THE POWERFUL TMD FOR BRIDGES COMPULSORILY SHAKEN
BY A REACTION FORCE FROM AN ABUTMENTⅠ. KANEKO, T. KATSUKAWA, S. YOSHITANI and T. SUZUKI

Technical Research & Development Institute, Kumagai Gumi Co･, Ltd･,
1043 0nigakubo, Tsukuba‑shi, Ibaraki 300‑22, Japan

ABSTRACT

A conventional TMD‑type device does not necessarily glVe Sufficient control force as the seismic control
device fbr bridges･ In order to highly improve damplng Capability of a conventional TMD, the Powe血1TMD is newly developed･ In this device, a TMD is compulsorily shaken by a reaction force from anabutment without any extemal energy･ In this paper, the outline of the new device is introduced in the firststep･ Modeling, formulation and optlmum tunlng technique forthe device are described next･ Shaking table
tests uslng a Simple bridge model, to which the proposed device can be applied,are carried out, and theresult of shaking table tests for a case in which the proposed device is applied is compared with other cases

in which no control device is applied and a conventional passive TMD device is applied, both experimentallyand numerically･ Then, the effectiveness and the applicability of the Powerful TMDare examined･

KEYWORDS
TMD‑type seismic control device; Bridge; Formulation; Shaking table test; Numerical Simulation･

INTRODUCTION

The fundamental theory and optimum tunlng technique of a TMD for harmonic vibration were presented byDen Hartog (Den Hartog, 1956). Then, they were established, having been extended to various vibration

conditions (Warburton, 1982). TMD‑type devices have been recently applied to many civil engineering
structures. Most of them are, however, adopted for structural control agalnSt Wind vibration. The applicationof this passive device to structures fわr the purpose of seismic control is limited, because the device is notcapable of fbllowlng initial rapid motion and mass of the device should be large enough to control stnlCtural

vibrations due to strong earthquake excitation･ In order to highly improve damplng Capability of a TMD so
as to be applied to seismic control for bridges, the Powerful TMD was developed･ This new device adopts
a prlnCiple of a lever, and is compulsorily shaken by a reaction force from an abutment without any extemalenergy･ Fomulation of equation of motion, optlmum tunlng and numerical simulations were conducted fわrrigorous 2‑DOF system (Kaneko et a1., 1994). Then, the effectiveness of the proposed device were verified･This paper focuses on the Powerful TMD for a simple 2‑DOF system developed for design purposes･ The
mechanism of the new device is described first･ Modeling, formulation and optlmum tunlng technique are
introduced next･ Then, the e胎ctiveness are verはed by vibration tests uslng a Shaking table and numericalsimulations.

OUTLINE OF THE POWERFUL TMD
Fig･ 1 illustrates a schematic diagram to represent a function of the Powerful TMD･ The device should belocated at the end of a bridge deck･ Although the concept of the device is based on a conventional passive

TMD, a TMD system is not simply fixed at a bridge deck directly but at a steel frame which is able to freely
slide on a bridge deck in the longitudinal direction･ The steel frame is connected to an abutment througha lever･ Letting an am ratio of the lever be 1 : α and the relative displacement between a bridge deck andan abutment due to earthquake excitation be u as shown in the figure, the TMD is compulsorily shaken with
an amplitude of (1+α)u, which is (1+α) times larger than that of a bridge deck･ When the control force, P

is given by the TMD excitation, it is transmitted to a bridge deck being amplified by (1+α) times･ Therefore,
We can obtain a seismic control fわrce (1+α) times larger than that obtained by a conventional TMD without

any extemal energy adoptlOn･Bridge ms, ksI cs

LeverTMq1 剪mT SteeIFrame(..]■■■璽劔剿末末末末睦A劔

A butmenthu劔剪I(1+α)u‑αPαP 劔冰(1+α)P(1+α)PT･鋳■■●●1AFig. 1.

