Kelvin Viscoelasticity and Lagrange Multipliers Applied to the Simulation of Nonlinear Structural Vibration Control

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1 964 Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural Vbraon Conrol Absrac Ths sudy proposes a new pure numercal way o model mass / sprng / damper devces o conrol he vbraon of russ srucures developng large dsplacemens. I avods he soluon of local dfferenal equaons presen n radonal convoluon approaches o solve vscoelascy. The srucure s modeled by he geomercally exac Fne Elemen Mehod based on posons. The nroducon of he devce's mass s made by means of Lagrange mulplers ha mposes s movemen along he sragh lne of a fne elemen. A pure numercal Kelvn/Vog lke rheologcal model capable of nonlnear large deformaons s orgnally proposed here. I s numercally solved along me o accomplsh he dampng parameers of he devce. Examples are solved o valdae he formulaon and o show he praccal possbles of he proposed echnque Keywords Nonlnear dynamcs, russ srucures, vbraon conrol, large sran, Kelvn/Vog rheology. Raphael Henrque Madera a Humbero Breves Coda b a Deparameno de Engenhara de Esruuras, Escola de Engenhara de São Carlos, Unversdade de São Paulo, Av. Trabalhador Sãocarlense, 4, CEP , Raphael.madera@usp.br b Deparameno de Engenhara de Esruuras, Escola de Engenhara de São Carlos, Unversdade de São Paulo, Av. Trabalhador Sãocarlense, 4, CEP , hbcoda@sc.usp.br hp://dx.do.org/.59/ Receved 6..5 In revsed form..6 Acceped..6 Avalable onlne 7..6 INTRODUCTION The developmen of new srucural maerals wh hgh srengh / densy rao leads o he desgn of srucures ncreasngly lghwegh and slender. Ths ype of srucure generally develops a level of dsplacemen amplude greaer han hose capable of beng analyed by lnear models. Furhermore, analyss of machnes and mechansms, or cvl srucures subjec o sesmc acons, are hghly mporan problems for whch he nonlnear dynamc analyss s mperave. Regardng hese subjecs one may consul he works of Sugyama e al. (), Bauchau and Boasso (), Lee e al. (8), Noron (), Spencer Jr and Nagarajaah (), Ramallo () and Moendarbar and Tagkhany (4). Thus s necessary o develop and ule geomercally accurae nonlnear formulaons ha consder he equlbrum analyss, or moon, a he curren confguraon of he

2 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural 965 srucure. In parcular, he formulaons based on he Fne Elemen Mehod are very acceped for solvng hs knd of problem. Updaed Lagrangan and co-roaonal formulaons can be seen, for example, n Bahe e al (975), Smo and Vu-Quoc (986), Armero (6), Jelenc and Crsfeld () and Yoo e al. (7). Toal Lagrangan formulaons based on posons and unconsraned vecors can be seen n Coda e al. (), Res and Coda (4) and Coda and Paccola (8). The sole advanage of updaed Lagrangan and co-roaonal formulaons s ha hey are exensons of lnear formulaons and, as a consequence, varous well known leraure soluons can be used o mprove resuls. Ths advanage s also a dsadvanage as n updaed formulaons fne roaons mus be lneared and he curren sress mus be updaed usng he Krchhoff-Jaumann formulae, whle for he above menoned oal Lagrangan formulaon none of hese rcks are necessary. Many lghwegh srucures, responsble for mechancal suppor of he desgned objec, conss of srucural elemens loaded predomnanly by normal force. Ths knd of srucure s usually called russ (Greco e al. (6)). The fne elemen used o model russes do no nclude bendng sffness brngng dffcules for modelng vbraon conrol devces of he ype mass / sprng / damper n nonlnear analyss. In pracce, hs ype of devce has a mass whch mus slde over a surface (over a fne elemen) and s conac force s orhogonal o he member, resulng n bendng sresses. Moreover, he dampers used n such devces can be represened by vscous consuve models ha, assocaed wh her sffness, resul n vscoelasc models smlar o he Kelvn/Vog one. The reamen of hese lnear and nonlnear vscoelasc formulaons s usually done by more or less complcaed radonal convoluon echnques or decayng funcons as explaned by Lemare and Chaboche (99), Lemare (), Lma e al. (5), Holapfel (996), Smo (987), Huber and Tsakmaks () and Peeau e al () used or no n vbraon conrol (Marko e al. (6) and Clough and Penen (985)). There are no sgnfcan dfferences regardng he radonal rheologcal vscoelasc approach of maerals n all consuled bblography. They use he same procedure, convoluon, o solve dfferen vscous maerals, developng small or large srans. In hs sudy, s presened: () a Toal Lagrangan russ fne elemen for geomerc nonlnear dynamc modelng of wo- and hree-dmensonal srucures. () a echnque based on Lagrange mulplers for he consderaon of passve mass / sprng / damper vbraon conrol mechansms and () an nnovave approach o he consderaon of vscous dampng n he form of he rheologcal model of Kelvn/Vog adaped o he Green nonlnear sran derved from lnear applcaons (Mesqua and Coda (), Mesqua and Coda () and Mesqua and Coda (7)). The man conrbuon of hs sudy s em () ha nroduces a pure numercal way o consder vscous maeral behavor n dynamc or sac applcaons. The proposed formulaon avods he analycal soluon of me dfferenal equaons for each consdered maeral law and he use of s soluon as a resdual n a convoluon echnque. Therefore, dfferenly from he exsen formulaons, he proposed sraegy can be easly exended o be used n any vscoelasc consuve relaon, lnear or no. The work s organed as follows. Secon presens he Energy Saonary Prncple on whch he formulaon s based. I also presens each par of he oal energy necessary o he developmen of he numercal process, deachng vscous dampng and s numercal reamen, he kernel of he work. Secon presens he equlbrum equaon and he organaon of he numercal soluon process. Secon 4 presens he mass / sprng / damper arrangemen for large sran applcaons Lan Amercan Journal of Solds and Srucures (6)

