Design of a Piezoelectric Actuator Using Topology Optimization

Transcription

1 Univerity of Tenneee, Knoxville Trce: Tenneee Reerch nd Cretive Exchnge Mter Thee Grdute School 5-23 Deign of Piezoelectric Actutor Uing Topology Optimiztion Jochim Drenckhn Univerity of Tenneee - Knoxville Recommended Cittion Drenckhn, Jochim, "Deign of Piezoelectric Actutor Uing Topology Optimiztion. " Mter' Thei, Univerity of Tenneee, Thi Thei i brought to you for free nd open cce by the Grdute School t Trce: Tenneee Reerch nd Cretive Exchnge. It h been ccepted for incluion in Mter Thee by n uthorized dminitrtor of Trce: Tenneee Reerch nd Cretive Exchnge. For more informtion, plee contct trce@utk.edu.

2 To the Grdute Council: I m ubmitting herewith thei written by Jochim Drenckhn entitled "Deign of Piezoelectric Actutor Uing Topology Optimiztion." I hve exmined the finl electronic copy of thi thei for form nd content nd recommend tht it be ccepted in prtil fulfillment of the requirement for the degree of Mter of Science, with mjor in Aeropce Engineering. We hve red thi thei nd recommend it cceptnce: J.A.M Boulet, Frnk H. Speckhrt, Chrle Collin (Originl ignture re on file with officil tudent record.) Arnold Lumdine, Mjor Profeor Accepted for the Council: Dixie L. Thompon Vice Provot nd Den of the Grdute School

3 To the Grdute Council: I m ubmitting herewith thei written by Jochim Drenckhn entitled "Deign of piezoelectric ctutor uing topology optimiztion." I hve exmined the finl electronic copy of thi thei for form nd content nd recommend tht it be ccepted in prtil fulfillment of the requirement for the degree of Mter of Science, with mjor in Aeropce Engineering. We hve red thi thei nd recommend it cceptnce: Accepted for the Council: Anne Myhew Vice Provot nd Den of Grdute Studie (Originl ignture re on file with officil tudent record.)

4 Deign of piezoelectric ctutor uing topology optimiztion A Thei Preented for the Mter of Science Degree The Univerity of Tenneee Jochim Drenckhn My 23

5 Acknowledgment I wih to thnk ll thoe who helped me in completing my Mter of Science in Aeropce Engineering. Specil thnk go to Dr. Lumdine without hi input, motivtion nd knowledge thi work would not hve been poible. He h been gret profeor, mentor nd dvier. I thnk my committee member Dr. Boulet, Dr. Speckhrt nd Dr. Collin for being in my committee. I thnk my girlfriend Atrid Gämlich for upporting me mentlly nd enduring tht I w gone for long time. ii

6 Abtrct Thi tudy invetigte the optiml topology for piezoelectric ctutor under ttic lod. The project conit of two mjor prt: implementtion of the control lw into the commercil finite element code ABAQUS nd tudie in topology optimiztion. The firt prt give thorough derivtion nd explntion of the implementtion of ttic feedbck control nd dynmic proportionl nd derivtive control. The reult i compred with reult publihed in the literture. The econd prt exmine the reult of topology optimiztion with different geometrie nd contrint. Thu, thi tudy develop fundmentl undertnding of dvntgeou hpe for optiml performing piezoelectric ctutor. iii

7 Tble of Content INTRODUCTION LITERATURE REVIEW FINITE ELEMENT MODELING AND CONTROL TOPLOGY OPTIMIZATION PROBLEM STATEMENT FINITE ELEMENT FORMULATION.8 2. THEORY FOR PIEZOELECTRIC MATERIAL STATIC CALCULATION NODE-TO-NODE CONTROL LAYER-TO-LAYER CONTROL IMPLEMENTATION FOR THE STATIC CASE VERIFICATION FOR THE STATIC CASE DYNAMIC CALCULATION PROPORTIONAL CONTROL IMPLEMENTATION OF PROPORTIONAL CONTROL VERIFICATION OF PROPORTIONAL CONTROL DYNAMIC CALCULATION DERIVATIVE CONTROL IMPLEMENTATION OF DERIVATIVE CONTROL VERIFICATION OF DERIVATIVE CONTROL OPTIMIZATION INTRODUCTION GENERAL TOPOLOGY OPTIMIZATION....5 iv

8 3.3 TOPOLOGY OPTIMIZATION IN THIS STUDY RESULTS GAIN STUDY ONE-LAYER GAIN STUDY TWO-LAYER GAIN STUDY DENSITY STUDY ONE-LAYER DENSITY STUDY TWO-LAYER DENSITY STUDY CONCLUSION AND FUTURE WORK 83 REFERENCES APPENDIX 92 A SURVEY OF SEVEN NON-LINEAR CONSTRAINED OPTIMIZATION PROGRAMS A. INTRODUCTION.93 A.2 DESCRIPTION OF EVALUATED OPTIMIZATION CODES..93 A.2. LINGO..93 A.2.2 MATLAB..95 A.2.3 NLPQL..95 A.2.4 EPOGY..96 A.2.5 OPTDESX.96 A.2.6 VISUALDOC 97 v

9 A.2.7 SOLVER DLL A.3 EXAMPLE PROPBLEMS A.4 EASE OF USE. A.4. LINGO..... A.4.2 MATLAB. A.4.3 NLPQL.2 A.4.4 EPOGY 3 A.4.5 OPTDESX... 4 A.4.6 VISUALDOC.. 5 A.4.7 SOLVER DLL A.5 RESULTS OF THE EXAMPLE PROBLEMS...6 A.5. PROBLEM 8 A.5.2 PROBLEM A.5.3 PROBLEM A.6 SUMMERY.4 A.7 RECOMMENDATION...7 VITA 9 vi

10 Lit of Tble Tble 2. Mteril Propertie for Simple Decoupled Bem Sytem...6 Tble 2.2 Vlidtion of Sttic Decoupled Sytem...9 Tble 2.3 Propertie of Tzou Bem (Sttic Ce)....2 Tble 2.4 Vlidtion of Proportionl Control (Decoupled Sytem) 33 Tble 2.5 Propertie of Tzou Bem (Dynmic)...36 Tble 2.6 Comprion Tzou v. FE Reult of Proportionl Control Clcultion...36 Tble 2.7 Comprion of Nturl Frequency of Tzou Bem (Proportionl Control) 38 Tble 2.8 Vlidtion for Derivtive Control (Decoupled Sytem)..44 Tble 2.9 Comprion Tzou v. FE Reult of Derivtive Control Clcultion..46 Tble 4. Propertie of Bem Ued for Optimiztion..58 Tble A. Progrm Fct Sheet 94 Tble A.2 Complete Lit for Problem..7 Tble A.3 Complete Lit for Problem 2..7 Tble A.4 Complete Lit for Problem 3..7 Tble A.5 Rnking for Problem... Tble A.6 Rnking for Problem Tble A.7 Rnking for Problem Tble A.8 Summrized Rnking...8 vii

