## 4 Data Analysis of the Tangential Contact Stiffness Measuring and Model Validation
### Experiments with Different Contact Materials  
These experiments mainly include but are not restricted to, Groups 1-6.  
In Figure 4, and in the following sections, \(k_{x}\) means the tangential contact stiffness of the \(x\)-axis, and \(k_{y}\) represents the tangential contact stiffness of the \(y\)-axis. For both of these two measurements, the relative error \(c_{relative}\) is:  
\[c_{relative}=\left|\frac{k_{measured}-k_{canculated}}{k_{measured}}\right| \tag{54}\]  
Figure 4: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Cu-Cu** & **Cu-Al** & **Cu-Fe** & **Fe-Fe** & **Al-Al** & **Fe-Al** \\ \hline F\({}_{\text{T}}\)/N & 0.8 & 0.6 & 0.8 & 1.0 & 0.8 & 0.5 \\ F\({}_{\text{N}}\)/N & 3.1 & 3.0 & 3.1 & 3.0 & 2.8 & 2.8 \\ u & 0.2581 & 0.2000 & 0.2581 & 0.3333 & 0.2857 & 0.1786 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The friction coefficient between selected materials.  
Figure 4: Cont.  
Figure 4: _Cont._  
In general, the relative error between the measured tangential stiffness and the predicted one is less than 10%. More specifically, to estimate the material mechanical parameter's role in the model's accuracy, six experimental groups, including three different materials, have been arranged. Logically, there should be 3 \(\times\) 3 = 9 groups to be set up. However, three of them are just changes in the material contact order, such as Cu-Al to Al-Cu, so there is no need to repeat testing the same material pair again, and only six groups are indispensable.  
In Group 1, the material of the contact pair is Cu and Cu, with radii of 8 and 9 mm. Relatively, Group 2 only changes half of this pair to another material, Fe, with the other parameters fixed. Comparing the results of these two groups, it is implicit to find out the difference. Because of the random data chosen, the results displayed do not stand for any tendency meaning, which is not the focus of the experiment.  
On the other hand, it could be concluded that the change of material has almost no influence on the model's accuracy. In Group 1, the largest relative error is 9.7%, while in Group 2, the largest relative error is 9.9%, which is almost the same as Group 1. This conclusion is also appliable to the rest of the experimental groups: from Group 2 to Group 3, the only changed parameter is the material from Fe to Al, but the largest relative error is 9.8%; from Group 2 to Group 4, the only changed parameter is the material from Cu to Fe, and the largest relative error is 10%; from Group 4 to Group 5, the only changed parameter is the material from Fe to Al, and the largest relative error is 9.9%; from Group 5 to Group 6, the only changed parameter is the material from Fe to Al, and the largest relative error is 9.9%. Above all, as our experiment schedule is constructed, in every group of experiments, only one parameter alters compared to its proximate one. While according to the data aforementioned, there is barely any difference in the accuracy of these groups, and this could verify the conclusion that the change of material has almost no influence on the model's accuracy.  
Figure 4: The comparison of tangential contact stiffness between measured ones and predicted ones among different materials.