Calculations on the steered motion of a ship under the action of external forces

7X,1(41.4 dt/fr,

9.

Calculations on the Steered Motion of a Ship under the

Action of External Forces

(Part I)

Coursekeeping and Turning of a Ship

in Uniform Wind and

Flow-Akihiro OGAWA*, Member

From J.S.N.A. Japan, Vol. 126, Dec. 1969)

Summary

Considering the possibilities of solving the non-linear equations of motion numerically on

a digital computor, the author contrived a method for numerical calcuration of the steered

motion of ships in uniform wind and flow.

The aerodynamic forces and moments and the longitudinal hydrodynamic force were taken in as the intermittent functions of relative wind direction and the Froude number respectively, and were interpolated at need. The so-called rotary derivatives were decided by a least-squares method, up to the third order of transverse velocity, turning rate and rudder angle, from the tested points at various values of the parameters. The equations of motion were described on the absolute motion of ship, and the external and inertia forces and moments caused by the relative motions were calculated at every small time

interval and integrated.

As an example, calculations were carried out on a mammoth tanker regarding the re-quired lowest speed and the course stability in a straight course and the turning behaviours

in wind and flow. Results of the calculation on the turning characteristics in calm sea were compared with the tested results on a free-running model and on the actual ship.

The following conclusions were obtained concerning the method of the calculation and the calculated results on the steered motions of a ship in wind and flow.

The non-linear equations of the un-steady motion of a ship can be calculated numeri-cally on a digital computor.

The required minimum speed and the course stability in a straight course within the limits of definite rudder and drift angles can be calculated.

In some cases the course stability index has a periodic solution.

At the above-mentioned minimum speed, the ship tends to be course-unstable main-ly in case of following wind, though the stability is improved by the higher ship speed.

The effect of wind and flow change remarkably according to the loading condition of ship.

The average direction of the macroscopic " drift " of the steadily turning ship does not necessarily coincide with that of the uniform wind or flow, and the average speed of the " drift " is smaller than that of wind.

tically, since the equations of motion usually

1. Introduction _{require non-linear treatment of hydrodynamic}

In general it is very difficult to solve the _{forces and moments on the ship. As is well}

unsteady motion of a steered ship analy-

_{known, however, by the application of an} 'lc Ship Dynamics Division, Ship Research Institute, electronic computor, there is a possibility of

124

Calculations on the Steered Motion of a Ship under the Action of External Forces 125

solving even a very complicated equation

without reforming the formulae.

Such a

method of numerical calculation can of

course be applied to that of ship's

manoeuvr-ing motion as, for instance, reported in 1).

In such a high speed numerical calculation

the formularization of the forces acting on the ship is not necessarily needed, and

ac-cordingly the external conditions such as

wind forces and so on can be freely taken

into the calculation.

By these methods a

higher accuracy of the calculated results can

be expected if the applied method was

suitable, and some applications, for instance

to the calculation of course-expectation for

preventing collision, can be thought of.

On the other

hand, though the

hydro-dynamic forces on ship's hull have been

investigated in the forms of wind force

coef-ficients and manoeuvrability derivatives by

the experiments in wind-tunnel, rotating arm and oblique-tow tests and/or forced oscillation

tests using restrained models, the results of these investigations do not seem to be fully

utilized by reason of the difficulty of the

method of applicational calculation. The

effect of wind are usually considered and

discussed in relation to the course-stability or coursekeeping ability as, for exmple, by Nakajima2), Eda, or Welnicki4), where the wind forces have been taken in under some

assumptions for the convenience of the calculations.

Considering the above state of affairs and

a favorable circumstances that the author

could use an electronic computor at will, he

contrived a method for numerical calculation

of the steered motion of a ship in uniform

wind and flow without reformation or approxi-mation of the formulae. The report describes the method of calculation and some examples

of the calculated results on a mammoth

tanker.

