## 2 Modeling
### Optimization Analysis  
The established optimization model was solved through the marine predator algorithm. The load of a single input gear was 800 Nm. The other dynamic parameters are listed in Tables 2 and 3. The number of prey and maximum number of iterations were set to 80 and 100, respectively, to reduce the computational scale of the optimization algorithm. Then, the computation was accelerated by parallel settings. The convergence curve of the algorithm is shown in Figure 21. We found a sufficient decrease in the function value around ten iterations, which also indicated that the marine predator algorithm had a fast convergence rate.  
Figure 19: Influence of support stiffness of lower face gear on load-sharing coefficient: **(a)** Input Gear 1; **(b)** Input Gear 2; **(c)** idler gear.  
Figure 20: Influence of support stiffness of upper face gear on load-sharing coefficient: **(a)** Input Gear 1; **(b)** Input Gear 2; **(c)** idler gear.  
Table 4 presents the optimized support stiffness coefficients for each gear. The load-sharing coefficients of Input Gears 1 and 2 and the idler before optimization were 1.375, 1.388, and 1.046, respectively; after optimization, they were 1.049, 1.052, and 1.000, respectively. We noted a marked reduction in the load-sharing coefficients of Input Gears 1 and 2. To more clearly show the effect of optimization, the meshing force of each pinion with the upper and lower face gears was calculated separately, as shown in Figure 22, with OPT indicating the optimized meshing force. As shown in Figure 22a, the meshing force between Input Gear 1 and the upper gear was reduced, and the average value was reduced from 14,830 N before optimization to 11,320 N. As the meshing force between Input Gear 1 and the lower face gear increased, the average value rose from 6730 to 10,240 N, leading the meshing forces Fn16 and Fn17 to be close. This indicated that more power from Input Gear 1 flowed to the lower face gear. Figure 22b presents the change in the meshing force of Input Gear 2 with the upper and lower face gears before and after optimization, with a similar law as that shown in Figure 22a. Figure 22c,d show that the mean value of the meshing force of the idlers with the upper and lower face gears appeared to increase after optimization, but its mean load factor slightly decreased. This is because the power transmitted from Input Gears 1 and 2 to the lower face gear increased, and the increased power was transmitted to the upper face gear through the idlers.  
To verify the validity of the optimized model, the meshing forces of the gear pair before and after optimization were compared with a previously established quasistatic finite element analysis model [38], as shown in Figure 23. The load of a single input gear was 1256 Nm. Similar to the dynamic meshing force, the static meshing force of Input Gears 1 and 2 with the upper face gear substantially decreased at the optimized support stiffness, while the meshing force with the lower face gear increased. In addition, the average meshing force of Idlers 1 and 2 changed from 6850 N and 7750 N to 10,560 N and 10,680 N, respectively. Thus, the load-sharing coefficient at the idler reduced. We  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Parameter** & \(\zeta_{in1}\) & \(\zeta_{in2}\) & \(\zeta_{id1}\) & \(\zeta_{id2}\) & \(\zeta_{t}\) & \(\zeta_{l}\) & \(\zeta_{u}\) \\ \hline Value & 0.10 & 0.10 & 6.32 & 9.99 & 3.82 & 0.58 & 5.93 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Values of optimized support stiffness coefficients.  
Figure 22: Dynamic meshing force curve before and after optimization: (**a**) Input Gear 1; (**b**) Input Gear 2; (**c**) idler gear–upper face gear; (**d**) idler gear–lower face gear.  
sufficiently demonstrated that the load was more evenly distributed among the gear pairs with the optimized support stiffness.