Taniguchi 2008 Engineering-Structures

Engineering Structures 30 (2008) 3478–3488

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Engineering Structures

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Effect of tuned mass damper on displacement demand of base-isolated structuresTomoyo Taniguchi a, Armen Der Kiureghian b,c,∗, Mikayel Melkumyan ca Tottori University, Tottori, 680-8552, Japanb University of California, Berkeley, CA 94720, USAc American University of Armenia, Yerevan, Armenia

a r t i c l e i n f o

Article history:Received 13 July 2007Received in revised form30 May 2008Accepted 30 May 2008Available online 2 July 2008

Keywords:Base isolationDisplacement demandFar-field motionsNear-field motionsOptimal designTuned-mass damper

a b s t r a c t

The third author of this paper has previously proposed the installation of a tuned-mass damper (TMD) toreduce the displacement demand on a base isolated structure. The TMDconsists of amass-dashpot-springsubsystem that is attached to the isolated superstructure, analogous to a pendulum. The present paperexamines the effectiveness of this scheme and determines optimal parameters for the design of the TMD.Both the base-isolated structure and the TMDaremodeled as single-degree-of-freedom, linear oscillators.The optimal TMD parameters are determined by considering the response of the base-isolated structure,with and without the TMD, to a white-noise base acceleration. Such an excitation is representative ofbroadband ground motions having a nearly constant intensity over a duration several times longer thanthe period of the base-isolated structure. It is found that, under such an excitation, a reduction of the orderof 15%–25% in the displacement demand of the base-isolated structure can be achieved by adding theTMD. Next, the responses of an example base-isolated structure with and without an optimally designedTMD to selected suites of far- and near-field recorded accelerograms are determined. The study showsthat for far-field ground motions the effectiveness of the TMD is more or less similar to that predicted bythe white noise model, whereas for near-field ground motions the effectiveness of the TMD is less, i.e. ofthe order of 10% or less. Reasons for this result are described.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Seismic base isolation has become one of the most effectivetechnologies in protecting structures against destructive earth-quakes. By providing flexibility in the base of the structure, theisolation system absorbs the bulk of the displacement demand ofthe earthquake, with the super-structure essentially displacing asa rigid body. Furthermore, base isolation drastically lowers the fun-damental frequency of the system, putting it outside the dominantrange of input frequencies and, thereby, reduces the accelerationat floor levels of the structure where sensitive equipment or non-structural systems may be located. In doing this, the base isolationsystem itself undergoes a relatively large displacement. One im-portant consideration in the design of the base isolation system isthis displacement demand.Our interest in this paper focuses on a base isolation system

made of laminated rubber bearings. The displacement capacityof such a system is directly related to its size and damping. One

∗ Corresponding author at: University of California, Berkeley, CA 94720, USA.Tel.: +1 510 642 2469; fax: +1 510 643 8928.E-mail addresses: [email protected] (T. Taniguchi),

[email protected] (A. Der Kiureghian), [email protected] (M. Melkumyan).

0141-0296/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2008.05.027

option for reducing the displacement demand on the isolationsystem is to provide supplemental damping. This, however, mayincrease the in-structure accelerations [5]. In this paperwe explorethe possibility of using a tuned mass damper to reduce thedisplacement demand of the base isolation system.The third author has been instrumental in initiating the

manufacturing of laminated rubber bearings in Armenia andusing them for retrofitting of existing buildings or for newconstruction [8,10]. Today, more than 30 buildings in Armeniaare built, retrofitted or under construction employing the baseisolation technology, mostly using locally manufactured bearingsmade of neoprene—thus putting Armenia at a top rank interms of the number of base-isolated buildings per capita. (Thepopulation of Armenia is around 3million.) At the present time themanufacturing technology is capable of producing bearings withneoprene compounds with low or medium damping. However,manufacturing of large-size (i.e. larger than 60 cm in diameter),high-damping (larger than 10%) bearings remains a technologicalchallenge in Armenia. For this reason, there has been an interestin exploring alternatives for reducing the displacement demand ofthe base-isolation system.With the above motivation in mind, the third author has

proposed a scheme to reduce the displacement demand ofa base-isolated structure by installing a tuned-mass damper

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http://dx.doi.org/10.1016/j.engstruct.2008.05.027

T. Taniguchi et al. / Engineering Structures 30 (2008) 3478–3488 3479

Fig. 1. Schematic of base-isolated structure with TMD.

