Damage detection of thin plates by fusing variational mode decomposition and spectral entropy

Abstract

This paper presents a new approach for damage detection in thin plates by fusing variational mode decomposition and spectral entropy (VMD-SE). In this method, after the received signal is decomposed into some intrinsic mode functions (IMFs) by variational mode decomposition (VMD), the spectral entropy ratio of the first and last IMFs is calculated for optimizing the VMD’s parameters and improving its decomposition performance. Moreover, the cross-correlation coefficient between the decomposed IMFs and the reference signal is computed to separate the desired IMF, which contains more damage information. Finally, the spectral entropy of the obtained IMF is calculated as an indicator for assessing the damage’s severity. The comparative analysis of the simulated signal clearly shows that only the proposed method can successfully separate the damage-related and reference signals. To verify the VMD-SE method, damage detection of two different types of damage on aluminum and composite fiber-reinforced polymer (CFRP) plates is conducted by using this new approach. The experimental results demonstrate that the parameters of VMD affect greatly its decomposition performance, and the best parameters are selected. The results also indicate that the normalized spectral entropy monotonically increases when the diameter of the through-hole or the length of the scratch increases. In addition, the correlation coefficients of the fitting lines of the plates are larger than 0.998. The experimental results of aluminum specimens demonstrate that the damage’s location has an influence on the normalized spectral entropy. At last, based on the linear relationship, the severity of damage in the fourth specimen is identified. The identification results demonstrate that the relative error of the aluminum and CFRP plates is less than 7.34%, which indicates that this new algorithm by fusing VMD and spectral entropy can detect the damage size in thin plates accurately and efficiently.

Keywords

Variational mode decomposition spectral entropy thin plate structures damage detection

Introduction

Recently, thin plate-like structures have been widely applied in a vast array of fields, such as aerospace, pipelines, and some other industrial productions.^1,2 Under their working conditions, various types of damage, including deformation,^3,4 cracks,^5,6 corrosion,^7,8 etc., usually occur in these structures. Thus, the structural health monitoring (SHM) of thin plate-like structures is of significant importance, providing an effective means to ensure structural safety and normal operation.^9,10

Benefiting from their advantages of a long propagation distance with a low energy dissipation,^11,12 Lamb waves have promoted the rapid development of SHM, and they have played an increasingly substantial role in early-stage detection of damage in thin plate-like structures.¹³ However, Lamb waves experience some complex and subtle changes such as mode conversion, energy attenuation, and scattering effect when encountering different types of structures, including the structural boundary, external loading, and structural damage.^14,15 As a result, this slight variation usually introduces a bottleneck in damage-related signal or feature extraction. Therefore, how to extract the damage-related component and feature from the complex Lamb waves more effectively has sparked enormous attention.¹⁶

Fortunately, the advancement of signal processing technologies, for instance, frequency-domain analysis, time–frequency analysis,¹⁷ wavelet transform,¹⁸ and empirical mode decomposition (EMD)¹⁹, opens up new prospects for damage-related component extraction. Li et al.²⁰ proposed a structural damage identification algorithm using EMD and wavelet analysis. Lu et al.⁹ employed EMD to obtain the component related to the structural damage. Xu et al.¹⁹ extracted impact damage information by using EMD. Considering that there is usually mode mixing in EMD, Dragomiretskiy and Zosso²¹ developed variational mode decomposition (VMD) based on EMD to overcome these drawbacks. VMD preserves the advantages of EMD, while it exhibits good adaptability to mode mixing and end effects. Moreover, numerous studies²² have demonstrated the strong applicability of VMD in desired signal extraction and separation. Chen et al.²³ utilized the intrinsic mode functions (IMFs) of VMD as a feature vector for fault diagnosis. Wang et al.²⁴ decomposed the vibration signal of pipe into various IMFs by VMD and used these IMFs for multisource information fusion to identify the leakage of water pipes. In addition, some investigators just selected a certain IMF of VMD as a feature for different applications. Li et al.²⁵ utilized the correlation energy fluctuation to screen the desired IMF, which contains more fault information. Li et al.²⁶ used the kurtosis of the signal as a criterion to select the IMF containing rich information on the fault. Liu et al.²⁷ adopted a feature selection algorithm to find out the useful IMF by considering its energy entropy and central frequency. In a word, these successful applications of VMD provide an innovative way of damage signal separation of Lamb waves. Moreover, the performance of VMD is greatly influenced by some parameters, including the decomposition number of IMFs and the penalty factors.^28,29 To address such issues, Xu et al.³⁰ introduced an algorithm based on mixed entropy to capture the dynamic changes in the complexity of IMF components, thereby obtaining the optimal parameter set for VMD. Jiang et al.³¹ obtained the optimal parameters by screening the changes in central frequencies in IMFs. However, there will be some new challenges in the best IMF selection of VMD in Lamb waves of thin plates, which needs further investigation.

