The radius a winch works at

A winch is quoted with a line speed, which makes it one more ratio that is not a number: so many metres per minute, at so many revolutions. The number is the drum’s circumference times its shaft speed, and the circumference is $2\pi r$.

The whole difficulty is in $r$. It is not the drum’s radius. It is the radius of the layer of line currently being wound, and that grows by one line thickness every time the drum fills across and starts again.

What the drift is #

Take a drum with a 30 mm core, 6 mm line, and room for twelve turns across. Each layer sits one thickness further out than the one under it, so its radius is the core plus a thickness for each completed layer, plus half a thickness for its own centre:

The sixth layer pays out 1.909 times as much line per turn as the first. That is not a small correction to a nameplate figure: it is nearly a two-to-one ratio change, delivered by a machine with no ratio-changing parts in it, and it happens in the middle of ordinary operation rather than at some limit.

The drum’s total capacity is 21.715 m in 72 turns, and more than a third of that is in the outer two layers.

Length is quadratic in turns #

Treat the winding as a spiral rather than as discrete layers — radius growing continuously at a pitch of $t =$ thickness ÷ turns per layer — and the length after $N$ turns is

$$L(N) = 2\pi\left(r_0 N + \tfrac{1}{2}tN^2\right).$$

Quadratic. Which means the inverse — how many turns to pay out a given length — is a square root:

$$N(L) = \frac{-r_0 + \sqrt{r_0^2 + tL/\pi}}{t}.$$

On this drum, one metre of line takes 5.089 turns, five metres take 22.360, ten metres take 39.831, and the full 21.715 m takes 72. Paying out the second five metres takes 17.27 turns where the first five took 22.36.

So a counter on the drum’s shaft is not a tape measure. Reading length off turns requires knowing the core radius and the line thickness and taking a square root, and reading it off shaft speed requires knowing where in the wind the drum currently is.

The two readings, and where they agree exactly #

The layered model and the spiral model are different objects. One holds the radius constant within a layer and steps it up at each boundary; the other lets it grow smoothly. They can be compared, and the comparison has a clean answer.

At every layer boundary they give the same number. Not nearly the same: the same, to $4 \times 10^{-12}$ mm across the whole drum. The half-thickness offset in the layered radius — the $(k + \tfrac12)$ that puts each layer’s working radius at its own centre rather than at its inner face — is exactly what makes the arithmetic close, and using $k$ or $k+1$ instead would leave a systematic error of half a layer’s worth of line at every boundary.

Inside a layer they differ, and the gap has a closed form. The largest disagreement is at the middle of a layer and comes to 56.5487 mm, which is $\pi T t / 4$ with $T$ the turns per layer and $t$ the thickness — $\pi \times 12 \times 6 / 4$, to twelve figures.

That is the useful form of the answer, because it says when the distinction matters. A drum with two turns per layer has a mid-layer discrepancy of 9.4 mm and a drum with fifty has 236. The spiral model is the better one for a drum whose line is fine and whose layers are many; the layered model is the better one for a drum with a handful of fat turns per layer, which is where the line visibly steps rather than climbing.

What a ratio means when there is no fixed arm #

The rest of this field measures a strand’s ratio as the derivative of the run’s length with respect to something. A drum fits that pattern exactly, and it is the case where the arm is a function of the past.

For a tendon over an idler on the joint axis, the arm is the idler’s radius and it is the same at every configuration. For a shaped drum, the arm is the support function, and it is a function of the body’s angle. For a winch, the arm is $r_0 + tN$ — a function of the total rotation since the drum was empty, which is not a configuration variable of the mechanism at all.

That is a third kind of dependence and it is worth naming. The drum’s ratio depends on its history: two identical winches at the same shaft angle, one holding four metres and one holding fifteen, have different ratios. Nothing about the position of any part distinguishes them.

Paying out a stated length #

The arithmetic above turns into a practical problem the moment somebody wants two metres of line.

Two metres takes 9.646 turns from an empty drum, 7.074 turns with two layers already on, and 5.585 turns with four. Nothing about the machine has changed between those three; the shaft has simply turned a different number of times before the question was asked.

