J Austral Math Soc Ser B 32 pp42--60, 1990.

On a periodically forced, weakly damped pendulum. Part 3: Vertical forcing

Peter J. Bryant and John W. Miles

Abstract

We consider the phase-locked solutions of the differential equation governing planar motion of a weakly damped pendulum forced by a prescribed, vertical acceleration egsin wt of of its pivot, where w and t are dimensionless, and the unit of time is the reciprocal of the natural frequency. Resonance curves and stability boundaries are presented for downward and inverted oscillations of periods T, 2T, 4T, ¼, where T (º 2p/w) is the forcing period. Stable, downward oscillations are found to occur in distinct regions of the (w, e) plane, reminiscent of the regions of stability of the Mathieu equation (which describes the equivalent undamped, parametrically excited pendulum motion). The regions are dominated by oscillations of frequencies (1/2)w, w, (3/2)w, ¼, each region being bounded on one side by a vertical state at rest in stable equilibrium and on the other side by a symmetry-breaking, period-doubling sequence to chaotic motion. Stable, inverted oscillations are found to occur also in distinct regions of the (w, e) plane, the principal oscillation in each region being symmetric with period 2T.

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Authors

Peter J. Bryant

Department of Mathematics, University of Canterbury, Christchurch, New Zealand.

John W. Miles

Institute of Geophysics and Planetary Physics, University of California at San Diego, La Jolla, California 92093, U.S.A.