Yesterday I came across this video series, put up by "EngineerGuy" Bill Hammack. It shows a mechanical analog computer originally designed by Michelson for building up Fourier series (sums of sinusoids) of up to twenty integer multiples of a fundamental frequency. Moreover, you could use this machine to go backwards, and mechanically do Fourier decomposition of periodic waveforms. It's really wonderful. I would love to have one to use as a teaching tool, and I'm sure some enterprising person will figure out how to 3d print all the relevant parts (except the springs and cables), or perhaps build one out of Lego.
I also wanted to point out Hammack's other videos. This is great outreach - really accessible, clear, well-produced content.