Control of Beam s Snapback Responses. Using Multiple Flexoelectric Actuators

Transcription

1 Cotrol of Bem s Spbc Resposes Usg Multple Flexoelectrc Actutors Bole Deg 1, Huyu L 1, d Horse Tzou 1 StrucTroc Systems d Cotrol Lb School of Aeroutcs d Astroutcs, Zhejg Uversty, Hgzhou, Zhejg, Ch Abstrct Flexoelectrc mterl uder cert homogeeous electrc feld c be utlzed to cotrol the vbrto of flexble structure bsed o the coverse flexoelectrc effect. The tomc force mcroscope (AFM) probe plced o the top of flexoelectrc ptch d the electrode uder t wth cert voltge ppled betwee them re used to geerte the homogeeous electrc feld. The ctuto membre force d bedg momet duced by flexoelectrc ptch tur fluece the vbrto of shells. I ths study, elstc ctlever bem excted by tl sttc mechcl force loctg t the tp ed s cosdered. As the mechcl force wthdrws the spbc respose ppers where the ctlever bem vbrtes wth ll mode prtcptg d ther mpltudes decyg expoetlly. Multple ctutors, cosst of AFM probes d flexoelectrc ptches, re used here to ccelerte the decrese of spbc respose. O ech probe voltges of multple frequeces re ppled to obt the cotrol effects o multple modes. The ctutor s postos mmzg the tp dsplcemet of the bem s selected s the optml oes. Two wy of determg the cotrol voltge,.e., ope loop d close loop, s troduced d the formto of postos of the ctutors re emphszed both methods. I cse studes, the free dmpg vbrto of the bem wthout cotrol s evluted d the frst mode domtes the vbrto. The ope loop method, wth costt mpltude of cotrol voltge, does t wor well reltvely log tme scle. Whle the close loop method, wth ts cotrol voltge relted to the tp dsplcemet detected by sesor, turs out to be more stble d effectve. The optml postos for the ctutos of both methods re clustered er the fxed ed, where the ctuto fctor reches ts mxml vlue. At lst, lmtto o voltge mpltude s mplemeted to vod the stress cocetrto, uder whch the cotrol effect does ot wee much. Ths study serves s troducto to flexoelectrc mterl & multple AFM probe bsed vbrto cotrol d theoretcl foudto for further reserches. Keywords: AFM probe, ope loop, close loop, ctuto fctor 1. INTRODUCTION Ctlever bem model s mog the most frequetly used structures egeerg pplctos,.e., wgs, tes, robot rms, etc. Studes o dymcs of ctlever bems hve bee thoroughly coducted usg modl expso method [1]. The sp-bc respose hs bee cosdered s mportt chrcterstc [-4] of the ctlever bem structure d served s bsc testg method [5].

2 Pezoelectrc mterls, developg rpdly the lst decdes, hve bee used o sesors, ctutors d ctve cotrol of flexble bem models [6-11]. Flexoelectrc mterls, bsed o the coverse flexoelectrc effect [1], re ewly developed mterls whch c be utlzed, s substtuto to pezoelectrc mterls, to cheve precso ctuto d ctve vbrto cotrol. Ths effect tht the stress sde the flexoelectrc mterl c be duced by cert homogeeous electrc feld hs bee studed both theoretclly [1] d expermetlly [14,15]. My reserches usg the drect flexoelectrc effect to buld flexoelectrc sesors hs bee crred out o vrous structures,.e., ctlever bems [16,17], rgs [18,19], cyldrcl shells [0] s well s the effect comprso wth pezoelectrc oes [1]. Actuto studes hve bee doe bsed o the homogeeous feld geerted by tomc force mcroscope (AFM) probes [] d symmetrc electrodes []. Comprg wth sgle-ctutor (cosst of AFM