Assume the mean instead — 301.59 mm per turn, the honest average over the whole drum — and twelve turns pays out:

The mean is exactly right over a full drum and wrong everywhere inside it, by nearly half at one end. That is the characteristic failure of averaging a ratio that varies: the total is preserved and every increment is wrong, and the errors do not tend to cancel over a short run — they are systematically one sign at the start and the other at the finish.

A drum is not a run #

It is worth saying plainly that the drum is the one mechanism in this field to which the field’s own machinery does not apply.

There is no tangency: the line lies on the previous turn rather than touching a body. There is no wrap angle in the sense the rest of the field uses, because the strand goes round more than once and the arithmetic that adds wraps to a turning number assumes each body is met once. There is no run, no hull, no closed form of the kind the tangent solve gives.

What the drum keeps is the thing that makes it belong here at all: one strand, one length, and the length is the constraint. Everything above is that length, accumulated. And the ratio is still the derivative — it is $2\pi(r_0 + tN)$, differentiated with respect to the turns rather than with respect to a position, which is the same operation performed on the same quantity.

The capstan, which is a different machine #

One mechanism looks like a drum, is not one, and is not in this field.

A capstan — which this field does not compute — takes several turns of rope round a barrel at a single radius. The turns sit side by side rather than stacking, so there is no layer growth, no drift and no quadratic — the rope’s speed per turn is $2\pi r$ and stays there. What a capstan is for is holding: the tension falls exponentially round the wrap, so a few turns and a light hand at the tail will hold a heavy pull.

That is a friction argument from beginning to end, and every number in it is a coefficient. It belongs to the subject that owns the capstan relation, and nothing in this field touches it. What is worth keeping is the geometric distinction, which is clean: a drum accumulates layers and its ratio drifts; a capstan accumulates wraps at one radius and its ratio does not. Confusing the two produces a winch whose stated speed is exactly right and a capstan whose stated speed is a fiction, or the reverse.

Where this compounds #

A powered hoist has both of the field’s ratio problems at once, and they multiply.

The tackle contributes a velocity ratio that rises with the block separation — 3.617 at 90 mm and 3.990 at 620, on a four-part reeving — and is worst when the load is highest.

The drum contributes a line speed per turn that rises with how much line is already wound — 207.35 mm on the first layer, 395.84 on the sixth — and is fastest when the load is lowest, since the line comes off the drum as the load goes up.

Multiply them and the hoist’s lift speed for a fixed shaft speed varies by roughly a factor of two across a full stroke, from two independent geometric causes with nothing in common except that neither is on the nameplate.

Nesting, and the layer pitch that is not the thickness #

One refinement is worth naming because it is the first thing a measurement of a real drum disagrees with.

The model above stacks each layer directly on top of the one below, so the layer pitch is the line thickness. A real wind does not do that: the turns of an upper layer sit in the valleys between the turns beneath, so the layer pitch is $t\sqrt{3}/2 = 0.866\,t$ — on this drum, 5.196 mm rather than 6.

That changes every radius after the first, so it changes every line speed and the capacity too, and it does so by 13% per layer of accumulated error. It is a straightforward substitution in the arithmetic, and the reason it is not made here is that it is only true of a wind that nests — which requires a level-wind mechanism, a line that does not deform, and a tension that keeps it seated. A drum that is wound carelessly does neither consistently, and the honest statement is that the layer pitch is between $0.866\,t$ and $t$ and is a property of how the machine is used.

What a winch is quoted at #

The practical consequence is a reading problem. A drum’s stated line speed is one of three numbers, and specifications rarely say which:

Bare drum. The fastest the machine will pull and the slowest it will run — the first layer, 207.35 mm per turn here. Quoting this makes the winch look strong.

Full drum. The last layer, 395.84 mm per turn. Quoting this makes it look fast.

Mean. The capacity divided by the turns: 21,714.69 ÷ 72 = 301.59 mm per turn, which is the fourth layer’s 320.44 rather than the third’s 282.74 — because there is more line on the outer layers, so the mean is pulled outward.

