# 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 the Contact Load  
The mechanical properties of the model presented in this research are compared to those of the MB contact model in Figure 11. The plastic yield stress of the material \(\sigma_{y}=235\times 10^{6}\) Pa, the elastic modulus \(E=2.06\times 10^{11}\) Pa, and the Poisson's ratio \(\
u=0.26\). The model in this paper and the revised MB model take the same fractal parameters, namely the fractal dimension \(D=D_{1}=D_{2}=1.46\), the fractal roughness \(G=G_{1}=G_{2}=1.0\times 10^{-10}\) m, and the eccentricity of the model in this paper \(e=0.2\). As can be seen, the mechanical  
Figure 10: Relationship between the ratio of the elastic contact area to the real contact area and fractal dimension \(D\) at different eccentricity \(e\). Figure 9: Variations of the dimensionless contact area \(A_{r}^{*}\) with fractal dimension \(D\): (**a**) different eccentricity \(e\); (**b**) different contact coefficient \(\lambda\). curves of the model in this paper deviate from those of the MB model, and the real contact load calculated by the model in this paper is less than the real contact load calculated by the MB model at the same real contact area. This is because the eccentricity consequences of the elliptical asperity are accounted for in the calculation model in this paper; hence, the dimensionless contact load of the model in this paper is reduced at the same real contact area. 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 relationship between the real contact area and the real contact load for different fractal dimensions can be seen in Figure 12a; it can be observed that the real contact load grows as the fractal dimension increases. The reason for this is that the fractal dimension \(D\) is positively associated with the rough surface's smoothness, and as the fractal dimension rises, the rough surface's topology performs more finely, decreasing the real contact load on the rough surface. 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. Figure 12b illustrates the relationship between the real contact area and the real contact load for various fractal roughness values, demonstrating that the real contact load reduces as the fractal roughness increases. This is because the fractal roughness \(G\) is inversely related to the rough surface's smoothness. When the fractal roughness \(G\) is raised, the projections and depressions in the rough surface topology increase, resulting in a reduction in the rough surface's real contact load. Figure 11: Comparison of the mechanical properties between the model in this paper and the MB contact model. Figure 12: Variations in the dimensionless contact area \(A_{r}^{*}\) with fractal dimension \(D\): (**a**) different fractal dimensions \(D\); (**b**) different fractal roughness \(G\). 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 relationship between the real contact area and the real contact load with increasing eccentricity \(e\) can be seen in Figure 13a. As seen in the figure, the real contact load reduces as the eccentricity \(e\) grows. This is because whenever the eccentricity \(e\) grows, the contact area of a single elliptical asperity diminishes, reducing the total contact area of the rough surface and, hence, reducing the rough surface's real contact load. 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.