APPENDIX
APPENDIX EQUATIONS OF MOTION AND ASSOCIATED FORMULAS The equations of motion used for this simulation project are
i~ = -g sin e + vr - wq + -
The following formulas were also used:
v = ("2 + v2 + w2) 1/2
a = tan-1 $
p = sin-1 E V
APPENDIX
APPENDIX
8 = q cos @ - r sin @
4 = p + q tan 0 sin @ + r tan 0 cos @
r cos @ + q sin @
$ =
cos I 9
h = u sin 8 - v cos 0 sin @ - w cos 0 cos @
+ - wp + ur - g cos e sin @
aY = g REFERENCES Ground Simulator Evaluations of Coupled Roll-Spiral Mode Effects 1. Newell, F. D.: on Aircraft Handling Qualities. AFFDL-TR-65-39, U.S. Air Force, Mar. 1965.
(Available from DDC as AD 617353.)
2. Anon.: Stability and Control. FTC.-TIH-64-2004, U.S. Air Force, 1964. (Available
from DDC as AD No. 443 699.)
3. Ashkenas, I. L.: Some Open- and Closed-Loop Aspects of Airplane Lateral-Directional AGARD Rep. 533, May 1966.
Handling Qualities.
4. Stapleford, Robert L.; and Tennant, J. Alford: Analysis of Several Handling Quality
Topics Pertinent to Advanced Manned Aircraft. Section I11 - Lateral-Phugoid
Mode Effects on Airplane Handling Qualities. AFFDL-TR-67-2, June 1967.
TABLE I. - SIMULATOR CONTROL CHARACTERISTICS
Force per 1 Gearings from cockpit control 1
Maximum ,
Breakout force 1 deflection
travel
to control surface I
Control , - ,
lbf /in. N/cm 1 deg/in. deg/cm 1
1 ; i cm lbf , N ~~ -
Stick (pitch) A2.7 1.0 to 1.5 4.45 to 6.66 3.5 Stick (roll) *3.3 k8.4 <1.0 c4.45 3.5 Pedal 3.3 8.4 5.0 2.22 14 *6,/6, for phase I1 was 12 deg/in. (4.7 deg/cm).
TABLE I I . - MASS AND DIMENSIONAL CHARACTERISTICS
OF HYPOTHETICAL AIRPLANE Weight. lbf (N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 000 (222 400)
Wing area. ft2 (m2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 (52.1)
Wing span. ft (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 (18.3)
Mean aerodynamic chord. ft (m) . . . . . . . . . . . . . . . . . . . . . . . 9.4 (2.9)
Ix. slug-ft2 (kg-m2) . . . . . . . . . . . . . . . . . . . . . . . . . . 78 000 (105 690)
IY. slug-ft2 (kg-m2) . . . . . . . . . . . . . . . . . . . . . . . . . . 260 000 (352 300)
. . . . . ' . . . . . . . . . . . . . . . . . . . . . 325 000 (440 375)
Iz. slug-ft2 (kg-m2)
TABLE III . . LATERAL-DIRECTIONAL AERODYNAMIC INPUTS AND DYNAMIC
STABILITY CHARACTERISTICS OF THE BASIC CONFIGURATIONS Phase I Phase 1 1 . . . . Aerodynamic inputs:
.-
~0.0573 -0.0573 CzB per radian . . . . . . . . . . . . . . . . . . . . . . . . . .
0.0573 0.0573 Czsa, per radian . . . . . . . . . . . . . . . . . . . . . . . . . .
0.0015 0.0015 Czsr. per radian . . . . . . . . . . . . . . . . . . . . . . . . . .
-0.5000 -0.5000 Cz p e r r a d i a n . . . . . . . . . . . . . . . . . . . . . . . . . .
P
0.1000 0.1000 . . . . . . . . . . . . .. . . . . . . . . . . . . .
2zr. per radian
0.1146 0.1146 Cn . per radian . . . . . . . . . . . . . . . . . . . . . . . . . .
P
Cnsa. per radian . . . . . . . . . . . . . . . . . . . . . . . . .
