APPENDIX A
APPENDIX A EQUATIONS OF MOTION This appendix presents the equations of motion for the physical system shown sche matically in figures 3 and 4 for the rigid-airframe option and the rigid-fuselage-elastic wing option.
Rigid-Airframe Option The equations of motion for the rigid-airframe option in the body-axis system and the gravity-axis system are presented in this section.
Airplane center-of -gravity (composite m a s s center) motion in body-axis system
p.5Lt sin a - 0.5(Df + Q) cos a - 0.5% + FG x cos (6 + a ) - FG z sin (6 + a )
1- (mf + mw + mh)g sin (e + a )
Xc,b =
mf + mw + mh
-0.5% - 0 . 5 h cos a - 0.5(Df + Dt) sin a - FA,^^ + (mf + mw)g cos (6 + a )
'ic,b = mf + mw Airplane center-of -gravity motion in gravity-axis system
zCpg = zc,b cos (8 + a ) - xC,b sin (6 + a )
Hub motion in body-axis system
FG,Z cos (6 + a ) + FG x sin (6 + a ) + FA,^^ + mhg cos (6 + a )
'ih,b = ~~ ~~ . .
mh APPENDM A Hub motion in gravity-axis system
sh,g = zh,b cos ( e + a) - x h b sin (e + a,)
Y Wing-mass -center motion in gravity-axis system 2w,g = xc,g + $dew sin , X c w + (3) + b2dcw cos j X c w + 8 ; - 2
'iw,g = Zc,g + i d c w cos :~kcw + 8 ) - e d,, sin (XCw + e'i
Fuselage -mass-center motion in gravity-axis system
xf,g = xc,g - Qdcf sin (Xcf + 0) - e ' 2dcf cos ( X c f + e )
Wing-gear -interface motion in gravity-axis system . 2
xwG,g = j;'c,g + edcwG sin (&cwG + e ) + e dcwG cos (XcwG + 0 )
Shock-strut motion in body-axis system g b = zwG,g cos ( e + a ) + x w ~ , g sin (e + a ) - zh,g cos (e + a ) - xh,g sin (e + a ) - e - 2 zSs Pitch angular acceleration about airplane center of gravity
-
- 0 . 5 h d c t + 0.5Dfdcf + 0.5Qdct + 0 . 5 L W d c ~ , ~ cos a, - 0. 5LwdCW,Z sin a, s i n a, - F ~ , ~ ~ d l - 0.5DwdCW,Z cos a, - 0 . 5 D ~ d ~ ~ , ~ + F ~ , x d 2 cos ( e + a,)
- F ~ , z d 2 sin ( e + a,) + Mn
~- - e = - I - .* -.
IYY 3 5
I
APPENDIX A
APPENDIX A Wheel angular acceleration about hub Brake -control sensor -mass acceleration about hub Rigid -Fuselage -Elastic - Wing Option The equations of motion for the rigid-fuselage -elastic -wing option in the body-axis system and the gravity-axis system are presented in this section.
Airplane center -of -gravity (composite m a s s center) motion in body-axis system
r0.5Lt sin a - 0.5(Df + Dt) cos CY - 0.5% + FG,X cos (e + CY) - FG,Z sin (0 + a )
- (mf + mw + mh)g sin (e + a )
1 i
~ ~ - -- ~- - - Xc,b =
mf + mw + mh
+ mfg cos (e + a ) - Fn cos 0 -0.5Lt cos a - 0.5(Df + Dt) sin CY - F w ~
Zc,b = - . . . . .
