Critical mode control of a wind-loaded tall building using an active tuned mass damper

Engineering Structures, Vol.

ELSEVIER

PII: S0141-0296(97)00172-1

19, No. 10, pp. 834-842, 1997 © 1997 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0141-0296/97 $17.(}0 + 0.(X)

Critical m o d e control of a windloaded tall building using an active tuned mass damper L. E. Mackriell Acer Wargon Chapman, Level 19, AIDC Tower, 201 Kent Street, Sydney, NSW 2000 Australia

K. C. S. Kwok School of Civil and Mining Engineering, Universi~ of Sydney, NSW 2006 Australia

B. S a m a l i Graduate School of Engineering, University of Technology, Sydno', NSW 2007 Australia (Received March 1996; revised version accepted October 1996)

The active control of the first mode of vibration of two slender, wind-loaded buildings, one 200 m tall, the other 400 m, is studied. The control is applied to the buildings via an active tuned mass damper located at the top of the building. An algorithm using acceleration feedback from the top of each building is used, as is a range of empirical algorithms using some combination of feedback from the first mode of vibration of the buildings. For both buildings, it is the more critical cross-wind response which is studied, and the response is reduced to a level acceptable for human comfort. All the algorithms provide the required control using achievable control forces; however, the algorithms using pure acceleration feedback produce the best control performance for both buildings. This is because acceleration feedback combines in the most efficient way with the passive control inherent in a hybrid control system. © 1997 Elsevier Science Ltd. Keywords:structural control, active control, wind loading, tall buildings, active tuned mass damper

1.

the use of simplified expressions for cross-wind loading, or its complete neglect. This last omission is perhaps the most serious, as the cross-wind direction is the most significant loading direction for very tall buildings. This study focuses on wind-loaded structures, and the authors hope to redress some of these deficiencies by using multidegree-of-freedom numerical models of tall buildings, which are loaded by digital time histories of cross-wind loads obtained from wind tunnel testing of scale models of the buildings. Several control devices have been proposed as appropriate for large civil engineering structures: cold gas pulse generators which disrupt the build-up of energy associated with resonant response have been suggested~6; as have aerodynamic appendages fitted to control along-wind responseS7; active tendons, through which actuators apply forces to counteract dynamic loadsr,~8; active mass drivers, which consist of a force actuator driving a mass connected

Introduction

In recent years, progress in the field of active structural control has been marked. A variety of control algorithms have been developed specifically for civil engineering structuresl-4; and many studies have been conducted using suitably adapted classical control algorithms 5-7. Several buildings in Japan, such as the Landmark Tower in Yokohama 8, Shinjuku Park Tower 9, the Ando Nishikicho m building, and the Kyobashi building in Tokyo now employ some form of active control device. Most work to date has concentrated on earthquakeloaded buildings; however, some work relating specifically to wind-loaded buildings has also been conducted 5:2 is. While some progress has been made in this area, those studies have been limited by one, or all, of the following: a restriction of the building model to one degree of freedom;

834

Control of wind-loaded tall building using ATMD: L. E. Mackriell et al. to the building by a stiffness element, and it is the reaction of this system acting on the system acting on the building which controls the buildingS9; and active tuned mass dampers (ATMD) which are a combination of a passive control device, the tuned mass damper (TMD) with a force actuator4.5,7,13.2(i. The aim of this work is the investigation of one type of these control devices, the ATMD, to control the cross-wind dynamic response of two tall buildings to levels acceptable for human comfort. The ATMD is a tuned device, making it particularly suited to the control of wind-loaded buildings, as their response is usually dominated by the first mode of vibration z~. The ATMD is also a hybrid device, augmenting the passive control of a TMD, which consists of a mass, spring and dashpot, with a force actuator situated between the mass and the building. For the control of the first mode of vibration, the ATMD is generally situated as close to the top of the building as possible, where the first mode response is greatest. Because the aim of the control is to reduce the first mode response only, a control approach used in this study is that of critical mode control 22. In this case, the control force is calculated using feedback from the critical, first, mode of vibration, and ideally controls only that mode, focusing control energy, and the response of the device, to the area of greatest need. For first mode control, the general form of the control force is of the form given in equation ( I ) where ~ is the first mode control force, rh is the first mode displacement, and Ga, G,, and G, are modal displacement, velocity and acceleration gains, respectively c~ = G,{qa + Gv;ol + G,,~l

(1)

In this study, a broad range of gain values have been used to calculate the control force using equation (1) to determine an efficient control system, where the control algorithm selected is that which works best with the control device. In particular, various combinations of first mode displacement and velocity, and first mode acceleration and velocity have been used. Complete feedback, and a combined displacement and acceleration feedback algorithm have not been considered because (as will be shown) the behaviour of displacement feedback and acceleration feedback is very similar. In addition, an algorithm using acceleration feedback from the top of the building, rather than modal feedback is also studied. Optimum ATMD parameters which minimize the control force are found for each algorithm via a parametric study, and the effect of inadvertent changes to the ATMD frequency is examined. The gain values used here have been obtained empirically via a parametric study to reduce the building response to a reference level, and the efficiency of the algorithms is then compared. A control :algorithm based on a mathematical optimization process, such as an LQR (Linear Quadratic Regulator) algorithm, has not been used. The motivation for an empirical approach is that a wide range of possible control solutions can be studied to determine which algorithm; or combination of feedback type; results in the least expenditure of control force and device movement for a wind-loaded building controlled by an ATMD. Mathematically optimal control algorithms, however, generally focus on a narrow range of control solutions, determined by weighting matrices chosen by the user, and the mathematical optimization process. They result in control

