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HULL VIBRATION AND MEASUREMENTS OF
PROPELLER-INDUCED PRESSURE FLUCTUATIONS
E. Huse').
Abstract
The paper presents a theo'retical and experimental investigation of the influence of hull vibrations
upon the accuracy of experimental determination of propéller-induàed pressure fluctuations. The
results indicate that in many cases it will be impossible to measure propeller-induced pressure
fluctuations on the afterbody of a ship by means of pressure transducers fitted in the hull plates.
Lab.
y. Scheepsbouwkunde
Technische Hogeschool87
Délff
Introduction.
Blade thickness and lift give rise to a pressure field surrdunding each blade of a rotating pro-peller. At a point fixed in space this prd'ssure
field will be felt as a
periodical pressure fluctuation of a fundamental frequency equal toblade frequency (propeller revolutions per second
multiplied by blade number). This pressure
fluctuation may be of importanôe in exciting hull vibration.
A considerable amount of information is
avail-able from model-scale measurements of pro-peller-induced pressures. A fairly good predic-tion of these pressures can also be obtained by present-day theories. Regarding the correlation between full scale and model scale measure-ments, however, our knowledge is more scarce. As far as the author is aware the only attempt at
correlating model and full
scale pressures
hitherto published is the measurements on the German research vessel 'Meteor', f
in this case the correlation was very good in
the lower speed ranges, the discrepancy increasingtoa maximum of about 40 percent for the highest ship speeds. However, the author knows of unpublished full-scale measurements where the amplitudes of the recorded pressure fluctuations have been far above those expected from theoretical considerations and model scale measurements. The discrepancy may amount to a factor of 10 or more.
Possible explanations for this large ')The Norwegian Ship Model Experiment Tank, Publication no. 103,
August 1969.
') Numbers in brackets refer to Reference listat end of artic1e
discrepancy may be for instance,
the effect of hull vibrations of the ship itself, the occurrence of propeller cavitätion in full scale.
This paper presents a theoretical and
ex-perimental investigation of the first item, the
effect of hull \ribrations on the measured pressure
fluctuations. If these'vibrations become excess-ive, the corresponding pressure fluctuations on
the hull will be considerably larger than the pressure fluctuations induced by the propeller. Experimental determination
of the
latter by means of pressure transducers on the hull is in this case impossible.It is the author 's hope that this pâper may be an aid tothose involved in experimental determina-tion of propeller -induced pressures, serving as a
guide to the vibration level that can be allowed
without affecting the measurements.
Theoretical considerations.
Consider a pressuretransducer fitted ina hull plate in the vicinity of the propeller of a full scale
ship. Our problem is to express the relationship
between the vibrations of the plate and the
resulting pressure fluctuations experienced by the transducer.
We shall here distinguish between two typés of
vibrations. First we shall investigate the case
where the afterbody of the ship is oscillating in vertical directionasarigici body. Later we shall discuss the effect of local vibrations of a single hull plate in the afterbody.
The first case can be approximately treated by
applying the theory of infinitely long cylinders
performing vertical Oscillations ma free surface,
[2J, [3] and [4]. Consider a ship section with a
submerged part of the form shown by the curve G in Figure 1. The x-axis is here coinciding with the
Figure 1. Mapping of contour into a circle.
mean free surface. For vertical oscillations of
small amplitude and high frequency it can be
shown, [3], that the flow field in this case is
equivalent to the cylinder G-G' performing the
same vertical oscillations in an unbounded fluid.
Suppose that the conformal transformation of the
exterior of the closed contour G-G' into the
exterior ofa circle of radius r0 in the c-plane is
given in the form
a1 a2 z + . .. (1) where z = X + jy and Ç = +
In [3] it is shown that for vertical oscillations the velocity potential becomes
= z (b e
+ b e'8)
n n n
where
b1 = -iV(r02 + a1),
b = -iV a. n = 2, 3,
By the formulae given above the potential cax in
principle be calculated numerically for any
section form. We shall here confine ourselves to
the so-called Lewis forms, which are those
derived from the unit circle (r0 1) in the -plane
by the transformation
a1 a3
z =Ç
+-+-where a1 anda3 are real. In this case the velocity potential on the surface of the cylinder (r = r0 = 1) becomes
- -v ((1 + a1) sine + a3 sin 3e).
