Vibration feature extraction based on the improved variational mode decomposition and singular spectrum analysis combination algorithm

Abstract

Extraction of the vibration characteristics of a flood discharge structure under the influence of intensive background noise is one of the main challenges in vibration-based damage identification. A novel algorithm called normalized central frequency difference spectrum is proposed to improve the variational mode decomposition algorithm for high-frequency noise filtering. To eliminate the errors caused by end effect, the waveform matching extension algorithm is used to further improve the variational mode decomposition. However, the vibration signal is still coupled in low-frequency noise. Thereupon, the singular spectrum analysis algorithm is applied to filter the low-frequency noise. In this article, a simulated signal and the measured signals from a dam model are analyzed by the proposed algorithm. The results indicate that the proposed algorithm is robust to noise and has high denoising precision. In addition, this algorithm can offer clues for damage identification and localization of a flood discharge structure.

Keywords

end effect flood discharge structure normalized central frequency difference spectrum singular spectrum analysis variational mode decomposition

Introduction

Due to the influence of external environment and human factors, the flow-induced vibration response of a discharge structure is easily disturbed by various noises. Noise affects the accuracy of vibration signal extraction and structural damage identification. Therefore, it is necessary to find an appropriate algorithm to reduce noise. A variety of algorithms have been applied to resolve the problem. Among these algorithms, the fractal geometry algorithm combined with modern nonlinear theories is popular to solve the chaotic phenomenon of signals, but it cannot be used to reduce the noise in flow-induced signals (Marvasti and Strahle, 1995). The higher order spectral analysis algorithm can quantitate nonlinear phase coupling relationship, and it is suitable to process non-Gaussian signals. However, it cannot be adopted to filter the flow-induced white Gaussian noise accurately (Liang et al., 2013). The modern spectrum analysis based on autoregressive moving average (ARMA) was focused on the prediction and estimation of signals, and thus it is not applicable to noise reduction (Sasikumar et al., 2009). Although the blind source separation (BSS) algorithm can recover the original signal from mixed signals, it is also difficult to implement and unable to meet the requirements of practical engineering applications (Zhang et al., 2007). Without pre-setting any basis function, the empirical mode decomposition (EMD) algorithm can be used to decompose the signal into series of intrinsic mode functions (IMFs), and it also has advantages in dealing with non-stationary and nonlinear signals. However, this algorithm has mode mixing problem, and it cannot completely extract the flow-induced vibration response from the low signal-to-noise ratio (SNR) and complex flow-induced signals (Boudraa and Cexus, 2007). For the ensemble empirical mode decomposition (EEMD) algorithm, the white Gaussian noise with uniform distribution is added to original signals, which makes the original signals have a uniformly distributed decomposition scale. Moreover, the EEMD can smooth abnormal signals such as pulse and discontinuous signals, and it can resolve mode mixing problems effectively. However, the added white Gaussian noise destroys the purity of the original signals (Lei et al., 2011). The basic idea of the variational mode decomposition (VMD) is to solve the squared L₂ norm of the gradient of demodulated signal variational problems. An iterative method is employed to search the optimal solution of the variational problem to realize signal decomposition. The VMD differs from the EMD and EEMD, and the variation is that there is no residual noise in the decomposed modes and the mode mixing phenomenon has been greatly alleviated, which is extremely helpful to seismic data processing and interpretation (Chen, 2016; Chen and Fomel, 2017). In addition, the VMD has a solid theoretical foundation. In essence, it is an extended improvement of the classic Wiener filter into multiple and adaptive bands (Dragomiretskiy and Zosso, 2014; Liu et al., 2017). By decomposing a large number of signals collected from the flow-induced vibration experiment as described in section “Experimental study,” it demonstrates that the VMD can extract the low-frequency signal which contains structural vibration information completely. Although it has potential in dealing with flow-induced signals, the VMD still has a drawback that the mode number needs to be set artificially before decomposition. The mode number is difficult to estimate when a signal is a complex nonlinear and non-stationary one. If the mode number is small, multiple components of the signal may exist in one mode simultaneously. Otherwise, one component will exist in multiple modes (Yang et al., 2017). Therefore, the VMD needs to be improved to make it capable of decomposing vibration signals adaptively.

In this article, a novel method based on the normalized central frequency difference spectrum (NCFDS) is proposed to improve the VMD, and the optimal mode number can be determined automatically. Furthermore, to alleviate the impact of end effect during the component decomposition process with the VMD, it is necessary to extend the signal to make the end effect occur at the extended parts of the signal. In this article, the VMD is further improved using the waveform matching extension method to eliminate end effect.

