Where the two analyses cross

Everything in the tolerance field rests on a number: the band on each link length. Give a four-bar’s four lengths ±0.01 each and the output band at a crank angle of one radian is 0.658°.

That number does not depend on how the parts are made, because nothing in the model records how they are made. This essay is the curve it should be.

The curve #

Hold every hole on every part to σ = 0.01 and vary the shared fraction f.

f = 0.00     band 0.931°     1.414 × the naive figure
f = 0.25     band 0.806°     1.225 ×
f = 0.50     band 0.658°     1.000 ×
f = 0.75     band 0.465°     0.707 ×
f = 0.90     band 0.294°     0.447 ×
f = 1.00     band 0.000°     0 ×

The ratio is √2 · √(1 − f), which the derivation gives: √2 for a length being made of two holes, √(1 − f) for the shared part of each hole’s error cancelling out of the distance between them.

The lengths-only figure is 0.658° at every value of f, because it has no f in it.

What is being compared #

It is worth being exact about what the two analyses are, because they are not two approximations to one thing.

The lengths-only analysis takes the four lengths as independent random variables with standard deviation σ and combines their contributions. That is a complete and correct calculation about a machine whose lengths are the things that vary.

The feature analysis takes the six hole positions as the random variables, derives the four lengths from them, and pushes the derivation through. That is a complete and correct calculation about a machine whose holes are the things that vary.

Both are right about their own model. The question is which model is a machine, and the answer is the second — a shop cannot make a length, and a length that came out right did so because two holes did.

So this is not a correction for an approximation. It is a different model, and the first one is right at exactly one point of the second’s parameter range.

Why the crossing is at one half #

The arithmetic is one line and it is worth having because it says the crossing is universal.

√2 · √(1 − f) = 1 when 1 − f = ½, so f = ½.

Nothing else appears. Not σ, not the mechanism, not the sensitivity of the output to that length, not the crank angle at which the band is evaluated. The crossing is a property of a length having two ends.

That is a stronger statement than it sounds. It means the lengths-only analysis is not conservative in some regimes and liberal in others depending on the machine; it is optimistic below f = ½ and pessimistic above it, for every mechanism, every tolerance and every configuration. The one number that decides which is a number about a factory.

The curve is a straight line in disguise #

A small observation that makes the whole thing easier to hold.

Plot the band against √(1 − f) rather than against f and the curve is a straight line through the origin with slope √2 σ times the sum of sensitivities. Every feature of it follows from that: the crossing at √(1 − f) = 1/√2, the zero at f = 1, the factor of √2 at f = 0.

So there is really only one shape here and it has been drawn against an awkward abscissa because f is the number a process sheet carries. The physically meaningful quantity is the independent part of each hole’s error, σ√(1 − f), and the band is simply √2 times that, times whatever the mechanism does with a length error.

That reframing also says how to measure f without measuring f. A shop can report the standard deviation of the distance between two holes over twenty parts — which is exactly √2 σ√(1 − f), one number, directly what the analysis wants — rather than reporting positions and a correlation. Measuring the derived quantity is easier than measuring its ingredients, and it is what the analysis needs.

Which side real shops are on #

The honest answer is that it varies, and that the variation is exactly why the number belongs on a process sheet rather than in a textbook.

Two holes bored in one setup on a jig borer or a machining centre, without unclamping, share nearly all of the machine’s positioning error: the table’s absolute position error is common, and what is left is the machine’s repeatability, which is often ten times smaller. That is f near 0.9 and a band less than half what the drawing assumes.

Two holes drilled from a jig plate share the jig’s errors and not the drilling’s. Somewhere in the middle.

Two holes in a large frame, positioned in two setups — because the frame does not fit the machine in one — share nothing but the frame’s own dimensional stability. That is f near zero and a band √2 worse than the drawing assumes.

And two pedestals bolted to a base plate, positioned by measurement, share nothing at all and add the bolted joint’s own error on top.

The distribution of real cases is therefore wide and straddles the crossing. There is no default value of f that would let this essay be replaced by a factor.

The band is a function of configuration too #

One axis this essay has held fixed and should not leave unexamined.

