A numerical solution for the diffraction of second order group-induced wave by floating body

(1)

a

P. 0 B0X 116, WUXI, JIANGSU

CHINA

CHINA SHIP SCIENTIFIC RESEARCH CENTER

A

ÑURICAL

SOLUTION FOR 1FF DIFFRAcTION OF

SECOND ORDER CROUP-tNDUCED WAVE BY FLOATING

ÖPY

_S un. -Qi

Gu Mao-Xiang

CSSRC Report

September 1985

English version-85007

(2)

A NUMERICAL SOLUTION FOR THE DIFFRACTION OF SF.CONJ') ORDER CROUP-INDUCF.D WAVE BY FLOATINC ROD?

Sun 8e-Q! and Cu Mao-Xiang

Abstract

This paper présents a 3-D numerical DIE me-thod for' calculating the forces and moments acting ot n floating body due to diffraction of second-order group-indiced waves of low fre-quency. The total second order wave yotential

is decomposed into three parts: the

-potential of the second order incident wave; the potential of the second order radiated wave and the

_P?5efltinl

of the second order diffracted wave. 4tY and cn he calculated by known methods, hu has been difficult to evaluate either analytically or hy n'umerical methods. The method presented here avoided the

direct calculation of 2)

, bit!: evaluated

ff-nids (j=l , 2, .

. .) using ilaskind rei ationsJiip

and developing Mohn's idea into a

numeri-cal method wich integrales over tite free-surface and over the body surface villi 11 its

non-linear terms taken into account of. Numerical results of second order diffraction loads for a semisubmersih1edill ship are presEnted. Mi

approxim1t concept to relate the second-order lod in bichromatic waves to wave damping ei-feci 'is proposed and a specially designed experiment' is conducted to vahidate'the concept. Qunhita-tive agreement between 'numerical and experimen-tal results are achieved.

i. Introduction

Large excursions of moored structures at low frequency

with

acconipanyilig high ninoringloads

have known t be import-tilt [rl offshore

engineer-ing. The forces that exci te lrge low' frequency hojzonta1 mitjon are known to he of second-order in magnitude as compared to first second-order wave excítinp forces of mormal- wave frequency.

Faltinsen, 0.1-T; and Loketi 1

, Papanikolnoit.. A2' Rabman , Eatock Taylor 'and S.M. Hung

Kyozuka, Y. , 1-li,ao c. r.

and Liu yz 6

have

.orked out on '2-D theories giving numerical nie-' thods for evaluating the diffraction wave poten-tials and loads oit cylinders. Triantafyllou

C. C. Hei put forth consi stetit theories using perturbation method of multiple scales for shi-ving second ordet' forc that give rise to zero

ordr motions. Pinkstpr Standing

Mohn

have worked out on 3-D theories and numerical methods for computing secòud-order excitation forces. However, a numbrical method treating generally the low frequency diffraction load on a floating body of arbitrary shape has not -been developed fully.

- At the san!e time, a governing factor for

evaluating the amplitude of excursion and moor-ing load on floatmoor-ing structur'ies hs'heeti the fluid damping coefficient of such a system.

Vichers. et 'al. , Tagaki' 13

, Cao , Qi

iniestigated experimentally into the damping co-' 'efficient of low frequency motions f moored

structures and toûnd thit damping coefficient of these motions associated with first order mo-tions of the structure in waves tire usually higher than tite calm-water damping coefficient of the same low frequency oscillations exet-utEci by the mOored svstmni in calm water. Tite

differ-ence is called wave damping. Cao and Qi 1 found that under certain frequencies damping c'i slow motions in waves are lighter than those in calm water, and may give rise to negative wave damping coefficient. The present report relates the in-phase component of second-order diffract load with the second-order incident wave .zontal velocity and found that the force

co-effiient so obtained is in general agreement with the wave damping coefficient ohtaned

ex-perimentahly. A'teiitatit.'e explanation o this effect is given in Section 4.

