Where a length comes from

The tolerance field takes the four lengths as the independent variables and gives each of them a ±. Everything it computes follows from that, and it says so in its own text: real parts are not toleranced that way; they are toleranced on features, and the lengths are derived from those.

Its boundary essay goes further and names the repair as the first thing the work after it should do. This is that.

A link is two holes #

There is no such thing as a link length on a real part. There is a piece of material with two holes bored in it, and the length is the distance between the holes’ centres.

Nobody makes a distance. A machinist positions a spindle and drills; positions it again and drills again. Each positioning has an error, the errors are properties of the machine and the fixture, and the distance between the two holes is what those two errors leave behind.

So a length is a derived quantity, and the variables underneath it are hole positions. That is not a pedantic reframing: it changes the arithmetic, and it changes it in both directions depending on a number the lengths-only analysis has no place for.

The part each hole’s error contributes #

Split each hole’s positioning error into two pieces.

A part shared with the other hole. If both holes are bored in one setup, on one machine, without unclamping, then whatever the machine’s positioning is off by, it is off by for both. The pattern of two holes is displaced as a unit.

And a part of its own. Backlash, thermal drift between the two operations, the spindle’s own repeatability. Independent between the two holes.

Now take the distance between them. A displacement of the whole pattern does not change the distance at all — that is what it means for the two holes to move together. The independent parts do change it, and being independent they add in quadrature.

If each hole’s error has standard deviation σ, of which a fraction f of the variance is shared, then the independent part at each hole is σ√(1 − f), and the distance’s band is

√2 · σ√(1 − f)

The √2 is for there being two holes. The √(1 − f) is for the common part cancelling.

Both directions are wrong #

A lengths-only analysis uses σ for the length’s band. Compare.

With f = 0 — the two holes located entirely independently, which is two separate castings bolted together, or two setups on two machines — the derived length’s band is σ√2 = 1.414 σ. The lengths-only analysis is optimistic by forty-one per cent.

With f = 1 — one setup, one machine, the whole error common — the derived length’s band is zero. The distance between two holes bored without unclamping is as good as the machine’s repeatability allows, however badly the pattern as a whole is placed. The lengths-only analysis is pessimistic by everything.

And at f = ½ they agree exactly, because √2 · √(1 − ½) = 1.

So the naive analysis is not conservative and it is not liberal. It is right at one value of a number it never asks about, and wrong on both sides of it. Which side depends on how the parts are made.

The shared fraction is a fact about a factory #

That number — how much of a hole’s positioning error is common to its setup — is the one input in this whole field that is not geometry, and it is worth being exact about why it is admissible.

The boundary this site holds is that every input must be readable off a drawing or an instrument, and that the excluded quantities are excluded because they are properties of materials: a friction coefficient, a stiffness, a wear rate. Those vary with conditions, are quoted with large uncertainties, and cannot be checked by anybody who builds the mechanism.

A shared fraction is not like that. It is a property of a process, it is written on a process sheet, and it is measurable directly: bore twenty pairs of holes in one setup, measure the twenty distances and the forty positions, and the ratio of the variances is the number. Anybody with the parts can check it.

So it enters as a stated parameter rather than as a fitted one, and every figure that depends on it says which value was used. That is the same treatment a clearance gets: a number read off a drawing, feeding an exact computation.

What it does to the mechanism’s band #

Push it through to the output and the effect is a uniform rescaling — with a proviso.

At a crank angle of one radian, with every hole on every part given σ = 0.01, the four-bar’s output band is 0.931° with the holes independent, 0.658° at the crossing, and 0.294° with ninety per cent shared. The naive lengths-only figure is 0.658° regardless.

The reason it is a uniform rescaling is that the transmission factor √2 · √(1 − f) is the same for every part when every part is made the same way. It divides out of the allocation and the ranking of which length matters most is unchanged.

