Wave excitation forces on two dimensional ship of arbitrary sections

Introduction

When the excitation forces on ships in

waves are calculated, the Strip Method is

usually available, because it is difficult to treat them in view of the three dimensional

ship form with forward speed. And the two dimensional calculations are required.

The classical theory has the Froude-Krylov

hypothesis, which assumes that waves are not disturbed by floating bodies. This theory considers only a velocity potential of incident waves including the Smith correction. The

calculations by Weinblum are based on this

hypothesis.'

On the other hand, Grim and Tamura

Institute of Industrial Science, University of

Tokyo.

Dept. of Marine Technology Mekelweg 2, 2628 CD Delft

The Netherlands

6.

_{Wave Excitation Forces on Two Dimensional Ship of}

Arbitrary Sections

Hisaaki MAEDA, Member* (From J.S N.A. Japan, Vol. 126, Dec, 1969)

Summary

Wave excitation has been calculated by Grim based on the diffraction potential for two dimensional bodies of Lewis form sections.

On the other hand, it has been suggested by Newman making use of Haskind relation that wave excitation could be induced from the radiation potential for which many works have been done by Ursa, Tasai and Porter on two dimensional bodies of Lewis form sections. In this paper, the author deals with the wave exciation due to beam seas on two dimen-sional ships of arbitrary sections making use of singularity distributions on the ship surface. The radiation potential induced by such singularity distribution was calculated, and the wave excitations for heave resolved into in-phase and out-of-phase components were obtained making use of the Kochin function.

The author also extended the same technique into calculation of wave excitation in oblique waves including head seas.

The calculation on wave-excitationless ship forms was performed, which showed satis-factory agreement with model experimental results. The calculation on three dimensional wave-excitationless ship forms was also done by means of the Strip Method. The results were in good coincidence with model experiment data.

63

developed the theory which was derived from

the wave resistance theory and used the

diffraction potentials.8,4) In-phase and out-of-phase components of wave excitation forces were calculated by this theory. By the way

the Haskind-Newman's relation shows that

wave excitation forces are in proportional to amplitudes of the radiation wave. And

Ursell-Tasai's calculation can be applied to

the wave excitation forces.7, But these methods

are available only on

n-parameter Lewis forms.

The author presents the method to calculate

the wave excitation forces and its in-phase and out-of phase components on arbitrary

ship forms, on which source singularities

are distributed.

This kind of method is

developed by W.D. Kim, Frank and Bessho.

64 Hisaaki MAEDA

But this method depends on Bessho's theory

and is independent of other theories.,-",")

And the usual methods can deal only with

excitation forces among lateral waves, but this method can be applied to oblique seas

by making use of radiation potentials.

Moreover the present paper deals with

wave excitationless ship forms which receive

no excitation heaving forces

at a certain

frequency in beam seas. This kind of ship

forms suggested first by Motora was

con-firmed by two dimensional experiments and

should be named as Motora's form.',", The

author shows that these excitationless forms

receive no excitation heaving forces at the same frequency in head seas either and the same results can be expanded to three dimen-sional ship forms.",

In this paper only the heaving force is dealt

with as the excitation forces, because the

same treatment is available to other kind of

forces.

1. Fundamental assumptions and formulae') 1.1 Coordinates and assumptions

The coordinate system is the Cartesian

co-ordinates as shown in Fig.

1 and 2.

The origin 0 in a midship section coincides with the undisturbed free surface.

Let a be the

angle between incident waves and the z-axis. The line normal to the ship surface is directed

outward in fluid.

"C" represents the

im-mersed ship surface. Other symbols are shown in Fig. 1 and 2.

The foundamental assumptions are

con-sidered as follows: The Strip Method

is

applicable and two dimensional problems are

Incident wave Fig. 1 Incident wave

< °

Fig. 2

dealt with. The fluid is inviscid and incom-pressible. Surface tension can be neglected.

The fluid is expanded to the semi infinite

region with the free surface. The amplitude of waves is small and the free surface condi-tion is linearized. Neglect higher order small quantity of pressure. Velocity potentials and conjugate stream functions exist. The ship

oscillates with small amplitude and without

forward speed. The ship and waves oscillate periodically.

Only the heaving motion is

treated as the oscillation in this paper. 1.2 Fundamental formulae

y), p and 72 stand for velocity potential, pressure and displacement of free surface of which time factors are eliminated, instead of 0, p, 7.7/ which have the time factor. And we use as a velocity potential of unit velocity amplitude and g5o,51 as a velocity potential

of unit amplitude, instead of the radiation

potential co, velocity potential of incident wave coo and diffraction potential col respectively.

We introduce the stream function 0 which corresponds to the velocity potential 0. If

the ship heaves with the amplitude Y and

with a certain frequency o about its

equili-brium position,

the radiation potential

is written in the following form,

0= R,{weiwt}, (1.1)

(p=io,Y5, (1.2)

where c/5

is the

velocity potential of unit

velocity amplitude. The velocity potential of

the incident wave and the diffraction poten-tial of the incident wave and the diffraction

potential are

(1.3)

iga

c.40,4= 00444 (1.4)

(.6

where "a" a is the amplitude of the incident

wave. Pressure and displacement of the free surface are

P=Re{pe''''}, (1.5)

72=1?,{T2ei-'}. (1.6)

Then the fundamental equation is derived

from the Euler's continuous equation, that is

a20 (3,95

+ =o. (1.7)

ax2 ay2

The boundary conditions are as follows; Introducing the Rayleigh's frictional cofficient a,

the linerized free surface condition

is

written as follows,

[K0-1- ay

-401

-=0, (1.8)

acb .

where K=w2Ig is the wave number. At infinite depth

grad 0-40 (y-4.0). (1.10) As the diverging waves exist at the infinite

distance on the free surface, introducing the Kochin function, the boundary condition is

±.9). (1.11)

And the boundary conditions on the ship

The pressure equation which is derived from the Euler's equation, for the radiation poten-tial and the diffraction potenpoten-tial respectively

are written as

P= -

1795, (1.15)

p= - pga(150+04). (1.16)

The surface displacement for the radiation

potential and the diffraction potential

respec-tively are as follows,

Fi/Y=K451v-o,

ida= (150 + 04)6=6

1.3 Pressure integral

When the ship heaves on the calm water, the hydrodynamic force F is produced, and

its normalized form f becomes

f=Flpw2Y= - ac6 ds

an ' (1.20)

where the suffix c and s stand for the real

and imaginary part. The normalized force

fe and fs mean the added mass A.M. and

the damping coefficient by diverging wave N respectively and these relations are

A.M. =ixofc,

(11.2234))

N=-p(efs.

The normalized exciting force by waves e

written as follows, from Eqs. (1.16), (1.12)

e-

-

-\ (00+04)ao ds. (1.24)

pga an

The suffix " + " sign means the radiation

waves in positive direction, the incident waves from positive direction and the exciting

forces by the incident waves, and the suffix

"-" sign corresponds to those in negative

direction. The velocity potential of the

in-cident waves is

The amplitude ratio A± of the

radiation

waves at infinity is written as follows

A-±=÷.1 =K1-1±(K). (1.28)

We consider the symmetric ship forms, then IH-,(K)I is equal to li-/-(K)1. It is, therefore, sufficient to consider only H(K), and we use

variable name e and A as the exciting force

surface are as follows, for radiation potential

acs ay

on C, (1.12)

=e"

n:

(1.25)

Making use of Green'stheorem,the exciting

force (1.24) is represented by Kochin function

an an

for diffraction potential

a on C, on C. (1.13) (1.14) e==-11±(K), (1.26) where acs a an (00+04)=0

H±(K)-\(

an -5b an )e-KvKrds (1.27) (1.17) (1.18) sb=x E 2

-66 Hisaaki MAEDA

and the amplitude ratio respectively.. The

Haskind's formula is derived from Eqs. (1.26), (1.28), which shows the relation between the

excting force and the amplitude ratio:

The Haskind-Newman's formula which shows the relation between the amplitude ratio and

the damping cofficient is written as follows

by means of Kochin function

f s=I-- I +(K)(.2,,, (11.30)

then the following relation iS derived from Eqs. (1.23), (1.28)

A

= _(1.31)

Po

Moreover the. Kramers-Kronies relation shows the relation between the added mass

A.M. and amplitude ratio A.

