- Email: [email protected]
A Lamb wave signal reconstruction method for high-resolution damage imaging
Chinese Journal of Aeronautics, (2019), 32(5): 1087–1099
Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
A Lamb wave signal reconstruction method for high-resolution damage imaging Xiaopeng WANG, Jian CAI *, Zhiquan ZHOU State Key Lab of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Received 25 January 2018; revised 13 September 2018; accepted 7 January 2019 Available online 21 March 2019
KEYWORDS Damage imaging; Dispersion compensation; High resolution; Lamb waves; Signal reconstruction
Abstract In Lamb wave-based Structural Health Monitoring (SHM), a high-enough spatial resolution is highly required for Lamb wave signals to ensure the resolution and accuracy of damage detection. However, besides the dispersion characteristic, the signal spatial resolution is also largely restricted by the space duration of excitation waveforms, i.e., the Initial Spatial Resolution (ISR) for the signals before travelling. To resolve the problem of inferior signal spatial resolution of Lamb waves, a Lamb Wave Signal Reconstruction (LWSR) method is presented and applied for highresolution damage imaging in this paper. In LWSR, not only a new linearly-dispersive signal is reconstructed from an original Lamb wave signal, but also the group velocity at the central frequency is sufficiently decreased. Then, both dispersion compensation and ISR improvement can be realized to achieve a satisfying signal spatial resolution. After the frequency domain sensing model and spatial resolution of Lamb wave signals are firstly analyzed, the basic idea and numerical realization of LWSR are discussed. Numerical simulations are also implemented to preliminarily validate LWSR. Subsequently, LWSR-based high-resolution damage imaging is developed. An experiment of adjacent multiple damage identification is finally conducted to demonstrate the efficiency of LWSR and LWSR-based imaging methods. Ó 2019 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Lamb waves are a kind of guided ultrasonic waves existing in thin-wall structures. With the ability of long distance travelling * Corresponding author. E-mail address: [email protected] (J. CAI). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
and high sensitivity to both surface and internal defects, Lamb waves have been widely used as a promising tool in Structural Health Monitoring (SHM) of plate-like structures.1–6 In practical applications, the spatial resolution of Lamb wave signals is undoubtedly a vital factor to determine the resolution and accuracy of damage detection. For a diagnostic signal of inferior spatial resolution, its useful defect-relevant wavepackets are easily overlapped and interfered with each other or by others. Signal interpretation would become less straightforward to make the extraction of damage information difficult. As a result, final detection results are badly affected. To ensure the resolution and performance of Lamb wave damage identi-
https://doi.org/10.1016/j.cja.2019.03.001 1000-9361 Ó 2019 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1088 fication, an adequately high spatial resolution of Lamb wave signals is required. Due to multi-mode and dispersion characteristics, the propagation of Lamb waves is extraordinarily complicated. Even for single-mode signals, dispersive wavepackets will spread out in time and space with their waveforms distorted as they propagate.7 This can definitely degrade the signal spatial resolution. To resolve the dispersion problem, a Time Reversal Process (TRP) was introduced to automatically compensate the dispersion effect without any priori-knowledge of Lamb wave dispersion properties.8 Unfortunately, the Time of Flight (TOF) of the compensated signals is also eliminated at the same time to result in much inconvenience to damage localization.9 Another strategy for Lamb wave dispersion compensation is Fourier-domain signal processing using wave dispersion relations. Wilcox performed dispersion removal by mapping Lamb wave signals from time to distance domains.10 The Time-Distance Domain Mapping (TDDM) method was then applied to advance the capacities of Lamb mode diversity imaging,11 phased array beamforming,12 and AE source locating.13 Some improvements for TDDM were also made considering signal waveform correction.14,15 Liu and Yuan proposed a Linear Mapping (LM) technique, in which the former nonlinear wavenumber relation is linearly expanded as a firstorder Taylor series for dispersion eliminating.16 From LM, approaches of Linearly-Dispersive Signal Construction (LDSC) and Non-Dispersive Signal Construction (NDSC) were developed and applied for dual neighboring damage imaging.17,18 Marchi et al. carried out Lamb wave dispersion compensation with Warped Frequency Transform (WFT) and extended the compensation procedure to irregular waveguides.19 Fu et al. presented a step pulse excitation-based WFT method for dispersion suppression.20 Using the above compensation approaches, every wavepacket in a Lamb wave signal can be well recompressed and recovered. Nevertheless, restricted by the time–space duration of its excitation waveform, i.e., the Initial Spatial Resolution (ISR) for the signal before travelling, the signal spatial resolution could remain inadequate for achieving a satisfying damage identification resolution. To further improve the Lamb wave signal resolution, Pulse Compression (PuC), initially developed for radar systems to address the trade-off between the resolution and SNR during signal transmission, was recently adopted together with dispersion compensation. Marchi et al. carried out PuC after WFT to increase the damage localization accuracy in an aluminum plate.21 Lin et al. investigated different Lamb wave excitation waveforms for attaining the optimal results of PuC and dispersion compensation.22 Malo et al. utilized dispersion eliminating and PuC to actualize 2-D compressed pulse analysis for ultrasonic guided waves in an aluminum rod.23 During these processing procedures, PuC is essentially equivalent to employing impulselike auto-correlation waveforms as new excitation signals with a narrower temporal width, for increasing the ISR. However, far from impulses, much broader auto-correlation waveforms were obtained in reality,22 even using carefully-designed PuC excitation signals. To efficiently enhance the resolution and capability of Lamb wave identification, a Lamb Wave Signal Reconstruction (LWSR) approach is presented in this paper. In LWSR, not only a new linearly-dispersive signal is reconstructed from the original Lamb wave signal for dispersion compensation,
X. WANG et al. but the group velocity at the central frequency is also decreased in multiples to sufficiently improve the ISR. Thus, an outstanding signal spatial resolution can be obtained in the reconstructed signal. The remaining content is organized as follows. Some fundamental analysis is performed for Lamb wave signals in Section 2. LWSR is proposed in Section 3, in which the basic idea and numerical realization are theoretically investigated and numerically validated. In Section 4, an LWSR-based high-resolution damage imaging method is developed. Experimental validation for LWSR and LWSRbased adjacent multiple damage imaging is arranged in Section 5. Conclusions are made in the last section. 2. Fundamental analysis for Lamb wave signals 2.1. Sensing model in frequency domain For the convenience of theoretical investigation, a Lamb wave sensing model is simplified in frequency domain before the signal spatial resolution analysis. With PieZoelecTric (PZT) wafers applied as actuators and sensors, a Lamb wave signal assumed of a single wavepacket can be represented in frequency domain as17,18 V0 ðxÞ ¼ Va ðxÞHðxÞ
ð1Þ
where x, Va ðxÞ, and V0 ðxÞ are the angular frequency, frequency-domain excitation signal, and sensor signal, respectively. HðxÞ, regarded as the transfer function of the whole procedure including Lamb wave exciting, propagating, and sensing, can be expressed as HðxÞ ¼ Aðr; xÞeiK0 ðxÞr
ð2Þ
where Aðr; xÞ is the amplitude spectrum of HðxÞ, r is the travelling distance, K0 ðxÞ is the wavenumber of the Lamb wave mode, and cp ðxÞ ¼ x=K0 ðxÞ; cg ðxÞ ¼ dx=d½K0 ðxÞ
ð3Þ
where cp ðxÞ and cg ðxÞ are the phase and group velocities of the mode, respectively For the narrowband excitation signal Va ðxÞ, Aðr; xÞ varying slightly within the limited frequency range can be simplified as ‘‘1” to facilitate the following analysis. The sensing model is derived as V0 ðxÞ ¼ Va ðxÞeiK0 ðxÞr
ð4Þ
Applying Inverse Fourier Transform (IFT) to Eq. (4), the time-domain Lamb wave signal v0 ðtÞ can be calculated. 2.2. Spatial resolution of Lamb wave signals The spatial resolution of v0 ðtÞ can be generally estimated as the Resolvable Distance (RD) of its signal wavepacket,7 i.e., ð5Þ RD ¼ cg0 T0 þ Tdisp where cg0 and T0 are the group velocity of the Lamb wave mode at the central frequency and the initial temporal duration of the wavepacket before travelling, respectively. Tdisp , the increased time duration of the wavepacket caused by dispersion, can be calculated as7 ð6Þ Tdisp ¼ r 1=cg min 1=cg max
A Lamb Wave Signal Reconstruction method for high-resolution damage imaging
1089
where r, cg min , and cg max are the travelling distance of the wavepacket and the minimum and maximum group velocities of the Lamb wave mode within the frequency range of the wavepacket, respectively. Clearly, the signal could possess a high spatial resolution with a short RD. Eqs. (5) and (6) indicate that, for v0 ðtÞ of severe dispersion, cg max differs much from cg min . Tdisp would rapidly rise as the signal wavepacket travels over increasingly long r, resulting in a large RD and a seriously-degraded signal spatial resolution. Hence, several dispersion compensation approaches have been introduced to remove Tdisp for improving the resolution of Lamb wave signals and damage detection.10–20 However, as expressed by Eq. (5), RD is not only dependent on Tdisp . When the dispersion effect has been well compensated, Tdisp ¼ 0 in Eq. (5). The signal spatial resolution is simplified as the RD of the pristine signal wavepacket at its exciting site, i.e.,
K0 ðxÞr=x to cause the signal wavepacket spreading out by Tdisp . Furthermore, from Eqs. (3) and (7), the ISR of V0 ðxÞ is limited as RD0 ¼ T0 cg0 ¼ T0 dx=d½K0 ðxÞjx¼xc , where xc is the central angular frequency of V0 ðxÞ. Hence, it is the former nonlinear wavenumber relation K0 ðxÞ that brings about the inferior spatial resolution of V0 ðxÞ. Regarding this problem, a signal V1 ðxÞ of an outstanding spatial resolution can be reconstructed in LWSR to replace the original Lamb wave signal V0 ðxÞ, by substituting the former wavenumber relation K0 ðxÞ in Eq. (4) with the newly designed one K1 ðxÞ. To remove the nonlinear K0 ðxÞ-induced dispersion, K1 ðxÞ requires linearizing as a first-order expansion of K0 ðxÞ around xc 17 as follows:
RD0 ¼ cg0 T0
where k0 and k1 are the parameters of zero- and first-order items of K1 ðxÞ, respectively. More importantly, to enhance the ISR by means of decreasing cg0 , K1 ðxÞ should satisfy
ð7Þ
where RD0 , actually the space duration of the original excitation wavepacket, can be regarded as the value of the Initial Spatial Resolution (ISR) for the Lamb wave signal. Eqs. (5) and (7) reveal that, besides the dispersion effect manifested as Tdisp , the signal spatial resolution is also highly restricted by its ISR. For a Lamb wave signal with an overhigh RD0 to have an insufficient ISR, the useful defectrelated wavepackets in the signal could still be seriously overlapped and interfered with each other or by other ones even without dispersion. Signal interpretation and flaw information extraction would become extremely hard. As a result, the resolution and accuracy of Lamb wave damage detection are severely impaired.
K1 ðxÞ ¼ k0 þ k1 ðx xc Þ
ð8Þ
K1 ðxc Þ ¼ K0 ðxc Þ
ð9Þ
cg1 ¼ dx=d½K1 ðxÞjx¼xc ¼ cg0 =m
ð10Þ
where cg1 is the decreased group velocity of the newly reconstructed V1 ðxÞ at xc , and m P 1 is the adjusting factor for cg1 . From Eqs. (8)–(10), the parameters of K1 ðxÞ can be derived as k0 ¼ K0 ðxc Þ; k1 ¼ m=cg0
ð11Þ
To theoretically investigate LWSR results, the exact expression of V1 ðxÞ under K1 ðxÞ can be deduced in time domain, with the excitation signal represented as an amplitudemodulated harmonic17,18 as follows:
3. LWSR method
va ðtÞ ¼ nðtÞeixc t
ð12aÞ
As analyzed above, to adequately improve the signal spatial resolution of Lamb waves, besides dispersion compensation, the enhancement of the ISR should be further made by decreasing RD0 . From Eq. (7), RD0 is proportional to cg0 and T0 . In practical applications, once the excitation waveform, for instance, the commonly-used windowed tone burst, is decided to acquire single-mode Lamb wave signals, T0 is basically fixed and inconveniently modified. Alternatively, the ISR could be enhanced through cg0 reduction, which is exactly executed by the proposed Lamb Wave Signal Reconstruction (LWSR). In LWSR, not only a linearly-dispersive version of the original Lamb wave signal is reconstructed for dispersion removal,17 but also the former cg0 is numerically decreased in multiples to make the ISR of the new reconstructed signal greatly surpass that of the original signal. Then, a satisfying signal spatial resolution can be achieved. In this section, based on the frequency-domain sensing model, the basic idea and realization of LWSR are theoretically investigated and preliminarily testified with numerical simulations.
