# Motion, Static Force, and Efficiency Analysis of Planetary Gear Transmission Based on Graph Theory
## 3 Performance Analysis Example of Automatic Transmission  
Figure 6 is a schematic diagram of a 6 HP26 automatic transmission developed by the German company ZF. The planetary wheel of the first planetary row is represented by a. The long planetary wheel and short planetary wheel of the third planetary row are represented by b and d, respectively. In Figure 6a, each gear is distinguished by a different color. A 6 HP26 automatic transmission will be taken as an example to analyze its motion, moment, power flow, and efficiency.  
Figure 5: Power flow diagram of a planetary row.  
A 6 HP26 automatic transmission can realize four reduction gears, two overdrive gears, and one reverse gear. The status table of control elements is shown in Table 2. The transmission diagram of a 6 HP26 automatic transmission in each gear is shown in Figure 7, and the gear ratio of each gear is shown in Table 3. In Table 2, black circle indicates that control element is in the engaged state.  
Figure 6: Schematic diagram of a 6 HP26 automatic transmission. (**a**) Structure diagram; (**b**) schematic diagram.  
Figure 7: Transmission diagram of each gear of a 6 HP26 automatic transmission. (**a**) First gear; (**b**) second gear; (**c**) third gear; (**d**) fourth gear; (**e**) fifth gear; (**f**) sixth gear; (**g**) reverse gear.  
The numbers of teeth of the sun gear, ring gear, and planetary gear for the first planetary row of a 6 HP26 automatic transmission are \(z_{\text{s1}}=37\), \(z_{\text{r1}}=71\), and \(z_{\text{p1}}=17\), respectively; the numbers of teeth of the sun gear, ring gear, and planetary gear for the second planetary row are \(z_{\text{s2}}=38\), \(z_{\text{r2}}=85\), and \(z_{\text{p2}}=23\), respectively; and the numbers of teeth of the sun gear, ring gear, long planetary gear, and short planetary gear for the third planetary row are \(z_{\text{s3}}=31\), \(z_{\text{s3}}=85\), \(z_{\text{p3,b}}=23\), and \(z_{\text{p3,d}}=28\), respectively. The input speed is \(n_{\text{i}}=4500\) r/min, and the input moment is \(T_{i}=400\) N\(\cdot\)m.  
The performance analysis model of a 6 HP26 automatic transmission is shown in Figure 8. The dotted line edge in Figure 8 indicates that the connection between components is controlled by control elements.  
In Figure 8, six connected branches can be obtained by deleting the connecting edges between basic components of each planetary row, and the connecting edges between vertices of the active and passive edges of the clutch, as shown in Figure 9. The degree of freedom of 6 HP26 automatic transmission W = \(6-3\) = 3.  
According to the engagement of control elements, the analysis model of each gear can be obtained, as shown in Figure 10.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Gear** & **C1,2,3,4** & **C4,5,6** & **C3,5,R** & **B1,R** & **B2,6** \\ \hline First gear & \(\bullet\) & & & & \(\bullet\) \\ Second gear & \(\bullet\) & & & & \(\bullet\) \\ Third gear & \(\bullet\) & & & & \\ Fourth gear & \(\bullet\) & \(\bullet\) & & & \\ Fifth gear & \(\bullet\) & \(\bullet\) & & & \\ Sixth gear & & \(\bullet\) & & & \(\bullet\) \\ Reverse gear & & & \(\bullet\) & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: The control element status table of a 6 HP26 automatic transmission.  
Figure 8: Performance analysis model of a 6 HP26 automatic transmission.  
Figure 10: Performance analysis model of various gears of a 6 HP26 automatic transmission. (**a**) First gear; (**b**) second gear; (**c**) third gear; (**d**) fourth gear; (**e**) fifth gear; (**f**) sixth gear; (**g**) reverse gear.  
Figure 9: Connected branch of performance analysis model of a 6 HP26 automatic transmission.