Document
- -
4kZ- Or, NASA TN D-81:
TECHNICAL NOTE
D-812
EFFECTS OF GYROsCOPIC CROSS COUPLING BETWEEN PITCH AND
ROLL ON THE HANDLING QUALITIES OF VTOL AIRCRAFT By John F. Garren, Jr.
Langley Research Center Langley Field, *.
.'ji
NATIONAL AERONAUTICS AND S P A C E ADMINISTRATION
WASHINGTON April 1961
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION TECHNICAL NOTE D-812 EFFECTS OF GYROSCOPIC CROSS COUPLING BETWEEN PITCH AND R O I L ON THE HANDLING QUALITIES OF VTOL AIRCRAFT By John F. Garren, Jr.
SUMMARY In order t o provide information r e l a t i v e t o the e f f e c t s of gyro- scopic cross coupling between p i t c h and roll on the handling q u a l i t i e s of VTOL a i r c r a f t , a f l i g h t investigation has been conducted during which cross coupling w a s simulated. Generality i s achieved by presenting the a c r i t e r i o n which r e s u l t s of the f l i g h t investigation i n the form of may be used t o predict the a c c e p t a b i l i t y of the l e v e l of cross coupling i n VTOL a i r c r a f t as a function of the a i r c r a f t design parameters. The c r i t e r i o n i s based on p i l o t s ' opinions of the a c c e p t a b i l i t y of t h e air- c r a f t motions f o r the range of cross coupling which w a s simulated during a maneuver i n which cross coupling i s p a r t i c u l a r l y objectionable. Theory i s used t o provide a basis f o r application of the c r i t e r i o n . The theory which i s developed i s shown t o predict accurately the a i r c r a f t motions.
The p i l o t s agreed t h a t the available control power determines t o a g r e a t extent the amount of cross coupling which can be tolerated. The t h e o r e t i c a l investigation indicates the extent t o which the damping and moments of i n e r t i a of the a i r c r a f t as well as the angular momentum of the engine influence cross coupling.
INTRODUCTION Recent j e t VTOL a i r c r a f t have exhibited undesirable motions attrib- utable t o gyroscopic moments a r i s i n g from the large angular momentum produced by the j e t engines. A s s t a t e d i n reference 1, even l a r g e r gyroscopic moments than those present i n conventional jet a i r c r a f t are l i k e l y t o be present i n VTOL a i r c r a f t . This s i t u a t i o n a r i s e s because VTOL a i r c r a f t generally require a g r e a t e r thrust weight r a t i o than con- ventional a i r c r a f t ; therefore, other factors being equal, the engine dimensions and weight are increased, and hence angular momentum i s increased. Furthermore, since the provision of high control power and high damping i n VT0L configurations i s l i k e l y t o involve s p e c i a l pen- a l t i e s , the VTOL a i r c r a f t may tend t o have low values of c o n t r o l power and damping as compared with high-speed airplanes. Thus, even i f the angular momentum of engines f o r VTOL a i r c r a f t were not greater, t h e r e l a t i v e l y poor control power and low damping i n the low-speed and hov- ering ranges would tend t o highlight the gyroscopic e f f e c t s f o r these f l i g h t conditions.
Coupling of a i r c r a f t motions due t o gyroscopic moments occurs when- ever an a i r c r a f t contains r o t a t i n g components having a n e t angular momen- tum i n any direction. If it i s assumed t h a t the angular momentum vector (as determined by the right-hand r u l e ) i s along one of the t h r e e P r i n c i p a l i n e r t i a axes, cross coupling w i l l be present between the o t h e r two axes.
This cross coupling i s characterized by a gyroscopic moment about one axis proportional t o the angular v e l o c i t y about the o t h e r axis. Specif- ically, i n the case of p i t c h - r o l l cross coupling, a pitching acceleration proportional t o the r o l l i n g v e l o c i t y i s produced and a r o l l i n g accelera- tion proportional t o the pitching v e l o c i t y i s produced.
