## 4 Frequency Estimation of Midspan Deflection Signal Acquisition
### Estimation Method Based on the Power Spectrum  
The selection method for the cutoff frequency depends on the spectrogram. Once the spectrogram is distorted or too complicated, it is difficult to find the accurate cutoff frequency [44]. Therefore, refer to the definition of the cutoff frequency in the sensor [45]: when the amplitude of the input signal is kept constant, the output signal is reduced to 0.707 times the maximum value by changing the frequency, which is expressed by frequency response characteristics; that is, the cutoff frequency is at the point of \(-3\) dB. The power spectral density of the signal describes the distribution of the power-limited signal with the frequency in the frequency domain. In order to highlight the main frequency components, the frequency at the \(-3\) dB point of the energy power spectrum function is selected as the cutoff frequency, and the amplitude spectrum corresponding to this frequency is about 0.707 times the maximum value; that is, the cutoff frequency \(f_{1}=0.62\) Hz and the acquisition frequency \(f_{s}=1.24\) Hz, as shown in Figure 13.  
The high-frequency components are removed by low-pass filtering, and the frequency-domain signal is converted into the time-domain signal and compared with the quasi-static curve, as shown in Figure 14.  
Figure 14: Comparison diagram of dynamic deflection reconstruction curve. Figure 13: Dynamic deflection power spectrum curve. (**a**) Upper layout plan. (**b**) Lower layout plan. (**a**) Cutoff frequency at the maximum value of the amplitude spectrum of 0.707. (**b**) Cutoff frequency at \(-3\) dB power spectrum. Through the statistical calculation of the discrete data of time domain signals, it is found that the signal with a truncated signal of \(f_{1}=0.62\) Hz has a mean value of \(u_{1}=-1.65968\), a standard deviation of \(\varpi_{1}=\ 0.86385\), a variance of \(\sigma_{1}^{2}\ =\ 0.74623\), and an absolute maximum value of \(S_{1}=2.88\). For this 30-m span simply supported beam bridge, the first-order natural frequency is 4.3019 Hz, the second-order natural frequency is 8.9381 Hz, and the third-order natural frequency is 13.061 Hz. The disturbance frequency \(\varpi_{1}=1\times\pi\times\
u/l=1.454\) Hz of the moving force, the cutoff frequency of the midspan deflection signal is \(f_{1}=0.62\) Hz, and the acquisition frequency \(f_{s}=1.24\) Hz, which is slightly less than the disturbance frequency of the first-order moving force. The error (absolute value) of the cutoff frequency reconstruction curve and various digital characteristics is as follows:  
Mean relative error:  
\[\alpha_{1}=\frac{u_{0}-u_{1}}{u_{0}}=\frac{1.65779-1.65968}{1.65779}=0.114\% \tag{13}\]  
Relative error of variance:  
\[\delta_{1}=\frac{\sigma_{0}^{2}-\sigma_{1}^{2}}{\sigma_{0}^{2}}=\frac{0.75695 -0.74623}{0.75695}=1.416\% \tag{14}\]  
Relative error of standard deviation:  
\[\psi_{1}=\frac{\varpi_{0}-\varpi_{1}}{\varpi_{0}}=\frac{0.87003-0.86385}{0.870 03}=0.710\% \tag{15}\]  
Maximum relative error:  
\[\Delta_{1}=\frac{S_{0}-S_{1}}{S_{0}}=\frac{2.73976-2.8810}{2.73976}=5.150\% \tag{16}\]  
Because the acquisition frequency is twice the cutoff frequency, i.e., \(f_{s}=1.24\) 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 (17):  
\[P(f\leq f_{s})=1-e^{-\frac{\pi}{b}}=95.860\% \tag{17}\]  
The meaning of the above formula is to determine the ratio of the amplitude spectrum area under the acquisition frequency of 1.24 Hz to the area of the original amplitude spectrum function. Draw the cumulative function distribution diagram of the structural acquisition frequency reliability under this working condition, as shown in Figure 15.  
Figure 14.0 Frequency reliability accumulation function based on power spectrum acquisition.