Study of Wave Motion on Tubulars Using Broad-Band Laser Ultrasound

8/7/2019 Study of Wave Motion on Tubulars Using Broad-Band Laser Ultrasound

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S t u d y o f W a v e M o tio n o n T u b u la r s U s in g B r o a d - b a n dLaser U l t rasoundby M.E. B al tazar -Lrp ez, C.P. Burger , R. Ch ona an d S. SuhA B S T R A C T - - W e i n v e s ti g a t e t h e t r a n s it io n c h a r a c t e r i s ti c so f l a s e r - g e n e r a t e d u l t r a s o n i c w a v e s p r o p a g a t i n g a l o n g t h eax ia l and c i r c um fe ren t i a l d i r ec t i ons i n t ubu la r s . The the rm o-a c o u s t o - p h o t o n i c n o n - d e s t r u c t i v e e v a l u a t i o n t e c h n i q u e i se m p l o y e d t o i n i t i a t e b r o a d - b a n d s u r f a c e a n d g u i d e d w a v e sin tubu la r s o f s ev era l d i f f e ren t ou te r d iam ete rs and wa l l t h i c k -n e s s e s . A n o n - c o n t a c t b r o a d - b a n d o p t i ca l s e n s o r k n o w n a sf i b e r- t ip i n t e r fe r o m e t r y i s u s e d f o r w a v e a c q u i s it io n . W a v e d i s -pe rs ion as a func t i on o f t h i c k nes s to m ean rad ius ra t i o i sr e s o l v e d u s i n g a G a b o r w a v e l e t t r a n s f o r m b a s e d a l g o r i t h m .T h e a l g o r i t h m e n a b l e s d i s p e r s i o n t o b e e s t a b l i s h e d t h r o u g ht h e d e p l o y m e n t o f j u s t o n e s e n s i n g i n t e r fe r o m e t e r . T h is i s i nc o n t r a s t to t e c h n i q u e s w h i c h r e q u i r e m u l ti p le d e t e c t i o n l o c a -t i o n s f o r t h e d e t e r m i n a t i o n o f w a v e d i s p e r s i o n .K E Y W O R D S - - L a s e r u l t ra s o u n d , c y li n d r ic a l s t r u ct u r e , Gaborw a v e l e t t r a n s f o rmI n t r o d u c t i o nWave mot ions in cy l ind r i ca l s t ruc tu re s have been wide lys tud ied and we l l documen ted . T heore t i c a l s tud ie s inc lud ingt h e f o r m u l a t i o n o f t h e o ry a n d c o m p a r i s o n o f a p p r o x i m a t i o n sto exa c t s o lu t ions a re p len ty . 1 -5 Ar me nak as e t a l . 5 p rov ide as e l f - c o n ta i n e d t r e a t m e n t o n p r o p a g a t i o n o f p l a n e h a r m o n i cwav es in c i rcu la r cy l inde rs , a ll w i th in the f ram ew ork o f th ree -d im ens iona l theo ry o f e la s t i ci ty , Us ing the s e theo r i e s , i t ispos s ib le to s tudy the f i rs t mo des o f p rop aga t ion a s func t ionso f the ra t io o f she l l t h i cknes s to mean rad ius . O the rs have fo -c u s e d t h e i r w o r k o n t h e e x p e r i m e n t a l w a v e m o t i o n s i n c y l i n -d e r s. F o r e x a m p l e , G o l d s m i t h e t a l . 6 o b s e r v e d t h e p r o p a g a t i o no f p u l s e s o n t h in a n d m e d i u m - t h i c k w a l l h o l l o w c y l i n d e r s i nwhich s t ee l s phe re s s t ruck the s pec imens and y ie lded pu l s ed u r a t io n s o f 2 0 - 4 0 I t s t h a t c o r r e s p o n d t o a f re q u e n c y c o n t e n to f u p t o 5 0 k H z . A s i m i l a r a p p r o a c h w a s c a r d e d o u t b y Y ina n d Y ue , 7 w h o u s e d d y n a m i c l o a d s o n t h e e n d s o f a m u l t i l a y -e red c i rcu la r cy l inde r to s tudy t rans ien t r e s pons es . T he re havebeen s tud ie s tha t have looked s pec i f i c a l ly w i th in the u l t r a -s o n i c r a n g e. O n e e x a m p l e i s t h e w o r k b y C a w l e y e t a l. 8 w h ou s e d e l e c t r o m a g n e t i c s e n s o r s t o g e n e r a t e c y l i n d ri c a l g u i d e d

