FINITE BEAM ELEMENT WITH PIEZOELECTRIC LAYERS AND FUNCTIONALLY GRADED MATERIAL OF CORE

Transcription

1 ECCOMAS Congrss 20 II Europan Congrss on Computational Mthods in Applid Scincs and Enginring M. Papadrakakis,. Papadopoulos, G. Stfanou,. Plvris (ds.) Crt Island, Grc, 5 0 Jun 20 FINITE BEAM ELEMENT WITH PIEZOELECTRIC LAYERS AND FUNCTIONALLY GRADED MATERIAL OF CORE ladimír Kutiš, Justín Murín, Juraj Paulch, Juraj Hrabovský, Roman Gogola and Jakub Jakubc Dpartmnt of Applid Mchanics and Mchatronics Institut of Automotiv Mchatronics Faculty of Elctrical Enginring and Information Tchnology Slovak Univrsity of Tchnology in Bratislava Ilkovičova 3, 82 9 Bratislava, Slovak Rpublic -mail: vladimir.kutis@stuba.sk, justin.murin@stuba.sk, juraj.paulch@stuba.sk, juraj.hrabovsky@stuba.sk, roman.gogola@stuba.sk, jakub.jakubc@stuba.sk Kywords: Pizolctric Matrial, FGM, FEM, Bam Elmnt, Transint Analysis Abstract. Nw smart matrials hav bn dvlopd in matrial scinc, that ar suitabl for mchatronic applications. Modrn mchatronic systms ar focusing on minimizing siz, activ control and low nrgy consumption. All this attributs can b incorporatd into trm Micro Elctro Mchanical Systms (MEMS). To improv prformanc of MEMS systm, nw matrials and tchnologis ar dvlopd - on of thm, which found broad application usag is Functionally gradd matrial (FGM). MEMS application usually contains multilayr structur and in som application class MEMS systms contain pizolctric layrs. Pizolctric structurs offr facilitis to mak motions. Pizolctric layrs can b also usd to damp vibrations as an activ damping or as an activ snsor. For bttr undrstanding ths multiphysical problms nw mathmatical modls ar dvlopd. Th papr dals with finit bam lmnt with pizolctric layrs and functionally gradd matrial of cor. In th papr homognization of FGM matrial proprtis and homognization of cor and pizolctric layrs is prsntd. In th procss of homognization dirct intgration mthod and multilayr mthod is usd. Thr is also prsntd th drivation of individual submatrics of local stiffnss and mass matrix, whr concpt of transfr constants is usd. Functionality of nw FGM finit bam with pizolctric layrs is prsntd by numrical xprimnts.

2 INTRODUCTION Nw matrials hav bn introducd in th ara of mchanisms and mchatronic applications. Contmporary mchanical systms ar focusing on minimizing siz, activ control and low nrgy consumption. All ths attributs can b incorporatd into trm Micro Elctro Mchanical Systm (MEMS). MEMS applications usually contain multilayr structur and som of ths layrs can hav pizolctric proprtis. Pizolctric structurs offr facilitis to mak motions or thy can b usd to damp vibrations as an activ dampr or as an activ snsor. 2 PIEZOELECTRIC CONSTITUTIE EQUATIONS Pizolctric constitutiv quations dscrib th rlationship btwn mchanical and lctrical quantitis []. This rlationship is drivd in tnsor notation, but for practical usag it can b rwrittn into matrics notation. 2. Tnsor notation of pizolctric constitutiv quations Th form of th constitutiv quations dpnds on chosn mchanical and lctrical quantitis and can b xprssd in two basic configurations [2]. Th first configuration is xprssd by mchanical strss tnsor componnts σ kl and vctor componnts of lctric intnsity E k and has a form D i = ɛ σ ike k + d ikl σ kl () ε ij = d ijk E k + s E ijklσ kl (2) whr ε ij ar strain tnsor componnts, D i ar componnts of lctric displacmnt vctor, d ikl ar tnsor componnts of pizolctric constants, ɛ σ ik ar componnts of prmittivity tnsor undr constant mchanical strss and s E ijkl ar componnts of complianc tnsor undr constant lctric intnsity. Th constitutiv quations can b also xprssd by strain tnsor componnts ε kl and vctor componnts of lctric intnsity E k and has a form σ ij = c E ijklε kl ijk E k (3) D i = ikl ε kl + ɛ ε ike k (4) whr nw quantitis ar componnts of stiffnss tnsor undr constant lctric intnsity c E ijkl and componnts of pizolctric modulus tnsor ijk. 2.2 Matrix notation of pizolctric constitutiv quations If w us symmtric proprtis of individual tnsor in constitutiv tnsor quations, w can rwrit constitutiv quations into matrix notation [3]. Thn quations () and (2) hav a form and quations (3) and (4) can b rwrittn in th form D i = ɛ σ ike k + d iq σ q (5) ε p = d pk E k + s E pqσ q () σ p = c E pqε q pk E k (7) D i = iq ε q + ɛ ε ike k (8) 2

