The lateral damping and added mass of an oscillating shipmodel

REPORT No. 65. S . December 1964

(S 2/73)

STUDIECENTRUM T.N.O. VOOR SCHEEPSBOUW EN NAVIGATIE

Netherlands' Research Centre T.N.O. for Shipbuilding and Navigation

SHIPBUILDING DEPARTMENT MEKELWEG 2, DELFT

*

THE LATERAL DAMPING AND ADDED MASS OF

A HORIZONTALLY OSCILLATING SHIPMODEL

(DE LATERALE DEMPING EN TOEGEVOEGDE MASSA VAN EEN HORIZONTAAL

OSCILLEREND SCHEEPSMODEL)

by

G. VAN LEEUWEN

(Scientific Officer Shipbuilding Laboratory University of Technology Deift)

Issued b the Council This report is not to be published unless verbatim and unabridged

CONTENTS

page List of symbols 4 Summary 7 1 Introduction 7 2 Measuring system 7

3 Particulars of the model 8

4 The test results 9

5 Calculation of course stability indices 17

6 General remarks 19

Acknowledgement 21

References 21

symbol typical non-dimensional formula L L2'

,r'=-',

V

/= 7

0 '12, YG 0,V0, V'o, 'o, TO, to U0 U0

Fn

%/g. L UR G g 11m1, Nmi Y1, Y2 Y11, Y21 8 1) F02 Wmax, 7max, max R0 L (== L9) T AR M M'=M/L8 Iz I5'=11/L5 definitions

coordinatesrelative tofixed axes coordinates relative to body axes

tEansverse velocity and acceleration of the origin Of body

axes relative to the undisturbed fluid

yaw angle, yaw angular velocity and yaw angular

accel-eration respectively

drift angle rudder angle

distance of for and aft fastening, point, and the centre of gravity, relative to the .-axis ( centre'line of the towing

tank)

amplitude of v, 1), ', i, , r, t respectively.

forward speed

estimated water velocity near the rudder distance of fastening points from G circular frequency

centre of mass of'the model' acceleration due to gravity

first harmonic'components ofmeasured'force and moment

respectively

'measured forces On for and aft fastening points first harmonic components of Y1 and Y2 respectively phase difference' between forces and accelerations very small value of velocity- or yaw angular velocity amplitude

very small value of cceleration- or yaw angular accel-erationamplitude

amplitudes of first harmonic components of forces Y11 and V21 respectively

phasedifference between the periodic motions of both the fastening points and the motion of G

half the' opening angle of a wave system, with respect to the direction of'the forward 'speed'

radius of turning circle . length ofthe model draught

rudder area

mass of the model

mass-moment of inertia of the model with respect to

z-axis

11L wU0

Partial derivatives of hydrodynamic forces and moments with respect to linear or angular velocity or acceleration.

symbol typical definitions

non-dimensional formula

- Yy

- Vi = -

added mass coefficient

1',,

- 1',,' = - Y,/4UOL2 sway damping force coefficient

-

- Y,' = - Yp/oUoL3 cross-coupling force coefficient, due to yaw damping distribution

- Y, = - Y1/4L4 cross-coupling force coefficient, due to added mass mo-ment of inertia distribution

Yj

1'6' = Yö/UO2L2

LI 2L2

- Yd* = - Yô' °

rudder force coefficient (JR AR

- N

- N' = - N/4L6

added mass moment of inertia coefficient

- Np

N' = - N/U0L4

yaw damping moment coefficient

N - N' = - N/4gL4 cross-coupling moment coefficient, due to theadded mass

distribution

- N,, - N,,' = - N,,/,3IUOL3 cross-coupling moment coefficient due to sway damping distribution

- Na - No' = Na/QUO2L° rudder moment coefficient

T1, T2 T1'

T1.-T2'= T2.

T

T'=T.

stability and manoeuvrability

indices as proposed by

THE LATERAL DAMPING AND ADDED MASS OF

A HORIZONTALLY OSCILLATING SHIPMODEL

*)

Summary

For a practical range of circular frequencies forced sway and yaw tests were carried Out, to obtain the values of mass and damping and their linear cross-couplingeffects.

Cross-coupling damping effectsare considerable, buttheapparent mass'centre of gravity appeared to be located close tothe

model's centre of gravity Ingeneral the results agree with:those of PAULLING [1]. The influence of the rudder and the propeller

in the low frequency range is shown clearly, except the influence on the moment of inertia coefficients. The results of the combined sway and yaw tests in general agree with the comparable pure motion results. The differences found will partly be caused by higher order cross-coupling effects, partly by the possible error of the test data. It seems that even for higher frequencies linear equations of motion may be applied.

As a secondary result the stability indices could be determined from the obtained values of the various coefficients. For a restricted frequency range negative stability was found. This result may be related to the critical state of the wave pattern generated by the oscillating model.

1

Introduction

The purpose Of the fOrced pure sway and pure yaw tests is to obtain some information about the effect of the frequency of periodical lateral motions on

added mass- and damping coefficients. The in-vestigations are divided into two parts: the first

part concerning the very low frequencies as occur

on a steered ship jn still water, the second pa't

extends to relatively high frequencies which may

occur when a ship runs in waves.

To separate the influences of the rudder as a part

of the lateral plane and those of the running

propeller, all tests were carried out three times viz.

without rudder and propeller, with rudder and

without propeller and finally with rudder and

propeller. The last series was performed for three numbers of propeller revolutions.

In the low-frequency range the influence of the amplitude of motion was investigated partly.

In order to obtain some information about the

influence of the speed of advance two Froude

numbers were considered.

Starting from linear equations of motion only first harmonic components of the forces were

determined. It was questioned whether at higher frequencies the linear equations of motion would satisfy to describe the lateral shipmotions. So, as a check, combined sway and yaw tests were carried out. See Fig. 1. The results of these tests could be compared indirectly with the pure motion results.

Using the measured values of the coefficients

of the linear equations of motion the stability

indices T1' and T2' were determined for the whoLe range of circular frequencies.

*) _{Publ. no 23. Shipbuilding Laboratory University of} Technology DeIft.

Pure swaying motion ( = 0)

Pure yawing motion (v = 0)

Combined, swaying and yawing motion (v = - U0.

Fig. 1. Definition of'motions

2

Measuring system

First harmonic components of sway and . yaw

forces were measured in the same way as has been

done by GERRITSMA and BEUKELMAN for

deter-mining heave coefficients [2]. The system is based on. EULER'S theory for finding the coefficients of

the n-th harmonic. of a periodical function by

multiplying the function by sin nwt and cos nwt respectively. This procedure. was performed by a mechanical-electronical FouRIERanalyser Further details of this system are to be 'found in [3]. Fig. 2

gives a scheme of the scotch-yoke mechanism

with which the forced motions were obtained. The model was fastened at two points at 'equal distances from the centre of gravity G by, means of two strain-gauge dynamometers which, were sensitive in a direction perpendicular to the models

plane of symmetry only. The maximum capacity was ten kilograms. One of the dynamometers (aft) was. fixed while the other admitted some longitu-dinal sliding.

Consequently the pure yawing

motion had to be corrected for a slight swaying

motion (See Appendix A.5).

3

Particulars of the model

The model used was one of the Todd' Sixty Series with block-coefficient GB = .70

and made of

polyester. The main particulars are given in

Table I.

Fig. 2.

