A ratio that is a function of the angle

The velocity ratio of a gear pair is a number, and this site has spent a good deal of effort on the fact that most quoted ratios are not. A gear pair’s really is: two circles roll on each other, the contact stays on the line of centres, and the output rate is the input rate times a constant.

The constancy comes from the circles. Take it away — ask for an output rate that varies through the turn — and everything else survives. The contact still stays on the line of centres, the two curves still roll without slipping, the pair is still exactly conjugate. What changes is that the rolling curves are no longer circles, and the demand decides what they are.

The demand is the shape #

The pitch point is where the two bodies’ material points have the same velocity, and for two bodies on fixed centres it lies on the line joining them. Call its distances from the two centres $r_1$ and $r_2$. Then $r_1 + r_2 = a$, the centre distance, and rolling without slipping means $r_1\omega_1 = r_2\omega_2$.

Write the demand as $g(\varphi_1) = \omega_2/\omega_1$ — how fast the output is to turn, as a function of where the input is. Those two equations then give both radii outright:

$$r_1 = \frac{a\,g}{1+g}, \qquad r_2 = \frac{a}{1+g},$$

and the output’s own angle follows by integrating the demand, $\varphi_2 = \int g\,\mathrm{d}\varphi_1$. That is the whole design.

There is no step in it where anybody chooses anything. The demand fixes the first pitch curve in polar coordinates about the first centre; the same demand, integrated, fixes the second in polar coordinates about the second. Teeth can then be generated on those curves by the same rack that cuts a circular wheel, because a rack rolling on a curve is the same operation whatever the curve is.

What is being computed here is a centrode. The pitch curves of a non-circular pair are the loci of the pitch point in each body’s own frame, which is exactly the definition of the centrodes the linkage field met when it found that a coupler’s motion is one curve rolling on another. A non-circular gear pair is a mechanism designed by writing down its centrodes and then building bodies with those boundaries — the only mechanism on this site that is specified that way round.

The pair that can be written down #

One non-circular pair has a closed form, and it is the one that gets built: two identical ellipses, each pivoted at one of its own foci, with the pivots a major axis apart.

The reason it works is the ellipse’s focal property. For any point of an ellipse, the two distances to the two foci add to the major axis $2A$. Pivot one ellipse at its focus and the other at its focus, put the pivots $2A$ apart, and at the contact the first curve’s radius $r$ and the second’s $2A - r$ add to the centre distance automatically. The contact is on the line of centres for free, at every position.

What is not free is the rolling, and that is the thing to measure. Two curves can be in contact on the line of centres and still slip; rolling requires that equal arcs pass the contact. Measured over a full turn at three eccentricities, the two rolled arc lengths agree to $2.5\times10^{-8}$ of their length, which is the integration step rather than a departure, and one turn of one wheel is exactly one turn of the other — the closure residual after a full revolution is $1.2\times10^{-14}$ radians.

The ratio range comes out of the eccentricity alone. The radius runs from $A(1-e)$ to $A(1+e)$, so the ratio runs between $(1-e)/(1+e)$ and its reciprocal, and the ratio of the extremes is the square:

$$\frac{g_{\max}}{g_{\min}} = \left(\frac{1+e}{1-e}\right)^{2}.$$

At $e = 0.3$ that is $3.448980$; measured on the generated pair, $3.448980$. At $e = 0.15$ and $e = 0.45$ the agreement is the same to six figures.

Reading the ellipse backwards #

It is worth going the other way round once, because it makes the “the demand is the shape” claim concrete rather than formal.

Take the elliptical pair as given and ask what demand it is delivering. Its first pitch curve has polar equation $r_1 = A(1-e^2)/(1 + e\cos\varphi_1)$ about the focus, so the demand it satisfies is $g = r_1/(2A - r_1)$ — a function of the input angle with a single maximum and a single minimum per turn, symmetric about them, and with a shape that is nothing like a sinusoid. At $e = 0.3$ it runs from $0.538$ to $1.857$.

Anybody wanting exactly that function would arrive at exactly these ellipses. Anybody wanting a sinusoid instead gets the curves in the figure above, which are visibly not ellipses — the maximum radius is flatter and the minimum sharper. The ellipse is not the general answer to “a varying ratio”; it is the answer to one particular demand, which happens to be the one with a closed form.

