Control of wind-induced vibration of long-span bridges by semi-active lever-type TMD

Journal of Wind Engineering

and Industrial Aerodynamics90(2002)111–126

Control of wind-induced vibrations of long-span bridges by semi-active lever-type TMD

M.Gu a,*,S.R.Chen a,C.C.Chang b

a State Key Laboratory for Disaster Reduction in Ci v il En g ineerin g,Ton g ji Uni v ersity,

Shan g hai200092,China

b Department of Ci v il and Structural En g ineerin g,Hon g Kon g Uni v ersity of Science and Technolo g y,

Clear Water Bay,Kowloon,Hon g Kon g

Received17April2001;received in revised form16July2001;accepted6September2001 Abstract

With the rapid increase of bridge spans,research on controlling wind-induced vibration of long-span bridges has been a problem of great concern.In view of the fact that the vibration frequencies and damping of long-span bridges vary with wind speed,and the variation characteristics could not be accurately known even if through wind tunnel experiments,a concept of semi-active(SA)control of wind-induced vibration of long-span bridges is put forward in this paper.A new SA lever-type TMD with an adjustable frequency and the corresponding control strategy are primarily developed.A case study of the Yichang bridge,a suspension bridge with a main span of960m,shows that the SA lever-type TMD device is much superior to passive TMD in control e?ciency and robustness.r2002Elsevier Science Ltd.All rights reserved.

1.Introduction

The main categories of wind e?ects on bridge decks are?utter and bu?eting. Flutter is a phenomenon of self-excited vibration,which may cause a bridge to vibrate continuously with increasing amplitude until the bridge structure https://www.360docs.net/doc/de3105552.html,rge bu?eting amplitude may cause fatigue damage in structural members or make drivers and passengers in moving vehicles feel uncomfortable.With the rapid increase of bridge spans,research on controlling wind-induced vibration of long-span bridges has been a problem of great concern.

*Corresponding author.Tel.:+86-21-6598-1210;fax:+86-21-6598-4882.

E-mail address:minggu@https://www.360docs.net/doc/de3105552.html,(M.Gu).

PII:S0167-6105(01)00165-9

The strategy of the control of the wind-induced vibration of long-span bridges could be classi?ed as structural countermeasure,aerodynamic countermeasure and mechanical countermeasure.Increasing the ?rst torsional frequency,which can be achieved using inclined cable-planes,can obviously improve ?utter behavior of bridges.This method was adopted in the structural design of the Yangpu cable-stayed bridge in China [1].The passive aerodynamic countermeasure can also e?ectively suppress the wind-induced vibrations of bridges [2,3].In the preliminary design stage of long-span suspended bridges,the ?utter-based selection method is usually adopted for selecting a deck cross-section con?guration with satisfying ?utter behavior [3,4].Wardlaw [3]summarized the countermeasures for improving the aerodynamic stability of bridge decks and noted that satisfying aerodynamic performance of bridge decks can be achieved by the use of shallow sections,closed sections,edge streamlining and other minor or subtle changes to the cross-section geometry.Furthermore,Gu et al.[4]proposed a concept of bu?eting-based selection and the corresponding analysis method,which has been adopted in the wind-resistant study of the Jiangyin suspension bridge with a main span of 1385m and other bridges in China [5,6].Active aerodynamic countermeasures have also been studied [7–10]for increasing the critical ?utter wind speed.

Studies on the mechanical countermeasure of the ?utter and bu?eting control of long-span bridges focus mainly on passive devices,such as tuned mass damper (TMD)[11–15]and tuned liquid column damper (TLCD)[16].Nobuto et al.[11]made a study on ?utter control using a couple of TMDs,and the numerical example indicated its e?ciency.On this basis,a more advanced parametric study was performed by Gu et al.[12]through a theoretical analysis and a wind tunnel test on the Tiger-gate bridge model.Tuned mass damper has also been theoretically and experimentally studied on the bu?eting control of long-span bridges [13,14].

