AEROELASTIC ANALYSIS OF BRIDGES UNDER MULTICORRELATED WINDS: INTEGRATED STATE-SPACE APPROACH

Transcription

1 AEROELASIC ANALYSIS OF BRIDGES UNDER MULICORRELAED WINDS: INEGRAED SAE-SPACE APPROACH By Xinzhong Chn and Ahsan Kam ABSRAC: In this pap, an intgatd stat-spac modl of a systm with a vcto-valud whit nois input is psntd to dsci th dynamic spons of idgs und th action of multicolatd winds. Such a unifid modl has not n dvlopd fo du to a num of innat modling difficultis. h intgatd stat-spac modl is alizd asd on th stat-spac modls of multicolatd wind fluctuations, unstady uffting and slf-xcitd aodynamic focs, and th idg dynamics. Both th quations of motion at th full od in th physical coodinats and at th ducd-od in th gnalizd modal coodinats a psntd. his stat-spac modl allows dict valuation of th covaianc matix of th spons using th Lyapunov quation, which psnts high computational fficincy than th convntional spctal analysis appoach. his stat-spac modl also adds tim domain simulation of multicolatd wind fluctuations, th associatd unstady fquncy dpndnt aodynamic focs, and th attndant motions of th stuctu. h stuctual and aodynamic coupling ffcts among stuctual mods can asily includd in th analysis. h modl also facilitats considation of vaious nonlinaitis of oth stuctual and aodynamic oigins in th spons analysis. An application of this appoach to a long-span cal-stayd idg illustats th ffctivnss of this schm fo a lina polm. An xtnsion of th poposd analysis famwok to includ stuctual and aodynamic nonlinaitis is immdiat onc th nonlina stuctual and aodynamic chaactistics of th idg a stalishd. INRODUCION Aodynamic focs on idgs hav convntionally n modld as th sum of motion-inducd slf-xcitd and windinducd uffting foc componnts. hs a in gnal functions of th gomtic configuations of idg sctions, th incoming wind fluctuations, and th ducd fquncy. In th wind vlocity ang of intst in stuctual dsign, th flow aound luff idg sctions is quit unstady and not amnal to quasi-stady analysis tchniqus, which a only valid at vy high wind vlocitis. h fquncy dpndnt chaactistics of aodynamic focs a gnally dscid in tms of xpimntally quantifid flutt divativs fo th slf-xcitd focs and in tms of admittanc and spanwis cohnc functions fo th uffting focs. Incopoating ths unstady aodynamic chaactistics is ssntial fo an accuat valuation of th focs and th attndant spons of th stuctus. hs chaactistics can asily incopoatd in th fquncy domain analysis famwok with and without th considation of intmod coupling ffcts (Davnpot 96; Scanlan 978a,; ain t al. 996; Katsuchi t al. 999; Chn t al. a). o account fo th stuctual and aodynamic nonlinaitis in th analysis, th quations of motion must cast in th tim domain and solvd using a tim domain schm. Most pvious tim domain studis concning idg uffting spons hav usd quasi-stady assumption whn modling th aodynamic focs. hs assumptions manifst thmslvs y nglcting th fquncy dpndnt flutt divativs, admittanc functions, and ffcts of spanwis colation. his inconsistncy with th fquncy domain appoach has n addssd y Chn t al. (), in which th fquncy dpndnt unstady aodynamic focs a accuatly modld in th tim domain analysis famwok. his tim domain Postdoct. Rs. Assoc., Dpt. of Civ. Engg. and Gological Sci., Univ. of Not Dam, Not Dam, IN Rot M. Moan Pof. and Chai, Dpt. of Civ. Engg. and Gological Sci., Univ. of Not Dam, Not Dam, IN Not. Associat Edito: Gog Dodatis. Discussion opn until Apil,. o xtnd th closing dat on month, a wittn qust must fild with th ASCE Manag of ounals. h manuscipt fo this pap was sumittd fo viw and possil pulication on un 6, ; visd Apil 4,. his pap is pat of th ounal of Engining Mchanics, Vol. 7, No., Novm,. ASCE, ISSN //-4 34/$8. $.5 p pag. Pap No appoach is gadd as a igoous psntation of th fquncy domain analysis of lina polms as long as th aodynamic focs a psntd y ational function appoximations (RFAs) xactly o with an accptal lvl of o. Rcnt dvlopmnts psntd in this pap may viwd as an xtnsion of this tim domain appoach y utilizing a stat-spac modling tchniqu that is ootd in lina systm thoy. Much sach has n pfomd in th aa of lina stat-spac modling of unstady slf-xcitd aodynamic focs in th aonautical fild y using RFA tchniqu (.g., Rog 977; Kapl 98). Among ths schms, Rog s RFA is th most widly utilizd caus of th accuacy, simplicity, and oustnss of th mthod, although diffnt foms of th appoximation, such as th minimum stat (MS) mthod, a availal with a focus on ducing th dimnsions of th augmntd aodynamic stats (Kapl 98). h application of RFAs to luff ody idg aodynamics can found in th psntation of th slf-xcitd focs (Scanlan t al. 974; Lin and Yang 983; Xi t al. 985; Buch and Lin 988; Matsumoto t al. 994; Wild t al. 996; Boonyapinyo t al. 999; Chn t al. a,). h modling of fquncy dpndnt uffting focs is givn in Matsumoto and Chn (996), Matsumoto t al. (996), and Chn t al. (). h modling of multicolatd wind fluctuations in a stat-spac famwok has also n addssd in Goßmann and Walt (98), Suhadjo t al. (99), Matsumoto t al. (996), and Kam (997). his is asd on factoization of th coss-pow spctal dnsity (XPSD) matix of th wind fluctuations. h spctal matix is fist xpssd in tms of RFAs and is thn dcomposd into a tansf function, which is thn utilizd to otain th stat-spac matics asd on th alization of th tansf function. h modling of a matix tansf function and susqunt opations a nontivial fo a lag siz wind fild simulation. In som cass, th mathmatical difficulty and numical o intoducd y th calculation pocdu pcludd th us of this tchniqu to alistic polms (Matsumoto t al. 996). Kam (997) has suggstd som simplifications, ut th appoach mains tdious as attstd y a lack of alistic fd-fowad modling of wind in th litatu. In Kam and Mi (999) and Bnfatllo and Muscolino (999), th stochastic dcomposition tchniqu is utilizd to dcompos a multicolatd andom 4 / OURNAL OF ENGINEERING MECHANICS / NOVEMBER

2 pocss into a st of indpndnt andom supocsss. Fo ach supocss th stat-spac modl is divd, and thn though a tansfomation th stat-spac modl of th oiginal multicolatd pocss is composd. Fo simplification of th stat-spac modling in th oiginal coodinat spac, an ignvcto matix valu at a fixd fquncy is chosn asd on th osvation that th ignvctos of th XPSD matix chang vy slowly with spct to th fquncy. his tchniqu quis th ignvalu analysis of th XPSD matix at ach disct fquncy, which may sult in lag computational ffots fo a lag siz wind-fild simulation. In addition, th assumption of constant ignvctos may intoduc os in som cass dpnding on th spctal matix of wind fluctuations. Chn and Kam () pointd out that, with th tuncation of high mods of wind fluctuations, this stochastic dcomposition tchniqu povids an fficint tool fo statspac modling of wll-colatd andom pocsss. Its ffctivnss in modling pooly colatd andom pocsss is ath limitd, paticulaly, fo psnting high-fquncy wind fluctuations. In this pap, an intgatd stat-spac modl of a multiinput and multioutput systm with a vcto-valud whit nois input is psntd to modl th dynamic spons of idgs und th multicolatd winds. Such a unifid modl has not n dvlopd fo du to a num of innat modling difficultis. his intgatd stat-spac modl is alizd asd on th stat-spac modls of th multicolatd wind fluctuations, th unstady aodynamic focs, and th stuctu. Both th quations of motion at th full od in th physical coodinats and at ducd od in th gnalizd modal coodinats a psntd. h full-od fom is mo appopiat fo nonlina polms y using tim-vaiant systm modls, whas th ducd fom is computationally mo fficint fo th lina polms. An application of this appoach to a long span cal-stayd idg dmonstats its ffctivnss. SAE-SPACE REPRESENAION OF RESPONSE UNDER WINDS h mathmatical modl fo dsciing th spons of wind-xcitd stuctu asd on lina systm thoy is schmatically shown in Fig.. h wind-inducd motions of th stuctu can psntd as th outputs of an intgatd multiinput and multioutput systm xcitd y a vcto-valud whit nois pocss. h multicolatd wind fluctuations a considd as th output of a systm with a vcto-valud whit nois xcitation, whos tansf functions can divd y factoizing thi pow spctal dnsity matix. Similaly, th uffting focs a divd as th output of a systm with wind fluctuations as input. hi tansf functions a dscid in tms of th admittanc function and spanwis cohnc of unstady uffting focs. Similaly, th slf-xcitd focs a modld as th output of a systm with th stuctual spons as input. hi tansf functions a dfind in tms of th flutt divativs. By augmnting th stat-spac quations of stuctual motion with th cosponding stat-spac psntation of th loading componnts and wind fluctuations, as statd aov, an intgatd stat-spac modl is stalishd that synthsizs th unstady chaactistics of multicolatd wind fild, fquncy dpndnt unstady aodynamic focs, and th dynamics of th idg. h intgatd stat-spac modl fo dsciing th spons of a stuctu und winds has sval significant mathmatical advantags. h casting of th ovall systm quations in th stat-spac fomat allows th us of tools asd on lina systm thoy fo spons analysis, optimization, and dsign of activ contol dvics to suppss flutt and uffting. By using this modl, th wind load infomation can incopoatd in a stuctual contol dsign as a fd-fowad link with th potntial to nhanc