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Increase of critical flutter wind speed of long-span bridges using tuned mass dampers
OOURN,~_ OF
~NNf~O Journal of Wind Engineering and Industrial Aerodynamics 73 (1998) 111 123
ELSEVIER
~/I]~[~(~
Increase of critical flutter wind speed of long-span bridges using tuned mass dampers M. G u a'*, C.C. Chang b, W. Wu a, H.F. X i a n g a aDepartment of Bridge Engineering, Tongji University, Shanghai 200092, China bDepartment of Civil and Structural Engineering, Hong Kong University of Science & Technology, Hong Kong, China
Received 5 November 1996; revised 8 May 1997; accepted 10 June 1997
Abstract In this paper increasing the critical flutter wind speed of long-span bridges by using tuned mass dampers (TMDs) is theoretically and experimentally studied. Equations governing the motions of a bridge with TMDs are established. The Routh-Hurwitz stability criterion is used to study the aerodynamic instability of the bridge based on the characteristic equation of the system of the bridge and TMDs. A sectional model wind-tunnel test on the Tiger Gate Bridge, a suspension bridge with a steel box deck and a center span of 888 m, is carried out to confirm the numerical results. Some new findings from the test and the calculation are presented. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Tuned mass damper; Long-span bridges; Critical flutter wind speed; Control;
Wind-tunnel test; The Tiger Gate Bridge
1. Introduction Nowadays, with the rapid increase of bridge spans, research on controlling flutter of long-span bridges has been a problem of concern. Flutter is a phenomenon of self-excited vibration, which may cause a bridge to vibrate continuously with increasing amplitude until the bridge structure fails. This phenomenon must be avoided during the construction and the operation stages of a bridge. The aerodynamic method is usually an effective countermeasure for flutter suppression, by increasing the flutter critical wind speed of a bridge. G o o d aerodynamic performance of a bridge deck can be achieved by using shallow sections, closed sections, streamline edges and other minor and more subtle changes to the crosssectional geometry [1,2]. * Corresponding author. 0167-6105/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0 1 67-6105 (97)00282- 1
112
M. Gu et al./J. Wind Eng. Ind. Aerodvn. 73 (1998) 111 123
Mechanical countermeasures, the main one of which is the passive tuned mass damper (TMD), can effectively improve the buffeting and vortex-induced vibration behavior of long-span bridges. There have been theoretical studies [3, 4] and a number of applications, such as for the construction stage of the Normandy Bridge in France [5]. But the use of T M D for flutter control has so far seldom been studied, eventhough, Nobuto et al. [6] indicated through numerical analysis and a simple wind-tunnel experiment that the T M D may also be effective in improving the flutter behavior of bridges. Aside from its effectiveness, other advantages of T M D may be lower cost and convenience. In this paper, the use of TMDs for increasing flutter wind speeds of bridges is theoretically and experimentally studied in much more detail than in Ref. [6]. First, the differential equations governing the motion of a bridge installed with T M D s are established, then the Routh Hurwitz stability criterion is used on the characteristic equation to determine the aerodynamic instability of the bridge. A sectional model of the Tiger Gate Bridge, a steel box deck suspension bridge with a main span of 888 m located in Southern China is carried out in a wind tunnel to verify the numerical results. The effects of the structural damping and the TMDs parameters, such as the frequency ratio, the mass inertial moment ratio of the T M D to the bridge, on the effectiveness of flutter control of bridges are investigated.
2. Basic theory Based on the classical method of flutter analysis [7], the use of the lowest modes relating to the vertical bending and the torsion of a bridge is sufficient, since they usually yield the desired lowest critical wind speed. It is assumed that these two modes are both symmetrical, so that only the lowest symmetrical vertical bending and torsion modes are considered in the current analysis. The phenomenon of flutter occurs when the frequencies of the lowest vertical bending mode and the lowest torsional mode merge as the wind speed reaches the critical value. Usually, a bridge in flutter would vibrate predominantly in the shape of the torsional mode. In this study, two identical TMDs, symmetrically located at the two sides of the cross section at the center of the main span of a bridge as shown in Fig. 1, are used to produce counter moments for the torsional vibration. The frequencies of the TMDs are tuned to the neighbourhood of the flutter frequency so as to increase the critical flutter wind speed. The schematic diagram of the bridge-TMD system together with some symbols used in this paper can be seen in Fig. I b, where Y1 and c~denote the vertical displacement and the torsion angle of the bridge deck, respectively; Y2 and Y3 denote the vertical displacements of T M D 1 and T M D 2 relative to Y1, respectively. These two T M D s are assumed to have the same mass, M2, the same spring stiffness coefficient, K2, and the same damping coefficient, C2. Let it be assumed that Yl(Z, t) = ~ l ( Z ~ l ( l }.
~l(z, t) =
tP(z)Tl(t),
(1)
(2)
M. Gu et al./J. Wind Eng. Ind. Aerodyn. 73 (1998) 111-123
113
cross section of the bridge deck
M,,L
Kt Ct ~.
g.
I
T
v Y2
I
K2 m:
C2
C2 Y L~
TMDt
L,
k
K2 ~, ,m TMD2 Y~
,t
Fig. 1. Schematic diagrams of T M D s and bridge deck system for flutter control analysis.
where q91(z)and ~Ol(z)are the first vertical bending and the first torsional mode-shape functions, respectively, ¢1(t) are the time-dependent generalized vertical and torsional modal amplitudes, respectively, z is the location along the bridge deck. Assuming that the aerodynamic forces and moment are as suggested by Scanlan and Tomko [7], the equations governing the motion of the bridge and the TMDs can be established as
[ M ] { f } + [C]{£} + [ K ] { Y } = O,
(3)
{Y}
(4)
where =
{~,, 7,, Y2, Y3},
m2~Px(Zo)
'Ms + 2m2~blZ(Zo) 0
I 0
[M]
=
m2~l(zo)
Lm24)l(zo)
[i 0
[K]
=
I, + 2m2LZtd)~(Zo) -- m 2 L t ~ l ( z o )
mzLt~ l(zo)
o K~ 0 0
- mzLt~l(Zo)
m2qgx(Zo)
mzLt~l(Zo)
m2
0
0
m2
-- p U 2 B K 2 A ' ~ C 2 2
0
K2 0
' K2I
, (5)
(6)
M. Gu et al./J. Wind Eng. Ind. Aerodyn. 73 (1998) 111 123
114
h
Ii
pUB2KH*CI
- pUB2KA*C12
[C] =
M~
pUBZKH'~C12
0
0
(_'~ - ,oUB2KA*C22
0
0
0
C2
0
0
0
C2
1
(s)
m(z)(p~(z)d:.
f
(7)
Mrtlcl
I, = f
l(z)tp~(z)dz,
(9)
slruct L
C11 =
(hi(z) d-,
(10)
o /.
(:12
t (pt(z)Ol(z)dz.