-0.0401 -0.0401 C q r . per radian . . . . . . . . . . . . . . . . . . . . . . . . .
0.0400 Cnp. per radian . . . . . . . . . . . . . . . . . . . . . . . . . .
-0.7000 -1.0000 Cnr. perradian . . . . . . . . . . . . . . . . . . . . . . . . . .
-0.2865 -0.286 5 CyB per radian . . . . . . . . . . . . . . . . . . . . . . . . . .
0.0229 0.0229 Cysa. per radian . . . . . . . . . . . . . . . . . . . . . . . . .
0.0573 0.0573 Cybr. per radian . . . . . . . . . . . . . . . . . . . . . . . . .
0.2000 0.2000 Cyp. per radian . . . . . . . . . . . . . . . . . . . . . . . . . .
0.3000 0.3000 Cy,. per radian . . . . . . . . . . . . . . . . . . . . . . . . . .
Dynamic stability characteristics: Roll mode:
t B s e c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.38 0.62
t1/2. sec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.26 0.43
Spiral mode:
t s s e c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
tl12. sec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Dutch roll: 0.98
OD. rad/sec . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 9 3
L 0.37
T D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.38
0.80
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 2 7
0.80
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.94
N CP TABLE 1V.- AERODYNAMIC INPUTS, DYNAMIC STABILITY CHARACTERISTICS, AND THE PILOTS' RATINGS AND MAJOR OBJECTIONS FOR EACH CONFIGURATION SIMULATED Dynamic stability characteristics
Aerodynamic inputs per radian Pilot rating Pilots, 1 4 numerator
primary Configuration clP cnp clP c n p clr cn, WD cD E $ / w j j 2 wRS cRS (@/P),, ~ ~ / w R S ] ~ Pilot A Pilot B objections -0.0573 0.1146 -0.5000 0.0400 0.1000 -1.0000 1.93 0.38 1.27 0.94 I- 1 2
1-2 10.31I 0.971 72.70 I 37.28 I 1 6
.04001 .10001 -1.0000 1.95 .91 I
1-3 .80001 .1000' -1.00001 2.30 1 il:i 1 .66 I .261 .96 1 10.40 1 52.10 I 7
1-4 -.0573 1 .1146 .5000 1.0000 .lo00 - l . O O O O 1 2.55' .79 1 1.32 .54 .24 .251 11.10 62.91 6 7 .39 1-5 -.0573 .1146 ' 1.5000 1.0000 .oooo -.zoo0 2.11 .I1 1.66 .77 .14 .34 18.40 110.60 10 10 .11 1-6 -.0573 .1146 -.5000 1.0000 .OOOO -.8000 2.55 1.38 .25 11.20 62.81 .47 .53 .24 .32 7 4 7 9 1-7 1.02 .24 8.50 37.59 -.0573 .1146 -.5000 1.0000 .OOOO -3.0000 3.76 .88 .25 .31 4L 1.09 1-8 -.8022 .0516 -.4000 .0400 .lo00 -2.0000 2.55 .46 1.56 1.39 .93 68.10 .I9 9 ; 1.07 .24 1-9 -.0802 .0516 -.4000 .3000 .lo00 -2.0000 2.68 .89 2.56 .23 .41 .19 11.20 9.71 5 1 . 0 6 1-10 -1.0028 .1146 -.4000 .0400 .loo0 -1.0000 3.35 - 3 1 10.90 .30 .83 .90 99.50 4.90 4 1 .36 1-11 -.0573 .1146 -.0902 .1638 .5220 -1.8420 2.00 . I O 1.39 .89 .30 .33 29.20 39.27 61.