Since a1 and a3 are constants which depend only
upon section form, the potential along a given section may be written
0 -V f(e). (2)
Tn[3] equations are given which make jt possible
to calculate f(e) at various positions along the
section. This has beèn done for the section shown
in Figure 2.
Having determined the velocity potential it is
now easy to calculate the pressure fluctuation due
to the oscillatory motion of the section.
Bernoullis equation is
- + 2. 1 ' q2 F(t). (3) at
At infinity 0. p. and q are independent of time. Therefore F(t must also be a constant and can thus be neglected here. Derivation of (2) with
respect to time yields
=-av f(8).
We are here dealing with oscillations of high frequency and small amplitude so that
I
_.»q?
as can be verified by simple numerical
calcula-tions. The oscillatory pressure may thus be written
p
= -oa.f(e).
(4)For practical purposes we are mainly interested
in the ratio between the pressure 'p and the
corresponding vertical acceleration
a.
We shall therefore express this ratio as a non-dimensional coefficient C1 defined byC
apb
pTaking p from (4) we obtain
f(8)
(5)
As mentioned above f(8) has been calculated
numerically at several positions along the section
shown in Figure 2. The resulting value of. C1 is showninFigure3. The value of C1 is C1 = 0.685
at the keel point of the section, increasing to a maximum value of 0.735 at x/b= 0.3, and then
decreasingto zeroas the waterline (x/b =1.0) is
approached.
It should be emphasized that this theory is based on the assumption of an infinitely long cylinder.
When applying the results to a vibrating ship a
discrepancy may be introduced by
'end effects' at the stern,
the fact that the various sections along the ship
are not vibrating in phase.
Until now we have only considered the case
where the afterbody
sections are vibrating
verticallyas a rigid body. Horizontal vibrations can be treated in a similar way.
We shall now derive the pressure field due to local vibrations of single hull plates in the after-body by means of the following mathematical
!nodel. The X-Y -plane is coinciding with a plane. solid boundary extending to infinity in
bothdirec-Figure2. Section form.
0 0.2 0.L6 0.8 t.D
'b
FIgure 3.. Theoretical value of C1 alongthe section .Shównin Figure 2.
tions. A part of this surface, denoted by H, is
vibrating at
high frequency and with small amplitude compared to its lateral dimensions. The space above the X-Y-plane is occupied by water. The vibratory motion of H can be re-presented by a source, distribution of oscillating density in the X-Y-plane. Consider a continuous source layer of density per unit area in an un-bounded fluid. The derivatives of with respect to coordinates parallel to the layer are supposed to be small so that 'elocity.components in these directions can be neglected near the layer. Thevolume of fluid emitted per unit time from the
imaginary closed volume indicated in Figure 4 is
then 2vH which must b
equal to4TrcH.(Relation between flux and source strength). Thus
(6)
2Tr
The vibratory motion of H may therefore be
re-presented byasource layer in the X-Y-plane with density given by (6) where y is flOW the
instant-\ctosed votume
Figure 4. Relation between source density and induced
velocity.
aneous velocity othe plate. The velocity potential
at a field point (X', Y',
Z') due to the source distribution is=ff.
H D5where Dsf is the distance between source point
and field point. The instantaneous pressure at the field point can be obtained from Bernoulli's
equa-tion by neglecting the, velocity-dependent term,
yielding
- P !2_( f v.dH
2,T&t
H Dsf
By carrying out the differentiation after the
integration, we get a dH
-p
rr
Z2ir
J)
DH
whera is now theinstantaneouS plate accelera-tion in Z-direcaccelera-tion.
As an example we shall make use of (7) to calculate the pressure field due to a vibrating
plate of the form shown in Figure 5. which could
for instance represent a hull plate. We shall
(7)
Figure 5. Vibratingplateclampedalong the sides Y.±A.
assUme that it. is clamped along the edges Y.A
and that it. is vibrating in such a mode that its
acceleration a (X;.Y, t) may be written
r
a
a(t).
(1_(X)2)2'Outside H, that is for tYI>A,. the acceleration is zerO. We shall here calculate only the pressure
contribution from the mòtion of the plate between,
X = -'-B and X = -CB.
Substitutión into (7) of.