Va ðxÞ ¼ Nðx xc Þ
ð12bÞ
3.1. Basic idea of LWSR It can been seen from Eq. (4) that the propagation characteristics of V0 ðxÞ are mainly determined by K0 ðxÞ. For a dispersive v0 ðtÞ, K0 ðxÞ is nonlinear with x.9,17,18 Different frequency components of V0 ðxÞ will have inconsistent time delays
where va ðtÞ is the time domain excitation signal. nðtÞ is the amplitude modulation function specifying the envelope of R va ðtÞ, NðxÞ ¼ nðtÞeixt dt is the Fourier Transform (FT) result of nðtÞ, and the carrier frequency xc corresponds to the central angular frequency of va ðtÞ. Note that, for the convenience of analysis, the practically real va ðtÞ is expressed in a complex one. Inserting Eqs. (12b) and (8) into Eq. (4) and applying IFT, It can be derived with the shifting property of FT as17,18 Z 1 v1 ðtÞ ¼ Nðx xc Þei½k0 þk1 ðxxc Þrþixt dx 2p Z eixc tik0 r ð13Þ NðxÞeixk1 rþixt dx ¼ 2p ¼ eixc tik0 r nðt k1 rÞ where v1 ðtÞ denotes the time-domain expression of V1 ðxÞ. With Eqs. (3), (11), and (12a), v1 ðtÞ can be ultimately reformulated as v1 ðtÞ ¼ va ðt k1 rÞeirðxc k1 k0 Þ ¼ va t mr=cg0 eixc r½m=cg0 1=cp ðxc Þ
ð14Þ
Eq. (14) proves that v1 ðtÞ is just a va ðtÞ delayed by the travelling time mr=cg0 corresponding to the decreased group velocity c . Due to the extra factor eixc r½1=ðmcg0 Þ1=cp ðxc Þ , the initial g1
1090
X. WANG et al.
phase xc of v1 ðtÞ is entirely shifted by of the carrier frequency xc r m=cg0 1=cp ðxc Þ , which fortunately produces no deformation on the signal envelope nðtÞ,17 i.e., the wavepacket shape. Therefore, under K1 ðxÞ, v1 ðtÞ can exempt from the influences of a nonlinear dispersion relation and too high cg0 . From Eqs. (7) and (10), the RD of v1 ðtÞ is successfully reduced to cg1 T0 ¼ RD0 =m. The whole spatial resolution of v1 ðtÞ is heightened to m times as just compared with the ISR of the original V0 ðxÞ. This means that a high spatial resolution is obtained in v1 ðtÞ via LWSR.
(1) After the inverse function K 1 0 ðkÞ is established from K 0 ðxÞ, frequency-domain interpolation is carried out to E0 ðr; xÞ at x ¼ K 1 0 ðkÞ, and Eðr; kÞ is gotten in wavenumber domain. (2) By implementing interpolation to Eðr; kÞ at k ¼ K 1 ðxÞ in wavenumber domain, the phase-delay factor Eðr; K 1 ðxÞÞ under the new linear dispersion relation K 1 ðxÞ, i.e., E1 ðr; xÞ in frequency domain, is obtained. According to the calculating procedure, E1 ðr; xÞ can be mapped into E0 ðr; xÞ as
3.2. Realization of LWSR
E1 ðr; xÞ ¼ E0 ðr; X1 ðxÞÞ
3.2.1. Computation of phased-delay factor under K1 ðxÞ
where
LWSR can be numerically realized based on the signal construction principle.18 With Va ðxÞ known in priori, from Eq. (4), the crucial problem for reconstructing V1 ðxÞ is how to pursue the phase-delay factor eiK1 ðxÞr , considering that the traveling distance r is probably unknown especially for a damage-scattered signal. The general expression of eiK1 ðxÞr can be rewritten in a composite function as
X1 ðxÞ ¼ K1 0 ðK1 ðxÞÞ
Eðr; xÞ ¼ Eðr; KðxÞÞ ¼ eikr jk¼KðxÞ
ð15Þ
where the subfunction is the wavenumber relation KðxÞ while the generating function Eðr; kÞ ¼ eikr . For a given r, Eðr; kÞ is merely a simple exponential function and irrespective of KðxÞ. This implies that Eðr; xÞ under various dispersion relations are subject to an identical Eðr; kÞ. As described by Eq. (15), Eðr; kÞ and Eðr; xÞ are the two distinct functions in wavenumber or frequency domain, but share the same functional values. Mathematically, performing wavenumber-domain interpolation to Eðr; kÞ at equally-spaced frequency points decided by k ¼ KðxÞ will result in Eðr; xÞ. Vice versa, Eðr; kÞ can be gotten by interpolating Eðr; xÞ in frequency domain with even wavenumber intervals in terms of x ¼ K1 ðkÞ, where K1 ðkÞ is the inverse function of KðxÞ. Thus, for V1 ðxÞ and V0 ðxÞ of identical propagation paths but different dispersion relations K1 ðxÞ and K0 ðxÞ, their phase-delay factors, rewritten as E0 ðr; xÞ ¼ Eðr; K0 ðxÞÞ ¼ eiK0 ðxÞr and E1 ðr; xÞ ¼ Eðr; K1 ðxÞÞ ¼ eiK1 ðxÞr , respectively, can be numerically converted to each other on the basis of the common Eðr; kÞ via the frequency- or wavenumberdomain interpolation process. This can definitely provide an approach to calculate E1 ðr; xÞ with K0 ðxÞ, K1 ðxÞ, and E0 ðr; xÞ, as illustrated by the dashed arrow lines in Fig. 1. That is:
Fig. 1
ð16Þ ð17Þ
is the interpolation mapping sequence from E0 ðr; xÞ to E1 ðr; xÞ. Eq. (16) suggests that E1 ðr; xÞ can be straightforwardly attained from E0 ðr; xÞ through only one time frequency domain interpolation with X1 ðxÞ, as illustrated by the solid arrow line in Fig. 1. By doing this, Eðr; kÞ and the wavenumber-domain interpolation for it is not demanded. Both the interpolation error and computation cost can be decreased. 3.2.2. Fast realization of LWSR v1 ðtÞ can be reconstructed by inserting Eq. (16) into Eq. (4) and applying IFT as Z 1 Va ðxÞE0 ðr; X1 ðxÞÞeixt dx ð18Þ v1 ðtÞ ¼ 2p Note that the travelling distance r, already presented in the phase-delay factor E0 ðr; xÞ, is not expressly needed in signal reconstruction. With Fast Fourier Transform (FFT) and Inverse Fast Fourier Transform (IFFT) operations, efficient LWSR for actually discrete Lamb wave signals can be expected. From Eq. (18), K0 ðxÞ, K1 ðxÞ, and the phase-delay factor E0 ðr; xÞ should be firstly determined to calculate v1 ðtÞ. K0 ðxÞ can be theoretically derived from the Relay-Lamb dispersion equation using structure material parameters. Then, K1 ðxÞ is decided by linearizing K0 ðxÞ with Eqs. (8) and (11). As for the phase-delay factor E0 ðr; xÞ, Eq. (2) indicates that E0 ðr; xÞ is equivalent to the transfer function HðxÞ with the amplitude Aðr; xÞ normalized. Essentially, HðxÞ is the frequency spectrum of the temporal impulse response captured
Procedure of calculating E1 ðr; xÞ with K0 ðxÞ, K1 ðxÞ, and E0 ðr; xÞ.
A Lamb Wave Signal Reconstruction method for high-resolution damage imaging by the sensor. Hence, HðxÞ as well as E0 ðr; xÞ can be gained by FFT to the impulse response, which can be acquired under impulse or step pulse broadband excitation.9,20 With Eq. (18) and using the convolution property of FFT, v1 ðtÞ can be finally calculated as v1 ðtÞ ¼ IFFT½ðVa ðxÞÞHðX1 ðxÞÞ ¼ va ðtÞ h1 ðtÞ
ð19Þ
where h1 ðtÞ ¼ IFFT½HðX1 ðxÞÞ is the newly calculated impulse response under K1 ðxÞ during LWSR, HðxÞ ¼ FFT½h0 ðtÞ, h0 ðtÞ is the impulse response of the original Lamb wave signal, and FFT½ , IFFT½ , and denote FFT, IFFT, and convolution operations, respectively. Eq. (19) is the fast calculation formula of LWSR. Stemming from the same signal construction principle,20 LWSR can be taken as the improved Linearly Dispersive Signal Construction (LDSC) approach.17 Their main distinction lies in that the previous cg0 maintaining unchangeable during LDSC is decreased by m 1 times in LWSR to further increase the ISR. Consequently, LWSR outperforms LDSC in enhancing the whole signal spatial resolution. Before implementing LWSR, it is necessary to note that: (1) v0 ðtÞ is supposed of a single wavepacket in the above analysis. When v0 ðtÞ contains N P 2 wavepackets of common dispersion characteristic K 0 ðxÞ, based on the time-invariant linear system theory, the transfer function P can be rewritten as H 0 ðxÞ ¼ Nn¼1 H n ðxÞ, where H n ðxÞ eiK 0 ðxÞrn is the transfer function of the n-th wavepacket with a travelling distance of rn . Substituting H 0 ðxÞ into Eq. (19) and using the linear property of FFT, it can be deduced that v1 ðtÞ ¼ IFFT½V a ðxÞ PN PN This n¼1 H n ðX1 ðxÞÞ ¼ n¼1 IFFT½V a ðxÞH n ðX1 ðxÞÞ. demonstrates that LWSR remains applicable to a signal composed of multiple wavepackets with identical dispersion relations. Moreover, the final processing result is equal to the sum of the results of LWSR on every wavepacket. (2) Because of the large reduction of cg1 during LWSR, much deviation between K 0 ðxÞ and K 1 ðxÞ can be produced even within the bandwidth of V 0 ðxÞ. The frequency range of the mapped points X1 ðxÞ decided with Eq. (17) will be unavoidably extended so large that ½ X1 ðx1 Þ X1 ðx2 Þ would be apparently broader than ½ x1 x2 , where ½ x1 x2 is the previous angular frequency range of V 0 ðxÞ. To achieve the desired outstanding signal spatial resolution through LWSR over abundant bandwidth, instead of narrowband excitation, only broadband excitation of impulse or step pulse is adopted here to availably acquire impulse responses within a frequency range no smaller than ½ X1 ðx1 Þ X1 ðx2 Þ . (3) The appropriate adjusting factor m should be carefully determined during LWSR. For one thing, the spatial resolution of v1 ðtÞ, evaluated as cg0 T 0 =m, can be improved with m increased. To achieve an excellent signal spatial resolution, m should be set as large as possible. For another thing, the higher m is, the more extension of the frequency range ½ X1 ðx1 Þ X1 ðx2 Þ of interpolation on H ðxÞ could be resulted in. Thus, m should be increased to a limited value, so that the
1091
desired Lamb wave mode could maintain dominated within the extended ½ X1 ðx1 Þ X1 ðx2 Þ to prevent a multi-modal effect on v1 ðtÞ.
