Virtual mass and moment of inertia of propellers

rAP

CEF

Lab.

.

Technische Hoscc,.o

Deift

VIRTUAL MASS AND MOMENT

OF

INERTIA OF PROPELLERS

(Determination of entraine4 water effects, for use in

connèxion with torsional and axial vibration calculations)

By PROFESSOR L. C. BURRILL, M.Sc.,

Ph.D., Fellow,

and W. ROBSON, B.Sc.

23rd March, 1962

SYNoPsIS.The paper describes the results of tests carried out inthe Research Laboratories of the Department of Naval Architecture and Shipbuilding at King s College, Newcastle upon Tyne, to determine the effect of entrained water on the mass moment of inertia and virtual mass of propellers oscillating both torsionally and axially in water These tests were carried out with a total of forty nine I6in model propellers, the chOracteristics of which were varied systematically, and the

resulls, which proved to be extremely consistent, reveal the effects of such important

variables as blade-area. ratio, pitch ratio, number of blades, etc., on the entrqined

mass effects.

From the experimental results, a series of simple empirical expressions have been derived for first estimation purposes, and finally a more detailed method of

calcula-tion is proposed which allows the entrained mass effects to be estimated directly

from the completed propeller drawing, and thus takes into account all the significant

variables, including the actual blade shape and pitch variation finally adopted, 1. Introduction

has for a long time been felt that our knowledge of the entrained

water effect on the mass moment of inertia

of propellers oscillating in

the torsional modes of vibration of main line

shafting, is very sketchy

and incomplete.

In this country, it is usual to allow for this

effect by

making a percentage addition

the calculated mass moment of inertia

of the metal propeller, varying from some 25 per cent to 30 per cent, but

it has been realized that this takes no proper account of the shape of

the blades, or the size of the boss.

In 1950, some preliminary

experi-mental work was undertaken on this subject with a limited range of

small propellers which were vibrated in torsion, in air

and in water and

the results, published in Ref. 1, showed that the percentages to be added

to the propeller mass moment of inertia varied considerably, and were

influenced by both the pitch angle of the blades

and by the amount of

blade surface area.

More recently, in connexion with a large passenger liner, it was decided to carry out a similar test with the l6in.diameter manganese bronze model of the 4-blãded propeller, as fitted to the ship, which had been made for the usual

cavitation tunnel experiments.

This test showed that the percentage addition to be madeinthis instance was

as high as 46 per cent, orabout half asmuch again as the normal allowance, the

pitch ratio m this instance being 1 17 and the

blade area ratio 73 This suggested the need for a systematic series of tests to determine, if possible, the

326 BURRILL & ROBSON VIRTUAL MASS AND MOMENT OF INERTIA OF PROPELLERS

effect of pitch ratio, bládè-area ratio, number of blades and other design features, on the proper allowances to be made for entrained water when carrying out the usual torsional frequency calculations, and, the test with the l6in. model propeller having proved to be successful, the present work was put in hand. As the work

proceeded, and the results obtained were found to plot in a very consistent

fashion; it was decided to extend the investigation to determine the axial entrained mass in addition to the entrained inertia effect in torsion, using the same experi-mental arrangement. This, at first, led to only a partial success, as the long thin

rod used in the torsional tests was found to oscillate in other than purely axial

modes; but this difficulty was overcome by using a much stiffer and shorter rod

in conjunction with a horizontal spring, and the results, as described in the paper, were found to be entirely successful. Various methods of plotting the experimental results were tried, and some of these are recorded in the several diagrams given in the paper. Finally, a method of calculation was devised whereby both the entrained inertia in torsion, and the added mass in axial translation may be estimated directly from a detailed drawing of the propeller.

2. Test Procedure

The method used for determining the polar mass moment of inertia of the

seveial propellers, in air and in water, was similar to that employed in the earlier tests described in Ref. 1, but owing to the smaller size of the model propellers, which were uniformly 1 6in. in diameter, several refliements were introduced in

order to secure the necessary accuracy. In the first place, the propellers were

suspended on a thinner rod, which was m m diameter and 14 5ft long and

the irieasureinents were calibrated by swinging three steel discs of known moment

Of inertia. Secondly, the frequency measurements were obtained by making an

extensive record of the oscillations on a paper strip by means of a capacitor

transducer and an accurate secondsclock, the propeller being displaced slightly

man angular direction and then allowed to oscillate freely in torsion, the lower end of the rod, which projected some 3m, below the propeller boss, being

sUpported in a socket to prevent sideways motion.

Changes in capacitance proportiOnal to the angular displacements, occurred when a horizontal centre plate, rigidly attached to the oscifiatisig rod, moved cyclically between two fixed plates attached to the stationary test-rig. The

changes of plate area and hence capacitance altered the frequency of a pre-set

oscillator, the output of which was converted from frequency to voltage variation by a lOw pass filter device operating at the centre of the cut-off frequency. In

this position, the slope of the response curve enables changes of input frequency

to re-appear as changes in output voltage, which, after suitable amplification, operate the armature coil of the pen-motor and the attached writing stylus. A

second pen, activated by the seconds clock, provides a time trace for calculation

of fEequency, and hence 1/f2. M the initial displacements lay well within the

elastic limit of the steel rod, the frequency of the recorded oscillations remained

independent of amplitude, throughout the test records.

