Field

As built

Every length here has been a number and every pin a point. Give the lengths ranges and the pins holes and the answers change shape: a curve becomes a band, and some of the mechanisms this site admires most stop working altogether.

The output is a band, not an angle. The rocker's angle through one turn of the crank, for a four-bar whose four lengths are each specified to ±0.01. The line is the nominal mechanism; the band is where the output of an actual one lies, found by building all sixteen extreme combinations of the four lengths at every crank angle and solving each. The band is not a constant width: it is 0.73° at its widest, near 30°, and 0.36° at its narrowest — a factor of 2.0. Which of those a designer is told depends entirely on where the mechanism was measured.

A length is a range

Every figure on this site so far has been drawn from four numbers. No four numbers were ever cut. Give each of them a tolerance of ±0.01 and the rocker's output stops being an angle and becomes a band 0.73° wide at one part of the turn and 0.36° wide at another — and which of those a designer is told depends only on where somebody measured.

Two routes to the same derivative. How much the output angle moves when the coupler length moves, through one turn, computed twice. One route rebuilds the mechanism at b ± 10⁻⁶ and solves both from scratch; the other differentiates the constraint equations and solves one linear system against the analytic Jacobian. They lie on top of each other — the strip beneath plots the difference on a four-decade log scale, and its largest value anywhere in the turn is 2.9e-10 against a sensitivity of order 0.41. That is the only independent check there is of the Jacobian itself, whose coupler rows carried four wrong signs from the foundation phase to 2026-08-12 without ever drawing anything wrong.

Two routes to a sensitivity

How far the output moves when a link length moves can be found by rebuilding the mechanism and solving it again, or by differentiating the constraint equations and solving one linear system. The two agree to two parts in a hundred million across a whole turn — and the second route is an independent test of the constraint Jacobian itself, whose coupler rows had carried wrong signs unnoticed.

The four lengths do not matter equally

Averaged over a whole turn, the coupler contributes 38% of a four-bar's output band and the rocker 11% — a factor of 3.4 between the ends of the ranking. A tolerance specified equally on all four therefore spends most of its money buying accuracy the mechanism cannot use, and the ranking that says so costs four linear solves.

Worst case and the square root

Four tolerances can be added as a straight sum or as a root-sum-square, and the second is smaller by between 1.42 and 1.96 through this linkage's cycle. The ceiling is √4 = 2 and no geometry can beat it. That factor is not found in the mechanism — it is bought entirely with an assumption of independence, and one fixture that locates two holes takes it straight back.

Where a tolerance stack-up stops meaning anything. The ratio between the first-order tolerance estimate and the band measured by building every corner linkage, for two four-bars at ±0.002 on each length. The crank-rocker's ratio is 1 at all 180 positions — a stack-up is exactly right for it, everywhere. The parallelogram is a change-point linkage, where all four bars can lie on one line, and at that position the estimate exceeds the measurement by a factor of 4.8e+5. The difference is not in the arithmetic, which is identical; it is that a derivative describes a map that can be inverted, and at a change point the map cannot.

Where a stack-up stops working

On a crank-rocker the first-order tolerance estimate matches the measured band at every one of 180 positions, to eight parts in ten thousand. On a parallelogram it exceeds it by a factor of 475,512. Same arithmetic, same tolerance, two linkages that differ only in their proportions — and nothing in the calculation says which one it is being run on.

A pin in a hole is a short link. Left: a pin of radius 0.86 in a hole of radius 1, so the clearance is 0.14. The pin's centre may sit anywhere within that of the hole's centre. Right: the same joint as it enters the kinematics — a binary link of fixed length 0.14 and free direction, with a revolute at each end. That is not an analogy. It is the same set of relative positions, so every count, every Jacobian and every solve on this site applies to it unchanged, and a four-bar with play at each pin is a mechanism with eight links and eight joints.

A clearance is a link

A pin in a hole is not a joint at a point. Its centre may sit anywhere within the difference of the two radii, so the two links it joins are connected by a body of fixed length and free direction — a binary link with a revolute at each end. That is not an analogy, and taking it literally makes a four-bar a mechanism with eight links, eight joints and five degrees of freedom.

How far the crank turns before the rocker does. With a clearance of 0.01 at each pin, the crank must be turned this far on reversal before the rocker moves at all. Through most of the cycle it is about 2.8°, and at its best 1.44°. At the two positions where the rocker reverses — marked — it is unbounded: the output velocity passes through zero there, so no amount of crank rotation moves the rocker out of its clearance band. The peak in a plot like this is therefore a property of the sampling and not of the mechanism; it reads 402° at 96 samples and grows without limit as the sampling is refined. The number worth quoting is the plateau.

