_{p1}=22\)\(\upmu\)m and \(C_{0}=\Delta r_{p2}=24\)\(\upmu\)m. The results of the clearance calculation are carried into the tooth profile Equation (31). \[\left\{ \begin{array}{l}x_{p}=e\cos(\alpha)+R_{b}\cos(\alpha+\beta)+r_{p}\cos (\alpha+\beta+\gamma)\\ y_{p}=e\cos(\alpha)+R_{b}\sin(\alpha+\beta)+r_{p}\sin(\alpha+\beta+\gamma)\\ \end{array} \right. \tag{31}\]  
The clearance variance caused by the tolerance can be calculated:  
\[d_{i}=\min\biggl{(}\sqrt{\left(x_{c}-x_{p}(i)\right)^{2}+\left(y_{c}-y_{p}(i) \right)^{2}}-r_{p}\biggr{)} \tag{32}\]  
Figure 10: Assembly dimension chain of the improved structure. Figure 9: Assembly dimension chain of the original structure. where \(x_{c}\) and \(y_{c}\) represent the coordinates of the discrete points of the cycloidal gear tooth profile. \(x_{p}(i)\) and \(y_{p}(i)\) represent the center coordinates of the \(i\)-th pin tooth, the calculation result is shown as follows:  
\[\left\{ \begin{array}{l}\chi_{p}(i)=R_{b}\sin\theta_{i}\\ y_{p}(i)=R_{b}\cos\theta_{i}(i=1,2\cdots z_{\text{b}})\\ \theta_{i}=2\pi(i-1)/z_{\text{b}}\end{array} \right. \tag{33}\]  
The addition of the sleeve visually causes a change in the meshing clearance. As shown in Figure 11, the clearance is overall symmetrically distributed. Overall, considering the economy of processing and manufacturing, the assembly dimensional chain clearance of the pin tooth sleeve housing is 2 \(\upmu\)m larger than that of the pin tooth housing. Thus, the clearance difference between the two displays a similarly periodically varying step function.