Acoustic Prism for Continuous Beam Steering Based on Piezoelectric Metamaterial

Transcription

1 Acoustic Prism for Continuous Bam String Basd on Pizolctric Mtamatrial J. Xu and J. Tang Dpartmnt of Mchanical Enginring Univrsity of Conncticut Storrs, CT 0669, USA Phon: (860) , and ABSTRACT This papr invstigats an acoustic prism for continuous acoustic bam string by a simpl frquncy swp. This ida taks advantags of acoustic wav vlocity shifting in mtamatrials in th vicinity of local rsonanc. W apply this concpt into th pizolctric mtamatrial consisting of host mdium and pizolctric LC shunt. Thortical modling and FEM simulations ar carrid out. It is shown that th phas vlocity of acoustic wav changs dramatically in th vicinity of local rsonanc. Th dirctions of acoustic wav can b adjustd continuously btwn to 16 dgrs by a simpl swp of th xcitation frquncy. Such an lctro-mchanical coupling systm has a fatur of adjusting local rsonanc without altring th mchanical part of th systm. Kywords: Acoustic mtamatrial, acoustic prism, LC shunt circuit, bam string, local rsonanc. 1. BACKGROUND Structural halth monitoring (SHM) aims at utilizing autonomous damag dtction stratgis to monitor a structur in ral tim. A SHM systm consists of snsors and actuators with data acquisition, computation and signal intrprtation moduls. Many typs of actuators and snsors hav bn adoptd including, for xampl, th pizolctric transducrs, fibr composits, magntostrictiv matrials and fibr optics. Among ths, pizolctric transducrs hav rcivd significant attntion du to its compactnss, wid bandwidth, and good linarity within functional rang [1-3]. Th pizolctric transducrs ar commonly usd in th impdanc basd and th guidd wav basd SHM systms. Towards th impdanc basd mthod, lctrical impdanc of th pizolctric transducr bondd to structur is masurd. Du to th two-way lctromchanical coupling ffct, th damag inducd mchanical impdanc changs can b rflctd in th lctrical impdanc shifting [4, 5]. Th impdanc basd mthod has advantags of simpl configuration, low powr consumption, and indpndnt on an analytical modl for implmntation [4-7]. Altrnativly, th guidd wav propagation approach shows bttr fasibility in far fild damag dtction [8-10]. This is bcaus th guidd Lamb wavs hav th advantag of wav propagation ovr long distancs with littl loss of amplitud. Consquntly, it is not rquird to plac th actuators or snsors in th vicinity of damags. By xtnding th guidd wav mthod into - or 3-dimnsional spac, th phasd array tchniqus aris and hav bn subjct to wid xploration. In th phasd array mthod, th surfac bondd actuators focus th wav nrgy on localizd dirctions or aras by controlld xcitation tim dlay of ach array lmnt [11-15]. Th bam forming dirction rorintation can b achivd by altring th xcitation tim squncs. Nvrthlss, this mthod nds complicat and prcis control coordination. Du to th dmand of low cost acoustic bam string, a frquncy-basd bam string mthod has bn proposd in a rcnt study by utilizing priodic array of pizolctric actuators [16]. Pizolctric actuators ar arrangd in rctangular array with distanc of on wavlngth btwn ach othr. Th proposd array can gnrats strong, frquncy dpndnt dirctional baming, and thrfor allows bam string through a simpl swp of th xcitation frquncy. On th othr hand, only limitd wav propagation dirctions can b obtaind du to constrain of th gomtry configuration. Ral-tim gomtrical adjustmnt of th frquncy-basd bam string dvic might b difficult.. RESEARCH OVERVIEW It has bn known in optics that light can b disprsd through a prism. Mor spcifically, light bams with diffrnc frquncis hav diffrnt disprsion angls whn travlling through a prism. Inspird by th working