M.E. Baltazar-lx~pez (SEM member; [email protected]) is a Professor Researcher, Departamento de lngenieria Mecdnica, Centro Nacionalde lnvestigacirn y Desarrollo Tecnolrgico (CENIDET), Int. del lnternadoPalmira S/N, Cuernavaca. Morelos 62490, Mexico. C.P Burger (SEM mereber) is Professor Emeritus, R. Chona (SEM member) is an Associate Pro-fessor, and S. Suh (SEM member) is an Associate Professor, Departmentof Mechanical Eagineering, Texas A&M Universit); College Station, Texas77843, USA.Original manuscript submitted: September 15, 2004.Final manuscript received: May 31, 2005.DO I: 10.1177/0014485105056899

wav es in tubu la r s. R os e 9 rev iewed exp e r ime n ta l t e chn iquesava i l ab le up to 2002 and d i s cus s ed the fu tu re d i rec t ions fo rp ipe in s pec t ion .B e c a u s e o f t h e v a r io u s p r o b l e m s a s s o c ia t e d w i t h t h e u s e o fcon tac t t r ans duce rs on cy l ind r i ca l geom e t ry , u l t r a s on ic wavemo t ions in cy l ind r i ca l s t ruc tu re s have been d i f f i cu lt t o s tudyexpe r im en ta l ly . T h e l a ck o f re s o lu t ion , t he invas ive na tu re o ft rans duce rs w i th re s pec t to the p ropaga t ing waves , t he s o -ph i s t i c a t ion a s s oc ia t ed w i th tubu la r cu rva tu re and re f l ec tedw a v e f o r m s , a n d p o o r a c c e s s i b i l i t y a r e a m o n g t h e d i f f i c u l -t i e s . By us ing a non-con tac t u l t r a s on ic gene ra t ion and de -t e c t i o n t e c h n i q u e k n o w n a s t h e r m o - a c o u s t i c - p h o t o n i c n o n -d e s t r u c ti v e e v a l u a t i o n ( T A P - N D E ) , B u r g e r e t a l . ] ~ d e m o n -s t ra t ed tha t mos t o f the d i f f i cu l t i e s cou ld be ove rcome . Incon junc t ion w i th an appropr i a t e fea tu re ex t rac t ion te chn ique ,t h e y s h o w e d t h a t T A P - N D E w a s i d e a l f o r c e r t a i n p r o b l e m si n v o l v in g c o m p l e x o r i r r eg u l a r g e o m e t r y .L iu and Qu " t p re s en ted a num er ica l s o lu t ion fo r c i rcum -fe ren t i a l waves tha t a re independen t o f the exc i t ing s ou rce .I n t h e i r w o r k , e i g e n f u n c t i o n e x p a n s i o n w a s e m p l o y e d t oana lyze the in i t i a t ed mul t imoda l , d i s pe rs ive gu ided waves ,A n o t h e r f e a t u r e e x t r a c t i o n t e c h n i q u e w i d e l y u s e d f o r a n a l y s i so f m u l t i m o d e , d i s p e r s i v e g u i d e d w a v e s i s F o u r i e r t r an s f o r m .Such a t e chn ique requ i re s mu l t ip l e de tec t ion po in t s in o r -d e r t o e x t r a c t f e a tu r e i n f o r m a t i o n a b o u t t h e s p e c i m e n b e i n gin te r roga ted . 