## 4 Cases Analysis
### S-Shaped Tooth Profile Spur Gear  
In order to explore the influence of the center distance error \(\Delta a\) on the contact performance individually, the control variable method was adopted to obtain the contact performance and transmission error of the tooth flank under the condition of a shaft deflection error of \(\delta_{x}=0\), \(\delta_{y}=0\). As shown in Figure 15a, the contact path (solid line perpendicular to the axis in the figure) and the contact path (white line in the figure) from the top of the gear tooth flank to the root (white line in the figure) for standard installation without error are given, which shows that the tooth contact is relatively stable. As shown in Figure 15b, the transmission error curves are given, which shows that the transmission error is zero when the pinion rotation angle range is \(\kappa_{1}\in[-96^{\circ},98^{\circ}]\). On the one hand, this proves the correctness of the above mathematical models; on the other hand, the S-shaped tooth profile gear meets the conjugate characteristics of the constant ratio transmission. In Figure 16, the tooth meshing path and contact trace are given under the condition that the error of the center distance is \(\Delta a=0.1\), \(\delta_{x}=0\), \(\delta_{y}=0\), and there are no other errors, which shows that the effective contact area moves to the top of tooth, but the tooth contact trace moves relatively steadily and the transmission error presents parabolic regularity. In addition, the maximum of error is \(0.078^{\circ}\) and the minimum is \(0.033^{\circ}\). Figure 14: Initial position of meshing point and tooth contact pattern of a spur gear and a helical gear. (**a**) Spur gear; (**b**) helical gear. Figure 15: Contact analysis of center distance without error. (**a**) Contact trace and meshing path; (**b**) transmission error. By using the above method, the change regularity in the transmission error varying with the center distance error from 0 to 0.7 mm was analyzed, and the maximum, minimum, and average error of the transmission error corresponding to an error period are given in Figure 17. For comparative analysis, under the condition of a given center distance error and derivation coefficient \(\lambda^{\prime}=0\) (i.e., involute gear pair with the same parameter), the involute transmission error has a constant value, and the change regularity is shown in Figure 17.  
Through contrastive analysis, the average transmission error of an S-shaped tooth profile gear pair and the transmission error of the involute gear pair with the same parameters both show the change regularity of an approximate proportional function, and their values are approximately equal to each other; but the difference in the transmission error between the maximum and the minimum of an S-shaped tooth profile gear, which is \(\Delta\psi_{\max}\), presents a trend of getting bigger, which shows that the increase in the center distance error of the S-shaped tooth profile gear which is \(\Delta a\) could lead to a larger transmission error fluctuation. Therefore, in the application of the S-shaped tooth profile gear, the transmission error caused by the center distance error needs to be analyzed, t  
Figure 16: Contact analysis with the value of center distance error being 0.1. (**a**) Contact trace and meshing path; (**b**) transmission error. Figure 17: Change regularity in the transmission error with varying center distance error. distance error range under the corresponding working conditions can be obtained so as to ensure the transmission accuracy and reduce the vibration. Considering that any installation error may exist in engineering practice, the contact pattern, meshing path, and transmission error of an S-shaped tooth profile spur gear with center distance error and shaft misalignments are given by Figure 18. At the same time, the involute gear with the same parameters is compared and analyzed.