## 4 Frequency Estimation of Midspan Deflection Signal Acquisition  
### Frequency Characteristics of Bridge Deflection  
Assuming that the highest frequency of the original signal is \(f\), the mean value is \(u\), the mean square value is \(\Psi^{2}\), and the variance is \(\sigma^{2}\), assuming that the cutoff frequency of the signal is \(f_{1}\) and the acquisition frequency is twice the cutoff frequency, that is, \(f_{s}\)\(=2f_{1}\), combined with the characteristics of the deflection response, the original signal is processed by low-pass filtering, and the signal is converted into a time-domain signal by inverse Fourier transform. The processed time-domain signal has a mean value of \(u_{1}\), a mean square value of \(\Psi_{1}^{2}\), and a variance of \(\sigma_{1}^{2}\). The reliability of the measured actual structural deformation condition of the bridge can be inferred from the consistent frequency data gathered by the sensors of the health monitoring system. The utilization of high-frequency sampling allows for the collection of accurate deformation data related to bridge constructions. Nevertheless, it is crucial to acknowledge that this methodology is accompanied by significant financial considerations and a surge in the amount of data being processed. If the approach of acquiring low-frequency and equal-frequency data is employed, it will result in the loss of crucial information and a significant discrepancy between the monitoring outcomes and the actual data [42]. Simultaneously, the quasi-static response predominantly manifests in the low frequency range of the overall dynamic response. In this part, the shortened time-domain signal will be fitted using the quasi-static deflection time-history curve of the bridge. The highest frequency of the quasi-static deflection response is \(f_{0}\), the mean value is \(u_{0}\), the mean square deviation is \(\sigma_{0}\), and the variance is \(\sigma_{0}^{2}\). Then, the relative errors of the digital characteristics of the structural response time-history curve and the quasi-static response time-history curve obtained through reasonable acquisition frequency are as follows:  
Figure 7: Fitting diagram of quasi-static deflection time-history curve. The concept of mean relative error pertains to the relative magnitude of the discrepancy between the mean values of two curves. A smaller mean relative error indicates a closer proximity between the mean values of the two curves. The mathematical representation of mean relative error is depicted in Equation (4). \[\alpha=\frac{u_{0}-u_{1}}{u_{0}} \tag{4}\]  
The mean square value relative error represents the relative size of the overall difference between the two curves; that is, the smaller the mean square value relative error, the closer the overall shape of the two curves is; its expression is shown in (5):  
\[\theta=\frac{\sigma_{0}-\sigma_{1}}{\sigma_{0}} \tag{5}\]  
The variance relative error, which represents the relative magnitude of the variation degree of the two curves, means that the smaller the variance relative error, the closer the variation degree of the two curves is, and its expression is shown in (6):  
\[\delta=\frac{\sigma_{0}^{2}-\sigma_{1}^{2}}{\sigma_{0}^{2}} \tag{6}\]  
The above three relative error indicators can effectively evaluate the similarity between the time-history curve after resampling with reasonable acquisition frequency and the quasi-static response time-history curve. Take the 30-m simply supported T-beam as an example. See Figures 1 and 2 for the section size and superstructure. The vehicle load travels at a uniform speed of 50 km/h, and the midspan time-history curve of the structure is obtained.