12 As repor t ed in Z u pan and He m ker 13 whe ret w o d e t e c t i o n p o i n t s w e r e e m p l o y e d f o r m e a s u r i n g i n - p l an es u r f a ce d e f o r m a t i o n s , w a v e f o r m s o b t ai n e d f r o m l a s e r - b a se ds e n s i n g s y s t e m s w e r e s u c c e s s f u l l y p r o c e s s e d u s i n g F o u r i e r -b a s e d t e c h n i qu e s . H o w e v e r , w h e n d e a l i n g w i t h o u t - o f - p l a n ed i s p lacem en t s , s eve ra l s ens ing loca t ions a re neces s a ry fo r thet w o - d i m e n s i o n a l f a s t F o u r i e r t r a n s f o r m ( F F F ) t e c hn i q u e . O n ea p p r o a c h f o r a l l e v i a ti n g t h e d e m a n d o n r e s o l v i n g d i s p e r s iv ewave s us ing Four i e r me tho ds wa s d i s cus s ed in Gao e t a l . 14whe re l inea r s pa t i a l cond i t ion ing a l lowing fo r d i rec t iona l lye n h a n c e d s u r f a c e a c o u s t i c w a v e p r o p a g a t i o n w a s u t i l i z e d ,T h e u s e o f o t h e r p r o c e s s i n g t e c h n iq u e s s u c h a s th e W i g n e r -Vil le d is t r ibut ion was a lso rep orted. 15, ] 6 T h i s p a p e r p r e s e n t s aw a v e l e t - b a s e d m e t h o d t h a t a ll o w s f e a t u r e i n f o r m a t i o n b e e x -t r a c te d t h r o u g h t h e d e p l o y m e n t o f o n l y o n e d e t e c ti o n p o i n t .T h e o b j e c t i v e o f t h e s t u d y i s t o i n t eg r a t e T A P - N D E w i t hG a b o r w a v e l e t tr a n s f o r m s f o r t h e g e n e r a t i o n a n d d e t e c ti o nof gu ided wa ves in cy l ind r i ca l t ub ing . By us ing wa ve le tt r ans fo rm fo r p roces s ing d i s pe rs ive waves , a r i ch s e t o f in -f o r m a t i o n a b o u t t h e w a v e s c a n b e r e a d i l y r e s o l v e d i n t h et i m e - f r e q u e n c y d o m a i n u s i n g o n e w a v e a c q u i s it i o n s e n s o r. 17T h i s i s i n c o n t r a s t t o t h e t w o - d i m e n s i o n a l F F F m e t h o d i nwhich mul t ip l e s ens ing loca t ions a re requ i red . A compar i s on

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0 . 2 ~

0 , 1 5

N!~- 0,~'

$1

Ot J0 0,01 0.02 00 3 0,04 OC ~ 0.01~ 0.07t ( m s e c )

F ig . 1 - - M- H d ispers ion p lo t ted as t ime versus f requency fo ra th in cyl indr ical shel l of m = h/R = 1/30, f lexural mode, n = 1.$1 = f irst ( lowest) f lexura l mode, $2 f irst ( lowest) torsion almode, and $3 = second f lexural mode

made wi th regard to the two signal processing techniques 18indicates the effect iveness of wavele t t ransf orm as a preferredtool for damage detect ion and heal th moni tor ing.Theory