3 966 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural makng use of Lagrange mulplers. Represenave examples are presened n secon 5 o valdae he proposed formulaon and descrbng s praccal possbles. Conclusons are presened n secon 6. TOTAL MECHANICAL ENERGY AND STATIONARY PRINCIPLE The prncple of saonary energy ndcaes ha he equlbrum of mechancal sysems occurs when he varaon of he oal mechancal energy s ero. When comes o dynamc applcaons, usng he D'Alamber prncple, he dynamc equlbrum s mean he equaon of moon. The prncple of saonary s usually employed when dsspave forces do no appear because, n hs case, sysem energy level falls connuously over me. However, s possble o apply he prncple of saonary consderng a larger energy sysem han he sysem suded, n whch he dsspaed energy s par of. Thus, he oal mechancal energy o be consdered n hs sudy s wren as. U K P Q () Where U s he sran energy K s he knec energy, P s he poenal of exernal forces (consdered conservave) and Q s he dsspave poenal resulng from he consdered dampng devce. The oal energy varaon can be expressed as: U K PQ () In russ srucure analyss s usual o consder exernal forces appled o he nodes of he srucure, as well as concenraed mass a nodes, oherwse would be admng he exsence of bendng sresses n srucural elemens, whch s no possble by he mechancal defnon of russ elemens. Thus, he prmary varables of he problem are nodal posons. Therefore, o esablsh he adoped fne elemen formulaon s necessary o wre equaon () as a funcon of nodal poson of he srucure.. Knec Energy and Exernal Forces Poenal Usng ndex noaon, he poenal of exernal appled forces s gven by: P F Y () n whch he exernal forces F are consdered conservave,.e., hey do no depend on posons. In equaon () represens drecon and russ nodes on whch he forces are appled or posons measured. The varaon of exernal forces poenal, aken n relaon o nodal posons, s wren as: F Y P Y F Y F Y F Y Y Y j j j j j Yj n whch assumes ero when node s no he same as node, oherwse assumes one. (4) Lan Amercan Journal of Solds and Srucures (6)

4 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural 967 One wres he knec energy for he analyed srucure as: nfe K fe y ydv (5) V fe where he over do ndcaes me dervave, fe sands for fne elemen and nfe s he number of fne elemens. To wre he varaon of knec energy one apples he prncple of conservaon of mass,.e.: nfe nfe nfe K d K d y y dvd y y ddv y y dv d (6) fe fe fe V V V fed fe fe Rememberng ha, for russes, mass s consdered concenraed a nodes, one wres: ( ) K M Y Y (7) n whch, M s he mass correspondng o node, calculaed as half he sum of he masses correspondng o fne elemens conneced o he node plus, when needed, he concenraed masses exsen n he srucure.. Sran Energy In he prevous em he mechancal energes ha depend drecly on he poson of he nodes of he srucure are presened. Ths secon shows he sran energy ha depends ndrecly on he nodal posons. Frsly, one defnes he adoped sran measure and s dependence of posons. Then he specfc sran energy s presened as a funcon of sran and he dependence of nodal poson becomes clear. Fgure shows a russ elemen before and afer he change of poson or movemen. The nal poson of he elemen s defned by he nal poson of nodes,.e., X n whch,, represens drecon and nodes. The curren poson s defned by he unknown varables,.e., he curren nodal posons called Y. Frs of all s mporan o nform ha unknown nodal posons are deermned by a predcor correcor numercal algorhm. Therefore, he followng expressons are wren conanng hese unknowns, bu ral values are always known. X Y L L x, y X f Y x, y Fgure : Truss elemen, nal and curren confguraons. Lan Amercan Journal of Solds and Srucures (6)

5 968 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural The nal and curren lenghs L and L are calculaed by: L X X X X X X (8) L Y Y Y Y Y Y (9) The one-dmensonal Green-Lagrange sran s chosen o descrbe he maeral behavor,.e.: L E L () From he calculaed Green's sran one can choose any Lagrangan consuve relaon o represen he maeral. In hs work he unaxal consuve relaon of San-Venan-Krchhoff s chosen and s expressed by he specfc sran energy and s dervave wh respec o he Green sran as u E u S E. E E () n whch E s he elasc modulus ha concdes wh he Young modulus for nfnesmal (or small) srans and S s he longudnal second Pola-Krchhoff sress relaed o he Cauchy sress by L / L. S as descrbed by Ogden (984) and Bone e al. (). Inegrang he specfc sran energy on he nal volume of he bar (Lagrangan quanes) one fnds he accumulaed sran energy on a fne elemen and srucure, such as: fe fe U fe udv fe V A L V E E and U nfe fe U () fe I s worh nong ha all quanes are assumed o be consan along he bar, faclang he negraon process. I s obvous ha expresson () depends on he curren posons of he nodes of he srucure. Then, one calculaes he varaon of sran energy regardng posons as: Usng expresson () one wres, U U U Y Y Y nfe fe () fe Y U E E E E (4) Y Y Y Y fe fe fe fe E AL EAL S. AL and consderng (9) and () one acheves: Lan Amercan Journal of Solds and Srucures (6)

