The rolling motions of cylinders of symmetrical sections at finite roll angles on the surface of a fluid

NORWEGIAN

SHIP MOOEL EXPERIMENT

TAN/(

THE TECHNICAL UNIVERSITY

_{OF NORWAY}

THE ROL L ING MO TIONS OF CYLINDERS

OF S YMME TR/CA L SECTIONS A T

FINITE ROLL ANGLES ON THE

SURFACE

OF. A

_FLUID

BY

T. SA BUNCO

N OR WEG iA N

_{SHIP MODEL EXPERIMENT TA NI( PUBLICATION}

_N°66

A UGUS T

₁₉₆₂

8

THE MOMENT OF HYDRODYNAMIC FORCES TIlE ADDED MASS MOMENT OF INERTIA

AMPLITUDE. OF THE OUTGOING WAVES NUMERICAL CALCULATIONS REFERENCES

Contents

16

19

20

21

24

page

i.

INTRODUCTION :1.

DESCRIPTION OF THE METHOD 2

THE BOUNDARY CONDITION ON THE CYLINDER 2

GENERAL COMPLEX POTENTIAL OF A MOTION CAUSED BY A CYLINDER OSCILLATING ON

THE SIJBFACE OF A FLUID

₅

THE FORCED HARMONIC ROLL MOTIONS OF AN ELLIPTICAL CYLINDER WHEN THE

FREQUENCY OF THE MOTION TENDS TO ZERO

13

e..

We consider the motion of a fluid of infinite depth,

which arises when a horizontal cylinder of arbitrary symmetrical

section performs simple harmonic

rolling

motions of finite amplitude about its roll

axis

which Is fixed in space

and

defined as the intersection undisturbed free water surface and

middle line plane of the section.

In this paper a two-dimensional method is developed

for determination of the fluid motion and the moment of

hydro-dynamic forces.

Calculations have been

restricted

to the

elliptical sections of

various breadth and

depth ratios when rolling frequency tends to zero.

1. INTRODUCTION.

In these types of problems it is generally accepted

that thè body is making some kind of forced simple harmonic

motions of small amplitude and consequently, it is permissible to satisfy the boundary condition on the contour of profile

when it attains its meán position. Therefore, In the subse.. quent analysis this supposition will be generalized and the

boundary condition on the contour will be satisfied at an

arbitrary

angle

of roll.

The sections considered here

are a

particular family of ship-like forms given by Lewis.

The

Lewis forms may be represented in

parametric

form by the

equatiöns

z=

_{(1a)co19#bcg39}

y.

(1-a)sIn2êbs.1Jr23&

or, 18 -16'

cty= e +ae b e

(1) where

o

O:?7

2 DESCRIPTION OF T}. )1ETdQD.

We assume that

The fluid motion is

incompressible,

homogeneous and inviscid.

The wave height is small in comparison to the

wave length.

o. - roll amplitude or the section is

sufficiently

small which allows us to make the following approximations

4U')2 9' = _and

2

(2)

d.

The motion is irrotational

and characterized

by

a complex potential

W()-_

_{.1y)Í '7"(,y).}

With these simplifying assumptions the roll motion

of a symmetrical section is a linear potential

problem.

We will develop, stream function with

the

help of a general complex

potential of the fluid

motion with

the infinite

numbers of

unknowns which will

be equated to the value of the stream

function on the contour or the section, that is, we will

expand both functions into the serles of a new

function ori

the

contour and equate the coefficients of the corresponding

functions,

which

process leads to sölving the infinite number of equations in the Infinite numbers of

_urilcnowna.

3. THE BOUNDARY CONDITION ON '1HE CYLINDER.

We wish to consider the boundary

condition

on the cylinder which is oscillating periodically in a roll. For this purpose wo intt'oduce two rectangular co-ordinate systems, the

first C'Y)- system is fixed in apace with the plane yo

in the undisturbed free surface and y, axis positive downwards, the other (z2,,y0)- system is fixed in the cylinder, such that

when there are no oscillations,

_x=

,

y=y

It ÇP

denotes the angular displacement at time t the relation

between two co-ordinate systems is given as (Fig. i)

-Y0$,incP

as tollOis

Fig. 1..

