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a o
v;i o * • *>» •o BIBLIOTHEEK TU Delft P 1294 2181 C 379654PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL TE DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS IR. H. J . DE WIJS,
HOOGLERAAR IN DE AFDELING DER MIJNBOUWKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP
WOENSDAG 7 APRIL 1965, DES NAMIDDAGS TE 4 UUR
DOOR
REirvnER WERELDSMA
WERKTUIGKUNDIG INGENIEUR GEBOREN TE HAARLEM.
INTERNATIONALE PERIODIEKE PERS - ROTTERDAM
\ • ' ' '
S u m m a r y .
I n t r o d u c t i o n 1 1. DESCRIPTION O F THE DYNAMIC PHENOMENA O F A SCREW
P R O P E L L E R . 5 1 . 1 . Application of t h e u n s t e a d y t w o - d i m e n s i o n a l a i r f o i l t h e o r y on s c r e w p r o p e l l e r s 5 1 . 1 . 1 . P r e l i m i n a r y r e m a r k s 1 1 . 1 . 2 . The p r o p e l l e r in a d i s t u r b e d inflow 6 1 . 1 . 3 . Dynamic effects of a v i b r a t i n g p r o p e l l e r in an u n d i s -t u r b e d inflow 11 1 . 1 . 3 . 1 . Axial m o t i o n s 11 1 . 1 . 3 . 2 . T r a n s v e r s e m o t i o n s 15 1 . 2 . C o n c l u s i o n s and r e m a r k s 18 1 . 2 . 1 . Effects t o b e t a k e n into a c c o u n t 18 1 . 2 . 2 . E q u a l i t i e s of the coefficients 20 1 . 2 . 3 . Additional c o n c l u s i o n s 20 2. THE A P P L I E D MEASURING TECHNIQUE AND INSTRUMENTS . . 21
2 . 1 . I m p r o v e m e n t of t h e s i g n a l n o i s e r a t i o 21 2 . 1 . 1 . G e n e r a l r e m a r k s 21 2 . 1 . 2 . Applied technique of n o i s e r e d u c t i o n 22 2 . 1 . 3 . A n a l y s i s of the p e r i o d i c s a m p l i n g s y s t e m 23 2 . 1 . 4 . P r a c t i c a l r e a l i z a t i o n of t h e s y s t e m 28 2 . 2 . The a x i a l d y n a m i c p r o p e l l e r p r o p e r t i e s 31 2 . 2 . 1 . The c o n s t r u c t i o n of the p r o p e l l e r e x c i t e r 31 2 . 2 . 2 . A p p r o x i m a t i o n s and l i m i t a t i o n s of t h e s y s t e m 34 2 . 3 . The p r o p e l l e r e x c i t e d v i b r a t o r y f o r c e s 36 2 . 3 . 1 . The c o n s t r u c t i o n of the m e a s u r i n g s y s t e m for the s i x
c o m p o n e n t s of t h e p r o p e l l e r . The m e c h a n i c a l p r o p e r
-t i e s of -t h e p i c k - u p 36 2 . 3 . 2 . A n a l y s i s of the d y n a m i c b e h a v i o u r in a x i a l d i r e c t i o n
2 . 3 . 3 . 1 . Effect of shaft elasticity and gyroscopic p r e
-cession 43 2 . 3 . 3 . 2 . Effect of the hydrodynamic properties of the
propeller 50 3. FULL SIZE PREDICTION FROM MODEL EXPERIMENTS . . . . 56
3 . 1 . Results of the experiments carried out on the ship and
propel-ler 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 be 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, neglections and analyses 70 3 . 2 . 2 . Full size 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
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 t r a n s v e r s e 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 c r o s s -correlation technique (periodic sampling). The same technique is applied to the full size measurements to eliminate disturbances due to ship motions and changes in propeller rpm etc.
INTRODICTION
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 p a r t s , 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 with troubles and problems, and the designer should study vibration problems in order to avoid them. In shipbuilding techniques several more or less severe vibration problems are encountered. The wavy surface, on which the ship is sailing gives rise to motions with very low frequencies, or, in the case of slamming, to high frequency vibrations. Another excitation is caused by the screw p r o -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 i n -vestigation - the prediction of thrust and torque vibrations of the propulsion system - is presented in this t h e s i s .
The investigations were started in order to obtain insight into the m e -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 in the propulsion system with 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 p a r t s . The presence of the ship's wake results in an instationary loading of the propeller. In addition to the basic components of average thrust and torque, we can distinguish average transverse forces, thrust eccentricity, fluctuations of the thrust and torque, transverse forces and thrust eccentricities. In the
6 compuni-nts
on the stt'm compunents
ship considered hydrodynamic coefficients
2 axial components
transverse mech. properties of the full size ship
equations of motions of:
hull considered as a beam (+) propeller •* shaft (shaft whirling)
axial hydrod>-n. coeff. of propeller equations of mutio axial mechanical prc^wrtiesofthefull size propulsion system
measurement
^•ibration of thi' stern of a full
full size bearing for-ces and transverse shaft-and propeller-motions
full s u e axial propeller
Figure 0 . 1 . Block diagram of the investigations into propeller excited vibra-tions.
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 a fluctuating p r e s s u r e field exists around the propeller, introducing fluctuating p r e s s u r e s on the hull of the ship, even when the p r o peller loading is stationary. This leads to a pure hydrodynamically t r a n s ferred excitation of the ship's afterbody, caused by propeller action. A r e -view of the three types of excitation is given in Fig. 0 . 1 .
Onlyapart 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 describedby linear differential equations. The coefficients of the equations are determined by the hydrodynamic properties of the propeller and the p r o -perties of the mechanical system.
, X
1"
The right hand side of the equations, the exciting forces in fact are p r e -sented by the propeller generated vibratory forces, due 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 e x -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. 0.2 the 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 a x i s -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 p r o -peller. ( Provisions are made for the application of right hand propellers).
C H A P T E R 1 .
DESCRIPTION OF THE DYNAMIC PHENOMENA OF A SCREW PROPELLER
1.1. Application of the unsteady two-dimensional airfoil theory on screw pro-pellers
1 . 1 . 1 . P r e l i m i n a r y r e m a r k s .
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]. Inthis chapter a description of the phenomena is given for all directions in which vibratory motions may occur. For the application of the two-dimensional theory, presented by von Karman 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 v e -locities and profile motions perpendicular to the inflow velocity and span directionof 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 a p -plied, supposing that linearity is present, due to the fact that the angles of attack have a value, not exceeding 5 °.
