A clearance inside a tolerance box

The practice field’s method is one idea applied everywhere: a tolerance makes a mechanism a set of mechanisms rather than one, every technique the site already has applies to each member, and the answers become intervals instead of numbers.

It has been applied to positions, to ratios, to coupler bands, to transmission angles and to lost motion. This is the same idea applied to a quantity none of those is: one with a sign.

The measurement #

A four-bar — ground 4, crank 1, coupler 3.5, rocker 3 — with a post bolted to its frame at (1.35, 2.5), links 0.16 wide. As drawn it clears by 0.104 at its worst instant.

Give each of the four lengths a band of ±0.02, which is an eighth of a link width and a two-hundredth of the frame. Sixteen corners of the box; at each one, build the mechanism, sweep it through a full turn, and take the smallest gap over every tested pair.

The interval that comes back runs from 0.076 to 0.132. A spread of 0.056 — 54% of the nominal clearance — from tolerances that would be considered ordinary on any of the four lengths.

Why an interval with a sign is different #

Read as a magnitude, that is unremarkable: a quantity of about 0.1 with an uncertainty of about ±0.02, which is the sort of thing every measurement on this site produces once the lengths have bands.

Read as a sign, it is a quarter of the way to a machine that does not go together, on tolerances nobody would query.

That is the whole distinction. A transmission angle with an interval of 40° ± 5° is a machine that works a little better or a little worse. A ratio with an interval is a machine that runs a little fast or a little slow. A clearance whose interval reaches zero is not a machine at all: on that side of it, two parts are in the same place, and there is nothing to run.

So the quantity to report is not the width of the interval. It is the margin: how far the worst corner is from zero, in units of the same tolerance that produced the interval. Here the worst corner is at 0.076 with a spread of 0.056, so the margin is about one and a third spreads — which is a statement a designer can act on and 0.104 ± 0.028 is not.

How large the effect is, in the units that matter #

A ±0.02 band on a length of 3 is a tolerance of seven parts in a thousand — loose for a machined part, tight for a fabrication, and unremarkable either way. It moves the clearance by 54% of its nominal value.

The amplification is worth naming because it is the practice field’s recurring finding arriving on a new quantity: a mechanism is a lever for dimensional error, and how much of a lever depends on which quantity is being watched. Here four tolerances of 0.02 produce a spread of 0.056 in a clearance of 0.104, which is a gain of a little under one from the sum of the bands and is large relative to the quantity itself, because the quantity is small.

That is the general shape of a clearance’s sensitivity and it is worse than most. A clearance is a difference between two positions that are each of the order of the mechanism’s size, so its relative error is the absolute error of a length divided by the clearance rather than by the length. A machine with a clearance of a hundredth of its size amplifies its tolerances a hundredfold in relative terms, and there is nothing to be done about it except to have a larger clearance.

Corners rather than samples #

The sixteen corners are enumerated rather than the box being sampled, and that is the same choice the practice field makes for the same reason: the extremes of a monotone quantity over a box are at its corners, and enumerating 2⁴ of them is cheaper and exact where sampling is neither.

Whether this quantity is monotone in the four lengths is the question that choice depends on, and it is not obvious. The gap is a minimum over a drive of a minimum over feature pairs of a distance between bodies whose positions come from a nonlinear solve. There is no reason in that description for it to be monotone in the frame length.

So it is checked: 24 random draws from the interior of the box, and none of them beats the worst corner. That is not a proof and it is the same evidence the practice field’s other corner enumerations rest on — with one difference worth noting, which is that the failure would be visible here. A quantity with a sign that is non-monotone would show up as an interior point going negative while every corner stayed positive, which is a machine that fails in the middle of its tolerance band and passes at its extremes.

What the corners say about which length matters #

The sixteen readings are not a cloud; they have structure, and the structure is the sensitivity that field measures for other quantities.

The sixteen readings fall into two blocks of eight, split by the sign of the rocker’s deviation: lengthening the rocker by 0.02 costs 0.031 of clearance, three times what any other length does. Then the frame, worth 0.016 in the other direction; then the crank, worth 0.009; and last the coupler, worth 0.0004 — which is to say nothing at all.

That last entry is the one worth stopping on, because the coupler is the part that touches. The witness pair at the worst instant is the coupler against the post, so a reader would expect the coupler’s own length to be what the clearance is most sensitive to, and it is the one length that does not matter.

