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PROPELLERLROTOR WHIRL FiXITER: // A STATE-OF-THE-ART REXIEh' By Wilmer H. Reed I11 NASA Langley Research Center Langley Station, Hampton, Va., U.S.A.
Presented at t h e Symposium on t h e Noise and Loading Actions on Helicopter V/STOL Aircraft and Ground Effect Machines GPO PRICE $ CFSTI PRICE(S) $
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ff 653 July 65 PROPELIER-R(TrOR WHIRL FLUTTER: A STATE-OF-THE-ART REVIEW
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>i y z h e r E. Reed III* NASA Langley Research Center Langley Station, Hampton, Va., U.S.A.
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- ABSTRACT
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The basic phenomenon of propeller whirl i n s t a b i l i t y i n connection with conventional and V/STOL a i r c r a f t i s described; theoretical and experimental investigations of the problem from t h e time it f i r s t became of concern on con- temporary turboprop a i r c r a f t t o t h e present a r e summarized; and some considera- t i o n i s given t o possible future configurations having hinged o r highly flex- i b l e propeller-rotors.
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INTRODUCTION -
Although t h e phenomenon known as propeller whirl f l u t t e r - a dynamic
i n s t a b i l i t y t h a t can occur i n a flexibly mounted a i r c r a f t engine-propeller combination - w a s discovered analytically by Taylor and Browne i n 1938 (ref. l ) , it w a s not u n t i l its "rediscovery" i n 1960 t h a t it became a problem of practi- ~ ,-" c a l concern. --- Following the loss of two turboprop a i r c r a f t i n fatal accidents it w a s established i n wind-tunnel investigations (ref. 2) t h a t propeller whirl f l u t t e r could have occurred if the nacelle stiffness was severely reduced, say by a s t r u c t u r a l f a i l u r e . I n t h e undamaged condition t h e a i r c r a f t had an adequate margin of s a f e t y from whirl f l u t t e r . I n addition t o t h i s wind-tunnel
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Assistant Head, Aeroelasticity Branch, Dynamic Loads Division.
~-4536 investigation f o r a specific configuration some generalized trend studies were a l s o conducted at NASA-Langley i n order t o identify and study the basic param- eters involved i n propeller whirl f l u t t e r (refs. 3 through 6).
As a r e s u l t of -these experiences on a turboprop a i r c r a f t , and the f a c t t h a t VTOL configurations are l i k e l y t o have unconventional propeller-rotor systems, whirl f l u t t e r has now become a design consideration on new propeller- driven a i r c r a f t . These considerations a r e reflected i n recent amendments t o U . S . Civil A i r Regulations ( r e f . 7) which require t h a t whirl f l u t t e r be included as a part of t h e dynamic evaluation of transport aircraf't, and that no f l u t t e r s h a l l occur as a result of f a i l u r e of any single element of an engine mount structure.
The purpose of t h i s paper is t o review some progress that has been made i n t h e area of propeller whirl f l u t t e r since the time the phenomenon became t h e subject of intensive study i n 1960. Following a description of the basic mechanism of propeller whirl f l u t t e r , t h e paper summarizes some principal findings of generalized trend studies f o r idealized systems, and then i l l u s - t r a t e s how these r e s u l t s can be a l t e r e d by the use of propeller-rotors with hinged blades and with highly f l e x i b l e twisted blades. I n addition, the paper reviews the s t a t u s of propeller aerodynamic coefficients used f o r the prediction of whirl f l u t t e r on conventional and VTOL aircraft; discusses some e f f e c t s of wing f l e x i b i l i t y on whirl f l u t t e r ; and, finally, c i t e s an example wherein w h i r l f l u t t e r of a Specific VTOL configuration i s studied by means of an aeroelasti- c a l l y scaled wind-tunnel model.
SYMBOLS Thrust thrust coefficient, CT C propeller chord lift-curve slope C 2U nacelle viscous damping factors i n pitch and yaw directions C e , C + e hinge offset distance on flappin$-blade propeller F + i G o s c i l l a t i n g l i f t function JtV
H propeller-tip- speed rat io, -
SlR I t o t a l moment of i n e r t i a of system about pivot R
mr(r - e)dr
13 = $ LR m ( r - e ) d r
mass moment of i n e r t i a of propeller about rotation axis, Ix
i = J - - i
JtV J adxance ratio, - 3 - , K = aerodynamic moments about nacelle pivot point Q,M,,, m propeller biade mass per unit length N number of blades R propeller radius r l o c a l blade radius R
m ( r - e ) d r
rotational spring constants of nacelle S O , E + f l i g h t velocity 2 hinge offset R’ viscous-damping r e l a t i v e t o c r i t i c a l damping of engine mount system pitch and yaw angles of propeller shaft r e l a t i v e t o s t a t i c thrust axis p + i v root of characteristic equation
damping r a t i o (+ unstable, - s t a b l e )
whirl frequency r a t i o (+ f o r forward mode, - f o r backward m o d e )
air density P
n propeller rotational frequency
propeller whirl frequency u) cantilever fundamental frequency of nonrotating propeller blade 9 3
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uncoupled pitch and yaw frequencies of system with nonrotating . U J e , v propeller uncoupled wing bending frequency for r i g i d nacelle and nonrotating % propeller fiiijje;;tsl - , . : . ; . , - ::AFL~~ be~fiir,rr-tsrsicn frpmipnmr for r i si d nacelle u1 a-- -- -u and nonrotating propeller MECHANISM O F P R O P E L U R WHIRL FLUTTER I n order t o introduce the basic ingredients of propeller whirl f l u t t e r it i s convenient t o reduce t h e problem t o i t s most elementary form as w a s done i n references 3 through 6. I n figure 1 t h e sketch represents an idealized system i n which t h e power plant o r nacelle is assumed t o be restrained by a set of springs and dampers at a pivot located behind t h e propeller disk. If t h e propeller blades and the nacelle structure a r e considered t o be rigid, the dynamic behavior of t h e system can be described i n terms of 8 and which represent small angular deflections i n pitch and y a w of t h e propeller axis r e l a t i v e t o t h e s t a t i c equilibrium position. The equations of motion, also indicltie ifie i-~t-ii-e of L1*- - r o n 4 r r - . n P n r m n a i n v n l v d . U U G "UI *VU" A "I ""Y & Y . --. --- Nnt.e shown i n fiwE 1 , that, i n addition t o t h e usual dynamic forces associated with inertia, damping, and e l a s t i c properties of t h e system, gyroscopic and aerodynamic forces are introduced by t h e rotating propeller.
