> There's another tide computer on display in the Science Museum London, which uses Cones moving on plates.
Here's how a mechanical harmonic synthesizer and analyzer work.[1] This is a very clear presentation of Michelson's design. It does forward and reverse Fourier transforms. Reverse (synthesis) is obvious enough, but the trick for analysis is clever.
The hardest part of the problem was the addition. Sine and cosine were easy. Multiplication by a constant was easy. But adding - that was hard.
The Michaelson machine has an elegant solution. The positions to be added apply tension to springs, which pull on a common pull bar opposed by a single spring. So the addition happens in force space.
The USGS tide predicting machine #2 [2] added up the terms by running a long chain over many pulleys that can slide. The position of the end of the chain sums the pulley positions. This is a true positional sum, and is more accurate. But the mechanism is much bulkier and heavier.
[1] https://www.youtube.com/watch?v=GyYflzRVu6M
[2] https://en.wikipedia.org/wiki/Tide-Predicting_Machine_No._2