STATIC-STABILITY MEASUREMENTS OF ASTAND- ON TYPE A COMPARISON WITH THEORY HELICOPTER by Jean LuLu

NASA TN D-189

TECHNICAL NOTE 0-189

STATIC-STABILITY MEASUREMENTS OF A STAND-ON TYPE HELICOPTER WITH RIGID BLADES, INCLUDING A COMPARISON WITH THEORY

By George E. Sweet Langley Research Center Langley Field, Va.

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON

Februaxy 1960

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION TECHNICAL NOTE D-189 ,

STATIC-STABILITY MEASUREMENTS OF A STAND-ON TYPE HELICOPTER WITH RIGID BLADES, INCLUDING

A COMPARISON WITH THEORY By George E. Sweet SUMMARY

Results of an investigation in the Langley full-scale tunnel of a small stud-on type helicopter in low-speed forward flight are presented. The rotor blades were fixed at the roots and had preset coning angles. The results include measurements of forces and moments and staticstability derivatives determined from these measurements. These tests indicate that forward speed of the helicopter would be limited to approximately 17 knots because of large stabilizing pitching moments. Pitching moments and static-stability derivatives for the isolated rotors as deduced from the helicopter measurements are compared with rigid-rotor theory. For the isolated rotor, the pitching moments and the static-stability derivatives with respect to angle of aitack may be estimated by meus of rigid-rotor theory provided that a linearlongitudinal-inflow variation is considered.

INTRODUCTION It has been demonstrated (ref. 1) that a man standing on a Jetsupported p .atform is capable of controlling the platform by bo'y motions. The-feasibility of apply& this concept of control to the helicopter was demoristrated by means of the rotor-supported platform described in reference 2. As a result of these investigations, several helicopter configurations incorporating this principle of control were built. One of these configurations was the subject of a concurrent flight and a theoretical study renorted in reference 3 which were undertaken to determine the static- and dynamic-stability characteristics (see figs. 1 and 2). Results of reference 3 indicated that the longitudinal control power was insufficient to trim the machine at speeds greater than abo,ut 16 knots and that the longitudinal oscillations of the aircraft were unstable throughout the speed range. Because of a decrease in the period of the

oscillations, control became more difficult as forward speed increased. At some speed near 16 knots, the dynamic motions became uncontrollable since there was no excess control power available. During a flight demonstration by the military, a crash occurred at near maximum level-flight speed. Since reference 3 was not available at that time, there was some question as to the cause of the crash. There was speculation as to whether the machine had become uncontrollable because of a longitudinal instability or whether there had been a collision between the blades of the coaxial rotors due to blade bending. Therefore, full-scale windtunnel tests of one of the aircraft were conducted. The purposes of the tests were to measure the overall aerodynamic characteristics, to determine the static-stability derivatives as an aid to the dynamic studies, and to determine the clearance between the rotor tips. This report presents the aerodynamic characteristics including static-stability derivatives of the aircraft as determined in the Langley full-scale tunnel. The thrust, H force (component of rotor resultant force in plane of rotor), and pitching moment of the rotor alone are compared with calculations based on rigid- and flapping-rotor theory. Equations for the H force and pitching moment of a rigid rotor are developed. Equations for the longitudinal static-stability derivatives of a rigid rotor with respect to angle of attack, tip speed, and forward velocity are also developed. Calculations based on these equations are compared with the measured derivatives. SYMBOLS b

slope of curve of section lift coefficient against section angle of attack in radians

constant term in Fourier series that expresses p; hence, the rotor coning angle, radians

number of blades per rotor

blade-section chord, ft

mean profile drag coefficient of rotor blade section

equivalent blade chord (on thrust basis),

d,0

, ft

rotor drag (H force) coefficient,

nR2p (4x3)

6 9

rotor thrust coefficient,

nR2p( m)2

aircraft torque coefficient, n

Q ~( m ~ 2~ p

C'Y

aircraft side-force coefficient based on tip speed,

C'D

aircraft drag coefficient based on tip speed,

Y nR2p( a?I2

D nR2p( srR)2

C'L

aircraft lift coefficient based on tip speed,

C'm,r

rotor pitching-moment coefficient about body axes based on tip

nR2p( a?)*

speed,

xR2p(QR) R

C'2,W

aircraft rolling-moment coefficient about wind axes based on tip speed,

%r nR2p (a?)%

CIItl,W

aircraft pitching-moment coefficient about wind axes based on tip speed,

cln,w

My, r

aircraft yawing-moment coefficient about wind axes based on tip speed,

Mz, r nR2p( a?)%

aircraft drag, lb

equivalent flat-plate area representing parasite drag, based on unit drag coefficient, D/q, sq ft

H force, component of rotor resultant force perpendicular to the control axis (in this instance rotor-shaft axis) in the plane of the airflow, positive when it opposes the translational motion, lb lift, lb rotor pitching moment, lb-ft unless otherwise noted aircraft rolling moment about wind axis, lb-ft unless otherwise noted

