Model tests concerning the damping coefficient and the increase in the moment of inertia due to entrained water of ship’s propellers

REPORT No. 31 M APRIL 1960

STUDIECENTRUM T.N.O. VOOR SCHEEPSBOUW EN NAVIGATIE

AFDELING MACHINEBOUW - DROOGBAK lA - AMSTERDAM

NETHERLANDS RESEARCH CENTRE T.N.O. FOR SHIPBUILDING AND NAVIGATION ENGINEERING DEPARTMENT - DROOGBAK la - AMSTERDAM

MODEL TESTS CONCERNING THE DAMPING COEFFICIENT

AND THE INCREASE IN THE MOMENT OF INERTIA

DUE TO ENTRAINED WATER OF SHIP'S PROPELLERS

MODELONDERZOEK NAAR DE DEMPINGSCOEFFICIENT EN DE VERGROTING VAN HET MASSATRAAGHEIDSMOMENT TENGEVOLGE VAN HET MEETRILLENDE WATER VAN

SCHEEPSSCHROEVEN

BY

N.J. VISSER

(HIEF ENGINEER - WERKSPOOR N.y. AMSTERDAM

JL1O

THIS REPORT IS NOT TO BE PUBLISHED UNLESS VERBATIM AND UNABRIDGED

RESEARCH COMMITTEE

Prof. Dr. Ir. J. J. Kocn

ProL Dr. Ir. W. P. A. VAN LAMMEREN

Dr. Ir. W. J. MULLER

Ir. W. H. C. E. RÖSINGH Ir. E. STRUYK

CONTENTS

page PART I. FESTS WITH NON-ROTATING PROPELLERS

Summary 4

Symbols 5

Purpose of the measurements 5

Test procedure 6

Model propellers used 7

Evaluation of the test data 7

Effect of the specific gravity of the propeller 8

Accuracy of the results 8

Correction of the results 8

Results and conclusions 9

Increase in the moment of inertia 9

Damping factor io

Water height above the propeller 10

PART II. TESTS WITH ROTATING PROPELLERS

I. Purpose of the measurements 12

II. Test procedure 12

1. Experimental arrangement and method 12

2. Model propellers used 13

3. Evaluation of the test data 13

Increase in the moment of inertia 13

Damping 13

4. Accuracy of the measurements 14

III. Measurement of the moment of inertia 14

i. Results 14 2. Conclusions 14 IV. Damping 15 I. Results 15 2. Conclusions 16 V. Discussion 17 VI. Appendix I 19

Comparison with the results obtained in part I 19

VII. Appendix II 20

SUMMARY

The experiments were performed in a towing tank with model propellers.

The experiments with non-rotating propellers were carried out

using 36 model propellers of the B-3 and B-4 series of the Netherlands Shipbuilding Experiment Tank at Wageningen. Model propellers of the B-4 series only were used in the rotating and towed propeller tests.

In the first case the torsional vibrations were generated by two

Philips exciters coupled together.

In the second case the excitation was generated by the driving motor with the aid of an alternating current adjustable in regard

to frequency and voltage.

The increase in the moment of inertia of the propeller due to

en-trained water is calculated in both cases from the decrease in the natural frequency of the torsional vibration system.

It is shown that the mass of the entrained water increases linearly with the pitch.

The non-rotating propellers indicated an increased mass coupling compared to the rotating propeller.

The mass increase appears to be independent of the r.p.m., the

towing speed and the slip.

The investigation does not give information about the influence of scale effect, flow inhomogenity and flow restriction.

Comparison with shipboard measurements is desirable.

In the first case the mean increase in the moment of inertia is

approximately 38%.

In the second case the corresponding increase is approximately 35%.

It is normal practice to consider a 25/ mean increase in the

moment of inertia.

In the first case the damping is determined from the decrease in

the angular amplitudes during the decay of the vibrations.

In the second case the damping is determined from the phase angle between the angular amplitudes of two points of the system. The methods used to determine the damping coefficients were not

particularly accurate but enabled the effect of various factors on the

kgcmsec' kgcmsec' kgcmsec2 kgcmsec' kgcmsec' kgcmsec' kgcmsec' kgcmsec' kgcmsec' sec sec-'

kgcmrad'

kgcmrad'

This investigation has been made to determine the

increase in the moment of inertia of a propeller submerged in water resulting from the vibration

of the surrounding water. This increase is

depend-erit on a number of factors, of which the

pitch-diameter ratio of the propeller, the number of

I. SYMBOLS

II. PURPOSE OF THE MEASUREMENTS

blades and the blade area ratio are the most im-portant. At the same time, the effects due to the

frequency, the amplitude and the height of water

above the propeller were examined.

Damping measurements were also carried out.

5

I. Moment of inertia of the

pro-peller

'1 Moment of inertia of the

pro-peller in air

'2 Moment of inertia of the

pro-peller in water

I

Moment of inertia at the

propeller due to the shaft etc.

I, '2 Moments of inertia before

deducting It

1m.

