Hydrodynamic behaviour of a huge floating body

a

International Conference In Ocean Engineering COE '96 UT Madras, India, 17-20 Dec. 1996

HYDRODYNAMIC BEHAVIOUR OF A HUGE FLOATING BODY

YoshyukiINOUE

Shigeru TABETA

Yokohama National University, Yokohama, Japan.

SYNOPSIS: A very large floating structure like an airport has relatively small rigidity, beuse it has a very small vertical dimon c..wdtO its horiwntal dimtcions. The elastic deformations cannot be negloeted in comparison with the motion as rigid body. mesi the fluid-structure interactions should be considered in the analysis of dynamic behaviours in waves. Interaction of a huge finiiting body and ocean environment is also important to estimate enviromnental forces. In this paper, a conceptual design of a floating airport is shown and environmental forces and hydroelastic responses are investigated for the very large floating body. A two-dimensional separated region method is used to calculate the dynamic forces and response in waves. Further, some numerical simulations by means of multilevel thodel are carried out to discuss the interaction of currents and a vary large floating structure in a bay.

INTRODUCTION

Although a lot of projects of ocean airport are being discussed in Japan, application of reclamation system in the deeper sea is very difficult to realize because of the necessity of the massive sand for the titling of the land. Long period of construction and environmental impacts are also obvious defects of this system. On the contrary, an airport with floating system clearly has advantages in some points. Especially, it is considered to give less damage to the environment than reclaimed one. Therefore, matters for investigation in designing a floating airport system are frequently discussed recently (e.g. moue, et al.(1995)).

A very large floating structure like an airport requires special response analysis techniques, because it has a very small vertical dimension compared to its horizontal dimensions. The elastic deformations due to the dynamic effects of waves may be of the order that cannot be neglected in comparison with the motion as rigid body. Then the hydroelastic interactions should be considered in the predictions. Further, when a normal size floating structure is constructed in the sea, the current forces acting on the floating body can be estimated by means of many empirical methods. On the other hand, there is no empirical data for estimating current force for a very large floating structure such as floating airport which has not been built in the world. However, the current force acting on the floating structure is important when à mooring system is designed.

In this paper, a conceptual design of a floating airport is shown and environmental forct and hydroelastic response are investigated for the floating structure. A two-dimensional separated region method is used to calculate the hydrodynamic forces and response in waves. Ijima et. al.(1972) analyzed motion responses of moored and freely floating rectangular bodies in waves at finite water depth by using the separated region method. This method can pmvide better accuracy in a short period of computer time, even including shallow water effect. In the present study the authors use this method combined the elastic analysis to calculate the responses of very large floating structure in waves. When elastic response is calculated, it is assumed that the elastic structure is divided into several equal components. These components are assumed to be connected with each other by elastic bais with bending rigidity. The continuous response of the elastic structure is expressed approximately as the motion response of every component. The effect of bending rigidity to motion responses is

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considered as the restorng forces in rigid motion equations of every divided component. The hydrodynarnic and ela.stit deformation problems can be solved simultaneously by this method.

Further, some numerical simulations are carried out to discuss the interaction of currents and a very large floating structure in a bay. A multilevel model is used in order to desçribe current field considering the influence of the floating structure. LCurrent ¡forces äcting on the structure are estimated as well by the humerical simulations.

CONCEPTUAL DESIGNS OF FLOATING AIRPORTS Descripiion of the Airport Model

In this paper, a box-shaped pontoon type floating airport is considered because of its simple shape and possibility of easier construction. According to both the investigation of existing airports and past studies about floating airports, the principal particulars of the airport model were determined as shown In Table.j..

Table 1. Principal jxirficuiais of the floating airport

The arrangement of the facilities sucb as terminal buildings which are necessaiy for the function of the airport was also mughly designed. The total weight of the structure was estimated which determine the displacement and the draft of the floating body. Fig.i shows the conceptual sketch of the floating airport úsed in the present study.

floatln Breakwater

Generally speaking, the responses of printoon type floating structures are bigger'than those of a semisubmersible ones in waves. The vertical responses of the floating, airport should not exceed the tolerance limit for safe lsnrting and takeoff of airplanes. Then a floating breakwater is planñcd to be used as a protection of the floating airport in waves.

