Dynamic Lateral Directional Modes Questions and Answers

Answer: A

Yawing inertia divides the yawing moment derivatives, so it sets the Dutch roll frequency through \( N_\beta = C_{n_\beta}\bar q S b/I_{zz} \) and its damping through the same division. A long, heavily loaded fuselage therefore wags more slowly. Roll subsidence depends on rolling inertia rather than yawing inertia. The spiral criterion is a ratio of products in which the inertias largely cancel, so it is far less sensitive. Saying none of them ignores the definition of every dimensional derivative in the problem.

Answer: B

A rudder doublet puts energy straight into sideslip, the state that drives the mode, and returning the pedals to centre leaves the aeroplane to oscillate freely. A slow aileron input excites the roll and spiral modes instead and gives a record dominated by bank angle. An elevator pull is longitudinal and would excite the short period. A throttle change excites the phugoid. Choosing the input to match the mode is the whole art of getting a clean modal record.

Answer: C

From the logarithmic decrement:

\[ \begin{aligned} \delta &= \ln(6.0/4.5) = 0.288 \\ \zeta &= \frac{\delta}{\sqrt{4\pi^{2}+\delta^{2}}} = \frac{0.288}{6.29} = 0.046 \end{aligned} \]

so about 0.046, a poorly damped mode that would fail most Level 1 requirements and calls for a yaw damper. 0.29 is the decrement itself, which differs from the damping ratio by a factor of about \( 2\pi \). 0.75 is the ratio of the peaks. 0.25 does not follow from the arithmetic at all.

Answer: D

The damping ratio is exactly at its limit, but the product is \( 0.08 \times 1.5 = 0.12 \ \mathrm{rad/s} \), short of the 0.15 required, so the aeroplane meets one criterion and fails the other. This is exactly why specifications state both: a low frequency mode can satisfy a damping ratio while still taking too long in absolute time to decay, and it is the product that fixes the decay rate. Raising the damping ratio to 0.10 would clear it.

Answer: A

A four-second lateral oscillation is the Dutch roll, and the rear cabin feels it worst because the yaw component gives the largest lateral displacement furthest from the centre of gravity. That asymmetry between cabin and cockpit is the classic reported signature. A spiral divergence does not oscillate. Roll subsidence is aperiodic and would not repeat. The short period is longitudinal and would be felt as pitching, not as side-to-side motion.

Answer: B

The bare airframe's Dutch roll damping is what the damper was supplying, so the crew sees a lightly damped yaw and roll oscillation that persists after every gust, usually with a procedural requirement to descend or reduce speed. The spiral is a separate root and is not held by the damper. Roll control is unaffected, since the ailerons are independent of it. The short period is longitudinal and cannot be influenced by a rudder loop at all.

Answer: C

It augments yaw damping, \( N_r \), because the rudder moment is proportional to and opposes the measured rate. The damping ratio rises while the Dutch roll frequency, set by fin stiffness, is barely touched. Directional stiffness would need feedback of sideslip rather than of yaw rate. Dihedral effect is a rolling moment with sideslip, so augmenting it means feeding sideslip to the ailerons. Roll damping is augmented by feeding roll rate to the ailerons, which is a roll damper.

Answer: D

Both are true and the reason explains the assertion. Feedback proportional to a displacement adds stiffness; feedback proportional to a rate adds damping. So sideslip to the rudder behaves like a bigger fin and raises \( \omega_n \), while yaw rate to the rudder behaves like more \( N_r \) and raises \( \zeta \). The same distinction runs through the longitudinal case, where incidence feedback adds pitch stiffness and pitch rate feedback adds pitch damping, and it is the quickest way to reason about any augmentation scheme.

Answer: A

The chain runs 1, 2, 3, 4, 5. The yaw produces sideslip; the fin turns the nose back; the same sideslip rolls the aeroplane through dihedral effect; the nose overshoots the trimmed heading because the fin's restoring moment is not damped out; and the sideslip reverses to start the other half of the cycle. Putting the fin's moment before the sideslip inverts cause and effect, since the fin can only respond to sideslip once it exists.

Answer: B

A wing leveller closes a feedback loop on bank angle, suppressing the spiral without touching the aerodynamics that set the Dutch roll, which is why it is the standard fix on light aeroplanes. A larger fin makes the spiral criterion worse, not better. A yaw damper addresses the Dutch roll and does little for the spiral, since the spiral involves almost no yaw rate to feed back. Reducing roll damping would degrade roll response and has no useful effect on the criterion.