RABALHOS - PAPERS Zjljij 1981
Lab. y. Scheepsbouwkund
ARCHIEF
Technische Hogeschool
PROPELLER EXCITED SHAFT AND HULL VIBRATIOJ1f
OF
SINGLE SCREW SHIPS
EIXO ATIVADO DO PROPULSOR E VIBRAÇÖES DO CASCO DE UM NAVIO MONO-HÉLICE
Introduction
The importance of vibration reduction on board
ships induced investigations in this field and
intensified the activities of the shipbuilders and the model research institutes in order to obtain
more knowledge on the vibration phenomena, to avoid failures due to fatigueness and overloading, and to improve comfort.
The first experimental investigations in this
field have been carried out about 10 years ago. Lewis and Tachmindji determined the
instation-ary effects of the propeller on the hull. Since
that time an increasing number of investigators have been working on experimental and
theore-tical research in the field of propeller excited
vibrations.
These vibrations can be divided into three parts:
Shaft vibrations in axial direction, loading the thrust block and gearing or crank shaft (thrust and torque variations).
Shaft vibrations in transverse direction, loading the stern tube bearing and the pro-peller shaft bearings (transverse forces and thrust eccentricity, resulting in shaft whir-ling).
e) Vibrations of the ship's afterbody, excited
by the variable pressure field of the pro
-pellers due to the finite number of blades. A more detailed description of these three types of excitation is given in this paper. Finally the
(x Head of the lnstrzzmenntion Department of ¡he Netherlands
Ship Model ftasin, Wageningen, Holland.
Sumário:
Nesse trabalho, o autor nos fornece, em primeiro lugar, urna descrição sucinta dos problemas ligados is vibra-çôes do exo do propulsor. Seguem uns dados aproximados sôbre a variação dos momentos e das fôrças atuantes do hélice. O autor nos dá também algurnas sugestöes no que diz respeito aos arranjos da pôpa, para efeito das vibraçöes na parte inferior do eixo. Finalizando, nesse trabaiho são discutidas as necessidades de conduzir pesquisas sôbre as propriedades mecânicas do sistema de propulsão.
by/por Ir. R. WERELDSMA (X)
possible methods to reduce the vibration level
are mentioned.
t. Description of different types of propeller generated vibrations
i: Axial shaft vibrations
Schuster[1] indicated that the axial behaviour of the propulsion system (engine, shaft, propeller)
can be described by two coupled equations of
motion. In these equations several important
parameters can be distinguished.
Propeller generated torque and
thrustfluc-tuations.
These exciting forces can not be etablished on board the full size ship, but can only be
measured by specially designed model
equipment. A result of systematic model
measurements is given in Fig. 1.
Hydrodynamic properties
of an
axiallyvibrating propeller.
Besides the well known added moment of inertia, we can distinguish torque damping,
thrust damping and mutual coupling effects,
proportional with accelerations and
velo-city.
Mechanical properties of the propulsion system i. e. mass of the propeller, elas-ticity of the thrust block, mechanical
im-pedance of the engine (turbine or diesel en-gine), elasticity of the shaft, etc.
With the knowledge of these properties the
equation can be solved. The solution gives the
control of the bearings in the main engine is carried out by
means of analog thermistor transmitters to a central electronic uniI alarm when the limit vatue set has been exceeded The
instrumentation as a whole includes uncomplicated components
and systems which as far as possible are independent of each other. The operational safety of the installation is partly
de-pending on the reliability of thc components, partly on the prin-cipal features of the system and last but not least of the
adjustment and trimming. There will as experience has shown
-be a very high operational safety if the original installation and
trimming has been mode carefully and if a fijial trininiing of instruments during the first voyage is made at the sanie tinte
as the engine room staff will he instructed and trained on hoard.
