(This note has several calculations I’ve noted errors in that need redoing.)
It’s often stated that one of the advantages of the Chinese windlass mechanism is that it’s self-locking, or not back-drivable. Another advantage is that, being a differential mechanism, its mechanical advantage can be arbitrarily large. A third advantage, rarely remarked upon perhaps because of its obviousness, is that since most of the mechanism is purely in tension, it can extend over a great distance while containing very little mass; it is practical to construct a Chinese windlass that applies substantial forces between points a hundred meters distant, weighing only 200 grams, half of which is 1-mm UHMWPE cord. A fourth is that most of the contact in the mechanism is not sliding contact and thus does not cause abrasive wear or frictional losses; only the bearing of the windlass drum has sliding contact, and possibly the bearing of the pulley.
In a sense the third advantage above is the opposite extreme from Reuleaux’s definition of a machine as something that imposes motion in desired degrees of freedom but prevents it in all others — that is, has effectively infinite rigidity in all other degrees of freedom. The differential pulley in the windlass mechanism is imposes motion in one desired degree of freedom, employing motion in a second degree of freedom (usually attached to a bearing so this can be ignored); it is somewhat restrained in a third degree of freedom, though not very rigidly; and has effectively infinite compliance in the remaining four degrees of freedom. In a well, that is, the bucket can swing back and forth and twist around while receiving only very small restoring elastic forces from the rope’s minuscule rigidity.
One disadvantage of the mechanism is that the differential mechanical advantage can be somewhat imprecise; as layers of rope build up on the drums, they change the drums’ effective radius and potentially their difference in radius. Grooved drums can prevent this from happening, but only if the drums are long enough.
The M.A. of the mechanism is the ratio between the crank arm and the difference in drum radii. This implies that the absolute drum radii can be as large or small as desired without changing the M.A. However, if the difference in radii is, say, 1mm, you only get 6.28mm of elongation per revolution, regardless of whether that revolution is running 100 mm of cord through the differential pulley or 100 m of it. So, this allows you to increase the rigidity of the drum, which might allow you to increase its length, thus permitting more unlayered cord, but not to use less layers of cord in the same length.
The wrapping of the cable on the drum can be protected from side-loadings by running the cable through a grommet in between the drum and the differential pulley. That way the angle at which the cable rolls onto the drum only depends on where the cable is on the drum and the tension on the cable, not on any other side-loadings. If the grommet is sort of hyperboloid-of-one-sheet-shaped, it will avoid kinking the cable there and avoid any concentrations of force at one point on the grommet, and if it is made of a hard material such as sintered sapphire, like sparkplugs, it will not suffer much from abrasion.
The self-locking nature of the mechanism is an advantage for some uses, but being able to use an arbitrarily large mechanical advantage in reverse would be useful in some situations. The reason it is usually self-locking is that the frictional torque is the side loading on the bearing, multiplied by the bearing’s radius, multiplied by the bearing’s coefficient of friction μ; and the frictional force resisting the differential pulley is that torque divided by the effective moment arm, which is the difference in radii. Typical coefficients of friction for dry journals are 0.2–1.6; for example, bronze on cast iron is 0.22, wood on dry wood is typically 0.25–0.5, steel on steel is 0.5–0.8, and copper on copper is 1.6. Usually the side loading on the bearing is the force on the differential pulley. So, if μ = 0.22, then 100 N of pull on the differential pulley will generate 22 N of friction at the bearing.
Suppose, for example, that the journal is 20 mm radius (40 mm diameter), and the radius difference is 10 mm (say, the drums are of radii 50 mm and 60 mm). So the torque on the shaft from the differential pulley is 100 N · 60 mm - 100 N · 50 mm = 100 N · (60 mm - 50 mm) = 100 N · 10 mm = 1 N m. And the torque from the friction is 22 N · 20 mm = 0.44 N m. So in this case the mechanism will not be self-locking. (It will be somewhat efficient: 1.44 N m from the crank will be converted to 1 N m at the drum and thus 100 N, for 69% efficiency.)
If we increase μ past 0.5, though, (the reciprocal of the M.A.), the frictional torque rises past 1 N m, the mechanism becomes self-locking, and its efficiency falls below 50% (assuming static and dynamic friction are equal), because the frictional force has exceeded the force from the slow/strong end of the mechanism, the differential pulley. The same thing happens if we leave μ at 0.22 and increase the M.A. past 4.55, for example by increasing the journal radius from 20 mm to 46 mm or by decreasing the difference in radii from 10 mm to 4.3 mm.
This is a case of a general phenomenon where mechanisms with efficiency above 50% are backdrivable, and those with lower efficiencies are instead self-locking, under certain simplifying assumptions.
But a M.A. of up to 5 is a far cry from an arbitarily large (inverse) M.A. What can we do if we want to use this mechanism to make things go fast instead of slow? Like, what if we want to pull a string to spin something at 15000 rpm by hand to generate electricity? Maybe something only 120mm in diameter, so it can be comfortably handheld?
First, consider the parameters of the problem: a person can pull about 200 N at about 3 m/s for about 1 m. (Think of someone trying to start a lawnmower or chainsaw with a pullcord.) The edge of the disc, which will be generating the alternating magnetic field that generates electricity, will be moving at about 94 m/s. So we need a M.A. of more than 62, probably at least 200 to be safe. We’re missing a factor of 40.
With modern UHMWPE fiber and its 3-GPa tensile strength, 200 N requires a 290-μm-diameter cord; better say 500 μm to be safe, which should be good to 580 N. This means that a single layer of wrapping on a 20-mm-long barrel holds 40 revolutions, and each layer adds 500 μm to the radius. An 8-mm-diameter barrel with its 25-mm circumference would hold about 1 m of cord per wrapping layer.
