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(1)

PAPERS

OF

SHIP RESEARCH INSTITUTE

On the Effect of the Forward Velocity on the Roll I)amping Moment

By

Iwao WATANABE

February 1977 Ship Research Institute

Tokyo, Japan

22 SEP. 1982

Lab.

y. Scheepbwwkundt

ARCHIEF

Technische Hogeschool

Duft

(2)

ON THE EFFECT OF THE FORWARD VELOCITY

ON THE ROLL DAMPING MOMENT*

By Iwao WATANABE:

NOMENCLATURES

x,y,z: Cartesian Coordinates

Dimensionless Cartesian Coordinates (= -, r= -y-, =

L,T,B,V: Length, draft, beam and forward velocity of the ship Rolling angle, amplitude and frequency

A,Th The projection of the hull into Oxz plane, the remaining Oxz plane g: Acceleration of gravity

p: Density of the fluid

1(x , y, z, t): Time dependent velocity potential 'P(x,y,z): Velocity potential

çl(x,y,z): Acceleration potential p(x,y,z): Pressure

v(x,y): y-component of perturbation velocity on Oxz plane Dimensionless form of v(x,z)

ç5(x,z): Acceleration potential ori Oxz plane Dimensionless form of ço(x,z)

vj(): E-component of v(e,) v2(): c-component of i-component of c-component of g/V2 ,L gL r: r 2V Q: M: Ci-: Cd: Cdt,Cd2 , Cj3: g

Aspect ratio of the ship centerplane (2T/L)

Hydrodynamic moment acting on a rolling ship Dimensionless hydrodynamic moment defined in eq. (44) Dimensionless roll damping moment

Roll damping components defined in eq. (48)

Received on November 4, 1976. Ship Dynamics Division.

1. INTRODUCTION

It is common to use the hydrodynamic forces calculated using the strip theory to estimate the ship motions in a sea way. We can get an accurate estimation of the forces corresponding to almost all modes of

(3)

2

motion by using this method. In the case of rolling, however, there is poor agreement between experiment and calculation. This discrepancy is due

to the fact that the assumed two dimensional flow in the strip theory is

not applicable in rolling. A new theoretical method is required therefore to give an accurate estimation of the hydrodynamic force.

It is known that the resonant rolling frequency varies little with

changes in the forward velocity, while on the other hand, the amplitude at the resonant frequency decreases remarkably as the velocity increases. These facts indicate that the velocity affects the damping component rather

than the inertial component of the hydrodynamic force. Therefore, in order to derive a practicable theory, initial investigations are necessary to the effect of the forward velocity on the roll damping. However, there

is a scarcity of reports relating to this problem.

Hishida has formulated the roll damping moment acting upon a

rolling ellipsoid moving at a constant velocity. His theory treats only

the damping component due to the wave making and it ignores the side force due to the asymmetry of the flow.

Hanaoka has presented a theory for the flow field around a ship

at a constant velocity in asymmetric motions such as rolling, yawing

and swaying under the assumption of thin and low aspect ratio ship. Although there are some limitations due to these assumptions and

ap-proximations, his theory allows the side force term to be taken into con-sideration.

In this paper, the author applies Hanaoka's theory to the estimation of the effect of the velocity on the roll damping. The theoretical formula is then transformed into a convenient form for numerical evaluation and the computed results are compared with experimental data from a model

ship. This comparison indicata good correspondence that the proposed theory seems to be adequate to predict the effect of the velocity on the

roll damping moment.

2. GENERAL FORMULATION

We are going to consider a flow field around a rolling ship moving at a constant velocity V. The problem is formulated within a framework

of the linearized thin ship theory. Assumptions used in this paper are

as follows:

The fluid is incompressible ideal fluid which is infinitely deep.

The beam of the ship is small compared to all of the other ship

dimensions and the ship's centerplane is rectangular.

The rolling motion is sinusoidal with angular frequency and an

infinitesimal amplitude . The axis of rolling is in the load water

line. Only linear quantities relative to the amplitude will be

(4)

AP

ML.

Oxz-pLane view

Fig 1. Coordinate system

To describe the problem, we shall adopt a coordinate system moving to the ship. The origin is located in the undisturbed free surface at

mid-ship. The axis Ox (0e) is directed toward the stern, the axis Oy (0v)

extends to starboard, and the axis Oz(0) is upward. The coordinate

system is illustrated in Fig. 1.

The projection of the centerplane on the Oxz plane is conveniently denoted by A, and the residual plane in the Oxz plane by A.

As the rolling flow field is asymmetric with respect to the Oxz plane, complete modeling of the whole domain can be obtained by analysis of only the half domain. Hereafter, the positive y domain will be discussed. The ideal flow assumption leads to the existence of a velocity poten-tial ø (x, y, z, t). Thus the problem is described by following conditions. (L): Since the fluid is incompressible, the potential satisfies the Laplace

equation in the entire fluid domain.

(F): On the free surface, the potential must satisfy the linearized free

surface condition.

(R): An outgoing dispersing wave far away from the ship exists.

(I); There is no disturbance in the infinitely deep region.

(H): Based upon the thin ship assumption, the boundary condition on the hull surface is replaced on the centerplane. Here the rolling motion

be denoted as

çû=

Re {iet}.

(1)

The boundary condition on the centerplane is

A

FP

(5)

4

-=çze

ay

We can expect the sinusoidal time dependence for the potential: P(x, y, z, t)=Re {ø(x, y, z)e'}.

