Zig-zag test technique and analysis with preliminary statistical results

Zig-Zag Pest Technique and Analy8ia with B.SR.PL. TRANSlATION NO. 2188.

Preliminary Statistical Resulta.

- by

N.H. Norrbin

- from '

Swedish State Shipbuilding Experimental Tank, Report No. 12.

Page

Summary _i

Introduction 1

Pecbnioue f the zigzag trial 2

PL short history

4

Prerequisites for the formaÍ analyzié ₉

The mathematical models and steering procese ₁₄

rho practical process of analysis 21

Some preliminary results ₂₄

Bibliograohy 30 List of figures

_3f

Spare pages . 35 37 Figures 38 Tables 64 Appendix A - Symbols Al

Appendix B - ReconÇrnended procedure for carrying out

B.S.R.L.

2188.

ZIG ZiG TE

TECRNIQUE AND A1LLYSIS

With Preliminary Statistical Results

Summary

The zig zag teat or ICempf manoeuvre is a teat of manoeuvring

qualities, which for thirty years now in marty places has been used

as a unique teat for comparing certain nautical measures, auch as

overawing angles, times to reach execute and periods of final

yawing

motion.

_{In later years its value has been still}

_{more accentuated as}

there are now new means of recording teat data and interpreting

_teat

results.

The manoeuvre is also suitable for testing modela of

new

design in an ordinary large-size towing tanic.

This report has been written to widen the knowledge of zig

zag

test techniques as well as of the possibilities for

_{using the}

statistical resulta arrived at in the design of rudders

_{and autopilots,}

etc.

Several new results are included in text and figures.

_{A list of}

figures in English is supplied.

An approximate method of analyzing teats with unstable

_{ships is}

set forth.

This method implies that only hull cross-flow drag

and

rudder force separation are retained as

_{sources of large-value-type}

hydrodyriam.ia non-linearities, and that trie loss of

_{speed and r.pam.}

during the manoeuvre will only affect control forces of rudders behind

screws.

_{Semi-empirical relations are introduced for rudder forces thus}

dependent on screw loading and separation, trie latter

_{here tied up with}

the aspect-ratio of the plan-form.

_{The final equation geverning the}

change of heading contains the static gain IC' and the time constants

T1', T

and T3' associated with the linear theory togetrier with

_{a new}

factor ¿appearing in a non-linear "correction term". All these

conatants, which are analytically built up of simple

_hydrodynsaic

derivatives, may be directly evaluated from zig-zag-teat records of

heading, drift angle, speed, and r.p.m., or, alternatively,

_from

For stable ships a linearisation of equations is permissible and in the simplest case the differential. equation relating ship heading response to rudder motion contains only the single time constant T'

introduced by Nornoto eiit years ago. This much more handy form has so far been the basis of the routine analysis of zig-zag tests per-formed at SSPA.

As a result of this analysis a new quality number P is suggested,

combining the two coefficients IC 'ani T'to form a standard measure of

change of heading in one 8hip length. A certain minimum value of this coefficient may be selected as a tentative design criterion and the corresponding value of K1then deduced.

A nomoam with supplementing diaams is given for a preliminary

estimate of K'of a certain new design, or of the ruddar area required to achieve the K'desired. It is to be understood that these diaams are based on a limited number of zig zag tests and that minor differ-ences appearing in designa of other origin may cause unexpected depart-ures from this prognosis. It is hoped that this presentation will stimulate future work of this kind, however.

It is important that at least one 200/200 zig zag manoeuvre be completed with every new ship, as is also recommended by the

Manoeuvrability Committee of the International Towing Tank Conference. As so is not now the case instructions for conducting the test are enclosed in an appendix.

B. S .R .A

2138.

1.

Introduction

Six years ago "Proposals for a proamme

r merchant thipa turning

circle and steering trials"

11/X)

was publïsned in this series of

repOrt8.

The basic details of the proamme were nàt new i the

pro-gamme comprised turnin( ofrole trials with the rudder hard over to

Port and Starboard and also at 200 to Port and Starboard, a so-called

zig-zag trial by putting the rudder over 20° (2JJ°/2c)°

trial), and if

the ship miit

thought to be dynamically unstable or just tender

a so-called Dieudonne 's spiral was included.

In a proposa]. made by

tne Manoeuvrability Committee of the 10th International Towing Tank

Conference in London 1963/2/ it.was recommended that similar tests

should be carried out or. all trial trips; it was proposed that 15°

steering trials be substituted for 20° trials.

At present only turning circle trials with the rudder hard over are

mandatory although some Swedish yarda as a rule also carry out 200/20°

'zig-zag trials, the

National Experimental Tank has received valuable

assistance in the

'oupa of trials from Deoca Navigator and Radar AB.

This tends to an increase in the une of this ìdnd of trial /3/, even

although Deoca records are not normally necessary when performing them.

However, the availability of Deoca records, as will be ahewn here,

enables more thorough analysis of the trials to be carried out.

Some

results of the statistical analysis of the records by computer are

presented in this report.

Now, the same as six years ago, the

manoeuvrability of ships is still incompletely understood and in the

design of a serieá of

naw constructions in the lar ge deadweight class,

it has often not been possible to make use of experience gained from

previous vessels.

It is very important, therefore, to stress once

again that the above mentioned trial proamme should really be applied

and also that the possibility of carrying out suitable

model trials

should be taken advantage of at a sufficiently early stage.

The

zig-zag test can, with advantage, be applied to standard modele in the

experimental tank.

Each of the proposed types cf cianoeuvring trials in the proamme

supplies information concerning certain characteristics, and together

they give a satisfactory picture of the manoeuvrability of the ship.

-1-The hard over turns, for example, give a direct measure of the advance

in an emergenoy turn and the spiral manoeuvres give a measure of the

deee of instability of an unstable ship but they require good trial

conditions and a relatively long time for the testa, say, about one and

a half hours. A zig-zag tria], carried out and ax.iysed in the correct

way can give corresponding information.

The method proposed by Nomoto for the analysis of zig-zag trials

also gives in coefficients (steering indices for the rodder effect and

the ship's reaction inertia) an impression for the so-called translation

functions for the simplest mathematical models with the help of which the

behaviour of a stable vessel can be calculated with a relatively

arbi-trarily prescribed rndder manoeuvre

The first order approximation

the complete translation functions is not satisfactory however, as

regards the correct tieatment of the automatic steering problem, and

the linear analrsis is completely unsatisfactory for unstable ships.

It is important, therefore, to develop new methods of analysis which

make possible better utilization of the available information.

After a description of the technique of the zig-zag-trial, a brief

historical survey is given in Chapter 3 with guidance on the formulation

of the problem.

In chapters 4 aM 5 an account is given of the theoretical

assumptions and the different mathematical models for the steering process.

These chapters can be omitted on the first reading.

Chapter 6 contains a

short description of theanalytical process with the S.S.P.A. old and nsw

computer proammes.

In Chapter

7,

some of the results obtained by

S.S.P.A. from the analysis of' standard trials by hitherto available,

methods are given.

The symbol

used are snmmized in Appendix A.

In Appendix B.

guidance is given for carrying out a zig-zag-trials.