The tilt the demand costs #

There is a quantity that decides how violent a demand may be, and it can be measured without a single force appearing.

On a circular wheel the pitch curve’s normal is radial: the contact normal runs through the centre, and the whole of the surface velocity at the pitch point goes into turning the wheel. On a non-circular wheel the pitch curve’s normal is tilted away from the radius by $\arctan(r'/r)$, and that tilt adds directly to the pressure angle — the same angle a cam’s follower has, and the same thing it decides.

Measured over a full turn:

• a demand of $1 + 0.4\sin\varphi$ tilts the normal by up to $13.5°$;
• the same amplitude at twice the frequency, $1 + 0.4\sin 2\varphi$, tilts it by up to $25.7°$, because the radius has to make the same excursion in half the angle;
• the elliptical pair at $e = 0.3$, whose ratio range is wider than either, tilts by $17.5°$.

Two things follow. A demand’s cost is set by how fast it changes, not by how far it goes — the second demand has the same ratio range as the first and twice the tilt. And a lobed pitch curve, which is how a designer gets several cycles of variation per turn, is expensive in exactly this currency: doubling the number of lobes at fixed amplitude roughly doubles the tilt.

What the tilt then does to the teeth needs a load, and belongs to somebody else. The geometry stops at the angle.

What a varying ratio is for #

Nobody builds a non-circular gear for the pleasure of it. Every application is a case where the output motion is the specification and a constant ratio cannot meet it.

A quick return. A shaper, a slotter or a press wants the working stroke slow and the return fast. The site has met that as a linkage property — a crank-rocker’s time ratio is set by the angle between its limit positions, and the designer takes whatever the linkage gives. An elliptical pair delivers a time ratio that was specified: pick the eccentricity that gives the required ratio of stroke times, and the wheels follow.

Flow measurement and pumping. A pair of non-circular rotors with a prescribed relation is how several positive-displacement meters work, and the shape is chosen to make the swept volume per turn constant while the instantaneous rate is not.

Compensating another mechanism. If a machine has a linkage that delivers the wrong law, a non-circular pair in front of it can be designed so that the composition is right. That is function generation with an exact answer, in a subject where three or four precision points is normally all anybody gets — and it is worth being precise about why the two problems are different. A four-bar has four free lengths and matches a demanded function at a handful of positions; a non-circular pair has a whole function’s worth of freedom and matches it everywhere.

That is the trade the field is really about. A linkage approximates a function with a few parameters; a shape reproduces it exactly and has to be manufactured. Which is better depends entirely on whether the shape can be made, and for two hundred years it could not.

More than one lobe, and more than one turn #

Two generalisations come free and are worth naming, because between them they cover most of what is actually manufactured.

Lobes. Nothing requires the demand to have period $2\pi$. A demand of $1 + 0.4\sin 2\varphi$ produces a pitch curve with two lobes and a mate with two lobes, and it closes exactly — the integral of a sinusoid over a whole number of its own periods is zero however many periods there are. Three, four and five-lobed pairs are made, and they are how a machine gets several working strokes per revolution.

Whole turns. The closure condition asks the output to come back to itself after one turn of the input, not to have turned once. If it has turned twice, the pair still lines up — the input wheel has one lobe pattern and the output has two, and the mechanism is a two-to-one reduction with a varying instantaneous ratio on top of it. What is forbidden is the output ending anywhere that is not a whole number of turns from where it started.

What has to be true of a demand #

Two conditions, and both are visible in the formulae above.

The demand must be positive. $g$ is a ratio of rates, and a $g$ that passed through zero would be a pitch radius of zero at the first centre — a mechanism whose output stops while the input turns, which is not a gear pair. A $g$ that changed sign would be an output that reversed, which no pair of rolling curves does.

The demand must close. After one turn of the input, the output must have turned a whole number of turns, or the teeth do not line up with themselves. That is the condition $\int_0^{2\pi} g\,\mathrm{d}\varphi_1 = 2\pi n$, it is one equation on a whole function, and almost no function satisfies it — which is why non-circular gearing is a search rather than a drawing exercise.

The elliptical pair satisfies both by construction, and that is the whole reason it is the one in the catalogues.

What the pitch curves leave open #

A pitch curve is not a tooth. It is the curve that rolls, and the teeth stand on it in the same way that a circular gear’s teeth stand on its pitch circle — which means everything this field says about tooth forms still applies, and applies more awkwardly.

