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Numerical Analysis of Tuned Liquid

Dampers for Structural Vibration Control

by

KamalenduGhosh

09CE3112

Under the Guidance of

Prof. Arghya Deb

Department of C!l Engneerng

"ndan "nsttute of #e$hnology

Kharagpur

2012%13


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Kharagpur%&21302

CERTIFICATE

#hs s to $ertfy that the pro'e$t ttled (Numerical Analysis of

Tuned Liquid Dampers for structural Control) s a bona*de

re$ord of the +or, $arred out by amalendu

!"os"#$%CE&''()*under my super!son and gudan$e for the

partal ful*llment of the re-urements for the degree of a$helor

of #e$hnology n C!l Engneerng/ "ndan "nsttute of #e$hnology/

Kharagpur.

Dr. Arghya Deb



Kharagpur

Date

2


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TA.LE /F C/NTENTS

C"apter '0 Introduction

1

C"apter (0 Literature Sur2ey

3

C"apter &0 4et"odolo-y

&5' T"e Approac"

6

&5( Some 4at"ematical formulations6

&5( Finite Element 4odel

&5&5' TLD

%

&5&5( Liquid

%

&5&5& Tan+

'$

&5&51 Interactions

'$

&5&53 !ra2ity Load

'$

&5&56 4odi7cations

'$

&5&58 4odelin- for a -i2en structure

'(

C"apter 90 Results

'3


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C"apter 10 Conclusion

'%

C"apter 30 Future :or+

($References

('

4


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'5 Introduction

#he $urrent trend of buldng stru$tures of e!er n$reasng heght and the use

of lght+eght/ hgh strength materal s often a$$ompaned by n$reasng

sus$eptblty to e$tatons su$h as +nd/ o$ean +a!es and earth-ua,es. #o

redu$e the rs, of stru$tural falures/ part$ularly durng a $atastroph$ e!ent/

t has be$ome mportant to sear$h for pra$t$al and e5e$t!e de!$es for

suppresson of these !bratons.

#he de!$es used for suppressng the stru$tural !bratons $an be

$ategor6ed a$$ordng to ther energy $onsumpton

Pass!e Control

A$t!e and 7em%a$t!e $ontrol

ybrd system $ontrol system.

Pass!e $ontrol de!$es are systems that do not re-ure eternal energy

supply. 7u$h systems are relable as they are una5e$ted by po+er outage/

+h$h s $ommon durng earth-ua,es. #hese de!$es dsspate energy usng

the stru$ture8s o+n moton to produ$e relat!e moton +thn the $ontrolde!$e. An ad!antage of su$h systems s ther lo+ $ost of mantenan$e

be$ause no a$t!aton me$hansm s re-ured. Eamples of su$h systems

n$lude !s$ous dampers/ tuned mass dampers/ tuned liquid dampers/ et$.

"n #Ds the sloshng moton of the :ud that results from the !braton of the

stru$ture dsspates a porton of the energy released by the dynam$ loadng

and therefore n$reases the e-u!alent dampng of the stru$ture. #uned

sloshng dampers $an be $lass*ed nto t+o $ategores/ shallo+ and deep+ater dampers based on the rato of +ater depth to tan, length n the

dre$ton of moton. #he #D system reles on the sloshng +a!e de!eloped at

the free surfa$e of the l-ud to dsspate a porton the dynam$ energy. #he

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gro+ng nterest n #Ds s due to ther lo+ $aptal and mantenan$e $ost and

ther ease of nstallaton n stru$tures.

(5 Literature Sur2ey

Pass!e dampng usng #Ds s a $hallengng area of resear$h and nno!aton

and as su$h a host of lterature +h$h $o!ers ths d!erse area ests. 7ome

referen$es are summar6ed here +th a gst of ther $ontent.

ds$usses the numer$al modelng of a tuned l-ud

damper=#D> as an e-u!alent tuned mass damper +th non%lnear st5ness

and dampng. #hs non%lnear st5ness and dampng =?7D> model $aptures

the beha!or of the #D system ade-uately under a !arety of loadng

$ondton. "n part$ular/ ?7D model n$orporates the st5ness hardenng

property of the #D under a large ampltude e$taton.

