Aerodynamic Derivatives Questions and Answers

Answer: A

The factor is \( 1 - 0.35 = 0.65 \), so the tail delivers sixty-five per cent of what its area and arm alone would suggest. That is why tail sizing is done with the downwash factor included from the outset rather than as a correction. 0.35 is the gradient itself, which is the fraction lost rather than retained. 1.35 adds where the physics subtracts. 0.88 does not follow from the expression at all.

Answer: B

Ackeret theory gives \( 4/\sqrt{M^{2}-1} \) for a thin section in supersonic flow, so the slope falls as Mach number rises well above unity. The second expression is the subsonic Prandtl-Glauert form, which diverges at \( M = 1 \) and is invalid above it. A Mach-independent \( 2\pi \) is the incompressible result. Sweep matters supersonically too, but it modifies the result rather than replacing the Mach dependence.

Answer: C

Aeroelastic deformation is the effect in question: under load the wing and tail twist in a direction that reduces the incidence change they achieve, so the real derivatives are softer than the rigid ones and the shortfall grows with dynamic pressure. Density is already accounted for by the dynamic pressure in the conversion. Reynolds number matters, especially near the stall, but it is a separate correction. Landing gear is a configuration change rather than a rigid-to-flexible one.

Answer: D

Stability derivatives sit in the system matrix, so they and the inertias fix the eigenvalues and hence the modes. Control derivatives populate the input matrix and decide how strongly a given control excites those modes. The output matrix simply selects which states are measured. The initial condition is the disturbance, which the derivatives act upon rather than form. Keeping the two matrices distinct is what makes it obvious why a bigger elevator changes response without changing the modes.

Answer: A

Frequency comes from the dimensional derivatives, and each moment derivative is divided by an inertia, so doubling the inertias lowers the frequencies by a factor of about the square root of two. Matching non-dimensional sets does not imply matching dynamics, which is exactly why a scaled model must match relative density and Froude number to be dynamically similar. Extra mass slows rotation rather than speeding it. Control derivatives affect the response, not the modes.

Answer: B

When a derivative has little influence on the recorded response, the fit cannot determine it, so common practice is to hold it at its analytical or tunnel value and fit the influential ones. The alternative is a set that matches the data with physically implausible values traded against one another. There is no need to abandon anything, certification does not turn on a single derivative, and forcing a real derivative to zero would introduce an error rather than avoid one.

Answer: C

Short-period frequency is set by pitch stiffness, so an under-predicted \( C_{m_\alpha} \) is the first thing to check, along with the pitching inertia that divides it. That the phugoid matches is informative: the speed derivatives must be about right, which points away from the drag term. The two lateral derivatives cannot affect a longitudinal mode at all, since the problems decouple. Reasoning from symptom to derivative like this is the core skill in model validation.

Answer: D

Both statements are true and the reason is why. Prediction has to model flexibility, engine installation effects and every geometric detail, and each of those is a source of error; the flight-identified set contains them all by construction. That is why training simulators are validated against flight test data within tolerance bands, and why a predicted set is used only until flight data exists. The predicted set remains essential before first flight, when nothing else is available.

Answer: A

Extending the flaps alters the aerodynamic shape, so the coefficients themselves change. Climbing at constant Mach number changes dynamic pressure and therefore the dimensional derivatives while leaving the non-dimensional ones nearly alone. Burning fuel changes mass and inertia, not aerodynamics. And the last option is self-contradictory as written, since at constant altitude an acceleration changes Mach number, but taken as a dynamic-pressure change it again affects only the dimensional set.

Answer: B

The first and third are the two central assumptions: small perturbations, and quasi-steady forces. The second is false, and importantly so - a derivative set is valid only near the condition at which it was evaluated, which is why flight simulation carries tables interpolated over Mach number, incidence and configuration. Believing otherwise is the single most consequential misunderstanding of the method, because it leads to a linear model being trusted far outside its range.