A clearance leaves one contact

What a second contact is for answered a question the rotors had left. Two identical rotors are exact conjugates and still cannot drive each other, because their one contact’s arm about the driven shaft passes through nought twice a lobe. A ring of pins and the disc they generate is just as exactly conjugate and can drive, because it has many contacts at once, and they never all lose their arms together.

All of that was about the disc as its pins generate it: the exact envelope, touching every pin that no neighbouring lobe has machined clear. A drive that is built has its disc cut a little undersize so that it can run, turn freely under temperature and take a film of oil. That essay ended by asking how many of the contacts survive the clearance, and whether the arm they guaranteed survives with them.

The measurement below says one contact survives, at every angle and in both directions, and that its arm is the best there is. Getting there needed a more exact picture of the disc than the one the earlier measurements used, and the better picture also corrected the exact drive’s own numbers.

The disc as its pins leave it #

A disc cut for pins of radius $r + \delta$ is the material that no such pin reaches at any point of the motion. Every pin’s centre traces one curve in the disc’s own frame. The curve is a trochoid, the same for every pin and a pin pitch along for each. The disc is therefore the region inside that curve which the tube of radius $r + \delta$ about it does not cover. Its boundary is the curve’s inner offset, trimmed wherever another stretch of the curve comes within $r + \delta$, which happens near every root.

The drive measured here, as in the essays before it, puts the eccentric at exactly the pin radius divided by the pin count, five on a ring of sixty for twelve pins. That proportion makes the trochoid an epicycloid with a cusp at every root, where the pin’s centre comes in along a line and goes straight back out. The reason is a speed. Seen from the disc, a pin’s centre moves because the eccentric carries the disc past it and because the disc turns. At the root the two motions are equal and opposite exactly when the eccentric times the pin count equals the ring’s radius, so the pin’s centre stops for an instant relative to the disc and turns back. With a smaller eccentric it never quite stops, and the root is a curve and not a point. The disc’s root there is a round socket, a circle of the generating pin’s radius about the cusp. A pin sitting at the root is held in a cup, not against a flank.

A pin’s gap is then its centre’s distance to that boundary, less its radius, computed from the definition to about a millionth of a unit.

The trimming is not a detail. Near a root the curve comes in to the cusp and goes straight back out, so its offset on the inner side crosses itself, and the part of the offset that loops through the socket is material the pins have already removed. The cutter takes back the tooth described undercut as an event of this kind: a later part of the generating motion passing back through material an earlier part had left, and taking it. An envelope computed without that check draws a profile the machine could not cut. Here the check is written into the definition of the material: a point is disc only if no position of any pin comes within reach of it.

The first result is a correction. Measured this way, the exact disc touches nine or ten of its twelve pins at every angle, not the eight to eleven the earlier essay reported. That essay’s test for contact read gaps against the ray table the disc is drawn from, which is out by as much as a tenth near the roots. A gap tolerance of 0.02 on it both admitted pins that stand clear and lost pins that touch.

The same test lost a contact outright at symmetric angles. At 60° of the eccentric, pin 4’s contact sits exactly on its circle’s parametric seam, where the root-finder opens no bracket. Its arm of 47.6 went missing, and the floor the earlier essay reported, 70.7% of the pitch offset, was that missing contact. With the seam sampled twice and gaps read against the true boundary, the exact disc’s largest arm in its worse direction never falls below 81.4% at one-degree steps. The earlier essay has been corrected to both numbers. Its argument, that many contacts keep an arm where one cannot, stands and is somewhat stronger.

Turn an undersize disc and one pin touches #

Now cut the disc 0.02 undersize, about a sixtieth of one per cent of the ring’s diameter. At the eccentric’s position every pin that was touching stands clear by 0.02. Turn the disc about its own centre, holding the eccentric still, until some pin’s gap reaches nought.

At every one of 72 angles of the eccentric, and turning either way, exactly one pin touches. At 0.08 undersize, the same. The reason is simple once seen. Turning the disc through ε closes each pin’s gap at its own rate, and the gaps close together only if those rates are equal. They are not: the earlier essay showed that the rates, the contacts’ moment arms about the disc’s centre, run smoothly from one sign through nought to the other. The first gap to close closes alone.

A rigid drive with any clearance at all therefore works through one pin at a time, and the nine or ten contacts that distinguished it from the rotors are, for a rigid disc, a property of a disc of exactly nominal size.

The rotors make the comparison sharper. The clearance that is the seal cut both of a Roots pair’s rotors back by the same amount and found the gap between them exactly twice that amount at every angle. Cut back, a Roots pair has no contact at all. It runs on its timing gears, and the gap is the seal. A cycloidal drive cut back keeps one contact, because its disc is turned by the pins and not held in phase by anything else, and the clearance becomes free turn. The same operation on two conjugate pairs removes all of one mechanism’s contacts and all but one of the other’s. In both, what is left is decided by the distribution of arms that where two shapes stop touching traced along each profile’s working arc.

The pin that touches has the best arm #

The earlier essay’s question was whether a clearance costs the drive its arm floor. The concern was that the contacts lost first would be the ones with the largest arms. The measurement says the reverse. The pin that touches first is the one whose gap closes fastest, and a gap closes as fast as its contact’s arm, so the first contact is the pin with the largest arm in the direction the disc is being turned.

The touching pin’s arm runs from 88.6% to 99.0% of the pitch offset over the turn, against the exact disc’s floor of 81.4%. It is never below the exact disc’s best, and usually above it, because the clearance changes which pins can be first. A pin at or near a root sits in the root’s round socket, and a pin in a socket closes its gap whichever way the disc turns, at up to the whole pitch offset. At the exact root angles, every 30° of the eccentric, that pin is first in both directions.

