Body Bending Modes — Theory & documentation

Overview

Whenever flexible-body effects matter — propellant slosh, axial vibration, body bending — Lagrange’s equations come into play. A rocket in flight bends as a free-free beam: nothing holds either end. Its bending frequencies must stay clear of the pitch frequency, the roll rate and motor pressure oscillations, and sensors placed near a node of the fundamental mode see little bending motion.

Uniform beam and Rayleigh–Ritz

For a uniform free-free beam of length L, stiffness EI and mass per length m, the modes are

\psi_j(\xi)=B_j\left(\sinh\beta_j\xi+\sin\beta_j\xi\right)+\cosh\beta_j\xi+\cos\beta_j\xi,\qquad B_j=\frac{\cos\beta_j-\cosh\beta_j}{\sinh\beta_j-\sin\beta_j}

with \xi=x/L and \beta_j the roots of \cos\beta\cosh\beta=1: 4.7300, 7.8532, 10.9956. Their frequencies are \omega_j=(\beta_j/L)^2\sqrt{EI/m}.

A real rocket is a chain of sections — nose, payload, motor case, skirt — with their own stiffness and weight. Its displacement is approximated by the three uniform shapes, y=\sum_jq_j\psi_j(x/L). Lagrange’s equations with the kinetic energy \tfrac12\int m\dot y^2dx and strain energy \tfrac12\int EI\,(y'')^2dx give \mathbf M\ddot{\mathbf q}+\mathbf K\mathbf q=0, with

K_{ij}=\int_0^LEI\,\psi_i''\psi_j''\,dx,\qquad M_{ij}=\int_0^Lm\,\psi_i\psi_j\,dx

and the natural frequencies satisfy \det(\omega^2\mathbf M-\mathbf K)=0. The calculator integrates each section (stiffness and weight varying linearly within it) by Simpson’s rule and solves the 3×3 eigenproblem directly. The lowest frequency is the most accurate; with three shapes the third is only a rough estimate.

Mode shapes and nodes

Each eigenvector gives a mode shape \sum q_j\psi_j, shown scaled to 1 at the nose tip. The fundamental mode has two nodes where the body does not move; an inertial sensor there sees the least bending.

Describing the body as tubes

Instead of entering each section’s bending stiffness and weight, you can describe it as the builder sees it: length, outer diameter, wall thickness, the tube material’s Young’s modulus, and the section’s mass with its contents. Each section is then a uniform tube:

EI=E\,\frac{\pi\left(D^4-(D-2t)^4\right)}{64},\qquad w=\frac{m}{\ell}

A thin wall gives EI\approx E\pi D^3t/8, so the stiffness grows with the cube of the diameter but only in proportion to the wall. To stiffen a body, a larger diameter or a stiffer material helps much more than a thicker wall. The section that matters most is the one carrying the largest share of the bending energy \tfrac12\int EI\,y''^2dx in the first mode; the drawing shades it and the result names it.

Verdict: keep the first bending frequency clear of the excitation

A flexible body is shaken by anything that repeats: the roll rate, the pitch frequency, a motor’s combustion, a control loop. If the first bending frequency f_1 is close to such an excitation f_{exc}, the bending grows and sensors on the body read the wobble. You give the highest excitation and the separation you want (default 3 times). The verdict (DECISIONS D33) is:

SAFE when f_1\ge r\,f_{exc};
MARGINAL between 1.25f_{exc} and that;
UNSAFE below 1.25f_{exc}, at or near resonance.

These are project choices, not published limits. The fixes are solved by bisection on the full Rayleigh–Ritz model:

the stiffness factor on the section that bends most, or on every section;
for tubes, the wall of that section;
the highest excitation that keeps your separation.

The nodes of the first mode, where the body does not move, are the places for accelerometers and IMUs.

Verification

For the original field tool’s Aries example (five sections, 365 in), the tests reproduce its three frequencies — 257.6, 585.5 and 1523.7 rad/s — and its stiffness and mass integrals within 0.1–0.2 %. They also check that the mode shapes are exact eigenvectors, and that a uniform body returns the exact uniform-beam frequencies.

Notes on the method

Mode shapes of the field tool. Its mode-shape macro misspells one matrix element (l13st for j13st), so the term -K_{13} is dropped and its mode shapes are wrong, although its frequencies are right. The calculator uses the full eigenvectors; the test confirms that the typo reproduces the tool’s shapes exactly.
Integration. The tool’s trapezoid sums omit the first small step of each section; the calculator integrates the whole section, a difference of about 0.01 %.
Uniform reference. The uniform-beam frequencies shown for comparison use length-averaged stiffness and mass.

References

C. P. Hoult, “Longitudinal Body Bending Dynamics,” RS&T, 5 August 2013. View in library

Body Bending Modes — theory