## 5 Case Study
### Power Flow Analysis Considering Power Loss  
The flow diagram of power loss along the transmission path is shown in Figure 9 and the results of the transmission efficiency calculations are presented in Table 4.  
Table 4 shows the transmission efficiency when power is transmitted along the transmission path. The transmission efficiencies of planetary transmission systems with 3, 4, 5, and 6 planetary gears are shown on the line graphs presented in Figure 10, and were calculated as 96.53%, 95.45%, 94.37%, and 93.32%, respectively.  
Transmission efficiencies of each transmission element along the transmission path as well as the total efficiencies are presented in Figure 11. As shown in Figure 11a, different power losses occur in different components of the planetary gear transmission system. The transmission efficiency from gear 2 to gear 3 and from gear 4 to gear 5 is lower because the sun gear meshes with multiple planet gears. Therefore, wind resistance loss and tooth surface friction loss will be large. As shown in Figure 11b, each transmission element produces some power loss and the total power loss is the sum of power losses of single elements.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number of Planetary Gears** & \(\eta_{IN-2}\) & \(\eta_{2-3}\) & \(\eta_{3-4}\) & \(\eta_{4-5}\) & \(\eta_{5-6}\) \\ \hline
3 & 99.98 & 98.83 & 99.09 & 98.89 & 99.23 \\
4 & 99.98 & 98.46 & 98.82 & 98.52 & 98.97 \\
5 & 99.98 & 98.08 & 98.55 & 98.16 & 98.71 \\
6 & 99.98 & 97.71 & 98.29 & 97.79 & 98.46 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Transmission efficiency when power transmitted along the transmission path.  
Figure 9: Flowchart of gear drive power loss calculation process.  
It can be noticed that tooth surface friction efficiency decreases and windage efficiency increases as speed increases, and bearing friction efficiency decreases very slowly. Figure 12 shows that 4.2 per cent of the total loss is owing to bearing friction, 5.3 percent of the total loss is owing to windage, and 90.5 per cent of the total loss is owing to tooth friction with an input speed of 4920 rpm. It further shows that 3.9 per cent of the total loss is owing to bearing friction, 15.1 per cent of the total loss is owing to winding, and 81 per cent of the total loss is owing to tooth friction with an input speed of 9348 rpm. [PERSON] studies the power loss in high-speed gears based on experiments. Analysing the distribution of each power loss, 46 per cent of the total loss is owing to windage [34]. Therefore, in the gear transmission system, distribution of each power loss is related to the structure and working conditions of the gear system. Different input conditions will also lead to the different power loss distribution.  
Figure 11: Transmission efficiency along the power transmission path. (**a**) Efficiency of each transmission element; (**b**) efficiency along the transmission path.  
Figure 10: Variation of efficiency along the transmission path using hypergraph method. (**a**) Three planetary gears; (**b**) four planetary gears; (**c**) five planetary gears; (**d**) six planetary gears.  
Under high-speed and heavy-load conditions, owing to the relatively high sliding speed of the meshing tooth surface, the instantaneous temperature rise directly affects the internal temperature of the gear system. As shown in Figure 13, the higher the tooth surface temperature, the smaller the power loss, because the viscosity of the lubricating oil decreases with the increasing temperature.