A ratio with no steps in it

Every mechanism in this field so far has had a ratio that is a fraction of integers. A gear train’s ratio is a count of teeth; a gearset’s ladder is that count arranged into a handful of exact numbers; even the enumeration of shift elements is a finite list. The whole subject has an arithmetic flavour, and the arithmetic is where the certainty comes from.

A continuously variable transmission gives that up entirely. There are no teeth in its ratio at all. Two pulleys, each made of a pair of conical sheaves that can slide together or apart on their shafts, and a belt between them; push the sheaves of one pulley together and the belt is forced to ride further out on its cones, which increases the radius it runs at. The ratio is the quotient of two radii and both of them are lengths.

Only one of the two radii is a design variable #

The temptation is to think of the two pulleys as two controls. They are not, and the reason is that the belt has a fixed length.

For a belt over two pulleys at a fixed centre distance $C$, the length is two straight tangent runs plus two arcs, and with $r_1 \ge r_2$ it comes to

$$L = 2\sqrt{C^2 - d^2} \;+\; (\pi + 2\beta)\,r_1 \;+\; (\pi - 2\beta)\,r_2, \qquad d = r_1 - r_2, \;\; \beta = \arcsin\frac{d}{C}.$$

That is one equation relating $r_1$ and $r_2$. Choose the primary’s radius and the secondary’s is determined; the mechanism has one degree of freedom of shift, not two, and the second pulley’s sheaves are pushed where the belt demands rather than commanded.

So the library solves for it. Given $r_1$, it brackets $r_2$ between the smallest and largest the pulley can produce and bisects until the belt length is right to $10^{-12}$ of itself. Two hundred bisections is a wildly extravagant way to solve a smooth monotone equation and it is used anyway, because a bracket that cannot fail is worth more than the two iterations a Newton step would save, and because this site has been caught before by a solver that converged confidently to the wrong branch.

The sum is not constant, and the departure has a closed form #

The rule of thumb, in every practical account of these transmissions, is that the two sheaves move by equal and opposite amounts, so that $r_1 + r_2$ stays put. It is a useful rule and it is almost right, and finding out exactly how it is wrong turned out to be the best thing in this essay.

Rewrite the belt length in terms of the sum $S = r_1 + r_2$ and the difference $d = r_1 - r_2$. The two arc terms combine:

$$L \;=\; \pi S \;+\; 2d\arcsin\frac{d}{C} \;+\; 2\sqrt{C^2 - d^2}.$$

Now differentiate the last two terms with respect to $d$. The two derivatives that involve $\sqrt{C^2-d^2}$ cancel exactly, and what is left is startlingly simple:

$$\frac{dS}{dd} \;=\; -\frac{2}{\pi}\arcsin\frac{d}{C}.$$

That is zero at $d = 0$. The sum of the radii is constant to first order and departs at second order, which is precisely why the rule of thumb is believed: at the middle of the shift range it is not merely approximately true, it is stationary. Expanding for small $d$,

$$S \;\approx\; S_0 - \frac{d^2}{\pi C},$$

and at the extreme of this variator’s travel, $d = 36.8$ mm on a centre distance of 155 mm, that predicts a fall of 2.78 mm against an exact 2.79.

Measured across the whole sweep, the sum runs from 110.00 mm at unity ratio down to 107.21 mm at the extremes — a swing of 2.56%, and a curve rather than noise.

What the rule of thumb costs #

A 2.5% error in a sum sounds harmless. It is not harmless in the ratio, because the ratio is a quotient of two numbers whose difference is doing the work.

Assume the sum is fixed at 110 mm and work out the secondary radius from the primary. At $r_1 = 46.5$ mm the assumption gives 63.5 against a true 62.94, and the ratio comes out 0.7323 instead of 0.7388 — 0.9% low. At the far end, $r_1 = 72$ mm, it gives 38.0 against a true 35.21, and the ratio is 1.895 instead of 2.045 — 7.4% low.

The error is worst exactly where a variator spends its most interesting time, at the extremes of its shift, and it is always in the same direction: the assumption under-states how far the ratio actually goes. A control system calibrated on it would think its transmission had a narrower range than it has.

This is a shape of mistake the site keeps meeting. An approximation that is stationary at a nominal point is very convincing near that point, and the whole of its error lives where the mechanism is being used hardest. The straight-line linkages were the first instance: Watt’s and Chebyshev’s are excellent near the middle of the stroke and 9% and 12% out over the full travel, and the symmetric pose everybody centres a sample on is a limit of the travel rather than its middle.

