## 3 Finite Element Simulation
### Result Analysis  
In this study, according to the theoretical derivation from Equations (4a) and (5a), it is hypothesized that the first two vibration modes dominate the bridge responses. Our approach involved estimating the acceleration at a particular point on the beam using acceleration data from two distinct locations. Adhering to the second key principle, we utilized the displacement histories captured at various points to generate two surrogate displacement mode shapes via the FastICA algorithm. Subsequently, in accordance with the first key principle as outlined in Step 4, these surrogate shapes were employed to deduce the target acceleration. For the initial scenario, the vehicle's speed was set at 5 m/s. A selection of seven observation points was made, with their respective coordinates detailed in Table 3. A representative displacement time history is illustrated in Figure 5. The FastICA algorithm was tailored to identify two independent signal sources, corresponding to the first two surrogate vibration modes of the bridge, with signals captured at seven designated receivers, each corresponding to the specific locations where displacement measurements were recorded. All seven displacement time histories were utilized as inputs for the FastICA algorithm. From the mixing matrix, the FastICA yielded two distinct surrogate displacement mode shape curves, which are presented in Figure 6. It is evident that these two curves deviate from the actual first and second mode shape functions of the bridge, which are sinusoidal functions represented by \(sinx\) and \(sin2x\), respectively. Subsequently, the proposed method's effectiveness was evaluated. The accelerations recorded at Points 7 and 4 were utilized to compute the surrogate inter-section transfer matrix of acceleration, which was formulated based on the two surrogate displacement mode shapes. Utilizing this matrix, the acceleration at Point 6 was deduced. The estimated acceleration at Point 6 was then compared with the results from the simulation, as displayed in Figure 7a for the time domain analysis, and in Figure 7b for the frequency domain analysis. In the time domain, as shown in Figure 7a, the estimated acceleration time history aligns closely with the simulation outcome. However, between 4.5 s and 5 s, when the vehicle is positioned above Point 7, it generates high-frequency numerical noise. This noise is transmitted to the estimation result via the proposed method, leading to a noticeable deviation between the estimated and simulated time histories at Point 6 during this interval. Figure 5: Displacement time history at Point 3. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline Points & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline Coordinate (m) & 1.2 & 6.9 & 9.7 & 13 & 16 & 19 & 23.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Coordinates of seven selected observation points. Furthermore, in the frequency domain comparison depicted in Figure 7b, the estimated and simulation results are in good agreement for the first and second vibration modes, thereby confirming the validity of the proposed algorithm. Nonetheless, the high-frequency components--highlighted with ellipses in Figure 7b--exhibit significant discrepancies. This discrepancy arises because the current study focuses solely on the first two vibration modes based on acceleration data from two specific locations, limiting the effective consideration of higher-order modes. The inter-section transfer matrix for bridge acceleration, derived from the proposed method, was further validated across various scenarios. As established in the previous study [41], the transfer matrix would remain invariant under diverse moving conditions. To validate the proposed method, two additional scenarios were modeled, one with a vehicle mass doubled and the other with the vehicle's speed doubled to 10 m/s. It should be noted that, for the speeds of 5 m/s and 10 m/s, they correspond to 18 km/h and 36 km/h, which are common vehicle running speeds in urban transportation systems.