A drum is a size, a wrap is a shape

A strand is a member that pulls and does not push, wrapping bodies and running straight between them. Its geometry is tangent lines and arcs, and both of those behave in a particular way under a scaling.

Tangency is a similarity #

Given two circles and a wrap sense there is exactly one tangent line, and the construction that finds it is ruler and compass.

Scale both circles and the distance between their centres, and the tangent line scales with them. The angle it makes with the centre line is unchanged; the arc it hands over to is subtended by the same angle; the whole picture is the original enlarged.

The step to check is the one about the wrap sense, because it is the field’s own subtlety. A tangent between two circles comes in two kinds — outer and crossed — and which one a strand takes is decided by the sense in which it wraps each body. That decision is combinatorial rather than metric, so it is unchanged by a scaling, and the tangent that scales is the same tangent rather than the other one.

Had it been otherwise, a scaled drive could route differently from its original, which would make the field’s figures figures of one machine each rather than of families.

So the field’s constructions are similarities, exactly as the curvature field’s and the synthesis field’s are, and for the same reason: every step is a line, a circle or an intersection.

Given two circles and a wrap sense there is exactly one tangent line, and the construction that finds it is ruler and compass.

What that makes a shape #

The list is long and it is most of the field.

The wrap angle — how far round each body the strand goes — is an angle. Unchanged.

The tangent directions are directions. Unchanged.

The velocity ratio of a belt drive is the ratio of the two radii, dimensionless. Unchanged, and it is what the drive exists to produce.

A tackle’s mechanical advantage is a count: how many strand parts support the load. In the third class, unchanged by anything.

And whether a strand can lie where it is asked to — whether the tangent exists, whether the wrap senses are compatible — is a condition on the geometry with no absolute length in it. Unchanged.

So a belt drive’s ratio, its wrap angles, whether it slips, how many parts its tackle has and whether its routing is possible are all shapes.

That is a striking proportion. Of the quantities a designer of a belt drive or a rope run actually asks for, nearly all are unchanged by making the whole thing bigger — which is why the same drawing conventions and the same rules of thumb serve a sewing machine and a mine hoist.

The list is long and it is most of the field.

The wrap angle — how far round each body the strand goes — is an angle. Unchanged.

The tangent directions are directions. Unchanged.

A tackle’s mechanical advantage is a count: how many strand parts support the load. In the third class, unchanged by anything.

What is a size #

Three things, and one of them is the field’s own practical difficulty.

The strand’s length. The sum of two tangent lengths and two arcs, all lengths, exponent one. That is what a person buying a belt needs and it is the one quantity that does not transfer.

The slack. How much longer the strand is than the path it must follow, a length, and the quantity a tensioner is for.

And the drum’s diameter, which sets everything else’s scale.

That is a short list against a long one, and it says the field’s output is mostly transferable — which is why a belt-drive design table can be printed with ratios and wrap angles and a separate column for the length.

It is also why the field’s own vocabulary is mostly angular. A drive is described by its ratio and its wrap; a rope run is described by its reeving; a tackle by its parts. The lengths appear at the end, when something has to be ordered, and everything before that is shape.

Three things, and one of them is the field’s own practical difficulty.

The slack. How much longer the strand is than the path it must follow, a length, and the quantity a tensioner is for.

And the drum’s diameter, which sets everything else’s scale.

The tangent has no residual #

A methodological point specific to this field and worth having, because it changes what the scaling probe is testing.

A tangent line is a construction rather than a solve: given two bodies and a wrap sense there is exactly one, and it is written down in closed form rather than found. There is no Newton iteration and no residual, which is a property the field’s own gotchas record — a construction that converges nothing has no residual, so nothing about it can fail loudly.

For the probe that has a consequence. The exponents here come back at machine precision rather than at a solver’s floor, because there is no solver: the quantities are closed-form expressions in the inputs and multiplying the inputs multiplies the outputs exactly.