Every number above is at a crank angle of one radian. The output band varies through the turn — that is the tolerance field’s own first result, and on this machine the band is a factor of a few wider near the limit positions than in the middle of the travel.

The feature factor does not vary through the turn, because it multiplies the length errors before the mechanism ever sees them. So the whole band-through-a-turn curve is scaled up or down bodily, its shape unchanged, its widest point in the same place.

That is convenient and it is worth confirming rather than assuming, which the second figure does: the curve at a crank angle of 2.2 radians is a different size and crosses at the same f. Two independent variables, and they separate cleanly — the mechanism decides the shape of the band through the turn and the process decides its overall size.

The one place they would not separate is a machine whose parts are made differently and whose sensitivities to those parts vary very differently through the turn. Then the four contributions are reweighted by the process and the reweighting changes which crank angle is worst. That case exists and this site has not measured it.

Where it bites hardest #

Two features of the four-bar make one of its four lengths much more exposed than the others.

The ground length is the distance between the two frame pivots, and a frame is the part most likely to be too big for one setup. So it is the length most likely to have f near zero.

It is also, on this site’s own machine, among the more sensitive: the output angle is read at O₄, so moving that pivot rotates the reading directly as well as through the mechanism, which is the term a naive derivation drops and which makes the ground length matter more than it looks.

The length most likely to be independently located is the one most likely to matter. That coincidence is not universal and it is common, and it is the reason a feature-based study is worth doing on a machine with a large frame and not worth much on one made of four small bars.

What it does not change #

The important null result, and it is what stops this being a general licence to redo every tolerance study.

If every part is made the same way — one f for the whole machine — then the transmission factor √2 · √(1 − f) is common to all four lengths. It multiplies the whole band and divides out of anything that compares one length against another.

So the ranking is unchanged. Whichever length a lengths-only allocation says to hold tightest, a feature-based one with a uniform process says the same. The band moves and the priorities do not.

That matters because it says where the effort should go. A feature-based analysis with one f is a scale factor on an answer already obtained. A feature-based analysis with four different f’s is a different answer, and that is the case that changes what a machinist is told.

Worst case and root-sum-square, both rescaled #

The tolerance field computes a band three ways — by enumerating the sixteen corners of the tolerance box, by summing the absolute contributions, and by combining them in quadrature — and the relationship between the three is one of its results.

All three rescale by the same factor here, and it is worth checking rather than assuming. The worst-case sum is Σ|∂ψ/∂ℓᵢ|·δᵢ, and replacing each δᵢ by √2·σ√(1 − fᵢ) multiplies the sum by that factor when the fᵢ are equal. The root-sum-square is √(Σ(∂ψ/∂ℓᵢ·δᵢ)²) and multiplies the same way. The corner enumeration solves sixteen mechanisms whose lengths are at their extremes, and the extremes move by the factor.

So the ratio between worst case and root-sum-square — which for four roughly equal contributions is about 2 — is unchanged by any of this. Feature-based tolerancing rescales the band and leaves its internal structure alone, under a uniform process.

Under a non-uniform one it does not, and that is where the interesting case lives: four different factors means the four contributions are reweighted, the ratio between worst case and quadrature moves, and the corner that produces the extreme can change which corner it is.

Reading the flat line #

One more thing to notice about the figure, since the flat line is doing more work than it looks.

It is not a bound and it is not an approximation. It is the answer to a different question — what is the band if the lengths themselves are the things that vary independently by σ — and that question has an answer whether or not any machine matches it.

Where it comes from is worth tracing. A drawing that says 3.500 ± 0.010 is an instruction about a length, and if a shop treats that instruction as a specification to be inspected — measure the distance, accept or reject — then the population of accepted parts really does have lengths distributed within ±0.010, whatever the holes did. Inspection to a derived dimension makes the naive model true by selection.

So the flat line is the answer for a shop that inspects to the length and scraps what misses. The curve is the answer for a shop that controls the process and does not inspect. Most shops are somewhere between, and the difference between the two models is also the difference between paying for inspection and paying for a better process.

That is a genuinely useful reframing of the figure: the vertical gap between the curve and the line at a given f is what inspection to the derived dimension is buying, or costing.

