249_ftp

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2003; 32:793–810 (DOI: 10.1002/eqe.249)

Experimental study on multiple tuned mass dampers to reduceseismic responses of a three-storey building structure

Genda Chen1;∗;† and Jingning Wu2

1Department of Civil Engineering; University of Missouri-Rolla; Rolla; MO 65401; U.S.A.2HBE Corporation; 11330 Olive Street Road; St. Louis; MO 63141; U.S.A.

SUMMARY

In this study, several mass dampers were designed and fabricated to suppress the seismic responsesof a 1

4 -scale three-storey building structure. The dynamic properties of the dampers and structure wereidenti�ed from free and forced vibration tests. The building structure with or without the damperswas, respectively, tested on a shake table under the white noise excitation, the scaled 1940 El Centroearthquake and the scaled 1952 Taft earthquake. The dampers were placed on the building �oors usingthe sequential procedure developed by the authors in previous studies. Experimental results indicatedthat the multiple damper system is substantially superior to a single tuned mass damper in mitigatingthe �oor accelerations even though the multiple dampers are sub-optimal in terms of tuning frequency,damping and placement. These results validated the sequential procedure for placement of the mul-tiple dampers. The structure was also analysed numerically based on the shake table excitation andthe identi�ed structure and damper parameters for all test cases. Numerical and experimental resultsare in good agreement, validating the dynamic properties identi�ed. Copyright ? 2003 John Wiley &Sons, Ltd.

KEY WORDS: multiple tuned mass dampers; optimum tuning frequency; optimal placement; shake tabletest; seismic performance

INTRODUCTION

Tuned mass dampers (TMD) have been extensively studied and applied to suppress wind-induced vibrations of building structures since the 1970s [1–4]. Much of the e�orts weredevoted to developing the design procedure and optimizing the TMD parameters for improvedperformance. In most applications, only a single TMD is installed on the top �oor and is tunedto the fundamental frequency of the structure.

∗Correspondence to: Genda Chen, Department of Civil Engineering, University of Missouri-Rolla, Rolla,MO 65401, U.S.A.

†E-mail: [email protected]

Contract=grant sponsor: University of Missouri Research Board, the Intelligent System Centers at the University ofMissouri-Rolla, and National Science Foundation; contract=grant number: CMS9733123.

Received 3 January 2001Revised 10 June 2002

Copyright ? 2003 John Wiley & Sons, Ltd. Accepted 15 August 2002

794 G. CHEN AND J. WU

However, several investigations [5; 6] have indicated that a single TMD often is not ase�ective in reducing seismic responses. There are mainly two reasons. First, earthquake loadsare typically impulsive and reach the maximum values rapidly. A TMD, subjected to a dy-namic load �ltered by the building structure, usually is not set into signi�cant motion in sucha short period. Second, earthquake ground motions include a wide spectrum of frequencycomponents and often induce signi�cant vibration in both the fundamental and higher modesof a tall building structure. A single damper tuned to the fundamental frequency of a struc-ture is unable to suppress the vibration of higher modes. In fact, it was reported that thesingle damper could even amplify the higher-mode responses due to coupling between thefundamental and higher modes [5].Recognizing the above shortcomings of a single TMD, several investigators introduced

multiple tuned mass dampers (MTMD) that are tuned to di�erent modes and placed at variouslocations to enhance the dampers’ seismic performance [6; 7]. Besides the improvement inperformance, such systems often do not require any dedicated space to house the distributedsmall dampers. Therefore, engineers can make full use of the spare space at di�erent �oorsand design the TMD system in a cost-e�ective way. Owing to their light weight, malfunctionof any damper will not cause detrimental e�ects on the structural responses so that the MTMDsystem can be more robust.MTMD systems were also studied by other investigators [8–12]. However, these studies

were mainly focused on the vibration control of a single mode or closely spaced modes of astructural system under a wide-band random input. Under certain circumstances, these systemsare equivalent to a larger TMD [8].Recently the study on MTMD systems for seismic applications was furthered by the au-

thors [13; 14]. In these studies, a MTMD system was divided into several groups, each cor-responding to one mode and consisting of several oscillators distributed on di�erent �oors.An approximate closed-form solution was derived for the optimal damper mass distributionamong the vibration modes and can provide general conclusion on the performance improve-ment by using the MTMD system rather than a single TMD. A sequential procedure has beendeveloped to sub-optimally place the dampers on the structure using acceleration reduction asan optimization objective.In this study, the performance of MTMD systems is compared with that of TMD systems

through shake table tests on a model structure. Several TMD systems were designed andfabricated in the laboratory. Their dynamic properties such as damping and frequency and themodel’s structure were identi�ed using free and forced vibration. Both white-noise processand scaled earthquake records were used in the shake table tests to experimentally verify theseismic e�ectiveness of the MTMD system.

STRUCTURE-MTMD SYSTEM AND TEST FACILITIES

The experimental structure used in this study is a 14 -scale, three-storey steel frame struc-

ture (48′′length× 24′′width× 100′′height) mounted on the shake table at the University ofMissouri-Rolla as shown in Figure 1(a). This structure was originally designed for activestructural control tests with a steel bracing supporting a hydraulic actuator installed on the 1st�oor [15]. For this study, the actuator is disconnected and the bracing has no e�ect on thestructural behavior.

