# A Signal Decomposition Method for Ultrasonic Guided Wave Generated from Debonding Combining Smoothed Pseudo Wigner-Ville Distribution and Vold–Kalman Filter Order Tracking

Authors: Junhua Wu; Xinglin Chen; Zheshu Ma
Journal: Shock and Vibration
 (2017)

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## Abstract

Carbon fibre composites have a promising application future of the vehicle, due to its excellent physical properties. Debonding is a major defect of the material. Analyses of wave packets are critical for identification of the defect on ultrasonic nondestructive evaluation and testing. In order to isolate different components of ultrasonic guided waves (GWs), a signal decomposition algorithm combining Smoothed Pseudo Wigner-Ville distribution and Vold–Kalman filter order tracking is presented. In the algorithm, the time-frequency distribution of GW is first obtained by using Smoothed Pseudo Wigner-Ville distribution. The frequencies of different modes are computed based on summation of the time-frequency coefficients in the frequency direction. On the basis of these frequencies, isolation of different modes is done by Vold–Kalman filter order tracking. The results of the simulation signal and the experimental signal reveal that the presented algorithm succeeds in decomposing the multicomponent signal into monocomponents. Even though components overlap in corresponding Fourier spectrum, they can be isolated by using the presented algorithm. So the frequency resolution of the presented method is promising. Based on this, we can do research about defect identification, calculation of the defect size, and locating the position of the defect.


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## Body

## 1. Introduction

Carbon fibre composite is widely used in modern industry, such as aerospace domain and military products, because of its high strength and light weight. At present, such a material has been generalized to automotive industry, obviously reducing the weight of automobile. Debonding defect is a major defect of the carbon fibre composites. A great number of investigations of the nondestructive evaluation and testing (NDE/NDT) have done research for this type of defect [1–5].Currently, ultrasonic guided wave (GW) testing has emerged as a popular NDE/NDT technique. The method can estimate the location, severity, and type of defects. Successful applications of defect identification of carbon fibre composites have been done [3, 6, 7]. However, dispersion effects and noise make ultrasonic testing waves as multicomponent signals, which results in that it is difficult to do NDE/NDT with raw testing waves. Therefore, isolating different components of GW and obtaining the corresponding time-frequency distributions (TFD) are vital for the inspection of the defect.A number of scholars have done investigations about signal processing methods of GWs. Kercel et al. [8] used Bayesian parameter estimates to isolate multiple modes in GW signals collected from laser ultrasonic testing on a manufacturing assembly line. Cai et al. [9] provided a time-distance domain transform (TDDT) method to interpret the dispersion of Lamb waves, which can result in high spatial resolution images of damage areas. Rizzo and di Scalea utilized Discrete Wavelet Transform (DWT) to extract wavelet domain features for enhanced defect characterization in multiwire strand structures [10]. Gangadharan et al. presented a time reversal technique using GWs to detect damage in an aluminum plate, and good results were achieved [11]. The wavelet analysis is widely used [12–17] in domains; many successful applications of wavelet transform (WT) for GW signals have been done. Li et al. [14] proposed a combined method employing empirical mode decomposition (EMD) and wavelet analysis to attain good time resolution of the response signals. Paget et al. [15] proposed a new damage-detection technique based on WT with a new basis. Yu et al. [16] used the techniques of statistical averaging to reduce global noise and discrete wavelet denoising using a Daubechies wavelet to remove local high-frequency disturbances. Y. Y. Kim and E.-H. Kim [17] evaluated the effectiveness of WT analysis for studying the wave dispersion.EMD, which can isolate adaptively different components, was proposed by Huang in 1998 [18]. At present, many investigations of theory and application have been done [19–24]. Li et al. [14], Osegueda et al. [20], and Salvino et al. [22] used EMD to process GW signals in plate structures. However, the frequency resolution of EMD is a limitation. Reference [25] reveals that when the ratio between a relatively low frequency and a relatively high frequency is greater than 0.75, two components of a signal cannot be separated.In 1993, Vold and Leuridan [26] proposed Vold–Kalman filter order tracking (VKF_OT) for the estimation of a single order component. In 1997, they [27] derived a scheme to simultaneously estimate multiple components. Instantaneous frequency of the isolated component is a necessary prior knowledge for VKF_OT. Therefore, we introduce Smoothed Pseudo Wigner-Ville distribution (SPWVD), which can remove the cross-term in frequency direction and time direction of the time-frequency panel, to get instantaneous frequencies of isolated components. We present a signal decomposition method for ultrasonic GWs combining VKF_OT and SPWVD in this paper.The rest of this paper is organized as follows. Section2 presents the theories of Smoothed pseudo Wigner-Ville distribution and Vold–Kalman filter order tracking. The principle of algorithm is illustrated in Section 3. Section 4 provides an illustration of the presented method. The details of the experiment are described in Section 5. Section 6 shows the application of the presented algorithm to the experimental signals. Finally, Section 7 concludes.