835

forces which are optimal for the particular set of weighting matrices chosen by the user. For example: classical LQR control; where gains are chosen such that the performance index J in equation (2) is minimized subject to the constraints of the state equation of the systemS J2; will always result in control using displacement and velocity feedback only, and the proportions of displacement and velocity feedback depend partially on the weighting matrices chosen by the user. In equation (2) y is the state space vector of displacements and velocities, u is the control force vector, and Q and R are user-defined weighting matrices. It should be noted that a mathematically optimal algorithm such as the LQR algorithm does not address which set of weighting matrices give the most efficient optimal result. If the user such an algorithm is already aware which type of solution is most efficient, from a study such as this one, considerable computation time may be saved.

J

=

o[yT(t)Qy(t) + uT(t)Ru(t)ldt

(2)

2. Analytical description of a controlled building If a building is modelled discretely by n degrees of freedom (DOF), then the equations of motion of the ATMD and the nth DOF (the point of attachment of the ATMD) are as shown below: where k,,, c,, and m,,; and ka, c,j and m~ are the stiffness, damping, and mass of the nth DOF and the ATMD, respectively; x,, is the displacement of the nth DOF relative to the ground, y is the displacement of the ATMD relative to the nth DOF, f,, is the wind load acting on the nth DOF, and a is the control force. mdY + C,~ + kay = -m,#,, + a

(3)

(m,, + m d ) ) , + C,~,, -- C,, ,k,,_.] + k,,x,, -k,, tx,, I = f , , - m,L~

(4)

Consideration of how the passive (that is, ~=0 in equation (3)) TMD works is an important first step in designing an efficient ATMD. A TMD primarily reduces the vibration of the building by transferring energy from the building to the device via the acceleration of the nth floor, and the term -moR, can be considered a 'passive input force'; and then dissipating that energy via the device damping. Thus, the term -m,~, which is the effect of the device on the building, or 'device output force' can be seen as an additional damping force, as it dissipates energy in proportion to the vibration of the building. An efficient way of increasing the passive control effect appears, from inspection of equations (3) and (4), to be to use a control force proportional to -)~,; and this is the basis of the acceleration algorithm proposed by Nishimura et a l . 4 in which an earthquake-excited building modelled by a single DOF was controlled. For example, if the response of the building is primarily sinusoidal, and the control force is given by c~ =-g;¢,, where g is a gain value; then the device output force will also be sinusoidal, although there will be a phase difference between L, and 5', dependent on the ATMD parameters. The amplitude of -m,~' will equal ( 1 + g) mamnP, where

Control of wind-loaded tall building using A TMD: L. E. Mackriell et al.

836

P is dependent on the amplitude of )~, and the ATMD parameters. For a TMD in the same circumstances the amplitude of -ma~ would equal mam,P, and so this active control force increases the passive control in a linear manner. If displacement feedback is used, then a control force of the same amplitude as the acceleration feedback case, given by Ol= --go)2x., where w is the circular frequency of the building response, will result in -m,£9 of amplitude (1 + g)mdm,P, identical to the acceleration feedback case. When c~ =-goat,, a velocity feedback algorithm resulting in a control force with the same amplitude to the acceleration case, -m~v has an amplitude of ( 1 + g)J/2 mam,,P, less than the acceleration case, and so the control effect on the building for the same level of control force will be less. However, the response of the building cannot always be approximated by sinusoidal motion, and the relatively simple argument given above needs to be verified in a more realistic setting. Indeed, in the cross-wind direction, response at the vortex shedding frequency may be significant compared to the response at the first mode frequency. In that case, the device is effectively partially detuned and the augmented TMD-like control action will be reduced in efficiency. It should be noted that a velocity feedback algorithm can add damping directly to the building (see Section 3.1.1 ), which can be of benefit in this case. The response of the top DOF of a wind-loaded structure is, as discussed previously, primarily composed of first mode response. Thus the effect of first mode control will be similar to that discussed above for the feedback from the top DOF. Except that any higher mode components will be excluded from the control force, focusing control energy, and the response of the device, to the area of greatest need. 2.1. Numerical model o f a controlled building For simplicity, the controlled building is assumed to be a linear structure, which can be modelled in one direction by an (n + 1 ) lumped mass model. The equation of motion is given by equation (5); where M, C and K are the ( n + l ) × ( n + 1) mass, damping and stiffness matrices, respectively, F,, is the (n + 1) element vector of wind forces, x is the vector of the displacement of each DOF relative to the ground, and U is the control force vector. U is shown explicitly in equation (6) M # + Ck + K x = F w + U

(5)

U = (0,...,O,-a,a) r

(6)

For modal control, it is convenient to transform equation (5) using the relationship x = ~r/, and pre-multiplying by ~r, where qb is the modal matrix, and r/the vector of modal displacements. This transformation will not, in general, result in the complete decoupling of equation (5) into n + 1 independent equations in modal space due to the damping of the ATMD, however, when damping is small, the coupling introduced is slight. For control design purposes only it is useful to closely approximate the equation of motion of the first mode of vibration as shown in equation (7) using the transformation described above; where m], c~ and kj are the first mode mass, damping and stiffness, respectively; and ut and f~ are the first mode control force and wind force, respectively. Equation (8) gives the relationship between ul and the physical control force a, where qSii is the ijth element of ~.

m,i)] + c,i'/, + kl'rh =fl + u] Ul

3. 3.1.

