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Aircraft Dynamics
Marcello R. Napoli tano
A I R C R A F T D Y N A M I C S
From Modeling to Simulation
AI
RC
RA
FT
D
YN
AM
IC
SFrom
Modeling to Sim
ulation
A I R C R A F T D Y N A M I C S
From Modeling to Simulation
Na
po
litan
o
www.wiley.com/college/napolitano
ISBN 978-0-470-94341-0
“Aircraft Dynamics:
From Modeling to Simulation”
- A textbook designed to take advantage of the extensive
computational resources commonly available to today‟s students.
…The majority of the textbooks in this discipline were written before
the introduction of Matlab® and Simulink®.
- A textbook designed to help students to be able to extrapolate from
low level formulas, equations, and details to high level
comprehensive views of the main concepts.
- A textbook with emphasis on teaching students the fundamental
skills of „basic modeling‟ of aircraft aerodynamics and dynamics.
- An „instructor friendly‟ textbook featuring: - An extensive variety of Student Sample Problems and Case Studies;
- An extensive variety of Problems;
- A number of sample Matlab ® codes;
- Detailed CAD drawings and geometric data for 25 different aircraft
from different classes;
- Complete aerodynamic, geometric, and flight conditions for 10
different aircraft;
- Approx. 500 Power Point-based instructor notes with instructional
videos
Sample figure from Ch. 1 showing the interaction of ALL the aircraft dynamic equations
2 2
X X
Y Y
Z Z
X A T
Y A T
Z A T
XX XZ XZ ZZ YY A T
YY XX ZZ XZ A T
ZZ XZ YY XX XZ A T
m U QW RV mg F F
m V UR PW mg F F
m W PV QU mg F F
P I R I PQ I RQ I I L L
Q I PR I I P R I M M
R I P I PQ I I QR I N N
CLME & CAME
1 sin tan cos tan
0 cos sin
0 sin sec cos sec
P
Q
R
KE
'
'
'
cos cos sin cos cos sin sin sin sin cos sin cos
sin cos cos cos sin sin sin sin cos sin sin cos
sin cos sin cos cos
X U
Y V
Z W
FPE
sin
cos sin
cos cos
X
Y
Z
g g
g g
g g
GE
, ,X Y Zg g g
, ,
, ,
, ,U V W
, ,P Q R
, ,U V W
, , ,
, ,
X Y Z
X Y Z
A A A
T T T
F F F
F F F
, ,
, ,
A A A
T T T
L M N
L M N
', ', 'X Y Z
Sample figure from Ch. 1 showing the sequential derivation of ALL the aircraft dynamic equations
'
'' ' '
:
:
A A A T
V V S
A A A T
V V S
d drCLME dV g dV F F dS
dt dt
d drCAME r dV r g dV r F F dS
dt dt
Aero Forces/Moments
Thrust Forces/Moments
Initial Conditions
X Y Z
' '
Pr r r :
:
PA T
A A T
V
dVCLME m mg F F
dt
d drCAME r dV M M
dt dt
, ,
d C CC
dt t
X Y Z X Y Z
X Y Z
:
:
P P A T
A A T
V
CLME m V V mg F F
CAME r r r dV M M
X Y Z
2 2
X X
Y Y
Z Z
X A T
Y A T
Z A T
XX XZ XZ ZZ YY A T
YY XX ZZ XZ A T
ZZ XZ YY XX XZ A T
m U QW RV mg F F
m V UR PW mg F F
m W PV QU mg F F
P I R I PQ I RQ I I L L
Q I PR I I P R I M M
R I P I PQ I I QR I N N
CLME & CAME
1 sin tan cos tan
0 cos sin
0 sin sec cos sec
P
Q
R
'
'
'
c c s c c s s s s c s c
s c c c s s s s c s s c
s c s c c
X U
Y V
Z W
sin
cos sin
cos cos
X
Y
Z
g g
g g
g g
Aircraft trajectory w/r X‟Y‟Z‟
0 .
0 .
P PV V const
const
1 1
1 1
1 1
1 1
1 1
1 1
1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1
2 2
1 1 1 1
1 1 1 1
sin
cos sin
cos cos
X X
Y Y
Z Z
A T
A T
A T
XZ ZZ YY A T
XX ZZ XZ A T
YY XX XZ A T
m Q W R V mg F F
m U R P W mg F F
m PV Q U mg F F
PQ I R Q I I L L
PR I I P R I M M
PQ I I Q R I N N
CLME & CAME
at steady state
Steady state
conditions
0
k
Steady state conditions
1 - Rectilinear flight
2 - Level turn
3 - Symmetric pull-up
Small perturbation
conditions
20, 0, 0, 0,...
