Estimating effective damping introduced by a Pendulum Tuned Mass Damper using the Extended Kalman Filter

Transcription

1 Pocdings of th 9th Intntionl Confnc on Stuctul Dynmics, EURODYN 4 Poto, Potugl, 3 Jun - July 4 A. Cunh, E. Ctno, P. Ribio, G. Müll (ds. ISSN: 3-9; ISBN: Estimting ffctiv dmping intoducd by Pndulum und Mss Dmp using th Extndd Klmn Filt Aon J. Roffl, Sim Nsimhn Stff II Stuctus, Engining Mchnics & Infstuctu, Simpson, Gumptz & Hg Inc., 4 Syon Stt, Wlthm, Msschustts, 453, Unitd Stts Associt Pofsso, Dptmnt of Civil & Envionmntl Engining, Univsity of Wtloo, Univsity Avnu Wst, Wtloo, Ontio, NL 3G, Cnd mil: joffl@sgh.com, snsim@uwtloo.c ABSRAC: Pndulum tund mss dmps (PMDs common uxiliy dmping dvics usd to ttnut xcssiv motion in tll stuctus. h concpt of ffctiv o quivlnt dmping is commonly fncd whn quntifying tund mss dmp (MD pfomnc. Effctiv dmping fs to dtmining th dmping in singl-dg-of-fdom (SDOF oscillto opting t th sm ntul fquncy tht would poduc n qul mn squd displcmnt spons s th combind min nd uxiliy systm. Dspit its simplicity, ffctiv dmping intoducd by MD hs xpincd ltivly littl us in dscibing th pfomnc of in-svic MDs, sinc its thoticl computtion is bsd on th displcmnt spons of th stuctu, which is sldom msud. Instd, ccltion spons msumnts tn fom which displcmnts nd to b infd. h poposd mthod pplis th xtndd Klmn filt fo combind stt nd pmt stimtion fo th pupos of stimting th ffctiv dmping intoducd by PMD. h ccltion spons msumnts fittd to th spons of SDOF systm, wh th unnown modl dmping is ppndd to th stt vcto nd stimtd. h ssumption of nown b stuctu ntul fquncy is subsquntly lxd nd th ntul fquncy of th stuctu without th PMD is stimtd longsid th ffctiv dmping. h lgoithm is fist dmonsttd using numicl xmpl nd compd to th thoticl clcultion. h mthodology is lso shown using full-scl ccltion spons msumnts collctd fom stuctu quippd with PMD. h sults of th study dmonstt th ppoch is n ccut nd libl mns of quntifying th pfomnc of in-svic PMDs without dict msu of th displcmnt spons of th ttnutd stuctu nd pioi nowldg of th undlying stuctu s ntul fquncy. KEY WORDS: Effctiv dmping, Equivlnt dmping, und mss dmps, Extndd Klmn filt INRODUCION und mss dmps (MDs dvics, usd in stuctus suscptibl to vibtions, which intoduc supplmntl dmping to ttnut th sponss. MDs consist of smll intil mss with stiffnss nd dmping lmnts. h stiffnss lmnt is gnlly sping usd to djust th fquncy chctistics of th dvic to closly mtch th min stuctu fquncy fo th mod of vibtion to b contolld; ltntivly, suspndd mss is usd wh fquncy djustmnt is pfomd by chnging th suspnsion lngth, nown s pndulum tund mss dmp (PMD. MDs dsignd to minimiz th oot mn squ (RMS displcmnt o ccltion spons of th stuctu, o to mximiz th ffctiv o quivlnt dmping intoducd. h ltt tm fs to th lvl of modl dmping in singl-dg-of-fdom (SDOF oscillto opting t th sm fquncy s th contolld mod of vibtion (gnlly th mod with th lgst contibution to th ovll spons tht would poduc th sm RMS displcmnt spons s th ttnutd systm. hoticl qutions lting th vious min nd uxiliy systm pmts (fquncy tio, mss tio, min mss dmping tio, nd uxiliy mss dmping tio to th ffctiv dmping [7,, 6] commonly usd in pdicting th pfomnc of MD duing th dsign phs; howv, th msu hs xpincd ltivly littl us in dscibing