Static and Dynamic Analysis of Bistable Piezoelectric- Composite Plates for Energy Harvesting

Transcription

1 5rd AIAA/ASME/ASCE/AHS/ASC Structurs, Structural Dnamics and Matrials Confrnc<BR>th AI - 6 April, Honolulu, Hawaii AIAA -49 Static and Dnamic Analsis of Bistabl Pizolctric- Composit Plats for Enrg Harvsting David N. Btts, H. Alicia Kim, and Christophr R. Bown Dpartmnt of Mchanical Enginring, Univrsit of Bath, Bath, BA 7AY, Unitd Kingdom and Danil J. Inman 4 Dpartmnt of Arospac Enginring, Univrsit of Michigan, Ann Arbor, MI, 489, USA Downloadd b UNIVERSITY OF MICHIGAN on April, DOI:.54/6.-49 This papr prsnts an arrangmnt of bistabl composit plats with bondd pizolctric patchs to prform broadband vibration-basd nrg harvsting from ambint mchanical vibrations. Ths bistabl nonlinar dvics hav th potntial to hibit improvd powr gnration compard to convntional rsonant sstms and can b dsignd to occup smallr volums than bistabl magntic cantilvr sstms. In this papr w initiall prsnt th rsults of an optimization stud to gnrat gratr lctrical powr b discovring th corrct gomtric configuration for nrg harvsting basd on th static stats of th dvic. Th rsults considr th optimal choic of dvic aspct ratio, laminat thicknss, laminat stacking squnc, and pizolctric surfac ara. Incrasd lctrical output is found for gomtris and pizolctric configurations which hav not bn considrd prviousl. This stud is thn tndd to includ dnamic considrations of both th static shaps and th snap-through transition. Optimum dsigns ar shown to b snsitiv to th vibration pattrn that is bing harvstd. Th optimum gomtric configurations basd on th static analsis alon ar not optimal undr all dnamic conditions. I. Introduction NERGY harvsting which convrts ambint mchanical vibrations into lctrical nrg is an ara of E considrabl rsarch intrst and has rcivd tnsiv attntion in th past dcad. A varit of mthods hav bn considrd including lctrostatic gnration, lctromagntic induction and th pizolctric ffct 4. Pria 5 dmonstratd that pizolctrics hav a numbr of advantags, including as of intgration within a sstm, highr strain nrg dnsitis compard to lctrostatic and lctromagntic sstms, and th simplicit of convrting strain nrg to lctrical nrg. In man cass harvsting dvics hav bn dsignd to oprat nar rsonanc to optimiz th powr gnration, for ampl simpl linar cantilvr bam configurations. Howvr, th ambint vibrations which ar intndd to b harvstd gnrall hibit multipl tim-dpndnt frquncis which can includ componnts at rlativl low frquncis. This can mak tpical linar sstms infficint or unsuitabl; particularl if th rsonant frqunc of th dvic is much highr than th frqunc rang of th vibrations it is attmpting to harvst 6. In ordr to improv th fficinc of vibrational nrg harvstrs, rcnt work has focusd on ploiting nonlinarit for broadband nrg harvsting. Encouraging rsults 7-8 hav bn obtaind using nonlinar or bistabl cantilvrd bams. Stanton t al. 8 modld and primntall validatd a nonlinar nrg harvstr using a pizolctric cantilvr. An nd magnt was placd on an oscillating cantilvr which thn intracts with oppositl pold, stationar magnts which inducs softning or hardning into th sstm and allows th rsonanc frqunc to b tund. This tchniqu was shown to outprform linar sstms whn citd b varing frquncis. Howvr, such a sstm would rquir an obtrusiv arrangmnt of trnal magnts and could gnrat unwantd lctromagntic filds. An altrnativ mthod 4 has rcntl bn prsntd whr a pizolctric lmnt is attachd to th surfac of an asmmtric and bistabl laminat plat, ploiting th larg structural Rsarch Officr, Dpartmnt of Mchanical Enginring, D.N.Btts@bath.ac.uk. Snior Lcturr, Dpartmnt of Mchanical Enginring, H.A.Kim@bath.ac.uk. Profssor, Dpartmnt of Mchanical Enginring, C.R.Bown@bath.ac.uk. 4 Profssor, Dpartmnt of Arospac Enginring, daninman@umich.du. Amrican Institut of Aronautics and Astronautics Copright b th Amrican Institut of Aronautics and Astronautics, Inc. All rights rsrvd.