probe d flexoelectrc ptch ths study) cotrol method, multple-ctutor cotrol gves more rrgemet possblty d thus more effcet. The effcecy of multple-ctutor cotrol method hs bee verfed by prevous pezoelectrcty-bsed multple-ctutor cotrol method [4,5] d dstrbuted ctutg systems [6,7]. However smlr reserches o flexoelectrcty-bsed method hve ot bee reported. I ths study, sp-bc s ppled to test the vldty of the multple flexoelectrc ctutors cotrol effect. The dymc respose of sp-bc ctlever bem s lyzed frst, followed by the cotrol theores o flexoelectrc cotrol usg multple ctutors. Two ds of cotrol method, ope loop d close loop, re preseted respectvely d ther cotrol effect re evluted. Lmtto o voltge mpltude s ppled close loop cse to vod stress cocetrto er the cotct pot (wth AFM probe). Ths study troduces pplcto for AFM probe & flexoelectrc mterl bsed ctutors.. THE BEAM MODEL OF FLEXOELECTRIC CONTROL A ctlever bem model, used to demostrte the cotrol effect of the sp-bc vbrto, s show Fg.1. Multple probes (geerlly probes) re ppled here, wth correspodg flexoelectrc ptches, to geerte the cotrol forces. Accordg to the coverse flexoelectrc effect, the flexoelectrc ptches eed to be ctuted by homogeeous electrc feld d such electrc feld s duced by AFM probes. I other word, the ctutors used here cosst of AFM probes d flexoelectrc ptches. The coordtes x d z deotes the logtudl d trsverse drectos; L s the legth, b s the wdth d h s the thcess of the elstc ctlever bem. Geerlly, the cotrol effect of ctutors, s demostrted Fg.1, s cosdered. For ech ctutor, the geometrcl sze of AFM probe d flexoelectrc ptch s ufed s probe rdus R, ptches thcess h d legth L. The posto of ech ctutor, whch plys mportt role evlutg the cotrol effect, s deoted by x, where the subscrpt dctes the -th ctutor. z ( ) x 1 (Not to scle) 1 st AFM probe y d AFM probe R th AFM probe x Flexoelectrc ptch x L x L Elstc bem b h h Fgure 1. The ctlever bem model of flexoelectrc cotrol.

3 Note tht t s ssumed tht the thcess of the flexoelectrc ptch h s much greter th the AFM probe rdus R whle much smller th the thcess of ctlever bem h,.e., R h h.. THE FREE VIBRATION OF SNAP-BACK CANTILEVER BEAM The free dmpg vbrto of the sp-bc ctlever bem wthout the cotrol of ctutors s lyzed frst. A mechcl force F s plced o the tp ed of the ctlever bem d cert sttc deformto s formed. It s ssumed tht the mechcl force s removed sttly t the tme t=0. The ctlever bem wll oscllte due to the tl codto of sttc deformto. Accordg to modl expso method, the trsverse dsplcemet c be expressed s the summto of ll the modes tht prtcptes the vbrto multpled the correspodg modl prtcpto fctors, s [1], ( ), (1) u x t t U x 1 where s mode umber, s the -th modl prtcpto fctor d U s the ctlever bem s -th mode shpe fucto whch c be wrtte s cosh L cosl U ( x) C cosh x cos x sh x s x sh L s L, where L s the root of the equto L L 1cosh cos 0 d the frst four roots re 1.875, 4.694, 7.855, respectvely d C s rbtrry costt. Utlzg the orthogolty of the mode shpe fuctos of ctlever bem, the modl prtcpto fctor equto c be wrtte s [1] () ˆ F t, () where s the -th turl frequecy; represets the modl dmpg rto of the rg, whch depeds o the equvlet dmpg costt c d the turl frequecy,.e., =c/(h ); Fˆ t s the modl force for the -th mode whch s bset from the lyss of the free dmpg vbrto of the