th pfomnc of in-svic MDs. his is bcus th computtion is bsd on th msud RMS displcmnt spons of th min stuctu, which is gnlly impcticl to msu dictly. his is pticully tu fo flxibl tll stuctus, which most suscptibl to wind-inducd vibtions nd commonly mploy MDs. h ccltion sponss usully msud, fom which th displcmnts nd to b infd. h noisy ntu of ccltion spons msumnts pclud dict intgtion fo th displcmnt spons, nd initil nd finl t-st ssumptions to coct cclomt bis gnlly not ppopit fo wind-xcitd stuctus. h issu is of pticul significnc whn it must b dmonsttd tht pscibd lvl of ffctiv dmping hs bn chivd. h cunt wo poposs using concpt nown s stt stimtion to dtmin th ffctiv dmping intoducd by th PMD. PMDs considd xclusivly hft, but th concpts cn b dily pplid to convntionl MDs. Stt stimtion is pfomd by modling pocss in od to povid n stimt of n intnl (gnlly unobsvd stt givn th msumnt of th ctul systm. A mthmticl modl of nown physicl systm is dvlopd lting th nown inputs, msud outputs, nd unnown intnl stts to b dtmind. Clssiclly, this hs bn ccomplishd using th Klmn filt [9, 8]. Estimting th ffctiv dmping is combind stt nd pmt stimtion poblm, which is inhntly nonlin, nd quis th ltd xtndd Klmn filt (EKF. h pp is ognizd s follows. Fist, bif thoticl bcgound of PMDs nd ffctiv dmping is intoducd. 543

2 Pocdings of th 9th Intntionl Confnc on Stuctul Dynmics, EURODYN 4 Scond, th concpt of combind stt nd pmt stimtion using th EKF is psntd. hid, th ffctiv dmping fo simpl numicl modl with nown ntul fquncy is stimtd nd compd with th thoticl clcultion. Finlly, th ffctiv dmping of full-scl stuctu is stimtd using ccltion spons msumnts; in this instnc, th quimnt of nown fquncy fo th contolld mod of vibtion fo th b stuctu is lxd. h findings dmonstt tht th poposd mthodology is ccut nd libl fo stimting th ffctiv dmping intoducd by PMD on n in-svic stuctu. Figu. Schmtic gomty of PMD ttnutd systm xpssd in modl coodints. HEOREICAL COMPUAION OF EFFECIVE DAMPING h concpt of ffctiv o quivlnt dmping intoducd by MD ws fist poposd by Vicy [7] nd McNm []. h mthod ss to mtch th mn squ displcmnt spons of th MD-ttnutd systm with SDOF oscillto opting t th sm fquncy of th contolld mod. h dmping in th SDOF systm is th ffctiv dmping of th ttnutd stuctu. Equtions lting th pmts of th min nd uxiliy systms of th MD-quippd stuctu to th ffctiv dmping hv bn dvlopd fo min mss xcitd [] nd bs xcitd stuctus [8] with convntionl tnsltionl MDs nd PMDs. PMDs considd hin, but th concpts cn b sily pplid to convntionl MDs. h qutions of motion fo PMD ttnutd stuctu (Figu xpssd in modl coodints s follows: M &&y(t + C &y(t + K y j (t + m && θ L = F (t m L θ(t && + c h θ(t & + ( m gl + h θ(t ( +m L&&y(t = wh M, C, K, nd F ( t th modl mss, dmping, stiffnss, nd foc nd yt ( is th modl coodint. θ ( t is th ottion of th uxiliy mss, m, with uxiliy dmping, c, nd stiffnss,. L is th lngth of th pndulum nd h is th lngth to th ttchmnt of th uxiliy sping nd dmping. h function i t Ft ( = ω is slctd in od to comput th complx fquncy spons functions, y H iω nd H ( i θ ω, wh ω is th focing fquncy. h sponss thn F ( t yt ( = Hy ( iω K ( F ( t θ( t = Hθ ( iω KL Computing th fist nd scond divtiv of th sponss in Equtions ( nd substituting into Equtions ( givs ω M / + m + iωc + K ω m H L y K = ω ml H / K L ω ml + iωch + h θ (3 ω = K M is th cicul ntul fquncy. wh n Eqution (3 is solvd simultnously fo th complx fquncy spons functions, plcing som of th tms with thi non-dimnsionl countpts. h non-dimnsionl focing fquncy is φ ω ω = (4 n h non-dimnsionl uxiliy to min mss tio is m μ = (5 M ζ nd ζ th min nd uxiliy dmping tios. h complx fquncy spons functions 544