(3)

As is usual in the sinusoidally oscillating phenomenon, we are going to drop the notation Re. The boundary condition is thus rewritten as

It is convenient to introduce an acceleration potential in the analysis. Let «x, y, z) be the acceleration potential, then we have

ax

The acceleration potential is equivalent to the pressure

p

3. MICHELL'S POTENTIAL

According to Hanaoka, the unsteady asymmetric flow can be formu-lated by using two types of potentials by applying Michell's method.

The first type is appropriate to expresse the flow when y (x, O, z), the y component velocity on the Oxz plane, has been specified. The formula has the same form obtained for the symmetric flow. The potential for the asymmetric flow is then written in the form of the acceleration potential as

(x.y,z)=_J dnÇ dmÇ dz' r

Jo /m+n

v(x',z')

(m+)

x exp {im(x - x') - + n2y} cos (nz' + E) cos (nz + e)dx'

onA.

(2)

onA.

(4)

x exp {im(x - x') - /m2- d'2y + d'(z + z')}dx'

V

dz' $

v(x', z') (m+

I +i

dm

J - J 2 - d2_ m2

X exp {im(x - x') - iE4,/d' - m2y + d'(z + z')}dx',

(7)

where m1)

,-1-2QT/1+4Q

m31

,1-2QF1-4Q

m2J 2 '

miIC

2 iV (Çm fm) ¡'O "O

vx' z'

/ '3 --- + I dm dz' I

'

' / f m+--7tiV' U Jrnj J - J - - d'2

\

V

(6)

g

d'=

(m+/V)2

e=tan' (d'/n),

V2 g

r-1

for oo<m<m1

1 for m2<m<oo.

The exact solution for the symmetric flow can be derived from this expression. However, in the case of the asymmetric flow, the

straight-forward substitution of the boundary condition on the centerplane into

eq. (7) does not give the exact solution, since the vortex generated by the hull in the asymmetric motion induces y (x, O, z) which is not known until the problem has been solved entirely. Therefore another formulation is required to obtain the exact solution.

The other type of the potential is formulated for the boundary con-dition on the centerplane specified by the acceleration potential Ø0(x, z)

instead of condition (4). The velocity potential is written by

b(x, y, z)= s: dn dm 50 dz' s: dx'50

xexp f - (x' - X)}ço(X, z')dX. exp {im(x - x') - ./m2 + n2y}

x cos (nz + e) cos (nz' + e)

,112 1114 X'

+

, f5

+J} dm

f dz' $

dx'J

x

exp f- (x'

X)}ç50(X, z')dX. (m+

x exp {im(x - x') - /m2 - d'2y + d'(z + z')}

exp f- (x'

X)}çbo(X, z')dX. (rn+

x exp {im(x_x/)_iep./d02_m2y+d(z+z/)}.

(8)

Differentiating eq. (8) with respect to y and equating to the bound-ary condition (4), an integral equation is obtained which gives the exact solution.

4. HANAOKA'S APPROXIMATE THEORY

In this section, instead of solving the integral equation, an approxi-mate solution will be given under the condition that y (z, O, z) on A has been estimated appropriately.

We may empirically assume that the shape of the wake and the

velocity distribution in the wake are not greatly affected by the existence

i

í

+ / r'I

+1 +

Ç' roo ro r r-"

(7)

6

of the free surface. The approximate theory is based on this assumption. First, an integral equation is solved under the rigid wall condition on z=O to determine ç5, on A. Next, y (x, O, z) is estimated from ç5,. And finally

by substituting y (x, O, z) into eq. (7), the potential containing the free

surface effects can be derived.

4.1 IntegraI equation under the rigid wall condition

The potential under the rigid wall condition is given by the limiting

form of eq. (8) as c'oo:

(x, y, z)=

v

s: dn s: dm 5°

dz' s: dx'f

x

exp {- (x'

X)}çbo(X z')dX.exp {im(xx') 4/m2 +n°y}

X COS flZ COS flZ'.

(9)

The integral equation for ç5, follows as

i

1' f0 ¡' ¡'s'

v(x, z)

-

dn dm I dz' I dx'

ay 22v .0

Jo

J-,

-x

exp (_ (x'

X)}ç50(X, z')dX. /m2+nbe1m(x_x)

X cos nz cos nz'. (10)

Eq. (10) is transformed into a more convenient form using the following parameters and variables:

L

_2x

,2x

y

2V'

L '

L '

T'

mL

'=nT,

v(x,z)=v(E, ), ç50(x,z) rrç50(, ),

rn=

, n

2 V V°

In a conventional hull form, the ship's draft is small compared to the ship length. Therefore, we can assume

/m2+n2=Ij(m')'+n'

(12)

Denoting the ratio 2T/L by 2, then eq. (10) can be expressed by:

-

s: dn

Ç

dm 5° dC'

Ç

dOE' f

xe1'530(8, ')dSe m(e_e')n cos n(ç')d', (13) where m' and n' defined in eq. (11) are replaced by m and n respectively.

Since the integrations with respect to ' and ?fl may be easily obtained

using the Fourier integral theorem, it follows that:

(8)

and

v2()= 1 f as,

d'

irA J-i

a: a-c'

Eq. (18) can be solved to give

= iwv) +

dv1

Eq. (19) is an integral equation 'which can be solved analytically. The

solution is

dçb22 c (21)

ir

\/1ç

-1

Now the boundary condition for the centerplane is substituted into

eqs. (20) and (21) and the solution of the rolling problem is obtained.

Recalling the dimensionless parameters defined previously, eq. (4) may be rewritten as

DC,)=ç15w2i

onA. (22)

irA

')dE j'

ndn s:

n(')d'.

