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An improved time reversal method for diagnostics of composite plates using Lamb waves
Accepted Manuscript An improved time reversal method for diagnostics of composite plates using Lamb waves Liping Huang, Liang Zeng, Jing Lin, Zhi Luo PII: DOI: Reference:
S0263-8223(17)30076-4 https://doi.org/10.1016/j.compstruct.2018.01.096 COST 9336
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
10 January 2017 15 December 2017 30 January 2018
Please cite this article as: Huang, L., Zeng, L., Lin, J., Luo, Z., An improved time reversal method for diagnostics of composite plates using Lamb waves, Composite Structures (2018), doi: https://doi.org/10.1016/j.compstruct. 2018.01.096
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An improved time reversal method for diagnostics of composite plates using Lamb waves
Authors Liping Huang PhD Candidate School of Mechanical Engineering Xi’an Jiaotong University Xi’an, Shaanxi Province, China, 710049 Email: [email protected] Liang Zeng PhD, Lecturer School of Mechanical Engineering Xi’an Jiaotong University Xi’an, Shaanxi Province, China, 710049 Email: [email protected] Jing Lin∗ PhD, Professor 1 State Key Laboratory for Manufacturing System Engineering, Xi’an Jiaotong University Xi’an, Shaanxi Province, China, 710049 2 School of Reliability and Systems Engineering, Beihang University, Xueyuan Road No. 37, Haidian District, Beijing, China, 100191 Email: [email protected]
Zhi Luo PhD Candidate School of Mechanical Engineering Xi’an Jiaotong University Xi’an, Shaanxi Province, China, 710049 Email: [email protected]
∗
Corresponding author
Abstract The time reversal method has been widely used to detect damage in composite plates. Based on the time reversibility of Lamb waves, the waveform of the original input signal will be reconstructed after the time reversal process. However, the time reversal operator of Lamb waves in composite plates is a function of frequency. Thus the shape of the reconstructed signal will not be identical to that of the original input signal. To alleviate the effects of the time reversal operator, an improved time reversal method (ITRM) is proposed. In this method, the response signal in the forward path and the reconstructed signal in the time reversed path are both modulated by a weight vector which is obtained as the product of a window function and the reciprocal of amplitude dispersion. The window function varies with the excitation signal adaptively, and its shape is also determined by a threshold. An experimental example is then introduced to illustrate the effectiveness of the ITRM. The results show that the time reversibility can be greatly enhanced and the impact damage in the composite plate can be exactly located by the ITRM.
Keywords: Lamb wave; Composite; Time reversal operator; Damage detection
1. Introduction Composite material has been widely used in many different fields (aerospace, civil infrastructure, energy engineering, and automobile industries, etc.) for its specific properties, such as high strength, high stiffness and light weight. However, composite materials are often damaged by low velocity impacts for their low transverse strength [1]. Low velocity impacts (e.g. bird strikes, tool drops or runway stones) loaded on composite materials may result in various types of damage such as delamination, indentation, fiber breakage or matrix cracking [2]. Such damages are often invisible and may lead to catastrophic failure. Hence, it calls for an effective and efficient inspecting technique which could detect the damage promptly and thus ensure the safety of these composite materials. Lamb waves are mechanical waves with wavelength that is in the same order as the thickness of the plate [3]. With the advantages including long propagation distance, low cost and good sensitivity to various defects, Lamb waves have attracted tremendous attention for testing and evaluating composite materials [4,5]. However, propagation of Lamb waves in composite materials are notoriously complicate due to their dispersive characteristic. For each mode, Lamb wave components at different frequencies will propagate at different velocities, which lead to wave deformation and amplitude decrease as they travel in composite plates. Therefore, the time reversal method has been widely used to compensate the velocity dispersion and improve the quality of Lamb waves [6,7]. The time reversal concept was first proposed in acoustics by Fink [8,9]. According to the principle of time reversal, an original input signal can be reconstructed at the transmitter location, if the response signal captured by a receiver is reversed in the time domain and
re-emitted back through the receiver to the transmitter. The time reversibility of Lamb waves is based on the principle of spatial reciprocity and time reversal invariance of linear acoustic wave equations [6]. However, damage will introduce nonlinearity into the structure and break down the time reversal invariance. Hence, it can be diagnosed by comparing the waveform of the reconstructed signal with that of the original input signal [10]. Sohn et al. [11] proposed a baseline-free damage detection method by applying the combination of consecutive outlier analysis and the TRM. Mustapha et al. [12] extended TRM to detect damage in reinforced concrete beams, and [13] proposed a new damage index (DI) to detect the location of the barely visible indentation damage in sandwich CF/EP composites. Agrahari et al. [14] developed a refined time reversal method to enhance the sensitivity of damage detection in isotropic plates. Due to the effects of the time reversal operator, the time reversibility may be a challenge that the TRM faces. To overcome this problem, comprehensive theoretical and experimental analysis of the time reversibility of Lamb waves was performed. Park et al. [10] proposed a wavelet-based signal processing technique to extract narrowband responses to enhance the time reversibility of Lamb waves in a composite plate. Xu et al. [15] presented that within-mode Lamb wave can be fully reconstructed when the frequency bandwidth of excitation signal is narrow. Poddar et al. [16] compared the time reversal reconstruct quality under different parameters, i.e., the excitation frequency and bandwidth, as well as the piezoelectric (PZT) geometry. Agrahari [17] investigated the effects of adhesive layer between the transducers and the host plate, the tone burst count of the excitation signal, the plate thickness and the piezoelectric transducer thickness on the time reversibility of Lamb waves. These studies could help alleviating the frequency dependency of the time-reversal
operator, but did not suggest a proper method to compensate that. The objective and motivation of this paper is to establish a methodology that can compensate the frequency dependency of the time reversal operator so that the time reversibility of Lamb waves could be enhanced. In the proposed method (i.e., ITRM), the amplitude dispersion compensation of the response signal in the forward path is achieved by placing a weight vector on its spectrum. The compensated signal is then reversed in the time domain and re-emitted through the receiver to the transmitter. The reconstructed signal captured at the transmitter location still suffers amplitude dispersion in the time reversed path. Thus the amplitude dispersion compensation is also applied to the reconstructed signal. The rest of the paper is organized as follows. The basic theory of time reversibility of Lamb waves is briefly reviewed in Section 2. The ITRM and its theoretical analysis are given in Section 3. Section 4 illustrates the process of the ITRM by an experiment example. In Section 5, the ITRM is applied to detect impact damage in a composite plate. Finally, the conclusions are drawn in Section 6.
2. Time reversal of Lamb waves in composite plates The concept of time reversal was first proposed in acoustics, and usually applied in the fields of medical, hydrodynamics and nondestructive examination [18]. An application of the time reversal concept to Lamb waves in solids is then studied [6,19]. In the time reversal process of Lamb waves, an original input signal can be reconstructed at the location of transmitter A if the response signal captured by receiver B is time reversed and re-emitted to the location of original transmitter A. The schematic diagram of TRM is shown in Fig.1.
During the process of TRM, to reverse the path direction, the transducer (e.g., PZT) changes its role from transmitter to receiver and vice versa.
Insert Fig. 1 here
The formula deduction procedures of TRM in the frequency domain are presented as follows. For the piezoelectricity of PZT materials, an electrical voltage VA(ω) excited at transmitter PZT A can be converted to mechanical strain ɛA(ω) [20].