3 D i and E k ar vctors with thr componnts, σ q and ε q ar vctors with six componnts, matrics s E pq and c E pq hav dimnsion, matrics d iq and pk hav dimnsion 3 and matrix ɛ ε ik has dimnsion 3 3. Constitutiv quations (5) and () writtn in a componnt form can b rwrittn as and quations (7) and (8) can b rwrittn as ε = s E σ + d T E (9) D = dσ + ɛ σ E (0) σ = c E ε T E () D = ε + ɛ ε E (2) 3 BASIC FEM EQUATIONS FOR PIEZOELECTRIC STRUCTURE To obtain basic FEM quations for pizolctric structur, th Hamilton s principl is usd and can b xprssd in th form t2 t (δl + δw ) dt = 0 (3) whr L is Lagrangian, W is work of xtrnal mchanical and lctrical forcs and t and t 2 dfind considrd tim intrval. Lagrangian of pizolctric structur is givn by L =T U + W = = 2 ρ ut ud 2 εt σd + 2 ET Dd whr T is kintic nrgy of structur, U is potntial nrgy of structur and W is lctric nrgy stord in pizolctric matrial. In kintic nrgy trm u rprsnts vlocity vctor. irtual work of xtrnal mchanical and lctrical forcs can b xprssd as (4) δw = ( δu T F ) ( δφ T Q ) (5) whr F and Q rprsnts discrt forcs and lctric chargs, rspctivly and u and φ ar displacmnt vctor and scalar lctric potntial, rspctivly. Equation (3) can b rwrittn using (4) and (5) t2 t [ δ 2 ρ ut ud δ 2 εt σd + δ + ( δu T F ) ( δφ T Q )] dt = 0 2 ET Dd + Aftr som manipulation and using constitutiv quations () and (2) Hamilton s principl for pizolctric systm can b xprssd in form t2 [ ρδu T üd δε T c E εd + δε T T Ed + δe T εd + t + δe T ɛ ε Ed + ( δu T F ) ( δφ T Q )] (7) dt = 0 3 ()

4 Using shap functions (N u, N φ ), w can writ rlationship btwn displacmnt of point and nodal displacmnt of finit lmnt and btwn scalar lctric potntial of point and nodal scalar lctric potntial of finit lmnt u = N u u (8) φ = N φ φ (9) Rlationship btwn componnts of strain and componnts of nodal displacmnts has form ε = B u u (20) Similarly rlationship btwn componnts of lctric fild intnsity and componnts of nodal potntial can b writtn as E = B φ φ (2) irtual strain and virtual lctric fild intnsity can b xprssd as δε = B u δu (22) δe = B φ δφ (23) Equation (7) can b modifid using quations (8)-(23) t2 [ ( ) ( δ(u ) T N T u ρn u d ü B T u c E B u d t ( ) ] B T u T B φ d φ + N T u F dt+ t2 [( ) ( + δ(φ ) T B T φ B u d u + B T φ ɛ ε B φ d t ) u ) ] φ N T φ Q dt = 0 From quations (24) w obtain finit lmnt quations for pizolctric analysis [ ] [ü ] [ ] [ ] [ ] [ ] [ ] M uu 0 C 0 0 φ + uu 0 u K 0 0 φ + uu K uφ u F K φu K φφ φ = Q (24) (25) whr M uu = N T u ρn u d (2) K uu = B T u c E B u d (27) K uφ = B T u T B φ d (28) K φu = B T φ B u d (29) K φφ = B T φ ɛ ε B φ d (30) F = N T u F (3) Q = N T φ Q (32) 4

