## 3 Application of Drive-by Methodologies on the Bridge Subsystem
### Bridge Modal Identification
#### 3.1.1 Natural Frequency Extraction  
The feasibility of extracting the dynamic properties of a bridge based on vehicle responses was initially raised in the study conducted by [PERSON] et al. [32]. For a simplified model (see Figure 2) with a sprung mass and a simply supported beam representing, respectively, the vehicle and the bridge, the authors derived a closed-form solution for the vertical displacement, velocity, and acceleration of the vehicle. In this solution, only the first mode of the bridge was considered, and the authors could demonstrate the presence  
Figure 1: Overview of the article’s layout. of the bridge's natural frequency in the vehicle's response. The authors also addressed the influence of bridge damping and vehicle speed on the responses. It was shown that the increase in bridge damping is responsible for reducing the magnitude of the bridge's contribution to the vehicle's response. Regarding the vehicle's speed, the authors noticed an increase in the response associated with higher vehicle speeds. This finding lead to the conclusion that higher speeds may help the identification of bridge frequencies; however, this preliminary study did not account for track irregularities which later studies have shown to cause difficulties for the identification when the vehicle is moving at high speeds. Further investigating the problem, [PERSON] and [PERSON] [33] derived closed-form solutions for the dynamic responses of both the vehicle and bridge by the modal superposition method and convolution integrals which are equivalent to the ones obtained in [32] if only the first bridge mode is considered. By comparison with a more detailed formulation, the authors demonstrated that considering only the first mode provides sufficiently accurate results. Slight differences between the two formulations were attributed to (i) the non-updating of the interaction forces between the vehicle and track and (ii) the assumption that the mass of the vehicle is much lower than that of the bridge, used to derive the analytical solutions. Using the Fourier transform of the vehicle's acceleration response, the authors demonstrated that it is governed by three main frequency components: (i) the driving frequency; (ii) the bridge's first and second natural frequencies shifted by the driving frequency, and (iii) the vehicle's natural frequency. The peaks corresponding to the contribution of each of these frequencies to the vehicle's acceleration response are indicated in Figure 3, as well as their mathematical expression. Among all these components, it was shown that the one from the bridge's first natural frequency dominates the vehicle's acceleration response, which can be clearly seen in Figure 3 by the prominence of the peaks associated with this frequency. The influence of the suspension damping was also evaluated in this study, demonstrating that it significantly reduces the magnitude of the vehicle frequency on the vehicle acceleration response; however, the bridge frequencies are much less affected as can be seen in Figure 3. This behavior regarding the suspension damping was studied in greater detail by [PERSON] and [PERSON] [34] who demonstrated that higher suspension damping tends not only to reduce the vehicle frequency but also improve identification in situations of higher amplitude track irregularities. These findings are quite encouraging in the context of the application of drive-by techniques in railway structures since railway vehicles tend to have suspensions with high damping ratios [35]. To validate the findings regarding natural frequency identification presented in the previous studies, [PERSON] and [PERSON] [36] conducted an experimental campaign on a 30 m span bridge in Taiwan. An instrumented trailer was pulled by a truck over the bridge at different speeds registering its vertical dynamic response. From the acceleration frequency spectra, the bridge's first natural frequency could be easily distinguished from other frequency components for low speeds of the testing vehicle.