Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



The figure-of-$ 8$ librations of the gravity gradient pendulum and modes of an orbiting tether

Author: Peter J. Melvin
Journal: Quart. Appl. Math. 46 (1988), 637-663
MSC: Primary 70M05; Secondary 70-04, 70K20, 70K40
DOI: https://doi.org/10.1090/qam/973381
MathSciNet review: 973381
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Abstract: An algorithm is presented for the Hill-Poincaré analytical continuation of the out-of-plane normal mode of the gravity gradient pendulum. The Poincaré-Lindstedt solution employs 17 Poisson series and 24 recursion relations and was evaluated to the 50th order on a CRAY. The trajectories of the nonlinear normal modes are figures-of-8 on the unit sphere which can be computed nearly to the orbit normal. Numerical integrations indicate further that 1) initial conditions computed at the nadir can be used to generate figures-of-8 over the pole, 2) the single hemispherical figures-of-8 appear to be stable at large amplitudes, and 3) 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.

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DOI: https://doi.org/10.1090/qam/973381
Article copyright: © Copyright 1988 American Mathematical Society

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