th contol ffctivnss (Suhadjo t al. 99). In addition, th stuctual and aodynamic coupling ffcts can automatically includd in th computation. Fo lina polms, convntional spctal analysis appoach quis intnsiv computational ffots in th stimation of th tansf function and spons spctal dnsity matix at ach disct fquncy with an intval that must vy small fo idgs with closly spacd natual fquncis. Susqunt intgation of th spctal matix ndd to dtmin th spons covaianc quis additional computational ffot. An intgatd lina tim-invaiant stat-spac modl of th spons with a vcto-valud whit nois input facilitats dict stimation of th covaianc matix of th spons though th Lyapunov quation and allows high computational fficincy. MODELING OF MULICORRELAED WINDS Consid a stuctu psntd y a finit-lmnt disctization. h longitudinal and vtical componnts of wind fluctuations at th cnts of lmnts, W(t), a psntd y a multicolatd andom pocss. hs can psntd as th output of a lina systm with input of a vcto-valud Gaussian whit nois pocss N(t) with a zo man and idntity covaianc matix. In this study, an autogssiv (AR) modl is usd to dsci this lina systm fo accuat modling and simplicity. h AR modl is considd as a spcial FIG.. Intgatd Modling of Dynamic Rspons of Wind-Excitd Stuctu OURNAL OF ENGINEERING MECHANICS / NOVEMBER / 5

3 cas of a gnal autogssiv moving-avag (ARMA) modl (.g., Samaas t al. 985; Mignolt and Spanos 987; Li and Kam 99a,). h AR modl is xpssd as p W(t) = PW(t kt) LN(t) () k k = wh t = tim intval; p = modl od; and P k (k =,,..., p) = cofficint matics satisfying th following Yul- Walk quations: p R W( jt) = R W(( j k)t)p k ( j =,,...,p) () k = wh R W ( jt) (j =,,...,p) = colation matix of th wind fluctuation vcto W(t). In this study, instad of dictly using (), an itativ pocdu is utilizd to dtmin th cofficint matics fo nhancing th computational fficincy (Ianuzzi and Spinlli 987). h colation matix R W ( jt) can valuatd fom th spctal dnsity matix S W ( f ) using th Win-Khintchin lationship: R ( jt) = S ( f )cos( jft) df (3) W and th matix L is givn y th Cholsky factoization: W LL= R (4) p R = R () PR (kt) (5) W k W k = Onc th AR modl is dvlopd, th a sval ways to xpss it in tms of a disct-tim stat-spac fomat; fo xampl, th contollal canonical fom, osval canonical fom, diagonal canonical fom, and odan canonical fom (Ogata 994). In this study, th following contollal canonical fom is usd: wh X (t) =A X (t t) B N(t) (6) w dw w dw W(t) =C X (t) D N(t) (7) dw w dw X (t) I w X w(t) I X w(t) = ; A dw = (8a,) X w( p) (t) I X wp(t) Pp Pp Pp P B dw = ; C dw =[PL p Pp L PL] (8c,d) I D = L (8) dw h quivalnt continuous stat-spac psntation can givn as with th lationship Ẋ = AX B N (9) w w w w W = CX D N () w w w A w t Awt A dw = ; B dw =( I)Aw B w (a,) C = C ; D = D (c,d) dw w dw w Similaly, th stat-spac modling of a multicolatd wind fild asd on a multivaiat ARMA modl can also psntd, ut is omittd h fo th sak of vity. Using such a paamtic appoach fo dsciing andom pocsss not only facilitats fficint simulation, ut it also povids an fficint and oust tool fo th stat-spac modling of andom pocsss compad with altnativ appoachs alludd to ali. MODELING OF UNSEADY BUFFEING FORCES h uffting foc componnts p unit lngth, i.., lift (downwad), dag (downwind), and pitching momnt (nosup), inducd y a sinusoidal wind fluctuation with cicula fquncy, a commonly xpssd (.g., Scanlan 978a,; Chn t al. a,) as u(t) w(t) L (t) = U B CL L (ik) (CL C D) L (ik) u w U U () u(t) w(t) D (t) = U B CD D (ik) (CD C L) D (ik) u w U U (3) u(t) w(t) M (t) = U B CM M (ik) CM M (ik) (4) u w U U wh = ai dnsity; U = man wind vlocity; B = is th idg dck width; C D, C L, C M = static foc cofficints; C L = dc L /d and C M = dc M /d; u and w = longitudinal and vtical wind fluctuations, spctivly; (ik) ( = L u, L w, D u, D w, M u, M w ) dnot th aodynamic tansf functions twn wind fluctuations and uffting focs p unit span, and thi asolut magnituds a fd to as th aodynamic admittanc functions; k = /U is th ducd fquncy; and i = pu imaginay unit. Accodingly, th uffting focs acting on a am lmnt of lngth l can givn y wh F (t) =(U Bl)A (ik)w (5) CL L (ik) L (ik) CL L (ik) L (ik) u u w w A = CD D (ik) D (ik) CD D (ik) D (ik) (6a) u u w w BCM M (ik) M (ik) BCM M (ik) M (ik) u u w w F =[L D M ]; W =[u /U w /U] (6,c) C L = C L; C L = (CL C D); C D = C D (6d f ) C D = (CD C L); C M = C M; C M = C M (6g i) u and w = wind fluctuations at th cnt of th lmnt; supscipt indicats th componnt on th lmnt; supscipt indicats matix tanspos; and (ik) ( = L u, L w, D u, D w, M u, M w ) dnots th joint accptanc functions that dsci th duction ffct of th uffting focs du to th loss of spanwis colation within th lmnt, as compad with th fully colatd cas, and xpssd as l l l = coh (x, x ; f ) dx dx (7) wh coh (x, x ; f ) = cohnc fluctuation of uffting focs; and x and x = coodinats of points and in th 6 / OURNAL OF ENGINEERING MECHANICS / NOVEMBER