Heavily loaded propellers are for instance excluded from the consider-ation made in this investigconsider-ation.
A strip theory will be 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 0 r , where 6 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
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 t h a t e . g . singing propellers are not considered inthis investigation. The torque on the profile caused by the asymmetrical attack of the p r o -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.
Itisfurther assumed that the wake of the hull is a stationary phenomenon, so that the fluctuating forces on 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, andconsequently the vibratory motions, are also synchronous with the shaft revolutions and a r e 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. T h e p r o p e l l e r in a d i s t u r b e d i n f l o w (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 in paragraph 1. 1. 1. these components can be describedby the following expressions:
\ " ^ I ^am 1^'" j '^ '® + P ^ ' + ' ^ a m f =axialwake
V = I Iv IsinJ m (0 + p ^ ^ )+({; I = tangential wake
Cz A F«
/ Fi
ez
t t - d L sin p cos 9
dTy«- dLr cos B cos 9
dTx\-dLrco» p sin 9
Figure 1 . 1 . Velocities and forces on a propeller blade with distui-bed inflow.
where p is the blade under consideration (p - 0 through z-1) (z = number of blades).
& r
Only the components perpendicular to the inflow velocity n will be
considered, so that the gust velocity Vg equals
rD = l
+ I V.
Vn = I I V ^ I s i n ^ m ( 0 + p ^ ) + 4 ) , ^ c o s B +
9 . ^ . 1 I Am I 1 '^ z Am ' *^
1 1 1 Tm
loaded. For the observation of variable phenomena this distinction can be neglected, if linearity is assumed.
In this case the angle P will be used to express the effect of the pitch. Now it follows for the lift force on the profile:
d l = Z n p c ^
m=l c o s p Am (Dm
2 «
Sin i m ( e + p — - ) + a r g ( S(^ ) +
°° Q r
+ <iJ, >cosp d r + I n p c ^— m=i
+ ar9 ( S j ^ J + . i ; ^ ^ | s l n p cos p
dr
Tm (Jüm
;in I m ( 0 + p ^ ) +
(1.01)
whereS Q, is the reduced frequency dependent part of the transfer c h a r a c t e r -istic of a profile with sinusoidal inflow as derived by von KarmSn 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 ) :
dTz = d l r sin p d F^ = d I cos p 2n dTx = - d l r cos p sin ( 6 + p - ^ ) dF = dl sin p sin ( 6 + p -^-S ) dTy = - d l r cos p c o s O + p ^-S ) dFy = d l sin p c o s ( e + p -^-^ ) \ > (1-02) /
T i _ n p é ^ Tx = - l n p é l
'-l^^'L
T , = - i R p 9 r Fy= 1 n p 9 ^ ' " 0 • " ' ' .-1 " ,"'H^J-h*JI * l " (e*PT)^V^i*+h •'"I^J-iVl.J ''"l-" (9*P7)*V*"H^
" ' - P N ^ H \ J i [-j(-*»PThv*"j- »{M(«-PT)^v«.l]* + l"''^fl=a.,l-hTj I h(-';p*pr)^^^'^^ - ««{(-IXB.P^!) *V^}1-'
I 0 '- -' R R
" '•''PKI-hAj|p'-(("-')(«-Pr)'*'>.^*".j- =«(M(e-PT)**»»,**"}|* + |" ^ W - I ^ r ^ i f-[("'-1^*PT)'*T„**„!- - { ( " - ' ^ ^ P T I - V * ™ ! " '
"' ="P|=%, H^->„l ^ [»-((-')(9-P ?)* *C*"i+ "" [H(9*PT)* V " J * + j " ' ""P|="J-|\I % [.i"{(-lJ(e*P^] •*,„* *„}+ sh((..,)(9.p^]* «.^..p^jdr
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 six expressions. 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„.
F o r a z-bladedpropeller the componentsT and F result from harmonic components of the wake field equal to multiples of the number of blades.
In the t r a n s v e r s e direction the fluctuating forces T^ and T , F^^ and Fy are built up by the components adjacent to multiples of the number of blades (m + l = 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, f our 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. N z 3 4 5 6 Harmonic components X X X X X X X X X X X X X X X X X X X X X x\ >> 3 4 5 6 0 X X X X 1 X 2 X 3 X X 4 X X X 5 X 6 =< X X 7 X 8 X X 9 X 10 X X X X 11 12 X X X 13 X X 14 X 15 X X 16 X X X 17 18 X X X X 19 X 2 0
In the transverse direction the first harmonic component (m = 1) of the wake causes an average transverse force and torque (F T F and T ),
x X y y independent of the number of blades.
In Table 1 . 2 a sui'vey is given of the sensitivity of various types of p r o -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 decreasingnumber 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. D y n a m i c e f f e c t s of a v i b r a t i n g p r o p e l l e r in a n u n -d i s t u r b e -d i n f l o w .
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 bladefrequency equals the product of revolutions per second of the propeller and the number of blades).
Forthisconsiderationwe make use of the results of the two-dimensional theory with the same assumptions and approximations as mentioned in p a r a -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
€2 = I Ez I sin ( nn 0 +(j) )
This motion is synchronous with the revolutions of the propeller and the frequency equals m times the number of revolutions per second of the p r o -peller. For the motion of a strip of a blade, perpendicular to the inflow velo-city of the strip, we obtain:
X "Cx A T , Ex A F X
H.