We can make some progress on the problem by using better bearing
materials. Steel on polyethylene has μ ≈ 0.2, on graphite, μ ≈ 0.1;
on teflon, 0.05–0.2; and on tungsten carbide, 0.4–0.6. Tungsten
carbide on tungsten carbide is listed as 0.2–0.25. Materials with
lower μ might be worse if they require a larger journal radius to
compensate for lower compressive strength; WC’s 3+ GPa compressive
strength would theoretically allow it to bear 200 N on a 250-μm-long
250-μm-diameter shaft, if it doesn’t bend too much, for example.
XXX I’ve confused circumferences and radii so there’s a missing factor of τ from some of the below
Or a 500-μm-long 125-μm-diameter WC shaft, maybe somewhat tapered to reduce the risk of breakage. That would give you a 62-μm bearing radius, so each 100 N of side load produces, say, 25 N of friction, but only 1.55 millinewtonmeter of friction torque. So if your difference in radii is, say, 30 μm, say because one drum is 8.00 mm in diameter and the other is 8.06, the 100 N will produce 3 millinewtonmeters of torque, which is twice as much as the friction, so it will be able to backdrive the mechanism. The M.A. to the magnet disc, then, will be 60 mm / 30 μm, or 2000. Our 3 m/s human pullcord would be able to spin the rim of the magnet disc at 6000 m/s, or Mach 18, spinning at 955000 rpm. Also, the ⅓ of the 200 J lost to friction, or 60 J, would be deposited in the tungsten carbide bearings, which are some 0.1 mm³ or about 1.6 mg of WC; with its room-temperature specific heat of 200–480 J/kg/K its temperature would rise to between 78000° and 188000°.
Although the bearings could withstand the mechanical pressure at room temperature, they would vaporize and the disc would explode. Also, any practical mass of disk would make the pullcord too hard to pull. And extending the pulley pull handle by 1 m at 30 μm of difference per revolution and 25 mm of cord per revolution would require 33000 revolutions, unwinding some 800 m of cord from one drum and winding it up on the other. Clearly this is taking things too far!
Let’s try backing off to our planned M.A. of 200: a difference of radii of 300 μm. And let’s consider ordinary cast iron on bronze: μ = 0.22. Cast iron can withstand 500+ MPa of compression, but UNS C93200 (SAE 660) bearing bronze only some 300 MPa, with a fatigue strength of only 110 MPa. Also, consider that there are two bearings, one at each end, so each can bear half the 200 N load. 100 N ÷ 110 MPa gives 950 μm diameter × 950 μm length, say, or 450 μm diameter × 2 mm length, giving 225 μm radius (probably, again, an average over a slight taper). 200 N · 0.22 · 225 μm = 9.9 millinewtonmeters of friction, and 200 N · 300 μm = 60 millinewtonmeters of applied differential torque. The leftover 50 mN m manifests as an 0.8 N resistance at the 60-mm-radius generator ring, which should be quite straightforward to produce by, for example, generating electricity through pancake coils.
This is an improvement, but we still face the dismaying prospect of 3000 revolutions of the main spindle unrolling 75 m of UHMWPE thread from one drum and rolling it onto the other. That’s a lot of revolutions! And the 17% power loss in the main journal bearings is worrisome not only because it’s wasteful but also because of the heat problem.
By adding an additional idler pulley behind one of the differential drums, so that the side loadings from the two strings are in opposite directions, you can reduce the side loadings on the drums’ bearings. The idler adds some friction, requiring a journal similar in stoutness to that of the main wheel, but it can be quite narrow (<1 mm) and large in diameter (40 mm, say). So the string running around the idler has a lever arm of 20 mm with which to counteract the 0.225-mm lever arm of the journal’s friction, an M.A. of 89, which when divided by μ = 0.22 gives us a factor of some 400. So the idler will consume about ¼% of the energy; and it ought to be able to reduce the friction in the bearings of the main differential wheel considerably, perhaps by a factor of 2, which matters a great deal more, because their friction has a great deal more M.A.
Instead of just having one idler pulley, one differential pulley on the pull handle, and one spindle, we can improve the situation further: by adding three more idlers in the body and three more in the pull handle, with an additional 1.5% efficiency loss, we can run the cable back and forth between the main body and the pull handle eight times instead of twice. This enables us to use a much smaller M.A. in the differential windlass mechanism itself — 50 instead of 200, realized by paying out 1.2 mm of differential cable per revolution of the drums rather than 0.3 mm. By pulling the two parts of the mechanism apart by 1 m, 8 m of cable is demanded of the differential windlass itself, which is still (!?) 6700 revolutions. But now each cable only bears 25 N (XXX, I was overspeccing it above by a factor of 2) and so can be hair-thin: 150 μm diameter at 3 GPa should withstand 50 N. And the side-loading imposed by these threads on the spindle, and the frictional losses in the journals, are correspondingly lower.
It’s important to design the handle so the pulleys can pivot to equalize tension among these threads; otherwise you will accidentally put all the tension on one thread while yanking the handle and break it.
Suppose our windlass barrels are 30 mm long and 7 mm in diameter. Each layer of this thread on the barrel is 200 revolutions, 22 mm in circumference, and thus holds 4.4 meters of thread. If this circumference remained unchanged, there would be 33 layers, totaling 4.95 mm in thickness, on a fully charged barrel. This is clearly a practical volume of thread.
I’m still concerned, though, about the thread thickness changing the barrel radius and thus the M.A. 1.2 mm of circumference difference is only 190 μm of radius difference: barely more than a single layer of thread.
It’s worth mentioning that in such a mechanism the electronics could be entirely sealed away from the rotor and pulleys, communicating exclusively through magnetic fields.