2. _{Expression of Motion and Method of}

Calculation

2.1 _{Coordinate Systems and Symbols}

A coordinate system as shown in Fig. I

was adopted according to the conventional

means which has been utilized generally for

the analysis of manoeuvring motion after

Hovgaard". Only the transfer and rotating motions of the ship in the horizontal plane

were considered. In order to avoid the con-fusion, some symbols were let coincide with

those in computor programs, regardless of

the usual ones. The applied notations are shown in Table 1, and the angles and direc-tions in Fig. 1 show the positive values.

Fig. 1 Coordinate Systems

Table 1 Nomenclature

0-X, Y Coordinate system fixed in space

Xo, Yo Position of C.G. of ship referred to

0-X, Y

00-x, Coordinate system fixed in ship, origin

at C.G.

UF, Absolute speed and direction of uniform

flow

UB, cbB _{Absolute speed and direction of uniform}

wind

U, /3 Apparent speed and drift angle of ship

14x, up _{x- and y-components of U (= Ucos}

Usin j3)

Turning rate of ship (=;)

Heading angle

,av Relative ship speed and drift angle to

water

vx, vp x- and y-components of V (=- Vcos tit,

126 Akihiro OGAwA

E, F, G

Ea FE, GE

Ew, Fw,G , t

Relative ship speed and angle to air Course angle (=0 19)

Rudder angle

Ship's mass and mass moment of inertia Added mass components and added moment of inertia

x- and y-components of force and moment on ship

Hydrodynamic forces and moment below waterline

Aerodynamic forces and moment above waterline

Propeller thrust and thrust deduction

coefficient Time

Distance of pivoting point before C.G. Ship length

2.2 Equations of Motion

The equations of planer motion of the ship

can first be expressed in an arbitrary

coor-dinate system moving with the uniformflow

as follows:

m+mr)13x(m+)novur=E

G=Ga+Gw+Gai,

The inertial terms were followed after the convensional expression as, if necessary, described in detail by Motora', or Watanabe.

Since the relation between the absolute

and relative motions to water are shown by: vx=ux+ Up cos (0

vyuv+ UF sin (OF 0)

z)z---.1tx U, sin (0E-0)S

/),-= tiy- Up cos (0,-0)S

Eq. (1) can be rewritten as: +

(111+ my)ityF,.r=Fy

(L+ I)i' Ggiy= G.

Ex= Err + +E(1 t)+E(m + m 5)v 5 (m + mr)Up sin (OF g5)Ir

Fy=FH+ Fm [(m+mx)vx (m+my)

(5)

Up cos (OF cb)Ir

G G Gw G,)U p cos (0 q5)r

for the convenience of following calculations. 2.3 Expression of Forces

Among the terms in Eq. (5), FR and Gil

were expressed by the third orderpolinomials

with the derivatives obtained from rotating

and oblique tow tests on a restrained model, as:

Fa= Fi+ Fry,' + Frr+ F03 +

+ Fr,r2 + F.52+ Fo,v 5r + Fro ro E,y5vy+ F.,yv53+

+ Foaa + Ftn,rvy'r+

Frror2a-Fooyii2vy+ yr + Froor52 Fom3v52 + F0,3v yro

= G1+ Gyvy+ Grr+Goo+ Gmv52

+ Goa' + Gorvyr + Grora

GoyOvy+ Gmyvy' + Gm-r' + y2 r + Grrar,o + Goa,,a,v+ GOD,.v 5r2 + Grr,oral

+GomOv + Gyrovyra

( 6 ) Since the function Eft to vy, r and ô had

not been decided, a resistance test result was applied as a function of Vi. The resistance coefficient was represented by an intermittent

function of the Froude number, and

inter-polated for a certain yr value at need, or

extrapolated for the lower speed range than

the lowest tested

point by a

quadratic formula. For ET and 1-t a similar method could be applied. Here, however, 1-t,

effec-tive power and propeller efficiency were

assumed constants for simplicity,

and a

relation:

ET(1 t)v, = (E110+ Ewo)vxo ( 7) for initial values Eno, Ewo and vA in straight

ahead condition of ER, E5 and yr was

as-sumed to be valuable.

