Theoretical Extension and Experimental Verification of a Frequency-Domain Recursive Approach to Ultrasonic Waves in Multilayered Media

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1 ECNDT Post 99 Thotical Extnsion and Expimntal Vification of a Fquncy-Domain Rcusiv Appoach to Ultasonic Wavs in Multilayd Mdia Natalya MANN Quality Assuanc and Rliability Tchnion- Isal Institut of Tchnology Haifa Isal Phinas DICKSTEIN Soq Nucla Rsach Cnt Yavn Isal Abstact. Th fquncy-domain Scott and Godon vitual-intfac modl fo th popagation of ultasonic wavs in multilayd mdia was xtndd analytically and vifid xpimntally. Rcusiv compact quations w dvlopd and intoducd that nabl to calculat th fquncy fatus of ultasonic wavs popagating in compatmnts composd of any abitay numb of lays. Th calculations a caid out though an itativ pocss. Th spcta of xpimntal ultasonic signals obtaind though a Foui Tansfom w compad to th fquncy fatus calculatd by th modl and w found to b in good agmnt.. Intoduction In this wok a modl is dvlopd in th fquncy domain fo th popagation of ultasonic wavs in multilayd compatmnts of an abitay numb of lays. Th motivation to dvlop th modl in th fquncy domain is that in many applications th fquncy spons and fatus a thos of intst and that som mathmatical opations a mo convnint in th fquncy domain. As an xampl consid th convolution intgal in th tim domain which bcoms a simpl multiplication in th fquncy domain. This wok xtnds analytically and xplicitly th Scott and Godon modl in th fquncy domain [] fo any numb of lays. Rcusiv quations w dvlopd and intoducd that nabl to calculat th fquncy fatus of ultasonic wavs popagating in multilayd compatmnts composd of any abitay numb of lays. Th quations a compact and th calculations a caid out though an itativ pocss. To vify and validat th modl numous xpimnts w caid out in which many compatmnts composd of diffnt lays w tstd ultasonically. Th spcta of th xpimntal ultasonic signals obtaind though a Foui Tansfom w compad to th fquncy fatus calculatd by th modl. Th xpimntal sults and th thotical simulations tund out to b in good agmnt. Th modl has povn to accuatly povid dtaild fatus of th ultasonic spcta of layd stuctus.

2 . Th Scott-Godon Appoach in th fquncy domain Unlik th fundamntal analysis caid out by Bkhovskikh () in th tim-domain Scott and Godon conductd thi analysis in th fquncy-domain. Consid th plana gomty of Figu. Lt wavs incidnt fom th lft upon mdium hav tansmission and flction cofficints T and R and lt thos incidnts fom th ight hav cofficints T and R. Th cofficints fo th bounday btwn and 3 a analogously dfind []. Incidnt Puls d MEDIUM ρ MEDIUM ρ MEDIUM 3 ρ 3 3 Fig.. Ultasonic puls incidnt upon mdium of finit thicknss []. A potion of th puls A(t) incidnt fom th lft onto mdium will b tansmittd with amplitud T A(t) and upon mging in mdium 3 will poduc a distubanc of th fom TT 3 At ( d/ ). In addition to th potion of th wav tansmittd into mdium th will b a numb of pulss mging which hav undgon multipl flctions within this mdium. Th total of all such wavs is givn by A () t T T A( t d/ ) T R R T A( t 3 d / ) L N (N ) d ( ) A () t = T T R R A( t ) wh =. Similaly th total wav flctd back into mdium is givn by 3 N = N ( ) 3 R () t R A() t T T R R R A( t Nd/ ). Taking th complx Foui tansfoms of xpssions () and (3) w hav: jωd / / 3( ) 3 ( 3 ) N jωnd A ω = T T R R A 4 jωd / / ( ) 3 ( 3 ) N jωnd R ω = R T T R R A 5 wh A 3 dnots th Foui tansfom of A 3 ( t) and R ( ω) is th Foui tansfom of R ( t ) tc. Th sum appaing in xpssions (.) and (.) can b adily valuatd fo any finit o infinit N as convgnc gomtic sis allowing th dfinition of gnalizd fquncy-dpndnt flction and tansmission cofficints (ω) and t(ω) fo th layd mdium. Fo N infinit w hav th xpssion jωd / A3( ω ) TT3 t ( ω ) jωd / A( ω ) = 6 R R 3

