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Wave Motion 46 (2009) 451–467
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Wave Motion
journal homepage: www.elsevier .com/locate /wavemoti
Understanding a time reversal process in Lamb wave propagation
Hyun Woo Park a, Seung Bum Kim b, Hoon Sohn c,*
a Department of Civil Engineering, Dong-A University, Busan 604-714, Republic of Koreab Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 85287-6106, USAc Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Republic of Korea
a r t i c l e i n f o
Article history:Received 8 May 2008Received in revised form 27 April 2009Accepted 30 April 2009Available online 9 May 2009
Keywords:Lamb waveTime reversal process (TRP)Within-mode dispersionMultimode dispersionReflectionsAmplitude dispersionRegular waveguidePiezoelectric transducerReference-free NDT
0165-2125/$ - see front matter � 2009 Elsevier B.Vdoi:10.1016/j.wavemoti.2009.04.004
* Corresponding author. Tel.: +82 42 869 3625; faE-mail addresses: [email protected] (H.W. Park)
a b s t r a c t
This study investigates the time reversal process (TRP) of Lamb wave signals which aretransmitted and received by piezoelectric transducers bonded on plate-like structures. Anumber of previous studies have paid attention to spatial and temporal refocusing capabil-ity of an original excitation through the TRP in highly dispersive and complex media. How-ever, when the TRP is applied to Lamb waves in a homogeneous regular waveguide, therefocusing capability is limited due to permanent residual side bands even if the durationof the time reversed signal increases. Based on the reciprocity of elastodynamics and linearpiezoelectricity, theoretical interpretation is conducted for the main and residual sidebands of the reconstructed signal in the time domain. In particular, the interpretationincludes the temporal effect of velocity and amplitude dispersions, the existence of multi-modes, and the reflections from boundaries during the TRP. Then, numerical and experi-mental tests are conducted to validate the theoretical findings of this paper. Practicalissues for the successful implementation of the TRP of Lamb waves are briefly addressedas well.
� 2009 Elsevier B.V. All rights reserved.
1. Introduction
According to conventional time reversal acoustics, an input signal can be focused at an excitation point if an output signalrecorded at another point is reversed in the time domain and emitted back to the original source point [1]. This time revers-ibility is based on the spatial reciprocity and time-reversal invariance of linear wave equations [2].
Time reversal acoustics was first introduced by the modern acoustics community and applied to many fields such as lith-otripsy, ultrasonic brain surgery, active sonar and underwater communications, medical imaging, hyperthermia therapy,bioengineering, and non-destructive testing (NDT) [1,3–5]. Then, Ing and Fink adopted the TRP to Lamb waves based NDTin order to compensate for the dispersion of Lamb waves and to detect defects in a pulse-echo mode [6–9]. The main interestof these studies was refocusing energy in the time and spatial domain by compensating for the dispersive characteristics ofLamb waves.
The authors advanced this TRP concept to develop a NDT technique where defects can be identified without requiringdirect comparison with previously obtained baseline data [10–12]. The authors’ work intends to reconstruct the‘‘shape” of the original input signal during the TRP. When nonlinearity is caused by a defect along a direct wave path, theshape of the reconstructed signal is reported to deviate from that of the original input signal. Therefore, examining the devi-ation of the reconstructed signal from the known initial input signal allows instantaneous identification of damage withoutrequiring the baseline signal for comparison.
. All rights reserved.
x: +82 42 869 3610., [email protected] (S.B. Kim), [email protected] (H. Sohn).
452 H.W. Park et al. / Wave Motion 46 (2009) 451–467
However, when the TRP is applied to Lamb waves in a homogeneous regular waveguide, the refocusing capability is lim-ited due to permanent residual side bands even if the duration of the time reversed signal increases. Draeger and Finkdemonstrated this limitation of the TRP through the cavity equation for the one-channel time reversal on chaotic cavities[13]. This approach utilizes the approximate eigenmode decomposition of wave fields in chaotic cavities which have manyadvantageous properties for modal analysis. Unfortunately, most structural waveguides used in civil, mechanical and aero-space engineering usually are not chaotic cavities. Therefore, the interpretation of the TRP based on eigenmode decomposi-tion is rather difficult because the modal properties vary according to the geometric shape and boundary conditions of thestructural waveguides. In this respect, interpretation of the TRP in terms of wave propagation is more appropriate becausedispersion characteristics are independent of the geometry and boundary conditions of a waveguide and determined by theproduct of the driving frequency and the thickness of a waveguide.
In this study, the time reversibility of Lamb wave signals generated and sensed using piezoelectric transducers is theo-retically investigated in the time domain, limitation in terms of full reconstruction of the input signal is discussed, and tech-niques to restore the shape of the input waveform are developed. In particular, attention has been paid to understanding ofthe following effects on the TRP in the time domain: (1) velocity and amplitude dispersion characteristics of Lamb waves, (2)the existence of multiple Lamb wave modes, and (3) reflections from structural boundaries. Numerical simulations andexperimental tests are conducted to validate the theoretical findings of this study.
2. A time reversal process for Lamb waves
2.1. Introduction to Lamb waves
All elastic waves including body and guided waves are governed by the same set of partial differential equations [14]. The pri-mary difference is that, while body waves are not constrained by any boundaries, guided waves need to satisfy the boundary con-ditions imposed by the physical systems as well as the governing equations. Lamb waves are one type of guided waves that existin thin plate-like structures, and they are plane strain waves constrained by two free surfaces [15]. The advances in sensor andhardware technologies for efficient generation and detection of Lamb waves and the increased usage of solid composites in load-carrying structures have led to an explosion of studies that use Lamb waves for detecting defects in composite structures [16–24].Because Lamb waves are guided and constrained by two free surfaces, the Lamb waves can propagate a relatively long distancewithout much attenuation. This long sensing range makes Lamb waves attractive for damage diagnosis.
Unlike body waves, the propagation of Lamb waves is complicated due to their dispersive and multimode characteristics[25]. Fig. 1 illustrates two distinct velocity dispersion characteristics of Lamb waves. The first dispersion characteristic isvelocity dispersion within a single mode, and it is referred to as the within-mode dispersion (often referred to as group velocitydispersion) in this paper. This within-mode dispersion is caused by the frequency dependency of a single Lamb wave mode.That is, the different frequency components in a single mode travel at different speeds, and this within-mode dispersion re-sults in the spreading of the wave packet as it propagates. The second dispersion characteristic is velocity dispersion amongmultiple modes and referred to as the multimode dispersion (often referred to as modal dispersion) hereafter. This multimodedispersion exists because different modes at a given frequency travel at different speeds. Therefore, when an input waveformwith a discrete driving frequency is applied to a thin medium, it is separated into multiple modes, traveling at different speeds.Finally, the amplitude attenuation of a Lamb wave is also frequency dependent, and it is called amplitude dispersion. Due tothese unique dispersion characteristics of Lamb waves, time reversibility of Lamb wave signals can be complicated.
