Time-varying modal identification using multi-channel measurements based on multivariate variational mode decomposition

Abstract

Structural modal parameters are essential to provide dynamic information for data analysis of structural health monitoring (SHM) system. Because of the variation of the physical properties or the influence of the environment, the structural modal parameters may change with time during the operation, showing time-varying characteristics. Therefore, accurate identification of time-varying modal parameters shows to be an important issue for SHM. In this paper, a time-varying modal identification method is proposed by improving the multivariate variational mode decomposition (MVMD) method with autoregressive power spectrum and windowed principal component analysis (PCA). Firstly, the method for determination of initial center frequency is proposed by autoregressive power spectrum to improve the efficiency of MVMD. Secondly, intrinsic mode function for each mode is extracted using multi-channel responses by MVMD. Subsequently, instantaneous frequencies are identified through detecting the ridgeline of the synchro-squeezed short-time Fourier transform (SSTFT). Moreover, identification method for time-varying mode shapes is proposed by using the windowed PCA of the multi-channel intrinsic modes. Finally, the proposed method is verified by the numerical and practical studies. The results of the numerical study show that the method is effective for continuously varying modal parameters under impulse and random excitations. Through the data analysis of practical bridge, the capability for practical application is demonstrated.

Keywords

structural health monitoring structural health monitoring time-varying modal identification multivariate variational mode decomposition synchro-squeezed time-frequency transform windowed principal component analysis

Introduction

With the rapid development of civil engineering construction, the structures are more and more diverse and complex. Due to the changes of load and external environment, the structure will be aged, damaged, and even destroyed during operation. To avoid economic losses and personal safety threats caused by damage and ensure the normal operation of the structure, structural health monitoring systems are widely installed on large-scale structures (Zingoni, 2005). Modal identification is an important part in the field of structural health monitoring, because the identified modal parameters can reflect the vibration characteristics of structures and evaluate the structural performance (Civera et al., 2021; Chinka et al., 2021; Liu et al., 2022). Since it is difficult to apply artificial excitation directly to large-scale buildings and bridges, output-only modal identification method is widely concerned. A large number of modal identification methods based on the assumption of ambient excitation have been developed, such as random decrement technique and natural excitation technique (Asmussen et al., 1999; Chang and Pakzad, 2013). Other methods have also been proposed, such as peak-picking approach and frequency domain decomposition (FDD) method in frequency domain (Gade, 2006; Yao et al., 2021), as well as Ibrahim Time Domain algorithm (Mohanty and Rixen, 2004), and stochastic subspace identification (SSI) technique in time domain (Peeters and De Roeck, 2001), etc.

Dynamic properties of civil structures, including mass or stiffness, always vary with the environmental effect, such as stiffness reduction under extreme load conditions or due to environmental erosion, time-varying modal properties of bridge-vehicle systems when the vehicle moves or the train crosses. Structures generally show time-varying characteristics under working conditions, leading to time-varying modal parameters. For example, the changes of temperature and humidity lead to thermal expansion and contraction of materials, causing alterations in stiffness. Besides, wind, earthquake and other environmental loads can also cause the time-varying modal characteristics of large-scale structures during operation conditions (Wang et al., 2018, 2021).

In recent years, time-frequency methods are commonly used to identify the modal parameters of time-varying structures. For example, the instantaneous modal parameters of structures are identified using the continuous wavelet transform based on the short-time invariant assumption (Wang et al., 2018). Liu et al. (2019) combined the extended analytical mode decomposition with a recursive Hilbert transform and a zoom synchro-squeezed wavelet transform to identify the instantaneous frequencies in low-frequency structures. Aied et al. (2016) applied ensemble empirical mode decomposition to capture sudden changes in bridge model stiffness. Gang et al. (2019) proposed a new technology based on synchrosqueezing transformation (SST), which has strong anti-noise performance and helps for bearing fault diagnosis.

In addition, new time-frequency analysis methods have also been developed. Variational mode decomposition (VMD) is a non-recursive modal analysis method proposed by Dragomiretskiy and Zosso (2014). By solving the variational constraint problem, the bandwidths and the central frequency of each component are determined by solving the constrained optimization problem. Based on this, all modal components can be obtained. Ni et al. (2018) combined VMD with Hilbert-Huang transform (HHT) to identify the instantaneous frequency of time-varying steel cable. Zhang et al. (2018) has optimized the VMD parameters by using the grasshopper-optimization algorithm, and the method was verified by the cases of fault diagnosis for machinery. The above studies of VMD focused on the decomposition of single-channel signal (Hou et al., 2021). Synchronous decomposition of multi-channel signals for identification of mode shapes has not been studied up now.

