On the sway, yaw and roll motions of a ship in short crested waves

b NOV. 1q72

ÂRCHEF

ocus&o

If

On the Sway, Yaw and Roll Motions of a Ship

in Short Crested Waves

By

Fukuzö TASAI

Reprinted from Reports of Research Institute for Applied Mechanics, Kyushu University

Vol. XIX, No. 62, July 1971

Lab. y.

Scheebobnde

Technische Hogeschool

With

Polirnents

i

ibliotbeeji van Onderafdeling.ri- e.sbouwkunâ. Iee4mische Hogeschoo, i DOCUMENIATIE _1:

/(i. 9/4

DAT UM:

/

ON THE SWAY, YAW AND ROLL MOTIONS OF A SHIP

IN SHORT CRESTED WAVES

By Fukuzö TASAI*

In this paper, first we have derived two kinds of the equations of the sway, yaw and roll motions in waves based on the strip theory.

One is according to the Ordinary Strip Method and the other according to the New Strip Method.

In the next place, we have measured these three motions for a model ship with full form self-propelling in short crested waves generated in _our long towing tank, and compared the results of computation by the above two equations with the results of experiments.

Through the above study, it is concluded that the computed values from the equation of motion by the Ordinary Strip Method differ very little from the ones by the New Strip Method. Comparisons between _computed values and experimental ones show satisfactory agreement for the ampli-tudes of the sway and yaw. As for the amplitudes of the roll in the

resonance condition, however, the computed values are larger than the

ex-perimental ones by about 10---20 percents.

1. Introduction

The coupled second-order diffential equations based _{on a strip theory for} the heave and pitch motions can predict their amplitudes and phase lags with a considerably good accuracy. Nowadays, these equations are widely used in the investigations on seakeeping qualities for longitudinal motions of _{a ship.}

Concerning the coupled motions of the sway and yaw in oblique waves, we have Eda's method' for an approximate computation, whose results show satis-factory agreement with experimental results.

It is, however, hardly investigated as to how effective the equations based on a strip theory are on three coupled motions which include the roll.

In this paper, we have measured these three motions for a model ship

self-propelling in short crested waves generated in a long towing tank; on the other hand, we developed two kinds of coupled equations of the sway, yaw and roll motions according to a strip method, and we compared and examined the results of these computations with experimental results.

*Professor, Research Institute for Applied Mechanics, Kyushu University. i

2 F. TASAI

2. Short Crested Waves in a Long Towing Tank

In a long towing tank, we can generate regular waves, irregular waves and transient waves2 by using the wave maker. They are usually, however, long crested waves.

In our long towing tank of Research Institute for Applied Mechanics of Kyushu University, the float of the wave-maker of the plunger type is divided into eight parts. If each of the eight floats is given the amplitude and phase based on computation and made to vertical motion in the same period, we can generate short crested regular waves which are composed of two kinds of one-dimensional waves advancing each other in different directions. Short crested irregular waves can also be generated by varying period of the wave-maker in every cycle.3>

In Fig. 1, let us introduce the coordinate system O-XYZ fixed in space. The velocity potential ç of the gravity waves progressing to the X direction can be expressed from the paper (4) as

Wall of Wafer Tank

\4A1

,/î\\\

At

_'.ol

i

A

/ liii; trt7 b est

\

\/

" _/

Y

'

/

A

A

/I

/ \ \ _/ \

/

Y

/ / \ \

/ y

Nod' line

A'

'&'od-

line

/\\

4'

AV

Y4Y' WA

Fig. i Three-node wave and coordinate System

w cosh k0(Z+h)<B00

k0 sinh k0h Acos 2 1 cos (\/k>7

(n7r)2

x-B0

where B0= tank breadth (2.1)

h = water depth

k0= wave number of one-dimensional wave

X

/

= w2/g-2or/Ào

)= wave length of one-dimensional wave

w circular frequency

and C0=w/k0 = °1/tanh 2'Th/A, _{is the phase velocity of one-dimensional wave.} The elevation of the progressing waves can be written, using the

equa-lion (2.1), in the form:

¡B0

n<B000/0

A,,cos

sin (k 2

/fl)2

Xwt+)

_(2.2)

B0

o

The above equation in the case of nO shows nodal waves varying

ampli-tude in the Y direction, that is, in the direction of tank breadth. _{In the case}

of n=O, it does, of course, long crested waves.

The equation (2.2) can also be transformed into the following form:

<Boko/A _r

-f-[Sifl(ko(J/1-_( flir\°

" y"wt+)

₊

B0k01 B0k0 )

(ko (Vi (air)2

_

_t

(2-3)

where if n is an even number, is equal to and if n is an odd number,

is equal to - or.

Putting nir/B0k0 nÀ2/2B0 = sinO. _(2.4)

we can obtain/i_

(n)O

=cos o,,.

The equation (2.3) _{presents the waves composed of two kinds of} one-dimensional waves, which progress at an angle of O,,=±sin' (nor/B0k0) to the X axis.

Putting A0=3rn, n=2 and B0=8m, we can get O2=sin°(3/8)= _{22 degrees.} Generally, the larger n and A0 become, the larger O,, becomes.

In the next place, we put

k0cosO,,=k,,. _(2.5)

Let N0 be the largest integer of n<B0k0/ir and assuming that h of _water depth is great enough to be regarded as deep sea waves, (2.1) and (2.2)_become

flor

e0Z _{A,, cos}

--

(4 + Y) cos(k,Xwt+r,,)

k0 N0 _flor Acos

_-

_{(° +}

sin(k,,Xwt+,,).

n-o o (2.6) (2.7)

4 F. TASAI

N0 is the maximum number of nodes of all waves in the water tank.

For example, the breadth of our tank is eight meters and therefore, if the maker is given 1.0 sec of period, we obtain 10 of N0. But the maximum num-ber of node is 5 at T,=1.0 sec, as far as the nodal waves can be generated in-dependently (without superposing of other nodal waves). The details about

these points are discussed in the papers (3) and (4).

k,, in the equation (2.5) is the apparent wave number. Letting wave length of nodal wave be A,,, we get

A=2r/k=A0/cosO,,. (2.8)

At the time of t=0, supposing that we put at O a wave node in the case of n in odd numbers or a wave crest in the case of n in even numbers,

Ç of

the sub-surface and , can be written, according to (2.6) and (2.7),

in the

form:

°ek0 Inir/B0

-

cos +

y)) sin(k,,Xwt)

(2.9)

flit IB0

- Cek0zcos f

B0 +

)} cos(kXwt)

(2.10)

where C,, is amplitude of nodal wave and 2C,,=H is wave height.

