### 4.1 Acceleration responses

Vertical acceleration responses at three observation points (as shown in Fig. 1) from 18:00 on September 18 to 24:00 on September 20, 2019 (as shown in Fig. 2) were selected as input signals for modal parameter analysis. Figure 7 shows the acceleration responses after the low-pass filtering process with a cut-off frequency of 4 Hz. The closer the measurement point to the cantilever end, the greater the vertical acceleration during the observation period. The vertical acceleration of the AC10 measurement point is significantly greater than that of the other two observation points, and it was selected to be input data for the next modal parameter analysis.

The displacement responses can be obtained by the frequency domain integration of the structural acceleration responses. Figure 8 gives the corresponding vertical displacement responses at a time interval of 10 min. The vertical displacement response of measurement point AC10 is the largest, AC8 is the second largest and AC6 is the smallest, which is consistent with the acceleration response results. The time during which the displacement exceeded 0.01 m for the whole observation period was only 7% of the total time, so the variation of the damping ratio in this case study can be considered to be unrelated to the displacement amplitude (Wang et al., 2020).

### 4.2 SSI-COV and EWT-HT identification methods

The SSI-COV is a traditional modal parameter identification method that is widely used in civil engineering (Van Overschee and De Moor, 1997; Zou et al., 2020). Figure 9 shows the process of identifying the modal parameters using the SSI-COV method.

In recent years, time–frequency signal processing techniques such as the empirical wavelet transform (EWT) have been widely used for the modal parameter estimation of civil structures. The EWT (Gilles, 2013) adaptively divides the Fourier spectrum of the analyzed multi-component signal. It constructs a group of wavelet filter banks to filter the divided spectrum and decompose the multi-component signal into a series of amplitude modulation and frequency modulation single-component components with a tightly-supported Fourier spectrum.

In order to improve the boundary estimation of Gilles’ method when dealing with low signal-to-noise ratio multi-component signals, the power spectral density spectrum was employed to divide the signal frequency bands in this study, rather than the Fourier spectrum. Figure 10 compares the results using the Fourier spectrum as the basis of the frequency band division with those using the power spectral density. The power spectral density spectrum is smoother than the Fourier spectrum. Even in noisy cases, it can still identify each mode order frequency. The power spectral density curve of every significant spectral peak can be the signal of a single modal, and the band boundary can be more accurately estimated to establish the corresponding wavelet filter bank and better separation signals for each order mode.

For the selection of the frequency band boundary, Gilles (2013) proposed the method of selecting the mean value of two adjacent peaks of the spectrum as the boundary, which is very effective for signals with no noise and simple signal components. However, for noisy signals, signal decomposition based on Gilles’ method is unsuitable due to the existence of small peak values caused by noise in the signal’s spectrum. In order to reduce the influence of noise on signal decomposition, Amezquita-Sanchez et al. (2017) proposed an improved frequency band selection method that uses two minima adjacent to the maximum peak of the spectrum as the boundary. The improved method can better isolate the main vibration frequency and noise, and reduce the influence of the noise to the greatest extent.

Through the EWT, the original signal sequence was decomposed into the modal response component of the structure, which was composed of the free vibration response and the forced vibration response. The free attenuation response was obtained by the random decrement technique (Ibrahim, 2001) for the single component. Its free attenuation response can be expressed as:

$$v\left(t\right)={A}_{0}{e}^{-\zeta {\omega }_{0}t}cos\left({\omega }_{d}t+{\phi }_{0}\right)$$

(9)

where \({\omega }_{0}\) is the circular frequency; \({\omega }_{d}\) is the damping frequency; \(\zeta\) is the damping ratio and \({A}_{0}\) is a constant. The analytic signal \(z\left(t\right)\) corresponding to \(v\left(t\right)\) is:

$$z\left(t\right)=v\left(t\right)+\mathrm{i}\widetilde{v}(t)=A\left(t\right){e}^{-i\theta \left(t\right)}$$