3 Pocdings of th 9th Intntionl Confnc on Stuctul Dynmics, EURODYN 4 wh H y H θ φ ( + iφ f iω = ζ + f φ 4 iφ 3 A φ B+ iφc + f φ ( iω = φ 4 iφ 3 A φ B+ iφc + f f ζ f + 4 f ζζ ( f ζ +ζ A f ζ A= ζ + + μ B = + + μ C = f D = ζ + + μ Fo th cs of whit nois with constnt spctl dnsity, S, th mn squ sponss [8] ω ns E y( t = H y i d K φ φ (8 ω ns E θ ( t = H ( i d θ φ φ KL Fo SDOF oscillto opting t th sm fquncy s th ttnutd systm with modl dmping, ζ, th mn squ spons is [6] n Kζ (6 (7 ω S E y( t = (9 Equting th Equtions (8 nd (9 fo th min stuctu modl sponss, th thoticl vlu fo th ffctiv dmping fo th combind min nd uxiliy systm is ζ = ( C BD C f D f BD C + A( ζ + C wh A, B, C, nd D dfind in Eqution (7. ( 3 EXENDED KALMAN FILER FOR COMBINED SAE AND PARAMEER ESIMAION Stt stimtion quis nown mthmticl modl to psnt physicl systm, nd ss to stimt unnown intnl stts by lting thm to msud outputs. h Klmn filt ddsss th poblm of stimting th stt x bsd on nowldg of th lin disct-tim pocss givn by x ( = A x + G u + w wh A, G, nd C th systm, input, nd msumnt mtics, u is th nown input, nd w nd v th unnown pocss nd msumnt noiss with covinc Q nd R, spctivly. Whn th modl pmts unnown (s is th cs whn th ffctiv dmping pmt is to b stimtd, th convntionl ppoch is to ppnd th unnown pmts to th stt vcto with constnt tnsitions, nd pfom tditionl stt stimtion. Howv, th systm of qutions bcom nonlin (dspit th fct tht undlying systm is lin du to th psnc of stts (ppndd pmts in th systm nd msumnt mtics, poducing poduct of stts within th tnsition nd msumnt qutions. h EKF is pplid fo nonlin stt stimtion. h EKF is Klmn filt tht linizs bout th cunt mn nd covinc. Fo wind-xcitd stuctus, th input is gnlly unnown. Fo th cs of ccltion spons msumnts of windxcitd stuctus, th is lso dict fd-though of th unnown input in th msumnt qution. Fo th cs of unnown fd-though stochstic distubnc nois, d with covinc S, th tnsition of th stts is govnd by th following nonlin diffnc qution: ( x d + x (3 = f, w with noisy msumnt givn by ( x d z = h, + v (4 Fo th s of bvity, th d is fd lswh fo dtild divtion of th EKF fo th bov systm [4]; summy is povidd nxt. Fo ch tim stp, th filt qutions f (, h(, J = E S F F S F + R = + J z P = A J C P A J C + E S E + Q J F S E J is on-stp pdicto gin mtix, x is th pioi (whn ll msumnts up to vilbl stimt of th stt x, nd ˆ (5 with noisy msumnt z givn by z C x + v = ( 545

4 Pocdings of th 9th Intntionl Confnc on Stuctul Dynmics, EURODYN 4 A E C F ( x, f = x ( xd, f = d ( x, h = x ( xd, h = d (6 th Jcobin mtics of ptil divtivs of f ( x,, f ( xd,, h( h,, nd h( xd, with spct to x nd d, vlutd t th cunt o pvious pioi stt stimt. P is th pioi stt stimt o covinc. Aft nw msumnt is tn, th msumnt updt qutions ( h( K = P C C P C + F S F + R ˆ ˆ = x + K z x, P = I K C P K is th Klmn gin, ˆ (7 x is th postioi (whn ll msumnts up to nd including vilbl stt stimt nd P is th cosponding stt stimt o