A strip method for the prediction of lateral drift force of a semi-submersible

ABSTRACT

The paper presents the results of a study of the usefulness of a strip technique for the prediction of the lateral drift forces on a semisubmersible platform floating in oblique regular waves.

The, strip method employs the well known Maruo's formula and sOurce distribution tech-nique, without taking account of the hydrodyna-mic interaction between the twin hulls and

col-umna of the seinisubmersjble. _{Thus the}

mono-hull hydrodynamic forces are used to evaluate the twin hull semisubmersjble forces with a correction for the transverse phase difference and by taking account -of the force and moment transfer due to the change of the coordinate

Systems.

The result of numerical calculation was compared with available experimental and three-dimensional theoretical data. It appears that

the strip technique shows overall more f

avor-able correlation with the experimental data than the three-dimensional theory.

It is, however, premature to conclude that the present technique has been fully validated. Further studies are needed in the theoretical and experimental areas.

NOMENCLATURE

a _{wave amplitude}

A± complex amplitude of scattered wave amp-litude at positive or negative infinity

B _{beam or 'restoring force coefficient} F _{wave-exciting force, drifting force} g _{gravitational constant}

G -Green's function

G _{Asymptotic expression of Green's function}

in the far field

h wave elevation

O?ps

_'t0iS

161

C. H. Kim and W. Bao Stevens institute of Tecl,nology Department of Civil and Ocean Engineering

Hoboken, New Jersey

i

L M M" Mink N Nmk INTRODUCTION

- The _{pplicability of a strip method for}

prediction of lateral drifting force was inves-tigated and the result is presented in this

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A STRIP METHOD FOR THE PREDICTION OF LATERAL DRIFT FORCE OF A SEMISUBMERSIBLE

length between perpendiculars

mass

added mass

mth mode motion induced added mass in kth direction

damping coefficient

mth mode motion induced damping coeffic!s-ent in kth direction

p _{hydrodynamic pressure} Q source intensity

50(m)

complex amplitude of mth mode motion displacement

x,y,z body coordinates of the single hull x0 _{transverse distance betweerL the main and}

local origins

X,Y,Z body coordinates of the .'s'misubmersible heave amplitude V _{displaced volume}

water density c phase angle

sway amplitude wave incidenc wave number

circular frequency of wave

rol]. amplitude _{potential for a single} hull velocity potentials pitch amplitude yaw amplitude Q n p V w

X

paper.

The study is the first endeavor to the writer's knowledge to apply Maruo's formula for evaluating the lateral drifting force on a semisubmersible in oblique regular waves.

Strip methods are in principle judged to be less accurate than the corresponding three-dimensional techniques. However, the present study exhibits that the strip method agrees overall more favorably with the experimental data than the three-dimensional analysis, especially in the practically important domain of wave frequency in which high wave energy is

contained.

There are two different approaches for e-valuating the drifting forces : the far-field and near-field methods. The former evaluates the momentum flux over the control surface in the far field, whereas the latter deals with the integration of the second order pressure distribution over the hull surface. Kochin

(1], Maruo [2], Newman [3] and others used the former approach, while Pinkster and van Oort-merssen [4] three-dimensionally and Kim and

Dalzell [5], Kyozuka (6] and Papanikolaou (7] and others two-dimensionally applied the latter

approach. Maruo's formula was applied by

Ogawa [8] for a fixed ship in oblique waves, by Kim and Chou [9) for an oscillating ship in

oblique

waves and by others for monohull ships. Pinkster [10] and Chakrabarti and Cotter Lili calculated the mean drifting forces on semisubmersibles. Hong [12] recently reported a result for three-dimensionally evaluated drifting forces on small water-plane twin hull

(SWATH) ships.

A hybrid strip method was also proposed by Le and Kim (13] for evaluating the drifting force of SWATH ships.

Bao [14] recently extended the str.p

meth-od, which was applied to a monohull ship in

oblique waves [9), to a semisubmersible in

beam seas. A further extension to oblique seas was carried out by the present writers.

The mean drifting force is derived from the first order velocity potential. The time varying drifting force is estimated using the quadratic frequency response function for the drifting force.

A unique technique using the quadratic frequency response function was develop-ed by Daizell (15) and Dalzell and Kim (16] and it was utilized for prediction of slowly varying drift force by Kirn and Breslin [17). A similar practical method was developed by Pinkster and Hooft (18).