The dynamic behavior of t h i s system can be i l l u s t r a t e d with t h e a i d of The sketches on t h e l e f t indicate t h a t with a non- t h e sketches i n figure 2.
r o t a t i n g propeller and with aerodynamic forces neglected, natural vibrations can occur independently i n e i t h e r t h e pitch plane o r t h e yaw plane. There is However, with a rotating propeller no coupling between these two modes.
- 5 - , (center sketch) the original pitch and yaw modes no longer occur independently, but a r e coupled by gyroscopic action of the spinning propeller. The natural modes i n t h i s case are referred t o as "whirl" ( o r precession) modes i n refer- i n which the propeller hub whirls about the s t a t i c t h r u s t ence t o the manner As t h e rotational speed of the propeller increases, t h e frequency of one axis.
whirl mode increases while t h a t of the other decreases. The higher frequency mode is known as the "forward w h i r l mode" because the direction of whirl is the same as that of t h e rotating propeller. Similarly, the lower frequency mode i s i t s rotational direction i s opposite known as the "backward whirl mode" because t o t h a t of t h e propeller.
It can be shown t h a t i f the propeller blades a r e r i g i d and there a r e no aerodynamic forces on the propeller t h i s mechanical system i s always stable.
blade elements However, since whirl modes produce angle-of-attack changes on of the propeller, aerodynamic forces a r e generated, and it i s these forces t h a t provide the mechanism f o r an i n s t a b i l i t y . Thus, j u s t a s i n c l a s s i c a l wing f l u t t e r , if the forward velocity of t h e system exceeds a certain c r i t i c a l value a dynamic i n s t a b i l i t y can be encountered. This i n s t a b i l i t y f o r rigid-blade systems invariably occurs i n the backward whirl mode.
The sketches on the right-hand side of figure 2 give an example of the manner i n which the system would respond i n the backward whirl mode following a disturbance such as a gust.
When t h e airstream velocity i s l e s s than the whirl f l u t t e r velocity, Vcrit, t h e path traced by t h e propeller hub i s a When t h e s p i r a l that converges t o t h e o r i g i n a l s t a t i c equilibrium position.
f l u t t e r speed i s exceeded, however, a s m a l l disturbance w i l l r e s u l t i n a diverging s p i r a l motion of t h e hub which w i l l continue t o build up u n t i l the structure f a i l s or i t s motion becomes limited due t o nonlinearities.
- 6 - PROPELLER-ROTOR SYSTENS: RIGID AND NONRIGID BLADES The generalized studies of c l a s s i c a l propeller whirl f l u t t e r i n refer- ences 3 through 6 were r e s t r i c t e d t o r i g i d propellers. This i s a reasonable assumption f o r conventional propeller-driven a i r c r a f t ; however, V/STOL designs often incorporate f l e x i b l e and ariicuiaiied p r u p e ~ l e ~ i - ~ t ~ i - s tkt 2';e c s q r c - mises between t h e long f l e x i b l e blades of a helicopter rotor and t h e short stiff blades of an a i r c r a f t propeller. It i s therefore of i n t e r e s t t o consider t h e manner i n which whirl f l u t t e r might be altered by the use of nonrigid propeller- rotors. O f equal i n t e r e s t here i s also the question of how rotor mechanical i n s t a b i l i t y - an i n s t a b i l i t y fed by energy of the rotating rotor rather than by
the airstream - might be a l t e r e d by the inclusion of propeller-whirl-type
aerodynamic forces.
For t h i s purpose we w i l l investigate some s t a b i l i t y characteristics of the three systems shown schematically i n figure 3. These systems each consist of a four-bladed propeller-rotor mounted, f o r convenience, on an axisymmetrical nacelle ( i n which s t i f f n e s s and i n e r t i a properties are the same i n the pitch The first system t o be considered here has r i g i d blades; and yaw directions).
the second system i s l i k e the first except the bia&es are liliiged SG 8 8 ts zlbv flapping i n t h e direction normal t o the propeller disk plane; and the t h i r d is a system with a f l e x i b l e twisted propeller-rotor. The vibration modes of importance f o r each of these systems a r e indicated by the sketches I n figure 3.
Rgsumi? of Rigid-Blade Cases The b a s i c phenomenon of propeller whirl f l u t t e r f o r systems t h a t comprise a r i g i d propeller and a flexibly mounted power plant, such as t h a t i l l u s t r a t e d i s now reasonably well understood. The s t a b i l i t y characteristics i n figure 1, - 7 - of such systems have been investigated analytically over a rather broad range of parameters by Reed and Bland (ref. 3), Houbolt and Reed ( r e f . 4), and Sewall (ref. 5 ) . I n addition, wind-tunnel studies have been conducted by Bland and Bennett ( r e f . 6) t o evaluate the theoretical prediction of propeller aero- dynamic derivatives as w e l l as whirl f l u t t e r s t a b i l i t y boundaries.
A general finding of these studies i s t h a t whirl f l u t t e r is strongly dependent on three basic system parameters: the stiffness, t h e damping, and The influence of these parameters on whirl f l u t t e r i s t h e pivot location.
typically as shown i n figure 4. For example, figure 4(a) i l l u s t r a t e s the e f f e c t of the r a t i o of pitch s t i f f n e s s t o yaw s t i f f n e s s on t h e s t a b i l i t y of a system. Note i n particular t h a t the w h i r l f l u t t e r boundary i s extended along the diagonal ray So = , 5 + ; t h i s indicates that i f a system had equal pitch and yaw stiffnesses it would be more prone t o f l u t t e r than if one of t h e s t i f f - nesses was appreciably reduced. The shape as well as t h e location of t h i s curve, however, may be a l t e r e d appreciably by the amount of s t r u c t u r a l damping i n t h e system (see ref. 4 ) . The l i n e s t h a t i n t e r s e c t each end of t h i s boundary denote the stiffnesses required t o prevent s t a t i c divergence of the system.
Figure 4(b) i s presented t o i l l u s t r a t e the powerful s t a b i l i z i n g influence of mechanical damping on whirl f l u t t e r . It i s interesting t o note t h a t if s t r u c t u r a l damping were assumed t o be zero i n whirl f l u t t e r analyses, as m y be done frequently i n wing f l u t t e r analyses, t h e s t i f f n e s s required t o prevent f l u t t e r would, i n many cases, be grossly overestimated. A s has been remarked by A. L . Head i n a discussion of propeller whirl considerations on t h e XC-142, "a l i t t l e damping goes a long way."
