## Pendulum Waves

I recently stumbled upon a hypnotic video of 15 out-of-phase pendulums from a physics experiment at Harvard University.

The pendulums have monotonically increasing lengths, designed to give each specific periods. Given that the pendulums start swinging at a relatively shallow angle, their periods do not vary significantly with their initial angle; hence their periods are known to a reasonable degree of accuracy. The longest swings at 51 oscillations per minute, with the shortest at 65 oscillations per minute,

This is a side-by-side video of a MapleSim model of the experiment, and the experiment itself.  I’ve tried to synchronize the movies so that the pendulums are released at the same time (but given the framerate of the original video, there’s an error of the order of 0.03s)

We have a relatively good match between the actual experiment and MapleSim, desipite the multiple sources of uncertainty and error! You see the same patterns and harmonics in both the real experiment and the model.  About 30s into the animation, half the pendulums are at either side of the swing, with a full cycle of the patterns taking one minute.

At times, the motion appears to be random, but that's simply because we're effectively sampling the motion at just fifteen points.

In creating the MapleSim model (attached to the bottom of this post), I removed the Izz term of the rotational inertia matrix of each rigid body to correctly model a simple pendulum (otherwise we'd have a compound pendulum).  I used the following power series expansion for the period of a pendulum with an arbitrary amplitude to determine the length of each model pendulum.

T is the desired period, θ is my chosen amplitude, L is the length, and g is the gravitational constant.

As a bonus feature, here's a video of the pendulums swinging at at ½ realtime

PendulumWaves.msim

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