## 3 Simulation Examples  
### Configurations  
The properties of a simply supported beam are given in Table 1. The beam damping property is neglected in this section and the following three sections (Sections 3.2.1-3.2.3). Its first three natural frequencies are 4.0202 Hz, 16.0850 Hz, and 36.1911 Hz. The beam is discretized by 25 equally long beam elements.  
The properties of the vehicles [41] are given in Table 2. The natural frequencies and damping ratios of this two-DOF vehicle are mostly (2.3884 Hz, 0.0365) for the pitch motion and mostly (3.1832 Hz, 0.0413) for the bounce motion. The natural frequency and damping ratio of the one-DOF vehicle are (1.1254 Hz, 0.0283).  
To ensure all vehicles vibrate adequately before entering the bridge, a 125 m long Class A (very good) road profile is generated. This road profile covers the bridge and two regular road parts (50 m) at both bridge ends. A 5 m long Class A road segment is plotted in Figure 4, as well as Class B (good) and Class C (average) road segments.  
For the tractor with the three trailers travels at a velocity of 5 m/s (18 km/h), the displacement responses of the three trailers' contact points are plotted in Figure 5a while traveling on the beam. The corresponding two residual accelerations between consecutive contact points are plotted in Figure 5b. By using a Hann window with a size of 128 and an overlap ratio of 99% under the sample frequency of 200 Hz, the STFT of the acceleration residual between the first two trailers (\(u_{1}\) and \(u_{2}\)) is plotted in Figure 5c. There are peak ridges marked as red dashed horizontal lines along the first two beam natural frequencies (4.0202 Hz and 16.0850 Hz). The third ridge is located around 33 Hz, which is slightly lower  
\begin{table}
\begin{tabular}{c c c} \hline
**Properties** & **Symbol** & **Value** \\ \hline Length & \(L\) & 25 m \\ Young’s modules & \(E\) & 30 Gpa \\ Width & \(B\) & 1 m \\ Height & \(H\) & 1.6 m \\ Moment of inertia & \(I\) & 0.34 m\({}^{4}\) \\ Mass per unit length & \(m\) & 4000 kg/m \\ \hline \end{tabular}
\end{table}
Table 1: The beam bridge properties.  
\begin{table}
\begin{tabular}{c c c} \hline
**Properties** & **Symbol** & **Value** \\ \hline The mass of a 2-DOF vehicle & \(M_{c}\) & 18,000 kg \\ The moment of inertia of a 2-DOF vehicle & \(l_{c}\) & 55,500 kg\(\cdot\)m\({}^{2}\) \\ Wheelbase & \(l_{c}\) & 3 m \\ Ratio to gravity center for the front axle & \(a_{F}\) & 0.65 \\ Ratio to gravity center for the rear axle & \(a_{R}\) & 0.35 \\ The spring of the front axle & \(k_{F}\) & \(2.0\times 10^{6}\) N/m \\ The dashpot of the front axle & \(c_{C}\) & \(1.0\times 10^{4}\) N/m \\ The spring of the rear axle & \(k_{R}\) & \(5.0\times 10^{6}\) N/m \\ The dashpot of the rear axle & \(c_{R}\) & \(2.0\times 10^{4}\) N/m \\ The mass of the 1-DOF vehicle & \(m_{1}=m_{2}=m_{3}\) & 1000 kg \\ The spring of the 1-DOF vehicle & \(k_{1}=k_{2}=k_{3}\) & \(5.0\times 10^{4}\) N/m \\ The dashpot of the 1-DOF vehicle & \(c_{1}=c_{2}=c_{3}\) & \(4.0\times 10^{3}\) N/m \\ \hline \end{tabular}
\end{table}
Table 2: The properties of vehicles.  
than the third natural frequency (36.1911 Hz). Additionally, at the third natural frequency, the magnitudes of the STFT components are lower than those at this peak ridge, which indicates that the beam is not adequately excited at this frequency. Despite the peak ridge frequency of around 33 Hz not being closely aligning with the natural frequency, it could still be effectively utilized to extract a promising third mode shape in Figure 5.  
Figure 4: The road roughness profile segments of three different road classes (A, B, and C).  
Figure 5: (**a**) The displacement responses of the contact points (\(u_{c1}\), \(u_{c2}\), and \(u_{c3}\)) of the three trailers while traveling on the beam; (**b**) the acceleration residuals between the consecutive contact points (\(\Delta u_{12}\) for \(u_{c1}\) and \(u_{c2}\), and \(\Delta u_{23}\) for \(u_{c2}\) and \(u_{c3}\)); (**c**) the STFT of the acceleration residual for \(\Delta u_{12}\).