2.2. Introduction to time reversal acoustics
The propagation of body waves in elastic media is a classical topic covered in many elasticity textbooks [14,26,27]. Bydefinition, the body waves propagate throughout a medium, which is not constrained by boundaries. Body wave character-
0
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7
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Fig. 1. A typical dispersion curve of Lamb waves in a thin aluminum plate.
H.W. Park et al. / Wave Motion 46 (2009) 451–467 453
istics are described by the Navier governing equations [26], and they can be further divided into longitudinal and shearwaves. In time reversal acoustics, an input body wave can be refocused at the source location if a response signal measuredat a distinct location is time-reversed (literally the time point at the end of the response signal becomes the starting timepoint) and reemitted to the original excitation location. This phenomenon is referred to as time reversibility of body waves,and this unique feature of refocusing has found applications in lithotripsy, ultrasonic brain surgery, active sonar and under-water communications, medical imaging, hyperthermia therapy, bioengineering, and non-destructive testing (NDT) [1,3–5].
2.3. Extension of time reversal acoustics to Lamb waves
While the TRP for non-dispersive body waves has been well-established, the study of the TRP for Lamb waves is still rel-atively new [10]. Previous works mainly investigated spatial and temporal focusing of the reconstructed input signal in thepresence of dispersive characteristics of waves in theoretical and experimental fashions [13,28–31]. Among the previousworks on interpreting the TRP of Lamb waves, the work done by Draeger et al. [13,28] is the most relevant to this study.In particular, the TRP was theoretically investigated through the cavity equation using the approximate eigenmode decom-position of wave fields in a strategically designed waveguide called chaotic cavities [13]. The refocusing capability of the TRPcould be simply expressed by the energy ratio of the main mode to the residual side bands in the reconstructed signal thanksto the modal properties of chaotic cavities.
However, most structural waveguides used in civil, mechanical and aerospace engineering are usually not chaotic cavi-ties. Therefore, the interpretation of the TRP based on eigenmode decomposition is rather difficult because the modal prop-erties vary according to the geometric shape of the structural waveguides. To alleviate this difficulty, interpretation of theTRP in terms of wave propagation is more desirable because dispersion characteristics are independent of the shape of awaveguide and determined by the product of the driving frequency and the thickness of a waveguide. Concerning theTRP of Lamb waves, the fundamental property of Lamb wave propagation is well known as described in Section 2.1. Becauseof dispersion and multimodal characteristics of Lamb waves, the interpretation of the time reversibility of Lamb waves be-comes complicated on the regular homogeneous plate, and it has limited the applicability of the TRP to Lamb waves. Thissubsection intends to describe how the TRP works for Lamb wave propagation.
The TRP is first formulated in the frequency domain incorporating the PZTs used for excitation and measurement of Lambwaves (Fig. 2). Based on the piezoelectricity of PZT materials, an electrical voltage V(x) applied to a PZT wafer is converted toa mechanical strain eðxÞ through the following electro-mechanical efficiency coefficient kaðxÞ [32]. Note that the dynamicsof a PZT wafer are neglected and it is assumed that the dynamic behavior of a plate is uncoupled from that of the PZT wafer[33]
eðxÞ ¼ kaðxÞVðxÞ ð1Þ
On the other hand, a voltage is generated when a PZT wafer is subjected to a mechanical strain. This conversion from themechanical strain to the voltage output is related by the other mechanical-electro efficient coefficient, ksðxÞ. Note that allfield variables in Eq. (1) are frequency dependent, and x denotes an angular frequency. Hereafter, the angular frequencyis omitted from the entire field variables for simplicity unless stated otherwise.
When an excitation voltage is applied to PZT A as shown in Fig. 2a, the corresponding response voltage at PZT B can berepresented by the following equation:
VB ¼ ksGkaVA ð2Þ
where G is the structure’s transfer function relating an input strain at PZT A to an output strain at PZT B. Here, the input volt-age applied at PZT A, VA, is first converted to a mechanical strain via ka. Then, the corresponding response strain at PZT B isconverted to the output voltage at PZT B, VB through ks (Fig. 2b).
(a) Input SignalPZT A
PZT B
ComparedTime Reversed
Backward PropagationForward Propagation
(a) Input Signal(a) Input Signal
(b) Response Signal
PZT A
PZT B
PZT APZT A
PZT BPZT B
ComparedComparedTime ReversedTime Reversed
(d) RestoredSignal
(c) ReemittedSignal
Backward Propagation
Backward Propagation
Backward PropagationForward Propagation
Forward Propagation
Forward Propagation
Fig. 2. A schematic outline of the time reversal process applied to a plate structure [10].
454 H.W. Park et al. / Wave Motion 46 (2009) 451–467
In the second step of the TRP, the measured response voltage, VB, is reversed in the time domain before reemitted back toPZT B (Fig. 2c). This TRP in the time domain is equivalent to taking a complex conjugate of the signal in the frequency do-main. In the final step, the reversed version of VB is applied back to PZT B, and the corresponding response is measured at PZTA. This final response at PZT A is referred to as the ‘‘reconstructed” signal. The reconstructed voltage signal at PZT A, VR, canbe related to the response voltage, VB, in the previous step as follows:
VR ¼ ksGkaV�B ð3Þ
where the superscript * denotes the complex conjugate operation. Note that the transfer function G in Eq. (3) is assumed tobe identical to the one in Eq. (2) based on the reciprocity of elastodynamics [32,34]. By inserting Eq. (2) into Eq. (3), thereconstructed signal, VR, is related to the original input signal, VA
VR ¼ ksGkaV�B ¼ ksGkak�s G�k�aV�A ¼ CKK�V�A ð4Þ
where K = kska, and C is the time reversal operator defined as C = GG*.If the time reversal operator and the mechanical-electro coefficients are assumed to be constant over the frequency range
of interest, Eq. (4) indicates that the reconstructed signal, VR, is a ‘‘time-reversed” and ‘‘scaled” version of the original inputsignal VA (Fig. 2d). That is, the ‘‘shape” of the original input signal should be at least reproduced by the reconstructed signalduring the TRP.
In reality, the time reversal operator is frequency dependent for Lamb waves [10,32] because signal components at dif-ferent frequencies travel at different speeds and varying attenuation rates. Therefore, the shape of the reconstructed signalwill not be identical to that of the original input signal for Lamb wave propagation.
3. Understanding various effects on the TRP of Lamb waves
3.1. Frequency dependency of the TRP
In time reversal acoustics, the time reversal can be considered an adaptive filter [35,36]. When it comes to Lamb waves,the main characteristics of this time reversal adaptive filter is its frequency dependency, causing amplitude dispersion ofLamb waves. Park et al. [10] investigated the time reversibility of Lamb waves on a composite plate and introduced the timereversal operator into the Lamb wave equation based on the Mindlin plate theory [37]. Because of the amplitude dispersionof Lamb waves, the time reversal operator varies with respect to frequency as shown in Fig. 3, and wave components at dif-ferent frequency values are non-uniformly amplified during the TRP. Due to this amplitude dispersion of the TRP, the originalinput signal cannot be properly reconstructed if the input signal consists of multiple frequency components such as a broad-band input signal. A narrowband excitation has been used to avoid this issue. Note that Park et al. [10] considered only thefundamental anti-symmetric mode. In reality, the existence of multimodes complicates the TRP of Lamb waves [38]. The ef-fects of within-mode dispersion, multimode dispersion, and reflections on time reversal are subsequently studied in thispaper.