In Fig. 2, 0

is the point of intersection of the load water line and the

vertical line through G of the center of gravity of a ship when it floats on the still water line. As shown in Figs. 1 and 2, Q xy0z1, is assumed to be a

coor-dinate system fixed in the ship which puts 01 at the origin and Gxyz which

j-.

puts G at origin to be a coordinate system fixed in the ship parallel with

Or-XbYbZ6.

Tn this paper, we will treat of the case of a ship advancing with constant speed of V in short crested regular waves under the head sea condition.

We assume that OY and O,y,, are on the sanie plane at t=0.

When a ship advances without oscillation, that is to say, under the restrained condition, the equation (2.10) can be expressed, using a body axis (x6, Yb, z6),

as

- Çek0cos (- _{(- - Yb)) CO5 (k,x6 + wet) (2.11)}

where w=w+k,,V=w+k0VcosO,. _(2.12)

The pressure p is given by

-

p pgC,ë0Z6cos

_{7 (}

-

Yb)) cos (kx9+wet) (2.13) 3. Equations of the Sway, Yaw and Roll Motions in Short Crested Waves

The author has already derived the coupled equations of the sway, yaw and

roll motions of a ship with zero speed in beam seas (1965)' and ones in

ob-lique waves (1966)6.

In the latter, we partially considered the effect of the ship speed, but

essentially neglected it as for the radiation forces and moments in these equa-tions of moequa-tions.

Regarding the pitch and heave motions, the Ordinary Strip Method7'

(here-inafter referred to as the

O. S. M.) explains results of experiments rather

well, but a more rational strip theory has been developed by Dr. Matao Takagi. Differences between the two are described in Tasai and Takagi*.

Recently, from another approach, N. Salvesen, E. O. Tuck and O. Faltinsen9' have developed equations of longitudinal motions and lateral motions (hereinafter

referred to as the S. T. F. method), where the end effect of the hull is taken

into consideration. As for longitudinal motions, Takagi's new equations

perfect-ly accord with the S. T. F. method's excepting the end effect. Furthermore, Salvesen and others9 have given coupled equations of the sway, yaw and roll motions in oblique waves.

In this chapter, first, we will derive the coupled differential equations of the sway, yaw and roll motions according to the O. S. M. and secondly ones based on Takagi's new strip theory. And they will be compared with those of

the S. T. F. method.

In Figs. i and 2, it is assumed that a ship advances with constant speed of V oscillating with small amplitudes.

6 F. TASAI

Let the angular displacements of the rotational motion about x and z axis be O and ç respectively, and be the sway displacement.

Assuming that all ship oscillations are small, we neglect effects of the pitch, heave and surge motions acting upon the sway, yaw and roll motions.

And moreover it is assumed that all viscous effects can be disregarded.

Because we let a model ship self-propel on the center line (Y=O) of the

water tank for the convenience of experimental equipments, short crested waves

with n in odd numbers were used in order to generate the lateral motions. Table i Nomenclature

L: length between perpendiculars B: breadth of ship

d: draft of ship mass of ship

W: weight of displacement of ship GM: metacentric height of ship

m : sectional mass per unit length w : sectional weight per unit length T : sectional draft

2b: sectional beam of water plane Sw : submerged sectional area G%1" : sectional metacentric height

m' : sectional added mass for horizontal motion m0TL: added mass of ship for horizontal motion

N0 : sectional damping coefficient for horizontal motion Ñ: damping coefficient of ship for horizontal motion

N0 : sectional damping coefficient for rolling motion about x, axis lever of sectional added inertia moment due to swaying motion lever of sectional damping moment due to swaying motion

lever of sectional added inertia force due to rolling motion about Xh axis J0' : sectional mass moment of inertia about x axis

J, : mass moment of inertia of ship about x axis

J'y: sectional added mass moment of inertia about x axis !.: added mass moment of inertia of ship about x7 axis

added mass moment of inertia of ship about x axis J, : mass moment of inertia of ship about z axis I: added mass moment of inertia of ship about z axis F',, : sectional exciting force for swaying motion

F,, : exciting force of ship for swaying motion M',0 : sectional exciting moment about z axis

M0e : exciting moment of ship about z axis M',, : sectional exciting moment about x axis

And therefore in both theoretical calculations and experiments, we will discuss

the sway, yaw and roll motions of a ship advancing in short crested waves

with three nodes under the head sea condition.

3.1. Equations of sway, yaw and roll motions based on the O. S. M.

In Fig. 2, let the horizontal displacement of 0 be , and j is given by

(3.1)

Hereafter the nomenclature in Table i will be adopted.

In Fig. 1, the momentum of the fluid between the control planes of X and

X+tx is

Vço). (3.2)

According to the O. S. M.7, therefore, the sway force dFfl/dxb acting upon

an unit length cross section of the hull is written, letting the wave exciting

force be F',7, in the form:

m(+x) (m') N1

(l'A)

-: O+F'.

(3.3) The above equation, being rewritten by using symmetric relations of ¡'ji,

=m'i and

becomes

=-m(+x,)--N,,(--O,G Ó+x,oVq')m'1ANlA

+ V(ñOG+xbVço)'+ VÒ

(m'l)+F'

(3.4)

Consequently, equations for the sway and yaw motions are given by

11(dXi'

'

iL dx,

I ¡dF,,)xbdxo=o.

In the next place, the roll moment acting on an unit length cross section

about the center of the gravity G is

dlvi,, d I'

- -

(1Y_OlG)O}

_t, - (1V0

-

(m'(iO,G)i,) N,,(l,,O,G)ij,+P,(l 0,G) wGM' O

= (JÇI'+m'17 0G) U + VU

(I'm'10)

Nl,,(i,, 0,G)Um',(i 0) --V,

(3.5)

8 F. TASA!

N(1

O1G)1

wGM'O+F',(1 01G).

(3.6)

Then the equation of the roll motion is given by

( _'dIvi6'

J(----)dx=O.