(11)
m/ 0
L p*
C22 = j~Pl(Z) dz,
(12)
,d 0
where m(z) and l(z) are the mass and the mass moment of inertia per unit length, respectively, Kh and K~ are the vertical bending and the torsional stiffnesses participating in the flutter (usually the first vertical bending and the first torsional stiffnesses), respectively, Ch and C, are the vertical bending and the torsional damping coefficients, respectively, L, the distance between the center of cross section of the bridge deck and one of the T M D s (see Fig. 1), L the main span length of the bridge, p the air density, U is the mean wind speed, B the bridge deck width, the reduced frequency K = B(o/U, H* and A* (i 1, 2, 3) are the flutter derivatives. The solution of f~Y] can be assumed as Y Io St
where ~X~ is a complex vector and S is a complex scalar. Substituting Eq. (13) into Eq. (3) yields ( [ M ] S Z + [ C ] S + [ K ] )• i Af ~ e~
St
=0.
14)
To ensure that ~A~ has a non-zero solution, the characteristic equation of Eq. 14) should be satisfied, IEM]S 2 +
[C]S + [ K ] J -- 0.
15)
115
M. Gu et al./J. Wind Eng. Ind. Aerodyn. 73 (1998) 111-123
Substituting Eq. (5)-(7) into Eq. (15), the characteristic equation can further be expanded as follows: 8 Z alS(8-1) = 0, i-O
(16)
ao = 4 A 2 A 2 o - 2 A ~ A v m 2 - 2A1A2om2 + A I A v m 2,
(17)
a l = -- 2 A Z A s m 2 -- 2 A Z A 7 C 2 + A1A8m22 + A z A 7 m2 + A 1 A v C z m 2 + A 1 A 7 m2 - 2 A 1 A 2 o C 2 - 2 A z A 2 o m 2 ,
(18)
a2 = - 2 A Z A 9 m 2 - 2 A 2 A 8 C 2 - 2 A z A 2 o C 2 - 2 A Z o m 2 K h + A a a , o m2 + A z A 8 m2 + Avm22Kh - A 3 A 6 m2 + 2 A I A s m z C 2 + 2 A 2 A T m z C 2 - 2A2ATK2 - 2AaA2oK2 + 2AIAvm2K2,
(19)
a3 = + + +
(20)
where
2 A 2 A 9 C 2 - 2 A 2 A s K 2 - 2 A 2 o C 2 K h - 2 A 2 A Z o K 2 + A z A 9 m2 A 8 K h m 2 -- A g A 6 m2 + 2 A 1 A 9 C 2 m 2 + 2 A 2 A 8 C z m 2 2ATKhC2m2 - 2A3A6C2m2 + 2A1A8K2m2 + 2A2AvK2m2 AIAsC 2 + A2ATC 2 + A1ATK2C2 + A1AsK2C2,
a4 = - A 9 K h m 2 + 2 A z A 9 C 2 m 2 + 2 A s C z m z K h - 2 A 4 A 6 C 2 m 2 + 2A1A9K2m2 + 2A2A8Kzm2 + 2AvK2m2Kh - 2A3A6Kzm2 "F a l Z 9 C2 -~- Z 2 A 8 C2 --F a T f 2 g h - Z 3 A 6 C2 - 2 A 2 A 9 K 2 + 2 A 1 A s C 2 K 2 + 2 A 2 A T C 2 K 2 + A 1 A T K 2,
(21)
0 5 = 2A9KhCzm2 + 2AzA9Kzmz + 2A8K2Khm2 - 2AgA6Kzm2 + A8KhC 2 - A3A6K2C 2 - AzAsKzC 2 + 2A1A9K2C2 + 2A2A8K2C 2 + 2ATK2KhC 2 + A1A8 K2 + A2A7 K2 ,
(22)
+ A 2 A 9 C2
a6 = A 9 K 2 K h m 2 + A 9 K h C2 + A a K 2 H h m 2 + 2 A 2 A 9 K 2 C 2 + 2 A 8 K 2 K h C 2 2A4A6K2C2 + A1A9 K2 + AzA8 K2 + AvK2Kh - A3A6K 2 ,
(23)
a7 = A 9 K Z K h + A 9 K z K h C 2 + A z A 9 K 2 + A 8 K Z K ~ - A 4 A 6 K 2 ,
(24)
a8 = A 9 K 2 K h
(25)
A1 = M~ + m21-ff~(Zo) + ~b~(Zo)-l,
(26)
A2 = Ch - p U B K H * C l l
(27)
-
-
and ,
A 3 = -- p U B 2 K H * C 1 2 ,
(28)
A4 = - p U 2 B K 2 A * C 1 2 ,
(29)
A5 = m2q~l(zo),
(30)
A6 = -- p U B Z K A * C I 2
,
(31)
A7 = ls + 2m2LZtp2(Zo),
(32)
A8 = C~, -- p U B 3 K A ~ C z 2 ,
(33)
Z 9 = K~, -- p U 2 B 2 K 2 A ' ~ C 2 2 ,
(34)
A t o = -- m2LZt qll(zo) •
(35)
116
M. Gu et al.:.l. Wind Eng. Ind. Aerodvn. 73 (1998) I l l
123
The Routh Hurwitz stability criterion is used to judge the stability of the bridge T M D system, and the critical flutter wind speed can be found according to Eq. (16). The numerical analysis results and the comparison between them and the experimental ones can be seen in Fig. 5.
3. Experimental study The sectional model test of the Tiger Gate Bridge with and without T M D s was carried out in the T J-1 Boundary Layer Wind Tunnel in the T o n ~ i University. The test aimed at verifying the above theoretical derivations and numerical analysis as well as studying the effects of using different structural and T M D parameters on the critical flutter wind speed.
3.1. Sectional model o[the lTger Gate Bridge The Tiger Gate Bridge is located in Southern China across the Pearl River. It is a suspension bridge with a steel box deck and a center span of 888 m. Fig. 2 shows the general view of the deck cross section. The TJ-1 BLWT has a working section of 1.2 m ( W ) × 1.8 m(ft) × 18 re(L), and the wind speed can be varied between 1 and 32 m/'s. Accordingly', the geometrical scale and wind speed scale were determined to be 7o and ~,, respectively, which correspond to a frequency ratio of 11.67. Accordingly, the torsional and vertical bending frequencies, 11,,,~and 1'~.~, of the model were 4.214 and 2.010 Hz, respectively. Mounting wind screens on a bridge deck can usually improve the traffic conditions for vehicles but quite often deteriorates the aerodynamic performance of the bridge, for example, by decreasing the critical flutter wind speed. Using the original model and the model with wind screens (see Fig. 2), the control effectiveness of the T M D s under the effect of different deck cross sections with different degrees of bhmtness was studied. The structural damping of the bridge model was produced by using a pendulumcontainer system. The pendulum consisted of two plates positioned in the vertical and
I
i /
Wind5~ctucl+
~
/
2(~()()it
I
A q
big. 2. General view of the deck cross section of the Tiger Gate Bridge.