6 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural 969 Y Y Y Y Y Y E L E ( ) Y L Y Y (5) (6) Subsung equaon (6) no (4) and usng he energy conjugae defnon (Ogden (984) and Bone e al. ()), ha s, he dervave of sran energy regardng poson (or dsplacemen) resuls nernal force, one wres: fe fe F EA L Y Y n ( ) E (7) L Equaon (7) represens he conrbuon of an elemen for he oal nernal force of he analyed srucure. The nernal forces are calculaed for all fne elemens and are combned no a sngle vecor of nernal forces conanng all nodes of he srucure by smply respecng her numberng. Consderng all fne elemens he varaon of sran energy, equaon (), resuls: (n) U F Y (8) where he ndex fe s omed as he sum ha corresponds o all nodes of he srucure has already been performed. I s of neres, as wll be shown nex, o calculae he hessan of he sran energy poenal as: H For one fne elemen one has: n whch U k Y Yk fe Y Yk nfe fe U (9) fe fe E E E E H k F A n L AL E E E () Yk Yk Y Y Yk Y Yk E ( ) ( ) k k Y Y L (). Dampng - Modfed Kelvn/Vog Model Despe he consuve model presened here s dedcaed o he devce mass / sprng / damper can be used n any fne elemen. The Kelvn consuve model ndcaes ha he dsspaon of power s generaed by a force proporonal o he speed of sran. As he Green sran s adoped, Lan Amercan Journal of Solds and Srucures (6)

7 97 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural he proposed model s called as Modfed Kelvn Model. Before descrbng he proposed model s mporan o noe ha, n general, s no possble o fnd an explc expresson for he dsspave poenal Q presened n equaon (), however, s possble o wre s nernal force and, herefore, he varaon of dsspaon Q. A schemac represenaon of he proposed model can be seen n Fgure. E, E S S E Fgure : Modfed Kelvn/Vog model adaped for Green sran. From Fgure he Pola-Krchhof sress s gven by: S. E. E E () for whch he frs erm s he dervave of he specfc sran energy n relaon o he Green sran, see equaon (). So he second erm corresponds o he dsspave par of he poenal energy. For a fne elemen one wres he varaon of he dsspave poenal regardng nodal posons as: q q E fe Q Y fe dv Y V dv Y V fe E Y fe V E fe E. E Y dv. EAL Y Y Y () where q s he dsspave poenal by un of volume (no wren) and all varables are consdered consan for a russ elemen. Usng equaon (6) one defnes he conrbuon of one elemen o he global dsspave force, as: ( ) (4) fe fe F. EA L Y Y vs L Consderng all elemens he varaon of he dsspave poenal resuls: ( vs ) Q F Y (5) where he ndex fe s omed as he expresson ncludes all nodes of he analyed srucure. Lan Amercan Journal of Solds and Srucures (6)

8 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural 97 To complee he consderaon of vscosy, should be menoned ha he soluon proposed here s pure numercal and hence he speed of sran s calculaed by he fne dfference mehod n soluon of nonlnear equlbrum equaon. Ths sraegy s orgnally descrbed here for nonlnear sran measure, bu was nspred n Mesqua and Coda (), Mesqua and Coda () and Mesqua and Coda (7) where lnear vscoelasc models are proposed. As far as he auhors knowledge goes s dfferen from all nonlnear vscoelasc formulaons presen n leraure, see for example, Lemare and Chaboche (99), Lemare (), Lma e al. (5), Holapfel (996), Smo (987), Huber and Tsakmaks (), Peeau e al (), Marko e al. (6) and Clough and Penen (985). To nroduce he numercal approxmaon one assumes a me sep and wres: Subsung equaon (6) no (4) resuls: E E (6) E ( ) ( ) L L fe fe fe F E AL Y Y EAL Y Y vs (7) Consderng he me sep suffcenly small one assumes: Y Y Y Y (8) Comparng equaons (7) and (7),one wres (7) as: fe fe fe fe fe F F F F F vs E n n n n (9) Therefore, he vscous nernal force s numercally relaed o he elasc nernal force. NONLINEAR EQUILIBRIUM EQUATION AND SOLUTION PROCEDURE Before nroducng he Lagrange mulplers s of neres o presen he srucure soluon consderng he dsspave erm concludng he descrpon of man orgnal conrbuon of hs work. Subsung equaons (4), (7), (8), (5) no equaon () one wres: (n) ( vs) F M Y F F Y () As he poson varaon Y s arbrary, equaon () can be rewren as: F F M Y F () (n) ( vs) Or makng he correspondence beween he node number and he movemen drecon wh a generc degree of freedom k d* (where d s he russ dmenson D or D) one can wre equaon () n a vecor form, as: Lan Amercan Journal of Solds and Srucures (6)

