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10, Equivalent AddedMassin CollisionsofShips

Seizo MoToRA*, A4ember,Masataka FuJiNo* Member,

Masanori SuGiuRA'*and

Matsuji SuaiTA'Meinber(Fbrm 1,S,N/A,
JaPan,VbL 126. Dec, 1969)

Summary
To calculate the absorbed
energyby the flacture of ship's structure
in case of collision,it.isnecessary
to estimate
accurately
the added
massof striken
ships.
Minorsky has proposed in his famous paperi) to use O.4 for the added
mass
coeficientof striken ships.
In this paper, the authors tried to obtain an exact expressionof the hydrodynamic force which
will
act
on
ships
durig a collisionby theoretical calculations andseriesof modelexperimentson the firstatomie-powered
ship of Japan wereconducted.
As a result, the authors have shownthat the ratio of external
forceto the accelerationofthe striken

of

the

the time elapsed
during the collision:i.e., the added mass
during the collision.
From this result, the authors introduced
an equivalent addedmasswhich,
dividingthe external
force by it,wirl give an acceleration
equal
to the exact value of acceleration
at the endof the collision.
It was also shownthat this equivalent
added
mass
changes
its valuewiththeof the collision.
If the durationis infinitelysrnall, the equivalent added mass is equal tothe added massfor infinite
high frequency. This agrees with Minorsky's assumption.
However,
if the duration is finite,
the equivalent
aclded mass
becomes larger as the duration
ship

is notship

varieswithcenstant"duration".Increases.

Sincethe duration isonly

relatedto the initial
speedof the striking
ship
andthe amountlower the initialspeed or deeper the penetration the longer the duration,
the equivalent
added
mass
coeMcientwi!1
be mucha softgreater than O.4 in the case whenstructuredislowresultingship
striken by aspeed ship
in a considerable
amountof penetof

penetration, andration.

1. Introduction
It is necessary
to take into account ofhydrodynamic effect of the surroundingwater
in dealingwith the absorbed
energyby collision of two ships. Minorskyr),in his
famous paper, has equatedthe absorbed
energyandthe lossof the kineticenergyofships involvedin the collision.
He has also
assumedthat the hydrodynamic effect wasrepresented
by an increasedinertiaforcedue
to added
mass

mass

coeMcienteffectofandestimatedthe colTided

the added
ship
to be

and

as the added
Johnson+)mass

coeficient

for lateralvibration.

In this apper,
theauthors treated the problem by hydrodynamic
Sincethe hydrodynamic force
pointof view.
in a transient conditionsuch
as a collisionisfunctionof time as well as acceleration,itis not possible to express the hydrodynamic
forceas a simple product of an added rnass(aconstant)

and

mass

i.e.,supposing

acceleration"a"

The authors

the acceleration.

proposed to int.roduceanofequivalent

an

exact

the

collidedsolution

ship

added
of

the

acted

by Kochi), Dieudonn63) by a prescribed force F, then rk" given by
the followingequation
is defined to be the***Universityof Tokyo.
Hitachi Shipbuilding &Engineering Co,, Ltd.
equivalent
addedmass for the acceleration.

O.4which

was

used

138

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F=:(m+nt")a
where

m

is the rnassoff(,)=u(,)=

itself.

the ship

139

Mass of Shipsin Collisions

Equivalent Added

I?l
flm)
t,>-.g(6)of U(t) isexpresadded
The FouriertransformIn the same sense, an equivalent
mass
forthe speed which gives an exact final sed as followsusing Dirac'sti function.
rnass for the
added
and
an equivalent
speed,
F<w)==n6(to)+-l(7)absorbed
energy can be defined.
Jw

mass

R(.)=

and

X<(v)

!

..x(t)e'j"tdt(2)

F(w)=!:.f(t)e-'intdt(8)

Substitutingthe firstequation
we(8)into (5)

of

get,
a(t)=-ii-!:..