Due to the multi-channel signals collected in the actual structure, people have great interest in extending existing methods to realize the processing of multi-channel signals, while VMD can only decompose single-channel signal. Therefore, Rehman and Aftab proposed the multivariate variational Mode decomposition (MVMD) in 2019 (Rehman and Aftab, 2019). Multivariate variational Mode decomposition, which inherits the advantages of VMD, extracts the multi-modulated oscillation set by solving the variational problem. So far, MVMD has been applied in fault diagnosis of mechanical structures (Gu et al., 2020), elimination of EEG and human eye artifacts (Gavas et al., 2020). However, the performance of MVMD for modal identification using multi-channel signals has not been studied.

This study proposes a time-varying modal parameter identification method using multi-channel signals by MVMD. Firstly, the initial center frequencies are determined by AR power spectrum to improve the accuracy of decomposition. Then, MVMD is used to decompose the responses into modal responses. Next, the instantaneous frequencies are identified by synchro-squeezed short-time Fourier transform (SSTFT). Besides, the method for identification of instantaneous time-varying mode shapes is proposed by using principal component analysis (PCA) under sliding time window. The effectiveness of the proposed method is verified by simulating a 4-DOF time-varying stiffness system under impulsive excitation and random excitation, and the practicability of the proposed method is verified by the data analysis of a simply supported steel truss bridge.

Multivariate variational mode decomposition

Multivariate variational mode decomposition is a completely non-recursive method for time-frequency analysis. By solving the constraints of the variational model, VMD decomposes the signal into a finite number of intrinsic modal functions (IMFs). However, it can only process the signal of each channel separately. Therefore, the obtained results do not consider the joint information among the channels. Multivariate variational mode decomposition extended VMD to decomposition and reconstruction at all data channels. Then, the multivariable modulation oscillations are obtained directly from the input data, so that the decomposition results preserve the mode-alignment property (Rehman and Aftab, 2019). The main purpose of MVMD is to decompose multivariate data into single-component modes.

Assuming that the signal contains K- modes, the signal with C- data channels can be decomposed as follows:

X (t) = [x_{1} (t), x_{2} (t), . . ., x_{C} (t)] = \sum_{k = 1}^{K} u_{k} (t)

(1)

where

u_{k} (t) = {[u_{1 k} (t), u_{2 k} (t), . . ., u_{C k} (t)]}^{T}

is the column vector composed of the single component modes of each channel at the k-th order mode.

To realize the decomposition of MVMD, the following hypotheses are established:

(1) The sum of the decomposed modal bandwidths is minimum.

(2) The sum of the extracted modes is restored to the original signal $u_{k} (t)$ .

Based on this, the variational constraint problem is established as shown in equation (2).

\begin{array}{l} \min_{{ω_{k, c}} {ω_{k}}} {\sum_{k = 1}^{K} \sum_{c = 1}^{C} {‖ \partial_{t} [u_{k, c}^{+} (t) e^{- j ω_{k} t}] ‖}_{2}^{2}} \\ s . t . \sum_{k = 1}^{K} u_{k, c} (t) = x_{c} (t), c = 1,2, . . ., C \end{array}

(2)

where

u_{k, c} (t)

is the single-component analytic signal of the k -th mode at the c-th channel.

u_{k, c} (t)

is combined with the Hilbert operator to obtain the analytical representation of each component vector

u_{k, c}^{+} (t)

u_{k, c}^{+} (t)

is multiplied by the exponential function

e^{- j ω_{k} t}

to tune the signal to the corresponding estimated center frequency. Then, the spectrum of the analytical signal is shifted to the baseband. Finally, the gradient squared parametrization is used to estimate the bandwidth of

u_{k, c}^{+} (t)

To solve the variational constraint problem, the Lagrange multiplier is adopted as follows.

\begin{array}{l} L ({u_{k, c}}, {ω_{k}}, λ_{c}) = α \sum_{k = 1}^{K} \sum_{c = 1}^{C} {‖ \partial_{t} [u_{k, c}^{+} (t) e^{- j ω_{k} t}] ‖}_{2}^{2} + \\ \sum_{c = 1}^{C} {‖ x_{c} (t) - \sum_{k} u_{k, c} (t) ‖}_{2}^{2} + \sum_{c = 1}^{C} 〈 λ_{c} (t), x_{c} (t) - \sum_{k} u_{k, c} (t) 〉 \end{array}

(3)

where λ_c is the Lagrange multiplier and α is the quadratic penalty term, both of which are used to reconstruct the fidelity. The constrained problem is solved using methods such as Wiener filtering and fast Fourier transform, then the alternating direction multiplier method is used for frequency-domain optimization to obtain the updated results for the intrinsic modes, the resulting center frequencies, and the Lagrange multipliers, alternatively. The updated parameters are deduced as follows:

