On the coupled motion of steering and rolling of a high speed container ship

5.

On the Coupled Motion of Steering anli Rolling of

a High-speed Container Ship

Kyoung-Ho S0N*, Member and Kensaku NoMoTo**, Member

(From J.SN.A. Japan. Vol. 150. Dec. 198!)

-Summary

Yaw-sway-roll coupling motion of a ship is investigated on the basis of captive

=0w

I

₀₀

2 :E C

w

.<

E C'4 cJ C) 0. -

o

C.) _W

jmodel tank tests. A single-screw, high-speed container ship has been chosen as a typical Itype for the study.

The smaller metacentric height naturally results in the heavier coupling of roil into yaw and sway, which affects manoeuvrability significantly: turning performance is im-proved by the coupling effect, and course-stability and quick response to steering are reduced. In short, the roll coupling lessens the hydrodynamic damping to yaw and sway

acting upon the hull.

When an automatic course-keeping device is introduced, as is quite popular in modern

navigation, another element of coupling is added: the rudder is activated in accordance with the yaw motion. This yaw-sway-roll-rudder coupling can become the cause of the heavy rolling often experienced on high-speed ships automatically steered in a seaway.

We maice use of a perturbation stability analysis of the problem to reveal the mechanics of

the unstable character of the coupled motion of a ship. Introducing rate-control to the

autopilot gives a remarkabie stabilizing effect.

1. Introduction

In recent years the yaw-roll coupling has drawn an increasing amount of attention1'"".

This phenomenon becomes particularly signi-ficant when:

A ship has rather a small metacentric

height and thus she tends to heel over

with steering.

A ship is operating at hioh-speed where

hydrodynamic heeling moment caused by yaw and sway becomes considerable. These situations often occur for modern

high-speed container carriers, RO-RO ships, and

some kinds of swift naval vessels.

We take a single-screw, high-speed

con-University of Osaka, Graduate School, Dept. of

Naval Architecture

University of Osaka, Dept. of Naval

Architec-ture, Professor

73

tamer ship as a typical type for the presenti

study. On the basis of captive model tank tests with varying heel angles, a set of equations of yaw-sway-surge-roil coupled motion are deriv-ed. The equations are employed to predict the

hard-over turning performance and zig-zag

steering behaviour of the ship, taking into

account the effect of roll motion. We also

use of the same equations,

together with

another equation for rudder control, to investi-oate the mechanics of the instability of the yaw-sway-roll-rudder couple&rnotion. This

instabil-ity can induce a self-exciting, heavy rolling

coupled with a considerable yawing of the same frequency.

2. Equations of Motion

Fig. I shows the co-ordinate system to be used. Neglecting the effect of pitch and heave, we obtain the fundamental equations of surge,

eL _{dk dû} (M) 1,2) (A7)

4r cos 6)

(AS) at ::iex integra-LC _{using the}

:')}

=l, 2) (A9)

4

y.

Fig. i Co-ordinate system

way, yaw and roll coupled motion:

rn(úvr)=(all the surge force)

rn('b±ur)=(all the sway force) T. =(all the yaw moment)

L=(al1 the roll moment)

m denotes the ship's mass, and I and

rer moment of inertia about the z and x

.x;:. respectively.

ccording to the established procedure of [ea:ng with hydro-inertial terms involved in

he ght-hand sides, and also introducing the

ta::c transverse stability moment included in

he roll moment, Eqs. (1) becomes

(m±rn)d(m±m)vr= X

(m+m)'b±(±m)ur

+m.fmlç= Y

(J±J» +m'b=N Yx

(J +J) - rn.1 l. 'bmir ur

WGMçi=K,

here X and Y denote the hydrodynamic

)rces (ex. hydra-inertial forces) in the x and

directions respectively, N the hydrodynamic aw moment about the midship, xa the distance

I C.G. in front of the midship, and K0 the

ydrodynamic roll moment about C.G.