On the other side, how to use the extracted signal for damage severity detection is also of importance, and it also needs further investigation. Recently, researchers have developed various algorithms for damage severity or size quantification, such as damage index,^32–35 image-based method,^36–38 and machine-learning-based damage size classification.^39,40 He et al.⁴¹ quantified the size of inclined cracks on an aluminum plate by using an amplitude-phase coupling index, and the simulation and experimental results validated their new method. To overcome the drawbacks of physical and virtual time-reversal technologies, Wang et al.⁴² proposed a physical–virtual time-reversal method for fatigue crack quantification by using Lamb waves, and this new method obtained a remarkable detection performance. Zeng et al.⁴³ combined a damage contour method and singular loci removal scheme to quantitatively estimate the damage size for composite fiber-reinforced polymer (CFRP) plates, and this method was successfully applied for circle- and rectangle-like damage detection.

Currently, as a quantifiable metric that describes the degree of disorder or uncertainty in a system or dataset, entropy, including sample entropy,⁴⁴ relative entropy,⁴⁵ and cross entropy,⁴⁶ has been widely utilized for characterizing the severity of damage. Castro et al.⁴⁷ introduced spectral entropy as an indicator to measure the influence of the weight and location of a mass on the waves, and they attempted to identify the mass-typed damage on a CFRP plate by directly using spectral entropy. Guan et al.⁴⁸ employed relative entropy to characterize the length of fatigue crack. Wang et al.³ used multiscale cross-sample entropy analysis to estimate the simulated damage’s severity in plates. In addition, based on the theory of nonlinear ultrasonics, the existence of the damage usually brings about nonlinearity in the structure and introduces higher order frequency components in the received signal.⁴⁹ Some studies demonstrate that the nonlinear change in the frequency domain is highly sensitive to small-size damage.^50–52 Therefore, to better reflect the change due to the structural damage, the spectral entropy, as one of the entropies that are sensitive to the change of the signal in the frequency domain, is introduced to characterize the change of the signal and to establish a relationship between the entropy and the severity of the damage.

Therefore, in this study, an innovative approach by fusing VMD and spectral entropy (VMD-SE) is proposed to settle the aforementioned issues. In this new avenue, some new techniques are employed to option the best component, which includes the large information on the damage with the help of VMD, and the spectral entropy of the separated component is used to quantitatively characterize the damage severity.

The upcoming tasks are planned as follows. Section Theoretical basis offers a theoretical overview of the proposed VMD-SE method. Section Analysis of the simulated signal demonstrates the efficiency and advantages of the proposed method by a comparative analysis. Section Experimental configurations outlines the experimental setup for identifying simulated holes in aluminum plates and scratches on CFRP plates using the VMD-SE method. Section Evaluation of the experimental results presents the experimental results. Section Conclusion provides a summary and outlook for the paper.

Theoretical basis

In this part, a detailed explanation of the theoretical foundation and the implementation process of the proposed VMD-SE algorithm are provided.

The flowchart of the proposed VMD-SE algorithm is displayed in Figure 1. As illustrated in Figure 1, firstly, the reference and damage signals x₀(t) and x(t) are collected by using the piezoelectric active sensing method. Secondly, VMD is performed on the damage signal to obtain some different components. Thirdly, the cross-correlation coefficient between the decomposed components and the reference signal is computed to find out the component, which is highly relevant to the structural damage. At last, the spectral entropy of the extracted component is calculated to estimate the size of the damage.