3 R ( ω ) T T R ( ). jωd / 3 ω = R jωd / A( ω ) R3R Claly fo th cas of layd mdia of finit thicknss a singl flction cofficint dos not xist fo th tim-dpndnt psntation of th wav A(t); howv w can dfin opatos Rˆ and Tˆ which whn opating on A(t) gnat th xpssions fo th tim-dpndnt flctd and tansmittd wavs spctivly []. Th appopiat dfinition fo ths opations is { ω ( )} ˆ ( ) ( ) ( ) RA t = F F A t 8 { ω ( )} ˆ ( ) ( ) ( ) TA t = F t F A t 9 wh F and F - a th Foui tansfom and its invs. Th us of this opato notation pmits a lay to b psntd as a bounday o vitual intfac having ths opatos as thi flction and tansmission cofficints... Ultasonic puls incidnt upon layd mdia th vitual lay appoach Consid th layd mdia in Fig.. This is ssntially th sam as Fig..xcpt that th bounday btwn mdia and 3 is now placd by a bounday mdium with tansmission and flction opatos ˆR 3 and T ˆ 3. Rwiting xpssions (.9) and (.0) in th fom of a sis and placing T 3 and R 3 by opatos w hav N ( ) ( ˆ N ) () ˆ d A t = T T R R A( t ) N ( ) ( ˆ N ) () () ˆ d R t R A t T T R R R A( t ). Using basic tansfom pai poptis and assuming appopiat Diichlt pincipl and absolut convgnc poptis it follows that T ˆRA ˆ ( t) = RTA ˆ ˆ ( t) 7 F( R N T N A( t)) = N t N F ( A( t)). Applying ths sults to tak th tansfom of xpssions 0 and w hav N j N d/ T A / A T t R ω = = 3 [ ] ( ) R = R ω T T ω ω R ω 3 ( ) [ ] N j N d/ ( ) ( ) ( ). Summing th sis as bfo w hav jωd/ Tt T = j d/ ω R jωd/ TT 3 R = R /. jωd R 4 5 3

4 Nomal incidnc flction and tansmission cofficints fo sval finit aays (fo mdium of -5 lays) hav bn calculatd using fomulas 45 by W. R. Scott and P. F. Godon in od to gt gaphical fom of dpndncs of th cofficints []. Incidnt puls d BOUNDARY MEDIUM MEDIUM MEDIUM MEDIUM 3 Figu : Ultasonic puls incidnt upon layd mdia []... Dvlopmnt of th Vitual Intfac Modl (VIM) in th fquncy domain... Mathmatical considations and dvlopmnt of th VIM Consid a multilayd compatmnt and th cas of nomal incidnc. Additional mathmatical calculations basd on th quations listd abov lad to th xpssions of tansmission and flction cofficints fo th fist lay of th multilayd mdium whn w count in vs od: n T ntn tn n = 6 n R R n n R R = 7 n n n n n n R nrn Wh ω d i Φ i = 8 i n is a numb of lays of th n-layd mdium whn lays a numatd fom i= to n and is an indx of th wat lay (abov and und th n-layd mdium). Th xpssion of tansmission and flction cofficints fo th scond lay of th multilayd mdium can b wittn as n T ntn n tnn = 9 n R n n n n R n n n nn n n n R n =. 0 Fo th thid lay of th multilayd mdium w can wit th xpssion of tansmission and flction cofficints in th fom n T n tn n tn n = n R nn n VIRTUAL INTERFACE 4

5 R = n n n n n n n nn R n And so on. Thn th xpssion fo th cofficints of th last lay can b obtaind as Tt3 n t n = 3 3 n R R 3 n n =. 4 3 n R Succssiv application of ths fomulas yilds a simpl itativ pocdu fo tabulating tansmission and flction cofficints fo mdia having an abitay numb of lays of vaying thicknss and acoustic paamts... Th cusiv quations of th VIM Consid multilayd mdium with abitay numb of lays n. Now w can asily xpss cusiv quations of th VIM by mans of th fomulation dvlopd abov. Not again that w count th lays backwads i.. in a vs od. Fo a singl lay whn n = k Tk ktk k tk n = 5 k R R k k k k k Rk k Rk k k n k Rk krk k = 6 Wh k=n i.. k= is th indx of th lay and k- is th indx of th wat lay. Fo any numb of lays whn n > k T ktk n tkn = 7 k R k n k k R k k n kn k k n R k = 8 t ( ) Wh k = n is th indx numb of th lay and n ω n a calculatd by mans of Equations 5 and Th algoithm of th VIM Accoding to th cusiv quations 5-8 w built an algoithm fo abitay numb of lays N and pogammd it by mans of Matlab 6. [4]. Fo xampl fo a thlayd compatmnt: Th xpssion of tansmission cofficint fo th scond lay (sinc th fist lay is wat) of th multilayd mdium was wittn as jωd/ TT 3 t = jωd/ 9 R R 3 And th xpssion of flction cofficint 5