With the improved VMD, a flow-induced signal can be decomposed into a series of IMFs adaptively. However, an appropriate criterion should be set to select useful components for noise reduction. Naudascher (2005) found that the flow-induced vibration possesses shock characteristics, and the kurtosis value of signals can reflect the shock characteristics (Aijun et al., 2012). Accordingly, the kurtosis criterion is applied to select the best IMF. The experimental results in section “Experimental study” show that after the flow-induced signals are decomposed by the improved VMD, low-frequency IMFs with the large kurtosis values can be obtained, which means that they contain most of the shock components. Therefore, the low-frequency IMFs are selected to reconstruct signal. Consequently, the white noise with high frequencies is filtered.

Usually, the low-frequency IMFs are complex. The effective vibration signal of a structure is coupled with simple harmonic waves, whose frequency varies from 0.5 to 50 Hz with the interval of 0.5 Hz. Besides, the simple harmonic waves vary with the operation conditions of the structure, and the coupling relationships between the simple harmonic waves and the structural vibration signal are complex, which makes low-frequency IMFs have a dense-frequency feature. However, the improved VMD can only be used to decompose the signal in a complete space, and it cannot separate the weak vibration signal from such a complex signal with dense-frequency noise. Therefore, the low-frequency structural vibration signal and low-frequency noise cannot be effectively separated by the improved VMD. To overcome the above-mentioned shortcomings, the singular spectrum analysis (SSA) algorithm is applied to further filter the low-frequency noise. The SSA has been widely used in seismic data for useful signal extraction, and it has better performance and lower computational cost for reconstruction and random noise attenuation (Huang et al., 2017; Zhang et al., 2017; Zhou et al., 2018). It can also describe the essential difference of the singular values between useful components and noise components, and a signal can be decomposed into an uncorrelated useful subspace and a noise subspace in time domain. Therefore, the low-frequency noise can be filtered by the SSA.

The rest of the article is organized as follows. In section “Methodology,” the characteristics of a flow-induced signal are analyzed, and the combination algorithm, which can remove high-frequency noise by the improved VMD and filter low-frequency noise by the SSA, is introduced. In section “Numerical example,” a simulated signal is constructed, and the improved VMD-SSA, VMD-SSA, EMD, and EEMD algorithms are used to process the signal, respectively, to verify the accuracy of the improved VMD-SSA. In section “Experimental study,” the improved VMD-SSA is used to process the experimental data from a dam model under different operation conditions. Finally, the conclusion is presented in section “Conclusion.”

Methodology

Characteristics of a flow-induced signal

The time-domain distribution of a flow-induced signal is shown in Figure 1. It shows that it is difficult to identify the characteristics from amplitude distribution of the signal in time domain.

Figure 1.

Time-domain distribution of the measured signal.

The power spectral density (PSD) function is often used to describe the characteristics of a vibration signal, which can reflect the distribution density of signal power in frequency domain (Kamkar-Parsi and Bouchard, 2009). Figure 2 shows the PSD of the flow-induced signal. There are five protruding peaks which are regularly distributed in the band of 0–500 Hz (as shown in Figure 2(a)). It represents the fundamental wave of alternating current (AC) and its odd harmonics. The PSDs of the fundamental wave and the strongest odd harmonic are 1520 and 450 mV²Hz⁻¹, respectively, which indicate that their energy is large, and it seriously affects the accuracy of signal processing. Therefore, they should be filtered first. Since the AC interference is regular and its frequency band is known, the adaptive filtering algorithm (Bernardi et al., 2017), which can filter the regular noise effectively, is used to eliminate this part of noise.

Figure 2.

PSD distribution of the measured signal.

To reveal the details of the low-frequency band, as shown in Figure 2(b), the band of 0–30 Hz is magnified. It shows that there exist low-frequency simple harmonic waves with relatively regular energy distribution in this band. The frequency interval of these waves is 0.5 Hz, which possesses the characteristics of dense frequencies. The flow-induced vibration of a hydraulic structure has low-frequency characteristics (Li, 2013). The useful low-frequency vibration signal of the structure is submerged in the dense low-frequency simple harmonic waves. Therefore, a suitable algorithm should be developed to extract the useful vibration signal. In this article, the improved VMD-SSA is adopted to denoise the signal.