mf Airplane center-of -gravity motion in gravity-axis system
xc,g = xc,b cos (0 + CY) + Zc,b sin (e + CY)
!ic,g = Zc,b cos (0 + CY) - xc,b sin (0 + CY)
Hub motion in body-axis system + i d c e sin Ace - i 2 d C e sin h c e
APPENDIX A
APPENDIX A P r i o r to shock-strut stroking Subsequent t o shock-strut stroking Hub motion in gravity-axis system Wing-mass -center motion in body-axis system FG,X cos (0 + a ) - FG,Z sin (0 + a ) - (mf + mw + mh)g s i n (0 + a ) - 0 . 5 F w ~ - 0.5(Df + Dt) cos a + 0.5Lt sin a
.. I
i
~~ ~ + &dew + 8dr-e sin Xce - d2dce COS hce Xw,b = mf + mw + mh P r i o r to shock-strut stroking cos (0 + a ) + FG,X sin (0 + a ) + (mw + mh)g cos (0 + a ) -0.5Lw + F,B
.. =w,b = rG9z ~ + Bdew
mw + “h Subsequent to shock-strut stroking - 0.5Lw + F,B + m,g cos (e + a ) zw,b = + 6dew m W 3 7
APPENDIX A
APPENDIX A Wing-mass -center motion in gravity-axis system
xw7g = xw7b cos (0 + a) + zw,b sin (e + a)
Zw,g = zw,b cos (0 + a) - xw,b sin (e + a)
Fuselage -mass-center motion in gravity-axis system
xf,g = xc7g - gdcf sin (Xcf + e) - h2dCf COS (Acf + e)
zf,g = zc,g - idcf COS (Xcf + 6) + d2dCf sin (Xcf + 6)
Wing-gear -interface motion in body-axis system
FA,^^ - 0.5Lw + F,B + mwg cos (8 + a)
- - ~ ~ _ .. _ - ‘wC,b = - + &(dew + &G) mW Shock-strut motion in body-axis system Motion of elastic axis a t spanwise location of wing m a s s center: P r i o r to shock -strut stroking Subsequent t o shock-strut stroking
FA,^^ - 0.5Lw + F,B + mwg cos (8 + a )
iew,b = mW
APPENDIX A
APPENDIX A Motion of elastic axis at spanwise location of wing m a s s center relative to fuselage Rotational motion of wing about elastic axis and relative to fuselage: P r i o r to shock-strut stroking Subsequent t o shock-strut stroking Wheel angular acceleration about hub Brake -control-sensor acceleration about hub Pitch angular acceleration about airplane center of gravity
1-0.5Qdg + 0.5Dfdq + 0.5Dtdq + 0.5Lwd5 cos a - 0.5Lwdg sin a 1
sin (0 + a ) + Mn
J
IYY I
APPENDIX B
APPENDIX B DESCRIPTION OF LANDING GEAR The landing gear simulated in this paper was originally designed for a small a i r plane having a g r o s s m a s s of approximately 2268 kg (5000 lbm). (See ref. 7.) The gear is a cantilevered type with a conventional oleo-pneumatic shock strut. The tire is a 0.69-m (27-in.) diameter type I (smooth contour) which was inflated to 221 k P a (32 psi).
(150 lbm) and the unsprung m a s s i s 59 kg (131 lbm).
