Aerodynamic Derivatives Questions and Answers

Answer: A

Non-dimensional form collapses the whole dynamic-pressure dependence into one table entry per Mach number, so a single set covers many speeds and altitudes. The equations of motion actually need the dimensional forms, which is why the conversion is always the next step. Inertias are still needed for that conversion. And dimensional derivatives are perfectly measurable; they are simply less portable between conditions.

Answer: B

\( C_{m_u} \) changes sign transonically as the aerodynamic centre moves aft, which is what produces Mach tuck and the need for Mach trim. Roll damping and side force with sideslip change in magnitude but keep their restoring signs. The lift curve slope rises towards \( M = 1 \) and falls beyond it, again without changing sign. A sign change is far more consequential than a magnitude change, because it can move a root across the imaginary axis.

Answer: C

A rotary balance spins the model about an axis aligned with the flow, which is how the derivatives that matter in a steady spin are obtained, at incidences far beyond the linear range. Static sweeps use a fixed sting. Hinge moments need an instrumented control surface. Structural stiffness is measured on the ground. Spin prediction is one of the places where a conventional derivative set is least adequate, which is exactly why the specialised rig exists.

Answer: D

Static margin fixes the stiffness, but damping scales with the square of the arm, so the longer-armed aeroplane is better damped at the same frequency. That is why two aeroplanes can share a static margin and feel quite different, and why the short period is specified with both a frequency and a damping ratio rather than a single number. The last option is the misconception the comparison is designed to break.

Answer: A

A workable longitudinal set runs to eight or ten: the speed, incidence, pitch rate and incidence-rate derivatives of axial force, normal force and pitching moment, plus the elevator control derivatives. That is few enough to tabulate and enough to reproduce both modes accurately. Two would not even fill the short-period approximation. Fifty overstates it for the rigid longitudinal problem, though a full six-degree-of-freedom model with configuration variations does run to hundreds of table entries.

Answer: B

The pitching moment derivative with incidence is written \( C_{m_\alpha} \), read as the derivative of the pitching moment coefficient with respect to angle of attack, and it is the static longitudinal stability derivative. \( C_{L_\alpha} \) is the lift curve slope, a force derivative on the same independent variable. \( C_{n_\beta} \) is the yawing moment with sideslip, the directional stiffness. \( C_{l_p} \) is roll damping. The convention is consistent throughout: the main subscript names the coefficient, the second names the variable differentiated against.

Answer: C

Rolling moments use the span, as yawing moments do, since it is the natural lateral reference length; pitching moments use the mean aerodynamic chord instead. The tail arm appears inside the estimating formulae for particular contributions but is never the reference for a whole-aeroplane coefficient. Fuselage length is not used as a reference at all. Because span and chord can differ by a factor of ten, using the wrong one produces an error far larger than any modelling uncertainty.

Answer: D

The wing supplies roll damping, because a roll rate changes the local incidence in proportion to the distance from the centreline, and the resulting lift difference is greatest at the tips where the moment arm is longest. That is why damping scales with the square of the span. The fin contributes a small amount through its own side force above the roll axis, the tailplane a little through its span, and the fuselage almost nothing.

Answer: A

The equivalent is \( C_{n_\beta} \): it provides the restoring moment in yaw just as \( C_{m_\alpha} \) does in pitch, and both must have the sign that turns the aeroplane back towards its trimmed condition. Rolling moment with sideslip is the dihedral effect, which is a lateral rather than a directional property. Yawing moment with yaw rate is the damping term. Side force with sideslip is a force rather than a moment, and contributes to Dutch roll damping rather than stiffness.

Answer: B

It indicates sideslip, so \( C_{y_\beta} \), \( C_{l_\beta} \) and \( C_{n_\beta} \) are the side force, rolling moment and yawing moment derivatives with sideslip - the three that govern the aeroplane's response to being pushed sideways. Incidence is \( \alpha \), roll rate is \( p \), and a control angle carries the surface's own symbol such as \( \delta_e \). Fluency in this notation is a prerequisite for reading any data sheet.