D5f =l-X')2+(Y-Y')2+Z'2«
. .and the above expression for a, yields
p a
A B .(j (.1.32)2 dX dYpC
.L1
2-A
_BX) +(Y-Y')
tZ'.If the field point' is in the X-Y-plane, we have
Z '=0 so that the integrand in (8) becomes infinite
atthe pointX =X', Y =Y'. However, the integral is still finite. (By introducing polar coordinates it is easy to show that the integral
ffdX
dy Sis finite when S is a circular area around the polè X - 0, Y = O).
In Appendix numerical evaluation of (8) bas been
carried out to obtain the pressure distribution along the Y-axis. (X.' O, Z' = O). The results
are presented in Figure 6 in the form of a
non-dimensional coefficient C2 defined by
p
a
pA
C
The calculations havé been carried out for B/A = 3 and B/A - 6. The results presented in
Figure 6 can now be used to calculate the pressure
amplitude caused by a given acceleration of the plate. Consider for instance a plate of half-width
A - 1 meter and half-length B
6 meters oscillating as shown in Figure 5. If theaccelera-tion amplitude at Y O is a - 1. 0 rn/s2 then the corresponding pressure amplitude at the field
- 2
point X - Y = Z O becomes p 143 kp/rn
decreasingalongtheY-axisto 9,2 kp/m2 at Y =A
and 65 kp/rn2 at Y 2A.
(8)
1.2 1.0 0.0 0.6 0. 0.2
Figure 6. Theoretical value of C2 along Y-axis of the plate shown in Figure 5.
Description of experiments.
In order to verify the theoretical calculation of
the coefficIent C1 some experimental investiga-tions have been carried out in model scale. Two
wooden-models, M-713 and M-885, were used in
these experiments. M-713 is the model of a
T2-tanker, scale 1:36. M-885 is the model of a
220 000 TDW tanker, scale 1:40. Body plans and
ligure 7. Body plan of M-713. Main dimensions of ship are Lpp = 153.3 displacement = 21520 m3, block coefficient 0.741.
main particulars of the ships are given in Figures
7 and 8. Pressure transducers were fitted flush
with the surface of the models at positions denoted
by Ti through T6 as shown in Figures 7 and 8. A sensitive accelerometer was fitted inside the
models directly above the transducérs to measure
acceleration in vertical direction. The
after-bodies of the models were forced to vibrate in
vertical direction by nieans of an electromagnetic
exciter with variable frequency and amplitude.
The signals from the accelerometer and the
pressure transducers were recorded
simultaneously on a galvanometer recorder as
shown in Figure 9. The models vere towed under the towing carriage with velocities corresponding
to ship speeds ranging from zero to 16 knots. The excitation frequency was varied from 20 to
80 c. p. s. (on the model) and the acceleration
amplitude from 0. 5 to 4. 0 rn/s2.
The ratio between recorded pressure and
acceleration may bè described by the coefficient C1 as defined by (5), where b is now taken to
mean the half-width at the waterline of the section
at which the pressure transducers are fitted. The results of the experiments may be summarized
as follows:
C1 is independent of model speed.
C1 is independent of the amplitude of the vibratory motion.
draught r 9 1 ni, beam =20.7 ni,
9!
p
Figure 8. Afterbodyplanof M-885. Maindimenslons of ship are Lpp = 312.0 m, draught = 19.5 m, beam = 46.4 m, displacement = 237782 rn3, block coefficient = 0.824.
C FA F GR
is about ±. 4 percent.
Discussion.
The theoretical calculation for the section form shown in Figure 2 gave values of C1 in the range 0. 68 to 0. 74 along the lower half of the section. This theory is based on an infinitely long cylinder.
andthus neglects 'end effects' to be expected at the afterbody of the ship. Further, the afterbody
section forms of the models are not exactly equal
to the one treated theoretically. In pite of these
approximations the theoretical value of C1
compares surprisingly well with the
experiment-al vexperiment-alues given in Table i. It therefore seems
that the theory of the infinitely long vibrating cylinder can be used to obtain an estimate of the pressure fluctuation due to a given vibration of
the afterbody.
The theoretical calculation of the pressure
Table 1. Experimental values of C1. M-713 M-885 Transducer position Ti T2 T3 T4 T5 T6 C1
0.44 0.82 0.82
0. 56 0. 58 066 Figure 9. Instrumentation. Ex = electromagnetic exciter, Ac = strain gauge accelerometer, T = pressure transducers,CFA = Hottinger KWS-6T carrier frequency amplifier,
F = 130 C. p.s. low-pass filters, GR = galvanometer recorder.
