## 3 TVMS Based on Load Tooth Contact Analysis
### Calculation of Time-Varying Mesh Stiffness
When three teeth are in contact, the mesh order of all element pairs on the three contact lines is that contact occurs first when the mesh clearance is minimal. The mesh clearances  
Figure 8: Mesh clearance of the slice tooth pairs in three-tooth mesh. of all elements on the three contact lines are placed in a matrix and sorted from smallest to largest to determine the order of contact between the element pairs. \[Z^{\prime}=sort[Z^{\prime}_{a},Z^{\prime}_{b},Z^{\prime}_{c}] \tag{19}\]  
where \(Z^{\prime}_{a},Z^{\prime}_{b},Z^{\prime}_{c}\) is the meshing clearance of the element pairs on the contact line of the front tooth, the current tooth, and the rear tooth, respectively. \(Z^{\prime}\) is the mesh clearance of any mesh element pair, and _sort_ is a sort function from small to large. The load \(F\) is equal to the sum of the mesh forces \(f_{q}\) of each slice mesh element that satisfies the contact condition, which can be expressed as:  
\[F=\sum\limits_{q=1}^{m}f_{q} \tag{20}\]  
where \(m\) is the number of slice element pairs that mesh at the same time. According to [PERSON]'s law, there are the following relationships among the mesh force \(f_{q}\), deformation \(\delta_{q}\), and mesh stiffness \(k_{q}\) of the slice element in the mesh state. \[f_{q}=k_{q}\delta_{q} \tag{21}\]  
\(\delta_{q}=\delta_{j}-Z^{\prime}\), where \(\delta_{j}\) is the deformation of mesh element pair \(j\) at the ridge point (max deformation). When \(\delta_{q}>0\), the mesh element pairs are in contact. Through the contact condition and the force equilibrium condition, the deformation compatibility equation can be derived as follows:  
\[F=\sum\limits_{q=1}^{m}f_{q}=\sum\limits_{q=1}^{m}k_{q}(\delta_{j}-Z^{\prime}) \tag{22}\]  
In the load state, it is known from the principle of minimum potential energy that the gear system will achieve equilibrium only when the energy stored in the mesh elements is minimized. The expression for minimum potential energy is as follows:  
\[W=\sum\limits_{q=1}^{m}f_{q}(\delta_{j}-Z^{\prime})=\sum\limits_{q=1}^{m}k_{q }(\delta_{j}-Z^{\prime})^{2} \tag{23}\]  
Therefore, the TVMS of the three-tooth mesh can be calculated as follows:  
\[K=\frac{F}{\max(\delta_{i},\delta_{j},\delta_{k})} \tag{24}\]  
By combining Equations (20)-(24), the expression for the TVMS of the internal helical gear pair can be derived. \[K=\frac{\sum\limits_{q=1}^{m}k_{q}}{1+\sum\limits_{q=1}^{m}k_{q}Z^{\prime}/F} \tag{25}\]  
In the load state, the calculation formula of LTE is:  
\[\text{LTE}=\frac{F}{K}+\min(e_{px}) \tag{26}\]  
The TVMS calculation flow of the internal mesh transmission based on load tooth contact analysis can be found in Figure 9. It can be observed from Figure 9 that the flow can be divided into three parts. The first step is to calculate the mesh stiffness of the slice gear pair at different positions on the tooth profile. The mesh stiffness expression with the mesh angle as the independent variable is obtained by fitting. Then, prepare for the subsequent TVMS calculation. The second step is to determine the initial mesh position under the load state. According to the parameters for tooth surface modification and the tooth surface equation, the tooth surface deviation data are obtained. The relative position relationship of the slice element pairs on different contact lines during three-tooth meshing is established to determine the meshing gap of these pairs. The mesh sequence of the slice element pair is determined based on the mesh clearance. The mesh stiffness of this element pair is determined based on its position on the tooth profile. The third step is to determine the number of slice element pairs involved in meshing, according to the deformation coordination principle and the minimum potential energy method, so as to determine the mesh stiffness corresponding to the rotation angle. \[Z^{\prime}=sort[Z^{\prime}_{a},Z^{\prime}_{b},Z^{\prime}_{c}] \tag{19}\]0 TVMS calculation flow.