## 2 Automatic Transmission Modeling
### Kinematics and Power Flows of a Transmission Structure
#### 2.1.2 Planetary Gear Train Power Flows  
Power flow analysis is important to calculate applied torques on central members of planetary gear sets in order to design these components in terms of strength and durability of gears. For analyzing power flows, moment and power equilibrium equations are needed for every planetary gear set. These two equilibrium equations for \(j\)-th planetary gear set are given as  
\[T_{sj}+T_{rj}+T_{cj}+\sum_{ \begin{array}{c}i=1\\ i\
eq j\end{array} }^{N_{PG}}\sum_{k}\sum_{m}T_{ki/mj}=0, \tag{6}\]  
\[T_{sj}\omega_{sj}+T_{rj}\omega_{rj}+T_{cj}\omega_{cj}+\sum_{ \begin{array}{c} i=1\\ i\
eq j\end{array} }^{N_{PG}}\sum_{k}\sum_{m}T_{ki/mj}\omega_{mj}=0, \tag{7}\]  
where \(N_{PG}\) and \(T\) denote the number of planetary gear sets and torque values, respectively. For subscripts, \(i\) and \(j\) mean a planetary gear set number (i.e., \(i\), \(j\ \in\ [1,N_{PG}]\)), and \(k\) and \(m\) are one of \(s\), \(c\), \(r\) (i.e., \(k\), \(m\ \in\ [s,r,c]\)). In Equations (6) and (7), \(T_{sj}\), \(T_{rj}\) and \(T_{cj}\) mean the external torques acting to \(s\), \(r\) and \(c\) of \(j\)-th planetary gear set respectively, and \(T_{ki/mj}\) stands for connection torque on \(m\) central member of \(j\)-th planetary gear set, which is applied by \(k\) central member of \(i\)-th planetary gear set. \(\omega\) in Equation (7) is calculated from kinematic analysis described earlier. In the power equilibrium shown in Equation (7), a power loss term is not included with the assumption that no mechanical power loss occurs in planetary gear decks. In addition, two groups of additional equations which involve constraints and torque assignments are necessarily prepared for completing the entire formulations of power flow analysis. Constraint equations formed by the connected members are torque balance equation of \(T_{ki/mj}+T_{mj/ki}=0\). Torque assignments present which external torques are applied to central members. If external torque \(T_{ex}\) is assigned to the central member \(k\) which are neither an output nor stationary member, \(T_{ki}=T_{ex}\) \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & & & & & & & & & & & & & & \\ \hline
**Speed** & **S1** & **C1** & **R1** & **S2** & **C2** & **R2** & **S3** & **C3** & **R3** & **S4** & **C4** & **R4** \\ \hline
1 st & 0 & 0 & 0 & 0 & 1.0 & 1.5 & 1.5 & 1.2 & 1.0 & 1.0 & 0.2 & 0 \\
2 nd & 0 & 0 & 0 & 0 & 1.0 & 1.5 & 1.5 & 1.5 & 1.5 & 1.5 & 0.3 & 0 \\
3 rd & 1.0 & 0.3 & 0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 0.5 & 0.3 \\
4 th & 1.8 & 0.6 & 0 & 1.8 & 1.0 & 0.6 & 0.6 & 0.6 & 0.6 & 0.6 & 0.6 & 0.6 \\
5 th & 2.2 & 0.7 & 0 & 2.2 & 1.0 & 0.4 & 0.4 & 0.8 & 1.0 & 1.0 & 0.8 & 0.7 \\
6 th & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 \\
7 th & 0 & 1.2 & 1.9 & 0 & 1.0 & 1.5 & 1.5 & 1.2 & 1.0 & 1.0 & 1.1 & 1.2 \\
8 th & 0 & 1.5 & 2.3 & 0 & 1.0 & 1.5 & 1.5 & 1.5 & 1.5 & 1.5 & 1.5 & 1.5 & 1.5 \\ REV. & 0 & 0 & 0 & 0 & 1.0 & 1.5 & 1.5 & \(-\)0.3 & \(-\)1.4 & \(-\)1.4 & \(-\)0.3 & 0 \\ \hline  & & & & & & & & & & & & & \\ \hline
**Speed** & **S1** & **C1** & **R1** & **S2** & **C2** & **R2** & **S3** & **C3** & **R3** & **S4** & **C4** & **R4** \\ \hline
1 st & 1.2 & \(-\)3.7 & 2.5 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 & \(-\)4.7 & 3.7 \\
2 nd & 0.8 & \(-\)2.5 & 1.7 & \(-\)0.3 & 1.0 & \(-\)0.7 & 0 & 0 & 0.7 & \(-\)3.1 & 2.4 \\
3 rd & 0.5 & \(-\)1.6 & 1.1 & \(-\)0.5 & 1.6 & \(-\)1.1 & 0 & 0 & 0 & 0.5 & \(-\)2.1 & 1.6 \\
4 th & 0.3 & \(-\)1.0 & 0.7 & \(-\)0.3 & 1.0 & \(-\)0.7 & 0.2 & \(-\)0.4 & 0.2 & 0.3 & \(-\)1.3 & 1.0 \\
5 th & 0.1 & \(-\)0.4 & 0.3 & \(-\)0.1 & 0.4 & \(-\)0.3 & 0.3 & \(-\)0.7 & 0.4 & 0.1 & \(-\)0.5 & 0.4 \\
6 th & 0 & 0 & 0 & 0 & 0 & 0 & 0.4 & \(-\)1.0 & 0.6 & 0 & 0 & 0 \\
7 th & 0 & 0 & 0 & \(-\)0.2 & 0.5 & 0.3 & 0.3 & \(-\)0.8 & 0.5 & 0 & 0 & 0 \\
8 th & 0 & 0 & 0 & \(-\)0.3 & 1.0 & \(-\)0.7 & 0.3 & \(-\)0.7 & 0.4 & 0 & 0 & 0 \\ REV. & \(-\)1.3 & 3.9 & \(-\)2.6 & \(-\)0.3 & 1.0 & \(-\)0.7 & 0.6 & \(-\)1.7 & 1.1 & \(-\)1.1 & 5.0 & \(-\)3.9 \\ \hline \hline \end{tabular}
\end{table}