A clearance is a link

The first half of this field gave the lengths ranges. This half gives the pins holes, and it turns out to be the same problem wearing a different hat.

The model, and why it is exact #

A revolute pair, as the constraint-counting field defines it, is two bodies sharing an axis: three planar freedoms minus two constraints leaves one relative rotation.

That definition requires the pin and the hole to have the same diameter. A pin that is exactly as big as its hole does not turn; it is an interference fit, which is a fastening rather than a joint. Every pin that turns is smaller than its hole, and the difference c = R − r is not an imperfection in a revolute pair — it is what makes the pair a pair.

Once c > 0, the set of relative positions the two bodies can take is: any rotation, together with any translation of magnitude at most c in any direction. Compare that with the set of relative positions produced by a rigid bar of length c with a revolute at each end: any rotation of the far body, together with a translation of magnitude exactly c in any direction.

Those are the same set, up to the difference between “at most c” and “exactly c” — which is the difference between the pin floating anywhere in the hole and the pin resting against the wall. A loaded joint has its pin against the wall, so the bar model is right for anything under load and conservative for anything not.

This is the standard clearance-link model, and its virtue is that it needs no new machinery. It is a mechanism made of links and revolutes, so the solver positions it, Grübler’s formula counts it, and the constraint Jacobian measures it, all unchanged.

What it does to the count #

Each clearance joint replaces one revolute by two revolutes and a link. In the plane that is one more link and one more joint, so Grübler’s count gains 3 − 2 = +1 per clearance joint.

A four-bar with play at all four pins therefore has eight links, eight joints and five degrees of freedom.

Nothing about that count is wrong. It is Grübler’s formula, applied correctly, to the mechanism that comes off the machine rather than to the one on the drawing. The four extra freedoms are perfectly real: a built four-bar can be pushed sideways at its coupler by a few hundredths of a millimetre with the crank held, and that displacement is one of them.

What the formula cannot say is that four of the five freedoms are a hundredth of a millimetre wide and one is a full revolution. A mobility count has no units. It reports the dimension of a space of motions, and the dimension of a space says nothing about its extent.

That is a limitation and not an error, and it is a useful one to have made explicit, because the whole of this field is about quantities that a count declares equal and a measurement does not.

In space it is worse, and the worse is the interesting part #

Counted in space, a planar four-bar is an overconstrained mechanism: six freedoms per body, five constraints per revolute, giving 6(4−1) − 5·4 = −2. The formula says it cannot exist, and the reason it does exist is that its four axes are exactly parallel, which is a non-generic condition the count cannot see.

Give those four joints clearance and the count changes completely, and by how much depends on what kind of pair a loose pin is.

A short pin rattling in a hole can tilt, so it is close to a sphere in a socket: three freedoms at each end of the clearance link. A long pin in a long bore cannot tilt; it is close to a cylindric pair, with two. The counts come out at 18 and 10 respectively, of which four in each case are idle spins of the clearance links about their own axes and count for nothing.

So the mechanism the spatial count declares impossible is, once its joints are real, wildly over-mobile instead. Both counts are arithmetically correct and neither describes a four-bar. What describes a four-bar is the one motion of full size and the fourteen or six of microscopic extent, and no formula that returns an integer can tell those apart.

The gap between the short-pin count and the long-bearing count is worth keeping, because it is a design lever rather than a modelling ambiguity. A bearing whose length is several times its diameter really does constrain a shaft against tilting, and a washer-thin one really does not. That is why a door hinge is a long tube rather than a thin ring, and it is the whole reason a hinge works at all.

Where the model stops being exact #

Three places, and naming them is the price of taking the model literally.

The pin is against the wall or it is not. The bar model fixes the clearance vector’s magnitude at exactly c; the real pin’s centre can be anywhere in a disc of radius c. Under load the pin is against the wall and the two agree. Unloaded — a follower on the return stroke, a linkage being positioned by hand — it is somewhere inside, and the bar model over-states the displacement. Which case applies is a question about force, so this site models the loaded case and says so.

Contact is a line, not a point. A pin and a hole touch along a line down the bore, and if the pin is slightly tilted they touch at one end of it. That is a three-dimensional effect and it is exactly the difference between the short-pin and long-bearing counts above. The planar model has no way to represent it.

The hole is not round. A bored hole is round to a tolerance of its own, and a worn one is not round at all — it wears into an oval along the direction the load comes from. The clearance is then a function of direction, which the model treats as constant. For a new joint this is a small correction; for an old one it is the dominant term, and it is what makes wear show up as a changing error rather than a growing one.

None of the three is a reason to prefer a vaguer model. They are the reason to say what the model is: it is the geometry of a round pin resting against a round hole under a load whose direction is not specified.

The output band it produces #

The kinematic consequence, computed the same way as a length tolerance: sample the clearance directions, re-solve, take the extremes.

A clearance vector at a joint splits into a component along the link it joins and a component across it. The along-component is exactly a length error — it makes the bar effectively longer or shorter — so the machinery of the first five essays applies to it unchanged. The across-component displaces the joint sideways, which for a bar of length L tilts it by roughly c/L, and that has its own sensitivity.

That decomposition is why this field is one field rather than two. A tolerance and a clearance are the same kind of perturbation; they differ in that a tolerance is fixed once the part is made and a clearance is free to take a different value every instant.

Measured at one crank position with a clearance of 0.01 at each pin, the achieved output deviation is 0.0092 radians and the first-order bound is 0.0230. The bound is loose by a factor of 2.5, and it is loose for a good reason: it assumes all four clearances can point their worst way at once, and the geometry does not always allow it. An inequality that is not tight is still an inequality, and here it is the difference between a mechanism that meets its specification and one that is rejected on paper.

