## 6 Conclusions  
In this paper, a TWR test on a level gear is analyzed by combining tests and simulations, and the stress distribution of the central level gear is predicted based on the simulation analysis, and the stress concentration problem of the tooth root and the back of the spoke caused by TWR is further expounded.  
1. The fatigue test of the driven gear is performed. The time-speed-sound pressure diagram of the prefabricated gear fracture process and the max stress at the position monitored in the test and the resonance speed are obtained. The results show that the new refined gear will not break even if it works in the resonant rotational speed range of BTW of the 4 th ND, but if the gear has an initial defect and works in the resonant rotational speed range of the BTW of the 4 th ND, the gear will break.
2. The modal analysis of driven level gears is performed; in view of the TWR theory, the influence of spoke plate thickness and working temperature on the TWR characteristics of the gears are studied. Compared with the measurement results, the error between the test value and the simulation value is 0.2-4.7%. According to the results, it is verified that the characteristics of TWR are sensitive to the spoke plate thickness, and with reduced spoke plate thickness, TWR speed shows a downward trend.
3. The transient dynamic analysis of the 3 rd ND and 4 th ND TWR are carried out, the error between the test value and the value achieved when using the transient dynamic finite element analysis model is 1.75-7.06%. The distribution form of contact stress obtained by simulation is in accordance with the law  
**Author Contributions:** Conceptualization, X.L. and Y.G.; methodology, X.L.; software, Y.G.; validation, X.L., Y.G., Z.Z. and Y.S.; formal analysis, Y.G.; investigation, X.L.; resources, G.L.; data curation, X.L. and Y.S.; writing--original draft preparation, X.L. and Y.G.; writing--review and editing, X.L.; visualization, Z.Z.; supervision, Y.S.; project administration, G.L.; funding acquisition, G.L. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by National Natural Science Foundation of China (NSFC), grant number 51579051; and Scientific Research Fund of the Liaoning Education Department, grant number JYT2020010.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Informed consent was obtained from all subjects involved in the study.  
**Data Availability Statement:** Not applicable.  
**Acknowledgments:** The authors would like to acknowledge the financial support of the Scientific Research Fund of the Liaoning Education Department (Grant No. JYT2020010); National Natural Science Foundation of China (NSFC) (Grant No. 51579051).  
**Conflicts of Interest:** The authors declare no conflict of interest. Funders are mainly responsible for the collection and summary of research literature.  
**Nomenclature**  
FTW Forward Traveling Wave BTW backward-traveling wave TWV traveling wave vibration TWR traveling wave resonance NC node-circle ND node-diameter \([M]\) the mass matrix of the node \([C]\) the damping matrix of the node \([K]\) the stiffness matrix of the node \(x(t)\) the displacement of the node \(F(t)\) exciting force \(\sigma_{H}\) contact stress \(F_{n}\) normal force acting on contact surface \(\rho_{1}\) curvature radius of contacted body \(\rho_{2}\) curvature radius of contacted body \(L\) length of contact line \(\mu_{1}\) Poisson's ratio of contacted body \(\mu_{2}\) Poisson's ratio of contacted body \(Z_{E}\) the elastic coefficient \(\rho_{E}\) comprehensive radius of curvature \(y\) lateral displacement \(B(r)\) amplitude at radius r \(\omega\) angular frequency of sector vibrations \(t\) time \(m\) the ND number \(f_{d}\) dynamic frequency of the gear \(f_{F}\) FTW frequency of the gear \(f_{B}\) BTW frequency of the gear \(i\) the gear ratio \(N_{1}\) rotational speed of the driving gear \(N_{2}\) rotational speed of the driven gear \(z_{1}\) teeth number of the driving gear