Ft E I Tl dTy.-dF,.r.cos(9+P^) 271, dTx\-dF,.r. sin(e+P —Figure 1.2. Propeller with axial motion
e^-The transverse velocity of the strip of the profile equals
WQ = I E^ I m 0 cos p cos (m 0 + ijj)
From [5] it follows:
d l = - n p c r ê | H ( ^ ^ | | E z | m 0 c o s ( m 0 + i | ) + ^ ^ ) d r ^^ ^^^
dT^ = - It pc m r 0 sin p H ^ ^ e ^ cos ( m 0 + ()j + ip ) dr m Adding over the blades results in
2 , 2
d T 2 = - i t p c z m r é s i n p l H l l e ^ l cos (m 0 + ({) + ip ) dr
2 . 2
= - z j r p c r 0 m s i n p H^,^ e U cos ( m 0 +4) ) cos vp + ^m
sin (m 0 + iJj) sin ip ^ dr m
dT = - z j t p c r ê s i n p H , (Jüm m 0 cos (m 0 + ( j j ) cos ip dr +
z *.2 m
+ z ji sin p H r r i ê ^ ^ r ' " ( m 0 + i j j ) s i n ip dr
Integration over the blade leads to: R Tz = - T t p z ê ez Acr s i n p | H y ^ | c o s i p ^ dr + R TlpZ „ /• 2 I I ~ "T^T ^ z / " ' ^ ' " P r w m s i n v p ^ dr -'n
because e z = m 0 £ 2 cos ( m 0 + ijj ) * « I I
and n r i e E z S i n ( m 0 + (|j ) 2 . 2 | I
Finally we obtain with 0 = 2 n n ( n = r p s of the propeller): T R Tz „ 2 f 2 I I T— = - 2 i r p z n / c r sin p H(jjm cos ^ dr «^Z JQ I I rn (1.10) TCpZ m c r sin p '(Jüm s i n i p „ dr ^ m (1.11)
Table 1.3. Axial coefficients of the screw propeller. Tz Cz ;^ ez H Ex Fz ËZ Tz ipz T z _ "i>z Fz_ - 2 n 2 p z n f c r 2 s i n p | H u ) „ 0 0 R - 2 r t 2 p z n | c r C O S P | H I Ü „ 0 ^— c r c o s p m J ' Hlü„ U p z ^ , sin^pi m J cos pi "" R - 2 T l ^ z n j c r ^ s i n p HUm 0 R Ttpz m [ cr^sinp|Hcü„ cosipn, dr cos<p„dr costPn, dr sinipmdr cosipmdr siniPn, dr (1.10) (1.11) (1.12) (1.13) (1.14) (1.15) (1.16) (1.17) Tx Tx Fx Fx Ty Ty Fy Fy Ez Ez Ez Ez Ez Ez Ez Ez Tx Tx Fx Fx Ty Ty Fy Fy *Pz <Pz * z 'Pz * z * z * z * z (1.18) (1.19)
In a similar way we can derive the axial propeller coefficients as listed in Table 1.3.
For the t r a n s v e r s e forces and moments we obtain from (1.02) and(l. 09):
2 . 2
dT^ = + jtpcnn r 9 cos p H ^ ^ e z C o s ( m 0 + tj; + ip ) s i n ( 6 + p ^ i ) d r
Adding the corresponding blade strips we obtain:
dTj^ = + l j t p c m r 0 cos p H y ^ II e ^ I Z Fsin J (m + D B + p — + i j ; + ip 1 +
- s i n / ( m - 1 ) 0 - p . ^ + ( j j + ( p ^ | d r
The part under the addition sign equals zero so that 1 ^ = 0 In an analogous way we can derive:
z "^z Tx _ Tx Fy Fy ^ Ty Ty Fy y _ r,
^z ^ z ^z '^'z "Pz 'f'z ^ z ^ z
See also Table 1.3.
1 . 1 . 3 . 2 . Transverse motions.
In transverse direction we assume a vibrational motion:
^ X ~ E x S i n ( m 0 + (j;)
The t r a n s v e r s e position of the strip of the blade equals:
| e ^ | s i n ( m 0 + ( j j ) s i n ( 0 + p - 2 - 5 ) s i n p
The transverse velocity becomes:
Table 1.4. Transverse coefficients of the screw propeller. Fx z } r siifBrm+ll 1 m - l l cos<j>„.i « " « C m * ! * — | H u „ . , | « n * „ Fx I n p f siri'prm+ll I m - l Tx ^ 2 ? J [""+11 I ""-Ij I - : - " + - T i p n I c r s i n B ^ H ^ cos<p_.,+ H^, kas<p„ . Ex 2 Q L m I " * ' ' m I ""m-l I ^"<-Tx z n p ? f m + l i m - 1 | — = + - — \ crsinB H u s i n i p „ j + Hoj sinip.^,
Fy z J ? s i n p r m + 1 | ~ = n p n \ c r — - H u sinip Ex 2 0 cospL m I "'m+ll i Fy z n p '? sin'prm+1| m - 1 | Ty z J R [mt-l I Ex 2 J L m I Ty_ z itp ? 2 r""*^ I ""^1 I
:: V c r s i n f l H(u cosiB-., H[o cosU)_ , m - 1 | , H(o sin<p m+1 m I m-1 I ^m dr (1.2 0 ) dr dr dr dr dr Fz Fz Tz Tz - - . - = - - - - 0 ^x ^x ^< ^
w.
sin p ( 1 . 2 1 ) dr (1.2 2 ) (1.2 3 ) ( 1 . 2 i ) (1.2 5 ) (1.2 6 ) (1.2 7 ) ( 1 . 2 6 ) e sin (m 6 + ( | j ) c o s ( © + p - ^ ) + + m 0 cos ( m 0 + ( i j ) s i n { e + p ^ - 2 1 ) |The lift force of the strip under consideration equals:
dl
f p c ^ ^
:os p 0 sin p'((JÜ m ) + 1 (m + 1)sinJ{m + 1)0+p-i-5 +
+ 'J'+'Prn4-1 f ~ ^^(4n - 1 ) ( m - 1 ) s i n | ( m - 1 ) 0 - p | ^ +ili +^p^_., !• With formula (1.02) and with the assumptions of paragraph 1.1.1 , the formulation ofthe t r a n s v e r s e coefficients as given in Table 1.4 can be made.