sin , cos 1
up wr pq p
x x x
1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1
1
cos
sin sin cos cos
cos sin sin cos
X X
Y Y
Z Z
A T
A T
A T
XX XZ XZ ZZ YY A T
YY
m u Q w qW R v rV mg f f
m v U r uR Pw pW mg mg f f
m w Pv pV Q u U q mg mg f f
p I r I Pq Q p I R q Q r I I l l
qI P
1 1 1
1 1 1 1
2 2XX ZZ XZ A T
ZZ XZ YY XX XZ A T
r pR I I P p R r I m m
r I p I Pq pQ I I Q r R q I n n
Small perturbations CLME & CAME
..from steady state, wing-level,
rectilinear flight conditions
1 1
1 1 1
1 1
1
1
cos
cos
sin
sin
cos
X X
Y Y
Z Z
A T
A T
A T
XX XZ A T
YY A T
ZZ XZ A T
m u qW mg f f
m v U r pW mg f f
m w U q mg f f
p I r I l l
qI m m
r I p I n n
p
q
r
, ,U V W
, ,P Q R
, , , ,X Y Zg g g
, ,
Sample figure from Ch. 3 showing the summary of the PITCHING modeling
1
1 1 1
22 2u q iE H
A m m m m m m E m H
P P P
u c q cm q S c c c c c c c c i
V V V
10mc at steady-state trimmed conditions
1
W
u
AC
m L
xc c
Mach
( ) (1 )( )WB HW H
Hm L CG AC L H AC CG
S dc c x x c x x
S d
( )HE H
iH
Hm L H AC CG E
m E
Sc c x x
S
c
( )i HH H
Hm L H AC CG
Sc c x x
S
Polhamus
formula
(Chapter II)
“Downwash” effect
(Chapter II)
2 2 2
0.5
2 2
2
1 tan2 1 4
1
L
ARc
AR Mach
k Mach
1.19
2
0.254.44 cos 1AR mr
Mach
dK K K Mach
d
Hm mc c
2( ) 2 ( )H HH H H
Hm L AC CG L H AC CG
S dc c x x c x x
S d
q q qW Hm m mc c c
3 2
/4
/4
3 20
/4
/4
tan 3
6cos0
tan3
6cos
q qW W
c
c
m mMach
c
c
AR
AR B Bc c
AR
AR
22
/ 41 cos cB Mach
/ 400
cosqW W
m q L cMachMach
c K c C
2
/ 4
3 2
/ 4
/ 4
0.5 2
2cos
tan1 1
24 6cos 8
W WAC CG AC CG
c
c
c
AR x x x x
CAR
AR
AR
22 ( )q HH H
Hm L H AC CG
Sc c x x
S Leading Edge
of wing MAC
Leading Edge
of tail MAC
Wing+Body
Aerodynamic
Center
Tail
Aerodynamic
Center
Aircraft CG
WBACX
HACX
CGX
Wing MAC
WB
WB
AC
AC
XX
c
CGCG
XX
c
H
H
AC
AC
XX
c
H. Tail
MAC
c
Hc
Leading Edge
of wing MAC
Leading Edge
of tail MAC
Wing+Body
Aerodynamic
Center
Tail
Aerodynamic
Center
Aircraft CG
WBACX
HACX
CGX
Wing MAC
WB
WB
AC
AC
XX
c
CGCG
XX
c
H
H
AC
AC
XX
c
H. Tail
MAC
c
Hc
Sample figure from Ch. 4 showing the summary of the ROLLING modeling
1 1 12 2 2p r A R
A l l l l l A l R
P P P
b p b r bl q S b c c c c c c
V V V
0lc
WB H Vl l l lc c c c
#3#1 #2( )( ) ( / )
WB WB WBWB
LE
l l l l Dihedral EffectDihedral Effect Dihedral Effectdue to Sweep Angledue to angle due to High LowWing
c c c c
1 0H WB
H Hl l H
H
S bdc c
d S b
1 1cos sin1
V V V
V VVl Y L V
Z XSdc k c
d S b
l A lc c
l
RME kc
22 2
l l
l A
Left Right
c cc
1 1 1 1cos sin cos sinR R V
VR R R Rl Y L V R R
SZ X Z Xc c c K
b S b
WB H Vl p l p l p l pc c c c
22
1, , 2
2WB W H W pV V
VH Hl p l p l p l p l Y
H
ZS bkc c RDP c c c c
S b b
W Vl r l r l rc c c
1
1
1
0
W
L
l r l r l rl r L W
L WMachC
c c cc c rad
c
1 1 1 1cos sin cos sin2
V V
V V V V
lr Y
X Z Z Xc c
b b
..except for substantial dihedral/anhedral angles for horiz. tail
A
l l lLeft Right
c c c
1
1 1/ 2
/ 4
/ 4
tantan
WB
c
W
l l l l
l L M f W M
L L W WAR
l
l W cZ
W c
c c c cc c K K K
c c
cc
Sample figure from Ch. 4 showing the summary of the YAWING modeling
1 1 12 2 2p r A R
A n n n n n A n R
P P P
b p b r bn q S b c c c c c c
V V V
0nc
W B H Vn n n n nc c c c c
57.3 S
B l
B Bn N R
S lc K K
S b
0Hnc
0Wnc
1 1cos sin1
V V V
V VVn Y L V
X ZSdc k c
d S b
1A A An n L lc K c c
1 1cos sinR V
V R Rn L V R R
S X Zc c K
S b
W Vn p n p n pc c c
1
1
0
W
L
n p n p
n p L W
L WMachC
c cc c
c
1 1 1 1cos sin cos sin2
V V
V V V V V
n p Y
X Z Z X Zc c
b b
r r rW Vn n nc c c
1 0
1 0
2
2
r r
W
n n
nr L D
L D
c cc c c
c c
2
1 1
2
cos sin2
V V
V V
nr Y
X Zc c
b
Sample figure from Ch. 7 showing the solution of the linearized LONGITUDINAL equations
