An experimental investigation of the dynamic forces and moments on one blade of a ship propeller

PREPRINT OF

An Expermental Investigation of the Dynamic Forces and Moments on one Blade of a Ship Propeller.

by E. Huse, Skipsmodelltanken, Norway.

ON TESTING fECHNQJES

IN SHIP CAVITATION

RESEARCH

31. MAY2.JUNE1967 THE NORWEGIAN SHIP MODEL EXPERIMENT TANK.CABLE:SKIpSTANK. PHONE: 28020

S Kl PS MOD E LLTAN KE N

AN EXPERIMENTAL INVESTIGATION OF THE, DYNAMIC FORCES AND MOMENTS ON ÓNE BLADE QF A SHIP PROPELLER.

E. Huse

The Norwegian Ship Model Experiment Tank, - Trondheim, Norway.

CONTENTS page

Introduction. 2

Relations between the dynamic forces on each 3

blade and the resuitil-ig propeller bearing forces.

Dynamometer fOr measurement of dynamic 12

proDeiler forces on one blade of a -biaded

propeller.

Three-hole spherical pitot tube for fast, 33

two-dimensional waké survey.

Investigations on a ship model with a clear- '43

water type of stern.

Investigations on a ship mod?]. with a rudder- 50

shoe type of stern.

Main conclusions. 56

References, Tables, Figures. 57

SEC. '. SEC. 5. SEC. 6. SEC. 7. SEC. 1. SEC. 2. SEC. 3.

provided a considerable amount of experimental data on the dynamic forces and moments to which a wake-operating propeller

is being subjected. Concerning the dynamic fOrces and moments

acting on each blade of the propeller, however, the amOunt of

ävailable experimental information is very limited. Such

information is of interest when designing the propeller blades to have sufficient mechanical strength and to avoid cavitation. The paper describes a dynamometer for model scale measurements of the statIc and dynamic components of the thrust, torque, and

bending moment ,on one blade of a -bladed propeller. Strain

gauges and an inductive transducer are fitted inside the boss. An analysis of the frequency response of the instrument is

presented. The added mass, moments o inertia, damping, and

coupling coefficients are calculated by unsteady, two-dimensional strip theory.

The paper also describes a spherical pltot tube for measuring the longitudinal and tangential components of the wake

field. Procedures for spherical pitot tubé wake urveys reporte.d

previously are very time-consuming. This disadvantage has been

övercome. by using differential pressure transducers to obtain

a fàst-response recording system. The flow velocity can thus be

measured at several positions in the wake field for each run of the towi-ng carriage, the number of positios béi1ng limited only by the turbulent fluctuations of the velocity.

The paper presents the results of measurements

with the dynamometer and pitot tube carried out on two tanker models, one of them having a clèar-water type öf stern, the

other a rudder-shoe type. Special attention is paid to the

influence of propeller-aperture clearances upon the dynamic forces

-2-SECTION 1:' INTRODUCTION.

Due to

ñon-uniformity

of the wake field

añd hydrodynamic interaction between the propeller and adjacent parts of' the rudder and Ship hull, the blades of a ship propeller are subjected to fluctuating forces. These. forces may ûnder certain circumstances induce

troublesome vibrations in the propéller Shaft and other

parts of the ship structure. During reàent years much

effort has been devoted to theoretical as well as experimental investigations on the problem of dynamic

propeller forces. 1n Ref. [i, an extensive list of

publicadons jn this field is given.

Many attempts have been made to measure

dynamic propeller forces and momentsin model scale. The

usual experimental procedure has been to measure strains

in thé pröpeller shaft by means of strain, gauges. Assuming

thé wake field to be statioñary (i.e. time-independent), it can be shown '(Sec. 1 of this report) that of all the

dynamic force components acting on each propeller blade, only those having 'a frequency equal to the blade frequency.

or one of its harmonics can be detected in the shaft. All

the òther components cancel by vectorial surnitation over the blades. The only diret experimental way of obtaining

information about these other components is to measure the

forces on one of the. propeller blades. Th'is method, of

course, will also yield the components at the blade freqLtency and its harmonics.

Since only the force components at blade frequency and its 'harmonics are able to induce shaft and hull vibrations, one might ask, why, bother about measuring ail the other

damage it is of interest to know the force, moment, and

circu]ation fluctuations on each blade. Secondly, measuring

all the force components will provide more knowledge about

the fundamental

hythodynamic effects giving rise. to dynamic

propeller forces. Por example, experimental testing of the

available theoretical methodS for evaluating propeller forces from wake field measurements, can be carried out more satisfactorily when all the Éorce components are

measured than when they are measured only at blade frequency

and 'its 'harmônics.

As part of 'a research projeèt ondynamic

propeller forces it was decided' at Norwegian Ship Model. Experiment Tank tö design á dynamöineter to measure

dynamic. forces. and moments 'on one single blade' and. with the

other blades present. As far ás the' author iS áware, this

type of measurement has not been published before.' This paper

iS a preliminary report of the project. It preseflts a

description of the dynamotheter and a f'ast-resonse. system fOr

two-dimensional wake.surveys. Results of measu'enents on

two tanker models are. described.

SECTION 2: RELATIONS BETWEEN THE DYNAMIC FORCES ON EACH BLADE AND THE RESULTING PROPELLER BEARING FORCES

'This secion döntains a deduction of the dynamic bearing forces and moments of a propeller expressed by the

moments measured on one blade, assuming the wake field to

be stationary For explanation of symbols, see Fig 2 1

a y(a) x,y F (a) tan Fm G(a) Gm MH ( a) Mv C a) C a) Q(c) C a) Rt a) T(a) Tm Te(a) z + e(a) i Notation in Section 2:

angular position of blade number one,

a.+.s 2irtZ,

direction of

phase angIe of y'th harinonicof the variable x, tangentiI force on b].ade nunber one,

m'th harmonic of

T(a) bending moment on blade number one,

m'th harmonic of G(a).,

instantaneous horizontal bending moment on Z-bladed propeller due tO thrust eccentricity,

instantaneoùs yertcal bending inomnt on Z-bladed propeller due to thrust eccentricIty,

instantaneous horizontal side force on Z-bladed. propeller,

instantaneous vertical si4e. force on Z-biaded propeller, contribution of blade number one to total propeller torque,

in'th harmonic of Q(a),

torque of Z-biaded propeller,

radial distänce to the point of application of the thrust force on blade number one,

thrust force on blade number.one, m'th harmonic of T(),

total thrùst force on Z-bladed propeller, number of blades,

radius vector giving instantaneous point of

applicatiófl of .T(a) integer number.

force

T()

measured on blade number one, will be periodic with

period 2ir. it can be expressed aS a Fourier series

T(cz) :'

Tm (2.1)

míO

where the Fourier components Tm and phase angles

T,m can be

measured, either by numerical Fouier analysis f the osciliogram

of the signal from the dynamornetér, or by some. suitable experi-mental equipment fór automatic spectral analysis.

The tOtal propeller thrust Te(a) is

the

sin of the

contribütions Of each blade

z

T()

T(a5)

' '. : (2.2)

s=i

FrOm Eqs. 2.1 and 2.2

z

TSlfl

(mc&5+

s:lmO.

z

(eS '

m:O sl

21 m:O s:l e_i.(ma _T,m e_1m52 ,mn e

e_(mas +

_T,m') ims2ir/,Z where

+ S

2ir/Z.

Now it can be deducèd that (see for eamplé [2]

Applying this relation when summing over S in the above

formula for Tr(a) gives V

where Te(a)

mO

Sm

imsir/Z

e:

s:l

i(in+

) f T,m -if _{in 4} kZ, k integer V

Froni this formula it. can be seen that of all the harmonic cofflpoient of the thrust on one blade, only those.

with frequency equal tò the blade frequency or a multiple thereof will contribute to the tota.1 propeller thrust.