Park et al. [10] described the frequency bandwidth effects of input signals on the TRP and justified the use of narrowbandexcitation as shown in Fig. 4. A Gaussian pulse (Fig. 4a) and a 100 kHz tone burst (Fig. 4d) input signals are employed toinvestigate the frequency dependency of the TRP. The distance between the PZT pair is assumed to be long enough so thatthe within-mode dispersion of Lamb waves can be observed at the response PZT (Fig. 4b and e). When the response signalsare reversed in the time domain and reemitted to the input PZT, the within-mode dispersion of Lamb waves is compensated(Fig. 4c and f).
As demonstrated here, the within-mode dispersion can be compensated during the TRP. Some wave components withinthe single Lamb wave mode travel at higher speeds and arrive at a sensing point earlier than those traveling at lower speeds.
0
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0.6
0.8
1
0 100 200 300 400 500
Magnitude of the time reversal operator
Frequency (KHz)
Nor
mal
ized
mag
nitu
de
Fig. 3. Normalized time reversal operator of the A0 mode [10].
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Fig. 4. Reconstruction of input signals using broadband and narrowband input signals through a numerically simulated time reversal process: (a) originalbroadband input signal-Gaussian pulse, (b) response signal-Gaussian pulse, (c) original input (dotted) and reconstructed input (solid) signal-Gaussian pulse,(d) original narrowband input signal-100 kHz tone burst, (e) response signal-100 kHz tone burst, and (f) original input (dotted) and reconstructed input(solid) signal-100 kHz tone burst [10].
H.W. Park et al. / Wave Motion 46 (2009) 451–467 455
However, during the TRP at the sensing location, the wave components, which travel at slower speeds and arrive at the sens-ing point later, are reemitted to the original source location first. Therefore, all wave components traveling at differentspeeds concurrently converge at the source point during the TRP, compensating for the within-mode dispersion of Lambwaves.
Due to the amplitude dispersion, the shape of the original pulse is, however, not fully recovered when the Gaussian inputis used (Fig. 4c). The various frequency components of the Gaussian input are non-uniformly scaled and superimposed dur-ing the TRP as indicated in Fig. 4c. On the other hand, the shape of the reconstructed tone burst waveform is practically iden-tical to that of the original input tone burst within a certain tolerance level because the amplification of the time reversaloperator is almost uniform for a limited frequency band (Fig. 4f).
This subsection illustrates that the frequency dependency of the TRP can be minimized by using a narrowband excita-tion signal rather than a broadband excitation. Although the use of a narrowband input can enhance the TRP, the shape ofthe reconstructed signal still will not be identical to that of the original input signal due to the multimode dispersion andreflections from the structure’s boundaries. Their effects on the TRP are theoretically described in the followingsubsections.
456 H.W. Park et al. / Wave Motion 46 (2009) 451–467
3.2. Understanding the effect of the within-mode dispersion on the TRP
In Section 3.1, it is shown that the amplitude dispersion of the TRP can be minimized by using a narrowband excitationsignal rather than a broadband excitation. In this subsection, the effect of within-mode dispersion on the TRP is mathemat-ically described by considering a single symmetric or anti-symmetric mode induced by a narrowband input excitation. InSection 3.3, this description is extended for general Lamb wave propagation where multiple symmetric and anti-symmetricmodes exist.
Considering the amplitude and within-mode dispersion of a single symmetric/anti-symmetric mode, the transfer functionG in Eq. (2) can be simplified as follows [39]:
G ¼ c expð�ikrÞ ð5Þ
where c, i and k denote the amplitude dispersion function, the imaginary number and the wave number of a specific modewhile r denotes a distance between actuating and sensing PZT wafers, respectively. In turn, the time reversal operator C inEq. (4) can be expressed as follows:
C ¼ GG� ¼ ½c expð�ikrÞ�½c� expðikrÞ� ¼ cc� ð6Þ
Eq. (6) shows that the within-mode dispersion in the forward wave propagation process, expð�ikrÞ, is automati-cally compensated during the time reversal operation, resulting in only amplitude attenuation at the end. By substi-tuting Eq. (6) into Eq. (4), the relationship between the reconstructed signal and the original input signal can besimplified
VR ¼ cc�KK�V�A ð7Þ
When a narrowband tone burst signal is applied around a central frequency �x, the amplitude dispersion (attenuation) func-tion and electro-mechanical transduction coefficients can be assumed to be constant over the narrow frequency band [10].Then, the reconstructed input signal in the time domain can be obtained by taking an inverse Fourier transform of Eq. (7)
VRðtÞ ¼1
2p
Z 1
�1VR expðixtÞdt ¼ 1
2p
Z 1
�1cc�KK�V�A expðixtÞdx ¼ C �jVAðT � tÞ ð8Þ
where C ¼ cð �xÞc�ð �xÞ, �j ¼ Kð �xÞK�ð �xÞ, and T denotes the total time duration of the measured reconstructed signal. Note thatthe complex conjugate of the input signal in the frequency domain is equivalent to the time reversed version of the originalinput in the time domain after the inverse Fourier transform.
Eq. (8) confirms again that, as long as a single Lamb wave mode is concerned, the reconstructed signal is simply a ‘‘time-reversed” and ‘‘scaled” version of the original input signal as previously described in Eq. (4). Note that the within-modedispersion of a single mode signal is compensated during the TRP, and it does not affect the time reversibility as also dem-onstrated in the previous works [8,10,11]. In the next subsection, the time reversibility is extended considering the multi-mode dispersion of multiple symmetric and anti-symmetric modes.
3.3. Understanding the effect of multimodes on the TRP
The effects of two independent wave modes (P and S modes) on the TRP have been theoretically investigated by Draegeret al. for the first time when both wave modes are simultaneously generated in an isotropic solid [28]. According to theirwork, the TRP produces not only the original input signal at the expected time but also sidebands that arrive before andafterwards. Similar to this case, the multimodal characteristics of Lamb waves complicate the TRP. The effect of the multi-mode characteristic is schematically shown in Fig. 5. When a tone burst signal is exerted to PZT A (Fig. 5a), multimodes arereceived at PZT B (Fig. 5b). In Fig. 5, the narrowband input frequency is selected so that only the first symmetric (S0) and anti-symmetric (A0) modes are generated. When the response signal is reversed in the time domain and reemitted to PZT B(Fig. 5c), each reemitted signal associated with the A0 or S0 modes at PZT B creates both S0 and A0 modes producing a totalof four modes in the reconstructed signal (Fig. 5d and e). In Fig. 5, S0/A0 denotes the S0 mode signal measured at PZT A due tothe A0 mode input at PZT B. A0/S0, S0/S0, and A0/A0 are similarly defined. After superposition of signals in Fig. 5d and e, thereconstructed signal consists of the main mode at the middle and two sidebands around the main mode (Fig. 5f). Note thatthe main mode signal in the middle is the superposition of the A0/A0 and S0/S0 and ‘‘symmetric” sidebands are produced as aresult of A0/S0 and S0/A0. Finally, the shape of the main mode will be practically identical to that of the original input signal.