That null result is worth as much as the effect. Feature-based tolerancing does not automatically change which dimension to hold tight; it changes it only when the parts are made differently, and a frame jig-bored in one setup beside a crank drilled twice on a drill press is exactly that case.

A frame is the case where it bites hardest #

The four-bar’s four lengths are not equally affected, and the frame is where the difference is most consequential.

Three of the four — crank, coupler, rocker — are distances between two holes on one part. A shop that makes those parts can decide to bore each pair in one setup, and a shop that cares will.

The ground length is not like that. It is the distance between two holes in the frame, and a frame is often the biggest part, often a casting or a weldment, and often too large for one setup on the machine available. So the ground length is the one most likely to have f near zero — and it is frequently the length with the largest sensitivity, since the output angle is read at one of its two pivots.

That combination is the practical warning of the whole essay. The length most likely to be independently located is the one most likely to matter, and a lengths-only analysis assigns it the same treatment as the crank.

It is also the length a designer has most freedom about. Two pivots on one machined plate is a different part from two pedestals bolted to a base, and the difference between them does not appear on a kinematic drawing at all.

Two routes, as always #

The formula above is a shortcut and this site does not trust a shortcut with a whole field on it.

The long route builds the covariance of all six hole positions explicitly — σ² on the diagonal, σ²f between two holes sharing a setup, zero otherwise — takes the derivative of the four lengths with respect to the six positions, and pushes the covariance through it. That is a six-by-six matrix and a quadratic form, and it makes no assumption about the parts being made alike.

The two agree to 1.3 × 10⁻¹⁵ over shared fractions from 0 to 0.95.

They did not the first time. The shortcut was applied to the wrong pair of holes on one part — the two ends of the crank rather than the crank and the frame — and the disagreement showed up immediately at f = 0.75, which is exactly what a second route is for. A formula that has been checked against a construction that shares none of its reasoning is worth having; a formula that has only been read carefully is not.

Why one half, exactly #

The crossing at f = ½ looks like a coincidence and it is arithmetic worth doing once, because it says what the two competing effects are.

Two independent errors combining into a distance give a factor of √2 — that is the cost of a length being made of two things. A shared fraction f removes a fraction f of each error’s variance from the distance — that is the saving of the two things being made together.

The two are equal when √2 · √(1 − f) = 1, which is f = ½. Below it the cost dominates, above it the saving does.

Two consequences follow and neither is obvious. The crossing does not depend on σ, on the mechanism, on the sensitivity, or on which length is being considered: it is a property of a length having two ends. And a process that shares only half its error is exactly as good, for the mechanism, as no analysis at all — so the effort of a feature-based study is wasted on a shop that happens to sit there and is worth a great deal on shops either side.

An analysis that has to know how the part is made #

The uncomfortable implication, stated plainly: a tolerance analysis of a mechanism cannot be done from a kinematic drawing.

Everything the tolerance field computes — the band through a turn, the allocation, the sixteen corners of the box — takes four numbers and four tolerances and produces an answer. All of it is correct arithmetic on its inputs, and its inputs are a model of a part nobody makes.

Doing it properly needs the process, and the process is decided by a manufacturing engineer who is usually not the person doing the analysis and often has not been appointed when the analysis is done. That is a genuine organisational difficulty and not a technical one, and it is worth saying because the technical fix is one line of arithmetic and the practical fix is not.

There are two honest responses and one dishonest one.

Compute both bounds. The band with f = 0 and the band with f = 1 bracket every process, and if the mechanism is acceptable at f = 0 then nothing about the process can spoil it. That is conservative, it is cheap, and it is available with no information at all.

Or state the assumption. This analysis assumes each part’s holes are located in one setup with at least sixty per cent of the positioning error common. That is a requirement on the process, it can be put on the drawing, and it can be verified by measuring twenty parts.

The dishonest response is the current default: tolerance the lengths, analyse the lengths, and let the shop do whatever it does. That is right only at f = ½ and the shop was never told.

