Aircraft Flight Dynamics Robert F. Stengel Lecture9 Aircraft Equations of Motion 2

Aircraft Equations of Motion - 2

Robert Stengel, Aircraft Flight Dynamics, MAE 331,
2012
"
•  Rotating frames of reference"
•  Combined equations of
motion"
•  FLIGHT 6-DOF simulation
program"
Copyright 2012 by Robert Stengel. All rights reserved. For educational use only.!
/>!
/>!
Euler Angle Rates
Euler-Angle Rates
and Body-Axis Rates"
Body-axis
angular rate
vector
(orthogonal)"
ω
B
=
ω
x
ω
y
ω
z
"
#
$

$
$
$
%
&
'
'
'
'
B
=
p
q
r
"
#
$
$
$
%
&
'
'
'
Euler-angle
rate vector"
Form a non-
orthogonal vector
of Euler angles"
Θ =

φ
θ
ψ
%
&
'
'
'
(
)
*
*
*


Θ =

φ

θ

ψ
%
&
'
'
'
(
)
*

*
*
≠
ω
x
ω
y
ω
z
%
&
'
'
'
'
(
)
*
*
*
*
I
Relationship Between Euler-Angle
Rates and Body-Axis Rates"
•  is measured in the Inertial Frame"
•  is measured in Intermediate Frame #1"
•  is measured in Intermediate Frame #2"
•  Inverse transformation [(.)
-1
≠ (.)

T
] "
€
˙
φ
€
˙
θ
€
˙
ψ

p
q
r
!
"
#
#
#
$
%
&
&
&
= I
3

φ
0

0
!
"
#
#
#
$
%
&
&
&
+ H
2
B
0

θ
0
!
"
#
#
#
$
%
&
&
&
+ H
2

B
H
1
2
0
0

ψ
!
"
#
#
#
$
%
&
&
&

p
q
r
!
"
#
#
#
$
%
&

&
&
=
1 0 −sin
θ
0 cos
φ
sin
φ
cos
θ
0 −sin
φ
cos
φ
cos
θ
!
"
#
#
#
$
%
&
&
&

φ


θ

ψ
!
"
#
#
#
$
%
&
&
&
= L
I
B

Θ


φ

θ

ψ
$
%
&
&
&

'
(
)
)
)
=
1 sin
φ
tan
θ
cos
φ
tan
θ
0 cos
φ
−sin
φ
0 sin
φ
sec
θ
cos
φ
sec
θ
$
%
&
&

&
'
(
)
)
)
p
q
r
$
%
&
&
&
'
(
)
)
)
= L
B
I
ω
B
"Can the inversion
become singular?"
"What does this mean? "
•  which is"
Euler-Angle Rates and Body-Axis Rates"
Avoiding the Singularity at θ = ±90°"

!  Don’t use Euler angles as primary
definition of angular attitude"
!  Alternatives to Euler angles"
-  Direction cosine (rotation) matrix"
-  Quaternions"
!  Propagation of rotation matrix
(9 parameters)"
-  From previous lecture"


H
B
I
h
B
=

ω
I
H
B
I
h
B


H
I
B
t

( )
= −

ω
B
t
( )
H
I
B
t
( )
= −
0 −r t
( )
q t
( )
r t
( )
0 −p t
( )
−q t
( )
p t
( )
0 t
( )
#
$
%

%
%
%
&
'
(
(
(
(
B
H
I
B
t
( )
; H
I
B
0
( )
= H
I
B
φ
0
,
θ
0
,
ψ

0
( )
Consequently"
Avoiding the Singularity at θ = ±90°"
!  Propagation of quaternion vector"
o  see Flight Dynamics for details"
e
1
e
2
e
3
e
4
!
"
#
#
#
#
#
$
%
&
&
&
&
&
=
Rotation angle, rad

x-component of rotation axis
y-component of rotation axis
z-component of rotation axis
!
"
#
#
#
#
#
$
%
&
&
&
&
&
!  Quaternion vector: single rotation from
inertial to body frame (4 parameters)"


e t
( )
=

e
1
t
( )


e
2
t
( )

e
3
t
( )

e
4
t
( )
!
"
#
#
#
#
#
#
$
%
&
&
&
&
&
&

=
0 −r t
( )
−q t
( )
−p t
( )
r t
( )
0 −p t
( )
q t
( )
q t
( )
p t
( )
0 −r t
( )
p t
( )
−q t
( )
r t
( )
0
!
"
#
#

