Dynamic behaviour of ship propellers

DYNAMIC BEHAVIOUR OF SHIP PROPELLERS

by

Dr; Ir. R. WERELDSMA

Publication no. 255

of the

NKFHERL4NDS SHIP MODEL BASIN

INTERNATIONAL PERIODICAL PRESS - ROTTERDAM

Archi ef

Mekeiweg 2, 2628 D Deift

Tel: 015-786873/Fa.x:781836 i

DYNAMIC BEHAVIOUR OF SHIP PROPELLERS

TECHNISCHE UNIVERSITEIT Laboratorium voor Scheepshydromechanca Archief Mekelweg 2, 2628 CD Deift Tel.: 015- 786873 - Fax: 015- 781835

by

I)r. Jr. IL WERELDSMA

Publication no. 255

of the

NETHERLANDS SHIP MODEL BASIN

CONTENTS

Summary.

Introduction

i

1. DESCRIPTION OF THE DYNAMIC PHENOMENA OF A SCREW

PROPELLER 5

1. 1. Application of the unsteady two-dimensional airfoil theory on screw propellers

1.1.1. Preliminary remarks 5

1. 1. 2. The propeller in a disturbed inflow G 1. 1. 3. Dynamic effects of a vibrating propeller in an

undis-turbed inflow 11

1.1.3.1. Axial motions 11

1.1.3.2. Transverse motions 15

1.2. Conclusions and remarks 18

1.2. 1. Effects to he taken into account 18

1.2.2. Equalities of the coefficients 20

1.2.3. Additional conclusions 20

2. THE APPLIED MEASURING TECHNIQUE AND INSTRUMENTS. . 21 2.1. Improvement of the signal noise ratio 21

2.1.1. General remarks. 21

2. 1. 2. Applied technique of noise reduction 22

2.1.3. Analysis of the periodic sampling system 23

2.1.4. Practical realization of the system 28

2.2. The axial dynamic propeller properties 31

2. 2. 1. The construction of the propeller exciter 31

2.2.2. Approximations and limitations of the system 34

2.3. The propeller excited vibratory forces 3G 2. 3. 1. The construction of the measuring system for the six

components of the propeller. The mechanical

proper-ties of the pick-up 36

2.3.2. Analysis of the dynamic behaviour in axial direction

Effect of shaft elasticity and gyroscopic

pre-cession 43

Effect of the hydrodynamic properties of the

propeller 50

3. FULL SIZE PREDICTION FROM MODEL EXPEffiMENTS . . . . 56

3. 1. Results of the experiments carried out on the ship and

propel-1er model of a single screw tanker 56

3. 1. 1. Experimental determination of the axial dynamic

pro-peller properties 56

3. 1. 2. Experimental determination of the propeller excited

thrust and torque fluctuations 62

3.1.3. Some considerations on scale effect 62

3.1.3.1. Introduction 62

3. 1. 3. 2. Correction to he applied and carried out for

the ship under investigation 64

3.1.3.3. Final remarks 69

3.2. The behaviour of the full size propulsion system 70 3. 2. 1. General lay out of the full size propulsion system,

esti-mations, neglect and analyses 70

3.2.2. Fullsize measurements of the thrust and torque

fluc-tuations 79

3.3. Prediction of the full size thrust and torque fluctuations and comparison of the predicted and measured values 83

4. FINAL CONSIDERATIONS 89

4. 1. Reduction of the propeller generated exciting forces 89

4.2. Critical considerations on the prediction problem 89

References 91 List of symbols 93 Acknowledgement 94 Overzicht 95 2. 3. 3. 1. 2. 3. 3. 2.

SUMMARY

By means of a method, based on a two-dimensional approximation, the hydrodynamic propeller coefficients that have to be taken into account for a description of the vibration phenomena, are determined.

The equations of motion of a propeller on an elastic support are given for the axial and the transverse direction.

The solution of these equations is given as a set of transfer functions,

applicable to the system designed for the measurements of the propeller fluctuating forces as well as for the description of the full size propulsion

system on board the ship.

The analysis shows that the transfer functions of the pick-up are hardly

affected by the hydrodynamic propeller coefficients. For the full size

be-haviour, however, the knowledge of the propeller coefficients is essential.

Results of model measurements of propeller excited vibratory forces are presented.

For the first approximation scale effects can be neglected, due to the

filtering effect of the propeller. The solution of the equation of motion for

the full size propulsion system gives the propeller generated thrust and

torque fluctuations and vibratory motions.

For the ship and propeller model under investigation the full size

be-haviour can be predicted in this way. This prediction is compared with full size measurements carried out on board the ship. The comparison shows

that a good agreement can be obtained if sufficiently accurate knowledge of the mechanical properties of the shaft and the thrust block, of the mobility

of the turbine and of the hydrodynamic propeller properties is available.

The measuring technique as applied to this investigation has to be of an

extremely low noise level, which is obtained by a special type of

cross-correlation technique (periodic sampling). The same technique is applied to

the full size measurements to eliminate disturbances due to ship motions

1NTROI)tCTtO

When beginning on the design of a technical construction, attention should be given to the static behaviour of the system in order to meet the

requirements of practical operation and to improve this behaviour from the point of view of economics and efficiency.

For constructions with moving and rotating parts, inconvenient vibra-tions, which are unfortunately always present, can be kept below an accep-table level, if the requirements are not too extreme.

When extreme static conditions are present, however, an inadmissible vibration level can be introduced. This effect is illustrated in the develop-ment of transport vehicles, whose speed tends to increase more and more.

In most cases vibrations are undesirable, not only from a point of view of comfort, but also for the dynamic loading and fatigue of the construction. The conception 'vibration' is therefore mostly associated withtroubles and problems, and the designer should study vibration problems in order toavoid

them. In shipbuilding techniques several more or less severe vibration problems are encountered. The wavy surface, on which the ship is sailing

gives risetomotionswithverylowfrequencies, or, in the case of slamming,

tohighfrequency vibrations. Another excitation is caused by the screw pro-peller. The tendency during the last ten years is to increase ship speeds, which leads to increased installed power and consequently to an increased vibra-tional output of the propeller, giving rise to problems of reliability and com-fort.

Due to serious problems encountered in existing vessels, the

Nether-lands Research Centre T. N. O. for Shipbuilding and Navigation initiated more basic investigations into the propeller vibration problem. A part of this in-vestigation - the prediction of thrust and torque vibrations of the propulsion

system - is presented in this thesis.

The investigations were started in order to obtain insight into the me-chanics of the dynamic behaviour of the propulsion system by means of scale effect corrected model measurements and by theoretical analyses, in order to predict the behaviour of a given design and to suggest improvements inthe

propulsion systemwith the intention of obtaining a more favourable behaviour from a point of stern vibrations and dynamic shaft loading.

The excitation, introduced by the propeller, can be divided into two parts. The presence of the ship's wake results in an instationary loadingof the propeller. In addition to the basic components of average thrust and torque, we can distinguish average transverse forces, thrust eccentricity, fluctuations

Geonipononta os the ate re

rail sise measurement seule effects

5h p eons Idered as a beam

r amps r Is n

t

vibration st the stern of a fail tice s hip

T

equatloss of motions of:

ball considered asia beam (al

peopeller

shall (shaft mhirling(

Etnitotion by the propeller

hydradynamir coelllelento

mecho:: I

h Iralasiirsl

lotisse bes iegfor- ces sad frsnsverseshafl- aad propeller-motions

comparison

equations at motions

(salati

toll sloe axial propelte

rompselsan

an o I mechan lesi propertlet elfte loll a lee reapoilslon system

Figure 0.1. Blockdiagramofthe investigations hilo propeller excited vibra- tions.

solai hydrodys.

mech. properties

nocif. nl

of the tali aise ship

ppppeflrP

2 aalal

components

axial direction (thrust and torque fluctuations) the excitation leads to a variable loading of the gearing and the thrust block. In the transverse direction,how-ever, the fluctuating forces introduce dynamic loadings of the shaft and the stern tube bearing of the ship. By way of the shaft and bearings the propeller action is mechanically transferred to the afterbody.

Onthe other hand afluctuatingpressure field exists around the propeller, introducing fluctuating pressures on the hull of the ship, even when the

pro-peller loading is stationary. This leads to a pure hydrodynamically

trans-ferred excitation of the ship's afterbody, caused by propeller action. A re-view of the three types of excitation is given in Fig. 0. 1.