It is well known that the performance of a TMD is sensitive to the frequency ratio between the TMD and the bridge.Slight deviation of the frequency ratio from its design value would render drastic deterioration of the TMD’s performance.In view of this situation,the multiple tuned mass damper (MTMD)system has been extended to control bu?eting of Yangpu bridges in Shanghai [15],a cable-stayed bridge with a main span of 602m.Even though,the robustness of the MTMD system is limited due to the probability of great drift of the controlled frequency or other dynamic parameters of the bridge under wind.Moreover,it has been indicated that the dynamic responses and dynamic parameters of bridges from ?eld measurements and wind tunnel tests and theoretical analyses are usually di?erent from each other because the phenomenon of the wind-induced vibration of bridges is very complex [17,18].The control e?ciency may be unsatisfactory even if a beforehand designed optimal MTMD system is adopted for the above reason.

In view of these situations,a more robust control system is necessary for suppressing the wind-induced vibrations of long-span bridges.The control objectives are ?rstly discussed and are constructed in this paper.Considering possible application in real bridges,a semi-active TMD (hereafter referred to as SA TMD)rather than an active device is thus proposed and studied.The frequency of the SA TMD is adjustable.Accordingly,the control strategy is discussed.Finally,the

M.Gu et al./J.Wind Eng.Ind.Aerodyn.90(2002)111–126

112

M.Gu et al./J.Wind Eng.Ind.Aerodyn.90(2002)111–126113 Yichang suspension bridge with a main span of960m is taken as an example to show the application of the SA TMD for controlling wind-induced vibration of long-span bridges.

2.Control objectives

For long-span bridges under wind action,?utter is a phenomenon relative to safety of the bridge https://www.360docs.net/doc/de3105552.html,rge bu?eting response may cause strength damage or fatigue damage in main structural members,such as the main girder.Thus,the safety objective is determined in the present paper to include increasing critical?utter wind speed and fatigue life of the bridge over their design values,and assuring structural strength,that is

? ;e1TU C X U C

? e2TL Fatig X L D

and

? ;e3Ts max

eTm p s m

? is the design critical where U C is the critical?utter wind speed of the bridge,U C

?utter wind speed,L Fatig is the shortest fatigue life of the critical member of the

? is the design life of bridge,which can be estimated with the method in Ref.[19],L D

the bridge,es maxTm is the maximum stresses of the critical members of the bridge and ? is the corresponding allowable stresses.

s m

In addition,large bu?eting response may make the drivers and passengers in vehicles uncomfortable.The comfort level criterion adopted in North America[20]is that the maximum peak acceleration should be smaller than50mg for wind speed below13m/s,and smaller than100mg for wind speed between13m/s and the design wind speed of the bridge;whereas that adopted in England is that the maximum peak acceleration should be smaller than40mg for wind speed lower than20m/s. Unfortunately,up to now no such criterion has been provided in Chinese code. Thus,the comfort level criteria used in North America and England are synthetically adopted in this paper to construct the serviceability objective.The synthetic criteria is written as

Ac peak p0:04g for U p20m=s;e4TAc peak p0:1g for20m=s o U p U d;e5Twhere Ac peak is the maximum peak acceleration(unit in m/s2)of the bridge deck and g is the acceleration due to gravity.

The comfort level criterion is schematically illustrated in Fig.1.In natural logarithm ordinate-linear abscissa system,the acceleration response usually appears in the form of a convex curve.Thus,on this basis of the characteristics of bu?eting acceleration with wind speed,the synthetic criterion for the comfort level can be

expressed as

ln eAc peak Tp

ln 0:1g eTàln 0:04g eT? U d à20U àU d eT?0:9163U àU d U d à20e6Tthat is

Ac peak p 0:1g e 0:9163eU àU d T=eU d à20T;e7Twhere U is the wind speed and U d is the design wind speed of the bridge.Eq.(7)is the straight line in Fig.1.

