## 3 Finite Element Modulation and Damage Detection  
In this section, the proposed damage detection method is implemented on a stationary vehicle response obtained from a FE model. The FE consists of 2D bridge made of shell elements in the commercial program of Ls-Dyna. Properties of the bridge are taken from the study of [PERSON], [PERSON] [5] which are shown in Table 1. The FE bridge model is divided equally into 1000 shell elements. Each element is of 0.25 m of length and 0.2357 m of width. The moving vehicle is modelled as a quarter car of two degrees of freedom, while the stationary vehicle is parked on the bridge and modelled as quarter car of single degree of freedom. The properties of the moving and stationary vehicles are shown in Table 2. The moving vehicle starts moving for an approach distance of 75 m to remove the effects of vehicle translational vibration effects.  
\begin{table}
\begin{tabular}{c c} \hline
**Property** & **Value** \\ \hline Bridge length & 25 m \\ Elastic modulus & 27.5 GN/m\({}^{2}\) \\ Moment of inertia & 0.12 m\({}^{4}\) \\ Bridge length density & 4800 kg/m \\ \hline \end{tabular}
\end{table}
Table 1: Beam Properties.  
\begin{table}
\begin{tabular}{c c} \hline
**Property** & **Value** \\ \hline Stationary vehicle body mass & 200 kg \\ Stationary vehicle spring stiffness & 1.2 \(\times\) 10\({}^{5}\) N/m \\ Stationary vehicle frequency & 3.98 Hz \\ Moving vehicle body mass & 14,000 kg \\ Moving vehicle axle mass & 1000 kg \\ Moving vehicle body stiffness & 1.2 \(\times\) 10\({}^{5}\) N/m \\ Moving vehicle axle stiffness & 2000 N/m \\ Moving vehicle body damping & 5\% \\ \hline \end{tabular}
\end{table}
Table 2: Properties of the moving and stationary vehicles.  
Road roughness is added in the FE model on the track where the moving vehicle is passing. Road roughness is simulated by the procedures introduced by ISO 8608 [45]. Road surface roughness can be obtained as follows:  
\[r(x)=\sum_{i}d_{i}\cos(n_{i}x+\theta_{i}) \tag{8}\]  
\[d=\sqrt{2G_{d}(n)\Delta n} \tag{9}\]  
\[G_{d}(n)=G_{d}(n_{0})\left(\frac{n}{n_{0}}\right)^{-2} \tag{10}\]  
where \(n_{0}\) is a constant that is equal to 0.1 cycle/m. The road surface profile \(G_{d}(n_{0})\), in this paper is selected for class B of ISO standards, which is 6 \(\times\) 10\({}^{-6}\) m\({}^{3}\). \(\Delta n\) represents the sampling interval for the spatial frequency, which is selected to be 0.04 cycle/m. The lower and upper spatial frequencies are selected as 1 and 100 cycle/m, respectively. Figure 3 shows the road surface profile of the class B roughness considering the mentioned values.  
The stationary vehicle acceleration response is taken from the top node of the stationary single degree of the freedom vehicle model. Then, the contact-point response of the stationary vehicle is computed using Equations (3) and (4). It is worthy to note that the bridge response is different from the stationary vehicle response, which is demonstrated in Figure 4. However, it is also demonstrated that the contact-point response shows good agreement with the bridge response. Therefore, the stationary vehicle can be used as a movable accelerometer instead of fixing the accelerometer directly on the bridge following the conventional method, which has several disadvantages as mentioned earlier.  
Figure 4: Dynamic response of the stationary vehicle, its contact-point, and bridge acceleration response at mid-span.  
Figure 3: A typical profile of road surface roughness.  
The damage on the bridge is modelled as stiffness reduction of shell elements of the bridge. Throughout this paper, the damage length is 1 m and the speed of the moving vehicle is 12.5 m/s, unless it is mentioned otherwise. The contact-point response of the stationary vehicle is decomposed into mono-component signals using the VMD method. Thereafter, the highest frequency mode is used to compute the IA.