Hydrodynamic force and moment produced by swaying and rolling oscillation of cylinders on the free surface

(1)

Reports of Research Institute for Applied Mechanics Vol. IX, No 35 1961

HYDRODYNAMIC FORCE AND MOMENT PRODUCED

BY SWAYING AND ROLLING OSCILLATION

OF CYLINDERS ON THE FREE SURFACE

By Fukuzo

TASAI

Ill. Conclusions

References

Appendix

Summary

Isso-dimensional values ot the hdrod namic force and _moment prod-iced by swaying and rolling oL;cillation o the suracc of a _{fluid were} exactly caleul,tted for cylinders with Lewis-form seclion.

Added m.tss coefficient Kr. progressive _{wave height ratio .4, the inertia}

coefficient of momnt K..- and th coeluiccnt of the da!npinz moment _(rs of swaying oscillation, and also the coetlicient of added moment of inertia Kr, wave height ratio .-1, the coefficients of swaying force KR, _of rolling oscillation are shown _{¡n several tables and figures.}

introduction

When a cylinder floating on the surface of a fluid oscillates horizontally (swaying oscillation) it suffers hydrodynamic force. This force is _{approximately} resolved into two components: I) the added inertia force in phase with the horizontal acceleration and 2) damping force in phase _{with the velocity.} _{In the case of} swaying oscillation of cylinders, since the _{motion of the fluid} _{is not symmetrical} the abovementioned force generally _{does not} _{act upon the center of gravity of}

CONTENTS - Introduction

1. Swaying oSCillation

1.1 Progressive wave height 1.2 Added mass

1.3 Rolling moment 1.4 Results of calculation Il. Rolling oscillation

2. I Progressive wave height Added moment of inertia 2.3 Coefficient of added moment 2.4 Coefficient of s\vaying force 2.5 Numerical calculations and swaying of inertia TECHNISCHE UNIVERSITET Laboratorium voor ScheepshydromeChaflica Archief Mekelweg 2, 2628 CD DeIft Te1 015-786873- Fax: 015 . 781838 force 91

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1'

i

I

i

92 _{F. TASAI}

the cylinder, so that rolling moment will be produced. This _{moment becomes the} coupled moment by swaying in the coupling motion between swaying and rolling.

These hydrodynamic force and moment are generally the function of frequency. O. Grim [lJ calculated the force and moment two-dimensionally on the three elliptic cylinders. Then he showed graphically the added mass and progressive

w2B

wave height ratio as the function of e =

-

2'

where a is the circular frequency

of swaying oscillation and B is breadth of the cylinder at the free surface.

O. Grim [2] has dealt with the added _{mass, rolling moment and progressive} wave height of cylinders with Lewis-form section in the neighbourhood _{of &i*O.} On the other hand, making use of several model ships _{S. MOTORA [3] measured} the added mass and added moment of inertia of _{swaying and yawing motion in} case of

@*.

_{But the above-mentioned data will not be} _{sufficient for calculating} the added mass, added moment of inertia _{and damping force of swaying and} yawing oscillation of a ship, by the strip method _{for example, as a function of} frequency.

The author treated of the added _{mass and damping force of heaving} oscilla-tion of cylinders with Lewis-form secoscilla-tion in [4].

In this paper, two-dimensional values of the added mass, progressive wave height ratio and rolling moment produced by swaying oscillation of cylinders with Lewis-form section were exactly calculated, applying above method.

The problem with respect to the hydrodynamic force and moment of a

cylinder rolling about the origin O could be dealt with in the same manner as done for swaying, and then the two-dimensional _{values of added mass moment of} inertia, wave-making damping and swaying force _{produced by rolling motion} were also obtained.

Making use of the hydrodynamic force and _{moments created by swaying and} rolling motion of the cylinder we are able _{to discuss the coupled oscillation} between swaying and rolling motion of the cylinder.

I. _{Swaying oscillation}

1. 1. Progressive wave height

In Fig. i let us take Cartesian coordinates with its x-axis taken horizontally

Z- plane

_plane

(3)

¡1}DROD}NAMIC FORCE ANA) MOMENT PRODUCED BY SWA YING AND ROLLING OSCILLATION OF CYLINDERS ON THE FREE SURFACE 93

on a still water surface and its y-axis vertically downwards and the mean position of the axis of the cylinder at the origin. It is assumed that the depth of water is

infinite, the fluid is invicid, incompressible and the fluid motion irrotational. Suppose now that the cylinder oscillates horizontally with a small displacement

X. -Scos(oH s) and velocity U-- So' sin(o1 s). As the cylinder is infinitely

long the motion of the water will be two-dimensional. The motion of the fluid is not symmetrical about y-axis and then the velocity potential has the following relation;

ç5(x,y)

(--x,y).

(1)

Linearised free surface condition is expressed as follows:

at (y=O

x:»).

(2)

The boundary condition on the surface of the cylinder is given by the next

equation

(u()

(3)

where ', is the outward normal of the cylinder surface.

As the amplitude of swaying oscillation is infinitesimally small the equation (3) holds, to the first order, at the rest position.

In this paper calculation was performed on the Lewis-form sectioa. which can be derived from the unit circle in the c-plane by th traiíformation

ZM(4.+').

(4)

where

z=x+iy,

M

.

ç=iee'0.

Putting a=O in the equation (4) the contour of the Lewis-form isixpressed as follows :

Xij M(l +a1)sin8a3 sin 3(1,

(5) yo--M(l--a1)cos() f-a3cos3O.

In consequence of the transformation (4), free surface condition reduces _to

'X -g

3 -3g \

-aie

a3e

)

o (e.-

± (6)

Using the stream function _{the boundary condition on the surface of the} cylinder reducQs to

(tÇt

-- )

u

(X

_a)au

UMR1ai)sinO±3asjn3O.

