Actuator Location and Voltages Optimization for Shape Control of Smart Beams Using Genetic Algorithms

Transcription

1 Actuators 3,, 8; doi:.339/act4 Articl OPEN ACCESS actuators ISSN Actuator Location and Voltags Optimization for Shap Control of Smart Bams Using Gntic Algorithms Gorgia A. Foutsitzi, *, Christos G. Gogos, Evanglos P. Hadjigorgiou and Gorgios E. Stavroulakis 3 3 Dpartmnt of Accounting and Financ, chnological Educational Institution of Epirus, EI Campus Psathaki, GR8 Prvza, Grc; E Mail: cgogos@tip.gr Dpartmnt of Matrials Scinc and Enginring, Univrsity of Ioannina, GR 45 Ioannina, Grc; E Mail: hadjig@cc.uoi.gr Dpartmnt of Production Enginring and Managmnt, chnical Univrsity of Crt, Institut of Computational Mchanics and Optimization Univrsity Campus, GR 73 Chania, Grc; E Mail: gstavr@dpm.tuc.gr * Author to whom corrspondnc should b addrssd; E Mail: gfoutsi@tip.gr; l.: ; Fax: Rcivd: 8 August 3; in rvisd form: 3 Sptmbr 3 / Accptd: Octobr 3 / Publishd: Octobr 3 Abstract: his papr prsnts a numrical study on optimal voltags and optimal placmnt of pizolctric actuators for shap control of bam structurs. A finit lmnt modl, basd on imoshnko bam thory, is dvlopd to charactriz th bhavior of th structur and th actuators. his modl accountd for th lctromchanical coupling in th ntir bam structur, du to th fact that th pizolctric layrs ar tratd as constitunt parts of th ntir structural systm. A hybrid schm is prsntd basd on grat dlug and gntic algorithm. h hybrid algorithm is implmntd to calculat th optimal locations and optimal valus of voltags, applid to th pizolctric actuators glud in th structur, which minimiz th rror btwn th achivd and th dsird shap. Rsults from numrical simulations dmonstrat th capabilitis and fficincy of th dvlopd optimization algorithm in both clampd fr and clampd clampd bam problms ar prsntd. Kywords: dsign optimization; placmnt optimization; gntic algorithm; grat dlug algorithm

2 Actuators 3,. Introduction Smart or adaptiv structurs with intgratd slf monitoring and control capabilitis ar of grat tchnological intrst du to th incrasing rquirmnts on structural prformanc. h slf monitoring capability of smart structurs has numrous applications in shap and vibration control of structurs, nois rduction, damag idntification, and structural halth monitoring. Notabl rfrncs among othrs ar [,]. Pizolctrics ar th most popular smart matrials, which can b usd both as snsors and actuators. h coupld lctromchanical proprtis of pizolctric matrials, along with thir possibility to b intgratd in various structurs, mak thm suitabl for us in advancd smart structurs. h rcnt advancs in smart structurs hav promptd intrst in modification and corrction of th shap of mchanical structurs,.g., for th corrction of th shap and curvatur of mirrors/antnnas for high pointing accuracy or for maintaining dsird shaps of arospac flxibl structurs, tc. h rviw articl by Irschik [3] dscribs rlvant applications of static and dynamic shap control of structurs by pizolctric actuation. On main objctiv of pizolctric shap control is to optimiz som control paramtrs (.g., th numbr, location and siz of th pizolctric patchs, th amount of lctric potntial to b applid, tc.) so that th dsird shaps ar achivd or bst matchd. Optimization of such paramtrs and configurations of pizolctric actuators for acquiring fficint and prcis shap control has bn an intrsting subjct of rsarch in rcnt yars. ong t al. [4] usd classical mathmatical programming mthods for dtrmining th optimal layout of actuators. Agrawal t al. [5] mployd th simplx sarch algorithm to find th optimal actuator locations and voltags, and found that sparatly optimizing actuator locations and voltags could produc rliabl rsults. Ch t al. [6] prsntd a huristic and intuitiv algorithm for dtrmining th orintation of pizolctric actuator patchs in shap control of smart structurs. Onoda t al. [7] usd a modifid gntic algorithm (GA) and th improvd simulatd annaling algorithm for optimal location of actuators in shap control of spac trusss. A systmatic and gnral mthodology, using a finit lmnt cod and gntic algorithms, for th shap control and/or corrction of static dformations of adaptiv structurs, was proposd and vrifid xprimntally by Silva t al. [8]. Hadjigorgiou t al. [9] invstigatd th shap control and damag idntification of a cantilvr composit bam using a gntic optimization procdur. A comprhnsiv rviw until 3, of th dsign mthodologis and application of formal optimization mthods to th dsign of smart structurs and actuators can also b found in []. In this work, th us of pizolctric actuators for shap control and corrction of static dformations is considrd. h modls widly usd for this kind of problms ar basd on th Eulr bam thory and th Kirchhoff Lov thory of plats with or without lctromchanical coupling (.g., [,]). hs ar considrd to b classical modls, suitabl for thin lastic structurs. An xtnsion basd on imoshnko thory and on inducd strain actuation thory has bn prsntd by Hadjigorgiou t al. [9], which is suitabl for rlativly thick structurs. In this work, a mathmatical modl, basd on th shar dformation thory, which incorporats th lctro-mchanical coupling ffcts, has bn dvlopd to charactriz th bhavior of th structur and th actuators. h mathmatical modl rprsnts an improvmnt ovr th modl prsntd in [9]. Mor prcisly, this modl accountd for th lctromchanical coupling in th ntir bam structur, du to th fact that