IP

-2.

Technique of the Zig_zaa.tial

(See also recommendations 'in Appendix a)

The so called zig-zag-trial is a steering trial in which there is a

eater or less deviation from a given mean course caused by zig-zag

steering forces impressed step by step.

The motion of the ship during

the trial is closely sinusoidal.

(See Fig.l showing the Décca recorde

B S.R.L.

2188

When manoeuvring with a conventional rudder the rudder is first put over to one side and then, when the change in course reaches

it is put over to the same angle -on the other side, etc. etc. As a result of the inertia of the vessel a characteristic overateering

- results, the magnitude of which is often taken to be a measure of the manoeuvrability of the ship. (See Fige. 2 and 3.) If the angles

6,

and

fr

are of the same maiitude the trial is usually talked of as Kempf's zig-zag trial. The trial is conveniently denominated as a 20°/20° zig-zag tria]. wmich indicates that the rudder angle used is nominally 2CP aid that the rudder effort is made when the deviation from t1 mean courue

is 200.

After a couple of rudder movements, both the rudder movement 6(t) and the Rn'lar velocity of the ship ('t). 4t fall into a periodic cycle, and the length of the period has also often been taken as another measure for the manoeuvrability, preferably in a dimensionless form such as the number of ships lengths traversed, viz, the periodic number

Z,

(See Fig.).) Insofar as the angles _£3 and-sb are equal the length of the period for a given ship is approximately the same irrespective of whether

6,

(and sfr,) is equal to, for example, 10° or 20°. (This applies only for the conditions that the function is essentially linear aid there-fore, in particular it does not apply to dynamically unstable ships). The periodic number corresponds with the circula' frequency içjr '=

It can be shown that neither the overawing angle nor the periodic number is determined unilaterally but that both are functions of pära-meters in the steering equations, i.e. in the simplest case they are

functions of the two indices K and T (See below.) It may therefore, be appropriate in certain respects to consider zig-zag trials as a "frequency teat" since the zig-zag formed disturbance function takes the place of the ordinary pure aine wave aigeal applied in control teclniquea. (If a suitably equipped steering gear were available, it oou]4 be assumed that an almost sinusoidal rudder movement would be produced by signals from a low frequency generator).

B. S.R.A.

2188'

S.S.P.A. has carried out both 100/100

2O/20° zig-zag scale

model teats.

Because of the large amplitude of onvement. the latter teat

can only be carried out with a radio controlled model in the sea.

For

testa with standard modela in the tank a special technique is being

developed

Lch aleo suite the general analytical method introduced here.

Pig. 4 gives an example of the recorda obtained with a mlti-chimnl

recording apparatus and a radio controlled model of a cargo liner.

For

the vessel itself the primary rösulte are only avail8ble in tabular term.

In Fig. 5., a comparison of the results between model and full scale tests

by simulation on the S.S.P.A analogue computer is given.

It should be

noted that the rudder amplitude to Port and the deviation from the comae

to Starboard with the sign convention adopted nowadays is given by.

positive S and -cfr

.

It is important to note the starting and stopping

time of the rudder movement. As a rule the corner points of the L (t)

curve do not fall exactly on the -fr(t) otave and no "cooking" of the

reault8 with this endifl view should be done.

In the processing of the

data in the S.S.P.A. computer proamne the trace of the

irvea can be

arbitrary.

Before dealing in detail with the fundamental theory and wa&yaie of

the zig-zag trial a short resum of literature on the subject will be

given.

3.

L short history.

-The zig-zag trial was probably first intorudoed by

aemer /4/, who

saw in it a method of comparing certain characteriatica of different

ships auch as, the time fbr the full speed of turning to be attained,

the maximum speed of turning, the duration required for the rudder

impulse and the overawing angle during manoeuvring.

ortly afterwards

Kampf proposed that the trial should be standardised in auoh a way that

the deviationa from the course

b during the manoeuvre should be equal

to the angle of helm 6, used.

In 1944 Kempf published a summary of 133 in number l0°/l0°

trials

on 75 different cargo ships, /5/.

The periods observed for the swing of

the vessels off course were distributed about a mean value of about 8

ships lengths.

For 6Q of the ships the corresponding periodic number

lay between 6 and 10 and the reduced frequency was therefore, between

0.6 and 1.0 according to Pig.6a.

In this connection it should be

-4--5

pointed out that Hattendorff, Mdckel and Gallin quite recently published a sinilar diagram based on modem mater:al and which has been shown in Kempf's form in Pig.6b, /6/. The periodic number 12.4 which occurs most frequently in the diagram, according to the aathors, is typical for fast modern vessels which, according to seafaring practice, can be considered as being easy to steer.

Kempf's data is compared with model resulto for some naval type vessels in a paper by Gimprich and Jacobs,

/7/,

in which the angle of

swing off course is considered for the first time as "tne frequency response" to the fundamental harmonio frequence of the rudder movement.

As a result of Kampf's paper the zig-zag trial became quite common and a large number of records of time to awing, overawing angles and periodic numbers were filed. In some quarters attempts were made, to

analyse the materia], collected systematically. A new paper has already been quoted. In 1959 and 1960 Cartier and Cover presented an attempt

to formulate certain criteria for the maximum recommended time 'of swing and overawing angles in a 200/200 trial (See ref. /o/ and Fig.7.)

Suerez and Breslin /9/ deny that the zig-zag trial has any great value insofar as concerns the possibility of evaluating from it any

unequivocal quantitative information regarding the stability characterist-ics of the vessel or of the unknown coefficients in the mathematical modelo However, the "first order theory" proposed by Nomoto for the analysis of zig-zag trials on dynamically stable ships offers new possibilities /io/, /11/. Assuming that the equations of motion can be witten in linear form Nomoto.wrote steering equations in an approximate form with one degree of freedom

3fr'

thus s

(3.la)

or

T'çV"+ «r'

K'(S-50)

where T (or the dimensionless coefficient T' T ) is an effective "time constant".

and where K (or K1- rK) with a term also taken from

is the angle of the helm to the straight äourse.

Both K and T are functions in a complicated

_{way of the}

hydrodynamio coefficients in the fundamental

_equations

of motion.

(The complete linear equation for the angle of

course

contains not anO but three time constants; see below).

The significance of the amplification factor and the effective

time constant is perhaps shown best by the diaam

of change in course

during an ordinary turning circle trial, in the upper part of Fig.8. K'

expresses the relationship

i---

_K'(S-S0)

between the stationary "angular velocity" (or curvature of trajectory)

and the angle of helm

_{on an easy turn, T' is the "time"}

(in ship lengths sailed) which the stationary tnning condition retarda

as a result of the inertia of the system.

_{It is obviously desirable}

that a ship should have a high K / value (good

_{turning qualities) and a}

low T' value(low inertia or high dynamic

_{stability; see Chapter 5).}

Nomoto analysed a large number of zig-zag

_{trials by means of the}

simplified steering equations and fòund that K'

_{and T' were both of the}

order of magnitude of 0.5 to 3.