Only apart of this review, indicated by heavy lines, is dealt with inthis

investigation, which is more or less focussed on axial propeller vibration

(thrust and torque fluctuations).

The first approach to the analysis of the problem consists in the setting up of equations of motion for the propulsion system, assuming that this can be described by linear differential equations. The coefficients of the equations are determined by the hydrodynamic properties of the propeller and the pro-perties of the mechanical system.

Lx

N

Figure 0.2. Frame of co-ordinates.

4

The right hand side of the equations, the exciting forces in fact are pre -sented by the propeller generated vibratory forces, clue to the non-uniform wake of the ship's hull.

The solution of these equations leads to the prediction of the full size vibrations based on the scale effect corrected model measurements of the ex-citingforces, generated by the propeller. On the other hand the same equa-tions lead to the determination of the transfer funcequa-tions of systems, designed for the measurement of the propeller properties and the measurement of the propeller excited forces.

In Fig. O. 2the frame of co-ordinates is indicated. The z-axis coincides with the centre line of the non-deformed propeller shaft. The positive x-axis is vertical upwards, fixed to the non-deformed hull. The y-axis is perpen-dicular to both other axes.

Positive motions and forces are in accordance with the positive axis-direction. The mentioned forces are exerted by the propeller on the shaft. In

order to obtain a positive average torque, the propeller is a left hand pro-peller. ( Provisions are made for the application of right hand propellers).

CHAPTER 1.

DESCRIPTION OF THE DYNAMIC PHENOMENA OF A SCREW PROPELLER

1.1. Application of the unsteady two-dimensional airfoil theory on screw pro-pel lers

1.1.1.

Preliminary remarks.

In the course of time various investigators have developed theoretical methods for the determination of the vibratory output of the propeller, based on quasi-steady or two-dimensional unsteady profile theory. More than 17 methods exist, each with its own refinements. All of them are applicable to

the axial behaviour only, however, in other words to thrust and torque

vibrations [1] [2] [3]. In this chapter a description of the phenomena is given for all directions in which vibratory motions may occur. For the application of the two-dimensionaltheory, presented by von Kármán and Sears [4] [5] on

screw propellers with disturbed inflow, or on vibrating propellers, we have had to make a number of assumptions and approximations. Only the gust ve-locities and profile motions perpendicular to the inflow velocity and span direction of the propeller blade section will be taken into consideration. Gust velocities and motions in other directions will not and can not be taken into account, due to the fundamental limitations of the theory and it is assumed that this approximation is acceptable. The principle of superposition is

ap-plied, supposing that linearity is present, dueto the fact that the angles of

attack have a value, not exceeding5°.

Heavily loaded propellers are for instance excluded from the consider-ation made in this investigconsider-ation.

Astriptheorywillbe applied for the analysis of a screw propeller. It is assumed that a small strip of one blade of the propeller at radius r has a

linear velocity r, where e equals the angular velocity in radians/sec of the propeller. It is assumed that the lift force on this strip is not affected by the differences of the adjacent strips and by the other blades. Helical wake ef-fects of the propeller blades are neglected. The wake of individual strips are

approximated by straight lines.

The addition of the results of all strips gives the behaviour of one blade and the addition of all blades results in the behaviour of the propeller.

For a propeller with a disturbed inflow (the wake field of the hull) the transverse inflow of the strip of a blade will have a periodic character due to propeller rotation. This periodic gust pattern can be analysed into harmonic

G

components, each of which represents a certain frequency, dependent on the rpm of the propeller and the order of the component. The reduced frequency depends, however, only on the chord length of the strip under consideration and the order of the component, but is independent of the number of propel-ler revolutions.

The assumption is made that the propeller is infinitely stiff in itself, which implies that e.g. singing propellers are not conside red in this investiga-tion. The torque onthe profile caused by the asymmetrical attack of the pro-file will not be considered for the analysis of the propeller vibrations. This torque acting on the propeller shaft is small in comparison with the torque

on the shaft generated by the lift of the blades.

It is further assumed that the wake of the hull is a stationary phenomenon, sothatthefluctuatingforceson the strip are synchronous with propeller rpm and composed of harmonic components equal to multiples of the rpm.

This implies that the vibratory forces generated by the propeller in the shaft, and consequentlythe vibratory motions, are also synchronous with the shaft revolutions and are stationary.

The analyses of the behaviour of a vibrating propeller are therefore

carried out only for a synchronous motion of the propeller.

1.1.2.

The propeller in a disturbed inflow (see Fig. 1.1).

The inflow of a strip of a blade at radius r can be resolved into three

components: the axial, the radial and the tangential component.

As pointed out inparagraph 1.1.1. these components can he deseribedby the following expressions:

{ m (ep

2Th z n

m (Op

=_{axial wake} =_{tangential wake} =radial wake VA = m=1 Vam V m =1 Tm V = 1 IV I R i Rm m=1

Figure 1. 1. Velocities and forces on a propeller blade with disturbed inflow. where p is the blade under consideration (p - O through z-1)

(z = number of blades).

Only the components perpendicular to the inflow velocity will be considered, so that the gust velocity Vg equals

y_g V

sin { m(+p

m1 Am

+E V

sin{m(e

2t -4-p - 1+ Z Tm} P Tm

8

In static analyses, a distinction is made between the pitch angle ¡3 of the

strip and the angle of inflow. The latter is smaller when the propeller is

loaded. For the observation of variable phenomena this distinction can be neglected, if linearity is assumed.

In this case the angle will be used to express the effect of the pitch. Now it follows for the lift force on the profile:

dl = mn 7tpc r cos _¡3 VAm 5Wm

+}

6.r cosp

drîtpc

mnl COS ¡3

-i-p __)+arg(Sm)+

sin {mco

2Tz V Tm SWm dTz dl r sin

dF= dl

coso dT = - dt r cos p sin ( + p dF = dl sin ¡3 sin (O p dT = -dt r cos ¡3 cos(®+p 2-zr dF = dl sin ¡3 cos (0+ p

{me+

) + (1.02) + arg (Sum) + _Tm}51fl p dr (1.01)

where S is the reduced frequency dependent part of the transfer character -istic of a profile with sinusoidal inflow as derived by von Kármán and Sears (4]. For the principal components on the shaft, resulting from a lift force dl generated by a strip at radius r of blade P,we can derive (see Fig. 1. 1 ):

Table 1 . 1.

Propeller excited vibratory forces.

Çcr 1 : n{ m ( Onp Çdr4 Çcr np . nH (o.p ) ++ In 1 npè ! cr2coppSI.iVAi :1 c + Çcr nps(.jv,1y [C(_1nP) Tnm -+ [co 19cP) c,+}_ 'nc e [cr0conPIs

iIi

' [nt(n-l)(eoP )o n E(+1)(O.p)+ 6ncdr+ cr' [in1(-1)(o.P) Tn° }+ oín{(.1)(9P)+

i.dr]

nPÓ[Çcr c.npjs : [In{(1)(OcP)o c+ n 1()(ocP)n

nfcr

ijii

[n{ 9nP) o+ w+ in[(.1)(9.P). ,nomdr1 np 9 cr2nnps,,I.I i (e F np O

nnl.j

i

(ø).

10

From the expressions (1. 01) and (1. 02) the contributions to the shaft forces of a strip at radius r of blade p can be determined.

Adding these contributions for each blade and integrating over the radius of the blade, we obtain sixexpressions. These expressions are given in Table 1.1

From the expressions for T and F it can be concluded that only for the

case m =z or its multiples there are contributions to T and F.

For a z-bladed propeller the components T and F result from harmonic components of the wake field equal to multiples of the number of blades.

In the transverse direction the fluctuating forces T and T. F and F

are built up by the components adjacent to multiples of the nuthber of blades

(m±i=n.z., where n=l, 2. 3...).

In an arbitrary wake field all harmonic components are present which implies that in addition to the normal thrust and torque, four other components can be distinguished, i. e. horizontal and vertical lateral forces and moments.

The propeller acts like a filter. The most important part of the wake

field (inflow fluctuations) causes internal forces on the propeller, which are not transferred to the shaft.

For the calculation of the force fluctuations, generatedby the propeller in the shaft, a very accurate knowledge of the wake field is necessary due to the filtering effect of the propeller.