5 4 FEM EQUATIONS OF FGM BEAM WITH PIEZOELECTRIC LAYERS 2D bam lmnt with pizolctric layrs, whr bam cor is mad from functionally gradd matrial is shown in Figur, whr all dgrs of frdom ar dpictd. Mchanical dgrs of frdom in ach nod ar two displacmnts (in dirction x a y) and rotation (in plan x y) []. Elctric dgrs of frdom ar lctric potntials φ i on 4 lctrods. Th hight of bam cor mad from FGM is h, th hight of pizolctric layr is h p, th dpth and th lngth of th bam lmnt ar b and L, rspctivly. Matrial proprtis of FGM cor ar function of longitudinal and transvrsal coordinat x and y, matrial proprtis of pizolctric layrs ar constants. Figur : Elctric DOF in 2D bam lmnt In ordr to driv individual submatrics of th bam lmnt with pizolctric layrs and FGM cor, two stps in homognization procss hav to b prformd. At first, homognization of matrial proprtis of FGM cor hav to b prformd, whr dirct intgration mthod is usd []. In th nxt stp, homognization of pizolctric layrs and homognizd FGM cor (from stp on) is prformd. Aftr ths two oprations, homognizd matrial proprtis of th bam vary through th lngth of th bam as a function of longitudinal coordinat x and ar constant in transvrsal dirction. 4. Equations for structural analysis Th structural submatrix K uu for th bam lmnt with pizolctric layrs can b xprssd in a form k u 0 0 k u 0 0 k v2 k v3 0 k v2 k v2 K uu = S k v33 0 k v3 k v3 Y k u 0 0 (33) M k v2 k v2 k v23 whr individual componnts contain th influnc of FGM cor stiffnss and also th influnc of pizolctric layrs stiffnss. Th calculation of componnts is idntical for classical multilayr or FGM bam without pizolctric layr and is dscribd in []. Mass matrix M uu can b calculatd numrically using classical shap functions and homognizd dnsity of FGM bam with pizolctric layrs according quation (2). 5

6 4.2 Equations for lctric analysis Elctric fild intnsity in pizolctric layr is constant and for top layr can by xprssd as [4, 5] E = φ y = φ 2 φ (34) h p and for bottom layr as E 2 = φ y = φ 4 φ 3 h p (35) Both componnts of lctric fild intnsity can b writtn in a form [ ] [ ] E /hp /h E = = p 0 0 E /h p /h p φ φ 2 φ 3 φ 4 = B φφ (3) For D problms th matrix of matrial proprtis for lctric fild prmittivity is rducd to only on matrial proprty ɛ ε, but bcaus th bam lmnt contains two idntical layrs, w can writ [ ] ɛ ɛ ε ε 0 = 0 ɛ ε (37) Thn th quation (30) has a form K φφ = B T φ ɛ ε B φ d = B T φ ɛ ε B φ A p dx (38) L whr A p is cross-sction of on pizolctric layr, i.. A p = bh p. Aftr som mathmatical manipulations th quation (38) can b xprssd as A plɛ ε A p Lɛ ε 0 0 h 2 p h 2 p A p Lɛ ε A plɛ ε 0 0 K φφ = h 2 p h 2 p 0 0 A plɛ ε A p Lɛ ε h 2 p h 2 p A p Lɛ ε 0 0 A plɛ ε h 2 p h 2 p 4.3 Coupling of structural and lctrical analysis Pizolctric matrial proprtis xprss coupling btwn mchanical and lctrical fild - matrics or d. Th rlationship btwn ths two matrics can b xprssd by lasticity matrix. In D problm in x y plan (in indx notation x x 2 ) w hav only on matrial proprty 2 or d 2, whr indx 2 rprsnts dirction of pizolctric layr polarization and also th dirction of lctric fild intnsity vctor and indx dfins dirction of mchanical dformation. Th rlationship btwn ths two quantitis is rducd to xprssion 2 = d 2 E p [7, 8], whr E p is Young modulus of lasticity of pizolctric matrial. In rality, rlationship btwn matrics and d is mor complicatd and th quantity 2 computd from matrix d and lastic matrix for 3D systm and th quantity 2 computd from d 2 and E p hav (39)