Finally, figure 4 ( c ) shows t h a t t h e f u r t h e r t h e pivot point i s moved from This fact, incidentally, the propeller disk t h e more stable the system becomes.
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can be a t t r i b u t e d t o t h e aerodynamic damping associated with transverse veloc- The effects of t h e parameters indicated i n fig- i t i e s of t h e propeller hub.
ure 4 and other parameters, such as advance ratio, thrust, propeller rotational
etc., are investigated i n d e t a i l i n the previously frequency, air density, mentioned generalized trend studies.
Flapping Blades The e f f e c t s of flapping hinged blades on whirl f l u t t e r have been examined i n several recent studies. The most comprehensive of these is the study by 8) i n which a considerable number of parametric Richardson and Naylor ( r e f .
variations were investigated analytically and some complementary test data pre- sented f o r a low-speed wind-tunnel model. Wind-tunnel experiments of t h i s type have a l s o been conducted by E. F. Baird of Grumman Aircraft (unpublished) and by Reed and Bennett i n reference 9. I n addition, a whirl f l u t t e r analysis f o r a specific V/STOL configuration which u t i l i z e s flapping-blade propellers i s presented by Gallardo and Flannelly i n reference 10.
The r e s u l t s which follow are based on a whirl f l u t t e r analysis of the This model shown i n figure 5 model t e s t e d by Reed and Bennett i n reference 9.
consisted of a windmilling propeller mounted on a spring restrained rod which could r o t a t e i n pitch and yaw about a set o f gimbal axes behind the propeller.
The blades are attached t o t h e hub by means of pins i n a manner such t h a t they can be e i t h e r fixed r e l a t i v e t o the hub ( t h e rigid-blade case) or allowed t o The hinges a r e oriented s o f l a p about one of two possible hinge locations.
t h a t t h e blades f l a p i n t h e direction perpendicular t o the plane of the propel- l e r disk.
- 9 - The theoretical analysis used i s based on equations developed i n refer- ence 8 wherein t h e dynamics of t h e system are expressed i n terms of four degrees of freedoms: pitch and yaw of t h e propeller disk about t h e gimbal axis and cyclic flapping of the blades i r ? t h e pitch an& y2-v directions nemal t o t h e propeller plane.
For an axisymmetric system, such as t h e one t r e a t e d herein, t h e number of degrees of freedom conveniently can be' reduced from four t o two.
This simplification i s made possible by t h e use of two complex coordinates t o represent t h e c i r c u l a r whirl modes of t h e system i n place of four real coordi- nates t o represent separately t h e pitch and yaw modes involved. For t h e sake of completeness, the f i n a l form of t h e flapping-blade system c h a r a c t e r i s t i c equation, based on derivations i n reference 8, i s presented i n t h e appendix of t h i s paper. The physical parameters of t h e model t e s t e d i n reference 9 are given i n table I.
Theoretical results f o r t h e model with r i g i d blades a r e presented i n f i g - ure 6. This figure shows i n a nondimensional form t h e variation of natural frequency and damping of t h e system (obtained from t h e roots h = p + iv of t h e characteristic equation i n t h e appendix) as a function of t h e frequency r a t i o R/oe, where 51 i s t h e propeller frequency and we t h e nacelle p i t c h o r a nonrotating propeller. i s wind- yaw frequency with Since t h e propeller milling, R is proportional t o t h e airstream velocity. I n t h e p l o t s of fre- quency ratio, obtained from t h e imaginary p a r t of a root, positive values denote forward whirl modes and negative values, backward whirl modes; i n t h e p l o t s of damping ratio, obtained from t h e real p a r t s of t h e roots, a negative value indicates a stable system and a p o s i t i v e value, an unstable system.
These r e s u l t s i l l u s t r a t e some c h a r a c t e r i s t i c features of propeller whirl f l u t t e r : the i n s t a b i l i t y develops i n the backward whirl mode and there i s no
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evidence of coupling between the two whirl modes involved.
Because of t h i s l a t t e r feature, it has been observed i n reference 8 t h a t propeller whirl may be regarded as a type of single-degree-of-freedom f l u t t e r s o long as the iiide eo:- sidered i s a whirl mode. For the system shown i n figure 6 whirl f l u t t e r is
predicted when the parameter & 2.9. The corresponding w h i r l f l u t t e r fre-
--u quency i s seen t o be about O.we.
Consider next the dynamic characteristics of t h e system when the blades A plot of the type a r e f r e e t o f l a p normal t o the propeller rotation plane.
shown i n figure 6 f o r r i g i d blades i s shown i n figure 7 f o r the same model hinged blades. The hinge offset i s O . l 3 R from the propeller rotation axis.
Motion of t h e system i s now characterized by four vibration modes. I n a l l but one of these modes t h e s t a b i l i t y increases continuously with increasing speed.
The mode i n which the i n s t a b i l i t y develops is, as i n t h e case of r i g i d blades, a backward w h i r l mode; however, the f l u t t e r velocity i s about two and one-half times higher than it w a s f o r fixed blades (Q/coe = 7.5 as compared w i t h 2.9).
A l s o shown i n figure 7 a r e the natural vibration mode shapes of t h e system a t Note that there i s relatively l i t t l e blade flapping present t h e f l u t t e r speed.
i n the f l u t t e r mode which i s identified as @ i n t h e figure. (These modes
were calculated f o r t h e propeller speed corresponding t o f l u t t e r but with aero- dynamic and damping forces ignored.)
