Modal Identification of Structures by Eliminating the Effect of the High Ocean Wind

Abstract

Tropical cyclone is a rapidly rotating storm system with severe excitation such as high wind. This severe excitation may change the performance of structures, such as bridges, wharves and wind turbine structures. It is very necessary to monitor this important change. Modal parameters are the ones to reflect the structural instinct behavior. However, many identification methods assume that the excitation is white noise, which is not the truth during high ocean wind excitation. Therefore, the modal identification method to deal with severe ocean wind excitation should be investigated. This paper proposes an innovative method to eliminate the effect of high ocean wind on modal identification. The formulation to generate an impulse response is described, where the effect of high wind is pointed. Then the elimination method is derived using the wind velocity spectrum and correlation function. After the wind field is simulated, the wind velocities and spectra at all accelerometer positions are obtained. The real impulse response form is obtained. Then, modal identification using the real impulse response is performed. Finally, a practical cable-stayed bridge is employed and modal identification is performed. The results show that the identified modes can reflect structural real behavior.

1. Introduction

The structures often withstand complicated and volatile excitation such as cyclic wave loading, high winds and sea quakes, which may change the structural performance and cause hidden safety problems. Therefore, it is very necessary to monitor the health condition of the structures under severe wind. The methods to monitor offshore structures, including long-span bridges, wharves [1,2], offshore platforms [3] and offshore wind turbine structures [4,5] attracted a lot of researchers. Some theoretical and experimental studies have been investigated. Russo et al. investigated the influence of waves and wind with a physical model of a spar buoy wind turbine with the angular motion of control surfaces implemented [6]. Kim et al. used static response and a factor accounting for dynamic amplification to estimate the dynamic peak response under extreme ocean environmental loads [7]. Li et al. proposed a new index called the total modal energy method, which shows strong robustness for the cases of multiple damage locations for offshore wind turbine structures [8]. Yang et al. investigated the low-frequency response characteristics under various regular and irregular wave sea states [9]. Yao et al. proposed a new modal identification method based on an autoregressive spectrum-guided variational mode decomposition method to develop accurate identification methods for structures under nonstationary excitations [10]. However, considering the problems of nonlinearity and time-variation, the frequency of excitation is included in the result of modal identification, which is also time-varying and not true.

Modal analysis is the common method to evaluate the structure performance in structural health monitoring, one of whose merits is that it can monitor the structure instinct dynamic behavior [11,12,13]. There are lots of identification methods, such as the stochastic subspace identification (SSI) method with the assumption of white excitation [14], the Eigensystem realization algorithm (ERA) which assumes that the responses are free decayed [15]. Qu et al. propose a new index and improved stabilization diagram for the eigensystem realization algorithm to distinguish spurious mode and improve the precision [16,17,18]. Recently, the blind source separation technique identified structural modes under the frame of the relationship between responses and coordinates [19,20,21].

The impulse response is very important for mode identification in the time domain, especially for ERA. For offshore structures, excitation, such as wind and waves, is always ambient. To transfer the ambient excitation response to impulse response, the natural excitation technique (NExT) is an effective way, which replaces the impulse response with a correlation function with the assumption of white noise excitation [22,23]. However, some loads, such as the high wind, are not white excitation. Some modal identification methods have been investigated for this issue. Qu et al. applied the virtual impulse response which is derived from the inverse Fourier transform of the ratio of the cross-power to auto-power spectral density functions of the measurement responses to deal with the non-white excitation [24]. Khan et al. identified the modal parameters of cable-stayed bridges by an improved ensemble empirical mode decomposition because the collected data is non-linear and time-varying [25]. However, the physical meaning is not clear. Lin applied the non-stationary correlation technique to perform modal identification [26]. However, it treated the non-stationary responses as a stationary random process with a slowly time-varying function, which cannot guarantee the white excitation. Dan Li et al. made the operational modal analysis by SSI with a delay index [27]. However, it is a little difficult to use when every time point is non-white. Guo and Kareem modified the short-time Fourier transform to deal with the non-stationary excitation [28]. However, when the non-stationary excitation is severe, the short time window is not available. For wind excitation, Sarlo et al. utilized the frequency method and SSI to identify the structural modes under wind and human-induced excitation [29]. However, it did not consider the non-white excitation. Li et al. identified modes under typhoons with NExT-ERA, which also did not consider non-white excitation [30].

This paper proposes an innovative method to generate real impulse responses. Then the real impulse responses are used to identify the modes of structures that are suffered from the severe wind. The formulation based on NExT is derived, and the drawback part of the correlation function is pointed out. The formulation to eliminate the effect of non-white noise is derived. Finally, a practical stayed-cable bridge is employed. The monitoring data during Typhoon Matmo passing by is analyzed.