Moment of inertia of the

driving motor rotor

Moment of inertia of the

pro-peller in air with Vi

J Moment of inertia of the

pro-peller in water with y,

I,,

Moment of inertia of the

pro-peller in air with V2

ISO Moment of inertia of the

pro-peller in water with y,

y1 Resonance frequency of the

torsional vibrations in air y, Resonance frequency of the

torsional vibrations in water

C Torsional rigidity of the shaft

between the motor and the propeller

C2 Torsional rigidity between

the first torsiograph and the propeller

Logarithmic decrement

n Number of vibrations sec-'

D Damping ratio with respect

to critical damping

Damping coefficient per unit

angular velocity kgcmsecrad

V Propeller volume cm3

r1 Inertia radius cm

a Amplification factor

Vi, V2 Specific gravities of the

pro-peller kgcm3

e Natural logarithm base

d Shaft diameter cm

g Acceleration due to gravity

cmsec'

M0 Excitation couple (maximum

value) kgcm

0, 0m, & Angular amplitudes at

the propeller, motor and first

torsiograph rad

, cpj, Phase angle between O and

M0 and between 0s0m

and M0 rad

w Circular frequency radsec

ß Phase angle between 0, and

eilt rad

HID Pitch-diameter ratio of the

III. TEST PROCEDURE

Fig. 1.

The increase in the moment of inertia of ship's propellers was determined by means of model

pro-pellers in a simple vibrating system. The decrease in the natural frequency of the torsional vibration of the first degree of this system, after submerging the propeller, is a measure of the increase.

The system was constructed as follows: a small

shaft is clamped in a block, the mass of which is assumed to be infinitely large. A model propeller

was attached to the free end of the shaft. The

block rests on a stand (see Figure 1).

The shaft with the model propeller is set into

torsional vibration by means of a butterfly-ring

by two exciters which are attached to the support.

A couple is

applied through both exciters so

that no bending vibrations are induced. The

exciters

are connected in series

and suitably

block

oscillator

exciter

are p life r pick op

dlsplacemenr

height

oscilloscope

matched in regard to amplification and phase

displacement.

A displacement recorder, fixed to the block, re-gisters the torsional vibrations via the

butterfly-ring. These vibrations are converted into an alter-nating voltage and are made visible on an oscillo-scope (see Figure 2).

The propellers used in these tests comprised six series each of six propellers, 36 in all. These series are referred to by the letter-figure combination:

B-3-35 B-3-50 B-3-65 B-4-40 B-4-55 B-4-70

V. EVALUATION O

The vibration system comprises one mass and one degree of freedom. The known formulae for this

system are:

42v12 = - and 4t2v2 . . . . (1)

which give:

'2 y2

'1 22

The increase in the moment of inertia due to the

entrained water is usually expressed as a

percent-age of the moment of inertia of the propeller in air. This is the reason for determining the ratio

The percentage increase is:

x 100%

J()2

i x 100% (3)

'1

I'2

J

To measure the damping coefficient, two ampli-tudes are measured on the photographic recording at a distance of n vibrations. The ratio of the two consecutive amplitudes, for the case of pure fluid

damping, which is here assumed, is equal to e, where 5 is the logarithmic decrement. The

am-plitude ratio after n vibrations is therefore e1ò. if

this is determined by the above-mentioned method,

6 can then be calculated.

The damping ratio D is given by:

2rD

6=

(4)

ViD2

(2)

IV. MODEL PROPELLERS USED

The damping ratio is determined by setting the propeller in resonance by means of the exciters.

The exciters are then suddenly switched off, after which the vibration dies away due to damping. If the signal is shown with a sufficient low timebase

on the oscilloscope, the amplitude behaviour of the

phenomena can be recorded photographically.

The first number indicates the number of propeller

blades, the second the blade area ratio. Each

series consisted of propellers with different

speed-diameter ratios. The corresponding HID ratios

for each series were:

0.5 0.6 0.8 1.0 1.2 1.4

F THE TEST DATA

Since D is very small and therefore D « i (see

Table 2) we get:

2r

(5)

For the derivation see TIM05HENK0, Vibration

problems in Engineering third edition page 72, which gives:

D=2

(6)

hence

(7)

To determine the increase in the moment of iner-tia as a function of the height of water above the

propeller the same system was used. However, two

other propellers were used as model propellers, of

which the moments of inertia were calculated from

a pendulum test as 39.5 x l0

kgcmsec2 and

28.3 x l0 kgcmsec2 respectively. The

calcula-tions were carried out as before. The height of the water was measured as the distance between the lower plane of the propeller normal to the shaft, and the water surface (see Figure 2).

The increase in the moment of inertia as a function of the water height is again obtained from the

decrease in

frequency with increasing water

The material of the model propellers is not always the same. If it is desired to compare the results of

the various tests, this factor must be taken into account.

The increase in the moment of inertia due to the

vibrating water is usually expressed as a

percent-age of the moment of inertia of the propeller in air.