{\hat{u}}_{k, c}^{p + 1} (ω) = \frac{{\hat{x}}_{c} (ω) - \sum_{i \neq k} {\hat{u}}_{i, c}^{p + 1} (ω) + \frac{{\hat{λ}}_{c}^{p} (ω)}{2}}{1 + 2 α {(ω - ω_{k}^{p})}^{2}}

(4)

ω_{k}^{p + 1} = \frac{\sum_{c} \int_{0}^{\infty} ω {| {\hat{u}}_{k, c}^{p + 1} (ω) |}^{2} d ω}{\sum_{c} \int_{0}^{\infty} {| {\hat{u}}_{k, c}^{p + 1} (ω) |}^{2} d ω}

(5)

{\hat{λ}}_{c}^{p + 1} (ω) = {\hat{λ}}_{c}^{p} (ω) + τ ({\hat{x}}_{c} (ω) - \sum_{k} {\hat{u}}_{k, c}^{p + 1} (ω))

(6)

where

{\hat{u}}_{k, c}^{p + 1} (ω)

is the Fourier transform of the p+1-th iteration of the mode, which will iterate until the convergence is satisfied.

\sum_{k = 1}^{K} \sum_{c = 1}^{C} \frac{{‖ {\hat{u}}_{k, c}^{p + 1} - {\hat{u}}_{k, c}^{p} ‖}_{2}^{2}}{{‖ {\hat{u}}_{k, c}^{p} ‖}_{2}^{2}} < ε

(7)

Time-varying structural modal parameters identification based on multivariate variational mode decomposition

According to the theory of structural dynamics, the vibration differential equation of the time-varying structure with n degrees of freedom can be written as:

M (t) X^{″} (t) + C (t) X^{'} (t) + K (t) X (t) = F (t)

(8)

where M(t), C(t) and K(t) are the mass, damping and stiffness matrices of the time-varying structure, respectively;

X (t)

is the displacement vector of the structure, and

F (t)

represents the excitation force acting on the structure.

According to the mode superposition principle, the acceleration of the structure can be expressed by the response mode superposition corresponding to the mode shapes:

X^{″} (t) = \sum_{i = 1}^{n} u_{i} (t) = \sum_{i = 1}^{n} φ_{i} (t) q_{i}^{″} (t)

(9)

where,

u_{i} (t)

is the acceleration response of the i-th order mode,

φ_{i} (t)

is the i-th order mode shape at time t,

q_{i}^{″} (t)

is its corresponding modal response. The above equations can be combined to obtain the n-th order modal equation.

q_{i}^{″} (t) + 2 ξ_{i} (t) ω_{i} (t) q_{i}^{'} (t) + ω_{i}^{2} (t) q_{i} (t) = φ_{i}^{T} (t) F (t) / m_{i}

(10)

where

ξ_{i} (t)

ω_{i} (t)

and

m_{i}

are the i-th order modal damping ratio, frequency and mass, respectively.

Applying an excitation to the z-th degree of freedom, the acceleration response of the i-th degree of freedom can be expressed as:

q_{i}^{″} (t) = \frac{F_{0} ϕ_{z i, t = 0} ω_{i} (t)}{m_{j} \sqrt{1 - ξ_{i}^{2} (t)}} e^{- ξ_{i} (t) ω_{i} (t)} \cos (ω_{d i} (t) t + v_{i})

(11)

where

ϕ_{z i, t = 0}

is the z-th element of the i-th order modal vibration vector

φ_{i} (t)

at t = 0, the i-th order instantaneous frequency

ω_{d i} (t) = ω_{i} (t) \sqrt{1 - ξ_{i}^{2} (t)}

, and

v_{i}

denotes the phase angle.

By comparing equations (1) and (9), it can be seen that the signal can be expressed as a series of superposition of single-component modes. Therefore, MVMD can be used to decompose the signal, and then time-frequency method can be used to obtain the time-varying modal parameters of the structure.

Selection initial center frequency based on AR power spectrum

Multivariate variational mode decomposition requires presetting the initial decomposition parameters, such as initial center frequency $ω_{k}^{1}$ and modal number K. Too many preset modes will lead to false modes, while too few number of modes will result in incomplete decomposition. The number of center frequencies is consistent with the number of modes, so the influence of the K value can be ignored as long as the preset center frequency is reasonable. In (Jiang et al., 2018; Yao et al., 2022), it is shown that the $ω_{k}^{1}$ which is close to true frequencies of IMFS can improve the decomposition efficiency. Similarly, determining the appropriate initial center frequency is a prerequisite for implementing MVMD.

