A Comb Transducer Model for Guided Wave NDE_ROSE

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ELSEVIER Ultrasonics 36 (1998) 163-169

A comb transducer model for guided wave NDEJ.L. Rose *, S.P. Pelts, M.J. Quarry

Engintwin~q Science und Mcchmnics. 114 Hullowell Building, The Pemsylvuniu State Univec~ity. University Pwk. PA 16802. USA

AbstractA countless number of guided wave modes at particular frequencies could be selected for a particular NDE problem, each point

producing special sensitivities by way of wave structure across the thickness of the component being studied and also specificpenetration powers as a result of interface and surface displacement values and subsequent energy leakage into neighboring media.The mode and frequency choice has a strong influence on NDE and flaw detection, classification and sizing potential as well asan ability to propagate guided waves over long distances, despite the presence of coatings and other surrounding media.

The approach to mode and frequency selection is therefore crucial, which can ultimately be based on theoretical and/orexperimental means. One aspect of a theoretical approach beyond dispersion curve analysis includes theory of elasticitycomputations of displacement distributions across a structure. Focus can be on achieving in-plane or out-of-plane optimal valueson a surface or at a specific location inside a structure in an attempt at flaw analysis or improved penetration power. From anexperimental point of view, an angle beam transducer at a specific angle can be used to achieve a particular phase velocity value.Unfortunately, the presence of a phase velocity spectrum due to a transducer source influence, size and velocity pattern, as wellas the frequency spectrum itself. often limits the ability to specifically achieve the particular mode and frequency of choice.Multiple modes can be obtained.

An alternate transducer choice to the angle beam transducer can be a multiple element array or comb based on various designchoices of element size, spacing and pulsing schedules to produce specific modes and frequencies. The purpose of this paper is topresent a model and subsequent solution to a boundary value problem that can evaluate the source influence as a function of thecomb transducer design parameters. Advantages of the comb transducer, the mathematical model and analysis, and sampleexperimental results are all presented in the paper along with an insight into future directions. 0 1998 Elsevier Science B.V.K~ywrr~s; Comb transducer; Lamb waves; Dispersion curves: RF signal

1. IntroductionThere are many obvious advantages of using ultra-

sonic guided waves for inspection purposes, most associ-ated with mode control, selection and utilization.Utilization of a small multi-element comb type ultra-sonic transducer is therefore proposed for guided wavemode control in Nondestructive Evaluation. Theoreticalmethods are developed and experimental results arepresented for guided wave generation and mode controlwith this efficient and versatile novel comb type ultra-sonic transducer. One of the first applications of a combtype transducer in the NDE field was discussed byViktorov [ 11, and some features of wave excitation bya comb type were studied in Refs. [2,3]. Fundamentalexperimental aspects and benefits of a comb transducer* Corresponding author. Tel: + I-814-863-8026:fax: + I-814-863-8164; e-mail: jlresm(@engr.psu.edu0041-624X.98/$19.00 0 1998 Elsevier Science B.V. All rights reserved.PII SO04 I -624X (97 )00042-S

design, and application concepts, can be found inRef. [4].

2. Theoretical modelingThe impulse excitation of Lamb waves in an elastic

isotropic plate using a comb transducer is studied. Thetransducer has the following parameters: element widthAW, number of elements N and comb sizing (elementwidth + gap) AS. The central excitation transducer fre-quency is Jb, transducer bandwidth is j and platethickness u (see Fig. 1).

The time duration of the input pulse is determined bythe number of cycles C,, of the tone burst pulse genera-tor. Every element of the transducer produces a loadingpressure on the elastic plate that one can model byusing a parabolic pressure distribution p(x), whereP(X) =I)~[ 1 -(_?/A W)] and p0 is a known constant.


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J. L. Rose et ul. / Ultrusonics 36 (1998) 163- 169

T ,Combnnnnnnnn r Plate

A(!+-- 1AW C

Fig. I. Problem geometry.