3.3. Numerical simulations To preliminarily validate LWSR, numerical simulations on an A0 mode signal composed of four wavepackets Ⅰ -Ⅳ in a 2024 aluminum plate with a thickness of 3 mm are arranged. Using the plate material parameters in Table 1, the theoretical wavenumber K0 ðxÞ of A0 mode can be derived based on the Relay-Lamb dispersion equation, as shown in Fig. 2. The narrowband excitation signal va ðtÞ is a modulated 3-cycle sine burst with the central frequency fc = 60 kHz, as illustrated in Fig. 3. h0 ðtÞ and v0 ðtÞ are synthesized within the frequency range [0 160] kHz as ( P h0 ðtÞ ¼ 4n¼1 IFFT earn eiK0 ðxÞrn ð20Þ v0 ðtÞ ¼ va ðtÞ h0 ðtÞ where the amplitude attenuation parameter a ¼ 1, the travelling distances for the four wavepackets are r1 = 150 mm, r2 =450 mm, r3 =800 mm, and r4 = 860 mm, respectively. Note that, with Eq. (3), the group velocity at central frequency, cg0 ð60 kHzÞ, can be estimated from K0 ðxÞ (see Fig. 2) as 2239 m/s.T0 ¼ 3=fc = 50 ls, and RD0 = 2239 m/ s 50 ls = 112 mm. The ISR under cg0 ð60 kHzÞ is too low to distinguish wavepackets III and IV with the spatial interval jr4 r3 j=j860 mm 800 mmj = 60 mm, even free of dispersion. The synthesized h0 ðtÞ and v0 ðtÞ are shown in Figs. 4(a) and 5(a), respectively. In v0 ðtÞ, as Fig. 5(a) illustrates, all the dispersive wavepackets I- IV are spreading out. Moreover, the last two neighboring wavepackets III and IV are severely overlapped and cannot be distinguished. The signal spatial resolution is affected by not only dispersion, but also the inadequate ISR. To overcome this problem, LWSR is performed on v0 ðtÞ with the procedure given in detail as follows:
Table 1
Material parameters of a 2024 aluminum plate.
Density q (kgcm3)
Poisson’s ratio l
Young’s modulus E (GPa)
2780
0.33
73.1
Fig. 2 Theoretical A0 mode wavenumber relation K0 ðxÞ in an aluminum plate with a thickness of 3 mm.
1092
X. WANG et al.
Fig. 3
Narrowband excitation signal with a central frequency of 60 kHz.
(1) Determining parameters. For the convenience of comparison, the adjusting factor m is set as 1 or 2, respectively. Using Eqs. (8) and (11), the linearized wavenumber curve K 1 ðxÞ or K 2 ðxÞ is computed with K 0 ðxÞ and cg0 ð60 kHzÞ, as Fig. 6(a) shows. Next, as illustrated in Fig. 6(b), the mapping sequence X1 ðxÞ or X2 ðxÞ can be determined by Eq. (17) with K 1 ðxÞ and K 0 ðxÞ, or K 2 ðxÞ and K 0 ðxÞ, respectively. Note that, with cg1 ð60 kHzÞ decreased to 1=m ¼ 0:5 times of
Fig. 4 Original impulse response and ones calculated during LWSR.
Fig. 5
Original and LWSR-processed A0 mode signals.
A Lamb Wave Signal Reconstruction method for high-resolution damage imaging
1093
Fig. 6 Linearized wavenumber curve K1 ðxÞ or K2 ðxÞ and interpolation mapping sequence X1 ðxÞ or X2 ðxÞ determined for LWSR with m ¼ 1 or m ¼ 2.
cg0 ð60 kHzÞ, the slope of K 2 ðxÞ becomes much steeper than that of K 1 ðxÞ to make the range of X2 ðxÞ largely extended to that of X1 ðxÞ, as Fig. 6 shows. (2) h1 ðtÞ or h2 ðtÞ. Applying FFT to h0 ðtÞ and interpolating the resultant H ðxÞ with X1 ðxÞ or X2 ðxÞ, H ðX1 ðxÞÞ or H ðX2 ðxÞÞ is obtained. Then, after IFFT to H ðX1 ðxÞÞ or H ðX2 ðxÞÞ, the new impulse response h1 ðtÞ or h2 ðtÞ under linearized wavenumber relation K 1 ðxÞ or K 2 ðxÞ is calculated, as illustrated in Figs. 4(b) or 4(c). Note that, since h0 ðtÞ is synthesized within the finite frequency range of 0–160 kHz, the four sampling functions, rather than the ideal impulses, are present in h1 ðtÞ or h2 ðtÞ with time-delays under cg0 ð60 kHzÞ or cg1 ð60 kHzÞ, respectively. (3) v1 ðtÞ or v2 ðtÞ. Based on Eq. (19), the final LWSR result v1 ðtÞ or v2 ðtÞ on v0 ðtÞ is finally computed with m ¼ 1 or m ¼ 2, as illustrated in Figs. 5(b) or 5(c). In v1 ðtÞ, though every dispersive wavepacket gets compensated with its waveform restored to the incipient va ðtÞ, as Fig. 5(b) shows, wavepackets III and IV are still overlapped because of an insufficient ISR under too high cg0 ð60 kHzÞ. By contrast, all the four recovered A0 mode wavepackets can be clearly recognized from v2 ðtÞ without any overlapping at the temporal positions corresponding to their time delays under cg1 ð60 kHzÞ ¼ cg0 ð60 kHzÞ=2, as illustrated in Fig. 5 (c). This proves that a satisfying signal spatial resolution has been accomplished via both dispersion removal and cg1 ð60 kHzÞ reduction through LWSR with m ¼ 2. Note that X2 ðxÞ cross an efficient signal bandwidth of [20 100] kHz (as shown by the main lobe of jVa ðxÞj in Fig. 3(b)) is about [4 154] kHz, as Fig. 6(b) illustrates. This should be the smallest frequency range of acquiring h0 ðtÞ to ensure a sufficient bandwidth of LWSR with X2 ðxÞ. Hence, h0 ðtÞ is particularly computed within [0 160] kHz in the above. On the contrary, when the original impulse response h00 ðtÞ (see Fig. 4(d)) is synthesized over an inadequate frequency range [0 100] kHz, the new impulse response calculated during LWSR (m ¼ 2) becomes h02 ðtÞ. In contrast with h2 ðtÞ (see Fig. 4(c)), the temporal width of every sampling function in h02 ðtÞ is apparently increased due to the frequency range decrease, as Fig. 4(e) illustrates, while in the final LWSR (m ¼ 2) result v02 ðtÞ, as Fig. 5(d) shows, some noise as well as wavepacket elongation is brought about. As a result, the efficiency of signal spatial resolution enhancement is influenced, which can be more clearly observed by comparing the envelopes of v2 ðtÞ and v02 ðtÞ in Fig. 7.
Fig. 7 Envelops of v2 ðtÞ and v02 ðtÞ obtained by LWSR (m ¼ 2) within 0–160 kHz and 0–100 kHz, respectively.
4. High-resolution damage imaging with LWSR Associated with the classic delay-and-sum imaging algorithm, LWSR can be employed for high-resolution imaging of adjacent multiple damages. The basic imaging algorithm can be illustrated in Fig. 8, where the monitored structure is integrated with a sparse transducer array of QðQ P 3Þ PZT wafers. For a PZT pair Pij ði–j; i ¼ 1; 2; ; Q; j ¼ 1; 2; ; QÞ composed of Pi at ðxi ; yi Þ and Pj at ðxj ; yj Þ, the scattering propagation path with respect to an arbitrary point O at ðx; yÞ can be geometrically determined. Assuming that only one Lamb wave mode exists with a constant group velocity cg0 at the central frequency, the relevant time delay is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxi xÞ2 þ ðyi yÞ2 þ ðxj xÞ2 þ ðyj yÞ2 t0 ij ðx;yÞ ¼ cg0 ð21Þ Take s0 ij ðtÞ as the original scattered signal measured by Pij . s0 ij t0 ij ðx; yÞ is related to the amplitude of the signal scattered from point O. All the scattered signals measured by PZT pairs Pij ði–j; 1 6 i; j 6 QÞ are time-delayed and summarized to get an average energy at point O, that is,
Fig. 8
Illustration of delay-and-sum imaging algorithm.
1094
X. WANG et al. "
E0 ðx; yÞ ¼
Q X Q X 2 s0 ij t0 ij ðx; yÞ QðQ 1Þ i¼1 j¼iþ1
#2 ð22Þ
With the energy E0 ðx; yÞ of each point normalized and greyscaled, a damage image over the whole structure can be generated. The local spots in the image with higher intensities will probably correspond to actual defects. The delay-and-sum imaging method is a simple but effective method that can automatically focus the damage scattered signal measured by every PZT pair to any real flaw points. However, the imaging resolution could most-probably suffer from a poor signal spatial resolution caused by dispersion and a low ISR. In the LWSR-based imaging method, LWSR is firstly applied for s0 ij ðtÞ during damage imaging, and the pixel value at point O is calculated as " #2 Q X Q X 2 E1 ðx; yÞ ¼ s1 ij t1 ij ðx; yÞ ð23Þ QðQ 1Þ i¼1 j¼iþ1 where s1 ij ðtÞ is the LWSR-processed s0 ij ðtÞ, and the time delay qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxi xÞ2 þðyi yÞ2 þ ðxj xÞ2 þðyj yÞ2 t1 ij ðx;yÞ ¼ m cg0 ð24Þ In Eq. (23), since each dispersive wavepacket in s0 ij ðtÞ is compensated with its RD0 simultaneously decreased by LWSR in s1 ij ðtÞ, a great enhancement for the resolution and performance of adjacent multiple damage imaging can be achieved. 5. Experimental validation 5.1. Experimental setup To validate LWSR and LWSR-based imaging methods, an experiment of adjacent multiple damage imaging with A0 mode Lamb waves is conducted on a 1000 mm 1000 mm 1.5 mm aluminum plate with material parameters given in Table 1. To monitor the entire plate, eight PZT wafers P1 -P8 (PSN-33, 8 mm in diameter and 0.48 mm in thickness) are bonded to form a square transducer array, as shown in Fig. 9. Four identical hexagonal hollow screws with a diagonal length of 8 mm, denoted as D1–D4, are successively mounted near each other on the plate to simulate adjacent damages. The exact positions
Fig. 9
of PZT wafers and damages in the orthogonal coordinate (see Fig. 9) are listed in Table 2. As illustrated in Fig. 9, the overall experimental setup is composed of a Lamb wave detection system, a matrix switch controller, a power amplifier, and the aluminum plate. The Lamb wave detection system can generate Lamb wave signals and amplify and collect sensor signals. The matrix switch controller controls the working sequence of all PZT pairs, and the power amplifier is applied to amplify the excitation signal to enlarge the monitoring area in the plate. Practically, only a step pulse signal with a raising time of 0.25 ls is generated by the Lamb wave detection system to acquire the impulse response of each PZT pair with 20,000 points at 10 MHz. The narrowband excitation signal va ðtÞ is a 3-cycle sine burst centered at 75 kHz, as shown in Fig. 10. By convoluting the impulse response with va ðtÞ, a desired A0 mode-dominated sensor signal is conveniently extracted. 5.2. Effect of LWSR The efficiency of LWSR for A0 mode dispersive sensor signals and damage-scattered ones is verified. From the impulse response h48 ðtÞ of PZT pair P48 obtained with step pulse excitation in the healthy plate, as Fig. 11(a) shows, the original sensor signal v0 48 ðtÞ under va ðtÞ can be extracted. As Fig. 11 (b) illustrates, the dispersive A0 mode direct arrival A is expanded in much more cycles with its time duration increased from 40 ls (see Fig. 10(a)) to about 90 ls, while the following boundary reflections denoted as B-L are severely superposed and hardly to be recognized. The spatial resolution of v0 48 ðtÞ is evidently influenced by severe dispersion effect and poor ISR.
Table 2
The coordinates of PZTs and damages.
PZT
(x, y) (mm)
PZT or Defect
(x, y) (mm)
P1 P2 P3 P4 P5 P6
(300, 300) (300, 300) (300, 300) (300, 300) (0, 300) (300, 0)
P7 P8 D1 D2 D3 D4
(0, 300) (300, 0) (80, 90) (105, 110) (125, 135) (150, 155)
Experimental setup.
A Lamb Wave Signal Reconstruction method for high-resolution damage imaging
Fig. 10
1095
Narrowband excitation signal with a central frequency of 75 kHz.
Fig. 12 Theoretical A0 mode wavenumber relation K0 ðxÞ in an aluminum plate with a thickness of 1.5 mm.
Fig. 11 Impulse response and original and LWSR-processed sensor signals of P48 .
