A drum for every law

A cone drive is not two cones took a stepped pulley to its limit: two drums on parallel shafts with the belt shifted continuously along them. It asked for the speed to change by the same proportion for every millimetre of shift, which is what a machine tool wants because cutting speed matters in proportion. A crossed belt needs the two radii to sum to the same value everywhere, and together those two conditions make each drum a logistic curve, not a cone. An open belt then needs each identical drum shaved in proportion to its radius.

That essay closed on the obvious generalisation. Other loads want other laws. A fan’s power goes as the cube of its speed, so a fan’s control might want the power, not the speed, to change evenly. A drive winding material onto a roll at constant surface speed, with the shifter following the roll’s radius as it grows, wants the reciprocal of the speed to change evenly. The question was whether every such law can be cut into drums, how far each set of drums is from a cone, and whether some law cannot be delivered at all.

The answer divides cleanly. What the belt needs is the same for every law. What the law changes is the drums’ shape, their symmetry and how steeply they taper.

One family of laws #

The named laws have a common form. Each asks that some power of the speed ratio $q$ change evenly with the shifter:

$$\frac{q^\lambda - 1}{\lambda} \text{ changes evenly along the travel},$$

with $\lambda = 0$ read as its limit, $\ln q$. The even proportion of the earlier essay is $\lambda = 0$. The speed changing evenly is $\lambda = 1$. A fan’s power changing evenly is $\lambda = 3$. The winding roll is $\lambda = -1$, since its demanded speed is proportional to the reciprocal of a radius that grows with the shifter. The drive is the earlier essay’s, measured with the same taut runs, each leaving its drum at a right angle: radius sum 160 mm, shafts 400 mm apart, a ratio from 1/2.56 to 2.56 over a travel of 300 mm, the range of the lathe one belt for every step sized. Every law in the family spans that same range in that same travel. They differ only in how they spend it.

Every law also has a mirror. The law $-\lambda$, run from the other end of the travel with the ratio inverted, is the law $\lambda$ again. That symmetry will turn out to matter for making the drums.

A crossed belt fits any law #

A crossed belt’s length depends on the two radii only through their sum, as the wraps add up to a turn showed. So given any ratio $q$ demanded at a station, the two radii are fixed at once:

$$r_1 = \frac{Sq}{1+q}, \qquad r_2 = \frac{S}{1+q}.$$

Both are positive for any positive ratio, and both are below the sum. So every law whose ratio stays positive and finite over the travel can be cut into drums for a crossed belt, and the belt fits them exactly. Built for all seven laws from $\lambda = -3$ to $3$, the radii sum to 160 mm at every station to the last bit of the arithmetic.

The one law that cannot be delivered is the one that asks for a ratio of nought or of infinity: a drive that takes its output to a standstill. Its driving drum would need a radius of nought at one end. That is the boundary, and it is a sharp one. A cone drive can be made to vary its ratio by any factor, but it cannot reach zero, which is why drives that must run down to a stop use a split power path, as a bounded ratio made unbounded found, rather than a belt alone.

Only the even proportion keeps the drums one shape #

The earlier essay’s drums had a property a workshop values: they were one shape. Mount it one way on the countershaft and the other way on the spindle, and one template cuts both. Straight cones share that property, and so do the logistic drums. It is not a property of drums in general.

The driving drum at station $s$ equals the driven drum at station $-s$ exactly when $Sq(s)/(1+q(s)) = S/(1+q(-s))$, that is, when $q(s)\,q(-s) = 1$: the ratio at a station and at its mirror must be reciprocal. The even proportion satisfies it, because its logarithm is odd about the middle. The straight cone satisfies it by construction. No other member of the family does.

The hero figure shows it for the speed changing evenly. The driven drum turned end for end, dotted, misses the driving drum by up to 30.7 mm. A drive cut for that law needs two different drums. The fan law needs two drums 54.5 mm apart, on drums whose radii run from 45 to 115 mm. The mismatch is not a small correction. The two drums are different objects.

A law and its mirror share one pair #

There is a consolation, and it comes from the mirror symmetry. The roll law is the mirror of the speed law, so its ratio at a station is the reciprocal of the speed law’s at the mirror station. Put that into the radii and the roll law’s driving drum is the speed law’s driven drum turned end for end, and its driven drum is the speed law’s driving drum turned end for end.

