# Protection of Buried Pipelines from High-Speed Train Vibrations with Expanded Polystyrene Geofoam
## 3 Results  
### Unprotected Pipeline Response  
Initially, the dynamic response of the buried pipeline without the EPS protection layer was examined for three typical HST passage velocities: a relatively low velocity for HST (240 km/h), the typical Thalys operational velocity (300 m/s), and a quite high velocity (360 km/h). Figure 8 illustrates the Fourier spectra on the top of the pipeline for the examined passing velocities. In the sequence, the peak frequencies of each spectrum are derived via the Dominant Frequency Method (DFM) [34]. According to the DFM, the Fourier spectrum is strongly dependent on the train's velocity and the geometry of its bogies. Hence, these parameters are responsible for the most dominant vibration peaks. The dominant frequencies for which these peaks occur in the Fourier spectrum are well known as bogie passing frequencies (\(f_{b,n}\)). Furthermore, apart from \(f_{b,n}\), there are two other dominant frequencies: the axle passing frequency (\(f_{a}\)), which depends on the distance of the HST's axles, and the sleeper passing frequency (\(f_{s}\)), which depends on the length of the sleeper's bay. The dominant passing frequencies of the bogies, the axles, and the sleepers can be calculated as follows:  
\[f_{b,n}=n\frac{V_{O}}{l_{b}} \tag{1}\]  
\[f_{a}=\frac{V_{O}}{l_{a}} \tag{2}\]  
\[f_{s}=\frac{V_{O}}{l_{s}} \tag{3}\]  
where \(l_{b}\): bogie distance (18.7 m); \(l_{a}\): bogie axles distance (18.7 m); \(l_{s}\): sleeper bay; n = 1, 2, 3,...; and \(V_{O}\): train passing velocity.  
The dominant frequencies are summarized in Table 2 for the three examined HST velocities. As illustrated in Figure 8, these frequencies are well captured in the results, which further verifies the validity of the developed numerical approach. In particular, the first six bogie passing frequencies have been captured quite accurately by the numerical model. The dominant peaks are placed at higher frequencies as the HST velocity increases from 240 km/h to 360 km/h. In the case of HST passing with low velocity (240 km/h), the most dominant frequencies are 3.5 Hz (\(f_{b,1}\)), 7.2 Hz (\(f_{b,2}\)), 46.4 Hz (\(f_{b,13}\)), and 49.9 Hz (\(f_{b,14}\)), and the vibration peaks vary between 1.3 mm/s and 2 mm/s. The peak at the axle passing frequency, close to 20 Hz, is also dominant and greater than 1 mm/s. In addition, the peak at the sleeper passing frequency has also been captured by the model, although it is quite low (3 mm/s).  
Figure 8: Fourier spectra at the unprotected pipeline top for HST passage with 240 km/h, 300 km/h, and 360 km/h.  
The same observations can be derived when the Thalys train is running with its usual operational velocity (300 km/h). As expected, all the dominant peaks have been slightly moved to the right in this case. The first, the second, and the thirteenth passing frequencies of the bogies are the most dominant in the frequency range. These peaks are significantly increased compared to the passage with 240 km/h. For instance, the peak at \(f_{b,1}\) has increased from 18 mm/s to 20 mm/s. The peaks at axle and sleeper passing frequencies are also slightly increased. For the highest passing velocity (360 km/h), except for the same dominant frequencies at lower velocities, several dominant peaks at other frequencies can be noticed. More specifically, at \(f_{b,1}\) (5.4 Hz) and \(f_{b,2}\) (10.7 Hz), the amplitude of the peaks is almost double, compared to the same values, in the case of Thalys' slower passage with 240 km/h. Nonetheless, it should be mentioned that the sleeper passing frequency has not been captured as it is quite high in this case (over 150 Hz).