6 By using: TTR jωd/ 3 = R jωd/ R3R. 30 R ik = R 3 ki TT RR =. 3 ik ki ik ki Now w can obtain fo th last lay (h w stat calculation fom th last lay i.. fom th thid lay instad of th fist) aft all substitutions as follows 4 TT 4 4 t4 = 33 4 R R 4 4 R R = R4R4 Fo th squnt lay (numb two) with th pio lay as a vitual intfac w can gt th cofficints as: 3 Tt 3 4 t34 = 35 3 R 4 3 R = R3 And finally fo th last lay (numb on) with th two pio lays as a vitual intfac w can find th cofficints as: Tt 34 t34 = 37 R 34 R = R Exampls of simulations of th fquncy spons fom sval multilayd compatmnts a dmonstatd blow: As an xampl a simulation of th fquncy spons fom a compatmnt composd of 3 lays: Aluminum Stl Aluminum is dmonstatd in Figu. 3. 6

7 Fig. 3. Th tansmission and flction cofficint of th 3-layd compatmnt (Aluminium Stl Aluminium). 3. Expimnt Th goal of th xpimnt is to validat th Vitual Intfac Modl xpimntally though ultasound popagation in vaious multilayd mdia composd of vaious typs of lays. Th diffnt compatmnts that w xamind ultasonically and compad to thos calculatd by mans of VIM tchniqu w: Al *Al 3*Al 4*Al St *St 3*St 4*St Al-St Al-St-Al Al-St-Al-St Al-St-Al-St-Al Al-St-Al-St-Al-St-Al Al-St-Al-St-Al- St-Al-St St-Al St-Al-St St-Al-St-Al-St-Al St-Al-St-Al-St-Al-St-Al. Th xpimntal st-up consistd of gula and standad quipmnt including a comput a digitiz an immsion tank a X-Y-Z position bidg a puls-civ and an ultasonic immsion focusd pob of 5MHz. Th Figus blow show Pow Spcta of diffnt compatmnts composd of sval lays of Aluminum Stl and a mixtu of both obtaind fom th xpimnt and fom simulations by mans of th thotical VIM tchniqu. Consid som xampls. A full dsciption of th sults is povidd in [3]. Figu. 4.A Th xpimntal gaph of pow spctum of th compatmnt composd of 4 plats of Stl and Aluminum: St-Al-St-Al 7

8 Figu. 4.B.Th thotical gaph of pow spctum of th compatmnt composd of 4 plats of Stl and Aluminum: St-Al-St-Al 4. Summay and conclusions Th Vitual Intfac Modl (VIM) was dvlopd to modl th intactions of ultasonic plan wavs with laminatd stuctus in th fquncy domain. Using this modl masuabl paamts such as th fquncy-dpndnt tansmission and flction cofficints can b accuatly calculatd fo laminatd compatmnts consisting of lays of lastic monolithic matials. W divd th cusiv quations of th VIM fo a compatmnt composd of an abitay numb of lays (3<N) in th fquncy domain. Succssiv application of ths fomulas yilds a simpl itativ pocdu fo calculating tansmission and flction cofficints fo mdia having an abitay numb of lays of vaying thicknss and acoustic paamts. Figu 5.A: Th xpimntal gaph of pow spctum of th compatmnt composd of 8 plats of Aluminium and Stl: Al-St-Al-St-Al-St-Al-St. 8

9 Figu 5.B.: Th thotical gaph of pow spctum of th compatmnt composd of 8 plats of Aluminium and Stl: Al-St-Al-St-Al-St-Al-St. By mans of th cusiv fomulas w simulatd th fquncy spons fom diffnt multilayd compatmnts and in od to vify and validat th modl w tstd ultasonically numous compatmnts consisting of vaious typs of lays. Th spcta of th xpimntal ultasonic signals obtaind though th Foui Tansfom w compad to th fquncy fatus calculatd by mans of th Vitual Intfac modl and tund out to b in good agmnt. List of fncs [] Scott W. R. Godon P. F. Ultasonic spctum analysis fo nondstuctiv tsting of layd composit matials. Jounal Acoustical Socity of Amica Vol. 6 No. July 977 pp [] Bkhovskikh L.M. Wavs in Layd Mdia. Applid mathmatics and mchanics. Scond Edition Acadmic Pss Inc. London 980 pp. -8. [3] Mann Natalya Wavs in Multilayd Mdia A final Pap towads M.Sc Quality Assuanc and Rliability (P. Dickstin Supviso) Tchnion Isal Institut of Tchnology 00. [4] Matlab Tch-nots No. 70 August 000 pp