The average value of the propeller inflow (m =0) gives the average torque and thrust. This is due to propeller action and is independent of the number of blades.

Table 1.2. Sensitivity of various propeller types to the harmonic components of the wake.

:

z Harmonic components 3 x x x x x x :

:

:

6 x x x x fr. 3 x X X X X X X X X X X X X X 4 x X X X X X X X X X 5 X X X X X X X X 6 x X x X X X X O 1 2 3 4 5 6 7 8 9 O 11 12 13 14 15 16 17 18 19 20

In the transverse direction the first harmonic component (m=1) of the

wake causes an average transverse force and torque (F. T, F and Tv).

independent of the number of blades.

In Table 1.2 a survey is given of the sensitivity of various types of pro-pellers to the harmonic components of the wake field.

This table illustrates the well known fact that increasing the number of blades reduces the fluctuating loading and forces of the propeller, due to the decreasing number of harmonic components, and the in general reduced am-plitudes of the higher harmonic components, to which the propeller is sensi-tive.

1.1.3.

Dynamic effects of a vibrating propeller in an

un-disturbed inflow.

Besides the non-uniform forces generated by the propeller, non-uniform motions are present due to its elastic support. Some considerations are made therefore of the forces and moments generated by a propeller vibrating in six directions. The motions and forces are in close relation to each other. The frequencies of the exciting forces and the motions coincide for the

sta-tionary case and are equal to the blade frequency and its multiples. (The

blade frequency equals the product of revolutions per second of the propeller and the number of blades).

For this consideration we make use of the results of the two-dimensional theory with the sanie assumptions and approximations as mentioned in para-graph 1. 1 1. Only vibrations synchronous with the blade frequency will be taken into consideration.

1.1.3.1. Axial motions.

Assume that the propeller carries out a sinusoidal linear vibration

zEz sin (m G

This motion is synchronous with the revolutions of the propeller and the frequency equals m times the number of revolutions per second of the pro-peller. Forthe motion of a strip of a blade, perpendicular to the inflow velo-city of the strip, we obtain:

Figure 1.2. Propeller with axial motion Ez.

The transverse velocity of the strip of the profile equals

WO = E m e cos cos (m e + , From [5] it follows: dL= -Tt perè HwmH EzLmeC0S(me+(4)+m)dr _(1.09) where = Org H (wm) 12 z X dTzdLr in 3 / cx Tx \ dt.

\

Together with formula (1.02) we obtain for dT:

2 .2

dT

= _rrpcmr e

_{pIHwmHCzcos(mO+()+p)dr}

Adding over the blades results in

22.

dT

= tpczmr è sin

2.2

= - Z 1t per e in sn Hwm

sin

(me+)

sin dr

m

dT7

_{= _Z1tpCOsinHwm mO}

Integration over the blade leads to: R T =.-7rpzÓ i TtZ m T7 -

2pzn

TEPZ in

smp

_rHwm o R

f

2 I

Cz! cr

sin 3 _Hwml jfl m dr l'o because = m e _Ez cos

(m O

and = _-

_rr2Ô2kzsin

_{(mO +}

Finally we obtain with = 2 r n (n=rps of the propeller):

R

2 Cr sifl

EzC0S (mO + 14.i+ ) dr

Ez

{cos(mO+)

cos

2 .2

m _{OIC7 Isn (mO +} )sin t.p dr

Ez05(in04-05m dr+

Hwm m dr (1.10) in dr (1.11) 2 cr sin p

dr +

R 2

cr sin

p Hwm o 2 TtC1 4-Z m

14

Table 1. 3. Axial coefficients of the screw propeller.

R

2it2pzn cr2sin P _HUm 'Pm dr (1.10)

o R m fcr2sinplHUm ISfl'Pm dr o R CZ 2it2pZn fcr COSPIHWm 1C0S'Pm dr o Ez m Fz itpZ Jcr c0s13!HWm dr R o Tz R _sirt2p

--2it2pzn fC_IHU)m ICOS'Pm dr

o cosp

R

m Jcr3cos

-

pum srn 'Pm dr

R FZ

2it2pzn cr2 sin Im lcO5'Pm dr

'Pz R !1f

---fcr2sir.pHw, tSin'Pm dr m o Tx Tx Fx Ex Ty Ty Fy Fy CZ CZ Z CZ Ez CZ z z (1 .18) Tx Ix Fx Ex Ty Ty Fy Fy (1.19) 'Z 'PZ

'z

'Pz 'z 'z 'z

2.2 dl =+i1tpcmrt3 cos_X 2 z_1

PHwm

H

PO [sin

sin

(m-1) e-p

+

+ mdr

The part under the addition sign equals zero so that T O In an analogous way we can derive:

i_

Ez E É2 _dz Ez Ez Ez dz

=

i=

==5=3L=

=

z z

'z

'z

z

See also Table 1. 3.

1.1.3.2. Transverse motions.

In transverse direction we assume a vibrational motion:

Ex Ex sin (m e+ )

The transverse position of the strip of the blade equals:

The transverse velocity becomes:

In a similar way we can derive the axial propeller coefficients as listed in Table 1.3.

For the transverse forces and moments we obtainfrom (1 . 02) and(1. 09):

211 _dr

m

Adding the corresponding blade strips we obtain:

{

(m+1)e+++}

(1. 18)

16

Table 1.4. Transverse coefficients of the screw propeller. Fx z

2

R _sirrm+l

m-1

---Ttpn

cr

Ex 2 ;ìI.;;- HWm,i COS (P,1 + HWm_i I COSQFS...I1 dr

xz

R siri3 m+1 m-li ]dr f cr 1---1 I sin.p,1+ Hwm_iI m cospi m msI R _{rm+ll m-1} Tx 2 ₂ crsini3[ _m 4m+, 0SÇ,1+ --H1 IcosPmi]dr + -Ttpfl £ 2 o R _msi mi crsinI _ISjfl91+IHw

+

4m o Lm1 m,iI ml m-1 Isinx.l] dr Fy 2 R _{sirp msi m-1} po cr k Isin H sin dr

2 cosp[ m m+il m+t m I m-1 m_i]

R 2 Fy z itp sin3[m+i

=t--

_{4 m}

f

Cr-o cosf3m Ty z ₂ R rm+il -= -f- Ttpfl S csnP[___ _{ftm.1 1Pm,1} 1 H1 IsincPm_IIIdr 2 R 2 rm+1 m-1

SCr srnI H ICO5m H _cospm..i]dr

E5 4m Lm

Fz Fz Tz Tz

CX E C, E

wo = L_X sin p sin (m C + cos (6 27t +

+m O cos(m 6 + ) sin (O +

z]

)

The lift force of the strip under consideration equals:

9. r

dL = - pc cosp

}HHm

m +1

HWmsi IcosWm,i H cosPi]dr

6 Sfl p H(Wm)+1

I

(m-q )sin

(1.27

(1.28)

With formula (1. 02) and with the assumptions of paragraph 1. 1. , the formulation of the transverse coefficients as given in Table 1 .4 can be made.

For the case of a rotational transverse vibration:

'-px = .Px sm (me +

(m l)sinm1)6 +p

P

L

Table 1. 5.

Transverse coefficients of the screw propeller.

= cr2snE IHlID,,Cl Jcr21 ltPf Cr2 i!j [r.i_1 CWm 2T2 ICWmll fl_dr (1 29)

= !f

cr2np[ + +IHW Isi,,,i]2+ ! 2r -Cw,iJco,_r]dr. Tx Z f fl+1 ni-1 Z ltPfl i z 2 22 rxml m-1 =--rt2pr j cr

+ I_

ICoCPnljdr+ ----Cr +IHwIcoSP,,_I]dC+ xx pnc r E- cw 1sn 1___jCw,,1 I5mp, ]dr. TxZ ? cr3cospI 1m,1 + IH ,In_x]dr+ f fcr3i HW,1Is)o +i.IHiIni]dr_

Ic.'

