Launcher Rail Deflection — Theory & documentation

Overview

A university sounding rocket that reaches several kilometres can land far from its launcher unless the launcher is set for the winds. The CSULB launcher was built to be adjustable in azimuth (a turntable) and elevation (a scissor jack), transportable, and stiff: its rail rides on a ham-radio tower truss. Its requirements included a rail bending deflection of no more than 0.05° at a quadrant elevation of 80°; Hoult’s memo takes 0.1° as a good general upper bound.

The rail must also not rotate as a rigid body when the rocket launches, and its lugs should drop free together (a “zero tip-off” rail-and-lug design, first developed at NACA Wallops). This calculator addresses the bending.

The rail as a cantilever

The rail is cantilevered from its base. The bending is largest just before the rocket leaves, when the greatest moment acts over the greatest length, with the transverse load F at the tip:

EI\frac{d^2y}{dx^2}=F(L-x)\quad\Rightarrow\quad\frac{dy}{dx}(L)=\frac{FL^2}{2EI},\qquad y(L)=\frac{FL^3}{3EI}

where E is the Young’s modulus, I the second moment of area, x the distance from the base and L the rail length. The rocket takes about as long to run up the rail as the rail’s own bending period, so the load is effectively applied suddenly and the rail overshoots: the memo doubles the static slope (dynamic factor k_d=2). The tip slope is, in effect, a change in quadrant elevation at launch.

Solving for the length at which the slope reaches the allowance \theta_{max} gives the longest acceptable rail, L_{max}=\sqrt{2EI\theta_{max}/(k_dF)}.

Lateral loads

In the pitch plane the load is the weight component across the rail plus the thrust misalignment:

F_{pitch}=W\cos QE+n_\sigma T\sigma_\delta,\qquad F_{yaw}=n_\sigma T\sigma_\delta

The weight component is fixed; the misalignment is random, so it is taken at n_\sigma standard deviations (Ammons and Hoult give standard values; 0.1° is typical). For Gold Rush II (55 lb, 400 lb thrust, QE 80°) the weight component is 9.55 lb while a 3σ misalignment gives only 2.09 lb: the weight dominates.

Tower truss stiffness

A ham-radio tower is three steel tubes at the corners of an equilateral triangle of face width d, tied by welded zig-zag rods that add little bending stiffness. The neutral axis passes through the centroid, twice as far from the single tube as from the pair, so

I=\frac{Ad^2}{2},\qquad A=\pi Dt\quad\Rightarrow\quad I=\frac{\pi Dt\,d^2}{2}

for tubes of outside diameter D and wall t (the memo’s typical tower, 12.5 in face and 1.25 × 0.065 in tubes, gives about 20 in⁴); the second moment about the perpendicular axis is the same, so pitch and yaw are equally stiff. The tip slope before the overshoot factor is then FL^2/(\pi EDt\,d^2). For other supports, enter I directly.

Where the rail bends

Along the rail the bending moment falls from FL at the pivot to zero at the tip, so the curvature is largest at the base. Integrating once gives the slope at a distance x from the base, and again the deflection:

\theta(x)=k_d\frac{F(2Lx-x^2)}{2EI},\qquad y(x)=k_d\frac{Fx^2(3L-x)}{6EI}

At mid-length the slope is already three quarters of its tip value: the lower half of the rail produces 75 % of the tip-off error. Joints, hinges and jacks near the base matter most; the model assumes they are rigid and the rail uniform.

Verdict, limit and fixes

The verdict compares the pitch tip slope \theta_{pitch} with the allowed slope \theta_{max} you set: safe up to \theta_{max}, marginal up to 1.25 times it, unsafe beyond. The default 0.1° is the bound Hoult chose for his memo; the CSULB launcher was specified to 0.05° at QE 80°. Neither is a published standard: the bands are this site’s choice.

Because the slope is proportional to L^2/(Dtd^2) (or L^2/I), each fix is solved in closed form with the ratio r=\theta_{pitch}/\theta_{max}: the longest rail L_{max}=L/\sqrt r, the face width d\sqrt r, the tube wall tr, or the second moment of area I_{req}=Ir. Ham-radio towers come in 10 ft sections, so the rail-length fix also gives the number of whole sections.

For an existing launcher, the same equation solved for the weight gives the heaviest rocket the rail takes within the limit:

W_{max}=\frac{2EI\theta_{max}/(k_dL^2)-n_\sigma T\sigma_\delta}{\cos QE}

Verification

The tests check the memo’s Gold Rush II load comparison, that every fix and W_{max} land exactly on the allowed slope, the truss second moment against an exact parallel-axis sum in both planes, and the slope and deflection against numerical integration of the beam equation. They also reproduce the memo’s plot of tip angle against rail length (weight load only, with the factor 2): about 0.053° at 240 in and 0.21° at 480 in for QE 80°, and 0.027° at 240 in for QE 85°. This confirms the memo’s recommendation that a 20 ft rail keeps the tip-off error below about 0.1°.

References

C. P. Hoult, “Launcher Tipoff Deflection,” RS&T, 28 August 2010. View in library
F. Calderon, A. Al Chamas, T. Wilson, A. Vore and B. Vite, “A New Launcher for University Sounding Rockets,” AIAA Region VI student paper, 2011. View in library
R. L. Ammons and C. P. Hoult, “Standardized Perturbation Values for Several Sounding Rockets to be Used in the Calculation of Dispersion and Structural Loads,” Aerojet-General memo 8110:M0658:ak, 1970. View in library

Launcher Rail Deflection — theory