L X6

The coupled equations of the sway, yaw and roll motions can be written as follows

a1ì-- a2, .-a+bllb+blZo+b3ço+clÖ+c2Ô±eO=F2.

a214+a22+a23ç+b2ï+b226+b23O+c21 +c22+c23

=M

(3.8)

a310+a320 +aO +b21

Coefficients in (3.8) derived from (3.4), (3.5), (3.6) and (3.7) are shown in Table 2, though; in the calculation of (3.5) and (3.7), we assume that the two-dimensional hydrodynamic force and moment at both ends of the hull are zero, as shown in [Vm']:==O. Sway : a11 : a12 ': a13 ç': b11 c': b12 4': b13 O: C11 O: C13 O : C13 Yaw 4': a21

c:

a22 4': a23 O: b21

Table 2 Coefficients of the Sway, Yaw and Roll Motions New Strip Method

mo(l+k,) SL N,dx6=!,, o fm'x6dx6+-- IL SL Nx6dx6_Vf o SL N,(1WOIG)dxh o Jz+J mxidx6+-!Ç-f N,x6dx

fN,xidx±

f

vJ N,x1dx;2f m'dx6

Lm'l' OjG)xbdx O. S. M. Jlfl'X,,dX1,L m'dx,

_VJNdX.

L fm'(4 O1G)dx,,

N1

O1G)dx6

i+ 5m'xdx

SLN, xdx6

v Ç

N,xdx6 _{v2 IL}m'dx6 SL m'(I, i7)xbdx6 (3.7)

Is Roll O: a31 O : a32 O: a33 : b31 '7: b32 27: _b,, 4°: C31 4°: C3, ç': c33

Vf

N,dxb Jx+ Ixo IL N,(1 ÖG)2dxb W- GM

Jm'(l,

-JN(1_ OIG)dxb o OTG)dxb + J m'(t, - O1G)x&dxb

J

N,(1--VJm'(lnOiG)dxx

o N,(1 OlG)xbdx,, L o JL IL N?xbdxb+VJm'dx o J+ Ixo IL N,(1,OlG)2dxb W-GM IL m'(l, - 07G)dx2 IL N,(IW01G)dxb o

JL'

OlG)xdxb JN,(1,OG)xbdxhL y L m'(l,ÓTG)dxb

- V J N,(1)dxb

In the next place, F',e is approximately given, letting the horizontal

com-ponents of mean orbital velocity and mean orbital acceleration of water particle be ii, and , in the form:

F',4rF',j + mT,)+N,Ü,=F2,,,±rnui,+ N,tJ, VJ, (3.9)

where F',1 is Froude-Kriloff's force.

Now, under the condition in head waves with three nodes, becomes

(2c)

CRI)Z CO5 j- (B _{+ y) } Sifl(k.,Xb+Û),t). (3.10)}

And therefore, y, of orbital velocity of water particle in the - Y direction

ê: b22 _{fN,(1 O1G)xbdxb} L

+VJm'(l ÖÒ)dxb

O : b3 vJ N,1(1, ÖT)dxh 27: C21 '7: C22

f

L Nxbdxb+VJ m'dxh L

10 F. TASAI

(3)

oekozsinO.3 cos (3i Y)sin(k3xb+o)l) (3.11)

Assuming that v and are approximately expressed as the value at Z= T/2 on the center plane of the ship, we obtain the following:

v==_we_k0n/2sin03sin(k3xb+wt)

-

wze_k0T/2sinô3cos_{(k3xb + wet)} (3.12) From (3.9) and (3.12), we can calcuate the whole sway exciting force

JF',, dx and the yaw exciting moment M=J F'fl,,xhdXb. Putting here =

Me/a

cco50).t + M'sSt flWt we can obtain F,,= - pg J kQS,SOcosk.xdxh - wwsin 03 J m' e_kün/2cosk3xbdxb -. wsin 03 J Nfle0T/2sink3xbdxb

-

PJkoSwSoSiflk3XbdXb ±wwsin03 JL m3e_AOnlzsink3xbdxb - oiS fl 03 J LNe_k0T12cosk3xdxb

.',

r-T

where S=

ek0sin(Cyb)dzb ,'oJJw o

and c3=3ir/B0(B0=8 m in our long towing tank). = - pg J

L

kS33S0x0cosk3x0dx0

- wwsin03 JL030

- wVsin 03 JLmoe0TIZsink3x0dx0

wsi nô3 JL N0e 0T/2xbsinkoxdxb,

'

gJ L kQSSoxbsink3xbdxb (3.13) (3.14) (3.15) (3.16)

where

+ wwsinO J m'e- 0n/2XbSI flk,XbdX

wVsinû, JL 3h b

- WSi n O

In the next place, putting the roll exciting moment as follows:

M,,M+M62+ M,3+M,,, (3.17)

we can approximately obtain such as:

M01=moment based on Froude-Kriloff's theory

M,= SL m'(1,7 O,G)dx, M,3= L N,v,(1,i1G)dx,,

M,= -

a (in'(1_-

_OIG))dxb J1. "aXa

=f

çam'

a G

-

(m'1) } vY,dx,. Consequently, putting Mg/C,Moccoswet + M0,sinwt (3.18) M. and M,,, are given by

M,, = pg L k,S, O,G + (P, - R,) T) coskx,,dx,,

- WW,SiflO:, IL m' (1,Ö)eoT 2cosk,xhdxb

- wsi nO J N,(1,, O,G)eont2sink3x,,dxb

(3.19)

M,,= - pgJ k,S,. (s O,G+ (P0_R0)T} sink,x,dx,, + ww,sinO, IL m' (1,, 0G) e0T/2sink,,x,dxh

12 F. TASAI

2 _{(1 e°0'by0sin}

CJybdyO ekobr,sin C4YbdZh) (3.20)

P0 R

_{k TS, tJ0}

_Jo

3. 2. Equations of the sway, yaw and roll motions based on Takagis new strip theory

Dr. Takagi° has developed the equations of longitudinal motions which

seem to be more rational than those of the O. S. M.. We will derive the equa-tions of lateral moequa-tions from Takagi's new strip theory.

In Fig. 2, a certain point (X, Y) on the space-fixed coordinate system can be expressed by (x,,, y) of the hull fixed coordinates as the following form:

-Yb=(Y+)OiG O+(X- Vt)ço,

xb=XVt.