11 Lit of Figure Figure. Modeled Cntilever Bem 7 Figure 2. Node-To-Node Control..2 Figure 2.2 Proce Digrm for Sttic Control 5 Figure 2.3 Decoupled bem Sytem 6 Figure 2.4 Tzou Bem.2 Figure 2.5 Bem Section. 22 Figure 2.6 Deformed Bem.25 Figure 2.7 Comprion of Tip-Diplcement of Tzou Bem for Different Gin...28 Figure 2.8 Reltive Difference Figure 2.9 Frequency Digrm for Proportionl Control (Decoupled Sytem)..35 Figure 2. Frequency Digrm of Tzou Bem (Finite Element Clcultion).38 Figure 2. Frequency Digrm of Tzou Bem (Proportionl Control)...4 Figure 2.2 Frequency Digrm of Tzou Bem (Derivtive Control)...48 Figure 3. Homogeniztion through Microcell with Rectngulr Hole (Hni nd Hinton 999)..5 Figure 3.2 Optiml Topology Deign (Pplmbro nd Dougl 2) 53 Figure 3.3 Optimiztion Proce Digrm...56 Figure 4. Bem ued for Optimiztion...58 Figure 4.2 Geometrie for One-Lyer Gin Study..6 Figure 4.3 Tip-Diplcement of One-Lyer Gin Study.62 Figure 4.4 Comprion One-Lyer Gin Study...64 viii

12 Figure 4.5 Comprion One-Lyer Gin Study gint % Mteril..64 Figure 4.6 Geometrie for Two-Lyer Gin Study..66 Figure 4.7 Tip-Diplcement of Two-Lyer Gin Study.68 Figure 4.8 Comprion Two-Lyer Gin Study Figure 4.9 Comprion Two-Lyer Gin Study gint % Mteril.69 Figure 4. Geometrie for One-Lyer Denity Study..7 Figure 4. Tip-Diplcement of One-Lyer Denity Study.73 Figure 4.2 Comprion One-Lyer Denity Study Figure 4.3 Comprion One-Lyer Denity Study gint % Mteril.75 Figure 4.4 Geometrie for Two-Lyer Denity Study...77 Figure 4.5 Tip-Diplcement of Two-Lyer Denity Study 78 Figure 4.6 Comprion Two-Lyer Denity Study Figure 4.7 Comprion Two-Lyer Denity Study gint % Mteril 82 Figure A. Objective Function Problem.. 9 Figure A.2 Contrined Function Problem..9 Figure A.3 Objective Function Problem Figure A.4 Contrined Function Problem 2..2 Figure A.5 Objective Function Problem Figure A.6 Contrined Function Problem 3..5 ix

13 Nomenclture A C Amplitude, Are Cpcitnce D ijkl Mechnicl contitutive tenor E j F Electric field Mechnicl force vector F (x) Objective function G d Derivtive gin mtrix G p Proportionl gin mtrix I Identity mtrix K uu Mechnicl tiffne mtrix K uφ Piezoelectric coupling mtrix K ΦΦ L M M Dielectric mtrix Length M mtrix Applied Moment P c Control Force Q U V Nodl electric chrge vector Nodl diplcement vector Volume x

14 b Width e ijk h h i Piezoelectric coupling coefficient tenor Height Equlity contrint g i Inequlity contrint h Thickne p ij Dielectric tenor q i r u x Electric diplcement vector Rdiu Diplcement Deign vrible Greek Symbol θ Φ α β δ Angle Nodl electric potentil vector Ryleigh dmping fctor Ryleigh dmping fctor Difference, Deflection ε ij η Strin tenor Lo fctor µ Poion rtio xi

15 ρ Mteril denity, Ditnce to neutrl xi σ ij ω Stre tenor Frequency Supercript b mx Actutor Bem Mximum Senor Subcript C HB T ve eq n out,2,3,..,n Compreion Hlf power bndwidth Tenion Averged Equivlent Nturl frequency Output point Node number xii

16 INTRODUCTION Piezoelectric mteril hve the unique cpbility of producing mechnicl trin when ubjected to n electric potentil or, converely, of generting n electricl chrge when ubjected to mechnicl trin. Thi cpbility h been ued in ctive control ytem utilizing piezoelectric element ctuting nd ening device (Tzou nd Teng 99). The reerch done in thi re h demontrted tht piezoelectric mteril hve the bility to modify the ttic nd dynmic chrcteritic of tructure. The tiffne nd dmping chrcteritic of tructure cn be modified by the ppliction nd control of piezoelectric mteril. Piezoelectric ctutor hve mny unique chrcteritic tht ditinguih them from other ctutor. A piezoelectric mteril exited with n ctivtion voltge to induce trin upon tructure i referred to piezoelectric ctutor. Severl of the fvorble chrcteritic of piezoelectric ctutor re tht they cn be formed to pecific hpe nd re reltively ey to control nd to intll. Therefore, piezoelectric ctutor hve fvorble chrcteritic tht mke them good choice for ue in compoite tructure where they cn be eily intlled. One of the didvntge of piezoelectric mteril i tht they produce very little trin. Lrger trin would incree the performnce of the piezoelectric ctutor chrcteritic nd i therefore deirble. For thi reon vriou reerch ttempt hve been mde to modify the microtructure of the piezoelectric mteril to incree trin (Silv, Nihiwki et l. 2). However, very little reerch h been done in detecting the optiml hpe of piezoelectric mteril to reduce diplcement or mximize the dmping chrcteritic of mechnicl tructure. Topology optimiztion uing

17 homogeniztion h been ued in vriou re of tructurl deign nd h been hown to be good method for olving thi kind of problem. For thi reon, it i the purpoe of the tudy to conduct topology optimiztion for piezoelectric ctutor. In thi tudy, the objective i to determine the bet topology of piezoelectric ctutor for cntilever bem under concentrted ttic lod t the tip. Thi i done by conducting topology optimiztion of the ctutor. A prmeter tudy i conducted with different mteril denitie nd different gin for the ctutor. The reult re compred nd bet topology for thi problem i determined. Furthermore, ttempt hve been mde to implement dynmic proportionl nd derivtive control for ctive vibrtion dmping.. LITERATURE REVIEW A review of relted pper i given to how the current tte of reerch in thi re. The introduced pper re divided into two mjor group, decribing () the finite element modeling of piezoelectric mteril nd implementing control lw nd (2) the topology optimiztion including pproche uing piezoelectric element... FINITE ELEMENT MODELING AND CONTROL Mechnicl nd electricl contitutive lw for piezoelectric mteril hve been well defined. Some commercil finite element code lredy hve the cpbility to ue uch element, uch ABAQUS (Hibbit, Krlon et l. 22), which h been ued for 2

18 thi tudy. A good overview of the generl topic of modeling piezoelectric mteril in finite element model h be done by Nillon et l. (983), who decribe method to nlyze piezoelectric tructure by uing the finite element method. Further work h been done by Lerch (99), who expnded the finite element theory for piezoelectric mteril to be ued in two or three dimenion. Furthermore, Hock et l. (99) decribe the finite element nlyi of -3 trnducer for different piezoelectric mteril nd compre the reult with experiment. Another pproch i hown by Guyn (965), who removed the electricl degree of freedom by uing condention mtrix. The control i pplied over the remining trnltionl nd rottionl degree of freedom. Much reerch h been done in implementing control lw for piezoelectric mteril to be ued enor nd ctutor in mrt mteril. Tzou nd Teng (99; 99) decribe ditributed piezoelectric enor/ctutor deign for piezoelectric mteril by uing finite element pproch. Tzou (993) h lo done more thorough derivtion of the theory of piezoelectric mteril nd it control in hi book Piezoelectric Shell. H et l. (992) did finite element nlyi of tructure contining piezocermic enor nd ctutor nd compred the computtionl reult with experiment. Similr reerch h been done by Bz nd Ro (995) who exmined the performnce chrcteritic of ctive contrined lyer dmping uing piezoelectric nd vicoeletic mteril by finite element method nd compred the reult to experimentl vlue. Vley nd Ro (996) built on thee reult to mke comprion of ctive, pive nd hybrid dmping in tructurl deign. Vrdn et l. (996) decribe 3