For Ew, Fw and Gw similar intermittent functions as E11 were applied using cubic

where: expressions of pw when interpolations were

(m+m5)z3,+(m+mx)v.r F ( 1 ) -=G where: E-= Ea + Ew + ET(1 t) F= FFI+ F w Ft ( 2 ) ( 3 ) ( 4 ) (L.+ + + + + + Grxr2 G60003 ( W, Pw m,, P2x,

needed,

As the calculations were carried out on the

motion relative to the ground, conditions of

external forces were given by UE, OF, U8 and II, and V, pv, W, pw etc. were calculated at every moment.

2,4 Method of Calculation

The numerical integration was carried out by an Euler's method. The ship motion was

calculated from Eq. (4) as following:

k 1) Linear and Angular Accelerations

V) (n+ iny)G+GtFu_

7

(in +in AL+ hz) FiG0 Linear and Angular Velocities,

(2)

Calculations on the Steered Motion of a Ship under- the Action of 'External Torce's 127

t-it U7(t)=142.(0)+ E a.x(r)At t - t try(t)= u(0)+ ,=0 t-KO= r(0)± E r(r)At ,=0

( 3 ) Position and Orientation

Table- 2 Principal Dimensions

3. Employed Data for Calculation

A 'mammoth tanker T-Maru was Selected as the object ship for the calculation, by the

reason that many experimental data were

available on her. The principal dimensions are shown in Table 2.

The model ship in

the table were employed for the experiments

on FH and GH by rotary tests., which con-.

t-Je

X4t)= X0(0)

, 0

lux(r) cos 0(r)--uy,(r) sin (p(r)1At

1-Jt Yo(t)= IT0(0)+

r=0

1u2(r)..Sin0(7)+14(7)cos 0(z-)1,At

I-Jt

OM= 0(0),+ E'r(r)A1

7=0

( 4 ) Drift angle and Pivoting point

p(t)=-ltan-'

uv(t).

ur(t) p(t)= U si13)(t)

In the computor grogram, both non-dimen-sional and dimennon-dimen-sional expressions were

ap-plied in order to decrease the number of

calculation times to let the accumulated

er-ror as less as possible.

(10)

Condition Full Load Ballast

Ship

,

' Actual Model Actual Model

Length b .., L (m) 290.00 1.5000 1 Breadth _(m) 47.50 0.2457 Draft at midship _(m) 16.08 0.0832 7.40 0.0383 Trim _(%,), =0.1 +1.33 L. c.. b. '(rn) =7.32 =0.0379

1.20

0.0062 Block coefficient 11 0.805 Displacement 4 (ma), 1 178,114 0.0246 ' 95,025. 0.0131 Prjoe'cted areas above waterline .(m2) Front 0.0275 00343 Side 0.0859 a.1386 8) in+ nix I( z+ hz)Fy+ Fi

(in + in5)(1x+ FiGt)

iiu(r)At

+

r(t) (

128 Aikihiro OGAWA

sisted of rotating-arm and Oblique tow tests,

and on Ew, Fw and Giv by tests in a tank

with wind-blower at Ship Dynamics Division,

Ship Research Institute., The rotary tests

were carried out at the model self-propulsion

point by the speed corresponding to about

13 knots of the actual ship. The derivatives

in Eq. (6) were decided by a least squares

method simultaneously in a non-dimensional

form from all of the measured forces and

moments, after being subtracted the

centri-fugal force component of virtual mass_ These

derivatives are shown in Table 3 F1 and GI were put to 0 in the calculation.