ti is difficult to determine the operational safety in figures but
as an indicative figure it can be said that separate circuits or lar-ger electronic units have 5 to 10 errors per 10(1,000 service hours This figure varies from installation to installation and it is diffi-cult to say whether there have beeil niaterial deficiences or errors in maintenance or handling. As a rule it can be said that
indica-ting instruments have a high degree of operational reliability
Also relay systems can work very well if they are correctly
di-mensioned and if it is certain that their operational temperatu-res will not be too high and that there will no be too much load on the contacts. Transistorised equipment has a very high de-gree of reliability in service if there is no overload in voltage.
o REMOTE CONTROL
Remote control of the main engine is but one of the compo.
nents of the "automation" system, hut it is one where lack of reliability ca,, make a wreck of a ship in a niatter of seconds. Fig 31 shows Gotaverkesi's standard system in which ail old
equipnient has been stripped fron< tise engine and tbe remote
control gear has been designed around the basic functions of the engine itself - fuel pump regulating shaft, starting air dis-tributor and speed governor - without a side-glance at any
pre-viously existing manoe.uvring gear.
Few of Gotaverken's engines are today equipped with a con-ventional control system, mounted directly ois the engine. Most installations have a separate control ruolo or bridge control and both these alternatives require remote contro,I system has been developed by Gotavcrken. The, .system us a whole and some of its components have been thoroughly tested in our experimentai shop. In addition some ships have bemi equipped with the sys-tem. The results from these installations have been most
encou-raging, and for this reason it has been decided that all our
mo-dern engine types shall have the system as standard equipment,
no matter whether the ship is intended for bridge control or
engine room control only.
The system includes the following main parts: (1)
manoeu-vring box (2) rotation direction sensor (3) speed regulating
de-vice (4) n<unoeuvring dede-vice.
/
The manoeuvring box contains two cylinders, one turning
Ihi- fuel regulating shaft to and from stop position and another one actuating the starting air distributor. The box also contains
two locks for the regalating shaft, actuated l'y the starling air
distributor and the rotation direction sensor res1)ectively. Thus the fuel injection cannot take place until the engine Is completely reversed and moving in the intended direction.
The m000euvring device includes valves for "ahead" and "astern" signals and a pressure regulator for the signal air to
Ihr hydraulic speed governor. This governor actuates the fuel
regulating shaft.
When the system is uscii for bridge control some alarm and supervision devices have to be added, bearing in mind the safety
of the machinery. An automatic load tinilng device is also
na-ressary.
llectrlc signals instead of impulse ai arc used between the
bridge control desk and the manieuvring box when the distance
exceeds fil) to 70 metres and when the connections have to beni
placed above deck.
VA ¡N ENGINE SER VICE CIRCUITS
The development of the engines towards higher ivutputs amid
(I,, development of automated installation, forces the engine
de-signer to take new interest in the auxiliary machinery around
the main engine. It is consequently not enough to specify how
nianv calories shall be rcn,ovcd from cooling water, lubricating
oil and piston cooiiog circuits, lt is necessary for the engin,
designer ti, lake a,, active l)art i,, the layout of these circuits
-specially when automated -- to ensure that the result gives the
best balance between: initial cost
brat economy
operational safety
simplicity.
Sii,,,.' 's,,,,, i< les nias Illustrate this statement
The a 1fr,,, pis Iii chas, calories for instanc,- by us og an
i' h;,ist as hoi er 'ir a frisli water eva pii rotor rising heal fro,,i lia' nia iii engine fresh ci,,, li rig water circuit has tended to n,ak,'
il,.' iii fi' ri-rit ci rcui Is lii lb. engine room mt,., ri- arid more con, pies
<rol (n sonic cases les,i rrli,il,le
Fig. 33
Fig 32a as an exanip.le shows part of Gotaverken's standard
fresh water system for the main engine. Originally water was forced through the evaporator by throttling a valve between the pipes to unit fron, Ihm evaporator (Fig 32b). The abolition
of this valve and the introduction of a small booster pump to
take care of the pressifre drop over the evaporator proved ti, be
a simplification of the system and a,, improvement of safety since it makes the cooling water volume to the engine stable which is essential In an automated plant.
