# Method for Determining the Susceptibility of the Track
## 2 Experimental Testing on a Real Object
### Vertical Stiffness Measurements of Straight Track
The resulting approximation results are shown in Figure 4.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{\(P\)-10\({}^{4}\) N} & \multicolumn{10}{c}{Mean Values z\(\cdot\)10\({}^{-4}\) m} \\ \cline{2-13}
**on a Wheel** & **p-p-1** & **p-p-2** & **p-p-3** & **p-p-4** & **p-p-5** & **p-p-6** & **p-p-7** & **p-p-8** & **p-p-9** & **p-p-10** & **p-p-11** \\ \hline
9.85 & 13.78 & 9.35 & 9.18 & 10.32 & 7.44 & 9.88 & 11.30 & 13.82 & 20.72 & 12.04 & 6.80 \\
10.4 & 14.33 & 10.33 & 9.28 & 10.26 & 7.76 & 10.50 & 11.32 & 14.10 & 20.74 & 12.24 & 7.65 \\
10.55 & 14.29 & 9.75 & 9.86 & 10.50 & 8.22 & 10.04 & 11.60 & 14.78 & 20.68 & 12.92 & 7.33 \\ \hline \multicolumn{13}{c}{Standard Deviation z\(\cdot\)10\({}^{-4}\) m} \\ \hline
3.1 & 0.71 & 1.15 & 0.23 & 0.92 & 0.30 & 0.35 & 0.27 & 0.62 & 0.40 & 0.20 & 0.64 \\
3.1 & 0.77 & 0.92 & 0.49 & 1.27 & 0.29 & 0.76 & 0.56 & 0.31 & 0.31 & 0.22 & 0.53 \\
4.15 & 0.61 & 0.62 For a track without a turnout, the second part of the measurements was carried out on a test stand at the Railway Institute in Warsaw. This experiment was conducted to determine stiffness and damping. The damping was assumed to be dry friction. The experiment consisted of measuring the displacement of the rail head when it was loaded and unloaded relative to the position when it was not loaded. Loads and unloads were carried out in two variants:  
* variant simulating the vertical forces acting between the railhead and the empty wagon (nominal force 2.5\(\cdot\)10\({}^{4}\) (N)),
* variant simulating the vertical forces acting between the head of a rail and a loaded wagon (nominal force 10\(\cdot\)10\({}^{4}\) (N)). The head displacement was measured under static conditions in 0.5\(\cdot\)10\({}^{4}\) N increments. In both variants, the linearity of the relationship between force and displacement under loading and unloading of the head was checked by determining the correlation coefficient--Table 4. Large values of the correlation coefficient justify the assumption of linear relationships. Straight lines were found to best describe the force-displacement relationships. The results relating to the left rail are shown in Figure 5. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Correlation Coefficients between:** & \(P_{i}\)**a \(z_{l}\)** & \(P_{i}\)**a \(z_{p}\)** & \(P_{i}\)**a \(z_{l}\)** & \(P_{i}\)**a \(z_{p}\)** & \(P_{i}\)**a \(z_{l}\)** & \(P_{i}\)**a \(z_{p}\)** \\  & **under Load** & **under Load** & **under Load** & **under Load** & **under Load** & **Medium** & **Medium** \\ \hline when simulating a lighter & 0.997 & 0.996 & 0.975 & 0.988 & 0.993 & 0.991 \\ wagon & 0.998 & 0.997 & 0.992 & 0.988 & 0.997 & 0.996 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Correlation coefficients. Figure 5: Results of checking the linear relationship between force and displacement under load and headstock relief: (**a**) variant I, (**b**) variant II; 1—load, 2—unload, 3—average of load and unload, 4—simple approximation. The hysteresis loop of the left and right rails was then determined, with the point determined by the nominal load and the mean deflection of the empty wagon taken as the common point of the loop. These results are shown in Figure 6.  
On the basis of the experimental results, the following analytical description of the track forces on the wheel was considered to be the most appropriate:  
\[\mathrm{P}(\mathrm{z})=\mathrm{k_{T}}\mathrm{z_{T}}+\mathrm{h_{T}}\mathrm{sgn} \dot{\mathrm{z}_{T}}, \tag{1}\]  
where \(\mathrm{k_{T}}\) and \(\mathrm{h_{T}}\) are constant coefficients that are different for the empty and loaded wagon (Table 5). Figure 6: Summary of the hysteresis loops of the left (1) and right (2) rails. The results obtained for the track stiffness \(\rm k_{T}\) are 1.5\(\cdot\)10\({}^{8}\)\(\rm N/m\). The determined damping coefficients \(\rm h_{T}\) vary from 3.2 to 7.3 \(\rm Ns/m\) depending on the load. The experimental results indicate that the damping in the track is of a dry friction nature.