Beating Response: Two-DOF system

You might recall the classical experiment in Freshman physics of two pendulums that are coupled together wtih a soft spring. This pendulum system is another example of a system that exhibits beating in the response.

As an exercise, you might choose to derive the equations of motion of this system and find the natural frequencies and mode shapes.(From your results, note how the closeness of the two natural frequencies is related to the softness of the coupling spring.) Find the response of the system with the system being released from rest with the left pendulum being rotated slightly.

Shown below is a short video clip of the beating motion. Observe the exchange of energy between the two pendulums during the beating response. Can you see this energy exchange in your solution??