Hydrodynamic reactions to propeller vibrations

1. INTRODUCTION

The mathematical description of the mechanical _part

of the propeller-shaft system in the vibratiòn analysis with

finite elements has established a powerful and a reliable calculation technique. For the hydrodynamic _interaction

with the environment, the mathematical description of the

propeller as a set of lifting surfäces is adequate. In the

calcülation of the unsteady fórces and moments that apply at the propeller, such models have been developed and were found to be applicable. These unsteady forces and moments

are either generated by the propeller operating

_{in a non.}

homogeneous wake field or by the pröpeller's _own

vib-rational motion.

Both types of loading participate in the

vibrating propeller-shaft system; the former type being called the excitation load, and the latter the hydrodyna.mic _reaction

load. Well-known aspects of the hydrôdynarnic reaction are

the added masses and moments of inertia, while further aspects include the damping and the coupling ofthe pro-peller vibrations.

Netherlands Ship Model Basin, The Ñeerisncjs 78 Paper C36 TECHI1!SCHE WIVER$ITEIT Laboratorium 'icor Scheepshydromechanjca ..Archlef Mekeiweg Z 2628 CD Deift Tela 015-786873- Fwc 015-78183$

HYDRODYNAMI C REACTIONS TO PROPELLER

_VIBRATIONS

Dr S Hylarides and Dr W

van Gent*

SYNOPSIS

Operafion of the propeller in a non-uniform wake field leads to unsteady hydrodynanijc loads which induce vibrations Depending on the magnitude of these vibrations, reactions

of the surrounding water will occur These reactions are effects of added _{masses and} added moments of inertia and of damping _{either parallel or normal to the vibrational} motion. For a number of propellers from the_{Wageningen B-screw series these reactions}

have been calculated, using the unsteady hftmg surface theory It is shown that no fixed

percentage of the propeller mass can be uséd to estimate the magnitude: of the added mass Based on basic considerations for airfoils an attempt is made to derive simple

expressions to characterize the hydrodynamic reactions using overall geometncal pro

peller data The effect of the reactions on the forced propeller vibrations _{is shown in}

some detail. For the transverse vibrations the hydrodynamic damping is found to be most

important: about 9% of the critical damping. For _{torsional vibrations a value of about}

6% is found, for the longitudinal vibrations the hydrodynainjc damping is about 3% of the critical dámping.

Due to the large masses and moments of inertia of the

propeller and the location of the propeller at a free end of

the shaft, the propeller plays a dominant role in the mech ariical aspects of the vibrational shaft system. Furthermore, it has been shown that hydrodynarnic reactions haue a very

important effect on the vibrations so that a careful deter

mination of their quantities is needed.

2. _{THE HYDRODYN&jJC REACTIONS TO PROPELLER.}

VIBRATIONS

In reference (I.) the authors have discussed the fact that the dynamic load on a popeller can be derived by

means of lifting surface theory as á response to the angle of

attack variations of the propéller blades. _{These vari tions} are twofold, on one sidé they originate from the _steady rotation of the propeller blades through the _{non uniform}

wake field, on the other side propeller oscillations_{give rise} to variations in the angle of attack As long as the vanations

in the angle of attack are sufficiently small, the_generated load will vary linearly with the angle of attack and the

principle of súperposition can be applied. This means that Trans ¡Mar Eid Vol 91, Conference No 4. Paper 06

independent evaluation of the excitation load

and the

reaction load is possible. To determine the excitationload of the propeller working in an inhomogeneous wakefield only the steady state rotation of the propeller is considered.

For the determination of the reaction load the propeller is assumed to perform a vibrational motion in a hothogeneous flow.

For the calculation of the reaction load the variations in angle of attack of the flow intothe propellçr are related to the vibrational motion ofthe propeller. In this respect it is assumed that the propeller behaves as a rigid body so that the motions in each point of the propeller are easily defined

by these rigid body motions. Considering each of the six

components of the rigid body motions separately,gives the six components of the reaction load. Because of coupling

effects, each component of the rigid body motions gives reaction forces and moments in each of the co-ordinate

directions. As a result of symmetry aspects the number of coupling terms is restricted.

Whereas the longitudinal and torsional vibrations ofthe

propeller do not lead to transverse reaction terms and the reverse does not hold, these axial and transverse motions

can be considered independently, with respect to these

hydrodynaniic effects.

Assuming the vibrations are sufficiently small, the

generated forces vary linearly with the vibrational motion

and therefore can be determined per unit of motion. The

quantities thus obtained are

called the propeller

co-efficients.

-TABLE I HYDRODYNAMIC PROPELLER COEFFICIENTS FOR VIBRATORY PROPELLER MOTIONS

In Table I the propeller coefficients are systematically

reviewed. The determination of the propeller coefficients

has to be performed by means of calculations in which the lifting surface theory has proved to be successful(').

