PASSIVE AERODYNAMIC CONTROL OF WIND INDUCED INSTABILITIES IN LONG SPAN BRIDGES

Transcription

1 WILDE KRZYSZTOF PASSIVE AERODYNAMIC CONTROL OF WIND INDUCED INSTABILITIES IN LONG SPAN BRIDGES POLITECHNIKA GDANSKA monografi 3

2 GDAŃSK UNIVERSITY OF TECHNOLOGY WILDE KRZYSZTOF PASSIVE AERODYNAMIC CONTROL OF WIND INDUCED INSTABILITIES IN LONG SPAN BRIDGES GDAŃSK

3 PRZEWODNICZĄCY KOMITETU REDAKCYJNEGO WYDAWNICTWA POLITECHNIKI GDAŃSKIEJ Zbigniw Cywiński REDAKTOR Andrzj Małasiwicz RECENZENCI Zbigniw Cywiński Kazimirz Szmidt PROJEKT OKŁADKI Jolanta Ciślawska Wydano za zgodą Rktora Politchniki Gdańskij Wydawnictwa PG można nabywać w księgarni PG (Gmach Główny) i zamawiać listowni lub pocztą lktroniczną pod adrsm: Wydawnictwo Politchniki Gdańskij ul. G. Narutowicza /, 895 Gdańsk, tl. (58) mail: wydaw@pg.gda.pl, PG Copyright by Wydawnictwo Politchniki Gdańskij Gdańsk ISBN WYDAWNICTWO POLITECHNIKI GDAŃSKIEJ Wydani I. Ark. wyd. 8,8. Ark. druku 9,5 Zamówini nr 4/ Druk: Zakład Poligrafii Politchniki Gdańskij ul. G. Narutowicza /, 8-95 Gdańsk, tl

4 CONTENTS Notation Introduction Wind loads on bridgs Mthods of analysis of arolastic rspons of bridgs Wind tunnl xprimnts Analytical mthods Supprssion of wind inducd vibration of long span bridgs Aim and scop of study Study on sctional modls of passiv arodynamic control systms Bridg-surfacs control systm Mathmatical modl of arodynamic forcs Equation of motion of bridg-surfacs control systm Numrical simulations Uncontrolld bridg Control systm Control systm Bridg flaps control systm Mathmatical modl of arodynamic forcs Equation of motion of bridg-flaps control systm Numrical simulations Uncontrolld bridg Control systm Control systm Conclusions Wind tunnl xprimnts on sctional modls of arodynamic control systms Exprimnt on bridg surfacs control systm Dscription of xprimnt Exprimntal rsults Exprimnt on bridg flaps control systm Dscription of xprimnt Exprimntal rsults Conclusions Study on thr dimnsional FEM modls of control systms Long span suspnsion bridg FEM modl of th bridg Numrical simulations Bridg surfacs control systm FEM modl Numrical simulations Control systm Control systm Bridg flaps control systm FEM modl Numrical simulations Control systm Control systm

5 4 Contnts 4.4. Conclusions Concluding Rmarks Acknowldgmnts Rfrncs Appndix A Appndix B Appndix C Abstract in English Abstract in Polish

6 NOTATION,, tosional displacmnt of bridg dck a, b, c, d,, f, l, m variabls dfining gomtry of wing-ailron-tab systm a, b, a hd, b hd, a vd, b vd, a hc, b hc, paramtrs for valuation of damping a, a i (i =,..., n ) cofficints of approximation of Thodorsn s function a h distanc btwn th cntr of rotation of th dck and th nd of hangr cabl A i (i =,..., 4) fluttr drivativs of arodynamic momnt A stat spac cofficint matrix angular displacmnt of pndulum; torsional displacmnt of lading flap b, b, b half chord width of th bridg-flaps systm, width of lading and trailing flap b c half of th distanc btwn main cabls b i, b i,i lag cofficints of rational function approximation B d, B, B width of th dck and control surfac c, c control gains c h, c, c, c damping cofficints of having, pitching and flap motions C(s) Thodorsn s function C D, C L, C M drag, lift, momnt stady forc cofficint C Dc stady drag cofficint of cabl C damping matrix C a matrix of arodynamic damping C a,i (i =,) matrix of arodynamic damping of lading and trailing surfac C a lmnt arodynamic damping matrix C s matrix of structural damping C, C, C, C damping matrics corrsponding to horizontal, vrtical, torsional shd svd s shc displacmnt of th dck and motion of main cabls d, d, d placs of connction of surfacs to th pndulum d c diamtr of main cabls DCk ( ( ), Ck ( )) distanc btwn points of curvs C(k) and Ck ( ) D c D d, D, D D r, L r, M r D D,i (i =,) D sc, D E, c, c drag forc on main cabl drag forc on bridg dck and control surfacs rlativ drag, lift and momnt forc matrix of rational function approximation of dck forcs matrix of rational function approximation of surfacs forcs global cofficint matrics of rational function approximation approximation rror matrix of rational function approximation position of hing lins of control surfacs with rspct to th cntr of th bridg dck location of th hings of flaps with rspct to th cntr of gravity of th bridg dck

7 6 Notation E,i (i =,) matrix of rational function approximation of surfacs forcs E a, R b, E b global matrics of cofficints of rational function approximation E a, R b, E b matrics of cofficints of rational function approximation for finit lmnt of arodynamic forcs rlativ angl of attack h, phass of having and pitching motion h, h inclination angls of hangr cabls Lxp, Mxp phass of xprimntally obtaind arodynamic lift and momnt F dissipation function F total arodynamic forcs acting on th bridg dck F d, F, F vctor arodynamic forcs on dck and control surfacs F s vctor of static arodynamic forcs torsional displacmnt of trailing flap h, h, h vrtical displacmnt of bridg dck and lading and trailing surfacs h,, amplituds of xcitation of having, pitching and rotation of lading flap h d, h c, h c horizontal displacmnt of dck and main cabls H dpth of bridg dck; tnsion forc in main cabls H, H i (i =,, 3) incrmnt in horizontal tnsion forcs H i (i =,..., 4) fluttr drivativs of lift forc H () n (i =,) Hnkl functions I mass momnt of inrtia I idntity matrix inclination of cabl lmnt k h, k, k, k stiffnss cofficints of having, pitching and flap motions k low, k up lowr and uppr bound for approximation of Thodorsn s function K, k rducd frquncy K a matrix of arodynamic stiffnss K a,i (i =,) matrics of arodynamic stiffnss of lading and trailing surfac K a lmnt arodynamic stiffnss matrix K b, K cc, stiffnss matrics du to supporting systm K sh cabl lmnt structural stiffnss matrix du to gomtric stiffnss of hangr cabls K sd lmnt structural stiffnss matrix K sc cabl lmnt gravity stiffnss matrix K s K sup i matrix of structural stiffnss stiffnss matrix du to supporting bam ignvalus L ordinary matrix diffrntial oprator L L ( i,, 3) lngth of cabl of i-th span i L xp, M xp L th, M th L d, L, L amplituds of xprimntally obtaind arodynamic lift and momnt amplituds of lift and momnt dtrmind thortically lift forc on bridg dck and control surfacs

8 Notation 7 L xp, M xp xprimntally obtaind arodynamic lift and momnt l h lngth of hangr cabl L h, M, M, M lift and momnt forcs acting of bridg-flaps systm L i (i =,, 3) lngths of sid and main spans L, U lowr and uppr constrains for optimization of lag cofficints M mass matrix m mass pr unit lngth M, M prstrssing momnts of control cabls M d, M, M lift forc on bridg dck and control surfacs M a lmnt arodynamic mass matrix M s lmnt structural mass matrix M ij, w ij wighting factor of approximation function M s matrix of structural mass N shap function matrix of structural lmnt N a shap function matrix for arodynamic stats n numbr of lag cofficints N c, N c shap function matrics for cabl lmnts p dimnsionlss Laplac variabl P vctor of wing-ailron-tab systm P i (i =,..., 4) fluttr drivativs of drag forc air dnsity Q matrix of cofficints of unstady arodynamic forcs q indpndnt dgrs of frdom of dck surfacs systm q, q vctors of absolut displacmnts of lading and trailing surfacs q d vctor of dck s displacmnts q vctor of lmnt structural dgrs of frdom q a vctor of nodal arodynamic stats of finit lmnt q f vctor of rlativ displacmnts of flaps q h, q, q, q displacmnts of wind-ailron-tab systm q t vctor of displacmnts of wind-ailron-tab systm R matrix of lag cofficints; matrix of gomtry of wind-ailron-tab combination r global dispalcmnt vctor structural dgrs of frdom s Laplac variabl S, S, S first momnt of inrtia of th dck, lading flap and trailing flap S, S coordinat transformation matrics; matrics of gomtry of windailron-tab systm S d coordinat transformation matrix t tim T, T q coordinat transformation matrix T d, T c, T c kintic nrgy of structural finit lmnt du to dck and main cabls T p priod of pndulum U man wind spd u c, h c, v c, longitudinal, horizontal and vrtical displacmnt of th cabl divrgnc wind spd U d

9 8 Notation U f fluttr wind spd U r rlativ wind spd V, V, V matrics of dynamic prssur V c potntial nrgy of cabls V potntial nrgy of cabls arising from gravity V c strain potntial nrgy of cabls V h cabl lmnt potntial nrgy du to stiffnss proprtis of hangr cabls v d, v c, v c vrtical displacmnt of dck and main cabls V d, V c, V c potntial nrgy of structural finit lmnt du to dck and main cabls v i ignvctors circular frquncy of oscillation x horizontal displacmnt of bridg dck X vctor of global nodal forcs x a vctor of arodynamic stats x c, x c points of connction of control cabls to th supporting bam x displacmnt vctor of lmnt cross-sction x a vctor of arodynamic stats of sction of bridg dck X sc vctor of forcs of lmnt cross-sction X U vctor of displacmnt indpndnt static wind forcs damping ratio y cabl dflction

10 Chaptr INTRODUCTION.. Wind loads on bridgs Bridgs hav always bn th gratst challng for nginrs. Construction of bridgs with longr span and nw structural and artistic solutions mark th progrss in civil nginring knowldg. Connctions cratd ovr th rivr banks, vallys or islands ar not only a practical achivmnt to hlp popls livs but also carry a philosophical aspct of ovrcoming th problm by bridging two sids. Rapid tchnological progrss, dvlopmnt of nw matrials and know-how using intllignt structurs hav cratd th possibility of xtnding th bridg structur up to an xtrm lngth. Today considrd bridg structurs my rach spans of 5 m (Gibraltar Strait Bridg) [94] or vn 3 km (Bridg btwn Hokkaido island, Japan, and Sachalin, Russia). With th span incras th structur bcoms vry flxibl and thrfor snsitiv to wind actions. Th suscptibility to wind ffcts bcoms, most oftn, th fundamntal critrion for th dsign of long span bridgs ([, 6). In th dsign of a wind rsistant bridg structur svral typs of wind xcitation hav to b invstigatd.. Air flow gnrats static arodynamic drag, lift and momnt forc on th bridg lmnts. Forcs acting on such flxibl structur (frquncy of fundamntal mod of a long span suspnsion bridg might b lowr than. Hz) can induc torsional divrgnc or latral buckling [6]. Th instability occurs whn th dformation of th bridg undr wind action incrass th wind forcs du to modification of th gomtry of th structur. This, in turn, dforms th structur furthr causing incras of th arodynamic forcs. Finally, a vlocity is rachd at which th magnitud of th wind-inducd forcs crats an unstabl condition and th structur twists and/or bnds to dstruction.. Bridg girdr, pylon lg or suspnsion cabls immrsd in th air flow can produc strong vortx-wak. Th shdding of vortics might b mor or lss rgular dpnding on th wind spd, cross-sction s shap and siz of th bridg lmnt. Approximatd formula ([8, 6]) indicat that shdding priod, and also priod of th arodynamic forc ar invrsly proportional to th wind spd. In rsonant conditions th oscillation of th structur control th rhythm of vortx shdding and harmonic oscillations of limitd amplitud can b xpctd to occur in th multipl rgims of wind spd. For long span bridg dcks, th lowst of ths rgions ar for wind spd of about m/s [8]. Th bridg rspons du to vortx shdding is not likly to caus th failur of a bridg in a short tim. Howvr, th fatigu damag may occur and th bridg movmnts may b unaccptabl to bridg usrs. 3. Stady wind gnrats on bridg girdr so calld slf-xciting arodynamic forcs associatd with an arolastic instability calld fluttr [48]. In th classical fluttr th torsion and vrtical bnding of th girdr is coupld through th unstady arodynamic forcs [55]. At crtain wind spd th coupld motion draws nrgy from th air flow so

11 Introduction that damping of on mod bcoms zro. At this wind spd, calld fluttr wind vlocity, bridg undrgos sustaind priodic vibrations. At wind spd highr than fluttr vlocity bridg oscillats with rapidly incrasing amplituds. Fluttr lads to th collaps of th bridg structur. 4. Fluctuations in th incoming wind xcit flxibl bridg structur with a considrabl powr (.g., [9, 3, 3]). This phnomnon is associatd with obsrvation that th pak portion of spctrum of natural wind turbulnc is concntratd at rlativly low frquncy rang and th magnitud of spctral dnsity dcrass rapidly with th incrasing frquncy. Th forcd movmnts of th structur du to turbulnc, calld buffting, ar stochastic in natur. Th turbulnc in th flow could b also inducd by bridg girdr itslf. A typical girdr with bluff cross-sction gnrats turbulnc in th fluid wak vn if th approaching flow is prfctly stady. This typ of xcitation mchanism is calld signatur turbulnc. Buffting rspons of th bridg can significantly shortn th fatigu lif of th bridg [6]. 5. Cabls of cabl-stayd bridgs can b svrly xcitd by wind bcaus of thir flxibility, small mass and small damping [49]. In addition to th vortx, galloping and wak galloping inducd vibrations [6] nw typ of xcitation mchanism has bn rportd, namly, vibrations of cabls du to wind and rain. Formation of watr rivult on th surfac of cabl modifis arodynamic proprty of th cabl and causs priodic oscillations with limitd amplitud. Th phnomnon has bn obsrvd for wind vlocitis from 5 m/s to 3 m/s and cabls with frquncis from Hz to 3 Hz ([63, 6, 85]). 6. Th tmporary lack of stiffnss of bridg lmnts during construction can xpos th structur to wind ffcts. Th most notabl risks occur du to buffting rspons of th cantilvr spans of a cabl-stayd bridg (.g., [99, 53, 64]) and fluttr instability during construction of th first girdr sgmnts of a suspnsion bridg. Snsitiv to buffting, galloping and vortx-inducd vibrations ar also fr standing bridg pylons ([, 65]). 7. Wind action has also influnc on cars passing th bridg, particularly influnc of modifid flow around th pylon on stability of th moving car [35]. Th dsign of th bridg must compromis th incrasd srvicability of th bridg with wind shilds and largr static forcs xrtd on th structur... Mthods of analysis of arolastic rspons of bridgs In th past many bridgs hav bn damagd du to action of wind, howvr, it was th collaps of th first Tacoma Narrows Bridg in 94 ([46, 47]) that triggrd rsarch on arolastic actions of wind. Wind flow around th bridg dcks is likly to b sparatd du to gomtry of th dcks which maks an analytical analysis difficult. Thrfor, th first studis of wind-bridg intractions hav bn conductd by mans of wind tunnl xprimnts. Th wind tunnl tsting tchniqus as wll as mathmatical modls for analytical study hav got considrabl imptus from aronautical nginring ([5, 55]). Th xprinc in analyzing and supprssing arolastic actions on airplan wings has bn gathrd sinc 96 ([55, 56]).