The source of gyroscopic cross coupling between p i t c h and roll i s mass r o t a t i n g about the v e r t i c a l a x i s - f o r example, j e t engines mounted v e r t i c a l l y . Although cross coupling may be eliminated by the use of counterrotating engines or by electronics, the a i r c r a f t must be control- lable i n the event of f a i l u r e of one o r more of s e v e r a l counterrotating engines or of the e l e c t r o n i c system.
A v a r i a b l e - s t a b i l i t y research helicopter w a s used t o simulate a wide range of cross coupling. An investigation of the angular momentum of t y p i c a l present-day j e t engines and of the probable s i z e of a i r c r a f t i n which these engines may be u t i l i z e d indicated t h a t the range of coupled responses simulated w a s adequate. The flight tests included several d i f f e r e n t f l i g h t conditions selected as representative of maneu- vers normally associated with VTOL a i r c r a f t , including maneuvers i n which cross coupling i s expected t o present the g r e a t e s t problems. The r e s u l t s of these tests, along with a discussion of t h e e f f e c t s on t h e coupled response (response produced by gyroscopic moment) of various parameters such a s t h e damping and moment of i n e r t i a of the a i r c r a f t , are presented herein. Generality i s achieved by presenting the r e s u l t s of t h e f l i g h t investigation i n t h e form of a c r i t e r i o n which may be used t o p r e d i c t the a c c e p t a b i l i t y of t h e l e v e l of cross coupling as a function of t h e a i r c r a f t design parameters. The c r i t e r i o n i s based on p i l o t s ' opinions of the a c c e p t a b i l i t y of the a i r c r a f t motions f o r the range of cross coupling which w a s simulated during a maneuver i n which cross coupling i s p a r t i c u l a r l y objectionable. Theory i s used t o provide a b a s i s f o r the application of the c r i t e r i o n . The theory which i s developed i s shown t o p r e d i c t accurately the a i r c r a f t motions.
Inasmuch as i n i t i a l t e s t s indicated t h a t t h e gyroscopic coupling never Produced an objectionable response about t h e longitudinal axis, t h i s investigation deals primarily with t h e coupled response occurring about t h e lateral axis, that is, the pitching acceleration proportional t o r o l l i n g velocity.
SYMBOLS damping moment proportional t o and opposing r o l l i n g velocity, *P ' lb-ft/radian/sec damping moment proportional t o and opposing pitching velocity,
%
lb-ft/radian/sec H n e t angular momentum of m a s s r o t a t i n g about v e r t i c a l axis, s lug-+/se c moment of i n e r t i a about body X - a x i s , slug-ft2 IX moment of i n e r t i a about body Y - a x i s , slug-ft2 =Y moment of i n e r t i a about body Z - a x i s , s l u g - f t *Z lateral c o n t r o l moment per inch s t i c k deflection, l b - f t / i n .
M6 r o l l i n g velocity, radians /sec P r o l l i n g accelerations, radians /se c
h
-
Laplace transform of p P pitching ve loc i t y , radians /se c pitching acceleration, radians/sec2
i
-
Laplace transform of q S Laplacian variable t time, sec 6 s t i c k displacement, in.
TEST EQUIPMENT AND PROCEDURES Helicopter All f l i g h t s i n which gyroscopic cross coupling between p i t c h and r o l l motions was simulated were performed with t h e v a r i a b l e - s t a b i l i t y helicopter shown i n f i g u r e 1. "he t e s t helicopter contains a vjxiable control system which makes it possible t o vary both the r a t i o of control moment t o s t i c k deflection, o r control power, and the apparent angular- velocity damping about each of the three p r i n c i p a l i n e r t i a axes. The components of which were used t o produce the variable control system, cross coupling, i s described i n reference 2. The helicopter i s equipped t o record angular v e l o c i t y about a l l three axes, airspeed, and a l l con- t r o l motions of the p i l o t . The general physical c h a r a c t e r i s t i c s of the helicopter are given i n t a b l e I.
Simulation The t e s t helicopter w a s the simulator. For t h e studies of r e f e r - ence 2, signals proportional t o helicopter r a t e operated continuously about a l l three axes t o provide additional angular-velocity damping.