3. THE EQUATIONS OF MOTION OF PROPELLER

VIBRATIONS

The vibratory motions of the propeller shaft system

can be split up into two independent systems: axial vibrations;

transverse vibrations.

Trans ¡Mar E (C) Vol 91, Conference No 4. Paper C36

The first type is formed by the torsional and longitudinal

vibrations, which are coupled to each other because of the hydrodynarnic coupling terms as generated and experienced by the propeller. The transverse vibrations are the coupled horizontal and vertical vibrations of the whirling vibrations. In thi case, the coupling has three causes:

the gyroscopy of the propeller;

the oil film, especially in the propeller bearing;

the hydrodynamic reactions of the water on the

propeller.

In principle, because of the hull structure, coupling is also possible. However, this is less important and more diffIcult

to describe and therefore not considered in this paper.

The interaction with the hydrodynamic phenomena

will be fully described for axial vibrations. For the trans-verse vibrations a similar derivation yields.

For the longitudinal vibrations, which are parallel with the X-axis (see Table I), the equation of motion is written:

F

mO+di O=F+ --

ox +

- O

+

-ox x

+

in which: r = propeller mass;

d1 =

dynamic stiffness of the shaft

as experienced by the propeller due to

its longitudinal vibrations.

=

the displacement of the propeller

along the X-aus with its time deriv-atives 0x and 0;

F = the thrust fluctuations, generated by

the operating propeller in the

in-homogeneous wake field.

= hydrodynamic reaction force

exper-ienced by the propeller due to its

acceleration per unit of acceleration. = hydrodynamic reaction force

exper-ienced by the propeller due to its

velocity per unit of velocity.

= hydrodynamic reaction force

exper-ienced by the

propeller

unit of

rotational acceleration.

= hydrodynamic reaction force on the propeller per unit

of rotationäl

vibration velocity.

The framed quantities are thus the hydrodynamnic

re-actions generated by the vibrations of the propeller and have been called propéller coefficients as given in the preceding section. Rewriting the above equation gives:

(m- F ox )

&,the propeller,

-F 'X

propeller vibrations with the torsional propeller vibrations.

In the same way the equation of motion for the

torsional vibrations is derived:

79 F

ox

F ox

peler encounters. i9ie other two propeller coefficients,

represents the added mass of the hydrodynamiC damping the

pro-represent the coupling of the longitudinal

components u vibratory motion

öz Px

2F

Fx I Fx F I

__J_Qy_y-Öy

____j_ôx

y Öt Öz I öz OzI Ot 'Py LPz ¼P

..±

r óz '.P, 'P 4 I I

ó,

I ôy öy i P-1 z

and now it is seen that - Fx ox F and F F 1,x =

-F ox F ¿X F

It has to be said that in comparing the above sets of

equations with those given in a pteviòus paper(i) it follows that the signs of the propellér coefficients have been

revers-ed'. This has been done because the above, derivation _is

thought to be more consequent.

The dynamic rigidity of the shaft systems, as

exper-lanced by the propeller, is given by the

_{terms d. For F}

the coupling between the horizontal and vertical vibrations is given. The shaft system 'itself can not give. these

coup-lings, unléss the gyroscopy of the shaft is accounted fori

which is not normal practice because' of its _{small effect.}

However, these dynamic terms also incorporate the effect

of the oil filin in the bearings, which strongly couples the

horizontal and vertical transverse vibrations. Moreover, the constructional coupling via the bearings can be accounted for by these rigidity terms.

4. _{THE PROPELLER SERIES}

To investigate the effect. of overall propeller parameters

on the propeller coefficients a number of propellers from

the Wageningen B4 screw series with zerO rake have been chosen. The blade area and the pitch have been varied, which variations are summarized m Table III The propeller outlines are given in Fig 1.

This series of Wageningen B-screws has been chosen

because of the general familiarity of this series and the

general use of the diagrams of these propellers. _Moreover, this series covers a great variety of parameters (2)

TABLE II. EQUATIONS OF MOTIONS OF TRANSVERSE PROPELLER VIBRATIONS

=

The framed quantities are the transverse propeller

co-efficients.

dynamic rigidity

of shafting

at propeller location.

