Wind - Induced Vibration Control of Long - Span Bridges by Multiple Tuned Mass Dampers

Transcription

1 Tamkang Journal of Scence and Engneerng, Vol. 3, o., pp. -3 (000) Wnd - Induced Vbraton Control of Long - Span Brdge by Multple Tuned Ma Damper Yuh-Y Ln, Ch-Mng Cheng and Davd Sun Department of Cvl Engneerng Tamkang Unverty, Tamu, Tawan 5, R.O.C. Abtract An analytcal model preented to examne the performance of multple tuned ma damper (MTMD) for the long pan brdge ubected to wnd exctaton. The reducton of dynamc repone and the ncreae of the flutter velocty by the attachment of the MTMD to the brdge are dcued. Through a parametrc analy, the charactertc of MTMD are tuded and the degn parameter ncludng ma, dampng, bandwdth, and total number of TMD are propoed. A comparon of effectvene between a ngle TMD and MTMD alo preented n th paper. The reult ndcate that the MTMD, degned wth the recommended parameter, not only more effectve but alo more robut than the uual ngle TMD agant wnd-nduced vbraton. The uperor robutne of the MTMD epecally gnfcant n the toronal drecton. Key Word : Tuned ma damper, Buffetng repone, lutter; Long-pan brdge. Introducton The development of vbraton control theore have led to the wde ue of tuned ma damper on many engneerng tructure uch a tall buldng, long-pan brdge, and o on. Intallng a TMD on a long-pan brdge ha been proven to be effectve for uppreng wnd-nduced vbraton both analytcally and expermentally [-4]. In recent year, the degn concept of a TMD extended to multple tuned ma damper (MTMD) whch are compoed of everal mall ocllator attached to the man tructure [,5,8]. The man dea of th degn to dtrbute the natural frequence of MTMD around the natural frequency of the uppreed mode of the tructure for leenng the reonant effect. rom the prevou tude [,5,8], we found that MTMD le entve to the offet of the tunng frequency than a ngle TMD, and the ma of each TMD can be made maller. The latter pecally mportant for large tructure, becaue the mave ze of the damper may caue dffculte wth brdge contructon and mantenance. The theore of MTMD have been dcued extenvely by ome reearcher [,5,8], whch have provded degn formula and recommendaton n ther paper. However, thoe formula were prmarly derved for general engneerng tructure and may not be drectly ued for flexble brdge ubected to wnd exctaton. or the nvolvement of aerodynamc dampng and aerodynamc tffne, the wnd-nduced repone of the long-pan brdge omewhat more complex than that of general tructure ubected to the harmonc load. urthermore, the aumpton of treatng wnd load a whte noe, adopted n ome paper [5], not completely vald, becaue the contrbuton of the background part to the total repone can not be gnored n mot cae. Hence, further tude of wnd-nduced vbraton control of flexble brdge by the MTMD are tll needed. In th paper an analytcal model preented to examne the performance of the MTMD ued n the long-pan brdge. The dynamc repone reducton and the ncreae of the flutter velocty of the flexble brdge are dcued. A cable-tayed brdge ubected to buffetng choen a the target for evaluatng the performance of the MTMD n th analy. Then, the degn parameter of the MTMD are propoed through th parametrc tudy.. ormulaton of Equaton of Moton

