## 4 Conclusions  
Due to the introduction of the regenerative braking system, the service life of the BEUB main reducer gears is reduced; this paper found that, after the introduction of the regenerative braking system, the braking mode of BEUB changes to electromagnetic braking, and the electromagnetic braking would cause the reverse torque of the driving axle during braking, thus changing the meshing condition of the main reducer gears. Therefore, when calculating the fatigue life of the main reducer gears, the factor of reverse torque is especially taken into account and compared with friction braking.  
Based on the UDDS urban cycle condition, the torque load spectrums of electromagnetic braking and friction braking are calculated and converted into bending stress and contact stress spectrums.  
Considering the influence of electromagnetic braking, the tooth root bending fatigue lives of the pinion convex and concave surface are 78.5% and 78.9%, respectively, of  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Braking Method** & **Stress Type** & & **Cycle Index** \\ \hline \multirow{3}{*}{electromagnetic braking} & bending stress & concave & 4.926 \(\times\) 10\({}^{6}\) \\  & contact stress & convex & 3.618 \(\times\) 10\({}^{9}\) \\  & bending stress & & 4.558 \(\times\) 10\({}^{4}\) \\ \multirow{3}{*}{friction braking} & bending stress & concave & 6.236 \(\times\) 10\({}^{6}\) \\  & contact stress & & 5.825 \(\times\) 10\({}^{4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Fatigue life of pinion dangerous nodes.  
those under the traditional friction braking, and the contact fatigue of the concave surface of the pinion is 78.2%.
3. The braking energy recovery system of the BEUB has a significant influence on the fatigue life of the main reducer gears. The main reason is that the introduction of reverse braking torque changes the gear working condition, thus leading to the main reducer gear meshing surface changing frequently during the BEUB operation, which reduces the main reducer gears' service life sharply. Therefore, more attention should be paid to the meshing performance of the nonworking faces when designing and manufacturing the BEUB main reducer gears. At the same time, the change in meshing would also have an impact, which should be taken into account in the design of the main reducer gears, so as to improve the comprehensive performance of the gear more comprehensively.
4. The proposed method is more consistent with the actual working conditions of BEUB and is also applicable to the gear transmission of other fuel vehicles fitted with retarders for braking, which is the same as that of the BEUB with the introduction of the regenerative braking system, so the method proposed in this article is equally applicable to the calculation of the fatigue life of the main reducer gears of a vehicle fitted with retarders.  
Conceptualization, J.D. and X.W.; methodology, J.D., X.W. and J.M.; software, J.D. and X.W.; validation, J.D. and J.M.; investigation, X.W.; resources, J.D.; data curation, X.W.; writing--original draft preparation, X.W.; writing--review and editing, J.D.; visualization, X.W. and J.M. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (grant nos. 51705419 and 61701397), China; The Natural Science Basic Research Program of Shaanxi (grant no. 2022 JM271), China; and the Postdoctoral Research Project of Shaanxi Province (grant no. 2018 BSHEDZZ10), China.  
Not applicable.  
We acknowledge the support provided to this study by Xi'an University of Technology, Xi'an University of Technology, in the form of time and facilities.  
The authors declare no conflict of interest.  
References