The 'second order force consists of six parts

f e

+ fPJJ j'4' I2dS

+ rJJ

"X°'ndS

+

R°'F"

+

(2) (21

, (2) io)

('hIs l:pt

ds

- p(r

'5.i

5,e

(Z)

+ 7

5,

je.3

(3)

TJf

ti

The first term is the integral of the first order wave pressure over the area between the

mann waterline ,oÌi _{the structure and the surface}

elevation. _{The second term is the integral}

of

the quardratic term in velocity over the mean submerged. area of the structure. _{The third}

term arises.

from _{the second order part of the}

integral

over the mean submerged area of the

first order wave

_{pressure wheri that pressure}

in evalutéd at

thé

position of the displaced

body.

.

The fourth

termarjses from the

_rotation

due to roll, pitch, and yaw df the total _first order fluid force' on the _{structure including} the'hydrostatjc restoring force. _Thefifth terni comês from second order _potential

The sixth term is second order _{hydrostatic restoring}

forcé.

When the first order potentials _{and motjoi} have, beeñ solved, the first _{four terms}

_{can be}

calcùlated without difficulty.

To calculate 'the fifth term and sixth. _term _{we mut évaluate the}

second-order potçnial _{and second order}

mo-tion' response ri

We decompose the second order potential (2)

Into thÑe parts, nàmely:

(2)'. 52)+ 2)

+ )

vhere th2)and

,(2)'

_{are potential of second order}

incident.waves and radiated waves respectively.

They can be calculated wtih

_{relative ease.}

_Fór

2)we have

_{defined a relation between}

ffE'J2)

'rmds(jl,2,.. .6) and nonlinear

_{terms in' the free}

surface condition ánd also in

the k'iñematic

boUndary condition on the mean submerged surface

E of the structure.

_{Using this relation,}

_the

second-or-derwave exciting force

_{have been}

ob--tamed which will be disc'rjbed

_{in detail in the}

fo1lwijrig.

2. _{Theoretical Formulation}

We make as usual the assumption

of a perfe:ct

and irròtational flow, which' allows

us to des-cribe the motion by a 'velocity potential (x.,

y, z, r), where (x, y, z') belongs to the

undis-turbed ilûid domain D.

We shall. .use a

right-handed, frame' of coordinate

f o -

xyz) with' oz

vertically upword and oxy plane

'correspondiñg

to the undisturbed free-surface, as shown in

figure 1.

The body coordinates is consistent

with the coordinates

{ o -. xyz

I at t=O.

F:ig 1

Let c 'be a perturbation parameter, identi-fied with the wave steepness k, _{where k'is the} wave number-, and 'ç the wave-amplitude, _{may be} Written as:

¿J

='f"'(z.5,, t)+ E'

''(

z.f.Z, t)

+ ...

and the fluid force

acting on the body as

F = F(°)

+ cF(1)

'+ cF(2) .

From (i) and (2) it is found that the se-cond-order force

F()

_{contains the term:}

f1,2)

ill

7t3S

.where n

= _n _, _n ₌ _n :1 x 2 _y n5zn1-xn3, (3) =

fl,T14-'

yfl3-zn2,

In, the case of difference feqtiency which is of main interest to us, expression(3) _becomes

(W -'1,3,

J]]

9(l

71j dS-

₍₄₎

For the sake 'of _conveniences _{we shall} here-after omit subscrirs tu of

_&1Ç).-

The second order potential (

-shou.l.d satisfy the follow-ing set. of conditions:

f

'1'(Z4,1)=. O' ' '

(z.y.

+2

cd,, v"'"

+ - ¿".)M =o

(x

°) E.,.

(A) ¿

= j;;i.

.!v +

1")'

(z.5.z)ez g', JWL

4,'4

')d.

+

t"

I'(°'x 7t'')

ds +

_eJJ1

_v4"

')d5 +'","

-f

f

JJii(

xn')d5

_{- f'V(}

_72'X?.8'+

_'s0/v

Sia/V +

_{522/V )}

_-

-+

-

Sii /v )

(2)

(4)

Radiaion condition

wherè D is

the

Fluid

dornajo,

î..

i.s the

undis-turbedfrèe surfce miñus the aterplane occu-pied by the body. Y is the mean weted

sur-face

of the body and 1p is the sea

bòttom.