4 acoss-wind diction. Fo tow and cal lmnts of calsuppotd idgs, only th dag componnt is gnally considd in th analysis of ovall idg spons. F h stat-spac modling of will accomplishd in two stps. h fist stp is to modl th admittanc function (ik), and th scond stp is to modl th joint accptanc function (ik). hs a xpssd as th following ational function appoximations (Chn t al. ), although oth foms can also utilizd: m j = m j = (ik)a, j (ik) =A, (8) ik d, j (ik)a, j (ik) =A, (9) ik d, j Accodingly, th stat-spac quations fo (Appndix I) as F a thn wittn Ẋ = AX BW () F = CX DW () Basd on th finit-lmnt pocdu, fo an assumd shap function of displacmnt within an lmnt, th nodal focs in th local coodinat systm can xpssd using th statspac modl. Susquntly, th total nodal uffting focs fo th nti stuctu can otaind y tansfoming th nodal focs in th local coodinats to th gloal coodinat systm and assmling th lmnt focs. h total nodal foc F (t) can finally xpssd in tms of stat-spac quations with th input of wind fluctuations, W(t), as Ẋ = AX BW () F = CX DW (3) Whn th uffting focs a simply xpssd using th quasi-stady thoy, i.., = =, th stat-spac modl of uffting focs thn coms wh F = DqsW (4) D qs =(U Bl) CD C D (5) Accodingly, F is xpssd as C BC L M C BC L M F = DqsW (6) Claly, th quasi-stady dsciption of uffting focs liminats th augmntd stats psnting th unstady poptis of th uffting focs. hfo, th cunt statspac modl mo accuatly psnts th fquncy dpndnt unstady chaactistics of th aodynamic focs, which a ssntial fo th spons analysis in oth tim and fquncy domains. An altnativ appoach fo th stat-spac modling of uffting focs is dictly asd on thi XPSD matix and susquntly xpssd in tms of an AR modl, simila to th cas of multicolatd wind fluctuations. Howv, th appoach psntd h is mo consistnt whn considing aodynamic nonlinaitis, such as th dpndncy of aodynamic paamts on th ffctiv angl of wind incidnc. In this cas, wind fluctuations must simulatd to accommodat th considation of th nonlina ffcts. It is woth noting that th spanwis colation of th uffting focs is gnally high than that of th incoming wind fluctuations (.g., Kam 99; Laos and Mann 998). o accuatly dsci th spanwis colation of uffting focs, th spatial poptis of uffting focs and not thos of th incidnt wind may usd in th valuation of th spctal matix of th wind fluctuations modld in th aov sction. his coction to th modl of wind fluctuations gnats what is fd to as th ffctiv wind fluctuations ath than a alization of th oiginal wind fild. In this study, this coction of spatial colation of wind fluctuations is usd to modl th mo complx gnation mchanism of uffting focs, which is cuntly mathmatically intactal. h cospctum twn th wind fluctuations on th cnts of ith and jth lmnts a givn as S ( f ) = coh ij( f ) S ( f )S ( f ) ( = u, w) (7) ij i j li lj ij i j ll i j coh ( f ) = coh (x, x ; f ) dx dx /( ) (8) wh l i and l j = lngths of ith and jth lmnts, spctivly; and and = joint accptanc functions of th ith and jth i j lmnts [(7)]. MODELING OF SELF-EXCIED FORCES h slf-xcitd foc componnts p unit lngth inducd y a sinusoidal motion with cicula fquncy a xpssd in tms of flutt divativs H i, P i, and A i (i =,,..., 6) (.g., Scanlan 978a,; ain t al. 996; Katsuchi t al. 999; Chn t al. a,) as h L s(t) = U () kh kh kh 3 U U h ṗ p kh kh kh U (9) ṗ 3 U U p h h h 3 U U h ṗ p D (t) = U () kp kp kp s kp kp kp U (3) M (t) = U ( ) ka ka ka s ka ka ka U (3) h slf-xcitd focs a commonly assumd to fully colatd in spanwis diction. Although a loss of spanwis colation of th slf-xcitd focs can affct th aodynamic damping and th flutt staility (Scanlan 997), a cnt xpimnt study has potd only slight tuulnc ffcts on slf-xcitd foc colation (Haan t al. 999; Haan ). In this study, full colation of th slf-xcitd focs is assumd. Futh xpimntal invstigation of this issu nds to undtakn. Onc a woking modl fo th spanwis colation of th slf-xcitd focs coms availal, th analysis famwok psntd h can incopoat it convnintly without futh modification. h slf-xcitd focs acting on a am lmnt of lngth l can givn as wh F s(t) = U A s(ik)y (t) A d(ik)y (t) (3) U OURNAL OF ENGINEERING MECHANICS / NOVEMBER / 7

