FEEDDBACK CONTROL OF PIEZO-LAMINATE COMPOSITE PLATE. Hafez Ave, Tehran 15914, Iran

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1 ICSV14 Carn Autrala 9-12 July, 2007 FEEDDBACK CONTROL OF PIEZO-LAMINATE COMPOSITE PLATE A. Yelagh Tamjan 1, M. Abouhamze 1, R. Mrzaefar 1, A.R. Ohad 1, M.R. Elam 1 1 Department of Mechancal Engneerng, Amrkabr Unverty of Technology (Tehran Polytechnc) Hafez Ave, Tehran 15914, Iran ayelagh@aut.ac.r Abtract Baed on the applcablty of mart materal n controllng the behavour of engneerng tructure, a new feedback control algorthm mplemented to control the dynamc repone of compote lamnate ung bonded pezoelectrc enor and actuator. Fnte element method baed on a hgher order theory of lamnated plate ued for analyzng the dynamc charactertc of the pezo-lamnate. Verfcaton of the code made by comparng the reult wth fnte element package ANSYS. Fnally, feedback control parameter, couplng the enor and actuator functon, contanng dplacement and velocty gan are changed and ampltude of dynamc repone thereby controlled. 1. INTRODUCTION Compote tructure and, n more practcally ued knd, lamnated component are beng ued vatly n aeropace and automotve applcaton due to ther hgh trength and tffne to weght rato. In addton, ue of them allow the degner to chooe between many poble tructural layout of the materal, n order to obtan hgh tructural performance. Pezo-lamnate a mart-ntellgent compote offer great potental for actve control of advanced aeropace, nuclear, and automotve tructural applcaton. From the prevou decade, many fnte element model have been propoed for modelng tructure along wth pezoelectrc actuator and enor. Mota et al. [1] have utlzed the CLPT for fnte element modelng of the actve vbraton control of a thn compote lamnated plate contanng pezoelectrc layer. However, Chandrahekhara and Tennet [2] and Banal and Ramawamy [3] have contructed ther formulaton on the ba of Frt-Order Shear Deformaton Theory of lamnated plate (FSDT). They have ued hear flexble four and nne node element for the ame purpoe to conder tranvere hear effect, repectvely. Furthermore, Hgher-Order Shear Deformaton Theory (HSDT) ext whch aume a parabolc dtrbuton of tranvere tran through the thckne, whch wa ntroduced by Reddy [4] for the analy of compote lamnated plate. Th theory ha been utlzed by Zhou et al. [5] for modelng a compote plate wth bonded pezoelectrc layer (wthout actve control). Dynamc behavor of lamnated plate mportant to be condered when

2 tranent load are appled. Dynamc repone of bmorph plate and beam are tuded by Wang [6]. In the author prevou work, fnte element formulaton were developed for the hape and vbraton control of Functonally Graded Materal (FGM) plate [7] baed on HSDT ung pezoelectrc enor and actuator. The effect of conttuent volume fracton (a a charactertc of the FGM materal), the confguraton of the enor/actuator par and the velocty and dplacement feedback control gan on the tatc and dynamc repone of the tructure were tuded n ther work. In th tudy, followng the tendency to tudy mart compote, actve control of compote lamnated plate beng to be condered. Fnte element method ha been choen to analyze the actve control of vbraton and dynamc repone of lamnated plate bonded to pezoelectrc actuator/enor patche. The utlzed element of four node econd order type. In order to nvetgate thck plate and alo the hear tranvere effect n the pezo-lamnate, fnte element formulaton derved baed on the hgher order hear deformaton theory of lamnated plate. In order to prevent hear lockng phenomenon n thn plate, C 1 contnuou element are ued n the preent work. A new feedback control algorthm, whch ntroduced by the author [7], mplemented to control the tatc deflecton, natural frequency and dynamc repone of compote lamnate ung bonded pezoelectrc enor and actuator. In the prevou work performed on the pezo-lamnate, ether no actve control mplemented (.e. Ref [3]) or only one control gan condered n the control trategy. For ntance, n the work publhed by Chandrahekhara and Tennet [2], actve control acheved ung only velocty control gan. However, n the feedback control ytem ued here, both dplacement and velocty gan are condered whch can be adjuted wth the correpondng parameter. Verfcaton of the developed fnte element code are carred out by comparng the oluton wth fnte element package ANSYS and proper correlaton nvetgated and hown. Fnally, feedback control parameter, couplng the enor and actuator functon, contanng dplacement and velocty gan are changed alternatvely and conequently the tatc and dynamc repone of the tructure are controlled. The numercal reult how how tatc deflecton, natural frequence and peak repone can be controlled by the dplacement control gan and actve dampng can be provded by adjutng the velocty control gan. 2. MATHEMATICAL FORMULATION 2.1 Governng Equaton Accordng to the lnear theory of pezoelectrcty [8], conttutve equaton for a pezoelectrc materal contanng drect and ndrect effect can be wrtten a below: σ = C ε e E (1) j jkl kl jk k D == e ε + K E (2) jk jk j j Whereσ j, D, E j, C jkl, e jk, and K j are the tre tenor, electrc dplacement, electrc feld vector, elatc modulu, electrc permttvty, and pezoelectrc tenor, repectvely. Dplacement-baed formulaton wll be made by the varatonal prncple [9] whch n the form: t1 t0 V.. t1 ( ρ u δu σ δε + f δu ) dvdt + D δe dv dt Tδu dsdt + Q δφdsdt = 0 (3) j j t0 Vp where S and V repreent the urface area and volume of both the compote and pezoelectrc materal, whle S p and V p reply for the urface area and volume of the pezoelectrc materal, p t1 t0 S t1 t0 S p

