The same part, dimensioned twice

A part with two holes has one distance in it and nothing to argue about. A part with three has three distances and only two of them can be dimensioned independently, so somebody has to choose which.

That choice is the datum scheme, it is made by whoever draws the part, and it changes the answer.

Two schemes #

Take a ternary link with holes at 0, 1.4 and 3.5, and a process that locates each hole to ±0.01.

Chain dimensioning measures each hole from the last. H1 is 1.4 from H0; H2 is 2.1 from H1. Each of those two dimensions carries the tolerance, so each gap is ±0.010 — and the distance from H0 to H2 is the sum of two independently-toleranced quantities, so it is ±0.010√2 = ±0.0141.

Baseline dimensioning measures every hole from H0. H1 is 1.4 from H0; H2 is 3.5 from H0. Each of those carries the tolerance, so the span H0–H2 is ±0.010 — and the gap H1–H2 is the difference of two independently-toleranced quantities, so it is ±0.0141.

Same part. Same process. The two schemes give the same bands to two of the three distances and swap the third.

The total is conserved #

Line the two up:

distance      chain        baseline
H0 – H1       ±0.0100      ±0.0100
H1 – H2       ±0.0100      ±0.0141
H0 – H2       ±0.0141      ±0.0100

Neither scheme is better. Each has exactly one distance at ±0.0141 and the other two at ±0.0100, and which distance gets the bad one is the whole of the difference.

That is not an accident of these numbers. Three holes have two degrees of freedom in their relative positions, a scheme tolerances two distances directly, and the third is derived — so there is always exactly one derived distance and it is always √2 worse, whichever two are chosen.

The scheme is therefore a decision about which distance to protect, and it is a decision with no cost attached: protecting one costs exactly the amount that protecting another would have cost.

Which distance the mechanism cares about #

That makes the choice easy in principle and it is routinely made by convention instead.

A ternary link in a six-bar has three pins and the mechanism cares about all three distances, but not equally: the sensitivities of the output to each are different, and the tolerance field computes them. The scheme should put the √2 on whichever distance the mechanism is least sensitive to.

That is a five-minute calculation and it is available before the drawing is made. What happens instead is that a drafting standard or a house style picks one scheme for everything, and the √2 lands wherever the numbering happened to put it.

A drawing convention is making a tolerance decision, and the decision is not recorded anywhere as a decision. It looks like a matter of drafting style, and its consequence is a distance that is forty-one per cent worse than it needed to be.

Through to the mechanism #

The bands above are on the part. What the mechanism sees is those bands multiplied by its sensitivities, and the swap therefore changes the output band by a computable amount.

Take a ternary rocker in a Watt six-bar, whose three pins are the two ends and the point the second dyad hangs from. The output angle’s sensitivity to each of the three distances is different — the site’s own six-bar has seven lengths and the tolerance field ranks them — and the scheme decides which of the three carries the √2.

The difference in the output band between the best scheme and the worst is the difference between multiplying the largest sensitivity by 1.414 and multiplying the smallest by it. On a part whose three sensitivities differ by a factor of three, that is a difference of about twenty per cent in the whole machine’s band, from a choice made in a drawing office.

Twenty per cent is more than most tolerance tightening buys, and it is free: no better process, no closer inspection, no tighter numbers on the drawing. Only a different arrangement of the same numbers.

Where the errors go under each scheme #

Behind the arithmetic there is a physical story and the two schemes correspond to two ways of working.

Chain corresponds to positioning each feature relative to the last: drill H0, move the table 1.4, drill H1, move 2.1, drill H2. Each move carries its own error and the errors accumulate. That is how a manual machine with a leadscrew and a dial works, and it is why chain dimensioning is the older convention.

Baseline corresponds to positioning every feature from one reference: set the table’s origin at H0 and go to 1.4 and to 3.5 in absolute coordinates. Errors do not accumulate; instead the reference’s own error is in every position and cancels out of the span.