2 Downloadd b UNIVERSITY OF MICHIGAN on April, DOI:.54/6.-49 chang which is nabld b th inhrnt bistabilit. Such harvsting structurs hav bn shown to hibit high lvls of powr traction ovr a wid rang of frquncis. This arrangmnt can b dsignd to occup a smallr spac and is potntiall mor convnint and portabl for broadband nrg harvsting. Bistabl laminats hav bn tnsivl studid for th dvlopmnt of morphing or adaptiv structur concpts 9-. Whn a composit laminat has an asmmtric stacking squnc th rsulting mismatch in thrmal proprtis btwn plis lads to a thrmall inducd strain. This lads to th laminat dvloping a curvd dformation as it is coold from its high tmpratur cur ccl to room tmpratur. Undr crtain gomtric conditions th thrmal strain can lad to th dvlopmnt of two stabl quilibrium stats. Such structurs ar of intrst for shap-chang applications sinc th 'snap-through' btwn stabl stats can rsult in a larg dflction but dos not rquir continuous nrg input to maintain th stabl shap. Figur shows an ampl of this bhavior for a squar [ n /9 n ] T laminat. For a low ratio of dg lngth to thicknss onl a saddl shapd singl stabl stat is obsrvd, point A, with - and -curvaturs of qual magnitud in opposit out-of-plan dirctions. As th ratio incrass th solution bifurcats, point B. Bond this point two approimatl clindrical stabl stats ar obsrvd, points C and D, whil th saddl stat bcoms unstabl (dashd lin). For morphing applications pizolctric matrials can b usd to induc 'snap-through' btwn points C and D. For harvsting applications, if a flibl pizolctric matrial is attachd to th laminat surfac and th structur is rpatdl actuatd btwn stabl stats C and D as a rsult of a mchanical vibration, th larg shap chang has th potntial to gnrat lctrical nrg b th dirct pizolctric ffct. Figur. Stabl (solid lin) and unstabl (dashd) shaps of a [ n /9 n ] T laminat with variation in gomtr. Btts t al. rcntl prsntd a stud for optimization of th dsign of bistabl laminats for morphing applications, nabld b th drivation of an analtical solution to isting numrical modling. Through variation of pl orintations and using a nonuniform laminat gomtr, laminats wr optimizd for diffring stiffnss charactristics, constraind b bistabilit and minimum dflction constraints. This work was tndd 4 to includ pizolctric lars into th isting analtical modl in a configuration similar to that considrd primntall b Arrita t al. 4. Most rcntl this optimization mthodolog 4 has bn adaptd for th dsign of bistabl pizolctric-laminat nrg harvstrs basd on th static shaps of th dvic onl 5. This papr will bgin b prsnting a summar of th corrct gomtric configurations of bistabl laminat nrg harvstrs basd purl on static considrations. Th variabls considrd ar th stacking squnc of th composit laminat, pl thicknsss, aspct ratio and pizolctric configurations. Ths rsults ar furthr discussd with a sris of dsign paramtr studis to provid a bttr undrstanding of th optimum rsults. Modling mthods ar thn tndd to includ dnamic considrations of th vibrational input and th tim dpndnc of th laminat during snap-through transitions. With considration of th amplitud of availabl actuation forc, optimal gomtric configurations ar shown to b snsitiv to th vibration pattrn which is bing harvstd. Amrican Institut of Aronautics and Astronautics

3 II. Dvic Configuration An isting primntal stud prsntd b Arrita t al. 4 suggstd a novl mchanism for broadband nrg harvsting of ambint mchanical vibrations, using composit laminats with an asmmtric laup. Whn subjctd to mchanical vibrations ths bistabl laminats can hibit larg amplitud oscillations. Arrita t al. 4 hav prviousl dmonstratd that this altrnating strss citation can induc high lvls of lctrical nrg in bondd pizolctric lars across a wid rang of frquncis. Figur shows th primntal arrangmnt considrd 4, using a [ /9 ] T laminat of.m squar dg-lngth with four bondd pizolctric lars for nrg harvsting. Th bistabl pizolctric-laminat combination was attachd to a shakr and sismic masss attachd to th cornrs of th laminat to facilitat snap-through btwn stabl stats. Downloadd b UNIVERSITY OF MICHIGAN on April, DOI:.54/6.