Deck 6: Principles of Stability and Control

Question 1

For a given wing-body combination, the aerodynamic center lies 0.03 chord length ahead of the center of gravity. The moment coefficient about the center of gravity is 0.0050, and the lift coefficient is 0.50. Calculate the moment coefficient about the aerodynamic center.

Answer 1

The expression for the moment coefficient about the center of gravity is expresses as follows,
Here,
and
are the moment coefficient about the center of gravity and about the aerodynamic center respectively,
,
and
are the lift coefficient, coordinates of the center of gravity in fractions of chord length and distance of aerodynamic center from the leading edge.
Rearrange the above expression of
in terms of
is as follows,
The aerodynamic center lies
chord length ahead of the center of gravity. Thus, the value of
is expresses as,
Now, the moment coefficient about the aerodynamic center is calculated as follows:
Substitute
for
,
for
and
for
in the above rearranged expression of
and solve,
Therefore, the moment coefficient about the aerodynamic center is
.

Question 2

Consider a model of a wing-body shape mounted in a wind tunnel. The flow conditions in the test section are standard sea-level properties with a velocity of 100 m/s. The wing area and chord are 1.5 m 2 and 0.45 m, respectively. Using the wind tunnel force and moment-measuring balance, the moment about the center of gravity when the lift is zero is found to be ?12.4 N · m. When the model is pitched to another angle of attack, the lift and moment about the center of gravity are measured to be 3675 N and 20.67 N · m, respectively. Calculate the value of the moment coefficient about the aerodynamic center and the location of the aerodynamic center.

Answer 2

The free-stream dynamic pressure is,
Here,
is the wind density and
is the free- stream velocity.
Substitute
for
and
for
.
The moment of coefficient about the center of gravity is,
Here,
is the moment about the center of gravity,
is the wing area, and
is the chord.
Substitute
for
,
for
, and
for
, and
for
.
For zero lift, the moment of coefficient about the center of gravity is equal to the moment of coefficient about the aerodynamic center.
Substitute
for
.
Therefore, the value of the moment of coefficient about the aerodynamic center is
.
The tail-lift coefficient is,
Here,
is the lift.
Substitute
for
,
for
, and
for
.
The moment of coefficient about the center of gravity when lift is not equal to zero is,
Substitute
for
,
for
, and
for
, and
for
.
The moment of coefficient about the center of gravity in terms of lift coefficient and moment coefficient about the aerodynamic center for the wing-body combination is,
Here,
is the location of the aerodynamic center for the chord length ahead of the center of gravity.
Rearrange the equation for
.
Substitute
for
for
, and
for
.
Therefore, the location of the aerodynamic center is
of the chord length ahead of the center of gravity.

Question 3

Consider the model in Prob. 7.2. If a mass of lead is added to the rear of the model so that the center of gravity is shifted rearward by a length equal to 20 percent of the chord, calculate the moment about the center of gravity when the lift is 4000 N.

Answer 3

7.3 From the results of Problem 7.2,
In the present problem, the e.g. has been shifted 0.2c rearward. Hence,
h - h ac = 0.02 + 0.2 = 0.22

Question 4

Consider the wing-body model in Prob. 7.2. Assume that a horizontal tail with no elevator is added to this model. The distance from the airplane's center of gravity to the tail's aerodynamic center is 1.0 m. The area of the tail is 0.4 m 2 , and the tail-setting angle is 2.0°. The lift slope of the tail is 0.12 per degree. From experimental measurement, ? 0 and ?? / ??= 0.42. If the absolute angle of attack of the model is 5º and the lift at this angle of attack is 4134 N, calculate the moment about the center of gravity.

Question 5

Consider the wing-body-tail model of Prob. 7.4. Does this model have longitudinal static stability and balance?

Question 6

For the configuration of Prob. 7.4, calculate the neutral point and static margin if h = 0.26.

Question 7

Assume that an elevator is added to the horizontal tail of the configuration given in Prob. 7.4. The elevator control effectiveness is 0.04. Calculate the elevator deflection angle necessary to trim the configuration at an angle of attack of 8°.

Question 8

Consider the configuration of Prob. 7.7. The elevator hinge moment derivatives are
Assess the stick-free static stability of this configuration.

Question 9

Consider the canard configuration as illustrated in Fig. 7.17 b and represented by the XB-70 shown in Fig. 7.18. You will sometimes encounter a statement, either written or verbal, that the canard configuration is inherently statically unstable. This is absolutely not true. Prove that the canard configuration can be made statically stable. What design condition must hold to ensure its stability?