12 .. Mthods of analysis of arolastic rspons of bridgs... Wind tunnl xprimnts Th most rliabl and gnrally accptd practical mthod to analyz th dynamics and arolastic actions of wind is th full arolastic modl wind tunnl tchniqu (.g., [7, 5, 65, 7, 89, 87]). In this typ of xprimnt, modl of th ntir bridg scald to usually /3 is tstd. Th modl, in addition to bing gomtrically similar to th full bridg must satisfy similarity rquirmnts prtaining to mass distribution, rducd frquncy, mchanical damping and shaps of vibrating mods. Du to high cost of full bridg tsts thy ar occasionally conductd for challnging long-span bridg projcts. Full bridg xprimntal tsting was conductd for dsign proposals of th Svrn Bridg in Grat Britain. This bridg modl duplicatd th obsrvd oscillations of th bridg prompting th accptanc of this mpirical tchniqu [59]. Th full modl can b applid to study many spcial problms lik th influnc of inclind winds on bridg during construction [53] and larg dflction ffcts on th rspons. A full bridg modl might not b appropriat for studying th vortx-inducd vibrations [69]. Sinc small scal is usd, th modling of fin dtails of th bridg is practically impossibl. Morovr, for long span suspnsion bridgs, th lock-in wind spd fall to vry low valu du to Froud numbr scaling [6]. As a consqunc som unwantd Rynolds numbr ffcts can tak plac. An altrnativ for full bridg wind tunnl tsts ar tsts on thr dimnsional partialbridg modls (.g., [9, 3, 64]). In modls of this typ th main span or half of it is modld in an conomical approximation. A support structur consisting of taut wirs or tubs, or of fin-wir catnary, that supports th gomtrically simulatd dck structural form. Usually only fundamntal vrtical and torsion mods of th bridg ar simulatd. In ths tsts th similarity law concrning Froud numbr can b rlaxd allowing highr flow spds to b usd in th xprimnts. Th most common and rlativly inxpnsiv ar xprimntal tsts on sction modls. Sction modls consist of rprsntativ span-wis scald sctions of th dck, supporting springs connctd to th sid nds allowing torsional and vrtical motion (.g., [39, 6]). Th sction of th bridg is nclosd btwn nd plats to rduc arodynamic nd ffcts. Sction modls ar constructd to scals of th ordr /5 to /5 so that th discrpancis btwn th full-scal and modl Rynolds numbr ar smallr than in th cas of full-bridg tsts. Sction modls ar usful for making initial assssmnt, basd on simpl tsts, to such xtnd to which a bridg dck shap is arolastically stabl. Th important advantag of sction modls is possibility of th masurmnts of th fundamntal arodynamic charactristics of th bridg dck on th basis of which comprhnsiv analytical studis can subsquntly b carrid out. Th stady forc [5] cofficints as wll as fluttr drivativs [8] ar dtrmind from sctional modl wind tunnl tsts. Thr ar vry limitd studis discussing th applicability of th rsults obtaind from th scal modl wind tunnl xprimnts to th full siz structur. Th similarity critria of th wind tunnl modl and th prototyp structur cannot b satisfid [6]. Thus, th xprimntal data ar thortically contaminatd by implications of th dissimilar scaling, lik that associatd with th Rynolds numbr. Study by Okauchi t al., compars th rsults of fluttr drivativs obtaind through larg scal sction modl tstd in th natural wind with data from wind tunnl tsts. Th two rsults agrd with ach othr. This has supportd th gnrally accptd viw that practical sction modls of bridg dcks with rasonabl scal can b dsignd such that th svr implications du to scaling problms ar not likly to occur.

13 Introduction... Analytical mthods Th difficulty in analytical analysis of wind ffcts on structurs is th mathmatical formulation of arodynamic forcs. Parkinson t al. ([3, 33]) and Novak ([3, 4]) hav proposd th galloping thory basd on th stady arodynamic cofficints obtaind from wind tunnl xprimnts on sctional modls. In thir approach, calld quasi-stady formulation of arolastic loads, it is assumd that th inclination of rlativ wind vlocity, accounting for th vlocity of th body, and th angl of attack, dscribing th stady wind dirction with rspct to th body at rst, is th sam. For dtrmind instantanous rlativ angl of attack and rlativ wind spd th arodynamic lift and drag forc could b computd through th stady forc cofficints. This xcitation modl, applid to th singl dgr of frdom oscillator, has yildd a nonlinar vibration systm with nonlinarity rlatd to th damping trm. By assuming that th amplitud is narly constant in th priod of on vibration cycl, thy wr abl to trat th amplitud dpndnc of th stability and th hystrsis. Dn Hartog ([6, 6]) showd that th singl-dgr of frdom systm xcitd by th arodynamic lift and drag can hav ngativ arodynamic damping for small amplituds oscillations. Dn Hartog critria stat that th systm bcoms unstabl if th sum of stady drag cofficint and slop of lift cofficint is ngativ. Sinc drag cofficint is always positiv, th slop of th lift cofficint charactrizs th stability. This instability is calld across-wind galloping. Analogous thory to th prsntd abov has bn proposd for rotational oscillations. A ngativ slop of th stady momnt cofficint, dscribing th arodynamic momnt gnratd on th structur, initiats th torsional galloping. Sinc th angl of attack, rlatd to th rlativ wind vlocity varis in diffrnt locations at th cross-sction, its valu should b dtrmind with rspct to som spcific point. For bluff cross-sctions th rfrnc point is th windward dg of th sction [6] and for flat plats or thin airfoils th analytical rsults indicat th chord lward quartr point [55]. Th applicability of two prsntd quasi-stady ngativ slop thoris for analysis of arodynamic stability of bridg girdrs has bn discussd ([47, 6]). It is gnrally accptd that ths formulations can b applid if th oscillations ar slow (th contribution of th vlocity of th structur to th rlativ wind spd is small) or th wind spd is far abov th rang whr flow unstadinss is dominant [7]. For bridg girdrs, formulation basd on th stady-stat thory suffrs from inconsistncy in th modling of th torsional damping but can b applid to othr load componnts and structural mmbrs [8]. Th modling of th arodynamic forcs on bridg dcks that taks into account thir unstady charactristics wr givn by Blich [5]. H simplifid formulas for lift and momnt forc drivd by Thodorsn [67] for thin airfoil by nglcting th trms rlatd to th arodynamic masss. Th Thodorsn s formulation, dscribs th unstady arodynamics lift and momnt gnratd on a thin plat undrgoing small amplitud sinusoidal oscillations in trms of plat s dflctions and thir tim drivativs. Th cofficints of th formulas contain frquncy dpndnt complx function calld th Thodorsn function. Scanlan and Tomko [48] proposd to rplac th trms containing th Thodorsn function by th xprimntally dtrmind fluttr drivativs. Fluttr drivativs ar frquncy dpndnt functions, in th form of th tabular data, dscribing th charactristic of th unstady arodynamic forcs on a moving bluff bridg girdr. Fluttr drivativs can b xtractd from th wind tunnl xprimnts on th sctional modl of th bridg. Thr ar

14 .. Mthods of analysis of arolastic rspons of bridgs 3 two tchniqus for th xprimntal xtraction: th fr vibration and forcd vibration tst. Th formr tchniqu is vry simpl to conduct in th wind tunnl sinc it rquirs only th masurmnts of tim historis of th bridg rsponss at slctd wind spds [8]. Howvr, it rquirs snsitiv modl idntification schms for fluttr drivativs stimation. Th lattr tchniqu uss actuators to induc sinusoidal motion of th sction of a bridg girdr. Fluttr drivativs ar dtrmind from forc masurmnts from a st of th tsts ovr th slctd frquncy and wind spd rangs. Sabzvari and Scanlan [43] (also [56, 6]) hav shown that thr ar significant diffrncs btwn arolastic wind loads on th flap plat and th considrd bridg dcks. Nvrthlss, sction modls rprsnting stramlind dcks of som modrn bridgs, including Svrn Bridg [7] and th Grat Blt East Bridg [39] ar satisfactorily analyzd for fluttr with us of thortical fluttr drivativs basd on Thodorsn s solution. A thortical dscription of th fluttr of airplan wings [67] as wll as xtnsion of this formulation to bluff bridg girdrs [48] dfins th unstady arodynamic forcs in trms of frquncy dpndnt functions, namly, Thodorsn s circulatory function or fluttr drivativs. Th rsulting arolastic quations of motion hav rducd frquncy dpndnt cofficints, thus, th convntional analysis rquirs itrativ sarch for a critical fluttr wind spd. Morovr, matrial and gomtrical nonlinaritis of th structural systm cannot b incorporatd into th analysis and th dpndnc on rducd frquncy limits th dsign of vibration supprssion systms to frquncy domain mthods. Tim domain modlling of th unstady arodynamic forcs has bn proposd by Scanlan ([47, 56]) and Bisplinghoff and Ashly [4]. Thy hav suggstd us of so calld arodynamic indicial functions obtaind by invrs Furir transform of frquncy dpndnt arolastic forcs. Howvr, this tchniqu could b applid to th problms whr frquncis and natural mods ar not gratly altrd by arodynamic forcs. Rogr [4] proposd a modling mthod which can transform th arolastic quation of motion of an airplan into tim domain. His mthod approximats arodynamic forc cofficints by rational functions of Laplac variabl. Th siz of th quation aftr approximation is xtndd, but th ovrall analysis is gratly simplifid. Th augmntd systm, in Rogr s formulation, has a rlativly larg numbr of nwly addd arodynamic stats, and modifications of his mthod wr proposd by Dunn [43] and Karpl [75]. A furthr improvmnt of approximation prformanc was obtaind [7] through optimization of both linar and nonlinar cofficints of th approximating functions. Th tim domain modlling of arolastic bhavior of bridgs with bluff girdrs with us of rational function approximation has bn conductd by Wild t al. [8]. Th analysis of svral bridg dcks rvald that th bridg fluttr can b dscribd in tim domain by th approximation function of th scond ordr. Th rational modl of th bridg-wind intraction has bn xtndd to covr th buffting rspons by Chn and Karm [6]. Th rapid dvlopmnt of th computr hardwar tchnology motivatd dvlopmnt of numrous mthods in computational fluid dynamics. Ths mthods ar basd on diffrnt discrtizations of th Navir-Stoks quation of fluid motion which dfins th intraction btwn th fluid and th structur. In such numrical xprimnts all th dtails of th flow movmnt around th body and th associatd forcs can b masurd. Howvr, such studis ar not widly in us. Thy rquir tim consuming calculations and th up-todat hardwar facilitis ar not powrful nough to solv th quations for th xtrmly fin computational grids ndd for accurat rprsntation of th small-scal turbulnc in

15 4 Introduction th analysis. As a compromis, various modls hav bn dvlopd to supplmnt th quations of motion and in such mannr includ th influnc that might b nglctd by working with coars grids (.g., [, 85, 9, 9]). An ovrviw of th computation fluid dynamic mthods applid in wind nginring has bn givn by Murakami t al. [6]..3. Supprssion of wind inducd vibration of long span bridgs Fluttr instability is of primary concrn for th wind-rsistant dsign of a long-span bridg, sinc it can lad to th total collaps of th structur. A numbr of analytical and xprimntal studis hav focusd on th supprssion of fluttr by mans of structural and arodynamic approachs (.g., [36, 7, 9]). Improvmnt of critical fluttr wind spd can b obtaind by structural modifications. To prvnt, charactristic for fluttr instability, coupling btwn torsional and vrtical bnding mods, it is suggstd to widn th diffrnc btwn th frquncy of thos mods. It can b achivd by th mploymnt of vry dp, torsionally rigid truss girdrs. Such torsionally stiff dck was usd for th rconstruction of th Tacoma Narrows Bridg in 95. Also, th longst bridg up to dat, Akashi Kaikyo, with main span of 99 m, utilizs this concpt. Stiff and dp truss girdr safguards th bridg against fluttr instability but shows svral disadvantags such as high static wind forcs, 5%-% highr wight compard to box girdrs, troublsom maintnanc and high costs [9]. In ordr to ovrcom ths drawbacks in mid-96s th concpt of a flat box girdr was introducd into th dsign and construction of Svrn suspnsion bridg in Grat Britain [4]. In this dsign a lightr structur with adquat torsional stiffnss and shap to rduc drag load was dvlopd. At th sam tim fabrication and rction procss was simplifid and maintnanc cost dcrasd. Th culmination of dvlopmnt of bridgs with flat box cross-sction has bn attaind with th compltion of th Grat Blt Bridg in Dnmark of th main span amounting to 64 m [87]. Othr structural mthod for th incras of fluttr wind spd is th modification of th cabl systm ([9, 9]). Us of additional suspnsion cabls or combind suspnsion and stayd cabls rsults not only in an incras of th structural stiffnss, particularly in torsion, but also nhancs structural coupling. Accordingly, coupld motion in thr dirctions rsults in an incras in th gnralizd modal mass which supprsss th coupld fluttr. Th most ffctiv mans of fluttr control is to improv th arodynamic prformanc of a bridg dck sction through th modification in its gomtric configuration. Although furthr stramlining of th bridg dck rducs drag load as wll as wind tndncy to shd vortics in th wak of th box, which rsult in rduction of undsirabl vibrations, th box girdrs ar pron to fluttr. Th critical wind vlocity for th onst of fluttr may b incrasd by approximatly % using longitudinal vntilation slots introducd into th girdr ([9, 9]). Th drawbacks of this solution ar incrasd drag forcs and construction costs. Arodynamic appndags such as guid vans, winglts or prforatd dg fairings (Fig..) showd thir profit as inxpnsiv safty prcautions [9]. For xampl th guid vans usd in th Littl Blt suspnsion bridg incrasd arodynamic damping in torsion and nhancd critical fluttr wind spd. Thir additional virtu is that thy may b applid as a rtrofit masur to allviat arodynamic ffcts unforsn at th dsign stag.

16 .3. Supprssion of wind inducd vibration of long span bridgs 5 Arodynamic shaping tchniqus ar rliabl and wll rcognizd in wind nginring. Nvrthlss, thy cannot provid allviation of wind inducd instability for bridgs with supr long spans [9]. a) Stramlind cross sction b) Cross sction with vntilation slots Railing Guid van c) Arodynamic appndags Fig... Arodynamic shaping for rduction of wind inducd rspons During construction of Humbr Bridg in England a mthod of dploymnt of ccntric mass was applid for th first tim in th history of long-span bridgs rction procss [88]. This mthod is most ffctiv for systms whr th natural frquncis of vrtical and torsional motions ar vry clos. Thus, it is most suitabl for rction stag, whn th dck tmporarily lacks torsional stiffnss. Rcnt studis of Phongkumsing t al. ([36, 37]) invstigatd application of th mthod to bridgs in srvic. Although th mthod turns out to b usful, it rquirs rlativly larg ccntric masss to satisfactorily incras th fluttr critical wind spd. Th drawbacks of th mthod ar nlargd dad load and static torsional dformation of th dck. Enhancmnt of th arodynamic stability can b also achivd by application of passiv damping dvics. Nobuto t al. [] studid th us of two Tund Mass Damprs (TMD) placd at two sids of th cross sction of a dck in ordr to affct both having and pitching motion. It has bn found that th improvmnt in critical wind vlocity is about 5%. Howvr, th control systm is vry snsitiv to th tuning condition which is vry difficult task for changing with wind spd frquncis of mods in coupld fluttr. In addition, th prformanc of TMDs, which ar vry ffctiv in damping purly harmonic oscillations, might b affctd by random buffting rsponss ([4, 58, 98]). Nw mthods of supprssion of wind inducd oscillations in long span-bridgs ar offrd by activ control systms (.g., [67, 84]). Such mthods, widly usd in arospac industry, drw rcntly considrabl attntion among bridg nginrs [7]. Dung t al. [4] xtndd th concpt of ccntric mass placd with fixd ccntricity to th ida of a mass movd by actuators (Activ Mass Damprs) with rspct to th motion of th dck.