MY,w

aircraft pitching moment about wind axis, lb-ft unless otherwise noted

MZ,w

aircraft yawing moment about wind axis, lb-ft unless otherwise noted

aircraft power required, Q!d/750, hp dynamic pressure, lb/sq ft rotor torque, ft-lb

radial distance to blade element, ft

blade radius, ft

rotor thrust, lb

component at blade element of resultant velocity perpendicular both to blade-span axis and UT, ft/sec

component at blade element of resultant velocity perpendicular to blade-span axis and to axis of no feathering, ft/sec

velocity, ft/sec unless otherwise noted

induced velocity at tip of rotor, ft/sec

momentum theory value of rotor-induced velocity,

-12 CTm

ft/sec V

rate of change of induced velocity per rotor radius along longitudinal axis of rotor at r = 0, ft/sec

side force, lb

rotor angle of attack, angle between axis of no feathering (in this instance rotor-shaft axis) and a line perpendicular to the flight path, positive when axis is inclined rearward, radians unless otherwise noted

blade flapping angle; angle between blade-span axis and plane perpendicular to axis of no feathering (in this instance rotor shaft axis), radians -

blade-section pitch angle; angle between line of zero lift of blade section and plane perpendicular to axis of no feathering (in this instance rotor shaft axis), radians

inflow ratio,

V sin a

tip-speed ratio,

- vo

V cos a

mass density of air, slugs/cu ft

rotor solidity, bCe/JrR

wake skew angle tan'l

blade azimuth angle measured from downwind position in direction of rotation, radians unless otherwise noted

c1 - -, radians A

6 R

rotor a n m a r velocity, radians/sec

Sub s c r i p t s : i

induced

profile ,

APPARATUS L

The t e s t apparatus i s shown i n s t a l l e d i n t h e Langley f u l l - s c a l e tunnel i n f i g u r e s 1 and 2. A complete d e s c r i p t i o n of t h e f u l l - s c a l e tunnel is given i n reference 4.

6 9 1

Model The p r i n c i p a l dimensions of t h e t e s t vehicle and t h e r o t o r blades a r e shown i n f i g u r e 3 . The a i r c r a f t i s an unducted stand-on type h e l i copter having two, two-blade, coaxial r o t o r s . Control about t h e pitching and r o l l i n g axes i s accomplished by simple body motions. Rotor t h r u s t i s varied by c o l l e c t i v e changes i n t h e speed of t h e coaxial r o t o r s while d i r e c t i o n a l control i s obtained by d i f f e r e n t i a l changes i n r o t o r speed. The rotors a r e powered by a 40-horsepower, reciprocating engine driving through a transmission having a speed-reduction r a t i o of 10.7 t o 1.0. The upper r o t o r t u r n s clockwise and t h e lower r o t o r counterclockwise as viewed from above. For these t e s t s t h e reciprocating engine w a s replaced by a 150-horsepower, variable-frequency e l e c t r i c motor ( f i g . 2 ) .

The r o t o r s have a diameter of 180 inches ( f i g . 3 ) and an equivalent s o l i d i t y of 0.071. The blades a r e fixed a t t h e root, and have p r e s e t p i t c h and coning angles. During these tests t h e upper r o t o r blades had go of p i t c h and a coning angle of 5' while t h e lower blades had 10' of p i t c h and a coning angle of 3'. The blades were constructed of wood wrapped with g l a s s cloth. They were untwisted and had 2-to-1 l i n e a r plan-form t a p e r r a t i o measured from t h e center of r o t a t i o n . The blade-section p r o f i l e varied from NACA 0018 a t t h e center of r o t a t i o n t o NACA 0009 a t t h e t i p . For these t e s t s a clothing-display dummy w a s i n s t a l l e d on t h e a i r c r a f t t o simulate t h e aerodynamic e f f e c t s of a human p i l o t . (See f i g s . 1 and 2 . )

. *

7 Instrument a t ion

The a i r c r a f t w a s mounted from the wind-tunnel support system by a six-component strain-gage balance which was used t o measure t h e forces and moments on t h e model ( f i g s . 2 and 3 ) . Motor torque was measured by means of a strain-gage beam torquemeter. The accuracy of t h e balance and torquemeter w a s as follows:

........................... ........................... .......................... Pitching moment, l b - f t . . . . . . . . . . . . . . . . . . . . . . Rolling moment, l b - f t . . . . . . . . . . . . . . . . . . . . . . Yawing moment, l b - f t . . . . . . . . . . . . . . . . . . . . . . . Motor torque, f t - l b . . . . . . . . . . . . . . . . . . . . . . . Lift,lb.. Drag, l b . . Side force, lb

i 6 9 1

k5 22

k75I

+5 +11 4 +2

The speed of each r o t o r w a s measured by magnetic pickups mounted at each r o t o r . The s i g n a l from these pickups w a s used t o d r i v e an e l e c t r o n i c pulse-counter frequency meter. Rotor-speed measurements a r e accurate t o