(14)

A further simplification of eq. (14) is possible by using the relation:

J: dn J

3(S, ')n cos n(_')d'J dn

¿3(, ç')n cos

r

dÇOd f

sinn(')dn= r

d0[

cos n(') l"d'

J -- d'

J J -- a:'

'

i=°

=f'l

x::S,

(15)

where the RiemannLebesgue lemma has been used to derive the last

relation. Consequently, a simplified integral equation is obtained

e«nc

a

d'

irA 1

a:' c-a'

We assume v(e, :) and c) separable,

)=v().v()

)=çi51().ç52(). Substituting them into eq. (16), we have

(9)

8

Without loss of generality, it can be also assumed that

= H( +1)1

v,()=

w2

where H(e) is Heaviside step function.

Substituting v1() of eq. (23) into eq. (20), we have

where ô() is Dirac delta function.

It remains to determine Ç12 Substitution of v,() into eq. (21)

aç, Ç9W2

/1 _Ç dc'.

r

'1

V-i

The integration can be replaced with the analytical formula*:

o' /1'

d

2_2/1_2 ln

1+1_2

(26) -i

Therefore, eq. (25) becomes

aç5, 2w2'

- /i

_2 ln

+ i/1_2 1>

/1

) j

and the c-component of the acceleration potential is given by:

22

{

/

4.2 Estimation of v(e, ) on A

The velocity distribution over A plane can be estimated from the results in the preceding section. The t-component of v(, ), v1(e) is ex-pressed for an arbitrary value of as

Here ç5' is given çb,=iU)+ô(e+1) onA çzl,O onA. The -component of v(, ), v2() is v2() 2Çw2

J

/1,j1_/1_c ln 1_c'2}

. (31)

When I<1, taking Cauchy's principal value, the integral reduces to * See Appendix A

(23)

leads to

(29)

(10)

v2(C)=

wA

J.

When >1, using the relations*:

Ç' ' dC'

j

/i'

=+sgn ()

and '1 I C'in 1+s/iC12 dC'

i

+C sin_I

J-i C'I

CC'

C

v2(ç) is written for the negative C as

v) =

2w2 I C

+Csin1.

4.3 The potential including free surface effects

The potential (7) on can be written in the dimensionless form as

ç,(e, O, C)=2-{_-

f

dC s:

C')e'

cos (n'+E)d'

><5dn$ (m+w) ehie cos (nC+)dm o - /(m2)2 + n2 where * See Appendix B X e°" e'dm

_IfdC'f

( i e1medm /d2m2 r rm fn4' m v(C', C')e- me'eui'dC I + I 2 2 Wm, JmsJ './m d

L}

flit m3 v(e',

C')e°'e'de'{_f

+5+

,LgL

d=(m+w)3,

=-tan'-,

2 2V2 ir n m,)

i-2Qp/1+4Q

> = K ?fl9J 2

m3i_2Qp/1_4Q

m.J 2 (35)

Substituting eqs. (23), (29) and (30), we have the velocity potential in-fluenced by the free surface. The formula can be integrated with respect

to e and C. The part related to C are

s

v1(Cf)e-mn'dC

=f dE f

e_i(m'+içi(5)dC

(11)

lo and and j' v2(c')e'd' =w2--F0(dA), where F.(n) =J1(n) + J0(n)H1(n) J1(n)H0(n)

F0(n) = Y1(n)+ r Y0(n)H1(n) Y1(n)H0(n)_,

ir F0(n) =K1(n)

2 dA

K0(dA)L1(d2) + K1(dA)L0(dA)

J, J1, Y0, Y1 denote Bessel functions, K and K1 denote modified Bessel functions,

H and H, denote Struve functions and

L0, L1 denote modified Struve functions. The derivation are shown in Appendix C. Therefore, the potential is given by:

ç30(e, O, )

{iÍ dn r

(2w sinm _ieim)

ir o

J-

/(mA)2+nh \ m

x e°° cos (n + E){F(n) coso+ F0(n) sin o}dm

2i ' f2 +

r'i (m +

w)2 (2w Sfl m je° , Um Jm3J %/m2_d2 \ m X eLEeF0(d2)dm {$m1

+Ç+Ç}

t'°2 (2w sinm

iei)

IC - in x

I+'

(m+w)

i

sinm - (m+w) m Ím} (36)

The integrations with respect to become

fv2(') cos (n'+e)d'= w2[F(n) cos o+F(n) sino]

(37)

(12)

5. ROLL DAMPING MOMENT

The hydrodynamic moment is obtained by integrating the product of

the pressure and the lever. Recalling the relation between the pressure

and the acceleration potential, the moment about x-axis, M becomes

Eq. (44) is evaluated by integrating with respectto and . Substituting

eq. (42) into eq. (44) and using the relations:

J'

d;$'

2sin m

{

sin (nE)

+

cos cos (nE)

}

m n n'

2

sinmf(n)

nl

' sinm {e-

1_e12}=2 sinm!(dA)

(46)

dJ

eímEed=2

-1 -1 m dA (dA) Then, we have

c

r w2' F-2i 100dn f00 1 (2w Sfl m sin r' I Jo J- s/(m2)2+n2 \ m

/\ m

xf,(n){F(n) cos s+F(n) sin s}dm

+ {f002+J04}

(m+w)' (sin m

iei;2)( sin m)

¿ mi ./m2_d2 ni m

Xfe(dA)Fo(dA)dm

001

+f003+J00} (rn+w)2 (2W sinm _iem)( sinm)

IC -00 002 m4 .,/d2_m m m

xfc(dA)Fo(dA)dm}. (47)

The roll damping moment Cd is defined as the real part of Cr. Thus, we have

wA'

F_ f dn

i / sin 2m )f,(nA)

2 j J - "(mA) + n2 \ m

x {F(n2) cos + F(n2) sin E}dm

Mr=JJ zPdzdx=_LT2V2 J' dJ°

(43)

2 -1 -1

The dimensionless moment is defined by

M

-

_J d $

O, (44)

(13)

12 and

+{-Ç'

+j

+J _í Ç"' f (m+w) (m+2w)( sinm )2fe(d2)(d2)dm] K - i2 m4 /d2_m2 [Ca, + C2 + Cd3], (48)

where C1, C, and Gdl correspond to the first term, the second term and the third term respectively.

