## 4 Calculating Time-Varying Mesh Stiffness
### Finite Element Preprocessing
Therefore, in actual operating conditions, using spiral bevel gears can accommodate larger loads. The deformation values of contact points with forces are averaged to represent tooth surface deformations, as illustrated in Figure 9.  
For the contact force, in the simulation analysis, a load of 300 N-m is applied. The force on a single tooth is zero when not in contact, and it increases when loaded. Once the contact is established, the force decreases back to zero, and the contact is no longer present. Therefore, the load on a single tooth follows a pattern of initially increasing and decreasing. In this state, the maximum contact force for a single tooth is 8525.05 N, and the contact time (\(T\)) for a single tooth is 0.56 s. The time difference between the contact of adjacent teeth is 0.33 s. Based on this data, the total contact stiffness is determined. Figure 10 represents the process of solving time-varying mesh force. Figure 9: Deformation solution graph. Based on the above results combined with Equation (37), the time-varying meshing stiffness for a single tooth is obtained, as shown in Figure 11. The time-varying meshing stiffness between individual teeth is coupled with a phase difference between the meshing stiffness of two adjacent teeth. Therefore, the total time-varying meshing stiffness is the superposition of the single-tooth meshing stiffness achieved by shifting the phase angle. The time from the generation to the disappearance of contact force is the actual engagement time T for a single tooth. This time can be obtained through simulation analysis. The time when adjacent teeth start or stop engaging is given by the reference [30], and the overlap ratio can be expressed as follows:  
\[\varepsilon=\frac{T}{t} \tag{40}\]  
By calculating the time-varying meshing stiffness of this spiral bevel gear model, it is found to be 1.956. Figure 11 shows the single-tooth meshing stiffness and total meshing stiffness of the gear. The relationship between single-tooth engagement time and overlap ratio is used to determine the comprehensive meshing stiffness. Finally, the time-varying meshing stiffness for spiral bevel gears with different fractal dimensions is solved. The comprehensive time-varying meshing stiffness exhibits clear periodicity. When the results are summarized, it is observed that as the fractal dimension decreases, the time-varying meshing stiffness continuously decreases. The coarser roughness of gear tooth surfaces leads to a reduction in the actual contact area. Ideally, gear tooth surfaces should be smooth to maximize the contact area, but this  
Figure 8:0 Meshing stiffness on the tooth surface. Figure 8:1 Meshing force on the tooth surface. is limited by the machining method. As roughness increases, the contact area decreases, resulting in increased deformation under the same contact force. Consequently, the time-varying meshing stiffness decreases. Figure 12 compares the time-varying mesh stiffness under different roughness levels. Therefore, reducing the fractal dimension to some extent helps decrease the time-varying meshing stiffness of the gears, thereby improving the smoothness of motion. Additionally, from the experimental results, it is evident that as the tooth surface fractal dimension decreases, the rate of change in the time-varying meshing stiffness increases. In other words, with a smaller tooth surface fractal dimension, the variation in time-varying meshing stiffness becomes more pronounced, indicating a more significant impact on vibration conditions.