# A Feasibility Study of the Drive-By Method for Damage Detection in Railway Bridges
## 3 Plan of the Simulations
### Data Processing Techniques
#### 3.4.1 Quasi-Static Method  
The quasi-static method derives from the works presented in [9; 10], which are focused on a bridge for road transportation. The method applied in these papers consists of filtering the signals with a moving average filter with a window length that is equal to the period of the first mode of vibration of the bridge, to cancel the dynamic components of the acceleration, and to extract the quasi-static component. The signal obtained for the healthy and damaged bridges are then compared, and the difference between the two signals (hereafter called damage component) is used as an indicator of the presence of structural damage.  
In this work, the same idea is developed, considering the peculiarities of the railway application, mainly consisting in longer vehicle and higher axle loads. Acceleration signals are low-pass filtered with a cut-off frequency depending on the first natural frequency of the bridge, to cancel the dynamic component due to bridge vibration. Figure 12 shows the accelerations of the damaged and healthy bridges after the low-pass filtering. Only the quasi-static component due to bridge deflection under the moving load is still present, with the highest acceleration levels corresponding to the damaged bridge. The difference of the two signals, representing the damage component, is filled with color in Figure 12a.  
Figure 11: Comparison between accelerations of bogie center of gravity, resulting from the damaged and healthy bridges. Bridge type #2, TSR train type, speed 100 km/h, damage level D40. (**a**) Raw signals. (**b**) Power spectral density.  
To identify the location of the damage, the acceleration corresponding to the healthy signal is subtracted from the acceleration acquired on the damaged bridge, obtaining the damage component. Figure 13 shows the damage component corresponding to the signals of Figure 12, which shows a maximum at position 0.505. This value is situated 0.7 m after the midpoint of the range [0.444; 0.5], in which the finite element with reduced Young modulus is located. The method is therefore sufficiently accurate for indicating the actual location of the damage.  
A fifth-order butterworth numerical filter is adopted. The cut-off frequency is a key parameter for the application of the algorithm, and it was set as a trade-off between the need of accuracy in damage localization, and the need of a high level of the damage component. The more that the signal is filtered, the better accuracy in positioning is obtained, but the acceleration level of the damage component is lowered. The cut-off frequency is obtained by reducing the bridge first natural frequency by a factor \(\rm{k_{cut}}\): \(\rm{f_{cut}}=\rm{k_{cut}}\times\rm{f_{bridge}}\). \(\rm{k_{cut}}\) value was set between 0.5 and 0.7, based on simulation results. For very short bridges with natural frequencies higher than 10 Hz, such as bridge number 1 (Table 1), the cut  
Figure 12: Acceleration of damaged and healthy bridges after low-pass filtering. (**a**) Low-pass filtered signals. (**b**) Power spectral density of filtered signals.  
Figure 13: Damage signal with its frequency spectrum. (**a**) Damage component. (**b**) Power spectral density.  
off frequency was upper-limited at 5 Hz, in order to filter natural frequencies that are related to the bogie vertical and pitch dynamics.