## 2 Calculation Method of Tooth Root Stress Based on Hierarchical Finite Element Technique
### Construction of a Secondary Sub-Model for Tooth Root Stress Analysis  
In the system-level model, the time-varying load lines on tooth surfaces are obtained by quasi-static mechanical analysis, and they are loaded on the tooth surfaces of the secondary sub-model as load boundary conditions. At the same time, the system elastic deformation results are extracted and loaded into the sub-model as displacement boundary conditions, so that the effects of elastic deformation and meshing misalignment of the whole system are naturally included in the tooth root stress analysis results of the sub-model. In addition, dynamic behavior factors such as gear rotation and meshing are also considered in the system model, and when these key factors are fully expressed in the system model, the overall structure of the sub-model can be considerably simplified. It can contain only a number of teeth and corresponding rim features (the overall configuration is shown in Figure 5c), and its modeling and computational costs will be more focused on the geometric details of tooth roots. In terms of model quality, the sub-model is more comprehensive than the system model in terms of tooth root geometric elements, and its accuracy level, mesh quality, and other quality indicators are higher. The secondary sub-model of tooth root stress analysis with high fidelity is established by a general finite element method. The parametrically defined detailed gear solid models in the system level model are as auxiliary modeling elements for the 3D finite element sub models, which solves the problem of difficult modeling for the general finite element method to a certain extent. In addition, in order to further alleviate the contradiction between calculation accuracy and operation speed, a first-order hexahedron element is adopted in the finite element sub-model. When meshes are relatively coarse, tooth top may appear \"jagged\" in appearance due to shear self-locking or hourglass effect from the first-order element, but this does not affect the analysis results of tooth root stress. The RM software platform here can recommend an economical adaptive mesh density that can be used to accurately capture the steep stress gradients at the root of teeth and is sufficient to allow the sub-model calculations to converge within an effective predetermined number of iterations. In the system-level analysis, the meshing forces on gear teeth are applied through the superconstriction nodes; whereas in the secondary sub-model, the meshing forces are applied by the tooth surface load lines obtained from the system-level analysis, which is a clear difference in the setting of the load boundary conditions between the two. Taking the sun gear as an example, the tooth root stress analysis program of its sub-model is run, and the maximum principal stress values of tooth roots are calculated according to the first strength theory. The calculation results of the system elastic deformation and tooth surface load lines are applied to the sub-model as input conditions, as shown in Figure 5, and the sub-model is subsequently solved according to the set rotation steps. According to the coincidence degree of the gear pair in the system, the meshing in and meshing out processes of gear teeth is divided into 16 rotation steps, and the sub-model performs a finite element solution for each step of the gear tooth revolution. Although this makes the stiffness decomposition and load vector inverse substitution process relatively complex (involving multiple recursive traversals at the substructure level), this significant reduction in computational cost by appropriately increasing the complexity of the procedure is worthwhile. Figure 5d shows the calculated tooth root stress history for the sun gear teeth at 16 rotation steps; from 12 to 14 steps the teeth are in single tooth engagement and the corresponding root stress values are higher.