Topology Optimization of Carbon Nanotube Reinforced Polymer Damping Structures

Transcription

1 Unversty o ennessee, Knoxvlle race: ennessee Research and Creatve xchange Masters heses Graduate School opology Optmzaton o Carbon Nanotube Renorced Polymer Dampng Structures Seshadr Mohan arma Damu Unversty o ennessee - Knoxvlle Recommended Ctaton arma Damu, Seshadr Mohan, "opology Optmzaton o Carbon Nanotube Renorced Polymer Dampng Structures. " Master's hess, Unversty o ennessee, hs hess s brought to you or ree and open access by the Graduate School at race: ennessee Research and Creatve xchange. It has been accepted or ncluson n Masters heses by an authorzed admnstrator o race: ennessee Research and Creatve xchange. For more normaton, please contact trace@utk.edu.

2 o the Graduate Councl: I am submttng herewth a thess wrtten by Seshadr Mohan arma Damu enttled "opology Optmzaton o Carbon Nanotube Renorced Polymer Dampng Structures." I have examned the nal electronc copy o ths thess or orm and content and recommend that t be accepted n partal ulllment o the requrements or the degree o Master o Scence, wth a major n Mechancal ngneerng. We have read ths thess and recommend ts acceptance: J.A.M.Boulet, Wllam.R.Hamel (Orgnal sgnatures are on le wth ocal student records.) Arnold Lumsdane, Major Proessor Accepted or the Councl: Carolyn R. Hodges ce Provost and Dean o the Graduate School

3 o the Graduate Councl: I am submttng herewth a thess wrtten by Seshadr Mohan arma Damu enttled "opology Optmzaton o Carbon Nanotube Renorced Polymer Dampng Structures" I have examned the nal electronc copy o ths thess or orm and content and recommend that t be accepted n partal ulllment o the requrements or the degree o Master o Scence, wth a major n Mechancal ngneerng. Arnold Lumsdane Major Proessor We have read ths thess and recommend ts acceptance: J.A.M.Boulet. Wllam.R.Hamel. Accepted or the Councl: Anne Mayhew ce Chancellor and Dean o Graduate Studes (Orgnal sgnatures are on le wth ocal student records)

4 opology Optmzaton o Carbon Nanotube Renorced Polymer Dampng Structures A hess Presented or the Master o Scence Degree he Unversty o ennessee, Knoxvlle Seshadr Mohan arma Damu December 2005

5 Acknowledgments I wsh to thank all those who helped me n completng my Master o Scence n Mechancal ngneerng. Specal thanks go to Dr. Arnold Lumsdane. Wthout hs nput, motvaton and knowledge ths work would not have been possble. He has been a great proessor, mentor and advsor. I thank Dr. J.A.M.Boulet and Dr. Wllam.R.Hamel or beng n my thess commttee and provdng me wth ther valuable suggestons. I would also lke to thank my amly and rends or supportng me throughout ths past two years.

6 Abstract opology optmzaton has been successully used or mprovng vbraton dampng n constraned layer dampng structures. Renorcng carbon nanotubes n a polymer matrx greatly nluence the mechancal propertes o the polymer. Such nanotube-renorced polymers (NRP) can be used to urther enhance the dampng propertes o the constraned layer structures. In ths work, topology optmzaton s perormed on constraned dampng layer structures usng NRP n order to maxmze the loss actor or the rst resonance requency o the base beam. In addton to the materal ractons o the NRP and elastc materal, the volume racton o the nanotubes n the polymer s also a desgn varable n the optmzaton process. he modal stran energy method s used or the loss actor calculaton. A commercally avalable nte element code ABAQUS s used or the nte element analyss. he structure s dscretzed usng 2-dmensonal 8-noded quadratc elements. Optmzaton s perormed wth a gradent based optmzaton code whch uses a sequental quadratc programmng algorthm. o make the optmzaton process more ecent, an analytcal method to calculate the gradents s derved to replace the prevously used nte derence method. he resultng structures show a remarkable ncrease n dampng perormance. o show the robustness o the optmzaton process, materal racton and base beam thckness parameter studes are also perormed.

7 able o Contents 1. Introducton Lterature Survey Problem Statement 7 2. Modellng Materal Modellng Composte Propertes scoelastcty System Loss Factor Modal Stran nergy Method Hal Power Bandwdth Method Optmzaton Problem Formulaton Analytcal Gradents Formulaton aldaton Results Parameter Studes Materal Fracton Parameter Study Base Beam hckness Parameter Study 46 v

8 5. Conclusons and Future Work 51 Bblography 53 IA 61 v

9 Lst o ables 2.1 Materal propertes Comparson o gradents calculated rom numercal and analytcal methods Comparson o results obtaned rom topology optmzaton usng the analytcal and numercal gradent calculaton methods Materal propertes (repeated rom table 2.1) Results obtaned rom modal stran energy method and hal power bandwdth method or materal racton parameter study Results obtaned rom modal stran energy method and hal power bandwdth method or base beam thckness parameter study 47 v

10 Lst o Fgures 1.1 Free layer dampng Constraned layer dampng llptcal hysteress loop or lnear vscoelastc materals Hal power bandwdth method Fnte element model producng local mode Optmzaton process low chart Intal conguraton Fnal shapes Intal conguratons Materal dstrbuton n the optmzed conguraton or materal racton parameter study Intal loss actor vs. materal racton Fnal loss actor vs. materal racton Percentage mprovement n the loss actor vs. materal racton Intal loss actor vs. base beam thckness Fnal loss actor vs. base beam thckness Percentage mprovement n the loss actor vs. base beam thckness 49 v

11 4.10 Materal dstrbuton n the optmzed conguraton or base beam thckness parameter study 50 v

12 Nomenclature NRP Nanotube renorced polymer CN Carbon nanotube ρ N Densty o carbon nanotube ρ v Densty o vscoelastc materal v olume racton o nanotubes N Stness modulus o carbon nanotube v Stness modulus o vscoelastc materal σ Stress ε Stran * Complex modulus lastc or storage modulus Loss modulus η c Core or materal loss actor * G Complex shear modulus x