17 6 Introduction Additional mass motion was dtrmind using th concpt of modrn H Optimal Control. For th sctional modl of a bridg dck of m main span lngth of th suspnsion bridg th critical wind spd for onst of fluttr was nhancd to m/s. Howvr, th torsional divrgnc was nglctd in this study. Wild t al. [8] studid th concpt of variabl ccntricity mploying Linar Saturatd Optimal Control. Thir study showd that a mass of about % of that of a dck is rquird to incras th critical wind spd for 4 m main span bridg to approximatly 75 m/s. Activ tndon control has bn analytically and xprimntally invstigatd by Fujino t al. ([53, 75, 76]). Th srvohydraulic actuators locatd in a dck of th bridg wr usd to control th cabl tnsion and to influnc th bridg rspons. Th mthod has bn found vry ffctiv in supprssing vrtical girdr motion with small cabl vibration, howvr, inffctiv for vibrations with larg cabl componnt. Th application of th first activ control dvic for supprssion of wind inducd oscillations of bridgs has bn rportd in 99 [66]. A dvic combining th Tund Mass Dampr (TMD) and control actuator was usd during rction of th towr of th Rainbow Bridg (Tokyo, Japan) to rduc vortx-inducd vibrations. At last 4 bridgs hav mployd activ systms lik Hybrid Mass Damprs (HMDs) with arch-shapd mass or multistp pndulum typ as wll as Activ Mass Damprs (AMDs) [54] to safguard th stability of th bridg lmnts xposd to actions of wind. Activ arodynamic control of fluttr utilizs th ida of using th air stram, which normally is rsponsibl for instability, to intract with th motion of th girdr in such a way, that it provids stabilizing ffct. Th concpt of th arodynamic control of bridg fluttr by additional activ surfacs was proposd by Ostnfld and Larsn [9]. Fig.. shows an activ control which uss additional control surfacs attachd bnath both dgs of th dck through arodynamically shapd pylons. Th rotational displacmnt of th control surfacs is activly adjustd by fdback control so that th gnratd arodynamic forcs provid a stabilizing action on th dck. In this mthod th stabilizing forcs ar not gnratd dirctly by th actuators but ar drawn from th air flow, and thir magnitud incrass proportionally to th wind spd squard, and thus, proportionally to th forcs acting on th bridg dck. actuator computr snsor control surfacs Fig... Activ arodynamic control of bridg dck fluttr by control surfacs attachd bnath th dck Th first rportd rsarch on this control systm was by Kobayashi and Nagaoka [83]. Thy conductd a wind tunnl xprimnt on th sctional modl of a bridg and

18 .3. Supprssion of wind inducd vibration of long span bridgs 7 obtaind an incras of fluttr wind spd by a factor of. Th control algorithm, usd in th xprimnt, was slctd as proportional to th rotational motion of th dck. Wild and Fujino [77] carrid out a thortical analysis of such systm. Thy applid a rational function approximation to modl unstady arodynamics and drivd a tim domain quation of motion for th control systm. Sinc th rsulting quation of motion was dpndnt on th man wind spd, thy proposd a variabl-gain output fdback law optimizd with rspct to th quadratic prformanc indx dfind ovr a slctd rang of wind spd [59]. Th suggstd controllr guarantd th systm s stability in th wind spd rang of intrst and allowd application of diffrnt control stratgis at low and high wind spd. Stabilizing action of activ flaps dirctly attachd to th sids of th bridg girdr (Fig..3) has also bn suggstd [9]. In this systm th flow pattrn around th dck is affctd by th motion of th flaps, and thus th stabilizing action coms not only from th arodynamic forcs gnratd on control flaps but can also b achivd through modification of th arodynamic forcs inducd on th bridg dck. Th mathmatical modl of a sction of th bridg quippd with activ flaps basd on th thortically dtrmind unstady arodynamic forcs [69] has bn givn by Wild t al. [8]. Th adoptd control stratgy coupld th motion of th flaps with both vrtical and torsional motion of th dck ([, ]). It has bn found that th most fficint action of th flaps was achivd whn th flaps hings wr locatd at th dck dgs. Such systm satisfis th rquird stability conditions in th considrd wind spd rang and rquirs flaps of small width. actuator computr snsor control surfacs Fig..3. Activ arodynamic control Activ control systms consist of rlativly complicatd hardwar and rquir powr supply. In addition, to safguard th rliability of activ control systm, its componnts should b duplicatd or triplicatd in th form of paralll systms, and must b constantly monitord and maintaind. This lads to high costs of activ control systms. Wild t al. [79] proposd and invstigatd, both analytically and xprimntally, th concpt of passiv systm utilizing control surfacs. Th control surfac motion was govrnd by an additional pndulum attachd to th cntr of gravity of th dck. Th numrical study on a sctional modl of th systm showd that th critical wind spd might b incrasd by 57%. Th prliminary xprimntal rsults showd vry good agrmnt with th thortical prdiction for small motions of th control surfacs. Howvr, for largr motions considrabl discrpancy has bn rportd and xprimntally obtaind critical wind spds wr highr than th thortically prdictd. Th ralization of th passiv control systm, basd on moving flaps attachd dirctly to th dgs of th girdr, has bn dvlopd by Omnzttr t al. ([7, 8]). Th rota-

19 8 Introduction tion of th flaps has bn coupld to th rotation of th bridg girdr through th additional cabls connctd to th main cabls. Th optimal paramtrs of th passiv control systms hav bn dtrmind on buffting rsponss of th sctional modl of th controlld bridg. Although, th passiv control systms cannot frly shap th dynamics of th bridg, thy can nhanc th arolastic prformanc of long-span bridgs to th rquird lvl..4. Aim and scop of study Th aim of th study is drivation of mathmatical modls of th proposd passiv arodynamic control systms and stimation of thir ffctivnss in th improvmnt of stability of long-span suspnsion bridgs xposd to action of wind. Th analyzd wind actions on bridg structur ar limitd to wind inducd instabilitis of th bridg structur.g., fluttr and divrgnc. Th influnc of th turbulnc on th unstady arodynamic forcs is nglctd [3]. Th bridg is assumd to undrgo small amplitud oscillations around th quilibrium position and thrfor th linar modl of th bridg structur is applid. Two typs of control systms ar proposd. Th first on, uss th control surfacs attachd bnath th bridg dck. Th motion of th surfacs is govrnd by a pndulum attachd to th cntr of gravity of th bridg. Th control surfacs ar connctd to pndulum, and thus, rotation of th surfacs is inducd by motion of th bridg girdr. This control systm is rfrrd to as th bridg-surfacs systm. Th scond considrd passiv systm consists of control surfacs dirctly attachd to th dgs of th bridg girdr. This systm is calld bridg-flaps control systm. Th motion of th flaps is inducd by additional cabls conncting th flaps with th main cabls is such a way that th rotation and horizontal dformation of th bridg induc rotations of th flaps. Th first part of th study is dvotd to drivation of th mathmatical modls of th sctions of th passiv control systms. Th tim domain formulation of th unstady arodynamic forcs, acting of th bridg quippd with control surfacs or flaps, is obtaind through rational function approximation. Th prliminary paramtric study is conductd to slct th configuration of th systms which ar th most ffctiv in stabilizing bridg structur xposd to actions of wind. Th scond part, prsnts th xprimntal study conductd in th wind tunnl on sctional modls of th systm. Th study of th bridg-surfacs systm was dsignd and conductd by K. Wild (formr Associat Profssor at th Bridg and Structur Lab., Univrsity of Tokyo, Japan). Th aim of th xprimnt was th stimation of th critical wind spds of th bridg-surfacs control systm. Th xprimntal tsts on th bridg-flaps systm wr conductd by T. Sato (formr Mastr studnt of th Bridg and Structur Lab., Univrsity of Tokyo, Japan). Th xprimnt was dvotd to masurmnts of th unstady arodynamic forcs acting on th moving bridg with rotating flaps. Th xprimntal rsults wr compard with th analytical rsults obtaind from th sctional mathmatical modls of th control systms. Th third part of th work prsnts th drivation of th full thr dimnsional FEM modls of th bridg and control systms. Th modls ar obtaind through th Finit Elmnt Mthod. Thn th dynamic analysis on FEM modls of th control systms with th slctd configurations is conductd. Th fficincy of th studid systms as wll as thir advantags and disadvantags ar discussd.

20 Chaptr STUDY ON SECTIONAL MODELS OF PASSIVE FLUTTER SUPPRESSION SYSTEMS.. Bridg surfacs control systm... Mathmatical modl of arodynamic forcs Th control systm consists of two control surfacs attachd bnath th girdr of th bridg (Fig..). Th stabilizing forcs countracting fluttr and divrgnc of th bridg ar gnratd on th control surfacs. Th rquird arodynamic forcs ar obtaind by rotation of th lading and trailing control surfac with rspct to th torsional motion of th girdr. U L d L M d M D air flow D d L D x B d h M lading surfac cntr of gravity H bridg dck trailing surfac B B Fig... Arodynamic forcs on bridg girdr and control surfacs Th width of th dck is dnotd by B d and th dpth by H. Th widths of th lading and trailing surfacs ar B and B. Th motion of th dck is dscribd by th vrtical displacmnt, h, rfrrd to as having and torsional displacmnt,, calld pitching. Th rlativ angls of rotation of th lading and trailing surfacs ar dnotd by and, rspctivly.

21 . Study on sctional modls of passiv fluttr supprssion systms Intraction of a vibrating structur with air flow crats unstady arodynamic forcs sinc flow has to adjust to a moving structural boundary. Th motion inducd forcs, L d, M d, L, M, L, and M may oppos th vibration, i.., crat an arodynamic damping, or may additionally forc th motion and, in xtrm cass, lad to th damag of th bridg. U x L d M r, M d, h U r L r D r D h B d x H Fig... Quasi stady forcs on a brid dck Th quasi stady formulation dtrmins arodynamic forcs through th stady forc cofficints [6]. Forc cofficints ar quantifid xprimntally for ach structur from stady forcs (tim-avrag forcs) obtaind from masurd discrt normal prssur distribution and strsss along th structural surfacs. Forc cofficints ar commonly xprssd in trms of non dimnsional shap factors. Th quasi-stady thory assums that arodynamic forcs can by computd through stady forc cofficints takn for th rlvant instantanous rlativ wind vlocity acting at th instantanous rlativ angl of attack (Fig..) [6]. Th arodynamic drag, lift and momnt rsulting from rlativ wind spd, U r, ar Dd Drcos Lrsin, Ld Drsin Lrcos, (.) Md Mr, whr Dr.5 UrHCD( ), Lr.5 UrBdCL( ), (.) M.5 U B C ( ). r r d M Th air dnsity is and th man wind spd is dnotd by U. Th instantanous wind vlocity, U r, and instantanous wind angl of attack,, ar U U x h r, h arctan. U x (.3)

22 .. Bridg-surfacs control systm Th us of stady forc cofficints and thir dpndnc on man incidnc can b simplifid by linarizing a portion of th curv, rprsnting rlation btwn forc cofficint and angl of attack, around th man incidnc, i.., dci Ci( ) Ci( ), ( i D, L, M). (.4) d Introducing (.3) and (.4) with th man incidnc into (.) yilds whr F Bd, Fd VKaqd Caq d Fs (.5) U T D L M q x B h B,, d d d d d d d T H dc H H dc D D CD CL Bd d B d Bd d dc H dc dc L D L Ka, Ca CL, (.6) d B d d d dc dc M M CM d d T.5U H H Fs C D CL C M, V.5U Bd. Bd.5U B d Th formulation of arodynamic forcs basd on th quasi stady thory is applicabl for problms with vibrations charactrizd by slow motion, small variations of wind angl of attack and curvs, of stady forc cofficints with rspct to th angl of attack, with slowly varying slops. A quasi-stady modl of th arodynamic forcs dos not modl th tim lags btwn th arodynamic forcs and motion of th structur that ar important in th analysis of fluttr. Formulation of unstady arodynamic forcs with us of th so calld fluttr drivativs has bn proposd by Scanlan and Tomko [48]. Th unstady arodynamic forcs, ar assumd to b linar in th structural displacmnts and thir first two drivativs i.., * x * B d * * x * h * h Dd U H KP KP K P3 K P4 KP5 K P6, U U Bd U U * h * B d * * h * x * x Ld U Bd KH KH K H3 K H4 KH5 K H6, U U Bd U Bd * h * B d * * h * x * x Md U Bd KA KA K A3 K A4 KA5 K A6, U U Bd U Bd (.7)

23 . Study on sctional modls of passiv fluttr supprssion systms whr th air dnsity is, and man wind spd is dnotd by U. Th nondimnsional rducd frquncy K is dfind as K B / U (.8) d *, and is th circular frquncy of oscillation. Th fluttr drivativs ar dnotd by P i H * i, A * i (i =,..,4), and ar functions of th rducd frquncy. For th bluff bridg dck, *, th functions P i H * i, A * i (i =,..,4) ar dtrmind xprimntally through th wind tunnl xprimnts on a sctional modl of th bridg dck. Th xprimntal fluttr drivativs ar dtrmind for a slctd st of rducd frquncy {K n } and ar givn in th form of tabular data. Transforming quation (.7) into th Laplac domain with th zro initial conditions givs (Appndix A) L(F d ) = VQL (q d ) (.9) whr * * * * * * KP4 pkp KP6 pkp5 KP3 p KP * * * * * * Q KH6 pkh5 KH4 pkh KH3 p KH. (.) * * * * * * K A6 pka5 K A4 p KA K A3 p KA Matrix Q is of siz n n. Th Laplac oprator is dnotd by L and th nondimnsional Laplac variabl is p s B / U. (.) Each gnralizd forc cofficint Q ij of Q is approximatd by a rational function of th nondimnsional Laplac variabl p ([7, 8]) d n ˆ Qp ( K ) ( C ) p D E, ij p b a ij a ij i j (.) whr Q ( p) dnots th approximation. Th lmnts of K a and C a rprsnt arodynamic stiffnss and damping, rspctivly. Th partial fractions, D i E j /p + b, ar commonly calld lag trms, bcaus ach rprsnts a transfr function in which th output "lags" bhind th input and prmits an approximation of th tim dlays inhrnt in unstady arodynamics. Th cofficints of th partial fractions b ar rfrrd to as lag cofficints. Th addition of ach partial fraction introducs into th rsulting stat-spac ralization nw stats, rfrrd to as arodynamic stats. Th numbr of partial fractions is dnotd by n and this numbr is found as a compromis btwn th prcision of th approximation and th siz of stat-spac ralization. Th unstady arodynamic data, Q, obtaind from th xprimnt, ar dtrmind only for purly imaginary valus of th nondimnsional Laplac variabl i.., p = ik. Thus, th approximation is prformd only for oscillatory motion. In ordr to obtain solutions on th Laplac domain for both growing and dcaying motion, it is ncssary to xprss th forcs as a function of p for th ntir nondimnsionalizd complx p-plan. To ovrcom this, th concpt of analytic continuation is oftn usd, which justifis xtnding ths functions, to th ntir complx plan by finding analytic functions which agr with th arodynamic forcing functions at all valus of frquncis [7]. Howvr, thr is only a finit numbr of frquncis at which tabular data ar availabl; hnc, this procss is at bst an approximat analytic continuation into th rgion nar th portion of th axis containing th tabular