Putting U for the fluid velocity in the - Y direction on the fixed cross

sec-tion of the hull, U is

U=(OlG6+xb,Vp).

(3.21)

Under the assumption that -(çbs+içLA) is the two dimensional velocity poten-tial when an arbitrary cross section oscillates with periodic velocity ett, çb,1

Now, let the whole velocity potential be 0, and we can calculate the pres-sure by the following linear equation derived from the Bernoulli equation

'a

pp0= - pg;

-

v--)

where

(3.25) of two dimensional radiation potential is given by

çb1=U (ts+i1).

(3.22)

For the sway, yaw and roll motions, putting as follows:

(3.23) q O=Oewe1o) becomes +

(n0s) WeÖl0 + OGçO.

(3.24)

0 =

(3.26)

çb= incident wave potential = radiation wave potential

ç= diffraction wave potential.

Fisrt of all we compute the pressure p) from çbri and obtain the following:

p«r)p

_{- (± V}

_{)+ Vw}

-

((

Abo)

-

(2v+oxn_

±

+ (v' + VWx,,t

+ VWeÇbA)) - OÔçi5O +

(wOG

A

+v')

-

Vw0OiGO]

(3.27)

In the next place, putting the following for the radiation potential when

the ship rolls about Xb axis

- (cYs+1/'A) Ô

-

+çYAwO (3.28)

and substituting this into (3.25), pressure p° becomes

p2=p ['sÔ çb'Aw

_VSaO

+ Va

0)01 (3.29)

Therefore, the whole radiation pressure p) is given by

p() =pt (3.30)

Herein, taking into consideration the following:

2pcbSdzb=_m',

2pJ0

_{(:) dz=}

-2 .10- _{çf.',dz0 =} -± am' ax0, N,, 2p I çbAdzb=

J-r

w,,

2pJ0

(aA)dZ_

w' 2p .Í1. çt/Aa'zb o

3çb'S\dz a _{(m'i0) 2pJ} (a'A )dz0= 1 a(N01,,)

2pJ(j

aX, ' - aX. , aX.

and making use of (3.27), (3.29) and (3.30), we can calculate the force dF,,0/

dxi, in the -Y direction acting upon an unit length cross section as shown in

the following form: - dx0

aN

aX0

14 F. TASAI

-- V

(m'1))

- v[a"

oG} O.

(3.32)

Coefficients for the sway and yaw motions in the left side of the equation (3.8) are computed from

r _{dF "} r

_{dF '}

j (ni -

---)

dx, and j (mì

-)xdx,.

L dxb L dx,,

These coefficients are also shown in Table 2.

The roll moment M, caused by p" is written, using the pressure difference

pC)=p)_p&) at the symmetric points on the cross section contour, as

f_01G

M,=

p01"y,,dy,,+ p,,"zhdzh.

Jo -T

Hereafter let us use the shortened signs of

p [] for p

[ jcbsy1dY + J' cbszo'].

Using these signs, we obtain the following:

p[ç,] = m'(1-01G),

PAI=

1r,

[:

i = -

(m' (1,, O,G)), p

[a4]

(N, (1,, O)),

p [i)',] = - (1's m'140,G), a

i = - aX,, (I's m'1 0,G),

[c'J

₌ N(1,.

-[a] =

a (N,,(1,,_ OLG) i,,)..

Taking the above into consideration, we will calculate the roll moment act-ing upon an unit length cross section.

That is as follows:

dM9,

- (1 '-2m'lO,G±m' 1,, 0,G2) O

dx,,

-+ [N(1w_ 0,G)° -+ Va

(I'0m'1O) + V0IGa

(m'(l O))] O

[vaa (N,,1(10,G)) + volGaa (N(1

_-

01G))] 0+

+ [

'

N(l,,

-

+x,,(m'(l,, 01G))]

(3.33)

X3,

(m'(l,,OÒ))

+ [1w 01G)

V

_102G))+v

a X;, ₂ ax;, We +

(m'(12 01G))VN(1, 01G) Vx;,

_aX,, (N (1m- b1Ö))1 ca

+ m'(1 O1G)i + [N(133_ O,G)- V_aX,a

(m'(l

01G))]

V (N(l,.

0G))

. _(3.35)

Coefficients of a.1, a23, C33 are also shown in Table 2. a33, that _s,

coefficient of restoring moment of the roll is assumed to be WGM.

The above radiation forces were calculated by assuming that the fluid is

perfect fluid. In the roll motion, however, viscous resistance is pretty large

because of the eddy-making resistance caused by bilge keel and the skin fric-tion of the hull. And consequently a2 should include viscous damping coeffi-cient in order to correctly predict the roll motion. This will be discussed in

chapter 5.

3.3. Differences in the coefficients of equations of motions

We will compare the coefficients of equations based on the O. S. M., Taka-gi's new strip method and the S. T. F. method with each other.

The end effect of the hull is considered in the S. T. F. method. Excepting this end effect, however, the coefficients of the S. T. F. method and those by Takagi's new strip method show agreement. For example, coefficients of i and ca can be written as in Table 3. Because ca

is a periodic function of w, we

obtain a21ca= Including a,3 in a2, we get

c': a,,

a21=J + J m'xhdx,,+

_JL""

JNx3dx3+

J ,n'dx2.

o

Table 3 Comparison of coefficients

S. T. F. Method New Strip Method

J m'x;,'dx;, f

w

j

J _{+ J m'x;,'dx,,--- -- jN3xdx2} VIL N3x,,dx,,V' Jm'dx And therefore, this accords with the one by the S. T. F. method.

In the next place, differences in the coefficients between the new strip theory and the O. S. M. essentially admitted only for the yaw motion. This says that there are no differences in coefficients for the sway and roll between both methods.

16 F. TASAI

Differences exist for a22, a23, b23 and c23 of the yaw. As shown on page 31 in the paper (8), differences in coefficients for the longitudinal motions be-tween Takagi's new strip method and the O. S. M. don't appear in the equation

for the heave but in the one for the pitch.

This resembles the case of the

above lateral motions.

3.4. Diffraction wave force

We will derive the diffraction wave force using Takagi's new strip method.

