Eleven lobes from twelve pins

A cycloidal reducer has three parts worth naming: a ring of round pins fixed in a housing, a disc with a lobed edge, and an eccentric on the input shaft that the disc rides on. Turn the input once and the disc rolls round the inside of the pin ring, coming back to where it started less one lobe. Eleven lobes inside twelve pins gives eleven to one.

The disc’s edge is the only shaped part in it, and it is not shaped by anybody. It is the envelope of the pins.

One pin does all the work #

The generation is worth watching, because it is not what one expects. Take one pin — a circle of radius 5 mm centred 60 mm from the housing’s axis — and hold it still. Move the disc: its centre orbits at the eccentricity $e = R/N$, and it turns backwards at $1/(N-1)$ of the input rate. Ask the meshing equation where the disc must be shaped for that pin to stay in contact through a full turn of the eccentric.

What comes back is one lobe, spanning $43°$ of the disc — a little more than a lobe pitch, which is $360/11 = 32.7°$.

The other ten lobes are the same curve. The pins are identical and evenly spaced, so the eleventh part of the disc that each of them cuts is congruent to the others — and the whole profile is the single generated lobe repeated at $360/11$ intervals, with the deeper cut winning wherever two copies overlap.

That the copies agree where they overlap is not arranged. Each is the envelope of a different pin at a different set of eccentric angles, and the fact that they meet cleanly at the lobe boundaries is a property of the geometry rather than of the construction.

What the numbers come out at #

Three quantities fall out of the generation and none of them was supplied.

The lobe count is eleven. Counted as radial minima on the generated profile, not taken from the pin count. The claim $N - 1$ is a statement about the shape, and the shape is what was counted.

The roots sit at 50.000 mm and the tips at 60.000 mm. Those are $R - r - e$ and $R - r + e$: the eccentricity is the lobe height, twice over. A designer choosing a lobe depth is choosing an eccentricity, and vice versa; they are the same decision under two names.

The reduction is eleven to one, and there are two independent routes to it. The first is the placement: the disc turns by $-\varphi/11$ per unit of input angle, which is how the motion was specified. The second is the shape: the disc has eleven lobes and the ring has twelve pins, so one turn of the eccentric walks the disc back by exactly one lobe. The two agree because the shape came out of the motion — which is the whole point of the field, and is the thing to check rather than to assume.

Why one pin is enough #

The claim that a single pin generates the whole disc deserves a paragraph, because it is doing a lot of work and it is a statement about symmetry rather than about envelopes.

The relative motion of the disc and the housing is periodic in the eccentric angle with period $2\pi$ — after one turn of the input everything is back where it was, except that the disc has rotated by one lobe pitch. So the material one pin removes over the second turn is the material it removed over the first, rotated by a lobe pitch. Over eleven turns it therefore removes eleven copies of the same lobe and the disc is complete.

The pins are also identical and evenly spaced, and by the same argument, what pin number two removes during the first turn is what pin number one removes during the second. The two facts together say that eleven pins over one turn and one pin over eleven turns cut the same disc, which is why the figures here can be built from a single generation and repeated.

Nothing about that is peculiar to twelve pins, and it is why the profile has no eleven-fold arithmetic anywhere in its derivation — a fact worth having when the alternative is a formula with an eleven in it that has to be right.

Half a dozen contacts, or twelve #

Here is the property the mechanism is sold on, and it needs care to state correctly.

At any instant, every one of the twelve pins is touching the ideal disc. Measured on the generated profile, all twelve gaps are within five hundredths of a millimetre at every eccentric angle tested, and most of them are within a hundredth — the residue is the resolution of the drawn curve rather than a real clearance.

That is not surprising once the construction is understood: the profile is the envelope of all the pins, so of course it touches all of them. It is the same statement as a gear tooth touching its mate — the shape was generated by the thing it is being compared against.

What a real reducer does is different, and the difference is deliberate. The disc is cut slightly undersize, so that only the pins on the loaded side make contact — typically about half of them — and the rest have a clearance. That is an allowance, in exactly the sense backlash is an allowance in a gear pair: without it, the mechanism has no room for the tolerances of twelve pin positions, a bore, an eccentric and a disc, and it jams.