*3 rpm. The a i r c r a f t angle of a t t a c k and d i f f e r e n t i a l r o t o r speed were cont r o l l e d remotely by means of e l e c t r i c actuators. Angle of a t t a c k w a s measured by means of a standard NASA control p o s i t i o n t r a n s m i t t e r having an accuracy of f0.1~. Tunnel airspeed w a s measured by a propeller-type anemometer located ahead of t h e model as shown i n figure 1. Airspeed w a s read t o t h e nearest 1/2 knot. Clearance between t h e t i p s of the coaxial r o t o r s w a s measured a t the probable point of minimum clearance, namely at 90' azimuth on t h e lower r o t o r and 270째 azimuth on t h e upper r o t o r ( f i g . l), by means of a modified o p t i c a l - e l e c t r o n i c blade-tracking instrument. Measurements are accurate t o t h e nearest 1/2 inch. TESTS

.. 4

Rotors Removed

The forces and moments on t h e aircraft fuselage with t h e r o t o r blades reiiioved were measured over a range of angle of a t t a c k and forward speed

which included t h e l e v e l - f l i g h t conditions of t h e a i r c r a f t . During these t e s t s t h e r o t o r hubs were operated a t about 500 rpm. These t e s t s were made t o determine t h e contribution of t h e fuselage t o t h e o v e r a l l chara c t e r i s t i c s of t h e machine.

Rotors I n s t a l l e d Trimed f1iRht.- The forces and moments, motor torque, r o t o r speed, angle of attack, and blade clearance of t h e a i r c r a f t were measured f o r trimmed, l e v e l - f l i g h t conditions a t several values of forward speed from hovering t o 33.6 knots. It should be noted t h a t , i n t h i s report, trimmed f l i g h t r e f e r s t o trimmed lift, drag, and yawing moment as indicated by t h e strain-gage balance. Pitching moments were r e s t r a i n e d by t h e balance and were measured r a t h e r than t r i m e d . A constant l i f t of approximately 483 pounds w a s set a t each value of forward speed by adjusting the motor speed. The rotor-shaft angle of a t t a c k was adjusted u n t i l the o v e r a l l drag force w a s zero as indicated by the strain-gage balance. The yawing moments were trimmed t o zero by adjusting t h e d i f f e r e n t i a l speed (i.e., torque) between t h e upper and lower rotors. S t a t i c - s t a b i l i t y derivatives.- A number of t e s t s were a l s o made t o determine t h e s t a t i c - s t a b i l i t y derivatives of t h e a i r c r a f t over t h e speed range. Derivatives were obtained with respect t o changes i n angle of attack, forward speed, and t i p speed. A trimmed condition w a s s e t a t a p a r t i c u l a r speed, and then small independent changes were made i n angle of attack, velocity, and t i p speed. The r e s u l t i n g changes i n forces, moments, and torque were then measured. These d a t a were p l o t t e d i n nondimensional form, and t h e slopes were measured t o obtain the d e r i v a t i v e s . Corrections.- Tunnel stream-angle corrections have not been applied t o t h e measured data because they a r e small, being of t h e order of -0.5O. Unpublished d a t a obtained from an e a r l i e r t e s t of t h i s a i r c r a f t i n t h e Langley f u l l - s c a l e tunnel with a ground r e f l e c t i o n plane i n s t a l l e d agree with data of t h e present investigation. Since t h e change i n t h e windtunnel configuration produced no observed changes i n experimental r e s u l t s , jet-boundary corrections must be negligible and therefore were not applied t o these data. Jet-boundary corrections f o r these tests would be expected t o be small since t h e r o t o r i s s m a l l with respect t o t h e wind-tunnel cross s e c t ion. Axis system.- Measured d a t a f o r t h e R i r c r a f t , both with and without r o t o r s , are presented with reference t o a center of g r a v i t y located 27 inches above the point midway between t h e coaxial r o t o r s . The data a r e presented f o r a system of wind axes, designated X, Y, and 2 which a r e shown i n f i g u r e 4(a). C h a r a c t e r i s t i c s of t h e i s o l a t e d r o t o r s a r e referenced t o a center of g r a v i t y located midway between t h e r o t o r s . The r o t o r axis system shown i n f i g u r e 4(b), commonly used i n t h e description of r o t o r aerodynamics, i s used f o r the presentation of t h i s portion of t h e data. I n t h i s system, r o t o r t h r u s t i s normal t o t h e t i p - p a t h plane,

L 6 9 1

Y 9 and rotor H force is normal to thrust in the longitudinal plane. Side force is orthogonal to both rotor thrust and drag. (See fig. 4(b).) RESULTS AND DISCUSSION

