## 2 Modeling
### Two Types of Meshing States  
The meshing process of the same meshing pair driven by the face gear and pinion is shown in Figure 8a. The tooth profile of face gear in Figure 8a is at the midpoint of the tooth width. Assuming that the middle plane of the pinion groove coincides with the middle plane of the face gear tooth, two meshing points, A and B, simultaneously exist on both teeth faces of the face gear if the clearance is not considered. If the pinion drives the face gear to rotate clockwise, the meshing point A moves toward A\({}^{\prime}\) at the root of the face gear. Conversely, if the face gear drives the pinion to rotate clockwise, meshing point B gradually moves toward B\({}^{\prime}\) at the tip of the face gear. The corresponding mesh stiffness curves are shown in Figure 8b. The figure shows that the two curves differ not only in phase, but also in curve direction. The meshing stiffness of the pinion-driving gear pair is denoted as \(k(t)\); then, the meshing stiffness of the face gear-driving meshing pair is denoted as \(\widetilde{k}\big{(}t+\widetilde{t}\big{)}\), where \(\widetilde{k}\) is the inverse of \(k(t)\), and \(\widetilde{t}\) is the phase difference. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(k_{x}\) (N/mm) & \(k_{y}\) (N/mm) & \(k_{z}\) (N/mm) & \(k_{bx}\) (Nmm/rad) & \(k_{gy}\) (Nmm/rad) \\ \hline Input gear & \(4.0\times 10^{5}\) & \(4.0\times 10^{5}\) & / & / & / \\ Idler gear & \(9.4\times 10^{5}\) & \(9.4\times 10^{5}\) & / & / & / \\ Tail gear & \(7.7\times 10^{5}\) & \(7.7\times 10^{5}\) & / & / & / \\ Upper face gear & \(3.2\times 10^{6}\) & \(3.2\times 10^{6}\) & \(1.8\times 10^{6}\) & \(4.8\times 10^{10}\) & \(4.8\times 10^{10}\) \\ Lower face gear & \(1.5\times 10^{7}\) & \(1.5\times 10^{7}\) & \(1.26\times 10^{7}\) & \(3.2\times 10^{11}\) & \(3.2\times 10^{11}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of gear support stiffness. Figure 8: Difference in meshing stiffness between face gear driving and pinion driving: (**a**) direction of meshing; (**b**) difference in meshing stiffness. Figure 7: Difference in meshing force between fixed and varying load: (**a**) lower face gear; (**b**) upper face gear. Due to nonlinear excitation, a face gear may have three contact states during the transmission process: regular meshing, disengagement, and tooth-back contact. In the dynamic studies of the face gear drive [12; 35], the meshing stiffness of the tooth-back mesh has not been differentiated from that of the regular mesh. Therefore, in this study, based on the meshing stiffness agent model, the meshing model considering the disengagement and tooth-back contact was established, denoted as:  
\[K(f_{\delta},t)= \begin{cases}k_{1}(t),f_{\delta}>0\\ 0,f_{\delta}=0\\ k_{2}(t),f_{\delta}<0\end{cases}  \tag{1}\]  
where \(k_{1}(t)\) is the meshing stiffness for regular meshing, \(k_{2}(t)\) is the meshing stiffness for tooth-back meshing, and \(f_{\delta}\) is the relative displacement of the gear pair. In the coaxial face gear split-torque transmission system, the distribution of the pinion around the circumference of the face gear and the difference in the number of pinion teeth may introduce meshing phase differences among the different meshing pairs. Figure 8a shows the initial meshing state of Input Gear 1 with the lower face gear, and the meshing stiffness for one cycle was calculated from this starting position and is expressed as \(k_{in1-L}=k(t)\). When the number of pinion teeth is even, no phase difference exists among the meshing pairs driven by the pinion. When the number of pinion teeth is odd, the phase difference between the rest of the meshing pairs and the Input Gear 1-lower face gear meshing pair is determined with the following method:  
(a) If pinion \(M\) and Input Gear 1 are distributed on the circumference of the lower face gear differing by \(n\) (\(n\) is an integer) face gear teeth, then the phase difference between the meshing stiffness \(k_{M-L}\) and \(k_{in1-L}\) is 0.