3 Actuators 3, 3 th pizolctric layrs ar tratd as constitunt parts of th ntir structural systm. In addition, th mathmatical formulation modls laminatd composit bam structurs, in which th pizolctric matrial may b locatd anywhr within th structur. his formulation is thn implmntd into a finit lmnt program. Bsids stablishing an accurat mathmatical modl for shap control applications, a critical factor for th succss and prformanc of th smart structur is th dtrmination of th optimal location of th pizolctric actuators togthr with th optimal actuation voltags. Nxt, th finit lmnt modl dvlopd is usd in static shap control. Shap control (SC) is dfind hr as th dtrmination of th applid voltags of actuators and thir layout, such that th structur that is activatd using ths paramtrs will conform as closly as possibl to th dsird shap. h problm is formulatd as mixd discrt continuous programming with a quadratic cost function as objctiv. A gntic algorithm is usd as th optimization tchniqu. Gntic algorithms (GAs) is a wll known optimization mthod [3] that blongs to th gnral class of volutionary computation [4], which rlis on th prmis that in a controlld population th individuals having bttr traits will finally stand out. Givn that in th actuator placmnt problm a chromosom ncoding capabl of capturing a solution is rlativly straightforward to construct, GAs sms to b a natural way to confront th problm. Howvr, GA basd optimization approachs hav som issus such as spd of xcution, proof of optimality and othrs that hav to b addrssd in ordr to b succssfully applid. In our approach, a numbr of simulations hav bn prformd in ordr to validat th fficincy of th dvlopd GA basd optimization algorithm in both clampd fr and clampd clampd bam problms.. Formulation of th Problm Considr a laminat formd from two or mor layrs bondd togthr to act as a singl layr matrial and sandwichd btwn two pizolctric layrs. h bond btwn two layrs is assumd to b prfct, so that th displacmnts rmain continuous across th bond. h classical formulation of laminatd matrials is followd [5] and complmntd with lctromchanical coupling trms. h whol continuum has lngth L, thicknss h and width b. h longitudinal and thicknss axs ar along x and z dirctions, rspctivly and th xy plan is th midplan of th bam. h pizolctric layrs hav poling dirction along z axis and th lctric fild is applid through th thicknss dirction. Elastic layrs ar assumd to b insulatd and ar obtaind by annulling th pizolctric constants... Strains and Elctrical Filds For a laminatd bam with midplan symmtry, th displacmnt fild, using th first ordr dformation thory, is xprssd as functions of two indpndnt nodal dgr of frdom of th middl axis, w and y, as:,,,,,,,,,, ux x y z z y x t uy x y z uz x y z w x t () whr w is th transvrs displacmnt of th bam middl axis and y is th rotation of th bam cross sction about th positiv y axis. Assuming small dformation, th strain displacmnt rlation can b xprssd as:

4 Actuators 3, 4 y x z, xz y x w x () A constant transvrs lctrical fild is assumd for th pizolctric layrs and th rmaining in plan componnts ar supposd to vanish. Consquntly, th lctric fild insid th p k th pizolctric layr is givn by whr E B pk k (3) B pk h pk pk and h, ar th thicknss and th lctric voltag of th p k th pizolctric layr. It should b notd that such formulation givs on lctric dgr of frdom pr layr pr lmnt of th lctric fild... Constitutiv Equations h linar constitutiv quations coupling th lastic and th lctric filds in a pizolctric mdium ar xprssd by th dirct and th convrs pizolctric quations, which ar givn as follows: D E, C E k k k k k k k k whr is th strss tnsor, is th strain tnsor, D is th lctric displacmnt, E is th lctric fild, C is th lastic stiffnss matrix, th prmittivity matrix. For non pizolctric layrs, and (4) is th pizolctric constant matrix, and is ar rducd to zro matrics. h constitutiv rlations, givn in Equation (4), ar with rfrnc to th global coordinat systm (x, y, z). For a on dimnsional bam whr th width in th y dirction is strss fr and by using th plan strss assumption, th gnral 3D constitutiv Equation (4) can b rducd to: Q D E E k k k k x x 3 k z 3x 33 z, z xz k Q 55 xz k k (5) whr Q, Q 55 ar th transformd plan strss rducd stiffnss cofficints, 3, 3 ar th transformd pizolctric moduli givn in [5] and 33 is th lctric prmittivity..3. Finit Elmnt Formulation o driv th quations of motion for th laminatd composit bam with surfac bondd snsor and actuator layrs, Hamilton s principl is usd: U W dt (6) h kintic nrgy, th potntial nrgy and th total work don du to virtual displacmnts ar givn as follows:

5 Actuators 3, 5 V, u u u u dv U D E dv x x z z x x xz xz z z V (7) F f F W u u u u ds u u ds q ds c S b x z 3 x z 3 x z 3 S Fc fs Fb S V S whr F c is th concntratd forc vctor, f S is th surfac forc vctor, F b is th body forc vctor, q is th surfac charg vctor, S is th surfac ara whr xtrnal forc is acting, and S is th surfac ara of pizolctric layr whr applid lctric charg is acting. A two nod finit lmnt is considrd with two mchanical dgrs of frdom, w and y, pr nod and on additional dgr of frdom,, pr pizolctric layr. Using standard discrtization tchniqus, whr X w, y, w, y u w, N X N N X, w (9) y w N is a cubic shap function and (8) N is a quadratic shap function. hs shap functions lad to a shar locking fr lmnt and thir xplicit xprssions ar givn in Rf. [9]. h strain fild is givn by:, B X () x xz whr B is th drivativ oprator btwn th corrsponding strain and th gnralizd nodal displacmnts. h lctric voltag vctor of th th lmnt can b xprssd as: pnpl,,..., () whr npl is th numbr of th pizolctric layrs of th th lmnt. Using th variational principl, givn by Equation (6), govrning quations of an lmnt can b writtn as: whr th mass matrix M X Kuu X u m K F K u X K FQ M, th lastic stiffnss matrix K Ku, th prmittivity matrix K load vctor F ar givn in th Appndix. m, th surfac lctric charg dnsity Q uu (), th lctromchanical coupling matrix F and th mchanical h global quations can b obtaind by assmbling th lmntal Equation (). Equation () can b usd in smart structurs applications such as vibration control, static or dynamic shap control, tc. In shap control applications, which is th cas in th prsnt study, th pizolctric layrs ar usd as actuators. hus, all th lctrical dgrs ar considrd as known quantitis and th coupld Equation () rduc to pur mchanical ons: K X F F (3) uu m l

6 Actuators 3, 6 NN N whr K uu R is th global stiffnss matrix, X N R is th mchanical forc vctor and F K F m l R is th nodal displacmnt vctor, is th lctrical forc vctor du to th actuation. A computr cod is dvlopd, basd on th aformntiond finit lmnt modl. A spcial numbring schm is usd to dnot th lmnts with pizolctric layrs. Elmnts with pizolctric layrs ar dnotd by, whil th rmaining layrs ar dnotd by for th idntification during th assmbly procss. 3. Optimal Shap Control h most gnral problm of SC of smart structurs considrs as dsign variabls th applid voltags of actuators, thir layout and thir numbr. h aim is to find th optimal dsign valus so that th diffrnc btwn achivd and dsird shap is minimizd. In ssnc, SC is an invrs problm whr th output, which is th dsird shap, is known and th input actuation paramtrs ar to b dtrmind. hrfor, itrativ huristic mthods ar quit suitabl to this task. In this work, this problm is solvd by a hybrid gntic algorithm. 3.. h Fitnss Function Considring bam lmnt, th shap of a structur is primarily dscribd by th shap of its middl axis, which itslf is dscribd by th transvrs displacmnt of th finit lmnt msh nods. hrfor, a rasonabl cost function is f, as givn by Equation (4), which is th sum of all th squard diffrnc of th transvrs displacmnts btwn th dsird (pr dfind) and th achivd (calculatd) shap at all nods. In th abov quation, r i d i i u f w w (4) d w i is th dsird nodal transvrs displacmnt valu and r is th numbr of concrnd displacmnts. Howvr, th static shap control critria in [9] is basd on th gnralizd displacmnts; that is, on both transvrs displacmnts w and rotations. hrfor, a fitnss function basd on th following cost function is usd in th aformntiond papr: r i d i i y f X X (5) It should b notd that th simultanous usag of displacmnts and rotations in th cost function f is vry rstrictd for bnding problms such as th ons studid hr. Nvrthlss, in this work, th two abov fitnss functions will b usd for comparison rasons. Rsults obtaind by using f as th fitnss function will show improvmnts ovr f. In following, a gnral symbol f is usd to dnot any fitnss function.