6

o,,. C218,, 6,,. C3ó,, 6,, 41 5Y +(m. az -F,. 62 + az+d225z + C 62 * d42 óz 'p,,- 'Dy + '0y 'Py C23Py d33'py I)'D.ø43 ,'( !i_ 'Dz Fy F2 M - M2 F',, F,, F,, _{F,, F,, F,, Fy} 6,, 5v 62 'Dy 'Dy _{'Dz 'bz.} 6,,-ay by" F2 ay F'2 F'2 F2 F2 Fz Fj F2 o,, _6z 62 'Dy M,, M,, _{M,, I1y} z ay ay 6,j _Dy 'Dz,'

i)oz+oz

C44'DZ -. M2 Mz M2 _M2 'Dy M2 'Dy, Mz ay '02 'Pt mn = mass Il = =

linear moment o inertia

polar moment of inertia

i

of the propeller

= _{angular shaft speed,}

(IP - )

-M

+dØ-

6

-:.'

6=M

in' which I, = polar moment of iñertia f the. propeller.

= added polar moment of inertia

=

the rotational rhotiòn of the propeller

along the X-axis with its time dérivation

øandO.

= hydrodynamic damping per unit of

torsional vibration velocity..

= dynamic shaft stiffness at the location of the propeller in torsional 'vibration. and

the hydrodynamic coupling terms

& with the longitudinal propeller

vibrations.

= _{the fluctuation in. tor4ue as experienced}

by the. propeller operating in a

non-unifòrm wake field.

In Table lithe equations of motion of the_{propeller in}

transverse vibration are given In these equations the _various propeller coefficients clearly show _{up, and their coupling} effect 'is also seen.

The gyroscopic coupling is given by the presence of in these equations, iii which I, is thé polar moment of the propeller and is the angular shaft speed.

80

Trans I Mar E (C) Vol 91, Conference No 4. Paper C36

ox

M

ï

I

_I

00 0.70 1 .00

1F

ItL 111

..

uI

_III

--L.

L

i?

iiui3j

_I _

-

t-I FIG I

Review of the Investigated B4-screw

TABLE III INVESTIGATED VARIATION OF THE 4-BLADED B-SCREW SERIES

Within the scope. of this paper a rather restricted

number of parameters has been varied, because it was not

clear how far general relations or expressions could be

indicated between overall propeller parameters and the pro-peller coefficients. Moreover, it has to be discovered how

far such general results are applicable iii

the vibrtioñ

analysis of the propeller-shaft system.

5. THE PROPELLER COEFFÏCIENTS

In this section the results of the calculated

hydro-dynamical propeller coefficients are presented. They are restricted to vibrational motions of blade frequency. The

mathematical description of the lifting surface model used can be fôuridin reference(7).

The results are presented in Table W and in Fig 2. The values given are made dimensionless in the following way:

¡n which the framed quantities are thç. physical propeller coefficients and the tirms F15 and F/S and so on are the dimensionless values, D is the diameter, n is the rate of

revolution and p is the mass density.

It must be realized that in a linearized approach to the

hydrodynarnica] effects, there is only one velocity scale,. viz, either the propélier rate of revolution of the forward speed, which are uniquely related to each other via the propeller pitch. Therefore, the above dimensionless

re-presentation means a reduction of the number of variables. In order to discuss the values of the coefficients, they

can be distinguished in three main groups, corresponding to:

axial vibrations, in which case the vectors for loads and motionsare always parallel;

transverse vibrations with parallel vectors for loads and motions;

transverse vibrations with mutually normal veçtors for loads and motions.

This same grouping is given graphically in Fig 2.

lt is remarkable that the added mass-like coefficients

(the 2nd and 4th column of Figure 2) exhibit a strong,

linear dependency on the expanded blade area ratio and a weaker dependency on the pitch ratio. The reverse holds for the damping like coefficients (the ist and 3rd column

of Fig 2). In the third group of coefficients these tendencies are less clear.

Both the geometry of the propeller and its vortex

system are rather complicated and make it difficult to inter-pret simply numerical values and dependencies on certain

propeller parameters. Basic understanding can be gained,

however, from a brief review of and a comparison with the analytical results available for non-steady airfoil theory, as discussed by Von Krmán and Sears(3.4).

The main parameter in that case is the so called reduc-ed frequency:

w =

vc/2U,

where y = circular frequency of vibration in radians per second,

c = chord length,

U =

flight velocity of airfoil.

For a propeller this quantity can be approximated as

follows:

t' = Zl for blade frequency vibration, where Z = nmber of blades

= propeller angular rate of revolution

2 D

c - (- - D ) / (Z) for

_{A0 4}

a rectangular approxi-2

mation of the blade area,

where AE/Ao = expanded blade area ratio, D = propeller diameter,

U ½7D/cosØ for the velocity along a blade section in

lightly loaded condition,

where Ø = pitch angle - arctan (P/xirD),

P = pitch,

x = representative radius.

In this way the reduced frequency for a propeller becomes:

.,.Ar.

/ P

w- cosØ.