5 k lh 4 k lh 6 k lh 3 s k la 4 k la 6 k la 3 A (ik) = klp k lp k lp (33a) A d(ik) = klp 5 klp klp (33) klh klh 5 klh 5 kla kla k la F s(t) =[L s(t) D s(t) M s(t)] (33c) Y =[h (t) p (t) (t)] (33d) and h, p, and = vtical, latal, and tosional displacmnt at th cnt of th lmnt, spctivly; and th ov-dot dnots th diffntiation with spct to tim. h tansf matix twn th slf-xcitd focs and th stuctual motion can psntd y th following RFA in tms of th ducd fquncy k. his is accomplishd y fitting th taula data H s(ik j) dfind at a st of disct - ducd fquncis k j ( j =,,...) fo which ths tansf function matics a availal (Rog 977; Chn t al. a,): H (ik) =A (ik)a = A (ik)a (ik) A s s d 3 m (ik)a j 3 j = ik d j (34) FULL-ORDER INEGRAED SAE-SPACE MODEL h govning quations of motion with spct to th static quiliium position of a idg a givn in matix fom y MY CY KY = F F (39) wh M, C, and K = mass, damping, and stiffnss matics, spctivly. Sustituting (9), (), (), (3), (37), and (38) into th aov quation, th following stat-spac quations a otaind: wh s Ẋ = AX BN (4) Y = GX (4) X A B C B D C ss ss ss ss w X = X ; A = A B C (4a,) w X A w w BssDDw Css B = BD w ; G = (4c,d) Bw wh A and ; j =,,..., m, A, A 3, A j 3, d j (d j )= fquncy indpndnt matics and paamts; and m is th od of RFA. Rplacing th Foui tansfom y a Laplac tansfom though analytic continuation with s [wh s =( i)k, and = damping atio of th motion) sustitutd fo ik, and y taking an invs Laplac tansfom, th slf-xcitd focs inducd y an aitay motion can xpssd in tms of th following stat-spac quations: du j sj sj j3 s 3 m X sj(t) X (t) = X (t) A Y (t) (j =,,...,m ) (35) F (t) = U AY(t) AY(t) AY(t) U U j = (36) wh X ( j =,,..., m sj ) = augmntd nw vaials psnting th aodynamic stats. Basd on th finit-lmnt pocdu, th total slf-xcitd focs can finally xpssd in tms of th nodal motion Y as du j X sj(t) = X sj(t) Aj3Y(t) (j =,,...,m) (37) m s 3 sj F (t) = U AY(t) AY(t) AY(t) X (t) U U j = (38) wh A, A, A 3, A j 3, and d j (d j ; j =,,..., m) = fquncy indpndnt matics and paamts; m is th od of RFA; and X sj (t) (j =,,..., m) = augmntd aodynamic stats. It is woth mntioning that, whn th slf-xcitd focs a modld using th quasi-stady thoy, only matics A and A a includd in th modl, thus liminating th augmntd aodynamic stats X sj ( j =,,...,m). Y Ẏ X ss = X s (4) X sm I M K M C U M U M A = U ss A4 di U A3m dmi (4f ) B ss =[ M... ] (4g) C ss =[I... ] (4h) M = M A 3; C = C UA (4i, j) K = K U A 3 (4k) h solution of aov quation can otaind (Soong and Gigoiu 993) y t A(tt ) A(t) X(t) = X(t ) BN() d (43) t h pvious quation in a disct fom is givn y At At X(t) = X(t t) ( I)A BN(t) (44) wh t = tim intval. 8 / OURNAL OF ENGINEERING MECHANICS / NOVEMBER

6 h covaianc matix R X can dictly calculatd y solving th following Lyapunov quation: Ṙ X = ARX RXA BB (45) Fo th tim-invaiant cas, it ducs to ARX RXA BB = (46) REDUCED-ORDER SAE-SPACE MODEL Fo lina stuctus, ducd-od quations of motion in tms of th gnalizd modal coodinats q can utilizd fo computational convninc: Mq Cq Kq= Q Q (47) s wh M = M, C = C, and K = K = gnalizd mass, damping, and stiffnss matics, spctivly; Q s = F s and Q = F a th gnalizd slf-xcitd and uffting foc vctos, spctivly; and = mod shap matix. h stat-spac quations of Q can givn as follows asd on th stat-spac modl of F [() and (3)]: Ẋ = AX BW (48) Q = C X D W (49) wh C = C, D = D. h stat-spac quations of Q s can givn as follows asd on th stat-spac modl of F s [(37) and (38)]: du j q sj (t) = q sj (t) Q j3 q (t) (j =,,...,m) (5) m s 3 sj Q (t) = U Qq(t) Qq (t) Qq (t) q (t) U U j = (5) wh Q = A ; Q = A ; Q 3 = A 3 ; Q j3 = A and q sj (t) = j3; X sj (t) (j =,,...,m). An altnativ appoach fo modling th gnalizd slfxcitd focs is to dictly fit th gnalizd modal aodynamic matics calculatd at disct ducd fquncis. If a small num of lag tms can thus otaind, it will lad to ducd aodynamic stats. Accodingly, th intgatd stat-spac modl of th systm is givn y wh Ẋ = AX BN (5) Y = GX (53) X A B C B D C ss ss ss ss w X = X ; A = A B C (54a,) w X A w B D D C ss w ss B = BD ; G = (54c,d) w B w q X ss = q q s (54) q sm w I M K M C U M U M A = U ss Q4 di U Q3m dmi (54 f ) B ss =[ M... ] (54g) C ss =[I... ] (54h) M = M Q 3; C = C UQ (54i,j) 3 K = K U Q (54k) Whn considing a lina aodynamic polm, th statspac modling of Q can futh simplifid using a multivaiat AR modl with a vcto-valud whit nois input N, which can divd asd on th XPSD matix of Q. Accodingly, it can xpssd as Ẋ = A X B N (55) Q = C X D N (56) and th intgatd stat-spac quations of th systm a wh EXAMPLE Ẋ = AX BN (57) Y = G X (58) Xss Ass BssC X = ; A = (59a,) X A BssD Css B = ; G = (59c,d) B In this sction an xampl is psntd to illustat th intgatd stat-spac analysis famwok dvlopd in this study. h xampl idg is a cal-stayd idg with a main