Where the travel actually is #

One more consequence of the secondary being a root rather than a setting: the shape of the control.

The primary’s radius is what an actuator moves, and it moves it linearly — a piston travels, the sheaves close, the belt climbs. If the ratio were linear in that travel, a controller could be a gain and nothing more. It is not, and the departure is large.

Across the sweep here the secondary’s radius falls at a rate that runs from 0.770 mm per mm of primary at the low end to 1.352 at the high end — a factor of 1.76 between the two ends of the same actuator’s stroke. Since the ratio is the quotient of the two, the ratio’s sensitivity to the actuator varies by more than that: the last millimetre of travel is worth several times the first.

The reason is visible in the belt-length identity. With $S$ the sum and $d$ the difference of the radii,

$$\frac{dr_2}{dr_1} \;=\; -\,\frac{\pi + 2\arcsin(d/C)}{\pi - 2\arcsin(d/C)},$$

which is $-1$ at $d = 0$ and grows in magnitude as the radii separate, because the two wrap angles are moving apart. At the extreme of this variator, $\arcsin(d/C) = 0.209$, and the ratio of the two brackets is 1.35.

So a variator’s control is nonlinear for a structural reason rather than through any imperfection, and the nonlinearity is worst exactly where the shift range ends. Everything a stepped gearbox knows about its ratios — that they are fixed, exact and known — a variator has to measure.

The belt, checked two ways #

The formula above is the one every reference prints, and it is the sort of thing that is easy to mis-transcribe. So the library computes the length a second time by construction: build the tangent points, measure the straight runs with Pythagoras, accumulate the wrapped arcs from the tangent angles, and add them up. It shares no line of algebra with the closed form.

They disagreed, the first time, by 0.11 mm on a 627 mm belt.

A tenth of a millimetre reads exactly like rounding, and it was a wrong formula. The angle from the line of centres to a tangent point has $(r_1 - r_2)/C$ for its cosine, not its sine — the tangent line is perpendicular to both radii, so projecting the centre-to-centre vector onto the radius direction has to return the difference of the radii. Using the arcsine puts the tangent points a couple of degrees round each pulley, which changes the split between straight and wrapped by about a part in a thousand.

Two things about that are worth keeping. First, the error grows with the shift: 0.11 mm at moderate radii and 1.23 mm at the extreme of the travel, which is a fifth of a per cent of the belt. Second, and worse, it would never have shown up as a wrong picture. The drawn belt would have looked perfectly taut, the pulleys would have been the right sizes, and every ratio the variator returned would have been very slightly wrong for ever. The check exists because the failure mode is invisible, which is the same reason the gates on this site break their own assertions on purpose.

The two routes now agree to $1.6 \times 10^{-16}$ of the length, which is arithmetic noise and nothing else.

The span, and where it comes from #

A variator is advertised by its span: the ratio of its tallest ratio to its shortest, which for the geometry here is 2.045 / 0.544 = 3.76. That number is worth taking apart, because it is set by three things and only one of them is obvious.

The obvious one is how far the sheaves travel, which decides the range of $r_1$. Here it runs 38 to 72 mm, a factor of 1.89 on its own.

The second is that the secondary moves too, and in the opposite direction, so the span is roughly the square of the primary’s range rather than equal to it: 1.89 squared is 3.58, which is most of the 3.76 observed. That is the whole reason a variator is built with two variable pulleys rather than one variable and one fixed — the second pulley doubles the exponent, not the range.

The third is the belt’s length relative to the centre distance, which decides how much of the sum is available to be traded. A longer belt on the same centres has a larger $S_0 = (L - 2C)/\pi$, so the radii can separate further before the geometry runs out, and the span goes up. A shorter belt has less room and a narrower span. That is a design lever with no moving parts in it at all.

None of the three is a free choice in a real transmission: the sheave travel is limited by the cone angle and the pulley’s width, the centre distance by the case, and the belt by what will run round the smallest radius without over-stressing. Which is why variator spans cluster where they do rather than being pushed as wide as anyone likes.

The actuator’s resolution is not uniform in ratio #

The secondary’s rate running from 0.770 to 1.14 mm per mm of primary is presented above as a fact about the control being nonlinear, and it is worth cashing out, because it decides what an actuator has to be able to do.

