On a periodically forced, weakly damped pendulum. Part 1: Applied torque

Abstract
Coplanar forced oscillations of a mechanical system such as a seismometer or a fluid in a tank are modelled by the coplanar motion of periodically forced, weakly damped pendulum. We consider the phase-locked solutions of the differential equation governing planar motion of a weakly damped pendulum driven by a periodic torque. Sinusoidal approximations previously obtained for downward and inverted oscillations at small values of the dimensionless driving amplitude ε are continued into numerical solutions at larger values of ε. Resonance curves and stability boundaries are presented for downward and inverted oscillations of periods T, 2T, and 4T, where T(≡ 2π/ω) is the dimensionless forcing period. The symmetry-breaking, period-doubling sequences of oscillatory motion are found to occur in bands on the (ω, ε) plane, with the amplitudes of stable oscillations in one band differing by multiples of about π from those in the other bands, a structure similar to that of energy levels in wave mechanics. The sinusoidal approximations for symmetric T-periodic oscillations prove to be surprisingly accurate at the larger values of ε, the banded structure being related to the periodicity of the J0 Bessel function.

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