-49 Figur. Eprimntal arrangmnt for actuation of a mm bistabl plat mountd from its cntr to a mchanical shakr. Adaptd from Arrita t al. 4. Whil this tp of dvic was shown to produc larg avrag powr from intrmittnt, limit ccl and chaotic oscillations, th dsign of th dvic was not optimizd to maimiz nrg outputs. In this work w look to ploit th man variabls availabl in th dsign of such a pizolctric-laminat structur to improv harvsting prformanc. An ampl arrangmnt considrd in this work is shown in Fig., with a similar pattrn of four pizolctric lars positiond on th top laminat surfac and mirrord on th opposit surfac b a furthr four pizolctric lars positiond orthogonall to thos on th uppr surfac. Figur. Actuation arrangmnt for a [ P / /9 /9 P ] T laminat with 4% pizolctric covrag. Suprscript P dnots pizolctric lmnt with º or 9º poling dirction. To approimat th ffct of ambint mchanical vibrations th dvic is modld as fid at th four cornrs (zro z-displacmnt) with a mchanical forc applid in th z-dirction at th gomtric cntr. Th dsign variabls which hav bn considrd in this stud ar th stacking squnc of th cross-smmtric laminat, th planform dimnsions of th rctangular dvic, pl thicknsss, and total pizolctric ara. Amrican Institut of Aronautics and Astronautics

4 Amrican Institut of Aronautics and Astronautics 4 III. Static Analsis In this sction th optimal gomtric configurations of th pizolctric-laminat nrg harvstrs ar considrd basd soll on th static shaps of th sstm. Pizolctric lmnts ar assumd to bhav as paralll plat capacitors, constraind b a maimum strain to avoid matrial failur, but with no considration for amplitud or frqunc of th sourc of th mchanical actuation. A. Modl for Static Laminat Shaps Th analtical modl usd to calculat th unloadd shaps of asmmtric laminats of arbitrar laup was first introducd b Dano and Hr and is a nonlinar tnsion to classical laminatd plat thor. Th co-ordinat sstm usd is that dfind in Fig., whr th gomtric cntr of th laminat sits at th origin and th pl orintations ar masurd from th -ais. Th out-of-plan displacmnt in th z-dirction, w is assumd to b of th following form. ( ) c b a w () Th midplan strains, including gomtrical nonlinarit according to th von Karman hpothsis, ar dfind as w w v u w v w u () whr u and v ar th in-plan displacmnts in th - and -dirctions, rspctivl. Th midplan strains ar approimatd b third ordr polnomials. Dano and Hr found that trms with powrs of and that sum to an odd numbr ar alwas zro. Thrfor th th midplan strains can b rducd to th polnomials of Eq. () () Using Eqs. (-) and introducing th additional shap cofficints 9- rsulting from intgration of th midplan strains, prssions for th in-plan displacmnts u and v can b dtrmind. ( ) ( ) , 8, b c bc v a c ac u (4) Th total strain nrg of th laminat, W lam can thn b prssd as th intgral of strain nrg dnsit ovr th volum of th laminat. / / / / / / ˆ L L L L H H ij ij kl ij ijkl lam dddz T c W α (5) whr c ijkl s ar lastic constants, ˆα ij s ar constants rlating to th thrmal pansion cofficints, L and L ar th planform sid lngths of th laminat, H is th total laminat thicknss, T is th tmpratur chang from th cur tmpratur and ij s and kl s ar th total strains dfind as, Downloadd b UNIVERSITY OF MICHIGAN on April, DOI:.54/6.-49

5 za zb zc (6) whr z is th out-of-plan distanc from th laminat midplan and a, b, and c ar out-of-plan shap cofficints dfind b Eq. (). To includ th additional stiffnss of th pizolctric lars an additional strain nrg trm W p is addd, dpndnt on th gomtr and matrial proprtis of th individual lars. Th pizolctric lmnts ar tpicall bondd to th laminat at room tmpratur aftr th cur ccl and to account for this th tmpratur chang is st to zro. Th total strain nrg of th pizolctric-laminat dvic W is thn th sum of th laminat and pizolctric componnts, W W lam W p (7) Downloadd b UNIVERSITY OF MICHIGAN on April, DOI:.54/6.-49 Epansion of Eq. (7) rsults in an prssion for th total strain nrg which is a function of th matrial and gomtric proprtis, th tmpratur chang from cur and th st of shap cofficints a, b, c,. For quilibrium, th minimum nrg stats rquir: W f i ; i... 4 (8) k i whr th k i s ar th st of 4 shap cofficints. In this form it is notd that throughout th 4 quations th shap cofficints - appar indpndntl and as linar trms onl. Howvr th quations ar not linar with rspct to th out-of-plan cofficints a, b and c. Th sstm can thrfor b rducd to just thr quations and th thr unknown curvaturs a, b and c. This procss rsults in a sstm dfining th quilibrium positions of th following form, which can b solvd fficintl using numrical solution mthods. W ( a, b, c) W ( a, b, c) W ( a, b, c) a b c B. Objctiv Function Th objctiv of this stud is to maimiz th lctrical nrg gnratd b th bistabl pizolctric-laminat structur during snap-through from ach stabl stat. At this stag w modl this basd purl on th static stabl stats of th sstm. Th objctiv function is thus th lctrical nrg, U inducd in th pizolctric matrial undr an applid strss