The results of the tests conducted in the Langley full-scale tunnel to determine the aerodynamic and static-stability characteristics of the complete aircraft are presented in the following sections. Force and moment characteristics of the aircraft with the rotor blades removed are presented first. The performance and static stability of the aircraft with rotors operating are then discussed. Finally, the longitudinal characteristics of the isolated rotors are presented and compared with calculations based on rigid-rotor (blade-element) theory. Tests With Rotors Removed The effect of forward speed on the aerodynamic characteristics of the aircraft fuselage-pilot combination with the rotor blades removed is summarized in figure 5. The results are shown for the angles of attack which correspond to the subsequent tests with the rotors operating. Drag and pitching moment are the only components which show significant variations with speed. The measured drag closely follows the equation D = qf (where f is a constant 10.3 square feet). The side-force component, which was not presented in figure 5, was essentially zero for the speed range investigated. Rotors Operating Trimmed flight.- The variation of the aircraft angle of attack, force and moment coefficients, rotor speed, and power required for trimmed level flight between hovering and 36 knots are presented in figures 6 to 8. Lift was held essentially constant at 483 pounds during these tests. Force and moment coefficients presented in this paper are based on rotor tip speed. It is noted in figure 6 that the angle of attack required for trimmed flight is linear and has a slope of -1.0' per 3 knots of forward speed. Since the lift was held constant, variations of lift coefficient with forward speed (fig. 6) are a function of rotor speed. (See fig. 8.) Probably the most significant aircraft characteristic is the pitching-moment coefficient (fig. 7). The pitching-moment coefficient required â&#x201A;Źor trim increases almost iinearly with forward speed. A 200-pound man leaning forward 12 inches from an erect position is capable

10 of applying about 2,400 inch-pounds of nose-down moment t o t h e a i r c r a f t . A s s u m i n g t h i s t o be the maximum a v a i l a b l e control moment, f i g u r e 7 indicates t h a t the s t a b i l i z i n g pitching moments of t h e a i r c r a f t could not be trirmned a t speeds above 17 knots. This agrees c l o s e l y with t h e l5.5-lmot maximum trimmed forward speed given by reference 3. The v a r i a t i o n of r o t o r I-otational speed and t o t a l power required f o r trimmed l e v e l f l i g h t a r e shown i n f i g u r e 8. The required power w a s computed from the measured motor torque and t h e r o t o r revolutions per minute by assuming t h a t t h e r e w a s no slippage i n t h e V-belt transmission. N o t a r e s were applied t o these data. The shape of t h e power-required curve i s t y p i c a l of helicopters. This f i g u r e i n d i c a t e s that approximately 40 horsepower i s required f o r hovering at a l i f t of 483 pounds.

Rotor-tip clearance.- Clearance between the t i p s of t h e coaxial r o t o r s was measured a t t h e probable point of minimum clearance, namely a t 90' azimuth on the lower r o t o r and 270째 azimuth on t h e upper r o t o r . Measurements were made a t speeds between hovering and 36 knots. A small portion of t h e data presented i n f i g u r e 9 was obtained from t h e previously mentioned wind-tunnel investigation of t h e a i r c r a f t . The clearance a t zero rpm w a s 10.2 inches. The clearance decreased t o approximately inches i n hovering. Only minor v a r i a t i o n s from t h i s value were

observed throughout t h e speed range. S t a t i c - s t a b i l i t y derivatives.- The v a r i a t i o n of a i r c r a f t s t a t i c s t a b i l i t y derivatives i n c o e f f i c i e n t form w i t h tip-speed r a t i o i s shown i n figure 10. These derivatives were measured from f a i r e d curves of t e s t data. Side-force derivatives, which a r e not presented, were essent i a l l y zero. Isolated-rotor c h a r a c t e r i s t i c s . - The forces and moments f o r i s o l a t e d r o t o r s i n the longitudinal plane were obtained by subtracting t h e d a t a with the r o t o r s removed from those obtained w i t h t h e r o t o r s i n s t a l l e d a t t h e same angles of a t t a c k and forward speed. This approach i s not s t r i c t l y correct as it i s assumed t h a t t h e a i r c r a f t - f u s e l a g e and p i l o t characteri s t i c s remain unchanged by interference when t h e r o t o r s a r e i n s t a l l e d . Nevertheless, the r e s u l t s obtained from t h i s approach w i l l be considered herein as c h a r a c t e r i s t i c s of t h e i s o l a t e d r o t o r s . Despite t h e f a c t t h a t t h e r e was an undetermined amount of flapping motion because of blade f l e x i b i l i t y , the calculated c h a r a c t e r i s t i c s a r e based on t h e assumption of completely r i g i d r o t o r blades. Forces and pitching moments f o r the r o t o r alone a r e a l s o compared w i t h calculations f o r a f u l l y a r t i c u l a t e d r o t o r . The flapping-rotor calculations, assuming t h e blades a r e hinged a t the center of r o t a t i o n , a r e based upon bladeelement theory given i n chapter 8 of reference 5 . Equations f o r t h e r i g i d - r o t o r c h a r a c t e r i s t i c s a r e derived i n appendixes A t o D. All