Dispe rs ion curves have been comm only used fo r t he ana l-ysis of wave mot ions in cyl indrica l s t ructures. Mirsky andHerr mann 3,19,20 developed a theory ( M -H theory) that de-scr ibed longi tudinal and c i rcumferent ia l wave mot ions incyl inders. Using the theory, dispersion curves were obta inedfor cyl inders of a wide range of thickness to diameter ra t ios.The inc lus ion o f a more gene ra l T imoshenko- type theory toaddress non-ax ia lly sym met r i c mot ions enab led a r i ch se t o fwave ph eno men a to be studied than could c lassica l shel l the-or ie s , which cons ide red o n ly memb rane and bend ing e f fec ts .The dispersion curves presen ted in thei r work s were obta inedby numerical ly solving a character ist ic wave equat ion. Fiveroo t s were found a s a r e su l t and each roo t cor re sponded toa dispersion curve . These curves were plot ted in terms of anon-d imens iona l wave ve loc i ty ( s ) and a non-d imens iona lwav enum ber (~) for a given cyl inder of thickness, h , andmean radius, R.Wave d i spe r s ion i s commonly repre sen ted a s e i t he rwavenu mber ve r sus f requency , f r equency ve rsus phase ve loc -i ty , o r non-d imens iona l w avenum ber ve r sus non-d imens iona lwave veloci ty. In the fol lowing, dispersion informat ion ispresented in terms of t ime and frequency, a representa t iontha t r e su l t s f rom app ly ing wave t r ansform to expe r imenta lwave fo rm process ing. T he d i spe r s ion in forma t ion p re sen tedin Fig. 1 of a wave acq uired in a stee l specimen shows the f i rstthree modes $1, $2, and $3, where each mode is associa tedwith a par t icular type of mot ion. Fo r example , m ode Sl i s thelowest f lexural mode. The non-dispersive $2 mode is asso-c i a ted wi th t he pure ly t o r s iona l mot ion k nown as t he l owertorsional mode, or the f i rst Lobar mode. $3 corresponds tothe second f lexural mode.The p rocess o f ob t a in ing the d i spe r s ion curves i s basedon e l a s t i c wave mot ion and ene rgy me thods . Cons ide r t hedifferent ia l e leme nt depic ted in the reference system in Fig. 2 .

Fig. 2--Reference system and stress element in a hol lowcy l inder

The d i sp l acement component s i n t he cy l indr i ca l coord ina t esys tem are des igna ted as 17x, t70, and ~z, whose dependenceson the z-coordinate normal to the middle surface are givenb y

ux(x, O, Z, t) = u(x, O, t) q- Zq)x(X, O, t)(tO(X, O, Z, t) = v( x, O, t) d- Z~oO(x, O, t)(tz(X, O, Z, t) = w( x, O, t)

(1 )

where t i s t ime, u , v , and w are the displaceme nt comp onen tsof a particle o n the m iddl e su rface, z = 0, and 99.~ and ~00 are

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the angles of rota t ion of the normal to the middle surface inth e x - z and x-O planes, respectively. Note that in eq (1) ~0xand ~00 represent the axia l and c i rcumferent ia l shear effec ts .Integra t ion of the st rain energy densi ty express ion across theshell thickness would then lead to the shell stress equationsof mot ion.

Consider the solutions for wave propaga t ion in the fol low-ing formu ( x , O, t) = U e i(mt-(xx) cos n0~Ox(X, O, t) = qJ e i(mt-ax) c o s n 0v ( x , 0, t ) = V e i (mt-ax) cos n0r O, t) = qb el(~176 co sn 0w ( x , 0, t) = W e i(mt-c~x) co sn 0

(2)

where n = 0, 1, 2 . . . i s an integer indicat ing the num ber ofwaves traveling circumferentially, m is the circular frequency,and c~ is the wave num ber defined as

m 2rtCp L

Substituting eqs (1) and (2) into in the equations of motion(not shown), a homogen eous system of l inear equat ions canbe found in the form of the f requency-equ at ion determinant :

IF] = O. (3)The experime nta l dispersion cu rves are plot ted as t ime versusfrequency a t a single detection point . Figure 3 shows the com-parison betw een theoret ica l dispersion curves and those plot -ted using experimental data. It can be seen that, for relativelow frequencies, the experimenta l G abor wavele t plots corre-spond wel l in t rend and shape to the theoret ica l values. In theexperimenta l case , the addit ional bands are not ext ra modes,but ra ther they are the resul t of using a high value of thet ime- f requency re so lu t ion pa rame te r i n t he Gabor wave le tfunct ion in order to be consistent wi th the zero-mean valuecharacter of the wavele t i t se l f. There i s a lso a discrepancy inthe arr ival t ime of the experimenta l plot compared wi th thetheory. This could be the result of imprecise mea surem ent ofthe propagat ion path. As shown in Fig. 4 , the discrepancy ismore evident when both plots are superimposed.