Torque.

The contribution

Q()

of blade number one to the total

propeller torque Q(a) can be expressed as a Fourier series,

Q(a) m sin (mOE +

Qm

m:O

Z ifm

kZ., k integer, O if in 4 kZ.

i(mu+T,m

if m kZ, (2.3)

Thus T(a) can be ritten

Tr(a)

k:O

is

Vertical bending moment.

The bending moment on blade number one due to thrust

Q(

° Z sin (kZa

+ Q,kZ k:O

G(a) T(u) _{. Rt(a)}

where _Rt(a) radius to the point of application

of the thrust on blade number one.

G(a) can be measured and expressed as a Fourier series

G(a) G sin (ma (2.6)

m0

The vertical bending moment Mv due to thrust eccentricity on a propeller with Z blades is

Z.

Mv(a) =

C(a) côs a

s;l

where a + s2ir/Z.

Z

Mv(a)

G sin (ma

+ _G,m cos a

s=l m:O

Z G i(mu + ) -i(ma + ) ia -ia

-11(e

G,m_e

s

G,m)(e

S)

m:O s=l

which ôan be witten where and Mv(a) s vi

-e

SV2 O

_i((m+1)a

. : )

_i(m+i)s2it/Z

G,m

_),

_,j((m!,1)a +

' )

_i(m_1)s2ir/Z

-e

G,m e ).

By using therelation (2.3) one obtains

O

if m + 1 + kZ, k

integer, G

-. (S1

i S)

LI'i

i((m+i)a +

4, ₎

i(rn+1)s27t/Z

Gm

(e ' e

i((m_1) +

4,,

si

)

i(m1)s2TrIZ

rn

in

e =

-i((m-1)ct +

4, ₎ G,m

m-i

kZ.

(2.7)

jf in-1 + kZ,

I

i((m-1)c' +4,

e

Substitution of these expressiöns for. S,,,1 and S,2 into Eq. 2.7 yields

Mv(a) =

Gz

.Z(e

G]1

e_k+Gk1)

k=l + GkZ+1 OZ(e G,kZ+1

eZG+1))

k=l

Lj

M()

k:l 2 (GkZl Sin (kZ + G,kZ-)) )

(2.8)

+ Gkz+l

sin

(kZa

+

G,kZ1

From this expression it can be seen that of all the bending

moment components acting on one blade, only those with frequency equal to the blade frequency harmonics plus and minus one, will contribute to the total vertical bending moment of the propeller.

Horizontal bending moment.

The horizontal bending moment of the propeller due to thrust eccentricity is

z

G(a) sin

a.

s=l

By a procedure similar to the previous deduction of Eq. 2.8 it can be proved that'

and its direction is given by (Fig. 2.2)

MM(a)

y(a) &rôtan C )

Mv(a) with the additional condition

- _{< y(a) < +}

_{f MÇa) >}

O 2 2 ir 3ir

-

< y(a) < -. 2 2 if

Mv()

< O. (2.9) cos .(kZa + .G,.kZ-1 Thrust eccentricity.

The instahtaneous thrust eccentricity of :the propeller

can be described by4 radius vector ¡(a) giving thé

instant-aneous position of tlie point of application öf the thrust force

T(a).

(Fig. 2.2).H The numerical value of ¡(a) s

(2.11)

Te(a), Mv(a), and M(a) in Eqs. 2.10 and 2.11 cati be ealcuÏated from Eqs. 2.4, 2.8, and 2.9.respectively.

=

: (Gkz+i cos (kZct + _G,kZ+l

Ig()I

(a)2 + MH(a)2

(2.10) Te(a)

included in the experimental investigation described later in

this report. For the sake of completeness, however, the

formulae for the total propeller side force, expressed by the tangentially directed force Ftan() on one bla, will be

included here

Ftan()

can be expressed as a Fourier series

e

Ft(a) =

Fm sin (ma + m=O

The instantaneous horizon-tal force PH(a) on the entire propeller is (Fig. 2.1)

e

PH(a) Fm sin (mae +

_{F,m05 as.}

s=l m:O

A procedure equal to the previous deduction of Eq. 2.8 yields

PH(a)

k:l

(FkZ_l sin(kZa

+ F,kZ-1

Vertical side force.

The instantaneous vertical fOrce is z

PV(a)

I

Z Fm

sin (m

+ _F,m

Sin a.

s=l m0

(2.12) + FkZ+l sin(kZa +

By analogy to

E4o 1.9

a 8 ru p

w.

Ag Aq A

A ,A qg gq P

12

-FkZ_i

cs (kZa+

_F,kZ-1

SECTION 3 DYNAMOMETER FOR MEASUREMENT OF DYNAMIC PROPELLER

FORCES ON ONE BLADE OF A LBLADED PROPELLR

Thi sec.on contains

a

escription of the dynamometer.

Calibration data are given and the frequency response of the

dynamome.ter is estimated by cä1cu1atn the added mass, ade

moments

of

inertia and hydrodynamic coupling effects from

unsteadytwò-diuieiona. airftil theory.

Notation in Section 3:.

angular ositiör of proe1ler blade.,

pitch ag1e,

prf ix

indicating standard deviation,

phase arie,

"reduced frequency",

phase, angle,

,densty

öf

fluid,

c1ibration constants,

kZ+l .cos (kZct + F,kZ+l (2 .

13)

T, Tex

Th, Trec

UgUqÚt

mechanical stiffnesses,, C1,D1.,E1,Ì calibration constants, F0,F1 J

Ftan tangential force,

G, Gex i

bending moment, Gh, Greci

H complex function,

Hre real part of H,

Him imaginary part of H,

H1 .thróugh H8 transfer functions,

I,

moments of inertia,

K(iv),

₁ sj

- . modified Bessel functi.ons of the second kind,

.1((iv)

J

through

e hydrodynamic propeiie ôöéffi.äients,

L 11f t force pr. unit span;

masses,

shaft revo1utons pr. second,

ex' '1 torque, "h' '<rec'J R R radius,

radial distance fromx-axis to the centre of gravity of blade and arm A,

radiai distance to he point of ppiicatiön

of tangential force,

effective radius at which the bending moment in arm A is measured,

radiai distance to the poi.nt of application

of the reslt4nthrus force on the propeller

blade,

thrust force,

signal from bending moment, torque, and thrust transducerè respectively,

- lL

-UdaUq'Ut1

VdaYgVtL

dynamic correction terms,

XdaXgXq

J

...

Vgu gust velocity,

W flow velocity, '

c chord,

f frequency of airfoil heaving motjon,

i

s.

iw,

.x, u, y coordinates.

Description oth'dy'athòmeté-'

The dynamorneter has.been designed to measure the static and dynamic parts of thefollowing three quantities:

the thrust T on one blade,

the contribution Q of one blade to the total propeller torque,

the bending thoment T.R where. Rt radial

distance from"propeller axis to the point öf a.pplioation òf the 'esuitañt thrust force on one blade.

Rt can be calculated from, the meàsüred values of T and G- and,

as shOwn iñSec. 2, the'thrust-eccentriòity of the propeller

can alsò be obtaiñed.

The

propeller has 'four blades whose data

are given in Fig. 3.2.. One of the blades has 'been cut off from the boss and then refitted by means of flexible arms and

membranes inside, the boss as shown in Fig. 3.1. The single

blade's contribution to the tOtal propeller torque appears

as a bending moment in the flexible arm A. This bending moment

is measured by means of strain gauges S1. There are two of them.,

The thrust force acting on the propeller blade causes

a deflection of the membranes Mb1 and Mb2 which Support B. This

deflection is measured by means of ari inductive displacement transducer consisting of the coils C connected in half bridge, and a ferrite core F (whose displacement relative to C is

measured).

