B.2 Dynamic Longitudinal analysis
B.2.1 Dynamic longitudinal analysis – Some preliminaries
○What do we mean by the dynamic analysis of an A/C?
We will analyze the A/C behavior when depart from an equilibrium state.
In particular, we will analyze the change in  , U and  in an A/C maneuver
--- In static longitudinal analysis, we focus on static values of  , U , etc.
--- In the dynamic analysis, we will examine the variations,  (t ) , U (t ) and
 (t ) , when the A/C maneuvers from an equilibrium.
The way  (t ) , U (t ) and  (t ) vary is determined by the longitudinal EOM:
y
x
z
Fext
 mu  mqw,
Fext
 mw  mqu,
M ext
 qI y
---  (t ) , U (t ) and  (t ) are solutions of the above equations.
y
x
z
--- In general, different A/C exhibits different Fext
, Fext
and M ext
. As a
result, different set of  (t ) , U (t ) and  (t ) will result for different A/C.
--- In other word, each A/C exhibits an unique characteristics of motion.
○Why do we want to analyze the dynamic behavior of an A/C?
Dynamic behavior of an A/C determines its maneuverability.
An A/C will be dangerous to fly if its maneuverability is poor.
--- We need to design the A/C so that it has an acceptable dynamics.
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《Dynamic Force Diagram in the X  Z Plane 》
 : Effective angle of attack of the aircraft
x


 U  : Pitch angle of the aircraft
 : Flight path angle of the aircraft (     )

T
mg : Gravitational force (aircraft weight W )
Local
horizon
D
L : Lift of the aircraft, perpendicular to U 
z
to U
D : Drag force of the aircraft, parallel

mg=W
T : Engine thrust, parallel to U
L
In dynamic analysis, we care only on the variations of  . As a result, installation
angle of the wing and trim angle of the tail are immaterial here.
《Equation set of the longitudinal motion》
x
Fext
 mu  mqw
--- The drag equation
z
Fext
 mw  mqu
--- The lift equation
y
M ext
 qI
 y --- The pitch moment equation
--- These equation assume a motion that is in a single vertical X  Z plane.---
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B.2.1 Linearization of the longitudinal equations
○Preliminary treatment of the equation set.
--- We will linearize the inertial terms, and expand the external force and external
moment into their gravitational and aerodynamic components..
--- We will assume a near level flight with small variations in u, and  .
--- We will proceed with the three equations separately.
1. For the Drag Equation:
x
a) Expansion of Fext
:
---
x
Fext
x
--- Fgrav
 mg sin   W

