effect of structural damping on the tuning between

Original Article

Journal of Intelligent Material Systemsand Structures1–15� The Author(s) 2018Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1045389X18758188journals.sagepub.com/home/jim

Effect of structural damping on thetuning between piezoelectric waferactive sensors and Lamb waves

Hanfei Mei and Victor Giurgiutiu

AbstractPiezoelectric wafer active sensors have been widely used for Lamb-wave generation and acquisition. For selective prefer-ential excitation of a certain Lamb-wave mode and rejection of other modes, the piezoelectric wafer active sensor sizeand the excitation frequency should be tuned. However, structural damping depends on the structure material and theexcitation frequency and it will affect the amplitude response of piezoelectric wafer active sensor–excited Lamb waves inthe structure, that is, tuning curves. Its influence on the piezoelectric wafer active sensor tuning reflects the effect ofstructural health monitoring configuration considered in the excitation. Therefore, it is important to have knowledgeabout the effect of structural damping on the tuning between piezoelectric wafer active sensor and Lamb waves. In thisarticle, the analytical tuning solution of undamped media is extended to damped materials using the Kelvin–Voigt damp-ing model, in which a complex Young’s modulus is utilized to include the effect of structural damping as an improvementover existing models. This extension is particularly relevant for the structural health monitoring applications on high-lossmaterials, such as metallic materials with viscoelastic coatings and fiber-reinforced polymer composites. The effects ofstructural damping on the piezoelectric wafer active sensor tuning are successfully captured by the improved model,with experimental validations on an aluminum plate with adhesive films on both sides and a quasi-isotropic woven com-posite plate using circular piezoelectric wafer active sensor transducers.

KeywordsStructural health monitoring, Lamb waves, piezoelectric wafer active sensors, tuning, structural damping, Kelvin–Voigtdamping model

Introduction

Structural health monitoring (SHM) is an emerginginterdisciplinary research area, which has shown greatpotential in aerospace, civil, mechanical, and navalstructures (Boller et al., 2009; He and Yuan, 2017; Qiuet al., 2013, 2014; Raghavan and Cesnik, 2007; Sohn,2007; Su et al., 2006; Yuan et al., 2016). Among theSHM technologies, Lamb wave, a type of ultrasonicguided waves propagating between two parallel surfaceswithout much energy loss, is suitable for the inspectionof large areas of complicated structures (Achenbach,1973; Graff, 1975; Rose, 1999). It has gained increasingattention from SHM communities in recent decades(Mitra and Gopalakrishnan, 2016).

Piezoelectric wafer active sensors (PWAS) are conve-nient enablers for generating and receiving Lamb wavesin structures for SHM applications (Giurgiutiu et al.,2002). Compared with conventional transducers, PWASare low profile, lightweight, low-cost, and unobtrusive tostructures (Giurgiutiu, 2014). They can be permanently

bonded on the host structures in large quantities andachieve real-time monitoring of the structural health sta-tus. PWAS can actively interrogate structures using avariety of Lamb-wave methods such as pitch-catch,pulse-echo, phased arrays, and electromechanical (E/M)impedance technique (Giurgiutiu, 2014).

Tuning between PWAS and Lamb waves in a plate

Under electric excitation, PWAS undergo oscillatorycontractions and expansions that are transferred to thestructure through the bonding layer to excite Lambwaves in structures. In this process, several factors

Department of Mechanical Engineering, University of South Carolina,

Columbia, SC, USA

Corresponding author:

Hanfei Mei, Department of Mechanical Engineering, University of South

Carolina, 300 Main Street, Columbia, SC 29208, USA.

Email: [email protected]

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influence the behavior of the excited Lamb waves:thickness of the bonding layer, geometry of the PWAS,and thickness and material of the structure. The resultof the influence of all these factors is the tuning betweenPWAS and Lamb waves (Giurgiutiu, 2005). The tuningis especially beneficial when dealing with multimodewaves, such as Lamb waves. The gist of the concept isthat manipulation of the PWAS size and the excitationfrequency allows for the selective preferential excitationof a certain Lamb-wave mode and the rejection of othermodes, as needed by particular Lamb-wave-basedSHM applications.

The analytical model of PWAS-generated Lambwaves and its tuning effect on isotropic metallic plateshas been well investigated (Giurgiutiu, 2005; Raghavanand Cesnik, 2005; Santoni-Bottai et al., 2007; Santoni-Bottai and Giurgiutiu, 2012; Sohn and Lee, 2009).Giurgiutiu (2005) first developed the theory of the inter-action of a rectangular PWAS with one-dimensional(1D) propagation, that is, straight crested Lamb waves,and presented a closed-form solution based on trigono-metric functions. Raghavan and Cesnik (2005) extendedtuning concepts to the case of a circular transducercoupled with two-dimensional (2D) propagation, thatis, circular crested Lamb waves, and proposed the cor-responding tuning prediction formula based on Besselfunctions. Sohn and Lee (2009) proposed a calibrationtechnique to address the discrepancy between experi-mental and theoretical tuning curves of Lamb waves bytaking into account the bonding layer and the energydistribution between various modes. Santoni-Bottaiet al. (2007) performed theoretical studies and experi-mental measurements of the Lamb-wave tuning andfound that the capability to excite only one desiredwave mode based on the tuning is crucial for theembedded ultrasonic structural radar. To improve themodel of tuning, an exact shear-lag solution of Lamb-wave tuning with PWAS was derived from first princi-ples using the normal mode expansion (NME) method(Santoni-Bottai and Giurgiutiu, 2012). A similar tuningeffect is also possible in anisotropic composite plates.However, the theoretical analysis is more complicateddue to the anisotropic characteristics of the wave pro-pagation in composite materials. Giurgiutiu andSantoni-Bottai (2011) derived the theoretical predic-tions of tuning using NME and the dispersion curves ofLamb waves in composite plates. These analyticaldevelopments of tuning facilitate the understanding ofPWAS-coupled Lamb waves for SHM applications.However, these solutions only apply to the tuningbetween PWAS and Lamb waves without consideringthe effect of structural damping.