.\/l +(_...)2

This dimensionless parameter depends on the geometry, but

not on the operating conditions as forward speed and rate of revolution. The dependency on the blade area ratio is

stronger than on the pitch to diameter ratio. In the range o

blade area ratios 0.4 <AE/Ao <1.0 and pitch ratio <1 2

the reduced frequency is given by:

O.6<w <.1.7.

In the corresponding range of values for non-steady

airfoil motions the added mass variation can be

approxim-atedby:

(0.56w - 0.16) irpcU. while the damping is constant:

0.55 irpcU.

These approximations are derived from references (3) and (4).

The chord cand the velocity U can be transferred again

to propeller quantities. The propeller thrust is the cos part of the blade lift, while for an axial translatory

vib-ration the normal cçmponent of the blade motion ¡s thecos Ø part of the propeller axial motiOn. Thus the dimensions added mass coefficient for the propeller can be derived as Nomenclature Blade area ratio

AE/Ao Pitch-diameter ratio P/D M-40-50 0.40 0.50 M-40-80 0.40 0.80 M-40-120 0.40 1.20 M-70-50 0.70 0.50 M-70-80 0.70 0.80 M-70-120 0.70 1.20 M-100-50 1.00 0.50 M-100-80 1.00 0.80 M-100-120 1.00 1.20 = pnD3 pnD4

F/,

F/i,

= pD3 F16

pD4 F/i.

M o = = pnD4 pnD5

M/6,=

M/Ø,

PD4 M/,

= pD5 M/.

00

FAIlLI IV

DIMI:Nll )NLFSS VALUIS OF 111F PROPFLLFIt COEFFICIFN1S

Propeller

type

B4-40-5O B4-40-80 B4-40-120 114-70-50 B4-70-80 134-70-120 B4-100-5O 134-100-80 B4100-120 AE/AO 0.40 0.40 0.40 0.70 0.70 0.70 1.00 1.00 1.00 P/D 0.50 0.80 1.20 0.50 0.80 1.20 0.50 0.80 1.20 . AXIAL VIBRATIONS

F/

-6.24 10 595 10 -5.42 10 -8.37 lO -7.27 10 -6.30 10 -8.53 10 -6.71 10 -6.12 10

F/

-2.91 i02 -2.74

io2

-2.31 io2 -8.37 io2 : io2 -6.01

io2

-1.34 i0 -1.18

io

-968 io2

F/4

4.96 io_2 75ß io_2 1.04

10.

6.66

o2

9.26

io2

1.20 l01 5.82 i02 8.54

io2

10_1

F/

2.31 1Ô3 3.48 1O 4.42 6.66 1'0 9.3,4 10 1.15 o_2 1.07 i02

i.so

o2

1.85 io_2 4.96 2.31

io-2

758 3.48 io_2 i0 1.04 4.42 l0 1O 6.66 6.66 io_2, 10 . 9.26 iø2 9.34 10 1.20 1.15 10

in2

5.82 1.07

io2

io

8.54 1.50 i0_2

io_2

1.17 1.85 10_i -3.95 10 -'9.65 10 -1.98 10_2 -5,. 30

io

-i. i

in_a 230 in_2 -4.63 -1.09

in_2

-2.23 in_2 -1.84 -4.43 i0 -8.44 -5.30 l0 -1.19 i0 -2.19 10 -8.48 ioT4 -1.91 .i:ü -3.53 i.o

TRANSVERSE VIBRATIONS, LOADS AND MOTIONS PARALLEL

F -2.69 in_2 -5.56

io2

112 10-1 367

io_2

-7.01 in-2 142

io'

5 ii in_2 -8.15 in-2 -1.62 10 Y Y --' -3 -3 -2 -3 -2 -2 F /6 -1.72 10 -3.56 10 -5.97 10 -3.94 10 -8.66 10 -1.51 10 -4.02 1:0 -1.46 iO -2.57 10 ' ' -2 -2 -2 -2 -2 -2 -2 -2 -2 F /4 -2.38 1:0 -3.55 iO -5.02 iO -2.92 10 -4.41 10 -6.33 10 -3.90 10 -5.28 10 -7.31 10 Y ..Y F /4 -1.35 -3 10 -1.93 -3 10 -2.33 -3 10 .-3.04 -3 10 --3 -467 10 -5.89 -3 10 -4.9.5 -3 1.0 -7.89 -3 10 -1.01 -2 .1:0 y y M / -2.60 -2 10 -3.64 -2 10 -5.06 -2 1.0 -3.18 -2 10 -2 -4.40 10 -6.28 -2 10 -4.36 -2 10 -5.23 -2 10. -7.17 -2 10

y y

3 3 3 -3 _ -3 -3 -3 -3 M /6 -.1.31 1:0 -1.88 10' -2.28 iO . -3.12 1:0 -4.64 10 -5.82 10 -5.06 iO -7.86 10 -9.97 10 ' ' -2 -2 -2 -2 -2 -2 -2 -2 -2 M / -3.51 10 -3,26 10 -3.09 10 -3.88 10 -3.79 10 -371 10 -4.90 10 -4.44 10 -4.20 1:0 y y -1.66 -3 10 -1.51 -3 iO -1.27 -3 1:0 -3.74 -3 1 -3 -3.58 iO -3.12 -3 10 -5.9,6 -3 10 -5.90 -3 10 -5.21 -3 l'O -.