2. Problem Description

For an $N$ degree-of-freedom structure, the equation of motion for a linear time-invariant system is found as follows:

$$M\ddot{y}\left(t\right)+C\dot{y}\left(t\right)+Ky\left(t\right)=f\left(t\right)$$

(1)

where $y\left(t\right)={\left[\begin{array}{cccc}{y}_1\left(t\right)& y_2\left(t\right)& \cdots & y_N\left(t\right)\end{array}\right]}^{\mathrm{T}}$ is the displacement vector; $f\left(t\right)={\left[\begin{array}{cccc}{f}_1\left(t\right)& f_2\left(t\right)& \cdots & f_N\left(t\right)\end{array}\right]}^{\mathrm{T}}$ is the load vector; $M$, $C$, $K$ are the mass matrix, damping matrix and stiffness matrix, respectively. When the structure is excited at position point $k$, the displacement response of point $i$ can be expressed as follow:

$$y_{ik}\left(t\right)={\displaystyle \sum _{s=1}^{2N}{\varphi }_{is}{\alpha }_{ks}{\displaystyle {\int }_{-\infty }^t{e}^{{\lambda }_s\left(t-q\right)}{f}_k\left(q\right)dq}}$$

(2)

where ${\varphi }_{is}$ is the value of the $s$-th modal shape at $i$-th measurement point; ${\alpha }_{ks}$ is a constant related to $s$-th mode and excitation point $k$; ${\lambda }_s$ are the eigenvalues of $s$-th modes; $f_k\left(q\right)$ is excitation at the excitation point $k$ at the $q$-th time step; $f_k\left(q\right)dq$ is a very short-duration impulse acting on the structure during the interval of time $dq$. The correlation function for the responses of point $i$ and $j$ can be expressed as follows:

$$\begin{array}{cc}{R}_{ijk}\left(\tau \right)& =\mathrm{E}\left[y_{ik}\left(t+\tau \right)y_{jk}\left(t\right)\right]\hfill \\ & ={\displaystyle \sum _{r=1}^{2N}{\displaystyle \sum _{s=1}^{2N}{\varphi }_{ir}{\varphi }_{js}{\alpha }_{kr}{\alpha }_{ks}{\displaystyle {\int }_{-\infty }^t{\displaystyle {\int }_{-\infty }^{t+\tau }{e}^{{\lambda }_r\left(t+\tau -p\right)}{e}^{{\lambda }_s\left(t-q\right)}\mathrm{E}\left[f_k\left(p\right)f_k\left(q\right)\right]dpdq}}}}\hfill \end{array}$$

(3)

where operator “$\mathrm{E}[·]$” represents expectation. In calculation, $R_{ijk}\left(\tau \right)$ is a vector, but its elements are computed independently. Then, the elements are recombined into a vector.

If the excitation $f\left(t\right)$ is ideal white noise, then the following expression is satisfied:

$$\mathrm{E}\left[f_k\left(p\right)f_k\left(q\right)\right]={\alpha }_k\delta \left(p-q\right)$$

(4)

where the operator “$\delta$” represents Dirichlet function; ${\alpha }_k$ is a constant related to excitation point $k$. Then Equation (3) is changed to:

$$R_{ijk}\left(\tau \right)={\displaystyle \sum _{r=1}^{2N}{\displaystyle \sum _{s=1}^{2N}{\varphi }_{ir}{\varphi }_{js}{\alpha }_{kr}{\alpha }_{ks}{\alpha }_k{\displaystyle {\int }_{-\infty }^t{e}^{{\lambda }_r\left(t+\tau -p\right)}{e}^{{\lambda }_s\left(t-p\right)}dp}}}$$

(5)

The integration part of Equation (5) can be simplified to the following expression:

$${\displaystyle {\int }_{-\infty }^t{e}^{{\lambda }_r\left(t+\tau -p\right)}{e}^{{\lambda }_s\left(t-p\right)}dp}=-\frac{e^{{\lambda }_r\tau }}{{\lambda }_r+{\lambda }_s}$$

(6)

Then Equation (5) is changed to:

$$R_{ijk}\left(\tau \right)={\displaystyle \sum _{r=0}^{2N}{b}_{jr}{\varphi }_{ir}{e}^{{\lambda }_r\tau }}$$

(7)

with

$$b_{jr}={\displaystyle \sum _{s=1}^{2N}{\varphi }_{js}{\alpha }_{kr}{\alpha }_{ks}{\alpha }_k\left(-\frac{1}{{\lambda }_r+{\lambda }_s}\right)}$$

(8)

It is obvious that Equation (7) is the form of the impulse response. Therefore, the correlation function $R_{ijk}\left(\tau \right)$ can be used as an impulse response to identify structural modes by some methods, such as ERA or SSI. This method to generate the impulse response is NExT.

However, there is an assumption of white noise excitation. For wind excitation, especially severe wind such as typhoons, it is not white noise. Therefore, Equations (4)–(8) are not satisfied. It is not the impulse response form, and cannot be used to identify structural modes.