ε A (ω ) = ka (ω )VA (ω )
(1)
where ka(ω) is the electromechanical coupling coefficient, and ω is the angular frequency. If Lamb waves are excited at PZT A and received at PZT B as shown in Fig. 1, the response signal captured by PZT B satisfies,
VB ( ω, r ) = k b−1 (ω ) ka (ω )VA (ω ) H (ω , r )
(2)
where r is the propagation distance of Lamb waves from transmitter to receiver, H(ω,r) is the structural transfer function of the sensing path, and kb(ω) is the electromechanical coupling coefficient of PZT B. If PZTs A and B are the same, ka(ω)=kb(ω) [13]. Hence, the Eq. (2) could be simplified as,
VB (ω, r ) = VA (ω ) H (ω, r )
(3)
It is noted that the time reverse operation on a signal is equivalent to taking the complex conjugate in the frequency domain. Consequently, the time reversal of the response signal captured by PZT B [i.e., Eq. (3)] can be written as
V B* ( ω, r ) = V A* (ω ) H * (ω , r )
(4)
where * represents a complex conjugate and V* B(ω) is the time reversed signal. Finally, V*B(ω) is re-emitted through PZT B and the corresponding response signal at the location of PZT A is the reconstructed signal VR(ω,r) which is defined as
VR (ω, r ) = VB* (ω ) H (ω, r )
(5)
Substituting Eq. (4) into Eq. (5), we obtain
VR (ω, r ) = VA* (ω ) H * (ω, r ) H (ω, r )
(6)
and the time reversal operator of Lamb waves HTR(ω,r) is defined as, H TR (ω , r ) = H * (ω , r ) H (ω , r )
(7)
The TRM used for damage detection is based on the principle of linear reciprocity [6,7], that is, the excitation signal can be fully reconstructed after the time reversal process. While sources of nonlinearity on the wave propagation path will break down the time reversibility of Lamb waves and the waveform of the reconstructed signal will be different from that of the original input signal. Damages in composite plates, such as low velocity impact, delamination, cracks and fiber breakage, may introduce nonlinearity, thus damages could be detected by comparing the waveform of the reconstructed signal with that of the original input signal. A damage index (DI), which is calculated to analyze the deviation between the reconstructed signal and the excitation signal, is defined as follows: [10] t1
DI = 1 − ∫ VA ( t )Vr ( t ) dt t0
∫
t1
t0
t1
VA ( t ) dt ∫ Vr ( t ) dt 2
t0
2
(8)
where VA(t) is the original input signal, and Vr(t) is the reconstructed signal, both of which are normalized with their maximum value. t0 and t1 denote the starting and ending time points of the signal for comparison. The value of DI falls within the range of [0,1]. DI = 0 represents that the two signals exactly coincide with each other. The value of DI increases and
approaches 1 with the growing difference between the reconstructed signal and the original input signal. From Eq. (6), it can be concluded that if the time reversal operator keeps constant over the whole excitation frequency range, DI will be zero when the structure is free of damage, and thus damage could be detected by evaluating the value of DI. However, for Lamb waves propagating in composite plates, the time reversal operator is generally frequency dependency, and the original input signal will not be fully reconstructed.
3. Improved time reversal of Lamb waves in composite plates
Referring to [21], the transfer function in composite plates is defined as follows: H ( ω , r ) = A (ω ) e −η (ω ) r
1 r
e − ik (ω ) r
(9)
where, A(ω) is the excitability of a particular mode [22], η(ω) is the attenuation coefficient and k(ω) is the wave number [21]. From the above equation, it can be observed that the transfer function of Lamb waves propagating in a composite plate is mainly affected by four factors, which are transducers, material damping, geometric spreading and wave dispersion. Among these, the excitability A(ω) is greatly influenced by the type and size of transducers and adhesive layers between transducers and composite plates. Material damping coefficient α(ω) describes how much energy dissipates of Lamb waves, which is induced by the viscoelasticity of carbon fiber reinforced materials. 1/√r represents geometric spreading which is caused by the growing length of a wave front departing into all directions from a source [23]. The wave dispersion e-ik(ω)r is a fundamental characteristic of Lamb waves and can be fully compensated by the TRM.
Thus, the time reversal operator HTR(ω,r) in a composite plate [i.e., Eq. (7)] can be represented by the following equation, H TR (ω , r ) = A2 ( ω ) e
−2η ( ω ) r
1 r
(10)
It means that the time reversal operator varies with the frequency components and the propagation distance of Lamb waves. In this case, wave components at different frequencies are non-uniformly scaled. As a result, the waveform of the reconstructed signal cannot exactly match that of the original input signal even in healthy structures. According to the definition of DI in Eq. (8), the deviation caused by the time reversal operator would also be captured by the DI. Thus a non-zero damage index arises and implies the existence of damage, which would contaminate the identification result and lead to a false alarm. To overcome this problem, an ITRM is proposed. Its schematic diagram is given in Fig. 2. The ITRM consists of five steps, i.e.,
Insert Fig. 2 here
(1) an original input signal is emitted from PZT A, and the response is captured by PZT B; (2) the response signal is modulated by a weight vector. (3) the modulated response signal is reversed in the time domain and then re-emitted through PZT B; (4) the reconstructed signal is captured at transmitter PZT A, and then modulated by the same weight vector as in step 2;
(5) the modulated reconstructed signal is then normalized with respect to its peak value and compared with the original input signal. The theoretical analysis of the ITRM is derived in the frequency field as follows. In the ITRM, the response signal VB(ω,r) in Eq. (3) is modulated by a weight vector, and the modulated response signal VBM(ω,r) is
VBM (ω , r ) = VB ( ω, r )W (ω , r )
(11)
where W(ω,r) is a weight vector. Ideally, if the Lamb wave signal is free of noise, a direct way to compensate the effects of the time reversal operator could be achieved by an inverse filter procedure, i.e., W(ω,r) =1/|H(ω,r)|. Here, |H(ω,r)| is the amplitude of the transfer function as Lamb waves propagating in a composite plate. It is a function of frequency, which is called amplitude dispersion. H (ω , r ) = A ( ω ) e−η (ω ) r
1 r
(12)
In practical applications, however, the Lamb wave signals are always contaminated by noises, and the estimated transfer function cannot accurately represent the actual one. In this case, the inverse filter procedure faces a challenge that it is instable and noise may be amplified during the modulation process, especially for the frequency range where the desired Lamb wave mode is quite weak [i.e., |H(ω,r)| ≈ 0]. To address this issue, the weight vector is modified as, W (ω , r ) =
1 H (ω , r )
T (ω )
(13)
where T(ω) is a window function which is defined as 1 T (ω ) = VA (ω ) / α
V A (ω ) ≥ α V A (ω ) < α
(14)
here, |VA(ω)| is the normalized magnitude spectrum of the original input signal, and ??∈[0,1] is a threshold which controls the shape of the window. If ?? = 0, the window function T(ω) becomes rectangular, and the weight vector W(ω,r) returns to the inverse filter. If ?? = 1, T(ω) = |VA(ω)|. Referring to [24], Hanning windowed tonebursts are often used as the input signals for Lamb wave inspections, and the operation frequency is selected within the frequency range where the desired Lamb mode is strong. If 0
VBM (ω, r ) = VA (ω ) H (ω, r )
T (ω ) H (ω, r )
= VA (ω ) T (ω ) e
− ik ( ω ) r
(15)
In frequency domain, time reversal of the modulated response signal is defined as, V BM* (ω , r ) = VA* (ω ) T * (ω ) e
ik ( ω ) r
(16)
Then the time reversed modulated response signal is re-emitted through transducer PZT B and received by PZT A, resulting in * VR ( ω, r ) = VBM (ω , r ) H (ω , r )
where VR(ω,r) is the reconstructed signal in the ITRM.
(17)
Substituting Eq. (16) into Eq. (17), we obtain ik ( ω ) r
VR (ω, r ) = VA* ( ω ) T * (ω ) e
− ik (ω ) r
H (ω, r ) e
= VA* (ω ) T * (ω ) H (ω, r )
(18)
Then the reconstructed signal is modulated by the weight vector and the modulated reconstructed signal VRM (ω,r) is defined as VRM (ω ) = VR (ω , r )W ( ω , r ) = V A* (ω ) T * ( ω ) T (ω )
(19)
Thus, we obtain V A* ( ω ) VRM ( ω ) = * V A ( ω ) VA ( ω ) / α
VA (ω ) ≥ α 2
VA (ω ) < α
(20)
Finally, the modulated reconstructed signal in time domain, i.e., Vrm(t), is obtained through inverse Fourier transform, and the DI in Eq. (8) is calculated by measuring the discrepancies between this modulated reconstructed signal and the excitation signal.
4. Experimental investigation In this section, an experiment is designed to illustrate the process of the ITRM. The experiment setup consists of an Agilent 33220A function/arbitrary waveform generator, a NF HSA4012 voltage amplifier and a NI PXIe-1082 data acquisition. The experiment is carried out on a T300/7901 16-ply quasi-isotropic carbon fiber-reinforced polymer (CFRP) laminates with [+45/-45/0/90]2s lay-up. The dimension of the specimen plate is 690 mm×690 mm×2 mm and the thickness of each laminate is 0.125 mm. The material properties of the composite specimen are given in table 1.
Insert Table 1 here
The schematic diagram of the composite plate is shown in Fig. 3. PZT A serves as the transmitter, while PZTs B and C serve as the receivers. Compared with the A0 mode, S0 mode has the advantage of high energy, low attenuation and high signal-to-noise ratio when propagating in composite plates [25]. Therefore, S0 mode is only concerned in this experiment.