@eed et al. =199> $on$luded that the dsspaton of !braton energy due to a

#D n$reases as the e$taton ampltude n$reases and the #D beha!es as

hardenng sprng system. #he authors also sad that to a$he!e a more robust

system the desgn fre-uen$y for the damper/ f t s $omputed by the

lnear6ed +a!e theory/ should be set at a !alue lo+er than that of the

stru$ture response fre-uen$y/ sn$e ths may result n the a$tual nonlnear

fre-uen$y of the dampng mat$hng the stru$tural response. #he authors also

found that e!en f there damper fre-uen$y s mstuned slghtly/ the #D

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al+ays performs +ell. #hey obser!ed no ad!erse e5e$ts due to slght

mstunng.

Kareem et al. =2009> elaborated on the nherent non%lnear beha!or of the

#Ds. #he non%lnear beha!or of the #Ds s $hara$ter6ed by an ampltude

dependent fre-uen$y response fun$ton/ +h$h translates nto $hanges n

fre-uen$y/ as +ell as $hanges n the dampng due to sloshng/ +th

ampltude. "n ths paper #Ds are modeled usng a sloshng%slammng =72>

analogy/ +h$h $ombnes the dynam$ e5e$t of l-ud sloshng and

slammngBmpa$t.

Kareem et al. =1994> ds$ussed the dynam$ $hara$terst$s and

e5e$t!eness of multple mass dampers =Ds>. A $olle$ton of se!eral mass

dampers +th dstrbuted natural fre-uen$es under random loadng +ere

n!estgated n ths paper. #he D atta$hed n parallel $on*guraton

mod*es the transfer fun$ton of the damper buldng +th a sngle #uned

ass Damper. #he D parameters $onsdered here n$lude the fre-uen$y

range of the Ds/ dampng rato of nd!dual dampers/ and the number of

dampers. A parametr$ study +as $arred out to $ompare the results +th a

sngle tuned mass damper. "t +as demonstrated that he D $on*guratons more e5e$t!e n $ontrollng the moton of the prmary system.

&5 4et"odolo-y

3.1 The Approach

In a Lagrangian finite element formulation the mesh motion is the same as the motion of the

material. In TLDs, the motion of the liquid, especially at the free surface, can result in extreme

deformations. This will result in severe mesh deformation as well. This will cause the Jacobian to

become very small or even negative. negative determinant of the Jacobian matrix would cause


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the analysis to terminate. In order to prevent this, and to permit simulation of large amplitude

sloshing motions, a coupled Lagrangian!"ulerian approach was adopted.

In contrast to a Lagrangian analysis, where the Lagrangian elements comprise a single material

only, in an "ulerian analysis the "ulerian elements may not always comprise of a single material#

many may be partially or filled with air i.e. void. The "ulerian material boundary must,

therefore, be computed during each time increment and generally does not correspond to an

element boundary. The recomputed material boundaries must ensure conservation of mass,

momentum and energy.

In addition, the nodes in an "ulerian finite element mesh are fixed in space, unli$e in the

Lagrangian case where the nodes move with the material. The "ulerian mesh is typically a

simple rectangular grid of elements constructed to extend well beyond the "ulerian material

boundaries, giving the material space in which to move and deform. If any "ulerian material

moves outside the "ulerian mesh, it results in loss of mass from the simulation, and in the

present application at least, in accurate results.

%nli$e the liquid in the tan$, the tan$ and its walls undergo limited. &e thus adopt Lagrangian

finite element analysis for these parts. 'ut at the liquid structure interface, it is necessary to

reconcile the two different formulations through "ulerian!Lagrangian (ontact. In this approach,at the interface the liquid motion results in traction forces at the boundary of the solid, while the

displacements of the solid result in displacement or velocity boundary conditions for the liquid.

3.2 Some Mathematical formulations

The governing equation in the fluid domain is based on potential flow theory. If the fluid is

assumed to be inviscid and incompressible, and the flow irrotational, the governing equation isgiven by Laplace)s equation,

*+-

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&here /x,y,t0 is the velocity potential function. 1owever in a moving tan$ the total potential

function can be split into two parts, the potential function tdue to the moving tan$ and the

disturbed potential function sdue to fluid motion.

- t2 s

"xpressing the Laplace equation in terms of s

*+s-

Hydrodynamic sloshing force

1ydrodynamic sloshing force is obtained by integrating the hydrodynamic pressure over the

pro3ected area of the tan$, which in turn acts on the structure as a base shear force. If the base

shear force is out of phase with the motion of the structure, it is li$ely to damp structural motion.

