## 4 Case Study
### The Effect of Randomness in the Generation of Different Synthetic Profiles on Vibration Levels  
Given the stochastic nature of the PSD function, one hundred simulations were run, each with a different synthetic unevenness profile, in order to quantify the randomness significance. Each unevenness profile can be generated by combining a different number of harmonics (three situations are considered: 25, 50, and 75). The influence of this latter aspect is analyzed in a second step.  
Figure 5: Schematic representation of the track-tunnel-ground system.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Element** & **Stiffness** & **Mass** & **Damping** \\ \hline Rail UIC 60 (\(\times\)2) & EI\({}_{\text{r}}\) = 6.1 \(\times\) 10\({}^{6}\) N\(\cdot\)m\({}^{2}\) & m\({}_{\text{r}}\) = 60.4 kg/m & (-) \\ \hline Railpad & k\({}_{\text{rp}}\) = 1.0 \(\times\) 10\({}^{9}\) N/m\({}^{2}\) & (-) & c\({}_{\text{rp}}\) = 22,500 Ns/m\({}^{2}\) \\ \hline Slab & EI\({}_{\text{s}}\) = 1.2 \(\times\)10\({}^{8}\) N\(\cdot\)m\({}^{2}\) & m\({}_{\text{s}}\) = 1800 kg/m & - \\ \hline \hline \end{tabular}
\end{table}
Table 3: Slab track properties.  
Figure 6: Train model: (**a**) geometrical configuration and axle loads of the Tram-Train; (**b**) multi-body train model.  
Based on that, Figures 7 and 8 show the mean curves at two different locations on the ground surface (plane of symmetry and 30 m away), in time (running RMS--Root Mean Square--values computed with a time window of 1 s as prescribed by the ISO 2631 standard [21]) and frequency domain (1/3 octave bands), as well as their upper/lower standard deviations and maximum/minimum curves. These curves are repeated for the different number of harmonics considered in the generation of each artificial unevenness profile (25, 50, and 75).  
Figure 7: Influence of the number of sinusoidal functions used to generate the rail unevenness profile on vertical velocity (observation point A): (**a**) 25 sinusoidal functions; (**b**) 50 sinusoidal functions; (**c**) 75 sinusoidal functions (left: running RMS curves; right: one-third octave bands). dB ref. \(1\times 10^{-8}\) m/s.  
It is seen that the different profiles cause changes in response, in both the time and frequency domains. In general, these are generated because the rail profile varies spatially (in a sinusoidal pattern), meaning that the rail is subject to varying interaction forces with respect to time. However, these variations are not so significant. As can be seen, a maximum  
Figure 8: Influence of the number of sinusoidal functions used to generate the rail unevenness profile on vertical velocity (observation point B): (**a**) 25 sinusoidal functions; (**b**) 50 sinusoidal functions; (**c**) 75 sinusoidal functions (left: running RMS curves; right: one-third octave bands). dB ref. \(1\times 10^{-8}\) m/s.  
around 3 dB is found in relation to the mean running RMS curve. In terms of frequency content, a maximum deviation around 5 dB can be found for the frequency range with the most significative frequency content.  
Despite that, further analysis reveals that for the time and frequency domain plots, the effect of randomness in the generation of different synthetic profiles on vibration levels is higher when considering a higher number of harmonics for the generation of the unevenness rail profile. This is particularly evident comparing the results from 25 sinusoidal functions to 50 and 75 functions.