In the case of a propeller of some material, differ-ent from that of the model propellers employed, the increase calculated as a percentage of its own

moment of inertia in the absence of water is

Y2

8

VI. EFFECT OF THE SPECIFIC GRAVITY OF THE PROPELLERS

The only error affecting the calculation of the

in-crease in the moment of inertia, is due to the

inaccuracy of the exciter. The maximum error of this apparatus is 0.25% of the induced frequency.

With vibration in air the resonance is very sharp, so that the maximum adjusting and reading error

is about 0.1 c.p.s. and therefore amounts for the

lowest frequency measured 41.1 c.p.s. in air and

29.9 c.p.s. in water with the B-3-65; HID = 1.4 to 0.244% of the natural frequency in air. With vibration in water the resonance peak is less sharp,

so that 0.4 c.p.s. or 1.35% of the natural frequency

in water must be accepted as the adjusting and reading error. According to formula (2):

'2

r2

Il

r2

In accordance with the method of calculating the

Some preliminary experiments were made with

three calibration discs instead of model propellers.

The dimensions of these discs were known

accurate-ly, so that the moments of inertia could be calcu-lated. These values were 0.032555, 0.025560 and 0. 015986 kgcmsec2 respectively. The shaft

dia-meter was 13 mm and the length 900 mm. For

the purpose of accurate calculation, the moments

of inertia of the shaft, the washer and the nut

were taken into account.

VII. ACCURACY OF THE RESULTS

VIII. CORRECTION OF THE RESULTS

times greater than that for an identical propeller made of the same material as the model propellers. A comparison of two similar propellers of different

materials gives: '21 = 111+a11, 122 I,2+a111 = V

= 112+ayi rt2

g vi V

'22 = Ii2+_cc_y2rt2

V2 g

+a

= I,2+a112 = 12(1

'

>'2 >2

For the torsioned section of the rod:

ly

_{-. - d41 kgcmsec2 =}

3 g 32

0.000066 kgcmsec2 . . . (10)

The moment of inertia of the threaded section, together with the washer and nut was estimated at 0.000100 kgcmsec2. Together, this produces

what will later always be called the ,,added"

moment of inertia:

= 0.000166 kgcmsec2.

errors, the error in is twice the maximum error

of r1 plus twice the maximum error of r2. The total error is therefore:

2(0.244+0.25)% + 2(1.34+0.25)% = 4.168% For the highest frequency measured, 58 c.p.s. in

air and 55 c.p.s.

in water, with the B 4 40;

HID = 0.5 the total error is:

2(0.174+0.25)% + 2(0.73+0.25)% = 2.8% The damping determination is inaccurate. The recordings of the damping in air do not indicate

an exponential but a linear decay of the vibrations. This is probably due to the fact that the damping

is not viscous. The damping in water also frequent-ly shows deviations and side effects. The results can

only be taken as a rough determination of the

damping coefficient.

The total moments of inertia determined from the measurements and calculated from

C C

enI2=

(2rvi)2 (2v2)2

with C = 2586 kgcmrad1

must now be reduced by the added moments of

inertia to obtain the values 1 and 12, thus:

1=1It and 12=12t1t

. . . (11) 20 20 1.0 20 H/D

IX. RESULTS AND CONCLUSIONS

The experiments with the calibration discs gave the following results:

The error is of order of 1%.

1. Increase in the moment of inertia

The results of the experiments are contained in

Table 1. The results of each series are summarised graphically (see Figures 3 and 4).

o o 4 20 o 60 o 40, 20 o 9

disc No. calculated frequency measured frequency

I II III sec' 44.7 50.4 63.6

sec'

44.4 50.0 62.8 0.5 06 08 10 12 1.4 HID Fig. 4. 0.5 06 08 10 12 14 Hf D 0.5 06 0.8 10 12 14 HID 0.5 0.6 08 10 12 1.4 HID 0.5 06 08 10 12 1.4 HID Fig. 3. 0.5 06 08 10 12 i 4 60 o o 40 o B-3-50 60 o o 40 0 80 o_e 60 0 60 o e 40 20 o

Conclusions

The graphs for each propeller series, show with very fair approximation a straight line. For both the 3-bladed and 4-bladed series, the entrained water increases with increasing blade area ratio and HiD ratio, only the values of all series for

H/D-0.5, show close agreement. Table I

Conclusions

Because of the inaccuracy of these measurements definite conclusions cannot be made. The values given in the following tables must be considered

as only approximate.

3. Water height above the propeller

The experiments were performed with three dif-ferent shaft diameters and two difdif-ferent propellers 10

2. Damping ratio

Table 2 shows the value obtained for the damping ratio D.

The side effects already mentioned were in some

cases so disturbing that no results could be

ob-tained

This is the reason for the gaps in the

table.

Table 2

with moments of inertia of 39.5 x I0 kgcmsec2

and 28.3 x 10-e kgcmsec2 respectively, (propellers

nos 930 and 1022). The results are summarised in Figures 5 and 6. The most marked change in fre-quency occurs with water heights between O and

30 mm, while with water heights exceeding 80 mm

the variation is practically nil.