The value -of the entrained inettia (4), reckoned as a percentage of (Ia), the

moment of inertia of the propeller in air, may be calculated from the experimental

frequency data by means of the formula:

Percentage (4)

(fwl_fiil)

< 100 = (4/li,) X 100 %

The numerical value of (4), however, cannot be determined unless the value of

(I,) is known.

The percentage figure derived from this simple formula includes a residual

frequency error, since fa (and fw) are the frequencies of oscillation of (propeller

plus rod and fairings), and are related to (l + 4) say, and (Ii, + I + 4), where 4 is the inertia of the light rod and fittings. In the numerator, the residual inertial0 cance1 out, but 4 persists in the dénominator

BIJRRILL & RoBsoN: vIRTUAL MASS AND MOMENT OF INERTIA OFPROPELLERS 327

denomihatorproduceS no significant error, but with the smaller model propellers

the residual value of I, cannot be ignored For this reason and also to deter-mine the numerical value of 1,, for the propellerin air the apparatus. win-

calF-brated by ineans of three discs of known momentof inertia.

The frequency of oscillation was measured for (Ia)

(I + 13) (I + I + 1)

and (I + 4 + 4 + 4) where 11 4 and 13 were the known

moments of inertia

of the discs and a calibration curve was then drawn as shown m Fig 1 which shows the actual calibration curve for torsional oscillation where (1 If') is

plotted to a base of moment of inertia in (lb.ft.2) units. The value of 4 is given

by the intercept on the "x" axir.

The example given in this diagram indicatçs the manner in which the thêasured

entrained inertia was derived, by interpolation -on theabove curve, i.e.,

(I + 1)

(4 + Ip) = (1 -

Je (lb.ft.2 units)

The smaller numerical values of 4 derived from the percentage formula may be observed by notmg the displacement of the dotted percentage value line from the actual calibration line shown in full.

The complete apparatus is shown on the right-hand side of Fig 16. The depth of immersion was limited to about l2in., beyond which depth no sigthficant changes in frequency were observed.

To determine the entrained mass in the axial direction, the propeller was rigidly attached at mid-span to a simply supported steel beam by means of a

short stiff vertical rod, of length 1 8in., sufficient to permit immersion to a depth

of l2in. The apparatus was first calibrated by the addition of knownweights to the free end of the vertical rod The calibration curve for axial vibration (Fig 2) shows (1/f2) plotted to a base of mass in lb weight The intercept on the "x" axis again indicates the residual mass of the beam plus fairihgs. The

accuracy of the apparatus is exhibited by the fact that the values of (1 /f2), for all the propellers tested, lie precisely along the imtial calibration hne when tested in air.

The example given on the corresponding calibration curve, Fig. 2, shows the

method of determining the entrained mass We from W in

the immersed

condition and W,, in air, i.e.

(W0+ W)(W0+ W)(W W)= Welb.

The recording of the vertical frequencies of vibration in air and in water was carried out by means of a condenser device similar to that used in the torsional oscillation tests. For this application, the centre plate was attached to the beam at mid-span and oscillated between two fixed plates attached to the stationary test-rig.

The apparatus used for axial vibration of thepropellers is shown on the left-hand side of Fig. 16.

A complete lit of the propellers tested and of thecorresponding measured values of Wp, Ip, We and Ic is given in Table 1, together with the calculated values obtained by applying the proposed new method of estimating these quantitiCs which is described in section (4) of this paper, and illustrated in Table 2.

As a matter of general interest, the values of We and 'e are also expresed in

terms of percentages of the cOn esponding Wp amid Ip figures.

3. Experimental Results

Torsional

The first use which was made of the experimentally derived values was to plot these against the major parameters (i.e. pitch ratio and blade-area ratio, etc.), to

find out whether these exhibited any consistent trends, or laws of variation,

328 BIJRI1.IIL & ROBSON: VIRTUAL MASS AND MOMENT OF INERTIA OF PROPELLERS

exaxñple; shows the measured values of Je the entrained inertia determined for the' KCA 3bladed series of Admiralty type propellers, plotted to a base of b.a.r., the blade-area ratiO. This shows that the measured values for each pitch ratio were readily separated, and that the variation was substantially linear with

blade-area ratio for each pitch zistio. Fig. 4 shows the same results plotted to a base of pitch ratio (P/D). This diagram shows that the law of variation with pitch ratio was also substantially linear for each blade-area ratio. This suggested that it might be useful to plot the results to a combined parametet (ba.r. x PJD)

and this is shown in Fig. 5. It was then found, that all the results could be represented with reasonable accuracy by a single straight line, which could be used as a basis for a simple empirical formula, for thst éstimatiön purposes.

Strictly speaking, this fOrmula, which is

Entrained inertia (Ic) = 1 37 (b.a.r. X P/D)-030 lb.ft.2'

can only be applied to propellers conforming to the basic 3-bladed design, but the diagram also includes the results obtained for the KCD 4-bladed series of merchant-ship type propellers, and several results for 5-bladed and 6-bladed screws, which, although not sulficient for a very complete analysis to be made for such propellers, would suggest the following additional formul for first

estimation purposes:

lb. ft.2

4 (four blades) = lO9 (b.a.r. x P/D) - O23

le (five blades) = O98 (b.a.r. x F/D) - O21

'e (six blades)

= 090 (b.a.r. x P/D) - O2O

It will be seen that the entrained inertia dftnihishes with increase in the number of blades it also increases steadily with increase in either the blade area ratio or pitch ratio.

