Full-scale measurement of frequency response characteristics of ships in steering - Parallel shift manoeuvre test as applied to actual ships

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Introduction

Highly centralized automatic control sys-tem of ships has been progressing rapidly, accompaniéd by successive appearance of specialized carrier and shortage of seamen. Along with it, course-keeping quality

be-comes the center of interst in the field. of

study on rnänoeuvrability of ships, especially the relation

between the

course-keeping

ability of ships with control systems in.

corporàting automatic steering device and

' Kobe University of Mercantile Marine

149 TECNJ

URIT

Labor

r

Spm

A Mekeiweg Z 2b C eÎN TeL O15.788813. FaIc O1S78183S

10.

_{Füll-scale Measurement of Frequency Response}

Characteristics of Ships in Steering

Parallel Shift Manoeuvre Test

as Applied, to Actual

Ships-Keiichi KARASUNO*, Member

(Fro,n J.S.N.A. Japan, Vol: 128, Dcc. 1970)

Summary

This paper shows the resdits obtained on actual ships by using the method presented before in A New Procedure of Manoeuvring Model Experiment i) The purpose of this paper is to determine these four characteristic constants K' Ti' T2' and r3' of actual ships

by means of frequency response analysis of a parallel shift manoeuvring and trapezoidal steering tests with fixed periods _{This procedure gives enough informations of frequency} response characteristics of a ship at practically important frequencies.

As an additional attempt this paper deals with a çb-çiphase plane analysis on a parallel

shift manoeüvre to determine the characteristic constants K', Ti', T2' and Ti'.

Full size experiments were carried out for eight actual ships ranging from a 360 GT

training ship to a 220,000 DWT oil tanker. The conclusiòns obtained are that:

the combination of the parallel shift manoeuvre and periodic trapezoidal steering is

a promising procedure for full size steering test to determine the steering characteristic constants K', Ti', T2' and T3'.

both steering tests mentioned above are also applicable to the phase plane analysis and this analysis can be used to determine the reasonable characteristic constants together with the frequency response analysis.

the steeriñg transfer futiction Y(iw), obtained from the frequency response analysis

in the intermediate frequency range provides useful data in connection with K'T3'/Ti'Tj'to

determine whether the course keeping ability, especially in the case of the automatic steer. ing, is sufficient or not.

proper course-keeping quality of the ships.

In

fact, many super-large ships showing

course-instability are operated sufficiéntly enough with auto-pilot systems, therefore one of the ship design problems is to find out the allowable course-instability limit and

con-sequently the least rudder area neçessary

for a given ship.

In general, the characteiistics of the

re-sponse behaviour of a ship to steering can be well represented by four characteristic

constants K, T, T2 and T3.

Usually, the simplified steering quality indices K and T

(2)

are widely used. These indices together

with the results of the spiral test turned out to be the common method of representing the manoeuvrability of ships. Hitherto, Numerous results of full-scale manoeuvre test to steering have, been summarized ac-cording to this simplified indices in the J apa-nese shipyards, and these indices, K and T become the criteria to judge the manoeuvra-bility of ships. Recently K and T were used also for the calculation of ship motion caused by automatic steering. However, when the ship is steered frequently by manual or by automatic steering and the course control is taken into account, K and T are inconvenient for the analysis and synthesis of the ship's motion, and all the four constants K, T1, T2 and.. T3 must be used.

This inconvenience is attributed to the

fact that the motion mentioned above sur-passes the applicable range of the first order system approximation in which the manoeu-vrability of ships are represented with just

a pair of indices K and T, and especially

due to the negligence of the stabilizing effect of the term K- T3á whiçh effects largely the ship motion.

Here, it is necessary to use

complete representation of the response

be-haviour of ships to steering with the four

characteristic constants K, T1, T2 and T3

described above.'

The aim of this study is to determine the constants K, T1, T2 and T3 through full-scale measurements of response behaviour of ships

to steering, other aim is to apply the new

procedure of manoeuvring model experi-ment'1 proposed by the author et al. to actual ships to examine its practicality. Besides,

the determination of the four characteristic constants was also attempted by the use of phase plane analysis"'3 which application has started in the treatment of naval archi-tecture problems.. Other advantages in this experimental procedure are that there is no loss of ship passage and the chance to en-counter with obstacles is small. It may be said that this is the only transient response technique applicable to full-scale experiments

in practice. Further, the comparison between the data of full-scale expêriments and, model experiments is possible, and useful

informa-tion can be obtained in regard to the

cor-relation of full-scale ships and model ships on manoeuvrability.

Principal particulars of given ships and the 'test conditions

Full-scale experiments were carried out in cooperation with some shipyards, Japan Ship-building Research Association and using a traning ship of Kobe University of Mercantile Marine. Results of the experiments on eight

ships, including the traning ship

"Fukae-maru" of 360G.T., a car carrier of 2,600G.T. up to a tanker of 220,000 D.W.T. are shown below.

Four ships were fully loaded, and

the other four were in ballast condition.

Particulars of these ships and the test con-ditions are shown in Table 1.

Parallel shift manoeuvre experiment

2.1

Procedure and measurement in 'the

parallel shift manoeuvre experiment

In the parallel

shift manoeuvre experi-ment, a ship is steered so that the eventual

course after the steering is parallel to the

initial course. The resulting ship's track is shown in Fig. 1. Considering the applica-tion öf two types of analysis technique de-scribed later on manoeuvring experiments, combined use of manual and automatic steer-ing was taken, in which at the beginsteer-ing of the experiment the ship was steered

manual-ly and then the helm was switched to the

automatic one halfway.

In carrying out

such an experiment, special attention should

be paid to start the experiment with

cau-tious course-keeping on straight course by best performance of the automatic steering or by skillful helmsmen so as to insure non-initial disturbance, also the course setting for the automatic steering equipment must be done accurately

o that the course after

steering become exactly parallel to the one at the start of the experiment.

An example of the procedure in the parallel

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Full-scale Measurement of Frequency Response Characteristics of Ships in Steering 151

¡(t) rudder anIe

Tabla i Particùiaro of the ShipG and Tact Condjtjonc

- g

I i

I ¡

t I

shift manoeuvre experin-ient is given belbw.

1) Advance- straight ahead as exactly as

possible by manual or automatic steering. Although in this state the ship advances

straight ahead, the yaw rate of the ship

varies around zero though slightly, but ex-periment must begIn with the ascertaining condition in which the yaw rate is perfectly zero.. Besides, it is convenient to take shIp's heading as round number of gyrocothpass as

s heodin

turn on CutomQtiC steerin

/ , / / stèering for t ¡ stroight course

I..

j

manut steering L autômotic steering

for full-scale experiments.

Order 50 starboard with the helm at

the instant of the zero yaw-rate mentioned in prócedure (1)

Order 50

_{pott with the helm}

when heading deviatión reached at 30

Order 5° starboard with the helm again, at 1° prior to approaching the original head-ing. Then, after overshooting the original headslightly,

the ship's heading starts to

hipNuabor ..). 2 4 .6

'7--8

9

Kind of Ship Training

Ship Oil Tanker Bulk Carrier Bulk Carrier Car Carrier Ore Cariier Oil Tanker Ore/Oli. Carrier

Length Between Perpendiculars 37 0 300 0 162 6 162 6 115 0 2490 307 0 251 0

L(m)

Length/readth, LIB 4.75 6.0 - 6.55 6.55 7.-1 5.94 6.36 6,15

Breadth/Draught, D/d 2.89 2.64 2.72 2.72 3.24 2.95 2.49 2.69

Block Coefficient, Cb 0.564 0.030 0.790 0.790 0.526 0.823 0.850 0.821

L.C.B. from Midohip, 1.cb. O.95%a. 368%f. 3.96%r. 3.99sf. 2.90$a. 270%f. 2.72%f. - 3.l0%f.