## 2. Smoothed Pseudo Wigner-Ville Distribution and Vold–Kalman Filter Order Tracking

### 2.1. Smoothed Pseudo Wigner-Ville Distribution

Wigner-Ville distribution has a fine time-frequency resolution and can reach the low boundary of Heisenberg uncertainty principle. It is defined as [28] (1)WVDst,f=∫-∞∞st+τ2st-τ2e-j2πfτdτ.However, for multicomponent signals, it suffers from inevitable interference of cross-terms. SPWVD can remove it in frequency direction and time direction of the time-frequency panel. And the formula of SPWVD is as follows [28]:(2)SPWsn,Θ=∑k=-L+1L-1hk∑l=M+1M-1glsn+l+ks∗n+l-ke-j2kΘ,where g(l) and h(k) are smoothing window functions in time direction and frequency direction, respectively. s is an analyzed signal, and n and Θ are time variable and frequency variables, respectively. The time resolution and frequency resolution of SPWVD are promising. Moreover, no interference is in the representation.


### 2.2. Vold–Kalman Filter Order Tracking

Isolation of different modes is important for defect identification by ultrasonic guided waves. On this basic, we can locate the defect and evaluate the defect size. Therefore, VKF_OT is employed to separate wave packages.In this paper, the angular-displacement VKF_OT techniques are adapted. The method is used to obtain the tracked components by minimizing the energy of errors for both the structural and data equations by mean of one of the least squares approaches [29].Thekth order component can be defined as(3)fkt=aktθkt+a-ktθ-kt,where ak(t) is the complex envelope and a-k(t) is the complex conjugate of ak(t) to make fk(t) a real waveform. It is noted that θk(t) is a carrier wave and defined as(4)θkt=exp⁡ki∫0tωudu,where du is the speed of the reference axle and ∫0tω(u)du is the elapsed angular displacement. The discrete form of (4) can then be written as(5)θkn=exp⁡ki∑m=0nωmΔT.


#### 2.2.1. The Structural Equation

As the tracked componentfk(t) can be written as (3), where the envelope ak(t) needs to be computed. Generally, ak(t) fulfills [29](6)dSaktdts=ψkt,where ψk(t) is a higher-degree term in ak(t). The corresponding discrete forms can be expressed,(7)∇Sakn=ψkn,where ∇ is the difference operator, the index s is the differentiation order, and ψk(n) physically is a combination of other spectral components and additional measurement noise.


#### 2.2.2. The Data Equation

A measured signaly(n) can be taken as a combination of several components, fk(t), and measurement noise,(8)yn=∑k∈jaknθkn+ξn,where the integral number j(=±1,±2,±3,…,and/or±K) is the order of spectral components to be tracked and ξ(n) is unwanted spectral components and measurement errors. Each component ak(n) of interest modulates with a carrier wave θk(n).


