A simplified analysis on ship motion under manoeuvre and proposed steering

A SIMPLIFIED ANALYSIS ON SHIP MOTION UNDER MANORSTRE AND

PROPOSED STEERING QUALITY INDICES. by .Kensaku besot°.

What measure of ship manoeuvrability 'should be reasonable has been an important problem, as ,mentioned in the decisions and recom-mendations at the 9th I.T.T.C. The present paper relates to a proposed measure of manoeuvrability based upon a simplified analysis . on ship motion under manoeuvre.

This measure, which consist4 of two indices, relates more to course-keeping and response to steering with moderate helm rather than hard-over turn. The indices for a ship can be determined

-by analysing proper type of manoeuvre (e.g. Kempf's zig-sag test). The analysis along this line were carried out for about a

hundred. actual ships and free-running models and it it: found that the indices are good measure of manoeuvrability for those cases.

These indioes are not merely a relative measure but represent 'a dynamic character of A ship under manoeuvre quantitatively, so

we can predict ship motion for a given steering within a practical accuracy, using them together with a simple differential equation. The indices can be related theoretically to hydrodynamic derivatives

which have been widely used in the analytical treatment, as well as to the empirical measures for manoeuvrability. like, as turning radious, reach and overewinging angle.

REDUCTION OF THE PRESENT ANALYSIS: _{The present analysis is based} upon the usual linear analysis on the ship motion, using the follow-ing form of simultaneous equations of motion,

1-)44-4

t

)/:.4

)4)11 1

"t(t)tii-4

N;

31

Ali

( 1)

where

S:

drift angle, cp: turning angular velocity, helm angle, V :

ship

speed, L skip length,

Act : nond.imensional virtual sass aad moment of inertia sad

rt.

: hydrodysaals drivative..

We can derive the transfer function from the equations, 'kith describes in this case the response character of a ship to steering, as follows,

20t

i(t) ,AAMYCti

,cakt, tared j observed

yo,_ Laplace tunsform of

t

"

Laplace transform of

0

(if

r3p)

cif rir)(ifT.p)

FIG.

_{1 LIGZAG TTRiZULT fuR A}

_{bALLA6MB CARt47u7BoAr.}

where

r :

parameter of the transform, which represents a frequency of rudder movement.

E, Ti , Ts and Ti are constants composed of the coefficients of

Eq. (1).

The transfer function may be simulated by the following simpler form if the frequencies of rudder-movement are adequately low (small

p

), that is

Ycp)

AC

where T = T, T, - Ts

Retransforming the above. approximate transfer function into the fora of equation of motion, we obtain

7-4A

-t- 02 ( 2 )

Sihce the actual movement of a rudder is not hasty and a ship ie theAess sensitive to the higher frequency steering, this approzi-natiOn may be valid for the ship motion under the usual manoeuvre.

Fig. 1 - 4 show how we can interpret. zig-zag test result. for various cases along thin line; the chain lines indicate com-puted ship motions by using Eq.(2) with the proper values of K and T, while the full lines the observed motion. The procedure of defining the values of K and T will be discussed later.

:Mare la a suffic1.st mount

so that it say be concluded that

&ascribed roughly by 411. (2) *ith

hermit in each individual ship.

TIO.

117.0-LAG

WEIS ram A nu,

WA= CIAX10-110AT

;

of this kind of %ha taut results,

the actual sotto& of a.ship say ha

a

the inicos K and T- which

ire-in-776. 3 -11134a/40 TEST

=MT FOR k

PULL-LOAD = ?ARM

,

/200

YU. 8 EWAN TNT

BIEULT FOB A WEALS

CA.

?BS MARINO

QUALITY INDIC'S IC LIED T; Nee let us consider

the sass that the rudder be put over by a certain angle

ir

when a ship is running straight. We can define motion of a ship from Sq. (2) as

The angular velocity yr increases exponentially

with a quickness depending upon T value and finally settles in a steady

va1ueAS4(Fig.5).-The larger K provides the rapider terminal turning and the smaller T the quicker response.

So K say be called as the -index of turning ability and ?

the index of quickness in res-ponding to steering.

It Should be noted here that how rapid turning a ship can sake tersinally and how early she can approach to the tersinal notion are quite different things from each other.(In the words of control engineering, the former should be called am static gain and

the latter time constant.)

Now let us consider thccase that a ship be stimulated to have a certain angular velocity }b., keeping her rudder asidship throughout. W. obtain ship notion in this case as,

(i

The angular velocity yr decays exponentially with a quickness

depending upon T

value;

the smaller ? providss the quicker decay and so the better stability on course. A negative ? represents insta-bility.

