Prediction of ship manoeuvrability

LABORATORIUM VOOR

S CH EEP S BOU WKU NDE

TECHNISCHE HOGEScHOOL DELFT

Prediction, of ship manoe.uvrabiiity making use of modeitests

by G. van, Leeuwen and J.M.J. Journ'e aprii 1970. Report No. 288

ii

CONTENTS

Sununary

i. introduction

2.. The equations of motion

introduction

the components of the hydrodynainic forces

some considerations concerning the similarityof conditions of motion

. hypothesi concerning the hydrodynamic forces acting,

on the manoeuvring ship

5. the set of equations of motion

3. Execution of thetests

1. the measuring equipment

2. determination of draught and trim

3. determination of theresistance- and propulsion coefficients determination of the ridder speed U

5. determination of the remaining coeficients

1. the static sway tests 2.. oscillatory swaying tests

3. oscillatory yawing tests with constant drift- and

rudderangle

6.

some experiments with a small model (a = loo)

7. some remarks concerning the computed coefficients

4. Computer programs

least squares 'analysis of measured data

solution of the differential equations

5. Comparison of computed and full scale manoeuvres turning circles zig-zag trials

6.

Final remarks 7. Recommandations

8.

Appendix List of symbols References

Summary

Static sway- and oscillatory yawingtests with a 1:55 model of the

50.. 000 DWT tanker "British Bombardier" are discussed..

The principal purpose of these tests was to determine the coefficients of a

non-linear mathematical model to predict a number of standard manoeuvres

which. were. earlier performed with the full scale ship. The results of these full scale manoeuvres are described in..reference [i].

The mathematical model chosen is based on the Abkowitz Taylor-expansion of the

hydrodynamic forces and moments [14]. However there is a principal differen-. ce with respect to the variaes involved, which enables. a more çorrçct

-description of some non-linear phenomena. Cbmparison of the predicted manoeuvres

with the corresponding full scale data shows a rather good agreement.

For comparison purposes also some experiments have been performed with a small

model of the same tanker (a = 1:00). However it is found that scale effects, due to the very low Reynolds number have a :cons.iderable influence. on the hydrodynamic derivatives.. Some interesting additional figures are. given

showing the contributions of each, term of the mathematical model during a

turning circle manoeuvre while also the change of the stability roots during

Introduction

In the Deift Shipbuilding Laboratory model tests are performed to

determine the coefficients of a non-linear mathematical model, which describes.

the still water manoeuvrability properties of a ship. Two types of tests are

performed.. First the static towing tests with a constant drift- and rudder

angle and second oscillation tests to détermine the added mass effect in the

swaying motions and the hydrodynamic derivatives of the yawing motion.

The prncipal purpose of the tests is to obtain information on the possibility

to predict the principal manoeuvres of a ship., using an adequate mathematical.

model and modelexperiments to determine the coefficients of this model. For comparison purposes also full-scale trials are perfrmed [i]

The modelexperiments have been performed at four different initial speed

conditions, corresponding with a constant propeller power each. Originally the results of these four sets of tests have been kept separated because it was assumed that some of the nondimensional coefficients could change with the

Froude number, based on the initial speed. In that case a set of coefficients which would be different for each initial speed would have been found. The results of this tentave. analysis are given in [ii].

During the analysis of the experimental data it was found that the principal differences between some of the nondimensional coefficients could be des-cribed effectively by considering the local watervelocity near the rudder.

In this way the apparent Froude-effect in these coefficients rather could be

calle,d a "power-effect", while the differences in the other coefficients were not considered significant, in view of both the. available information and the accuracy of the masurements..

It is not to be expected however that this method of describing the

pheno-men.a _{mentioned above will hold for other ships.. Especially when the Froude}

number becomes high e.g. .30 or higher it is not very likely that there will be no real Froude-effect in some hydrodynamic' derivatives, if they are com-pared with the corresponding values found at a Froude number of .10 e.g..

Provided a certain mathematical model has been adopted, there are several methods' to determine the coefficients of such a thodel by modeltests The

main problem is to find out how far it is allowed to uncouple the three motions in the horizontal piane. 0f course during an actual manoeuvre the motions

are aiways coupled. and! even if the mathematical model contains terms which

way these effects have to be determined. The most convenient method might be to perform free running model tests and find the coefficients of the mathematical model by analysing the data concerning position and course of a number of

representative manoeuvres. in. that case all variables remain coupled in the natural way.. In generai however too less room s available for these kiinds of

manoeuvres, sothat one is pressed to find an acceptable alternative. In this

respect the forced horizontal oscillation test provides an alternative solution.

On the other hand amongst others the problem of uncoupling the motions as

mentioned above is introduced. Another practical problem is to choose the right

combinations of oscillator frequencies and. -amplitudes.. Considering .an actual

oscillatory motion, due to the harmonic motion of the rudder, it is found that the combinations of frequences and amplitudes involved in these motions,

cannot easily be simulated by horizontal oscillation tests, because, the actual

ange of amplitudes is of the magnitude of one half to many shipiengths. In most

towing tanks sufficient. width ïs not aîailable.in this respect.. These problems are considered in greater detail in [2].

The conclusion. is that most horizontal oscillation tests, involve an unnatural

relation of amplitudes and frequencies. In other words the ratios of

velocity-and acceleration amplitudes are quite different from the actual values.

Apparently this is no problem, because most o± the mathematical models. which

are now in use do not contain any cross-coupling terms between velocities and

accelerations. This does not imply howeve.r that such cross-coupling effects. could not be introduced,, if the range of ratios of these variables is extended too far.

Finally some remarks considering, the mathematical model.

In the course of .the years a lot of studies have been devoted to this subject.

Starting with Davidson and Schiff [3], who described a model based on the linear equations of motion,. This .set of equations, which originally involves three equations, describing the' 'surging swaying and' yawing motion., has been used by several authors, though unfortunately omitting the' surge equation..

One of the most extended non-linear mathematical mOdels has been proposed by

Abkowitz [14].. In this model the, hydrodanainic part of the forces is expanded into a Taylor series of the variables concerned. This principié, is very useful,

particularlyif the constants of the model are to be determined by the analysis

of forced model tests, because all imaginable hydrodynaniic effects in principie.

can be described in this way. An important question involved in this

Taylor-expansion is u.pto which degree it has to be extended to be sure that the prin-cipal non-linear hydrodynamic effects are described correctly. On the. other

hand it is questioned to what extend it is necessary to retain a great number

of terms in such a model foi a reasonable' 'accurate description of manoeuvres, even if the separate hydrodyna.mic effects involved can be measured with forced model tests. In other words it is suggested that,, on the ground that during

an actual manoeuvre the ratios of the variables satisfy just one relation, it

might be possible to describe the joint effect of a number of terms by one term only. In this way a quite more simple mathematical model would arise, the

coefficients of. which had to be considered functions of the' coefficients of the original model.. Some simple non-linear models based on these grounds are. des-cribed in [5].

A disadvantage of such simplified' mathematical models is that its coefficients

cannot be determined by uncoupling the three motions which means that they can

only be derived from. free'. running, tests, either full scale or modeltests.. For practical purposes however, 'such. .as simulation studies and automatic piloting, these simplified non-linear models :can be applied succesfully.

The principle of the mathematical 'model used for the present model-tests is, apart from some details, the saine as has been used by Abkowit'z.. The way upon

which Abkowitz handles the influence of a change of the forward speèd however

brings about that no insight is gained' into the physical background of this influence..

In this paper the hypothesis is used that if the motions are simular, regarding

velocities and accelerations, the principal hydrodynamic forc.es òn the hull

are proportional with the square of the instantaneous forward speed, while

considering the forces wich mainly depend on the effective angle of attack of

th'e rudder, the proportionality with the square of' the' local, water velocity 'is assumed.