Insert Fig. 3 here
A series of 5-cycle sinusoidal toneburst signals with center frequencies increase from 50 kHz to 700 kHz in steps of 500 Hz are excited in turn at the transmitter. The maximum peak of the S0 mode Lamb wave response signals of different excitation parameters are divided by that of their excitation signals, respectively, to calculate amplitude dispersion curve. Fig. 4 shows the normalized amplitude dispersion curve of S0 mode Lamb wave transmitted from PZT A to PZT B.
Insert Fig. 4 here
For instance, a 240 kHz, 3-cycle toneburst is applied to PZT A as the original input signal. The response signal captured by PZT B is displayed in Fig. 5(a). It can be seen that the direct wave (sensing path ① in Fig. 3) and the reflected wave (sensing path ② in Fig. 3) are separated in time domain. Hilbert transform is then applied to the response signal, and the associated envelope is also given in Fig. 5(a) as the red dashed line. The waveform in the time
range where the magnitude of envelope is larger than 0.1 times that of the peak time (i.e., the waveform enclosed by the dashed rectangle in Fig. 5(a)) is time reversed and re-emitted by PZT B. Fig. 5(c) gives the comparison between the reconstructed signal captured at PZT A (i.e., the blue solid line) and the original input signal (i.e., the red dashed line). It is apparent that the two signals are quite different from each other, arising a DI value of 0.3458 in the undamaged case.
Insert Fig. 5 here
In the ITRM, the threshold α determines the shape of the window function T(ω), Fig. 6(a). Since T(ω) improves the robustness of the ITRM at the cost of compressing the bandwidth of the Lamb wave signal, the threshold α provides a compromise between the compensation degree and the noise resistance. In this paper, the threshold is determined as follows. The threshold α increases from 0 to 1 with a step of 0.05. At each threshold value, the modulated reconstructed signal in the ITRM is obtained. Subsequently, the DI is calculated to measure the discrepancy between the original input signal and the modulated reconstructed signal. Since the structure is free of damage, the smaller the DI value, the better the chosen threshold ??. Fig. 6(b) shows the evolution of DI value with the increase of threshold. It can be observed that the minimum DI value sits at the threshold α = 0.45.
Insert Fig. 6 here
As the threshold ?? takes 0.45, the response signal displayed in Fig. 5(a) is then modulated by the weight vector [i.e., Eq. (11)], and the modulated response signal is shown in Fig. 5(b). Since the direct wave is isolated in the time domain, it could be easily extracted as the wave-packet enclosed by the dashed rectangle. Subsequently, the direct wave is time reversed and applied to PZT B as the re-emitted signal. The reconstructed signal captured by PZT A is also modulated by the weight vector. Fig. 5(d) shows the comparison between the modulated reconstructed signal and the original input signal. It can be seen that the original input signal can be exactly reconstructed by using the ITRM. Accordingly, the DI value calculated from Eq. (8) reduces to 0.0018. Fig. 7(a) gives the comparison between the magnitude spectra (after Fourier transform) of the original input signal and the reconstructed signal in the TRM. It can be seen that compared to the original input signal, the peak frequency of the reconstructed signal shifts from 240 kHz to 294 kHz, and its frequency band is obviously narrowed due to the time reversal operator. That is the main reason why the reconstructed signal in the TRM is quite different from the original input signal, Fig. 5(c). In the ITRM, the time reversal operator is compensated by the modulations of the response signal and the reconstructed signal. As shown in Fig. 7(b), the spectra of the original input signal and the modulated reconstructed signal match well. Hence, the original input signal is fully reconstructed in the ITRM, Fig. 5(d).
Insert Fig. 7 here
For further illustration, the sensing path PZT A to PZT C is also considered. Fig. 8(a) shows the response signal captured by PZT C, where the direct wave (sensing path ③ in Fig. 3) and the reflected wave (sensing path ④ in Fig. 3) interfere with each other in time domain. In this case, the waveform in the time range where the magnitude of envelope is larger than 0.1 times that of the peak time [the waveform enclosed by the dashed rectangle in Fig. 8(a)] consists of both the direct wave and the reflected wave. Fig. 8(c) shows the reconstructed signal captured by PZT A as that waveform is time reversed and re-emitted through PZT C. It is apparent that the reconstructed signal deviates from the original input signal. Besides the time reversal operator, this deviation may also be attributed the overlap of multiple waves at the main lobe of the reconstructed signal [20]. For comparison, Fig. 8(b) displays the modulated response signal in the ITRM as the threshold ?? takes 0.45. It can be seen that the direct wave and the reflected wave are separated in the time domain. The reason behind this phenomenon may be explained as follows. At the operation frequency (i.e., 240 kHz), the velocity dispersion of S0 mode may be gentle, and the increase of duration of the direct wave mainly comes from the distortion of waveform caused by amplitude dispersion. In the ITRM, the effect of amplitude dispersion is compensated by the weight vector in Eq. (11). Thus, the duration of the direct wave decreases. Benefitting from that, it could be easily extracted in time domain, as the waveform enclosed by a dashed rectangle in Fig. 8(b). The comparison between the modulated reconstructed signal and the original input signal is given in Fig. 8(d). It can be seen that the original input signal can be exactly reconstructed by using the ITRM. That demonstrates another merit of the ITRM over the TRM.
Insert Fig. 8 here
5. Application time reversal for damage detection The application of the ITRM for damage detection of composite plates is investigated. The experiment setup and the composite specimen are the same as in the Section 4. Fig. 9(a) shows the composite specimen and sensor layout. 8 PZTs (circular, P-51, with dimensions of
Φ8 mm×0.5 mm) bonded on the CFRP plate are networked as a square senor array. In the active sensor network, the PZT elements take turns generating Lamb wave signals while the rest of them are listening. Totally 16 paths are available for defect imaging, as shown in Fig. 9(b). A rectangular Cartesian coordinate system is established with the center of the sensor array as the origin, and the coordinates of center of all PZTs are illustrated in Fig. 9(b). The monitoring area enclosing the impact damage is 280 mm×280 mm, and the distance between the adjacent PZTs is 140 mm. The composite plate was loaded in a quasi-static indentation apparatus with a 20 mm diameter indenter to introduce impact damage centering at [46, -46], i.e., the white-filled circle in Fig. 9(a).
Insert Fig. 9 here
A series of 5-count sinusoidal toneburst signals with center frequencies increase from 50 kHz to 600 kHz in steps of 500 Hz are excited in turn to obtain the amplitude dispersion curve of S0 mode Lamb wave of all sensing paths. For instance, the amplitude dispersion curves of path 4, 5 and 6 are displayed in Fig. 10.
Insert Fig. 10 here
In the ITRM, the key problem is to determine the weight vector. As mentioned, the shape of the window function depends on the original input signal and threshold ??. In this experiment, the original input signals are 3cycles, tonebursts centered at 250 kHz, 270 kHz and 290 kHz, respectively. Fig. 11 gives the relationship between the threshold ?? and the DI value. The abscissa is the threshold ??, which varies from 0 to 1 with a step of 0.05, and the ordinate is the average of the DI values of all 16 paths. In the healthy condition, the minimum DI value corresponds to the optimal choice of threshold ??. For illustration, as the center frequency of toneburst takes 250 kHz, 270 kHz and 290 kHz in turn, the optimal threshold ?? are 0.5, 0.3 and 0.2, respectively (Fig. 11).
Insert Fig. 11 here
Once the optimal choice of threshold is achieved, the DI values of all sensing paths could be calculated from the comparison between the original input signal and the modulated reconstructed signal. For instance, Fig. 12(a) shows the DI values as the input signal takes a 3-cycle 270 kHz toneburst. It can be seen that the sensing paths which cross the damage area (i.e., Path 1 and Path 10) are of high DI values, while the rest sensing paths are of low ones. This result verifies the effectiveness of the ITRM for damage detection. For comparison, in the TRM, the DI measures the discrepancy between the original input signal and the
reconstructed signal. Fig. 12(b) displays the DI values of all sensing paths calculated from the TRM. It can be seen that the result is chaotic, and the sensing paths which cross the damage cannot be correctly identified. The reason lies that in the TRM, besides the damage, the time reversal operator also changes the waveform of the reconstructed signal [i.e., Eq. (6)]. The combination of these two effects may not always leads to a larger DI value. As a result, the damage cannot be detected. In the ITRM, the effects of the time reversal operator are fully compensated by the modulations of the response signal and the reconstructed signal. Benefitting from that, the original input signal could be well reconstructed in the ITRM, and the deviation of the modulated reconstructed signal is only attributed to the presence of damage. Hence, its performance improves obviously.