The present study employs the linear hydrodynamic theory and assumes that the semi-submersible is of the column-pontoon type and no hydrodynarnic interaction exists between the submerged structures. Each hull section is of symmetric shape about its central vertical

axis. The close-fit source distribution meth-od and Maruo's drift force formula are

utili-zed.

The hydrodynaxnic characteristics of each hull section are employed to evaluate the twin hull hydrodynaxnic forces and moments about the center of gravity of the semisubmersible. EQUATION OF MOTION OF A SEMISUBMERSIBLE

The motion of a semisubmersible in waves

are described using the body and inertia1co-ordinate systems. We designate them by o-xyz and Ol-X1Y1Z1, which are in coincidence when the body is at the mid-position of osci1latior, under the assumption that the drifting motion is restrained without affecting the oscilla-tory motions. The XZ plane rests on the calm water surface and the Y- and Z-axis point ver-tically upward and longitudinally forward of the hull, respectively.

We write the equation for an oblique regu-lar wave in the body coordinates as shown be-low

h =

aeX5imCO5

₍₁₎

,where the time factor is omitted.

The a, w and y represent the amplitude, circu-lar frequency and wave number of the wave, re-spectively. i is the angle of wave incidence which is chosen so as to make 180 degrees in head seas and 90 degrees in beam seas progres-sing from starboard to port.

The response motions are surge, sway, heae roll, pitch and yaw, which are designated by

and x

Neglecting the surge we write the follow-ing linear coupled heave-pitch and sway-yaw-roll equations

{

'(_w2M_iwN) (2M

-iwN )

(-w2M-iwN)

xn xn M -juiN )(-w2M -iwN (_w2MnXiWNnx) (-w2 ,x

x

(-w2M -juiN )

ui2MiwN)

n, n.

where the time factor e"03t is omitted in both cases.

In the first matrix (square), B,M and N represent restoring, inertial and damping forces per unit displacement, acceleration,

X (3) a (2) = f

(B-wM) - iwN)((B-w2M )-iwN

_q. .) '. f(BçpW2Mçp) - iwN 1f(B_w2M,)_iwN,}II

and velocity, respectively. It is to be noted that the inertial M terms in the diagonal ele-mente represent virtual mass. The of

f-diagon-al elements indicate cross-coupling terms in which the first subscript represents the mode of motion, and the second, the mode of induced

force. The second matrix (column) indicates the response amplitude operators (RAOs) where-as the third matrix (column) the wave-exciting

forces. The formulas for the added masses and damping coefficients are given in Appendix 1. The sectional added mass and damping coeffici-ent for a semisubmersible will be calculated in the-following sections.

WAVE-EXCITING FORCE AND MOMENT

We calculate the wave-exciting force exer-ted on a restrained semisubmersjble in an obli que regular wave(l).

First we evaluate the wave-exciting force on a mono-hull on the left or right. To do

this we set local reference frames o-xyz on the left and right hulls, which are located +x0 apart from the Y-axis as shown in Fig. 1. The x0 is positive. _{We designate the vertical} coordinate of the center of gravity by

YG-Thus the main and local coordinates have

the following relations

x = ,

O>

-Y =y

(4)

dinate system is given by

The wave observed from a local body

coor-h0 + vzcosu) ₍₅₎

Substituting (5) in (1) , we obtain the follow-ing form

h = ae s1nU+ vzcosJ)ivxsiflu

=

h0e+Xo

(6)

This indicates that the oblique wave in reference to the main frame OXYZ is identical

to the wave

in reference to the local frame

oxyz with the transverse phase correction

which accounts for the transverse separation

of each hull from the main f raine. In other words the phase correction e1'Xo

_indicates

that the wave crest at the origin of the left local frame leads the wave crest at thç origin of the main coordinate system.

The e\Xo

in-dicates that the wave crest at the right local origin lags the wave crest at the main origin.

The longitudinal phase correction factor eiVZc0Si indicates that the phase of the inci-dent wave at a section at z lags in reference to the wave crest at the origin. And this factor remains the same in the main

and local

systems as it should be.

The corresponding incident wave potential is written in the following form

- _'e Sinj+jvzcosj

I_

O

=-

e' + i'vxsinu ₍₈₎

I w

In the above the same is applicable t each local hull.