The reason is that a length does two things and only one of them is local. Lengthening the coupler moves the coupler’s own material outward and moves the whole mechanism’s motion, and here the two nearly cancel. Lengthening the rocker does not move the coupler’s material at all and changes the motion decisively — it swings the coupler through a different arc past the post.

Which part touches and which length matters are different questions, and the second is not answerable by looking at the drawing. That is the same lesson the transmission angle’s sensitivity teaches about a different quantity, and the rankings are different: each quantity has its own ordering over the same four lengths, and a tolerance scheme tightened for one is not tightened for another.

Sixteen sweeps, and what they cost #

The arithmetic of this measurement is worth stating because it is the most expensive figure in the field and the cost has a shape.

Each corner is a complete measurement: build the mechanism at those four lengths, march it through ninety solved positions, compute the gap over every tested pair at each, and take the minimum. Sixteen corners is sixteen of those, plus twenty-four interior draws for the monotonicity check — forty sweeps for one number and one picture.

That is still under a second, and it scales badly in exactly the direction a real study goes. Four lengths give sixteen corners; add the link width, the boss radius and the post’s two coordinates and it is eight parameters and 256 corners. Add the fact that a real study wants a certificate at each corner and the sweep count multiplies again.

So the corner enumeration is the right method here and is not the right method everywhere, and the crossing point is around eight or ten parameters — which is where the practice field’s other tolerance studies already sit, and why they carry both a corner reading and a sampled one.

The worst instant moves #

A detail that matters for how the computation is done: the angle at which the worst gap occurs is not the same at every corner.

The nominal machine’s minimum is at 2.11 radians. At the corners it wanders, because changing the lengths changes the whole motion — the coupler passes the post at a different angle and with a different attitude. So each corner needs its own full sweep and its own minimisation; evaluating the sixteen mechanisms at the nominal machine’s worst angle would give sixteen numbers that are not the quantity.

That is the practice field’s standing warning in a new place: a tolerance study that fixes the configuration and varies the lengths is measuring a different thing from one that varies the lengths and re-optimises. Here the difference is that the second one is the answer and the first is a coincidence.

Where the interval reaches zero #

The measurement above has a comfortable margin, and the interesting case is the one that does not. It is easy to construct: move the post outward by a quarter of a unit, to (1.6, 2.5), and the nominal machine clears by about 0.02 — still positive, still a machine, and a drawing anybody would sign.

At that nominal clearance the same ±0.02 band on the four lengths produces an interval that straddles zero. Some corners of the box are machines that go together and some are not, and which one a given assembly is depends on where in its tolerance the rocker happened to land.

That is a qualitatively different verdict from a wide interval, and it has a name in manufacturing that this site does not otherwise need: a design with a yield rather than a design that works. A fraction of the parts assemble and a fraction do not, the fraction depends on the distribution rather than on the corners, and the root-sum-square reading is the one that answers it.

It also has a consequence for the earlier fields that is worth stating plainly. A clearance of 0.02 on a machine whose lengths are held to 0.02 is not a tight design; it is not a design at all, in the sense that whether it works is not decided by the drawing. And nothing in the nominal figure says so — 0.02 is a positive number and the machine in the picture runs.

What is being assumed #

Independence. Each of the four lengths varies over its own band, independently of the others, and the corners are the sixteen sign combinations. Real lengths on one plate share a datum and are correlated, and the practice field’s stack-up essays are about exactly that.

Rigid, exact bodies. The widths are nominal. In reality the link width, the boss radius and the post’s position all have tolerances of their own, and every one of them moves the gap directly rather than through the mechanism — so a full treatment has more than four dimensions and the shape of the answer does not change.

A fixed obstacle. The post is bolted to the frame at an exact place. In reality it is located by holes with their own tolerances, and its position enters the clearance directly rather than through the mechanism — so a millimetre of post position is worth more than a millimetre of any link length, and it is the cheapest thing in the assembly to get wrong.

And a worst case rather than a distribution. The corners give the extremes; what fraction of a real production run falls below any given clearance is a question about distributions, and the practice field has the root-sum-square reading for it. A worst-case clearance is the number to design to when the failure is does not assemble, because that failure is not something to be right about on average.

What it means for the earlier fields #

Every clearance quoted in the bodies field is a nominal one, and this essay is the correction factor.

The four-bar with a post clears by 0.104 nominally and by 0.083 at its worst corner: a real design margin about a fifth smaller than the drawn one. The width search reports the widest link a four-bar admits, and the same argument applies — the widest link a tolerance band of four-bars admits is narrower, by however much the worst corner moves the crossing.

