DYNAMIC SYSTEMS WITH HIGH DAMPING RUBBER: NONLINEAR BEHAVIOUR AND LINEAR APPROXIMATION. Andrea Dall Asta 1, Laura Ragni 2

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1 10 h Worl Conference on Seimic Iolaion, Energy Diipaion an Acive Vibraion Conrol of Srucure, Ianbul, Turkey, May 28-31, 2007 DYNAMIC SYSTEMS WITH HIGH DAMPING RUBBER: NONINEAR BEHAVIOUR AND INEAR APPROXIMATION Anrea Dall Aa 1, aura Ragni 2 1 ProCAm, Diparimeno i Progeazione e Coruzione ell'ambiene, Univ. Camerino, Ialy 2 DACS, Diparimeno i Archieura Coruzione e Sruure, Univ. Poliecnica elle Marche, Ialy ABSTRACT High Damping Rubber i wiely ue in eimic engineering an, more generally, in he paive conrol of vibraion. I coniuive behaviour i quie complex an i no imply non-linear wih repec o rain bu alo how a ranien repone uring which maerial properie change (Mullin effec). A number of recen work were eicae o analyzing an moelling maerial behaviour. The preen work inen o uy he conequence of uch non-linear behaviour in he ynamic repone of a yem where he reoring force i provie by iipaive evice bae on HDR (rucural yem wih iipaive bracing an iolae yem). Analye uner harmonic force an impulive exciaion were carrie ou in orer o characerize he able an ranien repone eparaely. Two ifferen linear equivalen yem are euce from he nonlinear analye in orer o eimae he upper an lower boun for he ynamic repone. The repone provie by linear an nonlinear moel are hen ue o uy he repone of he yem uner eimic exciaion in orer o compare oluion an evaluae he abiliy of linear moel in furnihing upper an lower boun of maximum iplacemen an force. 1. INTRODUCTION Rubber i a maerial characerize by hear iffne many ime maller han bulk iffne an exhibi large rain wihou failure an amage. A filler, generally carbon black, i ae o he naural rubber in orer o improve variou mechanical properie uch a he iipaion capaciy. The amping properie of he maerial may be paricularly beneficial in many iuaion an i i ue in many inurial an engineering applicaion. In he fiel of eimic engineering rubber wih enhance iipaing properie, uually known a High Damping Rubber (HDR), i exenively ue in bearing for he eimic iolaion of brige or builing (Gran e al. 2005). I i alo ue in iipaing evice, generally connece o he rucure by mean of brace, in orer o increae iffne an energy iipaion capaciy. Thi make i poible o conrol he ynamic rucural repone, reucing laeral iplacemen in he cae of mall remor an amage in he cae of rong moion (Fuller e al. 2000, Barera an Giacchei 2004, Dall Aa e al. 2006). The preence of filler, however, enail ome ifficuly in he ue of hi maerial, becaue he behaviour

