Roll Lock-In — theory

Fin design against vortex-induced roll lock-in. Based on C. P. Hoult and H. Tran, Sounding Rocket Fin Design to Mitigate Roll Lock-In.

Open the calculatorOriginal paper (PDF)

Overview

Roll lock-in (catastrophic yaw) is a persistent, nonlinear coning motion at high angle of attack. It usually starts in powered flight at pitch–roll resonance, when the roll rate equals the pitch natural frequency and the angle of attack driven by body-fixed perturbations is amplified. Nonlinear roll moments then hold the roll rate at the pitch frequency: the motion is lunar, one side of the rocket always facing the coning axis, and drag rises sharply for a long time.

Of the three causes — configuration asymmetries, centre-of-mass offset, and the interaction of nose vortices with the tail fins — only the last depends on fin design, and it is the one modelled here. The approach: size the fins for a required static margin, compute the induced roll moment, find the smallest angle of attack \alpha^* at which steady lock-in is possible, and estimate the probability that the trim angle of attack stays below it.

Nose vortex pair

The vortex pair separates at x_S, about where the zero-\alpha surface pressure is lowest. Wind-tunnel data give its strength and position at a station x, with \xi=(x-x_S)\alpha/R:

\frac{\Gamma}{2\pi RU\alpha}=0.6+0.08\,\xi^{1.52},\qquad \frac{y_V}{R}=0.7,\qquad \frac{z_V}{R}=1+0.21\,\xi

so the free vortex lies at radius r=\sqrt{y_V^2+z_V^2} and roll angle \phi_V=\tan^{-1}(y_V/z_V) from the leeward meridian, and its image inside the body at s=R^2/r. The cross flow is treated as incompressible, valid while \sqrt{M^2-1}\sin\alpha\ll1.

A vortex induces a velocity normal to a fin at span station y (eq. 9). With a straight-tapered chord c(y)=C_1+C_2y, the strip roll moment of one fin and one vortex integrates in closed form:

C_{Li}=\frac{C_{N\alpha}\Gamma}{2\pi U\,\pi R^2\,2R}\left[g(C_1,C_2,r,\phi_V,y,\phi)\right]_{R}^{R+b},\quad g=-C_1r\cos(\phi_V-\phi)f_1+(C_1-C_2r\cos(\phi_V-\phi))f_2+C_2f_3

with the integrals f_0\ldots f_3 of eqs. 13–16. The four vortices (two free, two images, with opposite signs) and all N fins at \phi+2\pi k/N are summed (eq. 19). The result is close to a sine wave in roll angle with period 2\pi/N; at small angles of attack its phase can flip by 180°.

Fin size from the static margin

The fin assembly normal-force slope needed for a static margin SM is

C_{N\alpha F}=C_{N\alpha N}\,\frac{x_{CG}+SM-x_{CPN}}{x_{CPF}-x_{CG}-SM}

and strip theory with body upwash (1+R^2/y^2) gives the root chord, with c_1=1+(R/b)(1-\lambda) and c_2=(\lambda-1)/b:

C_{N\alpha F}=\frac{NC_{N\alpha}c_R}{2\pi R^2}\left[c_1b+\frac{c_2(b^2+2bR)}{2}+\frac{c_1R^2b}{R(b+R)}+c_2R^2\ln\frac{b+R}{R}\right]

Lunar motion and roll damping

In steady lunar coning at rate \omega, p=\omega\cos\alpha and r=-\omega\sin\alpha. The yaw equation gives the coning rate and the pitch equation the trim angle of attack:

\omega\cong\frac{2\pi qR^4}{UI_p}(C_{Nr}-C_{Np\alpha}),\qquad \alpha\cong\frac{-T(l_T\delta_T+\Omega)}{2\pi qR^3\left[C_{M\alpha}-\frac{2\pi qR^5}{U^2I_p}(C_{Nr}-C_{Np\alpha})^2\right]}

Just before resonance the roll rate is the pitch frequency \omega_n=\sqrt{-2\pi qR^3C_{M\alpha}/I_p}, and the cant torque balances fin damping. The roll equation then reads

\frac{C_{Lp}}{U}\left(R\omega-R\omega_n\right)+C_{Li}(\phi,\alpha)=0

The derivatives follow the paper’s appendix: C_{M\alpha} from the nose and fin slopes, C_{Nr}=-\sum c_{N\alpha i}\Delta x_i^2/2R^2, the fin Magnus term C_{Np\alpha} (eq. 41) and C_{Lp} from strip theory, all on reference area \pi R^2 and length 2R.