Suppose that a synunetrical òylln4er 18 rolling about Its

centre fixed in space, the angular displacement at time t

being ₌

¿4(Wt

where

7

is the roll amplitude, and S the phase of the

motion of the

cylinder. The boundary condition is

=

_F(t)-w

(wtS)(+.2)

₍₎

on the cylinder', where F(t) is a function of only.

Using the two-parameter family of ship sections, so-called the Lewis forma the equations of the boundary of the contour is expressed in non-dimensional form in the (x,y)-aystem

where

tc

_a,

-=

= (ita)

c,sO-b C0s381

-=

(-a)th-&2381

or

₍₆₎

i

-Z91

-3O

.°zy0

e

-i-ae be

Consider the

points of'

intersection of the

profile with

the undlsturbed,tree surface when the profile turned at an

angle of

CJ'

these points are given in

_the

_parametric

equations by

9 =-9,

and

= 17-

consequently, wetted

contour of the profile changes between the interval -e0.JT-e0

(I-a)?280-bsizz 390

(i+a)co80#

(33

where,

9, is now a,

function of 9'

From Fig. 1

the

following relation can be written

(7)

using the series expansions

of tangent, sine

and

cosine

functions

and arranging

as

a function

of 4° we may obtain

the following approximate formula for, small

s°

values

O

f1ab

ab r

(1_aZi2z_4ab_272)(l+a+b)J

(-"I

4-a-3b

1_a_3bL

_(f-a-3b)3

(8)

Introducing

6,=8-O

then the wetted contour of the profile

changes

in

the range

0

O ÌT

, substituting this, into

equation

(6)

and then afterwards into the boundary condition

given by equation

(5)

we obtain

'= _íwAftz(wt4e)[(1#at#b2)* 2a(1+b)c2(8-O)2bc4(O-60)

F(t)

(9)

Put

90

and subtract the new equation

from (9),

thus

eliminating time depending

constants

-cc487

Or

= - fw

wt-)

f 2a(4+b)(c.S2 (c2r2O

-i) .sh2

+

L

In order to solve the problem we

_must

_{expand all the functions} into the sei'ies of a new function, in this respect preferably into the series of sine-functions,

vanishing

_at

₀₌₀

_{and 0=77}

Making use of Fourier series expansion we get

[cs(2P8)f]=

q6p2

&'fl(2k-l-1)0

_{O0< 7T}

k=O (2(kP)1J(2(k-P)i][2k-1/

consequently

Tk0

_(3J(2k+1JC2

k-1) 00 _(2k#))8 J[cos480

f2k.t5J(2k/J[2k-3J

where

One may observe

from

expression (12)

that

_Esczki)9J

are even functions of

O

,

while

[Siv2 28J

and [s-&'h0J

_are

odd functions of &

In

the

interval

O 0 ¡'T

This

consideration

is

important because

stream

function obtained

from general complex potential should have on the boundary even

and

odd parts.

4.

_{GENERAL COMPLEX POTENTIAL OP A MOTION CAUSED BY A}

CYLINDER OSCILLATING ON _{THE SURFACE OF A} FLUID,

We suppose that a symmetrical

_{section is}

_constrained

to oscillate sinusoidaily at frequency

_w

_{The system of}

°co-ordinatea

_{is fixed in space With the plane}

yo In the

undisturbed

free

surface and

y axis positive downwards.

The condition on 'the frée water. surface is

+ = (13)

when _y=

where v=

Let

_W(.)

be the complex pötentialof a motion

excited by a.double section in an..unbounded infinite fluid. It is known that such a tLinction

AJ()

which is

finite,

continuóus and single-valued', at all points of the space outside of a circle having the origin as ,còntre

can

be expanded in

the form

00

t't)= W()W(3)=

-.

