The wrap that walks along the axis

Six things a strand is not ends by predicting where this field’s next difficulty comes from: “a capstan’s several wraps and a multi-layer hoist are the places to look — and both are mechanisms this field already flags as sitting awkwardly with the rest of its arithmetic.”

The multi-layer hoist has its own essay. This is the other one, and the awkwardness is simpler than the prediction suggests: a strand that goes round twice cannot be in a plane, because the second turn has to lie somewhere and the only place left is beside the first.

What the field has been assuming #

Every construction in this field is planar. A wrap angle is computed from a pair of tangent lines; the wrap angles of a closed run sum to exactly one turn; the taut path is the shortest one in the plane. All of it is a statement about curves in a plane and none of it mentions a third dimension.

That was never stated as an assumption because no figure needed it. A belt over two pulleys goes round each once and lies in one plane; a tackle is drawn flat; a tensioner swings in a plane. The moment a strand takes a second turn on the same body, the assumption is being made rather than satisfied.

So the question is what it costs, and the answer has two halves that go opposite ways.

The half that costs nothing: the plan view #

The projection of a helix onto the plane perpendicular to its axis is a circle, traversed once per turn. That is elementary and it is the whole of the good news.

Measured on a four-turn wrap sampled at 4,800 points: the projected radius is constant to $2.8\times10^{-14}$ of the drum’s radius, and the projected angle sweeps 4.000000000 turns.

Every planar statement this field has made is a statement about the projection, so every one of them survives. The wrap angle of a helical wrap, measured as the angle subtended in plan, is exactly the planar wrap angle. The turning number of a closed multi-turn run, measured in plan, is exactly the integer the theorem demands. The tangency condition — the strand leaves along the tangent — holds in plan because it holds in space and projection preserves it for a line in the projection’s own plane.

That is a stronger result than it sounds, because it means the field does not need repairing. It needs a sentence: the wrap angles are plan-view quantities, and they always were.

The half that costs something: the length #

Length is not a projection quantity. The helix’s length per turn is

$$\sqrt{(2\pi r)^2 + p^2} = 2\pi r \sec\lambda, \qquad \tan\lambda = \frac{p}{2\pi r},$$

so the planar model under-reports by $\sec\lambda - 1$, and the only parameter in that is the ratio of the pitch to the circumference.

The pitch is not free. Successive turns lie against one another, so $p$ is the strand’s own diameter — which is what makes the correction small and is why nobody noticed it.

A ten-millimetre rope on a hundred-millimetre drum has a helix angle of 1.823° and runs 0.0507 per cent long — five hundred parts per million, against a rope whose own length tolerance is a per cent. The planar figures in this field were not an approximation anybody had to apologise for.

Where it stops being negligible is where the strand stops being thin. At a diameter equal to the drum’s radius the excess is 1.26 per cent; at twice, 4.94. Those are cords on bobbins and hoses on reels rather than ropes on winches, and for them the planar model is wrong by more than the quantities the field computes are worth.

It is worth being clear about which way that comparison runs, because the two figures differ only in one number and the number is not the drum’s.

Both drums have a radius of fifty and both carry four turns. What changes is the strand: ten units across in the first figure and forty in the second. So the second picture is not a smaller drum, it is a fatter rope — and the axial run is four times longer for the same four turns, which is why the second wrap reaches so much further up its barrel. A capstan with room for sixteen turns of one rope has room for four of the other, and it is the turn count rather than the length correction that a designer notices first.

Why the correction is quadratic and not linear #

The shape of the curve is worth a paragraph, because a reader who expected the error to grow in proportion to the pitch would read the figure as surprisingly flat rather than as correct.

The strand’s path per turn is the hypotenuse of a right triangle whose base is the circumference and whose height is the pitch. Adding a perpendicular component to a vector changes its length by nothing to first order — that is the whole reason a small angle costs so little — and the excess is therefore quadratic in the pitch from the start.