7 diffrnt valus. Thrfor if w hav quantitis 2 and d 2 computd from matrix rlationship, it is bttr to usd thm thn simplifid rlationship. Pizolctric matrial proprtis of both pizolctric layrs ar dfind as [ ] 2 0 y = 2 (40) 2 0 y 2 Th xprssion T E dfins mchanical strss causd by pizolctric ffct. In th bam lmnts, intrnal quantitis ar not mchanical strss but intrnal forcs and momnts, thn th first and th third column of matrix (40) multiplid by corrsponding componnts of B u and B φ as wll as corrsponding componnts of displacmnt u and potntial φ rprsnts axial forcs and bnding momnts, rspctivly. Dscription of pizolctric bhaviour by 2 is mor suitabl for snsor quation matrix K φu, dscription by d 2 is mor suitabl for actuator quation matrix K uφ, i.. [ ] d2 E = p 0 yd 2 E p (4) d 2 E p 0 yd 2 E p Using quations (40) and (4) w can writ (28) and (29) in form A pd 2 E p A p d 2 E p A pd 2 E p h p h p h p A y d 2 E p A yd 2 E p A y d 2 E p K h uφ = p h p h p A p d 2 E p A pd 2 E p A p d 2 E p h p h p h p A yd 2 E p A y d 2 E p A yd 2 E p h p h p h p A p d 2 E p h p A yd 2 E p h p A pd 2 E p h p A y d 2 E p h p and A p 2 A y 2 A p A y 2 h p h p h p h p A p 2 0 A y 2 A p 2 A y 2 0 K h φu = p h p h p h p A p 2 A y 2 A p A y 2 h p h p h p h p A p 2 0 A y 2 A p 2 A y 2 0 h p h p h p h p whr paramtr A y rprsnts first momnt of cross-sction of pizolctric layr (42) (43) A y = 2 A p(h + h p ) (44) 4.4 FEM quations for th bam lmnt with pizolctric layrs FEM quations for bam lmnt with pizolctric layrs and FGM cor for transint analysis hav classical form [ ] [ü ] [ ] [ ] [ ] [ ] [ ] M uu 0 C 0 0 φ + uu 0 u K 0 0 φ + uu K uφ u F K φu K φφ φ = Q (45) 7