Before ca lcula t ing the roots of the character ist ic eq (3) ,a change of var iables i s needed. The change is performedto have the dispersion be displayed as funct ion of t ime, t ,and frequency, m. In the following, wave velocity, s, andwavenumber, ~ , are expressed in t ime and frequency as

mh . / 2p (1 + v) ~oths = ~ S V E a n d 8 - - ( 4)2~xswith s being the t ravel distance of the propagat in g w ave fromthe excitation poin t to the detection point. D etails of the pro-cess of t ransforming dispersion curves and obta ining disper-sion informat ion from experimenta l data using Gabor wavele tt ransform can be found in Bal tazar e t a l . 21

Once the theoret ica l dispersion curves are t ransformed,i t is possible to com pare them direct ly to the wav ele t coeff i-c ients resul ted from processing experimenta l wave form. Notethat Gabor wavele t t ransform resolves a t ime waveform into

Fig. 3--Co mpa rison of theory (top) to experimental data(bottom)

Fig. 4~C omp aris on of theory (lines) to experimental data(bands)

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i t s jo int t ime- freq uen cy dom ains. The charac ter is t ic eq (3)is obta ined by ca lcula t ing the de terminant of the f reque ncyequat ion. Ther e are f ive roots , thus f ive possible m odes ex-is t for each case . The wave mot io n correspon ding to n = 0i s one o f ax i symmet ry and a ssumes no w aves p ropaga t ingci rcumferent ia l ly . When n = 1 , the case i s one of f lexura lmot ion w i thou t c i r cumfe ren t ia l l y p ropaga t ing waves .

One o f the abi l it ies of the M- H theory i s tha t it can beappl ied to a wide rang e of hol low cyl inders . Tubulars as th inas m < 1/10 and as thick as m > 7/1 0 were consideredby M irsky and He rrmann. 3,19,20 Addi t ional ly , the M -H the-ory can inc lude the specia l l imi t cases when m = 0 and wh enm --+ cx~. The form er case cor respond s to a f la t p la te in whic hthe theory reduces to the c lassica l p la te theory discussed byMind lin. 22 The latter l imit case wh en m --+ cx~ corr esp ond sto a so l id cyl indr ica l bar .Experimental Setup

A typ i ca l TAP-N DE was employe d fo r the s tudy . As show nin the schemat ics in Fig. 5 , the main com pon ents of the sys-t em inc lude a gene ra t ion subs ys t em cons i s t i ng o f an Nd:YAGlaser and focusin g opt ics , a fiber-t ip interferometer (F TI) de-t ec t i on subsys t em powered by an HeNe l a se r , a pho tode -tec tor , an osc i l loscope , and a PC-based signal acquisi t ion-p rocess ing subsys t em.

The physica l se tup in Fig. 6 shows tha t i t i s possible tohave a dual -FTI se tup. The addi t ional FTI seen in the f ig-ure served as a redundancy for va l ida t ion purpose . 1 i s thefocal lens, 2 are the posi t ioning stages, 3 and 4 are FTI 1and FTI 2 , respect ive ly , and 5 i s a s tee l test specimen. Twofiber-opt ic cables, labeled 6 , a re connec ted to the i r respect iveFBT couplers (not shown), which in turn are a t tached to thephotodetec tors . Acquired waveforms are registered as in ter-ferometr ic beams by the photodetec tors and then output tothe da ta acquisi t ion-processing uni t for fea ture ext rac t ion.