!
M. Gu et al./J. Wind Eng. Ind. Aerodyn. 73 (1998) 111-123
117
vibrating with the bridge model
tng plate
w
1 /
/
/
7
Fig. 3. Schematic diagram of the structural damper set-up.
spill
Fig. 4. Schematic diagram of the TMD model.
the horizontal planes, respectively, and a bar rigidly connected these two plates to the bridge model (see Fig. 3). The upper part of the bar was fixed to the sectional model of the bridge. The damping came from the relative motion of the two plates to the liquid in the container when the bar was in motion with the bridge model. Three damping ratios, 0.7%, 2.13% and 4.05%, were selected in the test to investigate the effect of structural damping on the control effectiveness. 3.2. Model of tuned mass dampers Fig. 4 shows the model of a tuned mass damper, which consists of two springs, two mass blocks and a damping device. The damping device was again modeled using
M. Gu et al.i.l. Wind Eng. Ind. Aerodyn. 73 (1998) 111~123
118
Tablc 1 P a r a m e t e r s of 10 g r o u p s of T M I ) No.
I
2
3
4
"~
6
7
8
9
10
/q (°'i,) /rr ~
0.9 0.98 0.03
2.5 0.98 0.03
5.0 0.98 ().03
10 0.98 0.03
0.9 0.98 0.085
2.5 0.98 0.085
5.6 0.98 0.085
10 0.98 0.085
5.6 0.918 0.085
5.6 1.04 0.085
a pendulum-container system where a horizontal plate with multiple holes was attached to a bar which was rigidly connected to the bridge sectional model. When the bridge model vibrated, the upper plate with the bar and the damping plate vibrated harmonically, and the lower plate with the lumped mass blocks and the container moved relatively to the model, respectively. The bridge sectional model weighted 5.5 kg, and each T M D mass m, was 40 g. The mass ratio between the total mass of the TMDs and that of the bridge model was l~m -- 1.45%. The ratio of the mass moment of inertia is defined as lq
.)
= -m, LU I m ,
(36)
where lm is the mass moment of inertia of the bridge model, Different values of#t can be obtained by adjusting the distance L,. Define the frequency ratio I~ as t', =./ii:']i-.
(37)
where ,/{ is the frequency of the TMD, .[j. the flutter frequency of the bridge model and was equal to 3.125 Hz from the wind-tunnel experiment on the model without control. Different frequency ratios and damping ratic, s of the T M D could be obtained by adjusting the springs and the damping of the model. Table 1 lists 10 sets of T M D parameters used in the present experiment.
4. The main results The critical flutter wind speeds of the original model (without the wind screens) were experimentally determined to the 16, 17, and 18 m/s for structural damping ratios of 0.7%, 2.13% and 4.05%, respectively. The corresponding wind speeds from the theoretical calculation are 16.5, 17.1 and 18.3m/s, respectively. The experimental critical wind speed of the model with the wind screens for a structural damping ratio of 0.7% was 12m/s, as compared to the calculated result of 12.7 m/s. It is noted that to improve the accuracy of the experimental results, several tests were conducted for the same modeling condition. These results were then averaged to give the critical flutter mean wind speed.
M. Gu et al./J. Wind Eng. Ind. Aerodyn. 73 (1998) 111-123
119
20 -~ 18 16, .=_ 14 numerical .=~ 12 .,...
•
experiment
~tf--0.98, ~t=0.085 t
10 2
4
6
I
I
J
8
I0
20
16 "O
.~ 14 "= 12 r...) .:-.
].tf=0.98, ~t=0.03
10 0
I
2 4 6 8 Ratio of mass moment of inertia ].tI (%)
10
Fig. 5. Comparisons of the experimental and numerical critical wind speed ((s = 0.7%, model without wind screens).
4.1. Comparison of the experimental and theoretical results Let the control efficiency be defined as
where U and Ut are the critical flutter wind speeds without and with control, respectively. Fig. 5 shows the main comparisons between the results obtained from the experiment and from the numerical analysis when the structural damping ratio was equal to 0.7%. It is seen that the calculated results agree well with the experimental results, with the maximum error of 7.6% among the 8 groups of results.
4.2. Effects of the mass moment of inertia Fig. 5 also presents the effects on the flutter wind speed of the ratios of mass moment of inertia. The experimental critical wind speed of the sectional model without T M D was 16 m/s for a structural damping ratio (s = 0.7% as mentioned
M. Gu et al./J. Wind Eng. Ind. Aerodyn. 73 (1998) 111-123
120
20
19
18
.r-
17
16 0.90
0.95
100
1.05
1. lO
Frequency ratio ~tf (%) Fig. 6. The critical wind speed as a function of the frequency ratio/~f between the T M D and the bridge (~ = 0.7%, ~i = 5.6%. (, = 8.5%, model without wind screens}.
above. When the T M D damping ratio ~'t was equal to 8.5%, the critical wind speeds from the experiment were t 7.0, 17.5, 19.5 and 18.5 m/s, and the corresponding control efficiencies were 6.3%, 9.4%, 21.9% and 15.6%, for the ratios of mass moment of inertia of 0.9%, 2.5%, 5.6% and t0.0%, respectively. When the T M D damping ratio was equal to 3%, the experimental flutter wind speeds were 16.5, 18.0, 18.5 and 18.5 m/s, and the corresponding control efficiencies were 3%, 12.5%, 15.6% and 15.6% for the ratios of mass inertial moment of 0.9%, 2.5%, 5.6% and 10%, respectively. It seems that the control efficiency increases as the ratio of the mass moment of inertia increases from 0.9% to 5.6%. The results also show that the control efficiency tends to saturate or even decrease when lq goes up to about 10%.
4.3. Effects ~?[the /?equency ratio Fig. 6 shows the experimental critical wind speeds versus frequency ratio Pt for the bridge model without wind screens for ~q = 5.6%, ff~(the structural ratio) of 0.7% and ~t = 8.5%. As seen from the figure, the critical wind speed was 19.5 m/s, with a corresponding control efficiency of 22% for lLf 0.98. The critical wind speeds became 17.4 and 18 m/s which corresponded to control efficiencies of 8.8% and 12.5% for j~f--0.92 and 1.04, respectively. There is a rather significant effect on the flutter control efficiency as the frequency ratio varies. The exact optimal frequency ratio between the T M D and the bridge would depend on the cross section shape of the bridge deck. =
M. Gu et al./J. Wind Eng. Ind. Aerodyn. 73 (1998) 111-123 20
121
40
- 3o~.
-~
nd speed ~. 16 -~ .7 "~ -=
no control i. ............... ....• . . . . . . . . . .....
14
•
gi=5.6%
• •
gi=2.5% gi=0.9%
~20 ~
control efficiency
"6 r.)
' " ' - . , . ,.
•
.....
................ I)... . . . .
"'l "'"'• ....... " - . . _ .....
............ o ...............