#### 2.2.3. Calculation of the Tracked Componentf

Lets = 2 and let data length be N; then the calculation matrix form can be expressed as [29](9)-21000⋯001-2100⋯0001-210⋯00⋮⋮⋮⋮⋮⋮⋮⋮00000⋯-21ak1ak2ak3⋮akN=ψk1ψk2ψk3⋮ψkN.To simultaneously track multiple orders and spectral components such as resonance, it can be extended to all order components of interest as well. Let(10)M⃡=-21000⋯001-2100⋯0001-210⋯00⋮⋮⋮⋮⋮⋮⋮⋮00000⋯-21,A⃡=ak1ak2ak3⋮akN,Z⃡=ψ~k1ψ~k2ψ~k3⋮ψ~kN,and then (9) becomes(11)M⃡000⋯000M⃡00⋯0000M⃡0⋯00⋮⋮⋮⋮⋮⋮⋮0000⋯0M⃡A⃡=Z⃡,where elements a~k in the matrix A⃡ are column vectors with a length N, which is the kth order component; ψ~k are error vectors with a dimension N×1; and M is a matrix with a dimension N×N.The terms with negative indexes in (8) assure fk(t) to be a real waveform. y~ is the measured signal with a length of N, ξ an error vector with dimension N×1, and B⃡k consists of carrier signals, as(12)B⃡k=θk100⋯00θk20⋯000θk3⋯0⋮⋮⋮⋮⋮000⋯θkNA⃡=Z⃡.Thus, (8) can be rewritten as(13)y~-B~1B~2B~3⋯B~ka~1a~2a~3⋮a~K=ξ~.As the angular-velocity VKF_OT scheme, we introduce a weighting factor and combine (9) and (13), and then(14)000⋮0y~-rM⃡00⋯000rM⃡0⋯0000rM⃡⋯00⋮⋮⋮⋮⋮⋮000⋯0rM⃡B~1B~2B~3⋯B~k-1B~ka~1a~2a~3⋮a~K-1a~K=rZ⃡ξ.Equation (14) can be symbolized as(15)Y⃡-P⃡A⃡=E⃡.The evaluation of tracked order components is exactly to find a vector A⃡ fulfilling(16)minA⃡⁡E⃡2=minA⃡⁡E⃡HE⃡=minA⃡⁡J;that is, ∂J/∂A⃡=0. The vector A⃡ can be written as(17)P⃡HP⃡A⃡=P⃡HY⃡.The matrix P⃡HP⃡ is written as(18)P⃡HP⃡=S⃡B⃡1,2B⃡1,3⋯B⃡1,KB⃡2,1S⃡B⃡2,3⋯B⃡2,KB⃡3,1B⃡1,2S⃡⋯B⃡3,K⋮⋮⋮⋱⋮B⃡K,1B⃡K,2B⃡K,3⋯S⃡,where S⃡=r2M⃡TM⃡+I⃡ and B⃡u,v=BuHBv. Moreover, P⃡H is written as(19)P⃡H=B⃡-1B⃡-2B⃡-3⋯B⃡-KT,where B⃡-K is the complex conjugate of B⃡K.


## 2.1. Smoothed Pseudo Wigner-Ville Distribution

Wigner-Ville distribution has a fine time-frequency resolution and can reach the low boundary of Heisenberg uncertainty principle. It is defined as [28] (1)WVDst,f=∫-∞∞st+τ2st-τ2e-j2πfτdτ.However, for multicomponent signals, it suffers from inevitable interference of cross-terms. SPWVD can remove it in frequency direction and time direction of the time-frequency panel. And the formula of SPWVD is as follows [28]:(2)SPWsn,Θ=∑k=-L+1L-1hk∑l=M+1M-1glsn+l+ks∗n+l-ke-j2kΘ,where g(l) and h(k) are smoothing window functions in time direction and frequency direction, respectively. s is an analyzed signal, and n and Θ are time variable and frequency variables, respectively. The time resolution and frequency resolution of SPWVD are promising. Moreover, no interference is in the representation.


