On a periodically forced, weakly damped pendulum. Part 2: Horizontal forcing

Abstract

AbstractWe 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 εg and dimensionless frequency ω. Analytical solutions for symmetric oscillations at smaller values of ε are continued into numerical solutions at larger values of ε. 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 ≡ 2π/ω or integral multiples thereof. Stable running oscillations with mean angular velocity pω/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 ω. The period-doubling cascades typically culminate in nearly periodic motion followed by chaotic motion or some independent periodic motion.