We consider the added mass, 'the damping coefficient, the radiation wave and the wave

exciting force as the hydrodynamic forces. We can derive any one of them from other hydrodynamic forces from the relations as

described above.

There may be two methods which make use of the radiation potential or diffraction

potential for calculating these hydrodynamic forces, The method by the radiation

poten-tial has two more way. One of them is usual method described as follows. The wave

ex-citing force

is obtained from Eqs.

(1.20),

(1.30), (1.28) and (L29). But the wave exciting force can not be devided into in-phase and

out-of-phase components, because the Kochin

function is obtained as the absolute value

from Eq. (1.30). Another method uses Eqs..

(1.26) which obtain the Kochin function directly and the in-phase and out-of-phase com-ponents are calculated through this method. Especially when source singularity is (1.29)

continuity of the partial derivative of the

radiation potential on ship surface.

The method by the diffraction potential

uses Eq. (1.24) which can calculate in-phase and out-of-phase components. When the

ex-citing force in head seas is calculated, this

method is as useful as the method by the

Kochin function in beam seas.

Z.. Integral equation as boundary value.

problem

2.1 ship motion represented by the singularity distribution

The Green functiOri G at point P (x, y) by

source of unit strength at point Q(x' y') can be expressed in the form

G(P, Q),=log ri log ral'2 L (2'.1)

G(P, Q)=G(Q, (2.2)

where

=(x x')2+(y

(2.3)

r22=(xx')2+(y-Fy'),2., (2.4) e-u(Y+7,') cos u(xx')

(

km 2.5)

re=.

uK+iç

It satisfies the boundary conditions (1.7), (1.8), (1.10), (1.11). Then the velocity potential at

point P is written .as follows

,cbcp 1 c ( ocis

0--9G(P Q)ds. (2.5)

27r an an

The other hand, we consider the velocity

potential 0' in the inner part of the ship

which satisfies the Lapiace's .equation and

the free surface condition. According. to the Green's theorem, taking account of that the point P is not in the inner region,

= 1 ( 895' G(P, Q)ds. (2.6)

c an an

The difference between Eq. (2.5) and Eq.. (2.6) is

11

distributed over the ship surface, the latter

method is

very useful, because the wave

S6(P)=-42_{7 o} a/5 a9Y G(P Q )

(On o n

exciting force is

obtained on the way to

solving the integral equation as the boundary

value problem, which decides the strength of .95') aG

(2.7)

(P, an

source distribution, making use of the dis- If we suppose,

K

P),

du.

then

sb=.75' on C,

(i6(p 1 c ao

2 ir

.)0 an_ an

₍ )G(P, Q)ds. (2.9)

This equation shows that the velocity

poten-tial at any point in hydrodynamical region

is represented by the source distribution over

the ship surface with the strength of (aolan

aoyan).

2.2 Radiation potential")

Supposing u(Q) as the strength of distributed sources over the ship surface at point Q, Eq. (2.9) is rewritten as follows

cb(P)= a(Q)G(P, Q)ds(Q). (2.10) Taking account of the discontinuity of the normal derivatives of the source singularity on the surface, the normal derivative of the potential ¢(P) on the ship surface is given by

d(P) )

ac(p,Q)

en

_,rw(P) + a(Q)

an ds, (2.11)

where suffix sign "+" represents the limit

which tends to the surface point P from the

positive side.

The left

hand side

of Eq.

(2.11) is given in Eq. (1.12) as the boundary

value on the ship surface.

Therefore Eq. (2.11) is the second kind Fredholm integral equation. We may consider that there is the

unique solution of this equation, because the

kernel function is not so much singular and

real physical phenomenon is present. Before we solve the integral equation, we rewrite

the Eq. (2.11) in the form, which uses the

stream function, T(P)10,1 c a(Q)- S(P,Q),ds(Q), where (P)Ion c =x, (2.13) S(P, Q)-= 01-112 -2 Is, (2.14) O. tan-' Y+11

x x' '

x x' (2.8) cu(v+v)sin u(x

Is= lim du,

(2.16) because the operation of the normal

deriva-tives on the surface in Eq. (2.11) is not good

for the accuracy of the calculation. If a(Q) is determined from this integral equation, the radiation potential

is calculated from Eq.

(2.10).

Next we show a key to solution of the

integral equation (2.12). Considering that the

complex values T(P), a(Q) and S(P.Q) are

expressed in the form

CP)=-Yre(P)+ iTs(P), (2.17)

a(Q)= ac(Q)+jas(Q), (2.18)

S(P, Q)=So(P, Q)+iSs(P, Q). (2.19)

Eq. (2.12) is rewritten as follows

4P)Ion c {ae(Q)Sc(P, Q) as(Q)Ss(P, Q)}ds(Q), (2.20) a (A) s(P)Ion c -=-- (ae(Q)Ss(P, Q) a8(Q)S,(P, 62)}ds(Q). (2.21) These equations are the fundamental

equa-tions which are available on the heaving,

rolling and swaying motions and even on

asymmetric ship forms.

In order to solve

this integral equations, we suppose that the strength of source is constant over discrete

small elements of ship surface the number of

which is N. Then the Eqs. (2.20), (2.21) are

reduced to simultaneous equations of 2 N

unknowns which give the solution ae, as.

If the ship form is symmetric, the problem

is made more simple and Eqs. (2.20), (2.21)

are rewritten into two kinds of simultaneous

equations of N unknowns and computing

time can be saved.

In the case of the heaving motion, boundary

conditions and ss are expressed in the form c =x, Ws(P)lo, c =0, (2.22)

Ss(P, Q)=2 re"" sin K(x x').

(2.23)

It is assumed that the ship form and a(Q) are symmetric, through the following equations

(2.15) (B), (C) Oi=tan-'

x')

(C)

1

o),c(x' , _y'),

Pc

, y')e-KY cos Kx'ds(x' , y'),

(2.24)

',=\ as(x' y')e-",' cos Kx'ds(x', y'),

(2.25)

y' )=a,( , y')-FPS , y'),

(2.26)

(2.27)

2.3 Exciting forces in beam seas")

The wave exciting force which is repre-sented by the radiation potential is

e-Kv-iKx

_c an an)

At the source singularity with the strength a is distributed over the surface, according to the potential theory, the discontinuity of

the velocity potential is expressed as follows

ao+ ao

an an

_

0+=q5 on C, (2.39) where the suffix sign "+" for the limit from

positive side and sign "" from negative side. Considering Eq. (2.39), the exciting force Eq.

(2.38) is rewritten in the form

e= (a25- e-Ku+'Kx ds

c an an

_22r\ ds. (2.40)

The first term of the right hand side is equal

to zero by the Green's theorem, because the

velocity potential 0 and exp (Ky+iKx)

satisfy the free surface condition and are

harmonic in the inner region of the ship.

Therefore, the exciting force e is

e= CI ds. (2.41)

If we define e=ec- ies,

the real and

imagi-nary components of the exciting force are

expressed as follows

ec= 27r aceK,' cos Kx ds, (2.42)

c

es= 27\ ase-"u cos Kx ds. (2.43)

Here, when we notice Eq. (B) and (2.42), (2.43), we realize that the exciting forces are solved

before the source strength a is obtained.