Lt is assumed that the equation (3.12) can approximately show the

com-ponent in the Y direction of the orbital velocity of water particles. The equa-tion is

_we_nÍ2 sinO3 sin(k3xh+wel).

Using v, the diffraction wave potential ç52 can be approximately given by

çba = i (ç, + içb).

From (3.36) and (3.25), we can obtain

pd) _{- pçL + pwFçb21+ p V(T,,}

-where

j

w°e-°°'2 sinû3cos (k3x0+ w01)

= Çe- OOT/2Sj n 03 cos (kaxb + Wet).

Using (3.31) and computing the sway exciting force and yaw moment, we

lind that the sway force agrees with the one by the O. S. M., though; as for

the yaw moment, the following terms are added to the one by the O. S. M.

M/=

Vo)

_sin

o _{L e_k0T/zNcosk3x0dx0}

(3.38)

Mes/

_-

_:.:::' sinû3f e_l0TI2Nflsink3xbdxt,

The roll exciting moment M0 computed by using (3.30) and (3.34) accords with the equation (3.19) obtained by the O. S. M..

As can be seen above, differences in the diffraction wave force between Takagi's new strip method and the O. S. M. exist only for M0. Concerning the sway exciting force and roll moment, both methods show perfect agreement.

The equations of the yaw moment added to those of the O. S. M. resemble the terms of on page 31 in the paper (8).

4. Model Experiments

We used the N-ship model which was described in the paper "A Study on

(3.36)

26

-Fig. 3 Bodyplan of the model ship Table 4 Principal particulars of the model ship

Description

Length between perpendiculars _L

Breadth _B

Depth _D

Draft _d

Displacement W

Block coefficient C8

Waterplane area coefficient

_c,

Prismatic coefficient C0

Center of buoyancy forward of midship Center of waterplane area forward of midship Radius of gyration for pitch and yaw

GM

01G

Natural rolling period T0

Without bilge keel. With rudder and propeller.

Model Ship 3.00 m 0.5012 m 0.2500 m 0.1686 m 206.48 kg 0.815 0.870 0.822 0.0655 m -0.001 m 0.235 L 0.072 m 0.0336 m 1.244 sec

Seakeeping Qualities of Full Ships".' The principal particulars of the model

are shown in Table 4 and the body plan in Fig. 3.

DULWARK

18 I-. TASAI

The experiments under the full load condition were carried out

in the

large tank (LxBxDxd=80mx8mx3.5mx3m) at the

Research Institute for

Applied Mechanics of Kyushu University.

4.1. Free rolling experiments of the advancing ship model

We made free rolling experiments of the model advancingin the still water. The experimental equipment are shown in Fig. 4. The sway displacement is perfectly restrained by the two guiding devices (the rolling axis is pivoted).

Towing Apporafus and Guide Devices

A Rolling guide B: Towing rod C Piano wire D Pivot port E Trimmer F: Vertical gyro

G Center of Gravity of the model

H:Ba// bearing J : Truck

Fig. 4 Towing apparatus and guide devices

The ship model was towed by the piano wire joined to the towing rod. A ball bearing was used where the piano wire was joined to the hull in order to lessen the effect of the twisting moment of the piano wire. The effect of the friction of each part is small enough to neglect.

The angular displacement of the roll was measured by the vertical gyro-scope.

Experiments were conducted by varying the initial heeling angle O, as

follows: 50, 15, 180. The extinction curve in the experiment of O=18° is

assumed to be AO-aOm+bO,,. In Fig. 5 are shown a and b (1/deg) of

coeffi-cients satisfying both O,=50 and O,,= 100 of the mean rolling angle.

Within the Froude number Fn used in experiments, a increased and b slightly decreased with the increase of the ship speed.

Furthermore, we calculated the mean value of Bertin's extinction coefficient N=4O/O2 for 0=8°, 100 and 120. It slightly decreased with the increase of

On the other hand, the value of N in the case of O,,,=10° at _{zero speed}

which was obtained from Watanabe and others12 _{was 0.0064.}

4.2. Experiments in head waves with three nodes

We let the model ship self-propel on the node line in three nodal waves and measured the amplitude of the sway, yaw, roll motions and the relative

bow motion.

We carried out experiments under the head sea condition, _{so that we might} get the roll resonance for each speed of our ship model with _{a short natural} rolling period. (In the case of a container ship model whose natural _rolling

period is relative]y long would be adopted the following waves.)

Motions of the ship model were measured by means of the equipment for measuring six motions of a ship model'3. _{Moreover, in order to measure the} amplitudes of the roll and yaw were used angular differential _{transformers.}

We used a manual steering apparatus in order not to let the model get off

the nodal line in three nodal waves and the model ship was steered as few

times as possible to lessen the effect of steering on the roll motion. The test conditions in waves were as follows:

A0/L=0.51 .7,

H/Aúzir1/50.

The wave height H was measured on the crest and trough line at the dis-tance of B0/6 from the center line of the tank.

Some illustrations of records of ship motions are shown in Figs. 6 and 7. In the analysis of the experimental results, we used the records while the model wasn't steered, as shown in Figs. 6 and 7.

5. Comparison between Experiments and Computations

5.1. Computations of ship motions

Computations of the sway, yaw and roll motions were carried out accord-ing to the O. S. M. and the new strip method, by usaccord-ing coefficients _{of a11, a7,}

c _{in Table 2 and the wave exciting force in (3.13)---(3.20) and (3.38).}

The two dimensional values of m', i, 4, and N, were calculated by ap-proximating the cross section of the hull to the Lewis form.

For the following items, we used experimental results. Table S Experimental values of N coefficient

F O 0.10 0.13 0.16 0.20

T=27r/(J,+I)/W.GM.

(5.1) 2) Damping coefficient ao for the roll

In order to consider the viscous roll damping moments in addition to the wave damping, we used, in this paper, the damping coefficient obtained from the free rolling experiments. (In principle, it is more rational to get it from the experiments by the forced oscillation method.)

As shown in Watanabe' andTasai5, equivalent linear roll damping is

2

a32/J+I==2c, = w0 (a+bAO,) (5.2)

ir

where w0=2ir/T0, A=T8/T,=a,,./w0 and T, is period of encounter. Putting

= otH/À0, (5.3)

O,Ei

j

(5.4)

the equation (5.2) can be written in the form

a22=(J+I)4-fa+b-'° (w+k3V)&,,

(5.5)

Using H/A3'l/5O in the case of experiments, that is, 6Ç=3.6; the equa-tion (5.5) becomes as

a32=g,a+g2b(w+k3V)iT (5.6) where g and g2 are known constants.