7 Actuators 3, Dsign Optimization Problms In gnral th displacmnt fild is a function of th lctric potntial, th layout, th gomtry of actuators and th numbr of actuators. In this framwork, two kinds of shap control (SC) problms of a bam with various boundary conditions ar studid. h Voltag Problm and th Location and Voltag Problm. h first SC problm (th Voltag Problm) consists in finding a st of actuation voltags i for a givn numbr and position of actuators, which minimizs th cost function f undr th constraint: whr i is th actuation voltag of th i th actuator and min and max (6) min i max th lowr and uppr saturation voltags. In this work, this problm is solvd by gntic algorithms. h MatLab softwar packag was usd for th dvlopmnt of an algorithm to optimiz actuator placmnt and voltag for a givn cost function and for givn numbr of actuators and bam dimnsions and proprtis. h computr cod dvlopd maks no assumption of linarity btwn th displacmnts and th lctric voltags, thus, it can b usd for non linar modls as wll. h scond SC problm (th Location and Voltag Problm) is mor gnral. It consists in finding th optimal position and lctric potntial simultanously for a givn numbr of actuators, which minimiz th cost function f. In this problm, it is assumd that vry actuator covrs xactly th lngth of on lmnt. h actuator position is modld using a Boolan typ discrt variabl for ach lmnt and th lctric potntial of th actuator using a boundd continuous variabl. h Mixd Intgr Problm that ariss is highly nonlinar and is solvd using a modifid gntic algorithm procdur in ordr to accommodat two diffrnt typs of information: th location of ach pizolctric lmnt and th voltag ndd to apply to ach of thm Gntic Algorithm and Grat Dlug Gntic Algorithms (GAs) ar a catgory of huristic optimization algorithms that mimics th way traits pass from parnts to offspring rsulting in th dvlopmnt of charactristics that giv an volutionary advantag to crtain mmbrs of th population. GAs ar an stablishd mthod of non xact optimization, maning that a GA is usually abl to find vry good solutions to hard combinatorial optimization problms whn it is difficult or vn impossibl for an xact optimization mthod lik Linar or Intgr Programming to addrss th sam problm within rasonabl solving tim. A dtaild tratmnt of GAs can b consultd in [3] whil succssful applications of GAs can b found in almost vry fild [6]. As GAs is a kind of simulation a grat numbr of function valuations ar rquird. For this rason, svral ways hav bn proposd so as to spd up th procss yt maintain quality of th acquird solutions. With this approach th GA is combind with th Grat Dlug mthod, which crats a part of th initial population consisting of good, yt divrs solutions that ar thn fd to th GA. h rst of th initial population gts gnratd using random valus. h choic of introducing individuals of rlativ high fitnss arly in th optimization procss sms to hlp th GA on finding vn highr quality solutions fastr. Grat Dlug Algorithm (GDA) is a local sarch optimization mthod, which was initially proposd by Duck [7]. It blongs to th gnral class of trajctory (singl point) sarch mthods for th rason

8 Actuators 3, 8 that a singl solution is continuously modifid so as to progrssivly achiv bttr rsults. Othr mta huristics that blong to th sam class ar Simulatd Annaling, aboo Sarch, Lat Accptanc Hill Climbing, and Grdy Randomizd Adaptiv Sarch, to nam a fw. In GDA bttr solutions ar always accptd whil wors solutions ar also accptd, providd that th computd cost is no wors than th cost of th currnt solution plus an artificial limit that gradually diminishs. h nam Grat Dlug was chosn in ordr to draw an analogy btwn th mthod and a situation whr a prson situatd in a landscap filld with plataus, paks, and dips tris to kp his ft dry whil a havy rain occurs causing th watr lvl to ris. GDA is similar to Simulatd Annaling with th addd bnfit that it rquirs tuning of a singl paramtr only. his paramtr known as Dcay Rat (DR) is th amount that th tolranc of accpting non improving solutions is rducd in ach itration of th mthod. GDA has bn applid to a numbr of optimization problms with promising rsults. Although GDA was originally applid to th ravlr Salsman Problm, most of th rsarch paprs that us this mthod ar in th ara of schduling problms and spcially cours and xamination timtabling [8,9]. GDA has also bn applid to othr optimization problms lik channl assignmnt in cllular communications [], prvntiv maintnanc optimization for multi stat systms [], constraind mchanical optimization [] and othrs. GDA is lss common than othr trajctory sarch mta huristics but its simplicity and singl paramtr tuning maks it a good candidat for svral optimization problms that occur in practic. h psudo cod for gnrating part of th initial population of th GA using GDA is shown in Algorithm. Each solution S is initially gnratd randomly, optimizd to a crtain dgr using GDA, and thn appndd to th population. h Dcay Rat paramtr DR is computd for ach individual of th population by dividing th initial fitnss valu of th individual F(S) by th numbr of itrations IER that GDA is allowd to prform. A nw solution S is gnratd from S using nighborhood functions similar to thos dscribd in []. hus, for a variabl rfrring to th voltag applid to a crtain position i th nw valu V i * is computd basd on th xisting valu V i and th following quation: * i i k V V rand() stp (7) Function rand() rturns a uniform random valu btwn and and stp is a paramtr with initially valu drawn randomly btwn 5 and,, which gradually diminishs. Stp is calculatd with Equation (8), whr function frac() rturns th fractional part of a ral numbr, φ is a paramtr that assums valu. and k is th itration countr. k frac k k k k stp stp stp (8) As voltag assums valus btwn a lowr and an uppr limit,.g., btwn Volts and 4 Volts, whn V i * gts a valu out of that rang, an adjustmnt occurs so as th valu to bcom valid again. In particular, whn th valu violats ithr th lowr or th uppr limit it is modifid so as to b spacd at th sam distanc from th limit that is violatd but in th fasibl rang of valus.