And the phase lag E is derived from Pd.

(2.38)

2.4 Other hydrodynamic forces

The added mass and damping coefficient e5= 27r Pe , es= 7.7 Ps , (2.44) (2.45)

E=tan-Pd.

(2.46) 68 Hisaaki MAEDA

then Eq. (A) is transformed into

x ai(x' , y')Sc(x, y; x', y')ds(x' , V),

_c

(2.28)

( I )

27r e-

sin Kx=5 ad(x', V)

Sc(x, y; x', y')ds(x', y'). (2.29) ad are solved from Eq. (I). Considering following equations

P92

Pi= Pc+

(D) Pc '

Pd= Psi Pc.

131, Pd are expressed in the form

'Pi=

x')e-'0' cos Kx'ds(x', y'),

Pd=\y')e"Y' cos Kx'ds(x', y').

.c

(2.33)

Pc, Ps are solved from Eq. (II)

131

+ P2)'

Ps= Pd.

Through (III) and (C), source

are solved in the form

{ac(x', y')=a1(x' , y')P8. us(x', Y')= Pe 6'.(xi, y'). In order to solve the boundary and obtain the source strength,

(I), (II), (III) and (IV) are

required. The treatment of the real part S, of S is described in the appendix. and (D) (2.34) (2.35) strength ac, as ad(x', (2.36) (2.37) value problem only equations (2.30) (2.31) or, ai(x' (H) (Hi) y'), (IV) , (B) =27w, \

-are obtained through Eq. (1.20)

f = Oa° ds

_c an

Now we define the auxiliary function follows

1

h=y

. (2.48)

Eq. (2.47) is transformed into

f

+15 al6 )ds+ h a° ds. (2.49)

c an an c an

Let the first term of right hand side be

and consider Eqs. (2.39), (2.48), (1.12)

ah'

\(h

)ds-27rc ah ds.

an an

(2.50)

According to the Green's theorem,

Ii is

rewritten in the form

1

A= 2 a (yK)ds

. (2.51)

Let the second term of Eq. (2.49) be 12 and

consider Eqs. (1.12), (2.48), then

12=c

hdx = (vA-)

K

where r is the volume and B is the breadth of the ship. From Eqs. (2.49), (2.51), (2.52),

the added mass fe and damping coefficient f s are expressed in the form,

B/2

(2.52)

27r \ ac(y

_{)ds+ --K-1,}

(2.53)

fs= 27I \ 0'3

7y

K1 )

Therefore, when the added mass and damping coefficient are required, we need not to obtain

the radiation potential and we need only the

spurce strength

3. Exciting forces in head seas and its approximation

3.1 Exciting forces in oblique seas

It is assumed that the velocity potential of the incident wave in oblique seas 0.0 and the diffraction potential 0.4 satisfy the following

conditions; that is (2.54) owor=e-,,,,Kssinc, 17,2g,w4=0, a anco.u+sbwo=0 on 01,14 ^'ill.±(K)eKVWCx(X-4 ± .0). (2.58)

These assumptions are equivalent to the one that it is possible to put the two dimensional radiation potential in place of the three

dimensional potential and that the normal

derivatives on the ship surface have no

components in the direction of z-axis.

Then the exciting force e in oblique seas

is expressed as follows

se=

S(0.o+

Ow+) _and (2.59)

Introducing Eq. (2.57) to the above equation,

and making use of the Green's theorem, ldçb

e= _an 0----a )0.0ds an

= an an e

(2.61)

In case of beam seas, the angle of the incident

wave a is equal to z/2 and the equation of exciting force corresponds with Eq. (1.26). In case of head seas, the exciting force is

e= 95--8 )e-"u ds. (2.62)

, an an

3.2 The approximation of the exciting forces in beam seas8,9,11,i8.1)

Here we write again the exciting force e in beam seas; that is

e=

(ana° 0-8an

Suppose the wave length is large as compared to the ship breadth, the approximation of the

incident wave potential can be expressed in

the form

e-Kin-tKx t 1+ K( y+ ix). (2.64)

Then in case of the symmetric ship form,

the exciting force is writtsn as follows

Kcan95--ds+ AKv, (2.65) (2.55) (2.56) C, (2.57) Ks s'n ds. e-Ku-"Kx ds . (2.63) (2.47) Ii as ,

70 .Hisaaki Ni A.EDA

where A is the water plane area (the breadth for two dimension).

Let the exciting force

with et be E and the displacement of the

incident wave be then the exciting force (2.65) is transformed from (1.20), (1.21), (1.22), (1.23) into

EpkFd2y,,,. 2 +

dt

Ndt ±(1.A7(1')PgAll'

(2.66)

where k is the added mass coefficient, (1,,,=FIA is the mean draft.

The third term of the

right hand side of Eq. (2.66) can be rewritten as e-KampgAy.. Eq. (2.66) is the approxi-mation by Motora.

3.3 The approximation of the exciting force

head seas21

The exciting force (2:62) in head seas is rewritten e-K, ,K-v e=K ,00 ds+iKc ds feKI .c an 1., an ds. an

and man may have constant sign for

ordinary ship form. Making use of the mean

value theorem for e'', then we have from

(1.20), (1.21), (1.22), (1.23> daYw _{+e Ica 2N}dVw E=e-Kaipkr_ dt2 dt where r+e-Kti.3pg A (2.169)

draft.

We can use. d. instead of db, d2, (13,,, then

E=e-K"'"'Ipkp

+N di +'ogA

161-(2.70)

This equation is exact in case of the box

type ship. This equation is the

approxima-tion by Watanabe.

We have calculated d1, d2, di for the Lewis form sections. (2.67) (2.68) d1=T{1.01 --1. 64(u 1. 06)2}, { d= 710,94(a 0.113)2+0.2591, (2.71) d3= T{1.08.0-0.08},

where 0,FIBT is the sectional area

coef-ficient. The range of 6 and KT is

19.55_6.<1.10,

0- _KT_.- 1.5..

3.4 The method of obtaining the exciting force in head seas from diffraction potential There are two methods which deal with the

exciting force in head seas on Lewis form section from the diffraction potential. One of them was treated by Abel and based on Grim. Another was done by Ganno which was based on Ursell-Tasai.

In either case

the velocity potential of the incident wave is

derived from Eq. (2.55) in which a is zero,

that is, Ow0=e-KY. These methods, however,

are applied only to the Lewis form sections.

Here we will show the method which can be

applied to arbitrary ship forms.

It

is the

same way as the hydrodynamic forces

in beam seas on arbitrary ship forms are

obtain-ed. We divide the exciting force into two

parts, that is, et and eT. Then from (2.62)

e=ei +ez, (2.73) where y ae, 13/2 ei ds--= e-Kudx, , an =11/.2, (2.72) ,

e2= Vw4---a95- as.

(2,7*

(2.75)

The first term ei

of Eq. (2.75) is

Froude-Krylov force in head seas which can be

calculated if the ship form is given.

The

strength of the diffraction potential

distri-buted over the ship surface is decided from

the boundary condition (2.57). In this case, the stream function grw4 is used

as the

boundary condition on the ship surface

in-stead of x in Eq. (2.27), where.

Cv4 cx Ke-fclidx on C. (2.76)

13/2

Moreover we apply the method which obtainS.

TI yw, in d2, T, dt2

-the hydrodynamic forces in -the chapter [2.3], then 1 , C2= IC

(y )as

B/2 ( _{1 )}

y

Ke-Kll dx. (2.77) - 8/2

If we define the real and imaginary part of

and a as follows, e=ec-Fies,

aw=a.cd-ichos,

from (2.73), (2.74), (2.75), the real and

imagi-nary part of the exciting force is

1 B/2

ec= 27 a.e(y--K)ds+K1

ye-x,

Jc

1

es=

-22r5 aws(y K)ds

4. Numerical examples

Two kinds of series models are chosen.