_m,

,

J and ..Ç denote the added

mass and

ded moment of inertia in the _{x and y} direc-ns and about the z and x axes, respectively.

denotes the x-co-ordinates of the centre of

,, and i and i. the :-co-ordinates of the

cen-(1)

(2)

Kyoung-Ho SON and Kensaku NoMoro

tres of m and m, respectively. The

hydrody-namic forces and moments are written down using hydrodynamic derivatives as follows:

X=-?E_LV

_{(X'(u')±(l - t)T'(J)±Xo'rv'r'}

±X v' ± X,Ç. '

±.3Ç ç3

r i

-r- 51fl

Y=P_L!V2[Y,vi± Yr'± Y''+Y

± Y,Ç,v'3 + Y,'rrT'3 ± Y..,'vrV'2r' ± YÇ v' rl! ± Y y!!

_{Yv' ç}

r Vi i_t r Vi

.T.Lrr#r p-1-A,r 9

±(l +aH)FÇ cos

- +NvV'3±i'Çrrt'3 ±N,,v'r'

+Nvrrvu'lz ±Nv' ±N6 '

cos

±KÇv'3 ±K r ± K:vrv''

±Kv'r'2 ±K0vtz ±Kv'ç

(I +aM):E.Ç cos 8}

(3)

Definitions of the hydrodynamic derivatives Y,... Y0',,,,, etc., are widely used nowadays

and nearly self-explanatory, but if there is

any ambiguity references should be made to the MMG Report6 and Reference 7 in that

order. It should be noted here that the hydra-dynamic derivatives Y,,', . . . Y,,',,0, etc., relate

only to forces acting upon the naked hull.

The forces caused by the rudder are represented

by the last term of each formula of Eqs. (3),

i.e., cFÇ sin 8, (1 ±a,,) F, cos 8, etc.

The rudder force FC, can then be resolved as:

FÇ=-63'1

_(U±V)5jflZR

(4)

(5)

u')=i4 T

where

J= uV/(n

u,=cos V'

±r«v'

= yv' + 0R

This rather

cori-the MMG Repo 7. We employ data from Refer this form. In t;

to rewrite the

acting upon the

in the same

fori-FC, terms. In derivatives Y,,',

their value to

i and rudder-hull 3. Test Model a

Table i and

screw container study. Since it

SR 108 project

Research Associ

have been done

extensive captiv rability predictic mitsu1'. To their resul (1) An oblique

of heel to

-Or , wt_ ¡2 A.P

UR=U;C ./l±8kKrI(J)

where

J= uV/(nD)

u=cosv'[(I-w)

+7 ((y' +XJprI)! +Cpril+Cprt'} I

V/7J±cjrr±cR,.rrr3 --cRrrvrv'

(7)

This rather complicated form is taken from

the MMG Report6> and partly from Reference

7. _{We employ many of the hydrodynamic}

data from Reference 7 and accordingly follow

this form. In the end, however, it is possible

to rewrite the whole hydrodynamic forces

acting upon the hull and rudder and propeller

in the same form as Eqs. (3) but without the

and rudder-hull interference.

3. Test Model and Experimental Results

(2)

Table i and Fig. 2 illustrate the

single-screw container ship we chose for the present study. Since it was originally designed for the

SR 108 project by the Japan Shipbuilding

Research Association3>, many investigations

have been done with this type, including an

extensive captive model test for

manoeuv-rability predictions by Matsurnoto and

Sue-mitsu7>.

To their results we have added:

(1) An oblique tow test with various angies of heel to define the sway-roll coupling

' cg.

On the Coupled Motion of Steering and Rolling of a High-speed Container Ship 75

A.R _{'iz J £}

(6)

Table I Pthcipal d>ensions of SR 108 container ship

F ro pr Il sr

.0.

_-0

Fig. 2 Lines of SR 108 container ship

derivatives.

Measurement of the roll moment exerted by rudder deflection to define the rudder-to-roll coupling derivatives.