EH was derived from a resistance test result

on a 6.5m model at Mejiro Tank. That of

the full load condition are shown in Fig. 2. As

the tested ballast conditions were different

from the calculated one, the resistance curve, was estimated by interpolation,

Table ,31 No"n-dimensional

Ew, Fw and Gpv were obtained from the

original test data by Tsuji and others", being non-dimensionalized with the sea water density for the convenience of the calculation.

These are shown in Fig. 3.,

On the added masses and so on, m5 and

./z. were derived from a mean value of the

test results on a 2m model by Fujino",

and L was let coincide with that of the

I

Fig_ 2 Resistance Coefficient of ship (Fun Load Condition) Rotary Derivatives

* Variables are taken in the non-dimensional form

Variables.* FRI1 -9pL2U2`x 105_{_} 1 GE/1-p/..3 U2 X 105 2 FuE 1 _ 1

Ballast Full Ballast

11. 1 10.37 29.95 -0.32 -8.18 ,yy 1 -.2,041.63 =4,197.31 =589.28 -821.19 I 841.79 393.04 -371.30 =172.37 a I -527.27 -416.00 209.56 176.98 I v7,2, r2 1 -48.93 -141.03 -68.19 -145.23 27.77 -83.64 0.29 -0.46 42 1 112.35 -69.63 -65.06 25.96 yvr 90.79 40.55 46.80. -24.15 ro -70.72 23.16 51.05 -13.09 ,Otry 27.09 -131.91 I -17.23 48.65 yin -4,825.21 -1,996.82 132.96 =11.91 rt -511.32 1 -169.45 11 5.108 -5.07 (33 vy2r '1 -150.78, 2,246.26" 19511 1 1,664_77 106.22 -1,647.93 -80.56 ----854.79 r2o -2.02 I 5.68 119_81 12.97 32vy -35.44 416.17 142.41 -160.44 vyr2 -2,996.94 -1,014.31 1,102.73 285.30 Mt 81.38 -80.28 -71.28 12.52 avy2 -596.27 1 -259.04 447.41 140.01 vyro 3,166.84 1,798.88 1 -1,583.73 =801.79' 0.010 -K II 1 it 11 IT FULL LOAD, 4:120 0.15

FROUDE NUMBER, Fn VArgt,-,

010

FH

r

Calculations On the Steered Motion of a Ship under the Action of External Forces 129

presented in Table 4.

4. Calculation on Straight Ahead- Condition

ill.ONGIT. FORCE

Fig. 3 Wind Force and Moment Coefficients (Non-dimensionalized with sea Water Density ps)

model, mx was :estimated from Motora's

chart"), and mx, my and Izz of tke ballast

condition from both the chart and those of the full load condition. .Since there were no

avail-able data on L of the ballast condition, the

same non-dimensional value of the radius of

gyration to that of the full load condition

was applied. Ft and Gi of the full load

con-dition were also based on 9). However, as these values, did not make any difference to

the calculated results, those of the ballast

condition were put to 9 .

These values are

Table 4 Added Mass Coefficients and Radii

of Gyration

41 Limiting Condition

As a basic state, the straight-ahead

condi-tion of a ship in uniform wind and flow was

considered at first.

In this case, the ship

advances in a straight course with a

combina-tion of drift and rudder angles, which are

usually smaller when her speed is higher

and increase rapidly with the decrease of her speed.

In general the problem of course

keeping quality in wind, when there are only

wind as the external force, is treated by the conception of the necessary rudder angle to

keep the course for a definite value of UBIU.

Here, from another point of view, the

mini-mum ship speed required to keep the straight

course within limited values of both rudder and drift angles were defined as the lowest speed navigable in the straight course. As the limit angles +10° both were chosen con-sidering that the ship was a mammoth tanker,

and that actually

the rudder angle was

ceaselessly fluctuating around its mean value, and further that she could change her course

by steering from that initial condition.

30

-30° -20° -10° 0 10° RUDDER ANGLE 6

Fig., 4 Example of the Solution of Eq. (12)

iN I LI L u , - ...

