MARIN
(NSMB/NMI)EXPERIMENTAL DETERMINATION OF
PROPELLER-INDUCED SHAFT FORCES.
by
J. van der Kooij
Publication No. 646
MARCH 1980
Technische Hogeschool
Dell t
2 CONTENTS ABSTRACT INTRODUCTION MEASURING TECHNIQUE 2.1. Introductory remarks
2.2. General description of the dynamometer
2.3. Description of the pick-up
2.4. Description of the measuring signals
SIGNAL ANALYSIS
3.1 Signal averaging
3.2. Analysis of the averaged signals
ALTERNATIVE TECHNIQUE FOR MEASURING THE STATIC
TRANSVERSE COMPONENTS
EFFECT OF CAVITATION
ACCURACY OF PREDICTED FULL-SCALE VALUES
ALLOWABLE FORCES AND MOMENTS
SUMMARY AND CONCLUSIONS
REFERENCES
EXPERIMENTAL DETERMINATION OF PROPELLER-INDUCED SHAFT FORCES
by
J. van der Kooij
ABSTRACT
A description is given of the measurement of the propeller excited shaft forces in the Depressurized Towing Tank of the NSMB. The mechanical properties of the six-component dynamometer and the technique of signal processing are discussed in detail. The factors influencing the accuracy of the predicted forces and moments are
reviewed and the effect of cavitation is indicated. Finally, some attention is paid to the judgement of the predicted excitation forces and to measures to reduce them.
1. INTRODUCTION
A screw propeller, operating in the non-uniform flow behind a ship, generates an unsteady load on the tail shaft. Six
load vector components can be distin-guished, viz, three forces and three moments, see Fig. 1. Starting from an idealized condition (a constant number of propeller revolutions and a steady
flow) , each load component can be consi-dered to be composed of a static part, or mean value, and a part which periodi-cally fluctuates. The fluctuating forces and moments cause shaft vibrations, local structural vibrations, deckhouse vibrations and vibrations of the
propulsion machinery /1/. In bad cases the vibrations can cause annoyance to the screw or damage to the bearings, the shafting, the gear wheels and to other structural parts /2/.
The vibration level strongly depends on, among others, the response
charac-teristics of the shafting. An important factor in the shaft response is the contact condition between shaft and bearings, which is greatly influenced by shaft alignment. Rational alignment can drastically reduce hull vibrations and minimize damage to the bearings
Z F M1
t
iz
LONGITUDINAL FORCE ( OR THRUST
Fy TRANSVERSE FORCE
F1 VERTICAL FORCE
LONGITUDINAL MOMENT ( OR TORQUE TRANSVERSE MOMENT
M1 VERTICAL MOMENT
HORIZONTAL THRUST ECCENTRICITY e1 r VERTICAL THRUST ECCENTRICITY
Fig. 1. Definition of the propeller-excited shaft forces and moments, and the resulting thrust
eccentri-cities.
Fx
etc. /2/. Today it is emphasized by an increasing number of investigators that the mean forces and moments, exerted by the propeller on the tail shaft, have to be taken into account in the shaft
alignment /2,3,4/. Therefore, the mean part and the fluctuating part of the propeller-excited shaft forces are both
important.
The propeller load fluctuations consist of two contributions, viz, a contribution caused by the non-uniform wake field and a contribution caused by the vibrational motion of the propeller. The relation between the vibrational motions and the
associated propeller loads is given by the so-called hydrodynamical propeller coefficients /5/. These propeller coefficients can be determined analyti-cally /6/, or experimentally /7/. The coefficients are included in the vibration analysis of the propeller vibrations. In this report no more attention will be paid to the vibration-induced propeller forces. Only the forces due to the non-uniform wake field will be considered, which is justified because the propeller vibrations are sufficiently suppressed during the measurements.
It is noted that besides the non-uniform flow, the finite number of propeller
blades plays an essential role in the occurrence of fluctuating shaft excitation forces. The greatest part of the flow variation in circumferential direction generates forces on the blades which are not transferred to the shaft due to
can-cellation. The propeller acts like a filter, and to get insight in this filter action it is necessary to decompose the variation of the flow into sinusoidal
(or harmonic) components. It can be shown
that for a Z-bladed popeller only the
harmonics of the order i.Z contribute to the thrust and torque fluctuations (i=1,
2, 3, ..), while only the harmonics of the order i,Z-i and i.Z+1 contribute to the fluctuations of the transverse forces and moments /8/.
4
The shaft excitation forces can be obtained by means of calculations /6/ and by means of experiments. For the
calcula-tion an accurate knowledge of the propellei inflow is necessary because of the
filter-ing effect of the propeller. In most cases the propeller inflow is deduced from the nominal wake field as measured with a model of the ship without the propeller mounted. This wake field has to be correc-ted for scale effects and for the effect of the propeller action. These corrections form the weakest point in the calculation procedure. An attempt to account for the effect of the propeller action has been made by Hoekstra /9/. To measure the propeller-hull interaction effects in the propeller plane, he simulated the propellez action by a diffuser. The difference
between the results of calculations starting from either the nominal wake or the wake in which the propeller action was accounted for is clearly demonstrated,
but the results obtained with the latter wake still show discrepancies with the
results of measurements.
In the experimental determination of the propeller forces the effect of the propeller action is automatically included provided some essential conditions are met, such as a geometrically scaled model and the absence of wall effects. Scale effects, however, still exist (see
Section 6).
At the NSMB the measurement of the propeller forces is always carried out in a towing tank, in most cases in the
depressurized towing tank, where the effect of cavitation, if any (see Section 5) , is taken into account. The test
condition is chosen such that the thrust coefficient for the model propeller is practically equal to that of the full-size propeller (see also Section 6) . The
air pressure in the depressurized towing tank is adjusted to a value which yields a cavitation number for the model
propeller which is equal to that for the full-size propeller.
2. MEASURING TECHNIQUE
2.1. Introductory remarks
The experimental determination of the propeller-excited shaft forces can be done either by measuring the forces exerted on the hub by one blade, followed by a
summation procedure, or by measuring di-rectly the resulting forces on the shaft. It seems logical to follow the latter method, since the summation procedure can be omitted then. Moreover, the following disadvantages of the first method can be mentioned:
The installation of a 6-component pick-up in the hub of a fixed-bladed propel-ler involves high additional costs for the propeller manufacture.
