# Motion, Static Force, and Efficiency Analysis of Planetary Gear Transmission Based on Graph Theory
## 2 Performance Analysis Model of Automatic Transmission
### Power Flow and Efficiency Analysis of Automatic Transmission  
The power transmission process ofan automotive automatic transmission is affected by factors such as gear meshing, oil droplet splashing, bearing friction, and air resistance, which can cause a certain amount of power loss. The greater the loss, the lower the power transmission efficiency and the poorer the fuel economy. When calculating efficiency in this article, only the power loss caused by gear meshing is considered, while other factors are ignored.  
During the power transmission process, there may be situations where the power transmitted by components is greater than the input power. This phenomenon is due to the formation of a closed power circuit inside the gear mechanism, known as cyclic power. This phenomenon will reduce transmission efficiency and increase meshing power loss, so it is important to avoid choosing solutions that are significantly affected by this phenomenon. Determining the power transmission path of automatic transmission through power flow analysis is beneficial for distinguishing and observing cyclic power and plays an important role in accurate efficiency evaluation.  
From the motion and moment analysis of an automatic transmission, the rotational speed and torque of each component can be obtained, respectively; thus, the power flow of automatic transmissions can be determined. If the component power is positive, it is input power; if the component power is negative, it is output power. The power flow diagram of a planetary row is shown in Figure 5. The arrow inward represents the input power and the arrow outward represents the output power.  
(2) Efficiency analysis  
Transmission efficiency is an important indicator for evaluating the performance of planetary gear trains. According to experience, the transmission efficiency of forward gear should not be less than 0.925, while for gears such as reverse gear and first gear that are relatively less frequently used, the transmission efficiency should be relaxed to not be less than 0.87.  
At present, there are many methods to determine planetary gear transmission efficiency. In this paper, the transmission ratio method is used to calculate transmission efficiency.  
For a transmission composed of \(k\) planetary rows, the efficiency of each gear is  
\[\eta=\frac{\widetilde{i}_{\alpha}}{i_{\alpha}} \tag{9}\]  
In Formula (9),  
\[i_{\alpha}=f(\alpha_{1},\alpha_{2},\cdots,\alpha_{k}) \tag{10}\]  
\[\widetilde{i}_{\alpha}=f[\alpha_{1}(\eta_{\mathrm{d}})^{y_{1}},\alpha_{2}(\eta _{\mathrm{d}})^{y_{2}},\cdots,\alpha_{k}(\eta_{\mathrm{d}})^{y_{k}}] \tag{11}\]  
In Formula (11), \(\eta_{\mathrm{d}}\) is the transmission efficiency of a planetary row. Denote the transmission efficiency of an external meshing gear pair as \(\eta_{\mathrm{w}}\) and the transmission efficiency of an internal meshing gear pair as \(\eta_{\mathrm{n}}\); then, for a single-star planetary row \(\eta_{\mathrm{d}}=\eta_{\mathrm{w}}\eta_{\mathrm{n}}\) and for a double-star planetary row \(\eta_{\mathrm{d}}=\eta_{\mathrm{w}}^{2}\eta_{\mathrm{n}}\). Usually, \(\eta_{\mathrm{w}}=0.97\), \(\eta_{\mathrm{n}}=0.99\).  
In addition, in Formula (11),  
\[y_{j}=\mathrm{sgn}(\frac{\alpha_{j}}{i_{\alpha}}\times\frac{\partial i_{ \alpha}}{\partial\alpha_{j}})(j\ =\ 1,2,\ldots,k) \tag{12}\]