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6. 0n Wave ExcitationlessShip Forms
SeizoMoToRA" andTakeo KoyAMA*, dember(FVom J,S.IV.A,
JaPan,Vot.117, June1965)

be a possibility that a body could bedesigned to have greater body-wave interaction in relation to the Froude-Krylov force
must

Introduction
It has been deduced theoretically by
Urselli)and Hishida2)that prisms of certain
create nosectional shapes
surface,roll ina stillwaterthat they

willcancelother.

each

As Motora has shown6),

they

when

waves

so

heaving forcethea summation
expressed
is approximatelyof an inertiaterrn, a damping term and aby McLeod andcheck has also been madein equation(1).
this theme into buoyance term as shown
HsiehS).Bessho has extended
anddevelmotions of six degree of freedom(1 )K.= rlk,pV2.+r2M2.+r3pgAz.
distribuoped
the theory ofasand experimental"Wave-freetions"D.

In addition, Newman5) has shown, on thebasisof Haskind relation, that the amplitude
wave
by an oscillating body in aof radiatedfree surface is directly related to the exciting force acting on the same body in waves.
These results indicatethat there must be
free from exciting forcesarebodies whichmwaves.
authors,

being interestedin the pos-existence

of

The
sible

such

wave-excitationress

bodies,have been carryingout experimental

have foundarethat there were a group of bodies which
inforce
heaving
free from wave-induced
researchwaves

for suchof specified

bodies,and

where

nil onlywhen

density of surrounding

z.:

wave

elevationM:

damping coeficient

If the wave

are4

volumeofcoeMcient

fluid

abody

elevationis of the form

Zw=zeitut

where

to is the circularfrequency,then,iib=: - (ti2zw .

Therefore, equation
lows:

is

However, the authors were aware thatis reverse in sign ofbody-wave interactionthat of Froude-Krylov force and that there
zero.

*

p :
V:

rnass

Correction Factor.

frequencres.

the waterplaneaddedof gravity
g : accelerationarea
A:waterplanefactors forthe
correction
r3
are
ri,
h,
and
the
wave.
motionoforbitalto the bodyThe firsttwo terms correspondthe third term correandwave
interaction,The
to the Froude-Krylov buoyancy.
spondstoaddfirstterm is also an inertia term due
r3 has been known as Smithed mass effect.

1. Approximate formula of heaying force
Froude-Krylov hypothesis has long been
induced heaving
wave
used in estimating
was clariinteraction
force until body-wave
fied in recent
years. According to the
Froude-Krylovhypothesis,heaving force becomes

kx :

'

(1)is rewritten

as

fo!-

L.;-rik,pi,2d22w+r21V12.+r3pgAzwC2)
The inertiaterm is rever$ed

Universityof Tokyo, Tokyo

in sign in rela-

75

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Seizo MoToRA

76tion to the buoyancy

term

and

will

and

increase

the frequencyw increases, In casetermof ordinary
ship form however, the firstisnot largeenough to cancel the buoyancy
term in a frequency range of ordinary wave
encounter.
However, if the form of a bodyis chosen so that the inertiaterm is large
enoughcomparedto the buoyancy term, it
the heaving
will be possible to eliminate
rapidly

Takeo KoYAMA

IVi=

as

pggh,

Fz.2

(two dimensionalcase)

(3)

!i"

FLiv(e)deIVI=.ptogK2A2(threedimensionalcase)where Koiswavenumberis encounter angleofanincident

wave

forcein a relatively low frequency range.
h is the wave height.
can
we
force
is
Substituteget approxl(3)into (2),
According to NewmanS), heaving
theamplitudeofthe heavingby the fol- mate value of
related to the damping coeficient
force R. as follows:
lowing formulae:
)i2

)22.i

.
ii (-hk,pVlv2+r3pgA)2.Vl+(rik.pl/Z,2+r3pgA)2(
prg2a2w#

Therefore,heaving force will vanish when
the frequency ev takes the following value:

;lkA.

tuo-V

strut

==

r3A
r,fe.v'

of

a

(6)

waves.

Therefore, let us

choosecircularofa verticalcylinderstrustofofdepth L andinbreadth B as showna,Fig. 1.

