On the motion of multihull ships in waves. Part I and II

6 NOV 197Z

ARCHIEF

Onderafdelin IDOCUmENIAMI ---J4Lphotheek van d epsbouwkunde nische Hogeschool,

DOCUMENTAT,E

I: Iry

_{_ dittrek}

DATUM:

lab. v.

Scheepsbouwkundg

Technische

Hopschool

Delft

With Compliments

15

6"

On the Motion of Multihull Ships in Waves (I)

tt

4

t 'Lb LI

kwilk Lk( Yet

By 1Ltfil-1/4/

Makoto OHicusu

Reprinted from Reports of Research Institute for Applied Mechanics, Kyushu University

Reports of Research Institute for Applied Mechanics,

Vol. XVIII, No_ 60, 1970'

ON THE MOTION OF MULTIHULL SHIPS IN WAVES (I)

By Makoto .OHKUSU*'

I,S

In this paper awe proposed an approximate method to calculate hydro-dynamic force and moment working upon multiple cylinders with an arbi-trary cross section which are forced to make a heaving, swaying and rolling_ oscillation about their mean position.

If we know the hydrodynamic coefficients (added mass and damping coefficient etc.) for a cylinder, we can comparatively easily calculate the coefficients for multiple cylinders composed of an arbitrary number of the cylinders by the approximate method. The accuracy of the coefficients obtained by the method was confirmed to be satisfactory by the comparison' with the exact solutions for two circular cylinders.

This simple method will be effective for the theoretical treatment of the motions of multihull ships in waves..

.i. Introduction

There have been reported many researches' upon the motion of catamaran due to waves, while there seem to be few examples that the motion is treated theoretically, for example, by the theory of ship motions which has been

succes-fully applied to ordinary ships. The floating rig which has been recently used

for the development of the ocean resources has common characteristic with

catamaran ship that it is composed of several simple hulls. And theoretical

researches may be also necessary for understanding its behaviour in the sea

and designing its performances.

This paper is concerned with the most fundamental problem for the theo-retical treatment of the motion of such multihull ships, that is, the theotheo-retical

calculation of the hydrodynamic forces, the amplitude and the phase of the

waves diverging at infinity, when two or more infinitely long cylinders, sub-merged or floating in a fluid, are given a swaying, heaving and rolling forced oscillation about their mean position, under the assumption of inviscid fluid, no

surface tension and the linearization of boundary conditions on the free surface of a fluid and the cylinders' surface. In the previous paper °}3 the theoretical

procedure to calculate the hydrodynamic pressure and the wave amplitude caused

by the heaving motion of two circular cylinders on the surface of a fluid was

developed and it was found that the results of numerical calculation agreed

* Lecturer of Kyushu University, Member of Research Institute for Appfied Mechanics.

33

3-

M. OHICUSU

with the experimental results., 'Just the same method can be applied to the

other motions, swaying and rolling, of two circular cylinders (Appendix 1).

The calculation by this method, however, is complicated and not convenient to solve the more general problem for obtaining the hydrodynamic forces and the waves generated due to the oscillation of two or more cylinders with an arbi-trary cross section. Then in this paper there is shown an approximate method suitable to treat such a general problem and its accuracy is examined :hrough the comparison of the numerical results by this approximation with the exact solution for two circular cylinders by the procedure described in the previous

paper.

If we know the hydrodynamic forces, the amplitude and the phase of waves outgoing at infinity when a body, whether it may be two dimensional or three

dimensional, makes a heaving, swaying and rolling oscillation, the phase as well as the amplitude of wave forces and moment which act upon the body

when it is held fixed on incoming plane waves can be easily completely derived

by Haskind relatioe extended by Besshoo (Appendix 2). Accordingly the

equations of the motion of multiple cylinders due to waves can be

mathemati-cally formulated through only the solutions of the radiation problem for the

multiple cylinders, which are the purpose of this paper, if the viscosity effect of a fluid is not taken into account.

2. The hydrodynamic force on- multiple cylinders"

Many analytical solutions are known for the radiation problem for one

-cylinder. The solutions are, for example, UrselVs one2)'> for Floating and

sub-merged circular cylinder and the solution by Tasaitm and others for a cylinder with a cross section derived from a circle by conformal mapping. In addition

Faltinsen,') Maedan have recently reported the results for cylinders with an

arbitrary cross section which have been calculated through solving numerically integral equation with respect to singularity distribution on the cylinder surface so that the boundary consition may be satisfied.

In this section is shown an approximate method to calculate the

hydro-dynamic coefficients (added mass and moment of inertia, damping coefficients)

and the diverging waves of N cylinders with an arbitrary cross section by

making use of these quantities for ,one cylinder given in the above mentioned

various solutions.

Let us consider the heaving motion y=Retehal, swaying motion x=Re[eiwt] and the rolling motion about the origin 0, 0 = Relef"t], of "a" cylinder composed

of two cylinders as shown in Fig. I, each of which need not have the same

cross section while they must be symmetrical with respect to their own center

line where the period of the motions is 27r/0).

The distances from the origin of the coordinate system to the center L of

the cylinder at left (hereafter referred to cylinder L) and the center Rof the

cylinder at right (cylinder R) are denoted respectively by PL, p. Let they have

ON THE MOTION OF MULTIHULL SHIPS IN WAVES (I) 35

PL Pa

Fig. 1 Coordinate system.

the same draft T for simplicity. The volume per unit length displaced by the cylinder L and R are respectively defined to be VL =k L. T2 and V=kT2.

Suppose that the added mass coefficients (the added mass moment of inertia

coefficients in the case of rolling oscillation) and the diverging waves at + co for each of the two cylinders oscillating without the existence of the other cylinder (the center of rolling is the point L or R for the cylinder L or R) are known and expressed as follows.

Table it. ('-'/ (j'nde,' S.,1

Motion Wave at X=+.0

x ,

y=e,o, , Tifyge,eVei..-K.L.R> 0 =elm, ,

Where K is the wave number (02/g(g is the gravity acceleration) and clockwise

roll-ing motion around L or R is taken to be positive and superscript L and R denote the quantities for the cylinder L and R respectively. XL means the X coordinate

with respect to the origin L for the case of the oscillation of the left cylinder

and X, with respect to the origin R for the right cylinder. Added mass

co-efficients and added mass moment inertia coefficient are defined to be the com-ponent of the hydrodynamic force per unit length of the cylinder 180° out of phase with the acceleration of motion/(pI/L, x the acceleration) and the com-ponent of the hydrodynamic moment per unit length 180° out of phase with the acceleration of rolling motion/(the accelerationx pV,,Rx T2), where p is the density of a fluid. In addition we introduce the following expression for the hydrodynamic cross coupling, that is the rolling moment exerted upon swaying cylinder, which is given by the summation of the components proportional to

d2xdx

and

dr

dt

Added mass coefficient

mkR

mi.,

L 2 ki (3) 36 M. OHKUSU dx2

PVL,T.M(_

_dr' + coPgz_' T. N L.' ( dx_dt (1) It should be noted, here, that for the application of the procedure described below it is not necessarily required that the left or right cylinder must be one cylinder but that the hydrodynamic coefficients of the left and right cylinder

are previously known. As a result if we know the hydrodynamic coefficients

for one cylinder and (N-1) cylinders with an arbitrary cross section we can

calculate the hydrodynamic coefficients for N cylinders with that cross section through the use of the procedure, because if the left cylinder is considered to be a bundle of (N-1) cylinders, the hydrodynamic characteristics of which are

known, and the right cylinder to be one cylinder, then we can calculate the

hydrodynamic force or moment for "two" cylinders made of the left and the right cylinder, that is, N cylinders. Summing up we can calculate the hydro-dynamic coefficients for a bundle of cylinders composed of an arbitrary number of cylinders if we know those of one cylinder.

i). Heaving

At first consider the heaving motion of two cylinders. f

n

7

7

0

The method adopted here to calculate the hydrodynamic coefficients of two cylinders is based upon an approximation that there exists the interference

be-tween two cylinders only with respect to the progressing waves generated by the oscillation of each of the cylinders, but the standing waves being mainly

in the immediate vicinity of one cylinder have no influence upon the other

cylinder. It is of course that the larger the distance between the cylinders is - as compared with the wave length and the dimension of the cylinders, we can

obtain the more exact solution under this approximation.

The velocity potential 0 which satisfies the boundary conditions on both

the cylinder L and the cylinder R can be written as follows.

0=CH-g+01.+Ce+46I+C,+..., (2)

where 0, 07, show respectively the velocity potentials that represent a fluid

motion as the cylinder L or the cylinder R oscillate without the other cylinder

and is determined as a diffraction potential of 07, due to the cylinder L 3

which satisfies the condition an (±)=O =0 on the surface of the cylinder

L and 0,, is a diffraction of due to the cylinder R under the similar

con-dition on this cylinder and so on.