One of them is the one of Lewis form for

comparing with comupted results by Tasai. Another is a wave excitationless ship form

which has not been calculated

yet. The computation method is as follows, we divide

the ship surface into N sections and

distri-bute the source singularity with constant

strength over each section. The source

strength ac, as are obtained by the procedure of (I), (II), (III), (IV) described in the preceding section 12.1]. If the ship form is symmetric,

the number of the section N must be odd.

Because, if N is even, only (N-1) independnt equations are given for N unknowns. After all from Eqs. (2.44), (2.45), (2.46), (2.53), (2.54), following six quantities are obtained, that is

the added mass,

the amplitude ratio of

radiation wave, wave excitation force with

its in-phase component and out-of-phase component, and the phase difference.

4.1 Accuracy of computation

It may be assumed that the computed value

converges to a certain value as the division

number N increases. In case of N=20, the computed value holds the accuracy of three

digits comparing with the exact solution of a of the circular cylinder in the case of K,

co. When the hydrodynamic forces of the

semi-submerged circular cylinder are

com-puted, the convergence of the added mass is

worst among the six kinds of the

hydro-dynamic forces.

But the results of N=20

agree with those of N=30 with an accuracy of two digits and the results of N=30 coin-cide with those of N=50 with an accuracy

of four digits. Then we may conclude that

computed value must be converged at N=30. Now we adopt N=30 in the following calcu-lation.

The method of dividing the ship

surface into N sectious, for example, dividing

closely near the water surface or near the

bottom, has no effects on the results. The larger the

breadth-draft ratio H or the

flatness of the eliptic cylinder of excitationless ship is, the more inaccurate the results are.

In the

_{latter case, the eliptic cylinder is}

similar to a bilge keel and the source

distri-bution on both side of the eliptic cylinder

tends to the doublet.

In this case other

treatment must be required. We used HITAC 5020 E as the electric computer.

4.2 Comparison with Tasai's results

The models are the Lewis forms with two parameters as follows

x=(1-kai) cos 0+a3 cos 3 0, y = (1 ai) sin 0 a3 sin 3 O.

The breadth-draft ratio H and the sectional

area coefficient a are defined as follows H= B/2 T=(1-1-ai+a3)/(1ai + a3),

= FIBT=714-(1a,2-3 a32)/((1+a3)2ai).

The calculations were done on nine types

models shown in the Table 1. Non-dimen-sionalized added mass coefficient K4, the

amplitude ratio of radiation wave A, and the

wave excitation force of heaving e', Cc' es',

are defined as follows

ofe/ 12 pr)(B where

Co =(1+ai +a3)21((1 +ai)2+3 a32), K4

72.

A = _If

e'=eIB12,

lec,=ecIB12,

es'=es11312..

Table I. Lewis Form

The computed results an shown in Fig.

3-11. In those figures, the frequency is

non-dimensionalized as KB/2. 2.0 1:5 e) eLe: AK. 1,0 -Hisaaki M AE14 90 3 0' to 'H 0.2 0,926 a, 0,6333 a, =0.05 0.4 003 1.2 ,_>1K13/2 Fig. 5 Lewis Forth

'90* '60' al 1, 02 i a 0.2 --0.6333 0.6666 0.06, 0.0 .0.926 0.785 --0.7 0.05 0,.644 I 0.0 --0.1 0.941 LD O.D 0.0 0.785 0.2 0.480 I 0.18 --0.1 0.947 I..5 H 0.282 0L1 0.623 H 0.2 I OM 0.785 1-11,0.2 L, A7:0:644 _.1132 a. ' °.7 TN 0, : 0.05 ' ' .. M. ...o.;

pr..

-104111hbft-ISIO' - ,A=X1-7,sok) .-11 I I 04, 10.8 11 2 --txs/2 Fig. 3 Lewis Form

04 0:8 11.2

KJ13/2

Fig... 4 Lewis Form

e; A }(4./ 10.5 0 20 e K4 K.(Tasei) B/2 3d 90° 60° 30° H 0.2 a- 0.765 -0.666 (Ta.i) 1.5 1.0 0.5

15

1.0

0.5

Fig. 6 Lewis Form

9 0° 6 0° 30° 1.5 1.0 0.5 L 0 or 0.9405 0.0 0.1 8/2 H 1.5 or 0.623 a 022 0,1 Ti 20 e' e,' 4 K4 2.0 K4 1-1 1.0 O 0.785 a, 0.0 a. 0.0 8/2 0.4 0.8 1.2 -> K B / 2 Fig. 8 Lewis Form

04 08 L2 KB/2 90° 6 0° 30° 0.8 1 2 KB/2 04 04 0.8 1.2 KB

Fig. 9 Lewis Form

Fig. 7 Lewis Form

Hisaaki MAEDA

The results of the amplitude ratio A agree almost completely with those by Tasai, but there is

a few difference about the added

mass coefficient Ict between the results by source distribution method and those by Tasai.

Paying attention to the in-phase component

of the wave excitation force e,',

it appears

a certain frequency at which force is to be zero in Figs. 9 and 10.

But this is not a

wave excitationless ship form, because the value of e' and es' exist. Therefore an

existence of finite frequency at which ee' is

zero is a necessary but not sufficient condition for a wave excitationless ship form.

Phase difference E is in proportion to non-dimensionalize frequency KB/2.

In case of

the bredth-draft ratio H=constant, if

the sectional area coefficient increases, the phase difference E decreases. In case of a=

constant, if H increases, E also increases.

Wave excitation force increases when H

increases, in case of a=constant, on the

contrary, it decreases when 47 increases in case of H=constant. In other word, V type model receives more vertical wave excitation force

than U type model. Among the V type

models, if

the draft

is constant,

a wide

breadth model receives stronger force than

narrow breadth one. Let us make a physical

interpretation on this phenomena with

sub-stituting the amplitude ratio

of radiation

wave in place of the wave excitation force

by Haskind's formula. When the model is

forced to oscillate up and down in a vertical

line in the still water, the fluid around ship

surface are diffused (or sucked in), hence the

radiation waves are generated.

If

a flat

board placed vertically with the infinite draft

se _{is moved up and down in a vertical line in}

the ideal fluid, no radiation waves will be

30'

generated. And yet if the flat board put in

inclined position is moved up and down in a

vertical line, some radiation waves will be occured. From the fact as described above,

we suppose that the radiation wave by the U type ship is small, because the radiation

wave is not generated at the place of the

2.0 e' e: e: 1.5 1.0 0.5 20 e K4

Fig. 10 Lewis Form

Fi 15 CT 0947 4 0.18 ch, 0.10

90'

90° 60° 50° 0,4 08 1.2 K B/2

Fig. 11 Lewis Form

eiz 1.5 cr 0.785 a, 0.2 a, 0.0 08 12 ---> KB/2 74

ship near the water surface and the bottom

which diffuses (or sucks in) the fluid has

little effect on the water surface.

On the contrary, the radiation wave by the V type

ship grows large, because the fluid is diffused

(or sucked in) near the water surface by the ship.

According to the considerations as

described above, an U type ship suffers less pitching moment and heaving force than the flared one.

4.3 Numerical results for wave excitation less ship forms

A shape of the wave excitationless ship is

shown in Fig. 12. The beam to maximum

breadth ratio BID, the draft to maximum

breadth ratio TID and the minor axis to

major axis ratio bla of the immersed eliptic cylinder are defined as parameters. Exam-plified computations are carried out as follows.

of immersed eliptic cylinder as shown in

Fig. 27. For B/D=0.50 and T/D=1.00, bla

turns 1.00, 0.50, 0.25.

