Aircraft Equations of Motion:
Translation and Rotation!
Robert Stengel, Aircraft Flight Dynamics, !
MAE 331, 2016
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Learning Objectives
What use are the equations of motion?
How is the angular orientation of the
airplane described?
What is a cross-product-equivalent matrix?
What is angular momentum?
How are the inertial properties of the
airplane described?
How is the rate of change of angular
momentum calculated?
Reading:!
Flight Dynamics!
155-161!
Copyright 2016 by Robert Stengel. All rights reserved. For educational use only.
http://www.princeton.edu/~stengel/MAE331.html
http://www.princeton.edu/~stengel/FlightDynamics.html
Lockheed F-104
1
Review Questions!
!! What characteristic(s) provide maximum gliding
range?!
!! Do gliding heavy airplanes fall out of the sky faster
than light airplanes?!
!! Are the factors for maximum gliding range and
minimum sink rate the same?!
!! How does the maximum climb rate vary with altitude?!
!! What are “energy height” and “specific excess
power”?!
!! What is an “energy climb”?!
!! How is the “maneuvering envelope” defined?!
!! What factors determine the maximum steady turning
rate?!
2
Dynamic Systems
Actuators
Sensors
Dynamic Process: Current state depends on
prior state
x
= dynamic state
u
= input
w
= exogenous disturbance
p
= parameter
t or k
= time or event index
dx(t)
= f [ x(t), u(t), w(t), p(t),t ]
dt
Observation Process: Measurement may
contain error or be incomplete
y
= output (error-free)
z
= measurement
n
= measurement error
y ( t ) = h [ x(t),u(t)]
z (t ) = y (t ) + n (t )
3
Ordinary Differential Equations
Fall Into 4 Categories
dx(t)
= f [ x(t), u(t), w(t), p(t),t ]
dt
dx(t)
= F(t)x(t) + G(t)u(t) + L(t)w(t)
dt
dx(t)
= f [ x(t), u(t), w(t)]
dt
dx(t)
= F x(t) + G u(t) + L w(t)
dt
4
What Use are the Equations of Motion?
•! Nonlinear equations of motion
–! Compute exact flight paths and
motions
•! Simulate flight motions
•! Optimize flight paths
•! Predict performance
dx(t)
= f [ x(t), u(t), w(t), p(t),t ]
dt
dx(t)
= F x(t) + G u(t) + L w(t)
dt
–! Provide basis for approximate
solutions
•! Linear equations of motion
–! Simplify computation of
flight paths and solutions
–! Define modes of motion
–! Provide basis for control
system design and flying
qualities analysis
5
Examples of Airplane Dynamic
System Models
•!
Nonlinear, Time-Varying
•!
Linear, Time-Varying
•!
–! Large amplitude motions
–! Significant change in mass
–! Small amplitude motions
–! Perturbations from a
dynamic flight path
•!
Nonlinear, Time-Invariant
–! Large amplitude motions
–! Negligible change in mass
Linear, Time-Invariant
–! Small amplitude motions
–! Perturbations from an
equilibrium flight path
6
Translational Position!
7
Position of a Particle
Projections of vector magnitude on three axes
! cos ' x
! x $
#
&
#
r = # y & = r # cos ' y
# cos '
#" z &%
z
#"
" cos ! x
$
$ cos ! y
$ cos !
z
$#
$
&
&
&
&%
%
'
' = Direction cosines
'
'&
8
Cartesian Frames of Reference
Two reference frames of interest
•!
–!
–!
I: Inertial frame (fixed to inertial space)
B: Body frame (fixed to body)
•!
Translation
•!
Rotation
–! Relative linear positions of origins
–! Orientation of the body frame with
respect to the inertial frame
Common convention (z up)
Aircraft convention (z down)
9
Measurement of Position in
Alternative Frames - 1
•!
Two reference frames of interest
–! I: Inertial frame (fixed to inertial
space)
–! B: Body frame (fixed to body)
Inertial-axis view
! x $
&
#
r=# y &
#" z &%
rparticle = rorigin + 'rw.r.t . origin
•!
