# A Fractal Prediction Model for the Friction Coefficient of Wet Clutch Friction Plates
## 2 Friction Plate Surface Roughness Contact Characteristics Research
### Rough Surface Fractal Prediction Model  
M-B fractal model concluded that, during the process of rough surface contact, the deformation of asperities follows a certain pattern: From the onset of asperity contact, the taller asperities begin to deform before the others. At this stage, due to the small amount of deformation, plastic deformation occurs. As the gap between the friction pair decreases, the asperities that have already undergone plastic deformation increase in deformation, and smaller asperities coalesce to form larger asperities. At the same time, the deformation area of asperities with larger deformation areas increases linearly. The relationship between the fractal dimension \(D_{\mathrm{s}}\) of the rough surface satisfying the fractal model and the fractal dimension \(D\) of the curve of the profile satisfies the equation:  
\[D_{\mathrm{s}}=D+1 \tag{4}\]  
Therefore, the fractal dimension of the profile curve of rough surfaces can be used to study the fractal characteristics of rough surfaces. Previous studies have shown that when the measured rough surface has statistical self-imitation fractal characteristics, the structure function measure and root mean square measure can be used to effectively calculate and characterize the surface fractal. The structure function measurement method of the surface contour is calculated as follows:  
\[S(\tau)=\Big{\langle}[Z(x+\tau)-Z(x)]^{2}\Big{\rangle}=G\tau^{4-2D}\] (Equations (5) and (6) can be written as consistent power-law relations:  
\[M(\tau)=G\tau^{a(D)} \tag{7}\]  
where \(M(\tau)\) represent structure function or root mean square equation, \(G\) is the scale coefficient, \(\tau\) is the measurement scale, and \(\alpha(D)\) is a function related to the fractal dimension. Formula (7) is a straight line in log-double coordinates, and the logarithm of both sides is taken to obtain:  
\[\lg M(\tau)=\lg G+\alpha(D)\lg\tau \tag{8}\]  
Therefore, the fractal parameters of the surface contour can be measured by the line made by the analytical Formula (8). The score dimension, \(D\), can be obtained from the slope, \(\alpha(D)\), of the line, and the scale coefficient, \(G\), can be obtained from the intercept, \(\lg C\), of the line on the axis \(y\). Thus, the features of the surface contour curve can be characterized by the fractal feature parameters \(D\) and \(G\). The fractal dimension, \(D\), represents the complexity of the rough surface, and the scale parameter, \(G\), represents the scale of the rough surface. However, neither of these two fractal parameters can reflect the characteristics of a rough surface alone and must have both fractal dimension \(D\) and scale parameter \(G\) in order to truly describe the shape and size of the rough surface. According to the hypothesis of M-B fractal theory, the curve that satisfies the fractal theory of rough surface is composed of countless different sinusoidal waves stacked together, thus forming a rough surface. By analyzing the constructor W-M function, we can see that the micro-convex curve before deformation can be expressed as:  
\[z(x)=G^{D-1}l^{2-D}\cos\Bigl{(}\frac{\tau x}{l}\Bigr{)},-\frac{l}{2}<x<\frac{l }{2} \tag{9}\]  
It can be concluded that the top curvature radius of the micro-convex body without deformation is:  
\[R=\left|\frac{1}{\left(\frac{d^{2}z(x)}{dx^{2}}\right)_{z=0}}\right|=\frac{l^{ D}}{\pi^{2}G^{D-1}}=\frac{a^{\frac{D}{2}}}{\pi^{2}G^{D-1}} \tag{10}\]  
The expression of the critical deformation in the M-B fractal model can be obtained as follows:  
\[\delta_{c}=\left(\frac{K\sigma_{y}}{2}\right)^{2}\frac{a^{\frac{D}{2}}}{G^{D-1}} \tag{11}\]  
The analysis Formula (11) shows that the critical deformation, \(\delta_{c}\), is positively correlated with the deformation area, \(a\), of the micro-convex body.