Flight Path Angle Dispersions — theory

How thrust misalignment and gusts rotate the velocity vector of a rolling rocket. Based on C. P. Hoult, Flight Path Angle Dispersions (rev. 1), 2011.

Open the calculatorOriginal memo (PDF)

Overview

Every perturbation in boost either changes the length of the velocity vector (an energy error) or rotates it. Sounding rockets fly nearly vertically, so apogee errors come mainly from energy errors and impact point errors mainly from rotations. Two perturbations rotate the vector: thrust misalignment, which rotates with the body, and wind gusts, which are fixed to the earth. Both act mostly at low altitude and low speed, just after the rail.

The analysis takes the phugoid (trajectory) part of the response, assumes the short-period oscillations damp promptly, and expresses the result as an equivalent launcher angle error: the launcher error that, with no perturbation, would give the same velocity tilt at burnout. It is conservative: the steady states it uses are upper bounds of the transient responses.

Thrust misalignment

A rocket rolling at roll wave number \lambda_R with a misalignment angle \delta_T and a thrust offset \Omega holds a steady angle of attack. Taking the two one-sigma effects as independent, its lift is constant along the boost:

L_A=qSC_{L\alpha}\alpha_{SS}=\frac{\rho SC_{L\alpha}T\sqrt{\ell^2\delta_T^2+\Omega^2}}{2I_p(\lambda_P^2-\lambda_R^2)}

It grows without bound as the roll wave number approaches the pitch wave number \lambda_P (roll resonance), which is why more static margin can cut the dispersion sharply. With speed U^2=2ah, acceleration a=T/m-g and the flight path angle \gamma_T measured from the vertical,

h\frac{d\gamma_T}{dh}=\frac{g}{2a}\gamma_T+\frac{L_A}{2ma},\quad \gamma_T(L)=0\quad\Rightarrow\quad \gamma_T=\frac{L_A}{mg}\left[\left(\frac{h}{L}\right)^{g/2a}-1\right]

Most of the turn happens close to the launcher of length L; a longer rail reduces it.

Roll

The lift rotates with the rocket, at roll angle \phi=\lambda_R(h-L), so each increment of turn points in a different direction. Resolved into east and north components and summed in altitude steps,

E=\int\cos\phi\,d\gamma_T,\qquad N=\int\sin\phi\,d\gamma_T,\qquad \gamma_D=\sqrt{E^2+N^2}

The calculator takes the exact \Delta\gamma_T of each step (4 ft, or 1/60 of a roll cycle if shorter) at its mid-step roll angle. Burning out after a whole number of roll cycles cancels the turn best, and one cycle is usually the minimum; a roll wavelength near 1000 ft is close to optimum over a broad range of pitch wave numbers.

Wind gusts

Above the first correlation distance the gust angle of attack has a nearly constant standard deviation (Hoult, Sounding Rocket Gust Angle of Attack):

\alpha_G=\sigma_G\lambda_P\sqrt{\frac{\lambda_G^*}{2a(\lambda_P^2+\lambda_G^{*2})}}

Its lift, L_A/2ma=\rho SC_{L\alpha}h\alpha_G/2m, drives the same equation as above, giving a planar one-sigma flight path angle

\gamma_G=K\left[\frac{h}{L}-\left(\frac{h}{L}\right)^{g/2a}\right],\qquad K=\frac{\rho SLC_{L\alpha}\sigma_G\lambda_P}{m(2a-g)}\sqrt{\frac{a\lambda_G^*}{2(\lambda_P^2+\lambda_G^{*2})}}

The model needs 2a>g, a thrust above 1.5 times the weight.

Equivalent launcher angle

The two errors come from independent causes, so

\gamma=\sqrt{\gamma_D^2+\gamma_G^2}

evaluated at burnout. Without lift, a launcher error \gamma_0 grows as \gamma=\gamma_0(h/L)^{g/2a}, so the equivalent launcher angle is

\gamma_0=\gamma\left(\frac{h_b}{L}\right)^{-g/2a}

Started from \gamma(L)=\gamma_0, a point-mass trajectory run gives one component of the impact displacement. Both components are normally distributed, unbiased and independent with the same standard deviation, because the misalignment phase is uniform in roll and the gust field has two independent components.

Verdict and fixes

The memo turns the dispersion into an equivalent launcher error \gamma_0 but sets no acceptable value: that depends on how far the impact may move on your range. The calculator asks for your limit \gamma_{max} (default 1°, this site’s choice) and compares: safe up to it, marginal up to 1.25×, unsafe beyond. To choose the limit, run the Trajectory Simulation with launcher errors of a few trial sizes and see how far the impact moves.

The fixes are solved on the full model by bisection: the misalignment \delta_T (with the offset \Omega scaled with it) that meets the limit, the pitch wave number (static margin), and the launcher length (shown only up to three times the present rail). The best roll wavelength is found by scanning 0.3–3 roll cycles to burnout, staying clear of resonance; Hoult notes that about 1000 ft is close to optimum over a broad band of pitch wave numbers. When the gusts alone exceed the limit, no rocket change helps and the verdict says so.

The steady misalignment lift grows as 1/(\lambda_P^2-\lambda_R^2). At or past 95 % of the pitch wave number the rocket rolls through pitch resonance (roll lock-in) and the steady model has no answer: the verdict is unsafe, with the roll wavelength or static margin that brings the roll wave number down to 70 % of the pitch wave number.

Verification

The tests reproduce the memo’s Figure 3 (g/2a=0.1, L_A/mg=0.2, L=40 ft: peak 0.042 rad at 0.4 roll wavelengths, 0.029 near one cycle) and Figure 1, the field tool’s lift, \gamma_T and gust columns, and, with the tool’s two slips put back, its tabulated equivalent launcher angle against pitch wave number at 1850 ft burnout to 0.5%.

Notes on the method

  • Field tool step. DISPERSION integrates E and N with the step (L_A/mg)(L_A/2ma)\,Z\,\Delta h/h (with a in g), which is L_A/m times the slope of eq. 3 and disagrees with its own \gamma_T column. The error grows near roll resonance: at \lambda_P=0.012 ft⁻¹ the tool gives 0.074 rad where eq. 3 gives 0.014 rad. The memo’s Figure 3 agrees with eq. 3, which the calculator follows.
  • Field tool gust factor. The tool uses \lambda_P^2+\lambda_R^2 where eq. 15 has \lambda_P^2+\lambda_G^{*2}; the gust field is fixed to the earth, so roll does not enter. The difference is a few percent.
  • Acceleration. The tool uses a=T/m; the calculator subtracts gravity, a=T/m-g, as the vertical-flight equations assume. Drag is neglected in both.

References

  1. C. P. Hoult, “Flight Path Angle Dispersions (rev. 1),” RS&T, 8 May 2011. View in library
  2. C. P. Hoult, “Thrust Misalignment Response,” RS&T, 30 December 2010. View in library
  3. R. L. Ammons, C. P. Hoult, “Standardized Perturbation Values to be Used in the Calculation of Dispersion and Structural Loads,” Aerojet-General, 1970. View in library
  4. C. P. Hoult, “The Theory Behind DISPERSION,” RS&T, 2007. View in library