# Fractal Contact Mechanics Model for the Rough Surface of a Beveloid Gear with Elliptical Asperities
## 5 Numerical Analysis and Discussion of Results
### Effect of Fractal Parameters on Normal Contact Stiffness  
The fractal dimension \(D_{1}=1.36\), the fractal roughness \(G_{1}=G_{2}=1.0\times 10^{-11}\) m, and we set the fractal dimension \(D_{2}\) as 1.36, 1.46, 1.56, 1.66, and 1.76, respectively. The influence of different fractal dimensions on the relationship between the normal contact stiffness and the real contact area can be seen in Figure 14a. As a consequence, when the real contact area is known, the normal contact stiffness is proportional to the fractal dimension, which suggests that the normal contact stiffness rises as the fractal dimension grows. This is because the fractal dimension does have a physical significance that pertains to the smoothness of the rough surface. On a macroscopic level, the larger the fractal dimension, the higher the roughness value of the rough surface, and the more asperities are present in the contact on the rough surface, enhancing the rough surface's normal contact stiffness. The fractal dimension \(D_{1}=D_{2}=1.46\), the fractal roughness \(G_{1}=1.0\times 10^{-10}\) m, and we set the fractal roughness \(G_{2}\) as 1.0 \(\times\)\(10^{-10}\) m, 3.0 \(\times\)\(10^{-10}\) m, 5.0 \(\times\)\(10^{-10}\) m, 7.0 \(\times\)\(10^{-10}\) m, and 9.0 \(\times\)\(10^{-10}\) m, respectively. The influence of the specific fractal roughness values on the relationship between the normal contact stiffness and the real contact area is depicted in Figure 14b. As can be observed, the normal contact stiffness is inversely related to the fractal roughness for a given real contact area, namely, the normal contact stiffness decreases as the fractal roughness grows. This is because when the fractal roughness \(G\) value improves, the rough surface morphology becomes rougher and the number of asperities in the contact diminishes, decreasing the normal contact stiffness of the rough surface. Figure 13: Variations in the dimensionless contact area \(A_{r}^{*}\) with fractal dimension \(D\): (**a**) different eccentricity \(e\); (**b**) different contact coefficient \(\lambda\). The fractal dimension \(D_{1}=D_{2}=1.46\), the fractal roughness \(G_{1}=G_{2}=1.0\times 10^{-11}\) m, and we set the eccentricity \(e\) of the elliptical asperity to 0.1, 0.3, 0.5, and 0.7, respectively. The influence of varying the eccentricity \(e\) on the relationship between the normal contact stiffness and the real contact area is illustrated in Figure 15. As shown in the figure, the normal contact stiffness is inversely proportional to the eccentricity for a given real contact area, which indicates that the normal contact stiffness diminishes as the eccentricity increases. This is because when the eccentricity value grows, the contact area of the single elliptical asperity decreases, resulting in a drop in the rough surface's total actual contact area and consequently a decrease in the rough surface's normal contact stiffness. The fractal dimension \(D_{1}=D_{2}=1.46\), the fractal roughness \(G_{1}=G_{2}=1.0\times 10^{-11}\) m, and we set the contact coefficient to 0.9, 0.7, 0.5, and 0.3, respectively. Figure 15 illustrates the effect of varying the contact coefficients on the relationship between the normal contact stiffness and real contact area. As shown in the graphic, the normal contact stiffness is proportional to the contact coefficients for a given real contact area, which suggests that normal contact stiffness reduces as contact coefficients decrease. This is because the contact coefficients' physical significance is to correct the contact area between two surfaces, and when the integrated curvature of the elastic surface approaches 0, namely, when the two elastic surfaces are in intimate interaction, the contact coefficient value approaches 1 at this time. As the integrated curvature of the elastic surface grows, the contact coefficients and contact area between the elastic surfaces decrease, reducing normal contact stiffness.