Fully automated modal tracking for long-span high-speed railway bridges

Abstract

Modal parameters are inherent structural characteristics that are valuable for model updating, condition assessment, and early warning of bridges. Operational modal tracking technology has been a popular research topic in bridge structural health monitoring (SHM) because of its output-only advantage; that is, only the vibration responses of the bridge are necessary for modal identification. The real-time objective of bridge SHM requires operational modal tracking to be fully automated. Because the loads acting on long-span high-speed railway bridges are various, the modal identification methods should be changed according to the excitation characteristics; otherwise, the results may be incorrect. However, there is no unified framework for simultaneously tracking the modal evolution of a bridge under different excitations. In this study, modal tracking strategies based on ambient loads, train loads and immediately a train moving past the bridge were developed to identify the operational modal parameters of the bridge. In addition, a unified tracking framework was established to automatically switch among the three-stage modal-tracking strategies, which utilizes the real-time positioning of the axle loads. Furthermore, the computational efficiency of the tracking strategies and the obstacles of operational modal analysis were analyzed to provide a reference for mode-based SHM of bridges, and the essential parameters in the tracking algorithms were suggested. The three-stage modal-tracking methods were validated through long-term monitoring data of a long-span high-speed railway bridge. The results indicated that the best tracking results were generated from free-vibration data, while the modal-tracking under ambient loads had best timeliness.

Keywords

Railway bridge modal parameter automated tracking train-induced vibrations free vibrations

Introduction

Long-span high-speed railway bridges are essential infrastructures constructed over rivers to connect railway networks, and performance degradation and damage of these bridges threatens human lives and the economy. Structural health monitoring (SHM) systems have been widely installed to track the performance evolution of long-span bridges (Meixedo et al., 2021; Xu and Xia, 2011). Because modal parameters (frequency, damping ratio and mode shapes) are inherent dynamic properties of a structure, their evolution can reflect the performance evolution of long-span bridges (Anastasopoulos et al., 2021). Therefore, considerable attention has been paid to tracking modal parameters from the SHM data of long-span bridges (Cabboi et al., 2017).

Because excitations (wind, train, scouring, etc.) acting on a bridge cannot be monitored, only the vibration responses of the bridge can be used for modal identification, which is known as operational modal identification (Vu et al., 2013). In previous studies, without any trains passing through, excitations were treated as Gaussian white-noise signals (Guo et al., 2021). Railway bridges can be analyzed through parametric methods, such as the stochastic subspace identification method (Reynders and Roeck, 2008), and non-parametric methods, such as the frequency domain decomposition technique (Brincker et al., 2001). Ozcelik et al. (2019) used data-driven stochastic subspace identification and enhanced frequency domain decomposition methods to analyze the ambient vibrations of a steel railway bridge, and the results indicated that the frequencies decreased slightly with an increase in the temperature. Considering that vibration data are often nonstationary, He et al. (2011) proposed an empirical mode decomposition-based random decrement technique to identify the modal parameters of the Nanjing Yangtze River Bridge. Because trainloads are not white-noise excitations, Matsuoka et al. (2021) developed a time-varying autoregressive exogenous model for the train and simply supported bridge system and adopted hierarchical Bayesian estimation to track the instantaneous frequencies of the bridge. Li et al. (2020) used a synchro-extracting transform to extract the instantaneous frequencies of bridges with vehicles passing through. In addition, a few studies have focused on modal analysis using free-vibrations of railway bridges. Kim et al. (2010) developed a time-domain decomposition technique for tracking modal parameters from free-vibration responses immediately after a train passes. Ülker-Kaustell and Karoumi (2011) extracted the natural frequency and damping ratio of the first vertical bending mode from free vibrations after the passage of freight trains through a continuous wavelet transform.

A common issue in operational modal tracking is that physical and spurious modes are difficult to distinguish, where physical modes are structural inherent characteristics and spurious modes arise due to the environmental noise and non-white noise excitation. Regarding parametric identification methods, the stabilization diagram (Afshar and Khodaygan, 2019) is widely used to classify physical and spurious modes, and clustering methods (Fan et al., 2019; Yang et al., 2019) are employed to automatically explain the stabilization diagram. He et al. (2022) proposed an automated identification process for bridges under ambient excitations, in which frequency uncertainty and density-based clustering were used to remove spurious frequencies occurring in the stabilization diagram. Regarding non-parametric methods, Yao et al. (2021) proposed a time-related modal assurance criterion to automatically select physical modes in frequency domain decomposition. Jin et al. (2021) developed an automatic peak-picking technique to identify the frequencies of cable-stayed bridges, whereby spectral peaks can be selected without prerequisites. Kim and Sim (2019) utilized a region-based convolutional neural network to automatically distinguish structural peaks from noise peaks. Another focus of mode-based SHM is to continuously track the evolution of modal parameters. Because the frequency bandwidth of excitation acting on the railway bridge changes over time, modes contributing to bridge vibrations also change, which leads to missing modes and misclassification. To track modal evolution without misclassification, He et al. (2021) connected the latter modes to the former modes by minimizing their difference. However, tracking fails if a missing mode exists. To overcome this obstacle, Yang et al. (2020a) constructed a modal subspace to update the reference mode list adaptively and cluster modes in the same order automatically through observability-vector similarity. Numerous automated operational modal identification methods have been developed to track the modal parameters of bridges under ambient loads. However, few studies have attempted to construct a unified automated modal tracking framework for long-span railway bridges affected by different excitations.

In this paper, a fully automated modal tracking approach is proposed for determining the evolution of the modal parameters of long-span high-speed railway bridges online. The stochastic, deterministic-stochastic, and free-vibration responses of the railway bridge are generated by ambient loads, train loads, and immediately after a train leaving the bridge. Therefore, a three-stage operational modal identification method was developed to track the modal parameters from different vibration responses. The key points include (i) determining the operational modal identification technique for each vibration type, (ii) constructing a unified automatic modal tracking framework, and (iii) evaluating the efficiency and uncertainty of operational modal tracking to provide a reference for mode-based SHM. The remainder of this paper is organized as follows. First, a unified modal tracking framework for a bridge under ambient loads, train loads, and immediately after a train leaving the bridge is described. Second, the efficiency of the method and parameter uncertainty are analyzed. Third, the tracking results for a railway bridge are presented. Finally, conclusions and prospects are presented.

