EFFECT OF LINKING FLUID DAMPERS ON WHIPPING EFFECT AND TORSIONAL RESPONSE OF TOWER-PODIUM SYSTEMS

Transcription

1 th World Confrn on Earthqua Engnrng Vanouvr, B.C., Canada August -6, 4 Papr No. EFFECT OF LINING FLUID DAMPERS ON WHIPPING EFFECT AND TORSIONAL RESPONSE OF TOWER-PODIUM SYSTEMS Zhn YANG, Xln LU, Tzong-Hr HWU and Youln XU 4 SUMMARY Wth th ams of undrstandng th nflun of lnng flud damprs on whppng ffts and torsonal rsponss of tall towrs wth podum strutur, ths papr prsnts a omprhnsv analytal study. To obtan an unvrsal rsult, a standardzd -D modl of two stors, surroundd by a -D on-story modl, s frst onstrutd to smulat a hgh towr wth a podum strutur basd on svral smplfd prnpls. In ths modl systm, varous of paramtrs an b tan nto aount, nludng story numbr rato and stffnss rato of th uppr part to ts lowr part for th towr, mass rato, translatonal stffnss rato and torsonal stffnss rato of th towr to ts podum, latral stffnss ntrts rsptvly for th podum, th uppr part and th lowr part of th towr, supplmntal dampng rsultd from lnng flud damprs, and plan dstrbuton of th flud damprs t.. Subsquntly, th quatons of moton of th towr-dampr-podum systm ar drvd usng th standardzd modl, wth abov-mntond paramtrs nvolvd n. Thr onntd ass, that s, onntd by flud damprs btwn towr and podum, onntd by rgd onntons, and wthout any onntons, ar onrd rsptvly. A dmnsonlss form of ths quatons s thn obtand through a normalzd transformaton for th varous rlatd matrs. Followd s th paramtr analyss of varous paramtrs to ssm rsponss through solvng th quatons of moton. Th analytal rsults show that ompard wth rgd onnton as, th vbraton prforman of a towr an usually b mprovd by nstallng flud damprs btwn towr and podum and aordngly whppng ffts an b fftvly allvatd. In addton, most of ssm rsponss nludng torson an b rdud for th towr and podum f th dampr paramtrs ar sltd appropratly. INTRODUCTION Du to nrasng of populaton, shortag of supply n land, and ntralzd srv rqurmnts, modrn ts oftn nd many tall buldngs. Som of tall buldngs ar bult as a towr strutur wth a larg podum strutur to ahv larg opn spa for parng, shops, rstaurants, and hotl lobby at th ground or lowr lvls. In most ass, th towr strutur and th podum ar bult togthr on thr a ommon box foundaton or a ommon raft foundaton. Thr ar no sttlmnt jonts and ant-arthqua Engnr, Aurot (Shangha) In., Shangha, P.R.C. Emal:zyang@6.om Profssor, Tongj Unvrsty, Shangha, P.R.C. Emal: lxlst@mal.tongj.du.n Post-Dotor Fllow, Aurot Corporaton, Tawan. Emal: hwu@robot.om.tw 4 Profssor, Th Hong ong Polythn Unvrsty, Hong ong, P.R.C. Emal: ylxu@polyu.du.h

