# Mesh Stiffness and Dynamic Modeling and Analysis of Modified Straight Bevel Gears
## 2 Time-Varying Meshing Stiffness Calculation Model
### The TCA Environment
#### 2.1.2 Contact Point Vector and Normal Vector  
To perform contact analysis for straight-bevel gears, the contact range of the straight-bevel gears is obtained as follows:  
First, place the obtained straight bevel gears in their assembled position. Initially, transform the coordinates of the midpoint of the tooth width on the pitch circles of both the large and small gears to the zx plane in the positive x-direction and negative x-direction. The transformation matrix is as follows:  
\[\mathbf{M_{i}}= \begin{bmatrix}\cos(\beta_{i})&-\sin(\beta_{i})&0\\ \sin(\beta_{i})&\cos(\beta_{i})&0\\ 0&0&1\end{bmatrix}  \tag{3}\]  
in the formula, \(i=p,g\) represents the pinion and the gear. \(\beta_{i}\) represents the angle of rotation of the gear projection onto the zx plane. Then, rotate the large gear to its assembled position. \(\sum\) represents the shaft intersection angle, and the transformation matrix is as follows:  
\[\mathbf{M_{\sum}}= \begin{bmatrix}\cos(\sum)&0&\sin(\sum)\\ 0&1&0\\ -\sin(\sum)&0&\cos(\sum)\end{bmatrix}  \tag{4}\]  
the point vector and the normal vector of one of the pinion can be represented as:  
\[ \begin{cases}r^{\prime}_{fp}=\mathbf{M}_{P}r_{fp}\\ n^{\prime}_{fp}=\mathbf{M}_{P}n_{fp}\end{cases}  \tag{5}\]  
the point vector and the normal vector of one of the gear can be represented as:  
\[ \begin{cases}r^{\prime}_{fg}=\mathbf{M}_{\sum}\mathbf{M}_{g}r_{fg}\\ n^{\prime}_{fg}=\mathbf{M}_{\sum}\mathbf{M}_{g}n_{fg}\end{cases}  \tag{6}\]  
The procedure described above is as shown in Figure 2. At this time, the driving and driven gears are very close or slightly interfering. By gradually rotating the driving gear, the two gears are made to approach or move away from each other until they meet the conditions of the meshing principle, thus obtaining the initial value of the contact point. the meshing trajectory is comprised of a series of meshing points. Starting from the initial meshing point, the rotation angle \(\theta_{p}\) of the pinion can be gradually adjusted, either increased or decreased. This adjustment allows for the calculation of the rotation angle \(\theta_{g}\). They are required for the large gear when the tooth surfaces are in contact. The iterative process is used to determine the contact area following Formula (7). \[ \begin{cases}\mathbf{M}_{\theta_{p}}r^{\prime}_{fp}=\mathbf{M}_{\theta_{g}}r^ {\prime}_{fg}\\ \mathbf{M}_{\theta_{p}}n^{\prime}_{fp}=-\mathbf{M}_{\theta_{g}}n^{\prime}_{fg} \end{cases}  \tag{7}\]The rotation angle matrices for the driving and driven gears are denoted as \(\mathbf{M}_{\theta p}\) and \(\mathbf{M}_{\theta qg}\), respectively. This process allows us to accurately identify the coordinates of the contact points vector (\(r_{i}=(r_{xi}r_{yi}r_{zi})\)) on the tooth surfaces of straight bevel gears, as well as their respective normal vectors (\(n_{i}=(n_{xi}n_{yiyi}n_{zi})\)), where \(i=P\), \(G\) represents the pinion and the gear, respectively. It also allows us to define the meshing region and the meshing line for the straight bevel gears, as shown in Figure 3, illustrating the meshing lines at the midpoint of the tooth width. Analysis based on the parameters in Table 1 reveals the calculated results of TCA for straight bevel gear transmission under no-load conditions. Appendix A and the red line in Figure 3 illustrate the meshing point at the middle position of the tooth width. The results indicate that the meshing line is arc-shaped, and the angler of action obtained through TCA calculations can be utilized to determine the contact ratio.