#
#
#
#
$
%
&
&
&
&
&
&
e
1
t
( )
e
2
t
( )
e
3
t
( )
e
4
t
( )
!
"

#
#
#
#
#
#
$
%
&
&
&
&
&
&
= Q t
( )
e t
( )
; e 0
( )
= e
φ
0
,
θ
0
,
ψ
0
( )

Rigid-Body
Equations of Motion
Point-Mass
Dynamics"
•  Inertial rate of change of translational position"
•  Body-axis rate of change of translational velocity"
–  Identical to angular-momentum transformation"


r
I
= v
I
= H
B
I
v
B


v
I
=
1
m
F
I

v
B

= H
I
B

v
I
−

ω
B
v
B
=
1
m
H
I
B
F
I
−

ω
B
v
B
=
1
m
F

B
−

ω
B
v
B
F
B
=
X
Y
Z
!
"
#
#
#
$
%
&
&
&
B
=
C
X
qS
C
Y

qS
C
Z
qS
!
"
#
#
#
$
%
&
&
&
v
B
=
u
v
w
!
"
#
#
#
$
%
&
&
&


r
I
t
( )
= H
B
I
t
( )
v
B
t
( )


v
B
t
( )
=
1
m t
( )
F
B
t
( )
+ H

I
B
t
( )
g
I
−

ω
B
t
( )
v
B
t
( )


Θ
I
t
( )
= L
B
I
t
( )
ω
B
t

( )


ω
B
t
( )
= I
B
−1
t
( )
M
B
t
( )
−

ω
B
t
( )
I
B
t
( )
ω
B
t
( )

#
$
%
&
•  Rate of change of
Translational Position "
•  Rate of change of
Angular Position "
•  Rate of change of
Translational Velocity "
•  Rate of change of
Angular Velocity "
r
I
=
x
y
z
!
"
#
#
#
$
%
&
&
&
I
Θ

I
=
φ
θ
ψ
%
&
'
'
'
(
)
*
*
*
I
v
B
=
u
v
w
!
"
#
#
#
$
%
&

&
&
B
ω
B
=
p
q
r
"
#
$
$
$
%
&
'
'
'
B
•  Translational
Position "
•  Angular
Position "
•  Translational
Velocity"
•  Angular
Velocity "
Rigid-Body Equations of Motion"
(Euler Angles)"

Aircraft Characteristics
Expressed in Body Frame
of Reference"
I
B
=
I
xx
−I
xy
−I
xz
−I
xy
I
yy
−I
yz
−I
xz
−I
yz
I
zz
"
#
$
$
$
$

%
&
'
'
'
'
B
F
B
=
X
aero
+ X
thrust
Y
aero
+ Y
thrust
Z
aero
+ Z
thrust
!
"
#
#
#
$
%
&

&
&
B
=
C
X
aero
+ C
X
thrust
C
Y
aero
+ C
Y
thrust
C
Z
aero
+ C
Z
thrust
!
"
#
#
#
#
$
%

&
&
&
&
B
1
2
ρ
V
2
S =
C
X
C
Y
C
Z
!
"
#
#
#
$
%
&
&
&
B
q S
Aerodynamic

and thrust
force "
Aerodynamic and
thrust moment "
Inertia
matrix "
Reference Lengths
b = wing span
c = mean aerodynamic chord
M
B
=
L
aero
+ L
thrust
M
aero
+ M
thrust
N
aero
+ N
thrust
!
"
#
#
#
$

%
&
&
&
B
=
C
l
aero
+ C
l
thrust
( )
b
C
m
aero
+ C
m
thrust
( )
c
C
n
aero
+ C
n
thrust
( )
b