/çe

aI4J!

-#. C4Zcwt ki()

('15)

whe re a "

are the

constants

to be determined

from

the

boundary condition on the cylinder, and where the motion in

all directions at infinity ceases. To satisfy the free

surface condition We must add a suitable new Í'unctión

_{W( )}

to _{W(3). .} Let us take, the first term of the expansion

of 'i.() , this represents, a horizontal doublet of Strength

a1

situated at

the origin in ax unbounded fluid. In order

to 8atisfy- the free aurfacC.'cond'ition we mustadd a new complex

'function. Th solution of, the problem may be obtained by

using different methods and is tound in thé related literature, for examplé,. following (reference 11, page 72)'

r

i

.

4í_,,/e.

_{dk(snwEJt1e'CSwt.(14).}

(3

Singularities of higher

order

may be found' by differéntiating

(li)

wIth respect to

_{Therefore, the solution of the}

The integrals in

the

expression (15) can be reduced to the simplest

form

by dividing the nöminator of the integrand

by its denominator

and

after some arrangements we obtain the

complex potential of the motion0

Thus for 'n

zm+i

w9 Go

()=&nwtZA

[1

ii,

[m:i21

27flI for

=m

0O,

kl '(s) = SWE]

A'

[-L

''

IL... ¿k

¿t

2m 32777 (J+

o

-vyr

Jki

o

f°

r-(k-)ikx

d,/.

oth

=

Goi (qe) -,-

cas'wt

LV0()

(17)

The merit of writing the complex potential in the above forms is that each term in the square

parentheses

satisfy separately free surface condition, where now the

A2m,

and A2m ' _are

_new

_{unknown constants. The functions}

W0 ()

and

will be determined from the radiation condition

and from the

boundary condition

_{on the cylinder.}

The evalutation of the integral occurring in the complex potential.

The following identity may be easily proved

cO

.dk

0O o +i112 '77 L

The first double integral term on the right of

equation (18) can be evaluated; first, by integrating

_with

respect to

d

_{and then expanding the Integrand into}

_power

series, and integrating again term

by' term with respect to y

s

namely

or

f ¡

(k-V)cW.#ÍkX

df e

ak

_{= - e}

₊

₊

£ (-'.'

_-

_Cix)

_{+t.& (v.g)}

'n1

'7?fl?

(1.9)

where C is Euler's constant.

In order to define the second integral term on the right of equation (18), we are considering the contour integral of the following type

where

£='kím),

for ..x

>û

we choose the contour

shown

in Fig. 2. In the limit when

'fl

_{radius of the circle}

f

-f

increases to Infinity and

+l")

the radius of the circle

/

tends to zero, we obtain the

(

r

following result from the

contour integral

FigS. 2

r

e

Jk-v

o

Combining the results obtained In (19) and (20) we

have

00

?

f e

_{dk= e}

[c+(v2)Z I

_{( iV3)-t(}

j k-v

n=i Thfl!

2J

In the limit when tends to Infinity we obtain

00

_[k-Y

L1m

.e

_{-a'k.. lue}

(22)

It is well

known

that at a distance of a few wave-lengths

troni

the cylinder, .the. motion on each side is described by a single regular wave train travelling away, from the

cylinder. In order to characterize the progressive waves at

infinity thé complex potentials must reduce to the following

fonns. for

2.rn#i

0?7d tor

7?:2fl7

¿Û.13.-wt)

4

_'W()=

illh14 e

(23) ' ¿('

W ()=-?7A

e

2)

I

This considération' and the other two boundary. conditions, namely free surface condition' and boundary condition òn the

cylinder, may suggest the font of the functions

_{v14 ()}

and

vv'(3.).

arising in the equatioñs (16) and (17), Therefore,

the following 'potentials have been tried to solve

the

problem

for

n 2.777+1

cswtfßL1!_

Tm)

1ff

Ji

(25) for

??22fl

t1;' (s.)

