This fun post illustrates the phenomenon of multiple uncoupled pendulums whose periods are all rational multiples of each other.
Click here to view a higher resolution version of this (12 Mb).
A pendulum ensemble consisting of nine simple uncoupled pendulums of increasing lengths dance together to produce visual travelling waves, standing waves, beating and seemingly chaotic motion.
The lengths of the pendulums are designed so that all of them complete a different whole number of swings every 60 seconds. The first (longest) pendulum swings 25 times in 60 seconds, the next one 26 times, the next one 27, and so on; the final (shortest) pendulum completes 33 swings in the same interval. This means that every 60 seconds, all the pendulums will swing return to be in sync with each other again.
That is, if the periods are all rationally related to each, then eventually the entire ensemble will repeat. Conversely, if there is at least one pendulum whose period is an irrational multiple of another one, then the ensemble will never repeat itself.
(Note that this animation plays at half speed compared to actual physics models.)
Source code: The source code to make the above animation was forked from this code: https://pastebin.com/CHkLUPyD
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