Aircraft Equations of Motion - 2!
"
Robert Stengel, Aircraft Flight Dynamics, MAE 331,
2014
•!
•!
•!
•!
•!
Learning Objectives!
How is a rotating reference frame described
in an inertial reference frame?"
Is the transformation singular?"
What adjustments must be made to
expressions for forces and moments in a
non-inertial frame?"
How are the 6-DOF equations implemented
in a computer?"
Damping effects"
Reading:!
Flight Dynamics!
161-180!
Copyright 2014 by Robert Stengel. All rights reserved. For educational use only.!
http://www.princeton.edu/~stengel/MAE331.html!
http://www.princeton.edu/~stengel/FlightDynamics.html!
1!
Euler Angle Rates!
2!
Euler-Angle Rates
and Body-Axis Rates"
Body-axis
angular rate
vector
(orthogonal)"
" !x
$
!B = $ !y
$
$# ! z
Form a nonorthogonal vector
of Euler angles"
Euler-angle
rate vector"
%
" p %
'
'
$
' =$ q '
'
$ r '
&
'& B #
% " (
'
*
!=' # *
' $ *
&
)
% "! ( % ,
* ' x
'
! = ' #! * + ' , y
!
' $! * '
) '& , z
&
(
*
*
*
*) I
3!
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"
•!
'!
0
0
$
!
&
#
& + H 2B #
&%
#"
0
(!
0
$
!
&
B 2#
& + H 2 H1 #
&%
#"
0
0
)!
$
&
&
&%
... which is"
! p $ ! 1
0
& #
#
# q & = # 0 cos )
# r & # 0 'sin )
% "
"
•!
!
! p $
#
&
#
# q & = I3 #
#"
# r &
%
"
'sin ( $!# )! $&
&
!
sin ) cos( &# (! & = LBI +
&
#
cos ) cos( &%" *! %
Inverse transformation
[(.)-1
#
(.)T]
$
&
"&
&
%
!!
"!
#!
"Can the inversion
become singular?"
"What does this mean? "
' $
) & 1 sin ! tan "
)=& 0
cos !
) &
( % 0 sin ! sec"
cos ! tan " '$ p
)&
*sin ! )& q
cos ! sec" )(&% r
'
) I
) = LB+ B
)
(
4!
Euler-Angle Rates and Body-Axis Rates"
5!
Avoiding the Singularity at " = ±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"
! IB h B = !
" I H IB h B
H
# 0
!r ( t ) q ( t )
%
! BI ( t ) = ! "" B ( t ) H BI ( t ) = ! % r ( t )
H
0
! p (t )
%
%$ !q ( t ) p ( t ) 0 ( t )
Consequently"
H BI ( 0 ) = H BI (!0 ," 0 ,# 0 )
&
(
( H BI ( t )
(
(' B
6!
Avoiding the Singularity at " = ±90°"
!! Quaternion vector: single rotation from
inertial to body frame (4 parameters)"
!
#
#
#
#
#
"
e1 $ !
Rotation angle, rad
& #
e2 & # x-component of rotation axis
&=
e3 & # y-component of rotation axis
#
e4 &% #" z-component of rotation axis
$
&
&
&
&
&%
!! Propagation of quaternion vector"
o! see Flight Dynamics for details"
!
#
#
e! (t ) = #
#
#
#"
'r (t ) 'q (t ) ' p (t )
e!1 (t ) $ ! 0
& #
0
' p (t ) q (t )
e!2 (t ) & # r (t )
&=#
0
'r (t )
e!3 (t ) & # q (t ) p (t )
& #
0
e!4 (t ) &% #" p (t ) 'q (t ) r (t )
$!
&#
&#
&#
&#
&#
&%#"
e1 (t ) $
&
e2 (t ) &
& = Q (t ) e (t ) ; e ( 0 ) = e ((0 ,) 0 , *0 )
e3 (t ) &
&
e4 (t ) &%
7!
Rigid-Body
Equations of Motion!
8!