The corresponding diffraction potential

is similarly given

, ivx 5inJ+ivzcosu

'D

DC,Y1 e

o

with

(10)

In the foregoing, the Green's function G is the two-dimensional pulsating _{source potem} tial of unit intensity at the point (,n) in

the lower half of the xy plane [19]. The C indicates that the integral is to be taken

a-long the hull-section contour below the calm-water surface. _{The unknown source intensitio} Q,which are discretely distributed along the

contour C,are determined by satisfying th condition of nonpenetration of the water

par.-tides across the contour:

onC

- fil)

Let the unit outward normal on the ith

elemen-tary segment along the contour C be

= (sinai, - cosa1) ₍₁₂₎

ith segmnt,where a. represents the direction of the

relative to the x-axis.

The kinematical boundary condition (li) is also written in accordance with Frank _[19) as follows:

jl

QjIjj+jiu+jJij=_e\)Yi

(sinucos

(ux1

sin.i) sine

-sin (\acsinj) cosa1)

N N

i

+s (vx.sS.ntj) cosa i] 1 3) Determining the source intensity Q by sol-ving the foregoing 2Nx algebraic equations, we can evaluate the velocity potential of the

fluid in the resultant diffracted wave fi1c.

- + +ivx Sinu+ivzcosu_o

The hydrodynamic pressure is

given by

p

p0e+VXo+1)05t1

with Po ipw(q)1 +

multiplied by

main frame is

The Po is the

hull contour.

hydrodynamic pressure on each

the transverse and longituthnal identical, to the pressure p9

Thus the p in reference to the

phase correction factors.

The wave-exciting force on a mono-hull section is as usual calculated by:

f(m)

Jc Po

um)dc

where -sinn cosa xcosa+ysinn

with respect to the main frame

(m) 1X-sinncosn+Ysjnn cosa

if the is to be evaluated with respect to its center of gravity, trie Y is to be given by

Y. ₌

y - (20)

Use of (4) , (18) , (19) , and (20) yields the following relation

(m) = (m)

u0

m=2,3

= ₊ (2)

o

Go

Lise of (15) and (21) gives the

wave-excit-ing force and moment of a twin section about its center of gravity G in the following

form,

(m) _(m)dC+J

dc

(m) eVXo1dc +jpu (m) eoSdc

=

2f(m)S(VXSifl),

_m=2,3

X

00

(17) (21)

L

(4)

=2{(f

rf)sx5fl)

+ix f

₀₀

sin(vxsjnj.j)}

I _{the foregoing expressions the}

longitudin-al phase correction factors are omitted. Integration of the f(m)over the hull length with the longitudinal phase correction yields

F(m)=

J

± Ç fC

zel\ZCOSUdz,L=2,3, m=6,5

IL

The previously used symbols for the wave-exciting fçrces in Eq. (2) and (3) are given by the y(m, (24).

ASYMPTOTIC EXPRESSION OF THE DIFFRACTION POTENTIAL

The asymptotic expression of the diffrac-tian potential for X-,.± with respect to the main origin is necessary to determine the

la-teral drift.ing .force using Maruo's formula.

In order to evaluate the asymptotic

expres-sion _D for X-± (or _{it is required to}

evaluate the _{D for x±= or D± with a phase} correction due to the transverse distance between the main and local origins _{at left} and right. _{When we consider the far-field}

ex-pression at negative infinity _{the left} body-induced diffraction potént.al

at

X--is _{(D_)lefte_1'o. Due to the right body}

we have (D_)right'0.

Since _{D-)left and}

D+riaht are ].dentical, we have from (14) the folïowing exoression

= 24,D cos (vxo(].sjnLi)]ei\c0sU ₍₂₅₎

Considering both ends

-

iVZCOSL.1

= 2+cos(o(l+s1nL1))

e

(26)

The calculation of the potential of

a mono-hull section is given in Appndix 2. Use of (A-7) , (A-8) and (A-9) in (26) gives

= A11 eX\

where

l)

=2A

cos (vx (].sin) ]

e1 ± +vzcosLi)

with

(1)

A(=+)2+DW2,e(].)=

-l_

The C

and D± are given in (A-8).

ADDED MASS AND DAMPING COEFFICIENT

The added masses and damping coefficients of a semisubmersible are also calculated us-ing the added masses and dampus-ing coefficients of the mono-hull.

We begin with definitions of the motion and radiation potentials of mono-and twin hull sections as in the following equations.