VMD of the damage signal

VMD is used to decompose the input signal into several discrete IMFs while maintaining the sparsity of the IMFs. This process allows for the decomposition and selection of the original signal, thereby obtaining information about damage.

After the decomposition, each IMF has an independent central frequency $ω_{k}$ , and it possesses orthogonality. Theoretically, the decomposed components satisfy

\begin{matrix} x (t) = \sum_{k = 1}^{N} u_{k} + residue \end{matrix}

(1)

where $u_{k} (t) = A_{k} (t) \cos (φ_{k} (t))$ is the kth IMF, $A_{k} (t)$ is the instantaneous amplitude of the kth IMF, φ_k(t) is the phase, and residue is the residual term. Moreover, the instantaneous frequency of the kth IMF is $ω_{k} (t) = \frac{d φ (t)}{d t}$ .

This decomposition process can be achieved by obtaining a minimum value:

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

(2)

where ${u_{k}} = {u_{1},, u_{N}}$ is the decomposed N IMFs, ${ω_{k}} = {ω_{1}, ω_{2},, ω_{N}}$ is the central frequencies of each IMF, δ(t) represents the impulse function, $\partial_{t}$ is the time derivative, and j represents the square root of −1. It should be noted that IMFs here represent signals with specific frequency bands.

It should be noted that the optimal solution of Equation (2) is usually obtained by the method of Lagrange multipliers,²¹ that is,

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

(3)

where α represents the penalty parameter, and λ(t) represents the Lagrange multiplication operator. As a result, with the help of the method of Lagrange multipliers, the optimization problem of Equation (1) is transferred to search the saddle point of Equation (3). Moreover, all the IMFs u_k(t) are directly obtained by implementing some suboptimizations, and the details of the implementation are referred to Dragomiretskiy and Zosso.²¹

Moreover, during the optimization of VMD decomposition, improper parameters, including the total number of IMFs N and the penalty parameter α, usually lead to an over- or underdecomposition. For an excellent decomposition of VMD, each IMF should only contain a specified narrow frequency band without overlap. Since each IMF has its independent central frequency, the spectral entropy of each IMF is quite different. In addition, a low mode usually contains a signal with a low-frequency band.^21,31 If the ratio of the spectral entropy of the first and last IMFs is much larger, their frequency bands are more different, and the frequency band of the last IMF is much higher than that of the first one. As a result, the whole frequency band of IMFs is wider, their frequency components are more dispersed, and the mode mixing effect is much weaker. If the ratio of the spectral entropy between the first and last IMFs is larger, the frequency components of the signal are more dispersed and diverse, and as a result, there is no mode mixing, and the damage signal can be separated effectively. Therefore, to obtain a good decomposition and to extract the damage signal, the best parameters N and α are determined by making sure that the ratio of the spectral entropy between the first and last IMFs gains a maximum value, that is,

\begin{matrix} I_{SEmax} = max (I_{SE}) = max (\frac{H (u_{1})}{H (u_{N})}) \end{matrix}

(4)

where H(u₁) and H(u_N) are the spectral entropy of the decomposed first and last IMFs, respectively. It should be noted that the calculation of the spectral entropy will be introduced in section SE-based damage index.

Criterion of the best IMF selection

Since VMD decomposes the original signal into a series of components with different frequency centers adaptively, therefore the decomposed IMFs of the reference signal x₀(t) and the damage signal x(t) are different, and their frequency centers of the corresponding IMFs are also not the same. Moreover, compared to the reference signal x₀(t), some new components in the damage signal x(t) are brought about by the existing damage. Therefore, to efficiently separate the best IMF that contains more information on the damage, the Pearson’s correlation coefficient R_k between the reference signal x₀(t) and the kth IMF of the damage signal x(t) is computed to determine the best IMF, and it is expressed as:

\begin{matrix} R_{k} = \frac{\sum_{i = 1} (x_{0 i} - {\bar{x}}_{0}) (u_{ki} - {\bar{u}}_{k})}{\sqrt{\sum_{i = 1} {(x_{0 i} - {\bar{x}}_{0})}^{2} \sum_{i = 1} {(u_{ki} - {\bar{u}}_{k})}^{2}}} \end{matrix}

(5)

where $x_{0 i}$ represents the ith data of the reference signal, and $u_{ki}$ represents the ith data of the corresponding kth component of the damage signal. ${\bar{x}}_{0}$ represents the mean of the reference signal, and ${\bar{u}}_{k}$ represents the mean of the corresponding kth component of the damage signal.

