Dynamics of Coupled Pendulums

Consider a system of two identical pendulums swinging in parallel planes and connected at the top by a flexible string. (A recently published Demonstration considered the related version of two pendulums with their bobs connected by a massless spring.) We describe here a more classic form of the problem, going back to the time of Huygens.
Each pendulum consists of a mass suspended from a fixed support by a massless string of length , with its coordinate described by the angle from the vertical. Under the action of gravity, the equation of motion is given by . Restricting ourselves to small-amplitude oscillations, we can approximate , leading to the elementary solution , where , the natural frequency of the pendulum, independent of . For the coupled pendulum system, to the same level of approximation, the Lagrangian can be written where is a measure of the coupling between the two pendulums mediated by the connecting string. The equations of motion are readily solved to give .
The system has two normal modes of vibration. In the in-phase mode, starting with , the two pendulums swing in unison at a frequency of . In the out-of-phase mode, starting with , they swing in opposite directions at a frequency of .
A more dramatic phenomenon is resonance. If one pendulum is started at and another at , then energy will periodically flow back and forth between the two pendulums.



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Snapshot 1: in-phase normal mode, starting with
Snapshot 2: out-of phase normal mode
Snapshot 3: two pendulums showing resonant exchange of energy
See this YouTube video for a real-life visualization of this Demonstration.


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