Built, the two pairs coincide to $10^{-9}$ mm. A shop that has cut the drums for a drive whose speed must rise evenly has also cut the drums for a winder whose speed must fall as the roll grows. The winder simply exchanges the shafts. The same holds for every $\lambda$ and $-\lambda$. The fan’s drums, turned round, are the drums of the law $\lambda = -3$, which would suit a load whose demand falls as the cube of its speed rises.

Every law spends the same range #

The laws cover one range, and the range sets how much change of speed the whole travel has to supply. The proportional change of speed per millimetre of shift is $d(\ln q)/ds$, and its integral over the travel is $\ln(q_1/q_0) = \ln 2.56^2$ for every law. A law can spend that total unevenly, but it cannot spend less of it.

The even proportion spends it flat, 0.627% per millimetre everywhere, and since the average of any law’s grading is that same number, no law can have a lower peak. The others pile the change into the slow end of the travel. The speed-even law changes speed by 0.28% per millimetre at the fast end and 1.85% at the slow end. With λ = 2 the slow end reaches 7%. The fan law runs from 0.11% to 31% per millimetre. That is what a cube does. The fan’s power runs over a factor of $2.56^6 \approx 280$, and giving each millimetre of travel the same share of that power leaves the first few millimetres to take the speed from its lowest value to a few times it.

A cone drive is not two cones found that the straight cone’s grading bends by 24% between the middle and the ends. Against the family, that is a small deviation. The straight cone and the logistic drum are the two laws with one drum shape, and they are also the two whose gradings are nearly flat.

What each law asks of the drums #

The grading becomes geometry at the drum’s surface, and the geometry is what a turner would notice first.

The even proportion’s drum departs from the straight cone by 0.95 mm. The speed-even and roll laws depart by 16 mm, λ = ±2 by 29 mm, and the fan law by 37 mm. The logistic drum was not a typical member of the family that happened to look like a cone. It is the member that looks most like one, and every other law’s drum is visibly curved.

The steepest taper tells the same story more sharply. Straight cones taper at 13.2° from the drum’s axis over their whole length, the least any law can manage. Their radius changes by a fixed amount over a fixed travel, and a constant slope has the smallest peak. The even proportion reaches 14.1° at its middle. The speed and roll laws reach 30.8° at their slow ends, λ = ±2 reaches 65°, and the fan law 82.3°, which is not a taper so much as a shoulder. The two optima are different, since the straight cone minimises the peak slope of the radius and the even proportion the peak slope of the ratio’s logarithm. They are a millimetre apart, and every other law is far from both.

What a steep law costs the belt #

The taper matters because a belt has width. A belt $w$ wide at a station spans a stretch of the travel $w$ long, and the ratio its two edges would each want differs by about a factor $e^{wg}$, where $g$ is the local grading. The belt can have only one ratio, since it is one piece, so the difference is taken up by the belt stretching unevenly across its width and slipping on the drum. The strand model this field is built on has no way to absorb it: a belt is a member with no length of its own, and a real belt’s small stretch is spent first on keeping its runs taut. A difference between its edges is a second demand on that same stretch, and it grows with the law’s grading, not with anything the belt’s tension can adjust.

For a belt 20 mm wide, the even proportion’s two edges want ratios a factor of 1.13 apart everywhere. The straight cone’s want 1.16 at its ends. The speed and roll laws reach 1.45 at their slow ends, λ = ±2 reaches 4, and the fan law 509. The last number is not a design, it is a proof that the law cannot be delivered by drums over this travel. The fan law’s first few millimetres ask for a change of speed that no belt of any useful width can straddle.

Even the best law is not gentle on this drive. A 13% difference between a belt’s edges is a great deal of uneven stretch to ask of a flat belt, which is why cone drives that were built had a longer travel for their range, or a much narrower belt, than the lathe-sized drive here. Doubling the travel halves every grading, which is the designer’s real lever. The law then decides how evenly a longer travel is used.

The open belt’s shave belongs to the ratio #

An open belt wraps both drums the same way, and its length depends on the difference of the radii as well as their sum. So on drums cut for a crossed belt, an open belt made for the middle station is too short everywhere else. The earlier essay fixed that for the even proportion with a proportional shave. At each station the ratio is held and both radii shrink together until the open belt is the middle station’s length, found by bisection on the belt’s own taut run.