Number of Eudder - - 1 1 - 1 1 1

ii,

-

i

Typo of Rudder Ordinary Ordinary t4arincr Mariner Mariner Ordinary Ordinary Ordinary

Rudder Area Ratio, A/L.d 1/34.15 1/61.8 1/55.1 l/55.11/44.3 1/61.7 1/69.1 1/66.7.

(with Horn) ith liorn)ith Horn)

- 1/66.4 1/66.4 1/59.2

(movable) (movable) Qaovable)

Rudder flight, h() 2.105 12.30 7.00 7.00 -

-lumber of Propeller 1 1 1 1 2 1 1 1

Propeller Diameter, O(a) 1.95 8.60 5.64. 5.64 3.285 6.90 8.40 6.50

Propeller Pitch Ratio Control 0 709 0 819 0 819 0 885 0 677 O 677 0 754

Number of Bladea. 3 5 5 5 4 5 5 .5

Direction of Rotation Left Right Right Right R,&L. Right Right aiht

Loading Öoniion Ful Full Ballast Ballast BaJ.1át Bàllhdt Pu11 - Ful1

-Dioplacement (ton) Tria, (d-df)/L 1.62%.454 243 954 0% 17 809 2.35% 17 405 220% 3 810 1.21% 66 055 0.21% 246 500 0% 126 300 0% -

-¡lean Draught, d,,,(m) 2.70 l9O2. 5.57 547 - 3.999 7.854 19.35 14.63

Ship Speed, V0(kt) 11.5 17.5 16.4 16.7 19.0 15.0 15.8 16.5

Remarka . equippc CutUp

-5amc Venu Lirge&

with .torn

. Lou9

Bow- _:-

-. Super

Thruotar S.No3 _StrucLurc

-Test

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152 , Keiichi KARASUNO

return to the original one again.

5) Turn on the automatic steering device

to keep the original heading, just before

reaching the original heading, usually Ö.5° before. Course-keeping with an automatic steering device is held up to approach to a regular damped or sustained yawing even-tually. Moreover it is desirable tJiat the rud-der angle unrud-der the operation of the auto-matic steering device is always proportional to the heading error, in addition continuous active rudder movement is carried out .so as

to make the sustained yawing motion'.

For adjusting the automatic course-keeping device, it is desirable that weather adjust-ment, dead band rate control and so on äre made as smalla s possible, and that the pro-portional steering of maximum helm adjust-ment is carried out.

'The above procedure is represented gra-phically in Fig. 1, but it is to be stated that

the experiments described in the present

paper must not necessarily follow the pro-cedure described above, instead, various procedures were attempted, such as the test in which the automatic steering devicewas turned on in place of procedure 4), test by bang-bang steering repeating procedure 4) and test by the automatic steering using it from the outset. The necessary time for any

of these experiments is usually about 15

minutes.

The rudder movement 5(t) was measuréd

with a potentiometer rotated by a piano

wire wound around a rudder stock, while yaw angle .çb(t) was measured with a

direc-tional gyroscope, both were recorded by

oscillographs. Measurement of yaw rate us-ing rate gyroscope was also employed partly.

The yaw añgle and the yaw' rate Were

caiibrated.by reading heading angle of the ship's gyrocompass. The ship forward veloci-ty was measured 'with a log, and the change in the forward speed was hardly observed except the parallel shift manoeuvre experi-ment ,with, 1arge motion.

2.2 Outline of frequency response analysis

on the parallel shift manoeuvre ex-perirnent'

The relation between the ruddér

move-ment 5(t) and the ship motion ç(t) in the

linearized motion of the ship's manoeuvre can be represented reasonably by the

equa-tion

d2ç1 ' dçt

TiT2--+(Ti+ T2)--+

=Kô+KT3.

Expressing Eq. (1) using the transfer func-tion for the rudder movement as input and the ship motion as output, we have

Y( )

K(l±PTs) ç1(p)

b()

p

(l+pT1)(1+pT2)

ô() - 5(p)/p

(1)

where =a+iw represents a complex num-ber.

In case of course-stable ships, both inte-grals of the numerator and the denominator converge witha-+O+, therefore we can write

K(1+iwT3) Ys(zw)-(1+ zwTi)Ys(zw)-(1+ zwT2)

5«t)e'te-1'tdt

=lim O

(3)

,--(l+ 5 5 3(r)dr (It

Measuring /(t) or

(t)

for an arbitrary

rudder motion 5(t) and substituting their

values in Eq. (3), the transfer function Ys(iw)

can be determined and then the response

characteristics of ships can be determined. The importance of using Eq. (3) in deter-mining Y in the range of small w from the

parallel shift motion has been described in literature', because no. eventual change of ship's heading is caused.

N11Tic

calculation of Y(iw) is

per-formed for several values of w according to

Eq. 3). But it is difficult to carry out the perfect parallel' shift manoeuvring

experi-L t Ptçíj(t)dt (2)

-LI \ 5(t)dt LJ0 ' J

-

r I I et \ 5(r)dr. dt Jo Jo

(5)

C-i/Ti

Full-scale Measurement of Frequency Response ;Çharacteristics of Ships in Steering 153

ment in full-scale test, because of the

ex-.istence of external disturbances acting upon

the ship in sea.

It 'is therefore necessary that some corrections are applied to the re-sults of measurements1, such as corrections

för initial motion, rudder neutral position

error,

residual steady turning, imperfect

parallel shifting of the truck, etc. Thus the result. may be converted to that for. the ideal parallel shifting, then the calculation accord-ing to Eq. (3) can be carried out.

2.3 Improvement of correction'calculation in the frequency response analysis

The outline of the frequency

response analysis.in the parallel shift manoeuvre

ex-periments is as described before. In case of small amount of, the correction due to the imperfect parallel shift motion, the error in

the. correction using the indices K and. T

described in literature' may be small. In

full-scale experiments, however, the time constant T is large in general and as a re-sult the amount of this correction is often not small .but. large amount of correction is required even forsmall initial motion. There-fore the application of the first order system approximation to this. case is improper, and exact correction with K, T1, T2 and T3 is

desirable.

In making this correction, the

four characteristic constants assumed before-hand can be òbtained either by finishing the frequency response -analysis. or by the use of phase plane analysis to be described later. The corrections based on K, T3, T2 and T3 turn out to be as follows,

ist correctioncorrection of the initial mo

tion-Assuming ço and ço as,, initial yaw rate

and yaw acceleration respectively, which the

ship has at the, time of begining the test,

"memory" of this initial motion is

con-tained in ship motion recorded, it comes to

be - TIç/o+TlT2o

=fi

T2q0+ T1T20

+T

_T1T2

+(Ti+T2)çi0+T1T20

AccordIngly, this value is subtracted from measured '.

2nd correctioncorrection of rudder neutral position error ¿5f'

The apparent rudder angle -ô, measured

is converted to the true rudder angle ô in

Eq. (3) (ô=Òm+ô). The ships advance straight at zero true rudder angle.

Namely, 'D=5rt is 'subtracted from the

measured D

3rd correctioncorrection of residual steady

turning ç

in full-scale experiments 9L is expected to become zero, because steering in this

pro-cedure is carried out on the bàsis of the

ship's heading measured with gyrocompass on board: Sometimes a case. occurs, however, in Which small is present dUe tò charac-teristics of the. automatic steering device in operation during short time of measurement.. At that time, stepped steering ô5-i(t)

induc-ing the residual turninduc-ing rate çii,,, and -its

motion response

-.

D=t

(TT

.et/rj

(T2 Ta)

+(Ti+ T Ta)ç&,ç&,.t

are added to the measured D and .b values

respectively.