3. Modal Analysis for High Wind Excitation

Kaimal spectrum is a method to calculate and analyze wind speed data in different areas. One of the characteristics of the Kaimal spectrum is that the spectral density varies along the height. According to the statistics, it is known that wind would only excite the structure at low-frequency range. Therefore, the effect of the non-white noise mainly caused by severe wind is only in the low-frequency region.

Denoting the truncated order of low-frequency range as $N_{tr}$, Equation (3) can be described as:

$$\begin{array}{cc}{R}_{ijk}\left(\tau \right)& =\mathrm{E}\left[y_{ik}^{low}\left(t+\tau \right)y_{jk}^{low}\left(t\right)\right]+\mathrm{E}\left[y_{ik}^{low}\left(t+\tau \right)y_{jk}^{high}\left(t\right)\right]+\mathrm{E}\left[y_{ik}^{high}\left(t+\tau \right)y_{jk}^{low}\left(t\right)\right]+\mathrm{E}\left[y_{ik}^{high}\left(t+\tau \right)y_{jk}^{high}\left(t\right)\right]\hfill \\ & ={\overline{R}}_{ijk}^{low}+R_{ijk}^{high}\left(\tau \right)\hfill \end{array}$$

(9)

with

$$\begin{array}{cc}{\overline{R}}_{ijk}^{low}\left(\tau \right)& =\mathrm{E}\left[y_{ik}^{low}\left(t+\tau \right)y_{jk}^{low}\left(t\right)\right]\hfill \\ & ={\displaystyle \sum _{r=1}^{N_{tr}}{\displaystyle \sum _{s=1}^{N_{tr}}{\varphi }_{ir}{\varphi }_{js}{\alpha }_{kr}{\alpha }_{ks}{\displaystyle {\int }_{-\infty }^t{\displaystyle {\int }_{-\infty }^{t+\tau }{e}^{{\lambda }_r\left(t+\tau -p\right)}{e}^{{\lambda }_s\left(t-q\right)}\mathrm{E}\left[f_k\left(p\right)f_k\left(q\right)\right]dpdq}}}}\hfill \end{array}$$

(10)

$$R_{ijk}^{high}\left(\tau \right)=\mathrm{E}\left[y_{ik}^{low}\left(t+\tau \right)y_{jk}^{high}\left(t\right)\right]+\mathrm{E}\left[y_{ik}^{high}\left(t+\tau \right)y_{jk}^{low}\left(t\right)\right]+\mathrm{E}\left[y_{ik}^{high}\left(t+\tau \right)y_{jk}^{high}\left(t\right)\right]$$

(11)

where the subscript “low” and “high” represent the low-frequency region and high-frequency region, respectively; $y_{ik}^{low}$ and $y_{ik}^{high}$ are the response with low frequencies and high frequencies, respectively.

For high wind excitation, $f\left(t\right)$ is not ideal for white noise. Then, the expectation of the product of two excitations at different time points is not zero. Therefore, Equation (4) cannot be satisfied and has the following form:

$$\mathrm{E}\left[f_k\left(p\right)f_k\left(q\right)\right]={\alpha }_k^{{\tau }_{tr}}$$

(12)

with

$${\tau }_{tr}=q-p$$

(13)

where ${\tau }_{tr}$ is the time difference; ${\alpha }_k^{{\tau }_{tr}}$ is a constant related to excitation point and time difference ${\tau }_{tr}$.

To make the derivation simple, only the integration of Equation (10) is considered, whose process is similar to Equation (6). Considering Equation (6), the integration part of ${\overline{R}}_{ijk}^{low}\left(\tau \right)$ in Equation (10) has three different expressions along with the different.

Case 1: For ${\tau }_{tr}\le \tau$, the integration part of ${\overline{R}}_{ijk}^{low}\left(\tau \right)$ has the following expression:

$$\begin{array}{cc}{\displaystyle {\int }_{-\infty }^t{\displaystyle {\int }_{-\infty }^{t+\tau }{e}^{{\lambda }_r\left(t+\tau -p\right)}{e}^{{\lambda }_s\left(t-q\right)}\mathrm{E}\left[f_k\left(p\right)f_k\left(q\right)\right]dpdq}}& ={\displaystyle \sum _{{\tau }_{tr}=0}^{\tau }\left[{\alpha }_k^{{\tau }_{tr}}{\displaystyle {\int }_{-\infty }^t{e}^{{\lambda }_r\left(t+\tau -p\right)}{e}^{{\lambda }_s\left(t-p-{\tau }_{tr}\right)}dp}\right]}\hfill \\ & =-{\displaystyle \sum _{{\tau }_{tr}=0}^{\tau }\left(\frac{{\alpha }_k^{{\tau }_{tr}}{e}^{-{\lambda }_s{\tau }_{tr}}}{{\lambda }_r+{\lambda }_s}{e}^{{\lambda }_r\tau }\right)}\hfill \end{array}$$