When Q exceeds 1/4, the formulation differs somewhat from eq. (48), since the roots m and m4 become imaginary. The integration from m2 to

?nc and that from m4 to infinity in the third term are joined to give the

integration from rn2 to infinity.

6. NUMERICAL CALCULATIONS

Eq. (48) contains the double integral and the singular integrals.

Generally speaking, the straightforward integration of these terms may lead to erroneous results irrespective of the large amount of computing

time. Therefore, it is better to transform eq. (48) into more convenient

formula for numerical computation. 6.1 Numerical formula

The term containing the double integral may be regarded as the

damp-ing component due to the wake produced by the hull oscillatdamp-ing asym-metrically. On the basis of the previous assumption that the wake is

influenced little by the existence of the free surface, it can be assumed that

C,11 is equal to the corresponding term of the moment computed with the

rigid wall free surface condition. Substituting eqs. (24) and (28) into

(44), we have C,1,:

Cdl=Re f_J' dJ° d{iw+ô(F1)}{1_C2+C2 ln

11}]=1.

-1 -1 C 2

(49) The integrands in the second and the third terms have the

singulari-ties of the order of 1/./ at the end of integration. In order to show the

way to remove these singular behavior, we will consider a simplified case:

(A) J" F(m) dm, (50) rn '/(m m,)(m - m) (B) G(m) ---dm (51) J-

s/(m,m)(mm)

+J

2 f K f"

f'f (m+w)2 (sin 2m

)fe(dFk(ddfl +

i

m 'ni n,

(14)

and

and

=sin 8,

the integral (50) can be transformed into regular integral:

F(m) C F (r) Jrn, /(m_mg)(me_m)dni

=j1

dr=J,2F2(0)dO. (53) By changing variables: m, - me rn, + me m 2 2 r=%1'l+tZ, we have G(m)

dm Ç

G(r)

Ç

G2(t) dt. (54)

L

(m3m)(mm)

i

r2_1 -

o

By substituting following formulae:

mem,

+ m3+m. 2 2 and we have H(m) dm= f' H1(r)

dr=

fl

H2(t) dt. (55)

Jme ./(mm8)(mm)

Ji ,/r2_1 Jo

/1+t

These results will be applied to eq. (48).

Then, the formula for C

becomes

(C)

Ç

i/(m m8)(m me)H(m) dm, (52)

where F, G and H are arbitrary functions which have no singularities in the range of integration.

By the change of variables:

me - m, r + m, + m

(15)

14

where

¡3=s71+4Q,

d' ' (/3 sin 8-1),

d"=

(r sin 0_1)2.

The formula for C3 is

C

-4Ff (ß/1+t+1)2(ß/1+t2+1_2Q) f sin

(/2)(ß./1+t2+1+2Q) 2

- U0

"(ß11 + t2 + 2) T2'1 + t2

i

ß1 + t2 + 1 + 2Q f xfC(e')Fk(e')dt +f (r1+t2+1)2(r1+t2+1+2Q) I sin (K/2)(I/1+t2+1_2Q) '2 J

r1+t2+2)2_ß21+t2

r1±t2+1_2Q

J xJ'e(e")Fk(e")dt + (ô1 sin 8 + ô2)2(ô1 sin 8 + ô2 + 2Q) J2 /(ö1 sinO+ô2+1+ß)(ô sin 5ò2+1+7)

x { (K/2)(ò1 Sin O+ô2-2Qsin O+Ô2-2Q

Cd22{1 (ßsin6-1)2 J -,f2 /(ß sin O_2)2_2 /2 (r sin O - 1)2 (r sin 8-2)2ß2 where

e'=

(ß1+t2+l)2,

e"=

(rf2+i)2,

ô1=i 2 Sfl K(ß sin 8-1-2Q) fe(d')Fic(d')dO (j9 sin O-1-2Q) Sfl A:(ï sin 8-1H-2Q) (r sin 8-1+ 2Q) =_P_ 2 K2

e"

- (ô1 sin O + ô2) 4 (56) (57)

When Q exceeds 1/4, the numerical formulae are obtained in the same manner as is used in obtaining eqs. (56) and (57).

The formula for Cd2 is given by

fr/2 (/3 sin 8_i)2 sin ic(ß sin O-1 2Q)

f(d')F(d')dO

GO=2J

- /2 /(ß sin 82)2 + 4Q 1 (/3 sin 8-1 2Q)

(58) C3 is obtained from the formula:

Cd3=4[J

(ß/1 +t2 + 1)2(ß/1+t2+ i-2Q) f sin (K/2)(ß/1 + t2 + 1 +2Q) '

O Ì J3/1+t2+1+2Q J

(16)

and

p"=

(/3%/1+t2_1). 6.2 Computed results

Before discussing the calculation, the integrands will be discussed. Figs. 2 through 6 show each integrands contained in eqs. (56) and (57).

Their parameters are 2=0.1, ,c=19.531 (Fn=0.16), w=4.O and Q0.205.

These figures represent the typical integrands in the case of Q<1/4. It

is seen that the integrands related to Cd3 (Figs. 4, 5 and 6) are dominated in magnitude.