9 Actuators 3, 9 Algorithm.: Grat Dlug Algorithm for initial population gnration. GA_POPULAION = N = dsird GDA gnratd population mmbrs for j:= to N crat a random initial solution S DR = F(S) / IER L = F(S) for k:= to IER gnrat a nw solution S basd on S using a nighborhood function if F(S ) max(f(s), L) thn S = S nd_if L = L DR nd_for appnd S to GA_POPULAION nd_for Chromosom Encoding A dcision dirctly rlatd to th succss of a GA in a spcific application is th chromosom ncoding, which is th ncodd form of ach individual blonging to th population. Chromosoms ar combind in ordr to brd nw individuals or mutatd so as to incorporat dirct changs. In any kind of problm xamind in this work, th bam is dividd in 3 qually spacd positions whr th actuators can b positiond. In th first kind of th problm (th Voltag Problm) vry six conscutiv positions bcom a group and th sam voltag applis to all actuators of th sam group. So, th chromosom ncoding is just a squnc of dcimal numbrs qual to th numbr of groups. Each valu of th chromosom is associatd with th voltag that will b applid to all actuators of th group in th sam plac. Givn that four diffrnt sttings ar tstd with two, thr, four, or fiv groups of actuators activ rspctivly th chromosom lngth bcoms two, thr, four, or fiv. Figur. Chromosom ncoding for th location and voltag problm. In th scond cas of th problm (th Location and Voltag Problm) th chromosom consists of two parts (Figur ). h lft on is a squnc of 3 binary valus carrying th information of prsnc () or absnc () of an actuator at th prdfind positions. h right part is th voltags that will b applid to ach actuator that is prsnt. h sum of ons in th lft part should qual to th numbr of dcimal valus in th right part. As, for ach run of th program, th numbr of actuators that will b activ is known in advanc, th chromosoms hav a constant lngth pr run. In this vrsion of th problm mor dgrs of frdom ar givn sinc diffrnt voltags can b applid to

10 Actuators 3, actuators that in th Voltag Problm blong to th sam group. So, bttr rsults ar xpctd and indd th GA manags to find thm as is shown in Sction GA Implmntation Issus hr ar numrous GA implmntations availabl in th form of callabl libraris, framworks or intgratd nvironmnts. In this papr, MatLab s Global Optimization oolbox Ra, which includs th Gntic Algorithm oolbox was usd. MatLab s Global Optimization oolbox starting from vrsion Rb has th capability of dfining intgr constraints out of th box. his was vry convnint in our cas givn that th Location and Voltag Problm is a Mixd Intgr optimization problm. 4. Numrical Rsults his sction prsnts numrical rsults from svral rprsntativ problms. First, a bnchmark problm is considrd in ordr to validat th prsnt optimization algorithm. Nxt, svral illustrativ optimization problms ar invstigatd using th dvlopd algorithm. All application xampls focus on bams with surfac bondd pizolctric patchs as actuators. h host bam is mad of 3/976 graphit/poxy and th pizolctric layrs ar PZ G95N. h lngth of th bam is qual to 3 mm, th dpth is qual to 9.6 mm and width is qual to 4 mm. h thicknss of th actuators is qual to. mm. h lastic constants of 3/976 graphit/poxy ar: E = 5 GPa, v =.3, G 3 = 7. GPa. h pizolctric matrial has th following proprtis: E = 63 GPa, v =.3, G 3 = 4. GPa, 3 = C/m and othr ntris in th pizolctric strss matrix ar zro. For comparison rason, in all th following xampls th fitnss valu is scald as [9]: f ln. f (9) It is notd that th gratr th valu of f, th gratr th shap controllability is. Figur. h smart bam structur. 4.. h Voltag Problm h problm studid by Hadjigorgiou t al. [9] is considrd hr in ordr to validat th optimization cod prsntd in Sction 3. h bam is dividd vnly into 3 finit lmnts and fiv groups along th x dirction as shown in Figur. Each group consists of six lmnts; on th uppr surfac of th lmnts actuators may b attachd. h bam is clampd at th lft hand sid and is