L 6 9 1

11 calculations and derivatives a r e based on t h e assumption that coaxial r o t o r s may be replaced by a s i n g l e rotor of equivalent s o l i d i t y ( r e f . 6 ) . The blade p i t c h and t h e preset coning angle of t h e upper r o t o r are assumed t o be those of t h e equivalent r o t o r . The blade-section-liftc o e f f i c i e n t slope ( a = 5.73) and t h e mean blade-section d r a g c o e f f i c i e n t = 0.012 were assumed t o be constant. (Cd,

)

Rotor thrust, H-force, and moment coefficients.- The measured t h r u s t , H-force, and pitching-moment coefficients f o r t h e r o t o r alone a r e compared with r i g i d - &d flapping-rotor theories i n f i g u r e s 11 t o 13. The rigidr o t o r calculations consider the presence of a l i n e a r longitudinal variat i o n of inflow as described i n reference 7. The use of t h i s inflow d i s t r i b u t i o n does not imply t h a t e i t h e r t h e a c t u a l or t h e e f f e c t i v e inflow d i s t r i b u t i o n over the r o t o r d i s k i s accurately pictured i n t h i s manner. Its use i s j u s t i f i e d simply on t h e grounds that previous work ( r e f . 8) has shown improved agreement between theory and experiment i n t h e case o f t h e l a t e r a l flapping of a f u l l y a r t i c u l a t e d r o t o r . This l a t e r a l flapping i s equivalent t o t h e longit u d i n a l pitching moment of a r i g i d rotor. Figure 11 shows a comparison of measured t h r u s t c o e f f i c i e n t f o r t h e r o t o r alone with t h a t calculated f o r r i g i d and flapping r o t o r s . Throughout t h e speed range t e s t e d , t h e measured t h r u s t c o e f f i c i e n t s a r e i n reasonable agreement with theory. Variation of thrust c o e f f i c i e n t , measured and computed, i s primarily a function of r o t o r speed since t h e t h r u s t changes a r e small. It should be noted that t h e average contribution t o thrust (over an e n t i r e r o t o r revolution) of e i t h e r first-harmonic flapping o r of a l i n e a r inflow d i s t r i b u t i o n i s zero. Consequently, t h e r e s u l t s of the calculations f o r both the r i g i d and t h e flapping r o t o r s a r e i d e n t i c a l ( f i g . 11). Figure 12, a comparison of measured H-force c o e f f i c i e n t s with r i g i d and flapping-rotor calculations, indicates poor agreement between measurements and theory. The l a r g e discrepancies shown here may be p a r t l y a t t r i b u t a b l e t o t h e assumption of no interference between t h e r o t o r and the remainder of t h e a i r c r a f t . Figure 13 shows a comparison of measured r o t o r pitching-moment coeff i c i e n t s with theory f o r a flapping rotor, a r i g i d r o t o r with longitudinal inflow variation, and a r i g i d r o t o r with uniform inflow. The pitching moment f o r a flapping rotor, hinged a t t h e center of r o t a t i o n , i s of course zero. The r i g i d - r o t o r calculation w i t h uniform inflow i s a funct i o n of r o t o r coning only (eq. (~16)),and it accounts f o r l e s s than half of t h e measured pitching-moment c o e f f i c i e n t . The c a l c u l a t i o n f o r a r i g i d r o t o r with inflew vari=%icn ( f i g . 13) i n d i c a t e s t h a t t h e magnitude of pitching moment of a t r u l y rigid-rotor system probably could be

12 estimated with reasonable accuracy f o r speeds t o a t l e a s t 35 knots. Note that pitching moments f o r t h e r o t o r system t e s t e d , which is o n l y semirigid, would be increased by blade deformations which tend t o increase t h e coning angle, and would be decreased by deformations which increase flapping.

Rotor-alone s t a t i c - s t a b i l i t y derivatives. - Figure 1 4 shows a comparison of measured and computed s t a t i c - s t a b i l i t y d e r i v a t i v e s f o r t h e r o t o r alone as a f'unction of tip-speed r a t i o . Derivatives of thrust, H-force, and r o t o r pitching-moment c o e f f i c i e n t s with respect t o velocity, r o t o r speed, and angle of a t t a c k a r e included. Generally speaking, t h e calculated derivatives with respect t o angles of a t t a c k show t h e b e s t agreement with measured derivatives.

L 61 91

Figure 14(a) shows a comparison between t h e measured and t h e c a l culated derivatives of t h r u s t c o e f f i c i e n t with respect t o t i p speed. Because of t h e l a r g e discrepancy between t h e measured and t h e calculated derivatives, a discussion of t h e significance and use of these derivat i v e s in calculating thrust changes i s warranted. Note that

where the subscripts o and 1 represent t h e values before and after a change i n t i p speed. A s a consequence of t h e d e f i n i t i o n of t h r u s t coefficient, To and T1 a r e To = C

stR2p(l;lR),

So t h a t , upon s u b s t i t u t i n g equations ( 2 ) and

( 3 ) i n t o equation (1)