(TMD) (denoted as ‘‘dynamic damper’’ in [7,9]) attached to theisolation floor. The effectiveness of an appended mass-springsystem in reducing the dynamic response of a structure has beenknown for a long time [1,12,11]. Numerous investigations andimplementations of this idea for fixed-base buildings have beenmade (see, e.g., [4,13,6,3]). In such buildings, the TMD is usuallyplaced in an upper floor in order to experience a larger accelerationfor efficiently mobilizing itself and absorbing the energy in thesystem. Usually a large portion of the floor or the entire floor of thebuildingmust be devoted to the TMD. In the case of a base-isolatedstructure, since the maximum relative displacement occurs at thelevel of the bearings, the TMD must be attached immediatelyabove the isolation system. It can be provided as a mass-springsubsystem attached either above or below the isolated floor of thebuilding, as shown in Fig. 1. One can imagine various functionsfor such a subsystem, e.g., an exercise room, a swimming pool,parking space, utilities room, as long as themass remains relativelyconstant in time and large displacement can be accommodated.As described by Melkumyan [9], the proposed TMD scheme alsohas the advantage of increasing the capacity of the base-isolatedbuilding against overturning forces.This paper aims at examining the effectiveness of the proposed

TMD scheme and determining the set of optimal parameters forits design. We assume the base-isolated structure can be modeledas a single-degree-of-freedom (SDOF) oscillator. This essentiallyassumes that the superstructure acts as a rigid body, which is areasonable assumption for a base-isolated structure. The TMD ismodeled as a SDOF oscillator as well, which is attached to thebase-isolated structure as an appendage. After formulating theequations of motion and defining key parameters, we considerthe stationary response of the base-isolated structure with andwithout the TMD when the system is subjected to a white-noise base acceleration. Such an excitation is representative ofa broadband ground motion having a nearly constant intensityover a duration several times longer than the period of the base-isolated structure. The simplicity of this excitation model allowsus to determine the effectiveness of the TMD in terms of a fewkey parameters. This then leads to the identification of optimalparameters (mass, frequency, damping) of the TMD for a givenbase-isolated structure. It is found that the optimally designedTMD can effect a reduction of 15%–25% in the displacementdemand of the isolators. Next, the responses of an example base-isolated structure with and without an optimally designed TMDto selected suites of far- and near-field recorded ground motionsare determined. The investigation shows that for far-field groundmotions the effectiveness of the TMD is more or less similar to thatpredicted by the white-noise model. Furthermore, the same levelof reduction is achieved in the acceleration response. For near-fieldgroundmotions, on the other hand, the effectiveness of the TMD isless, i.e., of the order of 10% or less. The reason has to do with theshort, pulse-type nature of near-field ground motions.

Fig. 2. Idealized model of base-isolated structure with TMD.

The practical implementation of a TMD in a base-isolatedbuilding obviously involves additional considerations, such asprovision of space for the displacement of the TMD and cost.These considerations, admittedly important ones, have not beenaddressed in this paper.

2. Equations of motion

Consider the combined system consisting of the base-isolatedstructure and the TMD, as shown in Fig. 1. We assume thebase-isolated structure alone behaves approximately as a SDOFoscillator having an effective massmp, a natural frequency ωp, anda damping ratio ζp, where the subscript p refers to ‘‘primary’’. Wealso assume the TMD by itself behaves approximately as a SDOFoscillator with an effective mass ms, a natural frequency ωs, anda damping ratio ζs, where the subscript s refers to ‘‘secondary’’.The combined system consisting of the base-isolated structure (theprimary subsystem) and the TMD (the secondary subsystem) is a 2-DOF system, as shown in an idealized form in Fig. 2. It is known (see[2]) that such a composite primary–secondary system is generallynon-classically damped, evenwhen the individual sub-systems areclassically damped. Hence, to properly model the system, accountmust bemade of the non-classical damping nature of the combinedsystem.The equations of motion of the combined system is described

byMu+ Cu+ Ku = −M1 xg(t) (1)

where u(t) = [up(t) us(t)]T is the vector of displacementsrelative to the ground, xg(t) is the ground acceleration and

M =[mp 00 ms

],

C =[2ζpωpmp + 2ζsωsms −2ζsωsms−2ζsωsms 2ζsωsms

],

K =[ω2pmp + ω

2sms −ω

2sms

−ω2sms ω2sms

], 1 =

11

.

