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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L. A. Month and R. H. Rand, An application of the Poincaré map to the stability of nonlinear normal modes, J. Appl. Mech. 47, 645–651 (1980)
G. Pecelli and E. S. Thomas, Normal modes, uncoupling, and stability for a class of nonlinear oscillators, Quart. Appl. Math. 37, 281–301 (1979)
Y. Nejoh, A nonlinear normal mode in an anharmonic lattice with a single defect, Physica Scripta 28, 561–564 (1983)
W. E. Ferguson, H. Flaschka, and D. W. McLaughlin, Nonlinear normal modes for the Toda chain, J. Comput. Phys. 45, 157–209 (1982)
S. Briere, Nonlinear normal mode initialization of a limited area model, Mon Weather Rev, 110, 1166–1186 (1982)
P. J. Melvin, On the construction of Poincaré-Lindstedt solutions: The nonlinear oscillator equation, SIAM J . Appl. Math. 33, No. 1, 161–194 (1977)
R. R. Dasenbrock, A FORTRAN-based program for computerized algebraic manipulation, NRL Report 8611, Naval Research Laboratory, Washington, D. C., 1982
A. Deprit and A. Rom, The main problem of artificial satellite theory for small and moderate eccentricities, Celestial Mechanics 2, 166–206 (1970)
T. A. Bray and C. L. Goudas, Doubly symmetric orbits about the collinear Lagrangian points, Astron. J. 72, No. 2 (1967)
R. H. Vassar and R. B. Sherwood, Formationkeeping for a Pair of Satellites in a Circular Orbit, J. Guidance, Control and Dynamics 8, No. 2, 235–242 (1985)
K. R. Symon, Mechanics, Second Edition, Addison-Wesley, Reading, Massachusetts, 1960
H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, Massachusetts, 1950
P. J. Melvin, The phase-shifting limit cycles of the van der Pol equation, J. Res. Nat. Bur. Standards 83, No. 6, 593–601 (1978)
A. Deprit, Celestial mechanics: Never say no to a computer, J. Guidance, Control, and Dynamics 4, 577–581 (1981)
A. Deprit and D. S. Schmidt, Exact coefficients of the limit cycle in van der Pol’s equation, J. Res. Nat. Bur. Standards 84, No. 4, 293–297 (1979)
M. B. Dadfar, J. Geer, and C. M. Andersen, Perturbation analysis of the limit cycle of the free van der Pol equation, SIAM J. Appl. Math. 44, No. 5, 881–895 (1984)
J. V. Breakwell and J. W. Gearhart, Pumping a tethered configuration to boost its orbit around an oblate planet, J. Astronaut. Sci. 35, No. 1 (1987)
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Article copyright:
© Copyright 1988
American Mathematical Society