19 the cloed loop finite element modeling of ctive dmping in ctive tructurl vibrtion control. In the me re two pper hve been publihed by Kim et l. (996; 997) which lo decribe the finite element modeling of mrt tructure including piezoelectric mteril. The vibrtion nd ctution chrcteritic of tructure with piezo-cermic ctutor hve been exmined by Hn (999) nd compred with experimentl reult. Other reerch h been done in the re of finding the bet controller for mrt tructure. Since thi i not the emphi of thi work, jut few publiction re decribed. Gudenzi et l. (2) compre the control of bem vibrtion between numericl nd experimentl reult. A more thorough pproch i tken by Gbbert et l. (22) who exmine the controller deign for mrt tructure. Chng et l. (22) decribe how to deign robut vibrtion controller for mrt pnel, lo by uing the finite element method...2 TOPOLOGY OPTIMIZATION Topology optimiztion i reltively recent field nd h been hown to be good method for finding optiml topologie for tructurl problem with given boundry condition. Bendøe nd Kikuchi (988) firt introduced the homogeniztion method for finding the optiml topology for tructurl problem. A more thorough decription of topology optimiztion uing homogeniztion i given in the book by Hni nd Hinton (999), Allire (22) nd Bendøe nd Sigmund (23). 4

20 Topology optimiztion h been pplied in tudie by Yi et l. (2) who ued topology optimiztion to optimize the hpe of vioeltic mteril to chieve bet dmping chrcteritic. Lumdine (22) did imilr work by uing topology optimiztion for finding the bet hpe of contrined lyer dmping mteril. The ide of finding the bet topology for given problem h further been pplied for mrt mteril, including piezoelectric mteril. Some reerch h been done in uing topology optimiztion for deigning piezo-compoite microtructure to improve the piezoelectric enor nd ctutor ttribute. In thi re, Silv et l. h to be mentioned, who publihed verity of pper to thi topic (Silv, Fonec et l. 997; Silv, Fonec et l. 998; Silv, Fonec et l. 999; Silv nd Kikuchi 999; Silv, Nihiwki et l. 999; Silv, Nihiwki et l. 2). Buehler et l. (22) ued topology optimiztion nd homogeniztion on piezoelectric enor to reduce the deflection of cntilever bem mde of piezoelectric mteril. Furthermore, comprion of mplified piezoelectric ctutor bed on topology optimiztion h been done by Lovedy (22). To the uthor knowledge no previou tudy h been done in finding the optiml topology for piezoelectric ctutor of bem under bending..2 PROBLEM STATEMENT The objective of thi tudy i to determine the bet topology of piezoelectric ctutor for cntilever bem under ttic lod to minimize the deflection of the bem t the free end. The bem conit of three lyer of mteril, piezoelectric enor, n 5

21 eltic bem nd piezoelectric ctutor (figure.). To determine the bet topology of the ctutor lyer, numericl optimiztion i conducted. Finite element re typiclly ued for topology optimiztion problem, nlytic formultion would be fr too complex to ue prcticlly. Thu, the bem i modeled uing finite element with two-dimenionl firt-order continuum element. Anlyi i done uing the commercil finite element code ABAQUS. A feedbck control loop i implemented in the finite element model by modifying the tiffne mtrice uing MATLAB. The developed finite element model i linked with the commercil optimiztion lgorithm ViulDoc by uing equentil liner progrmming lgorithm (SLP). Prmeter tudie hve been conducted by vrying the number of element nd the totl llowed mount of mteril in the ctutor lyer. The reult of optimiztion with different geometrie nd feedbck gin re hown. Chpter two give n overview of the finite element modeling nd implementtion of the feedbck control loop in the commercil finite element code ABAQUS. Verifiction clcultion nd comprion to publihed reult hve been mde to confirm the correct implementtion of the control lw. Attempt hve been mde to implement dynmic proportionl nd derivtive control for ctive vibrtion dmping. Chpter three explin the bic ide of optimiztion nd give detiled decription of topology optimiztion uing homogeniztion. The implementtion of the finite element model in the optimiztion lgorithm i decribed. Chpter four how the reult of the prmeter tudy for different optimiztion. The lt chpter how the concluion of thi work nd give n overview of poible future work. 6

22 Actutor Bem Senor F Figure. Modeled cntilever bem 7

23 2 FINITE ELEMENT FORMULATION The gol of thi tudy i to conduct topology optimiztion for mechnicl ytem, which include piezoelectric enor nd ctutor. Since there i no reonble nlyticl olution to thi problem, numericl repreenttion need to be determined. The finite element method h been choen for thi tudy, ince it i widely vilble nd i commonly ued for topology optimiztion tudie. For thi reon, thi Chpter give n overview over the bckground of piezoelectric mteril nd their contitutive eqution. Furthermore, the implementtion of the control loop in the finite element model will be decribed for ttic nd dynmic ppliction. Reult of verifiction clcultion of the implemented feedbck control re hown t the end of the pproprite ection. However, the derivtion of the finite element formultion of piezoelectric mteril will not be hown, ince it h not been necery for thi reerch, commercil oftwre h been ued for the finite element modeling. For detil in the undertnding the finite element formultion of piezoelectric mteril plee refer to Kim et l. (997), Nillon et l. (983) who give good overview over thi topic. 2. THEORY FOR PIEZOELCTRIC MATERIAL Piezoelectricity i n electromechnicl phenomenon tht couple the eltic nd electric field. In generl, piezoelectric mteril repond to mechnicl lod nd generte n electric field. Converely, n electric field pplied to the mteril induce mechnicl trin. Chnge in temperture nd mbient electric field re conidered negligible. Thee umption re comptible with the piezoelectric cermic, polymer, 8

24 nd compoite in current ue (Silv, Fonec et l. 999). For thi reon, the contitutive reltionhip for piezoelectric mteril cn be written : σ = D ε e E (2.) ij ijkl kl mij m q i = e ε + p E (2.2) ijk jk ij j where Eqution 2. decribe the reltionhip between the tree ( σ ij ), the trin ( ε kl ) nd the electric field ( E m), coupled through fourth-order eltic tenor ( D ijkl ) nd third-order piezoelectric tenor ( e mij ). The piezoelectric tenor define the coupling of the electric nd eltic field. The firt index decribe the direction of the electric field nd the lt two indice decribe the direction of the eltic field. For exmple, the e 2 field couple the electric filed in the 2-direction, with the eltic field in the -direction. Thi men, n pplied electric field in the 2-direction reult in trin in the -direction. Eqution 2.2 decribe the reltionhip between the electric diplcement ( q ), the trin nd the electric field, coupled through third-order piezoelectric tenor nd econdorder dielectric tenor ( p ij ). i 2.2 STATIC CALCULATION The finite element eqution for modeling liner piezoelectric medium re decribed by Nillon et l. (983) nd Lerch (99). For ttic nlyi, thee eqution cn be written : K K uu Φ u K uφ K ΦΦ U F = Φ Q (2.3) 9

