# A Fractal Prediction Model for the Friction Coefficient of Wet Clutch Friction Plates
## 3 Friction Coefficient Fractal Prediction Model
### Dynamic Prediction Model for Kinetic Friction Coefficient  
Assuming that a pair of rough surfaces, \(S_{1}\) and \(S_{2}\), are in contact, with the contact area denoted as \(A_{i}\), as shown in Figure 2, the heights of the micro-asperities before deformation can be represented as \(z_{1}(x)\) and \(z_{2}(x)\), respectively. Let \(h\) denote the gap between the two equivalent ideal planes of the rough contact.  
Based on the M-B (Mandelbrot-Benoit) fractal model theory, the rough surface profile curve exhibits self-similarity, which means that the rough surface has the same properties when measured at different scales. Due to the limitations of measurement accuracy and the requirements of actual engineering, we only need to perform calculations within a certain level of precision. Therefore, in the subsequent model establishment, we ignore the effects of micro-asperities at smaller scales.  
As shown in Figure 2, assuming the scale range of the curve represents the smallest scale of measurement accuracy, the asperity pair, \(i\), comes into contact at point \(B_{1}\). As the relative motion proceeds, the asperity pair, \(i\), undergoes a relative sliding distance, \(S\). Due to the relative motion and the resulting deformation of the asperities, points \(B_{4}\) and \(B_{5}\), which were initially in contact on the asperity pair, \(i\), now come into contact at point \(B_{2}\), and the point that was in contact at \(B_{1}\) moves to the position of point \(B_{3}\).  
The relative sliding and deformation of the micro-asperities due to physical contact can be quantitatively analyzed. Let the deformation of points \(B_{4}\) and \(B_{5}\) on the asperity pair be \(\delta_{1}^{(i)}\) and \(\delta_{2}^{(i)}\), respectively. These deformations can be resolved into the x and z directions, represented as \(\delta_{x1}^{(i)}\), \(\delta_{x1}^{(i)}\), \(\delta_{x2}^{(i)}\), and \(\delta_{x2}^{(i)}\). The change in the gap between the two surfaces can be expressed as:  
\[\Delta h=S\tan\alpha-\delta_{2}^{(i)} \tag{38}\]  
Figure 2: Schematic diagram of the contact between a single pair of micro-asperities.  
where \(\delta_{z}^{\rm(i)}\)= \(\delta_{z1}^{\rm(i)}+\delta_{z2}^{\rm(i)}\), \(\alpha\) is the angle of the contact surface in degrees; it can be approximated as:  
\[\tan\alpha=\partial z_{1}/\partial x_{1} \tag{39}\]  
Substituting this into Equation (38), and by differentiating Equation (38), we obtain:  
\[V_{z}=V_{x}\frac{\partial z_{1}}{\partial x_{1}}-\frac{\partial\delta_{z}^{ \rm(i)}}{\partial t} \tag{40}\]  
where \(V_{x}\) can be expressed as \(\partial S/\partial t\), and \(V_{z}\) can be expressed as \(\partial h/\partial t\).  
Considering the distribution and contact of micro-asperities over the entire contact surface, and since the majority of micro-contact surfaces have a small angle with the reference plane, it can be assumed that \(A_{rc}=A_{0}\), where the average slope satisfies \(0<c<1\). By integrating Equation (40) into all micro-asperities across the entire contact surface, the friction force generated by the micro-asperities can be obtained as follows:  
\[F=cF_{\rm N}+\left(1-c^{2}\right)\tau A_{\rm rp} \tag{41}\]  
where \(\tau\) is the adhesive shear strength of the lining material.  
Therefore, the dynamic friction coefficient is:  
\[f_{d}=\frac{F}{W} \tag{42}\]  
where \(W\) is the normal load in N.