2 Tamkang Journal of Scence and Engneerng, Vol. 3, o. (000) Structure MTMD K K M C M M C M U U Z L(t) Y Wnd D(t) L M(t) X = ~ g. Brdge - MTMD ytem The vertcal or toronal moton of the long-pan brdge generally domnated by the tructure frt mode n that drecton. Hence, t poble to model the brdge a a ngle degree of freedom (SDO) ytem and each TMD of the MTMD alo modeled a a SDO ytem. Provded that M M u&& = + ( C C ) u& C u& ( K K ) u K = = φ ( x ) m (4) = g. nte element model of brdge deck ubected to wnd load there are TMD ued n the tructure, then the brdge-mtmd ytem, hown n g., ocllate wth + degree of freedom. The equaton of moton of the tructural model can be expreed n the generalzed coordnate ytem a = u = T Φ e + M u&& C u& + C u& K u + K u = 0 ( =, ) () the ma of the th TMD. The other properte uch where u the generalzed dplacement, M, C, and K are repectvely the generalzed ma, dampng, and tffne, Φ the matrx contanng the frt mode of the brdge, e the elf-excted force matrx and ex the buffetng force matrx. The ubcrpt tand for the brdge and for the th TMD. Let φ (x ) be the component of Φ at the a dampng and tffne are defned n a mlar manner. The wdely accepted form of the elf-excted force, expreed by the flutter dervatve, were propoed by Scanlan and Tomko [6]. Thee form are adopted here to repreent the nteracton between flud and tructure. Snce only vertcal and coordnate x where the th TMD located. Then, toronal repone are concerned, the drag force the expreon of M and M are known a gnored. The elf-excted force actng on deck M = Φ T node n vertcal drecton MΦ (3) l and n toron e Φ drecton are n the followng form: where M the ma matrx of the brdge and m l y () ( )( ) ( ) ( t ) B ( ) ( t ) e t = * & * α * ρ U B K H K + H K & + KH 3 ( K ) α ( t ) L U U (5) t y () ( )( ) ( ) ( t ) B ( ) ( t ) e t = * & * α * ρ U B K A K + A K & + KA 3 ( K ) α ( t ) L U U (6) the trbutary length of the node (hown n g. ), y, α are the vertcal and toronal dplacement, where ρ ar denty, U wnd velocty, B deck * * wdth, K = Bω repectvely, Hl, Al (l=,3) are the flutter the reduced frequency, L dervatve. In th tudy the brdge deck aumed U nentve to the aerodynamc couplng, the couplng term n Eq. (5)-(6) are neglected. Thu, t e T ex ()

3 Yuh-Y Ln, Ch-Mng Cheng and Davd Sun: Wnd - Induced Vbraton Control of Long - Span Brdge by Multple Tuned Ma Damper Eq. (5)-(6) can be mplfed a l e t = ρ U * () ( B )( K ) H ( K ) y& U () t L (7) t e * () t ρ U ( B )( K ) A ( K ) ( t ) B = α & * + KA 3 α U ( K ) ( t ) L (8) Subttutng Eq. (7) or (8) nto () and makng ome manpulaton, we can rewrte the equaton of moton a follow: u&& + ξ ω + µ ξ ω ) u& µ ξ ω u& + ( ω + µ ω ) u = = = = ( µ ω u Φ T ex = M () u&& ξ ω u& + ξ ω u& ω u + ω u = 0 ( =, ) (0) where ξ, ω are the dampng rato and crcular Φ T ex ωt = 0e (6) frequency of the th TMD, repectvely, µ the M generalzed ma rato of the th TMD to the brdge, ωt u = Ae (7) ξ,ω are the effectve dampng and the effectve ωt u frequency of the brdge, repectvely. The = Ae (8) mathematcal form of µ hown n the followng: The tranfer functon of the brdge and the th TMD are then derved by ubttutng Eq. (6)-(8) φ ( x) m nto ()-(0) and ettng the generalzed force to be µ = () M unty. If we defne If the vertcal repone condered, the R = ω () mathematcal expreon of ω and ξ can be ω tated a ω = ω () ρb φ ( x ) L * ω ξ = ξ H M ω (3) If the toronal repone condered, the mathematcal form of ω and ξ can be expreed by 4 ρb φ ( x ) L * ω = ω ω A3 M (4) 4 ρb φ ( x ) L ξ ω ξω * = Aω M (5) To olve Eq. () and (0), the complex form of the generalzed dplacement and the external force are ued R = ω ω (0) D = R R () E = ξ R R () then, the tranfer functon of the brdge can be tated a: ( M ω ) H( ω ) = (3) Re( Z) + Im( Z) where Re(Z) and Im(Z) are defned by µ R( R D + E ) Re( Z) = R (4) = D + E µξ RR Im( Z) = ξ R + (5) = D + E Alo, the tranfer functon of the th TMD are obtaned a 3 R D + E ξ R R H ( ω ) = H ( ω) (6) D + E Thee reult are dentcal to thoe derved by Yamaguch & Harnporncha [8] except that the 5