Differences in frame orientations must
be taken into account in adding vector
components
Body-axis view
10
Measurement of Position in
Alternative Frames - 2
Inertial-axis view
rparticleI = rorigin!BI + H IB "rB
Body-axis view
rparticleB = rorigin!I B + H BI "rI
11
Rotational Orientation!
12
Direction Cosine
Matrix
" cos ! 11
$
H BI = $ cos ! 12
$ cos !
13
#
cos ! 21
cos ! 22
cos ! 23
cos ! 31 %
'
cos ! 32 '
cos ! 33 '
&
•! Projections of unit vector
components of one reference
frame on another
•! Rotational orientation of one
reference frame with respect
to another
•! Cosines of angles between
each I axis and each B axis
rB = H BI rI
13
Properties of the Rotation Matrix
" cos ! 11
$
H BI = $ cos ! 12
$ cos !
13
#
cos ! 21
cos ! 22
cos ! 23
B
" h11 h12
cos ! 31 %
'
$
cos ! 32 ' = $ h21 h22
$ h
cos ! 33 '
h
& I # 31 32
rB = H BI rI
B
h13 %
'
h23 '
h33 '
&I
s B = H BI s I
Orthonormal transformation
r
s
Angles between vectors are preserved
Lengths are preserved
rI = rB
;
sI = sB
!(rI ,s I ) = !(rB ,s B ) = x deg
14
Euler Angles
•! Body attitude measured with respect to inertial frame
•! Three-angle orientation expressed by sequence of
three orthogonal single-angle rotations
Inertial ! Intermediate1 ! Intermediate2 ! Body
•! 24 (±12) possible
sequences of single-axis
rotations
•! Aircraft convention:
3-2-1, z positive down
! : Yaw angle
" : Pitch angle
# : Roll angle
15
Euler Angles Measure the Orientation of
One Frame with Respect to the Other
•!
Conventional sequence of rotations from inertial to body frame
–!
–!
–!
–!
Each rotation is about a single axis
Right-hand rule
Yaw, then pitch, then roll
These are called Euler Angles
Yaw rotation (!) about zI
Pitch rotation (") about y1
Roll rotation (#) about x2
Other sequences of 3 rotations can be chosen; however,
once sequence is chosen, it must be retained
16
Reference Frame Rotation from Inertial
to Body: Aircraft Convention (3-2-1)
Yaw rotation (!) about zI axis
! x $ ! cos'
& #
#
# y & = # ( sin'
#" z &% #
0
"
1
sin'
cos'
0
0 $ ! x $ ! x I cos' + yI sin'
&#
& #
0 & # y & = # (x I sin' + yI cos'
zI
1 &% #" z &% I #"
$
&
&
&
%
r1 = H1I rI
Pitch rotation (") about y1 axis
! x $ ! cos'
& #
#
# y & =# 0
#" z &% #" sin '
2
0 ( sin ' $ ! x $
&# y &
1
0
&
&#
0 cos' %& #" z &%
1
r2 = H12 r1 = !" H12 H1I #$ rI = H 2I rI
Roll rotation (#) about x2 axis
! 1
! x $
0
0
#
&
#
0 cos ' sin '
# y & =#
# 0 ( sin ' cos '
#" z &%
"
B
$! x $
&#
& rB = H 2B r2 = !" H 2B H12 H1I #$ rI = H BI rI
&# y &
& #" z &%
%
2
17
The Rotation Matrix
The three-angle rotation matrix is the product of 3
single-angle rotation matrices:
H BI (! ," , # ) = H 2B (! )H12 (" )H1I (# )
# 1
0
0
%
= % 0 cos ! sin !
% 0 " sin ! cos !