11 Actuators 3, subjctd to a concntratd load qual to 4 N at th fr right nd. h uppr limit of th voltag is st d to b 5 V. h pr dfind displacmnt fild (dsird shap) is givn by X x and th aim is to calculat th actuator voltags rquird to induc this dsird shap. h Voltag Problm is studid in two cass. In th first cas, it is assumd that all th lmnts hav a pizolctric matrial layr bondd on its uppr surfac and th spcifid group is activatd by th usr. his is th sam situation of [9]. In th scond cas, ach lmnt is considrd as having no patch or bing fully covrd with pizolctric matrial. A spcial assmbly procdur was usd to account for th pizolctric actuator patchs instad of complt layrs of pizolctric matrial throughout th structurs. It is notd that in th formr cas, th stiffnss charactristics of th bam rmain constant throughout th shap control procdur, whil, in th lattr, thy chang dpnding on th numbr of actuators usd. Mor prcisly, th actuator patchs contribut lss to th bam stiffnss than th continuous actuator layrs in th first cas. In addition, it is pointd out that for fiv actuator groups, th two cass of th problm (cas and cas ) ar idntical to that of [9]. h optimization problm for both problm cass is solvd using th prsnt dvlopd gntic algorithm. h gntic algorithms wr run for gnrations and 4 individuals. Of svral tst cass run, th on xhibiting th bst fitnss is prsntd. h optimal valus of voltags for th most fficint combinations of actuator groups to shap control of th bam ar shown in abl. In th first row of abl th valus of optimal voltag prdictd by Hadjigorgiou t al. [9] ar includd. It can b notd that th valus of fitnss obtaind by th dvlopd algorithm ar qual (cas ) or smallr (cas ) than thos of [9] xcpt for th last cas of fiv groups of actuators, whr th fitnss valu is biggr. aking into account that th fitnss valus ar calculatd by Equation (9), w may conclud that: (i) th prsnt algorithm is abl to produc bttr rsults for th cas of problms involving fiv actuator groups, (ii) th smallr valus of fitnss obtaind in th scond cas ar du to th smallr stiffnss charactristics of th bam, lading to biggr displacmnt for th sam load condition. In addition, this xampl highlights th invrs natur of th shap control, whr uniqunss of solution is not guarantd in gnral. his ffct is also attributd to th inhritd randomnss of th GA. Most intrstingly, all thr modls prdict quit diffrnt voltag configurations but yt thy all manag to match th dsird shap quit wll. 4.. h Location and Voltag Problm A bam with similar matrial and gomtric proprtis as dscribd in Sction 4., is considrd to calculat th optimal location and applid voltag of actuators in ordr to modify its shap. h simulation compriss structurs with diffrnt boundary conditions: a clampd fr bam and a clampd clampd bam. All th thirty lmnts ar candidats for actuator locations. In addition, th pizolctric patchs ar symmtrically locatd on th uppr as wll as on th lowr sid of th bam. Fiv GA runs wr prformd, ach with diffrnt initial valus of th dsign variabls, and th bst rsults obtaind ar prsntd in th following paragraphs. In ordr to assss th bhavior of th algorithmic approach, svral runs wr prformd. For xampl, for th cas of th Clampd Fr problm, whr th numbr of actuators is 8, 55 runs wr prformd using initial valus gnratd with diffrnt random sd for ach run. Rsults showd that for this particular cas th maximum valu of fitnss function f was 3.45, th minimum valu was

12 Actuators 3, 7.35, th avrag valu was 9.35, th standard dviation was.97, and th rang of valus was 4.. Similar rsults wr obtaind for othr instancs of th problm dmonstrating that th approach is fairly robust giving consistntly good rsults. Inclusion of th Grat Dlug phas bfor th Gntic Algorithm addd valu to th approach. his is dmonstratd by th following xprimnt scnario: For th cas of th Clampd Fr problm, runs wr prformd using th Gntic Algorithm including th injction of solutions gnratd by th Grat Dlug and anothr runs wr prformd by dactivating th Grat Dlug phas and ltting th Gntic Algorithm form th initial population randomly. All runs whr xcutd on a Windows7/64 bit machin quippd with Intl i7 86 procssor and 6 GB of RAM. Each run took about ight minuts to complt. Rsults showd that th bst 4 gnratd fitnss valus during th xprimnt wr all achivd using th configuration of th solvr that includd th Grat Dlug stag. hus, a small numbr of good solutions (% or lss of th population siz) injctd to th initial population sm to driv th Gntic Algorithm to bttr solutions. abl. Optimal valus of voltags (clampd fr bam) for X d (x) =. Numbr of usd Actuator Groups (,3) (,,3) (,,3,4) (,,3,4,5) Mthod Voltag of th Actuator Groups (V) Fitnss f [9] Prsnt (cas ) Prsnt (cas ) [9] Prsnt (cas ) Prsnt (cas ) [9] Prsnt (cas ) Prsnt (cas ) [9] Prsnt (cas ) Prsnt (cas ) Clampd Fr Bam First th cas of a bam, which is clampd at th lft hand sid and is subjctd to a concntratd load qual to 4 N at th fr right nd, is considrd. In this cas, th lowr limit of th voltag is st to b V and th uppr limit is st to b 4 V (limit imposd du to dpoling of actuators). h dsird shap is givn by X d (x) =. abl shows th optimal solutions for placmnt of th actuators and th corrsponding optimal voltags for various numbrs of actuators. h gntic algorithms wr run using th following paramtrs: Gnrations =,, Population =, ElitCount =. Marginally bttr rsults can b obtaind in som cass by furthr fin tuning of GA paramtrs lik ElitCount and mutation rat. It should b notd that for a small numbr of actuators (8, ), th GA was trminatd bfor gnrations. W obsrv that for a small numbr of actuators (8, ) th