(7) (8I

Control algorithms First mode control

3.1.1. Displacement and velocity (DV) feedback algorithms For this family of algorithms, the first mode control force is given by equation (9), where g,l and g,. are the displacement and velocity feedback gains, normalized with respect to first mode stiffness and damping, respectively. Substitution of equation (9) into equation (7) shows that these algorithms increase the damping and stiffness (and hence the natural frequency) of the first mode, although it should be remembered that the control effect comes not only from the alteration of the building's eigenvalues, but also from the augmentation of the TMD-type control as discussed in Section 2. The physical control force is easily calculated from equation (8), using the transformation x = qb'0. This is a stable control solution as long as g,i and g,, are positive real numbers, and the error in assuming no coupling due to the damping matrix is small. The significance of the assumptions about the damping matrix can be checked after the control gains have been selected by transforming the equation of motion of the controlled system into state space and calculating the eigenvalues of the system to ensure that the real parts are less than zero. ul = -g,lkl Th - g,.cl qh

(9)

3.1.2. Acceleration and velocity (AV) feedback algorithms The general equation describing the first mode control force resulting from this family of algorithms is given by equation (10), where g, is the acceleration feedback gain. When equation (10) is substituted into equation (7) it is again seen that these algorithms increase both the frequency and damping of the first mode of vibration of the controlled structure. However, in this case, the frequency is increased by lowering the mass of the first mode, so the control will be stable if g, < 1 ul = -g,.cj qJ+t + g,,ml ~h

(10)

The term l/(~b(,+]>.rr,.~) in equation (8) is negative in this case, and so g, must be positive so that the resulting physical control force is close to being proportional to -ma2, in order to increase the TMD-type control in an efficient, linear manner. Similarly, ga in equation (9) is chosen as negative, as the first mode displacement is close to 180 ° out of phase with -m~,,. 3.1.3. Spillover The effect of the control on higher building modes can be assessed if U is transformed into model space, as described in Section 2.1. The modal control forces ui, for 1 --< i --< n + 1, are given in equation ( 11 ). The forces u2 to u,, + t represent control spillover, the unintentional effect of the control on the higher building modes.

u~ = a( 4)~,,+,,.i - 6,,.,

( 11 )

Control of wind-loaded tall building using A TMD: L. E. Mackriell et al. Observation spillover, the effect of unintentional feedback from higher modes on the control force, occurs when r h cannot be fully reconstructed from the measurements available. In this case, a is degraded by feedback from higher modes, potentially decreasing control efficiency, and the higher modes can receive, via the control spillover force, 'self-feedback'. This situation can be either beneficial or potentially destabilizing. The first mode lateral response is at a maximum at the top of the building, and so the sensor sites used in this study are assumed to be located on the nth DOF and on the ATMD itself, the (n + l)th DOF. The relationship between x,, x,+~, and the modal displacements can be expressed as shown in equations (12) and (13) x,,+l = 6~,,+,),,7h + 6i,,+1).<~+,~ + 6~,,+,).Rr/g

(12)

X. = 6,,.,~/, + 6,,.(,,+~)r/.+, + 6.,Rr/R

(13)

In equations (12) and (13), the ATMD mode displacement is assumed to be r/,,+t ;6~,,+~~.R and 6,.R are row vectors containing all but the first and (n + 1)th elements of the (n + 1 )th and nth rows of cb, respectively; and r/R is a column vector containing higher mode displacements. Manipulation of equations (12) and (13) yields an approximation of rh, using x, and x,+~, given by equation (14). Differentiation of equation (14) leads to approximations for modal velocity and modal acceleration vectors, ~ and ~h. The algorithm described by equation (10), with g~ = 0 for simplicity, can be used to illustrate the effect of observation spillover, and the effect on the other algorithms is analogous. The resultant A T M D force c~,pe, and the effect of the observation spillover on higher building modes, given by u2 to u,, are shown in equations (15) and (16). The second term on the right-hand side of equation (15) demonstrates that the higher building mode accelerations (but not the A T M D mode acceleration) have polluted the feedback to the ATMD, and hence the possibility of control degradation arises. The second term on the right-hand side of equation (16) represents the effect of observation spillover on the higher building modes. Each building mode receives 'self-feedback', as ~R contains all the higher building modes, and this situation can lead to instability in those modes. -I 6 , Z ( x . - 6 . . . . ,6.+,..+,x°+,)

rh"~'~' = 1

(14)

'-~ =i . . . . . - 6 . , , 6 .... ,6,,+,.,,+, 6,,+,.~

6:', ( 6.,.+, 6;',.,,+, 6,,+,,. - 6 . . . ) n .