which rsults in a transition btwn th two stabl stats dtrmind in th prvious sction. Whn oprating off-rsonanc a pizolctric lar bhavs as a paralll plat capacitor. Hnc th lctrical nrg gnratd b snap-through is givn b, U CV () (9) whr C is th capacitanc of th pizolctric lmnt and V is th opn circuit output voltag, as dfind b th following quations, Q dijσa C () V V V Etg ij σt p () whr Q is charg, d ij is th ffctiv pizolctric strain constant (charg pr unit forc), g ij is th ffctiv pizolctric voltag constant (lctric fild pr unit strss), σ is th strss, A is th surfac ara of th lar, and t p is th thicknss of th lar. Substituting Eqs. () and () into () givs, 5 Amrican Institut of Aronautics and Astronautics

6 U ( d g ) ( At ) ij ij σ () whr th trms hav bn split into th fid matrial proprtis of th pizolctric matrial (d ij and g ij ), th strss in th matrial which is a function of th laminat curvatur, and th matrial gomtr (A and t p ). In this analsis it is assumd that th pizolcitrc lmnt is polarizd along fibr ais of th laminat (as in Fig ). This is achivd using a Macro Fibr Composit pizolctric dvic with an intr-digitatd lctrod pattrn to align th lctric fild along th fibr ais. Whn attachd to th laminat surfac th pizo-lmnts ar straind in both th poling dirction ( dirction, Fig. ) and transvrs dirction ( dirction) du to th anticlastic curvaturs 9. Th strss varis across th volum of th pizolctric-lmnts as a function of th thrmall inducd strsss. Th lctrical nrg for pizolctric-lmnts positiond at (top surfac) and 9 (bottom surfac) is thrfor, p Downloadd b UNIVERSITY OF MICHIGAN on April, DOI:.54/6.-49 U 4 m ( dg dgσ ) dv ( dgσ dgσ ) v σ dv (4) v whr th factor 4 accounts for all pizolctric-lmnts on on surfac, m dfins th associatd shap, and v and v ar th volums of two lars on opposit laminat surfacs. Th associatd matrial proprtis and strss dirctions for ach componnt ar mor clarl illustratd in Fig. 4. It is intrsting to not that whil an incras in pizolctric-lmnt siz incrass th ara from which lctrical nrg can b harvstd, thir additional stiffnss rducs laminat curvatur and thus th rsulting strss σ. Othr potntial configuations of th pizolctric lmnt includ polarization in th through-thicknss dirction (z-ais in Fig. 4) so that th d cofficint is rsponsibl for harvsting in both and dirction; this is not considrd in this papr for brvit. Figur 4. Longitudinal and transvrs componnts of th total lctrical nrg, U. Not: subscript dnots th associatd dirction, suprscript dnots th top (t) and bottom (b) pizolctric lar. C. Rsults For th optimization stud th matrial proprtis d ij and g ij ar considrd to b fid as thir optimum valus hav bn dtrmind b Pria 4. Th surfac ara of th laminat is fid (.4m ) as rportd b Arrita t al. 4. A lowr bound on singl pl thicknss t is st as.5mm, consistnt with tpical minimum pl thicknsss. t p is fid for practical rasons. All othr variabls ar unboundd. Th optimum solutions ar subjct to constraints to guarant bistabilit 4 and limiting pizolctric strain to blow its failur strain 6 (~µstrain). Th optimization problm is solvd using a squntial quadratic programming mthod with multipl starting points uniforml distributd throughout th dsign spac to captur all local optima. Two optima ar found for this problm, shown in Tabl, with laups of [ P //9/9 P ] T (global optimum) and [ P /9//9 P ] T (local optimum) whr th global optimum outprforms th local solution b ~65%. 6 Amrican Institut of Aronautics and Astronautics

7 Tabl : Global and local optimum solutions. Local optimum solution Global optimum solution Stacking squnc [ P /9//9 P ] T [ P //9/9 P ] T Singl pl thicknss, mm Pizolctric ara, % Aspct ratio.. Maimum strain, µstrain 97 Actuation forc, N.9.4 Elctrical nrg, mj.4.7 Downloadd b UNIVERSITY OF MICHIGAN on April, DOI:.54/6.-49 For both solutions in Tabl, th major and minor curvaturs ar alignd with th polarization dirction of th pizolctric to utilis th d and d pizolctric ffct, and th ar intuitivl th optima. Th local solution is lss optimal du to th rducd curvatur (rlativ to th global solution) associatd with this stacking squnc. What ar lss obvious ar th gomtric configurations of pizolctric ara and dvic thicknss. Th optimum pl thicknsss ar.66mm and.69mm for th global and local solutions rspctivl, and ar not on an constraint boundar. Convnintl, this global optimum is almost actl 5 plis of.5mm maning th solution could b rdfind as [ P / 5 /9 5 /9 P ] T. Significant diffrncs in th pizolctric aras for th global (7.4%) and local (4.4%) solutions ar obsrvd. Ths rsults ar furthr