When Q exceeds 1/4, the dominating term is Cd also in this case. It can be seen from Figs. 7 to 9. Their parameters are 2=0.1, ic=12.S (Fn= 0.2), w=6.0 and Q=0.480.

In calculating C3 the integration to infinity must be truncated at a finite point. In order to minimize the error due to the truncation, it is necessary that the integrand beyond the point of the truncation should

decrease as rapidly as possible. The damping of the integrand is largely governed by icj3 or icr as is seen from the argument of the function Fk which dominates the damping of the integrand. The greater icß or ,c be-comes, the more rapidly the integrand decreases. Therefore, the

numeri-cal integration has been truncated at such a point as to be proportional

to icß or icr.

We should be careful about to the points of the numerical integration.

Every integrands is oscillatory owing to its sin term and its frequency

depends upon icß or icr. We should choose the interval between the inte-gration points enough to follow the oscillating characteristics of the in-tegrand. Thus in our calculation, we have chosen the interval to be

inversely proportional to icß or ,c- over the integration to infinity.

How-ever, the interval for the integration from ir/2 to ir/2 has been fixed,

say ir/200. The numerical method adopted here is the trapezoidal rule

using these intervals.

+Ç (ß/1+t2_1)2(ßs'1+t2_1+2Q) J sin (K/2)(ßs/1+t2_1_2Q) 2 Ju

/(ß/1+t2_2)2+4Q_11+t2

l fr%/'1+t2_1_2Q J Xfe(P")Fk(P")dt] (59) where

(ß/1+t+1)

(17)

90.00

A

AI

A

y°T

n7 Vin. on 60100 .90. nfl

A=O.100, = 19.531, w=4.000, [2=0.205

Fig. 2. ist integrand in Eq. (56)

2=0.100, K=19.531, w =4.000, [2=0.205

Fig. 3. 2nd integrand in Eq. (56)

2=0.100 = 19.531, w=4.000, [2=0.205

(18)

A=O.100, 'c=19.531, u=4.000, Q=O.205

Fig. 5. 2nd integrand in Eq. (57)

,O.00 Ofl.OtI O.D0 O.flfl 40,flfl GO.flO 0.DO

=O.1()O, =19.531. w=4.000, Q=O.205

Fig. 6. 3rd integrand in Eq. (57)

-SD. OD 60 O O. 00 4 .fl'. D_00

=O.1OO, ,r=12.500, w =6.000, Q=0.480

Fig. 7. integrand in Eq. (58)

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18 Cd 02 Q=114 w= 6 wr 4 O 0.1 0.2 ç 03 0.4 0 5

Fig. 10. Variation of Cd with Respect to ii

4.00 s. na 4.00 0.00

À=0.100, ,r=12.500, w=6.000, Q=0.480

Fig. 8. ist integrand in Eq. (59)

0 00 2.00 4.00 4.00 4.00 00.00 12.00

2=0.100, K=12.500, w=6.000, Q=0.480

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0.2 -A =0.1 Cd c, =8.0 0.1 dueto Cdl 0.05 Cd 0.025 A =0.15 2 4 e B 10 0 0.1 0.2 Fn 0.3 0.4

Fig. 12. Roll Damping Moment for A=0.10 against Froude Number

0 0.1 0.2 Fn 0.3 04

Fig. 13. RoIl Damping Moment for =0.15

against Froude Number

Cd 0.1 0.2 Fn 0.3 0.4 Fig. 11. Composition of Cd 0.75-0.5 0.25 0.5 25 0.75 A =0.1

(21)

20 Cd .5 1.0 Cd 0.5 1.0-0.5

7

o o 0 0.1 0.2 Fn 0.3 0.4

Fig. 15. Roll Damping Moment for 2=0.25 against Froude Number

0.2 0.4 0.6 0.8 1.0

f(Hz)

Fig. 16. Comparison of Calculated Roll Damping Moment with Experiment for Fn=0.20

0.5

0.4

Fn 0.2

o o with bilge keel

Cd 0.3 x without b.k. - present theory o o 0.2 O 0.1 X 0 0.1 0.2 Fn 0.3 0.4

Fig. 14. Roll Damping Moment for A=0.20 against Froude Number

(22)

Cd 0.4 0.3 0.2 0.1 0.2 0.1 o o 0.2 0.4 0.6 0.8 1.0

Fig. 18. Comparison of Calculated Roll Damping Moment with Experiment for Fn =0.30

The damping moment diverges at Q=l/4 because integrands in C42

and C43 become singular. Fig. 10 shows the behavior of C4 (=C41+C43)

to Q. Although C4 seems to be continuous across Q=l/4, we will discuss the damping moment after discarding the values near Q=l/4.

Fig. 11 shows the composition of the damping moment for w=4 and

u=1O in the case of 2O.l. The straight line correspond to Cd, and the

curved line, Cd. As C42 is negligibly small, the difference between these

lines is caused by C43. It is seen from this figure that the wave making

component once grows and then decreases as Froude number increases.

The roll damping moments are shown in Figs. 12 through 15 for

2=0.1,0.15, 0.2 and 0.25 respectively. The values near Q=l/4 have been

omitted since the confirmative values could not be identified. The roll

damping moment varies according to the Froude number for all 2 cases and it increases as 2 increases.

0.2 0.4 0.6 08 1.0

f(Hz)

Fig. 17. Comparison of Calculated Roll Damping Moment with Experiment for F=O.25

0.4

Fn 0.3

(23)

22

6.3 Comparison with experiments

Roll damping moments have been measured on a container model which has been towed at a constant speed with forced rolling motion.

The principal dimensions of the model are L=4.500 (m) B==O.653 (m)

T=O.243 (m)

The amplitude of rolling is 10 degrees.