covinc. An updtd stimt of th msumnt is givn by [7] zˆ = h ( ( ˆ h, + FS F FS F + R z x h lgoithm is initilizd s follows: P = E = E [ x ] [( x ( x ] (8 (9 Klmn filting quis nowldg of th unnown stochstic distubnc nois, pocss nois, nd msumnt nois covinc mtics S, Q, nd R. hs sttistics gnlly unnown fo mbint vibtion msumnts of wind-xcitd stuctus, nd thfo must b fist stimtd fom th msud dt. A nois covinc stimtion ppoch fist poposd by Bélng [3] nd dptd by Roffl [4] fo th cs of th dict fd-though of n unnown distubnc nois is usd in this nlysis. 4 ESIMAING EFFECIVE DAMPING USING HE EXENDED KALMAN FILER h ffctiv dmping of n in-svic PMD cn b computd using Eqution (9, povidd th displcmnt spons msumnts vilbl. Estimting th ffctiv dmping intoducd by th PMD using th EKF is poposd to ovcom th chllng of lc of vilbl displcmnt spons msumnts of th ttnutd systm. In th simplst cs, th ntul fquncy of th min stuctu (without th PMD nd min mss is nown, nd th only unnown pmt is th ffctiv dmping. 4. Effctiv Dmping Estimtion fo SDOF Oscillto with PMD h EKF lgoithm is fist pplid to synthtic ccltion spons dt gntd fo SDOF min systm quippd with PMD; th stimtd ffctiv dmping is compd with th thoticl vlu computd using Eqution (. Sinc th ffctiv dmping is stimtd in th modl domin, th msumnts fit to SDOF systm xpssd in modl coodints. h modl displcmnt nd ccltion slctd s th stts of th systm. x(t = x (t x (t x 3 (t = y(t &y(t ζ ( Following disctiztion, th nonlin tnsition qutions x[ + ] = x[ ] + x[ ] + w[ ] x[ + ] = x[ ] + x3[ ] x[ ] m m + d[ ] + w[ ] m x [ + ] = x [ ] + w [ ] wh = [ ] ( w [ ] [ ] 3[ ] w w w th dditiv pocss nois tms with covinc Q nd d [ ] is th unnown stochstic input with covinc S. h msumnt qution is z[ ] = x[ ] x3[ ] x[ ] + d[ ] + v[ ] ( m m m wh v [ ] is th dditiv msumnt nois with covinc R. h synthtic ccltion spons dt fom th PMD ttnutd stuctu is gntd using th mthmticl modl of th systm using whit nois xcittion with th following pmt vlus. h min nd uxiliy msss m = g nd m = g, spctivly. h min stuctu stiffnss is = N/m, sulting in ntul fquncy of f =.5 Hz ( ω n = 3.6 d/s. h dmping cofficint is slctd bsd on modl dmping tio of 546

5 Pocdings of th 9th Intntionl Confnc on Stuctul Dynmics, EURODYN 4 ζ =.. h fquncy tio is f =.996, sulting in n optiml pndulum lngth of L=.995 bsd on th slction of optiml PMD pmts vilbl in th littu [8]. h initil stimt of th ppndd pmt is ˆ ζ =.. h initil stt stimt o covinc is, slctd s % of th initil stt stimt. On hundd liztions of filt tht ws sconds long w un. h vg finl stimt of th ffctiv dmping ws ˆ ζ.33 = with cofficint of vition (COV of c ˆv = 9.55%. his psnts.3% o whn compd to th thoticl vlu of th ffctiv dmping computd li. his dmonstts tht th EKF lgoithm is cn ccutly nd consistntly stimt th ffctiv dmping intoducd by th PMD. h thoticl ffctiv dmping fo th ttnutd systm is ζ =.38 bsd on Eqution (. h stimtion of th ffctiv dmping is plottd with tim nd compd to th thoticl vlu in Figu. h mthodology is now xtndd fo unixil multi-dg-of-fdom systms quippd with PMD. x[ + ] = x[ ] + x [ ] + w[ ] x [ ] x[ ] x [ ] x [ ] + = ωn, j +, 3 + φ jd + w[ ] M, j x [ + ] = x [ ] + w [ ] wh ω n, j, φ j, nd, j ( ωn j (3 M th ntul fquncy, mod shp vcto, nd modl mss fo th contolld mod, spctivly. h msumnt qutions z (, x [ ], x3[ ] x[ ] = φ ω ω j n j n j + φ φ d + v j j M, j (4 h pfomnc of th ffctiv dmping stimtion fo th PMD-quippd MDOF systm is dmonsttd nxt using msumnt dt fom full-scl stuctu. 