airplane Linear dynamics

```Conventional airplane Linear differential equations
- Longuitudinal Dynamics and Lateral Dynamics
CDu
C
 D
u
CD0  CD0  CDF
W
CPX
u
CPX
0
SF
q S
q S
 CD0 H H  CD0 H V
H
V
S
q SW
q SW
CS
A
CS 
R
C
CD  D

CL0  Lift coefficient at zero AoA
CD

C
CDq  D
q
C
CD  D
E
 E
CD 
CS

C
 S
p
CS  
CS p
CS
r
C
 S
 A
CSr 
CLu 
CPZ
u
CPZ
0
CS
 R
CL
u
CL 
R
CL
 R
CM
u
 moment cofficient at zero AoA
CM u 
CL
CL 

CL
CL 

CL
CLq 
q
CL
CL 
E
 E
CL 
CL
p
CL
CLr 
r
CL
CL 
A
 A
CLp 
CL

CM 0
CPM
u
CPM
0
CM  
CM P

CM

C
CM   M

C
CM q  M
q
C
CM   M
E
 E
C N
CN 

C N
CN p 
p
C N
CNr 
r
C N
C N 
A
 A
C N
R
C N

 R
bW  main wing span
I xx 
 y
2
 z 2  V dV
2
 z 2  V dV
q  dynamic pressure
 y 2  V dV
m  vehicle ' s mass
Vol
I yy 
 x
Vol
I zz 
 x
2
cW  mean chord length
SW  main wing area
 0  zero lift angle of attack
Vol
I xy  I yx 
 xy 
V
dV
V
dV
U 0  trim reference velocity at x axis
Vol
I xz  I zx 
 xz 
Vol
I yz  I zy 
 yz 
V
dV
Vol
dT 
xT 
acW
XF
acH
X Re f
T 
xT
V
T
T
dT
Dynamics properies
m  u   Q0 w  W0 q   V0 r  R0 v    mg cos 0  f AX  f PX
m  v   R0 u  U 0 r    P0 w  W0 p    mg  cos 0 cos  0  sin 0 sin  0   f AY  f PY
m  w   P0 v  V0 p    Q0 u  U 0 q    mg  cos 0 sin  0  sin  0 cos  0   f AZ  f PZ


u    Q0 w  W0 q   V0 r  R0 v   g cos 0  f AX  f PX / m

cos      f

/m
v    R0 u  U 0 r    P0 w  W0 p   g  cos 0 cos  0  sin  0 sin  0   f AY  f PY / m
w    P0 v  V0 p    Q0 u  U 0 q   g  cos  0 sin  0  sin  0
f AX  f PX
m
f AY  f PY
m
f AZ  f PZ
m
f AX  f PX
f AY
m
 f PY
m
f AZ  f PZ
m

q SW
m
0
AZ
 f PZ
 
 
2
2
CD0    CPX 
CPX
    CDu 
u
0
 
U
U
0
0
 

  T cos T   0 

  u  CD  CL0   CD   CDq q  CD E  E  
m






q SW
CS   CS p p  CSr r  CS  A  CS  R
A
R
m
  T sin T   0 
 

q S  
2
2
  W     CLu 
CL0    CPZ 
CPZ   u  CL  CD0   CL   CLq q  CL  E  
u
0
E

m   
U0
U0
m
 




 X u u  X Pu u  X    X    X q q  X  E  E  X T  T
 Y   YP P  Yr r  Y A  A  Y R  R
 Z u u  Z Pu u  Z   Z   Z q q  Z E  E  ZT  T

 mA  mP  / I yy
Dynamics properies


p
I xz
1
r
I xz  Q0 p  P0 q    I yy  I zz   R0 q  Q0 r    l A  lP 
I xx
I xx
q
1
I yy
r
I xz
1
p
 I xz  R0 q  Q0 r    I xx  I yy   Q0 p  P0 q    n A  nP 
I zz
I zz
q
1
I yy
 I
zz
 I xx  R0 r  P0 r   2 I xz  R0 r  P0 p    m A  mP  

 I
zz

 1
1

 I 2 xz   I xz
1 

 I xx I zz   I zz

 I xx  R0 r  P0 r   2 I xz  R0 r  P0 p    m A  mP  

 1
 p
1

 
2
r
 I
  I xz
 
1   xz  
 I xx I zz   I zz
I xz   1

I xz  Q0 p  P0 q    I yy  I zz   R0 q  Q0 r    l A  lP  


I xx  I xx



 1
 I xz  R0 q  Q0 r    I xx  I yy   Q0 p   P0 q   n A  nP  
1 


I

zz





mA  mP q SW cW

I yy
I yy

 
2
2
CM 0    CPM 
CP
   CM u 
u
 
U0
U 0 M0
 


 L'   L  N  I xz / I xx  D

 L'p   L p  N p I xz / I xx  D
 '
 Lr   Lr  N r I xz / I xx  D
 '
 L A  L A  N A I xz / I xx D
 '
 L R  L R  N R I xz / I xx D




'
'
'
'
'
 L    L p p  L r r  L A  A  L R  R 
 '

'
'
'
'
 N    N p p  N r r  N  A  A  N  R  R 
N '   N   L I xz / I zz  D 

N p'   N p  L p I xz / I zz  D 

1
N r'   N r  Lr I xz / I zz  D  D 
I xz2

'
1

N A  N A  L A I xz / I zz D 
I xx I zz

'
N R  N R  L R I xz / I zz D 







 dT cos T  xT sin T 
  u  CM  CM P   CM   CM q q  CM  E  E    T
I yy


nA  nP q SW bW

C N    C N p p  C N r r  C N  A  C N   R
A
R
I zz
I zz



