# Stick-Slip Phenomena and Acoustic Emission in the Hertzian Linear Contact
## 4 Results and Discussion
In addition, Figure 5b shows the dependence between the kinetic friction coefficients on the driving speed, which decrease with increases in the driving speed of the surface, approximated by the logarithmic curves of the form y = a + (b ln (x)) as has been previously found by [PERSON] and [PERSON] [53; 54]. The Pearson correlation coefficients were calculated between the amplitude of the SS (\(\mu_{v}\)) as an independent variable and the normal force (\(F_{n}\)) and driving speed (\(v\)) as dependent variables. The obtained results can be found in Table 2.  
Figure 5: The influence of the driving speed and normal force on the friction coefficients: static friction coefficient (**a**), kinetic friction coefficient, including the fitted curve of the kinetic friction coefficient with a force of 20 N (**b**), the amplitude of the stick–slip phenomenon (**c**) with the frequency of the number of the stick–slip jumps (**d**). Regarding the acoustic emission, as a general trend, there was the continuous activity of low AE in the stick phase; however, there was a burst emission of high amplitude in the sliding phase, and both were dependent on the normal load. It is easy to see that the AE peaks appeared only at the jumps from the stick to the slip (the transition from static friction to kinetic friction), although they took place with a delay of a few milliseconds compared to the COF. This delay of the AE could be explained by the finite propagation speed of the AE elastic waves compared to the moment of the initiation of the slip phase (Figure 6). In the following, some characteristics of the AE, such as the AE amplitude, AE burst/continuous counts, and AE energy, are analysed in comparison to the characteristics of the stick-slip movement with the aim of investigating if they can be helpful for a comprehensive view of the identification of the stick-slip phenomenon by AE. The counts of AE in the stick and slip phases are presented in Figure 7. \begin{table}
\begin{tabular}{c c c c c} \hline \hline Coefficients & \(\begin{array}{c}\mu_{v}/F_{n}\\ v=0.2\text{ mm}/\text{s}\end{array}\) & \(\begin{array}{c}\mu_{v}/F_{n}\\ v=0.1\text{ mm}/\text{s}\end{array}\) & \(\begin{array}{c}\mu_{v}/F_{n}\\ v=0.05\text{ mm}/\text{s}\end{array}\) & \(\begin{array}{c}\mu_{v}/F_{n}\\ v=0.01\text{ mm}/\text{s}\end{array}\) \\ \hline Pearson correlation & 0.969 * & 0.986 ** & 0.976 ** & 0.995 ** \\ \hline Coefficients & \(\begin{array}{c}\mu_{v}/v\\ F_{n}=20\text{ N}\end{array}\) & \(\begin{array}{c}\mu_{v}/v\\ F_{n}=40\text{ N}\end{array}\) & \(\begin{array}{c}\mu_{v}/v\\ F_{n}=60\text{ N}\end{array}\) & \\ \hline Pearson correlation & \(-0.986\)** & \(-0.962\)* & \(-0.976\)** & \\ \hline \hline \end{tabular}  
* Correlation is significant at the \(p=0.05\) level (95% level of confidence). ** Correlation is significant at the \(p=0.01\) level (98% level of confidence).  
\end{table} 
Table 2: Correlation between SS amplitude and normal force, respectively, and driving speed. Figure 6: The variation in the friction coefficient and acoustic emission (peak extracted). In both the stick and slip phases, the counts of AE increased with the driving speed and decreases in the normal load. In fact, the higher the frequency of the stick-slip movement, the higher the AE counts. The reason for this result is due to the higher number of stick-slip movements for a single test and because the AE takes place basically at the beginning of the slip movement, as shown in Figure 8. For each slip phase sequence in a test, each AE count was calculated over the slip time, and finally, mediated for each test. Likewise, for the stick phase, the AE counts were determined for each sequence over the stick time, and finally, an average was determined for each test. Therefore, a low frequency, higher amplitude slip-stick movement gives a relatively low count of AE with the burst aspect, mainly occurring at the beginning of the slip movement.