10
10
*"..
12
-.11121
I
i
I
L
1
2
3
4
0 5
Structural damping ratio (%) Fig. 7. T h e effects of structural d a m p i n g ratio on the critical wind speed a n d the control efficiency of the T M D s (/if = 0.98, ~t = 8.5%, m o d e l without wind screen).
4.4. Effects of the TMD damping ratio Fig. 5 also shows the experimental critical wind speeds of the bridge model without wind screens versus two kinds of TMD damping ratios. The TMD damping ratio was designed to be 3% and 8.5°,/0. It can be seen from the figure that when the damping ratio varies from 3% to 8%, the variation of the critical wind speed is no more than 1 m/s for all the five values of #v Theoretical analysis gives slightly larger derivations. The TMD damping ratio seems to have less effect on the flutter control than the frequency ratio.
4.5. Effects of the structural damping ratio on the control efficiency Fig. 7 presents the effects of the structural damping ratio on the flutter control efficiency by using No. 5, No. 6 and No. 7 TMD designs listed in Table 1. For the bridge model without the TMD, the critical wind speeds were 16, 17 and 18 m/s for the structural damping ratios of 0.7%, 2.13% and 4.05%, respectively. When the No. 6 TMD (/h = 2.5%) was used, the critical wind speeds were 17.5, 18.2 and 18.5 m/s, which correspond to control efficiencies of 9.4%, 7.1% and 2.8%, for structural damping ratios of 0.7%, 2.13% and 4.05%, respectively. The critical wind speeds and the corresponding control efficiencies for the No. 5 TMD (#i = 0.9%) and the No. 7 TMD (#i = 5.6%) are also shown in this figure. These results show that the TMD is more effective for a bridge with a lower structural damping than for those with higher structural damping.
122
MI Gu et al./J. Wind Eng. Ind. Aerodyn. 73 (1998) 111-123 Table 2 Effect of the bridge deck cross-section shape Mode deck shape
Without w.s. With w.s.
No. 7 T M | )
No. 8 T M D
U, (m/s)
~l~-I%)
U, (m/s)
q~ (%)
19.5 17.5
21.9 45.8
18.5 18.3
15.6 52.5
w.s.: wind screen
4.6. Effbcts of the bridge deck cross-section shape on the control e~ciency To investigate the effects of bridge deck cross-section shape on the flutter control efficiency, experiments on the bridge model with and without wind screens were also conducted. Table 2 shows some of the experimental results. When the structural damping ratio was equal to 0.7%, the experimental critical wind speeds were 16 and 12 m/s for the model without and with the wind screens, respectively. As shown in Table 2, installing No. 8 T M D would increase the experimental critical wind speed form 16 to 18.5 m/s with a control efficiency of about 15.6% for the model without the wind screens; the same T M D can increase the critical wind speed from 12 to 18.3 m/s with a control efficiency of 52.5 % for the model with the wind screens. The effect of the No. 7 T M D on the control efficiency is similar to that obtained with No. 8 T M D . Similar trends can be seen for the other groups of T M D . These results indicate that the T M D is more effective for controlling the flutter of a bluffdeck than a streamlined deck.
5. Concluding remarks in this study, the effects of T M D s on the critical flutter wind speed of a bridge have been theoretically and experimentally analyzed. The theoretical approach was established by applying the Rough-Hurwitz stability criterion to the characteristic equation of the bridge and T M D combined system. The experimental studies were based on sectional model tests of the Tiger Gate Bridge performed in the T J-1 Boundary Layer Wind Tunnel in the Tongji University. Based on the results, the following conclusions can be found. 1. The T M D can increase the critical flutter wind speed significantly. The T M D with more than 5.6% ratio of mass inertia moment can increase the critical flutter wind speed of the Tiger-Gate Bridge sectional model with wind screens by more than 4O%. 2. Within a certain range, in general, the larger the T M D mass moment of inertia, the better the control performance. The numerical and experimental results show that for the Tiger-Gate Bridge the control efficiency tends to saturate or even decrease when the ratio of the mass moment of inertia goes up to about 10%.
M. Gu et al./J. Wind Eng. Ind. Aerodyn. 73 (1998) 111-123
123
3. T h e T M D is m o r e effective for c o n t r o l l i n g the flutter of a b r i d g e with low s t r u c t u r a l d a m p i n g t h a n with high s t r u c t u r a l d a m p i n g , t h a t is, it is m o r e efficient to use the T M D on steel bridges t h a n on c o m p o s i t e bridges, o r concrete bridges. 4. T h e T M D is m o r e effective for c o n t r o l l i n g the flutter of a b r i d g e with a bluff deck cross section t h a n with a s t r e a m l i n e d deck cross section. 5. T h e frequency r a t i o between the T M D a n d the b r i d g e as well as the d a m p i n g ratio of the T M D b o t h influence the c o n t r o l effectiveness. H o w e v e r , the effect of the frequency r a t i o is m o r e p r o n o u n c e d t h a n t h a t of the d a m p i n g ratio.
Acknowledgements T h e project is c o - s u p p o r t e d by the N a t i o n a l N a t u r e Science F o u n d a t i o n a n d N a t i o n a l Science F o u n d a t i o n for the O u t s t a n d i n g Youth, which are gratefully acknowledged.
References [1] R.L. Wardlaw, The improvement of aerodynamics performance, in: Allan Larsen (Ed), Proc. 1st Int. Symp. on Aerodynamics of Large Bridges, A.A. Balkema Publishers, Denmark, 1992, pp. 59-70. [2] M. Gu, Aerodynamic and mechanical countermeasures for suppressing wind induced vibrations of long-span bridges, in: X.Q. Na, Z.Y. Shen (Eds.), The Present Situation and Tendency of Structural Engineering, Tongji University Press, 1994, pp. 275 282 (in Chinese). [3] M. Gu, H.F. Xiang, A.R. Chen, A practical method of passive TMD for suppressing wind-induced vertical vibration of long-span cable-stayed bridges and its application, J. Wind Eng. Ind. Aerodyn. 51 (1994) 203-213. [4] A.R. Chen, H.F. Xiang, M. Gu, Vortex-excited vibration control of bridges using TMD, in: K.M. Lain, Y.K. Cheung (Eds), Proc. of APSOWE-3, The Central Printing Press Ltd., Hong Kong, Vol. 1 (1993) pp. 235-240. [5] V. Michel, Wind design and analysis for the Normandy Bridge, in: Proc. 1st International Symposium on Aerodynamics of Large Bridges, A.A. Balkema Publishers, Denmark, 1992, pp. 183-216. [6] J. Nobuto, Y. Fujino, M. Ito, A study on the effectiveness of T.M.D. to suppress a coupled flutter of bridge deck, Proc. Japan Society of Civil Engineering, No. 398/1-10, 1988, pp. 413416. [7] R.H. Scanlan, J.J. Tomko, Airfoil and bridge deck flutter derivation, J. Eng. Mech. Div. ASCE 97 (1971) 1712-1737.