AR power spectrum is a modern spectrum estimation method that calculates the power spectrum by using the AR coefficient of the established AR model. Compared with Fourier spectrum, the AR spectrum shows to have higher resolution and smoother spectral curve (Xin et al., 2019). AR power spectrum can characterize the energy fluctuation of unit frequency, so the initial center frequency of the signal can be determined by selecting the frequency value corresponding to each of its peak points. The autoregressive process of the output response of a linear system can be expressed as (Akaike, 1969):

X (n) = \sum_{k = 1}^{q} a_{q, k} X (n - k) + W (n)

(12)

where q is the order of the AR model,

a_{q, k}

is the AR coefficient, and

W (n)

is a Gaussian white noise sequence with a mean of zero and a variance of

σ_{k}^{2}

. The output power spectrum of the q-order AR model can be estimated as:

S (f) = \frac{σ_{k}^{2}}{{| 1 + \sum_{k = 1}^{q} a_{q, k} e^{- 2 π j k f} |}^{2}}

(13)

Parameters of AR model are solved by Yule-Walker equation:

[\begin{array}{l} R_{x} (0) & R_{x} (1) & R_{x} (2) & - & R_{x} (q) \\ R_{x} (1) & R_{x} (0) & R_{x} (1) & - & R_{x} (q - 1) \\ R_{x} (2) & R_{x} (1) & R_{x} (0) & - & R_{x} (q - 2) \\ | & | & | & \ & | \\ R_{x} (q) & R_{x} (q - 1) & R_{x} (q - 2) & - & R_{x} (0) \end{array}] [\begin{array}{c} 1 \\ a_{1} \\ a_{2} \\ | \\ a_{q} \end{array}] = [\begin{array}{c} {σ_{k}}^{2} \\ 0 \\ 0 \\ | \\ 0 \end{array}]

(14)

where R_x() denotes correlation function of the signal.

a_{k}

and

{σ_{k}}^{2}

can be obtained by Levinson-Durbin recursive algorithm (Akaike, 1969).

The determination of AR model order is the key to AR spectrum estimation. Error criterion is usually used to solve the order problem, including final prediction error (FPE) criterion, Akaike information theory and Criterion Autoregressive Transfer function. In this paper, FPE criterion is adopted as the basis for predicting the model order:

F P E (q) = σ_{k}^{2} \frac{N + q + 1}{N - q + 1}

(15)

where N is the number of sampling points, and q is the optimal order determined to minimize the right-hand side of the equation. After the AR coefficients are obtained, the smooth spectrum estimation of the signal can be carried out through equation (13). Considering all the output responses, averaged AR power spectrum is adopted to indicate the energies of the modes. The expression of the indication index is as follows:

\bar{S} (f) = \frac{1}{η} \sum_{i = 1}^{η} S_{i} (f)

(16)

where

S_{i} (f)

is the AR power spectrum of the i-th response;

η

is the dimension of the signal. The initial center frequencies can be estimated by selecting the peak frequencies of the AR power spectrum. The initial center frequency of the signal can be determined by picking the peak points of

{\bar{S}}_{Y}^{A R}

Instantaneous frequency identification by Synchro-squeezed STFT

The time-frequency analysis method can reflect the joint distribution information of the signal in the time-frequency domain, so it can analyze the time-varying signal effectively. After the time-varying signal is processed by MVMD, the resulting modal component needs further analysis to obtain the instantaneous frequency. Traditional short-time Fourier transform method is easy to implement, but there is problem of energy divergence in the time-frequency plane because of the constant size of time-frequency window and single resolution. As a post-processing method, synchro-squeezed transform redistributes the energy of the time-frequency plane in the frequency direction by extracting the instantaneous frequency information of the spectrum, and reconcentrates the energy of the signal time-spectrum near the instantaneous frequency to improve the frequency resolution (Sony and Sadhu, 2020). In this paper, the instantaneous frequencies of the signal are obtained by extracting the time-frequency ridges of SSTFT corresponding to each IMF. The theory of SSTFT is as follows:

Firstly, single harmonic signal is analyzed:

u (s) = A (s) e^{i φ (s)}

(17)

where

A (s)

is the instantaneous amplitude,

φ (s)

is the instantaneous phase. The first derivative of the instantaneous phase

φ^{'} (s)

is the instantaneous frequency.