Copyright ? 2003 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2003; 32:793–810

EXPERIMENTAL STUDY ON MULTIPLE TUNED MASS DAMPERS 795

30"

30"

40"

48"

Shaking Table

Damper base

(a) (b)

Figure 1. Three-storey steel frame structure: (a) overview; (b) schematics.

Figure 2. Tuned mass dampers (two in parallel).

Each tuned mass damper consists of a mass block, a set of extension springs and a slidingdual shaft bearing. Either one or two damper(s) can be installed on a steel damper base shownin Figure 2 for the two-damper con�guration. Supported on the bearing, the mass block isconnected to the wall of the damper base through the extension springs and it can move alongthe dual shafts only. The natural frequency of the damper can be adjusted by using di�erenttypes of springs. Since no damping element was intentionally added to the damper system,the friction action between the bearing and the shafts constitutes the main part of the system



damping. A total of three damper bases were fabricated, one on each �oor as schematicallyshown in Figure 1(b).The MTS dynamic testing facilities include the shake table, shown in Figure 1, the MTS436

control unit and the MTS406 controller. The 4′ × 7′ shake table can generate vibration in thelongitudinal direction with a maximum payload of 20 tons, an e�ective frequency range of0.01–10Hz and a maximum stroke of ±1:0′′. The MTS436 control unit was used to generatedisplacement signals of harmonic waveforms. Based on the displacement signals received, theMTS406 controller commands the shake table for the required motion at the base of the testedstructure. Random displacement signals such as white noise or earthquake ground motions wereexternally generated with a HP1415 workstation and directly sent to the MTS406 controller.The measurement system includes a HP1415 workstation, 4 accelerometers and 3 LVDTs.

The HP1415 can provide simulated earthquake ground motion signals and simultaneouslyrecord the measured responses of the structure up to 64 channels. Accelerometers are deployedat each �oor and at the structural base (shake table). Three LVDTs are used to measure thedisplacement of each damper relative to its supporting �oor.

PARAMETER IDENTIFICATION OF STRUCTURE AND DAMPERS

Structural parameters

The structure shown in Figure 1(b) is regarded as the uncontrolled structure and the dynamicparameters in the longitudinal direction are of interest to this study. The lumped masses of the1st, 2nd and top �oors are, respectively, estimated as 445 kg (981 lbs), 394 kg (868 lbs) and388 kg (855 lbs). Each lumped mass accounts for the �oor members, columns and additionalweights on the �oor.Forced vibration tests were conducted to identify the dynamic parameters of all three modes.

These tests were carried out in two steps. First of all, several swept-sine tests were performedto approximately identify the natural frequencies of the structure. A series of harmonic testswere then conducted with the excitation frequency varying around the natural frequencies ofthe tested structure. The transfer function of �oor accelerations can therefore be constructed ex-perimentally in di�erent frequency ranges [16]. By expressing the harmonic table acceleration,�xg(t), and the absolute acceleration, �xk(t), of the kth �oor as

�xg(t)= �Xg(!)e j!t and �xk(t)= �Xk(!)e j!t (1)

the acceleration transfer function H �Xk (!) can be de�ned as

�Xk(!)=H �Xk (!) �Xg(!) (k=1; 2; 3) (2)

in which j=√−1 is a complex unit; ! and t are, respectively, the excitation frequency

and the time instance; �Xg(!) and �Xk(!) denote the Fourier transform of the table and �ooraccelerations, respectively.On the other hand, the acceleration transfer functions of a three-storey structure with normal

modes can be theoretically derived as

�X (!)=3∑i=1

1 + 2�i�ij1− �2i + 2�i�ij

�i�i �Xg(!) (3)



Table I. Modal parameters identi�ed from forced vibration tests.

Mode no. i 1 2 3Frequency fi (Hz) 2.743 9.45 18.84

Normalized mode shape �i

0:01860:02990:0356

0:03520:0123−0:0316

−0:02680:0385−0:0158

Damping ratio �i (%) 0.48 1.15 1.45Participation factor �i 33.904 8.293 −2:863Orthogonality with respectto mass matrix �TMS�

1:0 0:001 0:0130:001 1:0 −0:0390:013 −0:039 1:0

where �X (!)= { �X1(!) �X2(!) �X3(!)}T; �i=!=!i; !i, �i, �i, and �i are, respectively, thecircular frequency, damping ratio, modal vector and modal participation factor of the ithmode. For lightly damped structures of sparsely spaced frequencies, the contribution of non-resonant modes to the total responses at resonance (!≈!i) is negligible and Equation (3)can be simpli�ed into

�X (!)≈ 1 + 2�i�ij1− �2i + 2�i�ij

�i�i �Xg(!) (4)

Both the natural frequency and modal damping ratio of the structure can then be determinedfrom the experimental transfer functions, |H �Xk (!)|, using the half-power method [17]. Themode shapes can also be determined from the transfer functions with the phase informationretrieved from the corresponding time histories. Since the structural damping is very small,the �oor accelerations are either in phase or 180◦ out of phase at the resonant frequencies.The modal parameters identi�ed from the forced vibration tests are presented in Table I. Itcan be seen that the identi�ed mode shapes satisfy the orthogonality requirement in terms ofthe lumped mass matrix.With the complete modal parameters, the sti�ness and damping matrix of the structure can

be calculated from the following equations:

KS = (�−1)T��−1 and CS = (�−1)T��−1 (5)

where � is the modal matrix; � and � are two diagonal matrices whose ith diagonal elementsare, respectively, !2i and 2�i!i. The damping and sti�ness matrices identi�ed, together withthe estimated mass matrix, are summarized in Table II.