24 .. Bridg-surfacs control systm 3 data. Sinc fluttr phnomna occur for points in th complx s-plan which li along th imaginary axis, approximations into th rgion nar th axis ar sufficint for most studis [44]. Th approximation (.) is calld minimum stat rational function approximation (MS RFA) [75]. Th matrix formulation of (.) is ˆ QpK C pd( pir) E, (.3) a a whr D and E ar rctangular matrics of siz n n and n n rspctivly, and R is a diagonal matrix of th form b R. (.4) b n Sinc R is of dimnsion n n th arodynamic dimnsion for th minimum-stat RFA formulation is n. (.5) Dfining th vctor of arodynamic stats, x a, of siz n, as na L (x a ) = (pi R) EL (q) (.6) and taking th invrs Laplac transforms in (.9) with approximation of unstady arodynamics (.3), givs th tim domain formulation of unstady arodynamic forcs n L d Bd Fd a d a d ixi, M VK q C q D d U i U U Eq x i bx i i i d i,, n. B B d d (.7) Arodynamic stats ar dnotd by x i and ar govrnd by first ordr ordinary diffrntial quations. D i dnots columns of matrix D and E i dnots rows of matrix E. For ach additional lag trm th siz of th rsulting stat-spac ralization is biggr only by on arodynamic stat. Thus, introduction of lagr numbr of lag trms dos not significantly incras th stat-spac ralization. Improvmnt of th approximation can b achivd by incrasing th numbr of lag trms. Howvr, it advrsly incrass th numbr of quations rquird to dfin th arodynamic systm. Minimization of approximation rrors can also b obtaind by rducing th frquncy rang ovr which th fits ar rquird, but this narrows th applicability of th approximation. Th additional improvmnts may b gaind by an optimization of th lag cofficints. In th minimum stat formulation (.3), th numrator cofficints for th lag trms ar th product lmnts of D and E, so th two stp itrativ linar optimization is mployd. First, for th slctd initial R and D, th matrics K a, C a and E ar obtaind through th last squars optimization such that th total approximation rror J w, (.8) j i ij ij

25 4. Study on sctional modls of passiv fluttr supprssion systms is minimizd. Th wighting factor is dnotd by w ij, and th masur of rror btwn th approximating curv and th actual tabular data is ˆ Q n ij ij p Qij p, M ij (.9) whr n is a siz of th st of xprimntal data and M max, Q ik. (.) ij ij n n Each trm in th sum in (.9) is a masur of th rlativ rror, if th maximum magnitud of Q ij (ik n ) is gratr than, but is an absolut rror for magnituds lss than. This rror function ssntially normalizs th arodynamic data prior to th nonlinar optimization. In th nxt stp, for th sam R and prviously dtrmind E, th matrics K a, C a and a nw on, D, is computd. Ths stps ar rpatd until th ovrall approximation rror (.8) convrgs or rachs th stopping critrion of a maximum numbr of itrations. Th lag cofficints, b, ar in th dnominator and ar found via th nonlinar nongradint optimizr proposd by Mldr and Mad []. Sinc th charactristic roots of th systm matrix corrsponding to th arodynamic stats must b stabl, th lag cofficints hav to b gratr than zro. In addition, it is dsird to rstrict th uppr bound of variation of b to th rang of frquncis ovr which th tabular data ar availabl, i.., L b U (for =,..., n ), whr L and U dnot th lowr and uppr limit, rspctivly. Ths sid constraints wr imposd by an invrs sinusoidal transformation. Th dtrmination of all th cofficints of th approximating function (.3) rquirs a multilvl optimization in th sns that linar last-squars optimization solution for linar paramtrs is prformd insid an itrativ sarch for paramtrs that ntr th problm in a nonlinar fashion. Th considrd bridg has a flat box girdr. This girdr cross-sction was considrd for th suspnsion bridg Akashi Kaikyo, Japan. Th xprimntal study on this flat box dck has bn conductd by Fujino t al. [5] in th wind tunnl of Univrsity of Tokyo, Japan. Th xprimnt dtrmind th stady forc cofficints for various angls of attack of th wind as wll as fluttr drivativs corrsponding to unstady arodynamic lift forc and momnt. Th fluttr drivativs for th considrd bridg sction wr dtrmind ovr th rang of rducd frquncis.3 K.. To widn th applicability of th rational modl of th bridg arodynamics and to covr also a static instability of th bridg calld divrgnc, th additional data corrsponding to th zro rducd frquncy hav bn addd. Th data for th K = is obtaind from th quasi static formulation of arodynamic forcs (.5) and ar found to b * * * dcl * KH, KH, KH3, KH4, d * * * dcm * K A, K A, K A3, K A4. d (.)

26 .. Bridg-surfacs control systm 5 For th considrd girdr cross-sction th static forc cofficints and thir drivativs at th zro man angl of attack ar [5] dcl d dcm 5.6,.5. (.) d Th rsarch of fluttr is focusd on arodynamic coupling btwn th having and pitching mod which is th sourc of an instability in th classical fluttr problm. Thus, th study in this sction is limitd to th two dgr of frdom systm with us of th xprimntal fluttr drivativs dtrmind for lift and momnt forc. Th xprimntal fluttr drivativs corrsponding to lift and momnt for th rducd frquncis from to ar shown in Fig..3. Linar and nonlinar optimization of th cofficints of rational function approximation of xprimntal unstady arodynamic data yilds Ka,, Ca n, b =.373, b = 6.988, D,,.45 D.33 (.3) E , E K A, K A, K A 3, K A 4, a) Fluttr drivativs of lift forc b) Fluttr drivativs of momnt K A 3 K A K (nondim.) K A 4 K A K H, K H, K H 3, K H 4, K H 3 K H 4 K H K (nondim.) xprimntal data approximation K H Fig..3. Exprimntal fluttr drivativs and thir approximations by MS RFA Th approximation of th xprimntal fluttr drivativs by th rational modl ar shown in Fig..3. Good approximations ovr th whol rang of rducd frquncy has bn obtaind with us of only two lag cofficints. Not that th applid approximation rsults in a vry good fit also for th zro rducd frquncy. Th control surfacs ar considrd to b flat plats. Thrfor, th matirx Q of unstady arodynamic forcs for control surfacs can b computd from thortical formulation of a flat plat providd by Thodorsn [67]. In this study, th rang of th rducd

27 6. Study on sctional modls of passiv fluttr supprssion systms frquncis ovr which th approximation has bn conducd is xtndd to includ th data for zro rducd frquncy. Th arodynamic forcs of th lading, F, and trailing surfac, F, with MS RFA ar n L i i Bi s Fi Vi Ca, iq i Ka, iqi i, lxi, l, M D i U l U U s E q x b x i, ; l,, n, il, il, il, il, i i Bi Bi (.4) wr th vctors of absolut vrtical and torsional displacmnts of th lading and trailing surfacs and corrsponding matrics V i ar q h B i /,,, T i i i i V i diag(.5 U Bi.5 U Bi ). (.5) K H, K H, K H 3, K H 4, a) Fluttr drivativs of lift forc b) Fluttr drivativs of momnt 7 thortical data 6 approximation K H thortical data K A 3 4 approximation 3 K H K H K (nondim.) K H 4 K A, K A, K A 3, K A 4,.5 K A K A 4 K A K (nondim.) Fig..4. Thortical fluttr drivativs and thir approximations by MS RFA Th approximation of unstady arodynamic forcs acting on th control surfac is prformd for th rang of rducd frquncis from to.. This rang of rducd frquncy is applicabl for th siz of th control surfac up to % of th width of th dck, i.., B B.B d. Th cofficint matrics of (.4) ar found to b Kai,,,, Cai s.6 s ni, bi, =.99, D, i,, i.9.63, ( i, )..4 E (.6) Th cofficints of matrix Q for control surfac and thir approximations by MS RFA ar shown in Fig..4. Good approximation of thortically dtrmind cofficints of matrix Q for th control surfac has bn obtaind only with on lag cofficint.

28 .. Bridg-surfacs control systm 7 This formulation of th arodynamic forcs of th dck and control surfacs assums that th arodynamic forcs on ach lmnt of th control systm ar indpndnt of ach othr. For xampl, th wak bhind th lading surfac modifis th flow nithr around th bridg dck nor th trailing surfac.... Equation of motion of bridg-surfacs control systm Th proposd passiv arodynamic control systm consists of additional control surfacs attachd bnath th bridg dck and th pndulum attachd to th cntr of gravity of th dck (Fig..5). Th control surfacs ar connctd to th pndulum by rigid bars in such a way so that th motion of th bridg and pndulum ar coupld with th rotations of th surfacs. U L d M d L M lading surfac trailing surfac h M L B d d pndulum control systm control systm d d B B U control systm control systm Fig..5. Cross sction of passiv bridg-surfacs control systm Two control stratgis ar considrd. Th control systm assums that, for not moving pndulum, th nos up torsional displacmnt of th dck causs nos down and nos up torsional displacmnts of th lading and trailing surfac, rspctivly (Fig..5). Con-

29 8. Study on sctional modls of passiv fluttr supprssion systms trol systm provids nos down torsional displacmnts of both control surfacs for nos up torsional displacmnt of th dck. Control systm rquirs a chang of th connctions btwn control surfacs and pndulum for wind spd coming from th opposit dirction, whras control systm has th symmtric connctions and is qually fficint for both dirctions of th incoming wind. Th gain of th control systm i.., th proportion of coupling btwn pitching of th dck and rotation of th control surfacs can b changd by shift of th point whr th connction bars ar attachd to th control surfacs as wll as by shift of th connction to th pndulum. Th quation of motion of th considrd sction of bridg dck of width B d with two control surfacs of width B and B and pndulum has thr structural dgrs of frdom: having, h, pitching,, and angular displacmnt of th pndulum,. Coordinat is assumd to b rlativ to th vrtical axis of th bridg dck. Th influnc of th bar systm, that coupls th motion of th pndulum with th control surfacs, on systm dynamics is nglctd. Th masss as wll as rotational inrtia of th control surfacs ar nglctd. Th horizontal motion of th systm is not considrd in th study on sctional modl. Th quation of motion of a sction of th dck-surfacs systm subjctd to th smooth flow is Mq+Cq+Kq=F, (.7) s s s whr th vctors of displacmnts, q, and forcs, F, ar T q hbd, T F [ L M R]. (.8) Th total lift forc, sum of th lift forcs acting on th dck and th surfacs, is dnotd by L and th sum of arodynamic momnts acting on th dck and control surfacs, with rspct to th cntr of gravity of th bridg dck, is dnotd by M. R rprsnts th torqu acting on th pndulum. For th slctd vctor of displacmnts, q, th cofficint matrics bcom: M diag( mb, I, I ), s d p m y hbd Cs I Ip p p, Ip p p Ip p p K s diag( m hbd, I, I p p). (.9) Mass of th dck pr unit lngth is dnotd by m. I and I p ar mass momnts of inrtia of th dck and th pndulum. Undampd natural frquncis and damping ratios of having, pitching and pndulum mod ar dnotd by h,, p and h,, p, rspctivly. Not, that th damping of th pndulum is obtaind from th linar dampr attachd from th pndulum to th bridg dck. Th total arodynamic forcs, F, acting on th systm, consist of forcs on th bridg dck, F d, on th lading, F, and trailing surfac, F. F F [ L M ], T d d d [ L M ], T F [ L M ]. T (.3)

30 .. Bridg-surfacs control systm 9 Th total arodynamic forcs ar whr F S F, ( i d,,), (.3) i i i Sd, S, S. (.3) Th distancs from th lading and trailing surfac hing lin to th cntr of th dck ar dnotd by and, rspctivly. Motion of th control surfacs is govrnd by th motion of th dck and pndulum. Th control law is c ( ), ( i,). (.33) i i Th gain of th control systm, c i (i =, ), dnots th way of coupling th motion of th pndulum and th dck with th motion of th control surfacs. As shown in Fig..5, th gain of th control systm can b slctd by changing th point of connction of th bar with th control surfacs and by chang of th connction point of th bars to th pndulum. Assuming that th connction bars ar rigid, th gain can b obtaind as follows d ci ( i,). (.34) d i Th gain c i is positiv for th rlativ motion of th surfacs in th clockwis dirction and ngativ for th opposit dirction. Th arodynamic forcs of th control surfacs can b xprssd in trms of vctor q. Th coordinat transformations btwn displacmnts usd for computing th arodynamic forcs on th dck, q T d = [h/b d ], and control surfacs, q and q (.5), and th global vctor of displacmnts, q (.8) ar qi Tq i, ( i d,,), (.35) whr T B, d Bi i Bd Bi d i, ( i,). ci c T (.36) i Substitution of th formulations of arodynamic forcs of th dck ((.7), (.3)) and control surfacs ((.4), (.6)), with corrsponding coordinat transformations (.35), into quation (.7), yilds th quation of motion of th passiv control systm x Ax. (.37) Th stat vctor bcoms T T T x q q xa xa xa xa x xn,,,,,,, d xa x x n xa x x n and matrix A rads, (.38)

31 3. Study on sctional modls of passiv fluttr supprssion systms whr A M C M K M S VD M SV D M S V D s s s s s d s s I U U ETd R B d B d, U U s E T R B B U U s E T R B B U U U B B, B, CCs SV CaTd S V Ca T S V Ca T, d K K SVK T S VK T S V K T. s a d a, a, (.39) (.4) Th modl of th dck-surfacs control systm with RFA of slf-xitd arodynamic forcs is dscribd by first ordr homognous matrix diffrntial quation. For givn man wind spd, U, quation (.37) dscribs th linar tim invariant systm. Th matrix of cofficints, A (.39), dpnds on wind vlocity U. Pols (ignvalus) and mod shaps (ignvctors) of th systm ar givn by solving th ignvalu problm, I A v (.4) whr i (i =,..., n; and n is th siz of A) ar th ignvalus, and v i ar th corrsponding ignvctors. Qualitativly th ignvalu problm (.4) yilds complx and ral ignvalus. Th complx ignvalus appar as complx conjugat pairs and corrspond to th oscillatory mods. Ral ignvalus corrspond to non oscillatory mods. Th valu of wind spd, U f, at which th ral part of a complx conjugat pair turns positiv, marks th onst of fluttr and is rfrrd to as th fluttr critical wind spd. Th valu of wind spd, U d, at which on of th ral ignvalus bcoms positiv rprsnts th divrgnc wind spd. Th xact valus of th pols of th systm subjctd to unstady arodynamic forcs can b also found through th frquncy domain analysis. In this approach th pols and th mod shaps of th systm ar givn as solution to th gnralizd ignvalu problm of th following form M CKFU, v, (.4) whr F is th Laplac transform of unstady arodynamic forcs. Th solution of ignproblm (.4) rquirs tdious itrativ sarch for ignvalus. Usually, th solution tchniqus attmpt to idntify only th critical fluttr wind spd, U f, which corrsponds to a purly imaginary ignvalu i [6]. Th mod tracing mthod [4] also nabls computations of changs of systm pols with wind spd. Th mthod is basd on th assumption that th dynamic charactristics do not chang radically with a small incras in wind spd. Th itration procdur starts at zro wind spd and assums known frquncy and damping of fr vibrations of th mod at hand. Thn th wind spd is incrasd by a small intrval and th ignvalu is

32 .. Bridg-surfacs control systm 3 adjustd to satisfy (.4). Th itrations continu till th dsird wind spd is achivd and all th mods of intrst hav bn tracd...3. Numrical simulations..3.. Uncontrolld bridg Th considrd bridg has a main span of 3 m, th sid spans ar m, th towrs ar 3 m high, th sag of th cabls is 3 m and th width of th dck is 3 m. Th dynamic paramtrs of th sctional modl corrspond to th first torsional and first vrtical bnding mod of th considrd bridg. Th natural frquncis and damping ratios of pitching and having of th sctional modl ar givn in Tabl.. Tabl. dck width Dimnsions, frquncis and damping rations of th bridg sction B d = 3 m mass of th dck m = kg/m mass momnt of inrtia of th dck I = kgm /m natural frquncy of having motion h =.47 rad/s natural frquncy of pitching motion =.97 rad/s damping ratio of having mod h =.83 damping ratio of pitching mod =.7 Th variations of dynamic proprtis of th uncontrolld bridg for wind spd from m/s up to m/s ar shown in Fig..6. Fig..6a shows variation of th pols of th bridg dck on th complx plan. Dots rprsnt th position of th pol on th complx plan for th conscutiv wind spd. Th initial wind spd of m/s is dnotd by th squar mark and th following wind spds of 4, 6, 8 and m/s ar dnotd by circl, cross, triangl and black circl, rspctivly. In gnral, thr ar no changs in pols position at spd from to m/s. For no wind condition mod is a purly having mod, whras mod is purly torsional. Th variation of mod is associatd with th dcras of th ral parts whil th variation of pols of mod for wind rang from m/s to 6 m/s is du to th dcras of th imaginary parts. At wind spd of U f = 54 m/s th pols of mod cross th imaginary axis marking th onst of fluttr. Th variation of pols is also prsntd in trms of th modal frquncy and damping. Th chang of th frquncy of th mods with th incras of wind spd is shown in Fig..6b. Th distanc btwn frquncis of mod and mod dcras with th incras of th wind spd. Such variation of frquncis is charactristic for th classical two dgrs of frdom coupld fluttr [67]. Th modal damping ratios of th mods ar givn in Fig..6c. Th damping of mod bcoms ngativ at wind spd of 54 m/s. This wind spd is considrd as th fluttr wind spd. For wind spd abov 54 m/s th damping ratio of mod quickly dcrass which indicats strong instability of this mod at high wind spd rang. Sinc th damping ratio of mod rapidly incrass with th wind, this mod is stabl for th considrd wind rang. Th instability du to th ral pol, corrsponding to divrgnc, occurs at wind spd of U d = 84.5 m/s.