The m a s s of the landing gear is 68 kg *2 Sketch (d) The important geometric characteristics of the landing-gear shock strut are illus trated in sketch (d), and the program input parameters are as follows: Shock strut:
Pneumatic area, A2 . . . . . . . . . . . . . . . . . . . . . 0.00535 m 2 (0.05762 ft2)
Hydraulic area, A 1 . . . . . . . . . . . . . . . . . . . . . 0.00437 m 2 (0.04708 ft2)
Primary orifice area, A0 . . . . . . . . . . . . . . . . . . 0.00005 m 2 (0.00056 ft2)
Pressurized pneumatic volume, V2 . . . . . . . . . . . . 0.0010 m3 (0.03545 ft3)
Charging pressure, po,a . . . . . . . . . . . . . . . . . . . . 299.9 kPa (6264 psfa)
Bearing separation f o r fully extended shock strut, I1 . . . . . 0.16828 m (0.5521 f t )
Axial length from hub to lower bearing for fully extended
shock strut, 12 . . . . . . . . . . . . . . . . . . . . . . . . 0.6785 m (2.22604 f t )
Mass of airplane acting on each main gear, W1 . . . . . . . . . 1094 kg (2411 lbm)
Unsprung gear mass, W2 . . . . . . . . . . . . . . . . . . . . . 59 kg (131 lbm)
I
APPENDIX B
APPENDIX B
Fully extended length of shock strut, lss,e . . . . . . . . . . . 0.9674 m (3.174 ft)
Friction coefficient (upper bearing), pss,u . . . . . . . . . . . . . . . . . . . . 0.15
Friction coefficient (lower bearing), pss,i . . . . . . . . . . . . . . . . . . . . 0.15
Volume between piston and cylinder, V3 . . . . . . . . . . . . . . . . o m3 (0 ft3)
Area between piston and cylinder, A3 . . . . . . . . . . . . . . . . . . 0 m 2 (0 ft2)
Specific weight of hydraulic fluid,
. . . . . . . . . . . 8226 N/m3 (52.36 lbf/ft3)
yH
. . 0.00862 N-sec,’m2 (0.00018 lbf-sec/ft2)
Dynamic viscosity of hydraulic fluid, p~
Mass density of hydraulic fluid, p~ . . . . . . . . . . . 838 kg/m3 (1.626 slugs/ft3)
Volume of hydraulic fluid in fully extended strut, VH . . . . -0.0014 m3 ( ~ 0 . 0 5 ft3)
Wheel and tire:
Wheel flange diameter, df . . . . . . . . . . . . . . . . . . . 0.39053 m (1.28125 ft)
Width of wheel rim, ww . . . . . . . . . . . . . . . . . . . . . . 0.227 m (0.745 f t )
Unloaded diameter of tire, d . . . . . . . . . . . . . . . . 0.686 m (nominal 2.25 ft)
Maximum width of undeflected tire, w . . . . . . . . . . . . . . 0.245 m (0.805 f t )
Unloaded rated inflation t i r e pressure, pr . . . . . . . . . 482.63 kPa (10 080 psf)
. . . . . . . . . . . . . 220.6 kPa (4608 psf)
Unloaded t i r e inflation pressure, po
P r e s s u r e -rise parameter, K . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.66
Vertical-force coefficient, C z . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.02
REFERENCES
Design Formulation and Analysis
1. Wignot, J a c k E.; Durup, Paul C.; and Gamon, Max A.:
AFFDL-TR-71-80, Vol. I, U.S. Air
Volume I. Analysis.
of an Active Landing Gear.
(Available f r o m DDC as AD 887 127L.)
Force, Aug. 1971.
A Feasibility Study of Active Landing
2. Bender, E. K.; Berkman, E. F.; and Bieber, M.:
(Available f r o m DDC as
Gear. AFFDL-TR-70-126, U.S. Air Force, July 1971.
AD 887 451L.)
Mechanical Properties of Pneumatic Tires
3. Smiley, Robert F.; and Horne, Walter B.:
NASA T R R-64, 1960. (Super
With Special Reference t o Modern Aircraft Tires.
sedes NACA T N 4110.)
4. Tung, C. C.; Penzien, J.; and Horonjeff, R.: The Effect of Runway Unevenness on the
NASA CR-119, 1964.
Dynamic Response of Supersonic Transports.
A Study of the Dynamics of Airplane Braking Systems as Affected
5. Batterson, Sidney A.: NASA TN D-3081, 1965.
by Tire Elasticity and Brake Response.
An Experimental
6. Milwitzsky, Benjamin; Lindquist, Dean C.; and Potter, Dexter M.:
(Supersedes
NACA Rep. 1248, 1955.
Study of Applied Ground Loads in Landing.
NACA T N 3246.
Analysis of Landing-Gear Behavior.
7. Milwitzsky, Benjamin; and Cook, F r a n c i s E.:
NACA Rep. 1154, 1953. (Supersedes NACA TN 2755.)
Sixth ed. Ronald Press Co., c.1936.
Engineering Aerodynamics.