Now, as mentioned above:

(5).

where the 4,2) satisfies conditions (z, Ç

c'

j.

') O) (B) .

\

(2)

(x,y.-f)EI,

satisfies the conditions.

.v

t

''

(2).=o 'z, =

_;;,

u,

/j

(.A_

+

iv')

-satisfies

conditions

(x,9,))ÇJJ

--';'

(I

4 w. 0 Cl) = C

u" - v") n"

(f'. ,)

") ;?,

-Radiation

condition

2)

i he second orde- incident wave potent-ial, 4'21 i the second order radiation potent-ial and is the second order diffractjcqi

potent-i-al. _{fl çan be seen that the free waves}

and waves- locked t-o the first-order wave sys-tern are iñcluded in the

Expression (4) may he rewritten as

Jj

Ç)

¿=JJ

+Jjç6"ri,,ds+Jf 9J (6)

where-the factor

i(wu

w) is omitted.

Fôr fini te water-depth the expression for

2)

has been.given by Bowers as:

(X,

y.

.i

ci

R '

L

=

b c4.2 i.(2)i 2(-",.t # 6z C6 f

4-E., ₎

+, (

(JJC)4L

(_

cvt-

:x&Le

+ Et)

(2

)J,(-wt

+

xCe

%;f4.,e f e;))

(7)

Here the subscripts for wave number and fre-quency ara defined as

-+ F. + - , F F, - FM and

3cL222

NCANC (4

N.a.2iM

,t )

C E + 2

(i--

tLE.(

44()

4. 't -. (Io-) = _g Ç(.i_ Jpj') -,4.;-_4,;L (II)

where aN is the amplitude of firs-order

md-dent wave, i.e.

-D&(C).it -(12) and

IL

=

* (13)

=

N' ("UM' , (u. (14)

The set of conditions (C) ref-erri,,ng

to

second order radiation potential ') is in

the same form as that for the first order ra-diation otentia1s.. - Its solution can he ob-tained by. the same- method as that used for-sol-ving first order radiation potentials.

Now we turn to discuss the integral fJ2) nids. .:First we shall investigate the -asympto-tic- behaviour of 2.)

Asymptotic behaviour of 2)

Using oZ as the veritcal aXis, we generate a circular cylinde-r Ec af radius R enclosing the structure. The circular cylinder

divid-es

the

fluid domain D -into two parts as shown

in ligure 2-. D1 i-s the inner domain enclosed. by

_c'

Fi' Ej-

and 1 where

TFi and

Bï are

free-surface and sèa bottom enclosed by 7r

-is.the oúter domain hounded by ),

_{Fo' 'Ro}

and infinity, where andTp,0 are. free-surface and

(5)

and sea bottom outside of circular cylinder E Then, the et of conditions (D) may be

trans-formed into two sets of conditions: For domain D1, r

'U

'L)

:0'

't'do Radiation condition 'Pd Fig 2

( xy,-&)E2,,

(Z,

2)Ç

(Z ,(,2)EJJ,

,

In order to investigate the behaviour of

2) at infinity, we need only to investigate that of

d2).

When the radjits R of the cir-cular cylinder Ec is large enough, _{we have form}

4-for first order radiation potentia1s

('J

_&77

-')C4t.L

i

(-iJ(+i

JT

where = iiJ2/g

k th kR =v

R"!,? +

a. is the source strength on the jth panel of tie Structure surface.

sj is the area of jth panel on body surface

(15) From formulae (16)-(20), it may be noted that a radius Rc can always he found for any arbitrary small positive number c, for which

i +

If r is small enough, then conditions _(F) can be approximated by the following set of conditions (C) (16) (2) ¿ 4dß

=0

i'L'

= y

(2) (2;

Pd0d1

_(li

(x.r,2)c2

I

¿L)j__o

(Z, y. - C ) E 2,..E 2,..