33 3. Study on sctional modls of passiv fluttr supprssion systms a) Variation of systm pols with th wind vlocity Imaginary part ~ m/s m/s 4 m/s 6 m/s 8 m/s m/s Mod Mod U f =54m/ s Arodynamic Mods U d =84.5m/ s Ral part b) Frquncis vs. wind vlocity c) Damping ratios vs. wind vlocity frquncy (rad/s). mod mod wind vlocity (m/ s) damp. ratio (nondim.) mod mod wind vlocity (m/ s) d) Amplitud ratios vs. wind vlocity ) Phas shifts vs. wind vlocity amp. ratio (nondim.) mod mod wind vlocity (m/ s) phas (rad) mod mod wind vlocity (m/ s) Fig..6. Variation of pols and ignvctors of uncontrolld bridg vs. wind vlocity Th normalizd with rspct to dck width amplitud ratio of th having componnt of th complx ignvctor, h/b d, ovr th amplitud of pitching,, is shown in Fig..6d. Du to th strong arodynamic coupling btwn pitching and having motion th contribution of th pitching componnt in mod incrass with th wind spd, whil in mod th having componnt is dominant for th whol considrd wind rang. For th wind spd ovr 6 m/s both mods bcom having dominant. Th contribution of th having and pitching in both mods is shown in Fig..6d. Th phas shift btwn th pitching

34 .. Bridg-surfacs control systm 33 componnt and having componnt for mod, h, and mod, h, ar shown in Fig..6. Th variation of th phas shifts ar larg ovr th considrd wind rang. It can b statd that for mod th pitching componnt is always lading th having componnt, whil for mod th pitching componnt lags th having on. With th incras of wind spd th valus of phas shift for mod ar approaching zro, whil phas shift of mod ar raching / Control systm Th hings of th control surfacs ar locatd in th middl of th control surfacs and th position of th hing lin is = = B d / for both control surfacs. Th siz of control surfacs is B = B = 3. m. Th position of th control surfacs as wll as thir siz was suggstd in [77]. 4 systm U Critical wind vlocity (m/s) systm systm 3 systm 4 U control gain c Fig..7. Critical wind vlocity of bridg-surfacs control systms vs. control gain Th optimal paramtrs of th control systms ar lookd for through sarch of th control gains which giv th maximum improvmnt in critical wind vlocity. It is assumd that th absolut valus of gain for th lading and trailing surfac ar th sam and th pndulum has vry long natural priod so that it rmains in th vrtical position during th vibration of th bridg. Th critical wind vlocitis vrsus control gains of th bridgsurfacs control systm ar shown in Fig..7. Th maximum critical wind vlocity of th bridg-surfacs control systm is obtaind for th following gains c 4., c 4.. (.43) Fig..7 shows critical wind vlocity of th control systms with all possibl connctions btwn th torsional displacmnt of th bridg and th rotation of th surfacs. Th control systm 3 rotats th surfacs in th opposit dirction than control systm. Th critical wind vlocity of this systm dcrass with th wind spd and instability is du to divrgnc. Th control systm 4 rotats th surfacs in th opposit dirction than control

35 34. Study on sctional modls of passiv fluttr supprssion systms systm and th stability of th bridg with this control law is also rducd. Th critical wind vlocity for all considrd gains is lowr than th fluttr wind spd of th uncontrolld bridg. Th instability of th control systm 4 is du to th oscillatory mod. Us of inappropriat control stratgy has an advrs influnc on th stability of th control systm. a) Variation of systm pols with wind vlocity Imaginary part ~ m/s m/s 4 m/s 6 m/s 8 m/s m/s Mod Arodynamic Mods Mod U f =3m/ s Ral part b) Frquncis vs. wind vlocity c) Damping ratios vs. wind vlocity frquncy (rad/s). mod.8.6 mod wind vlocity (m/ s) damp. ratio (nondim.) mod mod wind vlocity (m/ s) d) Amplitud ratios vs. wind vlocity ) Phas shifts vs. wind vlocity amp. ratio (nondim.) 5 mod 5 mod wind vlocity (m/ s) phas (rad) mod mod wind vlocity (m/ s) Fig..8. Variation of pols and ignvctors of bridg-surfacs control systm vs. wind vlocity

36 .. Bridg-surfacs control systm 35 4 a) Damping of pndulum p = % b) Damping of pndulum p = 5% 4 Critical wind vlocity (m/s) m p /m = % m p /m = 3% m p /m = 5% Critical wind vlocity (m/s) m p /m = % m p /m = 3% m p /m = 5% 5 5 Priod T p (s) 5 5 Priod T p (s) Fig..9. Influnc of dynamic paramtrs of pndulum on th prformanc of bridg-surfacs control systm Th variation of dynamic proprtis of th bridg-surfacs control systm, dscribd by quation (.37) with control law (.43), is shown in Fig..8. In this simulation it is assumd that th pndulum rmains in th vrtical position all th tim. Th chang of pols location with th incrmnt of wind spd is smallr comparing to th pols variations of th uncontrolld bridg. Variation of th pols of mod is du to th ral parts, whras of mod du to th imaginary parts. Th fluttr condition for th control systm is du to th instability of mod at wind spd of U f = 3 m/s. Th action of th control surfacs rsult in addition of arodynamic stiffnss to mod. Th frquncy of mod incrass with th incras of wind spd and th frquncy of mod is almost constant (Fig..8b). Thrfor, th stabilizing arodynamic forcs prvnt th approach of natural frquncis of oscillatory mods to ach othr. Th damping of mod is positiv for th considrd wind rang although its valu is small for th whol considrd wind rang (Fig..8c). For th control systm, th contribution of th having componnt in mod (Fig..8d) is small, i.., mod is torsional dominant dspit th incras of wind spd. Th phas shift btwn th pitching componnt and having componnt for mod, h, is about / and for mod, h, is for all considrd wind rang (Fig..8). Th control systm, through th action of th control surfacs, disturbs coupling btwn th having and pitching and prvnts th divrgnc of th bridg. Th paramtric study on influnc of dynamic proprtis of pndulum on th critical wind vlocity of control systm is shown in Fig..9. Th rsults ar prsntd for th pndulum mass qual to %, 3% and 5% of th mass of th dck and for damping of th pndulum qual to % and 5%. Th critical wind vlocity of th control systm for all considrd pndulum masss and damping ratios hav a small pak at th priod of pndulum, T p, of about 6, 8 sconds and th spik at priod of 4 sconds. Ths paks ar du to th strong coupling of th pndulum mod with th pitching and having mod, rspctivly. Th fluttr mod has a significant pndulum displacmnt componnt which is coupld with pitching in such a way that th amplitud of control surfacs motion is significantly incrasd. For T p bing mor than 3 sconds, th critical wind spd of th systm

37 36. Study on sctional modls of passiv fluttr supprssion systms monotonically incrass. Howvr, th pndulum with th mass ratio abov % and damping ratio 5% rsults in an undsirabl coupling btwn th pndulum and th torsional mods which lads to th dcras of th fluttr wind spd of th systm for T p xcding sconds. a) Variation of systm pols with wind vlocity Imaginary part ~ m/s m/s 4 m/s 6 m/s 8 m/s m/s Mod Mod. Arodynamic Mod U d = 8m/ s Ral part b) Frquncis vs. wind vlocity c) Damping ratios vs. wind vlocity frquncy (rad/s). mod mod wind vlocity (m/ s) damp. ratio (nondim.) mod mod vlocity (m/s) d) Amplitud ratios vs. wind vlocity ) Phas shifts vs. wind vlocity amp. ratio (nondim.) mod mod wind vlocity (m/ s) phas (rad) mod mod mod wind vlocity (m/ s) Fig... Variation of pols and ignvctors of bridg-surfacs control systm vs. wind vlocity

38 .. Bridg-surfacs control systm 37 a) Damping of pndulum p = % b) Damping of pndulum p = 5% Critical wind vlocity (m/s) m p /m = % m p /m = 3% m p /m = 5% 5 5 Priod T p (s) Critical wind vlocity (m/s) m p /m = % m p /m = 3% m p /m = 5% 5 5 Priod T p (s) Fig... Influnc of dynamic paramtrs of pndulum on th prformanc of bridg-surfacs control systm Control systm Th width of th control surfacs and position of hings in th control surfacs ar assumd to b th sam as for th control systm. Th critical wind vlocity of th control systm vrsus control gain is show in Fig..7. For gains from to 3.6 th incras of gains rsults in improvmnt of th critical wind vlocity associatd with th oscillatory mod. Howvr, for gains of 3.6 to 6 th critical wind vlocity rmains at th lvl of 8 m/s. Th instability is du to divrgnc. For gains xcding 6 th critical wind vlocity dcrass du to fluttr. Control systm givs robust prformanc for th following control gains c 4., c 4.. (.44) Th variation of dynamic proprtis of th control systm is shown in Fig... Th pols of mod do not significantly chang th position on complx plan with th incras of th wind spd, whras th ral parts of pols of mod dcras. Although th control systm dos not sparat th stiffnss of mods as ffctivly as th control systm (Fig..8b) it adds larg amount of damping to mod for all th considrd wind spds. Th instability of th control systm is not du to th oscillatory mods but du to th arodynamic mod. Th instability occurs at wind spd U d = 8 m/s du to divrgnc. Th influnc of th dynamic proprtis of pndulum on th prformanc of th control systm is shown in Fig... This systm bcoms unstabl du to th non oscillatory mod, namly, to th arodynamic mod at wind spd of 8 m/s. Th coupling of th arodynamic mods with th pndulum mod is vry small for damping of th pndulum of %. Thus, th chang of th dynamic proprtis of th pndulum dos not affct th critical wind vlocity of th systm. Th control systm with pndulum of larg damping, 5%, and mass ratio abov % bcoms unstabl du to th oscillatory mod in th sam fashion as th control systm (Fig..b).

39 38. Study on sctional modls of passiv fluttr supprssion systms Th paramtric study on th influnc of th pndulum dynamic proprtis on th control systms critical wind vlocity indicats that th input of damping to th control systms through th incras of th pndulum damping is not th solution for th improvmnt of stability of th control systm. Th most rliabl solution is th pndulum with small mass and vry long natural priod so that thr is no motion of pndulum. In such cas th rotation of th control surfacs is govrnd only by th torsion of th bridg dck... Bridg flaps control systm... Mathmatical modl of arodynamic forcs Th control systm consists of two flaps connctd dirctly to both dgs of th bridg dck (Fig..). Th stabilizing forcs countracting fluttr and divrgnc of th bridg dck ar gnratd on th lading and trailing flap rotatd with rspct to th motion of th girdr. lading flap b b bridg dck trailing flap b cntr of gravity h U M air flow M L h M Fig... Arodynamic forcs acting on bridg dck and control flaps Complx flow-structur intraction for bluff dck cross sctions prohibits th dvlopmnt of satisfactory solutions dparting from th fundamntal fluid flow principls. Howvr, for a stramlind bridg dck th thortical solutions, basd on potntial flow thory, giv a good agrmnt with th xprimntal rsults for small amplitud oscillations [67]. Th unstady arodynamic forcs acting on th dck and lading and trailing flaps (Fig..) ar T F Lb h M M M. (.45)

40 .. Bridg-flaps control systm 39 whr L h is a lift forc, and M, M, and M ar torsional momnts acting on th rspctiv dgrs of frdom. Th arodynamic forcs (.45) ar formulatd for th following vctor of variabls dscribing th motion of th systm T q h/ b. (.46) Th analytical formulation of th arodynamic forcs (.45) is drivd from th formulation of unstady arodynamic forcs for th wing-ailron-tab (Fig..3) combination drivd by Thodorsn and Garrick [69]. In thir solution th wing-ailron-tab systm is tratd as a flat plat and th analysis assums no lak of fluid in th gaps btwn th wing and th ailron and btwn th ailron and th tab. Th formulation was drivd from th potntial flow thory and undr th assumption of small amplitud oscillations of th systm mmbrs. Th wing-ailron-tab systm is considrd to hav a chord width of b, so that b is usd as a rfrnc lngth. Th gomtry of th systm is dscribd by th nondimnsional variabl such that th wing lading dg is at =, ailron lading dg at = c, tab lading dg at = d, and tab trailing dg at =. Th wing undrgos rotational motions about = a and ailron and tab hings ar locatd at = and = f, rspctivly. Th distanc btwn ailron lading dg and its hing, c, is dnotd as l, and that btwn tab lading dg and its hing, f d, is dnotd as m. b - a c d f cntr of gravity l m q q h ailron wing M q M q q tab L qh M q q Fig..3. Arodynamic forcs on a wing-ailron-tab systm du to Thodorsn and Garrick Having and pitching motion with rspct to th cntr of rotation of th wing ar dnotd by q h, q, q and q. Th angls of rotation of th ailron, q, and tab, q, ar assumd to b rlativ to th wing. Th vctor of displacmnts is chosn as and th corrsponding vctor of th gnralizd forcs is T q t qh / b q q q, (.47) T P Lqhb Mq Mq Mq. (.48)