The bending moment G gives rise to a bending moment

in the flexible arm B. This bending moment is measured with

the strain, gauges S2.

As will be seen from the paragraph dealing with frequency response later in this report, the maximum frequency at which dynamic forces can be measured is determined by the

stiffness of the membranes and arms supporting the blade. In

order to obtain a satisfactory compromise between stiffness and

signal to noise ratio, semiconductor strain gauges are used.

(BLM type SPB3-20-35, gauge factor 120). The signals from the

inductive displacement transducer and the strain gauges are transferred by a slip ring assembly mounted at the end of the

propeller shaft. This assembly (1DM Electronics Type PL-l2-02A)

has a built-in azimuth indicator contact to indicate propeller position.

In order to prevent the dynamic propeller torque from inducing fluctuations in the angular velocity of the propeller,

aflywhee]. with moment of inertia 0.06 kp in 2 is mounted on

the propeller shaft.

The main parts of the dynamometer are shown in

16

-Calibration with static forces and moments, infunof

gravity and centrifugal forçes.

The dyna.inometer has been calibrated by applying

static forces and moments to the propeller blade in air.

The corresponding

ign4ls from the induòtive transducer and

the strain gauges have been recordçd by carrier frequency

amplifiers'and a galvanometer recorder (Hottinger KWS/6T-5

amplifiers, CEC, Type 5-124 oscillograph).

The carrie

frequency of the amplifiers is 5 kc.

The signal from this

mèasuring equipment can be expressed jn milLivolts/volt'.

Thrust calibration:

A concentrated force. in thrust

direction has beeiapplied to the blade at a radial distançe Rt

from the propeller axis.

The resulting, signal U.

from the

induçtivedieplaceent transducer is given in Fig. 3.4.

The

signal is seen to

e iidependent of Rt.

Hysteresis and

deviation from linearity are negligible.

A simple test with

a force ap,plied.tangentially to the

propeller blade has proved

the thruCt signal

o be independent of propeller

torque.

From

observations with the propeller rotating in air it has been

concluded that gravity and centrifugal förces acting on the

propeller blade has no effect on the thrust signal.

From Fig., 3.4. the thrust signal U.

can be expressed

Ut

At

T

(3.1)

where At is a calibration constant that can be

determined from

Fig. .3.4.

The restílt is given in Table 3.1.

Torqué cabration:

The torque signal Uq from strain

gauge bridge S1 is,' unfortunately, not

only a function of the

.hydrodynamic torque Q.

« The.

following factors have to be

considered, and corrected for:

measured at a.certain distance R5 from. the propeller axis, this bending moment will not be equal to the. blade's contribution Q to the total propeller torque.

The center of gravity of the blade does not coincide

with the centerline of arm A. Therefore the

centri-fugal force will give rise to an additional bending moment in arm A, proportional to the square of the angular velocity, of the propeller.

The weight of the blade Will cause a bending iomen.t

in arm A, depending n, the angular position of thé,

blade.

Probably due to ináccurate positioning of the strin gauges S1 the torque signal Uq is also slightly

dependent. upon G.

Taking al]. the above-ientioned effects into acóöunt the torqUe signal Uq can. be expressed by the formula

Uq

Aq(l -)Q +Äqg G + E1n

1sin(a+n1),

(3.2)

where. Aq R5, Aqg E1, C1, and n1, are constants which have to be determined by calibration.

Rq radial distance from propeller axis to the

point of application of the resultant force in tangential direction (see Fig. 3.5).

R8 effective radius at which t.he bending moment

' in the flexible arm A (Fig. 3.1) is measured.

shaft revolutions pr. second.

a angular position of blade (a O when blade

18

-of propeller rotation, see Fig. 3.5).

From Eq. 3.2 Q may now be expressed as a function of calibration constants and quantities recorded by the dyna-mometer.

Results of the calibration are given in Fig. 3.6

which shows U versus R with the tangential force F as a

q q tan

parameter. Different tangential forces have been applied at

Rq 8.8, 7.2, and 3.6 cm. In Fig. 3.6 the lines through the

plotted points have been extrapolated to smaller radii. The

radius at which the extrapolated lines meet and make U O

q

is R.

The resulting R obtained from Fig. 3.6 is given in

Table 3.1. In Fig. 3.7 Uq is plotted as a function of Q with

Rq as a parameter. From this diagram Aq given in Table 3.1,

has been calculated.

By applying a few different bending moments C and recording the corresponding Uq the constant Aqg has been

deter-mined. The constant E1 has been determined by letting the

propeller rotate in air at different angular velocities. By

letting it rotate very slowly in air the constants C1 and n1

can be read from the oscillogram. The value of C1 obtained in

air is then corrected for the buoyancy of the blade in water by multiplying it by a factor

prop water

prop

where

prop = density of propeller material (btRa),

water density of water.

The final results are given in Table 3.1.

In order to obtain Q from Eq. 3.2, an estimate

has to be made of Rq for the propeller working in the wake field

examplê 60 .% of the propeller radius. However, it will probably

be a better approximation to assume that Rq _Rt. Since G and T

are measured, Rtis eaily cáiculatedas

1h the computer program fòr anay:sis of thé data from the

dynamometer oscillograms, the:approximation Rq Rt has been

applied.

The bendiñg moment

signai Ug from strain gauges S2 (Fig. 3.1) is mainly a function

of the hydrodynamic bending moment G.

HoweverUg is also

influenòed by the following effects which have to be corrected for:.

1. The weight of the blade.

2 The áetrifugal force.

3. _{A si:ight dependence upon the torqie Q.}

Taking these effects into accoi.tht Ug can be expressed by the formula

Ug AgG + AgqQ + F1n2 + F

+ D1 sin(a+1)

(3.3)

where Ag Agq'

F1, F0, D, and

are constants. to be deter4ned

by calibration. _Recor4in Ug _Q ñ, nd a, the hydrödynamic

bending momnt

can then le calculated from Eq. 3.3. ReSults of

the calibration are presented in Fig. 3.8, wheré Ug is iven as

a function öf G. From this curve the constant A is obtained.

The constants Aq F1, F0, D1, and

are determined

in a way .sini1ar to the ore dscibed for torque calibration. The results are given in Table 3.1.

Discussion of instrumént accuracy.

Table 3.1 includes a standard deviation for each.

of the calibration constants. The deviations given in Table 3.1

are nöt based on accurate measurements or calculations. They

represent a rough estimate of what the author would expect the singlé measurement standard deviations to be if the calibrations

were repeated many times. Although the given standard deviations

are not very accurate, they ma serve the purpose of gjvjng a

rough estimate of the accuracy of the dynamometer.

From Eq.3.1

U.

T -r

At

According to the statistical theory f error propagation

u )2 +

t t

U

((t)2

+

A _At

where T is the standard deviation of T, is the, standard

deviation of U etc, For a thrust force of the ordei of

1 kp, which is considered to be typical for the dynarnométer

in us?,te standard deviatjon in reading U. from the

oscillo-gram is suppped

be about .5

I

of U.,

Taking Aom

Table 3.1, thé standard deviatjon 4T computed from the preceding

fórmula amounts t _P..75 % of T.

From Eq. 3.2..

R Aq C Rq_R5

20

AA )2 )2 + _AR ?Aq q q _?Rs ' AA

)2+(AG)2+(_AE)2

q.