-4th correctioncorrection of the imperfèct

parallel shifting of the truck çbm

-Courses before and after the parallel shift manoeuvre must be made exactly in parallel, but eventual change of, ship's heading .about 0.5° is inevitable in full-scale experiments,

which is caused by error in. course-setting and

_{dead band and weather adjustment}

of autopiiot mechanism. Besides, sometimes these. parallel .courses may not materialize due to the ist and 3rd corrections.

Now assuming çb,1

_{as time avarãge of}

ship's heading measured from a base line

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154

.;

: : Keiichi KARASUNO

after the ist-.. and 3rd corrections are applied, correction is carried out by. adding impulse steering just sufficient for eliminat-ing this imperfect parallel shifteliminat-ing, and its motion response

Dçbrn,/Ke

/ (TiTs)çb

T1T2

(7'2 T3)çbmo,

T1T2

Summarizing the above four corrections, DDrn,o(ör+«3s)t

{D0+(T2-73)ç)

T3

{T2o+(T2--T3»

TT1T2o}]e/T1

T1o+(Tj T3 _¿

Tio+(Ti

T3)

T2o)]e/T2 + Ç& ± ç - t

where V

=no.±(ç1açi')(Ti+

T2 T3)

+ç1oTiT2+T3ç1'o, Dm,,=çbw,o/Ke

are the base lines for çb and D. Substitut-ing valus çb and D into Eq.. (3), Y5(0) can be

computed.

5th correctioncorrection due to the

dif-ference between assumed K and computed In general, final K.can be determined aç-cording to the equation,

--=(l/Ki + C/Ke)/(1 ± C)

Where

ICI <i, -

t/ (2.

dt)

çb:

yaw angle after

ist.,

3rd and 4th

correctiòns V

Ki: computed Ye(0) V V V

In the case of full-scale experiment,

some-times a case of ICI>1 may occur. When

C>i, then 1/K can be converged to a proper value if subsequent approximate calculation assuming [Ke+ Y(0)]/2 as new Ke is carried

out. When C<i, K comes to be zéro by

this method, and reasonable valuê of K can not be, obtained. Accordingly it should be

decided as satisfactory when there is no

much difference between Ye(0) and presumed! K attained by trial and error method. Be-sides, the situation of such ship motion of

C<-1 occurs when the initial yaw rate is

high and, in addition, hasthe same sign as

yaw rate given by the first swing of the

parallel shift manoeuvre. It

occurs also

when course error before and after steering is large, and the eve'ntual mean ship's

head-ing at the steady sustained yawhead-ing has añ

opposite sign to the ship's heading given at the first swing of the parallel shift motion. We should pay specially attention to avoid thses situations shown in Fig. 2.

Example 1 (I, Example 2 1' base line No.1.2.3 cor

bose line after No.123orrectiôns

Fig. 2 Examples of Erroneous Ship Motion at Parallel Shift Manoeuvre (C<-1)

When the exact correction based on the

above described characteristic constants K,

T1,

T2 and T3 is carried out, error due to

the corrections turns out to be less, and

re-asonable transfer function Y can be

ob-tained.

2.4 Results of harmonic analysis on fulI scale experiments

V

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Full-scale Measurement of Frequency sponse. Characteristics of Ships in Steering 155

periments and. its frequency response analysis was carried out for seven ships, namely S. Nô. 1, 2, 3, 4, 6, 7 and 9; are shown below. The procedure of the experiments for each

ship put to the test is different in detail.

be time histories of each rudder movement ô(t) and its ship response çL(t) is shown in

Figs. 3, 4, 5, 6, 7, 8 and 9. The transfer

function _{Y'(iw') obtàined as the results of}

Fig. 3 Record of Parallel Shift Manoeuïie

Test of S. No. 1

Fig. 4 Record of Parallel Shift Manoeuvre Test of S.No. 2

Fig. 5 Record of ParàIIel Shift Manoeuvre.

Test of S. No. 3

harmonic analysis on the parallel shift mo-tion is shown in the form of a Bode diagram in Figs. 10, 11, 12, 13, 14, 15 änd 16, and plotted with O marks in these figures. The

/ /

S No.4

Fig. 6 'Record of Parallel Shift Mänoeuvre Test of S.N. 4

Fig. 7 Record of Parallel Shift Manoeuvre l'est of S.No. 6

Fig. 9 Record of Parallel Sl'ift Manoeuvre

Test of S. No. 9

-50rn

Fig. 8 Record of Parallel Shift Manoeuvre

Test of S. No: 7 .5.Ño9 Sec sec., 5 5.- S.No.1

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Frequency Response Analysis. for Parallel Shift Monoeuvre o + Frequency Respanse Analysis for Periodic Steering 1og,lYl .0 - --0.5 Frequency Respörtse Analysis. for rttel 8 SI-if t Manoeuvre o d'. -10. 0.5 -50--0.5 + Frequency Response. 8 Analysis for Periodic Steering -10-- 1.0 log .lYl 0.2 log.lYl o o 0--Q5 K.O.8B T.05l.0.52i T0.51 -052i T0.64 Frequency Response Analysis for Periodic Steeriiig SN0 I tog.W _{!Q_} 1.5

Fig W Steering Transfer Function of .S. No. i as obtained by Frequency Response

Experiment SNo.2 log.W lVrl.1 6 T29 T;0.6.4 Iiif29Q K2.09 T5.93 T0.76 Tl.7B

Fg. 11 Steering Transfer Function of S. No. 2

a obtained by Frequency Response

Expfiinent S No.3 + Frequency ReSponse Anolysis br PeTrioalc Steering 02. log,IYI o logw k1.47 T;3.10 T;1.06 1.94 ioc! 1.5

Fig. 13 Steering Transfer Function of S. No 4 as obtained. by Frequency Response

Experiment. 0.2 log.lVl

0.

Frequency Response Analysis for Parallel Shift Manoeuvre. 0.2 log ,lYl o -1:5

.:.

Fig. 14 Steering Transfer Functiôn of S. No. 6

as obtained by Frequency Response

Experiment Frequency Response Analysis for Parallel Shift Manoeuvre + Frequency Response Analysis for Periodic Steering SNo7 -bc! -1.5

Fig. 15 Steering Transfer Function of S. Nô. 7

Experiment -li Frequency Respoñse olysis for Parallel hifI Mono.yr- l-th - -I

I..

..

+Frequency Response Analysis for Periodic Steering iM 156 Kêiichi KARASUNO log.w 5 Fig. 12 -1.5

Steering Transfer Function of S. No. 3

(9)

+Freei Response Jialysis for Periodic Steering 10 8 7 6 5 3 2 o

W 136

T;a89

T,,1 24

Figé 16 Steering Transfer Function of S. No 9 as obtained by Frequency Response

Experiment

Bode diagram reptesents common logarithms

of the noñdirnensional amplitude ratio

IogioIY5'(ïw')j

and the phase

lag ç1Arg

Ys'(iw') on its ordinate. Hereupon, w' mw/( VIL), Ys'(iw')m Y(iw)/(V/L), where V is forward speed. and,. L is ship length '

Moreover, Fig. 17 shows ali example of harnionic components comtained in tïre ne-gral of the rudder motion Dm

5

)(1)dt' in the

parallel shift manOeuvre arid give some

cri-tenon for judging up to what extent of fre-quency this analysis can be reliable. In this figure, the ratio of the harmonic component of D at frequency w to the zero-frequency harmonics of D is taken as thé ordinate, and

its change with the frequency . logiw' is

shown.

The outline of the results of each parallel shift manoeuvre experiment and its harmonic analysis for each ship is described below.

s.

