(Related to Differential spiral cam, which describes a way to use a similar mechanism to get a complex programmed sequence of motions from a simple mechanism.)
The standard Chinese windlass mechanism gets a very large mechanical advantage from a simple mechanism by using a pulley as a differential:
rope
__******__
|||||| | rope crank
|||||| |__******__ _____
|||||| |||||| | | ___|
|||||| |||||| |___| |
|||||| |||||| _____|
|||||| |||||| |
|||||| __||||||__|drums
|||||| | |*****
__||||||__| |
****** |
| |
| |
| ___ |
|/pul\|
||ley||
\\___//
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Rope is wound in the same direction around two drums of different sizes which rotate together, typically turned by a hand crank, or by another wheel with an endless chain or rope or belt around it; rotating the drums will always pay out rope from one and take up rope from the other. The rope runs down from one drum, around a free-hanging pulley, and back up to the other drum. So the pulley moves up or down by half the difference in the rope paid out from one drum and taken up by the other.
Through the magic of the differential mechanism, this very simple mechanism provides an arbitrarily large mechanical advantage, inversely proportional to the absolute difference in diameter of the two drums. If the two drums are equal in size, the mechanical advantage is infinite.
To take a more practical example, if you want to lift two tonnes (20 kN) by exerting 100 N on a 200-mm-radius crank, you need a mechanical advantage of 200: a 2-mm difference between the two drums will provide you with this. (That’s in radius; it’s a 4 mm difference in diameter.) Perhaps one drum is 50 mm in diameter, while the other is 54 mm in diameter, or perhaps one is 400 mm in diameter, while the other is 404 mm in diameter (although it may be difficult to find such a drum); the mechanical advantage is the same either way.
Let us suppose that we use a 50-mm and a 54-mm drum, or rather bar. Each lifting revolution lifts the pulley by 2πmm ≈ 6.3 mm, paying out π50mm ≈ 157 mm of rope from one bar and taking up π54mm ≈ 170 mm of rope on the other bar. Lifting the two-tonne weight by 2000 mm requires some 320 turns of the crank, paying out 50.24 meters of rope from one bar and taking up 54.4 meters of rope wrapped onto the other.
This has the somewhat annoying result that you will need 55 meters of one-tonne rope or webbing wrapped around your windlass to lift the two tonnes by only two meters. This rope might average 3 mm thick, and if we want predictable mechanical advantage, it cannot freely wrap in multiple layers — that would increase the effective diameter of the windlass by 6 mm. And we need about a meter of length of each drum, since we need 320 turns of 3-mm rope, one right next to the other. This has the rather dismaying result that our pulley hangs down by two meters but across by half a meter relative to its attachment point on each bar:
-------------__________________
)) )))))))))))))))|-_ crank
_____________------------------
\ big bar / little bar
\ / (with most of
\ / the rope)
\ /
\ / more rope
\o/
This is going to put rather significant side forces on the rope that will attempt to slide it along the length of the bar, add extra tension to the rope that isn’t part of the load it’s bearing, and change the mechanical advantage.
So, the alternative I was thinking about is that, instead of two drums, you can use a single tapered conical pulley with a single spiral groove on it, and run the pulley between two near-adjacent turns of the groove, one with a radius 2 mm greater than the other.
For example, suppose the groove is 3 mm wide, and there is a 16.5-turn blank spot with no rope on it in between the takeup part and the payout part. This makes the blank spot 49.5 mm long and means that the pulley tapers by 2 mm of radius every 49.5 mm of its length, an included angle of 4.6°. Because you still need 320 turns of rope, you still need a meter-long tapered cone, but now it’s only one meter long instead of two, or 1.01 m to be more exact. Its radius at the wide end is 40.8 mm greater than at the narrow end, and the pulley hangs from two points that are only separated by 49.5 mm of length and, say, 40 to 80 mm of width.
(You might be able to squish adjacent turns of rope together a bit if you skimp on making a full half-circle profile for the groove, or make it a bit elliptical in cross-section, deeper than a perfect torus, I don’t know. Maybe you could shorten the apparatus by a third that way.)
There are three different reasons for wanting very large mechanical advantages like the one provided by this mechanism. The first, discussed above, is to develop very large forces. (Tapered thread discusses another very simple mechanism that does this using a tapered helix.) A second is precisely the reverse — achieving very high speeds but low forces by applying force to backdrive the mechanism through its stiffer part. In this case this is not applicable because the frictional losses are too large, and this self-locking behavior is considered a feature in the original use of the Chinese windlass to lower buckets down wells. The third is to achieve very high precision by precisely translating a reasonable-sized movement, such as moving the crank by a millimeter, into a much smaller movement, such as moving the pulley by ten microns. Differential screws are an analogous mechanism commonly used for such high-precision movements.
With high-rigidity materials like
I worry, though, about the unknown and presumably load-dependent amount of squish in the transverse dimension of the cable as it wraps around the drum.