=

&n.w

JA2m(

im

_(:?4)

_32ñ1]

Ae1'3(ct

£

'j*17! (-

iv5'-[im(m(:1) 321]

V (2s)

Expression

given by (25) IB suitable to characterize a .motión which is unsymmetrical with respect to

x=o

'and

expression (26) is suitable for a motion symmetrical

with

respect to

(Z.O

. It we would satisfy the

boundary

10

instead of rather at an arbitrary angle of roll, then complex potential given by (25) would be a proper function to give

the fluid motion caused by a symmetrical section which is rolling. Consequently, velocity potential (25) gIves the

first approximation to the fluid motion excited by a rolling symmetrical section. For higher-orders approximations, the condition on the boundary given by equation (12) suggests the following form for complex potential of the fluid motion

caused by a rolling cylinder.

(-00 oo(n)

i'2wZ4oZA (1

'ty

f

4

_{(±) ilL}

lfl:I 2*I 327fl1

_{Th P'}

_{7:l ?flî}

₂

r,0ç)

(0)

4C4W)Z ÇDZ B

J ¿uvA e ¿ o

n,:

2r»1 2m 2fl? I

+

.s4?wqz sP

¡4

£_

_1

j_e

(c#.nv+Z

roo 'n ('n) (f) 1Y øo

(j1j

¿ 'il:! 32m

(2'n) 32fl7.l

p#

(

(9;) (1)

h'j

i

coswzL)Z coz

B

r-i-

iV

7M e

¿

?7:/ 2Ti 32m(zm_,) 2n1-/] o

(27)

('vi) _(4q) ("n)

where

A2,,,,

,

ß,,,,,

A2,,, and are infinite numbers

of unknowns which are to be determined from the

boundary condition

on the cylinder, and where

Suba ti tuting

-Z'c'O-,) -t'g-80)

-3t (9-),

=(#'y)=e (.x0+)e (e

--ce

i

Into

%()

_{given by (27) we get boundary value of the}

complex potential on the cylinder defined _{as the Lewis tonna.}

Independent of ¿

Coefficients of

2 't t,

the imaginary

part of the complex potential, that is, to

bring

the general stream function on the boundary to the

_saine

form with

the

boundary condition of (12) we put 9:0 _{on the}

general

stream function

and subtract the new equation from

the

first, namely

(29)

equating (29) to (12) we obtain the equation

''(9)= 7('R)

(30)

from which infinite sets of unknowns are to be determined. In doing this, equation of (30) must be arranged according to the powers of and equating the coefficients of the same powers of ¿ we obtain the equations of successive

approximations, namely

All '3't,,(6) include trigonometric functions ofe

which are symmetric with respeot to 9=? , namely

[(-8)s2koJ

_,

[c2ko-1J,

((-8)cs(2i..)e-

_{J ,}

LS(21)O3.

These functions can be

expanded into odd sine series in

the

interval O O 7

with the help of Fourier's theorem.

_On

the other hand, all

'3Ç,(9)

include trigonometric

functions of

e

antisynunetric with respect to

9=

-in the -interval

0

17

, namely

(e)

,

2ko],

t(Î_e)&,.(2i)oJ ,fcos(2k+,)O- _(-o)J .

Except

sine serles In the

range

O ¿9 17

Let us 'take the first

equation. öf (31). This relates to the first-order

approxi-mation and the infinite Set of

unknowns

areto be determined

from 'the known function '1Ç(e) ' , that is from the boundary

óondltlon defined when the profile takes its mean position.

For

this

case the system of equations can be obtained by. equating the coefficients' of the Successive Sine, functions,

namely

. . tO) rn (0) (0) (0) 12

A+

A' .---

ii_L- Ja(1+b). _._/

«q,If

_{°3 3}

_1,5

_{17c/ 3 .}

₁₅ f coeff.: , of sine

°A° JÁ°

' = ..L

coeff.

of

31ii39

.3,1 _{3.1 3 3.5 S C0Sj} (0) 'ai

(0))

₆₎ A° - - - -

f1-L i-L?

coeff

ot sin cO «5,1

i

5.3 .3 .55 5 CO j '337 7.3.31 0) ca _c.i (0) 1°) ' i

.f( 1 b).