+;I::;;(n,,(,)a.'),c.,,.td.

cos(9)

tutdtu"z2(.)C.(ric2(.)

the 2nd term, the range of integral
the integrandis converted (O,oo) considering
to be an even function. According to TasaiS),
to is suMcientlyof to5 whenc(to) is of order
evaluated
as
lst
term
can
be
small, the
where

'-O

=

fn2Cw)to'+c2(tu)I(a))- m2(tu)tu2+c2(w)denotedThe Fourier transform of x(t), f(t),followingintheby X(w),Rw) will be given
equatlon.

nm(w)to26(w)+c(te)xc(to)tob(w)-m(tu)o(1)m(w)th(t)+c(tu)fu(t)==f(t)glven asfollows

damping of aare functionsof
ship underperiodic motion
the circular frequency tu, letus denote thesevaluesas m(tu) and c(tu). Then the equationof a ship fortransverse motion x(t)of motion
forcef(t)is writtenundereffect of externa!
down as follows
As the apparent

R(bl)and I(w)of eq. (5)areThenof the equiyalent
2. Theoretical calculation
collision
mass
in a
added

at

to obtain
is te(t)(! follows
The item we are aiming
theFourier
A(w)
which
is
a(t)), Iet us obtainxM(to)to26(tu)LL
coswtdw=transform of a<t).
n-.m2(tu)to2+c2(tu)

!
co

Since

A(m)=(ju)2X(w),we
A(.)i! H<.) ･F<.)

get

mCO)

where

]w

mte)-jom(to)+c(tu)by the inverse Fourier
be obtainedtransformof A(to)a(t) =

Asa

a(t)=

(10)
is the(3) frequency. Thus wea(t) can

mlo)

mi(o7

mass

apparent

at+ -ll,'!:.2(.i(Y'llet usreference,Thea(O) of a(t).

c2(Js-cosa(t)=t!:..R(to)costotdto=-t!

.I(tu)sine,tdto

A(tu)=R(w)+j'I(tu)(5)a) In case of a step input
Let us firstconsider an external forcewhichfunctionU(t)concerning the time
is a step

examinevalue
a(O') ofinitialfollows
from
A(ca)
as
a(t) will
any
impulseprovided a(t) does not include
functionvalue℃

totdev

the initial(4)S).r!li..A(a,)ej"`d(vIn general, a(t) can be obtained either fromthe real part R(w) or imaginary part I(w)of
A(tu).

zero

obtainbeobtained(12)a<O')==limifteA(w)]--oe

On the other hand[j""(W']l,LM..
ILrn..
if,f.'g't,iil, )
,(to
=

=

"?goo

lw

(13)

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Masataka FuJmo, Masanori SUGIURA, Matsuji SUGITA

Seizo MoToRA,

Therefore a(O')=lfm(oo)apparent

whereni(oo)

is theEquating t==O at eq. (11),weget,n(i..)dw

m2(.)C.(Yi},2(.)

Thus,

size

Added

mo(to)

mass

darnping c(to) are

+

R,

R3-Rt

+

(V7-(V3

R2-R3+

toe-ct)3

wst)(COSto3t-COS

(V6t)

(16)

.iO)

+iS,OO.2(.)C.(?t,2(.)

･ sin wt dw

(19)

to

is obtained

followsasM*+rkv'=1"l}iocoln2(to)ca2+c2(to)sin tut
･

50

:6.o

'Eiu
's･o

If we

put t-O in eq. (20)and refering
eq. (14),
we know that the initialvalue
A
Lsi of ntv' is equal
to mo(co).
Another

ll
i/i

g

I

dw

ctit

(20)

'

equivalent
value

as

1

a6L

V(t) ==

C(a))

t
i-o

is the terminal velocity.

mlo)

--

S
La

v(t)

v(t)

7e

1.4
tAi

12

t

=v(t)

Since

inv'

a,Tt)

==z'lht)!if(r)dr(18)¢ os{vit)(cosct,3t-cos

rkv" as

m"

where

otdu

(est-COs(co$

WB-CV3massiit))(t>'o) (i7)==M'+fnv'=

3I,Oem2(w)glM,
lc,(to);lts[ (cosa,2t,,sR-O.,i

addedequivalentis the mass of the ship itself.
It should be noted that the initialvalue
ofntv" is equal to mo(oo).
In a same way, we
can define anequivalent
added
mass
nt,'
which
will be given in the followingformulaby strip method making
use of
Tasai'svalue of addedmass
and
damping
for each section. Results are as shown in
Fig. 1.' The integrand of eq. (11)is also
shownin Fig. 1 where we know that the
integrand ispositive function. In vertueofthe positivefunction,we may use an approximation
atinverse transformation of theintegrand (SeeAppendix). Resultisas follow

=

aJgt)

cv3t-cos

the

where

calculated

cos

(cos
f,iicR,3']m"+rkv"scale=1/58).
(L.,=2.0m,
and

toSt)

