Why a hinge works

Two essays ago this site had a problem it had not admitted to.

Overconstrained mechanisms need exact geometry. Bennett’s four-bar stops moving at a detune of one part in a hundred thousand. A planar four-bar, counted in space, is overconstrained too — its four axes must be exactly parallel — and exactly parallel is a condition no four bored holes have ever satisfied.

So by the site’s own argument, four-bars do not work. And there is one in every windscreen wiper.

The missing term #

The resolution is the thing the clearance essay put on the table and did not yet spend: a joint is not a point.

An overconstrained loop imposes some constraint more than once. When the geometry is exact, the copies agree and nothing happens. When a length or an axis is out by δ, the copies disagree by something of the same order — and the loop cannot close, because there is nowhere for the disagreement to go.

Give each joint a clearance and there is somewhere. The pins sit off-centre by whatever it takes, the disagreement is absorbed, and the mechanism moves.

That is the whole argument, and it is worth noticing how ordinary it is. Nothing exotic has been added: the clearance was already in every one of those joints, put there so the pin would turn, and it turns out to be doing a second job nobody specified it for.

How much play is needed #

The argument above is qualitative and could be told about any mechanism. The number is what makes it a design statement.

The misfit in Bennett’s linkage is measured as the closure residual the solver drives down to and then cannot improve on. Across four decades of detune it is exactly proportional to the detune — the ratio varies by 0.07% between δ = 10⁻⁶ and δ = 10⁻² — with a constant of 0.507.

Shared over the four joints that have play to give, each pin must absorb 0.127 δ of misfit.

Put a scale on that. A Bennett linkage with links of 1.6 cm, machined to one part in a thousand — which is 16 microns, an unremarkable milling tolerance — needs about two microns of play in each pin. A running fit on a pin of a few millimetres is tens of microns. The clearance available exceeds the clearance required by a factor of ten or more, without anybody having designed for it.

That is why the mechanism nobody can build is in every folding table, every scissor lift, every umbrella frame.

The proportionality is the part worth having #

A first-order argument predicts that the misfit should be proportional to the error, and predicting it is not the same as measuring it.

What theory cannot supply is the constant. 0.507 is a property of this linkage’s proportions — its lengths, its twists, which parameter was perturbed — and a different Bennett linkage has a different one. There is no way to obtain it except by computing the residual, and there is no way to convert “must be exact” into a specification without it.

The linearity is also the check that the computation is measuring what it claims. If the residual had grown quadratically, or saturated, or wandered, the interpretation as a first-order misfit would be wrong and the extrapolation to small δ would be unfounded. Over four decades to a part in a thousand it is a straight line of slope one, which is what a misfit looks like and is not what an artefact of the solver would look like.

Why the hinge is the right example #

A door hinge is the most overconstrained mechanism most people touch.

It has three knuckles, sometimes five. Each is a bearing on the same axis, so the hinge imposes the constraint “these two leaves share an axis” three times where once would do. In the counting of the spatial field, a three-knuckle hinge is 6(2−1) − 5·3 = −9: it should be impossible by a wide margin.

Three holes bored in a strip of brass are not exactly collinear, and they are not exactly parallel. Both errors are of the order of a hundredth of a millimetre in a hinge nobody would complain about.

The hinge works because the pin is a few hundredths smaller than the knuckles, and a few hundredths is more than a hundredth. That is the entire mechanism of it. Make the pin a press fit in all three and the hinge binds — as anybody who has hammered a slightly oversized pin into a hinge knows.

And the design detail that follows is exactly the one hinges have: the knuckles are long relative to their diameter, and the pin is long. That is the difference between the short-pin and long-bearing counts in the clearance essay — a long bearing constrains the pin against tilting, so the hinge is stiff against the direction it needs to be stiff in, while the clearance still absorbs the collinearity error. The geometry of a hinge is a solution to exactly the problem this essay is about, arrived at long before anybody wrote the count down.

The same argument, on the mechanism this site opened with #

It is worth running the argument on a four-bar rather than on a hinge, because the four-bar is what every other figure here is about and because the numbers are less forgiving.