COS1 Icw_i COPI. = -H+it1Cinw_x]dr_ Cr29PHwlIsnpnCCl _IHw,1 ._i]dr+ êr !_ ¡CoC4,. C jcosP_x]dr = _ Hw,1IcosPn_]dr_ !L rr2 tÏHWmrjIcoSPrCl_ HWx 1IcosPn.2]dr_f_ Çcr Cw1 r Cw1 Isnx ]dr ly z z r,x.-i m-1 z 1tpn 1 2 mxl m-1 --Tt Cr ___IHW,1S2n4¼1]dr+1 cr I {-ICumCl Cwx_i oxp, ]dr f lico ]dr_ [T1ICW i

IC.IICW

i

xJdr

o o u Ez Fz Tz x '4¼ ¼

18

we have to distinguish three contributions to the lift force of the strip,i.e.:

Transverse velocity of the profile The positional deviation of the angle 3

The angular velocity of the profile.

The same analysis as applied in the previous cases gives the relations presented in Table 1.5.

When the propeller has a fixed inclination ..p ('P a small angle) the

in-flow of the propeller can be resolved into a component in the z-direction and one in the y-direction, the latter being considered as a disturbance of the in-flow VT for the case m = 1 (see 1.1.2).

This disturbance gives rise to the transverse forces F, T, F, and T

(see Table 1. 6).

1.2. Conclusions and remarks

Bearing in mind the assumptions described in paragraph 1. 1. 1 we can make the following remarks:

1.2.1. Effects to be taken into account.

The hydrodynamic effects to be taken into account for a more detailed analysis are described by the coefficients given in Table 1.7.

Table 1. 6.

R

Transverse

sin3

coefficients of the screw propeller.

Fx 3 2Ç 2

-=

2z itpn cr '17X o smp, dr 1.38) T W-1 cos R _.2 Tx prcr

=-2z

a _cosp

-

S smp1 dr (1.39) O R Fy 3 3( 2 sìn3 -= 2z TE pn cr

-j-

Sw=1 COSp1 dr (140) 0 R slfl2P S Ty 2Ç

-

2zTtpn cr u COS dr 1.41) coso W1 Fz Tz (1.42) 'pX PX

Table 1.7. Propeller coefficients to be taken into consideration. E E E _'Px Py 'Pz F x Fx x y Fx y F ,px '45x Fx tX Fx (Py Py Fx F L): Ex Fy Ex Ey Fy y Fy 'px

!i

Px Fy 43x Fy 'Py !x Py Fy '5y F '5z T T

;

Ty Tx 'px ;px T x T .Py y T '5y T Ty EX Ty y Ty '.px x Ty

ly

py y Ty '5y Iz Tz

E;

T Tz

20

1.2.2.

Equalities of the coefficients.

From the formulae and considerations of symmetry the following equali-ties can be derived:

a) Ez 'P T F £z _'Pz

b)T

x Py x P Ex Fy Fx Fy Fx tPx _Py x _y 4x y Tx T Tx Ty Ex Ey _fx Fy F E Ly _x y 'Py Px LP Py _'Px E Fy Fx F Fx F Py Ipy

L

Ey _{x y Ex} Ey Ex Ey Ex 1.2.3.

Additional conclusions.

Coupling effects exist neither from the x- or y-axis to the z-axis nor the reverse and consequently the behaviour of the system in the z-axis can be investigated without regard to the behaviour and excitation in the x- and y-axis.

The rotational position of the propeller is correlated to the phase of

the exciting forces. For the dynamic properties of the propeller, however,

CHAPTER

2.

THE APPLIED MEASURING TECHNIQUE AND INSTRUMENTS

2.1. Improvement of the signal noise ratio

2.1.1.

General remarks.

The determination of a mass-exciting variable force by measuring the deflection of a spring supporting the mass, is restricted to exciting

frequen-cies which are small in comparison to the natural frequency of the

mass-spring system.

The acceptable ratio between the exciting frequency and the natural fre-quency depends on the required accuracy in measuring the amplitude and phase of the exciting forces. In general it can be stated that for an increased natural frequency the accuracy of the measurement (dynamic response of the system) will be improved. This increase can be obtained by decreasing the mass and increasing the spring stiffness.

For propeller model measurements a reduction of the vibrating mass is restricted to the choice of the propeller model material.

X i - - = static sensitivity F ç M> M2>M, sects itvity L -Q., V3

Figure 2.1. Frequency response of various systems.

-

V X Schematic M mechanicaL system 'I;

/

F. fcos Vt

r-The increase in spring stiffness canbe applied up to limitations specified by the maximum admissible dimensions and the minimum level of sensitivity (the sensitivity and the spring stiffness have a reciprocal relation, see Fig. 2.1)

For the construction of the pick-up for dynamic propeller forces a com-promise has to he found between the natural frequency and the required sen-sitivity.

The minimum sensitivity is specified by the signal noise ratio, depen-dent on the amplitudes of the exciting forces and the noise associated with the propeller generated signal.

For the design of the instrument the unknown forces have to be estimated. An artificial reduction of noise leads finally to an increase inthe bandwidth and improves the quality of the measuring system.

A specialtechnique of noise reduction will be discussed in this chapter. The design and the construction of the mechanical part of the pick-up will be considered below.

2.1.2.

Applied technique of noise reduction.

In paragraph 1. 1. 2 is pointed out that the signals resulting from the sta-tionary ship's wake field have a periodic character. The frequencies of the

signals to he recorded are multiples of the propeller rpm and are

syn-chronous with propeller rotation.

Tithe frequencies of the signal and the noise differ sufficiently, the signal noise ratio can be improved by means of filters, transferring the signal and eliminating the noise. When, however, the signal and noise frequencies are of the same order of magnitude, other techniques utilizing the advanced know-ledge of the signal frequencies can be applied, as are described in [6] [7] [8]. The applied technique is a special type of cross correlation without the con-siderable amount of equipment normally required for the time delay and multi-plication for the determination of the correlation function:

T

(1)_1im

T-..cD T (2.01)

Forthe casef2( t

t

)equals a periodic impulse with periodicityequal to that of f1Ct) (the unknown signal) the multiplication and integration can be carried outby periodically connecting during a short time the signal f 1(t) to an integrating circuit.

hitf101

Periodic command

(T = period time) to the ewitch.

Figure 2. 2. Illustration of the periodic sampling system.

The analysis can be made in the following way [9J [10] [11] [12]. The

block diagram in Fig. 2.2. can be converted into the diagram in Fig. 2.3..

v*

The transfer function -Y_,giving information on the value of the signals at the moment of sampling only, equals:

/ * vu vi *

i -e

Rc RC

iwT

1-e

The switch is controlled by an impulse which can be delayed with ease. The operation is, in fact, based on a synchronous sampling tecimique oper-ating in the same way as stroboscopic observation. The integration can be carried by a sample and hold circuit and a filter to smooth the signal. The static output equals the instantaneous value of the dynamic signal at the mo-ment of sampling.

When the moment of sampling is shifted slowly through the period, the variation of the output signal to a base t (the delay time) will be equal to the pattern of the input signal to a base t (the actual time).

Thegain inthe signal noise ratio will be analyzed in the next paragraph.

2.1.3.

Analysis of the periodic sampling system.

A block diagram of the system is given in Fig. 2.2. The system consists of an amplifier transferring the signal to a switch, which periodically con-nects the amplifier output to a condenser. The system acts like a sample and hold circuit. Smoothing is obtained by the output resistance of the amplifier, the parasitic resistance of the switch combined with the applied capacity.

Period e

24

vu vu *

V. *

and the effect of the sampler that can be seen as an impulse modulator vi

device, introducing additional higher and lower frequencies.

In the case that the input signal consists of noise with a constant power spectrum with frequencies ranging from O through (see Fig. 2.4 ), the

output signal of the system has a power spectrum according to the transfer function of the system

This is indicatedin Fig. 2.4.

The ratio between the output and input noise power equals:

- 2

co _{_JWT}

2 T

-iwT

iw

l-(1-E)

e

L J

Figure 2.3. Block diagram of the periodic sampling system.

vu

We are however, interested in the ratio - and must take into account vi

the transfer function:

i -e

WT st E 1-e duj hold circuit (2.03) (2.04) cr Vu Vi Vi o2 It T

and when p = WT holds: 2 25 2w5 w5 -2 (

-

E ) +

s'

J E:

i(iE) e

252 wI 1_e 1_e

Figure 2.4. Input and output spectrum of periodic sampling system. Thus the improvement of the signal noise power ratio equals:

2 - wT

i-e

_iwT ¡w 252 2 1-cosLp lt a i -cosLp 2 B cos d.p dw di.p

26

where:

B-

2 (1-e)

- 2(1-E)+E2

The solution of the expression:

(1 -ces z ) dz

f

2 / 2

p(1-Bcosp)

j

z(1-Bcosz)

o

-where:

0<9<1

and z is a complex variable, equals:

2(1_E)+E2)

2

(2+E)E

The signal noise power ratio improvement equals:

9_ E .