Effect of structural damping on PWAS tuning

In composite materials, the structural damping plays amajor role that cannot be ignored. Gresil and

Giurgiutiu (2015) performed PWAS tuning experimentson composite plates and found that higher frequencytuning seems virtually impossible because the Lambwaves strongly diminish above approximately 600 kHzfor the quasi-S0 mode and approximately 200 kHz forthe quasi-A0 mode. The structural damping dependson the structure material and the excitation frequencyand it will affect the amplitude response of PWAS-excited Lamb waves in structures, that is, Lamb-wavetuning curves. Its influence on the tuning reflects theeffect of SHM configuration considered in the excita-tion. Therefore, it is important to have knowledgeabout the effect of structural damping on the tuningbetween PWAS and Lamb waves.

To model structural damping, a damping coefficientis usually considered in a complex formulation of thematerial Young’s modulus, where the real part (storagemodulus) corresponds to the elastic behavior and theimaginary part (loss modulus) corresponds to the dissi-pative behavior (Christensen, 1982; Graesser andWong, 1992). This is known as the correspondenceprinciple (Auld, 1973). Several researchers have studiedthe damping models to predict the attenuation behaviorof Lamb waves in composite plates, including hystereticdamping model (Bartoli et al., 2006; Neau, 2003),Kelvin–Voigt damping model (Bartoli et al., 2006;Neau, 2003; Shen and Cesnik, 2016), and Rayleighdamping model (Gresil and Giurgiutiu, 2015; Ramadaset al., 2011). All these research studies indicate thatdamping models have the capability to capture theeffect of structural damping.

Overview of current paper

The main novelty of this article is to investigate theeffect of structural damping on the tuning betweenPWAS and Lamb waves. This study extends the analy-tical solutions of tuning to damped materials using theKelvin–Voigt damping model, in which a complexYoung’s modulus is utilized to include the effect ofstructural damping. This extension is particularly rele-vant for SHM applications on high-loss materials, suchas metallic materials with viscoelastic coatings (Mu andRose, 2008) and fiber-reinforced polymer composites(Lin et al., 2016; Sreekumar et al., 2015). The proposedmethod is validated on an aluminum plate with adhe-sive films on both sides and a quasi-isotropic wovencomposite plate using circular PWAS transducers. Itwill be shown in this article that structural damping canmodify the tuning effect considerably, both throughpeak shifts and through drastic diminishing of theamplitude at higher frequencies. This article containsanalytical developments and experimental measure-ments. The concepts of this article are applicable toboth metals (e.g. metallic plates with viscoelastic coat-ings) and composites.

2 Journal of Intelligent Material Systems and Structures 00(0)

Mathematical framework for straight andcircular crested Lamb-wave tuning withPWAS transducers

PWAS basis

PWAS are the enabling technology for active SHM sys-tems (Giurgiutiu, 2014). PWAS couple the electricaland mechanical effects (mechanical strain, Sij; mechani-cal stress, Tkl; electrical field, Ek ; and electrical displace-ment, Dj) through the following tensorial piezoelectricconstitutive equations

Sij = sEijklTkl + dkijEk ð1Þ

Dj = djklTkl + eTjkEk ð2Þ

where sEijkl is the mechanical compliance of the material

measured at zero electric field (E = 0), eTjk is the dielec-

tric permittivity measured at zero mechanical stress(T = 0), and djkl represents the piezoelectric couplingeffect. PWAS utilize the d31 coupling between in-planestrains, S1, S2, and transverse electric field, E3. Just likeconventional ultrasonic transducers, PWAS utilize thepiezoelectric effect to generate and receive ultrasonicwaves. However, PWAS are different from conven-tional ultrasonic transducers in several aspects(Giurgiutiu, 2014): PWAS are firmly coupled with thehost structure through an adhesive bonding, whereasconventional ultrasonic transducers are weakly coupledthrough the water, gel, or air. PWAS are non-resonant devices that can be tuned selectively into

several Lamb-wave modes, whereas conventionalultrasonic transducers are resonant narrow-band devices.PWAS are inexpensive and can be deployed in largequantities on structures, whereas conventional ultrasonictransducers are expensive and used one at a time.

Depending on application types, PWAS transducerscan serve several purposes (Giurgiutiu, 2014): (a) activesensing of far-field damage using pulse-echo, pitch-catch, and phased-array methods; (b) active sensing ofnear-field damage using high-frequency electromecha-nical impedance spectroscopy (EMIS) and thicknessgage mode; and (c) passive sensing of damage-generating events through detection of low-velocityimpacts and acoustic emission at the tip of advancingcracks. Figure 1 shows the schematic of PWAS applica-tion modes.

Straight and circular crested Lamb-wave tuningwithout damping

The analytical model of PWAS-generated Lamb wavesand its tuning effect in undamped isotropic plates havebeen well understood. In this section, straight and circularcrested Lamb-wave tuning without damping was discussed.

Straight crested Lamb-wave tuning without damping. Thestraight crested Lamb-wave tuning without dampingwas solved by applying the Fourier transform and thesymmetric and antisymmetric boundary conditions(Giurgiutiu, 2005). In the case of ideal bonding, the

Figure 1. Schematic representation of PWAS application modes (Giurgiutiu, 2014).