WI

t

5 ' .1 . - 0 Y -UF.-.12 Ikn tit=20°' UB .60 kn tpi3 =1140?' i :.

°

k

_IL

- ''. LUE,FZ661kn 1 c, ,' _Gnr0 h BALLAST C,ONDL.

Condition Full load Ballast.

Added mass

coefficients,

m,Im 0.060 0.035

mvini 0.810 0.500

Radius of gyration c011, 0.282 O. 282

Added radius of gyration_,

A-,..,/i... 0.205 0.150 Coupling term coefficients Fi 0.615 x 10-3 0.10 . GA) i' ,OL 100 x 10-4 0.0 20° 30° 3r 60. ao°

WtNch PREC11 ION if3iw

I ESCI 30° 20° UJI Er: 10 -20 0.6 -0.1 -0.2

\

FULL 0.4 0.2 120°

130 Akihiro OGAWA

4.2 Procedure of Calculation

Eliminating the terms of angular velocity,

linear and angular accelerations from the

2nd and 3rd equations of (4), the following

equations should be solved:

Fy(U, p, 5, Up, bF, UB, 013)=- FE+ Fpv =0

G.(U, /3,5, UF, OF, UBy OB)-= G11+ Gw.---0

FULL LOADI

(12)

First, fixing the ship speed U to a certain value, two curves which satisfy the each of

Eq. (12) is looked for in the range of drift

and rudder angles, p and 5, not more than

+ 300, and from the cross point of the two curves, a necessary combination of p and 3 at the speed can be obtained. The required

lowest speed UL is obtained thus from the

combination of p and 5

as a function of

IBALLAST I (3) 0 BALLAST I

VAIRP4

_&reign

440*

(c) (d)

Fig. 5 Required Lowest Speed in Wind and Flow

BY

2

UF

Calculations, on the Steered Motion of a Ship under the Action of External Force$ 131

various value of U.

An example of the

procedure is shown in Fig. 4. Of course, for

the successive calculation, it is necessary to

consider in the computor program how to

cope with the case when there are more than

two or no solutions and so on at every

calculating step. By way of example some results of such calculation are presented in

Fig. 5.

The arrows at the center of the

figures denote the direction of ship's course.

lithe lowest speed or the relative speed to

water at the lowest speed were less than

2 knots, as it did not seem to fit

in the

reality, 2 knots or absolute speed

corres-ponding to 2 knots of relative speed was

adopted as the required lowest speed.

4.3 Course stability

The above

obtained solution indicates

Merely a ballanced condition and not the

course stability. Here it is assumed that the

stability can

be judged by

a linearized

dynamic stability without considering the

variation of ux as the first approximation.

Putting the right side of the 2nd and 3rd

equations of (8) as P. and G and equating as:::

aP aP aP'

fo-au, - ar 110! IFLOW DIRECTION , ABC FLOW DIRECTION aC aG

aG I

,.gr

, go au, als

the stability equation becomes as'

d41,

_f.40

dzir

(it,gdiso+griir+goily5

Roots, of. the characteristic equation of Eq.,

(14):

+a(72'+,ba+c=0'

where,

a=

)

b= fug, frgugo

,e= ugo fog..

indicate the stability indices. The calchla-tions were carried out by a Cardano's method

simultaneously with the calculation of UL.

Some examples of the solution are shown in, Fig, 6. In order to observe the effect of the

difference of speed, roots at double the lowest

speed are also shown in Figs. 6 (a) and (b).. Here an attention should be paid that the roots of Eq. (15) are obtained in connection with Jo, and as the result, there exists very

often a root nearly equal to 0 as is seen in

Figs. 6 (a) and (b). Since the zero-root is out

Fig.. 6 Stability Indices Effected by Wind and Flow

°Or WINO DIRECTION, Ws, = In ,I-4l/17-1. FULL LOAD I U, 2 an at In =....