x
Faero

x
Fgrav
<== when   1
x
--- Faero  L sin   (T  D )cos   L  T  D.
L   1
D
x

U
T

z
b) Linearization of mu  mqw :
mg=W
--- u  U cos  U , w  U sin   U ,
and u  U cos   U sin   U  U .
--- Hence, mu  mqw  m(U  U  U ), q   .
c) The drag eq. becomes: L  T  D  W  m(U  U  U )
d) The lift equation will dictates that mU (   )   (L  W ).
The final drag equation thus becomes: T  D  W (   )  mU . ---※
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2. For the lift Equation:
z
a) Expansion of Fext
:
z
z
z
 Faero
 Fgrav
--- Fext
z
 mg cos  mq  W .
--- Fgrav
z
--- Faero   L cos   (T  D )sin    L  (T  D )   L.
 The term, (T  D ) , is usually small compared to L .
b) Linearization of mw  mqu :
  U sin   U cos   U  U . Also, both  and
--- u  U and w
U are small compared to U ; hence, w  U .
--- As a result, mw
  mqu  m(U  U ) .
The final lift equation thus becomes:  L  W  mU(   )
----※
3. For the pitch moment equation:
y
y
y
 M aero
a) Expansion of M ext
:
--- M ext
--- Gravitational field does NOT contribute to rotational moment.
b) Linearization of qI
 y:
--- qI
 y is already a linear term.
y
 qI
The final pitch moment equation: M aero
 y  I y
----※
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《The pre-treated longitudinal equation set》
L  W  mU (   )
--- The lift equation
T  D  W (   )  mU --- The drag equation
y
M aero
 I y
--- The pitch moment equation
《Digression》
---This set of equations can be used as performance equations. ---
    0
  0
  0,   0
○From the lift equation and assuming a static condition, i.e.   0 , then the
following conclusions can be drawn:
 If L  W , then       0 , and a straight flight path must result (left).
 If L  W , then       0 , and a curved flight path must result (right).
○The following conclusions can also be drawn from the drag equation:
 Excess thrust, T  D , can either cause an acceleration, i.e. U  0, along flight path,
or produce a climb, i.e.   0 (but   0 , center).
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○Expansion and linearization of the aerodynamic terms.
 To use these equations in their dynamic sense require accounting for the
changes in external forces and moments as the motion proceeds. This is
done by associating the aerodynamic forces and moments of the equations
with flight variables, as is shown as follows.
1. We will introduce perturbations of the major variables as follows:
U  U 0  U ,    0   , and    0  
where U 0 ,  0 and  0 are the values of U,  and  at the equilibrium
state, and U,  and  are the perturbed variables.
2. We will assume that the initial condition is steady, namely,
U 0   0   0  0.
3. We will expand, with respect to U,  and  , the aerodynamic forces
and moments in Taylor series about the steady condition.
4. We will also assume that  ,   1 and U  U 0 so that first order
Taylor expansion of the forces and moments are suffice for the analysis.
--- A brief outline on Taylor series expansion of analytic functions is included
in Appendix A of this note.
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1. Perturbation on the Lift equation: --- L  W  mU (   )
a) The lift force L is a function of  and U ; hence, we can write
L
L
L  L0  U U    .
E
E
--- The subscript E indicates that the derivatives,  L U and  L  , are
computed for the equilibrium state.
b) Also, mU (   )  m(U 0  U )( 0     0   )
c) In addition, we have L0  W ,  0   0  0 , and U 0  U ; hence, the
perturbed lift equation becomes as follows:
Lu U  L   U 0 (    ) .
where
Lu 
1  L
m U E
and L 
1  L .
m  E
2. Perturbation on the Drag equation: --- T  D  W (   )  mU
a) In general, the following expansions on D and T can be made:
D
D
T
D  D0  U U    and T  T0  U U .
E
E
E
--- In general, T does not depend on  , except for jet engine at high speed.
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b) For the inertial part, we have mU  m(U 0  U ) .
c) The steady equilibrium also imply that T0  D0  W ( 0   0 ) and
U 0  0 ; hence, the perturbed drag equation becomes as follows:
(Tu  Du )U  D   U  g(    ) .
D
1  T ,
m U E
D
1 
1 
and
.
D

Du  m

m  E
U E
y
 I y
3. Perturbation on the pitching moment equation: --- Maero
a) In general, the pitching moment M is a function of  ,  , ,U and  e ;
hence, we can expand M into as follows:
M
M
M
M
M
M  M 0           U U    e .
where Tu 
E
E
 E
E
b) Due to M0   0  0 , the perturbed pitching moment eq. becomes:
e E
M   M   M   MuU  M  e  
1 M , M  1 M ,

I y  E
I y  E
M
M
M  I1  , M u  I1 U ,
y  E
y
E
where M 
and M 
1 M .
I y  e E
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《Linearized equations of the perturbed longitudinal motion》
Lu U  L   U 0 (    )
(Tu  Du )U  D   U  g(    )
M   M   M   MuU  M  e  
--- As the above equations are concerned, Lu , L , Tu , Du , D , M ,
M , Mu , M and M , are constant parameters
--- For different equilibrium state,, different set of the aerodynamic derivatives
may be obtained for the equations.
《Comments on the linearized longitudinal equations》
 With these linearized equations, we attempt to find a set of time functions, u (t ),
 (t ) ,  ( t ) and  e ( t ) such that the above equations are stratified; this set of
u (t ),  (t ) ,  ( t ) and  e ( t ) are called the solution to the above equations.
 This solution of the above equations is determined by the values of the following
derivatives: Lu , L , Tu , Du , D , M , M , Mu , M and M .
 These derivatives reflect the aerodynamic properties of the A/C and are named
the aerodynamic derivatives of the perturbed longitudinal equations.
 These derivatives have the physical meaning of acceleration.
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