For the case of a rotational transverse vibration:
= 1 4 » cr'sinp ' ^ | H „ „ | C ( » * „ ^ , + ' ^ |HU^_^ |co«p„,_, 0 R z Tip , f'"*!! . ' " - 1 | I = _ _ c r ' s i n p | - ^ | H o ^ , | s , n ^ , „ . , + _ | H , , ^ ^ | s i n « „ . , ( R 0 R I T Ï P r J «p"-*-!.. . . " i - i i , . I . 0 R z 2 2 „In>*li.. , . f n - l = J- n pn cr S ' - ' P i y |Hu^„,,|sin*,„., " - ^ | H u v „ . , | » " * , n - l 0 0 m-1 lï-0 ' ^ - . - ' ^ " l - ' ^ J K ™ . , ^ " * , i z ttpn J 1 r 0 i ï p L ^ r i H u , , k + dr+
-^-^tp3K
z Itp'Si
H<D_ z Tipn 1 1 r ^^+2^=^c-Sp[l"<%» 0 (1.311 (1.321 H » „ _ , | c o s * „ , ] d r - J T t V - j . r ' i ^ p [ j S i l | C „ „ „ | . i n , „ , , - = i J - | c „ „ . , | . i n , „ . ] d r (,.2S 0 R I I • 1 . . z Ttp f , s i n f i r m + l , I m - 1 , , "n'P^,*|Ha,„.,|s,n^^,J*+,-S - "^^ S ^ ["ST |'='-™,1 | « » * - . i " " S T K - r l "«"m-ijo^ II * 1 R ='»*™», H""m-.h"*».-.]'''+ i «'P" \ " ^ h™.1 H"*".»1 - nS^ l^-^r^l h'"*-. ]•"•• 0 R »-*™.,*|"<^.,|^"'P^,]*-,-5 ^ - ^ [ v l'='"™.1 1'="*-' - ^ I'^'-m-l h°'*-l]*-0 R 0 R= « * n , . l - | H u „ . , | « > » f t „ j d r + ï ! ^ fcV !i!!2 Ü ; ! ! Co, Isintp^, t — | C u ^ _ , | B n ^ ) „ _ , dr (1.34)
I I lb m J cos|3 m I m i l 0 R '=''"Pn„l-|Hü(„.,|5""Pm-i dr— ^ n ' p o . ^ i ' l ï ^ l C u ^ J c o s i p ^ ^ + Ï Ï ^ I C u ^ J c o j * ^ . , dr (1.35) 0 R c<»>(W,-|Hoü„_,|c<»>p„^,dr- i ï f ' ? ' ^ ^ | C ü , „ „ | « n * „ „ , . ^ ' ^ | C u ^ , | » n « „ . , j d r (1.36) Fz Tr Tz •hi •fm 'fx (1.37) - J
we have to distinguish three contributions to the lift force of the s t r i p , i . e . : a) Transverse velocity of the profile
b) The positional deviation of the angle P c) 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 ^^ (vp j^is a small angle) the in-flow ofthe propeller can be resolved into a component in the z-direction and one inthey-direction, the latter being considered as a disturbance ofthe in-flow Vrp for the case m = 1 (see 1.1.2).
This disturbance gives rise to the transverse forces F,^, T^^, Fy,andT (see Table 1.6).
Table 1.6. Transverse coefficients of the screw propeller.
R Fx 3 2 f 2 Sin 61 I — = 2 z T i p n \ c r — r Pü)=1 sinip, dr (1.38) VX Q K R 2 Tx 3 j f 3 sin 31 I — = -2zn:^pri'Jcr _ - | S i , ^ i | s i n , p , dr (1.39) " 0 R , Fy 3 2f 2 sin G, . — = - 2 z Tt pn cr^ —^ \S^ü=^ cosm dr (1.40) ip ) cos^pl " ' - ' I ^1 " 0 R 2 Ty 3 2r 3 sin S I , — = 2 z n p n cr S ( J Ü - I costp, dr (1.41) 0 Fz Tz - - - - 0 (1.42) * x * x
1.2. Conclusions and remarks
Bearing in mind the assumptions described in paragraph 1.1.1 we can make the following r e m a r k s :
1 . 2 . 1 . E f f e c t s t o b e t a k e n i n t o a c c o u n t .
The hydrodynamic effects to be taken into account for a more detailed analysis a r e described by the coefficients given in Table 1.7.
Table 1.7. Propeller coefficients to be taken into consideration. Fx Fy Fz Tx Ty Tz Ex Fx Ex Fx è'x F^ ex Fi ex ex T x T y ex T y ^ y Fx ê y Fx ë y ^ y Fy ëy Tx é y T X T y éy Ty Ez Fz ez Fz ez T ^ ez Tz Ê-z ^ x Fx ' f x Fx ^ x Fx Fy * x F z >Px F y <Px T x ^ x Tx ^ x T x T l *Px Ty * x Ty ^ x Ty "Px * y Fx * y Fx ^ y Fx f y Fy ^ y Fz >Py Fy * y Tx 9 y Tx ^ y Tx *Py T y T y ^ y Ty * y vPr F z ^ z Fz * z Tz vf-z Tz * z
1.2.2. E q u a l i t i e s of t h e c o e f f i c i e n t s .
From the formulae and considerations of symmetry the following equali-ties can be derived:
b ) c ) ^ Tz ez Tz ^'z Tx ^x Fx 'Px Tx II x | > , LL 1 9 -Fz ^ z Fz ^ z f y 9y _"ry ^x Fy ^x Tx ^x Fx ^x Tx _ ex Fx . Ex Tx ^y Fx ^y Tx_ __ èy " •^x . ^y "~ 4>y Fy ^y ly éy ! y Éy Ty ^x Fy •Px Tl. ex Ex Tx >Px Fx \ Ex Fx E I Tx •^y"~ Fx _ T_x _ Ëy " Fx
ll
^ y «p-y ly ^•y ^ y Éy Ty <Px Fy * x Ty Ex Ex 1.2.3. A d d i t i o n a l c o n c l u s i o n s .1. 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.
2. The rotational position of the propeller is correlated to the phase of the exciting forces. For the dynamic properties of the propeller, however , the position is indifferent.
C H A P T E R 2 .
THE APPLIED MEASURING TECHNIQUE AND INSTRUMENTS
2.1. Improvement of the signal noise ratio
2 . 1 . 1 . G e n e r a l r e m a r k s .
The determination of a mass-exciting variable force by measuring the deflection of a spring supporting the m a s s , is restricted to exciting frequencies which are small in comparison to the natural frequency ofthe m a s s -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 ofthe exciting forces. In general it can be stated that for an increased natural frequency the accuracy ofthe measurement (dynamic response ofthe 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.
The increase in spring stiffness can be 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).
Forthe construction of the pick-up for dynamic propeller forces a com-promise has to be 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 ofthe exciting forces and the noise associated with the propeller generated signal.
Forthe design ofthe instrument the unknown forces have to be estimated. An artificial reduction of noise leads finally to an increase in the bandwidth and improves the quality of the measuring system.
A special technique 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 . A p p l i e d t e c h n i q u e of n o i s e r e d u c t i o n .
In paragraph 1.1. 2 is pointed out that the signals resulting from the sta-tionary ship s wake field have a periodic c h a r a c t e r . The frequencies of the signals to be recorded are multiples of the propeller rpm and are syn-chronous with propeller rotation.