Figs. 16 to 18 show some comparisons between the calculation and the experiments. The symbols denote the experimental values. The solid

line is the calculated results. It is seen that the theory agrees well with

the experimental values for the case of the bilge keels removed.

There-fore, for the container model, the roll damping moment acting on the

naked hull appears to be mainly governed by the draft. However, noticing

that the theory has based upon the assumptions which are not strictly applicable to a typical hull form, more detailed model experiments are

required to draw a definite conclusion as to the general applicability of this formulation to various ship forms.

7. CONCLUDING REMARKS

Based upon Hanaoka's theory, a method has been developed for the calculation of the roll damping moment of a rolling ship with a constant forward motion. The method makes it possible to evaluate the effect of

the forward velocity as well as the draft influences on the damping

moment. The calculated results indicate that the damping moment C,1

once grows and then decreases with increasing Froude number. This variation with the forward velocity is caused by changes in the wave

making component. As to the draft influences, the damping moment in-creases as the ratio of the draft to the ship length inin-creases.

The calculated results have been compared with model experiment showing good agreement. Therefore, it is possible to say that the present modeling, the thin and low aspect ratio ship assumption accompanied by the wave making phenomena is justified in the case of the rolling ship,

however, more detailed experiments are required to fully confirm this

conclusion.

REFERENCES

1) Hishida, T.: "Studies on the Wave Making Resistance for the Rolling of Ships

(Report 6. The effect of motion ahead on the wave resistance for the rolling)",

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Hanaoka, T.: "On the Velocity Potential in Michell's System and the Configura-tion of the Wave-ridges due to a Moving Ships", Journal of Zosen Kiokai, 93, 1953.

Hanaoka, T.: "On Michell's Method of Solving Non-Uniform Wave Making

Phenomena of a Ship", 5th Japan National Congress for Applied Mechanics, 1955. Hanaoka, T.: "Hydrodynamics Concerning the Ship Motions Among Waves", Unpublished, 1959.

Hanaoka, T.: "Non-Uniform Theory of Wave-Making on Low Aspect-Ratio Lift-ing Surface", 10th Japan National Congress for Applied Mechanics, 1960. Kondo, J.: "Integral Equations and their Applications", Corona-sha, 1959. Watson, G. N.: "Theory of Bessel Functions", Cambridge University Press, 1966.

Abramowitz, M. and Stegun, I. A.: "Handbook of Mathematical Functions",

Dover, 1964.

Hanaoka, T. and Ando, S.: "A Calculation on the Motions of a Ship in the Head Waves Part 1. The Numerical Calculation on the Wave Functions", Abstract Note of the 9th General Meeting of Ship Research Institute, 1967.

Appendix A Integration of eq. (26) The integration of eq. (25) is

Ç%/1_2

'L da'. (A-1)

Changing the variables using:

= cosO

and

'=cos 8',

eq. (A-1) becomes

Çsin2

8 cos O'j dû'= - Ç

COS

dû' +Ç cos2O' cos 6 I dO'

o - cos 6 cos 0' o cos û + cos O' o cos 6 + cos 8'

= {2 cos 0+(1cos2 8)I}, (A-2)

where

j=Ç

Icoso'I dO'. (A-3)

Jo cosO+cosû' When O<O<ir/2, i.e. C<O, I results in,

I=cosû

u

cosOcoso'

Jo cosû+cosO'

/2 do' Çr/2 dû'

_2°-lfl-c°50

(A-4)

sinO sinO

The same result is obtained in the case of >O.

(25)

24

The first term in the square bracket becomes

f' dC=2_2/1_2 in

(A-5)

for arbitrary values of .

Appendix B Derivation of eq. (33)

The second term of eq. (31) is

f1 ' in

1+/Ii_2

d'

J_i ic'i

= -

r in

1+/1_2 d'

1' in

1+/1_2 d'

J-'

' J_I

where "i I in

1+/1_t2

dc' J_i

The derivative of g() becomes

dg() In

+ s/12

d'

-

f1 1

d'

J_i

(_)2

J_i

f1ì

:'

-

'r sgn (se) C.%21 It follows by integration that

g()=

_7rJ'

/i

dC-7r Sifl

Substituting eqs. (B-4) into (B-1), we obtain eq. (33).

Appendix C Derivation of eqs. (37) and (38)

The integration (37) can be written in the separated form as

v2(C')

cos (n'+)d'=w2[Ç

' cos (n'+e)d'

-1

_--

J- 1i

f1 '

cos (n'+e)dc--- j'1

' sin

I

cos

(n'-_)d'].

(C-1)

(26)

f

' cos

{e

[( 1cosn sinn)

i(

.fjn

cos n

lflk

n

'n

The second term is expressed as

E

cos (n'+E)d'= Re

fe1 Ç

'

eC'd'}

= Re e- ' {J1(n) + iYi(n)}J (C-3)

where J1(n) and Y1(n) are Bessel functions.

