AOE 3104 Homework #5 Solutions
Problem 1.
A jetpropelled airplane has a parabolic drag polar with the following parameter values:
C
D
0
= 0
.
024
,
AR
= 7
,
e
= 0
.
85
,
S
= 120 m
2
,
m
= 75
,
000 kg
,
T
SL
= 150
,
000 N
.
Assume that
T
(
h
) =
T
SL
σ
(
h
) =
T
SL
ρ
(
h
)
ρ
SL
.
•
Compute the minimum and maximum equivalent airspeed (in m/s) for level equilibrium flight at
3000, 6000, and 9000 meters. Present your results in a table.
•
Develop a Matlab script to generate the aircraft’s flight envelope, as a plot of altitude versus equivalent
airspeed (for wings level, equilibrium flight). Use the resulting plot to determine the aircraft’s absolute
ceiling. (Include your Matlab script and the plot with your submission.)
Solution.
To compute the equilibrium speeds, we must compute the conditions for wingslevel equilibrium
flight at constant altitude. In this condition, thrust equals drag. The parabolic drag polar is
C
D
(
C
L
) =
C
D
0
+
1
πeAR
bracehtipupleft
bracehtipdownrightbracehtipdownleft
bracehtipupright
K
C
2
L
.
Since we are interested in equivalent airspeed, redimensionalize by multiplying by sealevel dynamic pres
sure times wing area:
D
=
parenleftbigg
1
2
ρ
SL
V
2
eq
parenrightbigg
SC
D
=
parenleftbigg
1
2
ρ
SL
V
2
eq
parenrightbigg
S
(
C
D
0
+
KC
2
L
)
Noting that
W
=
C
L
parenleftbigg
1
2
ρ
SL
V
2
eq
parenrightbigg
S
in wingslevel, constantaltitude equilibrium flight, we find
D
=
C
D
0
parenleftbigg
1
2
ρ
SL
V
2
eq
parenrightbigg
S
+
KW
2
bracketleftbiggparenleftbigg
1
2
ρ
SL
V
2
eq
parenrightbigg
S
bracketrightbigg
−
1
We assume that thrust exactly balances drag in equilibrium flight and that thrust scales directly with
altitude
T
(
h
) =
σ
(
h
)
T
SL
.
Setting
T
(
h
) =
D
and multiplying through by dynamic pressure times area, we obtain the following
quadratic equation in
z
=
V
2
eq
:
C
D
0
parenleftbigg
1
2
ρ
SL
S
parenrightbigg
2
z
2

σ
(
h
)
T
SL
parenleftbigg
1
2
ρ
SL
S
parenrightbigg
z
+
KW
2
= 0
.
The quadratic equation has real solutions if and only if the discriminant “
b
2

4
ac
” is nonnegative:
bracketleftbigg

σ
(
h
)
T
SL
parenleftbigg
1
2
ρ
SL
S
parenrightbiggbracketrightbigg
2

4
bracketleftBigg
C
D
0
parenleftbigg
1
2
ρ
SL
S
parenrightbigg
2
bracketrightBigg
bracketleftbig
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 Spring '08
 DR.CRAIGWOOLSEY
 Aerodynamics, Aviation terminology, Equivalent airspeed, Veq, parabolic drag polar

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