The greatest part of the generated blade force is not transferred to the
shaft. This means that the resolution available for the relevant force
contribution is restricted unnecessari-ly by measuring the total blade force, which affects the accuracy
unfavoura-bly.
The hydrodynamic force on the hub is
ignored.
On the other hand, the measurement of the resulting shaft excitation forces gives, in general, more problems in ful-filling the requirement to obtain a sub-critical dynamic system together with a sufficiently high signal-to-noise ratio.
At the NSMB both types of measurements are applied. If one is interested in the resulting shaft excitation forces, a dynamometer is used which gives these forces directly. This type of measurement will be considered in this report. The load on a single blade is only determined if one is really interested in this load, for example when a theory has to be verified, or when a controllable-pitch propeller has to be designed, for which the blade load is an important quantity.
The NSMB has two dynamometers for measu-ring the shaft excitation forces. The
older one was developed by Wereldsma /8/. It can be used for propellers with a
diameter of say 0.15-0.25 m. No provisions were made to use this dynamometer in low pressure conditions. The newer one is similar to the older one in many respects. It has been developed specifically for measurements in the NSMB depressurized towing tank. Hence, it can withstand low-pressure conditions and it can be used for propellers with a diameter up to say 0.40 m. The application for small propel-lers is restricted in view of the low magnitude of the forces, and the dimensions
of the dynamometer with respect to those of the propeller. In this report the larger
dynamorneter will be discussed.
2.2. General description of the dynamometer
The aft end of the dynamometer housing is tightly fitted in a conical bush, which
is mounted in the ship model beforehand (see Fig. 2). The other side of the dyna-is connected to an exdyna-isting frame in the model. The connection is made by means of resilient blocks in order to reduce the transmission of model vibrations.
The propeller forces are converted into electrical signals by means of strain-gauge bridges glued on a specially prepared part of the shaft (pick-up). Signal
conditioning is performed by AC carrier systems (Peekel, type 888 DNH), which are installed inside the towing carriage. The carrier systems are connected to the strain-gauge bridges via a multi-channel rotary transformer (Himmelstein & Co., type 2-16) . With the rotary transformer the electrical signals between the
-rotating part and stationary- part are coupled without any Contacte
In many cases the strain-gauge signals are very weak. In order to restrict masking of the signals by noise as much as possible, the signals are pre-amplified with a factor 200, as close as possible to the strain-gauge bridges.
FRAME FIXED TO THE SHIP MODEL
DRIVING SHAFT
/
DRIVING BELTAIR-TIGHT HOUSING FILLED WITH NITROGEN ( i ATM.)
PICK-UP (SEE FIG. 4)
HOLLOW SHAFT FLY-WHEEL
6
FRAME FIXED TO THE DYNAMOMETER
I\RUBBER RI NG
PLAIN BEARING
systems amounts to 1000 Hz, permitting an upper limit of the measurement bandwidth of about 250 Hz. At this frequency the signal attenuation is about 1 per cent. The carrier systems also introduce a phase shift of the signals. This phase shift occurs already from the lowest frequencies. In the bandwidth of zero to about 250 Hz the phase shift is almost linear with the frequency and mounts to 0.78 degrees per Hz.
Besides the signal pre-amplifiers and the rotary transformer, the housing of the dynamometer contains a pulse genera-tor (Gurley, type 8602) and two pulse amplifiers. The pulse of one channel appears with a frequency of one per
revo-lution, viz, when the directrix of one of the propeller blades is in upmost position. The pulses of the other channel appear with a much higher frequency, in most cases 120 or 180 times per revolution. They are equally spaced over a revolution, while one of them coincides with the one
PRE-LOADED BALL BEARING
1673 mm 6 MEASURING CHANNELS HOUSING WITH 6 PRE - AMPLIFIERS PULSE UNIT 2 PULSE AMPLIFIERS
Fig. 2. Lay-out of six-component dynamometer for measuring propeller-induced shaft forces and moments.
ROTARY TRANSFORMER
pulse per revolution. Both series of pul-ses are used for the analysis of the signals (Chapter 3).
The dimensions of the dynainometer, in particular those of the pick-up, have been carefully chosen. For the measure-ment of dynamic quantities the response
characteristics of the measuring system are crucially important. Above all it
should be tried to obtain a sub-critical system. In other words, the lowest natural frequency of the measuring system should be much higher than the highest signal frequency of interest. In that case the magnification of the signals by the
(unknown) dynamic response o.f the measu-ring system can be neglected. The realiza-tion of a sub-critical system requires a stiff suspension of the propeller to a sufficiently big mass. The big mass has been obtained by taking a very thick shaft (manufactured of stainless steel) and a thick-walled housing (manufactured of bronze). The total mass of the
dynamo-RESILIENT CONNECTION
meter is about 300 kg.
The stiffness of the pick-up cannot be taken arbitrarily high. A high stiffness conflicts with a high level of the measu-ring signals. The requirements of a high natural frequency and a high signal-to-noise ratio could not easily be combined. A solution has been found by applying
light propellers (manufactured of
alumi-nium) , by covering the stressed surface of the pick-up with strain gauges as much
as possible, and by signal enhancement (Chapter 3). The construction of the pick-up is considered in detail in the next
section. It is to be noted that the relatively large dimensions of the
dyna-Vx V 1.0 OS 06 04 02 o
-nR 0.42\
nR 0.73mometer form a favourable starting point for obtaining a relatively high signal-to-noise ratio. The gain in the so-called 'measurability" on larger models is due to the larger stressed surface available for gluing strain-gauges /10/.
The existing dynainometer shows a lowest natural frequency of about 250 to 300 Hz, permitting a maximum signal frequency of
80 to 100 Hz. At this maximum frequency the error due to the dynamic magnification amounts to about 10 per cent. For a 4-bladed propeller, rotating with 7 revolu-tions per second, the first three harmo-nics of the periodic propeller forces, having a fundamental frequency equal to
V MODEL SPEED
V LONGITUDINAL VELOCITY COMPONENT
Vtr r TRANSVERSE VELOCITY COMPONENT
r r RADIUS R r PROPELLER RADU)S WITHOUT DYNAMOMETER
-
WITH DYNAMOMETER Vtr r 0 0.2 160 180 40Fig. 3. Effect of dynamometer housing, mounted behind the port propeller of a fast twin-screw containership, on the nominal wake field.
nR r 1.04 1.0 08 06 0.4 02 10 08 06 04 02 o 90 180 270 360 (TOP) o
the blade frequency (=rotation rate times
the number of blades),can be determined
with acceptable accuracy. It should be noted that the third harmonics, and also the second harmonics, are often an order smaller than the first harmonics. In those cases the requirement with regard to the accuracy of these higher harmonics is less strong.