= rh -

1 - 6ZJi 6 . . . . , 6 . +- , . . + , 6 . + , , ,

o%,;, = g,,ml( ~l --

(~n,ll ( (~)n,n+l (~)n+lI ,n+l (])n+ I ,R -- (~)n,R)~g~R [ -- ~)n.]l(~n,n+l~Jnll,n+l~)n+l,i

(15) l~li : g.ml(

6,,+l,i

-I (

-- 6n,i)('~l

(16)

--

-I

6,,.! (h,,.,,+!~.+,.,,+L6,,+,.~ _- 6,.R_!~R I - 6 , -1 -I , . , 6 ...... ,6,,+,,..,6,,+,.,

3.2.

'

2 --< i --< n

Simple acceleration feedback algorithm

This control algorithm simply uses acceleration feedback from the top floor to drive the ATMD, in order to efficiently

837

increase the control effect of the device as outlined in Section 1. For this algorithm, the actuator force a is directly obtained from c¢ = - gm,fi,,

(17)

An advantage of this algorithm compared to those discussed in Section 3.1. is that knowledge of the mode shapes is not needed in order to calculate the control force; however, some control efficiency may be lost due to the presence of higher modes in the feedback. It is easy to show, using arguments similar to those in Section 3.1.3, for feedback from the nth DOF only, that this algorithm will introduce an observation spillover-type effect on the higher modes of vibration of the structure. In practice, the introduction of an online filter to remove higher mode components of the feedback is likely to be an effective control solution if control efficiency is lost.

4.

Numerical examples

Two buildings are used to illustrate the different algorithm performances. The first is 200 m tall, 34 m wide, square sectioned, and with first, second, and third mode frequencies of 0.18, 0.53 and 0.85 Hz, respectively, and with a total mass of 48 000 t. The building is assumed to be located in open country terrain, experiencing a 10-year return period wind, appropriate to its location. Cross-wind loading is used in this study, as this is the critical direction for this case, eliciting the greatest uncontrolled response. A study of the along-wind response of the same building using some of the algorithms studied here can be found in Mackriell et al. 23. The mean wind speed at the top of the building under these conditions is taken as 33.6 ms -~ 24, and the corresponding vortex shedding frequency is 0.10 Hz, 56% of the first mode value. The second building is 400 m tall, 40 m wide, square sectioned, has a mass of 128 000 t, and has first, second, and third mode frequencies of 0.10, 0.30, and 0.49 H, respectively. The location is assumed to be suburban terrain. A 10-year return period wind is experienced, and the mean wind speed at the top of the building is 33.9 ms -~, which gives rise to a vortex shedding frequency of 0.085 Hz, 85% of the first mode frequency. While these frequencies are not close enough for aerodynamic effects such as negative aerodynamic damping to have any effect 25, the cross-wind response will be influenced by the vortex shedding loading to a much greater extent than for the shorter building. The 200 m and 400 m buildings are modelled as having six and ten DOF, respectively. Building mass is modelled by equal lumped masses, and building stiffness is assumed to be provided by massless shear walls and columns of equal stiffness throughout each building's height. Damping of each of the building's modes of vibration is assumed to be 0.5% of critical for that mode, in line with serviceability criteria. The A T M D has a mass of 0.5% and 1.0% of the total mass for the 200 m and 400 m building, respectively. For each algorithm a range of different values of both ATMD frequency ratiofr, and A T M D percentage of critical damping ~ATMD was used, and the optimum parameters selected were those which minimized the control force standard deviation while keeping the first mode component of the nth DOF at a reference level based on serviceability criteria.

838

Control o f w i n d - l o a d e d tall building using A TMD: L. E. Mackriell e t a l. 0.50

for the 200 m and 400 m building, respectively. They clearly show the peaks which correspond to the vortex shedding frequencies of the buildings.

0.00 -0.50 -1.00 -1.50

v -2.00 -2.50 -3.00 -3.50 -4.00 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Freqmmcy (n), Hertz

Figure I 6th DOF cross-wind force spectrum, 200 m building 4.1. Experimentally obtained wind loading time histories The cross-wind force vector Fw was calculated using a time history of wind load coefficients obtained from wind tunnel tests of rigid pressure-tapped models of the buildings at a 1 : 200 and 1 : 400 scale for the shorter and taller buildings, respectively. Pressures acting on the tributary area of each lumped mass were measured using 24 taps per lumped mass, with 12 taps being equally spaced on two opposite faces in both cases. The pressures acting on each face were averaged using a 1 2 : l manifold, and measured with a Honeywell 160PC1 pressure transducer. The pressure measurement system had been tested at CSIRO Division of Building, Construction and Engineering in Melbourne, and was found to have a frequency response of +10% of the static response up to full-scale frequencies of 1.0 Hz and 0.60 Hz for the shorter and taller buildings, respectively. Time histories from each of the pressure transducers were recorded simultaneously, and combined to calculate a time history of the wind load coefficients for each mass, in the cross-wind direction. The mean wind speeds given in Section 4 were used in the force calculations. The testing was carried out in a 1.8 x 2.4 m boundary layer wind tunnel set up to simulate flow over open country terrain for the 200 m building, and suburban terrain for the 400 m building. For the shorter building, transducer output was sampled and lowpass filtered at full-scale frequencies of 2.3 Hz and 0.92 Hz, respectively. Figures 1 and 2 show the spectral density of the cross-wind force of the top DOF 0.50 0.00 -0.50 -1.00