(7)

Therefore we obtain

"(0) _{UM ( I--a1) cos í)+a3 cos 3fi +C(t),} ₍₈₎ where C(s) is a function of the time only.

(4)

.

g'

[V(K, x, y)cos cot+ !V(K, x, y)sin@1]

c==7re cos Kx

s=±esinKx

J

kcoskY_KsinkyC;kdkY

} +sin wt2QZ)fl(eB)[ o (12) where K==w2/g and ' is wave amplitude at infinity, and in equ. (12) the upper sign is for x>0 and the lower one for x<0.

Then we consider _{which satisfies the basic conditions in addition to the} horizontal and vertical velocity being continuous at x=O and

_y>T

_{and that it}

has a simple sine progressive wave at infinity.

Suppose the stream function expressed as follows:

i,ír =_{[!rc(eB, a, O)cos cùt + !F(E, a, O)sin wt]}

' gi I

cos 2mO

+COSCútP2,n(ei:)[e_ m+l)mcos(2m±

1)8

1 + ai +a3 i 2m

a 3a e_(2m+4)

-±

_{cos(2m+2)O 82m+0s(2m+4)0i}

where !F, have the sign given in the equation (12) according

_{to x0.}

Putting a=0 and using the condition in eq. (8) following _{equation is obtained.}

fr\

( J'fr(8)=co(eB,

_{\gl)) )}

O)cos at+!Fso(&B, O)sin út

±COS_{(Z)t2P2m(ei)[_cos(2m±1)O} +atcos(2?71±2)0

1+a1+a3( 2m _2m±2

i

:

94 F. TASAI

(a, -O) and the free surface condition (6).

-2+

2fl%=[e_m+sin(2rn+1)0+ Ï ±a1+a31 2m 2m+2

3s

_4)Ot1coswt _(m=1,2,3 ). (9) Jjsin z

Then the sets of conjugate stream functions are expressed as follows:

a1e2'2'

=[e

'cos(2m + 1)8

I+ai+a31 2m -cos 2mO + 2m +2 cos(2m +2)8

2m±4a! e_(2m+4cos(2mf4)OlC9S

)jsint

(m =1, 2,3

)

(10) These sets of functions tend to zero as a tends to infinity. As a stream function representing such a nain of _{waves at infinity we adopt the function of} a two-dimensional horizontal doublet at the origin. _{This stream function is given} by the following equations:

(5)

HYDRODYN4A!JC FORCE AV!) A'fOAIENT PROIJUCED BY SWA YING AN!)

ROLLING OSCILLATION OF CYLINDERS ON TIlL ERLE SURFACE

3a3 _{cos(2m±4)011} 2m+4 , {C0s2m0 ±a1 cos(2n±2)0 +Sifl(,Jt Qm(8)r_COS(2fll±1)0ii a3 2m 2m±2 In =I 3a3cos(2m+4)0 2m+4 (\gr

)UM(la1)cos 0-a3cos 30 ±C(t),

(14)

where !V and _!Vs) are the values at ar-O.

Putting O = we obtain the C(t). Thei1 substituting this equation of C(t) _in

(14) we have

[CI(R, 0).-_!Fei(n,

_)]

cos

()t±[.sa(, o)F.0(e,.,

)isin

fCOS 2mO a1 cos(2m±2)O

COS t 1 Pm(ii)[_cos(2m±1)0

1+a1+a31 2iz

- ±

2m-+-2

3a3cos(m2+4)01 _{±e,:(__1)m+I(} i a 3a3 "

2m4

1 l+a1±a3

2m2m±22m+4)J

(cos 2mO a1cos(2,n±2)O

+sin ot _{Qm(a)L_Cos(2m±1)0}

1 +a1 --à 2in ± 2m-r-2

3a3 cos(2m±4)0 fe8(_i)m+i( _i _a1 _3a3

2m + 4

f

1+ a1 ± a3 2m - 2m +2 2m ±4 Ii

=(\UM(1_-ai)cos 0+a3

cos

3û.

vi)

The right side of the equation _{(15) can be expressed as follows:}

(

'UM1(1a1)cos

O+a3 cos

3O h(0)P0

cos t±Q0 sin oij (16)

4 ,

where

gi)!

_';

¿t:

-h(0)=

J(1a1)cos0+a3cos 30

I ta1 --a3

7rS

rS

nsin , _r ,cos e

7)

Substituting (16) into (15) we get the following equation

'('(B, o)--PF(, ) f.m(iì) P'm !J'»,(',, O)-where

Jj1(0)h(0)

Jr 2 m

''

_jJnik I-UÇ (11i\ TI'!.tnz

I

(15) (18)

and f(0)

=-cos(2m+ 1)0 + jcos 2mO a1 cos(2m±2)0 3a3cos(2m+4)0L

1-fa1al

+

2m-s2 2m+4 J

I

(6)

96 F. TASAI

+ _{1+a1+a3 2m}e8(-1)m'J i

_{2m+2 2m+4}

a 3a3

(19) Assuming that the series of (18) converges uniformly

in the range of a>0,

-

and performing the numerical calculation by the same method as done for heaving [4] we can determine the Pm and Q:rn.

Using Po and Q, the amplitude ratio A is given by the following equation:

A

_{- S}

_/p02+Q02 20

In the case of the circular cylinder, when is very small P0 tends to zero

and Qo to respectively. Therefore it is resulted

into Air2.