Fig 33 shows the corresponding diagram for part of the sea water circuit. Here it is of importance to ensure ssUficieiit flow to the main engine interniediate air cooler under ail circumstan-ces and it is better to aLlow some waler to escape overboard after Ibis cooler than to force it through the lube oil cooler when the
sensor of the thermostat for the air cooler demands increased
cooling.
With the earlIer arrangement the capacity of the pumps could
be reduced to a minimum but it was very difficult to ensure that the automatic regulation of the circuit to fresh wuter arid
oil coolers would not starve the branch to the air coolers. The new arrangement with slightly larger water pumps and a separate discharge overboard is consequently the answer to the
call for operational safety and simplicity although It raises
initial cost slightly.
In connection with the sea water circuits it Is most interes-ting to study the experience drawn during the last years fron, ships well over say 50,000 tdw. Since these ships rarely enter
rivers or brackish water there is a marked inc-ease of the fouling
of all parts of the sea water circulation system, valves, pipes
and heat exchangers.
\
It was earlier natural to cool any unit which had advantage of low outgoing temperature with sea water since an indirect cooling with fresh water would entail a loss in temperature re-duction of say 8°C. However, if the salt water side of a heat exchanger Is fouled, part or the whole of tisis gaio may be lost
by the reduced efficiency of the heat exchanger. It has therefore
been suggested to have one double fresh water cooler in the
engine room capable of absorbing all heat which shall be removed
from waler-, air- and olI-ciruits. The double cooler gIves the possibility nf cleaning one half of the sea water side whenever
the ship is outside areas with tropical sea water temperature.
in this way sea water would he banned f roni all other heat
exchangers in Ihi' engine room arid one could count upon that all these fresh water cooled heat exchangers will 'e,nain clean on thr
water side for long periods thus promoting stability in the
se-crind;rn'y temperatures of all circuits served by this installation.
C f) N C L 1/ S I O N
It is evident that the Authors of this paper have not beemi
ahle to treat by far all engine components. They hope, however. that the examples given will clearly illustrate the importance of
research, not only in time manufacturers' own laboratories and test shops hut also through extensive contact wIth ships In
ser-sic,. This research - in close cooperation with owners willing io report their service results river prolonged periods and
prr-pared to talca special instruments and arrangements for research
purii,srs on board - will give ali parties concerned valuable
i:
resulting torque and thrust fluctations in the shaft
and the torsional and axial motions of the pro-peller,. the instantaneous loading of the gearing and of the thrust block. These quantities are of interest in order to determine the dimensions of the various parts of the propulsion system. As
pointed out above information of the mechanical construction cannot be avoided in this prediction.
2. Transverse shaft vibrations
fr. The approach of Jasper [2]. determining critical vibrations of the propulsion system in transverse direction (shaft whirling) is not suitable for the
prediction of the forced vibrations and the loading
of the stern tube bearing. These quantities can be predicted by the solutions of 4 simultaneous equations of motions, describing the vibrations
of the propeller on the elastic shaft in
trans-verse direction.
We can distinguish:
a) Variable propeller thrust eccentricity and transverse forces. These phenomena can not be established on board a full size ship, butcanonlybemeasuredby means of a spe-cially built propeller force pick-up.
b) The hydrodynamic properties of a propeller
vibrating in transverse direction (added mass, added moment of inertia, damping
factors and coupling terms).
e) Mechanical properties i. e. elasticity of the shaft in transversal direction, elasticity of the stern tube bearing and its support. The vibratory motions of the afterbody and of
VOL. I - N.° 2 - APRIL/JUNE 1968
Fgure t. Propeller vibratory forces.
Vr.EICAL 8C5b MOIe SI C P0001LLOP OEIL,' 001Lu000)
the shaft support affect the behaviour of the
pro-peller shaft system in transversal direction.
The above mentioned equations are coupled to other equations, describing the vibratory motion of the ship's afterbody.
3. The vibratory motion of the ship's afterbodv The afterbody is excited by the pressure fluc-tuations, resulting from the finite number of
blades of the propeller.