According to the above mentioned approximation, near the cylinder L the

velocity potentials Ce, 01R, g include only the progressing wave potentials

and so the velocity potentials 01, ... near the cylinder R. It should be noted that on the cylinder L.

(01±0%+...+0°R+95'R+951,+...)-o.

-ON THE MOTI-ON OF MULTIHULL SHIPS IN WAVES (I) 37

It follows that since in the neighbourhood of the cylinder R the wave CI due

to OD, is given by

_{al, hi}

11 0

ceLeL'

(4)

the wave corresponding to 01, which is a reflection of 0'," by the cylinder R becomes the progressing wave expressed by the equation (5) in the neighbour-hood of the cylinder L.

=iM,(K)e-12",PL,,, ctied.,+xxL), (5)

where A, P

HP,(K)=ie4 cos sine (6)

(Appendix 2).

Similarly CI due to 03 is derived from the reflection by the cylinder R of the reflection of V, by the cylinder L.

Hence

Cl=alee",+(xL)

=i1-11(K) iH-Pe(K)e-'2"(.1-,PR)al,et.t+Kx1.) (7)

where

111,(K)=ie'Elhcose.'ne'Elgsin4 (8)

Therefore by repeating this procedure we can obtain the coming wave upon the cylinder L, which is due to the potential (filk+03R+44+, as follows,

4(1+ re -I- - - + )ei.t+KiL)

=i1-1),(K)e-12"(PL+PR),Tie14(1+ a +

+)e"`""xL)

₍₉₎

where

re =EH; (1C)iH1(10e-kPo-P.)

Similarly the coming wave upon the cylinder L due to the velocity potential

On/2+ + + s expressed as

r CA

71f,e'Etlie-IK:PL*Po (1+a + )e"+Kxf.). (10)

The series (9) and (10) converge because I H .(K) is necessarily smaller

than the unity if the wave length

is not so small

in comparison with the

depthwise dimension of the cylinders.

It ma*ncluded that under our approximation the hydrodynamic force

upon the cylinder L is given by the sum of the one due to cb7, which is the

38 M. OHKUSU

of the cylinder R and the one due to the incident wave which is the sum of the series (9) and (10) such as

ilfl,"(K)e-""(PL"R)Afteefl±:41,ede-1K(Ri+Pe ei.:+xxL).

1iTh(K)tlf-j(K)e-,21(PL+PR,

The latter include, of course, the effect of the diffraction potential by the

cylin-der L of the wave C, that is, 01±0i-i-C-F according to the equation (3).

Referring to Appendix 2 the components of the force due to C in various di-rections are

Table 2.

direction force (or moment)

lag

_akAigeEe,.

KPg

-i'VT

a".74Lede",

K

By just the same procedure we can calcualte the hydrodynamic force upon

the cylinder R as the sum of the force by the potential and the incident

wave C1 upon the cylinder given by

1l-1):(K)e-'2"'RL"R)-2--§e'eA4-74-eeii.e-,R(Rf.+Pa) (12)

1iH;-,(K)i111,(K)e-`2"(PL*PR)

The force due to the latter is also shown in the following table

Table 3.

direction force (or moment)

a,7-11eiese"'

a,74-geide", Mt. about R , iPgr a .74-Reel:a",

K

The hydrodynamic force working on the two cylinders can be computed by adding to the forces shown in these tables the hydrodynamic forces due to

the potential ¢,(1, and which are the forces acting upon the cylinder L and

the cylinder R when they are making an independent heaving oscillation of

each other, and given by the solution for one cylinder as shown in Table I.

Then the components of forces, F, in X direction, Fy in Y direction and a

moment 11,1 about the origin 0, become

Mt. about L (11) a CL= a L iKag jag K L

ON THE MOTION OF MULTIMILL SHIPS IN WAVES Cl) 39

ii). Swaying

Just the same procedure as in the heaving motion of two cylinders can be

"

4°

= iPg-Cifke14- a , - Af e'es' De" , (13)

F,= (A-fiej4 ± . a L)e,,,,

+ p(-

dd'ic )(1/Lml-,+ VRm) + Pwc ddYt + ( 14 )

M=

(T kie'E h , - T71-.Z

Dei'

+ ipg(- P ine' 41

+ PR e'E;', ,.)e"ue

+ P c; -) - P VL mh+ R + P - di; )

x + CA1 C^..* CLEIYt

The added mass coefficient k of the heaving two cylinders.(added mass

/ p(VL+ V ,)) and the damping coefficient NV" are easily derived from Fy and they are kL k - 771", kL+k, k L + 1

+-

_{(KT)-2-(k + k ,)} Re [i(Af,e'd , + All, L) I , (16)

N h+ " = -P-92 (A1i2+7:11t) - eff_2_ 1m [i

(ile'd

+

e'd

L)]. (17)

co' co'

Nextly the progressing wave elevation :`% at X + co generated by the

heaving motion of the two cylinders may be regarded as the sum of the wave caused by the independent motion of the cylinder R and the transmitted

com-ponent of the incident wave by the cylinder R and it is given by

=[Aeid + (1+ (K)) .aLI ei "P R (18)

where

(K) = iej6Iii cos e+eiEf sin 4, (19)

and since X coordinate is measured from the origin 0, the term ei"PR is

multi-plied. And the wave at X = -03 can be obtained similarly.

(15)

40 _{M. OHKUSU}

applied to the computation of the hydrodynamic force upon the swaying two

cylinders. Then here are shown only the results derived from this application. Using the identical notation as in i),

= ak e"," KxL,

1111(K)e-i2,PL+PR,Ake'4 A, eon+ Kx) ₍₂₀₎

1+ H(K) Ift(K)e-12"cP1-+PR)

L= a L. eiCult-KX,)

1H-1(K)e-"K(PL+eR) Afe'd +21.e1e.1" e-,K(PL.PR, IH-H4R(K)111(K)e-i""P L+ PR)

The hydrodynamic force upon the two cylinders is given as the sum of the one which the wave a ei"" "'XL) acts upon the cylinder L and the wave aL.

upon the cylinder R, and the one to which they are respectively sub-jected when they are making a swaying oscillation independently of each other. We can calculate as well the moment upon the two cylinders as the force.

Then the force F, in x-direction, F. in y-direction and the moment Al, about

the origin 0 are given by

Pg(Ae'e. A'Se'd a

K

pug

F, KPg (Ali e'd a , + Ar, e'd ) eh",

6-41ket6

)e""

+ Pg- ( -a, +74-7,PRe'd a)e"",

roll 7, °

+VR.WO+ T

co.

Accordingly the added mass coefficient mT,- k of

ample, is

niVA, lc, in,: kk ink+ 1

k,±k,

k,+k

8 (KT)2(k,+kk)

Re [4:41.eie- a, :=1eief

)]

and the damping coefficient NV' is

pT d'x (Vt. M dt2 _ ddxt

(A'± A')

x (25) IsTk-+R (;112

-

Pe [i (i7L--

-

a L)] . co' (21) L (4.g dx _{(N +NR). (24) I al G 7)} dt

swaying motion, for

ex-(26)

± mk

then ' p(V,,+ V

AIL" (

+

TNL'"

dt2 e ( 2 4 1.-.16 mr.+E lc mR kL+k, +k (KT)2(k + k )

Re[i(Aikeid ,

Me'd

,. PL _A11eLidaR+ PR A"- R '611 )1 (28)

NL'RNL+N" im [i(Alke'd

,74'ei61; a,

p

A', e'EH + P " Ale'6,R, ,)] ,

In addition the progressing wave Cc., at X= +m

[Me + (1-Filli(K))

L] eiKPR ei,/- Kx' (30)

iii). Rolling

The rolling motion of this two cylinders about the origin 0 can be regarded to be composed of two motions, the rolling motion 0 Re re"] of the cylinder

L and R about their own centre L and R, and the heaving motion y Re [Pi_

.et] of the cylinder L and the motion y Re[Pke'""] of the cylinder R,

be-cause the swaying of each cylinder be-caused by this rolling motion of the two

cylinders is of secondary effect.