$
& # cos)
(%
(% 0
( %$ sin )
'
%
cos! cos"
'
= ' # cos $ sin" + sin $ sin ! cos"
' sin $ sin" + cos $ sin ! cos"
&
0 " sin ) & # cos*
(%
1
0
( % " sin*
0 cos) (' %
0
$
cos! sin"
cos $ cos" + sin $ sin ! sin"
# sin $ cos" + cos $ sin ! sin"
an expression of the Direction Cosine Matrix
sin*
cos*
0
0 &
(
0 (
1 ('
# sin !
(
*
sin $ cos! *
cos $ cos! *
)
18
Rotation Matrix Inverse
Inverse relationship: interchange sub- and superscripts
rB = H BI rI
rI = ( H BI ) rB = H IB rB
!1
Because transformation is orthonormal
Inverse = transpose
Rotation matrix is always non-singular
T
$% H BI (! ," ,# ) &' = $% H BI (! ," ,# ) &' = H IB (# ," , ! )
(1
H IB = H BI
( )
!1
= H BI
T
( )
= H1I H12 H 2B
H IB H BI = H BI H IB = I
19
Checklist!
"! What are direction cosines?!
"! What are Euler angles?!
"! What rotation sequence is used to
describe airplane attitude?!
"! What are properties of the rotation
matrix?!
20
Angular Momentum!
21
•!
Angular Momentum
of a Particle
Moment of linear momentum of differential
particles that make up the body
–! (Differential masses) x components of the
velocity that are perpendicular to the
moment arms
dh = ( r ! dm v ) = ( r ! v m ) dm
= #$ r ! ( v o + " ! r ) %& dm
•!
" !x
$
! = $ !y
$
$# ! z
%
'
'
'
'&
Cross Product: Evaluation of a determinant with unit vectors (i, j, k)
along axes, (x, y, z) and (vx, vy, vz) projections on to axes
r!v=
i
x
vx
j
k
vy
vz
y
z
= yvz " zvy i + ( zvx " xvz ) j + xvy " yvx k
(
)
(
)
22
Cross-ProductEquivalent Matrix
i
r!v=
x
vx
j
k
vy
vz
y
z
(
)
(
)
# yv " zv
z
y
%
= % ( zvx " xvz )
%
% xv " yv
y
x
%$
Cross product
= yvz " zvy i + ( zvx " xvz ) j + xvy " yvx k
(
)
(
)
&
# 0 "z y & # vx &
(
(
(%
( = r! v = % z
v
0
"x
(
%
y
%
(
(
%
% "y x
0 ( vz (
(
$
' %$
'(
('
" 0 !z y %
'
$
r! = $ z 0 !x '
$ !y x 0 '
&
#
Cross-product-equivalent
matrix
23
Angular Momentum of the Aircraft
Integrate moment of linear momentum of differential particles over the body
•!
# hx %
*
)
h = ' #$ r ! ( v o + " ! r ) %& dm = ' ' ' ( r ! v ) ( (x, y, z)dx dy dz = ) hy *
Body
xmin ymin zmin
) h *
)$ z *&
( (x, y, z) = Density of the body
xmax ymax zmax
•!
Choose the center of mass as the rotational center
h=
" ( r ! v ) dm + "
o
Body
= 0(
=(
"
Body
Body
$% r ! ( # ! r ) &' dm
$% r ! ( r ! # ) &' dm
" ( r ! r ) dm ! # ) ( " ( r!r! ) dm##
Body
Supermarine Spitfire
Body
24
Location of the Center of Mass
# x
xmax ymax zmax
% cm
1
rcm =
r dm = ! ! ! r" (x, y, z)dx dy dz = % ycm
!
m Body
%
xmin ymin zmin
$ zcm
25
The Inertia Matrix!
26
&
(
(
(
'
The Inertia Matrix
h=!
#
Bo dy
r! r! " dm = !
