What a second contact is for

Rotors that mesh and cannot drive each other is the sharpest negative result this field has. Two identical lobed rotors on a one-to-one pair are each other’s conjugates by construction, hold their ratio exactly at every instant, and cannot turn each other: their single contact’s moment arm about the driven shaft is $-R\sin(n\varphi)$, which passes through nought and changes sign every time a tip crosses the line of centres. For half of every turn the only contact there is pushes the driven rotor backwards.

The natural reading of that is a statement about conjugacy — that a pair which is its own mate is degenerate in some way, that being the same shape twice is what puts the dead points there. It is worth testing, because a great deal follows from it: if conjugacy between identical shapes were the culprit then a pair of unlike shapes would escape, and a designer wanting a self-driving pair would go looking for asymmetry.

The test is a mechanism already computed. Eleven lobes from twelve pins built a ring of round pins and the disc that is their envelope — a reduction of eleven to one with no teeth anywhere, and a pair every bit as conjugate as the rotors, generated by the same routine from the same equation — because the second shape is not a choice once the first and the ratio are given. It drives. It is what a cycloidal gearbox is, it carries load in both senses, and nothing about its geometry is less conjugate than the rotors’.

So conjugacy is not the culprit, and what is turns out to be a count.

An arm needs no force #

The quantity below is a moment arm and not a torque, and the difference is the reason any of this can be said at all in a collection that computes no forces.

A contact between two bodies is unilateral. Each can push the other along their common normal and neither can pull, because there is nothing holding them together. So the sense in which a given contact turns the driven body is fixed by geometry alone: it is the sign of

$$\text{arm} \;=\; (X - d) \times \hat{n},$$

with $X$ the contact, $d$ the driven body’s own centre and $\hat n$ the normal pointing into it. A positive arm turns it one way and a negative arm the other, whatever the magnitude of the push, so long as it is a push. Nothing about the size of the arm is a torque until somebody supplies a force, and nothing below supplies one.

That is enough to make the Roots pair’s failure a geometric fact rather than a mechanical guess. With one contact and an arm of nought, no force at all at that contact turns the other rotor: the mechanism is stopped by its shape. And it is enough to say what would rescue it. A mechanism with several contacts at once is stopped only if every one of their arms is nought together, which is a far stronger coincidence than one arm crossing zero.

It is also where the field’s own centrode result reappears. Two wheels in mesh slide everywhere but one point, and that point is the pitch point — the instant centre of the two bodies’ relative motion. The meshing equation says every contact normal passes through it, and that fact does a second job here that it does not do in a gear pair: since moving a point along a line changes no moment of that line, the arm may be evaluated at the pitch point instead of at the contact,

$$\text{arm} \;=\; (P - d) \times \hat n .$$

Every arm in the whole mechanism is therefore one fixed vector read against its own contact’s direction, and no arm anywhere can exceed $|P - d|$ in magnitude. That is a closed form with no contact in it, computed independently of the contact solve and required to agree with it — and it is what makes the comparison between two mechanisms of different sizes fair, since each one’s arms can be scaled by its own $|P-d|$.

Eight contacts, and none of them a choice #

At any angle of the eccentric, eight to eleven of the twelve pins are in contact with the disc. That is not a design decision either: the disc is the envelope of the whole ring, so every pin that the disc has not been machined clear of by a neighbouring lobe is touching it.

Sorting those contacts by their arms is what the argument rests on. At one representative angle the arms run from 54.2, smoothly down through nought, to −54.8 — with about half of the contacts on each side of the disc’s own centre. Four turn it one way and four the other. One of them is always near nought, which is exactly the situation the Roots pair is in; the difference is that here it is one of eight rather than the only one.

Two candidate contacts per pin come out of the meshing equation, on the near and the far side of the pin’s circle, with equal and opposite arms. Only one of them is a contact, and usually neither is: the disc is cut back by whichever pin reaches furthest at each place on its boundary, so most of a pin’s candidate contacts are in free space with the material that would have touched them already removed. Which candidate survives is a test against the generated profile rather than a choice, and the pins that fail it are drawn hollow.