High-frequency noise filtering based on the improved VMD

Improved VMD

First, the waveform matching extension algorithm is introduced to restrain the end effect. Assume that the original signal is ff, and its time-domain waveform is shown in Figure 3.

Figure 3.

Time-domain waveform of ff.

In Figure 3, S₁ is the left endpoint of ff, and the corresponding time t is 0. $M_{i} (i = 1, 2, 3, \dots)$ are the maximum points, and its corresponding time is $t m_{i}$ . $N_{i} (i = 1, 2, 3, \dots)$ are the minimum points, and its corresponding time is $t n_{i}$ . As shown in Figure 3, the three points S₁, M₁, and N₁ at the left end are used to form a triangle feature waveform $Δ S_{1} M_{1} N_{1}$ . Then, the remaining data are searched to find the waveform $Δ S_{i} M_{i} N_{i}$ which is matched with $Δ S_{1} M_{1} N_{1}$ . Finally, the left part before $Δ S_{i} M_{i} N_{i}$ is copied and connected to the left end of the original signal. In doing so, the extended signal can reflect the trend of the original waveform at the left end. Similarly, the right end can also be extended. The fully extended signal is denoted as f. After the above extension operations, the VMD is used to decompose the extended signal. The principle of the VMD is briefly introduced as follows.

Set initial mode number $K = 2$ , penalty factor $α = 2000$ , and fidelity coefficient $τ = 0$ . For the extended signal f, in the VMD algorithm, f is assumed to be composed of a given number of modes $u_{k} (t)$ . These modes are called the IMFs. The bandwidth of each IMF is estimated, and the constrained variational problem can be defined as follows (Dragomiretskiy and Zosso, 2014)

{\begin{matrix} min_{{u_{k}}, {ω_{k}}} {\sum_{k = 1}^{K} ‖ \partial_{t} [(δ (t) + \frac{j}{π t}) * u_{k} (t)] e^{- j ω_{k} t} ‖_{2}^{2}} \\ s . t . \sum_{k = 1}^{K} u_{k} (t) = f \end{matrix}

(1)

where $‖ \cdot ‖_{2}^{2}$ is the L₂ norm; $δ (t)$ denotes the unit pulse function; j, *, and $\partial_{t}$ are the imaginary unit, convolution, and partial derivative of time t, respectively; ${u_{k} (t)} = {u_{1} (t), \dots, u_{k} (t)}$ and ${ω_{k}} = {ω_{1}, \dots, ω_{k}}$ are the shorthand notions for the set of all modes and their central frequencies, respectively; and f is the extended signal.

To solve the constrained variational problem, the augmented Lagrange is introduced and the non-constrained variational problem is obtained

\begin{matrix} L ({u_{k}}, {ω_{k}}, λ) \\ = α \sum_{k = 1}^{K} {‖ \partial_{t} [(δ (t) + \frac{j}{π t}) * u_{k} (t)] e^{- j ω_{k} t} ‖}_{2}^{2} \\ + ‖ f (t) - \sum_{k = 1}^{K} u_{k} (t) ‖_{2}^{2} + 〈 λ (t), f (t) - \sum_{k = 1}^{K} u_{k} (t) 〉 \end{matrix}

(2)

where $α$ denotes the quadratic penalty factor, $λ$ is the Lagrange multiplier, and <,> is the inner product. The saddle point of equation (2) is the optimal solution of the original problem, which can be obtained using the alternating direction method of multipliers (ADMM; Boyd et al., 2011). All the modes can be obtained from equation (3) in the frequency domain by updating each mode $u_{k}$ and its central frequency $ω_{k}$ constantly

u_{k}^{n + 1} (ω) = \frac{f (ω) - \sum_{i = 1, i < k}^{K} u_{i}^{n + 1} (ω) - \sum_{i = 1, i > k}^{K} u_{i}^{n} (ω) + \frac{λ^{n} (ω)}{2}}{1 + 2 α {(ω - ω_{k}^{n})}^{2}}

(3)

The new central frequency $ω_{k}^{n + 1}$ is placed at the center of gravity of the corresponding power spectrum of mode, which can be updated by

ω_{k}^{n + 1} = \frac{\int_{0}^{\infty} ω {| u_{k}^{n + 1} (ω) |}^{2} d ω}{\int_{0}^{\infty} u_{k}^{n + 1} {(ω)}^{2} d ω} .