_{No. 1After the rudder was put to 3°}

port and then to 3° starboard by a

helms-man, the parallel shifting was carried . Out by turningon the automatic steering device (Fig. 3). The results of experiments of the

same procedure repeated. four times are

shown in the form of Bode diagram in Fig. 10 including re-experiments held three days later. The variation Of the amplitude ratio

I Y3'I for each experiment was found to be

within ±13% of their mean value, but the

phase lag was found to be in good agree.

ment. The analyzed frequency extends over

the range logiow'O.l, and it was found

that the harmonic component contained in

ship motion is larger as the frequency is

lower. The reliability of the results at low frequency range increases, as canbededuced from the diagram of the harmonic component

contained in the integral of rudder angle

(Fig. 7). Besides, in this ship,.the tendency of the, amplitude ratio lYs'! to decrease with the decrease of the frequency. was observed at the low frequency range, but it. seems to

be a special case in view of the previous

res ults.

S. No. 2After steering to

° port and 50

starboard .respectively, automatic. steering

device was. put in action. (Fig. 4). Experi-ments 'of the same procedure were repeated five times, and the results are shown iti Fig.

11 In these experiments the initial motion

were different slightly from each other. The variation of amplitude ratio I Ys'I for each ex'perimentwas within ±20% of their mean values, but' it can be said that the phase lag was almost in. agreement. The resúlts cover the frequency range of logiow'O. The re-ID(k')I/tD(o) - where ' o oTh 0.2 log,1Y1 S:No.9 Frequency Response Analysis for Parallel Shift Manoeuvre o q) cf-0 o 0.5 05 Iog,.Lv' o -10 -.Q5

Fig. 17 Amplitude Components of. _S(t)dt

against Frequency w' at Parallel Shift.

Manoeuvre

-10 -o 5

(10)

:158 _{Keiichi KARASUNO}

liability of the analyzed results are reflected by spectrum of rudder angle integral, and it can be seen that the reliability is best at logiuu'=O.l and decreases gradualy in lower frequency range than that value. While in higher frequency range than that value, the variation of Ys'(iw') for each experiment is very large, and the reliability of Y'(iw')

de-creases rapidly.

S. No. 3After the rudder was put once

to loo port and thento 100 starboard, the

parallel shift was carried out with course-keeping by bang-bang steering of 50 rudder angle. The. steering was carried out by a

helmsman throughout, and the automatic

steering device was not employed (Fig 5). The analyzed results are shown in Fig. .12

for the range of logow'0.l.

It was found that the experimental points of Arg Ys'(iw') at logiow'O.l are especially deviated from the smooth Bode diagram observed in thc common response characteristics, 'accord ing. ly the, response.' results at these points seem unreasonable. This may be due to the effect of inaccuracy of 'the analysis, because the course-keeping control in this experiment was made by the bang-bang steering, and also sufficiently steady sustained yawing mo-tion was notobtained. Besides, in the higher frequency range above Iogow'0.1, the har-monic component contained in motion de-creases rapidly, and it can be said that the

obtained Ys'(i')

_at

this frequency range

lacks reliability.

S. No.' 4After putting the rudder once

to 100 starboard, the parállel shift manoeu-vre was carried out with course-keeping by bang-bang steering of 5° rudder angle (Fig. 6). The analyzed results extending over the range of logioco'O.l are shown in Fig. 13. In spite of the fact that the test conditions were almost same and the types were also same for this ship and S. No. 3, the resu1ts of the harmonic analysis on the ship motiòn was found to be in great difference. Namely, comparing the amplitude ratio I Y'j for both tests, we can find differences of up to 25% in the low frequency range, 'while both _{I Y/I}

are in good agreement with the vicinity of logiocu'=O. On the other hand, comparing

the phase lag, we find that both

q. values are good in agreement at the low frequency range, and there are

differences of up

to 130

in the viòinity of logiow'=O. The

reasons of these differences seem to be that the reliability of the analysis is low because of considerably large initial motion in this experiment. Moreover, the accuracy of the experiment for S. No. 3 as mentioned before is insufficient, and that there are some dif-ferences in the magnitude of the motion for both ships. For all that the differences are too large to considered as éxperimental error for two ships of the same type.

S. No. 6After one cycle of triangular

wave-like steering of about 20° half ampli-tude was given, the parallel shift was carried out by putting automatic steering in action (Fig. 7). The results of the harmonic analysis

extending over the range of Iogw'O are

shown in Fig. 14. The amplitude ratio

de-creases with decreasing frequency in the

low frequency range, similarly to the

be-'haviotir observed in S. No. 1. Besides, the steady yawing motion at the final stage of the parallel shift manoeuvre in this experi-ment was short in time and was not steady enough, accordingl.y the period of the steady

yawing and transfer function of the

fre-quency are not exact.

Moreover, the

fre-quency at which the speòtra of the rudder

angle integral show maximum value is in

agreement with the period of steady

yaw-ing. As 'a result, it can be seen that judg-ing the accuracy of the transfer function in the vicinity of the frequency from the

spec-tral diagram, can lead to inaccurate

deci-sions.

S. No. 7Very large motion occured, since rudder angle of 35° was employed (Fig. 8). The test can be considered as an equivalent to the so-called parállel shift measuring test. Results extending over the range of logow'

0.3 are shown in Fig. 15.

It was found

that the phase lag at logow'0.15 seems to

(11)

obtained in the frequency range higher than that value is not good.

S. No. 9After steering once to 50 port

and then to 5° starboard, the helm was

switched to automatic steering (Fig. 9), and course-keeping by bang-bang steering of rudder angle was employed for the parallel shift motion (Fig. 31). Analyzed results of

repeated experiments for three times are

shown in Fig. 16 for frequency range of

logioco'0.1. The difference in Ys'(iw') for each experiment is large in the frequency

range of loga/-0.6, and there are about

±20% differences in the amplitude ratio j Y'j. In this respect, it may be one of the causes that accuracy of analysis in low frequency range on bang-bang parallel shift manoeu-vre is not satisfactory. On the other hand,

from the point of view of the accuracy of

analysis, it can be seen in view of spectra

of rudder angle integral that the accuracy

is maximum at

logiw'0,

and is

de-creases gradually in the lower frequency

range, while decreases rapidly in the higher frequency range, than that value of logiow'

=0.

Summarizing the results for the seven ac-tual course-stable ships described above, it

can be said that in general. the effective

analysis range of the frequency in the parallel shift motion is generally logiow'0. Needless to say,

it depends upon the experimental

procedure.

ç

¡(t):rudder ongte

-5.

3. Periodic steering experiment

3.1 Harmonic analysis on trapezoidal steer-ing experiment

The transfer

function Ys(iw) obtained

through the parallel shift motion is especially

lacking for reliability in the intermediate

and the high frequency ranges. As experi-mental technique to overcome this defect, sinusoidal steering has been carried out in model tests. However, it is difficult in

ac-tual ships to set the rudder in sinusoidal

motion, unless special devices are installed on the existing steering systems. Accord-ingly, as an alternative solution, trapezoidal steering is taken up, in which the rudder is changed port and starboard alternately with definite rudder angle and period. When the rudder movement ô(t) and the ship motion

çi'(t) in steady state are expanded into Fourier

series taking exciting period as basic period and the first terms are adopted, the rudder movement and the ship motion correspond-ing to the sinusoidal steercorrespond-ing can be obtained and then Ys(iw) can be computed. In ship manoeuvring, it is convenient to make the average course of motion close to the

origi-nal course at the time of test,

therefore

steering of the first half period is carried

out by being shortened to a quarter period

(Fig. 18). This corresponds to commence

the sinusoidal steering for model ships with cosine phase. The amplitude of the rudder

'I'(t):Ships heeding

Tr14+ 1,/2 ₊ 1,/2 ₊ 1,/2

+ 1,/2 .