"excitationlessiiBll!1

it will be necesprobable wave encounter,to make A!V smaller than usual proporef meastion. It will be noted that results
force to oscillate differentbowured external

E

indicatethe simby Paulling')
iEartendency though, in his case, the inertia
itself.
force includes the mass of a model

iC'/

L

sary

2.1 Submerged circular aylinder with a strut
1) Theoreticalconsideration
has relatively
In seeking
a body which
inertia
force,
one
may
aware
that
greaterisan extreme
case of such bodya completely
body. The main
submerged
part of the
body isheaving forceacting to a submerged

inertiaforcewhich is reverse in sign of
the wave elevation, Therefore,ifa vertical

t

h

1

rnodelsl

2. Two dimensionalcase

ana combination

radiusofsectionto a sub-it gives small arnountmergedbodyof vuoyancy,
it will be possibleto eliminateforce
at a specified
frequency
theheaving(s)frequency"
To bring this excitationlessfrequency towards low frequency range

Let us call wo as
for convenience.

is attached

width

that

so

or

Ko

of narrower

(4)

f

1

q

l,
i

Fig. 1. Geometry of circularcylindertype rnodels.

Though the heaving forceacting
cularminsS),
lnertlacylinderhason a circalculated by Cumfor us to calculate theitis necessary
andbuoyancy term

term

an approxlmate

calculation

wasseparately,

dQne making

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77

Ship Forms
Excitationlessgiven by Yamamoto9) .wasandfor that of a strut, values givenof the iner- used,that a summation
approximately
by TasaiiO)for fullsection was employed.
andfor a strut
tia term for a a cylinder
Calculatedinertia force and buoyancy forterm of the total body.
would givethe inertia
for the different breadth of the strut are as shownwas
appliedThe same assumptionin Fig. 2.
buoyancy term.
valuesof the heaving
The approximate
k, of a circular
For the inertiacoeMcientuseof

it was(4),i.e.,

the equation

cylinder,assumedvalues1,51.

.tt-if-INs}t):i(O,50'-'Q5o

o

O.5

.
.y-f

1,O

2.0

15

y

Fig. 2. Culculated inertia force andbuoyancy

fordifferent breaclth of the strut.

Ew-1pt>iq

e=a
E-ISO

o

O.5

1.0

i,5

20

!2. t

y

Fig.3. Calculatedheaving force of circular
der type models,cylin-NII-Electronic Library Service

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78

force is obtainedin Fig. 3.

andTakeo KOYAMA(4)as shownby equation

At relatively
low frequency range, thebueyancy is greater than the inertiaforce(orunder cancel), and the heaving forcewill
At tooelevation.
be in-phase of the wave
isheaving
force
by
the
equation
(5),
given
than
too,
the
zero,
and
at higher frequency
inertiaforce over cancels the buoyancy, i,e.,the heaving force at this frequency rangeof the wave elevawill be 180e out-of-phase

LINEARTRANSFORMER

7

ltZk-EN
PLATE:.

tion.

in Fig. 3 that wo shifts
It is also shown
the breadth
toward lower frequency whenof the strut
decreases.
2) To check the above described results,
experiments

were

conducted

on a rnodelof the followingsize.

Fig. 4, Set-up of heaving force measurement.

puted

The base of Fig.5 is non

one.

di-

frequency and the orwave
dinate is the non dimensionalized heaving
force. Thick solid line shows the heaving
circularcylinderforce of a semi-submerged
mensionalized

Diameter 2a=20cm
of radius a!2.
Breadth of the strut B= 15cm (1,5a),decrease of the heaving
A remarkable
10cm (a),and 5cm (O.5a)end
force at a frequency approxirnately equal to
Length=50cm (withplates).
the theoretical tuo wil! be recognized.
To prevent the sway and pitch of the modelFig. 6 is typical oscillogram of the heavto the model
as shown
a guide is attachedas (w21g)aforce and the wave
elevationwasrneasureding
in Fig,4. The heaving force
At
O.238 and O.402.
(w2!g)a=O.124,
by a rigid spring attached by a lineartransthe buoyancy isgreater than the inwhere
former as a pick-up.
A typical result for B=a and depth f==2a ertia force,the heaving force is in-phaseof
the waveelevation(undercancel). At (te2!
is shown in Fig.5 together with the com=O.124,

r""-

1,O

HEAV[NG

FORCE

FeR

A CIRCULAR CYLINDER OF RADIUS'rf2+N A FF"EE SURFACE

xxNNxcN.!.-:;1･,rl"i:LV

N,x
o.s

N

xxxNNxxxxxx

t.EASURED

NCALCULATEDro7x-.rF.x-

t'

Q5

1,O

os

:,'z,e6'

O.4

O.2

Ll

.'I
/t'.
,T

:･o

10

2E

z

i)

Fig. 5. Typical resu!t.