(2)

For the subsequent analysis, it is useful to introduce the massratio

γ =msmp

(3)

and the tuning parameter

β =ωp − ωs

ωave(4)

where ωave = (ωp + ωs)/2 is the average frequency. The massratio describes the size of the TMD;we consider values in the rangeγ = 0.01 to 0.10. The tuning parameter describes the proximity ofthe natural frequencies of the two sub-systems. It is well knownthat the TMD is more effective when γ is large and β is near zero.Results are presented below in terms of these two parameters, aswell as the individual system parameters defined earlier.

3480 T. Taniguchi et al. / Engineering Structures 30 (2008) 3478–3488

Fig. 3. Frequencies of the combined system.

3. Frequencies of the undamped combined system

As mentioned earlier, the combined system is non-classicallydamped. Nevertheless, it is insightful to examine its undampedfrequencies. Analytically solving the eigenvalue problemΩ2M8 =

K8 for the undamped system, we obtain the following expressionsfor the two natural frequencies of the combined system.

Ω1,Ω2 =1√2

ω2p + (1+ γ ) ω

2s

±

√ω2p + (1+ γ ) ω2s

2− 4ω2pω2s

12

. (5)

Fig. 3 shows plots of the two frequencies of the combinedsystem, normalized with respect to the frequency of the primarysub-system, as a function of the tuning parameter for differentmass ratios. The ratio of sub-system frequencies, ωs/ωp, is alsoshown for reference. It can be seen that for large negative β values,i.e., for ωp ωs, the first mode of the combined system hasa frequency that is close to but lower than that of the primarysub-system and the second mode has a frequency that is closeto but greater than that of the secondary sub-system. Conversely,for a large positive β value, i.e., for ωs ωp, the secondmode of the combined system has a frequency that is close tobut greater than that of the primary sub-system, while the firstmode has a frequency that is close to but smaller than that ofthe secondary sub-system. It is reasonable to expect that for thesevalues of β , the mode of the combined system that has a frequencyclose to that of the primary sub-system dominates its responsein the combined system. For β values close to zero, i.e., nearperfect tuning, the twomodal frequencies of the combined systemare symmetrically positioned relative to the frequency of theprimary sub-system. In this case, the two modes tend to equallycontribute to the responses of the primary and secondary sub-systems in the combined system. Aswewill shortly see, the TMD ismost effective in reducing the displacement demand on the base-isolated structure when β is in the range 0.1–0.2. In this range, thesecond mode of the combined system dominates the response ofthe base-isolated structure. It is also notable that the frequenciesof the combined system move further apart from the sub-systemfrequencies and from each other as the mass ratio, γ , increases.

4. Stochastic dynamic analysis

To examine the effectiveness of the TMD in reducing thedisplacement demand of the base-isolated structure, we firstconsider the stationary response of the combined system to azero-mean, broadband stationary stochastic base acceleration.To properly account for the non-classical damping nature of

the combined system, a frequency-domain approach using thefrequency-response matrix (FRM) (instead of the usual modalsuperposition approach) is used. The FRM of the system in (1) isgiven by

H(ω) =[−ω2M+ iωC+ K

]−1. (6)

LetΦxg xg (ω) denote the power spectral density (PSD) of the groundacceleration. The PSDmatrix of the response vector u is then givenby

8uu(ω) = H(ω)M1Φxg xg (ω)1TMTH(ω)∗T (7)

where the superposed asterisk denotes the complex conjugate.As measures of the responses of interest, we consider the mean-squares of the displacements up(t) and us(t) of the primary andsecondary subsystems, respectively. These are given by

σ 2p =

∫+∞

−∞

Φupup(ω)dω (8)

σ 2s =

∫+∞

−∞

Φusus(ω)dω (9)

whereΦupup(ω) andΦusus(ω) are the diagonal elements of8uu(ω).Closed form expressions for the elements of the 2 × 2 matrixH(ω)M are derived in Appendix. These are used in (7) to deriveexpressions for Φupup(ω) and Φusus(ω). Numerical integration isthen used to evaluate (8) and (9).To investigate the effect of the TMD in reducing the displace-

ment demand of the base-isolated structure, we consider the re-sponse ratio σp/σp0, where σp0 denotes the root-mean-square ofthe response of the base isolated structure without the TMD. Inorder to make the results independent of the specifics of the in-put ground motion, the excitation is assumed to be a white-noiseprocess having a constant PSD Φxg xg (ω) = Φ0. This model is agood approximation for broadband earthquake ground motionsand lightly damped structures. For thismodel, it is well known thatσ 2p0 = πΦ0/(2ζpω

