Bayesian-guided operational modal identification of a highway bridge considering non-uniform sampling

Abstract

During the structural health monitoring of bridges, it has been observed that the vibration data collected can sometimes be randomly lost or sampled non-uniformly. This leads to a low signal-to-noise ratio in the spectral functions of the measured data, making it difficult to identify weak modes. To address this issue, a framework for operational modal identification is proposed in this study. It utilizes the fast Bayesian fast Fourier transform (FFT) method to estimate the modal parameters of highway bridges considering the non-uniform monitoring data. The initial frequency parameters for the fast Bayesian FFT approach are automatically determined using the proposed autoregressive (AR) power spectral density (PSD)-guided peak picking method. This overcomes the challenge of capturing initial frequencies related to weakly contributed modes. Additionally, the bandwidth parameter for each mode is determined using the modal assurance criterion (MAC) of the first left singular vectors of PSD matrices. Furthermore, when analyzing non-uniform vibration data, it is recommended to use the non-uniform FFT (NUFFT) for calculating PSD functions in order to improve identification accuracy. The proposed method is validated using acceleration data from both a numerical model and a real-world bridge. The results demonstrate that the identification uncertainty of modal parameters increases with higher non-uniform levels.

Keywords

modal identification highway bridge health monitoring non-uniform sampling frequency

Introduction

The application of structural health monitoring techniques, which aim to implement non-destructive continuous monitoring solutions for preventive maintenance, has gained popularity in the field of bridge engineering (Du et al., 2023; Guo et al., 2021; Locke et al., 2022; Yang et al., 2018, 2022). Among the various monitoring strategies available, vibration-based structural health monitoring has become widely adopted due to its ability to provide global assessments, minimal intrusiveness, and relatively easy automation (Caicedo et al., 2022; Tomassini et al., 2023; Yang et al., 2018). Operational modal identification plays a crucial role in vibration-based structural health monitoring by extracting modal parameters solely from vibration signals, such as accelerations, measured during normal operating conditions of the bridge. Commonly used methods for operational modal identification (He et al., 2021; Yao et al., 2023) include time domain techniques like stochastic subspace identification (He et al., 2022), frequency domain methods like frequency domain decomposition (Brincker et al., 2001), and time-frequency domain methods like short-time Fourier transform. However, these methods typically assume that the vibration signals used for identification are uniformly sampled. In practice, non-uniform sampling of monitoring data, which can be considered as random loss of data, can occur due to various factors in wireless sensor networks, such as radio interference, long distances between sensor nodes and the fusion center, antenna orientation, weather conditions including rain and lightning, and hardware issues (Amini et al., 2021). Studies have shown that even small amounts of randomly missing data can have effects equivalent to significant measurement noise (Thadikemalla and Gandhi, 2018), leading to less reliable results in modal identification.

Bayesian operational modal identification methods have significant advantages in quantifying the reliability of identification results. One of their key strengths is the ability to estimate not only the most probable value (MPV) but also the posterior covariance (COV) matrix of modal parameters. As a result, Bayesian identification frameworks have gained popularity in the field of operational modal analysis for bridges. Katafygiotis and Yuen (2001) proposed a Bayesian spectral density method for structural modal identification. Kuok and Yuen (2016) applied the Bayesian spectral density approach to analyze vibration monitoring data from the Ting Kau Bridge and investigate its modal identifiability. To enhance computational efficiency, Au (2012a) introduced a fast Bayesian operational modal identification approach that utilizes the fast Fourier transform (FFT) of vibration signals. This method, known as the fast Bayesian FFT method, incorporates an efficient iterative procedure to expedite the determination of the MPVs of modal parameters. Additionally, a condensed form of the negative log-likelihood function is derived to reduce the computational effort required for the posterior covariance matrix (Au, 2012b). Zhou et al. (2022) employed the fast Bayesian FFT method to estimate modal parameters of a simply supported steel truss bridge for the purpose of bridge model updating. A Bayesian frequency domain method was also utilized to identify buried modes in vibration signals dominated by other modes, which was validated using ambient vibration data from the Jiangyin Yangtze River Bridge (Zhu et al., 2019). Matsuoka et al. (2020) developed a Bayesian time-varying autoregressive method to track time-frequency characteristics of a railway bridge using train-induced vibration signals. However, it is worth noting that the performance of modal identification using the aforementioned Bayesian operational modal identification approach heavily relies on the manual selection of empirical parameters.

In the fast Bayesian FFT method, the accuracy and completeness of modal identification heavily depend on the determination of initial frequencies and bandwidth parameters. Typically, the initial frequencies are identified by locating peaks in the power spectral density (PSD) functions of vibration signals (Ni and Zhang, 2019), while the choice of the bandwidth parameter involves a trade-off between the amount of data used for inference and the modeling error (Zeng et al., 2023). Alternatively, instead of using the PSD function, the singular value spectra of the cross-PSD matrices of vibration signals can be utilized to estimate initial frequencies. This approach is particularly useful for identifying closely-spaced modes. To automate the selection of peaks in PSD functions or singular value spectra, Yao et al. (2021) proposed a time-related modal assurance criterion (MAC) combined with a scale-space peak-picking approach. Kim and Sam (2019) developed a convolutional neural network-based peak detector to distinguish peaks from spectral data containing noise peaks. They trained possible peak regions using a deep learning framework. To automatically determine the initial frequencies and bandwidth parameter, Mao et al. (2023) proposed a cross-MAC-based frequency response interval determination method. This method denoises the cross-MAC matrix using a machine-learning algorithm. However, most existing methods primarily focus on automatically distinguishing structural frequency peaks from noise peaks, while neglecting the automatic detection of low-energy modes buried by noise. This is particularly challenging as the spectra of non-uniformly sampled vibration signals have low signal-to-noise ratios, making it difficult to detect the peaks and corresponding bandwidths associated with low-energy modes. Yang and Nagarajaiah (2015) focused on output-only modal identification considering non-uniform data, and took a compressive sensing technique to recover uniform data. Compressive sensing techniques have been employed to reconstruct vibration signals from randomly sampled data in many studies (An et al., 2024; Thadikemalla and Gandhi, 2018). However, these studies focused on signal reconstruction rather than modal identification.

In this study, the fast Bayesian FFT method is employed for operational modal identification of a highway bridge using non-uniformly sampled signals, where the empirical parameters are optimally selected. To overcome the challenge of capturing initial frequencies related to low-energy modes, an autoregressive (AR) PSD-guided peak picking method is employed to automatically determine the initial frequency parameters used in the fast Bayesian FFT approach. Furthermore, the bandwidth parameter for each mode is selected through a MAC-based method. Finally, the operational modal identification framework constructed in this study is verified through vibration signals of a numerical example and a real-world bridge.