Stated as a ratio: doubling the strand’s diameter relative to the drum quadruples the correction, and halving it quarters it. A rope at one part in thirty of the drum’s radius runs one part in two thousand long; at one part in three hundred it runs one part in two hundred thousand long. There is no regime between “utterly negligible” and “matters” that a designer has to think carefully about, which is a rarer situation than it sounds and the reason the assumption survived unstated for a whole field.

The same first-order insensitivity appears elsewhere in this field under a different name. A body joining a run costs length as the square of how far it intrudes, which is why a tensioner set at the edge of its own contact takes up almost nothing; the exponent there was measured at 2.0000 over four decades. This is the same statement about the same kind of geometry, and finding the exponent twice in one field is worth noticing: a strand’s length is stationary against any displacement perpendicular to its own run.

The small-angle form, and when it may be quoted #

The classical shorthand is $\sec\lambda - 1 \approx \lambda^2/2$, which in the field’s own variables is $\tfrac12 (p/2\pi r)^2$. It is worth checking rather than quoting, because an approximation used to argue that an error is small is an odd thing to leave unmeasured.

At $p/r = 0.2$ — the rope on the winch — the approximation is right to 0.03 per cent of itself. At $p/r = 1$ it is right to 0.6 per cent, and at $p/r = 2$ to 2.5. So the shorthand is good precisely in the regime where the quantity it estimates is negligible, and it degrades exactly where the quantity starts to matter. That is the usual behaviour of a small-angle expansion and it is worth stating the way round that makes it useful: the approximation may be used to show the correction is small and not to compute it when it is not.

The turn count, which is the quantity the field actually wants #

There is a quantity a capstan is bought for, and it is neither a length nor an angle: how many turns.

Everything about a capstan’s usefulness is proportional to its wrap angle, and its wrap angle is $2\pi n$ for $n$ turns — so the design question is how many turns will fit, which is the barrel’s length divided by the strand’s diameter. That is arithmetic with no geometry in it at all, and it is the reason a capstan is short and fat rather than long and thin: the turns are what is wanted and the barrel length is what buys them.

The helix makes one correction to that and it is in the useful direction. The turns are not $\ell/d$ but $\lfloor \ell/d \rfloor$, because a fraction of a turn is not a turn — and the wrap angle is therefore a staircase in the barrel length rather than a ramp. A barrel that is 3.9 strand diameters long gives three turns and so does one that is 3.1; lengthening it from 3.9 to 4.0 gains a whole turn.

That is the same shape of result as a chain’s loop having to contain a whole number of pitches, and it is the field’s third instance of a strand quantity that is an integer rather than a continuum. The first was the turning number, which is an integer because a closed curve’s winding is; the second was the chain’s pitch count, which is an integer because a chain has a smallest piece; and this is an integer because a wrap has a width.

Three integers, three reasons, and none of them visible in a planar figure of a single wrap.

What does change, and it is not a length #

The interesting consequence of leaving the plane is not the length at all. It is that a helical wrap has a direction the planar model has no room for.

The strand runs onto the drum at the helix angle to the plane perpendicular to the axis, and it has to come from somewhere — a lead sheave, fixed. As the wrap walks along the barrel, the angle between the incoming run and that perpendicular plane changes, and it is called the fleet angle:

$$\varphi = \arctan\frac{\text{barrel}/2}{\text{distance}}.$$

Past a degree or two the arriving turn stops lying beside its neighbour and starts climbing onto it, which is a winch that jams rather than a winch that is slightly long.

The rule that falls out is worth having because it is dimensionless. Imposing a limit of 1.5° gives

$$\text{distance} = \frac{\text{barrel}/2}{\tan 1.5°} = 19.09 \times \text{barrel},$$

the same multiple for every drum, checked to $10^{-16}$ across barrels from 200 mm to 1.2 m. A winch with a 300 mm barrel wants its lead sheave five and three-quarter metres away; one with a 600 mm barrel wants eleven and a half. That is a great deal further than anybody expects, and it is why winch installations are long.

It is also the field’s scale argument arriving in a new place. The fleet angle is a shape — an angle from a ratio of two lengths — so the condition transfers between drives of any size, and the distance it demands is a size that scales with them.