Thus it becomes clear that the stability on course and quick-ness in responding to rudder are two faces of.on character a ship, as it ham been recognised empirically.

I. conclusion it say be said that K is the index of turning ability and ? the index of quickness in responding to rudder and also of stability on course, and that sanoeuvrability ef a ship may be represented essentially by the two indices.

to t

Obviously from. the analytical. reduction of K _{and I as before.} these indices nay be written in terms of _{the hydrodynamic}

derivative's, as follows; (-L)Al4 )/,

r

rLiv,

r;

_-

_(-Pez -) Ai, -*;

r;

V.

T

I

where Oe Putting a t, we obtain

This expression _{leads us}

) x K. and T _{are composed} Eq. (1), and so may given ship. _There indices, however.

which the equation of motion ( under a certain manoeuvre. one for the _purpose;

be defined is another

It is to

a nondimensional form of

and T ( ) ? .

only of the _{Moadinensioaal}

the indices, that

is

coefficients of if provided these _{coefficients for}

a

and more _{practical way of defining}

the

find those values of the indices _with 2) may describe

an obServed ship _motion Kempf's zig-zag _{test may be a suitable}

ANALYSIS OF KEMPF'S _{ZIG-ZAG TEST (PRACTICAL}

PROCEDURE OF DEFINING _TEE IFIDICal( AND T FOR _{A SKIP):}

Usually a ship does not keep _{a straight}

course With her rudder apparently aMidshib by _{reason of miscellaneous}

factors. _Consideriag the fact we

put the angle of helm to be used _{for the analysis}

as

er,A(t) t

:

observed angle of helm : _{neutral helm}

correction that is an unknown _constant at the beginning.of

the analysis. At into Eq. (2) and

integrating the both sides

(r

4.)

t = ar.-,

(1) ate

_{-t- A'}

crrl-'her* V., : _{poseible initial}

angular rate at _{the start of} Defining a number _{of sampling}

time on a _{zig-zag test} usually at regular

intervals, we measure (i) , v.)

_41.

.4. Obtain

f cr..

_itt by numerical integration. _{A number of} . _ a test. record and then equation4 from t = 0 ( 3 )

6

of

the

type of gq. (5) are then made for each sanpling- time and

,

unknown quantities in thesa equations are T, K and

4

Employing the orinciple of the least minimum squares, we can solve these equations simultaneously to obtain K and T.

To use an electronic digital computer is practical for the calculation, though it is not so hard to carry on it y convensional hand calculator.

The indices thus obtained may be considered these which inter-pret, a ship motion under a-zig-sag test most closely on the whole.

A check

calculation may be carried on, if desired, putting _{the obtained}

indices and observed dr into Eq. (2) and integrating it. Fig.

1 - 4 illustrate 'the ip(t) _{thus obtained comparing with the observed} v(t).

APPLICATION OF THE PRESENT APPROACH ON NON-LINEAR _{MuTION UNDER} MANOEUVRE : Applying the linear analysis to small

perturbations

I

about steady turnihg, we can define IC _{in turning motion. This K} becomes-the. ratio of incremental turning curvature _{to incremental} angle of helm in a steady turn and is represented by the slope of

the LAns - d, curve. _{This diagram is drawn by plotting the steady}

turning curvature L / R _{obtained from the spiral test of Dieudonne} against angle of helm. employed Jr. , like as Fig.

6:

1

The K _{is.a function of turning curvature L / R by reason of} the non-linear changes of the hydrodynamic _{forces acting upon a ship's} hull.

Now considering the initial stage of turning, this K

varie

gradually with increase of turning _{curvature, and the ratio of final} turning curvature to the applied _{angle of helm represents}

some average value of the transient K . This average K _{is also indicated in} the figure.

In case of turning with _{a large helm angle, the variation of}

K

may increase considerably, _{so that no linear treatment}

can be adopted in a strict sense. _{It may be reasonable, however,}

to construct linear expression simulating _{the non-linear Phenomenon}

on the average; the average X _{would then be used for the pprpose.}

Linoar analysis on Lig-sag _{nanoeuvres should also be based} upoa this concept. _{In most cases it}

_ie

,/ average intensity of

notion (considerable non-linear

FIG.6 L/R6 DIAGRAM AND INDEX

Xi effect) by reason of poor

stability on sours' of the

super-tanker. Even in this

case each individual zig-zag

manoeuvre

can be not so poorly interpreted using properly

adjusted indices. According to a number of experiments, the

more intense is ship

motion, the smaller are the

indices, assuring that a ship

becomes more

stable on course when turning,

as it has been commonly recognized. It agrees with. the K - L/R6 relation

derived from the spiral test

as before. This tendency

appears the more remarkably for the less course-stable ships.