?A

qg q qg +

(2.An)2

+

(2Q_

_AC ?c1

By carrying out the differentiations and inserting the. following values, which are considered to be representative for the

dynamometer in normal use,

Q = 2.2 . io_2 kp m U (7.1 ± 0.71)10_2 mv/v n 12.3 ± _0.03 . cps G = (6 ± 0.O6)10 kp m Ug = (77 ± 0.77)10_2: . mV/V Rq = (6 ± 0.5)10_2 m1

the final reSu.t is

= 3 %. Q

A possiblé error in and n1 has been neglected here, since their

effect would sImply be to cause an error in the phase angles of

the different fourier components of Q. The standard deviatiOn

AQ

amounting

to 3 % of Q may seem to be very high. However, more than one half of it is due to the presumed standard deviation in

the quamtity Rq alone. When.the dynamometer is

_used

for

comparative purposes, for instance to study the influence of propeller clearances upon the dynamic propeller torque, then the

errors in R and the calibration constants will have to be

_q

considered as more or less ysternatic errors.

From

AG

- 22

- A

Q - F.

A

gq

g AA

)2

+ (_ AU

)2

+ AA

)2

g g -

gq

g g

.gq

F

)2

+

(ì.

_An)2

i

.

- D1 sin

More than one half äf this standard deviation is. due to the

assumed standard. deviatiOns of the different calibration

constants, and will therefore have to be considered as a

systematic error when the dynarnometer is used for çomparatjve

measurements.

i

BefOre the propeller blade wäs cut off from thé boss,

the propeller had been fi-ted to a certain ship model and

propulsion tests had been carried Out by means of a .nechariidal

dynamometer (type Dr. Gebers,).

The same tests have been repeated

with the electrical dynamometer described here.

The resUlts are

in good'agreernent.

i

G

(__ AD)).

F0.

1

By çarrying out the differentiatio

and

by:i:nseing the

numerical values that wee used in the calcuiationo

AQ

abovç, the fiflal resu

is

.

= i %.

G

Investigation of frequency response.

Since the intention with the dynamometer is to

measure dynathi

propeller forces up to the highest possible

excitation frequencies, it is of great importance to

evaluate its frequencyresponse.

The main intention with the

present analysis is to investigate the relation between the

dynamic excitation fçrce an

noments Tex

_ex'

and Gex acting

on the propeller blade, and the force and moments Trece arec'

and Grec which are actually recorded by the dynamometer.

For low excitation frequencies there will be no difference

between. the two.

At higher frequencies, however, they will

be different due to hydrodynamic çoupling effects and dynamic

amp1ification in the dînamometer.

The method to be uséd

here for theoretical investigation of the frequençy response

is in many respects similarto the method applied in

reference

[Li]

Let us first cönsider excitation frequencies

different from any blade frequenòy harmonics.

In this case,

assuming the wake field to be steady, the resi tant excitation

force on the propeller boss is zero due to vectorialcancellàtion.

among the blades.

Accordingly the boss itself will not vibrate.

In the case that the excitation frequency coincides with the 'blade

frequency there will be a resultant excitation force on the

boss.

However, the propeller shaft has been

signed sufficiently

stiff and has been fitted with flywheel 9f sufficient moment

f inertia, so that even in this case the vibrations of the

boss ca

be safely neglected.

-Fig. 3.9 is intendedto explain the coordinate

system.:

x

displaçement of membranes in axial direction,

u

angular deflection of. blade in bending moment

direätion,

.

- 24

The coordinate system is rotating with the propeller boss. The

equàtionsgoverning dynamic equilbrium of the blade are: Dynamic forde euilibrium is x-direction

+ m2)2:+ m2R0ü Th + T Cxx.

Dynamicmoment equilibrium of the blade with respect to the

X-axis:

+

ex - (3.5)

Moment equilibrium with respect tò the.Z-axis:

lU +

G +G

_{- Cu}

(3.6)

where

moment of inertia of blade and arm A with respect to the Z-axis (Fig. 3.9),

momént öf inertia of blade and arm A with respect to the X-axis,

= mass of arm B,

= mass of blade and arm A,

distañce fròm X-axis to the centre of gravity of blade and arm A.

C, C, and Cu are the mechanical stiffnesses of the blade in

thrust, torque, and bending moment direction respectively. _C

depends exclusively on the stiffness of membranes, while C,, and

are given by the stiffness of arms A and B. _Tex

ex' and

Gex are sinusoidal excitation force and moments iri thrust,

torque, and bending momènt direätion respectively. These

excitätions cause blade vibrations which give rise to Small oscillatory hydrodynamic force and momentSTh,

h' and Gh. The

directionsof positive force and moments are the same as the directiorof positive x, y, and u indicated in Fig. 3.9.

?Gh Gh

T;:;-Each of the quantities Th,

h' and Gh is a

function

of

*, , û, i, *, and U where one dot denotes the first derivative

with respect to time, two dots the second derivative. The small

force and moments Th,

h' and Gh can be approximately expressed

by T h

?û

?Qh

_h ?Qh

_'h

_h 'C,

+ - i + - * + - 5t +.- û + - u

û ?Gh

?Gh

'o'

+ - ) + - * +

K0(iv) + K1(iv)

2

where

K0(iv)

and

K1(iv)

are modified Bessel functions f the

,

Approximative expressions for the partial derivatives in Eqs. 3.7 can be deduced from two-dimensional, unsteady airfOil theory.

According to this theory, t5] , an airfoil of infinite aspect

ratio performing heaving motion in an otherwise uniform and steady flow field will be subjected to an unsteady lift force given by

L ir c W Vgu H (3.8)

where

L = lift force pr. unit span,

p _{desity of the fluid,}

c = chord,

w = velocity of the uniform flow,

vgu= velocity of heäving motion.

H is a complex, frequency-dependént function given by

K1 ( iv)

iv

-

26

-second kind with imaginary argument.

tC

2w where

w 2irf,

f frequency of heaving motion,

I

In the present casó (Fig. 3.10) the gust velocity is

gu

(*

+ Rù)cos B - R'' sin B (3.9) Expressing H by = H im and W by (Fig. 3.10) 2irnR cos B where n shaft frequency, B = pitch angle,

the following expression is obtained from Eqs. 3.8 and 3.9:

L = -2w2pc nR(*+Rû+R _{tan ß)(Hr+i Him)s}

(31O)

Now suppose that the sinusoidal excitation force makes the blade

vibrate in x-direction ata circular frequency w. The derivation

with respect to time is equivalent to multiplication by iw,

= iw*

or =

_i

w

Using these. identities, Eq. 3.10 may be written

L -2n2.pc flR(Hre(*+Rû+R'1 tan 8) * - (+Rü+RQ tan 8)).

The unsteady lift force on a narrow strip of the propeller blade (Fig. 3.10) can be approimate1y calclated by assuming it to be part of an airfoil of infinite aspect ratio. Integration over the radius yields the total unsteady force and

moments: .

Th

fL

cos 8 dR,

IL

R sin B dR,

¡L R cas B dR..

By,substitutingL from Eq.. 3.11 into Eqs. 3.12 the resulting

expressions for Th,

h' and Gh can nöw be used to obtain all the

partial derivativés in Eqs. 3.7. The resulting formulas for the

partial derivatives are tabulated in Table 3.2, where their

physical meaning is also indicated. They can now be

subs.ti-tuted into Eqs. 3.7. Further substitution of the resulting Th,

and Gh from Eqs. 3.7 into Eqs. 3.5, ank. 3.6 yields

after rearrangement of the terms

(mj+m2+K2)+.Ki*+Cx+K3ú+(K4+m2R)ü+K5+K6(

T,

ex'

(I+K8)ü+K7ù+Cu+K9+K10+K11*+(K12+m2R)

_Gex

where the coefficients K, through K18 represent the partial derivatives with opposite sign as indicated in Table 3.2.

Since H. is a. funòtiön of the exciation frequency, all the

coefficients will also be freqtiency-dependent. For increasing

(3.11)

(3.12)

y the coefficients tend to the assymptotic values given in Table 3.2.