ctlever bem whe t > 0. Usg Lplce trsformto, the soluto of the modl prtcpto fctor equto of subcrtcl cse c be expressed s s d t t t e 0cosdt 0 0 U x d 1 t ˆ t F e s d d, 0 t d where d s the dmped frequecy, whch s defed s d 1 0 re the tl codtos of the modl prtcpt fctors. Note tht the fluece of the modl force does ot ; (0) d exst s t > 0 for the vlue of tegrto the secod term s zero. Such result s resoble sce the vbrto s free dmpg wthout y mechcl force. (0) c be solved from the sttc modl prtcpto fctor fucto whch c be expressed s ˆ 0 F, where F ˆ 0 s the modl force duced by the mechcl force F t t=0, whch c be expressed, usg the orthogolty of the mode shpe fuctos, s (4)

4 1 L F U L F = F x LU xdx (5) ˆ 0. bhn 0 hbn Substtutg the expresso of mode force t t=0 (Eq.(6)) to the sttc modl prtcpto fctor fucto (Eq.(5)), the tl codto of modl prtcpt fctors c be expressed s 0 Fˆ 0 FU L bhn (6). Substtutg Eq.(7) to the soluto of modl prtcpto fctor equto,.e., Eq.(4) d otg tht 0 0 d Fˆ t 0 whe t>0, the dsplcemet of the bem c be wrtte, by the modl expso expresso (Eq.(1)), s t FU L U x u x, t e cos d t s d t 1 bhn 1 j dt e FU L U x e, t 1 bhn 1 (7) As show dsplcemet expresso of the ctlever bem,.e., Eq.(7), the tl codto of sttc force trggers every mode prtcptg the vbrto d ech mode decys expoetlly wth dfferet speed (geerlly hgher mode decys fster). 4. VIBRATION CONTROL BY MULTIPLE ACTUATORS As the prtcpto fctor of hgher mode c be smll eough to eglect, oly tl m modes re cosdered. To obt better cotrol effect o ll m modes, for every probe m hrmoc voltges wth dfferet frequeces re ppled. The expresso of the voltge o probe c be wrtte s m ˆ j p e p1 ptp. Applyg the ctuto voltge betwee the electrode of the -th flexoelectrc ptch d -th AFM probe, the trsverse electrc feld betwee them c be expressed ccordg to Abpllp s pproxmte [8,9] s E x x, z z R R h /h z x x R h /h z where E z deotes the trsverse electrc feld duced by the AFM probe. Note tht the postos of the ctutors re emphszed by regrdg them s depedet vrbles. The grdet of the electrc feld, s predcted by coverse flexoelectrc effect, c duces the stress o the flexoelectrc mterl, prtculrly the orml stress the logtudl drecto c be expressed s [], (8) where Ez Txx x x, z 1 z, (9) T deotes the logtudl orml stress duced by the -th ctutor, where the superscrpt xx

5 dctes the ctutor duced compoet. The membre force d cotrol momet c be clculted ccordg to the expresso of the duced stress s h h xx xx h,, M xx x x Nxx x x N x x T x x z dz h h, (10) where N xx deotes the membre force d M xx deotes the membre force duced by the -th ctutor. The dsplcemet of the ctlever bem duced by ctutors c be obted by ddg the effect of ech ctutor s [7] u ( x) 1 1 U x Fˆ x j( ) ( ) e, (11) where F ˆ s the -th modl force duced by the -th ctutor. The expresso of the ctutor duced modl force c be wrtte s [7] L 1 x M xx x x ˆ ˆ F = F = U, L x dx 1 1 hn x x (1) where N s L U 0 dx ; The modl ctuto fctor of -th probe d -th mode s defed ˆ A x F /, whch dctes the modl force geerted by the ctutor wth ut ctuto voltge. Accordg to the defto of the modl ctuto fctor d the expresso of the ppled cotrol voltge, the -th modl force duced by the -th ctutor c be wrtte s m ˆ j ˆ ptp F x A x A x pe. Note tht the modl ctuto fctor A x demostrtes the formto of the -th ctutor s posto d the ppled voltge cots the mgtude, frequecy d phse dfferece formto. By troducg the cocept of modl ctuto fctor, the fluece of the posto of the ctutor d the ctuto chrcterstcs of the ctutor tself re seprted, whch gves cler pcture of how the desg prmeters fluets the cotrol effect of the vbrto of ctlever bem. Substtutg Eq.