Buckling Analysis of FG Plate with Smart Sensor/Actuator

Transcription

1 Journl of Solid Mechnic Vol., No. (9) pp. - Bucling Anlyi of FG Plte with Smrt Senor/Actutor N.S. Vilini,*, S.M.R. Khlili,, H. Porrotmi Deprtment of Mechnicl Engineering, Ilmic Azd Univerity, Abhr Brnch, Abhr, Irn Fculty of Mechnicl Engineering, K.N. Tooi Univerity of Technology, Tehrn, Irn Fculty of Engineering, Kington Univerity, London, UK Received November 9; ccepted November 9 ABSTRACT In thi pper, the ctive bucling control of mrt functionlly grded (FG) plte uing piezoelectric enor/ctutor ptche i tudied. A imply upported FG rectngulr plte which i bonded with piezoelectric rectngulr ptche on the top nd/or the bottom urfce() ctutor/enor i conidered. When contnt electric chrge i impoed, the governing differentil eqution of motion re derived uing the clicl lminted plte theory (CLPT). The olution for the eqution of motion i obtined uing Fourier erie method nd the effect of feedbc gin on the criticl bucling lod for PZT-4 i tudied.the bucling behvior of mrt plte ubjected to compreive lod i lo invetigted. The enor output i ued to determine the input to the ctutor uing the feedbc control lgorithm. The force induced by the piezoelectric ctutor under the pplied voltge field, enhnce the criticl bucling lod. 9 IAU, Ar Brnch. All right reerved. Keyword: Smrt functionlly grded mteril; Piezoelectric; Bucling; Senor; Actutor INTRODUCTION T HE coupling of the eltic nd electric field in piezoelectric mteril i ued in mrt plte nd hell by bonding piezoelectric ptche for ening nd control. A foundtion wor on the bucling of tructure i preented by Bruh nd Almoroth []. They exmined the effect of initil imperfection on the criticl lod. Turvey nd Mrhll [] tudied the bucling nd potbucling of compoite plte due to the mechnicl nd therml lod. Mny invetigtion re conducted on the tbility nlyi of imperfect tructure. Eltic, pltic nd creep bucling of imperfect cylinder under mechnicl nd therml lod i tudied by Elmi nd Shriyt []. Movrli nd Elmi [4] tudied the thermoeltic bucling of iotropic nd orthotropic plte with imperfection. Murphy nd Ferreir [5] invetigted therml bucling nlyi of clmped rectngulr plte bed on energy conidertion. The firt ppliction of piezoelectric mteril i referred to the experimentl wor of Biley nd Hubbrd [6], in which thi mteril w ued ctutor for the vibrtion control. Recently, everl ttempt re mde to overcome thee problem by uing Functionlly Grded Mteril (FGM) [7]. FGM re novel nd microcopiclly inhomogeneou in which the mechnicl propertie vry moothly nd continuouly from one urfce to nother. It h mny fvorble performnce in engineering ppliction uch high reitnce to lrge temperture grdient nd reduction of tre concentrtion. Comprehenive wor on the pot-bucling of tructure of purely FGM or FG lminted plte hve been reported in the literture. Feldmn nd Aboudi [8] tudied the eltic bifurction bucling of FG plte under in-plne compreive lod nd preented the bucling lod of rectngulr plte with both imply upported nd clmped edge. Woo nd Meguid [9] derived n nlyticl olution expreed in term of Fourier erie for lrge diplcement of FG plte nd hllow hell under both trnvere mechnicl loding nd temperture rie field. Jvheri nd Elmi [] invetigted the therml bucling of rectngulr FG plte bed on clicl well higher-order her deformtion theorie nd obtined * Correponding uthor. Tel: nd E-mil ddre: vnvid@hotmil.com. 9 IAU, Ar Brnch. All right reerved.