Insert Fig. 12 here
Subsequently, the probabilistic imaging algorithm is introduced to visually pinpoint the damage [26,27]. DI is used to represent the severity of damage along a transmission pathway. The probability that a damage appears at a location can be estimated from the DI values of different sensor pairs and their relative position to the sensor pairs [26]. If the active sensing network consists of N sensing paths, the probability of damage occurring at the location (x,y) can be defined as: N
β − R j ( x, y ) DI j β −1 j =1 N
P ( x , y ) = ∑ Pj ( x , y ) = ∑ j =1
(21)
where Pj(x,y) indicates the damage occurrence probability estimated by the sensing path j, DIj(x,y) is the DI value related to sensing path j, and Rj(x,y) is defined as follows:
R j ( x, y) =
( xTj − x) 2 + ( yTj − y ) 2 + ( x Rj − x) 2 + ( y Rj − y ) 2 ( xTj − x Rj )2 + ( yTj − y Rj ) 2
(22)
where (xi,yi) and (xj,yj) are the coordinates of the transmitter and receiver, respectively, Rj(x,y) is the sum of distance from point (x,y) to (xi,yi) and (xj,yj), ?? is a scaling parameter which controls the size of the effective elliptical distribution area. As the scaling parameter ?? takes 1.02, the probabilistic imaging algorithm is used to reconstruct the location and size of damage. Fig. 13 shows the imaging results of the TRM (on the left hand side) and the ITRM (on the right hand side) as the original input signals takes 3-cycle tonebursts centered at 250 kHz, 270 kHz and 290 kHz, respectively. Each image is normalized by its maximum value. The range of actual damage is marked by a circular frame. It can be seen that the imaging results associated with the TRM are fuzzy, and the damage cannot be correctly identified. In contrast, in the results of the ITRM, the color at the damage area is much darker than that in other area, implies a higher probability value for the presence of damage. Especially, the grid with the highest probability value is considered as the central location of identified damage and marked by ‘+’. From Fig. 13(b), (d) and (f), it can be seen that the location of damage is correctly identified, which demonstrates the effectiveness of the ITRM for damage detection in composite plates.
Insert Fig. 13 here
6. Conclusion In this study, an improved time reversal method is proposed, and its applicability in health monitoring of a composite plate is investigated. In particular, a weight vector is
employed to alleviate the effects of the time reversal operator on the reconstructed signal and thus improve the time reversibility of Lamb waves. The weight vector is composed of a window function and amplitude dispersion. The shape of the window function is controlled by the threshold α and the excitation signal. And the amplitude dispersion is related to frequency components and propagation distance of Lamb waves. The effectiveness of the ITRM is verified by experiments. The results show that in the ITRM, the original input signal can be fully reconstructed in the healthy condition. Thus the deviation between the modulated reconstructed signal and the original input signal is only attributed to the presence of damage. Benefitting from that, the damage could be correctly identified by the ITRM. Ultimately, a transducer array is bonded on a CFRP plate, and a DI based probabilistic imaging method is introduced to help identifying the impact damage visually. The imaging results show that the ITRM obviously enhances the performance of Lamb waves for health monitoring of composite plates.
7. Acknowledgement The work is supported by the National Natural Science Foundation of China (Grant No. 51505365, 51421004) and the China Postdoctoral Science Foundation (Grant No. 2015M572552), which are highly appreciated by the authors.
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Technology 2014; 28(2):445~455. [2] Staszewski WJ, Mahzan S, Traynor R. Health monitoring of aerospace composite structures – Active and passive approach. Composites Science and Technology 2009; 69:1678–1685. [3] Park HW, Sohn H, Law KH, Farrard CR. Time reversal active sensing for health monitoring of a composite plate. Journal of Sound and Vibration 2007; 302:50–66. [4] Guo N, Cawley P. The interaction of Lamb waves with delaminations in composite laminates. Acoustical Society of America Journal 1993; 94(4):2240–2246. [5] Rose JL, Pilarski A, Ditri JJ. An approach to guided wave mode selection for inspection of laminated plate. Journal of Reinforced plastics and Composites 1993; 12(5):536–544. [6] Ing RK, Fink M. Time-reversed Lamb waves. IEEE Transactions on Ultrasonics. Ferroelectrics, and Frequency Control 1998; 45(4):1032-1043. [7] Ing RK, Fink M. Time recompression of dispersive lamb waves using a time reversal mirror - Application to flaw detection in thin plates. IEEE Ultrasonics Symposium 1996;1:659-663. [8] Fink M. Time reversal of ultrasonic fields - Part I: Basic principles. IEEE Transactions on Ultrasonic Ferroelectrics and Frequency Control 1992; 39(5):555–566. [9] Fink M. Time-reversed acoustics. Physics Today 1997; 50(3):34–40. [10] Sohn H. Park HW, Law KH, Farrar CR. Damage detection in composite plates by using an enhanced time reversal method. Journal of Aerospace Engineering 2007; 20(3):141–151. [11] Sohn H, Park HW, Law KH, Farrar CR. Combination of a time reversal process and a consecutive outlier analysis for baseline-free damage diagnosis. Journal of Intelligent
Material Systems and Structures 2007; 18(4):335-346. [12] Mustapha S, Lu Y, Li J, Ye L. Damage detection in rebar-reinforced concrete beams based on time reversal of guided waves. Structural Health Monitoring 2014; 13(4):1–12. [13] Mustapha S, Ye L, Dong X, Alamdari MM. Evaluation of barely visible indentation damage (BVID) in CF/EP sandwich composites using guided wave signals. Mechanical Systems and Signal Processing 2016; 76-77:497–517. [14] Agrahari JK, Kapuria S. A refined Lamb wave time-reversal method with enhanced sensitivity for damage detection in isotropic plates. Journal of Intelligent Material Systems and Structures 2015; 163(4):1429–1436. [15] Xu B, Giurgiutiu V. Single mode tuning effects on lamb wave time reversal with piezoelectric wafer active sensors for structural health monitoring 2007; Journal of Nondestructive Evaluation. 26(2):123–134. [16] Poddar B, Kumar A, Mitra M, Mujumdar PM. Time reversibility of a Lamb wave for damage detection in a metallic plate. Smart Materials and Structures 2011; 20(20):2501-2510. [17] Agrahari JK, Kapuria S. Effects of adhesive, host plate, transducer and excitation parameters on time reversibility of ultrasonic Lamb waves. Ultrasonics 2016; 70:147–157. [18] Fink M. Time-reversed acoustics. Scientific American 1999; 281(5):91–97. [19] Draeger C, Cassereau D, Fink M. Theory of the Time reversal Process in Solids. Journal of Acoustic Society of America 1997; 102(3):1289–1295. [20] Park HW, Kim SB, Sohn H. Understanding a time reversal process in Lamb wave propagation. Wave Motion 2009; 46(7):451–467. [21] Gresil M, Giurgiutiu V. Prediction of attenuated guided waves propagation in carbon
fiber composite using Rayleigh damping model. Journal of Intelligent Material Systems and Structures 2015; 26(16):2151-2169. [22] Belanger P, Cawley P. Lamb wave tomography to evaluate the maximum depth of corrosion patches. In: 34th Annual Review of Progress in Quantitative Nondestructive Evaluation, Golden, CO, 22-27 July 2007. P:1290-1297. [23] Schubert KJ, Herrmann AS. On attenuation and measurement of Lamb waves in viscoelastic composites. Composite Structures 2011; 94(1):177–185. [24] Su Z, Ye L. Identification of Damage Using Lamb Waves: From Fundamentals to Applications. Springer London, 2009. [25] Kassahun A, Larry H, Mannur S. Influence of attenuation on acoustic emission signals in carbon fiber reinforced polymer panels. Ultrasonics 2015; 59:86–93. [26] Zhao X, Gao H, Zhang G, Ayhan B, Yan F, Kwan C, Rose JL. Active health monitoring of an aircraft wing with embedded piezoelectric sensor/actuator network: I. Defect detection, localization and growth monitoring. Smart Materials and Structures 2007; 16(4):1208–1217. [27] Zeng L, Lin J, Hua J, Shi W. Interference resisting design for guided wave tomography. Smart Materials and Structures 2013; 22(5):955-962.
(1) Original input signal
Compared
(2) Response signal
Time reversed
PZT A PZT B
(3) Re-emitted signal
Fig. 1. Schematic diagram of time reversal process in a composite plate.
Modulate (1) Original input signal
Compared
(2) Response signal
PZT A
(6) Modulated reconstruct (5) Reconstructed signal signal
(3) Modulated response signal
Time reversed PZT B
(4) Re-emitted signal
Modulate
Fig. 2. Schematic diagram of improved time reversal process in a composite plate.
