# Motion, Static Force, and Efficiency Analysis of Planetary Gear Transmission Based on Graph Theory
## 2 Performance Analysis Model of Automatic Transmission
### Motion Analysis of Automatic Transmission  
The motion analysis of an automatic transmission mainly includes the degree of freedom of automatic transmission, the calculation of the transmission ratio in each gear, and the rotational speed of components in each gear. The degree of freedom is related to the control of shifting components, and the transmission ratio is an important indicator for evaluating the transmission performance ofan automatic transmission, which is directly related to the driving comfort and fuel economy of the car.  
The motion analysis model of each gear of an automatic transmission can be obtained by deleting the connection edges between basic components of each planetary row in Figure 4.  
(1) Motion characteristic matrix  
The motion characteristic matrix \(A_{1}\) describes the component motion relationship of each planetary row. The single-star planetary row satisfies the motion relationship:  
\[n_{s}+\alpha n_{r}-(1+\alpha)n_{c}=0 \tag{1}\]  
The double-star planetary row satisfies the motion relationship:  
\[n_{s}-\alpha n_{r}-(1-\alpha)n_{c}=0 \tag{2}\]  
For a transmission consisting of k single-star planetary rows, its motion characteristic matrix is  
\[\mathbf{A}_{1}=\left[\begin{array}{ccccccccc}S1&R1&C1&S2&R2&C2&\cdots&Sk&Rk&Ck\\ 1&\alpha_{1}&-(1+\alpha_{1})&0&0&0&\cdots&0&0&0\\ 0&0&0&1&\alpha_{2}&-(1+\alpha_{2})&\cdots&0&0&0\\ \vdots&&&&&&&&\\ 0&0&0&0&0&\cdots&1&\alpha_{k}&-(1+\alpha_{k})\\ \end{array}\right]\begin{array}{c}1\\ 2\\ \vdots\\ k\end{array} \tag{3}\]  
In the formula, \(\alpha_{1}\), \(\alpha_{2}\),..., \(\alpha_{k}\) are the characteristic parameters of each planetary row, respectively.  
(2) Connection matrix  
Connection matrix \(A_{2}\) is the matrix describing the component connection relationship of each planetary row. If one of the two connected components is marked as 1, the other  
Figure 4: Performance analysis model of various gears of an automatic transmission. (**a**) First gear; (**b**) second gear; (**c**) third gear; (**d**) reverse gear.  
is marked as \(-1\), and other components are marked as 0. Then, each two connected components will form a row vector containing only \(1\), \(-1\), and 0, and the resulting matrix is called a connection matrix.  
(3)  
Gear matrix \(\mathbf{C}_{i}\) (\(I=1,2,3,\ldots\)) is a matrix representing the brake and clutch engagement state of an automatic transmission in each gear. When the brake is engaged, the brake component is marked as 1; when the clutch is engaged, one of the two constant velocity components is marked as 1, the other is marked as \(-1\), and other components are marked as 0. The last line of the gear matrix is the input vector, and the input component is marked as 1; the others are 0.  
The transmission composed of \(k\) planetary rows satisfies the following motion equation:  
\[\begin{bmatrix}A_{1}\\ A_{2}\\ \mathbf{C}_{i}\end{bmatrix}\begin{bmatrix}n_{s1}\\ n_{r1}\\ \vdots\\ n_{ck}\end{bmatrix}=\begin{bmatrix}0\\ 0\\ \vdots\\ 1\end{bmatrix}(i\ =\ 1,2,3,\ldots) \tag{4}\]  
The speed of each component of an automatic transmission and the transmission ratio of each gear can be obtained by solving Formula (4).