8 whr individual submatrics ar dfind by (33), (39), (42) and (43), vctor of nodal unknowns is dfind as [ ] u = [ ] T u i v i ϕ i u j v j ϕ j φ φ 2 φ 3 φ 4 (4) φ and vctor of nodal loads is dfind as [ ] F = [ ] T F xi F yi M i F xj F yj M j Q Q 2 Q 3 Q 4 (47) Q whr Q, Q 2, Q 3 and Q 4 ar lctric charg on lctrods, 2, 3 and 4, rspctivly. 5 NUMERICAL EXAMPLES All drivd quations for FGM bam with pizolctric layrs wr implmntd in our FEM cod MultiFEM. In this sction thr numrical xampls ar prsntd: Exampl considrd simpl bam is mad from FGM with attachd pizolctric layrs, only static analysis is considrd Exampl 2 th sam simpl bam as in Exampl is considrd, but analysis is transint Exampl 3 considrd structur contains thr bam parts with partially attachd pizolctric layrs, two bam parts ar mad from FGM and on bam part has constant matrial proprtis 5. Matrial proprtis of FGM structur and pizolctric layrs In all thr xampls, th sam functionally gradd matrial is considrd. Matrial of matrix (indx m) is NiF with constant dnsity and Young s modulus and matrial of fibr (fibr indx f) is tungstn with constant dnsity and Young s modulus: Young s modulus: E m = 255 GPa, E f = 400 GPa dnsity: ρ m = 9200 kg/m 3, ρ f = 9300 kg/m 3 olum fractions of both constitunts v m (x, y) and v f (x, y) vary along th lngth (axis x) and hight (axis y) of bams according quations: v m (x, y) = x 3 y x x 2 y x y 2 +. [-] (48) v f (x, y) = x 3 y x x 2 y x y 2 [-] (49) Both functions of volum fractions for bam with lngth 0. m and hight 0.0 m ar shown in Figur 2. Effctiv matrial proprtis of FGM ar dfind by matrial proprtis of constitunts and thir variations and Young s modulus and dnsity of considrd FGM hav form E FGM (x, y) = x 3 y x x 2 y x y [Pa] ρ FGM (x, y) = x 3 y x x 2 y x y [kg/m 3 ] (50) (5) 8

9 ladimı r Kutis, Justı n Murı n, Juraj Paulch, Juraj Hrabovsky, Roman Gogola and Jakub Jakubc Figur 2: Lft volum fraction of matrix, right volum fraction of fibr In all thr xampls, hight and dpth of bam cross-sction is 0.0m. Homognizd matrial proprtis of invstigatd FGM bams can b calculatd by dfind cross-sction paramtrs of bams and by ffctiv matrial proprtis and hav following forms: N EFGM (x) = x x [Pa] (52) M EFGM (x) =9. 03 x x [Pa] (53) ρfgm (x) = x + 33.x [kg/m ] (54) N M EFGM (x) and EFGM (x) rprsnt homognizd Young s modulus for axial loading and for bnding, rspctivly. Pizolctric layrs in invstigatd bams ar mad from PZT5A pizolctric matrial. PZT5A is orthotropic matrial and has following matrial proprtis (dirction of poling has indx 3): mchanical proprtis: Young s moduli: E = GPa, E2 = GPa, E3 = 53, 2 GPa Poisson numbrs: µ2 = 0.35, µ3 = 0.38, µ23 = 0.38 shar moduli: G2 = 22. GPa, G3 = 2. GPa, G23 = 2. GPa dnsity: 7750 kg/m3 pizolctric proprtis: d3 = C/N, d33 = C/N, d5 = C/N, d24 = C/N rlativ prmittivity: σ = 728.8, σ22 = 728.8, σ33 = Exampl static analysis of pizolctric bam Figur 3 shows simpl cantilvr mad of FGM with pizolctric layrs, which is loadd by transvrsal forc F at fr nd. Elctrods on top and bottom pizolctric layrs ar short circuitd. Th goal of analysis is to invstigat static rsponds of structur on prscribd loading and compar rsults of D modl with rsults of 2D modl. Gomtry paramtrs of bam ar: 9