1 1
1 1
cos
sin
X X
Z Z
A T
A T
YY A T
m u qW mg f f
m w U q mg f f
qI m m
q
1 11
1 1
1 1 1
1 1 1
1 1
1 1
1 1
1 1
1
cos 2 2
sin 22 2
2 2
u X Xu E
u q E
u T Tu
D D T T D L D E
P P
P L L L D L L L E
P P P
YY YY m m m m m
P P
u umu mg q S c c c c c c c
V V
u c qcm w V q mg q S c c c c c c c
V V V
u uI I q q S c c c c c c c
V V
1 1
2 2T q Em m m m E
P P
c qcc c c
V V
, , , ,S S SX Y Z X Y Z
11 1, 0S SPU V W
w
q
1 1
,
,P P
q q
w V w V
1 11
1 1
1 1 1 1
1 1 1
1 1
1 1
11
11
1
cos 2 2
sin 22 2
2 2
u X Xu E
u q E
u T T Tu
D D T T D L D E
P P
P P L L L D L L L E
P P P
YY m m m m m m
P P
q S u uu g c c c c c c c
m V V
q S u c qcV V q g c c c c c c c
m V V V
u uI q S c c c c c c c
V V
1 12 2q E
m m m E
P P
c qcc c c
V V
1q
,c S
, YYm I
.aero coef
Longitudinal
Dimensional
Stability
Derivatives
1 1
1
1
cos
sin
u E
E
u E
u T E
P u q P E
u T T q E
u g X X u X X
V g Z u Z Z Z V Z
M M u M M M M M
1 1
1
1
( ) ( ) cos ( ) ( )
( ) ( ) sin ( ) ( )
( ) ( ) ( ) ( )
u E
E
u E
u T E
u P q P E
u T T q E
s X X u s X s g s X s
Z u s s V Z Z s s Z V g s Z s
M M u s M s M M s s s M s M s
t domain s domain
Laplace Transformation
1 1
1
1
( )
cos ( )
( )sin
( )
( )
( )
u
E
E
E
u
u T E
u P q P
E
u T T q
E
u s
s X X X g s X
sZ s V Z Z s Z V g Z
sM
M M M s M M s s M s
s
1
1
1
( )( )
( ) ( )
( )( )
( ) ( )
( )( )
( ) ( )
U
E
E
E
Num su s
s D s
Num ss
s D s
Num ss
s D s
Transfer
Functions
3 2
3 2
2
( )
( )
( )
u u u u uNum s A s B s C s D
Num s A s B s C s D
Num s A s B s C
4 3 2
1 1 1 1 1 1( )D s As B s C s D s E Routh-Hurwitz
Stability Analysis
4 3 2
1 1 1 1 1 1
2 2 2 22 2SP SP PH PHSP n n PH n n
D A s B s C s D s E
s s s s
1
1
1
1
1
1
( )( ) ( ); ( ) ( ) ( ) ( )
( ) ( ) ( )
( )( ) ( ), ( ) ( ) ( ) ( )
( ) ( ) ( )
( )( ) ( ), ( ) ( ) ( ) ( )
( ) ( ) ( )
uE
E E
E
E E
E
E E
Num su s u su s s u t L u s
s D s s
Num ss ss s t L s
s D s s
Num ss ss s t L s
s D s s
Short-Period
Approximation ( ) 0u t
1 1
1
1
( )
cos ( )
( )sin
( )
( )
( )
u
E
E
E
u
u T E
u P q P
E
u T T q
E
u s
s X X X g s X
sZ s V Z Z s Z V g Z
sM
M M M s M M s s M s
s
1 1
1
1
( )
cos ( )
( )sin
( )
( )
( )
u
E
E
E
u
u T E
u P q P
E
u T T q
E
u s
s X X X g s X
sZ s V Z Z s Z V g Z
sM
M M M s M M s s M s
s
1 1
( )
( )
( )( ) ( )
( )
E
E
P P E
q
E
s
sV Z V s Zs
s MM s M s s M
s
1
0, 0
sin 0, 0
q
T
Z Z
M
1
q
nSP
P
Z MM
V
1
1
2
q
P
SP
q
P
ZM M
V
Z MM
V
1 1
2 2 22q
q SP nSP nSP
P P
Z MZs M M s M s s
V V
Short-Period Char. Equation
Sample figure from Ch. 7 showing the solution of the linearized LAT-DIRECTIONAL equations
, , , ,S S SX Y Z X Y Z11S PU V
1q
,b S
, , ,XX ZZ XZm I I I
.aero coef
Lateral/Directional
Dimensional
Stability
Derivatives
t domain s domain
Laplace Transformation
2 2
2 2
2 2
( ) ( )( ) ( ),
( ) ( ) ( ) ( )
( ) ( )( ) ( ),
( ) ( ) ( ) ( )
( ) ( )( ) ( ),
( ) ( ) ( ) ( )
A R
A R
A R
A R
A R
A R
Num s Num ss s
s D s s D s
Num s Num ss s
s D s s D s
Num s Num ss s
s D s s D s
Transfer
Functions
,
,
,
3 2
2
3 2
( )
( )
( )
A R
A R
A R
Num s s A s B s C s D
Num s s A s B s C
Num s A s B s C s D
4 3 2
2 2 2 2 2 2( )D s s A s B s C s D s E Routh-Hurwitz
Stability Analysis
4 3 2
2 2 2 2 2 2
2 22DR DRDR n n R S
D s A s B s C s D s E
s s s s
1
2
1
2
1
2
( )( ) ( ); ( ) ( ) ( ) ( )
( ) ( ) ( )
( )( ) ( ), ( ) ( ) ( ) ( )
( ) ( ) ( )
( )( ) ( ), ( ) ( ) ( ) ( )
( ) ( ) ( )
Num ss ss s t L s
s D s s
Num ss ss s t L s
s D s s
Num ss ss s t L s
s D s s
Rolling
Approximation ( ) ( ) 0t t
Rolling Time Constant
1 1 1
1
1
cos
sin
cos
Y YA T
XX XZ A T
ZZ XZ A T
m v U r pW mg f f
p I r I l l
