Twist analysis of piezoelectric laminated composite plates

Transcription

1 wist analysis of pizolctric laminatd composit plats Mchatronics Enginring Dpartmnt, Faculty of Enginring, Intrnational Islamic Univrsity Malaysia, Malaysia ABSAC cntly scintists ar showing intrsts in smart structurs for thir capability in controlling structural bhavior and monitoring structural halth. wist control of hlicoptr rotors, micromirrors or shafts in torsional oscillation ar th activ aras of rsarch whr smart matrials li pizocramic crystal or shap mmory alloy can play vital rols. In this papr analysis of twisting of pizolctric laminatd composit plats using highr ordr shar dformation thory has bn prsntd. Kywords: Pizolctric laminatd plat, twist, highr ordr shar dformation thory.. INODUCION h nds for structurs with slf-monitoring and slf-controlling capabilitis spcially in arospac applications hav causd rmarabl growth in th rsarch and dvlopmnt of smart structurs. A smart structur can b dfind as a structur mad up of purly lastic matrials, calld th substrat, intgratd with surfac mountd or mbddd snsors and actuators that hav th capability to sns and ta corrctiv action ]. Laminatd composit plats ar found to b suitabl for us as substrats in th arospac applications for its high strngth-to-wight and stiffnss-to-wight ratios 2]. hus laminatd composit plat with mbddd pizolctric matrial that can act both as snsor and actuator is a good choic for activ control of structural shap. Howvr, this ind of activ structur still nds comprhnsiv analyss to b usd succssfully in applications. hr ar fw act solutions availabl to dscrib th inmatic bhavior of th pizolctric laminatd composit plats. Eact solution bcoms vry difficult whn th configuration of th structurs with mbddd pizolctric laminats bcoms compl. An altrnativ solution would b to us finit lmnt solutions. h wor of Alli and Hughs 3] can b considrd as on of th pionr in th fild of finit lmnt formulation, which includs pizolctric ffcts in structurs. A twodimnsional quadrilatral pizolctric plat lmnt with on lctrical dgr of frdom pr lmnt was dvlopd by Hwang and Par 4] to study vibration control of a pizolctric laminatd composit plat. h plat lmnt has four nods and ach of th nods has thr mchanical dgrs of frdom. Classical laminatd plat thory (CLP) with inducd strain actuation and Hamilton s principl wr usd to formulat th quation of motions. It was found activ and passiv control affct ach othr, thus should b considrd simultanously in dsigning fficint controlld structurs. h wors don using CLP wr found only suitabl to modl thin plat. Howvr to modl thic plat, it is gnrally accptd that highr-ordr shar displacmnt thory (HSD) can modl mor accuratly, sinc through HSD ffcts of transvrs shar strsss can b capturd. ay t. al. 5] dvlopd a finit lmnt modl for static analysis of a simply supportd rctangular intllignt plat using th HSD. An ight-nodd two-dimnsional quadratic quadrilatral isoparamtric lmnt was drivd to modl th coupld lctromchanical bhavior. h rsults obtaind in ths wor wr compard with act solutions don by th sam authors prviously 6] and found to b in good agrmnt. Chattopadhyay and Sly 7] dvlopd a rfind HSD and usd finit lmnt mthod to analyz laminatd composit plat surfac bondd or mbddd with pizolctric layrs. Non-linaritis wr introducd to th problm through th strain dpndnt pizolctric strain cofficints and th assumd strain distribution through th thicnss. sults obtaind from this modl wr shown to agr wll with publishd primntal rsults by Crawly and Lazarus 8]. All th abov wors ar mainly concntratd on load displacmnt of pizolctric plats undr