9 97 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural () (n) ( vs) F F M Y F ha s he nonlnear equaon of dynamc equlbrum, or movemen, of he analyed srucure. The adoped soluon process combnes: () he me negraon of he dynamc equlbrum by he Newmark mehod, () he orgnal vscous force approxmaon by equaon (9) and () he Newon-Raphson nonlnear solver for equaon (). Equaon () s rewren for a curren nsan ( ) as, n n n g F F F F MY CY () or n n g F F F MY CY (4) where he mass proporonal dampng s also nroduced, followng Coda e al. (), Coda and Paccola (4) and Coda and Paccola (8), n order o no loss he generaly. The Newmark approxmaons are used n he followng form, Y Y Y Y Y (5) Y Y Y Y (6) where s a me sep. Subsung equaons (5) and (6) n equaon (4) one fnds: n M gy F F Y Y C Y n MQ CR CQ F n whch vecors Q, R n and F represen he known dynamc conrbuons of he pas. The nernal force F n s calculaed n he prevous me sep and sored o be used n he curren nsan, values of Q and R are gven by: (7) Y Y Q Y (8) R Y Y (9) Lan Amercan Journal of Solds and Srucures (6)

10 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural 97 Equaon (7) s nonlnear regardng Y and can be summared by g Y. Is soluon s done by he Newon-Raphson procedure from a Taylor expanson runcaed a frs order, as: g(y ) g Y g Y Y g Y H Y (4) where U Q M C esa M C H gy H (4) Y S esa In equaon (4) H s he sac hessan gven, for each fne elemen, by equaon (9). In hs equaon he approxmaon descrbed by equaon (9) s employed, whch mples ha only he curren vscous force depends on he curren poson and has nfluence n he dynamc hessan, see equaon (7). From equaon (4) resuls he lnear sysem of equaon used o calculae a correcon for he ral curren poson,.e., g Y Y g Y H Y g Y (4) or where Y s he ral poson, usually adoped as Y a he begn of a me sep. Solvng Y a new ral poson Y s calculaed as Y Y Y (4) Ths value of Y s consdered he new ral poson Y. The acceleraon should be calculaed for each eraon by: Y Y Q (44) Equaon (44) s used o correc velocy n equaon (6) and he eraon connues unl, g(y ) TOL or Y / X TOL n whch TOL s a pre defned olerance. I s worh nong ha Q, R and n F (45) are updaed only a he end of he erave procedure,.e., a he begnnng of he nex me sep. For he frs me sep he acceleraon s calculaed by: U Y M F CY (46) Y Nex secon shows how he soluon procedure s compleed consderng he Lagrange mulpler o nclude vbraon conrol devces. Lan Amercan Journal of Solds and Srucures (6)

11 974 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural 4 INTRODUCTION OF THE LAGRANGE MULTIPLIER Despe he dampng descrbed above can be used n any bar of he srucure,n hs secon s specally adaped o smulae he passve mass / sprng / damper sysem of vbraon conrol shown n Fgure. Ths fgure shows a mass / sprng /damper sysem named for whch he sffness s K, he dampng parameer s and he mass s m. x, y K K EA/ L k EA/ L m r f k m r x, y Fgure : Vbraon conrol devce, D represenaon. As he fne elemen smple bar (russ) canno ress ransverse loads s necessary o consran he devce mass movemen o a sragh lne beween nodes k and r of he sysem. The wo equaons o mpose he necessary consran for a hree-dmensonal devce are: and r Y Y Y r Y k Y r Y k Y r Y (47) Y Y Y Y Y Y Y Y r r k k r k or r Y Y Y r Y k Y r Y k Y r Y (48) The nroducon of hese consrans on he equlbrum equaons s made by addng n he oal energy, equaon (), he sum of he followng Lagrangan poenals r r k r k r Y Y Y Y Y Y Y Y L L Y Y Y Y Y Y Y Y r r k r k r where vares from o he number of nsalled devces n,.e.: (49) U K PQ L (5) Lan Amercan Journal of Solds and Srucures (6)

12 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural 975 The search of he saonary s now made ncludng he above resrcon, as: U K PQL (5) Rememberng ha he Lagrange mulplers are also free varables, s wren L L ˆ L Y j F Y jj Y j (5) where forces F ˆ are acves when nodes belong o he devces and j are he assocaed Lagrange mulplers. Values of F ˆ and j are gven a appendx. Usng ndex noaon over equaon (5) and akng no accoun equaon (5), equaon () s rewren as: and (n) ( vs) ˆ F M Y F F F Y (5) j j Ths me, boh Y and j are arbrary, herefore, ˆ (n) ( vs) F F F M Y F (54) s s (55) j Equaon (54) s very smlar o equaon (), as he acve coordnaes of F ˆ concde wh he (n) coordnaes of F. Equaon (55) concdes wh he mposed consrans, see equaons (47) and (48). Thus, he addonal number of equaons s d * n where d s he dmenson of he consdered space (D or D) and n s he number of devces. The soluon process s dencal o ha descrbed n he prevous secon where boh he force vecor (equaons (A) o (A)) and he Hessan marx wll be changed. In he hree-dmensonal case, for a generc devce, one wres he erms o be added no he hessan marx, as: j L L ( ) Y Yj Y m H L L (56) ( ) ( ) ( ) n Yj n m or, developng he nvolved dervaves, one fnds: Lan Amercan Journal of Solds and Srucures (6)