(1) to the ctutors duced dsplcemet expresso,.e., Eq.(11) yelds u ( x) m 1 1 p1 ˆ p1 j( ptp p ) ( ) pe U x A x p p Note tht Eq.(14), deotes the mode umber; deotes the -th ctutor d p deotes the p-th hrmoc voltge ppled o the -th ctutor. To cotrol the sp-bc free dmpg vbrto of the. (1) (14)

6 ctlever bem, bsc method s to ccel out the dsplcemet of the vbrto by the dsplcemet duced by the ctutors d thus ccelerte the dmpg process. To me the method prctcble, the frequeces of the duced vbrto must be cocdece wth the orgl free dmpg vbrto. The p-th frequecy o the -th probe s settg to be the sme s the correspodg free dmpg frequecy for ech ctutor s, p d p. The ctutors duced dsplcemet expresso, cosderg oly frst m modes, c be expressed s u ( x) m m 1 1 p1 ˆ j U ( x) A x e p dpt p p dp dp where p s the phse gle lggg behd the p-th hrmoc exctto (voltge) of the -th ctutor o -th mode. Note tht ths phse gle deducted from p to p becuse t s rrelevt to the ctutor umber for the ctuto frequeces o ech ctutor re set to the sme. More th settg the ctutor frequeces hrmoc wth the free dmpg oes, the phse dfferece betwee them, f possble, should be zero to eble the mxml cotrol effect. Tht s p p for ech, d p whle oly the phse gle of ctuto voltge p s djustble. However, stsfyg the zero phse dfferece codto for ech, d p s mpossble, d lso, uecessry. The p crcumstce stds for the vbrto of -th mode duced by the p-th dmped frequecy. Comprg wth the vbrto of p= codto, whch s erly resoce, the p compoets re eglgble. Thus, oly the p= terms s esured to be of o phse dfferece. The phse gle for the -th ctutor c be determed s p p pp. The dsplcemet wth the cotrol of multple AFM probes c be obted by the superposto of free dmpg expresso (Eq.(7)) d ctutor duced expresso (Eq.(15)), d the phse gle of ech ctuto voltge o ech ctutor. Whe oly frst m modes re cosdered, the dsplcemet fter cotrol by multple AFM probes c be expressed s j d t e A x t m 1 bhn 1 ˆ j e dpt p pp p m U m x FU L p x, t e 1 bhn 1 1 p1 d p d p u t ˆ ( d ) U x FU L e A x ˆ j d e pt p pp p m j t A x p e 1 1 p1 1 p d p d p 4, (15) (16) Note tht the frst term Eq.(16) s the p= compoets where the free dmpg terms d cotrol terms re hrmoc wthout phse dfferece. As dscussed before, ths term lmost domtes the vbrto; the secod term s the p compoet where the terms of dfferet phse gle couplg together, whch s very dffcult to lyze. Fortutely, the cotrbuto of the secod term c be eglected s the vbrto duced by t s much smller th the frst term. So the frst term c be pcg s the expresso of respose uder cotrol. If costt mpltude of cotrol voltge s