To enhance the signal resolution by LWSR, the A0 mode theoretical wavenumber relation K0 ðxÞ is firstly derived with the plate material parameters in Table 1, as Fig. 12 shows. With the adjusting factor m set as 1 and 2.5 respectively for comparison, the linearized wavenumber curve K1 ðxÞ or K2 ðxÞ is computed from K0 ðxÞ with Eqs. (8) and (11), as Fig. 13(a) shows. Then, using Eq. (17), the mapping sequence X1 ðxÞ or X2 ðxÞ is decided with K1 ðxÞ and K0 ðxÞ, K2 ðxÞ and K0 ðxÞ respectively, as illustrated in Fig. 13(b). Using
Eq. (19), the LWSR (m ¼ 1) result v1 48 ðtÞ of v0 48 ðtÞ with X1 ðxÞ is attained. As Fig. 11(c) shows, all the dispersioncompensated A0 mode wavepackets get fully recompressed. The first six boundary reflections B-G are well separated, whereas, due to the limited ISR with m ¼ 1, the neighboring boundary reflections H-I and J-L remain overlapped respectively, as illustrated by the dashed box in Fig. 11(c). Much better improvement for the spatial resolution of v1 48 ðtÞ is demanded. Fig. 11(d) gives the LWSR result v2 48 ðtÞ of v0 48 ðtÞ with X2 ðxÞ. Since the ISR is improved to m ¼ 2:5 times, all the 12 recovered A0 mode wavepackets A-L, including the previously-overlapped ones H-L within temporal intervals of 2250–2500 ls and 2650–3000 ls, get apart and can be easily distinguished at temporal sites corresponding to their time delays under the newly decreased group velocity at 75 kHz, cg1 ð75 kHzÞ ¼ cg0 ð75 kHzÞ=2:5. The signal spatial resolution is sufficiently heightened by both dispersion compensation and cg1 ð75 kHzÞ reduction via LWSR with m ¼ 2:5. For LWSR of all the damage-scattered signals, their corresponding impulse responses, as shown in Figs. 14(a) and 15(a) for instance, should be obtained by subtracting the healthy impulse responses of all the PZT pairs from the damaged ones. The LWSR results of typical original scattered signals s0 18 ðtÞ and s00 18 ðtÞ from different numbers of adjacent damages in P18 are given in Figs. 14 and 15, respectively. In s0 18 ðtÞ of dispersion and low ISR, the scattered wavepackets from D1 and D2 are badly overlapped, as Fig. 14(b) shows. Even after dispersion removal by LWSR (m ¼ 1) without an ISR increase, the two recompressed scattered wavepackets in s1 18 ðtÞ are still mixed into a single one, as Fig. 14(c) illustrates. Conversely, after LWSR with m ¼ 2:5, they are successfully separated in s2 18 ðtÞ, as shown in Fig. 14(d). When the four flaws D1–D4 exist, their scattered wavepackets and other ones are completely or partly overlapped in the original s00 18 ðtÞ or its LWSR (m ¼ 1) result s01 18 ðtÞ, as Figs. 15
1096
X. WANG et al.
Fig. 13 Linearized wavenumber curve K1 ðxÞ or K2 ðxÞ and interpolation mapping sequence X1 ðxÞ or X2 ðxÞdetermined for LWSR with m ¼ 1 or m ¼ 2:5.
Fig. 14
Results of LWSR (m ¼ 1 or m ¼ 2:5) of the original scattered signal s0
Fig. 15
Results of LWSR (m ¼ 1 or m ¼ 2:5) of the original scattered signal s00
18 ðtÞ
18 ðtÞ
from D1–D2 with X1 ðxÞ or X2 ðxÞ.
from D1-D4 with X1 ðxÞ or X2 ðxÞ.
A Lamb Wave Signal Reconstruction method for high-resolution damage imaging (b) or 15(c) illustrates. Both s00 18 ðtÞ and s01 18 ðtÞ of a limited ISR are subject to the inadequate spatial resolution and the resultant signal superposition. With m raised to 2.5 during LWSR, the four separated damage-scattered wavepackets distinctly emerge with restored waveforms and enough time space in s02 18 ðtÞ, as illustrated in Fig. 15(d). Figs. 14 and 15 demonstrate that the signal spatial resolution can be adequately heightened by LWSR with m ¼ 2:5, making information extraction for adjacent multiple flaws become an easy work.
Fig. 16
In addition, to compare the performance of LWSR with those of other dispersion compensation approaches, the typical TDDM is performed for the original scattered signals s0 18 ðtÞ and s00 18 ðtÞ. Similar to the LWSR (m ¼ 1) results (see Figs. 14 (c) and 15(c)), though all the dispersive damage-scattered wavepackets get well compensated and compressed in the TDDM-processed scattered signals, as Fig. 16 shows, they keep grievously overlapped and interfere with each other, because of the insufficient ISR.
Results of TDDM of the original scattered signal s0
Fig. 17
1097
18 ðtÞ
from D1–D2 or s00
Different imaging results of D1 and D2.
18 ðtÞ
from D1-D4.
1098
X. WANG et al.
Fig. 18
Different imaging results of D1, D2, D3, and D4.
5.3. Results of damage imaging By using the original, LWSR-, or TDDM-processed scattered signals measured by all the 28 PZT pairs, several images can be generated, respectively. For clearer illustration, only imaging results for the 400 mm 400 mm central zone of the plate are given in Figs. 17 and 18, where every symbol ‘‘X” denotes an actual damage position. Due to the poor spatial resolution of the scattered signals caused by dispersion and inadequate ISR (see Figs. 14(b) and 15(b)), the diffused and blurred flaw spots in the images are combined together, leading to much difficulty for damage identification, as Figs. 17(a) and 18(a) illustrate. In the LWSR (m ¼ 1)-based imaging results, as shown in Figs. 17(b) and 18(b), the flaw spots get converged but still interfere with each other. Similar imaging results can be obtained with the TDDM-based imaging methods, as Figs. 17(c) and 18(c) illustrate. This is because the scattered signals cannot bear a good enough spatial resolution just after dispersion removal (see Figs. 14(c), 15(c), and 16). In comparison, with both dispersion eliminating and ISR increasing actualized in the LWSR (m ¼ 2:5)-processed scattered signals (see Figs. 14(d) and 15(d)), every defect can be expediently identified as a bright focalized spot without any interference, as illustrated in Figs. 17(d) and 18(d). Only the satisfying imaging
resolution and performance for different numbers of adjacent multiple damages can be achieved by the proposed imaging method based on LWSR with m ¼ 2:5. 6. Conclusions (1) To solve the problem of inferior signal spatial resolution caused by dispersion and poor ISR in Lamb wave damage detection, LWSR is developed in this study. In LWSR, a linearly-dispersive version of an original Lamb wave signal is reconstructed with its group velocity adequately decreased at the central frequency, so that both dispersion compensation and ISR improvement can be realized to achieve a satisfying super signal spatial resolution. (2) Associated with the delay-and-sum algorithm, LWSR is applied for high-resolution Lamb wave imaging of adjacent multiple damages. (3) Experimental results prove that all the A0 mode dispersive wavepackets in sensor signals or damage-scattered signals are properly recompressed to a pristine excitation waveform. Furthermore, with the adjusting parameter m raised from 1 to 2.5, the ISR is sufficiently heightened to successfully separate formerly compensated but super-
A Lamb Wave Signal Reconstruction method for high-resolution damage imaging posed neighboring wavepackets at their arrival times corresponding to the newly decreased cg1 ¼ cg0 =m. Using LWSR (m ¼ 2:5)-processed scattered signals with an adequate spatial resolution, different numbers of adjacent multiple damages can be successfully imaged and identified as bright focalized spots without any interference among them.
Acknowledgement This study was supported by the Fundamental Research Funds for the Central Universities, China (No. NS2016012). References 1. Yuan SF, Chen J, Yang WB, Qiu L. On-line crack prognosis in attachment lug using Lamb wave-deterministic resampling particle filter-based method. Smart Mater Struct 2017;26(8):085016. 2. Bhuiyan MY, Shen YF, Giurgiutiu V. Guided wave based crack detection in the rivet hole using global analytical with local FEM approach. Materials 2016;9(7):602. 3. Qiu L, Liu B, Yuan SF, Su ZQ. Impact imaging of aircraft composite structure based on a model-independent spatialwavenumber filter. Ultrasonics 2016;64:10–24. 4. Yuan SF, Ren YQ, Qiu L, Mei HF. A multi-response-based wireless impact monitoring network for aircraft composite structures. IEEE T Ind Election 2016;63(12):7712–22. 5. Hong M, Su ZQ, Lu Y, Sohn H, Qing XP. Locating fatigue damage using temporal signal features of nonlinear Lamb waves. Mech Syst Signal Process 2015;60–61:182–97. 6. Qiu L, Yuan SF, Chang FK, Bao Q, Mei HF. On-line updating Gaussian mixture model for aircraft wing spar damage evaluation under time-varying boundary condition. Smart Mater Struct 2014;23(12):125001. 7. Wilcox PD, Lowe MJS, Cawley P. The effect of dispersion on long-range inspection using ultrasonic guided waves. NDT & E Int 2001;34(1):1–9. 8. Park HW, Kim SB, Sohn H. Understanding a time reversal process in Lamb wave propagation. Wave Mot 2009;46(7):451–67. 9. Cai J, Shi LH, Yuan SF, Shao ZX. High spatial resolution imaging for structural health monitoring based on virtual time reversal. Smart Mater Struct 2011;20(5):55018–28.
1099
10. Wilcox PD. A rapid signal processing technique to remove the effect of dispersion from guided wave signals. IEEE Trans Ultrason Ferroelectr 2003;50(4):419–27. 11. Prado VT, Higuti RT, Kitano C, Martinez-Graullera O, Adamowski JC. Lamb mode diversity imaging for non-destructive testing of plate-like structures. NDT & E Int 2013;59(7):86–95. 12. Yu L, Tian ZH. Guided wave phased array beamforming and imaging in composite plates. Ultrasonics 2016;68:43–53. 13. Grabowski K, Gawronski M, Baran I, Spychalski WL, Staszewski T, Uhl T, et al. Time-distance domain transformation for acoustic emission source localization in thin metallic plates. Ultrasonics 2016;68:142–9. 14. Cai J, Shi LH, Qing XP. A time-distance domain transform method for Lamb wave dispersion compensation considering signal waveform correction. Smart Mater Struct 2013;22 (11):105024. 15. Luo Z, Zeng L, Lin J, Hua JD. A reshaped excitation regenerating and mapping method for waveform correction in Lamb waves dispersion compensation. Smart Mater Struct 2017;26(2):025016. 16. Liu L, Yuan FG. A linear mapping technique for dispersion removal of Lamb waves. Struct Health Monit 2010;9(1):75–86. 17. Cai J, Yuan SF, Qing XP, Chang F, Shi LH, Qiu L. Linearly dispersive signal construction of Lamb waves with measured relative wavenumber curves. Sens Actuators A 2015;221:41–52. 18. Cai J, Yuan SF, Wang TG. Signal construction-based dispersion compensation of Lamb waves considering signal waveform and amplitude spectrum preservation. Materials 2017;10(1):4. 19. Marchi LD, Marzani A, Miniaci M. A dispersion compensation procedure to extend pulse-echo defects location to irregular waveguides. NDT & E Int 2013;54(3):115–22. 20. Fu SC, Shi LH, Zhou YH, Cai J. Dispersion compensation in Lamb wave defect detection with step-pulse excitation and warped frequency transform. IEEE Trans Ultrason Ferroelectr 2014;61 (12):2075–88. 21. Marchi LD, Perelli A, Marzani A. A signal processing approach to exploit chirp excitation in Lamb wave defect detection and localization procedures. Mech Syst Signal Process 2013;39(1– 2):20–31. 22. Lin J, Hua JD, Zeng L, Luo Z. Excitation waveform design for Lamb wave pulse compression. IEEE Trans Ultrason Ferroelectr 2016;63(1):165–77. 23. Malo S, Fateri S, Livadas M, Mares C, Gan T. Wave mode discrimination of coded ultrasonic guided waves using twodimensional compressed pulse analysis. IEEE Trans Ultrason Ferroelectr 2017;64(7):1092–101.
Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
A Lamb wave signal reconstruction method for high-resolution damage imaging Xiaopeng WANG, Jian CAI *, Zhiquan ZHOU State Key Lab of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Received 25 January 2018; revised 13 September 2018; accepted 7 January 2019 Available online 21 March 2019
KEYWORDS Damage imaging; Dispersion compensation; High resolution; Lamb waves; Signal reconstruction
Abstract In Lamb wave-based Structural Health Monitoring (SHM), a high-enough spatial resolution is highly required for Lamb wave signals to ensure the resolution and accuracy of damage detection. However, besides the dispersion characteristic, the signal spatial resolution is also largely restricted by the space duration of excitation waveforms, i.e., the Initial Spatial Resolution (ISR) for the signals before travelling. To resolve the problem of inferior signal spatial resolution of Lamb waves, a Lamb Wave Signal Reconstruction (LWSR) method is presented and applied for highresolution damage imaging in this paper. In LWSR, not only a new linearly-dispersive signal is reconstructed from an original Lamb wave signal, but also the group velocity at the central frequency is sufficiently decreased. Then, both dispersion compensation and ISR improvement can be realized to achieve a satisfying signal spatial resolution. After the frequency domain sensing model and spatial resolution of Lamb wave signals are firstly analyzed, the basic idea and numerical realization of LWSR are discussed. Numerical simulations are also implemented to preliminarily validate LWSR. Subsequently, LWSR-based high-resolution damage imaging is developed. An experiment of adjacent multiple damage identification is finally conducted to demonstrate the efficiency of LWSR and LWSR-based imaging methods. Ó 2019 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Lamb waves are a kind of guided ultrasonic waves existing in thin-wall structures. With the ability of long distance travelling * Corresponding author. E-mail address: [email protected] (J. CAI). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
and high sensitivity to both surface and internal defects, Lamb waves have been widely used as a promising tool in Structural Health Monitoring (SHM) of plate-like structures.1–6 In practical applications, the spatial resolution of Lamb wave signals is undoubtedly a vital factor to determine the resolution and accuracy of damage detection. For a diagnostic signal of inferior spatial resolution, its useful defect-relevant wavepackets are easily overlapped and interfered with each other or by others. Signal interpretation would become less straightforward to make the extraction of damage information difficult. As a result, final detection results are badly affected. To ensure the resolution and performance of Lamb wave damage identi-
https://doi.org/10.1016/j.cja.2019.03.001 1000-9361 Ó 2019 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1088 fication, an adequately high spatial resolution of Lamb wave signals is required. Due to multi-mode and dispersion characteristics, the propagation of Lamb waves is extraordinarily complicated. Even for single-mode signals, dispersive wavepackets will spread out in time and space with their waveforms distorted as they propagate.7 This can definitely degrade the signal spatial resolution. To resolve the dispersion problem, a Time Reversal Process (TRP) was introduced to automatically compensate the dispersion effect without any priori-knowledge of Lamb wave dispersion properties.8 Unfortunately, the Time of Flight (TOF) of the compensated signals is also eliminated at the same time to result in much inconvenience to damage localization.9 Another strategy for Lamb wave dispersion compensation is Fourier-domain signal processing using wave dispersion relations. Wilcox performed dispersion removal by mapping Lamb wave signals from time to distance domains.10 The Time-Distance Domain Mapping (TDDM) method was then applied to advance the capacities of Lamb mode diversity imaging,11 phased array beamforming,12 and AE source locating.13 Some improvements for TDDM were also made considering signal waveform correction.14,15 Liu and Yuan proposed a Linear Mapping (LM) technique, in which the former nonlinear wavenumber relation is linearly expanded as a firstorder Taylor series for dispersion eliminating.16 From LM, approaches of Linearly-Dispersive Signal Construction (LDSC) and Non-Dispersive Signal Construction (NDSC) were developed and applied for dual neighboring damage imaging.17,18 Marchi et al. carried out Lamb wave dispersion compensation with Warped Frequency Transform (WFT) and extended the compensation procedure to irregular waveguides.19 Fu et al. presented a step pulse excitation-based WFT method for dispersion suppression.20 Using the above compensation approaches, every wavepacket in a Lamb wave signal can be well recompressed and recovered. Nevertheless, restricted by the time–space duration of its excitation waveform, i.e., the Initial Spatial Resolution (ISR) for the signal before travelling, the signal spatial resolution could remain inadequate for achieving a satisfying damage identification resolution. To further improve the Lamb wave signal resolution, Pulse Compression (PuC), initially developed for radar systems to address the trade-off between the resolution and SNR during signal transmission, was recently adopted together with dispersion compensation. Marchi et al. carried out PuC after WFT to increase the damage localization accuracy in an aluminum plate.21 Lin et al. investigated different Lamb wave excitation waveforms for attaining the optimal results of PuC and dispersion compensation.22 Malo et al. utilized dispersion eliminating and PuC to actualize 2-D compressed pulse analysis for ultrasonic guided waves in an aluminum rod.23 During these processing procedures, PuC is essentially equivalent to employing impulselike auto-correlation waveforms as new excitation signals with a narrower temporal width, for increasing the ISR. However, far from impulses, much broader auto-correlation waveforms were obtained in reality,22 even using carefully-designed PuC excitation signals. To efficiently enhance the resolution and capability of Lamb wave identification, a Lamb Wave Signal Reconstruction (LWSR) approach is presented in this paper. In LWSR, not only a new linearly-dispersive signal is reconstructed from the original Lamb wave signal for dispersion compensation,
X. WANG et al. but the group velocity at the central frequency is also decreased in multiples to sufficiently improve the ISR. Thus, an outstanding signal spatial resolution can be obtained in the reconstructed signal. The remaining content is organized as follows. Some fundamental analysis is performed for Lamb wave signals in Section 2. LWSR is proposed in Section 3, in which the basic idea and numerical realization are theoretically investigated and numerically validated. In Section 4, an LWSR-based high-resolution damage imaging method is developed. Experimental validation for LWSR and LWSRbased adjacent multiple damage imaging is arranged in Section 5. Conclusions are made in the last section. 2. Fundamental analysis for Lamb wave signals 2.1. Sensing model in frequency domain For the convenience of theoretical investigation, a Lamb wave sensing model is simplified in frequency domain before the signal spatial resolution analysis. With PieZoelecTric (PZT) wafers applied as actuators and sensors, a Lamb wave signal assumed of a single wavepacket can be represented in frequency domain as17,18 V0 ðxÞ ¼ Va ðxÞHðxÞ
ð1Þ
where x, Va ðxÞ, and V0 ðxÞ are the angular frequency, frequency-domain excitation signal, and sensor signal, respectively. HðxÞ, regarded as the transfer function of the whole procedure including Lamb wave exciting, propagating, and sensing, can be expressed as HðxÞ ¼ Aðr; xÞeiK0 ðxÞr
ð2Þ
where Aðr; xÞ is the amplitude spectrum of HðxÞ, r is the travelling distance, K0 ðxÞ is the wavenumber of the Lamb wave mode, and cp ðxÞ ¼ x=K0 ðxÞ; cg ðxÞ ¼ dx=d½K0 ðxÞ
ð3Þ
where cp ðxÞ and cg ðxÞ are the phase and group velocities of the mode, respectively For the narrowband excitation signal Va ðxÞ, Aðr; xÞ varying slightly within the limited frequency range can be simplified as ‘‘1” to facilitate the following analysis. The sensing model is derived as V0 ðxÞ ¼ Va ðxÞeiK0 ðxÞr
ð4Þ
Applying Inverse Fourier Transform (IFT) to Eq. (4), the time-domain Lamb wave signal v0 ðtÞ can be calculated. 2.2. Spatial resolution of Lamb wave signals The spatial resolution of v0 ðtÞ can be generally estimated as the Resolvable Distance (RD) of its signal wavepacket,7 i.e., ð5Þ RD ¼ cg0 T0 þ Tdisp where cg0 and T0 are the group velocity of the Lamb wave mode at the central frequency and the initial temporal duration of the wavepacket before travelling, respectively. Tdisp , the increased time duration of the wavepacket caused by dispersion, can be calculated as7 ð6Þ Tdisp ¼ r 1=cg min 1=cg max
A Lamb Wave Signal Reconstruction method for high-resolution damage imaging
1089
where r, cg min , and cg max are the travelling distance of the wavepacket and the minimum and maximum group velocities of the Lamb wave mode within the frequency range of the wavepacket, respectively. Clearly, the signal could possess a high spatial resolution with a short RD. Eqs. (5) and (6) indicate that, for v0 ðtÞ of severe dispersion, cg max differs much from cg min . Tdisp would rapidly rise as the signal wavepacket travels over increasingly long r, resulting in a large RD and a seriously-degraded signal spatial resolution. Hence, several dispersion compensation approaches have been introduced to remove Tdisp for improving the resolution of Lamb wave signals and damage detection.10–20 However, as expressed by Eq. (5), RD is not only dependent on Tdisp . When the dispersion effect has been well compensated, Tdisp ¼ 0 in Eq. (5). The signal spatial resolution is simplified as the RD of the pristine signal wavepacket at its exciting site, i.e.,
K0 ðxÞr=x to cause the signal wavepacket spreading out by Tdisp . Furthermore, from Eqs. (3) and (7), the ISR of V0 ðxÞ is limited as RD0 ¼ T0 cg0 ¼ T0 dx=d½K0 ðxÞjx¼xc , where xc is the central angular frequency of V0 ðxÞ. Hence, it is the former nonlinear wavenumber relation K0 ðxÞ that brings about the inferior spatial resolution of V0 ðxÞ. Regarding this problem, a signal V1 ðxÞ of an outstanding spatial resolution can be reconstructed in LWSR to replace the original Lamb wave signal V0 ðxÞ, by substituting the former wavenumber relation K0 ðxÞ in Eq. (4) with the newly designed one K1 ðxÞ. To remove the nonlinear K0 ðxÞ-induced dispersion, K1 ðxÞ requires linearizing as a first-order expansion of K0 ðxÞ around xc 17 as follows:
RD0 ¼ cg0 T0
where k0 and k1 are the parameters of zero- and first-order items of K1 ðxÞ, respectively. More importantly, to enhance the ISR by means of decreasing cg0 , K1 ðxÞ should satisfy
ð7Þ
where RD0 , actually the space duration of the original excitation wavepacket, can be regarded as the value of the Initial Spatial Resolution (ISR) for the Lamb wave signal. Eqs. (5) and (7) reveal that, besides the dispersion effect manifested as Tdisp , the signal spatial resolution is also highly restricted by its ISR. For a Lamb wave signal with an overhigh RD0 to have an insufficient ISR, the useful defectrelated wavepackets in the signal could still be seriously overlapped and interfered with each other or by other ones even without dispersion. Signal interpretation and flaw information extraction would become extremely hard. As a result, the resolution and accuracy of Lamb wave damage detection are severely impaired.
K1 ðxÞ ¼ k0 þ k1 ðx xc Þ
ð8Þ
K1 ðxc Þ ¼ K0 ðxc Þ
ð9Þ
cg1 ¼ dx=d½K1 ðxÞjx¼xc ¼ cg0 =m
ð10Þ
where cg1 is the decreased group velocity of the newly reconstructed V1 ðxÞ at xc , and m P 1 is the adjusting factor for cg1 . From Eqs. (8)–(10), the parameters of K1 ðxÞ can be derived as k0 ¼ K0 ðxc Þ; k1 ¼ m=cg0
ð11Þ
To theoretically investigate LWSR results, the exact expression of V1 ðxÞ under K1 ðxÞ can be deduced in time domain, with the excitation signal represented as an amplitudemodulated harmonic17,18 as follows:
3. LWSR method
va ðtÞ ¼ nðtÞeixc t
ð12aÞ
As analyzed above, to adequately improve the signal spatial resolution of Lamb waves, besides dispersion compensation, the enhancement of the ISR should be further made by decreasing RD0 . From Eq. (7), RD0 is proportional to cg0 and T0 . In practical applications, once the excitation waveform, for instance, the commonly-used windowed tone burst, is decided to acquire single-mode Lamb wave signals, T0 is basically fixed and inconveniently modified. Alternatively, the ISR could be enhanced through cg0 reduction, which is exactly executed by the proposed Lamb Wave Signal Reconstruction (LWSR). In LWSR, not only a linearly-dispersive version of the original Lamb wave signal is reconstructed for dispersion removal,17 but also the former cg0 is numerically decreased in multiples to make the ISR of the new reconstructed signal greatly surpass that of the original signal. Then, a satisfying signal spatial resolution can be achieved. In this section, based on the frequency-domain sensing model, the basic idea and realization of LWSR are theoretically investigated and preliminarily testified with numerical simulations.