(Note that K / with the convention

_{of signs used here is negative for}

stable vessels and positive for unstable

_{vessels aid also that the}

approximation (3.1) is not applicable to unstable ahipä for which it

should not be expected to get particularly large

_{effeotive time conatnta).}

By maldng certain assumptions regarding the bydodynamio coefficients he

showed that in principle for the expression

s-_= Q

L.E.

:

(3.2)

is relatively constant for vessels with similar

_{rudder arrangements.}

Arr = Area of the moveable part of the rudder.

L

Length çbetween perpendiculars

V'

Displaced volume

In Fig.8 Nomotos analytical resulta

_{are given in the Íbrm of}

1

LArr as a function of

_'.

_{The straight mean line corresponds}

_to

KV

4.7 and the moderate scatter of the primary

_{values indicates that}

a notable improvement of the relationship IC'

can only be obtained by a corresponding increase of the moveable part of

It has been pointed out above in several cases that the process of the manoeuvre for a dynamically unstable vessel is charactorised by non-linearity in tile hydrodynmiC forces. In the main it is only because of the adually increasing inemental force that an initially

unstable ship can be inanoeuvred. (See following chapter). It is

natural therefore, that non-linearity can not be completely neglected in the handling of a tender vessel. Norrbin /12/ examined the zig-zag tria]. on the basis of a quasi-linear frequency analysis of an equation

s. of the

type-T''+#'+c'5&'

i<'S6)

(3.3) with sinusoidal rudder effect. Prom the research material at SSPA it

is considered that the associated values

K'

i

land

0«.

i

are representative for a modern carp lixr and the calculated resulte

for different amplitudes of helm for this case are given inPig. 9. The expression, R for the amplitude ratio of the angular deflection

'o-is presented in the form of'

-w'R

(which describes direct the

amplitude ratio between the speed of turning and the angle of helm),

while the phase angle is defined as

(_V/P ),

viz the retardation of the curve relative to the curve. (Note convention of aigus).

In the same paper it is shown that a vessel with the &iaracteristice just quoted during a 200/200 zig-zag trial swings about the mean course with a basic frequency corresponding to -o.r'

0.7

According to the diaasn in Pig. ₉ the linear forecast in accordance with equation

(3.1) only involves an error of about 5 ùi the amplitude aid the error in the.shift of phase can be disregarded completely. However, for less stable types of vessels the linear theory gives worse results.

It is also proposed in ref. /12/ that the observations of the amplitude relationship and the shift of phase between the nearly

sinu-soida]. angle of oscillation off course and the zig-zag rudder movement can form the basis of a rapid preliminary evaluation of K' and T' by

means of a diaam carried on board The actual rudder movement is substituted in so doing by an equivalent sinusoidal oscillation defined

B.S.R.A. 7

I

a given way. Finally for complete information regarding the steer-ing characteristics of a atablO vessel, proposals are set out for

graphical analyai8 6f completed zig-zag trials at two higher frequencies obtained by a combination of s.ng1ea of the type 200/100 and

200/50

i.e. continuing with a helm angle of 20°. By means of this kind of test the coefficients of the complete linear

equation:-.T T

4Ç'+ (T,1)

_'+ jr'

K'(

-)+

K'T

S'

...

(34)

can be found relatively easily. The unknown time constants

I

I

T1, T and T occur in the "reciprocal frequencies" 1 , i and 1

I

.

T1T2

T3

which determine corner points in the series of polygane,

which in

turn approzimate the amplitude relationship between the speed of turning and

the angle of helm byfrccuency analysis. (See further about this in

Chapter

5.)

r the example given in Fig. 10, which is taken from ref /12/, the highest reciprocal frequency corresponda to the reduced

frequency - 6.

The coefficients in equation (3.4) are, as already pointed out, composed of the hydrodynamic coefficients in the fundamental equation of motion and are dependant variables like that of fronde 's Rumber and frequencies. It is, therefore, in order also to mention, in the

short survey of the history of the zig-zag trial, van Leeuwene compon-ent measuremcompon-ents on a model vessel which was towed with forced

oscilla-tions in a so-called "planar motion mechanism", /13/. The diagram

in

Pig. 11 has been based on van Leeuwena resulta, which, in agreement with Brard' s theory shows how the effects of the pulsating wave

forma-tion becomes critical when the product

O?

F

approaches . Under these conditions the wave fronts meet in front of the generating pressure point and the damping forces are radically modified. In this

case the corresponding value of r...r _{is 6.25. With very high} frequencies the effect of the frequency again disappears and all the values of the.coeffioients are at a different level.

B.S.R.L.

-8

It would appear that the frequency effect of the bydrodynamlo co-efficiente can be neglected in the calculation of the majority of normal manoeuvring processes but that without doubt the effect can complicate

the analysis of certain probleme in connection with automat io steering in waves of short frequency of approach, for example.

The ordinary manoeu'yring proceso can therefore, be predicted either on the basis of standardised zig-zag teats with a self propelled model or on the bacia of component measurements on a towed model which io made to perform forced oscillations of the same frequency. The advantage of the first method is, of course, that the result can be verified by similar testa during the sea trials without ai aàstunptione regarding

the mathematical relationship. _{The automatic steering problem may} require the introduction of frequency coefficients which, of com'se,

can be found from additional component measuremeuts at several higher frequencies but the mathematical model then becomes very complicated. The analysis of a nwnber of zig-zag trials of varying basis frequencies, proposed in ref. /12/, gives instead frequency functions of relatively simple analytical form with "effected" constants. _{The merits of both} methods have not been proven up to now.

The analysis of zig-zag tests of the ataMard type with full size ships must however, always be considered to be of predominating significance. _{The prerequisites for a practical analysis of the} trials on dynamically unstable vessels have not been dealt with in papers up to now. _{As there is reason to fear an increase in the} number

of

_{unstable ships, particularly among new supertankers with} rudders of conservative dimensions, a method for _{this 'kind of analysis} will be described in the following text.

4. PrereQuisites for the formal analysis

The movement of a vessel when manoeuvring can be deacribed by ita translations in the directions of the z and y axes of the body and f ita rotations about the vertical z axis (Pig. 12a).

B.S.R.&.

The translations and rotations are determined from the equilibrium equations for the corresponding forces and momenta in Eulera method

(see ref. /14/)

On a straight course, as when preparing for a manceuvring trial, the mass forces disappear and the external forces acting are qual. Normally it must be assumed that the trial will be hampered by a prevailing.wiM which together with the wind speed of the vessel gives rise to a more or

less constant turning moment and a similar constant cross force The tendency to fall away can be. counteracted by the rudder at an angle

but at trie same time the leeway cannot be neglected. The angle of leeway wider these conditions is indicated by

_ft

(see Fig. l2b). The angle

4

must not be confused with the angle of helm which in a single

screw vessel under ideal conditioiw may also be necessary in order' to maintain a straight course and which will be indicated by S0,, and ia

included in S0

As soon as the manoeuvre is started the speed dropa and the pro-peller load ri3ei (see below). Wit the change in the relative wind there is also an alteration in the internal disturbing forces. Ñot-withstanding these circumstances, it is assumed here that the angles

,ß and can represent the effect of wind and assymmetry throughout the whole of the zig-zag trial.