So the honest statement is the one with both halves. The ideal geometry has every pin in contact; the built mechanism shares the load over about half of them; and the reason a cycloidal drive is stiff and quiet is that “about half” is five or six contacts, where a gear pair has one or two. What each of those contacts carries is a force question and is not answered here.

Where the ratio comes from #

It is worth being explicit about the mechanism of the reduction, because it is the same trick as a hundred to one from a difference of one and it is easy to lose track of.

The disc’s pitch circle rolls inside the pin ring’s pitch circle without slipping. Their radii differ by the eccentricity, so one circuit of the eccentric rolls the disc round by an arc equal to the pin circle’s circumference — and the disc, being smaller by $e$, has turned by less than a full revolution by exactly the fraction $e/r_d$. With $N$ pins and $N-1$ lobes that fraction is one turn in eleven.

The reduction is therefore a difference of two nearly equal quantities, which is what makes it large from a small mechanism and also what makes it sensitive: change the pin count by one and the ratio changes by a whole step, and there is nothing in between. Eleven to one and twelve to one are neighbours in the catalogue and there is no eleven and a half.

The other consequence of the difference is that the output cannot be taken off the disc directly — the disc is orbiting as well as turning, and its centre goes round a circle of radius $e$. Real reducers take the output through a set of rollers in oversized holes, which cancels the orbit and passes on the rotation. That is a mechanism in its own right, it is not a shape question, and it is the part of the design that decides how large the thing has to be.

The eccentricity is the whole design #

Because the roots and tips come out at $R - r \mp e$, the eccentricity is not a free parameter sitting alongside the others — it is the lobe depth, and it is fixed by the pin count.

$e = R/N$: twelve pins on a 60 mm ring gives 5 mm; ten pins on the same ring gives 6 mm, deeper lobes and a coarser disc. So a designer who wants a higher reduction from the same ring gets shallower lobes automatically, and there is a point where the lobes are so shallow that the manufacturing tolerance on the profile is comparable to the lobe height. That is the geometric reason cycloidal reducers with very high single-stage ratios are hard, and it is visible in the arithmetic before any tolerance is quoted.

The pin radius is the one genuinely free number, and it trades two things against each other that the geometry can both state. A larger pin makes the contact patch broader and the pin stiffer; it also cuts a deeper hollow in the disc, since the profile is offset inward from the pin-centre path by the pin radius, and takes material out of the disc’s rim where it is thinnest. Neither of those is a force argument until a load appears — the geometry says how much metal is left, and stops.

The same construction as a pump #

Turn the picture inside out and the same pair of shapes is a gerotor: an inner rotor with $N-1$ lobes inside an outer with $N$, the spaces between them carried round from an inlet to an outlet as chambers. The mechanism is the same generation with the pins replaced by lobes of an outer rotor, and the chamber areas are computed the same way the three chambers of a rotary engine are.

That is not a coincidence of shape. A reducer and a pump are both machines in which two bodies stay in contact at many points at once, and the number of contacts is what the machine is for: the reducer wants many contacts because it shares load, and the pump wants many contacts because each pair of them seals a chamber.

How it compares with the alternatives #

A reduction of eleven to one from one stage, in a package about as deep as the pin ring is thick, invites comparison with the two mechanisms that do the same job.

A single gear pair at eleven to one needs a wheel eleven times the pinion’s pitch diameter — a large, mostly empty casting, with one or two teeth in contact at a time. A cycloidal stage of the same ratio is a disc that fits inside the pin circle.

A planetary stage gets to about eleven to one comfortably and is the usual answer. It shares load over three or four planets rather than five or six pins, and it needs proper gear teeth on four members rather than round pins and one shaped disc.

A harmonic drive gets much higher ratios from the same trick of a small integer difference, with a flexible member doing the work that the eccentric does here. It is not a rigid-body mechanism at all, which puts it outside this site: a deflected spline is an elasticity problem, and everything on these pages is bodies that keep their shape.

What the cycloidal stage buys, geometrically, is a shaped part that is easy to make well — a single closed curve, cut once, on one component — against a ring of pins that are cylinders. That is the same argument the lantern pinion won on for two centuries, at a different scale.

What the generation had to be told #

Two decisions in the computation are worth recording, because both were wrong first and neither showed up as an error.