(14)

Case 2: For $\tau <{\tau }_{tr}\le t$, the integration part of ${\overline{R}}_{ijk}^{low}\left(\tau \right)$ can be derived as follows:

$$\begin{array}{cc}{\displaystyle {\int }_{-\infty }^t{\displaystyle {\int }_{-\infty }^{t+\tau }{e}^{{\lambda }_r\left(t+\tau -p\right)}{e}^{{\lambda }_s\left(t-q\right)}\mathrm{E}\left[f_k\left(p\right)f_k\left(q\right)\right]dpdq}}& ={\displaystyle \sum _{{\tau }_{tr}=\tau +1}^t\left[{\alpha }_k^{{\tau }_{tr}}{\displaystyle {\int }_{-\infty }^{t-{\tau }_{tr}+\tau }{e}^{{\lambda }_r\left(t+\tau -p\right)}{e}^{{\lambda }_s\left(t-p-{\tau }_{tr}\right)}dp}\right]}\hfill \\ & =-{\displaystyle \sum _{{\tau }_{tr}=\tau +1}^t\left(\frac{{\alpha }_k^{{\tau }_{tr}}{e}^{{\lambda }_r{\tau }_{tr}}}{{\lambda }_r+{\lambda }_s}{e}^{-{\lambda }_s\tau }\right)}\hfill \end{array}$$

(15)

Case 3: For $t<{\tau }_{tr}\le t+\tau$, the integration part of ${\overline{R}}_{ijk}^{low}\left(\tau \right)$ can be expressed as follows:

$$\begin{array}{cc}{\displaystyle {\int }_{-\infty }^t{\displaystyle {\int }_{-\infty }^{t+\tau }{e}^{{\lambda }_r\left(t+\tau -p\right)}{e}^{{\lambda }_s\left(t-q\right)}\mathrm{E}\left[f_k\left(p\right)f_k\left(q\right)\right]dpdq}}& ={\displaystyle \sum _{{\tau }_{tr}=t+1}^{t+\tau }\left[{\alpha }_k^{{\tau }_{tr}}{\displaystyle {\int }_{-\infty }^t{e}^{{\lambda }_r\left(t+\tau -p\right)}{e}^{{\lambda }_s\left(t-p-{\tau }_{tr}\right)}dp}\right]}\\ & =-{\displaystyle \sum _{{\tau }_{tr}=t+1}^{t+\tau }\left(\frac{{\alpha }_k^{{\tau }_{tr}}{e}^{\left({\lambda }_r+{\lambda }_s\right)t-{\lambda }_s{\tau }_{tr}}}{{\lambda }_r+{\lambda }_s}{e}^{-{\lambda }_s\tau }\right)}\hfill \end{array}$$

(16)

Multiplying Equations (14)–(16) by the other parts of ${\overline{R}}_{ijk}^{low}\left(\tau \right)$, ${\overline{R}}_{ijk}^{low}\left(\tau \right)$ can be represented as follows:

$${\overline{R}}_{ijk}^{low}\left(\tau \right)={\overline{R}}_{ijk}^{low,1}\left(\tau \right)+{\overline{R}}_{ijk}^{low,2}\left(\tau \right)+{\overline{R}}_{ijk}^{low,3}\left(\tau \right)$$

(17)

where ${\overline{R}}_{ijk}^{low,1}$, ${\overline{R}}_{ijk}^{low,2}$ and ${\overline{R}}_{ijk}^{low,3}$ are the ${\overline{R}}_{ijk}^{low}$ for the three cases, respectively, and are expressed as follows:

$${\overline{R}}_{ijk}^{low,1}\left(\tau \right)={\displaystyle \sum _{r=0}^{N_{tr}}{\displaystyle \sum _{s=1}^{N_{tr}}\left[{\varphi }_{js}{\varphi }_{ir}{\alpha }_{kr}{\alpha }_{ks}\left(-\frac{1}{{\lambda }_r+{\lambda }_s}\right){\displaystyle \sum _{{\tau }_{tr}=0}^{\tau }{\alpha }_k^{{\tau }_{tr}}{e}^{-{\lambda }_s{\tau }_{tr}}{e}^{{\lambda }_r\tau }}\right]}}={\displaystyle \sum _{r=0}^{N_{tr}}\left({\overline{b}}_{jr}^1{\varphi }_{ir}{e}^{{\lambda }_r\tau }\right)}$$

