A new procedure of manoeuvring model experiment

Introduction

The main concern of the present paper is the response behaviour of a ship to steering in course-control, both manual and automatic;

what measures or

characteristic figures should be used to describe the response be-haviour and how to evaluate the figures for a given ship.

In

this subject, ship motion

is rather moderate and consequently the response be-haviour is described reasonably with a linear differential equation,

. .

Ti T2 .s.6-F(Ti-F T2).0+0=104 KT35., ( 1 ) * Faculty of Engineering, Osaka University.

** Kobe University of Mrcantile Marine.

112

0 denoting the yaw-rate and 3 the rudder

angle. This equation is obtained from the coupled equations of yaw and sway motions, excluding nonlinear terms and also coupling with forward motion, since both effects are negligible for small motions.

The first-order system approximation, sometimes called K-T analysis, starts from this equation to jive another and more simple equation,

( 1')

where T=

T2 T3. _This mathematical

model of ship response is quite simple, yet able to describe the basic characteristic of a ship in steering, thus having extensive utility

Lakereitdola mr Scheepshydimeghffitat

Archief

Mokalweg2, 2822 CD Ddft Tel.: 016 - 786873 Fax 016.; 781838

8. A New Procedure of Manoeuvring Model Experiment

Kensaku NOMOTO*, _{Member & Keiichi Kanmum**, Member} (From J.S.N.A. Japan, Vol. 126 Dec. 1969)

Summary.

Response behaviour of a ship to steering is described with the transfer function in

steer-ing Ye(10)=K(1 -1-iwT3)/(1 iw Ti)(1 _{iwT2)1). A simplified analysis with just a pair of indices} K and T is based upon this idea and has a wide utility to measure and to asses

manoeuvra-bility of ships. This simplification is, however, no more valid for the analysis and synthesis -.take of steering control of a ship, in which we should take an automatic steering device

and/or a human operator into account.2) Here the complete transfer function should be called

upon to describe the ship behaviour.

This paper relates to a new procedure of defining the transfer function and the tour characteristic constants K, Ti, Ti and T3 of a given ship, using a radio-controlled, free-sailing model. _{This principle can also be adopttd to a full-size experiment. The procedure}

employs two mancgivres; parallel shifting of the ship's track and periodic steering with a

numberof frequencies. Both manoeuvres can be performed in a conventional towing basin. The harmonic analysis of the parallel shift manoeuvre gives, together with the amplitude

and phase of yawing motion measured on the periodic steering, an information enough to

assess the transfer function of a ship, which facilitates to determine the characteristic

constants K, Ti, etc.

These constants are composed of eight stability derivatives (coefficients _{of the linear} equation of motion), so that we can define these derivatives through the present approach

if four of these derivatives are given byany other sources, for example, oblique towing test. The data involved were obtained at the Osaka University Experiment Tank No. 2, using

in practical applications.

For the analysis and synthesis of automatic

and manual course control, however, this

simplified approach is no more satisfactory

because the term Kna in Eq. (1) plays an

important role in this subject2). Here we should call upon Eq. (1) and the four charac-teristic constants K, T1, T2 and T3

These constants are composed of the linear coefficients of the coupled equations of yaw and sway. These coefficients represent the inertia of, and the hydrodynamic forces

act-ing upon a ship. We can thus evaluate

the constants of a given ship by defining

the hydrodynamic coefficients, often called "hydrodynamic derivatives.", from captive

model experiments with either the planar.

motion mechanism or rotating arm facilities.

Another method of evaluating these

con-stants, which the present paper relates to,

is to stimulate a radio controlled model by actuating the rudder and simultaneously

re-cording the time histories of yaw-rate (h

and rudder angle 8. The harmonic analysis of the recorded

and 5 gives the transfer

function of 'a ship in steering (we may also call it the frequency response) at a number of different frequencies. This makes it

pos-sible

to evaluate

the characteristic con-stants'). It should be noted that in this ap-proach we need not know what forces act

upon a ship but only know What

rudder movement induces what ship motion. As a consequence we can adopt it for a full size experiment as well as for models.