Considering this multimode effect on the TRP, the reconstructed signal will be composed of the following four modegroups:
VRðtÞ ¼ VSSR ðtÞ þ VAA
R ðtÞ þ VSAR ðtÞ þ VAS
R ðtÞ ð9Þ
where VSAR ðtÞ represents the symmetric mode signal in the reconstructed (one) generated by the anti-symmetric mode signal
in the forward (one), and VSSR ðtÞ, VAA
R ðtÞ, and VASR ðtÞ are defined in a similar fashion. Consequently, the time reversal operator C
in Eq. (4) is also decomposed to those associated with symmetric and anti-symmetric modes.
C ¼ CSS þ CAA þ CSA þ CAS ð10Þ
Fig. 5. The effect of multimodes on the time reversal process. (Note: S0/A0 denotes S0 mode produced at PZT A due to A0 mode input at PZT B. A0/S0, S0/S0,and A0/A0 are similarly defined.)
H.W. Park et al. / Wave Motion 46 (2009) 451–467 457
Initially, the coupling effect among symmetric mode signals (VSSR ðtÞ and CSS) is explained, and this description is extended
for multiple symmetric and anti-symmetric mode signals. For brevity, the superscription ‘‘S” denoting the symmetric modeis omitted until all multiple modes are included at the end of this subsection. Considering the coupling among symmetricmode signals, Eq. (6) can be extended as follows:
C ¼XnS
p¼1
XnS
q¼1
gpg�q ð11Þ
where gp denotes the transfer function of the pth symmetric mode, and nS, represents the total number of symmetric modesat a given excitation frequency �x.
Similar to Eq. (5), gp can be expressed as a function of the amplitude and velocity dispersions
gp ¼ cp expð�ikprÞ ð12Þ
where cp and kp denote the amplitude dispersion function and the wave number of the pth symmetric mode. Substituting Eq.(12) into Eq. (11) results in
C ¼XnS
p¼1
XnS
q¼1
gpg�q ¼XnS
p¼1
XnS
q¼1
cpc�q exp½irðkp � kqÞ� ¼XnS
p¼1
XnS
q¼1
Cpq expðihpqÞ ð13Þ
where, hpq ¼ r½kp � kq� and Cpq ¼ cpc�q. Subsequently, the reconstructed input signal associated with the coupling of symmet-ric mode signals can be expressed as follows:
VR ¼ CKK�V�A ¼XnS
p¼1
XnS
q¼1
Cpq expðihpqÞKK�V�A ð14Þ
Similar to Eq. (8), the reconstructed input signal in the time domain can be obtained by taking the inverse Fourier trans-form of Eq. (14) when an original input signal is a narrowband tone burst with a center frequency of �x
VRðtÞ ¼XnS
p¼1
XnS
q¼1
12p
Z 1
�1Cpq �jV�A exp½iðhpq þxtÞ�dx ð15Þ
where Cpq ¼ cpð �xÞc�qð �xÞ and �j ¼ Kð �xÞK�ð �xÞ. Because hpq in Eq. (15) equals to zero when p = q, Eq. (15) can be rewritten asfollows:
VRðtÞ ¼ VAðT � tÞXnS
p¼1
Cpp �jþXnS
p¼1
XnS
q¼1
ð1� dpqÞ1
2p
Z 1
�1Cpq �jV�A exp½iðxt þ hpqÞ�dx ð16Þ
where dpq denotes the Kronecker delta in which dpq = 0 if p – q and dpq ¼ 1 if p = q.
458 H.W. Park et al. / Wave Motion 46 (2009) 451–467
Note that there is no closed form solution for the integral term in Eq. (16). To get an approximate solution of this integral,hpq is expanded using a Taylor series near the driving frequency �x up to the first order term
hpq ¼ hpqð �xÞ þdhpq
dx
����x¼ �xðx� �xÞ þH:O:T: � hpqð �xÞ þ
dhpq
dx
����x¼ �xðx� �xÞ ð17Þ
By using the relationships among the angular frequency �x, the wave number k, the group velocity w, and the phase veloc-ity v, Eq. (17) is expressed as follows:
hpq � �x�spq þx�tpq ð18Þ
where
�spq ¼r
vpð �xÞ� r
wpð �xÞ
� �� r
vqð �xÞ� r
wqð �xÞ
� �� �; �tpq ¼
rwpð �xÞ
� rwqð �xÞ
� �ð19Þ
and
dx ¼ wpðxÞdkp; x ¼ vpðxÞkp ð20Þ
Using Eqs. (18)–(20), Eq. (16) is expressed as follows:
VRðtÞ ¼XnS
p¼1
Cpp �jVAðT � tÞ þXnS
p¼1
XnS
q¼1
ð1� dpqÞCpq �jfexpði �x�spqÞVA½T � ðt þ �tpqÞ�g ð21Þ
The first term on the right hand side of Eq. (21) indicates that both within-mode and multimode dispersions are compen-sated and Lamb wave modes converge to the main mode as long as the identical mode travels in both forward and backwardpropagation directions. On the other hand, the second term reveals that, when Lamb waves travel at two different groupvelocities in the forward and backward propagation directions, the corresponding modes in the reconstructed input signalare shifted in the time domain by ��tpq, creating ‘‘sidebands” around the main mode where the most of the energy converges.Note that the within-mode dispersion in the time domain is still fully compensated in the second term and the time shift��tpq depends only on the difference between the group velocities of the pth and qth modes. It is also noted that the ampli-tude of each sideband is proportional to expði �x�spqÞ, which depends on the phase velocity difference between the pth and qthmodes. Here, the term expði �x�spqÞ can be interpreted as the phase shift of each sideband. Because �spq ¼ ��sqp and �tpq ¼ ��tqp inEq. (18), it can be shown that the second term of the reconstructed signal in Eq. (21) is symmetric with respect to the mainpeak mode in the middle. Note that this symmetry of the reconstructed signal in Eq. (21) is valid regardless the symmetry ofthe structure or the PZT transducer layout as long as the input signal is symmetric.