7 selected for the ope loop cotrol, by crefully choose the postos of ctutors, oe c suppress the cpet vbrto effcetly. However otce tht the frst term the brcet s of trset stte, s the tme mrches whe the tl sttc force duced vbrto des dow the vbrto duced by the ctutor stll exst d such pheomeo wll be crefully lyzed cse studes. 5. THE FEEDBACK CONTROL As proposed prevously, f costt cotrol voltge mpltude s used, the cotrol effect s bd reltvely log tme scle. Thus the mpltude of the voltge should lso be of trset stte d ths c be relzed by troducg close loop cotrol,.e., ddg feedbcs to the cotrol system. The cotrol voltges c be determed by frequecy-dom lyss o the dsplcemet sgl detected s by the dsplcemet sesor, the sesed modl prtcpto fctor for -th mode c be obted s, where the superscrpt s deotes the sesg sgl. The the p-th ctuto voltge o -th ctutor, otg tht oly p= codtos s cosdered here, c be determed ccordg to the sesg sgl s ˆ s G, where G s the feedbc g fctor for p-th (-th) ctuto fctor o -th ctutor. Tg the expresso of ctuto voltge d cosderg oly the frst term, Eq.(16) c be wrtte s u U L FU L, e 1 4 t s e A x G jdt m L t 1 bhn 1 1 Idelly, the true tme respose s equl to the dsplcemet outputted by sesor. Usg modl s expso method to both true tme respose d sesor output dsplcemet yelds ˆ ˆ. Wth the flexoelectrc cotrol, the sesor output s multpled by g fctor G order to ehce the cotrol effect. Thus oe eeds to determe the modl prtcpto fctor by solvg the followg equto ˆ t t L e FU bhn 1 A x G ˆ The soluto of Eq.(18),.e., the true tme modl respose of the sp-bc bem uder the cotrol of multple ctutors c be wrtte s t FU t L e ˆ = 1 bhn 1.. (18) A x G (19) The modl sp bc respose of the ctlever bem wthout cotrol c be reduced from Eq.(19) by settg the g fctor G equl to zero d the result modl respose s detcl wth tht of the free dmpg sp-bc bem demostrted Eq.(7). The cotrol effect for ech mode of ctlever bem depeds o the vlue of the deomtor of the expresso of true tme modl respose of the bem. The deomtor s determed by both the postos of ctutors x d the g fctors G. Due to the stress cocetrto problems duced by AFM probe, the cotrol voltge must be lmted, whch tur lmts the g fctor G. Thus for the lmted g fctor set of optml postos must be foud frst to cheve better cotrol effect by mxmze the bsolute vlue of the deomtor for (17)

8 ech mode. Whle the optml postos for ctutors re dfferet from oe mode to other, oe should frst choose the prcple mode d the set the postos of ctutors ccordg to the optml postos for ths mode. Wth the prmeters for both elstc bem d ctutor lsted Tble.1, some detled studes re coducted cse studes. Tble 1. Prmeters d propertes of the model. Bem s prmeters vlues Other propertes Bem legth L, (m) Flexoelectrc ptches thcess, h c (m) 50 Bem wdth b, (m) Flexoelectrc ptches legth, L (m) Bem thcess, h (m) AFM probe tp rdus, R (m) 50 Youg s modulus of bem, Y (N/m ) Actuto voltge, c (V). Bem mss desty, (g/m ) 1100 Flexoelectrc costt, 1 (C/m) 100 Itl force, F(N) 0.05 G fctor, G (V/m) CASE STUDIES 6.1 Ope loop cotrol I the cse study, oly three ctutors re used d o ech ctutor oly three hrmoc voltges ˆ re ppled wth the sme mgtude p =.V. Note tht the voltge ctully ppled o the AFM probes re the superposto of voltges of dfferet modl frequeces, thus ctul voltge ppled o the ctutors c s lrge s 100V. To evlute the cotrol effect of ope loop method, tl sttc forces of dfferet mgtudes re ppled o the tp ed of the bem. The optml postos of the ctutors re selected to mmze the vbrto dsplcemet. Respose of the ctlever bem, wth d wthout cotrol, re show Fg.5 wth the tl force of 0.1N, 0.05N, 0.0N d 0.0N. () (b) (c) (d) Fgure 4. Tp dsplcemet of the bem wth tl sttc force of () 0.1N; (b) 0.05N; (c) 0.0N d (d) 0.0N.