(14) definedin the Introductionwill be obtained
as follows
the 2ndNow, let us calculate numericallyterm of eq. (11)forthe Japanese
firstatomic
ship at model(COStu3t-COS(V5-Ct)3+=ll,co

mlo-)

R,-R,

+

at tu==infinity.

mass

of

added mass which
gives exact
absorbed
energycan be defined

follows

3o
?o

o4L

2o

e2t

le

oL

Oo

[tf(.)v(T)dT{`v(T)dT
JOI

g
the

ls

lo

mo(w)

firstatomic

2o

and

damping

Ship

of

(21)

The initialvalue
caCsed)b) In case ofa
Next, letus
coeMcientde))Japan at2s

rnodel

3oO

scale

-

=Ji

iv2(t) iv2(t)toFig. 1 Added massof

in*+iMv=

. as

shown

in

of ni, is also

input

ramp
considereq.

mo(oo).

a

ramp

input

(23)

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The Fourier transform of f(t)isdti(.)
1--EJ7,
I<w)=i'xd.
Since R(w) and I(to)of eq. (5)areIf we put s(t)==a(t) in eq. (11),g(t) should be
the terminal acceleration
for a ramp input.
Therefore,rk,"will be given by the follow(25) ing formu!a.

"12(w)ldS+c2(to).to

M*+ntv"

We get a(t) as follows
c(cti)

-i!co

(26)

sinmtdd,

be deformed .a.sfollows

m(to)to!zfi'(w)sintot(lv

!-.

Sin

to

tdtu=

`t)t

of

of

eq.

(33)isin the samemanner.

eq.

(29)is exactlyis the matter
This
(19).
of course
because a ramp
input tlZl(t)is an
integalform of a step input U(t)6). If we
know

side

(29)

d.

to

as

hand

we
know
equal to that of eq. (20),
as niv"that ntv' for a step input is the sameforaramp input. rkv' and niv for a ramp

+3I:m2(.i(?ic2(to)

.[o)

. sM

We know

right

mt(o)

Since the integrandof the lst term
is even functien, we get
(26)

same

Since theinput can be obtained-.m2(to)w2+c2(bl)m(tu)w6(w)(33)(27) exactly[o6'(ew)=-O(to)m2(.).2+c2(to)

a(t)=

･

mlo)+-i}!,com2<.)C.(gelc2(.)m(w)to!rr6'(tu)

will

co

=

si:ttotd.

sin ldtde)

m2(.).S+c2(to).

m!(to)to2+ct(.)

the 2nd term

=

1

to

Refering that;(32)=Ils(T)dT

m2(te)te2+c!(to)-t!:.

for t<T, range of integral
to t. Finally we get,(24)-c(tu)+m(to)wSxh'(w)a(t) ==t!--141g(t)=!t..f'(T)S(t-T)dr==Ils(t-T)dT-m(to)-c(to)an6'(tu)I(.)=

Ships in Collisions(23) becomes -oo,,,,=,.,,,..[?j

R(.)-

of

that a(t) in eq.

v(t) ofeq.

the response

ponse g(t)to an
as follows.

s(t) to a step
arbitrary

input,

J

Fig. 2 Geornetrical approximation

the function

c(to)/m2(w)w2+c2(w)res-input f(t)is given

of

The

equivalent

addedmassesntv", ntv',

rkv thus obtained
are functions of time.
Since the initialvalues for these three kinds
g(t)=IH(O)1･f(-oo)+!:..f'(T)T(t-T)dr
to moCoa),ifof addedmasses are all equal(30) the duration of a collision is very short, thenis a valueof the Fourier
Where IH(O)1
rkv",ntv', and rkv are practically equal tofunction at
spectrumofa Iinear systemmo(oo).
To exarnine the dependancy of thefrequency ca=O
to the duration,theequivalent
added masses
For the ramp input,
ntv"/m',added
mass
coeficientsequivalentofthe lst.niv'lm', and ntlm* are calculatedf(-oe)-Oin Fig. 3,4, and 5.
(3i) atomic ship as shownf,(.)=[?lofthese
figuresisthe durationand
Abscissa;:8

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142
ff1.4H-H

129thu8i.o${QenawQ

as2sg
a4im

e,2

oo

02

o
on14

O.6

O.L
1,O

1.0

o,e

4.o
2.0

1.0 t.Csec)
tsCset),･.

ao

5.0

40

3.0

3Equivalent
inv[7m'(=m)addedmass,,xFig.

