MAE 5070
D. A. Caughey
March 9, 2011
Exercise Set III
1. Use the dimensions provided in Exercise Set II for the Boeing 747 aircraft to estimate the values of Cm q
and Cm α̇ for both flight conditions of Exercise Set I. Refer to Solution Set I for any additional parameters
needed for the M∞ = 0.80 flight condition. Compare your values with those given in the table below; note
that the flight conditions of Exercise Set I correspond to Conditions 2 and 9 in the table below. (You should
realize that you’ve already computed the first of these stability coefficients for Condition 1 in Exercise Set
II.)
2. Show that for a straight, untapered wing (i.e., one having a rectangular planform) having a constant spanwise load distribution (i.e., constant section lift coefficient), simple strip theory gives the wing contribution
to the rolling moment due to yaw rate as
(Clr )wing =
CL
3
3. Show that for a straight, untapered wing (i.e., one having a rectangular planform) having an elliptical
spanwise load distribution, simple strip theory gives the wing contribution to the rolling moment due to
yaw rate as
CL
(Clr )wing =
4
Explain, in simple terms, why this value is smaller than that computed in Exercise 2.
4. Show that for a straight, untapered wing (i.e., one having a rectangular planform) having an elliptical
spanwise load distribution, simple strip theory gives the induced drag contribution to the yawing moment
due to roll rate as
¡
¢
CL
Cnp wing = −
8
Note: For problems dealing with elliptic spanwise loadings the following integrals (which can be evaluated using
trigonometric substitution) are useful:
Z
0
1
p
1 − η 2 dη =
π
,
4
Condition
h (ft)
M∞
CL
CD
CLα
CDα
Cmα
CLα̇
Cmα̇
CLq
Cmq
CLM
CDM
CmM
CLδe
Cmδe
Z
1
0
2
SL
0.249
1.11
0.102
5.70
0.66
-1.26
6.7
-3.2
5.40
-20.8
0.0
0.0
0.0
0.338
-1.34
η
p
1
1 − η 2 dη = ,
3
5
20,000
0.500
0.680
0.0393
4.67
0.366
-1.146
6.53
-3.35
5.13
-20.7
-.0875
0.0
0.121
0.356
-1.43
7
20,000
0.800
0.266
0.0174
4.24
0.084
-.629
5.99
-5.40
5.01
-20.5
0.105
0.008
-.116
0.270
-1.06
Z
1
η2
0
9
40,000
0.800
0.660
0.0415
4.92
0.425
-1.033
5.91
-6.41
6.00
-24.0
0.205
0.0275
0.166
0.367
-1.45
p
1 − η 2 dη =
π
16
10
40,000
0.900
0.521
0.0415
5.57
0.527
-1.613
5.53
-8.82
6.94
-25.1
-.278
0.242
-.114
0.300
-1.20
Table 1: Longitudinal aerodynamic stability derivatives for the Boeing 747 at selected flight conditions; Data
from Heffley & Jewell, Aircraft Handling Qualities Data, NASA CR-2144, December 1972.