3p). Furthermore, the response ratio σp/σp0 is in-

dependent of the intensityΦ0 of the white noise. In fact, as can beverified from the expressions inAppendix, the ratioσp/σp0 only de-pends on themass ratio, γ , the tuning parameter,β , and the damp-ing ratios ζp and ζs of the two subsystems.Figs. 4(a)–(d) show plots of the response ratio σp/σp0 as a

function of β for the primary damping ratio ζp = 0.05 andsecondary damping ratios ζs = 0.05, 010, 0.15 and 0.20,respectively. Three curves are shown in each plot for themass ratiovalues γ = 0.02, 0.05 and 0.10. A fourth curve with diamond-shaped markers is also shown, which is described below.First consider Fig. 4(a), which is for ζp = ζs = 0.05. As

one would expect, the effectiveness of the TMD in reducing thedisplacement demand on the base-isolated structure depends onboth the mass ratio, γ , and on the tuning parameter, β . For a fixedγ , the optimal TMDoccurs at a positiveβ value,which correspondsto a frequency of the TMD that is smaller than the frequency of thebase-isolated structure. Thus, contrary to the notion of a ‘‘tuned’’mass damper, the optimal reduction in the response of the base-isolated structure occurs not at β = 0 but for 0 < β . The locusof these optimal points for all γ values is plotted in the figure(line with diamondmarkers) and is called the ‘‘design curve’’. Notethat the diamond marks on the design curve indicate γ valuesfrom 0.01 to 0.10 at increments of 0.01. It is seen that the optimalvalue of the tuning parameter β moves towards greater positivevalues with increasing mass ratio. For example, at γ = 0.05, theoptimal value of the tuning parameter is βopt = 0.10, which using(4) corresponds to the optimal TMD frequency of ωs = 0.905ωp,whereas at γ = 0.10 the optimal value of the tuning parameteris βopt = 0.18, which corresponds to ωs = 0.835ωp. At these


(a) ζs = 0.05. (b) ζs = 0.10.

(c) ζs = 0.15. (d) ζs = 0.20.

Fig. 4. Reduction in the seismic demand of the base-isolated structure caused by the TMD for ζp = 0.05.

mass ratios, the TMD reduces the demand on the base-isolatedstructure by about 21% and 22%, respectively. On the other hand,the reduction in demand for the optimal TMDwith mass ratio γ =0.02 is 17%. This confirms thewell known result that increasing themass ratio increases the effectiveness of the TMD (e.g., see [11]).Note, however, that even amass ratio of γ = 0.01 or 0.02 providesconsiderable reduction in the demand.Figs. 4(b)–(d) show results similar to those shown in Fig. 4(a)

but for secondary damping ratios ζs = 0.10, 0.15 and 0.20,respectively. It is seen that significant improvement in theeffectiveness of the TMD is achieved by increasing its dampingratio from 0.05 to 0.10. However, further increase in the dampingratio of the TMD provides marginal improvement or evendiminishes its effectiveness (e.g., compare the curves for γ = 0.02for increasing ζs). It appears that the TMD damping ratio ζs = 0.10is a good choice if the damping ratio of the base-isolated structureis ζp = 0.05.Figs. 5(a)–(d) show results similar to those described above but

for the primary damping ratio ζp = 0.10. It is seen that, comparedto the case of ζp = 0.05, the effectiveness of the TMD is reducedto no more than 5%–15%. Melkumyan [7] anticipated this effect bysuggesting the use of the TMD in conjunction with low-dampingrubber isolation bearings.The design curves in Figs. 4 and 5 can be used to select the

frequency of the TMD, for given frequency of the base-isolatedstructure, the mass ratio and the two damping ratios, to achievethe maximum reduction in the displacement demand on the base-isolated structure. It is important to observe, however, that thecurve for each fixed γ has a relatively flat bottom. Therefore, if asmall error is made in estimating the frequencies of the primary orsecondary subsystems, the reduction in the response of the base-isolated structure will not be greatly affected. Roughly speaking,relative variations δωp and δωs in the two frequencies lead to the

absolute variation δβ ∼= 2√δ2ωp + δ

2ωsin the tuning parameter.