In a similar manner, VSAR , VAA
R , and VASR terms in Eq. (9) can be calculated as follows:
VSAR ðtÞ ¼
XnS
p¼1
XnA
q¼1
CSApq
�jfexpði �x�spqÞVA½T � ðt þ �tSApqÞ�g ð22Þ
VASR ðtÞ ¼
XnA
p¼1
XnS
q¼1
CASpq
�jfexpði �x�spqÞVA½T � ðt þ �tASpqÞ�g ð23Þ
VAAR ðtÞ ¼
XnA
p¼1
CAApp
�jVAðT � tÞ þXnA
p¼1
XnA
q¼1
ð1� dpqÞCAApq
�jfexpði �x�spqÞVA½T � ðt þ �tAApq Þ�g ð24Þ
where superscripts S and A denote symmetric and anti-symmetric modes while nA denotes the total number of anti-symmet-ric modes at the excitation frequency �x. Eqs. (22) and (23) reveal that the VSA
R and VASR terms in the reconstructed input signal
do not converge to the main mode but only create additional sidebands. The same can be said about VAAR except that it con-
verges on the main mode when p = q. However, it is noted that the symmetry of the reconstructed signal is still preservedeven in the presence of multiple symmetric and anti-symmetric modes because the TRP is a temporal correlation betweenthe multiple Lamb wave modes.
The effects of modal dispersion on the time reversibility of Lamb waves described in Eqs. (21)–(24) is validated through anumerical simulation on an aluminum plate model with a pair of surface-bonded PZT transducers (PZTs A and B) in Fig. 6a.The geometric configuration of the aluminum plate and the location of PZT transducers are properly arranged so that onlythe effects of modal dispersion are included during the TRP without the interference of reflections. The parameters for thePZT transducers were adopted from PSI-5A4E type of PZT sheets (thickness = 0.0508 cm) which is commercially available[40]. The PZT transducers are assumed to be rigidly bonded on the host aluminum plate. The numerical simulation is con-ducted through ABAQUS 6.7-1 standard on IBM p595 in KISTI supercomputing center [41]. 1 mm � 1 mm 8-node quadraticplane strain solid elements are employed for the aluminum plate while 1 mm � 0.507 mm 8-node quadratic plane strain pie-zoelectric elements are used for the pair of PZT transducers. For the accuracy of solution, the Newmark-b method is em-ployed for numerical time integration with a time increment of 2.5 � 10�7 s (4 MHz).
The 100 kHz tone burst input signal in Fig. 7a is used to create the S0 and A0 modes as shown in Fig. 7b. Fig. 7b displays theoutput response signal at PZT B when the created Lamb wave modes arrive at PZT B, 60 cm away from PZT A. The relative
(a) An aluminum plate model with a pair of surface-bonded PZT transducers
0
1
2
3
4
5
6
0 100 200 300Frequency (kHz)
Gro
up v
eloc
ity (m
/mse
c) S0 mode
A0 mode A
1 mode
A
A'
(b) A dispersion curve of a 6 mm thick aluminum plate
100cm 60cm 100cm
PZT A PZT B
Plate6mm 1cm
PZT A PZT B
Plate6mm 1cm
Fig. 6. (a) An aluminum plate model with a pair of surface-bonded PZT transducers to validate the effects of modal dispersion on the time reversibility ofLamb waves (the thickness of a PZT pair: 0.507 mm) and (b) a dispersion curve of a 6 mm thick aluminum plate as a function of the input driving frequency.
H.W. Park et al. / Wave Motion 46 (2009) 451–467 459
differences of the group velocities of the S0 and A0 modes estimated from Fig. 7b are less than 1% compared to those calcu-lated in the theoretical dispersion curve in Fig. 6b. Note that the waveform of each mode in Fig. 7b deviates from that of theoriginal input signal in Fig. 7a due to the within-mode dispersion of Lamb waves.
Next, the effect of the time truncation point on the TRP is examined. For this investigation, the response signal measuredat PZT B is truncated at two different time points before reemitted to PZT B as shown in Fig. 7b. In Case 1, the response signalis truncated at t = 0.19 ms so that only the S0 mode is included in the TRP. On the other hand, the response signal for Case 2 istruncated at t = 0.30 ms so that both the S0 and A0 modes are used during the TRP. Fig. 7c presents the reconstructed inputsignals at PZT A for Cases 1 and 2, respectively. For the brevity of description, the time point corresponding to the main peakof the input signal is set to be zero in Fig. 7c. Using Eqs. (21)–(24), the reconstructed input signals for Cases 1 and 2 can berepresented as the following equations:
Case 1 : VRðtÞ ¼ �CSS11 �jVAðT � tÞ þ �CAS
11 �j expði �x�sAS11ÞVA½T � ðt þ �tAS
11Þ� ð25Þ
Case 2 : VRðtÞ ¼ ð�CSS11 þ �CAA
11Þ�jVAðT � tÞ þ �CSA11 �j expði �x�sSA
11ÞVA½T � ðt þ �tSA11Þ� þ �CAS
11 �j expði �x�sAS11ÞVA½T � ðt þ �tAS
11Þ� ð26Þ
The first term in Eq. (25) represents the main mode in the reconstructed input signal t = 0 at while the second term rep-resents the sideband that is shifted from the main mode by ��tAS
11 in the time domain. From Eq. (19), the time shift, ��tAS11 in Eq.
(25), is calculated as �rð1=wA0 � 1=wS0 Þ ¼ �0:6� ð1=2:994� 1=5:308Þ ¼ �0:0874 ms. From Fig. 7c, the time shift is esti-mated to be �0.0882 ms. The relative difference between these two time shift estimates is less than 1%. The first term inEq. (26) represents the main mode in the reconstructed signal at t = 0 while the second and third terms correspond to thesidebands that are shifted from the main mode by ��tSA
11 and ��tSA11, respectively. Note that �tSA
11 and �tAS11 are the same time shift
with the opposite signs (�tSA11 ¼ ��tAS
11), and this is clearly reflected on a symmetric pair of the side bands in Fig. 7c.The reconstructed input signal for Case 2 in Fig. 7c is magnified in the vicinity of the main mode and compared to the
original input signal in Fig. 7d. For the better comparison of the signal’s shape, the reconstructed and original input signalsin Fig. 7d are normalized so that their maximum values are equal to 1.0 at t = 0. The within-mode dispersion of the forwardsignal in Fig. 7b is compensated during the TRP, and the shape of the original input signal is recovered in the main mode ofthe reconstructed input signal.
The effect of modal dispersion on the time reversibility is also experimentally validated using a pair of PZT transducers(PZTs A and B) mounted on an aluminum plate shown in Fig. 8. The dimensions of the plate and each PZT transducer are122 cm � 122 cm � 0.6 cm and 1 cm � 1 cm � 0.0508 cm, respectively. The properties of the PZT wafer transducers are iden-tical to those used in the numerical simulation demonstrated above. The distance between the two PZTs is set to 40 cm inorder to avoid the effect of reflections, and a narrowband tone burst signal with a center frequency of 130 kHz is used as theinput signal.