T;,2

Fig. 18 Schematic Illustration of Trapezoidal Steering with a Constant Period e. / 5. I.-/ t.. / / s'

/

/ I I t / t / t / t / t

(12)

160 _{'..,Keiichi KARASUNO}

angle is decided so that to keep the yaw

rate small enough to suit 'the linier analysis, for example steering 'is carried out by taking the helm angle about '±3° in the low frequency range and ±10° in the high frequency range.

The above mentioned trapezoidal steering is active periodic, steering in which the period is given beforehand, and the trasfer: function

for any arbitraly frequency can be deter.

mined. 'On the other hand, the usual

zig-zag manoeuvre and the steady yawing mo-. tion in the parallel shift manoeuvre can be considered as passive periodic steering mo-tion from this viewpoint. When expansion into Fourier series is applied to these motions, they can shoùlder a role in determination of the transfoi- function Ys(iw). When the mo-tion is expressed by the measured yaw angle b(t), then ç is determined by differentiating

the basic harmonics of the yaw angle 'after expanding it. intO Fourier series, and Ys(iw) can be determined more accurately than the method of directly determining 'ç.

3.2 Results of harmonic analysis on full-scale experiments

A series of trapezoidal steering experi

ments by changing the period variously were carried out only for S. No. 1. As for other ships, the, usual zig-zag trials,, the modified zig-zag ,trials4 and the steady yawing motión of the parallel shift manoeuvre were con-' sidered substitutes' Of the trapezoidal

steer-ing experiments. Results of. the harmonic

analysis on periodic steering are denoted

with + marks in Figs. 10, 11, 12,13, 14, 15 and 16. The results for each tested ship are described below. '

S. No. 1The trapezoidal steering'

experi-ments were made. extending over the range of O.4log1Od'O.4 (Fig. 10); this frequency range corresponds to periods from about loo sec to 16 sec.

In order to keep the' yaw

rate small enough to suit the linear analysis, the amplitude of the rudder anglè was changed gradually from about ±3° in

the low frequency range to ±10° in high

frequency range.

The-results of the

har-monic analysis of the periodic steering shows

good agreement with the results of the

transient response analysis on the parallel shift motion around log0û/

0.2,

further, the amplitude ratio tends to decrease with

the decrease of the frequency in the low

frequency range.

S. No., 2Results of the harmonic analysis of the modified zig-zag manoeuvre of 5°-1° and 5°-0.5° are shown in Fig. 11 at the fre-quency around logiow'0.1 (period of 170 sec) and link smoothly, with the r,esults of the. harmonic analysis in the low frequency range ofthe parallel shift motion

S. No. 3Results of the frequency response analysis on the 50 zig-zag manoeuvre and the steady yawing motion in the parallel shift manoeuvre are shown in Fig. 12. The former is shown at frequency of logLow' = 0.2,(period

of 190 sec), and both amplitude ratio and

phase lag give values close to Ys' for the

parallel sh,i'ft motion. The latter is plotted

at the frequency of Iogioa'0.15 (period of

90 sec).

S. No. 4Results of the harmonic analysis on the steady yawing motion in the parallel shift manoeuvre are shown in Fig. 13 and are plotted at logow' 0.25, thus these 'results' offer valuable data in the intermediate fre-quency range.

S. No. 6The results of the. frequency re-sponse 'analysis' on the 10° zig-zag

manoeu-vre are given at logow'-0.3 (Fig. 14). The

motion is somewhat . small in comparison

with the transient motion in

the parallel shift 'manoeuvre experiment, consequently

the results are supposed to show poorer

course stability than, that of the transient

motion. On the contrary, the results come to be somewhat more stable only by little margin.

S.

No. 7The results of the harmonic

analysis on the 10° zig-zag trial are showñ in Fig. 15'. The motion is quite smaller than that in the parallel shift manoeuvre

experi-ment 'for the same ship.

Accordingly the results show poorer'course stability than that

(13)

of the harmonic analysis on the pararliel shift mOtion in Fig. 15.

S. No. 9The results

of the harmonic

analysis on the 5°-1° modified zig-zag trial are shown in Fig. 16, and the results show good agreement with Y/ for the parallel shift motion at the frequency of logjo w' 0 (period of 190 sec).

As a whole, the results of the harmonic

analysis, Ys(iw), on the trape2oidal steering trials, the zig-zag trials and the steady yaw-ing motion in the parallel shift manoeuvre seems to be in agreement with the results of the harmonic analysis on the parallel shift tion having almost similar magnitude of mo-tion to the periodic steering ones. Especially, it may be said that the trapezoidal steering, which is the active periodic steering, make up sufficiently for the defects of the parallel shift monoeuvre for the purpose of accurately determining Y3' in the intermediate frequency range which is important in the analysis or synthesis of coursekeeping steering.

4. Determination of the four characteristic

constants K', Ti', T2' and T3'

The combination of the harmonic analysis

on the parallel shift manoeuvre and the

periodic steering described in paragraphs 2 and 3 enable the determination of the trans-fer function Ys(iw) extending from the low

frequency range to the intermediate

fre-quency range.

The characteristics of the

high frequency range is not important in the actual manoeuvring motion, and when the response characteristics in the low and inter-mediate frequency ranges are known, they

can be practicaly used in cases such as

course-control motiön by the automatic

steer-ing and others.

From the standpoint of

control engineering, it

is not too much to

say that only this Bode diagram is enough to describe the response behaviour of a ship. On the other hand however, when the elu-cidation of physical properties of the ship manoeuvring motion and its application to naval architecture design are taken into con-sideration, it is convenient to represent the

response characteristics with the four con-stants K', T1', T2' and 7'3' using the transfer function of Eq. (2). Ftom the above stand-point, the four characteristic figures K', T1',

T2' and T3' can be determined from the

characteristics of Ys'(iw')w'. obtained by the experiments described above in low and inter-mediate frequency range by the use of Eq. (4) derived from Eq. (2).

T1'+T2' i K' _w'IYs'(iw') -(w'7'3 cosbrsinç5r) T3'.T2' 1 K' - w'2-K' _W?hIYS/(iWF)lTa +cosçb)

(4)

Hereupon ,K'mK/(V/L) and T'1,2,3(V/L)T1,2,3

The determination of the

four unknown quantities K', T1', T2' and T3' is possible in principle with four equations obtained by selecting any two different value of w', but

actually more exact characteristic figures

are; obtained by applying the least mean

sqüáfe method to Eq. (4).

In the examples of the test of the seven

full-scale ships described in this paper, the frequency response experiments were

repeated only few times for every ship;

Therefore numerical determination of the characteristic constant was carried out by

selecting frequency range - as. L0-logw'

and by the use of experimental point. The characteristic figures obtained are shown in Table 2, and alsO the response characteris-tics Ys'(iw') computed with the characteris-tic figures are drawn with full lines in the Bode diagrams. These curves fit well with the experimental points obtained from the harmonic analysis of the parallel shift mo-tion and the periodic steering momo-tion.

Among these characteristic constants. ob-tained above, time constants T' and T2' for S. No. 1 and S. No. 6 were found to be con-jugate complex numbers with positive real part. This fact shows that the free motion of ships becämes damped-oscillating yaw

(14)

motion, but in'view of the numerical _range. of hydrodynamic derivatives constituting _the characteristic constants, these phenomena can not occur in ships of normal types.5"

The results for each ship put to the test are described below,

S. No. 1-Starting from the phenomenon that the transfer function Y' in the low fre-quency range increases the amplitude ratio with the increase in -the frequency, T1' ànd 7'2' are calculated to be conjugate complex numbers; -The above-mentioned sequence

holds as far as

T1' and T2' are conjugate complex numbers: and in addition the con-dition of (T'-fTi')<2T1'.T2' is satisfied no matter how small T3' may be.

