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.


References [Enhancements On Off] (What's this?)

  • [1] G. W. Hill, Researches in the Lunar Theory, in The collected works of George William Hill, Carnegie Institute of Washington, Memoir No. 32, 284-335 (1905)
  • [2] P. C. Hughes, Spacecraft attitude dynamics, John Wiley and Sons, New York (1986)
  • [3] G. D. Birkhoff, Dynamical systems, Amer. Math. Soc. Colloquium Publications, New York, Vol. IX (1927) MR 0209095
  • [4] R. M. Rosenberg, On the existence of normal mode vibrations of nonlinear systems with two degrees of freedom, Quart. Appl. Math. 22, 217-234 (1964) MR 0163464
  • [5] P. J. Melvin, On deviations from linear wave motion in inhomogeneous stars, Quart. Appl. Math. 35, 75-97 (1977)
  • [6] R. H. Rand, A direct method for non-linear normal modes, Internat. J. Non-Linear Mech. 9, 363-368 (1974)
  • [7] 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) MR 586374
  • [8] G. Pecelli and E. S. Thomas, Normal modes, uncoupling, and stability for a class of nonlinear oscillators, Quart. Appl. Math. 37, 281-301 (1979) MR 548988
  • [9] Y. Nejoh, A nonlinear normal mode in an anharmonic lattice with a single defect, Physica Scripta 28, 561-564 (1983)
  • [10] W. E. Ferguson, H. Flaschka, and D. W. McLaughlin, Nonlinear normal modes for the Toda chain, J. Comput. Phys. 45, 157-209 (1982) MR 657490
  • [11] S. Briere, Nonlinear normal mode initialization of a limited area model, Mon Weather Rev, 110, 1166-1186 (1982)
  • [12] P. J. Melvin, On the construction of Poincaré-Lindstedt solutions: The nonlinear oscillator equation, SIAM J . Appl. Math. 33, No. 1, 161-194 (1977) MR 0441050
  • [13] R. R. Dasenbrock, A FORTRAN-based program for computerized algebraic manipulation, NRL Report 8611, Naval Research Laboratory, Washington, D. C., 1982
  • [14] A. Deprit and A. Rom, The main problem of artificial satellite theory for small and moderate eccentricities, Celestial Mechanics 2, 166-206 (1970)
  • [15] T. A. Bray and C. L. Goudas, Doubly symmetric orbits about the collinear Lagrangian points, Astron. J. 72, No. 2 (1967)
  • [16] 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)
  • [17] K. R. Symon, Mechanics, Second Edition, Addison-Wesley, Reading, Massachusetts, 1960
  • [18] H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, Massachusetts, 1950 MR 0043608
  • [19] 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) MR 525450
  • [20] A. Deprit, Celestial mechanics: Never say no to a computer, J. Guidance, Control, and Dynamics 4, 577-581 (1981) MR 639517
  • [21] 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) MR 544092
  • [22] 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) MR 759703
  • [23] 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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DOI: https://doi.org/10.1090/qam/973381
Article copyright: © Copyright 1988 American Mathematical Society

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