## 3 Peak Frequency Identification Method  
Judgment principles need to be defined primarily on how to identify the fundamental frequency of the bridge from the acceleration spectrum of the vehicle. The central idea is to select the peaks worth paying attention to according to specific indicators from the many peaks in the spectrogram within a specific frequency range. Then, known peaks are removed from the selected peaks and result in the isolation of the target peak. Combined with the issue at hand, the fundamental problem is to select the vehicle frequency item, bridge frequency item, and speed item from the spectrogram. When the vehicle frequency item and speed item are eliminated, the bridge frequency can be obtained. Firstly, the following parameters are introduced: Peak amplitude \(F_{pp}\), peak aspect ratio \(F_{hw}\), peak area \(F_{ae}\), and peak significance \(F_{sr}\). Their expressions are explained in the following section and in Figure 2.  
\[F_{pp}(i)=Y_{i}^{max} \tag{12}\]  
\[F_{ae}(i)=\frac{1}{2}\Big{[}Y_{i-1}^{min}+Y_{i+1}^{min}\Big{]}Y_{i}^{max} \tag{13}\]  
\[F_{hw}(i)=\frac{Y_{i}^{max}-\frac{Y_{i-1}^{min}+Y_{i+1}^{min}}{2}}{X_{i+1}^{min }-X_{i-1}^{min}} \tag{14}\]  
\[F_{sr}(i)=\min\left(\frac{Y_{i}^{max}}{Y_{i-1}^{min}},\frac{Y_{i}^{max}}{Y_{i+ 1}^{min}}\right) \tag{15}\]  
where, \(X_{i}^{max}\) or \(X_{i}^{min}\) represents the frequency value corresponding to the \(i\)-th extreme point and \(Y_{i}^{max}\) or \(Y_{i}^{min}\) represents the amplitude of the \(i\)-th extreme point. The superscript maxand min indicate the maximum value and minimum value, respectively. The physical meaning of \(F_{hw}\) is a quantity describing the peak shape. The larger the value of \(F_{hw}\), the more concentrated the frequency corresponding to the peak. \(F_{ae}\) and \(F_{sr}\) describe the ease in which the peak can be identified. The larger their value, the easier the peak is to be observed. \(F_{pw}\) indicates the degree of concern for the peak. Generally, it can be considered that the corresponding frequency with a large value of \(F_{pw}\) is the main component of the signal. Suppose there are \(n\) maximum points in the frequency range of interest, so the basic score of each maximum point can be defined as:  
\[V_{FB}(i)=F_{hw}(i)+F_{st}(i)+F_{ae}(i)/\max(F_{ae}(k)) \tag{16}\]  
where \(k=1,2,\cdot\cdot\cdot\cdot\cdot,n\). In order to improve identification accuracy of the peaks, it is necessary to make the following supplements:  
1. To ensure the significance of the maximum point between two adjacent minimum points, it is recommended that the amplitude of the maximum point should be at least 10% greater than that of the adjacent minimum point, which means \(F_{sr}(i)>1.1\). 2. In order to ensure the visibility of peaks, the area corresponding to each peak should not be too low. The ratio of the area corresponding to the peak in the same spectrogram to the maximum area should not be too low. To be conservative, take \(F_{ae}(i)/\max(F_{ae}(k))>0.01\) in this paper. Thus, the peak discrimination function is defined as: \[W_{pi}(i)=\left\{ \begin{array}{ll}1&F_{sr}(i)>1.1\quad\text{or}\quad F_{ae}( i)/\max(F_{ae}(k))>0.01\\ 0&\text{else}\end{array} \right..\] (17)  
A special case, as shown in Figure 3, has multiple successively decreasing small peaks (boxed in red) on the side of a major peak. This situation becomes more prominent as the vehicle crossing the bridge slows down, the number of sampling points increases, and the vehicle-to-bridge mass ratio increases. Preliminary analysis indicates that this phenomenon is caused by real-time changes in the frequency of vehicles and bridges in the VBI system as vehicles pass through the bridge. Applying Fast Fourier transformation alone to the signal cannot satisfactorily reflect the real-time frequency change in the spectrogram. Figure 2: Schematic diagram of peak parameters. For illustrative purposes, two cases are demonstrated briefly.