In Equation (5), a larger value of $R_{k}$ indicates that the kth IMF is more similar to the reference signal, and there is less information on damage in the kth IMF. Conversely, a smaller value of $R_{k}$ indicates that the correlation R_k between the kth IMF and the reference signal is low, and this IMF has more information on the damage. Therefore, the best IMF that includes more information on the damage has the lowest value of R_k.

SE-based damage index

After the best IMF u_b is determined, its spectral entropy $H_{SE} (u_{b})$ is evaluated by statistically analyzing the data in the frequency domain to assess the severity of the damage

\begin{matrix} H_{SE} (u_{b}) = - \sum [P (u_{b}) \log (P (u_{b}))] \end{matrix}

(6)

where $P (u_{b})$ represents the power spectral density of the best IMF u_b.

Finally, the obtained spectral entropy $H_{SE} (u_{b})$ is normalized by

\begin{matrix} H_{NSE} = \frac{H_{SE} (u_{b})}{H_{SE} (x_{0})} \end{matrix}

(7)

where $H_{NSE}$ represents the normalized spectral entropy, and $H_{SE} (x_{0})$ represents the spectral entropy of the reference signal. It should be noted that the normalized spectral entropy is used as an indicator for damage size quantification. It should be also noted that the whole algorithm is implemented in MATLAB 2016b with an open code of VMD.⁵³

Analysis of the simulated signal

To clearly illustrate the efficiency and advantages of the proposed method, a simulated signal is analyzed in this section.

Simulated signal

When Lamb waves reflect at the boundary of the structural damage, their amplitude is attenuated and the frequency distribution is changed.^50,54,55 Therefore, a simulated damage signal x(t) is synthesized, which is made up of three components x₀(t), u_b(t), and n(t). Among these signals, x₀(t) is the simulated reference signal, u_b(t) is the damage-related signal, and n(t) is white noise. Figure 2 shows the simulated signal. As shown in Figure 2, the simulated reference and damage-related signals are 10-cycle Hanning-modulated sinusoidal signals, their center frequencies are 280 and 290 kHz, respectively, and their amplitudes are 1 and 0.25, respectively. Moreover, since their propagation distance is different, there is a time-off between the two signals, and the time-off Δt is 0.0002 s.

\begin{matrix} x (t) = x_{0} (t) + u_{b} (t) + n (t) \end{matrix}

(8)

\begin{matrix} x_{0} (t) = Hann (t) \sin (2 π f_{c 1} t + s_{1} π) \end{matrix}

(9)

\begin{matrix} u_{b} (t) = 0.25 Hann (t - Δ t) \sin (2 π f_{c 2} (t - Δ t) + s_{2} π) \end{matrix}

(10)

where Hann(t) represents the Hanning window function, s₁ = 0.67, s₂ = 1.07, f_c₁ = 280 kHz, f_c₂ = 290 kHz, and Δt = 0.0002 s.

Performance comparison

To demonstrate the efficiency and advantages of the proposed method, EMD, wavelet packet analysis, and VMD are employed to extract the damage-related signal u_b(t) from the simulated signal x(t), and their performance is compared.

Figures 3 to 5 are the decomposed components or IMFs of VMD, wavelet packet analysis, and EMD. Tables 1 to 3 list the Pearson’s correlation coefficient R_k between the decomposed components and the reference and damage-related signals. It should be noted that the optimal parameters of VMD are obtained as section Theoretical basis described in advance.

Table 1.

The Pearson’s correlation coefficients R_k between different IMFs of VMD and x₀(t) and u_b(t).

Coefficient R_k	x ₀(t)	u_b(t)
IMF1	0.46151	0.57277
IMF2	0.96562	0.01637

IMF: intrinsic mode function; VMD: variational mode decomposition.

Table 2.