(4)

From the above decomposition process, the value of K must be determined a priori. Although various algorithms are proposed to choose the right value for K (Wu, 2016; Yi et al., 2016; Zhu et al., 2017), they suffer from different drawbacks. In this article, an algorithm based on NCFDS, which can optimize K automatically, is presented. The normalized central frequency $ω_{k}$ is arranged in ascending order as a vector $Q = {ω_{1}, ω_{2}, \dots, ω_{k}}$ ; the normalized central frequency difference $a_{i} = ω_{i + 1} - ω_{i} (i = 1, 2, \dots, k - 1)$ is calculated; and then, the vector $A = {a_{1}, a_{2}, \dots, a_{k - 1}}$ is formed. If $min (a_{i})$ is bigger than a small threshold $ε$ , it shows that mode mixing occurs and the signal can be further decomposed. Setting $K = K + 1$ , the signal is decomposed again until $min (a_{i})$ is less than or equal to the threshold $ε$ . In doing so, the best mode number K can be determined adaptively as $K = K - 1$ . Finally, the extended signal is decomposed into K components ${u_{k} (ω)}$ . The inverse Fourier transform is applied to ${u_{k} (ω)}$ , and the real parts of the solutions are the signals decomposed by the improved VMD, namely, ${u_{k} (t)}$ . After discarding the extended parts, the remaining parts are the decomposed components of the original signal ff. The steps of the improved VMD are summarized as follows:

Step 1. Extend the original signal to restrain end effect;

Step 2. Initialize $K = 2$ ;

Step 3. Initialize ${u_{k}^{1}}, {ω_{k}^{1}}, {λ^{1}}$ , and n;

Step 4. Update $u_{k}$ and $ω_{k}$ using equations (3) and (4);

Step 5. Update $λ$

λ^{n + 1} (ω) \leftarrow λ^{n} (ω) + π [f (ω) - \sum_{k} u_{k}^{n + 1} (ω)]

(5)

Step 6. Repeat steps 4 and 5 until

\sum_{k} \frac{‖ u_{k}^{n + 1} - u_{k}^{n} ‖}{‖ u_{k}^{n} ‖_{2}^{2}} < ε

(6)

Step 7. Arrange $ω_{k}$ the ascending order as a vector

Q = {ω_{1}, ω_{2}, \dots, ω_{k}}

(7)

Define $a_{i} = ω_{i + 1} - ω_{i} (i = 1, 2, \dots, k - 1)$ , and then form the vector $A = {a_{1}, a_{2}, \dots, a_{k - 1}}$ . Check $min (a_{i})$ : if $min (a_{i}) > ε$ , set $K = K + 1$ and go to step 3; otherwise, set $K = K - 1$ and go to step 8.

Step 8. Repeat steps 3–6 and the signal is adaptively decomposed into ${u_{k} (t)}$ ;

Step 9. Remove the extended parts and obtain the decomposed components of the original signal finally.

Performance evaluation of the improved VMD

To evaluate the performance of the improved VMD, a simulated signal $x (t) = \sin (40 π t) + \sin (80 π t) + \sin (120 π t)$ is constructed. The signal is decomposed by the improved VMD and VMD algorithms, respectively:

The improved VMD is used to determine the mode number K, and the result is $K = 3$ . Figure 4 shows the normalized central frequency distribution of the components decomposed by the VMD under different K. For $K = 4 - 10$ , plateaus appear in the curves, which means the normalized central frequencies of the adjacent components are close and mode mixing takes place. From Figure 4, the mode number K should be 3. It is just the result of the improved VMD. Hence, the accuracy of the improved VMD in determining the mode number K based on the NCFDS is verified.

To evaluate the effectiveness of the improved VMD in restraining end effect, one of the IMF components (e.g. the components at 40 Hz) is taken as an example. The waveform of this component in time domain is shown in Figure 5. It shows clearly that the end effect exists at both ends of the component decomposed by the VMD. However, no such phenomenon exists in the component decomposed by the improved VMD.

Figure 4.

Normalized central frequency distribution of components decomposed by the VMD under different K.

Figure 5.

Time-domain waveforms of the components at 40 Hz decomposed by (a) VMD and (b) improved VMD.

Criterion for high-frequency noise filtering

When the flow-induced signal is decomposed by the improved VMD, a corresponding criterion should be established to select the useful IMFs. As mentioned in section “Introduction,” the vibration induced by discharge possesses shock characteristics. The kurtosis is sensitive to the shock components, which can be used to reflect the content of shock components. The kurtosis is defined as

k_{u r} = \frac{1}{N} {\sum_{i = 1}^{N} (\frac{x_{i} - \bar{x}}{σ_{t}})}^{4}

(8)

where $x_{i}$ is the value of the signal, $\bar{x}$ is the mean of the signal, and $σ_{t}$ is the standard deviation. The experimental results in section “Experimental study” show that the obtained low-frequency IMFs have the large kurtosis, which shows that the signal contains shock characteristics. The low-frequency IMFs are selected for signal reconstruction. Hence, the high-frequency noise is filtered.

