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

Published online by Cambridge University Press:  17 February 2009

Peter J. Bryant  and
John W. Miles
Peter J. Bryant
Affiliation:
Department of Mathematics, University of Canterbury, christchurch, New Zealand.
John W. Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California at San Diego, La Jolla, California 92093, U.S.A.
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Abstract

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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.

Type
Research Article
Information
The ANZIAM Journal , Volume 32 , Issue 1 , July 1990 , pp. 1 - 22
Copyright
Copyright © Australian Mathematical Society 1990

References

[1] Bryant, Peter J. and Miles, John W., “On a periodically-forced, weakly damped pendulum. Part 2: Horizontal forcing”, J. Austral. Math. Soc. Ser B. 32 (1990) 23–41.CrossRefGoogle Scholar
[2] Bryant, Peter J. and Miles, John W., “On a periodically-forced, weakly damped pendulum. Part 3: Vertical forcing”, J. Austral. Math. Soc. Ser. B. 32 (1990) 42–60.CrossRefGoogle Scholar
[3] D'Humieres, D., Beasley, M. R., Huberman, B. A. and Libchaber, A., “Chaotic states and routes to chaos in the forced pendulum”, Phys. Rev. A 26 (1982) 3483–3496.CrossRefGoogle Scholar
[4] Feigenbaum, Mitchell J., “Quantitative universality for a class nonlinear transforma tions”, J. Stat. Phys. 19 (1978), 25–52.CrossRefGoogle Scholar
[5] Forbes, Lawrence K., “Periodic solutions of high accuracy to the forced Duffing equation: Perturbation series in the forcing amplitude”, J. Austral. Math. Soc. Ser. B 29 (1987) 21–38.CrossRefGoogle Scholar
[6] Miles, John, “Resonance and symmetric breaking for the pendulum”, Physica D 31 (1988) 252–268.CrossRefGoogle Scholar
[7] Miles, John, “Resonance and symmetry breaking for a Duffing oscillator”, SIAM J. Appl. Math. 49 (1989) 968–981.CrossRefGoogle Scholar
[8] Miles, John, “Directly forced oscillations of an inverted pendulum”, Phys. Lett. A. 133 (1988) 295–297.CrossRefGoogle Scholar