In this paper, we compute by using coefficient N and in this case, from the equation (5.6) a3, is given by the following:

a,2= 1.3356N(w +kT') (5.7)

We used the values of N shown in Table 5.

We will explain here the concrete computation method.

In the first place, we solve the equation (3.8) by putting a32=O and obtain

the value of-=o,/,,

=ç,/kÇ,

=O,/k,CO3 r,,, eQand r,.

Using this computed O and the equation (5.7), we can get a32, which is used in solving again the equation (3.8).

20 F. TASAI

I) Apparent moment of inertia for the roll

Because there's little knowledge about the three dimensional effect for I,

in

this paper we calculated J+I from

the following equation, using the

natural rolling period T0 obtained from the free rolling experiment in the case of V=O.

In this way, computing the equation (3.8) by the iterative method, we finish computing when the error of convergence of O, and others settles within 0.1 percents.

The computed results of s, and O according to the O. S. M. are shown in Figs. 8, 9 and lo.

The response curve of the sway amplitude in Fig. 8 has a hollow near the frequency of the roll resonance. This phenomenon can also be seen for the sway amplitude in beam seas5. Such phenomena are due to the coupling of the roll into sway in the roll resonance condition. We will call these

phe-nomena Coupled Resonance. This will be discussed in detail in the appendix. The response curve of the yaw ampiltude in Fig. 9 also has a small hump near the roll resonance condition for the same reason as the sway.

In Fig. 10, the roll amplitudes in the resonance condition become larger

with the increase of ship speed. For MC, there are few differences between the case of F,=O.l and the case of F,=0.2, and it can be inferred that the effect of the ship speed upon the roll amplitude in Fig. 10 is due to coupling of sway and yaw into roll.

In the next place, theoretical results and O computed by the new strip

theory show good agreement with those by the O. S. M.. There's little difference

only for the yaw. This can be seen from the curves of a solid line and a

dotted line in Figs. l5l8.

5.2. Comparison between computations and experimental results

Comparisons between Computations and experimental results for the ampli-tude of motion are shown in Figs. 11.22.

For the sway amplitude in Figs. 11-44, experimental results and computa-tions show satisfactory agreement.

For the yaw amplitude in Figs. 15-18, experimental values are considerably

scattered for A/L:l.3. But it can be generally said that the agreement between the computed and experimental results is satisfactory.

In the comparison of the roll amplitude in Figs. 19-22, the computed values are larger than the experimental ones by 10'-20 percents in the resonance con-ditions.

6. Concluding Remarks

In order to derive the rational method of prediction for the sway, yaw and

roll motions in oblique waves, we made such investigations as already

dis-cussed in this paper.

We have derived the coupled equations of the sway, yaw and roll motions not only by the O. S. M. but also by Takagi's new strip method.

22 _{F. TASAI}

viscous effect and the effect of the ship speed from the free rolling experiments in the still water. And we have let a full ship model self-propel in three nod-al waves and measured amplitudes of the sway, yaw and roll motions.

Subsequently, using coefficient N obtained from experiments and equations of motion based on the O. S. M., we have computed amplitudes of the _sway, yaw and roll motions in three nodal waves according to the iterative method, and the computation has been also carried out by the new strip method.

Finally, we have compared computed results with experimerntal results. These investigations may be reduced to the following conclusions.

The derivatives of the radiation forces given by the new strip theory show agreement with those by the S. T. F. method excepting end effect.

Difference in them between the former and the O. S. M. appears only for the yaw. This circumstance resembles the one for the longitudinal motions.

Difference in the wave exciting force between the new strip method and the O. S. M. also exists only for the yaw.

The computed values of the sway and roll amplitudes of the full ship model in three nodal waves differ very little between the new strip method and the O. S. M.. A few differences appear only for the yaw.

The agreement between computed and experimental amplitudes of sway and yaw is satisfactory.

The amplitude curves of the sway and yaw have a hollow and hump near the roll resonance condition. This is due to the coupling of roll into sway

and yaw.

The computed values of roll amplitude are larger in the resonance condi-tion than the experimental ones by about 10-20 percents.

Investigations to proceed further would be listed as follows:

Improving the computation method of wave exciting forces for an advan-cing ship, in particular, the roll exciting moment.

Measuring radiation derivatives by the forced oscillation method and check-ing up the equations of motions in this paper.

Developing a theoretical equation or an empirical formula to predict correct-ly the roll damping moment including viscous effect and the effect of advance speed.

Acknowledgments

The author wishes to express his deepest appreciation to Mr. M. Kurihara

who carried out numerical computation and he is also gratefall to Mr. J-i.

Arakawa, Mr. T. Kita, Mr. A. Tashiro and other members of the laboratory

who cooperated in the model experiments.

It is mentioned that we used in numerical computation FACOM 230-60 of the Computer Center of Kyushu University.

References

Eda, H. and Lincoln Crame, C., Steering Characteristics of Ships in Calm Water and Waves, S. N. A. M. E., Vol. 73, 1965.

Takezawa, S., Fukuhara, M. and Yamashita. S., On the properties of the Tran-sient Water Waves in Ship Model Basins, J. S. N. A. of Japan, Vol. 124, Dec.,

1968.

Kurihara, M., Tasai, F. and Suhara, T., On the Experimental Tank for Sea Disas-ters, J. S. N. A. of West Japan, No. 32, July, 1966.

Awaya, Y., Generation of Higher Nodal Waves in Long Basin, Bull. Res. Inst. Appi. Mech., No. 17, 1961.

Tasai, F., Ship Motions in Beam Seas, Rep. Res. Inst. AppI. Mech., Vol. XIII.

No. 45, 1965.

Tasai, F., On the Swaying, Yawing and Rolling Motions of Ships in Oblique Waves, Rep. Res. Inst. AppI. Mech., Vol. XIV. No. 47, 1966.

Tasai, F., Improvements in the Theory of Ship Motions in Longitudinal Waves, Appendix II of the Seakeeping Committee, 12th I. T. T. C., 1969.