Figure 8 shows a comparison between theory and experiment f o r t h e model For t h e rigid-blade case and f o r the 13-percent hinge t e s t e d i n reference 9.
o f f s e t case f l u t t e r occurred i n the backward whirl mode both i n the theory, as However, i n t h e experiment the indicated i n figures 6 and 7, and i n the t e s t s .
model with t h e 8-percent hinge offset f l u t t e r e d i n the forward whirl mode a t a much lower velocity than did e i t h e r of the other configurations. This forward
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whirl f l u t t e r w a s not predicted by the analysis, however nonlinearities may have been involved since t h e motion was amplitude limited and could only be i n i t i a t e d when t h e disturbing force exceeded a certain threshold level. It should be noted that, Richarbon's model with flqpl.ra blades a l s o encountered forwarri m e only way t h a t t h e analysis could be m a d e t o whirl i n s t a b i l i t i e s (ref. 8).
such an i n s t a b i l i t y w a s t o introduce large phase lags between the dis- predict placements of t h e propeller axis and the associated aerodynamic forces on t h e blades. (These aerodynamic lag e f f e c t s are related t o t h e unsteady h e l i c a l w a k e behind t h e propeller.) For example, the assumption of a 3 0 ° phase lag resulted i n forward-whirl f l u t t e r f o r the configuration t e s t e d i n reference 8.
For the configuration considered i n t h i s paper, however, aerodynamic phase lags as high as 4 5 O were assumed but an i n s t a b i l i t y i n the forward whirl mode (The phase lag assumed i n the calculations f o r could never be predicted.
f i g . 6 was 15O.)
Flexible-Twisted Blades Bending deformations of a twisted propeller blade d i f f e r from those of Whereas the t h e previously discussed flapping blade i n a fundamental way.
flapping motion of a hinged blade was assumed t o be normal t o the propeller
plane, the bending motion of a f l e x i b l e blade with t w i s t has components of ais-
placement i n t h e propeller plane a s well as components normal t o t h e plane.
The relative contribution of these two components t o a bending vibration mode of a blade w i l l depend on such f a c t o r s as i t s geometric pitch angle and t h e spanwise location of the root chord (i.e., t h e hub diameter).
These in-plane motions permitted by blade f l e x i b i l i t y make possible t h e unlike propeller occurrence of another class of self-excited vibrations which, whirl f l u t t e r , i s purely mechanical i n origin. This phenomenon, popularly
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called "ground resonance," has received considerable attention i n helicopter studies (e.g., refs. 1 1 and 12) and has a l s o been recognized as a potential problem on tilt-wing V/STOL configurations ( r e f . i3).
A recent study of t h e interaction between propeller whirl f l u t t e r and mechanical i n s t a b i l i t i e s on systems having flexible-twisted propellers has been I n t h i s study, as i n conducted by Richardson, McKillop, e t al. ( r e f . 14).
reference 8 f o r hinged propellers, a considerable number of parametric varia- t i o n s were analyzed theoretically and wind-tunnel t e s t s were conducted on a simple low-speed wind-tunnel model.
The mathematical and physical models considered by Richardson consisted of an axisymmetrical nacelle, and a f l e x i b l e twisted propeller having a r i g i d hub and three o r more uniform constant-chord blades. The relevant blade bending modes which can couple with whirl f l u t t e r a r e shown t o be cyclic "pitch" and cyclic "yaw" type motions i n which t h e t i p path plane i s pitched or yawed due t o blade bending. Each of these modes has associated with it an in-plane The flexible-blade mode i l l u s t r a t e d i n figure 3, component of displacement.
It i s f o r example, is t h e "pitch" mode accompanied by in-plane bending.
coning motions of the blades do not couple with the modes remarked t h a t involved i n t h e whirl i n s t a b i l i t i e s and consequently t h i s mode w a s not included i n t h e analysis.
Some p r i n c i p a l findings of the studies i n reference 14 are presented i n figures 9 and 10 of t h i s paper. Figure 9 shows the s t a b i l i t y characteristics a condition of zero damping and of one of t h e configurations analyzed f o r There is close s i m i l a r i t y without aerodynamic forces (e.g., a s i n a vacuum).
between these p l o t s and frequency diagrams f o r mechanical i n s t a b i l i t y obtained by Coleman (ref. 11). As the propeller rotational speed R increases a point
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i s reached where two of the frequencies coalesce, resulting i n two modes at t h e same frequency, one of which i s damped and t h e other unstable. This point, denoted by A i n t h e figure, marks t h e beginning of a region of mechanical i n s t a b i l i t y i n a fomard whirl rode.
Figure 1 0 shows t h e e f f e c t s of adding aerodynamic forces t o the s t a b i l i t y analyses by progressively increasing air density from zero, the value assumed i n figure 9. (Point A i n f i g . 9 lies on t h e zero density curve i n f i g . 10.)
S t a b i l i t y boundaries a r e presented i n figure 10 as p l o t s of blade frequency Boundaries against nacelle frequency f o r both forward and backward whirl modes.
f o r t h e forward mode originate from a mechanical i n s t a b i l i t y , and those f o r t h e backward mode originate from a whirl f l u t t e r i n s t a b i l i t y .
that i n t h e forward whirl mode as air density increases two regions Note of i n s t a b i l i t y develop. The region t h a t has t h e higher values of we/Q is not of p r a c t i c a l significance because the r a t e of growth of the unstable motions involved was found t o be extremely low and would be eliminated by a s m a l l amount of s t r u c t u r a l damping. The other region of i n s t a b i l i t y i n t h e but with forward w h i r l mode i s i n i t i a l l y made worse as density increases, f u r t h e r increase i n density t h e area of i n s t a b i l i t y i s reduced and eventually eliminated. It i s i n t e r e s t i n g t o note from t h i s f i g u r e t h a t t h e system encounters mechanical i n s t a b i l i t i e s only i f t h e nonrotating blade natural fre- quency, u t , , is l e s s than t h e r o t a t i o n a l frequency of t h e propeller.
With regard t o propeller whirl f l u t t e r - indicated by t h e s t a b i l i t y
boundaries f o r t h e backward mode - it appears t h a t blade f l e x i b i l i t y has rela- t i v e l y l i t t l e e f f e c t except i n a region where t h e blade frequency parameter i s near unity. The asymptotic boundaries f o r a r i g i d propeller a r e a l s o shown f o r comparison.
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PROPELLER AERODYNAMICS A s has already been mentioned i n a previous section of t h e paper the pitching and yawing oscillations t h a t accompany propeller-nacelle whirl motions produce aerodynamic forces on the propeller which i n t u r n provide the driving W b ~ ~ - w - * o l . " . . - A m n + h d a -1 "- ---- save -- - a v a f l n h l e -. . fnr nredicting the mc:&&,i~~,,, fvi- J - - - L - L * ' ~ * - ~ l l l U U B U I I L " J .