The general concept of this hypothesis is confirmed by the model experiments.

In' the next chapter this will 'be discussed in more detail.

2. The equations of motion

2. 1. introduction

If we are forming a. mathematical model and we start from the fact that the hydrodyna.mic forces are functions öf the. velocities and accelerations involved in a motion, we can exand these forces,, as has been done by Abkowitz

in a Taylor series of these velocities and accelerations. On the. ground

of considerations of magnitude we can ignore the terms the order of which is

higher then the third e.g. There are some objections to this procedure however..

Considering a term proportional to the third power of the angular velocity e g

then the omission of the fourthorder terms means that the contribution of this

term, regardless the forwar& speed, remains. proportional with the third power

of the angular velocity. From módeL experiments it is known that this -. and

simular terms - are reversed proportional with the forward speed. Neglecting this speed dependance consequently corresponds with 'underestimating the

non-linear efíects described by these third-order terms in the case of speed re-duction.. For .a speed reduction of 50 percent such a non-linear effect is under-estimated by a factor two..

Another objection, though of less importance,, is. that if considering, the. separate velocities as 'lateral-, forward and angular velocity, the particular role played by the forward speed hardly comes forward.

In section 2.3. a different. basis has been chosen for the, mathematica1 model, using th'e hypothesis mentioned in chapter 1. it will be shown that the non-dimensional variables involved are related to well-known' quantities, _like' drift-angle, rad'ius of' curvature and their change with respect to the. distance covered by the ship:.

The. effects of the fourth-degree terms, mentioned above,, is involved in _the third-degree Taylor expansion of the forces,, if they are considered functions of these cha]acteristic variables.

it is emphasized however that the. 'concept of this is 'not new, because also Davidson and 'Schiff .{3, Nomoto [6'jand othe.r investigators [7] already _paid attention, to the. importan'ce of these. variables, while both earl'iei' work arid the present investigation justify the adoption of' the hypothesis concerning the forces.

2.2. The components of the hydrodynamic forces

The equations of motion

des-cribing the balance of forces and moments during a still water manoeuvre can be written as' follows: (see also fig. i)

m(r+Ur)Y

(a')

I .

=N

(1b)

vr) = X (ic)

where Y represents the component of the hydrodynamic forces perpendicular to the

ship and N the corresponding moment,whiie. X represents the component f these fOrces acting in longitudinal direction.

The sum of the hydrodynamic forces can be devided into three groups. The first

group contains the components which depend' on the condition of motion of the

ship without propeller and rudder. The variables involved in this case will be discussed in section 2.3.

The second group, contains the forces, which act on the rudder. They depend _on the effective angle of attack of the rudder,'andas this quantity depends on the ship's condition _{of motion, these components will depend on the}

variables of the first group as well 'as on, the rudderangle, itself,.

The third group contains the forcecomponents which are, among others, caused

by the change of circulation around the ship, due to the rudderdeflecrti'on. In

general these components are to be considered the result of the fact that the

sum of the hydrodynamic forces is not to be obtained by the superposition o'f

the forces acting on the hull and those on the rudder, which may be'

approximately true for the side forces on sailing yachts.

Concerning the longitudinal force balance, a fourth group has to be considered,,

which involves the forces due to the resistance and the change of thrust caused by speedloss during manoeuvring. This group determines the difference between

the forward speed of the centre of gravity 'and the speed of the water near 'the rudder.

2.3. Some considerations' concerning the similarity of conditions of motion

in general the variables which describe the condition of motion of a

line-piece AtB in a horizontal plane are:

X

, X

, X ...

o o ₉

yo., Y0 Y'

, where and .

represent the displacements of the middle of the line-piece in

the direction of the x- and y axis respectively. Consequently these variables

and their derivatives determine the path while determines the change of the

angle between the line-piece and the x-axis.

The description of thé motion of a line-piece with these variables fixes this

motion in space as well as in time.

.itì the first instance we can, leave 'the: time- and.conseq.uentïy the velocities with which the motions are. executed - out 'of consideration and describe them as a

function of an infinitesimal displacement in the direction of the path.

The motion of the line-piece is then given by t'he equations:

,x

=x ()

y: y: (s)

)

=)

(s)

which is illustrated in the next figure. X0

Yo

If we compare the motions of two line-pieces which have different lengths L1

and L2,, we can cali these môtions similar in space' if both t'he 'motions can be described by the foilowing functions.:

f1 (s)

y0

f2(s)

= f

where x

= x/L, y

= y/L and s =

If thés.e functions and their derivatives are continuous' _{n a fixed interval of}

functions are also equal for both the motions: 2 3K 3 3K ax

ax

ax

o o o as

3'

as

'3E2'

a3'

2 s c a1

a1

a10

.3K'

_3(2'

3(3' as 3s as 3il) a2 _a3

32'

_3(3'

3E as as as X0 Yo Therefore we

same initial condition concerning 'the. coordinate system, 'is determined by the

above derivatives. On this ground we will further call them similarity

parameters.

If we also involve the time t as the fourth dimension in the description 'of motidns together with x., 'and il), we can speak of similarityin space' as well as of similarity in space and time.

Just as the length-ratio of the line-pieces could have an arbitrary valüe, also

the "time-ratio' can have any value.

Suppos.e the ratio of the times, necessary' to cover the unit

at1 at2

has the value i, sothat = _{If then T 'and T are}

a.s a's î 2

(auxiliary-) times,, in such _{a way that their ratio has also}

3 ti 3 t2 at

define t = and t

= - .

For both the motions now

-T1 2 T2

The motiois of both the line pieces are now defined simular if on a certain interval of 3, _{next to the functions}

., f2 and f3, also for

both the motions the function

t3 = f(s3E)

has the same value. Assuming this function and its derivatives _{are continuous} this also implies the equality of the following functions.

at3 a2t a3t

3'

3(2'

3(3

as

as

as

Bi

can also state that the similarity of motions, starting from the

of displacement, òonstant

the value p, then we

has the same value. in space and time

As it is more convenient to consider speed-ratios than"time-ratios" we keep in mind that the speed U has been defined to be Consequently from the

equality of it follows that the two speeds have a constant ratio: 1/U2 =

OE,1

= A where was the length-ratio and p the time-ratio. in the next. figure the meaning of the space and time similarity-conditions have been sketched'.

Similarity of motions in space and time

y X 01 01 L1,

== a= "L

y x 2 02 02

-.8-N

-s..

--X0,2

s.5--- s.5---

_{- -}

ti2 = 1-; U

1_a

u2

p;

X01' 5*

-2/U2

Anticipating the next chapter, we can imagine the two line pieces to be two

submerged hydrofoils, which rim along two-similar path-s. If in that case both the Reynolds numbers would he sufficiently high. and the angles of attack sufficiently small then,, regardless the ratio of the Reyflold-s numbers, which means regardless the valü'e of the product OEA, it might be expected that at

each corresponding moment the hdrod-ynamic forces would be in the proportion

of pU12L12 to -

oU2L22.

Provided this would be exactly true, then ve put

the question what amount of irEtantaneais similarity of both the motions is -necessary and sufficient so as to satisfy both the hydrodynamic forces the

same proportion.

The fac-t is that if -we could answer this question., we are- also able to form

a set of equations of motion for the hydrofoil, moving in the horizontai plane,

for an equatioÌÌ of motion describes the equilibrium of forces instantaneously. In the firs place- we will illustrate that if we compare two different

condit-ions of motion instantaneously,, there is a question of the aóünt of s-imilarity.