4 Yuh-Y Ln, Ch-Mng Cheng and Davd Sun: Wnd - Induced Vbraton Control of Long - Span Brdge by Multple Tuned Ma Damper dampng ratoξ and the frequency ω are replaced by the effectve dampng ξ and the effectve frequency ω n th tudy. 3. Buffetng Repone A the buffetng repone condered, the external force term ex n Eq. () ubttuted by the vertcal or toronal buffetng force that well-known a [7] l ( x t) U B C u dcl C w b, = ρ L + + D U d U L (7) α t ( x t) U B C u dcm w b, = ρ M + U d U L α (8) where C L, C D, and C M are repectvely the lft, drag, and moment coeffcent, u and w are the wnd peed fluctuaton n horzontal and vertcal drecton, repectvely. It hould be noted that the repone calculaton baed on the frt tructural mode n ether vertcal or toronal drecton, only one mode taken nto account for the analy. Ung the random theory and wnd velocty pectra, we can obtan the co-pectrum of buffetng force between deck node p and q, whch denoted C by S. The generalzed force pectrum S p q then obtaned C S = φ( x ) φ( x ) S () p q p q The dplacement pectrum of the brdge at node n the followng form: S ( x ) = φ ( x ) S H ( ω) (30) d Smlarly, the dplacement pectrum of the th TMD expreed by S ( x ) = φ ( x ) S H ( ω) (3) d p q we can obtan the mean quare of the repone of the brdge at node σ ( x ) = S ( x ) dω (3) 0 d Alo, the varance of the repone of the th TMD can be calculated from the followng equaton: σ ( x ) = S ( x ) dω (33) 0 d 4. Evaluaton of luttervelocty The man obectve of ung TMD or MTMD to uppre exceve repone nduced by buffetng. In addton, an accompanyng effect the ncreae of the flutter velocty epecally for the toron-retant damper. When the toronal retance of the brdge ytem concerned, the ue of MTMD not only can reduce the toronal repone but ncreae the crtcal velocty. In general, the moton of the brdge reaonably aumed to be both tructurally and aerodynamcally uncoupled, flutter n th cae wll be ngle-degree-of-freedom flutter. Snce th type of flutter domnated by the frt toronal mode, the flutter analy baed on the equaton of moton hown n Eq.()-(0) plauble. We conder Eq. () and (0) and drop the external force term n Eq. (), becaue the external force not relevant to the flutter analy. Then, by ubttutng Eq. (7)-(8) nto th ytem of equaton, a complex egen-value problem yelded and can be tated n a matrx form ([ G] [ ]){ A} λ = 0 (34) where [G] a quare matrx wth rank of +, [λ] a dagonal matrx wth rank of +, {A} an ampltude matrx. The defnton of thee matrce are Integratng Eq.(30) and (3) wth the frequency w, [ G] = ξ ω + µ ξ ω µ ξ ω µ ξ ω... µ ξ ω = ξ ω ξ ω ω ξ ω 0 ξ ω ξ ω ξ ω +

5 Yuh-Y Ln, Ch-Mng Cheng and Davd Sun: Wnd - Induced Vbraton Control of Long - Span Brdge by Multple Tuned Ma Damper ω + µ ω µ ω µ ω... µ ω = ω ω ω 0 ω ω ω [ λ] = [ ω ] { } T (36) A = [ A A A... A] (37) The quare root of the egenvalue obtaned from Eq. (34) the frequency n a complex form. The rato of the magnary part to the real part of th frequency nterpreted a the total dampng of the tructural ytem. The total dampng contrbuted by the brdge telf, MTMD, and aerodynamc. At ome wnd velocty, when the dampng beng zero, the tructural repone wll approach nfnty, that, flutter wll occur. The correpondng wnd peed called the flutter velocty, and the real part of the frequency called flutter frequency. It noted that Eq. (34) hould be olved by teraton for each wnd peed, becaue the matrx [G] cont of the unknown The teratve calculaton generally can yeld a convergent oluton by ung an approprate ntal value and a relable convergence crteron. 