The rudder movement used in this experi-ment can be of any type in principle. It will be useful, however, if a rudder movement is devised which causes no eventual change

of ship's heading, because it can be

em-ployed, for experiments on a conventional towing tank with a reasonable width. The practical advantage using a conventional

tank for a steering experiment needs, no

further emphasis. The present paper

pro-poses two manoeuvres of this kind. One is

the parallel shifting of ship's track using

an automatic coure-keeping device Rd board

113

A New Procedure of Manoeuvring Model Experiment

/with

and the other a pyliodic, or sinusoidal steer-ing commenced(ra a cosine phase.

The harmonic analysis on the result of the former experiment gives the frrouency

res-ponse of a ship at low and intermediate

frequencies and the latter provides directly the same response at high frequencies. Gathering the two together, the frequency response of a ship is defined at a reasonable extent of frequency range and this, in turn, facilitates to determine all the characteristic

constants K, Ti, etc.

1. Parallel Shift Manoeuvre A Transient

Response Technique

1.1 Mathematical Treatment

Taking the Laplace transform of the both

sides of

Eq,

(1), we define the transfer

function in steering,

MVP] K(1-Fpf3)

$(1')

L[o] _{(1+p7:1)i+PT2)'}

where p=a+io, is a complex number.

Since Eq. (2) is valid for every combination of ship motion ,fi(t) and the rudder movement

b(t) that induced the motiOn, it

can be

rewritten in a form to evaluate the transfer function from any combination of and 5

that induced the cb, that is;

oo

gy(f)aPidt

-\5(t)elgdt0

where sb is a ship moti..op, and ö the rudder movement that induced-Ali-lotion.

Both integrals of the denominator and

numerator of this equation converge with

a--,0+ in case of course-stable ships,

viz.

Ti) 0,

(note that

7'3

and Ti are always

positive), so that we can write

K(1 -I- ha T3) (1-1-iwn)(1+u,,T2)

=

o(or-,e-i" (It

a

To(t)e

e-i-iclt

(2)

App y ngIi

Eq. (3) to the rudder angle

_and

yaw-rate records obtained at a _{sequence of}

manoeuvre, we can compute

_{Y,(iw) at a}

number of w.

According to the theory of the

_transfer function, 11'4/(0)1 is the amplitude ratio of a yaw-rate harmonics with a frequencyw to a

rudder angle harmonics with the

_{same co,} both harmonics being involved respectively

in _{and 8 that caused the 0. Sirnilarl/Arg}

Ys(iw) represents the phase lag ofthe same yaw-rate harmonics after the rudder angle harmonics.

We can thus

determine. the frequency response of a ship in steering at a number of w from At) and o(t) recorded.

For the parallel shift manoeuvre, however, Eq. (3) gives indefinite form at w=0, _since the amounts

0(o0)=si,(t)dt and ö(t)d 1

0

must be null for this manoeuvre. _In conse-quence Eq. (3) fails to define Ys(icu) accurately

at low frequencies from the parallel shift

experiment.

In order to overcome this, we use a modi-fied form of Eq. (2), viz.

1,[$i

go]

L[a] L[avp

Correspondingly Eq. (3) is modified to be

cesb(t)e-a'e-"iclt

0

e-a' e-tat{ o(GP)dr.}dl

0 1

l's(iw). I i m

d-.0+

3 7

Eq. (4) gives a definite Ys(la,)

_{even at w=0}

and is able to define Ys(iw) accurately at low and intermediate frequencies from the paral-lel shift manoeuvre.

On the other hand, the integrals in Eq. (4)

as applied to the parallel shift

_manoeuvre becomes very small in value with increasing

w.