The hydrodynamic forces on the hull are subjected to a complicated series of maiitudes which describe the movement of the ship through the water. The functions of the forces can be developed formally in the form of Taylora series with several variables in which the higher terms represent the non-linearity of large disturbances and the effects of couples between forma of movement If the manoeuvre only involves small changes in the angle of helm it is often sufficient to consider the first order derivatives only, i.e. to assume that the forces result-ing from a given change in the movement are proportional to the mageitude of the change. With a iciowledge of some of these first order derivatives it is possible, inter elia, to determine whether the vessel is "dynamically stable" on a straight course or not and these derivatives are, therefore, also called "stability derivatives" (see ref. /14/). If the vessel io

B.S.R.A.

-B.S.R.A. 218$.

not dynamically atable the motion of the vessel can no longer be studied

on the basifl of a completely linear theory.

The hydrodynalnl3 functions of force and also the different derivatives cazi be determined from component measurements by towing and oscillating models in different ways, or by the analysis of the movement in self propelled models or s ves8el with a known rudder manoeuvre, for example, a zig-zag trial. However, for an analysis of this kind to be suitable for routine measurements the mathematical model should not involve more unknown constants than experience shows to be necessary.

On the basis of investitions carried out at the ship testing laboratory and an examination of published data, the dominant non-linearity in the hull forces at constant speed is to be found in the increase of the transverse force T with increasing angle of leeway

fi

_.

On the basis of physics it can be assumed that the function T ( ₎ contains a second degree term - as a result of the transverse flow resistance - and it can therefore be written s

Y(?)

='4j3i.

Y1p1

1Çi).

_(4.1)

Instead of the often preferred expression

s-.

Y

g

(owever, the latter expression is used below for the rudder force).

At a given speed suitable dimensionless coefficients can be intro-duced, indicated by the index

sigu

so that, for example s

-

y() y"().

V2L1

(y

+ Y2y

II)4

V2LT

il(r) N'(r).! V21...T

N.

_b'sfr

v2L2T. (4.2)

in which the reference area L T is the product of the length of the

ship

(L L) and the draught, and where

r'

₌

is the angular speed of the course. (The dimensionless speed of turning

r'=-ur

At a lower speed, that is to say at a lower Proude 8 number,

somewhat lower values are found for the dimensionless derivatives

, N. , etc. both from model tests and from the theory but it is assumed that these variatioñs, which are approximately equal for all of them, can be disregarded. In so doing it is also assumed that the hydrodynamic forces on the hull with unchanging relative motions vary

as the square of the change in speed. This does not apply of course, to the forces on trie rudder which functions entirely or partly in the wake of the propeller.

As the speed falls while manoeuvring the R.P.M. is adjusted to

suit the changed operating conditions in a manner which is determined by the characteristics of the propeller and the engine. With the

increased propeller loading (or slip) there ia an increased acceleration of the water past the rudder.

For a diesel ship with constant propeller torque it can be shown that the effective water speed past the rudder remains broadly constant and that therndsler is,therefore, effective as a control element in harmony with a reducing of the other hydrodynamic forces on the hull. The actual analysis, however, relates to trials with arbitrary engine

criaracteristica and the changea in trie rudder derivative should therefore, be expressed in the simplest possible manner as a function of repeated readings of the R.P.M.

In the absence of the propeller the rudder would function in the wa

abaft trie sterz.L of the ship and the forces (or force derivatives) would

be proportional to V. In the propeller wake (and actually a long way aft of the propeller) the mean speed in accordance with the simple actuator theory is n.P, where the geometrical pitch is substituted in trie relationship by the corrected hydrodynamic pitch. If trie height

of the rudder is represented by b, the rudder derivative is now approximately proportional to the expression

b - D 2

+ D (n p)

2 inwhioti the second term is usually

/

b Ve b

completely predominant. If the dimensionless rudder forces are substituted in the form we get the value for the run-in the general form

s-'n. V0 2

,

(wv)

(4.3)

B.S.R.A.

-12-With increasing loading, the working conditions of the propeller are altered and the rate of increase of the derivates becomes _somewhat

slower. The scanty research data in Fig. 13 yields the expression s-3,

(43)

w

which can be used for research analysis until further data is obtained. A similar result has been given by Okada, /15/.

The hull forces include the forces on the rudder in the zero position (amidships) and the expression Y' ( $ ) is reserved for changes in the force on the rudder and the adjacent hull as a result of the angle of

helm. The ignoring of higher pairing between the yaw, speed of turning and the angle of helm implies a certain restriction of the capabilities of the analysis when trials with s large angle of helm_{are involved on.} ships with a free water rudder fitted below a cut away free water after body, because the intensification of the thrust is no longer correctly described. (In. this case it may be expedient to supplement the standard trial with a 100/100 tria]). _{However, in genera]. the rudder is placed} in direct proximity to the rudder post _{or propeller aperture and the} flow past the rudder is essentially parallel to the longitudinal centre line of the ship. In auch a case the symmetrical function _{can be} written

:-The maximum value of the function is formally given by

s-.1_8

#3 '

L

i

The routine

analysis,

_{if possible, should not contain Y as an} extra unknown so for this reason an empirical

correction term is intro-duced below instead. _{(It should be noted that this only serves to find} the form of a minor correction which la otherwise generally completely neglected)

The intensification is conditioned primarily by the ener loases in the boundary layer on the suction side of the rudder which in turn depends on the pressure distribution,

Reynolds Number and the surface roughness etc. _{lbr a smooth model rudder and for a ships}

rudder

which works in the turbulent wake aft of the

propeller the differences in the maximum coefficient of lift CL _{balancee out but it is somewhat less} 13.S.R.A. 2188.

-13-Y'(6)

'Y6' 'rss

or w

2j

(4.4)

B.S.R.A. 2188.

for the model. The slope of. the lift curve, i.e. the derivative is the same in both cases and is a function of the effective aspect ratio of the rudder.

The effecji.spect ratio depends on the form of the after body. Because of the interference between the rudder and the hull it is considerably greater than the geometrical aspect ratio

IL

_{r -}

-A,.

62.

but in general it is less than the aspect ratio of the "flat mirror"

rudder.

ly'

¡y' In the diagram in Fig. 14 some model test results foi' '

"

have been plotted ainst the base

_r

and it can be seen

that the expression - Y'5 / Y

is approximately proportional

to 0.13 (

A.

_r 2 The constant of proportionality in

accordance with the above should probably be reduced to apply to full scale but in view of the lack of a basis for this the following approxr

imate formula is proposed' z

Y'(6)"Y

O"3(JLr)

'

_(4.5)

which appears to give good results for the normal Simplex or Mariner (open water) type of rudder arrangement. (The formula also shows that the angle of helm for a rudder with the normal aspect ratio

A 3

normally does not have to exceed 0.63 radians or 36!. With

J

n

_2.5

the corresponding hard over angle is

46°.

The

applicability of the formula is, of course, automatically limited to the same range of angle).