3 repectvely. Parameter T ndcate the appled urface tracton and Q the electrcal charge appled to the urface of pezoelectrc actuator. 2.2 Fnte Element Formulaton In order to mathematcally formulate the fnte element model, The dplacement feld aumed accordng to the Hgher order Shear Deformaton Theory (HSDT) [10]. By ung the HSDT, a four node plate element wth even degree of freedom contanng u 0, v 0, β, β (dplacement of a pont n the mddle or reference urface n x, y, drecton and x y the rotaton of normal to the md-urface about y and x axe, repectvely, nterpolated ung blnear Lagrangan nterpolaton functon) and w 0 ( dplacement z drecton), w0 / x, and w0 / y (nterpolated ung Hermte nterpolaton functon) choen. In addton, element contanng actuator and enor patche have two electrc potental degree of freedom. Structural and electrcal contnuou varable are etmated n term of nodal dplacement a: u = N u e (4) { } [ u ]{ } { φ} N { e φ φ } = (5) where N u ],[ N ] are the tructural and electrcal hape (nterpolaton) functon, repectvely. [ φ Notaton { e } u and { φ e } ndcate the tructural and electrcal generalzed nodal varable vector. Wth the above nterpolaton, tran component and the electrc feld can be obtaned through dervaton: e { E} = ϕ = [ B ]{ ϕ } (6) where B ] [ N ] = and [ B ] [ ϕ ϕ u ϕ e { ε } = [ B u ]{ u } (7) a matrx contanng the dervatve of nterpolaton functon. The knetc and tran energy of the panel can be readly found a: T = 1 2 n h k= 1 hk + 1 k k k k k k k T ρ du dv dw du dv dw k dz dxdy (8) dt dt dt dt dt dt n p hk + 1 T T n 1 U = { } { } { } { } 2 σ ε dz dxdy E D dv h (9) k V p k = 1 m = 1 where n number of layer of the lamnated plate, n p number of pezoelectrc layer, ρk denty, hk thckne, and k k k du dv dw th velocty vector of the k layer. The work done by dt dt dt the precrbed tracton T, body force F b and the appled electrcal charge denty Q on the actuator gven by: { } { } { φ } T T T W = u TdA + u F dv QdA b A V A p n whch φa refer to the electrc potental on the actuator. Now the Hamlton prncple appled to obtan the fnte element governng equaton of moton: t δ ( T U W ) dt = 0 0 (11 ) By ntroducton of Eq. (8)-(10) n Eq. (11) we wll obtan the equaton of moton of the pezo-lamnated plate a follow: a (10)