The Pearson’s correlation coefficients R_k between different components of the wavelet packet analysis and x₀(t) and u_b(t).

Coefficient R_k	x ₀(t)	u_b(t)
Node (9, 29) (Frequency band: 273.44–283.20 kHz)	0.1055	0.0362
Node (9, 30) (Frequency band: 283.20–292.97 kHz)	0.0474	−0.0005

Table 3.

The Pearson’s correlation coefficients R_k between different nodes of the EMD and x₀(t) and u_b(t).

Coefficient R_k	x ₀(t)	u_b(t)
IMF1	0.96878	0.15138
IMF2	0.06274	0.07315
IMF3	0.00963	0.01233
IMF4	0.00006	−0.00103
IMF5	−0.00011	−0.00011
IMF6	−0.00016	−0.00002
IMF7	0.00014	0.00006
IMF8	−0.00005	−0.00005
IMF9	−0.00003	−0.00007

EMD: empirical mode decomposition; IMF: intrinsic mode function.

Figure 3 and Table 1 clearly demonstrate that the Pearson’s correlation coefficient R_k between IMF1 and x₀(t) is 0.46, and the one between IMF1 and u_b(t) is 0.57. Moreover, the coefficient R_k between IMF2 and u_b(t) is only 0.016, while the one between IMF2 and x₀(t) is 0.97. Therefore, with the help of the proposed method, the damage-related and reference signals are successfully separated from the simulated signal.

In addition, to separate the two signals, the damage signal is decomposed into nine levels by wavelet packet analysis. Since the center frequencies of the reference and damage-related signals are within the nodes (9, 29) (frequency band: 273.44 –283.20 kHz) and (9, 30) (frequency band: 283.20–292.97 kHz) after the decomposition of wavelet packet analysis, only the components of these two nodes are plotted, and their correlation coefficient R_k is computed. Figure 4 and Table 2 show that the coefficients R_k of the nodes are very small, which indicates that after the decomposition of wavelet packet analysis, the damage-related signal is not separated.

As shown in Figure 5, EMD obtains nine IMFs, and their coefficients are also computed. Similarly, it can be observed from Figure 5 and Table 3 that R_k between IMF1 and x₀(t) is 0.97, and the one between IMF1 and u_b(t) is 0.15. Moreover, the coefficients of the left IMFs are less than 0.073, which demonstrates that the reference and damage-related signals are mixed in IMF1, and EMD cannot separate these two signals.

Therefore, the comparative results of VMD, wavelet packet analysis, and EMD indicate that due to its adaptive decomposition ability, the proposed method can extract the damage-related signal from the simulated signal efficiently.

Experimental configurations

To validate the performance of the proposed VMD-SE approach for damage detection of plates, some experiments are carried out on aluminum and CFRP plates to simulate real-world scenarios involving two types of damage, including the simulated crack and scratch. The experimental device primarily consists of the following components: a thin plate with a thickness of 1.5 mm (aluminum 6061 or CFRP T300), an NI PXle system with an NI PXIe-5423 signal generator and an NI PXIe-5172 oscilloscope, a piezoelectric amplifier of Trek 2100H, and a display device.

As depicted in Figure 6(a) and (b), the structural dimensions of the aluminum and CFRP plates are 450 × 200 × 1.5 mm and 400 × 250 × 1.5 mm, respectively. Considering the high piezoelectric effect, excellent stability and reliability, as well as the high energy density of PZT ceramics, two PZT transducers are fixed onto the plates by epoxy resin to achieve excitation and reception of Lamb waves. To guarantee the reliability and accuracy of the process, it is essential to ensure the random selection of the coordinates for the excitation and reception of the PZT transducers before conducting the experiments, and the location of the PZTs is displayed in Figure 6. Similarly, the positions of the simulated damage is also selected randomly, and it is shown in Figure 6.

Moreover, the aluminum plate has a simulated through-hole crack created by drilling with a tungsten steel drill bit. During the experimental process, the diameter of the through-hole is enlarged with an increment of 0.1 mm, ranging from 0.3 to 0.8 mm. Meanwhile, as shown in Figure 7, the scratch on the CFRP plate is formed by the help of a sliding trimmer cutting ruler with two clamps. The length of the scratch is extended incrementally from 0 to 8 mm with an interval of 2 mm, and its depth is kept the same as 0.2 mm.