5. The Mathematical models and steering process.

ulera equations for the eiuilibrium of forces in the Y axis and the equilibrium of moments with respect to the Z axis are reduced

here

to:-rn(.+

ru)=y

Ir1N

(5.')

or with coefficients in the dimensionless form axil the approximation

Sr

¡3 Co s i

-+

p--çI-)+

y'=o

rrn'k.

b.N'-.O

(5.2)

-With the assumptions given in the preceding chapter the motion of

the vessel in the horizontal plane can be described by the conjute

equations

z-(m'-Y+ (m'-)X+

_'OO)

-

{

Y$) - V '()J

NI3

(?-)-

=_?(Y'(6)_Y'(4)}

Y'()= '''3". EV'] {I+'S'S

(2

V's

(5.3)

plus an isolated equation, which regulates the fall in speed during

manoeuvring.

In the analysis of the zig-zag trial it is assumed that

the variation in speed is blown.

The factor

2...

stands for the

ratio N' (

) /y'(S) and it is therefore of the order of mageitude

of -

0.5

for a converitïonal vessel.

(Note other symbols in the

definitions of 2..

in ref. /14/.

The system of equations (5.3) may appear to be complicated but as

a matter of fact they constitute a much simplified mathematical model

of the general case of motion.

To be able to illustrate the steering

caaracteristios better it is oonvenient to consider the quasi-stationary

or periodic process in which the speed variations ('s') can be neglected.

(This does not mean that loss of speed has been neelected).

The last

phases of a zig-zag trial are typical of this kind of process.

After differentiating the second equation in (5.3) the angle of

yaw

can be eliminated from all terms and a non-linear third order

differen-tial equation is found with one deee of freedom

fr ..

This new

equation can, of course, be examined in principal by numerical

processes

in a computer but the process requires very

eat accuracy in the

recording of the angle of the course and ita differentials and, the

equations are not very easy to scan.

If on the other hand the p and

terms are eliminated but the

terms are retained in the form of a "correction term", we get

z-T1' T

(.b)z;.

(r,'+r$

b

+

=

K'

V 1nVO

LUit

Ye

S3}

+

Y'

+

_{3, YSk6}+}

'

(54)

2y

15

-in which the coefficients are given -in auth a form that the

correspond-ing

linear terms

are easily reooniaed.

_{(See Table I).}

_With

_the earlier assumptions for Y'

/

Y

all the unknown coefficients can

be evaluated from a zig-zag teat, see the next chapter.

In assessing the dynamic stability

of

a vessel small changes

in

the motion are investigated at constant speed which

may be called V

Instead of the time t the dimensionless

_time

_{factor t'} = V t, can be

L

used which is the number of

ship

_{lengths sailed. quation (5.4)}

_can

therefore be substituted by s

T

r+T;c

K'(°)(..5 )+K'T

₊

_C'K'lIp

₍₅₅₎

The

condition for "dynamic stability with locked rudder", i.e.

f

so that the vessel, after

a

disturbance, will again

be on a new

atraiit course,

is obtained

_from

the condition that the

algebraic

equation

s-s3+(,,)s211.,.

SO.

(5.6)

T1

T,

T1 T2

shall have no root in the positive semi-plane.

_{One root}

is S,,, O

which implies that the ship cannot sail on any definite course.

The two remaining roots are

S1

- 1,

and

₂

-

_{which again}

T T2

are

always

real

_{because the forces to put the vessel on a}

_definite

course are lacking

aid no

_{oscillating movement can therefore, take}

place.

A disturbance, say in

the

form

of an angular aoceleration dies out or increases in

_{accordance with the function.}

E'

{ C.

e''

.-

(I - e)

e'J

(5.)

which explains the nomenclature "time constante" for T1' and T

In

general the motion is dominated by one time constant, say

The root u'),

lies well to the riait on the true

axis but if

_the

T1

vessel is

_dynamically

stable it is still to the left of the origin.

.S.R.A.

-, / In the last named case T1> T? >0.

Il' the vessel is dynamically stable it canalso be stiewn that in

the approximation of the first order, a single effective time constant

I

# F

T T1 + T2 - T3 , which is of the same be used. Equation

_(5.5)

in this event

,

T'#'+*'K()

(5

I

or, if the definition of K is made lesa stringent, it reduces to Nomotos form

.T ';'+

4'.' = K '(s

-(See Fig.8a and ref. /10/ and /16/).

Equation (5.9) was applied by Nomoto to the analysis of a large number of zig-zag trials, /11/, and has also been used up to now as

the basis for the routine analysis of experimental resulta at the tank. (See Chapters 6 and

7.)

It is olear that the analysis gives different pairs of values for K'and T'if it is extended over the

whole test or is only applied to part of the test. As airealy remarked in the introduction the linear analysis is completely misleading if it io applied to tests on dynamically unstable vessels.

The analytical condition T'> T> O involves, in accordance with Table I, the plain inequality

I

I

Nr

>

Np

m'-W.

and it can be disposed so that the centre of pressure of the total forces, in turning movements of large radius without leeway, lies forward of the centre of pressure for motion in a straight line with

(little) leeway. In this way an hydraulic moment occurs after a

disturbance, which will bring the ship back to a new straight course.

(See Fig.l5).

If the ship is not stable on a straight course it is stable however on a gentle stationary turn of given minimum curvature in one direction or the other. In this stationary equilibrium condition the ang1e of

is

_F

O , and with a disturbance

_-

fl

the centre of pressure for the de-atabilizing transverse force is now given by

=

N/(VY1'p1pIj)

Yp

_pp

B...R.A.

2188.

-

17

-order of magnitude as T can reduces to the forai

(5.8)

(5.9)

B..R.A.

2188.

i.e. the point of application of the force is displaced much further aft with increasing R' and The, speed readings for

stat ionaxy phases of a spiral trial are plotted therefore as indicated in Fig.16. (See ref /11/.)

A zig-zag trial on an initially unstable vessel can be analysed by equations (5.3) or equation (5.5). In the latter case a netive T1' and at the same time a positive K, are obtained. Note that K' still gives the elope of the L () curve at the origin but that no stationary

R0

measurements can be obtained for the unstable (dotted) part of the curve in Fig. 16. An approximate forecast for the ap'..;. trial is obtained on the basis of equation (5.5) if it is noted that.the pivotal point in a turn is found by experience to be about 0.4 L forward of the centre of avity of the vessel. Therefore we can make ÇIr

- 2.5,B and hence

=Kl(flCVc)1'2(6_S

)

fl06

00

(5.11)

The reduction of speed in the turn is caused in particular by the component of the centripetal force

in

the forward aid aft direction aid

.,

&

it is therefore obtained primarily from D _R0. A semi-empirical mean curve published by Davidson /17/ is given in Fig. 18. The R.P.M. do not fall an equal amount and as an estimate it can be assumed that

NcI(1+ï)

Ti

2(

Vo).

As recommended in chapter 1 the turning circle trial should not only be carried

out with

the rudder hard over but also with angles of helm of

_15°

to starboard and to port As indicated in Pig. 16 it is often possible to determine from the resulta of such teats whether the vessel is dynamically stable or not. It may however be desirable to make an assessment while the trials are being carried out. _{One method} is to examine the rudder work.