"--~ -1.5o

4.2. 200 m Building results Time histories of the building's motion were obtained by numerical integration of equation (4) using a trapezoidal integration scheme. The uncontrolled response of the building in the cross-wind direction was calculated, and is presented in Table 1. The control criteria in all cases was to reduce the first mode component of the 6th DOF acceleration standard deviation to a level acceptable for human comfort which, based on the ISO 6897 serviceability criterion 26 adjusted for a 10-year return period, is 5.6 m-g at the first mode frequency. The first mode component was calculated by filtering the time history of the response through a 0.16-0.20 Hz digital bandpass filter. Preliminary studies indicated that a TMD of mass equal to 1% of the building mass with parameters set to those calculated optimum by Den Hartog 27, achieved the control criterion, whereas a TMD of half that mass did not, and these results are also presented in Table 1. The actively controlled response was calculated using an ATMD mass equal to 0.05% of the total building mass, in order to reduce the mass of the device compared to passive control. The two modal control algorithms described by equations (9) and (10), and the simple acceleration feedback algorithm described by equation (17) were all used to calculate an active control force. For each family of modal algorithms, several different gain combinations were used. For notational convenience, these algorithms have been given abbreviated names. AS refers to the simple acceleration feedback algorithm; the A1, DI, and VI algorithms use first mode acceleration displacement, and velocity only; AVI, and AV2 are different combinations of first mode acceleration and velocity feedback; and DVI and DV2 are combinations of first mode displacement and velocity feedback. The gain values, the equations used to calculate the control force, and the optimum values o f f , and ~ATMI)a r e shown in Table 2. Figure 3 compares the standard deviation of the control force required to achieve the reference response for each algorithm. In each case, the gains were adjusted by trial and error until the reference value was achieved. Results were obtained both with and without observation spillover, which was introduced by assuming that the first mode feedback was approximated using equation (14). Figure 4 shows the standard deviation of the ATMD displacement as a function of the control algorithm. For the same degree of control provided, it is clear that the lesser the energy expenditure and ATMD travel, the greater the algorithm efficiency. The introduction of observation spillover does not make a significant difference to the control efficiency, and it

-2.00

Table 1 Uncontrolled and passively controlled cross-wind 6th DOF response standard deviation

-2.50 -3.00

Control type -3.50 -4.00 0

I

~

I

I

I

I

I

I

I

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Freque~y (n), Hertz

Figure 2 10th DOF cross-wind force spectrum, 400 m building

No control 1.0% TMD 0.5% TMD

Model 1 acceleration, rn-g 13.4 4.94 5.95

Total acceleration, m-g 13.8 5.7 6.61

Control o f w i n d - l o a d e d tall b u i l d i n g using A TMD: L. E. Mackriell Table2

839

et al.

Control gains and ATMD parameters used, 200 m building

Algorithm

Equation

ga

g~

g~

g

~ArMD (%)

fr

D1

9

0.0157

0.0

--

--

5.33

0.99

DV1 DV2 Vl AV1 AV2 A1 AS

9 9 9,10 10 10 10 17

0.005 0.010 0.0 ----

1.34 0.755 1.96 1.37 0.771 0.0

--

--

--0.005 0.010 0.015-

------

--

--

--

5.23

3.24 3.87 2.72 3.24 3.87 5.33 5.33

0.95 0.96 0.94 0.95 0.96 0.99 0.99

~"

18[

=

" 14

m

I0

0

4

U

0

l i n e Obs, Spill [ll Ohs. Spill.

DI

DVI

DV2

Vl

AVI

AV2

AI

ii

AS

CoI~rROL ALGORITHM

Figure 3 Control force standard deviation, 200 m building

~"

0.55 0.5 0.45 0.4

I ~o.35 ~1 ~

0.2 0.15 0.! 0.05 0 DI

DVI

DV2

VI

AV1

AV2

AI

AS

CONTROL ALGORITHM

Figure 4

AT MD displacement standard deviation, 200 m build-

ing

causes no instabilities. As can be seen from Table I, the higher mode contribution to the response is slight, so the inclusion of those higher building modes in the feedback has little effect on the control. However, the choice of algorithm profoundly affects the efficiency of the control. As expected from the discussion in Section 1, pure acceleration feedback, using either the AS or A1 algorithms, is the most efficient, and they are closely followed by the pure displacement D1 algorithm. A significant decrease of efficiency is experienced as the velocity component of the feedback is increased in the AV2, AV 1 and finally the V 1 algorithms; as velocity feedback is added, the control requires significantly more force and ATMD displacement because the passive (--maX6) and active ( a ) input forces move out of phase and therefore combine in a less efficient manner. For example, the V1 algorithm required nearly 4.5 times the control force standard deviation as the A1 algorithm, while both the AV2 and DV2 algorithms (which had similar velocity components) required 2.5 times the force of the A I algorithm. The similarity of the AS and A1 results implies that, in this case, there is no loss of efficiency due to the use of the AS algorithm, and so its use may be preferred as no knowledge of the building mode shapes is required to implement it. For a TMD of the same mass as the ATMD, the optimum parameters were found to be ~rUO = 5.18% and fn'MO =