2 principl of optical prism, th hypothsis of this rsarch is that an acoustic prism may hav similar fatur, that is, string acoustic wav continuously by shifting th frquncy of an acoustic wav. Thr ar mainly two challngs to build such an acoustic prism. On challng hr is that w nd significant vlocity chang of acoustic wav. This is bcaus that th rfraction angl of a wav is dpndnt of th diffrnc of phas vlocitis of two mdiums. Th othr challng is that w nd phas vlocity chang within a small frquncy rang for bttr controllability in application. Acoustic mtamatrial shows good fasibility to build such a prim. Th mtamatrial, dfind as artificial structurs that xhibit physical proprtis not availabl in natural matrial, has xtraordinary capability in low-frquncy sound/vibration attnuation, ngativ rfraction, and acoustic/lastic lnss [17-4]. Not that phas vlocity of an acoustic wav in mtamatrial undrgos significantly shifting in th vicinity of local rsonanc [5, 6]. Hr w propos an acoustic prism for bam string. By utilizing th phas vlocity shifting, th dirction of acoustic wav can b adjustd continuously by a simpl swp of th xcitation frquncy. W apply this concpt into th mtamatrial with pizolctric shunt circuit. Our rational hr is that vlocity of acoustic wav would shift dramatically in th vicinity of th LC rsonanc which furthr yilds th bam string ffct. It is worth mntioning that such an lctro-mchanical coupling systm has a uniqu fatur of adjusting local rsonanc without altring th mchanical part of th systm. Th full intgration of adaptiv matrials, lctronics, computing rsourcs, and powr systms with passiv mtamatrials can form a hybrid activ mtamatrial systm whos matrial proprtis can b digitally and rmotly controlld. Th rst of this papr is organizd as follows. Sction 3 dscribs th concpt of th acoustic prism and prsnts th modling and analysis of dirctional guidd wav xcitation in thin plats. Sction 4 prsnts simulatd validations of th concpt, and Sction 5 summarizs th main rsults of th study and provids rcommndation for futur invstigations. 3. PIEZOELECTRIC METAMATERIAL BASED ACOUSTIC PRISM In this sction, w formulat analytical invstigation on th pizolctric mtamatrial basd prism. dscrib th basic ida, followd by a gnral mathmatical modl of th intgratd systm with illustration. W first 3.1 Concpt of mtamatrial basd prism Th acoustic prism taks advantags of wav rfraction. Rfraction is th bnding of a wav whn it ntrs a mdium whr its spd is diffrnt. Th rfraction phnomnon xists widly in optics, lctrics and acoustic. For xampl, th rfraction of light whn it passs from a fast mdium to a slow mdium bnds th light ray toward th normal to th boundary btwn th two mdia (Figur 1). Figur 1. Illustration of rfraction. Th rfractions of wav normally hav th following rlation: 1 1 sin1 sin (1) c ( ) c ( ) p1 p whr c ( ) and c ( ) ar th phas vlocitis of th wav in th plat and th prism, rspctivly; and p1 p ar th angl of incidnc and rfraction, rspctivly. Th rlation of th in-angl and th rfraction angl can b rwrit as: 1

3 Th phas vlocity chosn. cp1 ( ) cp( ) arcsin sin1 () cp 1( ) in th plat and th incidnc angl 1 ar constant whn th paramtrs of th systm ar On th othr hand, th phas vlocity in th prism cp ( ) can b modifid subjctd to th frquncydpndnt matrial charactristic. For xampl, th phas vlocitis of th light bam shifts for th light with diffrnt colors. This phnomnon yilds an intrsting optical lmnt calld prism, which can disprs th light bams in diffrnt dirctions (Figur a). Not that th phas vlocity of acoustic wav can b asily modifid in a mtamatrial du to local rsonanc ffct. Hr w propos a dsign of acoustic prism which can disprs th acoustic wav. Th disprsion ffct ultimatly yilds acoustic bam string continuously. Figur. Optical prism; prototyp of th acoustic prism. Th configuration of th mtamatrial basd prism is shown in Figur b. Th prototyp consists of unit clls arrangd in triangl array to form an acoustic prism. Th prism is bondd to a host plat mad of aluminum sht. Th unit cll consists of th host structur and a pizolctric transducr. An inductiv shunt circuit is connctd to th pizolctric transducr to crat th local rsonanc. A stand-alon pizolctric transducr is placd nar th acoustic prism as an acoustic wav sourc. Du to th local rsonanc, th triangl prism ara may hav significant acoustic vlocity chang in th vicinity of th local rsonanc. In such a cas, th acoustic wav travlling through th prism may hav diffrnc rfraction angl. Thrfor, th dirction of th acoustic wav can b asily changd by frquncy swp. Morovr, such an lctro-mchanical coupling systm has a uniqu fatur of on-lin tunability, that is, adjusting local rsonanc without altring th mchanical part of th systm. 3. Mathmatical modl In this sction, an lctro-mchanical modling is prsntd to illustrat th rsponss of th pizolctric mtamatrial. For simplicity and without loss of gnrality, hr w adopt th two-dimnsional mass-spring lattic, as shown in Figur 3a. This modl consists of priodic microstructurs with host mdium of mass M. W rfr th x and y dirctions as th principal dirctions. Th stiffnss of th host mdium is givn by th xtnsional spring K and th shar spring G. W assum that a pizolctric transducr is bondd to th host mdium. An inductor is connctd to th transducr (Figur 3b). Thrfor, this lattic modl can b usd to rprsnt a composit matrial with distributd pizolctric shunts. Figur 3. D lattic systm; rprsntativ unit cll.