(Z.y,1 )E1

- (*);

+ + ₌ + 2 4)j, L p4

2n'($) c(,

₁

Q O.77

(J

j

(2,

.O ) E

F.

(17)

(E) cL

(4 +L)

(Z, t, -'C)

v4'. ;"

_{= (' -}

_{" + (j'x}

"

.Z7T(t12-1) cUL ,1r

(18)

:'

-

_{ASJ C4t.J +L)}

For domain D0, o.778

(j1L4

N 2 '2) do = O (Z, L, 2) E

(4a

,p2) 4 ¡J

₍₁₉₎

(c)M

_)2I

+

= -¿2 c.j,

45j C,4'

+

,,r(pL

c L o.7?Z 1_ )

-

(20)

(F) ¡ p4 (Z.

y.o)E2

_'77

¿3jC4U(<+L)

i

11

Vdo =0

( z, y, 2 e z '2) U)

o (x.y,o)E2,-0

(6)

Tle. solution of ) can be expressed as follows iii 2?T(

e)

c.í(2CL

_r'

(.ç4 T(IR)

_{cJ.(R)Jdfc(7cc}

+ CM pA.. Ca

+ t)

+ i) K(

7'4 j='

.j

(21) where R o

a

3c)

are

the

source strength and area of thejth panel located on the circular cylinder of radius 1 respectively.

J0 is zero order Bessel function -of first kind.

Y0 is zero order Bessel function of second

kind.

K0 is zero order modified Bessel function of second kind.

Sjmilaç 1ormi4ae can be obtained for

thd'. and

4'd.

using the same method 17 as in the derivation of formulae

(16)-(20). Thus, we have

4, _1J

, Al,

¿'57'CA(

4i)

(22)

Expression (22) is an asymptotic expansion

of q1Ç) at infinity, which holds opproximately

as R

R.

Formula of calcuiçing

ff.

_2)ns

Following Hohn's idea

Il

and using the asymptotic expansion (22) , a relation similar to Haskind relation is àbt i ed. Based on this relatiOn the integrál 11E

2nds

may be

calcu-lated.

Let

Re (

eL(3$_c»)tJ.

(23)

wherew

'uj

corresponds to thedifference frequency of potential 2) and satisfies the following conditions:

=

O

(z,,2)E.D

(Z,(.2,)(1

(H)

(43)'j

(z,y,o)

(X,5.)lß

,1i

(Lu+j)_-_-Using oi as the vertical axis, we generate another circular cylinder 1R of radius R

en-closing the structure and the circular cylin-der of radius.P, . Let R > P . Again the cylinder E divides the fluid domain 1) into. twó 'parts denoted here by Di and D0. Domain Di is enclosed by ER, EFj, EBi nd , while

dam D0 is bounded by ER, EFO, Eg and

infinity

In domain D1, using Green's third identity we hâve

jJ

' njd

=

II!

ctS

=jjg'ct5 JJ1,.('_4'42fi)ds

(24)

From -free-surface conditions of (E) and (II), we may write

Jj

(-Fl2).)1

(25) Ire wherä

'free

-2'

--ci$u): :'#;'

' v4:;

+ 'UI' (26)

On the sea bottom, we have

JJ

(4Ç';'4i)d.so

(27) Note that 2) on R. we

have

JJ

('_q')ds

J°

¿2 I I

(*t_u%),(+v

e)

7T'

-'

'

_f

c-/ Y

I .ir (jJ1_

₄₎

_ft 'Jb (

-

zjc),( L, C4(

('$'

₎

_{I I X.}

.6 4'3Cf( I

0(f)

(28)

where 85h), O?) _{are sonirestrength,}

area, control point of the jth panel

_{and Nb,}

the- total number of pahes on tije structure surface respectively..

4c)

4c) are

source strength, area,- control point of

the

jth panel and Nc, the total number of panels on the surface of-the circùlar cylinder of ra-dius P respectively.