_{4 b}

i941B+,313B3/5i.----

7__._

co) (0)

where '

ánd

,a

are

the

functions

of

coétf. of sin

(o) IO) (0) (0) io)

+/3,38+/3358 -- - - 4

j

7is

coeff.

_{'of Sin3O}

to) (6) (0) (a)

, )

and every unknowns,

(o)

A2_fl?4I

The second approximation can be obtained from (31) by taking the coefficients of namely

'Ç(9)=

_c9) and

arranging

as the coefficients of (ii_e) and

@I-n2kê)

from which process we obtain again infinite sets of equations

equation Includes two sets of infinite number

(0) (ô) (0) (o)

namely

f

,q213 _{and where}

((JJ

and

_277J+/ are to be defined from the infinite

sets of equations associ.ated with the first order approximation given by (32) With the above given explanation, any desired degree of approximation can be achieved. It has been seen that thè evaluation of the unknowns A and B _requires the solutions of various groups of infinite numbers of

nknwns0

The number of groups of the unknowns increase with the order of approximations.. Even, for the first approximation,

to find

the unknowns Is a laborious process ±nvolving in

principle the solution of an infinite number of equations; in practice a finite number are sufficient. For slow motions, that is for small V values, this Is satisfactory, but for rapid motions it Is known that the process requires large

numbers of equations, which have to be solved to a high degree of accuracy.

5.

THE FORCED HARMONIC ROLL MOTIONS OF AN ELLIPTICAL

CYLINDER WHEN THE FREQUENCY 0F THE MOTION TENDS TO ZERO.

In order to apply the foregoing results to simple case and to see the. effect of the higher-order approximations

on the fluid motion we cônsider an elliptical cylinder

performing slow harmonic rol]. motions. We suppose that

may be taken as zero, which supposition leads to replace

the free water. surface by a rigid plate

In the limit when

V

tends to zero the general

complex potential given by (27) reduces to the following

fr)= W',547

-

1k

-(.fl)

()

A2m7

q4/

32mJ

-:3-(33)

On the other hand from equation (12), putting c

we get the

boundary

condition on an elliptical section which is obtained from a unit circle by transforming it. Thus

=

-qwA;nw

L COS2 ,Sin (2 _{± So?? 29 .SI'fl2O}

if

_{ko[23J(21JC2kuJ}

e

4s -(2n1+2

sí)n(2ko)

(36)

(34)

where 6 is given by equation(8) as the calculations

testricted to the second-order corrections, we

are

neglecting

the cube and higher powers of the roll amplitude therefore

9=

p=it0'E,coswt

For brevity omitting the intermediate calculations, the form of the stream function and velocity potential on the boundary of ellipse for successive approximations may be written as 9)

-awç &

'(9)

x)= -aw&'22w

' c,

s((2ì)

7J Ob ₇

D(l)

' p+I

p,

- W

_flwt1

_{:' (-i)}

(2fl7+P_1) _{5dM [2(m-tp)9j +} p.0 .4.

=û'"741

(_1)1(2r4P)a'j(2mI)_(2m+2p/)QJ.

v (2m42P1)2

-

15

-cv (0)

a

Pf-1 _p

)

+ A2m 1E E-i) (2m-4-P)a

((2mf-/)-(zm+2p,)

_{_]sn((2r172PI)9J}

-

(8)

2'

w(2)

CO P.4 p

=

-aw

A

(2n1+P)as.L(2

_pi)eJ

7,7=0 + _{in ]c'n7+P}

f

.t.