The boundary conditions for the plate, by using theabove-introduced notations, can be written ascrX,. x,y = + d/ 2,t) = 0 for Vx and Vlt> 0,o,,(x,):r) =P(x) sin(w&)

for XE G, y=d(2 and OCJJb,=0 for x $ G, y=d/ 2 and Vt>O,=0 for y= -d / 2 and Vx and Vt>O,

where G= UF= g, and o0 = 2rcf,,where g, is the platearea covered by the k,, element. The solution of theboundary value problem is obtained by applying adouble Fourier transform in the wavenumber and fre-quency domains [ 51 to the governing equation of motionand boundary conditions (Eqs. ( 1) and (2)).

The displacement can be written as1 xu,(x,y,t) = __

4in2 ss(x,?i,cr,o)e(~,w)G(o)

-m rx ei(ax-t) da dw (n= 1,2). (3)

LIP=u, u2 = w are respectively in-plane and out-of-plane displacements. Q(a,w) is a double Fourier trans-form over time t and variable x from loading pressuredistribution ~,,(x,d/2,r). The known functionK,,(x,y,c~,o) depends on the plate parameters and isindependent of comb transducer features. The functionG(w) = exp [-(w - 00)*/(47r2/j)] represents an influenceof the frequency spectrum of the transducer, by assumingthat the frequency spectrum has a Gaussian distributionwith center frequency Jb.

The integration contour r is chosen according toradiation conditions. The inner integral along r isevaluated by residue theory. The function K,,(x,y,a,w)has a countable infinite number of poles, which are theroots of the Rayleigh-Lamb dispersion equation

The far field solution can be represented using onlythe propagating waves. Therefore the summation in

Eq. (4) involves only real poles of the functionK,(x,y,cx,o). Eq. (4) shows that the far field solutioncan be represented as a Fourier integration in thefrequency domain over the A4 dispersion curves. Thefrequency integration is performed numerically by usinga fine frequency increment. The system response overthe frequency domain is limited by the transducer fre-quency bandwidth G(w). Integration over the interval(- x,m ) with respect to w in Eq. (4) can be transformedto the interval (0,x)).

3. Theoretical resultsLet us consider some theoretical results and explana-

tions for some practical situations (see Fig. 2).The mode activation lies on the line with a slope

depending on the comb spacing AS= 1.524 mm. Thepoint of interest (mode A 1) is a solid dot for a givenfrequency of excitation; the transducer also excites, butless efficiently, other points. They are marked with whitesquares (see Fig. 2(a)). One can see in Fig. 2(a) thatmodes Al and S2 are significantly separated from AO,SO and A2 which all have similar velocities. The goalwould therefore be to excite the design point, solid dot,

(a)

: Al A2 5204 _.. _0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

a(MHzmn)

(b)

6 I

-z$9 20

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15faW&

Fig. 2. Dispersion curves for an aluminum plate: v,. =6.3 km s-, V,=3.1 km SC. (a) Phase velocity dispersion curves show-ing mode activation for spacing AS= 1.524 mm. (b) Group velocitydispersion curves are shown with the excitation area along thedashed line.


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J. L. Rosr et al. ! Ulrrcrsoilic~s 36 f I YYH) 163-- I69

tunemlcrosec)(a) element width=90% of comb sizing

0.3 ,

(b) element width=50% of comb sizing

tune (microsec)(c) element width=33% of comb sizing

0.06 ,

-006 tune (mlcrosec) time (microsec)

(f) element width=8% of comb sizing (e) element width= 1% of comb sizing

- 0 1 5 time mnrosec)

(d) element width= 16% of comb sizing

Fig. 3. RF signal generated in a 2.25 mm thickness aluminum plate at a receiving distance of 100 mm. Transducer parameters: number of elements7. comb sizing (gap plus element width) 1.524 mm, central frequency 2.44 MHz, bandwidth 0.2 MHz. number of cycles 12. (a) Element width=90% of comb sizing; (b) element width = 50% of comb sizing; (c) element width = 33% of co mb sizing; (d) elem ent width = 16% of comb sizing; (e)element width = I % of comb sizing: (f ) element width = 8% of comb sizing.with as much energy as possible. Probe design parame-ters could be adjusted to accomplish this.

