# Fractal Contact Mechanics Model for the Rough Surface of a Beveloid Gear with Elliptical Asperities
## 2 A New Fractal Characterization Approach for a Rough Surface Texture  
[PERSON] and [PERSON] [38] established a three-dimensional fractal function model for rough surfaces in 1985 by improving the WM function and demonstrating that the fractal function model still satisfies randomness, multiscale, and self-similarity, and the expression of the three-dimensional fractal function model is:  
\[ \begin{split} z(x,y)=L\Big{(}\frac{G}{L}\Big{)}^{(D-2)}\Big{(} \frac{\ln g}{M}\Big{)}^{1/2}\frac{M}{\sum\limits_{m=1}^{M}\sum\limits_{n=n_{ \min}}^{n_{\max}}g^{(D-3)n}\\ \times\{\cos j_{m,n}-\cos[(2 pg^{n}f(x,y)/L)\cos(g(x,y)-a_{m})+j_{ m,n}]\}\end{split}  \tag{1}\]  
where, \(z(x,y)\) is the three-dimensional morphological height of the rough surface. \(D\) is the fractal dimension. \(G\) is the fractal roughness. \(n\) is the frequency index, \(n=\ln/l\ln\gamma\), \(l=1/\gamma_{n}\). \(\varphi_{m,n}\) is the random phase. \(\gamma_{n}\) is the spatial frequency of the rough surface. \(M\) indicates the number of overlapping elevated parts of the rough surface. When \(M=1\), the rough surface morphological characteristics show isotropic, when \(M\
eq 1\), the rough surface morphological characteristics show anisotropic. \(\alpha_{m}\) is the three-dimensional rough surface cosine direction, \(\alpha_{m}=\pi m/M\).\(f(x,y)\) and \(g(x,y)\) are the functions of independent variables related to the morphological characteristics of the rough surface, respectively. Roughness measurements, as seen in Figure 1a, showed that the asperities dispersed on the tooth surface may be approximated by an equivalent ellipsoid and that the distribution law of the elliptical asperity on the rough tooth surface exhibits some anisotropy, as seen in Figure 1b. To this purpose, this paper improves the W-M fractal function by making \(M=1\) and \(m=1\) in Equation (1), then the dual-autonomous W-M fractal function will degenerate to a single-autonomous W-M fractal function, whose expression can be expressed as:  
\[z(x)=L\Big{(}\frac{G}{L}\Big{)}^{(D-1)}(\ln\gamma)^{1/2}\sum\limits_{n=n_{ \min}}^{n_{\max}}\cos\varphi_{1,n}-\cos(2\pi\gamma^{n}/L-\varphi_{1,n})\gamma^{ (D-2)n} \tag{2}\]  
of a single elliptical asperity was established in this paper; \(p\) is the normal load applied to the elliptical asperity by a rigid plane, \(\omega_{n}\) is the actual deformation of the elliptical asperity, and its value is between 0 and \(\delta\). The profile curve \(z(x,y)\) of a single elliptical asperity with the major diameter \(l_{x}\) and the minor diameter \(l_{y}\) of the elliptical region as the base can be deduced from Equation (3):  
\[z(x,y)=(\ln\gamma_{1})^{1/2}G_{1}^{D_{1}-1}l_{x}{}^{2-D_{1}}\cos\left(\frac{2 \pi x}{l_{x}}\right)+(\ln\gamma_{2})^{1/2}G_{2}^{D_{2}-1}l_{y}{}^{2-D_{2}}\cos \left(\frac{2\pi y}{l_{y}}\right) \tag{4}\]  
The distance \(\delta_{n}\) between the tip of the elliptical asperity and the base can be presented as:  
\[\delta_{n}=z(x,y)\big{|}_{x=0,y=0}=(\ln\gamma_{1})^{1/2}G_{1}^{D_{1}-1}l_{x}{}^ {2-D_{1}}+(\ln\gamma_{2})^{1/2}G_{2}^{D_{2}-1}l_{y}{}^{2-D_{2}} \tag{5}\]  
The normal deformation \(\omega_{n}\) of the elliptical asperity can be expressed as:  
\[\omega_{n}=2(\ln\gamma_{1})^{1/2}G_{1}^{(D_{1}-1)}(2\tau_{x})^{(2-D_{1})}=2( \ln\gamma_{2})^{1/2}G_{2}^{(D_{2}-1)}(2\tau_{y})^{(2-D_{2})} \tag{6}\]  
The geometric relationships between the normal deformation of the elliptical asperity and the effective radii of curvature are illustrated in Figure 4 and the expressions are described in Equations (7) and (8):  
\[(R_{x}-\omega_{n})^{2}+r_{x}^{2}=R_{x}^{2} \tag{7}\]  
\[(R_{y}-\omega_{n})^{2}+r_{y}^{2}=R_{y}^{2} \tag{8}\]  
The semiminor and semimajor radius of the elliptical asperity could well be evaluated also as the normal deformation \(\omega_{n}\) is significantly smaller than the curvature radius of the elliptical asperity. \[2R_{x}\omega_{n}=r_{x}^{2} \tag{9}\]  
\[2R_{y}\omega_{n}=r_{y}^{2} \tag{10}\]  
Figure 4: The geometric model of a single elliptical asperity.