Response Mitigation on the Offshore Floating Platform System With Tuned Liquid Column Damper

Response mitigation on the offshore floating

platform system with tuned liquid column damper

H.H. Lee *, S.-H. Wong, R.-S. Lee

Department of Marine Environment and Engineering, National Sun Yat-sen University,

Kaohsiung 804, Taiwan, ROC

Received 21 March 2005; accepted 12 June 2005

Available online 1 December 2005

Abstract

In this study a typical tension-leg type of floating platform incorporated with the tuned liquid

column damper (TLCD) device is studied. The purpose is to find an effective and economic means to

reduce the wave induced vibrations of the floating offshore platform system. The floating offshore

platform has been widely applied for the offshore exploitation such as operation station, cross-strait

bridges, floating breakwater and complex of the entertainment facilities. For offshore platform being

employed as a public complex the stability and comfort to stay will be the major concern besides the

safety requirement. Therefore, how to mitigate the vibration induced from waves and similar

environmental loading becomes an important issue. The TLCD system utilizing the water sloshing

power to reduce the vibration of the main structure, a newly developed device that could effectively

reduce the vibrations for many kinds of structure is the first-time employed in the floating platform

system. In both the analytical and experimental results it is found that the accurately tuned TLCD

system could effectively reduce the dynamic response of the offshore platform system in terms of the

vibration amplitude and the resonant frequency.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: Tuned liquid column damper; Offshore structure; Response mitigation; Tension leg platform system

1. Introduction

A study on the vibration mitigation for an offshore floating platform system of tension leg

type subjected to the surge motion is presented in this study when the platform is incorporated

with the tuned liquid column damper (TLCD) device. The offshore platform has been widely

Ocean Engineering 33 (2006) 1118–1142

www.elsevier.com/locate/oceaneng

0029-8018/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.oceaneng.2005.06.008

* Corresponding author.

http://www.elsevier.com/locate/oceaneng

H.H. Lee et al. / Ocean Engineering 33 (2006) 1118–1142 1119

applied as the operation station for the offshore exploitation, floating breakwater, offshore

fish-farming platform and combination of the entertainment facilities. When the platform

located in the offshore marine environment many kinds of loading may exert on the platform

structure and result in tremendous vibration and instability. These vibrations not only

gradually damage the platform structural system but also cause the problem of uncomfortable

environment of staying. Therefore, the vibration stability of the platform becomes the major

concern besides the safety requirement particularly when the platform systems are being used

for public entertainment purposes. It is always a challenging task to try to reduce the vibration

induced from wave, flow or wind loading because of the difficulties of accurately monitoring

the platform motion and lack of the appropriate mitigation device applied to the system.

To monitor the platform motions many researches have been carried out based on the

platform type. For the tension-leg type platform system, a rather new development for the

offshore deep-sea exploitation, researches could be found in two major categories in

general such as those mainly focused on the wave-structure interaction problems and the

researches on the nonlinear effect of small body on the platforms. The problems of

structure-wave interactions were studied as early as 1970s in numerical method (Mei,

1978; Yamamoto et al., 1982), and also investigated in experimental test (Yamamoto

et al., 1982). Without considering the interaction between the structure and the wave,

numerical techniques were basically utilized such as the study on the coupled 6-degree

freedom of rigid-body motion for the platform (Jain, 1997) and the simulation for the

platform system by using a three-dimension beam element (Chatterjee et al., 1997).

While many efforts have been put in the study of the platform and waves, rare attention is

being paid on the vibration mitigation. In this study, however, the mitigation of vibration for

the platform system will be the purpose when the tuned liquid column damper (TLCD) is

incorporated in the platform system. The floating platform system employed for the study is

a platform supported by pontoons and anchored with pre-stressed tethers. When the

pontoon, the barge under the platform that provides the buoyant force has diameters smaller

than one fifth of the wavelength generally the wave structure interactions can be ignored.

Therefore, the small body in the waves will be assumed and the related theories of the waves

on the small body will be applied for the behavior of the platform system subjected to waves.

The TLCD system is a newly developed device by utilizing the water sloshing power to

reduce the vibrations of the main structure induced from the environmental loading. This

device has been widely applied to the traditional civil structures such as the application to

the structures in Sakitama Bridge and Shin Yokohama Prince Hotel in Japan (Fujino and

Sun, 1993) and the effectiveness for the vibration reduction was also verified. However,

the application of the TLCD to the offshore structural system has not been found in the

references yet. The TLCD was extended from the tuned liquid damper (TLD) device,

which has been widely discussed and studied in the late 1980s and early 1990s (Fujino

et al., 1988, 1992; Chaiseri et al., 1989). The TLCD is a U-shape tube-like device that

consists of water or similar liquid. The liquid passing through a small orifice opened at the

center of cross section of the tube causes the head loss of the liquid and then reduces

the vibration of the main structure attached. The head loss of the liquid induced from the

orifice at the cross section that has damping effect was firstly studied by Sakai et al. (1989).

In the study for the application of the TLCD system to the tower type structure Balendra

et al. (1995) provided a relationship between the tower height and the opening ratio for

H.H. Lee et al. / Ocean Engineering 33 (2006) 1118–11421120

the orifice. The application of TLCD system to the multi-degree-of-freedom structures

was also studied by Won et al. (1996) by using random vibration analysis for the

earthquake excitation.

Recently many improvement ideas for the TLCD system have been proposed such as

the variable orifice system or so called pressure control mechanism (Kareem et al., 1995)

and the studies on the characteristics of variable cross section between the horizontal and

vertical tube (Hichhock et al., 1997a,b; Gao and Kwok, 1997). In Gao and Kwok (1997)

study they found that the increase of the cross section of horizontal tube might reduce the

liquid column in vertical tube and the optimal parameters could be obtained to reduce the

amplification factor when the structure subjected to a harmonic vibration.