If the 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 o f t h e s a m e o r d e r of magnitude, other techniques utilizing the advanced know-ledge ofthe signal frequencies can be applied, as are described in [6] [7] [8]. The applied technique is a special tjT^e 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
4 j ( x ) = l ' m - - ^ / fi ( t ) . f 2 ( t-vx) d t (2.01) T - o o T J^
Forthe casef2( t -I-T )equals a periodic impulse with periodicity equal to that of f. ( t ) (the unknown signal) the multiplication and integration can be carried out by periodically connecting during a short time the signal f.,(t) to an integrating circuit.
The switch is controlled by an impulse which can be delayed with e a s e . The operation is, in fact, based on a synchronous sampling technique 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 X (the delay time) will be equal to the pattern of the input signal to a base t (the actual time).
The gain in the signal noise ratio will be analyzed in the next paragraph.
2 . 1 . 3 . A n a l y s i s of t h e p e r i o d i c s a m p l i n g s y s t e m .
A block diagram ofthe 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.
>
Ptriodic command to the switch.
Figure 2 . 2 . Illustration ofthe periodic sampling system.
The analysis can be made in the following way [9] [10] [11] [12]. The block diagram in Fig. 2 . 2 . can be converted into the diagram in Fig. 2.3..
V *
The transfer function —!^,giving information on the value of the signals at V j
the moment of sampling only, equals:
- Al_
Vu 1 - e Re (2.02) V i * (T = period time) At R e ^ - i ( j ü T 1 - e • ehold circuit
Figure 2 . 3 . Block diagram of the periodic sampling system.
We a r e , however, interested in the ratio and must take into accoimt
Vi
the transfer function:
- l ( j ü T
V * u
( 2 . 0 3 )
I (JÜ
V j *
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 (iJc
spectrum with frequencies ranging from 0 through—r^(see Fig. 2.4 ), the output signal of the system has a power spectrum according to the transfer
Vu function of the system -,
V j » This is indicated in Fig. 2 . 4 .
The ratio between the output and input noise power equals:
1 - e - i(jL)T d tü 1 - ( 1 - e ) e i (JÜT ICÜ Tt T ( 2 . 0 4 )
•/// y/A '//, '//A V/. •///, ^ ////,/. u 1 « ... ï « 1 2 a , , 1 S ü J , 2 ( D |
Figure 2 . 4 . Input and output spectrum of periodic sampling system.
Thus the improvement of the signal noise power ratio equals:
1 - e - itUT
1 - ( 1 - E ) e
and when ip = Cü T holds:
0 i _ 2 E^ IU)T I to d tü 1 - cos (p Jt ^ ^=— dip 2 (1 - E ) - H e I - 2 ( 1 - e ) cos ip 2 E ' 1 - cosip I t <^ 2 ( 1 - e ) + £'
T ^ ip'[l-Bcosip]
dipwhere:
B= 2 M - E ) . 2 ( l - e ) + e2
The solution of the expression:
CD f ( 1 - cosip ) dip J ip^(1 - B cosip) 1 2 •}- CD
J
z^d
- oo - cos z ) d z - B c o s z ) where: 0 < B < 1and z is a complex variable, equals:
„ { 2 ( 1 - E ) . E ^ ) 2 ( 2-t-e ) E
The signal noise power ratio improvement equals:
Oo! A e ^ £ ( 2 ( 1 - E ) - t - E 0 . e «'i^ • ' ^ | 2 ( 1 - E ) . E n 2 ( 2 + E)E " 2 - E
(2.05)
^ (2-1- E)E ^ +E
;
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 inputoutput relation in the frequency domain of the simplified r e -sponse results in the following ratio:
A l
1 _ e RC
H{iai)= M ^ ' • ° ' '
1 - e " RC . e - " ^ ' ^
where:
At = the time during the closure of the switch R = the parasitic resistance of the system C = capacity.
A graphical representation of H (jj^) represents the frequency response of the system.
The improvement ofthe signal noise ratio equals the ratio of the hatched areas of Fig. 2 . 6 .
input
output
At I At Lit At _ At
T.O 2T 3T 4T
a) Response of sampling system to a unit step function.
input
h(
h2
hi output
T . O 2T 3T 4T
b) Approximate response of sampling system to a unit step function. Figure 2 . 5 .
Figure 2 . 6 . Power spectrum of input signal (white noise) and output signal (ap-proximated) .
Now:
fl?
^f
""L
T f -. — /^ J
0 W, • ( l O O ) 2 do) = At 1 - e RC A t 1 + e RC 2 - e At RC ^ T (2.07)A rough approximation can be obtained by making the output signal as given in Fig. 2.5 equal to:
Vo = 1 - e " ^ A t = i _ e T f where X f is a fictive time constant.
The system is approximated by a simple first order RC filter. The sig-nal noise power ratio equals:
„ 2
OQ E
- 2 = 7 (2.08)
For the case E = 0 . 0 1 , a practical value , we obtain: according to (2. 05): =0.00490
according to(2. 07): j ^ = 0.005025
according to (2. 08): y = 0 . 0 0 5 .
It can be concluded that the approximations are sufficiently accurate.
2 . 1 . 4 . P r a c t i c a l r e a l i z a t i o n of t h e s y s t e m .
The switch is realized by means of a bridge of 4 silicon diodes giving a resistance in the conducting condition of 500 fi and in the non-conducting con-dition of 500 Mfi.
In combination with high quality condensers, the detailed response ofthe system to a imit 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 ofthe system is a dc signal, representing the
Bo+ ( B s - B „ ) ( 1 - . ^s ) B j = ( 1 - 3 . 1 0 * ^ ) Bn - 0 999 B(, - 0 999 1
1
1 " - ^ r conditionFigure 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:
4) = 2 Tt J 1 - T signal
T sampling
The relative phase shift ip and amplitude ratio between input and out-put signal is given in Fig. 2 . 8 , as a function of if with parameter:
_ At_
1 RC E = 1 - e
The required improvement of the signal noise ratio determines the value of e . The shift velocity ofthe sampling point expressed by (j) is determined from Fig. 2 . 8 , by an admissible amplitude reduction and phase shift ofthe highest harmonic component of practical interest.