13 * K Complex Bulk modulus * v Complex Posson s rato η System loss actor D nergy dsspated per cycle W otal energy per cycle U u Porton o stran energy attrbutable to the vscoelastc core (complex quantty) U Stran energy o the system [ M ] [ C ] [ K ] Mass matrx o the system Dampng matrx Stness matrx U Stran energy o vscoelastc elements calculated rom purely elastc analyss U Stran energy o elastc elements gen vector or mode shape x

14 [ K ] Stness matrx o vscoelastc elements obtaned rom purely elastc analyss [ K ] Stness matrx o elastc elements ω d Damped natural requency ω1 and ω 2 Frequences at hal power ponts v x Fracton o NRP materal o the element e x Fracton o elastc materal o the element v otal racton o NRP materal e otal racton o elastc materal n he number o NRP elements (whch s equal to the number o elastc elements). v olume racton o carbon nanotubes n the polymer materal x ( x) x Step sze Gradent o the objectve uncton wth respect to the th desgn varable x

15 ( x x) + Change n the objectve uncton due to a small change n the th varable. X [ x 1 x 2 x 3... x n ] where, x s the th desgn varable. Κ, Κ B Κ lemental stness matrces o the th vscoelastc element, th elastc element n the base beam and th elastc element n the desgn space respectvely. Part o the egenvector or th vscoelastc element B Part o the egenvector or th elastc element n the base beam element Part o the egenvector or th elastc element n the desgn space p Number o vscoelastc elements = number o elastc elements n the desgn space. b Number o elastc elements n the base beam. x

16 CHAPR 1: INRODUCION Unwanted vbratons n engneerng applcatons can have adverse aects rangng rom beng mldly annoyng to beng extremely dangerous. hese are a hndrance to perormance o machnery and cause human dscomort. xcessve vbratons also cause nose and materal atgue. bratons n structures wth nsucent dampng can result n loss o le and property. bratons n dynamc systems can be reduced by a number o means. Dampng by absorpton, solaton, ar dampng, magnetc hysteress, partcle dampng, lud vscosty and pezoelectrc dampng are a ew such methods. In structural applcatons, one common orm o dampng employed to reduce nose and vbraton s usng vscoelastc lamnates, usually n the orm o an add-on treatments appled to a structure. Dampng reers to the extracton o mechancal energy rom a vbratng system usually by converson nto heat. Internal dampng and structural dampng are two general orms o dampng n structures. Internal dampng or materal dampng reers to the mechancal energy dsspaton wthn the materal and structural dampng reers to dampng at supports, jonts, nteraces etc. Most engneerng structural applcatons have very lttle nternal dampng. In such cases, applyng a vscoelastc layer on the structure s one o the easest and most cost eectve methods o vbraton dampng. braton dampng usng vscoelastc materals can be classed as ether ree or constraned layer dampng treatment. Free layer dampng nvolves bondng the dampng materal to the structure, usually usng a pressure senstve adhesve. When the base structure deorms n bendng, 1

17 the vscoelastc materal deorms prmarly n extenson as shown n gure 1.1. he degree o dampng s lmted by the thckness and weght restrctons. In the vbraton analyss o a beam wth a vscoelastc layer conducted by Kerwn (1959), t was seen that the system loss actor o a ree layer system ncreases wth the thckness and loss actor o the vscoelastc layer. In constraned layer dampng treatment, there s an addtonal constranng layer on top o the vscoelastc layer as shown n gure 1.2. In ths case, the energy dsspaton occurs prmarly by shear. Ross et al. (1959) perormed analytcal and expermental studes o constraned layer dampng structures usng vscoelastc materals. hey showed that shear dampng (constraned layer dampng) s a more eectve method than ree layer dampng. 1.1 Lterature Survey One o the rst analytcal studes o unconstraned layer beams was conducted by Oberst and Frankeneld (1952). Commonly used methods or analyss o ree layer and scoelastc materal Base beam Deormed shape Fgure 1.1 Free layer dampng Constranng layer Base beam scoelastc core Deormed shape Fgure 1.2 Constraned layer dampng 2

18 constraned layer treatments were developed by Kerwn (1959) and Ross et al. (1959), respectvely. hese were ourth-order bendng equatons developed assumng snusodal expansons or the modes o vbraton applcable to smply supported beams. A more general sxth-order equaton o moton or arbtrary boundary condtons was derved by Daranto (1965) and Mead and Markus (1969). Rao (1978) developed a set o equatonso moton and boundary condtons or the vbraton o sandwch beams usng an energy approach. Numerous other studes were revewed by Nakra (1976, 1981 and 1984). Fnte elements have commonly been employed to characterze lamnated structures (or example, see Hwang et al. 1992). Ungar and Kerwn (1962) ntroduced the concept o dampng n terms o stran energy quanttes. he mplementaton o the stran energy method n nte element orm to predct the loss actor o composte structures was rst demonstrated by Johnson and Kenholz (1981). hey ntroduced the modal stran energy method whch s now wdely used. Son and Bogner(1982) used a nte element computer program, MAGNA-D to predct the response o damped structures to steady state nputs. Methods to predct the dampng n ber-renorced polymer compostes have also been nvestgated. Schultz and sa (1968) have expermentally determned the ansotropc, lnear vscoelastc behavor (or small oscllatons) o the ber-renorced composte. Abarcar and Cunn (1972) have ormulated a dscrete mathematcal model to predct the natural requences and ther correspondng mode shapes o xed-ree beams o general orthotropy. Huang and eoh (1977) perormed a theoretcal analyss o the vbratons o ber-renorced composte beams usng an energy approach. Hwang and 3