x 10

3

0.9927. These parameters affect the phase of the TMD output force so that it provides the most effective control to the building. The A 1, AS, and D 1 algorithms device output forces which are close to in phase with the TMD output force and so, as expected, Table 2 shows that their optimum ATMD parameters are close to the passive optimum parameters. As the velocity component of the feedback increases, the phase difference between the ATMD and TMD output forces increases, and the ATMD parameters change to compensate for this, as shown in Table 2. With the ATMD parameters at optimum, all the algorithms resulted in a stable first mode response; however, the robustness of the algorithms to inadvertent changes in those parameters is an important question to consider. An investigation into the sensitivity of the control to changes in ATMD frequency ratio was carried out for all algorithms, when no observation spillover was present, and when ATMD damping remained at the optimum value. The variation of the acceleration response (normalized with respect to the response at optimum f,) with changing f, is presented in Figure 5. The behaviour of the AI, AS and DI algorithms when fr is altered is very similar. All three algorithms lose efficiency when fr is changed by as little as 15%, however, instabilities are never introduced, and the response tends to the uncontrolled value when f, is both increased and decreased sufficiently. This behaviour is similar to that of a TMD when the optimum parameters are altered. The algorithms which contain velocity feedback behave very differently. At frequency ratios above optimum, the response increases markedly, and the algorithms which contain the most velocity feedback become unstable at frequency ratios not much above optimum. For example, for the V1 algorithm, response increases by 50%, and then becomes unstable, at frequency ratios only about 6% and 3.75 3.5 3.2.5 -2.75 m~

I~-"AVlandDVlresults

2.5

,.75 ~ 0 Z

~ ; J • .~ • I ~ I' ~, • ; ! ~ "~ [I ."; I:

[-- O'- - Vl Results ! ,it A1, AS, and D1 Results ]~-- O -- AV2 and DV2 Results

'9"

'

.....

"-::~:-t__

~

,~

i:¢~

,°.

1.5 1.25 !

0.75 0

I

I

i

I

I

I

I

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

ATMD FREQUENCY RATIO

Figure5 A l g o r i t h m building

200 m

sensitivity to ATMD parameter changes,

840

Control of wind-loaded tall building using A TMD: L. E. Mackriell et al. 2.00

Table4 C o n t r o l g a i n s a n d A T M D

parameters

used,

4O0m

building 0.00 -1.00

~ATMD

AIgorithm

Equation

g~

D1 DV1 DV2 V1 AV1 AV2 A1 AS

9 9 9 9,10 10 10 10 17

0.0337 0.0 0.0113 1.143 0.0225 0.388 0.0 3.66 1.234 0.467 0.0 -

g~

g.

g

(%)

f~

0.011 0.023 0.035 -

0.0405

6.54 3.32 4.81 2.88 3.25 4.57 6.42 6.65

0.93 0.91 0.92 0.86 0.91 0.92 0.93 0.94

-2.00 -3 . ~ -4.00

-5.00

~

I

I

I

I

I

I

I

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

FREQUENCY(n),HERTZ

Figure 6 6th 200 m b u i l d i n g

DOF acceleration

spectrum,

A1 algorithm,

10% above optimum, respectively. As discussed in Section 3.1.1, velocity feedback directly affects the damping of the first mode, and so if it is applied at the wrong phase, due to inadvertent changes to the ATMD frequency ratio, it can subtract from that damping, leading to instability. At frequency ratios below optimum, the decrease in control is lessened as velocity feedback is added to the control force. In this case, the algorithms containing some velocity feedback are adding damping directly to the first mode, even though the augmented TMD-like component of the control is reduced as the ATMD become severely detuned. Figure 6 shows a typical frequency response spectrum for the 6th DOF of the controlled building, using the A1 algorithm. The spectral distribution of the response is very similar for all algorithms used, with the response reduction concentrated entirely at the critical first mode.

4.3.

