The figureof8 librations of the gravity gradient pendulum and modes of an orbiting tether
Abstract
An algorithm is presented for the HillPoincare analytical continuation of the outofplane normal mode of the gravity gradient pendulum. The PoincareLindstedt solution employs 17 Poisson series and 24 recursion relations; it was evaluated to the 50th order on a CRAY. The trajectories of the nonlinear normal modes are figuresof8 on the unit sphere which can be computed nearly to the orbit normal. Numerical integrations indicate further that initial conditions computed at the nadir can be used to generate figuresof8 over the pole, that the single hemispherical figuresof8 appear to be stable at large amplitudes, and that the gravity gradient pendulum has chaotic solutions. A theory is developed for the linear normal modes of a tethered satellite, and the eigenvalues are found for the rosary tether.
 Publication:

Quarterly of Applied Mathematics
 Pub Date:
 December 1988
 Bibcode:
 1988QApMa..46..637M
 Keywords:

 Gravitational Effects;
 Librational Motion;
 Pendulums;
 Tethered Satellites;
 Degrees Of Freedom;
 Equations Of Motion;
 Modal Response;
 Recursive Functions;
 Astrodynamics