19 Gbson (1987) developed a mcromechancal model to descrbe the dampng n dscontnuous ber compostes usng stran energy approach. eh and Huang (1979) presented nte element models or the predcton o natural requences o xed-ree beams o general orthotropy. Alberts and Xa (1995) developed a mcromechancal model takng nto account the eects o ber segment lengths and relatve moton between neghborng bers. hey showed that ber-enhanced vscoelastc dampng treatment provdes sgncant dampng to a treated cantlever beam. Carbon nanotube (CN) renorced polymer compostes are beng wdely nvestgated or dampng purposes. Nkhl et al. (2003) studed the use o nanotube lms n structures or vbraton dampng. He used nanotube lms as nter-layers wthn composte ples. Hs expermental nvestgatons revealed that by ncludng nanotube lms there s a 200% ncrease n dampng levels. Zhou et al (2003) nvestgated the dampng characterstcs o polymerc compostes dstrbuted wth sngle-walled carbon nanotubes. hey demonstrated that sngle-walled carbon nanotube based compostes acheve hgher dampng than compostes wth other types o llers. Ros et. al. (2002) nvestgated the dynamcal mechancal propertes o sngle-walled nanotube renorced polymer compostes assumng a sngle lnear sold model. her work showed that there s a decrease n the loss actor wth an ncrease n the percentage weght o carbon nanotubes. Further research has manly ocused on mcromechancal modelng. Zhou et al (2004), descrbed the load transer between the CNs and the resn usng the concept o stck-slp moton. hostenson and Chou (2003) used the mcromechancal model used or modelng short ber compostes (Sun et al. 1985) to account or the structure o 4

20 nanotube renorced compostes. Lu and Chen (2003) demonstrated the boundary element method or modelng the mcromechancal behavour o CN based compostes. Many modcatons have been proposed to the topology o constraned layer structures n eorts to mprove ther dampng perormance. Multple constraned layer treatments were suggested by Ungar and Ross (1959). Plunkett and Lee (1970) developed a method to compute the optmal secton length o the constranng layer that provdes maxmum dampng. Ln and Scott (1987) optmzed the shape o a dampng layer usng a structural nte element model. Hajela and Ln (1991) used a global optmzaton strategy to maxmze the system loss actor wth respect to dampng layer lengths or a constraned layer beam. he role o bers n mprovng nherent dampng n composte structures has been studed extensvely by Gbson et al. (1982), Sun et al. (1985a) and Sun et al. (1985b). hese studes nvolved analytcal and expermental studes on algned short ber compostes, algned short ber o-axs compostes, and randomly orented short ber compostes. Fber aspect rato, angle between appled tensle load and ber drecton, stness rato between the ber and matrx materals, and the dampng rato between ber and matrx materals were optmzed to mprove dampng n the structure. Alberts and Xa (1995) derved optmal relaton between desgn parameters such as length, dameter, spacng and Young s modulus o bers and shear modulus o vscoelastc matrx to acheve maxmum dampng perormance. Bendsøe and Kkuch (1998) rst ntroduced the homogenzaton method or ndng the optmal topology or a structural problem. opology optmzaton has been shown to be an ecent tool or structural problems wth gven boundary condtons. ander Slus, et al (1999) have perormed topology optmzaton o heterogeneous 5

21 polymers usng homogenzaton, but ths study was purely statc and dd not examne dampng propertes. Y, et al (2000) perormed topology optmzaton usng the homogenzaton method to maxmze the dampng characterstc o a vscoelastc materal, but ths was not n the context o constraned layer dampng. Zeng et al (2003) perormed layout optmzaton o passve constraned layer dampng patches usng a genetc algorthm based penalty uncton method. hree-phase compostes have been studed n the context o optmzng thermal expanson or a composte (Sgmund and orquato, 1997), but not n the context o constraned dampng layer. Lumsdane (2002) successully used topology optmzaton to nd the optmal shape o a constraned vscoelastc dampng layer wth the objectve o maxmzng the system loss actor or the undamental requency o the base beam. A 325% mprovement n the loss actor was acheved due to the materal redstrbuton. Lumsdane and Pa (2003) extended ths work to perorm base beam thckness and materal racton parameter studes. hs nvolved perormng optmzaton studes or derent base beam thcknesses and materal ractons. Sgncant mprovements n the loss actor were obtaned. he varatons o the loss actor as the base beam thckness and materal racton were examned and the optmal base beam thckness and materal racton were determned. Pa et al, (2004) perormed expermental valdaton o the results obtaned rom topology optmzaton studes. In ths work, a conguraton smlar to the one used by Lumsdane (2002) was used. he optmzaton process requres calculaton o gradents n every teraton. In all o the prevous work (Lumsdane 2002, Lumsdane and Pa 2003 and Pa et al 2004), a nte derence based method was used whch requres a nte element run or each 6

22 gradent calculaton n an teraton. Wth a large number o desgn varables, ths consumed a consderable amount o tme (~30 hours) or the optmzaton. An am o ths study s to develop an analytcal method or the gradent calculaton n the optmzaton process. An analytcal method would mprove the ecency o the optmzaton process as t would sgncantly reduce the number o nte element runs requred per teraton. Addtonally, NRP materal s used n the core nstead o purely vscoelastc materal. Apart rom the materal ractons o NRP and elastc materals the volume racton o nanotubes n the polymer s also allowed to vary n the optmzaton process. 1.2 Problem Statement he objectve o ths work s to determne the best topology o a carbon nanotube renorced polymer dampng treatment so as to maxmze the system loss actor or the undamental requency. A constraned layer beam structure wth a NRP (Nanotube Renorced Polymer) core and an elastc constranng layer s used. he NRP core and the constranng layer consttute the desgn space n the optmzaton process. he beam s modeled usng nte elements wth two-dmensonal second-order plane stress contnuum elements. he materal ractons o NRP and the elastc materal n each o these nte elements and the volume racton o carbon nanotubes n the polymer are the desgn varables. Analyss s done usng the commercal nte element code ABAQUS. As the materal ractons and volume racton o CNs change n the optmzaton process, the rule o mxtures s used to determne the materal propertes (stness and densty) o the NRP core. A moded 7