Considering the two motions, skêtchéd above, for which at each correspondin

ax a-ya

3t-moment the variables and- _{and all their derivatives have} the -saine -value, it will be c-lear that. here- is- the question of the- highest-order

similarity. If we consider two motions and comparing them at a corresponding

moment, assuming only a. restricted number of derivatives 'of X3E, y3E, q) and t

to be equal at tat moment, than we. can state these motions to have a

restricted similarity., This will be illustrated with an example:

Provided that at the moment considered only the following, derivatives are equal for both the motions:

ax03

ay3

aq) t3E

3'

3E' 3E' 3E' as as

as

2:

.

2.

ax

ay

2

23

o o q)

at

2'

2'

2 and 2 3E. 3E 3E. 3 as

as

as

as

We then state there is a second-order similarity. Characterising a corresponding

moment by the equality of q), which means a specified choice of the coordinate system, and using the fôllowing equalities:

3E

23E

ax

ax

o - cos 4)

-

sin 4) '3E 2 3E

as

3

as

as

23

ay

ay

a

_____ -

sin 4) = cos

4).,

-3E 2 3E

as

3E

as

as

then at the corresponding moment also are equal:

= q) - 4)

L 4) and'-R 3E

as

X0 (see figure)

If we now consider the local "angle of attack" in P at distance x before G then is

find:

ConsequentlY this angle is equal for both the motions. The change of this

angle appears not to be equal however, as follows from the fact that

x

- L

2 _3xE

where

is a function of O _{which was not equal in both cases.}

3:5

_s3

So we can speak of restrictea-similar instantaneous conditions of motion indeed.

The principle of the restricted-similar conditions of motion is used to form the equations of motion of a body, moving in a horizontal plane, particularly

of the manoeuvring ship, in such a way. that: the:. drawbacks attending the

equations of motion which describe the .hydrodyna.mic forces being functions of the lateral, velocity and:acceleration..and the angular. velocity:and acceleration nondimensionalised Mith the initial speed or otherwise,, are..avoided.

2.J4. Hypothesis concerning the hydrodynamic forces acting on the manoeuvring ship.

As has been discissed in the preceding chapter the hydrodynainic forces

on two similar submerged bodies will be proportional to their force-units

1

22

1

22

pu1 L1 and pU2 L2 at a certain moment, if their conditions of motion

satisfy certain demands of similarity at that moment.

If these motions are rectilinear this similarity is implied in the same value

of the angle of attack and if the motions are non-stationary the principle of

the r'estricted. similarity of a certain order can be applied to the motions to be compared.

As a hypothesis concerning the hydrodynamic fòrces acting on two.similar ships,, we now state that, within a restricted range of length .and speed-Car time) ratios a and A (or p), these forces will be proportional to the force-units

pU1L12

and pU22L22 at a certain moment, if at the moment considered, both

the conditions o.f motion satisfy specified' similarity demands, which will have to be determined fürther.

We will now try, to form the equations of motion for the manoeuvring ship in

such a way that this hypothesis is expressed by them. In a first instance we

will only consider those forces which are independant of a rudder deflection

-and the instantaneous thrust, which means that only the first group forces (chapter 2.2.) are described.

The general form of the hyd'rodynamic. side-force _{equation can then be expressed} as follows:

Y ' p. U'L2

Where the equality of Y , _{if comparing two similar ships., has. tó.}

express that

the conditions of'moion have _{a certain amount of similarity. This}

means that. can only be a. function of a number of similarity-parameters, _thus:

.k

k..

(ax

ay

k k

''I

.

O '4J

o Y

= Y

e.

k

'

k

'

1,

k

(k '=

1,.

\ as

as

as

as'

where n has to be .determjned further.

The corresponding moment and longitudinal forces can he described in a similar way where it is noted that the latter contains' _{an. additional constant.,, expressing} the longitudinal resistance. to. be proportional _{to the square of the forward} speed in the considered range of speed-. and length ratios.

An esseñtial point involved in this _{matter is that in generai the hydrodynamic}

forces are assumed only functions of the velocities and accelerations involved which means that no higher derivatives' of displacements and angles play a role. As there is a reversible 'relation beiween the velocities, and accelerations at

the one side and the similarity _{parameters. on the other hand, this assumption}

contains a prescription of the amount of similarity.

It is not discussed here however _{to what extent this assumption is true.}

As can be. seen simply the amount of similarity _{concer.ned.cannot be greater than}

a 2

of the second order1 For a third-order similarity _{e.g.. the equality of}

is found., from which the equality of i follows, the tIme _{derivative of t'ha} lateral acceleration.

Consequently for the further description _{Of the.. hydrodynamic forces we assume}

22

that 'they are besides of' the for.ce unit _{.p U L , only functions .of the}

parameters which determine the second-order similariry _{of instantaneous}

con-ditions of motions.. Thus: _ax ayTM a.* TM

'

TM

'

TM' TM'

as

as

'

as

as

2 .2

₃

a x0

a10

a2tp

in the appendix it is shown that from the equality of the second order similarity parameters follows the equality of the. following variables

V Lr L L2? LÚ

from which follows the connection between these similarity parameters and the velocities and acc&Lerations.

The usefulness of this hypothesis, considering two identical ships,, will depend on the speed ratio X. On the ground of Froude's law it might be

expected that this value is sure to differ only little from unity. So it has

to be examined ithiñ what bounds this ratio can be chánged, without disturbing

the usefulness of the hypothesis. it also has to be examined to what extend the

lengths rati'o of the ships,, if they are. further similar, limits the use..

So it might be expected that the hypothesis. can only te applied in' .a restricted range of length- and speed ratios', in such a way that if wave generation can

not be neglected the Froude number should nearly have the same value, so as

to be sure that. the wave patterns of both ships would nearly be similar.

This means that 'the

ratio X/'Thshould nearly be equal to unity in that case. if the comparison concerned, two submerged bodies 'than the flow patterns would only be similar

if

the' Reynolds number wOuld b,e equal in both cases, which

means that the product .A would be equal to unity.

Surmnaizing it is to be expected that the usefulness of this hypothesis will

dépend on both Fraude- and Reynolds number.

It is noted however that if the hypothesis is applied to the longitudinal

force's on surface ships, i,t is very likely that the limit's of speed- and lengths ratios will differ from those due to the lateral forces and moment, because it is already known that the longitudinal resistance' is rather

sensitive to

scale

effects, which meais' to X and

values.

If the hypothesis is also. applied to t'he force' components of the second group 'than t'he local similarity is characterized by the actual angle of attack of

the. rudder and its "path derivative".

The actual angle of attack is assumed a linear function Of the three angles

involved,, thus

e.ff =

p2v + p3r

which indicates tha't the variables characterizing the similarity in this case

are 6, v, r, and if acceleration effects are considered, also the

path-derivatives of these variables. 'This set of variables, completed with the

local water velocity, determines the magnitudes of the force component's of the second group.

Concerning the components of the third' group it is assumed that they will

mainly depend on the variables of toth the first and second group while they

will partly be proportional t the square of the forward speed of the centre of gravity and for the rest to the square of the loca]i-(rudder4speed.

On this basis the mathematical mOdel can be build up., considering the three groups to be functions of the nondimensional. variables concerned. The expansion of the three groups in a third-degree Taylor series leads to a. number of terms

a part of which being proportional with the square of the forward speed, while

the remaining terms will be proportionaL to the square of the local rudder

speed.

Considering e.g.. the linear term in v, then the following expression is found:

The expressions for the, other terms will have similar forma. On this ground the mathematical model is to be written in the. following form:

m(ir + U r) X I r zz m(J -. r) X pU2L2 Y1 pU2L3 N1, PU2L2 x1

+

(a3 iU2 + a 2u)

vf

3rd group '(interaction rudder-hull; forces. due' to failing superposition-principle).

