# 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
### Modified Model of the Island Area Distribution Function for a Point Contact
#### 3.3.3 Contact Coefficient of the Elliptical Area of a Beveloid Gear  
As shown in Figure 7, the contact area of beveloid gear pairs is elliptical. Assuming that the principal curvatures of tooth surfaces are \(\rho_{11}\), \(\rho_{12}\), \(\rho_{21}\), and \(\rho_{22}\), respectively, the contact area distribution functions of the beveloid gear should satisfy the following relationship:  
\[n^{\prime}(A)=\lambda n(A) \tag{46}\]  
where \(\lambda\) is the contact surface coefficient, which can be expressed as:  
\[\lambda=\left(\frac{S}{\sum S}\right)^{\rho_{m}} \tag{47}\]  
where \(S\) is the theoretical contact area. \(\Sigma S\) is the sum of the surface areas of the two elastic tooth surfaces, and \(\rho_{m}\) is the integrated curvature coefficient, whose expression is:  
\[\rho_{m}=\rho_{11}+\rho_{12}+\rho_{21}+\rho_{22} \tag{48}\]  
According to [PERSON]'s theory, the contact area will be transformed from the theoretical point contact to an approximately elliptical contact under the action of a normal contact load on the tooth surface, and the semimajor radius \(r_{x}\) of the contact ellipse can be expressed as:  
\[r_{x}=\left[\frac{3 FE(e)}{2\pi E^{\prime}(1-e^{2})\rho_{m}}\right]^{1/3} \tag{49}\]  
where \(F\) is the contact load on the tooth surface.  
The elliptical contact area of the beveloid gear pair is:  
\[S=\pi r_{x}r_{y}=\pi r_{x}^{\ 2}\sqrt{1-e^{2}}=\pi\left[\frac{3 FE(e)}{2\pi E^{ \prime}\rho_{m}}\right]^{2/3}\left(1-e^{2}\right)^{1/6} \tag{50}\]  
Figure 7: Schematic diagram of a beveloid gear with the crossed shaft.  
The sum of the surface areas of the two elastic surfaces \(\Sigma\)S can be expressed as:  
\[\sum S=2\pi(\rho_{n1}+\rho_{n2})L \tag{51}\]  
where \(L\) is the contact length of the two gears, and \(\rho_{n1}\) and \(\rho_{n2}\) are the radii of curvature at the nodes of the gear and pinion, respectively, in the form of:  
\[\rho_{n1}=\frac{d_{1}\sin\alpha_{n}}{2\cos\beta},\ \rho_{n2}=\frac{d_{2}\sin \alpha_{n}}{2\cos\beta} \tag{52}\]  
Substituting Equations (50)-(52) into Equation (47), respectively, the contact coefficient of the elliptical contact area can be expressed as:  
\[\lambda=\left\{\begin{array}{c}\frac{2\pi\left[\frac{3 FE(e)}{2\pi E\rho_{m}} \right]^{2/3}\cos\beta}{\left(d_{1}+d_{2}\right)\sin\alpha_{n}L(1-e^{2})^{-1/6} }\right\}^{\rho_{m}} \tag{53}\]  
Figure 8 shows the trend of the contact coefficient \(\lambda\). The eccentricity \(e\) = 0.3, the number of teeth of the gear is 45, the number of teeth of the pinion is 29, the normal pressure angle \(\alpha_{n}\) = 20\({}^{\circ}\), and the modulus \(m\) = 4 mm. As seen in Figure 8a, the contact coefficient reduces as the integrated curvature of the elastic surface increases. When the integrated curvature of the elastic surface tends to 0, the two elastic surfaces are approximately in-plane contact and the surface contact coefficient \(\lambda\) tends to 1. As seen in Figure 8b, the contact coefficient rises when the contact load increases. This is because as the contact load increases, the contact area of the elliptical asperities grows proportionately, resulting in an increase in the surface contact coefficient.  
Therefore, the total elliptical contact area can be expressed as:  
\[A_{r}=\int_{0}^{d_{1}}\lambda n^{\prime}(a)ada=\lambda\frac{\zeta(D_{1}+D_{2}) }{2-\zeta(D_{1}+D_{2})}a_{l} \tag{54}\]