H.

UL IlPRO011 n calm, ,j.al 2UL I I BALLAST CONDITION U00 km LW =2.1m, (A)_______J.

R/-.In calm sea

at 20, \_{at UL} / I I lIBALLASI CONDITION, U, 'DOT Iis.20 W ,,---,U, -...-__ci,,, cairn a IL, sea (a) (b)' (c), (d) 0 + dt

fu+

(15)

/'

132 Akihiro OGAWA

of question when, as in general, so far as

only Juy and Jr are concerned, the stability can be judged by the other two roots in this

case. The fact is clearly compared with the case in calm sea, where JO term is not needed.

Though there are many ways to express

the stability,

it seems better

to present directly the characteristic roots in case of the numerical calculation on an electronic corn-putor. If necessary, a solution for a definite

initial condition can be easily calculated, or

even if the root was not decided, the motion

after a given initial condition can be

calcu-lated without any difficulty',.

repeatedly computing the Eqs. (8) to (11) at

each very small time interval. Though the

time interval can be defined by a

non-dimen-sional method, here it was given by the unit of second for certain reasons.

In case of

this calculation on the mammoth tanker, 1

or 2 seconds were suitable and 5 seconds

gave some difference to the calculated locus.

The following calculations were carried out

at each

1 second intervals. Besides, the

steering velocity was taken as 35°/15 sec. 5.2 In calm sea

First, calculations were carried out in case

of calm sea, where no wind nor flow were

considered. As a matter of course, forces

caused by the relative wind were taken into the calculation even in this case. In Fig. 7

a calculated turning locus at 15° starboard is shown with the corresponding one of a model

test. Since the calculated locus stayes on the same circle also after the last point, the

accuracy of the calculation can be considered

to be enough. The manoeuvring

characte-ristics obtained by such calculation are shown

in Fig. 8. The test results in Figs. 7 and 8

are derived from the free-running model tests

on a 4.5m model by Mori and others"' and

-40° ...30° -20° --- .. -° + .2 .5 6 9 1.0

Fig. 7 Calculated and Tested Turning Loci in Calm Sea 10 20° 30 40° ( STARBOARD ) .3 TEST CALC. TURNING RATE A - - MAXIMUM ADVANCE + SPEED REDUCTION PIVOTING POINT DRIFT ANGLE 1.0

FULL LOAD CONDITION .9

L5.290 m

u16 kn .8

V. .7

.6

FREE RUNNING MODEL .4.5 rn RESTRAIND MODEL

for STATIC DERIVATIVES=1.5 m for ACC. [JERI VAT I VES=2.0 m

for RESISTANCE TEST.6.6

.5 .3 a ° .2 6 ( PORT ) .1

5. Calculation of Turning Motion CALCULATION

30 min

min

=290 rn

5.1 Time Interval of Calculation, etc. ( Fn..0.145)1 5 kn FULL LOAD CONDITION

UF 0 or

The steered motion

of a ship can be

MODEL TE ST

1_,4 .5 m =0 kn

calculated by the method described in 2.4, Fn.r.0.16 6

Fig. 8 Calculated and Tested Turning

Characteristics

ACTUAL SHIP TEST .a

z2

)

Calculations on the Steered Motion of a Ship under the Action of External Forces 133

from actual ship test resultsla).

5.3 In Wind and Flow

Calculated examples of the turning loci in

wind, in

flow and in wind and flow are

shown in Figs. 9, 10 and 11 respectively. In Fig. 12, Calculated behaviours of speed, angular velocity, drift angle etc. in calm sea

and in wind and flow are shown.

6. Discussions on the results 6.1 On the method of calculation

The form of derivatives for FH and Gg only

was applied considering the accuracy of the

experiments and the capacity of the computor.

