A cone drive is not two cones

One belt for every step sized a lathe’s stepped pulleys so that one flat belt fits every pair of steps. A crossed belt’s length depends on the two radii only through their sum, so steps whose radii sum to the same value take the same crossed belt exactly. An open belt’s length depends on their difference as well, and to take one open belt every step’s radius sum has to be cut by about $d^2/\pi C$, with $d$ the difference of the step’s two radii and $C$ the centre distance.

The same idea taken to its limit is a pair of drums with the belt shifted continuously along them, giving a continuous range of speed. Some early variable-speed drives were built this way, and they are drawn as two cones of opposite taper. That essay closed by asking what profile the drums need for one open belt to fit at every station along them, and how far a straight cone falls short. The question turns out to have two parts. The first is about the open belt. The second, which the steps never raised because each step could be sized on its own, is about how the speed changes as the belt moves.

The drive #

The drums in the figures span the lathe’s range from the stepped-pulley essay: a speed ratio from 1/2.56 to 2.56, with the two radii summing to 160 mm at the middle of the travel and the shafts 400 mm apart. The belt travels 300 mm along the drums from one end of the range to the other. Every length below is the taut run the strand field builds — two tangent lines and two arcs, each run leaving its pulley at a right angle — at the station where the belt is. The belt is treated as lying in one plane square to the shafts at that station, which is what a flat belt on a gentle taper does.

Straight cones fit a crossed belt, and grade the speed unevenly #

Two straight cones of opposite taper keep the radius sum constant along their length: where one is larger by some amount the other is smaller by the same amount. A crossed belt’s length depends only on that sum, as the wraps add up to a turn showed, so it fits both cones at every station exactly. Built, its length differs between stations by at most $2 \times 10^{-16}$ of itself, the last bit of the arithmetic.

What a straight cone does not do is change the speed evenly. The speed ratio at a station is $r_1/r_2$, and with both radii linear in the station $s$ it is

$$q(s) = \frac{S/2 + ks}{S/2 - ks},$$

a bilinear function of position that bends away from a straight line as the belt nears either end. A machine tool’s speeds are wanted a constant factor apart for the same reason a stepped pulley’s are: a cutting speed matters in proportion, so each move of the shifter should change the speed by the same proportion wherever it is made.

Measured along the travel, straight cones change the speed by 0.584% per millimetre of shift in the middle and 0.723% at the ends: 24% faster at the ends than in the middle. A shifter moved a centimetre near the end of its travel changes the speed by a quarter as much again as the same centimetre near the middle.

The drums for an even speed are logistic curves #

An evenly graded drive needs $\ln q$ to grow in proportion to the station: $q = e^{\kappa s}$, with κ set by the range and the travel. A crossed belt still needs a constant radius sum, and the two conditions fix both radii:

$$r_1 = \frac{S}{1 + e^{-\kappa s}},\qquad r_2 = \frac{S}{1 + e^{\kappa s}}.$$

Each drum’s radius is a logistic curve along its length — the S-shaped curve that runs from one level to another — and the other drum is the same curve reversed. Over this drive’s range it gives 0.627% per millimetre at every station, and a crossed belt fits it exactly, since the sum is still 160 mm everywhere.

The logistic drum is not far from a cone. Over the lathe’s range it departs from the straight cone that spans the same radii by at most 0.95 mm. It is fuller than the cone towards its large end and slimmer towards its small end, crossing it in the middle, and a quarter of the way along it sits 0.93 mm inside it. That is too little to see in a drawing and far too much for a belt, which is a member with no length of its own to take up a millimetre with, but it is the difference between an even speed grading and an uneven one.

An open belt needs the drums shaved #

An open belt wraps both drums the same way round, and its length depends on the difference of the radii as well as their sum. At constant sum it grows as $d^2/C$ plus a much smaller $d^4/12C^3$, which the stepped pulleys measured. So on drums cut for a crossed belt, an open belt made for the middle station is too short everywhere else.