+

_pUL2

_Y2

+ ,

pUL3

N2

+

pUL2

+

(3a)

in these equations the components marked with a. star contain the terms.

originated from the Taylor expansions' of the first and second group,, while the

dashed components contain the corresponding terms of the second and third group, both star- and dash-marked components being functions of the rudder

.ag1e and the five variables determining the second-order similarity. 'The. longitudinal force.component' X3 describes the differencebe.tween the

ship's straight-line resistance and the change of thrus.t due to the speed-reduction.

On the ground of theoretical considerations and the experience from earlier

investigations a number of assumptions have, been made which sim1i'fy the

expressions for the various components. Some of these assumptions _{have been}

investigated particularly, while others. are not contradicted by the

measurements.

13

-y U + .a2v

'tR

i st group 2nd group

(forces on hull (forces on rudder without. rudder and due tO effective propeller) angle of attack.)

-i

_1

-i

The assumptions concerned are sunmarized as follows:

The forward speed U, as a variable of the equations (3), can be replaced

by its longitudinal component U. Consequently the variables vand rare

defined as V/U and '/U respectively, while the unit ds is defined as U dt/L.

X

The hydrodynamic lateral forces are independant of the longitudinal

acceleration,, while the hydrodynamic longitudinal forces are independant of the sway- 'and yaw accelerations.

Non-linear acceleration effects do not occur in the range of interest.

L. If the ship is on a straight course the forces due to a.certain rudder deflection are proportional to the square of the local. rudder-speed _UR. (This assumption may be considered the definition of the quantity UR for the present investigations).

5. The influence of the rudder-rate on added-mass effects is negligible for

practical purposes.

2.5.

The set of euatins of motion

Executing the Taylor-expansions of the three

groups of forces and moments

and applying the assumptions indicated above, the equations of motion can

be

written as follows: ) t

-

U t V

(1+u

y2

y

i

r

(1+u

)2

r

U

{ Y

y

(1+u') +

v1

, v' + {(Yr -m') (1+u') +

r1

, }r +

62

+

66o6

} u

15

-f. t

vvv

i L

vrr

i

+1

,2 U R

,3

1

-

U J

r

+ ,. 2 i

rrv

+

'

1+u 1

vyv

'

3J (1-i-u ) U . R

i''

.2

rvr +iY

L rrr rrr 1-i-u (1-i-u 1 U 1

-

R

i+u

vrr

(1+.0

'

)3.i

+Y

L

rvv

'

rvv

3 J: 1-i-u (1-i-u 1. Y + U

}t2

{

,x

&r +

6(1+u )2

tßrr

_(1+u')2

} t ÓI1V

_t5rr

{ i+u') +

_}v'2

+

{Y(1+u'

.. U R . r i-I-u ,, - róó , 1+u

fvro

2

,2}vr6+

+ Y6

{;

_(1+u')+

aB

+

U

{'

zz

-

C?

r

+ Ñ

r

_(i+u

)

-

{ y +

y

=

(1+u

U U { ?(1+u ) + _1+u

',

}, ' +

{

?(1+u

+ r '' } r' + 1+u r _{i -} ' r 1:

-H?

,IN

R Iv

+?

,+N

, Ir + L 1+u

_'(i+')3

L rrr 1+u rrr(l+ )3J U

{?

(1+u')+ }

'2

{ vôiS v66 r66

.(i+'y+

r66 i-4-u li-u

'2

{ N

+ N6

V r 6+ {

(i

+u y +

N UR

} +

{

+ + 2

-

16

-(b)

UR

}v'r'2±

'{ c i u vrr 3

rvv

rvv

3

(.l+u ) li-u (l-i-u

U } v' + { N + R Svv '

2

tßrr órr ' 2 (i-i-u ₎ (li-u

-

(x + R 2

)}'

U (i+u,

kr

VV

+m +X

vr { X.(1+u' -U

v'

_+{

u2

2 1+u { x (1+u') -} +

XT.0

(J4C)

in these equations ali variables

have been nondimensionailsed with the

initiai speed U sothat e.g.:

Ti

X

u

=-o

}

asymmetrical behaviour öf the

K. ' 2 - ,2

terms are Y (1+u ) + y u

a

aR

asymetrical rudder effectives

respectively.

d. If

the change

of

speed during

the hypothesis is valid,, than considered linear functions of

K

-as Y

and Y

should be added

vu

not necessary however,.

,2

y'2

+

_}

r'

+ rr rr (1+u 2 v'r' l-{XK.6(l+U') +

r1'' }

Some additional remarks concerning these equations::

a. The side force-

and

moment equations contain some terms to describe the

ró+

K X

y

= y

(1.+u ) and r =

r (u ).

ship. With zero rudder deflection these

and NK. (1+u')'2

+ N

U'

while the

a

aR

- 2 - 2

is described by the terms and

a manoeuvre is larger than the range in which

the star- and dash coefficients can be

the speed. in that case additional coefficiénts

For the present investigations this appeared

, 17

-b. The longitudinal force component X3 is devided into two components, the

first of which describes the. balance between the original thrust and the

straightine resistance while the second describes the (linearized)

increament of the thrust during a:manoeuyre.

e. The fourth-degree terms, which were mentioned in the introduction are

_due

to the factors i/(i+u) which may be linearized in the range

3. Execution of the tests

3.1. The measuring equipment

The principal property of the measuring equipment is, that only harmonic components of the forces are determined. Fbr the present investigation only the first harmonic component's were needed:. This

determination is achieved by multiplying the forces by sinwt and coswt

respectively while these products are integrated during, one or more periods

of the oscillation. The non-oscillatory components of the forces are determined

by integration of the force during a number of periOds.

A more detailed discussion on t'he oscillator and the measuring equipment is given

in ['8].

The static drift-angle 'adjustment 'during the oscillation tests is achieved by

turning the model with respect to the connecting line of the oscillator struts(the "pure yawing line"). This is sketched in figure 2.

-3.2.

Determination of draught and trim

As the model was restrained from heaving

and pitching the draught and trim, as dependant on the forward speed, and the rnnber of propeller revolutions had to e determined.

The revolutions adjusted fòr these tests were estimated from the available full scale data. A change of the rpm did not influence neither the mean draught nor the trim however. In the next table the various draughts fore and aft

are given for model and ship.

TABLE I

These draughts have been adjusted for the tests concerned. it is questioned however if it is correct to restrain the model in vertical direction during

oscillation- and other tests. A particular investigation might give the answer, but it is not expected that undesirable cross-coupling effects would

disturb the side-force measurements, due to the uncoupling of the swaying-and yawing motion. This is also expêcted with respect to the rolling motion. As it was not the purpose of this investigation to find an answer to this

question, it was considered a useful approximationto adjust the constant

draughts, derived from straight line tests and to restrain' the model also from rolling.

MODEL

SHIP

U Draught fore Draught aft : U O Draught fore Draught aft

(m/)

(mm) (mm) (kn) (ft) (ft)

1.080

36.0

229.3

5.5

42.6

.86b

232.1

2291

12.14 142.0 141.3 .6148

230.3

,

229.0

_H

9.3

141.5 r 141.3 .1432 _H

228.9

228.3

H

6.,2

141.3 141.2 O

221.2

227.2

0 141.0 141.0

3.3.. Determination of the resistance- and propulsion coefficients

The description

of the balance between longitudinal resistance and the propeller thrust is.

partly based on some full-scale measurements of rpm at

15.5

knots: and partly

on the propeller-characteristics. From these data the wake fraction was derived

while for the lower spe,ed', caused by manoeuvring, this wake fraction was consdered constant.. From the fact that during a manoeuvre the powei' does not

change, the increament of thrust and the decreament of rpm could he calculated

using the propeller characteristics For comparison purpose also the

full-sc:ale measurements of these quantities are given in figures 3' and 14 Concerning the other initial speeds , 12.14, 9.3 'and 6.2 knots, a linear relation between these speeds' and 'the rpm was 'adopted (aee figure

5).