-LSM-fltttO o ..ØØOQi

-0.0002

Fig. 3 Longitudinal force coecient due to roll angle

9'/z

Sai? 4500hZ. )iO.i1 Isagta e. P. I. (a) 175.00 3.00

ea..dtb I (a) 0.535 Praucho rar. 6 (s) 1.00 0.1371

. 00 0.1543

405e d (a) 1.50 o. usi

1.0.1. trae F.?.

d1U. Of gyration shoot a-aojo

aLlg. )..1 (a) OS1i 1. 0.24 45.70 O_7S bap4tt (va) 41.0 0.7754 Rudder Area a') 33.2376 0.0070H

height H (a) 7.7543 0.133

Aspect ratio A 1.0219

Area ratio 11(1.0

F, terms.

In this case,

the hydrodynamic

derivatives . .

. Y, etc.,

will change

their value to incorporate the rudder force

Oiater D

Pitrh ratio p

t,pand.d ars. ratio

00es ratio Noebcr of blades (a) 6.533 1.000 0.47 0.11 0.112

0sPl.cacC value. - (a')

height (roe n ita transverse

21.222 o. liss6

- t.CsOtr. 0M (a) so.3 0.1741 Height trae Oni to centre of

hoosoo-y 05 e) 5.6154 0.17912

iiaoh oacftii.sat 0.559

Priatic coo!. 0_S'O

w.t.rpl...conf. 0. Ç

hids4ip sentita ro.!. o.,"

hydrody-tten down >llows: y' r'

,t; '

(3)

:rivatives .:'.vadays ìere is

'ade to

-n that -, hydro-relate

¿.:j hull.

-escnted

qs. (3), e resolved n

(4)

(5)

76 0. 0 2 0+ ti

.A

Ç Di 0% 8 g

'v

lo G C -20' A Y'lHuuI otily) -,-0.004 Q

roll angle with drift angle

The oblique tow results

are shown in

Figs. 3, 4, 5, 6, and 7. Fig. 8 indicates the roll moment versus the lateral force, both of which are exerted by n.dder deflection.

The marks in the figures represent the meas-ured data and the curves the least square error :ittings. The regression formulae are noted by

the figures, where the coefficients define the

hydrodynamic derivatives.

Fig. 5 is perhaps particularly interesting

among these results for it suggests the key to the roll-to-yaw coupling mechanism. Suppose

a ship turning to starboard; she moves obli-quely to port and at the same time leans over

to port.

That means that a ship turning to

starboard has a positive fi and a negative

(cf. Fig. 1). _{Fig. 5 indicates that a positive fi} and a negative _{generates a starboard} turn-ing moment (positive N'); the greater the heel angle , the greater

the turning moment

becomes.

Now we can see the sequence: a

Kynung-Ho SoN and Kensaku NoMoro

10 G C

11 A A

12 .

-0028 5' *0 00 9*' i-O. OOD4

.O.019O5Ç,' _O.00SO%l,,8

Fig. 5 Yaw moment (hull only) coefficient due to roll angle with drift angle

e'.-000002t.

-0.0001"

---:1 D.t

Ie!cnt froto keel 00 center of roil ocoent 0.205 n

Fig. 6 Roll moment (hull only) coefficient due to roll

angle

ship makes turning, she heels over, and the heel generates even greater turning moment.

This is a sort of positive feed-back. By this

sequence the roll-yaw coupling genralIy

en--o. C Fig. 8 Relation der deec Courages turni yaw-damping, and quick steer

On the bas

we have estir

including the roll damping been employed j Eventually derivatives and 2 and 3. -0.0035 Fig. 7 Relation b oblique ru a k -0.002+ L SU-f lttlrtti C 13 I. 0' 0 y.006D'OC9O''O.00QC52 .O.O'6CS.O.lC3O4tt+'

20+

hi A 5

o.

7 .

Fig. 4 Lateral force (hull only) coefficient due to

S.c

_{g vv}

-0.001

(hiul I only>

- itEi,,q

ue to roll

: center

3.205 o

rst due to roll

;er, and the

g moment. ck. By tbk ;nerally en-- K' coio 05(y) -0.20 - o.