The purpose of this study is to investigate the vibration mitigation of a tension leg

offshore platform system of pontoon type when incorporated with the TLCD device and

subjected to surge wave motion. The stochastic analytical method in the frequency domain

was utilized and due to this the linearization scheme for the system was also applied. Since

the pontoon under the platform was small enough when compared to the wavelength the

Morison’s equation for small body was used for the calculation of wave forces. Numerical

examples were carried out and the results were discussed focusing on the platform motion

when subjected to waves of a range of periods. Along with the analysis, the parameter

effect such as the draft and the size of the pontoons for the floating platform was also

studied and discussed. It was found from the results that the accurately tuned TLCD

system could effectively reduce the dynamic response of the offshore platform system in

terms of the vibration amplitude and the resonant frequency. A preliminary experimental

test was also carried out for the feasibility study of the TLCD device applied to the floating

offshore platform. It was also found from the experimental test that after appropriate

design the TLCD could effectively suppress the wave-induced vibration.

2. Equation of motion for the platform with TLCD

For the TLCD device installed in a tension leg platform structure it is assumed that

there is no relative motion between the structure and the TLCD and when the structure

vibrates the liquid inside TLCD also sloshes. Fig. 1 is a typical tension leg strained

platform of pontoon type of two-dimension combined with the TLCD device that is

illustrated in Fig. 2, where a U-shape tube-like container has liquid inside and an orifice is

opened at the center of the cross section of the horizontal tube. The equation of motion for

the liquid in the tube is presented as (Sakai et al., 1989)

rdAdLd€H C

1

2rdAdhdj _Hj _H C2rdAdgH ZKrdAdBd

€X (1)

where hd is the factor of the head loss of the liquid, H _H and €H are the displacement,

velocity and the acceleration of the liquid in the vertical tube respectively. rd is the density

of the liquid in the tube. Ad is the inner area of tube cross section while Bd and Ld are the

length of horizontal portion and the total length of the tube occupied by liquid,

respectively. When incorporated with the TLCD device, in considering the 2-dimensional,

3-degree of freedom motion (surge, heave and pitch), the equation of motion for this

DECK

FLOATING BARREL

TENSION LEG

WAVESWL

Ls

TLCD

d D

X

Y

Fig. 1. Offshore floating platform system with TLCD device.


pontoon type platform system incorporated with the TLCD device may be presented as

M €X CCX_X CKXX Z Fw KrdAdBd

€H KrdAdLd€X (2)

M €Y CCY_Y CKY Y ZK2rdAdH €H KrdAdLd

€Y (3)

Mq€q CCq

_q CKqq Z ½2rdAdgH CrdAdðLdKBdÞ €H �1

2Bd C �h

� �C

1

2rdAdhdj _Hj _H �h

(4)

where X, Y and q are motions corresponding to surge, heave and pitch of the platform. M is

the mass of the platform system and Mq is the mass moment of inertia corresponding to

pitch motion while CX, CY and Cq are corresponding damping coefficients. Fw is the wave

force depends on both the waves and the structures submerged in the wave. KX, KY and Kq

are the stiffness corresponding to the surge, heave and pitch motion of the platform

X

H

H

Ld

orifice

ρd, Ad

Bd

Fig. 2. Illustration for the TLCD device.

em

Highlight

em

Highlight


respectively. �h is the distance from the center of mass moment of inertia to the bottom of

the structure.

It is noticed that except for Eq. (2) there are several nonlinear terms in above equations.

In the left-hand side of Eq. (1) and right hand side of Eq. (4) the nonlinear terms _Hj _Hj are

induced from the drag motion of the liquid in the tube. Since, a spectral analysis will be

employed in this study the technique of equivalent linearization (Wen, 1980; Xu et al.,

1992) is adopted here for those nonlinear terms in the equations. Assuming zero-mean

stationary Gaussian process for the velocity _H, the equation of motion for the liquid is

rewritten as

md€H Ccd

_H Cu2dmdH ZK

Bd

Ld

� �md

€X (5)

where the mass mdZrdAdLd and natural frequency udZffiffiffiffiffiffiffiffiffiffiffi2g=Ld

pwith g the gravitational

constant. The equivalent damping coefficient cd is obtained through the minimization of

the error 3 between the exact nonlinear drag damping and the equivalent one as

3 Zmdhd

2Ld

� �j _Hj _HKcd

_H (6)

After carrying out the minimization for this error the equivalent damping coefficient is

as

cd Z

ffiffiffiffi2

p

rrdAdhds _H (7)

where s _H is the standard deviation of the velocity of the liquid in the tube. Alternatively

Eq. (5) may be rewritten as

md€H Ccd

_H CkdH ZKmdad€X (8)

where kdZ2rdAdg is the stiffness coefficient for the TLCD system and adZBd/Ld is the

width-length ratio between the horizontal portion and the total length of liquid column. By

employing the same technique the first term at right-hand side of Eq. (3) can be written as

2rdAdH €H Z Q €H (9)

where QZ4ffiffiffiffiffiffiffiffi2=p

prdAdsH and sH is the standard deviation of the liquid displacement. Now

by substituting Eq. (9) back into Eq. (3) and employing the equivalent damping technique

of Eq. (7) for the nonlinear drag term in Eq. (4), equations of heave and pitch motion are

rewritten into

M €Y CCY_Y CKY Y ZKQ €H KrdAdLd

€Y (10)

Mq€q CCq

_q CKqq Z 2rdAdgð �h CBd=2ÞH Ccd�h _H C ½rdAdðLdKBdÞð �h CBd=2Þ� €H

(11)

Combining Eq. (8) with Eqs. (2), (10) and (11), equations of the motion for the

floating platform incorporated with TLCD subjected to wave force are given in


a matrix form as

md rdAdBd 0 0

rdAdBd M Cmd 0 0

Q 0 M Cmd 0

KrdAdBdðLdKBdÞð1=2Bd C �hÞ 0 0 Mq

266664

377775

€H

€X

€Y

€q

8>>>>><>>>>>:

9>>>>>=>>>>>;