Point-Mass
Dynamics"
•! Inertial rate of change of translational position"
r!I = v I = H IB v B
! u $
#
&
vB = # v &
#" w &%
•! Body-axis rate of change of translational velocity"
–! Identical to angular-momentum transformation"
v! I =
1
FI
m
v! B = H BI v! I ! "" B v B =
=
1
FB ! "" B v B
m
1 B
H I FI ! "" B v B
m
$
&
&
&
%
! X $ ! C X qS
#
& #
FB = # Y & = # CY qS
#" Z &%B # CZ qS
"
9!
Rigid-Body Equations of Motion"
(Euler Angles)"
•!
Rate of change of
Translational Position "
r!I (t ) = H (t ) v B (t )
•!
Rate of change of
Angular Position "
! I (t ) = L (t ) " B (t )
!
•!
1
Rate of change of
" B (t ) v B (t )
F t + H BI (t ) g I ! "
v! t =
Translational Velocity " B ( ) m (t ) B ( )
•!
Rate of change of
Angular Velocity "
I
B
I
B
" B (t ) I B (t ) ! B (t )%&
!! B (t ) = I B"1 (t ) #$M B (t ) " !
•! Translational
Position "
! x $
&
#
rI = # y &
#" z &%
I
•! Angular
Position " %
" (
'
*
!I = ' # *
' $ *
&
)I
•! Translational
Velocity"
! u $
&
#
vB = # v &
#" w &% B
•! Angular
Velocity "
" p %
'
$
!B = $ q '
$ r '
&B
#
10!
Aircraft Characteristics
Expressed in Body Frame
of Reference"
! Xaero + Xthrust
#
FB = # Yaero + Ythrust
# Z +Z
thrust
" aero
Aerodynamic
and thrust
force "
Aerodynamic and
thrust moment "
Inertia
matrix"
! Laero + Lthrust
#
M B = # M aero + M thrust
# N +N
aero
thrust
"
" I xx
$
I B = $ !I xy
$
$# !I xz
!I xy
I yy
!I yz
! CX + CX
$
aero
thrust
#
&
C
+
C
& = # Yaero
Ythrust
#
&
% B #" CZaero + CZthrust
(
(
(
$
! CX
& 1
#
2
& 'V S = # CY
& 2
# C
&%
" Z
B
$
&
& qS
&
%B
)
)
)
$
! C +C
laero
lthrust b
$
! Cl b $
&
#
&
&
#
& 1 2
#
& = # Cmaero + Cmthrust c & 'V S = # Cm c & q S
2
&
# Cb &
&
#
% B # Cnaero + Cnthrust b &
" n %B
%B
"
!I xz %
'
!I yz '
'
I zz '
&B
Reference Lengths
b = wing span
c = mean aerodynamic chord
11!
Rigid-Body Equations of Motion:
Position"
•! Rate of change of Translational Position "
x! I = ( cos! cos" ) u + ( # cos $ sin " + sin $ sin ! cos" ) v + ( sin $ sin " + cos $ sin ! cos" ) w
y! I = ( cos! sin " ) u + ( cos $ cos" + sin $ sin ! sin " ) v + ( # sin $ cos" + cos $ sin ! sin " ) w
z!I = ( # sin ! ) u + ( sin $ cos! ) v + ( cos $ cos! ) w
•! Rate of change of Angular Position "
!! = p + ( q sin ! + r cos ! ) tan "
"! = q cos ! # r sin !
$! = ( q sin ! + r cos ! ) sec "
12!
Rigid-Body Equations of Motion:
Rate"
•! Rate of change of Translational Velocity "
u! = X / m ! gsin " + rv ! qw
v! = Y / m + gsin # cos" ! ru + pw
w! = Z / m + g cos # cos" + qu ! pv
•! Rate of change of Angular Velocity "
(
{
(
{
} ) (I
xx zz
} ) (I
xx zz
p! = I zz L + I xz N ! I xz ( I yy ! I xx ! I zz ) p + "# I xz2 + I zz ( I zz ! I yy )$% r q
q! =
1 "
M ! ( I xx ! I zz ) pr ! I xz ( p 2 ! r 2 )$%
#
I yy
r! = I xz L + I xx N ! I xz ( I yy ! I xx ! I zz ) r + "# I xz2 + I xx ( I xx ! I yy )$% p q
I ! I xz2 )
Mirror symmetry, Ixz # 0!
