Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Stability of forced oscillations of a spherical pendulum


Author: John W. Miles
Journal: Quart. Appl. Math. 20 (1962), 21-32
MSC: Primary 34.51
DOI: https://doi.org/10.1090/qam/133521
MathSciNet review: 133521
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Abstract | References | Similar Articles | Additional Information

Abstract: The equations of motion for a lightly damped spherical pendulum that is subjected to harmonic excitation in a plane are approximated in the neighborhood of resonance by discarding terms of higher than the third order in the amplitude of motion. Steady-state solutions are sought in a four-dimensional phase space. It is found that: (a) planar harmonic motion is unstable over a major portion of the resonant peak, (b) non-planar harmonic motion is stable in a spectral neighborhood above resonance that overlaps neighborhoods of both stable and unstable planar motions, and (c) no stable, harmonic motions are possible in a finite neighborhood of the natural frequency. The spectral width of these neighborhoods is proportional to the two-thirds power of the amplitude of excitation. The steady-state motion in the last neighborhood is quasi-sinusoidal (at the forcing frequency) with slowly varying amplitude and phase. The waveform, as determined by an analog computer, is periodic but quite complex.


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

  • [1] Arthur Gordon Webster, The dynamics of particles and of rigid, elastic, and fluid bodies: Being lectures on mathematical physics. 2nd ed, Dover Publications, Inc., New York, 1959. MR 0103146
  • [2] J. H. Stoker, Nonlinear vibrations, Interscience Publications, New York, 1950
  • [3] Chihiro Hayashi, Forced oscillations in non-linear systems, Nippon Printing and Publishing Company, Ltd., Osaka, 1953. MR 0064248
  • [4] A. A. Andronow and C. E. Chaikin, Theory of Oscillations, Princeton University Press, Princeton, N. J., 1949. English Language Edition Edited Under the Direction of Solomon Lefschetz. MR 0029027

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DOI: https://doi.org/10.1090/qam/133521
Article copyright: © Copyright 1962 American Mathematical Society


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