(21)

1. 2. Added mass

In the next place, the velocity potential corresponding to the equation (13) is given as follows: a, 8)cos wt- a, 0)sin -. ± (e e m+)sLn(2m 1)0+ i sin 2mO 1--a1+a3' 2m 3a3e_(2m*4) +

_2m2

sin(2m+2)O

_2m±4 _sin(2m+4)0}i ±sin (a)t

Q() -

. fi (e

[

e m+I)lfl(2fl,+ 1)0 sin 2mO

m=1 I +a±a3 ( 2F??

a1 e-2 3a3 e (2m-l-4)a

sin(2nz ±4)0)i (22) + 2m-12 sin(2m±2)O- 2rn±4

where

1».____,.ß-KU in k'r

± eKVcos KxT cos ky±k sin kY Ádk

± K(2

}

o

in which the upper sign is for x>0 and the lower one for x<0. Flydrodynamic pressure on the surface of the cylinder can be calculated by the linearised equation

P- P)

Then the hydrodynamic force per unit length acting on the cylinder in the direction of x-axis is found by integrating the horizontal component of the

hydrodynarnic pressure on the surface of the cylinder, and the force becomes

F=B()[N0 sin tM0

cos t] ₍₂₄₎

provided that

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IIYDROD}'NAMIC FORCE AND MOMENT PRODUCED BY SWA_{YING AND} ROLLING OSCILL4TION OF CYLINDERS ON THE FREESURFACE 97

N0---

_{Iø((B,al,a3,O)----}

dû

(1+aI±a3) _1±al±a3 4

ea(l)»rÇ

I a1 _3a3

'

_{(F-f-a1 +as)2L4n2_ I - (2m+2)2-1}

_a1)

(

1

a1 _3a3

-±3a31409

_{± (2m ±2)2_9}

(2m+4)2-9

-M0 is obtained by converting into ø _{and P.m into} _Qm _{in eq. (25).}

«D are the values of ø, ø, at a --O, and p is the density of water.

The hydrodynamic force Fr is resolved into components in phase with the acceleration and with the horizontal velocity.

The added mass M. is the ratio of the the acceleration to the acceleration,

M 2 T2

.-

_p 2NOPO±MOQO

H0

- p2

_Q

where T is the draught and JJ= The added mass in case of

r-, C2

=[i

16(:3)2]

Therefore the coefficient of added mass

Kf(co)

_becomes hydrodynamic force

is given by Landweber [5J

(.oc)

C2

K.(tco)

- =

í,So irFfot,

According to M for a certain w _{in eq. (26). we have K.,.} K _- 4H2 N0P0±M0Q0

¿r _{P02 + Q} r

K

H0N0P9±MOQ0

(7 _P02_{± Qo2}

i is the area coefficient, namely, S0

1. 3. Rolling moment

Afs(wco)

K (w)

Expressing the coefficient in the ratio of added _{mass to the sectional} we obtain i 4OC2

and K:

(25) area So, (30) BT

In the case of (i,O, that

_{is, when the period of swaying oscillation} _is

extrenly long the free surface reduces to the similar boundary as that of the íìxed wall.

Then the added ma'.s is, _{as is well known, given by tite following formula:}

M.('O)

prTC3, C3 . ₍₃₁₎

(8)

98 p. TASAT

In the case of swaying oscillation, the symbols of the hydrodynamic pressure which act upon the right and left sides of the cylinder differ from each other,

so that the rolling moment is generally produced outside of Fr. Letting MR denote the rolling moment about origin O, we have

where

YR can be obta ed by converting (D into ø

and Pm into Q.

MR is resolved into two components in phase with the acceleration and velocity of swaying.

That is,

MR=pB2 [XRsin Ci)!- YR COS O)t]

jr

'R

a1, as, O) a1(1 +a)sin 20-2a3 sin 4DldO Ja

(l±ai+as)2

\

ne(aIP2a3Pj

₊ 8(l+a1±as (33)

MR=M(_)+Nsc(_J).

(32)

In the above equation, the first term is the inertia moment and the second is the damping moment produced by swaying oscillation.

Both are the coupled terms in the coupled oscillation between swaying and rolling oscillation of a cylinder.

Put

-3

Xp+

Ms4,= MsFsQ= MçKs,T. (34)

l'ben the ratio of the lever 1s to the draught leads to

I34

- 2H ('oXR + Q YR

Kb_T -

O

N0PfQ0

In the next place we have

pBco(P0 YR QOXR

2 Po2+Q02

On the other hand, the two-dimensional wave-making damping force per unit length of a cylinder and unit swaying velocity, that is, the coefficient of linear wave-making damping of swaying N becomes as follows:

NSA2.

-Putting now

(35)

we have

As mentioned above, the calculating method for Lewis-form section was given, and it may also be extended for the section of n-parameter family as given in case of heaving [6J.

i

N. = NTa8,

QOIR).

P27,1(e)(-1)' ( 2a(l +a5) 8a3

(9)

!J)DRODYNA.UC FORCE AN!) MOMENT PRODUCE!) BY SWA Y/NG AND ROLLING OSCILLATION OF CYLINDERS ON THE FREE SURFACE 99

1. 4. Results of calculation

In the equation (24), the component in phase with k is

Q0M0P0

F,,- ß(7)No (P0 cos r,,I± Qo Sin ùt).

t ±Qo

The rate of work done by the above force per unit time leads to

- (I) M0P0)

which equals the energy transmitted from both sides of the cyjinder per unit time, that is, Accordingly the following relation is given,

.4

¡.2

1.0 0.6 0.4

0.2

N0 Q M1P0--=

Ellipse

A.M

+J"tT2

I I I

02

04

0.6 0.8 Fig. 2. (39)

ANt\

viÇ

_(o

H0

0.4 H0.2

Ho:0.2.

.-

_o-0

H0=2/3 I

H - i.q7

H0=1.0

'VP

4m-I K (c-.00):

/H

L1.