r I p I n n
p
r
, , , ,B B BXX ZZ XZ XX ZZ XZI I I I I I
1
1 1
1 1
1 1
1
1
1
2 2
2 2
2 2
p r A R
p r A R
p r A R
P Y Y Y Y A Y R
P P
XX XZ l l l l A l R
P P
ZZ XZ n n n n A n R
P P
pb rbm v V r mg q S c c c c c
V V
pb rbI p I r q S b c c c c c
V V
pb rbI r I p q S b c c c c c
V V
2 2
1 1 1
2 2
1 1 1
1 1 1
cos sin sin 2
sin cos sin 2
0.5sin 2 0.5sin 2 cos 2
S B
S B
BS
XX XX
ZZ ZZ
XZXZ
I I
I T I
II
T
1
1
1
1
P
P
P
P
vv V
V
vv V
V
,
,
p p
r r
1 1
1 1
1 1
1 1
1
1
1
2 2
2 2
2 2
p r A R
p r A R
p r A R
P P Y Y Y Y A Y R
P P
XZl l l l A l R
XX XX P P
XZn n n n A n R
ZZ ZZ P P
q S b bV V g c c c c c
m V V
I q S b b bc c c c c
I I V V
I q S b b bc c c c c
I I V V
1 1 A R
A R
A R
P P A R
XZA R
XX
XZA R
ZZ
V V g Y Y Y Y Y
IL L L L L
I
IN N N N N
I
1 11
1
2
( ) cos ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
P p P r
p r
p r
sV Y s sY g s s V Y s Y s
L s s s L s s sI L s L s
N s s sI N s s s N s N s
1 1
1
2
( )
( )
( )
( )
( )
( )
P p P r
p r
p r
s
ssV Y sY g s V YY
sL s s L s sI L L
sNN s sI N s s N s
s
1 1
1
2
( )
( )
( )
( )
( )
( )
A
A
A
AP p P r
p r
A
p r
A
s
ssV Y sY g s V Y Ys
L s s L s sI L Ls
NN s sI N s s N s
s
1 1
1
2
( )
( )
( )
( )
( )
( )
A
A
A
AP p P r
p r
A
p r
A
s
ssV Y sY g s V Y Ys
L s s L s sI L Ls
NN s sI N s s N s
s
2 ( ) ( )( )
( ) ( ) ( )
A
AP
A A P
Ls ss L s L
s s s s L
1R
P
TL
1
2
4
p
xxR
l P
IT
c V S b
Sample figure from Ch. 7 showing the concept of SENSITIVITY ANALYSIS
Flight Conditions
1., , , ,Alt Mach q
Aircraft Geometry, , , ,
, , , ,..H
H H
H V AC
c c b b
S S S x
Aircraft Mass and
Inertial Properties
, , , ,XX YY ZZ XZm I I I I
Dimensionless
Stability and
Control Derivatives
0
0
0
, ,..,
, ,..,
, ,..,
, ,.., ,
, ,.., ,
, ,.., ,
E
E
E
p A R
p A R
p A R
D D D
L L L
m m m
l l l l
Y Y Y Y
n n n n
c c c
c c c
c c c
c c c c
c c c c
c c c c
Dimensional Stability and Control Derivatives
(Tables 7.1 & 7.3)
, ,.., , , ,.., , , ,..,
, ,.., , , , ,.., , , , ,.., ,
E E E
A R A R A R
u u u
p p p
X X X Z Z Z M M M
L L L L Y Y Y Y N N N N
Longitudinal and
Lateral Directional
Characteristic
Equations
1 2( ), ( )D s D s
Sample figure from Ch. 7 showing the key geometric parameters for SENSITIVITY ANALYSIS
Leading Edge
of wing MAC
Leading Edge
of tail MAC
Wing+Body
Aerodynamic
Center
Tail
Aerodynamic
Center
Aircraft CG
WBACX
HACX
CGX
Wing MAC
CGCG
XX
c
H
H
AC
AC
XX
c
Tail MAC
c
Hc
H
H
AC CG
S
S
x x
CRITICAL PARAMETERS
SX
SVX
SZ
SY
SVZ
VS
b
Vertical Arm of
Vertical Tail
Horizontal Arm of
Vertical Tail
CRITICAL PARAMETERS
VS
S
VS
,S SV VZ X
b b
Aircraft CG
Sample figure from Ch. 8 showing the STATE VARIABLE modeling of the aircraft dynamics
Dimensional Longitudinal
Derivatives
Dimensional Lateral
Directional Derivatives
, , ,
, , , ,
, , , , , ,
u E
E
u E
u T
u q
u T T q
X X X X
Z Z Z Z Z
M M M M M M M
, , , ,
, , , ,
, , , ,
E
E
E
u q
u q
u q
X X X X X
Z Z Z Z Z
M M M M M
Dimensional „Primed‟
Longitudinal Derivatives
Dimensional „Primed‟
Lat. Direct. Derivatives
, , , ,
, , , ,
, , , ,
A R
A R
A R
p r
p r
p r
Y Y Y Y Y
L L L L L
N N N N N
, , , , ,
, , , ,
, , , ,
A R
A R
A R
p r
p r
p r
Y Y Y Y Y Y
L L L L L
N N N N N
Dimensional
„Double Primed‟
Long. Derivatives
, , , ,
/
Eu qZ Z Z Z Z
and or others
Long Long Long Long Long
Long Long Long Long Long
x A x B u
y C x D u
Longitudinal SV Model
Dimensional
„Double Primed‟
Lat. Dir. Derivatives
, , , , ,
/
A Rp pY Y Y Y Y Y
and or others
. . . . .
. . . . .
Lat Dir Lat Dir Lat Dir Lat Dir Lat Dir
Lat Dir Lat Dir Lat Dir Lat Dir Lat Dir
x A x B u
y C x D u
Lat. Directional SV Model
. . .
. . . .
