## 4 Frequency Estimation of Midspan Deflection Signal Acquisition
### Estimation Method Based on the Amplitude Spectrum  
The finite element program is midspan to simulate the probability density function of the amplitude spectrum frequency curve. In conjunction with the original data distribution, it is evident that the distribution pattern of the amplitude of the midspan deflection signal at a given frequency bears resemblance to that of an exponential distribution. The probability density function of the exponential distribution is represented by Formula (7), and the corresponding fitting outcome is depicted in Figure 9.  
Figure 9: Simulation diagram of reconstruction curve at different acquisition frequencies. \[f(x)= \begin{cases}\frac{1}{\theta}e^{-x/\theta},&x>0\\ 0,&else\end{cases}  \tag{7}\]  
The basic parameter of the probability density function of the exponential distribution shown in Figure 10 is \(\theta\ =0.5817\). According to the characteristics of the graph, the acquisition frequency is \(f\ =2\) Hz, the high-frequency information is filtered by low-pass filtering, and the frequency-domain signal is converted into the time-domain signal by inverse Fourier transform, as shown in Figure 11.  
Statistical analysis of the time-domain-signal-related data obtained after statistical conversion shows that the mean value of the truncated signal deflection time-history curve is \(u_{1}=-1.65968\), the standard deviation is \(\varpi_{1}=0.87024\), the variance is \(\sigma_{1}{}^{2}=0.75733\), and the absolute maximum value is \(S_{1}=2.79337\). Compared with the quasi-static data, the error (absolute value) of each digital characteristic is as follows:  
Mean relative error:  
\[\alpha_{1}=\frac{u_{0}-u_{1}}{u_{0}}=\frac{1.65779-1.65968}{1.65779}=0.114\% \tag{8}\]  
Relative error of variance:  
\[\delta_{1}=\frac{\sigma_{0}^{2}-\sigma_{1}^{2}}{\sigma_{0}^{2}}=\frac{0.75695- 0.75733}{0.75695}=0.050\% \tag{9}\]  
Figure 11: Comparison diagram of dynamic deflection reconstruction curve. Figure 10: Simulation diagram of distribution function of amplitude spectrum diagram. Relative error of standard deviation:  
\[\psi_{1}=\frac{\varpi_{0}-\varpi_{1}}{\varpi_{0}}=\frac{0.87003-0.87024}{0.87003}=0. 024\% \tag{10}\]  
Maximum relative error:  
\[\Delta_{1}=\frac{S_{0}-S_{1}}{S_{0}}=\frac{2.73976-2.79337}{2.73976}=1.950\% \tag{11}\]  
Because the acquisition frequency is twice the cutoff frequency, i.e., \(f_{\rm s}=2\) Hz, when the bridge span is 30 m, the vehicle load is about 333 kN, and the vehicle speed is 50 km/h, the reliability calculation of the acquisition frequency of the structure is shown in Formula (12):  
\[P(f\leq f_{\rm s})=1-e^{-\frac{\pi}{\delta}}=96.790\% \tag{12}\]  
The meaning of the above formula is to determine the ratio of the area of the amplitude spectrum at the cutoff frequency of 2 Hz to the area of the original amplitude spectrum function. Similarly, when the acquisition frequencies are 2 Hz, 2.5 Hz, 3.0 Hz, and 3.25 Hz, the reliability of the acquisition frequency and the error of the truncated time-history curve are respectively selected, and the results are summarized as shown in Table 4.  
Draw the distribution diagram of the cumulative function of structural acquisition frequency reliability under this working condition, as shown in Figure 12.  
When the cutoff frequency is \(f_{1}=1.0\) Hz and the acquisition frequency is \(f_{1}=2.0\) Hz, the error of each digital characteristic of the signal is close to or less than 5%, and the reliability of the acquisition frequency is greater than 95%, which shows that it meets the requirements of practical engineering.