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79

On WaveExcitationlessShip Forms.r.h.e.8gL2>-dir-is!2!yg"
IDoID

42o24

u,=3,49

INERTIAFORCE<BUOYANCY:.ao8gL2fs--w".EgwW;
1.0o1.042o24

w=4,83INERTIAFORCE,-BUOYANCY.o8gL9>-sm'E'.v.IQ!y;

1.0oID42o24.

pm-pmpm-pa-pmmutumuveq"w=

6.28 INERTIA FORCE>BUOYANCY

Fig, 6. Typical oscillograrn
and

wave

of

the heavingforce

elevation.

quency as the breadth of the strut decreases.
g)a=O.238, the heaving forcealmost vanishes.
islessHowever, change of depth does not affect wo.
where
the
buoyancy
At (di21g)a
corT=30cm
is 5.e3 which
force,the heaving force is too for a=10cm,
than the inertia
for a fullscale ship of1800 out-phaseof the wave
elevation(over responds to too=O.873
140m
and
draft
10m. This frequency
valuableinlengthcancel). These oscillogram gave
towave
length 61m for beam
formationin determining a suitable breadth corresponds
waves,
seas. However, incase of longitudinalof the strut.
corresponds
towave
lengthbreadthof
thisfrequency
To examine
the effect of thethe strut as well as the depth of the cylin200m at ship speed 18kts,=O.4e2,

der,results are shown in Fig. 7 a), b),c). As
predictedby theory,coo shifts tQ IQwer fre-NII-Electronic Library Service

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80

and

Takeo

KoYAMA

1.S

k3,5a
B==O,5a1.0
B!=O.75a

scs-.[ChJt(

XL..

a)

B=1.5a'O.5
B==1.5a

O.6

O,4

O.8

1.0

1.0
1.5
JL]a(ts2a)

1rz

e"bea
1.D
""Qr"1pt)cb)8"O.75a

/

o.

×

A

O.4

O.2
o

glas

1.5-gL'r

1.0

e.s

2,O

2,5

1,o

y
1.5
[=2Sa

1.0
g9-kou1

2itQtSzs5.

c)

¢

/

o,s

Fig,7. Heaaving force ofthecircularcylindere=1,5a

x

type
08

G2

models.
oe.:"i.eyi.

1.5

t

2.0

r:.

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2.2 Eiliptic
cyiinderwith strut
To make the draftre!atively small,cylinderschosenwere

The proportion of

as

the

axis

major

elliptic

B

body.

main
mmor

and

axis

from 2 te 4, where the breadth
waskept to be a half of theof the strut
major axis.
(SeeFig. 8)
in Fig.9a) and b). DifResults are shown
resultof the former case,ferentfrom theldo seerns
to shift towards lower frequency
ofas the depth increases.Absolute value
as
the
the
same
the heaving force is about
former case.
were

81

Ship Forms
Excitationress

On Wavevaried-K"--r'X

Fig. 8. Geometry of elliptic
type

cylinderaJIOcm,6==5cm,e==10cm

ELLLPSEmodels.

B

-

-,

1.0

E

R

b2a
fLl.5q-------------･

pt

11'q'`N,.,x11e2.0e
T.25ots3,Q,

x<i
ts3,5e

×

M-t"

O.8

O,6

O.4

O.2

o

uJ2

i'aof
force

Fig. 9 a). Heaving

ELUPSEelliptica--IOcmtype

cylinder(alb=2).

S=t25cm B=!Ocm
fi

rrL.

1,O

J=#

-2a

"g-.1.CNi"

K
tsx
O,5

gx

kXlt

x

za.ss
b

6A

sk2,EtL5
x-

o

lts

"ny

o

o

02

Fig, 9b),Heaving

O.4

O.6

ferce of2"laellipticcylindertype08(alb=4).

