J Austral Math Soc Ser B 32 pp23--41, 1990.

On a periodically forced, weakly damped pendulum. Part 2: Horizontal 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 driven by horizontal, periodic forcing of the pivot with maximum acceleration eg and dimensionless frequency w. Analytical solutions for symmetric oscillations at smaller values of e are continued into numerical solutions at larger values of e. A wide range of stable oscillatory solutions is described, including motion that is symmetric or asymmetric, downward or inverted, and at periods equal to the forcing period T º 2p/w or integral multiples thereof. Stable running oscillations with mean angular velocity pw/q, where p and q are integers, are investigated also. Stability boundaries are calculated for swinging oscillations of period T, 2T and 4T ; 3T and 6T ; and for running oscillations with mean angular velocity w. The period-doubling cascades typically culminate in nearly periodic motion followed by chaotic motion or some independent periodic motion.

Browse the article

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.