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Effectiveness of tuned liquid column dampers for vibration control of towers
~
TTERWORTH I N E M A
Engineering Structures, Vol. 17, No. 9, pp. 66&-675, 1995 N
N
0141-0296(95)00036-4
Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0141 0296195 $10.(X) + 0.00
Effectiveness of tuned liquid column dampers for vibration control of towers T. Balendra, C. M. Wang and H. F. Cheong Centre for Wind-Resistant Structures, Department of Civil Engineering, National University of Singapore, Kent Ridge 0511, Singapore
The effectiveness of tuned liquid column dampers (TLCD) in controlling the wind-induced vibration of towers is studied. The nonlinear governing equation of the TLCD is linearized to obtain the stochastic response of the towers due to along-wind turbulence. Through parametric studies, the optimum parameters for maximum reduction in acceleration and displacement are presented for a wide range of towers. It is found that for any tower of practical interest, almost the same amount of reduction in acceleration can be obtained by choosing an appropriate opening ratio of the orifice in the TLCD. The same opening ratio would give almost the maximum reduction in displacement. Generally, the opening ratio needs to be varied between 0.5 and 1.0, with smaller opening ratios for shorter towers.
Keywords: tuned liquid column dampers, wind-induced vibration, towe rs 1.
sloshing frequency is tuned to the frequency of the structure. Recently Sakai et al. 5 proposed the tuned liquid column damper (TLCD), which suppresses the wind-induced motion by dissipating the energy through the motion of the liquid mass in a tube-like container fitted with orifices. The feasibility of TLCD for the Citicorp Centre in New York 2 and the Gold Tower in Japan 6 has been demonstrated by Sakai et al. 5. The purpose of this present investigation is two-fold. Firstly, to experimentally determine the characteristics of a TLCD. And secondly, to review the current development on the application of TLCDs and to assess the effectiveness of TLCDs for towers with different fundamental frequencies through random vibration analysis. A parametric study is carried out to determine the optimum parameters of the TLCD for maximum suppression of wind-induced acceleration and deformation of towers of practical interest. Since the governing equation of motion of the TLCD is nonlinear, the equivalent linearization technique is utilized to linearize the equation 7. Using the Harris spectrum 8 to model the along-wind turbulence, the stochastic response of the towers, modelled as a single-degree-offreedom system, is determined in the presence of the TLCD. The effectiveness of the TLCD for motion perceptions is quantified by the reduction in the RMS acceleration due to the TLCD. For structural deformation, the reduction in displacement is used to quantify the effectiveness of the TLCD.
Introduction
Recent advances in the development of lightweight, high strength material and construction techniques have led to more flexible and lightly damped structures. In such structures, the deformation and motion perception due to windinduced vibration become more prevalent and could lead to failure of the serviceability limit state. Thus in order to meet the serviceability requirements, the wind-induced vibration is suppressed either by active or passive control devices. In an active control device, an external force is applied to control the structural vibration, thereby keeping the structure safe and serviceable under extreme loading conditions. The amount of external force depends on the feedback from the structural response. Owing to the uncertainty of this infrequently used control system and the amount of power required to induce the control forces, in practice, the passive control system tends to be favoured over the active control system. Among passive control devices, the tuned mass dampers t.2 and tuned liquid dampers 3.4 are commonly used. The tuned mass damper consists of a large mass attached to the structure through a spring and dashpot. It dissipates energy when it is tuned to the frequency of the structure. The tuned liquid damper consists of a container which is partially filled with liquid. It dissipates energy through the sloshing action of liquid in the container. For optimum results, the
668
Tuned liquid c o l u m n d a m p e r s for vibration c o n t r o l o f towers: T. Balendra et al.
2.
Mathematical model
A TLCD with an orifice is shown in Figure 1. If x is the displacement of the liquid due to the horizontal motion y of the tube, then the equation of motion of the liquid column is given by Sakai et al. 5 as follows pAL x +
1
pA
+ 2pAgx = - pABy
( 1)
where 6 is the coefficient of head loss governed by the opening ratio of the orifice, L is the length of the liquid measured along the centreline of the tube, A is the crosssectional area of the tube, B is the width of the tube, p is the density of the liquid, and g is the gravitational acceleration. Equation ( 1 ) is nonlinear due to the presence of the liquid damping. However, the equation can be linearized using the equivalent linearization technique 7. Thus, consider an auxiliary system which is linear
where m (= pAL) is the total mass of the liquid, c is the equivalent damping coefficient and the natural frequency COd,or the natural period To, of the liquid column is given by or
where o~ is the standard deviation of the liquid velocity. In order to check the accuracy of the foregoing linearization, the Runge-Kutta-NystrOm method was used to integrate the second-order differential equation (1) with the initial conditions x ( 0 ) = dx(0)/dt = 0 and assuming a harmonic excitation y = y,, sin COdt where y, is the amplitude and COj the excitation frequency. The RMS velocity of the liquid determined is used in equation (7) to obtain the equivalent damping coefficient, c. This value of c is then used in equation (2) to determine the RMS displacement of the liquid for the same harmonic excitation. The RMS displacement of the liquid is compared with that obtained from equation (1), for different amplitudes of excitation, namely, Yo = 0.05 m/s 2, 0.2 m/s 2 and 0.4 m/s 2. The percentage difference is given in Table 1 for different head losses, 6. It is seen that the error increases with increasing head loss but the maximum error is about 10% for 6 -< 10. It should be noted that the values of 6 and the amplitudes of excitation considered would adequately cover the practical range.
(2)
mY + (B/L) mS~ + c.f + COdm x = 0
cou = \ 2 g / L
Tu = 27r\,L/2g
(3)
If the difference between equations (1) and (2) is denoted as e, then e = (mS/2L) [~12- c2
(411
The equivalent damping coefficient can be determined by minimizing the square of the error. Thus, we set E[e 2] = minimum
3.
In order to study the influence of the orifice on the motion of the liquid and to quantify the effect of the orifice opening on the damping coefficient, an experimental study was carried out. The TLCD shown in Figure 2 was constructed using perspex material. The size of the orifice was varied to have the following opening ratios: l, 0.75, 0.50, 0.25 and 0.10. Figure 2 shows an opening ratio of 0.25. The TLCD was then filled with water and placed on a shake table. The shake table was connected to a dynamic actuator which was controlled by a function generator. The displacement of the liquid was measured by a wave probe monitor, while the acceleration of the shake table was measured by an accelerometer. 3. I.
d d~-~ (El e2l) = 0
(6)
Substituting equation (4) into equation (6) yields 9
~ - ~r;
(7)
OrificeB
Y
./
Figure 1 Tuned liquid column damper (TLCD)
Experimental procedure
The TLCD was filled with water to a height of 270 mm, measured from the base of the tube. The natural frequency, wu of the liquid column calculated by equation (3) was found to be 0.71 Hz. A sinusoidal base motion was applied to the table. The excitation frequency was varied from 0.5 Hz to 0.9 Hz, at intervals of 0.02 Hz. Four amplitudes were chosen for the base motion: 6 mm, 8 mm, 9 mm and 10 mm. The experimental procedure adopted was as follows ( l ) Orifice opening ratio of 1.0 is selected first (2) The amplitude of base motion is set at 6 mm (3) For each frequency of excitation, the liquid displacement is measured (4) Step (3) is repeated for other amplitudes of base motion (5) Steps (2)-(4) are repeated for the other opening ratios: 0.75, 0.50, 0.25 and 0.10 3.2.
./_
Experimental investigation
(5)
where E[.] denotes the statistical average of mathematical expectation of a stationary Gaussian random process. The necessary and sufficient condition for equation (4) to be satisfied would be
c=
669
Head loss coefficient
For a particular value of orifice opening ratio and the amplitude of excitation, the relationship between the liquid displacement and the excitation frequency is investigated. Figure 3 shows the variations of liquid displacement amplitude with respect to the excitation frequency, for an orifice opening ratio of 0.50, and for a base excitation of amplitude
670
Tuned liquid column dampers for vibration control of towers: T. Balendra et al.
