The demand that cannot be met

A demanded output rate fixes both pitch curves of a non-circular pair completely, with nothing left over to choose. That makes the design of such a pair look like a transcription exercise, and it would be, except for one thing: almost no demand can be met at all.

The obstruction is not manufacturing and it is not accuracy. It is that a wheel has to line up with itself after a turn, and that requirement is an equation on the whole function the designer wrote down.

The condition #

The output’s angle is the integral of the demand: $\varphi_2 = \int_0^{\varphi_1} g\,\mathrm{d}\varphi$. Turn the input once and the output has turned

$$\Delta\varphi_2 = \int_0^{2\pi} g(\varphi)\,\mathrm{d}\varphi.$$

The pitch curve of the input wheel closes on itself automatically, because $g$ is periodic and the radius is a function of it. The output wheel’s does not. Its shape is drawn in its own polar coordinate $\varphi_2$, and if $\Delta\varphi_2$ is not a whole number of turns, the curve arrives back at a different angle from the one it started at, and the shape has a step in it.

So the condition is

$$\int_0^{2\pi} g\,\mathrm{d}\varphi = 2\pi n, \qquad n \in \mathbb{Z}^+,$$

which says the mean of the demand must be a whole number. Nothing else about the demand matters to it.

That last sentence is the useful one and it is worth confirming rather than deducing. Take a family $g = c\,(1 + A\sin\varphi)$ and measure the closure residual over a grid of $c$ and $A$: at $c = 0.95$ it is $-0.3142$ radians for every amplitude from zero to $0.6$; at $c = 1.05$ it is $+0.3142$; at $c = 1$ it is $10^{-14}$. The predicted value is $2\pi(c-1) = 0.3142$. Change the shape of the variation entirely — add a second harmonic, make it asymmetric — and the residual still depends only on the mean, to fourteen places.

The variation is free and the mean is not. A designer can ask for any excursion, any number of lobes, any asymmetry; what cannot be negotiated is the average, and it has to be an integer.

What a failure looks like #

Five per cent is not a large modelling error. A demand written as “one plus forty per cent of a sine” and then scaled slightly — because somebody adjusted a stroke, or because the mean was fitted to data rather than assumed — leaves the output eighteen degrees out after one turn of the input.

On a thirty-tooth wheel, eighteen degrees is a tooth and a half. The teeth cut on the first lap do not line up with the space they have to enter on the second, and the mechanism seizes or breaks on its second revolution. There is no “nearly”: a pair whose closure residual is a hundredth of a tooth is a pair that runs and wears; a pair whose residual is a tooth and a half is scrap.

That failure has a familiar shape on this site. The planetary gearset that could not be assembled fails an integer condition: unless the sum of the sun and ring tooth counts divides by the number of planets, the last planet has nowhere to go, and no amount of accuracy in the machining helps. A demand that does not close is the same kind of failure at the level of a whole function — an integer standing between a design and a mechanism, deciding not how well it works but whether it exists.

Why the mean and nothing else #

The result that the closure depends only on the mean is worth a paragraph of explanation, because it is both obvious in hindsight and easy to disbelieve.

The output angle is an integral. An integral over a full period picks out the constant term of a function and annihilates everything else that is periodic: every harmonic of the demand integrates to zero over its own whole number of cycles. So a demand of $1 + 0.4\sin\varphi$, one of $1 + 0.4\sin 2\varphi$, and one with a dozen harmonics in it all deliver exactly $2\pi$, and all of them close.

That has two consequences a designer feels immediately. The first is liberating: the shape of the variation is entirely free. Any excursion, any number of lobes, any asymmetry, any amount of dwell-like flattening — as long as it stays positive and manufacturable, it closes.

The second is unforgiving. There is no shaping the variation to fix a mean that is wrong. Adding a bump here and taking one away there does not help unless the two exactly cancel, in which case the mean did not change and the problem was never the shape. The one number that matters is the one number that cannot be adjusted by any local change.

The other three ways a demand fails #

Closure is the interesting condition and it is not the only one. Three more rule out demands that pass it.