TRANSVERSE VIBRATIONS, LOADS AND MOTIONS MUTUALLY PERPENDICULAR

F

R

2.65 1n 2.23 10 2.97 4.77 i0 9.99 iO 9.27 i0 1.77 in_2 5.26 10' 2.30 i0 6.18 10 -645 10 -8.26 iO . 2.96 1.42 '10 1.81 ko_6 5.74 1O 5.50 10 -2.14 10_5 1 23 10_2 1.24 in_2 1.63 in_2 ' 1.62 in2 1.70 in_2 2.76 in_2 2.70 in-2 2.52 in_2 4.20 in-2 F / 4.19 2.75 iO4 7.45 iO 1.13 i0 8.79 i0 5.56 10 1.80 1,0 1.49 i0 1.05 10 -6.22 10 -6.07 i0 -4.21 i0 -5.33 -6.10 i0 -1.27 i0 -1.40 10 -4.88 10 3.71 i0 '-4 79

io

-5.27 iO4 -4.91 i0' -9.ii IO -1.06 1O -1.01 -1,41 -1.67 -1.5,7 10 3:51 2.64 10 3.83 6.09 5.18 .44 1.13 in_2 8.91 1.64 in-2 -1.01 l0 -1.53 1O -1.89 1O . 6.1.9 in_6 -1.53 i0' -2.52 -1.74 1O -1.99 i0 -3.37 in4

-1 16- 152 12- 8-4-

---O.5

o .4 -8 -12 -16

F/.

iôY

(A/

O 1.2 -2-x1O _-2 hi o -2- 0.5 aol O -4- -6--8- z 10 8- z io 6-4- /uy .2 -o -2- -4-

-G-F/

ox 0.5 1.0/O -2- _-4 X io

\\o8

¶52

TRANSVERSE VIBRATIONS _{LOADS AND MOTIONS}

PARALLEL o -2 .4 -6 -8 o -4 2 1.2 1 2 G -1 -1 2- z i62 -2'-8 -3 X 10 Mx!

I:jA7AO

fax yI

'y

AXIAL VIBRATIONS 1.2 A6!

M/

¿AE/A ₀ 1.0 _0.5 1.0 -2 12 8 1.2 1 0.5

_/

x -12 -16 _-16 0.8 -1 2 o xt5 _{. 4-xIS} Mx! _3.

/'X

_2-. - 1A6/ ____

AL

-1- 23 -1.2 -.--- __._-0.5 z io2 1.2 0.5 f

A/

2- z io TRANSVERSE VIBRATIONS

LOADS AND MOTIONS MUTUALLY NORMAL

FIG 2 Dim ensloniess propeller coefficients_{as ifuncÑònof}

blade a,ea ratio for different pitch-diameter ratios

Mx /

/)z

F / f-0.5 2- z t52 _{z 10} - 1.2 31- Mzj' 4- 1?)' O5 AE/ o's

i/A0

O-

-I

M/

:

84

Trans ¡Mar E (C) Vol 91, Conference No 4. Paper C36 16

.--1.2

12

..-0.8

8 0.5 0.5 1.0

Fl

-a -6 -8 -3; Fz/._/P), -I .4 _-2 8- x 151 6 4 2 o -2 .4 -6 e -o -2 0.5 -4 -6 -6 0.6. -2 -8

F/6

= f

(0.56w - 0.16)

From the above considerations it follows that the factor f=

_{*2/8, but it}

is practical to fix a value for .f by

com-paring this latter equation cvith the results on Fig 2, at the same time estirnatmg a suitable value for the representative radius for the pitch angle Ø. Iriso doing the following values are obtained:

representative radius for pitch: x = 0.325,

proportionality factor

: f = 0.60.

Similarly for the damping coefficient

F/6.= . cos

in which f 0.92.

These latter equations fot added mass and damping show the already mentioned behaviour of the propeller coefficiónts viz. linear, dependency of the added mass on the blade area ratioand weak dependency of the damping

on the same parameter.

This simplified approach will also yield useful fòrmulae

for the other coefficients but this exercise has not beeñ undertaken in the paper. FOr the transverse vibrations the

procedure becomes more complicated because, iii this case,

phase differences between the loads on the blades occur.