Table 1 Normalized RMS displacement of liquid from equations (1) and (2) for various excitation amplitudes, Yo (m/s 2) c~= 0.7
c~= 0.9
Yo
6
~x~
~,a
% diff.
0.05
2 4 6 8 10
0.971 0.962 0.954 0.946 0.938
0.968 0.956 0.945 0.935 0.924
0.32 0.63 0.92 1.21 1.48
1.241 1.223 1.207 1.191 1.176
1.234 1.211 1.189 1.168 1.148
0.52 1.01 1.47 1.91 2.38
0.2
2 4 6 8 10
4.691 4.513 4.355 4.215 4.089
4.622 4.389 4.189 4.015 3.861
1.48 2.73 3.80 4.74 5.58
5.878 5.542 5.264 5.029 4.827
5.742 5.315 4.972 4.690 4.452
2.32 4.10 5.54 6.75 7.78
0.4
2 4 6 8 10
9.025 8.430 7.951 7.557 7.223
8.779 8.030 7.449 6.981 6.593
2.73 4.74 6.33 7.62 8.72
11.08 10.06 9.302 8.715 8.242
10.63 9.380 8.495 7.827 7.300
4.10 6.75 8.68 10.2 11.4
(rxl
a~
% diff.
gradient of the plot of the damping ratio versus the resonant liquid displacement, the head loss coefficients given in Table 3 are obtained. A plot of the head loss coefficient against the orifice opening ratio is shown in Figure 4. The general trend is that, as the orifice opening ratio decreases, the head loss coefficient increases. The increase in the head loss coefficient is substantial when the opening ratio decreases below 0.50.
L_ Figure2 Dimension 25% opening 5.0Q
zzo of tuned liquid column damper with
Orifice opening ratio 0.5
t,.O0
E 300 J
."
2.00 ],O0
4.
o;,o'5
o'.
o'B Qk
o92
Frequency(Hz) Figure 3 Variation of liquid displacement amplitude with frequency of sinusoidal base excitation amplitude of 6 mm
6 mm. The liquid displacement amplitude first increases with increasing frequency. It reaches a peak value at around 0.71 Hz which is the resonant frequency. It then gradually decreases with increasing frequency. The damping ratios ~= c/(2mwa) can be determined from the plot given in Figure 3 using the half-power bandwidth method. The damping ratios for the other opening ratios are similarly calculated and are shown in Table 2. The head loss coefficient ~5, is then calculated from
6 = 2L ~
3.3. Effect of orifice opening on the liquid displacement Due to the size constraint on the TLCD, there is an allowable limit for liquid displacement. The water might spill out of the TLCD, if the allowable limit is exceeded and hence it would fail to perform as an effective damper. Figure 5 shows the relationship between the liquid displacement at resonance and the orifice opening ratio for various amplitudes of excitations. It can be seen that as the orifice opening increases, the liquid displacement at resonance increases. The increase is significant when the opening ratio exceeds 0.75.
(~/Xr)
(8)
where xr is the resonant liquid displacement. Thus from the
Application of TLCD to buildings
The effectiveness of the TLCD in independently controlling the along-wind and across-wind motions of tall structures has been investigated by Xu et al. ~o.1~. A random vibration analysis utilizing the transfer matrix formulation was carried out to obtain the response statistics, when the alongwind turbulence and across-wind wake excitation were modelled as a stochastic process which was stationary in time and nonhomogeneous in space. A comparative study with the traditional tuned mass dampers '2 (TMD) using a 76-storey building which was 306 m in height, and a 370 m tall TV tower, revealed that the TLCD could produce a similar level of response reduction as the TMD provided that the parameters of the TLCD are properly selected. Samali and Kwok ~3 carried out a sensitivity analysis to optimize the parameters of the TLCD for maximum reduction in response for the 76-storey building. A displacement reduction of 45% was achieved with a damper mass equivalent to 0.7% of the building mass. To control both along-wind and across-wind motions simultaneously,
671
Tuned liquid column dampers for vibration control of towers: T. Balendra et al. Table 2 Damping ratios versus resonant liquid displacement for different opening ratios Opening ratio Forcing function amplitude (mm)
1.00
0.75
xr (cm)
~
xr (cm)
~
xr (cm)
~
xr (cm)
(
xr (cm)
6 8 9 10
8.18 9.65 9.99 10.64
0.052 0.060 0.061 0.066
3.96 4.52 4.80 4.90
0.061 0.073 0.077 0.083
3.49 3.96 4.10 4.32
0.080 0.087 0.093 0.099
1.73 1.91 2.00 2.10
0.130 0.172 0.195 0.197
1.22 1.34 1.42 1.51
Table3 Relationship between orifice opening and head loss coefficient Opening ratio
Head loss coefficient, 8
1.00 0.75 0.50 0.25 0.10
1.86 6.87 7.80 62.00 70.88
0.50
0.25
0.10
0.219 0.237 0.266 0.271
investigate the variation of the optimum parameters of the TLCD with respect to the structural properties, for maximum reduction in response. In order to overcome this deficiency in the current state-of-the-art of TLCDs, towers with different fundamental frequencies are considered in the next section.
5.
Analysis of towers with TLCD
A tower fitted with a TLCD is shown in Figure 6. In order to gain insight into the TLCD-structure interaction, the tower was modelled as a single-degree-of-freedom system. The governing equations of motion, in view of equation (2), become
80 cO
60
<,,,+[: o]{,}+[/ o]{x} ={o}
moe M JLy]
C
5;
K
y
(9)
F(t) 20
t,O
60
80
100
Opening rotio (%) Figure 4 Variation of head loss coefficient with orifice opening ratio
[ g
,0 8
G -,= (, 3
2 0
where M is the total mass of structure and liquid, K and C are the stiffness and damping coefficient of the structure, respectively, F(t) is the force acting on the structure due to wind, and t~ = B/L. Adopting the two-sided Harris spectrum for wind, the power spectral density of wind force is given by
J
-
Orifice opening rotio(%] ] ----tOrero -o-gram -,,-Smm --.-firnmI
U~
~
)
.
~--TLCD F(t)
Figure5 Variation of liquid displacement at resonance with
.~
-!
different amplitude of base excitation
D Turbulent
Zhang and Zhang ~4 proposed a TLCD in the form of crossed dampers. The along-wind turbulence was considered to be random while the across-wind loading was considered to be periodic due to vortex shedding. It was found that when a crossed damper is installed at the top of a multidegree-of-freedom system, the across-wind resonance response was reduced by 50%. All these investigations m-~4 pertained to a particular structure. No attempt was made to
(lOa)
(2 + X2) 5'6
component
'-
._j -
i
Meon component
711
f
f.//,,"
Figure 6 Tower fitted with a TLCD and subjected to turbulent wind
Tuned liquid column dampers for vibration control of towers: 7-. Balendra et al.
672 where XFo
wl~ 27rUle
m Xl = (~'~2 -- /32)( 1 _ /32) _ /32(1&0/2/32 + ~ C)
(20)
X2 =/3[C( f12 -/32) + ~(1 -/32)]
(21)
(lOb)
= p,~toCo~
(10c)
in which ko (= 0.03) is the ground surface drag coefficient, lx is the wavelength taken to be equal to 1200 m, Do is the density of air, Co is the drag coefficient, Ao is the frontal area of the structure, and Ulo and Uz are the mean wind velocity at 10 m and height z, respectively. In order to determine the response using the spectral method, let F(t) = Foei~°t
The power spectra of displacement and velocity can be expressed as S,,y = IH.,,I2 SFF
(22)
= IH2 S.-
(23)
S;x =/32 &~
(24)
where SFF is the nondimensional power spectral density of the wind force which may be written as
( 11 ) (I)
(25)
#q,/3
(26)
SFF(/3) "~- - -
where COis the excitation frequency and i = q--i. As equation (9) is linear, the solution can be obtained by setting x( t ) = 2 e i'°'
(12a)
y(t) = 33e i'°t
(12b)
where qb =
[2 + (rr3qj/3)2]s/6
Here Ur = U o/U~ and ~0= wplJ(2&Um). The mean square responses are determined from
Thus, we have ((CO~- w2) + i ~ ) 2 - ace2 33= 0
(13)
- mCOea2 + ((@ - CO2)M+ iCCO)33= Fo
(14)
-2~G--
where COp= K ~ is the natural frequency of the structure. For generality and convenience, the following nondimensional parameters are introduced m
2&
33&
~ = ~ ' ~=-k-' Y=-B' 8
60 COd 3 m"~. . -, COp a = -% -'
=
S.x d/3
(27)
o'~ = \ Bcop/ = ~
_~ ~2S~ d/3
(28)
-2O-y= ( V )
~ Syy d/3
(29)
2 = If
--2 = tO'YT#t2 0".~
c
\ no)2p /
= ~
1 f~
_~ [~4Svy d/3
(301
~=Z' c-- m.,~' c _ "c- mw.