The distribution of arms is the reason the mechanism has no dead point. Twelve contact directions spread round a ring, all read against one fixed offset vector, cannot all be perpendicular to it at once — there are only two directions perpendicular to a vector in the plane, and the contact normals are not free to stack on them. So an arm of usable size in each sense is always present, and which pins supply it changes smoothly as the input turns.

That is a different kind of argument from the one the rotors needed, and the difference is worth naming. The Roots pair’s failure was proved by a closed form: the arm is $-R\sin(n\varphi)$, a sine has zeros, and there is nothing further to measure. Nothing so tidy is available here, because the surviving contacts are decided by which lobe machined which pin clear and that is a property of the generated profile rather than of a formula. What replaces it is a bound and a sweep — the bound from the pitch point, which is exact, and the sweep over the input, which is sampled. The conclusion is correspondingly weaker in form and stronger in content: not the arm is never nought as an identity, but the arm is never less than seventy per cent of its bound as a measurement over a period.

The sweep that separates the two mechanisms #

Put both mechanisms on one axis, each scaled by its own offset, and the difference is not a matter of degree.

The drive’s largest available arm — the best any single contact can offer, in the more awkward of the two senses — never falls below 70.7% of what its geometry allows, through a whole turn of the input. The Roots pair’s falls to nought, exactly, twice in every lobe pitch, and it does so because its arm is a sine and a sine has zeros. Between those angles the pair is perfectly capable; at them it is stopped, and no tolerance, no clearance and no undersize moves the zeros, because they are where a tip crosses the line of centres and that is a statement about the lobes rather than about the fit.

The contact counts run alongside and say the same thing more plainly. The drive holds eight to eleven contacts. The pair holds one, and at the hand-over instants — when one tip has let go and the next has not yet taken up — it holds none at all. A mechanism with no contact is not transmitting anything by definition, and a mechanism with one is transmitting whatever that one contact’s arm allows, which twice a pitch is nothing.

It is worth saying precisely what is being claimed and what is not. The claim is that an arm of either sense is available at every angle. Which contacts actually carry, and in what proportion, is a statics problem with an elastic body in it and is genuinely outside: real cycloidal drives share load unevenly, and the pins near the ends of the engaged arc take less than the ones in the middle — an arc which, like every conjugate pair’s, has two ends somebody chose. But load sharing is a question that only arises once there is more than one contact to share between, and the geometric result above is the precondition for it rather than an approximation to it.

Where the offset comes from, and why it is fifty-five #

The bound on every arm is $|P-d|$, and it is worth computing rather than accepting, because it is the one number in the comparison that is not measured at a contact.

The disc’s motion is a rotation about its own centre at one eleventh of the input, and that centre orbits the machine’s axis on an eccentric of five. The instant centre of the disc relative to the fixed ring is therefore at a distance $e/\omega$ from the disc’s centre — the orbit speed divided by the spin rate — which is $5 \div \tfrac{1}{11} = 55$. It lies on the line from the axis through the disc’s centre, at radius sixty from the axis: the pitch point of a cycloidal drive sits on the pin circle itself, and it walks round that circle once per turn of the input.

That is the same construction every point of a moving plane has a centre reaches from the other direction, and it puts a useful reading on the pin count. Larger $N$ means a slower disc for the same input, so $e/\omega$ grows and the offset with it: six pins give an offset of 50, twenty give 57. The bound on the arm therefore rises slowly with the count while the share of it that survives does not, which is why the two columns of the table below run in different directions and why neither alone answers a design question.

There is a second reading of the offset that matters more than its size. Because the pitch point is on the pin circle, it is always at distance $R_p$ from the axis and always at the angle the eccentric points, so it is never at a pin and never between two particular pins for long. A mechanism whose pitch point sat at a fixed place relative to its contacts would have the same arrangement of arms at every instant, and the worst one would be permanent. Here it sweeps, which is why the contact that happens to have an arm near nought is a different contact a moment later — and it is the moving version of the thing a dead centre is: a coincidence between a line of action and a centre, which a mechanism survives by not letting it happen at the same place twice.