Fully automated modal tracking method

The modal identification process is based on the discrete-time state-space model of the structural system:

{\begin{cases} x (k + 1) = A x (k) + B u (k) + w (k) \\ y (k) = C x (k) + D u (k) + v (k) \end{cases}

(1)

where

x (k) \in R^{2 n \times 1}

and

y (k) \in R^{M \times 1}

are the state and measured vibration response vectors, respectively;

u (k) \in R^{n \times 1}

and

w (k) \in R^{2 n \times 1}

represent the deterministic and stochastic loads acting on the bridge, respectively;

v (k) \in R^{M \times 1}

represents the measurement noise;

A \in R^{2 n \times 2 n}

B \in R^{2 n \times n}

C^{M \times 2 n}

, and

D \in R^{M \times n}

are the discrete-time state, input, output, and feed-through matrices, respectively;

k

represents the discrete-time;

n

represents the number of degrees of freedom; and

M

represents the number of measurement channels.

The discrete-time state matrix $A$ comprises the following bridge physical parameters:

A = e^{[\begin{array}{c} 0 & I \\ - M_{b}^{- 1} K_{b} & - M_{b}^{- 1} C_{b} \end{array}] Δ t}

(2)

where

M_{b} \in R^{n \times n}

C_{b} \in R^{n \times n}

, and

K_{b} \in R^{n \times n}

are the mass, damping, and stiffness matrices of the bridge, respectively;

Δ t = 1 / f_{s}

represents the discrete-time step; and

f_{s}

represents the sampling frequency of the responses

y (k)

If the eigenvalues $λ_{i}$ and eigenvectors $ψ_{i} \in R^{2 n \times 1}$ of the discrete-time state matrix $A$ are obtained, the modal parameters, including the frequency $f_{i}$ , damping ratio $ξ_{i}$ , and mode-shape vector $φ_{i} \in R^{M \times 1}$ , can be obtained as follows:

f_{i} = \frac{| \ln λ_{i} |}{2 π Δ t}

(3)

ξ_{i} = \frac{- R e (\ln λ_{i})}{| \ln λ_{i} |}

(4)

φ_{i} = C ψ_{i}

(5)

where

i = 1,2, \dots, 2 n

;

| \cdot |

and

R e (\cdot)

represent the modulus and real part of the complex number, respectively.

Therefore, a necessary step of modal identification is extracting the system matrices $A$ and $C$ from the responses, that is, $y (k)$ . In this study, the responses $y (k)$ in operation are divided into three stages: the stochastic vibrations under ambient loads, the deterministic-stochastic vibrations under train loads, and the free vibrations that occur immediately after a train leaves the bridge. Correspondingly, a three-stage modal tracking technique based on the state-space model is proposed.

Modal tracking under ambient loads

Without a train passing through, the bridge is subjected to ambient loads such as wind, which are stochastic and assumed as white noise. Correspondingly, the state-space model in equation (1) becomes:

{\begin{cases} x (k + 1) = A x (k) + w (k) \\ y (k) = C x (k) + v (k) \end{cases}

(6)

To eliminate white-noise excitation $w (k)$ and measurement noise $v (k)$ , the natural excitation technique (NExT) (Caicedo et al., 2004) is used, where the correlation functions are calculated as follows:

r (τ) = \frac{1}{N} \sum_{k = 0}^{N} y (k + τ) y^{T} (k) = C A^{τ - 1} G

(7)

where

r (τ) \in R^{M \times M}

is the correlation matrix; the superscript T is the transposition operator;

N

represents the data length used to calculate the correlation; the next state-response correlation matrix

G = E [x (k + 1) y^{T} (k)]

denotes the mathematical expectation between state vector

x (k + 1)

and response vector

y^{T} (k)

;

E [\cdot]

denotes the expectation operator.

The block Hankel matrices $R (τ - 1) \in R^{M h_{r}^{A} \times M h_{c}^{A}}$ with $τ = 1$ and $τ = 2$ are constructed using the correlation matrices as follows:

R (τ - 1) = [\begin{array}{c} r (τ) & r (τ + 1) & \dots & r (τ + h_{c}^{A} - 1) \\ r (τ + 1) & r (τ + 2) & \dots & r (τ + h_{c}^{A}) \\ \dots & \dots & \dots & \dots \\ r (τ + h_{r}^{A} - 1) & r (τ + h_{r}^{A}) & \dots & r (τ + h_{r}^{A} + h_{c}^{A} - 2) \end{array}]

(8)

Then, the system matrices $A$ and $C$ can be extracted by performing singular value decomposition for the matrix $R (0)$ , which is known as the eigensystem realization algorithm (ERA) (Caicedo et al., 2004). Because mathematical errors and measurement noise cannot be eliminated well, spurious modes are included in the estimated state matrix. To distinguish spurious modes from structural modes, a state matrix $A \in R^{\bar{n} \times \bar{n}}$ with the model order $\bar{n} = {\bar{n}}_{\min}, {\bar{n}}_{\min} + Δ n, \dots, {\bar{n}}_{\max} - Δ n, {\bar{n}}_{\max}$ is estimated, and a two-stage clustering technique is used. Details were presented by Yang et al. (2019).

Modal tracking under train loads

When a train passes through, the bridge is simultaneously subjected to train and ambient loads. Due to the train-bridge interaction effect, the system becomes a time-variant discrete-time state-space model (Yang et al., 2021):

{\begin{cases} x (k + 1) = A (k) x (k) + B (k) u (k) + w (k) \\ y (k) = C (k) x (k) + D (k) u (k) + v (k) \end{cases}

(9)

where

u (k) \in R^{n \times 1}

is a distribution matrix of the quasi-static axle loads on the bridge nodes.

If the real-time positions of the axle loads can be measured, the distribution matrix $u (k) \in R^{n \times 1}$ of the loads can be virtually constructed, and then the deterministic-stochastic subspace identification (DSSI) method can be used for the time-variant system in equation (9). The first step in DSSI is to eliminate the deterministic loads $B (k) u (k)$ using an orthogonal projection of a Hankel matrix $Y (k) \in R^{M h_{r}^{D} \times h_{c}^{D}}$ (constructed by $(k)$ with $k = 1,2, \dots, h_{r}^{D} + h_{c}^{D} - 1$ ) onto the orthogonal complement of the virtual load Hankel matrix $U (k) \in R^{n h_{r}^{D} \times h_{c}^{D}}$ (constructed by $u (k)$ with $k = 1,2, \dots, h_{r}^{D} + h_{c}^{D} - 1$ ) as $Y_{p} (k) = Y (k) (I - U^{T} (k) {(U (k) U^{T} (k))}^{- 1} U (k))$ . The second step is to eliminate the effect of stochastic load $w (k)$ by calculating the correlation matrix $R \in R^{M h_{r}^{D} \times M h_{r}^{D}}$ of the projected matrix $Y_{p} (k)$ . By performing singular value decomposition for $R$ , the system matrices $A$ and $C$ can be estimated. The key step is to automatically construct a virtual distribution matrix based on the real-time positions of the axle loads (Yang et al., 2021). In addition, the modal parameters identified from the vibration responses $y (k)$ with $k = 1,2, \dots, h_{r}^{D} + h_{c}^{D} - 1$ are time-frozen and with increased bias because the time-variant characteristics due to train-bridge interaction is ignored.