That is cleaner arithmetic than a four-bar’s, whose null residual of 7.6 × 10⁻¹⁵ is one linear solve’s worth of rounding. It is also weaker evidence: an exponent that comes back exact from a closed form is confirming algebra rather than testing a computation, and the probe’s third job — catching a stray constant in the code — is the one it still does here.

A construction with no residual gives an exact exponent and a weaker test, which is worth knowing before reading the field’s numbers as more certain than a solver’s.

Two strands and a comparison #

The one place a length reappears in what looks like a shape is worth naming, because it is the field’s own hardest object.

A run over three or more bodies has to decide which side of each it passes, and some routings are impossible: the strand would have to pass through a body, or the tangent would not exist. Whether a routing is possible is a condition on the geometry, dimensionless, and unchanged by scaling.

But the margin — how nearly a routing is impossible, how close the strand comes to a body it is meant to clear — is a distance. So the feasibility is a shape and the clearance is a size, which is exactly the pattern a count and its margin produce everywhere else on this site.

That means a routing that just works at one size just works at every size, and a routing that just works with a strand of a given thickness may not with a thicker one. The strand’s own diameter is a length brought to the problem, like a bar’s half-width in the bodies field, and comparisons against it do not scale with the machine.

The field’s feasibility results are shapes and their margins are sizes, and the margins are what a real installation needs.

Reading two shafts and nothing else #

Apply the field’s own question.

Measure a belt drive’s two shaft angles as it runs. The reading is dimensionless, so the standing argument applies: the ratio comes back exactly and neither diameter does.

That is the same answer a gear pair gives and for the same reason, with one difference worth noting. A gear pair’s ratio is two integers and is exactly a count; a belt drive’s is a ratio of two continuous diameters and is a genuine shape, so it can be slightly wrong in a way a gear ratio cannot.

Measuring a belt drive’s ratio recovers a real number that could have been any real number, and measuring a gear train’s recovers a rational that was one of a discrete set. That is a distinction between the two fields that the scaling probe makes visible and that the fields themselves do not draw.

A tackle is entirely counts and shapes #

The field’s other object sorts even more cleanly and it is worth doing separately.

A tackle — a rope run through a set of blocks — has a mechanical advantage equal to the number of strand parts supporting the load. That is an integer, a count, unchanged by anything, and it is the whole of what the tackle is for.

Its rope length is a length. Its travel ratio — how far the hauling end moves per unit of load movement — is the reciprocal of the advantage, so also a count.

And whether a particular reeving is possible, which blocks the rope can reach in which order, is a combinatorial question with no numbers in it at all — a topology-field question arriving inside a mechanism made of rope.

So a tackle is a count, a count and a length, and the count is what a user cares about. That makes it the most transferable object in the field: a four-part tackle is a four-part tackle at any scale, in any material, on any load.

The one thing that is not a count is the friction, which multiplies at every sheave and is outside the site. The idealised advantage is exact and the real one is less, by a factor nobody here can compute.

Slip is a shape too #

The field’s characteristic failure deserves the same treatment because it is the reason a belt drive is not a gear drive.

Whether a strand slips is decided by the wrap angle and the friction coefficient, through the capstan relation. The wrap angle is dimensionless and the coefficient is dimensionless, so the condition is dimensionless: a drive that slips at one size slips at every size.

That is a strong statement and it is outside this site’s boundary, because a friction coefficient is a material property. What this field computes is the wrap angle; whether the drive slips depends on a number the site does not carry.

The geometric half is a shape and the material half is a coefficient, which is the same division the transmission field draws at its own boundary, and the two land in the same place: the part that is geometry transfers and the part that is outside is dimensionless anyway.

Where the length comes in #

The one quantity that does not transfer is worth dwelling on because it is the one a designer actually orders.

A strand’s length is the sum of two straight runs and two arcs. Both are lengths, so the total is a length, exponent one, and a drive at twice the scale needs twice the belt.