invstigatd in following dsign paramtr studis. Laminat Stacking Squnc Th first stud considrs th chang in objctiv function with laminat stacking squnc. Figur 5a shows th lctrical nrg gnratd for diffring valus of θ for laminats of [ P /θ/θ9/9 P ] T laup,.m squar dg lngth and.5mm pl thicknss. Optima for ach θ valu ar markd b black squars, whil th global and local optima ar markd b a whit circl and squar rspctivl. Diffrnt pizolctric orintations ar not considrd as onl th angls rlativ to pl orintations ar of intrst, but is somthing that warrants furthr invstigation. Th maimum lctrical nrg is obsrvd whn θ is, corrsponding to th global solution to th optimization problm (whit circl). As θ incrass from to 45 th major laminat curvatur rducs and th alignmnt of th pizolctric polarization dirction with laminat curvatur bcoms lss optimal, lading to a dcras in th lctrical nrg. For θ valus from 45 to ~6 th major curvatur continus to dcras but th pizolctric alignmnt improvs, th nt ffct bing a mor gradual dcras in nrg gnratd. At ~6 this pattrn switchs and th improvd alignmnt of th pizolctric with curvatur bcoms dominant compard to th rducd curvatur and th lctrical nrg incrass until th local solution at θ 9 (whit squar). Figur 5. (a) Variation in lctrical nrg with θ (shown on ach lin) and pizolctric surfac ara for [ P /θ/θ9/9 P ] T laminats, and (b) variation in lctrical nrg with n.5mm plis (shown on ach lin) and pizolctric ara for [ P / n /9 n /9 P ] T laminats. Black squars mark optima for ach θ or n, whit circls mark global optima, and whit squars mark local optima. 7 Amrican Institut of Aronautics and Astronautics

8 Downloadd b UNIVERSITY OF MICHIGAN on April, DOI:.54/6.-49 Dvic Thicknss Th scond stud considrs th ffct of pl thicknss on th problm. Figur 5b shows th objctiv function valus for diffring numbr of plis (n from to 6) for th full rang of pizolctric aras. All rsults corrspond to [ P / n /9 n /9 P ] T laminats of.m squar dg lngth. For ach diffrnt numbr of plis th optimum solution with rspct to pizolctric ara is highlightd b a black squar, with th global optimum markd b a whit circl. A consistnt pattrn of rsults is obsrvd for pl numbrs from to 5. For zro pizolctric ara th lctrical nrg gnratd is obviousl zro. Th lctrical nrg thn incrass with additional pizolctric ara to an optimum valu. This optimum ara is largr for ach additional numbr of plis up to 7.4% for 5 plis. Howvr, bond 5 plis this pattrn changs with both th optimum ara and th maimum lctrical nrg dcrasing for thickr laminats. This is a rsult of approaching th bifurcation point (btwn 6 and 7 plis dpnding on th pizolctric surfac ara), rsulting in dcrasing curvatur until th bistabl bhavior is lost (s point B in Fig. ). This pattrn confirms th finding that th bistabilit constraint is inactiv at th global optimum. This rsult suggsts that thickr laminats with lowr curvaturs ar mor optimal, which is not an intuitiv rsult. For th thickr dsign th pizolctric lars ar positiond furthr from th midplan and thus princ highr strain pr unit curvatur. Howvr, in this analsis th sourc and magnitud of th actuation forc has not bn considrd. Clarl, ths thickr laminat dsigns will hav highr stiffnss and rquir largr vibrational forc input to induc snap-through btwn th stabl stats. Th optimalit of ths dsigns is thrfor dpndnt on th sourc of vibration to b harvstd and will now b invstigatd. IV. Dnamic Analsis Th majorit of isting litratur for bistabl laminat analsis considrs th snap-through vnt to b an instantanous on, with th dnamics of th transition btwn th stabl stats rarl considrd. For an application such as nrg harvsting whr th dvic prformanc is dpndnt on its rspons to ambint mchanical vibrations, a bttr undrstanding of th dnamic transition is vital. In this sction th analtical modl of Sction IIIA is tndd using th mthod outlind b Diaconu t al. 7, providing a fast and robust mthod of valuating th tim-dpndnt rspsons of th bistabl pizolctric-laminat harvsting dvic. A. Modl for Dnamic Snap-through This sction prsnts an ovrviw of th dnamic tnsion to th analtical modl prsntd b Diaconu t al. 7. Th static stabilit of th bistabl structur subjctd to an trnall applid forc is initiall considrd. This simpl tnsion to th nrg minimization problm of Eq. (9) is prformd b solving th following sstm of thr nonlinar quations, T K ( ) F( ) (5) whr [ a, b, c], K is th vctor containing th thr unloadd quilibrium quations of Eq. (9), rprsnting a stiffnss function, T W W W K(),, (6) a b c and F is th forc function of a, b and c, giving th work don b th applid forc, whr f is th magnitud of th applid forc. If f is considrd to b constant, solution of Eq. (5) with rspct to th curvaturs a, b and c will rsult in th static shaps of th laminat whn subjctd to th static forc. In this cas w considr th forc f to b varing with tim, approimating th altrnating mchanical vibrations which ar to b capturd b th dvic. For all studis prsntd in this sction, f is assumd to b sinusoidal with a pak amplitud of f ma. Th dnamic quations of motion ar thn of th form, 8 Amrican Institut of Aronautics and Astronautics T L F L, f, (7)