Eqs 3.13 is the system of coupled, second order

differential equations describing the dynamic characteristics

of the dynamorneter. The most convenient way to analyse them

is by Laplaôe transformation, Drawing the bJØck diagram of the

system of equations and reducing it as much as posib1e yield the diagram shown in Fig. 3.11 where the transfer functions are

i H, - (I+Kj 2 (I+K8)s2 + K7s + Cu 1 1 28 -(m1+m2+K2)s2

+ Ks + C

' K18s' + 1<17 + K9s +rn2R)s2 + + K15s (3.114) H9 -' where s

According to the block diagram the equations can now be written.

where Xd X g T = ex

:iv+H8x+H11u,

(3.15) C X

rç

Cv C u

in Eqs. 3.15, and further stbstitu±ion of the transfèr functions from Eqs. 3.14, yield

T_ex

=T.

(l-X

+X +X),

rec da g q H1 ex rec {(m1 + m2 + T _Cu x + H6u + H9v, u + H5v + H7x.

The physical meaning of each term in Eqs. 3.15 may be seen

from the block diagram in Fig. 3.11. Substitution of

T rec , Vda + Vt + Vg) G _{Grec (J. - + Uq + Ut),}

Grec {(K1 +m2R

)2

+ K3s1 , + C

X'

}

\

(3.16) 1 H3 ex

G.

ex 1 H2 and Grec

q Qrec(1<6S2 + K T _C "da - {(i, K]4 = T (K16s2 + K5s) U

;1

da rec x

Grec<i8

+ 1<17 Q_rec C_u U Qrec(1(i G C rec V T _{«1<12. +}

mR)S2 + K11s J

G C rec X

With Eqs. 3.1:6 and 3.17 we have qbtained the final expressions describing the dynamic characteristics of the

dynamometer. According to Eqs. 3.16 we have to multiply the

recorded force and moments Trec

_Çec'

and Grec by a certain

factor (the expression in the parantheses of Eqs. 3.16) in

order to obtain thè true excitation force and moments Tex

ex' and Gex (which are. the quantities we want to

asure).

The terms Xäa Vda and Uda in Eqs. 3.16 have the physical

meaning of corrections due to dynamic amplification. The

terms X Xq Vg Uq and represent corrections due to

hydrodynamic coupling from the quantity indicated by the subscript

The correction terms all tend tö zero when the excitation + K8)s

J

-

30

-s + c,}

frequency tends to zerd.

Since they are ail complex quantities,

their effect is not only to correct the amplitudes of the

différent harmonics of the recorded T, Q, ánd G, but also to

change the phase angles of the harmonic components.

Correctiöns accórding to Eqs. 3.16 and

3.17

have been

included in the compùter program to be described 'later.

The stiffnesses Cx,

C, the masses m1, m2, and

the moments of inertia

which are all necessary for

computation of the correction terms (Eqs.

3.17),

are given

in Table

3.3.

In order to give an impression of the order of

mani-tude of the correction terms, their nurtterical values have been

coñiputed as function of excitation frequency for the following

conditions which are considered tö be typical for the

dynamo-meter in use:

Trec

O.25'4 kp,

grec

5.8

1O

kpm,

Grec

1.7 .

iü2

kpm,

n

= 11.5 cps.

The results are shown in Figs. 3.12,

3.13,

and 3.1k.

From Fig. 3.12 it can be seen that for the thrust measurement

the main cori'ectioti will be due to hydrodynamic cou1ing from

the bending moment.

.

Fig. 3.13 shows that for the torque

measurements the corrections due to dynamic amplification and

coupling from bending moment are ò

about equal importance.

According to Fig. 3.i' the correction of the measured bending

moient is Mainly 'due tO dynamic arn1ificatiòn alone.

_{In this}

numerical example the dynamic corrections will amount to about

20 % of the recörded values T,

_arec'

and Grec when the

excitation, frequency is

70

c.p.s.

At an excitation frequency

of 100 c.p.s. the. corrections will amount to about '40

%.

It is to be emphasized here that the corrections

are given

- 32

and not to the mean value of the quantity..

if, for example,

T

_rec

is 10 % of the mean thrust and the correction amounts to

-20 % of T, then the correction, will be 2

% of' the mean thrust.

The accuracy of thé present method for frequency response

investigation, is determined by. the accuracy that can be achieved

when computing thehydrodynaraic propeller coefficients

through K18 (defined by the partial derivatives in Table 3.2).

The formulae in Table 3.2 are based on an unsteady two-dimensional

stpthery. Previous investigations,

[Li)

and [6]

_,

indicate.

that the two-dimensional strip theory yields too high values for

the propeller coéffiòients.

Accordingly the present method for

calculating the dyamic correctiòns will probably overetimate

the corrections to a certain extent.

yen when the correctiòns are applied, the

dynamo-meter can not be cqnsidered capable of measuring dytiamic forces

and moments at excitation frequencies above 70 ör 80 c.p.s.

At

this frequency level the error due to dynamic amplification and

coupling effects may be of the order of & % of the amplitude of

the harmonic component in question.

There is á.nother possibl

source of error. in the

dynamic measurements.

Lateral Vibrations of the stern of the

hull model will be transmitted to the propeller by the shaft

bearings 'ànd give rise to ünwanted signals from the tranducers.

However, sincé the modelé are thade of wax, a material of high

internal damping, it is. béliçved that, at least for still-water

tests, the effect of hull vibratios .til1 be small compared to

the hydrodynamic force and moments.

Due to t1ç special design

of the dynamometer,,it will be inherently insensitive to

lateral hull Vibrations as far as thrust measurements are

'concerned.

Only the torque and bending moment

measurements

3. the radial velocity Vr

Suggestion for further improvements of the dynamometer.

The stiffness o the flexiblé arms A and B (Fig. 3.1)

has been chosen as a compromise between the stiffness require-ments from the view±-point of frequency response, and the

reuirement of tenperature insensitivity and: long-terM stability

of the strain gauges. The semiconductor strain gáuge bridges

have proved to be more stable and insensitive to temperature

changes than expected. From the 'experience ith the

dynamo-meter in use it can be concluded that the thicknéss of arms

A and B cou].d have been ijicreasêd to sorne extent,, thus iMproving

the frequency réspónse without reducing stability and signal to noise ratio to an intolerably low Ievel

Thé stiffness of 'arms A and B could also be further' increased by applying a nòtciing technique fo'strèss concen-tration and usingminiature semiconductor strain gaÙes 'in the

notches.

SECTION : THREE-HOLE SPHERICAL PITOT TUBE FOR FAST;

]WO-DIMENSIONAL WAKE SURVEY.

The instantaneous flow velocity at any point in the wake of a ship model can be ôompletély described by 'the

following three- cornDonents: '

1. _{the longitudinal velocity V1 parallel to the axisl}

of the.propeller shaft, '

By means of a fiv-hole spherical pitot tube all three

components can be measured, [7] . However, as far as dynamic

forces nd moments on a propeller are concerned, the radial

velocity component is of little interest. V1 and Vt can be

measurêd with a three-hole spherical pitot tube. This section

is a description of the three-hole spherical pitot tube

designed at the Norwegian Ship Model Experiment Tank for the purpose of measuring the lbngitudini and tangential flow

velocities n way of the propeller plañe of ship models.

Notatiôn in Seòtion 4:

y arctan (V/Vi),

prefix dnoting stañdard deviation,

6 angular distance from stagnation point,

c angular distance between holes of spherical head

pjtqt tube,

p density of fluid,

C,F,R pressures at various holes (Fig. 4.1),

U

V1 longitudinal flow velocity,

Vt tangential flowvelocity,

Vr radial flow velocity,

V (V12 ₊

p0,p1 pressures,

standard deviation of measúred pressure differences, ordinates of calibration curves.