4 .. [ M ] u + [ K ] u + [ K ] φ = { F } uφ m (12) [ K ]{ u} [ K ]{ } = { F } (13) φu φφ φ a q a [ K ] { u} [ K ] { } = 0 (14) φu φφ φ where [ M ],[ K ],[ K u φ ], and [ K φφ ] are ma, tructural tffne, coupled tructuralelectrc tffne, and electrc tffne matrce, repectvely. Vector{ F } the appled electrcal charge to the pezo-actuator patche and { Fm } the appled mechancal loadng. Subttutng Eq. (13) and (14) n Eq. (12) reult n the overall fnte element governng equaton of moton: [ M ]{ u} ([ K ] [ K u ][ K ] φ φφ [ K φu ]){ u} { Fm } [ K uφ ] a[ K φφ ] a { Fq } a + + = + (15) The nduced electrc potental n enor can be calculated from Eq. (14): 1 { φ} = [ K ] [ K ] { u} (16) φφ φu By ung the cloed loop feedback control algorthm that wll be ued n the actve control of the tructure:. { φ} = G { φ} + G { φ} (17) a d v wth Gd a the dplacement control gan and Gv a the velocty control gan. Now we ubttute Eq. (16), (17) to obtan: n whch:... * * [ M ]{ u} + [ C ]{ u} + [ K ]{ u} = { Fm } (18) * 1 1 [ K ] = [ K ] + Gd [ K uφ ] a[ K φφ ] [ K φu ] + [ K uφ ] [ K φφ ] [ K φu ] * 1 [ C ] = Gv [ K uφ ] a[ K φφ ] [ K φu ] + [ C p ] * { F } = { Fm } where C p the dampng matrx. q a (19) 3. RESULTS AND DISCUSSION 3.1 Verfcaton of the Fnte Element Code Wth the purpoe of valdatng the oluton obtaned from the developed fnte element code, verfcaton are made by comparng the reult obtaned for natural frequence and tatc deflecton under electrc and mechancal load wth the one ganed from the commercal fnte element code ANSYS. It hould be noted that comparon are made for no feedback control tuaton, nce mulatng the cloed loop control proce n ANSYS mpoble. Each pezoelectrc layer ha a thckne equal to m; however the core lamnated plate 0.005m thck wth lamnaton angle [0/90/90/0]. The plate of quare type wth length of 40cm and properte of both lamnated and pezoelectrc materal, condered n the preent tudy, are mentoned n Table 1 and Table 2, repectvely. Comparon of the frt fve natural frequence and centrelne 3 2 deflecton under unform load of N / m and n the cae of electrcal load, an addtonal voltage equal to 40 V appled on the actuator layer are llutrated n Table 3 and

5 Fgure 1 and from whch a good correlaton can be oberved. The well agreement of the oluton becaue of prece conderaton of tranvere hear deformaton effect n the hgher order hear deformaton theory ued n the preent tudy. Table 1: Properte of the lamnated materal. Graphte /Epoxy E =150GPa E 22 = E33=9GPa G 12 = G13 =7.1GPa 11 3 G =2.5GPa ν12 = ν13 = ν 23 = 0.3 ρ = 1600 Kg / m 23 Table 2: Properte of the pezoelectrc materal. Elatc modulu Poon rato Denty Pezoelectrc contant Delectrc coeffcent (Kg/m 3) (m/v) 9 (N/m) (F/m) Table 3: Comparon of the frt fve natural frequence (Hz) obtaned from HSDT and ANSYS package ANSYS HSDT Fgure 1: The centerlne deflecton of plate under unform dtrbuted load 2.5 m N / and actuator Voltage V = 40v.

6 Fgure 2: The centrelne deflecton of plate under unform dtrbuted load 2.5 m N / and actuator Voltage V = 40v. 3.2 Statc Deflecton Control In th ecton, a cantlever plate wth enor/actuator patch locaton condered for the analy. The materal properte are thoe gven n Table 1 and Table 2 and thckne of each patch and the lamnated part (the lamnaton of the compote core [0/90/90/0]) m and 0.01m, repectvely. A can be concluded from the mathematcal formulaton of the problem, the dplacement control gan value are conted n the tffne matrce and therefore change of t would have effect on the tatc deflecton of the lamnated plate. Wth the purpoe of obervng th effect, two dfferent dplacement gan value are condered n the analy and the reult for tp deflecton of the cantlever plate under mechancal and electrcal load are llutrated n Fgure 3. From the fgure, t can be concluded that reducton of the dplacement gan (Gd) value reult n decreae of deflecton. Therefore, wth varyng the Gd, tatc deflecton of the tructure can be controlled. Fgure 3: Tp deflecton of the pezo-lamnated plate under mechancal load 3.3 Dynamc Repone Control In th ecton we are gong to control the behavor of the lamnated plate n free vbraton and tranent dynamc condton. The properte and dmenon of the plate and pezoelectrc patche are the ame a the one mentoned n the prevou ecton. Frt the natural frequence of the plate are actvely controlled wth changng G d. Reult are ponted out n Table 4. A G d decreaed, natural frequency value are ncreaed. Table 4: Frt fve natural frequence (Hz) of the plate for two dfferent dplacement control gan