10 Figur 3: Exampl : static analysis of FGM bam with pizolctric layrs lngth 0. m hight of FGM cor 0.0 m, hight of pizolctric layr 0.00 m dpth of bam 0.0 m Matrial of bam: cor of bam is mad from FGM (NiF-tungstn) and its ffctiv and homognizd matrial proprtis ar dscribd by quations (50)-(54) pizolctric layrs ar mad from PZT5A rducd proprtis 3 = d 3 C/m 2 s E = 0.43 Using multilayrd mthod homognizd Young s modulus for axial loading and for bnding and homognizd dnsity of whol bam can b calculatd using homognizd matrial proprtis of FGM cor and constant matrial proprtis of pizolctric layrs and thy hav form E N (x) = x x [Pa] (55) E M (x) = x x [Pa] (5) Boundary conditions: ρ(x) = x x [kg/m 3 ] (57) lft nd fixd right nd transvrsal forc F = 0 N Th static analysis of systm was prformd by nw FGM bam lmnt with pizolctric layrs. Th analysis was prformd by, 2, 4 and 0 lmnts s Figur 4 lft. Dformd shap of bam is shown in Figur 4 right. Displacmnt in y dirction of fr nd is m. Elctric charg on top lctrods for FEM modls with diffrnt numbr of lmnts ar summarizd in Tabl. Th sam problm was analyzd by FEM cod ANSYS, whr two typs of plan lmnts wr usd PLANE223 with pizolctric capabilitis and PLANE83. Bcaus matrial proprtis of FGM cor vary along th lngth and hight of bam, discrtizd 2D modl contains 400 lmnts. Displacmnt in y dirction of fr nd was m and total lctric charg on top lctrod was C. As w can s from obtaind rsults, nw FGM bam lmnt with pizolctric layrs is vry accurat and ffctiv in static analysis, bcaus variation of matrial proprtis of FGM cor of bam is dirctly incorporatd into stiffnss matrix. 0

11 ykrslni prvkov Zobrazna vlastnost: matrial Figur 4: Exampl : lft discrtizd bam with nod and lmnt numbrs, right dformation of bam Numbr of lmnts Q lm [C] Q lm 2 [C] Q lm 3 [C] Q lm 4 [C] Q lm 5 [C] Q lm [C] Q lm 7 [C] Q lm 8 [C] Q lm 9 [C] Q lm 0 [C] Q SUM [C] Tabl : Exampl : Elctric charg on individual lctrods 5.3 Exampl 2 transint analysis of pizolctric bam In Exampl 2 transint analysis of FGM bam with pizolctric layrs with th sam gomtry and matrial paramtrs as in Exampl was prformd. Elctrods on top and bottom pizolctric layrs ar short circuitd. Th goal of analysis is to invstigat fr vibration of structur without considring damping. Boundary conditions: lft nd fixd right nd fr Initial conditions: initial displacmnt of nods initial dformation of systm is dfind by prscribd displacmnt of fr nd in vrtical dirction +0.0m initial vlocity of nods all nods hav zro initial vlocitis Th transint analysis of systm was prformd by nw FGM bam lmnt with pizolctric layrs. In th analysis Nwmark intgration schm was usd. Total simulation tim was 5 ms

12 and numbr of quidistant substps was 80. D modl of systm was discrtizd by 0 lmnts s Figur. 4 lft. Displacmnt of nods 4, 8 and in dirction y as function of tim ar shown in Figur 5 lft. Tim variations of lctric charg in top lctrod on lmnts 2 and 4 ar shown in Figur 5 right prvok 4 Gb prvok 2 Gb naboj Q uzol uzol 8 uzol 4 posunuti Y Figur 5: Exampl 2: lft Y displacmnt tim variation of nods (4, 8, ), right charg tim variation of top lctrods on lmnts (2, 4) 5.4 Exampl 3 transint analysis of simpl structur Figur shows simpl structur, which contains thr bams. Pizolctric layrs ar partially attachd to th bam and 2. Elctrods on top and bottom pizolctric layrs ar short circuitd. Th goal of analysis is to invstigat fr vibration of structur with considring damping. Figur : Exampl 3: fr vibration of structur with pizolctric lmnts Gomtry paramtrs of bams ar: 2