00
00
0 0
0 0
Long LongLong Long Long
Lat Dir Lat DirLat Dir Lat Dir Lat Dir
LongLong Long Long Long
Lat DirLat Dir Lat Dir Lat Dir Lat Dir
x Bx A u
x Bx A u
xy C D u
xy C D u
„Total‟ Aircraft SV Model
Sample figure from Ch. 9 showing the general architecture of a simulation code
Beaver dynamics
and output equations
16
rb/2V
15
qc/V
14
pb/2V
13
H dot
12
H
11
ye
10
xe
9
phi
8
theta
7
psi
6
r
5
q
4
p
3
beta
2
alpha
1
V
time
To Workspace
In To Workspace Out To Workspace
Mux
Double-click for info!
Mux
click
2x for
info!
Mux
Mux
Mux
Demux
Demux
Demux
Clock
FDC Toolbox
BEAVER, level 1
M.O. Rauw, October 1997
12
wwdot
11
vwdot
10
uwdot
9
ww
8
vw
7
uw
6
pz
5
n
4
deltaf
3
deltar
2
deltaa
1
deltae
xdot
y dl
x
uwind
uprop
uaero
Deflections of
Control Surfaces
Throttle Settings
Atmospheric
Turbulence
(optional)
Motion
Variables
Aircraft Equations of Motion
Pilot Inputs and
External Disturbance Aircraft
Outputs
FDC Toolbox
M.O. Rauw 1997
xdot
x
yhlp
Additional outputs
Aircraft equations
of motion (Beaver)
Wind forces
Gravity forces
Engine group (Beaver)
Aerodynamics
group (Beaver)
(co)sines of
alpha, beta,
psi, theta, phi
Add + sort
forces and
moments
Airdata group
18
yad3
17
yad2
16
yad1
15
yatm
14
Fwind
13
Fgrav
12
FMprop
11
FMaero
10
Cprop
9
Caero
8
yacc
7
ypow
6
yfp
5
ydl
4
yuvw
3
ybvel
2
xdot
1
x
hlpfcn
Gravity
Fwind
FMsort
BEAVER, level 2 (main level)
M.O. Rauw
-K-
-K-3
uwind
2
uprop
1
uaero
Modeling of the
Aerodynamic Forces
and Moments
Modeling of the
Propulsive Forces
and Moments
Modeling of the
Gravity Forces
Modeling of the
Atmospheric Turbulence
Forces
AIRCRAFT
EQUATIONS
OF MOTION
FDC Toolbox
M.O. Rauw 1997
xdot
x
yhlp
Additional outputs
Aircraft equations
of motion (Beaver)
Wind forces
Gravity forces
Engine group (Beaver)
Aerodynamics
group (Beaver)
(co)sines of
alpha, beta,
psi, theta, phi
Add + sort
forces and
moments
Airdata group
18
yad3
17
yad2
16
yad1
15
yatm
14
Fwind
13
Fgrav
12
FMprop
11
FMaero
10
Cprop
9
Caero
8
yacc
7
ypow
6
yfp
5
ydl
4
yuvw
3
ybvel
2
xdot
1
x
hlpfcn
Gravity
Fwind
FMsort
BEAVER, level 2 (main level)
M.O. Rauw
-K-
-K-3
uwind
2
uprop
1
uaero
Modeling of the
Aerodynamic Forces
and Moments
Modeling of the
Propulsive Forces
and Moments
Modeling of the
Gravity Forces
Modeling of the
Atmospheric Turbulence
Forces
AIRCRAFT
EQUATIONS
OF MOTION
Sample drawing and tables from Appendix C showing aircraft data for aerodynamic modeling
Samples of INSTRUCTOR NOTES from an
extensive set of approx. 500 slides freely available to the instructors !
“Aircraft Dynamics:
From Modeling to Simulation”
Chapter IV
21
V
V
LELE
V
V
LELE
V
Leading edge wing line
(right wing wrt. pilot)
Perpendicular to
leading edge wing line
nRV
V
V
V
Parallel to leading
edge wing line
nLV
Parallel to leading
edge wing line
Perpendicular to
leading edge wing line
RIGHT WING wrt. pilotLEFT WING wrt. pilot
V
Leading edge wing line
(right wing wrt. pilot)
Perpendicular to
leading edge wing line
nRV
V
V
V
Parallel to leading
edge wing line
nLV
Parallel to leading
edge wing line
Perpendicular to
leading edge wing line
RIGHT WING wrt. pilotLEFT WING wrt. pilot
(cont.) lc
Conceptual Modeling of
“Aircraft Dynamics:
From Modeling to Simulation”
Chapter IV
22
cos
cos
nR
nL
LE
LE
V V
V V
nR nLV V
R LL L
SX
V
RL
Negative rolling moment
#3
0WB
lc
Wing body dihedral effect #3
NOTE: „R‟ indicates
„RIGHT‟ wrt pilot
0
LL
R LL L
SX
V
RL
Negative rolling moment
#3
0WB
lc
Wing body dihedral effect #3
NOTE: „R‟ indicates
„RIGHT‟ wrt pilot
0
LL
R LL L
0WBl
IIIc
(cont.) lc
Conceptual Modeling of
“Aircraft Dynamics:
From Modeling to Simulation”
Chapter IV
41
Numerical Examples of lc
(cont.)
McDonnell Douglas F4 Aircraft
(see Student Sample Problem 4.3)
Term Value
[1/rad] % of lc
Wing contribution due to the
geometric dihedral angle
-0.014
16.6
Wing contribution due to the
wing-fuselage position
0.029
-34.4
Wing contribution due to the
sweep angle
-0.045
53.4
Wing contribution due to the
aspect ratio
-0.027
32.1
Wing contribution due to the
twist angle
-0.002
2.6
Body (fuselage) contribution -0.007 8.3
Horizontal tail contribution 0.0118 -14
Vertical tail contribution -0.030 35.6
TOTAL ( lc
) -0.0842 100
Key results - The predominant (53%) POSITIVE contribution to the dihedral
effect comes from the wing sweep angle.