Mei and Giurgiutiu 3

shear stress in the bonding layer is concentrated at theends and the pin-force model is used. The closed formis given by Equation (108) of Giurgiutiu (2014: 595),which gives the in-plane wave strain at the plate surfaceas

ex(x, t)jz= d=� iata

m

XJS

j= 0

ðsin jSj aÞ

NS(jSj )

D0S(jSj )

ei(jSj x�vt)

� iata

m

XJA

j= 0

ðsin jAj aÞ

NA(jAj )

D0A(jAj )

ei(jAj x�vt)

ð3Þ

where

Ns jð Þ= jb j2 +b2� �

cos ad cos bd;

Ds = j2 � b2� �2

cos ad sin bd + 4j2ab sin ad cos bd

NA jð Þ= � jb j2 +b2� �

sin ad sin bd;

DA = j2 � b2� �2

sin ad cos bd + 4j2ab cos ad sin bd

a2 =v2

c2p

� j2; b2 =v2

c2s

� j2; cp =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil+ 2m

r

s;

cs =

ffiffiffiffim

r

r; l=

n

1+ nð Þ 1� 2nð ÞE; m=1

2 1+ nð ÞE

where a is half the length of the PWAS size; d is thehalf thickness of the plate; cp and cs are the longitudi-nal (pressure) and transverse (shear) wave speeds,respectively; l and m are Lame constants of the struc-ture material; and E, n, and r are Young’s modulus,Poisson’s ratio, and material density of the structure,respectively. The terms sin (jS

j a) and sin (jAj a) control

the tuning between the PWAS and Lamb waves.Superscripts or subscripts S and A signify the sym-metric and antisymmetric Lamb-wave modes, respec-tively. The expressions D0S and D0A are derivatives of DS

and DA with respect to j evaluated at the correspondingjS and jA poles. The wavenumber j of a specific Lamb-wave mode at a given angular frequency v is obtainedfrom the solutions of the Rayleigh–Lamb equation

tan bd

tan ad=

�4abj2

j2 � b2� �2

" #61

ð4Þ

where +1 exponent corresponds to symmetric Lamb-wave modes and 21 exponent corresponds to antisym-metric Lamb-wave modes.

The expressions in equation (3) can also be used forless-than-ideal bonding conditions in which the shearstress t varies with x as t(x) by replacing ta and a withtheir effective values te and ae. At low frequencies, onlytwo propagating Lamb-wave modes exist, S0 and A0,and general solutions of equation (3) have only twoterms, that is

ex(x, t)jz= d = � iata

msin jS

0a� � NS(j

S0)

D0S(jS0)

ei(jS0x�vt) � i

ata

msin jA

0 a� � NA(j

A0 )

D0A(jA0 )

ei(jA0 x�vt)

ð5Þ

Circular crested Lamb-wave tuning without damping. The cir-cular crested Lamb-wave tuning without damping wasderived based on Bessel functions. In the case of idealbonding of a circular PWAS transducer that expandsand contracts radially, the shear stress in the bondinglayer is concentrated on the PWAS outer contour in theform of radially acting horizontal tractions. The closedform of the in-plane wave strain at the plate surface isgiven by Equation (166) of Giurgiutiu (2014: 607)

er(r, t)jz= d = � pia2ta

2me�ivt

XjS

J1(jSa)NS(j

S)

D0S(jS)

jSH(1)0 (jSr)� H

(1)1 (jSr)

r

!

� pia2ta

2me�ivt

XjA

J1(jAa)NA(j

A)

D0A(jA)

jAH(1)0 (jAr)� H

(1)1 (jAr)

r

!ð6Þ

where a is the radius of the circular PWAS transducerand r is the distance between the point of interest andthe transmitter PWAS. ta represents shear stressbetween the transducer and the host structure. l and m

are Lame constants of the structure material. J1 is theBessel function of order one, which captures the tuningeffect between PWAS and the host structure. H

1ð Þ1 and

H1ð Þ

0 are the first-kind Hankel function of order oneand zero, respectively, which represent an outward pro-pagating 2D wave field. j is the frequency-dependentwavenumber calculated from the Rayleigh–Lamb equa-tion in equation (4).

The expressions in equation (6) can also be used forless-than-ideal bonding conditions, in which the shearstress t varies with r as t(r) by replacing ta and a withtheir effective values te and ae. At low frequencies, onlytwo propagating Lamb-wave modes exist, S0 and A0, andthe solutions of equation (6) have only two terms, that is

er(r, t)jz= d = � pia2ta

2me�ivt J1(j

S0a)NS(j

S0)

D0S(jS0)

jS0H

(1)0 (jS

0r)� H(1)1 (jS

0r)

r

!

� pia2ta

2me�ivt J1(j

A0 a)NA(j

A0 )

D0A(jA0 )

jA0 H

(1)0 (jA

0 r)� H(1)1 (jA

0 r)

r

!ð7Þ


Damping effects on Lamb-wave tuningwith PWAS transducers

This section discussed damping effects on Lamb-wavetuning with PWAS transducers. Structural dampingdepends on the structure material and the excitationfrequency and it will affect the amplitude response ofPWAS-excited Lamb waves in structures, that is, tuningcurves. Its effect on the Lamb-wave tuning with PWAStransducers reflects the influence of the SHM config-uration considered in the excitation. Hence, it is impor-tant to have an understanding of the effect of structuraldamping.

Damping models for Lamb-wave propagation

The mathematical approach to predict the structuraldamping that attempts to model the deformation andflow of the viscoelastic material is known as the scienceof rheology. There are two commonly used models:Maxwell damping model, where a spring is in serieswith a dashpot, and Kelvin–Voigt damping model,where a spring is in parallel with a dashpot (Graesserand Wong, 1992). In this article, the Kelvin–Voigtdamping model based on the frequency-dependentdamping coefficient, which is more applicable to thefrequency-dependent attenuation of Lamb waves(Sreekumar et al., 2015), is adopted to investigate theeffect of structural damping on the tuning betweenPWAS and Lamb waves.