The housing of the dynamometer has a more of less streamlined shape. This is necessary since sometimes the dynamometer has to be installed behind the propeller instead of in front of it. This is the case for most twin-screw ships, but also for single screw ships with open stern. For those ships the dynamometer mounted in front of the propeller, would serious-ly disturb the propeller inflow. Also the installation behind the propeller has consequences for the propeller inflow, but in a less degree. An example of the effect of the dynamometer housing on the nominal flow field at port side of a twin-screw ship model is given in Fig. 3. Two effects have to be distinguisged. First, the change of the mean flow, which affects the mean propeller load. This effect is always compensated by a change of the model speed, so that the thrust
coeffi-cient becomes equal to the original value. In the example of Fig. 3, a speed decrease of about 6 per cent had to be applied. The second effect is the change of the
PROPELLER HUB
8
flow fluctuations. This effect should be as small as possible, since no corrections are possible. In the example of Fig. 3, the difference between the flow fluctua-tions without and with dynamometer housing is small. Starting from the results of
the harmonic analysis of the axial fow
field it is expected that the shaft forces are affected 5 per cent at most.
2.3.
Description of the pick-Up
The end part of the dynamometer including the pick-up, is shown in Fig. 4. The strains are measured in three crcss sec-tions, indicated with A, B and C. Cross section A is circular (see also Fig. 5) and is used for the determination of the fluctuating thrust and torque. Cross
sections B and C are square. Both sections are necessary for the determination of the transverse forces. At section C also the bending moments are measured. The position of the strain gauges, as well as their location in the bridges, are shown in Fig. 6. The more strain gauges are placed in a bridge, the higher the excitation voltage can be taken, resulting in a proportionally higher output voltage. The maximum allowable bridge excitation
is dependent on the possibility for heat dissipation.
For the determination of a transverse force the bending moments in two sections
Fig. 4. Longitudinal section of six-component force pick-up for measuring propeller-excited shaft forces and moments.
E
Fig. 5. Six-component force pick-up under construction.
at a known distance are measured (see relations can be deduced: Fig. 7). These bending moments form the
sum of three contributions, viz, a MB = MT + FH.a - W(a-d)
contribution MT due to the thrust
eccen-tricity, a contribution due to the Mc = MT + FH.c - W(c-d)
hydrodynamic transverse force, FH, and
a contribution due to the propeller The strain gauges in the sections B and weight (including a part of the shaft), W. C are arranged in a bridge in such a way
These latter two contributions are that the output is a measure for the
dependent on the location where they are difference of M and M3, which amounts measured. From Fig. 7 the following to (FH_W) .b. That means that the output
800 Ei SECT0NS B AND C (SEE F104) FE
ri
Ei B C WHEATSTONE BRIDGESFig. 6. Arrangement of strain gauges in six-component force pick-up.
00
3600 2 '0° 80° 900 0°
Ei
E QN
I
MT MB MT O TOTAL BENDING MOMENT lo
perpendicular to the shaft through a point at O.7R on the generator line (R=propeller radius) . Because of the dynamic character
of the forces and moments, the elimination of FH.0 is rather complicated, and there-fore it forms part of the standard analysis of the measuring signals (see Section 3)
The contributions of the propeller weight and the hydrodynamic transverse force to the measured bending moments are often relatively large. This is undesired, since in principle these contributions will unfavourably affect the accuracy of the bending moment due to the thrust eccentricity. It is possible to eliminate the contribution of the propeller weight in the output signal by distributing the strain gauges over two different longitudi-nal locations on the shaft, and by applying
a special circuit connection. The strain gauges should form two coupled bridges with separated, adjustable excitation voltages. The ratio of these voltages has to be adjusted in such a way that the out-put signal is zero during slow rotation of the propeller. Then the bending moment pick-up is no longer sensitive to
trans-verse forces which apply at the saine int as
the propeller weight (including a part of the shaft). Consequently, also the hydro-dynamic transverse force does not contri-bute anymore to the output signal if the distance d (see Fig. 7) is zero. In general, however, the distance d will differ from zero, so that FH still gives a contribution FH.d to the measured bending moment. For the elimination of this contribution the distance d must be known. An accurate determination of d is very difficult, however, so that
inaccuracies are introduced. In conclusion, the above-mentioned special way of circuit connection does not guarantee a real
improvement, and is therefore not yet
applied in the dynarnometer.
It is remarked that the technique for the measurement of the transverse forces is not very efficient. These forces are determined by the difference between
B C
LOCATION ON SHAFT
Fig. 7. Bending moment as a function of the location on the propeller
shaft.
of the bridge is independent of the point of application of the transverse forces. The contribution W is eliminated in the
final result. This elimination requires an additional measurement with very slowly rotating propeller, so that the hydrodynamic force, FH, is negligibly small, and only the contribution W is
measured.
To obtain the transverse and vertical bending moments explicitly, additional strain gauge bridges have been installed in cross section C. Just as described above, each bending moment consists of three contributions. The contribution due to the propeller weight is eliminated, so
that
Mc = MT + FH.c
It is often desired to know MT separately, which requires an estimation of the value c. This value is taken equal to the dis-tance between the centre of the strain-gauge bridges in section C, and the plane
B
relatively large signals. The signal-to-noise ratio of the resulting signal is always smaller than that of the separate signals. Today, more efficient techniques are used for the measurement of trans-verse forces (see for instance /11/) In view of an acceptable signal-to-noise ratio it is not necessary to replace the existing pick-up of the described dynamo-meter. There are plans, however, to apply the new technique for the smaller dynamo-meter which, as stated before, has a fun-damentally worse "measurability' than the
larger one.
The dynamometer is not suited for an accurate measurement of static signals. Nevertheless, the mean transverse forces
and moments can be determined, since these components are converted into dynamic quantities due to the shaft
rotation (see Section 2.4). Only the mean thrust and torque cannot be measured. In most cases, however, these components are known beforehand from propulsion test
results.
2.4. Deccription of the measuring signals
All forces and moments (see Fig. 1) consistof three contributions, viz, a static part, a periodically fluctuating part and a randomly fluctuating part. This latter contribution will be
elimina-ted (see Section 3). The fundamental fre-quency of the periodic contribution is equal to the blade frequency, i.e. Zw/27r,
in which Z=number of propeller blades,
and wangular velocity of the propeller
rotation.