Using relation of (25)-(28) , expression (24) may be rewritten as:

as R -' , we have -

JJ

'

njds

_Jj'

_t"

_y/, dS+ - (29)

f4

S6,"rt,dS

₌

_JJ

. +

--Jj4# j,ds

(lo)

Using relation (30) and Cnbstitujng in the known body -surface condition for _{from (F),}

(7)

-5-we c

ultimately evaluate 1h

integral

and heñce also

ff

:(4,2)

2))n.d&

3. .Numerical Results

We.consider the first-order wave elevation - in a regular wave grtup eonsisting-

of N

regu-lar waves with frequency, il2," ,

N.

=

a.a (oiAf - (L

In the numerical example thirty frequencies

are usd.

See

Tab. 1.

TIlgTE. l

The, second-order force associated with such

a

wave train haa

the

following form

''

"i

-

-F1 = X

_j

( Pritj t

t

C(c-ej) f

;'(Q;;1 +

+.

_{+ Q6f).f1(c-.1pt (-E )}

"Ç'ff

(Pj Pj) -

-(Ptj Pj)

+.-.*j(P,j

'PkJ)1

j

<'Ç'f(Qj

+(aÁTQ,,jj)

+ . .. f f ( Qij

QirjJ3 ..L

(Ç u'Uj)t

c((c-)t-j)+

Qi

((c..iL-c)J)t +

_.-

t1)

and

the

quatllratic transfer

function

Pj

2

Q;tj

the

phase

-' (

Pj /

where

Pij

=

j

CP.r ¿j

Pj t) + f ( P

¿j + Pj)

f ...+

( .. +

Q;j

=j.(Q

-Q;ijc) -f

...+

and subscript K.,

9.,

i

,j

(t1 2 ,. . . ,) denote

respectively second-order force(momínt) in the

kth direction, due-to th

Ith term oF. fornñila

(I) or (2) mid due

to the

(i ,j.)th pair of

hi-frequency, the supe.rscript

denotes difference

frequency. (Appeñdixl)

Computatioñ has been carried out for l/soth-model of a six. column , two-floater

seinisub-mersible, the principal particular of which

are-given -in table-land Fig.3

PROFILE

TOP VIEW

F.NP vjj

Fig. 3

Frotn(30), it is shown that calculating the se-cond crder diffraction force may he reduced to the evaluation of integral.

C,)

Ii=JJ

?'--z

J 211. (Ls and

12

_If

-To calculate integral

_11,

the wetted surfr.e of the hull is approximated hya total of 244 plâne facet elements. The facet scherrintisation of aquarter of the hull is shown in Fig. 4. Thus

I,

_{=ZjJ341fi'rt5}

.Pesipnation 'a1ue L 2.036 ?T. ¡.414 4.141 B- f...39 nT ¡.636

4.q7q

fl

1.427?')

- 1.159 -5.19.2 V .

o. 2F

1'

2.o7 5.414. TI- - -

4.0 "t

23°i

5.637

s o.

7'r

-«'

a.3.0

g.cf

s22

o.67

6-oli

11,

I'

0.00Ll.1Tt'/m

0oo7rqr,

.9io

3'

.3O3

.f

rt1/.t 3.414

673

7.

-o-I

't

3.937

9.970 -O..ZIT 77'

3.:!?

7.192 S

0.0

t3

4.0V

7.4'#

Io

₄₃₃

_¿

4.Slt

-7.3o

(8)

and the integral is approximated by

thesussna'-t i on.

(31)

denotes the ialue of the function at- the center of the kth panel and 6s is the area of kth panel-.

TOP VtFW'

BOTTOM

vtrw

Fig. 4

The second integral L2 =

1

_ffF.+jfFeedS

theoretically should be intr'grated over an infinite area of free surlare. This infinite area is divided into two fields b a circle of

- radius Ra. The near field free surface (i.e. R4R5) is approximated by Z28 plane facet ele-ments, and 12 is evaluated by summation as in

(31). Thé far field free surface (i.e. R'Ra-) is approximated-by the following -integral

'I23

JJ 4'j 1Fe'

fj2ja0

jfree

b4J ₍₃₂₎

where Cause approximate integral-formulae

g

I

' e

('

and-i:

LA'J(l')

are used. wLar'.the integrated coefficients

-and are integrated nodes.