Damper parameters

Six types of extension springs of di�erent constants were used during the shake table tests. Foreach type, three randomly selected samples were tested to determine their spring constants.To eliminate the slack e�ect on the springs, a pre-load ranging from 6.0 to 23.0 poundswas applied to each spring. With the speci�ed pre-load, all the tested springs were shownperfectly linear up to 1.0–1:5 in [16]. Both the manufacturer’s nominal and the measuredspring constants are listed in Table III. Their di�erences are within ±10% tolerance.



Table II. Structural property matrices.

Mass Ms (kg) Damping Cs (×102 N · sec=m) Sti�ness Ks (×106 N=m)445 0 00 394 00 0 388

7:770 −4:683 0:257−4:683 8:594 −4:2240:257 −4:224 4:057

2:669 −2:118 0:452−2:118 3:397 −1:6450:452 −1:645 1:260

Table III. Springs constants.

McMaster- Outside dia. Length Spring constant (lbs.=in)Carr no. (in) (in)

Nominal Measured

1 2 3

9654K302 23=32 4 30.78 30.74 29.21 31.509654K157 27=32 4 18.34 20.09 19.99 20.069654K274 5=8 4 70.00 66.89 65.24 67.169628K46 13=16 4 42.80 42.08 41.06 42.4294135K34 27=32 4 15.95 15.32 15.91 15.639654K324 9=16 4-1=8 14.21 14.23 13.80 13.94

It was planned in this study that only the �rst two modes of the structure would be con-trolled. Therefore, two types of dampers were fabricated, one tuned into the fundamentalfrequency and the other into the natural frequency of the second mode. They are referredto as the 1st- and 2nd-mode TMD, respectively. The weight of each damper consists of asliding bearing, steel plate(s), and accessories such as threaded rods and nuts as shown inFigure 2. Two plates and four springs were used in the 1st-mode TMD, and one plate andeight springs were used for the 2nd-mode TMD. For each damper, the change in frequencyis implemented through di�erent combinations of springs. Based on the estimated masses andthe nominal spring constants given in Table III, the natural frequencies of the dampers andtheir corresponding ratios between these frequencies and the 1st or 2nd natural frequency ofthe structure are listed in Table IV with various combinations of selected springs.To identify the damper parameters, one damper base was placed on the level ground and a

damper was assembled with the springs preloaded so that the initial extension is about 50%of the elastic range. For each combination of springs in Table IV, three free vibration testswere conducted to ensure the repeatability of test data. Shown in Figure 3 is an exampleof the acceleration attenuation curve of the 1st-mode damper. It is observed that the peakacceleration decreases almost linearly at the beginning and exponentially towards the end ofthe vibration. When the TMD undergoes signi�cant movement, the system damping mainlycomes from the friction between the dual shafts and the sliding bearing. Accordingly, the freevibration attenuates linearly [17]. As the vibration damps out, the friction e�ect from otherjoints as well as the spring material also contributes to the total damping. As a result, thesystem motion further attenuates exponentially and can be modeled with the viscous dampingmechanism. To simplify the analysis, the viscous damping mechanism is assumed for theTMD system regardless of the level of vibration. Obviously, the damping ratio will dependon the response amplitude. In general, the larger response amplitude used for calculation



Table IV. Nominal frequencies of TMD with di�erent combinations of springs.

1st-mode TMD 2nd-mode TMDm=39:3 kg (86:5 lbs:) m=20:9 kg (46:0 lbs:)

Springs Frequency �∗1 Springs Frequency �∗2!TMD (Hz) !TMD (Hz)

4× 9654K324 2.53 0.92 4× 9654K274 + 4× 9654K157 8.66 0.924× 94135K34 2.68 0.98 4× 9654K274 + 4× 9654K302 9.25 0.984× 9654K157 2.87 1.05 4× 9654K274 + 4× 9628K46 9.79 1.04

∗�i: tuning frequency ratio=!TMD=!i .

-0.4

-0.2

0

0.2

0.4

0 2 4 6 8 10

Time (sec)

Acc

eler

atio

n (g

)

Figure 3. Free vibration of the 1st-mode TMD (Springs: 4× 94135K34).

Table V. TMD parameters identi�ed from free vibration tests.

1st-mode TMD 2nd-mode TMDm=39:3 kg (86:5 lbs:) m=20:9 kg (46:0 lbs:)

Springs Frequency �∗1 � Springs Frequency �∗2 �!TMD (Hz) (%) !TMD (Hz) (%)

4× 9654K324 2.55 0.93 1.5 4× 9654K274 + 4× 9654K157 8.3 0.88 1.74× 94135K34 2.72 0.99 1.4 4× 9654K274 + 4× 9654K302 8.8 0.93 1.64× 9654K157 3.0 1.09 1.1 4× 9654K274 + 4× 9628K46 9.5 1.01 1.6

∗�i: tuning frequency ratio=!TMD=!i .

results in the smaller damping ratio. To be conservative, only the �rst ten cycles were usedfor the prediction of damping ratios. The identi�ed damping ratios are presented in Table Valong with the identi�ed frequencies. The di�erence between the measured frequencies andtheir nominal values in Table IV is within 5%.