23 modal stran energy method (Xu, et al, 2002) s used to compute the loss actor o the structure. A gradent based optmzaton code (NLPQL) s used. ach tme a new set o desgn varables are obtaned, the objectve (loss actor) and the gradents o the objectve uncton wth respect to the desgn varables are calculated analytcally. A varaton n the desgn varables aects the materal propertes (stness and densty) o the correspondng elements. Hence, a nte element analyss s perormed to obtan the new stness and mass matrces and the mode shape o the structure. hese are then used to compute the elastc and vscoelastc stran energes o the structure. he newly computed stness and mass matrces, mode shape and stran energes are used to compute the loss actor and the gradents. Parameters such as thckness o the base beam and the volume o the dampng materal and NRP materal are vared and the eect o these parameters on the optmal shapes s examned. It s seen that there s a remarkable mprovement n dampng o about 1000% n the structure. hs huge mprovement n the dampng levels s seen to be consstent or all the cases. hs demonstrates the robustness o topology optmzaton. Moreover, n all the cases, the volume racton o nanotubes ncreases to the maxmum allowable value. As a result the NRP materal becomes hghly st. hs hgh stness materal no longer dsspates energy by shear and changes rom beng a constraned layer to beng a ree layer dampng structure. Chapter 2 gves an overvew o the analytcal and nte element modelng, the theory behnd the problem and assumptons made, and also descrbes the mplementaton o the nte element model n the optmzaton algorthm. Chapter 3 explans the 8

24 analytcal gradent calculaton method. Chapter 4 shows the results a comparson o the results obtaned rom analytcal and numercal gradent calculaton methods and the results o the parameter studes. he last chapter conssts o the conclusons and a dscusson o possble uture work. 9

25 CHAPR 2: MODLLING hs chapter gves an overvew o the materal modelng or the vscoelastc materal and the NRP composte, the methods used to measure dampng n the structurethe modal stran energy method and the hal power bandwdth method, and the nte element modelng or the structure. Carbon nanotube based polymer compostes are beng wdely nvestgated or vbraton dampng purposes. Unlke purely elastc materals whch have a lattce structure, a polymer materal conssts o long chan molecules. Due to the mperect elastcty o these long chan polymers, the materal gves much larger energy dsspaton when deormed dynamcally. Carbon nanotubes have stness o the order o 1 Pa. When a polymer matrx s renorced wth such hgh stness materal, the resultng composte s assumed to exhbt greater stness due to the presence o nanotubes and greater energy dsspaton due to the vscoelastcty o the polymer. 2.1 Materal Modellng Composte Propertes Many mcromechancal models were used to descrbe the dampng propertes o nanotube renorced polymer compostes (see secton 1.1). None o these materal models are sutable to mplement n a dynamc F model as t requres both stness and dampng propertes o the composte whch none o these models provde. Snce a sutable model s not avalable n the lterature a smpled model s adapted or ths study. A better model wll be mplemented when one becomes avalable. It s assumed that the NRP composte behaves as a vscoelastc materal wth ts materal propertes determned by the rule o mxtures (equatons 2.1 and 2.2). Unorm dstrbuton and 10

26 random orentaton o nanotubes n the polymer matrx s assumed,.e. the equvalent vscoelastc materal s sotropc. ( 1 v ). ρ v ρ = v. ρ + (2.1) N N ( 1 v ). v = v. + (2.2) he materal loss actor vares wth the volume racton o the nanotubes. Agan, due to the unavalablty o a model whch sucently descrbes the eects o nanotube volume racton on the loss actor, a constant materal loss actor equal to the loss actor o the polymer matrx s assumed scoelastcty scoelastcty may be dened as materal response that exhbts characterstcs o both a vscous lud and an elastc sold. An elastc materal returns to ts orgnal poston nstantaneously when stretched and released, whereas a vscous lud such as putty retans ts extended shape when pulled. A vscoelastc materal combnes these two propertes,.e., t returns to ts orgnal shape ater beng stressed and released, but does t slowly enough to oppose the next cycle o vbraton. For elastc materals, σ = ε (2.3) And or vscoelastc materals under gong harmonc exctaton we have, * σ = ε (2.4) where ( + ) * = or, (2.5) * ( + η ) = 1 (2.6) c where 11

27 * s the complex elastc modulus s the elastc or storage modulus and s the loss modulus η c = / s the materal loss actor (2.7) Unlke elastc materals where the stress and stran are n phase, n vscoelastc materals, the stress leads the stran by a phase angle dependng on the loss actorη c. A plot o stress versus stran or one cycle o oscllaton s as shown n the gure 2.1. he area o ths loop gves the amount o energy dsspated per cycle o oscllaton. he loss actor s approxmately twce the dampng rato o the system ( η 2ξ ) or cases o lght dampng. Apart rom the elastc modulus, the shear modulus, bulk modulus and Posson s rato are also complex quanttes or a vscoelastc materal. hey are gven by equatons, ( G + G ) G * = (2.8) c σ ε Fgure 2.1 llptcal hysteress loop or lnear vscoelastc materals 12

28 K G * * * = ( K + K ) = (2.9) 3 * * ( v ) * ( 1 2v ) = (2.10) 2 1+ hese vscoelastc propertes are requency dependent. scoelastc propertes can be entered nto ABAQUS (nte element code used n ths work) n several ways. In the requency doman, tabular values o G, G, K, and K, sutably normalzed, can be entered as unctons o requency. Snce the elastc modulus and Posson s rato may be related to the shear modulus and bulk modulus usng equatons (2.9) and (2.10) (even n a dynamc analyss), a varyng Posson s rato can be taken nto account when enterng the shear and bulk modul. ery lttle Posson s rato data s avalable or vscoelastc materals n general. Oten, vscoelastc materals are assumed to be ncompressble (v = 0.5) n regons o rubbery behavor and v= 0.3 s assumed n regons o glassy behavor. he measurement o varaton o the Posson s rato wth requency s very dcult to obtan expermentally and s not avalable or most dampng materals. Hence, a constant Posson s rato o 0.4 s assumed n ths work. he materal propertes o vscoelastc materals are also dependent on the temperature. However these eects are not consdered here. 2.2 System Loss Factor Modal Stran nergy Method Ungar and Kerwn (1962) dened the loss actor o a vscoelastc system n terms o stran energy quanttes as, 13