!
"
#
#
#
#
#
$
%
&
&
&
&
&
B
1
2
ρ
V
2
S =
C
l
b
C
m
c
C
n
b

!
"
#
#
#
$
%
&
&
&
B
q S
Rigid-Body Equations of Motion:
Position"


x
I
= cos
θ
cos
ψ
( )
u + −cos
φ
sin
ψ
+ sin
φ
sin

θ
cos
ψ
( )
v + sin
φ
sin
ψ
+ cos
φ
sin
θ
cos
ψ
( )
w

y
I
= cos
θ
sin
ψ
( )
u + cos
φ
cos
ψ
+ sin
φ

sin
θ
sin
ψ
( )
v + −sin
φ
cos
ψ
+ cos
φ
sin
θ
sin
ψ
( )
w

z
I
= −sin
θ
( )
u + sin
φ
cos
θ
( )
v + cos
φ

cos
θ
( )
w


φ
= p + qsin
φ
+ r cos
φ
( )
tan
θ

θ
= q cos
φ
− r sin
φ

ψ
= qsin
φ
+ r cos
φ
( )
sec
θ
•  Rate of change of Translational Position "

•  Rate of change of Angular Position "
Rigid-Body Equations of Motion:
Rate"


u = X / m − gsin
θ
+ rv − qw

v =Y / m + gsin
φ
cos
θ
− ru + pw

w = Z / m + g cos
φ
cos
θ
+ qu − pv


p = I
zz
L + I
xz
N − I
xz
I
yy

− I
xx
− I
zz
( )
p + I
xz
2
+ I
zz
I
zz
− I
yy
( )
"
#
$
%
r
{ }
q
( )
I
xx
I
zz
− I
xz
2

( )

q =
1
I
yy
M − I
xx
− I
zz
( )
pr − I
xz
p
2
− r
2
( )
"
#
$
%

r = I
xz
L + I
xx
N − I
xz
I

yy
− I
xx
− I
zz
( )
r + I
xz
2
+ I
xx
I
xx
− I
yy
( )
"
#
$
%
p
{ }
q
( )
I
xx
I
zz
− I
xz

2
( )
•  Rate of change of Translational Velocity "
•  Rate of change of Angular Velocity "
Mirror symmetry, I
xz
≠ 0#
FLIGHT - 
Computer Program to
Solve the 6-DOF
Equations of Motion
/>FLIGHT - MATLAB Program"
/>FLIGHT - MATLAB Program"
Examples from FLIGHT
Longitudinal Transient
Response to Initial Pitch Rate"
Bizjet, M = 0.3, Altitude = 3,052 m!
•  For a symmetric aircraft, longitudinal
perturbations do not
induce lateral-
directional motions "
Transient Response
to Initial Roll Rate"
Lateral-Directional Response"
Longitudinal Response"
Bizjet, M = 0.3, Altitude = 3,052 m!
•  For a symmetric aircraft, lateral-
directional perturbations do
induce longitudinal motions "
Transient Response to

Initial Yaw Rate"
Lateral-Directional Response"
Longitudinal Response"
Bizjet, M = 0.3, Altitude = 3,052 m!
Crossplot of Transient
Response to Initial Yaw Rate"
Bizjet, M = 0.3, Altitude = 3,052 m!
Longitudinal-Lateral-Directional Coupling"
Alternative Reference
Frames
Velocity Orientation in an Inertial
Frame of Reference"
Polar Coordinates" Projected on a Sphere"
Body Orientation with Respect
to an Inertial Frame"
Relationship of Inertial Axes
to Body Axes"
•  Transformation is
independent of
velocity vector"
•  Represented by"
–  Euler angles"
–  Rotation matrix "
v
x
v
y
v
z
!