{21E2

C 2'l 2

(2+)s

o2 H ¡w)

--

_M

luiT

i-e

. e (2.05)

The behaviour of this system in the frequency domain can be approxi-mated by the response to a unit step function (see Fig. 2.5).

The input-output relation in the frequency domain of the simplified re-sponse results in the following ratio:

M

1-e

RC

(2.06)

where:

M =the time during the closure of the switch

R = the parasitic resistance of the system

C = capacity.

A graphical representation of H _{¡ w)} represents the frequency response of the system.

The improvement of the signal noise ratio ecjuals the ratio of the hatched

/

output_{At At}

¿T

T O

input

a) Response of sampling system to a unit step function.

n put

b) Approximate response of sampling system to a unit step function. Figure 2.5.

w

Figure 2. 6. Powe r spectrum of input signal (white noise) and output signal (ap-proximated). T 2T 3T ¿T At At At T.O T 2T 3T

/

_I

o iJ h2 _output

28 Now: It' 'T ci T It o 2 E 2 2 ai

For the case E = 0. 01, a practical value , we obtain:

E 2 +c = 0.00490 E = 0.005025

2 E

=0.005. according to (2. 05): according to(2. 07): according to (2. 08): H(iw)

A rough approximation can be obtained by making the output signal as given in Fig. 2.5 equal to:

t

T

vo =

i -

RC-t

whereT is a fictive time constant.

The system is approximated by a simple first order RC filter. The sig-nal noise power ratio equals:

2 dw = t

= l-e

Tf

-

At

i-e

RC _E

2E' 2

1+e RC (2.07) (2.08)

lt can be concluded that the approximations are sufficiently accurate.

I.

Practical realization of the system.

The switch is realized by means of a bridge of 4 silicon diodes giving a resistance in the conducting condition of 500 Q and in the non-conducting con-dition of 500 MQ.

Jncombinationwithhighquality condensors, the detailed response of the system to a unit step function is given in Fig. 2.7.

As can be concluded from this figure the behaviour of the system is suf-ficiently accurate. The overall inaccuracy is less than one per cent.

When there is an exact synchronism between the input signal and the sampling command, the output of the system is ade signal, representing the

= 2 (i input sigriat B0 = (1 3.1) B6 = 0999

/

= 0999 B0 0.999 r B= 0.900

Transient Steady condition

t5 tw t5

T stgnaL

T sampUrìg

B0

Figure 2.7. Response to a unit step function.

instantaneous value of the signal at the moment of sampling. A slow shift of the sampling point in the period results in a reproduction of the input signal at the output. however, with a change in time scale.

The accuracy of reproduction depends on the rate of shifting, which can be expressed as follows:

B+( B8B,)e

The relative phase shift tp and amplitude ratio between input and out-put signal is given in Fig. 2.8, as a function of with parameter:

M

n=

1-e

RC

The required improvement of the signal noise ratio determines the value

of c

. The shift velocity of the sampling point expressed by is determined

from Fig. 2.8,

y an admissible amplitude reduction and phase shift of the

- 10 - 20 - 30 - I.0 50 - 60 - 70 - 80 - 90 _lOO

H

cas 1.0 0.7 C .0.2

__uu..__....u.l

II

UUUUUUUU.-11.1_u .__....

111111

liii:

L.006 0.3 0.4 0.2 .0.1 10 0.7 0.5

Figure 2.8. Selectionchartforthe adjustment of the periodic sampling

sys-tem. o 2 3 4 5 6 7 8 9 lo Il) n degrees O 2 3 4 5 5 7 8 9 lo I) ¡n degrees 1.0 09 08 07 05 0.5 04 03 02 01

2.2. Axial dynani le propeller properties

2.2.1.

The construction of the propeller exciter.

As is pointed out in paragraph 1. 1. 3, the hydrodynamic properties of the propeller are of essential importance for the determination of the equations

of motion for the propulsion system.

Although a rough impression of these coefficients can be obtained by the formulae given in chapter 1, an experimental determination of the value of the coefficients is preferred as acheckon the theoretically analyzed approximate values. The instrument for the determination of the coefficients must be able to superimpose onthe static forward velocity and rotation sinusoida] motions in thrust and torque directions, synchronous with multiples of the number of revolutions of the propeller times the number of blades (blade frequency). The vibratory motions of the propeller must be recorded simultaneously with the torque and thrust fluctuations generated by the propeller on its support. Vibra-tory motions in torque and thrust direction can be applied alternatively. The

general idea is given in Fig. 2.9. The system consists of a flywheel

porting an interchangeable spring. The shaft connected to the spring

sup-0

i Watertight protecting cover S FLywheeL

2 Sliprings 7 Torque and thrust exciting units

3 tnterchangeable measuring springs 8 Air Lubricated beorngs

1. AxiaL and torsiatul undarmped ucceLerorneter Y Torque and thrust pick -up

S Axial and torsionut displacement pick - up 10 Propeller

32

Torque pick up Thrust pick up

Thrust spring

Figure 2.10. Essential parts of the propeller exciter.

ports the propeller shaft and the propeller. This shaft is air-lubricated (in

order to avoid frictional phenomena) and is provided with electromagnetic

torque and thrust exciting units. These units are based on the loudspeaker

voice-coil principle.

Thrust and torque pick-ups with negligible elasticity are mounted between the propeller shaft and the shafts with the exitationunits.

The support of the intermediate shaft can be either weak in thrust and

stiff in torque direction or the reverse. With torsional and axial displace-ment pick-ups, the vibratory motions of the propeller relative to the

fly-wheel canbe determined. The analysis of this system can be carried out ac-cording to the schematical representation in Fig. 2.10.

The flywheel is assumed to be an ideal immovable support for the springs. Due to the large mass and moment of inertia the errors made by this

assump-tion are negligible (see par. 2.2 ). Further, the elasticity of the thrust and

torque pick-up is neglected, which is admissible if the natural frequency of the propeller and the propeller shaft, by way of the pick-up infinitely stiff supported, is high in comparison to the frequencies of measurement (1,000

cps and 100 cps respectively, see the next paragraph). With these

as-sumptions the block diagram of Fig. 2. 11 can be made. The instrument

per-mits the detection of the eight propeller coefficients operating in the z-direction as mentioned in paragraph 1. 1.3. 1. The exciting inputs of the

sys-tem are F1 and T

. The outputs of interest are the propeller motions z

and t.p and the outpZut of the torque and thrust pick-up indicated in the block

diagram (F and. T ). For these outputs is valid, according to Fig. 2.11:

AxiaL accelerometer

F=( M1 +

M+Ï

_KF)Z+

o

Z =( Il + 'z +

I,pz

In the case that the shaft is sufficiently stiff in torsional direction (CT as high as possible) the output p7wíll be negligible as well as the signals

and

Then the equations(2. 09)and (2. 10) simplify to:

T7

Kî)p+

-Fz ₊ Fz

¼2Z

Figure 2.11. Block diagram of the propeller exciter.

IZ 'P7 (2.09) Ez (2.10)

F=(M1 + M7 +

F7 _{- <F ) +} F7 CZ (2.11) Ez -ro Tz

TI.

Z-

Cz + (2.12)

For operation in air we find the additional equation:

34

assuming thatthe density of air is negligible in comparison to that of water.

This expression is made equal to zero by adjusting the variable factor

(2. 15) can be solved and the coefficients can be determined.

When a sinusoidal excitation is applied, the ax ial excitation can be ampli -fiedbytuningthe axial system with a suitable value of CF. This value has to be determined experimentally for each individual condition of measurement.

Analogously the coefficients

can he determinedbyastiff support in longitudinal direction (CF) and a tuned system in torque direction (C

KF (and K1). This

=

means that

+

the equations 2.11 and 2.12 reduce to:

Ez (2. 14)

F7 F

T7 T7

= _{+ (2. 15)}

With the measurement ofE7and F and T in water the equations (2. 14)

X3 X1=Ez

Figure 2.12. Schematic representation of the propeller exciter.

2.2.2.

Approximations and limitations of the system.