Modal tracking immediately after train leaves

Immediately after a train leaves the bridge, free vibrations of the bridge occur. Because the initial state $x (0)$ of the bridge caused by the train is significantly greater than the vibration caused by ambient loads, the ambient loads $w (k)$ are ignored, and the discrete-time state-space model is as follows:

{\begin{cases} x (k + 1) = A x (k) \\ y (k) = C x (k) + v (k) \end{cases}

(10)

A correlation-based ERA is used for the responses in the form of equation (10) to obtain the system matrices $A$ and $C$ . Similarly, a Hankel matrix $Y (k) \in R^{M h_{r}^{F} \times h_{c}^{F}}$ is calculated, and singular value decomposition should be used for the correlation matrix $R \in R^{M h_{r}^{F} \times M h_{r}^{F}}$ of the Hankel matrix $Y (k)$ . The identification process is sensitive to the signal-to-noise ratio. Thus, an accurate detection technique has been proposed to automatically find effective free vibrations for modal identification (Yang et al., 2020b).

Three-stage fully automated tracking

The SHM of railway bridges requires that the modal parameters can be tracked online without supervision. Although automated modal identification methods based on three-stage vibrations have been implemented, switching between appropriate identification methods in real time remains a problem worthy of attention. In this section, a unified framework is first established to implement online modal tracking throughout the operational period. Second, the efficiency of the method is analyzed with regard to the computational time. Third, the modal uncertainty is analyzed to provide suggestions for mode-based warning of bridge safety risks.

Unified process framework

The automated switching of three-stage tracking methods mainly includes four steps.

Step (i): Determine whether the vibrations at the discrete-time $k$ are train-induced; that is, determine whether there is a train acting on the bridge. Because the SHM data used for modal identification are accelerations, which do not indicate the axle position directly, the strains of the girder are additionally used. A peak of the strain has been proven to occur if an axle load is located halfway between the strain gauge and sleeper (Deepthi et al., 2019). Therefore, the peak of the strain data of the girder can determine whether a train passes through the bridge, specifically, whether it passes through the bridge node where the strain gauge is installed. As ambient loads, environmental factors, and sensor noise exist, the strain data are nonzero (Figure 1) even if no train passes, which makes unsupervised peak searching difficult. In addition, as an environment-induced trend exists in the strain data (dashed line in Figure 1), determining the train-induced time through the larger strain values is also inappropriate, the main problem of which is that the strain threshold cannot be fixed. Compared with the peak and larger-value features, the significant increment in the strain data is a useful feature, which is defined as the primary difference in the strain data:

d ε (k) = ε (k) - ε (k - f_{s e})

(11)

where

f_{s e}

represents the sampling frequency of the strain gauges;

ε (k)

represents the strain at time instant

k

; and

d ε (k)

is the strain increment index.

Figure 1.

Strain data of the girder: (a) 1 month; (b) during a 16-carriage train passing.

Because the strain trend induced by environmental factors is eliminated after the primary-difference process, strain increments are induced by train loads and measurement noise, as shown in Figure 2. The strain increments induced by the measurement noise are slight and stochastic, whereas those induced by train loads have large values and are limited in number. Thus, the former can be easily distinguished through the following threshold:

ε_{T} = μ + 3 σ

(12)

where

μ

and

σ

are the mean and SD of the strain increments calculated from the historical strain data, respectively; and

ε_{T}

represents the threshold (dashed line in Figure 2).

Figure 2.

Strain increments: (a) 1 month; (b) during a 16-carriage train passing.

If the strain increment satisfies $d ε (k) \geq ε_{T}$ , the train passes through the node where the strain gauge is installed. Correspondingly, a train-induced time window $[k - K_{T}, k + K_{T}]$ is estimated, where the half length of the train-induced time window $K_{T}$ satisfies:

K_{T} \geq \frac{(L_{b, \max} + n_{c, \max} \times l_{c}) \times f_{s e}}{v_{\min}} + 30 f_{s e}

(13)

where

L_{b, \max}

is the maximum distance between the strain gauge and the bridge edges;

n_{c, \max}

is the maximum number of carriages and set as 16 because only 8-carriage and 16-carriage trains are put into operation;

l_{c}

is the carriage length which is set as 25 m for simplicity;

v_{\min}

is the minimum train speed estimated from historical data of train speeds; and

30 f_{s e}

aims to ensure that sufficient free-vibration data can be included.

If the strain increment satisfies $d ε (k) < ε_{T}$ and time instant $\bar{k} = k - K_{T} - 1$ has not been assigned to a train-induced time window, the time instant $\bar{k}$ is related to stochastic vibrations induced by ambient loads.

Step (ii): Classify the loading case to which the train-induced time window $[k - K_{T}, k + K_{T}]$ belongs. For a bridge with $γ$ lanes, the $2 γ$ loading cases of a single train and a loading case of multiple trains are generated. Because a loading case with multiple trains passing simultaneously is too complex to be analyzed, it is eliminated after the loading classification. For loading cases with a single train, the carriage number and running lane of the train should be determined using the strain data. In particular, the number of peaks in the shear strain data is related to the number of carriages of a train; that is, 17 peaks indicate 16 carriages, and nine peaks indicate eight carriages.

The details of the loading classification are as follows. First, except for the running lane $α$ , which has been selected as a train passing and used to determine the train-induced time window $\bar{k} \in [k - K_{T}, k + K_{T}]$ , other running lane $j$ whose strain increment satisfies $d ε_{j} (\bar{k}) \geq ε_{T}$ are selected. Second, the peaks in the strain data ( $ε_{α} (\bar{k})$ and $ε_{j} (\bar{k})$ ) are selected to determine the carriage numbers of trains running on lanes $α$ and $j$ , respectively. Third, the time instants ${\bar{k}}_{α, f i r s t}$ and ${\bar{k}}_{α, e n d}$ of a train entering and exiting lane $α$ of the bridge are roughly estimated using the time instants related to the first and last peaks of $ε_{α} (\bar{k})$ , the train speed $v_{α}$ , the distances between the strain gauge from the bridge edges, and the train length $n_{c} l_{c}$ . In addition, the time instants ${\bar{k}}_{j, f i r s t}$ and ${\bar{k}}_{j, e n d}$ of a train entering and exiting lane, respectively, are also estimated. If lane $α$ is unique within $\bar{k} \in [k - K_{T}, k + K_{T}]$ , it is characterized by single-train loading case. If lane $α$ is not unique but the estimated time instants of the train reaching and passing over the bridge are non-intersecting, that is, ${\bar{k}}_{α, e n d} < {\bar{k}}_{j, f i r s t}$ , various single-train loading cases occur, and the train-induced time window is divided into the corresponding single-train loading cases.