What is transferable is the length divided by the centre distance, or by a pulley’s circumference — a dimensionless number that says how much of the strand is straight and how much is wrapped. That ratio is a shape and it is a genuinely informative one: a drive whose belt is mostly arc behaves differently from one whose belt is mostly straight, and the difference survives scaling.

The field quotes the absolute length and the dimensionless form is the one that describes a design, which is the same recommendation the survey has produced in five other fields and is beginning to look like a fleet-wide reporting habit rather than a field-specific oversight.

A strand has no length of its own #

One of the field’s own findings interacts with all of this in a way worth stating.

A strand is a member with no length of its own: unlike a bar, which holds two points a fixed distance apart, a strand’s straight run is however long the geometry makes it, and what is conserved is the total around the loop.

So the field’s constraint is a statement about a sum of lengths being constant, which is a length, and every configuration it permits is one where that sum is met. Scale everything and the sum scales, so the constraint is satisfied at any size by the correspondingly scaled configuration.

A conservation of length is scale-covariant rather than scale-free, which is the same behaviour Bennett’s condition shows: the condition survives a scaling because both sides scale together, and it is not a dimensionless statement.

The drum that winds is the exception #

One object in the field genuinely resists the sorting and it is worth naming.

A winch drum on which a rope winds in layers has an effective radius that grows as the rope accumulates, so the velocity ratio changes as the drum fills. The ratio is dimensionless at every instant and it is a function of how much rope is on the drum, which is a length.

So the ratio is a shape and the schedule of ratios through a wind is not: it depends on the rope’s diameter compared against the drum’s, which is a comparison of two lengths and therefore a ratio — a shape again, but one built from an ingredient chosen independently of the machine.

That is the same structure the bodies field’s clearances have: a comparison between the mechanism and something brought to it. Scale the drum and hold the rope and the schedule changes; scale both and it does not.

A winch is the field’s one object whose behaviour depends on a length not its own, and it is correspondingly the one whose results transfer only if the rope is scaled with it.

Which of these numbers wants an error bar #

One object carries all three classes at once, which is unusual, and it makes the practical question sharp: given a number this field prints, should a reader expect it to be uncertain?

A count cannot be slightly wrong. How many parts of a tackle carry the load, how many turns are on the drum, whether a reeving closes: integers, recovered exactly from any observation that resolves them at all, and brittle rather than uncertain. What they can be is wrong, discontinuously, if the arrangement was misread. They want no error bar and they do want their margin — how nearly the wrap ran out, how close a turn came to overlapping — which the field computes and does not print.

A shape can be measured badly and transfers exactly. Wrap angles, tangent directions, the velocity ratio between two shafts, the slip fraction: all dimensionless, all recovered from angles alone, all identical on a model and on a mine winder. These want an error bar, and it is an error bar that means the same thing at every size.

A size is the number that does not travel. The strand’s length, the slack, the drum’s diameter. Each is a length, each is exactly as large as the machine it was measured on, and none of them says anything about a machine of another size unless it is divided by something first.

The distinction that matters in practice is between the second and the third, because the field prints them in the same units and the same typeface. A slack of 4.2 mm on this drive is not a fact about drives; a slip of 0.4% is. The dimensionless one is the transferable one and the field quotes the absolute one, which is a reporting convention rather than an error, and the same recommendation the survey has produced in five other fields.

There is a comparison worth ending on, because it separates two things a reader may think are the same. A gear train’s ratio and a belt drive’s ratio are both ratios of shaft speeds, both dimensionless, both recovered from a protractor on either end. They are not the same kind of number at all. The gear train’s is two integers divided: exact, brittle, and incapable of being 3.0001. The belt drive’s is a ratio of two diameters with a slip correction on it: continuous, measurable, and wrong by a little all the time.

Same reading, same instrument, same units, two different classes of answer — and nothing in a caption that quotes either of them says which. A reader shown ratio 2.400 has no way to know whether the fourth digit is a measurement or a rounding, and on this site the way to tell has been to know which field the figure came from. That is not good enough, and the repair is the one the counts want too: print what the number could have been instead.