So that the harmonic analysis

_{on this}

manoeuvre can not work well at high fre-quencies. _{For such a frequency range, the} other type of experiment,. viz, thesinusoidal (or periodic) steering provides _{a good source} for evaluating Ys(i co). This will be discussed later.

1.2 _{Instrumentation and Experimental} Practice

Fig. 1 shows the arrangement of apparatus aboard a radio-controlled, free-sailing model

equipped for frequency response steering

experiments. Fig. 2 illustrates schematically how a free-sailing model takes offVfrom the

towing carriage after being accelerated up to the specified speed and then how it makes the parallel shifting, of its track.

A 'pair of towing guides holds the _model by electro-magnetic force during accelerating, while heave and pitch remain free by

func-tion of rollers on each guide. When

the carriage has reached the specified speed, the operator turns off the electromagnet to free the model V and at the same time decelerates

the carriage for a few moments. A few

seconds after take-off, the rudder is put on to a small amount (usually 50) by radio

com-mand. The carriage follows theV model atV a

12

-1. Towing Guide (aft) V2. _{Steering Gear}

3. Automatic Steering Device 4. Rate

Gyro Amplifier _{5. Rate Giro 6. Recorder 7. Radio-control Receiver 8. V}

Speed-measuring Counter 9. Timer 10, Directional Gyro _{11. Speed Pick-Up 12. Towing} Guide (fore) 13. Batteries

A New Procedure of Manoeuvring Model Experiment . 115

reverse the rudder and turn on automatic

steer. system. (t): ship's heading

I(t): rudder angle

/ .

towelfree, /

in1

1

b., running_{- L____} //

I -well i. automatic steering steady 1

transient stage

Fig. 2

model ships trajectory

+state carria e enlarged Illustration of a Parallel Manoeuvre Test Shift

proper distance until recovering

it at the

tank end. As soon as the heading

devia-tion reaches a given angle (usually 30), a

photo-electronic switching circuit attached

to the directional gyroscope turns on the

automatic course-keeping device, the

com-mand course being the

original heading parallel to the tank centre-line. Hereafter the rudder is automatically controlled sothat its angle is always proportional to the

head-ing error. This results in the parallel shifting of the ship's track. The automatic course-keeping device should be adjusted to

be rather underdamped so that the ship

motion approach to a regular

damped or sustained yawing eventually.

It is because

such a periodic yawing is moredefinitive and therefore more suitable for the -integration up to infinite timeof Eq. (4), than the smooth but somewhat fickle motion which would occur if a well-damped course-keeping con-trol was employed.

In carrying out this experiment,

special

attention should be attached to guide

the model accurately parallel to the tank

centre-7

line during towing and also to see that the

c

carriage speed is kept exactly the same as

the free-sailing speed of the model at

take-off. Otherwise initial disturbances may

af-fect the ship motion and spoil the result:

As for measurement devices, care should be taken to minimize the stray drift of the di-rectional and rate gyroscopes.

1.3 Calculation

Numerical calculation of the integrals of Eq. (4) as applied for the parallel shift mano-euvre is carried out in two parts. Taking a time tc, after which the ghip motion is regard-ed to be a regular dampregard-ed or sustainregard-ed oscil-lation, we perform numerical integration of

the numerator as well as the denominator

from 1=0 to i.

The integrations from tg to 03 can be done formally,

for after the time

10 the integrands are both definitive functions

of the form of damped or sustained oscilla-tion whose frequency, amplitude and if any,'

damping ratio can be evaluated from the

records obtained.

If there was no initial disturbance and the parallel shifting of ship track was performed

perfectly, we would not need any further

calculation.

In practice, however, the

ob-tained records of ship motion and rudder

movement may contain some effects of these disturbances. Special care should be taken to minimize them, but small disturbance

ef-fects may be corrected

in the following

!manner.

In making this correction we need to know the response

behaviour of a given ship.

