## 3 TVMS Based on Load Tooth Contact Analysis
### Calculation of Time-Varying Mesh Stiffness  
The calculation of the time-varying mesh stiffness of the internal helical gear pair involves both the external and internal gears. The mesh stiffness of the internal mesh slice gear pair was determined using the mesh stiffness calculation method described in Section 2. When the load changes due to tooth surface deviation, the slice elements on the contact line are involved in the contact according to the order of the mesh clearance from small to large. At the same time, the number of elements involved in mesh contact is time-varying. Therefore, it is necessary for the comprehensive mesh stiffness of the internal helical gear pair to be determined through load tooth contact analysis. Figure 8 illustrates the mesh clearance of the sliced tooth pairs on the contact line when the tooth surface deviation exists under the load state. The red solid line and the black dotted line represent the theoretical and actual contact positions, respectively. \(e_{px}\) and \(e_{gx}\) rep  
Figure 7: The ease-off graph of the topological modification surface. resent the tooth surface deviations of the external gear and the internal gear, respectively (\(x\) is the sliced tooth pair on the contact lines \(i,j,k\). Only the external gear was modified, so \(e_{gx}=0\)). \(\delta_{i},\delta_{j},\delta_{k}\) are the deformations of the contact lines \(i,j,k\) under load. When three pairs of teeth are in contact, the sliced tooth pairs \(i\), \(j\), \(k\) at the ridge point of the contact line between the front tooth, current tooth, and rear tooth move distances \(e_{pi},e_{pj},e_{pk}\), respectively, under the load \(F\). This means that the point with the smallest tooth surface deviation on each contact line makes contact first. The geometric relationship between the tooth pair deformation and the tooth surface deviation is as follows:  
\[\left\{ \begin{array}{l}\delta_{i}+e_{pi}=\delta_{j}+e_{pj}\\ \delta_{j}+e_{pj}=\delta_{k}+e_{pk}\end{array} \right. \tag{16}\]  
As shown in Figure 8, the tooth surface deviation at the ridge point of the current tooth contact line is the smallest, so the contact occurs first when the three teeth mesh. The relative values of the tooth surface deviation at the contact line ridge points of the front and rear teeth relative to the tooth surface deviation at the contact line ridge point of the current tooth are \(\Delta e_{ij},\Delta e_{kj}\). Therefore, for contact to occur between the slice element pair at the contact line ridge point between the front tooth and the rear tooth, the relative value of the tooth surface deviation must be overcome. According to Equation (16), the relative value of the tooth surface deviation can be determined as follows:  
\[\left\{ \begin{array}{l}\Delta e_{ij}=\delta_{j}-\delta_{i}=e_{pi}-e_{pj}\\ \Delta e_{kj}=\delta_{j}-\delta_{k}=e_{pk}-e_{pj}\end{array} \right. \tag{17}\]  
Slice tooth pairs \(i,j\), and \(k\) represent the element pairs located at the ridge point on the contact line of the current, front, and rear teeth, respectively. In a three-tooth mesh, the mesh clearance between the front and rear teeth is influenced by the relative value of the tooth surface deviation. Therefore, it is essential to consider the relative value of the tooth surface deviation of the element pair along the contact line between the current tooth and the adjacent front and rear teeth. However, the mesh clearance of the current tooth is equal to the tooth surface clearance. Therefore, the mesh clearance of any slice element pair on the contact line of the front tooth, the current tooth, and the rear tooth can be expressed as follows:  
\[\left\{ \begin{array}{l}Z_{a}^{\prime}=Z_{a}+\Delta e_{ij}\\ Z_{b}^{\prime}=Z_{b}\\ Z_{c}^{\prime}=Z_{c}+\Delta e_{kj}\end{array} \right. \tag{18}\]  
where \(Z_{a},Z_{b},Z_{c}\) denote the tooth surface clearance of the slice element on the contact line of the front tooth, the current tooth, and the rear tooth, respectively.