Aerodynamic Derivatives Questions and Answers

Answer: C

Dimensional moment derivatives are directly proportional to dynamic pressure, so halving the dynamic pressure halves \( M_\alpha \) and lowers the short-period frequency by a factor of about \( \sqrt{2} \). The non-dimensional coefficient is unchanged at fixed Mach number, which is the whole convenience of the non-dimensional form. A quarter would follow if the relationship were quadratic, and twice inverts the proportionality altogether.

Answer: D

Both forms are dimensionless, so the factor of two is convention rather than necessity, arising from the use of a semi-chord or semi-span as the natural reference length in unsteady aerodynamics. What matters is consistency: a derivative quoted under one convention and used under another is out by a factor of two. The dynamic pressure factor is separate and already accounted for. And nothing here touches the choice of angular unit.

Answer: A

Oscillating the model and measuring the phase and amplitude of the response is how rate derivatives are obtained, since a static test cannot produce a pitch or yaw rate at all. Static derivatives come from a fixed-attitude sweep, which is much simpler and older. Control derivatives are measured by deflecting the surface at fixed attitude. Inertias are mass properties, measured by swinging the aeroplane or computing from the mass breakdown.

Answer: B

The extra divisor is a reference length: mean aerodynamic chord for pitching moments and span for rolling and yawing moments. Dynamic pressure and area appear in both force and moment coefficients, so neither is the difference. Density is already inside dynamic pressure. Mach number is dimensionless and cannot fix a dimensional imbalance. Using chord where span belongs is a common error, and it changes a lateral derivative by the ratio of span to chord, which can be a factor of four or more.

Answer: C

The stiffness term \( M_\alpha \) is much the largest single influence on the short period, and the three damping contributions are individually smaller and comparable with one another, with pitch damping the biggest of them. The downwash lag is usually the smallest of the three, contributing a tenth to a third of the total. Drag with incidence barely enters the short period. And elevator power governs response rather than the free frequency.

Answer: D

The subscript \( u \) denotes the forward speed perturbation, so \( C_{m_u} \) is the pitching moment change per unit speed change and is the derivative behind Mach tuck. Incidence is written \( \alpha \), sideslip \( \beta \) and pitch rate \( q \). The notation is worth learning as a set because it is used without explanation in every text and data sheet, and reading \( u \) as incidence would misattribute a speed-stability problem to the short period.

Answer: A

Stability axes are the usual choice: body axes rotated so that the x-axis lies along the relative wind in the trim condition, which makes the trim incidence zero and simplifies the equations. Earth axes are the reference for navigation rather than for aerodynamics. Axes that follow every gust would not give a fixed frame to linearise in. Principal axes remove the product-of-inertia terms and are sometimes used, but they are a separate choice and rarely coincide with stability axes.

Answer: B

Rudder side force matters for balancing an asymmetric thrust condition and for steady sideslip manoeuvres, both of which are trim problems rather than modal ones, and it feeds directly into the minimum control speed calculation. The short period is longitudinal. Roll performance depends on aileron effectiveness against roll damping. Longitudinal trim with speed involves the elevator and the speed derivatives, not the rudder.

Answer: C

The manoeuvre margin exceeds the static margin because a steady pull involves a pitch rate, and the pitch damping derivative adds a stabilising increment that a straight-and-level incidence change does not have. Pitching moment with incidence gives the static margin, which is the baseline the manoeuvre margin is compared with. Elevator lift is a small direct-lift term. Drag with speed belongs to the phugoid and to performance.

Answer: D

Tail volume coefficient is \( S_t l_t / S\bar c \) for the horizontal surface, and \( S_v l_v / S b \) for the vertical, a dimensionless measure of how much moment-generating capability the tail has relative to the wing. It is the single most useful number in tail sizing, since pitch stiffness and directional stiffness both scale with it. Thickness affects drag rather than moment capability, and dynamic pressure is not part of a geometric coefficient.