Ribner's aerodynamic forces and moments required i n a whirl f l u t t e r analysis.
analysis of yawed propellers ( r e f . 15) has long been used i n studies of a i r - c r a f t s t a b i l i t y and i s equally useful i n propeller whirl f l u t t e r studies (see,
Houbolt's s t r i p theory analysis i n reference 4 lacks
f o r example, r e f . 3 ) .
some of the refinements of Ribner's method, but i s simpler t o apply and appears Both of these theories a r e based on the assump- t o give comparable r e s u l t s .
t i o n t h a t the inflow angle is s m a l l and the aerodynamic forces a r e "quasi- s t a t i c , " i.e., o s c i l l a t i n g wake e f f e c t s are ignored.
Measured Derivatives To evaluate t h e o r e t i c a l methods f o r predicting propeller w h i r l f l u t t e r , Bland and Bennett (ref. 6) have conducted an experimental investigation on a model which resembles t h e simple mathematical models previously studied. The wind-tunnel model consisted of a windmilling propeller mounted on an isolated Measurements of nacelle which had symmetrical s t i f f n e s s i n pitch and yaw.
s t a t i c aerodynamic propeller derivatives and whirl f l u t t e r boundaries f o r t h e same propeller system were obtained over a range of t e s t conditions. Typical r e s u l t s from t h e study a r e presented i n figure 11which shows t h e nacelle damping required t o prevent f l u t t e r plotted against a nondimensional velocity.
The calculated f l u t t e r boundaries were determined on the basis of three s e t s the theoretical derivatives derived by t h e methods of aerodynamic derivatives:
- 15 -
of references 4 and 15, and the actual derivatives as measured on the model.
Note t h a t the calculations based on measured derivatives a r e i n excellent agree- ment with t h e experimental data, whereas those based on theoretical derivatives follow the same trends but tend t o underestimate the observed f l u t t e r speed.
Unsteady Flow These differences i n t h e f l u t t e r boundaries based on t h e o r e t i c a l and meas- It has ured derivatives may i n p a r t be due t o unsteady aerodynamic e f f e c t s .
been shown (e.@;., ref. 3 ) that aerodynamic phase lags associated with t h e oscillatory wake have a stabilizing effect on the usual backward-mode whirl f l u t t e r . The theoretical derivatives used i n figure 1 1 were modified t o account f o r phase lags on t h e basis of t h e Theodorsen circulation function F + i G f o r two-dimensional a i r f o i l s . This i s a good approximation f o r very large advance ratios; however, f o r smaller advance ratios, the circulation function i s sig- nificantly a l t e r e d due t o t h e h e l i c a l pattern of the wake.
I n reference 16 Loewy derives modified F + i G function f o r a propeller
with a helical wake. These r e s u l t s indicate t h a t f o r low advance r a t i o s t h e phase lag f o r a propeller can be much l a r g e r than would be predicted by the c l a s s i c a l F + i G function f o r two-dimensional a i r f o i l s . It i s interesting t o note t h a t f o r lowest advance r a t i o s investigated by Bland and Bennett i n reference 6 (J = 1 . 3 ) t h e measured phase lag w a s 24' as compared with a theo- r e t i c a l value of 1 3 ' based on Theodorsen's function.
Thrust It has been established t h e o r e t i c a l l y t h a t propeller t h r u s t has a rela- t i v e l y insignificant e f f e c t on whirl f l u t t e r s t a b i l i t y f o r conventional r i g i d propellers under high-speed f l i g h t conditions. This f a c t greatly simplifies
- 16 -
t h e construction and t e s t i n g of wind-tunnel models i n t h a t it permits the use of windmilling rather than thrusting propellers. I n a theoretical investigation of' the e w e c t s of propeller Ynrusi UKI w l i i ~ l fLUtAer, Rz-i-era (ref. 17) 8h~;;s t h a t t h r u s t causes large deviation i n t h e propeller derivatives a t low-speed high-thrust f l i g h t states, such a s take-off. However, at higher forward speeds where whirl f l u t t e r normally occurs, the deviation be.tween aerodynamic coeffi- cients f o r thrusting and windmilling propellers i s usually l e s s than 3 percent.
Similar conclusions are found i n reference 9 on the basis of experimental coef- f i c i e n t s obtained on a thrusting propeller.
It i s pointed out i n reference 17 t h a t before drawing general conclusions regarding t h e e f f e c t s of thrust on whirl f l u t t e r , one should examine the thrust characteristic curve of t h e propeller i n question. If it i s found, f o r example, t h a t at high forward speeds t h e thrust i s small r e l a t i v e t o m a x i m u m thrust, the e f f e c t s of t h r u s t on t h e propeller coefficients may be ignored. This case, O n the which i s t y p i c a l of most propellers, is i l l u s t r a t e d i n figure 12(a).
other hand, if the propeller delivers a significant percentage of maximum t h r u s t a t high forward speeds, such as i s i l l u s t r a t e d i n figure 12(b), thrust may well have an important influence on whirl f l u t t e r .
High Inflow A n g l e s VTOL a i r c r a f t from v e r t i c a l t o horizontal I n t h e t r a n s i t i o n maneuver of
flight t h e inflow angle. - i.e., t h e angle between the t h r u s t axis and the
airstream - may be as large as 90'. A t these high inflow angles t h e propeller
aerodynamic derivatives are l i k e l y t o have values t h a t d i f f e r markedly from those corresponding t o high-speed flight conditions (see refs. 18 and 1 9 ) .
Since whirl f l u t t e r is usually considered t o be a r e l a t i v e l y high-speed f l i g h t
- 17 -
phenomenon it is of i n t e r e s t t o explore t h e possibility of encountering whirl f l u t t e r i n low-speed f l i g h t during t r a n s i t i o n . This question w a s b r i e f l y examined i n reference 9. O n t h e basis of whirl f l u t t e r calculations which u t i l i z e d experimental propeller derivatives and a simple axisymmetric nacelle, angle-of-attack e f f e c t s were found t o be stabilizing i n t h a t the s t i f f n e s s required t o prevent f l u t t e r a t high inflow angles was s l i g h t l y l e s s than t h a t at low angles. These r e s u l t s a r e summarized i n figure 13 f o r a wind- required milling and a thrusting propeller.
WING FLEXIBILITY The propeller-nacelle systems considered i n previous sections of t h e paper were assumed t o have been flexibly mounted t o a r i g i d wing or backup structure.