This is obviously a paradox.. A reasonable procedure to solve it

is to apply the K-T

analysis" to the initial part of the parallel

shift manoeuvre thus evaluating the indices K and T, denoted here as Kg and Tg respec-tively, and to use them as the first approxi-mation of the response .behaviour to be em-ployed for the correction.

If the relative

amount of the correction is small then, this is adequate.

The correction is composed of five parts.

The first to the fourth are the

corrections

( 5 )

Base Line after No.123 Corrections Base Line after No.1,2,3,4Corrections,

Fig. 3 Notations used in Analysing the Parallel Shift Manoeuvre

This final adjustment of Dm., the last stage of the correction, becomes a step-by-step converging process starting with Ifd

as the

first approximation, and eventually it is per-formed with the equation that

0.+ Ta(0:00-00) 1+ CY:god Ke final Dm., Irscom 1+ C

c

+Tc(siicç/.o)}.to --2 .30a,rrecLori dt .o Su' ( 7 )

where Ife, Te, to 0. and cbcorreued _{are defined}

Eq. (4) using °mewl and n

previously a.nd Ygon is 178(0) first reached by

-,-corrected before the

final adjustment of Dm..

Using this Dm., the final Aorremod is _fixed and all the corrections_{are completed. Putting} these final Ocorrecwa _{and Ammmwd into Eq. (4),}

Ye(lo,) is computed at a series of frequencies. Incidentally, Ys(0), that is K, is determined eventually at this stage of correction,

seperately from Ye(iw) at non-zero frequencies;

the abovementioned step-by-step _converging process provides the final _{Ys(0) as well as}

Dm..

That is,

K= Yecon

-1+C

( 8 )

respectively for the initial motion, rudder

neutral position error, residual steady turning and imperfect parallel shifting of the track. All four of these corrections are carried out together, using the equation that,

Ocorreacd = Om sbie( 1 rt/Te)

Aorreculd =

where 'on,: measured 0(t),

: time average of Tit sbm for 1e<t<00

Om : _{time average of (OM t)}

for _tc<t<co

t, and Te being defined earlier in this section, and where

a(r)dr, , o(t) being

0

' the rudder angle,

Dm: D computed from measured O(t)

b.:

time average of Dm

dt for te<t<00

Dm.:

_{time average of (Dmi) ,1)}

for 1e<t<00.

These time average quantities

_4c<t<00

can be evaluated from the slope of, and the parallel offset from a straight line passing the origin with the same slope of the mean curves of Om and Dm in steady state, as is

shown in Fig. 3.

_{Dm., may, however, be}

somewhat modified at the next step of

cor-rection.

Now putting Ocorromed and Aorremcd into Eq.

(4), we first get

Ye(o). This Ys(0) should

satisfy the equation that

Yeco) =sbco 71(thoo Oo ). ( 6 )

This requires some adjustment of Dm., which at the same time also causes some change in the value of Ygo).

Eq. (6) is introduced by integrating Eq.(1)

from t=0 to co, taking the residual turning

rate Om and the initial yaw-rate Oo into

account. _{The relation that Yeco)= K is also} considered.

2. Sinusoidal Steering

Taking a periodic rudder movement,

30 COS cort

where do denotes the half amplitude of rudder motion and (0,..247; the frequency, being the period.

Eq. (2) gives us the steady state motion of a ship induced by this steering as follows;

si;=I Y.(i(0,)1 ao cos {fort+ Arg Y;(imr)}.

It

is then obvious that measuring the

amplitudes of yaw-rate and rudder motion on a sinusoidal steering experiment provides 178(iar,)1 and the phase lag of yaw rate after rudder angle, Arg Irs(icor).

This type of experiment is only practical for rather high frequencies because a vast extent of testing water would be required at low frequencies. It

is thus reasonable to

choose the parallel shift manoeuvre to define Y,(ia)) at low and intermediate frequencies and the sinusoidal steering, at high frequencies.