## 2.2. Vold–Kalman Filter Order Tracking

Isolation of different modes is important for defect identification by ultrasonic guided waves. On this basic, we can locate the defect and evaluate the defect size. Therefore, VKF_OT is employed to separate wave packages.In this paper, the angular-displacement VKF_OT techniques are adapted. The method is used to obtain the tracked components by minimizing the energy of errors for both the structural and data equations by mean of one of the least squares approaches [29].Thekth order component can be defined as(3)fkt=aktθkt+a-ktθ-kt,where ak(t) is the complex envelope and a-k(t) is the complex conjugate of ak(t) to make fk(t) a real waveform. It is noted that θk(t) is a carrier wave and defined as(4)θkt=exp⁡ki∫0tωudu,where du is the speed of the reference axle and ∫0tω(u)du is the elapsed angular displacement. The discrete form of (4) can then be written as(5)θkn=exp⁡ki∑m=0nωmΔT.


### 2.2.1. The Structural Equation

As the tracked componentfk(t) can be written as (3), where the envelope ak(t) needs to be computed. Generally, ak(t) fulfills [29](6)dSaktdts=ψkt,where ψk(t) is a higher-degree term in ak(t). The corresponding discrete forms can be expressed,(7)∇Sakn=ψkn,where ∇ is the difference operator, the index s is the differentiation order, and ψk(n) physically is a combination of other spectral components and additional measurement noise.


### 2.2.2. The Data Equation

A measured signaly(n) can be taken as a combination of several components, fk(t), and measurement noise,(8)yn=∑k∈jaknθkn+ξn,where the integral number j(=±1,±2,±3,…,and/or±K) is the order of spectral components to be tracked and ξ(n) is unwanted spectral components and measurement errors. Each component ak(n) of interest modulates with a carrier wave θk(n).


### 2.2.3. Calculation of the Tracked Componentf

Lets = 2 and let data length be N; then the calculation matrix form can be expressed as [29](9)-21000⋯001-2100⋯0001-210⋯00⋮⋮⋮⋮⋮⋮⋮⋮00000⋯-21ak1ak2ak3⋮akN=ψk1ψk2ψk3⋮ψkN.To simultaneously track multiple orders and spectral components such as resonance, it can be extended to all order components of interest as well. Let(10)M⃡=-21000⋯001-2100⋯0001-210⋯00⋮⋮⋮⋮⋮⋮⋮⋮00000⋯-21,A⃡=ak1ak2ak3⋮akN,Z⃡=ψ~k1ψ~k2ψ~k3⋮ψ~kN,and then (9) becomes(11)M⃡000⋯000M⃡00⋯0000M⃡0⋯00⋮⋮⋮⋮⋮⋮⋮0000⋯0M⃡A⃡=Z⃡,where elements a~k in the matrix A⃡ are column vectors with a length N, which is the kth order component; ψ~k are error vectors with a dimension N×1; and M is a matrix with a dimension N×N.The terms with negative indexes in (8) assure fk(t) to be a real waveform. y~ is the measured signal with a length of N, ξ an error vector with dimension N×1, and B⃡k consists of carrier signals, as(12)B⃡k=θk100⋯00θk20⋯000θk3⋯0⋮⋮⋮⋮⋮000⋯θkNA⃡=Z⃡.Thus, (8) can be rewritten as(13)y~-B~1B~2B~3⋯B~ka~1a~2a~3⋮a~K=ξ~.As the angular-velocity VKF_OT scheme, we introduce a weighting factor and combine (9) and (13), and then(14)000⋮0y~-rM⃡00⋯000rM⃡0⋯0000rM⃡⋯00⋮⋮⋮⋮⋮⋮000⋯0rM⃡B~1B~2B~3⋯B~k-1B~ka~1a~2a~3⋮a~K-1a~K=rZ⃡ξ.Equation (14) can be symbolized as(15)Y⃡-P⃡A⃡=E⃡.The evaluation of tracked order components is exactly to find a vector A⃡ fulfilling(16)minA⃡⁡E⃡2=minA⃡⁡E⃡HE⃡=minA⃡⁡J;that is, ∂J/∂A⃡=0. The vector A⃡ can be written as(17)P⃡HP⃡A⃡=P⃡HY⃡.The matrix P⃡HP⃡ is written as(18)P⃡HP⃡=S⃡B⃡1,2B⃡1,3⋯B⃡1,KB⃡2,1S⃡B⃡2,3⋯B⃡2,KB⃡3,1B⃡1,2S⃡⋯B⃡3,K⋮⋮⋮⋱⋮B⃡K,1B⃡K,2B⃡K,3⋯S⃡,where S⃡=r2M⃡TM⃡+I⃡ and B⃡u,v=BuHBv. Moreover, P⃡H is written as(19)P⃡H=B⃡-1B⃡-2B⃡-3⋯B⃡-KT,where B⃡-K is the complex conjugate of B⃡K.