LX BXD(m) 3000 X 800 X 4.5

Draft (m) 1.67

Dek Area (ha) 219.0

Fig.1 Conceptual sketch of the floating airport

Environmental Conditions

Environmental conditions for calculating the acting forces and dynamic responses of the airport are shown in Table 2. They are approximately corresponding to the environmental conditions of major inner bays in Japan. The water depth where the airport is located is assumed to be about 20 meters. The mean annual maximum values were used for the operational condition, and those of 100 years return period were used for the survival condition.

Table 2. Environmental conditions

METHODS FOR NUMERICAL ANALYSIS Force and Response in Waves

When the dynamic behavior of very large floating bodies are analyzed, it might not be appropriate to treat the floating structure as a rigid body because the rigidity is comparatively small. Then a two-dimensional separated region method was applied to analyze acting forces and motion responses of the floating body in waves;

lr the coordinate system chosen to describe the problem (see Fig.1), x is the horizontal coordinate that is coincident with the undisturbed .free-surface, and z is positive pointing upwards. The floating body and the fluid domain are divided into some partitions as described in Fig.1. In each fluid domain, incompressive and irrotational conditions which allow the existence of velocity potential

'

are assumed and infinite amplitude oflineartheory is assumed. The velocity potentials in each fluid domain are expressed as follows:

4, =(Ae''+Be

IJ)(OShk(Z+IZ)_{cosli k/i} +

Ce

LJ.rl) k,,(Z+/t)

« / ' COSknh

''

-Fe''

+ G L, ,,cosk,,(z+h)

cosh kit « i " cos kjz

coslt(Çx) siniz(S'x)

4,, = I, [(Da

/ +E,,,sinh .l' )cosS(z + qit) COShl.t (z + h)

+2z(-l) 7{zj wh cos(

_)}

(s)z)2sin ¡Tu ] (j = 1,2, ,N-1) (3)

where

In above expressions, ¡ is the half breadth of the floating body, 1' is the half breadth of the partitioned floating body, h is water depth, qh is the draft, x-0 denotes the coordinate of the center of each partitioned floating body, x1 is the local coordinate whose origin is x10, and k,k, are eigenvalues determined by the following formula:

9Çk=klztankh=k,,htank,ii

_(n=1,2,3,»)

(5)

In Eqs. from (1) to (3),A,B,C,,,F,G,,DSI,E,, are all complx constraints. A is representing the incident waves, B,the reflected waves F,the transmitted waves. C and D,,are non-propagating wave modes, D,, and E, arbitrary coefficients.

Fig.2. A sketch of a partitioned elastic box-shaped structure The continuous condition at the boundary of each partitioned fluid domain are as follows:

4,,, _4,--_-j

&4,,,, a4,

(-qh >z >-h)

at

1=!

(6) N'

--

(-qh >z > - h)

at x=-1

(7) 4,j

.inj,=

''(j=1,2,..,N-1)

(0>z>-qh)

at x=x1,-1' (8)

The rotating motion of the partitioned bodies are ignored because the floating body is very large. The force apting on the floating body are assumed to be composed of hydrodynamic force, restoring force due to hydrostatic pressure, and restoring force due to the rigidity. Then the motion equations of each partitioned floating bodies in the vertical direction are;

rnj,=fipW4,1Lx

-pgA,+F,

(9)

where Fi,,, is the the restoring force duc to the rigidity of the floating structure, z, is vertical displacement of each floating component, p is the density of sea water, and m,A,, are mass and water line area of the each partitioned component respectively. It is assumed that the restoring force due to the rigidity is proportional to the difference of the vertical displacementofadjacent components and the proportional coefficient is a functionofthe bending rigidity ofthe floating body EI.

(4)

Sirnfficant Values

Wind Wave Wave

Spee.d Height Period

(mIs) (m) (sec). Operational Condition 16.0 1.3 5.0 Survival Condition 50.0 5.0 9.6 o X N-1 N jn

i

1'out

-0

(0 >z> (0 >z > -qh),

The velocity potential and mótion of thç floating body can beobtained by solving the Eqs. of (6),(7),(8), and (9) simultaneously.

Interaction of Currents and Floating Bodies

The estimation of tidal cuneit force acting on a very largefloating body is important in designing mooring system and others. However, there is no established method to estitnate tidal current force in the case of very large floating structures. In the present study, the current force is estimated from numerical caláulation with multilevel model, which is a modification of the method by Tabeta and [noue (1995). When the floating body is divided into a finite number (N) of mesh components described in Fig.3, the static equilibrium of force acting on the. component

j

is;

mJg_FßJ-FRJ=O (10)

The gravity force mg, the buoyancy force F51, and the restoringforce due to the rigidity of the structure F are described as;

m1g'pgd&4,.

