## 2 Modeling
### Influence of Support Stiffness on Load-Sharing Coefficient under Different Loads  
To study the effects of the support stiffness and input load on the load-sharing characteristics of the coaxial face gear split-torque transmission system, the dynamics were analyzed under dual-input and dual-output operating conditions. The input torque \(T_{in}\) for a single input gear varied between 500 and 1500 Nm, and the support stiffness factor \(\zeta_{i}\) (\(i=in1\), \(in2\), \(id1\), \(id2\), \(t\), \(l\), \(u\)) for each gear ranged from 0.1 to 10. While analyzing the effect of the support stiffness of a single gear on the load-sharing factor for a given load, the support stiffness of the remaining gears was fixed. The influence of the support stiffness of Input Gear 1 on the load sharing under various loads was investigated. In Figure 14a, the value of the load-sharing coefficient \(\kappa_{in1}\) in the upper left region is the largest, which indicates that increasing the support stiffness and decreasing the load increase \(\kappa_{in1}\). Similar to prior results [38], the support stiffness of the input gear has a stronger effect on the load-sharing behavior of the input gear. However, the distribution of the load-sharing coefficient in Figure 14b is different from that in Figure 14a. As the support stiffness changes from 0.1 to 10, the load-sharing coefficient \(\kappa_{in2}\) shows an overall decreasing trend. In addition, Figure 14c shows that the smaller support stiffness of Input Gear 1 is more favorable for load equalization between the idlers. Figure 15 illustrates the effect of the support stiffness of Input Gear 2 on the load sharing at different input loads. Figure 15a shows that the value of the load-sharing coefficient \(\kappa_{in1}\) in the lower left corner is the largest and then gradually decreases with increasing support stiffness and load. The heatmap distribution of the load-sharing coefficient in Figure 15b is similar to that in Figure 14a, and the influence of the support stiffness of Input Gear 2 on \(\kappa_{in2}\) is more apparent. The effect of the support stiffness of Input Gear 2 on \(\kappa_{id}\) is similar to that shown in Figure 14c, indicating that \(\kappa_{id}\) decreases with a higher load and increases with a larger support stiffness. Figure 14: Influence of support stiffness of Input Gear 1 on load-sharing coefficient: (**a**) Input Gear 1; (**b**) Input Gear 2; (**c**) idler gear. \begin{table}
\begin{tabular}{c c c} \hline \hline
**System Parameter** & **Symbol/Unit** & **Value** \\ \hline \multirow{4}{*}{Supporting damping of face gear} & \(c_{68x}\), \(c_{68y}\), \(c_{68z}\)/N\(\cdot\)s\(\cdot\)m\({}^{-1}\) & 1.2 \(\times\) 10\({}^{4}\) \\  & \(c_{68x}\), \(c_{68y}\)/N\(\cdot\)m\(\cdot\)s\(\cdot\)rad\({}^{-1}\) & 0.9 \(\times\) 10\({}^{4}\) \\  & \(c_{78x}\), \(c_{78y}\), \(c_{72}\)/N\(\cdot\)s\(\cdot\)m\({}^{-1}\) & 1.2 \(\times\) 10\({}^{4}\) \\  & \(c_{78x}\), \(c_{78y}\)/N\(\cdot\)m\(\cdot\)s\(\cdot\)rad\({}^{-1}\) & 0.9 \(\times\) 10\({}^{4}\) \\ \hline Damping ratio & \(\underline{\varepsilon}\) & 0.05 \\ \hline Normal gear backlash & \(b_{\text{h}}\)/mm & 0.02 \\ \hline Static transmission error & \(e_{0}\)/mm & 0 \\ \cline{2-3}  & \(e_{\text{r}}\)/mm & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._Then, the change in the load-sharing coefficient with changes in the support stiffness of Idlers 1 and 2 is revealed in Figures 16 and 17. The thermal diagram and contour distribution of the load-sharing coefficient in Figure 16a, 16b, and 16c are similar to those in Figure 17a, 17b, and 17c, respectively, with only a slight difference in the numerical value. In other words, the variation in the support stiffness of Idlers 1 and 2 has a similar effect on \(\kappa_{in1}\), \(\kappa_{in2}\), and \(\kappa_{id}\). Notably, the effect of the idler support stiffness on \(\kappa_{id}\) is more prominent. Therefore, a smaller support stiffness of the idlers results in a more uneven load distribution.