Low-frequency noise filtering based on the SSA

Theory of SSA

Let the time series of a signal denoised by the improved VMD be ${h_{i}, i = 1, 2, \dots, N}$ , and then it is used to calculate a Hankel matrix with the expression

H = [\begin{matrix} h_{1} h_{2} \dots h_{K} \\ h_{2} h_{3} \dots h_{K + 1} \\ ⋮ ⋮ ⋮ ⋮ \\ h_{L} h_{L + 1} \dots h_{N} \end{matrix}]

(9)

where N is the number of sampling points and L represents the window length parameter, $1 < L < N$ ; and K is defined as $N - L + 1$ . The delay value is 1. The matrix H is referred to the trajectory matrix. The resulting trajectory matrix is then decomposed by means of singular value decomposition (SVD) and H can be rewritten as (Muruganatham et al., 2013)

H = \sum_{i = 1}^{d} H_{i} with H_{i} = \sqrt{λ_{i}} U_{i} {V_{i}}^{T}

(10)

where $d = rank (H)$ and it is the number of eigen components or modes with non-zero eigenvalue; $λ_{i}$ are the singular values sorted in descending order; and $U_{i}$ and $V_{i}$ are the associated left and right singular vectors, respectively. The group denoted by ${\sqrt{λ_{i}}, U_{i}, V_{i}}$ is called the ith eigentriple. Then, the elementary matrices $H_{i} (i = 1, 2, 3, \dots, d)$ are split into signal and noise groups. However, the important step is to determine a subset of eigentriple that encompasses the dominant variation in H. This amounts to approximating matrix $\tilde{H}$ by the summation of the first r elementary matrices, using the following equation: $\tilde{H} = \sum_{i = 1}^{r} U_{i} \sqrt{λ_{i}} {V_{i}}^{T}$ , where $\tilde{H}$ is attributed to signal and r denotes the number of eigentriples selected for signal reconstruction, and the residual $R = \sum_{i = r + 1}^{d} U_{i} \sqrt{λ_{i}} {V_{i}}^{T}$ is taken as noise (Muruganatham et al., 2013).

Signal reconstruction based on singular value difference spectrum

Selection of the eigentriple number r is the kernel of SSA-based signal extraction. If r is small, the part of the useful signal will be lost; otherwise, noise will be introduced. In this article, the singular value difference spectrum is applied to select r. When the signal is decomposed, the singular values are arranged in descending order and a matrix $S = {\sqrt{λ_{1}}, \sqrt{λ_{2}}, \dots, \sqrt{λ_{d}}}$ is formed. Let $b_{i} = \sqrt{λ_{i}} - \sqrt{λ_{i + 1}}, (i = 1, 2, \dots, d - 1)$ , and the vector $B = {b_{1}, b_{2}, \dots, b_{d - 1}}$ , called the singular value difference spectrum, is formed to describe the variation of every adjacent singular values. Zhao and Ye (2011) found that the sharp peak $max (b_{i}) = b_{r}, (i = 1, 2, \dots, d - 1)$ can represent the boundary between signal and noise, that is, the front r components can be selected for signal reconstruction.

The flow chart of the improved VMD-SSA is shown in Figure 6.

Figure 6.

Flow chart of the improved VMD-SSA.

Numerical example

When a discharge structure is subjected to shock loads of water flow, the vibration of the structure is an impulse vibration (Li, 2013). To evaluate the advantage of the improved VMD-SSA, the simulated impulse vibration signal $y (t)$ and the noisy signal $yn (t)$ are constructed

y (t) = 5 e^{- \frac{t * π}{2}} \sin (15 t) + 2 e^{- \frac{t}{3}} \sin (20 t)

(11)

yn (t) = y (t) + 0.5 randn (m) + y 0

(12)

where the sampling frequency fs is 100 Hz; the observed waveform signal time window is 100 s, and * is used for multiplication; $randn (m)$ means the white noise of standard normal distribution with the mean and standard deviation of 0 and 1, respectively; m represents the number of samples; $y_{0}$ is the white noise with the frequency bands of 0.25–0.75, 1.25–1.75, 2.25–2.75, 3.25–3.75, and 4.25–4.75 Hz to simulate low-frequency simple harmonic waves. The time domain waveform and the PSD of the original and noisy signals are shown in Figure 7.