span of appoximatly, m. Fo simplicity and without loss of gnality, only th aodynamic focs acting on th idg dck w considd. h von Kaman spcta w usd fo dsciing th pow spcta of th u and w componnts of wind fluctuations. Fo u and w componnts, tuulnc intnsitis and intgal lngth scals qual to and 7.5%, and 8 and 4 m, spctivly, w considd. High lngth scals w usd in th valuation of th cohnc function of wind fluctuations in od to account fo th stong spanwis colation in th uffting focs than thos of th wind fluctuations. Lngth scals w chosn as 6 and 8 m fo th uffting foc componnt associatd to th u and w componnts, spctivly, although ths can dtmind asd on wind tunnl tsts (.g., Laos and Mann 998). h idg dck was disctizd into 43 lmnts along th span. In this study, th u and w componnts w assumd to OURNAL OF ENGINEERING MECHANICS / NOVEMBER / 9

7 FIG.. Compaison of Simulatd and agt Auto- and Coss-Pow Spcta of Wind Fluctuations (U = 6 m/s): (a) Dck Elmnt, u Componnt; () Dck Elmnts and, u Componnt; (c) Dck Elmnt, w Componnt; (d) Dck Elmnts and, w Componnt indpndnt; thfo, ths can xpssd using two spaat stat-spac modls. h modling is staightfowad fo th cass considing th colation twn u and w componnts of wind fluctuations. Spaating wind fluctuations and cosponding uffting focs and sponss into two goups associatd to th u and w componnts will lad to computational fficincy ov a comind psntation, caus computational ffots latd to th tatmnt of coss tms twn th u and w componnts a liminatd. In this xampl, two stat-spac modls with 58 stats ach w usd fo th u and w componnts of wind fluctuations. Fig. shows th compaison of th pow spcta and cossspcta of wind fluctuations at lmnts and. h solid and dashd lins a th tagt and th calculatd valus fom th stat-spac modl, spctivly. h sults show an xcllnt agmnt, which dmonstats th accuacy of th stat-spac modl. Fo ach lmnt, diffnt admittanc functions fo u and w componnts w usd, which w all xpssd using RFAs with two ational tms. Davnpot s fomula with a dcay facto of 8 was usd fo dag, and th Sas function was usd fo lift and pitching momnt. wo diffnt joint accptanc functions w usd fo uffting foc componnts associatd with u and w componnts. hs w also xpssd using RFAs with two ational tms. h dimnsions of th stat-vcto fo th uffting focs acting on ach lmnt and th ovall stuctu w 6 and 58, spctivly, and w th sam fo oth componnts cosponding to u and w componnts of wind fluctuations. Fig. 3 shows th compaison of th pow spcta of th F uffting focs acting on lmnt. h dashd and solid lins psnt th sults fom th stat-spac modl and th spctal analysis, which is givn as follows, asd on (5): S (ik) =(U Bl) A (ik)s F W A (ik) (6) h dag componnt of th slf-xcitd focs du to latal motion was valuatd asd on th quasi-stady thoy, and th componnts lvant to th vtical and tosional motions w nglctd. h lift and pitching momnt componnts of th slf-xcitd focs w calculatd asd on th hodosn function. h gnalizd slf-xcitd focs on th fist idg dck dominatd natual mods w xpssd using RFA with two ational tms. A ducd-od stuctual modl was usd. h natual fquncis ang fom.7 to.6 Hz. h logaithmic dcmnt fo ach mod was assumd to.. h total dimnsion of th stat-vctos of th intgatd systm cosponding to u and w componnts w oth qual to 564. Fo ach of ths two intgatd systms, th covaianc matix was calculatd using th Lyapunov quation to otain th covaianc of th total spons. Fig. 4 shows th compaison of th oot man squa (RMS) uffting spons along th span in th vtical, latal, and tosional dictions at man wind vlocitis of 4, 6, and 8 m/s. h dashd lins and th dots a th sults fom th spctal analysis and th psnt appoach, spctivly. Rsults indicat that th stat-spac modl appoach givs sults that a vy clos to thos fom convntional appoach, whil th stat-spac modl is computationally mo fficint. Fo th xampl psntd h, computational ffot using th poposd schm is lss than half of that ndd fo convntional spctal appoach. Basd on th intgatd stat-spac modl, th wind fluctuations, th associatd aodynamic focs, and th uffting spons can simulatd in th tim domain y using Mont Calo simulation. Sampl alizations of this simulation a shown in Fig. 5, which psnt th u and w componnts of wind fluctuations at th midpoint of th main span, th dag, lift, and momnt componnts of th uffting and slf-xcitd focs acting on th am lmnt at th cnt of th main span, and th uffting spons in th latal, vtical, and tosional dictions at th midpoint of th main span. It is notd that th uffting dag foc is mainly contiutd y th u componnt of wind fluctuations, whil th uffting lift and pitching momnt a mainly contiutd y th w com- 3 / OURNAL OF ENGINEERING MECHANICS / NOVEMBER