l A  lP
 L   L p p  Lr r  L A  A  L R  R
I xx
mA  mP
 M u u  M Pu u  M    M P   M    M q q  M  E  E  M T  T
I yy
nA  nP
 N    N p p  N r r  N A  A  N R  R
I zz
I xz 
I xx    l A  l p  / I xx 


  nA  nP  / I zz 


1 



l A  lP q SW bW

CL   CLp p  CLr r  CL  A  CL  R
A
R
I xx
I xx
 M u u  M Pu u  M    M P   M    M q q  M  E  E  M T  T
  p  tan 0  sin  0 q  cos  0 r   Q0 cos  0  R0 sin  0      Q0 sin  0  R0 cos  0   0 sin  0 tan  0 
  cos  0 q  sin  0 r   Q0 sin  0  R0 cos  0  
   0 tan 0   sin  0 q  cos  0 r   R0 sin  0  Q0 cos  0    / cos  0
Dynamics properies at level flight

w  Z u u  Z Pu u  Z   Z    Z q  U 0  q  Z E  E  ZT  T

u   g  f AX  f PX / m


v  U 0 r  g  f AY  f PY / m

u   g  X u u  X Pu u  X    X    X q q  X  E  E  X T  T

q  M u u  M Pu u  M    M    M q q  M  E  E  M T  T
w  U 0 q  f AZ  f PZ / m
 q
u   g  X u u  X Pu u  X    X    X q q  X  E  E  X T  T
v  g  Y   YP P  Yr  U 0  r  Y A  A  Y R  R
v  g  Y   YP P  Yr  U 0  r  Y A  A  Y R  R
w  Z u u  Z Pu u  Z   Z    Z q  U 0  q  Z E  E  ZT  T
q   mA  mP  / I yy
 p
1
 
2
 r  I xx I zz  I xz
 I zz
I
 xz
I xz    l A  lP  


I xx   nA  nP  
p  L'    L' p p  L'r r  L' A  A  L' R  R
q  M u u  M Pu u  M    M P   M    M q q  M  E  E  M T  T
r  N '    N ' p p  N 'r r  N ' A  A  N ' R  R
p  L'    L' p p  L'r r  L' A  A  L' R  R
r  N '    N ' p p  N 'r r  N ' A  A  N ' R  R
p
u   g  X u u  X Pu u  X    X    X q q  X  E  E  X T  T
w  Z u u  Z Pu u  Z   Z    Z q  U 0  q  Z E  E  ZT  T
q  M u u  M Pu u  M    M    M q q  M  E  E  M T  T
 q


1
 Z u  Z Pu u  Z    Z q  U 0  q  Z E  E  ZT  T
U

Z
 
 0

 



X  Z u  Z Pu
u   X u  X Pu 
U 0  Z



  u   X






X  Z
U 0  Z

 U0  Zq

   g   X q  X  

 U 0  Z


1
 Z u u  Z Pu u  Z    Z q  U 0  q  Z E  E  ZT  T
 U 0  Z 
 



M  Z u  Z Pu
q   M u  M Pu 
U 0  Z


 q
h   w  U 0  w  U 0 
  u   M





 M P 

M  Z
U 0  Z
X  Z E


  q   X  E 
U 0  Z




X  ZT
E   XT 
U 0  Z



 T




    M q  M 


 U0  Zq

 U 0  Z
M  Z E


  q   M  E 
U 0  Z




M  ZT
E   MT 
U 0  Z



 T



X Z  Z Pu
  Xu  X P   u
u

U 0  Z


 Z u  Z Pu 



U 0  Z 


A
0


M Z  Z Pu
 M  M   u
Pu
 u
U 0  Z


0



X  Z E
X

  E
U 0  Z


Z E 
 
  U 0  Z 
B
0


 M  M  Z  E
  E U 0  Z

0




 



X  Z 
X

 

U 0  Z 

 Z 


 U 0  Z 
0
 
g
0
0

M  Z 
 M   M P 
 0
U 0  Z 
 

U 0
U0

X  ZT
 XT 
U 0  Z

 ZT 


 U 0  Z 
0
 
M  ZT
  MT 
U 0  Z
 
0



1

0



0
,
C




0


0





 X q  X 

 U0  Zq

 U 0  Z

 

 U0  Zq 


 U 0  Z 
1

 M q  M 

0 0 0 0
0

0
1 0 0 0

0 1 0 0  , D  0


0 0 1 0
0
0
0 0 0 1 
 U0  Zq

 U 0  Z
0
0
0 
0

0
0 

 


0



0


0

0


0 
u 
 
 
 
x  y    , u   E 
 T 
q 
 
 h 
v  g  Y   YP P  Yr  U 0  r  Y A  A  Y R  R
p  L'    L' p p  L'r r  L' A  A  L' R  R
 
 
 
 
x  y   p, u   A 
 R 
r 
 
 
r  N '    N ' p p  N 'r r  N ' A  A  N ' R  R
p

Y
Y
Yp
Y
Y

g
  
p   r  1 r  A  A  R  R
U0
U0
U0
U0
U0
 U0

   0 tan 0   sin  0 q  cos  0 r   R0 sin  0  Q0 cos  0    / cos 0
 Y

U 0
 0
A '
 L
N '
 
 0
Yp
g
U0
U0
0
1
0
L'p
0
N p'
0
0
 Y A
 Yr
 
 1 0 


 U0
 U0
 
 0
0
0
 '
B


L'r
0
 L A
N '
N r'
0
 A

 0
1
0 
Y R 

1
U0 
0

0 

0
C

L' R 

' 
0
N R 
0
0 
0 0 0 0
0

0
1 0 0 0

0 1 0 0  D  0


0 0 1 0
0
0
0 0 0 1 
0
0 
0

0
0 
```
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