## 2.2.1. The Structural Equation

As the tracked componentfk(t) can be written as (3), where the envelope ak(t) needs to be computed. Generally, ak(t) fulfills [29](6)dSaktdts=ψkt,where ψk(t) is a higher-degree term in ak(t). The corresponding discrete forms can be expressed,(7)∇Sakn=ψkn,where ∇ is the difference operator, the index s is the differentiation order, and ψk(n) physically is a combination of other spectral components and additional measurement noise.


## 2.2.2. The Data Equation

A measured signaly(n) can be taken as a combination of several components, fk(t), and measurement noise,(8)yn=∑k∈jaknθkn+ξn,where the integral number j(=±1,±2,±3,…,and/or±K) is the order of spectral components to be tracked and ξ(n) is unwanted spectral components and measurement errors. Each component ak(n) of interest modulates with a carrier wave θk(n).


## 2.2.3. Calculation of the Tracked Componentf

Lets = 2 and let data length be N; then the calculation matrix form can be expressed as [29](9)-21000⋯001-2100⋯0001-210⋯00⋮⋮⋮⋮⋮⋮⋮⋮00000⋯-21ak1ak2ak3⋮akN=ψk1ψk2ψk3⋮ψkN.To simultaneously track multiple orders and spectral components such as resonance, it can be extended to all order components of interest as well. Let(10)M⃡=-21000⋯001-2100⋯0001-210⋯00⋮⋮⋮⋮⋮⋮⋮⋮00000⋯-21,A⃡=ak1ak2ak3⋮akN,Z⃡=ψ~k1ψ~k2ψ~k3⋮ψ~kN,and then (9) becomes(11)M⃡000⋯000M⃡00⋯0000M⃡0⋯00⋮⋮⋮⋮⋮⋮⋮0000⋯0M⃡A⃡=Z⃡,where elements a~k in the matrix A⃡ are column vectors with a length N, which is the kth order component; ψ~k are error vectors with a dimension N×1; and M is a matrix with a dimension N×N.The terms with negative indexes in (8) assure fk(t) to be a real waveform. y~ is the measured signal with a length of N, ξ an error vector with dimension N×1, and B⃡k consists of carrier signals, as(12)B⃡k=θk100⋯00θk20⋯000θk3⋯0⋮⋮⋮⋮⋮000⋯θkNA⃡=Z⃡.Thus, (8) can be rewritten as(13)y~-B~1B~2B~3⋯B~ka~1a~2a~3⋮a~K=ξ~.As the angular-velocity VKF_OT scheme, we introduce a weighting factor and combine (9) and (13), and then(14)000⋮0y~-rM⃡00⋯000rM⃡0⋯0000rM⃡⋯00⋮⋮⋮⋮⋮⋮000⋯0rM⃡B~1B~2B~3⋯B~k-1B~ka~1a~2a~3⋮a~K-1a~K=rZ⃡ξ.Equation (14) can be symbolized as(15)Y⃡-P⃡A⃡=E⃡.The evaluation of tracked order components is exactly to find a vector A⃡ fulfilling(16)minA⃡⁡E⃡2=minA⃡⁡E⃡HE⃡=minA⃡⁡J;that is, ∂J/∂A⃡=0. The vector A⃡ can be written as(17)P⃡HP⃡A⃡=P⃡HY⃡.The matrix P⃡HP⃡ is written as(18)P⃡HP⃡=S⃡B⃡1,2B⃡1,3⋯B⃡1,KB⃡2,1S⃡B⃡2,3⋯B⃡2,KB⃡3,1B⃡1,2S⃡⋯B⃡3,K⋮⋮⋮⋱⋮B⃡K,1B⃡K,2B⃡K,3⋯S⃡,where S⃡=r2M⃡TM⃡+I⃡ and B⃡u,v=BuHBv. Moreover, P⃡H is written as(19)P⃡H=B⃡-1B⃡-2B⃡-3⋯B⃡-KT,where B⃡-K is the complex conjugate of B⃡K.