9 The cotrol effect from 0s to 1s, s show Fg.4 () d (b), ehces s the mgtude of tl sttc force decreses, whch s resoble becuse the cotrol blty,.e., ctuto voltge, eeps uchged durg the orgl vbrto become smller. Most mportt, the optml postos for frst two cses whe tl force s gve by 0.1N d 0.05N re ll 0.005m, 0.015m d 0.05m. For the reltvely bg tl deflecto, the optml rrgemet s to plce the ctutors s close to x=0 s possble, where the ctuto effect of frst mode cheves greter vlue. However, whe tl deflecto s smll the vbrto becomes out of cotrol ths tme scle, s show by Fg.4 (c) d (d). The optml postos here re o loger clustered er the fxed ed. Becuse s the tl sttc force decreses, the dsplcemet duced by the tl codto re outweghed by the vbrto duced by the ctutors, whch mes the ctutors chge ther fvorte postos to vod over-geertg the dsplcemet d substtute wth oe less effectve. Thus ope loop cotrol method s ot good eough for reltvely log tme scle, oe my solve ths problem by turg off the ctutors whe the tp dsplcemet s smll eough,.e., below cert threshold, or, s dscussed ext, ddg feedbc to the ctutors. 6. Feedbc cotrol To vod the emergece of chos demostrted ope loop cotrol, feedbc cotrol s lyzed here. The true tme respose of the ctlever bem c be obted by the dsplcemet expresso. Both the postos of the ctutors d the vlue of the g fctor wll fluece the effect of cotrol. The m s to obt optml cotrol effect uder the sme g fctor by djustg the postos of the ctutors. Recll tht the frst mode domt the others sp-bc vbrto, the postos wll be selected here ccordg the optml postos for frst mode,.e., 0.005m, 0.010m d 0.015m. The tp dsplcemets of both the free d cotrolled vbrtos, whe the tl sttc force s set to be 0.05N, re plotted Fg.5(). () (b) (c) (d) Fgure 5. Cotrol effect of bem () rel tme, (b) frst (c) secod d (d) thrd mode.

10 The frst mode respose lmost represets the true tme respose of the ctlever bem s show Fg.5(b) s ths mode s the domt oe sp-bc vbrto. The frst mode lso possess the best cotrol effect frst three modes (comprg to Fg.5(b) d (c)), becuse tht the postos of the ctutors s chose fvor of the frst mode rther th the others. To further lyze the feedbc cotrol of the ctlever bem, the feedbc voltges of three respectve modes re plotted Fg.6(). Uder ths g fctor the mxml feedbc voltge s bout 55V, whch s much lower th the ope loop cotrol s dscussed before. Note tht cotrol voltges re reltvely lrge t cpet stge, thus cert lmtto o mxml cotrol voltge c be troduced to vod the stress cocetrto problem. A lmtto of.v o the mpltude of cotrol voltge s posed here, d the feedbc cotrol voltge uder ths lmtto s plotted Fg.6(b). () (b) (c) Fgure 6. The cotrol voltges of the ctutors () wthout lmtto; (b) wth lmtto d (c) rel tme resposes of both cses. The lmtto forced the feedbc voltge to be smller th the mxml vlue,.e.,.v. The cotrol effect fter troducg the lmtto, before lmtto d free dmpg re show Fg.6 (c). Comprg wth the oe wthout voltge lmtto, the cpet cotrol effect s weeed becuse the cotrol voltge s reltvely smll. However, the overll cotrol effect suffcetly log tme scle s lmost the sme s the oe wthout voltge lmtto. Recll tht voltge lmtto s troduced to vod the stress cocetrto problem whch s of gret sgfcce, thus the feedbc cotrol uder voltge lmtto s cosdered superor th the oe wthout. 