(4eI't fr loe) to ceste, of roll nt 0.2%.

0X2 0.

Fig. 7 Relation between K' and Y' (hull only) under

oblique running test with heel angle

Fig. S Relation between K' and Y' induced by

nid-der dedection on the straight running

courages ttirning and reduces the effective yaw-damping, thus spoiling the course-stability and quick steering response.

On the basis of these experimental results

we have estimated sorne other derivatives

including the yaw-roll coupling ones. The

roll damping data of Reference 9 have also

been employed.

Eventually we obtain all the necessary

derivatives and coefficients as listed in Tables 2 and 3.

Table 2 Hydrodynamic derivatives and coefficients

94w nt . kr00 Id.h&p

loll -

).oO.od '50* of oo.o)Ey . C.C.

o Fn 3.2 lIsio resoto1.os XmO _{Table 3} Hydrodynamic derivatives around centre of

)I,t .

) 'ity (KG = 10.09 rn, GM =0.3 rn in full size)

of rol) 1t ) 0.2%

N

(4 '0) 054.3) (00))

o

7' ) t; Y. t;,, t;,, t;,, t;,, 'Xi 9. 0' ut: PL1.5 'Cotto 0. 3 1.4)47 0.000175 0. 000021. 0. 700 3520 3.0002205 -0. 0 1.20 35 0,00522 0.0 -.0.00007 04 0.240)64 0. 003 005 0.0093447 -0. 001.3 523 -0.002074 -0. 00243 -0.0034436

4. Manoeuvring Prediction Taking Roll Coupl-ing into Account

Solving Eqs. (2) and (3) with the numerical

data from Tables 2 and 3 makes it possible

to predict the ship motion induced by any

given rudder execution.

Fig. 9

illustrates the turning paths and

accompanying roll angle time histories with

a rudder deflection of 15°. The smaller

meta-centric height (GM) naturally results in the

greater roll angle with the same rudder

deflec-I oc.L4.00 I, 04-0.3 0.01 .00* I .4 40)0.0. 000.7 0.0070 O r; 0.0 4,, -0.010)I 0.0002)4 -0.00004) 0. 2053746 0.007040 V t -0.100 o;,, -0. 0034512 0.0000l4 0.00177 o;,, 0.0024 1,) J. 0.0000014 0.0214 I' 0.000)024 3. 0.000456 1. -0.0400 -0. 00006) .0 0.000419 0.00405 0.0. 1?n0 .11 0.00 0 0.0)13 z. 0.031) y

r,,

7. t',4 b.00. 0.000325 -0.001)41 E' 0.2 r,, (mo,2) -0,000021. ET 0.527-O. 450.3 4; -0.0031045 E.. 0.002443

r

-0.0004206 o; -0.00123 E. -0. 00 004 6) X. -0.00312. 0 0.000213 -0, 0005 II -0.00)44 o; -0,0071424 o,., 0.0010565 X. 0.00000 0' 0.001492 _o;', -0. 00 1201.2 X._o, -0.00020 N -0.052)4 044 -0.000079) t, -0.0114 Y. 0.00242 t;,, 0' -0.0424 0.001.54 o;,., e.,, -0. 0002 030.0000)009

b) 700PCLLER ANO 0)/OXEO

O 7.1.0 (F54 2.3) 'N 0.237 3.71). (op.) 1.14.04 (CO 0.3) -0.44 0.431. 154.19 (Fo 0.4) "z 0.70. 0.044 (050) (1-0) 0.125 4 0.3)) 0.113 (''CO) (1-o) 0.414 _C. 3.0 -0. 156 -0.5 _tpo 0.0 -0.275 o' -0.324 1.00 1.06

r,

o;,, 4;,, 74" o; p Ort the Coupled Motion of Steering and Rolling of a High-speed Container SOs

o;,, e;4, 0 3.30021.2 -0. 0001.0 41 -0.01.11.91 -0.000399 -0.003644 0.002)443 0.001:6 -0.000021. E 0,000314

,t'E)

-0.0000692 -0.0012094 -0. 0 000744 -0. 0 00)441 0.00003 524 0.0000155

tion; this in turn makes the hydrodynarnic

yaw damping the less and thus turning path

becomes the tighter.