25 where uu K, Φ u K nd ΦΦ K re the tiffne, piezoelectric nd dielectric mtrice, repectively. In ome publictio n u K Φ i decribed t u K Φ, which i identicl. F, Q, U nd Φ re the nodl mechnicl force, the nodl electric chrge, the nodl diplcement nd the nodl electric potentil vector, repectively. Both the enor nd the ctutor cn mthemticlly be decribed through eqution 2.3. Senor nd ctutor element re herby denoted with upercript nd, repectively. Feedbck control for the ttic ce i pplied through: { } [ ] { } p G Φ = Φ (2.4) where [ ] p G i gin mtrix the define the mgnitude of the gin for ech ctutor node. If eqution 2.4 i inert into eqution 2.3 for the ctutor the following my be derived: = Φ ΦΦ Φ Φ p u u uu Q F G U K K K K (2.5) = Φ ΦΦ Φ Φ u u uu Q F U K K K K (2.6) Senor nd ctutor eqution cn then be combined into: = Φ Φ ΦΦ Φ ΦΦ Φ Φ Φ p u u p u uu u uu Q Q F F U U G K K K K G K K K K (2.7) Thi pproch doe not llow olving directly for the electric potentil of the ctutor node, ince the fourth column i eliminted. If deired, the electric potentil of the ctutor node cn be recovered uing eqution 2.4.

26 For thi work two generlly different pproche of control hve been ued. The firt pproch pplie direct node-to-node control. Thi men, the electricl chrge of enor node i directly pplied to the correponding ctutor node. Thi pproch i difficult to implement in relity. In mot rel ytem the totl electric potentil of the enor i pplied to the ctutor. Therefore, in econd pproch the electricl chrge of ll enor node i verged nd fterwrd pplied to ll ctutor node, which decribe more commonly pplied lyer-to-lyer control NODE-TO-NODE CONTROL A mentioned bove, node-to-node control pplie the electricl chrge of enor node directly to correponding ctutor node. Figure 2. clrifie thi method. In thi tudy node-to-node control h been implemented in reding the tiffne mtrice of ech enor nd ctutor element out of the commercil finite element progrm ABAQUS into MATLAB. Following the tiffne mtrix, decribed in eqution 2.7, i embled for one enor nd ctutor element t time. Finlly, the new defined tiffne mtrice re red bck into ABAQUS nd the complete ytem i lo olved uing ABAQUS LAYER-TO-LAYER CONTROL Lyer-to-lyer control pplie the totl electric potentil of the enor to the ctutor. Thi cn be implemented in finite element clcultion by verging the electricl chrge of the enor node nd pplying thi chrge to the ctutor node. Thi

27 Φ = Gp Φ Φ 3 = Gp Φ3 Actutor Φ n = G p Φn G p Φ n Φ Φ 3 Bem Senor Figure 2. Node-to-Node control 2

28 3 h the effect tht even in region where there i no diplcement of the enor node, the correponding ctutor node will be loded with n electricl chrge. A mentioned bove, thi pproch i more relitic, ince in relity the meured electric potentil i typiclly n verge of the enor over the enor urfce nd the pplied potentil i pplied evenly on the ctutor urfce. To implement thi type of control it i necery to emble the complete tiffne mtrix for ll ctutor nd enor element. Thi llow multiplying u K Φ nd K ΦΦ of the complete tiffne mtrix with gin mtrix. The gin for node number one cn be decribed : n G n p Φ Φ + Φ Φ = Φ (2.8) where n decribe the number of node t the urfce where the electric potentil i meured. From thi conidertion the gin mtrix i concluded: Φ Φ Φ Φ = Φ Φ Φ Φ n p n n G M L M O M M M M M M L L L M (2.9) Now the complete tiffne mtrix cn be embled nd the ytem cn be olved IMPLEMENTATION FOR THE STATIC CASE The implementtion of the control lw into the finite element code ABAQUS i done in three tep. Firt, the tiffne mtrice re contructed in ABAQUS nd written

29 to file. In the next tep, thee mtrice re red with MATLAB, nd proportionl control i pplied in ccordnce to the theory decribed bove. Following, the modified tiffne mtrice re written out by MATLAB nd red in by ABAQUS gin, in which the finl clcultion i done. Figure 2.2 clrifie thi procedure VERIFICATION FOR THE STATIC CASE For imple bem with lod only in the xil direction it i eily poible to derive n nlyticl reult nd compre the reult with finite element clcultion. In order to vlidte the proportionl control implementtion imple decoupled ytem h been modeled with geometry hown in figure 2.3 nd mteril propertie hown in tble 2.. Anlytic reult re then compred gint reult from the finite element clcultion. A decribed bove the contitutive reltionhip for piezoelectric mteril i defined : σ = D ε e E (2.) ij ijkl kl mij m q i = e ε + p E (2.) ijk jk ij j For the one-dimenionl ce the eqution reduce to: σ (2.2) = D ε e2 E2 q 2 e2 + p22 E2 = ε (2.3) The enor h no electricl urfce chrge but i loded with force in the xil direction which led to: 4

30 Clcultion by ABAQUS & output of m nd tiffne mtrice for ech element Reding of ll mtrice by MATLAB & modifiction of the tiffne mtrice ccording to the control lw Output of the m nd modified tiffne mtrice from MATLAB Reding of the mtrice nd olving the eqution by ABAQUS Figure 2.2 Proce digrm for ttic control 5

31 Z Actutor Diplcement Actutor X Senor F Diplcement Senor Figure 2.3 Decoupled bem ytem Tble 2. Mteril propertie for imple decouple bem ytem Mteril property Vlue Young Modulu N D = D = 2E8 2 m Poion Rtio υ =. 3 Piezoelectric tre coefficient C e2 = e2 = m Dielectric contnt F p22 = p22 =.4E 9 m Length l = l =.m Cro ection re Force A = A = F = N Gin G = 24 p 2. m 2 6

32 F σ = (2.4) A q = 2 (2.5) Now, the electric field nd the diplcement in the x-direction of the enor cn be clculted : E 2 = e σ e + p D (2.6) σ p ε = (2.7) e p D The ctutor i not loded with mechnicl force nd therefore σ = (2.8) but n electric potentil i pplied to the ctutor through the control lw. Thi i defined in eqution 2.4 : { } = [ G ] { Φ } Φ (2.9) p which for the one-dimenionl ce reult in: E2 Gp E2 = (2.2) o tht the trin of the ctutor cn be clculted : e 2 ε = + Gp E2 p 22 (2.2) D The ytem hown in figure 2.3 h been modeled with ten element for the ctutor nd enor, repectively. The reult of the finite element clcultion h been checked gint the theoreticl vlue for both node-to-node nd lyer-to-lyer control. The 7