How far the crank turns first

Reverse the input of a four-bar with a hundredth of clearance at each pin and the output does not move for about 2.8° of crank rotation. At the two positions where the rocker reverses it does not move at all, however far the crank is turned — the lost motion is unbounded there, and the peak in any plot of it is a property of the sampling rather than of the mechanism.

The same error, twice, in two directions. A Sarrus linkage with one axis of one chain tilted off true, and the motion range that survives. Tilted within the plane the chain works in, it does not care: at 0.2 radians — eleven and a half degrees, which is not a manufacturing error by any standard — it still drives through a full turn. Tilted out of that plane, 0.001 radians stops it dead. Two hundred times the error, in the other direction, for no cost at all. What separates them is whether the perturbation lies in the screw system the mechanism leaves unconstrained — so "an overconstrained mechanism must be exact" is not merely crude, it is wrong about the case it is usually said of.

Fragility has a direction

Tilt one axis of a Sarrus linkage out of true by a thousandth of a radian and it stops dead. Tilt the same axis of the same mechanism by two hundred times as much, in the other direction, and it drives through a full turn with nothing measurably wrong. Three orders of magnitude between two errors of the same size — and the direction that matters is the one the reciprocal screw system names.

Why a hinge works

A door hinge with three knuckles is overconstrained — three axes imposed where one would do, and exactly parallel is a condition no bored hole has ever met. It works because the misfit is 0.507 times the error and the play in each knuckle is larger than that. The mechanisms this site called unbuildable are built every day, and the thing that builds them is the clearance that was already there.

The transmission angle through one turn. μ is the angle at B between coupler and rocker, computed from each solved position rather than from a formula. It runs from 54.3° to 100.3° for these lengths. The shaded band is the usual design rule — keep μ between 40° and 140° — and this linkage stays inside it throughout. The rule is about geometry alone: nothing here knows about friction, and a mechanism with a comfortable μ can still be a poor machine.

What is still outside

Eight essays here end by saying that clearance, backlash, friction or wear are not modelled. This field took two of those four, because a tolerance is a set of geometries and a clearance is a short link, and both are questions about where a mechanism can be. The other two are not, and this is the page that says exactly where the line falls and why it is where it is.

Where enumerating the corners stops being affordable. The two routes to a tolerance band, costed against the number of toleranced lengths. Enumerating every extreme combination is 2ⁿ mechanisms at every crank position; differentiating the constraints is n linear solves. A four-bar is 16 corners and a Watt six-bar is 128, which is still cheap — 142 solves for one position — and the curve is the point rather than either number: at twenty parameters, which is an ordinary spatial mechanism, the corner route is a million mechanisms and the derivative route is twenty. Both are drawn because the corner route is not merely slower, it is the one that assumes nothing, and its answer is what the cheap route has to be checked against.

Seven lengths and a hundred corners

Nothing in a tolerance analysis is about four. A Watt six-bar has seven lengths, its corner enumeration is 128 mechanisms rather than 16, and the two routes still agree to a hundredth of a per cent — but the costs have separated — 142 solves against seven. At twenty parameters, which is an ordinary spatial mechanism, it is a million against twenty.

A band with a direction in it

One whole direction of a four-bar's tolerance box does nothing. A machine made a quarter of a per cent too big all over has an output error of exactly zero — and an aluminium four-bar heated by a hundred degrees has an output error of exactly zero, while one with a steel frame has 0.076°.

Half a tooth, spent three ways. Each bar is one beat of the escape wheel: exactly half a tooth pitch, 6.0° on 30 teeth, whatever the faces are cut like. The dark part is the impulse, which is the only part that does anything to the pendulum; the pale part is the drop, in which nothing is touching anything; the short tail is the lock-in run, in which the arriving tooth drags the wheel backwards as it settles. On the arc with no draw that tail is exactly zero and the budget has two terms. On every other face it is not, and the three still sum to the half pitch to twelve figures — which is the check, since the three are computed from three different contacts.

What a drop cannot be smaller than

Two thirds of an escape wheel's travel is drop, and drop does nothing. The obvious economy is to cut it down, and it cannot be cut down, because every dimension it is made of has a tolerance and a drop smaller than the accumulated error is a tooth that does not clear the pallet it is leaving. The stack is 0.39°, and it barely moves when the tooth count triples.