The forces and moments, which have been stripped of random contributions, can be expanded into a Fourier series according to the following formula:
f(t) = ½A + (A sin nZut+3 cosnZut)
O n n n=1 (1) or f(t) C + I C sin(nZwt+9 o
n1
n n where: C = ½A0 C = n o=arctgA/B
n n nEquation (2) says that the periodic func-tion f(t) consists of a static component co, and an infinite number of sinusoidal
components (or harmonics) , which have amplitudes C and phases B. The
frequen-cies of the harmonics are multiples of the fundamental frequency.
The co-ordinate system in Fig. i is
fixed with respect to the ship hull. The pick-up, however, is rotating with angular frequency w. As far as the longitudinal force and moment (i.e. thrust and torque) are considered, the rotation does not cause a difference between the measured values and the values which are valid for the fixed co-ordinate system. The trans-verse forces and moments, however, undergo a conversion. If the components in the -rotative system of axes are distinguished
from the components in the fixed system (2)
Fig. 8. Decomposition of F and F into
components for the rotating system of axes.
of axes by adding the subscript r, the C = ½{(A +B
)2(3
-A )2}½Z y z
n n n
transverse forces for the rotative system n
B -A 'n Z e = arctg A + y1 y z n n n -B
)2(3
+A )2}½ = n Zn y z nSimilar expressions hold for the trans-verse moments.
With help of (1) Fy and F can be writ-ten as follows:
F = ½A + E (A sin nZut + B cos nZwt)
O n1
'nF=½A
+ E (AsinnZut+B cosnZut)
o
nl
n nSubstitution of (5) and (6) in (3) yields after some calculation:
F = ½A cosut + ½A
smut +
r o o + E I (½A
+½B
)sin(nZ+1)ut + n=l n n + (½A -½3 )sin(nZ-l)ut + y z n n + (½B -½A )cos(nZ+l)ut + n +(½B+½A)cos(nZ-l)utI
(7)
This formula can also be written as
fol-lows: F
=C
sin(ut+9 ) + y0 y0 C=½(A2 +A2
)½ Z Y0 Yo o Yo 8 = acrtg yo z o 12 B +A y Z ri n arctg A -B y z n n nSimilar formulea can be obtained for
Fz,M
andM.
r r r
Formula (8) says that the static components of the forces and moments in the fixed co-ordinate system are converted to sinusoidal components with a frequency equal to u/27r. In other words, the static transverse forces and moments are
experienced by the pick-up as periodical dynamic loads. Further it is shown in
fomulae (7) and (8) that the sinusoidal components in the fixed co-ordinate system with a frequency equal to nZw/21T, are converted to sinusoidal components with frequencies equal to (nZ+1)w/2rr
and (nZ-l)w/2rr.
Q°(TOP)
g0
1800ANGULAR BLADE POSITION
Fig. 9. Example of averaged signals of
and Mzr
for
a 5-bladed propeller.-270° 360'
become (see Fig. 8):
y z F = F coswt + F smut (3) ½{(Ay F = -F smut + F cosut
(4)
Z y Z + E C sin(nZ+1}ut+8 ) + n=1 'l n n + Csin(nZ-1}ut+O
(8) n ri where:The conversion of the signals by the propeller rotation hinders a judgement of the recordings in the time domain. In
Fig. 9 an example is shown of the signals of the torque fluctuations and the
dynamic bending moment from one of the two measuring channels. The results are valid for a 5-bladed propeller. The com-ponent with blade passage frequency dominates in the torque signal, as was expected. In the bending moment signal the component with a period equal to one propeller revolution appears. Further a component with a frequency equal to 4 times the rotation rate of the propeller, i.e. (Z-1)w,can be discerned.
3. SIGNAL ANALYSIS
3.1.
Signal averaging
If the propeller is operating in a per-fect flow, the generated shaft forces are periodic functions with a fundamental
frequency equal to the blade frequency. In reality, the flow is not completely steady, resulting in non-periodic contri-butions to the shaft forces. The relation between the unsteady flow components on model scale and on full scale is unknown. Therefore, the non-periodic force
contributions are eliminated. This elimination has no serious consequences for the prediction of the full-scale forces and moments since in most cases the non-periodic contributions are small with respect to the periodic contributions.
Cavitation can be another source of non-periodic force contributions. If low
frequencies are considered (say in the
order of blade frequency) , the non-perio-dic behaviour of the sheet cavitation has to be attributed for the greater part to the unsteady flow phenomena. As stated above, these phenomena are not understood, and the resulting cavitation behaviour and associated propeller forces can not be interpreted and should be left out. Besides low-frequency non-periodic
components, cavitation can also generate high-frequency non-periodic components due to the random collapse of a great amount of small vapour bubbles. This phenomenon will cause some noise on the measuring
signals. A more or less similar type of noise can be expected from the electronics,
specifically if the measuring signals are very small. This high-frequency noise disturbs the signals and has to be
eliminated.
The elimination of all these
non-periodic phenomena is realized b.y ensemble averaging. One énsemble represents a signal during one propeller revolution, being the fundamental period of the periodic part of the transverse forces and moments in the rotating system of axes (see Section 2.4). The ensembles are obtained by sampling and digitizing the analogue signals. A sample of each signal is taken every time when a pulse of the i pulses per revolution appears, where i amounts to 60, 90, 120, 180 or 360. The choice
is based on requirements regarding aliai-sing and on the storage capacity of the used computer (CDC, System 17). In most cases 120 or 180 samples per revolution are taken. The reduction of the non-periodic contributions strongly depends on the number of ensembles, k, used in the averaging process. If the non-periodic contributions to the sample values of an ensemble are uricorrelated (which
will
bethe case for the high-frequency noise) the signal-to-noise ratio will increase with a factor /k. In most cases k is taken about 200.