33 implementtion of the node-to- node control h furthermore been done with liner (fournoded) nd qudrtic (eight-noded) element. The reult of thee clcultion nd computtion time on SUN UltrSPARC IIe re hown in tble 2.2. A one cn ee, the reult of the FEM clcultion conform in ll ce to one hundred percent with the theoreticl vlue. A econd verifiction of the finite element clcultion h been done by developing n nlytic olution for controlled piezoelectric cntilever bem loded with moment t the free end. The geometry nd the propertie of the bem re hown in figure 2.4 nd tble 2.3, repectively. For the development of the nlytic olution the one-dimenionl piezoelectric eqution re conidered gin. Thi i reonble umption, ince the trin induced by the ctutor in the xil direction i the min cue for the increed tiffne. Eqution 2.2 nd 2.3 re rewritten for the ctutor nd the enor: σ (2.22) = D ε e2 E2 q2 e2 + p22 E2 = ε (2.23) σ (2.24) = D ε e2 E2 q2 e2 + p22 E2 = ε (2.25) where σ, σ, ε, pplied to the enor: ε, E 2, E 2 nd q 2 re ll unknown. Since no electricl chrge i q = 2 (2.26) The control lw i pplied decribed before through: Φ = Gp Φ (2.27) nd cn be retted through the definition of the urfce chrge denity: 8

34 Tble 2.2 Vlidtion of ttic decoupled ytem Tip-Diplcement Senor [m] Tip-Diplcement Actutor [m] Computtion Time [ec] Node-to-Node Liner element E E Qudrtic Element E E Lyer-to- Liner Lyer element E E Theory E E

35 Figure 2.4 Tzou Bem Tble 2.3 Propertie for the Tzou Bem (ttic ce) Plexigl Bem Young Modulu 9 N D = m Thickne 3 h =.6 m Width b =. m Length L =. m Poion rtio µ =. 3 Senor/Actutor Lyer Young Modulu 9 N D = D = 2. 2 m Senor/Actutor thickne h = h = 4µ m Dielectric contnt Piezoelectric contnt p = p =.648 e3 = e3 = V m.46 N F m 2

36 where q Q C Φ C Φ = 2 2 = = G p (2.28) A A A Q 2 i the electricl chrge of the ctutor nd C i the cpcitnce, repectively. The cpcitnce cn be written : p A C = 22 (2.29) h where h i the ditnce between the pplied electric potentil. Now eqution 2.28 nd 2.29 cn be combined into: q2 Gp p22 E2 = (2.3) which i ued to pply the control lw for the nlytic olution. Since not enough eqution re vilble to olve thi ytem more eqution need to be developed. The enor nd ctutor lyer re mll compred to the bem lyer nd it cn be umed tht the tree in the ctutor nd enor lyer re contnt For thi reon the um of force in the x-direction i clculted (figure 2.5). F A σ σ x σ da = P b A A A = A A + c σ da + σ da + σ da = Pc h ρ + σ 2 b σ mx T h ρ b mx T z da b ρ σ b mx C ρ 2 z da + σ A b ( h ρ ) σ mx C b ρ + σ A = Pc = P c (2.3) A new vrible ρ i introduced, which meure the ditnce from the top of the bem to the neutrl xi. Thi i necery, ince the loction of the neutrl xi chnge by pplying control force. 2

37 z σ h P c ρ b σ mx C h M b σ mxt h x σ Figure 2.5 Bem Section 22

38 To obtin nother eqution the um of moment round the neutrl xi i clculted (figure 2.5). Since the enor nd ctutor lyer re mll compred to the bem lyer, it cn gin be umed tht the tree in the ctutor nd enor lyer re contnt, which led to: M A NA z σ da = M b A A A σ = h eq z σ da + z σ da + z σ da = M h b + 2 ρ + σ σ h b mx C ρ eq 2 b + σ 3 h b h ρ + = M 2 2 eq b mx T b ( h ρ) 2 (2.32) where M eq i n equivlent moment tht cn be clculted through: h M = + eq M Pc ρ (2.33) 2 Now eight eqution re vilble for the unknown b σ mx C, b σ mx T, P c, M eq σ, σ, ε, ε, E 2, E 2, q 2,, ρ. Thu, more eqution re needed to olve thi ytem of eqution. From figure 2.5 nd imilr tringle reltion, the following cn be derived: σ b mx C ρ b σ mx T = h ρ (2.34) Euler-Bernoulli Bem theory tte furthermore tht plne ection remin plne, which led to: ε ρ + h ε = h h ρ + 2 (2.35) 23

39 Figure 2.6 how implified deformed bem. Uing thi figure trin reltionhip cn be derived. The mximum tenion tre in the be bem occur t the lower ide of the be bem nd cn be decribed : b b b σ mx T = ε mx T D (2.36) The rc length in generl i defined : L = r θ (2.37) nd cn be ued to clculte the trin t the low ide of the be bem by uing the trin definition: L ε = (2.38) L the trin become to: ( r + h ρ ) b L θ r θ h ρ ε mx T = = = (2.39) L r θ r Now eqution 2.39 cn be inerted into eqution 2.36 nd olved for r : r ( h ρ) b D = (2.4) b σ mx T Similr, formultion for the trin in the enor cn be derived to: h h ρ + ε 2 = (2.4) r nd eqution 2.4 cn be inerted into eqution 2.4, leding to: h h ρ + b ε = σ 2 mx T (2.42) b D ( h ρ ) 24

40 r θ ρ h L Figure 2.6 Deformed Bem 25

41 The lt eqution needed to olve thi ytem of eqution define reltionhip for the control force in the piezoelectric ctutor. Two effect produce the electric field in the ctutor: induced trin due to deformtion of the bem nd the pplied control. The electric field due to the induced trin cn be clculted uing the econd eqution of the piezoelectric contitutive eqution nd etting the electric urfce chrge denity of the ctutor ( q 2 ) equl to zero: = ε (2.43) e2 + p22 E2 Thi llow olving for the electric field induced by the trin: E 2 ind e = (2.44) 2 ε p22 The electric field of the ctutor produce tre in the ctutor tht reult in the control force. The induced electric field need to be ubtrcted, which led to: Pc = A = A e e 2 2 ( E E ) 2 E 2 e + 2 ind ε (2.45) p22 2 Eqution 2.22, 2.23, 2.24, 2.25, 2.3, 2.3, 2.32, 2.33, 2.34, 2.35, 2.42 nd 2.45 re ued to olve for ll unknown by uing the commercil oftwre MAPLE. The clculted equivlent moment cn thn be inerted into the diplcement eqution, which i defined : 2 M eq L δ tip = (2.46) 2 ( DI ) eq ) eq for the free end. ( DI be clculted through: b b ( DI ) D I + D I + D I eq = (2.46) 26

42 Figure 2.7 how the clculted diplcement of the nlyticl olution nd the diplcement clculted by FEM for different gin. It i viible tht the FEM olution how le diplcement for lower gin thn the nlytic olution. Thi cn be explined by two reon. The firt reon i tht different umption hve been mde for the clcultion. The nlytic olution w clculted by uing the Euler-Bernoulli Bem Theory, where the FEM clcultion ued the elticity theory. The elticity theory tke Poion rtio into ccount nd i bout % tiffer compred to the Euler- Bernoulli Bem Theory. Thi trend i in complince with the olution hown here. However, the difference between the nlytic olution nd the FEM reult i bout 7% to 24% nd i therefore higher then it i uppoed to be. Thi cn be explined with the fct tht the FE olution h not converged completely. The geometry h been modeled with 25 element in ech lyer, which reult in n pect rtio of ten to one for the piezoelectric element. A repreenttion of the geometry with more element would reult in le tiff tructure nd the diplcement would therefore be cloer to the nlytic olution. A clcultion uing more element h not been conducted, building new finite element model i very time conuming proce, in which completely new inputfile h to be written. Since clcultion for gin of zero need no implementtion of the control lw, finite element clcultion uing more element cn be done quickly. The olution uing piezoelectric element with n pect rtio of one to five i hown point t zero gin. One cn ee tht thi reult i much cloer to the nlytic olution. The difference i bout 7%, which how n excellent complince with difference in the different theorie. It i furthermore viible, tht the lope of both clcultion re lmot identicl. Thi i expected nd verifie the finite element clcultion. 27