8. Diehl, Walter Stuart: Takeoff and Landing Analysis (TOLA) Computer Program.
9. Lynch, Urban H. D.: Capabilities of the Takeoff and Landing Analysis Computer Program.
Part 1:
(Available f o r m DDC as
AFFDL-TR-71-155, Pt. 1, U.S. Air Force, Feb. 1972.
AD 741 942.)
loads induced / A loads
Input loads
f o r total
response analysis
'eedback
Shock strut
Perturbation
Airframe
.loads
response
, ,
-Tire
-
~~
Flexible airframe
Operational loads
0 Steering
0 Thrust reversal
and braking
0 Aerodynamic
0 Asymmetric
0 Modal frequencies impact
0 Modal displacements
Figure 1. - Variables considered significant for analyzing active control landing gear systems.
S e r ie s - hy d rau 1 i c
Parallel-hydraulic
Series-pneumatic
cont ro 1
control
control
m Sensor
Electronic /e=l-J
Figure 2.- Active control concepts for airplane landing gears.
X Y (into paper)
Gravity axes
Body axes
Shock- s t rut
FG, Zl
Figure 3. - Definition of axes and forces for rigid-airframe option.
Y (into paper)
Yb(into paper) g
zg
Body axes Gravity axes
---- Motion due to first wing
Shock-strut
bending and torsion
Figure 4 . - Definition of axes and forces f o r rigid-fuselage --elastic-wing option.
Fluid motion
1 1 1
Pressure force
t t t
Direction of motion
t
A2
A1
S - 1
*1
(a) Passive gear phase. Compress (b) Active gear phase. Low- (c) Active gear phase. High-
ing with wing force less than pressure operation; compress -
pressure operation; extending;
limit force. ing; wing force greater than
wing force less than limit force.
limit force.
Figure 5. - Operational phases of series-hydraulic active control gear.
Y (into paper)
Yb(into paper)
"T; '
Figure 6. - Rigid-airframe configuration used for study, (Dimensions a r e not to scale.)
I
0 Experiment (ref. 7) - .7 - 7 103
- TOLA
-6 - 5 -' Strut Strut Strut force, stroke,
16 c
force, Ibf ft kN - 3 - 2 - 1 - 0 Time, sec (a) Time histories of shock-strut force and stroke.
Figure 7.- Comparison of computed results obtained from ACOLAG and TOLA with experimental data for a vertical drop onto a flat stationary surface. 8 = Oo; iC,, = 2.7 m/sec (8.8 ft/sec); k c , , = 0 m/sec (0 ft/sec).
.7 I x lo3 0 Experiment ( r e f . 7 )
- TOLA
-
.20
- - -- -ACOLAG
.6 .16 .5 5
i
~ Vertical ground force 1 6 p e r gear,
12 L-
I - .2 - 2
8 r I
- .I - 1
4 1 I I I I 0 - 0 0 .05 .10 .15 2 0 .25 T i m e , s e c (b) Time histories of vertical ground force and t i r e deflection.
Figure 7. - Continued.
0 Experiment (ref. 7) -3.5 TOLA
-----
ACOLAG -3.0 - 2 . 5 -2.0 Mass-center acceleration, g u n i t s -1.5 -1.0 - .5 Tinre, sec (c) Time histories of mass-center acceleration.
Figure 7. - Concluded.
I I I I I
-2 Figure 8. - Comparison of computed results from ACOLAG and TOLA for landing-impact simulation.
0 = 10.5O; kc,, = 2.7 m/sec (8.8 ft/sec); kc,, = 45.7 m/sec (150 ft/sec).
32 -
- TOLA - . I I x lo3
_ _ - -
ACOLAG
- .20
r F o r c e
24 -
.16 Main gear Main gear 16 - strut force, s t r u t force, Ibf liN Time, sec (b) Time histories of shock-strut force and stroke.
