## 5 Conclusions  
Based on fractal geometry theory, this study establishes three-dimensional models of spiral bevel gears under different fractal dimensions and analyzes the dynamic characteristics of these gears. A modeling method for spiral bevel gears with different roughness is proposed by combining the W m function with the fractal dimension. Utilizing the finite element method, the single time-varying mesh stiffness is obtained, leading to a method for solving the total time-varying mesh stiffness of spiral bevel gears. By combining the time-varying mesh stiffness with dynamic equations, the vibration velocity and acceleration of spiral bevel gears with different surface roughness are revealed. A detailed analysis is conducted on the complex relationship between the microstructure of the gear surface and its mechanical behavior during meshing.  
As the fractal dimension decreases, the tooth surface becomes rougher, and the time-varying mesh stiffness decreases, and both vibrational velocity and vibrational acceleration increase. The specific results are as follows:  
1. Fractal dimensions are introduced into the involute equation. The MATLAB software is utilized to calculate the involute profiles under different roughness levels. By establishing the relationship between fractal dimensions and tooth surface roughness, the tooth profiles under varying roughness conditions are determined. The three-dimensional models of spiral bevel gears under different surface roughness levels are obtained using the principle of tooth cutting with a gear hob.
2. A novel model is proposed for calculating the mesh stiffness of spiral bevel gears under different fractal dimensions. Fractal dimensions are introduced into the differential dynamic equations of spiral bevel gears to determine the factors influencing the dynamic characteristics due to fractal dimensions and surface friction. Finite element analysis is employed to investigate the models of spiral bevel gears under various fractal dimensions, resulting in the determination of the time-varying mesh stiffness under different fractal dimensions. The results indicate that as the fractal dimension decreases, the time-varying mesh stiffness continuously decreases. Therefore, reducing the fractal dimension to some extent contributes to decreasing the time-varying mesh stiffness of the gears, thereby enhancing the smoothness of motion.
3. After establishing an experimental setup for spiral bevel gears, tests showed that the errors under two operating conditions were found to be 3.22% and 6.11%, respectively, indicating a small margin of error and validating the accuracy of the model. Through dynamic analysis using the developed model, the vibration characteristics of gears under different fractal dimensions were examined.  
The study can take insights into the variations in vibration characteristics under various fractal dimensions for spiral bevel gears, providing a new model for investigating the vibration behavior of spiral bevel gears under different roughness conditions.  
Methodology, X.-Q.Z.; Investigation, S.-W.Y.; Data curation, H.-Q.L. and L.-N.W.; Funding acquisition, Z.-J.C. and R.S. All authors have read and agreed to the published version of the manuscript.  
This work was funded by the China Aviation Development Innovation Special Fund Project (No. KY-1003-2021-0020) and the Development Funds of the Central Government for Local Science and Technology (No. 236Z1701G).  
Not applicable.  
The data presented in this study are available on request from the corresponding author. The data is not publicly available.  
The authors declare no conflict of interest.