The system can be represented by the mass-spring system given in Fig. 2 12

The following equations hold (see also [13]):

il-z! and

ri

F7 F7 z j l-z T F nd Lp7

It is assumed that X1 = X2 and X3 = 0. Although these assumptions are

admissible in the applicable frequency range an accurate information showing

the limitations of the system is given in Fig. 2.13. The range of operation

isdeterminedby--

=0.05 and -0.05. The amplification of the

excitation by tuning is also indicated.

==L

iiui

usuimuuuuus

_IlIIIlIUNIH!

=

R,ng. f p.ratrnn

- -

_.IIIIl

uiiiii

'!!!!!

=:::

=::::

- __...uu

- ...

_{II x}

uiiu.

iuuui

uii!ui

rAIIIIllI == 02

__...u.uI

UUuIIWAUIIII

_...uIIII

.uuIIII..uuuuuu!

!UflL___ !Ll!!!!--III!!!!

====I

_uuuuui

LiI::u;==:;:#i

_uuuuuiuuuuu

UUIIIP!

'iuuuniii

P4USUI

uiuuni

HHii

I iiuuinhi

uuuui F

iii

uuuuui IIIIH

liii

L

surni

uuuiiiui

1flh!!!

uiiiiSuiuii

Figure 2.13. Behaviour of the propeller exciter.

- F'-(

X3 - X2) C2 = M 2 X3 (2.16)

(X3 - X2

)c (X1 - X2) C1

= M2 S2 X2 _(2.17)

(X2 -X1 )C1 = (M1+ M2 + )S2X1 + S X1 (2.18)

36

2.3, The propeller excited vibratory forees

2.3.1.

The construction of the measuring system for

the

six components of the propeller. The mechanical

properties of the pick-up.

o

o

Arran9ement of the strain - gouges

Figure 2.14. Six component balance for the measurement of propeller fluc-tuating forces.

Figure 2.15. Six component balance under construction.

The design can be based on the required sensitivity and the forces to be measured. The required sensitivity depends on the acceptable signal noise ratio of the signal presented to the periodic sampling system and on the

mag-nitude of the signals to be measured. The signal can be estimated from a

theoretical analysis of the forces generated by the propeller (see chapter 1). Together with an estimate of the noise generated by turbulence, slip rings

and instrumentation, the signal noise ratio is obtained and examined for

acceptance. In this way a transducer is designed, having an acceptable

sig-nal noise ratio and bandwidth (natural frequency).

The dimensions of the final pick-up with an indication of the way in which thestrain-gauges are glued is given in Fig. 2.14. Fig. 2.15 (photo) gives an impression of the pick-up under construction.

Figure 2. 16.

Investigated 4-bladed propeller.

Jii.iî l-0 0 1.56 159 160 I 102 75 0.95 0.278 lOS. II 09 0.375 91.91 0.5 0.472 1.50 53.11 07 0.500 1.63 71.24 0.6 .510 14 59.35 05 0.475 LII

IL

-17.49 0.4 0.155 1.31 30.62 03 0.120 1.32

II4

--T

C 23.75 07 0.412 I 29 r x[. L[

i

Jj-2.3.2.

Analysis of the dynamic behaviour in axial

direc-tion with propèller model.

As canbeseenthe transfer functions depend on the geometry of the pro-peller i.e. among others on the dynamic and mechanical propro-peller properties. Fortunately it can be shown that the effects of the differences of these

pro.-perties on the transfer characteristics are negligible for a large number of

propellers normally applied to model tests of surface ships.

The analysis of the characteristics of the measuring system will be

car-ried out for a special propeller, which is representative for the

above

mentioned number of propellers. For this special propeller, a complete in-vestigation as mentioned in the introduction will be carried out (see Fig. 0. 1). The propeller is described in Fig. 2.16.

In this paragraph the axial behaviour of the pick-up with the propeller will be determined. Assumingthat the effect of the mass and moment of iner-tia of the propeller shaft on the measurement is negligible the behaviour in axial directioncanbe described by two coupled equations of motion, as given on the next page.

40 F, ci Fz s ₊

\/

C V M,.j + w +Ez

C= F

(2.19)

ligure 2. 18. Simplified diagram of the behaviour i z-direction of an elas-tically supported propeller.

z

'z

pz Tz

+ Tz + Ez Tz + E T

C=T

(2.20)

'pz Ez

The stiffnesses of the shaft are C and C in the longitudinal and torsional direction respectively.

The torsional and longitudinal motions are coupled by the working of the

propeller. These hydrodynamic effects are described in chapter 1 and

re-viewed in paragraph 1 .1 .3.1 Table 1. 3, and permit together with the mechanical

properties of a simple propeller-mass-spring system, the drawing of the

block diagram of the measuring system (see Fig. 2.17).

M2 + 'P, Fz Ez + + -r;---Z tÔ rz E-. +5 F2 +5 'p2

H1 H2 H3

Fz 1 - H1 H2 H3 H4 (2.24)

Foragood operation of the systemthe first two ratios are of importance, the lasttwo ratios, however, must be negligible. As H and H4 cannot be in-fluencedby the construction of the pick-up, but depend on the characteristics of the propeller, it is necessary to make the product H1 . H2 as small as

pos-sible. This means

H -

O

H2 -

O hence

1-H1 H2

H3 H4r-1

and E

H1 -r.

O z

.'

H2 _ O z

For the measurement of the exciting forces, not to be influenced by the

coupling terms of the propeller, the transfer functions H1 and H₂ must

be as small as possible, i.e. the support must be as stiff as possible. From the point of view of the dynamic response of the system an analogue require-ment can be made (see paragraph 2.1).

Assuming thL the differential equations are linear with constant coeffi-cients an acceptaMe approximation for the first approach can be made (see

Fig. 2.18).

For the characteristics of the pick-up we have to consider the transfer functions: H1 (2.21) F2 - 1-H1 H2 H3 H4 pz H2 (2.22) Tz 1-H1 H2 H3 H4 H1 H2 H4 T2 1-Hi H2 H3 H4 (2.23)

42

The final stiffness of the pick-up is determined experimentally by ex-citing the complete system in air.

The following data are of importance:

Mass of the propeller model

M= 410

kg sec2

m

Natural frequency in air in thrust direction 1000

C) Stiffness of the support in thrust direction _1.5V

d) Moment of inertia of the propeller

Natural frequency in air in torque direction 600 Ec. p. sI

Stifess of the support in torque direction = 835 kgm a d

From the formulae given in chapter 2 we can make an estimate of the properties of the vibrating propeller.

These model properties in axial direction can be expressed by 6

coeffi-cients, given in Table 2.1 and depend on the rpm of the propeller model.

The given values are calculated for full power absorption corresponding

to 10 revs. per second.

Four expressions can be distinguished. The transfer characteristics CZ _'pz Cz

and pz

Fz T _Tz

for the above conditions are given in Fig. 2. 19 A, B and C.

This table shows that the exciting forces can be determined up to fre-quencies of 200 cps with an error of 10 per cent.

Higher frequencies can be recorded with an increased error.

-5

_T

Table 2. 1. Hydrodynamic properties of the vibrating propeller.

2.3.3. Analysis of the dynamic behaviour in transverse

direction with propeller model.

In order to obtain an insight into the transverse behaviour of a propel-1er elastically supported by a shaft, we consider separately the different ef-fects to he taken into account.

2.3.3.1. Effect of shaft elasticity and gyroscopic precession. We define the following shaft properties (see also Fig. 0.2):

coefficient symbol numerical value

Hydrodynamic moment

of inertia

49 .

io'[igm sec1

T z

Hydrodynamic torsional

damping 1 . [kgm sed

z

Hydrudynamic mass Fz 5 . l0 kg sec1

E7

mj

Hydrodynamic axial damping 12.5 kg sec F7 m

Acceleration coupling

-

F 1.6 . 10-[kg seca]

z

T

-

_E

z

Velocity coupling 0.35 [ig sec]

F7 z

I

E

1.10

[]

u.... u.... uuui U..., 11111 11111

11111_iuniuiiiii

E!!!!-U...

....

uiui

U."

'I'll

¡111

iiui

--U.. u....

l'in

UI"

I'll' I'll'

E!!!!u.... u....