For example, if the primary frequency is known within a 4% error

and the secondary frequency is known within 2% error, thenthe estimated error in β is around 2

√0.042 + 0.022 = 0.09.

This formula can be used to estimate the range of variations inthe tuning parameter for given uncertainties in the sub-systemfrequencies.In summary, the following conclusions can be derived from the

above analysis of the stochastic response:

(a) The effectiveness of the TMD in reducing the displacement de-mand on the base-isolated structure increases with increasingmass ratio, provided the TMD is optimally tuned;

(b) The optimal TMD always has a frequency smaller than thefrequency of the base-isolated structure;

(c) The TMD is more effective for a lightly damped isolationsystem;

(d) For the damping ratio ζp = 0.05 of the base-isolated structure,a good choice for the damping ratio of the TMD is ζs = 0.10;higher TMD damping ratios do not significantly improve theeffectiveness of the TMD;

(e) A reduction of 20% or greater in the displacement demand ofthe base-isolated structure with damping ratio ζp = 0.05 canbe achieved by use of an optimally designed TMD having adamping ratio of ζs = 0.10 and mass ratio 0.05 ≤ γ ;

(f) Variations in the order of 2%–3% in the frequencies of thebase-isolated structure and an optimally designed TMD havea negligible influence on the effectiveness of the TMD.

Several of the above conclusions, including (a), (c) and (f), andthe fact that a higher damping for the TMD beyond a certain leveldoes not increase its effectiveness, are confirmations ofwell knownresults for TMDs in fixed-base buildings.The above analysis assumes that the TMD responds within

its elastic limit. Since the TMD is nearly tuned to the primarysubsystem, it may experience a large response, which may put itbeyond its yield limit. To investigate this possibility, we examinethe response ratio σs/σp, which is only a function of the parametersγ , β , ζp and ζs. Fig. 6 shows this response ratio as a function of


(a) ζs = 0.05. (b) ζs = 0.10.

(c) ζs = 0.15. (d) ζs = 0.20.

Fig. 5. Reduction in the seismic demand of the base-isolated structure caused by the TMD for ζp = 0.10.

Fig. 6. Amplification in the displacement of the TMD for ζp = 0.05 and ζs = 0.10.

the tuning parameter β for ζp = 0.05, ζs = 0.10 and the massratio values γ = 0.02, 0.05 and 0.10. It appears that, depending onthe tuning parameter and the mass ratio, the response of the TMDcan be 2–4 times larger than that of the base-isolated structure.This finding clearly calls for a careful design of the TMD andthe space it occupies in the building in order to accommodatethe large displacement demand. Alternatively, one may allow theTMD to dissipate energy through hysteretic action. This potentiallybeneficial effect is not considered in the present study.

5. Time history analysis

The analysis in the preceding section employed a stationarywhite-noise process as a model for the ground acceleration. Thisis convenient, since for this model the effectiveness of the TMDcan be assessed with the least number of system parametersand without involving any parameters that characterize the inputexcitation. However, one may question whether this idealizedmodel of the groundmotion accurately describes the effectivenessof the TMD, since real earthquake ground motions are neither

stationary nor have a uniform spectral content, as representedby the white-noise model. It is well known that, in order for theTMD to be effective, it is necessary that the energy input intothe system be gradual so that there is time for transfer of energyfrom the primary system (the base-isolated structure) into thesecondary system, the TMD. This suggests that the non-stationarynature of the ground motion may have a strong influence onthe effectiveness of the TMD. For this reasons, it was decidedto examine the effectiveness of the TMD by using time-historyanalyses with selected recorded accelerograms. The stand-alonebase-isolated structure considered for this analysis has a frequencyof ωp = π rad/s (2.0 s period) and a damping ratio of ζp =0.05. The TMD is assumed to have the damping ratio ζs =0.10, the mass ratio γ = 0.05, and the stand-alone frequencyωs = 0.910π rad/s (2.2 s period), which corresponds to theoptimally designed value of the tuning parameter, βopt = 0.094,as can be seen in Fig. 4b. For this system, a response ratio of 0.75(i.e. a reduction of 25% in the displacement demand) is expectedfrom the analysis with the white-noise excitation. The recordedaccelerograms are selected from the PEER strong-motion databaseat http://peer.berkeley.edu/smcat/index.html. To properly accountfor the non-classical damping nature of the combined system,direct numerical integral of the equations of motion in (1) iscarried out. The second-order Runge–Kutta algorithm is used forthis purpose.Considerable attention was given to the characteristics of