Theoretical outline.

The pressure distribution on the surf4ce öf a sphere in

an incompressible, non-viscous fluid with uniform flow velocity

V is givenby

P1 = Po - .- sin2 6 1 ,2 4

2P

(4.1)

where p1 pressure at a given point on the surface of the sphere,

P pressure jn undisturbed fluid,

desity of fluid,

6 angular distance of the given point measured from

the stagnation point.

From Eq. 11.1 thefollowing expressions can be derived.,

reference [7] , relating the pressures at three different.

points ona sphere in uniform flow, the flow velocity vector being parallel to the plane through the three holes and the center of the sphere,

F - R sin (2c) tan (2y)., ('1.2) (C-F) + (C-R) i cos (2c') C-R _{.2. (cos2} - cos2 (y+c)),

pV2

4

where the anglés c arid y and, the pressures F, C, and R are all

shown in Fig. 4.1. A small Velocity component normal to

will theoretically have no influence upon the pressure differences F-1, C-F, and C-R.

For a spherical pitot tube there will be deviations from the ideal case described by the above equations due to viscosity, finite size of the holes, devjations from spherical

shape, and other inaccuracies in the manufacturing of the pitot

tube. It must therefore be calibrated as described later. The chöice of sphere diameter has to be a compromise based on considerations of the following two effects.

In order to reduce the influence of the sphere upon the f.ow pttern, and to obtajn accurate measurements in äreas of high velocity, gradients, the diameter of the sphere should be as.small as possible.

- 36

2.

As will be shown later the calibration curves at low

velocities are veloci'ty-depèndent.

In order to avoid

this difficulty, Reynolds number should be higTi.

Thus, from the requirement of goperformance ãtlow

velöcities the diameter should be large.

Description of the instrument and data, recording procedure.

Fig. 4.1. shows the main dimensions of the sphere and

the holes

The sphere is fitted to a cylindrical arm A (Fig

4 2)

The pressures at the holes of the sphere are transferred through

axial holes in arm

4

to the flexible tubes (Fig.

1.2).

These

tubes are taken through the hollow shaft and connected to

electric differential pressure cells mounted inside the model.

By' means of the rotatiön

drive mechanism indicated in Fig. 4.3

the shaft can be rotated, thereby displacing the pitot tube

tangentially in steps of 7.5 degrees by just pushing a botton

on the instrument tàbleof the towing

carriage.

The main parts

of the drive 'mechanism are a»toöthed blocking wheel clamped on

the shaft, and á couple

f

electre-magnets.

Displacing of

pitöt tube in radial direction is done manually by depressing

the spring-loaded locking, piece (Fig. 4.2).

One of the main problems of Spherical head pitot tube

wake surveys by the conventional procedure, measuring the

pressures with Water-cplumri manometers, is that the procedure

is very .time-consuming.

For exampJ,e, according to

[8] it takes

several runs of the towing carriage to bujid up the water levels

in the manometers to their equilibrium positions.

Thus a

detailed survey of thewake field of:a ship model, requiring

the velocity components to be measured at some 100 positions

or morè,will be exireiely

time-consuming.

This problem has

been overcome at the Norwegian Ship Model Experiment Tank by

the use of electric differential pressure cells, arranged as

shown jn Fig. 4.3. 'Thrèe pressure cells (Sanborn Model

268B)

are used, eaòh of them

m'e'asuring'the pressu±'ê difference

between two of the holes of thé pitot tube.

The sigials from

the pressure cell-s are amplified by Sahborn carrier frequency

amplifiers (Model 150-1100AS) and recorded by a Sanborn

multichannel recôrder.

In this way, due to the "stiff"

of a frequency of up to' i c.p.s. or rnorecan be recorded. Since the pitot tube can displaced tangetially by just pushing a button, velocily' measu±eniènts can be taken at several positions in the wake field for each run of the

carriage. The time required for the pressure to reach its

equilibrium level can be adj üsted down to a few tenths of a second.

The number of measurements that can be taken for each run of the carriage is liMited by the turbulent

fluctuations of the velocity in the' wake field. In order to

obtain'the mean valuès of the presuredifferences they have

to b recorded over a certaIn period of timê, depending on the

amp].itude and frequency spectrum of the fluctuations. These

are, in turn,' dependent on the position in the wake field. It has been fóurd convenient to apply a certain amount of

damping tò the 'pressure signals, thus facilitating the reading

of mean values from the records. This damping is easily

achiev'êd by.. the use of closed volumes of air as indicated in

Fig. n.a. Air volumes of the order of 10 cm3 have been found

suitable. With this arrangement the number of measurements

that can be taken for each run of the Óarriage is usually between 2 and 6, depending on the position in the wake field. (Correspönding to 10 to 30 seconds recording time pr.. station.)

As an example of the time-fluctuations of thevelocity

described above, Fig. t _{shows the results of measurements}

with a rake pitot tube (yielding axial velocity only) in

connection with differential pressure cêlls. It is' to be

emphasized that the traces in, Fig. 'l have not been recorded

by the three-hole spherical pitot tube. They are included here

to show the necessity of taking the pressure. reading over a

certain period of time to obtain the mean value of the velocity.

Calibration of the three-hole spherical pitot tube.

According to Eqs. i.2 and Li.3 the calibration curves

38

-a cutve giving (F-R)/((C-F) + (C-R)) versus y, -and,

2. a curve giving (C-R)/V2 versus y.

These two curves should theoretically be independent of the magnitude of V.

The calibration has been carried out by mounting the instrument on a platform under the carriage, with the spherical

head submerged. Pressure difference readings have been taken

for different values of y and V.

The result is given in Fig. t5

It has been found that the curVe giving (F-R)/((C-F)

4 (C-R).)

versus y is practically independent of V in the velocity range

above ca. 0.35 rn/s. However, the curve (C-R)/V2 (or (C-F)/V2

for negative ) is independent of V only for veìociUes above

ca. 1.5 rn/s. Fig. L.5 shows ompiete calibrtion curves for

the velocities Ï.5, 0.7, and 0.35 rn/s. A few òontrol

measure-ments at velöci.ties above 1.5 rn/s coincide with the 1.5 rn/s curve.

Computer program for data processing.

The calculation

of

the longitudinal and tangential

velocities from the recorded pressure differences is a tèdious

and time-consuming procedure. Therefore a computer program

has been written for this purpose a The recorded pressure

différences are entered as input data for the program. The

expression (F-R)/((C-F)+(C-R)) is calculated first, and the angle y is obtained from calibrations data (the lowest curve

in Fig. '4.5). At this value of y the value of (C-R)/V2 (or (C-F)/V2 if y. is negative) is.taken from the uppermost curve,

and V is calcúlated. The value of V obtained in this way is

then used for interpolation bétweéh the three upper curves to

V is calculated. The longitudinal velocity V1 and the tangential velocity Vt are finally calculated from the relations

V1 V

cos y,

V. V sin y.

In the program the calibration curves are represented by a piece-wise approximation with second-order polynomials

Discussion of instrument accuracy.

The most important sources of error in the values öf

and V obtained by measurements with the pitot tubé are

I. inaccuracy in the reading f preséure differénáé,

2. inaccuracy of the ca1ibratio òurvês.

Formulae for calculation of the standard deviations of, V1

¡nd V expressed by assumed standard deviations of the

pressure difference readings and the calibration curves, can be deduced from the statistical theory of ero

propagation. The prefix A is used here to denote the

standard deviation of a single measuremént.

Since V1 V cos y

we have

AV1 .? V AV)2 +

(lA)2)

In the same way

((sin y V)2 + (V cos -r

The squaring of each term in Eq. ' is a slight violation

of the laws of error propagation because V and y are not

completely indepefident, but partly originate from the same. source of error, namely the inàccuracy in pressure reading. However, to avoid: unnecessary complications, it will be

used in the rough analysis presented here.