Four spec imens w i th di f ferent th ickness to mean radius ra-t ios were tested. Th e distance f r om exci ta t ion point to de tec-t ion point was 10 mm for a ll cases. For other parameters andconf igura tiona l details, refer to Balta zar et al .21. As als o notedin Bal tazar e t a l . , 21'23 the ini t ia ted ul t rasonic waves propagat -ing on a tubular surface can be surface waves i f the specim enis re la t ive ly thick, or pla te (guided) w aves i f the specim en isre la t ive ly thin . When analyzing stee l p ipes of severa l th ick-nesses, some genera ted ul t rasonic waves display behaviorstha t a re charac ter is t ic o f both su rface and guid ed wav es in di f -ferent ranges o f f requencies. Thus , of cer ta in cyl indr ica l con-f igura t ion , waves p ropaga t ing i n ho l lowed spec imens cou ldhave t ransi t ion charac ter is t ics tha t a re par t ia l ly dispersive andpar t ia l ly n on-dispersive .Results

After va l ida t ing the procedures for t ransforming theore t i -ca l d i spe rs ion cu rves t h rough a compar i son wi th t he wave l e tt ime-frequency plots , i t was possible to establ ish a corre la-t ion be tween theory and exper imenta l resul ts . In conjunct ionwi th i n fo rma t ion ava i lab l e f rom pas t expe r imen t s i n w h ichpipes of di f ferent curvatures w ere considered, i t was po ssibleto establ ish a guidel ine for performing test ing on specimensof di f ferent th ickness to mean radius (m) ra t ios. The fourspecime ns considered here in have va lues of m = 0.1575,0.1811, 0 .1871, and 0.2177, respect ive ly .

Fig. 5--Sche matic diagram of a typical TAP-NDE system

Fig. 6--Actu al experimental setup with dual FTI

Experimenta l resul ts a re presented as dispersion in te rmsof t ime versus f req uency f or each case . I t can be seen in Fig. 7tha t , as m is increased, wave dispersion t ransi t ion i s f rom apla te wave charac ter is t ic tha t i s of low-frequency content totha t o f a su r face wave rep re sen t ed by h ighe r - f r equency con-tent. Th e t ransi t ion effec t , noted as a band o f f requencies inwhich t he d i spe rs ion i s no l onge r o f pu re p l a te w ave o r su r -face wave charac ter is t ics , i s observed in each stage . I t shouldbe no ted tha t the s l ight inconsisten cy in the t ime o f a rriva l a taround 3 ~ts i s due to the measurement errors for l i f t ing thepropagat ion length.

For the specimen wi th m = 0.1575, the dispersion dem on-st rates a guide d wav e a t t r ibute a t re la t ive low freq uencies a t0 .4 MHz. From Fig. 7(a) , a t ransi t ion cover ing ini t ia l ly af requency spec t rum a t 0 .4 -0 .6 M Hz i s seen t o p rogre ss ive ly

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F ig. 7 - -D i spe rs i on spec t rum fo r f ou r cases o f t h i c kness t o med ium rad ius ra ti o (m = h/R): (a) m = 0.1575, (b) m = 0.1811, (c)m = 0 .1871, and (d ) m = 0 .2177

change to a more prominent surface wave characteristic pat-tern at 0.6-1.0 MHz. There are also other frequency compo-nents that arrive later in time. These are cons idered noise andare not taken into account for the transition effect. The disper-sion associated with the second specimen with m = 0.1811shows a plate wave attribute at relative low frequencies atapproximately 0.62 MHz. A transition covering frequenciesbetween 0.55 and 0.8 MHz is seen to progressively changeto a surface wave characteristic at 0.8-l.75 MHz. Similarobservations can also be made with the third and fourth spec-imens. A summary of results is shown in Table 1. Dispersioncontours of the four cases displayed in Fig. 8 show that thetransition characteristics fall below the 1 MHz range.Conclusions

