## 3 Results and Discussion
### Artificial Neural Networks
As expected, the highest value of the COF was observed for dry friction conditions. In lubrication conditions, initially with increasing oil viscosity, the COF value decreases and then begins to increase (Figure 11). This is due to the fact that the highly viscous oil adheres more to the rubbing surfaces, and consequently, the viscosity shear stability increases. The above conclusions are valid for the entire range of the analysed nominal pressures. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & **Architecture** & **Input Data** & **Quality of** & **Quality of** & **Quality of** & **MSE for** & **MSE for** & **MSE for** \\  & **of RBF ANN** & **Set** & **Training** & **Testing** & **Validation** & **Training Set** & **Testing Set** & **Validation Set** \\ \hline
1 & RBF-8-8-1 & ANN\_1 & 0.937774 & 0.986012 & 0.487998 & 0.000448 & 0.000507 & 0.000147 \\ \hline
2 & RBF-8-9-1 & ANN\_2 & 0.978036 & 0.989737 & 0.089184 & 0.000162 & 0.000292 & 0.000204 \\ \hline
3 & RBF-8-8-1 & ANN\_3 & 0.965571 & 0.980861 & 0.521025 & 0.000252 & 0.000435 & 0.000256 \\ \hline
4 & RBF-8-8-1 & ANN\_4 & 0.925234 & 0.982205 & 0.579015 & 0.000535 & 0.001283 & 0.000071 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Summary of the best networks for COF prediction. The ANN almost perfectly reflects the relationship between the nominal pressure and the root mean square height \(\mathrm{Sq}\) of sheet metal on the value of the COF (Figure 13). The highest value of the COF in the entire range of the analysed pressures was observed for the average value of the Rq parameter. In addition, there is a tendency to decrease the value of the COF with increasing nominal pressure. In general, there is good agreement between the experimental data and those predicted by the RBF neural network. The scatterplot of the actual and predicted COFs for the 95% confidence interval shows that the COFs are distributed correctly along the diagonal (Figure 14). The four points on the right side of this graph correspond to dry friction conditions where the COF has reached significant values in excess of 0.3. The graph in Figure 15a shows the comparison of the actual COF and the one estimated on the basis of the RBF-8-9-1 network. The first four points correspond to friction in non-lubricated conditions. So, initially, the COF has a much greater value than under lubrication conditions. In general, we can see that the forecasts are well matched to the experimental data. Figure 15b shows the performance of the ANN training. The \(y\)-axis shows the performance parameter (MSE). The individual epochs of network training were recorded on the axis of abscissa. The smallest MSE error in testing the neural network was achieved by the ANN for epoch no. 12 and was 0.009. It can be seen that the neural network continued the training algorithm for 52 more epochs in order to confirm the presumptive  
Figure 11: The effect of oil viscosity and nominal pressure on the COF: (**a**) experimental data, (**b**) prediction of the COF by the RBF-8-9-1 network. Figure 12: The effect of oil viscosity and root mean square height \(\mathrm{Sq}\) parameter on the COF: (**a**) experimental data, (**b**) prediction of the COF by the RBF-8-9-1 network. global minimum of the MSE. From epoch #1 to epoch #12, there is a downward trend in the MSE for the training and test sets. The use of the radial basis function ANNs to analyse the influence of the input parameters of the friction process on the value of the COF made it possible to determine the correlation between the selected surface roughness parameters and the value of the COF. It was found that the surface roughness parameter Sq is the most correlated with the COF. The research confirmed the effectiveness of RBF ANNs for the analysis of the friction phenomenon based on a limited number of experimental data points. Recognising the relationship between many parameters, often synergistically interacting, using analytical methods would be difficult and complex.