## 5 Conclusions  
In this study, the sensitivity of an S-shaped tooth profile gear pair with a few pinion teeth to shaft misalignments has been investigated, and theoretical derivations and numerical examples in support of the new gear pair have been presented. The theoretical analysis and numerical calculation of the spur gear and the helical gear showed that an S-shaped tooth profile gear with a few teeth has a good meshing performance under a certain range of center distance error, although its center distance does not exhibit separability of the center distance. It is not sensitive to the center-to-center distance error within a certain range of values, and the transmission error presents a certain rule. Specifically, the following conclusions can be drawn:  
1. For the spur gear, when there is only a certain range of center distance error \(\Delta a\) (0-0.7 mm), the transmission error of involute gear is smaller than that of S-shaped tooth profile, and the transmission error increases with the increase in the center distance error. When there is a central error \(\Delta a=0.1\) mm, \(\delta_{x}=0.02^{\circ}\), \(\delta_{y}=0.02^{\circ}\), the transmission error of the S-shaped tooth profile gear is obviously smaller than that of the involute gear, and the transmission error difference is a maximum of 0.5\({}^{\circ}\) and a minimum of 0.1\({}^{\circ}\) in one engagement cycle. Therefore, the meshing performance of the S-shaped tooth profile is less sensitive to shaft deviation than that of the involute gear, which has obvious advantages over the involute gear. 2. For the helical gear, when there is only a certain range of center distance error \(\Delta\) A (0-1 mm), the transmission error of the involute gear is greater than that of the S-shaped tooth profile. When the center error is \(\Delta a=0.1\) mm, \(\delta_{x}=0.02^{\circ}\), \(\delta_{y}=0.02^{\circ}\), the transmission error of the S-shaped tooth profile gear is smaller than that of the involute tooth profile, and the transmission error gap is stable at 0.02\({}^{\circ}\). It can be seen that the helical gear has fewer advantages in shaft deviation sensitivity than the straight gear, but similarly, the S-shaped tooth profile has a greater number of superiorities. Meanwhile, a visible, interactive, and high efficiency model of TCA has been proposed which can obtain the contact pattern and meshing path and display them intuitively. To summarize, applying an S-shaped tooth profile instead of an involute gear has advantages in bearing capacity and meshing performance, enjoying a good application prospect. The related studies on this innovative gear drive, which include experiments on prototypes and processing methods, are being carried out. Efforts putting this drive forward into practical application are also needed soon. Figure 22: Change regularity of transmission error with shaft misalignment. (a) Contact trace and meshing path; (b) transmission error. **Author Contributions:** Methodology, W.C.; software, Q.S. and Y.T. ; validation, W.C.; formal analysis, Y.T. and Q.S. ; resources, Q.S. and W.C.; data curation, Q.S. and W.C.; writing--original draft preparation, Y.T. and Q.S. ; writing--review and editing, P.Z., Q.S. and W.C.; visualization, Q.S. and P.Z. ; supervision, Q.S. and W.C. All authors have read and agreed to the published version of the manuscript. **Funding:** The authors gratefully acknowledge the financial support provided to this study by the Natural Science Foundation of Hebei Province under Grant No. E2021210066. **Informed Consent Statement:** In this study, all participants have been fully informed about the purpose, methods, and potential risks involved, and have provided informed consent by signing a consent form. Participants are aware of their voluntary participation and have the right to withdraw from the study at any time without any adverse consequences. We strictly adhere to principles of privacy protection and maintain confidentiality of participants' personal information. **Data Availability Statement:** Not applicable.