the selected ground motions, which are listed in Table 1. Sincethe analysis is linear, and the ratio of responses with andwithout the TMD is of interest, the intensity of the motion isimmaterial. However, the temporal evolution of the motion andits frequency content are important considerations. As proxiesfor these characteristics of the recorded motions, we selectedthe distance of the recording site from the fault rupture and thelocal site condition. The distance from the fault rupture tendsto influence the nonstationary character of the accelerogram. In

http://peer.berkeley.edu/smcat/index.html


Table1

Listofearthquakesandcomputedresponseratios

Earthquake

Recordcomponent

USGSsiteclass.

Distance(km)

PGA(g)

PGV(cm/s)

PGD(cm)

Accelerogram(g)

Responseratio

Disp.(%)

Acc.(%)

Far-fieldgroundmotions

Northridge

1994/01/1712:31

NORTHR116090

B41.9

0.208

10.3

2.67

107

103

LomaPrieta

1989/10/1800:05

LOMAPSFO090

C64.4

0.329

27.9

6.03

9092

ImperialValley

1979/10/1523:16

IMPVALL

H-DLT352

C43.6

0.351

33.0

19.02

8078

Kobe1995/01/16

20:46

KOBEKAK090

D26.4

0.345

27.6

9.6

7480

Chi-Chi,Taiwan

1999/09/20

CHICHITCU047-N

B33.01

0.413

40.2

22.22

6061

KernCounty

1952/07/2111:53

KERNTAF111

B41.0

0.178

17.5

8.99

7675

Near-fieldgroundmotions

Northridge

1994/01/1712:31

NORTHRNWH090

C7.10

0.583

75.5

17.57

9292

LomaPrieta

1989/10/1800:05

LOMAPLGP000

?6.10

0.563

94.8

41.18

9191

ImperialValley

1979/10/1523:16

IMPVALL

H-E08140

C3.80

0.602

54.3

32.32

9291

Kobe1995/01/16

20:46

KOBEKJM000

B0.60

0.821

81.3

17.68

9394

Chi-Chi,Taiwan

1999/09/20

CHICHITCU129-W

C1.18

1.01

60.0

50.15

9796

Tabas,Iran

1978/09/16

TABASTAB-TR

C3.00

0.852

121.4

94.58

7172


(a) Northridge. (b) Loma Prieta.

(c) Imperial Valley. (d) Kobe.

(e) Chi-Chi. (f) Kern county.

Fig. 7. Displacements of the base-isolated structure with (solid line) and without (dashed line) the TMD for far-field ground motions (Unit: m).

particular, near-field groundmotions tend to contain a long-periodpulse, the ‘‘fling’’, which directly results from the dislocation atthe fault, whereas far-field ground motions tend to have a fairlysmooth transition of temporal and spectral contents. To accountfor these effects, two sets of ground motions were selected: near-field ground motions with distances ranging from 0.6 to 7.1 kmfrom the closest point on the fault rupture, and far-field groundmotionswith distances ranging from26.4 to 64.4 km from the faultrupture. The local site condition influences the frequency contentof the ground motion. For the selected records, the site conditionis characterized by the USGS classification. Among the recordsconsidered, 4 have classification B (shear-wave velocity in therange 360–750 m/s), 6 have classification C (shear-wave velocity

in the range 180–360 m/s), and one has classification D (shear-wave velocity less than 180 m/s). One record has no classification.For each record, the ratios of the maximum absolute displacementandmaximumabsolute acceleration responses of the base-isolatedstructure with the TMD relative to the corresponding responsesof the stand-alone base-isolated structure are computed. Theseresponse ratios alongwith other characteristics of each earthquakerecord are listed in Table 1. Figs. 7 and 8 respectively showcomparisons of the computed displacement responses of the base-isolated structure with and without the TMD for the six far-field ground motions and six near-field ground motions, andFigs. 9 and 10 show similar results for the computed accelerationresponses.




