A hundred to one from a difference of one

An ordinary gear pair gives about five to one before the wheel is unreasonably larger than the pinion. Two stages give twenty-five, three give a hundred and twenty-five, and each stage is a pair of shafts, a pair of bearings and a case to hold them in. Getting a hundred to one out of a single stage is therefore worth some ingenuity, and three quite different mechanisms do it. All three work by the same trick — arranging for the output to be a small difference between two nearly equal quantities — and all three pay for it.

This essay is about what each one pays. The applied field already established that a compound epicyclic’s ratio is exact and fragile: 2176/106 for the catalogue drive, unchangeable by any measurement, and moved by forty per cent when one ring gains one tooth. What is new here is that the exactness has a geometric price as well as a numerical one, and that the harmonic drive pays a different price entirely.

The harmonic drive is an epicyclic with one mesh #

Start with the one that looks least like anything else. A harmonic drive has three parts: a rigid circular spline with internal teeth, a thin-walled flexspline with external teeth and two fewer of them, and a wave generator, which is an elliptical bearing inside the flexspline that pushes it out into contact with the circular spline at the two ends of the ellipse.

Turn the wave generator one revolution and the region of contact travels all the way round. The flexspline has two teeth fewer than the circular spline, so after that revolution it has slipped back by exactly two teeth relative to it. Two teeth out of two hundred is one hundredth of a turn, backwards.

Kinematically, that is a planetary with the wave generator as the carrier and one internal mesh:

$$z_f(\omega_f - \omega_w) \;-\; z_c(\omega_c - \omega_w) \;=\; 0.$$

Hold the circular spline, and the graph’s null space returns

$$\frac{\omega_f}{\omega_w} \;=\; \frac{z_f - z_c}{z_f} \;=\; \frac{200 - 202}{200} \;=\; -\frac{1}{100},$$

a hundred to one and reversed, as an exact fraction, from the same eight lines of elimination that answer a Simpson gearset. Nothing about the elasticity of the flexspline enters. The deformation is how a gear with two fewer teeth is persuaded to mesh with its ring at all; it is not part of the ratio.

Some checks the same machinery does apply:

The last row is a four-tooth difference, which means a four-lobed wave generator. The tooth difference has to equal the number of lobes, because the flexspline must be pushed into contact at as many places as it slips teeth, so the difference is always even for the usual two-lobe generator and the library refuses an odd one rather than returning a ratio for a mechanism that cannot be built.

A cycloidal drive is the same argument with a different way of making the mesh: an eccentric carries a lobed disc that rolls inside a ring of pins, with one fewer lobe than there are pins. Twenty-nine lobes in thirty pins gives $-1/29$. Same graph, same single internal mesh, same fraction.

What the harmonic drive pays #

Nothing geometric. Both splines are coaxial, there is one mesh, and there is no second centre distance to reconcile. What it pays instead is that the flexspline must flex, several times per revolution, for the life of the drive — which is a fatigue question and therefore a force question and is not on this site.

There is a kinematic consequence worth having, though, and it is about conditioning.

Perturbing one tooth of the flexspline is catastrophic in the same way it is for a compound epicyclic: the reduction is $z_f/(z_c - z_f)$, so its sensitivity to a change in $z_f$ is $|z_f/(z_c-z_f)| = 100$, and a single tooth would double the ratio. But nobody perturbs one tooth. A harmonic drive family is designed by keeping the difference at 2 and moving both counts together, and along that path

$$R = \frac{z_f}{2},$$

which is linear. A 160-tooth flexspline gives 80, a 200-tooth gives 100, a 320-tooth gives 160, and the design space is a plain arithmetic progression with no fragility anywhere in it.

That is a distinction worth being careful about, because “sensitivity” without a direction is meaningless. A quantity is sensitive along a direction in the parameter space, and the useful question is whether the directions a designer actually moves in are the sensitive ones. For the harmonic drive they are not. For the compound epicyclic they are: its reduction is $1/(1 - z_1z_4/z_2z_3)$, and moving any single count moves the whole answer, because there is no path through the design space that preserves the difference of the two products while changing their size.

What the compound epicyclic pays, and it is not only numbers #

Here is the part that had not been said, and it took drawing the mechanism to notice.