C

cd 0 0 0

0 CX 0 0

0 0 CY 0

Kcd�h 0 0 Cq

266664

377775

_H

_X

_Y

_q

8>>>>><>>>>>:

9>>>>>=>>>>>;

C

kd 0 0 0

0 KX 0 0

0 0 KY 0

K2rdAdgð �h CBd=2Þ 0 0 Kq

266664

377775

H

X

Y

q

8>>>><>>>>:

9>>>>=>>>>;

Z

0

Fw

0

0

8>>>><>>>>:

9>>>>=>>>>;

(12)

Stiffness KX, KY and Kq corresponding to the surge, heave and pitch motion of the

platform after modified from Jain (1997) and Lee et al. (1999a) for uncoupled motions

can be presented as

KX Z 2K* (13)

KY Z 2rgG C2AE

[(14)

Kq Z 2K� �h2C2

AE

[L2

s C1

2rgGL2

s (15)

where r, g and G are the water density, gravity acceleration and the cross section area of

the submerged pontoon respectively. d0 and d1 is the draft of the structure before and

after applying the pre-tension force. A and E are the section area and young’s modulus of

the tether respectively, [ the tether length and LS is the platform width. It is noticed that

the buoyancy difference, angle difference and the elongation of the tether were all taken

into accounts for the rotational stiffness component corresponding to the pitch motion of

the platform. K* is the equivalent stiffness of the platform system induced by one of the

pretension tethers (Lee et al., 1999b; Lee and Wang, 2000) related to the material

properties and dimension of the tethers, as

K� ZrgG

[ðd1Kd0Þ

m

m C1

� �(16)


where m is the proportional stiffness parameter defined as the ratio between the tether

stiffness and the buoyancy force of the pontoon structure submerged in the water of unit

depth.

The excitation force induced from the surge motion of wave exerting on the unit length

of pontoon member is given by a modified Morison’s equation (Newman, 1977; Isaacson,

1979) as

FWðtÞ Z rCm

pD2

4_uKrCa

pD2

4€X C

1

2rCdDjuK _XjðuK _XÞ (17a)

or alternatively, after employing the linearization technique for the nonlinear drag term, as

FWðtÞ Z rCmG _uKrCaG €X C1

2rCdDjujuK

1

2rCdD2hjuji _X (17b)

where hjuji is the time average of the fluid velocity u, r the water density and CaZCmK1 is

the coefficient of added mass while Cm is the inertia coefficient, Cd the drag coefficient and

D is the dimension of the structural member normal to the incoming wave. Now after the

integration of the wave force on the unit pontoon the equation of motion accounting for the

total wave force, added mass and drag effect on the damping can further be presented as

md rdAdBd 0 0

rdAdBd m� Cmd 0 0

Q 0 m* Cmd 0

KrdAdðLdKBdÞð �h C1=2BdÞ 0 0 Mq

266664

377775

€H

€X

€Y

€q

8>>>>><>>>>>:

9>>>>>=>>>>>;

C

cd 0 0 0

0 C�X 0 0

0 0 C�Y 0

Kcd�h 0 0 C�

q

2666664

3777775

_H

_X

_Y

_q

8>>>>><>>>>>:

9>>>>>=>>>>>;

C

kd 0 0 0

0 KX 0 0

0 0 KY 0

K2rdAdgð �h CBd=2Þ 0 0 Kq

266664

377775

H

X

Y

q

8>>>><>>>>:

9>>>>=>>>>;

Z

0

F

0

0

8>>>><>>>>:

9>>>>=>>>>;

(18)

where m*ZMCrCac, with c the submerged volume of pontoon. For the simplicity of

the damping effect C*Z2m*zun while the total wave force is obtained through integration

over submerged depth of the pontoon as

FðtÞ Z1

2rDCd

ðd1

d0

ujuj ds CrGCm

ðd1

d0

_u ds (19)


3. Response spectral density for the platform with TLCD

Now with the equation of motion modified as the linear form we may further find the

complex frequency response function for the floating platform incorporated with TLCD

and then solve for the response spectral density. When the wave force is harmonic the

response of the platform and the liquid motion in the tube could be assumed to be

harmonic. The complex frequency response function for the liquid column in the tube is

obtained as

H0 Zu2rdAdBd

ðKu2md C iucd C2rdAdgÞF0HXðuÞ (20)

where HX(u) is the complex frequency response function corresponding to the surge

motion of the platform and F0 is the amplitude of the harmonic wave force. The complex

frequency response function corresponding to the surge, heave and pitch of the platform

can further be written as

HXðuÞ Z1

Kðm� CmdÞu2 C iuCX CKX K

u4r2dA2

dB2

d

ðKmdu2CiucdC2rdAdgÞ

h i (21)

HY ðuÞ ZQu4rdAdBd

ðKu2md C iucd C2rdAdgÞ½Ku2ðm� CrdAdLdÞC iuCY CKY �HXðuÞ

(22)

HqðuÞ Zu2rdAdBd Ku2rdAdðLdKBdÞ

12

Bd C �h� �

C2rdAdg �h C iu �hcd

� �ðKu2mq C iuCq CKqÞðKu2md C iucd C2rdAdgÞ

HXðuÞ

(23)

According to the spectral analytical scheme the response spectrum of the system may

be obtained through the cross of the system transfer function and the spectrum of the wave

exerting force such as

SXðuÞ Z jHXðuÞj2SFðuÞ (24)

SY ðuÞ Z jHY ðuÞj2SFðuÞ (25)

SqðuÞ Z jHqðuÞj2SFðuÞ (26)

where SX(w), SY(w) and Sq(w) are the spectrum of the structural response and SF(w) is the

wave force spectrum while jHX(u)j2 the transfer function of the system is obtained from

the complex frequency response function. The wave force spectrum is obtained from the

wave spectrum when the Morison’s equation for the waves on small body was employed

and statistical independence between the velocity and acceleration was assumed


(Borgman, 1967; Sarpkaya and Isaacson, 1981) and presented as

SFðuÞ Z ShðuÞ8

p

uKd

sin hðkdÞ

ðd

0

suðsÞcos hðksÞds

24

35

2

Cu4K2

i

k2

8><>:

9>=>; (27)

where su is the standard deviation of fluid velocity, k is the wave number, sZyCd the

depth of the water from the free surface d. Kd and Ki are given by KdZ0.5rDCd and KiZrGCm Sh(u) is the wave spectrum. In this study a typical Pierson–Moskowitz spectrum

(Pierson and Moskowitz, 1964) is applied and given as

ShðuÞ Z2pag2

u5exp

KB

u4

� �(28)

where aZ8.1!10K3 is the Philip’s constant and BZ0.74 g/U with the characteristic wind

speed U over the water. By substituting Eqs. (27) and (28) into Eqs. (24)–(26) the response

spectra of floating platform incorporated with the TLCD are obtained.

4. Optimal design for the TLCD incorporated in the platform system

To have a good performance for this special design platform system incorporated with

the TLCD an accurate tuning-up process for the parameters of TLCD corresponding to the

structural characteristics is required that is so called optimal design process. Firstly the

parameters of TLCD that may have significant influence on the system behavior are

selected. According to previous studies the dimensions of the TLCD such as Ld, Bd and

area of cross section Ad are influenced by two important parameters, namely, the mass

ratio between TLCD and the main structure and the coefficient of the head loss of the

liquid. These parameters must be determined carefully so that the optimum performance

can be achieved. The coefficient of the liquid head loss is related to the opening ratio of the

orifice, the characteristics of the main structure and the setting-up conditions of the TLCD.

Fig. 3. The Pierson–Moskowitz wave spectrum.

0.00E+002.00E+034.00E+036.00E+038.00E+031.00E+041.20E+041.40E+041.60E+041.80E+04

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Frequency (Hz)

Rep

onse

spe

ctru

m (

m2 -

sec) B=3

B=4B=5

(a)

B=3B=4B=5

0.00E+00

5.00E+00

1.00E+01

1.50E+01

2.00E+01

2.50E+01

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Frequency (Hz)

Res

pons

e sp

ectr

um (

m2 -

sec.

)

(b)

B=3B=4B=5

0.00E+00

2.00E-03

4.00E-03

6.00E-03

8.00E-03

1.00E-02

1.20E-02

1.40E-02(c)

0 0.5 1 1.5 2 2.5 3 3.5 4

Frequency (Hz)

Rot

aion

al r

espo

nse

spec

trum

Fig. 4. Spectral densities of the platform responses without TLCD devices for various pontoon dimensions. (a)

Response in X-direction (surge motion); (b) response in Y-direction (heave motion); (c) response in rotation (pitch

motion).


By ignoring the high frequency mode and taking only the surge motion from the

equation of motion of the platform system incorporated with TLCD as shown in Eq. (19),

under the Gaussian white noise type of excitation the variance of the response spectrum

can be written as (Crandal and Mark, 1973, modified by Chang et al., 1998).

s2X Z pSF

ðb20=a0Þða2a3 Ka1a4ÞCa3ðb

21 K2b0b2ÞCa1ðb

22 K2b1b3ÞC ðb2

3=a4Þða1a2 Ka0a3Þ� �

a1ða2a3 Ka1a4ÞKa0a23

� �(29)

where SF is the spectral density of the exciting force while as and bs are parameters related

to the structural properties of the platform and also the TLCD parameters such as the

0.00E+00

5.00E+02

1.00E+03

1.50E+03

2.00E+03

2.50E+03

0 0.1 0.2 0.3 0.4 0.5 0.6

Frequency (Hz)

Res

pons

e sp

ectr

um (

m2 -

sec)

No Damper

With TLCD

(a)

No TLCD

With TLCD

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

3.00E+00

0 0.5 1 1.5

Frequency (Hz)

Res

pons

e Sp

ecru

m (

m2 -

sec)

(b)

No TLCD

With TLCD

0.00E+00

2.00E-03

4.00E-03

6.00E-03

8.00E-03

1.00E-02

1.20E-02(c)

0 1 2 3 4

Frequency (Hz)

Rot

atio

nal r

espo

nse

spec

trum

Fig. 5. Comparison of spectral responses for the platform with and without TLCD devices corresponding to

variation of pontoon dimension (3m ). (a) Response comparison in X-direction (surge motion); (b) response

comparison in Y-direction (heave motion); (c) response comparison in rotation (pitch motion).


width-length ratio of the liquid ad and the damping coefficient cd as:

a0 Z KXkd; a1 Z KXcd CkdC�X ; a2 Z ðm� CmdÞkd CmdKX CC�

Xcd;

a3 Z ðm� CmdÞcd CmdC�X ; a4 Z ðm� CmdÞmdKm2

da2d; b0 Z kd;

b1 Z cd; b2 Z md; b3 Z 0

(30)

0.00E+00

2.00E+03

4.00E+03

6.00E+03

8.00E+03

1.00E+04

1.20E+04

1.40E+04

1.60E+04

1.80E+04

0 0.1 0.2 0.3 0.4 0.5

Frequency (Hz)

Res

pons

e sp

ectr

um(m

2 -se

c)

No Damper

With TLCD

(a)

0.00E+001.00E+002.00E+003.00E+004.00E+005.00E+006.00E+007.00E+008.00E+009.00E+001.00E+01

0 0.5 1 1.5

Frequency (Hz)

Res

pons

e sp

ectr

um (

m2 -

sec) No TLCD

With TLCD

(b)

No TLCDWith TLCD

0.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

5.00E-03

6.00E-03

7.00E-03(c)

0 1 2 3 4

Frequency (Hz)

Rot

atio

nal r

espo

nse

spec

trum


variation of pontoon dimension (4 m). (a) Response comparison in X-direction (surge motion); (b) response



