C.
Lateral motion analysis
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《Governing Equation Set of Lateral Motion》
y
Fext
  Fi y  m( v  ru  pw)
--- Side force equation
x
Mext
  Mix  pI
 x  rI
 xz --- Rolling moment equation
z
Mext
  Miz  rI
 z  pI
 xz --- Yawing moment equation
--- Note that we have included ru and pw in Fi y for completeness, but we will
treat u and w as constants, i.e. u  U 0 and w   0U 0 , in lateral motion.
《Variables of lateral motion》
Sign convertions:

 p   : Roll rate in inertial space
Left aileron
 r   : Yaw rate in inertial space
p    0
 v: Sideslip velocity (bottom figure)
L0
--- Also define the sideslip angle as   v / U 0 .
v0
《Force and moments of lateral motion》
Y
L
Y0
and N 
《Control surfaces of lateral motion》
  a : Aileron deflection ( >0 for right aileron down )
  r : Rudder deflection ( same sign as r ).
y
Faero
,
x
Mext
z
M ext
r    0, N  0
U
y

v
x
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C.1 Linearization of the lateral equations
○We will linearize the inertial terms and expand the external force and external
moments into their gravitational and aerodynamic components. We will also
linearize the aerodynamic terms by perturbations in v , r and p from their
equilibrium values.
--- At equilibrium, v0  r0  p0  0 , therefore v , r and
p are perturbations as well as their absolutes values.
○For the side force equation
a) Linearization of Fi y   m( v  ru  pw) :
--- u  U 0 cos  0  U 0 and w  w0 ; both are considered constants.
y
--- Then, F   m( v  U 0r  w0 )
--- We have used p   .
i
.
y
b) Expansion of Fext
:
y
y
--- Fext
 Faero
 Fgy
y
mg
Y
Y
Horizon --- Aerodynamic force : Faero  Y  Ytrim   v v    v v
E
E

y
--- Ytrim  0 at trim.
y --- Gravitational force : Fg  mg
c) The final side force eq.: g  Yv v  v  U0r  w0
--- Yv 
1Y
.
mv E
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○For the rolling moment equation:
a) Inertial terms, Mix  ( pI
 x   I xz ) , are already linear in p and  .
x
x
b) Expansion of Mext
: --- Only aerodynamic terms appear on Mext
.
--- We normally denote this aerodynamic rolling moment as L .
--- L is a function of v , r , p,  a and  r ; hence, we can expand L into:
L
L
L
L
L
L  Ltrim   v v   r r   p p 
a 
 
a
r r
--- At trim condition, Ltrim  0 .
c) Normalize the equation by dividing by I x and define
L
L
L
L
L
Lv  1  v , Lr  1  r , L p  1  p , L a  1
, L r  1
,
Ix
Ix
Ix
Ix   a
Ix   r
the final rolling moment equation becomes:
Lv v  Lr r  L p p  p  r I x   L a  a  L r  r , I x  I xz / I x .
○For the yawing moment equation:
 xz ) , are already in linear form,
a) Again, the inertial terms, Miz  ( I z  pI
and only aerodynamic terms appear on the external yawing moment.
--- We normally denote this aerodynamic yawing moment as N .
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N
N
N
N
N
b) Expand N into: N  N trim   v v   r r   p p 
a 

a
r r
c) Normalize the equation by dividing by I z , set N trim =0, and define
N
N
N
N
N
N v  1  v , N r  1  r , N p  1  p , N a  1
, N r  1
,
Iz
Iz
Iz
Iz   a
Iz   r
the final yawing moment equation becomes:
N v v  N r r  N p p  r  I z p   N a  a  N r  r , I xz  I xz / I z
《Linearized equation set of the perturbed lateral motion》
g  Yv v  v  U0r  w0  0
Lv v  Lr r  L p p  p  rI
 x   L a  a  L r  r
N v v  N r r  N p p  r  I xz p   N a  a  N r  r
《Laplace transform of the lateral equation set》
 s  Yv

  Lv
 N
v

U0
 I x s  Lr
s  Nr
 g  w0 s   v   0
  
2
s  L p s  r   L a
 
2

 I xz s  N p s     N a

  a 
  
 
N r  r
0
L r
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C.2 Static Lateral Analysis
○We will analyze the equilibrium conditions of various steady lateral motions.
○By steady lateral motions we meant lateral maneuvers in which all variations
in variables have vanished; hence, the static equation set becomes as follows:
 Yv U 0  g   v   0 0 
  a 
 L
r    L

L
0
0
r
   , L r  1 .
 v
    a
 r
 N v  N r 0     N a N r 
○Steady turn:
Steady
 In an ideal turn, v=0.
trun
L  mg
 Side force equation gives U 0r  g  0 ;
向心力
hence, r  g / U 0 .
mg

Lr
Lr g
r

 We will need  a  
mU 0r
L a
L a U 0
離心力
N r r  N a  a
N g
 r
.
and  r  
--- Normally, N a  1
N
N U
r
r
0
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Discussions:
--- Require both  a &  r to maintain a turn.
--- Usually , Lr  0, L a  0, N r  0 and N r  0 .
  a is positive for positive   Aileron is held against turn
  r is negative for positive   Rudder is held with turn
 Both  a &  r will decrease with increasing U 0 .
○Straight side slip:
Yv v
L  mg

In a straight side slip, r  0 .
mg
Straight
Remaining equation set:

sideslip
U
 0 0 
 Yv  g 
  a 
x
  L 0 v    L
0

v

Side force
  

    a


r
Yv v
v
 N N 
 N v 0 
 a
r
Side wind

y
r
v
 Side force equation gives: Yv v   g ;
v
g
g
hence, v   ; or since    , we also have   
.
U0
YvU 0
Yv
--- Tilted lift to balance aerodynamic drag due to sideslip velocity v.
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L
 Rolling equation gives:  a   v U 0 .
 L  Lvv
L a
Dihedral
v
--- Lv is mostly the result of Dihedral, so we
need  a  0 to balance Dihedral.
N v v  N a  a
N
N
  v v   v U 0 .
 And yawing equation gives:  r  
N r
N r
N r
---  r  0 to balance the yawing moment from vertical tail due to sideslip angle  .
○Steady roll:
--- No yaw, no sideslip, or r = v = 0.
Remaining equation set:  L p p  L a  a
 N p p  N a  a  N r  r
--- Note that p=s.
From first equation:  a  ( L p / L a ) p
--- Normally,  a  0 for p > 0 to balance the aerodynamic drag
N p p  N a  a
Np

p.
From the second equation:  r  
N r
N r
--- We need  r  0 to balance the roll-induced yawing moment N p p .
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