These empirical formula have been derived directly from the experimental

values of I for the model propellers which were all 1 6in.. in diameter, so that in

order to obtain the corresponding values for full-size propellers it is necessary to multiply the results by the appropriate scale factor (Did)5, where D is the

diameter of the full-size propeller and d is the diameter of the model propeller. Furthermore, since the results were obtained in fresh water, it will be necessary

to apply a further correction for density, i.e.

I D'\5

Entrained inertia (full-size) = 4 (formula) x x 1O24 (lb.ft.2)

Axial

Fig. 6 shows the ineasured values of We the entrained axial mass for the KCA

3-bladed series, plotted to a base of b.a.r., and Fig. 7 shows the same values plotted to a base of P/D.'

It will be seen that while the values of We increase linearly with h.a.r., they

decrease steadily with increase in, pitch ratio, and it was therefore found

impos-sible to obtain a single line plotting by combining these two parameters. In view of this, a new parameter Cos2O was tried in place of F/D, where is the

pitch angle at two-thirds radius. This is 'shown in Fig. 8, from which it will be

seen that the values of W increase linearly with increase in this parameter. Finally, Fig. 9 .shows the same results plotted to a base of the combined

para-meter (b.a.r. x Cos282). It will be seen that all the results now lie substantially along a single line, which can be represented by

We = axial entrained mass

347,(b.a.r. x Cos2O) - 4-2 lb.

(3 blades)

an approximate formula which can therefore be used for first estimation

purposes.

The results for 4-bladed, 5-bladed and 6-bladed propellers are not sulTIcient to allow a similar analysis 'to be made, but as it is not expected. that this should

BURRILL & R0BS0N: VUiTIJAL MASS AND MOMENT OFINERTIA OF PROPELLERS 329

indicate any majOr differences from the 3-bladed analysis, the following

appr9x-irnate formul are- suggested for 4bladed, 5-bladed and 6-bladed screws,

respectively, based on the limited, data available.

(4-blades) We = 347 (b.a.r -x cos2

- 67 lb.

(5-blades) We

341 b.a.r. x Cos2 0

- 83 lb..

(6-blades) We

347 (b.a.r. x Cos2 0) - 96 lb.

As it is not anticipated that 5-bladed and 6-bladedpropellers will- be adopted

over a wide range of blade-area ratio or pitch ratio, and the propellers tested are

fairly representative of possible applications it is considered that the above

expressions should be quite adequate for present requirements.

These empirinal expressiOns have also been derived, directly from the

experi-mental values of We for the model propellers, and

therefore for full-size propellers the values will be obtained from

Axial entrained mass (full-size) - We:(formula)

_{X (}

_{j___3)}

x l024 lb.

In order to complete the'analysis, Fig. 10 has been prepared showing the variation of 1e'wjth respect to the çomplèrnentary function sin2 0a for torsional motion.

- 4. Cakidatioñ Method

Torsional

Having obtained the above data from experiments covering quite a wide range of variables, it was thought desirable to explore the possibility of establishing a method of calculation which would enable the designer to-estimate the entrained main and inertia for a particular propeller directly from the drawing, as is done; for example, in estimating the propeller weight and mass moment of inertia. As

a starting point, use was made of the knowledge gained in dealing with the similar problem of estimating the entrained mass of a. ship vibrating as a

free-free beam. . . .

Basically this is dealt with by using the simple expression forthe entrained

mass of an infinitely long cylinder of circular cross section oscillating sideways

or at right angles to its axis in which case the entrained mass per unit length

is given by

Entrained mass = iT/4 b2 x p per unit length where b beam (or diameter of the cylinder)

p density of the fluid

-and for ship calculations this expression is extended to entrained mass =

C;J. 'x n/4. b2 x p per'unit length, where C is a constant which allows for

the 'shkPe and proportions of the-actual cross-section a 'any point of the length,

and J'is a further

onstant which takes account of the finite length. The factor - is inti:odi.iced becaus,e the ship is floating at a free

surface.-In the case of a propeller oscillating m torsion, the problem is to some extent simplified by the fact that the blades are essentially flat plates, and the influence

of thickness and section shape is therefore probably quite small,

but, on the

other hand the sections are not moving in a direction perpendicular to the chord, the effect of interference due to overlapping of the-blades is unknown, and

there is no theoretical treatment which would assist in the assessment of the

three-dimensiOnal constant J. Nevertheless, as a first approach, it was decided

to make a preliminary attempt to resolve the problem by assuming that the

amount of the entrained volume of fluid at each radius. could be.represeited by drawing a circle round the extremities of the sections as projected on to a plane

at right-angles to the motion the area of this section being given

by iT/4

330 BURRILL & ROBSON: VIRTUAL MASS AND MOMENTOF INERTIA OF PROPELLERS

effect, of the finite blades this basic area at each radius was then reduced by applying a factor K which- varied elliptically from the axis to the tip,, and the

mass moment of inertia of the volume so described was calculated in the usual

way. This tentative procedure was found to give results which agreed very closely with the experimental figures for propellers with moderate pitch ratios and low blade-area ratios, but failed to predict those for the propellers -having

high pitch ratios and very wide blades, for which the calculated values were too

high. This suggested that, apart from the influence of blade interference, there was probably an aspect ratio or (blade-length/blade-width) effect, which was not

included in the calculations.