Since only the lowfrequency long-scale case is considered, the function _{jFee does not} have the character Of high frequency

oc-i1la-tion and'hence i-s amenable to such.methods of approximation.

The computation is carried out, on an IRM3O3I

cOmputer. Fig-.- 5 and Fig. 6 sbow the càlcuulat-ed results for qtiardratic transfer function of secondnrder 'forces in surge mid heave in a following sea. Thé- dOtted line takes into aCcount of the first four terms of eq. (-1) and

(2) only. The chain line inc'ludés -the above four terms and the' incident- wavé force- in

ad-'diti-on,, i.e. the integral J7 ,S2)njds,. The so1id-,ijne-futer 'takes into-account-of-the

integral ff$2'n:ds i.e. the force due to

diffraction',. lt is observed that the differ--ence betweén them is very small lot' secondary heave force response and -that the forCe due to inci'deit and diffracted second ordê.r wave force could for all practical purpose he neglected..

- However, this is not in the case of, surge, the

second order diffraction force ff 2nids - seen's -t-o be' 'important,' especilily for tYe

Ire-qúency range below 5.60. -Nevertheless, the force due to incident wave

42),i.e.

the term

;f.Ç2)nds

is still small and-could

heneg-- lect-ed. One should bear in mind, however,

that frequencies on a model Scale 'of less than 5.60 correSponds to a hill scale of 0.8 sec, which is- usually the energy gatherIng district of a wind wave spectrum.

'r-t,

'o

7

-4. Experimental Results »

i) A floating body executing free slow surge oacillations (we) in calm water under a soft spring is described by

A0 ' BJ( 4 C0X O ₍₃₃₎

where A0=m+a, m is the hodymass, and a is the added, mass at w0; is the calm water damping coefficiént', the potential part nf which is

-due to radiation waves frequencyw0.

Dividing -throughout by A0, we have

(9)

+z

+ o where

=

£/:_. , and ,

=

c, is determined experimentally by extinction

tests by Cao(1+J for the semi-submersible con-cerned and found to be 0.1025.

ii) The same mooring system and body is next placed in a regular wave of frequency IUj , and the saine extinction test performed. Another set of values u (wi, u1), w1

(hi1)

is deter-mined. In this case the motion is described by

(2)

.J)

A0(

+

F,, + P21

where F(2)is the mean drift force due to re-gular wave w

F) is the second order force in opposite phase with , and hence a damping force

(2)

_ii'

li

F(2) is the second order force in opposite phase with , and hence an added mass

(2)

(2) 3Fa1

I _u

-Let a , B be the added

mass and damping coeffIcient due to waves, then rearrangIng (2) and dropping out the mean force F

2 for it does not contribute to the

oscillajjn motion, we have

Allg + 4 C0X 0 ₍₃₅₎

where A1 A0 + = R0 +

Again putting it in rationalized form

g

2I (1)1* +

wX

O where

I

_aJE.

(34)

Define u

u1- u02_

_(for--«I)

and plot the experimental results nondimension-ally, the points in Fig. 7 _{are obtained.}

(Q.

L 2

In the figures _u' u

_a5.)

and the abscisea is

_W4T

, L being the length of the semisubmersible nd 2Ca the wave height.

iii) _{The same mooring system and body} _{may be}

placed in a brichromatjc wave field of frequen-cy (wj, Wj), j _{' i, propagating in the 4x} di-rection, and the same extinction test

conduct-ed. _{In this case whén the transient of freq.}

dies out, there will remain a stable forced

oscillation of frequencyw=w-w1. The

mo-tion may be described by

is, (LP .4*)

A.> +ß4CXo.F'2'+ FI2 F,2

+1

where A0, R, 'CO3 (2), carry the sanie meaning

-"

i.e. the second order hydrodynamic forces ex-perienced by a body in a bichromatic wave

(36) should be approximately the same, be it he streamedpast by an oscillating current or be-it be doing the oscillation itself.

