Model of F-16 · 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,

Model of F-16 Fighter Aircraft

- Equation of Motions -

Ying Huo Dept. of electrical Engineering - Systems

University of Southern California Los Angeles, CA 90007

[email protected]

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 ,