APPENDIX
APPENDIX ANALYSIS O F AIRCRAFT LATERAL MOTION CHARACTERISTICS WITH A SIMPLE STABILIZATION SYSTEM The function of a system of this type is to sense turning rate and apply aileron and rate. A limited theoretical analysis of system operation was under- rudder to null this taken to gain some insight into the factors which influence system performance.
It appears from the character of the system that the transfer function (the time relationship between sensed aircraft motion and opposing control movement) can be approximated by a first-order equation. When the equations of motion of the stabilized aircraft were derived, it was assumed that the motion consisted only of coordinated level turning and banking flight with no sideslip. Short-period modes were ignored in view of the relatively long period of the motion of interest. The stabilization system was assumed to act only through the ailerons. The dynamics of the basic control system were neglected. The motion of the aircraft can then be described by the equations: ( D 2 - a D ) @ - h V D + - c d , = O g where (V/b) c = - l C
C L S 2
For the type of motion considered (that is, no sideslip and moderate @) + and @ a r e
related by
APPENDIX
I APPENDIX The aileron deflection 6, was related t o the control force F by By its nature, the simple pneumatic stabilization system provides a substantial time lag between the sensing of an angular velocity by the gyro and the application of control force.
It is assumed that the effect of this lag in the action of the system can be approximated by the transfer function It appears that some lag in the reaction of the system is probably desirable to avoid the possibility of degrading the short-period lateral motion characteristics of the airplane.
T for the present system was not determined directly but was The time constant estimated to be about 4 seconds.
The lag in the system, as represented by T, has a destabilizing effect on the motion, which can be counteracted by tilting the gyro axis so that the gyro senses a com- ponent of rolling velocity as well as that of the turning velocity. The angular velocity sensed by the gyro u is then g Equation (Al) then becomes, upon substitution of equations (A2) to (A5),
b3+ ( : - a)D2+ (-h -: - $ D + (- -;I@= 0
where ‘ 2 , KFKS U a f =
APPENDIX
APPENDIX Solution of equation (A6) yields one large root hl, which represents a rapid subsidence and hence is of little concern, and a pair of roots X2 and X3 which describe the pri- mary characteristics of the motion.
I f the roots of the equation can be determined closely enough for conditions of interest as C (Ai)
'2'3 = 7
B '2'3
- -
h 2 + X = - X1 x1 from The damping ratio ( is then determined ( = - 1 '2 + '3
/-Ti
required for a desired damping ratio is then determined by sub- The gyro tilt angle
%
stituting varying tilt angles in the coefficients B and C, computing the corresponding
values of 5, and plotting these values against e These computations have been made
g' for two airspeeds and two values of 7, by using the coefficient values given in the sec- tion "Symbols," and the results a r e shown in figure 3.
The lateral motions of the airplane in hands-off recovery from a turn were com- puted, by using equation (A6), with gyro tilt angles of 0 ' and 35' for comparison with flight-test measurements. These results are given in figure 6.
REFERENCES 1. Brunstein, Alan I.: A Study of Fatal, General Aviation Accidents Involving Weather.
BOSP 7-4-1, CAB, July 1963.
2. Anon.: Summary Reports of Accidents - U.S. Civil Aviation. CAB, June 15 - Oct. 15, 1965.
3. Anon.: FAA Statistical Handbook of Aviation, FAA, Sept. 1964.
4. Williams, Alexander C., Jr.; Houston, Robert C.; and Wilkerson, Lowell E.: Simul- taneous Contact-Instrument Flight Training. Univ. of Illinois Bull., vol. 53, no. 42, Jan. 1956.
5. Campbell, John P.; Hunter, Paul A,; Hewes, Donald E.; and Whitten, James B.: Flight Investigation of the Effect of Control Centering Springs on the Apparent Spiral Stability of a Personal-Owner Airplane. NACA Rep. 1092, 1952. (Supersedes NACA TN 2413.)
6. Phillips, William H.; Kuehnel, Helmut A.; and Whitten, James B.: Flight Investigation of the Effectiveness on an Automatic Aileron Trim Control Device for Personal Airplanes. NACA Rep. 1304, 1957. (Supersedes NACA TN 3637.)
I !
Figure 1.- Lateral stability augmentation system.
Figure 2.- Aircraft flight instruments.
L-66-4524
5 0 1 -
I
-
4 0
/-- ' ! H a n d s OFF, system OFF Bank angle, deg Q,
20 -
Hands OFIF, system ON ( 0 10'-
- 35')
g i I I I I I I
0 2 4 6 8 10 12 14 1 6
Time, t, s e e Figure 3.- Lateral response with system on and off and with pilot-controlled recovery. Indicated airspeed, 135 knots; altitude, 5000 feet.
40 -
GPO tilt angle 55' 20 - 10 - .- - Aileron angle 30 - 4- Gyro tilt angle 35' 3 - M a
t ?
.I 2 - a 20 -Bank angle W a; rl rd 0 - d 2 1- d
/ \-Aileron angle
2 - 3- J I I 1 1 I I .
0 2 4 6 - 8 ~ 10 12 1 4
Time, t, sec Figure 4.- Lateral motion and aileron response following control release during a t u r n with various gyro tilt angles 8 as measured i n flight at 135 knots indicated airspeed and 5000 feet altitude.
4-
“r
Gyro tilt angle 35O 3 - 2 - M rd ., 1- cd ., 0 - aJ rl M
2 1 -
c k l 2 - rl 3 - 4- 5- 5OL 4-
40r
Gyro tilt angle Oo 3 - 2 - 1- M I) a M aJ 0 - a 1- .I a, rl M - c 2 - 20 cd 3 - A i l e r o n angle 4- 5 - 6 - 7- 70- I I I I I I I J I I I I I I I J 0 2 4 6 8 io 12 14 0 2 4 6 8 io 12 14 Time, t, sec Time, t, sec Calculated lateral motion and aileron response after control release during a t u r n with various gyro tilt angles 0 Figure 5.- Time constant, g‘ T = 4 sec; indicated airspeed 135 knots.
/ / T = 3 see; V = 135 knots V = 135 knots M 1.0- - C r i t i c a l damping- / .
c
0.5- -’// \
LT = 4 see; V = 90 knots
1 I I I 0 1 0 20 30 40 Gyro tilt angle, 0 deg g ’ Figure 6.- Calculated effect of gyro tilt angle on damping ratio.
lb newtons 91-
'i 30 -
I L a t e r a l
5 I -
control 20j- f o r c e , F
4-
I I
0 1 2 3 4 5 6
Rate of turn Y, deg/sec
I I I I 1
0 1 0 20 30 40
Angle of bank, a, deg Figure 7.- Lateral-control force d u r i n g a steady t u r n , at 135 knots indicated airspeed and 5M)O feet altitude. 0 = 3 5 ' .
D E '
-
0 1 0 Naut i ca 1 miles N Figure 8.- Flight pattern of noninstrument-rated pilot during cross-country flight.
r
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