(10)

where \(\widetilde{v}(t)\) is the Hilbert transform of \(v\left(t\right)\). For general engineering structures, the amplitude \(A\left(t\right)\) and phase \(\theta \left(t\right)\) in the formula can be further expressed as:

$$A\left(t\right)={A}_{0}{e}^{-\zeta {\omega }_{0}t}$$

(11)

$$\theta \left(t\right)={\omega }_{d}t+{\phi }_{0}$$

(12)

By using logarithms and differential operators, the amplitude function and phase function of Eq. 11 and Eq. 12 are respectively transformed to obtain:

$$lnA\left(t\right)=-\zeta {\omega }_{0}t+{lnA}_{0}$$

(13)

$${\omega }_{d}=\frac{d\theta \left(t\right)}{dt}$$

(14)

Obviously, the damping frequency \({\omega }_{d}\) can be obtained from the slope of the phase function \(\theta \left(t\right)\) corresponding to the equation of the line in Eq. 14. When the slope \(\zeta {\omega }_{0}\) of Eq. 13 of the line is determined, the \({\omega }_{0}\) and the damping ratio \(\zeta\) can be obtained by the following equation:

$${\omega }_{d}={\omega }_{0}\sqrt{1-{\zeta }^{2}}$$

(15)

Figure 11 shows the modal parameter identification calculation process of EWT-HT. The first step of this method is setting the maximum number of bands and calculating the power spectral density spectrum of the low-pass filtered acceleration response to determine the boundaries. Then, according to the divided frequency bands, the corresponding filter banks are built and multiple simple-frequency signals are obtained. The simple-frequency signals are treated by the random decrement technique to obtain the free damping vibration response, and the response amplitude and phase angle curves are found using the Hilbert transform. Finally, the least squares fitting method is used to extract the modal parameters.

### 4.3 Identification results and analysis

Based on the quasi-steady assumption, the aerodynamic force of the bridge deck can be expressed by the derivation of the static three-component coefficient and its first derivatives as follows (Kareem and Gurley, 1996; Kim et al., 2019). The theoretical aerodynamic damping ratio of the vertical bending motion is derived when the torsional motion is ignored:

$${\zeta }_{\text{aero }}=\frac{\rho BU}{8\pi {f}_{i}m}\left({\left[{C}_{L}^{^{\prime}}\right]}_{\alpha =0}+{C}_{D}\right)$$

(16)

where \(\rho\) is the air density, equal to 1.225 kg/m^{3}; \(B\) is the width of the bridge deck, which is 36.8 m; \(m\) is the mass per unit length, which is 34286 kg/m; \({f}_{i}\) is the natural frequency of the structure of *i* mode order (Hz); \({C}_{L}\) is the lift coefficient; \(\alpha\) is the wind attack angle; \({C}_{L}^{\mathrm{^{\prime}}}\) is the first derivative of the lift coefficient with respect to radians, which is 4.3154; \({C}_{D}\) is the drag coefficient, which is 0.2314 (Zhang, 2019).

According to Eq. 16, when the bridge structure is entirely determined, the aerodynamic damping ratio of the bridge deck is positively correlated with the mean wind velocity. This paper uses two methods, including the mentioned COV-SSI and EWT-HT, to identify the first three vertical bending modal parameters of the PSRB. The relationships between the mean wind velocity and modal parameters are linearly fitted by Eq. 17 and Eq. 18:

$${f}_{i}={a}_{1}\times U+{b}_{1}$$

(17)

$${\zeta }_{i}={a}_{2}\times U+{b}_{2}$$

(18)

where *U* is the 10-min mean wind velocity in this study (m/s); *f*_{i} is the modal frequency of the bridge (Hz); \({\zeta }_{i}\) is the modal damping ratio of the bridge; *a*_{1}, *a*_{2}, *b*_{1} and *b*_{2} are floating parameters that must be fitted using measured data.

