## 3 Dynamics Model of the Multistage Transmission System
### Calculation of Mesh-Load Factor  
For Equation (9) and at equilibrium, the dynamic force between the planet gear and sun gear is \(F_{spj}\), and \(F_{rpj}\) is the dynamic force between the planet gear and ring gear. The two forces are given as:  
\[\left\{ \begin{array}{ll}F_{spj}=k_{spj}\delta_{spj}+c_{spj}\delta_{spj}\\ F_{rpj}=k_{rpj}\delta_{rpj}+c_{rpj}\delta_{rpj}\end{array} \right.. \tag{12}\]  
A numerical integration method is used to solve Equation (10), which represents the dynamic transmission system, by making \(D_{1 spjk}\) and \(D_{1 rpjk}\) represent the load-sharing coefficients of the external and internal meshing, respectively, of all gear pairs in the first-stage planetary gear train; \(D_{2 spjk}\) and \(D_{2 rpjk}\) represent the load-sharing coefficients of the external and internal meshing, respectively of all gear pairs in the second-stage planetary gear train, as shown in Equation (13):  
\[\left\{ \begin{array}{ll}D_{1 spj\text{ks}}=\frac{1\text{N}\left(F_{1 spjk} \right)_{\text{max}}}{\frac{1\text{N}}{\text{N}}\left(F_{1 rpjk}\right)_{\text{ max}}},&D_{1 rpjk}=\frac{1\text{N}\left(F_{1 rpjk}\right)_{\text{max}}}{\frac{1 \text{N}}{\text{N}}\left(F_{1 rpjk}\right)_{\text{max}}}\\ D_{2 spj\text{ks}}=\frac{2\text{N}\left(F_{2 rpjk}\right)_{\text{max}}}{\frac{ 2\text{N}}{\text{N}}\left(F_{2 rpjk}\right)_{\text{max}}},&D_{2 rpjk}=\frac{2 \text{N}\left(F_{2 rpjk}\right)_{\text{max}}}{\frac{2\text{N}}{\text{N}}\left(F _{2 rpjk}\right)_{\text{max}}}\end{array} \right., \tag{13}\]  
where \(k\)s is the meshing cycle number for the external meshing of the planetary gear pair and \(kr\) is the meshing cycle number for the internal meshing of the planetary gear pair. Because the wind generator is often affected by irregular wind force, in order to more accurately reflect the load-sharing characteristic of each planet gear pair of the transmission system under the excitation of random wind speed, 100 points of input speed were randomly selected from Figure 4 as the external excitation and the fluctuation diagram of random input speed, as shown in Figure 5.  
Figure 4: Input speed curve of the transmission system in 100 s.  
The results of substituting the relevant parameters of the transmission system into Equation (10) for solution and using Equation (13) to obtain the dynamic load-sharing coefficients of each planetary gear pair in the same meshing period are shown in Figures 6 and 7. Figure 5: The fluctuation diagram of 100 points random input speed. Figure 6: Dynamic load-sharing coefficient of the first-stage planetary gear train under random input wind speed. (**a**) Dynamic load-sharing coefficient of the external mesh in the first-stage planetary gear train. (**b**) Dynamic load-sharing coefficient of the internal mesh in the first-stage planetary gear train. \(\text{D}_{\text{1 sp1}}\), \(\text{D}_{\text{1 sp2}}\), and \(\text{D}_{\text{1 sp3}}\) are the dynamic load-sharing coefficient of each external planet gear pair of the first-stage planetary gear train, respectively. \(\text{D}_{\text{1 rp1}}\), \(\text{D}_{\text{1 rp2}}\) and \(\text{D}_{\text{1 rp3}}\) are the dynamic load-sharing coefficient of each internal planet gear pair respectively. In order to further analyze the fluctuation degree of load-sharing coefficient for each planet gear pair of the transmission system, the relationship between the load-sharing coefficient curve of external meshing of the first- and second-stage planetary gear train is drawn in Figure 8.  
Through data analysis, the fluctuation degree of the load-sharing coefficient of each planet gear pair in the transmission system at 100 points random input speed can be calculated, as shown in Figure 9.  
Figure 7: Dynamic load-sharing coefficient of the second planetary gear train under random input wind speed. (**a**) Dynamic load-sharing coefficient of the external mesh in the second planetary gear train. (**b**) Dynamic load-sharing coefficient of the internal mesh in the second planetary gear train.