Figure 7.

(a) Time-domain waveform and (b) power spectrum density of the original and noisy signals.

As shown in Figure 7(a), the original signal is submerged by noise. Figure 7(b) shows that the noisy signal contains low-frequency and high-frequency noise. The signal $yn (t)$ is denoised by the improved VMD-SSA, VMD-SSA, EMD, and EEMD algorithms, respectively. The original and denoised signals are shown in Figure 8.

Figure 8.

Time-domain waveform of the original and denoised signals processed by the four algorithms, respectively: (a) improved VMD-SSA, (b) VMD-SSA, (c) EMD, and (d) EEMD.

As shown in Figure 8(a) and (b), the time-domain waveforms of the denoised signals by the improved VMD-SSA and VMD-SSA algorithms are similar to the original signal in amplitude distribution and waveform feature. Comparing the ends of the denoised signals shows that the signal denoised by the improved VMD-SSA is more similar to the original one at both ends, and the signal denoised by the VMD-SSA has certain attenuation at both ends. Thus, the improved VMD-SSA can reduce the errors caused by end effect, and it has a higher denoising accuracy. Figure 8(c) and (d) shows that the signals denoised by the EMD and EEMD differ a lot from the original signal in terms of amplitude distribution and waveform feature, which indicates that the denoising accuracy of the EMD and EEMD is low.

To evaluate the denoising effect quantitatively, the root mean square error (RMSE) is used as an evaluation indicator as follows

RMSE = \sqrt{\frac{1}{N} {\sum_{n = 1}^{N} [f (n) - \bar{f} (n)]}^{2}}

(13)

where $f (n)$ and $\bar{f} (n)$ are the original and denoised signals, respectively. Smaller RMSE means better denoising performance. A total of 20 groups of random noise with different intensities are added to the original signal $y (t)$ , respectively, and the SNR of the noisy signal is 0.5–10 dB with the interval of 0.5 dB. The above four algorithms are used to process the 20 groups of signals, and then, the RMSEs of the denoised signals are calculated. Figure 9 indicates that the RMSE of the signal denoised by the improved VMD-SSA is the smallest, thus accuracy of the improved VMD-SSA is the highest.

Figure 9.

RMSE of the denoised signals.

Experimental study

Experimental details

The Xin-ji Dam located in Da-xi River, Gansu Province, China, is taken as an example. The height of the dam is 39.5 m, and an experimental model for the flood discharge dam blocks with a scale of 1:60 is built as shown in Figure 10. The model is a thin-walled structure with the thickness of 13 cm.

Figure 10.

Experimental model.

Experimental setup

In the experiment, the dSPACE system is used for signal acquisition. The piezoceramic patches of type P5-3B are selected to fabricate piezoceramic sensors. In addition, the digital oscilloscope is used to display and analyse the information of waveforms.

Experimental process

The piezoceramic sensors are arranged in the dam blocks on the left and right sides of the overflow outlet. The layout is shown in Figure 11.

Figure 11.

Cracks of concrete dam.

Five sensors (P1–P5) are arranged on the right side of the dam block, and the other five sensors (P6–P10) are arranged on the left side. The experimental process is as follows: (1) Holes are drilled with an electro drill, and the piezoceramic sensors are put into the boreholes. The direction of polarization is along the flow, and then the gypsum is used to backfill the boreholes. Sensors P1–P5 are connected to the dSPACE interface from 1 to 5, respectively. (2) When the dam block is intact, the water gate is fully opened and signals from all the sensors are collected. (3) Cracks F1, F2, and F3 of 4 cm deep each is cut with the cutting machine. After each crack is cut, the water gate is fully opened and signals from all the sensors are collected. The openings of all the cracks are 1 cm. The locations of the cracks F1–F3 are shown in Figure 11.

Experimental results and analysis

Since P3 is located at the highest elevation and closest to the overflow outlet, it has large vibration response. The signals of P3 under four conditions are selected to demonstrate the improved VMD-SSA. First, the improved VMD is used to decompose each signal into four IMFs, and the dominant frequencies are shown in Table 1. From Table 1, no mode mixing is observed, so the improved VMD can separate the flow-induced vibration response completely.

Table 1.

Dominant frequencies of each IMF after improved VMD decomposition (in Hz).