2 jonts btwn th towr strutur and ts podum. Th prsn of th podum strutur, whos latral stffnss may b largr than that of th towr strutur, lads to a suddnly larg latral stffnss hang of th buldng at th top of th podum strutur. Consquntly, th ssm rspons of th uppr part of th towr strutur wll b sgnfantly amplfd, ladng to th so-alld whppng fft. Suh a problm an not b asly solvd usng th onvntonal strutural modfaton. In addton, th omplxty n plan onfguraton du to addng th podum strutur as a appndx of th towr possbly lads to th nras of torsonal rspons on aount of th hang of stffnss ntr and mass ntr for th whol towr-podum systm. Th onpt of lnng adjant buldngs or onntng podum struturs to a man buldng usng passv damprs, sm-atv damprs, or atv damprs has bn alrady proposd to mprov thr ssm rsstant prforman. Th nvstgaton of usng passv damprs to onnt adjant buldngs for nhanng thr ssm rsstant prforman has bn arrd out by obor t al.[], Luo t al. [], Xu t al [] among othrs. Th us of atv atuators to ln a group of buldngs to rdu thr ssm rsponss has bn xamnd by Yamada t al. [4], Sto and Matsumoto [5], and othrs. Th xprmntal nvstgatons of adjant buldngs lnd by flud damprs hav bn also arrd out by Xu t al. [6] for th buldngs undr harmon xtatons and by Yang t al [7] for th buldngs undr ssm xtatons through shang tabl tsts. All ths nvstgatons dmonstratd that th us of damprs to ln adjant buldngs of dffrnt fundamntal frquns ould sgnfantly rdu ssm rspons of thr buldng f th loatons and paramtrs of damprs ar appropratly sltd. Howvr, all ths nvstgatons ar lmtd to ontrol thr translatonal rsponss only. Th nvstgaton of usng ER/MR damprs to onnt a podum strutur to towr strutur to prvnt th whppng fft has bn prformd by Qu. t al. [8]. Thr rsults dmonstratd that th smart damprs wth propr y paramtrs ould not only prvnt th towr buldng from whppng fft but also rdu th ssm rsponss of both towr and podum struturs at th sam tm. Nvrthnss, only -D buldng modls ar usd and aordngly torsonal rspons an not b onrd n thr nvstgaton. Mor mportant, only on towr buldng wth fxd paramtrs and only El Cntro wav xtaton ar xplord n thr study and thrfor th unvrsalty of thr onlusons nd to vrfy furthr. Ths papr thrfor ams to mprov th undrstandng of how supplmntal vsous dampng nflun whppng fft and torsonal rspons of hgh-towr wth low podum systms. Th mhansm of whppng fft du to th suddn hang of stffnss s smply analysd usng a two-dgr-of-frdom systm pror to th man paramtr study. To obtan unvrsal rsults, a standardzd buldng modl s frst onstrutd to smulat th hgh towr wth a podum strutur basd on svral smplfd prnpls. Th standardzd modl s analogous to that n th nvstgatons of asymmtr on-story systms studd by Gol t al [9,, ] and Ln t al []. Thn, th quatons of moton of th towrdampr-podum systm ar drvd usng th standardzd modl, wth all nds of paramtrs nvolvd n. Thr onntd ass, that s, onntd by flud damprs btwn towr and podum, onntd by rgd onntons and wthout any onntons, ar onrd rsptvly. Nxt, a dmnsonlss form of ths quatons s obtand through a normalzd transformaton for th varous rlatd matrs. Th last s th paramtr analyss of varous paramtrs to ssm rsponss through solvng th quatons of moton. SIMPLE ANALYSIS ON MECHANICS OF WHIPPING EFFECT DUE TO SUDDEN CHANGE OF LATERAL STIFFNESS Bas onpts and assumptons To rflt th whppng fft of a two-part buldng rsultd from th suddn hang of stffnss, th maxmum rlatv dsplamnt rato of th uppr part to th lowr part s adoptd as wghng rtron.

3 For onvnn, t s alld whppng fft fator hrnaftr. Compard wth th maxmum rlatv dsplamnts of th uppr part and th lowr part, th maxmum rlatv dsplamnt rato s supror n avodng th nflun of th dvrsty of arthqua xtaton and thrfor an rflt whppng fft mor larly. Th nvstgaton on th mhansm of whppng fft s prformd by usng a two-story shar-typ analytal modl (Th bas paramtrs nlud m, m, and, whr m and ar, rsptvly, th mass and stffnss of th th story.). Four ass ar onrd. Th frst on s m = m and = (dntal both n stffnss and mass for two stors). Th sond on s m = m and = (suddn hang n stffnss at th top of th frst story). Th thrd on s m = m and = (suddn hang n mass at th top of th frst story). Th last on s m = m and = (suddn hang both n stffnss and mass at th top of th frst story). Th vbraton mods and tm hstory rsponss of th four ass ar all alulatd and gvn blow. st Vbraton Mod nd Vbraton Mod Bas m=m = M=M,= Bas m=m = M=M,= vrtal poston vrtal poston modal rspons modal rspons Fg.. Vbraton mod of two-story analytal modl for dffrnt stffnss rato and mass rato Modl analyss Compard wth th bas as (as on), th fft of suddn hang of stffnss or mass s shown larly n th lattr ass( shown n Fg.). In as two, modal rspons s nrasd gratly for both th st and nd vbraton mod. But for as thr, only th nd modal rspons s magnfd. Whl n as four, although th st and nd modal rsponss ar nrasd, th magnfd xtnt s rlatvly lmtd. It sms that th suddn hang of stffnss mas th vbraton manly our at th sond story no mattr n th st mod or n th nd mod. Th suddn hang of mass lads to th ntrstng rsult that modal vbraton manly tas pla at th frst story n th frst vbraton mod and orrspondngly at th sond story n th sond vbraton mod. As w now, th dsplamnt rspons s mostly ddd by th frst modal ontrbuton. It an thrfor b nfrrd that th suddn hang of stffnss s vry lly to lad to th rspons magnfaton at th nd story whl th suddn hang of mass dos not nssarly hav th sam fft. Sptrum analyss

