Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
EQUATIONS OF MOTION FOR A CONVENTIONAL
AIRCRAFT
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
PROBLEM DEFINITION
Roskam, J., Airplane Flight Dynamics and Automatic Flight Controls - Part I, Design, Analysis and Research Corporation, Lawrence, KS, 1995.
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
Note: These equations are only valid in Inertial Space
Newtons Law of Linear Motion
ForcesAppliedMomentumLinear
dsdVdVdtd
dtd
SVA
VA +=
Fgr
Eulers Law of Angular Motion
Moments AppliedMomentumAngular
+=
SA
VA
VdSdVdV
dtd
dtd Frgrrr
EQUATIONS OF MOTION general
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
+++=
++=
dmdtddm
dtddm
dtddm
dtd
dtdLHS
dmdtd
dtdLHS
V
P
VVP
P
VP
PV
'P
rrrrrrrr
rrrr
''
''
' )()(
dVr Ap =+= dm define also and :using gEliminatin rrr
Moments AppliedMomentumAngular
+=
SA
VA
VdSdVdV
dtd
dtd Frgrrr
ANGULAR MOTION EQUATION
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
Assumption 1: P is the Vehicle Center of Mass and rP is a constant for V. Assumption 2: Mass is a constant, i.e.,
0= V
dmr
0=dtdm
+=
+++=
+++=
dmdtddm
dtd
dtdLHS
dmdtd
dtddmdm
dtddm
dtd
dtdLHS
dmdtddm
dtddm
dtddm
dtd
dtdLHS
Vv
PP
V
P
VVP
v
PP
V
P
VVP
P
VP
rrrr
rrrrrrrr
rrrrrrrr
''
''
''
''
''
0 0
ANGULAR MOTION EQUATION
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
MASS DISTRIBUTION aircraft and spacecraft
! A constant mass distribution infers that the center of gravity remains constant during the period of interest. ? For stability and control purposes, aircraft responses are typically evaluated
for a duration of 30 to 60 seconds. ? For orbital spacecraft, the period of interest can range from several minutes
to several hours. ! Phenomena which violate the constant mass/moment of inertia
distribution include: ? Fuel sloshing ? Shifting payloads
i wandering passengers on commercial aircraft i solar panel deployment and communication antenna reorientation on spacecraft
? Expendable payloads i weapons i reconnaissance pods
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
CONSTANT MASS aircraft and rockets
17.2 19,300 - 100,000 112,000 Delta
27.7 1,800,000 - 4,500,000 6,500,000 Saturn 5
0.05 3.3 200 1,020 6,800 GA Twin
0.18 99 5,940 17,600 54,000 Fighter
0.22 1,500 90,000 291,000 675,000 SST
Mass Change % Takeoff weight
Mass Change (lbs)
Cruise Fuel Consumption
(lbs/hr)
Maximum Fuel Weight
(lbs)
Takeoff Weight (lbs)
Type
t = 60 seconds
Reference 2-1
! If the mass change is within 5% after 60 seconds, the constant mass assumption is considered acceptable. ? Reasonable for airplanes, but not for rockets.
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
+=
++=
dmdtd
dtddm
dtdLHS
dmdtd
dtddm
dtddm
dtd
dtdLHS
VV
PP
VV
PP
v
PP
rrrr
rrrrrr
2
'2'
2
'2'
''0
Special Identity:
0
0
0)(
0
0
2
2
2
2
2
2
=
=
+
=
=
V Spp
V
pp
V SVpp
V SVp
V SVp
dsdmdmdt
d
dsdmdmdtd
dsdmdmdtd
dsdmdmdtd
dtd
Frgrr
r
Fgrrr
Fgrr
Fgrr
ANGULAR MOTION EQUATION
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
Returning to the LHS:
++=
+=
dmdtd
dtddSdmLHS
dmdtd
dtddm
dtdLHS
VSP
VP
VV
PP
rrFrgr
rrrr
''
2
'2'
Now Looking at the RHS:
+++=
+++=
SSP
VVP
SP
VP
dSdSdmdmRHS
dSdmRHS
FrFrgrgr
Frrgrr''
'' ])[(])[(
0 Vover constant is V
== Vrggrg:3 Assumption dmdm
ANGULAR MOTION EQUATION
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
GRAVITATIONAL FORCES aircraft and spacecraft
! The Flat Earth Assumption implies that the gravitational vector is oriented along the positive Z axis. ? Aircraft
i not reasonable for intercontinental flight ? Orbital spacecraft
i not reasonable even for small vehicles ! Assuming constant gravitational acceleration over the entire volume
means that each mass element is subject to the same gravitational acceleration. ? Aircraft
i applies in virtually all cases ? Orbital spacecraft
i does not apply
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
Forming the Total Equation:
++=
++
SSP
VP
VSP
VP dSdSdmdmdt
ddtddSdm FrFrgrrrFrgr ''''
Therefore: =
SVdSdm
dtd
dtd Frrr
Under the assumptions: 1. P is the Center of Mass 2. Mass is constant 3. g is constant over V Note: The Rotational Motion Equations are only a function of r.
