Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



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
Full-text PDF Free Access

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] A. G. Webster, The dynamics of particles, Dover Publications, New York, 1959 MR 0103146
  • [2] J. H. Stoker, Nonlinear vibrations, Interscience Publications, New York, 1950
  • [3] C. Hayashi, Forced oscillations in nonlinear systems, Nippon Printing and Publishing Co., Osaka, 1953 MR 0064248
  • [4] A. A. Andronow and C. E. Chaikin, Theory of oscillations, Princeton University Press, 1949 MR 0029027

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 34.51

Retrieve articles in all journals with MSC: 34.51

Additional Information

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

American Mathematical Society