If hereafter the data at the small angular

velocities and

drift angles are

obtained,

higher accuracy of the calculation can be

expected in the vicinity

of the

straight

Fig. 9 Calculated Turning Loci in Uniform Wind

Fig. 10 Calculated Turning Loci in Uniform Flow

1 min U0 _/3.

1 5 kn O. 5° -1 .7°

- - 10 1.0°

134 Akihiro OGAWA

course, taking the data into account

in the

form of the intermittent functions of

r, p, and 3.

Though this method of calculation is in the

first step of developing the program and it is needed to calculate with various casesfor

instance on a small

boat or at

different (a)

BALLAST

U[3,-40 kn

Vti=99,e,

(b)

Fig. 11 Calculated Turning Loci in Wind and Flow

2

Yo/L

manoeuvres, estimating the calculation of

this time, it can be said that the method is

effective as a means of the calculation of the

steered motion of a ship especially in the

point that the external forces can be freely taken in, even when they are not uniform.

A difficult point is that many data are

re-6=35°P -3 -2 0 5 3 -J UF min>1 15 50° -2.2 --2

10

Calculations on the Steered Motion of a Ship under the Action of External Forces 135

2

3 4 5

TIME, t (min)

Fig. 12 Calculated Behaviour of Ship after

Execution

quired for the calculation on a ship, even

with one condition.

Since the theories are

not sufficient to afford such data in good

accuracies at present, massive experiment is needed. For instance, though the forces in x-direction were decided by a very simple assumption in this paper, in order to realize

the calculation, many important factors such

as the effect of drift angle and angular

velocity upon the

ship resistance, wake

distribution and so on, characteristics of the

main engine and propeller and mutual inter-ferences between such factors including the

rudder must be known.

However, the method of calculation itself is favorable in the point that if such condi-tions were fulfilled, since it is very easy to take them into account, a higher accuracy

too°

100°

200°

00°

can be expected and, moreover, the problem of the model-ship correlation could be solved

at the same time. In such sense, it seems

to be favorable to have more concern on the

researches on such method.

The similar approach will be effective and expected to

be applied also to the phenomena of broach-ing, capsize and so son.

6.2 On the calculated results

As the combinations of the external

condi-tions are infinite and the shown figures are

only a few of them, there are many facts

which are not shown. So the considerations

on the calculated results include them.

6.2.1

On the

lowest speed in the straight

course (Fig. 5)

When the effect of wind is small (In

full load condition as well as in

ballast

condition at low wind speed), the ship tends

almost toward the relative flow direction,

and the lowest speed is limited mainly by

the drift angle except when OF nearly equals to 00 or 180°, and the lowest speed is bigger

when the relative flow directs accrossways. When there is no flow, the effect of

wind appears nearly proportionally to the

wind speed. In full load condition Ur, is

decided by the limit rudder angle, and in

ballast condition, more part is restricted by

the

drift angle and the necessary lowest

speed is bigger.

It shows very complicated effect when there is wind and flow and it seems that there is no general rule in this case. Especially in ballast condition both effects of wind and flow appear almost equally in many cases, imply-ing the difficulty of the manoeuvrimply-ing.

6.2.2 On the course stability in the straight

course (Fig. 6)

Since the stability is here calculated on the above-described minimum speed,

the

ten-dency of the curves to the wind looks some-what different from usual ones. However if

the condition is taken as UB/U constant as

is

used in

general,

the tendency of the

stability curves agrees very well to 2) or 3).

(a) When there is no wind, the effect of

flow speed is very small.

2 3 4 T I ME 5 t (min) 10 9 4 -3 9 7 5 6 II 2 = =180° UB =40 FULL LOAD =15 kn

When there is wind only, the stability

varies with wind direction, is less than that

in calm sea and is unstable for the wind

from the stern. The change of the stability index according to the variation of the wind

speed is similar to the effect of the flow

speed. In some conditions, the stability be-comes periodic.

When there is wind and flow, the same description as 6.2.1 (c) will do. However in

the stable range in calm sea in full load

condition, for instance, the stability hardly varies for the wind from every direction.