These signals were generated by r a t e gyroscopes. For purposes of t h i s investigation, the simulation w a s accomplished by repositioning both the pitch-rate gyroscope and the r o l l - r a t e gyroscope so t h a t a s i g n a l proportional t o the r o l l i n g v e l o c i t y would produce a pitching accelera- t i o n and a signal proportional t o t h e pitching v e l o c i t y would produce a r o l l i n g acceleration.
F l i g h t Conditions The e f f e c t s of gyroscopic cross coupling on the handling q u a l i t i e s of the a i r c r a f t were investigated during hovering, low-speed instrument- landing-system (ILS) approaches, steep turns, and rapid r o l l reversals a t 30 t o 35 knots. The range of cross coupling covered during each f l i g h t condition, i n terms of i t s most s i g n i f i c a n t parameter - the r a t i o of angular momentum t o moment of i n e r t i a , i s l i s t e d i n t a b l e 11. For an i t s coupled axis ( a x i s about which a i r c r a f t with negligible damping about i s produced), t h i s r a t i o i s numerically equal t o the coupled response axis per u n i t angular v e l o c i t y acceleration produced about the coupled about the other axis.
TREORY The magnitude of the motions due t o gyroscopic cross coupling simulated i n t h e t e s t helicopter was predicted by using the equations developed i n t h e appendix. Equations ( 5 ) o r (6) of the appendix express the r o l l i n g and pitching angular-velocity responses r e s u l t i n g from a l a t e r a l s t e p input as functions of the moments of i n e r t i a and damping about each of the coupled axes, the control power about the l a t e r a l axis, and t h e n e t angular momentum of the r o t a t i n g components. Specifically, equation (6a) gives the lateral angular-velocity response t o a l a t e r a l step input and equation (6b) gives t h e simultaneous longitudinal angular-
v e l o c i t y response - the coupled response. Thus, with the use of these
equations, angular-velocity t i m e h i s t o r i e s may be obtained f o r any air- c r a f t f o r which the c i t e d parameters are available. By comparison of these computed time h i s t o r i e s with time h i s t o r i e s which have been cor- r e l a t e d with a i r c r a f t motions which were, i n turn, correlazed with p i l o t s ' opinions from f l i g h t t e s t s , a q u a l i t a t i v e estimate can be made of the a c c e p t a b i l i t y of the cross coupling present.
RESULTS F l i g h t Results Inasmuch as i n i t i a l f l i g h t t e s t s indicated t h a t a r o l l i n g response due t o pitching v e l o c i t y w a s not a problem i n any of t h e maneuvers, a t t e n t i o n w a s focused primarily on pitching response due t o r o l l i n g ve l o c i t y .
Hovering.- During the hovering maneuver, which involved only attempts at holding the a i r c r a f t absolutely motionless, the coupled response ( p i t c h acceleration due t o roll v e l o c i t y ) about the longitudinal a x i s w a s varied over the range given i n t a b l e 11. Even a t the maximum value of coupling, t h e p i l o t reported t h a t no coupled response w a s apparent.
hw-speed ILS approaches. - Low-speed ILS approaches were made by
u t i l i z i n g s e v e r a l longitudinal c o n t r o l powers, each with various amounts of cross coupling within the range indicated i n t a b l e 11. The coupled response became objectionable only f o r t h e condition i n which t h e cross coupling had the maximum value given f o r t h i s maneuver and the longitudi- n a l c o n t r o l power w a s simultaneously one-half t h a t of the basic t e s t helicopter. For t h i s extreme condition the p i l o t s t a t e d t h a t occasion- a l l y he w a s forced t o use m a x i m u m available longitudinal c o n t r o l t o cor- r e c t f o r t h e coupled response.
Steep turns and r o l l reversals.- Steep t u r n s and r o l l reversals are quite similar t o one another from the standpoint of cross-coupling e f f e c t s .
"he l a t t e r maneuver ( r o l l reversals) w a s selected as the f o c a l point of t h i s study because it permitted g r e a t e r roll r a t e s of longer duration than were f e a s i b l e o r even possible with any of the other maneuvers.