Figure 8. - Continued.
cn w .20 .16 .12 Vertical Vertical 16 - T i r e ground f o r c e ground f o r c e deflection p e r gear,
I
p e r gear, p e r gear, lbf m e t e r s kN l i .OI t
. oi
- 1 I I 1 I - 0 ( ( .05 .10 .15 .20 2 5 Time, s e c (c) Time histories of vertical ground force and t i r e deflection.
Figure 8. - Continued.
-
-3.0 TOLA
----
ACOLAG
-
-2.5
I I I I I
.5 0 .05 .10 .15 2.0 .25 Time, sec (d) Time histories of mass-center acceleration.
Figure 8. - Concluded.
Active gear
----
Passive g e a r
- 5 j6
, I Wing w i n g force, 1 6 L force,
-I4
kN I lbf - 3 I I
12 c
I , -I 2 8 L - 1 4 L I - 0 Wing displacement, meters
- Active gear
32 - - - - Passive gear
- I x io3
.7 Control flow rate, Control flow rate,
-428 l/min (-113 gal/min) - 4 5 8 l/min (+121 galjmin)
.2c 6 - 6 .16
5 -I 5
4 - 4 Strut .12 Strut force Strut 16 ' Strut per gear,.
ft stroke, force lbf meters per gear, .3 -3 kN .OE .2 . 0 4 .1 1
1 0 0
C a Time, sec (b) Time histories of shock-strut force and stroke.
Figure 9. - Continued.
I T x 103 .6 -5 3 Vertical groucd 4 Vertical .4 T i r e force ground deflect ic per gear, p e r geai force kN ft per gear, lbf .3 .2 2 .1 0 0 ) Time, s e c (c) Time histories of vertical ground force and tire deflection.
Figure 9. - Concluded.
-9 x lo3
- Active gear
----
Passive g e a r
- 3 6 -401 - -8
; A i r -7
; ‘ I
I I
I 1 36 %reduction ‘ I in wing force -6 I I I I I I I I Tolerance I I I I -5 Wing force, lbf -4 5 % i n c r e a s e in - I wing displacement -3 -2 - 1 I I I I I I 0 .04 .08 .12 .16 .20 .24 .28 0 Wing displacement, iiieters I I I I I I I I I I I 0 .1 .2 . 3 .4 .5 .6 .7 .8 .9 1.0 Wing displacement, f t (a) Wing force as a function of wing displacement.
i !
Figure 10.- Comparison of computed results for modified passive and active gears for a vertical drop onto a vertically oscillating surface. 8 = 0 0 . , z c,g = 2.7 m/sec (8.8 ft/sec); xc,g = 0 m/sec (0 ft/sec).
Q,
- Active gear
- -- - Passive g e a r
.24 2 0 .16 Strut Strut - Strut force per f o r c e stroke, p e r gear, gear, Ibf m e t e r s .12 kN 16 .08 .04 0 0 (b) Time histories of shock-strut force and stroke.
Figure 10. - Continued.
.28 - .9
-40 -
- -9 x lo3
- Active g e a r
- - - - Passive gear
-36 - .8 - - 8 .24 - ; - \
,’ ‘\
-32 - . I - -7 .20 -28
- .6 * -6
-24 .16 T i r e Vertical -5 Vertical . 5 T i r e deflection ground ground deflectic force -20 per gear, force p e r gea: p e r gear. meters p e r gear, kN .12 .4 f t -4 Ibf -16 .3 -3 -12 .08 -2 .2 -8 .04 .I -1 -4 0 0 Time, sec (c) Time histories of vertical ground force and t i r e deflection.
Figure 10. - Concluded.
Q, N
-9 lo3
- Active g e a r
- - -- Passive g e a r
-8
J -7
-36 - 3 2 ‘c
!
i Wing force, kN Wing displacement, m e t e r s
I I I
I I I I I I I .I I .8 .9 1 . 0 .3 .4 . 5 .6 0 .1 .2 Wing displacement, ft (a) Wing force as a function of wing displacement.
Figure 11. - Comparison of computed results for modified passive and active gears for a landing impact onto a smooth runway surface. 0 = 10.50; kc,, = 2.7 m / s e c (8.8 ft/sec); kc,g = 45.7 m/sec (150 ft/sec).