5. Parametrc Analy The tructure ued n th tudy a cable -tayed brdge. The brdge ha a total pan of 460m and a wdth of.5m. Two 00-m-hgh tower are upported by cable. A fnte element model, conted of beam element and cable element, ued to calculate the natural frequence of the tructure. The geometry of th brdge hown n g.3. Through the calculaton, the natural frequence of the frt vertcal mode and the frt toronal mode are 0.4Hz and 0.354Hz, repectvely. The flutter dervatve H and A ( =,3), modfed from Scanlan and Tomko [6], are hown n g.4. The drag, lft, and toron coeffcent C D, C L, and C M, ued for buffetng calculaton, are adopted from reference [] and hown n g. 5. The roughne length of.m ued. The ma and the dampng rato of each TMD are aumed the ame for practcal reaon. Snce the TMD are cloely mounted on the brdge deck, the locaton of each TMD connected to the brdge deck can be theoretcally aumed the ame wthout * * (35) any huge error. In th analy the degn parameter of the MTMD nclude the total ma rato, the dampng rato, the number of TMD, and the frequency bandwdth. To account for the mtunng problem, the effect of offet alo condered. The performance of a TMD wll ncreae n proporton to the ma rato. Generally ma rato le than % or larger than 4 % would be too lght or too heavy for practcal purpoe. Therefore, the range of total ma rato of the TMD choen from % to 4%. The dampng rato range of the TMD choen from % to 7%. The total number of TMD related to the bandwdth of the MTMD, and the range of thee parameter are tuded from to and from 0. to 0.5, repectvely. () Performance of the MTMD for Suppreng Vertcal Buffetng Repone Snce the charactertc of MTMD n the vertcal drecton wll generally not vary wth the wnd velocty, the followng tude are nvetgated at a ngle wnd peed only. (a) Effect of dampng rato When % total ma rato ued, the relatonhp of the dampng rato and repone reducton rato for varou number of TMD hown n g.6. or a ngle TMD, the performance ncreae wth the dampng rato and reache the maxmum at 5% dampng rato. Th reult concdent wth that n prevou tude [3]. or the MTMD the reult qute dfferent. The repone reducton rato harply ncreae to t optmum at a low dampng rato and then lowly decreae a the dampng rato ncreae. The comparon of the reult ndcate that the performance of the MTMD omewhat better than that of the ngle TMD. Th concluon can be expected becaue the frequency range of the dplacement pectrum at the reonant part wde band, whch reult n the uperorty of the MTMD. We can alo conclude that wth larger number of TMD, the maller the optmum dampng rato. However, th tendency

6 Yuh-Y Ln, Ch-Mng Cheng and Davd Sun: Wnd - Induced Vbraton Control of Long - Span Brdge by Multple Tuned Ma Damper wll not be obvou when the number of TMD more than. or degn purpoe, the dampng rato hould be elected on the afe de, that, the preferred value the one that larger than the optmum. By npecton from g. 6, we can ugget that % dampng approprate for the MTMD wth % total ma rato. (b) Effect of bandwdth The bandwdth one of the mportant degn parameter of MTMD. It degnate the range of the dtrbuted frequence of TMD and defned here a the rato of the dfference of the maxmum and the mnmum frequence of the TMD to the tructural frequency. g. 7 how the repone 00 m 0 m 30 m 80 m 30 m 460 m g 3 Geometry of the cable-tayed brdge.0.5 luter Dervatve H * A * A * 3 Coeffcent CD CL CM U/nB g. 4 lutter dervatve Reducton Rato ( % ) = = 3 = = 5 = Angle of low (degree) g. 5 Lft, drag, and toronal coeffcent Lft Drecton 30 m/ wnd peed Dampng rato = 0.0 Ma rato = Bandwdth g. 7 Repone reducton rato veru bandwdth for dfferent number of TMD

7 Yuh-Y Ln, Ch-Mng Cheng and Davd Sun: Wnd - Induced Vbraton Control of Long - Span Brdge by Multple Tuned Ma Damper 6.00 Reducton Rato ( % ) = =3 B=0. =5 B=0.5 = B=0. =3, B=0. Lft Drecton Wnd Speed = 30 (m/ec) Ma Rato= TMD Dampng g. 6 Repone Reducton Rato veru TMD dampng for dfferent number of TMD Reducton Rato ( % ) Wnd Speed = 30 m/ec = ; B = 0. ; u = Ma rato 6.00 u = u = 0.03 u = u = TMD Dampng g. 8 Repone reducton rato veru TMD dampng for dfferent ma rato Reducton Rato ( % ) Lft Drecton = Dampng = 0.05 Wnd Speed 30 ( m/ec ) = ; B = 0. Dampng = Offet g. Comparon of robutne between TMD and a ngle TMD Effectve requency ( Hz ) Toronal Drecton Effectve requency Effectve Dampng Wnd Speed ( m/ec ) g.0 Effectve frequency and dampng veru wnd peed for the typcal brdge Effectve Dampng Reducton Rato ( % ) B = 0. B = 0. B = 0.5 Toronal Drecton Wnd peed = 30 (m/ec) ; = TMD Dampng (a) n=3

