A tolerance shortens the drum

A drum for every law cut cone-drive drums for a family of demanded speed laws, from the even proportion a machine tool wants to a fan whose power should change evenly with the shifter. It found that a crossed belt fits any of them, that one table of open-belt shaves serves them all, and that what each law really costs is set by its grading, the proportional change of speed per millimetre of shift. A belt of width $w$ spans a stretch of travel $w$ long, and its two edges want ratios a factor $e^{wg}$ apart, where $g$ is the local grading. That factor must be absorbed by uneven stretch across the belt, and it is worst where the law is steepest.

So a law’s drums must be long enough that its steepest point is gentle enough for the belt. The even proportion grades evenly and so has the least peak of any law over a given range. Other laws pile their change into one end of the travel. The speed changing evenly needs about three times the even proportion’s travel for the same belt, and a fan’s power changing evenly about fifty times.

The essay ended by asking whether those steep ends are worth delivering exactly. A designer can usually accept speeds a little off the demand. If the law is allowed to depart from the demand by a stated tolerance, how much travel does that save, and is every practical drum then an even proportion in disguise?

The least steep law in a tube #

Write a law as $f(t) = \ln q$ against the share $t$ of the travel, from nought at one end of the drums to one at the other. A tolerance $\delta$ on the delivered speed, as a proportion, is a tube of half-width $\delta$ around $f$ in the logarithm. The two ends of the range are kept exact, since a drive that does not reach its stated top or bottom speed has changed its range rather than its law. The question is the least peak slope any law can have that stays in the tube from one end to the other.

That question has a short answer. Between any two places $t_1 < t_2$ a law in the tube must climb at least from the top of the tube at $t_1$ to the bottom of it at $t_2$, so its average slope there is at least

$$\frac{(f(t_2) - \delta) - (f(t_1) + \delta)}{t_2 - t_1},$$

with $\delta$ taken as nought at the two ends. Its peak slope is at least the largest such chord over all pairs of places. And a law exists that achieves exactly that: start at the exact end, and climb no faster than the largest chord allows. That law stays in the tube, because if it ever fell out a steeper chord would exist. The least peak, $m^*$, is therefore the steepest chord across the tube, the slope of a taut string pulled through it.

The picture is this field’s own object. A strand in a tube found that a Bowden cable pulled through its sheath takes the shortest path the sheath allows, straight across the clearance and wrapped round the inside of each bend. A law softened within a tolerance is a strand in a tube drawn on a graph of $\ln q$ against travel, and its least peak slope is the steepest straight run the tube forces on it. The taut path has more than one answer found two taut paths past one peg, one on each side. The least-peak law is not unique either, for a gentler reason: any law in the tube whose slope never exceeds $m^*$ is as good by this measure, and the tube leaves room for many. On the speed law within 5%, for instance, the string’s steepest straight run touches the tube’s lower edge at 7% of the travel, and any law that passes that point at the edge and climbs no faster than $m^*$ on either side is equally short.

For the laws in this family the grading only falls, or only rises, along the travel, so the steep part is all at one end. The taut-string law can then be built as a designer would cut it. It is the demanded law with its slope capped at $m^*$, running from the steep end. It lags behind the demand by at most the tolerance until it rejoins it, and from there it is the demanded law exactly.

Both halves were checked on every law and tolerance. The capped law’s largest slope equals the chord scan’s $m^*$ to a millionth of it, it stays inside the tube, and it keeps both ends exact. A looser tube never needs a steeper law, and the exact laws’ peak slopes, divided by the even proportion’s, give back the earlier essay’s factors.

What a tolerance buys #

The drum’s length is the peak slope’s partner. For a given belt width and a given spread allowed between the belt’s edges, the travel needed is proportional to $m^*$. Fixing both so that the even proportion needs the earlier drive’s 300 mm, every law’s travel can be read in millimetres.