Applying the same procedure as in i), ii) to each of these two motions we get the wave coming upon the cylinder L and upon the cylinder R as

follows, e",, Kx

1L

X ALet.13,-1 + (K) IPR(K)e-"K(PL' PR' e-1"(PL"R(LtAITe' ki.11-(K)e-'1"PL-1-:7 , x KX,), L___aL.eifiot-KXR)

1+ HI,(K)11)-?(K)e-'2,c(pk,p) x 1-4e4PR (K)e-121C,PL+ PR)

is (K) e21 PR) R L+ PR)] dx dr ) (27) (29)

./

(31) V krjfeld P K'PL +PR' +Al e'de--""PL P R' .T4T, e'6ZiHt (K)e- '2"(l', PR)]

(32)

41

H

iic I

1-7 p

° ot-e-% -et a_NC

_ _ t A

_ _dLt

-If

ON THE MOTION OF MULTIHULL SHIPS IN WAVES (I)

we express the moment M, as follows

+

=

=

x

42 M. OHKUSU

Similarly as in the case of i) or ii) the hydrodynamic force or moment on

the two cylinders is considered to be the totalof the ones which CR acts on

the cylinder L and on the cylinder R, and the ones due to the cylinder L's and R's individual heaving motion and rolling motion about the point L and

R, and accordingly the total force F2, F, and moment 1142 on the two cylinders

become 1.7

Reid aR 2}eidad

T(_ ft°21) (VLML ± V MR) + P

e

_{r(--- a )1 (NL+Nk)}

to' cit d2 dvt:- ) V L176 P d2yR ) VR/1 2

4

( dy

+ ()cog: ipit /Ted Li eiwt,

Mi= dc;i°2) ((1/2, I m)H- Pe at' dd2Yt

+P(

pg2 dyL) p Li-42 pg2 dy dtk) pR ;Ft dt

K [T -Ale'611 P L -iffie'611 R P R. .4. e'eg aL je""

From the equation (35) we obtain, for example, the added mass moment of inertia coefficient mi." of the two cylinders as follows (the added mass moment

of inertia/p(VL+ V R)T2)

rni; R kL (nik PL 2 m k (PR ) 2

kc±k,

T

"

kL+k

T

+ _{(Kr)2 (1c L-Ek JO} Re 1 erEIR' IaL

T R

PL LIR +AR. eel 42L

T T

and the damping coefficient is also

Nk Pg2 T2 CAT + Ar)± 40) (Pi. "Tf22+ P2R 411122)

2 !( ddOil (-42.± 1 + (34) 2 + ( ) + (33)

ION THE MOTION OF MULTIHULL SHIPs IN WAVEs (I) 43

'°4.a2--,0 lin fi(T71,1,:- ed.- aR -A7Zeiqa PL e'dak +PR XliterEll , (37)

The wave diverging from the cylinders, at x= + oo is given by

C.= 271-Re'sRe'w-KX)'= [7' + PR. Afaeid + (1+ if/ ,7?(K))c L] eiK P eawv-xn

As for cross coupling of sway and rolling the same result as in ii) are

obtained.

3. Numerical examples and discussion

Here are shown numerical examples of the hydrodynamic coefficients for some multiple cylinders which are obtained by.our approximate method.

1). Two circular cylinders

The hydrodynamic. coefficients, added mass and damping coefficient etc., for

two circular cylinders of radius a when they are forced to make a heaving, a swaying and a rolling oscillation with their axes on the surface of a fluid were

computed by the approximate method shown in the previous section. The results

for the heaving and the swaying motion are compared in Fig. 2, 3, 4, 5 and 6 with the exact solutions obtained with the procedure reported in the previous

paper 01 and Appendix I.

From these figures we can find immediately that the results by our approxi-mate method agree almost completely with the exact solution at least for 2P /a>31 unless Ka is very small. Then we can infer that for any multiple

cylin-ders our approximate method without complicated computation can give accurate

solution enough to be used in the equation of motions and in the result the fairly correct wave exciting force will be derived from this solution by the

relation shown in Appendix 2. Consequently the theoretical treatment of the motion of multiple cylinders with an arbitrary section is very easy.

Now substiltute into the equations (18) and (30) the following felation&

in the case of the two circular cylinders A=A=A,

=AL' = A3, Or ----8=20

7r

El's':= esl, V L= V R=

a2, and P=P,P.

and using the relations (6), (8)

2

and (19) reduce them into more simple form, then for the heaving

2(1-Fe"24-2")) .74-1 eidei( icx,e, 2+ (ea (2E3 -2_KP) 6,(24 -2KP))

and for the swaying

2(1ei(24-21P)) x

n Als". e Es ewt-K (40) 2_ (e1(24-2tiP) ei (26§- -2KP)I)

We can know from these expressions Of the waves at infinity that the zero.

,

T

(38)

C.= (39)

o Exact solution Approximate solution 3.0 iq 2 5 2P/0.6.0 2.0 _a ,1P/a5.0 9 1.5 i2P/a-40 2P/0 1.0 0.5 jt e , ge

tt

AA t.:1440t ,o pp g 0 .

.

1 ..

KO

44 _{M. OHKUSU} 2P/0 6 Cl 0 02 04 06 08 1 0 , .2 1.4 1.6 8 20 22 2.4 26

I lg. 2 AH, Ratio of wave amplitude to amplitude of heaving motion, re,, phase

lag of the motion behind wave motion for two circular cylinders.

amplitude of the waves diverging at infinity from the oscillating two cylinders occurs at Ka where 2(6 KP) ---(2n+l)7r are satisfied in the case of the heaving

and where 2(eKP)=2n7 in the case of the swaying.

Further since sg---te,;

when Ka is large as compared with unity, then the amplitude of the waves

tends to that of one cylinder A, Ay respectively unless the absolute values of the denominators of the equations (39), (40) are far from zero (this condition

is satisfied when 2(d KP) is far from (2n+1)7r for the heaving and 2(dKP) far from 2n7 for the swaying). On the other hand at Ka where 2(e 1,P) is

nearly equal to (2n+1)7r and therefore 2(dKP) nearly equal to (2n + 1)n, this

.

I

1,

ON THE MOTION OF MULTIHULL SHIPS IN WAVES (I) 60 40 ftlia--,^C, 20 1:-°_,.. t'4.,,. 1 A`i.

Si.A;

, .--. ..,,,,,..A \4\ ."-..--°-1!±!=1,1410.0.,044,...ititttvs/Al242.2 4451,9,4iLiskii: 1 2 1 4 I. I te 20 24 2 6 2P/0 6 0 2 P10 5.0 2 P10 4.0 28/0 3.0 -20 -40 -60 x 0 Exact solution Approximate solution

Fig. 3 mn. Added mass coefficient of heaving motion for two

circular cylinders.

amplitude due to the heaving two cylinders becomes very large. As mentioned

above near this Ka the zero amplitude also occurs and accordingly the variation of the amplitude with respect to Ka is ought to be abrupt (though continuous). Such an occurence is found in Fig. 2. As for the swaying it is similar. Since

e?>0, for the heaving two cylinders the maximum amplitude occurs at Ka a

little smaller than that where the zero amplitude are realized and for the

swaying two cylinders a little larger Ka.

As for added mass coefficient such an abrupt change happens but this

change may be not necessarily discontinuous known from the equations (16). (25) etc.

The reason why such a phenomenon occurs is that at comparatively large Ka reflection coefficient (the amplitude of reflected wave/that of incident wave)

is nearly equal to unity (not accurately) and the condition that 2(dKP)

4,0 3.0 2.0 .0 -2.0 0.2 to- LAI 1 & X w.4 crtz Ac if 1t Yp 0A7 1,

X.rx ,

IX & II 1I W-&° I I 2Pia -6.0 2P/0 5..0 1.4 1.6 2.0 XI &-MX & X

gir

' ' dr' ',IA I0.1r X & ,x 2P/a 41.0 2P/0 0 cr°9 1,ra o'er. o _{a . Exact sotunion} Approximate solution 2P/0. 6.0 2 Pio 5.0 2P/0 9.0 2.P/a 0 xN orr- ---K0 1 4 ;tttb I # ' :7412-tt-11.:. % I .4.&4.444131"" _'t *-&-. Attrt* , II .0

'

... t

-Wei' it ' ' e 1 , o , 6

KO

Fig. 4 A,, Ratio of wave amplitude to amplitude of swaying

motion, e.g, phase lag of the motion behind wave motion for

two circular cylinders.

(2n+1)7 for heaving and 2(eL) KP)=2nit for swaying are satisfied so that the effect of the reflected wave may be accumulated and in the resuts the very high wave comes upon each of the cylinders. Sometimes this high wave are cancel

out at infinity by the interference with transmitted wave by the cylinder of

this high wave. Even in this case, however, the very high wave exists between the cylinders (this is confirmed through the experiments in the previous paper,).

When such a high wave comes upon the cylinder and in spite of it the

force of .some phase is totally small, it is expected that the accuracy of the

4C, M. OHKUSU 0'.4 0.6 0.8,1 1.0 14.2 I #1 3 5 3.0 2 5 2 0 5 05 t 0 0.2 0.4 06 0 1.6 I 2.0 22 24 2.6

ON THE MOTION OF MULT IHU LL SHIPS IN WAVES (I) 47 X o Exac,t solution --- Approximate solution 2P/o..6.0 11,

Fig. .5 Ins, Added mass coefficient of swaying inotion for two

circular cylinders.

numerical integration of the hydrodynamic pressure along the surface of the

cylinder is very much lower. With this fact taking into account our approximate

solution is better than the exact solution at such Ka,

ir0. Four circular cylinders

Laying 2-cylinders of radius a, the hydrodynamic coefficients of which are

known from the calculation of i), at left and at right we can obtain the

co-efficients for four cylinders.