2 become rongly non-linear an boh he iffne an amping properie vary wih he ampliue of rain an epen on rain rae. Furhermore, he repone of he maerial i proce-epenen an how a ranien behaviour in which iffne an amping change remarkably. The phenomenon, uually known a he Mullin effec, i a conequence of he amage of he microrucure ha occur uring he proce (Govinnjee an Simo 1991 an 1992-a, Dorfmann an Ogen 2003). Recen uie (Govinnjee an Simo, 1992-b) how ha he ranien repone i relae o he maximum hear rain aaine by he maerial an i influence by he rain rae. The iniial properie of he maerial are however recovere in a ufficienly hor perio an hi apec of he problem canno be neglece in rucure unergoing eimic acion (Gran e al. 2005). In cienific lieraure, here are everal paper aempe o ecribe an moel HDR behaviour. Mo of he experimenal e an moel propoe however o no concern he eimic behaviour of rubber-bae evice (cyclic hear rain), an conier he enion-compreion behaviour uner loaing-unloaing pah (ion 1997, Haup an Selan 2001). Paper ha aim a ecribing he eimic behaviour of HDR evice uually propoe coniuive law bae on imple moel which are moifie in orer o ake ino accoun he rongly nonlinear rain-epenence an he rain rae-epenence. Only in a few work he proce-epenen behaviour i moelle wih ifferen approximaion level. In paricular Kikuci an Aiken 1997 preen a rain-rae inepenen elao-plaic moel, wih he aiion of elaic conribuion in orer o ake ino accoun he rain ampliue epenence an he ifference beween he fir an ucceive cycle. In Tai e al 2003 he auhor ue a moifie verion of he Bouc-Wen moel, aing a linear vicou erm. In boh cae he epenence on he loa hiory i neglece. An aemp o ecribe hi behaviour may be foun in Hwang e al. 2002, bu he moel propoe i bae on eparae parameer ienificaion proceure for ifferen loa pah (ifferen frequencie, ifferen emperaure, ranien an able behaviour). In aiion he reul furnihe are aifacory in he cae of inuoial loa only (Gran e al. 2005). Finally in Yohia e al., 2004 he auhor inrouce a amage parameer in orer o imulae he egraing of he elaic par, wherea he hyereic conribuion i elao-plaic an conequenly rain-rae an proce inepenen. A e program o fully inveigae he pure hear behaviour in a range of rain an rain rae o miigae eimic effec i ecribe in (Dall Aa an Ragni 2006). The ame paper alo propoe an analyical moel bae on a rheological, hermoynamically compaible, approach, in which inernal variable were inrouce o ecribe he inelaic phenomena an he amaging effec. Previou moel are uually quie complex an implifie moel, capable of furnihing aifacory reul in he eimic analyi of rucure wih iipaion evice, may be ueful in pracical eign. In hi regar ome inicaion are given by echnical coe. In paricular linear pring-ahpo moel are uggee by ome echnical coe (OPCM 3431, PrEN , FEMA 356) by inroucing he concep of equivalen iffne an equivalen amping coefficien. Elao-plaic moel are alo propoe in ome cae (FEMA 356), bu hee moel o no enure a more accurae ecripion of he rucural behaviour an moreover require a non linear analyi. The influence of nonlinear rain-ampliue epenence i aken ino accoun by uggeing ieraive proceure where he linear parameer are upae hrough o iplacemen converge. In aiion he coe recommen coniering he influence of he frequency (in a range aroun he eign perio), he influence of he emperaure an he ageing phenomena by aoping he mo unfavourable parameer. Pracical coe uually o no furnih clear inicaion on he ranien behaviour of he rubber. The OPCM 3431 coe for example recommen eermining he linear equivalen parameer coniering he hir cycle o ha he Mullin effec i

3 neglece. On he oher han, he FEMA 356 coe ugge performing muliple analye wih ifferen moel, relae o ranien an able behaviour, in orer o obain upper an lower boun for he ynamic repone. No accurae inicaion are however given on he proceure o follow o efine he implifie moel an on he error prouce when uing hee moel. Some uie were carrie ou in Hwang an Ku 1997 an Hwang an Wang 1998 bu hee neglece he ranien behaviour of he rubber. The cope of hi paper i o uy he ynamic behaviour of he non linear yem an o propoe a crierion for efining he linear moel equivalen o he non linear yem. Equivalence coniion, which may be ue in ifferen iuaion, are choen. In paricular wo reference moion are coniere: he fir i relae o he able behaviour a reonance coniion (uner inuoial exciaion), wherea he econ concern he ranien yem repone ubjece o an impulive exciaion, which i rongly influence by he Mullin effec. The crierion i applie o hree ifferen yem, howing a maximum repone in a range perio from 0.5 o 2.0., ubjece o hear rain value up o 200%. Reul may be of inere for rucural yem wih ifferen iffne, from frame wih iipaing brace o iolae rucure while he rain range coniere cover maximum value uually aope in he eign. Numerical reul are obaine by aoping he rheological, hermoynamically compaible, moel propoe by he auhor. In he cae of eimic inpu i may be oberve ha he wo equivalen moel, efine accoring he crierion propoe, furnih he exreme value of force an iplacemen wih accepable approximaion level. 2. DYNAMICA SYSTEMS 2.1 Nonlinear The S-DoF (Single-Degree of Freeom) ynamical yem coniere coni of a ma m an a iipaing evice, bae on HDR ubjece o hear rain, which furnihe he reoring force. The coniuive behaviour of rubber i ecribe by mean of he moel propoe in (Dall Aa, Ragni, 2006). Thi i a hermoynamically conien rheological moel in which he free energy per uni volume ( γ ) ϕ ; α i i expree by mean of he maerial ae ( γ, i ) coniing of he hear rain an a e of inernal variable α i ecribing he inelaic phenomena relae o Mullin effec an vicoiy. The hear re can be euce by he erivaive of he free energy, ϕ τ = (1) γ an he evoluion of he inernal variable i furnihe by ( γ α η) & α = g, (2) i, i once he ae an he proce η = & γ are known (uperpoe o enoe ime erivaive). The iipae power per uni volume w may be obaine from he erivaive wih repec o he inernal variable (repeae inexe enoe ummaion)