S. No. 2, 3, 4 and 9-T2' and T3' show con-siderably large figures in comparison with common practical figures decided from the results of model experiments at fully loaded and even keel conditions.

-S. No. 6-Though T1' and T2' areconjugate complex numbers in this ship, but the cön-dition óf (T'+'T2')2<2T1'T2' is not _satisfied, therefore these results differ from the results

of the S. No. 1:

it can not be said that the

peaks of j Y'j appeared for the same reason as S. No. 1. On the other hand, in this ship, Ta'. showed unusual large figures. in fact, if the relation of (T1'+T2')2<Ta'2+27'l'T2' _is satisfied among T1', _{T2' and T3', the peak} of _{Y'j appears at non zero frequency, and} this ship satisfies the relation. in terms of the above two reasons, it is clear

_{that the}

amplitude increasing effect of T3' is the cause of the phenomenon that the transfer func-tion in the low frequency range increases

the amplitude rätio with the

- increase of frequency.

S. No. 7-Although the results here _{are of} the parallel shift with large mòtion, charac-teristic figures of T2' and T3' do not differ from the common practical ones.

Summarizing the results for above seven ships, it can b said that the combinhtion of the frequency response analysis for the

parallel shift manoeuvre and the periodic

steering gives T2' and T3' considerablly large values when the method of determinatin _of the characteristic - constants is according to

Ship

No.

Frequency Response Analysis _{Zig Zag Test} Tufning

r'm - K ' -

T'

4-;-;

T;!'

+T.T3'

I

T'

K' 1 _{0 035} _{0 88}

-

_{0 64} _{1 04} _{0 84} 0 38 0 85 0 59 0 92 2 0.037 2.09 5.93 0.76 1.78 0.82 2.54 4.91 - 2.00 3.45 11.0 3 0.097 1.16 3.29 - 0.64 2.00 1.11 1.05 .1.92 0.88 0.92 1.13 4 0.083 1.47 3.10 1.06 1.94 0.87 1.70 2.22 6 0.103 0.69 _3.97 _0.49 _1.41 _0.36 _0.64 0.84 7 0 348 1 57 2 64 0 28 0 80 1 70 0 93 2 13 9 0.037 1.36 3.89 1.24 1.91 0.54 2.52 3.22 Ship N O.

Phase Plane Analysis - . -

-/ Tm K' T' i

T'

K' T' T'-

12

-

-+T2'-T3' Ti'T2' Ta' 9 . 0.046 1.24 2.46 0.14 0.24 0.87 1.42 2.36 8 0.045 5.50 6.40 0.16 0.36 1.90 - 2.90 6.20 162 _{Keiichi KARASUNO}

(15)

Eq. (4) and the above mentioned

experi-mental results of Ys'(iw')w' characteristics. This fact requires through examination here-after. Though the characteristic figures K', T1' and so on are important, from the control engineering point of view the determina-tion of the transfer funcdetermina-tion in the low and intermediate frequency ranges come to be

more important than those

characteristic constants. Summarizing, it may be said that if only Bode diagram is available, the ma-noeuvring characteristics of ships are re-presented thoroughly.

5. Comparative studies of the present results

with the results of spiral and zig-zag

trials

The results of spiral tests for S. No. 1, 2,

SNal

- 0.1

K': obtained from the Result of Frequency

Response Experiments

Fig. 19 Turning Test Results of S. No. 1

3 and 9 are shown with o marks in Figs. 19, 20, 21 and 22, and the results of zig-zag tests for S. No. 1, 2, 3, 4, 6, 7 and 9 are shown with o marks in Figs. 23, 24, 25, 26, 27 and

28.

Comparing the values of K' and T'(m T1' + T2' T3') derived from the harmonic analysis of the parallel shift motion and the periodic steering, with those derived from

the ordinary analysis on the spiral or the

zig-zag manoeuvre, it may be necessary to compare the indices K' and T' derived from motions having equivalent yaw rate, since K' and T' vary according to the magnitude

5.No.9

K': obtained from the Result of Frequency Response Experiments

0h

K': obtained from the .Result ofFrequency Response Experiments

Fig. 22 Turning Test Results of S.No. 9

S.No2 w'.

_r

0.3 0.2 o j1 K2.09 00 _-30° _-20° 0.1 100 1100 _{20°5 300} 4 o

7 nondimensionalized by Approach Speed

) ': obtained from the Result of Frequency Response Experiments

0.3 5.No.3 0.2

/

0.1 K_1.16 00 50 -15° -1 50 0 ₁₀₀ 1 -0.1 -02 03

(16)

164 _{Keiichi KARASUNO}

2

o

Response Experiments (T'= Ti'+ T2' - T3')

Fig. 23 Zig-Zag Test Results of S. No. i

+: obtained from the Result of Frequency

Response Experiments (T' = Ti' + T2'

7'3')

Fig. 25 Zig-Zag Test Results of S. No. 3

01 02 r., 03

Response Experiments (T'=T1'+T2'

Ti')

0

0 0.05 01 r, 0.15

Response Experiments _(T'=T1'+T2'

T3')

O

o

2

SNc.6

02

r,

04

Response Experiments (T'= Ti'+ T2'

7'3')

of ships. In the present paper, the magnitude of motion, the root

mean square value of

non-dimensional yaw rate rm', is defined as the average of extreme values of yaw rate

S.No.1 S,Na9 5.N3 o

.,.

_oK

iot'

2 o O 01

02 r02

o Response

_Ti')

Experiments (T'=T1'+T2'

o

1

r,

02

Fig. 28 _{Zig-Zag Test Results of S. No. 9} of motion because of non-linear motion

S.No.7 -I-2 1 3 2 03 01 02 r.,.

Nondimensjonaljzed by Approach Speed

(17)

Full-scale Measurement of Frequency Response Characteristics of Ships in Steering ₁₆₅ appeared on the first two swings of transient

part of the paralleí shift motion ovér Accordingly, it means putting weight on the initial part of the motion. The com-parison of the results of frequency respOnse experiments was made with K' of the spiral

test results (mean gradient of r'ô curves)

and with K' and T' of the zig-zag test

re-suits obtained by extrapolating or

interpolat-ing 1/K', 1/T'rm' curves at the same r' to

the parallel shift motion. These values are shown in Table 2, and also the results ob-tained through the frequency response analysis are recorded together with straight

lines of gradient K' in the results of the

spiral test (r'ö plane), and with + marks

in the results of zigzag test (1/K', 1/T'rm'

plane).

' According to the results of these

experiments:

S. No. 1The values of K' are almost in

agreement with the results of spiral tests

(Fig. 19) and zig-zag tests (Fig. 23). On the other hand, T1' and T2' are conjugate com-plex numbers, but the comparison of the re-sults of frequency response experiments with the zig-zag test results as for the tithe

con-stants can be made by the adoption of T',

since T'(T1'+T2'T3') comes to be a real

number. It is found, in Table 2 and Fig. 23 that T' derived from the frequency response experiments shows considerably smaller figures than those from the zig-zag tests.

S No. 2K' due to the frequency response

experiments is almost in agreement with

the results of the zig-zag tests (Fig.

24),

while T' due to the former experiments is

considerably larger than those of the latter experiments. Besides, K' according to the Spiral tests (Fig. 20) seems to be much larger than that of the ffequency response

experi-ments, but as the spiral characteristics at

sufficientlly small motion are not clear in Fig. 20, the comparison mentioned, above can not be made accurately..

S. No. 3 and 4It may be said that K'

obtained in the present approach is in

agree-ment with the results of' the spiral tests

rather 'than that of the zig-zag tests (Fig.