2P/0=5.0 .4 ₄ 2P10.4.0 t A !t.' ti I 4 111 ....° o se._{o ..4 .} 4..4, 2P./0.-3-0 0! At 0 0 . ..0, b..

4.'

CI 0' .... 0. .... .- w ' ./0-0- 0.0. 0. ./0-0- 0 ./0-0- ./0-0- ./0-0-12-2:k.'ay

°."

'C'-°

...

z..441

, ..

, ,..6-6,-4 . ,r4.4

.

. pA-4-AA ...,_,.. 2 0.4 0.6 0. 6 !I 0,1.21 /.4 ,AA 1 . 8 2. 0 ... 2.4 4v 0.

'

.KG 4 ,# 4 s 4

11

4 ?

t

.4 t 4 4

j

4 5 I el. 4 p S r ;11 4. _? 4 A

48 .3.0 2.0 3.0 -5.0 2 P/0 6. 0 2P/0.5.0 2P10. 4.0 2P/O 3. 0 2.0-o' .0 -2.0 -3.0 M. OHKUSU I. Jr :

acv

a 1...o-C S.5 e cr. 2 0.4 .0.6 0.81 L . 0 2 P/0 .4.0 /-2P/ 0-5.0

/

2P/0-6.0, 0--°- -0 2.4 2'0

-Ka

Exact sonction Approximate solution A Ar A- +I-",0411.- 1.-, , II 2P/03.0

,-*Ka

, I 6 6.8 .2'.0 2.2 2.4

Fig. 21-4 N, Coupling term of swaying for two circular cylinders_

Some examples of this calculation are shown in Fig. 7 and 8. It should be noted that the waveless condition are realized at two kinds of Ka. One

is the Ka where both left and right 2-cylinders do not make a progressing wave at infinity shown in i) and so this four cylinders composed of two of these

2-cylinders. Another is the Ka at which the interference between the left 2-cylin-ders and the right 2-cylin2-cylin-ders cause the zero amplitude of the wave,

A

Two and three cylinders with a Lewis forth cross section

Some of numerical( results for two cylinders and three cylinders with a Lewis form cross section (a,=0:0, a3= -0.1), where three cylinders are con-,_

2.0 .0 A 0 0 0.2 .r 4-0.6 0.8 1.0 I2 - 1 . 4 1.6 1.8 4 0 -4.0 4.0 3.0 ( . 0 0 -0-I I I + I 6 iii).

ON THE MOTION OF MULTIHULL SHIPS IN WAVES (I) 49 (.0 KO , , 0.2 0.4 016 0.8 1.0 12 1 4 1.6 1.8 2.0

Aieilibraik2Pak

VS VS VP

Fig. 7 AR, A11, Ratio of wave amplitude to amplitude of

heaving and swaying motion for four circular

cyli-nders.

sidered to be the combination of a left cylinder (two cylinders) and a right

cylinder (one cylinder), and the hydrodynamic coefficients are calculated from

those of two cylinders and one cylinder which are previously obtained, are

shown in Fig. 9, 10, 11 and Fig. 12, where kL, k, is considered to be r/2. Thus we can comparatively easily compute the hydrodynamic coefficients for multiple cylinders with a Lewis form cross section and accordingly applying the strip method we can treat theoretically as well the motions of catamaran

as ordinary ships. 2P/a 3.0 2 0-IQ 2P/0 6.0 o 0.2 0.4 0.6 0.8 1.0 1.2 1.4 /.6 (.8 2.0 0

1 5

1 0

-I 0

-1 5

-20

Fig. 8 mg. Added mass coefficient of heaving motion for four circular cylinders.

ON THE MOTION OF MULTIHULL SHIPS IN WAVES (I) 51 1.0 0.8 1:0 3-Cylinder 2-Cylinder 2P/T-3.0 Lewis form 2P 2P Of .0.0 0. 02 0.4 06 08 2 1!4 16 I 8 20

Fig. 9 A, Ratio of wave amplitude to amplitude of heaving motion for two and three Lewis form cylinders.

-I 0 -2 0 -3.0 40 3.0 20 I 0 0 Iq 02 "" 0.6 tt Ii 2.0 KT 12 1"6 1.8 3.0 2.0 1.0

M. OHKUSU

ME

OOP MCP OM

-8 0

Fig. 10 ma, Added mass coefficient of heaVing motioR fof two and

three Lewis form cylinders.

L ?P/T= Lewis form 0.0 2P 2P 0; = -0.11 52 6 0 4.0 2.0 0 0.2 041 2-Cylinder 3-Cylinder -2.0 -4.0 50

ON THE MOTION OF MULTIHULL SHIPS IN WAVES (I) 53

Y

2P/T.3.0 Lewis form

KT' Fig, II AR, Ratio of wave amplitude to amplitude of rolling

motionXT for two and three Lewis form cylinders.

- 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 2.0 3.0 2.0 10 -10 -2.0 -3.0 4.0 3.0 2.0 1.0 0 03.-o. 1.8

54 4 081 I O. 13 M. 'OHKUSU

--.2 II:4 it^ I . 8 2.0 2P/T=3.0 Lwis form 2P P KT 01=0.0 03=-0.r

Fig 12 mR, Added moment of inertia coefficient of rolling

motion for two and three Lewis form cylinders._

X /60 14.0 12.0 10.0 6.0 40 2 0 2.0 -4 0 -6 0 8.0 -10.0 -12.0

0

0 02 04 2-Cylinder 1.0

ON THE MOTION OF MULTIHULL SHIPS IN WAVES (I) 55

Acknowledgement

1 wish to express my gratitude to Prof. Fukuzo Tasai, Kyushu University, for his constant encouragement and interest in this work. Also I am indebted

to Mr. M. Yasunaga for his cooperation in card punching and Miss Y. Okazaki,

Miss M. Hohjo for their generous assistance.

I dedicate this paper to my daughter who passed away during I was

writing this paper.

References

Corlett, E. C. B.: -Twin Hull Ships". Quart. Transactions of the Royal Inst.

of Naval Archtect Vol. 111, No. 4, 1969.

Ursell, F: "On the Heaving Motion of a Circular Cylinder on the Surface of a Fluid." Quart. J. Mech. App!. Math. Vol. 2, 1949.

Ursell, F: "Surface Waves on Deep Water in the Presence of a Submerged

Circular Cylinder I." Proc. Cambridge Phil. Soc. 46, 1950.

Tasai, F: "On the Damping Force and Added Mass of Ships Heaving and

Pitching." Report of Research Institute for Applied Mechanics, Kyushu

Uni-versity,Vol. 7, No. 26, 1959.

Tasai, F: "Hydrodynamic Force and Moment Produced by Swaying and Rolling Oscillation of Cylinders on the Free Surface." Report of Research Institute for

Applied Mechanics, Kyushu University, Vol. 9, No. 35, 1960.

Faltinsen, 0: -A Study of the two-dimensional added mass and damping coef-ficients b the Frank Close-Fit Method." Det norske Veritas, Report No. 69-10-S,

1969.

Maeda, H: "Wave Excitation Force on Two Dimensional Ships of Arbitrary Section." J. of the Soc. of Naval Archtects of Japan. Vol. 126, 1969.

Bessho, M: On the Theory of Rolling Motion of Ships among Waves. Vol. 3,

No. 1. Report of Scientific Resetrch of Defence Academy.

Newman, J. N.: "The Exciting Forces on Fixed Bodies in Waves." J. of Ship

Research, Vol. 6, No. 3, 1962.

Ohkusu, M: -On the Heaving Motion of Two Circular Cylinders on the Surface of a Fluid." Report of Research Institute for Applied Mechanics, Kvushu

Uni-versity, Vol. 17, No. 58, 1969.

(Received April 9, 1970)

Appendix 1

Let us suppose that two infinitely long circular cylinders with their axes

on the surface of a fluid as shown Fig. A. I are forced to make a swaying

motion X Re [lei'] or a rolling motion 0 Re [a _{about the origin of the}

coordinate system and denote the velocity potential describing a fluid motion

by _{ew (here is assumed that a velocity potential exists and the boundary}

conditions on both the free surface of a fluid and the cylinder surface are

linearized). Of course the rolling motion may be regarded to be a heaving

Fig. A.1. Two circular cylinders

motion y= Re [Pe""] of the

left cylinder and y----Re [Pee""] of the right

cylinder.