ANGULAR MOTION EQUATION
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
frame. rotating Z)Y, (X,
in theobserver an by seen as of change of rate theis t
:Where
tdtd
:is frame coordinate fixed )Z,Y ,X( in theobserver an by seen as of change
of rate then theframe, coordinate fixed )Z,Y ,X( the torelative , rate,angular an at rotating is that frame coordinate a is Z)Y,(X, If
resides.observer hein which t framecoordinate on the depend will vector theof change of rate observed The
Z)Y,(X,
Z)Y,(X,)Z,Y ,X(
AA
AAA
A
A
+=
VECTOR DERIVATIVES
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
Eliminating r using: dmdVAP =+= and' rrr'
+=
+
SVVP dsFdmgdmrrdt
ddtd )( '
Again assuming: Mass is a constant, i.e., and g is constant over V. 0=dtdm
+=+SVV
P dsmdmdtddm
dtd
dtd Fgrr ][][ 2
2'
0
ForcesAppliedMomentumLinear
dsdVdVdtd
dtd
SVA
VA +=
Fgr
LINEAR MOTION EQUATION
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
LINEAR MOTION EQUATION
TAPP mm FFgVV ++=+
][
0
+=+SVV
P dsmdmdtddm
dtd
dtd Fgrr ][][ 2
2'
TAV
P mdmdtd FFgV ++= ][
:gives over constant are and , that Noting VPP VV
Notes: ! The Linear Motion Equations are only concerned with the motion of the Center of Mass
Defining new variables: t
ds P P T A S
'
and ; r V F F F = + = 2 '
2p
P t
=r
V
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
Returning to the Angular Motion Equation
=
SVdSdm
dtd
dtd Frrr
Expanding using the Chain Rule
=
+
SV
dSdmdtd
dtd
dtd Frrrrr 2
2
)(
0
Introducing the notation: TS
AdS MMFr +=
TAV
dmdtd MMrr += 2
2
ANGULAR MOTION EQUATION
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
Now accounting for the rotation of the coordinate frames:
rrrrr
rrr
+++=
+=
ttdtd
andtdt
d
2
2
2
2
Therefore:
TAVV
dmdmdtd
dtd MMrrrrr +=+=
)([22
Assumption 4: The body is rigid. i.e., 022
==tt
rr
ANGULAR MOTION EQUATION
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
ELASTIC STRUCTURES aircraft and spacecraft
! The Rigid Body Assumption implies that all mass elements maintain their relative distance to each other, except for mass elements which are a part of rotating machinery.
! Volume and surface integrals can only be evaluated if the external geometry is known. ? Aircraft
i geometry is known if the aircraft is rigid i if highly elastic (the usual case), then aeroelastic equilibrium must be
established before the external shape of the aircraft can be determined i an assumption which is nearly always made, but which is never actually valid
? Spacecraft i lightweight, elastic structures, often significantly more elastic than aircraft i validity varies from spacecraft to spacecraft i antennas and solar arrays large contributors to elasticity
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
i, j, and k are unit vectors along the (X, Y, Z) body fixed axes.
kjigkjiMkjiF
kjirkjiVkji
P
zyx
zyx
gggNMLFFF
zyxWVURQP
++=
++=
++=
++=
++=
++=
BODY AXIS unit vectors and components
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
BODY AXIS
Note: Positive Signs Shown
component definitions
Linear and Angular Velocities Aerodynamic and Thrust Moments
Aerodynamic and Thrust Forces Acceleration of Gravity
Ref. Roskams book
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
INERTIAS body axis moments and products
Y
X
Z
Convair B-36D Peacemaker
=
=
=
+=
+=
+=
Vyz
Vxz
Vxy
Vzz
Vyy
Vxx
dmyzI
dmxzI
dmxyI
dmyxI
dmzxI
dmzyI
)(
)(
)(
22
22
22
By Symmetry: Ixy = Iyz = 0
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
INERTIAS highly asymmetric aircraft
Blohm & Voss BV 141 B-0
Oblique Flying Wing
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
EQUATION SUMMARY Linear Motion
"")(
"")(
"")(
EquationLiftFFmgVPUQWm
EquationSideforceFFmgWPURVm
EquationDragFFmgWQVRUm
zTzAz
yTyAy
xTxAx
++=+
++=+
++=+
Drag Equation Sideforce Equation Lift Equation
Angular Motion Assuming the x-z plane is plane of symmetry, i.e., 0== yzxy II
TAxzxxyyxzzz
TAxzzzxxyy
TAyyzzxzxzxx
NNQRIPQIIPIRI
MMRPIPRIIQI
LLRQIIPQIRIPI
+=+++
+=++
+=+
)(
)()(
)(
22
Rolling Moment Equation Pitching Moment Equation Yawing Moment Equation
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
! Most airplanes are equipped with spinning rotors (propellers/turbine engines) that exert gyroscopic moments on the airframe.