The mechanical impedance of the afterbody and the hydrodynamic exciting forces (pressure fluc -tuations on the hull) give a complete description of this phenomenon. In order to come to a
pre-diction for the vibratory phenomena in transverse
direction, the equations of 2 and 3 must be dealt
with simultaneously.
As a first approach in this investigation on
predicting vibrations in transverse direction of
the propeller shaft and associated parts of the propulsion system analyses are made on the
loading of the propeller shaft and the stern tube loading with rigid afterbody [3] After full size measurements of the shaft vibrations and of the afterbody motions, a renewed analysis will be made, startingfromthe vibratory motions of the
afterbody as a boundary value and the force fluc -tuations, acting on the hull, measured on model as exciting forces.
H. Origin of the vibrations
Axial and transverse shaft vibrations are
in-troduced by the instationary operation of the pro-. 131
peller. Via the shaft bearings and the thrust block.
the afterbody of the ship is excited by the fluc-tuating shaft forces.
An other excitation of the afterbody occurs by
the variable pressure field, generated by the
finite number of propeller blades. Some
experi-mental investigations illustrate these effects
(Lewis, Pohl, Breslin, Tachmindji) [4-81. At the Netherlands Ship Model Basin experimental re-search on hull excitations is under preparation. Therefore no details can be given on this subject and our attention will be focussed on the origin of the fluctuating propeller forces. These forces
are highly related to the distribution of
theharmonic components of the peripheral wake
pat-tern on different radii. A simple analysis results in the conclusion that the thrust and torque fluc-tuations are generated by harmonic components of the wake having a number of fluctuations per
revolution equal to the number of blades and
multiples thereof.
Transverse force fluctuations and variations
of the thrust eccentricity are caused by blade
frequency harmonic cbmponents and multiples
thereof plus and minus one, so adjacent to the
components, generating the thrust and torque fluctuations.
For a conventional symmetric afterbody
con-struction, the even harmonic components are
re-latively important. Therefore an even bladed
propeller generates large thrust and torque fluc
tuations and an odd bladed propeller large trans
-verse forces and thrust eccentricity.
III. General results on exciUng forees obtained
from model experiments
A statistical investigation into the thrust and
torque fluctuations of 40 ship models leads to the following conclusions:
There is no systematic correlation between the amplitudes of the force fluctuations and important
hull parameters as block coefficient, prismatic coefficient and propeller diameter - ship length-ratio.
For a variation in the prismatic coefficient of
the afterbody from 0. 73 through 0. 79 holds with
a reasonable probability: a) For a 4-bladed propeller:
The amplitude of the first harmonic
com-ponent of the shaft torque fluctuations is
about 6. 5 per cent. of the average value.
The amplitude of the first harmonic
com-ponent of the thrust fluctuations is about 10 per cent. of the average value. The
ampli-tudes of the higher components are
con-siderably smaller.
b) For a 5-bladed propeller:
The amplitudes of the first and the second
harmonic component of the torque variations
amount toi. 5 and 1. Oper cent, respectively
of the average torque, those of the thrust variationstoabout2. Oandi. 5 per cent,
re-spectively of the average thrust.
In these indications deviations of 2 per cent.
may occur.
3) Fine and high speed ships give rise to higher fluctuations as indicated.
In the Figures 2 and 3 a typical experimental
result of a four and a five bladed propeller, oper-ating behind a single screw 15 knots tanker is given.
Dimensionless expressions are: Iorizontal transverse force Fy: Fy . O .7Di
mean 14-bladed _ 0.12 +0.073
sin(43 + 80°)
Vertical transverse force Fx:
ILFX .0.7DI
= 0.06 +0.076
mean 4-bladed
sin(4 + 126°) For the horizontal bending moment T due to
the thrust eccentricity holds:
f i
,Fzmean.O.7Dj4_bIaded = 0.007+0.008 sin(4 + 147°)
For the vertical ,ending moment
- Tv
{Pzmean.0.7Dj4-bladed
0.032+0.012Forthefive bladed propeller holds according-sin(4 + 130°) Sin(5p + 201°) I T, mean. O .7D 1 -bladed Ty
mean.0.7D 5-bladed
0.007 + 0.019sin(5+ 159°)
0.032 +0.053 Sin(513 + 155°)Fy.0.7D
= 0.12 +0.15 T mean 5-bladed sirt(5+ 100°)F.0.7D
- 0.06 + 0.13 mean 5 - hiadedr I : -s k s Io PL 0.4.1...., .,- '-,'°. î y__ ..0
General conclusions as regards the dynamic
behaviour in transverse direction of a 4- and
5-bladed propeller are:
The transverse force fluctuations of a five
bladed propeller are tvice as large as those of a four bladed propeller.