What a drawing should say #

The practical conclusion is a change to the drawing rather than to the analysis.

A drawing that dimensions a coupler as 3.500 ± 0.010 between hole centres is stating a derived quantity and leaving the process free. It is unenforceable in one direction — a shop that bores the two holes separately cannot hold it — and it gives away accuracy in the other, since a shop that bores them together could hold far better.

A drawing that dimensions the two holes from a datum, with a stated tolerance on each, is stating what is actually made. The derived length follows, and it follows differently depending on the process, which is information the designer wanted and did not have.

That is the whole content of feature-based tolerancing as this field uses it. Tolerance what is made, derive what is used, and the derivation is one line of arithmetic that depends on a number the process sheet already carries.

The same reasoning one level out #

A last generalisation, because the argument is not about holes.

Anywhere a quantity that matters is a difference of quantities that are made, the shared part of their errors cancels and the independent part survives. Two holes and a distance is the instance here. Two gear centres and a centre distance is another, and it decides backlash. Two faces and a thickness is a third.

And anywhere the quantity that matters is a sum rather than a difference, the shared part does not cancel — it doubles. A stack of three plates whose thicknesses were all cut on one machine that runs thick has a total that is three times thick, and no cancellation anywhere.

So the sign of the effect follows from whether the derived quantity adds or subtracts the features it is made of, and both cases are common in one machine. A link is a difference; a stack height is a sum. A single shared setup improves one and worsens the other, which is the sort of statement a lengths-only analysis cannot make because it has no place to record that the two came from one machine.

What the site had already, and what it did not #

The tolerance field carries a small version of this and it is worth saying exactly how small, because the difference is the content of this essay.

featureBand there takes one shared fraction and applies it to the ground length only, on the reasoning that the frame’s two holes are the obvious case. It shows the effect in both directions and it was enough to make the point in one figure at the end of that field.

What it does not do is treat the machine as a set of parts. Every one of the four lengths is a distance between two holes; the crank’s two holes share a setup or do not, independently of the frame’s; and a machine assembled from parts made in four different ways has four different transmission factors, which is exactly the case where the allocation changes.

So the extension is not a refinement of one figure. It is the difference between a mechanism with one derived length and a mechanism with four, and the second is what a mechanism is.

It also brings in two things the smaller version had no place for: the datum scheme, which is a choice about a part with three or more holes and has no analogue for two, and the covariance route, which is the second computation this site’s habit requires and which caught the shortcut being misapplied.

A length is not the only derived quantity #

Worth one more before the boundary, because the same treatment applies to something the site already computes and has not questioned.

A clearance is a hole radius minus a pin radius, and both of those are made. The clearance’s band is therefore a derived band, and whether the hole and the pin were made together — bored and turned in one operation, or matched on assembly — decides whether their errors cancel. A shop that matches pins to holes produces clearances far tighter than either dimension’s tolerance would suggest, which is exactly the mechanism this essay is about, one component down.

The site’s clearance model takes the clearance as an input, which is correct and is the same abstraction the length model makes. The extension is the same extension and it is not made here.

Recording it as a shortfall rather than doing it: the practice field’s clearance figures assume a clearance value, and where that value comes from is one layer further out in exactly the way a length’s is.

What this does not model #

Three things, named because the boundary should be as clear here as everywhere else.

Form. A hole is not a point; it has roundness, cylindricity and a position that varies down its depth. Everything here treats a hole as a point with a two-dimensional error, which is the same abstraction the practice field’s clearance model makes and is right for a question about where a mechanism can be.

Orientation. A hole can be perpendicular to the face or not, and in a planar mechanism that matters only through how it interacts with the pin’s length — which is a body question rather than a length question.

And any statistics beyond a variance and a covariance. The shared fraction is a ratio of variances and nothing here needs a distribution. Whether a real shop’s errors are Gaussian, and what happens in the tails, is somebody else’s subject and this field does not enter it.