9 M & D( & ) K( ) F(, t) (8) whr M is th mass matri as drivd b Diaconu t al. 7. Hr this includs th componnts rlating to onl th laminat, M lam, Downloadd b UNIVERSITY OF MICHIGAN on April, DOI:.54/ L H L LL H L ρhl L L L H L L H L M lam (9) L L 8 whr ρ is th laminat matrial dnsit, and th componnt rlating to th pizolctric lars M p with th corrsponding proprtis for th pizolctric matrial. Th mass matri M of th pizolctric-laminat combination is simpl th sum of M lam and M p. D is th damping trm with a mass damping cofficint η. D( ) & η M& () In this work th stiffnss damping is omittd as th snap-through vnt occurs in th low frqunc rang. Equation (8) can thn b solvd using Matlab s built in ODE solvr ODE5S, using ithr on of th two static shaps as an initial start point. B. Rsults As an ampl of th dnamic rspons of a bistabl pizolctric-laminat prdictd b th modl outlind in th prvious sction, th global optimum dsign outlind in Sction IIIC is initiall considrd. Th harvsting dvic has a laup of [ P //9/9 P ] T, with a singl pl thicknss of.66mm, a squar dg lngth of.m, and is citd b an altrnating forc with a maimum amplitud of 8N (s Fig. 6a). Th rsulting dformation of th pizolctric-laminat configuration ovr th first four sconds is shown in Fig. 6b, masurd as th displacmnt of on cornr of th dvic from th flat stat. Th dashd rd lins indicat th two stabl shaps as prdictd b th static modling of Sction IIIA, and th solid blu lin is th motion of th cornr displacmnt of th dvic with altrnating mchanical citation. Figur 6. (a) Sinusoidal forc citation with a maimum amplitud of 8N, and (b) static stabl shaps (dashd rd lins) and dnamic actuation path (solid blu lins). 9 Amrican Institut of Aronautics and Astronautics

10 Downloadd b UNIVERSITY OF MICHIGAN on April, DOI:.54/6.-49 Th laminat is initiall in th first stabl stat and has a ngativ cornr displacmnt. Th application of a positiv forc graduall rducs th laminat curvatur and at approimatl.7scs (s point A in Fig. 6b) a mor markd chang in th gradint of th displacmnt-tim curv is obsrvd. This corrsponds to a transition to th scond stabl stat with positiv cornr displacmnt. It is of intrst to not that th laminat ovrshoots th static stabl position (dashd rd lin), inducing a small amount of additional curvatur and thrfor additional strss in th pizolctric matrial. As th dirction of applid forc switchs th laminat rpats th sam ccl, transitioning from th scond stat to th first stat at approimatl.5scs (point B in Fig 6b), again ovrshooting th prdictd stabl position. This ccl thn rpats uniforml with th sinusoidal forc input. W now invstigat th optimalit of th fiv pl [ P / 5 /9 5 /9 P ] T global solution dtrmind in Sction III, with th additional considration of th altrnating forc citation. Figur 7 shows a masur of th harvstabl nrg (plottd on a logarithmic scal) for a rang of diffrnt forc amplituds from.n to N. Thr distinct rgions (I III, markd in Fig. 7) ar notd; an inst plot with a linar scal is shown to highlight th magnitud of th chang in nrg gnratd in th thr rgions. Initall, in Rgion I, th nrg harvstd for low forc amplituds (<6N) is found to b small but incrass with additional forc. This rgion can b invstigatd in mor dtail b considring a tpical actuation ccl with a maimum forc amplitud of 4N (point A). Figur 8a shows a phas plot for this actuation ccl, plotting th applid forc amplitud against th cornr displacmnt of th laminat. Th rd dots indicat th two stabl positions, whil th blu lin tracks th laminat displacmnt undr th cclic load. It is found that th nrg harvsting dvic is oscillating about th first stabl position at vr low displacmnts, with no snap-through vnt to th scond stat. Figur 8b confirms this bhavior, illustrating th dnamic path of on cornr of th laminat takn during th first si sconds of citation. Figur 7. Harvstabl nrg lvls (plottd on a logarithmic scal) for varing forc amplituds for a 5-pl laminat dsign corrsponding to th optimal solutions of Sction III, and a -pl dsign for comparison