# Drive-by Bridge Damage Detection Using Continuous Wavelet Transform
## 2 Numerical Models  
The efficacy of the proposed framework in detecting and locating damage will be developed and validated through numerical simulations. For this, we used a single-span bridge with a 25 m span length. The instrumented vehicle was modeled as a two-axle vehicle, commonly known as a 2 DOF half-car model. An overview of the numerical models of the bridge and the vehicle used in the numerical analysis is shown in Figure 1 The bridge was modeled using Bernoulli beam elements spaced at equal intervals of 0.5 m. The values of the bridge and vehicle parameters are presented in Table 1. In the first phase, we modeled the boundary conditions of the bridge as pin-supports, which completely restrain the vertical movement at the support nodes. However, virtually all bridges globally rest on bearings that attenuate the thermal and seismic effects. Therefore, in the parametric study, supports were modeled using elastic springs which emulate the behavior of bearings. The supports were assumed to be fixed in the longitudinal and transverse directions as the stiffness in these directions do not have significant impact in the vertical behavior of the bridge. Ref. [53] demonstrated that the supports in existing bridges can have some rotational stiffness due to several reasons such as aging. Nevertheless, this study ignored these effects, and the supports were simulated as free to rotate. Finally, rigid beams were modeled at both ends of the bridge to emulate the bridge approaches. Accordingly, the vehicle starts its trip with its front axle positioned at the first support and continues until the rear axle reaches the final support. In this situation, the rear axle moves during the initial phase of the car's motion on the approach, whereas the front axle undergoes a similar motion towards the end. Figure 1: Bridge and vehicle models. Among the parameters that adversely affect the accuracy of drive-by monitoring methods in e.g., estimating bridge frequencies and mode shapes or detecting damage, road roughness remains one of the most important because it is a major source of dynamic excitation in vehicles. While a vehicle traverses a rugged road surface, it experiences forces resulting from the uneven terrain, influencing its behavior and dynamic response, thereby contaminating the recorded accelerations of the vehicle. Thus, any drive-by monitoring method has to account for the adverse effects of road roughness. For this, we numerically generated a road roughness profile following the procedure described below and elsewhere [54]. [PERSON] and [PERSON] [55] proposed the Power Spectral Density (PSD) functions to generate the road surface roughness that is assumed to be a zero-mean stationary Gaussian random process. This function supplies the amplitudes of surface roughness based on its spatial frequency. Spatial frequency refers to the reciprocal of the wavelength of the roughness features. According to ISO-8608 [56], the one-sided PSD function, \(G(n)\), is described in Equation (1). \[G(n)=G(n_{0})\bigg{(}\frac{n}{n_{0}}\bigg{)}^{-2}\qquad n_{\text{min}}<n<n_{ \text{max}} \tag{1}\]  
where \(G(n_{0})\) is the roughness coefficient that represents the degree of roughness in the road classification ranging from Class A to Class H. In this study, \(G(n_{0})\) is adopted as \(0.01~{}\times~{}10^{-6}\) m\({}^{3}\), which accounts for Class A profile in the road classification [56], \(n_{0}=0.1~{}\text{ cycles}/\text{m}\) is the reference spatial frequency. \(n_{\text{min}}=0.01~{}\text{ cycles}/\text{m}\) and \(n_{\text{max}}=10~{}\text{ cycles}/\text{m}\) are the lower and upper spatial frequency limits, respectively. \(n\) is the spatial frequency value increased incrementally by \(\Delta n=0.01~{}\text{ cycles}/\text{m}\) in the range between \(n_{\text{min}}\) to \(n_{\text{max}}\). Road roughness irregularities are generated using the sum of a series of harmonics as described in Equation (2).