In the light of this result, a second attempt was made by applying the

three-dimensional inertia coefficient factors k' obtained by Lamb' (Ref. 2), ; 155, in-connexion with the calculation of the entrained inertia for prolate ellipsoids of various proportions, rotating about a minor diameter. These factors k' give the relationship between the moment of inertia of a volume of fluid described

by circles at each radius corrdsponding to the elliptical distribution, and the final inertia of the actual entrained fluid volume, and are therefore directly applicable to the basic volume used in the previous estimate.

-The new approach did, in fact, indicate an important aspect-ratio effect (in terms o.a/b where a = major axis radius, and b = minor axis radius of the ellipsoid), but the application of these k' factors led to the unexpected result that

while the calculated values again agreed closely for the moderate pitch ratios

and low blade-area ratios, those for the higher pitch ratios and blade areas now

worked out below the experimental values. This indicated clearly that the correction made for aspect ratio was too great for the wider propellers, and an examination was therefore made of the actual projections of the blades on to a transverse plane, in relation to those of the several ellipsoids, in an attempt to

find for a given propeller blade shape an equivalent ellipsoid which would have the same entrained inertia. Although this procedure was not successful, it did

draw attention to two factors which have an important bearing on the failure

of the ellipsoid analogy. The first of these is that while the projection of the

blades on to a transverse plane is approximately elliptical for the moderate blade area ratios, it is distinctly wide-tipped for higher blade-area ratios. The second

is that for a given diameter (2 a) there is a maximum value of 4 which can be achieved as the axis diameter (2 b) is increased, and that thereafter the value of 4 diminishes to zero when 2a = 2b, as the ellipsoid then becomes a sphere.

This is illustrated in Fig. 11 where the experimental values of 4 for the 3-bladed

series are plotted to a base of b, the equivalent mean value of the fore and aft

projection of the blade widths, extended to the axis.

As a further extension of this procedure, the values for the 4 derived from the

above mentioned work of H. Lamb have been calculated for the special case when the dimension c at right angles to b approaches zero, and the ellipsOids therefore become flat elliptical plates set at right angles to the direction of

motion. Itwillbe seen that this explains the failure of the previous analogy, as

the calculated values now lie within the region of the experimental results, but

that this method of approach still does not fully explain the effects of blade-area ratio and the influence of pitch ratio, as revealed by the experimental results.

As a result of this work, it was decided to approach the problem in a different

way, and to endeavour to derive the appropriate K factors from an analysis of

the actual measured results This was done by plottmg the values of a derived

K factor represented by the ratio

measured- value of I

-K(fl

_{- calculated value of 4}

for the propellers of the 3-bladed series against various parameters.

The

result of this examination is illustrated in Fig. 12 i,vhen it was found that the

majOr variable was undàubtedly the blade-area ratio, as the value of K1 for each

pitch ratio was sensibly constant over the middle part of the range, and only

-BURR1LL & RoBsoN: VIRTUALMASS AND MOMENT OEINERTIA OFPRQPELLERS 331 It was therefore decided to re-calculate all the valuesof 4 for the full range of propellers using a single line of K5 plotted to a base of b a r as shown in Fig 13

These new calculated values were then plotted against the residual parameter pitch ratio and the resultant errors. were so small that it wasP considered un

necessary to apply a further correction for pitch ratio. The final calculation method is illustrated in Table 2,, for the unity pitch ratio, 65 b.ar. propeller.

Axial Entrained Mass

The above described investigations were all undertaken in

relation to the

entrained mertia values, at a time when the axial entrained massvalues were unknown and a supreme test of the final method arrived at as a result of the

various speculations, was to calculate the entrained axial mass values in precisely

the same manner, but substituting the value of (chord x cosine 0) in place of (chord x sine 0) at each radius This was, in fact done m one day, and the results when plotted in the form of

- measured value of axial enttained mass

K axiaj)

-

_{calculated value}

were found immediately to fall in line

Having regard to the fact that the

quantities 4 (lb. x in.2) and We (lb.) are of a different order, this was considered

to be a sufficient proof that the basic method of calculation was at least sound in principle.

It was not expected that the values of K(i,tja) and K(iM)

would necessafily be equal, but it will be seen that they follow similar line or

trend with b.a.r., and in fact the rãtiô

_K(jacrtja)

= 1 O8, was found to be

constant over the full range of b.a.r. values tested.

The same procedure was later applied to the 4-bladedpropellers of the KCD series and the resultmg values of KA and K1 are shown in Fig 14 It will be

seen that these are slightly higher than the 3-bladed values, but that they

exhibit the same trend with b a r

and can, in fact be related to these by a

simple ratio, throughout the range of the tests. Similar curves for the 5-bladed

propellers are shown in Fig. 15, together with the correspondingvalues for the

single 6-bladed propeller tested. For calculation purposes, the values of K1 and ((A are listed at 02 intervals of b.a.r. in Table 3-for 3,4, and 5-bladed propellers

respectively.

5. Further Discussion of Experimental Results

It wifi be obvious from Table 1 that the method of estimating the entrained

water inertia by adding a fixed percentage to the propeller mass moment of inertia is totally inadequate excepting only for verysmall variations m b a r and pitch ratio from a given basic propeller. It so happens that there were three variations

of the KCA 110 propeller hiOh' were identical in all respects apart from the blade thicknesses, m that with respect to the normal propeller of 045 blade

thickness fractiOn, one model was deliberately made thinner,and another made

thicker, the corresponding blade-thiOkness fractions being, respectively, O3O

and 060.

