## 2 Numerical Simulation
### Simulated Dynamic Responses  
The simulated vehicle vibrations and the corresponding road, bridge, and input profiles in Scenario 1 are shown in Figure 5. The blue/green and orange/brown curves represent waves of front and rear wheels. Figure 5a shows \(r_{i}(t)\): the road profiles at the front and rear wheels. They are obtained by converting the road unevenness \(R(x)\) shown in Figure 3 by using \(x_{i}=x_{i}(0)+vt\). Figure 5b also shows \(\tilde{y}_{i}(t)\): the bridge profile of each wheel. The bridge profiles are the bridge deflection components \(y(x_{i}(t),\ t)\) that can be calculated by Equation (16). Figure 5c shows \(u_{i}(t)\): the input profiles. These input profiles generate dynamic vehicle responses. The generated vehicle vibration displacements, velocities, and accelerations are shown in Figure 5d-f, respectively. According to Figure 5a-c, the bridge profiles \(\tilde{y}_{i}(t)\) are much smaller than road profiles \(r_{i}(t)\). Since the road profiles \(r_{i}(t)\) are predominant in the input profiles \(u_{i}(t)\), road profiles \(r_{i}(t)\) predominantly affect the vehicle responses. The displacements of the unsprung-mass \(z_{ui}(t)\) shown in Figure 5d are similar to the input profiles \(u_{i}(t)\). Furthermore, the tendency of sprung-mass responses \(z_{si}(t)\) and unsprung-mass responses \(z_{ui}(t)\) are similar. In this numerical simulation, \(\tilde{z}_{si}(t)\) and \(\tilde{z}_{ui}(t)\) are assumed to be the measured data. \(\tilde{z}_{ui}(t)\) is much larger than \(\tilde{z}_{si}(t)\), but the same tendency is often observed in actual vehicles. While the unsprung responses are much larger than 1 G (=9.8 m/s\({}^{2}\)), the suspensions absorb strong vibrations and play an essential role in protecting passengers and cargo. Figure 6 also shows the vehicle vibrations in each case. There are four curves in each figure, but the differences are tiny. According to these figures, it is not easy to recognize the differences between the prepared scenarios. In SHM, low sensitivity of vibration to local damages like these results is a prevalent technical issue. \begin{table}
\begin{tabular}{c c c c c} \hline \multicolumn{2}{c}{**Newmark-\(\beta\) Method**} & \multicolumn{1}{c}{**Time**} & \multicolumn{1}{c}{**Convergence**} & \multicolumn{1}{c}{**Bridge**} \\ \(\gamma\) & \(\beta\) & **Increment** & **Tolerance** & **Element Number** \\ \hline \(1/2\) & \(1/4\) & \(\Delta t=0.001\) & \(10^{-3}\) & 7 \\ \hline \end{tabular}
\end{table} 
Table 4: The simulation parameters. Figure 4: The algorithm of PRE simulation. Figure 5: The vehicle vibration accelerations, velocities and displacements, and profiles: (**a**) road profiles \(\mathbf{r}(t)=\{r_{i}(t)\}=\{R(x_{i}(t))\}\); (**b**) bridge profiles \(\widetilde{\mathbf{y}}(t)=\{y(x_{i}(t),t)\}=\{\mathbf{N}(x_{i}(t))\cdot y(t)\}\); (**c**) input profiles \(\mathbf{u}(t)\); (**d**) vehicle vibration displacement \(\mathbf{z}(t)\); (**e**) vehicle vibration velocity: \(\dot{\mathbf{z}}(t)\); (**f**) vehicle vibration acceleration: \(\ddot{\mathbf{z}}(t)\). Enlarged figures of the part of the simulated waveforms are provided in Appendix A for readers' easy recognition. The curves shown in Figure 5e,f are separately shown in Figures 10 and 11, respectively. The boxed part of Figure 6a is also shown in Figure 12.  
The natural frequencies of the vehicle and the bridge are shown in Table 5. These natural frequencies are obtained by convergence calculation. According to this table, the natural frequencies of the bridge change after stiffness reduction. However, it is still quite difficult to distinguish the changes due to stiffness reductions in the frequency domain, as shown in Figure 7. This is one of the reasons why system identification is required in drive-by bridge monitoring. From Table 5 and Figure 7, the vehicle's third and fourth natural frequencies are close to the second natural frequency of the bridge. Thus, the dynamic response components of the vehicle around the bridge's second natural frequency from 12 Hz to 15 Hz tend to be predominant.