Applying the same procedure as in the previous paper' and defining the velocity potential 0, 0), respectively when the left cylinder is oscillating alone,

that is without the existence of the right cylinder, and the right cylinder is

oscillating alone, we can obtain the following expression for 0

7TCO 71-0) 0

(OA +0) + ci/itgss(KrA, 0-05(Kro, Of,)

gl gl

+

P[f(Ka, rA/a,

A) MICa. rade', 001)

+C2/1(sb,(KrA, 0A)+0,(Kr., OB)

QJg(Ka, rA/a, 0A)+g(Ka, re/a, 0,)])

(A. 1)

m-1

where for rolling motion 1 in this expression means .p.o and 05(Kr1, 04),

(Kr, 0) are the velocity potentials of a source at A and B and 0,(KrA, OA).

OB(Kr, 0) the velocity potentials of a dipole at A. and B. rA is a distance

from A to a point (X. Y) and r, a distance from B to the point. dB are

the angles that the rA, r make with a vertical line through A and a vertical

line through B respectively (counter-clockwise is positive). 955(Kr, 0), 0(Kr, 0) are given by

cbs(Kr, 0)ine-""" e-1rt

sin_(Krt cos0)tcos(Krt cos 0) eKrt ,sit1.9 di

Jo 1-1-t°

(A. 2)

56 M. OHKUSU

=

-ON THE MOTI-ON OF MULTIHULL SHIPS IN WAVES (I) 57

D(Kr, 0)=T-re-locosoeact Kr- Si n 0_(Kr)2

tsin(Krtcos0)+cos(Krtcos0) _{-e- Krt Isin01 di., for0 0,}

+

Further f,(Ka, r Ala, OA), f,(Ka, r,/a, OB) are wave free potentials symmetrical

about A and B respectively and g(Ka, r4/a, OA), g,(Ka, rH/a, 02) are

anti-symmetrical one about A and B respectively and they are

f,(Ka, r/a, 0)

g,(Ka, ria, 0)

F,, Q, are complex numbers and the wave elevation C due

X = +co

Re (C,,C2 _Licoet-KYTP)_ ( C,-LFK2),eiCcdt-KX-P)1

where C s the wave amplitude due to the potentials 011 or Q. The

stream function 0 derived from the velocity potential 0 is as follows

ItC0

gl

Kacos(2m-1)0 cos2m0

(2tn 1),(r/a)2"'-1

War'

7;MO)

7

(OA +0B) + (0s(Kr A VA)(Ps(Kr 13, LOB)

+ j 1), [$, (Ka, rA/O, 014)Em(Ka,. ritta, 0B)]) + (CbD(KrA, A)H-S)D(Krs.

-±E

,r4/a, OA)+77.(Ka. r8/a, 08))) (A.7) ort

cl,B(Kr, 0), OB(Kr, 0), e,(Ka, r/a; 0) and ti,(Ka, tia, 0) are the conjugate

har-monic functions of OB(Kr. c6B(Kr, 0), f,(Ka, la, 0) and g,(Ka. ,r/4, 0)

re-spectively.

If 0 satisfies the following condition on the surface of the cylmder A, then it satisfies the condition which should be satisfied on the cylinder B because 0 and this boundary condition for 0 is symmetrical] about Y-axis.,

Ii7r.KacosoA+C

for swaying,

inKa(P/a)sin0 A + C for rolling (A. 8)

where C is a constant independent of 0.

CI, of course, satisfies the condition (A.8) on the cylinder A and therefore the condition for 0 on the cylinder A becomes

cti (A. 3) (A. 4) (A. 5) to is at (A16) conjugate

Kasi n (2mmil)0 sin (2m ± 1)0

2m (r/a) 2n, --

(r/a)2"'

1 = 011)

Q,k(Ka,

0),

58 M. ,OHKUSU

where

Co=const.

0Bian_3( sin-2Pla

k cos°

(r3la)2= I 2 (2P/a)sine'± (2P/a)2' (A. II) The equations about the unknown coefficients _{Q, CO, C2/1 and Co are} solved by replacing this system of infinite numbers of equations with a finite number of equations and, for example, by the least square method..

Using the coefficients thus obtained we can calculate the hydrodynamic

pressure on the cylinder surface from the velocity potential given by the

equation (A.

The wave amplitude ratio A, for swaying and AR for rolling (wave ampli-tude// or a()) and the phase of the wave ER. ER are derived from the equation

(A. 6).

In addition added mass coefficient of swaying ins (added mass/p7ra2), added mass. moment coefficient mR of rolling (added moment/pn-a1) and cross coupling moment expressed as

dx

pna2.M(

d'x\

_{dt2 )} Pg.' a N(

(03 di

)

can be derived by integrating the hydrodynamic pressure on the cylinder surface:

Some results for the cases of 2p/a--10, 4.0, 5.0 and 6.0 are shown in Fig. 2, 3, 4, 5 and 6 as exact solutions.. .

Appendix 2

Let us restrict here the problem to two dimension for simplicity..

The so-called Haskind Newman relation91 is a formula by which we can compute the wave exciting force or moment in a direction upon a body when

it is held fixed on incoming plane waves from the wave amplitude diverging at infinity in a corresponding radiation problem. This relation, however, does

(A. 9)

(A. 10)

A. 12).

TC le

(clis(Ka, 0)-0s(KrB, R)

P.'{.(Ka I. 0)-6,(Ka, re/a, 08)1)

+ [OD(Ka 0) +0D(KrB, 0)_'

ON THE MOTION OF MULTIHULL SHIPS IN WAVES (I) 59

not mention of the phase of the exciting force or moment, but gives only its amplitude which is insufficient to be the term on the righthandside of the

equations of the motions.

BesshoK% showed extending this relation that if we know the amplitudes and the phase (the phase lag between the motion of the waves and the body) of the waves diverging at infinity in a radiation problem, we can immediately calculate the amplitude and the phase of the corresponding component of the wave exciting force or moment.

In addition he showed a simple formula that gives the amplitude and the

phase of reflected and transmitted waves tifi an incident wave coming upon a symmetrical body when the solutions for the swaying and heaving radiation problem of the body are previously known.

These relations are summarized in a little modified expression convenient for our use in this paper as follows.

When a symmetrical body shown in Fig. A.2 makes a swaying motion x

Re[e'l , a heaving motion y=Re[e"1 and a rolling motion 0 Re[e'l] about the origin 0, the progressing waves at X -For, are given respectively by the

following expression,

Fig. A. 2 Cylinder

A-sees ', for swaying, (A. 13)

A.e16n e".- Kx), for heaving, (A. 14)

A, .17 eic°1-", for rolling. 4iyed (A.15)

Suppose that an incident wave 1.ef 'F1C'1C'

comes upon the body, then the waves at X= +co and X=cc are respectively

C,=.4.,= 1. el't +KX) (K) 1. e'.'- t _{(A. 16)}

_1. eitout+ICX) H - (K) .1, et(.+ KX)

6 0 M. OHKUSU

where H (K) are the complex numbers dependent upon the body form and K and are given by 0 d_ ot

H (K) ei6 n cos e,T-e'Es.sin es, (A-18)

The incident wave and the diffraction wave of this incident wave by the

existence of the body exert upon the body the hydrodynamic force and moment which are as follows,

sie

x direction

y direction

K

moment around 0 g Ae,ER

K

where p is the density of a fluid and g the gravity acceleration.

1

I

1

6 NOV. 1972

ARCHIEF

Bibliotheek van de Onderafdeiirr-Schie-psbouwkunde Technisihe Hogeschool, --0...,CUMENTATtE

leg,

:5 DATUM:

Lab.

v.

Scheepsbouwkunde

Technische Hogeschool

Delft

With Compliments

Reprinted from Reports of Research Institute

for Applied Mechanics, Kyushu University Vol. XIX, No. 62, July 1971

On the Motion of Multihull Ships in Waves (II)

1,1,41L

kAAJA LAA,tk

By

t Loy

Makoto OIIKUSU and Mikio TAKAKI

Reports of Research Institute for Applied Mechanics

Vol, XIX, No. 62. 1971

ON THE MOTION OF MULTIHULL SHIPS IN WAVES (II)

By Makoto OHKUSLT* and Mikio TAKAKI"

This paper is the second report on the motion of multihull ships in waves and is especially concerned with the theoretical calculation of the seakeeping

qualities of twin hull ships in head seas and beam seas. The theoretical results obtained by using the method which was developed in the first paper in order to compute added mass etc. of twin hull cylinders and by apply-ing Strip method were found to be satisfactory in comparison with the

ex-perimental results.

The forces acting upon twin hull ship when it ran in waves were also

calculated.