Tests were therefore carried out with these three propellers and the

results are tabulated

below:=-Basic Propeller KCA 110 Blade-area ratio 80 Blades 3

Pitch ratio 1-0 Blade thickness fraction 045

Propeller Wp Ip

4

% additiOn- Calculated 4

KCA 110 (light) 1381

l083

-765 70-6% -746

KCA 110 (normal) l7l2

1685

-785

466%.

746

KCA llO(heavy) 20-12 2-004 -761 _38-0% 746

Propeller Wt (Wp) We % addition Calculated W

KCAIIO(light) 13-81 18-00 130-3% 17-78

KCA11O(normal) l7-12

.i798

1Q5.0.%

1778

KCA11O(heavy)

2Ol2..

1.7-70 _88-0% 17-78

332 BVRR]LL & ROBSON: VIRTUAL MASS AND MOMENT OF INERTIA OF PROPELLERS

It will be seen that the measured values of 'e and We are quite independent of the propeller mass moment of inertia and weight, and are governed only by the geometrical characteristics of the blades. It is also clear that the thickness of

the blade sections 'is of no importance in determining the entrained water mass. For the 4-bladed KCD series there were also thtee variations of the unity pitch

ratio, 587 b.a.r. propeller, in that, relative to the basic propeller, one had wide-tipped blades with narrow root widths and another had narrow-wide-tipped blades

with wide roots, all other characteristics remaining the same. This provided.the

possibility of makmg a check on the adequacy of the proposed calculation procedure to reflect correctly the effect of blade shape n the entrained inertia

and mass values..

The results obtained with Propeller KCD 4 (wide tip) KCD 4 (normal) KCD 4 (narrow tip) Propeller 'Wp We

KD 4 (wi4e tip)

.' 19, 38 '

978

KCD 4 (parent)

l938

.,

947

KCD4 (narrow tip)

2038

9.39

and it will be seen that, the measured values are completely calculated figures. '

Finally, it was decided to determine the effect of removing two opposite

blades from One of the four-bladed propellers, and the results are given below:

'Basic Propeller: KD 2

B.a.r. = 50 P/D = P846

Blades: 4 Propeller IJ) KCD 2 (4-blade) I o99 KcD 2 (2-blade)

559

Propeller Wp Kcp 2 (4-blade) .

l388

KCD 2 (2-blade) 9 12

these propellers are given below:

'P

4

l429

.393

l387

385

l440

' 381 Calculated 'e' 39l 381 .377 alcula1ed We 9.79 9.57 9 50 in accord with the

4

Galculated 4 C'orresponding K, -'226 ' ''235 ' -875

il8

'118

108

We Calculated 4 c'orresponding K4 7-83 7-88. -938 4-23 4-23. 1-00

It will be seen that in this special case of two opposite blades the factors K4 and Kj-approach to unity,so that the initial assumption as to the natifre of the entramed volume being descnbed by drawmg a circle round the extremities of

the sections as projected on to a plane at right angles to the motion is reasonably

correct; and no further correction is required.

-' ' 6. Final Comments and Suggestions

It is' not suggested that the results given by this series of, tests should be regarded as a complete and final resolution of the prOblem of estimating the

entrained mass effects to be included in the usual calculations for the torsional

and axial criticals of main hne shaftmg

It is, however thought that they

provide a much more fleAible and logical means 6f estimating these effects than the present method 'of making a percentage allowance, based on the calculated weight and mass moment of inertia of the metal propeller. The final adequacy

of- the proposed method of calculation, for full-size propellers, can, only be

assessed after it has been applied, in practice to a wide range of actual

installa-tions, but the consistency of the results Obtained on the model scale gives

sufficient confidence 'in the method to suggest that it should at least be tried Out

in practice. . . .

Th specimen calculation given in 'Table 2 has been' drawn up in terms of the standard radii used in defining the m'ödèl propellers, but can readily be adapted

BURRiLL & ROBSON VIRTUAL MASS AND MOMENT OP INERTIA OF PROPELLERS 333

to suit other methods of definingthe blade sections of the full-size. propellers (i.e., Ift. intervals, etc.) The same calculãtiön form can,. in fact, readily be extended to include the calculation of weight and polar moment of inertia for the metal propeller (Wy) and (I,,) by inserting the appropriate areas of the

bladesections at each radius and using the same Simpsons multipliers to find the corresponding volume and inertia functions, down to the crown of boss. The volume and polar moment of mertia of the boss can then be calculated iepãrately

and added to that for the blades, thus giving the radius of gyration and mass

moment of inertia of the completed propeller Experience has shown that the final figures thus obtained agree within one per cent with the results of full scale tests obtained by swinging large propellers in air, using the usual tri-filar method of suspension.

During the course of the investigation attention was directed to two paper (Ref 2 and Ref 3) which had been written on this subject in Athenca and in

particular to theeiiipiricalexpreSSiOris

-.