4 Th rsults of tm hstory rspons also provd th abov ponts. From Fg., th dsplamnt rato of th nd story maxmum dsplamnt to th st story maxmum dsplamnt ar prsntd. Compard wth th bas as (as on) n whh th rspons rato s about 5., as two, as thr and as four had th orrspondng ratos of about.,. and.5 rsptvly. Obvously, rsponss ar manly ourrd at th nd story for as two and at st story for as thr. As for as four, th rsponss may b vry larg for both st and nd story although th rspons rato s not magnfd. whppng fft fator as:whppng fft fator--th rato of th nd story maxmum dsplamnt to th st story's prod(s) whppng fft fator as:whppng fft fator--th rato of th nd story maxmum dsplamnt to th st story's prod(s) whppng ff fator as:whppng fft fator--th rato of th nd story maxmum dsplamnt to th st story's prod(s) whppng fft fator as4:whppng fft fator--th rato of th nd story maxmum dsplamnt to th st story's prod(s) Fg.. Whppng fft fator urvs of two-story analytal modl for dffrnt stffnss rato and mass rato Aordng to th abov-mntond analyss, th whppng fft du to th suddn hang of stffnss should b manly ausd by th sharp hang of th frst vbraton mod. Ths s on of th bass for th smplfaton to th standardzd modl dsrbd n th nxt ston. STANDARDIZED MODEL FOR MAIN TOWER-PODIUM STRUCTURE SYSTEMS In ordr to nvstgat th vbraton haratrsts n th most xtnsv fld for th systm formd by man buldng and podum strutur, ntrnal man buldng and surroundng podum ar smplfd to a standardzd modl, n whh varous paramtrs an b asly adjustd to orrspond to varous dffrnt typs. Standardzd Modl Th standardzd modl nluds an dalzd two-story buldng mod l (smply alld modl ) and a on-story buldng modl (alld modl ), whh both onssts of rgd ds supportd by strutural lmnts (walls, olumns, momnt-frams, brad-frams, t.). As Fg. shown, th ntrnal two-story modl (modl ) s usd to smulat man buldng and th surroundng on-story modl (modl ) s usd to rprsnt th podum strutur. In addton, th systm an nlud flud vsous damprs (FVDs)

5 nstalld btwn th two parts. Th mass proprts of th systm ar assumd to b symmtr about both th X- and Y-axs whras th stffnss and dampr proprts ar onrd to b symmtr only about th X-axs. Uy Uy(Uy) b Y X CM CM and CM CR CR CR CSD d Th ntr of mass (CM) of any d floor s dfnd as th ntrod of nrta fors of th d whn th systm s subjtd to a unform translatonal alraton n th drton undr onraton. Sn th mass s unformly dstrbutd about th X-and Y-axs, th CM of a d onds wth th gomtr ntr of th d. Th ntr of rgdty (CR) s dfnd Ug (t) as th pont on th d through whh a applaton of a stat horzontal for auss no rotaton of th d. For any story n ths modl, CR s also th Fg.. Sth map of th gnral modl dfnd n ths papr ntrod of rsstng fors n strutural lmnts at that story whn that story s subjtd to a unform rlatv translatonal dsplamnt n th drton undr onraton. Th la of symmtry n th stffnss proprts about th Y-axs s haratrzd by th stffnss ntrts,, dfnd as th dstan btwn th CM and th CR. Wth both CM and CR dfnd, th dg that s on th sam of th CM and th CR s dnotd as th stff dg and th othr dg s dsgnatd as th flxbl dg (S Fg.). Th ntr of supplmntal dampng (CSD) gnratd by th onntd damprs s dfnd as th ntrod of dampr fors whn th frst floor of modl s subjtd to a unform translatonal rlatv vloty wth rspt to th floor of modl n th drton undr onraton. Th la of symmtry n th dampr proprts about th Y-axs s haratrzd by th supplmntal dampng ntrty,, dfnd as th dstan btwn th CM and th CSD. Smplfaton Prnpls to Standardzd Modl For onvnn, th orgnal ral man buldng s dalzd as a shar-typ buldng wth th sam ( m + n) stors and th orrspondng podum had m stors. Th lowr m stors onnts wth podum. Th paramtrs of m and n / m an b hangd to smulat dffrnt pratal ass. To smulat th rlatonshp of th two parts of a tall man buldng wth a low podum, som paramtrs of th sond story of modl must b rstrand by th rlatd paramtrs of th frst story of modl so m that modl an smulat th ral man buldng to th most xtnt. Th rlatd paramtrs ar β =, m y γ = and η = y stons.) θr θr. (Not that all rlatd paramtrs wll b dtald xpland n th nxt