The characteristics of strain data related to different lanes should not interfere with each other. Otherwise, the multi-train loading case may be mistaken for a single-train loading case. For example, two eight-carriage trains running on the adjacent lanes and intersecting at the strain gauge, may be misjudged as a 16-carriage train. To avoid this misclassification, at least two strain gauges (or other sensors reflecting the location of the train, such as speedometers) should be installed in a lane, at a certain distance along the longitudinal direction.

Step (iii): For a single-train loading case, separate the vibration responses within the train-induced time window into deterministic-stochastic vibrations and free vibrations; that is, find a time instant ${\bar{k}}_{f r e e}$ that a train immediately leaves the bridge. Because the positions of the axle loads cannot be accurately determined through the peaks in the strain data, the time instants of a train reaching and leaving the bridge cannot be accurately determined. In this study, the modal responses of bridge accelerations were used to determine the time instant of the train moving past the bridge, that is, the time border ${\bar{k}}_{f r e e}$ between forced vibrations and free vibrations. Assuming that the vibrations within the time window $\bar{k} \in [k - K_{T}, k + K_{T}]$ are related to a single-train loading case, the time instant of the train moving past the bridge is described briefly as follows. First, the accelerometer $m$ closest to the bridge edge of a train leaving is selected. Then, the variational mode decomposition technique is used to obtain the modal response from the acceleration at the accelerometer $m$ . Finally, the starting time instant of a free-decay modal response with the longest duration is set as the time border ${\bar{k}}_{f r e e}$ . Details regarding free-vibration detection were presented by Yang et al. (2020b). Subsequently, the free-vibrations, that is, the vibrations within the time window $[{\bar{k}}_{f r e e}, k + K_{T}]$ , and the forced vibrations, that is, the vibrations within the time window $[k - K_{T}, {\bar{k}}_{f r e e})$ , are obtained.

Step (iv): Estimate the real-time positions of the axle loads to construct the virtual load matrix for the DSSI. Because the time instant of a train moving past the bridge was obtained in Step (iii), the real-time positions of the axle loads for each single-train loading case can be derived according to the train speed, time instant of the train moving past the bridge, and axle arrangements. The train speed is obtained from the speedometer installed in each railway lane, and axle arrangements of the train are determined according to the train schedule. Subsequently, a set of deterministic-stochastic vibrations is obtained for identification through the DSSI method.

For a time instant that exceeds the train-induced time window, the corresponding vibrations are stochastic. A batch of stochastic accelerations with data length $N$ is analyzed using NExT-ERA. The data length $N$ should be sufficient to ensure that the frequency evolution can be tracked well.

Because spurious modes always occur in operational modal analysis, regardless of the stochastic, deterministic-stochastic, and free vibrations, a two-stage clustering process is used to distinguish the physical modes from spurious modes. Furthermore, an automated mode-matching algorithm based on the correlation between modal observability vectors is used to track the modal parameters identified from different batches of vibration data. A flowchart of the unified modal tracking framework is shown in Figure 3.

Figure 3.

Flowchart of the unified modal tracking framework.

Method efficiency analysis

Because computational efficiency is a concern in bridge SHM, the computational complexity is qualitatively analyzed, and the computation time for a bridge example is presented to quantitatively evaluate the efficiency of the method. To compare the efficiencies of the three-stage modal tracking methods, they were divided into preprocessing, identification, and distinguishing processes. The computational complexities are presented in Table 1.

Table 1.

The computational complexity analysis.

Stage	Complexity
Stage	Preprocessing	Identification	Distinguishing
Amb-ient	O (MNlog₂ N)	$O ({(M h_{r}^{A})}^{3})$	$O (({\bar{n}}_{\min} + {\bar{n}}_{\max}) ({\bar{n}}_{\max} + Δ n - {\bar{n}}_{\min}) / Δ n \times κ)$
Train	$O (n h_{r}^{D} h_{c}^{D} \max (n h_{r}^{D}, h_{c}^{D}))$	$O ({(M h_{r}^{D})}^{2} \max (M h_{r}^{D}, h_{c}^{D}))$
Free	$κ^{V M D}$ and $α^{V M D}$	$O ({(M h_{r}^{F})}^{2} \max (h_{r}^{F}, h_{c}^{F}))$

According to the aforementioned description for modal tracking under ambient loads, preprocessing is used to calculate the correlation matrix $r (τ)$ in equation (7) through NExT, where the auto- and cross-power spectral density functions of $y (k)$ are calculated and then transformed into auto- and cross-correlation functions through an inverse Fourier transform. The fast Fourier transform, which has a complexity of O (MNlog₂N), accounts for the main computational burden. For the identification process, the computational complexity is mainly related to the singular value decomposition of a Hankel matrix $R (0) \in R^{M h_{r}^{A} \times M h_{c}^{A}}$ , as $O ({(M h_{r}^{A})}^{3})$ . In the distinguishing process, 2-means clustering is performed to distinguish the structural modes from spurious modes. The computational complexity of the 2-means clustering is $O (({\bar{n}}_{\min} + {\bar{n}}_{\max}) ({\bar{n}}_{\max} + Δ n - {\bar{n}}_{\min}) / Δ n \times κ)$ , where $κ$ represents the number of iterations.