(18)

$${\overline{R}}_{ijk}^{low,2}\left(\tau \right)={\displaystyle \sum _{r=0}^{N_{tr}}{\displaystyle \sum _{s=1}^{N_{tr}}\left[{\varphi }_{js}{\varphi }_{ir}{\alpha }_{kr}{\alpha }_{ks}\left(-\frac{1}{{\lambda }_r+{\lambda }_s}\right){\displaystyle \sum _{{\tau }_{tr}=\tau +1}^t{\alpha }_k^{{\tau }_{tr}}{e}^{{\lambda }_r{\tau }_{tr}}{e}^{-{\lambda }_s\tau }}\right]}=}{\displaystyle \sum _{s=1}^{N_{tr}}\left({\overline{b}}_{is}^2{\varphi }_{js}{e}^{-{\lambda }_s\tau }\right)}$$

(19)

$${\overline{R}}_{ijk}^{low,3}\left(\tau \right)={\displaystyle \sum _{r=0}^{N_{tr}}{\displaystyle \sum _{s=1}^{N_{tr}}\left[{\varphi }_{js}{\varphi }_{ir}{\alpha }_{kr}{\alpha }_{ks}\left(-\frac{1}{{\lambda }_r+{\lambda }_s}\right){\displaystyle \sum _{{\tau }_{tr}=t+1}^{t+\tau }{\alpha }_k^{{\tau }_{tr}}{e}^{{\lambda }_{r}t+{\lambda }_{s}t-{\lambda }_s{\tau }_{tr}}{e}^{-{\lambda }_s\tau }}\right]}}={\displaystyle \sum _{s=1}^{N_{tr}}\left({\overline{b}}_{is}^3{\varphi }_{js}{e}^{-{\lambda }_s\tau }\right)}$$

(20)

with

$${\overline{b}}_{jr}^1={\displaystyle \sum _{s=1}^{N_{tr}}\left[{\varphi }_{js}{\alpha }_{kr}{\alpha }_{ks}\left(-\frac{1}{{\lambda }_r+{\lambda }_s}\right){\displaystyle \sum _{{\tau }_{tr}=0}^{\tau }{\alpha }_k^{{\tau }_{tr}}{e}^{-{\lambda }_s{\tau }_{tr}}}\right]}$$

(21)

$${\overline{b}}_{is}^2={\displaystyle \sum _{r=0}^{N_{tr}}\left[{\varphi }_{ir}{\alpha }_{kr}{\alpha }_{ks}\left(-\frac{1}{{\lambda }_r+{\lambda }_s}\right){\displaystyle \sum _{{\tau }_{tr}=\tau +1}^t{\alpha }_k^{{\tau }_{tr}}{e}^{{\lambda }_r{\tau }_{tr}}}\right]}$$

(22)

$${\overline{b}}_{is}^3={\displaystyle \sum _{r=0}^{N_{tr}}\left[{\varphi }_{ir}{\alpha }_{kr}{\alpha }_{ks}\left(-\frac{1}{{\lambda }_r+{\lambda }_s}\right)e^{\left({\lambda }_r+{\lambda }_s\right)t}{\displaystyle \sum _{{\tau }_{tr}=t+1}^{t+\tau }{\alpha }_k^{{\tau }_{tr}}{e}^{-{\lambda }_s{\tau }_{tr}}}\right]}$$

(23)

It is clear that Equations (18)–(20) all have the impulse response form similar to Equation (7). However, they are not continuous, which means that the values of ${\overline{R}}_{ijk}^{low,1}$ and ${\overline{R}}_{ijk}^{low,2}$ at the boundary $\tau$ are not equal. This phenomenon also happens on ${\overline{R}}_{ijk}^{low,2}$ and ${\overline{R}}_{ijk}^{low,3}$. Therefore, ${\overline{R}}_{ijk}^{low}$ is not the ideal impulse response in the whole time history. It makes sense because the excitation is non-white. From Equations (18)–(23), it is found that ${\overline{R}}_{ijk}^{low}\left(\tau \right)$ is not only related to the variable $\tau$ as Equation (7), but also related to the variables $t$ and ${\tau }_{tr}$. This is the reason why ${\overline{R}}_{ijk}^{low}$ is not the ideal impulse response.

Comparing Equation (18) and Equation (7), the formulas are similar. However, Equation (21) is very different from Equation (8). The main different part is ${\displaystyle \sum _{{\tau }_{tr}=0}^{\tau }{\alpha }_k^{{\tau }_{tr}}{e}^{-{\lambda }_s{\tau }_{tr}}}$ in Equation (21). If this part can be eliminated, the response form can be changed back to Equation (7).

For the part ${\displaystyle \sum _{{\tau }_{tr}=0}^{\tau }{\alpha }_k^{{\tau }_{tr}}{e}^{-{\lambda }_s{\tau }_{tr}}}$, there are two variables that are not known, i.e., ${\alpha }_k^{{\tau }_{tr}}$ and ${\lambda }_s$. ${\alpha }_k^{{\tau }_{tr}}$ can be obtained by the calculated by correlation function of the wind velocity time history. The wind velocity can be monitored by the installed anemoscope. The velocities for other positions without anemoscopes can be simulated [26]. Then, the correlation functions for every position can be calculated. The values on the correlation function curves are the values of ${\alpha }_k^{{\tau }_{tr}}$.