'ts!
12-uLth8
1,e$!

oeflaao.6:g,,s8hl

tt

Q2

D

o

O.2

o

o

O.4
1,O

to

O,G

zo

LO

o.s

6,O

4o
30

4.e

t.tset)4 Equlvalent

tsCsec)s,erkvVm*(=x)addedmass,; ti[giFig,

x1,4
t'/s

7 t2N9Lth}.e8mveoeEogosa<-i04"{8tu
O.2
oo

02

o

o

O,4
20

T.Q

zo

30

Fig.5

es

O.6

1,O t.(sec.)ttCsetJt.-t,fgi

ao

L.O
40

so

Equivalent

rkvlm'(=N)addedmass

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EquivalentAdded Mass of Ships in Collisions
scale
and

time tm, fullscale time ts,
a non-dimensionalized
time t' are shownfor model

the reference.

asequivalent

addedexperiments

as

mass,

143
the authors

conducted

described in the following

sectlon.

A 2.0m modelof the lst atomic
ship
isFrom these figures,we can notice that all
increasesoitkindsof the equivalent added masses
connected
to a frictionlessthat isguideof theonly allowed
to sway (seeFig.6). A horizontal
as the duration increases. In casewasset on the model.
Astep input,the aforementioned
trend is mosttype accelerometer
folitsrernarkable.
For'any kind of input,thenylon
string,connected
to the mode!
atandlowing relation is-valid.
center of gravity, is lead horizontally
Tily"'> nty> ntv'

to aconnectedthat the model

addedweight.

with

be a･Iso noted that the equivalentchanges
itsvalue in accordance
type of the external
force.

3. Experiments

To examine

the theoretical valueof thethroughpulleyso

is pulled horizontallyby thestringis stretchedintheoppositeside
andis fixed to restrainmodel
to sway.
The principaldimensionsof
the model
are shownin Table 1. When the
fixednylon string is cut, the model would

It should

mass

weightAnotherTable 1 Principaldimensions of the model
usecl

at

the

experirnent

LppBDdriCbCpCw.CoG,pooB

2,OOO.Omm
327.6mm
227.6mm119.0mm53.0kg
O.664
O.672
O.826

O.988
O.804
Fig. 6 Frictional guidemeasurernent

of

39.6mmaft

lateralaccelerationfor

14

XQI, acot"12gg-10:-oge"o'H:.88E

"

,4.

Z/
･/
4

?P

o

O

02

O.4

Fig.7 Accelerationa(t) andequivalent

OS

O.6

added

mass

t(sec)

mv"lm'(==rc)

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Seizo MoToRA,

Masataka

FUJINO, Masanorl SUGIURA, Matsuji SUGITA

force and
are
also
shown
in Fig. 7 by solid lines.
The accelerationis There are slite differencebetween measured
recordedvaluesat initialstage,
butby an oscillograph. An example of and calculated
record with 1kg. weight'
isshown in Fig, 7. itcan be said that they showed reasonable
The accelerationagreement.
Speed of the ship v(t), as wellthus inducedby the stepasthetranslations(t) of the ship
are
also
wise
force by the weight is recorded
on anoscillograph.
Measured accelerationin caseobtained by integrating accelerationand theequivalent
added
mass
coethcient
based onof 1kg'weight is as shown in Fig. 7on time
the speed (i"v7m')andbased on the absorbed
basis. The equivalent added mass coeMcientobtained by measured
acceleraenergy
are also obtained
as shown
x (=ntv"lm')(i"vkn*)tion is also shownin Fig. 7 for different in Figs.8 and 9. Agreement with calculated
duration. Calculated values of the acceleravalues
werealso good.

be acted by a stepwise
be accelerated
to sway.

externartion and the equivalent

added

mass

coethcent7;gg.g-

RE,:t'8Un,vE:vvvE!ysev

G

g･s`

gg

O

02

Fig. 8 Velocityv(t) ando4-tcec)

equivalentD.6 added

rnassO.5 niv'lm*(=K)

x

A

E

e

t'

fi
s

ut

.