THEORETICAL PREDICTION ON OPTIMAL DAMPER LOCATION

Three sliding bearings (three dampers) were used for the shake table tests. Since each damperbase can accommodate one or two bearing(s), three dampers can be installed either on any



Total damper mass: 60kg



0

20

40

60

1 2 3

Floor

1 2 3

Floor

1 2 3

Floor

Mas

s (k

g)M

ass

(kg)

Mas

s (k

g)

1st mode

2nd mode

(a)

0

20

40

60

80

(b)

0

40

80

120

(c)

Figure 4. Sub-optimal damper placement (!g=!2): (a) total mass=60 kg;(b) total mass=80 kg; (c) total mass=120 kg.

Table VI. Optimum dampers’ parameters with a total mass of 60, 80, and 120 kg.

Mode no. Optimum parameters

Damping ratio (%) Frequency ratio

60 kg 80 kg 120 kg 60 kg 80 kg 120 kg

1 13 16 18 0.97 0.94 0.912 8 8 11 0.97 0.96 0.95

two �oors or all three �oors at the same time. The minimum and maximum total mass ofany three dampers de�ned in Table IV are about 60 and 120 kg, respectively.The sequential procedure developed by the authors [14] was used to determine the sub-

optimal damper locations when the dominant frequency (!g) of the excitation is near the2nd frequency of the structure. The seismic excitation is represented by the Kanai–Tajimispectrum [18]. Considering a 2-mode MTMD system with each oscillator weighing 20kg, thesub-optimal damper placement is presented in Figure 4 and, the optimum damping ratio andfrequency are given in Table VI for three total mass levels: 60, 80 and 120 kg. For example,



0.2

0.3

0.4

0.5

0.6

1 2 3Floor

Nor

mal

ized

RM

S ac

cele

ratio

n

MTMD: 60kg MTMD: 80kg MTMD: 120kg

TMD: 60kg TMD: 80kg TMD: 120kg

Figure 5. Floor acceleration of the controlled structure (!g=!2).

placing two 2nd-mode dampers on the 1st �oor and two 1st-mode dampers on the top �oorresults in the maximum square-root-of-the-sum-of-the-squared reduction in �oor accelerationswhen a total damper mass of 120kg is considered. The root-mean-square (RMS) accelerationresponses of the controlled structure normalized with those of the uncontrolled structure areplotted in Figure 5. The normalized RMS accelerations of the structure controlled with asingle TMD at the top �oor are also presented in the �gure. As one can see, the optimal2-mode MTMD is superior to the single TMD in mitigating the overall �oor accelerations.The MTMD can reduce the 1st �oor RMS acceleration response by an additional 20%. It isnoted that the 2nd �oor acceleration only decreases slightly from TMD to MTMD control.This is because the second �oor is near the node of the second mode. Installing the 2nd-modedampers on the structure would not a�ect the second �oor acceleration signi�cantly.The above performance is achieved with a MTMD system of optimum frequency, damping

and mass distribution. Since the optimum damping given in Table VI is considerably higherthan any TMD described in Table V can provide, this experimental study is mainly focusedon damper placement and frequency tuning. At the 60 kg level, the optimal MTMD systemwas implemented with one 1st-mode TMD (39:3kg) at the top �oor and one 2nd-mode TMD(20:9 kg) at the 1st �oor.

EARTHQUAKE EXCITATION AND SHAKE TABLE TEST PROCEDURE

Shake table output

Three types of simulated earthquake records were used in this study. They include the whitenoise acceleration process (WA), the S00E Component of the 1940 El Centro earthquake(EA), and the S69E Component of the 1952 Taft earthquake (TA). The time scales of the ElCentro and Taft earthquakes are, respectively, compressed to 1

2 and34 so that their dominant

frequencies are around the fundamental frequency of the structure. To avoid structural damage,the magnitude of each record is also scaled down to the level corresponding to a maximum



-0.06-0.04-0.02

00.020.04

0 5 10 15 20Time (sec)

Time (sec)

Time (sec)

Acc

eler

atio

n (g

)A

ccel

erat

ion

(g)

Acc

eler

atio

n (g

)

(a)

-0.06-0.03

00.030.06

0 5 10 15 20(b)

-0.06

-0.03

0

0.03

0.06

0 5 10 15 20(c)

Figure 6. Measured table accelerations: (a) white noise input (WA); (b) scaled ElCentro earthquake (EA); (c) scaled Taft earthquake (TA).

voltage=displacement signal of 4 volts generated with the HP1415 workstation. Therefore,there are a total of three loading cases. The absolute accelerations measured at the shaketable are presented in Figure 6.The MTS shake table is a displacement-controlled facility. It is composed of several me-

chanical and electrical components. Each component has its own dynamic characteristics.Therefore, the shake table as a whole can �lter out the high-frequency components in theinput voltage=displacement signals. It was observed that, without compensation, the expectedshake table accelerations obtained by di�erentiating the input displacement signals could bequite di�erent than those measured at the shake table [16]. However, the di�erence amongthe repeatedly measured table accelerations is insigni�cant. To accurately predict the seismicresponses of the building structure controlled with TMD or MTMD, the measured table ac-celeration time history will be used as the input in numerical simulation for each loadingcase.

Test procedure

The 14 -scaled building structure was tested on the shake table with or without the presence of

TMD=MTMD in three steps. They are (i) to test the building structure (uncontrolled) under thethree excitations, and measure its �oor accelerations; (ii) to install on the structure one or moresingle-mode dampers assembled with various extension springs and test the controlled structureto determine the optimal tuning frequency ratio; and (iii) to replace the single-mode dampers



Table VII. Damper placement plan.