6 For modl, ts frst story s th bas unt usd n ths papr and t rprsnts th part of a man buldng onntd wth th podum buldng substtutd by modl. Th bas frquny of th frst story y ω y = mbods th hang of m and th hang of n / m s ahvd usng th gudln on m gvn n th nxt paragraph. Th sond story of modl rprsnts th part of th man buldng abov th podum buldng. Thr ar total thr gudlns n smplfyng th man buldngs to th standardzd modl. Frst, th m mass rato of β = s st to b qual to story numbr rato n / m. Corrspondngly, th gudln two m s, for th modl and ral man buldng, to st th sam valu n th frst frquny valu. Thus, th y stffnss rato of γ = an b spfd. Ths pont also aords wth th rqurmnt of png th y frst modal vbraton as los as possbl for modl and ral man buldng. Th last gudln s to hoos th sam valus for translatonal stffnss rato and orrspondng torsonal stffnss rato, namly, θr y θr y = and =. Ths wll smplfy th rsarh wthout losng th unvrsalty n θ R y θr y pratal sgnfan. DERIVATION OF EQUATIONS OF MOTION Th quatons of moton of th man buldng modl lnd by flud damprs wth th podum buldng modl an b xprssd as: () ( ) () ( ) () U t + Cs + Cd U t + s + d U t = MΓ u g t whr U () t, U () t, and () t M () U ar, rsptvly, th dsplamnt vtor, th vloty vtor, and th alraton vtor of th systm rlatv to th ground ug() t s th translatonal ground alraton n th Y drton appld at th bas of th systm Γ s th transformaton vtor M s th mass of th systm and s, d ar th stffnss matrs of th systm and th onntor rsptvly. In ths papr, th onntor s vsous dampng dampr and thrfor th matrx of d s non C s, C d ar th dampng matrs of th systm and th onntor. Rlatd Varabls Usd n th Analyss α group (Gomtr dmnson group) a b a α = α = α = (5) d b α Aspt rato of th modl α Aspt rato of th modl α Gomtr dmnson rato of modl and modl β group (Mass group) m m β = β = (6) m m β Mass rato of th sond floor to th frst floor for modl n th standardzd modl

7 β Mass rato of th podum floor to th frst floor of modl n th standardzd modl γ group (Latral stffnss group) y y γ = γ = (7) y y γ Latral stffnss rato of th sond story to th frst story for modl n th standardzd modl γ Latral stffnss rato of modl to th frst story of modl n th standardzd modl η group (Torsonal stffnss group) θr θ R η = η = (8) θr θr η Torsonal stffnss rato of th sond story to th frst story for modl n th standardzd modl η Torsonal stffnss rato of modl to th frst story of modl n th standardzd modl ρ group (Normalzd radus of gyraton: dstrbuton) ρ ρ = And C R b ρ θ = (9) C y group (Normalzd ntrty group: plan poston) = = And = = () b b a b Rlatv loaton of th CR from CM for th frst floor of modl n th standardzd modl Rlatv loaton of th CR from CM for th sond floor of modl n th standardzd modl Rlatv loaton of th CR from CM for th floor of modl n th standardzd modl Rlatv loaton of th CSD from CM for supplmntal dampng dampr n th standardzd modl ξ group (Formal supplmntal dampng rato) C y ξ = () mω y Not that ξ s just an xprssv ndx of th amount of supplmntal dampng n th forms of dampng rato but t s not th ral dampng rato of th systm. Furthrmor, th valu of ξ ould xd. f nssary. ω group (Formal translatonal vbraton frquny) y ω y = () m Analogous to th tm of ξ, ω y s th angl frquny of th modl that s mad up of th frst story of modl. Although ω y s not th ral frquny of modl, t an rally ndat th magntud of th

8 ral of mod frquny baus thr s a dfnt proportonal rlatonshp btwn aordng to th smplfaton gudlns. Ω group (Rato of th formal torsonal and translatonal frquns) = ω θ θ ω y R Ω θ ωθ = () m ρ Ωθ s ndatv of th dgr of th ouplng of latral and torsonal motons n th last rang. Bas Formulas Mass radus of gyraton b + + α ρ = ρ = = b α + ad( a d ) b( b ) α α α + α α α + + ρ = = a = a f( α ) ( ) () ad b α α α ρ, ρ, ρ Th mass raduss of gyraton for th frst floor, th sond floor of modl and th podum floor, rsptvly Stffnss quantts for th frst story of modl y = y = x y + θ y + θr θ y x x y = And thrfor: y = () y, x Th latral stffnss and th dstan from th CM of th th rsstng lmnt along th Y-axs x, y Th latral stffnss and th dstan from th CM of th th rsstng lmnt along th X-axs y, θ, θr Th translatonal stffnss of th systm along th Y-axs, th torsonal stffnss of th systm about a vrtal axs at th CM and, th torsonal stffnss of th systm about a vrtal axs at th CR. D and d ar smlar to th d. Supplmntal dampng of th onntd flud damprs Supplmntal dampng: C y = y C = x y + θ y x x y = And thrfor: C y Cθ = C y + Cθ R (4) y, x Th latral dampng offnt and th dstan from th CM of th th FVD along th Y-axs x, y Th latral dampng offnt and th dstan from th CM of th th FVD along th X-axs C y, C θ, C θr Th translatonal dampng offnt of th systm along th Y-axs, th torsonal dampng offnt of th systm about a vrtal axs at th CM and th torsonal dampng offnt of th systm about a vrtal axs at th CSD