I..,'

11111

11111

AI!HIIII

_______!!!!!IIIIUUIII

_______I;;:,_______

.'II________

_- u

i lui__u...

iiiuiiui

LII11UIIi

uiiitii.uuui

IHIIIUIIII -lu-n---

rjI{UUuI..

Il UI UI__u....

'n,iiuuuuii

,iuu.mi

uuiu u_ui. u. u.... u...i

luu

iiui

uuuuuiu

I

I IIIII

ulIIIUuIII

I

I

II

I

IIIIP!11111

"I---"

I_______ 11111

IIlII

11111

ii

___________ -__... uuu.. u.... u....

lull,

U.U.! U.,,, Ululi NIH

liii

111H11111

1m] t.kg] 1.18 1.19 10000 000 O Frequency 100 O Frequency 100 c.p.0 1000 c.p.0 1000 1000010000

::::::::::

-..ui---U...

u.... u....

...i

.iuI..uuI Ez

Fz 11111

uIAIuIuI

11111

IIflIHIII

10 Frequency 100 c p.s. 1000

[]

iii ii

iiiiii

iii ji

liii I

11111 lUi

hifi I

ff10

I___

Iir______

:

--:1:'

_....

_.

I UIU u.. uui______

..Hr

.i.

I IJI

iiniiiiii

liii

iliii_iiiiiipiiiiiiiiiii

_____=;;;;;

_...

...0

..

uiuuuuni

iii!llhIuibl

III

-u.u.

.;...

iuuuuuuuiuu.i.

Uu111iuuii

uuu.'IIii..uiiu.iui

Figure 2.19. Transfer functions of the pick-up

10 Frequency 100 c.ps. 1000 10000

46 F

= C. E

=

F= C.(Py )

F= CxyPx

T = x =

Cy (-Ex)

= Cy = C

With these shaft stiffnes ses the equations of motion of the

non-rota-ting disk-shaft system can be given [14] [15]. The differential equations are:

MEx F)Ç C

Cy

Cxy C C Ex -2 F cx2y_c. c

c-c c

F T Lxy

LX 'x

cx_.c c

;

=

-x_cÇ c

F T Cxy Cy Cy 2 F T cxy_cy Cy c .Ex Cx_C C Ey +

-2 F T cxy_Cy Cy cxy C C 2 F T Cxy_Cy Cy 2 T Cxy Cy i.p (2.25) (2.26) (2.27) (2.28)

For the rotating propeller, gyroscopic effects have to be taken into ac-count.

The precession moments T and T equal respectively:

Tx -

_{'z 'y}

and T =

The block diagram of the propeller, rotating on the shaft, is given in Fig. 2.20. Tn this figure, however, the hydrodynamic effects are not

con-sidered.

e i c,cy.0 C C r r V.rtca mot's

f

c C Cy2__Cç c CxyC. C Cx)_C ; c.Cx c 'py Cxy.Cç ; cx_C

; K

Horizontal motions

Figure 2.20. Blockdiagramoftransversebehaviourof a rotating mass on a

shaft.

C cy_c C

48

Figure 2.21. Block diagram illustrating the hydrodynamic effects of a pro-peller vibrating in transverse direction.

Lx -Ex 2 Ex - + s+ s Ex 2 Ex Ex S+ S+

-+-,-±

-Ex 2 Ex Ex (x

s.

x

s.

Px * -o Ex 2 Ex

+ - s

+ s

--o

Dy i

_

ly 2 T -Lx Lx +-r---ly 2 ly Ty IL S + Y S * Y + T Ty 'Px Px Px

Ty2Ty

Ix-S1 Lx

+s2+

_tx _ Ix + ,y 1Dy 'Dy I x s2 -'Dx Px + Ix

S +5

x Tx s2 * s + y -y

+s2+

'Dx x 'Dx Fy 2 Fy S

+ -S

+ -C---ty

-Table 2.2. Four siniultaneous equations describing the transverse motion

the propeller

(Signs between brackets are valid for right hani

propellers). F. )., Ty. {

-},

r -L

i

--5- s

' J rTi - Liy s -- 1y s- øI,s. -n- -> _J r n-tn x.s'. -e-s

-

4. s 4. J I T - S L y T - E7 c, e'. e' C,y -Cy . C S C _Cy Cy J I)

s.s} s

{ii'iE s

]!i {ss iis s *

50

2. 3. 3. 2. Effect of the hydrodynamic properties of the propeller.

As is reviewed in Table 1 .2, 40 coefficients affect the vibratory behaviour inathwartshipdirection. The operation of these effects is illustrated in Fig. 2.21. This figure showsthateachmotion of the rotating propeller introduces coupling effects in all other possible directions. The equations of motions are given in Table 2.2. The following shaft elasticities are determined by a theore-tical approximation from the dimensions of the pick-up:

Table 2.3. Propeller model coefficients. E, F -O2781Q [j 5-0156 - -0 113-10' a=,agzoio.2_1 f=+O.13710'EgecTj .-D.139-10° !=-a1o.,o'

_:°-«

Fy t.O.15611c'[Z] _o.o.n1o'[d1

---092O-1O 9!!J ._oiiid' E...oi3gio° .+0.137- IO'

105 !..OE725-10' _..+oissiñ'[ku.j -,.o696lÓ'

F E!,o.i25ib !.OE3SO -1_O' [j

fo.1

T, iQ.598 1O kgp. _{i.a272taorI!2} L

+O.192-1O° _{j5_ol2O1cf EÇMI} !_o341 - -a2891

rnj

.Oi75 'Tö' =-139-ió'[kgrrn -.ø387 [w

Ty

!+0272 10' !.o sos io'

120-11f E°"j .?..+0192 -10' -0209-i1f r -O.301 -iö'

-ü175 10' Zooie10' IL.-0.387 IL.0.139

+o.350 10' L L,i. o 163 -' .+0.49

frnj

= Lx

fx

Ey = = cÇ

= 0.80.

rn 5px .Py = = 0.33 . 10 rad F Ex l-x s x_ E1, _Ex - 0.55 . 10

[k

The properties of the propeller model are calculated according to the

formulae given in section 1. These coefficients are given in Table 2. 3. Neces-sary mechanical information of the propeller is given below.

The moment of inertia _{1x = ly =} 1z = 2.9 1Ö

Egm se

2

_2 kg sec

Mass of the propeller M = 1 .10

m

From this information the coefficients of the four simultaneous equations can he determined (These equations are valid for the special propeller con-dition 10 rps only).

The solution of these equations can be represented by 16 transfer fune-tions given in Figs. 2. 22 and 2. 23. From these funcfune-tions can be concluded that the measurement of athwartships propeller excited moments and forces can be realized up to 200 cps with an error of 10 per cent.

10 Fr.quency 100 c.p. s. 1000 1.1öE 0000 00

J

Il

à

C 10 Friquincy 100 c.p.i. 1000 1 ii2 Ty Px _{Ix r-U 110 110} 10000 D Figure 2. 22

Transfer functions of the pick-up in transverse direction. With hyclrodyiìamic propeller coefficients. Without hydrodynamic propeller coefficients.

10

Frequency

100

C p.s.

1.10 Cy Fx Cs Fy 5 110 rml [kgj 6 1.10

I

-i_______

-....0

_...uu

i

1IH

=__________ u,'.'. 'l'uil

I

11111

i

,giII_________ 111111 I... i.u.uI

I

AIMU

_,_1.

-.1iuU

uiiuiur

lUlfill____ I

=,-- __;;;_______ I_w. wu... II r .ut_

iriiiuii

iii

..uhuI-II-

uuiuiiii

/

i

11 Lkg

\

/

_/

10 Fr.qu.ncy 100 c.p s 1000

j

I 10000 10 Fr.qu.ncy 100 c.p.s. 1000 10000

7

1.10

1.106 :..

.r

.

...IUI UU UUII

..u..i

Ill

UU U.U.. _UUI ______ill______

III

_________-U:ir

________ _..

-, U U

uiiiili

.

I

111111 III

ru ..I

U

IIII NUI .11

..

.ni._____ _uiuu uur

ii

ill.

..iil______

'1INI II

__________--u _______

.

lui UI

L U11IUu

U.0

l

u.0

u

juiuruiu

liii-II I

L IlillillIlk

iii________ iii Li______ Uil FAUl UI., lUI I

Il

41111

III

II. .u. iui i

i.,

/

III

III

C

/

-3 1.10 kgrn 1.10 10000

/

I IL

'liii

ti..

