# Analysis of Powertrain Loading Dynamic Characteristics and the Effects on Fatigue Damage
## 3 Dynamic Analysis of Powertrain System
### System Dynamic Model
#### 3.1.2 System Dynamic Equation  
Without regard to the frictions and mechanical losses, the system dynamic equation is  
\[\mathbf{j}\ddot{\mathbf{\theta}}+\mathbf{C}\dot{\mathbf{\theta}}+\mathbf{K}\mathbf{\theta}=\mathbf{T} \tag{2}\]  
where \(\mathbf{J}\) is the inertia matrix, \(\mathbf{C}\) is the damping matrix, \(\mathbf{K}\) is the stiffness matrix, \(\mathbf{\theta}\), \(\dot{\mathbf{\theta}}\), and \(\ddot{\mathbf{\theta}}\) are the angular displacement, the angular velocity, and the angular acceleration vector of the system, respectively, and \(\mathbf{T}\) is the torque vector. When the clutch is in separation, sliding, or engagement status, the system dynamic equations have different forms of expression. As a result, the following two sections were discussed in three cases. (1) Clutch sliding status  
The system is simplified to a model that consists of five rotor inertias and three spring-dampers in clutch sliding state, as shown in Figure 7. The rotor inertia \(J_{1}\) includes the inertias of the engine crankshaft, the flywheel, and the active part of the coupling. The rotor inertia \(J_{2}\) includes the inertia of the coupling passive part, the engine output shaft, and the impeller of the torque converter. The rotor inertia \(J_{3}\) includes the inertia of the turbine of the torque converter, the clutch, the transmission,  
Figure 6: 5-axle vehicle’s powertrain system schematic diagram. the transfer case, the bridge gear boxes, the main reducers, the differentials, the wheel reducers, the wheels, and the shafts among them. The rotor inertia \(J_{4}\) is the equivalent inertia of the vehicle. \(K_{1}\), \(C_{1}\), \(K_{2}\), \(C_{2}\), and \(K_{3}\), \(C_{3}\) are the equivalent stiffness and damping of the elastic coupling, the torsional damper, and the tires, respectively. The stiffness and damping of drive shafts are ignored. \(T_{\rm e}\) and \(T_{\rm R}\) are the engine torque and the equivalent resistant torque, respectively. The equivalent moment of inertia, stiffness, and damping of the transmission components are calculated as Equation (3),  
\[J_{i}=\frac{J_{i0}}{R_{i}^{2}},K_{\rm m}=\frac{K_{m0}}{R_{m}^{2}},C_{\rm n}= \frac{C_{m0}}{R_{n}^{2}} \tag{3}\]  
where \(J_{i}\), \(K_{m}\), and \(C_{n}\) are the equivalent moment of inertia, stiffness, and damping, respectively. \(J_{i0}\), \(K_{m0}\), and \(C_{n0}\) are the initial moment of inertia, stiffness, and damping, respectively. \(R_{i}\), \(R_{m}\), and \(R_{n}\) are the transmission ratios from the engine to the interests. According to the simplifying work above, we can give the matrixes and vectors in the Equation (2) as follows: the stiffness matrix is \(\mathbf{K}=\left[ \begin{array}{cccc}K_{1}&-K_{1}&0&0&0\\ -K_{1}&K_{1}+K_{2}&-K_{2}&0&0\\ 0&-K_{2}&K_{2}&0&0\\ 0&0&0&K_{3}&-K_{3}\\ 0&0&0&-K_{3}&K_{3}\end{array} \right]\); the damping matrix is \(\mathbf{C}=\left[ \begin{array}{cccc}C_{1}&-C_{1}&0&0&0\\ -C_{1}&C_{1}+C_{2}&-C_{2}&0&0\\ 0&-C_{2}&C_{2}&0&0\\ 0&0&0&C_{3}&-C_{3}\\ 0&0&0&-C_{3}&C_{3}\end{array} \right]\); the inertia matrix is \(\mathbf{J}=\left[ \begin{array}{cccc}J_{1}&0&0&0&0\\ 0&J_{2}&0&0&0\\ 0&0&J_{3-1}&0&0\\ 0&0&0&J_{3-2}&0\\ 0&0&0&0&J_{4}\end{array} \right]\); the angular displacement vector is \(\mathbf{\theta}=\left[ \begin{array}{c}\theta_{1}\\ \theta_{2}\\ \theta_{3-1}\\ \theta_{3-2}\\ \theta_{4}\end{array} \right]\); and the torque vector is \(\mathbf{T}=\left[ \begin{array}{c}T_{\rm e}\\ 0\\ -T_{\rm C}\\ -T_{\rm R}\end{array} \right]\). The torque of the clutch in sliding status is calculated by Equation (4),  
\[T_{\rm C}=\text{sign}(\omega_{\rm e}-\omega_{\rm c})z\mu_{\rm s}F_{\rm N}R_{ \rm C} \tag{4}\]  
where \(\omega_{\rm e}\) is the angular velocity of the active part of the clutch, \(\omega_{\rm c}\) is the angular velocity of the passive part of the clutch, \(z\) is the number of friction plates, \(\mu_{\rm s}\) is the friction coefficient, \(F_{\rm N}\) is the working pressure applied to the friction plate, and \(R_{\rm C}\) is the average friction radius of the friction plate.