Background of modal identification

Fast bayesian FFT method

Based on the modal superposition principle, the vibration signals $y (t) = {[\begin{array}{l} y_{1} (t) & y_{2} (t) & . . . & y_{M} (t) \end{array}]}^{T} \in R^{M \times 1}$ of a multiple degree-of-freedom (DOF) linear time-invariant system, subjected to Gaussian white-noise excitation, can be mathematically expressed as:

y (t) = \sum_{i = 1}^{n} φ_{i} q_{i} (t) + ε (t)

(1)

where

M

denotes the number of measurement channels;

n

denotes the number of modes;

φ_{i} \in R^{M \times 1}

denotes the

i^{th}

mode shape vector;

q_{i} (t)

is the

i^{th}

modal response;

ε (t) \in R^{M \times 1}

represents the prediction error vector including measurement noise and modeling errors; and the superscript T denotes the transpose operator.

Transform the vibration signals in equation (1) into the frequency spectra as:

Y (f) = \sum_{i = 1}^{n} φ_{i} Q_{i} (f) + Ξ (f)

(2)

where

Y (f)

Q_{i} (f)

, and

Ξ (f)

are the frequency spectra of vibration signals

y (t)

, modal response

q_{i} (t)

, and prediction error

ε (t)

, respectively.

The prediction errors associated with different measurement channels are assumed to have independent Gaussian white-noise characteristics. Correspondingly, the PSD matrix $S_{ε ε} (f)$ is a constant matrix as:

S_{ε ε} (f) = E [Ξ (f) Ξ^{T} (f)] = S_{ε} I_{M}

(3)

where

E [\cdot]

denotes the expectation operator;

I_{M} \in R^{M \times M}

is an identity matrix.

If the vibration signals $y (t)$ are accelerations, the corresponding frequency spectrum $Q_{i} (f)$ of the $i^{th}$ modal response can be expressed as:

Q_{i} (f) = H_{i} (f) P_{i} (f) = \frac{1}{1 - {(f_{i} / f)}^{2} - 2 j ξ_{i} (f_{i} / f)} P_{i} (f)

(4)

where

f_{i}

and

ξ_{i}

are the

i^{th}

natural frequency and damping ratio;

j = \sqrt{- 1}

is the imaginary unit;

P_{i} (f)

is the frequency spectrum of the

i^{th}

modal force; and the auto PSD function of modal force derived from white-noise excitation is a constant value as

S_{p_{i} p_{i}} (f) = E [P_{i} (f) {P_{i}}^{T} (f)] = S_{p_{i} p_{i}}

Assuming that modes contributed in vibration signals are well separated, that is, only one mode dominates in a frequency band $[\begin{array}{l} f_{i} - δ f_{i, L}, & f_{i} + δ f_{i, R} \end{array}]$ centered around the $i^{th}$ natural frequency $f_{i}$ of the structure, the corresponding frequency spectra and PSD matrices in the frequency band $[\begin{array}{l} f_{i} - δ f_{i, L}, & f_{i} + δ f_{i, R} \end{array}]$ respectively become as:

Y (f) \approx φ_{i} Q_{i} (f) + Ξ (f) for f \in [\begin{array}{l} f_{i} - δ f_{i, L}, & f_{i} + δ f_{i, R} \end{array}]

(5)

S_{yy} (f) \approx D_{i} (f) S_{p_{i} p_{i}} φ_{i} {φ_{i}}^{T} + S_{ε} I_{m} for f \in [\begin{array}{l} f_{i} - δ f_{i, L}, & f_{i} + δ f_{i, R} \end{array}]

(6)

where

δ f_{i, L}

and

δ f_{i, R}

are the bandwidth parameters for calculating the

i^{th}

mode; and the dynamic amplitude factor

D_{i} (f)

with

f \in [\begin{array}{l} f_{i} - δ f_{i, L}, & f_{i} + δ f_{i, R} \end{array}]

is expressed as:

D_{i} (f) = \frac{1}{{(1 - {(f_{i} / f)}^{2})}^{2} + {(2 ξ_{i} (f_{i} / f))}^{2}}

(7)

According to equations (6) and (7), the PSD matrix

S_{yy} (f)

is calculated from the vibration signals, while the parameters (the natural frequency

f_{i}

, damping ratio

ξ_{i}

, mode shapes

φ_{i}

, the spectrum parameter

S_{p_{i} p_{i}}

of modal force, and the spectrum parameter

S_{ε}

of prediction errors) should be identified. In fast Bayesian FFT approach, the above parameters are identified according to the following log-likelihood function:

L (f_{i}, ξ_{i}, S_{p_{i} p_{i}}, S_{ε}, φ_{i}) = - M N_{f} \ln 2 + (M - 1) N_{f} \ln S_{ε} + \sum_{f = f_{i} - δ f_{i, L}}^{f_{i} + δ f_{i, R}} \ln (D_{i} (f) S_{p_{i} p_{i}} + S_{ε}) + {S_{ε}}^{- 1} (d - {φ_{i}}^{T} A φ_{i})

(8)

where

N_{f}

is the number of spectral lines in the frequency band

f \in [\begin{array}{l} f_{i, 0} - δ f_{i, L}, & f_{i, 0} + δ f_{i, R} \end{array}]

; the variable

d

and matrix

A

represent as follows:

d = \sum_{f = f_{i, 0} - δ f_{i, L}}^{f_{i, 0} + δ f_{i, R}} (Re {(Y (f))}^{T} Re (Y (f)) + Im {(Y (f))}^{T} Im (Y (f)))

(9)

A = {\sum_{f = f_{i, 0} - Δ f_{i, L}}^{f_{i, 0} + Δ f_{i, R}} (1 + (S_{ε} / (D_{i} (f) S_{p_{i} p_{i}})))}^{- 1} (Re (Y (f)) Re {(Y (f))}^{T} + Im (Y (f)) Im {(Y (f))}^{T})

(10)

If the PSD matrices

S_{yy} (f)

in the frequency band

f \in [\begin{array}{l} f_{i} - δ f_{i, L}, & f_{i} + δ f_{i, R} \end{array}]

have high signal-to-noise ratios, the MPVs of parameters

{f_{i}, ξ_{i}, φ_{i}, S_{p_{i} p_{i}}, S_{ε}}

can be further estimated simply through the following steps.

