# Mesh Stiffness and Dynamic Modeling and Analysis of Modified Straight Bevel Gears
## 6 Conclusions  
This study addresses the meshing stiffness of straight bevel gears under various modification conditions and verifies the results using the finite element method. Additionally, a dynamic model of the straight bevel gear transmission system is developed using the finite element node method to analyze the effects of different modification conditions on the dynamic characteristics of straight bevel gears. The findings are as follows:  
1. The analytical algorithm for meshing stiffness of SBGs in this paper has been validated through comparison with the finite element method. The finite element calculation in this paper takes about 6 h (varying significantly based on computer configuration), while the analytical algorithm often requires only a few seconds, greatly improving efficiency while ensuring accuracy. Therefore, in dynamic simulation analysis, different modification schemes can be quickly designed according to vibration requirements.
2. With profile modification, the abrupt changes in nonlinear stiffness excitation are smoothed, reducing dynamic transmission error within a specific range. When designing profile modifications, it is crucial to consider the varying requirements of time-varying meshing characteristics and vibration response characteristics when selecting profile modification amounts.
3. Considering axial modification, the reduction in nonlinear time-varying meshing stiffness increases the dynamic transmission error amplitude, affecting the system's natural frequencies. Consequently, resonance peaks shift. In designing axial modifications, the axial modification amount can be adjusted according to actual operating conditions to move resonance peaks away from the operating speed.
4. When assessing the combined effects of profile and axial modifications, the change in vibration response is not merely additive, and their impacts on vibration response differ significantly. Therefore, when selecting modification parameters based on gear pair parameters and the vibration response induced by time-varying meshing stiffness excitation, minimizing or avoiding axial modification is advisable if resonance peaks do not require adjustment. Instead, control the profile modification amount to reduce system vibration while ensuring stability, thus achieving vibration reduction through modification.  
Although this paper accounts for the coupling effects of gears, bearings, and shafts, it does not address the role of the gearbox housing and the flexibility of the gear disc. This oversight could result in neglecting the lateral vibration of the system's web and the resonance of the housing under high-speed and heavy-load conditions. Future research will aim to refine the model further to more comprehensively consider all components of the entire transmission system.  
Validation, D.Z.; Formal analysis, Z.-Y.T.; Resources, J.-Y.T. and Z.S.; Writing--original draft, W.-T.L.; Funding acquisition, Z.-H.H. All authors have read and agreed to the published version of the manuscript.  
This work is supported by the National Natural Science Foundation of China (NSFC) through Grant No. 52375071. The authors also gratefully acknowledge the support of the Hunan Provincial Natural Science Foundation of China through Grant No. 2023 JJ20066 and No. 2023 JJ40748.  
Not applicable.  
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy.  
Author [PERSON] was employed by the Aero Engine Corporation of China. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.