In order to perform the sinusoidal steering

experiment in a towing basin, the model

launcher mentioned at 1.2 is conveniently

A New Procedure of Manoeuvring Model Experiment

M. P.: Model Point, S. P.: Ship Point

117

used;' accelerating the model With the towing carriage in the same manner as previouslY

mentioned, but keeping the rudder at the

extreme position of the sinusoidal motion namely 50, either to -starboard or port. As

soon as the model halbeen set free from the carriage, a microswitch fitted oil the towing

guide automatically turns on the periodic

rudder movement which is produced by a

scotch-yoke mechanism built in the steering

gear. R h

This procedure iOlies to commence the

periodic steering e a " cosine " phase and

as the result the mean track of the model

becomes parallel to the tank centre-line. A fine, adjustment mechanism for the middle position of rudder swing is advantageous to keep the model track exactly parallel to the tank centre-line, by eliminating miscellane-ous fasymmetric effects.

3. Examples of the Present Procedure with Two Todd Series Models

Table 1 indicates, in the two left columns, the principal particulars of the two 4.5 metre Table 1 Particulars of the Models

f

Present Data Leeuwen equiv. to (132) . Eda _Motora Fujin° 60 B 2(158)i (158) M. No.152 M. No.158 70 (152)

Length between Perpendiculars, L(m) 4:5- 4.5 2.258 1.525 _1.5251 2.0

Length/Breadth,

a

L/B 7.0 7.5 7.0 7. 0 7. 5 7.

Breadth/DT-MirIt B/d 2.5 2.5 2.5 2.68 2.5 2. 5

Length/Drt a

Lid - 17.5 18.75 17.5 18.75 _{18.75 18.75}

Block Coefficient, Cb 0.7 0.6 0.7 0.7' O.6 0. 6

L. C. B. from Midstiip, I. c. b. O. 5%f. 1. 5%a. O. 5%f. O. 5%f. 1. 5%a. 1. 5% a.

Radius of Gyration, O. 25 L O. 25 L O. 25 L O. 25 L O. 25 L 0. 20L

Rudder Area Ratio, A R/L d 1/66. 7 1/66. 6 1/67.8 1/62. 5 1/62. 5 1/62.5

Rudder Height 0. 755 d 0. 75-5 d 0. 756 d 0. 750 d 0. 750 d 0. 750d

Propeller Diameter 0. 70 d 0. 666d 0. 70 d 0. 70 d 1 0. 70 d O. 70 d

Propeller Pitch Ratio 1. 10 1.075 1. 10 0. 945 0. 945 1.073

Number of Blades 4 4 4 4 4

Direction of Rotation Right Right Right Right Right Right

Frotide Number 0.2 0.2 0.2 0.2 0.2 0.2

models out of the Todd 60 series, whose

re-sponse behaviour in steering was obtained through the present procedure at the Osaka University Experiment Tank No. 2, 80m x

7 m x 3.6 m. This table also carries the par-ticulars of some geometrically similar models

whose steering characteristics have been

given from the captive model experiments,

Tank Test Frequency Response Analysis Sin Steerin 1.0

I.

K.1.413 T:=3.414 Tir.0.452 T=1.01

by van Leeuweno, Eda anp Crane and

Motora and Fujino8) respectively.

Figs. 4 and 5 show the results of the

analysis in the form of a Bode diagram, the abscissa representing loglow' and the ordinate logiol Ysi(ial)1 and Arg Yial) respectively.

w is the freqnency of the harmonic com-ponents contained in ship motion and rudder

Fig. 4 Steering Transfer Funtion of M. No. 15ias obtained by Frequency Response Experiment

M.No.152 Tank Test

°Z. W

Experiment

Linear Analysis with no regard to cu-dependence

Fig. 5 _{Steering Transfer Function of M. No. 159 as obtained by} Frequency Response Experiment

A New Procedure of Mano movement and 175(i0)) the transfer function in steering, I It3(i0))1 being amplitude ratio of the

harmonic components of ship motion to that of rudder movement and Arg Ys(iw), denoted as 07 in the figures, the phase difference be-tween the two.

nI'= ) and 17.'(io,')=Ys(i0))/

are the non-dimensionalequivalents of a, and Ys(ico) respectively, where V denotes forward speed and L ship length.