Step (i): Preset the $i^{th}$ initial frequency $f_{i}$ as $f_{i, 0}$ , the $i^{th}$ initial damping ratio $ξ_{i}$ as $ξ_{i, 0}$ , and the $i^{th}$ bandwidth parameter as $δ f_{i, L}$ and $δ f_{i, R}$ .

Step (ii): Calculate the frequency spectra $Y (f)$ and the PSD matrix $S_{yy} (f)$ of measured vibration signals. Besides, calculate the sum of PSD matrices within the bandwidth $f \in [\begin{array}{l} f_{i, 0} - δ f_{i, L}, & f_{i, 0} + δ f_{i, R} \end{array}]$ as:

A_{0} = \sum_{f = f_{i, 0} - Δ f_{i, L}}^{f_{i, 0} + Δ f_{i, R}} S_{yy} (f)

(11)

Step (iii): Estimate the $i$ ^th initial mode shape vector $φ_{i, 0}$ as the eigenvector $\hat{φ}$ of $A_{0}$ corresponding to the largest eigenvalue $λ_{\max}$ , i.e., $λ_{\max} = {\hat{φ}}^{T} A_{0} \hat{φ}$ .

Step (iv): Estimate the MPVs of spectral parameters $S_{p_{i} p_{i}}$ and $S_{ε}$ as:

S_{p_{i} p_{i}} = \frac{1}{N_{f}} \sum_{f = f_{i, 0} - Δ f_{i, L}}^{f_{i, 0} + Δ f_{i, R}} \frac{{φ_{i, 0}}^{T} S_{yy} (f) φ_{i, 0}}{D_{i} (f)}

(12)

S_{ε} = \frac{d - λ_{\max}}{(M - 1) N_{f}}

(13)

Step (v): Estimate the MPVs of the $i$ ^th frequency and damping ratio through minimizing the following log-likelihood function:

{f_{i}, ξ_{i}} = \underset{f_{i}, ξ_{i}}{argmin} (\sum_{f = f_{i, 0} - δ f_{i, L}}^{f_{i, 0} + δ f_{i, R}} \ln (D_{i} (f)) + N_{f} \ln (\frac{1}{N_{f}} \sum_{f = f_{i, 0} - δ f_{i, L}}^{f_{i, 0} + δ f_{i, R}} \frac{{φ_{i, 0}}^{T} S_{yy} (f) φ_{i, 0}}{D_{i} (f)}))

(14)

Step (vi): Modify the MPV of the $i$ ^th mode shape vector $φ_{i}$ according to the eigenvector of $A$ with respect to the largest eigenvalue.

Step (vii): Calculate the posterior covariances of parameters ${f_{i}, ξ_{i}, φ_{i}, S_{p_{i} p_{i}}, S_{ε}}$ through the inverse of the Hessian matrix of the log-likelihood function, the details regarding the construction of the Hessian matrix can be found in Au (2011).

The aforementioned approach is well-suited for estimating modes that are clearly separated. In cases where closely-spaced mode pairs are present in the vibration response, the calculation method outlined in Li and Au (2019) or Zhu et al. (2021) can be employed.

Influence of non-uniformly sampling

For a uniform discrete-time vibration signal $y_{m} (k)$ , its discrete-time Fourier transform can be calculated as:

Y_{m} (l) = \sum_{k = 0}^{K - 1} y_{m} (k) e^{- j 2 π \frac{k}{K} l}

(15)

where

Y_{m} (l)

is the frequency spectrum of signal

y_{m}

at the

l^{th}

spectral line; the subscript

m

denotes the

m^{th}

measurement channel;

k

is the discrete time; and

K

is the total number of discrete-time samples.

For a non-uniform discrete-time vibration signal $y_{m} (k)$ , its non-uniform discrete Fourier transform is:

Y_{m} (l) = \sum_{k = 0}^{K - 1} y_{m} (k) e^{- j 2 π t (k) l}

(16)

where

t (k)

is the actual time instant that the

k^{th}

sample has been sampled. Besides, a non-uniform fast Fourier transform (NUFFT) (Wen and Houlihan, 2023) has been constructed for calculating frequency spectrum from non-uniformly sampled signals, based on the non-uniform discrete Fourier transform model in equation (16).

To visually illustrate the impact of non-uniform sampling on spectral analysis, a two DOF signal $y (t)$ in equation (17) is simulated:

y (t) = \sum_{i = 1}^{2} a_{i} e^{- 2 π ξ_{i} f_{i} t} \cos (2 π f_{i} t)

(17)

where the parameters are

a_{1} = 1

a_{2} = 0.05

ξ_{1} = ξ_{2} = 0.01

f_{1} = 1

Hz, and

f_{2} = 2

Hz, respectively.

In the case of uniform sampling, the sampling frequency is set as 100 Hz, and the total sampling time is 10 s. Some samples are discarded to simulate the non-uniform sampling cases (Figure 1(a)), where the time positions of the discarded samples are randomly generated. The non-uniform level $r_{non}$ is defined as the ratio of the number of discarded samples to the total number of samples in the uniform sampling cases, as:

r_{non} = \frac{K_{u} - K_{non}}{K_{u}} \times 100 %

(18)

where

K_{u}

and

K_{non}

denote the total number of original uniform samples and the total number of non-uniform samples after discarding process, respectively;

K_{u} - K_{non}

represents the total number of discarding samples.

Figure 1.

Schematic diagram of non-uniformly sampling case: (a) Time domain; (b) Frequency domain.

The spectra of simulated signals under different sampling cases are compared, in Figure 1(b). Two clear peaks appear on the frequency spectrum of the uniform sampling signal. As the non-uniform level increases, the corresponding frequency spectrum becomes increasingly unsmooth. When the non-uniform level reaches 10%, the peak associated with the second mode (a low-energy mode with a frequency of 2 Hz) is no longer easily visible in the frequency spectrum. Some peaks may go undetected, leading to difficulties in accurately determining the initial frequencies of certain modes.

Identification framework using non-uniform data

Autoregressive spectrum-based initial frequency determination

In the fast Bayesian FFT method, the initial frequency of each mode should be pre-selected, otherwise some modes would be lost during the identification process.

Classically, the first singular value (SV) spectrum has been widely studied for determining the initial frequencies. The process involves the following steps. First, the PSD matrices $S_{yy} (f)$ that spectral line $f$ ranges from 0 to $f_{s} / 2$ are calculated from vibration signals $y (t)$ through FFT. Second, the first SV $Σ_{1} (f)$ corresponding to each spectral line $f$ is extracted through performing the singular value decomposition on the PSD matrix $S_{yy} (f)$ . Third, taking the spectral line $f$ as X-axis and the first singular value $Σ_{1} (f)$ as Y-axis, the diagram of the first SV spectrum is plotted. Fourth, the initial frequencies are determined through locating the significant peaks in the diagram of the first SV spectrum. However, when non-uniform vibration signals are used, the first singular value spectrum may contain numerous noise peaks, making it challenging to detect the peaks associated with the structural modes.