8 0 Tamkang Journal of Scence and Engneerng, Vol. 3, o. (000) Reducton Rato ( % ) 5.00 B = 0. B = 0. B = Toronal Drecton Wnd peed = 30 (m/ec) ; = TMD Dampng Reducton Rato ( % ) B = 0. B = 0. B = 0.5 Toronal Drecton Wnd peed = 30 (m/ec) ; = TMD Dampng (b) n=5 (c) n= 6 g. Repone reducton rato veru TMD dampng at 30 m/ wnd peed (a) n=3 (b) n=5 (c ) n= Reducton Rato ( % ) 4 3 B = 0. B = 0. B = 0.5 Reducton Rato ( % ) 4 3 B = 0. B = 0. B = 0.5 Toronal Drecton Wnd peed = 60 (m/ec) ; = TMD Dampng Toronal Drecton Wnd peed = 60 (m/ec) ; = TMD Dampng (a) n=3 (b) n=5 6 5 Reducton Rato ( % ) 4 3 B = 0. B = 0. B = 0.5 (c) n= g. Repone reducton rato veru TMD dampng at 60 m/ wnd peed (a) n=3 (b) n=5 (c ) n= Toronal Drecton Wnd peed = 60 (m/ec) ; = TMD Dampng (a) n=3

9 Yuh-Y Ln, Ch-Mng Cheng and Davd Sun: Wnd - Induced Vbraton Control of Long - Span Brdge by Multple Tuned Ma Damper Reducton Rato (%) 4 3 = 3 Reducton Rato (%) 4 3 = 3 B = 0. B = 0. B= 0.3 B= 0.4 B=0. B=0. B=0.3 B= O f f e t Off e t 6 5 (a) n=3 (d) n=3 g. 3 Repone reducton rato veru offet at 60 m/ wnd peed (a) n=3 (b) n=5 (c) n= (d) n=3 6 Reducton Rato (%) 4 3 B = 0. B = 0. = Reducton Rato ( % ) = =, B=0. =, B= O f f e t O f f e t 6 (b) n=5 g. 4 Comparon of robutne between TMD and a ngle TMD n toronal drecton 5 8 u = 3 % Reducton Rato (%) 4 3 = Reducton Rato ( % ) u = % u = % = ; B=0. Dampng=0.0 B = 0. B = (c) n= Of f e t Effectve Dampng Rato g. 5 Repone reducton rato veru effectve tructural dampng for dfferent ma rato

10 0 Tamkang Journal of Scence and Engneerng, Vol. 3, o. (000) Table Suggeted bandwdth for dfferent number of TMD no. of TMD lower bound ( B n mn ) Suggeted value ( Bn ) upper bound ( B n max ) Table lutter velocty of the brdge lutter velocty w/o TMD = ( m/ec ) = no. of TMD; B = bandwdth;d=dampng rato µ = total generalzed ma rato;( ) = ncreae of the flutter velocty n percent µ = % µ = % µ = 3 % = D = (+3.4%) 8.0 (+0.0%) 86.6 (+6.% ) D = (+8.%) D = (+43.0%) = ; B=0. D = (+.%) 8.80 (+.7%) 8.77 (+.7%) D = (+3.%) 8. (+30.5%) 0.37 (+3.3%) D = (+7.0%) 7.7 (+4.4%) 0.8 (+4.%) reducton rato veru bandwdth for dfferent number of TMD. We can oberve that the optmum bandwdth ncreae wth the number of TMD but wll converge to a value of about 0.8 a the number equal to or larger than. A large bandwdth mean that ome of the frequence are far away from the tructural frequency and wll lower the effectvene of the MTMD. On the other hand, a mall bandwdth mple the charactertc of the MTMD are mlar to thoe of a ngle TMD and wll loe the beneft of MTMD. Therefore, the bandwdth hould be properly elected n the MTMD degn to enure better performance. or degn purpoe we take 5% of the optmum reducton rato a the degn target, and the correpondng larger and maller bandwdth are the upper and lower bound, repectvely. The uggeted value are hown n Table. (c) Effect of the number of TMD and ma rato An odd number of TMD often ued n the MTMD ytem, the central frequency tuned around the tructural frequency and the other are equally paced on both de of the central one. rom the reult n g. 7, the optmum number around. The ue of the larger number of TMD doe not ncreae the performance gnfcantly. There alo may be dffculte to tune the frequence precely for mall pacng of the frequence. Earler dcuon of degn parameter of the MTMD are baed upon % total ma rato. It known that the performance of MTMD ncreae wth the ma rato and the