Damper Placement Location Total masstype case no. (kg)

1st Fl. 2nd Fl. 3rd Fl.

1st mode 2nd mode 1st mode 2nd mode 1st mode 2nd mode

1st-mode 1 X 39.32 2X 78.63 X X X 117.9

2nd-mode 4 Y 20.95 2Y 41.86 Y Y 41.87 Y Y Y 62.7

2-modes 8 X Y 60.29 Y X Y 81.110 Y X 60.211 2Y X 81.112 Y 2X 99.5

X =one 1st-mode damper (39:3 kg),Y =one 2nd-mode damper (20:9 kg), 2X=2Y = two dampers.

with the optimally located two-mode MTMD and repeat the test in Step (ii). To ensure therepeatability of test data, three identical runs were conducted for each input and step.Twelve damper placement cases were considered in a series of tests. They are summa-

rized in Table VII, in which X and Y represent one 1st-mode damper and one 2nd-modedamper, respectively, and 2X or 2Y means two dampers. The total mass of all dampers foreach placement is listed in the last column of Table VII. Cases 1, 2, 4 and 5 represent theconventional single-mass dampers tuned into the fundamental and second frequencies of thestructure, respectively. Case 10 is designed to simulate the optimally placed MTMD systemdescribed in Figure 4 and Table VI.

TEST RESULTS AND NUMERICAL SIMULATIONS

The seismic performance of TMD=MTMD is investigated mainly based on the measuredstructural responses in this study. Numerical simulations were carried out to indirectly verifythe accuracy of the identi�ed parameters of the structure and dampers.

Uncontrolled structure

The building structure as shown in Figure 1 was subjected to the three simulated ground mo-tions. The absolute accelerations at the table and three �oors were measured. Their peakvalues for each test and the corresponding average of all three tests are summarized inTable VIII. Each test result is generally within 10% di�erence from the average acceler-ation. Numerical analysis was also conducted to calculate the structural response with themeasured table accelerations as seismic inputs. The measured and the numerically simulatedacceleration time histories of the structure under the scaled El Centro earthquake ground



Table VIII. Peak �oor accelerations of the uncontrolled structure.

Input Test Location Input Test Locationcases cases

1st Fl. 2nd Fl. 3rd Fl. 1st Fl. 2nd Fl. 3rd Fl.

Peak acceleration (g) Peak acceleration (g)

1 0.0567 0.0628 0.0776 1 0.0960 0.1201 0.1457WA-1 2 0.0487 0.0544 0.0645 WA-2 2 0.1369 0.1342 0.1841

3 0.0497 0.0599 0.0700 3 0.1057 0.1198 0.1507Ave. 0.0517 0.0590 0.0707 Ave. 0.1129 0.1247 0.16021 0.0680 0.0848 0.0920 1 0.1427 0.1992 0.2109

EA-1 2 0.0745 0.0974 0.0943 EA-2 2 0.1370 0.1905 0.20513 0.0696 0.0931 0.0965 3 0.1251 0.1888 0.2024Ave. 0.0707 0.0918 0.0943 Ave. 0.1349 0.1928 0.20611 0.0478 0.0705 0.0778 1 0.0962 0.1230 0.1494

TA-1 2 0.0525 0.0681 0.0790 TA-2 2 0.0806 0.1182 0.13633 0.0551 0.0717 0.0786 3 0.0900 0.1178 0.1380Ave. 0.0518 0.0701 0.0785 Ave. 0.0889 0.1197 0.1412

-0.4

-0.2

0

0.2

0.4

0 5 10 15 20Time (sec)

Acc

eler

atio

n (g

)A

ccel

erat

ion

(g)

Measured Simulated

(a)

-0.2

-0.1

0

0.1

0.2

0 5 10 15 20

Time (sec)

Measured Simulated

(b)

Figure 7. Responses of the uncontrolled structure under the EA excitation:(a) third �oor (top); (b) �rst �oor.

motion are presented in Figure 7. It is clearly shown that both results are in excellent agree-ment, indicating high accuracy in the identi�cation of structural parameters.

1st-Mode TMD e�ect

To �nd the optimal tuning frequency of a 1st-mode TMD, the building structure withdamper(s) installed according to the �rst three cases in Table VII was tested. The nominal



Table IX. Average acceleration reduction at 3rd �oor: Placement Cases 1–3.

Input Frequency ratio �1 =!TMD=!1

0.92 0.98 1.05 0.92 0.98 1.05 0.92 0.98 1.05

Case 1 Case 2 Case 3

Acceleration reduction (%)