43 Diplcement [m] 8.E-5 7.5E-5 7.E-5 6.5E-5 6.E-5 5.5E-5 5.E-5 4.5E-5 4.E-5 3.5E-5 3.E-5 Comprion of Anlytic nd FEM Reult Mgnitude of Gin FEM Anlytic Figure 2.7 Comprion of tip diplcement of the Tzou-Bem for different Gin 28

44 Figure 2.8 how the reltive difference between the diplcement for gin of zero nd other gin. It i viible tht the reltive difference of the diplcement i very mll for lower gin. The reltive difference of diplcement between the diplcement of gin nd gin 5 i bout 8% for the FEM olution nd bout 3% for the nlytic olution. Thi how good complince of the nlytic nd the FEM olution. Even t gin of the reltive diplcement of the FEM clcultion i bout 33% compred to 25% for the nlytic olution. Thi i difference of 2%, which i cceptble conidering tht different umption hve been mde for both clcultion. For thee reon it i to y, proper implementtion for the ttic ce i vlidted. 2.3 DYNAMIC CALCULATION PROPORTIONAL CONTROL The finite element eqution for modeling liner piezoelectric mteril were developed by Nillon et l. (983) nd Lerch (99). Conidering time hrmonic excittion, thee eqution my be written : 2 M Kuu Ku Φ U F ω + = (2.47) KΦu KΦΦ Φ Q Thi ytem of eqution decribe the finite element model for piezoelectric mteril without conidering dmping effect. Thi umption i pproprite for thi tudy, ince the piezoelectric enor nd ctutor lyer re very thin compred to the be tructure. Therefore, the piezoelectric mteril h lmot no influence on the dmping chrcteritic. The implementtion of proportionl control i biclly the me in the ttic ce. The control lw i formulted : 29

45 35 Reltive Difference of Diplcement t Different Gin to Gp= 3 Reltive Difference [%] Mgnitude of Gin FEM Anlytic Figure 2.8 Reltive Difference 3

46 3 { } [ ] { } p G Φ = Φ (2.48) nd inerted into the tiffne mtrix exctly in the ttic ce. Since the m mtrice hve no influence on the control, the m mtrice remin unchnged. Therefore, the finite element formultion for piezoelectric mteril including proportionl control loop cn be written : = Φ Φ + ΦΦ Φ Φ ΦΦ Φ p u p u uu uu u uu Q F Q F U U G K K G K K K K K K M M 2 ω (2.49) where M nd M repreent the enor nd ctutor m mtrice, repectively. A decribed in the ttic ce proportionl control cn be implemented in two wy, firt by pplying node-to-mode control or econd by pplying lyer-to-lyer control. For the lyer-to-lyer control the me method i ued in the ttic ce IMPLEMENTATION OF PROPORTIONAL CONTROL The implementtion of the control lw into the finite element code ABAQUS i done through n identicl proce in the ttic ce. For dynmic clcultion it i necery to include the m mtrice in the clcultion. Since the m mtrice hve no influence on the control they re written out by ABAQUS nd red bck in without ny modifiction. ABAQUS i not ble to red in tiffne mtrix for big model with gret number of element, ABAQUS require lrge mount of memory for thi proce. Even increing the memory work only up to limit of 2 GB ince ABAQUS i limited

47 in the SOLARIS 8 ytem to thi mount of memory. Since it i necery for lyer-tolyer control to red in the complete tiffne mtrix nother pproch h been tken by embling the complete tiffne nd m mtrix in MATLB nd olving the ytem in MATLAB VERIFICATION OF PROPORTIONAL CONTROL In the firt tep of the verifiction of the implementtion of proportionl control the ytem hown in figure 2.3 i ued gin. In thi ce the ytem i excited through tedy-tte hrmonic excittion, inted of ttic lod. Since the nlytic olution to thi problem i not trivil the reult will be compred t low frequencie only, the ytem i tiffne dominted for low frequencie nd m or dmping dominted ner it nturl frequency. Thi hould be ufficient for verifying the technique outlined bove, the m mtrice re not modified. For low frequencie the difference between the ctutor nd enor diplcement mut be very cloe to the ttic ce, ince frequency of zero correpond to the ttic ce. The difference between enor nd ctutor diplcement from the ttic ce cn be obtined from tble 2.2 by: δ = u u = = (2.5) where the upercript indicte the ttic ce. Tble 2.4 compre the reult for ytem modeled with element for the ctutor nd enor, repectively, for different frequencie nd proportionl gin of 24. It i viible tht the error incree with increing frequency. However, thi doe not indicte n ctul error in the clcultion. It i more tht the m which w neglected for the theoreticl comprion, become more 32

48 Tble 2.4 Vlidtion for proportionl control (decoupled ytem) Frequency Diplcement [m] Senor Actutor Difference in Diplcement [m] Error (%) gint the ttic ce Sttic E-6.273E E E E E E E E E E E E E E E E E

49 dominnt. Figure 2.9 how the frequency digrm for proportionl gin of 24. It how furthermore verifiction clcultion, which w done in ABAQUS jut for the enor to confirm tht the enor h the frequency-digrm remin the me fter implementtion of the control. It i viible, tht the reonnt frequency of the ctutor nd enor re identicl. It i furthermore viible, tht the diplcement of the enor in the new model i identicl to the verifiction clcultion. Thi indicte good ccordnce with the theory. In the econd tep, the enor nd ctutor element re coupled over be bem tructure. The modeled ytem i tken gin from Tzou (993) nd hown in figure 2.4. The propertie of the model re hown in tble 2.5. A dmping rtio of.2 w umed in the be bem ( in Tzou, 993). Thi w implemented in ABAQUS uing Ryleigh dmping, thi cn be eily implemented within ABAQUS. For thi tudy, the bem h been modeled with 25 element long ech lyer. Thi ure n pect rtio mller then one to ten for ll element, which i pproprite for liner element tht hve been ued to model thi ytem. The reult of the finite element clcultion re compred with the reult publihed by Tzou nd re hown in tble 2.6. It i viible tht the nturl frequency clculted by the FEM computtion without pplied control (Gp=) i higher thn clculted by Tzou. Thi cn be explined with the fct, tht the FEM modeled i tiffer thn the ctul ytem, ince n pect rtio of one to ten i the upper limit for liner element. The reonnt frequency of the FEM clcultion would decree with more element. It i furthermore viible tht the proportionl gin in the Tzou clcultion h lmot no influence on the reonnt frequency. Thi i different in the FEM clcultion. The reonnt frequency decree drmticlly with higher gin. Thi 34

50 Amplitude-Frequency plot Sytem modeled with -Element. Amplitude Frequency Verifiction Actutor Gp=24 Senor Figure 2.9 Frequency-digrm for proportionl control (decoupled ytem) 35