The maximum variation in the moment of inertia therefore also occurs between O and 30 mm water propeller

series

propeller

no. HID 's/hi _Ii .100%

B-4 40 648 0.5 1.113 11.3 150 0.6 1.135 13.5 151 0.8 1.222 22.2 152 1.0 1.274 27.4 153 1.2 1.348 34.8 154 1.4 1.431 43.1 B 4 55 649 0.5 1.120 12.0 155 0.6 1.132 13.2 156 0.8 1.283 28.3 157 1.0 1.359 35.9 158 1.2 1.411 41.1 159 1.4 1.543 54.3 B-4-70 666 0.5 1.125 12.5 665 0.6 1.191 19.1 664 0.8 1.294 29.4 663 1.0 1.403 40.3 662 1.2 1.581 58.1 661 1.4 1.651 65.1 B-3-35 650 0.5 1.120 12.0 300 0.6 1.146 14.6 301 0.8 1.224 22.4 302 1.0 1.312 31.2 303 1.2 1.354 35.4 304 1.4 1.459 45.9 B-3-50 651 0.5 1.138 13.8 290 0.6 1.151 15.1 291 0.8 1.229 22.9 292 1.0 1.397 39.7 293 1.2 1.490 49.0 294 1.4 1.549 54.9 B-3-65 660 0.5 1.138 13.8 659 0.6 1.198 19.8 658 0.8 1.300 30.0 657 1.0 1.402 40.2 656 1.2 1.494 49.4 655 1.4 propeller

series propeller_no. HID D in air D in water

B-4-40 648 0.5 0.0037 0.0052 150 0.6 0.0017 0.0055 151 0.8 0.0014 0.0061 152 1.0 0.0047 0.0056 153 1.2 0.0032 0.0052 154 1.4 0.0018 0.0099 B-4-55 649 0.5 0.0021 0.0069 155 0.6 0.0016 0.0058 156 0.8 0.0019

-157 1.0 0.0018 0.0069 158 1.2 0.0049 0.0068 159 1.4 0.0015 0.0071 B-4-70 666 0.5 0.0012 0.0053 665 0.6 0.0006 0.0058 664 0.8 0.0006 0.0059 663 1.0 0.0006 0.0067 662 1.2 0.0006 0.0069 661 1.4 0.0007 0.0091 B-3-35 _{650 0.5 0.0023 0.0046} 300 0.6 0.0032 0.0055 301 0.8 0.0019 0.0072 302 1.0 0.0026 0.0089 303 1.2 0.0013 0.0080 304 1.4 0.0016 0.0094 B-3-50 651 0.5 0.00 13 0.0053 290 0.6

-

-291 0.8

-

0.0065 292 1.0 0.0009 0.0066 293 1.2

-

0.0076 294 1.4 0.0025 0.0073 B-3-65 660 0.5 0.0017 0.0077 659 0.6 0.0013 0.0055 658 0.8 0.0018 0.0063 657 1.0 0.0019 0.0056 656 1.2 0.0026 0.0060 655 1.4 0.0036

-45 40 35 30 5 200 E E 150 100 propeller no. 930 shaft diameter 15.5 shalt diameter 13 shaft diameter 1Z.

Fig. . Relationship between frequency and water height.

height, while with a height of water greater than 80 mm practically no variation occurs.

In addition the excitation force was varied, by varying the current strength, which corresponds

to variation in the amplitude of the torsional

vibra-tions. These data were used to calculate the

average value of the frequency corresponding to a given water height. The frequency variation as

a function of the impulsive force is shown in

Figure 7 and appears to be small, i.e. about 0.6%

which lies inside the limits of accuracy of the measurements. 50 44 42 40 34

li

propeller shaft no. 930 shaft

diameter shaft diameter

dIameter 13 12 15.5

-'IP!_

i-50 29 30 31 32 33

increase in moment of inertia

Fig. 6. Relationship between the moment of inertia increase and the water height

o 50 100 150 200

water height in mm

o 0.05 0.10 0.15 0 20 0.25 0.30

Current in amps

Fig. 7. Variation in frequency with current variation.

60 t 55 50 56 54 152

The measurements described in this section can be considered as supplementary to these in Part I. Whereas previously the model propeller did not

rotate, it was now tested while rotating in

un-disturbed flowing water. The effect of the vibrating

1. Experimental arrangement and method

An attempt was made so far as possible, to

re-produce a two mass system. This was achieved by means of a system comprising the motor, which

The principal reason why this system does not

completely satisfy the requirements was the pres-ence of the heavy shaft emerging from the tube of the towing model. Although it was sturdily made

it was not entirely rigid, and because of its relative-ly great length, it displayed a certain elasticity.