5 EFFECIVE DAMPING ESIMAION FROM FULL- SCALE MEASUREMEN DAA h Apon ow t Pson Intntionl Aipot in oonto, Ontio, Cnd is considd s th tst bd fo dmonstting th lgoithm fo stimting ffctiv dmping intoducd by PMD (Figu 3. h tow iss 49 m bov th tminl oof blow. It is stl stuctu, with igid diphgm floos nd combintion of bcd nd momnt fms to sist ltl lods. h tow consists of six min columns nd sts on sis of lg tnsf gids t th oof lvl of th suppoting tminl. h fundmntl mod of vibtion is ppoximtly.67 Hz in th noth-south (y- diction. Figu. Effctiv dmping stimtion fo SDOF PMDttnutd systm. 4. Effctiv Dmping Estimtion fo MDOF Stuctu With PMD Fo MDOF systm, th ddd compliction is th gt numb of msumnts nd th incsd numb of mods in th stuctu without th PMD. h sttgy is to pfom modl tnsfomtion within th msumnt qution in od to lt th stts (modl displcmnts nd vlocitis fo th contolld mod of vibtion to th msumnts t th dgs of fdom (DOF using th mod shp vcto. hfo, th modl mss nd mod shp vcto fo th contolld mod, in ddition to th ntul fquncy, must b nown pioi. Fo th MDOF systm, th tnsition qutions s follows (using th sm slction of stts s pviously: Figu 3. Apon ow t oonto Pson Intntionl Aipot. Du to th stuctu s inhnt flxibility nd suscptibility to wind lods, it is quippd with pi of PMDs loctd within th tuss oof stuctu (Figu 4. h PMDs w instlld to duc us discomfot du to motion duing high wind vnts. Ech mss is 5, g, psnting mss tio of μ =.4%. 547

6 Pocdings of th 9th Intntionl Confnc on Stuctul Dynmics, EURODYN 4 Figu 4. PMD instlld in oof stuctu of Apon ow. h PMDs suspndd by sis of cbls fom th stuctul stl bov; ch cbl is xctly vticl whn th PMD is t st, pvnting ocing mod nd nsuing th PMD bhvs s point mss. It is gnlly difficult to pdict th ctul fquncis of th b stuctu duing th dsign phs to th dg of ccucy ncssy to tun th PMD. hfo, it is quippd with n djustmnt mchnism to djust th pndulum lngth. h msud lngth ws.57 m. Ech mss is quippd with fou doubl-cting fluid viscous dmps (two in ch hoizontl diction with p dmping foc of 3. N nd mximum sto of 78 mm. h dmping foc is vlocity-squd popotionl; th quivlnt lin viscous dmping cofficint ws clcultd 3 to b c =.3 N m/s [3]. 5. Msumnt Pogm An xtnsiv msumnt pogm ws conductd, wh th stuctu ws instumntd with sismic cclomts long th hight of th stuctu. Svl significnt wind vnts w msud, duing which high-fidlity msumnts w obtind contin ngy in svl mods. h stuctu ws instumntd with PCB Pizotonics high snsitivity sismic cmic flxul ICP cclomts. hs snsos idl fo low-fquncy vibtions nd povid stong output signl with high signl to nois tios. h dt ws msud continuously t smpling t of Hz. n snsos w instlld hoizontlly on th thid nd fist uppmost floos s wll s th top chod of th oof tuss stuctu. h snsos w ngd such tht th ltl motion in ch diction s wll s th ottion bout th vticl xis could b msud. h mining snsos w instlld in ch hoizontl diction on on of th PMDs to gth ltl ccltion spons msumnts of th uxiliy mss. A totl of ppoximtly 3 hous nd minuts of dt w collctd duing significnt wind vnt. h EKF ffctiv dmping lgoithm is pplid sptly in ch spons diction. Sinc th