29
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 d
ynamic analysis, we will examine the variations,
)
(
t
,
)
(
t
U
and
)
(
t
, when the A/C maneuvers from an equilibrium.
The
way
)
(
t
,
)
(
t
U
and
)
(
t
vary
is determined by the longitudinal EOM:
y
y
ext
z
ext
x
ext
I
q
M
mqu
w
m
F
mqw
u
m
F
,
,

)
(
t
,
)
(
t
U
and
)
(
t
are solutions of the above equations.

In general, different A/C
exhibit
s
different
x
ext
F
,
z
ext
F
and
y
ext
M
. As a
result, different set of
)
(
t
,
)
(
t
U
and
)
(
t
will result for different A/C.

In other word, each A/C exhibit
s
a
n 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.
30
《
䑹湡浩挠䙯牣攠䑩慧牡洠楮⁴桥
X
Z
偬慮攠
》
: Effective angle of attack of the aircraft
: Pitch angle of the aircraft
: Flight path angle of the aircraft (
)
mg
: Gravitational force (aircraft weight
W
)
L
: Lift of the aircraft, perpendicular to
U
D
: Drag force of the aircraft, parallel to
U
T
: Engine thrust, parallel to
U
《
Equation set of the longitudinal motion
》
F
m
u
mqw
ext
x

The drag equation
F
m
w
mqu
ext
z

The lift equation
M
q
I
ext
y
y

T
he pitch moment equation

These equation assume a motion that is in a single vertical
X
Z
plane.

Local horizon
x
z
L
D
T
U
mg=W
In dynamic analysis, we care only on the variations of
. As a res
ult, installation
angle of the wing and trim angle of the tail are immaterial here.
31
B.2.1 Linearization of the longitudinal equations
○
Preliminary treatment of the equation set.

We will linearize the inertia
l 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:
a) Expansion of
F
ext
x
:

F
F
F
ext
x
x
x
aero
grav

W
mg
F
x
sin
grav
<
== when
1

F
L
T
D
L
T
D
x
aero
sin
(
)
cos
.
b) Linearization of
m
u
mqw
:

u
U
U
w
U
U
cos
sin
,
,
and
cos
sin
u
U
U
U
U
.

Hence,
m
u
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
m
U
(
)
.

※
x
z
L
D
T
U
mg=W
1
32
2. For the lift Equation:
a) Expansion of
F
ext
z
:

F
F
F
ext
z
z
z
aero
grav

F
mg
mq
W
z
grav
cos
.

F
L
T
D
L
T
D
L
z
aero
cos
(
)
sin
(
)
.
The term,
(
)
T
D
, is usually small compared to
L
.
b) Linearization of
m
w
mqu
:

u
U
and
sin
cos
w
U
U
U
U
. Also, both
and
U
are small
compared to
U
; hence,
w
U
.

As a result,
m
w
mqu
m
U
U
(
)
.
The final lift equation thus becomes:
L
W
mU
(
)

※
3. For the pitch moment equation:
a) Expansion of
M
ext
y
:

M
M
ext
y
y
aero

Gravitational field does NOT contribute to rotational moment.
b) Linearization of
q
I
y
:

q
I
y
is already a linear term.
The final pitch moment equation:
M
q
I
I
y
y
y
aero

※
33
《
The pre

treated longitudinal equation set
》
)
(
mU
W
L

The lift equation
T
D
W
m
U
(
)

The drag equation
M
I
y
y
aero

The pitch moment equation
《
䑩杲敳e楯i
》

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 f
light 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).
34
○
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
U
0
0
0
,
,
and
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 initia
l 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 als
o 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 A
ppendix A of this note.
35
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
U
L
U
L
0
E
E
.

The subscript
E
indicates that the derivatives,
L
U
and
L
, are
computed for the equilibrium state.
b) Also,
mU
m
U
U
(
)
(
)(
)
0
0
0
c) In addition, we have
L
W
0
,
0
0
0
, and
U
U
0
; hence, the
perturbed lift equation becomes as follows:
L
U
L
U
u
0
(
)
.
where
L
u
m
L
U
1
E
and
L
m
L
1
E
.
2. Perturbation on the D
rag equation:

T
D
W
m
U
(
)
a) In general, the following expansions on
D
and
T
can be made:
D
D
U
D
U
D
0
E
E
and
T
T
U
T
U
0
E
.

In general,
T
does not depend on
, except for jet engine at high speed.
36
b) For the inertial part, we have
m
U
m
U
U
(
)
0
.
c) The steady equilibrium also imply that
T
D
W
0
0
0
0
(
)
and
U
0
0
; hence,
the perturbed drag equation becomes as follows:
(
)
(
)
T
D
U
D
U
g
u
u
.
where
T
u
m
T
U
1
E
,
D
u
m
D
U
1
E
and
D
m
D
1
E
.
3. Perturbation on the pitching moment equation:

M
I
y
y
aero
a)
In general, the pitching moment
M
is a function of
,
,
,
U
and
e
;
hence, we can expand
M
into as follows:
M
M
U
M
M
M
M
U
M
e
e
0
E
E
E
E
E
.
b) Due to
M
0
0
0
, th
e perturbed pitching moment eq. becomes:
M
M
M
M
U
M
u
e
where
M
M
I
M
I
M
y
y
1
1
E
E
,
,
M
M
I
M
u
I
M
U
y
y
,
1
1
E
E
,
and
M
I
M
y
e
1
E
.
37
《
䱩湥慲楺敤煵慴楯i猠潦⁴桥⁰敲瑵s扥搠汯湧b瑵t楮慬潴楯
》
L
U
L
U
u
0
(
)
(
)
(
)
T
D
U
D
U
g
u
u
M
M
M
M
U
M
u
e

As
the above equations
are concerned
,
L
u
,
L
,
T
u
,
D
u
,
D
,
M
,
M
,
M
u
,
M
and
M
,
are constant parameters

For different equilibrium state,, different set o
f the aerodynamic derivatives
may be obtained for th
e
equations.
《
Comments on the linearized longitudinal equations
》
With these linearized equations, we attempt to find a set of time functions,
),
(
t
u
)
(
t
,
(
)
t
and
e
t
(
)
such that the above equa
tions are stratified; this set of
),
(
t
u
)
(
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
:
L
u
,
L
,
T
u
,
D
u
,
D
,
M
,
M
,
M
u
,
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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