9 Rlatd Matrs for th Systm wth Dampr Conntons or Wthout Any Conntons Dgr of frdom Th on-way symmtr systm has sx dgrs of frdom (DOF) whn subjtd to ground moton along th Y-axs: on translaton along th Y-axs and rotaton about a vrtal axs at mass ntr for ah d. Th dsplamnt vtor U for th systm s dfnd by { bu u bu u au } T T U = u y θ y θ y θ whr u y rlatv to th ground for th th d th d. s th Y-drton horzontal dsplamnt u θ s th rotaton about th vrtal axs at th ntr of mass for th Mass matrx of th systm Lt m, m and m rprsnt th total mass of d, d and d, rsptvly. And lt ρ, ρ and ρ rprsnt th mass radus of gyraton. Thn, th mass matrx s gvn by: m m m M = (4) m44 m whr varous tms ar gvn as: m = m = m m 55 = m m m + α = m m44 m α α + α 55 m 66 = m66 f( α, α, α ) m = (5) Stffnss matrx of th systm = (6) s Whr varous tms ar gvn as: = y + y = y + y = y 4 = y b b b = = y θ R y θr = y b b b b b 4 = y + θ R = = = y 4 = y b b b 4 = = 4 = 4 44 = y + θ R b b

10 55 = y 56 = y 65 = 56 a 66 = y + θ R (7) a a Strutural dampng matrx of th systm Th proportonal dampng assumpton s gvn drtly n th nxt dmnsonlss stag. Dampr dampng matrx d d d5 d6 d d d 5 d 6 C = (8) d d 5 d 5 d 55 d 56 d 6 d 6 d 65 d 66 Whr varous tms ar gvn as: ρ d = C y d = C y d = d d = + C y b b b ρ d5 = C y d6 = C y d 5 = C y d 6 = C y a b + ab ab d 5 = d5 d 5 = d 5 d 6 = d6 d 6 = d 6 ρ d 55 = y d 56 = C y d 65 = d 56 a d = + C (9) 66 y a a Extaton vtor Γ = { } T Hn, T { } u g t = m m m u g t MΓ () Normalzaton to Dmnsonlss Form for Varous Matrs Th dmnsonlss formula an b obtand through dvdng th quatons of moton by m and substtutng th rlatd varabls nto thm. Th pross of th normalzaton s omttd to sav spa. CONTROLLING PARAMETERS OF THE SYSTEMS To nvstgat and valuat th ontrol prforman of supplmntal flud damprs on rduton of th whppng fft and th ssm rsponss of both man buldngs and podum struturs, thr ass ar nvstgatd. Th frst as s that modl s rgdly onntd to modl (B-Cas). Th sond as s modl totally sparatd from modl (B-Cas). Th last as (B-Cas) s modl onntd to modl by flud damprs as spfd abov. Equaton () ndats that th lnar last rspons of th systm dpnds on two sts of paramtrs f th xtaton s fxd. Th frst st of paramtrs orrspondng to th systm wthout FVDs onssts of