Tasai, F. and Takagi, M., Theory and Calculation of Ship Responses in Regular Waves, Symposium on the Seakeeping Quality, S. N. A. of Japan, July, 1969. Salvesen, N., Tuck, E. O. and Faltinsen, O., Ship Motions and Sea Loads, S. N. A. M. E., Vol. 78, 1970.

Tasai, F., On the Equation of Rolling Motion of a Ship, Bull. Res. Inst. Appl. Mech., No. 25, 1965.

Tasai, F., Kurihara, M., Arakawa, H., Kawasumi, K. and Kita, T., A Study on the Seakeeping Qualities of Full Ships, J. S. N. A. of West Japan, No. 37, Feb.

1969.

Watanabe, Y.. moue, S. and Murahashi, T., The Modification of Rolling Resis-tance for Full Ships, J. S. N. A. of West Japan, No. 27, March, 1964.

Tasai, F., Kaname, M. and Arakawa, H., Equipment for Measuring Six Motions of Ship Models, Rep. Res. Inst. Appl. Mech., Vol. XIII. No. 45, 1965.

Watanabe, Y., On the Motion of the Centre of Gravity of Ships and Effective Wave Slope, J. S. N. A. of Japan, Vol. 49, 1932.

Takaki, M., Arakawa, H. and Tasai, F., On the Oscillations of a Semi-Submersible Catamaran Hull at Ballast Condition, J. S. N. A. of West Japan, No. 42, May,

1971. (to be published)

Ueno, K., Theory of Free Rolling of Ships, Memoirs of the Faculty of Engineer-ing, Kyushu Imperial University, Vol . 9, 1942.

24 F. TASAI

ce,+be'

W 0.13 0.16 0.20

-F,

Fig. 5 Coefficients of the cuve of extinction

SteerIng S1i1

Thr,0o0e01Ve Hd 71 7»/L 1.770

12)V0 10)11

O)Re1at1 Ntthfl 15)Wse

Fig. 6 An illustration of records of ship motions and wave; 21/L=1.315 çwave was recorded on the line of trough and crest)

Ste.r1nS.1gn1

(1) 0w, 121 0. (3) 9011

(II) RelativeROCMotion )5)W.vn

1.0 0.8 0.6 0.4 0.2 Sway Amplitude (Ca/cu/at/on by O. S. M.)

Fig. 8 Calculated frequency characteristics of sway amplitude

2/'

I!

_'

",/ _{Yaw Amplitude}

,'//

(Calculation by O. S. M.)

//

'vi

Fn /1/ 0./O

0/

_/,i

_--0.20

Fig. 9 _{Calculated frequency characteristics of yaw amplitude}

0.3

0.2

0.5 1.0 _{'.5 2.0}

6.0 4.0 o o

¡.0

0.8

0.6 0.4 0.2 I Fn 0./O 0.13 0.16 - 0.20 0.5

Fig. lo Calculated frequency characteristics of roll amplitude

0./o

O Experiment

1.5 2.0

Fig. li Comparison of calculated and measured amplitude of sway; F,O.lO

26 F. TASAI

P0/I mp//rude (Ca/cu/ar/on by Q.S.M.)

I .0

0.8

0.6

0.4

0.2

Fig. 12 Comparison of calculated and measured amplitude of sway; F,=O.l3

0.5

O Experiment = 0.13 1.0

F = 0(6

Calculation

0.6

o _Experiment

0.4

0.2

0 o o 1.5 O o 2.0

Fig. 13 _{Comparison of calculated and measured amplitude of sway; F=O.l6}

e

1.0

0.8

0.3

0.2

0.!

1.0

0.8

0.6

0.4

0.2

o t 0.5 F O.20 Calculation 0 _Experiment 0.10

I.0

Fig. 14 Comparison of calculated and measured amplitude of sway; F,=O.2O

o o 1.5 o o 2.0 Calculation by 0.S.M. Calculation by

New Strip Method

O

Experiment

Fig. 15 Comparison of calculated and measured amplitude of yaw; F,=O.1O

28 F. TASAI

0.3

0.2

0. I

0.5 1.0 ¡.5 2.0

Fig. 16 Comparison of calculated and measured amplitude of yaw; F 0.13

o

0.3-

Il .._

F=0.16

¡

_-

_o

0.2

0.1 o Q

r

o o Calculation by 0.S.M. Calculation by

New Strip Method

o _Experiment

Calculation by 0.S.M. Calculation by

New Strip Method

O _Experiment

Fig. 17 Comparison of calculated and measured amplitude of yaw; F=O.16

30 0.3

0.2

0. I o

6.0

5.0

4.0

3.0

2.0 ¡ .0

Fn0.20

t

Fig. 18 Comparison of calculated and measured amplitude of yaw; F=O.2O F. TASAI

/

/

/

'

_{\__ /}

/

o Calculation by 0.S.M. Calculation by

New Strip Method

O

Experiment

Calculation

O _Experiment

Fig. 19 Comparison of calculated and measured amplitude of roll; F=O.1O

0.5 1.0 ¡.5 2.0

4.0

2.0

o

6.0

0.5 I.0 1.5

Fig. 20 Comparison of calculated and measured amplitude of roll; F=0.l3 Ca/culo tio n O Experiment /L Calculation O Experiment 2.0

Fig. 21 Comparison of calculated and measured amplitude of roll; F=0.l6

6.0

4.0

2.0

o

Appendix

On the Coupled Resonance

In the response curve of the sway amplitude, there appears a hollow near the roll resonance condition, as shown in Fig. 8. Although there is a coupling effect of yaw upon the sway, this phenomenon is mainly due to the coupling of roll into sway in the roll resonance condition.

In the swaying motion of a system with one degree of freedom without the coupling of roll and yaw doesn't exist the natural period of the sway.

Accordingly, even if we omit the damping force -in this case, it cannot be happened that the sway amplitude of the forced oscillation in waves becomes infinite.

On the other hand, when the sway motion has been coupled into by the roll motion, there appears the resonance peak of the sway amplitude in the

forced oscillation.

We will discuss the coupled oscillation of the sway and roll, excepting the yaw in order to make the discussion short.

The natural period of roll is usually calculated from the equation of roll motion of a system with one degree of freedom.