To mitigate the impact of noise peaks on initial frequency determination, a smoother spectral estimation method based on the AR model is employed in this study. A signal $y_{m} (t)$ affected by white noises can be modeled through the autoregressive model, as:

y_{m} (t) = - \sum_{γ = 1}^{ϒ} α_{m, γ} y_{m} (t - γ Δ t) + w_{m} (t)

(19)

where

α_{m, γ}

is the AR coefficient;

w_{m} (t)

represents the noise; and

Δ t

denotes the sampling time step;

ϒ

denotes the order of AR model, which can be determined by minimizing the prediction error criterion (Yao et al., 2022).

Correspondingly, the AR auto-PSD function of the signal $y_{m} (t)$ can be calculated as:

S_{y_{m}} (ω) = \frac{{σ_{w_{m}}}^{2}}{{| 1 + \sum_{γ = 1}^{ϒ} α_{m, γ} e^{- j ω γ Δ t} |}^{2}}

(20)

where

{σ_{w_{m}}}^{2}

represents the variances of noise

w_{m} (t)

;

| \cdot |

means the modulus.

Considering that the contributions of each mode may differ in signals obtained from various measurement channels, a simultaneous PSD index $S S$ is calculated as the sum of normalized PSD functions:

S S = \sum_{m = 1}^{M} \frac{S_{y_{m}} (f)}{S_{y_{m}, \max}}

(21)

where

S_{y_{m}, \max}

denotes the maximum value of the PSD function

S_{y_{m}} (f)

As the autoregressive PSD functions effectively smooth out the noise peaks, the peaks automatically selected from the simultaneous PSD curve $S S$ are only associated with the structural modes. Consequently, the frequencies corresponding to these selected peaks are considered as the initial frequencies to be used in the fast Bayesian FFT method.

Bandwidth parameter determination

Once an initial frequency $f_{i, 0}$ for a mode is selected, an effective frequency band $f \in [\begin{array}{l} f_{i, 0} - δ f_{i, L}, & f_{i, 0} + δ f_{i, R} \end{array}]$ centered around the initial frequency should be found, where the left and right bandwidth parameters $δ f_{i, L}$ and $δ f_{i, R}$ should be determined. Choosing excessive bandwidth parameters can render the single-mode assumption in equation (5) unreasonable, while selecting overly small bandwidth parameters may result in lower accuracy in mode estimation. Therefore, striking a balance and determining appropriate bandwidth parameters is crucial for achieving reliable modal identification.

In this study, the MAC in equation (22) is utilized to guide the selection of bandwidth parameters, which quantifies the correlation between vectors $υ_{i}$ and $υ_{j}$ . Generally, the number of elements in each vector should be greater than two. $MAC$ ranges from 0 to 1. The greater the MAC, the higher the correlation between two vectors.

MAC (υ_{i}, υ_{j}) = \frac{{υ_{i}}^{T} υ_{j}}{\sqrt{{υ_{i}}^{T} υ_{i}} \sqrt{{υ_{j}}^{T} υ_{j}}}

(22)

Because only one mode contributed in the frequency band

f \in [\begin{array}{l} f_{i, 0} - δ f_{i, L}, & f_{i, 0} + δ f_{i, R} \end{array}]

, the first left singular vector of the PSD matrix

S_{yy} (f)

when

f \in [\begin{array}{l} f_{i, 0} - δ f_{i, L}, & f_{i, 0} + δ f_{i, R} \end{array}]

is proportional to the

i^{th}

mode shapes. Therefore, the MAC between the first left singular vector of the PSD matrix

S_{yy} (f)

and the

i^{th}

mode shape vector can be used to find the effective bandwidth parameter. In particular, if the initial frequency of a mode has been effectively determined, the first left singular vector of the PSD matrix

S_{yy} (f_{i, 0})

can be considered as the potential mode shape vector

{φ_{p,}}_{i}

, and then the MAC between the first left singular vectors with respect to

f \in [\begin{array}{l} f_{i, 0} - δ f_{i, L}, & f_{i, 0} + δ f_{i, R} \end{array}]

and the first left singular vector of the PSD matrix

S_{yy} (f_{i, 0})

is calculated for bandwidth determination. The determination steps are as follows.

Step (i): Select the initial frequency $f_{i, 0}$ through picking peaks in the sum of normalized AR PSD functions in equation (21).

Step (ii): Calculate the potential mode shape vector ${φ_{p,}}_{i}$ through the first left singular vector of the PSD matrix $S_{yy} (f_{i, 0})$ .

Step (iii): Calculate the MACs between the first left singular vectors of PSD matrices $S_{yy} (f)$ with $f \in (0, f_{s} / 2)$ .

Step (iv): The frequency band around each initial frequency in which the corresponding MACs outnumber the threshold $ε_{MAC}$ is determined as the effective frequency band. Correspondingly, the bandwidth parameters can be determined.

Numerical verification

A three DOF system is simulated to verify the method, where the mass matrix and the stiffness matrix are listed in the following, and the damping matrix is $damp = 0.01 mass + 0.004 stiff$ . The impulsive loads are applied on the system to generate accelerations. The regular sampling frequency for uniform sampling case is set as 100 Hz. The system is sampled for a duration of 1000 s. The non-uniform sampling cases are simulated through discarding samples at random times, and the non-uniform level is defined in equation (18).

mass = 10^{6} \times [\begin{array}{l} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] kg, stiff = 10^{8} \times [\begin{array}{c} 1.941070 & - 1.170535 & 0 \\ - 1.170535 & 2.341070 & - 1.170535 \\ 0 & - 1.170535 & 1.170535 \end{array}] N / m

Initial parameter determination

The process for determining the initial frequencies and bandwidth parameters is demonstrated using a set of uniform sampling accelerations. To determine the initial frequencies, the sum of normalized PSD functions in equation (21) is calculated and presented in Figure 2(a), where the AR auto-PSD function for each acceleration is calculated. The AR model order is set as 80 through balancing the prediction error criterion. Obviously, three peaks are observed on the curve of the sum of normalized PSD functions. Using peak picking method, initial frequencies of three modes are determined, as depicted in Figure 2(a). For each mode, an effective frequency band around its corresponding initial frequency is determined according to the MAC requirements, where the threshold for MAC is set as $ε_{MAC} = 0.95$ . In Figure 2(b), the frequency band for each mode is highlighted with bright colors. The lower limit $δ f_{i, L}$ and upper limit $δ f_{i, R}$ for some modes are labelled.