Active g e a r
- - - -
Passiv.e g e a r
36 “1 .24
Control flow rate, Control flow rate, 4 4 7 I/min (+118 gal/niin) -360 Q/min (-95 gal/nim) - Control flow rate,
1 -397 U m i n (-105 gal/min)
S t N t Strut force per force per gear, kN stroke, gear, Ibf (b) Time histories of shock-strut force and stroke.
Figure 11. - Continued.
-
-40
---- Passive gear
- 8
-36 .24
-
-32 I
-
2.0
-
-28
- .6 - 6
-24 -
.16 Vertical ground force per gear, !04 (c) Time histories of vertical ground force and t i r e deflection.
Figure 11. - Continued.
- 2 0
-500 -
- Control flow rate
- 6 x I O - ' - -12 X 10
----
Control flow -8 -4 Control 4 flow rate, gal/min
.30 L
1 I I I 1 .15 2 0 .25 0 .05 'lo Time, s e c Figure 12. - Time histories of control flow rate and control flow for landing impact of .active gear onto a smooth runway surface. 0 = 10.50; zc,g = 2.7 m/sec (8.8 ft/sec); xc,, = 45.7 m/sec (150 ft/sec).
-32 7 x lo3 Active g e a r
-----
Passive g e a r I I -24 I I 4 5
-20 -
Wing force, lbf -' 3 -12 i- I
I
- 2
I
- 1
- 4 t I /
I I I I I I I
-
0 .04 .08 .12 .16 .20 .24 .28 0 Wing displacement, m e t e r s
L I I I I I
I I I I I 0 .1 .2 . 3 .4 .5 .6 .7 .8 .9 1.0 Wing displacement, ft (a) Wing force as a function of wing displacement.
Figure 13. - Comparison of computed results for modified passive and active gears for a landing impact onto a sinusoidal runway surface. 0 = 10.5O; iC,, = 2.7 m/sec (8.8 ft/sec); x c t g = 45.7 m / s e c (150 ft/sec).
40 r Active gear
_ - - - - -
Passive gear
I
Control flow rate, -
- Tire bounce
3 6 1 .24 .1154 ( min (-305 gal, m n )
r I
Strut Strut gear, kN meters
12 I- .08
.04
0 6I 0
Time, sec (b) Time histories of shock-strut force and stroke.
Figure 13. - Continued.
.28 Active gear .24 I I .2(
- -28
-
- 8 6 .6 \
-24 -
.16 \ - 1 .5 - 5 \ Vertical T i r e ! T i r e Vertical \ ground deflection deflection ground \ force -20 r p e r gear, per gear, I f o r c e per \ per gear, meters -4 ft - : 4 gear, Ibf
kN .12 -
-16 + - 3
i .3
-12 - .2 - 2
- -8
- 1
-4 -
- I-
0 O (c) Time histories of vertical ground force and t i r e deflection.
Figure 13. - Concluded.
- -1.00 lo3 -4 -
- Active gear
- -.75
_ - - - Passive gear
- -.50
- -.25
- 0 W i n g force, lbf - .25
- I I .50
I l l I .75 1.00
I
1 I I I I 1.25 0 .5 1.0 1.5 2 . 0 2.5 Time, sec (a) Time histories of wing force.
Figure 14. - Comparison of computed results for active and passive gears f o r a landing roll-out after traversing a vertically discontinuous bump. f3 = 00; zc,g = 0 m/sec (0 ft/sec); Xc,, = 44.5 m/sec (146 ft/sec).
-4 103
- Active Sear
_ _ - - Passivc g e a r
-201 -16
4 -3
-12
I
Vertical Vertical ground q round force, Ibf
force, liN - - 2
I
I
- -1 - 4 I I I I 0 0 .s 1 1.5 2 2.5 Time, sec (b) Time histories of vertical ground force.
r 4 m Figure 14. - Concluded.
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