1. Introduction

Recently the merits and demerits of twin hull ship have been discussed

from many points of view."--3) The twin hull ship does not seem to be able to get an economical advantage over the ordinary mono hull ship. The former, however, has several merits in its utility. Oceanographical research ships and

floating stations for ocean development having twin hull have been recently

built for such merits.

The research of twin hull ship has been carried out chiefly upon its resist-ance on calm sea and it is because the merit of the catamaran has been looked for from economical stand point. In the case of the research ships or the float-ing stations, however, the seakeepfloat-ing qualities are more important. There are a few examples'''' of the studies upon the motion of catamaran in waves and these studies conducted so far are not necessarily based upon the modern ship mo-tion theory which has been successfully applied to the momo-tion of the ordinary ship and they neglect the hydrodynamic interaction between two hulls of twin hull ships.

In this paper we calculated the damping force, added mass of twin hull

cylin-ders oscillating on the surface of a fluid and the wave exciting force upon these cylinders due to an incident wave by the method which was developed in the

first paper' on the motion of multihul I ships in waves and which took the

interaction effect into account as a matter of course, and solved the equations

* Lecturer, Research Institute for Applied Mechanics, Kyushu University.

" Research Associate, Research Institute for Applied Mechanics, Kyushu University. 75

761 M. OHKUSU and M.' TAKAKI

of the motIons of catamaran in head seas and beam seas which could be formu-lated by using the damping force, added mass thus obtained of twin hull cylin-ders. In addition the solutions were compared with the measurements on model

tests..

The forces:' exerted upon the twin hull ships Moving in wavesl were also

calculated.

2, The Motion of Twin Hull Ship in Head Sea

2.1.. Comparison of theory, and experiment

In the first paper° on the motions of multihull ships in waves one of the authors reported the method by which we can compute comparatively easily

the hydrodynamic force exerted upon multiple 2-dimensional cylinders when

they are oscillating on the surface of a fluid or they are in waves..

We can predict theoretically the motions of a catamaran in head seas by

applying the method to the calculation of added mass and damping force etc.

of its each cross section and with the aid of Strip method.

It may be safely

said that the accuracy of this prediction is not less satisfactory than it is in

the motion of ordinary mono hull ships.

Two twin hull ship models were selected for our study (it will be called

TWI and TW2 hereafter). These two models are composed of two mono hulls with the form longitudinally symmetrical with respect to midship as shown in

Fig. I. The principal dimensions of these models and their component mono

hull are given in Table 1. TW1 and TW2 are catamarans with 2p/T= 3 and 5 respectively where 2p is the distance between the longitudinal centers of their

Table Particulars of Models

In! 1 Description Single-hull 1 TW I ' 1 TW 2 2P/T

Length between perpendiculars

(4)

. 4. 00 m 3.00 4.00 m 5_00 4.00 in 1 Breadth. (B) 0..36 m 0.90 'm 1.26 rfi Draft (T) 0-18 m O. 18 m O. 18 ryl Displacement (v) i0207mi 0: 414m3 0.414m3

Radius of longitudinal gyration _{0.262, 0.262} 0:262 CR,/ Lpp)

Radius Of transverse gyration,

(Ro/T) ,=,,,-, .1. 772 2. 514 Metacentric height (GM /T)

1

1950._., 6. 272

Height of center of gravity from

L.w.L

ON THE MOTION OF MULTIHULL SHIPS IN WAVES (II) 77

Fig. I. Body plan

component mono hulls and Tis The draft.

'The heaving and pitching motions of twin hull ships TW1 and TW2

advanc-ing in regular head waves were theoretically calculated along the afore-mentioned

course, that is, to compute added mass and damping force of each cross section of the ships by our method and to apply the Strip method for connecting these

two dimensional results to get the equations of the motions of the ships in

head seas. In order to compute the added mass etc. of two-dimensional twin

hull cylinder by the method described in the first report it is necessary to solve previously the so-called radiation problem of heaving, rolling and swaying mo-tions of one cylinder which composes the twin hull cylinder. Then each cross

section of the mono hull cylinder composing TW1 and TW2 was

approximate-ly expressed by a Lewis form and the radiation problem of this section was

solved by Tasai method!'

The equations of the Motions of twin hull ship in head seas can be

formulat-ed as follows') by the Strip method, where a coordinate system is taken as

shown in Fig. 2. i

,

, 4

4

,

Ii

iti nil II Hi I

9

MILES111 FAIVAIMI

Al

WW.Porr

rrid

78

Taso;

(,6'1

2 Lf G, =f pg.2.bxdx, La,

- . M. OHKUSU and M, TAKAKI

2, Coordinate .systein

(m+ a)4+ b4+cZi+ do +ea+ g,O=Fen'' ,

(J o+ A)4+ BO + CO +Dki+ E4+4Z,G= mei.et

j.

Lf Lf

pSmidx, b=

n,lidx, c= pgAw,,

La La x, IL/ (nixYpSmi)dx,, J PSxm,da: La Lf = (pg- 2bx n Y)dx, x' pSmdx,, La L L f

B = f(x2ni+ (1±)2 n)dx,, C= pSmidx+ pgIw,

La co, La

D = rf

LaxpSmdx,, E= if (nx+VpSmi)dx,,L Lf F= 1(Pg'26. w LoepSm)+IconDe-KTm+iKx .dx,. La Li M=Cof [(pg.

x2bxto w. pSm)kx(on=VcopSinH)1

Lc x'eirr"'-"Kxd.X. 'Pr (3) Fig. d = e = f La =

where La and L, are the x coordinate of the fore and aft end of the ship, m: mass of the ship,

londitudinal mass moment of inertia of the ship, Aw: water plane of area of the ship,

1: londitudinal moment of the water plane area about the center of floatation.

S is the area under the water and 2b, T, and T, are the breadth, the draft and

half of the draft of a section at x of the ship. mil and n are added mass and

damping coefficient of this section. _{The ship runs in regular head waves of an}

elevation Cae."'"+"' with advance speed V where K is w' /g and coo=co+KV

and p, g are the density of a fluid and gravity acceleration. ZG is the heavng displacement of the center of gravity and 0 the pitching angle (bow down is

taken to be positive). It is of course that the equations have the same form

as those of ordinary ship, but mil, nit of each section at the coordinate x are

calculated as the values of twin hull cylinder by our method of the first paper. The amplitudes of heaving and pitching motions Z, 0 of TW1 and TW2 at the Froude number F0=0 and 0.1 which were obtained by solving the equations

(1), (2) and (3) are shown in Fig. 3, 4, 5 and 6 as theoretical values. In these

figures the amplitudes Z, G, are nondimensionalized by wave amplitude and

maximum wave slope respectively and A, L are the wave length and the

ship length. And the curves named as single hull in these figures give the am-plitudes of the motions of the mono hull ship which is a constituent element of

our twin hull ships. They were calculated by putting mil, etc. of this mono hull

into the equations (1), (2), and (3). If we neglect the interaction effect between ON THE MOTION OF MULTIHULL SHIPS IN WAVES (II) 79

0 0.5 Theoretical Measured 2 P/T 3 ₀ 2 P/T -= 5 Single-hull Pn= I 1.5 2.0 AIL

Fig. 3 Heaving motion of twin hull ships in head seas (F.--0)

1.0

0.5

J,.:

0

1.0 0.5 1.0 0.5 Theoretical 2 P/T 3 2 P/T = 5 Single-hull Measured a 4/L

Fig. 4 Heaving motion of twin hull ships in head seas (F,c=0.1)

0 5 1.0 1.5 2.0

-0-

AlL

Fig. 5 Pitching motion of twin hull ships in head seas (F=O)

80 M. OHKUSU and M. TAKAKI

0.5 1.0 1.5 2.0

FnF.:0.1 0

0.5

0.5 1.0 1.5 2.0

.1./L

Fig. 6 Pitching motion of twin hull ships in head seas (F=0.1)

two mono hulls of TW1 and TW2, we can consider the values of the single

hull curves as the amplitude of the motions of TW1 and TW2. From these comparison it is found that there is comparatively large difference between the motion amplitudes of catamarans with and without taking the interaction effect into account if Fn is not zero.

The experiments concerning the motions of our two catamaran models TW1 and TW2 in head waves were carried out at the large tank of Research Institute

of Applied Mechanics, Kyushu University.

In our experiments the amplitudes of regular head waves generated by the

wave maker at the end of the tank were made to be constant at any wave length, and they were measured at a fixed point in the tank before each

ex-periment started. The apparatus used in our experiments are shown in Fig.

7, and heaving displacement and pitching angle are measured by potentio

meters. The models are run by a propeller driven by a motor M and this

propeller is not behind the after perpendicular of the models but between two

hulls.