-100
-50
0
50
100
0.00 0.01 0.02 0.03 0.04 0.05 0.06
Orig
inal
inpu
t sig
nal a
t PZT
A (V
)
Time (msec)
-2.0
-1.0
0.0
1.0
2.0
0.00 0.05 0.10 0.15 0.20 0.25 0.30
Res
pons
e si
gnal
at P
ZT B
(V)
Time (msec)
S0 mode
Case 2
A0 mode
Case 1(0.19 msec)
(0.3 msec)
(a) 100 kHz tone burst input signal exerted at PZT A (b) Response signal at PZT B
-2.0
-1.0
0.0
1.0
2.0
-0.12 -0.08 -0.04 0.00 0.04 0.08 0.12
Case 1Case 2
Nor
mal
ized
am
plitu
de
Time (msec)
-1.5
-0.8
0.0
0.8
1.5
-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03
Reconstructed input signalOriginal input signal
Nor
mal
ized
am
plitu
de
Time (msec)
(c) Reconstructed input signal at PZT A through
the TRP (Case 1: only the S0 mode is truncated and
used for the TRP ; Case 2: Both S0 and A0 modes
are used for the TRP)
(d) Comparison of the main modes in the original
(dotted) and the reconstructed (solid) input
signals
Fig. 7. The effect of the multimode dispersion on the time reversibility of Lamb wave signals is numerically validated using the numerical setup in Fig. 6.
460 H.W. Park et al. / Wave Motion 46 (2009) 451–467
In Fig. 9, it is shown how selective truncation of specific mode(s) affects the reconstructed signal. In Case 1 shown inFig. 9a, the forward signal AB was truncated 0.105 ms after the input signal so that only the S0 mode could be reversedand resent to the original location. In this case, only the S0 mode contributed to the main mode, and there was only one side-band (Fig. 9b). In Case 2 shown in Fig. 9a, the TRP was repeated by truncating the forward signal AB at 0.160 ms so that boththe S0 and A0 modes could be included. In this example, the main mode was composed of the contributions from both the S0
and A0 modes, and two sidebands were created as expected. Furthermore, the sidebands were symmetric along the mainmode as illustrated in Case 2 in Fig. 9b because �CSA
11 ¼ �CAS11 and �tSA
11 ¼ ��tAS11. Note that the signals corresponding to Cases 1
and 2 are scaled in Fig. 9b so that the maximum peak of the main mode is equal to one.Finally, the time shift value between the main mode and the sidebands were measured experimentally from Fig. 9b and
compared with �tSA in Eq. (19). The experimental group velocities of the S0 and A0 modes were 5.316 m/ms and 3.115 m/ms,and the �tSA
11 was �0.053 ms (�tSA11 = 0.4/5.316 � 0.4/3.115 = �0.053 ms). This �tSA
11 value agreed well with the time gap betweenthe main mode and one of the sidebands observed from Fig. 9b (about 0.0531 ms). Therefore, it is successfully demonstratedthat Eq. (26) properly described the sidebands created by multimodes.
In conclusion, the main mode in the reconstructed signal is practically identical to the original input signal in spite of themodal dispersion of Lamb waves. The symmetry of the side bands with respect to the main mode of the reconstructed inputsignals is preserved as long as all symmetric and anti-symmetric modes of interest are included in the TRP.
3.4. Understanding the effect of reflections on the TRP
Similar to the multiple Lamb wave modes, the Lamb waves reflected from the boundaries of a structure create additionalsidebands in the reconstructed signal. The effect of the reflections on the TRP is illustrated in Fig. 10. For simplicity, it is
Fig. 8. An aluminum plate with PZT transducers used for experiment.
Fig. 9. Experimental verification of the multimodal effect by truncating the forward signals at different time point for the TRP: Case 1 truncated right afterthe S0 mode and Case 2 after S0 and A0 modes.
H.W. Park et al. / Wave Motion 46 (2009) 451–467 461
assumed that a single mode travels unidirectionally and the structure has only one finite boundary where the Lamb wave isreflected. When a single Lamb mode is generated at PZT A and travels to PZT B (Fig. 10a), the wave will take two differentpaths to arrive at PZT B (Fig. 10b). In Fig. 10, P1 and P2 denote modes traveling along direct and reflection paths in a forwardpropagation direction. P3 and P4 are defined in a similar fashion but in a backward propagation direction. When the P2 modeis emitted back to PZT A due to the reflection (Fig. 10c), this mode generates two modes, P3/P2 and P4/P2, in the recon-structed signal due to the two different wave propagation paths in the backward propagation direction (Fig. 10d). Similarly,when P1 is reemitted, it creates additional two modes, P3/P1 and P4/P1 (Fig. 10e). Finally, the reconstructed signal is com-posed of the main mode in the middle, which is the superposition of P3/P1 and P4/P2, and two symmetric sidebands due toP3/P2 and P4/P1 (Fig. 10f). Note that the symmetry of the reconstructed signal is independent of the symmetry of the struc-ture, the PZT layout and the boundary condition. In the example presented in Fig. 10, there is only one finite boundary wherewaves can be reflected, but the reconstructed signal is still symmetric. The number of sidebands will increase if there areadditional wave reflections.
To examine the effect of the reflections in a more theoretical manner, the time reversal operator C in Eq. (4) can bedecomposed to those associated multiple wave propagation paths between the actuating and sensing PZT wafers. For brevityof description, the discussion is first limited to a single Lamb mode with multiple wave propagation paths
C ¼ GG� ¼XnR
p¼1
XnR
q¼1
gpg�q ð27Þ
where subscript p of a field variable denotes a specific wave propagation path, while nR, and gp represent the total number oftraveling paths and an individual transfer function associated with the pth traveling path, respectively.
Considering the amplitude and velocity dispersion of Lamb waves, gp can be simply expressed as follows:
Fig. 10. The effect of reflections on the TRP. (Note: P1 and P3 are waves propagating along the direct path between PZTs A and B, and P2 and P4 are wavesreflected at one end of the plate in forward and backward directions. P3/P2 denotes a signal arrived at PZT A through a direct path, when the reflected signal,P2, is emitted back to PZT B after time reversal. P4/P2, P3/P1, and P4/P1 are similarly defined.)