## 3. Principle of the Presented Algorithm

As mentioned above, SPWVD has a promising time-frequency resolution. Therefore, we obtain frequencies and durations of modes from SPWVD distributions of testing guided waves. Furthermore, VKF_OT is adapted to realize isolation of different wave packages with obtained mode frequencies. Finally, the final mode waveforms are cut out from the wave packages of modes by durations of modes. The processing steps of the extension algorithm are shown in Figure1 and are as follows.Figure 1
The different steps of the algorithm presented in the paper.(1) Smoothed Pseudo Wigner-Ville Distribution. SPWVD is used for processing testing signal to get corresponding time-frequency distribution. And the promising time-frequency resolution of the method can lead to a high calculation accuracy of frequencies and durations of modes.(2) Extracting and Separating Valuable Areas of the Time-Frequency Panel. To extract and separate different modes, we do the following. (1) sum coefficients of each frequency-indexed group, and the calculation formula is(20)EΘ=∑n=1NSPWsn,Θ,where N denotes the number of the time variables of the time-frequency panel. (2) Calculate minimal values and maximum values of summations. Lines of the minimal values in the time-frequency panel can separate the valuable areas. And the maximum value evaluates whether the energy of a mode is strong enough to be kept. (3) Conduct threshold processing for the maximum values. And the equation is defined as(21)EΘmax=keepEΘmax>T0discardEΘmax≤T0,where E(Θ)max means the maximum of E(Θ). T0 denotes the threshold for sift valuable modes. A mode with a strong energy is more valuable for detecting the debonding defect, so a threshold process is introduced to improve calculation efficiency. Moreover, the noise can also be removed by this operation. After this step, the areas of the time-frequency panel corresponding to the significant modes can be extracted and separated. Moreover, corresponding frequencies at the maximum values are taken as the mode frequencies.(3) Vold–Kalman Filter Order Tracking. The mode frequencies have been calculated in Step (2). So we employ VKF_OT to filter the specific mode waveform with these frequencies. The different modes f0i(i=1,2,…,M) can be preliminarily isolated, and M is number of the maximum values obtained in Step (2).(4) Peak-Track Algorithm. Conduct the peak-track algorithm for the significant areas of SPWVD to obtain the primary IA of different modes. The principle of the peak-track algorithm can be found in [30]. And then, the IAs of significant modes can be obtained.(5) Constructing Filters in Time Domain. To remove calculation error of the mode waveform from result obtained in Step (3), we construct a corresponding filter in time domain based on the primary IAs of different modes obtained in Step (4). And the equation is as follows:(22)Ai=1A0i>TA0A0i≤TA,where Ai is the value of the ith filter in time domain and A0i denote the primary IAs of the ith modes. TA is a threshold to remove calculation error of the mode waveform.(6) Filtering in Time Domain for Calculation Result of Vold–Kalman Filter Order Tracking. Calculate the inner product between the mode waveforms obtained in Step (3) and the filter in time domain from Step (5). And then the final calculation result is obtained. The equation is(23)fi=Ai∗f0i.In Section4, the details of the present algorithm will be illustrated with a sample signal.