13 976 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural H Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y k k Y Y Y Y k k Y Y r k r SIM Y Y r r r r k r k k r (57) where he las wo rows and columns correspond o he addonal degrees of freedom per devce. The oher values are added o exsen posons of he orgnal hessan marx. I s worh nong ha n he soluon process vecors Y and Y also conan he updang of Lagrange mulplers, bu hese are no consdered o calculae he accepable error Y / X. 5 NUMERICAL SIMULATIONS Some examples are shown o confrm he varous aspecs of he formulaon. They are organed from very smple cases o more complcaed ones. 5. Mass / Sprng / Damper Devce Valdaon The example n fgure 4 s solved sacally, quas-sacally and dynamcally. In sac soluon he conrbuons of mass and dampng are negleced. An nclned force of F kn s appled as shown n fgure 4. The physcal properes of he bar are: E GPa, A 5cm and L m. From hese values resuls a sprng consan K 5 MN / m, hus f Hooke's Law were used a horonal dsplacemen of uh.m was expeced. As he developed sran s small he numercal value s praccally he same as he expeced one. Furhermore, he fnal value of he Lagrange mulpler s approxmaely 5 kn / m, showng ha n hs parcular case s value corresponds o he ransverse force per un lengh of he devce requred o keep he mass n ero vercal poson, accordng o equaon (54). Lan Amercan Journal of Solds and Srucures (6)

14 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural 977 K A L 45 F E / M K E A / L L L Fgure 4: Lagrange mulpler valdaon. For hs level of sran s also expeced ha he quas-sac problem, ncludng Kelvn/Vog vscosy and dscardng neral forces, approaches he lnear problem. Thus, he horonal dsplacemen as a funcon of me s gven by E / E u ( /) FL ( e )/ A. One may hnk h hs s he vscous dampng equaon however, for hs smple problem, soluons are smlar. Adopng 4MPa s, fgure 5 compares he acheved soluons for varous me seps, revealng he convergence of he proposed numercal vscoelasc approach o he analycal soluon...9 Dsplacemen (mm) Analyc Sac =.x - s =.x - s =.x - s Tme ( - s) Fgure 5: Quas-sac resuls for he devce. In order o valdae he damped dynamc model one employs bars densy equal o x kg / m. Thus, consderng he ohers physcal parameers one acheves he naural frequency of he devce as n rad / s. Consderng small srans he rao beween he maxmum dsplacemen amplude u and he sac dsplacemen u h s gven by / u / uh / ( / n) ( / E ). Table presens some resuls of u / u h as a funcon of for whch s he excaon angular frequency. For he dynamc es one consders wo / n force nenses F kn cos( ) and F kn cos( ) as depced n fgure 4, wh a me sep of 4 s for a oal me of f. s. Lan Amercan Journal of Solds and Srucures (6)

15 978 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural / n Anal Num. F Num. F Table : Maxmum amplude by he sac dsplacemen. When he load s small he numercal response approaches beer he lnear response, for larger loads was expeced very dfferen resuls, however, s a good surprse ha he dampng behavor connues presenng he same paern. For he purposes of hs sudy he resuls are more han sasfacory, moreover he small dfference ndcaes ha s possble o search beer consuve models o beer f he real devce o be used. 5. Opmaon of a Mass / Sprng / Damper Devce In hs and he nex example s used a hypohecal plane srucure for represenng a buldng wh floors, 6m n hegh and 5m n base. Is s used 49 russ fne elemens wh densy 7 kg / m, elasc modulus of E GPa and cross seconal area of A 75.4cm. The srucure s consraned a he base by wo fxed suppors and he vbraon conrol devce s fxed a he op floor as ndcaed n fgure 6. Fgure 7 shows a deal of he devce and he fxng sraegy. The supporng bars have he same properes of oher srucural bars, bu bars - and - of he devce have elasc modulus E GPa, nal lengh L.5m and cross seconal area A 75.4 cm. Fgure 6: Sac scheme and devce. Lan Amercan Journal of Solds and Srucures (6)

16 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural 979 Fgure 7: Deal of he devce. The naural frequences of he srucure are calculaed whou and wh he mass/sprng devce (no damped) by equaon M H I where H s he hessan marx for he nal poson ha concdes wh he lnear sffness marx of he srucure. The frs frequency whou he mass/sprng devce s.5h when he devce s coupled he frs frequency s.96 H. Applyng snusodal horonal base movemen n he same frequency of srucures (wh and whou he mass/sprng devce) and wh amplude of 5cm, as boh sysems are no damped, he resonance phenomenon akes place. The srucure whou he devce reaches a op dsplacemen amplude of 6m n less han 5s, whle he srucure wh he mass/sprng (no damped) devce reaches for he same perod of me an amplude less han m. Thus, he nroducon of he mass/sprng sysem whou dampng would have vbraon conrol effec only for exernal acons of shor duraon. Now s nroduced he modfed Kelvn/Vog dampng n he vbraon conrol devce as 4 shown n equaon (4). In order o opme he devce one vares from s hrough s wh a sep of s. From prevous analyss, was found ha wh he dampng varaon here was a small varaon n he naural frequency of he srucure. Due o hs reason s necessary o vary he excemen frequences from.96h hrough.4h wh a sep of F H n order o fnd he worse excaon frequency for each dampng parameer. Fgure 8 shows he maxmum dsplacemen of he op of he srucure as a funcon of he dampng parameer for he worse possble excaon frequency. Fgure 9 shows he worse frequency for each dampng parameer. Maxmum dsplacemen (m) Dampng Fgure 8: Maxmum dsplacemen amplude a he op as a funcon of. Lan Amercan Journal of Solds and Srucures (6)