.67 1-12 -.0573 .1146 -.0643 .1184 -.0937 -2.0281 2.00 . I O 2.26 .a9 .50 .30 29.60 14.10 4 1 .73 1-13 -.0573 .1146 -.0461 .lo83 -1.0751 -2.1904 1.99 . I O 5.62 .89 . I O .30 29.80 7.15 4 1 .79 1-14 -.0573 .1146 -.0206 .lo69 -2.3946 -2.3848 1.98 11.90 .70 .90 .90 .31 30.00 4.29 5 .86 1-15 -.0573 .1146 -.0442 .1379 .2686 -1.9511 2.00 . I O 1.43 .89 .40 .ll 30.50 22.07 5 . I O 1-16 -.0573 .1146 -.lo37 .1272 .2600 -1.9426 2.00 1.60 . I O .89 .40 .51 28.30 22.07 4 .70 1-11 -.0573 .1146 -.1645 .1190 .2489 -1.9292 2.00 . I O 1.76 .a9 .40 .91 25.80 22.08 .70 3 6 1-18 -.0573 .1146 -.0038 .0621 -.5910 -.2322 2.00 .05 2.50 .86 .40 .22 276.70 21.50 4 1 .10 1-19 -.0573 .1146 -.0645 .0501 -.6024 -.2185 2.00 .05 2.45 .86 .40 .62 220.60 21.50 4 .09 1-20 -.0573 .1146 -.1114 .0435 -.6134 -.2032 2.00 .05 2.37 .86 .40 .92 186.80 21.50 4 .08 1-21 -.0573 .1146 -.2161 .3998 .0047 -.lo02 1.97 .30 1.77 .88 .10 .87 61.60 334.20 7 .06 1-22 -.0573 .1146 -.2960 .5700 .3009 -.6933 1.97 .62 1.76 .10 .73 32.40 332.50 6 .89 .27 1-23 -.0573 .1146 -.3690 .IO00 .6101 -1.3124 1.94 .94 1.79 .93 .10 .93 20.60 328.30 6 .49 1-24 -.0573 .1146 -.0250 .0679 -.5312 -.3543 2.00 .10 2.44 .86 .40 .32 175.30 21.54 4 1 .14 1-25 -.0573 .1146 -.0465 .0986 -.2758 -.8657 2.00 2.21 .40 .32 67.40 .30 .87 21.71 4 1 .32 1-26 -.0573 .1146 -.0629 .1203 -.0108 -1.3963 2.00 .50 1.91 .88 .40 .32 41.00 5 21.89 .51 1-21 -.0573 .1146 -.OW1 .1346 .4061 -2.2308 2.00 .80 1.25 .89 .40 .30 25.80 22.17 5 .80 1-28 -.0573 .1146 -.6000 1.0000 .OOOO -.8000 2.40 .83 1.52 .61 .25 .93 9.20 56.21 4 1 .32 1-29 -.0573 .1146 -1.0000 1.0000 .OOOO -.8000 1.42 .54 .98 1.72 4 .32 1-30 -.0573 .1146 -1.0000 1.0000 .OOOO -2.0000 (e) (4 (e) (e ) 4 .74 (e) 1-3 1 -.0573 .1146 -1.0000 1.0000 .OOOO -.5000 1.41 .40 .95 1.74 .21 ,
4i
--------Ap--- 1 - - See footnote at end of table, p. 25.
L . .
I TABLE IV.- AERODYNAMIC INPUTS, DYNAMIC STABILITY CHARACTERISTICS, AND THE PILOTS' RATINGS AND MAJOR OBJECTIONS FOR EACH CONFIGURATION SIMULATED - Concluded Phase II , - I---- 7- n- 1 -0.0573 0.1146 -0.5000 o.oooo o.iooo -0.7000 0.98 0.37 1 0.80 0.88 0.35 0.80 i 3 ; 11-2 -.0573 .I146 -.0435 .1863 .2697 -.5696 1.00 .30 1.20 .77 0.10 0.30 11.29 77.54 7 8 .88 .31 11-3 -.0573 .1146 -.1574 .4113 .2695 -.5701 1.00 .50 1.20 .77 .10 .30 9.52 77.45 7 .88 .34 II-4 -.0573 -1146 -.3269 .7460 .2722 -.5765 1.00 .80 1.19 .78 .10 .31 7.26 77.83 8 .88 .39 l l - 5 -.0573 .1146 -.204& .4720, .3538 -.9187 1.00 ,70 1.13 .78 .20 .30 6.17 19.55 6 -88 .51 11-6 -.0573 .1146 -.1354 .2941 .2220 -1.3504 1.00 .70 1.20 .79 .40 .30 5.38 4.91 5 .89 .69 11-7 w.0573 .1146 -.1158 .2473 -.2374 -1.5747 1.00 .70 3.44 .80 .60 .30 5.25 2.17 5 .89 .79 11-8 -.0573 .1146 -.23098 .5844 .3337 -.6913 1.00' .70 1.16 .78 .10 .10 7.74 78.08 7 .88 .42 11-9 -.0573 .1146 -2960 .6712' .2279 -.4913, 1.00' .70 1.20 .77 .10 .40 8 . 