The increaxnent 'of thrust and t'hedecreainent of the rpm 'durng:manoeuvres with. these initial speeds were determined' as this was donê for the 1_{5.. 5} knots

initial speed.

in fgur.es 6 and 7. the calculated', thr.us.t nd torque: are:.piotted' i,b the initiai. conditions

It can be 'shown' that on a straight. course a para] olic. reiatdon. between' the thrust and the forward speed exists,,. provided the 'thrust-curve. of the: propeller diagram is liÌearized' in the range. of interest. Consequently the

assumption 'of a parabolic relation between the longitudinal resistance and

the speed is equivalent, to the assumption of 'a speed indeendan,t 'thrust-deduction fraction. This' number was estimated to b:e .20. an this basis the resisance

coefficient X was calculated:

X - 514.

io

while, for the, effective thrust-'increament coefficient was found:

XT ' -2..

ito-5.

H In the next 'table the computed rps, 'concerning the :varibus.condît'ions,, are 'summarized'.

it is noted however that these data are corrected for the differences between the full-scale and model propeller

(u

218 - B

_:5.60'

res:pectively)'o

TABLE II

The rps-values, adjusted during the modeltests, are not the.. same .as given in.

this table however, as originally these values were based on the full-scale

relation between thrust and speed as given in [i]. During the analysis of the test data this rélation appeared not to correspond with the full-scale data of rpm and

the propeller diagram,. consequently nor did the rpm-U relation. This is shown in figure 8. From this figure it. is assumed t.hat the full-scale

speed-measurements concerned are not correct.

As, in addition,. the coefficients, which descrile. the balance between resistance. and thrust during, a. manoeuvre based bn this relation., resulted in very large

differences between the computed manoeuvres and those executed on full scale, the linear relation between rpm and spòed was. adopted.

The còëfficients of the force- and moment components Y2 N2 and X2 have been

corrected. as far as necessary, which was possible because their relation with the rpm was known from the model experiments.

21 -U = 1.080 rn/sec o, . U .861 rn/sec o U .618 rn/sec o U = .32 rn/sec o U u rps _U u rps U u rps U u' rps 1.Ó80 O 12.85' .864 -.20 12.30 .86I 0 '110.23 .61'8 -»40 11.80

.68

-.25

9.70

.6148 0

7.73

3.14. Determination o the rudder-speed U

The quantity U has been derived from

the straight-line tests with constant rudderangie. These tests. were executed

for the ten combinations of speed and rps, while in each case the rudderangie

was varied from 36 degrees port to 36 degrees starboard. with steps of 9 degrees..

A or.l descr-iption of the watervelocity near the rudder has been based on the

irnpulse-theo.Y with respect to the propeller. According to this theory the

water velocity at a certain distance from the propeller disk can be written as:

U =V

_{R e}

+iiC'

a 22

-I

(5,)

Ve = U (';l+Ut '):( i-p,)

where.'the prime denotes the; non-dirnensionalising with the,, in'itial. speed' U...I'f'

t'he curves of the.pro.pelIertorque'and -thrust'

arê linearized, in the. speed range

of interest the expression for C.

can he written as' follows:

a

(ct/v' +

1)2

=

i

+

a2! A

a1,! A 2

(6)'

where the coefficients a1 and a2 describe the. linearized torque and thrust,.

Using the thrustdeduction fraction iji as derived in c.apter 3.2., the values

of Ca can be calculated for each of the ten speed-rpm combinations, given in

table II. If these values are applied to equation. C 6 ) the. unknown speed UR is replaced, by the' quantity which originally. i'ndiá'at'ed the distance from the propeller disk. It must be tioted.howeve'.that'in't'his case the, deviations

due to the assumtions used, 'culminate at' the' computation 'of p., sot'hat this quantity rather has, to be considerd' a .calcuÏation-qi.antity t'han.'an indication

för the distance' between' rudde.r. and propeller. The

saine is to. 'be applied' to

the. qüantity UR

Substitution of the expression for tiR into the formal description of the 'side force and moment. measurements at v '= .r =

o, for each of the rudderangies

applied., a value of p was, obtained. In the next figure the products p has been

FL 6

t

.2

.2 .4 .6

-I6I(rad)

Using this valué of i the values of UR were calculated. and plotted in figure

versus the relative speedloss u

infigure 9,for the ten.combinationsof. speed and rprnthe measured rudderforces and moments are plötted on the basis of while the rudderangle. is .a para-meter.

The values of Y and N were too small to. distinguish between star- and dash

a a

components. A mean value, derived from the swaying tests and the present tests is obtained. For the computation of the rudder coefficients the measurements

3.5.

Determination of the remaining coefficients

3.5.1.

The static sway tests

These tests, executed at the ten combinations of

speed and rpm, provided the coefficients of the following variables:

- 2I -2

v,v ,v6,v

ó 2 ii V

,v6

while also the Y and N

a a

In tableascheme of the

Using the testresults at6

mined while computing the

first mentioned, were .eons'idered known quantities.

In figures 10, 11 and. .12 the.sidè:fòr.ce, and moment.measuremen.s.concerned.are

plotted.whiie the longitudinal .force:measurementsareshown in fgures 20, 21

and 23.

As no information was available involving the influence of the rudder s.peed the dashcoefficients concerned could not be deterrnined.

(Y- and N equation.)

(X-equation)

coefficients were determined. test program concerned is given.

3

o the coefficients of y , y and y are

Q indicates that test has been executed

- sin 3

TABLE III (STATIC SWAY TESTS)

Actual Speed

1.08

rn¡sec

rn

.86

¡sec

rn

.65

/sec

- rn .43 ¡sec. R p s

12 38

_{11 95}

10 29

_{11 5+}

_{9 85}

8 31

_{11 20}

₉₅₃

_{8 00}

₆₅₀

V

+°

+80

+12°

±13..2385

.0698

.1392

.2079

e o e e e e e e e e e e e o e .e e e o e e 0 e e o e e -e e O e e e e e e e e e Combined drift and rudder tests 161

9 18 27 3691827 369 18 27 3

18 27 369 18 27369 18 27369 18 27 369 18 27 369 1827 369 18

_{27 36}

+

-000030e

e e e

eeo

oo e e e e e e e e e e o e e

e eoee e

e

e ee

o + -e e e

.oe

00

.090

e e eo e 03 e e 0O o e

ese

e e 00 e e 00 e e oe 0 e ee e e eo O 8 09 e .e OS 0 0 .0 .0 00 0e 0 0 0 e 03 03 0 e 0 e O e

3.5.2.

Oscillatory swaying tests

These tests were mainly executed to determine

the lateral added mass erfect. Only two initial speed conditions are considered while the influence of a change of rpm appeared negligible.. Consequently Y and

N. are set to zero. The range in which the ncndimensional acceleration. L/U2 was

V

changed was extended to . 25 though an estimation of the maximum full scale

value is about

.15.

Nevertheless the measured force appeared linear with the

acceleration in the whole range.

in figure 13 the data concerned are plotted where the speeds àre considered

para-meter.

-3.5.3.

Oscillatory yawing tests with constant drift- and r.udderangle

In .general these tests. were also. éxecuted for the;'ten.condit:ions given in table II.

The amplitude of the charact,eristic variable r was varied between .05. and 70 corresponding with turning, radii of: approximately. .20 and 3 shi.piengths

respectively.

The choice of the oscillator-frequency has been based on the following

considera-tions:

-Based on the tankwidth available and the properties of the oscillator the nondimensidnal frequency y = wU/g, which is the leading factor

deter-mining the oscillatory part of the wave pattern, was kept as small as possible.