The base shear is defined as#

4i-b5

!

6

where 78 and 7+are the free surface sloshing elevation on the two side walls of the tan$, p/!L9+,

y,t0 are the liquid pressures at x-/2:0L9+, and b is the width of the tan$, the dimension in the

other orthogonal direction.

1ydrodynamic pressure is obtained as#

;-!p5. 2 2 g.y 2 st 2 ?s 6

In the above equation stis the acceleration included in the tan$ due to structure response,

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?s is the damping term in liquid sloshing.

Shallow water and deep water Tuned Liquid Damper

Tuned sloshing dampers can be broadly classified into shallow and deep water dampers based onthe water depth to tan$ dimensions in the direction of motion. In shallow water dampers

structural vibration energy is mainly dissipated through the sloshing and wave brea$ing. The

sloshing of the fluid generates hydrodynamic force , primarily in the form of base shear that acts

to control structural response.

'ased on the linear wave theory /Lamb 8@A+0, fundamental natural frequency of the liquid

sloshing fw, is

fw=

1ereLis the length of tan$ measured in the direction of excitation and his the stationary water

depth.

In case of shallow water dampers the h/Lratio is limited to .8


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fw=

It is clear from the above equation that for deep water dampers the sloshing frequency is

insensitive to water depth ratio. 1ence deep water dampers can be tuned for short period

structures and hence there is a possibility that they can perform well for earthqua$e or wind

excitation.

3.3The Finite Element Model

The details of the finite element model are discussed in the next section.

3.3.1 TLD

The TLD was modeled using 'C%9"xplicit. ince liquid sloshing is essentially a +!D

phenomenon, a strip model of the tan$ was chosen. The dimensions of the tan$ and the "ulerian

domain are as given below#

Part Dimension( in metres)

Tan$ A.


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increment and thereby allowing the analysis to be completed in a realistic time frame. viscosity

of magnitude of 8."!


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4ollowing figures depict the final TLD model#

4ig. 8

The TLD model set up

1


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4ig. +

The TLD model with "ulerian domain.

4ig. A

The TLD model and "4 at the end of tep!8 /ravity tep0

Initially a liquid with density-8 $g9mA

was ta$en. 1owever the dampers with thisliquid did not show much damping. This was because the mass ratio 8was too small. &e

thus increased the density to +E8 $g9mAto obtain a reasonable mass ratio of .8 for the

shallow water damped tuned to the frequency of the structure.

3.3.7 Modein! for a !iven structure

structure was modeled using +!D beam elementsM so that it is first "igen frequency

would be small enough to allow us to model a shallow water tan$. The following

properties and dimensions were chosen#

'eam (olumn

1ass ratoass of damperBass of stru$ture

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"- A8.;a "-A8.;a

Density- +E$g9mA Density-+E $g9mA

Length- 8 m. Length- +


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4ig. !s #me

or the stru$ture +th shallo+ +ater damper/ dampng s $learly !sble.

o+e!er/ due to phase shft the dampng e5e$t of the deep +ater

damper s not $on$lus!e from ths *gure.

#he nternal energy of the stru$ture +as therefore eamned to ma,esome de*nt!e $on$lusons regardng dampng e5e$ts =g. 11>.

21


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g. 11#me stores of the "nternal Energy of the stru$ture

Agan nothng $on$lus!e $an be dra+n from ths *gure. #he nternal

energy +as therefore ntegrated to obtan g. 12

g. 12#otal "nternal Energy

#hus t s $lear that o!er the duraton of the analyss/ the total nternal

energy =stran energy> of the stru$ture for a shallo+ +ater damper s

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lesser than that n the $ase of an undamped stru$ture. 7n$e the total

nternal energy of the stru$ture s a measure of the deformatons

undergone by the stru$ture o!er the duraton of the loadng/ t s $lear

that the shallo+ +ater damper results n sgn*$ant dampng. o+e!er/ the deep +ater does not $ause dampng. @ather the

stru$tural stran energy s hgher at the end of e$taton and the

stru$ture s n dstress. #o understand the reason for ths the KE of the stru$ture and the +ater

for the shallo+ and deep l-ud $ases +ere plotted

g. 13Knet$ Energy of the stru$tures +th shallo+ +ater damper and deep%+ater damper

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g. 1Knet$ energy of the l-ud n the tan, n shallo+ +ater and deep%+ater damper

#he KE of the stru$ture/ the +ater and the stran energy are all larger

for the deep +ater damper $ase. o+e!er the total energy of the

e$taton s the same for both the shallo+ and the deep +ater damper

are the same. #hus t $an be $on$luded that the response of the deep

+ater damper s unstable. #o $on*rm ths a #2analyss of Koyna e$taton +as $arred out.