41 4. Study on sctional modls of passiv fluttr supprssion systms Th unstady arodynamic forcs acting on a wing-ailron-tab combination, according to Thodorsn and Garrick, may b rprsntd as U U a t a t a t Cp U t Cp U t. P M q b C q b K q b RS q b RS q (.49) Th man vlocity of th oncoming wind is dnotd by U. Th matrics M a, C a, K a, R, S, and S dpnd on th systm gomtry, namly th location of th wing rotation cntr, flaps siz and location of hings. Th xact formulas ar drivd from Thodorsn and Garrick [69] and ar shown in Appndix B. Th function C( ŝ ) apparing in (.49) is th gnralizd Thodorsn function [44], which is xprssd in trms of th Hankl functions H ( n,) of nondimnsional Laplac variabl, p, as n H p C sˆ, H p ih p (.5) whr p is normalizd with rspct to half of chord width i.., p = sb/u. Th formulation (.49) dscribs full arodynamic intraction btwn systm mmbrs. Th total unstady arodynamic forcs can b dividd into two groups: th noncirculatory ons and th circulatory ons. Th first thr trms of (.49) rprsnt th noncirculatory part which dos not dpnd on th vorticity in th wak. Th circulatory part (th last two trms) taks account of th vorticity of th wak gnratd at th trailing dg. Vctors dscribing th motion of th bridg-flaps systm and wing-ailron-tab combination ar rlatd through linar transformation q Tq. (.5) Consistntly, th rlationship btwn th slf-xitd arodynamic forcs is givn as whr t T F T P, (.5) ( dc a) ( a) T, (.53) and th location of th lastic cntr of th dck, nondimnsionalizd with rspct to b, is dnotd by d c. Introduction of (.5) into (.49) yilds th formula for th unstady arodynamic forcs of th bridg-flaps systm T U T U T a a a FT M Tq T C Tq T K Tq b b U T TRSTq. U T Cp TRSTq Cp b b (.54)

42 .. Bridg-flaps control systm 4 In cas of unstady arodynamic forcs obtaind from th wind tunnl xprimnts, th rational function approximation has bn applid to th forc cofficints consisting of fluttr drivativs. Howvr, in cas of thortically drivd formula for th arodynamic forcs (.54), th approximation can b don dirctly on th Thodorsn function. Thus, th approximation of th gnralizd Thodorsn function by rational functions of th nondimnsionalizd Laplac variabl is n l ai C pa, (.55) i p bi ~ whr C ( p ) dnots th approximation. Th partial fractions, ai /(p + b i ), ar calld lag trms and th cofficints of th partial fractions, b i, ar rfrrd to as lag cofficints as in Sction... Th Laplac transformation of (.54) with th approximatd formula for th Thodorsn function (.55) and zro initial condition yilds T L ( F) T M as U R b n l U b ( C ai s ( U a U ars) s ( K b S ) b bi U s S TL ( q). b a a RS ) (.56) whr s is th Laplac variabl. Not that L(F) and L(q) dnot th Laplac transforms of th ral tim dpndnt vctors F and q, rspctivly. Th arodynamic stats ar dfind in th Laplac domain as U U L ( xa) si Rb b EL (q) (.57) b Th matrics apparing in (.57) hav th following forms: T T bn l nms l E b S T S T, R b diag b,..., b. nl (.58) Taking th invrs Laplac transformation of (.56) and (.57) rsults in th tim domain formulation of slf-xitd arodynamic forcs givn as follows T U T FT MaTq T Ca arstq b U b U U x a Rbxa Eq. b b n l T U T T K a ars airstq T R a an x l a i b, (.59) Th arodynamic forcs dscribd by (.59) ar indpndnt of th rducd frquncy. Th tim domain formulation for arodynamic stats is a first ordr diffrntial quation.

43 4. Study on sctional modls of passiv fluttr supprssion systms Th Hänkl functions, which ar usd to dtrmin th gnralizd Thodorsn function, hav analytical continuation in th whol p-plan xcpt th branch cut-off along th ngativ ral axis [6]. Th gnralizd Thodorsn function is in fact dfind ovr a twodimnsional domain and its approximation is a fairly difficult task. Howvr, if our intrst is rstrictd only to purly imaginary valus of th nondimnsionalizd Laplac variabl, p = ik, wr k = b/u, th approximation problm is gratly simplifid. Sinc th fluttr phnomna occur for points in th p-plan which li along th imaginary axis, this rstriction may b justifid. Th approximation formula for th gnralizd Thodorsn function (.55) with th argumnt p = ik also has analytical continuation in almost all points in th p-plan. Thus, it is supposd that slightly dcaying or growing motions would b wll approximatd. Howvr, th approximation may not b appropriat for th prdiction of strongly dampd motions. ~ As th paramtr k varis in th rang from to, C(k) and C ( k ) trac two curvs in th complx plan. Th objctiv of th approximation procdur is to minimiz th rror of fitting th xact curv with th approximatd on. This rror can b valuatd as an intgral B D C k, C k dc k, (.6) A ~ ~ whr D(C(k), C ( k ) ) dnots th distanc btwn points of curvs C(k) and C ( k) corrsponding to th sam valu of paramtr k, and dc(k) is th lngth of an infinitsimal arch of C(k). A and B ar th trminal points of this part of th xact curv which is approximatd. Th intgral (.6) can b writtn in a paramtric form with rspct to k as whr kup klow dc k CkC k dk, (.6) dk / dc k H k H k H k k H k i dk H k ih k. (.6) and k low and k up ar th lowr and uppr bounds of th rducd frquncy intrval on which fitting is prformd. Th slction of th siz of th approximation function of th gnralizd Thodorsn function (.55) is a trad-off btwn th accuracy of th fit in th slctd rducd frquncy rang and th siz of th rsulting quation of motion. Hnc, th choic of th rducd frquncy intrval on which approximation is prformd rquirs carful considration. For larg siz of this intrval th rsulting numbr of lag trms would b also larg. On th othr hand, a narrow intrval dos not rquir many arodynamic stats, but applicability of th modl will b rstrictd. Th intrval of rducd frquncy for th considrd bridg modl that covrs th rspons of th bridg at wind spds of intrst is found to b k low =.7 and k up =.5. Th numrical mthod usd for approximation is th squntial simplx mthod proposd by Mldr and Mad [].

44 .. Bridg-flaps control systm 43.5 Thodorsn's function approximation lag trm lag trms Imaginary part k=.5 k=.7 3 lag trms Ral part Fig..4. Approximation of th Thodorsn function with diffrnt numbr of lag trms Th rsults of th approximation with, and 3 lag trms togthr with th Thodorsn function ar shown in Fig..4. Th Thodorsn function rprsnts a smooth curv in th complx plan. Thrfor, lag trm approximation givs rlativly good fitting. Howvr, th fitting with lag trm is not satisfactory for a low rducd frquncy. Th approximation with lag trms significantly improvs th fitting, rducs th approximation rror and provids a good compromis btwn th accuracy of th fit and th numbr of lag trms, and is hrin mployd. Th corrsponding approximation function is.3.8 C p.54. (.63) p.7 p.43 Notic that this approximation rsults also in small rror for k =, and hnc is also supposd to provid good assssmnt of nonoscillatory motion of th bridg dck rlatd to th divrgnc typ of instability.... Equation of motion of bridg-flaps control systm Th cross sction of a passiv bridg-flaps control systm is shown in Fig..5. A proposd passiv control systm consists of auxiliary flaps attachd dirctly to th bridg dck. Whn th dck undrgos pitching motion, th control flaps rotation is govrnd by additional control cabls spannd btwn th flaps and an auxiliary transvrs bam supportd by th main cabls of th bridg. Sinc cabls can only pull th flaps but not push thm, additional prstrssd springs ar usd to forc rvrs motion of th control surfacs.

45 44. Study on sctional modls of passiv fluttr supprssion systms b x c b c x c supporting bam main cabl U b bridg dck h control systm b control systm M prstrssd spring M L h c M c Fig..5. Cross sction of passiv dck-flaps control systm Two control systms ar considrd. Control systm assums that th nos up torsional displacmnt of th dck causs nos down and nos up torsional displacmnts of th lading and trailing surfac, rspctivly. Control systm provids nos down torsional displacmnts of both control surfacs for nos up torsional displacmnt of th dck. Control systm rquirs a chang of th connctions btwn control cabls and flaps for wind spd coming from th opposit dirction, whras control systm has symmtric connctions and is qually fficint for both dirctions of incoming wind. Th gain of th control systm i.., th proportion of coupling btwn pitching of th dck and

46 .. Bridg-flaps control systm 45 rotation of th control surfacs can b changd by shift of th point whr control cabls ar attachd to th flaps. A cross sction of th bridg-flaps control systm is assumd to hav four dgrs of frdom: vrtical and torsional displacmnt with rspct to th lastic cntr of th dck, dnotd by h, and rfrrd to as having and pitching motion, rspctivly, and rlativ rotational displacmnts of lading and trailing control flap, dnotd as and. Th dck togthr with flaps has a chord width of b. Th location within th systm is dscribd with rspct to th dck lastic cntr by variabl r, and th lading and trailing flap hings ar positiond at and, rspctivly. Th distanc btwn main cabls axs is dnotd by b c, locations of th support points of th lading and trailing flap by control cabls ar c and c. Locations of th support points of th additional cabls on th transvrs bam ar x c and x c. Th control cabls can only pull th flaps, and in ordr to govrn thir rvrs motion, th springs at th dck-flaps connctions must b prstrssd. It is assumd that aftr prstrssing th displacmnts of th systm ar zro and th vctor dscribing th motion of th systm is th sam as dfind in (.46), q T h/ b, and dnots displacmnts with rspct to th prstrssd configuration. Th horizontal motion of th dck and of th main cabls, as wll as changs in hangr lngth ar nglctd in th sctional study. Th quation of motion of th passiv systm is. s s s sup prs Mq Cq K K qff (.64) Mass and damping proprtis of th control cabls and th supporting bam ar nglctd, and hnc th matrics M and C ar s s whr mb Sb Sb Sb chb I S I S I c Ms, C s, (.65) I c sym. I sym. c m dm, S dm, S dm, S dm, Ad AlAt Ad AlAt Al At Ad AlAt Al At I dm, I dm, I dm. (.66) Th total mass of th systm is m, S is th first ordr momnt of inrtia of th systm about th dck lastic cntr, S and S ar th first ordr momnts of inrtia of th lading and trailing flaps about thir hings, rspctivly, I is th scond ordr momnt of inrtia of th systm about th dck lastic cntr, I and I ar th scond ordr momnts of inrtia of th lading and trailing flaps about thir hings, rspctivly. Th domains of intgration in (.66), A d, A l and A t, ar th dck, th lading flap and th trailing flap cross sctions, rspctivly.

47 46. Study on sctional modls of passiv fluttr supprssion systms Th stiffnss of th systm consists of two parts, K s and K sup. Matrix K s dscribs stiffnss of th dck-flaps systm without control cabls and supporting bam kb h k K s, (.67) k sym. k whr k i (i = h,,, ) rprsnt th stiffnss cofficints of corrsponding displacmnts. Matrix K sup rprsnting th stiffnss of th supporting bam and th cabls is found to b. K TK K K K T (.68) sup q b b cc cc q Matrix T q dscribs th transformation of displacmnts of th dck and flaps, q, into th total vrtical lastic displacmnts of th supporting bam and control cabls, i.., Th stiffnss matrix of th supporting bam is x c c c T q. (.69) c xc c kb kb K b, kb k (.7) b whr k bii (i =, ) ar stiffnss cofficints of th bam du to rotation of th lading and trailing flap. K cc is th stiffnss matrix of th control cabls. Th control cabls ar assumd to bhav linarly lastic undr tnsion forcs, and lack any stiffnss against comprssion forcs. Thrfor, K cc taks th following valus kcc k cc Kcc, cc, cc, cc. k K cc K k K cc (.7) Th stiffnss of th control cabl du to rotation of th lading and trailing flap ar k cc and k cc, rspctivly. Th first stiffnss matrix in (.7) is assumd whn both cabls ar in tnsion, th scond and th third matrix is takn whn only lading or trailing cabl is in tnsion, and matrix with zro cofficints is applid whn non of th cabls is in tnsion. Th condition for xistnc of tnsil forc is chckd by computing th sign of th summation of lastic displacmnts of th supporting systm, i.., th supporting bam and th control cabls, du to vibration and prstrssing. Th forcs F prs in th systm quation of motion (.64), ar du to prstrssing of th springs at th dck-flaps connctions and can b dfind in trms of th prstrssing momnts M k, M k, (.7) whr and ar initial displacmnt of th flaps. Th formulas for F prs also dpnd on th xistnc of tnsil forcs in th control cabls and ar givn as

48 .. Bridg-flaps control systm 47 F F x, F prs M M, c T T c c prs T prs x M M F M M c c T, prs. c (.73) Th first vctor F prs in (.73) is assumd whn both cabls ar in tnsion, th scond and third vctor ar takn whn only lading or trailing cabl is in tnsion, and th forth vctor in (.73) is assumd whn non of th cabls is in tnsion. Finally, th stat spac rprsntation of quation of motion (.64) with rational function formulation of unstady arodynamic forcs, F, acting on th control systm (.59) is whr q M C q I x a M ( K K sup ) M D q M q ( U b) E ( U b) R b xa nl T U T s a s a a ai b i M M T M T, K K T K RS RS T, U T U T s a a a an l CC T C RS T, D T R. b b F prs, (.74) (.75) Equation (.74) is nonlinar du to th variabl stiffnss of th control cabls which is dtrmind du to th condition of th xistnc of tnsion forc in th control cabls. Th cas whn on or both control cabls ar not undr th tnsil forc can rsult in th undsirabl motion of flaps. Thrfor, for practical rasons th control cabls should always rmain taut. Numrical simulations of th rspons of th bridg undr wind action [7], showd that for sufficintly larg prstrssing momnts, M and M, and initial angls of rotation, and, th supporting cabls may always rmain in tnsion. Morovr, for th supporting systm with adquatly larg stiffnss, th rotation of th flaps can b assumd to b proportional to th rotation of th dck, i.., c, c, (.76) whr control gains, c and c, ar dtrmind from th gomtry of th control systm as x c, c. c c c c c c (.77) Finally, th simplifid stat spac quation of motion drivd from (.74) bcoms q q z d d a M c C I c M c K ( U b) E c c M c Dc ( U b) R b q q z d d a. (.78) Th vctor of th prstrssing forcs is omittd sinc th bridg vibrations about th quilibrium point ar of intrst. Th matrics of (.78) ar dtrmind as follows

49 48. Study on sctional modls of passiv fluttr supprssion systms T T T M M M TTM TM T, C C C TTC TC T, c dd df fd ff c dd df fd ff T T T K K K TT K T K T, D D T D E E E T. c dd df fd ff c d f c d f (.79) Th abov matrics (.79) ar obtaind from th partition of th global systm matrics with rspct to th dgrs of frdom corrsponding to th dck, q T d = [h/b ], and th flaps, q T f = [ ], in th following fashion qd Mdd Mdf Cdd Cdf q, M, C, q f M fd M ff Cfd Cff (.8) Kdd Kdf Dd K, D, E Ed Ef. K fd K ff Df Matrix T dfins th rlationship btwn th controlld systm displacmnt vctor, q d, and th vctor of displacmnt of flaps, q f, q f Tq d, (.8) and is givn as c T. c (.8) Finally, a compact form of th simplifid stat-spac quation can b writtn in th following form x Ax. (.83) Th simplifid stat-spac quation of motion (.83) is linar for th slctd wind spd U...3. Numrical simulations..3.. Uncontrolld bridg Th dynamic charactristics for numrical simulations of th passiv arodynamic control rprsnt a sctional modl of a suspnsion bridg with a stramlind dck and th main span of 3 m (Tabl.). Th variations of pols of th bridg dck with th xact and approximatd formulation of arodynamic forcs for wind spd from m/s up to m/s ar shown in Fig..6. Th xact pols of th systm ar computd by th itrativ mod tracing mthod [4] and ar dnotd in gray color. Th variation of pols of th systm with th approximatd arodynamics ar plottd in black. Th pols computd with th approximation ar in a vry good agrmnt with th xact pols. Small discrpancis can b obsrvd btwn th rsults of th approximation and th xact formulation for mod at wind spd abov 55 m/s. Nvrthlss, th fluttr wind spd, for both mthods of modling of arodynamic forcs, is found to b U f = 53 m/s. At this wind spd th ral part of ignvalu corrsponding to th mod (pitching dominant at low wind spd) bcoms positiv. Th arodynamic pol of th bridg modl with RFA bcoms unstabl at a wind spd of U d = 7 m/s and marks onst of divrgnc. Th divrgnc wind spd for th systm computd by xact computations is also 7 m/s.