A schematic diagram to represent a function of the Powe血I TMD.

FORMULATION OF EQUATION OF MOTION
The dynamic model for the Powerful TMD is illustrated in Fig･2･ In this model, the displacement of thesteel frame, uT一, is constrained by that of a bridge structure, us, with a relation; uT, = (1+α)us. Using thisrelation, the formulation of equation of motion for the Powerful TMD canbe given as follows (Kaneko eta1.,1994):

ms.(1.a)2m,I oO

cs.a2cT/.(1.a)2cT ‑(1.a)cT‑(1mT+α)crcr(1)

ks.a2cT/.(I.a)2kT ‑(1.a)k,‑(I'cE)kT

kT

ms'(l'a)mT/ 00mT

where subscripts S, T, T'and G denote bridge structure, TMD, steel frame and ground, respectively, and u,m, c and k denote relative displacement, mass, damplng and stiffness, respectively･

Fig･ 2･A schematic diagram to represent a dynamic model for the Powerful TMD･

Having been discussed in the paper (Kaneko et a1., 1994), the natural frequency of a spring‑mass system

consistlng Of the steelframe, at which the Powerful TMD acts effectively as a tuned mass damper, is limited
in a range at most 2 or 3 times of that of a bridge structure. Therefore, the effect of the mass, stiffness anddamplng On the dynamic behavior of the whole system are considered to be negligible. Then, the equationof motion forthe Powerful TMD can be simplified as Eq.(2).
ms00 mTks十(1十α)2kT ‑(1十α)kT

cs.(I.a)2cT ‑(1十a)cT‑(I'a)cTCT‑(1'α)kT

kT(2)ms00 mT

OPTIMUM TUNING OF FREQUENCY AND DAMPING
The method for optlmum tuning for the Powerful TMD for hamonic excitations can be derived according

to the method by Den Hartog (Den Hartog, 1962). ne method of optimum tuning for Eq･(1) was socomplicated that it was not adequate for design purposes･ The optlmumfrequency, Vo,t, and damplng, ho,t,derived for Eq.(1), however, Canbe written in simple forms as shown in Eqs.(3) and (4), respectively.
1V optl･(lea)2ph opt

(3)

(4)

Both of them are detemined only by mass ratio,ド, which is the ratio of mass of a TMD to that of a bridgestructure, and am ratio, α. When α equals to 0, these fomulae are identical with those for a conventional

TMD.
Fig･3 Compares the optimum condition for the Powerful TMD derived from Eqs.(3) and (4) with that given
from the previous method, whenthe frequency ratio of the steel frame to a bridge structure, V', equals to2･ Within the range of mass ratiosand arm ratios shown in the figure, they can be regarded as identical･Then, Eqs･(3) and (4) can be applicable for engineering purposes. This result denotes, in other words, thatthe sprlng COnneCted between the丘･ame and a bridge deck can be omitted･

0000000 7 ′LU 5 4 3 2 1

L o o o o o o o 0 0 0ノ 0 7 ′LU 5 4 3 2Ill 劔

Strict+ Eq.(4)一･一一一‑ 劔lJ

G≡lId.(Lm∈N＼13): dol

号(S3＼ト3)=一dゞα

α=2a.
α=つ 劔lα=3 剪
誓言l I 劔Strict+ Eq.(3)‑ 劔ll11111

0･01

LA‑

mT/ms

O･1(a) Optimum frequency

Fig･ 3･0･01

p‑

mT/ms

(b) Optimum damping

Comparison of the optimum COndition fわr the Powe血l TMD derived触)m
Eqs.(3) and (4) with that given血)m the previous method (Kaneko et a1.,

1994), when the frequency ratio of the steel frame tothe bridge structure, V',equal to 2, and the mass ratio of the steel frame to the structure, p', is 0.1

times of that of the TMD to the structure, I.