In addition, the standard deviation (σ), coefficient of variation (*c*_{v}) and range (*R*) are used to judge the dispersion of the identified modal parameters using the two methods, as follows:

$$\sigma =\sqrt{\frac{\sum_{i=1}^{n} {\left({X}_{i}-\overline{X }\right)}^{2}}{n-1}}$$

(19)

$${c}_{v}=\frac{\sigma }{\overline{X} }$$

(20)

$$R=X_{max}-{\mathrm X}_{min}$$

(21)

where \(X\) is the modal frequency or modal damping ratio; \(\overline{X }\) is the mean value of *X*; \({X}_{max}\) is the maximum value of *X*; \({X}_{min}\) is the minimum value of *X*; *i* is the sample number; \(n\) is the total number of samples.

### 4.4 Modal frequency

Figure 12 shows the measured modal frequencies of the first-order vertical bending of the PSRB under various mean wind velocities using the SSI-COV and EWT-HT identification methods. The two identification methods can accurately identify the first-order vertical bending modal frequencies, but the SSI-COV identification method results in a few extreme values. The mean value of the difference between the two identification methods is very small and equal to -0.003 Hz. The difference decreases with the increase of the mean wind velocity. The greater the mean wind velocity, the more significant the difference between the two methods. The modal frequency identified using SSI-COV is quite large and has a few extreme values. This may be because there are many false modes in the application of the SSI-COV method. These false modes mainly arise from the system parameters and the input signal. The system parameters are also known as the system order, and this order, which must be empirically pre-determined, results in the generation of false modes. In addition, the input signal does not satisfy the white noise assumption, and the environmental influence on the output signal can also result in the generation of false modes.

Before the correlation analysis, the sample sizes of different wind velocity intervals were statistically analyzed. Figure 13 shows that the sample sizes are different in different wind velocity ranges. The samples are not evenly distributed across each entire wind velocity range, and the sample size in the wind velocity range of 8–20 m/s accounts for 85% of the total sample size. Since this study seeks the linear relationship between the modal parameters and the mean wind velocity, the linear fitting results using all the sample data cannot truly reflect the linear relationship between the sample data under such a distribution. Therefore, the median of the sample data within each wind velocity interval is set as 1 m/s for fitting, which can reflect the overall level and remove the influence of any extreme values.

Figure 14 shows the relationship between the measured modal frequency and wind velocity in the first three vertical bendings of the PSRB using the SSI-COV and EWT-HT identification methods. The first three vertical bending frequency results using the two methods are basically identical, and the fitting results show that the variation of the modal frequency with the mean wind velocity is relatively small. There is almost no correlation between the modal frequency and the mean wind velocity. When the mean wind velocity is zero, the structural frequency (the intercept of the fitting line) of the first-order vertical bending to the third-order vertical bending indicates that the identification results of the two identification methods are effective and accurate.

### 4.5 Modal damping ratio

Figure 15 shows the measured modal damping ratios of the first-order vertical bending of the PSRB under various mean wind velocities using the SSI-COV and EWT-HT identification methods. The mean value of the difference between the two identification methods is very small and equal to -0.587%, and the difference decreases with the increase of the mean wind velocity. Similarly, the SSI-COV method results in some extreme damping ratios and the identified damping ratios are more extensive than those identified using the EWT-HT method. Because the modal frequency and modal damping ratio of each sample always appear in pairs, the sample number distribution of the damping ratio in different wind velocity ranges is the same as that in Fig. 13. The identified modal damping ratios need to be processed using the same approach before the linear fitting.