0 7 77.71 9 .88 .34 11-10 -.0573 .1146 -.4235 .8828 -.0080 -.0313! 1.00 .70 1.25 .76 .10 .70 8.98 76.53 9 10 .87 .14 I 11-11
-.0573' .1146 -.0005 .1632'-.1912, -.48711 1.00 . l o ' 1.75 , .40 .31
.77 25.64 4.80 8 .88 .27 11-12 -.0573 .1146 -.0578: .2315 -.0791 -.7234! 1.00 ' .30 1.62 .77 . 4 0 , .31 12.61 4.83 5 6 .39 .88 11-13 -.0573 .1146' -.I038 .2771 .0565 -1.0070' 1.00 .50 1.45 .78 . 4 0 ' .30 7.61 4.87 6 .E8 .53
11-14 -.0573 1 .1146 1 -.1446' .2893
.3186 ~ -1.5496' 1.00 ' .80, 1.01 .80 , .40 .30 4.69 4.94 4 1 .89 .78
1 11-15 1 .1146 ~ -.0878< .3143
-.0573
.24181-1.3589! 1.00' .70' 1.07 , .79 1 . 4 0 ( .10 5.56 1 4.91 7 .89 !
, .70 1 . : I 1 11-16 -.0573 .I146 -.I853 .2782 .1977 -1.3327) 1.00 .70 1.31
.79 . 4 0 ' .50 5.15 I 4.91 4 .68
.89 1 11-17 -.0573 ~ .1146 ' -.3083, .2931 .lo05 -1.2003 I 1.00 , .70 1.42 .79 ' 4.33 4.91 4 6 .62 .89 11-18 -.0573 .1146 -.0023 .1183 -.2131 -.45621 1.00 . 0 5 ' 1.74 .I9 27.33 4.80 8 2 5 .88 11-19 -.0573 .1146 -.I899 -.5507' 1.00 ' .05 1.60 .77 .40 .70 -.0480 .0298 25.04 4.81 7 .88 .28 11-20 -.0006 .0046 -.2947 .lo64 -.6662 -.9481 1.00 .91 . 0 6 .IO .41 2.09 .74 5.34 10 1.81 .23 11-21 -.0573 .1146 -.ZOO0 2 9 4 1 1.25 .78 .39 .52 4.93 5.21 .69 .E9 11-22 -.0573 .1146 -.3000 2 9 4 1 1.34 A 1 .39 .89 3.90 3 1 5.31 .E9 .69 -.0573 .1146 -2000 .2315 1.45 9.20 .74 .37 .76 5.52 .E8 .39 4% -.0573 .1146 -.5000 1.0000 .96 .44 .26 .31 3.41 1.12 8 .88 .48 -.0573 .1146 -.5000 1.0000 1.20 4.79 .70 24.07 9 1 .88 2 4 .OOOO .96 .17 2.73 7.57 5 .91 3.05 .2697 -1.0000 .94 .47 1.21 .88 6.87 7 8.13 .88 .51 .2697
-3.OOOOi 2 : : ; I ::: -2.0000 1.01 3.83
.88 2.82 8 .90 .98 -
- -
- aLow roll damping.
bPoor roll response characteristics (too low or too high).
CPoor aileron yaw characteristics.
dStrong spiral stability.
modes are aperiodic.
TABLE V.- PILOT RATING SYSTEM Excellent, highly desirable.
SATISFACTORY 1 Good, pleasant, well behaved, Meets all requirements and expectations, good enough Fair. Some mildly unpleasant character - without improvement.
istics. Good enough for mission without ~~ ACCEPTABLE Some minor but annoying deficiencies.