~a-~;,
_ Fo#
-~;-
As the power spectra S~ and Syy are functions of ~;, equation (28) has to be evaluated first using an iterative procedure. The false position method was used for solving the transcendental equation (28) for ~;.
o~,,B
_ Fo~W
CO2pBM 2gaM
(15) 6.
Using these parameters, and in view of equations (12), equations (13) and (14) may be written, respectively, as [(fl 2 - / 3 2 ) + ~/3i] 2 - a/32 y = 0
(16)
-/xc~/32 2 + [(1 -/32) + ~/3i] y =
(17)
Solving the simultaneous equations (16) and (17) and squaring the solutions, yields the following transfer functions
X~, +)t~2 IHyl 2 =
where
~
2
(~-~2 -- /32)2 + ~'2/32 = ~ + ~
(18)
Effectiveness of TLCD
The effectiveness of the TLCD may be quantified by the ratio between the RMS acceleration and RMS displacement of the tower with and without the TLCD. The variances of acceleration and displacement of the tower without TLCD are given by - 2 2k,Ya o'y = - _ qb~i C O'y = ~
(31a)
-- /32)2 + j[~2C2 ~ 0/3
(31b) (19)
If the ratios between the RMS acceleration and RMS displacement with and without TLCD are denoted as ~ r and ~vr, respectively, then it can be shown that
673
Tuned liquid column dampers for vibration control of towers: 7-. Balendra et al. .
0.8(
d ~ } 1/2
Ioi
' ,o9o
ct=O.9
. . . . . Q =0.95 . . . . . . ~ =1.~ ~ =1.o5
\
~,,r =
" fi'( •
?
'
i l-/3:)~+/3:C:
}
0.71
~ \ ~ . .~
060
". '
....
f~ = 1.10
(32b)
*dis
~
. . . .
",~ .-.,.
.~---- ~-2--"2. . . .
Thus ~,. is a function of the.following parameters
,
0.50 ~fr =f(qA U r , P,, ~ , OL, ~!, (% F )
(33)
,
....
0.=0.6 or=07
.....
The fundamental period of the towers with a heavy mass at the top such as the Wuhan Television Tower of 115 m heighfl 5 or the central TV transmission tower of 256 m height ~6 can be estimated approximately by the following expression
T~= 2wlw n = z130
~= o02 ~:10
(b)
ct:O5
/
~=0.8
,_
0.70
/
.......~t=~...~....
~-~
f
(34a)
where Tn is the period in seconds and z the height of the mass above the base in meters. The period of lattice towers can be estimated by the following expression ~7 which has been validated from field measurements ~8
0.50
0.70
.
.
.
.
- I.,t= 0.01 . . . . . P. = 0.02 ......~: 0.03
---
.
(C)
~= o.o~
. . . . . ~: o.os
ct =09 ~ = 1,0
--j_./~/
/
TR = h21(750(wh + w,))
(34b)
where wh and w, are the width of the tower at the base and top of the tower, respectively, and h is the height of the tower. If the mass density of the tower is taken to be Pb = 2 k N / m 3, Po -- 1.2 kg/m 3, Co = 1.2, U~o = 20 m/s and the mean wind velocity profile is assumed to be of the form
060 0.50
__
"~.~ ...............
0.1.0
.
.
.
.
5
10
15
29
6
U:/U o = (z/10) °3
(35)
Figure 7 Influence o f head loss on r e d u c t i o n in a c c e l e r a t i o n f o r t o w e r w i t h q~= 0.5
then in view of equation (34a), it can be shown that
t~ = 58/z
(36a)
= (10/z) °'6 = (10~/58) °'6
(36b)
-
(36c)
D
where D is the diameter of the tower. A similar set of expressions can be obtained if equation (34b) is used instead of equation (34a). In view of equation (36), equation (33) reduces to
chosen, the performance of the TLCD can be controlled through 6, namely, by adjusting the orifice opening. The value of head loss coefficient which gives the maximum reduction in acceleration, denoted as 8,,~,, is shown in Figure 8 for values of + ranging from 0.2 to 1.0, which represent towers of practical interest. The corre10.0
8.0
--ct=0.5 - - - . a - " 0.6 ---a-'0.7 - - - - a - " 0.8
/, ,,,',• , pg,-',~/ -
~ - e t = 0.9
//s ° S °
~$:r f( ~b, /~, ~ , a, C, 6,) =
1
(37) ~:~ 6.0
The influence of 1), a, and/.t on O'er is shown in Figure 7 for various values of head loss coefficient, when D = 30 m, = 0.5 and C = 0.02. It is seen from Figure 7a, that for any given head loss, higher reduction in acceleration is obtained when ~ = 1.0 and for maximum reduction, there is an optimum value for 6. When the TLCD is not fully tuned (say l) = 0.95 or f~ = 1.05), by changing the orifice opening to an appropriate value, one can improve the performance of the TLCD. Figures 7b and 7c indicate that for better performance of the TLCD, one should choose higher values of a and /x, It can be seen that once a and /z are
,¢,~," 7"
//oS /
<;~/ / "
j
.,/7/
2.0 01
"
' 0.3
•
, 0.5
,
, 0.7
,
, 0.9
, 1.1
Figure 8 O p t i m a l head loss f o r m a x i m u m r e d u c t i o n in accelera t i o n f o r ~ = 1.0 and /~= 2%
Tuned liquid column dampers for vibration control of towers: T. Balendra et al.
674 0.70
0.65
----..... ---.....
0.01
(z= 0.5 a=0.6 0.= 0.7 0.=0.8 a= 0.9
0.06
--0.=0.5 ---ct=0.6 ..... o.=0.7
',
t~.
--.=....
--
o.:0.0 9
0.05
~ ° ~ . I
,~.b 0.60
,1~ _....~ O.Ot,
0.03
0.55
OO2o1 °%'.1 ' 013
o15
017
oJs
o'.7
0g
t.1
oJo ' H
qJ Figure 11 Velocity of liquid corresponding to o p t i m u m head Figure 9 Reduction in a c c e l e r a t i o n c o r r e s p o n d i n g to o p t i m u m
loss for D. = 1.0 and /~ = 2%
head loss for ~. = 1.0 and /~ = 2%
sponding reduction in acceleration is depicted in Figure 9. It is interesting to note that practically the same amount of reduction in acceleration can be obtained for any tower of practical interest by choosing the appropriate head loss coefficient. This optimum head loss coefficient increases with ¢ which means that a smaller opening ratio is required for shorter towers. In view of Table 3 and Figure 8, the TLCD could be used to control the wind-induced acceleration of towers with different fundamental frequencies, by varying the orifice opening ratio between 1.0 (fully opened) and 0.5. The liquid displacements and velocities corresponding to the optimum head loss coefficients are shown in Figures 10 and ll, respectively. The RMS displacement of the liquid expressed as the ratio of the length of the liquid (~rx/L = a-~x/Tr4) is found to vary between 2% and 6%, with smaller displacement for shorter towers. For sharp edge orifices, the separating shear layers are known to be independent of the Reynolds number, which may be computed using the approaching flow configuration as Re = Rtr~/u
(38)
where R is the hydraulic radius defined as the ratio of the wetted area to the wetted perimeter, u is the kinematic viscosity of water, and o-;,is the RMS velocity of the approaching flow which is given by
0.07 0.06
'
. . . . . .