At the t-th moment, $u (t)$ obtained by MVMD can be expressed as the harmonic form:

u (t) = u (s) |_{s = t} = A (t) e^{i φ (t)}

(18)

Assuming that

ε

is small enough for any time t to establish the correlation

| Α^{'} (t) | \leq ε

and

| φ^{″} (t) | \leq ε

, the corresponding Taylor expansion of

A (s) 、 φ (s)

at time t can be obtained respectively:

\begin{array}{l} A (s) = A (t) + o (s - t) \\ φ (s) = φ (t) + φ^{'} (t) (s - t) + o [{(s - t)}^{2}] \end{array}

(19)

After neglecting the higher order infinite minor items in the equation, the Taylor expansion of $u (t)$ can be written as:

u (t) = A (t) e^{i [φ (t) + φ^{'} (t) (s - t)]}

(20)

The STFT of $u (t)$ is expressed as:

\begin{array}{l} G (t, ω) = \int_{- \infty}^{+ \infty} g (s - t) A (t) e^{i [φ (t) + φ^{'} (t) (s - t)]} e^{- i ω (s - t)} d s \\ = A (t) e^{i φ (t)} \int_{- \infty}^{+ \infty} g (s - t) e^{- i (ω - φ^{'} (t)) (s - t)} d (s - t) \\ = A (t) e^{i φ (t)} \hat{g} (ω - φ^{'} (t)) \end{array}

(21)

where

\hat{g} ()

is the Fourier transform of the window function

g ()

, whose window length is

[- V_{ω}, V_{ω}]

According to the above formula, $G (t, ω)$ contains the time-frequency domain information. After short-time Fourier transform, the time-frequency spectrum $G (t, ω)$ contains the same instantaneous phase with the original signal. Taking the partial derivative of $G (t, ω)$ with respect to $t$ and ignoring the infinite minor terms, the expression by $φ^{'} (t)$ can be obtained as follows:

\begin{array}{l} \partial_{t} G (t, ω) = \partial_{t} [A (t) e^{i φ (t)} \hat{g} (ω - φ^{'} (t))] \\ = A (t) e^{i φ (t)} \hat{g} (ω - φ^{'} (t)) i φ^{'} (t) \\ = G (t, ω) i φ^{'} (t) \end{array}

(22)

When $G (t, ω) \neq 0$ , the instantaneous frequency of the time-varying signal is estimated in the time-frequency space, which can be written as:

\hat{ω} (t, ω) = \frac{\partial_{t} G (t, ω)}{i G (t, ω)}

(23)

Since the window function achieves the maximum value at the origin after Fourier transform, the highest point of $G (t, ω)$ in the frequency direction at any time $t$ is $G (t, φ^{'} (t))$ , and the ridgeline can be obtained by connecting the highest points at different times. The energy of $u (t)$ mainly concentrates near the ridgeline $φ^{'} (s)$ . However, since the time-frequency plane energy diverges along the ridge, which is not conducive to the identification of the instantaneous frequency, the time-frequency spectrum $G (t, ω)$ can be compressed along the frequency direction by using the selectivity of the Dirac function $δ$ . The expression is:

T_{s} (t, η) = \int_{- \infty}^{+ \infty} G (t, ω) δ (η - \hat{ω} (t, ω)) d ω

(24)

After the redistribution of $G (t, ω)$ , the resolution and time-frequency aggregation are enhanced. The ridge of the time-spectrum $T_{s} (t, η)$ is clearer and the energy it is more concentrated. By extracting the ridgeline, the accuracy estimation of instantaneous frequency can be obtained, which greatly improves the resolution of recognition results in the frequency direction.

Modal identification by windowed-PCA

Principal component analysis is a linear data dimension reduction method, which assumes that the original signal can be decomposed into unrelated components. According to equation (19), the i-th acceleration response $u_{i} (t)$ can be converted into the superposition of modal coordinates as $u_{i} (t) = φ_{i} (t) q_{i}^{″} (t)$ , so PCA can be performed to extract the main component of the modal response and the corresponding weighted coefficients (Wang et al., 2018).

Decentralize the IMF vector and transform it into normalized data with a mean of 0 and a variance of 1:

u_{i} (t) \leftarrow u_{i} (t) - E (u_{i} (t))

(25)

The following Eigen matrix V and eigenvalue matrix λ can be obtained by calculating the principal components:

R_{u_{i} (t)} = E (u_{i} (t) u_{i} {(t)}^{T}) = V λ V^{T}

(26)

where,

R_{u_{i} (t)}

is the covariance matrix; λ is the diagonal matrix composed of eigenvalues; V is the orthogonal matrix composed of eigenvectors, and each principal direction vector is the estimation of the mode shape.