Improvement is requested. Effect May have deficiencies which on performance is easily compensated warrant improvement, but UNSATISFACTORY for by pilot.
adequate for mission.
Reluctantly acceptable. Moderately objectionable deficiencies.
Pilot compensation, if required to achieve acceptable perfor- Deficiencies which warrant Improvement is needed. Reasonable mance, is feasible. improvement. Performance performance requires considerable CONTROLLABLE pilot compensation.
adeauate for mission with - Capable of being controlled pilot compensation* Very objectionable deficiencies. Major 6 o r managed in context of improvements are needed. Requires mission, with available best available pilot compensation to pilot attention.
achieve acceptable performance.
Major deficiencies which require man- 7 datory improvement for acceptance.
Controllable. Performance inadequate for mission, o r pilot compensation UNACCEPTABLE required for minimum acceptable per- formance in mission is too high.
Deficiencies which require mandatory - improvement. Inadequate perf o r mance
Controllable with difficulty. Requires a
for mission even with maximum feasible substantial pilot skill and attention to pilot compensation.
retain control and continue mission.
Marginally controllable in mission. 9 Requires maximum available pilot skill and attention to retain control.
UNCONTROLLABLE Uncontrollable in mission. 10 Control will be lost during some portion of mission.
__- -
4) deglsec
Rate command
___.
Attitude command,
h i g h O R S
$) deglsec
___ --__~ . __ F i g u r e 1.- Roll-rate response t o a n aileron step input.
I maginary axis(+)
Lines of constant
t i m e to half amDIitude
Lines of constant
- Period increasing
damping ratio r - - \
Lines of constant
damped period
uL-------
Lines of con
- 7
-e--- -+
undamped n a t u r a l
I I --
I
- Real axis(+)
I i
I
t~ increasing -
i i
I I
i f
Figure 2.- Features of the complex plane.
. . !
Dutch r o l l !
mode x i
L .
Roll-spiral coupled mode
Roll mode I
Dutch roll x
mode
Figure 3.- Formation of the roll-spiral coupled mode.
Figure 4.- Photograph of pilot's i n s t r u m e n t display.
L-68-7632
i, deglsec
6o c
bat deg
I I 1 I . . ! .I - I I---: 1
0 2 4 6 8 10 12 14 Time, sec Figure 5.- Time history of motion obtained for an aileron step input with configuration 1-1.
I
I
M o r e negative
-c- Less negative
- - - R e ( + )
(a) Variation of C + , .
More positive
- _ - Less positive
-Re ( + I
0) Variation of C noB' F i g u r e 6.- Root-locus sketches f o r v a r i a t i o n s of t h e six aerodynamic derivatives investigated.
I + I
More negative
--- Less negative
X J
-Re ( + I
(c) Variation of QP.
More positive !
/
-- -Less positive /
/ / / J L
fc
-Re (+) (d) Variation o f Cnp F i g u r e 6.- Continued.
.33
- M o r e positive
--_
Less positive
- R e ( + )
(e) Variation of Czr.
More negative
--- Less negative
_---
-* /x-
--Re ( + I
(f) Variation of Cnr.
Figure 6.- Concluded.
o Pilot A
0 Pilot B
'\\\ Acceptable l i m i t
m 4 0 0
c
0 0
0 6 0 0 .-
c,
m
L
.-
8 c
I I I I I I
I I I I I 1
I I I I
I I I 1 I
.6 .7 . 8 .9 1.0
0 .1 . 2 . 3 . 4 .5
Undamped frequency, ORs, rad/sec
F i g u r e 7.- V a r i a t i o n of pilot r a t i n g with WRS for phase I.
W cn w Q,
o Pilot A
0 Pilot B
- T T n Acceptable l i m i t
.- w m S 2 t 4
m L
0 .1 . 2 . 3 .4 .5 . 6 .7 . 8 .9 1.0
Undamped frequency, 0 rad/sec
R S'
Figure 8.- Variation of pilot rating w i t h W R S f o r phase I I.
c
-10 c
- -,700
-
-
- 350 h i m
~-
h, ft 1250
-
I I I I 0 (a) Configuration 11-1.