I ! I xz2 )
13!
FLIGHT -!
Computer Program to
Solve the 6-DOF
Equations of Motion!
14!
FLIGHT - MATLAB Program"
http://www.princeton.edu/~stengel/FlightDynamics.html!
15!
FLIGHT - MATLAB Program"
http://www.princeton.edu/~stengel/FlightDynamics.html!
16!
Examples from FLIGHT!
17!
Longitudinal Transient
Response to Initial Pitch Rate"
Bizjet, M = 0.3, Altitude = 3,052 m!
•!
For a symmetric aircraft, longitudinal
perturbations do not induce lateraldirectional motions "
18!
Transient Response
to Initial Roll Rate"
Lateral-Directional Response"
Bizjet, M = 0.3, Altitude = 3,052 m!
•!
Longitudinal Response"
For a symmetric aircraft, lateraldirectional perturbations do
induce longitudinal motions "
19!
Transient Response to
Initial Yaw Rate"
Lateral-Directional Response"
Longitudinal Response"
Bizjet, M = 0.3, Altitude = 3,052 m!
20!
Crossplot of Transient
Response to Initial Yaw Rate"
Longitudinal-Lateral-Directional Coupling"
Bizjet, M = 0.3, Altitude = 3,052 m!
21!
Aerodynamic Damping!
22!
Pitching Moment due to Pitch Rate"
(
)
M B = Cm q Sc ! Cmo + Cmq q + Cm" " q Sc
$
'
#C
! & Cmo + m q + Cm" " ) q Sc
%
(
#q
23!
Angle of Attack Distribution
Due to Pitch Rate"
•! Aircraft pitching at a constant rate, q rad/s, produces a
normal velocity distribution along x"
!w = "q!x
•! Corresponding angle of attack distribution"
!" =
!w #q!x
=
V
V
•! Angle of attack perturbation at tail center of pressure"
!" ht =
qlht
V
lht = horizontal tail distance from c.m.
24!
Horizontal Tail Lift
Due to Pitch Rate"
•! Incremental tail lift due to pitch rate, referenced to tail area, Sht"
(
!Lht = !C Lht
)
ht
1
"V 2 Sht
2
•! Incremental tail lift coefficient due to pitch rate,
referenced to wing area, S"
( !C )
Lht aircraft
(
= !C Lht
)
- " ( CL %
" Sht % *" ( C Lht %
" qlht %
ht
=$
!
)
=
,
/
$
'
$
'
$
'
'
ht # S &
,+# () & aircraft
/. # () & aircraft # V &
•! Lift coefficient sensitivity to pitch rate referenced to wing area"
C Lqht !
(
" #C Lht
)
"q
aircraft
% " C Lht (
% lht (
='
' *
*
& "$ ) aircraft & V )
25!
Moment Coefficient
Sensitivity to Pitch Rate of
the Horizontal Tail"
•! Differential pitch moment due to pitch rate"
! "M ht
1
%l ( 1
= Cmqht #V 2 Sc = $C Lqht ' ht * #V 2 Sc
&V)2
2
!q
,% ! CL (
% lht ( / % lht ( 1
2
ht
= $ .'
' * 1 ' * #V Sc
*
.-& !+ ) aircraft & V ) 10 & c ) 2
•! Coefficient derivative with respect to pitch rate"
Cmqht = !
" C Lht
"#
" C Lht $ lht ' $ c '
$ lht ' $ lht '
&% )( &% )( = !
& ) & )
V
c
"# % c ( % V (
2
•! Coefficient derivative with respect to normalized pitch rate is
insensitive to velocity"
Cmq̂ht
! Cmht
! Cmht
! C Lht
=
=
= "2
! q̂
!#
! qc 2V
(
)
$ lht '
&% )(
c
2
26!