2 N.S. Vilini et l. the cloed form olution of the problem under everl type of therml loding. From the vriou ind of mteril vilble for ue in the mrt tructure, only piezoelectric hve the unique cpbility to be ued effectively both ctutor nd enor element. Still vriou different piezoelectric mteril lie PZT-, PZT-4, PZT-5, PVDF nd PMN re under invetigtion for eropce nd other ppliction. Moreover, ince the PZT h higher piezoelectric, dielectric nd elticity coefficient it i widely ued in different indutril ppliction. The fundmentl wor of Tierten [] gve much of the necery theoreticl development for the ttic nd dynmic behvior of ingle-lyer piezoelectric plte. Lee nd Moon [], nd Lee [] ued the umption of Clicl Lminted Plte Theory (CLPT) to derive imple model for nlyi of piezoelectric lminte, ued primrily in deigning the piezoelectric lminte in bending nd torionl control. In the wor of Thompon nd Loughln [4], the ctive bucling control i exmined uing n nlyticl model nd experimentl procedure. In the wor of Vreli nd Srvno [5], the initil bucling of mrt bem nd plte i tudied uing n electromechniclly coupled formultion in combintion with n 8-node FE nd the criticl bucling lod i exmined by ltering the electricl condition. The iue of ctive bucling compention i lo tcled for mrt plte, however, the therml effect re not incorported. More recently, Yng [6] included higher order (qudrtic) electric potentil vrition through the thicne of the ctutor nd obtined two-dimenionl eqution for bending motion of n eltic plte with prtilly electroded piezoelectric ctutor ttched to the top nd the bottom urfce of thic plte. Due to the dded weight effect nd the difficultie in the fbriction conidertion inted of covering the entire urfce in the form of lyer, dicrete monolithic type of piezoelectric mteril cn lo be ued for ening nd control purpoe. In other word, when ingle piece of fully ditributed piezoelectric enorctutor i ued, there re ome obervbility nd controllbility deficiencie in monitoring nd controlling the plte nd the hell repone [7]. Sptil hping of ditributed enor nd ctutor, for exmple by egmenting them into number of mller piece, cn improve the controllbility nd obervbility. The effectivene of ditributed piezoelectric enor nd ctutor in the form of rectngulr ptch, bonded on imply upported rectngulr thin compoite plte nd circulr cylindricl hell hve been invetigted by Tzou nd Fu [8]. Krgrnovin et l. [9] invetigted the vibrtion control of FG plte ptched with piezoelectric ctutor nd enor under contnt electric chrge. In thi pper, the ctive bucling of mrt FG plte uing piezoelectric enor/ctutor ptche i tudied. The plte i imply upported nd i bonded with piezoelectric rectngulr ptche on the top nd/or the bottom urfce ctutor or enor. The clicl plte theory i ued to derive the governing differentil eqution of motion. The olution i obtined by Fourier erie nd the effect of feedbc of FG plte on the criticl bucling lod for PZT-4 piezoelectric ptche i tudied. The enor output i ued to determine the input to the ctutor uing the feedbc control lgorithm. MATHEMATICAL FORMULATION AND MATERIAL DESCRIPTION. Strin-diplcement reltion The diplcement field i bed on clicl lminted compoite plte theory nd i expreed follow [] w u( x, y, z, t) = u ( x, y, t) z x () w v( x, y, z, t) = v ( x, y, t) z y () w ( x, y, z, t) = w( x, y, t) () where u, v, w re the diplcement long the x, y, z coordinte direction, repectively for point on the midplne ( z = ). Auming the mll diplcement, the trin-diplcement reltion te the form ε ε x εyy = εyy + z y ε ε (4) IAU, Ar Brnch