25). '

K' due

' to the frequency respoñse experiments is, however, slightly larger than

that due to the spiral and zig-zag tests,

while '1"

due to

the former experiments

turns out to be considerably larger than

that due to the latter ones.

S. No. 6Though the cômparison' of the

results from this approach with the results from the zig-zag tests is difficUlt, because of the difference in the magnitude of motion be-tween the parallel shift motion and the zig-zag trial motiOn, it may be said that K' is almost in agreement in both cäses. T1' and T2' according to the frequency response ex-perinients are complex number and show considerably small values in comparisOn with ordinary ones by converting them to T'.

S. No. 7The parallel shift motion is very

large and

it _{is unreasonable to compare} this

results with the results of

100

zig-zag test which has much w'eaker motion

than that, hut presiming from the latter

results daringly, K' derived from the parallel shift' manoeuvre seems to' be proper. On

the other hand, T' due to the parallel shift motion turns out to be rather large.

S. No.' 9If the curves of the spiral test

results (Fig. 22) are regarded as intersect-ing ordinate (r axis)' at five 'points, K' ac-cording to the frequency response experi-ments seems to be proper. However, com-parison with this spiral tests is difficult

be-cause' of the insúfficiency of thé data of

spiral tests in the vicinity of r'=O. Besides, above peculiar spiral characteristics ap-peared recently also' in a' certain model ex-periments7) and this was realized when a course-stabilized ship by so-called unusual phenomena turned out to be unstable due to, the reduction of rudder area.

As described above, regarding K', the re-sults of frequency response experiments and the results of other experiments seems to be almost in agreement generally, but the formers show rather slightly larger figures. On the other band, the values of T' calcu-lated through the frequency response. tech-nique are considerably large. In cases when

(18)

166

T1' and T2' are conjugate complex. numbers, T' dèrived from the frequency response ex

periments are found to have small values

than that from the zig-zag tests.

As for this discrepancy, there are some problems in the method of defining mean magnitude of motion in the parallel shift manoeuvre, but it should be remembered also that T' T1'

+ T2' Ti', which is the,promise of first order system approximation, may. not be totally correct. When importance is placed on values of the tiansfêr function in the low frequency

range, T' can be obtained as T1'+T2'T3',

while when importance is placed on thehigh frequency range, T' becomes T1'T2'/T3'. As

shown in Table. 2, using the latter T', the

agreement between the results of the

pre-sent approach and the

results of zig-zag tests seem to be improved slightly..

6. Attempt to determine characteristic con-stants through the parallel shift mono-euvre by phase plane analysis

In harmonic analysis of. transient response motion as.shown here, when it is subjected to external disturbance even if partly dur-ing a. sequence of motion, the disturbances may afféct the ship motion and spoil the re-sult and give undesirable tendency Qf de-clining reliability of the result. This matter is especially rèmarkable in ships with poor cöúrse stability.

I.n order to make up for

this defect, determination of K', T1', T2',

and Ti' was attempted by the application

of phase plane analysis2'1

to the parallel

shift motion which enters into appikation recently to studies.on manoeuvrability.. There are some possibilities in this method that the influence of partial external disturbance'be-comes comparatively small. Moreover, since this analyzing method is suitable for steer-ing conditions, of repeated definite rudder angle., it is applicable to the trapezoidal steer-ing test also.

6.1 Analyzing method.

First, it . was intended to consider such small motion that non-linear terms could he

Keiichi KARASUNO

neglected, and Eq. (1) was employed. Now

selecting k points in time in a region in

which the rudder angle was kept constant, and writing conditions of the mçtion as ,

Çik

etc,

T1T2

ik+ T1+T2

cik+Çi,ikömi±ôr

_(5>

hereupon, 3,: measured rudder angle 5,.: correction of neutral rudder

angle (unknown)

ôôm+ör

Besides, suffix .i indicates the region z in

which the rudder angle was kept constant. Taking dfdçi. into consideration, the conditions of motion (çb, _{ç, ç,. öm) can be} re-presented on phase plane of ç.çt by the use of the time t as a parameter (Fig. 29).

Sub-stituting the various values read on this

phase plane loci into Eq. (5), T1T2/K, (Ti + T2)/K and 1/K can be determined by the methOd of indeterminate coefficient.

Fig. 29 çb Phase Plane Trajectory with

Con-stant Rudder Angie

1) Determination of T1 T2/K, (T1 + T2)/K and 1/K'

First, in order to eliminate' the unknown neutral rudder angle, qs. (5) for different i values are put to subtraction,

T1T2 ... ... T1+T2 '

(19)

Full-scale Measurement of Frequency Response Characteristics of Ships iñ Steering

k=1,2,3,---,n

by multiplying Eq. (6) by the factor l/(ç1ik

j). Besides, the unknown Sr can be

tained, if necessary, by. substituting the ob-tained T1 T2/K, (T1 ± T2)/K and 1/K into Eq. (5). Then, if T3 is determined, all the

manoeurv-ing characteristic constants turn out to be

determined. In principle, .Ta can be

deter-mined from the conditions of motion during steering according to procedure (1) in the present discussion, but as the exact reading of the conditions of the motion during steer-ing, (sb, , ç1, 5m, im), especially Sm and ¿m,

is difficult, it is intended to determine T3 actually by the f llowing method.

2) Determination of T3

Definite integration is carried out on Eq. (1) from t=ti to E=ta, and dividing both sides by K, we get the equation

T2T2 .. T1±T2 . i

[ /1]L2______[/,]t2

K K 1 K T ¿j

=s;5t+or(t2_11» Tl[S,]2

(7)

_=_s(T1T2

±s

T1T2) dS T1T2

J

Selecting ti and li as the time just before

₍₉₎

167

±($ikç1J)Ôrni

Ônij

(6)

And the second i is represented by j. Also because only one k is enough in the j region, representatián can be made by the suffix j

only.

Taking n=3, the coefficients. T1T2/K, (Ti

± T2)/K and 1/K can be determined by a

system of three linear simultaneous equa-tion

_{in three unknowns from adequately}

different conditions of the motion, but it is

better to use the method of least mean

squares for values of n thore than 3.

At this time, for the purpose of improving cal-culátion accuracy of i/K obtained from data in the vicinity of çÛ=O, the determination of the coefficients is carried out in the form of

T3T2 (k_J)± T1T2

_{(k-.J)+ i}

K

(çiçj)

K

_(kçiJ)

K (Ôm5mj)

and after steering respectively, the

deter-mination of T3. is carried out according to Eq. (7). As for '1T2/JC, (Ti±Ti)/K, 1/K and Sr, the values obtained in above procedure (1) are employed; Besides, when this method of selecting time is adopted, effects of the calculation errors for 1/K and Sr on T3 are

t2) are small.

small becaue the values of

5

2ô1dt and (t1

LI

By the above calculation in procedures (1) and (2), T1T2/K, (T1±T2)/K, i/K, T3 and Sr

are determined, and all the characteristic

constants are obtained. Among them, com-parison will be attempted for 1/K with re-sults determined by another method.

3) Examination of 1/K

T1 T2/K, (T1 ± T3)/K, 1/K, T3 and Sr are now

to be known amounts, but i/K, as an un

known, will be determined again according to equation (7).

Selecting ti and t2 as times when _Çi, ÇbL2

=0 since [bj

becomes maximum, Eq. (7) turns to be

TiT1..

1

=52ô74t±Sr(t2_tI)+T3ôm1 _(8). Substituting tjie results obtained in pro-cedures (1) and (2) into .T1T2/K, T3 and ô,-, 1/K can be determined. Thern accuracy of this calculation may be confirmed by

wheth-er this i/K diffwheth-ers largely from the values

determined in procedure (i) or not.