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82

TakeoKOYAMA

and2,3 FVns with strut
To makethe inertiaterm greater,it isnecessary
to increasethe added
mass
k.pU
The virtual mass for a thin plateof breadth
2a is pna! per unit length. Therefore,itwill
be possible to replace ellipse by adequatesize
of finsas shownin Fig. 10.
A model in which 2a=2'5'cm,
B=15cm, .f:'wastested in the same
technique.
Results are as shownin Fig. 11. In general,
it can be said that effectivenessof finsare=7cm

almostequivalent

2Q

Fig. 10. Geometry

to a thin ellipse.

type

fin

models.ofEl,O

as-w-s.:NIQ
X.x
O.5

35=
""-M-"rf.5ao

QlO,2O.3e.4

O.5

O.6

O.7

O.8O.9tu1

liaforce
Fig. 11. Heaving

3. Heaving
As

shown

variationsofamplitudein Section 2, there areof bodies which

are

not

severalacted

byfintype models.

frequency about cao. It will be interesting
to bring the resonant frequency equal to wo.
However this is proved not to be practicableby the following reason:

induced heaving forceat specified frequency. Howeyer, it will be premature toconcludethat these bodies do not heave at
the specified frequency.
wave

resonant(7)"s=V.fl4.,

As shown in equation(3),damping factor
isproportionalto the square of the heaving whereforce. This meansthat wave
excitationlessbody isalso a dampingless body.
In fact,in the case of the circular cylinder
as described in 2) of Section2.1, measured
heaving amplitudeis quite Iargeas shown in
12.
There
Fig,
appears a high peak at resonant frequency and a minimum
point at afrequency of heavingmis the mass

' excitationless

of

a

body.

frequencym,=Vrrz,g:.

(s)

As the ratio r31ri isalmost equal to the unity,a,. cannot be equal
to tuo unless the mass ofthe body is zero.

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On WaveExcitationlessShip Formscy"iklig5E6t9Lrzo(E

oz

4

>
<
u
=

4M
"
1
bl1ws

O.1

O.4

OS

factor of circularcylinder

O.3

02

08

L.9 a

y

Fig. 12. Heaving

magnification

type

model.

ofcal damping. Apparent critical damping
damping isconcerned,
factor diagram is due tohopeless to decrease the heaving the magnification
it seems
force as the
However, itshould be noted that rapid decreaseof the exciting
amplitude.
to too.
frequency approaches
eddy
damping or artificial damping are notincludedin Haskind-Newman relation.
inwaye
excitation
trial to eliminate
darnpmg or arti- 4. A
Therefore,eddy making
range
of frequency
wider
ficialdamping such as by a passive tank are
According to the above describedmethod,body, it will
given to a wave excitationlessthe exciting force is eliminated at only one
be possible to minirnize the heaving.
frequency. The following isa tria!
specifiedHeaving of a elripse (a/b=2)with a strut
the exciting forceat two or more
to eliminate
in Fig. 13a) in which
as shown
is measured
frequencles.
than the case of alessheaving amplitudein Fig. 15, a inner solid strut
As shown
due to eddy damping will
cylindercircularby a tank of bfeadthof breadth Bi iscovered
Thinner ellipse (alb=4)with
be recognized.
in Fig. 13b). B2. The tank is provided by small holes ofa strut is also testedas shown
total area S, depth H,
Remarkable decreaseof heaving clueto eddy
frequency
If we denote the exeitationlessdamping will be noted.
force wil! vanish
Heaving of a body with finsas described for Bi as tooi, the heavingarea
S is chothe
at frequency tooi provided
in 2.3 is also tested. Resultsare as shownlevel
in theso that the waterin Fig.14. Though the magnification factor sen adequately
equal to that of outer surface
tank isalways
diagram lookslikethat of critical dampmg,
critiat this frequency.
actual darnping is about one third the

As far as the wave

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SeizoMoToRAand TakeoKoyAMA

84

4
AMPLITUDE

t-2
ncoeo<LzoF<9L2oqE

2.

bl

1,4

oO,2O,4O.6O.81.01.2

1.6182,O

tolws

02

O,1

o

O,3

O.5

O,4

O,6 O.7

O.8 O.9-'a･yW2

factor ofFig. 13 a). Heaving magnification
type

model

an

elliptic cylinder(a/b=2).