Let us now consider a parametric study of varyingelement width as a certain percent of comb sizing whichis element size plus gap between comb elements. Sampleresults presented in Fig. 3 show that the results are notso strongly dependent on the element width except forpulse magnitude for unit input (see Fig. 4). By studyingtheoretical results at various distances, group velocities

0.25

00% 15% 30% 45% 60% 75% 90%

Element width expressed in % fmm comb spacing.Fig. 4. The RF magnitude value versus element width (expressed in %from fixed comb spacing) is shown for the same parabolic pressureloading. The maximum magnitude corresonds to the element widthequal to 50% of comb spacing. this means half of a wavelength.

can be calculated for mode identification purposes (seeFig. 5). Modes Al and S2 can be clearly identified inthe RF signal presentation by comparing the RF signalsfor two different observation distances. Mode superpo-sition AO, SO, Sl, A2 (with the similar group velocities)can be seen between Al and S2.

Because of a source influence etfect [6], it is generallynot possible to generate just a single mode. Note thatmodes in Fig. 2 are often very close together. Examplesof excitation are presented in Fig. 5. Mode Al is domi-nant showing the success of the mode generation pro-cedure. This is reasonable from an inspection point ofview since Al is so dominant. Other group velocitychanges can reduce the other mode influences becauseof a separation in the time domain (see Fig. 5 ). In orderto separate all other modes, the RF signal was calculatedfor larger distances of 200 mm (see Fig. 6).

The magnitude of the propagating modes distributionfollows concepts from Fig. 2 and can be estimated byusing Fig. 7 as follows. Consider a vertical line in thephase velocity dispersion curve and mode intersectionpoints at a particular ,fa value. The etrect on mode


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166 J.L. Rose et al. / Ultrasonics 36 (lYY8) 163-169

0 . 30 . 2

4 0 . 1 s2 e A'

f j 4 . :- 0 . 2

I - - + + - 2 0 6 0 8 0

- 0 . 3 1t i m( m i L ms e c )

0 . 3 ,0.23 0 . 1

g , , y

I - + 4 + s 2 L 4 - - A'

2 0 6 0 8 00 . 20 . 3 1

t i r n e @ L ms e c )

Fig. 5. The RF signal for two ditferent observation distances from thetransducer with element width =50% of comb sizing: (a) 90 mm and(b) 100 mm. Modes AI and S2 are identified by using group velocitydispersion curves. Mode superposition AO, SO, Sl. A2 (with the similargroup velocities) can be seen between Al and S2.

time (microsec)Fig. 6. All propagating modes are shown as part of the RF signal, thatis calculated for a 200 mm distance. The transducer element width =50% of comb sizing.

0.25

02

s 0.15.sc2 0.10.05

0A 0 S O A l S l A2 S 2

Fig. 7. Magnitude of the propagation modes versus mode number forthe transducer with element width = 50% of comb sizing.

-0.02 ~ cttme (microsec)(a) 1 element

-0 1 I time (mlcrorec)(b) 2 elements

-0 06 .Itome (mlcrosec)

(c) 3 elements0.15

3 0:o:$ .lJlfI-+0-0.15 time (mlcrosec)(d) 4 elements

0 . 13 0 . 0 5

i . , . , ;I - - +

5 0

- 0 . 1 time (microsec)(e) 5 elements

1 0 0

0 . 3. 0 . 2

B 0 . 18 - 0 . 1

- 0 . 2 I + -0 1 0 0

- 0 . 3 ' time (microsec)(0 6 elements

0 1 5Of 0.05

-cll;l-k- 100-015 time (microsec)(g) 7 elements

Fig. 8. The RF signal for the different number of elements for SO modeatfil=4.4. (a) I element, (b) 2 elements, (c) 3 elements. (d) 4 elements,(e) 5 elements, (f) 6 elements, (g) 7 elements.


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J. L. Rose rt al. Ultrasonics 36 (IYYX) 163-169 161

fI..._.---- Al A2 S20 1 2 3 4 S 6 7 8 9 10 11 12 13 14 15

B@j=WFig. 9. Activation line for experimental study of element width.