The demand must be positive. $g$ is a ratio of rates and $r_1 = a\,g/(1+g)$; a demand that passes through zero puts the pitch point at the input’s own centre, and one that changes sign asks the output to reverse while the input keeps turning, which no pair of rolling curves does. A quick-return demand can be slow — it cannot be a dwell, and it cannot be a reversal. Motion that stops and starts belongs to mechanisms with a discrete state, not to a pair of continuous curves.

The pitch radii must stay inside the centre distance. They do, automatically, for any positive $g$ — $r_1 = a g/(1+g)$ is between zero and $a$ — so this one is free. What is not free is the tooth space: a wheel whose radius drops very small has very little room for a whole number of teeth of the module the mate needs, and the pair fails on tooth count rather than on curve shape.

The curve must not be too sharply convex for the cutter. The undercut criterion for a circular wheel is about the pitch radius; for a non-circular wheel it is the same criterion applied point by point to the local radius of curvature. Where the pitch curve is tightest, a rack of the chosen module cuts into the flank it has already generated, in exactly the way it does on a wheel with too few teeth — and the fix is the same one, a profile shift, applied locally rather than to the whole wheel.

Each of those is a real limit on the amplitude of a demand, and together with the tilt of the pitch curve’s normal they are why a catalogue of non-circular gears has ratio ranges of two or three rather than of twenty.

How a designer gets round it #

The closure condition is one equation, so the standard manoeuvre is to write the demand with one free parameter and solve. That is worth setting out, because it is the difference between “non-circular gears are hard to design” and “non-circular gears are designed the way everything else is”.

Scale the whole demand. If the shape of the variation is what matters and the mean is negotiable, choose $c$ so that the mean is one. That is the family in the figure above and the solution is exact.

Add a constant. If the amplitude is fixed and the mean is not, the same equation says what constant to add.

Change the number of output turns. If the mean cannot be moved, look for an $n$ that fits: a demand whose mean is two is a perfectly good pair, with a two-to-one reduction on top of the variation, and the output wheel then carries two lobes for every one of the input’s.

Fit rather than specify. If the demand comes from data — a motion somebody measured, a curve somebody drew — then it will not close, and what is done is to fit the nearest function that does. That is a constrained approximation problem and it is the same shape as function generation with a linkage, with one important difference: here the approximation is in the specification, and once a closing demand is chosen the mechanism reproduces it exactly.

A worked case, end to end #

It is worth doing one all the way through, because the whole procedure is four lines and reads as though it must be harder.

Suppose a press wants its working stroke to take twice as long as its return. The stroke is half a turn of the output, so the demand has to spend half the output’s travel running slow and half running fast, in the ratio two to one. Write it as $g = c\,(1 + A\sin\varphi)$ and there are two unknowns and two requirements: the ratio of the extremes should be about two, and the pair must close.

The second requirement is instant: the mean of $g$ is $c$, so $c = 1$. That leaves the amplitude to carry the whole of the first, and $A$ is chosen so that $g_{\max}/g_{\min} = (1+A)/(1-A)$ is two — which gives $A = 1/3$.

The pitch curves then follow: $r_1 = 100\,g/(1+g)$ millimetres about the input centre, and the mate is $100 - r_1$ plotted against the integrated output angle. Nothing is iterated and nothing is fitted. The tilt of the normal reaches about eleven degrees, which is comfortable; the radius runs from 40 to 57 millimetres, which leaves room for teeth of a sensible module at both extremes; and the pair is done.

The point of the example is what did the work. The closure condition was not a check applied at the end — it chose one of the two parameters, immediately, before anything was drawn.

Why the ellipse is always the example #

The elliptical pair closes for every eccentricity, and it is worth seeing why in the terms of this essay rather than as a coincidence.

Its demand is $g = r_1/(2A - r_1)$ with $r_1$ the focal radius of an ellipse. Over a full turn, the ellipse’s focal radius sweeps the whole curve and comes back; the output wheel is the same ellipse traversed by the same total arc length, so it too has come exactly once round. The closure is a consequence of the two curves being congruent and of the arc length being the thing that matches — not of any property of the sinusoid or of the eccentricity.

That is why every book’s example is elliptical: it is the family in which the hard condition is satisfied identically, so a reader can be shown the geometry without also being shown a root-find. It is also why the examples are misleading about the difficulty. A general demand does not close, and finding the member of its family that does is the design.

What non-closure costs, in the units of the shop floor #

It is worth converting the residual into something a machinist would recognise, because “eighteen degrees” understates it.