A similar trend

is also seen in zig-zag

manoeuvre shown in Fig. 10: the roll coupling makes a ship less stable on course and slower fn response.

In both cases the effect of the roll coupling

.an be considerable. This is especially true

when the metacentric heihct is small.

5. Unstable Behaviour Induced by Yaw-Sway-Roll-Rudder Coupling

We have already pointed out the unstable

aratr of the

roil-yaw coupling: once

:ven a yaw motion, the yaw induces roll and

:he roll accelerates yaw even more. Together

the rudder movement in accordance with

aw motion,

this unstable character can erate a sell-exciting, roll-yaw coupling

;illation of an automatically steered ship. ::ause it is of a self-exciting type, this

oscil-:Lion can become really wild.

Taggart suggested this type of

yaw-iiuced roll as early as 1970 and recently Eda carried out a digital simulation study based on

captive model tests to indicate the feasibility

f this kind of coupling oscillation.

We will perform in this section a

mathe-matical analysis of this yaw-roll-rudder coupl-ng instability on the basis of Eqs. (2) and (3)

dnd the captive model data, both of which

were already introduced in the previous

sec-tions.

5.1 Equations of Motion

Suppose a ship sailing nearly straight with an automatic course-keeping device in opera-tion. _{We can assume a constant ship speed,}

so the first equation of Eqs. (2), the surge

equation, can be omitted.

Among hydrodynamic derivatives, the

third-order terms of yaw and sway can be

omitted

_{because yaw and sway velocities}

remain rather small in the situation under

73 Kyoung-Ho SON and Kensaku NoMoTo

20

I

20

Fig. 9 Turning trajectory and roil angle

Kt (Fn 1.31

quder

-k0 3/ne.

/ /T(z*

Fig. lo Z-Manoeuvre response and roll angle

consideration.

_{The terms rna.i' and m.xj'}

are generally so small that they can also be

omitted.

The higher-order terms of yaw-roll and

sway-roll couz.

ai This is bae

The roll can har -u As is indi small 5wa -roll-yaw implies a hieher-or±-Toe equat.1

o(rs section

___

(n'+rn

mL

-=Y

_xr'

_VV(

N-

(+J')p'--m'7---

v-7

K'

-'vieru _{p'= s =:} e wil

'adve

con-an exponL

4I1Tfmation ±ig gears. The uatioz is:

'

ç5'=ç(LT=

. r==rv/L),

b : 1 _{rmg geac} oruonai'

,, Spect\t.

XXLS differer-ZJysis is base± :

sway-roll coupling, however, should be includ-ed. This is because:

The roll angle may well reàch 200, which can hardly be considered small.

-As is indicated in Fig. 5, even a rather small sway velocity does accelerate the

roil-yaw coupling considerably, which implies a significant contribution of higher-order coupling terms.

The equations of motion for the analysis

of this section then become:

(m'±m)i,' Y0'v'±(m'+m-- Y)r'

inl,fr' Y'p' Yç Y.'0v'

- Y,v'çf Yr9'

Y,'r'ç

= Y'

- N,v'2 ç Nv'ç32

NrÇ* r'2

Nr'd2=N8

(I.±J.')ò'K, 'p'±(W'GM'K')

m lij'Kv' (mU--K) r'

K09v' çS K,9v'ç - K,Ç r'1 ç'

VP

- rr i - j

where p'=$'=ç5(L/V)

Next we will assume a

proportional-and-derivative control auto-pilot and a steering gear with an exponential lag. The latter is a good

approximation of current electro-hydraulic steering gears.

The equation of this rudder control mecha-nism is:

where L,'=c,&(L/V)r', b'=b(V/L), and where

T=TE(V/L), T being the time constant of

the steering gear, and a and b denote

the

"proportional" and "derivative" control para-meters, respectively, of the auto-pilot.