On straight cones the open belt is 12.32 mm short at each end of the travel, exactly the stepped pulleys’ figure, because the outermost station has the outermost step’s radii. It is short everywhere between, in proportion to the square of how far the radii differ. The lengths of belt a line-shaft machine could take up by stretching were a millimetre or two, and this is ten times that.

The stepped pulleys corrected each step on its own, cutting its radius sum by about $d^2/\pi C$. Drums have a constraint the steps did not. A lathe’s cone pulleys were made as one shape, mounted one way on the countershaft and the other way on the spindle, and a continuous pair of drums is most simply made the same way: two identical drums, one reversed. Then the correction cannot be put on one drum. It has to be shared.

It can be, in two ways, and only one of them keeps the grading.

The simpler is to take the crossed-belt profile at each station and shave both radii by the same amount $g$. At a fixed difference of radii the belt’s length falls by $2\pi g$, so the shave that restores the middle station’s length is about half the stepped pulleys’ correction, $g \approx d^2/2\pi C$ — 1.96 mm at the ends of this drive — and since $d$ is equal and opposite at mirror stations the shave is the same at both, so one profile serves both drums. But an equal shave takes a larger share of the smaller radius than of the larger one. At the end of the travel it moves the ratio from 2.560 to 2.631: the belt fits, and the even grading the logistic curve was chosen for is gone at the ends.

The shave that keeps both is proportional. At each station, hold the ratio the logistic law demands and shrink both radii together until the open belt is the middle station’s length — the stepped pulleys’ construction, done at every station. The radius sum falls by about $d^2/\pi C$ as it did for the steps, less a second-order $2d^4/\pi^2C^2S$ because shrinking both radii shrinks their difference too, and each radius gives up its own share of that in proportion to its size.

Found exactly, by bisection on the open belt’s own run at each station with the ratio held, a drum gives up 2.69 mm at its large end and 1.05 mm at its small end, and the law matches every station to half a per cent. Every shaved station takes the middle station’s belt to $10^{-9}$ mm, and the ratio at every station is the logistic law’s exactly.

The drums are still one shape. An open belt’s length does not care which of its two radii is which, so the solution at a station’s mirror, where the ratio is inverted, is the same pair of radii swapped. Measured, the driving drum’s radius at every station equals the driven drum’s at the mirror station to $6 \times 10^{-14}$ mm. One template cuts both drums, and the drums are identical and reversed, as a lathe’s cone pulleys were.

So an open-belt drive with an even speed grading has drums that are logistic curves, shaved by about a millimetre at their small ends and nearly three at their large ends. Neither the logistic curve nor the shave is visible in a drawing: each departure is a millimetre or two on a drum eighty millimetres in radius, and the drawings of such machines show cones for the good reason that nobody could see the difference. The belt could.

Sharing the correction, and what the other ways cost #

Putting the whole correction on one drum is possible too, and it is what a designer would do if the two drums did not have to match: leave the driving drum a pure logistic curve and shave only the driven one. Found by the same bisection, the driven drum then needs 3.54 mm off where it is the larger of the pair, at one end of the travel, and 4.44 mm off where it is the smaller, at the other. The correction is lopsided, the ratio moves wherever the shave does, and the two drums are different shapes.

The three corrections line up by what they keep. An equal shave keeps the drums identical and fits the belt, and spoils the grading at the ends. A one-drum shave fits the belt, spoils the grading and gives two shapes to make. The proportional shave fits the belt, keeps the grading and keeps the drums identical, and its only cost is that it is a curve rather than an offset. A drum is a size, a wrap is a shape found that a belt drive’s ratio is a ratio of radii and transfers across sizes, which is exactly why the proportional shave can change both sizes and leave the speed alone.

The grip along the travel #

An open belt grips its smaller drum least, and on drums the smaller drum changes as the belt moves. On the logistic drums shaved to keep the grading, the wrap on the smaller drum is 180° in the middle, where the radii are equal, 169.5° a quarter of the way out and 160.3° at the ends, where the larger drum gets 199.7°. The unshaved stepped pulleys gave 159.8° at their outer steps; the proportional shave closes the radii’s difference by a few per cent and hands back half a degree of wrap.