17 98 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural.4.. Frequency (H) Dampng.8. Fgure 9: Frequency for whch he maxmum dsplacemen occurs as a funcon of. 4 As deached n fgure 8 he parameer 8 x s mnmes he op dsplacemen. In hs case he frequency s approxmaely F.98H. Fgure shows he dsplacemen of he op of he srucure as a funcon of me for hese values of dampng and frequency. As can be seen, n addon o smaller amplude, he presence of dampng elmnaes, as expeced, he resonance profle. If a longer me s presened he maxmum dsplacemen sables a,7m as shown n fgure 8. Dsplacemen (m) Tme (s) Fgure : Top dsplacemen as a funcon of me for he opmum dampng. The proposed devce works very well and s use s no lmed o small dsplacemens as he formulaon presens a geomercally exac descrpon. One should noe ha he mass of he devce developed horonal and vercal movemens keepng he algnmen wh nodes and of fgure Srucure under Earhquake-D Model In hs example, he behavor of he prevous suded srucure s analyed when s submed o he effec of a real earhquake, see for nsance Res and Coda (4), Coda and Paccola (4) and Lan Amercan Journal of Solds and Srucures (6)

18 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural 98 <hp://peer.berkeley.edu/peer_ground_moon_daabase/specras/7/unscaled_searches/8457 /ed.>. The mposed base movemens n horonal and vercal drecons are depced n fgure. In addon o he mass / sprng / damper devce, he sldng base devce s creaed by changng he confguraon of he frs floor, see Fgure. As n he prevous example 5 russ elemens wh densy 7 kg / m, elasc modulus E GPa (wh he excepon of he dagonal 4 elemens) and cross seconal area of A 75.4cm are adoped. Dsplacemen (cm) Horonal Vercal Tme (s) Fgure : Base movemens n horonal and vercal drecons. Fgure : Sldng base devce. An opmaon process s used o fnd he elasc modulus ( E.5GPa ) of he cross dagonals of he frs floor lmng he base angle o be less han 5 (see fgure ). The same dampng 4 consan 8 x s s adoped for dagonal bars of he frs floor. In hs example he response of he srucure o he excaon presened n Fgure s suded. The horonal dsplacemen of he op floor, he normal force on he rgh column of he second Lan Amercan Journal of Solds and Srucures (6)

19 98 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural floor and he horonal acceleraon of he op floor are evaluaed. To perform hs analyss four suaons are consdered: () whou any vbraon conrol devce () wh vbraon conrol mass / sprng / damper () wh he sldng base devce Fgure and (v) wh he combnaon of he wo devces. Fgures o show resuls along me for he above descrbed cases. As can be seen he sldng base devce allows larger dsplacemen amplude, bu reduces he frequency of he horonal movemen of he op floor. Regardng he acceleraon of op floor, he devce mass / sprng / damper s he bes. Regardng he behavor of he normal force on he rgh column of he second floor, boh he sldng base devce and he mass / sprng / damper presen equvalen performances. Now he combnaon of devces, case (v), s analyed and resuls are shown n fgures, and Dsplacemen (m) Tme (s) Fgure : Dsplacemen a he op floor - whou devces. Dsplacemen (m) Tme (s) Fgure 4: Dsplacemen a he op floor - sldng base devce. Lan Amercan Journal of Solds and Srucures (6)

20 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural Dsplacemen (m) Tme (s) Fgure 5: Dsplacemen of op floor - mass / sprng / damper devce. 5 Acceleaon (m/s ) Tme (s) Fgure 6: Acceleraon a he op floor - whou devces. Acceleaon (m/s ) Tme (s) Fgure 7: Acceleraon a he op floor - sldng base. Lan Amercan Journal of Solds and Srucures (6)

21 984 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural.5 Acceleaon (m/s ) Tme (s) Fgure 8: Acceleraon a he op floor - mass / sprng / damper devce. - Normal force ( 5 N) Tme (s) Fgure 9: Normal force whou devces. Normal force ( 5 N) Tme (s) Fgure : Normal force - sldng base devce. Lan Amercan Journal of Solds and Srucures (6)

22 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural Normal force ( 5 N) Tme (s) Fgure : Normal force - mass / sprng / damper devce..5. Dsplacemen (m) Tme (s) Fgure : Horonal dsplacemen a he op floor - sldng base and mass / sprng / damper devces..6.4 Acceleaon (m/s ) Tme (s) Fgure : Acceleraon of he op floor -sldng base and mass / sprng / damper devces. Lan Amercan Journal of Solds and Srucures (6)