"
#
#
#
#
$
%
&
&
&
&
= H
B
I
u
v
w
!
"
#
#
#
$
%
&
&
&
u
v
w

!
"
#
#
#
$
%
&
&
&
= H
I
B
v
x
v
y
v
z
!
"
#
#
#
#
$
%
&
&
&

&
Velocity-Vector Components
of an Aircraft"
V ,
ξ
,
γ
V ,
β
,
α
Velocity Orientation with Respect
to the Body Frame"
Polar Coordinates" Projected on a Sphere"
•  No reference to the body
frame"
•  Bank angle, μ, is roll angle
about the velocity vector "
V
ξ
γ
#
$
%
%
%
&
'
(
(

(
=
v
x
2
+ v
y
2
+ v
z
2
sin
−1
v
y
/ v
x
2
+ v
y
2
( )
1/ 2
#
$
&
'
sin
−1
−v

z
/ V
( )
#
$
%
%
%
%
%
&
'
(
(
(
(
(
v
x
v
y
v
z
!
"
#
#
#
#
$

%
&
&
&
&
I
=
V cos
γ
cos
ξ
V cos
γ
sin
ξ
−V sin
γ
!
"
#
#
#
$
%
&
&
&
Relationship of Inertial Axes
to Velocity Axes"
Relationship of Body Axes

to Wind Axes"
•  No reference to
the inertial frame "
u
v
w
!
"
#
#
#
$
%
&
&
&
=
V cos
α
cos
β
V sin
β
V sin
α
cos
β
!
"
#

#
#
$
%
&
&
&
V
β
α
#
$
%
%
%
&
'
(
(
(
=
u
2
+ v
2
+ w
2
sin
−1
v / V

( )
tan
−1
w / u
( )
#
$
%
%
%
%
&
'
(
(
(
(
Angles Projected
on the Unit Sphere"
€
α
: angle of attack
β
: sideslip angle
γ
: vertical flight path angle
ξ
: horizontal flight path angle
ψ
: yaw angle

θ
: pitch angle
φ
: roll angle (about body x − axis)
µ
: bank angle (about velocity vector)
•  Origin is
airplanes center
of mass"
Alternative Frames of Reference"
•  Orthonormal transformations connect all reference frames"
Next Time:
Linearization and Modes of
Motion


Reading
Flight Dynamics, 234-242,
255-266, 274-297, 321-330 
Virtual Textbook, Part 10

Supplemental Material


r
I
= H
B
I
v

B


v
B
=
1
m
F
B
+ H
I
B
g
I
−

ω
B
v
B


ω
B
= I
B
−1
M
B

−

ω
B
I
B
ω
B
( )
•  Rate of change of
Translational Position "
•  Rate of change of
Rotation Matrix "
•  Rate of change of
Translational Velocity "
•  Rate of change of
Angular Velocity "
r
I
=
x
y
z
!
"
#
#
#
$
%

&
&
&
I
Θ
I
= fcn H
I
B
( )
v
B
=
u
v
w
!
"
#
#
#
$
%
&
&
&
B
ω
B
=

p
q
r
"
#
$
$
$
%
&
'
'
'
B
•  Translational
Position "
•  Angular
Position "
•  Translational
Velocity"
•  Angular
Velocity "
Rigid-Body Equations of Motion"
(Attitude from Rotation Matrix)"


H
I
B
= −


ω
B
H
I
B


r
I
= H
B
I
v
B


v
B
=
1
m
F
B
+ H
I
B
g
I
−


ω
B
v
B


ω
B
= I
B
−1
M
B
−

ω
B
I
B
ω
B
( )
•  Rate of change of
Translational Position "
•  Rate of change of
Rotation Matrix "
•  Rate of change of
Translational Velocity "
•  Rate of change of

Angular Velocity "
r
I
=
x
y
z
!
"
#
#
#
$
%
&
&
&
I
Θ
I
= fcn H
I
B
e
( )
"
#
$
%
v

B
=
u
v
w
!
"
#
#
#
$
%
&
&
&
B
ω
B
=
p
q
r
"
#
$
$
$
%
&
'

'
'
B
•  Translational
Position "
•  Angular
Position "
•  Translational
Velocity"
•  Angular
Velocity "
Rigid-Body Equations of Motion"
(Attitude from Quaternion Vector)"


e = Qe