Table I

Length between perpendiculars' L5 2.258 m

Length on the waterline 2.296 m

Breadth B .323m

Draught T .129 m

Volume of displacement .067 m3

L.C.'B. forward "/2L .011 m

Lateral plane area (L.T) .291 m2

Rudder area AR .0043 m2

Radiusof gyration k'5, k' .25

Diameter propeller (Hadler no. 3376) D .0903 m

Number of blades z 4

1.10

'E'A-/DA .50

Service speed .928 rn/sec

Number of propeller revolutions

4

Test results

4. 1 Pure motions

The tests without rudder and propeller are carried

out within the frequency range .2 < w < 6.0 rad/ sec while those with rudder arid propeller were

extended up to w = 15 rad/sec. The service speed

tests at Froude number .20 were carried out for three numbers of propeller revolutions (n), the first of which corresponds approximately to the service condition of the full size ship. The other

values of n were each 10% higher. For Fn .30

no tests with propeller Were carried out.

4.1.1 Low frequencies

Within the frequency range .2 <w. < 1.0 the

partial derivatives of forces and moments:. (See

Appendix A. 1)

(YÉ),o, (Y)6,, (1'Q)e,o, (1'ij')e,o,

(N41),

(N),

Were obtained for very snal1 valuesof velocity ()

and acceleration () amplitudes. No usable data could be obtained for lower values of w, partly because of the restricted tank length, partly b-cause the forces were too small to be measured.

Fig. 3 shows some examples how the coefficients were determined. Because of the linear behaviour

in this frequency range one is tempted to extra-polate these values to zero frequency, in other

words, when C represerits one of the coefficients, to suggest that

limC=

10 5 0 4 2 Table II

Non dimensional coefficients for very low frequencies Pure swaying motiOn

Table FF1

Non dimensional coefficients for very low frequencies

Pure yawing motion

20 10 0 15 10 kg

()

swaying motion (Foicosei+Fo2cose2)'4 swaying motion F01sine1+F02sin 62 yo 2.5cm o y0=10.Ocm o V0 t',= 2.5cm o y0==10.Ocm -yawing motion a(F01cos1+F02cose2) 0 yawing motion a(F01sinE,+F02sin62) o -(kgm)4° .!sin ' y0=1.Scm o y0=9.6cm .sln o y0=lScm _kgm o y0==96crn

(C')8,,, M'-Y1,' -Ye' -Ne' -N0'

Fn-.

.20 .30 .20 .30 .20 .30 .20 .30 A .0207 .0204 .0148 .0 162 .00034 .00022 .0066 .0076 D .0222 .0231 .0187 .0206 .00025 .00002 .0064 M074

a

.0229

-

0222 .00048

-

.0057

-(C')8,,, 12'-N,p' -Nrn'

-Y'

M'-Y,1,' Fn -*

.0

.30 .20 .30 .20 .30 .20 .30 A 0 .0012 .0012 .0012 .0012 .0012

-.0029 .0032 .0034 .0O31 .0034

-.00039 .00039 .00039 .00039 .00039 .0090 .0082 .0076 .0101 .0094

-0 2 3 2 rad.tec rad,ec

Fig 3. Examples of measured quantities

However, the condition v = ü = = = 0 is

not of interest for the present investigation.

Fastening point amplitude (Jo) variation

ap-peared to have no effect when the drift angle am-plitude (go) did not exceed the value of

approx-imately 10 degrees and the amplitude of the

angular velocity (TO') was kept below the value .3

(See Appendix A.2). Both these boundaries are

only estimations and are in fact coupled with the

amplitude of the accelerations and angular accel-erations respectively..

Rudder as well as propeller influence could be measured fairly well in the low frequency range. Both seem to act like an added part of deadwood.

10 Fn. = .20 screw .85 kg Fn. = .30 .135 m rudder 1.55 kg rudder .92 kg 1.06 m bare hull 3.55 kg bare hull 5.80 kg Fn. = 301 bare hull 1.25 kg

In Table II and III the principal results are

compared. The symbols in the Tables are the

same as those used in the Figures namely: tests with bare hull,

LI tests with added rudder,

C tests with added rudder and propeller.

The moment of inertia coefficient (I5'N')

could not be determined with sufficient accuracy to establish the influence of rudder and propeller.

A mean value for all cases

is given. Roughly

speaking it can be said that the mass coefficient increases some 5 to 10% due to rudder and

propel-ler each. For the sway damping coefficient the

increase is about 20 to 30% for the rudder as well as the propeller. For the yaw damping coefficient

this figure is 10 to 15%. The behaviour of the

damping cross-coupling terms may be illustrated by Fig. 4, from which it follows that the point of

application of the damping force moves aft in

both cases. The forces in this figure are relative

to v = 1 rn/sec and t) = 1 rad/sec.

Comparison between some sway- and yaw coefficients

Let p and q denote respectively the sway- and yaw

damping forces per unit lenght and per unit of

local transverse velocity. Then the sway damping force is obtained as:

Yv.v = fpdxu

[p =p(x,

_Uo,oi)]

Fn. = .20 bare hull 1.46 kg 4.51 m 2.43 m rudder screw .42 kg .15 kg rudder .53 kg

(

1.13 m

and the sway damping moment as:

+'/.L

= /pxdxv

- 'f.L

Likewise the yaw damping force (without the

mass-dependent part)

and the yaw damping

moment are

=fqdxxv

[q = q(x, Uo, cv)]

-'f.L

N.h =[qxdxxp

-'/,L

This means when p = q that also N =

-4

So from the difference between N and

-in the low frequency range (see Figs 8a, 8b and 12a, l2b) it follows that the sway damping

dis-tribution along the model's length is quite different from the yaw damping distribution.

It must be noted that the total yaw-damping

force has to be divided into three parts viz.

(MU0X. Uo?,).

being the centrifugal components due to the

model's mass and the added mass for the x-direc-tion and the mass-independent part respectively,

the last of which is accepted to be q. As X>O

the difference between - N,, and - Y, will be still larger than the difference between - N and

--Considering the added mass distributions for

sway and yaw respectively, it will be clear that in

the same way the ratio No/Y will provide some

swaying motion yawing motion

Fig. 4. Influence of rudder and propeller on the sway- and

information about the differences between these

added mass distributions. HORN '[4] also reports

about this problem, noting that the coefficients

- N' and -

are equal in the case of hydrofoil boats.

In the case of an equal damping distribution for theswaying (p)- and the yawing

(q)-motionrespec-tively, the ratio of (Nj,/Y)" would be a usable

quantity, indicating the character of this damping distribution because [q.x.cix.x

y

-

V Jp..dx

-It

is questioned, however, whether in case of

different damping distributions this ratio might

also be considered as a "damping radius of

gyra-tiori". Despite these considerations in Table IV

the ratio (N/Y)" has been given which only

differs a little from the ratio (N5j,/Y)" because the

yawing moment, caused by the added mass -X

will be very smalL Also. the locations of the

mass-and damping forces are given in this table. The ratio Of (J-No/M- Ye,)" may be compared with

Table IV

Location of forces etc. for the low-frequency range

k2

5 10 15

radhec

Fig. 5a. Total and added mass for swaying motion (Fn= .20)

the radius of gyration of the model Which is 0.25 L (0,56 m).

4.2.1 Higher frequencies

With respect to the applied amplitudes yo the

higher frequency, range has, been divided into two

parts viz. 1

< 6' and 6 < w < 14. In the

first range the amplitudes for sway and yaw were

respectively 2.5 and 1.5 cm and in the second range for both the motions the amplitude was

.75 cm, so as to keep the drift angle- and the

yaw-angular velocity amplitudes below the values

described in part. 4.1.1. The distance between the fastening points was 1 rn. 1'he connection with the results of the lower frequency tests was checked

by overlapping tests The differences observed could be neglected. in view of the'general accuracy;

so the relevant graphs are given as continuous

curves. (See Figs. 5 up to 12).