Figures 16 and 17 show the relationship between the identified modal damping ratio and the mean wind velocity using the SSI-COV and EWT-HT methods, respectively. According to the fitted curves and *R*^{2} results, the first-order vertical bending damping ratio has the strongest correlation with the mean wind velocity, and it increases with the increase of the mean wind velocity. The correlation between the second-order vertical bending damping ratio and the mean wind velocity is weaker than that of the first-order vertical bending, while the third-order vertical bending damping ratio can be regarded as having no correlation with the mean wind velocity. Figure 16 shows that when the mean wind velocity is zero, the intercept of the fitting curve of the damping ratio of the first three vertical bendings using the SSI-COV method (namely, the structural damping ratios), are 0.382%, 0.827% and 0.720%, respectively, which are larger than the structural damping ratio obtained by the theoretical aerodynamic damping ratio curve of the first three vertical bendings, i.e., 0.113%. 0.668% and 0.429%. Both the structural damping ratios obtained by the measured damping fitting curve and the structural damping ratio obtained by the theoretical aerodynamic damping ratio curve are different from the structural damping ratio of 0.5% recommended by the Chinese structural specification (JTG/T 3360–01-2018). Similarly, Fig. 17 shows that the structural damping obtained from the fitted curve of the damping ratios identified by the EWT-HT method is greater than that obtained from the theoretical aerodynamic damping curve. At the same time, regardless of the structural damping ratios obtained from the fitting curve of the measured damping ratios, or the structural damping ratios obtained from the curve of the theoretical aerodynamic damping ratios, there is no phenomenon through which the structural damping ratio decreases with the increase of the mode order, which is similar to the results in Li et al. (2011). Comparing the two identification methods, the fitting results of the damping ratios identified using the EWT-HT method are better than those found with the SSI-COV method in terms of *R*^{2} values. For the comparison fitted curve and theoretical curve, no matter which identification method is applied or which mode order is identified, the intersection point of the two curves is near the mean wind velocity of approximately 14 m/s. When the mean wind velocity is less than 14 m/s, the value of the fitted curve is greater than that of the theoretical curve. On the contrary, when the mean wind velocity is greater than 14 m/s, the theoretical curve value is greater than the fitted curve value.

Figure 18 shows the differences between the identified damping ratios of the first three vertical bendings using two methods. The mean differences between the identified damping ratios are -0.104%, 0.045% and 0.494%, respectively. The standard deviations of the identified damping ratios are 0.007%, 0.088% and 0.031%, respectively. To sum up, the difference between the two methods in identifying the first-order vertical bending damping ratio is small, while the differences in the identification of the damping ratios of the second-order and third-order vertical bendings are large.

The Pearson correlation coefficient is used to quantify the similarity between the modal damping ratios identified by different methods. The correlation coefficient \(r\) is defined as:

$$r\left({Y}_{1},{Y}_{2}\right)=\frac{Cov({Y}_{1},{Y}_{2})}{\sqrt{Var\left[{Y}_{1}\right]Var[{Y}_{2}]}}$$

(22)

where \({Y}_{1}\) is the modal damping ratio identified by the SSI-COV method; \({Y}_{2}\) is the modal damping ratio identified by the EWT-HT method; \(Cov ({Y}_{1},{Y}_{2})\) is the covariance of \({Y}_{1}\) and \({Y}_{2}\); \(Var[{Y}_{1}]\) is the variance of \({Y}_{1}\); \(Var[{Y}_{2}]\) is the variance of \({Y}_{2}\).

The correlation coefficients of the damping ratios of the first three vertical bendings identified by the different methods are 0.92, 0.16 and 0.17, respectively. The first-order vertical damping ratios identified by these two methods are highly consistent. In contrast, the similarities between the second-order and third-order damping ratios identified using the two methods are unsatisfactory.

In this study, the discreteness of the identification results of the two methods is analyzed to compare the methods’ engineering applicability and practicability. A mean wind velocity of 20 m/s is selected (for example, the mean wind velocity of 19–21 m/s is selected at 20 m/s) to calculate the standard deviation (σ), coefficient of variation (*c*_{v}) and range (*R*), as well as to draw a bar chart to analyze the dispersion of the results calculated by the different methods. Figure 19 shows that for the first- and second-order vertical bending modes, the three parameters related to discreteness using the SSI-COV method are all larger than those using the EWT-HT method, especially for the second-order vertical bending mode. However, it is just the opposite for the third-order mode. This is consistent with the results presented in Figs. 16 and 17. The identification results of the SSI-COV method appear to be more discrete. The EWT-HT method has better control over the calculation discreteness.