4 w ϕ & = α (3) i α i The ae of he ynamical yem i conequenly ecribe by he vecor x = [ u, v; α ] where u an v are he iplacemen an he velociy of he ma. Rubber rain γ = βu / h an rain rae η = βv / h are proporional o iplacemen an velociy an may be relae by inroucing he geomeric parameer β epening on he ype of connecion beween he ma an he evice. The parameer h i he hickne of he rubber layer. The reoring force per uni ma f f may be expree in he form βa = τ (4) m where A i he area of he HDR layer in he evice. The evoluion law of he ynamic yem may be wrien a follow u& v x & = ( ) v& = f u, α ; v + f = A( x) (5) i e α& g ( u, α ; v) i i i where f e i he exernal force per uni ma an he coniuive law are expree by mean of u an v inea of γ an η. i 2.2 iner equivalen yem In eign i i ofen ueful o approach he ynamic problem uing implifie linear moel ha permi eaier analye an a impler inerpreaion of reul. In he yem coniere only he relaionhip beween he reoring force an he ma iplacemen i non-linear o ha he earch for a implifie moel may be reuce o he evice only. Approximaion by a linear yem coniing of a pring an a ahpo ipoe in parallel i coniere. Thi yem i fully efine once wo parameer are aigne. Hereinafer i will be ecribe by he pring iffne per uni ma k an he ahpo iipaion properie exhibie in he able repone uner a inuoial eformaion hiory wih a circular frequency ω = k. Thee iipaion properie may be meaure by mean of he vicou amping coefficien efine a he raio beween he average energy iipae per uni of raian an he maximum rain energy. The ahpo conan c may conequenly be obaine by he relaion c = 2 ω (6) The reoring force of he linear yem i conequenly f = k u + 2 ω v. (7) The evoluion law of he linear yem ae = [ u, v] x ha he following form:

5 u& v &x = = ( ) (8) v& f u, v + fe A precie efiniion of he ynamic iuaion, where he equivalence i require, i neceary o efine a linear ynamic yem ha i equivalen o a non-linear yem. Accoringly, wo moion, relae o he linear an non-linear yem repecively, mu be elece, following from imilar exernal inpu. There mu be wo equivalence coniion, a a reul of he wo parameer ecribing he linear yem, an hee mu require ha he paricular quaniie oberve uring he wo moion are equal. I i evien ha ifferen choice may be mae in elecing he ynamic iuaion an equivalence coniion. The non-linear yem coniere how noable ifference beween he repone exhibie a he iniial age of moion, influence by he Mullin effec, an he ucceive age, when inernal amage reuce iffne an he iipaive properie. Two ifferen linear moel are conequenly eermine in orer o ecribe he wo limi iuaion. The former moel inen o ecribe he able behaviour of he maerial a he aigne rain an rain rae, once Mullin effec i over. Thi conier he po-ranien able repone uner inuoial exernal exciaion. The maximum perioic moion oberve in linear an non-linear yem were elece a ynamic iuaion. The econ moel inen o ecribe he par of he moion rongly influence by he Mullin effec an conier he iniial repone uner an impulive exciaion. In orer o obain coheren linear approximaion, equivalence coniion applicable o ifferen moion are efine. 3. HARMONIC RESPONSE In hi ecion, he able repone of he non-linear yem ubjece o inuoial exernal force i ecribe an ucceively aope o efine equivalen linear moel a ifferen level of rain. In he ee range of exernal acion he yem how a ranien repone an aain a able behaviour afer a cerain number of cycle. The repone relae o an exernal acion wih perio T i coniere o be able a he inan when he ifference beween he ae hiory oberve in he la perio an he ae hiory oberve in he previou perio i ufficienly mall. I houl be alo oberve ha he ynamical yem analyze exhibi a proce-epenen repone. In hi cae, conrary o elaic yem, he able repone obaine epen on he iniial coniion. Since he aim i o characerize he repone uner eimic even acing on he yem where he ae variable are iniially zero, he analye are performe by auming x = 0 for = 0. The exernal force ha he expreion () f in( 2 T ) f e = / (9) 0 π where f 0 i he ampliue of he force per uni ma an T i i perio. Three cae correponing o hree value of iffne are coniere in he analye. The inermeiae cae b wa obaine by a rubber layer wih an area A=78200 mm 2 an a hickne h=10mm; 5 a ma of 10 kg wa coniere. The oher cae a an c were obaine wih a evice area four ime larger an maller, repecively. Such value were choen in orer o obain ynamical yem ha how he iplacemen peak for exernal force perio of abou 0.5 (cae a), 1.0 (cae b) an 2.0 (cae c). The choen iffne value make i poible o uy