_I _¡ - - _{Values by GrimYH.'I.0)} i I

IL

H0=I5

t I 1.0 1.2 ¡.4 ¡.5 (38) H0=I.5

KA

(10)

loo F.

The relation was used to check the computation. Making

use of the same

method as done in the heaving oscillation by F. Ursell [7} we can prove the uniform convergence of the infinite series on the right-hand side of equation (18).

To obtain the coefficients P2,,,, _{Q,,,, it} should be solved using an infinite number of equations in an infinite number of unknowns. As a practical problem, the system of equations was replaced by a system involving only six polynomials f2m(0) as done in the case of the heaving [4J.

¡n this paper, calculations were carried out for fourteen sections of Lewis-form. The error was less than 2% in the case of Jf-0.2 and _{1% in the other sections.} In Fig. 2, K and A for five elliptic sections, H0 of which respectively equals

(,)2

0.2, 0.4, 2/3, 1.0 and 1.5

are given as a function of ç=

_{T, and the dotted}

line is the result by O. Grim [I]. Though he also calculated _{for other two elliptic} sections, H0 =2/3 and H0 _{1.5, we find a little error in the neighbourhood of the} maximum value.

Recently O. Grim and K. Tamura calculated again and obtained these _values for Lewis-form sections (unpublished).

As seen in Fig. 2,

_{the maximum of K and the OE,} _{corresponding to the}

maximum increase with the decrease of H0.

In a small A is _{larger in a section with larger H1, but this tendency} reverses with the increase of c.

H0=1-5 o- --O. 78 54 E---C-0.9469 Qs;::

njC

A C O 02 0.4 0.6 0.8 i.Q Fig. 3. 1.2

(11)

HYI)RODYNAMIC FORCE AN!) MOMENT PRODUCED BY SWAYING AND

?OLLING OsC!LL.4rIoN OF CYLINDERS ON THE FREE SURFACE 101

1.8 ¡.6 '.4 ¡.2 ¡.0 0. 0.6 0.4 0.2 H0=I.O Fig. 5. 6295 O--0=Q.7954 A-L7Q.94O4 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Fig. 4. H0= 2/3

o---0-06230

o---ç--0.7854 0. 9463

<J

=fT

Jì1

Kx(Wc0)

K(w-)

.4; 0.6 0.8 ¡.0 ¡2

¡4

02 0.4

(12)

102 F. TASA!

H0-O.4

Fig. 7. 0a0.6323

o-zO.7854

£...OzO.9377 O. 0. 92 52 o-- -r 0. 7854 0.6440

(13)

HYDRODYDAMIC FORCE AN!) MOMENT PRODUCED By SWA Y/NG AND ROLLING OSCILLATION OF C YLINDERS ON THE FREE SURFACE 103

Generally speaking, with respect _{to the Kf arid A of swaying oscillation the} variation with H0 is smaller than that of the heaving for the same magnitude of In Fig. 2, A of H -=0 is the _{one for the flat plate (See appendix). The effect}

of the area coefficient _{i in each H is shown in the Figures 3 to 7.}

_4o

_{.fíee',, (((}

As clearly seen in these figures, IC. for fuller sections are larger than the

--ones for fine sections till e10.6 and then the matters _{are adverse with increasing}

of &,. And also the fuller the section is, the larger _{A becomes.}

Results of the calculation are also given in Table l-1 to 1-5. 0. Grim [21

has given the A in the case of _{tending to O by the following equation}

--

r(1a1)

2 A= _(1+ai+a3)2

Rewriting the above equation _{as a function of}

Fig. 8.

(14)

104

we have

The equation (41) gives a good approximate value till e4_=O.2..O.3. A formula

of K, in the case of d-° is also given by O. Grim

[2].

4 12

K

I(w'O)

16aj111)+ a3a1a3+

2}±

_{5a3 Ïa32}

° _T

_{- 3r}

_{(l_aj)2l3a*9(T_ai+as)}

(42)

lt is shown in Fig. 8. for =0 shown in Table ¡ are that which were obtained from (42).

With regards to the K4,, as4, of two elliptic sections with H0=2/3 and 1.5, O. Grim gave his calculated values in Bild. 31 of [1], which are in good

agreement with the present ones. KsQ, a84, for full and fine sections are also

shown in Fig. 9 and 10.

_/

0.5 0.4 0.3 Ks,0.1 -a, -G3 0.4 a o o.' o -a, -a F. TASAI A==

L I

Fig. 9.

In this paper, calculations were carried out in the range of -<1.0 except for the circular section. From these calculations it was found that the variation of Ks,, as, with is generally small, and that in the elliptic sections aid in the ones with samli H0 the value of Ks, nearly equals as,.

11.4 Q463) JH.s'I.O(0 .09404) JH.-t.5(.a46'9) I I (41) 0.10.2 0.4 0.6 0.8 ¡.0 '.5 11.-02 (o-0.9252) H.Lf*r.a9463) 11.-IO (0-0.94 04) 02 0:4 O6 '.5 I I (o-a94a9)

II

t H..0.2(O. 0.9252)

(15)

Table I-3

Table l-4

H0l.0

Ho1.5

0 0.? 0.2 0.4 0.6 lO 1.5 2.0

'-e 0 0.2 0.4 0.6 ¡.0 ¡.5

c-06295 Z O 0.0260.0990.3320.5740.857 0 0.1330.267 0.4 0.667 ¡.0 o K.0.85I 0950?.0240.9640.7680456 0.423 A 0 0.0670.2340.43? 0.746 jç,O.0940O980 103 0.?0? -0.10? 0.102 Kx 1.0 ¡.208 ¡.2 0 1.025 0.609 0.405 -0.132-0.146 -0148 -0.152-0163 07854 K,,-Q530-0535-0530-Q524-Q556 0.7854 Z 0 0.032 0.123 0.417 0.686 ¡.056 ¡.365 ¡.605 -0.529-0515-350? -0486 Ka 1.0 1.163 ¡.248 I.? 41 0.813 3352 0.222 0.183 0.405 Z 0 0.0860.3080.5640926 1.20? 0.9404 Z 0 0038 0.159 0.540 0.865 ¡.23? Kx ¡.355 ¡.673 ¡.668 ¡.333 0.570 0.239 0.446 ¡.272 ¡.393 ¡.677 ¡.407 0.777 0197 0432 °9469M3fI35 -0.153 -0.142 -0.140 04 86 -0276 10.178 0.162 0174 0.174 0.180 0.19? SAs? -0210-0.185-0.16? -0.128-0.110 6$? OIS? 0.163 01700.1790.184 H0= 2/3

N.