The authors have given the formal relations for those

effècts in their earlier paper(1)

6. - PROPELLER VIBRATIONS

6.1 General

To show the importance of the propeller coefficients in the propeller vibrational response, the forced response of a

shafting system is

treàted. The dimensions have been

chosen so that the system corresponth to that of the in-stallation on a largé single screw tanker, of about 40,000 SF at around 70 to 90 rev/mm, at a service speed cf 16 knots.

The calculation refers to a short propeller-shaft,

supported by two point bearings and a thrustblock. The thrustblock prevents torsional vibrations at that location The shaft is taken as massless. All dimensions have been chosen so that thè natural frequencies of this rather artifi cial system are at about the sanie level as those met on

board the real ship.

Fig 3 shows the finite element breakdown of this shaft

system. Only the propellers with a pitch-diameter ratio of

0.80 have been coñsideted. The propeller weights run from

about 50 tonnes for the propeller with blade area ratio of 0.40 to about 120 tonnes fór a bladè area ratio of 1.00. In the next section the response of the pröpeller to an

excitation force of unit amplitude with varying frequency is

given. In this way the conibined effect of damping and

coupling is shown. The effect of added mass only manifests

itself in the natural frequency This effect is expected to appear most clearly by relating the added mass and

mom-ents of inertia to the corresponding values of the propeller itself, as shown in Fig 4 and Table V.

j

1 00 s--THRUST L0Ck WITH STIFFNESS 5 25 s io8 W/ io so

FIG 3 _{Finite element breakdown for propeller-shaft system} snztable to absorb 40,000 SHP a: aböut 70 rev/mini to ehicidate the importance of the propeller coefficients iñ the forced.propeller vibrations

Than: ¡Mar E (C) Vol 9!. Conference No 4. Paper C36

TABLE V ADDED MASS AND MOMENT OF INERTIA AS A PERCENTAGE OF THE PROPELLER MASS AND MOMENT OFINERTIA'

(PITCH-DIAMETER RATIO IS 0.80) al loo

AXIAL VIBRATIONS

0.4 07 10 AE/

TRANSVERSE ViBRATIONS

o/e loo 80 ADDED LINEAR MOM. OF iNERTIA 60 0.4 ADDED MASS ADDED POLAR MOM OF INERTIA 0.7 1.0

FIG 4 Added ma and moment of inertia as a fúnction of the blade area ratio in percentage of the propeller value

85 Propeller type Axial vibrations Added mass

1a

84-40-80 55% 33% B4-70-80 84% 50% B4-100-80 94% 56% Transverse vibrations Added linear Added mass _{mom. of inertia}

84-40-80 7% 55% B 70-80 10% 75% B4-lOO-80 12% 87% CONSTANT E 2 1 s 10's N/m2 DIAMETER 090 m 40 20 ADDED MASS 80 60 40 20

6.2 The axial vibrations

The axial vibrations are the longitudinal and totsional

vibrations. From a mechanical point of view theyare not

coupled (apart from some coupling in the crankshaft ofa diesel engine). For ships the torsional fundamental fre. quency is located in the range of 3 Hz, with the

funda-mentál longitudinal frequency for large ships with engine

aft being around 13 Hz. For the system treated iii

this paper the values are shown in Table VI.

TABLE VI FUNDAMENTAL FREQUENCIES FOR THE

AXIAL VIBRATIONS

Frequencies are given in Hz

Although these values to some extent meet the values

experienced aboard ship, they are also affected by the

artificial set-up of the chosen shaft system. For the

elucid-ation of the hydrodynamic effects these differences are thought to be acceptable.

In Fig 5 the torsiOnal response in rad/Nm_{is shown fôr}

the three propellers with different blade _{area ratios. The} fIrst peak in these response _{curves corresponds with the}

resonance frequency of the torsional vibrations;the second

peak, however, corresponds with the

_longitudinal vib-rations, (see also Table VI). This is because of the _hydro.

dynamic coupling terms. M

- ,

i-z- ,

---

and -

_{. This}

.i

. &

creases with the blade area ratio. According to Fig 4 the

terms and may be expected to be dominant

coefficients.