3 Bucling Anlyi of FG Plte with Smrt Senor/Actutor where T T yy ( ε ε ε ) u v u v = + x y y x T w w w ( x y ) = x y x y T (4) (4b). FG plte FGM re microcopiclly inhomogeneou mteril with the mechnicl propertie vry moothly nd continuouly through the thicne. FGM re typiclly mde from mixture of cermic nd metl or combintion of different metl. The cermic contituent of the mteril provide the temperture reitnce due to it low conductivity nd the metl prt prevent frcture cued by tree due to high temperture grdient in very hort period of time []. Furthermore, mixture of cermic nd metl with continuouly vrying volume frction cn be eily mnufctured. The coordinte ytem nd definition of the FG plte with piezoelectric ptche re hown in Fig.. In thi wor, it i poible to conider the FGM propertie, E, vrie ccording to the following expreion [] E = E( z) = E e β z (6) υ FGM = cont. (6b) where E = ln b E β (7) where E nd E re the vlue of eltic modulu t x= nd x=b, repectively, where b i the width of the FG plte. The Poion rtio υ, i conidered to be contnt... Stre trin reltion in FG plte The tre-trin reltion for the FG plte re imilr to thoe of orthotropic plte. The only difference i relted to the elticity module which i no longer contnt. Hence, the contitutive reltion for the FG lminte re [] G Actutor ptch FG plte z = h / z = h / z = h / + h A z = h / h B Senor Fig. Coordinte ytem nd definition of FG plte with piezoelectric ptche enor nd ctutor fixed to the lower nd upper ide long with the controlling circuit. 9 IAU, Ar Brnch

4 4 N.S. Vilini et l. σ c c ε σ yy c c ε yy τ = c ε yz 44 yz c55 τ xz ε xz τ c 66 ε E( z) E( z) in which c = c =, c c υfgmc, c 44 c 55 c 66 υ = = = = = fgm ( + υfgm ), τ ij re the her tree nd ε ij re the correponding trin. nd (8) σ ii re the norml tree.4 Piezoelectric mteril tre trin reltion The contitutive reltion in term of non-zero tre nd trin component for the ttched piezoelectric ptche re [] T { σ} [ C]{ ε} [ e] { E} { D} []{ e E} + []{ E} = (9) = (9b) or in the mtrix form σ c = σ yy c τ Dx = Dy DZ e c c ε ε yy c 66 ε ε + ε yy ε e e Ex e Ey Ez E x Ey Ez () (b) in which { σ } repreent the column of tre component, { ε } the column of trin component, { E } the column of electricl field component, { D } the column of the electricl diplcement component, [C] the mtrix of eltic contnt, [ e ] the piezoelectric tre coefficient mtrix nd [ ] the dielectric permittivity mtrix in which [ e ] = [ d][ C] () where [d] i the piezoelectric trin coefficient mtrix..5 Actutor nd enor contitutive reltion.5. Actutor contitutive reltion For piezoelectric ctutor ptch the Mxwell eqution re [5] D =. E. D = () in which D,, nd E re electric diplcement, dielectric contnt nd electric field, repectively. If the ptch i thin, then it cn be umed tht x nd y component of the electric diplcement field re contnt within the ptch long x nd y direction, hence the chrge Eq. () reduce to [4] IAU, Ar Brnch

5 Bucling Anlyi of FG Plte with Smrt Senor/Actutor 5 D D x y Dz ddz + + = = x y z dz () Moreover, it i umed tht the electric potentil inide the ctutor ptch vrie qudrticlly in the z-direction, i.e. + zφ + z φ φ = φ (4) Furthermore, it i umed tht the electric boundry condition for the ctutor ptch re φ = V φ = t t h z = z = + h h z = z = A (5) Uing thee condition in Eq. (4) yield to V ha h) ha φ = ( + φ (6) in which, = z z ) i thicne of ctutor. Now by combining Eq. (b) nd () nd conidering h A ( φ E z =, it cn be obtined z e w w φ = + (7) x y By ubtituting Eq. (6) nd (7) into Eq.(4), the electric field inide the ctutor ptch become E z ϕ V hm z w w e z h A x y = = + + (8) z + z h + ha in which, h m = =. In the next tep by ubtituting Eq. (8) into Eq. (), the tre- trin reltion for the ctutor ptch yield to e V e( z hm ) w w σ = cε + cεyy + + h A x y ev e( z hm ) w w σyy = cε + cεyy + + h A x y τ = c γ 66 (9).5.. Senor contitutive reltion From the reduced chrge eqution, the z component of the electric diplcement of the enor ptch hould be contnt long the thicne (z-direction, ee Eq. (4)). On the other hnd, ince there i no externl upply of 9 IAU, Ar Brnch