(t) = 5gm)

e1t

₍₃₀₎

s = z (m)e_iutm)

m°

= z

s0(m)etm)

amplitude, and wand 4m)represent the radia-tion potentials of a mono-secradia-tion in refereje to the local and ¡nain frame, respectively, both per unit motion displacement.

The radiation potential, as defined for the diffraction potential _D j (10) is

re-presented by

(m)

(x,y)=

JQ(m)

(,)G(xy)dc

(33)

The kinematical boundary condition on the mono-hull section contour with respect to a local body frame is given by

- (n)

- iwuo ₍₃₄₎

To solve (34) we use the same left-hand side as in (13). The hydrodynainic pressure is

(m) ₌ (m)

(35)

The integration of the pressure p over the section wetted contour at mean p8sition yields the motion induced hydrodynamic force.

where

=-L

im = Jc Re

For a symmetric ship-like section we have four independent pairs of the added masses and damping coefficients: (M1,N ), m= 2,3, 4 and (M24, N24); It is known tlre is one identity relation M4? = M24 and N42 = N24.

Prior to evaluating the added mass and damping coefficient of a twin-hull section, we formulate the relationship between the potentïals referred to the main and local

body frames.

The kinematical boundary condition on each mono-section contour, with respect to the main frame is

The result of the foregoing

_integration

_is given as follows

= 2

(2

_Mmm+icN )

= 2,3

F24 = 2{ (c2M24+iwN24)+y (2M +iwN) F42 = 2{(c2M24+iwN24)+y0(u2M22+iN22))s(4)

(36) F44 = 2{(w2M44+iwN44)+2yG(w2M24+jN)

In the above, the identity relations M 2= M24,N42 = N24 are used.

_{The remaining ter}

(37) F23, F32, F34 and F43 vanish.

The symbols of the sectional added masses and damping coefficients in Appendix 1 are now

represented by those derived in (43).

Use of (21) , (34) and (38) yields the

follow-ing relation (m) (in) = a m = 2,3 an 3) = _X0 + an on Since = _{we have}

an,,

+Y(w2M2a+iN2

m;5=2M22 ,N55=2N22 mHH=2M33 ,NHH=2N33

inSR=IflRS=2 (yM22+M24) SNsR=NRS=2(yGN22+N24)

mRR=2 (M44+?yGM24+yM22+M33) NRR2 (J44+2yN24+yN22.fX!3

Substituting (44) in

Eq.

(A-l)

and inte-grating the sectional added masses and

damp-ing coefficients, we determine the resultant values for the semisubmersible. Thus we can evaluate the response motions by solving (2) and (3).

2M33+iuN33) }s4 0

(43)

(39) _{ASYMPTOTIC EXPRESSION OF THE RADIATION} POTENT IAL

(44)

Referring to Eq. (32) and (40)we consider an evaluation of the asymptotic expression of the radiation potential with respect to

the r3in origin O : _{4R and} ,(m) for X+±.

(m)

(m)

m = 2,3 =

4)

X

₀₀

(2)

(40)

Substituting (40) in (32), _{we obtain the}

hydrodynainic pressure

(rn) (m) ₌

_50(m)(m,

¡n = 2,3

p( =

jp5(4)

(41)

Integration of the foregoing pressure with respect to the main coordinate system, over the two mono-section contours c and c', is

Fink =

p(m)u(dc+J01pmudc

(42) =

L

P(m)U(k)d (w2Mk+iuNI) sm) an = (38)

Since the is expressed in terms of ijm) as shown in (4O Sand the asymptotic ex-pression of the Cm) .s for we have to

cake account of te effect of phase differ-ences due to the bodies at left and right by +ivx0 and ±jvx0 for

X-P+.

It results to the following

(m)_ e o

eo

+ ivx =

2cosvx

ni =

2,3 - o,

2f-)The asymptotic expression of the potentiais o±L R represent those due to the source dis-tribition on a mono-section about its local coordinate systems. It is the same for the

local left or right sectiol)s. The formulas

for

evaluating the potentiàl are given in Appendix 2. Substituting f45) in (32), we have the sum of the radiation potentials for the three modes of motion of a twin-hull section

for

=

m=23,4

sm)et

(ni) (46)

Use of the formulas (A-7), (A-8) and (A-9) ior (45) and the use of (45) in (46) gives the following result.