! Gyroscopic moments are often negligible due to opposite rotations: ? Counter-rotating propellers ? Twin spool turbines
SPINNING ROTORS gyroscopic moments
Boeing V-22 Osprey Tiltrotor
iR
Assume one or more spinning rotors with total angular momentum
where each rotor has moment of inertia and spins with angular velocity
x y zh h h= + +h i j k
1
n
i ii=
= h h
1i i
n
i R Ri
I =
= h
iRI
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
( )( ) ( )
( )
2 2
XX XZ XZ XX YY z y A T
YY XX ZZ x z A T
ZZ XZ YY XX XZ y x A T
I P I R I PQ I I RQ Qh Rh L L
I Q I I PR IXZ P R Rh Ph M M
I R I P I I PQ I QR Ph Qh N N
+ + = +
+ + + = +
+ + + = +
A A TV
d d dr dvdt dt dt
+ = +r h M M
SPINNING ROTORS gyroscopic moments
The angular momentum equation can then be written
Taking the vector derivative of and assuming , 0iR
=ddth
Note that the gyroscopic moments appear with coupling.
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
The orientation of the body frame relative to the fixed frame can be specified using Euler Angles.
RELATIVE ORIENTATION inertial and body coordinate frames
Reference 2-2
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
EULER ATTITUDE ANGLES definition
Reference 2-2
yaw attitude anglepitch attitude angleroll attitude angle
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
=
RQP
seccossecsin0sincos0
tancostansin1
Notes: 1. This is a set of non-linear differential equations 2. In general, 3. These equation are not defined at i.e., there is a singularity.
RandQP
,.,deg90=
KINEMATIC EQUATIONS Euler angle and body axis rates
1 0 sin0 cos cos sin0 sin cos cos
PQR
=
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
EQUATIONS OF MOTION observations
! The Equations of Motion are a set of 9 differential equations: ? First order ? Nonlinear ? Coupled ? Ordinary
! The Variables are: ?
! The aerodynamic and thrust Forcing Functions:
?
are functions of: ? Velocity ? Angle-of-attack ? Sideslip angle
TATATATATATA NandNMMLLFFFFFF zzyyxx ,,,,,,,,,,
and , R, Q, P, W,V, U,
? Time ? Altitude ? Configuration
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
Angle-of-Attack
Sideslip Angle
Grumman F11F-1 Tiger
AERODYNAMIC ANGLES
XI
XB
VP
VP XB
definitions
Note: All Angles Shown Are Positive
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
Angle-of-Attack
Sideslip Angle
AERODYNAMIC ANGLES approximations
Grumman F11F-1 Tiger
v U1
u
1
1 1
sin v vU U
=
u
U1
w
1
1 1
sin w wU U
=
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
FORCES AND MOMENTS longitudinal perturbations
! With
! There are separate equations for the aerodynamic and thrust forces and moments.
1 1
and ; controls and :E Fw w
U U = =
X X X X X XX E F
E F
Z Z Z Z Z ZZ E F
E F
E FE F
F F F F F Ff u qu q
F F F F F Ff u qu q
M M M M M Mm u qu q
= + + + + +
= + + + + +
= + + + + +
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
FORCES AND MOMENTS lateral / directional perturbations
! With
! There are separate equations for the aerodynamic and thrust forces and moments.
RR
AA
RR
AA
RR
YA
A
YYYYYY
NNrrNp
pNNNn
LLrrLp
pLLLl
FFrr
FppFFFf
+
+
+
+
+
=
+
+
+
+
+
=
+
+
+
+
+
=
1 1
and ; controls and :A Rv v
U U = =
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
FORCES AND MOMENTS shorthand notation
! Dividing by mass, each term becomes a longitudinal linear or angular acceleration.
! Letting , dimensional derivatives are the linear or angular acceleration per change in the associated motion variable. ? is the pitch angular acceleration imparted to the airplane as the result
of a unit change in angle-of-attack.
! There will be separate equations for the aerodynamic and thrust forces and moments.
FEquYY
FEquZ
FEquX
FE
FE
FE
MMqMMMuMmI
ZZqZZZuZfm
XXqXXXuXfm
+++++=
+++++=
+++++=
1
1
1
M
variablemoment 1 momentvariable m
=
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
FORCES AND MOMENTS shorthand notation
! Dividing by mass, each term becomes a longitudinal linear or angular acceleration.