. F51 =pajM,,

F5 k5(ri_1 - ij) kR(77j+I -

r,)

where d is the mean draft of the floating body, Paj is the hydrostatic pressure at the, bottom of each floating component,AA,. is the bottom area of the each component, and i is the vertical displacement of each component. The restoring force due to the rigidity ofthe floating structure FRI is also assumed to be proportional to the difference of the vertical displacement of adjacent components and described by using the equivalent damping coefficient kR .The above equation isthe example of the case that the component contácts with two adjacent components. Now, the virtual surface level ç is introduced as follows;

P1P8cj't'7i)

(12)

ç. instead of Pdf, Eq.(1O) derives;

(13) pg(ç1-r)i4, +kR(7711 -r,1)+k5(771+ -i1)O

If the virtual surface level ç1 is known, the vertical displacement of each component is derived by solving the system of above equation.

oJj4

FR FB '7/ -iFR

jg

Fig3. Eqüilibrium of force acting on a separatedfloating component The following assurnptiöns are applied to take. a very large floating structure into the multilevel model. The subscript

j

for the index of partitioned component will be omitted iii the e,çpressionsbelow,

w=0

at zrj-d

(14)

2).Tentative pressure p at each grid point is calculated by

p

pg(ç-d)p,

where p denotes the atmospheric pressure.

3) Tentative

àriintalvelocities ', v

calculated by. horizontal momentum equations using the tentative pressure p. They arc not always satisfy Eq.(15).

(11) 4) Tentative vertical displacement of each floating component calculated by solving the system of Eq.(13).

5).'fl residual surface level Aç is calculated by substituting u, v, to the left hand of Eq.(15):

In the fluid domain where the floating body exists, horizontalvelocity componenti below the floating body are calculated as follows in order to satisfy the following, equation which is derived by integrating the equation of continuity vertically using Eq.(14):

û ,7-a û

'-'

r

vd2'=()

1) Tentative virtual surface level ç is assumed to 'be equal to the value of the previous time step.

(.15)

(16)

is

It is assumed that the force F due totidal current acting on thefloating body is composed.of two components, pressure force F andfrictional

förce Fr

FF9+Fr'

' ' (20)

In-the preent method, the pressure .resisnce force F is calculated by integrating the hydrinttatic pressure over the surface of the structure, in' which change of current speed is considered.

F .fpr1S

(21)

where S is the surface of the floating body and n' is'the inward normal unii vector, of S. The frictional resistance force is estimated by using the frictional coefficient C1:. .

Fr -

fCf'PoII'1b'15b (22)

a

udz--f

vdz()

(17)

The tentative virtual surface level ç is corrected by;

gAç-'8c'

'(18) where a is correction factor.

The, procedures from 2) to 6) are iterated until the maximumvalue of residual surface level in the domaih of floating body Aç becomes small enough..

If the iteration converges, the horizontal velocity components and vertical displacement of each component are determined:

y 77(I) 77 (19)

where Vb is the horizontal velocity vector at the layer just below the floating body, and S,, is the bottom surface of the floating body.. As the frictional coefficient C,, Prandtl-Schlichting formula is used;

cf-

, Re

log Re

0.455 ₍₂₃₎

V,

where L is the length of the floating body in the direction of the current and y is the coefficient of kinematic viscosity.

STEADY FORCES ANDMOORINGSYSTEM

Estimation of Steady Forces

Wave drift force, current force, and wind force acting on thefloating airport in the transverse direction were estimated by using the numerical methods mentioned above.

Wave Drift Force

Wave drift forces were calculated by the separate region method mentioned above. This method was also expanded to the problem of multi floating bodies and applied to the calculations for the floating airport accompanied with a floating breakwater.

At first, wave drift forces in regular waveswere calculated. The distance between the airport and breakwater was Set to be 10 meters. Fig.4 shows the wave, drift coefficient based on the wave period in the cases with breakwater and without breakwater.

Fig.4. Wave drift force in regular waves

Next, mean drift forces in irregular waves were calculated by the following expression:

= (24)

where R(w) denotes the coefficient of wave drift force in regular waves and denotes the spectrum of irregular wave. The ISSC spectrum was adopted for the irregular wave spectrum. The results were shown in Fig3. The horizontal axis denotes the visual observation wave period of [SSC spectrum and the vertical axis denotes the mean drift force. The design values of the wave drift force were determined by reading the values at the corresponding significant wave periods of both operational condition and survival condition from the figure. The result shows that the breakwater decrease the wave drift force by 50% in the operational condition and by 10% in the survival condition.

