Winding/Weight thoughts

I happened upon Mark Frank's Astronomical Skeleton Clock and was interested to see his use of a compound barrel which apparently has only be used once before in a Martin Burgess' design for a sculptural clock.

Well the idea is very ancient and used around 560AD.

It is a Chinese windlass, or a differential pulley, and has different diameter drums; one winding up and t'other down. It can give fantastic mechanical advantage, up to 50:1. He is using it for an 8:1 advantage.

The logic is completely counter intuitive to me.

Neglecting friction, the mechanical advantage = 2R/(R - r) and the weight drops by π(R-r)/2 per revolution. So as the difference between the two radii becomes small, the mechanical advantage increases! But at the other end, it is not possible to go as low as a 2x factor.

Now compare this to the Leverage Pulley I found - this diagram, for a turret clock, should explain how it works, and this is what was said about it...
The differential in cable lengths is due to the Leverage Pulley requiring two separate lengths of cable, while normal compounding can use a single length of cable. The Leverage Pulley was probably used because it would provide for a slightly longer running time, and would avoid the friction and cost associated with the two (extra) pulleys needed for conventional 3-fall compounding. Disregarding friction, and the half-diameters of the wire rope, this assumed 3:1 ratio will reduce the force felt at the clock winding barrel to a third of the actual weight, which will require that the weight be increased by a factor of 3. In this case, to reduce the weight drop required, 2 fall compounding would require a doubling of the weight, and would yield a total weight drop of about 22 .4 feet. Three fall compounding would yield a total weight drop of almost 15 feet and the driving weight would have to be three times as heavy.

Whoever designed this system had his head on his shoulders. He realized that three fall compounding was out of the question because of the limited weight drop available, and came up with an ingenious solution that used a little more rope, and achieved the same end, with less friction and expense for the extra pulleys. The need for a very short weight drop is, I suspect, the reason this odd system was used. Very simply, it reduces the overall complexity of a three-fall weight system.

In this system the advantage is easily calculated as the ratio of the two radii (R & r) and the drop is 2πr per revolution of the Leverage Pulley, but from the clock's winding drum/barrel (say radius B) the drop is 2πBr/R, which if we assume B is always smaller than R, this is quite good and better than a conventional compound pulley system.

Now this set me thinking (foolish thing to do!) - Clayton's design uses a 3/4" diameter drum and 5-6lbs of weight (on a 2 compound pulley). Lets assume at most I need 3lbs (w below) at the 3/8" radius drum. What if I increased the R:r ratio and drum radius? And how much total weight (W below) do I need?

B	w	R	r	factor	W (lbs)	days/drop (58")
3/8	3	4	2	2	6	2.05	basic 2 day clock, better than a compound pulley
3/4	1.5	4	1	4	6	2.05	will this give less strain on the Minute arbor?

3/8	3	3	1	3	9	3.07	bit longer to allow for weekends away
1/2	2.25	4	1	4	9	3.07

3/8	3	3	3/8	8	24	8.20	eight day clock, but will this be too much weight!
1/2	2.25	4	0.5	8	18	6.15

How about the Chinese Windlass idea?

B=r w (lbs) R factor W (lbs) days/drop (58")

3/8 3 6 2.13 6.4 0.27 Note from the formula, it is no good until x 3

3/4 1.5 1 1/2 4 6 2.05 same as Leverage Pulley

3/8 3 1 1/8 3 9 2.05 looses a whole days worth of running

1/2 2.25 1 4 9 3.07 same as Leverage Pulley

3/8 3 1/2 8 24 12.31 very interesting and OTT

1/2 2.25 2/3 8.02 18.1 9.27 better than Leverage Pulley

So for large factors the Windlass is better. I really need to do a bit more working of combinations and also include the total cord length (as it determines the depth of the pulleys). Overlapping cord will really make a difference to the Windlass factor and with large radii, as they will use a lot! But I wonder how much this matters on this inaccurate clock - but I'm sure that 3 days running is the realistic maximum.

However this is all Phase 2, but it has made me want to do a revision and abandon the 2nd winding pulley and use a key, inserted in between the Minute wheel spokes, instead. This should also reduce the friction in the train - why didn't I think of that earlier?.

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