HEAVINGAMPLITUDE

3

f-4

rT,

zQF<?Lze<E

2

1

o
bl

1.2

1.0

e02O.4O.608

1,41,6182.0-2.2

to!ws

O,1

o

O.3

O.2

O.4O,5O,6

O,7

O.8

O,9

.922a
y

Fig. 13b),
type

Heavingmocielrnagnification

factorof

anellipticcylinder(a/b=4),

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On WaveExcitationlsssShipForms,

HEAVINGAMPLITUDE
croFo<Lz9e`9Lrzo<E

B'=f

FINS

WITH

'2a=25cm
e=15cmr==3ocm2

f!22.5cm

2a
MEASURED-"---

1 ------------->N

CALCULATEDXNSs
s

o

2

1

3

7

5w6

4

9

8

10

oO,2O.4O.6O,81.01,2-1,41,61,82,e2,2
wlbls

O,1

o

o.2

o.3

o,4

o.6 o.7 o.s o.g l,o

e.s

.gla
y

Fig. 14. Heaving

factorof a fin-type rnodel.

magnification[mk2-=- it./=J
B,1

1

"

l

I･"
'

-T

K

"'.L

La1

1,O

HEAVINGFORCEFORSTRUT

Fig. 15.
'

B 2

of

type models.double
calssDnGeometrygg"..1.ouiqo.S

HEAVING FORCE

FOR

e,

COMBiNED(i=2)STRUTxx-

COMBiNED
o

x

uV

lt

(ie4)
o

O,2

O.8

O,5

O,4
e)1

li'a
Fig. 16. Heaving force ofadouble caissontype mode!.

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Seizo MoToRAancl TakeoKoyAMA

86

On the otherofofhand, according
to a theoryanti-pitchingtanki2),iS),the movement
water
in the tank, as the outer water

for three dimensional case(2)is applicablevaluesfor r are chosen,
provided adequateof heaving force of suchbody
the amplitude

becorneszero whenlevelchanges periodically,
the frequencycoincides to the followingfrequency;1'

coo'=V7'

Klt

Therefore,at the frequency a,o', tank water
is practicallysolidified and the breadth ofB2.
the strut is practically
excitationlessfrequency
Therefore, if thefor B2 is denoted wo2, heavingforcewill vanishat wo2. Thus, the heaving force will vanish at two differentfrequencieswoi, and evo!.
Experiments were conducted
for whichBi, B2 and H where

on

a

1

a

l
Fig. 17, Geometry
type rnodels.

of sphere

rnodelinis calculated by equation (2)as shownin Fig. 18.
shownin Fig. 15. Resultsare as shownof aFig. 16 in which three results for different The heaving force is also measured
size;of the following
model
size of the holes are indicated. By Fig. 16,
it will be easily seen that the heaving force
D=a anda=14.3cm
alrnost vanishes
at predicted woi, and wo!, andis very small in a frequency range founded
T=1.5a and 2a.
by uaoi and too2.
in Fig.
A typical record obtained is shown
chosenas

Aii'2'a,5.

Three dimensional problems

19

5,1 A spherewith vertical qylinder
As a most simple three dimensionalcase.

Ietus consider

a sphereder asin Fig. 17. As

shown

with

vertical

shows

which

the samealmosttendency

dimensionalcase shown in Fig.6.
20
In Fig. a), b),heaving forces for differReasonable agreement
ent depth are shown.
ascylin-

the equation

the twowiththeoretical calculation

will

be noticed.

O.5
O.4?g'-

H?.-03

3･

e

O.2

",-

s

O.1o*

o

odi]

p

r.･-.f2a
O.2

C.4

Fig. 18. Calculated heavlngforce of

'

/

e.6

Q8

spheretype

models.

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Ship Forms
On Wave Excltationless

87

I
l4

5.

,

ta
qo
L.J

V2
(
)4

.･--?,.S5'k･J

INETiTIA FO[･)CE..EUOYANCY

So][
'

g･
--

4

E

I2
p.
!o

Y2

;

4

k,=5.25

INERT[A FORCE==BUOYANCY/S".

