Module flutter 

Fin flutter: the speed at which a fin’s bending and twisting couple and grow, from D. J. Martin’s criterion (the fin-flutter milestone; the guide’s Fin flutter page).

Source. D. J. Martin, Summary of Flutter Experiences as a Guide to the Preliminary Design of Lifting Surfaces on Missiles, NACA TN 4197, 1958, appendix, eqs. 16 to 19, pp. 14–15, and figure 3, p. 19. Martin reduces Theodorsen and Garrick’s flutter speed for a bending-torsion wing (eq. 1) to a few planform numbers. With G_E the fin’s effective shear modulus, A the panel aspect ratio (span over mid-span chord), λ the taper ratio (tip over root chord), t/c the thickness ratio, p the static pressure and a the speed of sound, eq. 16 with his 1/(f₁² f₂²) ≈ (λ + 1)/2 reads

(V_f / a)² = G_E / D,   D = (24 ε / π) ρ a² · A³ / ((t/c)³ (A + 2)) · (λ + 1)/2

and with ρ a² = γ p (eq. 17), ε = 0.25 and γ = 1.4 it becomes eq. 18, whose constant 24 · 0.25 · 1.4 / π · 14.696 psi = 39.29 psi Martin prints as 39.3:

(V_f / a)² = G_E / (39.3 A³ / ((t/c)³ (A + 2)) · (λ + 1)/2 · p/p₀)

The constant is derived, not fitted. What is empirical is the aspect-ratio correction A/(A + 2), the best of those Martin tried, and where the line falls between wings that fluttered and wings that didn’t: his figure 3 plots D against G_E for missiles and wind-tunnel models, and a band separates them at D/G_E from 0.25 to 0.31 (FIGURE_3_BAND), a flutter speed of 1.8 to 2.0 times the speed of sound. His open points are wings that flew to at least Mach 1.3 without failing. So eq. 18’s V_f is a parameter his data calibrates, not a speed at which a fin is known to flutter.

Loft wrote the constant as 1.337 · (λ + 1)/2 psi, half of 39.3/14.696 = 2.674, so its flutter speed was √2 too high, on the unsafe side (Loft lesson L32).

A flutter dynamic pressure. Since ρ a² = 2q/M² for any gas, eq. 16 fixes the dynamic pressure at flutter, whatever the height:

q_f = ½ ρ V_f² = π G_E / (24 ε K (λ + 1)),   K = A³ / ((t/c)³ (A + 2))

and a fin flying at dynamic pressure q is below eq. 18’s flutter speed by the ratio V_f / V = √(q_f / q). The least ratio of a flight is therefore at its peak dynamic pressure, which crate::metrics::FlightMetrics finds on the dense output.

Readings. A trapezoidal fin’s A is 2s/(c_r + c_t) (span s over the mid-span chord) and t/c is the thickness over the root chord: Martin’s c is the root chord of his constant-thickness-ratio wing, and for a flat fin of constant thickness the root’s ratio is the smallest, so the flutter speed the least. G_E is Martin’s effective shear modulus: for a solid wing he takes the material’s own (his p. 6), which this module does. His definition, G_E = 6 J G / (c t³) (eq. 12), would give a flat plate, whose torsion constant is J = c t³/3, about twice that, and a flutter speed √2 higher; the lower reading is kept. For a NACA four-digit section it gives 0.946 G, so an airfoiled fin’s V_f here is up to 2.7% high.

Left out. Sweep, the fin’s mounting and the body’s own modes (Martin’s figure 8), stall flutter at high angles of attack, Mach number effects such as a transonic dip, and every other flutter type Martin lists. It is a screening number, not a flutter analysis.

Structs

FlutterMargin

How far below eq. 18’s flutter speed a fin flew, at the flight’s peak dynamic pressure.

FlutterPanel

The planform numbers Martin’s criterion takes from one fin.

Constants

CG_AFT_OF_QUARTER_CHORD

Martin’s ε: how far the section’s center of mass sits behind its quarter chord, as a fraction of the chord, assumed 0.25, which puts it at mid-chord (NACA TN 4197, p. 14).

FIGURE_3_BAND

Where Martin’s figure 3 separates wings that fluttered from wings that didn’t, as D/G_E, that is (a/V_f)²: its shaded band runs from 0.25 to 0.31.

HEAT_CAPACITY_RATIO

Martin’s ratio of specific heats for air, 1.4 (NACA TN 4197, eq. 17, p. 14).