A planar four-bar’s four axes must be parallel. Suppose one is out by an angle ε over a bearing of length L. The misfit that produces at the far end of the bearing is about εL — the pin and the hole are trying to be at an angle to one another, and the amount they must move apart to allow it is set by how long they are.

For a bearing 10 mm long and an axis out of true by 0.001 radians, that is 10 microns. A running fit on a 6 mm pin is around 20 microns. The mechanism assembles, and it assembles with room to spare.

Now make the bearing 100 mm long, as it might be on a large machine, keeping the same angular error. The misfit is 100 microns, five times the available clearance, and the mechanism binds — or, more usually, it is forced together and the bearings run hot on one edge.

That is a design rule falling straight out of the arithmetic: the longer the bearing, the tighter the angular tolerance must be, in exact proportion. It is why a long hinge has its knuckles bored in one operation, why a machine with widely-spaced bearings on one shaft has them line-bored rather than assembled from separate housings, and why nobody is surprised when a long shaft in two separate pillow blocks runs hot.

None of that is new to anybody who has built machinery. What is new is that it is the same computation as Bennett’s, with the same structure and a different constant, and that the constant is available before the part is made.

Exact constraint, as a design philosophy #

The trade in the previous section has a name and a literature, and it is worth pointing at because this field is most of the argument for it.

Exact constraint design — kinematic design, in instrument circles — says: constrain each degree of freedom exactly once, never twice. A body on six points is exactly constrained; a body bolted to a flat face is overconstrained many times over, and where it ends up is decided by which bolt was tightened first.

The case for it is precisely what this essay has been computing. An exactly-constrained mechanism has no redundant constraints, so there is no misfit to absorb, so the clearances can be chosen for their bearing duty alone — which means they can be small, which means the lost motion is small. An overconstrained one spends its clearance twice, and the accuracy is what pays.

The case against it is that overconstraint buys stiffness and load sharing, which is why a hinge has three knuckles and not one, and why a machine tool slide runs on two rails rather than on a kinematic mount. Three knuckles carry three times the load and are three times stiffer against the leaf twisting, and the price is that all three must be nearly collinear.

Neither is right in general. What this field adds is the exchange rate: the clearance required is 0.127 δ per joint for the mechanism measured here, and the lost motion it produces is proportional to it. Given those two, the trade is arithmetic rather than doctrine.

What the clearance costs #

Nothing is free, and the price is the subject of the clearance ladder.

The play that absorbs the misfit is the same play that produces lost motion, and it produces it whether or not it is needed. A mechanism given generous clearances to make it manufacturable is a mechanism with generous lost motion, and the two requirements pull in opposite directions in the most direct way possible: one wants the clearance large enough to swallow the manufacturing error, the other wants it small enough not to spoil the accuracy.

That is a genuine trade and it has a computable middle. The clearance required is 0.127 times the dimensional error; the lost motion produced is proportional to the clearance. So tightening the manufacturing tolerance lets the clearance be tightened, which reduces the lost motion — and the chain of proportionalities means the accuracy of an overconstrained mechanism is governed by its manufacturing tolerance twice over, once directly and once through the play it forces.

For a mechanism that is not overconstrained the second term is absent, and its clearances can be chosen on bearing grounds alone. That is a real advantage of an exactly-constrained design, and it is the argument behind kinematic mounts and the whole practice of exact constraint in instrument design: not that overconstraint does not work, but that it spends clearance on assembly which could have been spent on precision.

Two knuckles would have been enough #

A last look at the hinge, because the design has one more thing to say.

A hinge with two knuckles is exactly constrained if the pin is a proper fit in both: two bearings on one axis is the minimum that makes an axis, and a two-knuckle hinge does not need the argument in this essay at all. It would have no misfit to absorb and could use a closer fit, which would make it more precise.

Nobody builds them. The third knuckle is there because a hinge is loaded in a direction that tries to peel the leaves apart, and two bearings at the ends of a leaf carry that badly — the leaf between them bends. Three, with the middle one on the opposite leaf, halves the unsupported span and roughly quarters the deflection.