Normalizations are represented as follows.

K4 = fc

A=K/Ifs,

e' =el D. 4.3.1 Added mass K, 4.3.1.1 Parameter TID

Computed results have the tendency

de-pend on the breadth of strut. In case of

B/D=1.00, the deeper the draft, the larger

the added mass. In other cases, there is not

very much change of added mass in regard to the draft. But as seen from Fig. 21, in the case that the strut is narrow and the top of the immersed eliptic cylinder touches to the water surface, the added mass changes violently in regard to the frequency and as the draft becomes deep, the rate of change

decreases gradually and at last it tends to

constant value independent of the frequency.

4.3.1.2 Parameter BID

The added mass turns not monotonic in

regard to the breadth.

As BID becomes

narrower than 1.00 by degrees, the added

mass decreases and on the other hand,

it becomes narrower than 0.75, the added mass inclines to increase.

4.3.1.3. Parameter bla

It shows a tendency depend on the width of strut.

In case of B/D=1.00, the added

mass increases as the flatness of the immersed eliptic cylinder becomes large. On the other

hand, in case of the narrow breadth and of B/D=0.25, the added mass has little change in regard to the frequency as the immersed

eliptic cylinder becomes flat.

4.3.2 Amplitude ratio of radiation wave 4.3.2.1 Parameter TID

The deeper the draft, that is, TID increases, the more A decreases. It may be considered

that wave free frequency at which A tends

to zero is not depend on the change of draft. rc 132\

\2 /

Fig. 12 Waveless Form

Parameter BID:

This is the varation of

breadth of the strut of the wave excitationless

ship form as shown in Fig. 13. For TID=

1.35 and bla=1.00, BID varies 1.00, 0.75, 0.50,

0.25.

\\Iff) 0.75 0.50

Fig. 13 BID Series of Waveless Form Parameter 77D: This is the variation of

the draft as shown in Fig. 20. For B/D=0.25 and b/a =1.00, TID varies 1.00, 1.35, 1.70.

Parameter bla: This stands for the flatness

I.0 _0.5 10 T/D 135 RI, 0.25 05 1.0 15 ---> K D Fig. 14

Added Mass Coefficient

0.50 0 00

Fig. 15

Wave Amplitude Ratio

K D 2.0 0.5 10 0.5 Tip 1.35 0 5 1.0 15 KD Fig. 17

In-Phase Component of Heave Force

20 T/p 1.35 ' big 1.0 0 05 10 15 20 C.5 1.0 /.5 2.0 -> X ID Fig. 16 Heave Force /0 ec. ,b(0

>

scf 6 .sd 0 0.75 0 5 0. 50 10 1.5 --> k D Fig. 15

Out-Phase Component of Heave Force

Fig. 19

Phase Lag

K D

26

TID Series of Waveless Form

it0 _as 1.0 ₀₅ Kt= 0.25 b/0.1.00 05 7/0. 100 1.35 1.70 05 K D Fig. 21

Added Mass Coefficient

10

Fig. 22 wave Amplitude Ratio

15 K -, T/D 1.35 , , bia loo 1 IVO 1.00 7 0, 025 2.0 1.5 1.0 _0.5 0.5 1.0 ---> T/0 1.70 Fig. 1.0 15 T/0 1.35 1.00 B/0.1.00 0.75 D

05 0 10 0.5 05 10 Fig. 23 Heave Force I 5 K o 20 to 0.5 90 6 3 10 1.5

> K D

Fig. 25

Out-Phase Component of Heave Force

20 C. B/D = 0.25 b/0 = 1.00 T/D .1.00 1.35 1.70 05 10 I5 20 T/0.1.00 B/D= 0.25 = 1.00 170 1.35 0.5 10 1.5 2D K D K D Fig. 24

In-Phase Component of Heave Force

Fig. 26

Phase Lag _Co.5o)

Fig. 27

Na

Series of Waveless Form

9.25

1.35

0.5 _1.0 0.5 1.0 B/D 0.50 Tip 1.00 b/o. Loo 0_50 0.25 Fig. 28

Added Mass Coefficient

0.5

1.0

1.5

K D

Fig. 29

Wave Amplitude Ratio

K D 0.5 0 0.5 1.0 1.5 K D Fig. 30

Heave Force and its In-Phase Component

2.0 b4, 0.25 B/D 0.50 T/ 0 1.00 b/o. 1.00 0.50 'VII IIPIP 025 B/D. 0.50 T/D= 1.00 tvo = 1.00 0.50 0.25 20 05 10 1.5 B/D 0.50 D 1.00 0.5 0.50 20 0 i.00 05 1.0 1.5 K D Fig. 31

Out-Phase Component of Heave Force

60°

30°

4.3.2.2. Parameter BID

We cannot discuss the change of the added mass dependent on BID. But the wave free frequency grows large as BID increases.

4.3.2.3 Parameter bla

We cannot discuss the change

of the

amplitude ratio A in regard to the flatness

of the eliptic cylinder.

But when the ratio

of the flatness becomes large, that is,

bla

becomes small, wave free frequency grows large.

4.3.3 Wave excitation fore e, e,', e,'

Wave excitation force e have the same

tendency as A, because the absolute valueof

wave excitation force and the amplitude ratio of radiation wave A are in proportion to each other. The in-phase component coincides almost with e'. But has the same tendency

as e' in the case that BID is large, while in other cases of BID value, the absolute value

of e,' is very small one, and its indefinite.

4.3.4 Phase defference

E is almost unchangeable

in regard to the

change of TID and that of Ina. The larger

BID becomes, the more increases.

4.4 Excitation _forces in oblique seas

The numerical results of exciting heaving

force on circular cylinder in oblique seas are shown in Fig. 33. The in-phase component and out-of-phase component are shown in Fig.

34, and Froude-Krylov force

is shown in

Fig. 35. As an angle a of oblique seas, 900 (beam sea), 60°, 30° and 0° (head sea) are

chosen. Wave excitation force and its out-of-phase component decrease, as angle changes from 90° to 0°. But the in-phase component and Froude-Krylov force have contrary tendency.

We can realize these

tendency comparing with the corresponding

term of the approximate equation of beam

seas (2.66) and that of head seas (2.68).

B/D 1.00 T/D 0.50 tya 1.00 E1/0 T/D 0.50 1.00 b/a = 0.25 050 1 1.00 80 Hisaaki MAEDA 0.5 1.0 1.5 20 K

Fig. 32 Phase Lag

0.4 08 L2

K B

Fig. 33 Heave Force of Circular Cylinder in

Oblique Seas

2.0 e," 1.5 LO 0.5 2.0 e' 1.5 5. Experiment

Two kinds of experiment were carried out. One of them was two dimensional and

com-pared with the numerical results of two

dimensional theory. Another is three dimen-sional and compared with the result by Strip Method. The experiment of semi-submerged

circular cylinder was treated

in general oblique seas. Others were carried out only

in beam seas and head seas.

The three

dimensional experiment was done with only

in head seas, because we paid attention to

pitchingless ship forms. These experiments

0.5

Table 2 Two Dimensional Models

0.4 08 12

---> K 8/2 Fig. 35 Froude-Krylove Force in Oblique Seas

were carried at the water tank of the Ship

Motion Laboratory, University of Tokyo.

5.1 Two dimensional model")

As two dimensional models, three kinds of type are chosen. Table 2 shows the summary of these models.

Refer to Figs. 12 and 36

for symbols. Fig. 37 shows a plain figure

of two dimensional model. It

is 3m long

cylindrical wooden model. As shown in Fig.

36, it consists of the following three parts,

that is, the first half part (A), the middle

part (B) and the latter half part (C).