23 986 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural Normal force ( 5 N) Tme (s) Fgure 4: Normal force for sldng base and mass / sprng / damper devces. As shown n Fgures, and 4 he srucure wh wo devces presens nermedae dsplacemen n relaon o he ndvdual devces, moreover one can observe he beang profle. Furhermore, was found ha n hs case he oscllaons presen larger perods. On he oher hand, should be noed ha n erms of acceleraon, hs combnaon presens values lower han he masssprng-damper sysem, hus offerng more comfor o users. The devce combnaon presen an nermedae maxmum normal force amplude. Thus, he combnaon of devces offers a beer comfor for users and mproves he srengh of he srucure when compared o he sole mass / sprng / damper devce. Moreover, hs example shows ha he proposed vscoelasc rheologcal model s successfully mplemened, s very sable and useful for general analyss. 5.4 D Waer Thank Tower To confrm he possbles and usefulness of he proposed formulaon n praccal analyss, n hs example a D applcaon s carred ou wh he nenon of gvng obvous and conssen resuls. The example consss of he analyss of a lgh srucure ha suppors a waer ank subjeced o a horonal mpac wh he composon of forces F kn and F kn wh smulaneous duraon of.s. The force s appled as shown n Fgure 5, causng overall effec of bendng and wsng. The plan srucure s an equlaeral rangle of sde m. Vercal bars measurng m and dagonals, connecng he verces of he larger rangle wh he mdpons have lengh m. The connecon devce beween he waer ank and he srucure (represened n red) s, a s nal poson, parallel o y drecon. Each bar of he devce has ( /)m n lengh. The hank connecon bars are presen only a he op floor. Dsplacemens followng y, y and y drecons of he loaded node are shown n fgures 6 and 7 consderng and no consderng he modfed Kelvn/Vog dampng for vercal (drecon y ) bars of he frs floor. As one can see he proposed formulaon can be used o desgn dampers ha reduces dynamc effecs n srucures. The oal number of floors s, herefore he ower has m hgh. Lan Amercan Journal of Solds and Srucures (6)

24 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural 987 F y F M y Fgure 5: Geomery and loadng skech. All bars have he same cross secon, crcular ubes wh radus R 6cm and hckness.5cm. Srucural bars,.e., whou dampng, have 7 kg / m and E= GPa. For he hank connecon bars s adoped E.GPa, kg / m and.s. When base vercal bars are consdered damped, he adoped vscous parameer s.s. The waer ank mass s consdered M 6kg and s dead wegh P 6kN. Dsplacemen (m),8,6,4, -, -,4 -,6 -,8 y drecon y drecon y drecon Tme (s) Fgure 6: Dsplacemen of he loaded node whou base dampng. Lan Amercan Journal of Solds and Srucures (6)

25 988 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural Dsplacemen (m) y drecon y drecon y drecon Tme (s) Fgure 7: - Dsplacemen of he loaded node wh base dampng. Due o he Lagrange mulpler echnque, he waer ank s consraned o move along he sragh lne beween nodes and. Fgure 8 shows some seleced posons of he ower movemen. Fgure 8:- Some seleced posons. Lan Amercan Journal of Solds and Srucures (6)

26 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural 989 The proposed examples reveal ha he mass / sprng / dampng devce s workng properly for D and D applcaons. Example shows ha he rheologcal model and he Lagrange Mulpler echnque are approprae o sac and dynamc applcaon and unfy approaches. Examples, and 4 show ha he formulaon s able o model large dsplacemen suaons usng large sran mass / sprng / dampng devces. Furhermore, he success of numercally negrang he Kelvn- Vog rheologcal model opens he possbly of developng more realsc models for nonlnear dynamc dampng. 6 CONCLUSIONS Ths sudy presens a oal Lagrangan formulaon o perform nonlnear dynamc analyss of plane and space russes. For hs oal Lagrangan formulaon s proposed an orgnal Lagrangan mulpler and a modfed Kelvn vscoelasc consuve model approach o consder sldng mass / sprng / damper devces for vbraon conrol. The more mporan conrbuon s he proposed modfed Kelvn vscoelasc consuve model ha make possble assemblng dampng for sac and dynamc analyss from phenomenologcal vscosy a large sran usng he Green sran measure. The man advanage of hs formulaon, when compared o he classcal ones, s he possbly of s exenson o comprse more complex vscous behavor whou he necessy of analycally solvng dfferenal equaons. An neresng surprse of echncal neres s he same dampng behavor of he modfed Kelvn model for small and large dsplacemens and srans. The success of numercally negrang he Kelvn-Vog rheologcal model opens he possbly of developng more realsc models for nonlnear dynamc dampng and s exenson o D vscoplasc analyss. Acknowledgemens Auhors hank FAPESP (São Paulo Research Foundaon) for he fnancal suppor of hs research. References Armero, F. (6). Energy-dsspave momenum-conservng me-seppng algorhms for fne sran mulplcave plascy, Compuer Mehods n Appled Mechancs and Engneerng, 95 (7-4): Bahe, K.J., Ramm, E., Wlson, E.L., (975). Fne elemen formulaons for large deformaon dynamc analyss. Inerna. J. Numer. Mehods Engrg., 9, Bauchau, O.A., Boasso, C.L. (). Conac Condons for Cylndrcal, Prsmac, and Screw Jons n Flexble Mulbody Sysems. Mulbody Sysem Dynamcs, 5 (): Bone, J., Wood, R.D., Mahaney, J., Heywood, P. (). Fne elemen analyss of ar suppored membrane srucures, Compuer Mehods n Appled Mechancs and Engneerng, 9: Clough, R.W.,. Penen, J. (975). Dynamcs of srucures, MacGraw-Hll. Coda, H.B., Paccola, R.R. (8). A posonal FEM Formulaon for geomercal nonlnear analyss of shells. Lan Amercan Journal of Solds and Srucures, 5 (), 5-. Coda, H.B., Paccola, R.R. (4). A oal-lagrangan poson-based FEM appled o physcal and geomercal nonlnear dynamcs of plane frames ncludng sem-rgd connecons and progressve collapse. Fne Elem. Anal. Des., 9: -5. Coda, H.B., Paccola, R.R., Sampao, M.S.M. (). Posonal descrpon appled o he soluon of geomercally nonlnear plaes and shells. Fne Elem. Anal. Des., 67: Lan Amercan Journal of Solds and Srucures (6)

