The figure-of- 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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[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)

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
70M05,
70-04,
70K20,
70K40

Retrieve articles in all journals with MSC: 70M05, 70-04, 70K20, 70K40

Additional Information

DOI:
https://doi.org/10.1090/qam/973381

Article copyright:
© Copyright 1988
American Mathematical Society