In general the influence of the rudder and the

propeller in the higher frequency range could be fled fairly well. Within the range 10.4 r1p.s. < < 13.4 r.p.s. a change of the propeller revolutions did not affect the measured forces.

Though added mass and added mass-moment

20

10

0

Fn and

test,

. NV/YV Np/Y,,j, ' (N,p/1',,)'ia N/Y -N1/M-Y I.-N,/-Y1j (I0-N,,/M-Y0)'/' N1,/Y,,

symbol (m) (m) (m)

-

(rn) (rn) (m)

-.20 A 1.01 -2.43 .1.00 -2.46 .036 7.10 .55 .87 20 j .78 -2.10 .94 -1.85 .025 7.10 .53 .65 .20 o .58 -L88 88 ' -1.41 M48 7.10 .52 L25 .30 A 1.06 -4.51 .98 ' -493 .024 7.10 .55 56 .30 D ' 8l -3.49 . 92

-334

MOO 7.10 .51 MO WV01 .21 -Y X102

i:.

a tests without rudder and without screw a tests with rudder and without screw

M-Y M-Y M-Y

hgsec2/rn X 02 ratIo 1 hgsec2iw 4 g 4 5 10 15 a' radhec

Fig. 5b. Total and add'edrnass'for swaying motion ('Fn= .30)

2 4 6 a" X 02 14 2 20 x10 10

H

0

tests without rudder and wi hoot screw o tests with udder and without screw o tests with rudder and with screw 0

12

.or

10 5 15

--

radisec

Fig. 6a. Cross coupling moment coefficient due 'to added Fig. 6b; Cross coupling moment coefficient due to added mass distribution (Fn = .30) N; X 02 110 0 2 \ mass distribution (Fn = .20) lo 15 0) radhec x102 -N; io 10 6 10 15 w radAec wU0

IT

teats without otests with rudder

rudder and without and without screw screw

--wU0 - j t tests without o tests with rudder

o tests with rudder

rudder and without and without' and with screw

screw screw

-/

17' A tests without rudder and without screw

/'2 o tests with rudder and without strew

I I

(0U0 ,

g 4

/

/ A tests wlthou atdsts with rudder o.testn with rudder

'rudder and without and without and with strew

screw screw

(°Uol

A tests without o tests with

rudder and without rudder. and withàut

screw screw WU0 1 8- 4 -k A tests without o tests with -rudder a tests with

rudder and wi and without rudder and with

hout screw screw screw 2 4 0) 0)' 10 15 0) raddec

Fig. 7a. Sway damping force coefficient(Fn.= .20)

15

0) radisec

Fig. 7b. Sway damping force coefficient (Fn = .30)

15

0) radSec

Fig. 8a. Cross coupling moment coefficient due to sway Fig. 8b. Cross coupling moment coefficient due to sway damping distribution (F'n = .2U) damping distribution (Fn = .30)

2 0). 2 6 N' x kgsec2 10 5 0 0 N; 0 0 kgsecim 60 20 40 10 20

0

0

-r

x101 kg sec/n, 80 20 60 40 10 20 0 0 kgsec 40 20 0 kgsec 80 60 40 20 io

IN.. kgmsec1 6 4 2 0-I'N. x io 20 - 10 20 0

A tests without rudder and without strew o tests with radder and without screw o tests with rudder and with screw

2 4 4 N. kgm sec 20 10 x io 20 10 0 10 10

Fig. 9b. Total.and added.mass moment of inertia foryawing motion (Fn .30) 10'

.w

15 (0 rad,tec

Fig. Ila. Yaw damping mOment coefficient (Fn= .20) . Fig. lIb. Yaw damping moment coefficient (Fn =

I'

wU0 1 . .5 x104

I.

A tests without ' otEsts with --rudder and WI rudder nd withoUt-bout screw screw g 4 a tests without o teats 'with tests with \

rudder and without rudder and without rudder and 'with

screw screw screw wU0 1

I--=

a tests without otests with rudder and' w rudder and without

thout screw screw

/\

--a tests without a tests wIth o tests with

rudder and without rudder and without rudder and with

screw screw strew wU0 g 1

-v

atests without o;tests with rudder

rudder aed without and without

screw screw

10 15

U)

Fig. 9a; Total and.added mass moment of inertia for yawing motion (Fn = .20)

4 2 4 6

_...,.

- 10 .

'' .10

15

U)

rad/sec

Fig. 1 Oa. 'Cross coupling force coefficient due to added Fig. 1 Oh. Cross coupling force, coefficient due to added moment of inertia, distribution (Fn = .20) moment of inertia distribution (Fn = .30)

2 4 6 2 4 6 10 15 -cal 2 4 Y.. kgsec 0 Y kgsec2 x o 4 20 2 0 0 0 .0 kg m Sec ic 10 5 5 0

14 MU0Y kgsec 10 0 5 rad/sec 1 O 15 w00 1 g 4 wU0 1 g 4

a tests without rudder and without screw o tests with rudder and without screw o tests with rudder and with screw

\M_Y) x io' 10 (Sr 5 - 10 15 radhec

Fig. 12a. Cross coupling force coefficient due to yaw

damping distribution (Fn = .20)

of inertia respectively decrease to about 20% and

50% of the low-frequency values it should be noticed that the ratio of total moment of inertia

and total mass remains almost constant with

in-creasing frequency. (See Fig. 13). At the same

time Fig. 13 shows that the sway damping distribu-tion is subject to large changes when frequency

in--N I

!!

,///N\

\

i_ '1/ 'N

N

N

N---,J'\

_/

_N

--, I\,

5

A tests without rudder and wi hout screw a tests with rudder and without screw

Fig. l2b. Cross coupling force coefficient due to yaw

damping distribution (Fn = .30)

creases. Particularly at 0 = 1/4 g/Uo the point of

application of the damping force moves forward,

even to a distance greater than half the model's

length, which indicates negative damping aft

This phenomenon may be caused by the critical

form of the wave .pattern at this frequency. (See part 5). 4, / x lot

0'

curves derived from tests with rudder and with screw

Fig. 13. The location ofmass and damping forces for swaying motion and the ratio oftotal moment of inertia and total mass (Fn = .20) 4 6 2 15 radhec 10 MU0Y kgsoc 10 0-x1ou 20 10 Mr: x i' 20 10

in Table V the location of the various forces

and also the ratio (I'-N'/M'- Y')

concerning

the tests with rudder and propeller for Froude number .20 are summarized for the whole

fre-quency range tested.

4.2 Combined yawing- and swaying-motion

The definitionof this motionis given in the Appen dix (A.4).

According to mathematical considerations it is not to be expected that the results of the combined yawing and swaying tests will agree exactly with

the corresponding results derived from the pure motion tests. In the case of pure motions only

partial derivatives of forces of the form

and (F.

respectively are considered while

for the combined motion in addition, the deriv-atives of the form (F,,. etc. come into play.

(I-N)w'+

kgm 100

50

Fig. 14. Combined yawing and swaying tests. Moment coefficient in phase with yaw angularacceleration (Fn= .20)

Table V

Location of forces etc. for the higher frequency range

Figs 14,, 15, 16 and 17 show indeed that there are some discrepancies between the two results. Prob-ably these differences are partly due to the

deriv-atives mentioned above. On, the other hand the

accuracy of the tests will also have some influence.