4 Considr a lattic point at location ( xi, yi ) with rfrnc to th principal coordinats x and y. Th displacmnt in x dirction of th ith host mdium M, which is assumd to b rigid, is dnotd as is dnotd as u yi. Th quations of motion for this unit cll/microstructur ar Mu ( K G) u k Q K( u u ) G( u u ) xi xi 1 xi1 xi1 yi1 yi1 Mu ( K G) u k Q K( u u ) G( u u ) yi yi 1 yi1 yi1 xi1 xi1 LQ k Q k ( u u ) 0 1 xi yi u xi ; th displacmnt in y dirction whr is th lctrical charg on th surfac of th pizolctric transducr, is th invrs of th capacitanc of th pizolctric transducr, R rprsnts th rsistanc valu of th shunt circuit and L indicats th inductor in th shunt circuit. Lt l rprsnts th lngth of th unit cll. Th solution for a plan harmonic wav in an infinit lattic systm is givn Q W substitut Equation (4) into (3). u u xi yi i( tkxxkyy) x0 u i( tkxxkyy) y0 i( t) Q0 u Q For a nontrivial solution for wav amplituds k u x0 and u y0 (3) (4), trms associatd with tim must vanish. This lads to th disprsion quation, which can b solvd for th wav frquncy for givn valus of dimnsionlss wav numbrs and kl. Th disprsion quation for a unit cll with LC shunt circuit is givn as kl x y M K(1 cos k ) cos (1 cos ) cos yl G G kyl M K kxl G G kxl L k k1 0 M K(1 cos k l) G M K(1 cos k l) G 4G cos k l cos k l y x x y All propagating mods for lattic modls can b capturd by rstricting th dimnsionlss wavnumbr to th first Brillouin zon du to priodicity. Thus, w plot all th disprsion curvs basd on wavnumbrs within this zon. In th following thortical analysis, w choos th paramtrs as shown in Tabl 1. TABLE I. Paramtrs. Mass (kg/m) 0.08 Spring constant (GN/m) 1 Shar spring (GN/m) 8 Lattic spac (m) Capacitanc of transducr (nf).5 Inductor (H) 0.1 Rsonanc of shunt circuit (khz) 10 (5) Disprsion curvs for th unit cll with and without LC shunt circuit ar shown in Figurs 4a and 4b, rspctivly, in th first Brillouin zon. Without of gnrality, hr w slct th rsonant frquncy of th LC shunt circuit to b 10 khz and th systm lvl lctro-mchanical coupling cofficint is chosn to b Th systm lvl lctromchanical coupling cofficint is dfind k1 k (6) Kk Th systm lvl lctro-mchanical coupling cofficint k indicats th convrsion fficincy btwn lctrical and acoustic nrgy in pizolctric matrial. Th lctro-mchanical coupling cofficint is dirctly rlatd to th transducr matrial proprty, prdominantly th pizolctric coupling constant at th matrial-lvl. For xampl, on of th most commonly usd pizolctric transducrs, PZT5H, has limitd pizolctric coupling constant of 0.44 in th 31 dirction and 0.75 in th 33 dirction, rspctivly. Th dvic lvl lctro-mchanical coupling cofficint is