8 FIG. 3. Compaison of Pow Spcta of Buffting Focs Acting on Dck Elmnt (U = 6 m/s): (a) Dag Componnt; () Lift Componnt; (c) Momnt Componnt FIG. 4. Compaison of Buffting Rsponss: (a) Latal Displacmnt; () Vtical Displacmnt; (c) osional Displacmnt ponnt of wind fluctuations. It is also notd that th vtical and tosional spons xhiit coupling. h citical flutt vlocity was found to 3.8 m/s utilizing a staility analysis of this intgatd systm though th solution of th complx ignvalu polm. CONCLUDING REMARKS An intgatd stat-spac modl of a multiinput and multioutput systm with a vcto-valud whit nois input was psntd to dsci th dynamic spons of idgs und multicolatd winds (Fig. ). h stat-spac modling of multicolatd winds usd an AR modl, and th modling of unstady uffting and slf-xcitd focs was dvlopd using ational function appoximations of thi fquncy dpndnt chaactistics. h poposd appoach hlps to glan a cla insight into th modling of wind-inducd viation polms. It gins with a vcto of whit nois, which is succssivly tansfomd to colatd wind fluctuations, aodynamic focs, and th associatd stuctual motions. his appoach facilitats th us of tools asd on lina systm thoy fo th spons analysis and stuctual contol dsign. his pocdu allows th tim domain simulation of th spons to incopoat th fquncy dpndnt unstady aodynamic focs instad of invoking th gnally assumd quasi-stady focs. his novl fatu nhancs th accuacy of th pdictd sponss. his famwok can utilizd in a stuctual contol dsign y incopoating th wind loading infomation as a fd-fowad link, which has th pomis to impov th ffctivnss of contol. Dict calculations of th covaianc matix of spons using th Lyapunov quation offs high computational fficincy in compaison with convntional spctal analysis appoach. h ichnss of this analysis famwok lads immdiatly to th nxt lvl of analysis, i.., it can simply xtndd to th analysis of idgs/stuctus with stuctual and aodynamic nonlinaitis y using a tim-vaiant systm modl onc th nonlina stuctual and aodynamic chaactistics of a idg a stalishd. Both th nonlina ffcts and unstady fquncy dpndnt chaactistics of aodynamic focs can accuatly captud using this schm. Dtails will psntd in a futu psntation. Although mphasis in this study was placd on th spons of idgs und multicolatd wind xcitation, th poposd appoach offs immdiat applications to oth wind-xcitd stuctus as wll as systms with fquncy dpndnt paamts, such as thos involvd in soil-stuctu and fluid-stuctu intactions. OURNAL OF ENGINEERING MECHANICS / NOVEMBER / 3

9 FIG. 5. Simulation of Wind Fluctuations, Buffting and Slf-Excitd Focs and Bidg Rspons (U = 6 m/s, at Midpoint of Main Span): (a) Wind Fluctuations; () Buffting Focs; (c) Slf-Excitd Focs; (d) Bidg Rspons APPENDIX I. SAE-SPACE REPRESENAION OF BUFFEING FORCES F h stat-spac quations fo th uffting focs givn as Gnal Cas F a Ẋ = AX BW (6) F = CX DW (6) Assuming th admittanc function and th joint accptanc ( = L u, L w, D u, D w, M u, M w ) a uniqu fo ach uffting foc componnts. Fo ach lmnt, totally six admittanc functions and six joint accptanc functions nd to xpssd in tms of ational functions. hn, th matics of th stat-spac modl a A = diag[al AD AM AL AD A M ] (63) u u u w w w BL BD BM u u u B = (64) BL BD B w w Mw C =(U Bl) CLCL CLCL u w CDCD CDCD u w C C C C M Mu M M w (65) Spcial Cas CLDL CLD u Lw D =(U Bl) CDDD CDD D (66) u w CMDM CMD u Mw A B A = ; B = (67a,) BC A BD C =[DC C]; D = DD (67c,d) A = diag[d,u/... d,m U/] (68a) B =[A... A,,m ] (68) C =[d,u/... d,m U/] (69a) m,, j j = D = A A (69) A = diag[d U/... d,,m U/] (7a) B =[A... A,,m ] (7) C =[d U/... d,,m U/] (7a) m D = A, A, j j = (7) Consid a spcial cas 3 / OURNAL OF ENGINEERING MECHANICS / NOVEMBER

10 L = L = M = M = LM; D = D = D (7a,) u w u w u w L = D = M = u; L = D = M = w (73a,) u u u w w w In this cas, only two admittanc functions and two joint accptanc functions nd to xpssd in tms of ational functions. h ational function appoximations of LM, D, u, and w a givn y m (ik)a, j (ik) =A, ( = LM, D) (74) j = ik d, j m (ik)a, j (ik) =A, ( = u, w) (75) j = ik d, j and th matics of th stat-spac modl a givn y A = ALM BLMCu AD BDCu A u ALM BLMCw AD BDCw A w B = B D LM u D u B u B D B D LM w D w B D B w CLCLM CLCLM (76a) (76) C =(U Bl) CDCD CDCD BCMCLM BCMCLM C D C D L LM L LM (77) D =(U Bl) CDDD CDD D (78) BCMD LM BCMD LM A = diag[d,u/... d,m U/] (79a) B =[A... A,,m ] (79) C =[d,u/... d,m U/] (8a) m,, j j = D = A A = LM, D) (8) A = diag[d U/... d,,m U/] (8a) ACKNOWLEDGMENS B =[A... A,,m ] (8) C =[d U/... d,,m U/] (8a) m D = A, A, j ( = u, w) (8) j = h suppot fo this wok