462 H.W. Park et al. / Wave Motion 46 (2009) 451–467
gp ¼ c expð�ikrpÞ; ð28Þ
where rp denotes a traveling distance from the actuating PZT wafer to the sensing PZT wafer associated with the pth travelingpath, respectively. Using Eq. (28), Eq. (27) is expressed as follows:
C ¼XnR
p¼1
XnR
q¼1
gpg�q ¼XnR
p¼1
XnR
q¼1
cc�e½ikðrp � rqÞ� ¼XnR
p¼1
XnR
q¼1
C expðihpqÞ ð29Þ
where hpq ¼ k½rp � rq� and C ¼ cc�. From Eq. (29), the reconstructed input signal with multiple wave propagation paths can becalculated as follows:
VR ¼ CKK�V�A ¼XnR
p¼1
XnR
q¼1
CeðihpqÞKK�V�A ð30Þ
Similar to Eq. (6), the reconstructed input signal in the time domain can be obtained by taking the inverse Fourier trans-form of Eq. (30) if an original input signal is a narrowband tone burst with a center angular frequency �x
VRðtÞ ¼ VAðT � tÞXnR
p¼1
�C �jþXnR
p¼1
XnR
q¼1
ð1� dpqÞ1
2p
Z 1
�1
�C �jV�AðxÞ exp½iðxt þ hpqÞ�dx ð31Þ
Because the second term in Eq. (31) cannot be directly expressed in terms of VA(t), it is approximated by procedures sim-ilar to Eqs. (17)–(20)
VRðtÞ ¼XnR
p¼1
�C �jVAðT � tÞ þXnR
p¼1
XnR
q¼1
ð1� dpqÞ�C �jfexpði �x�spqÞVA½T � ðt þ �tpqÞ�g ð32Þ
where
�spq ¼ ðrp � rqÞ1
vð �xÞ �1
wð �xÞ
� �; �tpq ¼
ðrp � rqÞwð �xÞ ð33Þ
Note that Eq. (32) is identical to Eq. (21) except that the summation is performed over the multiple reflection paths in-stead of the multiple symmetric modes. Therefore, the reflections create additional sidebands similar to the ones created bythe multiple modes, but do not change the symmetry of the reconstructed signal regardless of the symmetry of the struc-ture’s boundary conditions. That is, the TRP can be interpreted as a temporal correlation among multiply reflected Lambwave modes at the boundary of the plate.
The effect of reflections on the time reversibility of Lamb waves described in Eqs. (32) and (33) is validated through anumerical simulation on an aluminum plate model with four identical surface-bonded PZT transducers (PZTs A, B, C, and
30cm 20cm 80cm
PZT A PZT B
PZT D PZT C6mm Plate1cm
PZT A PZT B
PZT D PZT C6mm Plate1cm
Fig. 11. An aluminum plate model with two pairs of surface-bonded PZT transducers to validate the effect of reflections on the time reversibility of Lambwaves (the thickness of a PZT pair: 0.127 mm).
H.W. Park et al. / Wave Motion 46 (2009) 451–467 463
D) in Fig. 11. The geometric configuration of the aluminum plate and the location of PZT transducers are properly ar-ranged so that only the reflections from the left boundary of the plate can be included during the TRP without interfer-ence with modal dispersion. The polarization of PZTs A and D is configured so that only the A0 mode is generated whenPZTs A and D are excited simultaneously. The polarization of PZTs B and C is configured identical to that of PZTs A andD. Reversing and re-emitting the response signal at PZTs B and C in the time domain, reconstructed input signals arereceived at PZTs A and D. The PZT transducers were made from PSI-5A4E type of PZT sheets (thickness = 0.0127 cm).The rest of the numerical setup here is identical to that in Section 3.3 unless otherwise mentioned.
Fig. 12a illustrates the 100 kHz tone burst input signal that is simultaneously exerted at PZTs A and D to create only the A0
mode in the forward propagation. The response signal at PZT B and C, 20 cm away from the excitation sources, is presented in
-1.0
-0.5
0.0
0.5
1.0
-0.30 -0.20 -0.10 0.00 0.10 0.20 0.30
Case 1Case 2
Nor
mal
ized
am
plitu
de
Time (msec)
-1.5
-0.8
0.0
0.8
1.5
-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03
Reconstructed input signalOriginal input signal
Nor
mal
ized
am
plitu
de
Time (msec)
-40
-20
0
20
40
0.00 0.01 0.02 0.03 0.04 0.05 0.06Orig
inal
inpu
t sig
nal a
t PZT
A (V
)
Time (msec)
-1.0
0.0
1.0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40Res
pons
e si
gnal
at P
ZTs
B an
d C
(V)
Time (msec)
A0 mode
Case 2
Reflected A0 mode
Case 1(0.20 msec)
(0.40 msec)
(a) The 100 kHz tone burst input applied at PZTs A (b) The response signal measured at PZT B
(c) The Reconstructed input signal at PZT A (d) Comparison of the main modes in the original a
through the TRP (Case 1: Only the direct A0 mode (dotted) and the reconstructed (solid) input
is truncated and used for the TRP ; Case 2: Both
the direct A0 and reflected A0 modes are included
for the TRP)
signals
Fig. 12. The effect of reflection on the time reversibility of Lamb wave signals is numerically validated using the numerical setup in Fig. 11.
464 H.W. Park et al. / Wave Motion 46 (2009) 451–467
Fig. 12b. In Fig. 12b, the A0 mode propagating through the direct path arrives at PZT B first followed by the A0 mode reflectedfrom the left boundary of the plate. Note that the polarization of PZT C is arranged so that the response at PZT C is identical tothe response at PZT B when there are only anti-symmetric modes [42]. The estimated group velocity of the A0 mode inFig. 12b is approximately 2.959 m/ms while the theoretical one is 2.994 m/ms at 100 kHz in Fig. 6b. The relative differencebetween the estimated and theoretical group velocities is less than 1%.
To investigate the effect of reflection on the time reversibility of Lamb waves, the response signal at PZT B is truncated attwo different time points as shown in Cases 1 and 2 of Fig. 12b, respectively. In Case 1, the response signal is truncated att = 0.20 ms so that only the A0 mode propagating along the direct path is included in the TRP. In Case 2, the response signal istruncated at t = 0.40 ms to include both the direct and reflected A0 modes in the TRP. Note that the response signal measuredat PZT B is reversed and applied to PZTs B and C simultaneously as before to excite only the A0 mode.
Fig. 12c shows the reconstructed input signal measured at PZT A through the TRP for Cases 1 and 2, respectively. For thesimplicity of description, the time point corresponding to the main peak of the input signal is set to be zero. For Case 1 inFig. 12c, only a single side band is observed in the left hand side of the main mode. From Fig. 12c, the time shift of the sideband from t = 0 is estimated to be �0.206 ms. On the other hand, it is estimated to be ��t12 ¼ �ðr2 � r1Þ=wA0 ¼�ð0:8� 0:2Þ=2:994 ¼ �0:2 ms (0.2 � 0.8)/2.994 = �0.2 ms from Eq. (33), and the relative difference error between the the-oretical and estimated time shifts is less than 3%.
For Case 2 in Fig. 12c, two identical side bands appear symmetrically with respect to t = 0. The time shift values of the leftand right hand sides are identical to that of Case 1. The reconstructed input signal for Case 2 is magnified near at t = 0 andcompared to the original input signal in Fig. 12d. In order to compare the shapes of the input and reconstructed signals, themaximum peak value at t = 0 is normalized to be 1.0. The within-mode dispersion of the forward signals observed in Fig. 12bis compensated during the TRP, and the shape of the original input signal is fully restored in the main mode of the recon-structed input signal.