Which branch of the envelope is the disc. Each pin gives two envelope branches — one on the near side of the pin and one on the far side — and only the near one is cut. The first version filtered them by radius, keeping everything inside $R - r$, which is a number that looks like the tip radius and is not one: the lobes reach $R - r + e$. That quietly truncated every tip, leaving a profile that agreed with the swept material at the roots and was two tenths of a millimetre wrong at the peaks. The fix is to test against the pin’s own centre rather than against a radius.

What “the disc” means where two lobes overlap. The generated lobe runs past its own share of the circle at both ends, into territory where the neighbouring pin is the one doing the cutting. The rule that settles it is not a tidying step: material is gone if any pin reaches it, so where two copies overlap the smaller radius wins. Checking that rule against a direct sweep of the pins over the blank — asking what survives, rather than where the envelope is — is what confirmed it.

Both mistakes produce a smooth, plausible, symmetric, wrong curve. Neither would have been visible in a picture at the scale a reader sees.

What is not in the geometry #

Three qualifications, all of them the same one this site has made in every field since the transmission.

No load sharing. The geometry says how many pins are within a given distance of the disc; how much of the torque each of them carries depends on the stiffness of the pin, the disc and the housing, and is a force calculation. The claim “many contacts” is geometric; the claim “therefore high capacity” is not.

No efficiency. Every contact slides, at the rate the pitch-point argument gives, and what that costs in friction needs a coefficient. What the geometry does say is where the sliding vanishes and how it varies round the ring.

No dynamics. The disc’s centre orbits, so the mechanism has a rotating imbalance built into it, and real reducers use two discs at a hundred and eighty degrees to cancel it. That is a mass argument, and the second disc changes nothing about the shapes.

The ratio and the imbalance are one choice #

The eccentricity coming out at $R/N$ rather than being chosen is described above as the design being decided, and the consequence is sharper than it looks, because $N$ is also the ratio.

Choose the reduction and $N$ is settled. Settle $N$ and the eccentricity is $R/N$, so the disc’s orbit radius is settled too — and the orbit radius is the mechanism’s rotating imbalance, since the disc’s centre goes round at input speed at that radius. The ratio and the imbalance are the same decision.

Run it the other way and the awkwardness is plain. A designer wanting a modest reduction gets a large eccentricity: ten pins on a 60 mm ring is 6 mm of orbit, deep lobes and a substantial mass swinging at input speed. A designer wanting a large reduction gets a small one: forty pins on the same ring is 1.5 mm, shallow lobes and very little imbalance. A low-ratio cycloidal drive is the badly-balanced one, which is the opposite of what an intuition about ratios would suggest.

There is no way to separate the two within one disc, and that is the structural difference from a planetary. A planetary’s ratio comes from the tooth counts and its balance comes from where the planets are; two independent decisions, and three equally spaced planets balance exactly. Here one integer decides both.

Which is why every cycloidal reducer ever built has two discs, at half a turn to each other. Two identical discs on the same eccentric shaft, phased 180° apart, put their centres on opposite sides of the axis at every instant, and the imbalance cancels exactly. That is not an optional refinement — it is the only remedy the geometry leaves, since the eccentricity cannot be reduced without changing the ratio.

The doubling has a second effect worth noting, and it is a bonus rather than a cost. Two discs at half a turn means the load is shared between two sets of contacts phased against each other, so the mechanism’s torque ripple is halved as well. One remedy, forced by the geometry, buying two things — which is the usual reason a feature turns up on every example of a machine without ever being presented as a choice.

What this shares with everything else here #

A cycloidal reducer is usually filed under “gearing without gears”, which is true and unhelpful. What it actually is, in this field’s terms:

• a pair of bodies in continuous contact, exactly like a gear pair;
• with one shape given — a circle, the easiest shape there is to make accurately — and the other computed as its envelope;
• under a relation that is not a plain ratio but an orbit and a rotation;
• and with the ratio coming out of an integer count of lobes on the generated shape rather than out of the relation directly.

The last of those is the interesting one and it is what it shares with a planetary gearset. In both, the useful reduction is a difference of two integers, the mechanism is small because the difference is small, and the price of the large ratio is that the ratios available are a sparse set rather than a continuum.