29 U u η = (2.11) U where η s the loss actor o a structure wth layered vscoelastc dampng.(system loss actor) U u s the porton o stran energy attrbutable to the vscoelastc core and U s the stran energy o the system he dscretzed equaton o moton o a dynamc system s, [ M ]{ x &} + [ C]{ x& } + [ K]{ x} = { F} where & (2.12) [ M ] [ C],[ K], are the mass, dampng and stness matrces (all real and constant) { x },{ x& },{& x } are the vectors o nodal dsplacements, veloctes and acceleratons { F } s the vector o appled loads For a system wth vscoelastc materal, [ C ] can be neglected snce the dampng due to vscoelastc materal s predomnant and s accounted or by usng [ K ] = [ K1]{ x} + [ K2]{ x} hereore the dscretzed equaton o moton or a vscoelastc system s, [ M ]{ x& } + [ K1]{ x} + [ K2]{ x} = { F} & (2.13) Solvng ths system gves complex egenvalues and egenvectors and s computatonally expensve. Johnson and Kenholz (1981) developed the modal stran energy method whch s an approxmaton to the complex egenvalue method. he modal stran energy method assumes that the damped structure can be represented n terms o the real normal modes o the assocated undamped system approprate dampng terms (the materal or 14

30 core loss actor) are nserted nto the uncoupled modal equatons o moton. Based on these assumptons the expresson or loss actor was gven as [ K ] U u U η = ηc = ηc (2.14) U U + U [ K + K ] where ηc s the loss actor o the vscoelastc core (materal loss actor) U s the stran energy o the vscoelastc elements obtaned rom purely elastc analyss U s the stran energy o the elastc elements s the egen vector o the structure whch s calculated rom purely elastc analyss K s the stness matrx o the elastc elements and K analyss s the stness matrx o the vscoelastc elements obtaned rom purely elastc Xu et al. (2002) revsed the modal stran energy method to nclude the requency dependence o the vscoelastc materal. he loss actor as gven by the revsed modal stran energy method can be wrtten as, η = η c 1 1 U + η U + 2 c η 2 c + U Rearrangng, we obtan ηc U η = (2.15) U + U 1 + η 2 c hs expresson or the loss actor o the structure s used n ths study. 15

31 2.2.2 Hal Power Bandwdth Method he system loss actor may also be computed rom the hal-power bandwdth method as shown n gure 2.2, whch requres obtanng the orced response over a wde requency range (see wns, 2000): ω1 2 2ω d ω η = (2.16) where ω1 and ω2 are the requences at the hal-power ponts. (.e., at A = A / 2 ). and ωd s the damped natural requency. In cases where the dampng s lght, the equaton (2.11) reduces to, ω2 ω η 1 ω d = (2.17) Unlke the modal stran energy (MS) method whch s an approxmate method to compute the loss actor, the hal power bandwdth method s an exact method. However, calculatng loss actor by hal power bandwdth (HPB) method requres calculatng results at many ponts n a gven requency range, whch n turn requres lengthy nte A A / ω 1 ω d ω 2 Fgure 2.2 Hal power bandwdth method 16

32 element calculatons and makes the HPB much more computatonally ntensve to mplement n an teratve process when compared to the MS method. Moreover, vscoelastcty need not be ncluded n the materal modelng or the F analyss whle usng the MS method because MS method uses the modes computed rom an equvalent elastc model. However, loss actor o the structure at the start and at the end o the optmzaton s computed usng the HPB method and compared wth the loss actor calculatons usng MS method. he structure analyzed n ths study s a cantlever beam modeled wth twodmensonal plane stress contnuum eght-noded quadratc elements. ABAQUS s used or the nte element analyss. Alumnum s used as the elastc materal or the base layer and the constranng layer. he propertes o the commercally avalable vscoelastc materal (ISD 112 rom 3M) are used or the polymer matrx materal. Materal propertes used n ths study are lsted n table 2.1. able 2.1 Materal propertes lastc Materal scoelastc Materal Carbon Nanotubes Stness Modulus (GPa) Densty (kg/m 3 ) Posson s Rato Core loss actor

33 CHAPR 3: OPIMIZAION hs chapter descrbes the optmzaton problem ormulaton, the low chart or the optmzaton process and the analytcal gradent ormulaton. 3.1 Problem Formulaton In ths work, a smple materal model s used, where the normalzed densty and modulus o the materal or each element are allowed to vary together rom 0% (n actualty, not zero but a very small value n order to prevent sngulartes n the stness matrx), whch would be a vod, to 100%, whch would represent 100% materal. hs s complcated by the act that there are two materal consttuents an elastc materal and a NRP materal. hs s handled by placng two elements n the same locaton n the constranng layer desgn space one that s NRP and one that s elastc. he densty (and thus the modulus) o each element s allowed to vary rom 0% to 100%, but the total densty n each locaton (the densty o the elastc element plus the densty o the NRP element) s not allowed to be greater than 100%. Although ths s artcal, n that t s unrealstc to consder manuacturng a structure wth propertes o two derent materals (elastc and NRP), the results o ths ntal study lead to nsght nto the optmal constraned layer conguraton, and could be used to develop a structure that s reasonable to manuacture. he objectve o ths study s to maxmze the system loss actor, measured usng the moded modal stran energy method (equaton 2.16). he desgn varables are the percentage o materal n each element, where 0% represents a vod, and 100% represents complete materal (elastc or NRP, whchever the case may be). he result s valdated by computng the loss actor usng the hal power bandwdth method. One constrant on the 18

34 objectve s that the total racton o each consttuent n the constranng layer s xed. (echncally, ths s ncluded as an nequalty constrant rather than an equalty constrant, but these constrants are vrtually always actve). For example, n one case, the NRP materal s lmted to be 20% o the total constranng layer desgn space, and the elastc materal s lmted to be another 20%. Furthermore, as mentoned above, the percentage o NRP materal plus the percentage o elastc materal must be less than or equal to 100% n each element locaton. o summarze, the optmzaton statement may be wrtten as Maxmze η (system loss actor) such that n = 1 n n = 1 n x x v e v e e + x x v 1 x 10 1x 10 1 x x v x v e = 1,2,..n x 10 2 = 1,2,..n = 1,2,..n = 1,2,..n where v x s the racton o NRP materal o the element e x s the racton o elastc materal o the element v s the total racton o NRP materal 19

