The ratio that has a tolerance

The practice field’s whole method is to take a mechanism that has been solved with exact numbers and give every number a range: a length becomes an interval, a pin becomes a hole, and the output that was a curve becomes a band. Eleven essays have done it to linkages, cams, platforms and arms.

It has never been done to a ratio, and the reason is worth an essay rather than a footnote. There has been nothing to do it to.

A count has no stack #

Take the simplest planetary on this site: sun 24, ring 72, ring held, carrier out. Its reduction is 4, and the fraction the null space returns is $1/4$ exactly.

Now try to put a tolerance on it. Where would it go?

• Not on the module. The module does not appear in the mesh relation at all — the relation is $z_a(\omega_a - \omega_c) + \sigma z_b(\omega_b - \omega_c) = 0$, and every entry is a tooth count. A gearset cut at module 1 and one cut at module 2.5 have the identical ratio, and there is nowhere in the matrix for a length to be.
• Not on the centre distance. A pair running at a centre distance away from its nominal changes its operating pressure angle and its contact ratio and its backlash, and does not change its ratio by anything at all; that is exactly what makes profile shift usable.
• Not on the tooth thickness, the bore, the runout or the helix angle. Every one of them is a length or an angle with a manufacturing tolerance on it, and none of them is in the ratio.

The ratio’s tolerance table is empty, and it is empty structurally rather than because the errors are small. A gearset made appallingly — teeth thick, bores eccentric, shafts bent — has precisely the ratio of a perfect one, averaged over any whole number of turns.

The word “averaged” is doing necessary work, and it is the boundary between this claim and its neighbours. A badly made gear pair has transmission error: within a turn the output is ahead of or behind where it ought to be, by an amount that depends on tooth spacing, runout and deflection. That is a positional error, it is real, it is what gear metrology measures, and it integrates to zero over a revolution because the teeth have to be got through one at a time. The chain drive is the same distinction in a starker form: its instantaneous ratio swings by 4% by design, and its mean over a whole turn is exactly $n_1/n_2$ to five decimal places, which the integration was never told.

So there are two different things a ratio might mean, and the count is exact for one and silent about the other.

A quotient of lengths has a stack #

Now the variator. Its ratio is $r_1/r_2$, both running radii, both in millimetres, and $r_2$ is the root of the belt-length equation rather than a setting. So the ratio depends on:

• the belt’s length, which is a manufactured dimension with a tolerance;
• the centre distance, which is a housing dimension with a tolerance;
• the sheave position, which is where an actuator put it, with a tolerance made of a position sensor, a hydraulic pressure and a cone angle;
• and on the cone angles and the belt’s own thickness, which are folded into the first and third.

Every one of these is in the ratio, so the ratio has a stack in exactly the sense the practice field means, and it can be computed the same way.

The belt-length sensitivity, in closed form #

The dominant term deserves its own treatment because it has an exact expression and the expression says where the trouble is.

Hold $r_1$ and vary $L$. With $S = r_1 + r_2$ and $d = r_1 - r_2$, the length is

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

and holding $r_1$ means $dS = dr_2$ and $dd = -dr_2$. The terms involving the square root cancel — the same cancellation that made the sum’s departure second order — leaving

$$\frac{dL}{dr_2} = \pi - 2\arcsin\frac{d}{C}, \qquad\text{so}\qquad \left|\frac{d\ln R}{d\ln L}\right| = \frac{L}{r_2\left(\pi - 2\arcsin\dfrac{d}{C}\right)}.$$

Two things are readable off that. The numerator is fixed, so the amplification grows as $r_2$ shrinks — at the tall end of the range, where the secondary is small. And $\arcsin(d/C)$ grows there too, shrinking the denominator further. Both effects push the same way, so the sensitivity is worst exactly where the mechanism is doing the most interesting thing.

The measured column is a central difference of the bisection solver, which shares no algebra with the formula; they agree to the step size, which is the honest floor and is quoted rather than driven to machine precision by shrinking $h$ until it looks impressive.

The stack #

Take three plausible manufacturing errors: a belt 0.3% long, a centre distance 0.2 mm out, and a sheave 0.15 mm from where the actuator was told to put it. Recompute the ratio with each, one at a time.