200
690
280
130 ╘
690
╗
A
╙
B
70 C ╚
Unite:mm
Fig. 3. Schematic diagram of the composite plate with PZTs.
Normalized amplitude Fig.4. Normalized amplitude dispersion of S0 mode.
(b)
Normalized amplitude
Normalized amplitude
(a)
(c)
(d)
Fig. 5 Comparison of TRM and ITRM for sensing path PZT A to PZT B: (a) response signal. (b) modulated response signal. (c) reconstructed signal in TRM (d) modulated reconstructed signal in ITRM.
Normalized amplitude
(a)
(b)
Fig. 6 The shape of window function at different threshold values (a), and the evolution of the DI value with the increase of threshold α (b).
Normalized amplitude
Normalized amplitude
(a)
(b)
Fig. 7 Comparison of the spectrum of original input signal and reconstructed signal (a) for TRM (b) and for ITRM.
(a)
(b) ITRM Reconstructed Signal Original Input Signal Envelope
1 0.5 0 -0.5 -1 0.08
(c)
0.1
0.12 Time (ms)
0.14
(d)
Fig. 8 Comparison of TRM and ITRM for sensing path PZT A to PZT C: (a) response signal. (b) modulated response signal. (c) reconstructed signal in TRM (d) modulated reconstructed signal in ITRM.
(a)
(b)
Fig. 9 Diagram of the composite specimen (the white circular is the location of the impact damage) (a) and coordinate of PZTs and monitoring area (b).
Fig. 10. S0 mode amplitude dispersion curves of paths 4, 5 and 6.
Damage Index Fig. 11. DI values for the ITRM varies with threshold.
0.3
Damage Index
0.25 0.2 0.15 0.1 0.05 0 0
5
10
15
Path
(a)
(b)
Fig. 12 DI Values of all sensing paths with a Fc=270 kHz, Cyc=3 toneburst excitation: (a) ITRM and (b) TRM.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 13 Imaging results of damage associated with the TRM (on the left hand side) and the RTRM (on the right hand side) as the 3-cycle tonebursts centered at different frequencies: (a) and (b) 250 kHz; (c) and (d) 270 kHz; (e) and (f) 290 kHz (the circular frame is the range of actual damage, ‘+’ is the central
location of the identified damage).
Table 1. Material properties of a single ply of the CFRP plate Material
ρ 3
E1
E2
G 12
properties
(kg/m )
(GPa)
(GPa)
(GPa)
Value
1560
135
10.9
4.7
v12
v 23
0.285
0.4
S0263-8223(17)30076-4 https://doi.org/10.1016/j.compstruct.2018.01.096 COST 9336
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
10 January 2017 15 December 2017 30 January 2018
Please cite this article as: Huang, L., Zeng, L., Lin, J., Luo, Z., An improved time reversal method for diagnostics of composite plates using Lamb waves, Composite Structures (2018), doi: https://doi.org/10.1016/j.compstruct. 2018.01.096
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An improved time reversal method for diagnostics of composite plates using Lamb waves
Authors Liping Huang PhD Candidate School of Mechanical Engineering Xi’an Jiaotong University Xi’an, Shaanxi Province, China, 710049 Email: [email protected] Liang Zeng PhD, Lecturer School of Mechanical Engineering Xi’an Jiaotong University Xi’an, Shaanxi Province, China, 710049 Email: [email protected] Jing Lin∗ PhD, Professor 1 State Key Laboratory for Manufacturing System Engineering, Xi’an Jiaotong University Xi’an, Shaanxi Province, China, 710049 2 School of Reliability and Systems Engineering, Beihang University, Xueyuan Road No. 37, Haidian District, Beijing, China, 100191 Email: [email protected]
Zhi Luo PhD Candidate School of Mechanical Engineering Xi’an Jiaotong University Xi’an, Shaanxi Province, China, 710049 Email: [email protected]
∗
Corresponding author
Abstract The time reversal method has been widely used to detect damage in composite plates. Based on the time reversibility of Lamb waves, the waveform of the original input signal will be reconstructed after the time reversal process. However, the time reversal operator of Lamb waves in composite plates is a function of frequency. Thus the shape of the reconstructed signal will not be identical to that of the original input signal. To alleviate the effects of the time reversal operator, an improved time reversal method (ITRM) is proposed. In this method, the response signal in the forward path and the reconstructed signal in the time reversed path are both modulated by a weight vector which is obtained as the product of a window function and the reciprocal of amplitude dispersion. The window function varies with the excitation signal adaptively, and its shape is also determined by a threshold. An experimental example is then introduced to illustrate the effectiveness of the ITRM. The results show that the time reversibility can be greatly enhanced and the impact damage in the composite plate can be exactly located by the ITRM.
Keywords: Lamb wave; Composite; Time reversal operator; Damage detection
1. Introduction Composite material has been widely used in many different fields (aerospace, civil infrastructure, energy engineering, and automobile industries, etc.) for its specific properties, such as high strength, high stiffness and light weight. However, composite materials are often damaged by low velocity impacts for their low transverse strength [1]. Low velocity impacts (e.g. bird strikes, tool drops or runway stones) loaded on composite materials may result in various types of damage such as delamination, indentation, fiber breakage or matrix cracking [2]. Such damages are often invisible and may lead to catastrophic failure. Hence, it calls for an effective and efficient inspecting technique which could detect the damage promptly and thus ensure the safety of these composite materials. Lamb waves are mechanical waves with wavelength that is in the same order as the thickness of the plate [3]. With the advantages including long propagation distance, low cost and good sensitivity to various defects, Lamb waves have attracted tremendous attention for testing and evaluating composite materials [4,5]. However, propagation of Lamb waves in composite materials are notoriously complicate due to their dispersive characteristic. For each mode, Lamb wave components at different frequencies will propagate at different velocities, which lead to wave deformation and amplitude decrease as they travel in composite plates. Therefore, the time reversal method has been widely used to compensate the velocity dispersion and improve the quality of Lamb waves [6,7]. The time reversal concept was first proposed in acoustics by Fink [8,9]. According to the principle of time reversal, an original input signal can be reconstructed at the transmitter location, if the response signal captured by a receiver is reversed in the time domain and
re-emitted back through the receiver to the transmitter. The time reversibility of Lamb waves is based on the principle of spatial reciprocity and time reversal invariance of linear acoustic wave equations [6]. However, damage will introduce nonlinearity into the structure and break down the time reversal invariance. Hence, it can be diagnosed by comparing the waveform of the reconstructed signal with that of the original input signal [10]. Sohn et al. [11] proposed a baseline-free damage detection method by applying the combination of consecutive outlier analysis and the TRM. Mustapha et al. [12] extended TRM to detect damage in reinforced concrete beams, and [13] proposed a new damage index (DI) to detect the location of the barely visible indentation damage in sandwich CF/EP composites. Agrahari et al. [14] developed a refined time reversal method to enhance the sensitivity of damage detection in isotropic plates. Due to the effects of the time reversal operator, the time reversibility may be a challenge that the TRM faces. To overcome this problem, comprehensive theoretical and experimental analysis of the time reversibility of Lamb waves was performed. Park et al. [10] proposed a wavelet-based signal processing technique to extract narrowband responses to enhance the time reversibility of Lamb waves in a composite plate. Xu et al. [15] presented that within-mode Lamb wave can be fully reconstructed when the frequency bandwidth of excitation signal is narrow. Poddar et al. [16] compared the time reversal reconstruct quality under different parameters, i.e., the excitation frequency and bandwidth, as well as the piezoelectric (PZT) geometry. Agrahari [17] investigated the effects of adhesive layer between the transducers and the host plate, the tone burst count of the excitation signal, the plate thickness and the piezoelectric transducer thickness on the time reversibility of Lamb waves. These studies could help alleviating the frequency dependency of the time-reversal
operator, but did not suggest a proper method to compensate that. The objective and motivation of this paper is to establish a methodology that can compensate the frequency dependency of the time reversal operator so that the time reversibility of Lamb waves could be enhanced. In the proposed method (i.e., ITRM), the amplitude dispersion compensation of the response signal in the forward path is achieved by placing a weight vector on its spectrum. The compensated signal is then reversed in the time domain and re-emitted through the receiver to the transmitter. The reconstructed signal captured at the transmitter location still suffers amplitude dispersion in the time reversed path. Thus the amplitude dispersion compensation is also applied to the reconstructed signal. The rest of the paper is organized as follows. The basic theory of time reversibility of Lamb waves is briefly reviewed in Section 2. The ITRM and its theoretical analysis are given in Section 3. Section 4 illustrates the process of the ITRM by an experiment example. In Section 5, the ITRM is applied to detect impact damage in a composite plate. Finally, the conclusions are drawn in Section 6.
