Model of F-16 · Nguyen, L.T., et al., Simulator study of stall/post-stall characteristics of a...
Transcript of Model of F-16 · Nguyen, L.T., et al., Simulator study of stall/post-stall characteristics of a...
Model of F-16 Fighter Aircraft
- Equation of Motions -
Ying Huo Dept. of electrical Engineering - Systems
University of Southern California Los Angeles, CA 90007
Ref : [1]. Brian L. Stevens, Frank L. Lewis, Aircraft Control and Simulation, John Wiley & Sons, Inc. 1992 [2]. Nguyen, L.T., et al., Simulator study of stall/post-stall characteristics of a fighter airplane with relaxed longitudinal static stability, NASA Tech. Pap. 1538, NASA, Washington, D.C., Dec. 1979
“ The mathematical model given here uses the wind-tunnel data from NASA-Langley wind-tunnel tests on a scale model of an F-16 airplane. The data apply to the speed range up to about Mach=0.6, and were used in a MASA-piloted simulation to study the maneuvering and stall/post-stall characteristics of a relaxed static-stability airplane.”
Nomenclature
State Variables:
V - true velocity, ft/sec α - angle of attack, radian ( range –10° ~ 45° )β - sideslip angle, radian ( range –30° ~ 30° ) φ - Euler (roll) angle, rad θ - Euler (pitch) angle, rad ϕ - Euler (yaw) angle, rad p - roll rate, rad/sec q - pitch rate, rad/sec r - yaw rate, rad/sec
disN - north displacement, ft
disE - east displacement, ft h - altitude, ft
powP - power
Control Variables:
Tδ - throttle setting, ( 0.0 – 1.0 )
Eδ - elevator setting, degree
Aδ - aileron setting, degree
Rδ - rudder setting, degree
Parameters:
ρ - air density, slugs/ft^3 M - Mach number T - total instantaneous engine thrust, N (lb) m - total airplane mass, slugs
tXC , - total x-axis force coefficient
tYC , - total y-axis force coefficient
tZC , - total z-axis force coefficient q - dynamic pressure, psf
sp - static pressure, psf
tLC , - total rolling-moment coefficient
tMC , - total pitching-moment coefficient
tNC , - total yawing-moment coefficient t - temperature, R u - velocity in x -axis direction, ft/sec v - velocity in -axis direction, ft/sec yw - velocity in -axis direction, ft/sec zW - vehicle weight (lbs) b - wing span (ft) S - wing platform area (ft2) c - mean aerodynamic chord (ft)
xI - roll moment of inertia (slug-ft2)
yI - pitch moment of inertia (slug-ft2)
zI - yaw moment of inertia (slug-ft2)
xzI - product moment of inertia (slug-ft2)
xyI - product moment of inertia (slug-ft2)
yzI - product moment of inertia (slug-ft2)
cgRX - reference cg location (ft)
cgX - center of gravity location (ft) g - gravitational constant (ft/sec2)
Eh - engine angular momentum (slug-ft2/s)
rd - radian-to-degree constant, 29578.57=rd
Table 1. Mass and Geometry Properties
Symbol Parameter Value
W Vehicle weight (lbs) 20500
b Wing span (ft) 30
S Wing area (ft2) 300
c Mean aerodynamic chord (ft) 11.32
xI Roll moment of inertia (slug-ft2) 9496
yI Pitch moment of inertia (slug-ft2) 55814
zI Yaw moment of inertia (slug-ft2) 63100
xzI Product moment of inertia (slug-ft2) 982
xyI Product moment of inertia (slug-ft2) 0
yzI Product moment of inertia (slug-ft2) 0
Table 2. Control Surface Actuator Models
Symbol Command name
Deflection limit
Rate limit
Time constant
Positive sign
convention Effect
Eδ Elevator ± 25.0˚ 60˚/s 0.0495sec lag
Trailing edge down
Negative pitching moment
Aδ Ailerons ± 21.5˚ 80˚/s 0.0495sec lag
Right-wing trailing edge
down
Negative rolling moment
Rδ Rudder ± 30.0˚ 120˚/s 0.0495sec lag
Trailing edge left
Negative yawing moment,
positive rolling moment
