Roll Rate History — theory

Spin from the launcher to apogee. Based on C. P. Hoult, Small Sounding Rocket Boost Phase Roll Rate (2013) and Small Sounding Rocket Apogee Roll Rate (2013).

Open the calculatorBoost-phase memo (PDF)Apogee memo (PDF)

Overview

Roll rate matters for the rocket’s dynamics — spin averages out thrust and fin misalignments, but a roll rate near the pitch frequency can lock in — and for the payload: cameras and sensors near apogee, where the payload lingers, and parachute deployment, where a fast spin can wrap the riser lines.

While the angle of attack is small, the roll equation is linear and decoupled from the rest of the motion. For vertical flight it is

I_R\frac{dp}{dt}=qSd\left[C_{Lp}\frac{pd}{2U}+C_{L\delta}\sum_i\delta_i\right]

where I_R is the roll moment of inertia, p the roll rate, q=\tfrac12\rho U^2 the dynamic pressure, S=\pi R^2 and d=2R the reference area and length, C_{Lp} the roll-damping derivative of all fins, C_{L\delta} the roll-driving derivative of one fin and \delta_i the fin cant angles. With N fins at the same cant, \sum\delta_i=N\delta.

Roll derivatives

The calculator takes C_{L\delta} and C_{Lp} from strip theory for a straight-tapered fin set, exactly as in the Fin Cant Angle calculator: each strip of fin carries a lift proportional to its local angle of attack, which is the cant angle for the driving moment and py/U at radius y for the damping moment. The fin geometry is shared between the two calculators.

Boost without inertia

For most of the boost the inertia term is small, and driving balances damping:

p\simeq-\frac{2UC_{L\delta}}{d\,C_{Lp}}\sum_i\delta_i,\qquad\lambda_{R0}=\frac pU=-\frac{2C_{L\delta}}{d\,C_{Lp}}\sum_i\delta_i

The roll wave number \lambda_{R0} is the roll per unit distance flown. Apart from Mach-number and aeroelastic effects it is constant, so the roll rate grows in proportion to speed, p=\lambda_{R0}U, and the roll angle in proportion to height, \Delta\phi=\lambda_{R0}(h-L). The rocket rolls along a helix that makes one turn every 2\pi/\lambda_{R0}.

Roll inertia and the lag

The roll rate is zero until the rocket clears the launcher, and just after it the dynamic pressure is still low, so inertia dominates and the spin lags behind \lambda_{R0}U. With constant acceleration, U^2=2ah, the memo shows that the lag is governed by

J=\frac{I_R}{2\pi\rho(h-L)R^4C_{Lp}}=\frac{k}{h-L},\qquad k=\frac{I_R}{2\pi\rho R^4C_{Lp}}

(J and the roll-lag length k are negative because C_{Lp} is). Summing the asymptotic series gives the closed-form estimate

\lambda_R\simeq\frac{\lambda_{R0}}{1-J},\qquad\Delta\phi\simeq\lambda_{R0}\left[h-L+k\ln\frac{h-L-k}{-k}\right]

which the memo compares with Prospector-18D flight data. The estimate is accurate while |J| is small — within about 1 % for |J|\lesssim0.05 — but it overstates the rate when the inertia is large: by about 20 % at |J|=0.3. The calculator therefore integrates the roll equation itself, with the same constant-acceleration boost (U=at, h=at^2/2, p=0 at launcher exit), and reports the integrated values; the closed-form estimate is shown alongside for comparison. Very stiff cases (tiny inertia) switch to an exponential integration step that is stable for any step size.

Roll rate at apogee

After burnout, near apogee, gravity is the main force changing the speed, dV/dt\simeq-g, and with the density at apogee \rho_A the roll equation becomes

\frac{dp}{dV}-GVp=-HV^2,\qquad G=-\frac{\rho_ASd^2C_{Lp}}{4gI_R},\quad H=\frac{\rho_ASd\,C_{L\delta}\sum\delta_i}{2gI_R}

With the integrating factor e^{-GV^2/2} and p_B=HV_B/G the memo arrives at

\frac{p_A}{p_B}=e^{-GV_B^2/2}+\frac{G}{V_B}\int_0^{V_B}V^2e^{-GV^2/2}\,dV

and evaluates the integral numerically for its Figure 1. The integral does have a closed form; with a=GV_B^2/2,

\frac{p_A}{p_B}=\frac{\sqrt\pi\,\mathrm{erf}\sqrt a}{2\sqrt a}

which falls from 1 at a=0 to 0.50 at GV_B^2=6, as in the memo’s figure. If the roll rate at burnout has not reached \lambda_{R0}V_B because of the lag, the same integration gives

p_A=p_Be^{-a}+\lambda_{R0}V_B\left[\frac{\sqrt\pi\,\mathrm{erf}\sqrt a}{2\sqrt a}-e^{-a}\right]

The approximation is meant for small rockets whose apogee is well inside the troposphere.

Coast to apogee

After burnout the fins keep driving the roll while the air damps it, but the rocket slows and climbs into thinner air. Hoult’s “Small Sounding Rocket Apogee Roll Rate” (2013) takes the coast as slowing at dV/dt\approx-g with the air density at apogee:

\frac{dp}{dV}-GVp=-HV^2,\qquad G=-\frac{\rho_AS d^2C_{Lp}}{4gI_R}

The memo integrates this numerically. It has a closed form, which the calculator uses: the apogee roll rate relative to burnout is p_A/p_B=\sqrt\pi\,\mathrm{erf}(\sqrt a)/(2\sqrt a), with a=GV_B^2/2. A large spin parameter GV_B^2 (a light, fast rocket with big fins) means the roll dies away more before apogee. The coast is drag-free, so apogee height and time are upper estimates.

Your limits and the verdict

The memos give no roll-rate limit: it depends on what the rocket carries (a camera, an antenna, a parachute that must not wind up). So you set two limits: the highest roll rate in flight, and the highest at apogee, where recovery begins. The verdict (DECISIONS D32) takes the worse of the two ratios, roll rate over your limit:

  • SAFE up to 1;
  • MARGINAL up to 1.25;
  • UNSAFE beyond.

The roll equation is linear in the cant, with no spin at launcher exit, so every roll rate scales with the cant. The largest cant within your limits is therefore exact: your cant divided by the worse ratio.

Verification

No spreadsheet exists for this topic. The tests compare the integrated burnout rate with an independent fine integration of the roll equation (to 10^{-8}), confirm that the closed-form estimate agrees within 0.5 % when |J| is small, check the closed-form apogee ratio against direct quadrature of the memo’s integral and against its Figure 1, check the apogee rate against a numerical solution of the coast equation started from the lagged burnout rate, and test the error function against tabulated values.

Notes on the method

  • Upper stages. The boost memo also sketches the case of an upper stage that ignites already spinning (its eqs. 5f–5i). As printed those equations are dimensionally inconsistent (the roll inertia is missing), so they are not used here.
  • Constant acceleration. The boost is modelled with constant acceleration from the launcher base to burnout and a single representative density. Use a trajectory simulation for detailed histories.

References

  1. C. P. Hoult, “Small Sounding Rocket Boost Phase Roll Rate,” RS&T, 28 September 2013. View in library
  2. C. P. Hoult, “Small Sounding Rocket Apogee Roll Rate,” RS&T, 15 January 2013. View in library
  3. C. P. Hoult, “Strip Theory for Setting Fin Angles (rev. 3),” RS&T, 2014. View in library
  4. B. Etkin, Dynamics of Flight: Stability and Control, Wiley, 1959.