The road a wheel carries with it

The rolling field opens with a wheel that may not slide sideways, and with the observation that its condition is a statement about velocities which does not integrate into one about positions. That is the whole reason a wheel is interesting: it cannot go sideways and it can be parked anywhere at any heading, and the gap between those two sentences is worth an exponent.

A taut strand on a pulley is also a rolling contact — precisely, and in the same sense. And its constraint integrates perfectly.

The difference between those two facts is worth an essay, because it says what makes a rolling constraint nonholonomic, and the answer is not “rolling”.

A strand rolls #

Take a strand lying on a body and a mark on each. As the body turns, the tangency point travels: along the surface, and along the strand. The material at the tangency does not travel at all relative to the surface — a taut inextensible strand does not slip on what it wraps, so the piece of strand in contact and the piece of surface under it are momentarily at rest with respect to each other.

That is the definition of a rolling contact, and it puts the instant centre of the body relative to the strand at the tangency point.

Everything a belt drive does follows from that. Two bodies sharing one straight span each have an instant centre on it; the span is rigid in length; so the two bodies’ surface speeds at their tangencies are the same, and the ratio between their rotations is the ratio of their arms — the perpendicular distances from each axis to the strand.

For circles the arms are the radii, which is why the ratio of a belt drive is a ratio of radii and why nobody notices that a separate quantity has been used. Measured directly, as a distance from a point to a line, on the two base circles of a 24 : 36 gear pair: 45.105246 and 67.658 mm against radii of 45.105246 and 67.658, to $1 \times 10^{-10}$ mm.

Where the arms are not the radii #

The moment the body is not round, or does not turn about its own centre, the arm and the radius part company and the instant-centre argument is the one that survives.

A shaped drum has an arm that varies with its angle: the support function $h(\psi)$, which is exactly the perpendicular distance from the axis to the tangent. Measured off a drawn profile and compared against the strand actually paid out, the two agree to two parts in a thousand million.

An eccentric idler — a round body whose centre is off the axis it turns about — has an arm of $r + e\cos\psi$, and the same measurement gives 8.00 to 16.52 mm per radian on a 14 mm idler mounted 6 mm off.

So rolling without slipping and constant ratio are different claims, and the strand separates them cleanly: the first is about the contact, the second is about the arms being constant, and only a circle on its own centre has both.

Two rates, and the one that is not the ratio #

There is a second quantity here that is easy to conflate with the arm, and the shaped drum makes them visibly different.

Strand paid out per radian is $h$, the arm.

Surface engaged per radian is $\rho = h + h''$, the radius of curvature — because the tangency point travels along the body’s own surface at that rate.

For a circle those are the same number and the distinction has no name. On the profile drawn here they are not: the arm runs from 23.80 to 44.20 mm, a swing of 1.86, while the surface engaged per radian runs from 3.40 to 64.60 mm, a swing of 19. The contact crawls round the body at wildly varying speed while the drive’s ratio changes by less than a factor of two.

Neither of those is a slip. The strand and the surface are still at rest against each other at the contact; what varies is how much new surface is being brought into contact per radian, and that is a curvature rather than a slipping.

Why this constraint integrates and a wheel’s does not #

Now the point of the essay.

A rolling wheel’s no-slip condition says the contact point’s velocity is zero, which is two scalar conditions in the plane and cannot be written as the time derivative of any function of the configuration. That is what nonholonomic means, and it is why the wheel reaches everywhere despite being unable to go sideways.

A strand’s no-slip condition looks identical at a point, and it does integrate: the total strand paid out is a function of the configuration, full stop. The rest of this field is built on that — one scalar equation, run(configuration) = length.

The difference is not in the contact. It is in the dimension of the surface being rolled on.

A strand is a curve, and a body rolling on a curve has only one way to go: forward along it. Arc length on the body and arc length along the strand match, so the accumulated rolling is a difference of two arc lengths, which is a function of position. Rolling of a curve on a curve always integrates.