Concerning, the influence of this variable the reader is referred to

[9]

The forces involved in the lowest speed,Hcorresponding with-a Froude number

.07.,

had to be reasonab3y measurable.

In order to judge the frequency rangez used at these oscillatory motions, the quantity .271 /w can. he used, indicating the nimber if ship lengths sailed during one period.

The restricted tank width and oscillator-amplitude involves a disagreement between the forced motions of the model and full-scale manoaivres e.g. sinus-response tests. These full-scale manoeuvres involve, rather large amplitudes in

the range of practical fuli-scále frequencies. It is not'known however how far

this discrepancy between model- and full-scale manoeuvres influences the hydrodynamic derivatives. In references

[2]

and [io] some more details

concerning this matter are discussed.

-N)

TABLE IV (YAwING TEST)

e Indicates that test has been executed

i) test only executed at = = o

L L

r =-.r

w

-.w

U U g

X X

= number of hiplengths sailed .uring one period of oscillation.

Actual Speed 1 08 rn/sec _{86 rn/sec} 65 rn/sec _{43 rn/sec}

-R.p..s. 12.38 11.95 10.29 11.5)4 9.85 8.3)4 11.20 9.53 8.00 _6.50

r*

o 2ir I .05 5.03 .037 .10 3.53 .053 .15 _{.2.87 .Ò65} .019 .019 .019 .20 _{2.47 .075} .0)48 .0)48 _{.027 .027} .027 .012 .012 .012 .012 .25 2.20 .013 .013 .013 .013 .30 1.99 .060 .060 f033 .033 .033 .015 .015 .015 .015 .35 1.83 .016 .016 .016 .016 .40 _{1.70 .070} .070 _{.039 .039} .039 .017 .017 .017 .017 .50 1.51 _{.079 .079} .044 .0)44 .044 .60 1.36 .087 .087 .049 .04.9

0491)

.70 1.24 .096 .096 .05)4 .05)4 .05)41) 1iftang1e (degr, 0 4 8 12 04 8 12

b 4

8 12 0 4 8 12 0 4 8 12 0 .4 8 12 0 4 8 12 0 4 8 12 0 8 12 0 4 8 12 rudder- O

_{eo e}

e e e e e e e e e o o e o o e e angle

120

8 e e o o o o o e (degr.) 18 24 e e o e e o e e e e e

eeee

e e

AAAA

e e

In table IV ascheme.of the complete yawing program is given.

The measured forces and moments concerning these tests are to be divided into

three components:

a) proportional to the. angular-velocity (sine -component)

b.) proportional. to the angular-acceleration (cosine -component) e) the constant.component.

n the next table the various components are summarized while: the ariabIes concerned are mentioned in the sequence of determination.

TABLE V

Due tO the citeria given in the preceding chapter, conceThing the ranges of

oscillator frequencies and - amplitudes, the maximum value of the angular-acceleration amplitude exceeds the corresponding value ever occuring at the

full-scale ship,, the latter being estimated about 1.3.

Consequently the coefficients concerned have been determined, in this full-scale range the more: so as outside this range 'the. model experiments showed a conside-rable. nonlinear effect.

in figures 14 upto i9 the 's'ide. force;and moment measurements are plotted,, while in figures 22., 2I and .25.the coi'responding..longitudina1force measurements are given.

sin O component

cosine

comp. _Hconstant component variables.

varied Y, N X Y, N Y, N X i

r,r'

2 r.v

ró

rv

r6

' 2., 2 L'r'/U X : ' . . .2

vr

tSr 'i2' r , r .

r,ó

r, v,tS

3.6.

Some experiments with a small model (a =

too)

For comparison purpose the results of a restricted number of tests with a small. modél are given.

These tests have been executed before those with the larger model. _{Because of}

the very low speeds involved the results are considered not _{very trustworthy.}

The testprogram consisted of:

static sway tests

oscillatory swaying tests

oscillatory yawing tests (without rudder- and driftangle). The influence of the s.peedreduct ion and consequently of the thrust increaznent

was not examined _{in particular. It was assumed that the hypothesis mentioned in}

chapter 2.4. would hold,, which means that the tests only had to be execùted for the. initial speed consitions.

in.. the next table. the results of thesetest.s _{are 'summarized and are compared.} with those of the large model.. in figures 26 _{upto'.3.5 the measurements are shown.}

TABLE VI

29 -Cbeffj.cjen.tsof small model (1:100)

important coefficients _{less, important coefficients}

!coeff.

1:55

model H

1:100

L , model diff. ¡o , coeff. 1:55 moael 1:100 mouel diff. o'

_/0

Y'

_l797

-16314 L

-8867

_-11911

₃₅ Ï -m r - 7711.

- 863

, 12 , Y rrr .+ 14014

+ 303

25

+ 330

+ 291.1. 11 '

-

-

Ito 16 N

- 1473

-

1473 ₀

_N'

_{- 620}

_{+ 360}

150 N r -

252

- 200

21 N_.rrr - 270

.- 323

20

-

1614

- 132

19 N + 27 +

23

16 m -Y _L

zr

+2277

+ 128

+2303

+ 111

1 8 . N r .- 140 _H

-

65

-

27 , - 28

33

57

As follows from table Vi for the important coefficient's the magnitude of the

differences between the coefficients o the large and the small model is about lo percent while this percentage for the less important coefficients is about 25.. The importance of these differences is partly shown in table VII,,

in which the results of a turning circle manoeuvre are given computed with

the 1:100 model coefficients, completed with some of the large model.

TABLE VII

Concerning the range of variables applied with this model It is noted that

nearly the same maximum values of rudderangle, driftangle and angular velocity

were adjusted. The difference between the frequency-ranges of the two mòde'ls

..

2ir

is expressed by the two quantities ¡w , denoting the number of shiplengths covered during one period of the oscillation., and y which quantity governs the wave pattern during the oscillatory motion.

The first quantity varies from 3.0 at r = .10 to 1.1 at r = .50 which is

nearly the same range as is applied for the large model. The maximum value

ofat r

= .50 (.17). is about two timesthe.'corres.pondingcvalue of the large

model however. Nevertheless.. thisvalue. is. .considereds.ufficiently.low as to. avoid disturbing influences of this parameter.

Putting theresu]Lts of the two models in. the.lIght.of the:,hpothesis:, mentioned.

in chapter 2.1, it appears that the.hypothesis:holds forboth models serately

but not ifcomparing both. models. Asinboth casesthe Froude numbers had the

sante values the differences between the corresponding coefficients can be traced

to the small Reynolds number of the small model.

ôo = - 19°- .

scale 1:100 _1:55 full scale:

advance (m) 993 985 972 transfer (m) 665 687 660 tactical 'disin. (rn.) _{1255 1276} 1233 'diameter (m) 1086. _.1071 1100 r (degr.,/sec.) _.45,3

.66

..J90

Uc(kfl)

_8.I8 8.59 9.10 ?(degr.) . 10.5 9.9 9.5

31

-3.7.

Some remarks concerning the computed coefficients

it is found that the

rpm -effect on the linear tems is very small, compared to the magnitude of

these terms, while concerning the non-linear terms it was, not possible to

distinguish between the normal scatter of the data and this rpm-effect, due

to' the restricted accuracy of the measurements and the relatively small values of thus non-linear terms. This. does not apply to the pure rudderangle-dependant terms of course, the. change of which with rpm is' considerable.

Another important result is the usefulness of the hypöthesis., concerning the

proportionality of the forces with the square of the instantaneous speed, and

the characteristic varables v and r. This is clearly shown e.. g. in figures

10, '11 and 11L. Though it has also been tried to distinguish between the results concerning the four initiaI speeds [i]!, it .foflcws from the' figures. just

mentioned. that the. differences, .jhjch;'couId' be considered a. Froude-number effect are. not significant however.