7. CONCLUSION Ths study cocetrted o the optml postos of the ctutors d cotrol strteges of flexoelectrc-bsed vbrto cotrol o ctlever bem structure. The sp-bc vbrto of the bem, excted by wthdrwg the tl sttc force t t=0 momet, ws cosdered d the gol s to ccelerte ts dmpg process. Multple ctutors cosst of flexoelectrc ptches d AFM probes

11 were used d o ech AFM probe voltges of dfferet frequeces re ppled. To obt mxml cotrol effect, the frequeces of cotrol voltges re set equl to the turl frequeces of the ctlever bem model d the phse gle of them s selected to dmsh the phse dfferece domt compoets whle eglect the osgfct oes. Two ds of cotrol method, ope loop d close loop, ws troduced bsed o the former theoretcl results. A problem of ope loop cotrol s tht the ctutors wll cotue to ctute the bem fter the tl trset sp-bc vbrto dmshes. Oe possble soluto s to tur dow the ctutors whe the mpltude of the vbrto s below cert threshold (smll eough). The close loop cotrol, detectg the dsplcemet formto from the bem s tp ed, feedbc the vbrto codtos to the ctutors. Ths method obts smlr cotrol effect wthout duces extr vbrto, d furthermore, less voltge mgtude s requred. The voltge c be further lmted close loop to vod the stress cocetrto. The voltge lmted oe, by gvg lmtto o the voltge mgtude ppled, wors lmost s well s the ulmted oe but wth much less stress cocetrto. ACKNOWLEDGEMENTS The reserch ws supported by the Ntol Nturl Scece Foudto of Ch (No d ). REFERENCES 1. Sodel, W., 199, Vbrtos of Shells d Pltes, Deer, New Yor.. Bolz, L., Cgo, S., Togo, G., d Crpter, A., 1989, Sp-bc Softeg Istblty Hgh-stregth Cocrete Bems, Mterls d Structures, (6), pp Hellweg, H. -B., d Crsfeld, M. A., 1998, A New Arc-legth Method for Hdlg Shrp Sp-bcs, Computers & Structures, 66(5), pp Crrr, P., Ferrett, D., Fredd, d F., Rost, G., 011, Sher Tests of Crbo Fber Pltes Boded to Cocrete wth Cotrol of Sp-bc, Egeerg Frcture Mechcs, 78(15), pp Peder, M. J., Alge, T. B., Orese, R. P., d Wotherspoo, L. M., 011, Sp-bc Testg for Estmto of Noler Behvor of Shllow d Ple Foudtos, Proceedgs of the Nth Pcfc Coferece o Erthque Egeerg, PCEE Tzou, H. S., d Gdre, H., 1989, Theoretcl Alyss of Mult-lyered Th Shell Coupled wth Pezoelectrc Shell Actutors for Dstrbuted Vbrto Cotrols, Jourl of Soud d Vbrto, 1(), pp Tzou, H. S., d Ye, R., 1994, Pezothermoelstcty d Precso Cotrol of Pezoelectrc Systems: Theory d Fte Elemet Alyss, Jourl of Vbrto d Acoustcs, 166(4), pp Tzou, H. S., d Hollmp, J. J., 1994, Collocted Idepedet Modl Cotrol wth Self-sesg Orthogol Pezoelectrc Actutors (Theory d Expermet), Smrt Mterls d Structures, (), pp Tzou, H. S., d Ye, R., 1996, Alyss of Pezoelstc Structures wth Lmted Pezoelectrc Trgle Shell Elemets, AIAA Jourl, 4(1), pp Bo, Y., Tzou, H. S., d Veyy, V. B., 1998, Alyss of No-ler Pezothermoelestc Lmted Bems wth Electrc d Temperture Effects, Jourl of Soud d Vbrto, 09(), pp Wg, D. W., Tzou, H. S., d Lee, H. J., 004, Cotrol of Noler Electro/Elstc Bem d Plte Systems (Fte Elemet Formulto d Alyss), Jourl of Vbrto d Acoustcs,

