## 3 Peak Frequency Identification Method
The value of each parameter is shown in Table 1. The Hilbert transform was used to analyze the vehicle and bridge terms in the vehicle acceleration signal, as well as to obtain their time-frequency curve during the passage of the vehicle on the bridge, shown in Figure 4. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & **Case 1** & & **Case 2** & \\ \hline \(\mu_{1}\) & & 1.5 & & & 1.5 & \\ \(\alpha_{nm}\) & 0.01 & 0.15 & 0.35 & & 0.15 & \\ \(\alpha_{v}\) & & 0.3 & & 0.15 & 0.3 & 0.45 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Value of parameters. Figure 4: Frequency variation of vehicle and bridge when vehicle passing the bridge. (**a**) With different vehicle mass; (**b**) with different speed. Figure 3: Vertical acceleration spectrum of vehicle with different (**a**) \(\alpha_{nm}\) and (**b**) \(\alpha_{v}\). In order to minimize the interference of these small peaks and highlight the frequency of the vehicle and bridge, gain factors are defined for each major peak. The approach of assigning gain coefficient is as follows: (1) The top three with the largest value of _Fpv_ are given 5, 2, and 1 in turn:  
\[W_{G1}(i)=\left\{ \begin{array}{ll}5&\text{Rank}\big{(}F_{pv}(i)\big{)}=1\\ 2&\text{Rank}\big{(}F_{pv}(i)\big{)}=2\\ 1&\text{else}\end{array} \right.\quad. \tag{18}\]  
For the maximum point in the maximum point, assign 5:  
\[W_{G2}(i)=\left\{ \begin{array}{ll}5&F_{pv}(i)\text{ is the maximum point among the maximum points}\\ 1&\text{else}\end{array} \right.\quad. \tag{19}\]  
In summary, the score of the maximum point is:  
\[V_{F}(i)=V_{FB}(i)\times W_{pi}(i)\times\text{max}(W_{G1}(i),W_{G2}(i)). \tag{20}\]  
Peaks in the spectrogram were sorted according to the above scoring principle. The acceleration of the vehicle contains the vehicle frequency term, bridge frequency term, and speed term, which correspond to the vehicle frequency, the bridge frequency, and the speed frequency in the vehicle vertical acceleration spectrum, respectively. In this paper, most of the top three scoring peaks correspond to the bridge frequency, vehicle frequency, and speed frequency. The intuitive performance on the spectrogram is three significant peaks. After excluding the vehicle frequency \(\Omega_{bex}\) and speed frequency \(\Omega_{sex}\) from these three frequencies, the resulting bridge frequency \(\Omega_{bex}\) can be obtained successfully. However, changes in the vehicle-to-bridge mass ratio \(\alpha_{mv}\), vehicle-to-bridge frequency ratio \(\mu_{1}\), and vehicle speed \(\alpha_{v}\), pose difficulties in bridge frequency recognition. 1. When \(\alpha_{mv}\) is small and \(\alpha_{v}\) is large, the vehicle-bridge interaction is weak. At the same time, the number of collection points is small. Although \(\Omega_{bex}\) can be identified, the bridge frequency term cannot be fully reflected in the vehicle acceleration signal, which leads to the failure of extracting \(\Omega_{bex}\). 2. When the vehicle speed gradually increases to meet \(\alpha_{v}>2\pi\Omega_{v}\), which is \(L/v<1/2\pi\omega_{v}\), the time for the vehicle to cross the bridge is less than its vibration period. The vehicle has left the bridge before finishing a complete vibration cycle. This will result in that the vehicle acceleration signal does not include the vehicle frequency term, and \(\Omega_{bex}\) cannot be obtained. 3. When \(\mu_{1}\) is small, the vehicle frequency is too low to distinguish between vehicle frequency and speed terms in the spectrogram. 4. The vehicle frequency coincides with the high-order frequency of the bridge. While the overlap between the vehicle frequency and bridge fundamental frequency is avoided in this paper, the vehicle frequency term coincides with the \(n\)-th order bridge frequency term in the spectrogram when \(\mu_{1}\) is close to \(n^{2}\) (\(n>2\)). As a result, it is impossible to solve for \(\Omega_{bex}\), however it generally does not affect the identification of \(\Omega_{bex}\).