of actuation forcs. Inst plot with a linar scal to highlight th magnitud chang in nrg gnratd. In Rgion II btwn approimatl 6 and 7N in Fig. 7 thr is a sharp incras in th harvstd nrg. Again, b slcting a point in this rgion (6.8N at point B) w can invstigat th dnamic bhavior of this dvic. Figur 8c shows th associatd phas plots. Whil th displacmnt of th dvic dos not rach th scond stabl stat, th cornr displacmnt dos crossovr into th positiv rgion, marking a transition into th scond stat rgion. Figur 8d dmonstrats that this rgion rprsnts a finl balancd stat of th sstm whr snap-through has occurrd but th dirction chang of th applid forc at snap-through immdiatl forcs th pizolctric-laminat back towards th first stat without vr raching th scond static stabl position. Howvr, this ampl dmonstrats that, compard to Rgion I, highr lvls of nrg ma b harvstd at forc lvls which ma not b sufficintl larg to induc a full stat chang. At highr forc lvls gratr than 7N (Rgion III) th lctrical nrg gnratd with forc continus to incras but at a rducing rat (s Fig 7). If w considr point C at 8N it can b obsrvd that this rgion marks th point at which th dvic princs a complt snap-through ccl ncompassing both static stabl stats. Figur 8 shows th phas plot for this actuation ccl and confirms that th dvic is oscillating btwn th two stabl Amrican Institut of Aronautics and Astronautics

11 Downloadd b UNIVERSITY OF MICHIGAN on April, DOI:.54/6.-49 stats, as indicatd b th rd dots. Figur 8f shows th dnamic path takn indicating a small ovrshoot bond th static shaps and this bhavior plains wh th nrg harvstd continus to incras as th forc is incrasd furthr. If th altrnating forc applid to th harvstr continus to incras it will vntuall rsult in straining of th pizolctric matrial bond th prviousl prdictd stats. Th maimum strain constraint in th pizolctric matrial, prviousl found to b inactiv, is thrfor likl to b an important considration for optimization studis basd on th dnamics of th sstm; spciall sinc th failur strain of th pizolctric matrial is much smallr that th carbon-fibr rinforcd laminat 8. Figur 8. Dnamic rspons of a [ P / 5 /9 5 /9 P ] T laminat with an osciallating forc with maimum amplitud of (a-b) 4N, (c-d) 6.8N, and (-f) 8N. Figurs a, c, and show th oscillating forc displacmnt bhavior around th two stabl stats (rd dots). Figurs b, d, and f show th tim varing pattrn of cornr displacmnt whr stabl positions ar dnotd b rd dashd lins. Amrican Institut of Aronautics and Astronautics

12 For comparison, a scond laminat dsign is also includd in Fig. 7 to dmonstrat an important considration whn using th optimal dsigns dtrmind basd purl on th statics of th sstm (Sction III). Th rd lin indicats th sam nrg gnraton plot for varing forc amplituds for a thinnr laminat dsign; [ P / /9 /9 P ] T. With rfrnc to Fig. 5b this rprsnts th optimal pl dsign with.% pizolctric ara and th sam.m squar dg lngth. W obsrv that whn th dnamic rspons of th sstm is takn into account th thickr dsign is confirmd to b mor optimal than th thinnr dsign, producing mor harvstabl nrg in th low forc rgion whr snap-through dos not occur (Rgion I), and th high forc rgion whr bistabilit is full ploitd (Rgion III). Howvr, th transition btwn ths rgions (Rgion II) occurs at a lowr forc for th thin laminat, rsulting in th thinnr laminat gnrating mor lctrical nrg in this rlativl narrow forc rang. In practical applications thr is likl to b a constraint on th vibrational forc amplitud and th high lctrical nrgis of th thickr dsign ma not b achivd. Constraints basd on th ambint mchanical vibrations to b harvstd ar thrfor an important considration for optimization of th pizolctric-laminat combination basd on th dnamics of th sstm. Downloadd b UNIVERSITY OF MICHIGAN on April, DOI:.54/6.-49 V. Conclusion Th optimization stud prsntd in this papr has invstigatd th maimization of th lctrical nrg harvstd of a bistabl pizolctric-laminat dvic du to th altrnating strss citation inducd b rpatd mchanical actuation. Constraints hav bn imposd to nsur bistabilit and to limit th strain in th pizolctric matrial to blow its failur strain. Th rsults obtaind in this stud can aid in th dtrmination of th optimum gomtric configurations for nrg harvsting and aid in th undrstanding of th undrling mchanism for high lctrical nrg gnration in bistabl pizolctric-laminat basd dvics. Through variation in pl orintations, laminat gomtr and pizolctric ara it was found that cross-pl laminats [ P //9/9 P ] T offr th largst nrg outputs sinc th laminat curvaturs ar maimizd and