6 ifferen iuaion referable o ifferen rucural yem where evice are inrouce in orer o reuce eimic effec, like frame wih iipaing bracing (T= ) or iolae rucure (T= ). For each cae, i wa aume β = 1 an he maximum ineniy of he exernal force f 0 wa calibrae o provie maximum value of hear rain (γ = u/h) equal o 2.0, 1.5 an 1.0, which are he uual maximum rain value ue in eign. Figure 1 repor he reul for cae b. I ecribe he maximum value of iplacemen an reoring force per uni ma oberve in he nonlinear yem (oli line) ogeher wih he reul of he linear approximaion (ahe line). The iagram are in a non-imenional forma an were obaine by iviing iplacemen, force an perio by reference value efine wih he crieria inicae below. Diplacemen u ref an he reference force f ref, he value of he maximum iplacemen an he value of he maximum evice force aaine wih he exernal force f 0 m are aume a reference. Since he reference perio T ref i aume o be he perio a which he maximum value of he repone are reache. Similar reul were oberve in he oher cae a an c. By oberving he nonlinear reul i i evien ha he main iplacemen peak occur a abou T=T ref only when he maximum iplacemen i u ref (he hear rain i γ=2). For he oher curve, relae o he ifferen force ineniy value, he yem i iffer an he perio a which he main peak occur are remarkably maller, a i uual for ofening yem. Oher non linear phenomena may be oberve, uch a he non proporional relaion beween exernal force an maximum iplacemen an he econary peak exhibie for perio abou 1.8 ime hoe of he main peak. u/um f/fm T/T ref T/T ref Figure 1 Harmonic analyi: non linear (oli line) an linear (ahe line) yem cae b The equivalen linear yem i obaine by aoping he following rule: i. The perioic moion of he wo yem a he frequencie a which he maximum iplacemen are aaine are elece a imilar ynamic iuaion o apply he equivalence coniion ii. The iffne of he linear yem i obaine a he ecan iffne oberve in he nonlinear moion a he inan in which he iplacemen aain he maximum value: f ( T ) m ( T ) k = (10) u m

7 where Tm i he perio of he exernal force proucing he maximum repone in he nonlinear yem. An equivalen linear iffne G may be euce for he rubber a follow h G = k (11) A iii. The amping coefficien i obaine by requiring ha he energie iipae by he wo yem in he wo reference moion are he ame: W T ) = W ( T ) (12) ( m m where T i he exernal force perio proucing he maximum iplacemen in he m linear yem. The amping coefficien ha he expreion ( T ) W = (13) 2πk u m 2 In general, he curve obaine for he approximae linear yem a ifferen level of exernal force aequaely ecribe he maximum value of iplacemen an force (Fig. 1). In he cae of repone curve involving mall rain (γ =1.0), he perio a which he repone i exreme almo coincie in linear an nonlinear yem. arge ifference may be oberve for higher rain level (γ =1.5 e γ =2.0). Far from he repone peak he ynamic repone prouce by he linear yem i quie ifferen from he nonlinear repone: he iplacemen are uually overeimae an he force unereimae. The ifference are larger for cae c (he mo eformable yem) an maller for cae a, (he iffe yem). The econary peak can obviouly no be ecribe by he linear moel. The value of he equivalen iffne an he equivalen amping raio are repore in Table 1 for he hree cae analyze an for ifferen level of rain. From he numerical reul i may be oberve ha he equivalen iffne ignificanly ecreae by paing from cae a, which vibrae rapily an ha a larger vicou repone, o cae c, which vibrae lowly. The equivalen iffne i alo influence by he rain ampliue an ecreae when he rain increae. Conrary o he equivalen iffne, he equivalen amping raio oe no how a remarkable variaion an he value oberve are boune by an Thi mean ha he iipae energy approximaely varie proporionally o he iffne. Table 1 Equivalen linear parameer able repone γ=2 γ=1.5 γ=1 G G G (N/mm 2 ) (N/mm 2 ) (N/mm 2 ) Cae a Cae b Cae c