3.Motion equations of bridge and SA TMD

For most long-span suspended bridges,the ?rst vertical and torsional mode response components usually dominate both the ?utter and bu?eting responses.Thus,a couple of SA TMDs is arranged in the similar way as in Refs.[12,13]to control the vertical vibration and torsional vibration simultaneously.The mechan-ical model for the analysis is shown in Fig.2,where Y 1and a are the vertical displacement and torsional angle of the deck,respectively,Y 2and Y 3are the vertical displacements of the SA TMD1and SA TMD2relative to Y 1;respectively.SA TMD1and SA TMD2have the same mass,M 2;sti?ness,K 2;and damping coe?cient,C 2:

10100

1000

Wind velocity U (m/s)

A c P e a k (m i l l i -g )Fig.1.Criterion for comfort level.M.Gu et al./J.Wind Eng.Ind.Aerodyn.90(2002)111–126

114

Considering the e?ects of wind direction on the bu?eting response [15],the equation governing the motions of the bridge deck and the SA TMDs can be derived in matrix style as follows:M ? .Y èétC e ? ’Y èétK e ? Y f g ?F f g ;e8Twhere Y f g ?x 1;g 1;Y 2;Y 3èé;

e9TM ? ?M s t2M 2f 21ez 0T0M 2f 1ez 0TM 2f 1ez 0T0I s t2M 2L 2t c 21ez 0TàM 2L t c 1ez 0TM 2L t c 1ez 0TM 2f 1ez 0T

àM 2L t c 1ez 0TM 20M 2f 1ez 0TM 2L t c 1ez 0T0M 2

66664377775;e10T

K e ? ?K h àr eU cos b T2BK 2H n 3C 110000K a àr eU cos b T2BK 2A n 3C 220000K 200

00K 2

266664377775;e11T

Fig.2.A couple of SA TMDs on a bridge deck.M.Gu et al./J.Wind Eng.Ind.Aerodyn.90(2002)111–126115

C e ? ?C h àr eU cos b TB 2KH n 1C 11àr eU cos b TB 2KH n 2C 1200àr eU cos b TB 2KA n 2C 12C a àr eU cos b TB 2KA n 2C 220

000C 2

0000C 2

666437775;e12T

M s ?

Z struct m ez Tf 21ez Td z ;e13TI s ?

Z struct I ez Tc 21ez Td z ;e14T

C 11?Z

L 0f 21ez Td z ;e15T

C 12?Z

L 0j 1ez Tc 1ez Td z ;

e16T

C 22?Z

L 0c 21ez Td z ;e17T

where x 1et Tand g 1et Tare the ?rst generalized vertical and torsional coordinates of the bridge deck,respectively,f 1ez Tand c 1ez Tare the ?rst vertical and torsional mode shape functions,respectively,L and B are length and width of the main bridge,respectively,r is the air density,m ez Tand I ez Tare the mass and inertial moment of mass per unit length of the bridge,respectively,M s and I s are the ?rst generalized mass and inertial moment of mass,respectively,K h and K a are the ?rst modal vertical bending sti?ness and the torsional sti?ness,respectively,C h and C a are the vertical bending and torsional damping coe?cients,respectively,H n i and A n

i (i ?1;2;3)are the ?utter derivatives,L t is the distance between the center of the

cross section of the bridge deck and one of the SA TMDs (see Fig.2),z is the location along the bridge deck,z 0is the location of the SA TMDs along the bridge deck and b is the wind yaw angle,i.e.,the wind direction in the horizontal plane.F f g is the bu?eting force,which equals to zero for the ?utter control analysis,that is,

F f g ?0000

8>>><>>>:9>>>=>>>;e18TM.Gu et al./J.Wind Eng.Ind.Aerodyn.90(2002)111–126

116

and for the bu?eting control,it is written as F f g ?F ex ;t T00M ex ;t T8>>><>>>:9>>>=>>>;;e19T

where F ex ;t Tand M ex ;t Tare the ?rst generalized bu?eting force and moment,respectively,and can be found in Ref.[19].