Free direction is what makes it different #

The one property a clearance has that a tolerance does not is worth stating on its own, because everything in the rest of this ladder follows from it.

A length error is fixed. Once the part is cut, that bar is 3.51 and it is 3.51 for the life of the machine. The output error it causes is a fixed offset, repeatable, and correctable by adjusting something else.

A clearance direction is not fixed. It changes every time the load reverses, every time the mechanism passes a position where the force through that joint changes sign. So the error it causes is not repeatable, and no adjustment removes it.

That distinction is the difference between a mechanism that is inaccurate and one that is imprecise, and they are addressed by completely different means. Inaccuracy is calibrated out. Imprecision is not — it is designed out, by preloading a joint so that the pin is always against the same side of the hole, which is what a spring-loaded follower or a split anti-backlash gear is for.

It is also why the coupler point of a built linkage does not trace a curve. It fills a band, and the band’s width varies around the curve by a factor of two — the mechanism is stiff against its clearances at some positions and not at others.

That factor of two matters more than it looks for the straight-line linkages. Watt’s and Chebyshev’s are judged on the flattest part of their stroke, which is the part a designer uses — and it is not necessarily the part where the band is narrowest. A linkage whose deviation from a straight line is 9% of its span, built with joints whose clearance contributes a band of comparable size, has an accuracy that is not the accuracy of its coupler curve.

The magnification in the figure is the honest part #

A detail of presentation that is really a matter of method.

A clearance is between a hundredth and a thousandth of a link length. Drawn to scale in a figure of a four-bar it is less than one pixel, which means every drawing of a clearance ever published is magnified. The question is only whether it says so.

The hero figure above draws the pin and hole at a ratio of 0.86 to 1, which is a clearance of 14% — nothing like a real fit, which is nearer 0.1%. It is drawn that way because a figure of a joint at a real fit is a picture of a circle. The generator takes the magnification as a parameter and states it in the caption, so the reader can see the geometry and knows they are not seeing the proportion.

That distinction runs through this whole field. The band round a coupler curve is drawn at a clearance of 0.012 on links of order 1 to 4, which is about ten times a good fit and about right for a worn one; at a new-machine clearance the band would be a line. Every such figure names its clearance, and the arithmetic is linear, so a reader who wants the band at a tenth of that can divide by ten and be exactly right.

What is not honest, and does not appear here, is a figure drawn at an exaggerated clearance with the exaggeration unstated. That is the picture that teaches somebody that a linkage rattles, when what a linkage does is move a hundredth of a millimetre.

Play is not the same thing as slop #

A word about vocabulary, because two ideas share a name in ordinary speech and separating them is most of what an engineer knows about joints.

Clearance is designed. It has a number on the drawing, it comes from a fit table, and a joint with too little of it seizes when the shaft warms up or the housing is bolted down slightly out of true. Every running fit in every machine has it, and the amount is chosen against the shaft diameter, the speed, the lubricant and the expected temperature swing.

Wear is what happens to it afterwards. It is not designed, it grows, and it grows fastest where the load is highest and the sliding is longest — which for a linkage is usually not the joint with the largest clearance to begin with.

The kinematics of the two are identical. A joint worn to 0.03 behaves exactly as a joint made with 0.03 of clearance, and every figure in this ladder applies to both. What differs is that one is a specification and the other is a history, and a mechanism’s accuracy over its life is set by how fast the second overtakes the first.

This site computes the geometry and cannot compute the history. Wear rate depends on contact pressure, surface finish, hardness and lubrication, none of which is kinematic. What can be said kinematically is which joints matter — a clearance in the joint with the largest sensitivity costs the most output error — and that is the same ranking the tolerance allocation produced, computed the same way, for a different reason.

The practical consequence is a design rule that falls out of the arithmetic rather than out of experience: put the best bearing where the sensitivity is highest, not where the load is highest. Those are frequently different joints, and only the second is obvious from the drawing.

One more consequence: the mechanism has no repeatable zero #

A short observation that follows from everything above and is easy to miss.

A mechanism with no clearance has a definite output for a definite input. A mechanism with clearance has a set, and which member of the set it is in depends on the history of the loads — which way it was last driven, whether it has been stopped and restarted, whether the load reversed during the last stroke.

That means a built linkage does not have a repeatable relationship between input and output, even in principle, and no amount of care in manufacture removes it. Two identical machines given identical commands can be in measurably different places, and the same machine given the same command twice can be too.

For a mechanism doing bulk work this is irrelevant. For anything that positions — a machine tool axis, a robot, an instrument — it is the dominant error term, and it is the reason positioning mechanisms are built around preloaded joints, recirculating-ball screws with an interference fit, and elastic flexures with no sliding contact at all.

A flexure is worth a moment because it is the limiting case of this whole essay: replace every pin with a thin elastic strip and the clearance becomes exactly zero. The mechanism then has a perfectly repeatable input-output relation, no lost motion at all, and no wear — and in exchange it has a very limited range of motion and a restoring force it did not ask for. That is a straight trade of range against repeatability, and the fact that both ends of it are occupied by real machines says the trade is a real one.

What comes next #

Three consequences of the free direction, each with its own essay.

The band is not the same at every crank angle, and at the two positions where the output velocity passes through zero the input can be turned a long way with no output motion at all — lost motion, which is unbounded at those positions and about 2.8° everywhere else.

For a mechanism that is overconstrained, the clearance is not a defect but the thing that makes it work: the redundant constraints stop agreeing when the geometry stops being exact, and the play is where the disagreement goes.

And in a gear mesh the same quantity is called backlash, is specified on the drawing, and is bought with a centre distance rather than tolerated.