~NNf~O Journal of Wind Engineering and Industrial Aerodynamics 73 (1998) 111 123
ELSEVIER
~/I]~[~(~
Increase of critical flutter wind speed of long-span bridges using tuned mass dampers M. G u a'*, C.C. Chang b, W. Wu a, H.F. X i a n g a aDepartment of Bridge Engineering, Tongji University, Shanghai 200092, China bDepartment of Civil and Structural Engineering, Hong Kong University of Science & Technology, Hong Kong, China
Received 5 November 1996; revised 8 May 1997; accepted 10 June 1997
Abstract In this paper increasing the critical flutter wind speed of long-span bridges by using tuned mass dampers (TMDs) is theoretically and experimentally studied. Equations governing the motions of a bridge with TMDs are established. The Routh-Hurwitz stability criterion is used to study the aerodynamic instability of the bridge based on the characteristic equation of the system of the bridge and TMDs. A sectional model wind-tunnel test on the Tiger Gate Bridge, a suspension bridge with a steel box deck and a center span of 888 m, is carried out to confirm the numerical results. Some new findings from the test and the calculation are presented. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Tuned mass damper; Long-span bridges; Critical flutter wind speed; Control;
Wind-tunnel test; The Tiger Gate Bridge
1. Introduction Nowadays, with the rapid increase of bridge spans, research on controlling flutter of long-span bridges has been a problem of concern. Flutter is a phenomenon of self-excited vibration, which may cause a bridge to vibrate continuously with increasing amplitude until the bridge structure fails. This phenomenon must be avoided during the construction and the operation stages of a bridge. The aerodynamic method is usually an effective countermeasure for flutter suppression, by increasing the flutter critical wind speed of a bridge. G o o d aerodynamic performance of a bridge deck can be achieved by using shallow sections, closed sections, streamline edges and other minor and more subtle changes to the crosssectional geometry [1,2]. * Corresponding author. 0167-6105/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0 1 67-6105 (97)00282- 1
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M. Gu et al./J. Wind Eng. Ind. Aerodvn. 73 (1998) 111 123
Mechanical countermeasures, the main one of which is the passive tuned mass damper (TMD), can effectively improve the buffeting and vortex-induced vibration behavior of long-span bridges. There have been theoretical studies [3, 4] and a number of applications, such as for the construction stage of the Normandy Bridge in France [5]. But the use of T M D for flutter control has so far seldom been studied, eventhough, Nobuto et al. [6] indicated through numerical analysis and a simple wind-tunnel experiment that the T M D may also be effective in improving the flutter behavior of bridges. Aside from its effectiveness, other advantages of T M D may be lower cost and convenience. In this paper, the use of TMDs for increasing flutter wind speeds of bridges is theoretically and experimentally studied in much more detail than in Ref. [6]. First, the differential equations governing the motion of a bridge installed with T M D s are established, then the Routh Hurwitz stability criterion is used on the characteristic equation to determine the aerodynamic instability of the bridge. A sectional model of the Tiger Gate Bridge, a steel box deck suspension bridge with a main span of 888 m located in Southern China is carried out in a wind tunnel to verify the numerical results. The effects of the structural damping and the TMDs parameters, such as the frequency ratio, the mass inertial moment ratio of the T M D to the bridge, on the effectiveness of flutter control of bridges are investigated.
2. Basic theory Based on the classical method of flutter analysis [7], the use of the lowest modes relating to the vertical bending and the torsion of a bridge is sufficient, since they usually yield the desired lowest critical wind speed. It is assumed that these two modes are both symmetrical, so that only the lowest symmetrical vertical bending and torsion modes are considered in the current analysis. The phenomenon of flutter occurs when the frequencies of the lowest vertical bending mode and the lowest torsional mode merge as the wind speed reaches the critical value. Usually, a bridge in flutter would vibrate predominantly in the shape of the torsional mode. In this study, two identical TMDs, symmetrically located at the two sides of the cross section at the center of the main span of a bridge as shown in Fig. 1, are used to produce counter moments for the torsional vibration. The frequencies of the TMDs are tuned to the neighbourhood of the flutter frequency so as to increase the critical flutter wind speed. The schematic diagram of the bridge-TMD system together with some symbols used in this paper can be seen in Fig. I b, where Y1 and c~denote the vertical displacement and the torsion angle of the bridge deck, respectively; Y2 and Y3 denote the vertical displacements of T M D 1 and T M D 2 relative to Y1, respectively. These two T M D s are assumed to have the same mass, M2, the same spring stiffness coefficient, K2, and the same damping coefficient, C2. Let it be assumed that Yl(Z, t) = ~ l ( Z ~ l ( l }.
~l(z, t) =
tP(z)Tl(t),
(1)
(2)
M. Gu et al./J. Wind Eng. Ind. Aerodyn. 73 (1998) 111-123
113
cross section of the bridge deck
M,,L
Kt Ct ~.
g.
I
T
v Y2
I
K2 m:
C2
C2 Y L~
TMDt
L,
k
K2 ~, ,m TMD2 Y~
,t
Fig. 1. Schematic diagrams of T M D s and bridge deck system for flutter control analysis.
where q91(z)and ~Ol(z)are the first vertical bending and the first torsional mode-shape functions, respectively, ¢1(t) are the time-dependent generalized vertical and torsional modal amplitudes, respectively, z is the location along the bridge deck. Assuming that the aerodynamic forces and moment are as suggested by Scanlan and Tomko [7], the equations governing the motion of the bridge and the TMDs can be established as
[ M ] { f } + [C]{£} + [ K ] { Y } = O,
(3)
{Y}
(4)
where =
{~,, 7,, Y2, Y3},
m2~Px(Zo)
'Ms + 2m2~blZ(Zo) 0
I 0
[M]
=
m2~l(zo)
Lm24)l(zo)
[i 0
[K]
=
I, + 2m2LZtd)~(Zo) -- m 2 L t ~ l ( z o )
mzLt~ l(zo)
o K~ 0 0
- mzLt~l(Zo)
m2qgx(Zo)
mzLt~l(Zo)
m2
0
0
m2
-- p U 2 B K 2 A ' ~ C 2 2
0
K2 0
' K2I
, (5)
(6)
M. Gu et al./J. Wind Eng. Ind. Aerodyn. 73 (1998) 111 123
114
h
Ii
pUB2KH*CI
- pUB2KA*C12
[C] =
M~
pUBZKH'~C12
0
0
(_'~ - ,oUB2KA*C22
0
0
0
C2
0
0
0
C2
1
(s)
m(z)(p~(z)d:.
f
(7)
Mrtlcl
I, = f
l(z)tp~(z)dz,
(9)
slruct L
C11 =
(hi(z) d-,
(10)
o /.
(:12
t (pt(z)Ol(z)dz.
