# A Fractal Prediction Model for the Friction Coefficient of Wet Clutch Friction Plates
## 4 Experimental Verification
### Experiment on the Variation of Friction Disc Surface Topography and Coefficient of Friction  
For the friction discs subjected to a mating pressure of 1.2 MPa, a rotational speed of 3000 rpm, and a mating interval of 30 s, the machine is disassembled and the friction discs are removed after every 200 matings. The friction discs are then cleaned thoroughly to remove any lubricating oil, and the surface topography of the friction discs is measured using the RTEC 3D profilometer. The obtained data are processed to create surface topography maps. Figure 12 illustrates the change in the surface topography of the friction discs with the number of matings.  
Figure 12: Variation of friction disc surface topography with the number of engagements. (**a**) Engagement times, 0; (**b**) Engagement times, 200; (**c**) Engagement times, 400; (**d**) Engagement times, 600; (**e**) Engagement times, 800; (**f**) Engagement times, 1000.  
From Figure 12, it can be seen that the darker the color in the image, the higher the point's height, and the lighter the color, the deeper the point's depth. It can be observed that as the number of engagements increases, the surface gradually becomes smoother. The data obtained from the experiment are calculated based on the structure function measurement dimension method to obtain the fractal dimension, \(D\), and the scale factor, \(G\)[25], as shown in Figure 13a,b.  
The average friction coefficient and engagement time for each engagement are extracted. Based on the surface topography parameters, the friction coefficient fractal prediction model (Equations (30)-(42)) and the wet friction clutch motion differential equation (Equation (29)) can be solved to obtain the predicted change curves of the engaging time and the average friction coefficient, and then compared with the above experimental data, as shown in Figure 13c,d.  
For the input parameters, only the surface topography parameters (\(D\), \(G\), rms) of each point change, and the rest are fixed: for each input condition of the differential equation of motion, circumferential (\(I_{\text{i}}\), \(I_{1}\), \(I_{2}\), and \(I_{0}\) are the moment of inertia at the input end, the active end, the driven end, and the load end, respectively; kg\(\cdot\)m\({}^{2}\), \(k_{1}\), and \(k_{2}\) are the torsional stiffness of the input and output axes, N\(\cdot\)m\(\cdot\)s/rad; \(c_{1}\) and \(c_{2}\) are the torsional damping of input shaft and output shaft, N\(\cdot\)m\(\cdot\)s/rad; \(T_{\text{i}}\), \(T_{\text{f}}\), and \(T_{\text{o}}\) are, respectively, input torque, friction clutch transmission torque, and output torque, N\(\cdot\)m), axial (mass, number of pieces), and working parameters (speed, pressure, power).  
By analyzing Figure 13c,d, it can be seen that the motion state obtained by the simulation analysis has the same trend as that obtained by the actual experiment. It is verified that the fractal prediction model of the friction coefficient of the wet friction clutch has certain accuracy in predicting the contact state of the friction plate.  
Figure 13: Comparison of experimental and simulation data. (**a**) Fractal parameter, \(D\); (**b**) Scaling coefficient, \(G\); (**c**) Comparison of engagement time; (**d**) Comparison of friction coefficient.