The drum that is not round

A strand leaving a body runs along a tangent. That is the whole of its contact with the body’s shape, and it has an immediate consequence that is worth stating before anything else: the only thing about a body a strand can feel is the perpendicular distance from the axis to that tangent.

Not the radius. Not where the tangency point is. Not the curvature. A perpendicular distance, one number per direction, and the function that gives it is called the shape’s support function $h(\psi)$, with $\psi$ the direction of the tangent’s outward normal.

For a circle turning about its own centre, $h$ is the radius, at every angle, which is why nobody notices that it is a separate quantity from the radius at all.

The arm is the rate #

Turn the body by $d\psi$ and the tangent line sweeps; the strand pays out $h(\psi)\,d\psi$ of length. So the support function is not merely a description of the shape — it is the strand paid out per radian, which for a drive is the ratio.

That makes the design problem an integration. A demand for a rate is a demand for $h$, which is the strand’s ratio; and a convex shape is recovered from its support function by

$$\mathbf{p}(\psi) = h(\psi)\,\hat{\mathbf{u}} + h'(\psi)\,\hat{\mathbf{u}}^{\perp},$$

with $\hat{\mathbf{u}} = (\cos\psi, \sin\psi)$. There is no iteration, no envelope and no solve: the shape is two evaluations and a derivative.

Checking a construction with no residual #

A closed form has nothing to converge, so nothing about it can fail loudly. A sign error in the $h'$ term produces a shape that is smooth, closed, plausible and a different mechanism, and a figure of it looks entirely correct.

So the shape is asked to give the demand back. The support function is recomputed from the drawn profile — as the largest projection of the sampled outline onto each direction, which is a statement about a polyline rather than about the formula that made it — and compared with what was asked for.

That is one route. The second is better, because it goes through the strand.

Put an anchor 260 mm from the axis, run a strand from it to a mark on the body’s surface, and let the body rotate. The strand’s length is the tangent from the anchor plus the arc from the tangency to the mark, both measured off the drawn outline. Differentiate that length with respect to the rotation and the answer should be the perpendicular distance from the axis to the tangent — which is measured off the drawing too, as a distance from a point to a line.

The two agree to $2 \times 10^{-9}$ of the arm, over thirty-six positions. Neither of them has been told what a support function is: one is a difference of two lengths, and the other is a distance from a point to a line.

The demand that cannot be met #

$h$ and $h''$ together give the radius of curvature of the shape: $\rho = h + h''$. So a demand is a shape only while $h + h'' > 0$ everywhere. Where that expression goes negative there is no convex body with that support function — not a difficult one, none — and the strand would have to leave the surface and come back.

For a demand of the form $h_0(1 + a\cos n\psi)$ the arithmetic is immediate: $h + h'' = h_0(1 - a(n^2 - 1)\cos n\psi)$, which is positive everywhere exactly when

$$a < \frac{1}{n^2 - 1}.$$

Bisecting on whether the construction refuses gives, at each harmonic:

The shape of that result is what matters more than the numbers. A designer may ask for a great deal of variation slowly and almost none quickly. A third of the mean arm at two cycles per turn; an eighth at three; a twenty-fourth at five. A demand that wants the ratio to change sharply somewhere in the turn contains high harmonics whether or not it was written down that way, and the ceiling applies to each of them.

The first harmonic is not a shaped pulley at all #

The demand that looks easiest — one cycle per turn, $h_0(1 + a\cos\psi)$ — has no ceiling, and the reason is worth the essay’s most surprising figure.

Its curvature is $h_0(1 - a(1 - 1)\cos\psi) = h_0$, constant. The shape is a circle, of radius $h_0$, with its centre moved off the axis by $h_0 a$.

So the answer to “make me a pulley whose ratio varies sinusoidally once per turn” is: bolt a round pulley on off-centre. It is a part any workshop can produce and any bearing can carry, and no shaped profile is involved anywhere. The first harmonic of any demand can always be absorbed that way, which means the shaped part of a shaped pulley is everything from the second harmonic up — and that is exactly the part with the ceiling on it.

Read the other way round, this also says what an eccentric mounting costs. A pulley bolted 1 mm off-centre on a 34 mm radius is not a pulley with a 1 mm run-out; it is a variable-ratio drive with a 2.9% swing, once per revolution, and it is indistinguishable from one that was designed.

