## 3 Results and Discussion
### The Effects of Tooth Number and Module on Scaffing Load Capacity
#### 3.1.2 Comparison of Contact Ratio, Specific Sliding, and Load-Carrying Capacity  
First, the contact ratio and tooth root bending safety factor for the five aforementioned gear sets were calculated. Figure 5a illustrates that the total contact ratio increases linearly with decreasing module and increasing number of teeth (Denser Teeth Design), ranging from 3.884 to 4.814, which is a 23.94% increase. The most notable enhancement is observed in the overlap ratio \(\varepsilon_{\beta}\), while the transverse contact ratio remains largely stable. This indicates that the design strategy of Denser Teeth Design is effective for enhancing the gear contact ratio. Consequently, this can positively influence the smoothness of gear motion and other NVH (Noise, Vibration, and Harshness) performance. Following the design of a gear pair with denser teeth, the tooth root bending stress and safety factor become the critical constraints, especially when the module is reduced and the external load remains constant. The bending fatigue safety factor at the tooth root is generally required to be no less than 1.2 to ensure reliability. Figure 5b demonstrates that, with the speed ratio kept essentially constant, the bending safety factor under peak load conditions decreases from 1.71 to 1.24 as the number of teeth on the pinion increases from 22 to 30. This trend clearly indicates that the combination of tooth numbers 30 and 87 represents an extreme value for this specific center distance, highlighting the need for careful consideration in gear design to maintain safety margins. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Kinematic Viscosity** & **Kinematic Viscosity** & **Density @15** & **FZG Scaffing Test** \\
**@40 \({}^{\circ}\)C** & **@100 \({}^{\circ}\)C** & \({}^{\circ}\)C** & **Grade** \\ \hline
18 mm\({}^{2}\)/s & 4.6 mm\({}^{2}\)/s & 850 kg/m\({}^{3}\) & 12 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The parameters of gear lubrication oil. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  
The curves in Figure 6 illustrate the variation in sliding velocity and specific sliding with respect to the module. It is observable that below the pitch circle, during the entry phase, the sliding velocity decreases with an increase in the number of teeth. Conversely, above the pitch circle, during the exit phase, the sliding velocity increases. The specific sliding is the same trend as the sliding velocity. However, there is a distinction: the sliding velocity at the instantaneous contact point increases linearly with the diameter distance from the point to the pitch circle. In contrast, the sliding rate at the instantaneous contact point shows a nonlinear increase with the diameter distance from the point to the pitch circle. Furthermore, the sliding rate's rate of change is more significant below the pitch circle and less pronounced above it. Tooth surface load distribution is a critical indicator for assessing the load-carrying capacity of gears. Following the design modifications that increase tooth number and reduce module, it is essential to examine the resulting changes in tooth surface load. This examination also explores the impact of various parameters on the load-carrying capacity of gears, thereby providing a foundation for the subsequent analysis of TSF. Table 5 presents the tooth surface pressure distribution for five gear sets, with the left column indicating the highest speed condition and the right column indicating the peak torque condition. In the highest speed condition, the tooth surface pressure remains largely stable despite the reduction in the module, with a minimum of 877.9 MPa and a maximum of 894 MPa and fluctuations not exceeding 1.8%. In contrast, under the peak torque condition, there is a reduction in tooth surface pressure with module reduction,  
Figure 5: The tooth contact ratio and bend safety factors for variable modulus. (**a**) Contact ratio; (**b**) tooth bend safety factors. Figure 6: The relative sliding velocity and specific sliding of tooth surfaces.