11 () transvrs vbraton prod of th bas unt, whh s th frst story of modl, Ty = π / ω y () mass rato of th sond story to th frst story of modl, β (two nd stffnss ratos, γ and η, ar dpndnt on β aordng to th standardzd gudln of th standardzd modl) () mass rato of modl to th frst story of modl, β (two tams of stffnss ratos, γ and η, ar ndpndnt on β ) (4) stffnss rato of modl to th frst story of modl, γ and η (5) normalzd stffnss ntrty,, and (6) rato of torsonal and transvrs frquns, Ω θ (7) varous aspt ratos, α, α and α (8) natural dampng ratos, ξ to ξ 6, whh ar usd to spfy strutural dampng matrx. Th aspt ratos ar nludd as on st of th systm paramtrs baus t faltatd a mor appalng dfnton of th stffnss ntrty as a prntag of th plan dmnson. Th sond st of paramtrs orrspondng to supplmntal dampng onssts of () supplmntal dampng ndx n trms of dampng rato, ξ () normalzd supplmntal dampng ntrty, and () normalzd supplmntal dampng radus of gyraton, ρ. ξ s ndatv of th amount of addtonal dampng, as a fraton of th rtal valu, whh s provdd by FVDs wth th sam valu as n th as that a strutur, sam as th frst story of modl, s onntd to a fxd objt by th sam FVDs. s ndatv of how vnly FVDs ar loatd wthn th systm n th Y drton. A zro valu of mpls that FVDs ar loatd symmtrally about th CM, whras non-zro valus ndat unvn dstrbuton. ndats how muh farthr apart from th CSD th FVDs ar loatd. ρ Ths paramtr s also ndatv of th dampng n th torsonal mod of vbraton. Zro valu of mpls that all FVDs ar loatd at th CSD and that thy provd zro dampng n th torsonal mod, whras larg valus ndat that FVDs ar loatd farthr from th CSD and that dampng s nrasd n th torsonal mod. SELECTED SYSTEM PARAMETERS Rsponss of th systm ar prsntd for th abov-mntond thr ass. Th rlatd paramtrs ar ntrodud blow. Valus of Ty ar sltd n th rang of.. to rprsnt many low-rs and mdrs buldngs for whh supplmntal dampng s xptd to vdntly nflun th rspons. Th mass ratos of β =, 4, 6 and 8 rprsntd that th story numbr rato of uppr part to lowr part for th man buldng ar, 4, 6 and 8 rsptvly. Furthrmor, th stffnss ratos of γ and η ar spfd orrspondngly as.764,.48,.48 and.7 rsptvly aordng to th smplfaton gudlns of th standardzd modl. Th mass rato of modl to th frst story of modl β s gnrally st as. whl th stffnss ratos of γ and η ar sltd as.5,.,. and 4. suh that th ffts of suddn stffnss hang wth dffrnt dgrs an b nvstgatd. In ordr to show ffts of th ouplng of latral and torsonal motons, th typal valu of Ω θ = s adoptd whn stffnss ntrts xstd. Th normalzd stffnss ntrts vars from.5 to.4 for and and. to. for n most ass. Th aspt ratos, α, α and α, ar fxd at on, on and two rsptvly. Varous modal dampng ratos, from ξ to ξ 6, ar sltd as.5 to alulat th strutural dampng matrx. ρ

12 Although th dampng offnt of FVDs dpnds on th frquny and ampltud of moton as wll as on th opratng tmpratur, th dampng for of flud damprs ar onrd to only dpnd on th rlatv vloty and th dampng offnt s only rlatd to th FVDs thmslvs n ths papr. It should b notd that th usd ξ s just a substtut of dampng offnt and not ral dampng rato. Th valu of ξ s sltd at.5 for most rlatd ass. For a lmtd numbr of ass, howvr, varatons of ξ n th rang of.-. ar onrd. In gnral, four valus of = -., -., -., and ar sltd. In all rlatd ass, th valu of ρ s fxd at.5 to p onrabl dampng ffts on torsonal mods. N-S 94 El Cntro arthqua wav s usd as th ssm xtaton and nputtd n th Y drton only. Th pa valu of El Cntro xtaton s sltd as.g and th tm ntrval of nputtng pont s.s. INVESTIGATION ON TRANSLATIONAL WHIPPING EFFECT In ths ston, whppng fft of th translatonal moton rsultd from th only Y drton xtaton s frst nvstgatd for th systm wthout any stffnss ntrts. Thn th ontrol ffts of supplmntal vsous dampng on th whppng ffts ar studd. Lastly, th nfluns of supplmntal vsous dampng on rsponss of th systm ar also nvolvd. Whppng Efft of Translatonal Moton Whppng fft ndx usd n ths part s dfnd as th rato of whppng fft fators of B-Cas to B- Cas. Th whppng fft fator, B-Cas and B-Cas ar alrady dfnd bfor. Fg.4 dpts th whppng fft ndx aganst th prod T y for four dffrnt stffnss valus of γ =.5,.,. and 4.. Th rsults show that whppng fft dpnds sgnfantly on th stffnss rato of mod to th frst story of modl ( γ ). Whn γ s qual to.5, thr s narly no any sgn of whppng fft. Whnγ s largr than., whppng fft boms larg wth th largst valu of.5 and t s vdnt for almost all T y. Obvously, th largr γ s, th mor vdnt whppng fft s. Ths dnots that th whppng fft s rsultd from th suddn nras of th latral stffnss. As a ontrast, Fg.5 gvs th whppng fft ndx aganst th sam prod T y for four dffrnt mass valus of β =.,., 4. and 8.. It shows that whppng fft dos not nssarly xst for th whol ntrstd sop of T y, spally for th ass of sltng rlatv larg valu of β. Ths also vrfs th rason ausng whppng fft, whh s dsussd bfor. rato th rato of whppng fft fator(as to as),beta=,beta=4 GA M A =.5 GA M A= GA M A = GA M A= prod(s) Fg.4. Whppng fft ndx for dffrnt stffnss rato valus of GAMA rato rato of whppng fft fator( B-Cas to B- Cas),GAMA=,BETA=4 BETA =. BETA =. BETA =4. BETA = prod( s) Fg.5. Whppng fft ndx for dffrnt mass rato valus of BETA rato rato of whpp ng fft fator( B-Cas to B- Cas),BETA=,GAMA= BETA= BETA=4 BETA=6 BETA= prod ( s ) Fg.6. Whppng fft ndx for dffrnt mass rato valus of BETA