- The second POSITIVE contribution (36%) comes from the vertical
tail.
- The low-wing configuration provides a substantial NEGATIVE
contribution (anhedral).
- UNIQUE FEATURE OF THIS AIRCRAFT: the high geometric
anhedral angle of the horizontal tail provides a substantial
NEGATIVE contribution (anhedral).
“Aircraft Dynamics:
From Modeling to Simulation”
Chapter IV
47
nc
Conceptual Modeling of (cont.)
..Starting from: WBnc
SX
SY
V V
VV
0
Vfrom right
of the pilot
SX
SY
V V
VV
0
Vfrom right
of the pilot
SX
SY
V
V
0
Vfrom right
of the pilot
Resultant of the
Lateral Side Force
in front of CG
Resultant of the
Lateral Side Force
behind CG
Moment Arm
in front of CG
Moment Arm
behind CG
Negative
Yawing Moment
Positive
Yawing Moment
SX
SY
V
V
0
Vfrom right
of the pilot
Resultant of the
Lateral Side Force
in front of CG
Resultant of the
Lateral Side Force
behind CG
Moment Arm
in front of CG
Moment Arm
behind CG
Negative
Yawing Moment
Positive
Yawing Moment
SX
SY
V
V
Moment Arm
in front of CG
Moment Arm
behind CG
Negative
Yawing Moment
Positive
Yawing Moment
0WB
nc
SX
SY
V
V
Moment Arm
in front of CG
Moment Arm
behind CG
Negative
Yawing Moment
Positive
Yawing Moment
0WB
nc
0WBnc
For most aircraft
“Aircraft Dynamics:
From Modeling to Simulation”
Chapter IV
49
nc
Conceptual Modeling of (cont.)
..next, on: Vnc
SX
SZ
SVXMoment arm
Point of application of the lateral
force on the vertical tail
SX
SZ
SVXMoment arm
Point of application of the lateral
force on the vertical tail
SX
V
Positive yawing moment
0V
nc
0
SVX
Lateral force
SY
SZSX
V
Positive yawing moment
0V
nc
0
SVX
Lateral force
SY
SZ
SX
SY
V V
VV
0
Lateral force on
the vertical tail
SX
SY
V V
VV
0
Lateral force on
the vertical tail
0Vnc
“Aircraft Dynamics:
From Modeling to Simulation”
Chapter IV
87
Ranking of Stability Derivatives
Relative
Importance
Ranking
(0 to 10)
Stability
Derivatives
Group #1 10 , , ,L m l nc c c c
Group #2 9 , , ,q p rm m l nc c c c
Group #3 7-8 0,D Dc c
Group #4 6 0 0,L mc c
Group #5 4 ,qL Lc c
Group #6 3 , , , ,r p rY Y p Y n lc c c c c
Group #7 0-1 0, 0,
0, 0, 0
qD D
Y l n
c c
c c c
The stability derivatives in Group #1 and Group #2 have MAJOR
implications on the aircraft static & dynamic stability (as discussed
in chapter VI and VII)
“Aircraft Dynamics:
From Modeling to Simulation”
Chapter V
6
Review of Basic Aircraft Performance
Conservation of Linear Momentum Equations (CLMEs) along X and Z:
cos sinT
W dVT D W
g dt
2
sin cosT
W VT L W
g R
Assuming: 0,cos 1,sin 0T T T
sinW dV
T D Wg dt
2
cosW V
L Wg R
At steady state rectilinear conditions: , 0dV
Rdt
sin 0T D W
cos 0L W
with: cos
sin
x V
h V
“Aircraft Dynamics:
From Modeling to Simulation”
Chapter V
7
Review of Basic Aircraft Performance :
Power at Level Flight
Level Flight: 0 0T D T D
0L W L W
with: 2 21 1
,2 2
D LD V S c L V S c
2
O
LD D
cc c
AR e where:
Maximum Aerodynamic Efficiency
2
2
1
21
2
LL
DD
V S ccL
ED c
V S c
Definition:
DMax
Max Min L Min
cL DE
D L c
OMax
L DEc c AReGoal: to evaluate:
2
0O
LD
D
L L L L
cc
cd d AR e
d c c d c c
2 2 22
2 2
2
0O OO
L L LLD DD
L L L L
c c ccc cc
AR e AR e AR ed AR e
d c c c c
2
20O O
LD L D
cc c c ARe
ARe
OMaxL DE
c c ARe
O
O
D
D
DMax
L Min
c AR ec
cE
c
AR e
22O O
O O
D D
D D
Min
c c
AR ec AR e c AR e
“Aircraft Dynamics:
From Modeling to Simulation”
Chapter V
8
Review of Basic Aircraft Performance :
Power at Level Flight (cont.)
