## 3 Application of Drive-by Methodologies on the Bridge Subsystem
### Bridge Modal Identification
#### 3.1.2 Mode Shape Extraction  
The application of the drive-by concept for the dynamic characterization of bridges is not limited to natural frequency assessment. Accurately extracting a bridge mode shape with sufficient spatial resolution usually requires a large number of sensors or multiple experimental setups. These issues make very attractive the idea of using the vehicle as a moving sensor to scan virtually every point along the bridge's length. [PERSON] et al. [48] were pioneers in the development of a methodology for mode shape assessment from dynamic data of a moving vehicle. They proposed the use of a vehicle equipped with an exciter capable of applying a sinusoidal force. For this special vehicle moving over a bridge modeled as a plate, the authors analytically derived a solution for its vertical acceleration and proved that, under the assumption of a very stiff vehicle, the square of the modal ordinate amplitude is proportional to the amplitude of the acceleration spectra. The method was tested and demonstrated both numerically and experimentally in a reduced-scale experiment. However, the necessity to install a shaker on the vehicle imposed extra difficulties on practical applications. [PERSON] et al. [30] first introduced a formulation for assessing the mode shapes of a simply supported bridge in which the excitation is provided only by the passage of the vehicle. The authors demonstrated analytically that the mode shape can be extracted from the instantaneous amplitude, which is computed as the modulus of the analytical signal derived from the Hilbert transform of the part of the vehicle response related to the natural  
Figure 7: Drive-by natural frequency identification of the Schmutter bridge: (**a**) experimental setup and (**b**) natural frequency results [27]. frequency of the bridge. This method implies the use of dedicated techniques to identify the frequency of the bridge and filter the signal to isolate only the portion related to the bridge. The authors applied the method to a case study involving a simply supported bridge and a sprung mass vehicle. From this study, two major drawbacks of the method became evident. Firstly, its accuracy, especially for higher order modes, seems to strongly rely on the use of very low speeds, on the order of 2 m/s, which would impose operational constraints from the practical point of view. Besides that, the results for higher-order modes have shown to be deeply affected by track irregularities which is another obstacle to be overcome to enable practical applications if the identification of higher-order modes is required. A recent study conducted by [PERSON] et al. [29] further investigated the methodology proposed by [PERSON] et al. [30] for different values of the bridge's damping ratio. As presented in Figure 8a, the damping significantly distorts the mode shape, shifting it to the left. Based on an analytical formulation of the bridge vehicle interaction model considering the damping, the authors proposed a correction factor for the original formulation which is a function of the bridge's natural frequency and damping ratio. As can be seen in Figure 8b, the proposed correction can significantly improve the results; however, some distortion is still present for high damping values, which the authors claim can be corrected by applying a smooth operation to the instantaneous amplitudes. [PERSON] and [PERSON] [31] introduced the Short Time Frequency Domain Decomposition (STFDD) technique for mode shape assessment. This method is based on the application of the Frequency Domain Decomposition (FDD) [49] method to the signals measured in consecutive axles of the vehicle. The bridge is divided into equal-length segments (see Figure 9) and the measurement is performed in stages where each axle is over one of the segments. For each stage, a modal ordinate vector with two elements is obtained by the FDD, and since each stage has overlapping segments, the global mode shape vector can be reconstructed by a progressive rescaling procedure.