The third term is transformed into

ï

1

sin'

cos (n'+)d'=Re

[e-1g Ç ' sin

4e;'d']

= Re ie { - --e + --i{J0(n) + iY0(n)}

i

/2_1

= -- Re [

{_sin n+i

cos n+J0(n)+iY0(n) COS n sin n

2 n n n

+ F(n) + G(n)

H (C-4)

n n

where J0(n) and Y)(n) denote Bessel functions, H((n) and H1 (n) denote Struve functions and

F(n) = i - nJ0(n) - T_n{J1(n)H0(n) J0(n)H1(n)} (C-5)

G(n) = - nY0(n) - ?_n{ Y1(n)H0(n) - Y0(n)H1(n)}. (C-6)

Here, in order to get the last equation, we have applied the following

relations: r°

e1'

d'=Ji_J H1)(t)dt}

J1

f/2_i

2 . o

=

Fi - -_n{H" (n)H0'(n) + H' (n)Ho(n)}] 2 2 = -- Fi - -f-n{H1 (n)

-

Hi(n)} + H'(n) HU)} J 2 2 = -- [F(n) + iG(n)]. (C-7) (C-2)

(27)

26

Thus, substituting eqs. (C-2), (C-3) and (C-4) into (C-1), the

inte-gration is evaluated using:

50

v2(') cos (n'+)d'= wA[F(n) cos+F(n) sin

(C-8)

where

F3(n) = J1(n) + ir J0(n)H1(n) - J1(n)H0(n) (C-9)

F0(n) = Y0(n) + ir Y0(n)H1(n) - Y(n)H0(n) (C-10) The process to obtain eq. (38) is quite similar.

The integration (38) can be written as

j'0

vo(')e'd'=cw2[f 'e'd'

$

ed'd

$1

sin

ie'd'].

(C-11) The first term in the square bracket becomes

S

U-d

e

-i dA (dA)2

The second term is expressed as

J'

e"'d'= -f

e''d'= K1(dA),

(C-13) where K0(dA) is modified Bessel function.

The third term is transformed into

-1 Ied2c'd

i

[ir

-Jc'sin'

d2

e dAK(dA)

2 + 1

1--e' r°

e1'

dA2

Ji

f21

=1.

[e_

+K1(dA1(dA)+K1(dA)L0(dA)], (C-14)

where K0(dA) denote modified Bessel function and L0(d2), L(dA) denote modified Struve functions. Here, in order to obtain the last equation, we

have applied the relations:

Pd2

f

e' d'=1(1_

I K0(t)dt)

2 \ Jo

(28)

= - [i - -- id2{H1) (id2)H(id2) + H' (id2)H)(id2)}]

- id2{_ _iKu(d2){_ + L1(d2)}- _iK1(d2)L0(d)}j

= [i-d2{K0(d2)+ K0(d2)L(d2) + Kl(d2)L(dA)}

j.

(C-15)

Therefore, substituting eqs. (C-12), (C-13) and (C15) into (C-11),

we have the formula:

fv2(/)ed2C'd = ç5w2-Fk (C-16)

where

F(d2) = K1(d2) K0(dA)L(d2) + K1(dA)L0(dA)

2 d2

(29)

PAPERS OF SHIP RESEARCH INSTITUTE

29

No. 10 Studies and Considerations on the Effects of Heaving and Listing upon

Thermo-Hydraulic Performance and Critical Heat Flux of Water Cooled

Marine Reactors, by Naotsugu Isshiki, March 1966.

No. 11 An Experimental Investigation into the Unsteady Cavitation of Marine Propellers, by Tatsuo Ito, March 1966.

No. 12 Cavitation Tests in Non-Uniform Flow on Screw Propellers of the Atomic-Powered Oceanographic and Tender ShipComparison Tests on Screw Pro-pellers Designed by Theoretical and Conventional Methods, by Tatsuo Ito,

Hajime Takahashi and Hiroyuki Kadoi, March 1966.

No. 13 A Study on Tanker Life Boats, by Takeshi Eto, Fukutaro Yamazaki and Osamu Nagata, March 1966.

No. 14 A Proposal on Evaluation of Brittle Crack Initiation and Arresting Tem-peratures and Their Application to Design of Welded Structures, by Hiroshi Kihara and Kazuo Ikeda, April 1966.

No. 15 Ultrasonic Absorption and Relaxation Times in Water Vapor and Heavy Water Vapor, by Yahei Fujii, June 1966.

No. 16 Further Model Tests on Four-Bladed Controllable-Patch Propellers, by Atsuo Yazaki and Nobuo Sugai, August 1966.

Supplement No. 1

Design Charts for the Propulsive Performances of High Speed Cargo Liners with CB= 0.575, by Koichi Yokoo, Yoshío Ichihara, Kiyoshi Tsuchida and Isamu Saito, August 1966.

No. 17 Roughness of Hull Surface and Its Effect on Skin Friction, by Koichi Yokoo, Akihiro Ogawa, Hideo Sasajima, Teiichi Terao and Michio Nakato, Septem-ber 1966.

No. 18 Experiments on a Series 60, CB=0.70 Ship Model in Oblique Regular Waves, by Yasufumi Yamanouchi and Sadao Ando, October 1966.

No. 19 Measurement of Dead Load in Steel Structure by Magnetostriction Effect, by Junji Iwayanagi, Akio Yoshinaga and Tokuharu Yoshii, May 1967. No. 20 Acoustic Response of a Rectangular Receiver to a Rectangular Source, by

Kazunari Yamada, June 1967.

No. i No. 2 No. 3 No. 4 No. 5 No. 6 No. 7 No. S No. 9 ;

Model Tests on Four-Bladed Controllable-Pitch Propellers, by Atsuo Yazaki, March 1964.

Experimental Research on the Application of High Tensile Steel to Ship

Structures, by Hitoshi Nagasawa, Noritaka Ando and Yoshio Akita, March

1964.

Increase of Sliding Resistance of Gravity Walls by Use of Projecting Keys under the Bases, by Matsuhei Ichihara and Reisaku moue, June 1964. An Expression for the Neutron Blackness of a Fuel Rod after Long Irradia-tion, by Hisao Yamakoshi, August 1964.

On the Winds and Waves on the Northern North Pacific Ocean and South

Adjacent Sea of Japan as the Environmental Condition for the Ship, by

Yasufumi Yamanouchi, Sanae Unoki and Taro Kanda, March 1965.

