Lecture #15 Topics
Linear Flight Dynamics Analysis
• Develop perturbation set in state space form
• Review linear systems analysis
• Traditional aircraft modes
Linear Equations Governing Perturbations
(In Fuselage Fixed Reference Frame)
u  Q 0w  qW 0  V 0r  vR0  g cos 0   f AX  f PX  / m
v  R0u  rU 0  P0w  pW 0  g  cos 0 cos  0  sin  0 sin  0    f AY  f PY  / m


w  P0v  pV 0  Q 0u  qU 0  g  cos 0 sin  0  sin  0 cos  0    f AZ  f PZ  / m


p
q
r

I yy  I zz
I xz
I
r  xz  P0q  pQ 0  
I xx
I xx
I xx
 I zz  I xx 
I yy
 P0r  pR0  
 Q r  qR   l
0
0
A
 l P  / I xx
2I xz
  pP0  rR0   m A  m P  / I yy
I yy

I xx  I yy
I xz
I
p   xz Q 0r  qR0  
I zz
I zz
I zz
  P q  pQ    n
0
0
A
 n P  / I zz
  p  tan  0 q sin  0  Q 0 cos  0  r cos  0  R0 sin  0  
Q 0 sin  0  R0 cos  0   0 sin 0 tan 0 
  cos  0q  sin  0r  Q 0 sin  0  R0 cos  0  
   0 tan  0  q sin  0  r cos  0   R0 sin  0  Q 0 cos  0    sec 0
8 Linear Equations Governing Perturbations
(In Stability-Axes Reference Frame)
u  Q 0w  qW 0  V 0r  vR0  g cos 0   f AX  f PX  / m
U 0   R0u  rU 0  P0w  pW  g  cos 0 cos  0  sin  0 sin  0    f AY  f PY  / m


U 0  P0v  pV 0  Q 0u  qU 0  g  cos 0 sin  0  sin  0 cos  0    f AZ  f PZ  / m


p
q
r

I yy  I zz
I xz
I
r  xz  P0q  pQ 0  
I xx
I xx
I xx
 I zz  I xx 
I yy
 P0r  pR0  
 Q r  qR   l
0
0
A
 l P  / I xx
2I xz
  pP0  rR0   m A  m P  / I yy
I yy

I xx  I yy
I xz
I
p   xz Q 0r  qR0  
I zz
I zz
I zz
  P q  pQ    n
0
0
A
 n P  / I zz
  p  tan  0 q sin  0  Q 0 cos  0  r cos  0  R0 sin  0  
Q 0 sin  0  R0 cos  0   0 sin 0 tan 0 
  cos  0q  sin  0r  Q 0 sin  0  R0 cos  0  
   0 tan  0  q sin  0  r cos  0   R0 sin  0  Q 0 cos  0    sec 0
Substitute Linear Models of the Forces and
Moments
f PX  T cos(T   0 )
f PY  0
f PZ  T sin(T   0 )

l A  q  SW bW Cl    Cl p p  Clr r  Cl A  A  Cl r  r




2
2
  C M u  C M 0  C PM u  C PM 0  u

U
U
0
0





m A  q  SW cW  C M    C MP   C M    C M q q  C M  e  e 


 T dT cosT  xT sin T 






n A  q  SW bW C N    C N p p  C N r r  C N  A  A  C N  r  r

Equation (8.20)
Eqn 8.14
Straight and SLUF Condition
P0  Q 0  R0  0
V0  W0  0
0  0
u  Q 0w  qW 0  V 0r  vR0  g cos 0   f AX  f PX  / m
U 0   R0u  rU 0  P0w  pW  g  cos 0 cos  0  sin  0 sin  0    f AY  f PY  / m


U 0  P0v  pV 0  Q 0u  qU 0  g  cos 0 sin  0  sin  0 cos  0    f AZ  f PZ  / m


Straight and SLUF Condition
P0  Q 0  R0  0

I yy  I zz
I xz
I xz
p
r
 P0q  pQ 0  
I xx
I xx
I xx
q
 I zz  I xx 
I yy
 P0r  pR0  
0  0
V0  W0  0
 Q r  qR   l
0
0
A
 l P  / I xx
2I xz
 pP0  rR0   m A  m P  / I yy

I yy

I xx  I yy
I xz
I xz
r
p    Q 0 r  qR 0  
I zz
I zz
I zz
  P q  pQ    n
0
0
A
 n P  / I zz
Longitudinal and Lateral-Directional Equations Decouple!
Straight and SLUF Perturbation Description
Substitute the linear force and moment model
Longitudinal Description
 u  X    X u
U  Z    

 0
   Zu
 q  M   M u

 


 
 X Pu
X
Xq
 Z Pu
Z
Zq U 0
 M Pu
M   M P
M q U 0
0
1
0
Lateral-Directional Description
 U0 


Y p Yr  U 0
Y
Ixz
 p
r   

L L p
Lr
I xx 



Ixz
Np
Nr
 N

p r  
 I xx

0
1
tan  0



0
sec 0


  0





g cos 0
0
0
0
0
 g cos 0   u   XT
 g sin  0     ZT
 
0
  q  MT
  
0
    0
0  
 Y a
0  p  
 
L a
0  r   
   N  a
0    
0




0   
X E 
Z  E  T 

M  E   E 

0 
Y r 
L r   a 

N  r   r 

0 
Lockheed C-5A Galaxy
Longitudinal State-Space Description
***************************************************
Longitudinal State Space Model
Alon =
-0.1407
-0.0002
-0.0000
0
Blon =
0
-0.0378
-1.5616
0
17.0992
0 -32.1763
-0.6076 0.9591 -0.0020
-3.1928 -0.6971 0.0002
0 1.0000
0
Lateral-Directional State-Space Description
***************************************************
Lateral/Directional State Space Model
AlatD =
-0.0208 0.0000 -0.9942 0.0517
0
-1.8718 -0.6576 0.0396
0
0
0.8650 -0.0383 -0.0163
0
0
0 1.0000 0.0384
0
0
0
0 1.0007
0
0
BlatD =
0.0016
0.6076
0.0266
0
0
0.0010
-0.0168
-0.6031
0
0