Pitch-Rate Derivative Definitions"
•! Pitch-rate derivatives are often expressed
in terms of a normalized pitch rate"
•! Then"
Cmq̂ =
q̂ =
qc
2V
! Cm
! Cm
" 2V %
=
=$
C
# c '& mq
! q̂ ! qc
2V
(
)
Pitching moment sensitivity to pitch rate"
C mq =
! Cm " c %
=$
'& Cmq̂
#
2V
!q
But dynamic equations require $Cm/$q"
2
# "VSc 2 &
!M
# c & # "V &
2
Sc = Cmq̂ %
= Cmq ( "V 2 ) Sc = Cmq̂ %
$ 2V (' %$ 2 ('
!q
$ 4 ('
27!
Roll Damping Due to Roll Rate!
2
" !V 2 %
" b % " !V %
Sb
=
C
Sb
Cl p $
l p̂ $
# 2V '& $# 2 '&
# 2 '&
" !V % 2
= Cl p̂ $
Sb
# 4 '&
•!
( )
Vertical Tail
( )
+ Cl p̂
Horizontal Tail
pb
2V
( )
+ Cl p̂
Wing
Wing with taper"
(C )
l p̂
•!
< 0 for stability!
Vertical tail, horizontal tail, and wing
are principal contributors"
Cl p̂ ! Cl p̂
•!
p̂ =
Wing
=
! ( "Cl )Wing
! p̂
Thin triangular wing"
NACA-TR-1098, 1952!
NACA-TR-1052, 1951 !
=#
C L$ & 1 + 3% )
(
+
12 ' 1 + % *
(C )
l p̂
Wing
=!
" AR
32
28!
Roll Damping Due
to Roll Rate!
•! Tapered vertical tail"
(C )
l p̂
vt
=
CY
! ( "Cl )vt
= # $vt
! p̂
12
% Svt (% 1+ 3+ (
' *'
*
& S )& 1+ + )
p̂ =
pb
2V
•! Tapered horizontal tail"
( )
Cl p̂
ht
=
CL
! ( "Cl )ht
= # $ht
! p̂
12
% Sht (% 1+ 3+ (
' *'
*
& S )& 1+ + )
•! pb/2V describes helix
angle for a steady roll"
29!
Yaw Damping Due to Yaw Rate!
2
" !V 2 %
" b % " !V %
Sb = Cnr̂ $
Sb
Cnr $
# 2V '& $# 2 '&
# 2 '&
" !V % 2
= Cnr̂ $
Sb
# 4 '&
r̂ =
< 0 for stability!
rb
2V
30!
Yaw Damping Due
to Yaw Rate!
( )
Cnr̂ ! Cnr̂
Vertical Tail
( )
+ Cnr̂
r̂ =
Wing
rb
2V
•! Vertical tail contribution"
( )
! ( Cn )Vertical Tail = " Cn#
(C )
nr̂ vt
=
Vertical Tail
! " ( Cn )Vertical Tail
(
! rb 2V
)
( )
rlvt
=
( )
V = " Cn#
Vertical Tail
! " ( Cn )Vertical Tail
! r̂
$ lvt ' $ b '
&% )( &% )( r
b V
% lvt (
' *
Vertical Tail & b )
( )
= #2 Cn$
•! Wing contribution"
(C )
nr̂ Wing
= k0C L2 + k1C DParasite, Wing
k0 and k1 are functions of
aspect ratio and sweep angle"
NACA-TR-1098, 1952!
NACA-TR-1052, 1951 !
31!
Next Time:!
Aircraft Control Devices and
Systems!
Reading:!
Flight Dynamics!
214-234!
32!
Supplemental
Material
33!
Airplane Angular Attitude (Position)!
(! ," ,# )
•! Euler angles "
•! 3 angles that relate one
Cartesian coordinate frame to
another"
•! defined by sequence of 3
rotations about individual axes"
•! intuitive description of angular
attitude"
•! Euler angle rates have a
nonlinear relationship to bodyaxis angular rate vector"
•! Transformation of rates is
singular at 2 orientations, ±90°"
34!