alignd with th pizolctric polarization ais. A local solution is also found [ P /9//9 P ] T with optimal pizolctric alignmnt but rducd laminat curvatur. Th optimum pl thicknsss ar found not to b on a constraint boundar, with thickr laminats that hibit a lowr curvatur bing mor optimal than thinnr high dflction dsigns. This pattrn rvrss as th gomtr approachs th loss of bistabilit (s Fig. ). Similarl, incrasd pizolctric siz with incrasd laminat thicknss is found to b optimal, with th pattrn rvrsing nar th loss of bistabilit. Whn considring th dnamic transition btwn stabl stats as th pizolctric-laminat combination is posd to an oscillating mchanical forc, th optimum dsigns ar found to b snsitiv to th magnitud of th applid forc. Whil th thickr laminats ar confirmd to produc highr lvls of nrg whn snap-through is full inducd, th forc rang in which snap-through occurs is mor limitd than for th lss optimal, thinnr dsigns. Furthrmor, th strain in th pizolctric matrial rachs highr lvls than prdictd basd soll on th static stats, suggsting th pizolctric matrial failur constraint ma indd b activ. In futur optimization studis whr th problm is constraind b dnamic considrations, tuning of th dvic for a spcific vibration lvl is thrfor ssntial. Acknowldgmnts W acknowldg th support of Enginring and Phsical Scincs Rsarch Council (grant numbr EP/J489/) for partiall funding this work. Rfrncs [] Anton, S. R., and Sodano, H. A., A rviw of powr harvsting using pizolctric matrials (-6), Smart Matrials and Structurs, Vol. 6, No., 7, R-R. [] Mitchson, P. D., Miao, P., Stark, B. H., Yatman, E. M., Holms, A. S., and Grn, T. C., MEMS lctrostatic micropowr gnrator for low frqunc opration, Snsors and Actuators A: Phsical, Vol. 5, No. -, 4, pp [] Glnn-Jons, P., Tudor, M. J., Bb, S. P., and Whit, N. M., An lctromagntic, vibration-powrd gnrator for intllignt snsor sstms, Snsors and Actuators A: Phsical, Vol., No. -, 4, pp [4] Arrita, A. F., Hagdorn, P., Erturk, A., and Inman, D. J., A pizolctric bistabl plat for nonlinar broadband nrg harvsting, Applid Phsics Lttrs, Vol. 97, No.,, pp. -. [5] Pria, S., Advancs in nrg harvsting using low profil pizolctric transducrs, Journal of Elctrocramics, Vol. 9, No., 7, pp [6] Erutrk, A., and Inman, D. J., An primntall validatd bimorph cantilvr modl for pizolctric nrg harvsting from bas citations, Smart Matrials and Structurs, Vol. 8, No., 9, pp. -8. Amrican Institut of Aronautics and Astronautics

13 Downloadd b UNIVERSITY OF MICHIGAN on April, DOI:.54/6.-49 [7] Erturk, A., Hoffman, J., and Inman, D. J., A pizomagntolastic structur for broadband vibration nrg harvsting, Applid Phsics Lttrs, Vol. 94, No. 5, 9, pp. -. [8] Stanton, S. C., McGh, C. C., and Mann, B. P., Nonlinar dnamics for broadband nrg harvsting: Invstigation of a bistabl pizolctric inrtial gnrator, Phsica D: Nonlinar Phnomna, Vol. 9, No.,, pp [9] Hr, M. W., Calculations of th room-tmpratur shaps of unsmmtric laminats, Journal of Composit Matrials, Vol. 5, (Jul) 98, pp [] Dano, M. -L., and Hr, M. W., Thrmall inducd dformation bhavior of unsmmtric laminats, Intrnational Journal of Solids and Structurs, Vol. 5, No. 7, 998, pp. -. [] Tawfik, S. A., Dancila, D. S., and Armanios, E., Planform ffcts upon th bistabl rspons of cross-pl composit shlls, Composits: Part A, Vol. 4, No. 7,, pp [] Gud, M., Hufnbach, W., and Kirvl, C., Pizolctricall drivn morphing structurs basd on bistabl unsmmtric laminats, Composit Structurs, Vol. 9, No.,, pp [] Btts, D. N., Kim, H. A. and Bown, C. R., Optimization of stiffnss charactristics for th dsign of bistabl laminats, AIAA Journal, (Accptd). [4] Btts, D. N., Kim, H. A. and Bown, C. R., Modling and optimization of bistabl composit laminats for pizolctric actuation, Journal of Intllignt Matrial Sstms and Structurs, Vol., No. 8,, pp [5] Pria, S., Critrion for matrial slction in dsign of bulk pizolctric nrg harvstrs, IEEE Transactions on Ultrasonics, Frrolctrics and Frqunc Control, Vol. 57, No.,, pp [6] Guillon, O., Thibaud, F., and Prru, D., Tnsil fractur of soft and hard PZT, Intrnational Journal of Fractur, Vol. 7, No.,, pp [7] Diaconu, C. G., Wavr, P. M., and Arrita, A. F., Dnamic analsis of bi-stabl composit plats, Journal of Sound and Vibration, Vol., No. 4-5, 9, pp [8] Bown, C. R., Dnt, A. C., Nlson, L. J., Stvns, R., Cain, M. G., and Stwart, M., Failur and volum fraction dpndnt mchanical proprtis of composit snsors and actuators, Procdings of th IMchE, Part C: Journal of Mchanical Enginring Scinc, Vol., No., 6, pp Amrican Institut of Aronautics and Astronautics