0

.

o 0.05 0.? 0.2 0.4 0.6 co 0 0.075 0.15 0.3 0.6 0.9 cc 0.6230 A 0 0.050 0.194 0.58! 0.9?? k 0.87? I.00l 1.100 0219 0.85705620.4:8 0.220 0.196 0217 0.208 cLs 0.180 0.174 0.163 0460 07854K A 0 O-015 0.0600.236 0.705 ¡.029 Kz ¡.0 ¡.088 1.196 ¡.287 0892 0458 0.405 0236 0234 0.23? 02260217 0.215 0.233 0.23? 0.225 0.210 0202 0.9463

¡

0 0.0730284 0835 0.867iJ540.424 1.157 K ¡.219 1.478 ¡.539 ,c 0.323 0.309 0305 Q32 I 0.309 0.3000.30402950298

Ho 0.4

O' 0 0.05 Ö.? 0.2 0.3 0.4 oO 0 0.125 0.25 0.5 0.75 ¡.0 00 0.6323 Z 0 0.03! 0.122 0.459 0.797 Ki Q.9070.998j.055 ¡.015 0.714 0.4!? Ñ,Q343 0.324 0.289 0.275 0.254

4'!

0.32? 0.274 0.259 0.237 0.7854 Z 0 0.0360.1500.575 1.149 ¡. 1.137 ¡.272 1.185 03500405 0.364 0.377 0.358 0.327 0.290 0.3650.3440.3?? 0.270 Z O 0.cl 0.1750.661 ¡.315 K 1.13 7 1.289 1.498 1.271 0.2300.413

0.9377O387 0373

0.3620.342 0.33? 0382 0369 0.3 42 0.330

Table l-I

Table l-2

(16)

106 F. TASA! -H..0.2 (0.0.6440)

r L[ï

H..0.4 v.0.6323)

Ij--

_{"-H..-3 (0'. 0.6230)} t..0.2(O. 0.6440) (0.0.6323) ¡

y4104

a o., 0.6 0.8 IO 6295) I _.L_._ I i i i I Fig. IO.

Table I-5

II. Rolling oscillation

2. 1. Progressive wave height

In the case of rolling motion, suppose now that a cylinder performs rolling oscillation about the origin O with a small angular displacement _{O.-Oocos(wt±r)} and velocity O=-Ooosin(ct+r) in the clockwise direction (See Fig. 11).

The fluid motion, in this case, is quite similar _{to the swaying oscillation, and}

H0=O.2

C'

.

0 0.05 0.1 0.15 d 0 0.25 _0.5 _0.75 00 06440K'.

¡

Q 0.126 0.5370949 0946 1.223 1.2980811 0.407 Ks 0.401 0351 0.356 0343 0342 0.33 7 0317 Q 754

¡

0 0.135 0606 ¡.5O K. ¡.0 1.350 ¡.390 0.760 0.405 ¡ 0.407 0.383 0356 0343 0.374 0.3430.325 09252Kx

¡

1.0671.454 l.430Q6600.4070 0.155 0661 ¡.122 V'.sç 0415 0399 0.373 0347 Sf _{0.390 0.360 0.333} 0.5 0.4

ií T

_i 0.3 02 0.l o 0.1 02 0.6 -a, 04 i I I I 0.3 0.8 1.0 0.62 95)

(17)

H )DROD YNAIIC H)RCE .4 NI) _{MOMENT PRODUCED BY SWA YING AND} ROLLING OSCILLI TJON OF CFLINIJERS ON THE _{FREE SURFACE} ₁₀₇

I--equation (1) holds of course on the velocity potential

_{in the rolling. The}

boundary condition on the surface of the cylinder is given by the following equation

(\_ (d8dr

accordingly

f\ fdo\a çl

_S)=1(X0 +Yo )j

2 2

where is the velocity potential and 1r the stream function of the fluid _{in rolling} motion respectively.

Therefore on the surface of the cylinder we obtain

i_-

(,xo2

f-y02)±C(t). (44)

By the similar way as done for swaying oscillation _{we obtain the following} equation which corresponds to (14):

(wr).(0)

_gij

=V(e,

O)coswi+ !Vs;(OEjt, O)sin (i)t

+COS (i)t _{P2m[_cos(2m+1)O_} e {COS 2g

1+ai±as

2171

a1cos(2m+2)O 3a3 ccs(2m+4)O

2m+2 2m+4 e _cos2mû +SIfl(útjQmLCOS(2flt+l)O _1+aj±a3{ 2m a1 cos(2m+2)O _{3a8co(2,n+4)O} + 2m+2

-

2m+4 X (43)

(18)

108 F. TASAI

=_()()(xo2+yo2)

+C(t) (45)

in which !F and !F

are the same ones as given in (II) and (12), and

ì is of

course the amplitude of progressive wave.