R±Thi=23,

4Am)

e

(±vx-iy-t)

(47)

,where

Am)_2B Cm) (ni) ejt±(ni) +cm)),m=2, 3 - - ± IS0 jcosvx

A4=2JS4 l{B4)ei(±)S

G8 e ± S cosvx (4)

ix0B+(3)e±CS

_sinux.0J ,and where (ni) CS (ni) B4 (m)

= the phase angle of S0

=Jc±(m)2

D±(m)

D

= tan ±

Cm)amd Dim) _{are given in (A-8)}

The foregoing motion amplitudes 5m) now iave to be expressed in terms of the sernisub-aersible motion about its center of gravity

2)

= +

The n, C, ch andx are determined by solving (2) and (3).

THE SCATTERED WAVE AND DRIFTING FORCE The scattered wave at negative infinity is the sum

of the waves due to both

diffrac-tion and rad4t.on. Designating these waves

by h(1) arid ni = 2,3,4, we have the fol-lowing

hW=

+ i x0 invx0] (45) - g

-ni =

2,3,4 (51)

A1 and A(m}

,

m=2,3,4 are

given in (28) and

(48)

The sum of these is

A

=A(m)

-

gm=l

The mean drifting sectional force is, ac-cording to Maruo

2 2

f= ½ pga A_I (53)

Integrating the foregoing sectional forces

over the hull length, we obtain the resultant

mean lateral drifting force of a

semisubmersi-ble in an oblique wave,

F ½ pga2

f

_{IA_I2dz} (54)

)L

NUMERICAL CALCULATION AND DISCUSSION

A numerical calculation was carried out for

a semisubmersible which was previously investi-gated by Pinkster (103. The principal parti-culars are shown in Table 1. The model is an ordinary column-pontoon type which is symmetric

fore and aft and each hull is symmetric about xy-plane. _{There are two small transverse bars} that connect the pontoons. _{The forces due to} the bars were neglected.

The six columns are identical and the

pon-toons have uniform transverse sections.

There-fore the two-dimensional source distribution

procedure was applied to one

column together

with the, pontoon portion attached to it, and

one section of the pontoon hulls alone to

avoid unnecessary computation. The one column together with the pontoon portion attached to

it, was sliced by eleven sections and the

nu-merical datá for the hydrodynamic

characteris-tics of the

column

were employed to evaluate

thehydrodynamic forces on the remaining

col-umns. _{The hydrodynamic characteristics of the}

pontoon section were also repeatedly utilized to

estimate other sectional hydrodyriamic

charac-teristics. In the calculations, the phase corrections were made using the previously de-veloped formulas in the foregoing sections.

The contour of each section was divided

in-(50)

to twenty segments to evaluate the source

strengths. The sharp cornets were also

round-ed by short inclinround-ed segirtents. But the re-sults showed negligible differences from those with the sharp corners.

The. calculated response motions and drift-ing fOrces are shown in the figures and corn-pared with the experimental data and the

three-dimensional analysis .

Theresponse motions of the semisubmersi-ble in the beam seas are shown in Figs. 3-5 and those in the quartering seas in Figs.6-lO.

The lateral drifting forces in. the beam arid

quartering seas are illustrated in Figs. li

and 12.

The response motions calculated by both methods agree well in general with each other and with the experimental data. There are

some exceptions. The sway and yaw motions in the low frequency region show large

discre-pancies. The three-dimensional results exhi-bit the trend to increase to infinity as the frequency approaches zero, whereas the pre-sent results reach finite values such as uni-ty for the sway in the beam seas and less thai one in the olbique waves, as expected on

phy-sical grounds.

The couplïng effects between the sway and roll are seen in the sway responses in the

form of dips.

The estimates of the resonance frequencies of all the response motions show slight dif-ferences in both calculations.

For the driftin.g force in the beam seas both theoretical results have a similar be-havïor except in the very high frequency

re-gion. In the region of frequencies between 0.8-2.0 the present results more favorably agree with the experimental data than the three-dimensional values, whereas in the very high frequency region this situation is

re-versed.

The present results show a tendency toward rapid fluctuation in the very high frequency

domain. This may be due to the éffect of the. phase correction with an argument of y x0 which is contained in each term of the expression for the' amplitudes of scattered

waves.

In the very low frequency domain, both re-sults show that the drifting forces are very

small. This is due to the fact that most of the diffracted waves are transmitted and the response motions and consequently the radiat-ed waves, become infinetesirnally small at negative infinity.