Which pin's play costs the most. Each pin's clearance taken one at a time, at 0.01 on links of 1 to 4, with the direction swept rather than assumed. The ranking runs A 32%, B 25%, O₂ 22%, O₄ 21% — a spread of 1.52 against the 3.43 the four lengths spread over. Clearances are more evenly weighted than length tolerances because each pin joins two links and so enters two of the four sensitivities, which is why the best bearing buys less than the best-held length does — and why it still goes somewhere the load path does not suggest.

Which pin to buy

The four lengths of a four-bar contribute 38, 26, 25 and 11 per cent of its output error — a spread of 3.4. Its four pins contribute 32, 25, 22 and 21 — a spread of 1.5. Clearances are more evenly shared than length tolerances, because every pin joins two links and so appears in two of the four sensitivities, and that changes what a better bearing is worth.

Where the boundary moved again

The practice field's inventory ended by naming what the work after it should do first: take the feature positions as the variables and derive the lengths. That is done, and it turned out not to be an extension of the tolerance field but half of a different one — because where a length comes from and what a measurement determines are the same question.

What preload takes away, and what it leaves. The same four-bar with 0.01 of clearance at each pin, driven so that the load through every joint keeps one sign. The upper line is the lost motion it had before: about 2.80° of crank rotation thrown away on every reversal, unrepeatable, and not removable by calibration. The lower curve is what is left once the pins are held against one side of their holes — a fixed offset of between -0.330° and 0.013°, which is a dimensional error rather than play, so its average of -0.136° comes out in a calibration and only the 0.343° of variation survives. A factor of 8.2, bought with a permanent parasitic load that this site does not model.

Taking up the play

Preload does not make a clearance smaller. It takes away the clearance vector's direction, which is the property that made the error unrepeatable — so 2.80° of lost motion becomes a 0.343° offset, of which 0.136° is a constant that calibrates out. A factor of eight, bought with a permanent parasitic load this site does not model.

Tolerancing the holes

A drawing does not tolerance link lengths. It tolerances holes, and the lengths are derived from them — so what a length's tolerance really means depends on whether its two holes were bored in one setup. Located separately, a lengths-only stack-up is optimistic by 11%; bored together, it is pessimistic by 34%. Neither is a correction to apply blind.

What a belt ratio's tolerance is made of. The three ways a variator's ratio can be wrong when the parts are wrong, at a nominal ratio of 1.848: a belt 0.3% long, a centre distance 0.2 mm out, and a sheave 0.15 mm from where it was commanded. Worst case they add to 2.98% and combined in quadrature they are 2.04% — the two conventions the practice field already argues about. The last row is a gear train's ratio under the same treatment, and it is empty: there is no length in a tooth count for a tolerance to be on.

The ratio that has a tolerance

Every dimension in this collection has been given a range at some point, and the ratios never were, because a gear ratio is a count and a count has no tolerance. A belt ratio is a quotient of two solved lengths, so every length in the mechanism is in it — and the amplification from belt length to ratio runs from 2.7 to 6.4 across the travel, which puts a whole per cent on a mechanism whose gearbox equivalent has none at all.

The square a robot thinks it drove, and the two it drove. A differential drive with wheels one per cent apart in radius, sent round a four-metre square twice — once clockwise and once anticlockwise. What it believes is the square; what it did closes 1.60 m out one way and 1.40 m out the other, and the errors point different ways. A wrong track width instead gives 0.177 m and 0.177 m — the same both ways round. That difference is why the test is run in both directions: one run cannot tell the two errors apart and two runs can.

The error that is an integral

A tolerance on a link length moves an output by a bounded amount. A tolerance on a wheel radius moves a vehicle by an amount that grows with how far it has driven: one per cent of mismatch between two wheels bends a commanded straight line onto a 30 m radius, and a four-metre square comes back 1.60 m from where the machine thinks it is.

The error that is repeated

Thirty-two units cut on one setting of one machine are thirty-two copies of one error, not thirty-two draws from a distribution — so a tong's span is out by thirty-two times a unit's, not by the square root of thirty-two times it. The two estimates differ by a factor of 5.66, and the second one is the comforting one.