86

M

ax

coupling effect

in-TABLE VII _{DIMENSIONLESS DAMPING FOR THE AXIAL} PROPELLER VIBRATIONS

From the responses in Fig 5 it seems that with an

in-creasing blade area ratio the two peaks in the

response curves come closer to each other ¡ri an absolute sense,

though this is hardly the case in a relative sense with regard to the fundamental frequency (Table VI). Therefore, it

cannot be stated that because of a smaller distance in

resonance frequency the coupling becomes stronger. This is also confirmed by the increase in _{and iii Fig 2.}

The response to the thrust fluçtuations is shown_iii

Fig 6 as a functiOn of the blade area ratio. Remarkably, the

coupling of the longitudinal vibrations with the tòrsional vibrations at the fundamental torsional frequencies is less

than reverse. This is easy to understand if one realizes that

the propeller coefficients are constant, but they _aîe

multi-plied with velocity or acceleration in the set of equations of vibration of the propeller (see sectiOn 3), so that in a higher frequency range the coupliñg becomes stronger.

From the forced response it is further seen that the

resonance frequency is higher for the system with coupling

than for the system without hydrodynaniic coupling _and

damping effects. This is explained by ccnsidering the polar

moment of inertia cf the propeller to act as an effective

mass for the longitudinal vibrations, as. indicated in refer-ences (5) and (6). In the absence of damping for a double-mass-spring system, as shown in Fig 7, the equivalent mass as experienced by the driven mass becomes:

=

I -w2/

m2

in which w,

is

_{the natural frequency of the attached}

system 2, (see Fig 7). From this value of equivalent _{mass it}

10.I,à--180 - Pr.,. Un

tt4

9°

- -90-iL leo' ¿ 13'

- - ...Un t..tIcI.,t. .aa..é.

12 H, 1Q leo leo ¿ e P, nu t ei It IO hl

FIG 5 Torsional propeller responseas a .funcrion of blade crea ratio, P/D = 0.80

Trans J Mar E (C) Vol 91. Conference No 4. Paper C.36 Propeller type B440-80 B4-70-80 - B4-l0O-80 Blade area

ratio 0.40 0.70 1.00

Longitudinal 0.037 0.032 0.026

Torsional 0.054 0.063 0.057

Propeller type B4-40-80 B4-70-80 B4-lOO-80 Blade area ratio 0.40 0.70 1 .00 Longitudinal I 2.8 8.9 7.2 Torsional 6.6 4.8 4.0 90- 90 - .fl.UUT eUih.,t. .XL..d..

:o

-I

:o

Trans ¡ Mar E (C) Vol 91, Conference No 4. Paper C36

FIG 6 Longitudinal propeller response as á function of blade area, P/D 0.80

these lower frequencies. To quantify the damping, refer-ence is made to Fig 8 in which the definition of damping

for a singiC mass-spring system is given. The maximum magnification gives information of the inagmtude of the dimensionless damping ß. The results are shown in Táble VII. For longitudinal vibrations the damping is thus

FORCE EBERTED BT ON 2

-IS REPLACED BY THE INERTIA FORCE OF AN EFFECTIVE MASS -W2 l"

EOUATION OÇ MOTION FOR '"2 READS

-W2 '"22 2 2 -ô,)3 0 21 THUS 52: 82-W '"2 SUBSTITUTWG YIELDS '"2 - _i WITH W, _\J1

FIG 7 Replacement of mass-spring Íyrtezn by an equivalen; mass follows that, for frequencies above the natural frequency of

the secondary system, the equivalent mass has negative values so that, apparently, the main mass becomes lighter, leading to a higiier natural frequency. A similar fact holds

for the investigated shaft system, the torsional behaviour acting as a secondary system excited, above resonance. In the same way th resonances of the torsional vil,-ratio as in the coupled system are smaller than would be

de-ducted from the uncoupled system. The change here is smaller because of the smaller effect of the coupling at TABLE VIII NATURAL FREQUENCIES AND

DIMENSION-LESS DAMPING FOR THE TRANSVERSE

PRO-PELLER VIBRATIONS IN WHICH THE EFFECT

OF GYROSCOPY ISOMITFED 300 SO 'O 'O 0 30 IO O W 3'I,

3IJ2

SS? BU On 20 WO.EB WOO ST .352

lOO-00 '

0.05 0.1 COSSOIPO-OR YO .30505 SCG OP COSILLYI 013R53"

/

02 O IO 0.1 S-0 lUI lOO 103 OS OS lO 13 13

is.

FIG 8 Dynamic responses of a iingfr-mass-spring system variòus values of the dim ensionlëss damping

Propeller type B4-40-80 B4-7Ò-80 B4-1 00-80 Blade area _{0.40 0.70 1.00} Nat. frequency Hz. 10.7 7.5 -6.0 Dimensionless _{0.087 0.092 0.105} for 87

about 3% of the critical damping, the torsional vibrations

having about double this value.

The magnification for such small damping is given by the reverse of twice these percentages, thus for longitudinal vibration about

-

17, for the torsional vibrations

about 9. 2X3%

6.3 The transverse vibrations

In Fig 9 shows the transverse responses for propeller with different blade area ratios. In these calculations the effect of gyroscopy was accounted for, which led to the two peaks in the figure. The short vertical line on top in-dicates the natural frequency without gyroscopy (Table VIII).