fundmntl mod of vibtion is nown to b in th noth-south (y- diction, only th y-diction spons is considd futh. 5. Filt Initiliztion h ffctiv dmping stimtion hs so f lid on pioi nowldg of th fquncy nd modl mss fo th contolld mod of vibtion. Altntivly, th qutions of motion fo PMD-quippd stuctu cn b cst in such wy tht th modl chctistics of th b stuctu (ntul fquncy, dmping tio, nd mod shp vctos of th stuctu without th PMD cn b stimtd ith in dvnc of o simultnously with th ffctiv dmping. A dtild ttmnt of this ppoch hs bn dvlopd by Roffl [4], nd is usd in th psnt nlysis fo poviding th cunt stimt of th ntul fquncy nd mod shp vcto fo th ffctiv dmping stimtion filt. hfo, two simultnous EKFs un. h fist filt stimts th b stuctu ntul fquncy, dmping tio, nd mod shp. h scond filt uss th cunt stimt of th ntul fquncy nd mod shp vcto fo th b stuctu nd stimts th ffctiv dmping of th combind min nd uxiliy mss systm. A sonbl stimt of th modl mss fo th contolld mod usd thoughout th filt option, s wll s th initil ntul fquncy nd mod shp stimts, ws stblishd by dvloping finit lmnt modl of th stuctu using softw pcg SAP [5]. h initil stimt of th ntul fquncy of th contolld mod is.656 Hz; th initil dmping stimt fo th b stuctu ws slctd s.5%. h nois covinc mtics w stblishd using th stimtion ppoch mntiond li. h dt sts w split into non-ovlpping dt sts ch with totl lngth of minuts. Ech st ws smpld t 4 Hz in od to limit th ffct of th ppoximtion intoducd by th disctiztion ppoch. 5.3 Effctiv Dmping Idntifiction Rsults h fist mod ntul fquncy, vgd ov th liztions of th filt, is shown in Figu 5. h ws sonbly fst convgnc on th finl stimt of f n, =.676 Hz with high dg of confidnc ( c ˆv =.98%. h initil nd finl convgd stimt of th mod shp vcto fo th contolld mod is shown in Figu 6. h finl stimt of th b stuctul dmping tio is ˆ ζ =.8. Figu 5. Ntul fquncy stimt fo th contolld mod of vibtion. 548

7 Pocdings of th 9th Intntionl Confnc on Stuctul Dynmics, EURODYN 4 pfom sttisticl nlysis using vious vnts ov piod of tim in od to btt stblish th pfomnc of th PMD. h poposd ppoch is vy mnbl to this, s th lgoithm cn sily b implmntd in l tim nd cn continuously povid n stimt on th ffctiv dmping of th systm. Figu 6. Initil nd finl stimt of th mod shp vcto fo th contolld mod of vibtion. h ffctiv dmping EKF is un simultnously with th filt stimting th b stuctul modl chctistics. At ch tim stp,, th cunt stimt of th cicul ntul fquncy, ω n, j, nd th mod shp vcto, φ j, usd to updt th nown modl pmts in th ffctiv dmping filt (Equtions 3 nd 4. h initil stimt of th ffctiv dmping is 7.5% ( ˆ ζ, =.75. h nois covinc mtics, S nd R, th sm usd fo th b stuctu modl idntifiction filt. h pfomnc of th ffctiv dmping stimtion, vgd ov th liztions of th filt, is shown in Figu 7. h finl convgd vlu is ˆ ζ =.36 with COV of c ˆv = 8.