Where this leaves the field’s boundary #

Two things about a capstan are outside what is computed here and they should be separated, because one of them is out for a good reason and the other is merely undone.

Friction is out. Whether a strand slips is the capstan relation, and it needs a coefficient, which is a material property not carried here. The geometric half — the wrap angle — is what the field computes, and the result above says that half is a plan-view quantity and is unaffected by the helix. So the capstan’s geometry is now complete and its statics is still somebody else’s.

The laying pattern is undone. Whether the turns actually lie beside one another, or climb, or cross, is a question about where each turn is put rather than about where it could be — a sequencing question with a geometric answer that nobody here has computed. The fleet-angle rule is the engineering proxy for it, and a proxy is what it is.

What was measured, and what it could have refused #

Four checks, and two of them are exact statements while two are quadratures, which is a distinction worth keeping.

Exact: the projected turning number. 4.000000000 against 4, held to $10^{-9}$. A helix whose projection was not a closed circle would fail it, and so would a sampling that lost a turn.

Exact: the projected radius. Constant to $2.8\times10^{-14}$ of the radius. A projection along the wrong axis would give an ellipse and this is the check that says the axis is the right one.

A quadrature: the length. The sampled polygon is short of the true helix by a term in the square of the step — 1.1 parts per million at 4,800 samples over four turns — so it is compared against the closed form at $10^{-5}$ rather than at $10^{-9}$. Holding it to the same tolerance as the two exact claims would have refused a correct measurement, and the tolerance is set by the polygon rather than by the geometry.

A direction: the planar model is short. Not merely different — the excess must be positive and the helix must be longer than $2\pi r$ per turn. A sign error in the pitch term would have produced a plausible correction with the wrong sense, and a check on the magnitude alone would have passed it.

What this changes about the field’s figures #

Nothing, and saying so precisely is the honest end of the argument.

Every figure in this field draws a planar run, and the result above says that a planar run is the plan view of the real one, exactly. So the figures are not simplifications of the truth — they are projections of it, and a projection is a different relationship to have with a fact than an approximation is.

The distinction shows up in what each licenses. An approximation licenses statements with an error bar on them. A projection licenses a stated subset of statements with no error bar at all, and refuses the rest outright. So:

• Every wrap angle in this field is exact, at any number of turns, with no correction.
• Every turning number is exact, for the same reason.
• Every tangency point is exact, in plan.
• Every length is short, by a factor of sec λ per wrapped body, and the correction is computable from the strand’s own diameter.

Only the last item needs anything, and only for the bodies that are wrapped more than once — a belt over two pulleys wraps each less than once, so its length is exact as drawn. The field’s own tensioner solve, which finds an arm angle from a strand length to $10^{-13}$, is therefore exact for the drives it is run on and would need the factor for a capstan.

That is a small correction with a narrow scope, and the reason this essay is worth its place is not the correction. It is that the field now knows which of its claims are projections and which are lengths, which it did not before, and the two behave completely differently when the strand leaves the plane.

Still open: the wrap whose turns are not beside one another #

The pitch above is the strand’s own diameter, which is the neatest case and the one a winch is designed for. Two other cases exist and this field has computed neither.

A crossing wrap, where the strand deliberately returns over itself — the way a rope is belayed, or a sail’s halyard is figured on a cleat — has a path that is not a helix at all, and its wrap angles in plan are no longer the whole story because the strand now bears on itself rather than on the drum. The contact is strand-on-strand, which is a body this field has never given a strand.

The strand’s own thickness is the other thing quietly introduced above and not followed. Every body in this field is a shape and every strand is a curve with no width — which is exactly the relationship the bodies field found between a link and a bar, one field over, and it changed everything there. A strand with a diameter has a bend radius it cannot go below, a contact patch rather than a tangency point, and a minimum drum size. None of that is computed here; what is computed is the one consequence of thickness that the wrap forced, which is the pitch.

And a multi-layer wrap has both difficulties at once: the working radius changes with the layer and the helix angle changes with it, so the length per turn and the fleet angle both move as the drum fills. The first of those is measured in this field already; the second is the product of the two arguments and is one line of arithmetic away.