correpondng optmum dampng rato alo ncreae. To tudy th effect, we ue TMD and bandwdth of 0. for the analy. The relatonhp between repone reducton rato and dampng rato for dfferent ma rato hown n g. 8. The reult ndcate that % dampng rato can yeld a good performance when the ma rato fall between % and 3%, but a hgher dampng rato (about 5%) more utable for a ma rato of 4%. Another fndng that the ncreae of repone reducton rato gnfcant when the ma rato raed from % to or 3%, but t not obvou when 4% ma rato ued. or degn purpoe, the approprate value of the upper bound of total ma rato about 3%. (d) Robutne g. how the comparon of the robutne between the MTMD and the ngle TMD. The defnton of the offet ued here the rato of the dfference between the tuned and peak frequence to the peak frequency. The reult how that the MTMD better than a ngle TMD but the dfference not gnfcant. The reaon that the repone relevant to the encloed area bounded by the dplacement pectrum and frequency, and the change of th area due to mtunng not entve.

11 Yuh-Y Ln, Ch-Mng Cheng and Davd Sun: Wnd - Induced Vbraton Control of Long - Span Brdge by Multple Tuned Ma Damper urthermore, the tructural frequency n the vertcal drecton wll not change wth the wnd peed and the mtunng problem not o erou. However, th effect wll become more mportant n the toronal drecton and wll be dcued later. () Performance of the MTMD for Suppreng Toronal Buffetng Repone There are ome tructural charactertc n the toronal drecton that are dfferent from thoe n the vertcal drecton. rt, the effectve dampng n the toronal drecton decreae wth wnd peed and may nduce flutter. Second, the effectve frequency alo change wth wnd peed and the tunng frequence of MTMD become more mportant. Thee relatonhp between wnd peed, the effectve dampng, and the effectve frequency for a typcal brdge are hown n g. 0. (a) Effect of dampng rato and bandwdth Wth % ma rato, the relatonhp between performance and dampng rato correpondng to 30m/ and 60m/ wnd velocte are hown n g. -, repectvely. We can ee that a % dampng rato for the MTMD can yeld a good performance n all cae. Th uggeted dampng rato the ame a that n the vertcal drecton. Therefore, we can conclude that the optmum dampng rato ndependent wth the effectve frequency of the tructure. The reult n g. - alo how that the choce of bandwdth nearly ndependent of wnd velocty (or effectve dampng rato), and a maller bandwdth often reult n better performance. However, a maller bandwdth may caue a tunng problem for the MTMD. The reaon that the frequency nterval of the MTMD may be too mall to be tuned precely. or th concern t eem that the bet choce to ue 3 TMD wth a bandwdth of 0. and a dampng rato of %. The other choce to ue TMD wth a bandwdth of 0. and a dampng rato of % that alo produce a good performance. However, the choce of the bandwdth alo dependent on the robutne that may be the domnatng factor and wll be dcued n the next ecton. (b) Robutne The effectve frequency of the brdge ubected to wnd exctaton wll be changed by the aerodynamc tffne. or th reaon t not poble to exactly tune the TMD frequency to the frequency of the peak repone for each wnd peed and there wll be ome offet to the peak value. urthermore, the natural frequency dcrepance between the real tructure and the prototype are, n practce, nevtable. Therefore, the offet hould be taken nto account for determnng the degn parameter to enure the MTMD performance. Becaue the total number of TMD and bandwdth wll affect the robutne, thee factor are nvetgated n the followng analy. To mplfy the tudy, % ma rato and % dampng rato are ued n th analy. At 60 m/ wnd peed, the relatonhp of repone reducton rato and offet for 3, 5,, and 3 TMD are hown n g. 3. or 3 TMD, the curve of bandwdth