Before the experiments, to obtain a larger response of the structures by PZT transducers, a frequency-swept signal is firstly excited in the plates, and the frequency response of the received signal is shown in Figure 8. As shown in Figure 8, when the frequency is about 280 kHz, the frequency response is relatively larger, and therefore, the best center frequency is selected as 280 kHz. During the experiment, in order to minimize the modal interference in the thin plate, an arbitrary waveform generator of National Instruments (NI) is used to generate a 5-cycle Hanning-modulated sinusoidal pulse; its center frequency and voltage are 280 kHz and 1.2 V, respectively. The sinusoidal pulse waveform is shown in Figure 9. The amplified signal is then applied to a PZT patch to produce waves on the thin plate. With the help of the piezoelectric effect, the other PZT patch is used to receive the Lamb wave signals. Finally, the time-domain waveforms are acquired and recorded by using an NI oscilloscope. The sampling frequency is set at 2 MHz, and a total of 10,000 data points are collected.

In the experiment, seven identical aluminum specimens and four CFRP specimens are prepared. For the aluminum specimens, the location of the through-hole of specimens 1–4 is (243, 105 mm), and the one of specimens 5–7 is (210, 128 mm). The location of the scratch of CFRP specimens 1–4 is (219, 115 mm). Among the aluminum specimens, specimens 1–3 and 5–7 are used to investigate the influence of the location of damage on the normalized spectral entropy. Moreover, three specimens of the aluminum and CFRP specimens 1–4 are employed to study the relationship between the normalized spectral entropy and the damage severity, while the remaining one is employed to verify and test the performance of the proposed algorithm for damage’s severity assessment.

Evaluation of the experimental results

Determining the parameters of VMD

To avoid over- or underdecomposition of VMD, the total number of IMFs N and penalty parameter α are the two key parameters, which need to be determined in advance.^21,26,56,57 Moreover, according to the theory of alternate direction method of multipliers (ADMM),⁵⁸ the whole optimization can be implemented by two suboptimizations, that is, the total number of IMFs N is optimized firstly, and then, the penalty parameter α is obtained.

Figure 10 shows the 3D plot of the spectral entropy ratio I_SE of aluminum specimens 1–4 versus different value of N and α.

From Figure 10, it is evident that the spectral entropy ratio I_SE initially increases and then decreases as the value of N grows, and it reaches the maximum value when N is 4, which indicates that the decomposition is sufficient to avoid mode mixing during the decomposition, and the damage signal is separated effectively. Therefore, the optimal value of N for aluminum specimens 1–4 is determined as 4. Figure 10 demonstrates that the spectral entropy ratio I_SE reaches its maximum value when α is 1000. Therefore, the optimal value of α of aluminum specimens 1–4 is determined as 1000.

Figure 11 shows the 3D plot of the spectral entropy ratio I_SE of CFRP specimens versus different values of N and α. Similarly, as shown in Figure 11, the changing trend of the spectral entropy ratio I_SE is similar to the spectral entropy ratio I_SE of aluminum specimens 1–4, and the optimal two parameters are selected as 3 and 1000.

After the parameters of VMD are obtained, the recorded signal is decomposed by VMD, and the decomposed IMFs are shown in Figure 12.

Selection of IMF for damage detection

To obtain the best IMF for damage detection, the correlation coefficients R_k of different IMFs are calculated, and they are listed in Tables 4 and 5. According to Table 4, it can be observed that the coefficient of IMF1 is the minimum one, which implies that IMF1 is less similar to the reference signal, and it contains more information on the damage. Therefore, IMF1 is selected to obtain the spectral entropy for the measure of the severity of damage in the aluminum plate.

Table 4.

The cross-correlation coefficients R_k of different IMFs of aluminum specimens 1–4.

IMFs	IMF1	IMF2	IMF3	IMF4
Coefficient R_k	0.437	0.651	0.644	0.526

IMF: intrinsic mode function.

Table 5.

The cross-correlation coefficients R_k of different IMFs of CFRP specimens 1–4.