The final result of the averaging process is, for each force component, a mean curve of the force during one
revolution, consisting of i discrete values. Examples of these curves are given in Fig. 9. The arrays of discrete values are used for further analysis of
3.2. Analysis of the averaged signals
The averaged signals, which represent the periodic propeller-induced shaft forces as experienced by the pick-up, have to be converted to forces and moments valid for the fixed system of co-ordinates shown in Fig. 1. First, the calibration factors are taken into account, and the transverse forces and moments are corrected for the contribution due to the propeller weight. Then the signals are analysed into harmonic
components according to:
f(t) = I C sin(mut+
m m
rn= 1
where: f(t) = periodic function of t
t = time (t=0 if the one pulse per revolution appears) w = angular velocity of the
propeller
As discussed in Section 2.4, the periodic forces and moments in the fixed system of co-ordinates generate specific harmonics in the rotating system. For the thrust and torque only the harmonics m=n.Z are generated, while for the transverse forces and moments the harmonics m=l and m=nZ+l are created, where n=1,
2 ...In
practical cases n ranges from i to 3. In reality the remaining harmonics of the measuring signals will not be zero due
to disturbances. These remaining harmonics, however, should be small with respect to the relevant harmonics. An example of the harmonic content of a torque signal and one of the bending moment signals is shown in Fig. 10. For this example the signals shown in Fig. 9 have been used. It is clearly demonstrated that the relevant harmonics have significantly larger amplitudes than the remaining harmonics, although the differences become smaller at higher frequencies. It is plausible that
the errors in the relevant harmonics correspond roughly to the amplitudes of the respective adjacent harmonics. Therefore, the harmonic analysis is also
14 lo HARMONIC COMPONENT 15 15 5 io HARMONIC COMPONENT
Fig. 10. Harmonic components of M and
Mzr for a 5-bladed propeller.
used as a check on the accuracy of the measurement results.
For the Conversion of the transverse forces from the rotating system of axes to the fixed system of axes, the following formulae are used (see also Section 2.4):
F = F coswt - F
smut
z ' 'r r F = Fsmut + F
cosut z y z r rexpressions hold. In the past these con-versions were realized, before signal
averaging, by means of a special type of rotary transformers, which were directly coupled to the propeller shaft. Today the conversions are performed off-line by means of the computer.
After elimination of the contribution of the transverse forces to the bending moments, all six components are converted
to full-scale values, using the following extrapolation factors for the amplitudes of the forces and moments respectively:
The subscripts s and m indicate full-scale and model values respectively.
If Froude-number similarity is observed, the factors simplify to respectively
M e = X F X and and
From the six components the thrust eccentricity is calculated, using the following formulae for the components in x and y direction respectively
M
z
e
Y F
The thrust eccentricity is also periodic with a fundamental frequency equal to the blade frequency. The static value as well as the harmonic components cannot easily be expressed in the static values and harmonic components of the bending
moment and thrust. Assume that the moment,
M, and the thrust, T, consist of a static part and a sinusoidal part with a frequen-cy equal to the blade frequenfrequen-cy, Zu, hence
M = M0 +
M1sin(Zwt +F = F + F1sin (Zut +
o
Then the thrust eccentricity, e, amounts
to
e = e0+e1sin(Zut+y1) +higher harmonics
Only for e0 an exact expression can be
derived: M M o e
=-
+ o ' i_Ii
2 cos(a1-1) (F-F'Y2 1 (F-F1)2 1It can be shown that the second term -O
M0 F1
if -- O, so that e
= -. If - -
oF0 o F0 F0
then it further follows that M1
e1 = 11 = al
higher harmonics = O.
In practice, the harmonic components of the thrust eccentricity are derived
from the time function. The time function is calculated from the known time func-tions of the bending moment and the thrust.
It should be noted that the thrust eccentricity is only useful if a normal operating condition is calculated. At zero thrust, which can occur for example with a controllable pitch propeller, the thrust eccentricity becomes infinitely large, which, however, is not saying anything about the bending moment, nor about the character of the inflow into the propeller.
4. ALTERNATIVE TECHNIQUE FOR MEASURING THE STATIC TRANSVERSE COMPONENTS
The two components of the static transverse force can be measured, in principle, with a 1-component pick-up if besides the force signal the angular position of the propeller is recorded. This is also the case for the static transverse moment, although for the
determination of the moment due to thrust eccentricity only, also the transverse
F o p N 2 ps N
s25
and-pm m m m
where: p specific mass of water
N X
= rotation rate of propeller length scale ratio
Fig. 11. Decomposition of F and F into
components contributing to Fr
force has to be known, including its point of application, just as in the case of the dynamometer described in the
previous sections.
If the transverse forces are purely
static then the measured force, Frl amounts to (see Fig. 11):
F = -F smut + F cosut
r y0 z0
It can easily be shown that
F = -2(F smut)0
y0 r
F = 2(F coswt)
z r o
o
where the subscript o indicates the time average over a long period.
In reality the transverse forces also contain components which vary in time. Assume the transverse forces consist of a static part and a sinusoidal part, hence
F = F + F sin (Çt+a
y y0 y1 1 F = F + F sin (2t+ß
z z z i
o i
The question is whether or not the
ROTATING
AXIS 16
sinusoidal part also contributes to the
terms (F sinwt) and (F cosuit) . If only th
r o r o
sinusoidal part is considered, Fr becomes
Fr = -F
sin(St+c1) smut +
y1
+ F sin(Ç2t+ ) coswt
z1 1
Then the following expression should be
zero:
2
um
f -F cosa1 sinnt sin ut +T- O
-F sina cost sin2wt +
y1 1
+F cose sin2t coswt smut +
z1 1
+F sine cos2t coswt smnut} dt
z1 1
Each term of this integral is always zero, exceptthe second one. By replacing sin2wt by ½(l-cos2wt) the integral becomes
T
lirnF
sincx f cos2t cos2wt dt2T y i
T-= i o
This expression is also zero, except if Ç2=2u. In that case it is found that
(Fsinwt)
¼Fsina1
In the same way it can be derived that (FrCOSWt)o=O except for Ç2=2w. Then
(F coswt)r = ¼F
sin1
o z1
In conclusion it can be stated that the alternative method for determining the static transverse forces and moments can be applied except for a 2-bladed propeller
A further important question is which requirements have to be fuif-illed with respect to the dynamic behaviour of the measuring system. It has been shown above that static transverse forces are experienced by the pick-up as sinusoidal
forces with a frequency equal to
Contributions with the same frequency, but from other sources, introduce errors in the results. Such contributions can be
caused by vibrations of the propeller-pick-up-shaft system. In general their effect will be negligibly small, except at resonance. One should take care, there-fore, that the natural frequency of the bending vibration of the system is not equal to . But also the natural
frequency of the torsional vibration should be different from in view of interaction phenomena. The safest way is constructing the shaft so stiff that the lowest natural frequency is always higher than r-. This requirement is much less strong than that for the dynamometer
discussed previously, for which the natural frequency should be about a factor 50
higher.