I n a sample case ( f o r t h e t e s t r o t o r a t p = 0.08), of equation ( 4 ) contributes only about 10 percent t o t h e change i n t h r u s t . Thus, the l a r g e discrepancies between urement shown by f i g u r e 1 4 ( a ) a r e of minor Fmportance i n

t h e f i n a l term t o t a l calculated theory and measpredicting

changes i n t h r u s t f o r changes i n t i p speed. of t h e derivative zero,

&T -

The physical significance

may be seen i n equation ( 4 ) .

becomes proportional t o differences i n

(srR)2.

a c ~i s s e t am Thus acT am

only a measure of t h e degree by which T departs from a p e r f e c t proporFigure 15 shows a comparison of t h e measured and t h e t i o n t o (sLR)2. calculated d e r i v a t i v e s of t h r u s t with respect t o m. Note t h e apparently improved agreement over t h a t shown f o r t h e d e r i v a t i v e

aCT am

ure 1 4 ( a ) .

Notice that

i s expressed as

aT -

i n fig-

can be found more simply i f t h e derivative

instead of

since

(5)

AT=AflR-

The foregoing discussion indicates t h a t it may o f t e n be simpler t o s e t up t h e equations of motion f o r r o t o r s i n dimensional r a t h e r than nondimensional terms. In t h i s manner, t h e equations may be considertibly shortened. Note, f o r example, t h e additional terms required i n order t o describe a change i n t h r u s t i n equation ( k ) , which uses t h a

*T -

rather

aT -

am'

CONCLUDING RFMARKS

The r e s u l t s of t h i s wind-tunnel i n v e s t i g a t i o n of t h e aerodynamic c h a r a c t e r i s t i c s of a stand-on helicopter i n forward f l i g h t i n d i c a t e t h a t t h e forward speed of t h e complete a i r c r a f t would be l i m i t e d t o below 17 knots because the s t a b i l i z i n g pitching moments would become g r e a t e r than t h e estimated available p i l o t control moment. Also measurements show t h a t t h e t i p clearance between t h e coaxial r o t o r s of t h e a i r c r a f t w a s never l e s s than about 5 inches during any phase of t h e t e s t s . An analytical. study of t h e i s o l a t e d r o t o r system i n d i c a t e s t h a t r i g i d - r o t o r pitching moments and s t a t i c - s t a b i l i t y d e r i v a t i v e s may be predicted w i t h reasonable accuracy, provided a longitudinal inflow

14 variation is assumed. Omission of the longitudinal inflow variation in some cases leads to large errors. For example, if onlythe effects of rotor coning are considered, less than one-half of the moment coefficients are predicted.

Langley Research Center, National Aeronautics and Space Administration, Langley Field, Va., September 23, 1959.

6 9 1

APPWM A

DEYELOPMENT OF DRAG AND PITCHING-MOMENT EQUATIONS FOR A R I G I D ROTOR Rotor Drag Coefficient

L 6 9

The equation f o r t h e profile-drag component of t h e r o t o r drag force H, as given by equation ( 5 0 ) on page 197 of reference 5 , is

The rotor-induced drag component, as given by equation (52) on page 197 of reference 5, i s

Hi =

k2x

-$ pac

f3 cos Jr + (BUTUP + Up2)sin

dq dr

where

and

cos

v 1 cos

cos

r @ dt

-v

cos a cos 9 s i n

(A41

equation f o r

vâ&#x20AC;&#x2DC; cos Jr cos p which has been added t o t h e R Up given i n reference 5 describes a l i n e a r v a r i a t i o n of

The expression

16 rotor induced velocity in the longitudinal plane (fig. 16). The expression for v' (ref. 7 ) in the terminology of this report is v' = Vt

X vo = vo tan 2

where X, the wake skew angle, is defined as

tan-1

-x

6 9 1

By definition

For a rigid rotor dB is zero and

angle

/3 equals the rotor coning dt A s coning is usually small, less than 8O, it is assumed that

and

Substituting expressions ( A 5 ) , ( A 6 ) , and ( A 7 ) into equations ( A 3 ) and (Ab), UT and Up may be written as

and up =

A ' Q r cos

* - qom

cos

Upon substituting expansions for UP and UT into equations (Al) and (A2) and integrating around the azimuth, the equation for H force is

which when integrated with respect to

r becomes

L 6 9 1

or, in coefficient form h'ao

cH =

oat': - + - o

pao2

+--

Rotor Pitching-Moment Coefficient C Im, The pitching moment of an element of a rigid rotor blade, illustrated in figure 16, may be expressed as

where nose-up moments are considered positive. The equation for the thrust increment dT as given on page 189 of reference 5 is

Substituting the above expression into equation (+U2), the differential moment equation becomes

18 A f t e r substituting equations (A8) and (Ag) i n equation (Al3) and performing the indicated integrations, t h e equation f o r r o t o r pitching moment becomes

or, i n coefficient form,

6 9

and with uniform inflow Cfm,,

E?(!) 4 3

APPENDIX B DEXJ3LOPMENT OF EQUATIONS FOR T”!3 STATIC-STABILITY DERIVATIVES

OF A R I G I D ROTOR FOR CHANGES I N SHAFT ANGLF: a, FORWARD

V, AND ROTOR TIP SPEED i2R

VELOCITY

Thrust Coefficient Derivatives The equation f o r t h r u s t coefficient as given on page l9O of r e f erence 5 is