The same links and pins, and between 12 and 15 link lengths in the stack-up. Every closed loop in a mechanism is one equation a tolerance analysis has to satisfy, and the equation involves every link the loop passes through. All 16 chains here have the same number of independent loops — 3, which is pins minus links plus one and is fixed by the two totals — but not the same shortest set of them. The bars are the total length of a minimum cycle basis, and they run from 12 to 15. So the smallest number of link dimensions that any stack-up on this mechanism can involve is decided by the graph, before a single dimension has been chosen, and two topologies a count cannot tell apart differ by 3 of them.

Where the shortest loops are

A tolerance stack-up goes round a loop, and every link the loop passes through is a dimension in it. Two eight-link chains with the same links, the same pins and the same number of loops can need twelve link lengths in their shortest independent set or fifteen — decided by the graph, before any dimension is chosen.

Every surface tried, and the group it permits. The census the six lower pairs come out of. Each row is a surface, sampled at 240 points; the freedoms column is six minus the rank of a matrix with one row per point, saying that the velocity a twist gives that point is tangent to the surface. Nothing is fitted and no shape is recognised — the surface's own normals write the matrix down. Three different surfaces of revolution give the same group, which is the content of the classification; two surfaces give nothing, which is what almost every surface gives. Eleven surfaces, six groups. The last column is the ratio of the smallest singular value kept to the largest discarded, so a row reading 10¹⁵ is not near being reclassified by anybody's tolerance.

The pair a catalogue sells

A plain bearing is a cylindrical pair and a catalogue calls it a bearing. Add two thrust faces and it is a revolute pair, which is a different joint and changes every mobility count downstream. The kinematic identity of a bought part is decided by which surfaces touch, and the catalogue's word for it is not the same information.

One bar made 0.0001 too long, one bar at a time. Every bar of the machine compiled from a lemniscate lengthened by 0.0001 in turn, the machine re-solved, and the polynomial read at the tracing point. It is no longer zero anywhere. The worst bar takes it to 9.1e-3 — an amplification of 91 — and the median bar to 2.8e-4. The bars that matter are the reflectors, which are the cheapest part of the machine; the translators, which are most of it, barely move the answer at all. Size and fragility live in different parts.

Exactness a micron destroys

Lengthen one bar of a compiled machine by a ten-thousandth and its tracing point leaves the curve. On the smallest machine the error comes out smaller than it went in; on a fifty-bar one it comes out ninety times larger — and the bars that matter are the reflectors, which are the cheapest part of the machine.

a crank rocker with a post: the closest pair at one position. The same four-bar with a post bolted to the frame, just clear of the coupler's path. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: 0.1809 here, between coupler · post. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour.

Where the boundary moved

Three phases ago this site drew a line around what it computes and listed one thing on the far side as a gap rather than a boundary: interference between links, which needed no new physics, only a body and a test. Here is what that turned out to cost and what it turned out to open.

A clearance that has a sign, over a tolerance box. The same machine at each of the sixteen corners of a ±0.02 band on its four lengths, with the smallest gap over a whole drive computed at each. The nominal machine clears by 0.1041; the worst corner clears by 0.0763 and the best by 0.1321, a band 0.0558 wide from a tolerance of 0.02 on each length. Every other quantity computed from the lengths becomes an interval when the lengths do. This one has a sign, and an interval that reaches zero is not a wider answer to the same question — it is a different answer, because on that side of it the parts do not go together. 24 random draws inside the box beat no corner, which is the check that the extremes are where they are assumed to be.

A clearance inside a tolerance box

Every quantity derived from the lengths becomes an interval when the lengths become ranges. This one has a sign, and an interval that reaches zero is not a wider answer to the same question — it is a different answer, because on that side of it the parts do not go together.

The bores, and the one line that has to pass through all of them. A hinge of 6 knuckles, its bores drawn at the distance each was made from the nominal axis in units of the bore tolerance. The leaf is a rigid body, so its pins are on one straight line — two parameters of position and two of direction — and it assembles when some line passes within the clearance of every bore. The line drawn is the one whose largest miss is smallest, and that miss is 0.875 of the tolerance. 2 of the 6 bores are at that distance and hold the fit; the rest are slack and could have been bored anywhere inside it without changing the answer.

A piano hinge is not forty door hinges

A three-knuckle hinge works because the misfit its bore errors create is smaller than the play already in its pins. A piano hinge has forty knuckles and thirty-nine of them are redundant, so the obvious reading is that it needs thirteen times the play. It needs two and a half times, and it can never need more than the bore tolerance itself — because a rigid leaf has one axis and a line through the middle of the errors misses every bore by at most the largest of them.

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