	C1	C2	C3	C4
IMF1	0–80	0–100	0–90	0–100
IMF2	1800–2200	1500–1800	1500–1800	1800–2300
IMF3	2800–3200	1900–2600	2000–2300	2400–2800
IMF4	4500–5000	2800–3200	2500–3800	2900–3400

Under the fluctuation pressure of water flow, the flow-induced vibration response of the discharge structure has low-frequency characteristics while the dominant frequencies of IMF2–IMF4 are all above 1500 Hz. Hence, there is no useful vibration signal in the components of IMF2–IMF4. From Table 2, the kurtosis values of each IMF1 are the biggest, and the ones of IMF2–IMF4 are much smaller than IMF1. It also shows that useful components are concentrated in IMF1. From the result of dominant frequency distribution and the kurtosis values, the component of IMF1 should be selected for signal reconstruction.

Table 2.

Kurtosis values of each IMF decomposed by improved VMD.

	C1	C2	C3	C4
IMF1	30.6652	20.7880	16.1894	37.2124
IMF2	7.2115	3.1916	6.5443	10.0917
IMF3	8.1480	3.1800	8.2605	7.9813
IMF4	9.0196	3.2603	8.9822	4.9819

The improved VMD filtering–based denoised signal of P3 is processed with the phase space reconstruction, and the trajectory matrixes are constructed, then the trajectory matrixes are decomposed by SVD. The first 20th singular values and their difference spectrum are shown in Figure 12. According to the singular difference spectrum theory, the first 2 components are selected for signal reconstruction. As a result, the low-frequency simple harmonic waves are filtered.

Figure 12.

(a) Singular value and (b) singular value difference spectrum of P3 for all the conditions.

The vibration signal waveforms of P3 in time domain are shown in Figure 13. Comparing Figure 13(a) with Figure 1, the characteristics of the flow-induced signal in amplitude distribution are revealed by the improved VMD-SSA.

Figure 13.

Time-domain waveforms of P3 under all the operation conditions: (a) C1, (b) C2, (c) C3, and (d) C4.

The amplitude of a vibration signal reflects the vibration degree of a structure under the fluctuation pressure of the water. It has a strong ability to characterize the vibration state of the structure. As shown in Figure 13, under the same discharge excitation and at the same measuring point, as damage aggravated, the extreme amplitude of vibration signal is reduced from 0.5 to 0.15 mV. This phenomenon can be attributed to the introduction of new interfaces and discontinuities by cracks, which may lead to energy attenuation of stress wave during propagation. The dominant frequencies can reflect the flow-induced vibration characteristics of a structure in frequency domain. Therefore, the PSDs of vibration signals of P3 in frequency domain are calculated, and the results are shown in Figure 14.

Figure14.

Power spectrum density of P3 under all the operation conditions: (a) C1, (b) C2, (c) C3, and (d) C4.

From Figure 14, dominant frequencies of the dam model are distributed in the range of 1.7–4.6 Hz. Under the different conditions at the same measuring point, the dominant frequencies and energy distribution of the vibration are different. As damage increases, the stiffness of the structure decreases, thus the dominant frequency of vibration is reduced from 4.6 to1.7 Hz. Due to the existence of cracks, the PSD of the dominant frequencies decreases from 1.25 to 0.03 mV²Hz⁻¹.

To study the relationships of vibration signal at different positions, the extracted signals of P1, P2, P4, and P5 are used to calculate the cross-correlation coefficients between P1 and P5. The results are shown in Tables 3 to 6. Under the intact condition C1, the cross-correlation coefficients between the adjacent measured points are all large and close to 1. It shows that the vibration of each measuring point of the dam model almost synchronises and the integrity of the dam is good.

Table 3.

Cross-correlation coefficients between P1 and P5 under C1.

C1	P1	P2	P3	P4	P5
P1	1	0.9997	0.9986	0.9953	0.9988
P2	0.9997	1	0.9995	0.9969	0.9988
P3	0.9986	0.9995	1	0.9976	0.9983
P4	0.9953	0.9969	0.9976	1	0.9941
P5	0.9988	0.9988	0.9983	0.9941	1

Table 4.

Cross-correlation coefficients between P1 and P5 under C2.

C2	P1	P2	P3	P4	P5
P1	1	0.7821	−0.0021	−0.0037	−0.0040
P2	0.7821	1	0.0117	−0.0169	−0.0046
P3	−0.0021	0.0117	1	0.0041	0.7773
P4	−0.0037	−0.0169	0.0041	1	0.0021
P5	−0.0040	−0.0046	0.7773	0.0021	1

Table 5.