Eqs. (8) and (9) compose the set of simul-taneous differential equations which the present analysis is based upon.

5.2 Stability AnalysisRoot Locus and Range of Stability Diagrams

When the yaw and sway velocities and the roll angle are all very small, the stability analy-sis is simple: the third-order terms of the roll-yaw and roll-sway couçling can then be omitted and Eqs. (8) become linear. We can define the characteristic roots, or eigen-values, which govern the stability.

The results are indicated in Figs. (11) and

(12) by the x-marks.

There are six

eigen-values and all the real parts of them are

nega-tive in these two cases. The whole system

(ship and rudder control device) is then stable,

and any small deviation from upright (=0), straight sailing (r'=v'=O) will decay out with

time.

Next we will consider the case when the

yaw, sway, and roll are not very small. The

third-crder coupling terms can not be omitted, then. We will employ the principle of

perturba-tion stability around an arbitrary equilibrium

situation.

Assuming an equilibrium of r'=r, v'=v,

and ç5=ç, a small perturbation around it is

described by the following equations:

(m'±m,)i' Y:v'±Y3r' Yß'

Y5p'Y6ç=Yò'

(1±T)

Nr '- N3v'N4p'

N5ç5=N

(10)

(I+J)p'Kp' ±1(3 ç K4i'

Ksv'Ker'=K

T±= açt'b'çb'

It should be here noted, however, that all

the motion parameters y', r',

, ,

in this

equation are ones of small perturbation, not the whole amount of the motion parameters. For example, whole sway velocity is v0±v,

y being the perturbation. The new coefficients

Y's, N's, and K's are:

Y=Y±2YÇ,vç

ol, " A2_t'#*'*'

I

O

hr,,r'

I

On the Coupled Motion of Steering and Rolling ofa High-speed Container Ship 79

I ngie z Dii angie

nd nzj,'

an also be -rolI and

y4=ml

K3= (W'GM'K)K:rV2-2K:#*vo

K,=nij

V - Vf

_{0T vO'i'0T vQ}

t 1V' .,J A V' K6= (m1±K) ±2Kr# r ç (11) Eqs. (10) are obviously linear, and there-ore the stability of small perturbation around

he situation v, r, and

, can be examined

,y the same procedure already mentioned in

he stability analysis for upright, straight

sail-n .

The algebraic equation to define the eigen-values is

À6±A2A5 ±A 2t' ±A3À3 ±A 4À

±A5A±A=0

(12)

vhere A's are composed of the coefficients of Eqs. (10).

Eqs. (11) tell us that the coefficients Y's,

r'5, _{and K's do change their values with the}

initial motion v,

r,

and ç, so the same is

true of the A's of Eq. (12). The stability

eigen-values thus change with the initial motion v,

r, and

.

Figs. Il and 12 illustrate this. Starting at

the X-mark that corresponds to straight sailing, the characteristic roots move along the arrow-headed solid curves with increasing initial yaw velocity r . In this figure, the initial sway

velocity is

given to be proportional to r,

Kyoung-Ho SON azd Kensaku NoMom

Fig. 11 Root locus diagram of stability characteristic

equation; o0, without yaw-rate control

11,10 00'S r.z..t (Fo0.23

00 0.3.

*.Jt00llot o-1,0. 0.1.0 1010r1M c-c- lo.2.5toc.

1Cl tIll Cat. FnO.2

O r:-c.0

Fig. 12 Root locus diagram of stability characteristic

equation;Ço=O,yaw-rate gain b'=l.O

v=-0.45r

(13)

This means that the pivoting point is located

0.45 ship length in front of C.G.

