Fin flutter

In short

• What it models: whether a fin may flutter, meaning its bending and twisting feed each other until the fin shakes itself apart. hpr gives two readings of one criterion. The first is Martin’s own chart check: is the fin on the flutter side of the line his flight data draws? The second is the ratio of the criterion’s flutter speed to the rocket’s airspeed at the flight’s peak dynamic pressure. The first decides; the second alone is not a safe line.
• Sources: D. J. Martin’s criterion in NACA TN 4197 (1958), eq. 18, and his figure 3, which separates missile and wind-tunnel wings that fluttered from those that didn’t (References). The fin’s stiffness enters as its shear modulus; hpr has one, with its source, for 14 of its built-in materials.
• How well it is validated: hpr reproduces Martin’s formula and both of his worked examples, including his verdicts on three metals and his titanium design (Tests). His safe/unsafe line is a band that hpr measured off his printed chart: 0.25 to 0.31 (The line in Martin’s data). His unfailed wings flew to at least Mach 1.3; nothing here is checked against a hobby rocket. Where the source allows two readings, hpr takes the one with the lower flutter speed. That doesn’t make the whole result conservative: nothing shows that it is.
• What it leaves out: sweep, and how the fin is mounted: Martin’s wings were clamped at the root, and a fin glued to a tube is less stiff there. Also left out are stall flutter at high angles of attack, Mach-number effects such as a dip near Mach 1, the rocket body’s own bending, and fins that aren’t trapezoids. G10/FR-4, the commonest fibreglass fin sheet, has no built-in modulus (Shear moduli).

What flutter is

Air pushing on a fin twists it a little. The twist changes the fin’s angle to the air, so the push changes, and the fin bends. Below a certain speed the fin’s stiffness damps this out. Above it, each cycle feeds the next, and a fin can fail within a second. The speed depends on the fin’s shape and thickness, how stiff its material is in shear, and the air it flies through.

The criterion

Martin starts from Theodorsen and Garrick’s flutter speed for a wing that bends and twists (his eq. 1, from their NACA Report 685 of 1940). He reduces it to a few numbers from one fin’s outline. For a trapezoidal fin with root chord c_r, tip chord c_t, span s and thickness t:

Eq. 18 gives a flutter speed V_f through a denominator D, in pascals:

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

Martin prints the constant in pounds per square inch (psi) at sea-level pressure p₀: 24 · 0.25 · 1.4 / π · 14.696 psi = 39.29 psi, which he rounds to 39.3. His X (eq. 19) is 39.3 K psi, and D = X · (λ + 1)/2 · p/p₀. The constant is derived, not fitted. Stiffer or thicker fins flutter faster: V_f grows as √G_E and as (t/c)^1.5. At a fixed speed of sound, thinner air raises it as 1/√p.

A flutter dynamic pressure. The air enters only through ρ a² = γ p, and the dynamic pressure is q = ½ ρ V². So the criterion fixes a dynamic pressure at V_f, the same at every height:

q_f = π G_E / (24 ε K (λ + 1))

A fin flying at dynamic pressure q is below V_f by the ratio V_f / V = √(q_f / q). That makes the flight’s least ratio the one at its peak dynamic pressure, “max q”, which Flight metrics already finds between the integrator’s steps.

The line in Martin’s data

Eq. 18’s V_f is not the speed at which a fin is known to flutter. Martin plots D against G_E for missiles and wind-tunnel models (his figure 3). Wings that fluttered or failed lie above a shaded band, and wings that flew to at least Mach 1.3 without known failure lie below it.

hpr measured the band on a 250 dots-per-inch scan of the figure. Both log axes were calibrated on their tick marks, and the band’s edges were traced in 69 pixel columns from G_E = 0.05 to 10 × 10⁶ psi. The band runs at D / G_E = 0.25 to 0.31 all along that range, from wood to steel. In eq. 18’s terms that is (V_f/a)² = 3.2 to 4.0, so Martin’s line sits where V_f is 1.8 to 2.0 times the speed of sound. hpr calls this ratio D / G_E the figure 3 ratio:

Martin takes p where the wing flies. At the launch site’s pressure, the highest a flight sees, the ratio is at its largest. Figure 3’s axis runs from 0.05 to 20 × 10⁶ psi (0.34 to 138 GPa) in G_E; balsa’s modulus is left of it, so its ratio is an extrapolation.