If a vessel is dynamically unstable it requires continuous rudder movements to keep it on a straight course. If therefore, for example, during a trial the position of the helm is measured by a rule in the steering gear compartment at equal distances run,one could. expect that the bulk of the readings would be concentrated about the atable

-B.S.Rj,. 2188.

equilibrium position, i.e. in the proximity of the zero position for a stable vessel and about two angular values on either side of the

zero position for an unstable ship. (See the histograms at the bottom of Fig. 16.) It has been observed i.a. by C.A. Lyster that

this is actually the case on a number of large British tanloers. The effective time constant T for the stable vessel in equation (5.9) can most simply be defined from the condition that the angle of coursé for large t'(t) shall be the sane as that in accordance with the corresponding complete Ïinear equation

i-T1'T

_{ç'+(Ti'+T)*'+}

=

K'(S-S00)+

I'

(5.12)

The same result is Obtained by an approximation of the frequency contour of the transfer operator. The transfer operator

Ypg (s)

is defined as the complex fraction between the LA.PLLCE transformation for the course angle and the time functions of the rudder movement. The frequency form _{ffr$ (}

j DI)

is obtained by substituting the complex variable s in the operator by the imaginary factor i-u/where W'is the reduced frequency.

With the sign convention used the frequency function is

s-I

I

I

_'p

(j'urt)

(5.13)

The function is convniently depicted graphically by means of a log-decibel diagram as shewn in Fig.lO, viz, the amplitude relation-snip in the form of -o/R is expressed in decibels (db) on a base of 1og The product

sIR

forms, as is easy to see, the actual relationship between the amplitudes of the sinusoidal time functions of the turning speed and the rudder movement The phase angle '

still gives the lag of the

fr

curve relative to the S curve. We now have therefore

z-,2

,2

2OIoq (-K')+ 20 Io

J+ilftirI2_

20

Ic ,/i+T1

ti,'

-d10

2O1°)

,2 j

--

_{+ £rc Ian 13}

_{- arc}

_Ian

_{- arc tan 1}

.crr'

-

19

B.S.R.A.

2188.

For low frequencies the amplitude relationship tends asymptotically

to the giorizontal axis

T'R)db = 20 log10 ( Ic') while the phase angle

tends to Ø'

in the function

s-0

For hi

frequencies

db

has the asymptot

'1)db

20 log (-K')

20 logsjw- 20 log10

, whicn intersects the right hand asymptot

T'

at the point

at the frequencym.Çu j

_T

The left hand asyinptot has

a slope of 20 db per decade oi'tut

The polygon througt the points

Bi, 33 and B2 approximates closely to the frequency function throughout

the whole frequency range.

The"reciprocai frequencies" are

i

, i

and 1

respectively.

I

I

T1 T.3 T2

It can be seen direct from equation (5.13) that the reciprocal

points of B1 and B2 correspond to the real roots

5 D

_and

=

_{of the characteristic equation. Each factor of.the type}

T2

+

_{in the denominatorsiifjea a downward flexure of the}

polygon contour of 20 db per decade wiich corresponda to a phase shift

of

%0

_{In the same way a factor of this type in the numerator}

indicates an upward flexute o±' the polygon contour of 20db per. decade

and a phase shift of 90°.

The simple frequency equation

a-I

K

J'(i+jT'-w')

_(5.15)

corresponda to (5.9) the contour of whose polygon embraces only the

two asymptotes which meet at the point B at the reciprocal frequency

i

.

_{Equation (5.15) approximates accurately the reciprocal}

T'

frequency (5.13) to the rigtit of this reciprocal frequency.

However,

it can not be used for the analysis of automatic steering

characterist-ics.

_{In this connection reference should be made to refa. /14/ and}

/18/ and to the completely new paper by Bru /19/ in which the control

process la examined on the basis of equation (15.13).

-20-The dynamics of the vessel + rudder system forms one of the blocks in the closed control circuit in Fig. 17. The others are the

rro compass, automatic pilot and the steering engine. Each block is characterised by its transmission function which can be determined by frequency analysis of testa on the open part circuit. The zig-zag test can be considered as a frequency test of this kind. See further on this subject in ref. /12/.

0f the routine manoeuvring trials carried out on new vessels, the zig-zag trial probably gives the most valuable information..

6. The practical process of analysis.

As pointed out above the zig-zag tria], is especially suitable for analysis and for obtaining the coefficients in the equation of motion -for stable ships - with respect to the transmission operator. Alter-natively it can be regarded as a purely "functional" trial for measur-ing the primary comparative characteristics auch as the time of swmeasur-ing

on to course and the angle of overawing. With access to only a limited number of trial recorda there is probably a greater possibility of finding an accurate correlation between the geometrical properties of the vessel and the different coefficients or "steering indices" than between the geometrical propeties and the different nautical

parameters. The majority of the results of zig-zag tests received at the Tank have therefore been analysed to find the amplification factor K and the time constant T in the appromimate course angle equation

z-SSPA computer programme No.10 is used for this purpose (See ref. /20/.) Equation (6.1) gives a completely misleading result if those

coefficients are applied to the results of a trial on a dynamically unstable ship. _{It is ksown that many new large tankers as well as} some small coasters are unstable on a atraiit course and it i

_a

matter of urgency tò have access to an adequate method of analysis

for this case, especially as a Iaow1edge of the limiting values for rudder aizos etc. is of the utmost importance in desiga work. _{At the} present time the possibilities offered in this direction by the

B.S.R.A.

2188. 21

B.S.LA. 2188.

equation derived in the previous chapter are being investigated at

ssPA

#l(L)Zm#,JL.

_{+K'(4°)(s-s)+}

.4. K'T

-i

(.2)

For this analysis not only are readings of the course angle required tt the poeition must also be determined by the Decca method, for example, or by direct measurement of the leeway. The

correspond-ing cmputer proamme is No.44.

Both in proamme No.10 and in No.44 the unknown magnitudes in the equation of motion are determined so that, in the meaning of the method of leant squares, they approach most closely to a].l the measured points. A similar process has also been proposed by Nomoto /21/ with respe'ct to

equation (6.1).

roamme No.10 (for FACIT DB) will take readings of up to 76

successive course angles at constant time intervals and 26 angles of helm at arbitrary instants in time. The angular values are fed in for the whole of the recorded manoeuvres but the computer can be made to analyse only a limited part, for example the first curve of the course - so that the analysis can be said to correspond to transient analysis - or the periodic movement etc The values fed in are used for t he calculation of the mean speed in the interval considered

which, in its turn, is used as the basis for forming the dimensionless constants If both the speed readings and the Decca records are lacking, the speed curve can b? approximated by means of the empirical curves in Pigs. 18 and 19. In so doing the mean value of the maximum angular velocity at instants t7 and t9 is determined first so the mean speed during the periodic motion can be estimated by means of the auxiliary diaam at the bottom of Fig. 18.

The R.P.M. are not fed in, in their entirety, but they are of value, of course, in the continuous analysis of a given case.

(See equation (5.8).

-8.S.R.A.

2188.

It must be pointed out again that the programme is not restrictej to standardised zig-zag trial but that it can be adapted to an

arbitrary (linear) manoeuvre in which the rudder movement is defined by the corner points of a polygon series as described aboye. The values obtained for K'and T' can however only be used in connection with a standard test as the basis for an accurate statistical

assess-ment.