The three differ by nearly two to one, and the difference is not a tolerance or an approximation. It is the machine doing exactly what it is supposed to do.

The drum as a shaped body, seen sideways #

There is a way to read the whole of this essay in the vocabulary of the shaped drum, and it makes the difference between the two mechanisms sharp.

A shaped drum’s arm is $h(\psi)$: a function of the body’s angle, periodic, coming back to the same value every turn. Its ratio varies within a revolution and repeats for ever.

A winch’s arm is $r_0 + tN$: a function of the accumulated turns, not periodic, never coming back. Its ratio is very nearly constant within any one revolution — it grows by $2\pi t$ per turn, which is 3.14 mm per turn here against an arm of 33 to 63 — and drifts monotonically over the machine’s whole stroke.

So the two are opposite in the frequency they vary at, and the arithmetic that catches one misses the other completely. Measure a winch over one revolution and it looks like a perfect constant-ratio drive. Measure a shaped drum over its whole travel and its mean looks like a constant-ratio drive too. Both are wrong at exactly the timescale the other is right at.

Between them those three exhaust the ways a strand’s arm can fail to be a number: it can depend on the driven body’s own angle, on another body’s configuration, or on the history. A drive whose arm depends on none of them is a round pulley on its own axis, and that is the only one.

The shaft angle is not a coordinate #

There is a structural oddity in this mechanism that the rest of the field does not have, and stating it explains both the counter and the encoder.

Every other mechanism on this site has a configuration space in which the state is where the parts are. Turn a four-bar’s crank through a full revolution and it is back where it started, in every respect: same positions, same velocities, same everything. Turn a winch’s drum through a full revolution and it is not — the line has moved on by a turn, the radius is a fraction of a line-thickness larger, and the next revolution will pay out more line than the last.

So the drum’s shaft angle is not a coordinate of the mechanism. The coordinate is the accumulated turn count, of which the shaft angle is the fractional part, and the configuration space is a line rather than a circle. The mechanism is a mechanism whose state cannot be recovered from where its parts are, because two states with every part in the same angular position differ in how much line is wound.

That is the exact reason the counter beats the encoder, and it is a stronger statement than a counter is more convenient. An absolute encoder on the drum shaft measures the angle, the angle is the fractional part of the state, and no amount of resolution recovers the integer part. A machine fitted with one and no counter genuinely does not know how much line is out, and it cannot be made to know by any measurement at the shaft — the information is not there. Something must count revolutions, or measure the line, or see the layers.

It also explains why the field’s other ratio failures feel different from this one. A shaped drum’s arm varies with the body’s angle and returns to itself every revolution, so its ratio is a periodic function of a genuine coordinate and an encoder settles it completely. A tendon’s arm varies with a joint angle, likewise a coordinate. The winch’s varies with something that is not a coordinate of any part, and that is what makes it the one mechanism in the field where the ratio depends on the machine’s history.

Which puts the quadratic in its proper place. Length going as the square of the turns is the arithmetic; the reason there is a quadratic to have is that the turns are a coordinate in their own right and the radius is a function of them. On a mechanism whose state repeats, no such function exists and the ratio is periodic instead.

What is not modelled #

The line has a thickness and no other property: it does not flatten under the layers above it, does not bed into the ones below, and does not stretch, all of which change the effective radius on a real drum and none of which are geometry. The wind is assumed to fill each layer completely and start the next at the same end; a real level-wind reverses direction and produces a crossover on every layer, where the turns climb over each other and the local radius is briefly larger. Nothing here computes where the line comes off the drum — the fleet angle, which decides whether the wind is neat — and nothing here knows about the drum’s flanges. There is no tension anywhere in this, so nothing says whether the wind will stay where it was put.

One consequence worth recording, since it is a fact about the machine rather than about its instrumentation. A mechanism whose configuration space is a line has no periodic behaviour to speak of, so nothing about a winch ever comes back — which is why the honest specification is a table over the layers rather than a single figure, and why every quantity in this essay had to be quoted with the fill it was measured at. That is not a shortcoming of the model. It is the only shape a specification of this mechanism can take.