12 16(1), pp Todorov, A. T., Petrov, A. G., d Fedler, J. H., 1994, Frst Observto of the Coverse Flexoelectrc Effect Blyer Lpd Membres, The Jourl of Chemcl Physcs, 98(1), pp Mdl, R. D., d Eshel, N. N., 1968, O Frst Str-grdet Theores Ler Elstcty, Itertol Jourl of Solds d Structures, 4(1), pp Fu, J. Y., Zhu, W., L, N., Cross, L. E., 006, Expermetl Studes of the Coverse Flexoelectrc Effect Iduced by Ihomogeeous Electrc Feld Brum Strotum Ttte Composto, Jourl of Appled Physcs, 100(), pp Zubo, P., Ctl, G., Bucley, A., Welche, P. R. L., d Scott, J. F., 007, Str-Grdet-Iduced Polrzto SrTO Sgle Crystls, Physcl Revew Letters, 99, p Hu, S. D., L, H., d Tzou, H. S., 015, Dstrbuted flexoelectrc structurl sesg: Theory d expermet, Jourl of Soud d Vbrto, 48, pp Hu, S. D., L, H., d Tzou, H. S., 01, Sesg Sgl d Eergy Geerto Alyss o Flexoelectrc Bem, Proceedgs of the ASME 01 IMECE, IMECE , pp Hu, S. D., L, H., d Tzou, H. S., 01, Flexoelectrc Resposes of Crculr Rgs, Jourl of Vbrto d Acoustcs, 15(), p Hu, S. D., Tzou, H. S., 010, Sgl Alyss of Flexoelectrc Trsducers o Rgs, Proceedgs of the 010 Symposum o Pezoelectrcty, Acoustc wves, d Devce Applctos, SPAWDA10, pp Ro, Z., Hu, S. D., Tzou, H. S., 011, Dgol Flexoelectrc Sesor o Cyldrcl Shell Substructure, Proceedgs of the 011 Symposum o Pezoelectrcty, Acoustc wves, d Devce Applctos, SPAWDA10, pp Hu, S. D., L, H., d Tzou, H. S., 014, Comprso of Flexoelectrc d Pezoelectrc Dymc Sgl Resposes o Flexble Rgs, Jourl of Itellget Mterl Systems d Structures, 5(7), pp Hu, S. D., L, H., d Tzou, H. S., 011, Sttc No-Cotrol of Ctlever Bems Usg the Iverse Flexoelectrc Effect, Proceedgs of the ASME 011 IMECE, IMECE , pp Bsr, S., Thruvml, S., Heo, H., Lee, H. J., Frcs, S. M., Rmchdr, N., d Fu, J. Y., 010, Coverse Pezoelectrc Resposes Nopezoelectrc Mterls Implemeted v Asymmetrc Cofgurtos of Electrodes, Jourl of Appled Physcs, 108(6), p Cruso, G., Gle, S., d Me, L., 00, "Actve Vbrto Cotrol of Elstc Plte Usg Multple Pezoelectrc Sesors d Actutors," Smulto Modelg Prctce d Theory, 11(5-6), pp, Clr, R. L., Fuller, C. R., 199, "Expermets o Actve Cotrol of Structurlly Rdted Soud Usg Multple Pezocermc Actutors," The Jourl of the Acoustcl Socety, 11(4), pp Bley, T., d Hubbrd, J. E., 1985, "Dstrbuted pezoelectrc-polymer ctve vbrto cotrol of ctlever bem", Jourl of Gudce, Cotrol, d Dymcs, 8(5), pp Tzou, H. S., 199, Pezoelectrc Shells: Dstrbuted Sesg & Cotrol, Kluwer Acdemc Publshers, Dordrecht, Bosto, Lodo. 8. Abpllp, M., 001, Pezorespose Scg Force Mcroscopy of Ferroelectrc Doms, Ph.D. Dssertto, Swss Federl Isttute of Techology, Zürch. 9. Agro, A., Molots, M., Rosews, Y., Rosem, G., Rodrguez, B. J., Kgo, A. I., d Gruverm, A., 006, "Dymcs of Ferroelectrc Dom Growth the Feld of Atomc Force Mcroscope," Jourl of Appled Physcs, 99(10), p