Smallest lock-in angle of attack

At small angles the induced moment is too weak to satisfy the roll equation at any roll angle. Steady lock-in first becomes possible at the angle where its largest value over roll angle just reaches the damping term:

\left|\frac{C_{Lp}}{U}\left(R\omega-R\omega_n\right)\right|\le\left|C_{Li\,max}(\alpha^*)\right|

The calculator scans \alpha up to the chosen limit, finds the largest |C_{Li}| over one fin period at each (scan plus golden section), and bisects the first crossing. Below \alpha^* lock-in cannot happen; above it, whether it does is a matter of statistics.

Probability of no lock-in

The misalignment \delta_T and offset \Omega are random, unbiased and independent, so each transverse component of the trim angle has standard deviation

\sigma_\alpha=\frac{T\sqrt{l_T^2\sigma_{\delta_T}^2+\sigma_\Omega^2}}{2\pi qR^3\left|C_{M\alpha}-\frac{2\pi qR^5}{U^2I_p}(C_{Nr}-C_{Np\alpha})^2\right|}

and the total angle of attack is Rayleigh distributed. The probability of no lock-in is the probability that it stays below \alpha^*:

P=1-\exp\left(-\frac{\alpha^{*2}}{2\sigma_\alpha^2}\right)

Implications for fin design

No fin design guarantees freedom from lock-in: a large enough perturbation always succeeds. But the probability can be made very small. Static margin is more important than generally realized — more than the classical two calibres extends the region where lock-in is dynamically impossible. Three and four fins differ little; five or more help dramatically, six much more than four and eight more still (mortar bombs fly with eight). Taper ratio and span have minor effects. Lower dynamic pressure (higher altitude) makes lock-in easier.

Verdict and fixes

The memo gives the probability of no lock-in and recommends static margins above two calibres and more than four fins, but no acceptable risk. The calculator asks for yours, r_{max} (default 1 %, this site’s choice), and compares the lock-in risk 100-P with it: safe up to it, marginal up to twice it, unsafe beyond. When no lock-in angle is found up to the largest angle examined, the risk shown is an upper bound.

The trim angle of attack is Rayleigh distributed, so the lock-in angle your risk needs is

\alpha_{need}=\sigma_\alpha\sqrt{-2\ln(r_{max}/100)}

The fixes: the static margin whose risk equals your limit (bisection on the full model, the fins re-sized for each margin), six or eight fins at the same margin when that is enough, and the alignment — since \sigma_\alpha\propto\sqrt{l_T^2\delta_T^2+\Omega^2}, scaling the misalignment and the offset by \alpha^*/\alpha_{need} meets the limit exactly.

A static margin that the nose and fin positions cannot give, or fins ahead of the vortex separation, are moved to the nearest buildable value with a note instead of being refused.

Verification

The tests reproduce the field tool’s fin slope, pitching and yaw-damping derivatives and pitch frequency, its vortex circulation and position, the roll moment of each fin and of all fins at 9°, and its table of |C_{Li}|_{max} against angle of attack, once its slips below are fed back in. The closed-form strip integral g is checked against Simpson quadrature of eq. 12, and the induced moment is odd in roll angle with period 2\pi/N.

Notes on the method

  • Units of the paper’s example. The paper and the field tool use inches, psi and I_p=WL^2/g “slug·in²”. In \omega_n^2=-2\pi qR^3C_{M\alpha}/I_p with q in psi the inertia must be in lbf·s²·in, so the tool’s pitch frequency is low by \sqrt{12}; WL^2/g is also 12 times the inertia of a uniform rod. The calculator works in consistent units and its example uses mL^2/12. The paper’s figures therefore are not reproduced numerically.
  • Root chord. The tool takes Excel’s LOG (base 10) of (b+R)/R in eq. 33, where the integral of 1/y is the natural logarithm; its chord, and with it every roll moment, is 12% small for the example.
  • Vortex radius. The tool uses R\sqrt{1+0.21\xi^2+0.49} where eqs. 2–3 give R\sqrt{0.49+(1+0.21\xi)^2}.
  • Magnus derivative. The tool’s C_{Np\alpha} cell holds a stale value its own formula (which also misplaces a parenthesis) does not give. It only enters the small coning rate.
  • Normalization. The paper writes C_{Nr}=-2\sum c_{N\alpha}(\Delta x/R)^2 and C_{Lp} over \pi R^4; on the reference \pi R^2\cdot2R and rates R\omega/U used in its moment equations (eqs. 21–25) these are -\sum c_{N\alpha}\Delta x^2/2R^2 and 2\pi R^4, as the tool has them.
  • Probability. Eq. 28 writes 1-\exp(-\alpha^2/\mathrm{var}(\alpha)) with the per-axis variance of eq. 27; for a Rayleigh-distributed magnitude the exponent is -\alpha^2/2\sigma^2, which the calculator uses.

References

  1. C. P. Hoult, H. Tran, “Sounding Rocket Fin Design to Mitigate Roll Lock-In,” RS&T. View in library