According to the description for modal tracking under train loads, preprocessing involves constructing a virtual load Hankel matrix $U (k) \in R^{n h_{r}^{D} \times h_{c}^{D}}$ and projecting a response Hankel matrix $Y (k) \in R^{M h_{r}^{D} \times h_{c}^{D}}$ onto the orthogonal complement of the virtual load Hankel matrix. The projection is time-consuming, and its computational complexity is $O (n h_{r}^{D} h_{c}^{D} \max (n h_{r}^{D}, h_{c}^{D}))$ . For the identification process, the correlation matrix $R (τ) \in R^{M h_{r}^{D} \times M h_{r}^{D}}$ of the projected response matrix $Y_{p} (k) \in R^{M h_{r}^{D} \times h_{c}^{D}}$ should be calculated with a computational complexity of $O ({(M h_{r}^{D})}^{2} \max (M h_{r}^{D}, h_{c}^{D}))$ , and singular value decomposition is performed on the correlation matrix with a computational complexity of $O ({(M h_{r}^{D})}^{2})$ . Accordingly, the computational complexity of the identification process is $O ({(M h_{r}^{D})}^{2} \max (M h_{r}^{D}, h_{c}^{D}))$ . In addition, the computational complexity of the distinguishing process is consistent with that under ambient loads.

According to the description for modal tracking immediately after a train leaves, preprocessing aims to detect free vibrations through the variational mode decomposition technique. The computational complexity of the variational mode decomposition technique depends on the number of iterations $κ^{V M D}$ and the regularization parameter $α^{V M D}$ . For the identification process, the correlation-based ERA is used to analyze a response Hankel matrix $Y (k) \in R^{M h_{r}^{F} \times h_{c}^{F}}$ . The most time-consuming procedures in the correlation-based ERA are the correlation calculation and singular value decomposition with the computational complexity of $O ({(M h_{r}^{F})}^{2} \max (h_{r}^{F}, h_{c}^{F}))$ . The computational complexity of structural mode distinguishing is $O (({\bar{n}}_{\min} + {\bar{n}}_{\max}) ({\bar{n}}_{\max} + Δ n - {\bar{n}}_{\min}) / Δ n \times κ)$ .

The vertical accelerations of a bridge were taken to quantify the computational time, and the results are presented in Table 2. The algorithms were tested by MATLAB R2020a, installed on a computer with Intel(R) Core(TM) i5-10500 CPU @3.10 GHz and 16.0 GB RAM.

Table 2.

The computational time.

Stages	Time (s)
Stages	Preprocessing	Identification	Classification	Summary
Ambient	1.54	8.51	13.34	23.39
Train	80.01	32.80	19.81	132.62
Free	29.73	24.37	13.61	67.71

Under ambient loads, the data length was set as

N = 3600 f_{s}

; the number of measurement channels was

M = 6

; the numbers of rows and columns in the Hankel matrix in ERA were

h_{r}^{A} = 600

and

h_{c}^{A} = 400

, respectively; the model order parameters were

{\bar{n}}_{\min} = 50

Δ n = 2

, and

{\bar{n}}_{\max} = 200

; the stopband frequency of the low-pass filter was 4 Hz; and nine modes were identified, as shown in Table 3. The computational time of the entire process was 23.39 s, and the most time-consuming procedure was automatically distinguishing the structural modes.

Table 3.

Frequencies of identified modes.

Stages	Frequency (Hz)
Stages	Mode t	Mode 1	Mode 2	Mode 3	Mode 4	Mode 5	Mode 6	Mode 7	Mode 8	Mode 9
Ambient	—	0.695	0.756	0.961	0.996	1.221	1.312	1.413	1.978	2.053
Train	—	0.690	0.737	0.987	1.016	1.226	1.312	1.404	1.972	2.042
Free	0.501	0.666	0.753	0.964	0.999	1.209	1.312	1.399	1.948	2.020

Under train loads, the data length was automatically calculated as $N = {\bar{k}}_{f r e e} - (k - K_{T})$ ( $N =$ 11,021 for the group of samples to obtain the modal parameters in Table.3); the numbers of rows and columns in the Hankel matrix in the DSSI method were $h_{r}^{D} = 800$ and $h_{c}^{D} = N - 2 h_{r}^{D} - 1$ , respectively; and the model order parameters were ${\bar{n}}_{\min} = 200$ , $Δ n = 2$ , and ${\bar{n}}_{\max} = 300$ . The identified frequencies are presented in Table 3. Because the data length was insufficient, the effects of stochastic loads and noise on the vibration responses could not be eliminated well by calculating the correlation matrix of the projected response matrix. Considering that spurious modes due to noise and load interference appear in the lower model order while the bridge modes appear in the higher model order, the model orders ${\bar{n}}_{\min}$ and ${\bar{n}}_{\max}$ exceeded those under ambient loads for ensuring that structural modes could occur in the stabilization diagram. The computational time of the entire process was 132.62 s, which was longer than that of the tracking process under ambient loads. Preprocessing was the most time-consuming process.

For free vibrations, the data length was automatically calculated as $N = (k + K_{T}) - {\bar{k}}_{f r e e} + 1$ ( $N =$ 9351 for the group of samples to obtain the modal parameters in Table 3); the numbers of rows and columns in the Hankel matrix in the correlation-based ERA were $h_{r}^{F} = 600$ and $h_{c}^{F} = N - h_{r}^{F} - 1$ , respectively; the model order parameters were ${\bar{n}}_{\min} = 50$ , $Δ n = 2$ , and ${\bar{n}}_{\max} = 200$ ; and 10 modes were identified, as shown in Table 3. A mode with a frequency of 0.501 Hz was identified only from the free-vibration responses of the bridge, which is labelled as Mode t in Table 3. This is because the 0.501 Hz mode contributed significantly to the transverse vibration accelerations of the bridge but did not contribute significantly to the vertical vibration accelerations.

As indicated by the computational times presented in Table 2, the computational efficiency of the algorithm was the highest under ambient loads. If a sliding window-based tracking method is used for the modal analysis of a bridge under ambient loads, the results can be generated every 30 s. The frequencies listed in Table 3 indicate that more modes can be identified from the vibration responses of the bridge immediately after a train leaves, but the preprocessing to obtain the free-vibration responses is more time-consuming than that under ambient loads.

Modal uncertainty analysis

Because the operational monitoring data of the bridge cannot strictly satisfy the assumptions of modal identification methods, modal uncertainty was involved. In this section, the frequency and mode-shape uncertainties are analyzed according to the long-term tracking results, and algorithm parameters are recommended. Additionally, issues related to operational modal parameters are explained.

Frequencies are the most reliable modal parameters of the bridge; their uncertainty is closely related to the algorithm parameters, including the data length $N$ and the numbers of rows and columns in the Hankel matrix. The numbers of rows and columns in the Hankel matrix were automatically determined if the number of stable modes in the stabilization diagram changes slightly (Yang et al., 2019). In this section, the data length is discussed.