Part

(A) and (C) are connected with each other by

Model bla BID _(cm)B D

(cm) b (cm) a (cm) _(m)L 1 (cm) h (cm) E-1 1.0 1.0 30 30 15 15 3.8 10 45 E-2 1.0 0.5 15 30 15 15 3.0 10 35 E-3 0 . 5 0.5 15 30 7.5 15 3.0 10 35

Froude - Krylov Force

8/D 1.00 T/D 0.50 bia 1.00 08 04 12 ---> K 2/2

Fig. 34 In-Phase and Out-phase Component in Oblique Seas

6b.

Fig. 36 Two-Dimensional Model

ILinear

Transformer

the band plate. The clearance between part (B) and part (A) or part (C) is 3 mm. Portion

B is a movable part for the purpose of

measuring the two dimensional wave exciting force in oblique seas. Portion A and C are

fixed to external part and portion B is fixed to external part through a canti-lever (refer

to Fig. 38). The wave excitation force was

measured by a canti-lever attached by a linear transducer as a pick up. The entrance form of the water plane of the model is expressed

ii the following form,

y =B/2 Z/50)2),

where

y in cm,

OZ5O cm.

5.2 Three dimensional model",

As a three dimensional model, three types of models are chosen as follows, wave

excitationless type named F--1, U type of

these two named FU-1.

The last type has

form of U shape at midship section and near the end of the model it has a narrow breadth

on water plane but it has the constant major

E-3

Fig. 37 Two-Dimensional Model

Table 3 Three Dimensional Models

Fig. 38

Cantilever

axis of the submerged eliptic cylinder. The

principal dimensions are refered to Table 3

and Fig. 39. Water plane form is shown as follows,

B (( Z

2 L/2 ) .

The reason why the length of model L is 1.1

meter is to adjust the wave excitationless

frequency of this type to the frequency which bring the maximam pitching moment obtained by the Froude-Krylov's method. Wave

excita-tion forces were measused by the apparatus

shown in Fig. 40, 41 in regard to heaving

force and pitching moment respectively.

5.3 Experimental method

The model fixed to the towing carriage

was placed in center of the water tank.

The experiment in oblique waves was carried out by means of varying the angles a between the z-axis of the ship and longitudinal

direc-tion of the tank. a=0° means head seas and

a=90°, beam seas.

In the theory, the amplitude of the incident wave was assumed

to be infinitesimal, but in the experiment, it

bla BID _(cm)B _(cm)L _(cm)D _(cm)h F-1 0.5 0.5 10 1.1 20 45 FU-1 0.5 1.0 10 1.0 10 40 U-1 (4=-0.633 as=-0.5 10 1.0 H=0.2 40 a =0.926 82 Hisaaki MAEDA :Model E-2

Linear Transformer

'hook-fli

-H.-200

Fig. 39 Three-Dimensional Model

Fig. 40

Model

Fig. 41

was 3 cm high on all occasions.

The ex-periment was carried out by measuring the wave exciting forces caused by waves with

period 71,=0.7 2.0 sec. For each models the following experiment were carried out.

5.3.1 Model E-1

The condition of the model was summarized as follows,

when T=15 cm a varies 0°, 30°, 600 and 90°,

when T=30 cm a varies 00 and 90°, when T=37.5 cm a varies 0° and 90°.

Results are shown in Figs. 42, 43 and 44. 500

Z/',/,.///////./(/////

Cantilever

5.3.2 Model E-2, E-3

The heaving force was measured. The

condition of the both models was summarized as follows,

when T=30 cm, 45 cm, a varies 0° and 90°. Results are shown in Figs. 45, 46, 47, 48, 49 and 50.

5.3.3 Model F-1

The heaving force and pitching moment

were measured. The condition of the model was summarized as follows,

when a=0°, T varies 20 cm, 25 cm and 30cm. Results are shown in Figs. 51 and 52.

5.3.4 Model U-1, FU-1

The heaving force and pitching moment were measured. The condition of the model

was summarized as follows,

when a=0°, T=20 cm (both models)

when a=0°, T=25 cm (U-1 only). Results are shown in Figs. 51 and 52.

Normalizations for heaving force E of two dimensional model are shown as follows,

e, 1E1 pgaBI

-10 20 30

--> K B

Fig. 42 Heave Force of E-1 Model ,/

2.0 e' e' 1.5 1.0 0.5 2.0 1.0 0.5 0 K B KB

Fig. 44 Heave Force of E-1 Model

Fig. 43 Heave Force of E-1 Model

2.0 e' 1.5 1.0 0.5 0 _{1.0 2.0} ---> K D

Fig. 45 Heave Force of E-2 Model

1.5

Jo

0.5

K 0

Fig. 46 Heave Force of E-2 Model

0 B/D Loo Tip 1,00 bia LOO a 0 so* 41, 0' a 60. a 00 TI -A-D .. o.., c..00. 0 _o a B/D T/D tvo 0.50 1.00 1.00 L a 90° I Bee. 0° (Head A 5.0) Sia ) a . 0 ' o 9 B/D -(/c. 0/a 1.00 1.25 1,00 0 a 0'

so

-7, C . a a a 0. G, a B/0 0.50 Tio 1.25 b,c, Loc, T i,24

3

_A D 0 L a 90' O. e 0 a 0° a 90° ° 84 Hisaaki MAEDA 1.0 20 I0 20 1.0 2.0 0 A,

1.5 ID 0.5 B/D 0.50 T/D 1.00 b/a 050 1 0 20 KID

Fig. 47 Heave Force of E-2 Model

1.0 20

K D

Fig. 48 Heave Force of E-3 Model for three dimensional heaving force E5(21L) and pitching moment Eo(AIL),

1E1

E#(2/L)

EY (2IL)= pgaBL

Non-dimensionalized frequency wave was represented by the

length A to the wave length

length L. IMol 112pgaBL2 of the incident ratio of wave A to the ship 1.0 2.0 K D

Fig. 50 Heave Force of E 3 Model

5.4 Results and discussion

Results are shown in Figs. 42-52.

The solid line in the figures shows the computed value. In Figs. 51, 52, the computed value by Weinblum which was based on the

as-sumption of Froude-Krylov. 5.4.1 Model E-1

From the case of semi-submerged circular cylinder of T=15 cm, it is seen that the wave

B/D T/D b/a 050 1.25 MO e c, LI a se 0* o o Cop '. m ,,, p o I . a ...so' 84, T/0 11/4 0.50 1.50 050 T a 0 9 0* 4 0* 0 0 0 A

.

' a .t. ,. ° --... 1.0 2.0 K D

Fig. 49 Heave Force of E-3 Model

10 0.5 1.5 e' 0.5 1.5 1.0 as Beam sea) (Heti Seal 19/0 050 -0

0.3 ,0.2 OA 1r, a 2 11 _{o o} 0 Q a: -0 a 0 0 0.2 0- 0.926 Oy -0633 02 -05

excitation force becomes small as the incident wave turns gradually from beam sea to head sea. Theoretical value shows this tendency qualitatively.. Theoretical value for beam sea

is in good agreement with the experimental value, but as it goes to head sea, the theore, tical value is greater than the experimental

one. In the same manner as mentionedabove,

in case of other different draft, the wave

excitation force caused by head sea is smaller

than that by beam sea.

This difference,

however, contracts as the draft increases.

The theoretical value, is in good agreement

with the experimental one with regard to

ia 8

-Pitching_{Moment of F-1- Moment, of U-1}Pitching

at000 16/o 0.50 a 2.0. 2.5 , 30, 0 0

Fig. 52' liewie Force of Three.Dimensiona Models

Lewis Form 0.2 a 0926 01 -0.633 .3-05 T/B 2.0 25 Pitching Moment of FU-1 B/D L0 by0 _0.5. 0 2.0 0 2:5 beam sea.