An actuator moves the primary sheave and it does so in steps — a valve’s resolution, a stepper’s increment, whatever the smallest commanded movement is. The ratio change that step produces is not the same everywhere. Near unity a step of the primary moves the secondary by 0.77 of it, and near the extreme of travel by 1.14, so a fixed actuator increment buys a different ratio increment depending on where the variator already is.

That is the same trouble the ladder of a stepped gearbox does not have, arriving in a mechanism whose whole selling point is not having a ladder. A stepped box has known steps and the control problem is which one to be in. A variator has no steps and a control resolution that varies by roughly half across its travel, so it has a ladder after all — an invisible one, with rungs of unequal spacing set by the actuator rather than by the gears.

The direction of the variation is the awkward part. The rate is largest at the extremes, which is where the ratios are furthest from unity and where a small ratio error matters most — the tallest overdrive and the lowest crawl are exactly the settings a driver notices. So the mechanism’s control resolution is worst where its ratio is most valuable, and an actuator sized for adequate resolution at unity is under-resolved at both ends.

None of that is a fault to be designed out, since it comes from the belt-length identity and the belt has to have a length. What it is, is a term that has to be in the control’s model. Scheduling an actuator on the assumption that ratio moves linearly with sheave position mis-schedules it by up to a half at the ends of the range, and the mis-scheduling has exactly the shape of the rule-of-thumb error the essay has already measured — right in the middle and wrong where it matters.

There is a second consequence the falling sum carries with it, and it belongs to the neighbouring field. The radii sum falls from 110.00 mm to 107.21 mm across the sweep, so the belt runs on a smaller total at the extremes — which means the two wrap angles are not what they were either, and the wraps still add to one turn while their distribution shifts. A variator at full shift has less wrap on one pulley than the nominal geometry suggests, and that is a strand-field quantity with its own consequences.

Two mechanisms that are not this one #

Two neighbours are worth separating out, because “continuously variable” is a description of an effect rather than of a mechanism.

A toroidal variator uses rollers tilting between two toroidal discs, and its ratio is the quotient of two contact radii, both of which are trigonometric functions of one tilt angle. That is a cleaner problem than the belt’s — there is no length constraint tying the two radii together, so the ratio is a closed form in the tilt rather than a root — and it is a different mechanism with a different sensitivity. Nothing in this essay transfers to it except the conclusion that the ratio is a quotient of lengths.

A chain on stepped sprockets is not continuously variable at all, however many sprockets there are. Its ratios are counts, exact, and its steps are as much a ladder as an automatic’s. What a derailleur has that an automatic does not is many steps, and the difference between fourteen steps and infinitely many is a difference of degree; the difference between a count and a length is a difference of kind. The chain drive essay belongs on the count side of that line, despite the chain’s ratio varying by four per cent inside every tooth — because that variation is periodic, has zero mean, and is a design property rather than an error.

What is gained and what is lost #

What is gained is obvious and is the reason the mechanism exists: there is no ladder, so nothing has to be chosen and then lived with, and the whole argument about geometric progressions and progressive spacing simply does not arise. The ratio is a continuous function of one position, and the control problem — pick the ratio that puts the engine at the wanted operating point — has an exact answer at every road speed instead of the nearest of four.

What is lost is the arithmetic. Three things go with it:

• Exactness. A gear ratio is a count and carries no error. A variator’s ratio is a quotient of two solved lengths and carries every error that went into those lengths. That is the next-but-one essay’s whole subject, and it is a genuine change of kind rather than a matter of precision: there is no number in the mechanism that a manufacturing error cannot move.
• The lever. A variator has no members with fixed proportions, so there is no lever diagram and no plane of motions with lines in it. It is a one-freedom mechanism with a parameter, not a two-freedom mechanism with a constraint.
• Linearity. The ratio is not linear in the sheave position, and the secondary’s response is not linear in the primary’s. A control that assumes either is calibrating against a curve.

There is one more thing worth saying about what a variator is not, because it is easy to file it under the same heading as the differential. A CVT is not a two-degree-of-freedom mechanism. Its shift is a parameter, changed slowly by an actuator, not a second motion the mechanism permits at speed. That is why it appears in this field at all — because the ratio is the object, and the field is about ratios that are not numbers — and not because it has more than one input.

The one genuinely two-input CVT is the arrangement that puts a variator on one input of a planetary and a fixed drive on the other, so that the output is a difference of two ratios and passes through zero. Which is the same trick as the three difference mechanisms, applied to a continuously variable quantity, and it inherits their conditioning exactly.