5 rlatd to not only th matrial proprty of th transducr, but also th spcific dsign faturs of th harvstr. b modifid by choosing diffrnt pizolctric matrial or structural lvl optimization [7-30]. It can Figur 4. Disprsion curv without shunt circuit; with LC shunt circuit. First, w analyz th influnc of LC shunt towards th disprsion curv of th unit cll, as shown in Figurs 4a and 4b, i.., th disprsion curvs of th unit cll without and with LC shunts. It can b obsrvd that th LC shunt circuit inducs a bandgap around th LC rsonant frquncy. As a rsult, th phas vlocity of acoustic wav changs in th vicinity of this bandgap. Th phas vlocity of acoustic wav can b obtaind through th disprsion curv, whr C p ( ) (6) k x Figur 5. Phas vlocity without shunt circuit; with LC shunt circuit. W may also obtain rspctivly th phas vlocity of acoustic wav within unit cll without and with LC shunt circuit, as shown in Figurs 5a and 5b. It can b obsrvd that whil th phas vlocity with LC shunt circuit gnrally follows th tndncy of th on without shunts, it shifts significantly in th vicinity of th LC rsonanc. It indicats that th acoustic wav, whn passs from th plat ara into th mtamatrial ara, may dramatically chang its dirction du to th shifting of wav spd. This phnomnon furthr yilds th frquncy dpndnt bam string. Morovr, w obtain continuous phas vlocity shifting during th abovmntiond frquncy rang. It indicats that th angl of th acoustic wav can b tund continuously. This capability has an advantag ovr th priodic array of pizolctric actuators [16] du to its continuous adjustability and compact siz. It is also worth mntioning that, th local rsonanc du to LC shunt circuit is dpndnt upon th paramtrs of th capacitanc and th inductor. On can asily chang th frquncy of th LC rsonanc by simply modifying th valu of th inductor, i.., without altring th mchanical part of th prism. In th nxt stp, w analyz th systm charactristics subjctd to diffrnt lctro-mchanical coupling cofficints. Figurs 6a to 6d show th disprsion curvs of th unit cll with diffrnt systm lvl lctromchanical coupling cofficints. It can b obsrvd that LC shunts inducs a bandgap for all of th four cass. Morovr, incrasing th coupling cofficint can ffctivly incras th width bandgaps. For xampl, th unit cll

6 with k has bandgap width of khz. On th othr hand, whn th lctro-mchanical coupling of th unit cll is incrasd to, th bandgap width is incrasd to 70 khz. This is bcaus that incrasing lctromchanical coupling cofficint may incras th portion of acoustic nrgy that convrtd to lctrical nrgy. Subsquntly th LC rsonanc of th shunt circuit may gnrat a largr racting forc that applid to th pizolctric transducrs that prvnt wav propagation within th unit cll. Thrfor th bandgaps can b xpandd through incrasing th systm lvl lctro-mchanical coupling cofficint. k 0.1 (c) Figur 6. Disprsion curv k ; k 0.01 ; (c) k 0.05 ; (d) k 0.1. (d) Figurs 7a to 7d show th phas vlocity of acoustic wav within th unit cll with diffrnt lctro-mchanical coupling cofficint. It can b obsrvd that, th phas vlocity has significant changs in th vicinity of local rsonanc. Th bandgap of th phas vlocity xpands with th incrasd lctro-mchanical coupling cofficint. On th othr hand, th lctro-mchanical coupling cofficint may rduc th phas vlocity of acoustic wav in low frquncy rang. It indicats that w may incras th rfraction angl by incras th lctro-mchanical coupling cofficint. Manwhil, th phas vlocity changs littl du to th coupling cofficint in high frquncy rang.

7 (c) Figur 7. Phas vlocity k ; k 0.01 ; (c) k 0.05 ; (d) k 0.1. (d) 4. FEM SIMULATION AND DISCUSSION Finit lmnt mthod (FEM) simulations ar carrid out to vrify th proposd concpts. Th configuration of th unit cll is shown in Figur 8a. Th unit cll consists of aluminum substrat (siz mm 3 ) and a pizolctric transducr (siz mm 3 ) bondd on its surfac. An inductor is connctd to th top and bottom surfacs of th 3 pizolctric transducr as th shunt circuit. Th paramtrs for th matrials ar: 7500 kg/m, 3 b 700 kg/m, ar d 31 = 30pC/N E b = 6 GPa, and E p = 106 GPa. Th matrial constants of th pizolctric transducr, PZT-5H, 8 8 h N/C and Vm/C. p Figur 8. Configuration of th unit cll; Configuration of th mtamatrial basd prism. Th unit clls ar arrangd into a triangl-shapd array to form th acoustic prism, as shown in Figur 8b. Th lngths of th two sids of th triangl prism ar 160 and 80 mm, rspctivly. A stand-alon pizolctric transducr