For ${\lambda }_s$, the value of ${\lambda }_s$ in ${\displaystyle \sum _{{\tau }_{tr}=0}^{\tau }{\alpha }_k^{{\tau }_{tr}}{e}^{-{\lambda }_s{\tau }_{tr}}}$ would be chosen at every frequency points in the low-frequency region. Then, the value of the part ${\displaystyle \sum _{{\tau }_{tr}=0}^{\tau }{\alpha }_k^{{\tau }_{tr}}{e}^{-{\lambda }_s{\tau }_{tr}}}$ is calculated, which Equation (18) is divided by to generate the response form as shown in Equation (7). A similar process can be implemented for Equations (19) and (20). Then, Equations (18)–(20) eliminated the effect of ${\alpha }_k^{{\tau }_{tr}}$ which is caused by the non-white wind loads. After the elimination, Equation (17) is performed to obtain the ideal impulse response form.

When the monitoring data of the structural vibration is collected, the response $y\left(t\right)$ cannot be used directly, and should be separated into the response $y_{}^{low}$ with low frequency and $y_{}^{high}$ high frequency by Fourier transform and inverse Fourier transform. Then the correlation function for the response $y_{}^{high}$ is implemented to obtain the form of impulse response in the high frequency region.

For the response $y_{}^{low}$, $\mathrm{E}\left[y_{ik}^{low}\left(t+\tau \right)y_{jk}^{low}\left(t\right)\right]$ in Equation (9) can be performed, which is not an impulse response due to the non-white wind loads. The correlation function of wind velocity is performed to obtain ${\alpha }_k^{{\tau }_{tr}}$. In the low-frequency region, every frequency point is used, e.g., ${\displaystyle \sum _{{\tau }_{tr}=0}^{\tau }{\alpha }_k^{{\tau }_{tr}}{e}^{-{\lambda }_s{\tau }_{tr}}}$ in Equation (21). Then, the effect of non-white excitation can be eliminated by the division between ${\displaystyle \sum _{{\tau }_{tr}=0}^{\tau }{\alpha }_k^{{\tau }_{tr}}{e}^{-{\lambda }_s{\tau }_{tr}}}$ and Equation (18). Through some similar process, ${\overline{R}}_{ijk}^{low}\left(\tau \right)$ is cleaned to be an ideal impulse response form denoted as $R_{ijk}^{low}\left(\tau \right)$. Then the impulse response form $R_{ijk}^{new}\left(\tau \right)$ in the whole frequency region can be obtained by adding $R_{ijk}^{high}\left(\tau \right)$ as described in the following equation:

$$R_{ijk}^{new}\left(\tau \right)=R_{ijk}^{low}\left(\tau \right)+R_{ijk}^{high}\left(\tau \right)$$

(24)

For other excitation points, the wind field can be simulated. Then the wind excitation time history can be estimated. $R_{ijk}^{new}\left(\tau \right)$ for other excitation points can be obtained. The excitation vector can be obtained. An ERA is then performed to identify the modes of structures.

The procedures to identify the modes of structures considering the non-white noise feature of wind are summarized as follows:

Step 1: According to the measurement data from the anemoscope on the structure, the wind velocity and wind velocity spectra are measured. The power spectrum density function of the response of the accelerometer near the anemoscope is plotted.

Step 2: Comparing the regions of wind velocity spectra and power spectrum density function of the accelerometer, if the regions are not overlapped, then the procedure Step 8 is directly performed. If the regions are overlapped, the truncated order of low-frequency range $N_{tr}$ in Equation (9) is determined through the wind spectra.

Step 3: The measured response of the accelerometer is separated into the response $y_{}^{low}$ with low-frequency region and the response $y_{}^{high}$ with high-frequency region.

Step 4: The correlation functions of $y_{}^{low}$ and $y_{}^{high}$ are calculated. ${\overline{R}}_{ijk}^{low}\left(\tau \right)$ and are then obtained.

Step 5: Calculating the correlation function of the wind velocity record, the values of ${\alpha }_k^{{\tau }_{tr}}$ are determined. Using every frequency point in the low frequency region, $e^{-{\lambda }_s{\tau }_{tr}}$ is obtained. $R_{ijk}^{low}\left(\tau \right)$ is then calculated by removing ${\displaystyle \sum _{{\tau }_{tr}=0}^{\tau }{\alpha }_k^{{\tau }_{tr}}{e}^{-{\lambda }_s{\tau }_{tr}}}$ in Equation (18). ${\overline{R}}_{ijk}^{low,1}\left(\tau \right)$ is divided by ${\displaystyle \sum _{{\tau }_{tr}=0}^{\tau }{\alpha }_k^{{\tau }_{tr}}{e}^{-{\lambda }_s{\tau }_{tr}}}$ to eliminate the effect of non-white wind load. A similar process is performed for Equations (19) and (20), $R_{ijk}^{low}\left(\tau \right)$ is calculated.