Va ðxÞ ¼ Nðx xc Þ
ð12bÞ
3.1. Basic idea of LWSR It can been seen from Eq. (4) that the propagation characteristics of V0 ðxÞ are mainly determined by K0 ðxÞ. For a dispersive v0 ðtÞ, K0 ðxÞ is nonlinear with x.9,17,18 Different frequency components of V0 ðxÞ will have inconsistent time delays
where va ðtÞ is the time domain excitation signal. nðtÞ is the amplitude modulation function specifying the envelope of R va ðtÞ, NðxÞ ¼ nðtÞeixt dt is the Fourier Transform (FT) result of nðtÞ, and the carrier frequency xc corresponds to the central angular frequency of va ðtÞ. Note that, for the convenience of analysis, the practically real va ðtÞ is expressed in a complex one. Inserting Eqs. (12b) and (8) into Eq. (4) and applying IFT, It can be derived with the shifting property of FT as17,18 Z 1 v1 ðtÞ ¼ Nðx xc Þei½k0 þk1 ðxxc Þrþixt dx 2p Z eixc tik0 r ð13Þ NðxÞeixk1 rþixt dx ¼ 2p ¼ eixc tik0 r nðt k1 rÞ where v1 ðtÞ denotes the time-domain expression of V1 ðxÞ. With Eqs. (3), (11), and (12a), v1 ðtÞ can be ultimately reformulated as v1 ðtÞ ¼ va ðt k1 rÞeirðxc k1 k0 Þ ¼ va t mr=cg0 eixc r½m=cg0 1=cp ðxc Þ
ð14Þ
Eq. (14) proves that v1 ðtÞ is just a va ðtÞ delayed by the travelling time mr=cg0 corresponding to the decreased group velocity c . Due to the extra factor eixc r½1=ðmcg0 Þ1=cp ðxc Þ , the initial g1
1090
X. WANG et al.
phase xc of v1 ðtÞ is entirely shifted by of the carrier frequency xc r m=cg0 1=cp ðxc Þ , which fortunately produces no deformation on the signal envelope nðtÞ,17 i.e., the wavepacket shape. Therefore, under K1 ðxÞ, v1 ðtÞ can exempt from the influences of a nonlinear dispersion relation and too high cg0 . From Eqs. (7) and (10), the RD of v1 ðtÞ is successfully reduced to cg1 T0 ¼ RD0 =m. The whole spatial resolution of v1 ðtÞ is heightened to m times as just compared with the ISR of the original V0 ðxÞ. This means that a high spatial resolution is obtained in v1 ðtÞ via LWSR.
(1) After the inverse function K 1 0 ðkÞ is established from K 0 ðxÞ, frequency-domain interpolation is carried out to E0 ðr; xÞ at x ¼ K 1 0 ðkÞ, and Eðr; kÞ is gotten in wavenumber domain. (2) By implementing interpolation to Eðr; kÞ at k ¼ K 1 ðxÞ in wavenumber domain, the phase-delay factor Eðr; K 1 ðxÞÞ under the new linear dispersion relation K 1 ðxÞ, i.e., E1 ðr; xÞ in frequency domain, is obtained. According to the calculating procedure, E1 ðr; xÞ can be mapped into E0 ðr; xÞ as
3.2. Realization of LWSR
E1 ðr; xÞ ¼ E0 ðr; X1 ðxÞÞ
3.2.1. Computation of phased-delay factor under K1 ðxÞ
where
LWSR can be numerically realized based on the signal construction principle.18 With Va ðxÞ known in priori, from Eq. (4), the crucial problem for reconstructing V1 ðxÞ is how to pursue the phase-delay factor eiK1 ðxÞr , considering that the traveling distance r is probably unknown especially for a damage-scattered signal. The general expression of eiK1 ðxÞr can be rewritten in a composite function as
X1 ðxÞ ¼ K1 0 ðK1 ðxÞÞ
Eðr; xÞ ¼ Eðr; KðxÞÞ ¼ eikr jk¼KðxÞ
ð15Þ
where the subfunction is the wavenumber relation KðxÞ while the generating function Eðr; kÞ ¼ eikr . For a given r, Eðr; kÞ is merely a simple exponential function and irrespective of KðxÞ. This implies that Eðr; xÞ under various dispersion relations are subject to an identical Eðr; kÞ. As described by Eq. (15), Eðr; kÞ and Eðr; xÞ are the two distinct functions in wavenumber or frequency domain, but share the same functional values. Mathematically, performing wavenumber-domain interpolation to Eðr; kÞ at equally-spaced frequency points decided by k ¼ KðxÞ will result in Eðr; xÞ. Vice versa, Eðr; kÞ can be gotten by interpolating Eðr; xÞ in frequency domain with even wavenumber intervals in terms of x ¼ K1 ðkÞ, where K1 ðkÞ is the inverse function of KðxÞ. Thus, for V1 ðxÞ and V0 ðxÞ of identical propagation paths but different dispersion relations K1 ðxÞ and K0 ðxÞ, their phase-delay factors, rewritten as E0 ðr; xÞ ¼ Eðr; K0 ðxÞÞ ¼ eiK0 ðxÞr and E1 ðr; xÞ ¼ Eðr; K1 ðxÞÞ ¼ eiK1 ðxÞr , respectively, can be numerically converted to each other on the basis of the common Eðr; kÞ via the frequency- or wavenumberdomain interpolation process. This can definitely provide an approach to calculate E1 ðr; xÞ with K0 ðxÞ, K1 ðxÞ, and E0 ðr; xÞ, as illustrated by the dashed arrow lines in Fig. 1. That is:
Fig. 1
ð16Þ ð17Þ
is the interpolation mapping sequence from E0 ðr; xÞ to E1 ðr; xÞ. Eq. (16) suggests that E1 ðr; xÞ can be straightforwardly attained from E0 ðr; xÞ through only one time frequency domain interpolation with X1 ðxÞ, as illustrated by the solid arrow line in Fig. 1. By doing this, Eðr; kÞ and the wavenumber-domain interpolation for it is not demanded. Both the interpolation error and computation cost can be decreased. 3.2.2. Fast realization of LWSR v1 ðtÞ can be reconstructed by inserting Eq. (16) into Eq. (4) and applying IFT as Z 1 Va ðxÞE0 ðr; X1 ðxÞÞeixt dx ð18Þ v1 ðtÞ ¼ 2p Note that the travelling distance r, already presented in the phase-delay factor E0 ðr; xÞ, is not expressly needed in signal reconstruction. With Fast Fourier Transform (FFT) and Inverse Fast Fourier Transform (IFFT) operations, efficient LWSR for actually discrete Lamb wave signals can be expected. From Eq. (18), K0 ðxÞ, K1 ðxÞ, and the phase-delay factor E0 ðr; xÞ should be firstly determined to calculate v1 ðtÞ. K0 ðxÞ can be theoretically derived from the Relay-Lamb dispersion equation using structure material parameters. Then, K1 ðxÞ is decided by linearizing K0 ðxÞ with Eqs. (8) and (11). As for the phase-delay factor E0 ðr; xÞ, Eq. (2) indicates that E0 ðr; xÞ is equivalent to the transfer function HðxÞ with the amplitude Aðr; xÞ normalized. Essentially, HðxÞ is the frequency spectrum of the temporal impulse response captured
Procedure of calculating E1 ðr; xÞ with K0 ðxÞ, K1 ðxÞ, and E0 ðr; xÞ.
A Lamb Wave Signal Reconstruction method for high-resolution damage imaging by the sensor. Hence, HðxÞ as well as E0 ðr; xÞ can be gained by FFT to the impulse response, which can be acquired under impulse or step pulse broadband excitation.9,20 With Eq. (18) and using the convolution property of FFT, v1 ðtÞ can be finally calculated as v1 ðtÞ ¼ IFFT½ðVa ðxÞÞHðX1 ðxÞÞ ¼ va ðtÞ h1 ðtÞ
ð19Þ
where h1 ðtÞ ¼ IFFT½HðX1 ðxÞÞ is the newly calculated impulse response under K1 ðxÞ during LWSR, HðxÞ ¼ FFT½h0 ðtÞ, h0 ðtÞ is the impulse response of the original Lamb wave signal, and FFT½ , IFFT½ , and denote FFT, IFFT, and convolution operations, respectively. Eq. (19) is the fast calculation formula of LWSR. Stemming from the same signal construction principle,20 LWSR can be taken as the improved Linearly Dispersive Signal Construction (LDSC) approach.17 Their main distinction lies in that the previous cg0 maintaining unchangeable during LDSC is decreased by m 1 times in LWSR to further increase the ISR. Consequently, LWSR outperforms LDSC in enhancing the whole signal spatial resolution. Before implementing LWSR, it is necessary to note that: (1) v0 ðtÞ is supposed of a single wavepacket in the above analysis. When v0 ðtÞ contains N P 2 wavepackets of common dispersion characteristic K 0 ðxÞ, based on the time-invariant linear system theory, the transfer function P can be rewritten as H 0 ðxÞ ¼ Nn¼1 H n ðxÞ, where H n ðxÞ eiK 0 ðxÞrn is the transfer function of the n-th wavepacket with a travelling distance of rn . Substituting H 0 ðxÞ into Eq. (19) and using the linear property of FFT, it can be deduced that v1 ðtÞ ¼ IFFT½V a ðxÞ PN PN This n¼1 H n ðX1 ðxÞÞ ¼ n¼1 IFFT½V a ðxÞH n ðX1 ðxÞÞ. demonstrates that LWSR remains applicable to a signal composed of multiple wavepackets with identical dispersion relations. Moreover, the final processing result is equal to the sum of the results of LWSR on every wavepacket. (2) Because of the large reduction of cg1 during LWSR, much deviation between K 0 ðxÞ and K 1 ðxÞ can be produced even within the bandwidth of V 0 ðxÞ. The frequency range of the mapped points X1 ðxÞ decided with Eq. (17) will be unavoidably extended so large that ½ X1 ðx1 Þ X1 ðx2 Þ would be apparently broader than ½ x1 x2 , where ½ x1 x2 is the previous angular frequency range of V 0 ðxÞ. To achieve the desired outstanding signal spatial resolution through LWSR over abundant bandwidth, instead of narrowband excitation, only broadband excitation of impulse or step pulse is adopted here to availably acquire impulse responses within a frequency range no smaller than ½ X1 ðx1 Þ X1 ðx2 Þ . (3) The appropriate adjusting factor m should be carefully determined during LWSR. For one thing, the spatial resolution of v1 ðtÞ, evaluated as cg0 T 0 =m, can be improved with m increased. To achieve an excellent signal spatial resolution, m should be set as large as possible. For another thing, the higher m is, the more extension of the frequency range ½ X1 ðx1 Þ X1 ðx2 Þ of interpolation on H ðxÞ could be resulted in. Thus, m should be increased to a limited value, so that the
1091
desired Lamb wave mode could maintain dominated within the extended ½ X1 ðx1 Þ X1 ðx2 Þ to prevent a multi-modal effect on v1 ðtÞ.