%. his psnts n incs in dmping intoducd by th PMD of Δ ζ =.88. h sults within th ng of th xpctd pfomnc of PMDs [5]. Figu 7. Estimtion of ffctiv dmping intoducd by th Apon ow PMDs. h vition in th finl convgd stimt of th ffctiv dmping coss th dt sts is du to th fct tht dmping in stuctus is blivd to b dpndnt on th lvl nd ntu of th xcittion. Although th concpt of viscous dmping is indpndnt of th mplitud of th xcittion [], it is usd in this cs to modl n inhntly mo complx phnomnon. his vlu is xpctd to vy dpnding on th tun piod of th spcific wind vnts. A mo comphnsiv ssssmnt of th pfomnc would 6 CONCLUSIONS h EKF lgoithm is n ccut nd libl mns of stimting th ffctiv dmping intoducd by th PMD. Whn th opting fquncy of th ffctiv dmping systm is nown, th o in th stimt ws only.3% whn compd to th thoticl vlu. h lgoithm ws xtndd to stimt ffctiv dmping whn th modl chctistics of th undlying stuctu w unnown. h lgoithm cn b implmntd onlin, nd povid l tim msu of th ffctiv dmping fo th pticul loding vnt. h pimy dvntg of th ppoch is ovcoming issus ltd th lc of libl mns of msuing o infing th displcmnt spons of th ttnutd stuctu. ACKNOWLEDGEMENS h uthos gtful to th Ntul Scinc nd Engining Rsch Council of Cnd (NSERC nd th Ontio Cnts of Excllnc (OCE fo poviding finncil suppot fo this wo. h uthos lso thn th Gt oonto Aipots Authoity (GAA nd Rown Willims Dvis nd Iwin, Inc. (RWDI who sv s industil ptns in this sch. Insights on th pcticl dsign spcts of MDs by Gg hompson, Scott Gmbl, D. Pt Iwin, nd vo Hstt of RWDI gtfully cnowldgd. his wo ws md possibl by th fcilitis of th Shd Hichicl Acdmic Rsch Computing Ntwo (SHARCNE nd Comput/Clcul Cnd [4]. REFERENCES [] J. Simpson nd E. Win, Eds., h Oxfod English dictiony, Clndon Pss, Oxfod, UK, scond dition, 989. [] Buu Intntionl ds Poids t Msus, h intntionl systm of units (SI, BIPM, Pis, Fnc, ighth dition, 6. [3] P. R. Bélng, Estimtion of nois covinc mtics fo lin tim-vying stochstic pocss, Automtic, (3:67-75, 974. [4] Comput/Clcul Cnd, Shd Hichicl Acdmic Rsch Computing Ntwo (SHARCNE,. [5] Computs nd Stuctus Inc., SAP, Vsion..8, 8. [6] S. H. Cndll nd W. D. M, Rndom Vibtions in Mchnicl Systms, Acdmic Pss, 963. [7] G. A. Einic nd L. B. Whit, Robust xtndd Klmn filting, IEEE nsctions on Signl Pocssing, 47(9: , 999. [8] R. R. Ggs nd B. J. Vicy, Optiml dsign of pndulum-typ tund mss dmps, h Stuctul Dsign of ll nd Spcil Buildings, 4(4: , 5. [9] R. E. Klmn, A nw ppoch to lin filting nd pdiction poblms, nsctions of th ASME Jounl of Bsic Engining, 8(Sis D:35-45, 96. [] A. Km nd K. Guly, Dmping in stuctus: its vlution nd ttmnt of unctinty, Jounl of Wind Engining nd Industil Aodynmics, 59(-3:3-57, 996. [] K. C. S. Kwo nd B. Smli, Pfomnc of tund mss dmps und wind lods, Engining Stuctus, 7(9: , 995. [] R. J. McNm, und mss dmps fo buildings, ASCE Jounl of Stuctul Division, 3(9: , 977. [3] G. Pcn, J. B. Mnd, nd S. S. Chn, Fundmntl considtions fo th dsign of non-lin viscous dmps, Ethqu Engining nd Stuctul Dynmics, 8(:45-45,

8 Pocdings of th 9th Intntionl Confnc on Stuctul Dynmics, EURODYN 4 [4] A. J. Roffl, Condition ssssmnt of in-svic pndulum tund mss dmps, Doctol dissttion, Univsity of Wtloo, Wtloo, Ontio,. [5] M. P. Scs nd J. C. Swllow, und mss dmps fo tow nd buildings, Pocdings of th Symposium of Stuctul Engining in Ntul Hzds Mitigtion, , 993. [6] H. n nd C. Y. M, Effct of tund mss dmps on wind inducd spons of tll buildings, Jounl of Win Engining nd Industil Aodynmics, 4(-3: , 983. [7] B. J. Vicy nd A. G. Dvnpot, An invstigtion of th bhvio in wind of th poposd Cntpoint ow in Sydny, Austli, Engining Scinc Rsch Rpot, Univsity of Wstn Ontio, Fculty of Engining Scinc, 97. [8] G. Wlch nd G. Bishop, An intoduction to th Klmn filt, IEEE nsctions on Rlibility,. 55