+

L

71)a'°f(2m#;)...(2 ?fl 2 P41)

)V? f(2m+ 2PI) ej

= (39)

cìs w[

f(_/j'2

n P)

a'°c,os((2m#2P1-I)J

cv(i)

P?'.' _p

42A2m')

(2flP_I)C(?(n7#p)aJs2(n-,p)9j+

cv (o) P..i p z. +

.o'2mZf')

(277+P)a ((2n/)-(2m.2P+!)Lt

where

( vs'

_PI-

-

m.(nli). ...(nS)

P!

From the expression

(35),

one may obtain an infinite

number of equations in infinite number of unknowns from which

any desired number of unknowns related to the first order

approximàtion can be defined

correctly due tö

the forms of the

equations, this

may be seen well from the following

(0)

-A = L!

f

I

'T 3.1.(-i) /6 3 -n-(o) (0) (oj 16

i 75.3

(o).

(8)awcocs1jnwt

2!±_S4.fl2G

(3 7) w P

--P(YdGY#

dx)=

f

ìe

at

. (43)

Using the same process, other sets of infinite numbers

of unknowns associatéd with the higher-order apprcxirnaticns

can be determined from the expressions (37) and (39).

Therefore

the problem is

solved.

6. T}. MOMENT OF HYDRODYNAMIC FORCES.

The pressure at any point in the fluid is given by

Bernoulli's

equation

where F(t) is a function of time to be deteniined from

the

condition that at the

free

surface: the pressuré is ¿ezo and at

the infinity on each side of the cylindér regular

waves

are

travelling away. Hence withOut loss of generality the

variable part of

the pressure is

3t

(42)

and the, moment of the hydrodynamic

forces about i'he:prigin of ellipse may be written as

substituting the values of

given by (28) into (143) we get

(44) o

where.

is .to be taken from the equation (36),. (38) and (1o)

for Sucoessivè approximations. CAfter Integration with respect

to 0

the part

of the total moment contributed by

_OgC)

can

be written as

M.

I

-

17

-2 2

kw(o)

p p

,1= ga flcswlL E

(21

_ØJfrii,Z'

A.

j)(2fl+ )a

k.O

i-a

(2)/

o2m/p=of2m+2p.3]Em-2p_,J

(45) where

f

A direct method of evaluation of (45) might be the

calculation of any number.of unknowns from (41) and inserting

thése constants into (45). However, instead of this,;:we

will follow a different procedure. It may easily be. observed.

that in the equation system (41) every röw, that is every equation has been

obtained from

(35) by taking

?= (.ntP)

combination as a constant Integer. If we òalculate double summation of (45) by taking

4î=(97'P)

as constant;integers

and varying fl from zero to infinity we get the expressions

which are

equal

to the left hand side of the equation system (41) and every row Is multiplied by a constant of 1

-L2#3J(2fl-1J

Using the right hand sides of

equátionsystent (41)

we obtain immediately the value of t10 _as

2k k _i..'

"o=

.2P1COsw E(2

c,oj

?k)! (27?+3)2(Zfl+))(2V-/)2

1

L?aPr{c4swt(1-

()2

(4e)

2

(zn3) 2(2fl!)(2 )2

After integration

with: respect

to 8 , the other parts of

the totalmoment contributed by

and

can

be

written, as follows oo (0) o P+lp

-

2a2f'7' c4s2WtS1/122 ±ß A c-o a

r

(4rn-p+2)

i(2rn2P#3J(fl7+2P-/3

(

8)

since the first two

-

18

-r

p

f (2)

(_,)

(2mP)

a

M2=

ga2prq(coswt 3c otJcs(2_cp)

ZA2m,

_{(2rn+2P+3)(27fl-2P-1)}

0

(0) P1 p

ita

12

1 Z A

_{2 inO 2 mi p=0}

2 (-i) (2mP)a

_{P (2m 2P 33(2 m 2P- I]}

((2rn#i)-(2m4-2P#I) vf

(49)