By s u b s t i t u t i n g f o r

and

A, t h e preceding equations become

and

T =

1 pabc(QR)R(; 4

2 ~ a

+2 v

s i~n a -~

The p a r t i a l derivative of equation (Bl) with respect t o shaft angle of attack a is

The p a r t i a l of equation ( B 1 ) with respect t o forward v e l o c i t y

t h e p a r t i a l derivative with respect t o r o t o r - t i p speed

1 The p a r t i a l of t h r u s t given by equation (B2) with respect t o t i p speed i s

Rotor H-Force Derivatives After expanding p, equation ( A l l ) becomes

CH =

V cos a

%v'

A, and

a. 2V cos a

+-+

by means of t h e i r d e f i n i t i o n s ,

0V cos a ,

I V

sin a

vo) _.

The p a r t i a l derivative of equation (B7) with respect t o shaft a i s then

angle

The p a r t i a l d e r i v a t i v e with respect t o forward v e l o c i t y V

-1

039) L

The p a r t i a l derivative w i t h respect t o r o t o r t i p speed

S1R

6 9 1

Pitching-Moment Derivatives After s u b s t i t u t i n g the d e f i n i t i o n s f o r t h e r o t o r moment equation becomes

hâ&#x20AC;&#x2122;

and

i n equation (Al5)

The p a r t i a l d e r i v a t i v e of equation (B11) with respect t o s h a f t angle is

i The p a r t i a l d e r i v a t i v e with respect t o forward v e l o c i t y

and t h e partial derivative with respect t o r o t o r t i p speed

SZR

0314)

Expressions f o r t h e derivatives of vo and v â&#x20AC;&#x2122; with respect t o V, and ilR a r e developed i n appendixes C and D, respectively.

L 6 9 1

23 APPENDIX c DEVELOPMENT OF MPRESSIONS FOR INDUCED VELOCITY AND INDUCED VELOCITY DERIVATIVES FROM MOMENTUM AND BLADE-EIXMENT CONSIDERATIONS Expression for Average Rotor Induced Velocity vo From momentum theory

cTm

iFT2

but from the blade-element theory of reference

5 (eq.

(21), p. 190)

Therefore, by substituting equation ( C 2 ) into equation (Cl)

8(h2

p2rI2

from which the induced velocity may be found by iteration. Derivatives of Average Induced Velocity Equation (C3) may be written as

0 = O(v0,a,S1R,V)

24 where

e+ cp =

'VO

ep2

+ 1/2

8(A2 + p2)

+ ua

from which t h e desired derivatives may be obtained by i m p l i c i t d i f f e r e n t i a t i o n as follows:

6 9 1

where by p a r t i a l d i f f e r e n t i a t i o n of equation ( C 3 )

. and

a0 = --\A 2 ' m

p2>1/2

1/2

+ -ua

(m)2(A2+ v')

h 0

2VOh

By s u b s t i t u t i n g the above expressions i n t o equation (C>), t h e p a r t i a l vo with respect t o a becomes r

+ -aa

The p a r t i a l of manner i s

vo with respect t o V determined i n t h e same

or L

6 9

V~(V

m2h2 + Cr4

and t h e p a r t i a l of

. .

vo s i n a)

1/2

s ( 2 e p cos a

with respect t o

C!R

s i n a)

APPENDIX D DEVELOPMENT OF EQUATIONS FOR THE DERIVATIVES OF

For t h e l i n e a r longitudinal v a r i a t i o n of induced v e l o c i t y which i s assumed herein, v ' , as a consequence of i t s d e f i n i t i o n , i s t h e difference between the induced velocity a t t h e t r a i l i n g edge of t h e r o t o r disk and t h e mean induced velocity. (See f i g . 16.) From reference

7, i n t h e present notation

6 9 1

v ' = vo t a n

x 2

01)

= cp(v0,~)

Thus,

where

A s a consequence of equation (D3)

?E(?!-

ax\&, \

aa + ./' avo

The p a r t i a l d e r i v a t i v e of vo with respect t o a i s given by The following terms a r e obtained by e x p l i c i t d i f f e r equation ( C 6 ) . e n t i a t i o n s o f equations ( D l ) and (D2):

-h =

cos

Similarly an expression for the partial of v' with respect to V is obtained from equations (Dl) and (D3) and i s

Expressions for ?E,

avo

(C7), (D6), to

and

3 2 av

*ax'

and

are given by equations (D5),

(D7)' respectively. The partial of CP with respect

Making use of equations (Dl) and (D3), the partial of v' with respect to s2R is

where

is zero and the terms

by equations ( D 5 ) ,

(CS), (D6),

and

(D7), respectively.

are given

28 REFEEENCES 1. Zimmerman, C. H., Hill, Paul R., and Kennedy, T. L.: Preliminary Experimental Investigation of the Flight of a Person Supported by a Jet Thrust Device Attached to His Feet. NACA RM L52Dl0, 1953. 2. Hill, Paul R., and Kennedy, T. L.: Flight Tests of a Man Standing on a Platform Supported by a Teetering Rotor. NACA RM L54B12a,

1954.