The Kelvin–Voigt damping model is presented inFigure 2, in which the stress acting on the body isassumed to be proportional to the strain and its timederivative (the strain rate)

s =Ee+ g _e ð8Þ

For wave propagation, the time-dependent behaviorof stress and strain is assumed to be harmonic e�ivt,given by the expressions

s =s0ei jx�vtð Þ; e= e0ei jx�vtð Þ ð9Þ

where j is the wavenumber and v is the angular fre-quency. Substituting equation (9) into equation (8)yields

s0 = E � igvð Þe0 =E�e0 ð10Þ

where E� is referred to the complex Young’s modulus

E�=E � igv ð11Þ

Define the storage modulus, E0, corresponding to theelastic behavior and the loss modulus, E00, correspond-ing to the dissipative behavior as

E0=E; E00= gv ð12Þ

Substitution of equation (12) into equation (11) yields

E�=E0 � iE00 ð13Þ

The damping coefficient h can be introduced as

h=E00

E0ð14Þ

Hence, the complex Young’s modulus of equation (13)can be rewritten as

E�=E0 1� ihð Þ ð15Þ

From the derivation of the complex Young’s modulusfor the Kelvin–Voigt damping model, it can be foundthat the damping coefficient h is linearly dependent onfrequency. In the literature (Castaings and Hosten,2000; Neau, 2003), the reference damping coefficient h0

was measured at a reference frequency f0 correspondingto the frequency of the transducer used in conventionalultrasonic interferometry. In our study, the referencefrequency f0 is initially chosen at the upper limit of thefrequency range (f initial0 = 2MHz) and the dampingcoefficient h0 is initially taken as 5% (hinitial

0 = 5%).Then, the reference frequency f0 and the dampingcoefficient h0 are modified by repeated updating ofthe model to achieve an acceptable match with experi-mental tuning cures. Finally, the complex Young’smodulus E� at a generic frequency, f, can be obtainedby scaling the damping coefficient at the reference fre-quency f0

E�=E0 1� ih0

f

f0

� �ð16Þ

The Kelvin–Voigt damping model results in a smallattenuation below f0 and a larger attenuation above f0.The complex Young’s modulus in equation (16) can besubstituted in equations (5) and (7) to modify the ana-lytical solutions of tuning by introducing the effect ofstructural damping.

1D tuning of straight crested Lamb waves withdamping

In this study, we focus our attention on fundamentalLamb-wave modes (S0 and A0), which find the widest

Figure 2. The Kelvin–Voigt damping model.


application in Lamb-wave-based SHM applications.However, higher modes (S1, A1, S2, A2, etc.) in thepractical application may also exist. The analyticalframework can be easily extended to consider thesemodes under the same principle. Equation (5) has beencoded into MATLAB, which provides the tuning curvesof the structures for a given PWAS length, materialproperties, material thickness, and frequency range.

A quantitative study of the damping effect on Lamb-wave tuning using three various damping coefficientsh0 (0, 1%, and 10%) was conducted. In the simulation,the reference frequency f0 is fixed to 2 MHz, which isthe upper limit of the frequency range. First, 1D tuningcurves without damping (h0 = 0) of a 9 mm PWASand a 1.016 mm 2024-T3 aluminum alloy plate areshown in Figure 3. It can be noted that the normalizedstrain curves follow the general pattern of a sine func-tion. As indicated in Figure 3, the high-frequencyresponse has a relatively large amplitude, comparedwith the amplitude at low frequencies. However, it isnot true for the materials with damping. To improvethe prediction by including damping effect, the Kelvin–

Voigt damping model is utilized to investigate dampingeffects using the complex Young’s modulus with vari-ous damping coefficients.

Figure 4 presents 1D tuning curves with small damp-ing (h0 = 1%) using the Kelvin–Voigt damping model.It can be noticed that the amplitudes at higher frequen-cies are relatively diminished by introducing the struc-tural damping, whereas the low-frequency responsealmost remains the same, which is in agreement with the-oretical predictions of the Kelvin–Voigt damping model.

To investigate the larger damping effect, the damp-ing coefficient is increased from 1% to 10%. The corre-sponding 1D tuning curves with large damping(h0 = 10%) are shown in Figure 5. It should be notedthat the amplitudes at higher frequencies are signifi-cantly reduced compared with the results of smalldamping. At the same time, the low-frequency responsestill has a large amplitude. It can be concluded that thestructural damping can considerably modify 1D tuningcurves at higher frequencies, and the damping modelcan capture the effect of structural damping.

The attenuation phenomenon of Lamb waves notonly depends on the damping coefficient but alsodepends on the propagation distance. A numericalstudy was performed to investigate the effect of propa-gation distance on Lamb-wave tuning. The 1D tuningcurves at three various propagation distances (100, 120,and 140 mm) were calculated using our proposedmodel. Figure 6 shows the corresponding 1D tuningcurves with the damping coefficient of 10% and refer-ence frequency of 2 MHz. It can be noted that theamplitude of the tuning curves decreases slightly withthe propagation distance.

2D tuning of circular crested Lamb waves withdamping

In this section, fundamental Lamb-wave modes (S0 andA0) of the circular PWAS were employed to investigate

Figure 3. One-dimensional tuning curves without damping(h0 = 0 and 9 mm PWAS, 1.016 mm 2024-T3 aluminum alloyplate, and straight crested Lamb waves).

Figure 4. One-dimensional tuning curves with small damping(h0 = 1%, f0 = 2 MHz, 9 mm PWAS, 1.016 mm 2024-T3aluminum alloy plate, and straight crested Lamb waves).