Minimum Aerodynamic Drag
0T D T D
0L W L W
with: 2
2 2
2 2 4 2
1 2 4,
2L L L
W WV S c W c c
V S V S
2 22 2
2 4 2
22
2
1 1 1 4
2 2
1 1 2
2
O O
O
LD D
D Parasite Induced
c WD V S c V S c
AR e AR e V S
WV S c D D
AR e V S
leading to:
Goal: to evaluate: MINIMUMD
V
2 22
2 3
1 1 2 1 40
2 O OD D
dD d W WV S c V S c
dV dV ARe V S ARe V S
2 2
3 4
1 4 1 4O OD D
W WV S c S c
ARe V S ARe V S
24 1 4
Minimum
O
D
D
WV
SARe S c
2 2 1
Minimum
O
DD
WV
S ARe c
4
2 1
Minimum
O
DD
WV
S ARe c
“Aircraft Dynamics:
From Modeling to Simulation”
Chapter VI
42
Applications of the Trim Diagram
10.3, 2CGx
“Aircraft Dynamics:
From Modeling to Simulation”
Chapter VI
58
Numerical Example:
2 Engines-Out Condition for Boeing B747
FAA Worst Case Scenario
Loss of 50% of installed
thrust (same side)
1
1 2,4 1,21 1
1
0
1 1
4 4
10.75 0.4
4 2 2
13786 112.7 1,553,682.2
T
T T T
L
N T y T y
b bT
lbs ft
1st step – Analysis of the Yawing Moment 1
1
1
1.76T
EOn
N
c q S b
2nd step – Analysis of the Rolling Moment
1
1
1
111.98
A
T
l EO
AEO
l
Lc
q S b
c
3nd step – …back to the Yawing Moment
4th step – …back to the Rolling Moment
1
1
1
1
1
0.35
A
R
T
n n AEO EO
REO
n
Nc c
q S b
c
1
10.13
R
A
l REO
AEO
l
c
c
1 1 1 112.10
FINAL INITIAL
A A A AEO EO EO EO
“Aircraft Dynamics:
From Modeling to Simulation”
Chapter VI
27
Pilot elevator doubletPilot elevator doublet
NUMERICAL EXAMPLE:
F-104 Longitudinal Dynamics (Approach Conditions) – (cont.)
poles=roots(den)
poles =
-0.4514 + 1.3967i
-0.4514 - 1.3967i
-0.0205 + 0.1465i
-0.0205 - 0.1465i
“Short Period” response:
- High natural frequency;
- High damping.
“Phugoid” response:
- Low natural frequency;
- Low damping.
“Short Period” response:
- High natural frequency;
- High damping.
“Phugoid” response:
- Low natural frequency;
- Low damping.
“Phugoid” response:
- Low natural frequency;
- Low damping.
“Phugoid” response:
- Low natural frequency;
- Low damping.
“Short Period” response:
- High natural frequency;
- High damping.
“Phugoid” response:
- Low natural frequency;
- Low damping.
“Short Period” response:
- High natural frequency;
- High damping.
“Phugoid” response:
- Low natural frequency;
- Low damping.
“Short Period”:
- high
- highSP
nSP
“Phugoid”:
- high
- highPh
nPh
“Short Period”:
- high
- highSP
nSP
“Phugoid”:
- high
- highPh
nPh
“Aircraft Dynamics:
From Modeling to Simulation”
Chapter VI
33
SPECIAL CASE:
Short Period Approximation (cont.)
Numerical Example : Cessna 182
Cessna 182 (Altitude=5,000 ft, Mach=0.21)
4 3 2
1 1 1 1 1 1
4 3 2222.05 1985.95 6262.29 329.88 180.58
D A s B s C s D s E
s s s s
FULL-BLOWN Characteristic Equation
-4.4498 + 2.8248i, -4.4498 - 2.8248i
-0.0220 + 0.1697i, -0.0220 - 0.1697i
0.844, 5.27 , 0.129, 0.171sec sec
SP nSP Ph nPh
rad rad with:
SHORT-PERIOD APPROXIMATION Characteristic Equation
1
1 1
2
1
2 464.71 464.71 4.337220.1 2.5428 4.337 19.26
220.1 220.1
SP
q
P q
P P
Z MZD V s s M M s M
V V
s s s
2 2 28.99 28.41 2 SP nSP nSPs s s s
. .0.843, 5.33
secAPPROX APPROXSP nSP
rad
…Approximation error: < 1% !!
“Aircraft Dynamics:
From Modeling to Simulation”
Chapter VI
67
Rolling Approximation (cont.)
Mc Donnell Douglas F-4 roll response following a +2 deg. aileron deflection
(sec)t
p
( / sec)rad
0.9
. 40,000 .
Mach
Alt ft
deg0.2765 15.84
sec secSS
radp
63% 0.1742sec
SS
radp
. 0.813secRRoll Time Const RTC T
Boeing B747-200 roll response following a +2 deg. aileron deflection
(sec)t
p
( / sec)rad
0.9
. 40,000 .
Mach
Alt ft
deg0.0119 0.682
sec secSS
radp
63% 0.0075sec
SS
radp
. 1.98secRRoll Time Const RTC T
Boeing B747-200 roll response following a +2 deg. aileron deflection
(sec)t
p
( / sec)rad
0.9
. 40,000 .
Mach
Alt ft
deg0.0119 0.682
sec secSS
radp
63% 0.0075sec
SS
radp
. 1.98secRRoll Time Const RTC T
Comparison of Roll Responses between a F4 and a Boeing B747
“Aircraft Dynamics:
From Modeling to Simulation”
Chapter VIII
8
Transfer Function-Based vs.
State Variable-Based Modeling (cont.)
SUMMARY
Differential Equations (DEs)
“Transfer Functions” Model
11 12 1 1
21 22 2 2
1
1 21
1 2
( ) ( ) ... ( ) ... ( )
( ) ( ) ... ( ) ... ( )
( ) ... ... ... ... ... ...( )
( ) ( ) ... ( ) ... ( )( )
... ... ... ... ... ...