Average r o l l rates of 0.5 radian/sec were maintained during a t t i t u d e 3 0 ' r i g h t t o 3 0 ° l e f t while the cross coupling changes from bank angles of was varied over the range indicated i n t a b l e 11. For t h i s maneuver, ratings of various amounts of gyroscopic cross coupling were obtained from three p i l o t s . The p i l o t s ' r a t i n g s f o r the d i f f e r e n t values of cross coupling were i n substantial agreement. The r e s u l t s are tabulated as follows : Gyroscopic cross coupling,
- H radians/sec2
P i l o t s ' r a t i n g Iy' radianslsec ~ ~~~~~~~ 0.11 A c cep t able .22 Marginal Poor ' 33
.44 Unacceptable
Analytical Results The a i r c r a f t motions f o r a l a t e r a l s t e p input were correlated with computed angular-velocity time h i s t o r i e s predicted by equations (6) i n the appendix. Figure 2 contains a p l o t of equations (6) f o r the known helicopter parameters and a selected value of angular momentum H. The test-point symbols, which indicate reasonably good agreement with the theory, represent the experimental angular-velocity response about the r o l l axis and t h e p i t c h axis f o r a l a t e r a l s t e p input f o r t h e same value of angular momentum H s e t i n t o t h e simulator.
Although the p i l o t s ' r a t i n g s of various amounts of cross coupling f o r a sustained r o l l - r e v e r s a l maneuver have been tabulated i n terms of H/Iy, i n essence, however, t h e p i l o t i s a c t u a l l y r a t i n g t h e amount of a given l a t e r a l input.
coupled response which he experiences f o r 4/p The r a t i o t / p i s a function both of H / I y and M&y. For the r o l l - reversal maneuver the input resulted i n an average r o l l r a t e of about 0.5 radian/sec; t h e coupled response w a s the pitching velocity, which the p i l o t attempted t o n u l l i f y with a marginal longitudinal control sys- tem. O n the b a s i s of the p i l o t s ' r a t i n g s f o r the d i f f e r e n t amounts of gyroscopic coupling and the known physical c h a r a c t e r i s t i c s of t h e t e s t helicopter, the curves i n figure 3 were p l o t t e d by using equations ( 5 ) i n t h e appendix. The dashed curve represents the l a t e r a l angular veloc- i t y r e s u l t i n g from the l a t e r a l input. The two s o l i d curves, which form t h e cross-coupling boundaries, represent the coupled response about the axis f o r d i f f e r e n t values of gyroscopic coupling but f o r the p i t c h same lateral input.
DISCUSSION The r e l a t i v e insignificance of gyroscopic e f f e c t s during the precision-type maneuvers of hovering and low-speed ILS approaches i s a t t r i b u t e d t o the absence of both large pitching and large r o l l i n g motions of t h e a i r c r a f t . For a similar reason, that is, absence of large pitching v e l o c i t i e s i n a l l maneuvers, the p i l o t s reported t h a t there w a s l i t t l e or no apparent r o l l i n g due t o cross coupling during any of the maneuvers.
Only pitching due t o r o l l w a s of s u f f i c i e n t magnitude t o cause a problem.
Since it w a s not possible t o vary t h e a i r c r a f t damping about the coupled axes during the simulation (the variable damping system was being used t o provide t h e coupling s i g n a l ) , t h e e f f e c t of angular-velocity damping on the coupled response was calculated by using the equations
presented i n the appendix. Figure 4 contrasts the coupled response -
longitudinal response f o r a l a t e r a l input - of an a i r c r a f t with zero
longitudinal damping and the coupled response of the same a i r c r a f t with i t s longitudipal damping increased t o t h a t found desirable f o r hovering and low-speed flight i n reference 3. A more q u a n t i t a t i v e measure of the e f f e c t of damping may be obtained by comparing the steady-state angular v e l o c i t y about t h e input axis with t h a t about the coupled a x i s f o r a s t e p input. With the use of equations ( 5 ) or (6) i n the appendix, the r a t i o of steady-state angular v e l o c i t i e s (q/p)t=m i s given by H P q . Hence, f o r a given input angular v e l o c i t y about t h e l a t e r a l axis, t h e steady- state angular v e l o c i t y about the longitudinal axis i s inversely propor- t i o n a l t o t h e longitudinal damping.