Though it seems meaningless to decide

the lowest speed in the unstable range, it

will still be of use since it is considered that

the ship can change course to an arbitrary

direction by steering from this state (for that

purpose the limit rudder angle was chosen smaller), and that even an unstable ship can keep her course by a suitable manoeuvring.

The course stability indicies are decided by UFIU and UBIU, and the smaller are those

ratios, the nearer to the value in calm sea

become the indicies. The unstable range in

Fig. 6 is caused by the big value of the above ratios.

6.2.3 On the turning in calm sea (Figs. 7 and

8)

(a)

In Fig. 8 the calculated r'd at small

rudder angles appears more stable than that of the tests. The reason is that the

hydro-dynamic forces are expressed in the form of third-order derivatives and the characteristics

of the ship in such condition is not

repre-sented perfectly.

(1)) The difference between calculation and

test in the range of big turning rate seems

to be caused by the imperfect estimation of

E. The improvement of this point will be

an important subject in the following step.

(c)

The reason of the difference of the

speed drops is that the calculation is based

on the assumption of constant power, and on

the contrary the model test

is carried out

on the basis of constant propeller revolution using D.C. shunt motor. This also seems to

be a reason of the shift of the position of

the turning circle in Fig. 7.

(d)

The turning

becomes steady after

about 2700 turn regardless of the approach

speed.

6.2.4 On the turning in wind and flow (Figs. 9-12)

(a) The average direction of the

macro-scopic " drift " of the steadily turning ship

does not necessarily coincide with that of the

uniform wind or flow.

And the average

speed of the " drift " is smaller than that of

wind.

(b)

These effects vary with the loading

condition, ship speeds, speed of wind and flow

and so on.

In general the effect of the

external condition is bigger at lower speed,

and the effect of wind is bigger in ballast con-dition and that of flow in full load concon-dition.

(c) The linear and angular velocities and

the other items of ship vary periodically

during the turning, though the forms of the variation are not simple.

7. Conclusions

The following conclusions were obtained

concerning the method of the calculation and the calculated results on the steered motions

of a ship in wind and flow.

( 1 ) The non-linear equations of the

un-steady motion of a ship can be calculated

numerically on a digital computor.

( 2 ) The required minimum speed and the

course stability in a straight course within

the limits of definite rudder and drift angles

can be calculated.

( 3 )

In some cases the course stability

index has a periodic solution.

( 4 )

At the above-mentioned minimum

speed, the ship tends to be course-unstable

mainly in case of following wind, though the stability is

improved by the higher ship

speed.

( 5 ) The effect of wind and flow change remarkably according to the loading condition of ship.

( 6 ) The average direction of the macro-scopic " drift " of the steadily turning ship

does not necessarily coincide with that of the

CalcuTations on the, Steered Motion of a Ship under the Action of External Forces 137

uniform wind or flow, and the average speed

of the "drift" is smaller than that of wind.,

8. Acknowledgement

The author expresses. his gratitude to Dr.

Y. Yamanouchi, Vice Director, Ship Research

Institute and Dr.. T. Hanaoka, Head, Ship

Dynamics Division for their continuous

guidance on this work, to Messrs N. Mori, T. Tsuji and M. Kan for their great deal of

experiments, to. Mr. K. Nonaka for his

corn-putor programming on the derivatives, to

Messrs. T. Yamamoto, T.. Saruta and T..

Sato for their collaboration to the experiments

and to members of committee II of JTTC

, and the members of Ship Dynamics Division for their timely discussions on this research.

The calculations, were carried out on a

digital _{computor FACOM 270-20 at Ship}

Dynamics Division programmed in

FOR-TRAN. The author is also indebted to Mr.

S. Ando of the Division for his frequent

discussion on the

programming and the

appliCation of the computor, since majority

of the research times were spent for the

development of the computor programs-.

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