Following in principle the saine procedures as employed in the calculation of M0 , one finds after. sorne long and

oomplióated manipulations the whole moment of the hydrodynamic forces calculated through two approximations, thus

00 = a'°

I53G-ç Zt

¡-a

iio =

PO/Ç(k_p)3Jf2(k_P)_IJ L2+31[2-1 - (1a)(7-ga) 2

(,_a)Z

(2r3)(2r_7)

co

2r+3)(2r_/)

0

2

s-i

r =16

q

,

a

2 a

i-a

s=a(2r#:+2s(2rl-2s,)

P012(7_P)3]L2(7_P._I]

L2r3112r_1JJ

r f Z + 512

r

2 a ¿j (k'ì) S

3/

_{ko P:4f2(k-P)#3)[2(k-)-IJ}

L+3(24)J r4(s)!(2k1)

143t(2kd.l)2JJ ?E2I2521_2s3

ro(2r+3)(2r+1)(2r-I)

(2r3)(2r_ì)

484+0

p ¡

-

(iv'

i _P)+33f2(r.aP)13

(r3](.r_I3

2

ir

, a 72 ( (2k-t/)S ) Jf4(5-t) fs.2i*i)'JJ r+I+2cL2'-zsJJ

(50)

In the above we have given the moment of the

hydroa

dynamic forces for an ellipse obtained by the transf'oxnation of the

unit,

circle. In order to apply the

reSult

to the

£ _I

(_3a)(1a)

4-1-tlZ

_(4-a)

21

p

s a

_2L±.g J

2r,4JI2r25J

f-á

lt

ellipses defined by their major and minor axes, we should

transform a circle of radius

R

and multiply the expression

of (50) by R4

. Using the parametric formulas

the major axis of ellipse

R(i-a) B

the minor axis of ellipse

.2

from where we get

=

(si)

Therefore thé moment of hydrodynamic forces experienced by an ellipse of any breadth and depth ratio may be given in the

following förm'

222

2.

2

' [(

)coswt

t,

ec's3wJ

The aböve formula corresponds to' the value of the moment calculated through two approximations from a general formüla. of the type

2

22v

2fl

(k-B

_1z

7T o _'27k

It

Is known

that in

an unbounded fluid the hydrodynamic forces

and moments experienced by a moving body are proportional with

the acceleration of that body at the instant consIdered. _When the body is on the surface of a fluid this Is no longer the

case.

Even the motion is time-periodic and the wave motion

also periodic, the forces and moments on the

body are

compli-cated functions of

wt

.

This fact may be 'well represented

by the formula (53) even under the assumption that the free

water surface may

be replaced by

a rigid plate..

TI ADDED MASS MOMENT 0F INERTIA.

(53)

(52)

. The coefficient of COSwt in (52) gIves the added

mass moment of inertia per unit length of an elliptical

20 -//

= t'

2) 2 ¿ 2 IT o:,1 1.O J ₍₅₄₎

Fìrst approximation to the added mass moment of inertia for two parameter family of Lewis foins, when the frequency of the roll motion tends to zero, can be obtained by using a

similar process as employed in the calculation of t40

The result is 4b 'I

-

q

r

q a(1-b) i q _{nO (2m#/)}

_I

_{(2m3)(2n7-/) + (2m-3)(25)]}

(55)

where

B the half breadth of the Lewis profile

D

the depth of the Lewis profile

The above given result can be appl.ed to an ellipse of axes 2A and 2B by putting in the parametric equations

Thus

J"= .Q:

pc

I -

fc4

ç ¡r

inO

-

₁₇ (5e) 'Z

/A2 r2

where C

(10

In order to see to what extent the added mass moment of inertia changes with the frequency of the roll motion we

have obtained the ratio //

(57) lT2

which pictures the condition on the free surface

c;ae

and

CxO)=O

respectively.

THE AMPLITUDE OF THE OUTGOING WAVES.