By some reduction, the right-hand side of the equation (45) becomes Br

_{p(0)sin(at+) +C(t)}

where h,==B50/2

and u(0)=[(1+al2+aa2)-2al(1+a3)cos2O+2a3 cos4û]/(1±ai+a3)2. (46) Putting o

in the equation (45) and eliminating the constant C(z) we

obtain the following equation:

h4Jt(0)-1Sjn(ot+r).

Put

ji(0)i=.g(0).

Then the right-hand side of the equation (47) reduces to

g(0)P0cos o1+ Qo sin oi1,

where P0=

_{h1sinr, Q0-=---h,cosr}

Finally, we obtain !V(eB,

_{0)_lV((eB, )=}

2 ,,, + (49) o)!VQ(eB4) nO provided that

fog(0) =a(0)-1

f21COS(2fl1+ 1)0+ +a1cos(2m+2)0 3acos(2m±4)0

1+a1±a3 2m

¿a(_l)m+f i

a1 3a3

i±1

2m _{2í-i-2 - 2m+4}

From the equation (49

(50)

oo

({!V(B4O)!Vc(eB4)}+

nl-I±al±a31 2m

P..m[_cos(2m+1)O (cos2mO a1cos(2,n+2)O 3a3cos(2m+4)O

eB(1)m' I

a1 3a3 ₁

2m+4

fÏ

a(2mm+2_)J)COt

+ 2,n±2

¿B fCOS 2mo

1+a1+a31 2m

a1cos(2m±2)O 3a3cos(2m+4)O Çß(__I)m 1 a1 3a3

i

+ _2,n+2 _2m+4

I

(19)

HYDRODYNAMIC FORCE AND MOMENT PRODUCED BY SWA Y/NG AND ROLLiNG OSCILLATION OF C YLINDERS ON THE FREE SURFACE 109

VP,,2+Q02-Therefore we have

e

A---

-- hr

2V'p02 ± Qo

When the values of P0 and Qo are obtained by solving the equation (49) the progressive wave height created by the rolling motion is given, accordingly the amplitude ratio A is found as a function of or c,,.

2. 2. Added moment of inertia and swaying force

Hydrodynamic pressure on the surface of the cylinder is given as follows: =

([ø, 0) + 2P..m{sin(2m± 1)0 +

_f a1sin(2rn+2)0 3a3sin(2m+4)0ì i 2m-2 - 2m±4 )çJsinwt ç sin 2mO

o)±1Q.m

{sin(2m+ 1)0±

i+ai±a3(

2m a1 sin(2rn-f 2)0 3a3sin(2rn+4)O)}Ï (52) 2mf-2 2m+4

Hydrodynamic moment M,, in clockwise direction and hydrodynamic force Fi.' in the direction of ---x are evaluated using the above pressure (52).

That is,

M - -

2J

p (x0dx0 dyo

do +Yo do ) do

These are rewritten:

M+ - pgB2?; [X,?Sifl (1)1- YR cos

ir

[N0 sin (,)t---- A'!0 COS (i)t]

pgB

ir

(51)

where XR, YR, N0 and Al) are the same equations as were used in the swaying. Therefore added moment of inertia IF: iS

Ih_PB4(QOYR+PDXP)

8 (p02+Q02) '

The component of moment in phase with Ô is

QOX,POYR_{(Qo sin (1)1±P0 cos} _r)pgB2" (56)

By some reduction as done in the swaying we have the following relation which were used to check the computation,

}

(53)

}

(20)

110 _{i. TASAI}

P1iY,: Q0A

8 (57)

Letting N, denote the ¡ine wive-making damping which is proportional to the rolling velocity, N, leads to

N=A2(.

(58)

and

2.3. Coefficient of added moment of inertia Added moment of inertia in case of

ú-for the Lewisú-form section, that is,

pzß'

a12(1+a,)2+2a39

"°° 16 _(1+a1+a3)

For a ellipse it will be

pirÇ/B'2 ₂ 2

161\2)

On the other hand, added moment of inertia in

_{case of O is obtained by}

O. Grim [21.

pir/B\4r128

ai2(1+a3)2±aias(1±aa)+

16

= 8)

L2

_(1+a1±a3)' _]

For a ellipse it will be

fi {(B2T}

and then we have the following relation for the elliptical _sections !,o/I'. =16hr241.621.

Generally, for the Lewis form sections

_1,,>I,.

₍₆₁₎

B4

Writing the ratio of the added _{moment of inertia to}

8

(2) and KT

the ratio of that to

pT4 we obtain

16 a12(1+a3)2+aiaa(1+a3)± 9 a3-(1+ a1+ a3)4 8ai2(1±a3)2±2as2I

K,(cu)--In general for a certain w

16 _QOYR+PDXR

KQB

= -.

r

p2i2

I.

K,= KßHo4

is given by Professor Kumai [8]

(59)

(60)

2.4. Coefficient of swaying force

(21)

FIYDRODYNAMIC FORCE AND MOMENT PRODUCED BY SWAYING AND ROLLING OSCILLATiON OF CYLINDERS ON THE FREE SURFACE 111

in phase with i?.

l'ha t is,

/ d20 _/ dO\

Fz'FisÇ'

dz2) -f-NR.s'i

_d»

For the FR and NRS we have

pB3 (Qo?vfo+P0N0) FRS ₈ _{02 ±} Qo2 pwB3 (poMoQ0N0) NRS 8 p02 ±_Qo2 Put _Fk.='A /'kS, 'RS KRS T. From eqs. (55) and (66), we get

KRS=2H

OpNQ).

In the next place putting

NRS X 'RS NR, ¡'RS = aR5 T

we obtain

H0

a&=2J

_(P0M0Q0N0)

Accordingly, we can evaluate the coefficients of swaying force in rolling oscillation by the following equations:

NR

F,.=

- -

and N,= ---g,.