O2lpD(WR)2fl

e -

[1 + (p/d)2] (03 F MW1)

-

..()

Ic= O02 W .p2

...(2)

which had been suggested. An attempt was made to interpret the results

obtained in the present series of tests by applying these formuhi, but it was found that although they had a linlited application to the propellers of small b.a.r. and

low pitch ratio they failed to predict themeasured results for the wider blades

and higher pitch-ratios. For example,

One important point did, however, emerge from this examination, and that is

that the authors of Ref. 3 showed by their analysisthat the apprOpriate value

of W,, to be Used in the axial vibration calculation was not that obtained from

the tests with rotational restraint, as described in this paper, but an equivalent entrained mass W' given by

We

We'

₌

W lb

...(3)

+\2)

i,

fp\2

We

wherep = pitch in feet, and the term (j x

_\.71J

1p

is a coupling factor to allow for a propeller inlongitudinal (i.e., axial) motion, but unrestrained by the shaft in rotation. The writers of the present paper are unable to comment with authority on this matter, as they have no extensive

experience of the results of axial vibration calculationsin relation to measured values obtained from actual ships, but having studied the analysis presented in

Ref 3 it would appear that this correction is fullyjustified

Itis hoped that

marine engineers who have had practical experience of the results of axial

vibration calculations will be able to comment on this point. Propeller

Ref.No. P.R. B.A.R.

Exp.W

Formula

w

Exp.Ie Formula -Ic 1 -60

50

11-21 11-89 190 -152 2 -60

65

16-04 17-42 -256 -223 4 80 -50 10-44

986

-288

225

5 -80 -65

l534

14-44

427

329

28 1-60 95

1748

il56

l85

1-05 32 2-00 95 15-28

822

2-33

117

334 BIJRRILL & ROBSON; VIRTUAL MASS AND MOMENT OF ThJERTIA OF PROPELLERS

A further matter of interest is the relationship vetween 4 and We which is

implicit in equation (2) above, which may be re-written_as

= OO2p2

This has been examined in relation to the measured values obtained in the

present series of tests, and has been found to be substantially correct, although

the values of the constant vary from about OO23 to 0O25 and a more suitable

mean value appears to be about 0024.

Acknowledgments

The authors wish to acknowledge their indebtedness to Mr. R. H. Curry, B.Sc., for his valuable assistance in connexion with the development of the calculation method described in the paper and the preparation of the various diagrams, and

also to Mr. R. Greenwell who carried out the tests in the

laboratory. Dr. R. L. Townsin is also to be thanked for his exploratory workon the ellipsoid analogy described in Section 4 of the paper.

BURRILL & RoBsoN: VIRTUAL MASS AND MOMENT 01' INERTIA OF PROPELLERS 335

REFERENCES

BURRILL, Prof. L. C. and BoGGIs, A. G.," Electronics in NavalArchitecture,"

N.E.C.Inst., 67, 1951.

KANE, J. R. and MCGOLDRICK, R. T., ' Longitudinal Vibration of Marine

Propulsion-Shafting Systems," S.N.A.M.E., 57, 1949.

LEWIS, F. M. and AUSLAENDER, J.," Virtual Inertia of Propellers."

Detailed drawings of the KCA and KCD propellers are to be found in the

following:-KCD Series: Buiuw.i, Professor L. C. and EMERSON, A., "Propeller Cavita-tion: Some Observations from l6in. Propeller Tests in the New King's College Cavitation Tunnel," N.E.C. Inst., 70, 1954. KCA Series: GAWN, Dr. R. W. L. and BuRIuLL, Professor L. C., "Effect of

Cavitation on the Performance of a Series of l6in. Model

TABLE IList of Propellers Tested w Pro eller Number P/D B.A.R. Number Of Blades We! hr wp. lb.

Measured Values Calculated Values Percentage Values

Entrained Mass We..lb. Moment oJI,zeriia lb. ft.a.

jp.

-Entrained Inertia 'e lb.ft:2 -Entrained Mass W. lb. Entrained Inertia 'e lb:,(t.2 (W 1W_{e0, P} /0 (1 /1 0

I '60 _'50 3 1I'19 I1'21

983

. 9Ø

.1094. -'-164 1002

I93

2 -60 65 3 14'56

l604

1'290 -256 1599 '240 I10'2

198

.3

'60 80 3 1806

2074

161I

'299

2059.

-311

114-8 .

185

4 .50 3

1I06

10-44 .983

..

288

1020

.27-3 94'4 29'3 5 80

65

3

143l

15'34

i'3d4

427

14.92 399 107.2 -327 6 '-80 80 3 1825 19'75' 1.604

.. 562

1922

'517

1082

350

7 '80 '95 3 22'OO 24'75 1955 - '7O5

'2458

'660

112'5

360

8 -80

110

3 2650- 29'90- 2'282 . -832 .29-74 795 112'8 . 36'4 9 I'OO 50 3 I0'94

946

_{960 .357 9.45 -395}

_865'

₃₇₂

10 i'OO

65

3 1475 13"64 1'300 -'581 -1382- ;577 _{92'S 44'7} 111(L)

100

3

i381

1800

1"083 765 17'78- '.746 130'-3

706

12(P)

i00

80 3 1712 17'98. 1'685 . 785 1778.-. '746 105'0

466

,13(H) 1.00 . '-80 3 20 12 17'70 2'004 - '761 17.78:

.746

.88'O

380

14 1.00' . "95 - 3 22'12 22'88 1'893 ,94-3

2275

954

I034

49'-8 15 1.00

110

3 -27'12

2668

2391 -F095 2753- ,1.'148

98'4

45.7 16

120

_{'50 3 't1'12}

₈₂₈

_{'976 51-3 8;68 :52! 74.5}

₅₂₆

17 1'20 '65 3 14'56 12'32 1'-267 -'765 1269 '763

-84'6

604

18 1'20 '80 3 1756 16'62 1-584 ,'993 -16'34 '986

946

62'7 19

120

.95 3

2100

,20'30 1"901 - F22'9 20'90 1-259

.967

65'i

20 1'20 1'lO 3 25'38 25'OO 2'293 1'495

2529.