And the certificate is about one mechanism at a time. A certified sweep proves that this machine, at these lengths, misses nothing; the tolerance box needs the certificate at every corner, which is sixteen certificates rather than one.

Sixteen certificates, and what the sixteenth says #

Run them. Each corner is a different machine, each needs its own dense sweep to measure the gap’s speed bound on, and each returns a floor below which the gap provably did not go between the samples. All sixteen certify.

The three numbers that come back are worth setting side by side, because they are the same quantity measured with three different amounts of rigour and they are not close.

sampled minimum over the sixteen corners     0.07627
certificate for the NOMINAL machine          0.06035
certificate for the whole box                0.03101

The box’s guarantee is less than half what a certificate on the middle of the box says, and less than a third of what the sampled study reports. A design admitted on the sampled figure has been admitted on a curve nobody bounded between its samples; a design admitted on the nominal machine’s certificate has been admitted on a machine that will not be built, since the machine that gets built is somewhere in the box and the box’s worst corner is what it might be.

There is a second thing the sixteen say and it is about the sweep, not the design. The certificate reports how many samples would have been needed to certify, and across the corners that runs up to 108 where the study itself used 90. The nominal machine certifies comfortably at 90; four of its corners do not have that margin, because a corner with a longer crank sweeps its parts past each other faster and a faster gap needs a finer grid to bound. The sample count a tolerance study needs is a property of its worst corner rather than of its nominal machine, and nothing about the nominal machine’s sweep suggests it.

So the honest cost of a certified tolerance study is sixteen dense sweeps at the resolution the worst corner demands, which is more than sixteen times the nominal machine’s — and on this machine it is still four-tenths of a second. The expense is a reason to automate it, not a reason to skip it.

Two routes, in a study whose answer is an interval #

The site’s standing habit is that no quantity is published until two routes have produced it, and a tolerance study makes that awkward: the interval is not a number to be computed twice but a set to be characterised.

There are two routes here and they check different halves.

The corners against the interior. Sixteen exact evaluations at the extremes, against twenty-four random draws inside — and the claim being checked is that the extremes are at the corners. Twenty-four draws is not a proof and it is a test that could fail: an interior point beating a corner would say the quantity is not monotone in the lengths, which would invalidate the whole method rather than merely the number.

And the nominal against the box. The nominal machine’s clearance is computed by the same code, at the box’s centre, and must lie inside the interval. That is a trivial-looking check and it catches the commonest coding error in a study like this, which is a corner enumeration whose sign bits do not correspond to the lengths they are supposed to.

Neither is as strong as the two independent computations the site prefers, and the reason is stated rather than hidden: a worst case over a box has no second route that does not amount to enumerating the box differently. What can be checked is the assumption the method rests on, and that is what these two do.

The general form #

A quantity with a sign turns a tolerance study from an error analysis into a feasibility analysis, and the two want different things reported.

An error analysis reports a centre and a spread, because the answer is a number and the question is how well it is known. A feasibility analysis reports a margin and the corner that produced it, because the answer is yes or no and the question is how close the no is.

This site’s practice field has been doing the first for fifteen essays, correctly, on quantities that are magnitudes. The clearance is the first one that is not, and the reporting has to change with it: worst corner, distance from zero, and which length moved it — rather than nominal, band, and sensitivity.

That is a small change in what is printed and a large change in what it means. 0.104 ± 0.022 invites a reader to think the answer is 0.104. Worst corner 0.083, two spreads clear, coupler-dominated tells them what would have to go wrong, and by how much, for the parts not to go together.

An interval that reaches zero being a different answer rather than a wider one deserves a general statement, because this site produces intervals constantly and most of them do not behave this way. A tolerance band on an angle, a length, a ratio or a curvature is a quantitative interval: every value in it is an answer of the same kind, and the width says how well the quantity is known. A band on a signed clearance is not. It straddles a qualitative boundary, and the two sides of that boundary are the parts fit and the parts do not, which are not two values of one quantity. So the honest report changes shape at the crossing: below zero the right answer is not a number at all but a refusal, and averaging across the interval mixes measurements with non-existence. The general rule is worth having wherever a tolerance study is run. Ask whether the quantity’s zero is a boundary or a value, and if it is a boundary, report the fraction of the box on each side rather than the extremes — because a mean that spans an impossibility is a number describing nothing.