7 Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 G d = G d = Dynamc repone of the lamnated plate alo controlled wth pezoelectrc enor/actuator patche. The plate ubjected to a unt force at the tp n vertcal drecton and then releaed. Th would provde an ntal dplacement for the plate to be aumed a an ntal condton for the dynamc tranent problem. The reult are plotted n Fgure 4 n whch two dfferent dplacement gan value and a velocty gan value equal to 0.01 are condered and can be oberved from that reducng G d caue a reducton n the peak repone of the tructure. Now we want to ee the effect of G v on the tranent repone of the plate. The repone of the lamnated plate for dplacement gan value ame a n Fgure 4 and a dfferent velocty gan value equal to 0.1 plotted n Fgure 5. A can be een by comparng the two fgure and recallng the contrbuton of Gv to the dampng matrx of the tructure, varaton of velocty gan value ha affected the dampng proce of the tranent repone. It can be concluded that G d ha effect on the peak repone and G v ha effect on the dampng of the lamnated plate. In order to oberve the nfluence of a dfferent orentaton of fber n the compote plate on the dynamc repone, the ame reult are obtaned for lamnaton angle [30/45/60/90]. Dynamc deflecton of th plate llutrated n Fgure 6. The ame effect of G d and G v can be concluded on the peak repone and dampng peed of the repone. Agan, the reducton of dplacement gan reduce the ampltude of dynamc repone and the nfluence of velocty gan obervable. Fgure 4: Dynamc deflecton at the tp of the cantlever plate wth dfferent dplacement gan value and velocty gan value of 0.01 (wth lamnaton angle [0/90/90/0])

8 Fgure 5: Dynamc deflecton at the tp of the cantlever plate wth dfferent dplacement gan value and velocty gan value of 0.1 (wth lamnaton angle [0/90/90/0]) Fgure 6: Dynamc deflecton at the tp of the cantlever plate wth dfferent dplacement gan value and velocty gan value of 0.1 (wth lamnaton angle [30/45/60/90]). 4. CONCLUSIONS Fnte element formulaton are preented bae on HSDT for actve control of a pezolamnated plate under electrc and mechancal load. Verfcaton of the code carred out by comparng the reult wth poble oluton made by ANSYS fnte element package. Influence of dfferent orentaton of fbre n the lamnated plate oberved on the oluton. Then actve control of tatc and dynamc repone of the tructure made and the nfluence of dplacement and velocty control gan oberved. REFERENCES

9 [1] J.M.S. Mota, I.F.P. Correa, C.M.M. Soare, C.A.M. Soare, "Actve control of adaptve lamnated tructure wth bonded pezoelectrc enor and actuator", Computer and Structure 82, (2004). [2] K. Chandrahekhara, R. Tennet, "Thermally nduced vbraton uppreon of lamnated plate wth pezoelectrc enor and actuator", Journal of Smart Materal and Structure 4, (1995). [3] A. Banal, A. Ramawamy, "FE analy of pezo-lamnate compote under thermal load", Journal of Intellgent Materal Sytem and Structure 13, (2002). [4] J.N. Reddy, "A mple hgher order theory for lamnated compote plate", Journal of Appled Mechanc 23, (1984). [5] X. Zhou, A. Chattopadhyay, R. Thornburgh, "Analy of pezoelectrc mart compote ung a coupled pezoelectrc-mechancal model", Journal of Intellgent Materal Sytem and Structure 11, (2000). [6] S.Y. Wang, "A fnte element model for the tatc and dynamc analy of a pezoelectrc bmorph", Internatonal Journal of Sold and Structure 41, , (2004). [7] A.Yelagh Tamjan, R. Mrzaefar,A.R. Ohad, M.R. Elam, Vbraton Control of FGM Plate wth Pezoelectrc Senor and Actuator Ung Hgher Order Shear Deformaton Theory, 8th Bennal ASME Conference on Engneerng Sytem Degn and Analy, July , Torno, Italy. [8] H.F. Terten, "Lnear pezoelectrc plate vbraton", New York: Plenum, [9] J.N. Reddy, "Energy and varatonal method n appled mechanc", New York: Wley,1984. [10] J.N. Reddy, "Mechanc of lamnated compote plate and hell", New York: CRC Pre, 2004.