Firstly,the critical ?utter wind speed can be found by solving the above relative equations.The lowest critical ?utter wind speed usually takes place at zero wind direction angle.Let us assume that the solution has the following style:

Y f g ?X f g e st ;e20Twhere S is a complex number.The characteristic equation can thus be derived as

X 8i ?0

a i S e8ài T?0;e21Twhere a i (i ?128)are constants relative to the parameters of the bridge and the SA TMDs,and are not shown here due to their very complex and super?uous expressions but can be found in Ref.[12].The Routh–Hurwitz stability criterion is used here to judge the stability of the bridge.

Considering the e?ects of wind direction on the bu?eting response,the power spectra of the vertical bu?eting displacement and the torsional angle are derived and shown,respectively,as

S b eo T?f 21ez 0TS F eO TM s

A 11eo Tj j 2;e22TS a eo T?c 21ez 0TS M eO TI s A 44eo Tj j 2;e23T

where O is the vibration frequency with the e?ects of the aerodynamic sti?ness.Their largest mean square values can be approximately computed using the formulas:s 2b ?f 21ez 0TS F eO h TM 2s Z N 0

A 11eo Tj j 2d o ;e24Ts 2a ?c 21ez 0TS M eO a TI s Z N 0

A 44eo Tj j 2d o ;e25Twhere S F eO h Tand S M eO a Tare the values of the power spectra of the ?rst generalized bu?eting force and moment,which have the following forms:S F eo T??r U ecos b T

B 2Z L 0Z L

e àc z 1àz 2j j =L

f 1ez 1Tf 1ez 2Td z 1d z 2S L eo T;e26T

S M eo T??r U ecos b TB 2 2Z

L 0Z L 0e àc z 1àz 2j j =L c 1ez 1Tc 1ez 2Td z 1d z 2S m eo T;e27T

M.Gu et al./J.Wind Eng.Ind.Aerodyn.90(2002)111–126117

S L eo T?C 2L ?S u eo Tcos 2b tS v eo Tsin 2b teC 0L tC D A =B T2S w eo T=4;

e28TS m eo T?C 2M ?S u eo Tcos 2b tS v eo Tsin 2b tC 02M S w eo T=4;e29T

where c ?lo L =e2p U cos a T;l is the parameter re?ecting the spatial correlation of the ?uctuating wind speed,S u eo T;S v eo Tand S w eo Tare the power spectra of longitudinal,lateral and vertical components of the ?uctuating wind speed,C L and C M are the lift force coe?cient and moment coe?cient of the bridge deck,normalized using the bridge deck width B and the bridge deck height A ;respectively,C 0L and C 0M are the ?rst derivatives of C L and C M with respect to wind attack angle in the vertical plane,respectively,O h is the ?rst vertical bending vibration frequency with the e?ects of the aerodynamic sti?ness,O a is the ?rst torsional vibration frequency and A 11eo Tj j and A 44eo Tj j ;the transfer functions of the vertical bending and torsion responses of the bridge,SA-TMD system,respectively,are very complex in form,and are not shown here but can be found in Ref.[13].

Correspondingly,the maximum root mean square values of the vertical bending and torsional bu?eting accelerations can be approximately expressed,respectively,as

s .b ?s b O 2h ;

e30Ts .a ?s a O 2a :e31T

Now the fatigue life of the critical member can be estimated with Eqs.(22)and

(23)and the fatigue life estimation method given in Ref.[19];and the comfort level for the drivers and passengers in vehicles can be distinguished with Eqs.(30)and (31)and Fig.1.

The control e?ciencies for the bu?eting response and the critical ?utter wind speed are ?nally de?ned,respectively,as follows:R ?1às #

s 100%e32Tand

R U C ?U C #U C à1 100%;e33Twhere #s is the RMS of vertical bending displacement or torsional angle of the bridge deck without control,s is the corresponding value with control,#U

C and U C are the critical ?utter wind speeds of the bridge without control and with control,respectively.

4.SA TMD device and control strategy

It is common knowledge that the frequency ratio between the passive TMD and the structure is the dominant factor for the control of wind-induced vibration of long-span bridges,while e?ects of the other factors,such as damping of the TMD,

M.Gu et al./J.Wind Eng.Ind.Aerodyn.90(2002)111–126

118

M.Gu et al./J.Wind Eng.Ind.Aerodyn.90(2002)111–126119

Fig.3.Principle of the SA TMD.

on the control e?ciency are much less.Thus,in the present paper only the frequency of the SA TMD is taken as the design variable.