For modal tracking under ambient loads, the data length $N$ in NExT should be sufficient. Because the tracking methods for railway and highway bridges under ambient loads are consistent, guidelines for the data length in highway bridge modal identification can be referenced. For example, Load Test Methods for Highway Bridge (JTG/T J21-01-2015, 2016) suggested a data length of $N \geq$ 1800 $f_{s}$ for suspension and cable-stayed bridges with low frequencies. In NExT, the frequency uncertainty decreases with an increase in the frequency resolution used in the power spectral density calculation. Therefore, the recommended data length is $N \geq 2^{n e x t p o w 2 (f_{s} / δ f)}$ . In addition, we set $δ f = 0.01 f_{\min}$ to ensure that a frequency variation of 0.01 $f_{\min}$ can be observed, where $f_{\min}$ is the fundamental frequency of the bridge. The frequency results obtained with different frequency resolutions are compared in Figure 4, including the power spectral densities, the stabilization diagrams, and frequency evolutions. As shown in Figure 4(a), the peak frequencies of the power spectral densities with high and low resolutions differed. For the high resolution ( $δ f = 0.01 f_{\min}$ ), the power spectral density had so many spectral lines that the peak related to the bridge frequency could be precisely located. Owing to the lack of analytical modal parameters of the bridge, the frequency uncertainty could not be quantified, and different frequencies were identified from the stabilization diagram in Figure 4(b). However, the degree of frequency evolution in Figure 4(c) for the high resolution is smaller than that for the low resolution, indicating that the frequency has less uncertainty at a higher resolution.

Figure 4.

Frequency resolution effects: (a) comparison of power spectral densities; (b) comparison of stabilization diagrams; (c) comparison of frequency evolutions.

For modal tracking under train loads, the stochastic vibration components induced by the ambient loads, irregularities, and dynamic loads of a train should be eliminated via the correlation method, similar to NExT. Thus, the data length $N$ also should be sufficient. However, the data length under train loads was limited, leading to the largest frequency uncertainty among the different types of loads. For modal tracking under free vibrations, the structural vibration components have a signal-to-noise ratio and can be directly used for modal identification via the ERA. Therefore, the data length could not be excessive but sufficient to construct the Hankel matrix, that is, $N > h_{c}^{F} + h_{r}^{F} + 1$ . Because of the high signal-to-noise ratio, the frequency uncertainty identified from the free-vibrations was the smallest.

The mode shapes, which reflect the spatial characteristics of a bridge, have potential for bridge early-warning and abnormal positioning. However, the mode shapes identified from SHM data have not been widely used, because of their large uncertainty. Owing to the non-proportional damping of the bridge, non-uniform or asynchronous sampling of the sensor, measurement noise, etc., the identified mode shapes are complex. In particular, the phase of the complex mode shape is sensitive to these factors. For example, if the multi-channel vibration responses were slightly asynchronous, the complex mode shapes changed significantly, as shown in Figure 5 (‘Syn’ and ‘Non-syn’ represent mode shapes identified from the synchronous and asynchronous data). The real and imaginary parts of the complex mode shapes are compared in Figure 5(a), while the amplitudes are compared in Figure 5(b).

Figure 5.

Complex mode shapes: (a) real-imaginary part; (b) amplitude.

The amplitude of a complex mode shape coefficient $φ_{m i}$ was calculated as $| φ_{m i} | \times R e (φ_{m i}) / | R e (φ_{m i}) |$ , where the subscripts mean the measurement channel $m$ and the mode $i$ . Clearly, the amplitudes of the mode shapes changed significantly, even when there was no damage. In fact, the mode shapes identified from the operational SHM data should be called operating deformation shapes but not structural mode shapes. For high-quality operational SHM data, the operating deformation shapes are nearly real values and can be regarded as mode shapes. For low-quality operational SHM data, it is inappropriate to use the operating mode shapes for early warning else if the phase evolution of complex mode shapes can be explained completely.

Case study

The operational SHM data of a long-span high-speed railway bridge are analyzed in this section. First, the bridge and its SHM are described briefly. Subsequently, the modal tracking results under ambient loads, train loads, and free vibrations are discussed. Finally, the modal evolutions are analyzed.

Bridge description and its monitoring data

Long-term monitoring data were recorded for a long-span high-speed railway bridge with six spans. The bridge had six lanes, including two high-speed railway lanes (Lanes 1 and 3 in Figure 6), two normal-speed railway lanes (Lanes 2 and 4), and two metro lanes, respectively. The running direction of Lanes 1 and 2 was from north to south, and that of Lanes 3 and 4 was from south to north. Monitoring data from 2013 were analyzed in this study, and only the high-speed and normal-speed lanes were in operation at that time.

Figure 6.

Sensors installed on the long-span railway bridge.

A unidirectional accelerometer was installed in the middle of each span to measure the vertical vibration response of the girder with a sampling frequency of 200 Hz. In addition, four speedometers with a sampling frequency of 10 Hz were installed on each of the high-speed and normal-speed lanes. Two strain gauges with a sampling frequency of 50 Hz were installed on the high-speed and normal-speed lanes. The sensor locations are shown in Figure 6. The strain and speedometer data were used to classify the loading cases, and the accelerations were used for modal identification.

Discussion of modal tracking results

For the classification of loading cases, the threshold $ε_{T}$ of the strain increment in equation (12), the maximum distance $L_{b, \max}$ in equation (13), and the minimum speed $v_{\min}$ of each railway lane in equation (13) should be determined. The historical monitoring strains from March 1 to 31, 2013 were analyzed, and the threshold $ε_{T} = 9.70$ $μ ε$ is presented in Figure 2. In addition, the distances between the strain gauge and the north and south edges of the bridge (Figure 6) are 468 m and 804 m, respectively. Therefore, the maximum distance $L_{b, \max}$ in equation (13) equals 804 m. The minimum speeds $v_{\min}$ were determined according to the speeds shown in Figure 7. Considering the effect of sensor noise on the measured speed values, the minimum speed related to each lane was selected as $v_{\min} = μ_{v} - 3 σ_{v}$ , where $μ_{v}$ and $σ_{v}$ represent the mean and SD of the measured speed values, respectively. As shown in Figure 7, the train speeds in the high-speed railway lanes ranged from 180 to 260 km/h, with estimated minimum speeds close to 200 km/h, and the train speeds in the normal speed railway lanes ranged from 80 to 220 km/h, with estimated minimum speeds of 99.3 and 167.2 km/h.

Figure 7.

The statistical distribution of train speeds: (a) Lane 1; (b) Lane 2; (c) Lane 3; (d) Lane 4.