Figure 5. One-dimensional tuning curves with large damping(h0 = 10%, f0 = 2 MHz, 9 mm PWAS, 1.016 mm 2024-T3aluminum alloy plate, and straight crested Lamb waves).


2D tuning of circular crested Lamb waves with damp-ing. Equation (7) has been coded into MATLAB, whichprovides the tuning curves of the structures for a givenPWAS length, material properties, material thickness,and frequency range.

First, 2D tuning curves without damping of a 9-mm-diameter circular PWAS and a 1.016 mm 2024-T3 alu-minum alloy plate are presented in Figure 7. It can benoted that the normalized strain curves follow a similarpattern in 1D case, and the amplitudes at higher fre-quencies do not reduce significantly, which is not truefor the materials with damping.

To investigate the damping effect, the Kelvin–Voigtdamping model is utilized to calculate the 2D tuningcurves of damped materials using different damping

coefficients. Figure 8 presents 2D tuning curves withsmall damping (h0 = 1%) of a 9-mm-diameter circularPWAS and a 1.016-mm 2024-T3 aluminum alloy plate.It should be noticed that the high-frequency response isreduced, to some extent, by the structural damping,whereas the amplitude at lower frequencies almostremains the same.

Similarly, the damping coefficient is increased from1% to 10% to study the larger damping effect onLamb-wave tuning. The corresponding 2D tuningcurves with large damping (h0 = 10%) are shown inFigure 9. It can be observed that the response at higherfrequencies is strongly reduced compared with theresults of small damping, whereas the low-frequency

Figure 6. One-dimensional tuning curves of A0 and S0 modes at various propagation distances (h0 = 10%, f0 = 2 MHz,9 mm PWAS, 1.016 mm 2024-T3 aluminum alloy plate, and straight crested Lamb waves).

Figure 7. Two-dimensional tuning curves without damping(h0 = 0, 9-mm-diameter circular PWAS, 1.016 mm 2024-T3aluminum alloy plate, and circular crested Lamb waves).

Figure 8. Two-dimensional tuning curves with small damping(h0 = 1%, f0 = 2 MHz, 9-mm-diameter circular PWAS,1.016 mm 2024-T3 aluminum alloy plate, and circular crestedLamb waves).


response remains a large amplitude. It can be con-cluded that the structural damping can modify signifi-cantly the amplitudes at higher frequencies, and itseffect on 2D tuning curves can be predicted using theKelvin–Voigt damping model.

To investigate the effect of the propagation distance,2D tuning curves at three various propagation dis-tances (100, 120, and 140 mm) were calculated usingour proposed method, as shown in Figure 10. It can beobserved that the amplitude decreases obviously withthe propagation distance. This is because the waveenergy contained in the wavefront is distributed over alarge radius and the wave amplitude decreases. In a

plate, the wave amplitude decreases inversely with thesquare root of the propagation distance.

Validation of Lamb-wave tuning withPWAS transducers in the presence ofstructural damping

In this section, we present and discuss experimentalresults used to validate the numerical predictions. Thestudy will demonstrate that the Kelvin–Voigt dampingmodel is capable of capturing the effect of structuraldamping on Lamb-wave tuning with PWAS transdu-cers for metallic plates and composite plates by com-paring with experiments.

Damping effects on Lamb-wave tuning for aluminumplates

Experimental setup. To investigate damping effects onLamb-wave tuning for metallic plates, pitch-catchLamb-wave experiments on an aluminum plate wereperformed, as shown in Figure 11. The specimen is a2024-T3 aluminum alloy plate with corrosion-resistantcoatings and adhesive films on both sides. Figure 11(a)presents the thin polyvinyl chloride (PVC) film, whichis utilized to protect the plate surface by the manufac-turer. In this experiment, it plays as a viscoelastic mate-rial to increase the structural damping of the plate.

The plate dimension is 600 3 600 3 1.016 mm3.Two 9-mm-diameter circular PWAS transducers(STEMINC SM412, 9-mm-diameter and 0.5 mm thick)

Figure 9. Two-dimensional tuning curves with large damping(h0 = 10%, f0 = 2 MHz, 9-mm-diameter circular PWAS,1.016 mm 2024-T3 aluminum alloy plate, and circular crestedLamb waves).

Figure 10. Two-dimensional tuning curves of A0 and S0 modes at different propagation distances (h0 = 10%, f0 = 2 MHz,9-mm-diameter circular PWAS, 1.016 mm 2024-T3 aluminum alloy plate, and circular crested Lamb waves).


were installed on the front side of the plate to performLamb-wave tuning analysis, as shown in Figure 11(b).The experimental acquisition was performed betweenthe transmitter PWAS (T-PWAS) and the receiverPWAS (R-PWAS). The frequency range from 10 to2000 kHz in steps of 10 kHz was explored. Thedistance between T-PWAS and R-PWAS is 100 mm.An HP 33120A function generator was used to gener-ate three-count tone burst excitation signals to theT-PWAS. A Tektronix TDS 5034B digital oscilloscope,synchronized with the function generator, was used tocollect the response signals from the R-PWAS. At eachfrequency, the wave amplitudes of A0 and S0 modeswere extracted using the Hilbert transform to generatethe experimental tuning curves.

Evaluation of results. Figure 12 presents the experimentalLamb-wave tuning curves of A0 and S0 modes in thealuminum plate with PVC adhesive films. It can beobserved that the first maximum of the A0 mode hap-pened at around 100 kHz, at which the amplitude of S0mode is very small. The first minimum of the A0 modewas found around 240 kHz. At this frequency, the S0mode amplitude is nonzero and placed on an increasingcurve. Thus, the S0 signal is dominant. It should benoticed that the S0 maximum happened at 330 kHz,

which is the same frequency as that of the second A0maximum. Moreover, it can be noted that the struc-tural damping can considerably reduce the wave ampli-tudes at higher frequencies (1000–2000 kHz).