With regard to the drifting force in the quartering seas, it is rather difficult to explain the results. The analysisshow large discrepancies with each other ánd also with the experimental data. In the domain of fre-quency between 0.9 and 1.2 the two-dimensional result agrees better with the experimental data, whereas in the region between 1.2 and

1.7 the three-dimensional. calculation agrees better with the experimental data. In the

rest of the higher frequency domain it is seen that both analysis disagree with the experi-mental data. Similar trends are observed in the .very high and low frequency regions.

Since the strip theory is practically valid in beam. seas and long oblique waves, it is

167

expected that the present analysis will not predict well in the short oblique wàves.

The interaction effect generally appear for the semisubmersible having small hull separation and large columns. Therefore the seznisubmersible under consideration should have adequately large hull separation and small colunns. A further study is necessary to determine the critical valués of separa-tion of Pontoon hulls and size of columns in order to apply the present method for the prediction of drifting force.

Both two and three-dimensional methods predicted rapidly fluctuating drift forces in certain regions of the frequency domain, where the number .f experimental points are inade-quate for comparison. Therefore it is neces-sary to have more experimental points to ac-hieve the objective of this study.

Of course the study with the same model i.c not adequate to fully validate the theoreti-cal technique. Other model test data on column-pontoon type senisubmersibles and cor-responding correlation studies are highly desirable. Finally, it is recommended that the near-field technique be applied to the present study with the same assumption that. the hydrodynamic interaction between the twin hull sectiOns is negligible.

Table 1 - Principle Dimensions of the Catamaran

REFERENCES

[li Kochin, N.E., "The Theory of Waves Gener-ated by Oscillation of a Body Under th' Free Surface of. a Heavy Incompressible Fluid", SNAME, Technical and Research Bulletin No.. 1-10, Apr. 1952.

[21 Maruo, H., "Thè Drift of a Body Floating

on Waves", JSR, VoI. 4, No.3, Decembe" 1960, pp. 1-10.

[3) Newman, J.N., "The Drift Force. and Momht on Ships in Waves", JSR, March 1967.

[4] .. Pinkster, J.A. and van Oortmerssen ,

"Computation of the First and Second.C:-. der Wave Forces on Oscillating Bodies -.

Designation Unit Value

Length (between m 100.00 perpendiculars) Breadth in 76.00 Draft n 20.00 Displacement m3 35,925 Center of Gravity m 7.92 above Baséline Transverse Meta- mn _17.48 centric Height Roll Gyradius m 30.89 Pitch Gyradius m 30.55 Yaw Gyradius m 41.74

in Regular Waves", The Proceedings of

the Second International Conference on

Numerical Hydrodynamics,Sept. 1977,

University of California, Berkeley,

PP 136-156.

5]

Kim, C.H. and Dalzell, J.F., "An Analysis

of the Quadratic Frequency Response for

Lateral Drifting Force and Moment", JSR,

Vol. 28, No. 2, June 1981.

6)

_{Kyozuka, Y., "Experimental Study on Secor}

Order Forces Acting on a Cylindrical By

in Waves", Fourteenth Symposium on Naval

Hydrodynamics, Aug. 1982, University of

Michigan, Ann Arbor, Michigan.

7]

Papanikolaou, A., " Über Offene Fragen bei

der Hydrodynamischen Theorie und von

Berechnungen Zweiter Ordnung für in Einer

Welle Oszillierende Zylinder", Techniscle

Universität Berlin, Institut für

Schiffs-und Meerestechnik TUB/ISM, Bericht Nr.

82/12.

L)

Ogawa, A. "The Drifting Force and Moment

on a Ship in Oblique Regular Waves",

In-ternational Shipbuilding Progress, Vol.

14, Jan

1967, No. 149.

t]

Kim, C.H. and Chou, F., "Prediction of

Drifting Force and Moment on an Ocean

Platform Floating in Oblique Waves",

In-ternational Shipbuilding Progress, Vol.

20, Oct. 1973, No. 230.

jIO)

Pinkster, J. A.

"Low Frequency Second

Order Wave Exciting Forces on Floating

Structure", Publication No. 650,

Nether-lands Ship Model Basin, Wageningen,

Netherlands.

[il] Chakrabarti, S.K. and Cotter, D.C.,

"In-teraction of Waves with a Moored

Semisub-mersible", Proceeding of the 3rd.