Iñ tables V1i1DC&id X in the left.colurnns the'coeffic'ients of the set of

equatiOns

(3)

are given while in the right. columns the corresponding 'coefficients of the equations ('LL) are summarizéd', the latter being derived from the preceding ones. This derivation is partly based on the following approximations':

(1+u ) = .9LLO + i..LL'oo u

1'

= .837 - 2.300 u i+u

= .709 + .632 u

The accuracy of these approximations is:shown inf.igure 36 in. which these quantities are plotted. Concerning, the signs

of.

the coefficients in these

tables

the rule holds' that ali 'terms are transported to the right hand sides

32

-TABLE VIII

These coefficients include the ship's mass m'. LATERAL FORCE COEFFICIENTS (io)

"Star"-model

e9(3a)

"Prime"-modeeq,.lia)

var. Cvar. var. var.' Cvar.

varu

, C'var.0

-1797 0 y'

-V97 v'u'

_-1197

r _- o r' i) r'u' - 774 o +330 6 + 23I óu' + 208 -8867 Ó _V' -71423 v'3u' +2039h

r3

+ 14014 0 r'3 + 339 r'3u' - 930 o _ - 314 _{63u' - 30}

vr2

-2208 0 v'r'2 -18148 v'r'2u' +5078 +

₂₇₇

0 v'62 + 277 v'62u' + 277

rv2

+2562» 0 r'v'2 +21145

r'v'u' -5893

r62

O O i"62 o r'62u' O

6v2

+ 652H 0 6v'2 + 652 ôv'2ìf O

6r2

+ 218» 0 6r'2 + 218 ôr'2u' O y r 6 + 1412 0 v'r'ó + 1412 v'r'u' O H + 140 62 + 28 62u' 4 ₂₅ 1 - 114 0 1 - 13 u' - 20 Lv,02 1 ₂₂₇₈ O ) 2278 _{'u' O} X

L r,2

- 65 0 r' _{- 65 r'u' O}

TABLE IX

1These coefficients _{include ship's moment of inertia I'}

=33-MOMENT COEFFICIENTS (x 10

"St ar"-model

eq.(2b)

"Prime"niodei

eq.(3b)

var.o

M'

M

C

var. Cvar. var. C var. var. u, CTvar.0

- I73 _O

_y'

kî3

vu't

73

- 252

1 0 _r' - 252 r"uT - 252 o

-

1614 o

- ii6

:0 u' - 103 vM3 _{- 620 0}

_V'3

_H -519 v'3u! +11425

r3

- 27O: O r'3 - 226 r'3u' 1+ 620 O _i- 2 o _i- 19 &3u' 17 + 395 O v!r!2 _{Hi- 331 v'r'2u' - 908} - 69 O

V'O2 H_

_{69 v'02u! -}

₆₉

-2191 0 r'v'2 '-18314 r'v"2u' +50140 O O r'02 O r'02u' o

-

235 0 0v'2 - 235

ó'y'2ti'

O

- 1014

0

Or'2

- 1014

ôr'2u'

o

- 232

0

'v'r'O

- '232

v'r'óu'

O O

-

12

o2

8 62u' 1- ï O O i 'O

u'

0

Lir/u2

_-

_° ' - 14o !u" _'0

1L2,2

1/

- 128

-

128

''U

_H O X L

- 31L

-TABLE X

LONGITUDINAL FORCE COEFFICIENTS

Cx io)

"STAR"-MQDEL

(eq.2c)

"PRIME"-MODEL (eq.3c)

3

var Cvar var var' C'var var'u' var.0

J + 88 o y'2 + 88 v'2u' O r O O r'2 O r,2u' O i) 0

-177

i) - 125 62u' -112 y r +2149 O v'r' +211L9 v'r'u' O r3 - 50 0 r'6 - 50 r"óu' - 50

6v3 + ii7 O 6v' + i17 6v'u' +117

Lù/u21)

x

-1329 O i) -1329

''

o

resistance -. and thrust coefficients

- 51t _{u' - 133}

u'U'

-

514

L

-

25

i

l4 Computer progrwns

14.1. Least squares analysis of measured data (IBM 360/65 -program nr. SBSLX M03)

This program has been developed to compute the coefficients of a fourth

order polynomial of four variables: p

k i

m

n

F(x1, x2, x3, x14)

>j C X1 x2 X3 X14 t=1

using the least squares criterion. Herein is p 70 and k + i + m + n 14.

If some of the coefficients are already known they can be given and the

corresponding terms are subtracted from the measured value of the function.

If the distance between a measurement and the computed value of the polynomial exceeds two times the R.M.S.-value the data concerned are dropped while the coefficients are computed again. This "data points selection procedure" prima-rily serves to locate measuring - writing - or punching errors. The factor 2

used in this criterion has been found. experimentally. In statistics usually

a factor 3 is applied, though it has been found that if the number of

measure-ment is relatively small even large errors are not located in that case.

This data points seloetion procedure comes into operation again unless:

the number of selected data exceeds ten percent of the total number,

the R.M.S.-value is already smaller than a boundary value given

beforehand.

In the next stage the maximum value of each term is computed. If these maximum contributions of a number of terms is smaller than the boundary value just mentioned the procedure is repeated, though without the coefficient the maximum contribution of which was the smallest.

Also this ticoefficients selection procedure" is repeated until the contributions of all terms remained are sufficient.

Further the standard deviation and the correlation coefficient of each

coefficient are computed. The first quantity is plotted in figures 37 and 38

concerning the large and the small model respectively.

-14.2. Solution of the d'iferential equations. (IBM

360/65 -

program nr. SBSLMò2)

In this computer-program the differential equations are solved for given

time-de,1 endant rudd.ersi,gn'als, where the Runge-Kutta procedure is applied. Two cas;es are considered. In the first the rudder rate is constant or zero while in

the second a sinus-oidal rudder input can be given to determine the

frequency-characteristics of a. ship.

The output-quantities can also be required on punch-charts só as to obtain the

input for the coefficients-program M03 to determine the coefficients' of a mathe-matical model with 'a rêduced number of coefficients.

Further for each step the value of each term öf the set öf equations 'of' motion is computed which' enables to get an insight in the importance of the various components. An illustration of this is given in ±.igu'res

_{39, :140}

and '141.

Another way to observe, the process of a manoeuvre is to. 'linearize the equations

of motion at each step to the set:

= a1, 1.Av + 'a12Ar + a1,3A.6

i- aAu

A; = 'a21.v + a24r + a23A& + a214Au

Au = a31Av + a32Ar _{+ a334t5} +

Using the coefficients of this .set of euatioxis. the stability of the system can be observed.. The time-constants o.f the system can be found from the roots of the set:

. 36

-a31. a32 a314-À

An example o' the change of these constants,, .i'ndict'i.ng the change of stability, during a turning circle manoeuvre,, is given in figure 42.

Concerning the time in'ter\aI between two steps. of the computation, for all manoeuvres ten steps per .shi.plength., based on the initial speed,, was applied. The time. intervals following from this ar.ê given in the next table:.

TABLE XI rn/sec _{At (sec)} 8.0 2.7625 6.8 3:.2500

.6.

3. 914614 14.8 4.60141 3.2 6.9063 loo)

a11-A a1,2 a1,14

a21 a22-A a214

5.

Comparison of Computed and full-scafl manoeuvres

5.1.

Turning circles

The principal data of the turning circles are summarised

as follows:

TABLE XII

The results of the computed turning circles are shown in figures

h3 upto 16.

As follows from these figures the (final) rates of turn are somewhat smaller than the full-scale values though, combined with the final speed, the turning

diameters agree very well however.