The reason why this length had to be so great was that the propeller had to be towed. This was

nec-essary to allow the propeller to rotate in a flow

PART II

TESTS WITH ROTATING PROPELLERS I. PURPOSE OF THE MEASUREMENTS

II. TEST PROCEDURE

Fig. 8.

water is determined as before i.e., by determining the increase in the moment of inertia of the pro-pefler and the damping effect due to the

surround-ing water.

also served as the exciter, the propeller shaft, which

formed the elastic element, and the model

propel-1er (see Figure 9). This system was accommodated in a towing model which was towed through the water (see Figure 8).

field as homogeneous and undisturbed as possible. If the propeller were pushed the flow field would be of such a disturbed character that reproducible measurements would be very difficult.

The shaft was driven by an adjustable speed elec-tric motor; this motor also served as the vibratory torque exciter, the intensity and frequency of the torque being adjustable.

armature by means of which it was possible to combine the driving motor and vibratory torque

exciter in a single machine, thus producing a very simple vibrating system.

The stator winding of the motor was supplied with a variable frequency three phase alternating

cur-rent. The rotor was supplied with direct current

and in addition, with an alternating current which could be adjusted both in regard to frequency and

phase. This alternating current provided the tor-sional excitation of the system.

As described in the experiments in Part I, the in-crease in the moment of inertia of the propeller in water is calculated from the natural frequency of

direction of travel

Model propellers used

Of the propellers used in Part I only the series B-4-40, B-4-55 and B-4-70 were used for these

experiments i.e., three series of six propellers. As previously explained the figure 4 indicates the number of blades of the propeller, the figures 40, 55 and 70 the blade area ratios. Each series con-sists of propellers with different HID ratios, i.e.,

HID = 0.5 0.6 0.8 1.0 1.2 1.4.

Evaluation of the test data

a. Increase in tile moment of inertia

For purposes of calculation the vibration system

Fig. 9.

the system. The results are determined by means

of two identical torsiographs, which produce a

true phase record of the torsional vibrations. The

outputs from the torsiograph were registered on

a pen recorder after amplification. In order to

determine their phase - relation the combined

sig-nal was recorded in the same way. The frequen-cy of the vibrations was read directly from the oscillator, the propeller revolutions were measured by means of an optical system and an electronic counter.

The towing speed of the ship's model and there-fore of the propeller was also measured with an

electronic counter.

was considered as an ideal two mass system. The

natural frequency was:

==j/1,

(1)

These values are found by comparing the moments of inertia of the propellers in air with the values determined in Part I (see Appendix I).

The maximum error for these experiments in air

is 2.1%; the mean error is less than I %.

b. Damping

The damping was determined by a method entire-ly different to that used in the experiment with a

non-rotating propeller, being calculated on the

basis of propeller equilibrium.

The theoretical derivation is given in Part II of

the Appendix.

4. Accuracy of the measurements

The precision of this method is not great. The sig-nals have been recorded simultaneously. One of

the signals was very small so that its error cannot

1.

Results

The natural frequency was determined for each

propeller for different towing speeds and propeller

r.p.m. The moment of inertia of the propeller in

water '2 was then calculated with the aid of

formula I.

Table 3 shows the propeller series, the HID ratio,

the natural frequency r1 of the system with the propeller in air, the moment of inertia '1 of the propeller in air, the natural frequency r2 of the

system with the propeller in water, the moment of inertia '2 of the propeller in water and the percent-age increase in the moment of inertia.

The natural frequencies were determined as the mean of a few experiments with different r.p.m.

and towing speeds. The scatter was not more than

0.2 c.p.s. The results for the three series of

pro-Table 3

The damping coefficient g in kgcmsecrad' is calcu-lated from the following formula (see Appendix II):

g

= tan

(q22-ç1)

Jw

be less than 5%. Although for this reason consid-erable scatter of the results was to be expected and did indeed occur, nevertheless important

conclu-sions can be derived.

III. MEASUREMENT OF THE MOMENT OF INERTIA

pellers are given in Figure 10. For comparison the results of the previous experiments with non-rotat-ing propellers are reproduced.

2. Conclusions

The increase in the moment of inertia appears to

be independent of the propeller revolutions, the towing speed and the slip in the region of maximum

output.

The time available in the tank did not permit this to be established for all values of propeller

revolu-tions, towing speeds and slip. The scatter of the

results was as already stated 0.2 c.p.s. The curves clearly indicate that the increase in the moment of

inertia increases linearly with the HID ratio, in agreement with the results obtained with

non-rotating propellers.

