Notes on manoeuvrability analysis

(1)

i

NOTES ON M4/VØE1'R4B/UrY

ANALYSIS )

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ÛÌt-OSAKA UNIVERSITY

ERIMmr MA

1. Th INPPOVD SIMI1LPÁ1OUS TAPION MTHO1 OP ANALYZINQ ZTGZAÇ NotJ ( TABL.g II ) Applying the are derived, £ equation at t = te ,

4 & t

-

xJ,t

(2)

the following equations

Solving (k and (5) simultaneously, K in

later half and

are

obtained. Then using this K in earlier half is obtained from

(3),.

These procedures are the sane with the forger one described in

Ref-erence I, eccept the correction for a possible initia], heading rate _8o This correction is made merely by a alight modficatioz in reading up te, _et

6,

¿9 & ₍₉ , as is shown iii the figure

above. he principle is clear from the equatión (2). Integrating the equation from t = O to t, we obtain

/(1dt

i-

íJ,te

(2)

Next, integrating the equation of motion (1) _{from t} _t to te, t 4' to t arid t _{to t} _, _{we get}

¡a-2

(

e -

- X

Jite-q

( 6 )

(8

_-

_4')

_-

_/J

Ji4(z'-4(/

₍₇₎

.7 e

(8

2 1,

ti'

/ 2 t2'

,

4'

ad

are selected a little

earlier from, or

just the sane as t.2 t _and _t1

respectively. _Thia

selectj

depende upon consideration of easiness _{and reliability}

in.

reading u _{the heading rate}

8

at these instants. In this calculation, _I

in earlier half is to be used for _Eq.

₍₆₎

and K in later half _{for Eq.}

₍₇₎

and (8).

This new procedure of obtainin T is generally

more reliable than the former

it is hard to be affected by a possible local _distU.rbce in a heading angle _{record becauseof}

its short integration period and also by a possible error in Ö,

_{valúe, Since 4.term}

in the procedure does not became

₈₀

_large.

2. Tg LEAST _QUAR _{METHOD ( TABLE}

III )

The equation of _{motion. of integrated form}

(2) is employeä again

i _{the procedure.}

lodifying the _{equation, we get}

T().*f

Applying

the

_{equation to successive}

instants of a certain interval over whole manoeuvre, _{we obtain numerous}

simultanéons equatjo, the

unknowns of which are , K and _.

,

These

three may be determined 1rough the usual tachnics of the least _{square method.}

The

heading rate _{at an arbitrary instant is obtained}

by

the procedure of finite difference, _{if necessary, with}

some fairing.

t-Jat those instants when

_Q _{the rudder is at rest is}

calculated successively

_{following the}

_formulae

below.

Jv4 ea. Ç

,s

_{e-/Iz, kept}

1-t .3

'2

1 _e,Pt,

(3)

=

¿t

-;_

d&//ds

L/R and where s =

iCa-

or

dc

V

So the similar analysis applied to the transformed record yields

no.ndìmensional indices K' and T' immediately. The procedure is more reasonable than the usual tinie base analysis, since it takes account of instantaneous speed variation during _{a manoeuvre.}

In case of a model test, the procedure is crried out conveniently using a vane-wheel current meter with a resolution pick-up that

produces a pulse signal by each revolution. The- relation between

the revolutional rate of a current meter, rps' and shIp speed is linear as is shown below. If the straight line- obtained from trial

runs passes the oriîn, each one revolution of the vane corresponds -exactly tea certain ship travel, that is, V/rpst in metre.. An

actual rps'- V line passes not exactly through the origin but near the origin. For a well-manufactured currentmeter, however, its

rps'- V line may be simulated by an ideal rps'-.V' line through the

origin with a gradient of _{rpsmean)/ V(mean)} _{in a fair aécuracy.} rps'(meah) is a mean rpst over a-whole manoeuvre and. V(mean) _2.8 a speed correspond-ing rps'(mean)

3

J

2

I_1

Ìt2t1) 2 3 ₍₇

rT)

when ¿ is-being kept,

¡I

-

¡Z

(tf4»

(rt)

-

2

Z

when is being kept.

