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Pendulums Coupled by a Torsion Spring

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Two identical frictionless pendulums coupled together by a torsion spring oscillate with equal displacements and velocities. Unstable synchronized states lead to chaotic behavior.

Contributed by: Enrique Zeleny (April 2013)
Open content licensed under CC BY-NC-SA

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The equations of the system are

,

,

where is the spring constant, proportional to the stiffness of the spring. The total momentum of the system is conserved because the forces are equal and opposite, so the system satisfies Newton’s third law.

References

[1] J. C. Sprott, Elegant Chaos: Algebraically Simple Chaotic Flows, Singapore: World Scientific, 2010 pp. 121–123.

[2] E. J. Banning and J. P. van der Weele, "Mode Competition in a System of Two Parametrically Driven Pendulums; the Hamiltonian Case," Physica A 220(3–4), 1995 pp. 485–533. doi:10.1016/0378-4371(95)00153-X.

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