A code and Some Results of a Numerical Integration Method of the Photon

Transport Equation in Slab Geometry, by Iwao Kataoka and Kiyoshi

Takeuchi, March 1965.

On the Fast Fission Factor for a Lattice System, by Hisao Yamakoshi, June

1965.

The Nondestructive Testing of Brazed Joints, by Akira Kann, November

1965.

Brittle Fracture Strength of Thick Steel Plates for Reactor Pressure Vessels, by Hiroshi Kihara and Kazuo Ikeda, January 1966.

(30)

No. 21 Linearized Theory of Cavity Flow Past a Hydrofoil of Arbitrary Shape, by Tatsuro Hanaoka, June 1967.

No. 22 Investigation into a Novel Gas-Turbine Cycle with an Equi-Pressure Air Heater, by KOsa Miwa, September 1967.

No. 23 Measuring Method for the Spray Characteristics of a Fuel Atomizer at Various Conditions of the Ambient Gas, by Kiyoshi Neya, September 1967. No. 24 A Proposal on Criteria for Prevention of Welded Structures from Brittle

Fracture, by Kazuo Ikeda and Hiroshi Kihara, December 1967.

No. 25 The Deep Notch Test and Brittle Fracture Initiation, by ICazuo Ikeda, Yoshio Akita and Hiroshi Kihara, December 1967.

No. 26 Collected Papers Contributed to the 11th International Towing Tank Con-ference, January 1968.

No. 27 Effect of Ambient Air Pressure on the Spray Characteristics of Swirl Atomizers, by Kiyoshi Neya and Seishiro Satö, February 1968.

No. 28 Open Water Test Series of Modified AU-Type Four- and Five-Bladed Pro-peller Models of Large Area Ratio, by Atsuo Yazaki, Hiroshi Sugano, Michio Takahashi and Junzo Minakata, March 1968.

No. 29 The MENE Neutron Transport Code, by Kiyoshi Takeuchi, November 1968. No. 30 Brittle Fracture Strength of Welded Joint, by Kazuo Ikeda and Hiroshi

Kihara, March 1969.

No. 31 Some Aspects of the Correlations between the Wire Type Penetrameter Sensi-tivity, by Akira Kanno, July 1969.

No. 32 Experimental Studies on and Considerations of the Supercharged

Once-through Marine Boiler, by Naotsugu Isshiki and Hiroya Tamaki, January

1970.

Supplement No. 2

Statistical Diagrams on the Wind and Waves on the North Pacific Ocean, by

Yasufumi Yamanouchi and Akihiro Ogawa, March 1970.

No. 33 Collected Papers Contributed to the 12th International Towing Tank Con-ference, March 1970.

No. 34 Heat Transfer through a Horizontal Water Layer, by Shinobu Tokuda,

February 1971.

No. 35 A New Method of C.O.D. Measurement Brittle Fracture Initiation Charac-teristics of Deep Notch Test by Means of Electrostatic Capacitance Method, by Kazuo Ikeda, Shigeru Kitamura and Hiroshi Maenaka, March 1971. No. 36 Elasto-Plastic Stress Analysis of Discs (The ist Report: in Study State of

Thermal and Centrifugal Loadings), by Shigeyasu Amada, July 1971. No. 37 Multigroup Neutron Transport with Anisotropic Scattering, by Tornio

Yoshimura, August 1971.

No. 38 Primary Neutron Damage State in Ferritic Steels and Correlation of

V-Notch Transition Temperature Increase with Frenkel Defect Density with Neutron Irradiation, by Michiyoshi Nomaguchi, March 1972.

No. 39 Further Studies of Cracking Behavior in Multipass Fillet Weld, by Takuya Kobayashi, Kazumi Nishikawa and Hiroshi Tamura, March 1972.

No. 40 A Magnetic Method for the Determination of Residual Stress, by Seiichi

Abuku, May 1972.

No. 41 An Investigation of Effect of Surface Roughness on Forced-Convection

Surface Boiling Heat Transfer, by Masanobu Nomura and Herman Merte, Jr., December 1972.

No. 42 PALLAS-PL, SP A One Dimensional Transport Code, by Kiyoshi Takeuchi, February 1973.

No. 43 Unsteady Heat Transfer from a Cylinder, by Shinobu Tokuda, March 1973. No. 44 On Propeller Vibratory Forces of the Container Ship Correlation between

(31)

31

by Hajime Takahashi, March 1973.

No. 45 Life Distribution and Design Curve in Low Cycle Fatigue, by Kunihiro lida and Hajime moue, July 1973.

No. 46 Elasto-Plastic Stress Analysis of Rotating Discs (2nd Report: Discs

sub-jected to Transient Thermal and Constant Centrifugal Loading), by

Shigeyasu Amada and Akimasa Machida, July 1973.

No. 47 PALLAS-2DCY, A Two-Dimensional Transport Code, by Kiyoshi Takeuchi, November 1973.

No. 48 On the Irregular Frequencies in the Theory of Oscillating Bodies in a Free Surface, by Shigeo Ohmatsu, January 1975.

No. 49 Fast Neutron Streaming through a Cylindrical Air Duct in Water, by

Toshimasa Miura, Akio Yamaji, Kiyoshi Takeuchi and Takayoshi Fuse,

September 1976.

No. 50 A Consideration on the Extraordinary Response of the Automatic Steering System for Ship Model in Quartering Seas, by Takeshi Fuwa, November 1976. In addition to the above-mentioned reports, the Ship Research Institute has another

series of reports, entitled "Report of Ship Research Institute". The "Report" is

Cytaty

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