13 cross-sction all bams hav hight and dpth of cross-sction 0.0 m, hight of pizolctric layrs is 0.00 m and its dpth is 0.0 m lngth bam and 2 hav lngth 0.05 m and bam 3 has lngth 0. m, lngth of pizolctric layrs is 0.0 m Matrial paramtrs of bam ar: cor of bam bam and 2 ar mad from FGM, homognizd matrial proprtis ar dfind by quations (52)-(54), bam 3 has constant matrial proprtis: E = 39.4 GPa, ρ = kg/m 3 pizolctric layr PZT5A Thr diffrnt typs of bam finit lmnts wr usd in transint analysis of systm: FGM bam lmnt, FGM bam lmnt with pizolctric layrs and bam lmnt with Hrmit shap functions. Discrtizd D modl of systm is shown in Figur 7, whr nods and lmnt typs ar dpictd ykrslni uzlov ykrslni prvkov 7 4 Zobrazna vlastnost: typ Figur 7: Exampl 3: lft nods, right lmnts with lmnt typs numbring Boundary conditions: lft nd fixd right nd fr Initial conditions: initial displacmnt of nods initial dformation of systm is dfind by prscribd displacmnt of fr nd in horizontal dirction +0.0m initial vlocity of nods all nods hav zro initial vlocitis In th analysis Nwmark intgration schm was usd. Total simulation tim was 7.5 ms and numbr of quidistant substps was 400. In th analysis Rayligh damping was usd. Massproportional damping cofficint and stiffnss-proportional damping cofficint had th sam valu 0 5. Dformd shap of systm at th nd of simulation tim (7.5 ms) is shown in Figur 8 lft. Figur 8 right shows tim variation of lctric charg in top lctrod on lmnts and. Displacmnts of nods 8 and 2 as function of tim in dirction x and y ar shown in Figur 9 lft and right, rspctivly. 3

14 Figur 8: Exampl 3: lft X displacmnt of structur at tim 7.5ms, right charg tim variation of top lctrod on lmnt and Figur 9: Exampl 3: lft X displacmnt tim variation of nods (8, 2), right Y displacmnt tim variation of nods (8, 2) CONCLUSIONS Th papr prsnts nw bam finit lmnt with pizolctric layrs, whr cor of th bam can b mad of FGM matrials. Such combination of matrials is vry attractiv for mchatronic applications, bcaus matrial composition of FGM cor can b optimizd for dsign strss stat and dformation can b controlld by voltags on lctrods. Th bam finit lmnt can b usd for analysis of such systms vry ffctivly and accuratly. ACKNOWLEDGEMENT This work was supportd by th Slovak Rsarch and Dvlopmnt Agncy undr th contract No. AP-4-03 and AP-024-2, by Grant Agncy EGA, grant No. /0228/4 and /0453/5. REFERENCES [] Mohimani, S.O.R. and Flming, A.J.. Pizolctric Transducrs for ibration Control and Damping. Springr, (200). [2] Ballas, R.G. Pizolctric Multilayr Bam Bnding Actuators, Springr, (2007). 4

15 [3] Barntt, A.R., Plamdu, S.M., Dukkipati, R.. and Naganathan, N.G. Finit Elmnt Approach to Modl and Analyz Pizolctric Actuators. JSME Intrnational Journal, (200) 44: [4] I. Bruant, G. Coffignal, F. Ln, M. rg. Activ control of bam structurs with pizolctric actuators and snsors: modling and simulation. Smart Matrial Structurs (200) 0: [5] Q. Luo, L. Tong. An accurat laminatd lmnt for pizolctric smart bam including pl strss. Computational Mchanics (2004) 33: [] Kutiš,., Murín, J., Blák, R., Paulch, J. Bam lmnt with spatial variation of matrial proprtis for multiphysics analysis of functionally gradd matrials. Computrs & Structurs, (20) 89: [7] Prumont, A. Mchatronics Dynamics of Elctromchanical and Pizolctric Systms, Springr, (200). [8] Prumont, A. ibration Control of Activ Structurs, Kluwr, (2002). 5