6 6 N.S. Vilini et l. electric chrge to the ptch, the totl chrge which pper on the enor ptch electrode urfce hould be zero. Thi condition cn be repreented D z z= z + D = z z= z. From thi condition nd the reduced chrge eqution which i zero i.e. D z =, everywhere inide the ptch, it i poible to get D = e ε + e ε + E () z yy z Then, the electric field in the enor cn be obtined E z e u v w w = + z + x y x y () Now by ubtitution Eq. () into contitutive Eq. (), the tre-trin reltion for the enor ptch cn be obtined σ = + e e yy c ε + c + εyy e e σ = c + ε + c + εyy () τ = c 66 y The verge enor potentil (the voltge tht pper between enor electrode) i he u v h+ h w w V = A = A A A x y x y ( φ z = z φ ) d d z = z () When the tructure ocillte, rel-time voltge (V ) ignl will be generted in the enor. The ignl i fed to the control lgorithm tht determine the power input of the ditributed ctutor on the top plte hown in Fig.. If V i the ctuting voltge determined by the control lgorithm, nd the enor ignl i V for the contnt gin control lgorithm, then, V i obtined by the following e Ghhm w w V = GV, V = + x y (4) in which h m h + h = nd G i feedbc gin. GOVERNING EQUATIONS OF EQUILIBRIUM The eqution of motion for FG rectngulr plte with n ctutor nd enor ptche bonded to the upper nd lower fce, repectively nd under ditributive lod of q, cn be obtined uing the clicl plte theory nd Nvier eqution. The motion eqution re (ee Fig. ) σ + σ + σ = ρ, x, y xz, z u σ + σ + σ = ρ (5) yx, x yy, y yz, z v σ + σ + σ = ρ zx, x zy, y zz, z w IAU, Ar Brnch

7 Bucling Anlyi of FG Plte with Smrt Senor/Actutor 7 Now, by integrting through the thicne of ech lminte, the equilibrium eqution cn be obtined N + N = ρu, x, y N + N = ρv (6) yy, y, x M + M + M = ρw q,, yy, yy in which q repreent the pplied uniform ditributed lod over the plte urfce. Collection of the equilibrium eqution of ctutor nd enor ptch nd rerrnging them give N N N,, +, x, y =,, yy, y N, x,,,,, yy, yy + = (7) M + M + M = nd the equilibrium eqution for FG plt (Hot plt) re N N H H, x + N, y = H H yy, y N, x + = (8) H H H H H H, +, + yy, yy =, + yy, yy +, M M M ( N w N w N w ) nd the equilibrium eqution for mrt plt re N N, x + N, y = yy, y N, x + = (9) H H H, +, + yy, yy =, + yy, yy +, M M M ( N w N w N w ) Furthermore, the membrne reultnt force cn be found H N σ σ σ z z z H N yy = σyy d z + σyy R d z + σyy R dz z z z H N σ σ σ () nd the reultnt moment re H M σ σ σ z z z H M yy = σyy z d z + σ yy R z d z + σyy R z dz z z z H M σ σ σ () ij ij H σ ij Hereσ, σ nd indicte the tre component of the, ctutor, enor nd FG plte, repectively. R repreent the loction function (Box cr function) defined [6] L< x < L, Lb< y < Lb R( x, y) = [ H( x L) H( x+ L) ][ H( y Lb) H( y+ Lb) ] = () elewhere 9 IAU, Ar Brnch