Figure 2.

Determination of initial parameters: (a) Initial frequencies; (b) Bandwidth.

The impact of the AR model order on spectra is explained through the sum of normalized PSD functions calculated from the simulated accelerations without noise, as depicted in Figure 3. When the AR order is too small, such as the subplot (order = 2, without noise) in Figure 3, the curve of the sum of normalized PSD functions is underfitted, with only two peaks appearing in the curve, and one of the peaks is unrelated to the structural frequency. As the AR order increases, the curve becomes better fitted, and three peaks corresponding to structural frequencies emerge. However, if the AR order becomes excessively large, the curve becomes overfitted, resulting in numerous spurious peaks unrelated to structural frequencies, as observed in the subplots (orders 120 or 300, without noise) in Figure 3.

Figure 3.

The sum of normalized PSD functions in different AR model orders.

Furthermore, the noise signals with a mean value of zero and a standard deviation of $σ_{n o i s e} = 5 % σ_{a c c e l e r a t i o n}$ are added into the simulated accelerations, where $σ_{n o i s e}$ and $σ_{a c c e l e r a t i o n}$ represent the standard deviation of noise and the standard deviation of simulated acceleration without noise, respectively. The sum of normalized PSD functions for accelerations with noise is also depicted in Figure 3. When the AR model order exceeds 50 but is below 100, only three peaks corresponding to structural frequencies are observed in the curve. In comparison to the subplots corresponding to accelerations without noise, the effective range for the AR model order becomes narrower.

Comparison between FFT and NUFFT results

Set the non-uniform level as 2%, the non-uniform sampling case is simulated. Using the fast Bayesian FFT technique, both uniform and non-uniform accelerations are analyzed. The analytical and identified modal parameters are presented in Tables 1 and 2. The analytical frequencies and damping ratios are directly calculated from the mass, stiffness, and damping matrices of the system. When the accelerations are used for modal identification, modeling errors are considered in the fast Bayesian FFT process, where the MPVs and COVs of the identified frequencies and damping ratios are simultaneously given. When the uniform accelerations are analyzed, the MPVs of identified frequencies and damping ratios are closest to the analytical values, demonstrating the effectiveness of fast Bayesian FFT method. When the non-uniform accelerations are analyzed, the NUFFT process should be performed, otherwise the identification errors will significantly increase. For instance, non-uniform accelerations may be mistaken for uniform accelerations if the sampling times are unknown. If non-uniform accelerations are erroneously considered, the FFT suitable for uniform accelerations might be mistakenly applied to them. This could result in an increased modeling error in the fast Bayesian FFT method, thereby leading to a larger gap between the analytical solutions and the MPVs of the identified frequencies and damping ratios, as shown in Tables 1 and 2 When the NUFFT process is utilized for the non-uniform accelerations, the MPVs are closer to the analytical solutions, that is, the identified errors become smaller. Therefore, when employing the fast Bayesian FFT method, it is crucial to perform the NUFFT on non-uniform accelerations.

Table 1.

Frequencies of the numerical system.

Mode	Analytical (Hz)	Uniform		2% non-uniform (FFT)		2% non-uniform (NUFFT)
Mode	Analytical (Hz)	MPV (Hz)	COV (%)	MPV (Hz)	COV (%)	MPV (Hz)	COV (%)
1	0.675	0.675	0.094	0.689	0.099	0.674	0.095
2	2.011	2.007	0.136	2.046	0.203	2.007	0.135
3	3.052	3.044	0.160	3.099	0.252	3.044	0.162

Table 2.

Damping ratios of the numerical system.

Mode	Analytical (%)	Uniform		2% non-uniform (FFT)		2% non-uniform (NUFFT)
Mode	Analytical (%)	MPV (%)	COV (%)	MPV (%)	COV (%)	MPV (%)	COV (%)
1	0.966	0.983	10.353	1.114	10.944	0.987	10.448
2	2.566	2.572	9.521	2.835	11.625	2.575	9.367
3	3.861	3.871	8.329	4.366	9.356	3.889	8.504

To further illustrate the impact of performing the FFT process on non-uniform accelerations, different non-uniform levels are simulated, while the first SV spectrum of accelerations are compared, as shown in Figure 4. The first SV spectrum is calculated from the cross-PSD matrices, which are calculated through performing the FFT on the accelerations. In general, the peaks in the first SV spectrum can be used to locate frequencies of the system. When the non-uniform accelerations are mistakenly treated as uniform data and the classical FFT is applied, the resulting SV spectra are shown in Figure 4(a). It is evident that as the non-uniform level increases, the peaks in the first SV spectrum shift, rendering the initial frequencies determined based on these peaks unreliable. When the NUFFT is performed on the non-uniform accelerations, the first SV spectra are presented in Figure 4(b). As the non-uniform level increases, the smoothness of the first SV spectrum decreases, but the peaks with respect to the structural frequencies remain unchanged. In this case, the initial frequencies determined using peak picking methods are still effective. Comparing Figure 4(a) with Figure 4(b), it becomes necessary to use NUFFT instead of direct FFT when dealing with non-uniform signals.

Figure 4.

Comparison of SV spectra: (a) FFT-based spectrum; (b) NUFFT-based spectrum.

Comparison between AR and welch spectra

Accelerations with increasingly non-uniform levels are simulated to illustrate the influence of non-uniform data on the initial frequency determination. The spectral characteristics of accelerations with different non-uniform levels are presented in Figure 5, respectively. In Figure 5(a), (c), (e) and (g), the first SV spectra calculated from uniform accelerations are presented as solid lines, while those calculated from non-uniform accelerations are presented as dashed lines. As the non-uniform level increases, the smoothness of the first SV spectrum decreases. In particular, the peaks with respect to the second and the third structural frequencies are buried by noise peaks when the non-uniform level reaches 50%, as Figure 5(g) shows. Perform the peak picking technique on the first SV spectra in Figure 5(g) to find initial frequencies. The peaks (dots in Figure 5(g)) selected from the first SV spectrum with respect to non-uniform accelerations are different from the peaks (square in Figure 5(g)) selected from the first SV spectrum with respect to uniform accelerations, illustrating that the accurate determination of initial frequencies becomes challenging as the non-uniform level increases.

Figure 5.