This simplifying assumption i s very useful, especially i n preliminary design studies, i n t h a t it permits one t o readily i s o l a t e the e f f e c t s of various basic parameters. However, dynamic coupling between t h e propeller-nacelle system and the wing on which it i s mounted can a l t e r t h e whirl f l u t t e r boundaries predicted f o r a r i g i d w i n g . In general it has been found t h a t wing aeroelastic e f f e c t s have a stabilizing influence on whirl f l u t t e r .
it can best be i l l u s t r a t e d with the This general trend and an exception t o a i d of figure 14. Stiffness boundaries of t h i s type show the combination of \ pitch and yaw nacelle s t i f f n e s s e s t h a t would be required t o avoid whirl f l u t t e r Of a system f o r a given s e t of f l i g h t conditions. When t h e nacelle s t i f f n e s s e s are greater than the minimum values indicated by these boundaries, propeller i n whirl f l u t t e r would not be encountered by the system. The curves presented figure 1 4 are based on r e s u l t s obtained by Zwaan and Bergh i n a n analog com- puter study of whirl f l u t t e r ( r e f . 20). These curves show the influence of wing f l e x i b i l i t y on w h i r l f l u t t e r f o r three cases: (a) a r i g i d wing, (b) a wing having freedom t o t r a n s l a t e vertically, and ( c ) a wing having freedom t o t r a n s l a t e and rotate about a n p l a 8 t . i ~ axis- Ir? +_he lztter t u c CQSCG, = e m - dynamic forces are assumed t o a c t on t h e wing a s w e l l as on t h e propeller.
Consider first the case of a r i g i d wing. Features of t h i s s t a b i l i t y
boundary are typical of those found i n reference 4 and discussed here i n
it i s t o be noted that the bovlndary i s extended along the figure 4(a). Again, diagonal ray S8 = E + , such t h a t the symmetrical system, indicated by point A, is t h e most c r i t i c a l from t h e standpoint of whirl f l u t t e r . The points a t which a s well as other curves i n the figure, terminate a r e the s t a t i c t h i s curve, divergence boundaries f o r t h e system.
When t h e wing has freedom t o o s c i l l a t e i n v e r t i c a l translation it can be This increased seen t h a t i t s e f f e c t on whirl f l u t t e r i s always stabilizing.
s t a b i l i t y can be a t t r i b u t e d t o aerodynamic damping forces on t h e w i n g . It i s of p a r t i c u l a r i n t e r e s t t o note t h e "necked down" portion of t h e curve where the s t a b i l i z i n g influence of t h e wing i s most pronounced. A t t h e points labeled B, t h e whirl frequency o coincides with the wing bending frequency q so t h a t whirl motions of the propeller-nacelle tend t o drive the wing a t a resonant amplitude. Thus, Zwaan and Bergh suggest t h a t the wing i n t h i s case m y be considered as a type of tuned damper which absorbs greatest energy when the system frequency is close t o the tuned frequency of t h e damper.
I n t h e t h i r d case shown i n figure 14 the wing vibration m o d e involves coupled bending and torsion motions. The frequency of the f i r s t coupled wing mode 0 1 i n t h i s case is approximately the sane as the uncoupled bending fre- discussed previously, but t h e mode shape indicates strong coupling quency % between t h e bending and t w i s t i n g motions h and a . The s t a b i l i t y boundary
- 19 -
f o r t h i s condition i l l u s t r a t e s an exception t o the generally observed trend t h a t wing f l e x i b i l i t y s t a b i l i z e s whirl f l u t t e r . It should be noted t h a t t h e m a x i m u m s t i f f n e s s required occurs i n t h e v i c i n i t y of point C where t h e So w i n g fundamental coupled frequency cu1 and t h e nacelle yaw frequency a , , , are
-
Since t h e w i n g mode involves pitching of t h e propeller, point C may t h e same.
be regarded as a coincidence of "pitch" and yaw frequencies f o r t h e f l e x i b l e as point A was f o r t h e r i g i d w i n g . I n both cases t h e region of wing j u s t i n s t a b i l i t y is extended at these points where t h e p i t c h and yaw frequencies coincide.
Thus, a general conclusion reached i n reference 20 i s t h a t a f l e x i b l e wing has a s t a b i l i z i n g e f f e c t on propeller whirl f l u t t e r except possibly i n a region where t h e uncoupled yaw frequency i s close t o a wing t o r s i o n frequency. Benefi- c i a l effects of a wing on whirl f l u t t e r have a l s o been reported i n references 21 and 22.
AEROELASTIC MODELS Simplified mathematical and physical models, such as those discussed i n t h i s paper, provide valuable aids f o r gaining insight i n t o a complex phenomenon However, as t h e and a r e useful i n guiding t h e course of preliminary design.
f i n a l design of an a i r c r a f t evolves, it i s customary t o employ more refined a n a l y t i c a l and experimental techniques t o insure t h a t the f l u t t e r margin i s The a n a l y t i c a l adequate for any f l i g h t condition within t h e f l i g h t envelope.
techniques used may involve separate consideration of whirl f l u t t e r and wing o r t h e propeller-nacelle system m y be included as additional ingredi- f l u t t e r , a n a e r o e l a s t i c s t a b i l i t y analyses of t h e complete system. Similarly, e n t s i n
- 20 -
model t e s t i n g techniques may vary i n complexity from an isolated propeller- nacelle t o an aeroelastically scaled model of t h e complete a i r c r a f t .
A specific example wherein propeller whirl f l u t t e r was a design considera- A review of t i o n from t h e outset i s provided by the X C - l 4 2 A - a VTOL a i r c r a f t .
t h e wind-tunnel investigation of aerUelastic .zt+ili+.y f o r the X C - 1 4 2 A is reported by Head and Smith i n reference 22. This investigation involved an aeroelastic model i n which each of four engine-gearbox-propeller systems was The propellers were nonpowered but were shafted together dynamically scaled.
t o insure t h a t a l l turned at the same speed. The remarkable degree of d e t a i l achieved i n t h e dynamic simulation of the engine-gearbox system i s apparent i n figure 15. The gearbox is connected t o the engine by a multiredundant strut arrangement on t h e model i n t h e same manner a s it i s on the a i r c r a f t . The f l e x i b l i t y of each strut as well as the overall f l e x i b i l i t y between the engine and t h e gear box a r e accurately scaled i n the model. The authors of refer- ence 22 stated t h a t the simulation of t h i s component represented t h e most d i f - f i c u l t design problem on the model.