10 20 - 40 - SO
7 -
70 80 - 90 _100 ^p
\ ^ \ \ \ \V
0.00 ^ \ \ • ^ • \ a ^ ^\ J
0.01 ^ •-.^ \ —. ^s.^ ao5 =1: — • " = ^--, h - . , 1 — —-• ^ --—. • J ._^ • ' ^^ _ | • e - 0 5 - 0 . 7 OS - 0 . 4 ro.2 4 5 6 7 •It in d e g r e e s 1.0 0 9 0 7 0 6 0 5 0 4 0.3 n 7 0.1r
\ \s
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i \ \ \ \s
\ \ \v.,
N \ \ , ^X
\ 0.01 0 0 0 ! ss
\ —N
\ \ \s ,
N
> ^ \ N ' X = = • ~ ~ ~ -^ • ^ •V — ' • •«^ X ^ , — ' " ^ " ^ ~" ^ N ^ E v ^ • 0.0^ — 0.4 0.3 •0.2 .0.1 1.0 0.7 0.5 3 4 5 6 7 4J in d e g r e e sFigure 2 . 8 . Selection chart for the adjustment ofthe periodic sampling sys-t e m .
2.2. .Axial dynamic propeller properlies
2 . 2 . 1 . T h e c o n s t r u c t i o n of t h e p r o p e l l e r e x c i t e r .
As is pointed out in paragraph 1.1.3, the hydrodynamic properties ofthe 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 a check on the theoretically analyzed approximate values. The instrument for the determination of the coefficients must be able to superimpose on the static forward velocity and rotation sinusoidal 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-1 W a t e r t i g h t protecting cover 6 F l y w h e e l
2 Sliprings 7 Torque and t h r u s t e x c i t i n g units 3 Interchangeable measuring springs 8 Air Lubricated bearings 4 A x i a l and torsional undamped a c c e l e r o m e t e r 9 Torque and t h r u s t picl<-up 5 A x i a l and t o r s i o n a l d i s p l a c e m e n t p i c k - u p 10 P r o p e l l e r
Torque pick up Thrust pick up
Torsional displacement pick up
Torsional occelerometer Axial displacement pick up
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 a r e 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 r e v e r s e . With torsional and axial displace-ment pick-ups, the vibratory motions of the propeller relative to the flywheel can be determined. The analysis of this system can be carried out a c -cording to the schematical representation in Fig. 2.10.
The flywheel is assumed to be an ideal immovable support for the springs. Due tothe large mass and moment of inertia the e r r o r s 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 a s sumptions the block diagram of Fig. 2.11 can be made. The instrument p e r -mits the detection of the eight propeller coefficients operating in the zdirection as mentioned in paragraph 1 . 1 . 3 . 1 . The exciting inputs ofthe s y s -tem are F ' and j ' . The outputs of interest a r e the propeller motions e^ and ip^ and the output of the torque and thrust pick-up indicated in the block diagram (F ^ a n d T °). For these outputs is valid, according to Fig. 2.11:
F ° = ( M., + Mz-f 1 ^ I ^ z K F ) ë , + Fz éz Êz + Fz ^?z •P^* * z ^P^ (2.09) T ^ { I i * I : Tz >P, K T ) ^ ; + ^ Tz ^ z ^ z + Tz ëz ëz + Tz éz E z (2.10)
hl the c a s e t h a t the shaft is sufficiently stiff in t o r s i o n a l d i r e c t i o n (C™ a s high a s p o s s i b l e ) the output ip will be negligible a s well a s t h e s i g n a l s ip^ and ip^ .
Then the e q u a t i o n s ( 2 . 09)and (2.10) simplify to: F ; = ( N Tz° = 1 + Tz ëz Mz + ëz + Fz ez Tz ez Kp ) e ^ + (2.11) (2.12)
F o r o p e r a t i o n in a i r we find the additional equation:
F ° = { M i -h M^ - Kp ) ë ^ (2.13)
a s s u m i n g t h a t t h e d e n s i t y of a i r is n e g l i g i b l e in c o m p a r i s o n to that of w a t e r . This e x p r e s s i o n i s m a d e equal to z e r o by adjusting t h e v a r i a b l e f a c t o r Kp (and K j ) . T h i s m e a n s t h a t t h e e q u a t i o n s 2 . 1 1 and 2 . 1 2 r e d u c e t o :
! z
Ez
(2.14)
(2.15) With the m e a s u r e m e n t of e J, and F 2 and T ° in w a t e r t h e e q u a t i o n s (2.14)
(2.15) c a n be solved and the coefficients and
can be d e t e r m i n e d .
When a s i n u s o i d a l e x c i t a t i o n is a p p l i e d , the axial e x c i t a t i o n can be amjDli-fied by tuning t h e axial s y s t e m with a s u i t a b l e value of Cp . Tliis value h a s to be d e t e r m i n e d e x p e r i m e n t a l l y for e a c h individual condition of m e a s u r e m e n t .
Analogously the coefficients -rr-
Iz
'P'z
' • Fz ^ z
and Fz
^ z
can be d e t e r m i n e d by a stiff s u p p o r t in longitudinal d i r e c t i o n (Cp) and a tuned s y s t e m in t o r q u e d i r e c t i o n ( C j ) . Fi Exciting unit Cz F i g u r e 2 . 1 2 . S c h e m a t i c r e p r e s e n t a t i o n of the p r o p e l l e r e x c i t e r 2 . 2 . 2 . A p p r o x i m a t i o n s a n d l i m i t a t i o n s of t h e s y s t e m The s y s t e m can be r e p r e s e n t e d by the m a s s - s p r i n g s y s t e m given in F i g . 2 . 1 2 .
F i - ( X3 - X J ) C2 = M3 S'^ X 3 (2.16)
F^ + ( X 3 - Xg ) C 2 + ( X l - X J ) C i = M j S ^ X J ( 2 . 1 7 )
S X l ( 2 . 1 8 ) ( X j - X T ) C I = ( M , + M2 + S^ X ,
-t-It is assumed that X = X and Xg = 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
2<J.
is determined by 0.05 and ^ l - ^ a 0.05. The amplification of the excitation by tuning is also indicated.
+1 n n " v,o-^. i i n ' ,.10-' vii'. ,.,5». v i 5 ' M0-'. nn'. 1 — V =— ^ = = = ^ =NN :: ^= ^= = ^= — F ~ =NN -~ — = — — = ^ 1 — y := = = ^ -M h Rana* or OMrotion r ^= — = -— = ^ ^ — = H = = ^ = _
M
=W — _ X S X ^ 1 " = p t ^ -.-1 4 '^1 =
, / ^—^ ^ = = •s • • ^= ~ = 3d
E \j
c 0 • 0 . 0 ^ ^ N^ V -1 'rl --j -.:: :..=, / 'Jn ' = -/ 1 «2 1 = ; = ^ ^ ' ^ = ^ \ = -= r-=EÊ 7 7 EEEEE il -Fi _ ^f^ = % = = ..J -. i n 1 ^ ^ ^ttH
\ ==— x 11
'1^ttH
1 /.11,
1 Frequency 10 c.p-s 1002.3, The propeller exeited vibratory forces 2 . 3 . 1 . T h e c o n s t r u c t i o n of t h e me a s u r i n g s y s t e m f o r t h e s i X c o mp o n e n t s of t h e p r o p e l l e r . T h e m e c h a n i c a l p r o p e r t i e s of t h e p i c k - u p . 20 25 A r r a n g e m e n t of the s t r a i n - gauges
Figure 2.14. Six component balance for the measurement of propeller fluc-tuating forces.