So the scheme is not merely a drawing style; it also expresses an expectation about how the part will be made. Where the expectation matches, the analysis is right. Where they disagree — a baseline drawing made on a manual machine by chaining, or a chained drawing made on a machining centre from absolute coordinates — the analysis describes neither.

Neither scheme is the average #

A tempting third option is to tolerance all three distances at some intermediate value and hope. It does not work, and seeing why is instructive.

Three distances on a line satisfy one exact relation: the span is the sum of the two gaps. Whatever the errors are, that relation holds identically for every part ever made. So the three bands cannot be chosen freely — the derived one is determined by the other two, and no drawing can assign all three independently.

A drawing that tries produces an inconsistency: it demands three things of which one is implied by the other two and does not follow from them at the stated tolerance. In practice the shop resolves it by inspecting to two of them and ignoring the third, which is a datum scheme chosen by the inspector rather than by the designer.

The constraint is arithmetic and no convention escapes it. What a drawing can choose is which distance is derived, and what it cannot choose is whether one is.

That is worth stating because it defeats the natural instinct, which is to think the √2 is a defect of the schemes rather than a property of the geometry. It is the second: three collinear points have two free distances and the third is a sum.

The relationship to the shared fraction #

This is the same mechanism as the two-hole case, and seeing that unifies the two.

A common error that displaces a group of holes cancels out of distances inside the group. Under baseline dimensioning, every hole is placed from H0, so the reference’s error is common to all of them and cancels out of every internal distance — which is why the span is ±0.010 rather than ±0.0141.

Under chain dimensioning there is no common reference, so nothing cancels globally; what cancels is only the accumulated error at each intermediate hole, which cancels out of the gap it bounds and not out of the span.

The datum scheme is a statement about which errors are shared, expressed in the language of drafting rather than of statistics. The shared fraction of the previous essays is the same idea for a pair; a datum scheme is that idea for a pattern.

A scheme also decides what is inspected #

Beyond the analysis, the scheme decides what a measurement of the finished part reports, and that has its own consequence.

Inspecting a chain-dimensioned part means measuring H0–H1 and H1–H2 and comparing each against ±0.010. A part whose derived span is out by 0.013 passes, because the span is not a dimension on the drawing.

Inspecting a baseline-dimensioned part means measuring H0–H1 and H0–H2. A part whose derived gap H1–H2 is out by 0.013 passes, for the same reason.

So the two schemes accept different sets of parts from one process. Neither is wrong — each accepts exactly what its drawing asked for — and a batch inspected under one scheme and used in a mechanism that cares about the other’s distance has been inspected against the wrong thing.

A tolerance is a contract about what will be measured, and the datum scheme is the clause that says which measurements those are. Reading it as a statement about the part’s geometry rather than about its inspection is where the confusion starts.

Three holes is the smallest case #

Two holes have no scheme — the one distance is the one dimension, and there is nothing to choose. The interesting cases start at three, and this site has several.

A ternary link in a six-bar carries three pins. A bell crank carries three: a pivot and two arms. The frame of a six-bar carries three ground pivots, which is the case where the ground length was already the exposed one and now has two companions. A Stephenson chain’s two ternary links carry three each.

With four holes it gets worse rather than more of the same: four holes have three independent distances and six distances in total, so three of the six are derived and their bands depend on the scheme in a way that is no longer a simple swap. The site has one such part — a quaternary link — and this field has not computed its schemes.

What happens when the holes are not collinear #

The three-hole part above has its holes on a line, which is a bell crank straightened out. Real ternary links are triangles, and the arithmetic changes in one respect worth naming.

Three points in the plane have six coordinates, of which three are a rigid placement, so a triangle has three shape parameters — three distances, all independent. Unlike the collinear case there is no exact relation among them: a scheme can tolerance all three side lengths and nothing is derived.

What is derived instead is anything else: an angle, an altitude, the distance from one vertex to the opposite side’s midpoint. Those follow from the three sides and carry bands that combine the three sides’ errors, with weights that depend on the triangle’s shape and can be large for a thin one.