0.125 0.1- 0.075- 0.050.025 -0 s i0 15 Period (sec)

Fig.5. Mean drift force in irregular waves with ISSC spectrum Current force

The tidal current force were estimated numerically by means of multilevel model. The horizontal scale of major inner bays in Japan where floating airports could be located is several tens kilometers. Therefore, the calculations were carried out in the simple bay described in Fig 6. The structure was placed at about the center of the bay which is 40km long, 20km wide and has constant depth of 20m. The tidal wave given at the open boundary had the amplitude of 0.5m and the period of 12 hours. The horizontal mesh size was taken to be 400m (longitudinal direction of the bay) X 500m (transverse direction of the bay) and the number of vertical levels is five. The Coriolis force was ignored. The calculation was carried out for five tidal periods (60 hours) and discussion was däne about the result of the last one cycle. Further, restoring force due to the rigidity of the structure was ignored because the structure is supposed to deform along the shape of the surface elevation in the waves whose length is much longer than the

length of the structure.

40km

l2kmJ

:0km

k

20km

30km

Fig.6. The rectangular basin and positions of the floating airport used for calculation of current forces

Fig.7 shows the time variation of pressure force ¡, frictional force Fft, and total force ¡, in the longitudinal dfrectionof the bay (the positive values denote the force toward the inner bay). The phase of the pressure force is different from that of the frictional force whose phase is almost same as that of the velocity. It indicates the rate of the component due to the gradient of ses level, whose phase is almost same as that of the sea level variation, in the pressure force should be significantly large. Because the gradient of the sea level due to tide is usually very small, the force due to it can be ignored when the size of the structure isn't very large. However, when a very large floating structure is considered, the effect of the gradient of tidal level should be take intó account.

200 300

loo

Fig.7. Time variation of tidal current force

Fig. 8 shows the maximum values of the pressure force, frictional force, and total force due to the position of the floating structure. Because there are phase lags between the pressure force and frictional force, the simple sum of two doesn't give the total force. The force acting on the floating structure becomes lager when it is located offshore, because the maximum values of the current velócity and the surface gradient becomes larger.

-L SU - .- 1CflONAL R -. TOTAL

-4000 3000

...

/

2000 z

--X -, io H

--Pai RRE

-1000 _{- -- - FRIC11CNAL} - -TOTAL -2000 0 10 20 30 DISTANCE 1km]

Fig.9. Forces by wind-driven current at various locations in the bay of coefficient of pressure resistance and frictional resistance were chosen to be same as the values used in moue et. al. (1995). The wind forces acting on the airport buildings on the floating structure were also estimated. The coefficient of pressure resistance for each building is also determined according to moue et. al. (1995). The effect of the wash board which is attached only in the case without the breakwater is also included. The vertical distribution of wind speed is assumed to be based on the law of 1/7th power.

Total force

The calculated environmental forces on the floating airport are shown in Table 4. It indicates that the wave drift force is the largest component in both of the operational and survival conditions. Accordingly the breakwater must be effective for the purpose of decreasing the total force. The simple sum of wave drift force, current force, and wind force is used for the design of the mooring system.

Table 3. Environmental forces in the transverse direction of the floating 4..

1. FloatingBody 2. AIrportBuildings

Design of Mooring System

A feasible design of mooring system is shown against the estimaitd steady forces. The dolphins and fenders system is adopted because it is considered to be most adequate in coastal zone. The type of fender was assumed to be C3000H which has already used for actual oli storage bases. The fenders were planned to have initial compressive strain of 1 percent (= 3cm). The minimum number of dolphins and

NOperational

Condition Survival - Condition j Without Breakwater With Breacwater Without Breakwater With Breakwater 'I WaveDrift Forer

4.1 x103 2.0x10

75x104

6.8x104

curreniForea

08x103 08x103

60x103

60x103

Wmd Force F.B.1

2.4x103 2.4x103

2.3x104 2.3x104 A.B.2

30x103 - 15x103

29x104

1.5x104 TolalFor

103x103 6.7x103 13.3x104 11.2x10'

0 10 20 30 DISTANCE 1km]