Assuming that the standard deviation of the flow velocity of the liquid in the tube is

constant and ignoring the structural damping the optimal total length Ld and the damping

coefficient cd can be obtained through the differentiation of the variance of the response

spectrum and given as

LOPT Z2gm�

KX

(31)

0.00E+00

2.00E+03

4.00E+03

6.00E+03

8.00E+03

1.00E+04

1.20E+04

1.40E+04

1.60E+04

0 0.1 0.2 0.3 0.4 0.5

Frequency (Hz)

Res

pons

e sp

ectr

um (

m2 -

sec) No Damper

With TLCD

(a)

0.00E+00

5.00E+00

1.00E+01

1.50E+01

2.00E+01

2.50E+01

0 0.5 1 1.5

Frequency (Hz)

Res

pons

e sp

ectr

um (

m2 -

sec) No TLCD

With TLCD

(b)

No TLCD

With TLCD

0.00E+00

2.00E-03

4.00E-03

6.00E-03

8.00E-03

1.00E-02

1.20E-02

1.40E-02(c)

0 1 2 3 4

Frequency (Hz)

Rot

atio

nal r

espo

nse

spec

trum


variation of pontoon dimension (5 m). (a) Response comparison in X-direction (surge motion); (b) response



cOPT Z md mda2du2

d Cu2n C

u4d

u2n

K2u2n

� �1=2

(32)

where mdZmd/m* is the mass ratio, un and ud is the natural frequency of the platform in

surge motion and the TLCD respectively. When the natural frequencies of the TLCD and

the platform in surge motion are the same thus in the resonance the damping effect induced

by the TLCD device achieves the optimal performance and at this resonant stage the loss

73.2

99.7

0

73.367

81.5 84.4

6755.4

37.1

94.5

55.3

0

20

40

60

80

100

120

Surge Heave Pitch Total

Corresponding platform motion

Dis

sipa

ted

ener

gy (

%)

B=3 B=4 B=5

Fig. 8. Comparison of the energy dissipation for various pontoon dimension.


of the water head is optimum given by

hOPT Z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipmdKX

2ðm�Þ2

rBd

s _H(33)

When the structure is under the same exciting force the mitigation effect of TLCD is

determined by its effective damping presented as (Sun et al., 1993)

ze ZpSF

2u2nðm

�Þ2s2X

(34)

When this effective damping ratio is determined the area of the cross section of the tube

is obtained by substituting Eq. (28) into Eq. (32) and after rearrangement it is given as

Ad ZLdðm

� CmdÞ

B2rd

K2zeu3

nðm�Þ2 b2

0=a0

� �a2a3 Ca3 b2

1K2b0b2

� �Ca1b2

2Ka1a2a3 Ca0a23

� �B2

drd 2zeu3nðm

�Þ2 b20=a0

� �a1Ka2

1

� �� (35)

With Eqs. (29), (31) and (33) the optimal parameters for the TLCD such as the total

liquid length Ld, loss of the water head hd and cross section area Ad are determined and

then the optimal mitigation for the platform is also obtained. However due to the

interdependence of the standard deviation sH of the liquid velocity and the damping

coefficient as shown in Eq. (7) and then Eq. (31) the iteration scheme for the convergence

is needed for the solution.


5. Numerical results and discussion

Numerical examples for the offshore floating platform of pontoon type incorporated

with the TLCD system were carried out in this section. The investigation was focused on

the spectral density of the response of the platform and the effectiveness of mitigation

when the platform system was subjected to waves with various periods. The influence on

the mitigation effect from the pontoon diameter, draft and mass of the platform structure

was also examined. A typical Pierson–Moskowitz wave spectrum to simulate the wave

condition was adopted in this study and presented as Fig. 3.

0.00E+00

2.00E+03

4.00E+03

6.00E+03

8.00E+03

1.00E+04

1.20E+04

1.40E+04

1.60E+04

0 0.1 0.2 0.3 0.4 0.5

Frequency (Hz)

Rep

onse

spe

trum

(m

2 -se

c)

D1=2

D1=3

D1=4

D1=2

D1=3

D1=4

0.00E+00

5.00E+00

1.00E+01

1.50E+01

2.00E+01

2.50E+01

0 0.2 0.4 0.6 0.8 1

Frequency (Hz)

Res

pons

e sp

ectr

um (

m2 -

sec)

D1=2

D1=3

D1=4

0.00E+00

2.00E-03

4.00E-03

6.00E-03

8.00E-03

1.00E-02

1.20E-02

1.40E-02

(a)

(b)

(c)

0 1 2 3 4Frequency (Hz)

Rot

atio

nal r

espo

nse

spec

trum

Fig. 9. Spectral densities of the platform responses without TLCD devices for various pontoon drafts. (a)


motion).

0.00E+00

2.00E+03

4.00E+03

6.00E+03

8.00E+03

1.00E+04

1.20E+04

1.40E+04

1.60E+04

0 0.1 0.2 0.3 0.4 0.5

Frequency (Hz)

Res

pons

e sp

ectr

um (

m2 -

sec)

No TLCD

With TLCD

(a)

No TLCD

With TLCD

0.00E+00

5.00E+00

1.00E+01

1.50E+01

2.00E+01

2.50E+01

0 0.2 0.4 0.6 0.8 1

Frequency (Hz)

Res

pons

e sp

ectr

um (

m2 -

sec)

(b)

No TLCD

With TLCD

0.00E+00

2.00E-03

4.00E-03

6.00E-03

8.00E-03

1.00E-02

1.20E-02

1.40E-02(c)

0 1 2 3 4

Frequency (Hz)

Rot

atio

nal r

espo

nse

spec

trum


variation of pontoon draft (2 m). (a) Response comparison in X-direction (surge motion); (b) response comparison

in Y-direction (heave motion); (c) response comparison in rotation (pitch motion).