0.254 mm width01 1

generation also depends on the number of elements. Inorder to analyze the influence of the number of combtransducer elements on mode generation, an RF signalis calculated for a central frequency excitation of1.9 14 MHz. The points of interest are modes A0 and SOfor a given frequency of excitation. The transducer alsoexcites other modes, but less efficiently. Sample theoreti-cal results for this problem are presented in Fig. 8.

0.127 mm width

b) S2 mode at fd=8.2

Fig. 10. Element width experimental study RF waveforms. (a) Al mode at.@=5.4; (b) S2 mode at ,fiI=8.2: (c) SO mode at ld=4.0; (d) SI modeat .fJ= 6.4; (e) A2 mode at ,jd= 7.5.


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168 J. L. Rose rt al. J Ultrusonic:s 36 (1998) 163-169

Fig. 11. RF waveforms demonstrating the etTect on the generation of the S2 mode from increasing the number of elements in the comb transducerarray (comb transducer element width: 0.127 mm; spacing: 1.524 mm; frequency x thickness: 7.8 MHz mm; mode: S2). (a) 1 element at 3.46 MHz;(b) 2 elements; (c) 3 elements: (d) 4 elements: (e) 5 elements; (f) 6 elements; (g) 7 elements.

Note the improvement in the dominant mode S/Nratio with respect to the other modes as the number ofelements is increased.

4. Experimental resultsFig. 9 presents the activation line for experimental

study of element width.A comparison of two combs of different widths is

shown in Fig. 10. Each has 7 elements and a 1.524 mmspacing. The widths are 16% and 8% of comb spacing,respectively. An angle beam shoe was used to receivethe signals. The larger width has a larger amplitude.

In a pulse-echo setup, a comb transducer with0.127 mm element width and a 1.524 mm spacing wasused to analyze the influence of changing the numberof elements on the modes produced. The modes wereproduced in a 2.25 mm thick aluminum plate. Theactivation of modes lies on the line with the slope ofthe spacing divided by the thickness.

The influence of changing the number of elements

was analyzed experimentally by producing SO, Al andS2 modes (see Fig. 11).

5. Concluding remarksA comb probe model has been developed that can be

used to study transducer design parameter influence onthe resulting guided wave field for inspection purposes.Theoretical and experimental results are presented forsuch items as element width and numbers of elements.Methods of achieving a dominant mode are discussed.

AcknowledgementThanks are given to Dr. Vinod S. Agarwala of the

Naval Warfare Center, Aircraft Division, PatuxentRiver, MD, for technical support of this project and tothe Office of Naval Research for financial support.


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References[I] I.A. Viktorov, Rayleigh and Lamb Waves: Physical Theory and

Applications, Plenum Press, New York. 1967.[2] J.J. Ditri. J.L. Rose. A. Pilarski. Generation of guided waves in

hollow cylinders by wedge and comb type transducers, in: D.O.Thompson. D.E. Chimenti (Eds.). Review of Progress in Quantita-tive Nondestructive Evaluation. vol. 12. 1993, pp. 21 l-218.[3] T. Demol. P. Blanquet, C. Delaberre. Lamb wave generation usinga flat multi-element array device. in: 1995 IEEE Ultrasonic Sympo-sium Proceedings, pp. 791 794.

[4] J.L. Rose, S.P. Pelts, J.N. Barsinger. M.J. Quarry. An UltrasonicComb Transducer for Guided Wave Mode Selection in MaterialsCharacterization. presented at the Eighth International Conferenceon Nondestructive Characterization of Materials. Boulder, CO.June 15-20. 1997.

[5] L.M. Brekhovskikh. Waves in Layered Media. Academic Prea,New York. 1960.

[6] S.P. Pelts. D. Jiao, J.L. Rose. A comb tramducer for guided wavegeneration and mode selection, in: l9Y6 IEEE llltrasonic Sympo-sium Proceedings Vol. 2 ( 1996) pp. 857-860

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A Comb Transducer Model for Guided Wave NDE_ROSE