35 e s the total racton o elastc materal n s the number o NRP elements (whch s equal to the number o elastc elements) and v s the volume racton o carbon nanotubes n the polymer materal hus the materal propertes or a NRP/polymer materal are (usng equaton 2.1 and equaton 2.2) ρ ( v ρ + ( 1 v ) ρ ) v v N v = x (2.19) v v ( v + ( 1 v ) ) = x (2.20) N he materal propertes or an elastc element are, v e e ρ = x ρ (2.21) e e e = x (2.22) e he lower bounds on the materal ractons o NRP elements are derent rom that or the elastc elements because the stness o the vscoelastc materal vares by several orders o magntude rom that o alumnum. he volume racton o nanotubes n the commercally avalable NRP s generally n the range o 0.1 to 5%. Hence n ths study an upper lmt o 5% s used or the volume racton o nanotubes. One dculty n the optmzaton process s n ndng the rst bendng mode. As the denstes and the stnesses o the dampng layer elements change, t s possble or new modes to appear locally. hs happens when an elastc element s loatng n space, (as shown n gure. 2.3) connected to the rest o the structure by elements that are at a very low stness. hs results n a hghly damped, low requency mode that has no mpact on 20

36 Fgure 2.3 Fnte element model producng local mode the rst mode o the base beam. Lumsdane (2002) developed a heurstc n order to ensure that the mode whose loss actor s beng optmzed s truly the rst mode o the beam, and not a local spurous mode. he rst mode o the beam s one where the normalzed dsplacement o the tp o the beam s large, whle or the local spurous modes, dsplacements n the constranng layer are much larger than dsplacements at the tp o the beam. Addtonally the rst mode o the base beam generally does not change drastcally between teratons, whle the local spurous modes oten have very low requences. hese two quanttes (normalzed tp dsplacement and natural requency) provde the most obvous clue as to whch mode s the correct mode. Both need to be examned to denty the correct mode. I only the natural requency s examned (as was done ntally by Lumsdane, 2000), then spurous modes develop at natural requences close to the requency o the base beam. hus, a crteron s developed that examnes both the natural requency and the normalzed dsplacement. he nverse o the normalzed tp dsplacement s added to the derence between the natural requency o the mode n the current teraton and the natural requency o the structure n the prevous teraton o the optmzaton process. he rst ten modes are examned, and the mode wth the lowest value s chosen as the bendng mode o the base beam. It should be noted that ths quantty has no physcal meanng. It has proven eectve, however, n a varety o cases, to denty the proper mode o the beam. he optmzaton low chart s shown n gure 2.4. he ntal values o the desgn 21

37 Input ntal values o desgn varables to ABAQUS Determne natural requences and mode shapes rom ABAQUS Calculate the loss actor and the gradents o the loss actor wth respect to the desgn varables New values o desgn varables Pass the loss actor and the gradent values to NLPQL or optmzaton Soluton converged? No Yes Save results Fgure 2.4 Optmzaton process low chart 22

38 varables are used to compute the materal propertes o each o the nte elements. hese materal propertes are then gven to the nte element sotware (ABAQUS). he egenvalue and egenvector o the bendng mode are obtaned rom the egen analyss perormed by ABAQUS. hese values along wth the stness and mass matrces are then used to compute the loss actor and the gradents o the loss actor wth respect to each o the desgn varables. he values o the objectve uncton (loss actor) along wth the gradents are then nput to NLPQL whch s a gradent based optmzaton code. NLPQL then checks or convergence. I convergence s acheved, the process ends. I the convergence s not acheved, then t perorms a lne search to determne the next set o values or the desgn varables. hese varables are then used to compute a new set o materal propertes to nput to the nte element code. 3.2 Analytcal Gradents Formulaton he most tme consumng part o a gradent-based optmzaton process s the gradent calculatons. Wth the ncreasng number o desgn varables, tme taken or the gradent computaton ncreases dramatcally. In the prevous studes (Lumsdane, 2002 and Lumsdane and Pa, 2003 Pa. et al., 2004) gradents were computed usng the nte derence method. x ( x + x) ( x) x where x s the step sze x s the gradent o the objectve uncton wth respect to the th desgn varable 23

39 ( x + x) s the change n the objectve uncton due to a small change n the th varable x s the vector [ x 1 x 2 x 3... x n ] where x s the th desgn varable Wth n desgn varables, ths nvolves n loss actor (objectve) computatons per teraton. ach loss actor computaton requres a nte element run snce the stran energes requred to calculate the loss actor are obtaned rom the nte element analyss. Hence, each teraton n the optmzaton process requres as many nte element runs as the number o desgn varables. Moreover, a ew more gradent calculatons are requred durng the lne search n the optmzaton process. Wth a large number o desgn varables, ths consumes apprecable amount o CPU tme. An alternatve s to compute the gradents analytcally so that only one nte element run s necessary per teraton (plus a ew more runs or the lne search). he loss actor o a constraned layer dampng structure usng the moded modal stran energy method (Xu, et al, 2002) s gven by equaton 2.16 where η η c s the core loss actor U s the vscoelastc stran energy and U s the elastc stran energy η U c = (3.1) 2 U + U 1 + ηc he stran energes n terms o the elastc and vscoelastc stness matrces and the mode shape (Johnson and Kenholz, 1981) may be wrtten as U = Κ (3.2) 24

40 U = Κ (3.3) where Κ s the vscoelastc stness matrx o the structure Κ s the elastc stness matrx o the structure and s the egenvector or the bendng mode he stness matrces and the mode shape can be obtaned rom the nte element analyss o the structure. And, snce the core loss actor s assumed to be a constant value, the loss actor o the system can be computed. hese stness matrces Κ and Κ are respectvely the elastc and vscoelastc stness matrces or the structure.(he order o these matrces s equal to the number o degrees o reedom o the structure). quatons (3.2) and (3.3) can also be wrtten as U = p = 1 Κ (3.4) U b B = BΚ B + = 1 p = 1 Κ (3.5) where B Κ, Κ and Κ are the elemental stness matrces o the th vscoelastc element, th elastc element n the base beam and th elastc element n the desgn space, respectvely s the part o the egenvector or th vscoelastc element B s the part o the egenvector or th elastc element n the base beam element s the part o the egenvector or th elastc element n the desgn space p s the number o vscoelastc elements = number o elastc elements n the desgn space 25