Spectral characteristics under different non-uniform levels: (a) SV spectra ( $r_{non} = 0 %$ and 2%); (b) AR and Welch’s spectra ( $r_{non} = 2 %$ ); (c) SV spectra ( $r_{non} = 0 %$ and 10%); (d) AR and Welch’s spectra ( $r_{non} = 10 %$ ); (e) SV spectra ( $r_{non} = 0 %$ and 30%); (f) AR and Welch’s spectra ( $r_{non} = 30 %$ ); (g) SV spectrum ( $r_{non} = 0 %$ and 50%); (h) AR and Welch’s spectra ( $r_{non} = 50 %$ ).

To enhance the reliability of initial frequency determination in cases of higher non-uniformity, the sum of normalized PSD functions is utilized, where the AR-based spectrum estimation is employed for calculating the PSD functions. To illustrate the advantage of AR-based PSD functions, the AR-based and Welch’s spectra with consistent frequency lines are compared, as shown in Figure 5(b), (d), (f), and (h). Obviously, the curve of the sum of normalized AR PSD functions is smoother than that of normalized Welch’s PSD functions, regardless of the non-uniform level. Even when the non-uniform level reaches 50%, only three smoother peaks appear on the curve of the sum of normalized AR PSD functions. Performing the peak picking method on the curve in Figure 5(h), three peaks can be found from the curve with respect to AR PSD functions, and the frequencies of these peaks are consistent with the structural frequencies. However, if only three peaks are picked from the curve with respect to Welch’s PSD functions, represented by squares in Figure 5(h), the corresponding frequencies are different from structural frequencies, particularly for the second mode. Hence, when analyzing signals with higher non-uniform levels, the sum of normalized AR PSD functions outperforms in accurately determining the initial frequencies.

Based on the determined initial frequencies and frequency bands, the fast Bayesian FFT method is performed on the accelerations with different non-uniform levels. The identified MPVs of frequencies and damping ratios are compared with the analytical solutions, as shown in Figure 6. Besides, the initial frequencies estimated from two types of PSD functions (i.e., the sum of normalized AR PSD functions and the sum of normalized Welch’s PSD functions in Figure 5 (b), (d), (f), and (h)) are utilized, and the corresponding identified MPVs are compared in Figure 6. The MPVs of frequencies identified from accelerations with non-uniform level below 30% are basically consistent with the analytical solutions, regardless of the initial frequency determination method used. When the non-uniform level becomes 50%, the second frequency cannot be accurately estimated because its initial frequency has not been determined well from the sum of normalized Welch’s PSD functions. As Figure 6(b) shows, the damping ratios cannot be identified well in the case of 50% non-uniform level, regardless of the initial frequency determination method used. It is because that the damping ratio is strongly interfered by noise peaks (caused by non-uniform data) in the PSD functions.

Figure 6.

MPVs of identified parameters: (a) Frequencies; (b) Damping ratios.

Case study of an actual bridge

The monitoring data of Dongbaohe Xin’an Bridge are analyzed to illustrate the effectiveness of the proposed modal identification process. The bridge has a total length of 1040 m, and its girder is a three-span continuous composite box beam with span lengths of 88 m + 156 m + 88 m. To measure the vertical vibrations, three accelerometers have been installed at the mid-span of the girder. The locations of these accelerometers are shown in Figure 7. The expected sampling frequency for each accelerometer is 50 Hz.

Figure 7.

Locations of accelerometers.

Based on the time information of the monitored accelerations, it is observed that there is a non-uniform sampling phenomenon. Taking the monitoring data from 0:00 to 1:00 on September 1, 2022 as an example, the sampling frequency of acceleration samples in each second inferred from the time information is presented in Figure 8(a). It is evident that there are fewer or more than 50 samples per second, indicating a certain degree of non-uniform sampling during this time period. The probability distribution of those sampling frequency samples is presented in Figure 8(b) to illustrate its non-uniform level. In Figure 8(b), the shaded area represents the uniform condition where the sampling frequency is 50 Hz, that is, the expected uniform sampling frequency. By examining the probability of the sampling frequency not being equal to 50 Hz, it can be observed that the non-uniform ratio during the period from 0:00 to 1:00 on September 1, 2022 exceeds 25%.

Figure 8.

Sampling feature of accelerations in accelerometer A1: (a) Sampling frequency variation in 1 h; (b) Probability distribution of sampling frequencies; (c) Non-uniform ratios.

By considering 1 h as the time unit, the non-uniform ratio of the monitored accelerations has been calculated as the probability of the sampling frequency not being equal to 50 Hz. The calculated non-uniform ratios of the monitored accelerations from September 1 to 12, 2022, are presented in Figure 8(c). It can be observed that more than half of the time during this period exhibits non-uniform sampling of accelerations, with a maximum non-uniform ratio reaching up to 45.7%.

The effects of non-uniform accelerations on the modal identification process are discussed in the following section by comparing the identification steps using uniform and non-uniform data. The non-uniform acceleration data used in this section are obtained from 0:00 to 1:00 on September 1, 2022. On the other hand, the uniform acceleration data used in this section are obtained from 12:00 to 13:00 on October 11, 2022.

The initial frequencies used in the fast Bayesian FFT method need to be pre-determined through the sum of normalized PSD functions. In this case, the AR-based and Welch-based PSD functions are used to calculate the PSD functions of uniform accelerations, and the results are shown in Figure 9(a). Since the accelerations are uniformly sampled, both curves of the sum of normalized Welch’s PSD functions and the sum of normalized AR PSD functions appear smooth. On the other hand, the sum of normalized Welch’s PSD functions and the sum of normalized AR PSD functions for non-uniform accelerations are presented in Figure 9(b). It is evident that the curve of the sum of normalized Welch’s PSD functions is less smooth, with numerous noise peaks crowded together. To determine the initial frequencies, six peaks on the sum of normalized AR PSD functions are selected and indicated as dots in Figure 9.

Figure 9.

The sum of normalized PSD functions: (a) Uniform data; (b) Non-uniform data.

After determining the initial frequency, the frequency band for each mode needs to be established. Taking the uniform case as an example, the determination results of the frequency band for the first mode are presented in Figure 10(a), with the MAC threshold set as 0.95. In Figure 10(a), the solid line represents the MAC values between the first left singular vector at the initial frequency and the first left singular vectors at other frequencies around the initial frequency. By identifying the MAC values that exceed the MAC threshold, the frequency band for the first mode can be determined, as illustrated by the shaded area in Figure 10(a). Additionally, Figure 10(b) illustrates the relationship between the MAC threshold and the frequency’s COV, with the MAC threshold ranging from 0.9 to 1. As the MAC threshold approaches 1, the effective frequency band narrows, and the COV of frequency gradually increases. However, when the MAC threshold is below 0.98, the COV of frequency increases slowly and remains a lower level. In this study, setting the MAC threshold at 0.95 (less than 0.98) could ensure that the COV of frequency remains at a lower level.

