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: 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.
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J. H. Stoker, Nonlinear vibrations, Interscience Publications, New York, 1950
- Chihiro Hayashi, Forced oscillations in non-linear systems, Nippon Printing and Publishing Company, Ltd., Osaka, 1953. MR 0064248
- 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
A. G. Webster, The dynamics of particles, Dover Publications, New York, 1959
J. H. Stoker, Nonlinear vibrations, Interscience Publications, New York, 1950
C. Hayashi, Forced oscillations in nonlinear systems, Nippon Printing and Publishing Co., Osaka, 1953
A. A. Andronow and C. E. Chaikin, Theory of oscillations, Princeton University Press, 1949
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Article copyright:
© Copyright 1962
American Mathematical Society