Since Vis calcuÏatedfrom the expressiön

.( (C-R)/y,)

we have

y2

whiäh, after sothe manipulation, can .be written

_.].

_(&(c-R2

2Vy2

+

V

('4.7)

in the equations above is the value of (C-R)/V2 finally

obtained by nterpòlatio between the three upper curves in

Fig. '4.5. Two factors contribute to y2,

1. the standard deviation of y, causing y2 to be read

from the upper curves in Fig. '4.5 at an cct1rate

abscissa,

2 the inaccuracy 'y2 resulting from inaccurate

calibration curves and 'the inaccuracy involved in the ir±erpolatìon between them.

= r is given by Av (AtJ)2 +(Aty1)2)k yi where F-R (4.10) (C-F) + (C-R)

and y1 is the lowest curve in Fig. 4.5 (the curve giving U versus y).

Eq. 4.lO gives

¿U (( ¿(F-R))2 + ( U ¿(C-R))2 ?(F-R) _?(C-R) (

;J

¿(C_F))2) C-F)

For the sake of simplicity we shall make use of the

approximation that the measured préssure differences all

have the same:standard deviation Ap,

¿(F-R) ¿(C-R) ¿(C-F') ¿p.

After some manipulation the above expression for ¿U, may then be written

AU tJ Ap (i + 2U2) (F-R)

Now, values of y1 arid y2 .ari be estimated from the

calibration data. _p is estimated from the accuracy with

which the pressure differences are recorded. Calculation of

ÊU from Eq. '4.11,and insertion into Eq. '4.9 yield ty.

Further substitution into Eq. '4.8 and Eq.'4.7 gives

V,

and.

then the resulting standard deviations and are

calculated by using Eqs. '4.5 and 4.6.

Results of this. calculation are given in. Fig.'4.6. Here, p, t'y1, and ty2 have been given the following values,

which are considered to be ypical for the. pitot tube in use:

From Fig 4.6 it can be concluded that for V 1.5 mis the

standard deviation of V1 will be of the order o.f 2 % of V.

For V = 0.7 rn/s and V = 0.35 rn/s the standard deviation will

be about 2.5% and 5 %respectively. Further it

is

inter-estingto nó.te that increases for decreasing

Iii.

The standard déviation of Vt shows the opposite effect,

Vincreasing with increasing

Iii.

Vt is from 0.5 tó 3.5

per cent of V, depending on V and

Iii

as shown in Fig. 4.6.

In the above discussion the following sources of error have been neglected:

The finie size of the sphere, giving rise to a

disturbanceof the flow field,and introducing an additional error in areas of high velocity

gradients y(degrees) 0 10 20 30 &'y., (mmw (m/s).2) 0.03 0.5 0.0'4 0.6 Ó.06 0.8 Ó.20. 1.0 V (mis) 1.5 O.7 Ó.35 p(mmW) 0.5 0.3 0.2

2.. The pressure difference C-R i proportional to the

square of the flow velocity. Therefore, if the

velocity is fluctuating with time, the velocity components measured with the pitot.tube will be their root mean square values and not their mean values.

A radial velocity component, if any, may ha.ve some influence on the calibration, curves.

SECTION 5: INVESTIGATIONS ON A SHIP MODEL WITH A

CLEAR-WATER TYPE OF STERN.

This section contains the results o an experimental

investigation, of the dynamic thrust, torque, and bending moment on one blade of a propeller working behind a tanker model with

a stern of the clear-water type (Fig. 5.1). Special attention

is paid to the influence of propeller clearanóes upon the

-dynamic forces. Two-dimensional wake field data ar also given.

Notation in Section 5:

a angular position of propeller blade (Fig. 2.1),

phaseangle of w

l,m l,m

t,m phase angle of

phase angle of rn'th harmonic of thrust,

B breadth,

CB block óoefficient,

T draught,

N shaft r.p.m,

peak-to-peak value öf thrust fluctuations on one blade,

T(a) instantaneous thrust on one blade,

Q(a) contribution of one blade to total propeller torque,

Vi

Vt

V VM X a b tm g1 W1 wt w l,m Wt,m

DesöriPtiqflof model and test conditions..

Experimental InveStigations wre cärried out, on twO tanker models, one of them having a clear-water type of stern,

the other having the rudder-shoe type o This sectiöml deals

exclusively with the former. The latter is to be dealt ith in

Section 6 of this zepört.

Main particûlars of ship with clear-watar Stern:

L. 250.00 m pp B

42.00m

T l5. 2.4 m CB 0.80 ¿V 128552 300Ò0 hp N = L1Or.p..m. V 17 knots.

longitudinal flow velocity, tangêt'±al flOw velocity, ship speed,

modl s,eed,

rad.ius/tp radius,

ciearance be.tween propeller and rudder, in percent

of pope1e diameter (Fig. 5.2),

clearance between propelle and sternframe, in

percent pf prope.er diameter (Fig. 5.2.),

single amplitude of mTth harmonic of T(a), Q(), ad G(a) respectiviy, in percent of mean value,

i _(Vi/VM),

Vt/VM,

sngle amplitude of m'th of w1 and

where

VM model speed.

The direction of positive Vt is, by definition,

counterclock-wise when looking forwards. Figs. 5.6 and 5.7 show w1 and w

fOr the extreme aft position, b 7.9% (position D, Fig. 5.2).

Fig. 5.1 shows the lines of the model (SMT Model. Mo. 699).

The model scale was A O:l. The dynamorneter pròpéller

described in Sec. 3 of this report was originally designed

for this hull model.

Ail

measurements described in the

subsequent paragraphs were carried out at DWL.

Wake surveys.

Mapping of the longitudinal and tangential components of the wake field was carried out with the three-hole spherical

pitot tube and pressure cells described in Sec. of this

report. Measurements were taken at four different axial positions A, B, .C, and D as shown in Fig. 5.2, corresponding to clearances b of 13.7, 23.5, 33.2, and L79 percent of

propeller diameter. For each axial position measurements were

taken at 5 different radii, and for each radius at 28 angular

positions (steps. of 150). The measurements weré carried Out

with the rudder removed.

Figs. 5. and 5.5 show the lôngitudinal flow velocity

V1 _{nd the tangential flow velocity Vt for axial position A,}

corresponding to b 13.7 %, the velocities being presented in

the dimensionless forms V, w1 i

-VM

A detailedand more useful picture of how the wake field depends upon the clearance b is given in Figs. 5.8 through

5.13.

These diãgrãms show the results of a

intervals

númeriòal Fourier analysis of the dirdumferential

dstri-butioñ of w1 arid w..

The Fourier séries represenl:ationS

of w1 and w.

are

m

Wi,m

sin (fia.

+

m0

- 46

Wt,rn

sin (ma +

An examination of the diagrams shows that

the even harmonics

of both w1 and w

geherally decrease with increasing b.

The

odd harmonics do not show thé same régular

dependence on b.

In Fig. 5.14 the magnitde of the

various harmoniçs can be

compared directly for öne single value of

b.

Since the results given in Figs. 5.8

through 5.14

are not based on'repéatd

measurements, the repro

c'eabi]i:y

can not be estitha'ted

directly.

However, judging from the

regularity of the plotted points and the

discussion of the

pitot tube accuracy given

n Sec. 4, the

standard devj'tions

of thé w1

and

valueé plptted in Figs. 5.8 through

5.14

+

are believed to böf the

ordér of

0.01, or less.

Dynarriic forces and moments.

Réóbrding of static and dynamic thrust, torque,

and

bending moment wa.

carried out as ordinary still-water

propu1s'iòntstS tdth the dythometer described in Seò. 3.