5.4.2 Model E-2 and E-3'

These two models are so called wave ex-citationless ship forms, which generate no

radiation wave at a certain frequency as

they oscillate on the water surface. In other words, according to Haskind's formula, these

type of ships do not receive the wave

ex-citation force at the frequency. The wave

excitationless frequency of head sea appears smaller than that of beam sea in every cases.

In the figures, the forces are treated in only

a positive region, but in fact, the

forces,

appear in also a negative region when

the

86 Hisaaki IVIAEDA,

2.0 3.0

/L Fig.. 51 Pitching, Moment of Thee-Dimensional Models

2, III A'rno,_Method a 0 4.0 1.0 2.0 3.0 40 1.0 0.5

by varying the draft of either model. As to

the reason why the experimental value of

pitching moment does not disappear at wave

excitationless frequency, following

circum-stances are considered. The heaving force

occured on both ends of the model contributes to the pitching moment of the model, yet the

strip method does not hold at both ends and

the three dimensional effect may be included.

And the surging motion was allowed to

prevent the intermixture of surging force into

pitching moment, therefore the condition of the theoretical method is different from that

of the experiment. Moreover being allowed

to surge, the inertia force caused by surging

was mixed into pitching moment. After all at _{the wave excitationless frequency the} heaving force and pitching moment of model F-1 shows smaller value than that of general ship form model U-1. Then the existence of wave excitationless three dimensional model

is shown.

5.4.4 Model FU-1

The sectional shape of this model is U type at the midship section, but excitationless ship form at both ends. The wave exciting forces at both ends are overcanceled at a frequency

above a certain

cycles, compared to the exciting forces at the midship. Even if the

overcanceled forces are small, overcanceled pitching moments will be large, because the

lever from the midship is very effective on

pitching moments. The experimental results show this fact in Figs. 51, 52. The heaving

force is close to that of the model U-1, but

the pitching moment is located between that

of the model U-1 and the model F-1. When we modify the shape of the water plane and

increase the overcanceled forces at both ends of the model, it will be possible to develop a

pitching momentless ship form.

6. _{Chart for wave excitationless ship form}

Taking account of the fact that the

theore-tical value of the wave free frequency in

beam seas is in good agreement to experi-mental value in head seas, we tried to draw

up the chart for wave excitationless ship form

frequency passes through the wave

excita-tionless frequency. According to this fact, it is seen that the experimental result of the

wave excitation force by head sea is smaller

than that by beam sea in all cases.

There-fore the wave excitationless frequency in

regard to head sea is lower than beam sea. The reason why the wave excitation force

of beam sea appears larger than that of head

sea, is considered from the point of view of

physical meanings, that is, taking account of

the orbital motion of water particles, the

beam sea has the components in the direction

of both swaying and heaving, but the head

sea only in the direction of heaving. Model

E-3 receives larger excitation force than

model E-2.

For this

fact, the following reasons can be considerd. When these two models have the same draft, the position of the center of the submerged eliptic cylinder of model E-3 is deeper than that of model

E-2, and so the less radiation waves are

generated by model E-3 than by model E-2.

It can be said the theoretical value is

in

good agreement with experimental value in

regard to excitation force of beam sea in the case of both models. The theoretical value

of the wave excitationless frequency for beam

sea will be able to use as that for head sea.

5.4.3 Model F-1 and U-1

Model F-1 is a wave excitationless ship

form. As to heaving force, the theoretical

value is in good agreement with experimental

one, and at excitationless frequency, almost

no excitation force appears. As to pitching moment on these models, the theoretical value

is not in good agreement with the

experi-mental one quantitatively but good in qualitatively. And that model F-1 comparing

with a general ship form model U-1, the

pitching moment of model F-1 decreases by approximately one-half. As to model U-1,

the theoretical results are in good agreement

with the experimental results in regard to

heaving force and pitching moment too.

Therefore, as to excitation forces, the strip method can be considered to hold. We can-not recognize the significant difference caused

88 Hisaaki MAEDA

using the theoretical value in beam seas. Fig. 53 shows this chart. The horizontal axis in Fig. 53 means non-dimensionalized wave ex-citationless frequency KB

and the vertical

axis is non-dimensionalized length BID.

Parameter 10 represents the state of the

flatness of ellipse. As a ship form, however, Motora's form with immersed elliptic cylinder

on which the wall sided strut is attached is

kept in mind. From this chart,

it can be

understood easily that when BID

tends to

zero, the wave excitationless frequency KB

also tends to zero. And when BID tends to

1, then wave excitationless frequency KB

seems to tend to infinite value.

This fact

can be analogized from the result of wave

excitationless ship form given by Bessho and from the fact

that Tasai's method uses

radiation potentials added at a center of Lewis forms whose BID is equal to 1.0. Now let's design a three dimensional wave excitational ship form. The pitching moment based on

the assumption of Froude-Krylov has maxi-mum value at AIL=1.3-1.5 among any water

plain form. Then if the ship form that won't suffer the excitation

force on each

hull

section is chosen in above frequency range, the pitching moment will disappear in case

K 1.5 1.0 05 05

-0

B/0

Fig. 53 Chart of Waveless Form

1.0

0.3

0.2

0.1

05 10

Fig. 54 Chart for Three-Dimensional

Waveless Form

of wave excitationless ship form, yet it will show a maximum value in case of general ship form. In the former case, heaving force

will also disappear.

Hence as it may be

quite all right to consider 2/L=1.4, Fig. 53

can be transformed into Fig.

54. On the

occasion of designing the wave excitationless

ship form, decide the length L and breadth B first, draw a line parallel to the horizontal

line

and passing through the

B/L point, choose the proper point crossing with line on

this chart, and finally choose BID and bla which satisfy with a demand.

Conclusions

The results mentioned above are

sum-marized as follows.

( 1 ) Hydrodynamic forces on two

dimen-sional arbitrary ship forms were calculated by means of source singularities distributed on the ship surface.

( 2 ) In-phase component and out-of-phase

component of wave excitation force are ob-tained from Kochin function which uses

ra-diation potentials.

It seems to be a first

example which uses radiation potentials.

( 3 ) Practical theoretical equations which

represent

the excitation force caused by

oblique waves are presented. And Watanabe's approximate equation for head

sea and

Motora's approximate equation for beam sea are derived from this theoretical equations.

( 4 )

From theoretical and

experimental

results, it was shown that the wave excita-tion force for heaving moexcita-tion decreased, as beam sea turned to head sea.

( 5 ) Existence of three dimensional wave

excitationless model for heaving motion was

confirmed. A chart for wave excitationless

ship forms was presented.

It remains to be proved to clear up the

limit of application of the source distribution method on arbitrary ship forms and develop

the way to be able to deal with bilge keel.

The author wishes to express his acknow-ledgements to Professors S. Tamiya, S. Motora and M. Bessho for their invaluable advices

and continuous encouragements.

He is also indebted to Messrs. M. Sugita

and other members of the

Ship Motion Laboratory,

University of Tokyo and

S.

Eguchi, technician of Institute of Industrial

Science, University of Tokyo for the

ex-perimental work and the drawing of the

manuscript.

He is

deeply indebted to

his respected

friend Mr. H. Isshiki graduate student of

University of Tokyo for his advices and

discussions.

He used the

electric computer HITAC 5020E which belonged to the Computer Center of University of Tokyo.