According to the aforementioned parameters, vibration data can be divided into a unified modal tracking framework. It was observed that more than 200 trains passed over the bridge per day, and the loading cases for a day are presented in Table 4. Cases C1 to C8 involved single trains and C9 involved multiple trains.

Table 4.

Loading cases of the railway bridge in a day.

Case	C1	C2	C3	C4	C5	C6	C7	C8	C9
Lane	1	2	3	4	1	2	3	4	—
Carriage	8	8	8	8	16	16	16	16	—
Number	21	3	15	2	38	38	48	44	56

The tracking results for February 2, 2013 were presented to compare the efficiencies of the three-stage methods, the algorithm parameters of which were consistent with those mentioned in previous section. The frequencies identified from accelerations of the bridge under ambient loads (Case A), eight train load cases (Cases C1 to C8), and free vibrations (Case F) were compared in Figure 8, respectively. In terms of frequency fluctuation, the minimum is Case A, the second is Case F, and the maximum is Cases C1–C8. The frequencies under ambient loads have the smallest fluctuation because the number of them is smallest. The frequencies related to train-loading cases have the greatest fluctuation because the interference factors on vibrations are too complex to be eliminated. In addition, the frequencies in Case F are less than the frequencies in Case A. One possible reason is that free vibrations still reflect the dynamic characteristics of the train-bridge interaction system.

Figure 8.

Tracking results of frequencies in a day: (a) Mode 1; (b) Mode 2.

The mode shapes related to the identified modes are presented in Figure 9, where the black dots represent bridge bearings, and the red squares represent the identified mode shape coefficients. Because the accelerometers were installed at the centers of the cross-sections of the girder, only the symmetric vertical bending mode was identified. Owing to the limited number of accelerometers, the mode shapes of the entire girder could not be presented completely, and the mode shape coefficients of Mode 3 were proportional to those of Mode 7. In addition, Modes 8 and 9 were local modes, and their mode shape coefficients at the side spans were significantly larger than those at the main spans.

Figure 9.

Mode shapes: (a) Mode 1; (b) Mode 2; (c) Mode 3; (d) Mode 4; (e) Mode 5; (f) Mode 6; (g) Mode 7; (h) Mode 8; (i) Mode 9.

Analysis of long-term modal evolution

The long-term modal tracking results were analyzed in this section, including the evolution trends and statistical characteristics.

Taking the tracking results under ambient loads as an example, the evolution trends of the frequency, damping ratio, and mode shapes were presented in Figure 10, where the mode shape evolution was quantified by the modal assurance criterion (MAC), defined as:

M A C (φ_{i}, φ_{j}) = \frac{φ_{i}^{T} φ_{j}}{\sqrt{φ_{i}^{T} φ_{i}} \sqrt{φ_{j}^{T} φ_{j}}}

(14)

where

0 \leq M A C \leq 1

and

M A C = 1

means that the mode shape vectors are completely proportional.

Figure 10.

Tracking results under ambient loads: (a) frequency; (b) damping ratio; (c) MAC of mode shapes.

Frequencies were compared in Figure 10(a), where Modes 1–4 had the lower identifiability than Modes 5–9, because the signal components related to Modes 1–4 were less contributed in ambient vibration accelerations of the girder. The damping ratios and MACs of mode shapes were more stochastically scattered than frequencies.

The lognormal distribution function was adopted to fit the statistical distribution of the damping ratios, and the results are presented in Figure 11. The damping ratios of Modes 4 and 5 were significantly larger than those of other modes.

Figure 11.

Statistical distribution of damping ratios: (a) Mode 1; (b) Mode 2; (c) Mode 3; (d) Mode 4; (e) Mode 5; (f) Mode 6; (g) Mode 7; (h) Mode 8; (i) Mode 9.

To quantify the evolutions of frequencies, damping ratios and mode shape MACs, the frequency evolution index $e_{f_{i}}$ , SD of damping ratios $σ_{ξ_{i}}$ , and SD of MACs $σ_{{M A C}_{i}}$ were used. The frequency evolution index was defined as:

e_{f_{i}} = \frac{f_{i, \max} - f_{i, \min}}{f_{i, m e a n}}

(15)

where

f_{i, \max}

f_{i, \min}

, and

f_{i, m e a n}

represent the maximum, minimum, and mean values, respectively, of the frequencies of Mode.

The modal evolutions of frequencies, damping ratios, and MACs were listed in Table 5. The frequency evolution index of Mode 2 was 9.83%, which was twice or three times those of the other modes. In addition, the damping ratios of Mode 2 had the largest SD. Therefore, the Mode 2 had the largest uncertainty for frequencies and damping ratios under ambient loads. However, the mean of MACs of Mode 1 was smallest, indicating that the mode shapes of Mode 1 changed significantly under ambient loads.

Table 5.

Modal evolutions under ambient loads.

Mode		1	2	3	4	5	6	7	8	9
Frequency	$f_{i, m e a n}$	0.695	0.770	0.978	1.001	1.230	1.312	1.419	1.976	2.051
Frequency	$e_{f_{i}}$ (%)	3.19	9.83	2.16	4.22	2.93	4.45	4.64	4.10	4.69
Damping ratio	$ξ_{i, m e a n}$	0.74	0.96	0.80	2.24	2.68	0.99	1.63	0.84	0.63
Damping ratio	$σ_{ξ_{i}}$ (%)	0.44	0.57	0.26	0.49	0.51	0.35	0.39	0.27	0.24
MAC of mode shapes	${M A C}_{i, m e a n}$	0.78	0.97	0.93	0.89	0.88	0.93	0.95	0.93	0.98
MAC of mode shapes	$σ_{{M A C}_{i}}$ (%)	2.42	4.51	5.34	4.78	6.39	5.02	4.20	5.22	2.16

Taking Mode 1 for example, the long-term tracking results related to different cases were compared through the statistical analysis. The statistical distributions for frequencies of Mode 1 were presented in Figure 12, where frequencies in Cases C1–C7 and Case A were fitted through the t-Location-Scale distribution function and frequencies in Case F were fitted through the Burr-type-XII distribution function. The frequencies in Cases C1–C7 ranged from 0.64 to 0.72 Hz, the frequencies in Case F ranged from 0.67 to 0.69 Hz, and the frequencies in Case A ranged from 0.68 to 0.72 Hz. In addition, the frequency evolution index in equation (15) was calculated for each Case, as shown in Figure 13(a). The frequency evolution index in Case F was the smallest. Therefore, the frequencies identified from free vibrations (Case F) were the most concentrated, that is, with the minimum uncertainty.

Figure 12.