13 Actuators 3, 3 optimal actuation voltags ar clos to th uppr saturation limit and th optimal positions ar closd to th clampd nd. A graphical prsntation of ths rsults is givn in Figur 3. abl. Optimal Location and voltags of actuators within th 3 finit lmnt msh for Clampd Fr Bam. Numbr of Actuators in Us Νumbr of Elmnts f f f f f Fitnss As can b shown in Figur 3, in th last thr cass a vry good agrmnt was found btwn th dsird shaps and th numrical rsults, showing that mor actuators can control th dformation mor fficintly. Compard to rsults in Figur 3, w can s that th maximum nodal displacmnt for th optimal solution obtaind using fitnss f is smallr than th on obtaind using f (.g., m compard to.93 5m for 8 actuators). Hnc, it can b concludd that th dflction controlld by fitnss f is closr to th dsird shap than th on controlld by f.

14 tranvrs displacmnt (mm) tranvrs displacmnt (mm) tranvrs displacmnt (mm) tranvrs displacmnt (mm) tranvrs displacmnt (mm) tranvrs displacmnt (mm) Actuators 3, 4 Figur 3. h cntrlin of th cantilvr smart bam undr th action of various numbrs of actuators for X d (x) = with th optimal location of th actuators and th optimal valus of actuation voltags. x -5 numbr of actuators= 5 x numbr of actuators=8 - dsird shap no-control x numbr of actuators= x numbr of actuators= x numbr of actuators=4 x numbr of actuators= dsird shap no-control - fitnss fitnss Clampd Clampd Bam Scond, th cas whr th bam clamps on both sids and is subjctd to a concntratd load qual to 4 N at th cntr is considrd. h lowr limit of th voltag is st to b 4 V and th uppr limit is st to b 4 V. h pr dfind displacmnt fild (dsird shap) is givn by X d (x) =. h optimal valus of voltags for th most fficint combinations of numbr of actuators to shap control of th bam ar prsntd in abl 3. h GA runs using th following paramtrs: Gnrations = 3,, Population = 5, ElitCount =. It should b notd that, for a small numbr of actuators (8, ), th GA was trminatd bfor,5 gnrations. W obsrv that for a small numbr of actuators (8, ) th optimal positions of th actuators ar clos to th clampd nds whr th optimal actuation voltags ar clos to th uppr saturation limit and at th middl of th bam whr th optimal actuation voltags ar clos to th lowr saturation limit. A graphical prsntation of ths rsults is givn in Figur 4. By comparing th curvs in Figur 4, it can b sn that th dflction

15 Actuators 3, 5 controlld by fitnss f is closr to th dsird shap than th on controlld by f (th maximum displacmnt from m is rducd to.7 6m for actuators). Again, th rsults indicat that incrasing th numbr of actuators has a bnficial ffct on controlling th shap of th bam. 5. Conclusions A mathmatical modl of a laminatd composit bam with bondd pizolctric patchs usd as actuators is considrd in this study. h modl is built using finit lmnt mthod and is applid as a platform for th invstigation of shap control of th bam. Shap control was applid to a bam structur with diffrnt boundary conditions. h optimal valus for th locations of th pizo actuators and optimal voltags for shap control ar dtrmind for clampd fr and clampd clampd bams by using a gntic optimization procdur. A two stp procss including Grat Dlug and thn a Gntic Algorithm has bn prformd in ordr to improv sarch fficincy. h rsults prsntd abov dmonstrat th capability of th proposd hybrid GA approach in dtrmining optimal voltags and locations of control actuators within a larg numbr of possibl positions. Numrical rsults on a bnchmark problm validat both th finit lmnt cod bing usd as wll as th optimization algorithm. Exampls that dmonstrat th capabilitis and fficincy of th dvlopd optimization algorithm in both clampd fr and clampd clampd bam problms wr prsntd. In th nar futur, our rsarch tam plans to apply th proposd hybrid GA to mor ralistic nginring problms such as plat structurs. abl 3. Optimal Location and voltags of actuators within th 3 finit lmnt msh for Clampd Clampd Bam. Numbr of Elmnts Numbr of Actuators in Us f f f f f f f f f f

16 tranvrs displacmnt (mm) tranvrs displacmnt (mm) tranvrs displacmnt (mm) tranvrs displacmnt (mm) tranvrs displacmnt (mm) tranvrs displacmnt (mm) Actuators 3, 6 abl 3. Cont Fitnss Figur 4. h cntrlin of th clampd clampd smart bam undr th action of various numbrs of actuators for X d (x) = with th optimal valus of actuation voltags. x numbr of actuators= x numbr of actuators=8 - dsird shap no-control x numbr of actuators= x numbr of actuators= x numbr of actuators=4 x numbr of actuators= dsird shap no-control fitnss fitnss Acknowldgmnts his rsarch has bn co financd by th Europan Union (Europan Social Fund ESF) and Grk national funds through th Oprational Program Education and Liflong Larning of th National Stratgic Rfrnc Framwork (NSRF) Rsarch Funding Program: ARCHIMEDES III Invsting in knowldg socity through th Europan Social Fund. h authors gratfully acknowldg this support.