(e) Chi-Chi. (f) Tabas.

Fig. 8. Displacements of the base-isolated structure with (solid line) and without (dashed line) the TMD for near-field ground motions (Unit: m).

The following observations can be made from the computedresponse ratios listed in Table 1 and the time-history results inFigs. 7–10: It is first noted in Table 1 that in all but one case(far-field Northridge record) reductions in the displacement andacceleration responses of the base-isolated structure are achievedby adding the TMD. Secondly, the response ratios are nearly thesame for the displacement and acceleration responses for eachground motion. This indicates that the TMD has similar influenceson the displacement and acceleration responses of the base-isolated structure. Thus, the reduction in the displacement demandis not achieved at the expense of increasing the accelerationresponse, as may be the case with other alternatives, such as theprovision of supplemental damping [5]. Thirdly, no correlation isobserved between the response ratios and the site classifications.

Evidently, the site classification is too crude a measure to have adirect relation with the effectiveness of the TMD.For the far-field ground motions (the first six rows in Table 1),

the response ratio is around the expected 75% for four outof the six records. Among these records, the ones of ImperialValley, Kobe and KernCounty earthquakes have nearly stationarycharacters during their respective strong motions phases. The Chi-Chi earthquake has a distinctly nonstationary behavior. However,the largest pulses in the accelerogram occur after a period ofgradual increase in the intensity of the motion (see Figs. 7 and 9bottom left). This gives opportunity to the TMD to be mobilizedand absorb energy from the primary structure. For the far-fieldLoma Prieta record, the response ratios are 0.90 and 0.92. The smallreduction in the response can be attributed to the fact that the peak




(e) Chi-Chi. (f) Kern county.

Fig. 9. Accelerations of the base-isolated structure with (solid line) and without (dashed line) the TMD for far-field ground motions (Unit: g).

responses to this record are primarily due to a large accelerationpulse happening around 11s (see Figs. 7 and 9 top right). Sincethis large acceleration pulse occurs early in the excitation, theeffectiveness of the TMD is similar to the case of near-field groundmotions described below. The most puzzling result is that ofthe far-field Northridge record, which appears to have a fairlystationary character during its strong-motion phase, but the effectof the TMD in this case is a slight amplification of the response ofthe base-isolated structure. This case is further examined below.For the near-field ground motions (the last six rows in Table 1),

the reductions in the displacement and acceleration responses ofthe base-isolated structure are less than 10% in all but the case ofTabas record. We attribute this to the fact that, with the exception

of Tabas, all these motions contain large acceleration pulses earlyin their time histories, which produce the peak responses of thebase-isolated structure (see Figs. 8 and 10). As mentioned earlier,this is a typical characteristic of near-field ground motions, whichare directly affected by the fault slip with little influence fromthe dispersive effect of waves traveling long distances. For suchrecords, there is no time for transfer of energy from the base-isolated structure into the TMD and, hence, the TMD does notbecome fully mobilized to achieve its effectiveness, as predictedby the stationary analysis in the preceding section. Although thisphenomenon is generally known,we have not found a quantitativeanalysis of its effect in the TMD literature, particularly in relationto near-field ground motions. In contrast to the other near-field




(e) Chi-Chi. (f) Tabas.

Fig. 10. Accelerations of the base-isolated structure with (solid line) and without (dashed line) the TMD for near-field ground motions (Unit: g).

records, since large acceleration pulses in the Tabas record occurin the middle of the record, there is enough time for the TMD tobemobilized to achieve its effectiveness (see Figs. 8 and 10 bottomright). In this sense, the Tabas record behaves like a far-field groundmotion, while the far-field Loma Prieta record behaves like a near-field ground motion.We now return to examine the case of the far-field Northridge

record. As mentioned earlier, this record has a nearly stationarystrong-motion phase, yet the response ratios for both thedisplacement and acceleration responses are greater than 1. Thatis, for this ground motion, attaching the optimally designedTMD actually enhances both the displacement and accelerationdemands on the base-isolated structure. To understand the reason

for this behavior, we examine the response spectra of the selectedfar-field ground motions. Fig. 11 shows the 5% damping responsespectra for these motions, all normalized by their values at0.5 Hz, which is the frequency of the base-isolated structurewithout the TMD. A thicker line highlights the spectrum for theNorthridge record. It can be seen that the Northridge record hasa sharp dominant peak at around 0.4 Hz frequency, which hasa much larger value than the spectral displacements at all otherfrequencies. Thus, the energy in thismotion ismostly concentratedaround 0.4 Hz frequency. As can be verified in Fig. 3, this frequencyhappens to coincide with the first undamped modal frequency ofthe base-isolated structurewith the TMD. Thus, by adding the TMD,the first mode of the combined system, i.e. the mode dominated