Table 3. Other parameters used in the model
Symbol Parameter Value
cgRX Reference CG Location (ft) c35.0
g Gravitational constant (ft/sec2) 32.174
Eh Engine Angular Momentum (slug-ft2/s) ( assume fixed ! )
160.0
rd Radian-to-degree constant 57.29578
Six-degree-of-freedom Motion Equations The equations used to describe the motions of the airplanes were nonlinear, six-degree-of-freedom, rigid-body equations referenced to a body-fixed axis coordinate system. ------------------------------------------------------------------------------------------------------------ Force Equations ------------------------------------------------------------------------------------------------------------
222
cossinsin
coscos
wvuV
VwVvVu
++=
===
βαβ
βα
)(1sin , TSCqm
gqwrvu tX ++−−= θ&
tYCmSqgrupwv ,sincos ++−= φθ&
tZCmSqgpvquw ,coscos ++−= φθ&
VwwvvuuV&&&& ++
=
------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------
)(sin
)(tan
1
1
Vvuw
−
−
=
=
β
α
2)cos( βα
Vuwwu &&
&−
=
2)cos(coscosβ
βββV
VvvV &&& −=
------------------------------------------------------------------------------------------------------------ Kinetics ------------------------------------------------------------------------------------------------------------
)cossin(tan φφθφ rqp ++=&
φφθ sincos rq −=&
θφφϕ
coscossin rq +
=&
------------------------------------------------------------------------------------------------------------ Moments ------------------------------------------------------------------------------------------------------------
tLxx
xz
x
zy CISbqpqr
IIqr
III
p ,)( +++−
= &&
rhCIcSqpr
IIpr
IIIq EtM
yy
xz
y
xz −+−+−
= ,22 )(&
qhCISbqqrp
IIpq
III
r EtNzz
xz
z
yx ++−+−
= ,)( &&
or
}])([)({1 22 qhIILINIqrIIIIpqIIIIIII
p Ezxzzxzxzzyzzyxxzxzzx
+++−−++−−
=&
rhIMpr
IIpr
IIIq E
yy
xz
y
xz −+−+−
= )( 22&
})(){(1 222 qhIILINIqrIIIIpqIIIIIII
r Ezxzzxxzyxzxzyxxxzzx
+++−−++−−
=&
where tLsbCqL ,= , tMCcsqM ,= , tNsbCqN ,=
------------------------------------------------------------------------------------------------------------ Navigation ------------------------------------------------------------------------------------------------------------
)sinsincossin(coscossin)sincossincos(sinsincoscoscoscos
ϕφϕθφβαϕφθϕφβϕθβα
++−+=
VVVNdis&
)cossinsinsin(coscossin)coscossinsin(sinsinsincoscoscos
ϕφϕθφβαϕφθϕφβϕθβα
−+++=
VVVEdis&
θφαθφβθβα coscossincossinsinsincoscos VVVh −−=&
Coefficients
14.453 )10703.00.1(10377.2 h−− ×−×=ρ
≥<×−
=−
0.350000.3900.35000)10703.00.1(519 5
hhh
t
=
=
pressure static1715
pressure dynamic21 2
tp
Vq
s ρ
ρ
tVM
××=
3.17164.1
),()(2, EdxdXqtX CqCVcC δαα +=
0.30086.0
0.20021.0])()([
202.0
])()([2
),,(,
RAdYpdYrd
dYpdYrRAdytY
pCrCVb
pCrCVbCC
δδααβ
ααδδβ
++++−=
++=
0.2519.0)(
2),(
)(2
),,(
1,
,
EdZqddz
dZqEddztZ
qCVcC
qCVcCC
δαβα
αδβα
−+=
+=
30),(
0.20),(])()([
2),(
])()([2
),,,(
3,2,1,
,
Rddl
AdldLpdLrddl
dLpdLrRAddltL
CCpCrCVbC
pCrCVbCC
δβαδβαααβα
ααδδβα
++++=
++=
),()()(2 ,, EdmcgcgRtZdMqtM CXXCqCVcC δαα +−+=
30),(
0.20),(
)(])()([2
),(
)(])()([2
),,,(
3,2,
,1,
,,
Rddn
Addn
cgcgrtYdNpdNrddn
cgcgrtYdNpdNrRAddntN
CC
XXCbcpCrC
VbC
XXCbcpCrC
VbCC
δβαδβα
ααβα
ααδδβα
++
−−++=
−−++=
Table 4. Source of aerodynamic coefficients
Coefficients Source Independent
variables ( ββαα rdrd dd == , )
XqC Table dα
xC Table Ed δα ,
YrC Table dα
YpC Table dα
1,ZC Table dα
ZqC Table dα
1,lC Table dd βα ,
LrC Table dα
LpC Table dα
2,lC Table dd βα ,
3,lC Table dd βα ,
MqC Table dα
mC Table Ed δα ,
1,nC Table dd βα ,
NrC Table dα
NpC Table dα
2,nC Table dd βα ,
3,nC Table dd βα ,