Airplane Angular Attitude (Position)!
B
$ h11 h12
&
H BI (! ," ,# ) = & h21 h22
& h
h
% 31 32
h13 '
)
h23 )
h33 )
(I
H BI (! ," ,# ) = H 2B (! )H12 (" )H1I (# )
T
$% H BI (! ," ,# ) '( = $% H BI (! ," ,# ) '( = H IB (# ," , ! )
*1
•! Rotation matrix "
•! orthonormal transformation"
•! inverse = transpose"
•! linear propagation from one attitude to
another, based on body-axis rate vector"
•! 9 parameters, 9 equations to solve"
•! solution for Euler angles from parameters
is intricate"
35!
Airplane Angular Attitude (Position)!
Rotation Matrix!
H BI (! ," ,# ) = H 2B (! )H12 (" )H1I (# )
H BI (! ," ,# ) =
% 1
0
0
'
' 0 cos ! sin !
' 0 $ sin ! cos !
&
( % cos"
*'
*' 0
* '& sin "
)
0 $ sin " ( % cos#
*'
1
0
* ' $ sin#
0 cos" )* '
0
&
sin#
cos#
0
0 (
*
0 *
1 *)
H BI (! ," ,# ) =
%
cos" cos#
'
' $ cos ! sin# + sin ! sin " cos#
' sin ! sin# + cos ! sin " cos#
&
cos" sin#
cos ! cos# + sin ! sin " sin#
$ sin ! cos# + cos ! sin " sin#
$ sin "
(
*
sin ! cos" *
cos ! cos" *
)
H BI H IB = I for all (! ," ,# ), i.e., No Singularities
36!
Airplane Angular Attitude (Position)!
Rotation Matrix = Direction Cosine Matrix!
Angles between
each I axis and each
B axis"
" cos !11
$
H BI = $ cos !12
$ cos !
13
#
cos ! 21
cos ! 22
cos ! 23
cos ! 31 %
'
cos ! 32 '
cos ! 33 '
&
37!
Airplane Angular Attitude (Position)!
!
#
#
#
#
#
"
e1 $ !
Rotation angle, rad
& #
e2 & # x-component of rotation axis
&=
e3 & # y-component of rotation axis
#
e4 &% #" z-component of rotation axis
$
&
&
&
&
&%
!! Quaternion vector "
!! single rotation from inertial to body frame"
!! non-singular, linear propagation of attitude based on bodyaxis rate vector"
!! 4 parameters; more compact than the rotation matrix"
!! solution for rotation matrix and Euler angles from
parameters is intricate"
38!
Airplane Angular Attitude (Position)!
Rotation Matrix from Quaternion Vector!
H BI =
" e2 ! e2 ! e2 + e2
2
3
4
$ 1
$ 2 ( e3e4 ! e1e2 )
$
$ 2 ( e1e3 + e2 e4 )
#
2 ( e1e2 + e3e4 )
e12 ! e22 + e32 + e42
2 ( e2 e3 ! e1e4 )
2 ( e2 e4 ! e1e3 )
%
'
2 ( e2 e3 + e1e4 ) '
'
2
2
2
2
e1 + e2 ! e3 + e4 '
&
Euler Angles from Quaternion:
see p. 186, Flight Dynamics!
39!
Euler Angle Dynamics!
%
'
!
!='
'
&
"!
#!
$!
( % 1 sin " tan #
* '
cos "
*=' 0
* ' 0 sin " sec#
) &
cos " tan # ( % p
*'
+ sin " * ' q
cos " sec# * '& r
)
(
*
I
* = LB, B
*
)
LIB is not orthonormal
LIB is singular when ! = ± 90°
40!
Rigid-Body Equations of Motion!
(Euler Angles)!
r!I ( t ) = H IB ( t ) v B ( t )
! ( t ) = LI ( t ) " ( t )
!
I
B
B
v! B ( t ) =
H IB , H BI are functions of !
1
FB ( t ) + H BI ( t ) g I ! "" B ( t ) v B ( t )
m (t )
!! B ( t ) = I B "1 ( t ) #$ M B ( t ) " !" B ( t ) I B ( t ) ! B ( t ) %&
41!