2. Time reversal of Lamb waves in composite plates The concept of time reversal was first proposed in acoustics, and usually applied in the fields of medical, hydrodynamics and nondestructive examination [18]. An application of the time reversal concept to Lamb waves in solids is then studied [6,19]. In the time reversal process of Lamb waves, an original input signal can be reconstructed at the location of transmitter A if the response signal captured by receiver B is time reversed and re-emitted to the location of original transmitter A. The schematic diagram of TRM is shown in Fig.1.
During the process of TRM, to reverse the path direction, the transducer (e.g., PZT) changes its role from transmitter to receiver and vice versa.
Insert Fig. 1 here
The formula deduction procedures of TRM in the frequency domain are presented as follows. For the piezoelectricity of PZT materials, an electrical voltage VA(ω) excited at transmitter PZT A can be converted to mechanical strain ɛA(ω) [20].
ε A (ω ) = ka (ω )VA (ω )
(1)
where ka(ω) is the electromechanical coupling coefficient, and ω is the angular frequency. If Lamb waves are excited at PZT A and received at PZT B as shown in Fig. 1, the response signal captured by PZT B satisfies,
VB ( ω, r ) = k b−1 (ω ) ka (ω )VA (ω ) H (ω , r )
(2)
where r is the propagation distance of Lamb waves from transmitter to receiver, H(ω,r) is the structural transfer function of the sensing path, and kb(ω) is the electromechanical coupling coefficient of PZT B. If PZTs A and B are the same, ka(ω)=kb(ω) [13]. Hence, the Eq. (2) could be simplified as,
VB (ω, r ) = VA (ω ) H (ω, r )
(3)
It is noted that the time reverse operation on a signal is equivalent to taking the complex conjugate in the frequency domain. Consequently, the time reversal of the response signal captured by PZT B [i.e., Eq. (3)] can be written as
V B* ( ω, r ) = V A* (ω ) H * (ω , r )
(4)
where * represents a complex conjugate and V* B(ω) is the time reversed signal. Finally, V*B(ω) is re-emitted through PZT B and the corresponding response signal at the location of PZT A is the reconstructed signal VR(ω,r) which is defined as
VR (ω, r ) = VB* (ω ) H (ω, r )
(5)
Substituting Eq. (4) into Eq. (5), we obtain
VR (ω, r ) = VA* (ω ) H * (ω, r ) H (ω, r )
(6)
and the time reversal operator of Lamb waves HTR(ω,r) is defined as, H TR (ω , r ) = H * (ω , r ) H (ω , r )
(7)
The TRM used for damage detection is based on the principle of linear reciprocity [6,7], that is, the excitation signal can be fully reconstructed after the time reversal process. While sources of nonlinearity on the wave propagation path will break down the time reversibility of Lamb waves and the waveform of the reconstructed signal will be different from that of the original input signal. Damages in composite plates, such as low velocity impact, delamination, cracks and fiber breakage, may introduce nonlinearity, thus damages could be detected by comparing the waveform of the reconstructed signal with that of the original input signal. A damage index (DI), which is calculated to analyze the deviation between the reconstructed signal and the excitation signal, is defined as follows: [10] t1
DI = 1 − ∫ VA ( t )Vr ( t ) dt t0
∫
t1
t0
t1
VA ( t ) dt ∫ Vr ( t ) dt 2
t0
2
(8)
where VA(t) is the original input signal, and Vr(t) is the reconstructed signal, both of which are normalized with their maximum value. t0 and t1 denote the starting and ending time points of the signal for comparison. The value of DI falls within the range of [0,1]. DI = 0 represents that the two signals exactly coincide with each other. The value of DI increases and
approaches 1 with the growing difference between the reconstructed signal and the original input signal. From Eq. (6), it can be concluded that if the time reversal operator keeps constant over the whole excitation frequency range, DI will be zero when the structure is free of damage, and thus damage could be detected by evaluating the value of DI. However, for Lamb waves propagating in composite plates, the time reversal operator is generally frequency dependency, and the original input signal will not be fully reconstructed.
3. Improved time reversal of Lamb waves in composite plates
Referring to [21], the transfer function in composite plates is defined as follows: H ( ω , r ) = A (ω ) e −η (ω ) r
1 r
e − ik (ω ) r
(9)
where, A(ω) is the excitability of a particular mode [22], η(ω) is the attenuation coefficient and k(ω) is the wave number [21]. From the above equation, it can be observed that the transfer function of Lamb waves propagating in a composite plate is mainly affected by four factors, which are transducers, material damping, geometric spreading and wave dispersion. Among these, the excitability A(ω) is greatly influenced by the type and size of transducers and adhesive layers between transducers and composite plates. Material damping coefficient α(ω) describes how much energy dissipates of Lamb waves, which is induced by the viscoelasticity of carbon fiber reinforced materials. 1/√r represents geometric spreading which is caused by the growing length of a wave front departing into all directions from a source [23]. The wave dispersion e-ik(ω)r is a fundamental characteristic of Lamb waves and can be fully compensated by the TRM.
Thus, the time reversal operator HTR(ω,r) in a composite plate [i.e., Eq. (7)] can be represented by the following equation, H TR (ω , r ) = A2 ( ω ) e
−2η ( ω ) r
1 r
(10)
It means that the time reversal operator varies with the frequency components and the propagation distance of Lamb waves. In this case, wave components at different frequencies are non-uniformly scaled. As a result, the waveform of the reconstructed signal cannot exactly match that of the original input signal even in healthy structures. According to the definition of DI in Eq. (8), the deviation caused by the time reversal operator would also be captured by the DI. Thus a non-zero damage index arises and implies the existence of damage, which would contaminate the identification result and lead to a false alarm. To overcome this problem, an ITRM is proposed. Its schematic diagram is given in Fig. 2. The ITRM consists of five steps, i.e.,
Insert Fig. 2 here
(1) an original input signal is emitted from PZT A, and the response is captured by PZT B; (2) the response signal is modulated by a weight vector. (3) the modulated response signal is reversed in the time domain and then re-emitted through PZT B; (4) the reconstructed signal is captured at transmitter PZT A, and then modulated by the same weight vector as in step 2;
(5) the modulated reconstructed signal is then normalized with respect to its peak value and compared with the original input signal. The theoretical analysis of the ITRM is derived in the frequency field as follows. In the ITRM, the response signal VB(ω,r) in Eq. (3) is modulated by a weight vector, and the modulated response signal VBM(ω,r) is
VBM (ω , r ) = VB ( ω, r )W (ω , r )
(11)
where W(ω,r) is a weight vector. Ideally, if the Lamb wave signal is free of noise, a direct way to compensate the effects of the time reversal operator could be achieved by an inverse filter procedure, i.e., W(ω,r) =1/|H(ω,r)|. Here, |H(ω,r)| is the amplitude of the transfer function as Lamb waves propagating in a composite plate. It is a function of frequency, which is called amplitude dispersion. H (ω , r ) = A ( ω ) e−η (ω ) r
1 r
(12)
In practical applications, however, the Lamb wave signals are always contaminated by noises, and the estimated transfer function cannot accurately represent the actual one. In this case, the inverse filter procedure faces a challenge that it is instable and noise may be amplified during the modulation process, especially for the frequency range where the desired Lamb wave mode is quite weak [i.e., |H(ω,r)| ≈ 0]. To address this issue, the weight vector is modified as, W (ω , r ) =
1 H (ω , r )
T (ω )
(13)
where T(ω) is a window function which is defined as 1 T (ω ) = VA (ω ) / α
V A (ω ) ≥ α V A (ω ) < α
(14)
here, |VA(ω)| is the normalized magnitude spectrum of the original input signal, and ??∈[0,1] is a threshold which controls the shape of the window. If ?? = 0, the window function T(ω) becomes rectangular, and the weight vector W(ω,r) returns to the inverse filter. If ?? = 1, T(ω) = |VA(ω)|. Referring to [24], Hanning windowed tonebursts are often used as the input signals for Lamb wave inspections, and the operation frequency is selected within the frequency range where the desired Lamb mode is strong. If 0
VBM (ω, r ) = VA (ω ) H (ω, r )
T (ω ) H (ω, r )
= VA (ω ) T (ω ) e
− ik ( ω ) r
(15)
In frequency domain, time reversal of the modulated response signal is defined as, V BM* (ω , r ) = VA* (ω ) T * (ω ) e
ik ( ω ) r
(16)
Then the time reversed modulated response signal is re-emitted through transducer PZT B and received by PZT A, resulting in * VR ( ω, r ) = VBM (ω , r ) H (ω , r )
where VR(ω,r) is the reconstructed signal in the ITRM.