Cross-correlation coefficients between P1 and P5 under C3.

C3	P1	P2	P3	P4	P5
P1	1	0.7825	0.0019	−0.0042	−0.0010
P2	0.7825	1	0.0103	−0.0482	−0.0083
P3	0.0019	0.0103	1	0.0029	0.7199
P4	−0.0042	−0.0482	0.0029	1	0.0006
P5	−0.0010	−0.0083	0.7199	0.0006	1

Table 6.

Cross-correlation coefficients between P1 and P5 under C4.

C4	P1	P2	P3	P4	P5
P1	1	0.7799	−0.0038	0.0009	0.0128
P2	0.7799	1	0.0107	−0.0459	−0.0069
P3	−0.0038	0.0107	1	0.0008	0.6987
P4	0.0009	−0.0459	0.0008	1	0.0011
P5	0.0128	−0.0069	0.6987	0.0011	1

Comparing C2 with C1 shows that the cross-correlation coefficients between P1 and P2, P2 and P3, P2 and P4, P3 and P5, P1 and P4, and P4 and P5 are reduced by 0.2716, 0.9878, 1.0138, 0.221, 0.999, and 0.992, respectively. It can be concluded that damage must be located in the trapezoidal region formed by P1, P2, P3, P4 and P5. Since the cross-correlation coefficient decreases the amplitude between P2 and P4 is the biggest, the area between P2 and P4 should be the most seriously damaged. Actually, between P2 and P4, there exists a crack F1.

Comparing C3 with C2 shows that the cross-correlation coefficients between P1 and P2, P1 and P4, and P2 and P4 are reduced by 0.0004, 0.0005, and 0.0007, respectively, as shown in Table 5. The range of the decreased amplitude is [0.0004, 0.0007], which is small. Therefore, the damage should not be in the triangular region formed by P1, P2, and P4. The cross-correlation coefficients between P2 and P3, P3 and P5, P4 and P5, and P2 and P4 are reduced by 0.0014, 0.0574, 0.0015, and 0.0313, respectively. The range of the decreased amplitude is [0.0014, 0.0574], and the magnitude of decrease is relatively large. Thus, damage should be located in the rectangular region formed by P2, P3, P4 and P5. The change in cross-correlation coefficient between P3 and P5 is the biggest, which shows that the damage exists between P3 and P5. In fact, a crack F2 exists between P3 and P5.

Under C4, compared with C3, the cross-correlation coefficients between P1 and P2, and P3 and P5 are reduced by 0.0026 and 0.0212, respectively, as shown in Table 6. Conversely, the cross-correlation coefficients between P2 and P3, P4 and P5, and P1 and P4 are increased by 0.0004, 0.0005, and 0.0051, respectively. Hence, the regular pattern of the cross-correlation coefficients is not obvious, and no damage should be located in the area of the sensors. In fact, crack F3 is outside the area formed by P1, P2, P3, P4 and P5. Therefore, when the damage is outside the region of the sensors, damage localization cannot be achieved by analyzing the variation of the cross-correlation coefficients.

From the above-mentioned results, the structural damage degree can be judged by the time- and frequency-domain characteristics from the extracted signal. When the damage exists in the region among the sensors, it can be localized by analyzing the variation of the cross-correlation coefficients. In the future, we will employ some other parameters for damage localization when no nearby sensors are used in the damage-prone area.

Conclusion

In this article, the VMD algorithm is improved and the improved VMD based on the NCFDS is proposed. To eliminate the errors caused by end effect, the waveform matching extension algorithm is used to further improve the VMD to filter high-frequency noise. Since the principal component of the structural vibration is still coupled in low- and dense-frequency noise, the SSA algorithm is applied to filter low-frequency noise from the denoised signal by the improved VMD. The numerical example shows a clear advantage of the improved VMD-SSA over the VMD-SSA, EMD, and EEMD algorithms. The improved VMD-SSA is also used to process the signals of a dam model experiment. The experimental results indicate that the improved VMD-SSA can offer a condition for damage identification and localization of a flood discharge structure.

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 was partially funded by the National Natural Science Foundation of China (Grant Nos. 51579086 and 51739003), the National Key R&D Program of China (2018YFC1508603), and the Priority Academic Program Development of Jiangsu Higher Education Institutions (Grant No. YS11001).

ORCID iD

Hui Li

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