## 4 Calculating Time-Varying Mesh Stiffness
### Finite Element Preprocessing  
Table 3 contains the parameters for generating the three-dimensional model of the spiral bevel gear. Based on the fundamental data of the spiral bevel gear, the three-dimensional model of the gear is created. The tooth surface roughness is obtained by simulating cutting with different involute curves. Based on the relationship between the gear model diagram and the tooth surface fractal dimension, various models of spiral bevel gears with different surface fractal dimensions are obtained. Analyzing the models with different fractal dimensions allows for an exploration of the dynamic characteristics of spiral bevel gears under varying fractal dimensions. The established gear model is subjected to FEA. Proper preprocessing steps are crucial for the analysis, shown in Figure 7. The key steps in preprocessing include material definition, meshing, analysis type specification, contact pair definition, and boundary condition definition. These steps should be determined based on the actual working \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Pinion** & **Gear** \\ \hline Number of teeth & 16 & 27 \\ Module (mm) & 4.25 & \\ Face width (mm) & 12 & \\ Pressure angle (\({}^{\circ}\)) & 20 & \\ Spiral angle (\({}^{\circ}\)) & 30 & \\ Shaft angle (\({}^{\circ}\)) & 90 & \\ Hand of spiral & Left hand & Right hand \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the spiral bevel gear. conditions. Figure 8 shows the tooth contact force cloud diagram by taking the middle tooth of the five teeth as the analysis object. From Figure 8, it can be observed that, during the loading, the tooth surface contact region is formed by an elongated ellipse tilted on the tooth surface. The output of the contact force magnitude over time is obtained by setting the total force on the middle tooth as a historical variable. The elastic modulus of the material is defined as 210 GPa, and the Poisson's ratio is 0.377. The analysis type is defined as implicit static analysis. The analysis is conducted with a time step of 0.01. The contact pairs are defined between the pinion and gear, with a friction coefficient of 0.15 in the tangential contact. Boundary conditions are defined in accordance with the analysis steps. The analysis is conducted in three steps: Step 1 introduces a small rotation to the pinion to eliminate the gap between gear surfaces, ensuring mutual contact between the pinion and gear. Step 2 applies torque to the gear, representing the operational  
Figure 8: Contact stress. Figure 7: Process diagram of FEA. torque. In Step 3, torque is applied to the gear while simultaneously imparting a rotation to the pinion, thereby simulating the entire process of meshing motion between the gear pair. To simplify the analysis model, the five-tooth model is used as the research model. Upon completion of the run, the tooth surface contact stress and contact positions are obtained, as shown in Figure 8. Analysis variables are output as historical variables to calculate the normal contact force between tooth surfaces, with results presented in Figure 7. Tooth surface deformations are extracted from the deformation of unit points on the tooth surface. The deformation values of contact points with forces are averaged to represent tooth surface deformations, as illustrated in Figure 9.  
Simulation analysis was conducted using the commercial software Abaqus 2016. Initially, historical variables were set for the normal force of a single tooth. The analysis step was set as static general analysis. Deformation of the specified tooth surface was obtained through historical variable output. Stiffness calculation formulas were applied to compute the time-varying meshing stiffness for a single tooth. The overall meshing stiffness was obtained based on the relationship between gear overlap and the time of single-tooth meshing. The stress results are displayed, as shown in Figure 8, and the stress is lower than the stress in the case of single-tooth contact.