5.1. Mitigation effect with various pontoon diameters

Since the pontoons attached under the deck provide the buoyant force for the platform

and also take the pounding waves their dimensions will have influence on the initial draft,

the added mass and the wave force exerted on the platform system. Therefore it is interested

to examine the mitigation effect when the parameters of TLCD are designed optimally

corresponding to the pontoon diameter variation. Presented in Fig. 4(a, b and c) are spectral

densities of the platform responses in X-, Y- and rotational direction corresponding to the

frequency of wave exerting on the platform without TLCD device. These figures showed


that corresponding to the increase of the pontoon dimensions the response of the platform is

increased generally and shifted to a higher resonant frequency while for the heave motion

the response shifted to a lower frequency range. Figs. 5–7 are the comparisons of the

responses for the pontoon-type platform before and after the installation of a TLCD device,

corresponding to various pontoon dimensions of 3, 4 and 5 m, respectively. In the figure, (a),

(b) and (c) indicated the response of X-, Y- and rotational motion. It is observed from part (a)

of Figs. 5–7 that the TLCD could effectively reduce the response of the platform in surge

motion as the peak value is dropped while the resonant frequency almost remains

0.00E+00

1.00E+03

2.00E+03

3.00E+03

4.00E+03

5.00E+03

6.00E+03

7.00E+03

8.00E+03

9.00E+03

0 0.1 0.2 0.3 0.4 0.5

Frequency (Hz)

Res

pons

e sp

etru

m (

m2 -

sec)

No TLCD

With TLCD

(a)

No TLCD

With TLCD

0.00E+002.00E+004.00E+006.00E+008.00E+001.00E+011.20E+011.40E+011.60E+011.80E+012.00E+01

0 0.2 0.4 0.6 0.8 1

Frequency (Hz)

Res

pons

e sp

ectr

um (

m2 -

sec)

(b)

No TLCD

With TLCD

0.00E+00

2.00E-03

4.00E-03

6.00E-03

8.00E-03

1.00E-02

1.20E-02(c)

0 1 2 3 4

Frequency (Hz)

Rot

atio

nal r

espo

nse

spec

trum


variation of pontoon draft (3 m). (a) Response comparison in X-direction (surge motion); (b) response comparison

in Y-direction (heave motion); (c) response comparison in rotation (pitch motion).


unchanged. However, for the heave motion the response was dropped in some cases such as

Figs. 5(b) and 6(b) but for case as Fig. 7(b) the peak remained about the same while the

resonant frequency was shifted to a much lower range in all three cases. For the rotational

motion the TLCD device mostly reduced the response amplitude as indicated in part (c) of

Figs. 6 and 7 while the resonant frequency was not affected.

The mitigation effect of the platform incorporated with the TLCD may be represented

as dissipated energy through the integration for the spectral density of the response. Fig. 8

is the comparison for the effectiveness of the TLCD for various pontoon dimensions

represented as the percentage of the energy dissipated corresponding to the surge motion

(X-direction), heave motion (Y-direction), pitch motion and finally the total amount of

dissipated energy were also illustrated. It was found in this comparison that although the

energy dissipated in the heave and pitch motion is in a large variance it would still be a

small percentage for the total amount of energy. The most paramount energy is found in

the surge motion and thus overall, the energy is mostly dissipated in the surge motion. The

amount of the energy being dissipated is decreased with respect to the increase of the

pontoon dimension from 73 down to 55%. This means that the vibration mitigation

effectiveness (mostly for the surge and heave motion) reduces corresponding to the

increase of the pontoon dimension even though the suppression on the pitch motion seems

to be more effective correspondingly.

5.2. Mitigation effect with various pontoon draft

When the initial pontoon draft varies the pretension force of the tether and the wave

force will be influenced. Therefore it is interested to know the variation of the pontoon

draft on the mitigation effect induced from the TLCD. Fig. 9(a, b and c) are spectral

densities of the platform responses in X-, Y- and rotational direction corresponding to

the frequency of wave exerting on the platform without TLCD device while draft of the

pontoon is varied from 2 to 4 m. It is observed that corresponding to the increase of the

pontoon draft the response of the platform is reduced. The resonant frequency shifted to a

Surge Heave Pitch Total

Corresponding platform motion

0

20

40

60

80

100

120

Dis

sipa

ted

ener

gy (

%)

d1=2 d1=3 d1=4

55.4

37.1

94.4

55.354.6

33.5

98.5

54.546.5 44.2

99.4

46.4

Fig. 12. Comparison of the energy dissipation for various pontoon drafts.


higher range for both the surge and pitch motion while for the heave motion similar to the

case with various pontoon dimensions the response shifted to a lower frequency range.

Figs. 10 and 11 present the spectral density of the platform responses when the platform

is incorporated with or without the TLCD devices and the pontoon draft varies from 2 to

3 m, where (a), (b) and (c) are in the surge, heave and pitch motion correspondingly. Again

the mitigation effect induced from the water in the TLCD was realized in both cases. It was

found that the reduction on the peak value of the spectral density is very significant for the

surge and pitch motion as indicated in (a) and (c) of Figs. 10 and 11. For the heave motion

0.00E+00

2.00E+03

4.00E+03

6.00E+03

8.00E+03

1.00E+04

1.20E+04

1.40E+04

1.60E+04

0 0.1 0.2 0.3 0.4

Frequency (Hz)

Rep

onse

spe

ctru

m (

m2 -

sec)

No TLCD

With TLCD

(a)

No TLCD

With TLCD

0.00E+005.00E+001.00E+011.50E+012.00E+012.50E+013.00E+013.50E+014.00E+014.50E+015.00E+01

0 0.2 0.4 0.6 0.8 1

Frequency (Hz)

Res

pons

e sp

ectr

um (

m2 -

sec)

(b)

No TLCD

With TLCD

0.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

5.00E-03

6.00E-03(c)

0 1 2 3 4 5 6 7

Frequency (Hz)

Rot

atio

nal r

espo

nse

spec

trum

Fig. 13. Comparison of spectral responses of the platform with and without TLCD devices (MZ10 T). (a)


motion).