It was pointed out in Ref.[15]that the large static stretching of the spring of the classical hanging-type TMD may cause the spring to perform nonlinearly and the TMD may not be?tted into the space within the bridge deck available for installation.As a result,a passive lever-type TMD was raised in Ref.[15].The SA TMD proposed in the present paper has the similar style as the passive lever-type TMD,but with an adjustable frequency.The principle of the SA TMD is schematically illustrated in Fig.3.The mass block can be controlled to move along the horizontal rigid bar.With the motion of the mass block,the distance between the mass block and the support point,L;varies,thus the frequency of the SA TMD varies.However,the motion of the mass block will lead to an unbalance between the moments of the mass and the tension force of the spring about the support point.That is to say,in order to keep the rigid rod horizontal all the time,the base of the spring should be correspondingly adjustable. In brief,when the structural frequency varies,the mass block should be moved correspondingly,and the base of the spring should be raised or dropped correspondingly too.

Now we discuss how to tune the distance between the mass block and the support point as well as the height of the base of the spring in terms of required optimal frequency of the SA TMD.Fig.4shows the schematic diagram for the analysis. Keeping the rigid rod horizontal requires

H total?L0tL StL1e34T

and

KL S ?M 2g L L 0 ;e35T

where L 1is the height of the riser,L 0is the original length of the spring,L S is the static stretch of the spring under the action of the mass block of the SA TMD,L 0is the distances between the spring and the support point and L is the mass block and the support point.In Fig.4,L d is the dynamic stretch of the spring.

It is easy to derive the equivalent sti?ness,K 0;at the position of the mass block.Obviously,K and K 0have the following relationship:

K 0?K L 0L

2:e36TDe?ning that at time i the optimal frequency of the SA TMD is f T eTi ;the distance between the mass block and the support point is L i ;and the height of the base of the springs is L 1eTi :The corresponding values at time (i t1Tare de?ned to be f T eTi t1;L i t1and L 1eTi t1;respectively.When the controlled frequency of the bridge varies from f B eTi to f B eTi t1with wind speed,the corresponding optimal frequency of the SA TMD should vary from f T eTi to f T eTi t1according to the optimal algorithm,which will be illustrated in the Yichang bridge example in Section 5.Thus,the revised quantities of L and L 1can be derived,respectively,as

D L ?b à1eTL

e37T

and

D L 1?àb à1eTM 2g L 0K L ;e38T

Fig.4.Analysis model for parameter adjustment of the SA TMD.

M.Gu et al./J.Wind Eng.Ind.Aerodyn.90(2002)111–126

120

where

b ?f T eTi f T eTi t1:e39T

When Eq.(37)has a positive value,the mass block should be moved away from the support point;in contrast,when the result of Eq.(37)is negative the mass block should be moved towards the support point.

5.An example

The Yichang bridge over Yangtze River is taken here as an example to show the application of the SA TMD system for controlling the wind-induced vibration of bridges.The bridge has a steel deck and a main span of 960m and two side spans of 245m.The main parameters of the bridge given in Ref.[21]are listed in Table 1.The ?rst symmetric vertical bending and torsional frequencies are 0.161and 0.337Hz,respectively.Flutter derivatives are not shown here but can be found in Ref.[21].Kaimal’s and Panofsky’s expressions [22]are adopted for the power spectra of the longitudinal and vertical components of ?uctuating wind speed,respectively,in the analysis.A couple of SA TMDs are mounted at the center of the main span to control the ?rst symmetric vertical bending and torsional modal coupling vibration.So only the ?rst symmetric vertical bending and torsional modes are taken into consideration for the analysis.