:li["V
VX-/V----`hN

'<-.

a

f.
'2
I

:-

o

=

ge2
(

Y4

u.Lur"J.LbJ.LbLL.u.LLJJLuJLLLLLLt.r")'JLLL".LLt.tJSLulftuLLjuJJ"JLttY"LnLS･"=7.85 :NERTIA

Fig, 19. Typical record

obtaineds.2 Api'isjnin iongitudinalwave

In relation to the possibilityof applying
the strip method
for three dimensionalcases,
itis important to know how a wave act to

HnyISec"'･/

FoRcE.BuoyANcysowasfor a spheretypemodel.

that heaving force acting on this segment
measured.

Resultsare as shown in Fig. 22. In Fig.
forces in beam seas as well as
wave
longitudinal
are comparedwith theoreticalvalue.
Reasonableagreem'entbetween
23, heaving

length of' a prism fixed inthem will suggest that the strip methodwil!
longitudinaltrain.
be applicablefor obtaining heaving force asThis problem was treated by Okumura andof ships
in longiin well as pitching rnoment
Sugiuraii)for a priSm of a section shownwascut
tudinalwaves.
Therefore,
there
will
be a
Fig.13b). An unit length of prism
off andkept free from the main body, andof
possibilityto eliminate pitching momentby the same technique.
was
supportedby a rigid spring attached
ships
with
lineartransformer as shown in Fig.21,asection

unit

wave

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of
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SeizoMoToRAand Takeo KoyAMA

88

SPHERE
T-r3.5a' .
.];t;] il' -i

O.5

1'"tEv-<Ni"

s

+
..tt:

i1-.fi1,

O.25

/

"O--i

.t-.tv

-p

,IJ=
.y2d,
t

1

e

O.4w!-･ay

e.2

2.0

1.0

o
Fig. 20 a).

o.s

O.6

].o

b.:t
'L'"-Xof

Heavingforce

spheretype

models(T=3.5a).

SPHERE

o.s

P3a
Sr

tt'R-(tNiQ

025
e==o

D==J'2a

o

O,2

O.6'

O.8

'O.4.!!ly]
e

1.0

2.0

JgL.Zlyof

Fig.20 b). Heavingforce

spheretype

models(T=3a).

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of
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NavalArchitects
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On WaveExcitationlessShipForms
TRANSDUCER/SPRING

:

1.0

RfL

Fig. 21. Set-up of heaving
force rneasurement.

Ee-iCN>

1

O.5
¢

x,Xxx
r==2.0a

x

o

e,2

O,1

t--.----x------------'-Xfe2.5a:r='-'-'--`-N-----trk3,5afo3,Oa

PT=-'$N

O.3

O.5.yL2aylength
O.6

O.4

-

Fig. 22.prlsm
Heaving force per unlt
in longitudina1 wave.

of

an

O,8

O.7

elliptic

type

1,O

le2,5aEw"[ouo.sljiL:lxX.
Xxx)X>io.

-

//

xXxx

//

×
XN.
Xs.

O,4
6

4

3

2

O.5
1.5
A(m)

1

Co:nparison of heaving force in longitudinal waves
Fig. 23.withthat in transverse

waves.

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Seizo MoToRAand

6. Comparison withtion " theory

theTakeoKOYAMA

oW.L

distribu-"waye-free

Bessho has given in his paper
distribution"i),a group of two dimensionalas they heave inbodieswhich are wave-free
a free surface.
They are derived by the

1.0

2.0

"wave-free-1.0

followingprocedure:
Supposing m(t) is a functionwhich is regis vertically skew-symular at infinityand
metry.
Then a function
d

-2.0

(10)

Km(t)+i-dtm(t).f<t)==

h--oo
K=

-3.0

satisfies
give a velocity potential which
notthe free surface
condition,and should
associatesurfacewaves.
Therefore,f(t)isisa velocity
a body which
potential around
free from surface
wave
as it heaves in awill

free surface.
If apoint at depth h is taken to

singularbe m(t),

viz,

(ii)

m(t)-iog(l!l.Z)

Fig. 24. Typical stream
two-clirnensional+Kiog(lli.fi
f(t):=
t.'ih ,.i,,dis-tributions.

linesgiven by this function are as
in Fig. 24. As seen in Fig. 24, oneof themis similar to the circular cylinder
streamshown

type rnodel of the present paper.
It isquite interesting
to see that two

dif-

ferentapproaches
wasconducted
to measure
An experimentbody. Results
the heaving force for such(i2) are as shown in Fig. 25 where heaving force
reachedtherr,

lines ofwave-freesimilarresults.