8 4. TRANSIENT RESPONSE The Mullin effec rongly influence he yem behaviour in he iniial pah of he repone uner exernal force. In orer o analyze hi iuaion he yem wa ubjece o iniial coniion coniing of an iniial velociy v 0, o which an iniial value of kineic 1 2 energy ( W 0 = mv 0 ) i aociae, while he oher ae variable were aken a being 2 equal o zero. The hree previouly efine cae were coniere an he iniial velociy wa calibrae in orer o aain he ame maximum iplacemen aope in he previou ecion an correponing o he limi hear rain (γ =2.0, 1.5, 1.0). Figure 2 repor he reul of cae b an ecribe he ime hiorie of iplacemen an he evice force per uni ma oberve in he nonlinear yem (oli line) ogeher wih he reul of he linear approximaion (ahe line). Here again he reul are preene in a non-imenional forma. The ame value of he reference iplacemen, evice force an ime inerval were aope in orer o implify he comparion wih he reul. Comparing, he cyclic repone an he free vibraion moion correponing o he ame limi rain, i i evien ha he free yem vibrae more rapily a he iniial age an he maximum value of he evice force i noably higher. In oher wor he yem i remarkably iffer. A he iniial velociy reuce, he yem become iffer an iffer. u/um f/fm /T ref /T ref Figure 2 Tranien repone: non linear (oli line) an linear (ahe line) yem cae b The equivalen linear yem i obaine by aoping rule imilar o hoe ue in he previou ecion: i. The fir par of he moion of he wo yem up o maximum iplacemen are elece a imilar ynamic iuaion for applicaion of he equivalence coniion ii. The iffne of he linear yem i obaine a he ecan iffne oberve in he nonlinear moion a he inan in which he iplacemen aain he maximum value: ( ) 1 ( ) f k = (14) u 1 where 1 i he ime a which he maximum iplacemen of he nonlinear yem i reache. An equivalen linear iffne G may be euce for he rubber a follow

9 h G = k (15) A iii. The amping coefficien i obaine by requiring ha he energie iipae by he wo yem in he wo reference moion are he ame: W ( 1 ) = W ( 1 ) (16) where 1 i he ime a which he maximum iplacemen of he linear yem i reache. The energy iipae a he ime 1 by he linear yem i W ( ) = W W ( ) (17) 1 0 e 1 where W ( ) = ω u ( ) (18) e Conequenly from equaion (17) i i poible o fin he coefficien once he iplacemen expreion of a linear yem ubjece o an iniial velociy v 0 i inrouce in he equaion. The iplacemen expreion i u v = (19) ω 0 ω () e inω 2 where ω = ω ( 1 ). The curve obaine by uing he linear approximae yem aequaely ecribe maximum value of iplacemen an force (Fig. 2) in he cae of repone curve involving mall rain (γ=1.0), wherea large ifference may be oberve for higher rain level (γ =1.5 e γ =2.0). Generally he linear yem are iffer wih repec o non linear yem an are no able o ecribe he repone afer he fir quarer of a cycle. The value of he equivalen iffne an he equivalen amping raio are repore in Table 2 for he hree cae analyze an for ifferen level of rain. From he numerical reul i may be oberve ha he parameer obaine are ignificanly ifferen from hoe obaine by coniering he able repone. In paricular, he equivalen iffne are much larger epecially in cae a wih large rain value (when he Mullin effec i maximum). Differenly he equivalen amping raio remarkably ecreae by paing from he cae c o he cae a an by ecreaing he rain. Table 2 Equivalen linear parameer ranien repone γ=2 γ=1.5 γ=1 G G G (N/mm 2 ) (N/mm 2 ) (N/mm 2 ) Cao a Cao b Cao c