C1 is approximately independent of the
frequency of the vibratory motion.
The experimental values of C1 for the various
transducers are given in Table 1.
Regarding item 3 above a deviation of C1 of up
to 20 percent from the mean value was observed at certain frequencies. This effect is most likely due to the various modes of vibration of the woodenhulibeingexcited. Thevalues of C1 given in Table
i
are mean values obtained in thefrequency range 20 to 80 c. p. s. (model scale).
The measurement accuracy (in terms of standard
fluctuations due to local vibrations of a single hull plate of the afterbody has not been verified experimentally and should therefore be applied with some reservation.
In additioi to the experiments described above,
the pressure fluctuations induced by a 6-bladed propeller fitted to model M-885 were measured
during propulsion tests inthe towing tank. In this case the electromagnetic exciter was not applied, so that only the pressure fieldinduced by the pro
-peller was measured. When referring the results to the full scale ship the single amplitude of the propeller-induced pressure fluctuatIon p was 120, 45, and 20 kp/rn2 for transducer positions T4, T5, and T6 respectively.
Withthe known value of C1 we may now use (5)
to calculate the maximum acceleration am that
can be allowed without affecting the measurement
of Pp Ofl the full scale ship. As a criterion we shall here require that the vibration-induced pressure p,, is less than 10 percent of p. In this
way we obtain
0. 1 pp
a
m pbC1
-Usingthe values of PP mentioned above we obtain
am = 0. 026 rn/s2 for transducer position T4,
a = 0. 009 rn/s2 for transducer position T5,
a = 0. 004 rn/s2 for transducer position T6.
Thus, in order to measure the pressure
fluctua-tioi induced by the propeller at for instance
positionT4 with an accuracy better than 10 per-cent, the acceleration of the vibratory motion of
the afterbody has to be less than 0. 026 rn/s2. Now it should be noted that this requirement is strictly
necessary only for acceleration cothponents of
frequencies equal to the frequencies of the
components in the propeller-induced pressure
signal. If the vibratory frequency is different from those of interest in the pressure signal, its effect can be filtered out by spectral analysis.
The values of am obtained in the numerical
example above are indeed very small. The
vibratory level of many ships may be orders of
magnitude higher. This means that in. many cases it is impossible to measure the propeller-induced
pressure fluctuations on the afterbody of a ship by means of pressure transducers fitted in the ìull plates. If such an attempt is to be made, it is absolutely necessary to measüre the vibration
of the afterbody, especially the hull plates in the vicinity of the pressure transducers, by sensitive accelerometers. By the theories outlined in this paper it is then possible t. obtain an estimate of the magnitude of the pressure signal due to the vibration, lithe pressure signal recorded by the transducer is of about the same magnitude, one
must conclude that the propeller-induced
pressure cannot be measúred. If it is considerably
higher, however, itis an indication that the pro-peller-induced pressure has been measured.
The same reasoning also applies
toex-perimental investigations of propeller -induced pressures in model scale. It is the author's ex-perience that wax models usually represent no problem. Wooden models, however, will often vibrate sufficiently to reduce the possibility of
measuring accurately the propeller -induced
pressure fluctuation. One remedy that has been
successfully applied at the Norwegian Ship Model Experiment Tank, is to isolate the propulsion unit
and propeller shaft from the model so that no bearing forces are transferred to the model.
Co nc lus ion s.
Pressure transducers fitted in hull plates of the
afterbody of a ship will record the sum of the following two high-frequent pressure signals:
Thepressurepinducedby the propeller, and thepressurepdue to the vibratory motion of
the afterbody and the hull plates in the vicinity
of the transducers.
It has been verified by model scale experiments
that the pressure due to the vibratory motion of the afterbody may be calculated by the theory of
an infinitely long vibrating cylinder. A method for
calculating the pressure field in the vicinity of a vibrating hull plate is also shown.
It is shown by numerical calculations that in many cases
Pv » Pp
so that experimental measurement of the
pro-peller -induced pressure fluctuations on the after-.
body of a ship by means of pressure transducers fitted in the hull plates becomes impossible.
Experimental investigations of
propeller-induced pressure fluctuations on ships as well as
accelerometers. Calculations by the methods
presented in this paper will then confirm whether
the recordèd pressure fluctuation is mainly due to vibrations or induced by the propeller.
A ekn owl edge ni en t.