! Letting , dimensional derivatives are the linear or angular acceleration per change in the associated control variable. ? is the sideforce linear acceleration imparted to the airplane as the result
of a unit change in rudder deflection.
! There will be separate equations for the aerodynamic and thrust forces and moments.
controlforce 1 force
control m =
RY
RArpZZ
RArpXX
RArpY
RA
RA
RA
NNrNpNNNnI
LLrLpLLLlI
YYrYpYYYfm
+++++=
+++++=
+++++=
1
1
1
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
LIFT AND DRAG FORCES definitions
Grumman F11F-1 Tiger
Lift
U1
Drag
XB
ZB
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
In body axes,
1
211 12
21 11 1 1 1 1 12 2
1
cos cos
cos cos (2 )
X
X
A D
A DD
F D U SC
F CU S SC UU U
= =
=
1 1
1
211 1 1 112
1 1 11
1
1111
2 21cos cos
cos 2
X
X
u
A D DD D
AD D
F C CC Cu U S u q S uU U U U UU
U
F uu q S C CU U
= =
=
Then
Or
NON-DIMENSIONAL DERIVATIVES
UCU
UUCC DDDu
=
= 1
1
Define the derivative:
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
In body axes:
1
211 12
21 11 1 1 1 1 12 2
1
cos cos
cos cos (2 )
X
X
A D
A DD
F D U SC
F CU S SC UU U
= =
=
uUC
UUC
USqu
UC
UCSUu
UF DDDDAX
=
=
1
1
11
1
2112
1
1
11212Then:
Or [ ]1
1
11
2UuCCSqu
UF
DDA
u
X =
NON-DIMENSIONAL DERIVATIVES
UCU
UUCC DDDu
=
= 1
1
Define the derivative:
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
DIMENSIONAL DERIVATIVES longitudinal linear accelerations
)sec(
)sec(2
)sec(2
)sec()2(
)(sec)2(
)sec()sec()2(
)(sec)2(
)(sec)2(
21
1
1
11
1
1
211
1
1
2121
1
1
11
1
1
11
1
11
=
=
=
+
=+
=
=+
=
+=
+=
ftmSCq
Z
ftmU
CcSqZft
mU
CcSqZ
ftm
CCSqZ
mUCCSq
Z
ftmSCq
Xftm
CCSqX
mU
CCSqX
mUCCSq
X
E
E
q
u
E
E
XuXu
L
L
q
L
DLLLu
DLD
TT
uTDD
u
Note: All Quantities are Linear Acceleration per Unit of Applied Variable
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
)(sec
)(sec2
)(sec
)sec()2(
)sec()2(
21
1
1
2
1
21
11
1
1
11
1
1
1
1
=
=
=
+
=
+
=
YY
M
YY
M
YY
M
YY
TT
uT
YY
MMu
ICcSq
M
UI
CcSqM
ICcSq
M
ftUI
CCcSqM
ftUI
CCcSqM
E
E
MuM
u
DIMENSIONAL DERIVATIVES longitudinal angular accelerations
Note: All Quantities are Angular Acceleration per Unit of Applied Variable
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
)(sec)(sec
)(sec2
)(sec2
)(sec)sec(
)sec()sec(2
)sec(2
)sec(
2121
1
1
211
1
21
2121
211
1
1
1
1
121
==
==
==
==
=
=
XX
L
XX
L
XX
Lr
XX
L
p
XX
LY
YYr
Y
p
Y
ICbSq
LI
CbSqL
UICbSq
LUICbSq
L
ICbSq
LftmSCq
Y
ftmSCq
YftmU
CbSqY
ftmU
CbSqYft
mSCq
Y
R
R
A
A
rp
R
R
A
A
r
p
DIMENSIONAL DERIVATIVES lateral linear accelerations
Note: All Quantities are Linear Acceleration per Unit of Applied Variable
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
)(sec)(sec
)(sec2
)(sec2
)(sec)(sec
2121
1
1
211
1
21
2121
==
==
==
ZZ
N
ZZ
N
ZZ
Nr
ZZ
N
p
ZZ
N
TZZ
N
ICbSq
NI
CbSqN
UICbSq
NUICbSq
N
I
CbSqN
ICbSq
N
R
R
A
A
rp
T
DIMENSIONAL DERIVATIVES lateral angular accelerations
Note: All Quantities are Angular Acceleration per Unit of Applied Variable
Dr. Kamesh Subbarao Department of Mechanical and Aerospace Engineering, UTA
STATIC STABILITY CRITERIA
r q p w v u Forces and Moments wU1
=vU1 =
xx TAF F+
yy TAF F+
zz TAF F+
TAL L+
TAM M+
TAN N+
Du 0C >
y 0C
l 0C nr 0C