Next, the appearance of sidebands due to reflections from boundaries and the symmetry of the resulting reconstructedsignal were experimentally demonstrated. The test was conducted using a pair of PZT patches attached to the aluminumplate shown in Fig. 8. PZT A was mounted 10 cm away from the left side edge of the plate and PZT B was placed 32 cm awayfrom the right side boundary of the plate. In this configuration, multiple Lamb wave modes arrived at PZT B in the followingorder as shown in Fig. 13a: (1) the S0 mode along the shortest (direct) wave path, (2) the S0 mode reflected from the leftboundary, and (3) the direct and reflected A0 modes. Because the direct and reflected A0 modes arrived later than the directand reflected S0 modes, it was possible to truncate the forward signal including only the S0 modes to study the effects of the
Fig. 13. Experimental investigation of the effect of reflections on the TRP by truncating the forward signals at different time points during the TRP: Case 1includes only the direct S0 mode and Case 2 includes the reflected S0 mode as well in the TRP.
H.W. Park et al. / Wave Motion 46 (2009) 451–467 465
reflections. Note that the dynamic range of the data acquisition system is reduced here to improve the effective resolution ofthe direct and reflected S0 modes in Fig. 13a. As a result, the A0 modes which had higher amplitudes than S0 modes werepartially saturated. The center frequency of the tone burst signal was set to 130 kHz.
To see the effect of the reflection, the truncation time point in the forward signal was varied. First, the forward signal inFig. 13a was truncated at 0.17 ms so that only the direct S0 mode was reversed and resent to the original location (Case 1).Next, the truncation was done at 0.2125 ms to embrace both the direct and reflected S0 modes (Case 2). In Fig. 13b and c, thesidebands created by the reflections are shown. When the TRP was conducted including only the direct S0 mode, a singlesideband appeared only on the left hand side of the main mode (Fig. 13b). On the other hand, two symmetric sidebands weredeveloped when both S0 modes were included during the TRP (Fig. 13c).
Next, it was investigated if the phase shift of these sidebands matched with the theoretical prediction. Based on the for-ward signal shown in Fig. 13a, the group velocity of the S0 mode was 5.239 m/ms. The arrival time of the right sideband inFig. 13c was estimated to be 0.0396 ms, and it was close to the ��tDR value obtained from Eq. (33)ð��tDR ¼ �ðrD � rRÞ=wS0 ð �xÞ ¼ �ð0:80� 1:00Þ=5:239 ¼ 0:0382 msÞ. Furthermore, it must be noted that the shapes of the leftand right sidebands were almost symmetrical along the main mode as illustrated in Fig. 13c.
In conclusion, the main mode in the reconstructed signal is practically identical to the original input signal in spite of themodal dispersion of Lamb waves. The symmetry of side bands with respect to the main mode of the reconstructed input sig-nals is preserved as long as all direct and reflected modes of interest are included in the TRP.
3.5. Summary of practical issues to be addressed for the TRP of Lamb waves on a plate
So far, various effects of Lamb waves on the TRP such as multimodes and reflections has been studied and validated bynumerical simulations as well as experiments in a laboratory environment. When the applicability of the TRP to real fieldstructures is sought, however, a number of additional practical issues need to be resolved in advance. For instance, it canbe very challenging to control the bonding condition of each PZT transducer. Also, the performance of the TRP under oper-ational and environmental conditions should be investigated. Therefore, additional experiments have been designed andconducted in laboratory and field environments to resolve the practical issues. Due to the page limitation, only the test re-sults are briefly discussed in this section.
First, the effects of PZT size, orientation, shape and bonding condition on the TRP have been examined. When the bondingconditions and areas of two PZT wafers were identical, the TRP could be successfully conducted regardless to their shapesand orientations. When the TRP was tested using two PZTs with different sizes, the time shift of the main peak in the recon-structed signal was observed. This is because the impedance values of the two PZTs are different. Assuming two identicalPZTs are used for the TRP, the amount of time delay due to the impedance of the exciting PZT in the forward signal is iden-tical to that of the backward signal, resulting in no time shift of the main peak. On the other hand, the time delay appearswhen PZTs with different impedance values are used. However, in spite of the time delay, the shape of the main peak waspreserved in the TRP. Note that variation in the PZT bonding conditions has a similar effect as change in the PZT size, and thetime shift of the main peak in the reconstructed signal was observed when PZTs with different bonding conditions were usedfor the TRP.
Next, the effects of environmental and operational variations on the TRP are investigated. In spite of temperature, bound-ary and surface condition changes, and ambient loading, the TRP could be successfully achieved. For instance, the timereversibility of Lamb wave signals is tested by instrumenting a steel bridge girder under normal traffic load [43]. The tem-perature effect is also tested through temperature chamber experiments. It is noteworthy that Ribay et al. demonstrated thesame theoretical result in which the effect of temperature change on the time reversibility was negligible in the metallicmedia such aluminum used in this study [44].
4. Summary
Guided waves such as Lamb waves have recently received much attention in NDT applications. NDT techniques usingguided waves are based on the premise that, when propagating waves encounter a defect, wave patterns are altered bythe defect. Therefore, the damage can be identified by comparing the baseline signal obtained from an intact condition ofthe structure and the current signal obtained from an unknown condition of the structure. However, it has been reportedthat this type of pattern comparison with the baseline signal can be susceptible to false alarms because other natural vari-ations of the system can produce various changes in the signal. To relax this dependency on the baseline data, the authorshave been developing a damage detection technique that does not require a direct comparison of the test signal with thebaseline data [12]. The pivotal concept of this reference-free NDT technique is the TRP. While the TRP has been widely usedin modern acoustics, its use in Lamb waves has been limited due to the amplitude and velocity dispersion characteristics ofLamb waves.
In this study, the applicability of the TRP to Lamb wave propagation is theoretically investigated. In particular, the pri-mary interest in the TRP is to match the shape of the final response signal (the reconstructed signal after the TRP) with thatof the original input. Because the shape of the reconstructed signal deviates from the original input waveform when there isa certain type of defect along the wave propagation path, this feature can be utilized for damage diagnosis. However, the
466 H.W. Park et al. / Wave Motion 46 (2009) 451–467
interpretation of the TRP becomes complicated for Lamb wave propagation due to its dispersive characteristics. Based on thetheoretical, numerical and experimental studies presented in this paper, it has been shown that (1) within-mode dispersionof a single mode is fully compensated during the TRP; (2) a narrowband input signal should be employed to minimize ampli-tude dispersion and to restore the shape of the input signal; (3) the existence of multiple Lamb wave modes and reflectionscreate additional sidebands around the main response mode in the reconstructed signal; and (4) the reconstructed signal issymmetric along the main mode as long as a symmetric input signal is exerted regardless of the topology and boundary con-ditions of the structure. Finally, practical implementation issues relevant to field deployment are briefly discussed.
Acknowledgements
This work was supported by Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Re-search Promotion Fund) (KRF-2008-331-D00590), in which main calculations were performed by using the supercomputingresource of the Korea Institute of Science and Technology Information (KISTI), and the Radiation Technology Program underKorea Science and Engineering Foundation (KOSEF) and the Ministry of Science and Technology (M20703000015-07N0300-01510). The second author would like to acknowledge the graduate fellowship program from Samsung Scholarship in Seoul,Korea.
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