IMFs	IMF1	IMF2	IMF3
Coefficient R_k	0.036	0.998	0.260

CFRP: composite fiber-reinforced polymer; IMF: intrinsic mode function.

Similarly, according to Table 5, it can be seen that IMF1 exhibits the minimum value of the CFRP plate, which indicates that IMF1 contains the most significant damage information. Therefore, IMF1 is selected for the CFRP plate.

Relationship between the damage severity and the normalized spectral entropy

After all the parameters of VMD-SE method are optimized, the normalized spectral entropy of aluminum specimens 1–3 and CFRP specimens 1–3 under different damage is obtained, and the influence of the damage severity on the normalized spectral entropy is analyzed.

The normalized spectral entropy of different damage in the aluminum specimens

Figure 13 plots the curve of the normalized spectral entropy with different damage sizes of aluminum specimens 1–3.

Figure 13 clearly shows that the normalized spectral entropy of the three specimens increases from 1.00 to 1.51 as the through-hole diameter changes from 0 to 0.8 mm. This changing trend can be explained as follows: as the damage severity increases, the complexity of Lamb waves in thin plates also goes up; thereby, the normalized spectral entropy eventually increases.

Moreover, the fitted line of the average of the three specimens is also obtained and displayed in Figure 13. The correlation coefficient for the fitted line is 0.999, signifying that there is a robust linear correlation between the spectral entropy and the damage’s severity. Therefore, this linear relationship can be utilized to detect the damage’s severity in aluminum plates later.

The normalized spectral entropy of different damage in the CFRP plate

Figure 14 plots the curve of the normalized spectral entropy with different damage sizes of three CFRP specimens.

Figure 14 clearly illustrates that the normalized spectral entropy of the three specimens grows linearly from 1.00 to 2.11 as the length of the scratch increases from 0 to 8 mm. The reason for this changing trend is the same as that of the aluminum specimens in section the normalized spectral entropy of different damage in the aluminum specimens.

Similarly, the fitted line of the average of the three specimens is also obtained and displayed in Figure 14. The correlation coefficient for the fitted line is 0.998, which indicates that there is a good linear relationship between the spectral entropy and the scratch length of the three CFRP specimens. This linear relationship can be used to detect the damage’s severity in CFRP plates later.

Influence of damage’s location on the normalized spectral entropy

To investigate the influence of damage’s location on the normalized spectrum entropy, aluminum specimens 5–7 with a through-hole at a different location are tested and compared with aluminum specimens 1–3. Figure 15 is the 3D plot of the spectral entropy ratio I_SE of aluminum specimens 5–7 versus different values of N and α, and Figure 16 plots the curve of the normalized spectral entropy with different damage sizes.

From Figure 15, it is easily found that the optimal values of N and α for aluminum specimens 5–7 are 5 and 800, respectively.

Figure 16 clearly shows that the normalized spectral entropy of aluminum specimens 5–7 still increases linearly from 1.00 to 2.01 as the diameter of the through-hole changes from 0 to 0.8 mm, and the correlation coefficient for the fitted line is 0.997. Moreover, compared to the curve of aluminum specimens 1–3 in Figure 13, the slope and intercept of the two fitted lines are different, which indicates that the location of the damage has an influence on the normalized spectral entropy. That is because the location of damage changes the propagation path of Lamb waves, and eventually, their frequency characteristics are changed.

Damage severity identification by using the linear relationship

After the linear relationship between the damage’s severity and the normalized spectral entropy is achieved, the fitted lines of Figures 13 and 14 are utilized for damage detection of the fourth specimen. Tables 6 and 7 display the relative error between the actual and identified severity of the fourth specimens of the aluminum and CFRP plates, respectively.

Table 6.

The relative error between the actual size of the aluminum plate and the estimated size.

Damage’s diameter of the fourth aluminum specimen/mm
Actual size	0.4	0.6	0.7	0.8
Estimated size	0.417	0.607	0.724	0.787
Relative error	4.06%	1.11%	3.35%	1.63%

Table 7.

The relative error between the actual size of the CFRP plate and the identified size.

