## Transcribed Text

1. What modifications that can be done to the wing of the aircraft shown in the figure below to
make it more statically laterally stable?
2. In the 2-1-2 Euler angle sequence, the XYZ frame is brought to the xyz frame through the
following successive rotations: a rotation of magnitude ψ about Y-axis, then a rotation of
magnitude θ about
x
-axis, and finally a rotation of magnitude ϕ about axis
y
-axis.:
a. Determine the kinematic expressions relating the Euler-angle rates to the rates in the xyz
frame (p, q, r) for this 2-1-2 rotation sequence.
b. For small Euler angles, what are the relationship between p, q, r and the Euler angular
rates?
3. The longitudinal characteristic roots of an aircraft in complex plane is given below:
a. Identify the roots associated with the phugoid and short-period modes in the figure. Is the
aircraft dynamically stable longitudinally?
b. Estimate the undamped natural frequency and damping ratio of the modes. Describe the
motion associated with each of the modes.
c. Determine the longitudinal characteristic equation of the aircraft.
2
4. The following set of differential equations below describes the longitudinal dynamics of an
aircraft:
q
dt
d
u q
dt
d
uX
dt
ud
e
u e
)(
.0 002 .1 05 .1 02 .1 04 )(
21 .08.9 05 )(
where ∆u, ∆α , ∆θ and ∆q are deviations of forward speed, angle of attack, pitch angle and
pitch rate from the nominal flight conditions, respectively. ∆δe is the elevator deflection. Xu is
the longitudinal-direction force derivative with respect to u. During the motion, the angle of
attack can be assumed constant at its nominal value.
a. Derive the 2nd order state-space representation of the longitudinal dynamics above.
b. Which characteristic mode of motion does the equations of motion above represent? State
clearly your reasons.
c. Determine the characteristic roots of the equations above in terms of Xu and determine the
range of Xu values to achieve dynamic stability.
d. Will these characteristic roots be affected by elevator deflection ∆δe ?
e. If Xu = −0.1, describe the aircraft natural motion characteristics, i.e. whether the natural
motion is oscillatory or non-oscillatory, stable or unstable, and calculate the relevant
motion parameters such as damping ratio, frequency, or time constant.

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