27 99 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural Greco, M., Gesualdo, F.A.R., Venurn, W.S., Coda, H.B., (6). Nonlnear posonal formulaon for space russ analyss, Fne elemens n analyss and desgn 4 (): Holapfel, G.D., (996) On large sran vscoelascy: Connuum formulaon and fne elemen applcaons o elasomerc srucures, Inernaonal Journal for Numercal Mehods n Engneerng, 9, 9-96 Huber, N., Tsakmaks, C. (), Fne deformaon vscoelascy laws, Mechancs of maerals,,-8. Jelenc, G., Crsfeld, M.A. (). Dynamc analyss of D beams wh jons n presence of large roaons, Compuer Mehods n Appled Mechancs and Engneerng,9: Lamare, J., () Handbook of Maerals Behavor Models, vol., Academc Press. Lamare, J., Chaboche, J. L. (99). Mechancs of Solds, Cambrdge Unversy Press. Lee, S.H. e al. (8). The developmen of a sldng jon for very flexble mulbody dynamcs usng absolue nodal coordnae formulaon, Mulbody Sysem Dynamcs,, (): -7. Lma, A.M.G., Bouhadd, N., Rade, D.A., Belons, M. (5). Tme-doman fne elemen model reducon mehod for vscoelasc lnear and nonlnear sysems, Lan Amercan Journal of Solds and Srucures, (6): 8-. Marko, J., Thambranam, D., Perera, N. (6). Sudy of vscoelasc and frcon damper confguraons n he sesmc mgaon of medum-rse srucures, Journal of Mechancs of Maerals and Srucures, (6): -9. Mesqua, A.D., Coda, H.B. (). Alernave Kelvn Vscoelasc Procedure for Fne Elemens, Appled Mahemacal Modellng 6 (4): Mesqua, A.D., Coda, H.B. (). New mehodology for he reamen of wo dmensonal vscoelasc couplng problems, Compuer Mehods n Appled Mechancs and Engneerng, 9 (6-8): Mesqua, A.D., Coda, H.B. (7), A boundary elemen mehodology for vscoelasc analyss: Par II whou cells, Appled Mahemacal Modellng. (6): Moendarbar, H., Taghkhany, T. (4). Sesmc opmum desgn of rple frcon pendulum bearng subjeced o near-faul pulse-lke ground moons. Sruc Muldsc Opm, 5 (4) Noron, R. L. (). Desgn of machnery: an nroducon o he synhess and analyss of mechansms and machnes. 5ª ed. Boson: McGraw-Hll. Ogden, R. W. (984). Nonlnear elasc deformaons, Chcheser: Ells Horwood. Pascon, J. P.; Coda, H. B. () A shell fne elemen formulaon o analye hghly deformable rubber-lke maerals. Lan Amercan Journal of Solds and Srucures, (6) Peeau, J.C., Verron E., Ohman, R Le Sourne, H., Sgrs, J. F., Barras, G.. () Large sran rae-dependen response of elasomers a dfferen sran raes: Evoluon negral vs nernal varable formulaon, Mech Tme- Depend Maer (), p Ramallo, J.C., Johnson, E.A., Spencer JR, B.F. (). Smar Base Isolaon Sysems, Journal of Engneerng Mechancs, 8: Res, M. C. J.; Coda, H. B. (4). Physcal and geomercal nonlnear analyss of plane frames consderng elasoplasc sem-rgd connecons by he posonal FEM. Lan Amercan Journal of Solds and Srucures, (7): Smo, J.C. (987), On a fully hree-dmensonal fne-sran vscoelasc damage model: formulaon and compuaonal aspec, Compu. Mehod7 Appl. Mech. Eng., 6, 5-7. Smo, J.C., Vu-Quoc, L. (986). On he dynamcs of flexble beams under large overall moons he plane case: par, Journal of appled mechancs, ASME, 5: Spencer JR., B.F., Nagarajaah, S. (). Sae of he Ar of Srucural Conrol. Journal of Srucural Engneerng, ASCE, Sugyama, H., Escalona, J.L., Shabana, A. A. (). Formulaon of hree-dmensonal jon consrans usng he absolue nodal coordnaes. Nonlnear Dynamcs, (): Lan Amercan Journal of Solds and Srucures (6)

28 R.H. Madera and H.B. Coda / Kelvn Vscoelascy and Lagrange Mulplers Appled o he Smulaon of Nonlnear Srucural 99 Yoo, W. S., Km, K.N., Km, H.W., Sohn, J.H. (7). Developmens of mulbody sysem dynamcs: Compuer smulaons and expermens. Mulbody Sysem Dynamcs, 8 (): Appendx For each devce one calculaes he conjugae forces, F ˆ and j, as: ˆ L F k Y r Y Y r k Y Y (A) ˆ L F k Y r Y Y r k Y Y (A) ˆ L F k Y r k Y Y ˆ L k r k r F Y Y Y Y (A) Y (A4) ˆ L k r F Y Y Y (A5) ˆ L k r F Y Y Y (A6) ˆ L F r Y k Y Y k r Y Y (A7) ˆ L F r Y k r Y Y ˆ L F r Y k r Y Y r r k r k r ( ) (A8) (A9) L Y Y Y Y Y Y Y Y (A) L Y Y Y Y Y Y Y Y r r k r k r ( ) (A) When a D russ s consdered one dsregards he componens ˆF, and he erms Y. Lan Amercan Journal of Solds and Srucures (6)