The oldest solution to this problem is to make the pendulum very long, and have it describe a very small angle, so that the system is locally linear and the period is independent of amplitude for all practical purposes. Furthermore, every clock will turn out to have a preferred pendulum amplitude, so it's really only important to get the time constant right for that amplitude. There is another solution, which is to make the pendulum arm of a flexible material, like a thin wire or string, and to confine its upper reaches between two convex-downward guides which come to a cusp at the fulcrum. This has the effect of shortening the arm for large excursions. The correct guide shape will direct the pendulum bob to trace a cycloid. I don't remember the correct shape: it's something simple, like a circle or another cycloid. But it doesn't matter how complicated the guide shape is: it's fairly easy to carve or bend it to within tolerance by trial and error. It turns out not to be worth the effort in increased accuracy, though. On 11/25/12, Henry Baker <hbaker1@pipeline.com> wrote:

All this talk about antikytheras led me to think about a cycloidal escapement for a clock.

The problem is that the rolling circle has to roll on the _ceiling_, not on the floor, according to Lawlor's Brachistochrone paper.

Ok, so we put a "rack" (a linear gear) on the ceiling & parallel to the ceiling.

Then we _hang_ a gear wheel from a rail which runs parallel to the ceiling. The hanger has a small rolling mechanism similar to that in a "sliding" door.

The gear teeth of the hanging gear wheel engages the teeth in the linear gear rack on the ceiling.

Now, if we push the entire gear wheel slide forward, the engaged gear teeth will rotate the gear wheel.

So far, so good.

We now make the wheel extremely light with as low a moment of inertia as possible -- e.g., magnesium.

We also make the sliding hanger mechanism as light as possible, with as little friction as possible.

We then mount a lead weight on the gear wheel.

With small-to-medium displacements of the wheel, the lead weight will now move in a cycloidal path, similar to Huygens's path.

We can now arrange for some other mechanism (light actuated, e.g.) which notices when the amount of motion becomes too small, and gives the wheel a small additional kick to add energy.

This is what I came up with in a few minutes.

Perhaps someone here can come up with a more elegant version.

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