So the grip falls away towards both ends of the travel, where the speed ratio is most extreme and — at the end where the small drum is driving — the belt’s tension has the least wrap to work with. A crossed belt would give both drums the same wrap at every station, 227° on this drive, and fit straight cones exactly. The open belt’s price is a profile and a grip that fades; the crossed belt’s is the rubbing where its runs cross and the twist in each run, which is why it was the rarer arrangement even though it was the simpler one to make.

The half-way measure #

A drum is easier to turn as a cone than as a curve, and there is a compromise between the drawn cone and the exact profile. Shave both straight cones by one constant amount instead of a profile, chosen so the belt’s error is centred.

The open belt’s error on unshaved cones runs from nought at the middle to 12.32 mm at the ends. A constant shave of 0.980 mm off each drum shortens the belt the middle station needs by $2\pi \times 0.980 = 6.16$ mm, and the error then runs from −6.16 mm at the middle to +6.16 mm at the ends: half as much at its worst, to be taken up by the belt’s stretch or by a tensioner across the whole travel. That is the Chebyshev compromise for a parabola fitted by a constant — the same equal-and-opposite error where the precision points go found in function generation — and it halves the error for exactly that reason.

It also shows the scale of the choice. Cones shaved by a constant need a belt with ±6 mm of take-up. Cones shaved by the profile need none. The profile’s own departure from the cone is two millimetres at the ends, so a turner cutting cones by eye and a template for the profile are separated by two millimetres of steel and twelve of belt.

Why the steps did not show the grading #

One belt for every step chose its steps a constant factor apart, and a geometric series of ratios at constant sum puts each step’s radii at $S q/(1+q)$ and $S/(1+q)$ — exactly the logistic drum sampled at evenly spaced stations. The steps were already on the logistic curve. They just did not need to be joined, and nobody had to notice that joining them with straight lines, as a cone does, would bend the grading.

A drum does need them joined, and the join decides the grading along the whole travel, not only at a few stops. That is the part of the problem the continuous drive adds: with steps the designer chooses speeds, and with drums the designer chooses a law. The drum that is not round found a drum’s profile fixed by a demanded rate in a winding problem, and this is the same kind of result: once the rate along the travel is demanded, the drum’s shape is not a choice.

What the drums do not settle #

Whether the belt stays put. A flat belt on a tapered surface tends to climb towards the larger diameter, which is why cone drives have a shifter fork holding the belt at its station. Whether an open belt on shaved drums is steered more or less strongly than on cones, and whether the fork has to fight it, is a friction question not modelled here.

The belt’s own plane. Everything treats the belt as lying in a plane square to the shafts at its station. On drums tapering in opposite directions the two runs meet the drums at slightly different angles, and a real belt twists a little between them. The error is second order in the taper and is not measured.

Crowning and width. A real belt has width, and on a taper its two edges run on different radii. The station here is the belt’s centre line.

Chains. A roller chain cannot run on a drum at all, since a chain is not a strand: its length comes in whole links and its sprockets are polygons. Continuously variable chain drives exist, but they use split pulleys that close on the chain’s sides, a different geometry.

Slip. The wrap on the smaller drum falls towards the ends of the travel for an open belt, as it did step by step; where the belt starts to slip, at a given tension, is not computed.

Still open: a belt drive whose ratio law is chosen for the load #

An evenly graded drive changes speed by the same proportion per millimetre everywhere, which suits a machine tool whose cutting speed matters in proportion. Other loads want other laws. A winch or a fan wants its power, not its speed, to change evenly with the control; a drive feeding material at a constant surface speed onto a growing roll wants its ratio to follow the roll’s radius.

The distinct argument would be that inverse problem, solved generally: given any demanded ratio law along the travel, the two drum profiles that deliver it with a crossed belt, and the shave that makes one open belt fit them, found by the same bisection. The measurement worth making is how far each law’s drums are from straight cones and from each other — whether the logistic drum is an outlier or typical — and whether some demanded law cannot be delivered at all, because it asks one drum’s radius to fall below nought or the other’s to exceed the radius sum.