Which reading decides. The figure 3 ratio does: it is Martin’s own check, and a fin must be below the band. V_f / V alone is not a safe line, at 1 or at any fixed number. At the flight’s max q the two are tied by D / G_E = 1 / (M · V_f/V)², with M the Mach number there, so the band is at V_f / V from about 1.8/M to 2.0/M. A fin is below the band only if V_f / V is above about 2.0/M: 1.5 at Mach 1.3, but 4.0 at Mach 0.5. Martin’s data show nothing about a fin above the band on a rocket slower than Mach 1.3: treat it as not shown to be safe.

Loft’s mistake. Loft, hpr’s predecessor, wrote the constant as 1.337 · (λ + 1)/2 per psi, half of 39.3 / 14.696 = 2.674. So its flutter speeds were √2 too high, about 41%, on the unsafe side (L32, a lesson from Loft).

Martin’s worked examples

Martin gives two examples (pp. 6–7). The first is a wing with A = 2, 4% thick, untapered and ground-launched. He reads its X off his figure 4 as “about 1.25 × 10⁶ psi” and judges the wing by material. The second picks titanium and holds figure 3’s ordinate to 0.8 × 10⁶ psi, well under titanium’s modulus, “to allow a reasonable margin of safety”. It then asks how thick the wing must be at each aspect ratio.

Martin read these off a log-scale chart and printed them on grids of 0.05 × 10⁶ psi and half a percent (his thicknesses all end in .5). Each of hpr’s values rounds to his on that grid.

His verdicts on the first wing are the margin half of the example; the titanium row is his second example, at the ordinate he chose. hpr checks them with the moduli Martin marks on figure 3’s axis, each a small box read off the same scan:

A rocket’s fins

The example program fin_flutter (in crates/hpr/examples/) takes the repository’s synthetic 54 mm rocket. Its three fins are 3.2 mm thick, with a 150 mm root, a 60 mm tip and a 70 mm span. It flies the rocket on an I175 motor from a site 200 m up, and prints:

Panel: aspect ratio 0.667, taper ratio 0.400, thickness ratio 0.0212
Max q: 72455 Pa at 2.09 s, 432 m above the pad
Top speed: 355 m/s at 2.10 s; top Mach number: 1.05 at 2.10 s
Martin's band (figure 3): D/G_E from 0.25 to 0.31

material            G (GPa)   q_f (kPa)   V_f 0 m (m/s)   V_f 3 km (m/s)   V_f/V at max q   D/G_E
aluminum_6061        26.200       836.3            1169             1356             3.40   0.083
carbon_fiber          4.826       154.1             502              582             1.46   0.450
birch_plywood         0.750        23.9             198              229             0.57   2.893
basswood              0.511        16.3             163              189             0.47   4.246
balsa                 0.138         4.4              85               99             0.25  15.680

The columns V_f 0 m and V_f 3 km are eq. 18’s flutter speed in standard air at sea level and 3 km above it. D/G_E is the figure 3 ratio at the launch site’s pressure.

• Aluminium passes both readings: its figure 3 ratio is well below the band, and at max q the rocket flies at under a third of V_f.
• Carbon fibre at 3.2 mm has V_f / V of 1.46, but its figure 3 ratio, 0.45, is above the band, where most of Martin’s wings fluttered or failed: not shown to be safe. Its modulus is a unidirectional ply’s of one aerospace prepreg; a ±45° layup of it would be stiffer, but wet-laid or woven hobby sheet may be softer.
• Plywood, basswood and balsa fins of this size fail both: at max q the rocket flies at 1.75 times plywood’s V_f.

CI checks that the program still prints exactly this.

In code, with a FlutterPanel (its API page has a worked example that CI runs):

// `fin_set` is a design's `FinSet`; `summary` a flight's `FlightSummary` (Flight metrics).
let panel = FlutterPanel::of_fins(&fin_set)?;
let g = materials::shear_modulus_of(&fin_set.material).unwrap().shear_modulus_pa;
let chart = panel.figure_3_ratio(g, launch_pressure_pa)?; // against FIGURE_3_BAND
let margin = panel.margin(g, &summary)?; // V_f/V at max q; `None` if it never flew

FlutterPanel::new takes A, λ and t/c directly. Martin’s figure 4 covers A from 0.5 to 3 and t/c from 1% to 10%; the example’s 0.667 is inside, and outside that range the numbers are an extrapolation.

Readings where the source leaves room

Most of these give the lower of the flutter speeds the source allows. Two can go the other way: an airfoiled fin’s modulus, and a booster’s max q.