The computation indicates that the comfort level criterion cannot be satis?ed when no control device is used.This can be seen from Fig.5.The critical ?utter wind speed of the bridge under wind with t31and 01attack angles are 47and 70m/s,respectively,whereas the design ?utter wind speed of the bridge is 44m/s [21].This means that the ?utter behavior of the bridge is satisfactory.The shortest fatigue life of the critical member is estimated to be much longer than the design life of the bridge using the method in Ref.[19].In addition,the stress strength objective is also satis?ed.In view of these results,only the maximum bu?eting acceleration is

Table 1

Main parameters of the Yichang bridge

Main span (m)

960Width of the deck (m)

30Clearance above water (m)

50Equivalent mass per length (t/m)

15.07Equivalent inertial moment of mass per length (t m 2)

1111Design wind speed (m/s)

29Lift coe?cient at 0o attack angle

à0.12Moment coe?cient

at 0o attack angle 0.023q C L =q a àá a ?01 4.43q C M =q a àá a ?01 1.018Structural damping ratio 0.005

M.Gu et al./J.Wind Eng.Ind.Aerodyn.90(2002)111–126121

required to be mitigated for the serviceability criterion.However,in order to investigate the advantages of the SA TMD over passive TMD,the SA TMD is studied for the serviceability objective and the safety objective increasing critical ?utter wind speed.Thus,the control objectives can be given as

Ac peak p e 0:1018U à29eTeU p 29m =s T

for serviceability objective e40T

and

max eU C TeU X 29m =s Tfor safety objective :e41TIn the process of design of the SA TMD system,its total mass is ?rst determined in terms of the serviceability objective.The variations of not only the ?rst vertical frequency and the ?rst torsional frequency but also their ratio with wind speed known from the wind tunnel experiments are usually di?erent from the actual variations of these parameters,as mentioned above.In the present analysis,only the frequency ratio from the wind tunnel test is assumed to be the same as the actual one,whereas the ?rst vertical and torsional vibration frequencies from the wind tunnel tests are still considered to be di?erent from those of the actual bridge.Thus,the ?rst vertical bending frequency and the ?rst torsional frequency can be treated as the variables for the analysis.Here,the vertical bending frequency is taken as the design variable at lower wind speed and the ?rst torsional frequency as the variable at the higher wind speed according to the phenomenon that at lower wind speed,the vertical bending component in the total bu?eting response greatly exceeds the torsional component,while at higher wind speed the contrary is the case.Fig.601020304050

6070

101001000

Wind velocity U (m/s)

A c P e a k (m i l l i -g )Fig.5.Peak acceleration of the Yichang bridge deck without control.

M.Gu et al./J.Wind Eng.Ind.Aerodyn.90(2002)111–126

122

shows the relationship between the optimal frequency of the SA TMD system,which is determined by maximizing the control e?ciency (Eq.(32)),and the dominant vibration frequency of the bridge.In the process of actual control of the bridge,the dominant vibration frequency of the bridge can be identi?ed on line from the recorded acceleration signals from ?eld measurements,and the frequency of the SA TMD system could thus be optimally determined according to Fig.6.

From the computation,the variation of the optimal frequency of the SA TMD system ranges from 0.17to 0.28Hz due to the e?ects of the ?utter derivatives.If the classical hanging-type TMD is used,the largest static stretching,d ;of its spring is

8.59m in terms of the equation of d ?g =o 2;where o is the circular frequency of the TMD and g is the acceleration due to gravity.Since only 2m height is available within the deck of the Yichang bridge for the installation of the SA TMDs,the allowable static stretching of the spring could only be limited to 0.7m.Thus,the ratio of L to L 0is correspondingly determined to be 12.3according to the principle of the lever-type TMD [15].The variation ranges of L and L 1were further determined to be from 3.0to 5.0m and from 0.253to 0.68m,respectively,in terms of the optimal frequency range of the SA TMDs.The total mass is computed to be 160t for the serviceability objective and the damping ratios of the SA TMDs take about 5%.The result of the bu?eting control for the comfort level objective is illustrated in Fig.7.From this ?gure,it is seen that the comfort level objective has been achieved.In addition,the critical ?utter wind speed is increased from 70to 86m/s with the SA TMDs.

