## 3 Application of Drive-by Methodologies on the Bridge Subsystem
### Bridge Modal Identification
#### 3.1.1 Natural Frequency Extraction
The combined use of the residual spectrum and the contact response is an efficient way to take advantage of the aforementioned potentialities of both and which the authors showed to be essential for the correct distinction of bridge modes under conditions of track irregularities. The authors developed an analytical formulation of the bridge-vehicle interaction problem from which the contact-point response was derived. A parametric study was conducted testing the performance of the method under varying irregularity levels, vehicle design parameters and velocities, bridge damping ratio as well as support conditions, and the number of spans. Under all these conditions, the method provided reliable results for at least the first three natural frequencies of the bridge with data from only a single vehicle. A different approach for dealing with the vehicle and driving frequency disturbance was adopted by [PERSON] et al. [23] which is schematically presented in Figure 4. Firstly, the vertical acceleration at the bogies from a few time steps before entering the bridge to a few moments after leaving it needs to be measured and its FFT computed. Then, some peaks of the Fourier amplitude spectrum are selected as candidate frequencies, and for each candidate frequency, the Continuous Wavelet Transform (CWT) coefficients are investigated over time. The frequencies which only presented significant energy levels during the bridge crossing are considered to be bridge natural frequencies. The validation of the technique was performed based on simulation data generated through detailed 3D models of the bridge, track, and vehicle. It was demonstrated that the first natural frequency of the bridge can be properly distinguished from contributions due to the vehicle and driving frequencies and track irregularities for speeds up to 90 km/h. Aiming to reduce the negative influence of track irregularities, [PERSON] et al. [24] applied the time-domain subtraction method (TSM), proposed by [38], to the assessment of natural frequencies of a one-span simply supported railway bridge. The performance of the TSM was evaluated for the six classes of irregularities established by the American standards [44] in which 6 is the best grade. As stated in Figure 5a, Class 4 irregularities are enough to make bridge frequencies practically indistinguishable from background noise while applying the TSM (Figure 5b); the first natural frequency becomes very evident, and even the second natural frequency can be detected. The authors also conducted studies varying the design parameters of the vehicle and the bridge as well as the driving speed. The method performed well under different design parameters; however, the increase in vehicle speed seems to be an issue. For speeds around 20 m/s, errors of approximately 5% are associated  
Figure 4: Flowchart of the methodology proposed by [23]. with the estimative of the first natural frequency which can be a problem in applications involving high-speed trains. In a recent study involving high-speed trains, [PERSON] et al. [25] proposed a very robust methodology for detecting the natural frequency of bridges based on the dynamic response of the vehicles. The authors minimized the influence of the irregularities of the rails and the frequency components of the vehicles calculating the residual response computed by the subtraction between the acceleration response of consecutive vehicles. Additionally, the robustness of the method was significantly improved by combining the residual responses from several pairs of consecutive wagons to calculate the residual spectrum, increasing the amount of temporal information available. The authors numerically demonstrated the accuracy of the method for different levels of irregularities, vehicle speeds, bridge stiffness, damping, and measurements of noise levels. The technique performed well in distinguishing the bridge's first natural frequency; however, the second natural frequency already appeared with very little amplitude, and the others could not be identified.