The compound epicyclic’s planet is one shaft carrying two gears — 30 teeth meshing the output ring of 68, and 32 meshing the fixed ring of 69. One shaft means one orbit radius. But at a common module the two meshes ask for

$$a_{\text{out}} = \frac{68 - 30}{2} = 19, \qquad a_{\text{fixed}} = \frac{69 - 32}{2} = 18.5 .$$

Half a module apart. The mechanism as specified by its tooth counts cannot be assembled at a common module at all.

There are two ways out and both are used:

• Profile shift. Cut one mesh with the cutter moved out, so that the pair runs at an operating centre distance away from its cutting value. This is the technique the teeth field already owns, where it is used to cure undercutting on small pinions; here it is used for a completely different reason, to reconcile two centre distances that arithmetic will not reconcile. The tooth thicknesses change, the operating pressure angle changes, and the ratio does not — which is the property that makes it available.
• Two modules. Cut the two stages at modules in the ratio $19/18.5 = 1.0270$, so that each mesh is standard at its own module and the two agree on the radius. Real compound reduction gearboxes do this, and it is why their two rings are not interchangeable parts even though they differ by one tooth.

Either way, the exactness is bought. The reduction is 2176/106 and no measurement will disagree, and the price is a pair of meshes that are 2.7% away from being ordinary. That is the invoice the earlier essay did not print.

And it explains something that essay left as an observation. It noted that a hundred-to-one reduction fits in a coffee mug and that one tooth moves the answer by tens of per cent, and treated the tooth counts as though they could be picked from a wide field. They cannot: the counts have to make the products nearly equal and the centre distances nearly equal at the same time, and those are two conditions on four integers. The catalogue of buildable compound reductions is far shorter than the catalogue of arithmetically pleasing ones, and this is the reason.

Why a single stage can do this at all #

There is a question underneath the three mechanisms that is worth asking directly: why is a large reduction hard in an ordinary train and easy in these?

An ordinary pair’s reduction is $z_2/z_1$, and both counts are bounded — the pinion below by undercutting at about seventeen teeth and the wheel above by how large a gearbox anybody will accept. So a stage is limited to something like five or six to one, and there is nothing clever to be done about it: the reduction is a ratio of two sizes and sizes are bounded.

A planetary does slightly better by adding the carrier, but only slightly, and the reachable set tops out around nine.

The difference mechanisms escape the bound by making the output depend on a quantity that is not a size. The harmonic drive’s output per turn of the wave generator is two teeth — a count that has nothing to do with how big anything is, and which stays at two while the flexspline grows to two hundred teeth. The compound epicyclic’s output is $z_2z_3 - z_1z_4$, a difference of products that can be made as small as 1 while the products themselves are in the thousands.

That is the whole trick, stated in one line: the reduction is large because the output is a small integer and the input is a large one, and nothing about the mechanism’s size enters either. It is why these drives are compact, and it is why they are fragile, and the two are not separate facts.

The other difference mechanism is a graph of the same kind and is worth reading in its matrix form, because that is where the small integer is visible as arithmetic rather than as a claim about teeth.

Three mechanisms, one idea #

Set the three side by side and the common structure is unmistakable.

All three compute $1/(a - b)$ for two nearly equal quantities, and that is a shape this site has now met in several places. It is met again, in three mechanisms that are not gearboxes at all, in three mechanisms, one subtraction, where the conditioning number turns out to be literally the same expression in a differential screw, a chain hoist and this drive.

There is one more member of the family worth naming and not building here. Put a variator on one input of a planetary and a fixed drive on the other and the output is a difference of two ratios that can be driven through zero — an infinitely variable transmission, with a genuine standstill in the middle of its range and no clutch. It is a two-degree-of-freedom mechanism used as one, its ratio is $1/(a-b)$ with $a$ continuously adjustable, and its conditioning blows up exactly where the output is most interesting. Everything this file computes applies to it; what it does not have is a tooth count anywhere, so its ratio has a tolerance on it rather than being a fraction.

Where the drawing has to be honest #

One note about how these mechanisms are drawn here, since two of the three cannot be drawn faithfully at all.

A harmonic drive’s flexspline is elliptical. Its shape is the result of a wave generator pushing it out of round, which is an elastic deformation, and this site does not compute elastic deformations. So the drive is not drawn as a mechanism; it appears as its graph and as its ratio, both of which are exact and neither of which needs the shape. Drawing an ellipse of an assumed eccentricity with teeth sketched onto it would be a picture of an assumption.