4.3 Determination of the sway damping- and rudder coefficients with static tests (w = 0)

Sway tests with constant driftangle were carried

out with the model equipped with rudder for

Froude numbers .20 and .30 and alsowith running propeller for a Froude number .20. (See Figs. 18 and 19). The largest possible driftangle that could

be obtained with the oscillator was about 10

degrees. Within this driftangle range damping forces appear to be almost independent of the

various testing conditions as opposed to the cross-coupled damping moments which increase with

5

0

radhec

Fig. 15., Combined yawing and swaying tests. 'Farce coef-ficient in phase with yaw angular acceleration (Fn = .20)

w ,' rad/sec N,,/Y,, (m)

-N/M-Yô

(m) N,j,/Y,,j, (m) 1-No/-Y,p (rn) (i_N,e/M-'YeY" (m) .2 575 .048

- L80

7.10 5l9 .530 .042. - 1.85 6.74 .520 2 1.170 .009 - 1.60 7.36 .538 3 1.216

-026

- 1.58 19:20 577 4 350 -.095 -24.86 5.06 .541 5 048 -.109 4.63 4.03 .523 6 - .037 -.098 2.33 7.89 .539 7 -.041 -.057 2.28 ' -22.67 .566 8 -.030 -.022 2.98 - 8.18 .601 9 - .019 - .018 4.86 - 6.50 ' .617 10 .009 - .019 7.63 - 6.00 .612 11 .026 -.019 10.47 - 6.04 .601 12 .044 -.012 13.64 - 6.38 .597 13 .060 -.012 27.81 - 7.13 .593 14 074 - .012 -43.32 - 7.62 . 587

tests with rudder derived measured

and without screw

wuo g tests with rudder and

._...rneasuredwithout screw

Au4 'WI-Jo g 4 Q 2 6 (U radec

16

N-U0N

kgmsec

5

0

Fig. i6. Combined yawing and swaying tests. Moment Fig. l7..Combined yawing and swaying tests. Force coefficient in phase with yaw angular velocity (Fn_{= .20) coefficient in phase with yaw angular velocity (Fn - 20)}

o tests with rudder and without screw. Fis .30 tests with rudder and without screw. Fn .20 o tests with rudder and with screw. Fn .20

a) radeec

the Froude number and decrease on account of

.the action of the propeller. Qualitatively this

agrees with the results of the dynamic tests but quantitatively there is a remarkable difference

with respect to the location of the lateral force, due to the propeller.

In both cases the moment caused by the propel-ler is about 20 per cent of the total moment, but the lateral orce due to the action of the propeller is in the dynamic case about 16 per cent and in the

static case only 3 per cent of the total damping

force. This can be seen in Fig. 20.

UOY)+; kgsec U0 xio4 kgsec 10 5 10 5 0 .05 .10 .15 .20 0 00

Fig. 18. Static value of the sway damping force coefficient _{Fig. 19. Staticvalueoftheswaydarnpingmomentcoeflicient}

the forces in this figure are relative to v =1 mfsec screw .14 kg 2 dynamic tests static tests 4 4.50 m .67 m bare hull + rudder screw _{4.47 kg} .85 kg bare hull + rudder 491 kg

Fig. 20. Sway damping coefficients. Cothparison of static and dynamic test results (Fn .20).

tests with rudder arid without screw measured

derived

-

tests with rudder

.______measuredand without screw. derived

a tests with rudder tests with rudder o tests with rudder

and without and without and with screw.

screw. Fn .30 screw. Fn .20 En .20 .05 .10 15 .20 U0 x kgseclm 40 20

In Table VI the results of static tests and dy- that the rudder force as well as the rudder moment are almost linear with the rudder angle for values

of âr smaller than approximately 20 degrees In

this linear range the relation between the rudder force- and moment coefficients and the propeller

speed was obtained as a subsidiary result. (See

Fig. 23). namic tests are summarized.

Table VI

Rudder force- and moment coefficients were obtained with zero driftangle only and with the same set-up as was used for the pure oscillatory

motions. 'From Figs. 21 and 22 it can be concluded

Fig 2 1. Rudder forceasa function of rudder angle and r.p.s.

Fig. 22. Rudder moment as a function of rudder angle

and r.p.s.

2

From the propeller speed the water velocity

'near the rudder UR, could be estimated, using the

momentum theory. So' the relation between the,

rudder force coefficient and UR was obtained ex-pressed by the equation:

y

_Ya*.4eUR2.AR

In Table VII the results are given for a Froude

number .20. Within the investigated r.p.s.-range the above equation appears to hold fairly well.

Table VII

Comparison of non dimensional rudder fOrce coefficients

5

Calculation of the stability and

manoeu-vrabiity indices

The results of the pure motion tests can be used to find the exponents T1 and T2 of the equation:

Aie_/TI+A2e_t/T = r

being the solutiOn of the second order differential equation mentioned in the Appendix A.3, when

âr = 0.

From the expressions for T1'T2' and Ti'±'T2'

(Appendix A.3 eq. l3a, l3b) it follows that the sign

of T1' and T2' in the first place depends on the

denominator:

-Ni,'. -Y'--(M'-Y'p). -Nt'

Fn and

test

Static test results (in = 0)

Dynamic test results

(0.2 <w < 1.0) symbol

_-Ne'

(-Y'),o

(N')eo

.20 Lii .0205 .00608 ' .0187 .00645 .20 0 .0211 .00488 .0222 .00567 .30 .0208 .00794 .0206 .00737 -N (location rudder force) s( N6 static results tests -Yxs1 x104

110' 4

0 -40° _200, 10 200 400

with rudder and tests with rudder and

m n5=10.4r.p.s. n,11.4r.p.s. n,=12.5r.p.s. n,=1 3.5r.p.s. 3600 without with screw Fn .20 n (r.ps;) I U. (rn/see)

-

10 -Yã 10.40 0.980 2.11 2.16 '11.44 1.065 2.58 2.12 12.47 1.150 285 , 2.11 13.51 ' '1.240 3.30 2.11 Nxã x1o4

-4-.

4Q0 _{-20°c 20° 40° 360°}

of_____

_{tests with rudder and}

Fn .20

without

screw

Fn.30

with rudder and with screw

'tests

5 _{a n,10.4rps} Fn.20

'V

is=1i.4r.p.s. - n,=12.5rp.s. V -n=l35r.p.s. 0 5 10 15 (r.p.s.

Fig. 23. Rudder force and moment coefflcientas,a function of r.p.s. (Fn = .20)

18

Fig. 25. Wave system boundaries of a pulsating source

- U

I--

I

.20 tests without rudder and without screw Fn. = .20

.30 tests without rudder and without screw Fn = .30

.20o tests with rudder and without screw Fn. = .20

.30 0 tests with rudder and without screw Fn = .30

.20 o tests with rudder and with screw Fn. = .20

4 _{negative stability} positive' stabliity -180° _9ax 125°15'52° 044r 19°216° 351552T T .1 .2 .3

.4 .5y

----

1 2 b

'_'i

arcsin = osculatIng pressure point (ship)

\

\

' 4) qc70 I

.

\

_\

WJflqJ

.1 .2 .25 .3 .4 .5 .6 7 =

Fig. 24. DirectiOnal stability curvesfor various test conditions

0° 0.1 .2 .3 .4 3 .6 .7 .8 .9' 10 11 1.2 -1 3 1.4 15 16 7 t 1200 110° 1000 90° 80° 70° 60° 500 400 30° 20° 10°

as the numerator of both the expressions will

aIwaysb positive. So the unequality:

---N'.