'0.=o15 ---0.=0.6 . . . . c~=0.7
'\ ~
---0.=0.8
Using equation (34a) and Figure 11, the Reynolds number computed for various configurations of the liquid damper for a tower of height 300 m is given in Table 4. It can be seen that the Reynolds number corresponds to turbulent flow. Since the head loss coefficients are independent of the Reynolds number for a turbulent flow, the designer has the option of adopting the most suitable dimension for the TLCD As in the case of acceleration, the reduction in displacement due to the presence of the TLCD, computed according to equation (32b), is found to be largest when the frequency of the damper is tuned to that of the tower. Furthermore, a higher reduction is obtained for larger values of a and /x. The maximum reduction in displacement achievable is shown in Figure 12 for towers with different fundamental frequencies. The corresponding head loss required is shown in Figure 13. It can be seen from Figure 12 that a higher reduction in displacement is possible for taller towers. Comparing Figures 13 and 8, the head loss required for maximum reduction in displacement is approximately the same as that required for maximum reduction in acceleration. It can be seen from Figures 9 and 12 that a TLCD would provide a maximum reduction of 50% in acceleration and displacement when the structural damping is 2%. The percentage of reduction could vary slightly with different structural damping. However, the trend would be similar. Thus, the TLCD provides a convenient method for controlling both the acceleration and the displacement of towers. Because of nonlinear damping of TLCDs, in general the results presented in Figures 9-13 would depend on the amplitude of excitation. However, as the head loss coefficient corresponding to the practical range of the orifice
O.OZ,
Table4 Reynolds numbers for TLCD in 300 m high tower
0.03
Length of liquid, L (m)
Thickness
of tube (m)
Width of tube (m)
Reynolds number
20 10 10 5
1.0 1.0 0.5 1.0
1.27 2.54 5.08 5.08
211 000 135 000 86000 79 000
0.020.1
. . 0.3
. . 0.5. . . 0.7 .
0.9
1.1
Figure 10 Displacement of liquid corresponding to o p t i m u m head loss for ~ = 1.0 and /~ = 2%
Tuned liquid column dampers for vibration control of towers: 7-. Balendra et al.
.....o
0.65
--
~-=0.5
0.60
--..... --.....
ct=0.6 ct=0.7 ct-08 ct=O.
jJ'
0.50
,
.-"
0.55
,f
.~ /"
a higher ratio of width to liquid length and higher mass ratio are better for higher reduction in acceleration and displacement. However, economic considerations will govern the value of the mass ratio. For best performance, the frequency of the TLCD must be tuned to the frequency of the tower. But, even if the TLCD is not perfectly tuned, by controlling the opening ratio, one could easily obtain a satisfactory performance from the TLCD. This feature is advantageous to the designer, as it is often difficult to assess the structural frequency accurately at the design stage. Moreover, the frequency of the structure would not remain the same during the lifetime of the structure.
'
-
/x,.~'
a
......-...-----
...~
.-,-
s"
i'
o.'3
o;s
o19
q~
1.1
Figure 12 R e d u c t i o n in d i s p l a c e m e n t o f t o w e r c o r r e s p o n d i n g t o o p t i m u m h e a d loss f o r m a x i m u m r e d u c t i o n in d i s p l a c e m e n t f o r O. = 1.0 a n d /x = 2% 10.0 c~-'0.5 8.0
--
- - - - ct--0.5
,,,I"
. . . . ct=07 - - - ct=O.8 . . . . . a=O 9
~" -" .I "~-"" -" -'.~"
6.0
t :-~.
../.:~-
om"
~,'.;Y L,.O
1
1"
i"
,,...,~.ti.-
¢:;,"
-
2.0
0,00
Figure 13
L 0.3
,
0.'5
' 0.7
O p t i m u m h e a d loss f o r m a x i m u m p l a c e m e n t of t o w e r f o r f~ = 1.0 a n d /x = 2%
' 0.9
'
1.1
r e d u c t i o n in dis-
opening ratio would not exceed 8, in view of Table 1, the effect of nonlinearity is small, less than 10%. Hence, the results presented would be practically independent of the excitation level.
7.
675
Conclusions
The effectiveness of a TLCD in suppressing wind-induced acceleration has been demonstrated for towers with different fundamental frequencies. It has been found that the amount of reduction in acceleration is almost the same for any tower of practical interest, provided one chooses an appropriate opening ratio for the orifice. The same opening ratio would give almost the maximum possible reduction in displacement. In practice, the opening ratio needs to be between 1.0 and 0.5 for optimal performance. TLCDs with
References 1 Wargon, A. 'Design and construction of Sydney Tower', Struct. Engr 1983, 61A (9) 2 Peterson, N. R. 'Design of large scale tuned mass dampers', in 'Structural control', (Ed. H. H. E. Leipholz) North-Holland, Amsterdam, The Netherlands, 1980, pp. 589-596 3 Modi, V. J. and Welt, F. 'Vibration control using nutation dampers', Proc. Int. Conf. on Flow Induced Vibration, Bowness-on-Winderwere, UK, 1987, pp. 369-387 4 Fujii, K., Tamura, Y., Sato, T. and Wakahara, T. "Wind-induced vibration of tower and practical applications of tuned sloshing damper', J. Wind Engng Ind. Aerodyn. 1990, 33, 263-272 5 Sakai, F., Takaeda, S. and Tamaki, T. 'Tuned liquid column damper - new type device for suppression of building vibrations', Proc. Int. Conf. on Highrise BuiMings, Nanjing, China, 1989, pp. 926-931 6 Noji, T. 'Study on vibration control damper utilizing sloshing of water', J. Wind Engng, No. 37, 1988 7 Iwan, W. D. and Yang, I. 'Application of statistical linearization techniques to nonlinear multidegree of freedom systems', J. Appl. Mech. 1972, 39, 545-550 8 Walshe, D. E. J. 'Wind excited oscillations of structures', National Physical Laboratory, London, 1972 9 Sun, K., Cheong, H. F. and Balendra, T. 'Effect of liquid dampers on along-wind response of structures', Proc. Third Asia-Pacific Syrup. on Wind Engineering, Hong Kong, 1993, pp. 835-840 10 Xu, Y. L., Samali, B. and Kwok, K. C. S. 'Control of along-wind response of structures by mass and liquid dampers', J. Engng Mech., ASCE 1992, 118 (2), 20-39 11 Xu, Y. L., Kwok, K. C. S. and Samali, B. 'The effect of tuned mass dampers and liquid dampers on cross-wind response of tall/slender structures'. J. Wind Engng Ind. Aerodyn. 1992, 40, 33-54 12 Weisner, K. B. 'Tuned mass dampers to reduce building wind motion', ASCE Convention and Exposition, Preprint 3510, 1979, ASCE, New York 13 Samali, B. and Kwok, K. C. S. 'Vibration control of wind excited tall buildings with passive dampers', Proc. Third Asia-Pacific Symp. on Wind Engineering, Hong Kong, 1993, pp. 829-834 14 Zhang, X. T. and Zhang, R. C. 'On control of along-wind and acrosswind vibration of structures by crossed dampers', Proc. Third AsiaPacific Symp. on Wind Engineering, Hong Kong, 1993, pp. 859-864 15 Cao, H., Li, Q., Ou, S. and Li, G. 'Dynamic behaviour of high-rise structures', Proc. Third Asia-Pacific Syrup. on Wind Engineering, Hong Kong, 1993, pp. 347-352 16 Liang, S., Le, J. and Qu, w. 'An analysis of galloping oscillation for the mast of the central TV transmission tower', Proc. of Third AsiaPacific Syrup. on Wind Engineering, Hong Kong, 1993, pp. 397-402 17 Standards Australia, 'Design of steel lattice towers and masts', Australian Standard AS 3995, 1994 18 Holmes, J. D. 'Along-wind response of lattice towers: part 1 - derivation of expressions for gust response factors', Engng Struct. 1994, 16 (4), 287-292
TTERWORTH I N E M A
Engineering Structures, Vol. 17, No. 9, pp. 66&-675, 1995 N
N
0141-0296(95)00036-4
Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0141 0296195 $10.(X) + 0.00
Effectiveness of tuned liquid column dampers for vibration control of towers T. Balendra, C. M. Wang and H. F. Cheong Centre for Wind-Resistant Structures, Department of Civil Engineering, National University of Singapore, Kent Ridge 0511, Singapore
The effectiveness of tuned liquid column dampers (TLCD) in controlling the wind-induced vibration of towers is studied. The nonlinear governing equation of the TLCD is linearized to obtain the stochastic response of the towers due to along-wind turbulence. Through parametric studies, the optimum parameters for maximum reduction in acceleration and displacement are presented for a wide range of towers. It is found that for any tower of practical interest, almost the same amount of reduction in acceleration can be obtained by choosing an appropriate opening ratio of the orifice in the TLCD. The same opening ratio would give almost the maximum reduction in displacement. Generally, the opening ratio needs to be varied between 0.5 and 1.0, with smaller opening ratios for shorter towers.