(71)

"kSl

aR5.,

In the case of (o*O, KRS is given by O. Grim [2], and it is as follows:

6[aj2(1+a)2± aia3(1+a)

+ i96a321

Kftç

_{- ir[a1(1a1)(l±a3)±a1a(1+a1)xO.6+as(1ai)xO.8-1.714as2](1ai±as)}

(72) For a ellipse it reduces to

KF:s__6(11)2.

(73)

The formula (73) is the same one as given

by F. Ursell in [13]. Á(-0)

given by O. Grim [2] is also

A=16 3 a1(1+a3)+ 5a (1 +a1 +as)3 i

j .

(65) (66) (67) (68) (69) (70) (74) 2. 5. Numerical calculations

For the ten Lewis-form sections numerical calculations were carried out. In Figures 12 to 14, A is shown as a function of &.

The coefficient of added moment of inertia K, is given in Figures 15 and 16. General tendency of K,,7 with respect to is quite similar to the one of swaying oscillation.

(22)

112 0.6 0.3 0.3-Fig. 13. H0=10 F. TASAI -Fig. 12. o.e ai 0.2 03 0.4 0.5 0.6 07

(23)

HYDROD YNAM1C FORCE AND MOMENT PRODUCED BY SWA YING AND ROLLING OSCILLATION OF CYLINDERS ON THE FREE SURFACE 113

0. 7 0.4 0.2-V t o .4 1.2 s. 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.! -o.9252

I

Q7854

I

iru

.

.0623. O7854 Ø 9404 p.0.62 Fig. 16. O! 2 0.3 04 05 0.6 07 0.8 0.9 w G! 02 0.3 04 0.5 06 0.7 0.8 Fig. 15. &., 9252 440 Q0.7854 ¡-dQ9463 6230 Q 7854 I.. H0= ¡.5 1.6 12 1.0 Q

(24)

-2.0--1.8 1 6 1.4 --I 2 -L0 -0.8 -0.6 -0.4 -02 02 - 0.2 -06 O, 02 0.3 04 C.' 02 03 0.4 0.5 0.6 0,7

H0I5

9469

Im---Fig. 17. H0=IO 0.9404

i

b 0. 6295 Fig. 18. 0. 78 54 05 06 07 114 I. F. TASAI 06 C.4

(25)

HYDRODYNAMIC FORCE AND MOMENT PRODUCED BY SWAYING AND ROLLING OSCILLATION OF CYLINDERS ON THE FREE SURFACE 115 Generally speaking, the values of KRS differ considerably from the a,,.. In the case of Jf=O.2, however, the both values are fairly near.

Numerical results are also given in Tables 2-I to 2-4.

Wave-making resistance of rolling motion of ships was investigated in detail by Hishida [9}-[1O}. First, the problem is dealt with as a two-dimensional wave problem to develop an approximate solution of wave motion produced by

0.8 0.6 0.4 0.2 0.6 0.4 0.2 0 0.1 0.2 03 H0=O.2 04 05 06 07 08 0.9

J-

.

___L,

-=-- 0.64400 7854 0. 9252 O Ql 02 03 04 05 06 0' 08 Fig. 19. 9252 1.06440 O Q 0.2 0.4 0.6 ¡.0 d 0 0.1333 0.2667 0.4 0.6667 s 0.78 A 0 0.0229 0.0755 0.1398 0.2321 K?T ¡.2656 1.4671 ¡.3922 1.1527 0.8120 0.7776

KR$ -o5967-a32 -5700-0.5583 -0.6381

dR, -0.5122 -0.4970 -04864 -0.4650

0.9469

0 0.0116 0.0347 0.0511 0.0531 K,T ¡.4291 1.467g 1.4286 1.3745 ¡.3279 1.3770 KRS -1.9441 -1.5056 -18390-3.1599-23.5290 dAS -0.200C-0.I660-0.1278-0.0873

Table 2-I

H0

¡5

(26)

O'-e 0 0.1 0.2 0.4 0.6 1.0 s d 0 0.1 0.2 0.4 0.6 1.0 0.6295 0 0.0136 0.0476 0.1102 0.1221

KT 0.1574

0.1711 0.I700 0.1600 0.1088 KRS -0.4906 -0.3896-0.4218 -0.4158 ORS 0.1396 -0.1428 -0.1967

0.9404

0 0.0063 00260 0.0925 0.1564 KT 0.3513 0.3700 0.4003 0.3815 0.3035 0.2432 KR$ 0.3885 03274 0.3326 0.3800 0.4922 01630 01623 0.1704 0.1823 0 0.05 0.1 0.2 0.4 0.6 0 0.075 0.15

03

0.6

09

0 6230 0 0.0135 0.0519 0.1415 0.1976 K,T 0.2342 0.2795 0.2910 0.2548 0.2218 0.2070

K.

0.3462 0.3461 0.3291 0.3744 04585 O(s 0.1808 0.1718 0.1568 0.1427 07854 A 0 0.0053 0.0207 0.0793 0.2260 0.3173 K,7 0.2502 0.2711 0.2871 0.2988 0.2106 0.1321 0.1542 KRS 0.2653 0.2613 0.2584 0.2541 0.2623 0.3037 oCts O2324 0.2282 0.2208 0.2113 0.2044 0.9463 0 0.0347 0.1328 0.3895 0.5297 K,7 0.5568 0.6427 0.6745 0.4079 0.1928 0.2127 KR, 0.3526 0.3436 0.355 I 0.4 395 0.7409 oC,s 0.3101 0.3100 0.3059 03034 116 F. TASAI

rolling of a body floating on a water surface. Using the conformal transformation he obtained the amplitude of two-dimensional progressive waves of various section formrepresentirig ship's section.

Comparison between the amplitude ratio obtained by Hishida and the present one will be performed inthenear future.