(17)
Substituting Eq. (16) into Eq. (17), we obtain ik ( ω ) r
VR (ω, r ) = VA* ( ω ) T * (ω ) e
− ik (ω ) r
H (ω, r ) e
= VA* (ω ) T * (ω ) H (ω, r )
(18)
Then the reconstructed signal is modulated by the weight vector and the modulated reconstructed signal VRM (ω,r) is defined as VRM (ω ) = VR (ω , r )W ( ω , r ) = V A* (ω ) T * ( ω ) T (ω )
(19)
Thus, we obtain V A* ( ω ) VRM ( ω ) = * V A ( ω ) VA ( ω ) / α
VA (ω ) ≥ α 2
VA (ω ) < α
(20)
Finally, the modulated reconstructed signal in time domain, i.e., Vrm(t), is obtained through inverse Fourier transform, and the DI in Eq. (8) is calculated by measuring the discrepancies between this modulated reconstructed signal and the excitation signal.
4. Experimental investigation In this section, an experiment is designed to illustrate the process of the ITRM. The experiment setup consists of an Agilent 33220A function/arbitrary waveform generator, a NF HSA4012 voltage amplifier and a NI PXIe-1082 data acquisition. The experiment is carried out on a T300/7901 16-ply quasi-isotropic carbon fiber-reinforced polymer (CFRP) laminates with [+45/-45/0/90]2s lay-up. The dimension of the specimen plate is 690 mm×690 mm×2 mm and the thickness of each laminate is 0.125 mm. The material properties of the composite specimen are given in table 1.
Insert Table 1 here
The schematic diagram of the composite plate is shown in Fig. 3. PZT A serves as the transmitter, while PZTs B and C serve as the receivers. Compared with the A0 mode, S0 mode has the advantage of high energy, low attenuation and high signal-to-noise ratio when propagating in composite plates [25]. Therefore, S0 mode is only concerned in this experiment.
Insert Fig. 3 here
A series of 5-cycle sinusoidal toneburst signals with center frequencies increase from 50 kHz to 700 kHz in steps of 500 Hz are excited in turn at the transmitter. The maximum peak of the S0 mode Lamb wave response signals of different excitation parameters are divided by that of their excitation signals, respectively, to calculate amplitude dispersion curve. Fig. 4 shows the normalized amplitude dispersion curve of S0 mode Lamb wave transmitted from PZT A to PZT B.
Insert Fig. 4 here
For instance, a 240 kHz, 3-cycle toneburst is applied to PZT A as the original input signal. The response signal captured by PZT B is displayed in Fig. 5(a). It can be seen that the direct wave (sensing path ① in Fig. 3) and the reflected wave (sensing path ② in Fig. 3) are separated in time domain. Hilbert transform is then applied to the response signal, and the associated envelope is also given in Fig. 5(a) as the red dashed line. The waveform in the time
range where the magnitude of envelope is larger than 0.1 times that of the peak time (i.e., the waveform enclosed by the dashed rectangle in Fig. 5(a)) is time reversed and re-emitted by PZT B. Fig. 5(c) gives the comparison between the reconstructed signal captured at PZT A (i.e., the blue solid line) and the original input signal (i.e., the red dashed line). It is apparent that the two signals are quite different from each other, arising a DI value of 0.3458 in the undamaged case.
Insert Fig. 5 here
In the ITRM, the threshold α determines the shape of the window function T(ω), Fig. 6(a). Since T(ω) improves the robustness of the ITRM at the cost of compressing the bandwidth of the Lamb wave signal, the threshold α provides a compromise between the compensation degree and the noise resistance. In this paper, the threshold is determined as follows. The threshold α increases from 0 to 1 with a step of 0.05. At each threshold value, the modulated reconstructed signal in the ITRM is obtained. Subsequently, the DI is calculated to measure the discrepancy between the original input signal and the modulated reconstructed signal. Since the structure is free of damage, the smaller the DI value, the better the chosen threshold ??. Fig. 6(b) shows the evolution of DI value with the increase of threshold. It can be observed that the minimum DI value sits at the threshold α = 0.45.
Insert Fig. 6 here
As the threshold ?? takes 0.45, the response signal displayed in Fig. 5(a) is then modulated by the weight vector [i.e., Eq. (11)], and the modulated response signal is shown in Fig. 5(b). Since the direct wave is isolated in the time domain, it could be easily extracted as the wave-packet enclosed by the dashed rectangle. Subsequently, the direct wave is time reversed and applied to PZT B as the re-emitted signal. The reconstructed signal captured by PZT A is also modulated by the weight vector. Fig. 5(d) shows the comparison between the modulated reconstructed signal and the original input signal. It can be seen that the original input signal can be exactly reconstructed by using the ITRM. Accordingly, the DI value calculated from Eq. (8) reduces to 0.0018. Fig. 7(a) gives the comparison between the magnitude spectra (after Fourier transform) of the original input signal and the reconstructed signal in the TRM. It can be seen that compared to the original input signal, the peak frequency of the reconstructed signal shifts from 240 kHz to 294 kHz, and its frequency band is obviously narrowed due to the time reversal operator. That is the main reason why the reconstructed signal in the TRM is quite different from the original input signal, Fig. 5(c). In the ITRM, the time reversal operator is compensated by the modulations of the response signal and the reconstructed signal. As shown in Fig. 7(b), the spectra of the original input signal and the modulated reconstructed signal match well. Hence, the original input signal is fully reconstructed in the ITRM, Fig. 5(d).
Insert Fig. 7 here
For further illustration, the sensing path PZT A to PZT C is also considered. Fig. 8(a) shows the response signal captured by PZT C, where the direct wave (sensing path ③ in Fig. 3) and the reflected wave (sensing path ④ in Fig. 3) interfere with each other in time domain. In this case, the waveform in the time range where the magnitude of envelope is larger than 0.1 times that of the peak time [the waveform enclosed by the dashed rectangle in Fig. 8(a)] consists of both the direct wave and the reflected wave. Fig. 8(c) shows the reconstructed signal captured by PZT A as that waveform is time reversed and re-emitted through PZT C. It is apparent that the reconstructed signal deviates from the original input signal. Besides the time reversal operator, this deviation may also be attributed the overlap of multiple waves at the main lobe of the reconstructed signal [20]. For comparison, Fig. 8(b) displays the modulated response signal in the ITRM as the threshold ?? takes 0.45. It can be seen that the direct wave and the reflected wave are separated in the time domain. The reason behind this phenomenon may be explained as follows. At the operation frequency (i.e., 240 kHz), the velocity dispersion of S0 mode may be gentle, and the increase of duration of the direct wave mainly comes from the distortion of waveform caused by amplitude dispersion. In the ITRM, the effect of amplitude dispersion is compensated by the weight vector in Eq. (11). Thus, the duration of the direct wave decreases. Benefitting from that, it could be easily extracted in time domain, as the waveform enclosed by a dashed rectangle in Fig. 8(b). The comparison between the modulated reconstructed signal and the original input signal is given in Fig. 8(d). It can be seen that the original input signal can be exactly reconstructed by using the ITRM. That demonstrates another merit of the ITRM over the TRM.
Insert Fig. 8 here
5. Application time reversal for damage detection The application of the ITRM for damage detection of composite plates is investigated. The experiment setup and the composite specimen are the same as in the Section 4. Fig. 9(a) shows the composite specimen and sensor layout. 8 PZTs (circular, P-51, with dimensions of
Φ8 mm×0.5 mm) bonded on the CFRP plate are networked as a square senor array. In the active sensor network, the PZT elements take turns generating Lamb wave signals while the rest of them are listening. Totally 16 paths are available for defect imaging, as shown in Fig. 9(b). A rectangular Cartesian coordinate system is established with the center of the sensor array as the origin, and the coordinates of center of all PZTs are illustrated in Fig. 9(b). The monitoring area enclosing the impact damage is 280 mm×280 mm, and the distance between the adjacent PZTs is 140 mm. The composite plate was loaded in a quasi-static indentation apparatus with a 20 mm diameter indenter to introduce impact damage centering at [46, -46], i.e., the white-filled circle in Fig. 9(a).