1'520 - 98'S 65"2 21 - 1'20 1'25 3 27'12

2870

2622

1741

2950

1'7-75 105'8 66'4 22 23

I'40

' 65 3 14'19 1141 1'264 P893

i162:

-951 .. 80'4

706

140

80 3 17'19 14'81 1'547 - '1231 '1496 1'232 - 86'2 79-5 -24 1'40 '95 3 2-1.75 18"95 1'915 1'555 19'15 1.574 87'I

.811

TABLE 1List of Propellers Tested (continued) Propeller Number ,

'"

.

'"

NumberDi.4es . Weight Wp lb

Measured Values Calculated Values Percentage Values -Entrained, Mass We. lb. Moment of Inertia lb. 11.2 Jp Entraüed Inertia le lb.ft.2 Entrained Mass W. lb. Entraized inertia

Ie lbfe.2

-(WIW

_0/ 0 (Je/Ip)_0/ 0 26 27 28 29 30 31 32 33 34 p35(P) 36(N) 38 39 40 41 42 4.3 44 45 46 47 48 49

160

160

60

160

200

200

200

60 983 983

983

120

140

1'60 983 983 983 846

846

965

117

1181 .73 65 80 95

110

50

65

95

587

587 587 587 587 P587 587

587

587

x587

x587

50

25

588

73 802

.4

3 3 3 3 3 3 3 4 '4 4 '4 4 4 4 4 3 3 4 4 2 5 4 . 5 6 1438

1819

2156

2650

1131 1512

2262

2006

.1.956 1938

2038

1938

1962

1862

1875

2100

1819

1'638

''

1318

912

1925

2325

2250

1.225

i022

1381 .1748

2050

609

878

1414

1164

1022 .947

939

978 '

868

758

685

10.60 1'41

602

783

423

863

12'75.

1310

9.95 1285 1591 1917 2351 .973

1317

1994

1405 .

1366

1:387 1.440 1429 1401

1350

i386

1'383

1219

1149

1099

.559

-1405

i'598

1679 1.256 1081 1423

1856

2239

963 1342.

2328

.197 297 385 381 393: 499 P594 .738

517

.334

263

-226 118 '326 .743 752 . 225

1062

1367

1748

2i16

604

813

1454 1137 1051

957

950

979

.851

7.73

696

1014

681

622

788

423

8'oS

1264

1311

995

1125 1'457 1''862

2242

10O9 F474

2438

P168 P276 381 , 377 391 510,

627

738

450

-298

254,

P235 P118 ' '330 741 763 225 71-1 75.9

811

774

531

.581

656

58Q

522

489

461

.,507

:442

407

365

505

407

368

564

464

441

-,

548

58'2 81'2 841

894

961

952

99O 1O19 1:16.8 14O

2!7

28'4

264

274.

35:.6 44'Q

532

374

274

228

206

211

23'2 46'5 44'8 17'9

K.C.A. 410 PJD

10

_{B.A.R. = 0.65.}

From Fig 13 (orTable 3)

_{K, =0599andK4}

_O646

Entrainedinertiaje =1

R2 x

_{x N x , x K1 x}

jb.fj.2

=

_{x 64 x}

! x625

>

x 381164 x0599

Calculated

_{le = 0577 lb. ft.2}

Measured: value I = O58O lb. ft.2

Axial entrained mass We

₌

N X X L4 ' K4 X lb

We =

>< 3

IT x625

x 375 9367 x 0646x1

Calculated

We = i382ib.

Measured value

We = 1364 lb.

TABLE 2Specimen Calculation for

_{4 and W}

Number of Blades

_{- 3 = N}

/R Pitch iii.

Tzo,

=

00 _Chord in in. (Ozord x Sin 01 (Giord XSifl 0)2 / R2 S M. Function (C'hord

x Cos 0) x C'os 8)2(Chord Function

o 250

0375

O5OO 0 625 Ø75Q

0875

1 000

-l6in l6in. l6in. l6in l6in. 16in l6in 1 2731

08489

06365

0 5093

04244

03638

0 3183

51 51mm

40°-20min. 320-29min.

27° 0mm

23° 0mm.

20° Omia.

17 39mm 4 38

593

725

8 26

879

8'25

-3 444

3838

3893

3 750

3434

2822

-0 7414 2O714 37889 5 4932 66332. 6O972

-2 1 2 1 2 0 3707 4 1428 37889 10 9864 66332 12.1,944 2 706

4520

61I5

7 360 8091

7753

7 3224 2O4304 3:73932 54 1696

65464

60'1090 3 6612 408608 37-3932 108 3392 654643 i'202'1'80

= 381164

_3759367

BURRIL.L & RoBsoN: VIRTUAL MASS AND MOMENT OF INERTIA OFPROPELLERS 339

TABLE 3K, and KA Factors for 3, 4 and 5bladed Propellers

B.A.R. K, KA 3 Blades 4 Blades 5 Blades 3 Blades 4 Blades 5 Blades 0-50 0696 0875 0752 0938 2 0-683 0-840 0737 0-905 4 0670 0811 0899 0723 0878 0-977 6 0-657 0788 0-873 0700 0853 0950 8 0-644 0-768 0-850 0-695 0-832 0926

