J Austral Math Soc Ser B 32 pp1--22, 1990.
(Received 12 May 1989; revised 18 October 1989)
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 e are continued into numerical solutions at larger values of e. Resonance curves and stability boundaries are presented for downward and inverted oscillations of periods T, 2T, and 4T where T(º 2p/w) is the dimensionless forcing period. The symmetry-breaking, period-doubling sequences of oscillatory motion are found to occur in bands on the (w, e) plane, with the amplitudes of stable oscillations in one band differing by multiples of about p 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 e, the banded structure being related to the periodicity of the J0 Bessel function.