2 wist analysis of pizolctric laminatd composit plats, diffrnt actuation voltag. wisting or torsional dformation has almost bn disrgardd so far. In th prsnt rsarch, pizolctric matrial that has th capabilitis to act as actuators and snsors du to dirct and convrs pizolctric ffct is chosn and intgratd with a laminatd composit plat to study twisting ffct of th plat undr diffrnt actuation voltag. Laminatd composit plat is chosn as th substrat for its high strngth-to-wight and stiffnssto-wight ratios. hs charactristics ma th laminatd composit plat suitabl to b usd in many applications spcially in arospac applications. HSD dvlopd by Prvz 9] is usd in ordr to accuratly dscrib th inmatic bhavior of th laminatd composit plats and th pizolctric layrs. h lastic fild and th lctric fild ar coupld through th linar pizolctric constitutiv quations. In th finit lmnt modl, an ight-nodd two-dimnsional isoparamtric lmnt is usd. Each nod has svn mchanical dgrs of frdom and on lctrical dgr of frdom. 2. MAHEMAICAL MODEL Kinmatics and Constitutiv lations Figur- shows th gomtry of a laminatd composit plat with surfac bondd pizolctric actuator layrs on th top and th bottom surfacs. h plat considrd hr has total thicnss h, lngth a, width b and numbr of layrs n. Figur Gomtry of a laminatd composit plat with surfac bondd pizolctric layrs. h highr-ordr shar displacmnt filds considrd in this rsarch wor ar 9]: u 3 (, y, z, t) = uo (, y, t) + zθ + z ζ (a) rspctivly; and ζ,ζ y ar th third ordr displacmnts or warping functions. h strain associatd with th displacmnt filds can b prssd as follows ]: { ε} = ' { ε} whr, Z (2a) { } { ε ε γ γ γ } ε = (2b) { ε } = y uo yy vo w θ y + uo vo + o yz 3ζ y z y + y y wo θ + 3ζ y + (2c) 3 z z 3 z z Z ' = 2 z (2d) 2 z 3 z z Hr ε, ε yy and γ y ar th in plan strain componnts. γ yz and γ z ar th transvrs shar strain componnts. In th prsnt rsarch, quasi-static loading and plan strss formulations ar assumd. h constitutiv quations for th th orthotropic layr prssd in th matrial coordinat systm ar givn as follows: σ σ 2 σ 23 σ 3 σ = ε ε 2 γ 23 γ 3 66 γ (3) For th pizolctric layrs, th linar pizolctric constitutiv quations that coupl th lastic fild and lctric fild ar givn as follows: v 3 (, y, z, t) = vo (, y, t) + zθ y + z ζ y (, y, z, t) = w (, y t) w o, (b) (c) whr u, v, w ar th displacmnts of a gnric point (, y, z) in, y and z dirctions rspctivly; u o, v o, w o ar th displacmnts of mid-plan in, y and z dirction rspctivly; z is th coordinat in thicnss dirction; θ, θ y ar th rotations of normals to rfrnc surfac about th y and as

3 Procdings of th st Intrnational Confrnc & 7th AUN/SEED-Nt Fildwis Sminar on Manufacturing and Matrial Procssing σ σ 2 σ 23 σ 3 σ D D2 D 3 p + = 5 = p E E 2 E E E2 p E ε ε 2 γ 23 γ 3 γ h lctric filds ar givn as ]: E E E yy zz whr φ φ = φ z φ is th lctric potntial. 66 ε ε 2 γ 23 γ 3 γ (4b) (5) (4a) h lctric potntial is assumd to hav linar variations across th thicnss of th actuator/snsor layrs as: φ z h = o (6) h h + (, y, z) φ (, y) + whr φ o can b tratd as th gnralizd lctric potntial similar to th gnralizd displacmnt at any point on th surfac of th actuator and th snsor layrs. 3. FINIE ELEMEN FOMULAION h ovrall plat is mshd into a finit numbr of lmnts by an ight-nodd two-dimnsional isoparamtric lmnt. frring to Eq.(), th gnralizd displacmnt vctor for nod i (i=,2,..,8) of th lmnt can b writtn as: { u } { u v w θ θ ζ ζ } i = (7) oi oi oi i yi i yi From th abov quation, th nodal gnralizd displacmnt vctor of a typical ightnodd lmnt can b prssd as: { } {( u ) ( u ) ( u ) ( u ) } u = (8) hus, th gnralizd displacmnt vctor at any point within th lmnt can b obtaind by intrpolating th nodal gnralizd displacmnt as: { } N ]{ u } u = (9a) whr th shap function matri is givn as: ] N ] N ] N ] ] N = (9b) N8 and Ni ] = ni I ]. Hr I ] is an idntity matri of siz 77 and n i ( i =,2,3...,8 ) is th shap function of natural coordinats ( ζ, η) associatd with th i th nod ]. frring to Eq.(2a), by substituting Eq.(9a) th gnralizd strain vctor { ε } at any point within th lmnt can b prss as: { ε } = B]{ u } (a) whr, th nodal strain-displacmnt matri is givn as: ] B ] B ] B ] ] B = (b) B8 h lmnts of ach submatri B i ] of B] ar givn plicitly in Mahmood ]. Similarly, lctric potntial can also b gnralizd at any point within th lmnt as: { } ]{ } φ = (a) o N p φo whr, { } { } o φo φo2... φo7 φo8 ] N N N ] φ = (b) N p = (c) N8 and { }( i =,2...8 ) φ oi is th lctric potntial at th ith nod. Using Eq.(a), lctric fild in Eq.(5) can b writtn as follows: { E } Z p ] Bp ]{ φo } whr Z ] and ] = () p Mahmood ]. B ar givn plicitly in p Variational principl in conjunction with Eq.(7) to () yilds th following two sts of matri quations: K ]{ u } + K ]{ φ } = { F } uu uφ o S (3a)