II

.iiiirn

1111u

I

II UU

a.l.1.

UI UIUIt? 1u1u

iiiIIJiIlM

II IP,

Figure 2.23. Transfer functions of the pick-up in transverse direct Oil.

With hydrodynarnic propeller coefficients.

-Without hydrodynarnic propeller coefficients.

10 F equency 100 CP.S 1000 10 Frequency 100 C P.S 1000 110

Ill

[kgJ 110 -7 110 1.10 110

56

CHAPTER 3.

FULL SIZE PREDICTION FROM MODEL EXPERIMENTS

3.1. Results of the experiments carried out on the ship and propeller model of a single screw tanker

3.1.1.

Experimental determination of the axial dynamic

propeller properties.

The dynamic properties of the propeller model given in Fig. 2.16, have to be determined, in order to enable the prediction to he made mentioned in the introduction. Theoretical analyses of the propeller properties illustrate a weak dependency of the frequency (see Table 3. 1) which, it is assumed, can be neglected [16].

By means of the propeller exciter described in paragraph 2. 2 , the dy-namic properties of the propeller model are experimentally determined. The advance ratio of the propeller behind the ship model and during these tests is kept constant at 0.45. The experimental and theoretical results are compared in Table 3. 2, which shows relatively important differences between the calculated and the measured values. The effect of these differences on the transfer functions and predictions have to be considered.

Table 3.1.

Frequency dependency of theoretically obtained propeller model coefficients.

Order of vibration +5

.iü

.10 2 +2 .10 -1

.10

103 1 .10 z T z Fz

r-E z Fz

-

_E Z F Z F

-«p_Z 4 4.9 1.00 5.00 1.25 1.60 3.5(1 8 5.43 0.96 5.45 1.20 1.76 3.36 12 5.55 0.95 5.60 1.19 1.80 3.33 16 5.60 0.95 5.63 1.19 1.81 3.32

Table 3. 2. Compari son of the calculated and the measured hydrodynamic pro-peller model coefficients.

Quantity Symbol Calculated Measured Units

Hydrodynamic moment of inertia

4.9 10'

7.6 10 kgni seca T2 Hydrodynamic torsional damping - 10 0.86 10 kg m sec Tz

Hydrodynamic mass 5 . 10 3.2 . lO kg seca

F2 Cz m Hydrodynamic axial damping 12.5 2.7 kg sec F2 m

Acceleration coupling F2

-

1.6 . 10 1.4 10' kg seca

tz

Acceleration coupling 1.6- 10' 1.48. 10 kg se&

T2

Cz

Velocity coupling 0.35 0.23 kg sec

Fz

-tpz

Velocity coupling 0.35 0.13 kg sec

Tz Cz

__. u..

III

__. .n. e.. ...

u..

....0 III

III

1111111

III ____________Ei________ __________¡ii I I i.. III 11L1Uf UI 1AI1luI _I.I

I

I

III

I!IìÍII

I _.

.n..

US s.l.. u..

u..

.. li. UU UI. uiui lui

I Iill

A____

_____________________________ U.0

s..

lu

...

l

UL 1'

III U.. i U.0 ,_ ________uuu

III

II

IIIIIIH

i.iO8 5uu1

..uuii

uiuui

u..."

'lull

--u.u. ----UI

-u...

u...'

u:::::_:_::::::::::

.

..uul____....

uuu..uuui

u...,.

u uuui u..uuu U.U.,, u iuui

.u.ui

I I 11111 9

..iuii

IIIIII

---u..

uuuui

Ililli

----U.

iiIji

I'll'________

u uuH

UIII

i

....,

u...

i-_11111_111111

______u::::: i.

...

_..uUU

I_IiIl_111111

____-\u 1I1

UIU

. ui

uulll I. JI, II

iui

I_AIIIIIIH

i.lu

..U11UUU1

I.

uuul.l.11

u

iiui.uuuuul

I

JIlllIIIIH

10 Frequency lOE) c.p s 1000 1OEIOO 1.10 10 Frequency 100 C .S 1000 100 Fz -5 110 1ml Lkgj 110 110 Tz

['I

110 1108 ,ia...0 u...______ ...uI ..ul.

uzuiuuuu

_iuiuuui

11111

IUfl

III " 11111

11111 AhIH

::u

___... ___uu. u... .uuI

...

u..l.

liii

11111 111111 111111

III

_____

______lii._1111K

I_____________

liii

11111

1Liii

uhu

u...

u...

u iiii.

UIUI UI II 1ILUU UUUIII

liii

iilii____

111111 _... __.... u...

...

...

UUUU uUH uulIU UhU

UUUUiIUUII

UhIflU

liii

IIIIIIlli

____________ __________-....---....

liii

1111W

..._

liii

IhIIl

UlUli

uili

I1IUi!uIiI

Figure 3.1. -10 Frequency 100 i000 10000

5= APP

19.5

19

18

Figure 3.2. Body plan of the ship under investigation.

c.w.L

FulL size values ) M.Ton ) Ton '3 q) > o o o w o C e u o-10

i.

i

_r_.. ia

t!__llHW4UUIIII

.r. u

tp A

iii

______,I1II

\

/ /

-_________

-VJ

N

IN

-VA Propeller - position LI

il

4)

f.'

_{I I}

-Torque vario Thriart vnri g,

Figure 3.3. Thiust and torque variations measured on the model.

t Ion tion

Torque = - 75.32 -6.23 sin (I. e + 128 ) -1.85 sin (8 0i 85 (-0.6 sin(12 0 264 ) -0.26 sin (160 286 Thrust =

98.51 + 8.82 sin(4 0124 )+2.L5sjn(8O+153 ) +O.62sin(120+2O4 ) 0.24 sin(16 O + 255

+15 + 10 /

I

e

t.,

lin

a)

I

62

The effect on the transfer functions of the six component pick-up. The added mass and the added moment of inertia are corrections to be made on

the mechanical properties and a small change in these corrections is

as-sumed to be negligible. The damping coefficients and the coupling terms have little influence on the bandwidth of the instrument.

The effects, negligible in the frequency range of operation, are illustrated

in Fig. 3. 1, where the transfer functions with the theoretical and

experi-mental coefficients are given. It is assumed that for the transverse

direc-tion the same conclusions can be drawn.

The effect on the prediction of the full size behaviour. The full size behaviour will certainly be influenced by the change in propelle r coefficients.

Small shifts in the value of the natural frequency may occur. The mutul

coupling effects will introduce additional corrections. The corrections to be applied are indicated in Figs. 3.10 and 3.11.

3.1.2.

_{Experimental determination of the propeller}

ex-cited thrust and torque fluctuations.

With the instrumentation described in paragraph 2.3, the thrust and

torque fluctuations generated by the propeller were determined for a turbine-driven tanker given in Fig. 3.2, with apropeller indicated in Fig. 2.16. The experiments were carried out on a model to a scale of i 27 . 5 at equal Froude numbers for model and full size. The results of the measurements,

expres-sed as percentages of the average values, are given in Fig. 3.3. The thrust

and torque fluctuations show roughly the same pattern and are opposed to one another. The mathematical expressions of the fluctuations are:

= - 75.32 - 6.23 sin (4e-t 128°) - 1.85 sin (8e+ 185°) -F - 0.6 sin (12e-F 244°) - 0.26 sin (160-F 296°) mTon = 98.61 -F 8.82 sin (40+ 124°) + 2.45 sin (80+ 153°) +

+ 0.62 sin (12e+ 204°) ± 0.24 sin (160+ 255°) Ton.

3.1.3.

Some considerations on scale effect.

3.1.3.1. Introduction.

numbers ( for ship and model, because the free surface ph

As already mentioned the model tests were carried out at equal Froucle

e-C

Figure 3.4. Distribution ofthe axial wake pattern for the ship under

investi-gation.

A and B: measured behind the model. C and D: scale effect corrected.

B

p.

AW

Jjffi

lo nomena should be taken into consideration.

The wake of the ship, however, is influenced by the boundary layerofthe hull and, therefore, by the friction effects of the medium, sothat it depends on the Reynolds number. ( Re =

The Reynolds numbers for the full size and for the model differ with about a factor io2. This implies that the boundary layer of the model does not

corres-pond to that of the full size ship. This effect can be studied according to

reference [17] where for different scales the wake pattern of the same hull is given.