Rotation Matrix Dynamics!
! IB h B = !" I h I = !" I H IB h B
H
# 0
%
!! = % ! z
%
%$ "! y
"! z
0
!x
!y &
(
"! x (
(
0 (
'
! IB = !" I H IB
H
! BI = ! "" B H BI
H
42!
Rigid-Body Equations of Motion!
(Attitude from 9-Element Rotation Matrix)!
r!I = H IB v B
! BI = !"
" B H BI
H
v! B =
1
FB + H BI g I ! "" B v B
m
!! B = I B"1 ( M B " !" B I B! B )
No need for Euler angles to solve the dynamic equations!
43!
Quaternion Vector Dynamics!
!
#
#
e! ( t ) = #
#
#
#"
e!1 ( t ) $
&
e!2 ( t ) &
& = Q (t ) e (t )
e!3 ( t ) &
&
e!4 ( t ) &
%
! 0
'r ( t ) 'q ( t ) ' p ( t )
#
# r (t )
0
' p (t ) q (t )
=#
0
'r ( t )
# q (t ) p (t )
#
0
#" p ( t ) 'q ( t ) r ( t )
$!
&#
&#
&#
&#
&#
&% #"
e1 ( t ) $
&
e2 ( t ) &
&
e3 ( t ) &
&
e4 ( t ) &
%
44!
Rigid-Body Equations of Motion!
(Attitude from 4-Element Quaternion Vector)!
r!I = H IB v B
e! = Qe
H IB , H BI are functions of e
1
v! B = FB + H BI g I ! "" B v B
m
!! B = I B"1 ( M B " !" B I B! B )
45!
Alternative Reference
Frames!
46!
Velocity Orientation in an Inertial
Frame of Reference"
Polar Coordinates"
Projected on a Sphere"
47!
Body Orientation with Respect
to an Inertial Frame"
48!
Relationship of Inertial Axes
to Body Axes"
•! Transformation is
independent of
velocity vector"
•! Represented by"
–! Euler angles"
–! Rotation matrix "
! vx $
! u $
&
#
&
#
B
# v & = H I # vy &
&
#
#" w &%
v
#" z &%
! vx $
! u $
&
#
&
I #
# vy & = H B # v &
&
#
#" w &%
v
z
&%
#"
49!
Velocity-Vector Components
of an Aircraft"
V, ! , "
V, ! , "
50!
Velocity Orientation with Respect
to the Body Frame"
Polar Coordinates"
Projected on a Sphere"
51!
Relationship of Inertial Axes
to Velocity Axes"
•! No reference to the body
frame"
•! Bank angle, %, is roll angle
about the velocity vector "
! vx $ ! V cos ' cos (
& #
#
# vy & = # V cos ' sin (
& # )V sin '
#
#" vz &% I "
$
&
&
&
%
#
vx2 + vy2 + vz2
%
# V &
%
( %
)1 #
2
2 1/2 &
!
%
( = % sin $ vy / vx + vy '
% " ( %
$
' %
sin )1 ( )vz / V )
$
(
)
52!
&
(
(
(
(
(
'
Relationship of Body Axes
to Wind Axes"
•! No reference to
the inertial frame "
! u $ ! V cos ' cos (
& #
#
V sin (
# v &=#
#
#" w &%
V sin ' cos (
"
$
&
&
&
%
#
# V & %
%
( %
% ! (=%
%$ " (' %
$
&
u 2 + v2 + w2 (
sin )1 ( v / V ) (
(
tan )1 ( w / u ) (
'
53!
Angles Projected
on the Unit Sphere"
•! Origin is
s center
airplane
of mass"
" : angle of attack
# : sideslip angle
$ : vertical flight path angle
% : horizontal flight path angle
& : yaw angle
' : pitch angle
( : roll angle (about body x ) axis)
µ : bank angle (about velocity vector)
54!
Alternative Frames of Reference"
•! Orthonormal transformations connect all reference frames"
55!