The larger points indicate the results of

harmonic

analysis on the

parallel shift manoeuvre. A number of repeated experi-ments have resulted in some scatter seen in

the figure, from which we can say that

the repeatability of this procedure is rather fair.

The result of the .harmonic analysis of the parallel shift connects reasonably, with the result of the sinusoidal steering experiments, which is indicated with smaller points.

At the high frequency range where logiom' >0.3, the harmonic analysis is less

reli-able and tends to result in bad scatter, as

just a small amount of such a high

fre-quency component is contained in the ship motion and rudder movement. The sinusoidal steering should then 'called upon to define accurately 11,(ia)) in such a high frequency range.

Refering to Figs. 4 and 5, 17,(ico) obtained from sinusoidal steering with frequencies for

which logia>0.5 shows a particular

be-haviour against 01. This behaviour can not be interpreted with the transfer function of the form of Eq. (2) with any constant K, T1, 7'2 and T3.

Indeed the same tendency is

found for the Ys(ico) computed from

frequency-dependent K, T1, etc. at very high frequencies,

which have been given by van Leeuwen,

using the forced yaw experiment for 2.2

metre model of the same 60 series formo.

It

is therefore concluded that the particular ten-dency of 11;(i(0) at very high frequency results from frequency dependence of the

charac-teristic constants K, T1, etc.

euvring Model Experiment 11

In

I

carryijig out the sinusoidal steering, the rudder 11jIL amplitude is choosen to be 4° for log (0'.0 0.3, 8° for log 0,'=0.3-05, and 16° and 32° for log 0/.0.5-1.0, in order to keep the yaw amplitude small enough to suit the linear analysis, yet not too small to cause in accuracy.

In order to examine the tank wall effect a series of sinusoidal steering experiments

for the same models were performed on a

wide, open-air basin. The comparative study

showed that the wall effect was not

signi-ficant in this case, that .is, 4.5 metre model at a reasonable forward speed in a 7 metre wide towing tank.

in

4. Determing K, T1, T2 and T3

A

The harmonic analysis on the parallel shift manoeuvre gives, together with the results

of a series of sinusoidal steering

experi-ments, the transfer function

17,(iw)

at all

significant freqencies. This gives a complete description of the response behaviour of a

ship in steering, that is to be used in the

analysis and synthesis of course-control. It will be sometimes convenient, however, to use the constants K, T1, etc., which are

the coefficients of the transfer function,

instead of a complex function Y3(ini) itself.

These constants can be determined by

apply-ing a

" curve-fitting " procedure to Ys(ia,

obtained from model experiment.

At very high frequencies, the aforemen-tioned frequency dependence becomes

pro-minent. From the viewpoint of describing

the response behaviour of a ship in course-control, however, this frequency dependence

is not of great importance, because very

little high frequency harmonic component is contained in any real ship motion and rudder movement.

We can thus take Ys(i0) obtained at the low and intermediate frequency, say logiow.

= and determine the characteristic constants which represent the response be-haviour of a given ship except at very high frequency motion.

non-dimen-sional equivalent,

/f1(1 +la,' T3')

YsVal)

.

(1+1w' Ti')(1+ icy' 7y) where ys, ys

i( V

/( V

L L

=

1 (1-7)

and 2, 3= V Ti, 2, 3,

V being the forward speed and L ship length. From this equation we get

7.2'+ T2' 1

K' lYs'r x

(al T3' cos Arg Arg Ys')

7'2' T2' 1 1

K' co" _{w"1 1's' x}

(alTs' sin Arg Ys' + cos Arg Ys')

(8 )

This pair of equations can be made at every w', the unknown quantities being K', T1', 7'2' and T3'. Any two pairs of these equations made at two different d could define these unknowns in principle. A more practical and reasonable way is to apply the least mean square methods to a number of pairs of the equations set up at the same number of co, properly distributed over the low and inter-mediate frequency range.