A wheel rolls on a plane, which has two directions. The no-slip condition holds instantaneously in both, but the accumulated rolling depends on the path taken across the plane — go round a loop and the wheel does not come back to the same rotation. There is no function of position to differentiate, and that is the whole of it.

So the sentence to keep is: a rolling constraint is nonholonomic when the surface rolled on has more directions than the contact does. The wheel and the strand are the two sides of that, and the ball rolling on a plane — which comes back turned by the area it went round — is the extreme case.

The two rolling constraints, side by side #

Setting the wheel and the strand against each other makes the structural difference concrete rather than definitional.

Every row but the first differs, and every difference comes from the second row.

The line worth dwelling on is the fifth. A wheeled vehicle’s configuration cannot be written down as the solution of anything — it has to be reached, by a path, and the manoeuvring depth is a real cost with an exponent attached to it. A strand mechanism’s configuration is the root of run = length, computed in closed form, and the only thing that has to be reached is the taut set: which strands are pulling, and which route each is on.

So the two mechanisms are non-classical in opposite directions. The wheel has a constraint that is well defined everywhere and does not integrate. The strand has a constraint that integrates perfectly and is not always there.

The centrodes are the body and the strand #

The curvature field describes any planar motion by its two centrodes: the locus of the instant centre in each of the two frames, which roll on each other without slipping.

For a body wrapped by a strand, those two curves are named without any computation. The moving centrode is the body’s own outline. The fixed centrode is the strand’s path. The instant centre is the tangency, and the two curves roll on each other because the strand does not slip.

That is why the two classical roulettes are strand constructions.

Rolling a line on a circle gives the involute — which is a strand unwound from a drum, and is a gear tooth’s flank, agreeing with the gears field’s own involute to $1.5 \times 10^{-14}$ mm.

Rolling a circle on a line gives the cycloid — the same pair of centrodes with the roles exchanged.

A mesh slides; a strand does not #

The comparison worth making is with the mechanism a belt is most often replaced by.

Two involute flanks in mesh are not rolling on each other. They roll at exactly one point — the pitch point, where the relative velocity vanishes — and slide everywhere else, by up to 12 mm per radian at the ends of the contact. That sliding is where the wear is, where the friction losses are, and why a gearbox needs oil.

A strand slides nowhere. Its whole wrap is a rolling contact, and it stays one at every position.

Both statements are geometric and neither is about friction. What they explain is a division of labour that is usually justified on other grounds: a belt is quiet and needs no lubrication because its contact rolls, and it cannot hold a phase relationship because nothing in a rolling contact prevents the whole strand creeping round. Teeth solve the second problem by giving up the first.

The intermediate case is the chain, and it has both. A chain’s rollers seat on the sprocket teeth without sliding — that is what the rollers are for — and the chain’s pins sit on a polygon rather than on a circle, so the effective arm swings by 4.05% within each tooth of an eleven-tooth sprocket. Rolling contact, and a ratio that varies anyway.

The one place a strand’s rolling is not obvious #

There is a case where the no-slip claim needs stating carefully, and it is the tackle.

A sheave in a block has a rope running over it, and the rope’s two parts are at different points of the same body. The rope rolls on the sheave, so the sheave turns; but the sheave is not driving anything and nothing is being transmitted through its rotation. What the sheave does is redirect a length, and its rotation is a consequence rather than a purpose.

That is why a tackle’s ratio comes out of differentiating a length rather than out of any arm. Ask what arm a sheave presents and the answer is its radius; ask what that arm does to the ratio and the answer is nothing at all, because a sheave with a rope over both sides pays out on one side exactly what it takes up on the other. The whole of a tackle’s ratio is in the directions of its parts of line, which is why the second route to it is a sum of cosines.

So the rolling reading explains a belt drive completely and a tackle not at all, and the division is clean: rolling matters where a strand’s contact with a body transmits the body’s rotation, and not where it merely turns a corner.

What the rolling reading buys #

Three things that were separate results in this field become one.

The ratio is a ratio of arms, whatever the shapes. Circles, shaped drums, eccentrics and base circles are all the same statement, and each was measured off the drawn run rather than taken from the body that produced it.