SHAKING TABLE TESTS AND NUMERICAL SIMULATIONS
Outlt'ne and Testing Method
In order to verify the validity of fomulation and the seismic control effect for the Powerful TMD, shaking
table tests were caqied out･ Table 1 summarizesthe similarity of a prototype and a shaking model of thebridge of concems in this study･ The model for shaking table tests were designed based on the similarity
in Table 1 and the result of tuning. Parameters for the shaking table model are summarized in Table 2. TTle

bridge model was consisted of a concrete slab andrubber bearlngS･ Which constitutes.a SDOF system･ The

Powerful TMD was optimally tuned using the previous method based on a ngorous 2‑DOF system for I =0.043 and α茸2.

Four different strong motion records listed in Table 4 includingthe record at JMA Robe station during thesouthem Hyogo earthquake of January 17, 1995, Were used as the input motion of the test. Time durationof each input motion was shortened by 2/3 and the maximum acceleration was detemined in considerationof the limitation of measurlng lnStmmentS and capability of the model. Fig.4 illustrates the outline of themodel and location of measurlng lnStrumentS.

Table l･ Similarity of a prototype and a shaking model of the bridge･A･ Prototype structure B･ Shaking model B / AWeight7,500

Sti托IeSS

16,000Period

1.6

sec

0.923

tftⅣm

sec

Remarks A prestressed concrete bridgewith the deck length of 250m.

5.75

tf

1/

1,300

27.2

t〟n

1/600/

32

Table2.

Parameters fわr the shaking table model.

Bridge

model

TMD

Steel

frame

W5 ‑ 5･75 tf WT ‑ 0･25 tf WT, ‑ 0･065 tf

Weight
Parts
Mass

Stiffness

Ks

Concrete

ratio

‑

127.2

II

tf/m

SlabSteel

Lump

Light0･043

KT‑0･611

tf/mKT′

Steelp'‑0.01

1‑

tf/m

0.307

Parts Lubber Bearing Coil Spring Coil SpringFrequency

fs‑Frequency

Damplng

Ratio

hs‑

1･08

ratio

1Ratioofthev

fT

≡

=0･78

hT

Oil‑

0･32

免F

ﾆWfW&③steellumpofTMDmass@coilspringofTMD

6ﾗ&匁v

W&eDﾔB

宙皮7FV

棉7‑

1･08

hT′

ﾆg&

eDﾔB

‑

1.00

‑

0.01

2

⑨LeverofPowe血lTMD@outersupportofthelever@
ﾖR

g7FVVﾆg&@Innersupportofthelever@

ﾖR

Direcdon of the Excitations

Fig･ 4･Hz

Damperα@Rubberbearing

fT,V′‑

Lever(丑concreteslabHz0･72

0･03

PartAmHz

Outline of the model and location of measurlng lnStrumentS･

Table3･Strong motion records used as the input motion of the test･

Name of strong ground motion record Maximum acceleration
1Ta允(EWcomp.)

2EI

3JMARobe

4Kobe

UniversityCentro(NS(NS1952･7･21150galcomp.)

1940･5･18

150galcomp.)

1995･1･17150gal

1995･1･1750gal(NScomp.)

Compan'son of Test Results alZd Anatyscs
Numerical simulations for shaking table tests were also conducted solving Eq.(1) with parameters as shownin Table 2 for 4 cases in which the Powerful TMD was installed･ Fig･5 comparesthe response time historiesof relative displacement and acceleration between the shaking table test andthe numerical analysis, whenthe Taft‑EW component was applied as the input motion･Asshown in the rlgure, the analysis successively
simulates the test result, which validates a fbmulation of the dynamic model fわr the Powe血I TMD. ¶ecomparison between the test result and the analysis subjected to other input motionsare summarized in terms

or maximum values in Table 4･ The table denotes that the analysis glVeS the almost complete simulationof the shaking table test.