(11)
m/ 0
L p*
C22 = j~Pl(Z) dz,
(12)
,d 0
where m(z) and l(z) are the mass and the mass moment of inertia per unit length, respectively, Kh and K~ are the vertical bending and the torsional stiffnesses participating in the flutter (usually the first vertical bending and the first torsional stiffnesses), respectively, Ch and C, are the vertical bending and the torsional damping coefficients, respectively, L, the distance between the center of cross section of the bridge deck and one of the T M D s (see Fig. 1), L the main span length of the bridge, p the air density, U is the mean wind speed, B the bridge deck width, the reduced frequency K = B(o/U, H* and A* (i 1, 2, 3) are the flutter derivatives. The solution of f~Y] can be assumed as Y Io St
where ~X~ is a complex vector and S is a complex scalar. Substituting Eq. (13) into Eq. (3) yields ( [ M ] S Z + [ C ] S + [ K ] )• i Af ~ e~
St
=0.
14)
To ensure that ~A~ has a non-zero solution, the characteristic equation of Eq. 14) should be satisfied, IEM]S 2 +
[C]S + [ K ] J -- 0.
15)
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M. Gu et al./J. Wind Eng. Ind. Aerodyn. 73 (1998) 111-123
Substituting Eq. (5)-(7) into Eq. (15), the characteristic equation can further be expanded as follows: 8 Z alS(8-1) = 0, i-O
(16)
ao = 4 A 2 A 2 o - 2 A ~ A v m 2 - 2A1A2om2 + A I A v m 2,
(17)
a l = -- 2 A Z A s m 2 -- 2 A Z A 7 C 2 + A1A8m22 + A z A 7 m2 + A 1 A v C z m 2 + A 1 A 7 m2 - 2 A 1 A 2 o C 2 - 2 A z A 2 o m 2 ,
(18)
a2 = - 2 A Z A 9 m 2 - 2 A 2 A 8 C 2 - 2 A z A 2 o C 2 - 2 A Z o m 2 K h + A a a , o m2 + A z A 8 m2 + Avm22Kh - A 3 A 6 m2 + 2 A I A s m z C 2 + 2 A 2 A T m z C 2 - 2A2ATK2 - 2AaA2oK2 + 2AIAvm2K2,
(19)
a3 = + + +
(20)
where
2 A 2 A 9 C 2 - 2 A 2 A s K 2 - 2 A 2 o C 2 K h - 2 A 2 A Z o K 2 + A z A 9 m2 A 8 K h m 2 -- A g A 6 m2 + 2 A 1 A 9 C 2 m 2 + 2 A 2 A 8 C z m 2 2ATKhC2m2 - 2A3A6C2m2 + 2A1A8K2m2 + 2A2AvK2m2 AIAsC 2 + A2ATC 2 + A1ATK2C2 + A1AsK2C2,
a4 = - A 9 K h m 2 + 2 A z A 9 C 2 m 2 + 2 A s C z m z K h - 2 A 4 A 6 C 2 m 2 + 2A1A9K2m2 + 2A2A8Kzm2 + 2AvK2m2Kh - 2A3A6Kzm2 "F a l Z 9 C2 -~- Z 2 A 8 C2 --F a T f 2 g h - Z 3 A 6 C2 - 2 A 2 A 9 K 2 + 2 A 1 A s C 2 K 2 + 2 A 2 A T C 2 K 2 + A 1 A T K 2,
(21)
0 5 = 2A9KhCzm2 + 2AzA9Kzmz + 2A8K2Khm2 - 2AgA6Kzm2 + A8KhC 2 - A3A6K2C 2 - AzAsKzC 2 + 2A1A9K2C2 + 2A2A8K2C 2 + 2ATK2KhC 2 + A1A8 K2 + A2A7 K2 ,
(22)
+ A 2 A 9 C2
a6 = A 9 K 2 K h m 2 + A 9 K h C2 + A a K 2 H h m 2 + 2 A 2 A 9 K 2 C 2 + 2 A 8 K 2 K h C 2 2A4A6K2C2 + A1A9 K2 + AzA8 K2 + AvK2Kh - A3A6K 2 ,
(23)
a7 = A 9 K Z K h + A 9 K z K h C 2 + A z A 9 K 2 + A 8 K Z K ~ - A 4 A 6 K 2 ,
(24)
a8 = A 9 K 2 K h
(25)
A1 = M~ + m21-ff~(Zo) + ~b~(Zo)-l,
(26)
A2 = Ch - p U B K H * C l l
(27)
-
-
and ,
A 3 = -- p U B 2 K H * C 1 2 ,
(28)
A4 = - p U 2 B K 2 A * C 1 2 ,
(29)
A5 = m2q~l(zo),
(30)
A6 = -- p U B Z K A * C I 2
,
(31)
A7 = ls + 2m2LZtp2(Zo),
(32)
A8 = C~, -- p U B 3 K A ~ C z 2 ,
(33)
Z 9 = K~, -- p U 2 B 2 K 2 A ' ~ C 2 2 ,
(34)
A t o = -- m2LZt qll(zo) •
(35)
116
M. Gu et al.:.l. Wind Eng. Ind. Aerodvn. 73 (1998) I l l
123
The Routh Hurwitz stability criterion is used to judge the stability of the bridge T M D system, and the critical flutter wind speed can be found according to Eq. (16). The numerical analysis results and the comparison between them and the experimental ones can be seen in Fig. 5.
3. Experimental study The sectional model test of the Tiger Gate Bridge with and without T M D s was carried out in the T J-1 Boundary Layer Wind Tunnel in the T o n ~ i University. The test aimed at verifying the above theoretical derivations and numerical analysis as well as studying the effects of using different structural and T M D parameters on the critical flutter wind speed.
3.1. Sectional model o[the lTger Gate Bridge The Tiger Gate Bridge is located in Southern China across the Pearl River. It is a suspension bridge with a steel box deck and a center span of 888 m. Fig. 2 shows the general view of the deck cross section. The TJ-1 BLWT has a working section of 1.2 m ( W ) × 1.8 m(ft) × 18 re(L), and the wind speed can be varied between 1 and 32 m/'s. Accordingly', the geometrical scale and wind speed scale were determined to be 7o and ~,, respectively, which correspond to a frequency ratio of 11.67. Accordingly, the torsional and vertical bending frequencies, 11,,,~and 1'~.~, of the model were 4.214 and 2.010 Hz, respectively. Mounting wind screens on a bridge deck can usually improve the traffic conditions for vehicles but quite often deteriorates the aerodynamic performance of the bridge, for example, by decreasing the critical flutter wind speed. Using the original model and the model with wind screens (see Fig. 2), the control effectiveness of the T M D s under the effect of different deck cross sections with different degrees of bhmtness was studied. The structural damping of the bridge model was produced by using a pendulumcontainer system. The pendulum consisted of two plates positioned in the vertical and
I
i /
Wind5~ctucl+
~
/
2(~()()it
I
A q
big. 2. General view of the deck cross section of the Tiger Gate Bridge.
!