( ) ( ) ... ( ) ... ( )
j m
j m
l x
l xmi i ij im
m x
l l lj lm
G s G s G s G s
G s G s G s G s
Y sG s
G s G s G s G sU s
G s G s G s G s
l xm
DYNAMIC SYSTEM
“State Variable” Model
1 11
1 1 1
n x n n x mn x m xn x
l x n l x ml x n x m x
x A x B u
y C x D u
1( )
( )( )
Y sG s C s I A B D
U s
„s‟ domain„time‟ domain
Differential Equations (DEs)Differential Equations (DEs)
“Transfer Functions” Model
11 12 1 1
21 22 2 2
1
1 21
1 2
( ) ( ) ... ( ) ... ( )
( ) ( ) ... ( ) ... ( )
( ) ... ... ... ... ... ...( )
( ) ( ) ... ( ) ... ( )( )
... ... ... ... ... ...
( ) ( ) ... ( ) ... ( )
j m
j m
l x
l xmi i ij im
m x
l l lj lm
G s G s G s G s
G s G s G s G s
Y sG s
G s G s G s G sU s
G s G s G s G s
l xm
DYNAMIC SYSTEMDYNAMIC SYSTEM
“State Variable” Model
1 11
1 1 1
n x n n x mn x m xn x
l x n l x ml x n x m x
x A x B u
y C x D u
“State Variable” Model
1 11
1 1 1
n x n n x mn x m xn x
l x n l x ml x n x m x
x A x B u
y C x D u
1( )
( )( )
Y sG s C s I A B D
U s
„s‟ domain„time‟ domain
“Aircraft Dynamics:
From Modeling to Simulation”
Chapter VIII
State Variable Modeling of the
Longitudinal Dynamics
Starting from Chapter VII longitudinal equations:
1 1
1
1
cos
sin
u E
E
u E
u T E
P u q P E
u T T q E
u X X u X g X
V Z u Z Z g Z V Z
M M u M M M M M
Using the relationship: ,q q
1 1
1
1
cos
sin
u E
E
u E
u T E
P u q P E
u T T q E
u X X u X g X
V Z Z u Z g Z V q Z
q M M u M M M M q M
q
NOTE: The 2nd equation is nested within the 3rd equation through the
ALPHA_DOT term. Therefore, neglecting: ( , )uT TM M
1
1 1 1 1 1
1 1 1
1
1
1
cos
sin
sin
u E
E
u T E
q Pu
E
P P P P P
uu
P P P
u X X u X g X
Z V ZZ Z gu q
V Z V Z V Z V Z V Z
Z Z gq M M u M M M
V Z V Z V Z
1
1 1
E
E
q P
q E
P P
Z V ZM M q M M
V Z V Z
q
“Aircraft Dynamics:
From Modeling to Simulation”
Chapter VIII
10
State Variable Modeling of the
Longitudinal Dynamics (cont.)
with:
1
1 1 1
1 1
1
1
, , cos , 0,
, , ,
sin,
,
,
u E E
E
E
E E E
u u T q
q Pu
u q
P P P
P P
u u u
q q q
X X X X X X g X X X
Z VZ ZZ Z Z
V Z V Z V Z
ZgZ Z
V Z V Z
M M Z M M M Z M
M M Z M M Z M
M M Z M
Define:
Long Long Long Long Longx A x B u
0 0 1 0 0
Long Long E
u q E
u q E
E
u q E
u u
A Bq q
X X X X u X
Z Z Z Z Z
M M M M q M
STATE EQUATIONS
,T
Long Long Ex u q u
“Aircraft Dynamics:
From Modeling to Simulation”
Chapter IX
4
Introduction to the “Flight Dynamics
& Control” (FDC) Toolbox
Matlab®/Simulink®-based flight simulation package freely available on
“www.dutchroll.com”
Beaver dynamics
and output equations
16
rb/2V
15
qc/V
14
pb/2V
13
H dot
12
H
11
ye
10
xe
9
phi
8
theta
7
psi
6
r
5
q
4
p
3
beta
2
alpha
1
V
time
To Workspace
In To Workspace Out To Workspace
Mux
Double-click for info!
Mux
click
2x for
info!
Mux
Mux
Mux
Demux
Demux
Demux
Clock
FDC Toolbox
BEAVER, level 1
M.O. Rauw, October 1997
12
wwdot
11
vwdot
10
uwdot
9
ww
8
vw
7
uw
6
pz
5
n
4
deltaf
3
deltar
2
deltaa
1
deltae
xdot
y dl
x
uwind
uprop
uaero
Deflections of
Control Surfaces
Throttle Settings
Atmospheric
Turbulence
(optional)
Motion
Variables
Aircraft Equations of Motion
Pilot Inputs and
External Disturbance Aircraft
Outputs
Beaver dynamics
and output equations
16
rb/2V
15
qc/V
14
pb/2V
13
H dot
12
H
11
ye
10
xe
9
phi
8
theta
7
psi
6
r
5
q
4
p
3
beta
2
alpha
1
V
time
To Workspace
In To Workspace Out To Workspace
Mux
Double-click for info!
Mux
click
2x for
info!
Mux
Mux
Mux
Demux
Demux
Demux
Clock
FDC Toolbox
BEAVER, level 1
M.O. Rauw, October 1997
12
wwdot
11
vwdot
10
uwdot
9
ww
8
vw
7
uw
6
pz
5
n
4
deltaf
3
deltar
2
deltaa
1
deltae
xdot
y dl
x
uwind
uprop
uaero
Deflections of
Control Surfaces
Throttle Settings
Atmospheric
Turbulence
(optional)
Motion
Variables
Aircraft Equations of Motion
Pilot Inputs and
External Disturbance Aircraft
Outputs
FDC 1st Level
12 inputs (aerodynamic control surfaces, throttle settings,
atmospheric turbulence)
27 outputs (12 aircraft states + 12 derivatives of aircraft states,
3 dimensionless angular velocities)