Because the mode shapes of the time-varying system vary with time, sliding windows with a certain window length can be added to realize time-varying mode identification. The sliding window method is to select a short window length for the IMFS decomposed by MVMD, and the data window moves from left to right to ensure certain data in the window. Assuming that the data sequence length is L and the window length is l, then for the i-th order mode, L-i+1 instantaneous vibrational modes can be identified, and the data in each window can be decomposed as:

u_{L}^{j} = φ_{L}^{j} q_{L}^{″ j}, j = 1,2, . . ., L - i + 1

(27)

where,

φ_{L}^{j}

is the modal shape, and

q_{L}^{″ j}

is the modal coordinate. According to the time-invariant assumption in a short time window, the mode shapes can be considered time-invariant within the scope of window length (Wang et al., 2018). Therefore, the modal shape at each time in each window can be obtained through windowed PCA with sliding windows, and the instantaneous modal shape identification of time-varying structure can be realized through connecting the windows.

The window length can be determined by the speed of time-variation. If the modal parameters of the time-varying structure change fast, the window length should be reduced. The speed of time-variation is measured by:

γ_{l}^{i} = \frac{f_{l E}^{i} - f_{l B}^{i}}{f_{l E}^{i}}

(28)

where

γ_{l}^{i}

represents the frequency variation rate in the analyzed time window

l

for the i-th mode;

f_{l E}^{i}

and

f_{l B}^{i}

are the end-frequency and the begin-frequency of the i-th mode.

The steps of the proposed modal identification method for time-varying modal parameters are summarized as follows:

(1) Establishing the AR model of the response, and calculate the AR power spectrum by equations (12)–(16);

(2) Selecting the initial center frequency of each order of MVMD by picking the peak frequencies of the AR power spectrum;

(3) Decomposing the responses into IMFs by equations (1)–(7) at the selected initial center frequencies;

(4) After obtaining the IMFs, the SSTFT is used to extract the ridgeline and identify the instantaneous frequencies of each IMF through equations (17)–(24);

(5) Time-varying mode shapes are then identified by applying windowed PCA for the IMFs of each mode.

Numerical simulations

A 4-DOF shear building with time-varying stiffnesses is used to verify the effectiveness of the proposed method. The differential equation of the motion is described in equation (8). The mass and damping of each floor is $m_{1} = m_{2} = m_{3} = m_{4} = 4 kg$ and $c_{1} = c_{2} = c_{3} = c_{4} = 0.5 N / (m^{- 1} s)$ , respectively. k₁, k₂ is the time-varying inter-storey stiffnesses with $k_{1} = (1 - 0.04 t) \times 10^{4} N / m$ and $k_{2} = (1.2 - 0.03 t) \times 10^{4} N / m$ . Other inter-storey stiffnesses are $k_{3} = k_{4} = 1.2 \times 10^{4} N / m$ . Simulations including two conditions of impulse excitation and random excitation are investigated.

Numerical study on the 4-DOF time-varying structure under impulse excitation

Impulse excitation at the fourth layer of the structure is applied to obtain the accelerations using Newmark-β method with sampling frequency of 1000 Hz and a duration of 10 s. Figure 1 shows the AR power spectrum of the structural response. The four frequencies corresponding to the peaks are 2.8 Hz, 8.15 Hz, 12.3 Hz and 14.9 Hz. Therefore, the initial center frequencies for MVMD are determined. Subsequently, the responses are decomposed by MVMD to obtain the IMFs of each mode. The IMF is processed by SSTFT to obtain the instantaneous frequency of each mode. For purpose of comparison, STFT is also used to analysis the instantaneous frequency.

Figure 1.

AR power spectrum of the structure response under impulse excitation.

Figures 2 and 3 show the identified results by STFT and SSTFT, respectively. In the figure, the blue line indicates the theoretical value of the frequency derived from the eigenvalue analysis, and the red dotted line represents the identification values. It can be seen from the figures that the frequency components of each order are well separated after the MVMD decomposition. Both methods can recognize instantaneous frequencies. Compared with the results of STFT, the instantaneous frequency results of signals processed by SSTFT are more concentrated, and the identified results at both ends are more satisfactory. Hilbert-Huang transform is adopted as the comparative method to identify the instantaneous frequencies (Bahar and Ramezani, 2014). Figure 4 shows the identified results by HHT. From the figure, it can be seen that the accuracy of the identified frequencies is low for the fourth mode. Through the comparison, the proposed method provide more accurate results.

Figure 2.

Instantaneous frequencies obtained by STFT under impulse excitation.

Figure 3.

Instantaneous frequencies obtained by synchro-squeezed short-time Fourier transform under impulse excitation.

Figure 4.

Instantaneous frequencies obtained by HHT under impulse excitation.