WA 34.2 35.7 37.2 54.7 27.8 48.1 51.1 13.7 37.6EA −11:5 −20:5 17.6 −16:1 −1:5 4.3 −24:5 −9:2 −4:4TA −45:8 −7:2 42.0 −21:5 17.3 30.0 −14:0 9.1 35.8

damper frequency ratios (�1 =!TMD=!1; !TMD is the frequency of the mass damper) for eachcase are, respectively, chosen to be 0.92, 0.98 and 1.05. The average reduction in peak ac-celeration at the top �oor is summarized in Table IX for three tuning frequency ratios.Under the white noise excitations, all three damper con�gurations exhibit excellent perfor-

mance in suppressing �oor acceleration. For example, the top �oor acceleration can generallybe reduced by 35–55% when �1 = 0:92. The optimum frequency ratio �1 to minimize theacceleration of the fundamental mode can be determined theoretically using the formula de-rived by Warburton [4] based on the modal mass ratio. For the three damper con�gurations:Cases 1–3, it is, respectively, estimated to be 0.94, 0.92 and 0.89. The experimental resultspresented in Table IX indicates that Case 2 with �1 = 0:92 is most e�ective, thus validatingthe theoretical expectation under the white noise excitation. It is also observed from the ex-perimental results that Case 2 can reduce the top �oor acceleration more than Case 1 due tothe increase in damper size (mass). However, Case 3 leads to a larger acceleration than Case2 even though its total mass is 50% larger. This is mainly because the dampers in Case 3are not optimally placed on the structure.For all three con�gurations, dampers with �1 = 1:05 are signi�cantly more e�ective when

the building structure is subjected to the Taft earthquake than subjected to the El Centroearthquake. The maximum acceleration under the Taft earthquake can be reduced by 30–42%.The test results also indicate that a damper of �1 = 1:05 rather than the theoretical optimumfrequency ratio (�1≈ 0:92) leads to the minimal acceleration at the top of the building. Thediscrepancy in optimum frequency of the damper and the signi�cant di�erence in performanceare mainly attributable to the impulsive nature of the earthquake excitation. Unlike the whitenoise excitation, the EA excitation reaches its maximum within a short period and has ashort duration of strong motion as illustrated in Figure 6. The rapid increase in magnitudeof the excitation makes the dampers unable to respond to the structural response and thusthe damper performance degrades substantially. The short duration e�ect on the structuralresponse is equivalent to an additional structural damping. As a result of increased damping,a higher frequency of the damper is required to minimize the acceleration responses [19].Compared to the El Centro earthquake, the peak acceleration of the Taft earthquake increasesmuch slowly and therefore, the damper’s performance is improved under this loading.Table IX also implies that a damper system could even amplify the �oor acceleration up to

40% if improperly tuned and placed on a building in earthquake applications. This result maybe due in part to the non-optimum damping property of the dampers. For a classical shockabsorber used to control the harmonic responses of an undamped structure [20], the optimumfrequency of the absorber is determined to make two �xed points of the transfer function



Table X. Average acceleration reduction at 1st �oor: Placement Cases 4–7.

Input Frequency ratio �2 =!TMD=!2

0.92 0.98 0.92 0.98 0.92 0.98 0.92 0.98

Case 4 Case 5 Case 6 Case 7


WA 27.2 21.1 34.3 33.0 40.6 38.6 48.0 46.6EA 0.4 0.7 8.3 9.9 3.2 3.3 12.6 9.9TA 18.2 22.0 31.6 33.1 26.8 29.2 32.0 31.2

equal in height. The optimum damping of the absorber is chosen to minimize the two peaksof the function. Controlled with the optimal absorber, the transfer function of the structurevaries with the excitation frequency very slowly around the peaks and no ampli�cation willbe observed. However, the TMD system used in this study is lightly damped. The transferfunction is expected to reveal two sharp peaks. Consequently, the structural response may beampli�ed when the dominant frequency component of the excitation corresponds to the peaks.In the following tests on MTMD systems, the frequency ratio for those dampers tuned to thefundamental mode of the structure is selected as 0.92 under the white noise excitations and1.05 under the El Centro and Taft earthquakes.

2nd-Mode TMD e�ect

Four damper con�gurations were considered as de�ned in Table VII, Cases 4–7. The per-formance of the TMD systems under various excitations is compared in Table X at twofrequencies ratios, �2(=!TMD=!2)=0:92 and 0.98. Since the 1st �oor of the building struc-ture corresponds to the largest contribution of the 2nd vibration mode, Table X only givesthe reduction in maximum acceleration of the 1st �oor.Unlike the 1st-mode TMD, dampers nearly tuned to the second natural frequency of the

structure can suppress the 1st �oor acceleration for every damper con�guration considered.This result indicates the e�ectiveness of controlling higher vibration modes with mass dampersand the necessity to introduce multiple dampers for reduction of earthquake-induced responses.The optimum frequency ratios can be estimated theoretically using the modal mass ratio of

the 2nd mode [4]. For the four damper con�gurations, the optimum frequency ratios are 0.97for Case 4 and 0.95 for Cases 5 to 7, respectively. The test results listed in Table X show thatthe two tuning frequency ratios lead to almost the same response of the controlled structure.There are two possible reasons for this. First, the fundamental mode constitutes a large, usuallythe largest, portion of the structural response. The structural responses are less sensitive tothe change in dynamic characteristics of the 2nd mode than those of the 1st mode. Secondly,the seismic excitation used in the tests has a dominant frequency equal to the fundamentalfrequency of the structure, resulting in more uniform energy distribution around the secondnatural frequency of the structure. Therefore, the sensitivity of �oor acceleration to the changein frequency ratio is lower for the 2nd-mode TMD. In the following tests on MTMD systems,�2 = 0:98 is chosen as the frequency ratio for dampers tuned into the 2nd vibration mode.It can be clearly observed from Table X that all dampers perform signi�cantly better when

the building structure is subjected to the white noise excitation or the scaled Taft earthquake



Table XI. Average acceleration reduction at 1st and 3rd �oors: Placement Cases 8–12.