According to Fig. Il, the ship is stable at

upright and straight sailing because all the

roots remain on the left-half plane Any small rolling will decay with time. With increasing

yaw velocity r, however, a pair of complex roots move to rightward and cross over the

imaginary axis when r reaches ±0.167. This means that the ship (and the whole system)

becomes

unstable when her yaw velocity

exceeds 0.167. Now any small rolling will build up and the ship develops heavy rolling

accompanied by considerable zig-zag yawing

U$S'TA.ILE Onthc o.

i

Fig. 13 Stability re tions, ro' Fig. 14 Stability tions, ro' a:

Yo11 Y±

±2 Yv0± Y,r2

±2Yr0

ç N,

=

N ±

v o

±N

ç5 N,

=

N'

N5=N' ±N0v ±2Nv

c

±N

r

±2NÇ r ç

K.=K'

20 r.-o0ct -c.0 -2.0 2.2 .0 Ocol n10 Ilote d .201 Fn 0.00 on 0.3 0l,ttOl 02.0 -Sleet,, 00« 'c-25o.C. Iloltlol Cc-C. -0.0 o »0.0 o o -C -20 -lo.

rdinate is the nsional), and le, ç5. The ne stability. ical (boarder-by Fig. 11. t (negative ₎ tial starboard ritical. The heel is quite heel is much aptive model I US the basic 'Fig. 13 also out 20° may it any initial erivative con-i-pilot, is zero

-rate control

whole system Fig. 14. This al simulation -cr surprising .te auto-pilot would

ict that it has

;.

Thisisa

-roll coupling. f the steering whole system. e constant is gear speed is e: the critical ens to 0.106 :raight sailing Again it is

ar speed has

ea. Together he auto-pilot of a common s that skillful

steering is essential to avoid heavy rolling at sea.

6. Conclusions

The important conclusions we obtained are: The captive model tank tests revealed that yaw moment and sway force induced by

roll depend much upon the yaw and sway

velocities. This is

particularly true for an

outward heel, the lean over to the opposite

side of the ship's turning. Accordingly the

yaw-roll and sway-roll coupling hydrodynamic forces have essentially a non-linear character. The third-order, cross-coupling hydrodynamic

derivatives play an important roll as well as

the linear terms in the mathematical modelling of the hydrodynamic forces acting upon a hull. The yaw-sway-roll coupling has a

destabilizing effect on the yaw motion of a

ship: improving turning performance and spoiling directional stability and quick response.

The smaller the metacentric height and the

higher the ship speed, the more prominent

this tendency becomes.

The yaw-sway-roll coupling can induce a self-exciting, heavy rolling accompanied by a considerable yawing of a ship under automatic course-keeping. Since this phenomenon depends much upon the higher-order yaw-roll

cross-coupling, an accidental heel

over of

moderate degree will not last long if a ship is sailing really straight. Once she begins to yaw

and sway, however, even an infinitesimally

small heel can develop into heavy rolling ac-companied by yawing.

The performance of an auto-pilot has

a great effect on this

unstable behaviour.

Yaw-rate control proved very

effective in

suppressing this type of heavy yaw-roll motion.

Slowing down the steering gear speed spoils

the overall stability considerably.

On the Coupled Motion of Steering arid Rolling of a High-speed Container Ship

Acknowledgement

Acknowledgement is made to Prof

H.

Eda of Davidson Laboratory as well as to

Prof. M. Hamamoto of Osaka University for their influential discussions and valuable ad-vice. We are also indebted to Mr. H. Tatano and the staff of the Manoeuvrabiiity

Labora-tory of Osaka University for their help at

experimental tank.

References

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Mer-chant Ship Steering Systems, Marine Technology, Vol. 7, No. 2 (1970)

,, 2) H. EDA: Rolling and Steering Performance of

High Speed Ships, 13th O.N.R. Symposium

(1980)

M. HxP-No and J. TAXASRINA: A Calculation of Ship Turning Motion Taking Coupling Effect Due to Heel into Consideration, Trans. of

West-Japan S.N.A., No. 59 (1980)

J. N. NEWMAN: Marine Hydrodynamics, MIT

Press, 1978

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MMG Report VOn the Experimental

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MA5IJYAMA: Free Rolling Test at Forward Speed, J. of Kansai S.N.A., Japan, No. 146 (1972) (in Japanese)

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