0.00E+00

5.00E+03

1.00E+04

1.50E+04

2.00E+04

2.50E+04

0 0.1 0.2 0.3 0.4

Frequency (Hz)

Res

pons

e sp

ectr

um (

m2 -

sec) No TLCD

With TLCD

(a)

No TLCD

With TLCD

0.00E+00

5.00E+00

1.00E+01

1.50E+01

2.00E+01

2.50E+01

3.00E+01

3.50E+01

0 0.2 0.4 0.6 0.8 1

Frequency (Hz)

Res

pons

e sp

ectr

um (

m2 -

sec)

(b)

No TLCD

With TLCD

0.00E+00

2.00E-03

4.00E-03

6.00E-03

8.00E-03

1.00E-02

1.20E-02(c)

0 1 2 3 4 5 6 7

Frequency (Hz)

Rot

atio

nal r

espo

nse

spec

trum

Fig. 14. Comparison of spectral responses of the platform with and without TLCD devices (MZ30 T). (a)


motion).


as indicated in (b) of Figs. 10 and 11 the peak of the spectrum seems to be not reduced but

shifted to a lower resonant frequency.

When we examined the mitigation effect by applying the integration over the

spectral density the dissipation of the energy was obtained as shown in Fig. 12. The

mitigation effect on the platform with three different draft of the pontoon from 2 to 4 m

was studied. It is noticed that the mitigation effect is reduced corresponding to the

increase of the pontoon draft from 55 down to about 46%. Corresponding to the

increase of the pontoon draft the TLCD system may have negative influence on

mitigation effect. However the correspondence is not very significant.

Fig. 15. Comparison of the energy dissipation for various platform mass.


5.3. Mitigation effect with various platform mass

The mass of the platform will influence the vibration frequency and the amplitude. Figs.

13 and 14 show two sets of spectral density of the platform response corresponding to the

wave frequency while the mass is varied from 10 to 30 ton. The mitigation effect is quite

pronounced for both the surge and pitch motion in both cases of 10 and 30 ton as indicated

in (a) and (c) of Figs. 13 and 14. However, the vibration was amplified a bit for the heave

motion and also shifted to a lower resonant frequency as indicated in (b) of both figures.

Fig. 15 shows a comparison of the mitigation effect in terms of the dissipated energy. It

seems that there is not much relevance between the mass variation and the effectiveness of

the TLCD shown in this comparison. This is clear that for the design of the TLCD system

the mass ratio between the TLCD and main structure is the most important factor.

6.42m10.67m

0.7m

Video recorder

Wave damper

Wave maker

Wave meter

Testing model-platform

Fig. 16. Testing set-up for the model-platform equipped with TLCD.

Fig. 17. The model-platform under test in the water tank.

0

20

40

60

80

100

120

140(b)

0 0.2 0.4 0.6 0.8 1

Frequency(Hz)

Res

pons

e Sp

ectr

um(c

m2 -s

ec) No Damper

With TLCD

–12

–7

–2

3

8

0 5 10 15 20Time(sec)

Dis

plac

emen

t(cm

)

(a)

No DamperWith TLCD

Fig. 18. Testing results for the model-platform. (a) Response comparison for the model-platform in the time

domain; (b) response comparison for the model-platform in the frequency domain.


0

20

40

60

80

100

120

140

0 0.2 0.4 0.6 0.8 1

Frequency(Hz)

Res

pons

e Sp

ectr

um(c

m2 -s

ec)

No DamperWith TLCD

Fig. 19. Analytical response comparison for the model-platform in the frequency domain.


Therefore, once the mass ratio has been determined through the optimum design process

the mitigation effect is not likely to vary significantly.

6. Experimental test and results

An innovational feasibility experimental test was performed for the floating platform

system incorporated with the TLCD devices. For this rather preliminary experiment the

schematic sketch for the test set-up is illustrated in Fig. 16, where a scaled down model of

the platform incorporated with the TLCD device is installed in a water tank. The water

tank is 30 m long, 1 m wide and 1.2 m high. At the left hand-side a hydraulic wave maker

is installed while at the right end of the tank the energy dissipation device is available to

reduce the reflecting waves. The wave meters and video recorders were both employed

and connected to the computers to collect experimental data. The testing model was

constructed with woods as well as the polymer material and a U-shape glass-wise tube was

attached on the top of the platform as a TLCD device in which the water was filled up to

designed level. The dimension of the platform is 0.91 m!0.91 m and the pontoon leg is

0.75 m long by 0.1 m!0.1 m section-area anchored to the bottom of the tank by strained

tethers. The parameters for the model of TLCD are as: LdZ1.20 m; BdZ0.84 m; AdZ2.027!10K3 m2. All of these parameters were through optimal design process for the

best suppression of vibration. The platform-model under test in the water tank is shown

in Fig. 17.

Two sets of test were carried out, one for platform without installation of TLCD device

and the other with TLCD system. The test conditions for the height of the incident waves

and the water depth were set to be constant as 10 and 70 cm while the period of wave was

varied around 1.9 s. Fig. 18(a) showed a typical response comparison for the platform

model in the time domain, where the response of the platform was suppressed obviously

while the TLCD system was installed. The corresponding response spectrum also

indicated a similar suppression effect as shown in Fig. 18(b). However, when compared to

the theoretical prediction as shown in Fig. 19, this mitigation phenomenon is a little less

significant. From this rather preliminary test the result still illustrates qualitatively that


the TLCD system can quite effectively suppress the wave induced vibration for the floating

platform of pontoon type.

7. Conclusions

In this study a typical offshore floating platform of pontoon type incorporated with the

TLCD mitigation system was studied. The mitigation was evaluated corresponding to the

variation of the diameter and the draft of the pontoon and the mass of the platform system.

It was found that as long as the parameters were tuned-up appropriately according to the

properties of main structure the TLCD could have a good performance on vibration

mitigation of a floating platform system.