The higher harmonic components of a five bladed propeller are equal to those of a four
bladed propeller.
The fluctuations in thrust eccentricity for a five
bladed propeller are considerably larger than
those of a four bladed propeller.
Besides the static thrust and torque, the pro-peller generates a static thrust eccentricity and transverse force.
The static transverse force is negligible in re -gardtothe propeller weight as can be seen from the Figures 2 and 3.
The thrust eccentricity, however, affects the bending moment of the propeller shaft
consider-ably. This is illustrated by Figure 4. which shows
the steady shaft loading for a rotating propeller. With the knowledge obtained up till now it is
possible to suggest some special constructions in
order to improve the behaviour of a propeller
from a point of view of vibrations.
One of the first ideas was that of Grim, to make an elastic support of the propeller shaft in .trans
-VOL. I - N.° 2 - APRIL/JUNE 1968
Fgsr 2. Dvnanc phe,sornosa of a 5-blodesS propeller.
4
verse direction by means of a long elastic tube E9].
When the propeller is operating over critically,
the generated excitation forces are for the greater
part consumed by the acceleration of the
pro-peller mass. In this way the dynamic loading of the support of the propeller shaft is considerably reduced. This method can be seen as 'vibration
isolation
On the other hand a reduction of the vibration level can be obtained by reducting the exciting forces. The application of a six bladed propeller will give a gain in vibration reduction, but this
includes a lower efficiency.
The four bladed propeller generates large
thrust and torque variations and the five bladed propeller large bending moments. These aredisadvantages of both types of propellers. An accept able compromise can be obtained by the construc
-tion of an asymmetric stern, or an eccentric
pro-peller shaft location. In general. improvement is obtained when symmetrical constructions are avoided.
With respect to the afterbody excitation, a
special stern construction must be mentioned in
this paper. The special construction, indicated
in Fig. 5, is chosen from an efficiency point of view. This construction, however, resulted in a noise free afterbody with a low vibration level. Due to the open. very lai'ge. screw aperture, the
__0
exciting forces could be kept small.
Another practical approach to vibration re-ductionisthe appliation of contra-rotating
pro-pellers. In particular propellers with different
number of blades will generate very low vibration
levels, although the generated frequencies are
increased. A ducted propeller with a Hognerstern
arrangement is also favourable from a vibration
point of view. However. realisation implies prac
-tical difficulties.
The theoretical approach to the problem of the
propellerexcited vibrations is reviewed in 110]
where 17 existing methods are presented. all
based on quasi steady or two-dimensional
in-stationary profile theory.
All
these analyses start from the unequal
peripheral wake pattern as introduced by the
ship's hull. Only specific harmonic components of this pattern contribute to the force variations in the propeller shaft.
Due to the lack of accuracy in the wake field measurements. the results of these analyses are
open to doubt. More attention should be payed for
improving the determination of the wake pattern. The scale effect should also be seriously
con-sidered in these investigations. The wake pat-terns for the model and the full size ship differ due to the different Reynolds number for these
two conditions. A more detailed study shows,
however, that by the filtering action of the
pro-peller, the scale effect and corrections are of
I
'0O
I t
L a
--Figure 3. tfyaoamc phanaooaeao of a 4-baded propefler,
90
minor importance and can, for the first approach.
be neglected in regard to other uncertainties that
unfortunately exist in the prediction problem [11].