13 Prsntd n Fg.6 ar whppng fft ndxs aganst th prod of T y for four dffrnt mass rato valus of β =., 4., 6. and 8.. Th whppng fft ndxs gnrally p nar. or so xpt th as wth mass rato valu β of., n whh th whppng fft ndx an rah 5.. Ths dmonstrats that whppng fft wll xst unvrsally for varous story numbr ratos of n / m n a m + n -story man buldng. ( ) rato th st story of modl:maxmum dspamnt rato(rgdly onntd:sparatd),beta=,beta=4 GA M A =.5 GA M A = GA M A = GA M A = prod( s ) rato th nd story of modl:maxmum dsp amnt rato(rgdly onntd:sparatd),bet A=,BETA=4 GAMA=.5 GAMA= GAMA= GAMA= prod( s) rato th st story of modl:maxmum dspamnt rato(rgdly onntd:sparatd),beta=,beta =4 GAM A=.5 GA M A = GAM A= GA M A = prod( s) Fg.7. Varous rlatv dsplamnts of modl and modl for dffrnt stffnss of th st story It should b notd that whppng fft ndx ovr. just mans th ourrn of whppng fft but t dos not dnot th nssary nras for dsplamnt rsponss at dffrnt floors. Ths s baus whppng fft ndx s just a rlatv wghng ndx of uppr dsplamnt to lowr dsplamnt. Du to th suddn nras of latral stffnss at th frst story for modl, dsplamnt rsponss of both th frst floor and th sond floor of modl hav th tndny to b mtgatd although th tndny s mor ntns for th frst story. Thrfor, t s possbl that th dsplamnt rsponss of th frst floor and sond floor ar both drasd whn th whppng fft fator s nrasd. In ordr to larly show ths pont, Fg.7 prsnts th varous rlatv dsplamnts of modl and modl aganst th prod of T y for th four dffrnt stffnss ratos of γ =.5,.,. and 4.. Whn latral stffnss rato γ s largr than., a gnral rul of rlatv dsplamnts an b obsrvd: th rlatv dsplamnt () s almost drasd at varous prods for th frst floor of modl () s oftn drasd but somtms nrasd at dffrnt prods for th sond floor of modl () s almost magnfd at varous prods for modl. Thrfor, only for som prods of T y, dsplamnt of th sond floor of modl wll b magnfd du to th suddn nras of latral stffnss although whppng ffts wll wdly our for varous prods. Influn of Supplmntal Damprs on Whppng Efft In ordr to wdly show th ffts of supplmntal dampng that s rsultd from th flud damprs onntd at th frst floor btwn modl and modl, th whppng fft ndxs of B-Cas to B- Cas ar prsntd aganst th dampng rato ξ for th four dffrnt prods T =. y,.4,.6 and.8 [(S Fg.8). Svral ruls an b obsrvd. Frst of all, whppng fft ndxs ar gnrally lss than. and thrfor whppng fft an b mtgatd n most ass. Sondly, ontrol ffts on whppng fft ar dffrnt for dffrnt prods. From th prod of. to.8, th ontrol fft s gradually dtroratd wth th orrspondng maxmum rduton ratos from 58% to 4%. Thrdly, th ontrol ffts ar gradually dtroratd wth th nras of ξ whn ξ ar mor than rtan valus. As a omparson, Fg.9 prsnts th whppng fft ndxs of B-Cas to B-Cas. Wth th nras of ξ, th whppng fft ndxs ar gnrally nrasd but th nras s slow. Furthrmor, th ndxs an b lss than. at a fw valus of prod.