400 m building results

A time history of the building's response was obtained by numerical integration, as for the smaller building. The uncontrolled, and passively controlled, response standard deviations are shown in Table 3. The control criterion is again based on ISO 6987, adjusted for a return period of 10 years. The maximum allowable standard deviation of acceleration is 7.75 m-g at the first mode frequency, and 87.26 m-g at the vortex shedding frequency. Responses at the first mode and the vortex shedding frequencies were calculated by digitally bandpass filtering the time history at 0.09-0.12 Hz and 0.07-0.09 Hz, respectively. The passive, TMD, mass used was 1% of the total mass of the building, and it was not sufficient to adequately control the building. It is generally not feasible to increase the mass of a passive damper beyond 1% of the total building mass, so in this case the additional control required can only be provided by active means. The same control algorithms, and notation, used for the

smaller building were used for the taller building. Table 4 shows the gains, ATMD parameters, and equations that were used to obtain the control forces. Figure 7 shows the standard deviation of the control force, both with and without observation spillover, for all algorithms when the first mode standard deviation of the 10th DOF acceleration was reduced to the reference value, and the response at the vortex shedding frequency was kept at, or below, 8.3 m-g. Observation spillover was introduced using equation (15), and as expected given the very small contribution from higher modes as seen in Table 3, observation spillover was not significant. The results show a broadly similar trend to those of the smaller building: as velocity feedback was introduced, generally more control force was required; and acceleration feedback can be replaced by displacement feedback to achieve similar results. However, in this case it is not the pure acceleration A I or AS algorithms which use the least force. The introduction of a small amount of velocity feedback, in the DV2 or AV2 algorithms, results in a slightly lower force than that of the AI algorithm. It is well known that the performance of a TMD deteriorates when the response of the building is not at the frequency for which it is tuned. In this case, the non-first mode response at 0.085 Hz is of some significance, and so the augmented TMD-like control, supplied by the acceleration feedback algorithms was not as effective as when some damping was applied directly to the first mode, in the AV2 and DV2 algorithms. The results change slightly when the standard deviation of the ATMD displacement is examined, shown in Figure 8. In this case, the A1, AS and D1 algorithms do show the best performance, and the trend of the results is the same as for the smaller building. The response at the vortex shedding frequency is shown in Figure 9 for the DVI, AVI and V1 algorithms, the vortex shedding response was a control design constraint, and the ATMD parameters had to be altered from the values which minimised the control force standard deviation in order to keep 150

Table3 U n c o n t r o l l e d and passively c o n t r o l l e d lOth DOF response standard deviation

cross-wind,

I I N o Ob~. Spill • Obs. Spill

125

Vortex

Mode 1 acceleration

shedding acceleration

Total

Control type

(m-g)

(m-g)

(m-g)

No control 1.0% T M D

23.75 9.60

5.66 6.83

24.46 12.23

accel-

eration 0 DI

DVI

DV2

VI

AVI

AV2

AI

AS

CONTROL ALGORITHM

Figure 7 C o n t r o l force standard deviation, 4 0 0 m b u i l d i n g

841

Control o f wind-loaded tall building using ATMD: L. E. Mackriell et al. 5

4~, AI, AS, ~ d DI Results 1 -- ~ - VI R~ults

4.5

U~ t~

Z [-

,"

/" "O"" "AVI and DV! RecruIts / 4

:

[~_'O -- AV2 ~tud DV2 R ~ l ~ j

. . -0 -~

.~ / "

3.5

,e,

e~

3

~t'

2.5! DI

DVI

DV2

VI

AVI

AV2

AI

......

;~

2]

AS

CONTROL ALGORITHM

O Z

Figure 8 ATMD displacement standard deviation, 400 m build-

1.5 I

ing

0.5 0

8.5

0.2

0.4

0.6

0.8

1

1.2

1.6

1.4

ATMD FREQUENCY RATIO

t~

Figure 10 Algorithm sensitivity to ATMD parameter changes, 400 m building

8< ~

7 2

6.5

1

0 ~

6 DI

DVI

DV2

VI

AV1

AV2

AI

%

AS

t. . . . . . Noco.t~]

'!

' ,

o

CONTROL ALGORITHM

Figure 9 10th DOff acceleration at vo~ex shedding frequency,

-2

400 m building

the vortex response at 8.3 m-g. The response was lowest for the A I and AS algorithms, and increased with the amount of velocity feedback present. This result is most likely linked to the amount of damping applied directly to the building's first mode, via the velocity feedback term in equations (9) or (10). The AV2, AVI and V1 algorithms result in first mode damping values that are 114%, 149% and 360% of the A 1 first mode damping values, respectively. As damping increases, the first mode mechanical admittance function broadens, decreasing the response at the first mode, but increasing it at frequencies close to the peak, hence the tendency of the response at 0.085 Hz to increase as damping is added to the first mode at 0.10 Hz. The optimum passive TMD parameters at this mass are: frrMO 0.98 and ~rMo= 7-4%27. From Table 4 it can be seen that the ATMD parameters move away from these values as velocity feedback is introduced, as for the smaller building; however, in this case, all the algorithms result in a frequency ratio less than the passive optimum value. The theoretical optimum TMD parameters are calculated assuming the building was loaded sinusoidally 27, hence implying a sinusoidal response. This assumption is not true in this case, due to the significant response at the vortex shedding frequency. It is likely that it is this forced response which has caused f, to deviate from the trends shown by the smaller building, where vortex shedding was not as significant. The sensitivity to changes in the optimum ATMD frequency ratio while ATMD damping remains constant, shown in Figure I0, follows the same pattern as for the smaller building. The AI, AS and D1 algorithms perform in a similar way to a TMD, their performance decreases, but no instabilities are introduced. Again, the introduction of velocity feedback increases the likelihood of instabilities being introduced due to an increase in frequency ratio, and it increases the control performance at lower frequency ratios. Figure 11 shows the frequency response spectrum of the 10th DOF acceleration with the A1 algorithm used. The spectral distribution is similar in all cases, with the =