Fig8. Maximum values of tidal current force at various locations in the bay

Ncxl, the forces duc to wind-driven currents are also calculated by the some method. The calculations were carried out under the constant wind which blows in the direction from the entrance to the interior of the bay. Fig.9 shows the results under the operational condition at 61) hours after the beginning of the calculation when the currents reach almost steady state. The surface elevation and its gradient becomes larger at the interior part of the bay because of wind set up. Then the pressure forces mainly due to the surface gradient work in the directioñ frotn the interior to the entrance of the bay and it becomes (arger at the interior part of the bay. On the other hand, the directions of the frictional forces axe from the entrance to the interior because currents flow in that directionnear the surface. When the floating body is located offshore, the total force acting on the floating body works in the direction toward the interior part of the bay because the frictional component is dominant. The magnitude of the total force becomes small at the interior part of the bay because the pressure component and frictional component cancel each other.

Although the ,ifinite location of the floating airport isn't determined in the present study, the sum of current forces due to tidal current and wind-driven current is adopted as the design value of current force. Wind force

Wind force was also assumed to be composed of pressure resistance and frictional resistance, and they were calculated separately. The values

48 51 54 57 60

fenders were calculated under both with a brcakwater and without it so that the strain of a fender due to steady force under the survival condition might keep less than lo percent. The numbrs of dolphins and fenders along the longitudinal direction of the floating airport is;

Without breakwater 38 (the intervals is about 80 meters) With breakwater 32 (the intervals is about 95 meters) Then the maximum displacement of the floating body by the steady force under the operational condition becomes 3.7cm, which does not disturb the normal operation of the airport.

MOTION ANALYSIS

Because the length of the floating airport is much larger than wave length and the draft is very small than its horizontal dimensions, the analysis including the hydroelastic effects is necessary. Then the vertical motions in regular waves were investigated by using the separate region method.

Fig. 10 shows the vertical responses of each component when the floating body is divided into 5 components. The results when the floating structure has several different rigidity of

El - X,

EI 1.0 x 1O"kg m3 El - O are shown in the figure. The line noted rigid denotes the results when the rigidity is , which corresponds to the response as a rigid body. The notation j denote the position of

the divided component and, the j=1 corresponds to the weather side component. Generally speaking, the smaller the rigidity of the floating structure, the larger the amplitude of the vertical motion becomes. The responses at the both ends of the structure tend to be [arger than those at the center part of the structure.

0.4 0.2

N.

0.2 0.4 t 02 XI B = 1 .0

---

---XI B=0.5 XI B =0.125 J

Fig.11. The instantaneous transverse distribution

of the

vertical displacement

CONCLUSIONS

In this study, the environmental forces acting on a very large floating structure and hydrodynamic response in waves were investigated. The main conclusions obtained thus are as follows.

The wave drift forces was estimated by the separate región method including the effect of breakwater. The results shows that the breakwater decrease the wave drift force especially under the operational condition.

The current forces acting on the very large floating body was estimated by means of numerical simulation with multilevel model. The result indicated that the force caused by the gradient of tidal level which is ignored in the conventional estimation might be taken into account in the case of a very large floating structure.

The motion responses of the floating airport in waves have been investigated. Lt i clarified that the rigidity of the structure has great influence on the hydroelastic responses.

Acknowledgments

This research was supported by the Grant-in-Aid for Scientific Research of the Ministry of Education, Science and Culture.

REFERENCES

moue, Y. et. al. (1995). Dynamic Behaviours of a Floating Airport and its Effects on Ocean Current. Proc of Inter. Offshore and Polar Eng.Conf., Hague, Vo13, pp. 406-413.

ljima, T., Tabuchi M., and Yumura Y. (1972). Scattering of Surface Waves and the Motions of a Rectangular Body by Waves in Finite Water Depth, Proc. of The Japan Society of Civil Engineers, No.202, pp.33-48

Tabeta, S. and moue, Y.(1995). Interaction of Ocean Current and a Huge Floating Structure in Restricted Sea. Proc. of Inter. C'onf on Technology for Marine Environment Preservation. Tokyo, Vol.1, pp505-512

o 0.2 0.4 0.6 0.8

Fig.10. The amplitude of vertical motion of each partitioned component in regular waves

The motion responses ofthe whole floating body in long crested irregular waves were investigated in the both cases with breakwater and without it. The significant values of heaving amplitude of the floating body is 0.8cm in the operational condition, and 9.1cm in the survival condition. In spite of overestimation because of two-dimensional assumption, they are not so large. On the other hand, when the elasticity of the structure is considered, the significant values of heaving amplitude at the weather side end is 2.6 cm in the operational condition and 34.6 cm in the survival condition and when E1-l.OxlO"kgm3/s2. The motion amplitude changes significantly due to the rigidity of the structure. lt indicates the estimation of elastic response is more important than that of rigid body itself.

