## 5 Results  
The two approaches presented in this paper are compared by considering two different runs of the instrumented vehicle on a single track section between two stations. The second run is performed six days after the first. On the analysed site two curves are present: the first is a right curve (radius of 133 m) and the second is a left curve (radius of 173 m). It is known that corrugation appears on the low rail of both the curves. The data relative to Day 1 are shown in Figure 10a, while Figure 10b refers to the data acquired on Day 7. The curves are highlighted by vertical dashed lines for a better understanding. All the data are represented as a function of the position of the leading wheelset of the train on the line, measured in kilometres. In the upper plot the vehicle speed profile (solid line) and the estimated horizontal curvature of the track (dashed line) are represented. The latter is equal to the inverse of the curve radius and it is obtained from the combination of the signal by the gyroscope mounted on the carbody with the vehicle speed. If this quantity is equal to zero the vehicle is running on a straight track, if it is positive the train is negotiating a left curve, while if it is negative the train is negotiating a right curve. The moving rms of the acceleration measured at the axleboxes of the leading wheelset of the instrumented bogie are shown in the lower plots. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value (s)** \\ \hline Time bin duration & \(T_{bin}\) & 1 \\ Sub-window duration & \(T_{win}\) & 0.5 \\ Overlapping time & \(T_{over}\) & 0.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters used in the algorithm for the estimation of the power spectrum of rail irregularity. At about km 0.32 there is a switch, responsible for the high rms levels. A high acceleration is experienced by the right axlebox in correspondence to the first right curve (from km 0.02 to km 0.12). This may indicate the presence of a severe corrugation on the low (inner) rail. When the vehicle is negotiating the second left curve (from km 0.14 to km 0.28), there is an increase in the rms level relative to the left axlebox accelerometer from Day 1 to Day 7. This increment may be related to the rapid development of rail corrugation on the low rail of the curve. The power content of the rail irregularity is estimated in terms of Roughness Level (RL) as a function of the wavelength of the irregularity, which is defined by EN ISO 3095 [25] as  
\[RL\bigg{(}\frac{1}{\lambda}\bigg{)}=10\log_{10}\left(\frac{P_{irr}\Big{(}\frac {1}{\lambda}\Big{)}}{r_{0}^{2}}\right)\qquad\qquad r_{0}=1\;\mu m, \tag{24}\]  
As an example, the spectrograms of the estimated rail irregularity (by using the flexible wheelset approach) relative to Day 1 are shown in Figure 11. The spectrogram represents the Roughness Level (colormap) as a function of the wavelength (\(y\)-axis) and the absolute position of the wheelset along the line (\(x\)-axis). In correspondence to the two curves, the rails manifest a high RL restricted to a narrow wavelength band, indicating the presence of rail corrugation, which is almost mono-harmonic, with a wavelength of about 130 mm in the first right curve (highlighted in black) and a 170 mm wavelength in the second left curve (highlighted in red). In-field observations show that on both the curves rail corrugation tends to develop on the low rail. From the spectrogram, it is not clear which rail shows corrugation, since a high RL is present on both of them. The reason is that in the model of the vertical dynamics of the wheel-rail interaction the two sides of the wheelset are considered dynamically decoupled. Actually, the irregularity on a rail generates vibrations also measured by the accelerometer placed on the axlebox on the opposite side. Thus, a more sophisticated model that considers this coupling effect will allow to better identify which rail is corrugated.