Where the tangency actually is #

For a circle the tangent whose normal points along $\psi$ touches at the point in direction $\psi$, and the two are so obviously the same thing that the distinction has no name. For a shaped body they are not the same, and the gap is the second term of the reconstruction.

The tangency point is $h\hat{\mathbf{u}} + h'\hat{\mathbf{u}}^{\perp}$: displaced sideways from the direction of the normal by $h'(\psi)$. For the demand drawn here, $h' = -h_0 a n \sin n\psi$, which reaches 20.4 mm on a mean arm of 34 — so the strand can be touching a point more than half an arm’s length round the body from where the normal points.

That matters for two reasons and neither is aesthetic. The tangency point is where the strand’s contact is, so it is where the surface has to be finished, and it sweeps a different range of the body than a naive reading of the demand suggests. And the tangency point is what moves when the body turns: the contact does not travel at a constant rate round the surface even when the pay-out is smooth, because the arc between successive tangencies is $\rho\,d\psi = (h + h'')\,d\psi$ rather than $h\,d\psi$.

So the two rates a shaped drum has are different functions. Strand paid out per radian is $h$; surface travelled per radian is $h + h''$. On the profile above the first runs from 23.8 to 44.2 mm and the second from 3.4 to 64.6 — a nineteen-fold swing in how fast the contact crawls round the body, on a drive whose ratio only varies by a factor of 1.86.

Adding harmonics is not adding ceilings #

The per-harmonic table above is exactly right for a demand that is a single harmonic, and it is worth saying what happens when a real demand is not.

The condition is on the whole function: $h + h'' > 0$ everywhere. For $h = h_0(1 + \sum a_n \cos n\psi)$ that reads

$$1 - \sum a_n (n^2 - 1)\cos n\psi > 0 \quad\text{for every } \psi,$$

and the sum is what has to stay under one. So the individual ceilings are necessary but not sufficient: two harmonics each at four fifths of their own limit will fail wherever they line up, and a demand built from a handful of them can fail while every single term is comfortably legal.

That gives the design rule its final form. The budget is one, and each harmonic spends $a_n(n^2 - 1)$ of it, at whatever angles it happens to peak. A demand written as a shape in the mind — “fast here, slow there, and hold it for a while” — spends most of that budget on the corners, which are the high harmonics, and the smooth part costs almost nothing.

The same problem, done two entirely different ways #

This site has met “the shape is the unknown” before, in the meshing field, and the two machineries are worth putting side by side because they answer different questions.

A conjugate pair is solved. Two bodies must stay in contact under stated motions; the second shape is the envelope of the first one’s positions; finding it means solving the meshing equation at every position, and the answer exists because a root exists. The first shape is given and the second is deduced.

A shaped drum is integrated. One body must give a stated rate to a strand; its support function is that rate; the shape falls out of $h$ and $h'$ in closed form. Nothing is given but the demand, and there is exactly one shape.

The most useful difference is in what can go wrong, and undercutting is the pair’s version of it. A conjugate pair can fail by undercutting — the generated shape cuts away material it needs — which is a property of the pair and shows up only when the generation is run. A shaped drum fails by non-convexity, which is a property of the demand alone and can be checked before anything is drawn.

The second difference is closure. A pair of non-circular gears must close: the demanded ratio’s mean has to be a whole number or the wheels do not come back into mesh after a turn, and the residual is $2\pi(\bar{k} - 1)$ whatever the shape. A cable on a shaped drum has no such requirement, because the cable ends: it is unwound and rewound and never has to arrive back anywhere.

What this is for #

Three uses, and all three are ordinary.

A drive whose ratio changes through the stroke. A cable pulling a mechanism through an arc where the required rate varies — a hatch, a window regulator, a leg — can be given the right rate at every angle by shaping the drum rather than by adding linkage. The variation available is bounded by the arithmetic above, and it is generous at low harmonics.

Cancelling a mechanism’s own variation. A cable driving a four-bar sees the four-bar’s transmission function, which is a long way from linear near its limit positions. A drum shaped as the inverse of that function turns the pair into a constant-rate drive, and the demand is a smooth once- or twice-per-turn function — comfortably inside the ceiling.

Reading a rate off an existing part. Any convex part a strand runs on has a support function whether it was designed or not, so measuring one is measuring a drive’s actual ratio. A worn drum, a pulley on a bent shaft, a spool with a flat: each is a demand that somebody did not intend, and its first harmonic is an eccentricity and its higher ones are a ratio error that varies within a revolution.