Neglecting the damping moment for the roll, the equation of free rolling is

a3 Ñ+a330==O. (Al)

From (Al), the natural circular frequency w0 and natural period T0 of the roll are given by

Calculation

o Experiment

o o

___

0.5 ¡.0 ¡.5 2.0

Fig. 22 Comparison of calculated and measured amplitude of roll; F,=O.2O

w32=a03/a31

and T6=2/a3/a

. _(A2)

Putting ç=O in the equation (3.8) and considering a03=c1=b33=O, the coupled equations of free oscillation are as follows:

+ Cil C130 0

a31O+a32O+a33O+b31 +b327=O (A3)

where c11=b and c12=b32.

The natural period of the coupled oscillation can be approximately calculated from the following equations, omitting the damping forces,

a30 a3 + c11= O.

a1

+cï=o

(A4) Assuming that both 7) and O oscillate harmonically with the circular

fre-quency w, it results that a330= - Ü.

And (A4) becomes as

a + C11O = O c21i+a30O=O a33

where a30 =a31

-The natural circular frequency w of the coupled oscillation can be got

from the following equation:

c11a30 = O. (A7)

Consequently, in the case of the undamped oscillation, the amplitudes of 77

and O become both infinite if the frequency of the wave exciting force and

that of the free oscillation coincide.

For such a phenomenon of resonance as a motion originally without the natural period comes to have by the coupling effect of another motion, we will call the Coupled Resonance.

In both computations and experiments of the investigation in this paper

did not appear a large resonance amplitude for the sway, however, it is predicted that if the roll damping moment is very small, a large peak of resonance comes out.

The investigations for the coupled motion of the sway and roll for a two dimensional cylinder have been carried out by Mr. M. Takaki and others15 who omitted the viscous damping. Their calculation shows that a very large resonance amplitude of sway is generated in the resonance condition of the roll. It is reasonable that the same phenomenon as the sway should consist in

3-4 F. TASAL

We see no examples that a large coupled resonance peak appears for the sway or yaw in the case of an usual ship.

There would be, however, the need to consider the coupled resonance in the case of a marine structure with the multi-hull.

Now, let us suppose V=O. In Table 2 can be presented the following:

ao=bjo==cz=b,3==c33==0, C21 b21,

c22=b12, b21 =c31, b22=c32, b31 =c11 and b32 = e2,

Then, the equation of the free oscillation can be written, using (A6) in the form:

a+a12+ b1( +b2,+ c110 ± c120=Ø

b11+b12o +a'21±a2+b'2ï +b22Ó= ø (A9) c-l- c12+ b21ç +b22,+ a300±a320=0

where a'21=a2

a and

b'21=b21-Coefficients of a11, , a12 in the above equations are generally functions

of circular frequency w, and therefore, it is very hard to compute the coupled natural frequency from (A9).

Omitting items of damping forces, that is, items of , and Ö in (A9),

we can get graphically the coupled natural frequency of the undamped oscilla-tion by the following:

a b1, c11

-4=

b11 a'21 '21

'

=0. (Alo)

e12

b,

a30

Hereafter, we will discuss the coupled natural frequency of the sway and roll.

The coupled free oscillation of the sway and roll has been studied in

de-tail by Dr. Ueno.

In the first place, he derived the following coupled equations of the sway and roll.

(I+11h)ü+110hfi+ WGMO (MI--MlO)hl=O (M+ M1) y + M0t' - (I O + I O) = 0.

Solving the above equations of motions, he showed that the roll motion O is approximately expressed by the following form of the dampedoscillation:

O=O,secce"'cos(wtcr) (A 12)

and that the sway motion y is composed of one aperiodic damped horizontal translation and one damped horizontal oscillation, as shown in the following:

(A8)

}

sec

[e_- sin±e_tcos(w,,t_a_e)]

_{(A 13)} According to him, the center of gravity of the ship translates transversely toward the opposite direction to the side to which a ship is initially forced to list by external moment, oscillating horizontally.

The oscillation gradually damps with the horizontal shift of the center of gravity.

In the next place, he calculated the roll motion of the ship, whose horizontal motion of the center of the gravity was strictly restrained.

Furthermore, usingthe model ship (Lpp=189.2 cm, H0=B/2d=l.08,with bilge keel), he carried out two kinds of experiments; one is the coupled free

oscil-lation and the other is the free rolling of the ship model whose horizontal

translation is strictly restrained. I-Je obtained the following results.

J, The coupling effect of the horizontal motion on the damping coefficient of the roll decreased with the increase of the value of O : about 7 percents at O=O, about 4 percents at O=35 degrees.

2) The coupling effect of the horizontal motion on the natural rolling period showed as small a value as about 2 percents irrespective of O.

From the above he concluded that the influences of the horizontal motion upon the damping power and the period of free rolling are small enough to be neglected.

Now, in the case of zero speed, from the Table 2 we obtain

a, =m(I

+k),

W.GM

a30=(J+I0) _Wz

Using the above, the equation (A7) is transformed into the following equa-tion:

[ILm'mn_ Öi) dxb]

m (1+K,,)

we can obtain the natural circular frequency by the graphical method _as

shown in Fig. Ai.

In the case of an usual ship, because the value _{of K, would be positive,} w is always larger than w0 obtained from a30=Ø.

(J + I,)

-O) (A 14)

m'(l 01G)dx1'

Putting

_{(.J+I0) -}

'1

N

N

(D 3 (D 4f 36 F. TASAI

U)

2 Fig. Al Herein, putting [JLm'(ln_ OiG)dx8]2/mo(l+7<)=â (A 16) the difference between w0 and w,, becomes larger as c increases.

For example, we will compute the approximate value of 8 for a two

dimensional body, whose section is the Lewis form (H0=2.o and

From the appendix of the paper8 we obtain K0=m'/pS=0.7 and LSR/T=-0.8 in the small circular frequency. And therefore, we have m'= 1.26 pbT, 1 =-0.8T and m4=1.8 pbTL.

Assuming the 01G is much smaller than 1, and neglecting the value of 01G we obtain

8 0.332 pbT3L.

Putting ô = m(K2b) we get K = 0.107.

Accordingly, in this case, it results that the radius of gyration is 0.214b less than the one computed from the equation of roll of a system with one degree of freedom.