3 4NALYSIs IN RESPECT TO SHIP TRAVEL _{COMP'TSA'ION FOR SPEEP ARIATION}

When ship speed is measured during a manoeuvre, we can transform

₈

¡

record against time into against ship travel by integrating speed record;

of course nondimensional ship travel s is more reasonable than travel

itself. _{The equation of motion against ship travel has the similar}

form with Eq. (1), that is

'dra

'r

-ds

-t Ç2

K (d,,

(4)

/

k'1-

1?

j,

y I

7--According to this principle, all analyses are c-arried out against current meter revolution pulse signal instead of time signal as is commonly used. The obtained X and T are nondimensionalize as follows,

k. INDICES K' AND T' 'AS FUNCTIONS OF TURNING CURVATURE LEVEL refer to the attached figures )

In most cases it is possible to simulate well an observed ship motion in the zig-zag manoeuvre utilizing the linear equation with

K and .i derived by the linear analysis (eference I). Those

indices 1erived from the manoeuvres with different angles of helm employed, however, show sometimes a considerable difference to each

other. This suggests that ship motion in the manoeuvre is no essentially linear1 but ' linear on the average " if linear indices are adjusted

adequately. The indices in this definition may of coue depend

upon a level of ,turning curvature. -From this point of view, it is

recóiidble to carry out the manoeuvre not only by a gle helm angle bt by several one-s and then to formulate the indies'obtained as a function of average turning curvature.

The turning curvature level may be defined by the root mean square

value of _-Q over a whole manoeuvre. For the

least square analysis ou zig-zag.manoeuvre, this value has been.2 provided _{during the procedure, i.e. Z?2 is.a summation of} ₍₉ ₍ _or

.Q2 in. the travel base analysis). The simultaneous eqs. method

does not give te value op the oher hand. Accordînz to experiences.,

mean value of /t94/ and

/8/

multiplied by 0.7, is nearly the root

mean square of .

This ilay be used as a measure of turning

curvature leve], for convenience.

The attached figures illustrate two examples of the representation

for a cargo li'ner with 1?iariner stern arrangement.

5.

THE LEAST SUAP ANALYSIS ON TURNÚSG TRIAL

( refer to the attached firures ) The'following procedure of the analysis is now in use at our organization, while several procedures have been tried up to the

present. Ref erng to the attached figures,,pip,econd order

expression oflf#1tseems t be more successful. 'Yurther comparative examination for other ship types is now underway.

(1) Steady turning radious for the angles of helm of about±5 ± 10:

± 15, ± 20, I 25, 1 30, ± 35

and if necessary ± 4O are obtained -.

through usual turning trials.

()

Steady turning angular velocities and ship speeds for helm angZes of about f 1° and helm midship are also defined by running trials with those angles f helm. Then we get steady turning radious as follows, _L,'

-=

If the turning paths for smaller angleä of helm may run out of the manoeuvring basin, the procedure of (2) of dàfining Q is employed

(5)

also

for the turñing trials.

a-(3)

The neutral helm angle, with

_which

_{a ship}

_runs

_{exactly straight,} is defined

using

_{only the results for ±} 10 _{helm angle}

_{and helm}

midship. in this small range,

Q - J

_{relation may be regarded} _aa linear.

(1f) Using

helm

that is

Picking up X2.

- ä

relations for the helm

± 15° , the coefficient

a

i _determined

square method with a describing equation of the

r

4.12.

ò

aQ

±

b2

é

aQ

(5)

Employing this

_a

value, the cdefficient

_b

_{for stay-board turn} is obtained from to normal _{equations derived from}

_Q

_-

¡

relation ii starboard turn only. _{Since -these equations produce}

somewhat different. _b _{value, the mean value of them is taken as the}

final

value.

The similar procedure is repeated for port

_turn

so that

b

vale

for port turn is determined.

the neutral helm angle,

-äp,

_aildetected

¿n

- angle of

are transformed into effective angle of helm

6,

A

or

I Nomoto; _{Analysis of the standard manoeuvre of Kempf and} proposed steering, quality indices,

Trans. of ship manoeuvrability simposium at Washington, 1960

II Nils H. Norrb.in; Model tests and ship correlation for _{a cargo liner} Tyrans. Royal Insti.tution of Nay. Arch. (advance

_COpy;for

_Gothenburg meeting, 1961)

angles within about

through the 1eat

form,

(6)

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The space here is usually

&fo'

for calculation of solving the simultaneous equations derived from

the least square method. .Lhe coefficients of the

equations are described below.

-

r/

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