## 4. Illustration of the Presented Algorithm

We construct a sample signal to illustrate the presented algorithm,(24)st=s1t+s2t+s3t,s1t=00≤t≤0.000054000t-0.00005sin⁡300000×2πt0.00005<t≤0.0003-4000t-0.00055sin⁡300000×2πt0.0003<t≤0.000550,0.00055<t≤0.001s2t=00≤t≤0.000254000t-0.00025sin⁡55000×2πt0.00025<t≤0.0005-4000t-0.00075sin⁡55000×2πt0.0005<t≤0.000750,0.00075<t≤0.001s3t=00≤t≤0.000454000t-0.00025sin⁡50000×2πt0.00045<t≤0.0007-4000t-0.00075sin⁡50000×2πt0.0007<t≤0.000950,0.00095<t≤0.001.The sampling frequency is 2 MHz. The sample signal consists of three modes at frequencies of 50, 55, and 300 kHz. The curve of the sample signal in time domain is shown in Figure 2.Figure 2
The curve of the sample signal in time domain.Firstly, we employ SPWVD for the sample signal to obtain the corresponding time-frequency panel, which is shown in Figure3. As can be seen in Figure 3, the time resolution and frequency resolution of the time-frequency are promising. Besides, no interference is in the representation. However, the difference between the component at 50 kHz and the component at 55 kHz is so little that the two components cannot be separated in the SPWVD distribution of the sample signal, so is in Fourier spectrum, as shown in Figure 4.Figure 3
The representation of the sample signal by using SPWVD.
(a)
(b)Figure 4
The Fourier spectrum of the sample signal.After that, (20) is used on the SPWVD distribution. As shown in Figure 5, three maximum values at 50, 55, and 300 kHz are kept. So VKF_OT is employed on the sample signal with 50, 55, and 300 kHz to get the corresponding filtering results.Figure 5
The different frequency-group summations of the time-frequency panel of the sample signal.And then, the primary IAs of different modes are obtained by the peak-track algorithm. The filters in time domain of different modes are obtained by employing (22) on the primary IAs, as shown in Figure 6. We can adjust the threshold in (22) to get filters in time domain of different modes with a high accuracy in time resolution.Figure 6
The filters in time domain of different modes: (a) mode at 300 kHz, (b) mode at 55 kHz, and (c) mode at 50 kHz.
(a)
(b)
(c)Finally, we conduct time-domain filter for the result of VKF_OT, and the results are shown in Figure7. As presented in Figure 7, the decomposition results almost overlap with the corresponding original modes. To further validate the effectiveness of the presented algorithm, we calculate the error of the decomposition result, as shown in Figure 8. The absolute error is less than 0.1 for the modes at 50 and 55 kHz and is less than 0.02 for the mode at 300 kHz, which reveals that the calculation accuracy of the presented algorithm is promising.Figure 7
The decomposition result and the original modes of the sample signalx by the presented algorithm: (a) mode at 300 kHz, (b) mode at 55 kHz, and (c) mode at 50 kHz.
(a)
(b)
(c)Figure 8
The errors of decomposition result of the sample signalx by the presented algorithm: (a) mode at 300 kHz, (b) mode at 55 kHz, and (c) mode at 50 kHz.
(a)
(b)
(c)To compare with EEMD, the sample signal is also processed by this decomposition method. Figure9 shows the coefficients of correlation between different IMFs and the sample signal. We can learn that IMFs 1–4 are vital components of the signal, as the coefficients of correlation are relatively greater. So these IMFs are shown in Figure 10. It is visible that the mode mixing occurs in EEMD for modes at 50 and 55 kHz, as shown in Figure 10(b). Because the ratio between modes at 50 and 55 kHz is greater than 0.75, they cannot be separated, as revealed in [25]. Therefore, the poor frequency resolution is a limitation of EEMD for its application.Figure 9
Correlation coefficients between IMFs and the original signal.Figure 10
The IMFs 1–4 in the EEMD of the sample signal and the corresponding Fourier spectrums: (a) the IMFs and (b) the corresponding Fourier spectrums.
(a)
(b)