M. Gu et al./J. Wind Eng. Ind. Aerodyn. 73 (1998) 111-123
117
vibrating with the bridge model
tng plate
w
1 /
/
/
7
Fig. 3. Schematic diagram of the structural damper set-up.
spill
Fig. 4. Schematic diagram of the TMD model.
the horizontal planes, respectively, and a bar rigidly connected these two plates to the bridge model (see Fig. 3). The upper part of the bar was fixed to the sectional model of the bridge. The damping came from the relative motion of the two plates to the liquid in the container when the bar was in motion with the bridge model. Three damping ratios, 0.7%, 2.13% and 4.05%, were selected in the test to investigate the effect of structural damping on the control effectiveness. 3.2. Model of tuned mass dampers Fig. 4 shows the model of a tuned mass damper, which consists of two springs, two mass blocks and a damping device. The damping device was again modeled using
M. Gu et al.i.l. Wind Eng. Ind. Aerodyn. 73 (1998) 111~123
118
Tablc 1 P a r a m e t e r s of 10 g r o u p s of T M I ) No.
I
2
3
4
"~
6
7
8
9
10
/q (°'i,) /rr ~
0.9 0.98 0.03
2.5 0.98 0.03
5.0 0.98 ().03
10 0.98 0.03
0.9 0.98 0.085
2.5 0.98 0.085
5.6 0.98 0.085
10 0.98 0.085
5.6 0.918 0.085
5.6 1.04 0.085
a pendulum-container system where a horizontal plate with multiple holes was attached to a bar which was rigidly connected to the bridge sectional model. When the bridge model vibrated, the upper plate with the bar and the damping plate vibrated harmonically, and the lower plate with the lumped mass blocks and the container moved relatively to the model, respectively. The bridge sectional model weighted 5.5 kg, and each T M D mass m, was 40 g. The mass ratio between the total mass of the TMDs and that of the bridge model was l~m -- 1.45%. The ratio of the mass moment of inertia is defined as lq
.)
= -m, LU I m ,
(36)
where lm is the mass moment of inertia of the bridge model, Different values of#t can be obtained by adjusting the distance L,. Define the frequency ratio I~ as t', =./ii:']i-.
(37)
where ,/{ is the frequency of the TMD, .[j. the flutter frequency of the bridge model and was equal to 3.125 Hz from the wind-tunnel experiment on the model without control. Different frequency ratios and damping ratic, s of the T M D could be obtained by adjusting the springs and the damping of the model. Table 1 lists 10 sets of T M D parameters used in the present experiment.
4. The main results The critical flutter wind speeds of the original model (without the wind screens) were experimentally determined to the 16, 17, and 18 m/s for structural damping ratios of 0.7%, 2.13% and 4.05%, respectively. The corresponding wind speeds from the theoretical calculation are 16.5, 17.1 and 18.3m/s, respectively. The experimental critical wind speed of the model with the wind screens for a structural damping ratio of 0.7% was 12m/s, as compared to the calculated result of 12.7 m/s. It is noted that to improve the accuracy of the experimental results, several tests were conducted for the same modeling condition. These results were then averaged to give the critical flutter mean wind speed.
M. Gu et al./J. Wind Eng. Ind. Aerodyn. 73 (1998) 111-123
119
20 -~ 18 16, .=_ 14 numerical .=~ 12 .,...
•
experiment
~tf--0.98, ~t=0.085 t
10 2
4
6
I
I
J
8
I0
20
16 "O
.~ 14 "= 12 r...) .:-.
].tf=0.98, ~t=0.03
10 0
I
2 4 6 8 Ratio of mass moment of inertia ].tI (%)
10
Fig. 5. Comparisons of the experimental and numerical critical wind speed ((s = 0.7%, model without wind screens).
4.1. Comparison of the experimental and theoretical results Let the control efficiency be defined as
where U and Ut are the critical flutter wind speeds without and with control, respectively. Fig. 5 shows the main comparisons between the results obtained from the experiment and from the numerical analysis when the structural damping ratio was equal to 0.7%. It is seen that the calculated results agree well with the experimental results, with the maximum error of 7.6% among the 8 groups of results.
4.2. Effects of the mass moment of inertia Fig. 5 also presents the effects on the flutter wind speed of the ratios of mass moment of inertia. The experimental critical wind speed of the sectional model without T M D was 16 m/s for a structural damping ratio (s = 0.7% as mentioned
M. Gu et al./J. Wind Eng. Ind. Aerodyn. 73 (1998) 111-123
120
20
19
18
.r-
17
16 0.90
0.95
100
1.05
1. lO
Frequency ratio ~tf (%) Fig. 6. The critical wind speed as a function of the frequency ratio/~f between the T M D and the bridge (~ = 0.7%, ~i = 5.6%. (, = 8.5%, model without wind screens}.
above. When the T M D damping ratio ~'t was equal to 8.5%, the critical wind speeds from the experiment were t 7.0, 17.5, 19.5 and 18.5 m/s, and the corresponding control efficiencies were 6.3%, 9.4%, 21.9% and 15.6%, for the ratios of mass moment of inertia of 0.9%, 2.5%, 5.6% and t0.0%, respectively. When the T M D damping ratio was equal to 3%, the experimental flutter wind speeds were 16.5, 18.0, 18.5 and 18.5 m/s, and the corresponding control efficiencies were 3%, 12.5%, 15.6% and 15.6% for the ratios of mass inertial moment of 0.9%, 2.5%, 5.6% and 10%, respectively. It seems that the control efficiency increases as the ratio of the mass moment of inertia increases from 0.9% to 5.6%. The results also show that the control efficiency tends to saturate or even decrease when lq goes up to about 10%.
4.3. Effects ~?[the /?equency ratio Fig. 6 shows the experimental critical wind speeds versus frequency ratio Pt for the bridge model without wind screens for ~q = 5.6%, ff~(the structural ratio) of 0.7% and ~t = 8.5%. As seen from the figure, the critical wind speed was 19.5 m/s, with a corresponding control efficiency of 22% for lLf 0.98. The critical wind speeds became 17.4 and 18 m/s which corresponded to control efficiencies of 8.8% and 12.5% for j~f--0.92 and 1.04, respectively. There is a rather significant effect on the flutter control efficiency as the frequency ratio varies. The exact optimal frequency ratio between the T M D and the bridge would depend on the cross section shape of the bridge deck. =
M. Gu et al./J. Wind Eng. Ind. Aerodyn. 73 (1998) 111-123 20
121
40
- 3o~.
-~
nd speed ~. 16 -~ .7 "~ -=
no control i. ............... ....• . . . . . . . . . .....
14
•
gi=5.6%
• •
gi=2.5% gi=0.9%
~20 ~
control efficiency
"6 r.)
' " ' - . , . ,.
•
.....
................ I)... . . . .
"'l "'"'• ....... " - . . _ .....
............ o ...............
10
10
*"..