-5 0

I

t

I

I

I

I

I

I

I

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

FREQUENCY (n), HERTZ

Figure 11 10th DOF acceleration spectrum, 400 m building response reduction being restricted to the first mode region, however, the increase in response at 0.085 Hz is greater for the algorithms with velocity feedback, and an increase in first mode frequency is seen as acceleration and displacement feedback is increased. Overall, the A I and AS, and to a lesser extent, the D l, algorithms are the most efficient. This is the expected result when the hybrid nature of the ATMD is considered, as discussed in Section 1. While the introduction of a small amount of velocity feedback decreases the control force required for the taller building, it increases the amount of space needed to contain the ATMD stroke length, increases the response at the vortex shedding frequency, and increases the risk of instability if the ATMD parameters are inadvertently altered.

5.

Conclusions

All the algorithms tested adequately controlled the crosswind response of both a 200 m and 400 m tall building, reducing the response to levels acceptable for human comfort using achievable control forces. The use of critical mode control, combined with an ATMD tuned close to the first mode frequency, focused the control energy where it was most needed, and the observation spillover associated with modal control did not adversely affect the control quality for any algorithm. There were significant, and important, differences in the behaviour of the algorithms. Those algorithms which used a feedback which was close to being in phase with the first mode acceleration performed much better than those which did not use such a feedback; and the deterioration increased as the phase difference increased. This was because the

842

Control of wind-loaded tall building using A TMD: L. E. Mackriell et al.

passive component of the force driving the ATMD is proportional to the acceleration of the top floor, which is primarily due to the first mode response in both cases, and the passive and active components of the ATMD force combine in the most efficient manner when they are in phase. For both buildings, two algorithms were found to be the more efficient: a pure first mode acceleration feedback algorithm; and the simple direct feedback acceleration algorithm, suggesting that complete knowledge of the building's mode shapes may not be strictly necessary. The first mode pure displacement algorithm also performed well, but algorithm performance deteriorated as velocity feedback was introduced. The worst algorithm was found to be the pure first mode velocity algorithm, which required the most control force and actuator displacement, as well as being the most prone to dangerous instabilities introduced as a result of inadvertent changes to the ATMD frequency ratio. The response of the taller building was livelier than the smaller building, and active control was needed in order to control it adequately, whereas for the smaller building, active control allowed the damper mass to be reduced. The differences between the buildings' responses were due to the greater influence exerted by the vortex shedding forces on the taller building; for a very tall and slender building it is important to consider the cross-wind response not only at the first mode frequency, but also at the vortex shedding frequency. The variability in the performance of the different control algorithms tested here shows that the choice of control algorithm is of utmost importance in developing an efficient, safe active control scheme. When the control device has both active and passive (hybrid) characteristics, as is the case for an ATMD, the best choice of active control algorithm will be one which works best with the inherent passive characteristics of the device. For an ATMD this choice is an acceleration feedback algorithm. Acknowledgments The authors would like to acknowledge the assistance of a University of Sydney Postgraduate Research Award, a Roderick Bequest Scholarship, and an Australian Wind Engineering Society Postgraduate Scholarship in providing funds to enable this study to be carried out. Thanks are due in particular to Dr John Holmes at the CSIRO Division of Building, Construction and Engineering for his helpful advice with regard to the pressure measurement system. References 1 Yang, J. N., Akbarpour, A. and Ghaemmaghami, P. 'New optimal control algorithms for structural control', J. Engng. Mech., ASCE 1987 113(9), 1369-1386 2 Yang, J. N., Li, Z. and Liu, S. C. 'Instantaneous optimal control with velocity and acceleration feedbacks', J. Probabilistic Engng Mech. 1991 16(3), 204-211 3 Yang, J. N., Li, Z. and Liu, S. C, 'Stable controllers for instantaneous optimal control', J. Engng Mech., ASCE 1992, 118(8), 1612-1630

4

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I1

12 13

14

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17 18

19

20 21

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22 23

24 25 26

27

Kareem, A. 'Wind excited response in higher modes', J. Struct. Engng, ASCE 1981, 102, 701-706 Mackriell, L. E., Kwok, K. C. S. and Samali, B. 'Active control of along-wind and cross-wind acceleration of a tall building', Proc. 91CWE New Delhi, India, 1995, pp 1667-1678 Standards Australia 1170.2-1989 'SAA loading code part 2: wind loads', Standard Australia, Sydney 1989 Kwok K. C. S. and Melbourne, W. H. 'Wind induced lock in excitation of tall structures', J. Struct. Div., ASCE 1981, 107(1), 57 72 ISO 6897 Guidelines for the evaluation of the response of occupants of fixed structures, especially buildings and off-shore structures, to low frequency horizontal motion (0.063-1.0 Hertz), International Standards, 1984 Den Hartog, J. P. Mechanical vibrations (4th edn) McGraw-Hill, New York, 1956