# Motion, Static Force, and Efficiency Analysis of Planetary Gear Transmission Based on Graph Theory
## 2 Performance Analysis Model of Automatic Transmission
### Moment Analysis of Automatic Transmission  
The mechanical transmission part of an automatic transmission is a key part of the power transmission, so the stress situation of each component has a significant impact on the working performance and service life of an automatic transmission. The mechanical analysis of an automatic transmission is aimed at analyzing the applied torque of each component, mainly including the external torque applied to the components and the internal torque suffered by the interaction between planetary rows. When conducting moment analysis, the influence of friction between components is ignored, and it is assumed that each component rotates at a uniform speed without considering the influence of inertia torque.  
In Figure 4, by deleting the connection edges between the basic components of each planetary row, the connection edges between the vertices of active and passive edges of the engaged clutches, and the vertices of disengaged clutches, the moment analysis model of each gear of an automatic transmission can be obtained.  
(1)  
The moment characteristic matrix \(\mathbf{U}\) describes the component moment relationship of each planetary row. The single-star planetary row satisfies the moment relationship:  
\[\frac{T_{s}}{1}=\frac{T_{r}}{\alpha}=\frac{T_{c}}{-(1+\alpha)} \tag{5}\]  
The double-star planetary row satisfies the moment relationship:  
\[\frac{T_{s}}{1}=\frac{T_{r}}{-\alpha}=\frac{T_{c}}{\alpha-1} \tag{6}\]  
For a transmission consisting of k single-star planetary rows, its moment characteristic matrix is \[\mathbf{U}=\left[\begin{array}{ccccccccc}\text{S1}&\text{R1}&\text{C1}&\text{S2}& \text{R2}&\text{C2}&\cdots&\text{Sk}&\text{Rk}&\text{Ck}\\ \text{1}&\text{0}&\text{1}/(1+\alpha_{1})&\text{0}&\text{0}&\text{0}&\cdots& \text{0}&\text{0}&\text{0}\\ \text{0}&\text{0}&\text{0}&\text{1}&\text{0}&\text{1}/(1+\alpha_{2})&\cdots& \text{0}&\text{0}&\text{0}\\ \text{0}&\text{0}&\text{0}&\text{0}&\text{1}&\text{\alpha}_{2}/(1+\alpha_{2})& \cdots&\text{0}&\text{0}&\text{0}\\ \vdots\\ \text{0}&\text{0}&\text{0}&\text{0}&\text{0}&\text{0}&\cdots&\text{1}&\text{ 0}&\text{1}/(1+\alpha_{k})\\ \text{0}&\text{0}&\text{0}&\text{0}&\text{0}&\cdots&\text{0}&\text{1}&\text{ \alpha}_{k}/(1+\alpha_{k})\\ \end{array}\right]\begin{array}{c}1\\ 2\\ 3\\ 4\\ 2k-1\\ 2k\end{array} \tag{7}\]  
In the formula, \(\alpha_{1}\), \(\alpha_{2}\),..., \(\alpha_{k}\) are the characteristic parameters of each planetary row, respectively.  
The independent member matrix without external force \(\mathbf{D}_{i}\) (i = 1, 2, 3,...) is a matrix representing the stress state of each independent member without external force. The independent member without external force is not an input and output member, and it is not a braking member. The independent member without external force is recorded as 1, the others are recorded as 0, and the final matrix is called the independent member matrix without external force.  
Record all input members as 1 and others as 0 to obtain the moment input matrix \(\mathbf{I}_{i}\) (i = 1, 2, 3,...).  
A transmission composed of \(k\) planetary rows satisfies the following torque equation:  
\[\begin{bmatrix}\mathbf{U}\\ \mathbf{D}_{i}\\ \mathbf{I}_{i}\end{bmatrix}\begin{bmatrix}T_{s1}\\ T_{r1}\\ \vdots\\ T_{ck}\end{bmatrix}=\begin{bmatrix}0\\ 0\\ \vdots\\ 1\end{bmatrix}(j=1,2,3,\ldots) \tag{8}\]  
Solving Formula (8) can provide the component moment of an automatic transmission under each gear.