During the experiment, it was noted that the bestagreement between experiment and prediction wasachieved using the effective PWAS length, a theoreticalPWAS length smaller than that of the real PWAStransducer (Santoni-Bottai et al., 2007; Sohn and Lee,2009). In the development of the theory, it is assumed

(a)

(b)

Figure 11. Experimental setup on 2024-T3 aluminum alloy plate with PVC adhesive films: (a) PVC adhesive films on both sides and(b) PWAS on the front side.

Figure 12. Experimental Lamb-wave tuning curves withdamping (9-mm-diameter circular PWAS and 1.016 mmaluminum plate with PVC adhesive films).


that there is an ideal bonding between the PWAS andthe host structure. This assumption means that thestresses between the PWAS transducers and the plateare fully transferred at the PWAS ends. In reality, thestresses are transferred at a region adjacent to thePWAS ends (Santoni-Bottai and Giurgiutiu, 2012).Therefore, the effective PWAS length of 8.4 mmfor the 9-mm-diameter circular PWAS transducer isused in the improved analytical model of tuning toachieve a good agreement between experiment andprediction.

To predict theoretical 2-D tuning curves with damp-ing and compare it with experimental results, theimproved analytical model of tuning using the Kelvin–Voigt damping model was utilized. First, the referencefrequency f0 was initially chosen at the upper limit ofthe frequency range (f initial0 = 2MHz) and the dampingcoefficient h0 was initially taken as 5% (hinitial

0 = 5%).Next, the reference frequency f0 and the damping coef-ficient h0 were adjusted by repeated updating of themodel to achieve an acceptable match with experimen-tal tuning cures. Eventually, we found that the mostappropriate values for the damping coefficient h0 andthe reference frequency f0 are 8% and 1.5 MHz,respectively.

The comparisons between experimental and analyti-cal 2D tuning curves of A0 and S0 modes are presentedin Figure 13. Theoretical predictions with damping ofthe two modes agree well with experimental measure-ments. However, a difference at the frequency rangefrom 350 to 1000 kHz for A0 mode can be observed.The error may be attributed to the effect of the bonding

layer. As shown in Figure 13, both experimental andtheoretical results demonstrate that the damping intro-duced by PVC adhesive films can considerably modifythe tuning curves, both through peak shifts andthrough drastic diminishing of the amplitude at higherfrequencies.

Damping effects on Lamb-wave tuning for compositeplates

In the case of viscoelastic fiber–reinforced polymercomposites, one of the main complexities involved inthe use of Lamb waves is the attenuation caused bystructural damping. This is essentially due to viscoelas-tic properties of the fiber and matrix constituents(Asamene et al., 2015; Treviso et al., 2015). In this sec-tion, damping effects on Lamb-wave tuning for compo-site plates were investigated.

Experimental setup. The pitch-catch Lamb-wave experi-ments between T-PWAS and R-PWAS on a wovencomposite plate were conducted to investigate thedamping effect on Lamb-wave tuning. The specimenunder investigation is a woven carbon fiber–reinforcedpolymer composite plate. The plate dimension is3503 3503 2 mm3 and the layup is [0�, 45�, 45�, 0�]s.

A large number of PWAS transducers (STEMINCSM412; 9-mm-diameter and 0.5 mm thick) wereinstalled on the top surface of the plate, as shown inFigure 14. Three-count tone burst excitation signalswere applied to the T-PWAS. The frequency range

(a) (b)

Figure 13. Comparison between experimental and theoretical 2D tuning curves: (a) A0 mode and (b) S0 mode (h0 = 8%,f0 = 1.5 MHz, 9-mm-diameter circular PWAS, 1.016 mm 2024-T3 aluminum plate, and circular crested Lamb waves).


from 10 to 2000 kHz in steps of 10 kHz was explored.The response signals were collected from the R-PWAS1, 2, and 3 in the 0� direction and R-PWAS 4 in the45� direction. The distances between T-PWAS andR-PWAS 1 and 4 are 100 mm. The interval of theR-PWAS 1, 2, and 3 is 20 mm. Since Lamb waves havea much stronger attenuation in composite plates, apower amplifier (NF Corporation HSA4014) was usedto amplify the excitation signal to 140 V peak to peak.At each frequency, the wave amplitudes of A0 and S0modes were extracted using the Hilbert transform togenerate the experimental tuning curves.

Evaluation of results. Experimental Lamb-wave tuningcurves of A0 and S0 modes at 100 mm in the 0� direc-tion are shown in Figure 15. A similar pattern can beobserved from the 2D tuning curves, where the firstmaximum of the A0 mode happened at around 80 kHz,at which the amplitude of S0 mode is very small. The

first minimum of the A0 mode was found around210 kHz. At this frequency, the S0 mode amplitudeplaced on an increasing curve, which means that the S0signal is dominant. It should be noticed that the S0maximum happened at 300 kHz. Moreover, it can benoted that the amplitudes at higher frequencies (850–2000 kHz) were reduced significantly due to the struc-tural damping of the woven composite plate.

To capture the effect of structural damping, theimproved analytical model of 2D tuning was used to pre-dict the theoretical tuning curves. Based on the assump-tion of quasi-isotropic material for this woven compositeplate, the theoretical tuning curves can be calculatedusing the effective Young’s modulus and Poisson’s ratio(Gresil and Giurgiutiu, 2015). This is because the wovencomposite with the layup is quasi-isotropic, which pos-sesses the same material properties in all the directions.Gresil and Giurgiutiu (2015) performed pitch-catchexperiments to measure group-velocity curves in differentpropagation directions (0�, 30�, 45�, 60�, and 90�) on thewoven composite plate and found that the quantitieswere the same in all the directions.