2 # ast ourer #ransform

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g. 14Koyna a$$eleraton data =tme hstory>

g. 1;re-uen$y Doman

rom g. 1; t s $lear that the Koyna e$tatons $ontan sgn*$ant

spe$tral densty n lo+ fre-uen$y bands $lose to the natural fre-uen$y

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of the stru$ture. Fhen the shallo+ +ater damper s repla$ed by the

deep +ater damper/ the addtonal mass of the +ater results n a

redu$ton n the natural fre-uen$y of the $omposte stru$ture%damper

system. "n the present $ase/ +hat appears to be happenng s that the

redu$ton n fre-uen$y of the stru$ture%damper system results n near%

resonan$e $ondtons/ sn$e no+ sgn*$ant spe$tral densty s present

at or near the natural fre-uen$y of the stru$ture. #hs results n energy

gro+th or Pnegat!e dampngQthat results n unstable response.

#he e5e$t of n$reasng the number of #Ds/ +th fre-uen$es spannng

a range of fre-uen$es/ +as studed by atta$hng t+o #D s to the

stru$ture.

95( Sub;ectin- t"e structure to El-Centrobase e +ere sub'e$ted to El-Centro base

e$taton and analy6ed. #he follo+ng results +ere obtaned

g5 1&

Dspla$ement =of the topmost node> !s #me

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oth n the $ases of shallo+ and deep +ater dampers dampng +as

obser!ed. Agan the nternal energy of the stru$ture +as eamned to get a more

a$$urate estmate of the amount of dampng.

g 1

#me stores of the "nternal Energy of the stru$ture

#he tme hstory of the nternal energy sho+s that the nternal

energy of the stru$ture n the damped stru$ture =both shallo+ and

deep%+ater damper> s $onsderably smaller than the undamped

stru$ture.

#he total nternal energy presents a $learer p$ture

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g. 19

#otal "nternal Energy

#hus both the deep +ater damper and the shallo+ +ater damper/

tuned to the stru$tural fre-uen$y/ damp the stru$ture $onsderably. # +as performed on the El-Centro e$taton +th the am of

desgnng a damper tuned to the domnant e$taton fre-uen$y and

studyng ts e5e$ts on stru$tural !bratonal mtgaton.

g. 20El%Centro a$$eleraton data

2


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g. 21re-uen$y Doman

#he deep +ater damper thus modeled tuned to the most domnant

e$taton fre-uen$y ho+e!er dd not ehbt any sgn*$ant dampng.

g. 22Dspla$ement of topmost node !s. tme

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g. 23"nternal Energy of the stru$ture

95( Sub;ectin- t"e structure to periodic #sinusoidal) base

e


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g. 24

Dspla$ement of the topmost node !s. tme

g. 2;

"nternal Energy tme hstory

31


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g. 2&


95(5'5( AL >$5$&&

g. 2

Dspla$ement of the topmost node !s. tme

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g. 29


g. 30


95(5'5& AL > $5$1

33


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g. 33


95(5(95(5( Sinusoidal e$5$(

34


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g. 3

Dspla$ement of topmost node !s. tme

g. 34


g. 3;


3;


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95(5(5( AL >$5$&&

g. 3&

Dspla$ement of topmost node !s./ tme

g. 3


3&


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g. 39


95(5(5& AL>$5$1

g. 0

Dspla$ement of topmost node !s. tme

3


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g. 1


g. 2


95(5& Bercenta-e dampin- for all t"e cases discussed abo2e

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Type ofe


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not bear out ths trend the per$entage dampng de$reased as +as

n$reased to 0.04. #hus the n:uen$e of the ampltude of e$taton

does not appear !ary monoton$ally.

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References

1> AAHI7 ;.9 Jnlne Do$umentaton.2> Dorothy @eed/ arry eh/

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