Keywords: tuned liquid column dampers, wind-induced vibration, towe rs 1.
sloshing frequency is tuned to the frequency of the structure. Recently Sakai et al. 5 proposed the tuned liquid column damper (TLCD), which suppresses the wind-induced motion by dissipating the energy through the motion of the liquid mass in a tube-like container fitted with orifices. The feasibility of TLCD for the Citicorp Centre in New York 2 and the Gold Tower in Japan 6 has been demonstrated by Sakai et al. 5. The purpose of this present investigation is two-fold. Firstly, to experimentally determine the characteristics of a TLCD. And secondly, to review the current development on the application of TLCDs and to assess the effectiveness of TLCDs for towers with different fundamental frequencies through random vibration analysis. A parametric study is carried out to determine the optimum parameters of the TLCD for maximum suppression of wind-induced acceleration and deformation of towers of practical interest. Since the governing equation of motion of the TLCD is nonlinear, the equivalent linearization technique is utilized to linearize the equation 7. Using the Harris spectrum 8 to model the along-wind turbulence, the stochastic response of the towers, modelled as a single-degree-offreedom system, is determined in the presence of the TLCD. The effectiveness of the TLCD for motion perceptions is quantified by the reduction in the RMS acceleration due to the TLCD. For structural deformation, the reduction in displacement is used to quantify the effectiveness of the TLCD.
Introduction
Recent advances in the development of lightweight, high strength material and construction techniques have led to more flexible and lightly damped structures. In such structures, the deformation and motion perception due to windinduced vibration become more prevalent and could lead to failure of the serviceability limit state. Thus in order to meet the serviceability requirements, the wind-induced vibration is suppressed either by active or passive control devices. In an active control device, an external force is applied to control the structural vibration, thereby keeping the structure safe and serviceable under extreme loading conditions. The amount of external force depends on the feedback from the structural response. Owing to the uncertainty of this infrequently used control system and the amount of power required to induce the control forces, in practice, the passive control system tends to be favoured over the active control system. Among passive control devices, the tuned mass dampers t.2 and tuned liquid dampers 3.4 are commonly used. The tuned mass damper consists of a large mass attached to the structure through a spring and dashpot. It dissipates energy when it is tuned to the frequency of the structure. The tuned liquid damper consists of a container which is partially filled with liquid. It dissipates energy through the sloshing action of liquid in the container. For optimum results, the
668
Tuned liquid c o l u m n d a m p e r s for vibration c o n t r o l o f towers: T. Balendra et al.
2.
Mathematical model
A TLCD with an orifice is shown in Figure 1. If x is the displacement of the liquid due to the horizontal motion y of the tube, then the equation of motion of the liquid column is given by Sakai et al. 5 as follows pAL x +
1
pA
+ 2pAgx = - pABy
( 1)
where 6 is the coefficient of head loss governed by the opening ratio of the orifice, L is the length of the liquid measured along the centreline of the tube, A is the crosssectional area of the tube, B is the width of the tube, p is the density of the liquid, and g is the gravitational acceleration. Equation ( 1 ) is nonlinear due to the presence of the liquid damping. However, the equation can be linearized using the equivalent linearization technique 7. Thus, consider an auxiliary system which is linear
where m (= pAL) is the total mass of the liquid, c is the equivalent damping coefficient and the natural frequency COd,or the natural period To, of the liquid column is given by or
where o~ is the standard deviation of the liquid velocity. In order to check the accuracy of the foregoing linearization, the Runge-Kutta-NystrOm method was used to integrate the second-order differential equation (1) with the initial conditions x ( 0 ) = dx(0)/dt = 0 and assuming a harmonic excitation y = y,, sin COdt where y, is the amplitude and COj the excitation frequency. The RMS velocity of the liquid determined is used in equation (7) to obtain the equivalent damping coefficient, c. This value of c is then used in equation (2) to determine the RMS displacement of the liquid for the same harmonic excitation. The RMS displacement of the liquid is compared with that obtained from equation (1), for different amplitudes of excitation, namely, Yo = 0.05 m/s 2, 0.2 m/s 2 and 0.4 m/s 2. The percentage difference is given in Table 1 for different head losses, 6. It is seen that the error increases with increasing head loss but the maximum error is about 10% for 6 -< 10. It should be noted that the values of 6 and the amplitudes of excitation considered would adequately cover the practical range.
(2)
mY + (B/L) mS~ + c.f + COdm x = 0
cou = \ 2 g / L
Tu = 27r\,L/2g
(3)
If the difference between equations (1) and (2) is denoted as e, then e = (mS/2L) [~12- c2
(411
The equivalent damping coefficient can be determined by minimizing the square of the error. Thus, we set E[e 2] = minimum
3.
In order to study the influence of the orifice on the motion of the liquid and to quantify the effect of the orifice opening on the damping coefficient, an experimental study was carried out. The TLCD shown in Figure 2 was constructed using perspex material. The size of the orifice was varied to have the following opening ratios: l, 0.75, 0.50, 0.25 and 0.10. Figure 2 shows an opening ratio of 0.25. The TLCD was then filled with water and placed on a shake table. The shake table was connected to a dynamic actuator which was controlled by a function generator. The displacement of the liquid was measured by a wave probe monitor, while the acceleration of the shake table was measured by an accelerometer. 3. I.
d d~-~ (El e2l) = 0
(6)
Substituting equation (4) into equation (6) yields 9
~ - ~r;
(7)
OrificeB
Y
./
Figure 1 Tuned liquid column damper (TLCD)
Experimental procedure
The TLCD was filled with water to a height of 270 mm, measured from the base of the tube. The natural frequency, wu of the liquid column calculated by equation (3) was found to be 0.71 Hz. A sinusoidal base motion was applied to the table. The excitation frequency was varied from 0.5 Hz to 0.9 Hz, at intervals of 0.02 Hz. Four amplitudes were chosen for the base motion: 6 mm, 8 mm, 9 mm and 10 mm. The experimental procedure adopted was as follows ( l ) Orifice opening ratio of 1.0 is selected first (2) The amplitude of base motion is set at 6 mm (3) For each frequency of excitation, the liquid displacement is measured (4) Step (3) is repeated for other amplitudes of base motion (5) Steps (2)-(4) are repeated for the other opening ratios: 0.75, 0.50, 0.25 and 0.10 3.2.
./_
Experimental investigation
(5)
where E[.] denotes the statistical average of mathematical expectation of a stationary Gaussian random process. The necessary and sufficient condition for equation (4) to be satisfied would be
c=
669
Head loss coefficient
For a particular value of orifice opening ratio and the amplitude of excitation, the relationship between the liquid displacement and the excitation frequency is investigated. Figure 3 shows the variations of liquid displacement amplitude with respect to the excitation frequency, for an orifice opening ratio of 0.50, and for a base excitation of amplitude
670
Tuned liquid column dampers for vibration control of towers: T. Balendra et al.