So the swap this essay is about is a feature of collinear or over-determined patterns, and a triangle escapes it — at the price that the quantity a mechanism cares about may not be a side. A ternary link’s three pin distances are exactly what a mechanism cares about, which is why the triangular case is the comfortable one and why a bell crank whose pivot lies between its arms is not.

What a drawing cannot express #

One genuine limitation, and it is why this essay does not end with always choose the best scheme.

A scheme tolerances two distances and derives the third. What a designer often wants is to tolerance all three — to say that every pin-to-pin distance must be within ±0.010. That is not a scheme; it is a requirement, and it is satisfiable only by a process good enough that the derived distance also lands inside.

Stating it as a requirement rather than as a scheme moves the problem to the shop, which is sometimes right: a shop that knows its process can decide how to hold all three. Stating it as a scheme tells the shop how to measure and gives away one of the three.

Geometric dimensioning and tolerancing exists largely to express requirements that a chain of ± dimensions cannot, and this is one of them. What this field can say is what a chain of ± dimensions actually implies, which is the arithmetic above, and it is different from what most people reading such a drawing believe it implies.

What the site’s own parts do #

Worth checking the collection rather than only the principle, since this site has several three-hole parts and has never drawn one.

Its figures draw links as centrelines between joints, which is the right model for everything the other fields ask and carries no datum information at all. The bodies field gave links a width and an outline, which brings a part closer to being a part, and still says nothing about how its holes are located relative to one another.

So a datum scheme is a piece of information the site does not carry for any of its parts, and could. A ternary link’s three pins are in its links list; adding a scheme would be one field naming which two distances are toleranced.

Whether that is worth doing depends on whether anything would read it, and at present nothing would — the tolerance field works in lengths and this field’s feature model works in holes without reference to a drawing. It is recorded as an absence rather than a gap: the analysis in this essay is complete for a part whose scheme is stated, and no part on this site states one.

The scheme and the setup are two different questions #

It is worth separating the two ideas in this half of the field before they get run together, because they are independent and they multiply.

The shared fraction is about a process: how much of two features’ positioning error is common. It is measured by making parts and measuring them, it is a number between zero and one, and it scales the whole band.

The datum scheme is about a drawing: which distances are toleranced and which are derived. It is chosen, it is discrete, and it decides which distance gets the √2.

A part has both. A ternary link bored in one setup with a chain-dimensioned drawing has a high shared fraction and a chain scheme; the sharing makes every band smaller and the scheme puts the worst of them on the span. Change either and the other is unaffected.

So the answer for a part is a product of a process number and a scheme choice, and the two are decided by different people at different times, usually without either knowing what the other did. That is the practical reason feature-based tolerancing is not routine: the information it needs is distributed across a design office, a drawing office and a shop floor, and the analysis is the only place it would ever meet.

What a calibration would see #

The other half of this field measures a machine rather than predicting it, and the datum scheme leaves a signature there too.

Calibrate twenty machines built from parts made to one drawing and look at the spread of the recovered shapes. The spread is the process’s variation pushed through the mechanism, and the pattern of it — which combinations of the recovered parameters vary together — carries the correlation structure of the parts’ errors.

A chain-dimensioned ternary link produces one correlation pattern among its three distances; a baseline-dimensioned one produces another. Those patterns are different and both are visible in a batch of twenty calibrations, so in principle the scheme is recoverable from measurements of finished machines.

Whether that is worth doing is another matter — it is twenty calibrations to learn something written on a drawing — and it is worth knowing that the drawing’s choice is a physical fact about the parts rather than a bookkeeping convention, and physical facts can be measured. That is the sharpest available answer to anybody who thinks a datum scheme is only paperwork.

The rule that comes out #

Two sentences, and they need no new machinery.

A part with n holes has n − 1 toleranced distances and the rest derived, and every derived distance is √2 worse than a toleranced one. Choose the scheme so that the derived ones are the distances the mechanism is least sensitive to, using sensitivities the tolerance field already computes.

And record that the choice was made. A drawing that says nothing about why its datums are where they are has hidden a tolerance decision inside a drafting decision, which is exactly the kind of quiet substitution this whole field exists to find.