The count, not the shapes #

The two mechanisms differ in exactly one structural way, and it is not their conjugacy.

A pair of rotors with the same number of lobes has, at any instant, one place where a tip of one is engaged in a root of the other. A ring of $N$ pins and a disc of $N-1$ lobes has $N-1$ places where the counts fail to line up, and at each of them the mismatch puts a lobe flank against a pin at a different phase of the same engagement. That is what a difference of one in the counts buys: it stops all the engagements being the same engagement.

The same reading explains the machines. A Roots blower’s rotors are two lobes against two lobes and it needs timing gears; a screw compressor’s rotors are four lobes against six and, in oil-flooded machines, one drives the other with no timing gears at all. The usual account of that difference is the asymmetry of the screw rotors’ profiles. The measurement here says the profiles are not doing the work — a cycloidal drive’s pins are as symmetric as a shape can be — and the count is.

What more pins do not buy #

The obvious next expectation is that more contacts is better, and over six pin counts it is false.

Six pins keep half of their available arm at the worst angle, eight keep 70.7%, ten keep 72.5%, twelve keep 70.7%, sixteen keep 75.2% and twenty keep 62.9%. The count rises monotonically and the share does not. What decides the share is how the surviving contacts’ directions happen to lie relative to the offset vector at the worst angle of the input, and adding pins shifts that arrangement about rather than filling it in.

The claim the whole set supports is the weaker and more robust one: every count from six to twenty keeps more than one contact at every angle, and every one keeps an arm of both senses worth at least half of what its geometry allows. A designer choosing a pin count is choosing lobe size, load per pin and manufacturability — not choosing whether the drive has a dead point, because none of them does.

The row that does not belong to that family is the last one. Two lobes against two, one contact, and a worst arm of nought.

What is not modelled #

Force, friction and load sharing. Every statement here is about which senses of turning are available, which is geometry. How a real drive divides its load between eight contacts is an elastic problem with contact stiffness in it, and nothing in this collection computes one.

Clearance is nought. The pins and the disc are drawn at their nominal sizes, so every candidate contact that the profile allows is a contact. A real drive has backlash and a disc cut slightly undersize, and both reduce the count — the engaged arc shortens and the pins at its ends come out of contact first. Whether the count can be driven down to one by clearance alone is not measured here, though the drive’s own profile suggests it cannot without also losing the ratio.

One tolerance decides the count. A pin is called in contact when its candidate contact sits within two hundredths of a unit of the generated profile, on a ring of radius sixty. The count moves by a pin or two either way at neighbouring tolerances; the floor on the available arm does not, because the contacts near the boundary of the criterion are the ones with the smallest arms.

The disc’s profile is sampled. It is generated on 1200 steps and stored on 1440 rays, and the material test that decides whether a candidate contact survives is a comparison of two radii interpolated between those rays. The pitch-point identity is checked against the analytic route and agrees to $10^{-13}$; the count is the quantity that depends on the sampling.

Two mechanisms, not a survey. A cycloidal drive and a two-lobe Roots pair are one case of many contacts and one of a single contact. A screw pair’s four-against-six is named above as the machine the reading explains and is not computed here.

Still open: the count a clearance leaves #

Everything above is at nominal size, where a contact exists wherever the generated profile allows one. A drive that is actually built has the disc cut undersize and the pins a little small, and the engaged arc then shortens from both ends — which is the whole of how a cycloidal drive is given backlash, and it removes contacts. Backlash is an allowance rather than a defect, and the question is what this one costs.

Its distinct argument would be the count as a function of that undersize, and the arm floor with it. Two things would come out of it. The undersize at which the count first falls to one somewhere in the turn, which would be the point at which a real drive acquires the Roots pair’s problem and is a number a manufacturing tolerance can be checked against; and whether the arm floor falls smoothly with the undersize or collapses at that threshold — since the contacts lost first are the ones at the ends of the arc, which are the ones with the largest arms rather than the smallest. If the floor collapses, then the clearance a drive is given is buying backlash at the direct expense of the property that makes it drive at all, and the two cannot be traded independently.