3.3. Numerical simulations To preliminarily validate LWSR, numerical simulations on an A0 mode signal composed of four wavepackets Ⅰ -Ⅳ in a 2024 aluminum plate with a thickness of 3 mm are arranged. Using the plate material parameters in Table 1, the theoretical wavenumber K0 ðxÞ of A0 mode can be derived based on the Relay-Lamb dispersion equation, as shown in Fig. 2. The narrowband excitation signal va ðtÞ is a modulated 3-cycle sine burst with the central frequency fc = 60 kHz, as illustrated in Fig. 3. h0 ðtÞ and v0 ðtÞ are synthesized within the frequency range [0 160] kHz as ( P h0 ðtÞ ¼ 4n¼1 IFFT earn eiK0 ðxÞrn ð20Þ v0 ðtÞ ¼ va ðtÞ h0 ðtÞ where the amplitude attenuation parameter a ¼ 1, the travelling distances for the four wavepackets are r1 = 150 mm, r2 =450 mm, r3 =800 mm, and r4 = 860 mm, respectively. Note that, with Eq. (3), the group velocity at central frequency, cg0 ð60 kHzÞ, can be estimated from K0 ðxÞ (see Fig. 2) as 2239 m/s.T0 ¼ 3=fc = 50 ls, and RD0 = 2239 m/ s 50 ls = 112 mm. The ISR under cg0 ð60 kHzÞ is too low to distinguish wavepackets III and IV with the spatial interval jr4 r3 j=j860 mm 800 mmj = 60 mm, even free of dispersion. The synthesized h0 ðtÞ and v0 ðtÞ are shown in Figs. 4(a) and 5(a), respectively. In v0 ðtÞ, as Fig. 5(a) illustrates, all the dispersive wavepackets I- IV are spreading out. Moreover, the last two neighboring wavepackets III and IV are severely overlapped and cannot be distinguished. The signal spatial resolution is affected by not only dispersion, but also the inadequate ISR. To overcome this problem, LWSR is performed on v0 ðtÞ with the procedure given in detail as follows:
Table 1
Material parameters of a 2024 aluminum plate.
Density q (kgcm3)
Poisson’s ratio l
Young’s modulus E (GPa)
2780
0.33
73.1
Fig. 2 Theoretical A0 mode wavenumber relation K0 ðxÞ in an aluminum plate with a thickness of 3 mm.
1092
X. WANG et al.
Fig. 3
Narrowband excitation signal with a central frequency of 60 kHz.
(1) Determining parameters. For the convenience of comparison, the adjusting factor m is set as 1 or 2, respectively. Using Eqs. (8) and (11), the linearized wavenumber curve K 1 ðxÞ or K 2 ðxÞ is computed with K 0 ðxÞ and cg0 ð60 kHzÞ, as Fig. 6(a) shows. Next, as illustrated in Fig. 6(b), the mapping sequence X1 ðxÞ or X2 ðxÞ can be determined by Eq. (17) with K 1 ðxÞ and K 0 ðxÞ, or K 2 ðxÞ and K 0 ðxÞ, respectively. Note that, with cg1 ð60 kHzÞ decreased to 1=m ¼ 0:5 times of
Fig. 4 Original impulse response and ones calculated during LWSR.
Fig. 5
Original and LWSR-processed A0 mode signals.
A Lamb Wave Signal Reconstruction method for high-resolution damage imaging
1093
Fig. 6 Linearized wavenumber curve K1 ðxÞ or K2 ðxÞ and interpolation mapping sequence X1 ðxÞ or X2 ðxÞ determined for LWSR with m ¼ 1 or m ¼ 2.
cg0 ð60 kHzÞ, the slope of K 2 ðxÞ becomes much steeper than that of K 1 ðxÞ to make the range of X2 ðxÞ largely extended to that of X1 ðxÞ, as Fig. 6 shows. (2) h1 ðtÞ or h2 ðtÞ. Applying FFT to h0 ðtÞ and interpolating the resultant H ðxÞ with X1 ðxÞ or X2 ðxÞ, H ðX1 ðxÞÞ or H ðX2 ðxÞÞ is obtained. Then, after IFFT to H ðX1 ðxÞÞ or H ðX2 ðxÞÞ, the new impulse response h1 ðtÞ or h2 ðtÞ under linearized wavenumber relation K 1 ðxÞ or K 2 ðxÞ is calculated, as illustrated in Figs. 4(b) or 4(c). Note that, since h0 ðtÞ is synthesized within the finite frequency range of 0–160 kHz, the four sampling functions, rather than the ideal impulses, are present in h1 ðtÞ or h2 ðtÞ with time-delays under cg0 ð60 kHzÞ or cg1 ð60 kHzÞ, respectively. (3) v1 ðtÞ or v2 ðtÞ. Based on Eq. (19), the final LWSR result v1 ðtÞ or v2 ðtÞ on v0 ðtÞ is finally computed with m ¼ 1 or m ¼ 2, as illustrated in Figs. 5(b) or 5(c). In v1 ðtÞ, though every dispersive wavepacket gets compensated with its waveform restored to the incipient va ðtÞ, as Fig. 5(b) shows, wavepackets III and IV are still overlapped because of an insufficient ISR under too high cg0 ð60 kHzÞ. By contrast, all the four recovered A0 mode wavepackets can be clearly recognized from v2 ðtÞ without any overlapping at the temporal positions corresponding to their time delays under cg1 ð60 kHzÞ ¼ cg0 ð60 kHzÞ=2, as illustrated in Fig. 5 (c). This proves that a satisfying signal spatial resolution has been accomplished via both dispersion removal and cg1 ð60 kHzÞ reduction through LWSR with m ¼ 2. Note that X2 ðxÞ cross an efficient signal bandwidth of [20 100] kHz (as shown by the main lobe of jVa ðxÞj in Fig. 3(b)) is about [4 154] kHz, as Fig. 6(b) illustrates. This should be the smallest frequency range of acquiring h0 ðtÞ to ensure a sufficient bandwidth of LWSR with X2 ðxÞ. Hence, h0 ðtÞ is particularly computed within [0 160] kHz in the above. On the contrary, when the original impulse response h00 ðtÞ (see Fig. 4(d)) is synthesized over an inadequate frequency range [0 100] kHz, the new impulse response calculated during LWSR (m ¼ 2) becomes h02 ðtÞ. In contrast with h2 ðtÞ (see Fig. 4(c)), the temporal width of every sampling function in h02 ðtÞ is apparently increased due to the frequency range decrease, as Fig. 4(e) illustrates, while in the final LWSR (m ¼ 2) result v02 ðtÞ, as Fig. 5(d) shows, some noise as well as wavepacket elongation is brought about. As a result, the efficiency of signal spatial resolution enhancement is influenced, which can be more clearly observed by comparing the envelopes of v2 ðtÞ and v02 ðtÞ in Fig. 7.
Fig. 7 Envelops of v2 ðtÞ and v02 ðtÞ obtained by LWSR (m ¼ 2) within 0–160 kHz and 0–100 kHz, respectively.
4. High-resolution damage imaging with LWSR Associated with the classic delay-and-sum imaging algorithm, LWSR can be employed for high-resolution imaging of adjacent multiple damages. The basic imaging algorithm can be illustrated in Fig. 8, where the monitored structure is integrated with a sparse transducer array of QðQ P 3Þ PZT wafers. For a PZT pair Pij ði–j; i ¼ 1; 2; ; Q; j ¼ 1; 2; ; QÞ composed of Pi at ðxi ; yi Þ and Pj at ðxj ; yj Þ, the scattering propagation path with respect to an arbitrary point O at ðx; yÞ can be geometrically determined. Assuming that only one Lamb wave mode exists with a constant group velocity cg0 at the central frequency, the relevant time delay is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxi xÞ2 þ ðyi yÞ2 þ ðxj xÞ2 þ ðyj yÞ2 t0 ij ðx;yÞ ¼ cg0 ð21Þ Take s0 ij ðtÞ as the original scattered signal measured by Pij . s0 ij t0 ij ðx; yÞ is related to the amplitude of the signal scattered from point O. All the scattered signals measured by PZT pairs Pij ði–j; 1 6 i; j 6 QÞ are time-delayed and summarized to get an average energy at point O, that is,
Fig. 8
Illustration of delay-and-sum imaging algorithm.
1094
X. WANG et al. "
E0 ðx; yÞ ¼
Q X Q X 2 s0 ij t0 ij ðx; yÞ QðQ 1Þ i¼1 j¼iþ1
#2 ð22Þ
With the energy E0 ðx; yÞ of each point normalized and greyscaled, a damage image over the whole structure can be generated. The local spots in the image with higher intensities will probably correspond to actual defects. The delay-and-sum imaging method is a simple but effective method that can automatically focus the damage scattered signal measured by every PZT pair to any real flaw points. However, the imaging resolution could most-probably suffer from a poor signal spatial resolution caused by dispersion and a low ISR. In the LWSR-based imaging method, LWSR is firstly applied for s0 ij ðtÞ during damage imaging, and the pixel value at point O is calculated as " #2 Q X Q X 2 E1 ðx; yÞ ¼ s1 ij t1 ij ðx; yÞ ð23Þ QðQ 1Þ i¼1 j¼iþ1 where s1 ij ðtÞ is the LWSR-processed s0 ij ðtÞ, and the time delay qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxi xÞ2 þðyi yÞ2 þ ðxj xÞ2 þðyj yÞ2 t1 ij ðx;yÞ ¼ m cg0 ð24Þ In Eq. (23), since each dispersive wavepacket in s0 ij ðtÞ is compensated with its RD0 simultaneously decreased by LWSR in s1 ij ðtÞ, a great enhancement for the resolution and performance of adjacent multiple damage imaging can be achieved. 5. Experimental validation 5.1. Experimental setup To validate LWSR and LWSR-based imaging methods, an experiment of adjacent multiple damage imaging with A0 mode Lamb waves is conducted on a 1000 mm 1000 mm 1.5 mm aluminum plate with material parameters given in Table 1. To monitor the entire plate, eight PZT wafers P1 -P8 (PSN-33, 8 mm in diameter and 0.48 mm in thickness) are bonded to form a square transducer array, as shown in Fig. 9. Four identical hexagonal hollow screws with a diagonal length of 8 mm, denoted as D1–D4, are successively mounted near each other on the plate to simulate adjacent damages. The exact positions
Fig. 9
of PZT wafers and damages in the orthogonal coordinate (see Fig. 9) are listed in Table 2. As illustrated in Fig. 9, the overall experimental setup is composed of a Lamb wave detection system, a matrix switch controller, a power amplifier, and the aluminum plate. The Lamb wave detection system can generate Lamb wave signals and amplify and collect sensor signals. The matrix switch controller controls the working sequence of all PZT pairs, and the power amplifier is applied to amplify the excitation signal to enlarge the monitoring area in the plate. Practically, only a step pulse signal with a raising time of 0.25 ls is generated by the Lamb wave detection system to acquire the impulse response of each PZT pair with 20,000 points at 10 MHz. The narrowband excitation signal va ðtÞ is a 3-cycle sine burst centered at 75 kHz, as shown in Fig. 10. By convoluting the impulse response with va ðtÞ, a desired A0 mode-dominated sensor signal is conveniently extracted. 5.2. Effect of LWSR The efficiency of LWSR for A0 mode dispersive sensor signals and damage-scattered ones is verified. From the impulse response h48 ðtÞ of PZT pair P48 obtained with step pulse excitation in the healthy plate, as Fig. 11(a) shows, the original sensor signal v0 48 ðtÞ under va ðtÞ can be extracted. As Fig. 11 (b) illustrates, the dispersive A0 mode direct arrival A is expanded in much more cycles with its time duration increased from 40 ls (see Fig. 10(a)) to about 90 ls, while the following boundary reflections denoted as B-L are severely superposed and hardly to be recognized. The spatial resolution of v0 48 ðtÞ is evidently influenced by severe dispersion effect and poor ISR.
Table 2
The coordinates of PZTs and damages.
PZT
(x, y) (mm)
PZT or Defect
(x, y) (mm)
P1 P2 P3 P4 P5 P6
(300, 300) (300, 300) (300, 300) (300, 300) (0, 300) (300, 0)
P7 P8 D1 D2 D3 D4
(0, 300) (300, 0) (80, 90) (105, 110) (125, 135) (150, 155)
Experimental setup.
A Lamb Wave Signal Reconstruction method for high-resolution damage imaging
Fig. 10
1095
Narrowband excitation signal with a central frequency of 75 kHz.
Fig. 12 Theoretical A0 mode wavenumber relation K0 ðxÞ in an aluminum plate with a thickness of 1.5 mm.
Fig. 11 Impulse response and original and LWSR-processed sensor signals of P48 .