Whereas the aim of the present study is to show the

effect of the surrounding water, in a second calculation run the gyroscopy has been eliminated, l'bis was done by

putt-r r

2-Ii

/ Q'.

ji

ji

i

ti 0 2 14Hz 160

'rrT

T - ._...P,n.tUu tStItC,..,. t*dfl 90-. O 2 ¿ 6 B 10 12 11. Hz 180

,,.,,,,

-r-T-L pt.flUflt,.ffiC,.,i. ni,d,d

P'.nU.q_{.a.d a...I at},iin.,tfl .nI.d.d

90 - -90-iQ i Ii i ii i Ii II

ii

-90 I.. i;

_i'

4 .6 10 -p u C çC,. il '..I P,.flUn .tf,.,flt. .nL,.öfl

snot tS.d atti ata.a.,t

aL 1900 L 0L -180

it

/i

Ii

I'

I I I , / t o'. -unE ndlànat.

----n.nuo ..4ttj'.t. .ii'._{s... Ifla.'.}

-90 Ci ii It i. ¡ i / t 6 B 10 12 14 HO 10 nnt« snita..'.. a.n. fl.U.r atn,t. .tS

tant Inn tC ad a.,,...t

ti an,.. 'F 'r 0 r 190 90-E leoI ¿ i ...ttcnøtt tt.d'. ____p,.tn .ti.,,..t, .5±5t'._{.t.pt out n.. .14}

¡ng the polar moment of inertia of the propeller, which is the cause of the whirl together with the shaft speed (see

equation for the transverse vibration in section3),equal to zero. The resulting responses are shown in Fig

lo-in Table VIII the dimensionless damplo-ing ¿3 is also given, obtained from the maximum magnification. These values

are about three times the values for longitudinal vibrations. The hydrodynamnic effects cause a coupling between the athwartship vibrations and the vertical vibrations. This is found back in the calculated responses. The horizontally

excited vibrations also showed a vertical response, at

resonance with an amplitude of about 23% of the hori-zontal amplitude. Therefore it was thought that the direct response should give a good insight into the main

para-meters.

Comparing the responses in Fig 10 with those in Fig 9

it follows that, due to coupling, the maximum response is smaller. However, the high response is extended over a

larger frequency range.

FIG 9 7)ansverse response of the propeller for different blade area ratio P/D = 0.80

FIG 10 7)'ansverse of the propeller for different blade area, without accounting for gyroscepy P/D = 0.80

88 Trans ¡Mar E (C) Vol 91. Conference No 4. Paper C36

7. CONCLUSIONS

For the investigated, propellers it has bec;1 shown that

added masses and moments of inertia hve n important,

more or léss linear, linear relation with blade irea ratio and

a weaker dependency on pitch. For the axiJ translatory

vibrations, a simple expression for the added mass, as well

as for the dampmg has been derived The geea validity

has to be investigated further. More work is also required

on the other hydrodynarnic reaction effects.

The relatiOns between added masses and moments of inertia with respect to the corresponding pripeller values

are neither linear nor conStant.

The added masses arid moments of inertia have such values

that they essentially influence the natural fre.

quencies. Thé effect of the hydrodynamic _{actions with} respect to damping is smallest for the loflZjtudinal vib.

rations, about 3% of the critical damping, a-fld highest for the transverse, vibrations, about 9%.

An accurate determination and an adeque accounting 7)

of these hydrodyriarnic reactions is essentii in mòdern

vibration analysis.

REFERENCES

HYLARIDESS and VAN GENT W, 1974, "Propeller

hydrodynamics and shaft dynamics" Symposium on.

"High Powered Propulsion of. Large Ships",

Wagenin-gen, The Netherlands, December Organized by the Netherlands Ship M9del Basin

VAN LAMMEREN W P A, VAN MANEN .1 D. and

OOSTERVELD M WC; 1969, "The 'Wageningen B-screw series" SNAME, proceedings of the annual meeting, November

VON KARMEN T H and SEARS W R, 1938, "Airfoil

theory for non-uniform motion" Jo. of the Aeronaut-ical Sciences

SEARS W R, 1941, "Some aspects of non-stationary airfoil

theory and its practical application" Jo.

_of

A erönautical Sciences

DEN HARTOG J P, "Mechanical _{Vibration" McGraw} Hill Book Co., Inc

MCGOLDRICK RTand VLRusso, 1955, _"Hull

vibration investigation on SS Gopher Marin er" SNAME Vol 63

VAN GENT W, 1975, "Unsteady Lifting Surface

Theory for Ship' Screws, Jo. Ship Research, Vol 1g