A cycloidal drive’s disc has a real profile — an epitrochoid, offset by the pin radius — and it could be computed. It is not drawn here either, and the reason is different: the disc’s profile is a tooth geometry question and belongs to the teeth field, whose whole subject is what shape makes a conjugate mesh. Putting it in this field would be putting a profile essay in a ratio field.

What both of them get instead is the arithmetic, which is the part this field is about, and a statement of what is missing. That is the same treatment the escapement field gave the pallet’s angular budget: the term that needs a quantity the site does not have is returned as absent rather than estimated.

The mesh a harmonic drive is really making #

One last kinematic point, since a two-tooth difference sounds as though it ought to be impossible.

An ordinary internal pair with two teeth’ difference cannot mesh: the pitch circles differ by one module in radius, so the teeth would have to occupy the same space. The harmonic drive gets round it by putting the flexspline in contact with the circular spline at only two places — the ends of the ellipse — and out of contact everywhere else. Fifteen or twenty per cent of the teeth are engaged at any moment and the rest are clear, so there is no requirement for a full conjugate mesh all the way round.

That is the geometry that makes the arithmetic legal, and it is also why the tooth difference must equal the lobe count: the flexspline slips by one tooth per lobe per revolution of the generator, so a two-lobe generator gives two, and a three-lobe would give three and would need an odd difference and a flexspline pushed into a triangle.

Kinematically, all of this is outside the relation the null space uses. That relation says only that the two splines’ relative rotation is in proportion to their tooth counts, which is exactly the statement that the teeth have to be got past one at a time — and getting past a tooth is a fact about counting, not about how many teeth are touching while it happens.

What all three pay, in one line #

The three mechanisms compute $1/(a-b)$ for two nearly equal quantities, and the price of that shape can be stated once rather than three times — and stating it once shows which half of each mechanism the price falls on.

Differentiating gives it immediately. If the reduction is $R = a/(a-b)$, then a relative error $\epsilon$ in either quantity produces a relative error of about $R\epsilon$ in the reduction. The mechanism amplifies every relative error in its members by its own reduction factor. A hundred to one amplifies by a hundred; a thousand to one by a thousand.

That sounds ruinous and it is not, and the reason is the whole point of these mechanisms. The quantities being differenced are tooth counts, and a tooth count cannot be slightly wrong. There is no $\epsilon$ to amplify: the flexspline has two hundred teeth or it does not, and the reduction is the exact rational the counting gives. The conditioning is real and has nothing to act on.

Where it does act is on everything continuous, and that is the honest statement of what these mechanisms pay. The exactness of counting protects the ratio and nothing else. The eccentricity of a wave generator, the elastic deflection of a flexspline, the clearance at a cycloidal drive’s pins, the centre distances of a compound epicyclic’s two meshes — every one of those is a length, every length has an error, and every error arrives at the output multiplied by $R$.

That is why a harmonic drive’s reputation is for exact ratio and imperfect positioning at the same time, and the two are not in tension. Turn the input a whole number of revolutions and the output has turned the exact fraction the counts demand; ask where the output is mid-revolution and the answer carries the amplified sum of every compliance in the mechanism. The first is arithmetic and the second is geometry.

It also explains the compound epicyclic’s half-module problem in the same terms rather than as a separate defect. Two meshes whose centre distances disagree by half a module is a length error, so it is amplified by the reduction like every other one — which is why it cannot be lived with and has to be removed by profile shift or by two modules rather than absorbed. A mechanism with a reduction of twenty amplifies half a module twenty-fold, and there is no clearance anywhere that will take that.

So the family’s common structure gives a common design rule: spend accuracy on the continuous quantities and none at all on the counts, because the counts cannot be improved and the lengths are the only place the conditioning can reach.

The check that has to be able to lose #

The compound epicyclic’s reduction is asserted against the closed form the applied field published two phases ago, at four sets of tooth counts, requiring agreement to a part in $10^{12}$. That check can fail: the topology here is the one place in this library where a mesh could be attached to the wrong body — the fixed ring is a gear whose link is the frame — and getting it wrong produces a perfectly reasonable-looking number. The first version of the differential in this same file made exactly that class of mistake and reported that a differential’s two wheels always turn together.

The evenness of the harmonic drive’s tooth difference is refused rather than warned about, and so is a circular spline with fewer teeth than its flexspline. Neither refusal is decorative: both describe mechanisms that would return a ratio and could not be assembled, and this field has enough of those already.