8 is placd in th vicinity of th prism as th wav sourc. Th substrat plat is mad of aluminum, and th thicknss of th plat is 10 mm. ANSYS 14.5 is usd for th FEM simulation. In th simulation, th capacitanc of th pizolctric transducr is usd as.5 nf. Without of gnrality, w choos th rsonant frquncy of th LC shunt circuit to b 10k Hz. Thrfor, th inductor in th circuit is chosn to b H. Simulations ar prformd undr xcitation of diffrnt frquncis around th local rsonanc of th shunt circuit. Th lctro-mchanical coupling cofficint is hingd upon th matrial paramtrs and systm configuration,.g., th thicknss ratio and siz of th pizolctric transducr. On th othr hand, activ circuit can also incras th systm lvl lctro-mchanical coupling cofficint. For xampl, th ngativ capacitanc circuit (Figur 9) shows xtraordinary ability in th improvmnt of th coupling cofficint [31]. Th bandwidth of th tuning rang is xpandd by incrasing th lctro-mchanical coupling cofficint. This is bcaus that incrasd lctro-mchanical cofficint of th pizolctric matrial incrass th systm lvl lctro-mchanical coupling cofficint, which yilds th rduction of th anti-rsonanc frquncy. An improvd coupling cofficint may induc xpandd tuning frquncy rang. Th rsulting bnfit is that th dirction of th acoustic bam can b turnd with bttr rsolution. k Figur 9. Unit cll with inductor & ngativ capacitanc shunt circuit. In th nxt stp, FEM simulations ar carrid out that th ngativ capacitanc circuit has compnsatd 90% of th stiffnss of th pizolctric transducr. In othr words, th lctro-mchanical coupling cofficint of th pizolctric transducr is incrasd by 10 tims for nhancd bam string ffct. Figur 10 shows rprsntativ xampls of th wav string ffct Figur 10. Illustration of bam string ffct. It can b obsrvd from Figur 10 that whn th frquncy of th wav incrasd from 1050 Hz to Hz, th dirction of th wav is bndd to th lft. Morovr, whn th frquncy is furthr incrasd, th angl of rfraction is dcrasd. This is bcaus that in th vicinity of th local rsonanc, as aformntiond, w hav significant changs of th phas vlocity. Th phas vlocity shifting ultimatly yilds th chang of th wav travlling dirction. Th rlation of th string angl and frquncy of th acoustic wav is shown in Figur 11 which shows th string angl vrsus th frquncy of th wav. Th rsults ar obtaind by FEM simulation in stady-stat condition. It can b obtaind that whn th frquncy of th wav incrass from 10.5 khz to 11 khz, th string angl of th wav changs from 6 to 16 dgr. This is bcaus th prism with LC shunt circuit has dcras th phas vlocity of th wav dpndnt on th frquncy. Thrfor, frquncy dpndnt wav string ffct is achivd. Morovr, du to th local rsonanc, th LC shunt circuit rducs th phas vlocity of th wav to ngativ and crat a bandgap zon btwn to 1.3 khz. In this zon th wav from th sourc has bn rflctd back. In th

9 rang btwn 1.3 and 13 khz, th string angl dcras from 1 dgr to 1.8 dgr. Th tndncy of th string angl matchs th thortical prdiction in gnral. Figur 11. String angl vrsus frquncy of th wav. 5. CONCLUDING REMARKS Our rsarch finds that th proposd mtamatrial basd prism offrs th capability of string th acoustic wav du to th local rsonanc from th LC shunt circuit. Hr w hav illustratd th bnfits of th capability of acoustic bam string by choosing th rsonanc of th LC shunt circuit to b 10 khz. Th angl of th wav can b tund btwn to 16 dgrs in our simulation. It is worth noticing that th rsonant frquncy of th shunt circuit can b modifid by changing th valu of th inductor without mchanical tailoring. For xampl, w may utiliz a simulatd inductor with on-lin controllability. Th concpt proposd hr simplifis th configuration and control stratgy of wav string. It can b applid in structural halth monitoring and ultra-sonic wav gnrators. 6. ACKNOWLEDGMENT This rsarch is supportd in part by NSF undr grant CPS REFERENCES [1] Wilki, W. K. Robrt, G., Bryant, Jams W. High, Robrt L. Fox, Richard F. Hllbaum, Anthony Jalink, Jr., Bruc D. Littl, Paul H. Mirick, 000, Low-cost pizocomposit actuator for structural control applications, Proc. SPIE 3991, Smart Structurs and Matrials, 33. [] Giurgiutiu, V., 005, Tund Lamb wav xcitation and dtction with pizolctric wafr activ snsors for structural halth monitoring, Journal of intllignt matrial systms and structurs 16, [3] Giurgiutiu, V., Zagrai, A., and Bao, J., 004, Damag idntification in aging aircraft structurs with pizolctric wafr activ snsors, Journal of Intllignt Matrial Systms and Structurs, 15, [4] Park, G., Sohn, H., Farrar, C. R., Inman, D. J., 003, Ovrviw of pizolctric impdanc-basd halth monitoring and path forward, Th Shock and Vibration Digst, 35, [5] Grisso, B. L., Inman, D. J., 006, Impdanc-basd structural halth monitoring of thrmal protction systms, Nondstructiv Evaulation for Halth Monitoring and Diagnostics, 6176,

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