Test: Max. ‑12.891mm (5.236sec)

Test

Analysis : Max. 13.61 1mm (4.656sec)
2 1

1 つー

: Max. ‑27.246mm (5.256sec)

Analysis : Max. 27.223mm (4.656sec)
3 2 1 1 2 30 0 0 0 0 0 000000(u ) uau3 t!tds!Q(∈∈)tua∈auqlds!QA冤▼i

̲A".. 伜(爾苒Vy"I嵐

YYYV 遅1,^まｲr

ﾂ襁

̲̲I̲ヽ

I. 苓XーVｲ礼問rA.A 帝<r粐粐

ﾅd‑...

rV 鋲eeb粐裴.bÅユニユ̲ 凵｣ヽYVl 覧bR‑l一一r‑Test ‑.‑AnalyslS ll】

I 冤 劔0510152025303505l10Time (see)G2253035

and the bridge model.

: Max. ‑58.203mm (5.483sec)

Test

Analysis : Max. ‑56.649mm (5.463sec)
605胡302̲0̲0230 506

2 1(u∈) uau3 t!tdsSQ

1 2: Max. 187.578gaI (5.496sec)

Analysis : Max. ‑193.647gaI (2.535sec)

0 0 o0 0(t亀)uo!t !.t313 3V

凵｣

J 番ﾂ l20(b) Relative displacements between the Steel frame

and the shaking table.

一ﾇ

5Time (see)(a) Relative displacements between the bridge model

Test

1J 白 ﾆﾂl鳴I牝I∫ 敦ﾂ1 緬白l 迫DﾂヽA

L1､一 緬f 白rr粨

侘2lハll 錨ﾄ蛭Fﾂ..tllii 冶8VVlHyl 勇ge優8爾▲■■一八一一一yyY Yー〔ソ標嵐∴迫ﾄﾄｨ而"山AA^VYYY 容藍E

補VYVVVA.l^儺rTJ'T
僊.■̲LJ‑ﾂ 抱r[IIIIIー馴

白lT白‑.,Y白'ーlI 白 Il
一一J≠ 白

I‑Test ド.‑Analysts. 一lーIl 白
05101520253035

Time (see)(C) Relative displacements between the TMD mass

0白5白1015

Test AnatyslS l2025(d) Accelerations at the TMD mass.

and the bridge model.

Fig･ 5･Comparison of the response time histories between the shaking table test andthe numerical analysュs, When the TafトEW component was applied as the input

motlOn.

30

Time (sec)35

Table4.

Ratio of the maximum values or the test result to that or analysts.

Taft EI Centro JMA Kobe Kobe Univ.
Acceleration

Displacement

TMD

Bridge

mass

1.032

0.987

modeトshaking

table

1.056

Steel舟ame ‑ bridge model

TMD

mass‑bridge

model

1.004

10.999

･145

1.063

0.973

0.935

1.010

0.904

0.970

0.91 6

0.971

0.908

0.884EjfecLiveness of the Powcdut TMD
Table 5 summarizes the maximum displacement of the bridge deck model among three cases which are nocontrol, a conventional TMD installation and the Powe血I TMD installation. The results of two cases ofno control and the Powe血l TMD installation are glVen by shaking table tests and that of a conventional

TMD installation which was optimized uslng the method by Den Hartog was given by the numericalsimulation･ The effect of the Powerful TMD on the reduction of a bridge deck displacement is 30through60 % Compared tothe case in which no device is installed･ On the contrary, a conventional TMDgives onlyTable

5. Comparison of the maximum displacements of the bridge deck model, whichwere normalized bythat of the case of no control, between the cases of thePowerful TMD installation and the conventional TMD installation.