To enhance the signal resolution by LWSR, the A0 mode theoretical wavenumber relation K0 ðxÞ is firstly derived with the plate material parameters in Table 1, as Fig. 12 shows. With the adjusting factor m set as 1 and 2.5 respectively for comparison, the linearized wavenumber curve K1 ðxÞ or K2 ðxÞ is computed from K0 ðxÞ with Eqs. (8) and (11), as Fig. 13(a) shows. Then, using Eq. (17), the mapping sequence X1 ðxÞ or X2 ðxÞ is decided with K1 ðxÞ and K0 ðxÞ, K2 ðxÞ and K0 ðxÞ respectively, as illustrated in Fig. 13(b). Using
Eq. (19), the LWSR (m ¼ 1) result v1 48 ðtÞ of v0 48 ðtÞ with X1 ðxÞ is attained. As Fig. 11(c) shows, all the dispersioncompensated A0 mode wavepackets get fully recompressed. The first six boundary reflections B-G are well separated, whereas, due to the limited ISR with m ¼ 1, the neighboring boundary reflections H-I and J-L remain overlapped respectively, as illustrated by the dashed box in Fig. 11(c). Much better improvement for the spatial resolution of v1 48 ðtÞ is demanded. Fig. 11(d) gives the LWSR result v2 48 ðtÞ of v0 48 ðtÞ with X2 ðxÞ. Since the ISR is improved to m ¼ 2:5 times, all the 12 recovered A0 mode wavepackets A-L, including the previously-overlapped ones H-L within temporal intervals of 2250–2500 ls and 2650–3000 ls, get apart and can be easily distinguished at temporal sites corresponding to their time delays under the newly decreased group velocity at 75 kHz, cg1 ð75 kHzÞ ¼ cg0 ð75 kHzÞ=2:5. The signal spatial resolution is sufficiently heightened by both dispersion compensation and cg1 ð75 kHzÞ reduction via LWSR with m ¼ 2:5. For LWSR of all the damage-scattered signals, their corresponding impulse responses, as shown in Figs. 14(a) and 15(a) for instance, should be obtained by subtracting the healthy impulse responses of all the PZT pairs from the damaged ones. The LWSR results of typical original scattered signals s0 18 ðtÞ and s00 18 ðtÞ from different numbers of adjacent damages in P18 are given in Figs. 14 and 15, respectively. In s0 18 ðtÞ of dispersion and low ISR, the scattered wavepackets from D1 and D2 are badly overlapped, as Fig. 14(b) shows. Even after dispersion removal by LWSR (m ¼ 1) without an ISR increase, the two recompressed scattered wavepackets in s1 18 ðtÞ are still mixed into a single one, as Fig. 14(c) illustrates. Conversely, after LWSR with m ¼ 2:5, they are successfully separated in s2 18 ðtÞ, as shown in Fig. 14(d). When the four flaws D1–D4 exist, their scattered wavepackets and other ones are completely or partly overlapped in the original s00 18 ðtÞ or its LWSR (m ¼ 1) result s01 18 ðtÞ, as Figs. 15
1096
X. WANG et al.
Fig. 13 Linearized wavenumber curve K1 ðxÞ or K2 ðxÞ and interpolation mapping sequence X1 ðxÞ or X2 ðxÞdetermined for LWSR with m ¼ 1 or m ¼ 2:5.
Fig. 14
Results of LWSR (m ¼ 1 or m ¼ 2:5) of the original scattered signal s0
Fig. 15
Results of LWSR (m ¼ 1 or m ¼ 2:5) of the original scattered signal s00
18 ðtÞ
18 ðtÞ
from D1–D2 with X1 ðxÞ or X2 ðxÞ.
from D1-D4 with X1 ðxÞ or X2 ðxÞ.
A Lamb Wave Signal Reconstruction method for high-resolution damage imaging (b) or 15(c) illustrates. Both s00 18 ðtÞ and s01 18 ðtÞ of a limited ISR are subject to the inadequate spatial resolution and the resultant signal superposition. With m raised to 2.5 during LWSR, the four separated damage-scattered wavepackets distinctly emerge with restored waveforms and enough time space in s02 18 ðtÞ, as illustrated in Fig. 15(d). Figs. 14 and 15 demonstrate that the signal spatial resolution can be adequately heightened by LWSR with m ¼ 2:5, making information extraction for adjacent multiple flaws become an easy work.
Fig. 16
In addition, to compare the performance of LWSR with those of other dispersion compensation approaches, the typical TDDM is performed for the original scattered signals s0 18 ðtÞ and s00 18 ðtÞ. Similar to the LWSR (m ¼ 1) results (see Figs. 14 (c) and 15(c)), though all the dispersive damage-scattered wavepackets get well compensated and compressed in the TDDM-processed scattered signals, as Fig. 16 shows, they keep grievously overlapped and interfere with each other, because of the insufficient ISR.
Results of TDDM of the original scattered signal s0
Fig. 17
1097
18 ðtÞ
from D1–D2 or s00
Different imaging results of D1 and D2.
18 ðtÞ
from D1-D4.
1098
X. WANG et al.
Fig. 18
Different imaging results of D1, D2, D3, and D4.
5.3. Results of damage imaging By using the original, LWSR-, or TDDM-processed scattered signals measured by all the 28 PZT pairs, several images can be generated, respectively. For clearer illustration, only imaging results for the 400 mm 400 mm central zone of the plate are given in Figs. 17 and 18, where every symbol ‘‘X” denotes an actual damage position. Due to the poor spatial resolution of the scattered signals caused by dispersion and inadequate ISR (see Figs. 14(b) and 15(b)), the diffused and blurred flaw spots in the images are combined together, leading to much difficulty for damage identification, as Figs. 17(a) and 18(a) illustrate. In the LWSR (m ¼ 1)-based imaging results, as shown in Figs. 17(b) and 18(b), the flaw spots get converged but still interfere with each other. Similar imaging results can be obtained with the TDDM-based imaging methods, as Figs. 17(c) and 18(c) illustrate. This is because the scattered signals cannot bear a good enough spatial resolution just after dispersion removal (see Figs. 14(c), 15(c), and 16). In comparison, with both dispersion eliminating and ISR increasing actualized in the LWSR (m ¼ 2:5)-processed scattered signals (see Figs. 14(d) and 15(d)), every defect can be expediently identified as a bright focalized spot without any interference, as illustrated in Figs. 17(d) and 18(d). Only the satisfying imaging
resolution and performance for different numbers of adjacent multiple damages can be achieved by the proposed imaging method based on LWSR with m ¼ 2:5. 6. Conclusions (1) To solve the problem of inferior signal spatial resolution caused by dispersion and poor ISR in Lamb wave damage detection, LWSR is developed in this study. In LWSR, a linearly-dispersive version of an original Lamb wave signal is reconstructed with its group velocity adequately decreased at the central frequency, so that both dispersion compensation and ISR improvement can be realized to achieve a satisfying super signal spatial resolution. (2) Associated with the delay-and-sum algorithm, LWSR is applied for high-resolution Lamb wave imaging of adjacent multiple damages. (3) Experimental results prove that all the A0 mode dispersive wavepackets in sensor signals or damage-scattered signals are properly recompressed to a pristine excitation waveform. Furthermore, with the adjusting parameter m raised from 1 to 2.5, the ISR is sufficiently heightened to successfully separate formerly compensated but super-
A Lamb Wave Signal Reconstruction method for high-resolution damage imaging posed neighboring wavepackets at their arrival times corresponding to the newly decreased cg1 ¼ cg0 =m. Using LWSR (m ¼ 2:5)-processed scattered signals with an adequate spatial resolution, different numbers of adjacent multiple damages can be successfully imaged and identified as bright focalized spots without any interference among them.
Acknowledgement This study was supported by the Fundamental Research Funds for the Central Universities, China (No. NS2016012). References 1. Yuan SF, Chen J, Yang WB, Qiu L. On-line crack prognosis in attachment lug using Lamb wave-deterministic resampling particle filter-based method. Smart Mater Struct 2017;26(8):085016. 2. Bhuiyan MY, Shen YF, Giurgiutiu V. Guided wave based crack detection in the rivet hole using global analytical with local FEM approach. Materials 2016;9(7):602. 3. Qiu L, Liu B, Yuan SF, Su ZQ. Impact imaging of aircraft composite structure based on a model-independent spatialwavenumber filter. Ultrasonics 2016;64:10–24. 4. Yuan SF, Ren YQ, Qiu L, Mei HF. A multi-response-based wireless impact monitoring network for aircraft composite structures. IEEE T Ind Election 2016;63(12):7712–22. 5. Hong M, Su ZQ, Lu Y, Sohn H, Qing XP. Locating fatigue damage using temporal signal features of nonlinear Lamb waves. Mech Syst Signal Process 2015;60–61:182–97. 6. Qiu L, Yuan SF, Chang FK, Bao Q, Mei HF. On-line updating Gaussian mixture model for aircraft wing spar damage evaluation under time-varying boundary condition. Smart Mater Struct 2014;23(12):125001. 7. Wilcox PD, Lowe MJS, Cawley P. The effect of dispersion on long-range inspection using ultrasonic guided waves. NDT & E Int 2001;34(1):1–9. 8. Park HW, Kim SB, Sohn H. Understanding a time reversal process in Lamb wave propagation. Wave Mot 2009;46(7):451–67. 9. Cai J, Shi LH, Yuan SF, Shao ZX. High spatial resolution imaging for structural health monitoring based on virtual time reversal. Smart Mater Struct 2011;20(5):55018–28.
1099
10. Wilcox PD. A rapid signal processing technique to remove the effect of dispersion from guided wave signals. IEEE Trans Ultrason Ferroelectr 2003;50(4):419–27. 11. Prado VT, Higuti RT, Kitano C, Martinez-Graullera O, Adamowski JC. Lamb mode diversity imaging for non-destructive testing of plate-like structures. NDT & E Int 2013;59(7):86–95. 12. Yu L, Tian ZH. Guided wave phased array beamforming and imaging in composite plates. Ultrasonics 2016;68:43–53. 13. Grabowski K, Gawronski M, Baran I, Spychalski WL, Staszewski T, Uhl T, et al. Time-distance domain transformation for acoustic emission source localization in thin metallic plates. Ultrasonics 2016;68:142–9. 14. Cai J, Shi LH, Qing XP. A time-distance domain transform method for Lamb wave dispersion compensation considering signal waveform correction. Smart Mater Struct 2013;22 (11):105024. 15. Luo Z, Zeng L, Lin J, Hua JD. A reshaped excitation regenerating and mapping method for waveform correction in Lamb waves dispersion compensation. Smart Mater Struct 2017;26(2):025016. 16. Liu L, Yuan FG. A linear mapping technique for dispersion removal of Lamb waves. Struct Health Monit 2010;9(1):75–86. 17. Cai J, Yuan SF, Qing XP, Chang F, Shi LH, Qiu L. Linearly dispersive signal construction of Lamb waves with measured relative wavenumber curves. Sens Actuators A 2015;221:41–52. 18. Cai J, Yuan SF, Wang TG. Signal construction-based dispersion compensation of Lamb waves considering signal waveform and amplitude spectrum preservation. Materials 2017;10(1):4. 19. Marchi LD, Marzani A, Miniaci M. A dispersion compensation procedure to extend pulse-echo defects location to irregular waveguides. NDT & E Int 2013;54(3):115–22. 20. Fu SC, Shi LH, Zhou YH, Cai J. Dispersion compensation in Lamb wave defect detection with step-pulse excitation and warped frequency transform. IEEE Trans Ultrason Ferroelectr 2014;61 (12):2075–88. 21. Marchi LD, Perelli A, Marzani A. A signal processing approach to exploit chirp excitation in Lamb wave defect detection and localization procedures. Mech Syst Signal Process 2013;39(1– 2):20–31. 22. Lin J, Hua JD, Zeng L, Luo Z. Excitation waveform design for Lamb wave pulse compression. IEEE Trans Ultrason Ferroelectr 2016;63(1):165–77. 23. Malo S, Fateri S, Livadas M, Mares C, Gan T. Wave mode discrimination of coded ultrasonic guided waves using twodimensional compressed pulse analysis. IEEE Trans Ultrason Ferroelectr 2017;64(7):1092–101.