Aerodynamic Derivatives Questions and Answers

Answer: C

Substituting the section value and the aspect ratio:

\[ \begin{aligned} \frac{a_0}{\pi A} &= \frac{6.283}{25.1} = 0.25 \\ C_{L_\alpha} &= \frac{6.283}{1.25} = 5.03 \end{aligned} \]

so about 5.0 per radian: the finite wing loses a fifth of the section slope to induced downwash. 6.3 is the section value returned unchanged, which would need infinite aspect ratio. 3.1 halves it. 8.0 is the aspect ratio quoted back as a slope, which is a units error rather than an arithmetic one.

Answer: D

The wing's downwash takes \( l_t/V \) seconds to reach the tail, so while incidence is changing the tail is immersed in the downwash of an earlier incidence. That lag makes the tail's contribution depend on the rate as well as the value, and it is the one unsteady effect retained in the otherwise quasi-steady rigid-body set. Circulation build-up is a genuinely unsteady effect but is not what this derivative models. Actuator and engine lags are control-system dynamics, not aerodynamics.

Answer: A

Tuck follows from \( C_{m_u} \), the pitching moment derivative with speed, turning negative as the aerodynamic centre moves aft with Mach number. That drives the last coefficient of the longitudinal quartic through zero and produces an aperiodic nose-down divergence. Pitch damping affects the short period's decay, not the trim with speed. Elevator lift is a control derivative. The drag derivative with incidence is small and matters mainly in the phugoid, which tuck is not.

Answer: B

All four are negative in the usual stability-axis convention. The first is static stability, the second pitch damping, and the fourth says that a trailing-edge down elevator gives a nose-down moment. The third is the one that catches people: \( Z_\alpha \) is negative because the normal force axis points downward in the standard body-axis set, so an increase in incidence produces a more negative \( Z \) even though it produces more lift. Sign conventions are worth pinning down before any derivative is used in anger.

Answer: C

Frequency comes from \( M_\alpha \) and damping from \( M_q + M_{\dot\alpha} + Z_\alpha/u_0 \), so the first option names them correctly. Drag and lift with speed are the phugoid's derivatives. Elevator effectiveness and hinge moments concern control and stick force rather than the free motion. The last pair are lateral derivatives entirely, which cannot appear in a longitudinal mode because the two problems decouple for small perturbations.

Answer: D

The factor is \( 1/\sqrt{1-0.36} = 1/0.8 = 1.25 \), so

\[ C_{m_\alpha} = -0.9 \times 1.25 = -1.13 \ \mathrm{rad^{-1}} \]

giving -1.13 per radian: compressibility strengthens the derivative and so raises the short-period frequency. -0.72 divides by the factor instead of multiplying, which gets the trend backwards. -1.50 uses a factor of \( 1/(1-M^{2}) \) without the root. -0.90 leaves the derivative uncorrected, which is only valid well below \( M = 0.3 \).

Answer: A

Both statements are correct, and the second is the derivation of the first. The incidence increment is \( q\,l_t/V \), so the tail force is proportional to the arm, and the moment is that force times the arm again - hence the square. It is why stretching a fuselage improves pitch damping so effectively, and why a tailless configuration struggles for it. The same argument, with the span in place of the arm, explains why roll damping scales with the square of the span.

Answer: B

It is the dihedral effect, \( C_{l_\beta} \), negative on a stable aeroplane so that sideslip from the right rolls the aeroplane to the left, away from the sideslip. Geometric dihedral, wing sweep and a high wing position all contribute to it. Weathercock stability is the yawing moment with sideslip, a different derivative on a different axis. Roll damping is the rolling moment with roll rate. Adverse yaw is the yawing moment produced by rolling, which is \( C_{n_p} \) together with aileron drag.

Answer: C

The yawing moment with yaw rate, \( C_{n_r} \), does most of the damping, supported by the side force derivative with sideslip. The fin supplies nearly all of it. Rolling moment with sideslip is the dihedral effect, which sets the roll content of the mode and generally makes the damping ratio worse. Aileron effectiveness is a control derivative. Yawing moment with sideslip is the stiffness term and sets the frequency rather than the damping.

Answer: D

In a yaw, the outer wing moves faster through the air and generates more lift, rolling the aeroplane into the turn. That term multiplies the fin's stiffness in the spiral criterion, which is why a large fin is spirally destabilising. The fin's side force does produce a rolling moment, but that is part of the dihedral effect term rather than this one. Aileron float is a stick-free control matter, and a stall is a large-disturbance phenomenon outside the linear derivative model.