14 rato rato of whppng fft fator(b-cas to B- Cas),BETA=,GAMA=,BETA=4 Ty=. Ty=.4 Ty=.6 Ty= supplmntal dampng Fg.8. Influn of supplmntal damprs on whppng fft(b-cas to B-Cas) rato rato of whppng fft fator(b-cas to B- Cas),BETA=,GAMA=,BETA=4 OMIGA=. OMIGA=.4 OMIGA=.6 OMIGA= supplmntal dampng Fg.9. Influn of supplmntal damprs on whppng fft(b-cas to B-Cas) Influn of Supplmntal Damprs on Strutural Rspons To assss th prforman of flud damprs on th systm, strutural rsponss bs whppng fft ndx should b nvstgatd. Fg. and Fg. (Th two fgurs ar abbrvatd du to spa lmtaton ) dpt th rspons ratos of B-Cas to B-Cas and B-Cas to B-Cas rsptvly. Compard wth B-Cas, th dsplamnt rsponss of modl and th sond floor of modl ar drasd gnrally but thos of th frst floor of modl ar not always drasd n B-Cas. Whl ompard wth B- Cas, th dsplamnt rsponss of th frst and th sond floor of modl ar always mtgatd and thos of modl ar gnrally nrasd xpt T =. y 8. Ths dnot that supplmntal dampng an mtgat rsponss of th systm not only for B-Cas but also for B-Cas on most oasons. Dung to th spa lmtaton, th nvstgaton on torsonal rspons s abbrvtd n ths manusrpt. Th full ontnt of th manusrpt wll b submttd to a journal soon latr. CONCLUSIONS Whppng fft and torsonal rspons of towr-podum systm wth and wthout flud damprs onntd hav bn nvstgatd usng a standardzd mthod. Th mhansm of whppng fft s smply analysd. A st of standardzd modl systm s albratd to smulat gnral hgh towrs and thr podum struturs. Th quatons of moton of th systm ar drvd and dvlopd to faltat th paramtr analyss of towr-podum systm. Th ffts of a srs of y paramtrs on whppng fft and torsonal rspons ar nvstgatd through solvng th dvlopd quatons of motons for th systm. Th study on mhnsm of whppng fft du to th stffnss hang dmonstrats that whppng fft s rsultd from th sharp hang of th frst vbraton mod othr than th sond vbraton mod. Th rsults of th omprhnsv paramtr analyss show that ssm rsponss and whppng ffts ar nflund by varous paramtrs. In usual as (hrnaftr th ass of stffr podum ar rfrrd), ompard wth rgdly onntd as, th vbraton prforman of a towr an b mprovd by usng lnng flud damprs and aordngly whppng ffts and ssm rsponss of th towr an both b fftvly allvatd. In addton, th ssm rsponss of th podum an also b rdud. Whl ompard wth th as of wthout any onntons, usng lnng flud damprs an rdu ssm rsponss of th towr but at th sam tm possbly nras thos of th podum. Th dtroraton of th ssm prforman of th podum s owng to th lmtd nrgy-dsspatd apablty of th flud damprs baus th damprs an not b xtd ffntly du to thr low nstallng loatons. In that as, th fft of drt mutual ntraton btwn towr and podum prdomnats ovr th fft of damprs nrgy dsspaton. Thrfor, although usng lnng flud damprs an vdntly allvat whppng fft of towr, t s not always fftv to rdu th ssm

15 rsponss of both towr and podum by usng ths approah. Smlar onlusons an b drawn for th ass whh ar rlvant to torsnal rsponss of th systms. ACNOWLEDGEMENTS Th wrtrs ar gratful for th fnanal support from th Natonal Natural Sn Foundaton of Chna through a NNSF grant (558). Thans ar also du to Mr. Cody Y.C. Chang, th prnt & CEO of AUROTE CORPORATION, for hs nouragmnts to th frst author. REFERENCES. obor T, Yamada T, Tanaa Y, Mada Y and Nshmura I. Efft of dynam tund onntor on rduton of ssm rspons: applaton to adjant off buldngs. Pro. 9 th World Conf. On Earthqua Engng., Toyo-yoto, Japan. 5: , Luo JE, D Barros FCP. Optmal dampng btwn two adjant last struturs. Earthqua Engng. Strut. Dyn : Xu YL, Yang Z, Lu XL. Inlast ssm rspons of adjant buldngs lnd by flud damprs. Strutural Engnrng and Mhans 5(5): Yamada Y, Iawa N, Yooyama H, Tahbana E. Atv ontrol of struturs usng th jont mmbr wth ngatv stffnss. Pro. st World Conf. On Strut. Control, Los Angls, Calforla, USA., TP: 4-49, Sto T, Matsumoto Y. A strutural vbraton ontrol mthod of flxbl buldngs n rspons to larg arthquas and strong wnds. Pro. nd Int. Worshop on Strut. Control, Hong ong , Xu YL, Zhan S, o JM, Zhang WS. Exprmntal nvstgaton of adjant buldngs onntd by flud dampr, Earthqua Engng. Strut Dyn : Yang Z, Xu YL, Lu XL. Exprmntal study of ssm adjant buldngs wth flud damprs. J. Strut. Engng, ASCE 9(): Qu WL, Xu YL. Sm-atv ontrol of ssm rspons of tall buldngs wth podum strutur usng ER/MR damprs. Th Strutural Dsgn of Tall Buldngs : Gol R. Effts of supplmntal vsous dampng on ssm rspons of asymmtr-plan systms. Earthqua Engng Strut. Dyn 998 7: Gol R. Ssm bhavor of asymmtr buldngs wth supplmntal dampng. Earthqua Engng Strut. Dyn. 9: Gol R. Ssm rspons of asymmtr systms: nrgy-basd approah. Journal of Strutural Engnrng, ASCE : Ln WH, Chopra A. Undrstandng and prdtng ffts of supplmntal vsous dampng on ssm rspons of asymmtr on-story systms. Earthqua Engng Strut. Dyn. :