At the NSMB preparations are made to in-vestigate the above described measuring technique is practice. The pick-up will be constructed in such a way that also the mean thrust and torque are obtained.
5. EFFECT OF CAVITATION
In the beginning of the 1970's it was found that cavitation can have a dramatic effect on the propeller-induced pressure fluctuations on the hull (see for instance /12/ and /13/). The dynamic behaviour of sheet cavitation was found to be the most important mechanism /14/. A characteristic feature of the hull-pressure fluctuation, generated by a non-cavitating propeller,
is the approximately sinusoidal behaviour. The frequency of that harmonic fluctuation is equal to the blade frequency. If sheet cavitation is present the sinusoidal character disappears. Higher harmonics, specifically those with two, three and even four times blade frequency, can become of the same order of magnitude as the first harmonics.
Also with regard to the shaft force fluctuations the sinusoidal component with blade frequency predominates usually,
if cavitation is absent. The question arose whether or not these shaft forces are affected by cavitation in a similar
way as the hull-pressure fluctuations. Therefore, in 1974 the NSMB started an
investigation in the depressurized towing tank. A series of tanker models was tested under depressurized condition as well as under atmospheric condition. It appeared that the effect of cavitation on the shaft excitation forces was rather small, also when the pressure fluctuations on the hull were affected tremendously /15/. The inves-tigation showed that cavitation can cause an increase as well as a decrease of the shaft forces. The effect on the mean
values amounted to 10 per cent at maximum. The harmonics with blade frequency were affected somewhat more, viz. 35 per cent at most. The ratio between the higher harmonics and the first harmonics did not change substantially.
A type of ship which is currently under investigation at the NSMB is the Ro-Ro vessel. The single-screw Ro-Ro vessel is characterized by a flat stern at a rather small distance above the propeller, a relatively small draught, and a high power absorption. These properties often result into extensive sheet cavitation ori the propeller and large pressure fluctua-tions on the hull. An example of an extreme-ly extensive cavitation pattern is shown in Fig. 12. The cavitation was present over a restricted rotation angle. Moreover
Fig. 12. Cavitation pattern on a CP-propel-1er tested behind a Ro-Ro vessel.
it broke up irregularly. For this propel-1er the shaft excitation forces were mea-sured. Two test conditions were considered, viz, the condition for which the cavitation pattern in Fig. 12 was observed, and a condition which was similar to the first condition, except for the environmental pressure. This pressure was taken so high that cavitation did not occur anymore.
It was found again that the mean forces and moments did not change substantially due to cavitation (less than 10 per cent). The effect on the fluctuations, however, was considerable (see Fig. 13). The most alarming effect was the strong increase of the higher harmonics. Also the first harmonics changed, but not always an increase was observed. Besides the anIplitudes also the phase angles of the first harmonics were affected
significant-ly.
The above mentioned case studies do not give practical information for the
prediction of the effect of cavitation on the propeller-excited shaft forces. From these studies only the conclusion can be drawn that the ignorance of cavitation
WITH CAVITION
fi
WITHOUT CAVITION0.04_ Fz Cm 0.02 F0 o Fx
RL
2 3 1h
2 3 1 HARMONIC COMPONENT m 2 3 I 2 3 HARMONIC COMPONENT, miq. 13. Example of the effect of cavita-tion on the amplitudes of the shaft excitation forces and
moments.
18
may lead to unacceptable errors prediction of the force fluctuat as a consequence in the calculat vibrational behaviour of the sha the afterbody.
6. ACCURACY OF PREDICTED FULL-SCALE VALUES
The accuracy of the predicted full-scale forces and moments is determined by various factors, which can be classed in two categories, viz.
factors related to properties of the measuring system,
factors related to scale effects on hydrodynamic phenomena.
The most important properties of the measuring system concern the vibrational behaviour of the system and the signal-to-noise ratio. Both aspects have been consi-dered already in previous sections, althouc considerations on the vibrational behavioui were restricted to properties of the
dynamometer itself. It was concluded that mainly the higher harmonic components were
affected. The error in the third harmonics can amount to 10 per cent due to dynamic magnification. The error due to noise can
even be larger if these higher harmonics are very small.
However, also ship model vibrations can
fluctuations and have, therefore
in the ions and, ion of the fting and
introduce inaccuracies. The solid connec-tion between the dynaniometer and the ship hull near the propeller implies that vibrations of the model are transferred
to the dynamometer. As long as the frequencies of these vibrations differ from the frequencies of the propeller-excited shaft forces, the forces generated by vibrations of the model are reduced drastically during signal processing. The main vibrations, however, are caused by
the propeller-induced hull-pressure
the same frequencies as the shaft excitation forces. The vibrations can contribute in two
different ways to the measured forces, first by inertial effects of the mechanica] system and, second, by hydrodynarnical
2 3
effects. It can be shown that the latter effects are an order of magnitude smaller than the first effects, at least if trans-verse vibrations are considered. In
longi-tidunal direction both effects can be of the sanie order of magnitude. Several experiments have shown that with our measuring system the model vibrations are no source of serious inaccuracies, although it is worthwhile to check the vibrations during each test.
Another source of errors in the measuring system might be due to interaction pheno-mena. Interaction caused by imperfections of the pick-up (its geometry, the material homogeneity, the location and properties of the strain gauges) can be eliminated by careful calibrations. Another form of interaction is the cross-talk between the electric signals. This phenomenon is minimized by a perfect isolation and a correct location of the wiring. Cross-talk between signals belonging to different channels can be avoided by switching on only one channel at a time. The error due to interaction phenomena of the described dynamometer amounts to a few per cent at
most.
In the bending moments, and as a conse-quence in the thrust eccentricities, an additional error exists which originates from an error in the estimated point of application of the transverse forces (see Section 2.3). In most cases the bending moments will not be affected more than 2 or 3 per cent, but in exceptional cases
(propeller with large rake for instance) the error may amount to say 5 per cent.
Scale effects on hydrodynamic phenomena are associated with the propeller inflow and with cavitation. In the NSMB depres-surized towing tank Froude-number
similarity is observed, which is the basis for a correct simulation of phenomena in which the free surface plays a role. A correct Froude number implies, however, that the Reynolds number is a factor
(about 100) smaller for the model than for the ship, so that phenomena in
which viscosity plays a role, are scaled incorrectly. Viscosity is particularly important in boundary layer formation. An incorrect boundary layer on the hull results into deviations from the correct wake field.