Ta氏

Powe血lTMD

Conventional

0.498

TMD

EI

Centro

0.691

0.876

JMA

Robe

Robe

0.366

1.052

0.463

0.668

No control Device

0.699(Test) Max.‑46.484mm (5. 1 69sec)

ConventionalTMD (Analysis) Max.‑31.039mm (5.176sec)
Powerful TMD(Test) Max.‑ 1 6.992mm (4.235sec)

50 30 2 川 0̲ 20 3 仙 5I 一l 冤t .I:;(u ) uaEaUt21ds!Q〜 l一一 冩2 ●一 一● !∧AA

ﾘ ﾈ耳耳 "

9 豆陳ﾂrrﾂvﾂ粐窈B 停 ｣ｳ･ nr.

ー̲れ∫l 秒‑lI‖Tｦﾂ

1^逃坪軏ﾆ唯篦I 冢Xﾖ%Cr粐粐縒''..'t..㌔̲ t^.̲t!t^'.̲'､.｣､ 剃ﾇG%

I'｢iY..Y‥T., lILrt,. l一一一(ﾈ､ﾂﾒIlllt‖書 剳日...'t,.:Ht.' 冤J宇¥

｢謦粭討tI ヽ 冤 一 ｣ 棉ﾆﾂ

示ﾂ苒

･‑‑NocontrolDevlCe ‑ConventionalTMD ‑PowerrulTMD llー■一 ∫510152025Time (sec)Fig･ 6･Comparison fわr time histories ofdisplacement of the bridge deck model among

three cases which are no control, the conventional TMD installation and thePower仙TMD installation in case of the excitation using the JMA Robe input

motlOn.

Univ.

0 through 30 % reduction in tens of the bridge displacement･ Therefわre, much higher seismic control canbe attained by the proposed device than by a conventional device･ Fig･6 illustrates an example of comparison
for time histories of relative displacement between the shaking table and the bridge deck model among thethree cases in case of the excitation uslng the JMA Robe input motion･ Although the peak of displacement
response was glVen at 5 second in case of a conventional TMD, It was glVen One Cycle earlier at 4 secondin case of the Powerful TMD. Therefore,the proposed device acts effectively ln the early stage of theearthquake excitation･CONCLUSIONS

Conclusions derived inthis paper are summarized inthe followlngS:1) Simplification of the dynamic model for the bridge on which the Powerful TMD is installed was madefor design purposes･ Then, the optlmum tuning for the model was proposed･ TYle Validity of thesimplification and the tuning technique was verified bythe comparison withthose for ngorous model
previously presented by the authors.

2) The formulation for the dynamic model for the bridge on which the Powerful TMD is installed wasverified by the comparison of the response time histories between the shaking table tests and numericalsimulations.
3) Shaking table tests revealed that the seismic response of bridges can be reduced up to almost a half by
the application of the Powe血l TMD, in a case the mass ratio is 0.043 and the am ratio is 2.

ACKNOWLEDGEMENTS
This research was made as a part of the joint program on 'lDevelopment of seismic control technology using

high damplng materials applicable to long‑span bridgesH between the PWRI,the PWR'c and 19 pnvate firms
in Japan･ The authors would like to acknowledge the helpful guidance of Dr. H. Ohtsuka. TYlanks are also
glVen tO the members of the joint research program for their fruitful discussion.

REFERENCES

J.P. Den Hartog (1956). Mechanical VL'bratL'ons, 4th edn., McGraw‑Hill, New York.
G･B･ Warburton ( 1 982)･ Optimum Absorber Parameters for Various Combinations of response and Excitation
Parameters, Earthquake Eng. Struct. dyTZ., 10, 381‑401.
Ⅰ. Kaneko, Y. Adachi, M. Tanaka and T. Suzuki (1993). A seismic control device based on TMD with thesimple mechanism for control force amplification･ Proc･ 9th JEES, 1 861‑1866. (in Japanese)

I･ Kaneko, T･ Katsukawa, S･ Yoshitani and T. Suzuki (1995). Shaking table test of the Powerful TMD. Proc.
23rd JSCE Earthq･ Eng･ Symp., 525‑528. (in Japanese)