# Geometric Design of a Face Gear Drive with Low Sliding Ratio
## 6 Conclusions  
This paper presented a design method for face gear drive associated with a low sliding ratio. Based on the analysis and results, the conclusions are as follows:  
(1) Based on the provided contact path function \(y=a_{1}x+a_{2}x^{3}\), general mathematical models of a pair of the conjugated spur gear and rack with low sliding ratio were given, including the working and the fillet profiles;  
(2) Using the established spur gear as the shaper to envelope the face gear, the mathematical models of the working and the fillet surfaces of the face gear were obtained, and the tooth width restriction conditions were established, including the undercutting of the inner and the pointing of the outer. The results show that the tooth width of the proposed face gear was larger than that of general face gear, due to the smaller inner radius. Furthermore, as the coefficients \(a_{1}\) or \(a_{2}\) increased, the inner radius and the outer radius increased; conversely, the tooth width of the face gear increased.  
(3) An explicit solution method for the contact point of the tooth surface of the face gear pair was provided, which is more convenient than the general solution method.  
Figure 13: Motion simulation of the proposed face gear pair.  
Additionally, this method is also suitable for solving the meshing points of the general face gear drive, in which case, let \(a_{2}=0\);  
(4) The calculation method of the sliding ratio of the spatial meshing of the face gear pair was studied, and the effects of design parameters on the sliding ratio were analyzed. The sliding ratio of the proposed face gear pair was lower than those of the general face gear pair. Moreover, as the coefficient \(a_{1}\) or \(a_{2}\) decreased, the sliding ratio of the face gear drive decreased.  
(5) The rationality of the proposed face gear was demonstrated by accurate 3D modeling and motion simulation. However, to determine the optimum design and precision machining, further studies are needed.  
Conceptualization, W.S. and J.Z.; methodology, W.S.; software, J.Z.; validation, H.Z. and R.Z.; formal analysis, W.S. and J.Z.; investigation, R.Z.; resources, Z.L.; data curation, W.S.; writing--original draft preparation, W.S.; writing--review and editing, J.Z. and H.Z.; visualization, H.Z.; supervision, Z.L.; project administration, Z.L.; funding acquisition, Z.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Key R&D Program of China, grant number 2019 YFB2004700; National Natural Science Foundation of China, grant number 52175053; National Science and Technology Major Project, grant number 2019-VII-0017-0158.  
Not applicable.  
This research was supported by National Key R&D Program of China (Grant No. 2019 YFB2004700), National Natural Science Foundation of China (Grant No. 52175053), and National Science and Technology Major Project (Grant No. 2019-VII-0017-0158). The authors also sincerely appreciate the comments and suggestions for modifications made by the editors and anonymous referees.  
The authors declare no conflict of interest.