## 5. Details of the Experimental

The material of the specimen is a specific composite material. The size is 400 mm × 300 mm × 3 mm and contains 15 layers. The corresponding size diagram is presented in Figure11. Two debonding defects are in the specimen. The side length of the defects is, respectively, 20 mm and 30 mm.Figure 11
The size diagram of the specimen.Figure12 is the diagram of the testing principle. In testing process, the exciting probe sends an exciting wave, and then the receiving probe will receive excited lamb waves. The excitation wave is a sine wave with a frequency of 100 kHz. The sampling frequency is 20 MHz. Three sets of signal are collected in the experiment, responding to three situations, that is, no defect, 20 mm defect, and 30 mm defect, respectively. The corresponding GWs collected in the experiment are presented in Figure 13.Figure 12
The diagram of the testing principle.Figure 13
The GWs collected in the experiment: (a) no defect, (b) 20 mm, and (c) 30 mm.
(a)
(b)
(c)


## 6. Result and Discussion

Figure15 presents the decomposition result of experimental signals by using the presented algorithm. As shown in Figures 14 and 15, two modes exist in the experimental signal of no defect and are at 50.3 kHz and 34.5 kHz. The signal of 20 mm defect consists of modes at 49.2 kHz, 41.2 kHz, and 34.1 kHz. And the signal of 30 mm defect consists of modes at 49.8 kHz, 44.8 kHz, and 33.9 kHz. Figure 16 shows these modes are decomposed by the presented algorithm. This indicates the method is effective in isolating different modes. Besides, considering that 50.3 kHz, 49.2 kHz, and 49.8 kHz pose little differences and this phenomenon is similar to 34.5 kHz, 34.1 kHz, and 33.9 kHz, it seems that defects stimulate new modes and 41.2 kHz for 20 mm defect and 44.8 kHz for 30 mm defect. On the basis of this phenomenon, we can try to detect the defect. Moreover, the frequency of the new mode becomes greater along with the defect size (41.2 kHz for 20 mm and 44.8 kHz for 30 mm). Maybe we can try to evaluate the size of the defect, according to this relationship. Finally, as we have isolated different wave packets, the location of defect can be obtained by the decomposition results.Figure 14
The SPWVD representations of the GWs in the experiment: (a) no defect, (b) 20 mm, and (c) 30 mm.
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(b)
(c)Figure 15
The different frequency-group summations of the time-frequency panel of the experimental signals: (a) no defect, (b) 20 mm, and (c) 30 mm.
(a)
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(c)Figure 16
The decomposition result of experimental signals by using the presented algorithm: (a) no defect, (b) 20 mm, and (c) 30 mm.
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(b)
(c)


## 7. Conclusion

This paper presents a decomposition algorithm aiming to analyze the characteristics of ultrasonic GWs generated in a NDT for the debonding in a type of composite material by combining SPWVD and VKF_OT. The presented method succeeds in isolating different GW modes. On the basis of the presented algorithm, the characteristics of the experimental signals were investigated. Some conclusions, which are valuable for identification of defect, calculation of defect size, and locating defect, are obtained. The technique also can be applied in analogue NDTs and NDEs on the basic of the ultrasonic GWs. Further research will be done to validate the feasibility for locating defects by the algorithm.



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*Source: 7283450-2017-09-14.xml*