To validate the quasi-isotropic assumption of thewoven composite plate, experimental 2D tuning curvesobtained from R-PWAS 4 in the 45� direction werecompared with the results in the 0� direction, as shownin Figure 16. It can be found that the tuning curves inthe two various directions are almost the same. Itmeans that the specimen considered in this article is aquasi-isotropic plate, which possesses the same materialproperties in all the directions.

To determine the reference frequency f0 and thedamping coefficient h0 of the Kelvin–Voigt dampingmodel, the same procedure used in section ‘‘Evaluationof results’’ was employed. Eventually, we found that themost appropriate values for the damping coefficient h0

and the reference frequency f0 are 12% and 1.5 MHz,respectively.

The comparisons between experimental and theore-tical 2D Lamb-wave tuning curves of A0 and S0 modesare shown in Figure 17. A good match between experi-mental measurements and theoretical predictions withdamping was achieved for the two modes. It should benoted that the theory also predicts that the S0 maxi-mum would happen at 300 kHz, A0 maximum wouldhappen at 80 kHz, and the rejection of the A0 wouldoccur at 210 kHz; these predictions were validated bythe experiments. However, it can be observed that thedamping coefficient of the woven composite plate(h0 = 12%) is larger than that of the aluminum plate(h0 = 8%), which means that the woven compositeplate has a much stronger damping effect. It can beconcluded that both experiment and prediction demon-strate that the structural damping of the woven compo-site plate can significantly change the tuning curves,both through peak shifts and through drastic diminish-ing of the amplitude at higher frequencies.

Figure 14. Experimental setup on the composite plate (Gresiland Giurgiutiu, 2015).

Figure 15. Experimental Lamb-wave tuning curves withdamping at 100 mm in the 0º direction (9-mm-diameter circularPWAS and 2 mm woven composite plate).


In addition, a further experimental measurementwas conducted by looking at the tuning curves at vari-ous propagation distances (100, 120, and 140 mm). Theexperimental 2D tuning curves were obtained byextracting the wave amplitudes of the received signalsfrom R-PWAS 1, 2, and 3, as shown in Figure 18.Theoretical 2D tuning curves at different propagation

distances were calculated by our model and comparedwith the experiment. Normalized amplitudes using themaximum value of the tuning curves at 100 mm wereplotted.

A good match between the experiment and predic-tion can be observed. Figure 18 (a) shows the compari-son of A0 mode at different locations, and two peaks

Figure 16. Comparison between the tuning curves in the 0º direction and in the 45º direction (9-mm-diameter circular PWAS and2 mm woven composite plate).

(a) (b)

Figure 17. Comparison between experimental and theoretical 2D Lamb-wave tuning curves: (a) A0 mode and (b) S0 mode(h0 = 12%, f0 = 1.5 MHz, 9-mm-diameter circular PWAS, 2 mm woven composite plate, and circular crested Lamb waves).


and one valley were predicted by our model, whichwere validated by the experiments. Figure 18 (b) pre-sents the comparison of S0 mode at various propaga-tion distances, and only one peak around 300 kHz canbe observed. Some small differences between the experi-ment and prediction can also be observed; this may beattributed to the variation in the R-PWAS transducersand the corresponding bonding layers. In general, bothexperiment and prediction demonstrated that the

amplitude of the tuning curves decreases with the pro-pagation distance.

Summary, conclusions, and future work

In this article, the analytical solution of Lamb-wavetuning for undamped media is extended to dampedmaterials using the Kelvin–Voigt damping model, inwhich a complex Young’s modulus is utilized to

(a)

(b)

Figure 18. Comparison between experimental and theoretical 2D Lamb-wave tuning curves at various propagation distances:(a) A0 mode and (b) S0 mode (h0 = 12%, f0 = 1.5 MHz, 9-mm-diameter circular PWAS, 2 mm woven composite plate, and circularcrested Lamb waves).


account for the effect of structural damping. To vali-date the proposed method and demonstrate its capabil-ities, experimental measurements were performed on analuminum plate with PVC adhesive films on both sidesand a quasi-isotropic woven composite plate using 9-mm-diameter circular PWAS transducers. Experimentalresults show that structural damping can modify thetuning effect considerably, both through peak shifts andthrough drastic diminishing of the amplitude at higherfrequencies. Moreover, the amplitude of the tuningcurves would decrease with the propagation distance. Itwas concluded that the improved model achieved goodagreements with experimental measurements and theeffects of structural damping were successfully captured.This is an important improvement over the conven-tional solution of Lamb-wave tuning in the literature.The concepts of this article are applicable to bothmetals (e.g. metallic plates with viscoelastic coatings)and composites.

For future work, it is desirable that the improvedanalytical solution should be extended to capture thecomplicated and direction-dependent damping beha-viors in general composite materials.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest withrespect to the research, authorship, and/or publication of thisarticle.

Funding

The author(s) disclosed receipt of the following financial sup-port for the research, authorship, and/or publication of thisarticle: This work was supported by the National Aeronauticsand Space Administration (NASA; grant no. NNL15AA16C)

and by Air Force Office of Scientific Research (AFOSR; grantno. FA9550-16-1-0401).

ORCID iD

Hanfei Mei https://orcid.org/0000-0002-8921-494X

References

Achenbach J (1973) Wave Propagation in Elastic Solids. Lon-

don: North-Holland Publishing Company.Asamene K, Hudson L and Sundaresan M (2015) Influence

of attenuation on acoustic emission signals in carbon fiber

reinforced polymer panels. Ultrasonics 59: 86–93.Auld BA (1973) Acoustic Fields and Waves in Solids, vol. II.