The two right-hand columns are the two conventions this field already argued about: worst case adds the magnitudes and assumes every error is at its limit and pulling the same way, while the root-sum-square treats them as independent and gives a figure about one and a half times smaller. Everything that essay said applies unchanged — including its warning that the square root is a statement about a distribution and is wrong for a systematic error, and a belt supplier whose whole batch runs long is a systematic error.

What is new is that this table exists at all for a ratio. The gearbox equivalent of it is a table with no rows.

The stack a variator’s control adds #

The three terms above are manufacturing errors: they are fixed once the unit is built. A running transmission has a fourth kind, and it is bigger than any of them.

The sheave’s position is commanded by an actuator, and the actuator has a control error as well as a manufacturing one — the pressure it is holding, the friction in the slide, the resolution of whatever measures where it has got to. That error moves the ratio in exactly the same way a manufacturing error does, and it moves it continuously rather than once.

So the sensitivity computed above has a second use. A control system that can position the primary sheave to $\pm 0.1$ mm holds the ratio to about 0.36% at the low end of the travel and 0.49% at the high end, which is a specification on the transmission derived entirely from a specification on the actuator. A stepped gearbox has no such derivation to do, because its ratio is not a function of a position at all: the clutch is applied or it is not, and the ratio is $37/54$ either way.

That distinction is what makes a variator’s control problem different in kind rather than merely harder. In a stepped box the controller decides which ratio; in a variator it decides the ratio, continuously, and every error in the deciding is an error in the ratio.

The term that is not a tolerance at all #

The three-term stack is a statement about the unit as it leaves the factory, and a belt ratio has a fourth contribution that no stack of manufacturing tolerances contains: the belt does not stay the length it was made.

A stack is a distribution over units. Build ten thousand variators and the belt lengths scatter about a nominal, and the table above says what that scatter does to the ratio. Take one variator and run it, and its belt lengthens — bedding in over the first hours, then creeping under tension for the rest of its life. That is a drift, not a scatter: it is the same unit at two times rather than two units at one time, and it moves in one direction only.

The sensitivity computed above applies to it unchanged, which is the useful part. An amplification running from 2.7 to 6.4 across the travel converts belt growth into ratio error at the same rate whatever caused the growth, so the closed form already says what a given amount of stretch is worth. What the stack cannot say is how much stretch there will be, because that is a materials question and nothing in this field measures it.

The distinction matters because the two behave oppositely under every remedy. Scatter is reduced by tighter manufacture and is then fixed for the life of the unit. Drift is unaffected by tighter manufacture and grows monotonically, so a variator that met its ratio specification when new is not thereby a variator that meets it at ten thousand hours. A gear train has neither term: teeth do not lengthen, and the count is the count.

Which is why a variator can be calibrated and a gear train cannot #

There is a consolation hidden in the same sensitivity analysis, and it inverts the usual reading of a large sensitivity coefficient.

The dominant term in the stack is the belt length, and a belt length is one unknown constant per unit. It is not a noise source that differs from measurement to measurement; it is a single number that happens not to be known. So it can be found: command the sheave to a known position, measure the ratio that results, and invert the closed form to recover the effective belt length of this particular unit. One measurement, one number, and the largest term in the stack has been removed from it.

That is a general property of a deterministic sensitivity and it is worth naming as such. A large coefficient on a quantity that is constant per unit is an opportunity, because the same coefficient that amplifies the error amplifies the measurement of it — a ratio that moves 6.4 times as fast as the belt length is a ratio from which the belt length can be read 6.4 times as precisely. A large coefficient on a quantity that varies within a unit, like the sheave position the actuator is holding against friction, is simply a cost, and no measurement removes it.

So the three terms sort into two kinds. Belt length and centre distance are per-unit constants and are calibratable in principle; the sheave’s control error is not, and it is what remains after any calibration whatever. The residual stack is therefore smaller than the built one, and the field’s usual worst-case-against-root-sum-square argument should be run on the terms that survive rather than on all of them.