3 . Townsend, M. W., Jr.:

Stability and Control of Unducted Stand-On Helicopters: Preliminary Theoretical and Flight Test Results. Rep. No. 404 (Contract DA 44-177-TC-392), Dept Aero. Eng., Princeton Univ., Nov. 1957. (Available From ASTIA as Doc. No. AD 120524.)

4. DeFrance, Smith J.: The N.A.C.A. Full-scale Wind Tunnel. NACA Rep. 459, 1933.

5. Gessow, Alfred, and Myers, G m r y C., Jr.: Aerodynamics of the Helicopter. The Macmillan Co., c.1952.

6. Harrington, Robert D.: Full-Scale-Tunnel Investigation of the StaticThrust Performance of a Coaxial Helicopter Rotor. NACA TN 2318, 1951-

7. Coleman, Robert P., Feingold, Arnold M., and Stempin, Carl W.: Evaluation of the Induced-Velocity Field of 811 Idealized Helicopter Rotor. NACA WR L-126,1945. (Formerly NACA ARR L5ElO.)

8. Wheatley, John B.: An Aerodynamic Analysis of the Autogiro Rotor With a Comparison Between Calculated and Experimental Results. NACA Rep. 487, 1934.

6 9 1

l I

a -a

Figure 2.- Test apparatus.

L-37-2916.I.

(a) Schematic diagram of aircraft.

Z I

(a) Wind axis system.

Figure

4.- Axis

and force notation showing positive directions of forces and moments.

(b) R o t o r axis system.

Figure

4.- Concluded.

D, l b u)

---

Figure 5.- Variations of a i r c r a f t l i f t , drag, yawing moment, pitching moment, and r o l l i n g moment with v e l o c i t y f o r t r i m angles of attack. Rotors removed, simulated p i l o t i n s t a l l e d .

35 0

knots

Figure 6.- Variations of a i r c r a f t angle of a t t a c k , lift, drag, and sideforce c o e f f i c i e n t s with velocity at trimmed f l i g h t conditions. Rotors operating.

. 0

knots

Figure 7.- Variation of aircraft pitching-, rolling-, and yawing-moment coefficients with velocity at trimmed flight conditions. Rotors operating. 70

0 , radians/sec

le v,

knots

Figure 8.- Variation of aircraft rotor speed and total horsepower with velocity at trimmed flight conditions. Rotors operating.

0 w

I n

s 0 )r\

0 (u

0 PI

aci anR â&#x20AC;&#x2122; aec/it

120

acl

- 9

1 radian8

-40 0

.04

.06

.12

.16

(a) Aircraft lift-coefficient derivatives.

Figure 10.- Measured static stability derivatives.

. -008

ac; aa

1 radlane

.08

cr (b) Aircraft drag-coefficient derivatives. Figire 10.- Continued.

.os

.12

* 16

( c) Aircraft pitching-moment coefficient derivatives. Figure 10.- Continued.

5Y 41

eec/ft

-06

LLLu_uII

(a) Aircraft rolling-moment coefficient derivatives. Figure 10.- Continued.

-.4

80X

40 a&, aa 1 radians

-40 0

-04

.08

.le

.16

CI.

(e) Aircraft yawing-moment coefficient derivatives. Figure 10.- Continued.

acQ, aa

1 radians

- 100 0

.04

.os

.12

w (f) Aircraft torque-coefficient derivatives. Figure 10.- Concluded.

v, knots Figure 11.- Comparison of experimental rotor-alone thrust coefficient with the calculated thrust coefficient for rigid and flapping rotors.

knots

Figure 12.- Comparison of experimental rotor-alone H-force coefficient with calculated H-force coefficient for rigid and flapping rotors.

v, c

knots

Figure 13.- Comparison of experimental rotor-alone pitching-moment coefficients with moment calculations f o r rigid and flapping rotors.

.04

.Od

.12

.16

(a) Rotor thrust-coefficient derivatives. Figure 14.- Comparison of experimental rotor-alone static stability derivatives with calculations based on rigid rotor with nonuniform inf10.~.

acH, av

eec/rt

.01

-*02

40x

-20 0

.04

.12

Ir.

(b1 %tor

&=a& coef ficient derivatives. Figure 14.- Continued.

aec/ft

200:

,-5

160

120

radians

-40 0

.04

.08

.12

.16

Figure 14.- Concluded.

2.0

1.6 *

,aT, lb-eec

afLR

1.2

.16

Lc.

Figure 15.- Comparison of experimental and celculated r o t o r - t h r u s t deriva t i v e s with respect t o rotor t i p speed.

\ Figure 16.- Thrust moment and induced velocity notation.