Y'...(l

-

M'-_Y'b)>

,

may be considered as the principal stability crite-rion ]5]. Defining

Np'

_{= x,, and:}

N',

M' -

=

the unequality is equivalent to:

Xv' XtQ

which means that the dynamic stability is negative when the point of application of the yaw damping

force lies alt of the corresponding point of the sway damping force, assuming that both these

points are located in front of the centre of gravity.

In Fig. 24 the ratio x/x, has been plotted

against y = soUo/g. This figure shows that for

each of the five cases investigated there is a region

around the value y = , where the stability

be-comes negative.

With BRARD'S theory the wave systems,

generated by an oscillating pressure point and the shape of these systems have been analized. Partic-ularly the boundaries of the various systems have

been fixed and plotted in Fig. 25 from which it appears that for y half the opening angle of the principal wave system. (q3"max) becomes 180 degrees, which means that for those frequencies the generated waves run in front of the oscillating source. It also follows from this figure that sidewall

4.

effects will occur when q"max exceeds the value of .pw = arctg b/L, where .b is the width of the

towing, tank and L the model length, which means

in this case y < 0.365.

The negative stability also follows from Fig. 26,

showing T1' versus w. For both the roots of the

equation Xv/Xs,h = 1, T1' has vertical asyrptotes. In the same figure also T2', T3' and.K' (See Appen-dix A.3) are plotted. LI.t must be emphasized,

how-ever, that for the calculation of T3' and. K' the

static rudder test results' were used. To what extent rudder force. and rudder moment coefficients will be affected by the state of motion of the model has not been investigated. Further investigations with respect to this problem would be useful'.

2

derived from with rudder and

curves

tests with screw

oLl01

g -'

K -TT c_2cr3 T I curves

testswith rudder and with screwderived from

negative T vaIuesofT 0 Or 10 15 0) radhec

Fig. 27. Some ratios derived from pure yawing and pure swaying test results (Fn = .20)

In Fig., 27 the values of Ti'T2'/K'Ta',I'N'/N'6

'and (T1'+T2'T3")K' are plotted iersus w for a

Froude number .20 '(See Appendix A.3).

Among other things the effect of the mass cross-coupling coefficients appears to be small.

Finally the difference between T1' T2'/K' T3' and (T1' + T2' T3') /.K' has to be considered as an indication for the validity of the first order approx-imation, mentioned in the.Appendix A.3.2 eq. (14).

6

General remarks

Though the main purpose of the present investi-gations was to determine the coefficients of 'the

linear equations of motion as a function of the

circular frequency, the derived results' are also of interest. Particularly Fig. 24 shows'some interesting

results.

5 10 15

0)

radhec

Fig. 26. K' and T' indices as a function of frequency

(Fn = 2O)

K

T

2

20

In the first-place it. follows from this figure that rudder and propeller are important for the course stability of the ship. For very small values of the

circular frequency and consequently for small

values of the velocity- and acceleration amplitudes the bare hull curves show negative stability,

where-as 'the rudder makes stability positive for both

Froude numbers. Another increase of stability is caused by the, action of the propeller. In general

the same trend is asserted for higher values of

frequency.

The maximum values of the negative stability for all test conditions are of further interest. These maximum values are found in a narrow range of frequencies. This matter has already. been

consid-ered in part 5. it is not clear, however, to what

extent tank wall effects are due to this result.

As noted already in part 4.1.1 one is tempted

to accept that the values of the various coefficients for zero-frequency will be the same as those for very low frequencies, but particularly with respect to the added mass and the added mass-moment of

inertia there seem to be objections to this

extra-polation.

Concerning the damping coefficients the prob-lem seems to be less complicated. The very good

agreement between the low frequency values of (_ Yv)8,,, (- N)8, and the corresponding values obtained from the static sway tests may be an

indication that in this case the extrapolation will. be permitted.

The effect of the forward speed appeared to be important for the damping. coefficients only. In the low-frequency range this effect is very smali but at higher frequenciesthe values of - Y'v and - N'

decrease with increasing Froude number. This

seems to be in accordance with the behaviour of the dampingcoefficients NEWMAN described for ellipsoids

[8]. The mass-coefficients Y' and

N',p only change a little with increasing

for-ward speed just as the cross-coupling coefficients do.

A problem that has been considered partly is the influence of the applied amplitudes_yo, related to -drift- and angular velocity amplitudes. The main results of these investigations are given in Fig. 28. A similar trend as follows from this figure concern-ing the dampconcern-ing coefficients exists for the added mass coefficients, though to a smaller amount.

Though for theoretical investigations [7], [8], [9], in general only the parameter w is considered assuming the amplitude of motions to be. infini-tesimal, it seems that there are some objections to the use of the independent parametersJo and w.

It might be better to carry out the tests for

con-stant values of the velocity- and acceleration

am-plitudes respectively for these are the variables

which are considered in the equations of motion.

Yv

.05

.005

U?

Fig. 28. Influence of motion-amplitude on the damping coefficients (Fn = .20)

In that case suitable 'combinations ofyo and (0

have to be choosen. A plot of the various coefficients on a base of velocity and acceleration amplitudes respectively could be useful to judge the 'separate effects of both these variables.

Another point that should be investigated is the

effect of the lateral motions of the model on the

rudder force- and' moment coefficients. One could expect that these coefficients are affected by the

driftangle and the turning radius of the ship. For

a ship having periodical lateral motions, in

addi-tion the influence of lateral and angular

acceler-ations should be considered. In other words it

might be necessary to know Y' and N', as a

function of the frequency of the models periodical motion.

'With respect to the composed graphs in Figs.

26 and 2-7 it will be clear that because of working

up several of the measured

coefficients, the relative error will 'have a cumulative eflect in this

case. This effect does not change the general

character of these graphs however.

Finally it has to be remarked' that within a

restricted frequency range (w 6) additional pure swaying tests were performed for Froude numbers .15 and .25 The results of these tests, not given in this paper, agree in general with those of the for-ward speeds mentioned.

tests with rudder and without

- screw

2.5cm o lOOcm

tests with rudder and without y0= 1:5cm screw oy0 9.6cm

o y0=18.Ocm

N

0 .010

Acknowledgement

The ãüthor is indebted' t the' staff of the Deift

Shipbuilding Laboratory for their readiness to

assist him by word and deed.

Particularly he wished to thank Prof. Jr. J.

GERRITSMA, whose constructive criticism guarded him from overdetailed investigations.

Further the author thanks Ir H J ZUNDERDORP and Mr. M. BUITENHEK for d'esining and con-structing the electronical' facilities and Mr; j. M j. JOURNEE for the patience and accuracy with which he worked up' the great quantity of test data into the graphs.

References

PAULLING, J. R; and L. W.' WooD, The dynamic

prob-lemof two ships, operating on paralIelcourses inclose proximity. University of California, July 1962.

GERIUTSMA, J. and BEUKELMAN, W., Distribution of

damping and added mass along the length of a ship-model. Netherlands Research Centre T.N.O. Report No 49S, March 1963.

ZUNDEItDORP, H. J. and M. BUITENHEK, Oscillatory,

techniques at the,Shipbuilding, Laboratory. Report No. 11.1 of the 'Shipbuilding Laboratory Ofthe Tech-nological University - Delft, 1963.

HORN, F. and 'E. A. WALINSKY, Untersuchungen uber

DrehmanOver und Kursstabilität von Schiffen.

Schiffstechnik, Heft 29, November 1958.'

DAVIDSON, K. S. M. and L. 1. SCmFF, Turning and course keeping qualities. S.N.A.M.E. 1946.

6., BRARD, R., Introduction a l'Ctude théorique du.tangage

en marche. A..T.M.A. 1948.

MOroa, S., On the measurement of added mass and added moment of inertia of ships in steering motion.