).N`oue-'1ov

LO

/
O..5

04

O,2

O,6

Fig. 25. Comparison between the heaving forceof
by Bessho and that of thea
free body obtained
cylindertypegwave-circularmodel.

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On Wavefortype is also shown
comparison.
From Fig.25, it wi!1 be noticed
body shows quite similarthat the wave-freefora circularwith the circularcylindertype

compared"Experimental

Summarizingforegoings,the followingcan･
be concluded;
1) The Froude-Krylov buoyancy and the inis induced by the added
ertia forcewhich
in sign. Therefore,
mass effect are reverse
themcancel each otheritispossibleto makeso that wave

excitationis reduced.

frequency at which the absolute
2) At
value
of the inertiaforceis equal to that ofbuoyancy,the heaving forcebecomes exactly
to be zero theoretically, and is proved experimentally to be almost zero.
3) To bring the excitationless frequency mto low frequency range such as ordinary
it will be necesfrequency,
wave encounter
large
volumeary
to make the underwater
water
to
the
in relation
plane area.
enoughbyreplaced
can bevolume
The underwater
has the same amount ofa fiatplate whichabody.
as the underwater
as the heav4) Waye-darnping also vanishes
ing force vanishes. Therefore,'it will bemass

to give damping other than wavenecessaryheaving or
damping to a body to reduce
pitching motion.
body, for5) In the case of two-dimensionalof

to beam seas

the RollingMotion ofCylinders in the Surface of a Fluid," Quart,Journalof Mech. & Applied Math., Vol, II, 1949.
Study on the Wave-Making
2) T. HIsHIDA:
Resistance for the Rolling of Ships II," Journal of Zosen Kiokai, Vol. 85, 1952.
3) W. McLEoD and T, HsrEH:
Makingof Ursells Tbeory Wave
Investigation
Bd. 10,by a Rolling Cylinder,"Schiffstechnik,

1) F. URsELL;

L'On

``A

model.

Conclusiona segment

References

cylinderheaving forcevs. frequency curve

added

91

Ship Forms
Excitationlesslength,heaving forcedue
is almost equal to that due to

unit

Therefore, ship method
wave.
longitudinalcan be applied for three-dimensional bodies.
It will be also possible to apply the present
technique to the pitching motion.
6) It is also possible to extend this techniqueinto three-dimensional problem.

1963.M.
the Wave-free Distribution
BEssHo:Problemof the Ship," Jourin the Oscillation
117,
1965.
nal
of Zosen Kiokai, Vol.
"The
ExcitingForce on Fixed
5) J.N, NEWMAN:
Bodies in Waves;' Journalof Ship Research,
Vol. 6, 1962.
Calculationof Hydro6) S. MoToRA:
dynamic Forces due to Beam Seas," Davidson
Lab. Notes No. 656, 1962, Journalof Ship Re-

4)

"On

"Stripwise

search,Vol. 8, 1964.
andPAuLLrNG

R. K. RIcHARDsoN:
Pressure,Forces,and Radiating Waves for CylindersOscillating in a free
Report of Instituteof Eng. Research,
Surface,''
Univ. Calif., 1962J
Trans, of Symposium on Ships
8) W. CvMMms:
on Wave Research, SNAME,
andWavesCounc!17) J. R,

"Measurernent

9)

of

1954.
Y. YAMAMOTo:der the Water

the Oscillating Body unSurface,'' Journal of Zosen"On

Kiokai,Vol. 77, 1955.
the Damping10) F, TAsAI:
ed Mass of Ship'sHeaving and``On

Force and AddPitching,''
Jour-

Zosen Kiokai, Vol. 105, 1959.
Wave
and M. SVGIURA:
11) Y. OKuMURA
Referenceto
Forms
SpecialExcitationless ShipThree Dimensional Cases,''Graduation thesis,
Tokyo Univ., 1966.
M. KENGAKu, T. IDA
12) E, TAsAKA, S. MoToRA,``A Experimental Study ofand
M. KoyANAGI:of Anti-pitching Tanks," Jourthe Effectivenessnal of Zosen Kiokai, Vol. 117, 1965.
Discussionto Gidding's paper.
13) S. MoToRA:on Naval Hyclrodynamics,Ber5th Symposium.
gen, 1964.
nal

of

"On

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