12
-.11121
I
i
I
L
1
2
3
4
0 5
Structural damping ratio (%) Fig. 7. T h e effects of structural d a m p i n g ratio on the critical wind speed a n d the control efficiency of the T M D s (/if = 0.98, ~t = 8.5%, m o d e l without wind screen).
4.4. Effects of the TMD damping ratio Fig. 5 also shows the experimental critical wind speeds of the bridge model without wind screens versus two kinds of TMD damping ratios. The TMD damping ratio was designed to be 3% and 8.5°,/0. It can be seen from the figure that when the damping ratio varies from 3% to 8%, the variation of the critical wind speed is no more than 1 m/s for all the five values of #v Theoretical analysis gives slightly larger derivations. The TMD damping ratio seems to have less effect on the flutter control than the frequency ratio.
4.5. Effects of the structural damping ratio on the control efficiency Fig. 7 presents the effects of the structural damping ratio on the flutter control efficiency by using No. 5, No. 6 and No. 7 TMD designs listed in Table 1. For the bridge model without the TMD, the critical wind speeds were 16, 17 and 18 m/s for the structural damping ratios of 0.7%, 2.13% and 4.05%, respectively. When the No. 6 TMD (/h = 2.5%) was used, the critical wind speeds were 17.5, 18.2 and 18.5 m/s, which correspond to control efficiencies of 9.4%, 7.1% and 2.8%, for structural damping ratios of 0.7%, 2.13% and 4.05%, respectively. The critical wind speeds and the corresponding control efficiencies for the No. 5 TMD (#i = 0.9%) and the No. 7 TMD (#i = 5.6%) are also shown in this figure. These results show that the TMD is more effective for a bridge with a lower structural damping than for those with higher structural damping.
122
MI Gu et al./J. Wind Eng. Ind. Aerodyn. 73 (1998) 111-123 Table 2 Effect of the bridge deck cross-section shape Mode deck shape
Without w.s. With w.s.
No. 7 T M | )
No. 8 T M D
U, (m/s)
~l~-I%)
U, (m/s)
q~ (%)
19.5 17.5
21.9 45.8
18.5 18.3
15.6 52.5
w.s.: wind screen
4.6. Effbcts of the bridge deck cross-section shape on the control e~ciency To investigate the effects of bridge deck cross-section shape on the flutter control efficiency, experiments on the bridge model with and without wind screens were also conducted. Table 2 shows some of the experimental results. When the structural damping ratio was equal to 0.7%, the experimental critical wind speeds were 16 and 12 m/s for the model without and with the wind screens, respectively. As shown in Table 2, installing No. 8 T M D would increase the experimental critical wind speed form 16 to 18.5 m/s with a control efficiency of about 15.6% for the model without the wind screens; the same T M D can increase the critical wind speed from 12 to 18.3 m/s with a control efficiency of 52.5 % for the model with the wind screens. The effect of the No. 7 T M D on the control efficiency is similar to that obtained with No. 8 T M D . Similar trends can be seen for the other groups of T M D . These results indicate that the T M D is more effective for controlling the flutter of a bluffdeck than a streamlined deck.
5. Concluding remarks in this study, the effects of T M D s on the critical flutter wind speed of a bridge have been theoretically and experimentally analyzed. The theoretical approach was established by applying the Rough-Hurwitz stability criterion to the characteristic equation of the bridge and T M D combined system. The experimental studies were based on sectional model tests of the Tiger Gate Bridge performed in the T J-1 Boundary Layer Wind Tunnel in the Tongji University. Based on the results, the following conclusions can be found. 1. The T M D can increase the critical flutter wind speed significantly. The T M D with more than 5.6% ratio of mass inertia moment can increase the critical flutter wind speed of the Tiger-Gate Bridge sectional model with wind screens by more than 4O%. 2. Within a certain range, in general, the larger the T M D mass moment of inertia, the better the control performance. The numerical and experimental results show that for the Tiger-Gate Bridge the control efficiency tends to saturate or even decrease when the ratio of the mass moment of inertia goes up to about 10%.
M. Gu et al./J. Wind Eng. Ind. Aerodyn. 73 (1998) 111-123
123
3. T h e T M D is m o r e effective for c o n t r o l l i n g the flutter of a b r i d g e with low s t r u c t u r a l d a m p i n g t h a n with high s t r u c t u r a l d a m p i n g , t h a t is, it is m o r e efficient to use the T M D on steel bridges t h a n on c o m p o s i t e bridges, o r concrete bridges. 4. T h e T M D is m o r e effective for c o n t r o l l i n g the flutter of a b r i d g e with a bluff deck cross section t h a n with a s t r e a m l i n e d deck cross section. 5. T h e frequency r a t i o between the T M D a n d the b r i d g e as well as the d a m p i n g ratio of the T M D b o t h influence the c o n t r o l effectiveness. H o w e v e r , the effect of the frequency r a t i o is m o r e p r o n o u n c e d t h a n t h a t of the d a m p i n g ratio.
Acknowledgements T h e project is c o - s u p p o r t e d by the N a t i o n a l N a t u r e Science F o u n d a t i o n a n d N a t i o n a l Science F o u n d a t i o n for the O u t s t a n d i n g Youth, which are gratefully acknowledged.
References [1] R.L. Wardlaw, The improvement of aerodynamics performance, in: Allan Larsen (Ed), Proc. 1st Int. Symp. on Aerodynamics of Large Bridges, A.A. Balkema Publishers, Denmark, 1992, pp. 59-70. [2] M. Gu, Aerodynamic and mechanical countermeasures for suppressing wind induced vibrations of long-span bridges, in: X.Q. Na, Z.Y. Shen (Eds.), The Present Situation and Tendency of Structural Engineering, Tongji University Press, 1994, pp. 275 282 (in Chinese). [3] M. Gu, H.F. Xiang, A.R. Chen, A practical method of passive TMD for suppressing wind-induced vertical vibration of long-span cable-stayed bridges and its application, J. Wind Eng. Ind. Aerodyn. 51 (1994) 203-213. [4] A.R. Chen, H.F. Xiang, M. Gu, Vortex-excited vibration control of bridges using TMD, in: K.M. Lain, Y.K. Cheung (Eds), Proc. of APSOWE-3, The Central Printing Press Ltd., Hong Kong, Vol. 1 (1993) pp. 235-240. [5] V. Michel, Wind design and analysis for the Normandy Bridge, in: Proc. 1st International Symposium on Aerodynamics of Large Bridges, A.A. Balkema Publishers, Denmark, 1992, pp. 183-216. [6] J. Nobuto, Y. Fujino, M. Ito, A study on the effectiveness of T.M.D. to suppress a coupled flutter of bridge deck, Proc. Japan Society of Civil Engineering, No. 398/1-10, 1988, pp. 413416. [7] R.H. Scanlan, J.J. Tomko, Airfoil and bridge deck flutter derivation, J. Eng. Mech. Div. ASCE 97 (1971) 1712-1737.