Figure 10.

Analysis of frequency band determination: (a) The first mode; (b) The change of frequency COVs.

The MPVs for frequencies and damping ratios identified from uniform and non-uniform accelerations are compared in Table 3. It is observed that the MPVs under the uniform and non-uniform cases exhibit slight differences. These differences can be attributed not only to the non-uniform accelerations but also to the fact that the uniform accelerations and non-uniform accelerations are obtained from different time durations.

Table 3.

MPVs for frequencies and damping ratios of the bridge.

Mode		1	2	3	4	5	6
Frequency (Hz)	Uniform	0.947	1.724	2.306	2.856	4.402	5.775
Frequency (Hz)	Non-uniform	0.945	1.723	2.307	2.854	4.431	5.637
Damping ratio (%)	Uniform	1.43	1.51	0.87	1.02	1.45	3.56
Damping ratio (%)	Non-uniform	1.41	1.75	1.54	1.34	0.90	2.97

The COVs for the identified frequencies and damping ratios are compared in Figure 11 to illustrate the effects of non-uniform data on the identification results. The COVs reflect the uncertainty of the identification results, with higher COVs indicating higher uncertainty. As depicted in Figure 11, it can be observed that the uncertainty of the identification results obtained from non-uniform sampling data is generally greater than that from uniform sampling data, regardless of frequencies and damping ratios.

Figure 11.

Comparison of COVs: (a) Frequencies; (b) Damping ratios.

The identified mode shapes are presented in Figure 12. In Figure 12(a), the mode shapes identified from uniform accelerations are represented by squares, while in Figure 12(b), the mode shapes identified from non-uniform accelerations are depicted as triangles. Because there is a lack of analytical mode shapes of this bridge for illustrating the reliability of identified mode shapes, the analytical mode shapes of a three-span continuous beam with equal cross-section are simulated, as shown in Figure 12(c). The marker of square in each subplot of Figure 12(c) is used to describe the mode shape value at mid-span. The marker of circle in each subplot of Figure 12 is used to locate a bearing between two spans. Because the number of accelerometers is limited, the actual mode shapes of higher order mode cannot be constructed through the identified mode shapes, illustrated as the difference between dash lines of Modes 4∼6 in Figure 12(a) or Figure 12(b) and the solid lines of Modes 4∼6 in Figure 12(c). However, if only the mode shape value at each mid-span is considered, the identified mode shapes are basically consistent with the analytical mode shapes of a three-span continuous beam, illustrated through the vibration direction difference between squares in Figure 12(a) and (c).

Figure 12.

Mode shapes: (a) Uniform case; (b) Non-uniform case; (c) Analytical solutions.

Different non-uniform levels (ranging from 1% to 10%) are simulated through randomly discarding samples of uniform accelerations from 12:00 to 13:00 on October 11, 2022. Correspondingly, the identified frequencies and their COVs are respectively compared in Figure 13. If the non-uniform accelerations are mis-regarded as uniform accelerations for direct modal identification, the identified frequencies and their COVs are presented as dots in Figure 13. For each mode, the identified frequencies and their COVs increase overall with the rising levels of non-uniformity. If the non-uniform accelerations are known, and the NUFFT could be employed for these non-uniform accelerations for modal identification, the results are presented as triangles in Figure 13. For each mode, the identified frequencies and their COVs exhibit a slight variation with the increasing non-uniformity levels. This highlights the crucial need to distinguish whether the vibration data is uniform or not, and misinterpreting non-uniform vibration data as uniform can lead to errors in modal identification.

Figure 13.

Identified results under different non-uniform cases: (a) Frequencies; (b) COVs.

Conclusions

In this study, an operational modal identification framework is constructed that utilizes the fast Bayesian FFT method to estimate modal parameters of highway bridges using non-uniform monitoring data.

• One of the challenges in analyzing non-uniform signals is the presence of noise peaks, which significantly disturb the signal spectrum and make it difficult to identify weakly contributed modes. To address this issue, an autoregressive PSD-based peak picking method is developed that enables to capture initial frequencies related to low-energy modes.

• It is important to note that when dealing with non-uniform monitoring signals, the spectral analysis process should employ the NUFFT instead of the traditional FFT. This is because using FFT directly can cause the peaks associated with structural frequencies to shift, leading to increased errors in frequency and damping ratio identification.

• The monitoring accelerations of a real-world bridge are analyzed through the identification framework. The results demonstrate that as the non-uniform level of accelerations increases, the identification uncertainty of modal parameters also increases. This emphasizes the importance of considering non-uniformity in the analysis of monitoring data for accurate modal parameter estimation.

Nevertheless, the MAC-based bandwidth parameter determination method still requires an adequate number of measurement channels and a relatively high signal-to-noise ratio in the vibration signals, which should be further improved in the future.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Grant No. 52108270), the Foundation for High Level Talent Innovation Support Program of Dalian (Grant No. 2022RQ013), the Natural Science Foundation of Liaoning Province (Grant No. 2023-BS-064), and the Open Project Program of Guangdong Provincial Key Laboratory of Intelligent Disaster Prevention and Emergency Technologies for Urban Lifeline Engineering (Grant No. 2022ZA02).

ORCID iDs

Xiao-Mei Yang

Ting-Hua Yi

References

Amini

Hedayati

Zanddizari

(2021) Exploiting the inter-correlation of structural vibration signals for data loss recovery: a distributed compressive sensing-based approach. Mechanical Systems and Signal Processing 152: 107473. DOI: 10.1016/j.ymssp.2020.107473.

Xue

(2024) Deep learning-based sparsity-free compressive sensing method for high accuracy structural vibration response reconstruction. Mechanical Systems and Signal Processing 211: 111168. DOI: 10.1016/j.ymssp.2024.111168.

(2011) Fast Bayesian FFT method for ambient modal identification with separated modes. Journal of Engineering Mechanics 137(3): 214–226. DOI: 10.1061/(ASCE)EM.1943-7889.0000213.

(2012a) Fast Bayesian ambient modal identification in the frequency domain, Part I: posterior most probable value. Mechanical Systems and Signal Processing 26: 60–75. DOI: 10.1016/j.ymssp.2011.06.017.

(2012b) Fast Bayesian ambient modal identification in the frequency domain, Part II: posterior uncertainty. Mechanical Systems and Signal Processing 26: 76–90. DOI: 10.1016/j.ymssp.2011.06.019.