To further investigate the control e?ciency of the SA TMDs on the wind-induced vibration of the Yichang bridge with uncertain vibration frequencies,the ?rst vertical vibration frequency and the ?rst torsional vibration frequency of the bridge are assumed to have 715%errors.Fig.8shows the numerical results of the control 0.1600.1650.1700.1750.1800.185

0.14

0.16

0.18

0.20

0.220.240.26

0.28

0.30

f o p t (H z )f(H z)

Fig.6.Relationship between optimal frequency of the SA TMD and the dominant vibration frequency of the bridge.

M.Gu et al./J.Wind Eng.Ind.Aerodyn.90(2002)111–126

123

e?ciencies.Three curves are drawn in the ?gure.The upper one represents the control e?ciency of the SA TMDs,which is almost the same as the control e?ciency corresponding to the case where no frequency errors occur;the other two curves are the control e?ciencies of the optimal passive TMDs.The total mass of the SA TMD

1001000

010********

6070

W ind velocity U (m /s)

A c P e a k (m i l l i -g ) Fig.7.Peak acceleration of the Yichang bridge deck under

control.R b (%)U (m/s)

Fig.8.Control e?ciencies with wind speed for the three cases.

M.Gu et al./J.Wind Eng.Ind.Aerodyn.90(2002)111–126

124

M.Gu et al./J.Wind Eng.Ind.Aerodyn.90(2002)111–126125

system is the same as the mass of the optimal passive TMD.The optimal parameters of the passive TMD are determined in terms of the wind tunnel test results but without considering the frequency errors.This means that the optimal passive TMD is only‘‘optimal’’for the ideal condition but not‘‘optimal’’for the bridge with uncertain vibration frequencies.The results shown in this?gure indicate that the SA TMD system has much better control e?ciency than the passive TMD system for the bridge with uncertain vibration frequencies.In addition,all the control e?ciencies are seen to descend with wind speed at the lower wind speed.This is because the?rst vertical mode dominates the total bu?eting response at the lower wind speed and the vertical vibration damping,which is the sum of the structural damping and the aerodynamic damping,increases with wind speed.When wind speed increases,the torsional bu?eting component increases,while the torsional damping decreases.When wind speed arrives around50m/s,the optimal frequency of the SA TMDs could jump close to the?rst torsional vibration frequency of the bridge from its former optimal value near the?rst vertical vibration frequency.Moreover,the torsional damping is much smaller than the vertical one at this time.As a result,the control e?ciency of the SA TMDs gradually becomes higher for wind speed higher than50m/s.As for the passive TMD,it is not able to follow the track of the optimal frequency,so its control e?ciency continues to decrease when wind speed is over50m/s.

6.Concluding remarks

This paper aims at developing a semi-active control system for the control of wind-induced vibrations of long-span bridges.The control objectives are discussed ?rstly in terms of the characteristics of the wind-induced vibrations of bridges.The critical?utter wind speed and the strength and fatigue damages due to bu?eting are chosen as the safety objectives,while the comfort level of the drivers and passages in vehicles is the serviceability objective.A new lever-type TMD with an adjustable frequency and the corresponding control strategy are primarily developed.A case study of the Yichang bridge,a suspension bridge with a main span of960m,shows that the semi-active TMD device has much better control e?ciency and robustness for the control of wind-induced vibration of long-span bridges than passive TMD. Acknowledgements

This project is supported by the National Science Foundation for the Outstanding Young and Foundation for University Key Teacher by the Ministry of Education of China,which are gratefully acknowledged.

References

[1]Wind-resistant study on the Yangpu bridge,Research Report,Tongji University,1992.

[2]K.Wilde,Y.Fujino,T.Kawakami,Analytical and experimental study on passive aerodynamic

control of?utter of a bridge deck,J.Wind Eng.Ind.Aerodyn.80(1999)105–119.

[3]R.L.Waldlaw,The improvement of aerodynamic performance,in:https://www.360docs.net/doc/de3105552.html,rsen (Ed.),Aerodynamics of

Larger Bridges,Balkema,Rotterdam,1992,pp.59–70.