5 4 3 2 2 5 4 3 0.4 _{5 4 3 2} 04

j 02

. o N -02 .0.4 0.9 0.8 0.7 0.6 0.4 0.3 0.2 0.1 o

International Conference In Ocean Engineering COE '96 lIT Madras, India, 17-20 Dec. 1996

ACFIVE CONTROL OF TENSION LEG PLATFORM UNDER RAM)OM SEA

SyedlllaleeqAhmad'

SuhailAhnwd2

I Lecturer in Civil Engineerin Jamia Millia Islamia, New DeUil-25, India.

2 Assistant Professor in Applied Mechanica, Indian Institute of Technology, Delhi, New Delhi-16, India

SYNOPSIS: Tension Leg Platform (TLP) is a deep water semi-submersible_{typé complismt, positive buoyant oore strutune.}

In hostile envfronmejai conditIons the dynafliic response of T12 hull and tendons may go beyond permissible limits leading to major catastrophic failures. This paper deals with an active control of coupled TLP, tendon response and proved to be a promising alternative to mitigate structural damage. _The desirable control of the significant surge response and the resulting_{excessive tendon strain is presented. The operability is improved}

and the tendon and user are protected from damage. Response time histories and

power spectra without control are generated for random sea and compared with the controlled time histories and power spectra The present study suggests the possibility that active control can provide additional_{position-keeping} ability in emergency cases to avoid detrimental UPresponse.

INTRODUCTION

A Tension Leg Platform(TLP) shown in fig.1 is an Important compliant offshore structure used for oil

exploration and drilling operations in the sea.

It

is a

positive buoyant structure, subjected to heavy

environmental loads and moored by vertical taut cables

called tendon or tether. In a deepwater TLP the restoring force becomes smaller and the natural frequency of the system decreases with increasing water

depth due to increase in the mass of tendon including

added mass. In hostile environmental conditions the dynamic response of TLP hull and tendon may exceed the

permissible limits leading to the failure. Hence, there

¡s a neèd to control the dynamic response of TLP In

order to improve its serviceabifity and survivability.

This can be done by the conventional DPS thrusters,

working on the principles of Active Control, a feedback

system designed to sense the structural motions and to

generate a corrective force which alleviates the undesirable motion characteristics.

Various investigators have contributed in the area of

active control of compilant offshore structures. Yoshida

et

aL (1990) verified the applicability of activé

control of a tower-like offshore structure by

the experiments conducted on a model.

Suzuki et al. (1993),

actively controlled the reentry operations of a

marine riser.

Su±uki

et

al. (1994) have also shown

experimentally the effect of active control on a 1/100

scale model of a TLP, considering the surge response of

UP hull and tendon strain.There is a need to study the

controlled TI..? response considering realistic model of

environmental loads, associated nonlinearities and coupling of various degrees of freedom. This paper

presents an active control of hull and direct strain of

TI.? prototype subjected to actual environmental conditions.

MAThEMATICAL FORtvU..ATIONS

The equation of motion of TLP hull with six

d.o.f. (surge, sway, heave, yaw, pitch and roll) is given as,

[M1( + [Cl(V + IKJ(V) = (F(t» (i)

where, 1Ml is the structural and added mass matrix, [Cl damping matrix, [Kl the coupled stiffness matrix,V. 'c',

and i are the displacement,velocity and acceleration

vectors of TI..?.. F(t) Is the force induced on UP due

to current and waves(the wave parameters taken do not

violate the validity

criteria of Morrison's equation) given as,

F(t) = 0.5p DC (Û +Û - '')Û +1) -''

+ 0.25irD

w D w e w c

+0.Z5irDCM(U_ V) (2)

where, p

is the density of sea water, D. the diameter

of cylinder, CM the inertia coefficient, C the drag

coefficient, Û and U are respectively the wave

parti-cle velocity and acceleration, U is the current velo-city. The random sea-state is represented by Plerson-Moskowitz sea-surface elevation spectrum given below.

S =( H2 T/8ic2)(Tfi expl(TfY4/irl (3)

where,

f

= frequency in cycles/sec

H = significant wave height in meter

T = wave period In sec

S = P-M (single-sided) sea surface elevation spectrum