## 3 Quasi-Static Analysis Model  
The accurate calculation of the meshing force is the prerequisite for the study of the load-sharing characteristics of the concentric face gear split-torque transmission system. Generally, the smaller the mesh cell size is in the FEA, the more continuous the contact stress distribution on the tooth's surface and the more accurate the meshing force results, but it leads to a higher computational cost. For this issue, some methods were proposed to balance the computational accuracy and computational cost [22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33]. Accordingly, the coupled elastic support and mesh non-coordinated face gear model was established based on the multi-point constraint method, as shown in Figure 4.  
Figure 4 shows the FEA model of the upper face gear. The teeth of the upper face gear were separated from the hub, the teeth were discretized using fine meshes, and the rim was discretized using coarse meshes. The nonconforming mesh interface was connected using a multi-point constraint algorithm. In addition, the driving and coast sides of the face gear were meshed with grids of different sizes, and the coupling connection was carried out at the tooth root segmentation based on the MPC method. Meanwhile, to reduce the number of computational nodes, just the teeth engaged in the meshing were retained.  
In finite element analysis, solid elements generally have no rotational degrees of freedom. An additional node needs to be created when applying the boundary conditions of torque and angular displacement. The node \(P_{f}\) in Figure 4 was coupled with the internal surface of the face gear hub to apply the boundary conditions. The node \(P_{b}\) was coupled to the bottom surface of the upper face gear rim and was used as a moving node of the spring element. In Figure 4, \(k_{2x}\), \(k_{2y}\), \(k_{2z}\), \(k_{6x}\), and \(k_{gy}\) are the \(x\)-radial, \(y\)-radial, \(z\)-axial, \(x\)-angular and \(y\)-angular stiffnesses of the spring element in the local coordinate system of the upper face gear, respectively. The red dot in the figure is considered to be a fixed node of the spring element.  
Figure 5 shows the finite element model of the cylindrical gear coupled with elastic support. Six teeth were reserved at the meshing with the upper and lower face gears. \(P_{c1}\) was an auxiliary node connected to the inner surface of the cylindrical gear. First, it allowed the cylindrical gear to apply torque and angular displacement, and additionally, it acted as a moving node for the spring element. In the figure, \(P_{c2}\) and \(P_{c3}\) are the fixed nodes of the spring element. \(k_{ix}\) and \(k_{iy}\) are the radial stiffnesses of the cylindrical gear support. The schematic diagram of the transmission system model is shown in Figure 6.  
In addition, applying the convergence criteria of force and displacement in the solution and reducing the convergence values were beneficial for the accuracy of the results. The basic geometric parameters of the concentric face gear split-torque transmission system are shown in Table 1 and the support stiffness values are listed in Table 2. The input torque of the single input gear was 1256 Nm and the output torque of the tail transmission was 251 Nm. The elastic modulus of the gear material was E = 206 GPa and the Poisson's ratio was \(\mu\) = 0.3. The coordinate system of the model was a Cartesian coordinate system. The \(x\)-axis and \(y\)-axis of the face gear were radial, and the \(z\)-axis coincided with the axis of the face gear pointing to the upper face gear. The \(x\)-axis and \(y\)-axis of the cylindrical gear were radial, the \(z\)-axis was axial, and the \(y\)-axis pointed to the lower face gear. The contact forces of each meshing pair were compared under elastic support and rigid support of the system in Figure 7, respectively. From the figure, it can be seen that when considering  
Figure 4: Schematic diagram of the finite element model of the upper face gear.