The scale effect on the mean wake is compensated by choosing a slightly higher model advance speed than would follow from
Froude scaling:
1-w V
-m 1-w s
ni
where: V = advance speed w = wake fraction X = length scale
The indices m and s refer to the model and ship values respectively. In this way the thrust coefficients for the model is almost equal to that for the
ship.
A correct mean wake does not necessarily imply a correct wake distribution. The wake distribution influences the static part
of the transverse forces and moments, as well as the fluctuating part of all six
components. In /8/ an estimate is given of the influence of the wake scale effect on the fluctuating thrust and torque for
a tanker with V-shaped afterbody and
fitted with a 4-bladed propeller. The ratio of the thrust amplitudes derived from the corrected and uncorrected model wake respec-tively, amounted to 0.85 for the first
harmonic, 1.20 for the second harmonic and 1.27 for the third harmonic. The ratio of the torque amplitudes amounted to 0.84, 1.16 and 1.27 respectively. In comparison with the applied correction to the model wake field, the change of the thrust and torque fluctuations appeared to be rather small, which has to be
attributed to the filtering action of the
propeller.
Deviations from the correct Reynolds number can also affect the cavitation behaviour. Specifically cavitation-incep-tion problems can arise /16/. To overcome these problems the leading edge of the
propeller blades is roughened. Further, timely inception is stimulated by seeding artificial nuclei in the water by means of electrolysis. Deviations from the correct cavitation behaviour can still occur due to the scale effect on the propeller inflow. The error in the shaft forces caused by these deviations is expected to be one order smaller than the total effect of cavitation on the forces.
Overlooking all possible errors it is expected that the error due to scale ef-fects on the wake field will dominate in most cases and will amount to say 10 to
20 per cent for the blade rate frequencies. The error in the higher harmonics can
even be greater, say 20 to 30 per cent, whereas the error in the static part of the transverse components is expected to be somewhat less, say 10 per cent. It is clear that the above figures are only indicative. Unfortunately, a comparison of the model test results with full-scale data will hardly provide more accurate figures, since the determination of the forces and moments on full scale contains inaccuracies of at least the same order of magnitude as the inaccuracies in the predicted forces and moments. The full-scale forces and moments cannot be measured directly. Either the stresses
somewhere on the shaft surface, or local shaft vibrations are measured. The
stres-ses as well as the vibrations are the result of the excitation forces, the structural response of the shafting and the vibration-induced hydrodynamic forces and moments. If the structural response and the so-called hydrodynamical propeller coefficients (see the Introduction) are known, a comparison can be made between the predicted and measured stresses or vibrations. Such an exercise has been carried out in the past with respect to the thrust and torque fluctuations /8/. A reasonable agreement between the predic-ted and measured values was found.
20
7. ALLOWABLE FORCES AND MOMENTS
Guide-lines for allowable propeller-shaft forces hardly exist, which is not surpri-sing, however, since the relation between these forces and the phenomena of interest
(vibrations, bearing forces, stresses on material, etc.) is very complicated and completely different for different ships and even for one ship at different opera-ting conditions. It can be stated therefore that the shaft excitation forces alone havE not so much value and in literature quite rightly more attention is paid to guide-lines for allowable vibrations of vital parts of the propulsion system than for allowable excitation forces (see for instance /1/)
Nevertheless, the magnitude of the excitation forces forms an essential link in the generation of vibrations, stresses, etc. A low excitation level is always favourable and therefore it may be
worthwhile to judge the excitation forces separately. A judgement can be based on the formulae given in /17/: AF AM X < 0.25 F M
-X X o o iF+AF
AM +AM
y z y Z<0.35
FxO.3RF
x o oin which F and M are the various forces and moments (see Fig. 1) , R is the propeller radius, A means peak-to-peak value, and the subscript o means static value. Most of the results of the measure-ments carried out at the NSMB in the last two decennia meet the requirements given by the above formulae. The t-ransverse components more often exceeded the limit than the longitudinal components did. In practice, however, the dynamic thrust
seems to be the most important component in the generation of severe vibrations /1/.
The static transverse forces and moments are used for rational shaft alignment and
a low magnitude of these forces and moments is favourable, although it is imaginable that hydrodynarnic forces and moments are welcome if they have an effect opposite to
the effect of the propeller weight. A bad situation may arise when the forces and moments vary strongly at different
opera-ting conditions, for example at two diffe-rent draughts.
From experiments carried out at the NSMB it has been found that the static forces and moments differ considerably from ship to ship, the differences being concerned with the magnitude as well as the direction
If the propeller operates in a strong oblique flow, while the axial flow is more or less uniform, the resulting transverse force will be directed roughly in the direction of the transverse velocity components. If the transverse flow velo-cities are small, the direction of the resulting transverse force is determined by the location of the highest wake. If the wake peak is located in the upper part of the propeller disc, the transverse force will be directed roughly to port side (for
a right-handed propeller). In most cases the magnitude of the resulting static trans-verse force will amount to say 1 to 10%
of the static thrust, whereas the resulting static transverse moment
will correspond to a thrust eccentricity (at normal operating conditions) of say i to 5% of the propeller diameter.
As pointed out in the Introduction,only a selected number of harmonics of the wake contribute to the propeller-excited shaft forces. As far as the fluctuations of the forces and moments are concerned, the selected wake harmonics are strongly related to the number of blades. The static forces and moments are determined by the first harmonic of the peripherial wake distribution only, irrespective of
the number of blades. In the past it was often found, specifically for ships with closed stern and a small clearance between propeller and rudder, that the even
harmonics of the wake field were
much greater than the odd harmonics /18/. This tendency is no longer met in modern ships. Roughly speaking the amplitude
decreases, if the harmonic order increases. Therefore it is nearly always found that the blade-rate forces and moments are lar-ger than the forces and moments at two or more times blade frequency. Moreover, it can be expected that a propeller with a high number of blades generates smaller forces than a propeller with a low number of blades. This no hard rule, however, since the wake field of modern ships still often shows one or more predominant
harmonics and the number of blades should be chosen such that these harmonics do not contribute to the shaft forces. The
choice is not always easy, because in general the longitudinal force and moment
are sensitive to other wake harmonics, than the transverse forces and moments. Moreover, the choice of the number of blades
is limited in practice to 4, 5 or 6,
and determined also by the requirements of avoiding coincidence of excitation
frequency and natural frequency of the
shafting.