The arm floor, in other words, is not a casualty of the clearance. A clearance selects the contact that turns the disc best. What it removes is every other contact.

The backlash is not one number #

A clearance also buys backlash: the angle through which the disc can turn between touching a pin one way and touching one the other way. Backlash is an allowance treated backlash as a deliberate spend, and this is what the spend buys on a cycloidal drive.

The free turn is the clearance divided by the forward pin’s arm, plus the clearance divided by the backward pin’s arm. It runs from 0.0367 to 0.0410 radians per unit of clearance, so a clearance of 0.02 on a ring of 60 leaves the disc 0.042° to 0.047° of free turn. It is least at the root angles, where one pin in its socket stops the disc both ways with the full pitch offset. It repeats every pin pitch.

The first-order account makes a prediction that can be checked against this. If each gap closes at its contact’s arm, the free turn is $\delta(1/a_+ + 1/a_-)$, with $a_+$ and $a_-$ the exact disc’s best arms each way. At mid-pitch, 15° from a root, the measured free turn at 0.02 undersize agrees with that to two parts in ten thousand. At 10° and 20° it falls 5.6% below it, and at the roots 12.5% below. The difference is the socket. A flank closes its gap at its arm, but a pin near a cusp moves across the socket’s round wall and closes faster than any flank can, so the disc stops sooner than the flanks alone would stop it. The first-order law is exact where the drive’s arms are smooth and an overestimate where they are not. This is the same division a flat holds the wheel to second order found in a Geneva’s lock, between a face whose gap closes at first order and one that does not. Here the socket is the face that closes faster than first order would say.

The two clearances do not give quite the same curve. At 0.08 the free turn per unit of clearance is smoother and slightly smaller between the roots. That is the sockets again. A larger clearance means a larger turn before contact, and a larger turn carries pins near the root further into the socket’s round wall, where they close fastest. The backlash of a cycloidal drive is not proportional to its clearance, though it is close to it, and it breathes by about a tenth twelve times per turn of the eccentric.

Deflection brings the contacts back #

A real disc and real pins are not rigid. Under load the first contact deflects, the disc turns a little further, and the next gaps close. How many pins end up carrying is an elastic question that is not computed here, but its geometric half is a count. If the contacts can give by λ before the load is taken, the pins within reach of carrying are those whose gap at the first contact is less than λ.

With a give of a tenth of the clearance, one or two pins are within reach. With a quarter, two to four. With half, three to five, and with the whole clearance, six or seven. That is still short of the exact disc’s nine or ten, because at the first contact the other pins’ gaps are spread from nought to more than the clearance.

The two clearances give nearly the same counts at the same share — the same range at a tenth, three-quarters and the whole clearance, and never more than two pins apart at any single angle — so what sets the count is, to that accuracy, the deflection measured in clearances. That has a consequence which runs against intuition. A drive cut four times more accurately shares its load among as many pins only if it also deflects four times less. Better accuracy on its own leaves the ratio unchanged unless the elements are stiffer, and the stiffer they are, the more accurately the drive has to be cut to share the load. The many-contact property of the exact profile is real, but for a real drive it is bought with the combination of fit and compliance, not with the profile alone.

What a designer can do with one contact #

The measurement points to two ways to make a drive share its load again, and they are not alternatives.

The first is to make the clearance small against the deflection, by cutting the disc closer to nominal size or by making the contacts more compliant. The counts above say how far that has to go. To bring nine pins within reach the contacts must give by more than the whole clearance, so the clearance has to be smaller than the elastic approach of a loaded pin on the disc. That is a statement about manufacturing and materials together, and neither alone settles it.

The second is to accept one contact and make it the right one. The pin that touches first is always the best-arm pin or a pin in a root socket, so the drive’s worst case under a clearance is better than its exact profile’s worst case. For a drive whose job is to position, not to carry, one well-placed contact is the requirement, and the clearance’s cost is the free turn, which is small and measured above.

What the measurement rules out is the reading in which the exact profile’s contact count is a property of the drive as built. It is a property of the drive as drawn. The second shape is not a choice established that a profile’s conjugate is determined, and it is determined at nominal size. At any other size, which is every size a drive is made at, the count of contacts is set by the fit and the load, not by the curve.

What this does not settle #

The load. Which pins carry, and how much, is a statics problem with the contacts’ stiffness in it. The count here is of pins close enough to carry at a stated give, and says nothing about how the load divides among them.

The standard modifications. Production discs are cut with an equidistant offset, which is the undersize measured here, and often a second modification that shifts the pin circle as well, which opens the gaps unevenly round the ring. Only the equidistant offset is measured.

The proportion. Everything here has the eccentric at exactly the pin radius over the pin count. That proportion gives the disc its round root sockets, and the sockets are what make the root pin first both ways and pull the first contact’s arm towards the pitch offset. It is the proportion the earlier essays used, and it is one choice among several.

Still open: a disc whose roots are not sockets #

Cycloidal drives are also made with an eccentric smaller than the pin radius over the pin count, so that the pin’s centre traces a curtate trochoid with no cusps. Their roots are smooth curves, not sockets. There the argument above changes in one place. With no socket, no pin closes its gap both ways, and the first contact each way should be exactly the exact disc’s best-arm pin in that direction.

The distinct argument there would be that drive measured the same way: the eccentric as a fraction of the pin radius over the pin count, from about a half up to one. Three things would come out of it: the exact disc’s contact count and arm floor as the proportion falls, the first contact’s arm under a clearance, and the backlash per unit of clearance. The question is whether the sockets are a benefit of the exact proportion or a peculiarity of it. If the first contact’s arm falls away from the pitch offset as soon as the cusps disappear, the round root was doing real work, and a designer choosing a smaller eccentric to avoid sharp roots gives something up that no count of contacts shows.