So the hinge is overconstrained for a stiffness reason, pays for it in required clearance, and gets the clearance for free because the pin needed some anyway. That exchange is the whole subject of this field in one everyday object, and the fact that it comes out favourably is why hinges have three knuckles and not two.

It is also a warning against reading the exact-constraint argument too hard. An exactly constrained hinge would be more precise and would sag; the overconstrained one is what the application actually wants. Precision is one requirement among several, and the reason to compute the exchange rate is to make the trade deliberately rather than to always choose one side of it.

Why nobody had to know any of this #

A closing observation about how the knowledge actually travelled, because it is unusual.

None of the mechanisms in this essay was designed with a screw system in mind. Hinges predate the mobility formulas by several thousand years. Line-boring predates Kutzbach. The rule that a long bearing needs a tighter angular tolerance was known to every millwright who ever ran one hot.

What happened instead is that the practice was arrived at by failure, kept by tradition, and only much later given a reason. The reason is what this field computes, and it changes nothing about the hinge.

What it does change is the new case. A designer meeting an unfamiliar overconstrained arrangement — a redundant leg on a platform, a mechanism with two parallel guides, a linkage with a fifth bar added for stiffness — has no tradition to draw on, and the choice is between guessing and computing. The computation is the same one, and its output is a required clearance and a set of directions that must be held.

That is what a theory is for. It does not improve the cases where experience already had the answer; it supplies an answer where there is no experience, and it does so with a constant that can be checked.

What this does not rescue #

The argument has a limit and it is worth marking it, because “clearance absorbs the misfit” could be read as a licence.

The misfit grows with the error. Play does not. So there is a perturbation beyond which the available clearance is not enough, and past it the mechanism binds, is forced together, or wears itself a clearance in the first hours of running. Where that threshold sits is exactly the arithmetic above: δ such that 0.127 δ exceeds the play.

Two things push a design towards it. Scale, because the misfit is proportional to the link length while the clearance is set by the pin diameter, and large machines have long links on modest pins. And bearing length, because a longer bearing turns the same angular error into a larger misfit — which is the same trade the previous section described from the other side, and which is why the hinge’s long knuckles are a compromise rather than a free win.

So the useful statement is not that overconstraint is fine. It is that overconstraint costs a computable amount of clearance, that ordinary joints usually have it to spare at small scale, and that the margin shrinks as the machine grows.

The count was never wrong #

One last thing to put back in place, because this field has been rough on the mobility formulas and they do not deserve it.

Kutzbach’s count of a planar four-bar in space is −2. That is arithmetically correct, and its meaning is precise: a generic mechanism with that topology, whose axes are in general position, has no motion. The four-bar escapes it by not being generic — its axes are parallel — and the formula never claimed to cover that case.

What this field adds is that the escape is not exact either. Real axes are neither generic nor exactly parallel; they are nearly parallel, which is a third case the formula also does not cover. And the mechanism that results is not the −2 one and not the +1 one; it is the +18 or +10 one with all but one of its freedoms microscopic.

So there are three mechanisms wearing one name — the generic one that cannot move, the ideal one that moves in one way, and the real one that moves in fourteen — and the formula is right about all three. What it cannot do is say which one is on the bench, and that is not a fault in the formula. It is what a number with no units cannot say.

The proportionality is the part worth having and it is worth spelling out what makes it a design tool. The misfit is 0.507 times the alignment error — a constant of the geometry, not of the hinge’s size or its material — so the whole question of whether an overconstrained assembly works reduces to comparing two lengths: the misfit the geometry produces, and the play the joints provide. Both are numbers a drawing already carries. That turns will this overconstrained thing work from a matter of experience into an inequality, and it is an inequality either side of which the answer is definite: play larger than misfit and the assembly is fine, play smaller and it binds. The reason nobody had to know this is that ordinary bearing clearances are generous relative to ordinary bore alignments, so the inequality holds by a wide margin on everything anybody builds. It stops holding when either term is pushed — a precision assembly with tight fits, or a large assembly whose alignment error grows with its span — and those are exactly the cases where overconstraint suddenly becomes a problem after a lifetime of being harmless.