2

A point of interest may be drawn to find the amplitude of the progressive waves at infinity, prom equations (23) and

(3)

the velocity potential of the progresáive waves may be written as

-

21

-_,7va,we

,fl(ì1WZL)

since the elevation of the surface

-í

s

at

therefore ' cQs(L1.X-wt)

where amplitude of the waves at infinity

for small frequency ratios; 1found from equaticn system (111)

might be satïsfactory for calculating the amplitude of

waves2

hence

_2

2= --ac°V

the mp1itude of the waves produced by an ellipse of -xes 2A and

28

can be written as

,=

2

where

(f42

₈₂₎

and

the above result may be generalized for two parameter Lewis

forms, namely

i 4b 7

;=

(8-D)2(8D)L-52(1b)]

3

where 8= the half breadth of the section O = the draft of the setion.

9. NUMERICAL CALCULATIONS.

To see the effect of the second order approximation

to the hydrodynarnic moment, calculatIons were made for ellipses

of different breadth and depth ratios by using the formula

(50). The variation of dimensionless coefficients and

of formula (52) with breadth

and

depth ratios is shown in

figure 3, from which one may easily conclude that discrepancies

between the results obtained by using the first and second order theories increase with the breadth

and

depth ratios. For

zero breadth, that is for a vertical plate first order approxi-rnation is sufficient and the higher order approximations

22

-contrIbute

nothing

to the value of

hydrodynamic moment.

If O

is the depth of such a vertical plate the added mass moment of Inertia is

'I

7çp_

-i-

(5g)

In order to make a clearer

comparison, the added moment of inertia calculated

according to

the first and second order theories has been plotted on a base of various breadth and depth ratios for a roll amplitude 1O. _{The results are} presented in (figure k).

- 23

of' ',o;

H

/1

Scq/e oc'J(0)

...-L . .

'-a.

'.7'

. --...

:«.

-

...:;L11..

-

L

r.,".: .' ,_: 4 . . t 'IT '3 . t1 : II

o

f I . .. 4 ,

..

. . . . --.--.--. .

. ..

H

-..-.-..-.,--1 . . . : .

. j.._..

-t , -» ----. . -Tt

L

r

1T1J1

1H

j II

141

_1]TITr

4I4I

-Grim, O. Grim3 O. Grim, O. . Grim, 0. Ursell, F. Ursell, F. Ursell, F. 21 -References.

Berechnung der durch Schwingungen eines

Schiffskörpers erzeugten hydrodynamischen

Krf te. Jahrbuch der Schiffbautechnischen

Gesellschaft, 1953, 277-295.

Durch Wellen an einem Schiffskörper

erregte Krf te. Proceedings of the Symposium of Ships in a Seaway, 1957,

232-265.

Die durch eine 0berf1àchenwe11e erregte Tauchbewegung. Forschungshef t fUr

Schïffstechnik, 1957, 98-100.

A Method for a more precise Computation

of Heaving and Pitching Motions both in Smooth Water and in Waves.

Third Symposium on Naval Hydrodynamics 1960, 271-326.

Lewis, F.M. The inertia of the water surrounding

by a vibrating ship. Transactions of the Society of Naval Architects and Marine

Engineers, 1929.

Mllne,Thomson, L.M. Theoretical Hydrodynamics, second edition.

On the Heaving Motion of a circular Cylinder

on the Surface of a Fluid. Quart. J. Mech. and Applied Math.,

19k8, 218-231.

On the Rolling Motion of Cylinders in

the Surface of a Fluid. Quart. J. Mech. and Applied Math., 19?8, 335-353.

On the Virtual Mass and Damping of Floating

Bodies at Zero Speed Ahead. prcceedlngs

Symposium of Ships in a Seaway, 1957,

- 25

References

Weinbium, G. Contribution 6f

Ship Theorytò the

Seaworthiness Problem. Symposium Naval Hydrodynamics,1956, 61-98.

Wehausen, J.V. Water Waves 1.

University of Californiá

Institute of Engineering Research,