In conclusion it may be said

that K-T representation of aanoeu-vrability can be adopted even for the

nonlinear range by

introducing

the concept of "linear on

the average". Considering a

wide utility

of linear treatment, it is more

convenient than an exact non-linear one, especially for practical purposes.

In this connectiOn, to

. conduct the

sig-sag trials applying several

angles of heln and to

define the indices as

functions of the average intensity of motion nay be recommendable.

7

the actual ship motion

observed in the trials,

utilising the linear

equation with X and T derived by the

linear analysis. The indices

derived Syoirthe trials

with different angles of

helm employed,

however, show sometimes a considerable

difference from each other.

This suggests that ship

motion in the trials is not

essentially linear,

but is "linear on the average"

if linear indices are properly

adjusted

depending upon the average

intensity of motion ( rather than upon the helm angle, because a rudder force

is aaually linear up to

large

angle of helm).

1-,4q5

Fig. 7 illustrates the

diTerent, a

indices of a super-tanker fr

and her model against the

wo-age

average angular rate in a

t

h-J. nondisensional

fora.

In this case the indices vary very hardly with the

I

-4 T

/01

/5%1 ot

0/

AgOWW-WMUNN

---t--

mmmt 0 1 . 2451. I . fLm-wermwaN I

404400rE diF °lit

2 0.3 A

12-FIG'. 7 _{EIG-LAG TEST RESULTS FOB A SUPER-TANKER}

nopkt

(1)

351

USUAL MEASURES OF MANOEUVRABILITY AND, TEX PRESENT INDICES:

The steady turning radio:am

_Ri

_{has an immediate -kelatioa} ith the indax K. as before, that is,

where the average K' should be mod..

Th "turning lag" and "roach", _{which has boon usold as a} easure for quicknesa in responding to ta:wing, may be Written is tars. oi T. as follows,

turning lag

= 7

t,

reach Y(7 .t 2),EKT.L.

_-f

.

'where Y, : _{time spent to set a holm (fairly saaller than}

T except -smaller crafts)

: ship aimed.

-The.ovarswinging angle has been widilly :Wad as a

of controllability obtaisd from the sig-sag trials. _{This angle is} usually oasured from -holm _{reversiauto extreme course deviation,} but it is affected by a.apeed of a stooriag gear, which is sot a

1

FIG NOT AT ION OF OVER'S* ING ING ANGLE COURSE CHANGING LAG.

46 Ta: )

I-el

,44/./41

wn61,1 AINI4S OF HIE 1,pq

THIS VL A TIM'S An" REMICth ale THE WAWA,

A SIMPER TANK( It 3/4/P MOCEL Id, I I 4. I 0

L

S. I 1 / 0' i

_Ii.i

701

1 A / 07 . it 3

251

0.4

PIG. 9 ZIG..4AG TEST RIB GUS FOR A S UM-TAXI= & lEN ))DEL.

301.1

!2.R21 3.

T,

dhip'e

_{charactor; the slower}

is the speed,

the larger is the over-swinging. _{SO m4difying}

the definition of the eagle _{so as to be} measured from the time _{when a rudder}

passes amidship (see _Fig.

_8),

we can find the _{following relation}

through integrating _{eqa. (2) for} a sig-sag manoeuvre, that is,

the overswinging angle is nearly _{proportional to} where : angle, of helm

employed. It should be _{noted that the}

overswinging angle _{as a measure}

of

manoeuvrability has a shOricomming that _{it can not}

discriminate rood turning

ability

_{with quick}

response to rudder

(large K and small 7) from

poor turning ability with sloe _response

_(meant(

and large 1)1 the corker yields _{good andeuvrability}

and the latter poor One. In order to _cover

this

shortcoming, another _{measure of gale&} response, "course changing lag" TL, _{so termed temporarily,}

should be used together with _{the overswinging}

angle. _{TL is defined as a time}

duration from rudder pawning _{amidship to extreme}

heading deviation, as is shown in Fig.

_8.

Using

_{the equation of motion (2), we}

can find that the course changing lag 1', is _{nearly proportional}

to T. _{So we obtain} the relation that, where, is nearly proportional to O.

end _{may be obtained}

immediately from _{the sig-sag}

trial

and be used

as relative measdres of turning

ability

_and

quickness is

responding ,to _steering.

These two

quantities are nearly _proportional to K' and 7'

respectively and change _{their values}

with

the average intensity of ship motion as similar

as K'and-T', _{as is shown in}