When fins flutter

Air pushing on a fin twists it a little. The twist changes the angle at which the air meets the fin, so the push changes, and the fin bends. Below some speed the fin’s stiffness damps each wobble out. Above it, each cycle feeds the next, and the wobble grows until the fin breaks, sometimes within a second. That speed is the flutter speed.

What resists it is the fin’s stiffness against twisting, which comes from its material’s shear modulus and, far more, from its thickness. A fin that feels stiff when you bend it by hand can still be soft in twist.

Martin reduced the theory of a wing that bends and twists to a few numbers from the fin’s outline, and checked it against the flutter record of missiles and wind-tunnel models (The criterion). His flutter speed is

V_f = a · √(G_E / D)

with a the speed of sound, G_E the fin’s shear modulus, and D a denominator that holds the rest: the air’s pressure and the fin’s shape. Read in words:

• Thickness counts most. The flutter speed grows as the thickness ratio (thickness over root chord) to the power one and a half: a fin twice as thick flutters at nearly three times the speed.
• Stiffness counts less. It grows as the square root of the shear modulus: a material four times as stiff in shear doubles it.
• Shape. A fin with a longer span for its chord (a higher aspect ratio) flutters sooner. On a fin of one root chord, span and thickness, a narrower tip lowers the flutter speed too: it raises the aspect ratio by more than the narrower tip’s taper helps.
• Air. Thinner air raises the flutter speed, so the same fin can fly faster higher up.
• Where it comes closest. The air’s push at the flutter speed, the dynamic pressure there, is the same at every height. So a flight comes closest to its fins’ flutter speed where its own dynamic pressure peaks, at max q.

Martin’s constant comes from the theory, not a fit to data: it assumes the fin’s mass sits at mid-chord. His data decide only where his line falls, below.

The guides’ rocket, a small rocket for a first certification flight that the repository keeps for its how-to guides (its design file), has 3 fins of birch plywood, 3 mm (0.12 in) thick, with a 130 mm (5.12 in) root chord, a 60 mm (2.36 in) tip chord and a 75 mm (2.95 in) span. The design file’s plywood is OpenRocket’s, which HPR Sim doesn’t match to one of its materials, so this guide takes HPR Sim’s built-in birch plywood, which has a cited shear modulus, 0.75 GPa, from a plywood maker’s handbook. In Martin’s terms the fins have an aspect ratio of 0.79, a taper ratio of 0.46 and a thickness ratio of 2.3%. The criterion leaves out one more thing about them: their leading edges are swept back by 70 mm (2.76 in) over the span.

In the launch site’s air, standard air at sea level, the criterion puts their flutter speed at 175 m/s (574 ft/s), Mach 0.51. On the faster of its motors, the rocket reaches 286 m/s (938 ft/s), Mach 0.84. Max q comes 298 m (978 ft) above the pad, in air close to the launch site’s. At its max q, 48.5 kPa, the flutter speed is 0.62 times the airspeed there: by this criterion, the fins fly faster than their flutter speed. No measurement known to this project sizes the criterion’s error on fins like these, so this reading marks the fins for a closer look; it doesn’t say they will fail. For the flutter speed to reach the airspeed at max q, the plywood would have to be 4.2 mm (0.166 in) thick.

The hpr command doesn’t print these yet; in code, HPR Sim’s FlutterPanel gives the flutter speed and the ratio V_f / V at max q (A rocket’s fins has a worked example that CI runs).

The ratio V_f / V is the first reading. The second is Martin’s own check: where the fin plots on his chart against the line his data draw. His chart gives each wing one number, the denominator D over its shear modulus G_E: the higher it is, the less stiff the wing for the air it flies in. Wings that fluttered or failed lie above a band, and wings that flew to at least Mach 1.3 without known failure lie below it. The band runs at a ratio of 0.25 to 0.31, a flutter speed 1.8 to 2.0 times the speed of sound (The line in Martin’s data). So the band isn’t a speed to stay under: it is how stiff Martin’s unfailed wings were, all of them faster than sound.

The guides’ rocket’s fins plot at 3.79, above the band; to plot below it with the same outline, they would have to be 7.5 mm (0.296 in) thick. On the figure below, the same fact reads the other way up: the fins’ flutter speed sits below the band. That says less than it seems to: Martin’s wings below the band flew past the speed of sound, and the rocket tops out at Mach 0.84. His data show nothing about a fin above the band on a slower rocket: not that it is safe, and not that it will flutter.

Of the two, Martin’s chart is the stricter, and it is his own check: a fin below the band is among wings that flew faster than sound unharmed. A ratio V_f / V above one says only that the rocket stays below the criterion’s speed, whose error on hobby fins no measurement known to this project sizes. Neither is a line to fly by.

A flutter speed that reads high flatters: it says a fin is stiffer, or the rocket further from flutter, than it is. The figure draws the guides’ rocket’s flutter speed by thickness, with the best-known such error above it.

Apogee Components’ Peak of Flight newsletter printed Martin’s formula in issues #291 and #411 with its constant halved, so its flutter speed reads 41% high; issue #615 corrected it. A calculator or a spreadsheet that copied the printed form gives the higher speed, and Loft, HPR Sim’s predecessor, made the same mistake (L32, a lesson from Loft). On the guides’ rocket, the halved constant puts the guides’ rocket’s flutter speed at 247 m/s (810 ft/s), and the ratio at max q at 0.88.

The figure marks the halved constant’s line in the caution color. The test the_halved_constant_reads_a_higher_flutter_speed pins which way that error goes, on three fins in two kinds of air: the halved form’s flutter speed is the square root of two times HPR Sim’s, never below it, and its ratio V_f / V higher by the same factor. Only the constant differs, so the factor is the same on every fin in any air. The guide’s own test pins that the marked line is that speed.

Where Martin leaves a choice, HPR Sim takes the one that gives the lower flutter speed (Readings where the source leaves room):

• the thickness ratio at the root, the smallest on a fin of one thickness;
• a solid fin’s shear modulus as its material’s, not the flat plate’s reading, which is twice as stiff;
• Martin’s simpler taper factor, which lowers the speed for most tapers.

Two choices can go the other way: a fin with an airfoil section reads a few percent high, and a booster’s fins are held to the whole flight’s max q, without a check of the booster’s own after it separates. And the correction in #615 reads where the fin’s mass sits from the whole fin’s outline, not from Martin’s section; on a swept fin like the guides’ rocket’s, that puts the mass further aft and the flutter speed lower still. HPR Sim keeps Martin’s reading.

What the criterion leaves out may matter more than any of these. Martin’s wings were clamped at the root; a fin glued to a tube twists more there, so its real flutter speed can be lower, on the flattering side. Sweep, which the guides’ rocket’s fins have, is left out, and no test pins which way it moves the speed. So a flutter speed only a little above the top speed is not a margin anyone has measured.

• Fiberglass and printed fins. G10, the commonest fiberglass fin sheet, has no cited shear modulus in HPR Sim, and nor do the common printing plastics; pass your own from a measurement or your supplier (Shear moduli).
• How the fin is built and attached: fillets, tip-to-tip layups, through-the-wall tabs, and the body tube’s own bending.
• Speed effects near the speed of sound, where flutter speeds can dip, and stall flutter at large angles of attack.
• A rule for how much margin is enough. That is the safety code’s, the RSO’s and the fin maker’s, not this guide’s.