In our experiments some remarkable phenomena were observed. One is that the height of waves between two hulls caused by their motions and the

incident waves is rather large even when their heaving motion is small. The

others are that at a certain wave length the motion amplitudes make

com-paratively large change with a little variation of the wave length and that the measured amplitude of the motions does not become stationary but varies with

ON THE MOTION OF MULTIHULL SHIPS IN WAVES (II) 81

Theoretical Measured 2 P/T = 3 2 P/T --- 5 A 1.0 _Single-hull 0 0

Fig. 7 Ship model and experimental apparatus

a time like a phenomenon of so-called "beat". The reason why this beat phe-nomenon occurs seems to be that the incident wave is not a sinusoidal wave

with one component of frequency but includes other components of small

amplitude or that a small drift speed of the models due to head waves causes

a small variation in the frequency of encounter and as a result

a change of

motion amplitude because of the characteristic of catamarans, that is, the large variation of motion with the small change of wave frequency at some frequency. But it is not confirmed now.

It can be concluded from the comparison of theoretical and measured

heav-ing motion in Fig. 3 and 4 that both are

in good agreement. The

heav-ing motion of twin hull ships TW1 and TW2 is very large compared with that of ordinary mono hull ships at resonance period and it is because the damping

force of heaving motion of the twin hull ships become smaller than that of

the mono hull ships at this period. When we neglect the interaction effect

between two hulls, theory does not predict this large motion.

In the case of pitching motions of Fig. 5 and 6, there is some difference

between theoretical and measured amplitude, though the theoretical values cal-culated by taking into account the ineraction effect are better. This difference

seems to increase when the advance speed of the ships becomes larger.

2.2. The force exerted upon twin hull ship in head seas.

The wave bending moment at an arbitrary section of twin hull ship in

head seas which is caused by the motion and the wave force can be also cal-culated') by applying our procedure to solve the radiation problem of twin hull

cylinder.

Some examples of the moment at midship of TW1 and TW2 are compared

C C 0 s - .

in Fig. 8, where F is 0.1. The moment upon the twin hull ships when the

hydrodynamic interaction between two hulls of the ships is not taken into

account is also shown as single hull curve. It is, of course, twice of that which

acts upon the midship of one element hull of TWI and TW2 when the element hull runs in head seas as a mono hull ship.

Le"-°yin-2

0

where

ON THE MOTION OF MULTIHULL SHIPS IN WAVES (II) 83

2 P/T-= 3 2 P/T = 5

Single-hull

H* (KT) ieft2cos e-Te'elsin e,.

Theoretical

0 0.5 1.0 1.5 2.0

Fig. 8 Wave bending moment of twin hull ships in head seas (F,i=0.1)

It can be concluded from Fig. 8 that it is sometimes dangerous to consider the wave bending moment upon twin hull ship as only twice of those values of one mono hull ship which constitutes the twin hull ship.

As stated in the first paper when a twin hull cylinder as shown in Fig. 9

makes a heaving oscillation, a progressing wave CR given approximately by the equaion (4) comes upon the left cylinder. And also a wave C, with the same

amplitude and the same phase as C, but progressing rightward comes upon

the right cylinder.

cr_

A2e,,,64") _ei(rot+Kx)

I ill+ (KT)e-i2"

A,i(E2.1 1CP)

84

L

M. OHKUSU and M. TAKAKI

0

Fig. 9 Coordinate system

and A, and e, are the amplitude and the phase of progressing wave at infinity when one cylinder of the twin cylinders makes j oscillation about the origin L

or R (j=1 is swaying y =e"" and j=2 is heaving ze) without another cylinder.

Accordingly each cylinder receives a hydrodynamic force in Y direction and a moment due to these incident waves respectively. The forces and the

moments upon the right and left cylinder have the same amplitude and the

opposite direction with each other. The force of Y direction upon the right

cylinder is given by

WAVE

pSA

A2eife2 tot)

s

Inils(-2) + Pg2 n ( =

i K(7)

Pg

1 iii+ (KT)e-2KP

It is to be supposed immediately that the similar force and moment act

upon each mono hull of twin hull ships TW1 and TW2 when they run in head

waves.

The formulations" to calculate vertical force on a cross section of a ship

running in head waves which is, of course, based on Strip method can be also applied to the calculation of such transverse force or moment upon each mono hull of twin hull ship in head seas.

In this application the terms of inertia force and static water pressure force which appear in the equation of the vertical force must be removed, because these forces do not act in the transverse direction.

The force F of Y direction upon x section of the right hull of a twin hull

ship running in head waves with advance speed V is expressed as

, V

yPg

vP a (y cm

F=[pg-

5 nfis+ xp

m,]0 +[-

v2

ji

-

s)is. COe 2 we 2

2 3 x

+

_{2 ax}

(Sm,1s)0+ P2S MILS±C+[wPge:nBS V (S HS)]1G L = 2

t,A..4,01.-ON THE MOTIt,A..4,01.-ON OF MULTIHULL SHIPS IN WAVES (II) 85

PS

[( wPg23 nlis V 2P a ax (S inns)) icy wwl

2

s "

Fig. 10 shows the total of such forces acting upon the right hull of TW2 at F=0.1 in the non-dimensionalized form, where Ca is the amplitude of head

waves. Such force or moment will have important meaning when it is necessary to know the stress at the bridge structure of a twin hull ship.

2 P/T = 5

A./L

Fig. 10 Transverse force of twin hull ships in head seas (F=0.1)

3. The Motion of Twin Hull Ship in Beam Seas

3.1. The equations of motions

The added mass and damping coefficient of twin hull cylinders calculated

by our method can be also used for formulating the equations of motions of

twin hull ship in beam seas. The equations are formulated by integrating with

respect to x the hydrodynamic forces (added mass and wave exciting force

etc.) upon cross sections of the ship which are considered to be 2-dimensional. Added mass (moment) coefficients m, (j=1, 2, 3) and M, N of each

sec-tion are calculated, where mt and m, are added mass coefficients of swaying and heaving motion of twin hull cylinder shown in Fig. 9, and are defined to be added mass divided by pS (S is sectional area under the water of the twin

hull cylinder) and m, is a coefficient of rolling motion given by _{a moment}

around the origin 0 divided by pST. M and N are also defined by the following

equation f= PS M + wPg: ₍₉₎ 1.5 2.0 (8) 0.04 0.03 0.02 0.01 0 0 0.5 1.0

86 M. OHKUSU and M. TAKAKI

where f is a moment about 0 due to hydrodynamic force when the cylinder makes a swaying motion y=e.

In addition we can calculate twin hull cylinder's A, and e, by which the

wave at y = + co is expressed as when the cylinder makes the

oscilla-tion j (j= 1, 2 mean the swaying and the heaving mooscilla-tion, and j=3 is the rolling motion about the origin 0). Accordingly the wave exciting force when a plane

wave with the elevation e"'"') comes upon the twin hull cylinder from rightward

as shown in Fig. 9 can be exactly derived by Haskind-Bessho relation.' These wave forces (moment) are expressed by

where j=1 and 2 mean the forces in x and y direction, and j=3 the moment

around the origin 0.

Then the equations of motions of the center

e"''

are given by ; ce (

(,6,

fLi (1+ nz7) Sdx+ Pg: fLf A.22dX1-z. pgr 2bdx o.) La La g eiwt Lf Ae"Vx (10) K _JLa fLfLa(1-Ftn,)Sdx+Y Pg.' f Li T (N co' L 0.2

f

/1,2dx+ p 0 f

ST(M-I- 1G CO- La La

1; A,z)dxiPg

e' _{114 A.,e'sidx,} dx tpro2 LfSdx+ pjLf ST' [m3+ ()2 m, 2 ( 1G- )Midx La La

+[(1 3)2 + (1 ,)2 A7+ 2(4-) Nidx+0- pgGM

sax

(.0 L T JLa La TfS (M+ m,)dx +5, f (N+ Al2)dx IL- co LaT jPg_e

f

LLT. le 3+' A3( ,s -T, e'6') dx, (10)

of gravity G_in beam seas

C +2,S

C Lts C C

(12)

where y and z are a swaying and a heaving displacement of the point G and

13 rolling angle. And r,, is the radius of gyration, GM the transverse metacentric

height and 1G the z coordinate of the center of gravity of each cross section of the ship. Another symbols have the same meaning as in the section 2.

Substituting z

y= Ye'' and 0=0 et into the equations

(11), (12), +

20

1.0

Fig. 11 Heaving motion of twin hull ships in beam seas

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 KT (=-71T

ON THE MOTION OF MULTIHULL SHIPS IN WAVES (II) 87

and (13) we can obtain the amplitudes and the phases of the motions.

3.2. Comparison of theoretical and experimental ship motions

The heaving, swaying and rolling motions in beam seas of TW1 and TW2 were measured at zero advance speed. The experimental apparatus and regular

waves used in the measurement are the same one as used for the experiment in head seas.