Table 1 Normalized RMS displacement of liquid from equations (1) and (2) for various excitation amplitudes, Yo (m/s 2) c~= 0.7
c~= 0.9
Yo
6
~x~
~,a
% diff.
0.05
2 4 6 8 10
0.971 0.962 0.954 0.946 0.938
0.968 0.956 0.945 0.935 0.924
0.32 0.63 0.92 1.21 1.48
1.241 1.223 1.207 1.191 1.176
1.234 1.211 1.189 1.168 1.148
0.52 1.01 1.47 1.91 2.38
0.2
2 4 6 8 10
4.691 4.513 4.355 4.215 4.089
4.622 4.389 4.189 4.015 3.861
1.48 2.73 3.80 4.74 5.58
5.878 5.542 5.264 5.029 4.827
5.742 5.315 4.972 4.690 4.452
2.32 4.10 5.54 6.75 7.78
0.4
2 4 6 8 10
9.025 8.430 7.951 7.557 7.223
8.779 8.030 7.449 6.981 6.593
2.73 4.74 6.33 7.62 8.72
11.08 10.06 9.302 8.715 8.242
10.63 9.380 8.495 7.827 7.300
4.10 6.75 8.68 10.2 11.4
(rxl
a~
% diff.
gradient of the plot of the damping ratio versus the resonant liquid displacement, the head loss coefficients given in Table 3 are obtained. A plot of the head loss coefficient against the orifice opening ratio is shown in Figure 4. The general trend is that, as the orifice opening ratio decreases, the head loss coefficient increases. The increase in the head loss coefficient is substantial when the opening ratio decreases below 0.50.
L_ Figure2 Dimension 25% opening 5.0Q
zzo of tuned liquid column damper with
Orifice opening ratio 0.5
t,.O0
E 300 J
."
2.00 ],O0
4.
o;,o'5
o'.
o'B Qk
o92
Frequency(Hz) Figure 3 Variation of liquid displacement amplitude with frequency of sinusoidal base excitation amplitude of 6 mm
6 mm. The liquid displacement amplitude first increases with increasing frequency. It reaches a peak value at around 0.71 Hz which is the resonant frequency. It then gradually decreases with increasing frequency. The damping ratios ~= c/(2mwa) can be determined from the plot given in Figure 3 using the half-power bandwidth method. The damping ratios for the other opening ratios are similarly calculated and are shown in Table 2. The head loss coefficient ~5, is then calculated from
6 = 2L ~
3.3. Effect of orifice opening on the liquid displacement Due to the size constraint on the TLCD, there is an allowable limit for liquid displacement. The water might spill out of the TLCD, if the allowable limit is exceeded and hence it would fail to perform as an effective damper. Figure 5 shows the relationship between the liquid displacement at resonance and the orifice opening ratio for various amplitudes of excitations. It can be seen that as the orifice opening increases, the liquid displacement at resonance increases. The increase is significant when the opening ratio exceeds 0.75.
(~/Xr)
(8)
where xr is the resonant liquid displacement. Thus from the
Application of TLCD to buildings
The effectiveness of the TLCD in independently controlling the along-wind and across-wind motions of tall structures has been investigated by Xu et al. ~o.1~. A random vibration analysis utilizing the transfer matrix formulation was carried out to obtain the response statistics, when the alongwind turbulence and across-wind wake excitation were modelled as a stochastic process which was stationary in time and nonhomogeneous in space. A comparative study with the traditional tuned mass dampers '2 (TMD) using a 76-storey building which was 306 m in height, and a 370 m tall TV tower, revealed that the TLCD could produce a similar level of response reduction as the TMD provided that the parameters of the TLCD are properly selected. Samali and Kwok ~3 carried out a sensitivity analysis to optimize the parameters of the TLCD for maximum reduction in response for the 76-storey building. A displacement reduction of 45% was achieved with a damper mass equivalent to 0.7% of the building mass. To control both along-wind and across-wind motions simultaneously,
671
Tuned liquid column dampers for vibration control of towers: T. Balendra et al. Table 2 Damping ratios versus resonant liquid displacement for different opening ratios Opening ratio Forcing function amplitude (mm)
1.00
0.75
xr (cm)
~
xr (cm)
~
xr (cm)
~
xr (cm)
(
xr (cm)
6 8 9 10
8.18 9.65 9.99 10.64
0.052 0.060 0.061 0.066
3.96 4.52 4.80 4.90
0.061 0.073 0.077 0.083
3.49 3.96 4.10 4.32
0.080 0.087 0.093 0.099
1.73 1.91 2.00 2.10
0.130 0.172 0.195 0.197
1.22 1.34 1.42 1.51
Table3 Relationship between orifice opening and head loss coefficient Opening ratio
Head loss coefficient, 8
1.00 0.75 0.50 0.25 0.10
1.86 6.87 7.80 62.00 70.88
0.50
0.25
0.10
0.219 0.237 0.266 0.271
investigate the variation of the optimum parameters of the TLCD with respect to the structural properties, for maximum reduction in response. In order to overcome this deficiency in the current state-of-the-art of TLCDs, towers with different fundamental frequencies are considered in the next section.
5.
Analysis of towers with TLCD
A tower fitted with a TLCD is shown in Figure 6. In order to gain insight into the TLCD-structure interaction, the tower was modelled as a single-degree-of-freedom system. The governing equations of motion, in view of equation (2), become
80 cO
60
<,,,+[: o]{,}+[/ o]{x} ={o}
moe M JLy]
C
5;
K
y
(9)
F(t) 20
t,O
60
80
100
Opening rotio (%) Figure 4 Variation of head loss coefficient with orifice opening ratio
[ g
,0 8
G -,= (, 3
2 0
where M is the total mass of structure and liquid, K and C are the stiffness and damping coefficient of the structure, respectively, F(t) is the force acting on the structure due to wind, and t~ = B/L. Adopting the two-sided Harris spectrum for wind, the power spectral density of wind force is given by
J
-
Orifice opening rotio(%] ] ----tOrero -o-gram -,,-Smm --.-firnmI
U~
~
)
.
~--TLCD F(t)
Figure5 Variation of liquid displacement at resonance with
.~
-!
different amplitude of base excitation
D Turbulent
Zhang and Zhang ~4 proposed a TLCD in the form of crossed dampers. The along-wind turbulence was considered to be random while the across-wind loading was considered to be periodic due to vortex shedding. It was found that when a crossed damper is installed at the top of a multidegree-of-freedom system, the across-wind resonance response was reduced by 50%. All these investigations m-~4 pertained to a particular structure. No attempt was made to
(lOa)
(2 + X2) 5'6
component
'-
._j -
i
Meon component
711
f
f.//,,"
Figure 6 Tower fitted with a TLCD and subjected to turbulent wind
Tuned liquid column dampers for vibration control of towers: 7-. Balendra et al.
672 where XFo
wl~ 27rUle
m Xl = (~'~2 -- /32)( 1 _ /32) _ /32(1&0/2/32 + ~ C)
(20)
X2 =/3[C( f12 -/32) + ~(1 -/32)]
(21)
(lOb)
= p,~toCo~
(10c)
in which ko (= 0.03) is the ground surface drag coefficient, lx is the wavelength taken to be equal to 1200 m, Do is the density of air, Co is the drag coefficient, Ao is the frontal area of the structure, and Ulo and Uz are the mean wind velocity at 10 m and height z, respectively. In order to determine the response using the spectral method, let F(t) = Foei~°t
The power spectra of displacement and velocity can be expressed as S,,y = IH.,,I2 SFF
(22)
= IH2 S.-
(23)
S;x =/32 &~
(24)
where SFF is the nondimensional power spectral density of the wind force which may be written as
( 11 ) (I)
(25)
#q,/3
(26)
SFF(/3) "~- - -
where COis the excitation frequency and i = q--i. As equation (9) is linear, the solution can be obtained by setting x( t ) = 2 e i'°'
(12a)
y(t) = 33e i'°t
(12b)
where qb =
[2 + (rr3qj/3)2]s/6
Here Ur = U o/U~ and ~0= wplJ(2&Um). The mean square responses are determined from
Thus, we have ((CO~- w2) + i ~ ) 2 - ace2 33= 0
(13)
- mCOea2 + ((@ - CO2)M+ iCCO)33= Fo
(14)
-2~G--
where COp= K ~ is the natural frequency of the structure. For generality and convenience, the following nondimensional parameters are introduced m
2&
33&
~ = ~ ' ~=-k-' Y=-B' 8
60 COd 3 m"~. . -, COp a = -% -'
=
S.x d/3
(27)
o'~ = \ Bcop/ = ~
_~ ~2S~ d/3
(28)
-2O-y= ( V )
~ Syy d/3
(29)
2 = If
--2 = tO'YT#t2 0".~
c
\ no)2p /
= ~
1 f~
_~ [~4Svy d/3
(301
~=Z' c-- m.,~' c _ "c- mw.
