Mathematical Modeling and Numerical Simulation of Smart Structures Controlled by Piezoelectric Wafers and Fibers

Transcription

1 Mathematcal Modelng and Numercal Smulaton of Smart Structures Controlled by Pezoelectrc Wafers and Fbers U. Gabbert #, W. Kreher*, H. Köppe # # Insttut für Mechank, Otto-von-Guercke Unverstät Magdeburg *Insttut für Werkstoffwssenschaft, Technsche Unverstät Dresden 1 Introducton Smart structures are characterzed by a synergstc ntegraton of actve materals nto a passve structure connected by a control system to enable an automatc adaptaton to changng envronmental condtons 1. The ncreasng engneerng actvtes n the development and ndustral applcatons of structronc (structure + electronc) systems requre effectve and relable smulaton and desgn tools n ths feld 2,3. In vbraton and nose control as well as n shape control pezoelectrc wafers and fbers embedded n composte materals are very common smart structural components n structronc systems. The global behavor of such hgh ntegrated complex engneerng smart structures can be smulated and desgned wth suffcent accuracy on the bass of the fnte element method extended by control and optmzaton strateges. In pezoelectrc controlled smart composte structures the coupled electrothermo-mechancal feld equatons have to be solved, where the parameters of the materal tensors are needed as nput parameters 9,1. These macroscopc (homogenzed) materal tensors of such compostes are non-lnearly dependent on the propertes as well as the arrangements of the consttuents n the composte. The measurement of these data s tme consumng and expensve. Alternatvely, analytcal ( e.g. based on the Mor-Tanaka-type mean feld approach) as well as numercal (e.g. based on the fnte element analyss of a representatve volume element) methods can be used to calculate homogenzed materal tensors of a heterogeneous materal. The paper frst presents the applcaton of the Effectve Feld Approxmaton (EFA) for the analytcal determnaton of homogenzed materal tensors of electro-thermo-mechancal materal systems, consstng of a matrx materal wth pezoelectrc nclusons 7 (e.g. n form of thn pezoelectrc fbers 4 ). Then a bref descrpton of a general purpose fnte element package for the smulaton of electro-thermo-mechancal coupled feld problems s gven 9,1. The presented analytcal homogenzaton strategy can be smply ncorporated n the fnte element smulaton tool. Fnally, as a smple test example an actvely controlled beam structure s gven, where pezoelectrc fber wafers are used as actuators. The homogenzed materal tensors are calculated frst based on the EFA. These data are than used n the fnte element smulaton.

2 2 U. Gabbert, W. Kreher, H. Köppe 2 Consttutve equatons The general lnear relatonshp between the feld quanttes stress σ, stran γ, electrc feld E and delectrc dsplacement D are wrtten here n the form E E σ k = cklmγ lm elk El λkθ, k, l, m = 1,2,3 (1) γ γ D = elmγ lm + εl El + p θ where we have ntroduced the standard notaton for the lnear materal property tensors (θ denotes the temperature change). These consttutve equatons can be wrtten n the compact form wth the 9 9 materal property matrx j = L f + j p, q = 1,2, K, 9 (2) p pq q s p E t c αβ eα j L = (3) γ e β ε j Ths type of consttutve equatons can be appled to descrbe the materal behavor of sngle components as well as of pezoelectrc compostes. 3 Homogenzed propertes of compostes In order to derve homogenzed propertes of fber renforced materals, we apply here the effectve feld method. Ths method s especally useful f we are concerned wth a matrx materal L () whch contans nclusons L (k) ( k=1,2,...,n ). In the theoretcal scheme every ncluson s embedded n an nfnte matrx medum, and the nteracton between dfferent nclusons s projected nto an effectve feld whch acts on the consdered ncluson. Ths effectve feld s then determned n a self-consstent manner 5,6,7. The result for the effectve (homogenzed) materal propertes (here denoted by an upper astersk) can be wrtten nto the followng concse form * () 1 () L = A( L ) A( L ) L (4) * () 1 () s = A( L ) A( L j (5) j ) where the 9 9 matrx () ( I + ( L L ) ) 1 () A ( L ) = P (6) depends on the matrx propertes L (), the ncluson propertes and on P whch s the generalzed Eshelby Tensor formed wth the matrx propertes (see e.g. Huang and Kuo 8 ). Ths Eshelby tensor depends also on the shape of nclusons whch are approxmated here as spherods, where the aspect rato s used to model dfferent ncluson geometry. The angular brackets denote averagng over all types of components.