In Table II an example is given of a record sheet for the analysis of a zig-zag trial The sheet from the computer has been supplemented with some general particulars of the vessel such

as

the Owner, Ship Builder and Ship Number and data regarding the model tests at SSPA. The sheet gives, in the first section, the ship's name, date of trial, actual drau&its and the calculated volumetric displacement in metric units, the approach speed and shaft speed in knots and R.P.M. respectively, the approach course in degrees, nominal angle of helm and deviation off course in degrees for defining the type of manoeuvre, the length of the ship between perpendiculars in m. and the area of the moveable part of the rudder blade in I71.

The second part of the sheet suminarises the analytical results for the given interval of time Data are given for the mean speed and neutral rudder angle, the values of K and T , the dimensionless comparative values _{T', and the EMS value for the dimensionless} speed of yaw during the time interval. _{This value is. of importance} especially for a more detailed analysis of the non-linear effects

_aM

it is also used for the auxiliary diagram in Fig. 18. _{On the lagt} line is given the constant Q2 which is defined in equation (3.2).

The computation of the coefficients for the time interval (ta, t3) is done after integration of equation (6.1)

s-T(i/ _Sg)P

_{SI - 60) itt.}

(6.3)

and sigaifies that the computer uses the measured values at every

instant t _{within the interval for equal numbers of equations of}

condition in accordance with (6.3). _{If K is assumed to be constant} over the interval each equation can be written in the form

i--SL.)T+(

5d.t).K_(tL_t.J_KS.b...

_.j.p

(6.4)

-where the three magnitudes T, K and KS,,are the unknowns in the above

system of equations.

(Only under very favourable conditions

is

S0= s00

).

The method only requires differentiation of

the curve of course angles which is obtained in this

_{case for each}

recorded point by means of a parabola through this

poirZt and the

nearest adjacent points.

When the analysis is done' in accordance with equation (6.2)

the

time function of the course angle has to be differentiated

twice

and this is approximated with great accuracy by a trignornetrical

polynomial.

_{In model tests the angular leeway can be measured}

direct and can be approximated by the

_{same method.}

_{In the analysia}

of an ordinary zig-zag trial the angular leeway

_{must therefore be}

determined from the readings of the course angle and the positions

from the Decca navigator records which are subsequently approximated

with an arithmetical aiid a trignometrical

polygón.

7.

Some preliminary results.

In this chapter some of the results will be

given which have been

arrived at as a result of work carried out at SPA

_{on the analysis of}

zig-zag trials of Swedish vessels in

_{accordance with the simplified}

linear theory.

_{No vessels have been considered regnrding}

_which

there have been any doubts concerning

_instability.

_{As regarde the}

statistical distribution of the primary values

of K' and T' obtained

from the analysis, Nomotos results from ref. /11/

_{have also been used}

here.

_{The histogram for K' in Fig. 20 is therefore}

_{constructed on}

this basis.

The value of the median in the K' distribution shewxi is

approximate-ly

K'.

- 1.2, and for

- K

_{the corresponding value is about 0.86.}

T

(In accordance with Fig. 8b the same relationship gave

s-- K' a

_{and therefore for a ship with the}

repre-T'

Spp

¡iP/Arr

sentative figures s-

a 0.70, 'IB a

7.0

_{and 1(2/Arr}

₅₅

_{we have}

-

K'

= 0.86 exactly).

F'

. S.R.A.

2188.

-The diaaxn in Fig. 21 gives a convenient interpretation of K' and T' with respect to the periodic number i". The value of the median

mentioned above corresponds to to 10 and the result can therefore be compared with Kempf's arid Hattendorff's analyses in

Fig. 6. The shift to hiier values of

'V

is not so marked as with Hattendorff's analysis.

om Fig. 21 it would also appear that the Kempf periodic number for a stable ship is the quickest means of finding the value of IÇ/' but that it only gives indirect information regarding the deee of stability of course holding. In principle the value of - K' will

T'

be high and the periodic number will be low, although

Z'

12

according to Hattendorff would apparently be quite acceptable. (See page 6.). - K' increases for a given project directly as the

T'

movable rudder blade area Azr Two vessels with the same value of - K do not necessarily have the same characteristics of manoeuvrability

T

however and it has been found to be particularly desirable to limit the absolute value of T', the effective time constant, in order to ensure a given speed of response of the vessel in answering the rudder

move-ment.

The quality numeral P is introduced here as a convenient factor of comparison

s-whicn expresses the deviation off course i/P in ship lenGths after the momentary rudder movement

_S

The relationship between P and the two steering indices K'and T' is clearly illustrated by the level curves in Fig. 22. The results of Nomoto's and SSPA'.s

analyses are also shewn in the same dia'am as a broad band of reducing P values to tne right of the figure. The distribution of

the P values is illustrated in addition by ttie,bar chart a.nd the

cumulative distribution diaam in Fig. 23. The median value for

P is 0.36. Until further notice

P> O3

(7.2)

will be taken as the "desige criterion".

21e0. ₂₅

-P=-E

.

oc

If special importance is attached to the manoeuirrability of the vessel it should be possible to specify that the quality numeral P places the vessel among the lQ' which have the best P value, i.e. P should be ofthe order of magnitude, 0.5. This means that with an angle of helm of 200, for example, the deviation off com'se rises to

loo _{when the ship sails a ship's length measured from the position}

when the rudder passed the 100 position, in accordance with Fig. 8a. The requirement for P is obviously associated with certain re-quirements for K / and T' in accordance with Fig. 22.

From the above

z-T'=-O21

.

L/8

_Arr

can also be used as a guidance value for T / for a "norma]." vessel.

It now appears to be easier to find a relationship between K'

and the geometry of the ship than between T'and the ship. (See Table I). $SPA data, which is still limited, has been used as the basis for the investigation.

K'cari be written as a function of the hydrodynamic differential coefficients in the linearized equation of motion and in principle a guidance value for it can be obtained on the basis of, for example, analogies between the plane reflection of the underwater body of the vessel and a slender aerofoil but geod aeement can not be expected from several semi-empirical corrections.

For a rectangular aerofoil with a low aspect ratio 2T we have

s-(See, for example, ref /22/ and /23/). Tests with naked ship modela shew; that at least N' can be predicted quite well by the formulae _in

(7.4); that _B and NB are Somewhat high; arid that is low arid varies in a way which is difficult to assess. _{The esenoe of the}

propeller, rudder and bilge keels have an effect on all the differential

coefficients. 21u6. 26 -/

T

L

N' =.I

(7.4.)

N'

_4L

B. S. R a

21.88.

It may also be of interest to note that an uncorrected aerofoil ana1or in conjunction with the stability condition (5.10) gives

s-LIIB_L2T> Z

L

7

T

Since the length of the vessel increases with unchanged draught and fullness;

I/B

should be increased so as to maintain a margin against instability. In accordance with the above inequality the stability condition for ordinary hull forma, with I/B = 7, 0.8 and a Suez draught of 38', is not fulfilled for lengths above 160 ui. so that special stabilizing fins would be required. These fina are always present of course in the form of rudder and"propeller but it is obvious that the new wider and fuller vessels need proportionately greater rudder areas tkian has been usual hitherto. SSPS hopes in a

later re)ort to be able to return to the question of the required rudder area. K'< I for ships which satisfy the stability condition and, by the nature of things, equation (7.1) can only be applied under this condition.