The experimental investigation was carried out using theTAP-NDE system for the non-contact and non-invasive gen-eration and detection of ultrasonic waves in tubular con-figurations of several different thicknesses to mean radiusratios. By using a Gabor wavelet transform based featureinformation extraction scheme, it was possible to establishwave dispersion with the deployment of only one FTI sen-sor. Features characteristic of both surface and plate waves,thus referred to as the transition waves, were studied usingtime-frequency dispersion curves for pipes o f four differentratios of thickness-to-mean radius. It has been demonstrated

F ig. 8 - - V ie w o f tr ans i t ion ranges f o r f ou r cases

that time-frequency dispersion better represents the transi-tion effect. Although transition effects were observed in allpipe specimens considered in the study, it is concluded thattransition become non-negligibly prominent when the tubularthickness-to-mean radius ratio is close to 0.18.

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T A B L E 1 - - F R E Q U E N C Y CHARACTER IST ICS ( M H z /S p e c i m e n M = h /R P l a t e w a v e T r a n s it io n S u r f a c e w a v e

1 0 . 1 5 7 5 B e l o w 0 . 42 0 . 1 8 1 1 B e l o w 0 . 6 23 0 . 1 8 7 1 B e l o w 0 . 5 54 0 . 2 1 7 7 B e l o w 0 . 6 5

0 . 4 - 0 . 6 0 . 6 - 10 . 6 2 - 0 . 8 0 . 8 - 1 . 7 50 . 5 5 - 0 . 8 0 . 8 - 1 . 9 50 . 6 5 - 0 . 9 0 . 9 a n d a b o v e

Re f e r e n c e s1. Naghdi , P M. and C ooper, R .M., "Propagat ion of Elast ic Waves inCyl indrical Shells , Includ ing the Ef fects of Transverse S hear an d RotaryInertia ," Journa l of the Acoust ical So ciety of America, 28 , 56-63 (1956) .2. Gazis, D.C., "Three-dimensional Investigation o f the Propagation ofWaves in Hollow Circular Cylinders. [. Analytical Foundation. II . Numeri-cal Resul ts ," Journal of the Acoust ica l Society of America, 31(5), 568 -578(1959).3 . Mirsky, L an d Herrmann, G., "N on-axial ly Symm etric Motions ofCyl indrical Shel ls ," Journal o f the Acoust ical S ociety o f America, 29(10),1116-11 23 (1957).4. Greenspon, J.E., "Vibrations of a Thick -walle d Cylindrical Sh ell -Comparison of the Exact Theory wi th Approxim ate Theories," Journal o fthe Acoust ica l Society of America, 32(5), 5 71-5 78 (1960).5. Armenakas, A.E., G azis, D.C., and Herrman, G., Free Vibrations ofCircular Cylindrical Shells, Pergamon Press, New York (1969).6. Goldsmith, W., Lee, P.Y., and Sackm an, J.L., "Pu lse Propagation inStraight Circular Elast ic Tubes," Jour nal o f App l ied Mechanics, 39 , 1 0 1 1 -1018 (1972) .7 . Yin, X .C. and Yue, Z .Q ., " 'Transient Plane-strain Response of Mu l-t ilayered Elast ic Cyl inders To Axisymm etric Impulse," Journal o f Appl ie dMechanics, 69(6), 8 2 5 4 3 5 ( 20 0 2) .8. Ca wley, P, Lowe, M _J.S., Alleyne, D .N., Pavlakovie, B, an d Wilcox, P.,"Pract ical Long Rang e G uided Wave [Ultrasonic] Test ing: Appl icat ion s ToPipes and Rai l ," Materials Evaluat ion, 61(1), 66-7 4 (2003) .9. Rose, J.L., "A Basel ine and Vision of Ul trasonic Guided Wave In-spection Potential," Journ al of Pressure Vessel Technology, 124(3), 273 -28 2(2002).10. Burger, C.P, Dudderar, T.D., Gilbert, J.A., Peters, B.R., Smith, ZA.,and Raj , B . , "Therm al Acousto -opt ic Exci tat ion fo r Non-contact ing NDE,"Proceedings o f the 1986 SEM Spring Conference on Experimental Mecha n-ics, New Orleans, LA, Societyfo r Experimental Mechanics, Bethel, CT, 680-