The tangency is the instant centre, so the whole apparatus the curvature field built for planar motion applies to a strand contact directly — the centrodes, the velocity distribution, the pole.

The constraint is a length, and now for a reason rather than by observation: the rolling integrates because the contact is confined to a curve.

Where the two meet: a wheel driven by a strand #

One mechanism has both constraints at once, and it is an ordinary one.

A bicycle’s rear wheel rolls on the road — nonholonomic, two directions, no function to differentiate — and is driven by a chain, whose constraint is a length and integrates. The two are in series: the chain fixes a relation between the crank’s rotation and the wheel’s, and the wheel’s rolling turns that rotation into a path across the ground that depends on the whole history of the steering.

So the pedals determine the distance travelled along the path exactly, through an integrable relation with a measurable ratio — subject to the four per cent of chordal variation — and determine nothing whatever about where the machine ends up. Two constraints in one drive train, one of which is a length and one of which is not, and the boundary between them is the wheel’s contact patch.

The rule applied to every rolling contact here #

The criterion the essay arrives at — a rolling constraint is nonholonomic when the surface rolled on has more than one dimension — is worth running over the whole collection, because it sorts every rolling contact on the site and the sorting is not the one intuition offers.

A strand on a pulley. The strand is a curve, one-dimensional, so the contact integrates and the constraint is a length. Holonomic, and the whole belt-drive apparatus follows.

A gear mesh. Two involute flanks touch along a line of action, and the relative motion is confined to that line. One dimension again, so the constraint integrates and a gear train’s ratio is a relation between angles rather than between rates — which is why a ratio is a count and why a gear train has no manoeuvring problem of any kind.

A wheel on a rail. The rail is a curve, so a flanged wheel on a track integrates: the train’s position along the line is the wheel’s rotation times its radius, exactly, and a train has no sideways freedom to manoeuvre into and no bracket to climb.

A wheel on a plane. Two dimensions, so the constraint does not integrate. Gap of one, an $\varepsilon^2$ manoeuvre, and the whole of this field.

A ball on a plane. Two dimensions again, and additionally the roller has an orientation of its own that the plane does not constrain — so the gap is three rather than one, and the manoeuvring is two brackets deep. The surface’s dimension decides whether; what the rolling body carries decides how deep.

Read down that list and the intuition that gets overturned is the one about contact. All five are the same physical situation at the point of touching — material at rest against material, no sliding, a tangency that is an instant centre — and they land on both sides of the deepest division this site draws. Nothing about a contact decides whether its constraint integrates, which is the sentence this essay exists to establish, and the list is what makes it more than an assertion about two cases.

It also says where to look on a mechanism nobody has classified. The question is never is this rolling — it always is — but what is being rolled on, and how many directions does it have. A ball in a straight groove rolls on a curve and integrates; the same ball on the flat does not. The mechanism has not changed and the surface has, which is the whole content of the rule.

What is not modelled #

A real belt creeps. Its tension differs between the tight and slack sides, so it stretches by different amounts on the two spans and must slip somewhere on the wrap to make up the difference — which is the creep that gives a flat belt drive a percent or two of speed loss. That is an elastic argument and needs a modulus, and nothing in this essay survives it: the no-slip claim here is a claim about an inextensible strand. A real cable also has a bending stiffness, so it does not lie exactly on the surface it wraps, and the arm is measured to a line that is a fraction of the cable’s diameter off the surface. And nothing here computes what a rolling contact costs — rolling resistance is a material property, and this field’s entire content survives with every force unknown.

One consequence of the sorting is worth recording because it inverts a piece of received wisdom. Rolling contacts are routinely presented as the difficult case in mechanism analysis and sliding contacts as the easy one, and the list above says the opposite as often as not: four of the five rolling contacts here integrate perfectly and produce some of the most exactly behaved mechanisms on the site, while the gear mesh that slides is among them. What makes a mechanism hard to reason about is not whether its contacts slide but whether its constraints integrate, and the two questions are independent — which is the reason this field had to be built separately rather than inheriting the constraint field’s machinery unchanged.