Table 2-2

H0 =1.0

(27)

HYDROI)YNAIIJC t)RCL AN!) .fO!tIF,\TT PRODUCED BY SWA YING AND

ROLLING OSCILLATION OF CYLINDERS ON ¡HL IRLE SURr1CF 117

Table 2-4

H0=O.2

Ill. Conclusions

Two-dimensional hydrodynamic force and moment produced by swaying and/or rolling oscillation were calculated for the Lewis-form cylinders floating _{on the} surface of a fluid. We could gain the following conclusions within the limit

from the results mentioned above.

First, with respect to the swaying oscillation

As regards K and A the effect of the variation in H0 is smaller than that of the coefficient of added mass and amplitude ratio in case of the heaving.

A, in case of the same H0, of the fuller sections are larger than the ones of the fine sections. This is opposite to the case of the heaving. Within the limit 5==O.6, the fuller the section is

_{the larger the K}

becomes, however, it has the adverse tendency beyond this limit.

The formula of given by O. Grim [2J gives fairly good approximate values till 4=O.2-'--O.3.

As for K

and a

the effect of the variation in is small. Second, with respect to the rolling oscillation.

The tendency of K40T with regard to , is quite similar to the K.

The values of KRS differ considerably from the a,.

Throughout these works, the writer is very grateful to Messrs. Tomioka and Arakawa for their help in numerical calculations as well as in presenting the manuscript. 0

0.05

0.1 0.15 00 d 0

0.25

0.5

0.75 0.6440

4

0 0.2481 1.0142 1.8705

K T

0.7083 0.8202 0.8200 0.5683 0.4656

KRS

0.4627 04348 0.4322 04664

QRs

0.3940 0.3836 0.3756

0 7854

4

0 0.2 716 1.145 I

20000.

K,1 0.7498 0.9400 0.9400 0.5380 0.4624

KAS

0.4584 0.4430 0.4451 0.4772

¿RS

0.3923 0.3897 0.3803

0.9252

0

0.3369 ¡.3260 2.1645

1<_?1

0.8500 1.1098 1.0019 0.4838 0.4890

KAs 0.4585 0.43990.4428 05.015

0.4248 0.4100 0.4022

(28)

where T and Ki(d) are the modified Bessel functions, and Ll(ed)

2.0 ¡.6

¡.2

cta

05 ¡.0

2.0

3.0

vX2!12(ed) ± K12 (e4)

m=or(m+3')r(m+

) (See [14]).

As tends to zero in the equation (i), A tends to This is the same

with the approximate formula which results from putting a1 = - 1.0, a3 =0 in the equation (41). In Fig. 20, A is shown as a function of .

2.0

4.0

5.0

6.0

7.0

8.0

Fig. 20. 0.05 0.1 0.2 0.3 - 0.4 0.5 - 0.6 0.7 0.8

A o.o

001670.0720.175 0.3330.5380.7650.975 1.142 0.9 1.0 1.5 2.0

3.0

4.0

5.0 6.0

8.0

1.265 1.3551.5971.7341.897 1.9671.991 1.999 2.0 118 F. TASA! Appendix

The progressive wave height produced by swaying oscillation of a flat plate with draught T can be obtained by solving the integral equation after the method developed by F. Ursell [11] and [12].

That is, the amplitude ratio A is given by the following equation:

(29)

HYDRODYNAMIC FORCE AN!) MOMENT PRODUCED BY SWA YINU AND ROLLING OSCILLATION OF CYLINDERS ON THE FREE SURFACE 119

References

[1 ] O. Grim: "Berechnung der durch Schwingungen eines schiffskörpers erzeugten hydrodynamischen Kräfte ". J. S. T. G., 1953.

[2 1 0. Grim: "Die hydrodynamischen Kräfte beim Roliversuch" Schiffstcchnik, Bd. 3, 1955/56.

S. Motora: "On the measurement of Added Mas and Added Moment of Inertia for Ship Motions, Part I & 2" J. S. N. A. in Japan, 1959.

F. Tasai: "On the Damping Force and Added Mass of Ships Heaving and Pitching" J. S. N. A. in Japan, 1959, or Reports of Research Institute for Applied Mechanics, Vol. VII, No. 26, 1959.

L. Landweber and M. C. de Macagno: "Added Mass of Two-Dimensional Forms Oscillating in a Free Surface"

E 6] F. Tasai: "Formula for Calculating the Hydrodynamic Force of a Cylinder Heaving on a Free Surface (n.Parameter Family)" Reports of Research Institute for Applied Mechanics, Vol. VIII, No. 31, 1960.

[7 J F. Ursell: "Short surface waves due to an oscillating immersed body" Proc. Roy.

Soc. A, 220, 1953.

T. Kumai: "Added Mass Moment of Inertia Indui by Torsional Vibration of Ships ". Reports of Research Institute for Applied Mechanics, Vol. VII, No. 28,

1959.

T. Hishida: "A Study on the Wave-making resistance for the rolling of ships, Part I & 2" J. S. N. A. in Japan, Vol. 85, Dec. 1952.

T. Hishida :" A Study on the Wave-making resistance for the rolling of ships, Part 3 & 4" J. S. N. A. in Japan, Vol. 86, March 1954.

F. Ursell: "The Effect of a fixed Barrier on Surface Waves in Deep Water" Proc.

Camb. Phil. Soc. 43, 1947.

F. Ursell: "On the waves due to the Rolling of a Ship ". Q. J. M. A. M.. Vol. 1, 1948.

F. Ursell; "On the rolling motion of cylinders in the surface of a fluid" Q. J. M. A. M., Vol. II, Part 3 (1949).

G. N. Watson: "Theory of Bessel Functions, p. 329.