• Thickness ratio at the root. Martin’s wings keep one thickness ratio from root to tip. A hobby fin keeps one thickness, so its ratio grows toward the tip. hpr takes the root’s, the smallest.
• A solid fin’s G_E is its material’s G. Martin says a solid wing of aluminium plots at aluminium’s modulus (p. 6), and hpr does the same. His definition, G_E = 6 J G / (c t³) (eq. 12), assumes a thin airfoil’s torsion constant J ≈ c t³/6 (eq. 10). A flat plate’s is c t³/3, which would double G_E and raise the flutter speed by √2. hpr keeps the lower reading. The exception runs the other way: for a fin with an airfoil section, eq. 12 gives 0.946 G, so hpr’s V_f for such a fin is up to 2.7% high.
• Martin’s taper factor. His derivation carries a factor 1/(f₁² f₂²). Here f₁ corrects the twisting frequency for taper (his eq. 8) and f₂ gives the chord three-quarters of the way out (his eq. 14). He replaces the factor with (λ + 1)/2. The two agree at λ = 1 and are 3% apart at λ = 0. Between, (λ + 1)/2 is up to 47% larger (at λ ≈ 0.31), which lowers V_f there by up to 17.5%. hpr keeps his form, since his figure 3 was drawn with it.
• Every fin set sees the whole flight’s max q. A booster’s fins leave at the separation, so the flight’s max q can come after they are gone. Their true V_f / V is then at least the one given, as long as the booster’s own dynamic pressure after the separation stays below the flight’s peak. hpr doesn’t check that.

Shear moduli

The criterion needs the fin’s shear modulus in its own plane. hpr_design::materials::SHEAR_MODULI gives it for 14 built-in materials, each with its source, page and the web address it was read from, and shear_modulus_of finds a design’s copy of a built-in material by its name and density. Where a source gives a range, the lower value is kept. Metals are given in ksi (1000 psi) and Msi (10⁶ psi); 1 Msi is 6.895 GPa.

How the derived values are made:

• Woods: G_LT is the shear modulus in the plane along the grain and along the growth rings. hpr takes the Wood Handbook’s ratio G_LT / E_L (Table 5-1) and multiplies it by the wood’s bending modulus at 12% moisture raised by 10%, which is how the table’s footnote says to estimate the stiffness along the grain, E_L. G_LT is the smaller of the two in-plane ratios for every wood listed. Eastern white pine is not in Table 5-1, so it has none. Martin marks solid wood at 0.070 to 0.120 × 10⁶ psi (0.48 to 0.83 GPa); the handbook’s birch and oak are above that.
• Plastics: an unfilled plastic taken as isotropic (the same in every direction), from its data sheet’s tensile modulus E and Poisson’s ratio ν. Nylon soaks up water. Its conditioned value, measured after it has, is less than half the dry one.
• Carbon fibre: a unidirectional ply’s in-plane G₁₂. That is a 0/90 laminate’s in-plane shear modulus. Plies at ±45° raise it several times, so for such a laminate this reads low.

No source found states a shear modulus for G10/FR-4, the commonest fibreglass fin sheet: its data sheets give flexural moduli only. There is none for PLA, ABS, PETG, polycarbonate or acrylic either. For those, pass your own G_E from a measurement or your supplier. A woven glass/epoxy laminate is not isotropic, so E / (2(1 + ν)) doesn’t apply to it.

Tests

In crates/hpr-sim/src/flutter.rs, unless named otherwise:

References

• 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. 1 to 19, pp. 11–15; examples, pp. 6–7; figures 3 and 4, p. 19.
• T. Theodorsen and I. E. Garrick, Mechanism of Flutter: A Theoretical and Experimental Investigation of the Flutter Problem, NACA Report 685, 1940. Martin’s ref. 6, the source of his eq. 1; not read for hpr.
• MIL-HDBK-5J, Metallic Materials and Elements for Aerospace Vehicle Structures, 2003.
• Forest Products Laboratory, Wood Handbook, FPL-GTR-190, 2010, Tables 5-1, 5-3a and 5-5a.
• NCAMP, Hexcel 8552 AS4 Unidirectional Prepreg Qualification Statistical Analysis Report, NCP-RP-2010-008 Rev D, 2011, Table 3-3.
• Riga Wood, Plywood Handbook, 2022, Table 4.11.
• Celanese, Zytel 101L NC010 data sheet, 2023; Delrin 100P NC010 data sheet.

The decision behind these choices is ADR-078, from the M1.10b milestone.