4 wist analysis of pizolctric laminatd composit plats, K ]{ u } + K ]{ φ } = { } φ u φφ o S (3b) whr, + + n = ] Kuu = B] Z ' ] Z ' dz B] dt Jdςdη = h+ + + n = φ ] K B] Z ] Z ] dz u = ' p B p ] dt Jdςdη = h+ + + n Kφ ] B ] ] = u = Z p ] Z ' dz B] p dt Jdςdη = h+ (3c) (3d) (3) activating ithr P or B schm producs an asymmtric twist in th cantilvr plat. It is found that activating th P-B schm producs a symmtric twist in th cantilvr plat as shown in Figur-5. Unli in th cas of P and B schms whr th plat cntrlin dflcts, th cntrlin of th cantilvr plat undr P-B schm rmains flat as can b sn in Figur-6 and Figur-7. Figur-8 prsnts furthr invstigation on symmtric twist, whr th actuator voltag is varid from V to 3 V. h dgr of twist is found to incras as th actuator voltag incrass. + + n φφ ] ] K B Z ] p] Z ] dz p p B p ] = = p dt Jdςdη = h+ ] FS = N ] Z ] z h { } ds = S (3f) (3g) ] ] = N p q ds S S 2 In th abov quations ] matri, ] ] K uφ ' (3h) φ u K is th lastic stiffnss uu = K is th coupling stiffnss matri btwn mchanical and lctrical ffcts, F is th K φφ ] is th dilctric stiffnss matri, ] mchanical forc as a rsult of th surfac forc and S ] is th lctric forc as a rsult of th applid surfac charg on th actuators. 4. VALIDAION OF HE MAHEMAICAL MODEL h rsults obtaind using th mathmatical modl and th computr cod dvlopd in this rsarch wor wr validatd comparing with thos availabl for transvrs dflction of laminatd plats 3] and pizolctric bams 4]. In both th cass maimum rror wr found to b limitd to lss than.4%. It is to b notd hr rsults obtaind in 3] and 4] ar basd on First Ordr Shar Dformation hory (FSD), whras th prsnt analysis uss HSD. wist Analysis his rsarch is mainly targtd to analyz twist activation in a pizolctric laminatd plat undr diffrnt actuation voltag. o do so, sgmntd actuators ar placd on a cantilvr plat in th configuration as shown in Figur-2. Svral st of activation schms as listd in abl. ar prformd on th cantilvr plat to furthr analyz this capability. Figur-3 to Figur-8 prsnt shaps of th cantilvr plat producd undr ths activation schms. frring to Figur-3 and Figur-4, S Figur 2 Sgmntd actuator configuration to produc twist. abl St of activation schms to produc twisting. Schm Actuator sgmnt A B C D P 3 V V -3 V V B V 3 V V -3 V P-B 3 V -3 V -3 V 3 V Figur 3 Shap of a cantilvr plat undr P schm of activation undr constant actuator voltag of 3 V

5 Procdings of th st Intrnational Confrnc & 7th AUN/SEED-Nt Fildwis Sminar on Manufacturing and Matrial Procssing Figur 4 Shap of a cantilvr plat undr B schm of activation undr constant actuator voltag of 3 V. Figur 7 Effct of sgmntd actuator on fr nd ( = a) dflction of a cantilvr plat undr a constant actuator voltag of 3 V. Figur 5 Shap of a cantilvr plat undr P-B schm of activation undr constant actuator voltag of 3 V. Figur 8 Effct of diffrnt actuator voltag on fr nd ( = a) dflction of a cantilvr plat undr P- B schm of activation. 5. CONCLUSIONS It is found that, symmtric as wll as antisymmtric twist can b introducd in a cantilvr plat by activating sgmntd actuators undr diffrnt schm of activation. h dgr of th twist can b incrasd by just incrasing th applid voltag to th sgmntd actuators. his capability prsnts a good opportunity to b usd in individual blad control (IBC) of hlicoptrs. A scald down schm of such a plat may b usd in micromirror orintation for oprating optical switchs. Figur 6 Effct of sgmntd actuator on cntrlin (y = b/2) dflction of a cantilvr plat undr a constant actuator voltag of 3 V. EFEENCES ] ddy, J.N, (999), On laminatd plats with intgratd snsors and actuators, Enginring Structurs 2, ].Jons, M. (975), Mchanics of composit matrials. Cambridg: Cambridg Univrsity Prss. 3] Alli, H. and Hughs,.J. (97), Finit lmnt mthod for pizolctric vibration, Int. J. for Numrical Mthods in Enginring 2,

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