Fig.' S.3 shows thé friction correction

applied.

FIg.. 5.16

shows

setup a

The

.gna-ls from the

dynéinömetr weré recorded by a

galvanometer recorder.

Pulses generated by the azimuth contact

of the slip ring

assembly inditéd the instantaneous angular position öf the

propeller on a separate channel of the

recorder.

Time,

indicating pulses,from a frequency-stable generator were

The recorded signals from th,e dynamometer were analysed by selecting one propeller revolution on the

oscillogram. Readingsof signal amplitudes were taken at

2L1 equidistant stations (steps of 15°), and analysed on

a UNIVAC 1107 computer. The amplitudes and the model

speed and shaft c.p.s. were entered as input data for the

computer program. Thrust, torque, and bending moment were

first calculated at each of the 2 stations, taking into

account all the calibration constants described in the

paragraph dealing with calibration in Sec. 3. Numerical

Fourier analysis was then performed and the harmonic

components corrected in accordance with the dynamic correction

terms derived in Sec. 3. The computer output gave the

corrected amplitudes and phase angles of the harmonic compo-nents, and the thrust, torque, and bending moment at each of

the 2 stations.

In order to study the influence of propeller

clearances upon the dynamic forces and moments, measurements were taken with the rudder and propeller in various positions. The clearances were changed by moving propeller and rudder

parallel to the propeller axis. Fig. 5.17 shows the propeller

and rudder arrangement. For the lowest values of

propeller-rudder clearance a cut-out for the propeller boss had to be

made in the rudder. The combinations of clearances for which

measurements were taken, were (with a and b given in percent of propeller diameter):

For

a9.8%:

b 33.2, '43.9, 57.7 %.

For .a = 19.6 %:

b 23.5, 33.2, '47.9 %.

For a (rudder removed):

F,igs. .5.] hrough 5,.,2 show the. resulting Fourier

components (single amp itudes) o.thrusttm, torque and

bending. moment on one blade., expressed in percent of mean thrust, iean torq.u, and mean bending moment on the blade..

With this

notation.

the. instantaneous thrust T(a) is

T

T(a) mean 100

Corresponding expressions can be written for the contribution Q(a) of one blade to the total propeller torque, and the

bending moment G(a) dueto the..thrust on one blade.. m=1

corresponds to shaft c.p.s

For each combination of a and b measurements were

taken at 'model. speedsof 1.18, 1.30, l.'46, and 1.61 mIs. This was originally done in order, to measure any influence of model

speed. upon the: dynamic. forces. ...However, the reproduceability

of the nasuremexvts was not goqd enough to justify and definite conclusions regarding the influence of model speed upon:'the

various q, and. . . -.

The Fourier components plotted in. Figs... 5 f18 through 5.2t, are. themea values obtained by. averaging, the results of the. four measurements (one measurement at,each model speed). The standard deviations indicated by arrows in.. the diagrams have been calculated from the formula

(Y(deviat.ion from mean vaiue)2

L ('t-l)

Standard deviations statistically calculated on the bàsis of only four measurements are not very acòuráte, of course.

However, they still represent a quantitative indication of

the reproduceability of the measurements The relatively

large spread obtained for thèse measurements is partly due to the fact that the measurements were taken at different

ments were taken at different speeds. _{the main reasàn for the}

spread indicated in Figs 5.18 tlirough 5.24 is therefôre möst

likely the time fluctuatiQns of thé wake field. _{By examination}

of Fig. 4.4 it becomes evident that the dynamic forces may

ch.nge considerably from one revolution of the propeller ±0

another.

Figs. 5.18 and 5.19 show the results of the thrust

measurements, each harmonic component being given as a

function of thé clearance b ànd. ith the. ciearänce a as a

parameter. The. first harmonic t1 (the shaft frequency

component) seems to be reduced by the _{presence öf the}

rudder. For t2 it is clear that a 9.8 % gives a higher

amplitude than a 19.6 % and a (rudder removed). For

the higher harmonics it is hard to draw any definite

conclu-sions conccrnì.ng the influence of clearances. The diagrams

only give a rough indication of thé magnitude of each component.

The results of the torque measurements are shown

in Fig. 5.20, 5.21, and 5.22, _{The conclusions to be drawn}

from these diagrams are generally the _{same as for the}

corre-sponding thrust components, q1 tends to dècrease while q2

definitely increases with decréasing _{rudder clearance a.}

Increasing the clearance b _{seems to reduce q1 and q2 slightly}

while the influence upon the higher harmonics is more uncertain.

The. same can be said about the bending _{moment fluctuations shown}

in Figs. 5.23 and 5.24. -It is _{noteworthy that, except the}

third and fifth harmonics, the components of .bending moment

fluctuations are considerably ldgher than the _{corresponding}

thrust components. This means that "the bending _{moment f1uc.tua.}

tions are not only due to variationS in _{the thrust force,}

but also to the fact that the point of application of the

thrust force on each blade is moving.. Fig. _{5.25 shows an}

example of the locus curve of this point of application

50

-of the circumferential variation -of thrust, torque, and

bending moment, de to the first six harmonics.

The following table shows the phase angles of the

thrUst, torque, ana bending moment components for b: 33.2 %,

a : 19.6 %. The angles given in the table have been obtained

by averaging the results at the four model speeds. The

standard deviations of the tabu-lated angles are of the order

of 10 degrees for m]. and inceasing to about 40 degrees for

m:6. The variations Of phase angles with varying clearances

are in most cases too small tO be detected due to this relatively high spread.

SECTION 6: INVESTIGATIONS ON.A SHIP MODEL WITH A RUDDER-ShOE TYPE OF STERN.

This section contaiñs the results of a experimental

investigation of thé wake field añd the dynamic thrust, torqué,

and bending moment or one blade of a propeller working béhind

a tanker model with a stern of the rudder-éhöe type (Fig. 6.1).

Secia1 attention is

paid

to the iñfluence of propeller

clear-ances upon the dynamic forcés.

Notation in Section 6: Same as seOtiot 5.

m 1 2 3 4 5 6 T,m (degrees) 13 95 253 102 218 53 Q,m 3 89 2146 86 198 58 C,m 33 110 2914 1'4l 296 160

Description of riode1 and test cqnditions.

The model of a 33.850 T.D.W.. tanker was chosen for

object of the investigations described here. The main

particulars öf the ship were.

L 198.73 in pp B 26.21 rn T l0.'47 m CB 0.79 'p3073

Fig. 6.1 shows thê shape of the afterbody. The model scale

was À 29:1. The measurements described n the following

paragraphs wee takena.t DWL and at a model speed VM: 1..'47 in/s

corresponding to a hip speed of knots... At this model

speed the number of shaft revolutions pr. séç. were aböut

11.5 cps, corresponding to N 128 r.p.m. fòr the siip.

.The friction öorrection. applied to the model during,.

propulsion tests was 1.16 kp. it i to be noted, that the

propeller has not been des.ignedspecially for this ship model.

It is the propeller described in Sec. 3 of this report.

Wake surveys.

Mapping of the longitudinal and tangential components of the wake field was carried out with the three-hole spherical pitot.tube and pressure cells described in Sçc. 4 of this

report. Measurements were taken at the three axi.l posi.tioñs,

A, B, and C, slown in Fig. 6.2. corresponding to propeller

clearànces b of 8.8, J,6.Q, and 25.8 percent of propeller

diameter.. For ea1axiai position. measurements w..r takén at

5 different radii, 40, 55, 70, 80, and 95 % of propellér tip radius, and for each radius at 28 diffe'ent angular

positions (steps of 1.5°). All wake measuremeits.were carried

out with the rudder removed.

Figs. 6.4, 6.5, and 6.6 shpw the longitudinal flow

velocity V1 and the tangential flow velocity \/ for the three