A wheel is cut by rolling a rack round the pitch curve, and the teeth are laid down as the rolling proceeds. If the curve does not close, the tool arrives back at the start of the second lap displaced by the residual — so the last tooth and the first tooth are not a tooth pitch apart. On a thirty-tooth wheel with a residual of eighteen degrees, they are a tooth and a half apart: there is a gap where a tooth should be, or a tooth where a gap should be, depending on the sign.

Nothing downstream can repair it. A different module does not: the residual is an angle and the tooth pitch scales with it. A different cutter does not, since the fault is in the curve. Adjusting the centre distance does not, because the pitch curves were computed for a centre distance and changing it changes them both.

The one thing that does work is a change to the demand, which is why the condition has to be checked before anything is drawn, and why it is written as an integral over the specification rather than as a check on the drawing.

The condition is about the wheel being a wheel #

There is a way of hearing the closure condition that makes it sound like an accounting rule, and a better way that makes it the same statement as everything else in this field.

A pitch curve is a centrode — the locus of the pitch point in one body’s frame — and the pair of them is the mechanism. For a pair of centrodes to be the boundaries of two bodies that turn, each has to be a closed curve traversed once per revolution of its own body. Rolling is what makes the arc lengths match; closure is what makes both of them curves rather than spirals.

Drop the requirement that the bodies turn indefinitely and the condition goes away. A sector gear — a piece of a pitch curve, used through part of a turn and then returned — is under no obligation to close, and non-circular sectors are used exactly where a full turn is not wanted: quadrant drives, oscillating feeds, mechanisms that go out and come back. That is the same escape the timing field uses when it lets a gear with its teeth cut away drive for part of a turn and rest for the rest, and it costs the same thing: the mechanism now has a beginning and an end.

One equation on a function is a mild condition #

The closure condition looks severe when it is first met — a demand that fails leaves the output eighteen degrees out after a turn — and it is worth placing it against the other conditions of its kind on this site, because by that comparison it is the mildest one anywhere.

Count what it constrains. The demand is a whole function, so the design space is infinite-dimensional; the condition is a single scalar equation on its mean. One equation on a continuum, which leaves a set of solutions of codimension one — enormous, and reachable from anywhere by moving one number.

Better than that, it is reachable by a move that is always available. Divide the demand by its own mean, and the condition is satisfied exactly, at the first attempt, for any positive demand whatever. There is no search, no failure case, and no demand that resists it — the only thing spent is the overall ratio, and even that can be recovered by choosing which $n$ to close at.

Set that beside the site’s other feasibility conditions and the contrast is stark. A reverted train’s coaxial condition is two equations in four integers, and a twelve-to-one reduction has no solution at all below sixty-three teeth. A planetary’s assembly condition is a divisibility, and two thirds of drawable gearsets fail it. A clock’s train needs a factorisation that may simply not exist. Every one of those is a condition on integers, and a condition on integers can be unsatisfiable.

A condition on a continuum cannot be. That is the structural reason non-circular gearing’s hard-sounding requirement is the easy one: the quantity it constrains is a real number and there is a real parameter to move. The demand’s mean is not an integer by accident of the design; it is made one, by a division, at no cost but a scale.

Which is the useful place to leave the condition. It is not a constraint that limits what a designer can ask for — it is a normalisation that any demand passes through on its way to becoming a pair of wheels. What it is worth checking for is not feasibility but arithmetic: a demand that reaches the cutter without having been normalised is a wheel that will not close, and eighteen degrees is what forgetting the division costs.

What the field takes from this #

The pattern is worth naming because it recurs wherever a mechanism is specified by a function rather than by a handful of numbers.

A specification that is a function has a space of solutions, and the constraints that decide whether any of them exist are usually not the ones the specification is written in. Here the specification is a rate and the constraint is on its mean; in planetary assembly the specification is a ratio and the constraint is on the divisibility of tooth counts; in synthesis with precision points the specification is a set of positions and the constraint is on how many a linkage’s parameters can absorb.

In each case the useful move is the same: find the condition, write it as an equation on the specification, and use it to choose the one free parameter the design has left. The alternative — draw the mechanism, discover it does not close, and adjust — never converges, because the thing that has to change is the demand.