The pressure angle varies, because the pitch curve’s normal is not radial. Where the radius is changing fastest, the line of centres and the curve’s normal are furthest apart, and the force on the teeth is furthest from the direction that turns the wheel. That is a design limit on how violent a demand can be, and it is a force argument, so what this field can say about it is only the angle.

The tooth count is not free. Teeth have to fit round the pitch curve a whole number of times, and the curve’s length is fixed by the demand — so the module is decided by the demand and by an integer, rather than chosen.

And the undercut condition is local. The place where a non-circular wheel is most likely to undercut is where its pitch curve is most sharply convex, because that is where the local equivalent radius is smallest, and it is the same criterion as for a small circular wheel applied point by point.

Why they were rare, and why they are not now #

The mathematics of non-circular gearing is old — the elliptical pair is in the eighteenth-century literature, and the general theory was worked out in the nineteenth — and almost nothing was built. The reason is entirely manufacturing, and it is the same reason the involute beat the cycloid.

A circular gear can be cut by rolling a straight rack along a circle, and a circle is what a machine tool does when it is not asked to do anything. A non-circular gear needs the rack rolled along a curve that varies, which means two coordinated axes whose relationship is a function rather than a ratio. On a mechanical hobbing machine that meant a master cam or a set of change gears per design, made to the same accuracy as the part; the tooling for one pair cost more than the pair was worth unless a great many were wanted.

A numerically controlled machine has no such difficulty: the coordination is a table, and a table for an ellipse costs the same as a table for a circle. What changed was not the geometry.

The mean ratio is a count #

The closure condition is stated above as a requirement the demand must satisfy, and reading it as an integral says something surprisingly strong about what a non-circular pair can average.

After one turn of the input, the output must have turned a whole number of turns, so

$$\int_0^{2\pi} g(\varphi_1)\,\mathrm{d}\varphi_1 = 2\pi N .$$

The left-hand side divided by $2\pi$ is the demand’s mean, so the mean of $g$ is exactly $N$: an integer. A non-circular pair with one lobe on each wheel has a mean velocity ratio that is a whole number, and no shape of demand can change that.

The generalisation covers the multi-lobed case and is the same statement. A pair with $n_1$ lobes on the input and $n_2$ on the output closes after the input turns through one of its own lobes, and the mean ratio comes out at $n_2/n_1$ — a ratio of two integers, exactly, whatever the demand does within a lobe.

So a non-circular gear pair has the property this site has been collecting since the gears field: its average ratio is a count, in precisely the sense a circular pair’s ratio is. The shape decides everything about the variation and nothing about the mean; the lobe counts decide the mean and nothing about the variation. The two halves of the design are cleanly separated, and only one of them has any latitude in it.

That is a considerably better position than it first appears, and it explains a fact about what gets built. An elliptical pair averages one to one, because both wheels have one lobe — which is why every picture of non-circular gearing shows two identical ellipses turning at the same average rate. Asking for a pair averaging 1.5 to one with a prescribed variation is asking for lobe counts of three and two, which is a different mechanism with three lobes on one wheel and two on the other, and the demand then has to close over three input turns.

It also puts a hard limit where a designer would want latitude. A demand averaging 1.4 cannot be met by any non-circular pair at all, because 1.4 is not a ratio of small lobe counts — the nearest are 7/5 exactly, which needs seven lobes against five, and a wheel with seven lobes is a very different object from an ellipse. The mean is quantised and the quantisation gets coarse fast, which is the same trade a compound epicyclic makes and arrives here from a completely different direction.

The measurement that keeps it honest #

Everything above rests on the pitch curves rolling rather than slipping, so the check that matters is the arc-length one — and it is a check rather than an identity, because the two curves are computed from the demand by different routes. The first is a polar plot of $r_1$ against $\varphi_1$; the second is a polar plot of $r_2$ against $\varphi_2$, where $\varphi_2$ came out of a numerical integration of the demand.

If the integration were wrong, the second curve would be a distorted version of the right one, the contact would still be on the line of centres, and the pair would look perfect. What would give it away is that the arc lengths would not match: the wheels would have to slip to stay in contact.

They match to $2.5\times10^{-8}$, which is the trapezoidal step. That is the arithmetic saying the same thing the geometry does, by a route that would have failed if any of it were wrong.