Moreover, in case of dS/dç0 (Sd/dçf)

determination of T1T2/K and (T1.±T2)/K äc-cording to procedure (1) is difficult, but in

this case S= lIT1 or i/T3 comes óut

and an approximate value of 1/K can be

determined by the use of Eq. (9) which geii-erally holds true when ôm=const.

T1T2 T1+T3 Ôm+ôr

K

[S]0+

K

(20)

168 ' _{Keiichi KARASUNO}

At the tinie of executing the

_above

de-scribed numerical calculation, it _{is proper to} make phase piane first. _{Yaw rate} ç

and yaw accelaration ç required for this

were determined by averaging çb(t) measured

with rate

_{gyroscope and afterwards by}

numerical treatment method3. Namely suc-cessive - seven values of çi'

read at proper

time intervals were taken _{up and} approxi-mated, with a cubic expression, and then _the

ç1 and _{at the median point were determined} according to this cubic expression. This operation was repeated by shifting the time intervals by one interval each.

6.2 Practical examples of phase plane analysis

Phase plane analysis described _{above, was}

A

f'

(deg).'

\"t8

/ \,

\!

i

_r

,

(degIs'/

/

'i»

¿

AH

f-...'

/

\

----,\

/

\

attempted for the expetiments of S.

_{No. 8}

and 9. _{The records of time histories of}_each parallel shift motion are shown in _{Figs. 30} and 31. On

_{ç phase planes, only the}

conditions of ship motion 'with constant rud-der angle are shown in Figs. _{32 and 33.}

Below are the records of the time histories of motion and the details of _{analysis for the} two ships,

S No. 8After steering twice by definite

rudder angle, the parällel shift manoeuvre

was carried out by putting the automatic

steering device in action (Figs. 30 and 32'),

but the. parallelism of courses before and after the steering was destroyed largely, and

in addition, the test

was finished without attaining steady yawing motion. Accord-ingly the harmonic analysis _{was impossible}

12 -16 Fig. 31 I..

/

A ¿00 _---..-_..__ ,-00 (sec) ¿ S No9

,

F-H b--1

Record of Parallel Shift Manoeuvre _{Testof S. No. 9}

(sec) 500

Fig. 30 _{Record of Parallel Shift Manoeuvre Test of S. No. 8}

(21)

I

-4

I..

Fig. 32 ççb Phase Plane Trajectbry with Con-stant Rudder Angle at Parallel Shift

Manoeuvre of S. No. 8

- Sio

-2 -3

--5

Fig. 33, çb Phase Plane Trajectory with Con. stant Rudder Angle at Parallel Shift Manoeuvre of S No. 9

and then phase plane analysis was executed. The figures of T1T2/K, (Ti+ Ta)/K, 1/K and Ò were determined by the use of part ® and point ® in Figs 30 and 32. Next, T3 was determined With pöints (D) änd CE. Besides,

for checking up 1/K, this value Was also

determined from points cA) and ® and points

®and©.

S Nô. 9Bang.bang parallel shift motion

of.5° rudder angle, for which the results of frequency response analysis had been ob-tained (Figs. 31 and 33), was analyzed. For the determinatiön of T11'2/K, (Ti+T2)/K, 1/K

Full.scale Measurement öf Frequency Response Characteristics of Ships in Steering 169

and ô, the combination of part ® and point

31 and 33 were employed and mean values of both results were adopted. T3 was deter-mined with points © and . Moreover i/K

up-As the results of the above described analy-sis, K', T1', T2' and T3' thus obtained are

re-corded also in Table 2. For S. No. 8,

a

straight line with gradient K' obtained by

this approach is drawn in the spiral test

re-suIts Fig. 34, and 1/K' and lIT' due to this approach are recorded with + marks in the results. of the zig-zag trials of Fig. 35.

N8

K 5.50

.30° 4

2

K': obtained from the Phase Plane Ana ysis of Parallel Shift Manoeuvre Test Fig. 34 Turning Test Results of S. Nô. 8

1.0

00

01

02 r03

+: obtained from the Phase Plane Analysis of Parallel Shift Manoeuvre Test

Fig. 35 _{Zig.Zag Test Results of S. No. 9}

.102 eg/s) 8 1 2 S.Noß

/

o

(22)

170 - Keiichi KARASUNO

The results of S. No. .8 obtained by the above described analyzing technique seem to show almost close values in comparison with K' due to the spiral test results. There are no means to make comparisons for T1', T2' and T31, however, it may be said that

and T3' ate slightly too small in com-parison with values according to common model experimental results and theoretical calculation results. For S. No.. 9, T2' and obtained here show consi,derably small values in comparison with the frequency re-sponse resúlts, and are not very plausible. However, major constants K' and T' seem to have almost reasonable values, though they show slightly smaller values than that due to the frequency response analysis.

Regarding discrepancy between both re-suits of the frequency response analysis and the phase plane analysis especially the dis-crepancy for TI' and T3', examination- must

be made on analyzing and. experimental

techniques. But anyhow, this phase plane analysis may b an auxiliary means for the frequency response analysis in the present discussion at the time of determinig the four characteristic constants K', T1', T2' and T3' through the parallel shift manoeuvre experi-ments

Conclusion

Combination of the parallel shift ma-noeuvre and the periodic steering may be suitable for full-scale manoeuvre experi-ments, and it may be promising as a means to determine four characteristic constants K', T11, T2' and 1V on course-keeping

mo-tion.

Both manoeuvre experiments described above are suitable also for the phase plane analysis technique, and this analyzing tech-nique may be promising for reasonable deter-mination of the four characteristic constants in parallel with the frequency response analysis technique.

3).. The transfer function Ys'(It'n')

in the

intermediate frequency range obtained by the frequency response technique _{may give} useful data for deciding course-keeping capa-blity on coursekeeping steering, espécially by automatic steering device, in relation to. K'T3'/T11. T2'.

Acknowledgement

At the time of the present study, guidance

was given throughout by Prof. Kensaku

Nomoto of Osaka University, and hereupon deep sense of gratitude is extended to him.

Also heartfelt thanks are offered

to those concerned in each shipyard and the Japan Shipbuilding Reseach Association who afford-ed convenience for fúll-scale measurment tests and data. _{Thanks are offered} respect-fully to Captain Keinosuke Honda and each crew member who cooperated at the time of the experiments on the traning ship "Fukae-maru" and Minoru Hirota who whole heartily gave data concerning manoeuvrability of the Fukae-maru.

References

-1) _{NOM0TO and KARASUNO: A new Procedure of}

Manoeuvring Model Experiment, Selected Papers from .LS.N.A. Japan Vol. 7, (1971)

) M. BECH and L. W. SMITT: Analogue

Simula-tion of Ship Manoeuvres Based on full scale trials or free sailing model tests, Hydro-og

Aerodynamisk Laboratorium, Nov. (1968)

Y. TANAKA and K. KOSE: _{Phase PÌane}

Analy-sis on Manoeuvring Motion of Ships, Spring lecture meeting 1970, J.S.N.A. Kansai Japan

(in Japanese)

M. FUJINO and S. MOTORA: On the Modified Zigzag Manoeuvre and its Application, J.S.N.A. Japan Vol. 128, Dec. (1970)

K. NOMOTO: _{60th Anniversary Series Vol.}

11-Researches on Manoeuvrability of Ships in

Japan, Çhap. 2, J.S.N.A. Japan, (1966)

S. MOTORA: Course Stability of Ships, J.S.N.A. Japan, Vol. 77, Jul. (1955) (in Japanese)

M. FUJINO: _{The Results 'of Reversed Spiral}

Tests of a Ship, reported at Japan Towing