Where the arm and the radius part company #

It is worth being explicit about the two ways a body’s arm stops being its radius, because they are different and both are common.

The body is not round. Then $h$ varies with $\psi$ and the profile is the object above.

The body is round but does not turn about its own centre, which is what an off-axis idler is. Then $h$ is constant in the body’s frame and varies in the frame that matters, which is what the eccentric figure shows and what an off-axis idler does.

Both give a rate that varies within a revolution, and no measurement of the part distinguishes them — only a measurement of the assembly does. That is the practical form of the first-harmonic result: an eccentric mounting and a first-harmonic profile are the same drive.

One more check, because the first one was not enough #

The round trip through the drawn profile catches an error in the reconstruction. It cannot catch an error in the demand — if $h$ is wrong, the shape is faithfully wrong and gives the wrong arm back perfectly.

What catches that is the strand. Running an actual strand from an anchor to a mark on the body, and differentiating the length it pays out, tests the claim the whole essay rests on: that the rate is the perpendicular distance. Neither side of that comparison is the formula. One is a difference of two measured lengths — a tangent from a point to a polyline, plus an arc along the polyline — and the other is the distance from the axis to the line joining the anchor to the tangency.

They agree to two parts in a thousand million, at thirty-six rotations. That number is the finite difference’s own floor rather than a departure, and its being the floor is the point: there is nothing left in the disagreement to be a mistake.

The strand sees a hull, so the ceiling is not about convexity #

The construction refuses a demand exceeding $1/(n^2-1)$ because $h + h''$ goes negative, and $h + h'' > 0$ is exactly the condition that a support function describes a convex body. That makes the refusal look like a limitation of the construction — build a non-convex drum, the reasoning goes, and the ceiling is somebody else’s problem. It is not, and the reason is worth stating because it turns a construction’s refusal into a statement about every possible body.

A strand pulled taut over a body touches its convex hull and nothing else. Where the profile is concave the strand bridges the gap, running straight from one side of the concavity to the other, and the material inside is not touched at any point of the rotation. So the arm the strand feels is the hull’s support function, not the body’s.

Which means a non-convex drum’s strand behaviour is identical to that of its hull, and the hull is convex, and a convex body’s support function satisfies $h + h'' \ge 0$ by definition. The ceiling is therefore not a property of the construction and not a property of convex drums. No body whatever can deliver an arm profile violating it, because the only thing a strand can feel is a convex body, and every convex body obeys the inequality.

That is a considerably stronger result than the one the bisection reports, and it comes free with the observation about hulls. A demand for a second-harmonic variation exceeding one third is not merely hard to make or outside this library’s construction; it is a demand for a rate of length change that no shape can produce, and the right response to it is to change the demand.

It also says what the concave parts of a real shaped drum are actually doing, since the essay’s own note concedes that many of them have concavities. They are not shaping the strand’s motion — the hull does that — so they must be doing something else: clearing another part, providing a groove to keep the strand from wandering sideways, reducing mass, or accommodating a fixing. Every one of those is a reason that has nothing to do with the arm, and a designer who cuts a concavity expecting it to change the ratio has cut a decoration.

There is one case where the concavity does matter and it is worth marking so the claim is not over-read. A strand that is wrapped rather than merely tangent can be made to follow a groove into a concavity if something holds it there — a keeper, a second strand, or the tension of a wrap greater than half a turn against a suitably shaped channel. Then the contact is no longer the free tangency this field computes, and the mechanism is not a shaped drum in this field’s sense at all.

So the ceiling stands, and it stands for a better reason than the one it was found by. The bisection measured where a construction fails; the hull argument says the construction fails exactly where reality does.

What is not modelled #

The profile is convex by construction, and a real shaped drum is often not — a spiral cam with a step in it, where the cable is picked up at a fresh radius, is a perfectly good machine and has no support function at all. Nothing here handles a strand that leaves the surface and returns, which is what a concave region means. A closed belt over two shaped pulleys is not built: it needs the total length to stay constant at every angle, which is a genuine constraint coupling the two shapes, and this field has only the one-body case. The strand has no thickness, so the arm is measured to the surface rather than to the strand’s centreline, and on a small drum with a thick cable the difference is a substantial fraction of the arm.