A f l u t t e r design requirement f o r t h i s a i r c r a f t is t h a t no f l u t t e r s h a l l occur as a result of f a i l u r e of any single s t m c t u r ~ l element (e.g., see r e f . 7). Therefore, i n t h e model t e s t s a failure of various s t r u t members w a s The strut failure condi- simulated by simply removing t h e strut i n question.
t i o n s t h a t were actually simulated on t h e dynamic model were selected on t h e b a s i s of analysis. It i s interesting t o note t h a t i n some cases the calculated whirl f l u t t e r speed w a s increased as a result of a strut f a i l u r e .
- 21 -
CONCLUDING REMARKS This paper has attempted t o review t h e state-of-the-art of propeller whirl f l u t t e r and some recent contributions t o it. O n t h e basis sf this review t h e following observations can be made: 1. Classical propeller whirl f l u t t e r i n the backward whirl mode is amenable t o analysis providing the propeller aerodynamic coefficients and the damping of t h e engine mount are known with reasonable accuracy. I n most instances strip-theory methods appear adequate f o r determining t h e propeller aerodynamic coefficients.
2. The use of nonrigid propeller-rotors can have a s i g n i f i c a n t influence on t h e whirl s t a b i l i t y of a system. For instance, flapping blades introduce t h e p o s s i b i l i t y of f l u t t e r i n the forward mode and f l e x i b l e blades give rise t o mechanical i n s t a b i l i t i e s , however it w a s found t h a t t h e propeller aerodynamic forces which create whirl f l u t t e r tend t o mitigate mechanical i n s t a b i l i t i e s .
3 . The e f f e c t s of high inflow angles and large t h r u s t coefficients asso- ciated w i t h VTOL t r a n s i t i o n maneuvers a r e r e l a t i v e l y unimportant from t h e standpoint of whirl f l u t t e r .
4. Wing f l e x i b i l i t y generally has a s t a b i l i z i n g e f f e c t on whirl f l u t t e r except when the nacelle uncoupled yaw frequency i s close t o t h e wing t o r s i o n .
frequency Finally, it should be remarked t h a t t h e p a r t i c u l a r parameters selected f o r discussion i n t h i s paper, although representative, are not necessarily the only ones of significance with regard t o whirl f l u t t e r .
O n s p e c i f i c configurations such f a c t o r s as t h e interaction of whirl modes with thrust-control system dynamics, the operation of propeller-rotors as "pushers" which would have upstream pivot locations, o r t h e coupling between w h i r l modes and in-plane
- 22 -
wing bending modes are l i k e l y t o be important. If these f a c t o r s a r e not impor- t a n t the dynamicist concerned with propeller-rotor systems should have no d i f - f i c u l t y i n discovering some others t h a t are.
- 23 -
APPENDIX ~AI3IL;T'I'Y DETERJUNm FOR WHIRL FLUTTER O F PR0pELI;ERS WITH HINGED BLADES The equations used t o calculate s t a b i l i t y of the hinged-blade propeller model i n figures 7 and 8 a r e based on derivations by Richardson and Naylor i n The equations of motion f o r the system a r e expressed i n terms of reference 8.
four degrees of freedom: pitch and yaw of t h e propeller hub about a point aft of t h e hub, and cyclic (antisymmetric) flapping of the blades i n t h e pitch and yaw directions. When t h e system has a x i a l symmetry, as does the model under consideration, the whirl modes a r e c i r c u l a r (i.e., pitch and yaw motions are 9 0 ' out of phase and are of equal amplitude). For such conditions the equa- a s e t of four equations with r e a l coeffi- t i o n s of motion can be reduced from cients t o a set of two equations with complex coefficients.
The s t a b i l i t y determinant of t h e system i n t h e notation of reference 8 is:
IA2A + A(B + D) +
where A = p + i v is a solution (eigenvalue) of t h e equation. The sign of p
determines whether the system i s stable ( - 1 or unstable (+), and the sign of
v y = w/Q) determines t h e direction of (the nondimensional whirl frequency whirl. A positive sign represents forward whirl and a negative sign, backward whirl. The coefficients A, B, C, etc., are matrices t h a t contain the i n e r t i a , gyroscopic, and aerodynamic parameters of t h e system. For t h e axisymmetric system under consideration these matrices a r e defined a s follows:
- 24 -
c
I- 1
-FaH 3 A 1 + GH 2 A 3 FaH(A3 - EA^) - G(% - €4)
+ i( FH2A3 + GaH3Al) - i [F(9 - EA^) - GaH (A3 - CAP]
[ C ] = K -&(F + i G )
GH2(A3 - EA^)
The above matrices are of the form of those presented i n reference 8; how- l ever additional terms, not e x p l i c i t l y given i n reference 8, have been included
here t o represent unsteady aerodynamic effects (F + i G ) and viscous-type
-25 -
s t r u c t u r a l damping 5 . The parameters t h a t make up the matrix elements a r e
defined i n the l i s t of symbols and t h e numerical values of parameters used i n t h e present calculations a r e given i n t a b l e I.
- 26 -
.
REFERENCES 1 . Tayiujr, E. E., e=& B r w n e , K. A.: Vibration Isolation of Aircraft Power Plants. Jour. Aero. Sci., vol. 6, no. 2, Dec. 1938, pp. 43-49.
2. Abbott, Frank T., Jr., Kelly, H. Neale, and Hampton, Kenneth D.: Investiga- t i o n of Propeller-Power-Plant Autoprecession Bounaaries FGX- G 2 J - c ~ i c - Aeroeiastic Model of a Four-Engine Turboprop Transport Airplane. N A S A TN D-1806, 1963.
3 . Reed, Wilmer H., 111, and Bland, Samuel R.: An Analytical Treatment of NASA TN D-639, 1961.
Aircraft Propeller Precession I n s t a b i l i t y .
4. Houbolt, John C., and Reed, Wilmer H., 111: Propeller-Nacelle Whirl F l u t t e r .
Jour. Aerospace Sci., vol. 29, no. 3, Mar. 1962, pp. 333-346.
5 . Sewall, John L.: An Analytical Trend Study of Propeller Whirl I n s t a b i l i t y .
NASA TN D-996, 1962.
Wind-Tunnel Measurement of 6. Bland, Samuel R., and Bennett, Robert M . : Propeller Whirl-Flutter Speeds and Static-Stability Derivatives and Com- parison With Theory. NASA TN D-1807, 1963.