Nomenclature C= ship surface

ds =line element on ship surface a,=encounter frequency li=w2/g=wave number

INK)=Kochin function

N=damping coefficient, number of divisions on ship surface

0o=velocity potential of incident wave

E= phase difference

a=amplitude of incident wave n= normal line

a=intensity of source distribution v=volume

Ki =added mass coefficient

0= radiation potential

A =amplitude ratio of radiation wave E,e=wave excitation force

04=diffraction potential

0,00, c)4=velocity potential of oblique wave

Xy(AIL)= non-dimensional heaving force

a=incidence angle of incident wave T=draft of ship

yw=displacement of incident wave suffix c,s=-real part and imaginary part

a, b=-major axis and minor axis fo(2/L)=non-dimensional pitching moment

L=length of ship B=breadth of ship

2= wave length

mean draft D= maximum breadth A =water plane area A.M. =added mass

Reference

G. WEINBLUM, and St. DENIS: On the Motions

of Ships at Sea, Trans. S. N. A. M. E., Vol. 58,

1950.

Y. WATANABE: On the Theory for Heaving and Pitching Motion of Ships, Mem. of Eng., Univ. of Kyushu (in Japanese).

0. GRIM: A Method for a More Precise Com-putation of Heaving and Pitching Motions both

in Smooth Water and in Waves, 3rd Symposium of Naval Hydrodynamics.

K. TAMURA: The Calculation of Hydrodyna-mical Forces and Moments acting on the Two

Dimensional Body, (Unpublished) (in Japanese).

J.N. NEWMAN: The Exciting Forces on Fixed Bodies in Waves, Jour. S.R., Vol. 6, No. 3,

1962.

F. URSELL: _{On the Heaving Motion of a}

Circular Cylinder on the Surface of a Fluid, Quar. Jour. Mech. App. Math., Vol. 2, Pt. 3,

1949.

F. TASAI: On the Damping Force and Added Mass of Ships Heaving and Pitching, J.S.N.A. Japan., Vol. 105, 1959 (in Japanese).

M. BESSHO: A Memorandum on the Wave

Excitation Forces and so on, 1964. (Unpublished)

(in Japanese).

M. BESSHO: On the Theory of Rolling Motion in Waves, Mem. Defense Academy, Vol. 3, No.

1, 1965. (in Japanese).

M. BESSHO: On the Wave Free Distribution

dm=

90 Hisaaki MAEDA

in the Oscillation Problem of the Ship, J.S.N.A. Japan, Vol. 117, 1965 (in Japanese).

M. BESSHO: On the Theory of Ship Motions in Waves (with zero forward speed), Mem. Defense Academy. Vol. 3, No. 2, 1965. (in

Japanese).

M. BESSHO: On the Theory of Rolling in

Waves (continued report), Mem. Defense

Academy, Vol. 3, No. 3, 1966. (in Japanese) M. BESSHO: On the Two-Dimensional Theory of the Rolling Motion of Ships, Mem. Defense Academy, Vol. 7, No. 1, 1967.

M. BESSHO: A Memorundum on the Two

Dimensional Wave Motions, 1968. (Unpublished)

M. BESSHO: On Boundary Value Problems of an Oscillating Body Floating on Water, Mem.

Defense Academy, Vol. 8, No. 1, 1968. W. FRANK: The Heave Damping Coefficients of Bulbous Cylinders partially immersed in

Deep Water, DTMB, 1966.

S. MOTORA: Stripwise Calculation of Hydr-dynamic Forces due to Beam Seas, Jour. S.R., Vol. 8, No. 1, 1965.

S. MOTORA, and T. KOYAMA: On the

Excita-tionless Ship Forms on Heaving and Pitching Motions in Waves, J.S.N.A. Japan, Vol. 117,

1965. (in Japanese)

S. MoToRA, and T. KoYAmA: On Wave Ex-citationless Ship Forms, 7th Sym. of

Hydro-dynamics, 1967.

M. SUGIURA, and Y. OKUMURA: On the Wave Excitationless Ship Forms of Heaving and

Pitching Motions, Graduation Thesis, Univer-sity of Tokyo, 1966. (in Japanese)

H. ISSHIKI, H. SASAKI, and T, NISHIWAKI: Relation between Excitation force and Damping

for Rolling Motion, Graduation Thesis, Univer-sity of Tokyo, 1964. (in Japanese)

J. KOTIK, and V. MANGULIS: On the

Kramers-Kronig Relations for Ship Motions, I.S.P.' Vol. 9, 1962.

W.D. KIM: On the Harmonic Oscillations of a Rigid Body on a Free Surface, Jour. Fluid.

Mech., Vol. 21, 1965.

M. GANNO: Excitation Forces by Head Sea,

1968. (Unpublished) (in Japanese)

F. ABELS: Die Druckverteilung an einem

festgehaltenen Schiffsmodel im regelmaessigen Seegang. Jahr. Schiff. Gesrll., 1959.

Y. YAMAMOTO: On the Oscillating Body Under

the Water Surface, J.S.N.A. Japan., Vol. 77,

1955. (in Japanese)

Appendix 1 Calculation of lc and Is

Generally speaking, we can use x, y in

place (x- x'), (y + y') of le and /s, which appear in Eqs. (2.4), (2.16). Here we transform Eqs. (2.4) and (2.16) into the following equations,

L=.11

c" cos ux

. du (A 1)

o u A+ 1 p

Is= tims= c's C" sin ux. du. (A.2)

p-o.co 11+ 42

If we define 7rJ as follows,

,cc.o iuy

du'

(A.3)

O u tn.

then Eqs. (A .1), (A.2) become

lc= &WI re---Kw sin Kix' ire--Kv cos Kix!,

(A.4) h=sgn(x)[/7.{7,J}-Hre-K, cos Kix'

sin Klx1}. (A 5) Therefore we must calculate Eq. (A.3), when /c and Is are required. If we introduce z=x

-Fly, then Eq. (A.3), is transformed into e-v

dv .

= (A 5')

Kz V

This equation represents evidently the

ex-ponential integral. Let the integral term of the right hand side of Eq. (A 5) be replaced by

-v

E(iKz)= e

v EciEs,

(A -6)

ilrz V

then we have

Re{aj} -=(Ec cos KI xi +Es sin Klxpe-Kv ,

(A 7) Ini{rJ}=(E, sin Kix' Es cos Iflxpe-K'.

(A.8) The constant r' is defined as follows

7- =log ,

where r

is the Eulerian constant, then Eq.

(A 6) is rewritten in the form

(iKz)

n- n! .

(A 9) If we use the following complex variable z,

E(iKz)= log(iKzy) w =1 '13) I . El

Z--lx1-1-iy=r2e 2 , (A 10)

where

r2=N/x2+ y2,02= tan-1 (A.11) Ix!'

then Eq. (A.9) becomes

E(iKz)= log(r/Kr2)i 02 +

(Krz)ein(e2-0)

. (A .12) 71= n !

This equation is devided into real and imaginary part as follows,

(Kr2)' Ec= log(11Kr2) n! cos n(02 (A.13) 7r `'" (Kr2)"

Es= (02+)

772-1 n sin n 02 - . (A.14) \ 2

By substituting these eqations into Eqs. (A.

14), (A. 7), (A. 8) and then into Eqs. (A. 4), (A. 5), we obtain le and 19 as follows

I0= {E cos K1 + (Es 7r)sin KI x }e-"Y

ire-KY cos xl, (A.15) L=sgn(w)[{Ee sin Klxl(Esr)

cos Klx!}e-KY _{sin Klx1].}

(A. 16)

Appendix 2 Kernel function So

Taking account of Eqs. (2.14), (2.19), (2.23), (A. 16), the kernel function Se which appear

in the integral equation (I) of chapter

2 is

rewritten as follows,

Sc= 0

02-2 sgn(xx')

[1.-(70+7re-K,v+?") cos Kix el].

(A -17) As shown in Fig. 1A, we should note that the discontinuity of a gradient of a tangent which touches the ship surface at point P is z.

_{In other words, we must consider that}

the kernel function S, has the discontinuity

of TC when the point Q passes over the point P.

Fig. 1A

7r