## 3 Application of Drive-by Methodologies on the Bridge Subsystem
### Bridge Modal Identification
#### 3.1.1 Natural Frequency Extraction
However, with the increase in the speed, high frequency components from structural components of the vehicle and track irregularities start to couple with the natural frequency of the bridge, making both completely indistinguishable for speeds of approximately 50 km/h. These influences  
Figure 2: Simplified vehicle bridge model [32]. had not been taken into account in the previous analytical and numerical studies [32; 33]. Besides that, higher speeds results in shorter measurement times, and, consequently, a lower frequency resolution. All these issues impose extra difficulties for applications involving high-speed vehicles. These preliminary studies [32; 33; 36] formed the basis for the development of drive-by methodologies for identifying the natural frequencies of bridges. The issues raised by these studies regarding contributions due to vehicle frequencies and track irregularities have been treated in several other studies. [PERSON] et al. [37] introduced the concept of the residual spectrum, defined as the difference between the frequency spectra calculated from measurements taken in two consecutive vehicles. The authors demonstrated that this technique reduces the contribution of undesirable frequency components from the vehicle and track irregularities. On the other hand, [PERSON] et al. [38] proposed the calculation of a residual response in the time domain by subtracting the responses of two identical consecutive vehicles corresponding to the instants of time in which both were in the same position on the bridge. Since in these time instants the vehicles are subject to the same excitations arising from the irregularities, this subtraction allows eliminating their effect. Taking a different approach, [PERSON] et al. [39] proposed the use of bandpass filters, singular spectrum analysis, or a combination of both (which has proved to be the most effective) to remove undesirable frequency components. The combined use of these two techniques allowed the vehicle frequency to be removed without introducing spurious peaks in the signal. [PERSON] et al. [40] introduced the idea of using the contact-point response to assess the bridges' natural frequencies. Since the contact point is below the suspension components, it will not be affected by the vehicle frequency [41]. However, the application of this technique requires precise knowledge of the vehicle dynamics since the contact-point response is derived from the responses of other parts of the vehicle. In addition, this specific response is not easily measured from the practical point of view. [PERSON] et al. [42] derived analytical expressions for assessing both vehicle and contact-point responses considering a simplified single-DOF vehicle model, such as the one in Figure 2, running over a simply supported railway bridge. The track structure was modeled by a beam, representing the rail, attached to the bridge by a set of spring-dashpot assemblies representing the sleepers and ballast interfaces. The derived analytical expressions were successfully validated by comparison with a FEM-based simulation. The Fourier transform  
Figure 3: Influence of suspension damping on the vehicle vertical acceleration (\(\
u=\) vehicle speed, \(\omega_{\rm{p}}\) vehicle natural frequency; \(L=\) bridge span length and \(\omega_{\rm{b}}\) bridge frequency) [33]. of the responses was performed and the one from the contact point has shown to be much more suitable for estimating the natural frequency of both bridge and track since it was not influenced by vehicle and driving frequencies. In a parametric study, the method showed to decrease in performance for high values of speed, track irregularities, and track damping. [PERSON] and [PERSON] [43] proposed the use of the residual spectrum calculated from the contact response of each one of the axles of a two-axle vehicle.