8 8 N.S. Vilini et l. in which, H i the Heviide function. In ddition, ha, h nd L nd Lb re the thicnee of the ctutor nd the enor ptche, the length nd width of the enor nd ctutor ptche, repectively. Note tht h / I H = ρ ( z ) d z i the m denity of FG plte nd h repreent the thicne of the FG plte nd nd b re h / the length nd the width of FG plte, repectively. 4 EQUILIBRIUM EQUATION OF SMART PLATE By ubtituting Eq. (4), (4b) nd (4c) into the tre-trin reltion of the ctutor, the enor nd FG plte i.e. Eq. (8), (9) nd () the tre component of the ctutor, the enor nd the FG plte will be obtined. Then, by ubtituting the tre component of the mrt plte into Eq. () the moment reultnt eqution given below M M M yy c = c4 c7 c c c 5 8 w c x w c 6 y c 9 w x y b + b 4 b7 b b b 5 8 w b x w b 6 y b 9 w x y H w x w 6 y 9 w x y () in which c i, b i nd i component re eg( h + hs ) hsa er B ( h + ha ) A c = cbr R + ha eg( h + hs ) hsa er B ( h + ha ) A c = cbr R + ha c7 = c8 = c6 = c = c4 = c c5 = c c66b c9 = R BA D b = R = υ BA D b = R = υ υ b6 = b7 = b8 = b = = 6 = 7 = 8 = b4 = b 4 = b5 = b 5 = c66b D b6 = R 9 = + υ (4) By ubtituting Eq. () into Eq. (9) nd by further implifiction, the equilibrium eqution of the mrt plte i obtined nd then for the rectngulr FG plte with enor nd ctutor ptche, the generl equilibrium eqution reduce to the following IAU, Ar Brnch

9 Bucling Anlyi of FG Plte with Smrt Senor/Actutor AeGhS ( h hs ) Ae( h ha) eb e D 4 w cb + C + B R + w = N x ha υ x (5) where h = + + = / 4 w w w w, D 4 4 z E( z)dz x x y y h /, e e A = C, A = C (6) 5 SOLUTION METHODOLOGY To obtin the criticl bucling lod of the mrt plte, the feedbc gin i et to be zero. The Nvier olution for the imply upported rectngulr FG plte with ctutor nd enor ptche cn be obtined by expnding the diplcement in Fourier erie. Aume double Fourier erie for the lterl diplcement follow [4] w mπx nπy = Amn in in b (7) in which nd b re the length nd width of FG plte. Subtituting expreion (7) into Eq. (5), the criticl bucling lod of the FG mrt plte with piezoelectric ptche, where the in-plne force re N = N cn be obtined π Ae Gh ( h + h ) Ae ( h + h ) e B e D m m n n N ( m, n) [{ c B ( c ) B } R ]( ) 4 4 S S A = m ha υ b b nd by further implifiction, it i obtined (8) π D N ( mn, ) = FR ( m + mn + n) m b υ (9) where b =, h / ( )d, h / D = z E z z Ae Gh ( h + h ) Ae ( h + h ) e B e F = c B + + ( c + ) B S S A ha (4) nd R repreent the loction function (Boxcr function). 6 RESULTS AND DISCUSSION In the preent mrt FG plte, piezoelectric ptch fixed to the bottom urfce i conidered to ene the trin nd generte the electricl potentil field nd piezoelectric ptch fixed to the top urfce, wor n ctutor nd control the bucling of the tructure. For thi FG plte, the mteril nd the geometricl propertie re given in Tble. The mteril nd geometricl propertie of piezoelectric ptche for two type of PZT re given in Tble. Further, it i umed tht e ij =e ji for i j. It hould be noted tht the firt derivtion of R, with repect to x or y yield to the delt function, i.e. R x = [ δ ( x L) + δ ( x + L)][ H ( y Lb) + H ( y + Lb)] nd ince, δ ( y Lb) = δ ( x L) = then R = R = R =. x y 9 IAU, Ar Brnch