of 0. a bell-lke hape; the repone reducton rato reache the peak (57%) at zero offet and reduce rapdly wth the ncreae of offet. In th cae, robutne mlar to that of a ngle TMD and the allowable offet mall. A a bandwdth of 0. ued, the repone reducton rato fluctuate wth offet and produce three peak. The allowable offet larger but the maxmum repone reducton rato drop to 53%. A the bandwdth ncreaed to 0.3 and 0.4, the peak are more obvou and the allowable offet larger but the maxmum repone reducton rato maller. Th explan that the robutne ncreae wth the bandwdth but the performance decreae wth t. To atfy both robutne and performance requrement, a bandwdth of 0. eem to be a bet value for 3 TMD. or 5 TMD, the relatonhp between offet and repone reducton rato mlar to that of 3 TMD. In th cae, the curve of bandwdth of 0. more flat and the repone reducton rato about 55% whch lghtly larger than that of 3 TMD. We then can conclude that for a gven bandwdth more TMD are more robut and produce better performance. Th concluon can be verfed further for or 3 TMD n whch the performance of a bandwdth of 0. even better. However, the comparon of the reult between and 3 TMD ndcate that the maxmum performance acheved a TMD are ued. The performance of 3 TMD almot the ame a that of TMD. In the cae of or 3 TMD, another fndng that the allowable offet almot a half of the bandwdth. or degn procedure, the allowable offet hould be determned frt and then the bandwdth. Due to the change of the effectve frequency wth wnd velocty, the allowable offet S hould be controlled by the followng:

12 0 Tamkang Journal of Scence and Engneerng, Vol. 3, o. (000) n n f Δ S = (38) n where n the tructural frequency, n f the flutter frequency. After the offet evaluated from the above equaton, the requred bandwdth twce of the offet. rom the comparon of the reult, hown n g. 4, we can fnd that the MTMD wth TMD uperor to the uual ngle TMD. (c) Effect of the number of TMD and ma rato The effect of the number of TMD on the performance of the MTMD can be clearly explaned n g. 3. The reult ndcate that the requred mnmum number of TMD to obtan bet performance n the toronal drecton, whch the ame a that n the vertcal drecton. The performance of the MTMD defntely ncreae wth ma rato but t doe not gan much at the low effectve dampng rato a hown n g. 5. or degn purpoe, a % ma rato can obtan a good performance and recommended n the toronal drecton. Alo, g. 5 can be a ueful tool to predct the repone reducton rato when the effectve dampng rato of the brdge known. (d) Increae of flutter velocty The crtcal velocty of the typcal brdge wthout damper m/. A ngle TMD and TMD wth varou combnaton of dampng rato and ma rato are analyzed to tudy the ncreae of the flutter velocty by the addton of the tuned ma damper. The frequency of the central TMD tuned to the natural frequency of the frt toronal mode. The reult, llutrated n Table, ndcate that the ncreae of flutter velocty about 3-43% for a ngle TMD and -4% for the MTMD. Generally the flutter velocty ncreae wth dampng and ma rato. or a fxed dampng rato, a hgher ma rato yeld a hgher flutter velocty n the cae of a ngle TMD. However, n the cae of MTMD wth a fxed dampng rato, the ncreae of flutter velocty due to the ncreae of ma rato not obvou. In th cae, a hgher dampng rato hould be ued for a hgher ma rato to effcently rae the flutter velocty. Comparon of a ngle TMD and the MTMD, both degned wth the ame ma, how that the MTMD more effectve than a ngle TMD for ncreang the tructure tablty. It hould be noted that the reult hown n Table are baed on the aumpton that the frequency of the central damper tuned to the natural frequency of the frt toronal mode. If the ncreae of toronal tablty the maor concern, th frequency