Pitch-Yaw Wave Number — Theory & documentation

Overview

Sounding rockets are nearly symmetric about their roll axis: the pitch and yaw moments of inertia are equal, and so are the aerodynamic moments in pitch and yaw. Whatever is said about pitch therefore holds for yaw as well.

A statically stable rocket that is disturbed — by a gust, by leaving the launcher, by a misaligned thrust — swings about its centre of gravity like a weathervane until its own damping settles it. The pitch/yaw wavelength is the distance it travels through the air during one of these swings. It sets how quickly the rocket turns into the wind off the launcher, how much a thrust misalignment deflects it, and whether its roll rate can lock in with the pitch motion.

Derivation

Picture the rocket in a wind tunnel on an axle through its centre of gravity. If the nose is tweaked, the model oscillates according to

I\frac{d^2\alpha}{dt^2}=\tfrac12\rho S d\,U^2C_{M\alpha}\,\alpha

where I is the pitch moment of inertia about the CG, \rho the air density, S=\pi d^2/4 the body cross-section area, d the body diameter, U the airspeed, C_{M\alpha} the pitching-moment slope about the CG (per radian) and \alpha the angle of attack. Substituting \alpha=\sin\omega_N t gives the natural frequency

\omega_N^2=\frac{-\rho S d\,U^2C_{M\alpha}}{2I}

The period is P=2\pi/\omega_N, and in that time the rocket moves the wavelength L=UP:

L=2\pi\sqrt{\frac{2I}{-\rho S d\,C_{M\alpha}}},\qquad \lambda_p=\frac{2\pi}{L}=\sqrt{\frac{-\rho S d\,C_{M\alpha}}{2I}}

The minus sign is harmless: C_{M\alpha} is negative for a stable rocket. The wave number \lambda_p is a frequency in space rather than time, in radians per metre or per foot.

When the normal-force slope C_{N\alpha} (from the centre-of-pressure calculation) and the static margin are known, the pitching-moment slope about the CG is

C_{M\alpha}=-C_{N\alpha}\,\frac{X_{CP}-X_{CG}}{d}

If no measured inertia is available, the calculator can treat the rocket as a uniform slender rod, I=m\ell^2/12. Real rockets carry their mass at the motor and in the nose, so this is only an estimate; the Bifilar Pendulum calculator turns two stopwatch timings into the measured value.

Frequency and speed

The speed cancels from the wavelength: the restoring moment grows as U^2, so the swing gets faster in exactly the proportion that the rocket covers more ground. The frequency is therefore simply

f_N=\frac{\omega_N}{2\pi}=\frac{U}{L}

The memo notes that the same wavelength holds when the speed is not constant but increases under constant acceleration, as during boost. Density does enter: the wavelength grows as \rho^{-1/2}, so a rocket coasting into thin air weathercocks more and more slowly.

Scaling

For two geometrically similar rockets of lengths in the ratio \ell, the density and C_{M\alpha} are unchanged, Sd\propto\ell^3, and I\propto\ell^5 (the mass grows as \ell^3 and the radius of gyration as \ell). So

L\propto\ell

Bigger rockets have longer wavelengths, but the wavelength measured in body lengths, L/\ell, is the same for similar rockets of any size. For typical student sounding rockets the initial wavelength is of order 200 m, or about 40–60 body lengths.

Roll against the pitch frequency: the verdict

A rolling rocket carries its misalignments around with it. Hoult’s “Strip Theory for Setting Fin Angles” (rev. 3, p. 6) shows that the angle of attack and the loads they cause grow by

\frac{\lambda_P^2}{\lambda_P^2-\lambda_R^2}=\frac{1}{1-(f_R/f_N)^2},\qquad \lambda_R=\frac{2\pi f_R}{U}

This grows without limit when the roll reaches the pitch frequency (lock-in; see the Roll Lock-In calculator). The memo keeps the damage small with \lambda_R below 30–40 % of \lambda_P, and designs for 30 % at burnout (p. 4). The verdict uses the same bands as Fin Cant Angle (DECISIONS D31):

SAFE up to 30 %;
MARGINAL up to 40 %;
UNSAFE beyond.

Since \lambda_P\propto\sqrt{-C_{M\alpha}/I}, the fix is either the highest roll rate for 30 %, or the factor by which the restoring moment (static margin) must grow, or the inertia shrink.

The launcher and the trajectory simulation

The wavelength is the distance the rocket flies during one wobble, so it sets how quickly the rocket turns into the wind after leaving the rail. The rail length in wavelengths, L_r/L, says how much of a wobble happens on the rail. A point-mass trajectory simulation has no rotational inertia: it needs a longer simulated rail to turn into the wind like the real rocket. That rail comes from Hoult’s “Launcher Length for Sounding-Rocket Point-Mass Trajectory Simulation” (1976), using \lambda_P=2\pi/L. The Trajectory Simulation takes this wavelength as its pitch/yaw wavelength input, and “Fill from the Trajectory Simulation” brings its burnout speed here.

Verification

No spreadsheet exists for this topic. The tests check the memo’s relations between \omega_N, P, L and \lambda_p, integrate the pitch equation numerically over twenty periods and recover the period to one part in a million, and confirm the speed independence, the \rho^{-1/2} dependence and the L\propto\ell scaling law.

References

C. P. Hoult, “The Pitch-Yaw Wave Number (rev. 3),” RS&T, 5 January 2013. View in library

Pitch-Yaw Wave Number — theory