41 b s the number o elastc elements n the base beam Note that, Κ, B Κ and Κ are 16x16 matrces (snce the elements are eght noded elements wth 2 degrees o reedom per node) and, and are vectors o B length 16. Moreover, = or = 1..p, snce the vscoelastc elements and the elastc elements n the desgn space are dened on the same nodes. Also, the base beam element stnesses are not aected by any o the desgn varables, snce the base beam s not ncluded n the desgn space. In other words, B Κ or = 1..b are all constant throughout the optmzaton. However, changes. B changes as the mode shape o the structure he gradent o the loss actor can be computed by derentatng equaton (3.1) wth respect to the th desgn varable, η x = η c 1 + η 2 c U U x 2 [ U ] 2 + U 1 + ηc - U U x (3.6) Hence, to obtan the gradent o the loss actor wth respect to the th desgn varable, U U and x x need to be computed. Derentatng the equatons (3.2) and (3.3) wth respect to desgn varable x yelds U Κ = 2 Κ + x x (3.7) x U Κ = 2 Κ + x x (3.8) x 26

42 , x Κ and x Κ stll need to be determned to nd the gradent o the loss actor. o x nd the dervatve o the egenvector o a matrx wth respect to a desgn varable ( ), x a method proposed by Jung and Lee (1997) has been used and s descrbed later n ths chapter. he terms Κ x and Κ x take derent orms dependng on whether the varable x s (a) materal racton o a vscoelastc element; (b) materal racton o elastc element; or (c) volume racton o carbon nanotubes. ach o these possbltes s outlned below. Case (a): x = x, materal racton o th vscoelastc element. In ths case, Κ x = 0 snce, a change n the materal racton o a vscoelastc element does not eect the elastc stness matrx and Κ x Κ x ( ) ( ) ( ) Κ Κ Κ 1 = p can be expanded as, 1 1 p p (3.9) x x x ach o these elemental stness matrces ( Κ, =1..p) vares lnearly wth the stness modulus o the correspondng element. he dmensons are constant throughout the optmzaton process but the stness modulus vares lnearly as the materal racton as shown by equaton (2.20) 27

43 ( v + ( 1 v ) ) = x (3.10) N v hereore, the stness matrx o the th element s drectly proportonal to the materal racton o that element and the matrx can be wrtten as, Κ = x Κ (3.11) where II Κ II s the stness matrx o a 100% vscoelastc element. Usng equaton (3.11) n equaton (3.9), we obtan Κ x = 1 ( x Κ ) ( x Κ ) ( x pκ II ) 1 II x II x ( x II ) All the terms n the above expanson goes to zero except Κ ( x Κ ) Κ = = x II x Usng equaton (3.12) n (3.7) yelds Κ II x p x p.hereore, (3.12) U = 2 Κ + Κ II x x (3.15) Case (b) :x = x, materal racton o th elastc element. In ths case, Κ x Κ = 0 and x can be expanded as Κ x b B = Κ + B B = x 1 ( Κ ) ( Κ p ) ( Κ p ) p (3.16) x x x + 1 p 28

44 Here, b = 1 B Κ x B B = 0 snce the base beam elements are ndependent o the materal racton o the elements n the desgn space. hereore, Κ x ( ) ( ) ( ) Κ Κ Κ 1 = p 1 1 p p (3.17) x x x Agan, each o these elemental stness matrces ( Κ, =1..p) vary lnearly wth the stness modulus o the correspondng element. he dmensons are constant throughout the optmzaton process but the stness modulus vares lnearly as the materal racton as shown by equaton (2.22) = x. (3.18) e hereore, the stness matrx o the th element s drectly proportonal to the materal racton o that element and the matrx can be wrtten as Κ = x Κ = 1..p (3.19) where III Κ III s the stness matrx or a 100% elastc element n the desgn space. Usng equaton (3.19) n equaton (3.17) gves Κ x ( ) ( ) ( III ) x Κ III x x III =.. Κ.. p Κ x 1 ( x III ) All the terms n the above expanson goes to zero except Κ ( x Κ ) Κ = = x x III Usng equaton (3.21) n (3.8) yelds Κ III 29 x x p x p (3.20). hereore, (3.21)

45 U Κ x x = 2 + Κ III (3.22) Case (c): x = v, volume racton o nanotubes n the vscoelastc materal. In ths case, Κ v = 0 snce volume racton o nanotubes does not aect the elastc stness matrx and Κ v Derentatng Κ v can be expanded as ( ) ( ) ( ) Κ Κ Κ 1 = p 1 1 p p (3.23) v v v Κ partally wth respect to volume racton o nanotubes gves Κ v = v Κ s gven by, equaton (2.0) as, ( v + ( 1 v ) ) = x hereore, Κ = x v N Κ ( N - ). quaton (3.11) gves Κ = x Κ II. Substtutng ths n the above equaton and expandng leads to Κ v = x ( - ) N. x ( v + ( 1 v ) ) x N Κ II Smplyng, we nd 30

46 31 ( ) ( ) ( ) II N N. v 1 v x v Κ Κ - + = (3.24) akng ( ) ( ) ( ) N N K v 1 v C - + = gves II K. C x v Κ Κ = (3.25) Usng ths n equaton (3.23) yelds Κ v = p II K x C Κ = 1 Usng ths n equaton (3.7) gves p II K x C v v Κ Κ = + = 1 2 U Summarzng all the three cases, we have [ ] 2 2 c 2 c c η η η η 1 U U U U - U U = x x x (3.26) Κ Κ x x x + = 2 U (3.27) Κ Κ x x x + = 2 U (3.28) For x = x =materal racton o th vscoelastc element. II x x Κ Κ + = 2 U (3.29)

47 U Κ x x = 2 (3.30) For x = x, materal racton o th elastc element. U Κ x x = 2 (3.31) U Κ x x = 2 + Κ III (3.32) For, x = v, volume racton o nanotubes n the vscoelastc materal U x = 2 x Κ + C K p = 1 x Κ II (3.33) U Κ x x = 2 (3.34) o nd the partal dervatve o the egenvector wth respect to a desgn varable, descrbed below., a method proposed by Jung and Lee (1997) s used. hs method s x Step 1: Dene K K = * λ M M M 0 (3.35) where K s the global stness matrx, M s the global mass matrx and λ s the egenvalue o the undamped system. Here, K = K + K (3.36) and M = M + M (3.37) 32

