## 3 Dynamic Analysis and Problem Description during Engine Starting Process  
### Dynamic Analysis during Engine Starting Process  
Figure 6 shows the lever diagram in the engine startup process. The black solid line in this figure is the lever balance line. The speed on the balance line is defined as zero. The speed below the balance line is negative, and the speed above the balance line is positive. The up arrow = indicates that the torque is positive, and the down arrow indicates that the torque is negative.  
As shown in Figure 6, in the process of engine starting, MG1 torque pluses \(T_{\text{MG1}}\) to start the engine and bring the engine to its ignition speed. The output power of MG1 is negative, indicating that MG1 is in the power generation state. MG2 adopts a compensation torque \(T_{\text{MG2}}\) on the basis of \(T_{\text{MG2}}\) to ensure the vehicle runs forward. Ignored were the stiffness-damping elastic effect on the MG1 shaft, MG2 shaft, engine output shaft, drive shaft, as well as the dynamic meshing force and efficiency loss between gears. Based on the lever method [35], the torques of MG1 and MG2 during the engine startup can be expressed as follows:  
\[\left\{\begin{array}{ll}T_{\text{MG1}}=&\frac{(i_{2}-1)T_{\text{r},\text{rg }}+i(T_{\text{frag}}-(I_{\text{frag}}+L_{\text{i}})\tilde{\theta}_{\text{ frag},\text{des}})}{i_{1}-i_{2}}\\ &+(I_{\text{s1}}+I_{\text{MC1}})(i\dot{0}i\dot{1}\tilde{\theta}_{\text{WH}}+(1 -i_{1})\tilde{\theta}_{\text{Eng},\text{des}})\\ T_{\text{MG2}}=&\frac{(i_{1}-1)T_{\text{r},\text{rg}}+i_{1}(T_{\text{frag}}-( I_{\text{frag}}+L_{\text{i}})\tilde{\theta}_{\text{Eng},\text{des}})}{i_{2}-i_{1}}\\ &+(I_{\text{s1}}+I_{\text{MC2}})(i\dot{0}i\dot{2}\tilde{\theta}_{\text{WH}}+(1 -i_{2})\tilde{\theta}_{\text{Eng},\text{des}})\end{array}\right. \tag{14}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Engine maximum torque/speed (Nm/rpm) & 145/3600 \\ MG1/MG2 maximum torque (Nm) & 96/246 \\ Planetary parameter of front row \(i_{1}\) & \(-\)3.179 \\ Planetary parameter of rear row \(i_{2}\) & 2.342 \\ Final drive ratio \(i_{0}\) & 4.044 \\ Carrier inertia \(I_{\text{C}}\) (kg\(\cdot\)m\({}^{2}\)) & 0.0039 \\ Ring inertia \(I_{\text{R}}\) (kg\(\cdot\)m\({}^{2}\)) & 0.0017 \\ Inertia \((I_{\text{MG1}}+I_{\text{S1}})\) of MG1 and \(S_{1}\) (kg\(\cdot\)m\({}^{2}\)) & 0.041 \\ Inertia \((I_{\text{MG2}}+I_{\text{S2}})\) of MG2 and \(S_{2}\) (kg\(\cdot\)m\({}^{2}\)) & 0.0723 \\ Engine inertia \(I_{Eng}\) (kg\(\cdot\)m\({}^{2}\)) & 0.18 \\ Vehicle equivalent mass \(m_{\text{v}}\) (kg) & 1538 \\ Wheel radius \(r_{\text{v}}\) (m) & 0.31 \\ Vehicle frontal area \(A\) (m\({}^{2}\)) & 2.19 \\ Air resistance coefficient \(C_{d}\) & 0.307 \\ Tire rolling resistance coefficient & 0.0137 \\ Rated capacity of battery (Ah) & 37 \\ Rated voltage of battery (V) & 308 \\ Equivalent TI torsional stiffness \(k_{TI}\) (Nm\(\cdot\)rad\({}^{-1}\)) & 2864 \\ Equivalent TI torsional damping \(c_{TI}\) (Nm\(\cdot\)s\(\cdot\)rad\({}^{-1}\)) & 15 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Main PS-PHEV Parameters.  
Figure 6: Lever chart during engine starting process.  
where \(T_{MG1}\) is the MG1 torque; \(T_{MG2}\) is the MG2 torque; \(T_{r,req}\) is the ring torque required; \(T_{Eng}\) is the engine torque; \(I_{MG1}\) is the moment of inertia of MG1; and \(I_{MG2}\) is the moment of inertia of MG2. Additionally,\(I_{S1}\) is the moment of inertia of the little sun wheel; \(I_{S2}\) is the moment of inertia of the big sun wheel; \(i_{0}\) is the speed ratio of the main reducer; \(\hat{\theta}_{WH}\) is the angular acceleration of the wheel; \(I_{Eng}\) is the moment of inertia of the engine. \(I_{C}\) is the moment of inertia of the planetary frame; \(\hat{\theta}_{\text{Eng\_des}}\) is the engine acceleration; \(i_{1}\) is the front-row speed ratio; and \(i_{2}\) is the rear-row speed ratio.