According to the analytical results the energy dissipated from the TLCD device on the

floating platform system may reach a value higher than 70%, in many cases over 50%. It is

also evident that the variation of the parameter of draft and dimension of the platform

structure will influence the TLCD performance. The amount of the energy being dissipated

is decreased from 73% down to 55% with respect to the increase of the pontoon dimension

in the cases studied. It is also noticed that the mitigation effect is reduced corresponding to

the increase of the pontoon draft from 55% down to about 46%. For the mass variation the

relevance between the mass variation and the effectiveness of the TLCD is not significant.

Furthermore, in the experimental results from a preliminary test for the feasibility of the

TLCD application, this device could be effective on the vibration suppression for the

floating platform.

Therefore, it is concluded that according to this study the application of the TLCD

device to the offshore platform system to ease the wave-induced vibrations is encouraging

in terms of its economy, effectiveness and easiness to apply.

References

Balendra, T., Wang, C.M., Cheong, H.F., 1995. Effectiveness of tuned liquid column dampers for vibration

control of towers. Engineering Structures 17, 668–675.

Borgman, L.E., 1967. Spectral analysis of ocean wave forces on piling. Journal of Waterways and Harbor

Division 93 (WW2), 129–156.

Chaiseri, P., Fujino, Y., Pacheco, B.M., Sun, L.M., 1989. Interaction of tuned liquid damper (TLD) and structure:

theory, experimental verification and application. Structural Engineering and Earthquake Engineering 6, 273–

282.

Chang, C.-C., Hsu, C.T., Swei, S.M., 1998. Control of buildings using single and multiple tuned liquid column

dampers. Earthquake Engineering and Structural Dynamics 23, 388–417.

Chatterjee, P.C., Das, P.K., Faulkner, D., 1997. A hydro-structural analysis program for TLPs. Ocean Engineering

24, 313–333.

Crandal, S.H., Mark, W.D., 1973. Random Vibration in Mechanical Systems. Academic Press, San Diego, CA.

Fujino, Y., Sun, L.M., 1993. Vibration control by multiple tuned liquid dampers (MTLDs). Journal of Structural

Engineering 119, 3482–3502.

Fujino, Y., Pacheco, B.M., Chaiseri, P., Sun, L.M., 1988. Parametric studies on tuned liquid damper (TLD) using

circular containers by free-oscillation experiments. Structural Engineering and Earthquake Engineering 5,

381–391.


Fujino, Y., Sun, L.M., Pacheco, B.M., 1992. Tuned liquid damper (TLD) for suppressing horizontal motion of

structures. Journal of Engineering Mechanics 118, 2017–2030.

Gao, H., Kwok, K.C.S., 1997. Optimization of tuned liquid column dampers. Engineering Structures 19, 476–

486.

Hitchcock, P.A., Kwok, K.C.S., Watkins, R.D., Samali, B., 1997a. Characteristics of liquid column vibration

absorbers (LCVA)—I. Engineering Structures 19, 126–134.

Hitchcock, P.A., Kwok, K.C.S., Watkins, R.D., Samali, B., 1997b. Characteristics of liquid column vibration

absorbers (LCVA)—II. Engineering Structures 19, 135–144.

Isaacson, M., 1979. Nonlinear forces on bodies. Journal of Waterways and Harbors Division ASCE WW3, 213–

227.

Jain, A.K., 1997. Nonlinear coupled response of offshore tension leg platforms to regular wave forces. Ocean

Engineering 24 (7), 577–592.

Kareem, A., Kline, S., 1995. Performance of multiple mass dampers under random loadings. Journal of Structural

Control 3–5, FP5-19–FP5-28.

Lee, H.H., Wang, P.-W., 2000. Analytical solution on the surge motion of tension leg twin-platform structural

system. Ocean Engineering 27, 393–415.

Lee, H.H., Wang, W.-S., Wang, P.-W., 1999a. Dynamic motion of tension leg platform with wave–large body and

wave–small body interactions. International Offshore and Polar Engineering Conference 1, 309–314.

Lee, H.H., Wang, P.-W., Lee, C.P., 1999b. Dragged surge motion of tension leg platforms and strained elastic

tether. Ocean Engineering 26, 575–594.

Mei, C.C., 1978. Numerical methods in water wave diffraction and radiation. Annual Review of Fluid Mechanics

10, 393–416.

Newman, J.N., 1977. Marine Hydrodynamics. MIT Press, Cambridge, MA.

Pierson, W.J., Moskowitz, L., 1964. A proposed spectral form for fully developed wind seas based on the

similarity theory of S.A. Kitaigorodskii. Journal of Geophysical Research 69, 5181–5190.

Sakai F., Takaeda S., Tamaki T. 1989. Tuned liquid column damper—new type device for suppression of

building vibration, Proceedings of International Conference on High-rise Buildings, pp. 926-931.

Sarpkaya, T., Isaacson, M., 1981. Mechanics of Wave Forces on Offshore Structures. Van Nostrand Reinhold,

New York.

Sun, K., Cheong, H.F., Balendra, T., 1993. Effect of liquid dampers on along-wind response of structures.

Proceedings of the Asia-Pacific Vibration Conference ’93, Kitakyushu, Nov. 1993, Symposium on FIVES,

pp. 835–840.

Wen, Y.K., 1980. Equivalent linearization for hysteretic systems under random excitation. Journal of Applied

Mechanics 47, 150–154.

Won, A.Y.J., Piers, J.A., Haroun, M.A., 1996. Stochastic seismic performance evaluation of tuned liquid column

dampers. Earthquake Engineering and Structural Dynamics 25, 1259–1274.

Xu, Y.L., Samali, B., Kwok, K.C.S., 1992. Control of along-wind response of structures by mass and liquid

dampers. Journal of Engineering Mechanics 118, 20–39.

Yamamoto, T., Yoshida, A., Ijima, T., 1982. Dynamics of elastically moored floating objects. In: Kirk, C.L. (Ed.),

Dynamics Analysis of Offshore Structures. CML, Southampton.

Documents

Response Mitigation on the Offshore Floating Platform System With Tuned Liquid Column Damper