51 Tble 2.5 Propertie of the Tzou bem (dynmic) Plexigl Bem Young Modulu 9 N D = m M Denity kg ρ = 9. m 3 Thickne h =.6 Width b =. m Length L =. m Poion rtio µ =. 3 Ryleigh dmping fctor 4 α =, β.35 Senor/Actutor Lyer Young Modulu 9 N D = D = 2. 2 m M Denity kg ρ = ρ = 8. m 3 3 m Senor/Actutor thickne Dielectric contnt Piezoelectric contnt h p = h = p = 4µ m =.648 e3 = e3 = V m.46 N F m Tble 2.6 Comprion Tzou v. FE reult of proportionl control clcultion Nturl Frequency Firt Mode Proportionl Lo Fctor [Hz] Feedbck FEM FEM Gin Tzou Error (%) Tzou Error (%) Clcultion Clcultion

52 influence the lo fctor well, ince it i clculted by the hlf power bndwidth method. According to thi method, the lo fctor i defined : ω ω η 2 ω n = (2.5) where ω n decribe the nturl frequency. ωnd ω 2 cn be found by clculting: An A HB = (2.52) 2 nd finding the correponding frequencie to thee mplitude. Figure 2. how the mplitude-frequency digrm for proportionl gin of,, 5 nd, repectively. Since no dditionl dmping i dded to the ytem by pplying proportionl feedbck control, the chnge in the lo fctor i only due to the chnge in nturl frequency. The reult of thi clcultion differ from imple one-dimenionl pring-m-dmping ytem, where the lo fctor i independent of the reonnt frequency. The me clcultion hve been done with proportionl lyer-to-lyer control. Since the clcultion hd to be done completely in MATLAB, fewer point hve been choen to plot the mplitude-frequency digrm, ince MATLAB require high computtionl power to olve big mtrix eqution. For thi reon, comprion of the lo fctor i not meningful, ince it i not poible to clculte n ccurte lo fctor without enough reolution. A comprion of the nturl frequencie i hown in tble 2.7. It i viible, tht the nturl frequency doe not decree ft uing lyer-to-lyer control by uing node-to-node control. However, the difference of the nturl frequency t proportionl gin of between the Tzou olution nd the FEM olution 37

53 2 Tzou-Bem 25 Element Proportionl Node-to-Node Control in ABAQUS Amplitude Frequency ABAQUS Gp= Gp= Gp=5 Gp= Figure 2. Frequency digrm of Tzou bem (finite element clcultion) Tble 2.7 Comprion of the nturl frequency of the Tzou bem (proportionl control) Proportionl Feedbck Gin Tzou Node-to-Node Lyer-to-Lyer Comp. Comp. Frequency Error (%) Time Frequency Error (%) Time (hour) (hour)

54 i too lrge. For eier viuliztion figure 2., how the frequency-mplitude digrm for proportionl gin of for node-to-node nd lyer-to-lyer control. We were unble to obtin tifctory reult for the dynmic clcultion uing proportionl control, ince publihed reult could not be repeted. 2.4 DYNAMIC CALCULATION - DERIVATIVE CONTROL Tzou (993) how, tht proportionl control i not well uited for ue in ctive dmping tructure. Proportionl control only h influence on the nturl frequency, but leve the dmping rtio nerly unchnged. Since the gol of ctive dmping i to incree the dmping effectivene, proportionl i not good choice. For thi reon, derivtive control h lo been implemented for dynmic clcultion IMPLEMENTATION OF DERIVATIVE CONTROL The feedbck ignl from the ctutor to the enor i now repreented : { } = [ G ] { } d Φ Φ & (2.53) where [ G d ] repreent the derivtive control gin mtrix nd the undercore indicte tht the vrible i time dependent. For hrmonic excittion the enor potentil cn be written : i t { } = { Φ } e ω Φ (2.54) nd the ccording derivtive : iωt { } = iω{ Φ } e Φ& (2.55) 39

55 Tzou Bem Element Comprion of control lw Amplitude Frequency Gp = Gp = (lyer-to-lyer) Gp = (node-to-node) Figure 2. Frequency digrm of the Tzou bem (proportionl control) 4

56 4 Now, eqution 2.55 cn be inerted into eqution 2.53, which led to: { } [ ]{ } t i d e G i ω ω Φ = Φ (2.56) For Node-to-Node control the gin mtrix cn be implified to: [ ] [ ] I G G d d = (2.57) where [ ] I i the identity mtrix. Since the mtrix term ocited with Φ re now ninety degree out of phe, new dmping mtrix mut be included in eqution The complete finite element formultion for piezoelectric mteril with derivtive feedbck gin cn be written : = Φ Φ + + ΦΦ Φ Φ ΦΦ Φ ΦΦ Φ u u uu uu u uu d d u Q F Q F U U K K K K K K K K G K G K i M M 2 ω ω (2.58) where the middle mtrix decribe the dmping mtrix of the piezoelectric mteril. To implement lyer-to-lyer control eqution 2.9 mut be chnge to: Φ Φ Φ Φ = Φ Φ Φ Φ n p n n G & M & & & L M O M M M M M M L L L M (2.59) nd the verged vlue cn be inert into the dmping mtrix in eqution ABAQUS h no wy of defining the dmping mtrix directly. The eiet method in ABAQUS to include dmping i Ryleigh dmping, in which the dmping mtrix i defined : [ ] [ ] [ ] K M C β α + = (2.6)

57 where [ M ] nd [ K ] re the m nd tiffne mtrix nd α nd β the Ryleigh dmping coefficient, repectively. Since the finite element code ABAQUS offer no poibility to define dmping mtrix directly, novel pproch hd to be developed in order to crete tiffne mtrix derived in eqution Thi pproch cn be ummrized in three tep: *. An dditionl element i creted with tiffne [ K ] 2. The dmping mtrix i defined in term of thi mtrix * [ C] β [ K ] where β = = (2.6) ** 3. A third element i creted [ K ] with the property: ** * [ K ] [ K ] = (2.62) * to ubtrct out the tiffne of [ K ], it i not prt of the tiffne of the ctul ytem VERIFICATION OF DERIVATIVE CONTROL To verify the correct implementtion of the derivtive control two-tep pproch i ued gin. In the firt tep the mechnicl ytem hown in figure 2.3 i ued nd excited hrmoniclly. A w done for proportionl control, the nlytic olution to thi ytem i not trivil nd therefore the m influence i neglected for the verifiction. For thi reon, the theoreticl vlue re compred with the finite element clcultion t low frequencie only, ince the ytem i tiffne dominted for low frequencie. 42

58 The contitutive lw for the one-dimenionl ce from eqution 2. nd 2.2 i vlid well for the ttic ce for the dynmic ce. For hrmonic excittion, the tre i function of time nd cn be written : σ F iω t = e (2.63) A The control lw decribed in eqution 2.53 to 2.56 cn now be implified to: = & (2.64) E 2 Gd E2 where the undercore indicte tht the vrible i time dependent. The electric field of the enor for hrmonic excittion cn lo be written : E iω t 2 = E 2 e (2.65) nd ccordingly the time derivtive of the electric field: & iω t 2 = i ω E 2 e (2.66) E which i inerted into eqution 2.64 nd become: E iω t 2 = i ω Gd E2 e (2.67) With thi reltionhip nd the contitutive eqution it i now poible to clculte the trin of the ctutor : ε e2 Gd ω E2 iω t = i e (2.68) D where the middle prt of eqution 2.68 decribe the mplitude, which i compred in thi verifiction. Tble 2.8 how the theoreticl vlue nd the reult of finite element clcultion for the decoupled mechnicl ytem hown in figure 2.3. It i viible tht the 43