3 Modellng and Smulaton of Smart Structures Controlled by Pezoelectrc Wafers and Fbres 3 4 Fnte element analyss for coupled problems The basc equaton for the fnte element analyss of an coupled electromechancal feld problem can be obtaned as vrtual energy formulaton that combnes the Cauchy s equaton of moton σ j j + ρ B = ρ a (7), wth the electrcal balance equaton D, = (8) n form of G = ( σ j, j + ρb ρa ) δvkdv + ( D, ) δkdv =. (9) V V 1 Wth the consttutve equatons (1), the stran dsplacement relaton γ = u + u ) j 2 ( j j and the relaton between the electrc feld and the electrc potental E = we can express equaton (9) by the unknown feld varables u and. We approxmate these feld varables n a fnte element by shape functons = (u) L u L = ( L ) ( L ) 1 u N N ( ) L L (1) where N (u), N () are the shape functon for the dsplacements and the electrc potental, respectvely. Followng the standard procedure for the development of fnte element equatons we get the sem-dscrete form of the equatons of moton for a coupled electro-mechancal problem n the form 9 M uu u&& Ruu + && u&? + &? uu u? u K u f = f Based on the above gven theoretcal background, a lbrary of pezoelectrc fnte elements has been developed and tested. Ths lbrary ncludes sold elements, plane elements, axsymmetrc elements, rod elements as well as specal multlayer composte shell elements 9,11 and has been mplemented n the general purpose fnte element system COSAR. For statc solutons a specally optmzed sub-matrx orented Cholesky solver can be used. For the soluton of egenvalue problems the subspace teraton method by Mc Cormck&Noe s used. In transent problems modal based technques as well as tme ntegraton schemes such as the Newmark-Method, the Wlson- Method, or the Central-Dfference-Method can be appled. The fnte element code has the capablty to use a substructure technque. Ths can be utlzed as an excellent precondton to solve optmzaton problems 12 and nonlnear problems effectvely. The fnte element software can also be used to smulate controlled structures. A data nterface between the fnte element software and control desgn tools such as MAT- LAB/SIMULINK has been developed to desgn controller for real engneerng applcatons. u (11)

4 4 U. Gabbert, W. Kreher, H. Köppe 5 Numercal results Frst, homogenzed materal propertes of compostes made of pezoelectrc fbers embedded n a polymer matrx are calculated on the bass of the effectve feld method presented. The electromechancal materal propertes of each of the two consttuents of the composte as well as the calculated homogenzed data for a specal composte consstng of a 3% fber volume fracton and a fber aspect rato of 5 are gven n the Table 1. Table 1. Electromechancal propertes of components and a specal composte (fber volume fracton 3%, aspect rato 5) c E 11 c E 12 c E 13 c E 33 c E 44 e 31 e 33 e 15 ε γ 11 ε γ 33 GPa C/m 2 nf/m PZT-fbers Polymer Composte In Fgures 2a and 2b results are presented for dfferent volume fractons and fber (a) 6. Pezoelectrc constant e 33 [ C m-2 ] Fber aspect rato = Volume fracton fbers 3. (b) Pezoelectrc constant d 33 [ 1-12 m/v ] Fber aspect rato = Volume fracton fbers Fgure 1. Predcted pezoelectrc constants e 33 (a) and d 33 (b) for a composte made of pezoelectrc fbers n a polymer matrx