In Table 2,

the right upper columns,

K, T1,

etc. obtained through the present

approach for the 4.5 metre 60 series models are listed. The Ti' and T3' shown are determined by the least mean square method applied for the CO range where loglool= 1.0

0.3. K' is seperately obtained at the final

stage of correction stated at 1.3.

The solid lines in Figs. 4 and 5 indicate

the I Ys' (la/)j and Arg YAiol) computed with these K', T1', etc.,

fitting well with the

Ys'(ico') points obtained from the harmonic

analysis of the parallel shift and the

sinu-soidal steering, except at the very high fre-quency range.

5. _{Comparative Studies of the Present Results}

With Other Data

5.1 _{Comparison with Turning and Zig-zag}

Test Results

Figs. 6 and 7 indicate the results of steady

turning and spiral tests, and Figs. 8 and 9

zig-zag trial results of the same 4.5 metre models. The experiments were carried out at the Osaka University Manoeuvring Ex-perimental Pond, using the radio-controlled, free-sailing model technique.

The index K' at an infinitesimal motion, as extrapolated from the spiral and zig-zag

test results in the manner shown in the

figures, agree fairly well with K' derived

from the harmonic analysis of the parallel shift motion. 0.5 0.4 0.3 0.2 0.1 !,40-30 -20' -70' 0,3 0.2 0.1 -40' -30* -20' -10*

Fig. 6 Turning Test Results of M. No. la

0.5

Fig. 7 Turning Test Results of M. No. 15

o./ 10* 20' 30' 4(

M

1=FA NV 0.2 0.3 0.4 05

14i

r.zocoir

r

r'

0.05 0.10 015 0.20

Fig. 8 Z-Test Results of M. No. 1

2 11(

2

A New Procedure of Manoeuvring Model Experiment

0.0.5 0.10 0.15 0.20

Fig. 9 Z-Test Resu ts of M. No. 151 On the other hand, another index T' at an infinitesimal motion from the zig-zag test is much less than T' obtained from the present approach with the assumption that T= T1+

7'2 T3; the former is roughly as much as

70% of the latter. It should be remembered however that the assumption T= T1+ n- T3

may not be totally correct.

At the same time one can argue that the ship motion on the parallel shift and sinusoidal steering is not exactly infinitesimal. The root mean square value of the non-dimensional yaw-rate,

on the first two swings of

parallel shift is in this case approximately 0.04 and at the later stages, much less. The K' and T' at rmi=0.04, as derived from the spiral and zig-zag tests, are also indicated in Table 2.

5.2 Comparison with Captive Model

Experiments

The same ship forms of the Todd 60 series

/21

that were used in the present survey had

been tested with the captive model technique by a number of authors'', 43.'', to yield the values of the hydrodynamic derivatives. The constants K, T1, etc. can be simply calculated from these derivatives for each case and they are also listed in Table 2. For the forced-yaw results, the derivatives at (0=-0 are used to compose K, T1, etc:.cApparently there is still some inconsistanc-19 in results among different models, which may be caused partly by difference in model size, propeller loading and also relative rudder size (cf. Table 1). It seems, however, that as for the two domi.-nant constants K and T1, the different ap-proaches with different model size give roughly similar figures. On the other hand, T2 and T3 given from the present approach seem to be too large, compared with the ones from captives model experiments.

Recalling here how T3 is composed of the hydrodynamic derivatives'), we get

Ya

in' +mu' T3 ' Yfii ₎

Determining Y/, N A', "17,;' and Na' from the

oblique tow experiment for the same models

+my' are derived using this

equation utilizing T3 which is already determined m'+;ny' thus obtained is indicated in Table 2 at the columns of the present data.