The IMFs corresponding each mode are processed with windowed PCA to obtain the instantaneous mode shapes. The length of the sliding window is selected as 0.3 s. Therefore, instantaneous mode shapes can be identified in the segment of data. Modal assurance criterion (MAC) is used to measure the reliability of mode shape identification. The expression of MAC is:

MAC ({\hat{φ}}_{i}, φ_{i}) = \frac{{({\hat{φ}}_{i}^{T} φ_{i})}^{2}}{‖ {\hat{φ}}_{i}^{T} {\hat{φ}}_{i} ‖ ‖ {\hat{φ}}_{i}^{T} φ_{i} ‖}

where

{\hat{φ}}_{i}

and

φ_{i}

are the i-th order theoretical and recognized mode shapes, respectively. The MAC value represents the correlation between the mode shapes. If the MAC value is close to 1, the correlation between the referential and identified mode shape vector is high (Yao et al., 2021). That is to say, the accuracy of the identified mode shape is reliable.

Figure 5 shows the comparison diagram of the theoretical mode shapes and the identified ones at t = 2 s obtained by windowed PCA. It can be seen that the estimated mode shapes are in good agreement with the theoretical ones. The results of the instantaneous mode shapes obtained by windowed PCA for all time instants are shown in Figure 6. In this figure, the trend of the mode shapes which varies with time is displayed by lines with gradual changed color. The time-varying property of the mode shape can be seen visually. Figure 7 shows the MAC values between the identified and theoretical mode shapes at each time instant. It can be seen that the MAC values are basically close to 1. The results indicate the effectiveness of the method for time-varying mode shape identification.

Figure 5.

Comparison of instantaneous theoretical and identified mode shapes at t = 2 s under impulse excitation.

Figure 6.

Identification results of instantaneous mode shapes under impulse excitation.

Figure 7.

MAC values between instantaneous identified and theoretical mode shapes under impulse excitation.

Numerical study of the 4-DOF time-varying structure under random excitation

Random excitation is used to generate responses. The simulation and calculation of accelerations are the same as those of impulse excitation. The initial center frequencies of the structure are determined as 2.8 Hz, 8.8 Hz, 13.4 Hz and 16.6 Hz through the AR power spectrum. MVMD is used to decompose the acceleration responses into IMFS with the aid of the selected initial center frequencies. The instantaneous frequencies are identified by SSTFT. The comparison between the identification and the theoretical frequencies are shown in Figure 8. It can be seen that the IMF component decomposed by MVMD retains the frequency characteristics well, and the instantaneous frequencies identified by SSTFT corresponds with the theoretical frequencies well. Comparing Figures 3 and 8, it can be seen that the accuracy is slightly reduced under random excitation. Figure 9 shows the identified frequencies by HHT. It can be seen from Figure 9 that the accuracies of HHT for the third and fourth modes are low. The proposed method shows better results than HHT for impulse and random excitation cases.

Figure 8.

Instantaneous frequencies obtained by SSTFT under random excitation.

Figure 9.

Instantaneous frequencies obtained by HHT under random excitation.

The windowed PCA is used to process the IMFS and identify the instantaneous mode shapes. The comparison between the identified mode shape of each order and the theoretical one when t = 2 s is shown in Figure 10. Figure 11 shows the instantaneous modes at all-time instants. The identified modes are basically consistent with the theoretical ones. The MAC values between the identification results and the theoretical mode shapes are shown in Figure 12. The MAC curve fluctuates to some extent, indicating that the random excitation has a certain influence on the instantaneous mode shape identification. However, the general trend of the identification results matches with the theoretical mode shapes because the MAC values are all close to 1. Therefore, the performance for random excitation condition is demonstrated.

Figure 10.

Comparison of instantaneous theoretical and identified mode shapes at t = 2 s under random excitation.

Figure 11.

Identification results of instantaneous mode shapes under random excitation.

Figure 12.

MAC values between instantaneous identified and theoretical mode shapes under random excitation.

Study of a practical bridge

In this section, an actual simply supported steel truss bridge is selected to further verify the practicability of the proposed method. This bridge is located in Japan with a main span of 59.2 m, a mid-span of 8.2 m, and a deck width of 3.6 m (Chang and Kim, 2016). The bridge was approaching its designed service life in 2012, so artificial damage tests were carried out on the bridge before dismantlement. Artificial damages were employed to simulate the actual damage caused by corrosion or overload. The vertical members of the test bridge were artificially cut and welded for repair, and five working conditions were created. The first condition was the bridge in good situation. The second one was to cut the unilateral connect of the vertical component at the middle of the bridge. The third condition was to fully cut the member section of working condition 2. The fourth condition was recovering the structural component by welding the cut member. The fifth condition was a full section cut of the vertical truss member located at span 5/8 (Chang and Kim, 2016). The location of the vertical components for the simulated damages are marked by blue, as shown in Figure 13.

