## 3 Finite Element Analysis
### Validation of the Derived Physical Characteristics  
To validate the derived physical properties stated in Section 2, the simulation result of a series of finite element models were analyzed and are illustrated as follows. Three observation points located at 18 m, 10 m, and 5 m along the beam were selected for analysis. Their positions are represented in Figure 9. The direct simulated displacement time histories for these points are presented in Figure 10. The high-frequency local fluctuations observed on these curves represent the high-frequency displacements as described by Equation (9), while the remaining, smoother curves correspond to the low-frequency displacements as delineated by Equation (8). Given that the first natural bending frequency of the bridge was approximately 5 Hz, a bandpass filter was strategically applied to differentiate between high-frequency and low-frequency displacements, using a cutoff threshold of 1 Hz. The separated high-frequency and low-frequency displacement components are \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Unit mass** & **Bending Stiffness** & **First Natural** & **Second Natural** \\
**(kg/m)** & **(N\(\cdot\)m\({}^{2}\))** & **Frequency** & **Frequency** \\  & & **(Hz)** & **(Hz)** \\ \hline
2303 & \(2.87\times 10^{11}\) & 5.18 & 20.81 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of the beam. \begin{table}
\begin{tabular}{c c} \hline \hline
**Interaction** & Penalty method \\ \hline
**Calculation Framework** & Implicit (static analysis followed by dynamic analysis) with Hilber–Hughes–Taylor (HHT) method \\ \hline
**Time step** & 0.002 s \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Details of simulation. Figure 8: Finite element model of VBI system. depicted in Figures 11a and 11b, respectively. Notably, the high-frequency displacement appears to approximate a free-vibration response, which is consistent with the theoretical derivations presented in Equation (9) and corroborated by findings from previous studies [19]. Furthermore, the low-frequency displacement closely resembles the quasi-static bridge displacement, a phenomenon that has been previously illustrated and discussed in the literature [20]. We initially confirmed the theoretical feature that the transfer matrices for both low-frequency and high-frequency components were identical. For this simple VBI system, the  
Figure 11: Separated high\(-\)frequency and low-frequency bridge displacement at three observation points with the bandpass filter of 1 Hz: (**a**) high\(-\)frequency displacement; (**b**) low\(-\)frequency displacement. Figure 10: Simulation result for bridge displacement at three observation points. Figure 9: Three observation points. transfer matrix associated with the high-frequency responses was estimated using MATLAB's multiple linear regression function [26]. As indicated by Equations (8) and (9), the first two modes predominantly influence the bridge's responses. Thus, in this investigation, we focused exclusively on the primary vibration modes. Accordingly, a 2 \(\times\) 2 matrix as formulated in Equation (13) was employed as the transfer matrix. The high-frequency displacements at the 18 m and 10 m positions were designated as the two inputs, with the high-frequency displacement at the 5 m position set as the sole output. Utilizing MATLAB's multiple linear regression function (_'regress'_), the transfer matrix for high-frequency displacement was calculated. Subsequently, the two inputs were substituted with the low-frequency displacements at the 18 m and 10 m positions. By applying the determined transfer matrix to the low-frequency displacement inputs, the low-frequency displacement at the 5 m location was successfully reconstructed. Comparison of the low-frequency displacement time histories, as depicted in Figure 12, reveals a close correspondence between the reconstructed and simulated results.