E

.

z
gv

f-

b-

.z
=
g

Fig.9 Translations(t) andequivarent

added

rnass

niylm*(=rc)

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Mass

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Ships in Collisions

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4. Estimation of the durationof collisions
duration is a function(37)shows that
of
only
the
speed
Since the equivalent
added
mass
increases
penetration and the initialof the striking
ship.
It is proportional toas the durationof a collision.
the penetration, and inverselyproportionala) In case of step input
to the initialspeed".
If a comparatively
Let us assume
the external forceisa stepship
deeplypenetrates a soft structured
wise
oneof magnitude
fo. Denoting the slow
ship, the durationshould
be appreciable.
durationT, the equivalent
addedmassesiizx,"therkvandsuMx

the striken

expressing

A andthe striking

by sub-shipby sub-sufixshipb) In case of a ramp wise force.
Replacing fo in (34)by f(t)=t,
weget

jl'(UB-v.)dt()

B, we can write the equationof motionsoftransverse motion
ofthe A ship andlongitudinalmotionof the B ship as follows:S==- 'iiS-

du.

(MH+ Mnx) dt
dv.
(MA+ rkAV)dt=+
infi. MAt

mBt

=-fo

(34)
=

=-5-

2

U.,T

rkAV

3S
2u.,

UhOT,'=

fO

(381'

we
know that for
Comparing (38)and (37),
the
same
S
and
the
durationof caseUko,
On the other hand, denotingthe penetratlonb) is 3f4 times the duration of case a).
of the B ship into the A ship as S we get
For instance,assuming
the penetration to
be4m,
the
initialspeed
to be 18kts, theS=Uh,.
durationof stepwise input isO.865sec.whileithe duration of ramp input isO.649sec. The
fbT2'+(35) equivalent
rkBxmA+"Jz.,)
added
rnass
coefucient
basecl ontheabsorbed
energy
for
both
cases
are O.57
where
UBo is the initial
speedof the B ship.
and O.47respectively
which
are fairlygreater
Then, if we denote the terminal velocity ofthen mo(oo)lm"(=O.382).
Values of equivalentboth $hips as U
at i=T), U can be

Sl(uB-v.)dt(mBl=(speedwritten

as

fOT
MD+

added

follows
fiIB=

fOT
UEo- U, MA+MAY

U

:=

(36)
Therefore substituting

UE]oT

S=2'

we
(36)into (35),

mass

U,,

get

ramp

mput.

wlse

5. Conclusion

ing the absorbedbyenergyathis paper, the authors
showedno time-independent constantmass.

coeficient.

Incollision.

that there isvalue

of added

However, it is possible tointroducean equivalent
added
mass
coethcientwhich
is a constant and will give exact valueof terminal valueof acceleration
when
it is
used

as

equation

the
of

added

mass

added

mass
of

equivalentmass

coeMcientterminal velocity
the absorbed

addedenergyin thecoerncient

In the same

motion.

equivalent

Lateral Corlisioncom-(1) Minorsky has used O.4 for the added(37) mass coedicient of a striken ship in calculat-

2ST=

1

Fig. 10to various

corresponding

binationsof the penetration and the initialspeed
are shownin Table 2 for stepwise andthe striken

senes,

anbased on
ship

and

an

based onbe defined.

coeMcient
will

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146

Table 2 Durationof collision
ship

'x.

(atactual

Penetration

Initial

Input
T

and

equivalent

added

mass

coerncient

X of the firstatornic

ship)

4m

2m

8m

6m

10m

Ramp3,89O.700
Step6.48O.9303.89O.935
Rarnp4.86O.8282.92O
Ramp
Stepl1.94O.5531.17O.448
Ramp2.92O,6831.76O.5261.25O.459O.975O.424O.795
Step5.18O.9343.12O,9052.23O.7631.73O.6
Step1,30O,562O.777O.460O.557O.504
RampO,975O,424O.583O.392O.418O.388
Step2.59O.8301.56O.6231.11O.522O.865O.475O.708O.449O.598O.435

speed

6

T

3.o.899352.o.347871.67O.650Lo.305621o.06.511(sec)
xT(sec)

2,34O.610

10xr

1.67O.514
2.79O.864

O,833O.410

.14

(sec)

18

O.649O.396
O.324O.385O.266O.384(sec) O.432e.415O.354O,409O.299O.403

22

(sec)