I n l i g h t of the foregoing statements, the r a t i o of n e t angular momentum t o a i r c r a f t moment of i n e r t i a i s not a r e l i a b l e c r i t e r i o n f o r determining t h e magnitude and acceptability of the cross coupling present i n a given a i r c r a f t , since it does not take i n t o account t h e a i r c r a f t damping. Therefore, f o r an a i r c r a f t i n which cross coupling i s a n t i c i - pated, it i s advisable t o compare computed angular-velocity time h i s t o r i e s with those given i n f i g u r e 3 i n order t c de+,ernine t h e a c c e p t a b i l i t y of cross coupling. However, it should be noted t h a t the r a t i n g s presented i n f i g u r e 3 were established f o r a specific maneuver and f o r t h e basic For a n a i r c r a f t performing maneuvers c o n t r o l power of the t e s t vehicle.
involving roll rates g r e a t e r than 0.5 radian/sec the requirements would The p i l o t s agreed t h a t with g r e a t e r available c o n t r o l be more s t r i n g e n t .
power they would be able t o t o l e r a t e more cross coupling. Conversely, with l e s s available control power, t h e amount of cross coupling which could be t o l e r a t e d would be correspondingly decreased.
The f a c t t h a t recent VTOL a i r c r a f t have exhibited adverse cross- coupling e f f e c t s while performing even mild maneuvers such as precision hoyering i s a t t r i b u t e d i n p a r t t o t h e i r negligibly low damping and t o t h e i r low control power during t h i s f l i g h t condition. The low control power and low damping, which are present i n many of the p a s t VTOL air- c r a f t designs during hovering and t r a n s i t i o n phases, are traceable t o a lack of a ready source f o r producing desirable control and damping moment during t h i s f l i g h t condition. Although increases i n both control power and damping are expected, the increases are not l i k e l y t o eliminate e n t i r e l y the problems associated with cross coupling f o r a l l maneuvers.
Thus a means should be sought t o minimize t h e gyroscopic moment i n i t s own r i g h t .
CONCLUSIONS As a r e s u l t of f l i g h t t e s t s and a t h e o r e t i c a l investigation of gyro- scopic cross coupling between p i t c h and r o l l i n VTOL a i r c r a f t , the f o l - lowing conclusions are drawn: 1. For a i r c r a f t which meet requirements f o r control power and angular-velocity damping, gyroscopic cross coupling between p i t c h and r o l l i s not l i k e l y t o present a problem f o r mild, precision maneuvers such as low-speed instrument-landing-system approaches o r hovering. The r o l l r a t e s encountered during these maneuvers might be s i m i l a r t o those encountered i n commerical-type operations such as t r a n s p o r t or passenger .
carrying .
2. Cross coupling i s a problem i n sustained r o l l maneuvers performed at low speed such as might be encountered i n m i l i t a r y operations, f o r example, where the a i r c r a f t i s used as a weapon platform and considerable maneuvering m i g h t be required f o r moving i n t o and away from t h e combat area.
3. A coupled response i s more l i k e l y t o occur and t o be a problem about the p i t c h a x i s r a t h e r than about the r o l l axis, because of the absence of large pitching v e l o c i t i e s i n the majority of maneuvers.
4. The magnitude of the coupled response which w i l l occur i n a p a r t i c u l a r a i r c r a f t i s inversely proportional t o the a i r c r a f t moment of i n e r t i a about the coupled axis and decreases s i g n i f i c a n t l y with increased damping moment about the 'coupled axis. The amount of coupled response which can be tolerated i s increased with an increase i n longitudinal control power inasmuch as it enables the p i l o t t o compensate more readily f o r the coupling.
Langley Research Center, National Aeronautics and Space Administration, Langley Field, Va., February 13, 1961.
APPENDIX DERIVATION OF ANGULAR-VELOCITY RESPONSE FOR A LATERAL STEP INPUT WITH CROSS COUPLING PRESENT For a lateral displace-and-hold type of input, t h e moment equations about the l a t e r a l and longitudinal axes, respectively, are Mq H q + - q - - p = o
J
=Y =Y Assuming the i n i t i a l conditions p(0) = q(0) = 0 and taking the Laplace transform of equations (1) y i e l d s MP H -
S P + - 5 + - q = 6
Ix IX IX
Solving a l g e b r a i c a l l y for and gives transform i s taken and the After the inverse terms are rearranged, equa- t i o n s ( 3 ) become I
r
,.