More attention has
to be payed to, the
determination of the dynamic properties of the me
-chanical part of the propulsion system and the
.afterbody. i. e. thrust block flexibility mechanical impedance of the engine (turbine or diesel engine). flexibility of the propeller shaft and bearing sup-ports and the mechanical impedance of the ship's
hull.
On the other hand knowledge must be obtained
of
the dynamic properties of the
propeller.Theoretical analyses, based on the two -dimens-ional strip-wise approximation, are available for
/
Figure 4. Transverse loading of the shaft due to the pro-peller action.
the determination of these properties to be ex-pressed as propeller coefficients. Experiments
on these coefficients are carried out by Lewis
[12] and Visser [13]. These investigations give
information on the added moment of inertia. Both investigators have determined this value. expres
-sed as a percentage of the corresponding me-chanical quantity. Their results do not coincide (25% respectively 15%). The two-dimensional theory gives roughly 15%. As this added inertia depends on the density of the medium and the dimensions of the propeller, a better method is to express the added momept of inertia in these
quantitiesi.e. asapercentageof p.D5 (D =pro-peller diameter).
A specially designed instrument for the mea-surements of dynamic propeller coefficients is very recently available at the Netherlands Ship
Model Basin.
This exciter enables an experimental
deter-mination of the added mass and the added moment
of inertia, the damping in torsional and axial
direction and the mutual coupling terms. In
part-icular the coupling effects play an important
Figure 5. Special ship stern.
role in the determination of the forced vibrations of the propeller shaft and the propulsion system.
As is indicated in [14] there is a tendency to
improve the methods for the theoretical
de-termination of the dynamic phenomena of a pro-peller, not only for the determination of the shaft excitation, but also for the instationary cavitation.
phenomena.
When more rel(able information can be obtained on these hydrodynamic phenomena, it is
neces-sary to have available accurate knowledge on the
VOL. I - N. 2 - APRIL/JUNE 1968
mechanical properties of the propulsion system
and of the afterbody in order to complete the
equations of motion for the propeller and to pre
-dict important information, such as the loading of the stern tube bearing, thrust block, propeller shaft, gearing, etc., in order to design a reliable
structure, in accordance with the requirements. Full size measurements should be carried out in order to check the predicted phenomena and when necessary to be able to improve the pre-diction technique. In this way predictions on
vibrations can be made with increased accuracy.
Full size measurements are required, otherwise much of our effort and research into propeller vibratory phenomena will be fruitless.
H fe re n e
Schuster S. 'Beitrag zur Frage des fünfflügeligen
Propellers'. Jahrbuch S. T. G. . 1955.
Jasper, N. H. 'A design approach to the problem of critical whirling speeds of shaft-disk systems'.
D.T.M.B. Report 890, 1954.
Wereldsma, R. 'Determination of the dynamic
pro-perties and propeller excited vibrations of a
special ship stern arrangement'. Intern. Shipb. Progr. . 1964.
Tachmindji, A.J. and McGoldrick, R.T. 'Note on
propeller-excited hull vibrations'. .J. of Ship
Res. , June 1959.
Breslin, J. P. 'Ship vibration. Part I:
Propeller-gene rated excitations'. Appi. Mech. Rev., 1960.
.6. Pohl. K. H. 'Die durch eine Schiffschraube auf
benachbarten Platten erzeugten periodischen hydro
-dynamischen Drücke'. Schiffstechnik, 1960.
Lewis, F.M. 'Propeller vibration'. Trans.
S. N.A. M.E. , 1935, 1936.
Lewis, F.M. and Tachmindji, A. J. : 'Propeller
for-ces exciting hull vibration ',Trans. S. N.A. M. E.
1954.
Grim, O. 'Lagerung der Propellerwelle in einem
elastischen Stevenrohr'.Schiff und Hafen, 1960. Stern. R. and Ross. D. : 'The calculation of
alter-nating forces of wake -operating propellers'. Bolt
Beranek and Newman Inc. Report 1133. 1964.