The investigations presented in this paper have
been financially supported by The Norwegian Council for Scièntific and Industrial Research.
The author wishes to express his cordial thanks
to the Director and staff of The. Norwegian Ship
Model Experiment Tank for their cooperation and
support. Special appreciation is extended to Mr.
Svein Eggen for his assistance in carrying out and
analysing the experiments, and to Mr. Thomas Nordbakke for his assistance in the numerical
calculations.
References.
Keil, H., 'Messung der Druckschwankungen an der
Aussenhaut liber dem Propeller', Schiff und Hafen,
Heft 12/1965, 17 Jahrgang.
Ursell, F., 'Slender oscillating ships at zero forward speed', Journal of Fluid Mechanics, Vol. 14,
No. 4, December 1962.
Landweber, L, and Macagno, M. C. de, 'Added mass Of two-dimensional forms oscillating In a free surface', Journal of ShipResearch, VOl. 1, No. 3,
November 1957.
UrselI, F., 'On the heaving motion of a circular
cylinder on the surface of a fluid Quarterly Journal
of Mechanics and Applied Mathematics, Vol. 2,
1949.
List of symbols.
A half -width of vibrating plate,
B half-length of vibrating plate,
C2
C1
p/(a.p.b),
p/(ac. p.A),
distance between source point and
Dsf
field point,
general time-dependent function in F (t) Bernoulli's equation, V
X, Y, z
x',.Y', z'
am,.
a a(X, Y) b bn f( 8) i n p pp pv qr
r0 s t V x, y z e Ç , Ti evertical velocity vibrating
cylinder,
source-point coordinates on
vibrating plate,
field -point coordinates near vibrating plate,.
maximum acceleration that can be
allowed without affecting measure-ment of propeller -induced pressure fluctuation,
coefficients of conformal mapping,
acòeleration of vibrating plate at
Y=o,
Vertical acceleration of vibrating
cylinder,
acceleration of vibrating plate,
half-width of section,
coefficients in series expansion of
velocity potential,
complex conjugate of b,
(l+a1) sinO + a3 sin3ø,
/ -1
summation index in expression
for ,
pressure,
oscillating pressure induced by
propeller,
oscillating pressure induced by
vibration of afterbody or hull plates,
flow velocity in Bernoulli's
equa-tion,
field. point radius in Ç-plane,
radius of circle in Ç-plane,
(Y-Y ')/B,
time,
velocity of vibrating plate, coordinates,
x+iy,
small, positive number, coordinates,
angular coordinate of field point in
Ç-plane,
density of water,
density of source layer,
Integration of (8) in X-direction yields
p=
p af(1_(42)2.
.10g(B_X+((B_X)+(Y_Y)+Z)y
-B -X'+((B +X')2 +(Y -Y')+Z2)2For field points on the Y-axis we have X' = O and
Z' = O. Then by introducing
s_Y-Y'
B we obtain p ac B Ii 2r where A-Y' B11= f
Q(s)ds -A-Y' B and Q(s) =(1-( Aiog
-1+(1+s2For numerical evaluation of I one has to consider
the following three cases differently.
O <Y' <A.
In this case we have Il = 12 + 13 14 where 12 = 1 Q(s) . ds, -A-Y' B A -Y' B 13=
f
Q(s). ds.sB+Y'22
Appendix Case 3. 14 = _j Q(s). ds.12 and 13 contain no singularities and may there-fore be evaluated by ordinary numerical
procedures. By neglecting higher order terms one obtains by series expansion
2
-1+ (1+ s2) =
For small values of E we therefore have
I=f
(1which can be solved analytically, yielding
14 =4(1 ...(f42)2.(1 +log(4). E
Case 2.
Y' =A.
In this case we have
Ij 15 + 16 where -E 15= J.
Q(s) ds,
-A-Y' B15 is here regular, and
I
=ll
6 4 Y' > A In this case may thus be procedures. The results obtained by 80 I2 13. and 15 0. 005. 16 is seen to becontains no singularities and evaluated by ordinary numerical presented in Figure 6 have been intervals' Simpson integration of The value of E has been chosen
95
SPECIAL SUMMER SESSIONS OFFERED BY THE DE PARTMENTOF NAVAL ARCHITECTURE AND MARINE ENGINEERING, MASSACHUSETTS INSTITUTE OF TECHÑOLOGY.
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