Figure 13.

Layout of the sensors on the steel truss bridge.

Eight accelerometers were installed on both sides of the bridge to collect structural response signals. Five of the accelerometers were installed on the artificial damage side (A1–A5) and the remaining 3 on the other side (A6–A8). The sampling frequency of the sensors is 200 Hz and the duration is 50 s. The layout of the bridge structure and sensors is shown in Figure 13. The tested data under ambient excitations for the five conditions are analyzed. The time histories of the locations from A1 to A5 and their corresponding fast Fourier transform (FFT) spectra are displayed in Figure 14.

Figure 14.

Time histories of locations A1 to A5 and their corresponding fast Fourier transform spectra.

The sampling data of five working conditions are sequentially connected to simulate the time-varying characteristics of the structure. The data duration is 250 s in total with the sampling frequency of 200 Hz (Kim et al. 2021). The AR power spectrum of acceleration response is shown in Figure 15. Seven peak frequencies are identified as 2.97 Hz, 6.23 Hz, 6.63 Hz, 9.68 Hz, 10.48 Hz, 13.27 Hz, and 19.27 Hz, which are taken as the initial center frequencies. Subsequently, MVMD is used to decompose the responses into IMFs. Synchro-squeezed short-time Fourier transform and windowed PCA are then applied to obtain instantaneous frequencies and mode shapes, respectively. Figure 16 shows the identified instantaneous frequencies. Obvious changes of the frequencies are shown for the five damage conditions.

Figure 15.

AR power spectrum for the acceleration response of the bridge.

Figure 16.

Instantaneous frequencies of the bridge obtained by SSTFT.

For the instantaneous mode shapes identification, the IMFs obtained from the decomposition is processed by windowed PCA. The time-varying mode shapes are displayed in Figure 17 using lines with gradual changed colors. The results are highly consistent with the identified results given in (Chang and Kim, 2016), but the time-varying property is shown by the proposed method. Besides, the weak-energy modes are also identified.

Figure 17.

Identified mode shapes of the bridge.

For comparative purposes, FDD and SSI are adopted. Because these methods are used for invariant modal identification, the data of undamaged case are considered. The instantaneous frequencies and mode shapes are averaged to compare with FDD and SSI. The results are shown in Table 1. It can be seen from the table that the frequencies of the three methods are close and the MAC values of the mode shapes are high. In conclusion, the capability of the proposed method for practical applications are demonstrated.

Table 1.

Modal parameters of the practical bridge.

Parameters	Method	Mode
Parameters	Method	1	2	3	4	5	6	7
Frequency (Hz)	Proposed	2.98	6.20	6.78	9.64	10.56	13.38	19.41
	FDD	2.97	6.12	6.85	9.68	10.48	13.36	19.44
	SSI	2.97	6.22	6.85	9.65	10.48	13.37	19.42
MAC	FDD	0.99	0.99	0.96	0.99	0.99	0.94	0.98
MAC	SSI	0.99	0.99	0.98	0.99	0.99	0.94	0.97

Conclusions

In this paper, MVMD is introduced into the field of civil engineering, and a new modal parameter identification method for time-varying structures is proposed by combining AR power spectrum, SSTFT, and windowed PCA. The validity and practicability of the proposed method are verified by a simulated 4-DOF time-varying structure under impulse and random excitations and an actual bridge.

• The peak frequencies of the AR power spectrum can be selected as the initial center frequencies of MVMD, and the decomposition order can be determined incidentally, which improves the accuracy and efficiency of MVMD to convergent to the real frequencies quickly.

• Synchro-squeezed short-time Fourier transform uses the selectivity of Dirac function to compress the time-frequency spectrum along the frequency direction to make the ridge clearer and improve the accuracy of instantaneous frequency identification. The results of numerical examples and the actual structure show that the instant frequencies identified by SSTFT match the referential values well.

• Windowed PCA is performed to calculate the principal components of the IMFs by a sliding window for identification of time-varying mode shapes. The results of the 4-DOF time-varying structure simulation under impulse and random excitations show that the MAC values of the instantaneous mode shapes identified by the proposed method are close to 1, which indicates that performing windowed PCA on MVMD results can accurately and effectively identify the mode shapes of time-varying structures. Moreover, the performance of windowed PCA on application of actual bridge structure is verified.

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. 51908183, 52108270), the Natural Science Foundation of Hebei Province (Grant No. E2020202056), and the Foundation for High Level Talent Innovation Support Program of Dalian (Grant No. 2022RQ013).

ORCID iDs

Xiao-Jun Yao

Xiao-Mei Yang

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