[4]M.Gu,H.F.Xiang,Z.X.Lin,Flutter-and bu?eting-based selection for long-span bridges,J.Wind

Eng.Ind.Aerodyn.80(1999)373–382.

[5]Wind-resistant study on the Jiangyin bridge,Research Report,Tongji University,1995.

[6]Wind-resistant study on the Tiger-gate bridge,Research Report,Tongji University,1995.

[7]S.Predikman,D.T.Mook,A new method for actively suppressing ?utter of suspension bridge,

J.Wind Eng.Ind.Aerodyn.69–71(1997)955–974.

[8]H.Hansen,P.T.Christensen,Active control of long bridges using ?aps,Proceedings of the Second

World Conference on Structural Control,1998,pp.851–858.

[9]S.Miyachi,M.Yoneda,K.Edamoto,E?ects of leading edge stabilizer on the aerodynamic stability

of cable-stayed bridge section,Proceedings of the Second World Conference on Structural Control,1998,pp.827–834.

[10]K.Wilde,Y.Fujino,Aerodynamic control of bridge deck ?utter by active surfaces,J.Eng.Mech.

ASCE 124(7)(1998)718–727.

[11]J.Nobuto,Y.Fujino,M.Ito,A study on the e?ectiveness of T.M.D.to suppress a coupled ?utter of

bridge deck,Proceedings of the Japan Society of Civil Engineering,No.398/I-10,1988,pp.413–416.

[12]M.Gu,C.C.Chang,W.Wu,H.F.Xiang,Increase of critical ?utter wind speed of long-span bridges

using tuned mass dampers,J.Wind Eng.Ind.Aerodyn.73(1998)111–123.

[13]M.Gu,H.F.Xiang,Optimization of TMD for suppressing bu?eting responses of long-span bridges,

J.Wind Eng.Ind.Aerodyn.42(1992)1383–1392.

[14]M.Gu,H.F.Xiang,A.R.Chen,A practical method of TMD for suppressing wind-induced vertical

bu?eting of long-span cable-stayed bridges and its application,J.Wind Eng.Ind.Aerodyn.51(1994)203–213.

[15]M.Gu,S.R.Chen,C.C.Chang,Bu?eting control of the Yangpu bridge using multiple tuned mass

dampers,in:https://www.360docs.net/doc/de3105552.html,rsen,https://www.360docs.net/doc/de3105552.html,rose,F.M.Livesey (Eds.),Proceedings of the 10th International Conference On Wind Engineering,Balkema,Rotterdam,1999,pp.893–898.

[16]S.D.Xue,Torsional vibration control of suspension bridge decks with tuned liquid column damper,

Ph.D.Thesis,The Hong Kong Polytechnic University,Hong Kong,1999.

[17]E.Conti,G.Grillaud,J.Jacob,N.Cohen,Wind e?ects on Normandie cable-stayed bridge:

comparison between full aeroelastic model tests and quasi-steady analytical approach,J.Wind Eng.Ind.Aerodyn.65(1996)189–201.

[18]G.Diana,F.Cheli,A.Zasso,A.Collina,J.Brownjohn,Suspension bridge parameter identi?cation in

full scale test,J.Wind Eng.Ind.Aerodyn.41–44(1992)164–176.

[19]M.Gu,Y.L.Xu,L.Z.Chen,H.F.Xiang,Fatigue life estimation of steel girder of Yangpu cable-

stayed bridge due to bu?eting,J.Wind Eng.and Ind.Aerodyn.80(1999)383–400.

[20]A.I.Peter,Motion criteria,Technical Notes of the Rowan Williams Davies &Irwin Inc.(RWDI),

Ont.,Canada,1999.

[21]Wind-resistant study on the Yichang bridge over Yangtze River,Research Report,Tongji University,

1998.

[22]E.Simiu,R.H.Scanlan,Wind E?ects on Structures:An Introduction to Wind Engineering,Wiley,

New York,1986.M.Gu et al./J.Wind Eng.Ind.Aerodyn.90(2002)111–126

126