hould be tuned le. A utable value for achevng th purpoe the flutter frequency wthout ung TMD. 6. Degn Recommendaton of MTMD A ummary of the parametrc analy on the MTMD n the vertcal and toronal drecton can be tated a follow: ()or the vertcal MTMD, the uggeted frequency of the central damper the frequency of the frt vertcal mode of the tructure. or the toronal MTMD, the uggeted tunng frequency of the central damper the average between the frt toronal mode frequency and the flutter frequency of the brdge wthout ung TMD. If the ncreae of toronal tablty the maor concern, th frequency can be mply tuned to the flutter frequency wthout ung TMD. ()or both the vertcal and toronal MTMD, the uggeted number of damper. or the vertcal MTMD, the total ma uggeted to be % or more to enure the vertcal performance and the correpondng dampng rato %. or the toronal MTMD, the ma uggeted to be % whch uffcent for obtanng good performance and the correpondng dampng rato alo %. or the vertcal MTMD, the uggeted bandwdth hown n Table. or the toronal MTMD, the uggeted bandwdth the larger one between the value obtaned from Eq. (38) and Table. 7. Concludng Remark A parametrc analy of the MTMD ued for uppreng aerodynamc repone of long-pan brdge preented. Through th analy the uggeted degn parameter ncludng dampng, ma, number, and bandwdth of the MTMD are propoed. The reult how that the MTMD more effectve than an optmzed ngle TMD for uppreng buffetng repone and ncreang the crtcal flutter peed. Wth a proper degn, the ue of MTMD can be advantageou to mprove the aerodynamc behavor of a long-pan brdge. The charactertc of the MTMD ued n the vertcal and toronal drecton are almot the ame except the tunng frequency and bandwdth that are

13 Yuh-Y Ln, Ch-Mng Cheng and Davd Sun: Wnd - Induced Vbraton Control of Long - Span Brdge by Multple Tuned Ma Damper affected by the aerodynamc tffne. The brdge and the aerodynamc coeffcent ued here are choen arbtrarly, due to the fact that thee tructural properte and aerodynamc effect have been normalzed. Therefore, the value recommended n th paper are tll ueful for other brdge. It hould be mentoned that the relatve dplacement between TMD and the tructure not ncluded n th tudy; f th dplacement not allowable, the uggeted dampng rato of TMD hould be ncreaed. Earthquake Eng. Struct. Dyn., Vol., pp. 5-6(3). [] Wnd Tunnel Invetgaton of the Kao Png H Brdge, Tawan Area atonal Expreway Engneerng Bureau, Tape, Tawan(4) (n Chnee). Manucrpt Receved: Apr. 8, 000 Revon Receved: May., 000 And Accepted: May,, 000 Acknowledgement The wrter gratefully acknowledge the fnancal upport of part of th work by the atonal Scence Councl (R. O. C.) under the grant SC E Reference [] Abe, M. and uno, Y., Dynamc Characterzaton of Multple Tuned Ma Damper and Some Degn ormula, Earthquake Eng. Struct. Dyn., Vol. 3, pp (4). [] Gu, M. and Xang, H.., Optmzaton of TMD for Suppreng Buffetng Repone of Long-Span Brdge, J. Wnd Eng. Ind. Aerodyn., Vol. 4-44, pp (). [3] Gu, M., Xang, H.. and Chen, A. R., A Practcal Method of TMD for Suppreng Wnd-Induced Vertcal Buffetng of Long-Span Cable-Stayed Brdge and It Applcaton, J. Wnd Eng. Ind. Aerodyn., Vol. 5, pp. 03-3(4). [4] Honda, A. et al., Aerodynamc Stablty of Kana Internatonal Arport Acce Brdge, J. Wnd Eng. Ind. Aerodyn., Vol. 4, pp (3). [5] Igua, T. and Xu, K., Vbraton Control Ung Multple Tuned Ma Damper, J. Sound Vb., Vol. 75(4), pp (4). [6] Scanlan, R. H. and Tomko, J. J., Arfol and Brdge Deck lutter Dervatve, J. Eng. Mech. Dv., ASCE, Vol. 7(EM6), pp (7). [7] Smu, E. and Scanlan, R. H., Wnd Effect on Structure, nd ed., John Wley & Son, ew York(86). [8] Yamaguch, H. and Harnporncha,., undamental Charactertc of Multple Tuned Ma Damper for Suppreng Harmoncally orced Ocllaton,