# Transmission Efficiency of Cycloid-Pinion System Considering the Assembly Dimensional Chain
## 5 Comparison of Transmission Efficiency of Different Structures
### Transmission Efficiency Comparison  
The presence of the pinion sleeve affects the location of the sliding friction. Specifically, without the pinion sleeve, the main manifestation is the sliding friction between the cycloid gear and the pin tooth. When there is one pinion sleeve, it becomes the sliding friction between the pin tooth and the pin tooth sleeve. In summary, \(dx_{1}\) represents the movement of friction work when there is no pinion sleeve, i.e., the relative displacement of the cycloid and pin tooth. Additionally, \(dx_{2}\) indicates the movement of the work performed by the frictional force in the presence of the pinion sleeve. By taking the derivative of the two with respect to time, the corresponding velocity ratio may equal the radius ratio of the pinion sleeve to the pin tooth. \[\frac{dx_{1}}{dx_{2}}=\frac{v_{p1}}{v_{p2}}=\frac{R_{tp}}{R_{rp}} \tag{34}\]  
where \(R_{tp}\) and \(R_{rp}\) are the radii of the pinion sleeve and the pin tooth, respectively. The addition of a pin tooth sleeve affects the distribution of friction in the cycloid system. Inputs such as the equivalent radius of curvature and contact point velocity will also change at this point. The result will be a different mixed friction coefficient. The specific situation is shown in Figure 12. In the case of the two structures with and without pin tooth sleeves, the corresponding mixed friction coefficients of the two have a similar variation pattern at the initial moment. After considering the assembly error, the number of simultaneously meshing teeth is reduced to seven pairs for the structure without sleeve. Further, for the structure with sleeve, the number is one more pair. When t = 200 ms, the number of extra meshing teeth is two. Besides, the fluctuations of the non-zero values of the two were analyzed with the standard deviation (STD). Figure 12 shows that the  
Figure 11: Pattern of meshing clearance distribution (w = 20 r/s, t = 0 ms). pin tooth sleeve structure allows for less fluctuation in the friction coefficient of the entire system. The overall force on the cycloid is smoother. The average power consumption and transmission efficiency for two moments (t = 0 ms and t = 10 ms) are shown in Figure 13. In the instant of starting, the direct contact power consumption between pin tooth and cycloid gear is 72.3321 W and the transmission efficiency is 99.9%, whereas after replacing the structure of pin tooth sleeve, the power consumption reduced to 27.2156 W, and the transmission efficiency increased to 99.96%. As t = 10 ms, the power consumption reduced from 35.9208 W to 16.9709 W, and the transmission efficiency improved by 0.03%. The above experiments only illustrate the transient transmission characteristics at two different moments. Afterwards, the power consumption characteristics corresponding to  
Figure 12: Mixed friction coefficient comparison (w = 20 r/s, Ra = 0.3 \(\upmu\)m). Figure 13: Average efficiency and average power comparison. (**a**) t = 0 ms; (**b**) t = 10 ms.  
the different structures will be analyzed from a continuous time period. As in Figure 14, the power loss curves corresponding to both structures oscillate up and down periodically. The curve in red, which is the structure containing the pin tooth sleeve, always remains at a lower level. Further, the box shows the variation of the first-order derivatives of the two curves, and it can be seen that the red curve has a smaller amplitude of variation. The corresponding system power transmission is more stable. After exploring the transmission characteristics of different structures over the same settings, the variable parameters, i.e., speed as well as surface roughness, are introduced. Figure 15 shows the variation of the average efficiency for different input speeds. The rotational speed was set to 10 r/s, 20 r/s, 30 r/s, and 40 r/s. The transmission efficiency decreased from 99.97% to 99.84% as the speed increased without the pin tooth sleeve.