Wereldsma. R. : 'Dynamic behaviour of ship propel-lers'. (to be published)
Lewis, F M. and Auslaender. J. : 'Virt,ual inertia of propellers'. J. of Ship Res. . March 1960.
Visser. N.J. : 'Model tests concerning the damping coefficient and the increase in the moment of inertia due to entrained water of ship's propel-lers'. Netherlands'Research Centre T.N.O. for
Shipbuilding and Navigation Report No. 31 M, 1960.
Manen, J .D .van :'Durch die Schraube erregte Schiffs -schwingungen'. Symposium on the occasion of the
50th Anniversary of the H.S. V.A. . Hamburg, 1964.
L
DETERMINA
1. ENUNCIATION OF ThE METhOD
Our aim is to determine the sensitivity to
cavi-tation of a hydrofoil cascade without formulating any
restriction as concerns the geometrical and hydro-dynamic parameters.
in accordance with the definition stated in some previous papers fil] , [2] , the sensitivity to cavitation
of ahydrofoil cascade means the functional dependence between the absolute minimum coefficient of pressure -k0, the cri ticsl angle of incidence c, and the
relative abscissa at which and where this absolute minimum is achieved.
Analytically, the sensitivity to cavitation is
expressed by the following two functions
k0 = k0(ct)
OEC
= 1.2)
In the above quoted papers, two hypotheses have
been accepted:
The incipience point of the cavitation phenomenon of relative abscissa X,11 practically coincides with
the point at which the distribution of velocities
realizes an absolute maximum.
The investigation of the maxima of the velocity function can be carried out, observing the hypothesis
of the potential flow.
The hypotheses which have been set forth are used and maintained only in the process of the expansions of first approximations.
As a rule, as the experiment proves, the incipience points of the cavitation phenomenon are being
distri-buted in three zones: a zone corresponding to the
()
Engineers and scientists of Polytechnic institue Timisoara of136
TION OF THE SENSITIVITY TO CAVITATION OF A CASCADE OF
HYDROFOILS OF ARBITRARY SHAPE
PREDETERMINAÇAO DA CAVITAÇAO NUMA ASA DE AEROBARCO EM FORMA ARBITRARIA DE HIDROFÓLIO
by/por ION ANTON and OCTAVIAN POPA *
Resumo:
Um método teórico é apresentado neste trabalho pa ra a predeterminação da cavitaçäo corn a água circulan -do em volta de urna asa em forma de hidrofólio geométrica e arbitràriamente molda-do. Urna dependência fun-cional entre as magnitudes criticas correspondentes à verificaçäo do fenômeno da cavitação é o objeto principal dessa tese, sendo que as mesmas magnitudes dizem res peito ao ângulo de incidéncia, ao coeficiente de pressáo e sua relativa abscissa, O raciocinio fundamenta-se na hipótese segundo a quai o ponto inicial da cavitaçio prati-camente coincide corn o ponto de máximo absoluto verificado pela distribuição da velocidade potencíal no contôrno do perfil da asa do aerobarco em forma de hidrofólio. O método proposto se orienta na solução de algumas
equaçôes diferenciais do tipo Fredhoim de segundo grau. Empregando os teoremas de Fredhoim, ficam comprova-das a existéncia e a univoquacidade da soluçäo obtida, na hipótese de que o tipo de circulação scia determinado.
upperside surface of the hydrofoil and two zones
corresponding to the leading edge, on its upperside and lowerside. Fig.l represents the theoretical
curves of sensitivity to cavitation for th Clark Y hydrofoil, which have been obtained by means of an original theoretical method in a previous paper. The
experimental results obtained by F.Numachi [3] for the
same hydrofoil are superposed by us on these curves
in order to illustrate the validity of the hypothesis and theoretical method proposed.
44
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-the Socialist Republic of Romania.
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-a [Aper/,z'enla/ -(,4'uiriach,2 CILARA' V8 -5 -4 -? 0 2 4 5 8Ci'///ce/ angle ofu7ci6ence cK. =
Fig.
la
TECNOLOGIA NAVAL