060

0-631 0750 0830 0681 0-814 0905 2 0618 0.734 0-812 0667 0798 0-887 4 0605 0720 0797 0653 0783 0-870 6 0592 0708 0-783 0-639 0-770 0-855 8 0579 0695 0-769 0-625 0757 0840 070 0567 0685 0757 0612 0-745 0826 2 0555 0674 0745 0-600 0733 0813 4 0544 0663 0734 0588 0-721 0801 6 0-533 0723 0576 0789 8 0522 0-712 0564 O777

080

0-511 0701 0552 0765 2 0500 0-690 0540 0753 4 0490 0680 0529 0-742 6 0-480 0518 8 0-471 0-508

090

o462 0498 2 0453 0488 4 0-444 0479 6

0435

0470 8 0-427 0461 1-00 0419 0-452 2 0-412 0444 4 0-403 - 0436 6 0396 0428 8 0-389 0420

1iO

0-382 0413

0+ 04 03 0 02 U 'U 02C 2

0

Lil 015 a-010 005

340 BURRILL& ROBSON: VIRTUAL MASS AND MOMENT OF INERTIA OF PROPELLERS

,

F /

,/

.. -x 1, BY :AL,eRATIo 0f /

- !f'BJLk__

I 4

-41

'f .020

H

_.0 25 3 35 +0 MOMENT OF INERTIA IN L FTZ

0020 0016 0012 000 000 kU

r

C -4

r

> z C C z - C z -4 > C '1 C -1'l

r

C, ._!PQ1!U041 N'Il.

.

-CALlbRA1IOH ,411614T

N'II._

CALIBRATIOkI JEIIfl -W.IM1ER:EPr O -3CAIJ. I W N'!l W1 4-4 7geIb.

-W..31.I I-Fol N'U.

1 + B 16 20 24-PROPELLER WEIGHT IN LB.

342 BURRILL & RoBsoN: VIRTUAL MASS AND_MOMNTOF INERTIA OF PROPELLERS LU

z

LU

z

z

LU

BLADE REA RATIO.

Fig. 3Measured Valuesof 4to Bzse

of

B.AR. (3-bladed Popellers)

06 04 10 12

PITCH RATIO.

Fig. 4Measured Values of 4 to Base ofPitch Ratio (3-bladed Propellers)

----.0 .

A

d4L4

05 06 07 O8 09 .10 24 1:20

-I6 LU 12 LU I.- ₀₈ LU

04

16 lB

30

BURRILL & ROBSON: VIRTUAL MASS AND MOMENT OF INERTIA OF PROPELLERS 343

S 24 0 II 'i-I , 3øLAE RLAE5. 0+ o.s a 16 BAR. x P..

Fig. 5Meàsurëd Values of 4 to Base of

(B.A.R x P/D)

to

05

06. 07 05

03

BLADE AREA RATIO:

30 25 2 20 LJ

z

z

-J

< 10

>c 5

344 _{BURRILL& ROBSON: VIRTUAL MASS AND MOMENT OF INERTIA OF PROPELLERS}

30

25

5

PITCH RATIO.

Fig. 7Measured Values of W to Base P/D

06 07 05

VALUES OF CO5Gz.

Fir. 8Measured Va/ties of Wç to Ba.e cos2o _,

09

- 06 08 tO lZ 14

t6

- IA En

--30

25

S

BURRILL, & R0BSON:. VIRTUAL MASS AND MOMENT OF INERTIA OFPROPELLERS 345

02 04 Ob b

BAR. % COS29a

Fig. 9Measured Values of W to Base B.A . R.

Cos2 0213

20

01 02 03 0

VALLJESOF 51N29

Fig. 10Measured Values of

4

to Base

5j,12 0 2/3

I,-'I

s4 I

It.-

SBl.ADE -J4ADES.

fiNs

s/f...-o'

r#A

..-a w

z

w

2

z

346 AURRWL &Ronsoi:vijftuAL MASS AND MO?kENT_{OF INERTIA OF PROPELLERS}

2-16 pa 08 0+ 1 b iN JNCI4Z5.

Fig. lIC'omparison oJ Measured_- 'e Valus with Theoretical Values

- ---- for Prolate Ellisojd

/

-- /

/

- - '-a b rtho,. C &AbIu ANGLES T b. CONWrANT Aii 1ADiU AT QuilT

I,

-

_1/

--f

-..-- __i

-:

07

06

05 ow LU 0 0

02

0I

BURRILL & RoBsoN: vIRTUAL.'MASS AND MOMENT OF INERTIA OF PROPELLERS 347

050 8A.R.!065 S 080 O B.AR .O95,

Il0,

S 06 08 (0 I2. I PITCH RATIO. measured va/tieof 4 Fig. 12--Derivation ofFactor K,

calculated va/tieof 4

348 _{BURRJLL & ROBSON VIRTUAL MASS AND MOMENT OF INERTIA OF PROPELLERS}

05 06 07 08 -

09

10 II

BLADE AREA RATIO.

P

BLJRRILL & RoBsoN: VIRT1JAL MASS AND MOMENT OF INERTIA OF PROPELLERS 349.

09

06

05

Fig. 14Curves

of

K1 and KA to Base B.A .R. (4-bladed Propellers)

Fig. 15Curves of K1 and KA to Base B.A .R. (5-bladed Propellers)

09

N

k

k 06 050 055 0 60

05

070 075

BLADE AREA RATIO.

K,-GBLAES.

K-6 BLADES.

055 0-60 065 070 075 080

BLADE AREA RATIO. 08

-o

07 08 2 < 07

(a) Axial Vibration (b) Torsional Vibration