% H L
+ b - - t I
Ix IY
Equations (4) , when simplified, become
c
The l a t e r a l and longitudinal angular-velocity responses t o a l a t e r a l step input are given by equations ( 5 ) when IXIY
k-$-T& 4H2 < 0, equations ( 5 ) become
+ + J REFEFENCES 1 . Roy, R . E., and Carmichael, R . P. : Propulsion Problems f o r Vertical
Take-Off Aircraft. 5 6 ~ ~ ~ - 1 6 5 7 (AD 83376), Power Plant Lab., Wright
Air Dev. Center, Mar. 1956.
2. Salmirs, Seymour, and Tapscott, Robert J.: Instrument Flight Trials With a Helicopter Stabilized in Attitude About Each Axis Individu- ally. NACA TN 3947, 1957.
3 . Salmirs, Seymour, and Tapscott, Robert J.: The Effects of Various Combinations of I)amping and Control Power on Helicopter Handling Qualities During Both Instrument and Visual Flight. NASA TN D-58, 1959.
TABLE I.- PHYSICAL CHARACTERISTICS OF THE TEST HELICOPTER
Gross weight, l b . . . . . . . . . . . . . . . . . . . . . . . . 5,500
Moments of i n e r t i a :
Pitch, IY, Slug-ft . . . . . . . . . . . . . . . . . . . . . 7,000
Roll, Ix, Slug-ft 2
. . . . . . . . . . . . . . . . . . . . . . 2,000
Yaw, IZ, slug-ft 2
. . . . . . . . . . . . . . . . . . . . . . 5,000
Number of blades i n main r o t o r 3 . . . . . . . . . . . . . . . . .
Rotor r o t a t i o n a l speed, radians/sec . . . . . . . . . . . . . . 19.4
Rotor diameter, f t . . . . . . . . . . . . . . . . . . . . . . . 48
. . .
Height of r o t o r hub with respect t o center of gravity, f t 6.5
Blade mass f a c t o r . . . . . . . . . . . . . . . . . . . . . . . 9
Control t r a v e l :
Longitudinal cyclic, in. . . . . . . . . . . . . . . . . . . 13.6
Lateral cyclic, in. . . . . . . . . . . . . . . . . . . . . . 13.6
Pedal, i n . . . . . . . . . . . . . . . . . . . . . . . . . . 4.75
Basic control power:
Pitch, f t - l b / i n . of control t r a v e l . . . . . . . . . . . . . . 508
Roll, f t - l b / i n . of control t r a v e l . . . . . . . . . . . . . . 474
Yaw, f t - l b / i n . of control t r a v e l . . . . . . . . . . . . . . . 4,140
Basic damping:
Pitch, ft-lb/radian/sec . . . . . . . . . . . . . . . . . . . 2,495
Roll, ft-lb/radian/sec . . . . . . . . . . . . . . . . . . . . 2,495
Yaw, ft-lb/radian/sec . . . . . . . . . . . . . . . . . . . . 10,600
TABLE 11.- RANGE OF CROSS COUPLING SIMULATED Pitch, R o l l , H radians/sec2 H radians/sec2 Flight condition
-
Iy’ radians/sec Ix’ radians/sec
Hovering 0 to 1.40 o to 1.05
Low-speed ILS approaches 0 to .60 o to 1.05
Steep turns 0 to .44 o to 1.05
0 to 1.05
Roll. reversals 0 to .44
c i d M I rl 0 20 Theory Exp
R o l l -- 0
P i t c h 0 10 Figure 2.- Comparison of predicted and experimental angular-velocity time histories f o r a lateral step input.
\ I
\
\
\
\
\
\
\
\
\
\
\
\
\
0 \ \ \ I I I 1 \ In In 0 In
? t
rl ?
/ :
/ A Damping found desirable
d e
by reference 3
cd 1 . 0 1 . 5 2 . 0 Time, see
Figure 4.- The effect of longitudinal damping on the longitudinal
response for a lateral input.