First Symposium on' Ship Manoeuvrability. D.T.M.B.

Rep. 1461, October 1960.

NEWMAN,J. N., Thedamping ofan oscillating ellipsoid

near a free surface. D.T.M.B. Report No. 1500, February 1962.

T#sM, F., Hydrodynarnic force and moment produced by swaying and rolling oscillation of cylinders on the free surface. Reports of Research Institute for

Applied Mechanics. Vol. IX, No. 35, 1961.

DAVIDSON, K. S. M., On the turning and' steering of ships. S.N.A.M.E 1944.

NoMoTo, K., Frequency response research on steering qualities of ships. Technology report of the Osaka University. Vol. 8, No. 294, November 1957. NoMoTo, K., Analysis of the' standard manoeuvre test

of Kempf and proposed steering 'quality indices. Symposium ShipManoeuvrability, Washington 1960.

A Appendix

A. I Basic formulae

Using EULER'S equations for the forces and mo-ments, with respect to a coordinate system, fixed

to a body, moing in a horizontal plane with

constant forward speed U0, the following

equa-tions of lateral moequa-tions are obtained: (see Fig. 29)

M( U0ji) = Y (v, i, i,

, 5r) + external force

N (u,i, iji, , r)±external moment

'Considering only first harmonic components of

the forces and moments the linear equations of

motion are consequently:

(MY)YvvY+(MUoYp)ibYoôr = Ymi

(1)

(I-N N N Nv N,är

= Nmi

where' Ymi represents the first harmonic component

of the measured lateral force and Nmi the anal

ogous component of the measured moment.

2a

d2yu

dt2 dt2\ 2

d2

=

(yo cos cot) =.y0c02 coswt,

thus 3o =yow2 and Vo =JOw where Jo is the am-plitude and cv the circular frequency of the motion. Substituting in equation (1) we obtain

(M Yj)yoco2 cos cot - Yv .y sin cot = Ymi

-

.yow2 cos cot - N .yico sin cot Nmi

Let:

F0 cos (cvtei)

and:

121 = F02 cos (cotc2)

be the first harmonic compOnents of the measured

forces Y1 and 1'2. As:

Yrni

'li+2i

and:

Nmj = a.(YuY21),

where 2a is the distance between the fastening

points, we find from equation (3):

M

F1 cos'ci ±F2cos e2

. (4a)

.yFiSflC1±F25flE2

. (4b)

Joco

a(Fi cosei - F2 cos e2)

=

. (4c)

JoW2

_NV=a(F15md1F25ie2)

. (4d)

JOW

A.l.2 The pure yawing motion

In the same way for the pure yawing motion is

started from the angular acceleration:

=0coswt

v=O

(5)

J00J2Slfl 92

cos cot = cos cot a

23

Fig. 29. Definition of,symbols

A. 1.1 The pure .swaying motion

The first of these equations will be satisfied by

substituting:

This motion is defined by:

= yosin(cvt+ç)

₍₆₎

= Josin(cotql)

J

=

o COS cot

I-.

(2) because:

,1)2

Jo cos cot, tg

= - sin 72.

2a a

Consequently the motion of both the fastening

points of the model can be chosen as: so that:

Joc05ifl97

12 COS cot, (7)

a

smwt=osinwt,

The model's centre line being tangent to the path of G leads to the condition.:

dja

=tg'p

dx

From equation 6 we have:

-2

-

Jo COS Sin wt, thus: dj3a yowCbsq7

=

coswt dx Uo

So the pure yaw condition is satisfied if:

yow

yosin

U0 a or aw tg

=

U0 (8)

Substituting equations (5) and (7) into (1):

(IN)JowSl

_{os wt +} = Nmi yow2sin q' CosO)t+ a = Yrni Defining in this case:

Nmi = a(Yu+Y21) *)

Ymi

the pure yaw coefficients are found as:

a2(Foicos 61 + Fo2cos 82) y0w2sjn_q a2(Foisin 81 + Fo2sin 82)

Ny,

-Joe) Sin l)

-

a(Foicos 81 - Fo2cos 62) Joe) S1fl

-MU0

a(Foisin 61 - F02sin 62)

Yoo) Slfl?9

A.2 Some notes-on amplitude variation

Excepting the tests. concernitig Fig. 28 all tests were

carried out for driftangle amplitudes and yaw angular velocity amplitudes which were kept

below a specified value. These values were chosen

in such a way that the ranges are in accordance

with practical values for actual ships. For the

-maximum driftangle amplitude is- taken fib = 10 degrees and for the maximum 'yaw angular

veloc-ity amplitude ro' = .3. The latter value

corre-sponds to a turning radius Rc = 3*L [10].

Be-sides, these values are approximately the bound-aries of linearity in the case of static tests

With respect to the pure yawing motion there

is another point that had to be considered, viz, the yawangle amplitude o, which may not exceed

the value of about 10 degrees, as he approxima-tion: vri tg (See equation (7)), will not 'hold with sufficient accuracy for larger values of o.

Thus, for pure swaying the secondary condjt'ions are:

v0 2iv Joe) 1

= U0 360 1,00 or

<

18 (1-1)

and 'for pure yawing:

L Lyow ,

=---sinq-<3

(12a) 11m1n u a and: 1110 = - sin q < a 18

A.3 Calculations of stabilit_y and manoeuvrability

indices

A.3. 1 Second order differential equation

Starting from the linear equations of motion as

mentioned in A.l and eliminating the lateral

velocity v the following second order differential

equation is obtained:

Ti'T2'?'+(T1'+ T2')t'+r' -- K'r+K'T3'5r'

The coefficients T11T21, Ti'+T2', K' and K'T3'

will be subject to a small correction with respect to the corresponding forms 'used by NoMoTo [11], because in our case the mass cross-coupling

coeffi-cients Y',, and N' are included. Thus:

(I5'N'p) . (M' Y')'( Y'. N',)

T11T2'

=

N',1,.

Np'

(-13a)

T'

i+T2

N'j1 (M'. Y'ü) -+ (I'5N'1) . -

_{N' YM'Y').--N'}

+

(M'Y'j).N'+Y'.N'5

_(13b)

N',,j, Y'(M'Y'1p)

N8'. - y,,', -v-- Y8'. - N'

*) Note that for the pure yawing motion the sign of V,, has K' T3' =

been changed.

-N'j,.Y'(M'Y'p) Ne'

N81. (M'Y'1,)+ - Y61. N'1,

N'1p.Yv'---(M'Y'1,b).

'(1 2b) (9) (lOa') (lob) (lOc) (lOd)

A.3.2 Comparison of second order - and first order differential equation coefficients In many cases the second order differential

equa-tion menequa-tioned above can be substituted by the

first order approximation [11], [12]:

T'i'+r'

K'r

(14) This implies- that the difference between:

T1'T2'

and T1'+T21Ts'

13

is small. Particularly when this difference equals zero both the expressions can be replaced by T'.

In that case:

T1'T2'

Tj'+T2'Ts'

T'

K'T31 K'

(See Fig. 27) The first of these equalities implies a certain rela-tionship between the coefficients of the equation (1) This relation is obtained by substituting the right hand sides of the equations (13).

As:

T1'T2'

(J5'N')

.(M'Y')(Y'. N')

K'T3'

(16)

the difference between: T1'T2'

d

Iz'N'Q

K'T3' an

N'

may provide some information about the effects of the mass cross-coupling coefficients. (See Fig.

27).