Exact, the speed law needs 884 mm, 2.95 times the even proportion’s travel. Allowed a 2% error in the speeds it delivers, it needs 721 mm; 5%, 635 mm; 11%, 547 mm; 22%, 437 mm. The first few per cent of tolerance buy the most, because they remove the sharpest part of the peak, the first few millimetres of the slow end where the speed law changes speed by nearly 2% a millimetre. The law with $\lambda = 2$ falls from 3,290 mm exact to 1,651 at 11%. The fan law falls from 13.4 m, a drive nobody would build, to 6.1 m at 11% and 3.9 m at 22%, still a drive nobody would build.

The roll law, the speed’s reciprocal changing evenly, needs exactly what the speed law needs at every tolerance. It is the speed law run from the other end with the ratio inverted, the mirror the earlier essay found, and a tube is mirrored with it.

What each law costs to make buildable #

Turned round, the same curves say how much tolerance a law needs to fit a drive of a given length. For the speed law, drums twice the even proportion’s length, 600 mm, need a tolerance of 7% on the delivered speed; one and a half times, 450 mm, needs 20%. Those are tolerances a designer might accept for a drive whose speeds are set by eye. For $\lambda = 2$, twice the even proportion’s length needs 57% and three times needs 34%. For the fan law, three times needs 86% and twice needs 112%: no tolerance worth the name makes a fan’s power law buildable on drums of a practical length.

The first per cent is worth the most. One per cent of tolerance takes the speed law from 884 mm to 767, the fan law from 13.4 m to 11.5, a saving of 13% and 14%, because a steep end’s very steepest stretch is short and a small tube is enough to straighten it. Each further per cent buys less, since the flat stretch the string pulls across lengthens and the grading it caps grows gentler. A designer asked for a speed law should therefore ask first whether a percent or two is acceptable, because that question is worth an eighth of the drum.

The drum that is not round found a winding drum’s profile forced by a demanded rate, with no freedom left to the designer. The cone drive has one freedom the winding drum lacked, its length, and the tolerance converts a demand for exactness into a choice about that length.

The steep end is all that changes #

The softened law is not a compromise spread over the whole travel. Within 5%, the speed law’s peak grading falls from 2.95 times the even proportion’s to 2.12, held flat over the first 16% of the travel, and from there the law is exactly as demanded. Within 11% the flat stretch is at 1.82 and lasts 26% of the travel; within 22%, 1.46 over 47%. Everywhere beyond the flat stretch the demanded grading is already below the new peak and needs no help.

That is what a taut string through a tube does. It is straight where the tube forces it to be and touches the tube elsewhere, and here touching the tube means following the demand. A designer who softens a law in this way changes the speeds delivered only at the slow end, and only by the tolerance allowed, at most at the place where the flat stretch is furthest behind the demand.

The drums #

The three drums share their end radii, since every law runs over the same range from 1/2.56 to 2.56 with the radii summing to 160 mm. They differ in length and in where they change. The exact speed law’s drum is 884 mm long and almost straight-sided over its fast half, rising most steeply at its slow end. The softened one is 547 mm, its slow end replaced by an evenly graded run. The even proportion’s is 300 mm, the logistic drum the cone-drive essay found.

A cone drive is not two cones found that the straight cone’s grading departs from even by only a quarter between the middle and the ends, and the softened speed law at 22% is no gentler than that at its peak. A designer wanting the speed law’s behaviour over most of the travel can have it at a tolerance of a tenth to a fifth, with a drum somewhat under twice the even proportion’s length.

The ledger #

The ledger puts the five laws side by side. Each pair of mirrors, $\lambda$ and $-\lambda$, needs the same travel at every tolerance, as the mirror requires. The rows are far apart. The speed and roll laws are buildable at modest tolerances. The laws with $\lambda = \pm 2$ need a tolerance of a fifth before their drums come under four times the even proportion’s. The fan law stays over three metres at any tolerance a specification could state. The last column is the same story read the other way: the further a law’s own shape departs from the even proportion, the larger the tolerance needed before the two are interchangeable, and the departures run from half the speed to more than one and a half times it.

