## 2 Modeling of DTS
### Modeling of the Planetary Gear Sets Based on the Lagrange Method  
Instead of traditional dynamics analysis, the bond graph method, and lever analogy, the Lagrange method does not need to consider the actual gear structure and calculate the interaction force and torque on each element [15; 22], so it is a good way to solve complex dynamic equations of planetary gear sets in DTS. Figure 1: \(6+2\) speed automatic transmission (AT) powertrain system. A high-power 6 + 2-speed AT is composed of four planetary gear sets--where _Si_, _Ri_, _Ci_, and _Pi_ respectively represent the sun gear, ring gear, planet carrier, and planetary gear (_i_ = 1, 2, 3, 4)--and six electro-hydraulic actuated clutches (_CS_, _BS_, _CH_, _BM_, _BL_, and _BR_). The combined clutch/brake schedule is shown in Table 1. The fourth planetary set and the clutch _BR_ are ignored in the following modeling because they only participate in reverse gear. Based on the velocity relationship of a mesh point, the kinematics equations of each planetary set are  
\[\left\{ \begin{array}{c}\omega_{Ci}R_{Si}-\omega_{Pi}R_{Pi}=\omega_{Si}R_{Si} \\ \omega_{Ci}(R_{Si}+2R_{Pi})+\omega_{Pi}R_{Pi}=\omega_{Ri}R_{Ri}\end{array} \right., \tag{1}\]  
where \(\omega\) is the angular velocity of the component and \(R\) is the radius. Therefore, the planetary gear system has six constraint equations and nine motion parameters, and the system is a three-degree-of-freedom system. \(\omega_{S1}\), \(\omega_{C1}\), and \(\omega_{C3}\) are selected as independent variables to facilitate calculation, and other coordinate parameters can be expressed by these three independent variables as  
\[\left[ \begin{array}{cc}\omega_{R15253}&\omega_{C2R3}&\omega_{R2}&\omega_{P1 }&\omega_{P2}&\omega_{P3}\end{array} \right]^{T}=A\left[ \begin{array}{cc} \omega_{S1}&\omega_{C1}&\omega_{C3}\end{array} \right]^{T}, \tag{2}\]  
where \(A\) is the constant coefficient matrix determined by the transmission structure. Since the planetary gears are symmetrically arranged, the potential energy change is considered to be zero \(E_{p}=0\). Thus, the total energy of the system \(E\) can be regarded as the sum of the kinetic energy \(E_{k}\). \[E=E_{p}+E_{k}=E_{k}, \tag{3}\]  
Each component kinetic energy \(\kappa\) can be expressed as  
\[E_{k}=\sum_{i=1}^{3}\kappa_{Si,Ci,Ri,Pi}=\sum_{i=1}^{3}\frac{1}{2}I_{Si,Ci,Ri,Pi }\omega_{Si,Ci,Ri,Pi}^{\alpha}=E_{k}\Big{(} \begin{array}{cc}\omega_{S1}& \omega_{C1}&\omega_{C3}\end{array} \Big{)}, \tag{4}\]  
where \(I\) is the rotational inertia. The Lagrange equation of the system is written as  
\[\left\{ \begin{array}{l}Q_{S1}=\frac{dt}{dt}\Big{(}\frac{\partial E}{\partial \omega_{C1}}\Big{)}-\frac{\partial E}{\partial\omega_{C1}}\\ Q_{C1}=\frac{dt}{dt}\Big{(}\frac{\partial E}{\partial\omega_{C1}}\Big{)}- \frac{\partial E}{\partial\omega_{C1}}\\ Q_{C3}=\frac{dt}{dt}\Big{(}\frac{\partial E}{\partial\omega_{C1}}\Big{)}- \frac{\partial E}{\partial\omega_{C3}}\end{array} \right., \tag{5}\]  
where \(Q\) is the generalized torque of the system on independent variables. According to the principle of virtual work, the virtual work of the system \(\delta W\) is the sum of the products of the virtual angular displacement \(\delta\alpha\) and each component torque \(T\). \[\delta W=\sum_{i=1}^{3}T_{Si,Ci,Ri,Pi}\delta\alpha_{Si,Ci,Ri,Pi} \tag{6}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Gears** & _CS_ & _BS_ & _CH_ & _BM_ & _BL_ & _BR_ & **Gear Ratio** \\ \hline
1 & \(\surd\) & & & & \(\surd\) & & 4.00 \\
2 & & \(\surd\) & & & \(\surd\) & & 2.67 \\
3 & \(\surd\) & & & \(\surd\) & & & 2.00 \\
4 & & \(\surd\) & & \(\surd\) & & & 1.33 \\
5 & \(\surd\) & & \(\surd\) & & & & 1.00 \\
6 & & \(\surd\) & \(\surd\) & & & & 0.67 \\ R1 & \(\surd\) & & & & \(\surd\) & & \(-\)5.00 \\ R2 & & \(\surd\) & & & \(\surd\) & & \(-\)3.33 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Combined clutch/brake schedule. Based on the connection relationship between the clutch and each component from Figure 1, it can be summarized as  
\[