10 N.S. Vilini et l. Tble Mteril propertie of the FG plte FGM E ( N / m ) Totl Thicne(m) Size(m) Aluminum oxide(cermic).e Ti-6A-4V(metl).57e.6e-.5.5 Tble Mteril propertie of the piezoelectric ptche (PZT) [6, 7] Piezocermic C C d d d e e e υ Thicne (m) PZT-4.8e 77.8e e-8..5e- To obtin the bucling lod for n iotropic plte under unixil compreive force, vlue of β in the model of FGM nd R in Eq. (9) hould be equl to zero ( β = R = ). In thi wy, the piezoelectric nd the FGM effect on the plte re ignored. By doing o, the bucling lod cn be obtined follow π D 4 4 N ( m, n) = [ ]( m + m n + n m b υ 4 ) (4) in which Eh D = (4) ( υ ) Eq. (4) in the non-dimenionlized form cn be written D N = N b /( π ) (4) After implifiction nd reduction of the model for iotropic plte, Eq. (4) gree well with the reult given in Ref. []. Fig. how the vrition of non-dimenionlized bucling lod for n iotropic plte v. pect rtio S (/b). The trend of vrition i gin imilr to the reult obtined by Ref. []. In Fig. (), the vrition of the bucling lod v. feedbc gin for different pect rtio S i illutrted for PZT-4. By increing S, the plte dimenion incree, which men tht the plte tend to become more flexible nd it cn be oberved, the feedbc gin i more dominnt in increing the criticl bucling lod. Fig. 4 how the effect of feedbc gin on bucling lod of FG plte. It cn be een tht for higher control gin, the bucling lod of the plte i increed. Fig. Non-dimenionlized bucling lod of n iotropic plte v. S (/b) for imply upported plte (m=, m=, m=), (β=, R=). IAU, Ar Brnch

11 Bucling Anlyi of FG Plte with Smrt Senor/Actutor Fig. Vrition of bucling lod v. feedbc gin for different vlue of S=/b for PZT-4 (R=). Fig. 4 Vrition of bucling lod v. S=/b for different vlue of feedbc control gin G nd (R=, PZT- 4) () Gin=, (b) Gin =, (c) Gin=. 7 CONCLUSION In thi pper, FG rectngulr plte which i bonded with piezoelectric rectngulr ptch on the top nd the bottom urfce() ctutor/ enor i conidered. Under contnt electric chrge, the governing differentil eqution of the motion for the plte re derived uing clicl lminted plte theory (CLPT). The olution for the motion eqution i obtined uing Fourier erie method nd the effect of feedbc gin nd pect rtio S on the criticl bucling lod i tudied. Bed on prmetric tudy performed on thi mrt FG plte, it i concluded tht: -The vrition of criticl bucling lod v. feedbc gin indicte tht by increing the feedbc gin, the bucling lod incree. -By increing S, the plte dimenion incree. It men tht the plte i more flexible nd the feedbc gin i more dominnt in increing the criticl bucling lod. REFERENCES [] Bruh D.O., Almroth B.O., 975, Bucling of Br, Plte, nd Shell, McGrw-Hill, New Yor. [] Turvey G., Mrhll I., 995, Bucling nd Potbucling of Compoite Plte, Chpmn-Hll, New Yor. [] Elmi M.R., Shriyt M., 997, Eltic, pltic nd creep bucling of imperfect cylinder under mechnicl nd therml loding, ASME Trnction, Journl of Preure Veel Technology 9 (): 7-6. [4] Movrli A., Elmi M.R.,, Thermoeltic bucling of plte with imperfection bed on higher order diplcement field, Journl of Therml Stree 5(8): [5] Murphy K.D., Ferreir D.,, Therml bucling of rectngulr plte, Interntionl Journl of Solid nd Structure 8 (-): [6] Biley T., Hubbrd J.E., 985, Ditributed piezoelectric-polymer ctive vibrtion control of cntilever bem, Journl of Guidnce, Control nd Dynmic 8: [7] Hue T., Kouvtov A., Steinhuen R., Seifert W., Beige H.,, Bending behvior of functionlly grdient mteril, Ferroelectric 8: IAU, Ar Brnch

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