Figure 2 . 1 5 . 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 ofthe 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 the strain-gauges are glued is given in Fig. 2.14. Fig. 2.15 (photo) gives an impression of the pick-up under construction.
2 . 3 . 2 . A n a l y s i s of t h e d y n a m i c b e h a v i o u r in a x i a l d i r e c -t i o n w i -t h p r o p e l l e r m o d e l .
As can be seen the transfer functions depend on the geometry of the p r o -peller i . e . among others on the djmamic and mechanical pro-peller properties.
Fortunately it can be shown that the effects of the differences of these p r o -perties on the transfer characteristics are negligible for a large number of propellers normally applied to model tests of surface ships.
The analysis ofthe characteristics of the measuring system will be c a r -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. Assumingthattheeffect of the mass and moment of iner-tia of the propeller shaft on the measurement is negligible the behaviour in axial direction can be described by two coupled equations of motion, as given on the next page.
"'
^ r
^--^—\_ ,/XX
''^^^^^^y^y r
j-'—\u^-—\_r^—^^ \ 1 -'' *' r
L
( . 1 Fz T" - i '*' r
L
i Tz FT Tz"1 ''
J
"1 *'
J
1 c 11 ' 1
1 r I , Ez 1 ' J " f Thrust systtm | Torqut s y s t t m 1 r , 1 . •, 1 ' 1 1 r. 1E M -t- E ^Z Z "^Z ^ z Iz + ^z ^ 7 -t- E . + ^ , [_z
éz
^ 7 + * , + E. z z Fz «Pz Tz ^z '^z ^z Fz ^ z Tz éz + £2 Cz (2.19) + ^2 C^= Tz (2.20)The stiffnesses ofthe shaft are C 2 and C z in the longitudinal and torsional direction respectively.
The torsional and longitudinal motions are coupled by the working of the propeller. These hydrodynamic effects a r e described in chapter 1 and r e -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).
Iz +
* z
Figure 2.18. Simplified diagram of the behaviour in z-direction of an elas-tically supported propeller.
Assumingthut the differential equations are linear with constant coeffi-cients an acceptable 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 fimctions: (2.21) 1 — Hi Ho HQ Hi ^ z Tz Ez Tz ^ . F z 1 - H T H I 1 - H i H i 1 - H T H 2 H2 H2 Hz H2 H2 H3 »u H3 H3 H3 H4 H4 H4 (2.22) (2.23) (2.24)
For a good operation ofthe system the first two ratios are of importance , the last two ratios, however, must be negligible. As H 3 and H^ cannot be in-fluenced by the construction of the pick-up, but depend on the characteristics ofthe propeller, it is necessary to make the product Hi . H2 as small as pos-sible. 0 0 This hence and means H ^ H2 1 - Hi H2 i ^ « Hi -•^z ^z ^ ^ H2 -1=» r=» H3 H, -^ 0 - ^ 0
For the measurement of the exciting forces, not to be influenced by the coupling terms of the propeller, the transfer functions Hi and H 2 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 r e q u i r e -ment can be made (see paragraph 2,1).
The final stiffness of the pick-up is determined experimentally by ex-citing the complete system in a i r .
The following data are of importance:
_ ^ LT n
dar-a) Mass ofthe propeller model 1^^= 4-10 k g s e c '
b) Natural frequency in air in thrust direction 1000 fc.p.s"!
c) Stiffness of the support in thrust direction C^ = 1 . 6 - 10 k g m
d) Moment of inertia ofthe propeller F^ = 5 . 8 - 1 0 k g m sec
3
e) Natural frequency in air in torque direction 6 00 l a p s I
f) Stiffness of the support in torque direction C^ = 83 5 k g m r a d
From the formulae given in chapter 2 we can make an estimate of the properties of the vibrating propeller.
These mod'el properties in axial direction can be expressed by 6 coeffi-cients, given in Table 2.1 and depend onthe rpm ofthe propeller model.
The given values are calculated for full power absorption corresponding to 10 r e v s , per second.
Four expressions can be distinguished. The transfer characteristics
Ez Fz ' ^z Tz ' c 1 ^ 1 and ^ z Fz
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 e r r o r of 10 per cent.
Table 2 . 1 . Hydrodynamic properties of the vibrating propeller. coefficient Hydrodynamic moment of inertia Hydrodynamic torsional damping Hydrudynamic m a s s Hydrodynamic axial damping Acceleration coupling Velocity coupling symbol Tz Tz ^ z Fz Fz
Ez
Fz Fz = Tz = Tz numerical value 4.9 • IQ-'Ikgm sec" 1-10"' [kgm sec 5 • 10-= kg s e c ' m 12.5 kg sec m 1.6 • 10-' kg sec' 0.35 kg sec 2 . 3 . 3 . A n a l y s i s of t h e d y n a m i c b e h a v i o u r i n t r a n s v e r s e d i r e c t i o n w i t h p r o p e l l e r m o d e l .In order to obtain an insight into the t r a n s v e r s e behaviour of a propel-ler elastically supported by a shaft, we consider separately the different ef-fects to be taken into accoimt.
2 . 3 . 3 . 1 . Effect of shaft elasticity and gyroscopic precession.
i - i ;
-ff-X
r ^ \\1
, . Uxf
4 IL.
• ' 1 — ••'u
Xf-^
Xn
:: : ^"R °°
— J
- -4-' J -4-' -4-' = — : T•
Xtt
_ — T •1 '
"li' i 1 1 i 1 I I-ttx
=td."t^ = ^^ï-
- " ^ ^ TLl -1 =^ '~^~ = =^ = N 1 — L \N\
\ • \~"\' <\\ = ^ ^^ , '— ï J. •X-j
XX
^ = 3 — 1 —#u
,4?
c
/m—gË
___:: 1 TT1
wrm
1
^ ^ — / . — 1 1i ^^
V -z==:r::::
\ 1V
-\:::;;—5EEE:::;
1 10 Frequency 100 c.p.s. 1000 10000Figure 2.19. Transfer functions of the piek-up in axial direction.