50 .. Bridg-flaps control systm 49 Imaginary part a) Variation of systm pols with wind vlocity Mod Mod U f =53m/ s ~ m/s m/s 4 m/s 6 m/s 8 m/s m/s. Arodynamic Mod U d =7m/ s Ral part b) Frquncis vs. wind vlocity c) Damping ratios vs. wind vlocity frquncy (rad/s). mod mod wind vlocity (m/ s) damp. ratio (nondim.) mod mod wind vlocity (m/ s) d) Amplituds vs. wind vlocity ) Phas shifts vs. wind vlocity amp ratio (nondim.) mod mod wind vlocity (m/ s) phas (rad) mod mod wind vlocity (m/ s) Fig..6. Variation of pols and ignvctors of uncontrolld bridg vs. wind vlocity Mod shaps of th oscillatory mods ar shown in Fig..6d. At zro wind spd mod and mod ar purly having and pitching, rspctivly. Howvr, as wind spd incrass both mods bcom having dominant (Fig..6d). Although phas shifts btwn having and pitching componnts xhibit larg variations in th considrd wind vlocity rang (Fig..6), for mod th pitching componnt always lads th having componnt, whras for mod it always lags.

51 5. Study on sctional modls of passiv fluttr supprssion systms..3.. Control systm Th dynamic analysis of control systm is conductd in this sction on th systm with optimal configuration. It has bn found that th bridg-dck control systm is th most ffctiv in nhancing th critical wind vlocity whn th hings of th flaps ar attachd dirctly to both dgs of th bridg [8]. Th optimal flap siz and control gains has bn obtaind in [8] with rspct to th prformanc indx basd on systm dgr of stability [9]. Absolut valus of th control gains, c and c, wr confind to intrval c. a) Variation of systm pols with wind vlocity Imaginary part ~ m/s m/s 4 m/s 6 m/s 8 m/s Mod Mod Arodynamic Mods Ral part b) Frquncis vs. wind vlocity c) Damping ratios vs. wind vlocity frquncy (rad/s) mod mod wind vlocity (m/ s) damping ratio (nondim.) mod mod wind vlocity (m/ s) Fig..7. Variation of pols of bridg-flaps control systm vs. wind vlocity Th prstrssing momnts hav bn dtrmind from th rspons of th control systm du to action of wind of vlocity of 8 m/s and turbulnc intnsity of vrtical componnt qual to 4.%. Th wind conditions hav bn gnratd from th fild masurmnts during typhoon [78]. Th obtaind prstrssing momnts and rsulting tnsion forcs in th control cabls yild th ncssary stiffnss of th supporting bam and control cabls. Th paramtrs of th optimal control systm ar givn in Tabl.. Fig..7a shows th pols of th controlld dck govrnd by th simplifid quation of motion (.83) with th configuration corrsponding to optimal paramtrs (Tabl.). Th action of flaps pushs both mods away from th imaginary axis i.., th damping

52 .. Bridg-flaps control systm 5 ratios of th considrd mods ar significantly incrasd (Fig..7c). Th stabilizing forcs ffctivly kp frquncis of both mods far from ach othr (Fig..7b). Also th arodynamic mods ar stabl in all considrd wind spd rang i.., divrgnc of th bridg dos not occur. Th changs of shaps of mod and mod ar similar to changs of th corrsponding mods of th bridg-surfacs control systm (Fig..8d,). Paramtrs of th optimal control systm Tabl. width of flaps b = b = 3.5 m prstrssing momnts M = M = 3 KNm/m gain of lading flap c = 6.7 gain of trailing flap c =. stiffnss of flaps k = k =.4 KNm/m ( = = 3 ) damping of flaps = =. axial stiffnss of control cabls EA =.3 GN/m bnding stiffnss of supporting bam EI = 3. GNm connction of control cabls to th bam x c = 7.5 m, x c = 7.5 m Control systm Th dynamic analysis of bridg-flaps control systm is conductd on th systm with optimal configuration. Th hings of th flaps ar attachd dirctly to both dgs of th bridg [8]. Th optimal flap siz and control gains has bn obtaind [8] with rspct to th prformanc indx basd on th systm dgr of stability and prformanc indx dscribing th rlativ improvmnt of critical wind vlocity. Sinc control systm has symmtric cabl connction, absolut valus of th control gains ar assumd to th sam i.., c = c = c. Thn, th ncssary prstrssing momnts that guarant that th control cabls ar always in tnsion, hav bn dtrmind [8] for this control systm. Th prstrssing momnts hav bn dtrmind from th rspons of th control systm du to action of wind of vlocity of 65 m/s. This wind spd has bn chosn during th optimization procss bcaus control systm bcoms unstabl du to divrgnc at approximatly such wind spd for any flap siz. Th obtaind prstrssing momnts and rsulting tnsion forcs in th control cabls yild th ncssary stiffnss of th supporting bam and control cabls. Th paramtrs of th optimal control systm ar givn in Tabl.3. Not that this systm rquirs flap widths of only. m. Fig..8a shows th pols of th bridg-flaps control systm govrnd by th simplifid quation of motion (.83) with th optimal paramtrs (Tabl.3). It can b sn that th action of th control surfacs incrass th dumping of mod at highr wind spd (Fig..8c), and thus prvnts th onst of fluttr. Stiffnss of this mod dcrass with th incras of wind vlocity (Fig..8b). For wind vlocity from m/s to 6 m/s th pols of mod slowly mov away from th imaginary axis with almost no chang of th imaginary parts. For wind vlocity mor than 6 m/s th stiffnss of mod rapidly dcrass (Fig..8b). Th instability of th systm occurs du to th divrgnc U d = 7 m/s whn th ral mod turns unstabl. Mod at wind vlocity abov 6 m/s bcoms strongly

53 5. Study on sctional modls of passiv fluttr supprssion systms pitching dominant (as in Fig..d) which indicats that th control ffort of this control systm is put on th supprssion of having motion. a) Variation of systm pols with wind vlocity. Imaginary part.8.4 ~ m/s m/s 4 m/s 6 m/s 8 m/s Mod Mod Mod -.4 Arodynamic Mod U d =7m/s Ral part b) Frquncis vs. wind vlocity c) Damping ratios vs. wind vlocity frquncy (rad/s)..8.4 mod mod wind vlocity (m/ s) damping ratio (nondim.) mod mod wind vlocity (m/ s) Fig..8. Variation of pols of bridg-flaps control systm vs. wind vlocity Paramtrs of th optimal control systm Tabl.3 width of flaps b = b =. m prstrssing momnts M = M = 8 KNm/m gain of lading flap c = 4. gain of trailing flap c = 4. stiffnss of flaps k = k = 5.3 KNm/m ( = = ) damping of flaps = =. axial stiffnss of control cabls EA = MN/m bnding stiffnss of supporting bam EI = MNm connction of control cabls to th bam x c = 7.5 m, x c = 7.5 m

54 .3. Conclusions Conclusions Th drivation of th quation of motion for th sction of th bridg-surfacs and bridg-flaps control systms hav bn prsntd in this chaptr. Two typs of control stratgy, i.., th way of coupling th motion of surfacs and flaps with motion of th bridg dck, ar considrd. Th dynamic analysis of th control systms with th slctd paramtrs is conductd. Although th formulations of th arodynamic forcs on th bridg-surfacs and bridg-flaps control systms ar diffrnt, thr ar similaritis in th dynamics of thos systms. Control systm (antisymmtric connctions of th surfacs or flaps with torsion of th bridg) is ffctiv in supprssing th torsional motion of th girdr. Th arodynamic forcs gnratd on th surfacs and flaps countract th arodynamic momnt gnratd on th dck. This control stratgy also prvnts th divrgnc of th bridg by liminating th pitching componnt form th divrgnc mod. Control systms (symmtric connctions btwn th surfacs or flaps and dck torsion) rduc th having vibrations of th bridg. Th arodynamic forcs gnratd on th surfacs supprss th vrtical vibrations prvnting fluttr. Howvr, th control systms hav not influnc on th divrgnc wind vlocity. Th arodynamic forcs gnratd on th dck, lading and trailing surfacs, of th bridg-surfacs systm, ar assumd to hav no influnc on ach othr. Th motions of th lading and trailing surfac do not altr th flow around th bridg dck. Th formulation of th arodynamic forcs on th bridg-flaps systm taks into account full intraction btwn th forcs on flaps and dck. Not that th bridg-surfacs control systm satisfis th fluttr condition vn whn th flaps of a vry small width ar usd.

55 Chaptr 3 WIND TUNNEL EXPERIMENTS ON SECTIONAL MODELS OF AERODYNAMIC CONTROL SYSTEMS 3.. Exprimnt on bridg-surfacs control systm 3... Dscription of xprimnt Wind tunnl tsts on th bridg-surfacs control systm wr prformd in th boundary layr wind tunnl of th Univrsity of Tokyo. Th tstd sction of th bridg dck modl is shown in Fig. 3.. Th gomtry of th sction is idntical to th box sction proposd for th Akashi Kaikyo bridg by Fujino [5]. Th tstd modl was of a rigid timbr construction with stiffning aluminum vrtical ribs. Th modl had a lngth of 7 cm and width of 58.5 cm (Fig. 3.). Th control surfacs wr mad from stl plats and thir width was 6 cm. Th surfacs wr attachd to th dck through a stl bar systm so that th surfacs hing lins wr alignd with th dgs of th dck. Each surfac was mountd on four barings. Th bars conncting th surfacs with th pndulum wr attachd to th bar of th pndulum so that th point of connction could b changd (Fig. 3.3). Th allowabl variation of th gain was from.5 to. 585 (mm) ~ vrtical rib L** connction bar pndulum Control systm 87 6 Control systm 3 5 Fig. 3.. Modl of th passiv arodynamic control systm Two control systms wr invstigatd. Control systm had th control surfacs connctd to th pndulum in such a way that th phass of th lading and trailing sur-

56 3.. Exprimnt on bridg-surfacs control systm 55 facs, with rspct to th dck torsional displacmnt, wr both 8 dgr (Sction..). For control systm th lading surfac had a phas of 8 dgr whil th trailing surfac had a phas of dgr with rspct to th torsional displacmnt of th dck. Th lngth of th pndulum was limitd by th siz of th wind tunnl and was st to 9 cm. Th dimnsions and masss of th lmnts of th modl ar shown in Tabl 3.. Fig. 3.. Photograph of th sction of th bridg-surfacs systm in th wind tunnl Fig Connction btwn th pndulum and control surfacs

57 56 3. Wind tunnl xprimnts on sctional modls of arodynamic control systms Dimnsions and masss of th lmnts of xprimntal modl Tabl 3. width of th dck hight of th dck width of control surfacs position of surfacs hing lins lngth of pndulum diamtr of pndulum rod mass of dck sction mass of control surfacs mass of pndulum mass of pndulum s rod mass of connction bar systm mass of supporting systm addd mass du to actuators B d = 585 mm H = 67 mm B = B = 6 mm = = 9.5 mm l p = 9 mm d p = mm.56 kg.7 kg.4 kg.57 kg.3 kg.3 kg.6 4 kg Th modl was mountd on two flxural pivots supportd by two hlical springs which providd th rquird stiffnss for having motion. Flxibl stl strips wr usd for adjusting th torsional stiffnss of th modl. Th supporting and actuation systm of th tstd modl ar shown in Fig. 3.3 and Fig cm 58.5 supporting wirs Wind 7 Tst modl springs for vrtical motion 5 pndulum color targts Torsional spring 7 supporting systm camra Elctro-magntic actuators Fig Exprimntal stup Th displacmnt rsponss of thr points of th modl of th control systm, namly, both dgs of th dck sction and th nd of th pndulum, wr monitord by a

58 3.. Exprimnt on bridg-surfacs control systm 57 camra, Multi Truckr MUB-8, with sampling frquncy of 3 Hz and prcision of. mm. Th rcordd displacmnt data wr convrtd to digital signals. Th frquncis abov 3 Hz wr filtrd out from th masurd rcords by mans of th Fast Furir Transform and invrs Furir transform. Fig Viw on th xprimntal stup from outsid of th wind tunnl Th xprimntal program consistd of thr parts. Th first part consistd of fr vibration tsts in th still air. Th purpos of ths xprimnts was to thoroughly dtrmin th natural frquncis and damping ratios of having and pitching. Th dpndnc of th frquncis and damping on vibration amplitud and gain has bn indicatd in th prliminary study on th passiv arodynamic control systm [79]. Th tsts wr conductd for gains qual to.5,,.5 and.. Th initial condition for th having mod was th vrtical displacmnt of th modl by 7 mm and for th pitching mod th initial torsional angl of.7 radian. Th frquncy and damping of both mods wr computd as an avrag of thr cycls. Th scond part of th xprimnt consistd of fr vibration tsts of bridg dck without control surfacs undr action of wind of vlocity from 3 m/s to 9 m/s. Th purpos of ths tsts was to vrify th fluttr drivativs usd for numrical simulations. Th fluttr drivativs wr dtrmind for th sctional modl of a diffrnt scal [5]. Th surfacs wr rplacd by quivalnt lumpd masss addd to th supporting systm so that th dynamic paramtrs of th having and pitching mod of th dck wr idntical to th frquncis of th bridg dck with control surfacs and pndulum. Th initial condition for th tsts was, applid to torsional motion of th dck, fw cycls of forcd sinusoidal oscillations of amplitud of.3,.5 and.7 radians and frquncy Hz. Th sinusoidal oscillations wr forcd by four lctro-magntic actuators (Fig. 3.4). Th wind spd at which th systm undrwnt sustaind oscillations wr considrd as a fluttr wind vloc-

59 58 3. Wind tunnl xprimnts on sctional modls of arodynamic control systms ity. Th wind vlocity that causd non oscillatory incras of th displacmnts was considrd to b a divrgnc wind vlocity. Th third part of th xprimnt consistd of fr vibration tsts of th control systm and control systm undr th action of wind of vlocity from 3 m/s to 9 m/s. Both control systms wr tstd with both control surfacs attachd to th dck as wll as only lading and trailing surfac. Th rmovd control surfac was rplacd by quivalnt lumpd mass addd to th supporting systm so that th dynamic paramtrs of having and pitching mod of th dck wr idntical to th frquncis of th bridg dck with control surfacs and pndulum. Th initial xcitation was th sam as in th scond part of th xprimnt. Th gains varid from.5 to. with th stp of.5. All th tsts wr conductd for zro angl of attack. Although th static momnt cofficint, C M, of th tstd bridg sction is vry small, th bridg dck with th surfacs xhibitd considrabl torsional dformation du to forcs gnratd on th surfacs and pndulum. To rduc th static dformation of th pndulum du to a drag forc a splittr plat of lngth 6 cm and width cm was attachd to th rod of th pndulum. Such plat quitd th wak flow [6] and rducd th displacmnts of th pndulum to ngligibl valus. In addition, th initial angl of th control surfacs was slightly adjustd during ach tst so that th bridg dck subjctd to th slctd wind spd had zro angl of attack Exprimntal rsults Th rsults of th first part of th xprimnt conductd on th control systm ar shown in Fig. 3.6 and Fig Fig. 3.6a shows th rlation btwn th frquncy and amplitud of vibration of th having mod for all considrd gains. It was found that th frquncy of th having mod was indpndnt of th amplitud of vibration and gain. Th avrag valu of f h is.4 Hz. Fig. 3.7a shows that th damping ratio of th having mod for amplitud largr than mm is constant and th avrag valu is found to b h =.5. Th damping ratio is lagr than th avrag valu for amplituds smallr than mm. Th tim rsponss of ths tsts showd that thr was no coupling btwn having and pitching motions. Thus, th additional damping cam from th friction btwn th lmnts of th supporting systm. Sinc most of th xprimntal tsting was conductd for largr amplituds, this variation of damping was nglctd in th analytical studis. Fig. 3.7 shows th frquncy and damping of pitching vrsus amplitud of torsional vibration for th control systm with diffrnt gains. Excitation of th pitching mod rsults in motion of th control surfacs. Th rcordd tim historis showd that th torsional motion of th dck did not induc any vibration of th pndulum. Although thr ar som variations of frquncis (Fig. 3.7a) for diffrnt gains, th avrag valu of f =.55 Hz has bn assumd to b rprsntativ for all configurations of th control systm. Fig. 3.7b shows th variation of th damping ratio for th considrd gains. Th control systm govrnd by largr gains had largr damping ratios. Sinc th proportion of coupling btwn th pitching of th dck and th rotation of th control surfacs is govrnd by gain, th largr gain rsultd in largr damping that cam from ight barings usd for th control surfacs. Nvrthlss, th diffrncs in damping ratios for diffrnt gains ar ngligibl sinc th arodynamic damping is much highr than th structural on. For th simulations th avrag valu of damping ratio of =.7 has bn usd.

