Watching the “Clickspring” series of videos on hobby clockwork, I came across the section about the “stopwork”, which stops the mainspring from being wound by more than, say, five turns.
The way this is done is very simple. A pair of defective gearwheels mesh; one is held in place only by light friction applied by a spring, and has five teeth, with the rest of the gear solid out to the tip of the teeth (the addendum circle), so if it were to be meshed with another, non-defective wheel, it would only be able to rotate a fraction of a rotation before locking, as the teeth of the other wheel crash into the edge of the solid disc where no teeth have been cut.
However, it is instead meshed with a gearwheel that has been filed down to just the base circle, except for a single tooth that remains protruding. So, for most of each rotation, this single-toothed wheel doesn’t contact the counter wheel at all, but when it does, it advances the counter wheel by a single tooth — unless the counter wheel has already rotated all five teeth, in which case the single tooth crashes into the solid disc much as the teeth of an ordinary gearwheel would have done.
Thus an up/down counter is provided, one which blocks further motion upon reaching the end of its count. Its memory is retained in an entirely analog fashion by friction, although the count being remembered is essentially digital.
It occurred to me that a different approach to solving this problem is to use a “Geneva drive” mechanism (aka “Maltese cross”, “Geneva stop”, or sometimes “Geneva wheel”) which is similarly defective, with one of the slots for the drive pin being blocked, so the drive wheel can only spin three, or four, or six rotations, or whatever.
This is apparently the original use of the Geneva drive!
The Geneva drive does not depend on friction and thus is invulnerable to vibrations, and moreover is susceptible to being chained in a way that ordinary gear wheels are not, which requires further explanation.
The defective gear wheels in the stopwork mechanism demonstrated by Clickspring have the property that the “mechanical advantage” is fairly accurately 1 during the moment when they are engaged, since they happen to have the same pitch diameter, 0 when they are not engaged, and then ∞ when they are locked. This contrasts with the simplest straightforward stopwork mechanism in which a drive pinion spins a larger wheel which encounters a stop at some point in its rotation; if the drive pinion is to be allowed to turn 8 times, for example, we might drive a 73-tooth wheel with an 8-tooth pinion, occupying 9/73 of its rotation with a stop. But this stop needs to resist 8 times the torque applied to the drive pinion. The defective-wheel mechanism does not have this problem.
But the Geneva drive permits carrying this further: not only can we arrange for its mechanical advantage to average 1 during the driving part of the cycle, but we can use one such wheel to drive another, which drives another. (Hmm, maybe that’s not such a big difference after all; the one-tooth driver can do the same if it’s driving an ordinary gearwheel, after all.)
A single Geneva wheel can be made with arbitrarily many slots, at the cost of pushing the duty cycle up toward 50% with a single drive pin. In the limit of 50% you have an intermittent-motion version of a rack and pinion, with the possibility of endstops.
In the Geneva wheel’s original use as a stopwork, it was in fact a limit on differential rotation: it limited not the absolute rotation of the inner shaft of the mainspring or its outer barrel but their relative rotation.
By rotating one or more disc sectors in a plane parallel to the Maltese cross, it is possible to obstruct the entry of the drive pin into the slot. But perhaps a more interesting possibility for logic is axial displacement; in the usual construction, the drive wheel has a ward in the form of a partial circle that nestles into the Maltese cross to keep it from turning when the drive pin is not engaged, with a cutout in the circular ward around the drive pin to allow the Maltese cross to rotate when the drive pin is engaged. But the circular ward could be made from part of a solid round shaft, along which the Maltese cross can be slid; if the cutout does not extend along its entire length, then sliding the cross up it by the thickness of the cutout and enough to clear the pin, the cross can be exempted from being incremented or decremented by the next passage of the pin.
This potentially gives us a clocked-logic system similar to Drexler’s rod logic or Merkle’s buckling-spring logic, though probably less suitable for miniaturization than the latter, since it relies heavily on not only contact but sliding contact.