Damage’s length of the fourth CFRP specimen/mm
Actual size	2	4	6	8
Estimated size	2.065	4.208	6.399	8.634
Relative error	3.16%	4.95%	6.24%	7.34%

CFRP: composite fiber-reinforced polymer.

Tables 6 and 7 clearly demonstrate that the fitted curves obtained from the experiments can accurately identify the damage severity of aluminum and CFRP plates within an error range of approximately 7.34%. Therefore, the proposed VMD-SE method can be utilized for damage detection of both aluminum and CFRP plates.

Discussions

In this section, the VMD-SE algorithm is used for damage severity identification of both aluminum and CFRP plates. Experimental results show that the parameters N and α have a significant effect on the performance of VMD, and they are determined by maximizing the value of the spectral entropy ratio between the first and last IMFs. After the optimal parameters are obtained, the Pearson’s correlation coefficient between different IMFs and the reference signal is used to determine the best IMF that contains more damage information. Moreover, the expreimenal results of aluminum and CFRP plates also demonstrate a linear increase in normalized spectral entropy as the size of the simulated crack on the aluminum plate and the scratch on the CFRP plate grows. After this linear correlation between the normalized spectrum and the damage’s severity procured, it is applied for detecting damage size in the test specimens, and the identification errors are consistently below 7.34%.

Conclusion

This paper proposes an innovative approach by fusing VMD-SE for damage size estimation in thin plates. In this approach, the received signal is decomposed into a succession of IMFs by VMD. To improve the decomposition performance of VMD, the spectral entropy ratio of the first and last IMFs is computed, and the optimal parameters of VMD are obtained when the ratio reaches the maximum value. Moreover, the correlation coefficient R_k between the decomposed IMF and the reference signal is applied to extract the desired damage signal from the complex waves. The larger the R_k is, the more similar the decomposed IMF is to the reference signal. Therefore, the best IMF that contains more information on the damage is selected when the coefficient is the minimum value. Finally, the spectral entropy of the desired IMF is calculated as an indicator for assessing the damage severity.

To show the efficiency and advantages of the proposed method at damage-related signal extraction, the proposed VMD, wavelet packet analysis, and EMD are employed to extract the damage-related signal from a simulated signal, and their performance is compared. The extraction results clearly show that wavelet packet analysis only obtains the reference signal, EMD cannot extract the target component, and only the proposed method separates the damage-related and reference signals from the simulated signal.

To verify the VMD-SE algorithm in this study, damage detection of two different types of damage on aluminum and CFRP plates is conducted by using the new approach. The experimental results of the two plates reveal that the parameters, including the total number of IMFs and the penalty parameter of VMD, significantly affect the effectiveness of the method. Moreover, based on the experimental results, the optimal number of IMFs is 4 and 3 for aluminum specimens 1–4 and CFRP plates, respectively, and the optimal penalty parameter is 1000 for them. In addition, the experimental results of aluminum specimens 1–3 show that the normalized spectral entropy of the three specimens grows linearly from 1.00 to 1.51, as the diameter of the through-hole increases from 0 to 0.8 mm. Similarly, as the length of the scratch in CFRP plates grows from 0 to 8 mm, the normalized spectral entropy grows linearly from 1.00 to 2.11. In addition, the correlation coefficient of the two fitted curves of the two plates is over 0.998. The experimental results of aluminum plates also demonstrate the location of damage has an influence on the normalized spectral entropy. At last, the damage size of the fourth specimen is evaluated by using the linear association between the normalized spectral entropy and the damage severity. The identification results show that the relative error of the aluminum and CFRP plates is less than 7.34%, which indicates that the proposed method by fusing VMD-SE can detect the damage size in thin plates accurately and efficiently.

In the future, the proposed algorithm will be further used for early-damage’s severity and location detection. Additionally, it will be extended for damage detection of various types of damage in different materials.

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 work was supported in part by the National Natural Science Foundation of China (Grant No.: 51808417) and Hubei Key Laboratory of Disaster Prevention and Mitigation (Grant No.: 2021KJZ10). Partial work of numerical calculation is also supported by High-Performance Computing Center of Wuhan University of Science and Technology.

ORCID iD

Guangtao Lu

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