60 3.. Exprimnt on bridg-surfacs control systm 59 a) Frquncy.3.8 frquncy (Hz).6.4 c=.5 c=. c=.5 c=... b) Damping ratio.5 damping ratio (nondim.) amplitud(mm) c=.5 c=. c=.5 c= amplitud(mm) Fig Variation of frquncy and damping of having mod vs. amplitud of vibration Th variation of th frquncy of pndulum mod was found to b ngligibl and th avrag valus was f p =.566 Hz. Th structural damping of th pndulum was vry small and considrd as p =. Th arodynamic damping of th pndulum was calculatd as for th bar with th circular cross-sction [6]. For vibration of th pndulum with small amplitud, th formula for th damping cofficint has bn drivd as c a p dbulp CD.5. (3.) Th dnsity of air is =. kg/m 3, d b dnots diamtr of th pndulum bar, l p is a total lngth of th pndulum, and th drag cofficint of th circular cross-sction is assumd to b C D =. [6]. Th natural frquncis and damping ratios of th considrd mods of th xprimntal modl ar summarizd in Tabl 3..

61 6 3. Wind tunnl xprimnts on sctional modls of arodynamic control systms a) Frquncy.65.6 frquncy (Hz).55.5 c=.5 c=. c=.5 c= amplitud(rad) a) Damping ratio.5 c=.5 c=.. c=.5 c=. damping ratio (nondim.) amplitud(rad) Fig Variation of frquncy and damping of pitching mod vs. amplitud of vibration Frquncis and damping ratios of th modl in th still air Tabl 3. Having mod Pitching mod Pndulum mod f h =.4 Hz y =.5 f =.55 Hz =.7 f p =.566 Hz p =.63 U In th scond part of th xprimnt, dvotd to th tst on bridg dck without control surfacs and pndulum, it was found that th fluttr wind vlocity of th bridg dck modl dpnds on th amplitud of initial xcitation. Th fluttr vlocitis du to thr amplituds

62 3.. Exprimnt on bridg-surfacs control systm 6 of initial xcitation as wll as th thortical prdiction ar givn in Tabl 3.3. Th thortical prdiction is vry clos to th xprimntal fluttr wind vlocity obtaind for initial amplitud of.5 radian. a) tim rsponss at wind spd of 3 m/s.6 xprimnt nondim. disp. having pitching 5 5 tim (s) nondim. disp thory having pitching 5 5 tim (s) b) tim rsponss at wind spd of 6 m/s. xprimnt nondim. disp. having pitching tim (s) nondim. disp having pitching thory tim (s) Fig Exprimntal and thortical tim rsponss of modl without control surfacs

63 6 3. Wind tunnl xprimnts on sctional modls of arodynamic control systms Fluttr wind vlocitis of bridg dck without control surfacs Tabl 3.3 Amplitud of initial xcitation (rad.) Fluttr wind vlocity (m/s) Analytical prdiction 4. a) tim rsponss at wind spd of 3 m/s.6 xprimnt nondim. disp. nondim. disp having pitching 5 5 tim(s) thory having pitching 5 5 tim (s) b) tim rsponss at wind spd of 6 m/s.6 xprimnt nondim. disp. nondim. disp having pitching 5 5 tim (s) thory having pitching 5 5 tim (s) Fig Exprimntal and thortical tim rsponss of control systm with gain c =

64 3.. Exprimnt on bridg-surfacs control systm 63 Critical wind vlocity (m/s) Thory.3 rad.5 rad.7 rad.5.5 gain c(nondim.) Fig. 3.. Critical wind vlocity of th bridg-surfacs control systm Exprimntal and thortical tim rsponss of th dck du to initial xcitation of amplitud.5 radian at wind spd of 3 m/s ar plottd in Fig. 3.8a. Th rspons of th having motion is normalizd by th width of th dck. Th initial conditions for th thortical prdication ar obtaind from th xprimntal rcord for th tim t = scond. At this wind spd th rational modl slightly ovrstimats th damping and th amplitud of pitching motion of th systm. Th xprimntal rspons of th systm at wind spd of 6 m/s is shown in Fig. 3.8b. At this wind spd th dck is unstabl and th motion could not b inducd by initial conditions usd for th stabl dck, sinc th quick build up of vibration amplitud could hav damagd th modl. Th motion was inducd by a small turbulnc prsnt in th flow without any initial conditions. Th vibrations wr stoppd whn th amplitud of th pitching motion xcdd.6 radian. Th corrsponding thortical rspons at wind spd of 6 m/s (Fig. 3.8b) indicats that th ngativ damping as wll as th participation of th pitching motion ar undrstimatd by th mathmatical modl. Nvrthlss, th ovrall agrmnt btwn th xprimnt and analytical prdiction is fairly good and it can b concludd that th arodynamics of th bridg dck without control surfacs is wll dscribd by th rational function approximation of th applid fluttr drivativs. Th third part consistd of tsts on th control systm and. Th xampl of th xprimntal and thortical rsponss for gain c = du to th initial xcitation amplitud of.5 radian at wind spd of 3 m/s and 6 m/s ar shown in Fig Gnrally spaking, th rational modl accuratly prdicts th rspons of th systm for all gains of intrst at low wind spd. Th critical wind vlocity of th systm and thir thortically obtaind valus ar shown in Fig. 3.. Th critical wind vlocity of th systm with gain c = could not b dtrmind sinc th systm was stabl for all allowabl wind spds. For this control pattrn, th instability of th systm for th whol rang of gain is du to th oscillatory mod. It can b sn that th analytical fluttr wind vlocitis slightly ovrstimat th xprimntal rsults. Th xampl of th xprimntal and thortical rsponss of th control systm with th gain c = at wind spd of 3 m/s and 6 m/s ar shown in Fig. 3.. Th prdiction of rsponss by th rational modl dtriorats for th highr wind vlocity. Th xprimntally and thortically obtaind critical wind vlocitis ar prsntd in Fig. 3.. In this cas, th thortical prdiction dos not modl corrctly th oscillatory mods of th

65 64 3. Wind tunnl xprimnts on sctional modls of arodynamic control systms systm. For th gains in th rang of.5 to th modl lost th stability du to th fluttr mod whil thory indicats a divrgnt typ of instability. Sinc th fluttr phnomna occurrd for lowr wind spd it was not possibl to dtrmin, if at this rang of gain, th prdiction of divrgnc is corrct. Howvr, th xprimnt showd that th divrgnc wind vlocity for th gain highr than, was in a vry good agrmnt with th thortical rsults. a) tim rsponss at wind spd of 3 m/s.6 xprimnt nondim. disp. nondim. disp having pitching 5 5 tim (s) thory having pitching 5 5 tim (s) b) tim rsponss at wind spd of 6 m/s.6 xprimnt nondim. disp. nondim. disp having pitching 5 5 tim (s) thory having pitching 5 5 tim (s) Fig. 3.. Exprimntal and thortical tim rsponss of control systm with gain c = To clarify th sourc of th discrpancy of th modling of th control systm with both surfacs, tsts wr conductd on th systm with only lading or only trailing surfac,

66 3.. Exprimnt on bridg-surfacs control systm 65 movd in th sam as wll as in th opposit dirction to th pitching motion of th dck. Th critical wind vlocitis of th systm with only th lading surfac movd in th dirction opposit to th pitching of th dck and in th sam dirction as th pitching ar shown in Fig. 3.3 and 3.4, rspctivly. Th action of th lading surfac movd in th dirction opposit to th torsional displacmnt of th dck causd th dcras of th fluttr wind spd. Motion of th lading surfac according to th stratgy of systm incrasd wind spd from 4 m/s (c = ) to 5 m/s (c = ). Th thortical prdictions of th fluttr wind spds of systm with only th lading surfac ar in a vry good agrmnt with th xprimntal rsults. Critical wind vlocity (m/s) Thory.3 rad.5 rad.7 rad.5.5 gain c(nondim.) Fig. 3.. Critical wind vlocity of th bridg-surfacs control systm Critical wind vlocity (m/s) Thory.3 rad.5 rad.7 rad.5.5 gain c(nondim.) Fig Critical wind vlocity of th systm with only th lading surfac movd in th dirction opposit to th pitching of th dck Th xprimntal and thortical rsults of th control systm with only th trailing surfac ar shown in Fig For this cas, th agrmnt btwn th xprimnt and thory is fairly good. Howvr, th rspons of th xprimntal modl of bridg-surfacs control systm with only th trailing surfac for gain from to could not b rproducd

67 66 3. Wind tunnl xprimnts on sctional modls of arodynamic control systms from th thortical formulation. Exampls of th xprimntal and thortical tim rsponss for gain c = at wind spds of 3 m/s and 6 m/s ar shown in Fig Critical wind vlocity (m/s) Thory.3 rad.5 rad.7 rad.5.5 gain c(nondim.) Fig Critical wind vlocity of control systm (or ) with only th lading surfac Critical wind vlocity (m/s) Thory.3 rad.5 rad.7 rad.5.5 gain c(nondim.) Fig Critical wind vlocity of control systm with th only trailing surfac Th tim historis obtaind from th xprimnt and dtrmind numrically for wind spd of 3 m/s ar vry similar (Fig. 3.6a). But, th vibration of th xprimntal modl at wind spd of 6 m/s (Fig. 3.6b) consists of sustaind larg amplitud pitching vibration with significant having componnt. Such rspons of th xprimntal modl indicatd th onst of fluttr. Th rspons at 6 m/s computd from th thortical modl is rprsntd by th dcaying motion with approximatly th sam amplituds of pitching and having. Fig. 3.7 shows th xprimntal and thortical critical wind vlocitis of th control systm with gains varying from to. For gains bing from.5 to th xprimntal modl lost th stability du to an oscillatory mod, whil th thortical rsults indicatd divrgnc. Nvrthlss, th xprimntal critical wind vlocitis match th thortical ons for gains from.5 to. At this rang of gains th sourc of th instability was du to divrgnc obsrvd in th xprimnt and dtrmind in th analytical analysis.

68 3.. Exprimnt on bridg-surfacs control systm 67 nondim. disp. a) tim rsponss at wind spd of 3 m/s.6 xprimnt nondim. disp. nondim. disp. nondim. disp having pitching 5 5 tim (s) thory having pitching 5 5 tim (s) b) tim rsponss at wind spd of 6 m/s xprimnt having pitching 5 5 tim (s) thory having pitching 5 5 tim (s) Fig Exprimntal and thortical tim rsponss of control systm with th only trailing surfac and gain c = Notic that th discrpancis btwn th xprimntal and thortical critical wind vlocitis of th control systm with only th trailing surfac (Fig. 3.7) ar vry similar to th rsults of th control systm with both surfacs (Fig. 3.). It indicats that th main sourc of rror for crtain rangs of gains and wind spd is du to modling of th arodynamic forcs of th trailing surfac. Th thortical modl of th control systm assums indpndnt formulation of th arodynamic forcs of th dck and th surfacs,

69 68 3. Wind tunnl xprimnts on sctional modls of arodynamic control systms i.., th flow around th dck dos not modify th arodynamic forcs of th surfacs and vic vrsa. Th xprimnt indicats that for this gomtry of th xprimntal modl th intraction btwn th dck and th trailing surfac for highr wind spd should b incorporatd into th analysis. Critical wind vlocity (m/s) Thory.3 rad.5 rad.7 rad.5.5 gain c(nondim.) Fig Critical wind vlocity of control systm with only th trailing surfac 3.. Exprimnt on bridg flaps control systm 3... Dscription of xprimnt Wind tunnl tsts on bridg dck-flaps control systm wr prformd in th wind tunnl of th NKK Corporation in thir rsarch facilitis in Kawasaki. Th tstd modl was of a rigid timbr construction with stiffning vrtical ribs and aluminum plats (Figs. 3.8, 3.9, 3.). Th modl had th lngth of 5 cm and width of 54.6 cm. Th flaps wr mad of woodn plats and had th width of 5. cm. Th hings of th flaps wr locatd in th middl of th flap. Th flaps hing lins wr locatd at th distanc of 3.9 cm from th dgs of th dck. Th gap btwn th flaps and th dck of width 3 mm was sald with flxibl plastic plats fastnd to th dck and th flaps. To rduc th influnc of th wind ffcts coming from th supporting systm, two innr walls wr built in th wind tunnl and two nd plats wr attachd to both sids of th modl. Th dimnsions and th masss of th xprimntal modl ar givn in Tabl 3.4. Tabl 3.4 Dimnsions and masss of th lmnts of xprimntal modl width of th dck b = 546 mm hight of th dck H = 5 mm lngth of th dck L = 5 mm width of control surfacs b = b = 5 mm position of surfacs hing lins = = 47 mm mass of dck sction 8.98 kg mass momnt of inrtia. kgm

70 3.. Exprimnt on bridg-flaps control systm mm 494 aluminum bar vrtical ribs 5 mm nd plat flap flxibl plastic plat puls motor flap hing mm Fig Cross sction of th bridg-surfacs control systm Fig Photograph of th bridg flaps control systm Th dck was supportd on four stl bars connctd to four actuators locatd bnath th bottom of th wind tunnl (Figs. 3., 3.). Th actuators wr usd to induc having and pitching motion of th dck modl. Th rlativ motion of th lading flap with rspct to th dck was inducd by two puls motors locatd insid th dck modl (Fig. 3.). Th actuators wr controlld by a prsonal computr. Gnratd sinusoidal functions wr snt to th motor drivrs controlling th work of th actuators (Fig. 3.3). Phas shifts btwn

71 7 3. Wind tunnl xprimnts on sctional modls of arodynamic control systms th motion of th dck and th flap wr obtaind through phas changs of th input signals to puls motor govrning th motion of th flaps. Fig. 3.. Dck of th bridg modl Fig. 3.. Photograph of th xprimntal stup