Fig. 11. Normalized response spectra of far-field ground motions for 5% damping.

by the displacement of the TMD, is subjected to this large spectralamplitude, resulting in an amplification of the response of thebase-isolated structure relative to its response without the TMD.In other words, in this case, because of resonance of the TMD withthe input excitation, the TMD further excites the base-isolatedstructure instead of absorbing its energy. It is noted that none ofthe other response spectra in Fig. 11 has this particular feature.The response ratios predicted based on stochastic analysis

shown in Figs. 4 and 5 represent ensemble averages. Theeffectiveness of the TMD for individual realizations of thestochastic ground motion would, of course, vary around theseaverages. Therefore, in examining the time-history results, it isappropriate to consider the averages over the selected samples.Considering all six samples, the average response ratios for thefar-field ground motions are 0.81 for both the displacement andacceleration responses. If the Northridge record is not included,the averages reduce to 0.76 and 0.77, respectively, which are inline with that predicted by the stochastic analysis. For near-fieldground motions, the average response ratios are around 0.89 forboth displacement and acceleration responses. If the Tabas recordis excluded, both averages are around 0.93. It is clear that theTMD is not effective in reducing the demand on the base-isolatedstructure for near-field ground motions.

6. Conclusions

The effectiveness of a TMD to reduce the seismic demand on abase-isolated structure is investigated. Using stochastic dynamicanalysis based on a white-noise model of the ground motion, theoptimal parameters of the TMD that maximally reduce the seismicdemand on the base-isolated structure are determined. Thisinvestigation reveals that, depending on the mass, damping andfrequency characteristics of the TMD, the displacement demandon the base isolated structure can be reduced by 15%–25%. It isshown that the TMD is more effective for lightly damped isolators.Furthermore, the effectiveness of the TMD increases with its mass,but not necessarily with its damping.To account for the nonstationary and non-white nature of

ground motions, a series of time history analyses with far- andnear-field recorded ground motions are carried out. The analysesshow that for far-field groundmotions the effectiveness of the TMDis in concordance with the predictions of the stochastic analysis,except for one particular record, which happens to have a sharpspectral peak in resonance with the TMD. Importantly, the TMD

produces virtually identical reductions in the displacement andacceleration demands of the base-isolated structure. For near-fieldground motions, the effectiveness of the TMD is no more than7%–10%. The reason is that for such motions the peak responseusually is due to a large pulse early in the record, so that sufficienttime is unavailable for the TMD to be mobilized.One may question the viability and cost-effectiveness of

installing a TMD with a mass ratio as large as 5% or more of thebuildingmass to effect a reduction of nomore than 15%–25% in thedisplacement demand of a base isolation system. However, if theTMD is designed as an integral part of the base-isolated buildingand it serves a useful function, then such a scheme may prove tobe both beneficial and economical. We note that, even though theTMDmay experience large displacements, thesemotions will havelow frequency and, hence, will be tolerable by humans and certainequipment (similar to wind-induced motions in top floors of tallbuildings). In any case, the results presented in this paper providevaluable information to any engineer contemplating the use of aTMD in a base-isolated building.

Appendix

The product of the frequency response matrix and the massmatrix in (7) is given by

H(ω)M

=1D

[ω2s − ω

2+ 2iζsωsω ε

(ω2s + 2iζsωsω

)ω2s + 2iζsωsω ω2p + εω

2s − ω

2+ 2iω

(ζpωp + εζsωs

)] (A.1)where D is

D = ω2ω2 − ω2p − ω

2s (1+ ε)− 4ζpζsωpωs

+ ω2pω

2s

+ 2iωωpωs

(ζpωs + ωpζs

)− ω2

[ζsωs (1+ ε)+ ζpωp

]. (A.2)

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Taniguchi 2008 Engineering-Structures