~a-~;,
_ Fo#
-~;-
As the power spectra S~ and Syy are functions of ~;, equation (28) has to be evaluated first using an iterative procedure. The false position method was used for solving the transcendental equation (28) for ~;.
o~,,B
_ Fo~W
CO2pBM 2gaM
(15) 6.
Using these parameters, and in view of equations (12), equations (13) and (14) may be written, respectively, as [(fl 2 - / 3 2 ) + ~/3i] 2 - a/32 y = 0
(16)
-/xc~/32 2 + [(1 -/32) + ~/3i] y =
(17)
Solving the simultaneous equations (16) and (17) and squaring the solutions, yields the following transfer functions
X~, +)t~2 IHyl 2 =
where
~
2
(~-~2 -- /32)2 + ~'2/32 = ~ + ~
(18)
Effectiveness of TLCD
The effectiveness of the TLCD may be quantified by the ratio between the RMS acceleration and RMS displacement of the tower with and without the TLCD. The variances of acceleration and displacement of the tower without TLCD are given by - 2 2k,Ya o'y = - _ qb~i C O'y = ~
(31a)
-- /32)2 + j[~2C2 ~ 0/3
(31b) (19)
If the ratios between the RMS acceleration and RMS displacement with and without TLCD are denoted as ~ r and ~vr, respectively, then it can be shown that
673
Tuned liquid column dampers for vibration control of towers: 7-. Balendra et al. .
0.8(
d ~ } 1/2
Ioi
' ,o9o
ct=O.9
. . . . . Q =0.95 . . . . . . ~ =1.~ ~ =1.o5
\
~,,r =
" fi'( •
?
'
i l-/3:)~+/3:C:
}
0.71
~ \ ~ . .~
060
". '
....
f~ = 1.10
(32b)
*dis
~
. . . .
",~ .-.,.
.~---- ~-2--"2. . . .
Thus ~,. is a function of the.following parameters
,
0.50 ~fr =f(qA U r , P,, ~ , OL, ~!, (% F )
(33)
,
....
0.=0.6 or=07
.....
The fundamental period of the towers with a heavy mass at the top such as the Wuhan Television Tower of 115 m heighfl 5 or the central TV transmission tower of 256 m height ~6 can be estimated approximately by the following expression
T~= 2wlw n = z130
~= o02 ~:10
(b)
ct:O5
/
~=0.8
,_
0.70
/
.......~t=~...~....
~-~
f
(34a)
where Tn is the period in seconds and z the height of the mass above the base in meters. The period of lattice towers can be estimated by the following expression ~7 which has been validated from field measurements ~8
0.50
0.70
.
.
.
.
- I.,t= 0.01 . . . . . P. = 0.02 ......~: 0.03
---
.
(C)
~= o.o~
. . . . . ~: o.os
ct =09 ~ = 1,0
--j_./~/
/
TR = h21(750(wh + w,))
(34b)
where wh and w, are the width of the tower at the base and top of the tower, respectively, and h is the height of the tower. If the mass density of the tower is taken to be Pb = 2 k N / m 3, Po -- 1.2 kg/m 3, Co = 1.2, U~o = 20 m/s and the mean wind velocity profile is assumed to be of the form
060 0.50
__
"~.~ ...............
0.1.0
.
.
.
.
5
10
15
29
6
U:/U o = (z/10) °3
(35)
Figure 7 Influence o f head loss on r e d u c t i o n in a c c e l e r a t i o n f o r t o w e r w i t h q~= 0.5
then in view of equation (34a), it can be shown that
t~ = 58/z
(36a)
= (10/z) °'6 = (10~/58) °'6
(36b)
-
(36c)
D
where D is the diameter of the tower. A similar set of expressions can be obtained if equation (34b) is used instead of equation (34a). In view of equation (36), equation (33) reduces to
chosen, the performance of the TLCD can be controlled through 6, namely, by adjusting the orifice opening. The value of head loss coefficient which gives the maximum reduction in acceleration, denoted as 8,,~,, is shown in Figure 8 for values of + ranging from 0.2 to 1.0, which represent towers of practical interest. The corre10.0
8.0
--ct=0.5 - - - . a - " 0.6 ---a-'0.7 - - - - a - " 0.8
/, ,,,',• , pg,-',~/ -
~ - e t = 0.9
//s ° S °
~$:r f( ~b, /~, ~ , a, C, 6,) =
1
(37) ~:~ 6.0
The influence of 1), a, and/.t on O'er is shown in Figure 7 for various values of head loss coefficient, when D = 30 m, = 0.5 and C = 0.02. It is seen from Figure 7a, that for any given head loss, higher reduction in acceleration is obtained when ~ = 1.0 and for maximum reduction, there is an optimum value for 6. When the TLCD is not fully tuned (say l) = 0.95 or f~ = 1.05), by changing the orifice opening to an appropriate value, one can improve the performance of the TLCD. Figures 7b and 7c indicate that for better performance of the TLCD, one should choose higher values of a and /x, It can be seen that once a and /z are
,¢,~," 7"
//oS /
<;~/ / "
j
.,/7/
2.0 01
"
' 0.3
•
, 0.5
,
, 0.7
,
, 0.9
, 1.1
Figure 8 O p t i m a l head loss f o r m a x i m u m r e d u c t i o n in accelera t i o n f o r ~ = 1.0 and /~= 2%
Tuned liquid column dampers for vibration control of towers: T. Balendra et al.
674 0.70
0.65
----..... ---.....
0.01
(z= 0.5 a=0.6 0.= 0.7 0.=0.8 a= 0.9
0.06
--0.=0.5 ---ct=0.6 ..... o.=0.7
',
t~.
--.=....
--
o.:0.0 9
0.05
~ ° ~ . I
,~.b 0.60
,1~ _....~ O.Ot,
0.03
0.55
OO2o1 °%'.1 ' 013
o15
017
oJs
o'.7
0g
t.1
oJo ' H
qJ Figure 11 Velocity of liquid corresponding to o p t i m u m head Figure 9 Reduction in a c c e l e r a t i o n c o r r e s p o n d i n g to o p t i m u m
loss for D. = 1.0 and /~ = 2%
head loss for ~. = 1.0 and /~ = 2%
sponding reduction in acceleration is depicted in Figure 9. It is interesting to note that practically the same amount of reduction in acceleration can be obtained for any tower of practical interest by choosing the appropriate head loss coefficient. This optimum head loss coefficient increases with ¢ which means that a smaller opening ratio is required for shorter towers. In view of Table 3 and Figure 8, the TLCD could be used to control the wind-induced acceleration of towers with different fundamental frequencies, by varying the orifice opening ratio between 1.0 (fully opened) and 0.5. The liquid displacements and velocities corresponding to the optimum head loss coefficients are shown in Figures 10 and ll, respectively. The RMS displacement of the liquid expressed as the ratio of the length of the liquid (~rx/L = a-~x/Tr4) is found to vary between 2% and 6%, with smaller displacement for shorter towers. For sharp edge orifices, the separating shear layers are known to be independent of the Reynolds number, which may be computed using the approaching flow configuration as Re = Rtr~/u
(38)
where R is the hydraulic radius defined as the ratio of the wetted area to the wetted perimeter, u is the kinematic viscosity of water, and o-;,is the RMS velocity of the approaching flow which is given by
0.07 0.06
'
. . . . . .