F i g u r e 9.- Typical approaches of configurations 11-1, 11-10, a n d 11-17.
W
2o r
2 * 5 r
-
-2.5
-
-. 5
c I I I
I 0 60 120 180 240 Time, sec la) Concluded.
Figure 9.- Continued.
,
e , deg 0
-
-100 t
-10
-
. -
- -350 h p m h, ft 1250 -
-
- I 1 I 0 I I I I (b) Configuration 11-10.
u Figure 9 . - Continued.
W -20
2.5 r
-0.5 ' \
0 60 120 180 240 Time, sec (b) Concluded.
Figure 9.- Continued.
P
t
-25
t
-10 I- 1700 -
-
- - 350 h, m
h, ft 1250 -
I I I I I I I . o
(c) Configuration 11-17.
Figure 9.- Continued.
cp N -20 L Egs, deg 0 - (c) Concluded.
Figure 9.- Concluded.
KP
o 0.350
0 0 080
O e 009
(a) Configuration 1-1.
Figure 10.- Root-locus sketches of the closed-loop bank-angle transfer function for variation in pilot gain.
KP
o 0.350
n * 080
0 .009
X k
(b) Configuration 1-22.
Figure 10.- Continued.
KP
o 0.350
N
0 . 080
0 . 009
(c) Configuration 1-17.
Figure 10.- Concluded.
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. 20546 FIRST CLASS MAIL OFFICIAL BUSINESS POSTAGE A N D FEES PAID NATIONAL AERONAUTICS A N D SPACE ADMINISTRATION I 0 3 U 0 0 1 2 7 5 1 3 D S 7 0 0 5 8 00903 A I K F O R C E WEAPONS L A B O R A T O R Y / W L O L /
K I R T L A N D A F B , N E W M E X I C O 8 7 1 1 7
A T T E. UOU B O W M A N , C H I E F , T E C H - L I B R A R Y P ~ ~ T h I A ~ T E R : If Undeliverable (Section 158 Postal Manual) Do Nor Return "The nesoizriiiticnl and spnre nctisities of the United Stntes shnll be
coizrliicted so ns t o coiztribiitr . . . t o the expnasion of hzfiiiaiz k i z o i d -
edge of phhzoijieiin in the cittiiosphere nizd space. The Adiiiiaistrntio?~ rhdl proi'ide for the widest pr(icticnble ciizd npproprinte disseiiihzntioii of ii~foriii(itioa coizrerniiig its :ictii'ities nizd the resiiltr thereof."
-NATIONAL AERONAUTICS A N D SPACE ACT OF 195s
NASA SCIENTIFIC AND TECHNICAL PUBLICATIONS
, I * I TECHNICAE.REPORTS: Scientific and TECHNICAL TRANSLATIONS: Informarion technical infoj"on considered important, ptiblished in a foreign langiiage considered complete, and a lasting contribution to existing to merit NASA distribution in English.
knowledge. . _. * *
SPECIAL PUBLICATIONS: Information TECHNICAL NOTES: Information less broad derived from or of value to NASA activities.
in scope but neveitheless of importance as a Publications include conference proceedings, contribution t6 existing knowledge. monographs, data compilations, handbooks, sourcebooks, and special bibliographies.
TECHNICAL MEMORANDUMS: TECHNOLOGY UTILIZATION Information receiving limited distribution PUBLICATIONS: Information on technology because of preliminary data, security classifica- used by NASA that may be of particular tion, or other reasons.
interest i n commercial and other non-aerospace CONTRACTOR REPORTS: Scientific and applications. Publications include Tech Briefs, technical information generated under a NASA Ttchnology Utilimtion Reports and Notes, contract or grant and considered an important and Technology Surveys.
contribution to existing knowledge.
Details on the availability o f these publications may be obtained from: SCIENTIFIC AND TECHNICAL INFORMATION DIVISION
NATI 0 NA L AERON AUT1 C S AND SPACE ADM I N I STRATI 0 N
Washington, D.C. 20546