New York: Wiley.Bartoli I, Marzani A, di Scalea FL, et al. (2006) Modeling wave

propagation in damped waveguides of arbitrary cross-sec-

tion. Journal of Sound and Vibration 295(3): 685–707.Boller C, Chang FK and Fujino Y (2009) Encyclopedia of

Structural Health Monitoring. New York: John Wiley &

Sons.Castaings M and Hosten B (2000) Air-coupled measurement

of plane wave, ultrasonic plate transmission for

characterising anisotropic, viscoelastic materials. Ultraso-

nics 38(1): 781–786.Christensen RM (1982) Theory of Viscoelasticity: An Introduc-

tion. 2nd ed. New York: Academic Press.Giurgiutiu V (2005) Tuned Lamb wave excitation and detec-

tion with piezoelectric wafer active sensors for structural

health monitoring. Journal of Intelligent Material Systems

and Structures 16(4): 291–305.Giurgiutiu V (2014) Structural Health Monitoring with Piezo-

electric Wafer Active Sensors. 2nd ed. Amsterdam: Elsevier

Academic Press.Giurgiutiu V and Santoni-Bottai G (2011) Structural health

monitoring of composite structures with piezoelectric-

wafer active sensors. AIAA Journal 49(3): 565–581.Giurgiutiu V, Zagrai A and Bao JJ (2002) Piezoelectric wafer

embedded active sensors for aging aircraft structural

health monitoring. Structural Health Monitoring 1(1):

41–61.Graesser EJ and Wong CR (1992) The relationship of tradi-

tional damping measures for materials with high damping

capacity: a review. In: Kinra and Wolfenden (eds) M3D:

Mechanics and Mechanisms of Material Damping. West

Conshohocken, PA: ASTM International, pp. 316–343.Graff KF (1975) Wave Motion in Elastic Solids. Oxford: Clar-

endon Press.Gresil M and Giurgiutiu V (2015) Prediction of attenuated

guided waves propagation in carbon fiber composites

using Rayleigh damping model. Journal of Intelligent

Material Systems and Structures 26(16): 2151–2169.He J and Yuan FG (2017) Lamb-wave-based two-

dimensional areal scan damage imaging using reverse-time

migration with a normalized zero-lag cross-correlation

imaging condition. Structural Health Monitoring 16(4):

444–457.Lin J, Gao F, Luo Z, et al. (2016) High-resolution Lamb

wave inspection in viscoelastic composite laminates. IEEE

Transactions on Industrial Electronics 63(11): 6989–6998.Mitra M and Gopalakrishnan S (2016) Guided wave based

structural health monitoring: a review. Smart Materials

and Structures 25(5): 053001 (27 pp.).Mu J and Rose JL (2008) Guided wave propagation and

mode differentiation in hollow cylinders with viscoelastic

coatings. Journal of the Acoustical Society of America

124(2): 866–874.Neau G (2003) Lamb waves in anisotropic viscoelastic plates:

study of the wave fronts and attenuation. PhD Thesis, Uni-

versity of Bordeaux, Bordeaux.Qiu L, Liu M, Qing X, et al. (2013) A quantitative multidam-

age monitoring method for large-scale complex composite.

Structural Health Monitoring 12(3): 183–196.Qiu L, Yuan S, Chang FK, et al. (2014) On-line updating

Gaussian mixture model for aircraft wing spar damage

evaluation under time-varying boundary condition. Smart

Materials and Structures 23(12): 125001 (14 pp.).

Raghavan A and Cesnik CE (2005) Finite-dimensional piezo-

electric transducer modeling for guided wave based struc-

tural health monitoring. Smart Materials and Structures

14(6): 1448–1461.Raghavan A and Cesnik CE (2007) Review of guided-wave

structural health monitoring. Shock and Vibration Digest

39(2): 91–116.


Ramadas C, Balasubramaniam K, Hood A, et al. (2011)Modelling of attenuation of Lamb waves using Rayleighdamping: numerical and experimental studies. Composite

Structures 93(8): 2020–2025Rose JL (1999) Ultrasonic Waves in Solid Media. New York:

Cambridge University Press.Santoni-Bottai G and Giurgiutiu V (2012) Exact shear-lag

solution for guided waves tuning with piezoelectric-waferactive sensors. AIAA Journal 50(11): 2285–2294.

Santoni-Bottai G, Yu L, Xu B, et al. (2007) Lamb wave-modetuning of piezoelectric wafer active sensors for structuralhealth monitoring. Journal of Vibration and Acoustics

129(6): 752–762.Shen Y and Cesnik CE (2016) Hybrid local FEM/global

LISA modeling of damped guided wave propagation incomplex composite structures. Smart Materials and Struc-

tures 25(9): 095021 (20 pp.).

Sohn H (2007) Effects of environmental and operationalvariability on structural health monitoring. Philosophical

Transactions of the Royal Society of London A: Mathemat-

ical, Physical and Engineering Sciences 365(1851): 539–560.Sohn H and Lee SJ (2009) Lamb wave tuning curve calibra-

tion for surface-bonded piezoelectric transducers. Smart

Materials and Structures 19(1): 015007 (12 pp.).Sreekumar P, Ramadas C, Anand A, et al. (2015) Attenuation

of A0 Lamb mode in hybrid structural composites with

nanofillers. Composite Structures 132: 198–204.Su Z, Ye L and Lu Y (2006) Guided Lamb waves for identifi-

cation of damage in composite structures: a review. Journal

of Sound and Vibration 295(3): 753–780.Treviso A, Van Genechten B, Mundo D, et al. (2015) Damp-

ing in composite materials: properties and models. Compo-

sites Part B: Engineering 78: 144–152.Yuan S, Ren Y, Qiu L, et al. (2016) A multi-response-based

wireless impact monitoring network for aircraft composite

structures. IEEE Transactions on Industrial Electronics

63(12): 7712–7722.


effect of structural damping on the tuning between

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