For comparison with the spiraLmanoeuvre results' some additional turning circles

have been computed. Ouly the final values' of thevarious, quantities .ha'e been used to obtain complete curves.

The computation of the. ₃₇ degrees port rudder turning.circle hasbeen repeated

omitting certain coefficients,, which significancy was 'very little, according to the model tests.

The results are shown in figure _45.

As, appears from this figure these. coefficients have also little importance in the mathematical model.

-

37

-TURNING CIRCLE DATA

nr. 9A 9B 9D2

ó(degr)

+

37.0

-' 19.0

rpm H 1:00.0 1:00.0 100.0

U(m/sec)

ß(o)(degr) 8.00

.358

8.00

+

.358

8.00

+

.358'

p(o)(degr) -

_.858

.+

.758

-x(o)(m)

O

-

.06

+

_.09

y(o)(m)

+

.03

+

.03

_'+

_.37

i(o)(degr/sec)H-i- .10 -

.20

+

_'.05

( t! )

2.500

2.500L

2.500

5.2. Zig-zag trials

In table XIII the principal data of the zig-zag. trials are summarized. Concerning the initial conditions, only the course i. was considered while no other data were available. The values of the rudder rate. of turn

were derived from the data and figures given in [i].

The computed zig-zag manoeuvres are plotted in figures 7 upto 51 and compared with the fuJl-,s:cale measurements. The overswinging angles are somewhat smaller,

but the mean period-times agree very well. These two quantitites are plotted

.j figure 56.

Concerning the initial speed conditions of these, manoeuvres the adopted linear relation: U0 = 1.2.5 X rpm has been applied, while the rpm values for both full scale trials arid computations.are the. same'.

-Table XIV. Zig-zag triàl data

TABLE XIII

I)_{The manocuvres concerned have bccn cornputcd with the-accuracy of these data given here.}

initial conditions i lE III IV V VI

nr. rpm (J

'p-

o 'lex Ò (io 'l'ex v'2

oip

¿ (5ø 'Pèx

--

Aß _'l'ex.

(mis) (degr) (degr/ (degr) (del,r) (degr/ (dcbr) (dcl,r) (degr/ (dcgr) (dcl,r) (debr/ (dcgr) (degr) (degr/ (degr) (degr) (dcgr/ (degr) (dea)

sec) sec) scc) sec) sec) sec)

7A 97.5 7.80

-2.6

2.108') 19.0 -20.8 3.105

-19.0

14.-2 3.850 18.2 -19.6 3.482 -19.2 19.9 3.591 18.6

-18.6

3.482 -19.3 9.5 7C 97.2 7.78

- .4

2.500 8.9 -15.7 2.375

- 9.7

23.2 2.436 8.8 -17.2 2.632

- 9.9

22.4 7D 98.1 7.85 0 2.982 29.3 -16.9 3.929

-31.0

21.1 4.522 28.5 -16.1 3.719 -31.3 21.5 4.000 29.2 -16.2 3.067 -31M 2.-5 7E 87.0 6.96 0 - 1.020 9.5 -22.8 1.681 -10.1 23.5 1.887 9.3 -23.0 2.330 -=10.3 24.0 1.944 9.3 -17.8 2.143 -10.2 20.0 7F 87.0 6.96 0 2.635 19.1 -16.7 3.565 --19.3 20.7 4.149 18.7 -16.0 3.023 -19.9 16.1 4.115 18.4

-I62

3.185

-l9.4

20.0 7G 86.4 6.91 0 2.305 29.6 -19.-5 3.041 -31.3 19.-5 4.020 29.2 -19.7 3.400

-309

20.1 4.727 28.7 -1.4.3 3.199

-30.9

20.0 71 675 540

- 5

2232 ISS

-200

2950

-192

192 2875 187

-198

23l

-195

229 2980 188

-192

2760

-192 200

7L 59.7 4.78

-

.7 3.488 29.3 -20.8 4.050

--0.2

19.7 4.997 28.8 -22.0 4.173 -30.5 19.5 4.355 28.8

-20.6

4.539

-30.2

0

:6. Final remarks

To judge the result Of the modl experiments discussed in this paper, figures 55 and 56 may serve in the first place. They provide an overall

picture of the principal parameters of turning circle-, spiral- and zig-zag, triais:

Figure 55a rc against 6 relation between turning circle diameter (rca -and rudderangle.

Figure 55b _Uc against 6 final speedreduction of turning circle manoeuvres.

Figure 56a t against relation between period tmes of zig_zag trials and

i:nitial speed.

Figure 56b ijmax/6 against

176 _{reation;between overswinging. angle and, nominal} rudder angIe.

From these figpres, in whïh the:computed:quant:ities. are.!compared with those, measured, during the fullscale trials,..it appears that it ±s.possibie. to predict

the manoeuvring properties. of theship concerned by means of oscillation tests

with reasonable accuracy.

1t must be noted however that both full-scale data and computed data have their uncertainties. In particular this may be important if x-y plots are compared, because these plots are obtained very indirectly.

Concerning the zig-zag trials it is found that a littlechange of the "execution course" has a relatively large effect on the maximum course deviation and the

peribd time. if we finally keep inmind thatthe determination of the«

manoeuvring pröperties of a ship: via, horizontal oscillation tests is rather

indirect at least, while the motions of the model diring these tests are rather

:unrealistic, from the four figures above mentioned,, it may then be concluded

that if scale effects playa role inthe:present investigation their importance

'is not verylarge and of the.samemagnitude. as the accuracy of both the

7. Recommandations

The mathematical description of the manoeuvring properties is based on a hypothesis .which describes the relation between the forces acting on the ship and the forward speed. The application of this hypothesis is fully justified by the present model-experiments. This does not mean however that

all hydrodynaic effects which play a part in this mathematical model are

realy necessary -for a sufficient: description of the horizontal motions. Therefore it might. be interesting to find: -a simple mathematical model, which properties

are not to give and accurate description of the hydrodynamic phenomena but

8. Appendix

The relation betweên the second-order similarity parameters and

the velocities and accelerations

The second-order similarity parameters are:

3E 3E ax, ay 3E o o at

,

-as3 as3 ds3 as .2 2 i a a

y0

.a2 a2t

2'

2 ' 2 ' 2 3 3E 3E 3E as as as as * X0

Choosing the coordinate system at the moment of.cornparison in such a way that:

(i)

3E

ay

then from the equality of °

andthat:of

3E

3

as sothat also V As = - sin ß vi V2

from (3) it follows that

1 2

_14 *

Yo

2

a

From the equality of

and

that of

it follows that: S as

a1

2 M

3

as

as

at3

From the equality of

-i

it

follows that

as

i

,ti

1'

at

T

3

as 2

as

UL.

and

because i 2 1

(8)

:UL

ap1

it. follows from the. equality -

= -

.

(9)

as.

L.

L..

that

li

2 2

(110)

1ir

u2

Fibm'eq.uality (6..): together. .wiJth.'t'he.equaiities:(T,), .(,8):,. () .and::(5'), it. follows

that:

L11

y1 L1Ú1

L2r2

y2

'L2

il,

2ThJ'

2 u2u2

U1

.1.

2 2 2

From the equality

of

together

with the equalities (T) and ('8) it follows.

that:

w

as

(and

2

a y o 2 M

as

2.

L1 r1 U,1 2 a

a2

where and are denoted as. r and' respectively..

at

theeqa1ity of

at2,

togetherwith'ÇT)and.'('8)itfoìlows that.:

sothat from equalities (ii) and (i2) it follows that:

2.

2.

L1v1

L2v2

L.

r1

L2 r2

.2. 2

and ₂

-2 U (13) -

143

-(6)

('7)