Input Damper placement

Case 8 Case 9 Case 10 Case 11 Case 12

1st Fl. 3rd Fl. 1st Fl. 3rd Fl. 1st Fl. 3rd Fl. 1st Fl. 3rd Fl. 1st Fl. 3rd Fl.


WA 52.2 50.9 58.9 56.9 61.8 55.0 58.8 52.7 53.6 42.5EA 25.9 16.5 34.7 24.2 22.6 17.3 27.3 20.3 10.3 5.4TA 47.8 48.3 45.3 46.1 45.4 45.5 45.5 45.7 37.7 28.1

for the same reasons as discussed for the 1st-mode dampers. The 1st-�oor acceleration can bereduced by 20–47% under these loadings. Table X also implies that the damper con�gurationin Case 5 (equivalent to a single TMD) generally results in smaller acceleration than inCase 4 due to the increase in damper mass and the uniformly-placed dampers in Case 7perform slightly better than in Case 6 for the same reason.

Two-mode MTMD e�ect

To illustrate the e�ect of damper location on control performance, �ve damper con�gura-tions were considered in this investigation: Cases 8–12. Case 10 represents the theoreticallypredicted optimal placement. The reductions in maximum accelerations at the 1st and 3rd�oors are shown in Table XI. It is observed from the table that all damper con�gurationsare signi�cantly more e�ective under the white noise excitation and the scaled Taft earth-quake. The accelerations at the 1st and 3rd �oors can be reduced by 45–60% except forCase 12. Under the scaled El Centro earthquake, however, the maximum reduction in �ooraccelerations becomes less than 35% due to the impulsive nature of the earthquake input.For various loadings, all damper con�gurations except Case 12 lead to comparable reductionsin maximum acceleration. Since the dampers in Cases 8 and 10 (60:2 kg) are substantiallylighter than the dampers in Cases 9 and 11 (81:1kg), Cases 8 and 10 are more e�ective. Bothdamper con�gurations signify the simultaneous control of the �rst two vibration modes. Thecomparable e�cacy of these damper con�gurations is due to the fact that the values of thesecond vibration mode of the structure are close at the �rst and top �oors as listed in Table I.Even though Case 12 has dampers that are heavier than for other cases, the reduction in �ooraccelerations is smaller due to its non-optimal placement.The measured and the numerically simulated response time histories of the structure con-

trolled with Case 10 are compared in Figure 8. In general, they are in good agreement. Incomparison with Figure 7, the �rst peak acceleration of the uncontrolled structure is sup-pressed at the �rst and top �oors. This result indicates the e�ectiveness of the theoreticallyoptimal damper con�guration in reducing the �rst peak.The advantage of using multiple dampers can be seen from the comparison of the seismic

e�ectiveness of various con�gurations under the same number of dampers. The performanceof the damper con�gurations of Cases 2, 5, and 10 are presented in Table XII. As indicatedin the table, Case 10 can mitigate all �oor accelerations signi�cantly more than Case 5. Italso performs better than Case 2 even though the former is lighter. Another comparison to



-0.4

-0.2

0

0.2

0.4

0 5 10 15 20Time (sec)

Time (sec)

Acc

eler

atio

n (g

)A

ccel

erat

ion

(g)

Measured Simulated

Measured Simulated

(a)

-0.2

-0.1

0

0.1

0.2

0 5 10 15 20(b)

Figure 8. Responses of the controlled structure with optimal damper con�guration Case 10 under theEA excitation: (a) top �oor; (b) �rst �oor.

Table XII. Comparison of average acceleration reductions: Placement Case 2, 5 and 10.


Case 2 Case 5 Case 10 Case 2 Case 5 Case 10 Case 2 Case 5 Case 10

Acceleration reduction Acceleration reduction Acceleration reductionat 1st Fl. (%) at 2nd Fl. (%) at 3rd Fl. (%)

WA 47.1 33.0 61.8 51.6 10.5 49.7 54.7 17.5 55.0EA 10.9 9.9 22.6 17.0 14.1 31.2 4.3 11.9 17.3TA 32.3 33.1 45.4 30.0 26.5 43.9 30.0 27.0 45.5

Table XIII. Comparison of average acceleration reductions: Placement Case 3, 7, 9 and 11.


Case 3 Case 7 Case 9 Case 11 Case 3 Case 7 Case 9 Case 11

Acceleration reduction at 1st Fl. (%) Acceleration reduction at 3rd Fl. (%)

WA 46.3 46.6 58.9 58.8 51.1 32.7 56.9 52.7EA 7.3 9.9 34.7 27.3 −4:4 11.7 24.2 20.3TA 32.6 31.2 45.3 45.5 35.8 23.6 46.1 45.7

show the superiority of the multiple dampers is made in Table XIII among Cases 3, 7, 9,and 11. Both cases of 9 and 11 can reduce the �rst �oor acceleration by 10–15% more andthe top �oor acceleration by 5–10% more than Case 3 even though their masses are 30%



smaller. They can even suppress the top �oor acceleration up to 24% more than for Case 7.Therefore, it can be concluded that MTMD is superior to the conventional TMD strategy.The e�ectiveness of multiple dampers can also be illustrated by the performance comparison

