Stability in flight

A rocket’s weight acts at its center of gravity (CG), the point it balances on. When a gust or a tilted rail turns it so that the air meets it at an angle, the angle of attack, the air pushes on its side. That push, the normal force, acts at the center of pressure (CP). If the CP is behind the CG, the push turns the nose back into the oncoming air, as a weathervane turns; if it is ahead, the push turns the rocket further off, and it tumbles.

The stability margin is the distance from the CG back to the CP, in body diameters, called calibers. Each part of the rocket pushes on its own: the nose near the front, the fins at the back. The CP is the average of the parts’ stations, each weighted by how hard that part pushes per unit of angle, its normal-force slope C_Nα:

x_CP   = Σ (C_Nα,i · x_i) / Σ C_Nα,i
margin = (x_CP − x_CG) / d

with x_i each part’s own center of pressure, measured back from the nose tip, and d the body’s diameter. So anything that makes one part push harder than the others moves the CP toward that part, and so does anything that moves a part’s own center of pressure. The margin HPR Sim prints as static margin is the one this gives at low speed, with the air straight along the rocket: the number Barrowman’s method gives by hand. Check stability for a certification flight reads it.

Below the speed of sound, air squeezes a little as it flows past, more the faster the rocket goes. In HPR Sim’s model each fin pushes harder for the same angle as the speed rises, through the Prandtl–Glauert factor, the classic correction for that squeezing; on fins as short as most rockets’ the gain is well below the factor itself. The nose’s push, in the slender-body theory that sets it, doesn’t change with speed. So in the model, as a rocket with fins at its tail speeds up, the fins gain on the nose, and the CP moves aft: the margin grows. The wind tunnel below doesn’t bear this out on the rocket it tested.

Faster than sound, the air can’t move aside ahead of a fin. HPR Sim uses supersonic linear theory there, where a fin’s push falls as the speed climbs, while a smooth body’s push grows as shocks form along it. No closed-form method covers the speeds between, so HPR Sim joins its subsonic and supersonic fin methods with a straight line. This guide calls that stretch the join (Fins through the speed of sound); it starts just below the speed of sound and ends where supersonic theory starts for the fins, which depends on their shape. Across the join each fin’s own center of pressure also slides aft, from a quarter of the way along its chord toward the middle. Once supersonic theory takes over, the CP turns and moves forward: the margin shrinks.

The guides’ rocket, a small rocket for a first certification flight that the repository keeps for its how-to guides (its design file), shows it, as HPR Sim computes it: no measurement of this rocket exists. Straight into the wind, its center of pressure sits 13.99 calibers behind its nose tip at low speed. By Mach 0.8 it has moved 0.19 calibers aft, by Mach 1.2, 1.04 calibers aft, and by Mach 3 it sits 1.88 calibers forward of where it started (report). Supersonic linear theory starts for this rocket’s fins at Mach 1.37; the speeds just below that are in the join, where the tunnel below finds HPR Sim’s center of pressure too far aft. The CG doesn’t depend on speed; it moves only as the motor burns, which the flight margin follows. So each caliber the CP moves is a caliber of margin.

So the margin printed at low speed is not the margin the rocket has at its top speed. hpr sim --json also gives the flight margin: the small-angle margin at the flight’s own Mach number and CG at each moment, and its least value from the rail to apogee (min_flight_margin_cal, Flight metrics).

NASA tested half-scale models of the Arcas Robin, a sounding rocket with a long nose and swept fins, in wind tunnels from Mach 0.6 to 4.63 (comparison). From Mach 0.6 to 0.8 the tunnel’s center of pressure moved forward by 0.69 calibers, while HPR Sim’s moved aft by 0.10. From Mach 0.8 to 1.2 the tunnel’s center of pressure moved aft by 2.60 calibers, then forward by 5.23 calibers by Mach 4.63: from there on, the shape the section above describes, larger. The figure sets HPR Sim’s center of pressure against the tunnel’s; its axis doesn’t start at the nose tip.

Each of the tunnel’s points was read off the reports’ plots to within 0.05 to 0.11 calibers either way, the band drawn around its dot. At Mach 0.6, HPR Sim’s sat 0.12 calibers aft of the tunnel’s, just outside that band. At Mach 0.8, still on its subsonic fin method, HPR Sim’s sat 0.91 calibers aft of the tunnel’s. In the join it didn’t follow either: it sat up to 1.35 calibers aft, at Mach 0.9, where the tunnel’s center of pressure moved aft later and more steeply than the straight line between Mach 0.8 and 1.2 puts it. And from Mach 1.2 up, it stayed within 0.35 calibers of it. On the model drawn, 9 of the 12 speeds are within 0.50 calibers, the target the comparison set before measuring. On the longer model the tunnel also tested, it sat up to 2.37 calibers aft, at Mach 1, and 5 of its 10 speeds are within the target (Aerodynamics has the tables and the slopes).

A CP that reads aft of the true one makes the margin read high: the rocket is less stable than the number says. That is the flattering side for stability, the side to leave extra margin on. A CP that reads forward costs margin: the rocket is more stable than the number says.

The figure marks the flattering side: the build marks each speed where HPR Sim’s error is positive. The test a_cp_aft_of_the_reference_reads_a_higher_margin pins what that sign means: a CP aft of the reference’s is a positive error, and HPR Sim’s own margin calculation reads higher by that much at any CG. It pins the sign, not how often HPR Sim errs aft; the marks show that. At 7 of the 12 speeds HPR Sim’s center of pressure sat aft of the tunnel’s, the largest from just below the speed of sound through the join. So for a rocket that flies there, read HPR Sim’s flight margin as possibly high; the printed static margin, taken at low speed, says nothing about those speeds. These are one rocket’s errors, not an upper limit: a rocket unlike the Arcas Robin may err by more.

At an angle of attack the body itself pushes sideways too, as a cylinder does in a side wind. That push is the body lift. It grows with the square of the angle’s sine, so it is small at small angles and large at large ones, and it acts near the middle of the body’s outline seen from the side. On most rockets that point is ahead of the fins, so as the angle grows the CP moves forward, and the margin shrinks.

At Mach 0.3, the guides’ rocket’s center of pressure moves forward by 0.25 calibers at 2°, 0.57 calibers at 5°, 1.01 calibers at 10°, and 1.34 calibers at 15°, each from where it sits at the same speed with the air straight along the rocket (report). The printed margins, static and flight, are the small-angle ones, so they don’t show this. The simulated flight does: HPR Sim flies the body lift at each moment’s angle, so a rocket leaving the rail in a crosswind, or knocked by a gust, is turned back toward the wind by a smaller margin than the printed one.

HPR Sim’s body lift has been checked against wind-tunnel tests of bodies without fins (Body lift), but no measurement here checks a whole rocket’s center of pressure at a large angle, and no test pins which way it errs there. So this section marks no side.

• The center of gravity’s own error. The margin is only as good as the CG. HPR Sim takes the masses from the design file; weigh and balance the built rocket, motor in (Check stability for a certification flight).
• Dynamic stability: how quickly a disturbed rocket settles, which also depends on its mass, its length and the air’s damping, and coning or roll. HPR Sim flies these, but this guide covers only where the air’s push acts.
• Parts the tunnel comparison doesn’t cover. Tube fins, pods and freeform fins are checked only against OpenRocket; compare the margin in both programs (Check stability for a certification flight).
• A rule for how much margin is enough. That is the safety code’s, the RSO’s and the published guidance’s (Compare with published guidance), not this guide’s.