(37)

8

as before.

'2'

F ' _{hydrodynamic force in}

phasç yith ; the hydrodynamic force being due

to 2) and .62) of !requencyf, and being the horizontal orbital velocity induced by

2)

F22)

91r

hydrodynamic force in

phase with ; tie hydrodynamic force bing due

to (2) and +62 of frequencyw, and f being

the tfiorizontai orbital acceleration induced by

2)

Fc(2) second order exciting forces of

fre-quencyw, which are the sum of the first four terms of C f ). These components are related

only to the products of first order potential, motions and their derivatives, and hence with-out di,rect bing to the second order

potent-ial 2),

D

However, it is not easy to carry out such an experiment accurately and besides, it is im-possible to break up the various components on the RUS experimentally.

Therefore, a special case for our numerical model is taken, i.e. the body is restreined from second order motion - but free to execute first ordçr motions - and the forces Fr),

F2(22), F2) obtained from numerical computa-tion.

The basic assumption nov is

(2) (*2 (2)

? F, Fu

'F,z

lFu

With this in mi9d, the numerically calculat-ed forces due to and .2) in the present example are reduced to the same form as Cao, vize

o.IL

fL (

Fu )(_)

The constant 1000 converts tons into kilograms, while all computations, are based on r.1 ,r.,j of

unit amplitude, andw 0.222 sect, a figure convenient for calculation hut being near to

o and w1 of the experiment.

The plot of u ¿ between numerical and ex-perimental results agree reasonably veli.

5. Concluding Hemarks

This paper presents a numerical solution to the practical problem of calculating second order oscillating diffraction loads on a semi-submersible. The diffraction component is trivial with respect to heave force, hut is considerable for certain frequencies of first

(10)

order waves in surge force. _{These frequencies,} furthermore3 are usually the frequencies of high. energy density in a moderate wind wave spectrum. Hence second order diffraction load

in generai shôuld not be neglected.

A preliminary comparison nf h'ydrodynami.c co-efficient-retted to second order orbitnl..wave

velocity. Is .cornpare _{with the "wave damping"}

coefficient obtained from extinction experi-ment. Reasonable agreement is obtained, how-ever, further, vork:is required in future..

6. Acknowledgement

This research has besn sponsored by the

-China State Economic Coninission under a gener-alcontract relatíñg to investigation ofbe-haviours of-moóred systems on high sea.

.7. Notations

Unit normal to structure sur-face

Location vector ou structure surface

First-order displacement .Tot-al first-order force Total first-ordèr moment Amplitude of wave elevation

Wate rp lane

Waterline

Displaced volume

f5jxids il 2

f5ixxds

i ,jl,2 Relative wave elevation

Ù) -(rip)

# r)y)

Motion response in the mode Height of the center of bucyan-n Si0-Sii Cr

nl

X38 H R(1)

1'k;ij 'Qk115

Pk21J iQk.ij

'k3ij 'Qk3ij

ls

_{.j 'Qk1J}

t'k5ij'QksIJ

1'k6 ti 'Qk6IJ

Tkij

C r I: o cy Waiter depth

I

-t

(I.)

.

o

%n5()

Tl in-phase nents of

In-phase and out-phase

nents of F(l)

in-phase and out-phase nents of

_offr

_2)nkds

n5

(I)

o

and out-phase

Compo--!t gfc.(1)2nii

in-phase and out-ç,hase compo-nents of ff1 Iv+(I)Fnkds Iii-phase and nut- hase corn-ponents o. off1 vl)X(l)flkdq

compo-in-phase d out-phase corapo-nents of second-order

hydrasta-tic restoring force

Quadratic transfer function for second-order force

Ph as e

Structure's mean vetted surface water density

supe.recripts(0) (1) (2) _{denote reroth,} first and fecond V

order..quart-ities.-

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qR4-.

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