of various damper con�gurations of the same mass. The maximum accelerations at the 1stand 3rd �oors are, respectively, compared among the con�guration cases of 2, 9 and 11. Itis observed from Tables XI and XII that Cases 9 and 11 can reduce the 1st and 3rd �ooraccelerations by 13–22% under the EA and TA earthquakes and up to 12% under the WAexcitation more than Case 2 can. Although the total mass of Cases 9 and 11 (81:1 kg each)is slightly larger than that of Case 2 (78:6 kg), the performance of the dampers is improvedwith the two-mode MTMD system. Similarly, the maximum accelerations at the 1st and 3rd�oors of the structure controlled with Cases 7, 8 and 10 are, respectively, compared betweenTable XI and XIII. Cases 8 and 10 can reduce the 1st and 3rd �oor accelerations by 5–23%more than Case 7 even though the former is 2:5 kg lighter than the latter. This result clearlyindicates that the seismic performance of dampers can be enhanced with application of thetwo-mode MTMD system.

CONCLUSIONS

The seismic e�ectiveness of a two-mode MTMD system is studied on a 14 -scale, 3-storey

steel frame structure through shake table tests. With the structural and damper parametersidenti�ed, the sequential procedure proposed in our previous study is applied to design theMTMD system for control of the model structure. Several tuned mass dampers were fabricatedand installed on the structure. The control performance of various dampers of di�erent tuningfrequency and placement is investigated under three simulated ground motions. Based on theexperimental study, the following conclusions can be drawn:

1. Using the identi�ed structural and damper parameters and the measured table accelera-tion, numerical simulations can accurately predict the responses of the uncontrolled andTMD=MTMD-controlled structures.

2. Under the white-noise excitation, the TMD=MTMD systems can e�ectively reduce thestructural responses up to 62%. The theoretical optimum TMD parameters give rise tothe best performance of the damper systems.

3. When the structure is subjected to real earthquake excitations, the optimum TMD tuningfrequency ratio (�1 = 0:92) derived from the case of stationary white-noise excitationsmay not lead to the minimal structural response. The structural response can even beampli�ed with the installation of such a single-mode damper. Experimental results indi-cate that the optimum tuning frequency ought to be �1 = 1:05 under the scaled El Centroand Taft earthquakes.

4. The performance of a 2nd-mode TMD is less sensitive to the change in tuning fre-quency ratio than a 1st-mode TMD. The 2nd-mode TMD can e�ectively mitigate the�oor acceleration of structures, especially at lower �oors.

5. Although the MTMD system used in the test is sub-optimal in terms of frequency,damping and mass, it is signi�cantly superior to a conventional single-mode TMD inacceleration control under di�erent excitations. Even when the dominant frequency ofthe excitation coincides with the fundamental frequency of the structure, the structuralresponses can be further reduced by the MTMD system.



6. The sequential procedure proposed in the previous study is validated to render the sub-optimal design of MTMD systems. It is straightforward and easy for engineers to followin practical applications. The optimally placed MTMD system has better performancethan other non-optimum systems.

ACKNOWLEDGEMENTS

This study was supported in part by the University of Missouri Research Board, the Intelligent Sys-tem Centers at the University of Missouri-Rolla, and National Science Foundation under Grant No.CMS9733123 with Drs Shih-Chi Liu and Peter Chang as Program Directors. These sponsorships aregreatly appreciated.

REFERENCES

1. Wirsching PH, Campbell GC. Minimal structural response under random excitation using the vibration absorber.Earthquake Engineering and Structural Dynamics 1974; 2:303–312.

2. McNamara RJ. Tuned mass dampers for buildings. Journal of the Structural Division (ASCE) 1977; 103(9):1785–1798.

3. Luft RW. Optimal tuned mass dampers for buildings. Journal of the Structural Division (ASCE) 1979;105(12):2766–2772.

4. Warburton GB. Optimal absorber parameters for simple systems. Earthquake Engineering and StructuralDynamics 1982; 8:197–217.

5. Chowdhury AH, Iwuchukwu MD, Garske JJ. The past and future of seismic e�ectiveness of tuned mass dampers.In Leipholz HHE (ed.), Proceedings of the 2nd International Symposium on Structural Control. MartinusNijho� Publishers: Dordrecht, 1987; 105–127.

6. Clark AJ. Multiple passive tuned mass dampers for reducing earthquake induced building motion. Proceedingsof the 9th World Conference on Earthquake Engineering, 2–9 August, Tokyo-Kyoto, Japan, V, 1988;779–784.

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8. Xu K, Igusa T. Dynamic characteristics of multiple substructures with closely spaced frequencies. EarthquakeEngineering and Structural Dynamics 1992; 21:1059–1070.

9. Igusa T, Xu K. Vibration control using multiple tuned mass dampers. Journal of Sound and Vibration 1994;175(4):491–503.

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14. Wu J, Chen G. Optimal mass distribution of multiple tuned mass dampers. Proceedings of the 14th EngineeringMechanics Conference, 21–24 May, Austin, TX, 2000.

15. Tian P. Generalized optimal control of elastic and inelastic structures subjected to earthquake excitation. Ph.D.Dissertation, University of Missouri-Rolla, 1997.

16. Wu J. Seismic performance, design and placement of multiple tuned mass dampers in building applications.Ph.D. Dissertation, University of Missouri-Rolla, 2000.

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18. Kanai K. Semi-empirical formula for the seismic characteristics of the ground. Bulletin of the EarthquakeResearch Institute, Tokyo University 1957; 35(2).

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