59 Tble 2.8 Vlidtion for derivtive control (decoupled ytem) Frequency Diplcement [m] Theory FEM Clcultion Difference [m] Error (%).5839E E E E

60 error for low frequencie i negligible. However, by pproching the nturl frequency the m become dominnt for the olution nd the theoreticl vlue hown here re no longer ccurte. In the econd tep the enor nd ctutor lyer re coupled by be bem gin. The me ytem h been ued, which w ued for the verifiction of the proportionl control nd i hown in figure 2.4 with the ccording propertie in tble 2.5. The reult of the FEM clcultion re compred gin gint the olution publihed by Tzou (993). The reult of thi comprion re hown in tble 2.9. It i viible, tht the derivtive gin ued by Tzou i much higher then the derivtive gin ued in the FEM clcultion. Gin in the order of mgnitude ued by Tzou reult in over dmped olution for the FEM clcultion. To chieve reult with reonble lo fctor the gin w reduced. Tzou how furthermore, tht derivtive gin of i lmot n optiml vlue for dmping of the firt mode, bigger vlue re le effective nd the lo fctor i decreing. Tzou explin thi effect with the fct, tht higher gin chnge the boundry condition t the free end to liding-roller boundry condition. Thi effect cn only be oberved if lyer-to-lyer control i pplied, ince node-to-node control h lmot no effect t the free end. It w not tried to implement lyer-to-lyer control for derivtive control, ince the implementtion of lyer-to-lyer control for proportionl control reulted in extremely long computtion time. The FEM clcultion how tht increing derivtive gin incree the lo fctor nd i therefore contrry to the Tzou reult. It i lo oberved, tht increing gin in the Tzou reult incree the firt nturl frequency. Thi could not be repeted by the FEM clcultion, which how n identicl nturl frequency for ll derivtive 45

61 Tble 2.9 Comprion Tzou v. FE reult of derivtive control clcultion Derivtive Feedbck Gin Tzou Nturl Frequency Firt Mode [Hz] Derivtive Feedbck Gin FEM Nturl Frequency Firt Mode [Hz] Lo Fctor Lo Fctor

62 gin. For clrifiction, the frequency repone how for the FEM clcultion of the Tzou bem for different derivtive gin i hown in Figure 2.2. In concluion, I w not ble to obtin tifctory reult for the implementtion of proportionl nd derivtive gin. The reult of the ttic clcultion re the only vlidted reult nd re therefore ued for the tudy. 47

63 Tzou-Bem 25 Element Derivtive Node-to-Node Control in ABAQUS Amplitude Frequency ABAQUS Gd= Gd=.5 Gd= Gd=2 Figure 2.2 Frequency digrm of the Tzou bem (derivtive control) 48

64 3 OPTIMIZATION The concept of optimiztion i bic ide in engineering. The deire to improve deign, for exmple, to mke product better, lighter, cheper or more relible, h been mjor ide ince erly engineering yer. Numericl optimiztion h been proven to be ueful tool for improving complex deign. Thi chpter give n overview of the generl ide of optimiztion routine ued nd how the optimiztion problem i et up for the given ce. 3. INTRODUCTION In generl, n optimiztion problem begin with et of independent deign vrible nd uully include condition or retriction tht define cceptble vlue for thee vrible. For given vlue of deign vrible it mut be poible to compute n objective function, which give meure for the goodne of the deign. In mthemticl term, optimiztion i the minimiztion or mximiztion of n objective function within given contrint on the deign vrible. The generl form of n optimiztion problem cn be expreed in mthemticl term : r min F ( x) n x R ubject to r h i ( x) =, i =, 2,..., m r g i ( x), i =, 2,..., l (2.) where it i umed, tht the objective nd contrint function re continuou rel-vlued clr function. 49

65 3.2 GENERAL TOPOLOGY OPTIMIZATION The trditionl wy of finding the bet hpe or topology for mechnicl tructure i n itertive tril-nd-error pproch. The deign engineer ue hi experience nd intuition to find olution to given problem. Mechnicl or numericl tet then how if the deign meet the pecified criteri. If the deign fil, the deign engineer enhnce the deign until tifying reult i found. Thi ytem require both pecil kill nd experience for truly good deign nd it doe not gurntee tht the bet poible deign h been found. Structurl hpe optimiztion cn be thought of determintion of the optiml ptil mteril ditribution. In other word, for given et of lod nd boundry condition, the mteril i reditributed in order to minimize the objective. Therefore, the generl hpe optimiztion problem cn be conidered point-wie mteril/no-mteril pproch. However, implementtion of thi on-off pproch to n optimiztion problem require the ue of dicrete optimiztion lgorithm. Such pproche hve been hown to be time conuming nd untble, unle compoite mteril re introduced (Hni nd Hinton 999). Conidering compoite coniting of n infinite number of mll hole, which re periodiclly ditributed, cn olve thi problem (figure 3.). In fct, uing cellulr body with periodic microtructure move the on-off pproch of the problem from the mcrocopic cle to the microcopic cle (Bendoe 989). One pproch to introduce thee microtructure i homogeniztion. The theory of homogeniztion i ued to determine the mcrocopic mechnicl propertie of thee mteril. In prctice, fter chooing the deign domin nd the finite element dicretiztion, it i umed tht ech element conit of cellulr mteril with pecific microtructure. The geometricl 5

66 compoite mteril micro tructure Figure 3. Homogeniztion through microcell with rectngulr hole (Hni nd Hinton 999) 5

67 prmeter of thee microtructure re the deign vrible of the optimiztion problem. In the implet ce the microtructure h rectngulr hole or void ( hown in figure 3.) nd the mechnicl propertie become proportionl to the denity of the mteril. Figure 3.2 illutrte thi proce for the deign of brcket uing homogeniztion, where the deign domin nd the boundry condition re hown in Figure 3.2(). The optimiztion proce vrie the denity of ech finite element, which i hown through gry cle in the picture. A cell tht i completely blck correpond to denity of %, where cell tht i completely white correpond to denity of %. The optiml mteril ditribution for thi problem ( tiff lightweight deign) i hown in Figure 3.2(b). Since mteril with intermedite denitie re rtificil nd cnnot be produced, thi olution need to be interpreted. Thi i done in Figure 3.2(c). Furthermore, generl mnufcturing rule cn be pplied, which led to the finl olution of thi problem, hown in figure 3.2(d). 3.3 TOPOLOGY OPTIMIZATION IN THIS STUDY Topology optimiztion uing the homogeniztion method cn be ued for piezoelectric mteril hown by Silv et l. (999). A homogeniztion pproch uing microcell with rectngulr void i ued decribed bove, where the piezoelectric propertie need to be conidered dditionlly. The finite element formultion of piezoelectric mteril h tiffne tenor D ijkl, piezoelectric tenor e ijkl nd dielectric tenor p ij decribed in chpter 2. If the bic mteril conidered in the nlyi i teel, the piezoelectric coefficient re zero, nd the electric effect i not 52

68 () (b) (c) (d) Figure 3.2 Optiml topology deign (Pplmbro nd Dougl 2) 53