Insert Fig. 9 here
A series of 5-count sinusoidal toneburst signals with center frequencies increase from 50 kHz to 600 kHz in steps of 500 Hz are excited in turn to obtain the amplitude dispersion curve of S0 mode Lamb wave of all sensing paths. For instance, the amplitude dispersion curves of path 4, 5 and 6 are displayed in Fig. 10.
Insert Fig. 10 here
In the ITRM, the key problem is to determine the weight vector. As mentioned, the shape of the window function depends on the original input signal and threshold ??. In this experiment, the original input signals are 3cycles, tonebursts centered at 250 kHz, 270 kHz and 290 kHz, respectively. Fig. 11 gives the relationship between the threshold ?? and the DI value. The abscissa is the threshold ??, which varies from 0 to 1 with a step of 0.05, and the ordinate is the average of the DI values of all 16 paths. In the healthy condition, the minimum DI value corresponds to the optimal choice of threshold ??. For illustration, as the center frequency of toneburst takes 250 kHz, 270 kHz and 290 kHz in turn, the optimal threshold ?? are 0.5, 0.3 and 0.2, respectively (Fig. 11).
Insert Fig. 11 here
Once the optimal choice of threshold is achieved, the DI values of all sensing paths could be calculated from the comparison between the original input signal and the modulated reconstructed signal. For instance, Fig. 12(a) shows the DI values as the input signal takes a 3-cycle 270 kHz toneburst. It can be seen that the sensing paths which cross the damage area (i.e., Path 1 and Path 10) are of high DI values, while the rest sensing paths are of low ones. This result verifies the effectiveness of the ITRM for damage detection. For comparison, in the TRM, the DI measures the discrepancy between the original input signal and the
reconstructed signal. Fig. 12(b) displays the DI values of all sensing paths calculated from the TRM. It can be seen that the result is chaotic, and the sensing paths which cross the damage cannot be correctly identified. The reason lies that in the TRM, besides the damage, the time reversal operator also changes the waveform of the reconstructed signal [i.e., Eq. (6)]. The combination of these two effects may not always leads to a larger DI value. As a result, the damage cannot be detected. In the ITRM, the effects of the time reversal operator are fully compensated by the modulations of the response signal and the reconstructed signal. Benefitting from that, the original input signal could be well reconstructed in the ITRM, and the deviation of the modulated reconstructed signal is only attributed to the presence of damage. Hence, its performance improves obviously.
Insert Fig. 12 here
Subsequently, the probabilistic imaging algorithm is introduced to visually pinpoint the damage [26,27]. DI is used to represent the severity of damage along a transmission pathway. The probability that a damage appears at a location can be estimated from the DI values of different sensor pairs and their relative position to the sensor pairs [26]. If the active sensing network consists of N sensing paths, the probability of damage occurring at the location (x,y) can be defined as: N
β − R j ( x, y ) DI j β −1 j =1 N
P ( x , y ) = ∑ Pj ( x , y ) = ∑ j =1
(21)
where Pj(x,y) indicates the damage occurrence probability estimated by the sensing path j, DIj(x,y) is the DI value related to sensing path j, and Rj(x,y) is defined as follows:
R j ( x, y) =
( xTj − x) 2 + ( yTj − y ) 2 + ( x Rj − x) 2 + ( y Rj − y ) 2 ( xTj − x Rj )2 + ( yTj − y Rj ) 2
(22)
where (xi,yi) and (xj,yj) are the coordinates of the transmitter and receiver, respectively, Rj(x,y) is the sum of distance from point (x,y) to (xi,yi) and (xj,yj), ?? is a scaling parameter which controls the size of the effective elliptical distribution area. As the scaling parameter ?? takes 1.02, the probabilistic imaging algorithm is used to reconstruct the location and size of damage. Fig. 13 shows the imaging results of the TRM (on the left hand side) and the ITRM (on the right hand side) as the original input signals takes 3-cycle tonebursts centered at 250 kHz, 270 kHz and 290 kHz, respectively. Each image is normalized by its maximum value. The range of actual damage is marked by a circular frame. It can be seen that the imaging results associated with the TRM are fuzzy, and the damage cannot be correctly identified. In contrast, in the results of the ITRM, the color at the damage area is much darker than that in other area, implies a higher probability value for the presence of damage. Especially, the grid with the highest probability value is considered as the central location of identified damage and marked by ‘+’. From Fig. 13(b), (d) and (f), it can be seen that the location of damage is correctly identified, which demonstrates the effectiveness of the ITRM for damage detection in composite plates.
Insert Fig. 13 here
6. Conclusion In this study, an improved time reversal method is proposed, and its applicability in health monitoring of a composite plate is investigated. In particular, a weight vector is
employed to alleviate the effects of the time reversal operator on the reconstructed signal and thus improve the time reversibility of Lamb waves. The weight vector is composed of a window function and amplitude dispersion. The shape of the window function is controlled by the threshold α and the excitation signal. And the amplitude dispersion is related to frequency components and propagation distance of Lamb waves. The effectiveness of the ITRM is verified by experiments. The results show that in the ITRM, the original input signal can be fully reconstructed in the healthy condition. Thus the deviation between the modulated reconstructed signal and the original input signal is only attributed to the presence of damage. Benefitting from that, the damage could be correctly identified by the ITRM. Ultimately, a transducer array is bonded on a CFRP plate, and a DI based probabilistic imaging method is introduced to help identifying the impact damage visually. The imaging results show that the ITRM obviously enhances the performance of Lamb waves for health monitoring of composite plates.
7. Acknowledgement The work is supported by the National Natural Science Foundation of China (Grant No. 51505365, 51421004) and the China Postdoctoral Science Foundation (Grant No. 2015M572552), which are highly appreciated by the authors.
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(1) Original input signal
Compared
(2) Response signal
Time reversed
PZT A PZT B
(3) Re-emitted signal
Fig. 1. Schematic diagram of time reversal process in a composite plate.
Modulate (1) Original input signal
Compared
(2) Response signal
PZT A
(6) Modulated reconstruct (5) Reconstructed signal signal
(3) Modulated response signal
Time reversed PZT B
(4) Re-emitted signal
Modulate
Fig. 2. Schematic diagram of improved time reversal process in a composite plate.
200
690
280
130 ╘
690
╗
A
╙
B
70 C ╚
Unite:mm
Fig. 3. Schematic diagram of the composite plate with PZTs.
Normalized amplitude Fig.4. Normalized amplitude dispersion of S0 mode.
(b)
Normalized amplitude
Normalized amplitude
(a)
(c)
(d)
Fig. 5 Comparison of TRM and ITRM for sensing path PZT A to PZT B: (a) response signal. (b) modulated response signal. (c) reconstructed signal in TRM (d) modulated reconstructed signal in ITRM.
Normalized amplitude
(a)
(b)
Fig. 6 The shape of window function at different threshold values (a), and the evolution of the DI value with the increase of threshold α (b).
Normalized amplitude
Normalized amplitude
(a)
(b)
Fig. 7 Comparison of the spectrum of original input signal and reconstructed signal (a) for TRM (b) and for ITRM.
(a)
(b) ITRM Reconstructed Signal Original Input Signal Envelope
1 0.5 0 -0.5 -1 0.08
(c)
0.1
0.12 Time (ms)
0.14
(d)
Fig. 8 Comparison of TRM and ITRM for sensing path PZT A to PZT C: (a) response signal. (b) modulated response signal. (c) reconstructed signal in TRM (d) modulated reconstructed signal in ITRM.
(a)
(b)
Fig. 9 Diagram of the composite specimen (the white circular is the location of the impact damage) (a) and coordinate of PZTs and monitoring area (b).
Fig. 10. S0 mode amplitude dispersion curves of paths 4, 5 and 6.
Damage Index Fig. 11. DI values for the ITRM varies with threshold.
0.3
Damage Index
0.25 0.2 0.15 0.1 0.05 0 0
5
10
15
Path
(a)
(b)
Fig. 12 DI Values of all sensing paths with a Fc=270 kHz, Cyc=3 toneburst excitation: (a) ITRM and (b) TRM.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 13 Imaging results of damage associated with the TRM (on the left hand side) and the RTRM (on the right hand side) as the 3-cycle tonebursts centered at different frequencies: (a) and (b) 250 kHz; (c) and (d) 270 kHz; (e) and (f) 290 kHz (the circular frame is the range of actual damage, ‘+’ is the central
location of the identified damage).
Table 1. Material properties of a single ply of the CFRP plate Material
ρ 3
E1
E2
G 12
properties
(kg/m )
(GPa)
(GPa)
(GPa)
Value
1560
135
10.9
4.7
v12
v 23
0.285
0.4