Besides wake field and number of propel-ler blades, some other factors influence the magnitude of the shaft excitation forces, for instance the clearance between the propeller and the hull and rudder /19/, and the propeller geometry. Specifically skewed propellers can give a considerable reduction of the fluctuating shaft forces and moments, /20,21/.
8. SUMMARY AND CONCLUSIONS
For a trouble-free operation of the ship it is necessary to include the propeller-excited shaft forces and moments in the ship design. For high-powered ships it is absolutely necessary to take the static part of the forces and moments into
account in shaft alignment procedures. The fluctuating part of the forces and moments is particularly important with regard to vibrations of vital parts of
22
the propulsion system.
For the measurement of all six components of the shaft excitation (three forces and three moments) the NSMB has developed and constructed a dynamometer which is employed in the Depressurized Towing Tank. This dynamometer is capable to measure the static and fluctuating part of all forces and moments, except the static thrust and torque. The forces and moments are measured for a stiff construction, so that the
contributions due to the vibrational motion of the real full-size propeller is not simulated.
In order to obtain reliable measurement results the lowest natural frequency of the measuring system is about one order of magnitude higher than the blade rate
frequency. The realization of such a high natural frequency, in addition to an ac-ceptable signal-to-noise ratio, requires the propellers to be manufactured of a light material, for example aluminium. The signal-to-noise ratio of the rough measuring signals is improved by ensemble averaging. By this technique the
contributions which are not periodic with one propeller revolution are reduced
dras-tically.
If only the static part of the forces and moments has to be determined, a much simpler measuring technique can be applied, at least for propellers with three or more blades.
The accuracy of the test results is determined by a lot of factors.
Overlook-ing all possible errors it is expected that the error due to scale effects on the wake will dominate in most cases, and amount to say 10 to 20% for the blade rate harmonics. The error in the higher harmonics can be somewhat greater, where-as the error in the static part of the transverse components is expected to be
less.
The effect of cavitation on the shaft excitation forces and moments is much smaller than that on the pressure
fluctuations on the hull surface. If the
cavitation behaviour is normal, the
effect on the mean forces and moments will not exceed 10%, and the effect on the fluctuations in most cases will not be higher than say 35%. In extreme cases, however, the effect of cavitation can be much greater, specifically on the higher harmonic components.
A judgement of the magnitude of the propeller-excited shaft forces and moments
alone is hardly meaningful. For a predic-tion of the vibrapredic-tion level the dynamic behaviour of the excited structure is at least as important. Moreover, for a final determination of the vibration level it is also necessary to account for the forcez and moments generated by the vibrational motion of the propeller.
The static transverse forces and moments are hardly affected by propeller design parameters. They are governed by the first harmonic component of the circumferential wake distribution. The fluctuating part
of the forces and moments (also in
longitudinal direction) is caused by wake harmonics which are related in a prescribec way to the number of blades. Therefore,
this fluctuating part is strongly affected by the propeller design, specifically the number of blades and skew, but also other parameters play a role.
REFERENCES
Seminar on Ship Vibration (specifically papers Nos. 1, 5 and 6), Det norske
Ventas, June 1977.
Volcy, G.C.: 'Interaction of compatibi-lity between machinery and hull from a static and vibration point of view". Ship Vibration Symposium,- Arlington, October 1978.
Larsen, O.C.: "Some considerations on shaft alignment of marine shafting'. Norwegian Maritime Research, Vol. 4, No. 2, 1976.
Vassilopoulos, L.: The influence of propeller mean loads on propulsion shaft alignment". Propellers '78
Sympo-siurn, Virginia Beach, May 1978.
Hylarides, S., Gent, W. van: "Propeller hydrodynamics and shaft dynamics'. Symposium on High Powered Propulsion of Large Ships, Wageningen, December 1974.
Gent, W. van: "Unsteady lifting-surface theory for ship screws. Derivation and numerical treatment of integral
equa-tion". Journal of Ship Research, Vol. 19, No. 4, December 1975.
Wereldsma, R.: "Experiments on vibrating propeller models". International Ship-building Progress, Vol. 12, No. 130, June 1965.
Wereldsma, R.: "Dynamic behaviour of ship propellers". Publication No. 255 of the NSMB, 1965.
Hoekstra, M.: An investigation into the effect of propeller-hull interaction on the structure of the wake field".
Symposium on Hydrodynamics of Ship and Offshore Propulsion Systems, Oslo, March 1977.
Wereldsma, R.: "Effect of model size on the measurability of dynamic
phenomena on ship models". Proceedings 12th ITTC, Propeller Committee, Rome, September 1969.
Gommers, C.M.J.: "Enige nieuwe vormen voor rekstrookmeetelementen". Mikroniek, 16e Jaargang, No. 6, June 1976.
Takahashi, H., Ueda, T.: "An experimen-tal investigation into the effect of cavitation on fluctuating pressures around a marine propeller". Paper No. 33 of Ship Research Institute, Tokyo, March 1970.
Manen, J.D. van: "The effect of cavita-tion on the interaccavita-tion between
propeller and ship's hull". IUTAM Symposium, Leningrad, June 1971.
Huse, E.: "Pressure fluctuations on the hull induced by cavitating propellers". Publication No. 111 of Norwegian Ship Model Experiment Tank, March 1972. Huse, E.: "Effect of cavitation on propeller-induced vibratory forces". Proceedings 14th ITTC, Propeller
Committee, Ottawa, September 1975. Kuiper, G.: "Scale effects on propeller cavitation inception". 12th Symposium on Naval Hydrodynamics, Washington D.C., June 1978.
Proceedings 15th ITTC, Propeller Committee, The Hague, September 1978. .18. Werelsma, R.: "Comparison between the
vibratory forces of regular and irregu-lar propellers behind a single screw ship". International Shipbuilding Progress, Vol. 9, No. 96, August 1962. Hadler, J.B.: "The effect of variations of several parameters on the propeller alternating thrust and torque for series 60-0.60 CB model". Proceedings 13th ITTC, Propeller Committee, Berlin/Ham-burg, September 1972.
Valentine, D.T., Dashnaw, F.J.: "Highly skewed propeller for San Clemente
Class ore/bulk/oil carrier design considerations, model and full-scale evaluation". STAR Symposium, Washington D.C., August 1975.
Hammer, N.O., McGinn, R.F.: "Highly skewed propellers - full-scale vibra-tion test results and economic conside-rations". Ship Vibration Symposium, Arlington, October 1978.