Why this was not in the tolerance field #

The tolerance field already carries a version of this — one figure, at the end, with the shared fraction applied to the ground length only — and knowing why it stopped there says what changed.

That field’s question is what does a tolerance do to a mechanism, and the answer needs a tolerance as an input. Where the tolerance comes from is one layer further out, and stopping at the layer the question needs is the right decision rather than an oversight; the field says so in its own boundary essay and names the extension as the next thing to do.

What makes the extension a field rather than a figure is that it is the same subject as identification. Both ask where a number on a drawing comes from and what could confirm it; both answer with a derivation from things that were actually made or actually read; and both find that the obvious quantity — a length, a parameter — is not the thing that exists.

A length is derived from holes, and a parameter is derived from readings. The two halves of this field are one question asked of the two ends of a machine’s life, which is why they sit together and why neither is a subsection of the tolerance field.

The measurement behind the curve #

Every point on the curve is computed twice, because a curve with a whole essay resting on it should be.

The shortcut applies the factor √2 · √(1 − f) per part and combines the four contributions. The covariance route builds the six-by-six covariance of the hole positions — σ² on the diagonal, σ²f between two holes sharing a setup — takes the derivative of the four lengths with respect to the six positions, and evaluates the quadratic form.

They agree to 1.3 × 10⁻¹⁵ at every f tested. They did not initially: the shortcut was applied to a wrong pair of holes on one part and the two routes parted at f = 0.75, which is what a second route is for and is the reason the site insists on one.

The covariance route is also what makes the non-uniform case computable at all. The shortcut assumes each part’s two holes are the only correlated pair; the covariance handles any pattern of shared setups, including two holes on different parts machined in one operation — which happens, on a frame whose pivots are bored with the base.

What f = 1 really means #

The right-hand end of the curve says the band is zero, and a band of zero deserves scepticism.

It is correct within the model and the model is a limit. f = 1 says the entire positioning error is common to the two holes, so the pattern is displaced rigidly and the distance between the holes is exact. That is the idealisation of boring both holes without moving anything, and what it neglects is everything that is not a positioning error: the spindle’s runout, the drill’s wander, the hole’s own form, the measurement of the hole’s centre.

Those set a floor, and the floor is not zero. In practice a pair of holes bored in one setup has a centre-distance error of the order of the machine’s repeatability, which is small compared with its absolute positioning error and is not nothing.

So the curve should be read as approaching a floor rather than reaching zero, and the floor is a separate number that the model here does not carry. Adding it is one term — an uncorrelated contribution that does not scale with f — and it changes the right-hand end and nothing else.

That is worth saying because the zero is the most quotable point on the curve and it is the least trustworthy. The useful part is the middle, where the factor runs between 1.4 and 0.5 and the model’s assumptions are not being pushed.

A curve that a calibration could measure #

A closing connection between the two halves of this field, because they look like separate subjects and are not.

Everything above predicts a band from a stated process. The other half of this field measures a machine’s actual dimensions from its motion. Put the two together and the prediction becomes testable: calibrate twenty machines built to one drawing, look at the spread of the three recovered invariants, and compare it against the band the feature analysis predicts.

That is a real experiment and it would settle the shared fraction empirically, for that shop, on those parts, without measuring a single hole. The calibration recovers a shape; the spread of shapes over a batch is the process’s own variation pushed through the mechanism; and the feature model predicts exactly that spread as a function of f.

One unknown, one measurement, and the measurement is one this field already knows how to make. It is not run here — twenty machines is not something a computation supplies — and it is the natural test of the whole feature half.

The one thing to be careful of is what a calibration recovers. It recovers the shape and not the size, so the spread it measures is a spread of the three invariants, and the prediction has to be pushed into those coordinates rather than left in the lengths.

The one number to ask for #

If this essay is reduced to a single instruction it is this.

Before a tolerance analysis of a mechanism, ask the shop how much of each part’s hole-positioning error is common to the pair. It is one number per part, it is measurable in an afternoon, and it moves the answer by a factor between zero and 1.414.

Without it, the analysis is a calculation about a machine whose lengths vary independently, which is a machine nobody makes, and it is right only if the shop happens to sit at f = ½.