#
Bo dy
r! r! dm " = I"
"
" !x
$
! = $ !y
$
$# ! z
%
'
'
'
'&
# 0 !z y & # 0 !z y &
%
(%
(
I = ! " r! r! dm = ! " % z
0 !x ( % z
0 !x ( dm
Body
Body %
!y x
0 ( % !y x
0 (
$
'$
'
where
# (y 2 + z 2 )
!xy
!xz
%
= " % !xy
(x 2 + z 2 )
!yz
Body %
!yz
(x 2 + y 2 )
%$ !xz
&
(
( dm
(
('
Inertia matrix derives from equal effect of
angular rate on all particles of the aircraft
27
Moments and Products of Inertia
" (y 2 + z 2 )
!xy
!xz
$
2
2
I = ( $ !xy
(x + z )
!yz
Body $
2
!yz
(x + y 2 )
$# !xz
" I xx
%
$
'
' dm = $ ! I xy
$
'
$# ! I xz
'&
! I xy
I yy
! I yz
! I xz %
'
! I yz '
'
I zz '
&
Inertia matrix
•! Moments of inertia on the diagonal
•! Products of inertia off the diagonal
•!
•!
If products of inertia are zero, (x, y, z)
are principal axes --->
All rigid bodies have a set of principal
axes
! I xx
#
# 0
# 0
#"
0
I yy
0
0 $
&
0 &
I zz &&
%
Ellipsoid of Inertia
I xx x 2 + I yy y 2 + I zz z 2 = 1
28
Inertia Matrix of an Aircraft
with Mirror Symmetry
" (y 2 + z 2 )
0
!xz
$
I= ( $
0
(x 2 + z 2 )
0
Body $
0
(x 2 + y 2 )
$# !xz
%
" I xx
'
$
' dm = $ 0
'
$
$# ! I xz
'&
0
I yy
0
! I xz %
'
0 '
I zz ''
&
Nose high/low product
of inertia, Ixz
Nominal Configuration
Tips folded, 50% fuel, W = 38,524 lb
xcm @0.218 c
I xx = 1.8 ! 10 6 slug-ft 2
North American XB-70
I yy = 19.9 ! 10 6 slug-ft 2
I xx = 22.1! 10 6 slug-ft 2
I xz = "0.88 ! 10 6 slug-ft 2
29
Checklist!
"! How is the location of the center of
mass found?!
"! What is a cross-product-equivalent
matrix?!
"! What is the inertia matrix?!
"! What is an ellipsoid of inertia?!
"! What does the “nose-high” product of
inertia represent?!
30
!is"rical Fac"ids
•!
•!
Technology of World War II Aviation
1938-45: Analytical and
experimental approach to design
Spitfire
–! Many configurations designed and
flight-tested
–! Increased specialization; radar,
navigation, and communication
–! Approaching the "sonic barrier
Aircraft Design
–! Large, powerful, high-flying aircraft
–! Turbocharged engines
–! Oxygen and Pressurization
B-17
P-51D
31
Power Effects on Stability and Control
•! Brewster Buffalo: overarmored and underpowered
•! During W.W.II, the size of
fighters remained about the
same, but installed
horsepower doubled (F4F
vs. F8F)
•! Use of flaps means high
power at low speed,
increasing relative
significance of thrust
effects
Brewster Buffalo F2A
GrummanWildcat F4F
Grumman Bearcat F8F
32
World War II Carrier-Based Airplanes
•!
•!
•!
•!
Takeoff without catapult, relatively low
landing speed
http://www.youtube.com/watch?
v=4dySbhK1vNk
Tailhook and arresting gear
Carrier steams into wind
Design for storage (short tail length, folding
wings) affects stability and control
Douglas TBD
Grumman TBF
Chance-Vought F4U Corsair
F4U
33
Multi-Engine Aircraft of World War II
Boeing B-17
Consolidated B-24
Boeing B-29
Douglas A-26
•! Large W.W.II aircraft had
unpowered controls:
–! High foot-pedal force
–! Rudder stability problems
arising from balancing to
reduce pedal force
•! Severe engine-out problem
for twin-engine aircraft
North American B-25
Martin B-26
34
WW II Military Flying Boats
Seaplanes proved useful during World War II
Martin PB2M Mars
Martin PBM Mariner
Saunders-Roe SR.36
Lerwick
Lockheed
PBY
Catalina
Boeing XPBB Sea Ranger
Grumman JRF-1 Goose
35
Managing Control Forces!
Chapter 5, Airplane Stability and
Control, Abzug and Larrabee!