was povidd in pat y NSF Gants CMS 9496, CMS , and CMS his suppot is gatfully acknowldgd. h wits a thankful to D. Fd Haan,., fo his commnts on th manuscipt. REFERENCES Bnfatllo, S., and Muscolino, G. (999). Filt appoach to th stochastic analysis of MDOF wind-xcitd stuctus. Poailistic Engg. Mch., 4, 3 3. Boonyapinyo, V., Miyata,., and Yamada, H. (999). Advancd aodynamic analysis of suspnsion idgs y stat-spac appoach.. Stuct. Engg., ASCE, 5(), Buch, C. G., and Lin, Y. K. (988). Stochastic staility of idgs considing coupld mods.. Engg. Mch., ASCE, 4(), Chn, X., and Kam, A. (). On th application of stochastic dcomposition in th analysis of wind ffcts. Poc., Int. Conf. on Advancs in Stuct. Dyn., Hong Kong, Chn, X., Matsumoto, M., and Kam, A. (a). Aodynamic coupling ffcts on th flutt and uffting of idgs.. Engg. Mch., ASCE, 6(), 7 6. Chn, X., Matsumoto, M., and Kam, A. (). im domain flutt and uffting spons analysis of idgs.. Engg. Mch., ASCE, 6(), 7 6. Davnpot, A. G. (96). Buffting of a suspnsion idg y stom winds.. Stuct. Engg., ASCE, 88(3), Goßmann, E., and Walt, H. B. (983). Analysis of multi-colatd wind-xcitd viations of stuctus using th covaianc mthod. Engg. Stuct., 5, Haan, F. L. (). h ffcts of tuulnc on th aodynamics of long-span idgs. PhD thsis, Univsity of Not Dam, Not Dam, Ind. Haan, F. L., Kam, A., and Szwczyk, A. A. (999). Influnc of tuulnc on th slf-xcitd focs on a ctangula coss sction. Wind Engg. into th st Cntuy: Poc., th Int. Conf. on Wind Engg., Balkma, Rottdam, h Nthlands, Ianuzzi, A., and Spinlli, P. (987). Atificial wind gnation and stuctual spons.. Stuct. Engg., ASCE, 3(), ain, A., ons, N. P., and Scanlan, R. H. (996). Coupld flutt and uffting analysis of long-span idgs.. Stuct. Engg., ASCE, (7), Kapl, M. (98). Dsign fo activ flutt suppssion and gust allviation using stat-spac aolastic modling.. Aicaft, 9(3), 7. Kam, A. (99). Masumnts of pssu and foc filds on uilding modls in simulatd atmosphic flows.. Wind Engg. and Indust. Aodyn., 36, Kam, A. (997). Modling of as-isolatd uildings with passiv damps und winds.. Wind Engg. and Indust. Aodyn., 7, Kam, A., and Mi, G. (999). Stochastic dcomposition fo simulation and modal spac duction in wind inducd dynamics of stuctus. Application of statistics and poaility, Balkma, Rottdam, h Nthlands, Katsuchi, H., ons, N. P., and Scanlan, R. H. (999). Multimod coupld flutt and uffting analysis of th Akashi-Kaikyo Bidg.. Stuct. Engg., ASCE, 5(), 6 7. Laos, G. L., and Mann,. (998). Gust loading on stamlind idg dcks.. Fluids and Stuct.,, Li, Y., and Kam, A. (99a). ARMA systm in wind ngining. Poailistic Engg. Mch., 5(), Li, Y., and Kam, A. (99). Rcusiv modling of dynamic systms.. Engg. Mch., 6(3), Lin, Y. K., and Yang,. N. (983). Multimod idg spons to wind xcitations.. Engg. Mch., ASCE, 9(), Matsumoto, M., and Chn, X. (996). im domain analytical mthod of uffting sponss fo long-span idgs. Poc., 4th Nat. Symp. on Wind Engg., apan Association fo Wind Engining, 55 5 (in apans). Matsumoto, M., Chn, X., and Shiaishi, N. (994). Buffting analysis of long-span idg with aodynamic coupling. Poc., 3th Nat. Symp. on Wind Engg., apan Association fo Wind Engining, 7 3 (in apans). Matsumoto, Y., Fujino, Y., and Kimua, K. (996). Wind-inducd gust spons analysis asd on stat spac fomulation.. Stuct. Mch. and Eathquak Engg., 543/I-36, (in apans). Mignolt, M. P., and Spanos, P. D. (987). Rcusiv simulation of stationay multivaiat andom pocsss. Pat I.. Appl. Mch., 9, Ogata, K. (994). Disct-tim contol systms, Pntic-Hall, Englwood Cliffs, N.. Rog, K. L. (977). Aiplan math modling mthods fo activ contol dsign. Stuctual Aspcts of Activ Contols, AGARD-CP-8, OURNAL OF ENGINEERING MECHANICS / NOVEMBER / 33

11 Samaas, E., Shinozuka, M., and suui, A. (985). ARMA psntation of andom pocsss.. Engg. Mch., ASCE, (3), Scanlan, R. H. (978a). h action of flxil idgs und wind. I: Flutt thoy.. Sound and Viation, 6(), Scanlan, R. H. (978). h action of flxil idgs und wind. II: Buffting thoy.. Sound and Viation, 6(),. Scanlan, R. H. (997). Amplitud and tuulnc ffcts on idg flutt divativs.. Stuct. Engg., ASCE, 3(), Scanlan, R. H., Blivau,.-G., and Budlong, K. S. (974). Indicial aodynamic functions fo idg dcks.. Engg. Mch. Div., ASCE, (4), Soong,.., and Gigoiu, M. (993). Random viation of mchanical and stuctual systms, Pntic-Hall, Englwood Cliffs, N.. Suhadjo,., Spnc, B. F.,., and Kam, A. (99). Fquncy domain optimal contol of wind-xcitd uildings.. Eng. Mch., ASCE, 8(), Wild, K., Fujino, Y., and Masukawa,. (996). im domain modling of idg dck flutt.. Stuct. Engg., okyo, 3, / OURNAL OF ENGINEERING MECHANICS / NOVEMBER