## 5 Structural Dynamic Deformation Monitoring of Rainbow Bridge
### Vibration Signal Analysis  
As depicted in [PERSON] sampling theory, when the sampling frequency is greater than twice the highest frequency of the signal, the digital signal completely retains the information of the original signal; thus, the sampling signal can be reconstructed as the initial continuous form without the loss of information. According to the numerical simulation results in the previous section, the first order natural frequency (i.e., 0.1632 Hz) is much less than 10 Hz. Therefore, this sampling frequency meets the theoretical requirements. Dynamic deformation monitoring was conducted for 10 consecutive hours from 9:00 a.m. to 7:00 p.m. on 24 February 2017. The collected data and its PSD function at the monitored point C3 are shown in Figure 11. The amplitude of the vertical vibration displacement is within \(-\)0.0564-0.0652 m. FFT was employed to identify the structural natural frequency. There is an obvious peak corresponding to the structurally dominant frequency (i.e., 0.1835 Hz). Compared with the result of the numerical analysis, the identified modal parameter of the bridge using the FFT method is larger. We obtained noise distribution characteristics according to the stability experiment. The proposed combined denoising method was employed to deal with both low-frequency and high-frequency noise. At first, the original signal was decomposed by the CEEMDAN method and 14 IMFs were obtained. Figure 12 illustrates the 14 IMFs and their corresponding PSD functions, from which we can  
Figure 10: (**a**) The first order of the mode shape, (**b**) the second order of the mode shape, (**c**) the third order of the mode shape, (**d**) the fourth order of the mode shape, (**e**) the fifth order of the mode shape, and (**f**) the sixth order of the mode shape. Figure 9: The FEM of Rainbow Bridge. conclude that the components containing the structural natural frequency is mainly distributed within IMF6-IMF8. IMF1-IMF3 and IMF11-IMF14 components should be removed because they contain a lot of high-frequency and low-frequency noise, respectively. This operation is the equivalent of denoising with a band-pass filter. IMF6-IMF8 components were reconstructed as Signal R1. Signal R1 and its PSD function are shown in Figure 13. It is obvious that the amplitude of the displacement (i.e., \(-\)0.0349-0.0257 m) was significantly reduced following reconstruction. In addition low-frequency noise was effectively eliminated. Subsequently, the three-layer WP technique was used to deal with Signal R1, and thus Signal R2 was derived. As shown in Figure 14, the amplitude of displacement (i.e., \(-\)0.0235-0.0198 m) was further reduced. Moreover, a single WP technique was adopted to denoise the original signal from Figure 11, with the result (i.e., Signal R3) indicated in Figure 15. Meanwhile, the RMS of the original signal, Signal R3, Signal R1, and Signal R2 were obtained, being 0.0153 m, 0.0147 m, 0.0064 m, and 0.0056 m, respectively. In view of the above discussion, it may be concluded that the combined method used in this paper is more effective than the individual application of CEEMDAN or WP. Figure 11: Original signal of Point C3 and its PSD function. Figure 12: The IMFs of the original signal and their corresponding PSD functions. In general, the damping ratio selected for design is estimated empirically. Therefore, it is of great importance to use the RDT method to obtain a damping ratio from the field measured data. In some research, the value of subsample \(N\) has been in the range of 400 to 2000 [42]. In this paper, the value of \(N\) is defined as 1500, and different values of the time interval were chosen as \(\tau=100\) s, \(\tau=150\) s, and \(\tau=200\) s. Figure 16 illustrates the computed results of the RDS, and the structural damping ratios corresponding to different \(\tau\) values can be obtained via Equation (23) (i.e., \(\tau=100\) s \(\xi=1.77\%\); \(\tau=150\) s \(\xi=1.70\%\); and \(\tau=200\) s \(\xi=1.73\%\)). The results indicate that the damping ratio is close to 1.73%.