A.4 Combinedswaying and yawing motion

This motion is defined by the following relations:

= o cos oit 'ow2cos wt,

-

_{being.rectilinear;see Fig. 1)'}which means the path of G

(17)

As j9

-

v/U0 equations (I) are to be written as:

(INo).i?i+(N7j,Uo. NbU0.

=

= a(Yii+Y2i)

a2

(INj)co2--N. UO = - (Foicos-ei + Faicos-ea)

Jo _-(18a)

y2_ Yv U0.=

(Fncos ei -F02cose2)

yo _(18b)

*) The positive direction of Y, is defined in the same way as is done for the pure yawing motion.

a2

NN.U0

== (Foisinei ±Foisine2)

yaw _(18c)

Y Y.U=

a

(FoisineiFoisine2)

yaw _(18d)

A. 5 Pureyawing. course deviation

The kinematic conditiOn for the pure yawing

motion is

based on the supposition that the

model's centre of gravity is located mid-length between the two fastening points. In the actual experiments the model has a fixed pivot at' A,

the other fastening point is sliding in longitudinal direction. consequently, the centre of gravity (C) is subject to a small swaying motion: (Fig. 30).

Jo2.

1 + sin2qcos2wt

a2

Fig. 30.

v = GM.b

(19) (the positive directiOn of v is to starboard). From equation (7) we have:

=

arct(_

sin q cos

thus:

sin sin wi

yaw

sin cos2 sin wt (20)

As AG = a and- AM = a , we have:

cos'

GM =

a(

- l

(21)

\cosp I

thus:

-v = --yow sin

q (cos-' -

cos2) sin wt . . . (22)

Let Vi be the first harmonic component of v, then we have:

1)1 = Vosifl wt,

so that the- equations (1) become-:

25

(U0.YuYUo.Y.--Y =

= Y11Y2 1)

2'oo'

F01sine1 +Fo2sine2 2'ow Foicos ei - Foacos e2

Np(N).

Vo (25a) p0W 1)0 (25b) )ow

(25)

'çvo(o "POW (25d) fsjnwt 1 +tg2ocos2wt

The first is an elliptic integral of the second kind with modulus sin o. So it can be written as:

11.

/i\

3 (1.3\2 5

cos voj +

.SjflZio

sinpô±...

(27) and the elementary solution of thç second integral is found as: 1 1 1

coso

+ tg2

'o

(281) thus: ,i/1\2 3

=

cos'PoI) S'fl211'O+...

or, as a. suitable approximation:

1 1 yo2.

=

_{-- sin2}

16

16a

As sin q < 1 and for all casesyo < .30 rn:

From equations 25 combined with Figs. 5 up to 12

it can be 'concluded 'that only the mass crOss

coupling coefficient Y,1 has to be considered.

As in general yo < .10 m, from Y, has to be

subtracted one quarter per cent of the total mass

coefficient M Yü. It will be clear that this correc-tion can be neglected with respect to the general accuracy.

In the case of the combined sway and yaw tests

smaller values for the corrections are found, as for these tests q = 900 andyo = .0075 m.

For the tests, concerning the influence of the

amplitudeyo (see Fig. 28) the maximum value of

was .30 m corresponding with a maximum correction for - Y1 of 3%. d(wt) 1 1 tø2

22

o'J (29) {_Y,p.,ow2_(M_Y)vow}cosoit +

+{(MUoY)'ow(Yv)vo} sin wt =

(Foicos.ei - Fo2cos e2) cos at +

+ (Foisin ei Fo2sin e2) sin wt (23a) and:

{(Iz_N)'pow2_(_N)Vow}coswt +

+{

N.ow(Nv)'.vo}sin wt =

{ (Foicos 'Cl + Fo2ccsea) cos cot +

+'Foisin Cl ± Fo2sin E2) sin wt} . (23b)

The corrections, which are necessary fOr the deter-mination of the pure yawing coefficients, can be Obtained from the following, equations, where:

2yosin (a = .5m). (24)

Foicos;ei ± F02cos e2

)0W2

Foi sin Cl - Fo2 sin C2

= MUoY,,,(Y)

v,0w

when --- = ro' it follows from the equations (22)

11)0W

and (23) that

2m

=

- / (cos - cos2) sin2a,t d(wt) . (26)

1

f

sin cot

do

₊

PUBLICATIONS OF THE NETHERLANDS' RESEARCH CENTRE T.N.O. FOR SHIPBUILDING AND NAVIGATION

Reports

No. 1 S The determination of the natural frequencies of ship vibrations (Dutch).

Byprof. ir H. E. Jaeger.May 1950.

No. 3 S Practical possibilities of constructional applications of aluminium alloys to ship construction.

By prof. ir H. E. Jaeger. March 1951.

No. 4 S Corrugation olbottom shell plating in ships with all-welded or partially welded bottoms (Dutch).

By prof. ir H. E. Jaeger and ir H. A. Verbeek.November 1951.

No. 5 S Standard-recommendations for measured mile and endurance trials of sea-goingships (Dutch).

By prof. ir J. W. Bonebakker; dr ir W. J; ltlullèr and ir E. J. Diehl. February 1952.

No. 6 S Some testson stayed and unstayed masts and a comparison of experimental resuitsand calculated.stresses (Dutch).

By irA. Verduin and ir B. Burghgraef.June 1952.

No. 7 M Cylinder wear in marine diesel engines (Dutch). By ir H. Visser.December 1952;

No. 8 M Analysis and testing of lubricating oils (Dutch).

By ir R. N. M; A. Malotaux and irj. G. Smit.July 1953.

No. 9 S Stability experimentson.modelsofDutch and French standardized lifeboats.

By prof. ir H. E. Jaeger, prof. ir J. W; Bonebakker and J. Pereboom, in collaboration with A. Audige.October 1952; No. 10 S On collecting ship service performance data and their analysis.

By prof. ir j. W. Bonebakker.January 1953;

No. 11 M The use of three-phase current for auxiliary purposes (Dutch).

ByirJ.C.G.van Wjjk. May l953

No. 12 M Noise and noise abatement in marineengine rooms (Dutch).

By "Technisch-Physische Dienst T.N.O.- T.H." April 1953.

No. 13 M Investigation of cylinder wear in diesel engines by means of laboratory machines (Dutch).

By ir H. Visset.December 1954.

No. 14 M The purification of heavy fuel oil fordiesel engines (Dutch). By A; Bremer;August 1953.

No. 15 S Investigation of thestress distribution in corrugated bulkheads with vertical troughs.

By prof. ir H. E. Jaeger, ir B. Burghgraef and I. van der Ham. September 1954.

No. 16 lvi Analysis and testing of lubricating oils II (Dutch).

By ir R. N; M. A; Malotaux anddrsJ. B. Zabel. March 1956.

No. 17 lvi The application of new physical methods in the exaniinationoflubricating Oils. By ir R. N. M; A. Malotaux and dr F. van Zeggeren.March 1957.

No. 18 M Considerations on the application of thrëé phase current on board ships for auxiliary purposes especially with regard to fault protection, with a survey of winch drives recently applied on board of these ships and their in. fluence on the generating capacity (Dutch).

ByirJ. C. G. van Wijk.February 1957.

No. 19 M Crankcase explosions (Dutch).

By ir J. H. Minkhorst. April 1957.

No. 20 5 An analysis of theapplication of aluminium alloys in ships' structures.

Suggestions about the riveting between steel and aluminium alloy ships' structures.

By prof. ir H. E. Jaeger.January 1955.,

No. 21 S On stress calculations in helicoidal shells and propeller blades.

BydrirJ. W. Gohen.July 1955.

No. 22 S Some notes on the calculation of pitching and heaving in longitudinal waves.

By ir J. Gerritsma.December 1955.