(2)

propeller series HID r1 c.p.s. I kgcmsec2 r2 c.p.s. I kgcmsec2 ( 100%

B-4--40 0.5 35.4 0.01927 35.0 0.02007 4.2 0.6 34.7 0.02070 34.0 0.02224 7.4 0.8 35.6 0.01888 34.3 0.02157 14.2 1.0 34.8 0.02049 33.1 0.02439 19.3 1.2 34.2 0.02179 32.1 0.02704 24.1 1.4 34.5 0.02113 32.0 0.02733 29.3 B-4--55 0.5 32.1 0.02704 31.7 0.02819 4.3 0.6 31.6 0.02849 30.8 0.03098 8.7 0.8 31.7 0.02819 29.9 0.03446 22.2 1.0 32.0 0.02733 29.3 0.03637 33.1 1.2 31.5 0.02879 28.5 0.03970 37.9 1.4 31.7 0.02819 28.1 0.04151 47.3 B-4-70 0.5 29.4 0.03598 28.3 0.04059 12.8 0.6 29.8 0.03446 28.3 0.04059 17.8 0.8 29.8 0.03446 27.6 0.04391 27.4 1.0 29.8 0.03446 26.9 0.04758 38.1 1.2 30.6 0.03164 26.7 0.04870 53.9 14 29.7 0.03483 25.3 0.05758 65.3

I.

Results

Table 4 shows the propeller series, the H/D ratio,

the propeller revolutions n, the model towing

speed r and the damping coëfficiënt .

The propeller vibration amplitude was

approxi-mately 2° for all the measurements, except the one

series B-4-55. The damping coefficient was always

determined at the resonance frequency. The

pro-peller r.p.m. value n is always the same as the mean

normal r.p.m. of the actual propeller (in two cases

of propeller B-4-70 this did not apply because The increase in the moment of inertia is greater

with a non-rotating propeller than with a rotating

one; this difference decreases as the blade area ratio increases.

When applying the results of this investigation to

actual propellers the scale effect must be taken into account, together with the effects due to ro-tation in an inhomogeneous and restricted flow region.

The mean value of the increase in the moment of inertia used in practical calculation is 25%.

In some cases the increase appears to be less and

did not vary in accordance with increase in the

HID ratio and blade area ratio.

With adjustable propellers however, some

experi-ences showed a strong dependence on the HID ratio.

It would be most useful if actual shipboard meas-urements of the increase in the moments of inertia of propellers due to the vibration of the

surround-ing water, could be assembled, classified and

com-pared with the results of the present investigation.

60 e o 40 60 20 40 20 40 Fig. IO. Rotating propeller - -: Non-rotating propeller

IV. MEASUREMENTS OF THE DAMPING COEFFICIENT

HID

measurement did not appear possible at this pro-peller speed). The towing speed r is always taken

as O or "normal", i.e. just equivalent to or less than the maximum propeller efficiency.

The results of the 3 series are shown in Figures 11

and 12. Because the scatter of the points is large

the curve has not been drawn.

A few measurements were made with propeller

B-4-55 with HID = 0,8 with varying r.p.m., speed

and frequency. Table 5 and Figure 13 give the

results for different r.p.m. but always for the same

point of propeller operation, i.e. in correspondence 15 B-4-40

I.-B-4-70

III

0.5 06 08 10 12 1.4 HID 0.5 06 0.8 10 12 1.4 HID 0.5 06 08 10 12 14 o 80 o o 60 20 o

Table 4

16

with the towing speed of the model. Table 6 and Figure 14 give the results obtained with different

amplitudes, obtained by not adjusting the frequen-cy to the resonance frequenfrequen-cy; table 7 gives the results obtained with different towing speeds r,

constant propeller r.p.m. n and constant vibration

amplitude of 2°.

Table 5

2. Conclusions

Although the method used did not possess a high degree of accuracy and the results indicate consid-erable scatter, it is possible to draw some

conclu-sions concerning the behaviour of the damping coefficient as a function of various quantities.

Figures 11 and 12 show that the damping

coeffi-cient increases with increase in HID, therefore with increase in the couple M.

Figure 13 indicates that the damping coefficient increases with increasing propeller r.p.m. if the

conditions otherwise remain unaltered. Since the propeller moment increases with the propeller this phenomenon is to be expected.

Figure 14 gives an unexpected picture of the

dependence of the damping coefficient on

fre-quency and amplitude. At frequencies below the resonance frequency the damping coefficient

de-creases with increase infrequency. Under conditions of resonance, apparently because of the increase in displacement, the damping coefficient decreases at

a slower rate. It appears therefore that the increase

in frequency has a diminishing influence on the

damping coefficient and increase in displacement

the opposite effect.

Table 7 gives the values of the damping coefficient as a function of the towing speed for constant

propeller revolutions. The values are such that no conclusions can be drawn.