The crossed belt’s two radii come straight from the ratio at each station, as the wraps add up to a turn showed, so every softened law in the ledger is a pair of drums the moment its ratio is known. Nothing about the belt’s geometry limits the softening; only the belt’s width does, where a strand leaves a body at a right angle on each drum whatever the law.

Never an even proportion in disguise #

The second half of the question has a clean answer. The even proportion is the straight line in $\ln q$ from one end of the range to the other. It lies in a law’s tube exactly when the tolerance reaches the law’s largest departure from that line, and only then can the least-peak law be the even proportion, its peak equal to the average.

For the speed law that departure is 0.42 in the logarithm, a speed error of 52%. For $\lambda = 2$ it is 0.75, 112%; for the fan law 0.98, 165%. No designer specifying a speed law would accept speeds off by half, so no practical softening of these laws reaches the even proportion. At every tolerance short of the departure the chord scan’s peak stays above the average. At the departure, and beyond it, the scan returns the average to a thousandth.

So the earlier essay’s suspicion is answered no. A practical softened drum is the demanded law with a shorter, gentler run at its steep end. It needs less travel than the exact law and more than the even proportion, and for the speed law at a tolerance of 11% its length lies just over halfway between the two in the ratio of their travels. The law’s character survives everywhere but where it was hardest to deliver. An operator who set the shifter by the demanded law’s markings would find every marking correct over three quarters of the travel at 11%, and the slow end’s markings out by at most a tenth, always on the slow side, since the capped law lags the demand rather than leading it.

The formula a designer can carry #

For a belt of width $w$ and an allowed spread $E$ between the speeds its two edges want, the shortest drums that deliver a law within a tolerance $\delta$ are

$$L = \frac{w\,m^*(\delta)}{\ln E},$$

with $m^*$ the steepest chord across the law’s tube in $\ln q$ against the share of travel. For the even proportion $m^*$ is the range’s logarithm, $2\ln 2.56$, which gives the earlier drive’s 300 mm for a 20 mm belt with its edges 13.4% apart. For any other law the chord scan gives $m^*$ in a fraction of a second over a thousand stations, and the capped law it describes is the drum to cut.

One belt for every step sized a stepped pulley, a cone drive with only a few stations, and needed no tolerance because its steps were chosen to fit. The continuous drum has every station, and the tolerance is what lets the designer decide which ones are worth delivering exactly.

What the tolerance does not settle #

The belt’s own stretch. The edge spread allowed, $E$, stands for a belt’s ability to absorb a difference of ratio across its width by stretching unevenly. The essays in this field treat a belt as a member with no length of its own, so the right value of $E$ for a real belt is not measured here. The travels scale with it.

Exact ends. The tolerance is applied everywhere but at the range’s two ends. Allowing the range itself to shrink a little would let a law dodge its steep end more cheaply. That is a choice about the drive’s specification rather than its law, and it is not explored.

One kind of tolerance. The tube is a constant proportional error. A tolerance that tightens towards the speeds used most, or loosens towards the extremes, gives a different tube and a different string, and the chord scan handles any tube unchanged.

The open belt. All drums here are for a crossed belt. An open belt needs the shaves the earlier essays found, which depend only on the ratio at each station, so a softened law’s shaves are read off the same table at its own ratios.

Still open: a drum that changes its own law #

A cone drive’s law is fixed when the drums are cut. A drive that served two duties, a lathe that also wound material, would want two laws, and one pair of drums gives one.

The distinct argument would be a cone drive whose belt is guided by a second shifter, a cam on the shifter’s own motion, so that the belt’s station is a chosen function of the operator’s lever. The drums could then be cut for the even proportion, the gentlest any range allows, and the law delivered by the lever’s cam instead. The measurement would compare that arrangement’s peak grading on the drums with the softened laws here, and ask what the cam must do to deliver the speed law exactly. Since the drums would then always be graded evenly, it would also ask whether the belt’s limit has simply moved from the drum to the operator’s lever, as a steep end of the cam instead of the drum.