With hydrodynamic propeller coef-ficients .
Without hydrodynamic propeller coefficients.
Fy - ^ X • ^X F y - Cy • Ey F x = C x y ( - ^ y ) Fy = C x y . ^ ^ Tx = C j • <p^ Ty = C x y ( - E j Tx = Cxy • Ey Ty = Cy • iPy
With these shaft stiffnesses the equations of motion of the non-rota-ting disk-shaft system can be given [14] [15].
The differential equations a r e :
M Ë , = F x - ^' ^'^ ' € . - ^ " y '^^ ^'^ -^v (2.25) 2 F T 2 F T ' ^ x y - ^ x • ^x Cxy-Cx Cx I y ^ p y = T y _ L _ __ . £ ^ _ _ _ ^ " ^ - ^ y (2.26) - x y - C x Cx C x y - C x C X MËy = Fy - / ^ \ -Ey . '^^ ; ';--Px (2.27) C x y - C y Cy C x y - C y Cy
Ix^p-x = T^ + Si^VL^LBi.. E ü L ^ i ^^ (2.28)
2 F T 2 F T Cxy-Cy Cy C x y - C y CyFor the rotating propeller, gyroscopic effects have to be taken into a c -count.
The precession moments Tx and Ty equal respectively:
Tx = - é Iz ^Py
and Ty = 0 Iz ip
The block diagram of the propeller, rotating on the shaft, is given in Fig. 2.20. Inthis figure, however, the hydrodjoiamic effects are not con-sidered.
-©—G>
U 1 "
c S . c , / - : x / - c ^ c ;-l/h-
—/xJ
\ -®—© motions \ \ " 1- / h
Cy . Cxy n 2 n F pT C x y - C y . C y— v ^
, i -^ r Horizontal motions /Figure 2.20. Block diagram of t r a n s v e r s e behaviour of a rotating mass on a shaft.
^ ly • S ' Ex Cx 37 T7 _ i I x S ' 1 M - S '
•i
Figure 2 . 2 1 . Block diagram illustrating the hydrodynamic effects of a pro-peller vibrating in transverse direction.
propellers). Fx = T » . Tx = Fy = tx f 1 \FX] t [FITI CX - C - ' - 1
'Xi="*l^)"
'Xi="l=}'-• y I liJ [id la '»-'^' '=' J (-irrrri , rrri . rnn \ •x
I liJ iijl c,;-c;.c; Liii / \ y [iJ KJ c,;.c;.c;/ =»
'-(l=''IX='
w fiïn 1 (Til 1 l [ y ih] i:,;-c;.c; ƒ 'X-'-|^"*|=*^^}-CD
2 . 3 . 3 . 2 . Effect of the hydrodynamic properties of the propeller.
As is reviewed in Table 1.2, 40 coefficientsaffectthe vibratory behaviour in athwartship direction. The operation of these effects is illustrated in Fig. 2 . 2 1 . This figure shows that each motion of the rotating propeller introduces coupling effects in all other possible directions. The equations of motions a r e given in Table 2 . 2 . The following shaft elasticities are determined by a theore-tical approximation from the dimensions of the pick-up:
' X Ex I>L ^ x
ll
^x 1 = C^ = Cy = 0.80 10i^=
c:
Ty T 'y Fx ^ . Cy = 0.33 . 10 = 0.56 . 10 m k g m r a d[-]
Table 2 . 3 . Propeller model coefficients.
t o r c t \ Fx Fy F l Tx T» Tz Cx f<-0.11310*' fe?l tx L"" J p.-0.725.10""' ^32^ .Q._a92o.io' r ^ Ex L"" J |i:.+0105.1ö' Cx ks»?| Iï.+0,l921(f [kgsJ =^=+091810' kgse£ ^«•O-^O-IO" p s s " ! %-o.ns m |kgs.l] Ey ^...0,920 1 0 - 1 ^ ] | = - 0 . 1 0 S . 1 0 " ^ [ ^ Q=-o.ii3.iiJ' [^aüfl Cy L " ' J 5=+0.725 lO"' Cy kjsiJj _ m J ^=-0.120 icf [l<Qs«c~| p!.»ai75.1ö' kgsK. fy L J
l^.+o.i92 icf [i<gs«n 5^.+0.918 1Ö' Us«/~| Cy L -1 Cl f ' . . 0 . 1 2 5 l S ' M El L "^ J ^.*0.5 .10 M l 8z L m J ^.»0.350 lo'fkgsecl | . t 0 . 1 6 3 . 1 ^ [ k o » [ | •« 5; =-0.278 10*' •x /?od| ? ! . • 0.137-lo'lkgs.rl j^.tO.SSS 10"* LgsecJ a=.0.15S.,0*'[^,] fü.+o.iag-io" [kgMtl .B.t0.l5a.1ö' kgse^ •x L J 7."°"«""'fe] !ï=-0.341 iS Cgms«3 Ü.=-0139 « kgms4 •x L J !£.,0.272 « ' f e l »x L-^'^J ^=-0.289 lö' L m s J ÏÏ.-0.387 lö'|^ms«c »y &=.01S8.10*[^^^ £!—0.13910' [kgs«r| T^s-O.ISBIÖ* kgs*^ fy L J ^"0™-»*'M ^'.4.0.13710" [kgjtcl *, L J .Q!=t0.696-1ö' kgsei? i*y L J !ï.=-0 272.lrfrK87"l fy L -'^''J Ii=-0.299l5'Lms«l lU*0 387 10' 1^ k^m» J a,.„,S98.10'[X] S„0.3<1 lö' Uttis.^ IÏ.-0.139 10 t»ns.ï] <f! i;.+0.350.1oLs«:n —.*0,163 10 kgse? •z .:-M+1 0 • lölkgmsec. ^.+0.49 - r a N r n J
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.
1 -•
The moment of inertia I x = T y = — ^ z = 2.9 . 10
Mass of the propeller M = A • 1Ö
From this information the coefficients of the four simultaneous equations can be 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 func-tions given in Figs. 2.22 and 2 . 2 3 . From these funcfunc-tions can be concluded thatthe measurement of athwartships propeller excited moments and forces can be realized up to 200 cps with an e r r o r of 10 per cent.
k g m sec 2
kg sec m
I I 1 , j 1 — ^ .—1 —
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