Statistical distribution of frequencies of Mode 1: (a) Case C1; (b) Case C2; (c) Case C3; (d) Case C4; (e) Case C5; (f) Case C6; (g) Case C7; (h) Case F; (i) Case A.

Figure 13.

Statistical parameters for frequencies of Mode 1: (a) the frequency evolution index; (b) frequency with the highest probability.

In addition, the frequency with the highest probability in each statistical distribution diagram in Figure 12 was shown in Figure 13(b). Obviously, the frequency with the highest probability under ambient loads (Case A) was larger than those under train loads (Cases C1–C8) and free vibrations (Case F). This phenomenon is consistent with the frequency variations shown in Figure 8(a). The possible reason is that frequencies identified from train-induced and free vibration responses are related to the train-bridge interaction system while frequencies identified from ambient vibration responses are only related to the bridge system.

Conclusions

A fully automated modal identification framework was developed for tracking the modal evolution of a long-span high-speed railway bridge online under operational conditions, which is useful for bridge performance evaluation and early warning.

• A three-stage modal identification method was developed to identify modal parameters from the stochastic, deterministic-stochastic, and free vibrations of the bridge, in which the automatic separation of spurious and structural modes, adaptive modal matching, and efficient detection of free vibrations without supervision have been achieved.

• An automated switching technique based on the simultaneous information of strain peaks, strain increments and train speeds is proposed, which can be used to transform modal tracking methods for various vibration types in real time.

• The algorithm efficiency and parameter uncertainty were analyzed according to long-term modal tracking results for a long-span high-speed railway bridge. The highest computing efficiency and the lowest modal evolution were respectively implemented on stochastic and free vibrations. The results also show that the operating mode shapes become complex especially for the low signal-to-noise ratio or asynchronous responses.

Footnotes

Declaration of conflicting interests

The author(s) 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 research work was jointly supported by the National Natural Science Foundation of China (Grant Nos. 52108270, 51978128, 52078100), the National Postdoctoral Program for Innovative Talents (Grant No. BX2021052), the China Postdoctoral Science Foundation Funded Project (Grant No. 2021M700673), and the Fundamental Research Funds for the Central Universities (Grant No. DUT22ZD213).

ORCID iDs

Xiao-Mei Yang

Ting-Hua Yi

Chun-Xu Qu

References

Afshar

Khodaygan

(2019) Enhanced stabilization diagram for automated modal parameter identification based on power spectral density transmissibility functions. Structural Control and Health Monitoring 26(7): e2369.

Anastasopoulos

De Roeck

Reynders

EPB

(2021) One-year operational modal analysis of a steel bridge from high-resolution macrostrain monitoring: influence of temperature versus retrofitting. Mechanical Systems and Signal Processing 161: 107951.

Brincker

Zhang

Andersen

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

Cabboi

Magalhães

Gentile

, et al. (2017) Automated modal identification and tracking: application to an iron arch bridge. Structural Control and Health Monitoring 24(1): e1854.

Caicedo

Dyke

Johnson

(2004) Natural excitation technique and eigen system realization algorithm for Phase I of the IASC-ASCE benchmark problem: simulated data. Journal of Engineering Mechanics 130(1): 49–60.

Deepthi

Saravanan

Meher Prasad

(2019) Algorithms to determine wheel loads and speed of trains using strains measured on bridge girders. Structural Control and Health Monitoring 26(1): e2282.

Fan

Hao

(2019) Improved automated operational modal identification of structures based on clustering. Structural Control and Health Monitoring 26(12): e2450.

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.

Hua

Chen

, et al. (2011) EMD-based random decrement technique for modal parameter identification of an existing railway bridge. Engineering Structures 33(4): 1348–1356.

10.

Liang

Obrien

, et al. (2021) Continuous modal identification and tracking of a long-span suspension bridge using a robust mixed-clustering method. Journal of Bridge Engineering 27(3): 05022001.

11.

Yang

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

12.

Jin

Jeong

Sim

, et al. (2021) Fully automated peak-picking method for an autonomous stay-cable monitoring system in cable-stayed bridges. Automation in Construction 126: 103628.

13.

JTG/T J21-01-2015 (2016) Load Test Methods for Highway Bridge [In Chinese], Ministry of Transport of the People’s Republic of China. Beijing: China Communications Press.

14.

Kim

Sim

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

15.

Kim

Lee

(2010) Extracting modal parameters of high-speed railway bridge using the TDD technique. Mechanical Systems and Signal Processing 24(3): 707–720.

16.

Zhu

Law

, et al. (2020) Time-varying characteristics of bridges under the passage of vehicles using synchroextracting transform. Mechanical Systems and Signal Processing 140: 106727.

17.

Matsuoka

Tokunaga

Kaito

(2021) Bayesian estimation of instantaneous frequency reduction on cracked concrete railway bridges under high-speed train passage. Mechanical Systems and Signal Processing 161: 107944.

18.

Meixedo

Santos

Ribeiro

, et al. (2021) Damage detection in railway bridges using traffic-induced dynamic responses. Engineering Structures 238: 112189.

19.

Ozcelik

Yormaz

Amaddeo

, et al. (2019) System identification of a six-span steel railway bridge using ambient vibration measurements at different temperature conditions. Journal of Performance of Constructed Facilities 33(2): 04019001.

20.

Reynders

Roeck

(2008) Reference-based combined deterministic-stochastic subspace identification for experimental and operational modal analysis. Mechanical Systems and Signal Processing 22(3): 617–637.

21.

Ülker-Kaustell

Karoumi

(2011) Application of the continuous wavelet transform on the free vibrations of a steel-concrete composite railway bridge. Engineering Structures 33(3): 911–919.

22.

Thomas

Lafleur

, et al. (2013) Towards an automatic spectral and modal identification from operational modal analysis. Journal of Sound and Vibration 332(1): 213–227.

23.

Xia

(2011) Structural Health Monitoring of Long-Span Suspension Bridges. London: Spon Press.

24.

Yang

, et al. (2019) Automated eigen system realization algorithm for operational modal identification of bridge structures. Journal of Aerospace Engineering 32(2): 04018148.

25.

Yang

, et al. (2020a) Continuous tracking of bridge modal parameters based on subspace correlations. Structural Control and Health Monitoring 27(10): e2615.

26.

Yang

, et al. (2020b) Modal identification of high-speed railway bridges through free-vibration detection. Journal of Engineering Mechanics 146(9): 04020107.

27.

Yang

, et al. (2021) Performance assessment of a high-speed railway bridge through operational modal analysis. Journal of Performance of Constructed Facilities 35(6): 04021091.

28.

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.