17 Actuators 3, 7 Rfrncs. Chopra, I. Rviw of stat of art of smart matrials structurs and intgratd systms. AIAA J., 4, Srinivasan, A.V.; McFarland, D.M. Smart Structurs: Analysis and Dsign; Cambridg Univrsity Prss: Cambridg, UK,. 3. Irschik, H. A rviw on static and dynamic shap control of structurs by pizolctric actuation. Eng. Struct., 4, ong, D.; Williams, R.L.; Agrawal, S.K. Optimal shap control of composit thin plats with pizolctric actuators. J. Intll. Matr. Syst. Struct. 998, 9, Agrawal, B.N.; ranor, K.E. Shap control of a bam using pizolctric actuators. Smart Matr. Struct. 999, 8, Ch, C.; ong, L.; Stvn, G.P. Pizolctric actuator orintation optimization for static shap control of composit plats. Compos. Struct., 55, Onoda, J.; Hanawa, Y. Actuator placmnt optimization by gntic and improvd simulatd annaling algorithms. AIAA J. 993, 3, Da Mota Silva, S.; Ribiro, R.R.; Rodrigus, J.D.; Vaz, M.A.P.; Montiro, J.M. h application of gntic algorithms for shap control with pizolctric patchs An xprimntal comparison. Smart Matr. Struct. 4, 3, Hadjigorgiou, E.P.; Stavroulakis, G.E.; Massalas, C.V. Shap control and damag idntification of bams using pizolctric actuation and gntic optimization. Int. J. Eng. Sci. 6, 44, Frckr, M.I. Rcnt advancs in optimization of smart structurs and actuators. Int. J. Eng. Sci. 3, 4, Prumont, A. Vibration Control of Activ Structurs: An introduction; Springr-Vrlag: Brlin Hidlbrg, Grmany,.. homas, O.; Dü, J.F.; Ducarn, J. Vibrations of an lastic structur with shuntd pizolctric patchs: Efficint finit lmnt formulation and lctromchanical coupling cofficints. Int. J. Numr. Mth. Eng. 9, 8., Goldbrg, D. Gntic Algorithms in Sarch, Optimization, and Machin Larning; Addison Wsly Profssional: Nw York, NY, USA, Back,.; Fogl, D.B.; Michalwicz, Z. Handbook of Evolutionary Computationl; st d.; IOP Publishing Ltd.: Bristol, UK, Rddy, N.J. Mchanics of Laminatd Composit Plats: hory and Analysis; CRC: Nw York, NY, USA, Man, K.F.; ang, K.S.; Kwong, S. Gntic algorithms: Concpts and applications. IEEE rans. Ind. Elctr. 996, 43, Duck, G. Nw Optimization Huristics th grat dlug algorithm and th rcord to rcord travl. J. Comput. Phys. 993, 4, McMullan, P. An xtndd implmntation of th grat dlug algorithm for cours timtabling. Lct. Not. Comput. Sci. 7, 4487, Ozcan, E.; Misir, M.; Ochoa, G.; Burk, E.K. A Rinforcmnt larning grat dlug hypr huristic for xamination timtabling. Int. J. Appl. Mtah. Comput.,,

18 Actuators 3, 8. Kndall, G.; Mohamad, M. Channl Assignmnt in Cllular Communication Using a Grat Dlug Hypr Huristic. In Procdings of th IEEE Intrnational Confrnc on Ntworks, Brlin, Grmany, 6 9 Novmbr 4; pp Nahas, N.; Khatab, A.; Ait Kadi, D.; Nourlfath, M. Extndd grat dlug algorithm for th imprfct prvntiv maintnanc optimization of multi stat systms. Rliab. Eng. Syst. Safty 8, 99, Baykasoglu, A. Dsign optimization with chaos mbddd grat dlug algorithm. Appl. Soft Comput.,, Appndix Dtaild xprssions of th mass and stiffnss matrics as wll as loading vctors that appar in th papr. nlayr M k N N dv (A.) k Vk nlayr K B Q BdV (A.) uu k k Vk K u B B dv B B dv B B dv p p p Vp Vp V k (A.3) npl Ku K u (A.4) B B p dv Vp B B dv p K V p B B dv p npl V p npl (A.5) Q F B q ds S Fm N fs ds N Fc S (A.6) (A.7) 3 by th authors; licns MDPI, Basl, Switzrland. his articl is an opn accss articl distributd undr th trms and conditions of th Crativ Commons Attribution licns (