5 Modellng and Smulaton of Smart Structures Controlled by Pezoelectrc Wafers and Fbres 5 aspect ratos whch show that a hgh fber volume fracton and a large aspect rato s requred to get the desred performance of the composte. Ths s especally mportant when the e 33 -effect has to be utlzed, but s not so crtcal for the d 33 -effect. Note that all calculatons have been performed wth an dentcal polng state of the fbers as shown n Table 1. If the polng of the composte s done "n-stu", the resultng poled fber propertes may vary n the same manner as the effectve propertes of the composte, whch can be also calculated wthn the same theoretcal approxmaton scheme. The man advantages of fber compostes are the better structural conformty n comparson wth PZT wafers (see the low stffness propertes n Table 1) as well as the an-sotropc sensng and actuaton behavor. The homogenzed materal propertes were used for a numercal smulaton. As test case a beam s used wth two attached pezoelectrc fber composte layers at top and bottom of the beam (see Fgure 2), where these layers consst of the homogenzed materal propertes gven n Table 1. The actuators are controlled wth a tme dependng electrc potental ( t) = sn Ωt wth 1 = 1V, Ω = 1s. In Fgure 3 the dsplacement response of a pont at the free end of the beam s shown. X, u 2 x 1 X 3 7,5 mm Beam : Young s modulus E = N/mm 2 Posson s rato ν =.345 Densty ρ =2.69 g/cm 3 42,5 mm Pezoelectrc layer 11 mm + + sn( t),2 mm,5 mm 2 7,5 mm Fgure 2. Elastc beam controlled by pezoelectrc composte fber paches Dsplacement response for a beam wth tme dependng electrcal potental (Dsplacements for a pont at the end of the beam),2 3 Dsplacement u [mm],15 2,1 1,5,1,2,3,4,5,6,7,8,9,1 -,5-1 -,1-2 -,15 -,2-3 tme t [s] u [mm] tme dependng potental Fgure 3. Response of the beam n the tme doman

6 6 U. Gabbert, W. Kreher, H. Köppe 6 Conclusons A smulaton concept for pezoelectrc controlled smart structures has been presented, where n the frst step a homogenzaton method based on the Effectve Feld Approxmaton s gven. In the second step the homogenzed materal tensors are used n the fnte element smulaton of the global structural behavor. In the near future ths soluton wll be used to optmze smart materal systems wth respect to the overall performance of structronc systems. Acknowledgment Ths work has been supported by the German Mnstry for Scence and Technology (BMBF ) and the German Research Socety (DFG). Ths support s gratefully acknowledged. References 1. H. S. Tzou, G. L. Anderson (Eds.), Intellgent Structural Systems, Kluwer Tzou, H.-S., Guran, A. (Eds.), Structronc Systems: Smart Structures, Devces and Systems, Part 1: Materals and Structures, Part 2: Systems and Control, World Scentfc, U. Gabbert (Ed.), Modellng and Control of Adaptve Mechancal Structures, Fortschr.- Ber. VDI Rehe 11, Nr. 268, Düsseldorf, VDI Verlag D. Sporn, A. Schoenecker, Compostes wth pezoelectrc thn fbers frst evdence of pezoelectrc behavour, Mat Res Innovat (1999) 2: S. K. Kanaun, V. M. Levn, Effectve feld methods n mechancs of matrx composte materals, n: Advances n mathematcal modellng of composte materals (ed: K. Z. Markov), Seres on Advances n Mathematcs for Appled Scences, vol. 15, pp. 1-58, World Scentfc Publ. Comp., Sngapore, V. M. Levn, The overall propertes of pezoactve matrx composte materals, n: Contnuum Models and Dscrete Systems (ed: K. Z. Markov), Proc. of the 8th Int.Symposum, Varna 1995, pp , World Scentfc Publ. Comp., Sngapore, V. M. Levn, M. I. Rakovskaja, W. S. Kreher, The effectve thermoelectroelastc propertes of mcronhomogeneous materals, Int. J. Solds & Structures, 36 (1999) J. H. Huang, W.-S. Kuo, Mcromechancs determnaton of the effectve propertes of pezoelectrc compostes contanng spatally orented short fbers, Acta mater., 44 (1996) 12, H. Berger, X. Cao, H. Köppe, U. Gabbert, Fnte Element Based Analyss of Adaptve Structures, n Fortschr.-Ber. VDI Rehe 11, Nr. 268, Düsseldorf, VDI Verlag 1998, pp H. Köppe, U. Gabbert, H. S. Tzou, On Three-Dmensonal Layered Shell Elements for the Smulaton of Adaptve Structures, n Fortschr.-Ber. VDI Rehe 11, Nr. 268, Düsseldorf, VDI Verlag 1998, pp U. Gabbert, A. Görnandt, H. Köppe, F. Seeger, Benchmark Problems for the Analyss of Pezoelectrc Controlled Smart Structures, ASME 1999, Desgn Engneerng Techncal Conferences DETC 99, Las Vegas, Nevada, September 12-16, 1999, paper: DETC99/VIB U. Gabbert, C-T. Weber, Analyss and Optmal Desgn of Pezoelectrc Smart Structures by the Fnte Element Method, Proceedngs of the European Conference on Computatonal Mechancs, August 31 September 3, Munch, 1999, 14 pages