Step 6: The impulse response form $R_{ijk}^{new}\left(\tau \right)$ in the whole frequency region is combined as shown in Equation (24).

Step 7: The wind field is simulated, and the wind velocity at each position of the accelerometer is obtained. The correlation function and wind spectra of wind velocity at each position of the accelerometer are calculated. Run Steps 3 to 6. When $R_{ijk}^{new}\left(\tau \right)$ for all the accelerometer positions is obtained, go to Step 8.

Step 8: ERA is performed to identify the modes of the structures using the impulse response form.

4. Modal Identification for a Cable-Stayed Bridge under Typhoon

4.1. Structure and Monitoring System

Compared with other offshore structures, long-span bridges are easier to be excited by wind load, especially for typhoons. Therefore, a cable-stayed bridge is preferred in this paper. The cable-stayed bridge is located in mainland China as shown in Figure 1. It is a two-tower two-cable plane cable-stayed bridge, which the main span of 510.0 m. This bridge was opened to traffic in December 2004.

To monitor the structural performance, a structural health monitoring system was built, which contains an anemoscope, temperature sensors, a global positioning system and accelerometers. There are 38 accelerometers installed on the bridge, which are denoted as Accs. The positions of the accelerometers on the bridge are displayed in Figure 1 and

Table 1. The specific position of the acceleration sensor on the bridge.

Sensor	Position	Sensor	Position
1	The midspan of top desk of box girder in the upstream side (vertical)	14	The lower beam of north tower (longitudinal)
2	The midspan of top desk of box girder in the upstream side (lateral)	15	The lower beam of north tower (vertical)
3	The midspan of top desk of box girder in the downstream side (vertical)	16	The top desk of box girder at the south tower in the upstream side (vertical)
4	The midspan of top desk of box girder in the downstream side (lateral)	17	The top desk of box girder at the south tower in the downstream side (vertical)
5	The upper beam of north tower (longitudinal)	18	The lower beam of south tower (lateral)
6	The upper beam of north tower (lateral)	19	The lower beam of south tower (longitudinal)
7	The upper beam of south tower (longitudinal)	20	The lower beam of south tower (vertical)
8	The upper beam of south tower (lateral)	21	The top desk of box girder in the upstream side (vertical)
9	The top desk of box girder in the upstream side (vertical)	22	The top desk of box girder in the downstream side (vertical)
10	The top desk of box girder in the downstream side (vertical)	23–26	The cable of north tower in the upstream side
11	The top desk of box girder at the north tower in the upstream side (vertical)	27–30	The cable of north tower in the downstream side
12	The top desk of box girder at the north tower in the downstream side (vertical)	31–34	The cable of south tower in the upstream side
13	The lower beam of north tower (lateral)	35–38	The cable of south tower in the downstream side

. Accs 1–4 are installed in the middle of the main span. Accs 1–2 measure the upstream vertical direction accelerations and the upstream lateral direction, respectively. Accs 3–4 measure the downstream vertical direction accelerations and the downstream lateral direction, respectively. The sampling frequency for all the accelerometers is 20 Hz. The anemoscope is installed on the upstream bridge deck in the middle of the main span as located as Accs 1.

4.2. Typhoon Monitoring

On $23$ July 2014, Typhoon Matmo landed on the coast of Changbin Township, Taiwan province. On $24$ July 2014, the cable-stayed bridge is affected by this typhoon to a certain extent. The anemoscope installed on the bridge records the wind history, as shown in Figure 2.

Kaimal spectrum is a classical standard wind power spectrum. To investigate the features of the typhoon, the wind velocity spectrum is plotted and compared with Kaimal spectrum as shown in Figure 3.

4.3. Extraction of Bridge Frequencies

As mentioned above, the typhoon acted on the bridge on $24$ July 2014. To compare the performance of the bridge with a typhoon passing by and without a typhoon, we monitored the data from the preceding three days to three days after. To be specific, the data of seven days are applied, from $21$ to $27$ July 2014. For the accelerometer Acc 1, the time history in $24$ is shown in Figure 4.

The power spectrum density function of the responses of Acc 1 is shown in Figure 5.

From Figure 5, it is found that the first four frequencies are smaller than 1 Hz. Compared with the wind velocity spectrum in Figure 3, the frequencies of the response of Acc 1 identified from the power spectrum density function are in the region of wind excitation frequency region. From the wind velocity spectrum, as shown in Figure 3, wind load is not white noise. Then the expectation of multiplication of two wind loads at different time steps is not a Dirichlet function as described in Equations (4) and (12). Therefore, the wind loads affect the impulse response form generated by the accelerometer response correlation function. Before modal analyses of the bridge, the effect of non-white noise caused by wind loads should be eliminated.