26

(sec)

xT
1.30O.4651.04O.431O.9ooO.416
2.16O,751

xr

1.77O.6701.50O.608
1.33Oa4681.13O.4

O.531O.390O.446O.389

xT

oo.898,480

O.225O,384

x(2) Thedefinedarecollision,

addedequivalentandfunctionsof the duration of a
the Ionger the duration theaddedgreater the equivalent
When the duration is zero,

added

equiva!ent

to m,(oo)which(3) The

mass
was

mass

coethcient.

allkinds of thecoethcientsusedequivalentcalculated

by superposition

of

based

on

is obtainedbyThe function Ri"(tu) whichwithrespecttofrequency
differentiatingRi(bl)
into, has four impulse functionsas shown
Fig. A-2.

coincideRi"(to)=by Minorsky.

speed,
acceleration,coethcients
added
mass

motionthus

massesofw2-(el

and

the

which

are

-

R
Ml-teS

linearequation

the frequency agreed{6(to-a)-j(ev-w2)}(A-2){6(ev-ws)-fi(o-or}

By the way, ifwe denote the inverseFouriertransform of Ri(w) as ri(t), -t2ri(t)stands forthe inverse Fourier transferm of R2"(tu).

This
results.
quite well with experimentalof equafact showsthat linearsuperposition
dependent
hasfrequencytion of motion which
is valid for this kind of problerns.
coeficients
AppendixSome approximate

R

Namely,-t2rt(t)=;!:Ri"(w)

cos

of dw

(A-3)

of calculating

methods

describedin a few Substitutingthe eq. (A-2)into this equation(16)
books6),')but for conandusing
the next relationship,
published reference
venience
we shall dare to describethe methods
thisappendix.
Let us consider the inverseincos wtdw=cos
.,t
(A-4)
Fouriertransform of Ri(w) of which the form
is trapezoid as shownin Fig.A-1.
wernay obtain the inverseFouriertransformcos(A-1) ri(t) as follows.
,r,(t)the eq.

werealready!,oob(w-loi)r;!,e"R,(.),,td,,NII-Electronic
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Equivalent AddedMassofShips in Collisions

147

Rflaij
R

o

tu4wh

co3

LLij

o

Fig. A--2

Fig. A-1

er@
Rvuo

o

R

o

di2

wtFig.

al

A-3

ri

(t) =

x2te

Fig. A-4

(cos
[{lan

'

(cosct)stHcosa)4R- m32
r2(t)=

ant-cosw2t)

an

wgt)

]

R

'

rrt2

.2-.1

(cos{vlt-costu2t)(A-8)
Sincethe functionc(to)!(m2(w)tu2+c2(w))

iscom-(A-5) posed of two trapezoids and five triangles,

Similarlyin case of R2(tu),of which the form
istriangle as shownin Fig. A-3, we canobtain the inversetransforrn r2(t).
r2(t)-;[,coR2(a,)cosotdto

we

can

obtain(16)byaddingthe desiredinversetransformthe abovestatedinversetransforms.
Reference

(A-6)

"An
1) V. U. MINoRsKY:
Analysisof Ship Collisionswith Reference to Protectionof Nuclear
has two irnpulsefunctions
The functionR2"(ua)Power Plants",Journalof Ship Reserch, Vol.
3, No. 2, (1956),
pp. 1-4.
R
KoCH:
Method forDeter2)R!"(o,)=J.J.
{ti(e,-a,i)-e"(a)-tt,E)}.

"Experimentala)2-tu1

VirtualMass forOscillationsof Ships
Ingenieur Archiv, Vol. 4, part 2, (1953).

'',

mining(A-7)

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Japan

148

Seizo MOToRA,

3) J.DIEVDONNE:

FUJINo, MasanoriSVGIURA,MatsujiSUGITA

Masataka"VibrationinShips'ZT.I.NA.,on(1959).

4) A.J.JOHNsoN:
and
all Riveted 10,OOO ton
Dry Cargo Ship",Trans. N.E.C.I.E.S.,
Vol. 67,(1950-1951).
"HydrodynamicForceandMoment
5) F.TAsAI:"VibrationTestsofallWelcled

Produced by Swaying

Oscillation of

the SurfaceofaFluid",

Journalef Zosen

Kiokai,Vol. 110, (1961).
6) A. PApouus:
Fourier Integral and Its
Applications'', MCGRAw-HILL,
7) V. V. SOLODOVINrKOV:
to theStatistical
Dynamics of Automatic Control Systems", Dover Pub. ING.
(196e).
"The"Introduction

Cylinders'

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