propeller series HID propeller revolutions per sec e rn/sec kgcrnsec rad-1 M kgcm B-4-40 0,5 7,5 0.72 0.109 18 0,6 7,5 0 0.338 42 7.5 0.54 0.265 31 7.5 0.72 0.190 27 7.5 0.90 0.122 21 0.8 7.5 0 0.485 69 7.5 0.86 0.422 46 7.5 1.04 0.367 40 7.5 1.22 0.294 32 1.0 7.5 0 0.626 99 7.5 1.22 0.484 59 7.5 1.40 0.468 49 7.5 1.58 0.488 38 1.2 7.5 0 0.824 137 7.5 1.53 0.653 78 7.5 1.71 0.593 67 7.5 1.89 0.689 53 1.4 7.5 0 0.955 175 7.5 1.71 0.915 103 7.5 1.98 0.890 86 7.5 2.25 0.851 66 B-4--55 0.5 6.3 0 0.433 24 6.3 0.48 0.465 16 6.3 0.60 0.373 12 0.6 6.3 0.48 0.426 23 6.3 0.60 0.323 19 6.3 0.72 0.273 16 0.8 6.3 0 0.566 56 6.3 0.98 0.505 23 6.3 1.13 0.404 16 6.3 1.29 0.512 7 1.0 6.3 0 0.879 83 6.3 0.98 0.591 46 6.3 1.13 0.527 38 6.3 1.28 0.528 30 1.2 6.3 0 1.274 113 6.3 1.28 0.766 55 6.3 1.44 0.835 44 6.3 1.59 1.191 34 1.4 6.3 0 1.593 142 6.3 1.36 1.138 73 6.3 1.59 1.135 62 6.3 1.81 1.095 46 B-4-70 0.5 6.3 0 0.259 21 10 1.13 0.357 24 10 1.32 0.457 14 0.6 6 0.68 0.308 13 6 0.79 0.338 10 6 0.91 0.265 6 0.8 6 0 0.599 52 6 0.65 0.354 18 1.0 6 0 0.949 81 6 0.94 0.487 41 6 1.22 0.433 24 1.2 6 0 1.150 122 6 1.15 0.644 54 6 1.31 0.616 43 6 1.47 0.613 32 1.4 6 0 1.903 162 6 1.44 1.873 67 6 1.58 1.215 55 6 1.73 0.983 43 pro-pelier series pro-peller no. HID propeller revolutions pr sec rn/sec kgcmsec rad B-4-55 156 0.8 2 0.31 0.285 2.3 3 0.47 0.254 5.2 4 0.62 0.486 9.2 5 0.78 0.550 14.4 6 0.94 0.364 20.7 7 1.09 0.441 28.2 8 1.25 0.693 36.8 9 1.40 0.580 46.6 10 1.56 0.762 57.5

1.0 E 0.8 0.6 0.4 0.2 o 0.8 0.6 0.4 0.2 B-4-40 X X X X o

As in the case of the increase of the moment of

inertia little is known concerning the influence of

the scale effect on damping. It may be expected

that this effect will not be great, since in the case of propeller moment and output the effect is very small. Special experiments would give more

pre-E 1 .4 1.2 ¡ 1.0 0.8 0.6 0.4 0.2 '1 0.8 E 0.6 0.4 0.2 o o 160 z 120 80 4° 0 6o 401 20 rs revs/sec O 02 0.4 0.6 0.8 1.0 1.2 1.4 16 e rn/sec

cise data on this point.

The surface roughness of the propeller models na-turally has a very marked influence on the

damp-ing.

A closer investigation into the effect of the

am-plitude would also provide useful information.

17 F/F propeller no. 156 = 0.55 X H/D 0.8 X ₁₆₀ 80 o 120 o: 80 40 5? X X X X Mfor V=0 X X 0.5 06 08 10 12 14 HID B-4-55 X X X X X X Mfor V=0 Fig. 11. Fig. 13. V. DISCUSSION 03 06 08 10 12 1.4 HID 0.8 Fig. 12. 0.5 _{06 10} 12 14 HID 4 8 10 1.4 E , 1.2

o 23F Fig. 14. f) amplitude Table 6 Table 7 propeller series propeller no. HID n revs/sec u rn/sec vibrations per sec amplitude in degrees kgcmsecrad-1 B-4-55 156 0.8 6.3 0.98 24 0.96 1.293 25 1.12 1.186 27 1.20 0.803 28 1.57 0.595 29 2.26 0.485 29.2 2.52 0.539 29.4 2.75 0.526 29.6 2.78 0.552 29.8 2.99 0.544 30.0 3.18 0.613 30.2 3.08 0.652 30.4 3.02 0.606 30.6 2.85 0.548 30.8 2.75 0.579 31 2.34 0.199 36 0.30 0.841 40 0.11 2.473 resonance / / 3 3 o _/ / __.5_ _, 42 J cl

00

O o D D _o O FI F propeller 0.55 no. 156 HID = 0.8 r = 6.3 revs/sec 0.98 rn/sec pro-peller series pro-peller no.

HID _revs/sec' _rn/sec kgcmsec rad-' B-4-55 156 0.8 6.3 0.1 0.443 0.2 0.450 0.3 0.506 0.4 0.603 0.5 0.341 0.6 0.351 0.7 0.541 0.8 0.452 0.9 0.519 1.0 0.753 1.1 0.352 1.3 0.361 24 25 26 27 28 29 30 32 cpn 1.6 1.4 0.8 0.6 0.4 02 3.2 0 2.8 1.6 1.2 0.8 0.4

COMPARISON WITH THE RESULTS OBTAINED IN PART I

The vibrating system is conceived as an ideal two mass system, with a natural frequency:

c

V1=-r=--- 1/

+

2r

2r

' 11+1t

I

The moment of inertia of the propellers is known from previous experiments. The circular

frequen-cy w of the system is measured.