48 where M s the mass matrx o the vscoelastc elements and M s the mass matrx o the elastc elements. he elemental mass matrces M and M (16x16 matrces) or the vscoelastc and elastc elements, respectvely, vary lnearly as the densty o the correspondng elements. he denstes are lnear unctons o the materal racton and are gven by equatons (2.19) and (2.21) as v v ( v ρ + ( 1 v ) ρ ) ρ = x N v e ρ = x e ρ e hereore, as n the case o stness matrces, we dene M II as the mass matrx o a vscoelastc element wth 100% materal and M III as the mass matrx o an elastc element n the desgn space wth 100% materal such that M = x M (3.38) II M = x M (3.39) III and the partal dervatve o a vscoelastc mass matrx o the th element wth respect to the volume racton o nanotubes s, M = x CM. M II (3.40) v where C M = ( ρ -ρ ) ( v ρ + ( 1 v ) ρ ) N N Step 2: Compute, K M λ x x = M 0. 5 x (3.41) When the desgn varable s the materal racton o vscoelastc elements, 33

49 = ( K λ M ) II 0. 5 M II II (3.42) When the desgn varable s the materal racton o elastc elements, = ( K λ M ) III 0. 5 M III III (3.43) When the desgn varable s the volume racton o nanotubes, ( C K λ C M ) K = 0. 5 C M M M (3.44) he rst element o the vector as gven by equaton (3.41) s a vector o length equal to the number o degrees o reedom (ndo) o the structure and the second element s a scalar. In the equatons (3.42) and (3.43) the vectors [ ( K λ M ) ] [ ( K λ M ) ] III III and are vectors o length 16 and need to be expanded to the length o ndo. hs s done derently or each element dependng on the poston o the element n the structure. II II Step 3: Compute = K λ x x * 1 (3.45) Jung and Lee (1997) also proved the non-sngularty o the matrx K *. Substtutng obtaned rom equaton (3.45) n equatons (3.29) through (3.34), we x can obtan U U and x x or all the desgn varables. Usng U U and x x n equaton (3.6) gves all the gradents. 34

50 he analytcal calculaton o gradents has been coded n FORRAN. Usng the analytcal gradents n the optmzaton process resulted n a much aster convergence An optmzaton process whch took approxmately 30 hours to converge when the gradents are calculated numercally, takes approxmately 2 hours to converge when analytcal gradents are used aldaton o valdate the above procedure, optmzatons were perormed usng both the numercal and analytcal gradents on a conguraton smlar to one used n Pa et al., (2004) (shown n gure 3.1). hs s a cantlever beam usng vscoelastc materal- ISD 112 rom 3M (not an NRP) or the constraned layer and alumnum or the elastc constranng layer. he desgn space s dscretzed nto 80 elements (5 rows o 16 elements each). An elastc and a vscoelastc element are dened n each o these 80 locatons. he materal ractons o each o these elements are the desgn varables. So, we have 160 desgn varables n the optmzaton process. he ntal conguraton shown n gure 3.1 conssts o 20% materal racton. hs means that the total amount o vscoelastc materal n the desgn space amounts to 20% o the total desgn space. Here, the rst 16 elements (rst layer) are 100% vscoelastc. hs mples that n the optmzaton process, the materal actons o the rst 16 vscoelastc elements are one and the materal ractons o the remanng (80-16) 64 vscoelastc elements are zero ( In act, a very small number, 1x10-7 s used to avod sngulartes n the stness matrx). 20% materal racton also means that total amount o elastc materal n the desgn space s 20%. 35

51 Desgn Space Fgure 3.1 Intal conguraton able 3.1 shows a comparson o the gradents calculated rom numercal and analytcal methods. Snce we have 160 gradent calculatons per teraton, only the rst 25 gradents or the rst teraton are shown. hese are the gradents o the objectve uncton wth respect to rst 25 vscoelastc elements. It can be seen that the error or the rst 16 values s less than 1%. From 17 to 25 (and urther) the error s observed to be somewhat larger. hs can be explaned as ollows. In the rst teraton, the desgn varables 1 to 16 have a value 1 and the desgn varables 17 to 80 have a value 1x10-7. he step sze n the numercal method s Perturbng a desgn varable whch has a value o 1 (desgn varables 1 tll 16) by gves an error close to zero, whereas perturbng a desgn varable whch has a value close to zero (1x10-7 ) by make ( x + x) >> x and hence the error. Usng all x =0.2, an error close to zero or all the gradents has been observed. Gradents computed wth all x =0.3, have also been examned and the error was ound to be close to zero or all the desgn varables. 36

52 able 3.1 Comparson o gradents calculated rom numercal and analytcal methods Desgn varable no. Gradents usng analytcal method Gradents usng numercal method rror % % % % % % % % % % % % % % % % % % % % % % % % % 37

53 CHAPR 4: RSULS hs chapter contans a comparson o the results obtaned rom the optmzaton usng analytcal and numercal gradent calculaton methods and the results or materal racton and base beam thckness parameter studes. A comparson o the results rom the topology optmzaton o the cantlever beam wth the ntal conguraton shown n gure 3.1 usng the analytcal and numercal gradents methods n the optmzaton are as shown n the table 4.1. he vscoelastc and elastc materal dstrbuton at the end o the optmzaton are as shown n the gure 4.1. It can be seen that these are two derent nal shapes (although contanng smlar eatures). hese could be two derent local optma. Both the methods gve approxmately 1500% mprovement n dampng. he optmzaton runs were perormed on a Lnux OS wth Pentum dual core processor (3GHz) usng ABAQUS6.5 or the nte element analyss. he optmzaton run usng the numercal method completed n 31 hours 20 mnutes and took 91 teratons, whereas the optmzaton run usng the analytcal method or gradent calculatons completed n 2 hrs 30 mnutes and took 581 teratons to converge. able 4.1 Comparson o results obtaned rom topology optmzaton usng the analytcal and numercal gradent calculaton methods Analytcal Numercal Intal Fnal Intal Fnal Natural requency Loss actor % Imp me taken 2 hrs 30 mn 31 hrs 20 mn No. o teratons