Brincker

Zhang

Andersen

(2001) Modal identification of output-only systems using frequency domain decomposition. Smart Materials and Structures 10(3): 441–445. DOI: 10.1088/0964-1726/10/3/303.

Caicedo

Lara-Valencia

Valencia

(2022) Machine learning techniques and population-based metaheuristics for damage detection and localization through frequency and modal-based structural health monitoring: a review. Archives of Computational Methods in Engineering 29(6): 3541–3565. DOI: 10.1007/s11831-021-09692-6.

, et al. (2023) Advances in intellectualization of transportation infrastructures. Engineering 24: 239–252. DOI: 10.1016/j.eng.2023.01.011.

Guo

Zhu

, et al. (2021) Monitoring-based evaluation of dynamic characteristics of a long span suspension bridge under typhoons. Journal of Civil Structural Health Monitoring 11(2): 397–410. DOI: 10.1007/s13349-020-00458-5.

10.

Liang

, et al. (2021) Fully automated precise operational modal identification. Engineering Structures 234: 111988. DOI: 10.1016/j.engstruct.2021.111988.

11.

Yang

(2022) A three-stage automated modal identification framework for bridge parameters based on frequency uncertainty and density clustering. Engineering Structures 255: 113891. DOI: 10.1016/j.engstruct.2022.113891.

12.

Katafygiotis

Yuen

(2001) Bayesian spectral density approach for modal updating using ambient data. Earthquake Engineering & Structural Dynamics 30(8): 1103–1123. DOI: 10.1002/eqe.53.

13.

Kim

Sim

(2019) Automated peak picking using region-based convolutional neural network for operational modal analysis. Structural Control and Health Monitoring 26(11): e2436. DOI: 10.1002/stc.2436.

14.

Kuok

Yuen

(2016) Investigation of modal identification and modal identifiability of a cable-stayed bridge with Bayesian framework. Smart Structures and Systems 17(3): 445–470. DOI: 10.12989/sss.2016.17.3.445.

15.

(2019) An expectation-maximization algorithm for Bayesian operational modal analysis with multiple (possibly close) modes. Mechanical Systems and Signal Processing 132: 490–511. DOI: 10.1016/j.ymssp.2019.06.036.

16.

Locke

Redmond

Schmid

(2022) Evaluating OMA system identification techniques for drive-by health monitoring on short span bridges. Journal of Bridge Engineering 27(9): 04022079. DOI: 10.1061/(ASCE)BE.1943-5592.0001923.

17.

Mao

Wang

, et al. (2023) Automated Bayesian operational modal analysis of the long-span bridge using machine-learning algorithms. Engineering Structures 289: 116336. DOI: 10.1016/j.engstruct.2023.116336.

18.

Matsuoka

Kaito

Sogabe

(2020) Bayesian time–frequency analysis of the vehicle–bridge dynamic interaction effect on simple-supported resonant railway bridges. Mechanical Systems and Signal Processing 135: 106373. DOI: 10.1016/j.ymssp.2019.106373.

19.

Zhang

(2019) Fast Bayesian frequency domain modal identification from seismic response data. Computers & Structures 212: 225–235. DOI: 10.1016/j.compstruc.2018.08.018.

20.

Thadikemalla

VSG

Gandhi

(2018) A data loss recovery technique using compressive sensing for structural health monitoring applications. KSCE Journal of Civil Engineering 22(12): 5084–5093. DOI: 10.1007/s12205-017-2070-z.

21.

Tomassini

García-Macías

Reynders

, et al. (2023) Model-assisted clustering for automated operational modal analysis of partially continuous multi-span bridges. Mechanical Systems and Signal Processing 200: 110587. DOI: 10.1016/j.ymssp.2023.110587.

22.

Wen

Houlihan

(2023) Application of the non-uniform Fourier transform to non-uniformly sampled Fourier transform spectrometers. Optics Communications 540: 129491. DOI: 10.1016/j.optcom.2023.129491.

23.

Yang

Nagarajaiah

(2015) Output-only modal identification by compressed sensing: non-uniform low-rate random sampling. Mechanical Systems and Signal Processing 56-57: 15–34. DOI: 10.1016/j.ymssp.2014.10.015.

24.

Yao

Zhao

, et al. (2021) Fully automated operational modal identification using continuously monitoring data of bridge structures. Journal of Performance of Constructed Facilities 35(5): 04021041. DOI: 10.1061/(ASCE)CF.1943-5509.0001614.

25.

Yao

(2022) Autoregressive spectrum-guided variational mode decomposition for time-varying modal identification under nonstationary conditions. Engineering Structures 251: 113543. DOI: 10.1016/j.engstruct.2021.113543.

26.

Yang

, et al. (2018) Correlation-based estimation method for cable-stayed bridge girder deflection variability under thermal action. Journal of Performance of Constructed Facilities 32(5): 04018070. DOI: 10.1061/(ASCE)CF.1943-5509.0001212.

27.

Yang

, et al. (2018) Monitoring-based analysis of the static and dynamic characteristic of wind actions for long-span cable-stayed bridge. Journal of Civil Structural Health Monitoring 8(1): 5–15. DOI: 10.1007/s13349-017-0257-0.

28.

Yang

Zhou

, et al. (2022) Joint deterioration detection based on field‐identified lateral deflection influence lines for adjacent box girder bridges. Structural Control & Health Monitoring 29(10): e3053. DOI: 10.1002/stc.3053.

29.

Yao

Wang

(2023) Structural modal identification and enhancement of low-energy modes by successively variational extraction of high-energy modes. Structures 54: 1360–1370. DOI: 10.1016/j.istruc.2023.05.147.

30.

Zeng

Xie

Kim

, et al. (2023) Automation in Bayesian operational modal analysis using clustering-based interpretation of stabilization diagram. Journal of Civil Structural Health Monitoring 13(2): 443–467. DOI: 10.1007/s13349-022-00644-7.

31.

Zhou

Kim

Zhang

, et al. (2022) Vibration-based Bayesian model updating of an actual steel truss bridge subjected to incremental damage. Engineering Structures 260: 114226. DOI: 10.1016/j.engstruct.2022.114226.

32.

Zhu

Brownjohn

JMW

(2019) Bayesian operational modal analysis with buried modes. Mechanical Systems and Signal Processing 121: 246–263. DOI: 10.1016/j.ymssp.2018.11.022.

33.

Zhu

, et al. (2021) Bayesian operational modal analysis with multiple setups and multiple (possibly close) modes. Mechanical Systems and Signal Processing 150: 107261. DOI: 10.1016/j.ymssp.2020.107261.