•! What are the principal subject and scope of the
chapter?!
•! What technical ideas are needed to understand the
chapter?!
•! During what time period did the events covered in
the chapter take place?!
•! What are the three main "takeaway" points or
conclusions from the reading?!
•! What are the three most surprising or remarkable
facts that you found in the reading?!
36
Rate of Change of
Angular Momentum!
37
Newton
s 2nd Law, Applied
to Rotational Motion
In inertial frame, rate of change of angular
momentum = applied moment (or torque), M
! ) dI
dh d ( I!
d!
=
= !+I
dt
dt
dt
dt
" mx %
'
$
!! + I!! = M = $ my '
= I!
$ m '
$# z '&
38
Angular Momentum and Rate
Angular
momentum and
rate vectors are
not necessarily
aligned
h = I!
!
39
How Do We Get Rid of dI/dt in the
Angular Momentum Equation?
Chain Rule
!) !
d ( I!
= I!
! + I!!
dt
... and in an inertial frame
I! ! 0
•! Dynamic equation in a body-referenced frame
–! Inertial properties of a constant-mass, rigid body are
unchanging in a body frame of reference
–! ... but a body-referenced frame is non-Newtonian
or non-inertial
–! Therefore, dynamic equation must be modified for
expression in a rotating frame
40
Angular Momentum
Expressed in Two
Frames of Reference
•! Angular momentum and rate
are vectors
–! Expressed in either the inertial
or body frame
–! Two frames related algebraically
by the rotation matrix
h B ( t ) = H BI ( t ) h I ( t );
h I ( t ) = H IB ( t ) h B ( t )
! B ( t ) = H BI ( t ) ! I ( t );
! I ( t ) = H IB ( t ) ! B ( t )
41
Vector Derivative Expressed
in a Rotating Frame
Chain Rule
Effect of
body-frame rotation
! IB h B
h! I = H IB h! B + H
Alternatively
Rate of change
expressed in body frame
h! I = H IB h! B + ! I " h I = H IB h! B + !" I h I
Consequently, the 2nd term is
! IB h B = !" I h I = !" I H IB h B
H
... where the cross-product
equivalent matrix of angular rate is
# 0
%
!! = % ! z
%
%$ "! y
"! z
0
!x
!y &
(
"! x (
(
0 (
'
42
External Moment Causes
Change in Angular Rate
Positive rotation of Frame B w.r.t.
Frame A is a negative rotation of
Frame A w.r.t. Frame B
In the body frame of reference, the angular momentum change is
! BI h I = H BI h! I ! " B # hB = H BI h! I ! "" B hB
h! B = H BI h! I + H
= H BI M I ! "" B I B " B = M B ! "" B I B " B
Moment = torque = force x moment arm
! m $
! m $ !
$
# x &
# x & # L &
B
M I = # my & ; M B = H I M I = # my & = # M &
#
#
&
& # N &
%
#" mz &%I
#" mz %&B "
43
Rate of Change of BodyReferenced Angular Rate due to
External Moment
In the body frame of reference, the angular momentum change is
! BI h I = H BI h! I ! " B # hB
h! B = H BI h! I + H
= H BI h! I ! "" B hB = H BI M I ! "" B I B " B
= M B ! "" B I B " B
For constant body-axis inertia matrix
h! B = I B !! B = M B " !" B I B ! B
Consequently, the differential equation for angular rate of change is
!! B = I B "1 ( M B " !" B I B ! B )
44
Checklist!
"!Why is it inconvenient to solve momentum rate
equations in an inertial reference frame?!
"!Are angular rate and momentum vectors
aligned?!
"!How are angular rate equations transformed
from an inertial to a body frame?!
45
Next Time:!
Aircraft Equations of Motion:
Flight Path Computation!
Reading:!
Flight Dynamics!
161-180!
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
46
Supplemental
Material
47
Moments and Products of Inertia
(Bedford & Fowler)
Moments and products of inertia tabulated for
geometric shapes with uniform density
Construct aircraft moments and products of inertia from
components using parallel-axis theorem
Model in CREO, etc.
48