Sympo-sium on Offshore Mechanics and Arctic

Engineering,Vol. 1, Feb. 1984.

ji21

Hong, Y.S., "Improved Prediction of Drift

Forces and Moment", David W. Taylor Naval

Ship Research and Development Center,

DTNSRDC-83/069. Seot. IQR3.

[13)

Lee, C.M. and Kim, Y.H., "Predication

of Drift Forces on Twin Hull Bodies in

Waves", International Symposium on

Hydro-dynamics in Ocean Engineering Vol. 1,

The Norwegian

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Aug. 1981.

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Semisubrnersible Catamaran Floating in

Beam Seas", Thesis for Master of Ocean

Engineering, Stevens Institute of

Tech-nology, Hoboken, New Jersey 1983.

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Sympo-sium on Naval Hydrodynamics, London,

March, 1976.

[16)

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Sept. 1979

117]

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APPENDIX 1.

ADDED MASS AND DAMPING COEFFICIENT

The formulas of added masses and damping

coefficienjn Eqs. (2) and (3).

M=5m"dz

N=JNssdz

Mn jm"zdz

N =5 NS5ZdZ

M= M=Jm"dz

Nç Nfl=,ÇNsRdz

M"=

IL

m5z2dz

N=

JL Nssz2dz

M=

'L

IPRZdZ JL

N=

JL

2th

The sectional added mass and damping such as mR and

denote the addal roll and damping roll nrinents due

to swaying rrction.

APPENDIX 2

ASYMPTOTIC EXPRESSION OF THE VELOCITY ¡'QTENTIL

Since, the diffraction and radiation

poten-tials (10) and (30) have the saine form, we

re-present them as

1234

(A-2)

where m=l stands for the diffraction potential

and 2,3,4 for the radiation potentials of mode

= 2,3,4.

By introducing the suffix ± for

X.±

we

M=jm'dz

N=JNdz

Mj m"dz

Ncc=J Ndz

Mç-j m"zdz=M

Ncq,=_j Nzdz=N

write the asymptotic expression as

±=J Q,n±,,ds

(A-3)

where the superscript (m) is temporarily omitted.

The evaluation of the asymptotic expres-sion is similar to the previous work (9]. We assume here that the shape of the section may be asymmetric about its central vertical axis.

The asymptotic expression of the Green's function Gt (20] is given by

G(x,y;,ri)=±Re (ie ]_i.Re(e1\

,where z = x+iy,

c=+in

Use of (A-4) in (A-3) gives

o±=ji. (Qj+iQN

Gc±ds+i5(_GsdsJ

with

Gc±=±Re[ie_iV(z_)]i

G+=Re(e

)] (A-6)

Calculating the foregoing integrals, we obtain the far field expression (A-3), in

the following

form

0±=(C± where N C = N =

jl

±QjLjQNjKj)

(A-7) (A-8) 169 h o w

000

y X 7 z1

Fig. 2 Hull

Form of

the Semisubmersible

o X

()

z, K.

cos(v+1+)_e )cos(v+)]

C i2.6

-L] vn.

s1n(v1+)_e

sin(v+n)]

o I ¿

(A-9)

I..

Io

100.0

_oJ

(A-4)

C -Xe Xe

1.0

efl's

'-I 1.29 2.07 - 2.0 o utrip theory 3-0 theory 0. 1 O o joent otrlp theory 3-0 theory eeperiosot1 Itrip theory 3-D theory

'oj

O opecieent 'n 110° 1,00 90° lag 00 900load

aig. 5

Rolling Motion ofa Semisubmersible

in Beam Seas

1.00

iig. 4

Swaying Motion of a

Semisubmersible-in Beam Seas

lag 900 900 lead L 1. 0.S IlL a - 130°

- drip theory

3-0 theory 11101 O eepsrioent

Fig. 6

Heaving Motion of a Senisubmersible

in Quartering Seas

1.20 0.07 0.510

00

a - 125 100° lead 200 100 00 lead 200

Fig. 7

Pitching Motion of a Semisubrnersible

in Quartering Seas

-

theory 30 theory 1)00) O ep.rth.nt lag

Fig. S

Swaying Motion of a Semisubmersible

in Quartering Seas

0.29 2.07 0.511

Fi;. 3

Heaving Motion of a Semisubinersible

in Seam

Seas