No.23 S Second series of stability experiments on models of lifeboats.

Byir B. Burghgraef.September 1956.

No. 24 M Outside corrosion o and slagformation on tubes in oil-fired boilers (Dutch).

Bydr W.J. Taat.April 1957.

No. 25 5 Experimental determination of damping, added mass and added mass moment of inertia of a shipmodel.

By ir J. Gerrilsma. October 1957.

No. 26 M Noise measurementsand noisereduction in ships. By ir G.J. van Os and B; van Steenbrugge.May 1957;

No. 27 S Initial.metacentric height of small seagoing ships and the inaccuracy and unreliability of calculated curves of righting levers.

By Prof. ir J. W. Boñebakker.December 1957.

No. 28 M Influence of piston temperature on piston fouling and piston-ring wear in diesel engines using residual fuels.

Byir H. Visser.June 1959.

No. 29 M The influence of hysteresis on the value of the modulus of rigidity of steel. By ir A. Hoppeandir A. Al. Hens.December 1959.

No. 30 5 An experimental analysis of shipmotions in longitudinal regularwaves.

By irJ. Gerritsnia. December 1958.

No 3l M Model tests concerning damping coefficients and the increase in the moments of inertia due to entrained water

on ship'sprOpellers.

By N. J; VLmser. October 1959.

No. 32, S Theeffectof a keel on therolling characteristics of a ship.

By irJ. Gerritsma. July 1959.

No. 33 M The application of new physical methods in the examination of lubricating oils. (Continuation of report No. 17 M.)

By ir R. N. M. 4. Malotaux and dr F. van Zeggeren. November 1959. No. 34 5 Acoustical principles in ship design.

By ir J. I-I. Janssen. October 1959. No. 35 S Shipmotions in longitudinal waves.

By ir J. Gerrilsma. February 1960;

No. 36'S Experimental determination of bending moments for three models of different fullness in regular waves.

By Prof. ir H. E. Jaeger and ir.7..7. W. Nibbering. January 1961.

No. 39 M Crankshaft coupled free torsional-axial vibrations of a ship's propulsion system.

By ir D. van Dort and N. .7. Visser. September 1963.

No. 40 5 On the longitudinal reduction factor for the added mass of vibrating ships with rectangular cross-section. By ir W. P. A. Joosen and dr J. A. Sparenberg. April 1961.

No. 41 S Stresses in flat propeller blade models determined by the moire-method. By ir F. K. Ligienberg. June 1962.

No. 42 S Application of modern digital computers in naval-architecture. By ir H. 7. Zunderdorp. June 1962.

No. 43 C Raft trials and ships' trials with some underwater paint systems.

By drs P. de Wolf andA. M. van Londen. July 1962.

No. 44 S Some acoustical properties of ships with respect to noise-control. Part I. By ir J. H. Janssen. August 1962.

No. 45 S Some acoustical properties of ships with respect to noise-control. Part II. By ir.7. H. Janssen. August 1962.

No. 46 C An investigation into the influence of the method of application on the behaviourof anti-corrosive paint systems

in seawater.

-By A. M. van Londen. August 1962.

No. 47 C Results of an inquiry into the condition of ships' hulls in relation to fouling and corrosion.

By ir IL.C. Elcama, A. M. van Londen and drs P. de Wolf. December 1962.

No 48 C Investigations into the use of the wheel abrator for removing rust and millscale from shipbuilding steel (Dutch) Interim report.

By ir.7. Remmelts and L. D. B. van den Burg. December 1962.

No. 49 S Distribution of damping and added mass along the length of a shipmodel.

By prof. jr 7; Gerritsma and W. Beukdman. March 1963.

No. 50 S The influence of a bulbous bow on the motions and the propulsion in longitudinal waves. By prof. ir 3. Gerrilsma and W. Beukelman. April 1963.

No. 51 M Stress measurements on a propeller blade of a 42,000 ton tanker on full scale.

By ir R. Wereldsma. January 1964.

No. 52 C Comparative investigations on teh surface preparation of shipbuilding steel by using wheel-abrators and the

application of shop-coats.

By 1, H. C. Ekania, A. M. van Londen and ir J. Remmelts. July 1963.

No. 53 S The braking of large vessels.

By prof. ir H. E. Jaeger. August 1963.

No. 54 C A study of ship bottom paints in particular pertaining to the behaviour and action of anti-fouling paints.

By A. M. van Londen. September 1963.

No. 55 S Fatigue of ship structures.

By ir 3.3. W. Nibbering. September l963

No. 56 C The possibilities of exposure of anti-fouling paints in Curaçao, Dutch Lesser Antilles. By drs P. de Wolf and Mrs M. Meuter-Schriel. November 1963.

No. 57 M Determination of the dynamic properties and propeller excited vibrations of a special ship stern arrangement.

By ir R. Wereldsma. March 1964.

No. 58 5 Numerical calculation of vertical hull vibrations of ships by discretizing.the vibration system.

By .7. de Viies. April 1964.

No. 59 M Controllable pitch propellers, their suitability and economy for large seagoing ships propelled by conventional, directly-coupled engines.

By ir C. Kapsenberg. June 1964.

No. 60 S Natural frequencies of free vertical ship vibrations.

By ir C. B. Vreugdenhil. August 1964.

No. 61 S The,distribution of the hydrodynamic forces on a heaving and pitching shipmodel in still water. By prof. ir.J. Gerritsma and W. Beukelman. September 1964.

No 62 C The mode of action of anti fouling paints Interaction between anti fouling paints and sea water

By A. M. van Londen. October 1964.

No. 64 C Barnacle fouling on aged anti-fouling paints; a survey of pertinent literature and some recent observations.

By drs P. de Wolf. November 1964.

No. 65 S The lateral damping and added mass of a horizontally oscillating shipmodel. By C. van Leeuwen. December 1964.

Communications

No. 1 I'l Report on the use of heavy fuel oil in the tanker "Auricula" of the Anglo-Saxon Petroleum Company (Dutch).

August 1950.

No. 2 S Ship speedsover the measured mile (Dutch).

By ir W. H. C. E. RJsingh. February 1951.

No. 3 5 On voyage logs of sea-going ships and their analysis (Dutch).

By prof. irj. W. Bonebalcker and ir 3. Gerritsma. November 1952.

No. 4 S Analysis of model experiments, trial andservice performance data of a single-screw tanker.

By prof. ir J.:W. Bonebakker. October 1954.

No. 5 S Determination of the dimensions of panels subjected to water pressure only or to a combination of water pressure and edge compression (Dutch).

By prof. ir H. E. Jaeger. November 1954. V

No. 6 S Approximative calculation of the effect of free surfaces on transverse stability (Dutch). By ir L. P. Herfs. ApEil 1956.

No. 7 S On the calculatiOnof stresses in a stayed mast.

By ir B. Bw'g/igraef. August 1956;

No. 8 S Simply supported rectangular plates subjected to the combined action of a uniformly distributed lateral load and

compressive forces in the middle plane. By ir B. Burg/zgraef. February 1958.

No. 9 C Review of the investigations itito the prevention of corrosion and fouling of ships' hulls (Dutch). By ir H. C. Ekama. October 1962.

No. 10 S/M Condensed report of a design study for a 53,000 dwt-class nuclear powered tanker

By the Dutch International Team (D.I. T.) directed by ir A; M. Fabery de Jonge. October 1963.

No 11 C Investigations into the use of some shipbottom paints based on scarcely saponifiable vehicles (Dutch)

By A. M. van Londen and drs. P. de Wolf. October 1964.

M = engineering department V

S shipbuilding department