7. U.S. C i v i l A i r Regulations 4b.308 Amendment 4b-16, e f f e c t i v e October 5 , 1964.
Whirl F l u t t e r of Fropellers !:Tith 8. Richardson, J. R., and Naylor, H. F. W.: Hinged Blades. Report No. 24, Engineering Research Associates, Toronto, Canada, March 1962.
9. Reed, Wilmer H., 111, and Bennett, Robert M.: Propeller Whirl F l u t t e r Con- siderations f o r V/STOL Aircraft. Cal/Trecom Symposium Proceedings,
vol. I11 - Dynamic Load Problems Associated With Helicopters and V/STOL
Aircraft, June 1963.
.- 27 -
10. Gallardo, V. C., and Flannelly, W i l l i a m : Propeller-Nacelle Whirl F l u t t e r Analysis of K - 1 6 ~ Amphibious VTOL/STOL Aircraft. Rept . No. G-113-41, Kaman Aircraft Corp., AM. 1962.
11. Coleman, Robert P . , a d Feirsold, Arnoid M.: Theory of Self-Excited Mechanical Oscillations of Helicopter Rotors With Hinged B l a d e s . NACA TR 1351, 1958.
12. Brooks, George W.: The Mechanical I n s t a b i l i t y and Forced Response of Rotors on Multiple-Degree-of-Freedom Supports. PhD Thesis, Princeton University, October 1961.
13. Loewy, Robert G ., and Yntema, Robert T. : Some Aeroelastic Problems of
Tilt-Wing VTOL Aircraft. Journal of the American Helicopter Society, 3, no. 1, Jan. 1958.
vol.
14. Richardson, J. R., McKillop, J. A., Naylor, H. F. W., and Blandler, P. A.: Rept. No. 43, Whirl F l u t t e r of Propellers With Flexible Twisted Blades.
1963.
Engineering Research Associates, Toronto, Canada, Dec.
15. Ribner, Herbert S . : Propellers i n Yaw. NACA Rep. 820, 1945. (Supersedes NACA ARR 3L09.)
16. Loewy, Robert G.: A Two-Dimensional Approximation t o t h e Unsteady Aero- dynamics of Rotary W i n g s . Jour. Aeronautical Sci., vol. 24, no. 2, Feb. 1957, pp. 81-92.
17. Ravera, Robert J.: Effects of Steady State B l a d e Angle of Attack on Propeller Whirl F l u t t e r . Rep. No. ADR 06-01-63.1, Grumman Aircraft Eng.
Corp., July 1963.
18. De Young, J. : Propeller a t High Incidence. AIAA Jour. of Aircraft, vel. 2 , no. 3, May-June 1965, pp. 241-249.
- 28 -
19. Shenhman, Albert M.: Generalized Performance of Conventional Propellers Hamilton Standard Rept. No. HS-1829, March 1958.
f o r VTOL-STOL Aircraft.
20. Zwaan, R. J., and Bergh, H.: Restricted Report, ~ . 2 2 8 , N a t . Aero- Astronautical Research Inst., N.L.R., Amsterdam, Feb. 1962.
21. Bennett, Robert M., urd El&, Cy~~~2e.1 E?=! Eqerimental and Analytical Investigation of Propeller Whirl Flutter of a Power Plant on a Flexible Wing. NASA. TN D-2399, 1964.
Dynamic Model Testing of t h e XC-142A 22. Head, A. L., and Smith, W. D.: Aircraft. Proceeding of Symposium on Aeroelastic and Dynamic Modeling Technology, RTD-TDR-63-4197, Part I, March 1964.
TABLE I . - SYSTEM PARAMETERS USED I N CALCULATIONS O F FLAPPING-BLADE MODEL FLUTTER BOUNDARIES (SEX F I G S . 5 AND 8) a , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.25
c, ( a t 3 / 4 R ) , f t . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.0835
c 2 , per rad. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2J(
U eS, slug-ft 2 4
. . . . . . . . . . . . . . . . . . . . . . . . . . 0.0495 x 1 0 '
* F + i G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.67- i 0.18
J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 I, slug-ft2 -4
. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.310 x 10
11, slug-ft 2 4
. . . . . . . . . . . . . . . . . . . . . . . . . . 0 . 3 8 1 6 ~ i o -
12, slug-ft . . . . . . . . . . . . . . . . . . . . . . . . . . 0.2386 x
13, slug-ft . . . . . . . . . . . . . . . . . . . . . . . . . . 0.2090 x 10-4
N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
R , f t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.70
E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.137
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.04
v o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Varied
*
The F + iG functions employed here combine aspect r a t i o and phase lag
corrections.
The phase lag i s based on experimental data.
- 30 -
a , rl rl a ,
0 " /
x x H H
z
r: e , rl
I +
rl a , PI k 0 3 PI
%
v,
m a
+
+
H H ..
I ..
* 3 I -
i i 9
- /
al PI k PI rd
> \
d
> \
bo d k l - a :
-
n
a :
a
a :
e
a : OL
a . I
cu
I
a .
FRONT SIDE
a -
@ RIGID BLADES
0 HINGED BLADES
ri
0 FLEXIBLE-TWISTED
BLADES
NASA Figure 3 . - Propeller-rotor systems considered.
I k a, d rl aj PI k Y n pc W
%
' 3 01
\a, ffi l- v, 0, a , k a, k a , L in a , a , W W
\
\
\
\
P
\
LT d
w k
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\
/ I- /
\
LI rl cd N
cu 0 N
u I I I w 2 - a , k
W O
n H Q, d k a3
/'
M d II k
\ rcl
/
\
r
/
t Y
\
w
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I
L L
i
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I
I i
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I
i I
L
>
CK W
I
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I
I
cu 0
E l k W c d k 0,
Z cn
rl
g+g 3
rl Q, P!
0 c(9
In I I n
-
v
->
k a rl rl Q, PI PI
-
n /-
-* -----
-
I
-----------
(3 \
m \
__y \ k %I ... " d e, L
N > S Ilr
in
1 I I I rl l - i e, [r W J k W
n
El
>
k k
n
n
W W
n
v) I
I
a
-
n
I
o
W k a , LT rl rl W a , J PI J k W PI
>
n
I cu [r rl II a , J
rl i ! l
a
n Frc
o
- 0
Y
n
I -
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