## 5 Bridge Influence-Line Identification
### Bridge Influence-Line Identification Update  
Using the derived first-order vertical bending frequency of the cable-stayed bridge as shown in Equation (5), the theoretical analytical frequency of the test bridge is calculated to be 0.707 Hz, and the error quantization formula is introduced as shown in Equation (17) [26], which shows that the relative error between the vertical bending frequency derived from the empirical formula and the measured bridge fundamental frequency and the computed frequency of the finite-element model is controlled at about 15%. The error accuracy is not high. To obtain a more accurate vertical bending frequency of the bridge, the empirically derived frequency is used as the threshold value of _IMF_ decomposition, and the E-VMD method is used to strip the _IMF_ components, whose main frequency is larger than the fundamental frequency, and Fast Fourier Transform is performed on the stripped power components [27], as shown in Figure 11. The operational modal frequency of the bridge is identified to be 0.84034 by stripping the power components that are doped in the influence line. The relative error between the identified operational modal frequency and the theoretical frequency is improved to 0.16%, and the error with the measured frequency is 0.32%.  
\[d=\frac{\sum\lvert\varepsilon_{a}-\varepsilon_{m}\rvert}{\sum\lvert\varepsilon_ {a}\rvert} \tag{17}\]  
where \(d\) denotes the relative error to the measured data, \(\varepsilon_{a}\) denotes the measured data, and \(\varepsilon_{m}\) denotes the data identified by the method in this paper.  
From the error analysis in Table 2, it can be seen that the method proposed in this paper identifies the bridge vibration fundamental frequency. The accuracy meets the general engineering needs, avoiding the actual measurement steps of the fundamental frequency at the engineering site and providing a new idea for the rapid testing of bridge inspection vehicles.  
Figure 11: Extraction of vibration fundamental frequency from dynamic response.  
The measured deflection response is preprocessed, and the empirically derived value is used as the cutoff frequency of the E-VMD method to obtain the deflection curve, which is combined with the Tikhonov regularization to construct the pathological equation for the identification of the influence line. The quasi-static deflection influence line is identified through the relative error analysis with the measured base frequency as the decomposition threshold to identify the deflection line of influence. It is found that the error is 14.02%, and the accuracy also does not meet the requirements of load-carrying capacity assessment, model correction, and other research accuracy, with the above identification of the bridge-operation modal frequency to re-use as the threshold value of _IMF_ decomposition, re-peeling the dynamic components, restoring the quasi-static time curve, and then removing the axle effect, to obtain the updated deflection line of influence, at the same time, the measured deflection influence line is compared. The relative error of the measured deflection line is only 0.83%. The recognition accuracy is greatly improved compared with the previous period, which meets the subsequent research requirements, as shown in Figures 12 and 13.