If the experiments had been completely free from error and if the system was in fact ideal then:

w2_11

=0

(2)

't, trn and C were determined by reducing the

function: C

C2

Ii+It

I)

APPENDIX I (1) (3) to a minimum.

Putting = q, from the method of least squares:

m

f(C, It, q) =

(w2

)2

The minimum conditions for f (C, I, q) are:

of

o

OC ' OI

'ôq

This results in the following three equations:

I1i(c2

_jj

C

q)=o

(Il+l)2(w2

'i't

C

q)=o

(&

-

Il+CI, q)

= o

whereby the w and I measurements for the model

propellers are summated, i.e. the results for the

3 series of 6 propellers (18 propellers).

An exact solution of the equations is very compli-cated and laborious; the solution is therefore ob-tained by "trial and error", which comes down to

"trying and interpolating".

The results thus obtained are:

C = 1349 kgcmrad' It = 0.01160 kgcmsec2

q = 5771 and therefore Im = 0.23375 kgcmsec2 From this correlation it appears that:

1) the values of C agrees well with the calculated

value 1/C = 742 x 10-6 therefore C = 1347

(see Table 8);

the values of I lie higher than expected from

the calculated value;

the value of im is lower than the value given by the makers.

Table 9 shows the measured frequencies r and the

corresponding values of I, also the values of I found in the earlier experiments with the difference

expressed as a percentage.

The maximum error is only 2.1%; the mean

error 0.92%. Table 8 Tabel 9 19 section I1 kgcmsec2 1/C rad/kgcm propeller (+ entrained water) to be

measured

-streamlined nut and -streamlined

connecting piece on the shaft 0.00360 -.

shaft section between the

propel-ler and the first torsiograph 0.00615 l29.l0 first torsiograph+pulley 0.00066

-shaft section between the first and

second torsiograph 0.00162 569.10_6

second torsiograph+pulley 0.00066

-shaft section between the second

torsiograph and the motor 0.00108 44.10_6

motor (according to the maker's

data 0.33000

-total 742.10_6

pro-peller

series

HID _{c.p.s. kgcmsec2}

I

previously

'

determined deviation % B 4 40 0.5 35.4 0.01927 0.01912 0.8 0.6 34.7 0.02070 0.02058 0.6 0.8 35.6 0.01888 0.01905 0.9 1.0 34.8 0.02049 0.02080 1.5 1.2 34.2 0.02179 0.02165 0.6 1.4 34.5 0.02113 0.02111 0.1 B-4--55 0.5 32.1 0.02704 0.02657 1.8 0.6 31.6 0.02849 0.02839 0.4 0.8 31.7 0.02819 0.02791 1.0 1.0 32.0 0.02733 0.02791 2.1 1.2 31.5 0.02879 0.02911 1.1 1.4 31.7 0.02819 0.02803 0.5 B 4 70 0.5 29.4 0.03598 0.03611 0.4 0.6 29.8 0.03446 0.03446 0 0.8 29.8 0.03446 0.03446 0 1.0 29.8 0.03446 0.03446 0 1.2 30.6 0.03164 0.03162 0.1 1.4 29.7 0.03483 0.03478 0.1

Figure 1 5a is a diagramatic representation of the two mass system with the two torsiographs drive arrangement. The determination of the propeller

damping and the moment of inertia of the

pro-peller in water is performed with the aid of a

vector diagram (see Figure 15b).

The following equations of motion can be derived

for the two mass system, where the excitation is provided by the motor and the damping by the

propeller:

0m2+C(9m08)

=

M0

C(OsOm) -08I50)2+O3eico = 0

91w2 -

+ Oeiw = M0

In the case of resonance the following conditions

apply. The amplitudes of the two torsiographs and the difference in _the signals form a vector

triangle with the sides @. and a. The

deforma-tion of the secdeforma-tion L1 can now be calculated from

the elasticity ratio of L2 to L1. The side a of this

triangle can now be extended by the calculated section b and gives directly the magnitude of 9. Determination of this value permits the couple 0snIsnO2 to be set out, and since the direction of

O5ew

and 0Ico2 is known, the entire diagram

can be completed on the basis of the previously mentioned equations of motion. q1 is now the phase

angle between 9 and A1' = 90e, while 2 is the

phase angle between s0m and M0. The

damp-ing is given by

tan (92l) =

_{0513w2 18w}

20

DETERMINATION OF THE DAMPING APPENDIX II

Figure 1 5c gives a three dimensional representation of the phase difference of the propeller mass as

compared with the motor mass.

or'2x2 LI dann p ing

k

T

to ra sg raph ea i9n, (.)f9,nC excitation Fig. 15.

Two mass system with and without phase displacement with torsiograph drive and vector diagrams