And a gear train has nothing to calibrate, which is not an advantage so much as an absence. There is no measurement that would improve a ratio of 24 to 96, because there is no unknown in it — the exactness that made the tolerance table empty makes the calibration table empty too. That is the honest full statement of what a count buys: not a smaller error than a length, but a mechanism with no free parameter and therefore no adjustment, for better and for worse.

Which of this field’s ratios are counts #

It is worth going through the field and sorting them, because the answer is not what the mechanism looks like.

The pattern is not “gears are exact and belts are not”. It is whether the mechanism’s output is decided by an incidence or by a dimension. A tooth has to be got past; there is no partial tooth; so a count of them is a count. A belt sits wherever the geometry puts it, and where the geometry puts it is a length.

And the two difference mechanisms are the interesting middle case. A differential screw’s output is a difference of two pitches — lengths — so it has a stack, and the stack is multiplied by the conditioning number $\kappa = |a/(a-b)|$, which is 10 for the screw here and 20 for the chain hoist. A differential screw with threads held to 0.1% has an advance held to 1%. That is the same arithmetic as the variator’s, arrived at from the opposite direction: the variator’s amplification comes out of a belt-length derivative and the screw’s out of a subtraction, and both are “the ratio is a small difference of lengths so it inherits their errors ten times over”.

What a tolerance on a count would even mean #

It is worth spending a paragraph on the objection, because it is a reasonable one: surely a gear can be made wrong enough to change its ratio?

It can be made wrong enough to fail. Teeth can be cut so badly that the mesh skips, or the pair can be assembled at a centre distance so far out that the teeth do not engage, or a tooth can break. Every one of those changes what the mechanism does, and none of them changes its ratio — a pair that skips a tooth has not acquired a new ratio, it has stopped satisfying the relation the ratio came from.

That is the honest form of the claim, and it is worth stating in the site’s own vocabulary. The ratio is a property of a state of the mechanism, the state being “in mesh, teeth engaged, nothing skipping”. Within that state the ratio is the tooth ratio exactly. Outside it, there is no ratio to talk about.

Which is the same structure the timing field found for intermittent mechanisms: the configuration includes a discrete label, the continuous quantities are exact inside a label, and what a manufacturing error can do is move the mechanism to a different label. A gear pair has one label and a very wide margin around it, so the label almost never changes and the exactness is almost always available.

A variator has no labels. Its ratio is a continuous function of a continuous state, and there is nowhere for exactness to hide.

The difference mechanisms are the interesting middle #

Three of this field’s mechanisms sit between the two extremes, and they are the ones where the practice field’s usual machinery bites hardest.

A differential screw advances by $p_1 - p_2$, two lengths. So it has a stack — and the stack is multiplied by the conditioning number $\kappa = |p_1/(p_1-p_2)|$, which for the pitches used here is 10. Hold both threads to 0.1% and the advance per turn is held to 1%. A chain hoist’s $\kappa$ is 20 and its lift per turn is held to 2% by the same thread quality.

That amplification is the same quantity the tolerance essays call a sensitivity coefficient, and this field supplies an unusually clean instance of it: a mechanism whose sensitivity coefficient is its gain, chosen deliberately, at the moment the gain was chosen.

It is also where the two conventions come apart most sharply. Worst case and root-sum-square differ by about $\sqrt{n}$ for $n$ comparable terms; but in a difference mechanism the terms are not comparable — one of them, amplified by $\kappa$, dominates everything else. When one term dominates, the two conventions converge, and the square root buys nothing. The variator’s stack has three terms of similar size and the square root buys 35%; the differential screw’s has one term worth ten times the others and it buys almost nothing.

What this says about where the field’s certainty came from #

Eleven fields on this site have carried an implicit assumption that is worth making explicit now that it has been broken.

When a mechanism’s behaviour is decided by lengths, everything about it is negotiable: the straight line is straight to 9.8 × 10⁻¹⁶ in theory and to whatever the pins allow in practice, and the as-built field exists to say how much. When it is decided by counts, the behaviour is a fact about integers, and manufacturing cannot touch it.

Gearing is the only place on this site where the second happens, and it is why the gears field has a different feel from the linkage fields — an exactness that does not have to be argued for. This essay is the boundary of it. Put a belt in the same gearbox and the exactness is gone, not degraded: there is no longer a number in the mechanism that a badly made part cannot move.