These values of the lateral virtual mass

are unreasonably large and therefore we are

forced to conclude that the

T3' obtained through the present procedure is too large.

As an alternative approach, we may

( 1 ) first determine Ta' from in' ± u

estimated and Ili', N,', Yo' and Na' pro-vided by the oblique tow and

( 2 )

then determine K', T1' and T2' from

11;(i0/) derived from free model results using Eq. (8).

The dotted lines in Figs. 4 and 5 indicate

the Y

) Composed of K', T1, etc. thus

determined. In this way a reasonable value

of T3' is given, but the simulation of the

Ys(i0) obtained becomes worse in the

inter-mediate frequency range, as shown in the

*: Unreasonably large, cf. Sec. 5.2.

PMM: Planar Motion Mechanism. RA: Rotating Arm.

figures.

_{This paradoxial results will need}

further studies, while it is of the first

im-portance from the standpoint of

course-control to determine the transfer function

Y g(i co) at all significant frequencies, with less

regard to evaluating each individual coef-ficient, K', T11, etc.

Acknowledgement

The experiment involved in this paper was 'Carried out_{as graduation work at the Osaka} 'University by Mr. H. Kashima and Mr. 0. Yamamoto in 1968 to 69. Thi4

acknowledge-ment is made for their contribution in this work.

References

1). K. NOMOTO et al.: On the Steering Qualities

of Ships, J. Soc. Nay. Arch. Japan, Vol. 99,

101, (1956-57), English version published in

International Shipbuilding Progress, (1957).

K. NOMOTO: _{Directional Stability of}

Automa-tically Steered Ships, J. Soc. Nay. Arch. Japan,

Vol. 104, (1959). English version published in

DTMB Rep. 1464, (1960). on the,

S. MOTORA and M. FUJINO: Measurement of

the Stability Derivatives by Means of Forced Leeuwen

PMM

Eda & Crane .

RA Motora& Fujino Present Data -Remarks for Present (152) 70 (152) 60B2 (158) PMM(158) M. No.152 M. No. 158 Data 1.11 1.60 1.12 K' 1.26 1.41 2.80 3.60 2.59 3.12 Ti' 3.12 3.41 Freq. 0.31 0.30 0.23 0.09 Ti' 0.52 0.45 Resp. 0.71 0.70 0.66 T3' 1.33 1.01 (1) 2.40 3.20 2.21 T' 2.30 2.85 rm.' K' 0 1.24 1.43 0.04 1.07 1.08 Zig-Zag. T' 0 1.56 2.00 0.04 1.33 1.39 K' 0 1.29 1.41 0.04 1.14 1.15 Spiral 0.388 0.324 0.270 0.330 Yp' 0.339 0.318 _Oblique 0.100 0.104 0.108 0.115 Np' 0.091 0.099 Tow 0.037 0.052 0.046 - Ye' 0.052 0.042 0.019 0.025 0.022 No' 0.024 0.021 (2) 0.401 0.380 0.330 0.335 ne +mi,' 0.718* 0.523* 0.0213 0.0237 0.0215 0.0104 /2-1-../Y 0.023 0.023 Derived 0.133 0.125 0.078 0.107 nz'± tn.' - Y, 0.040 0.061 from 0.060 0.066 0.067 0.060 - Nr' 0.041 0.043 (1)± (2) 0.0125 0.0125 0.0100 0.0064

Il

0.0125 0.0100

A New Procedure of Manoeuvring Model Experiment 123

Yawing Technique, (written in English), J. Soc. Nay. Arch. Japan, Vol. 118, (1965).

4) G. VAN LEEIJWEN: The Lateral Damping and

Added Mass of Horizontally Oscillating Ship.

model. Tech. University Delft, Pub. No. 23,

(1964).

5) H. EDA and L. CRANE: Manoeuvii-trg

Charac-teristics of Ships in Calm Water and,j,ii Waves,