The measured amplitudes of swaying, heaving and rolling oscillations of TWI and TW2 are shown and compared with the theoretical ones obtained by solving the equations (11), (12) and (13) in Fig. 11, 12 and 13. The transverse

radius of gyration r of TWI and TW2 used in the equations was determined

experimentally from free rolling test.

It should be noted that the amplitude of swaying motion shown in Fig. 12

is not of gravity G but of the point 35.3 cm above the point G. Ship models

drift due to the wave force in beam seas and accordingly the abscissa of these

w

figures is T, where co, is the circular frequency of encounter of incident

waves.

Theoretical and measured values of the motions are generally in good agree-ment. Since damping force given rise to by the generation of progressing waves

is larger, the theoretical and the measured values are also in comparatively

good agreement even in the case of rolling in spite of the neglect of the force due to viscosity of a fluid.

Theoretical Measured

2 P/T= 3 2 P/T= 5

0

88 4.0 3.0 2.0 1.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 A/29 0.1 c

frac-11

7,

/

i...

,...t..,,,_44

,,

M. ofikusu and M. AIT7r-d 'kei

C9

0.1 0.2 0.3 0.4 0.5 0 7 0.8 0.9 1.0

--"1" KT (=T

ips in beam seas

Fig. 12 Swaying motion of twin hull st

0.6

0.2 03 0.4 0.5 0.6 0 0.8 0.9 1.0

KT (=-T)

Fig. 13 Rolling motion of twin hull ships in beam seas P/T 3 :24)/T=5 Theoreticil or tical 0 Meaiur

ON THE MOTION OF MULTIHULL SHIPS IN WAVES (II) 89

In the experiments it was observed that the waves between two hulls began to progress in the lengthwise direction of the ships especially at the frequency when the waves became very high and the heaving motion was larger. This

fact means that three dimensional effect is strong at this frequency of motion. And at such frequency the measured amplitude of the heaving motion was not stationary but varied with a time like a beat.

It will be due to these facts

that at the resonant frequency the measured amplitude of the heaving motion does not reach to the level of the amplitude predicted by theory.

Next let us study the details of Fig. 11. Theory predicts that there are three

resonance point of heaving motion for TWI in the range of KT shown in this figure, where a resonance point of heaving motion means a frequency at which the sum of inertia force and restoring force upon a ship making a heaving mo-tion vanishes. One of these three points is a frequency where the amplitude

of the heaving motion of TWI becomes the largest as shown in Fig. 11 and the

other two points are KT*0.4 and 0.55. At the latter two points, the amplitude does

not show a peak because A that is, damping force is large there (vertical

wave exciting force becomes larger with A, but damping force is much larger). Although the resonance frequency of heaving motion of a mono hull ship

which constitudes TWI or TW2 is known to be at KT--.'F0.7, the resonance point

of TW1 and TW2 differs from the mono hull's one as a result of an interaction between two hulls. When 2p/T is from 3 to 10, it can be said that the larger the distance between two hulls is, the smaller the difference becomes. By the theory developed in the first paper, added mass, damping force of heaving motion and vertical wave exciting force of twin hull are known to be functions of KT and

as far as KT is not so small. Accordingly if a twin hull ship composed

of some two hulls is at resonance of heaving motion with KTa and 2KP= 3,

then another twin hull ship which is composed of the same two hulls but has 2KP=i3+271- at KTa makes just the same heaving motion at this frequency. The heaving motion of TW1, for example, is in resonance at KT---0.9 and that

of a twin hull ship with widened breadth as 2P/T-10 is also in resonance

at KT*0.9.

It is shown in the first paper that there are some frequencies at which A.

of twin hull cylinder is zero and then vertical wave exciting force upon the

cylinder is also zero. In the heaving motion of TW1 and TW2 there is the frequency at which the wave exciting force become almost zero and as a result the ship does not make a heaving motion as shown in Fig. 11 at this frequency.

3.3. Miscellaneous

Let us compare the motion amplitudes of twin hull ships in beam seas

calculated approximately without taking the interaction effect between two hulls into consideration with the exact ones obtained in 3.2.

When the interaction is neglected the added mass, damping force and the

wave exciting force of a twin hull ship in heaving and swaying motion are considered to be the sum of the ones for the mono hulls which are elements

90 _{M. ,OHKUSU and M. TAKAKI}

of the twin hull ship. In rolling, added mass moment and damping coefficient are derived from the sum of the hydrodynamic forces acting upon each mono

hull when it makes a heaving motion and a rolling motion about its center.

The wave exciting forces of Y and Z directions (j=1 is Y direction and j=2 Z direction), for example, are expressed as follows

4i go_ .1 - f [Ajeigjegwt- + efejeacot + K Pldx, and the wave exciting rolling moment about the origin is.

I Pgf Li [ P.,,,T2e44egtot - P) p.Tieinel(rat +1CP) +.743eie3ei(a)t- KP), ;4;e43e1(eatr+KP) _dx,,

K L.

(15)

where A,, e, are the values of the mono hull cylinders..

The added mass and the wave exciting force thus calculated are substituted into the equations (11), (12) and (13), then we can obtain the motion amplitudes of twin hull ship when the interaction is neglected.

In Fig. 14 and 15, there are shown the heaving and the rolling motion am-plitude of TWI with and without the interaction, where approximate denotes the values without the interaction. As seen in Fig. 14, if the interaction is not taken into account, the amplitude of the heaving motion is not only smaller than the exact one, but also the amplitude is rather large when the exact calculations

show the zero amplitude. Both the exact and the approximate calculations give

almost the same resonant frequency and the wave exciting moment in the rolling,

but the damping force without the interaction is larger and the approximate

213/T-3 Exact Approximate -2.0 , 0

Fig. 14 Comparison of approximate and exact solutions (heaving)

0_4 0.5 0.6 0.7 0.8 0.9 1.0 NT (=IT ) (14) r + + 1.0 0.1 0.2 0.3

5.0

4.0

3.0

1,0

ON THE MOTION OF MULTIHULLSHIPS IN WAVES (II)

P/T = Exact

Approximate

KT (=-`2g4T

Fig, 15 Comparison of approximate and exact solution (rolling)

.amplitude is much smaller the real value.

91

Each element hull of twin hull ship making oscillation in beam seas receives also a hydrodynamic force with Y direction and a hydrodynamic moment about its center line. This force and the moment upon right and left hull have the

s

same magnitude but the opposite direction with each other. Such a ftrce4pLoir:,,e,,L...ia

twin hull hip moving in beam seas is expressed by

L f g pg2 Lf -

LfW1

g

A ( 2 ) pi

m dx+

i) nHsdx+ PvR wi L dx, HS 0)3 L. La 2

Cop

t.. -c (16) A.A.ve

where myys and nffs of each section are calculated from the equation (7) W1R

and W1L are wave exciting forces of Y direction upon the sections of right and left hull respectively and are given as follows.

When the. wave e' comes upon the twin hull cylinder shown in Fig. the exchange of diffraction 'waves between two cylinders described in the first

paper produces the, incident wave upon the right cylinder given by

(1 4- iH- (KT)) iH-E (KT) ,my, t.7

Ev(KT)e"`"-KY R) -=

1+H+ (KT)2e-i4KP e . wtK-XR);, (17)

,and the incident wave upon the left Cylinder given by tiA-^C.-:

(1 +1TH- (KT)) !PP 41(cot + K E2 (KT) emot+Koy _{e e} , _K18) 1+H+ (KT)2e-i 0 . 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ( 9, 7.0 6.0 2 3 2.0 0

92 M. OHKUSU and M. TAKAKI

where 1-1±(KT) is given by the equation (6) and yR and yi, are the Y coordinate measured from the center of the right and left cylinders. The wave exciting

forces, VV,L and WiR upon the right and the left cylinder are calculated as the exciting force due to these incident waves by

WiL=

(19)

W 41)1 Aie"i[ei"±E,(KT)]ei'l

..41,21,,t,

Hydrodynamic moment upon each hull can be calculated by the same

procedure.

Fig. 16 and 17 give the amplitudes of such force and moment of TWI and

TW2. At small heaving amplitude the force or moment becomes rather large

in some cases.

4. Conclusion

The method to solve the radiation problem of twin hull cylinder developed in

the first report was applied to the theoretical calculations of the various motions of twin hull ships by Strip method and it was found that the theoretical predic-tions gave satisfactory results in comparison with the measurements on model

tests. 5.0 4.0 3.0 -2 P/T = 3 2 P/T - 5 I 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 KT (=V T )

Fig. 16 Transverse force of twin hall ships in beam seas 1 1 1 I I I, , I , I. ,1 V I 1 V 2.0 1.0