685 (1986).11. Liu, G . and Qu, J., "'Transient Wave Propagation in a Circu lar An -nulus Subjec ted to Transient Exci tat ion on i ts Ou ter Surface," Jou rnal of theAcoust ical Society of America, 104(3), 1 210-1 220 (1998) .12. Valle, C., Niethammer, M., Q u, J., and Jacobs, L.J., "Crack Charac-

terizat ion Using Guide d Circumferent ial Waves," Journal of the Acoust icalSociety of America, 110(3) , 1 282-12 90 (2001) .13. Zupan, M. and Hemker, K.J., "Application of Fourier Ana lysis to theLase r Based Interferometric Strain~Displacement Gage," EXP ERIME NTALMECHANICS, 42 , 214- 2 20 ( 2002).14. Gao, W , Glorieux, C., and Thoen, J., " Stud y of CircumferentialWaves and The ir Interact ion wi th Defec ts on Cyl indrical Shel ls Using Line-source Laser Ultrasonics," Journa l of Appl ied Physics, 91(9), 6 114 -611 9(2002).15. Niethammer, M, Jabobs, L.J., Q u, J., and Jarzynski, J., "Tim e-f requency R epresen ta tions o f L am b W aves," Journa l o f t he A coas t i ca l Soc i e tyof America, 109(5), 1 841-1 847 (2001) .16. Prosser, WH ., Seale, M. D, and Smith, B .T. , " l~me - frequency Anal-ysis o f the Dispersion of Lamb M odes," Journal of the Acous t ical Society ofAmerica, 105(5), 2669 -267 6 (1999) .17. Wooh, S .C. an d Veroy, K., "Spectrotemporal Analysis of Guided-wave Pulse-echo Signals: Pa rt 1 . Dispersive Systems," EX PERIM ENTA LMECHANICS, 41 , 224-331 (2001) .18. Kim, H. and Melhem, H., "Fourier and W avelet Analyses for Fa-t igue Assessm ent of Concrete Beams," EXP ERIME NTAL MECHANICS, 43 ,131-1 40 (2003).19. Herrmann, G. and Mirsky, I., "Three-dimensional and Shell-theoryA na lys i s o f A x ia ll y Symm et r i c M ot ions o f Cy l inders ," Journa l o f A pp l i edMechanics, 23, 563 -568 (1956) .20. Mirsk~, 1. and Herrmann, G. "A-riaI ty Symm etric Motions of ThickCyl indrical Shel ls ," Journal of Appl ied M echanics, 25 , 97-102 (1958) .21. Baltazar, M.E., Chona, R., Suh, C .S., and Burger, C.P, "Stud y onLaser-generated Ultrasonic Waves on Cylindrical Surfaces," N ove l App li-cat ions of Experimental Metho ds in Mechanics, Proceeding s of the SEMA nnua l Con ference and E xpos i t ion o f E xper imen ta l and A pp l i ed M echan-ic.v, June 2-4 , Charlotte, NC, 1 63 -16 7 (2003).22. Mindl in , R .D., "Influence of Rotatory Inert ia and Shear on Flexu -ral Motions o f l sotropic Elast ic Plates," Journal o f Appl ie d Mechanics, 73 ,31-38 (1951) .23. Baltazar, M.E., Suh, C. S, Chona, R., and Burger, C.P, "Applica-t ions of TAP-NDE Technique to No n-contact Ul trasonic Pipe Inspect ion,"Proceedings of the ASN T 2000 Fal l Conference an d Qual i ty Test ing Show,Nove mber 14-17 , Indianapol is, IN, 161-1 66 (2000).

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