10 5. SEISMIC RESPONSE In hi ecion he repone of he previou hree cae ubjece o exernal inpu ecribing eimic even are analyze, wihin he previouly coniere range of maerial rain an rain rae. More pecifically, even arificial groun moion are coniere. The accelerogram are generae o mach he elaic repone pecrum given by he OPCM3431 coe for zone 1 an groun ype B-C-E, accoring wih he rule furnihe by he ame coe. In orer o obain reul which are comparable wih oher, iuaion wih he ame average value of maximum rain equal o γ = 1.5 were coniere. In hi regar i houl be noe ha he ynamic properie of he hree yem analyze o no change by moifying area A an hickne h of he evice an ma m of he yem, if he raio A/hm i conan. Conequenly, in he hree cae analyze, area A* an hickne h* were aigne o a o obain he eire maximum rain γ = 1.5 uner he inpu ineniy coniere an by mainaining heir raio conan. Table 3 repor he value of he area an he hickne obaine in he hree cae an he value of he equivalen iffne an he equivalen amping coefficien of he repecive linear moel. In paricular he parameer of he linear moel relae o he able repone (k, ), he parameer of he linear moel relae o he ranien repone (k, ) an he average value of previou parameer (k m, m ) are repore. Table 3 Geomeric characeriic of HDR evice an equivalen linear parameer HDR in in in m h* A* k k k m m (mm) (mm 2 ) (N/mm) (N/mm) (N/mm) Cae a Cae b Cae c The exreme value of iplacemen an force obaine by uing he non linear moel an he hree linear moel were compare. A ueful repreenaion of he reul of he ifferen analye carrie ou i repore by Fig.3 (cae b). For each accelerogram he maximum value of iplacemen an force obaine by he non linear analyi an he hree linear analye are repore in he abcia an orinae axe repecively. In hi way he oe line, which i he biecing line, inicae when he iplacemen or force obaine by uing he linear moel coincie wih hoe obaine from he non linear analyi. I may be oberve ha he linear moel referring o he able behaviour (in 1) how a enency o overeimae he iplacemen an unereimae he force, wherea he linear moel referring o he ranien behaviour (in 2) ha he oppoie enency. Thi i confirme by he average value repore by Table 4. The reul obaine how ha he wo equivalen moel, efine accoring he crierion propoe, can furnih he exreme value of force an iplacemen an o no iffer by more han 10% wih repec o hoe obaine by he non linear analyi.

11 IN (cm) in in m in HDR (cm) IN (kn) in in m 100 in HDR (kn) Figure 3 Maximum iplacemen force for he even groun moion coniere cae b Table 3 Average reul obaine wih non linear an linear yem Non inear in in in m ipl. force ipl. force ipl. force ipl. force (cm) (kn) (cm) (kn) (cm) (kn) (cm) (kn) Cae a Cae b Cae c CONCUSIONS inear equivalen moel are uggee by curren echnical coe o moel HDR behaviour. Thee moel are generally efine by coniering able behaviour an neglecing he ranien repone of he maerial, which rongly influence he eimic behaviour of rucure wih HDR-bae evice, ince he exreme value of iplacemen an velociie are reache only a few ime uring a eimic even. Thi paper propoe a meho o efine wo equivalen linear moel, bae on able an ranien behaviour. The effecivene of he propoe moel o preic he eimic repone wa checke by carrying ou linear an non linear analye of imple one-egree-of-freeom yem, in he rain ampliue an rain rae fiel ha are of inere in eimic eign. The reul obaine how ha he wo equivalen moel, efine accoring he crierion propoe, furnih he exreme force an iplacemen value wih accepable level of approximaion. REFERENCES Barera F. an Giacchei R., Seel iipaing brace for upgraing exiing builing frame. J. Conrucional Seel Reearch, 60(3), Choi H.a, Kim W.-B., ee S.-J., A meho of calculaing he non-linear eimic repone of a

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