The acceleration response as shown in Figure 4 is separated into the response $y_{}^{low}$ with low-frequency region and the response $y_{}^{high}$ with high-frequency region. Then, considering Equations (10) and (11), ${\overline{R}}_{ijk}^{low}\left(\tau \right)$ and $R_{ijk}^{high}\left(\tau \right)$ are calculated. It should be noted that ${\overline{R}}_{ijk}^{low}\left(\tau \right)$ is not the real impulse response form in low frequency region. The reason is that ${\overline{R}}_{ijk}^{low,1}\left(\tau \right)$ has the multiplier ${\displaystyle \sum _{{\tau }_{tr}=0}^{\tau }{\alpha }_k^{{\tau }_{tr}}{e}^{-{\lambda }_s{\tau }_{tr}}}$ in Equation (18). ${\overline{R}}_{ijk}^{low,1}\left(\tau \right)$ in Equation (19) and ${\overline{R}}_{ijk}^{low,1}\left(\tau \right)$ in Equation (20) have their own multipliers.

To eliminate the effect of ${\displaystyle \sum _{{\tau }_{tr}=0}^{\tau }{\alpha }_k^{{\tau }_{tr}}{e}^{-{\lambda }_s{\tau }_{tr}}}$, the correlation function of wind velocity at Acc 1 is shown in Figure 6.

Then the curve values in Figure 6 are ${\alpha }_k^{{\tau }_{tr}}$. From Figure 3, the truncated order of low-frequency range $N_{tr}$ in Equation (9) is chosen to be 1 Hz. ${\lambda }_r$ is every frequency point. Then ${\displaystyle \sum _{{\tau }_{tr}=0}^{\tau }{\alpha }_k^{{\tau }_{tr}}{e}^{-{\lambda }_s{\tau }_{tr}}}$ can be obtained. The effect of non-white wind loads on ${\overline{R}}_{ijk}^{low,1}\left(\tau \right)$ can be obtained by eliminating ${\displaystyle \sum _{{\tau }_{tr}=0}^{\tau }{\alpha }_k^{{\tau }_{tr}}{e}^{-{\lambda }_s{\tau }_{tr}}}$. The ideal impulse response form $R_{ijk}^{low}\left(\tau \right)$ can be calculated by performing a similar process for Equations (19) and (20).

Combine the responses $R_{ijk}^{low}\left(\tau \right)$ and $R_{ijk}^{high}\left(\tau \right)$, then the real impulse form $R_{ijk}^{new}\left(\tau \right)$ at the excitation point $k$ = 1 without the effect of wind is obtained. Simulating the wind field, then the wind velocity and wind velocity spectrum at each position of the accelerometer are calculated [31]. The simulated wind velocity and spectrum at the position of Acc 21 are shown in Figure 7.

Using the same procedures mentioned above, $R_{ijk}^{new}\left(\tau \right)$ for each position point $k$ is obtained. Then the modal identification of the bridge is performed. The identified frequencies along time with the unit hour are shown in Figure 8.

In Figure 8, the x-axis is the time with unit hour, and the y-axis is the frequency. To identify mode one, the time period for acceleration data is one hour. Therefore, one day contains 4 frequency points for one order frequency. Moreover, there are 168 points total for one order frequency. It is obvious that the horizontal frequency line is almost straight. At the region from 72 to 96 in the x-axis, there are some vibration points, especially for 2nd frequency. The frequencies of the second and third modes are close to each other, which affects the accuracy of modal identification. Nevertheless, the frequencies are low, and the deviation is acceptable. After yphoon Matmo passes by, the frequencies go to be stable.

5. Conclusions

To identify the modes of offshore structures during typhoon passing by, this paper proposes an innovative method to generate a real impulse response by eliminating the effect of typhoons. There are some conclusions are summarized as follows:

(1) The impulse response excited by white noise is derived. The drawback part which would be effective by non-white noise is pointed out. It is that the correlation function of excitation is not the Dirichlet function.

(2) To eliminate the effect of non-white noise on the impulse response, this paper analyzes the wind velocity spectra and their correlation function. Then the correlation function of excitation is calculated and accumulated, which is the effect of non-white wind loads and needed to be eliminated. The procedures are then summarized.

(3) A practical stayed-cable bridge is employed. The features of Typhoon Matmo are analyzed. Following the proposed procedures, the bridge modes are identified. It is obvious that the bridge instinct dynamics changed during a typhoon passing by.

(4) The method to separate the vibration response $y\left(t\right)$ into $y_{}^{low}$ and $y_{}^{high}$ by the method of Fourier transform and inverse Fourier transform can be influenced by the frequency mixing problem. This topic should be investigated more in the future.