'0.=o15 ---0.=0.6 . . . . c~=0.7
'\ ~
---0.=0.8
Using equation (34a) and Figure 11, the Reynolds number computed for various configurations of the liquid damper for a tower of height 300 m is given in Table 4. It can be seen that the Reynolds number corresponds to turbulent flow. Since the head loss coefficients are independent of the Reynolds number for a turbulent flow, the designer has the option of adopting the most suitable dimension for the TLCD As in the case of acceleration, the reduction in displacement due to the presence of the TLCD, computed according to equation (32b), is found to be largest when the frequency of the damper is tuned to that of the tower. Furthermore, a higher reduction is obtained for larger values of a and /x. The maximum reduction in displacement achievable is shown in Figure 12 for towers with different fundamental frequencies. The corresponding head loss required is shown in Figure 13. It can be seen from Figure 12 that a higher reduction in displacement is possible for taller towers. Comparing Figures 13 and 8, the head loss required for maximum reduction in displacement is approximately the same as that required for maximum reduction in acceleration. It can be seen from Figures 9 and 12 that a TLCD would provide a maximum reduction of 50% in acceleration and displacement when the structural damping is 2%. The percentage of reduction could vary slightly with different structural damping. However, the trend would be similar. Thus, the TLCD provides a convenient method for controlling both the acceleration and the displacement of towers. Because of nonlinear damping of TLCDs, in general the results presented in Figures 9-13 would depend on the amplitude of excitation. However, as the head loss coefficient corresponding to the practical range of the orifice
O.OZ,
Table4 Reynolds numbers for TLCD in 300 m high tower
0.03
Length of liquid, L (m)
Thickness
of tube (m)
Width of tube (m)
Reynolds number
20 10 10 5
1.0 1.0 0.5 1.0
1.27 2.54 5.08 5.08
211 000 135 000 86000 79 000
0.020.1
. . 0.3
. . 0.5. . . 0.7 .
0.9
1.1
Figure 10 Displacement of liquid corresponding to o p t i m u m head loss for ~ = 1.0 and /~ = 2%
Tuned liquid column dampers for vibration control of towers: 7-. Balendra et al.
.....o
0.65
--
~-=0.5
0.60
--..... --.....
ct=0.6 ct=0.7 ct-08 ct=O.
jJ'
0.50
,
.-"
0.55
,f
.~ /"
a higher ratio of width to liquid length and higher mass ratio are better for higher reduction in acceleration and displacement. However, economic considerations will govern the value of the mass ratio. For best performance, the frequency of the TLCD must be tuned to the frequency of the tower. But, even if the TLCD is not perfectly tuned, by controlling the opening ratio, one could easily obtain a satisfactory performance from the TLCD. This feature is advantageous to the designer, as it is often difficult to assess the structural frequency accurately at the design stage. Moreover, the frequency of the structure would not remain the same during the lifetime of the structure.
'
-
/x,.~'
a
......-...-----
...~
.-,-
s"
i'
o.'3
o;s
o19
q~
1.1
Figure 12 R e d u c t i o n in d i s p l a c e m e n t o f t o w e r c o r r e s p o n d i n g t o o p t i m u m h e a d loss f o r m a x i m u m r e d u c t i o n in d i s p l a c e m e n t f o r O. = 1.0 a n d /x = 2% 10.0 c~-'0.5 8.0
--
- - - - ct--0.5
,,,I"
. . . . ct=07 - - - ct=O.8 . . . . . a=O 9
~" -" .I "~-"" -" -'.~"
6.0
t :-~.
../.:~-
om"
~,'.;Y L,.O
1
1"
i"
,,...,~.ti.-
¢:;,"
-
2.0
0,00
Figure 13
L 0.3
,
0.'5
' 0.7
O p t i m u m h e a d loss f o r m a x i m u m p l a c e m e n t of t o w e r f o r f~ = 1.0 a n d /x = 2%
' 0.9
'
1.1
r e d u c t i o n in dis-
opening ratio would not exceed 8, in view of Table 1, the effect of nonlinearity is small, less than 10%. Hence, the results presented would be practically independent of the excitation level.
7.
675
Conclusions
The effectiveness of a TLCD in suppressing wind-induced acceleration has been demonstrated for towers with different fundamental frequencies. It has been found that the amount of reduction in acceleration is almost the same for any tower of practical interest, provided one chooses an appropriate opening ratio for the orifice. The same opening ratio would give almost the maximum possible reduction in displacement. In practice, the opening ratio needs to be between 1.0 and 0.5 for optimal performance. TLCDs with
References 1 Wargon, A. 'Design and construction of Sydney Tower', Struct. Engr 1983, 61A (9) 2 Peterson, N. R. 'Design of large scale tuned mass dampers', in 'Structural control', (Ed. H. H. E. Leipholz) North-Holland, Amsterdam, The Netherlands, 1980, pp. 589-596 3 Modi, V. J. and Welt, F. 'Vibration control using nutation dampers', Proc. Int. Conf. on Flow Induced Vibration, Bowness-on-Winderwere, UK, 1987, pp. 369-387 4 Fujii, K., Tamura, Y., Sato, T. and Wakahara, T. "Wind-induced vibration of tower and practical applications of tuned sloshing damper', J. Wind Engng Ind. Aerodyn. 1990, 33, 263-272 5 Sakai, F., Takaeda, S. and Tamaki, T. 'Tuned liquid column damper - new type device for suppression of building vibrations', Proc. Int. Conf. on Highrise BuiMings, Nanjing, China, 1989, pp. 926-931 6 Noji, T. 'Study on vibration control damper utilizing sloshing of water', J. Wind Engng, No. 37, 1988 7 Iwan, W. D. and Yang, I. 'Application of statistical linearization techniques to nonlinear multidegree of freedom systems', J. Appl. Mech. 1972, 39, 545-550 8 Walshe, D. E. J. 'Wind excited oscillations of structures', National Physical Laboratory, London, 1972 9 Sun, K., Cheong, H. F. and Balendra, T. 'Effect of liquid dampers on along-wind response of structures', Proc. Third Asia-Pacific Syrup. on Wind Engineering, Hong Kong, 1993, pp. 835-840 10 Xu, Y. L., Samali, B. and Kwok, K. C. S. 'Control of along-wind response of structures by mass and liquid dampers', J. Engng Mech., ASCE 1992, 118 (2), 20-39 11 Xu, Y. L., Kwok, K. C. S. and Samali, B. 'The effect of tuned mass dampers and liquid dampers on cross-wind response of tall/slender structures'. J. Wind Engng Ind. Aerodyn. 1992, 40, 33-54 12 Weisner, K. B. 'Tuned mass dampers to reduce building wind motion', ASCE Convention and Exposition, Preprint 3510, 1979, ASCE, New York 13 Samali, B. and Kwok, K. C. S. 'Vibration control of wind excited tall buildings with passive dampers', Proc. Third Asia-Pacific Symp. on Wind Engineering, Hong Kong, 1993, pp. 829-834 14 Zhang, X. T. and Zhang, R. C. 'On control of along-wind and acrosswind vibration of structures by crossed dampers', Proc. Third AsiaPacific Symp. on Wind Engineering, Hong Kong, 1993, pp. 859-864 15 Cao, H., Li, Q., Ou, S. and Li, G. 'Dynamic behaviour of high-rise structures', Proc. Third Asia-Pacific Syrup. on Wind Engineering, Hong Kong, 1993, pp. 347-352 16 Liang, S., Le, J. and Qu, w. 'An analysis of galloping oscillation for the mast of the central TV transmission tower', Proc. of Third AsiaPacific Syrup. on Wind Engineering, Hong Kong, 1993, pp. 397-402 17 Standards Australia, 'Design of steel lattice towers and masts', Australian Standard AS 3995, 1994 18 Holmes, J. D. 'Along-wind response of lattice towers: part 1 - derivation of expressions for gust response factors', Engng Struct. 1994, 16 (4), 287-292