## 3 Dynamics Model of the Multistage Transmission System
### Dynamical Equations of the Transmission System  
The power-flow route of the transmission system is: planetary carrier of first stage \(\rightarrow\) sun gear of first stage \(\rightarrow\) planetary carrier of second stage \(\rightarrow\) sun gear of second stage \(\rightarrow\) driving gear of parallel-shaft gears \(\rightarrow\) driven gear of parallel-shaft gears. In order to facilitate the analysis and derivation, the dynamic model is appropriately simplified, and the following assumptions are made: each member is rigid, the planetary gears are evenly distributed around the circumference and the torsional stiffness of supporting members and the friction of the transmission system are ignored. Supposing that the meshing of gear teeth is simulated by spring and damper, a dynamic model (Figure 2) is established and shown in Figure 3, based on lumped parameter theory. The linear displacements of all components of the transmission system are shown in Equation (7):  
\[\left\{ \begin{array}{l}u_{1c}=r_{1c}\theta_{1c},u_{1s}=r_{1 bs}\theta_{1s}\\ u_{1 pj}=r_{1 bpj}\theta_{1 pj},u_{2c}=r_{2c}\theta_{2c}\\ u_{2s}=r_{2 bs}\theta_{2s},u_{2 pj}=r_{2 bpj}\theta_{2 pj}\\ u_{3g1}=r_{3 bg1}\theta_{3g1},u_{3g2}=r_{3 bg2}\theta_{3g2}\end{array} \right., \tag{7}\] \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Driving Gear** & **Driven Gear** \\ \hline Teeth number & 98 & 29 \\ Module & 6 & 6 \\ Pitch radius/mm & 294 & 87 \\ Base circle radius/mm & 276.27 & 81.75 \\ Mass/kg & 1096 & 102 \\ Moment of inertia \(J_{3}\)/kg-m\({}^{2}\) & 152 & 1.5 \\ Pressure angle \(\alpha\)/(\({}^{\circ}\)) & 20 & 20 \\ Mesh stiffness K\({}_{361g2}\)/N-m\({}^{-1}\) & \(1.48\times 10^{10}\) & \(1.48\times 10^{10}\) \\ Coupling stiffness & \(7.53\times 10^{8}\) & - \\ K\({}_{23g1}\)/N-m\(\cdot\)rad\({}^{-1}\) & \(7.53\times 10^{8}\) & - \\ \hline \end{tabular}
\end{table} 
Table 2: Primary parameters of the parallel-shaft gears. Figure 3: Dynamic model of the transmission system. k\({}_{1 spj}\) (\(j\) = 1,2,3) and k\({}_{1 rpj}\) (\(j\) = 1,2,3) are the mesh stiffness of the sun gear and planet gear, the ring gear and planet gear in the first-stage planetary gear train respectively, c\({}_{1 spj}\) (\(j\) = 1,2,3) and c\({}_{1 rpj}\) (\(j\) = 1,2,3) are the coupling stiffness of the sun gear and planet gear, the ring gear and planet gear respectively; k\({}_{2 spj}\) (\(j\) = 1,2,3) and k\({}_{2 rpj}\) (\(j\) = 1,2,3) are the mesh stiffness of the sun gear and planet gear, the ring gear and planet gear in the second-stage planetary gear train respectively. c\({}_{2 spj}\) (\(j\) = 1,2,3) and c\({}_{2 rpj}\) (\(j\) = 1,2,3) are the coupling stiffness of the sun gear and planet gear, the ring gear and planet gear, respectively. where \(\theta_{1i}\) (\(i\) = s, c, r, p\({}_{j}\)) is the angular displacement of each component in the first-stage planetary gear train around its rotation center and \(\theta_{2i}\) is the angular displacement of each component in the second-stage planetary gear train around its rotation center. Relative displacement of gear-meshing pairs of the transmission system along the meshing line are shown in Equation (8):  
\[\left\{ \begin{array}{l}\delta_{1 spj}=u_{1s}+u_{1 pj}-\cos\alpha_{1s}u_{1c}-e_{ 1 spj}\\ \delta_{1 rpj}=u_{1 pj}-u_{1r}+\cos\alpha_{1r}u_{1c}-e_{1 rpj}\\ \delta_{2 spj}=u_{2s}+u_{2 pj}-\cos\alpha_{2s}u_{2c}-e_{2 spj}\\ \delta_{2 rpj}=u_{2 pj}-u_{2r}+\cos\alpha_{2r}u_{2c}-e_{2 rpj}\\ \delta_{3g1g2}=u_{3g1}+u_{3g2}-e_{3g1g2}\end{array} \right., \tag{8}\]  
where \(e_{1 spj}\), \(e_{1 rpj}\) (\(j\) = 1,2,3) is the meshing error of sun gear, ring gear and planetary gear in first-stage planetary gear train, respectively; \(e_{2 spj}\), \(e_{2 rpj}\) (\(j\) = 1,2,3) is the meshing error of sun gear, ring gear and planetary gear in second-stage planetary gear train, respectively; \(e_{3g1g2}\) is the meshing error of driving gear and driven gear in parallel-shaft gears.