In addition to the effect of the rudder and propeller the magnitude of the hydrodynamic differential coefficient is influenced of course by the hull form and to a lesser extent by the speed. In the preliminary

statistical examination it has therefore been assumed that K 'can be expressed by means of a fundamental value

i.7o

for a vessel of a

given standard form and that small deviations from this form cause small deviations of K'.

The standard form is comprised of a ship in whidi 1/3 7.0 and = 0.70 but which has the same slenderness coefficient

L/V '

the same

lIT

and the same LT/A,.

as

the actual vessel. However, the rudder of the standard vessel is assumed to be completely moveable and the speed is assumed to be so low that wave making can be neglected. The fundamental value K';70 can be obtained from the nomogram in Fig.24 from the approximate formula

s-R'

=-5.25

vtr3

7.70

27

which

has been derived from the analysis of zig-zag trial results _ai data obtained from model tests. The formula is preliminary and will be modified when more data is available.

Linear corrections are made for the actual ship; for the partly fixed rudder area; for deviations in the block coefficient and the breadth/length ratio; for incomplete loading down to the marks; for

trim

and

for the speed; in accordance with the expression

z-R'

ad

L\.{I+K5(6.K2(.)+

).K4(.)i.Ç(.)}

(.r.j)

7.70

tA )

r

The corresponding correction factors and correction terms are conveniently taken from the diaains in Figs. 25 and 26. It is

also known that the length of the bugs keel may modify the value of K' but the limited amount of research information available has not revealed any data regarding this. The above correction terms already involve some deee of uncertainty but they give the correct

trend.

A dry careo vessel for a service speed of 17 knots is taken as an example of the calculations. The principle dimensions are;

L=l4Om, B=l9.2m,T=8.0mandV.l48O0m3.

Asa

preliminary measure the vessel is assumed to have a Mariner type (open water) rudder having a total area Ar = 17.8 m1 of which Arf 4.0 mt corresponds to the fixed fin. This data forms the basis for interpolating in the nomoam in Fig. 24, viz.

LT/Ar 63,

VT =

17.5 and

iv"

= 5.7 from which the corresponding fundamental value 1'7.7O - 1.12 is obtained. Thé fixed rudder fin causes a loss in K' of about 11. om the above corrections

taken fröm Figs. 25 and 26 the final forecast IC':. 0.90 is obtained,

or for the sèa trials at a speed of entry to the turns of 19 knots with a mean drauit Tm - (Ta + T) (5.5 3.8) _4.65 , we have K' - 0.70. Fiom equation (7.5) T'' approx 1.45, corresponds in this case to K' = - 0.90. From Fig. 22 a P value of about 0,25 is obtained and the forecast does not therefore fulfil the specified

requirements of P> 0.3. On examination it is estimated that the moveable part of the rudder area should be increased by 2Q from

whicti a new forecast gives K'c _{- 0.98 and T" 1.32, from which the}

-28-desired P value is just exactly obtained In this case, therefore, a moveable rudder area is proposed which amounts to 1.47 of the product VP, or a total rudder area of l.83 of VP.

It is interesting here to make a comparison with Det Norske Varitas requirements. This, as is known, is the first Classification Society which has endeavoured to formulate rules. According to the olaasifica

tion rule, ref /24/ we have

fAr\

25

I

_{+ (L/B)t}

(7.8)

in which half the area of a streamlined rudder post is included. Fbr the careo vessel considered above the formula gives an "effective"

area of 16.5 n, or

_Arr

14.5 and Ar 18.5 n2 . The corresponding

percentages are

1.3

and 1.65 so that the ölassification rule permits about lQ" less rudder than the proposed method.

Formula (7.8) shews clearly that a broader vessel should be equipped

with

a bigger rudder. However, it neglects the effects of

fullness and appears to give areas which are much too low for modern tankers, for example.

Another example is given therefore. This case is of a 16 knot tanker in which L = 240 m, B

= 37.5

m, T 11.58 in and _6pp

0.81.

A Simplex rudder is to. be fitted. From experience a rudder area of

l.9

appears to be satisfactory. The nomoam in Fig.24 gives

K7.7 = - 2.20 and the corrected K is 2.00. Formula (7.3) gives

T' 2.79 and from Fig.22 P 0.31. The area selected therefore

appears to be suitable. On the other hand Det Norske Ventas requires for this tanker a relative rudder area of only 1.6l. The other Classification Societies do not give any recommendations at all.

It has to be pointed out once again that the results given here may have to be modified to some éxtent in the lieht of new zig-zag trials and of fresh analysis by the process outlined in chapters 5 and 6 which can also be applied to unstable ships. At present however there does not appear to be any other comparable result on method of calculation whïch can be used in design work like the above

simple ne tbod.

B..R.A.

B.s.a.A.

2188

30

8. Bib1io,aphy.

Norrbin, N.z "Fra1ag till proain fSr gir-och atyrningaprov med handelafartyg", SSPA AlimAn Rapport b

5, 1959.

(Proposed proamnie for turning circle and steering trials on merchant ships).

ITTCz "Report of the Manoeuvrability Committee", 10th International Towing Tank Conference, London

_1963.

Decca Navigator och Radar AB: "Decca Navigator. Fart- och mandver prov", Demonstrationaprm, Giteborg

1963.

(Decca Navigator speed and inanoeuvring trials).

Kraemer, O.: "Richtlinien fUr Steuerversuche", Werft-Reederei-Hafefl,

Heft 1,

_1934.

(Code of practice for steering trials.)

Kempf, G.: "ManUveriernorm fUr Schiffe", Hansa, Heft 27/28,

_1944.

(Manoeuvrability Standards for Ships.)

Hattendorff, H.G., Mdckel, W., and Gallin, C.: "Versuche mit dem Modell eines schnellen 10 000-tdw-Frachtschiffes", Schiff und Hafen, Heft 2,

_1965.

(Model tests of fast 10000 T.DW Cargo ships.)

Gimprich, M., ani Jacobs, W.R.: "Application of Kempf Manoeuvrability Test to Six Naval Vessels", TT TM No.

_{89, 1948.}

Gertler, M., and Cover, S.C., "Handling Quality Criteria for Surface

Stiips", Chesapeake Section Paper, SNABIE 1959, Pirat Symposium

on Ship Manoeuvrability, DTMB Report

1461, 1960.

Suarez, A., ¿n& Breslin, J.: "Comments Concerning Zig.Zag and Spiral 7anoeuvre Data", Davidson Laboratory Note No.

_596,

Fbrina]. Contribution to 9th International Towing Tank Conference, Parie

1960.

Nomoto, K., et al.: "On the Steering Qualities of Ships", Intern. Shipb. Prop., Vol.

4, No. 35, 1957.

Nonato, K. "Analysis of Kempf's Standard Manoeuvre Test and Proposed Steering

_Quality

Indices", First Symposium on Ship Manoeuvrability, r1W Report

1461, 1960.