The question for other laws seemed to be whether that correction has to be worked out afresh for each. It does not, and the reason is short. The open belt at a station knows nothing about the law. It meets two radii at that station, and those two radii are fixed by the ratio there and the sum. So the shave at a station is a function of the ratio at that station alone, whichever law put that ratio there.

Built for four laws, every station falls on one curve. The bisection agrees at every station with $(r/S)(d^2/\pi C - 2d^4/\pi^2 C^2 S)$, written in the ratio alone, to within 1%, and identically across laws. At a ratio of 2.274 the fan law’s station 60 mm from the middle and the even proportion’s station 131 mm from the middle each shave 2.068 mm, to $10^{-9}$ mm. One table of shaves, indexed by ratio from 1/2.56 to 2.56, cuts the open-belt drums for any law. The law only decides where along the drum each entry of the table is used.

That also settles the earlier essay’s figures for every law at once. The ends of the travel have the same ratio in every law, so every law’s drums lose 2.69 mm at the large end and 1.05 mm at the small, and an open belt wraps the smaller drum by 160.3° there in every law.

What a designer can take from it #

There are three rules, one for each of the belt, the drums and the travel.

The belt does not care about the law. A crossed belt fits any positive law exactly, and an open belt fits any law after the same ratio-indexed shave. The drive can be designed for its load first and corrected for its belt afterwards, in that order. That is the continuous form of what a drum is a size, a wrap is a shape found for fixed pulleys: the belt sees radii, and the law is only the order in which it meets them.

The law decides whether one template will do. Only laws whose ratio at mirror stations multiply to one give one drum shape. Of the loads named here, only the machine tool’s even proportion is such a law. A speed-even drive and a constant-surface-speed winder can share a pair of drums between them, exchanged.

The travel pays for the law’s peak. The even proportion has the least peak grading any law can have, so any other law needs more travel for the same belt width, in proportion to its peak. The speed law needs about three times the travel of the even proportion to be as gentle on its belt at its worst point, and the fan law about fifty times. The drum that is not round found a winding drum’s profile forced by a demanded rate. Here the demanded rate forces the profile too, and the designer’s remaining freedom is how long a drum to make.

What the laws do not settle #

Whether the belt stays at its station. A flat belt on a taper climbs towards the larger diameter, and a steep taper pushes harder. How strongly each law’s drums steer the belt, and how hard the shifter fork has to hold it, is a friction question not modelled here.

Take-up. An open belt on unshaved drums needs its error taken up somewhere, and the tensioner is the unknown in any strand drive that has one. Because the open-belt error depends only on the ratio, a tensioner sized for one law’s drums is sized for every law’s, and the choice between shaving and a tensioner is the same choice whatever the load.

The belt’s own plane. As in the earlier essay, the belt is treated as lying in a plane square to the shafts at its station. On drums tapering at 30° or more that is no longer a small approximation, and the laws beyond λ = ±1 are measured here as geometry, not as working drives.

Slip. The wrap on the smaller drum is the same at the same ratio in every law, so grip at the ends does not depend on the law. Where each law’s belt starts to slip under a given load at intermediate stations, where the loads differ, is not computed.

Other families. The laws here are one family. A law with a flat stretch, such as a speed held constant over part of the travel, has drums that are cylinders there, and nothing in the rules above excludes it.

Still open: the shortest drum for a belt of given width #

Every law’s cost came down to its peak grading, and the peak can be traded against the travel. That makes the question a designer would ask next a minimum.

The distinct argument would be the shortest pair of drums that delivers a demanded law with a belt of a stated width, under a stated limit on the ratio difference between the belt’s edges. For the even proportion the answer is a formula, the range’s logarithm times the width over the limit’s logarithm. For every other law it is set by the law’s worst point. The measurement would go further than scaling the travel. It would let the law be modified near its steep end within a stated tolerance, a speed law with its slow end softened for example, and find how much travel the softening saves against how far the delivered speeds depart from the demanded ones. It would say whether the steep ends of the named laws are worth delivering exactly, or whether every practical drum is an even proportion in disguise.