Which pin to buy

A clearance is a link, so it has a sensitivity, so the four of them can be ranked exactly as the four lengths were.

The ranking comes out flatter, and the reason is structural rather than incidental.

Why a pin is not a length #

A length belongs to one link. Change the coupler and only the coupler has changed.

A pin belongs to two. The pin at A joins the crank to the coupler, so a clearance there lets the crank pin sit anywhere within c of the coupler’s eye — which shortens the crank a little, or lengthens the coupler a little, or does some of each depending on which way the pin has moved.

So each clearance enters two of the four length sensitivities rather than one. Four clearances spread across four lengths with every pin touching two of them, and the result is an averaging: the extremes of the length ranking get pulled towards the middle, and the spread falls from 3.4 to 1.5.

That is a fact about the topology of a closed chain and not about this linkage’s proportions. Any mechanism whose joints each connect two links will have a flatter joint ranking than length ranking, and the flattening is worse — meaning more complete — the more evenly the sensitivities are distributed to begin with.

What that is worth knowing #

Two consequences, in opposite directions.

A better bearing buys less than a better-held length. Halving the clearance in the worst joint removes 16% of the clearance-induced error. Halving the tolerance on the worst length removes 19% of the tolerance-induced error. Neither is dramatic, but the second is the better purchase and it is usually the cheaper one — a tolerance is a setting on a machine and a bearing is a part.

And there is less to lose by getting it wrong. With a spread of 1.5 rather than 3.4, choosing the worst joint to improve rather than the best costs a third of the benefit rather than two thirds. Joint selection is a lower-stakes decision than tolerance allocation, which is worth knowing before spending a week on it.

The honest summary is that this ranking matters less than the other one — and establishing that is a result, because the two are usually spoken of as though they were the same kind of choice.

The ranking that is usually used instead #

The usual answer to where the good bearing goes is: where the load is highest.

That is a completely reasonable answer to a different question. Load decides wear, and wear decides how fast the clearance grows over the machine’s life. It says nothing about what the clearance costs while it is there.

On this linkage the two rankings disagree. The rocker-to-frame pin carries the largest force through most of the cycle — it is the output pivot, taking the whole reaction — and it is last in the error ranking at 21%. The crank-to-coupler pin is first at 32% and carries much less.

So the two criteria point at different joints, and which one should win depends on what the mechanism is for:

• A mechanism that positions — an instrument, a machine-tool axis — wants the error ranking, because its specification is an accuracy and it is probably not running enough hours to wear out.
• A mechanism that works — a pump, a press, anything running continuously under load — wants the load ranking, because its accuracy requirement is loose and its clearances will double before anything else fails.
• A mechanism that has to stay accurate for years wants both, and the product of them: the error a joint’s clearance costs, multiplied by the rate at which that clearance grows.

This site can compute the first factor exactly and cannot compute the second at all. Wear rate depends on contact pressure, sliding distance, hardness, finish and lubrication, of which only the sliding distance is geometric — and the sliding distance is available here, from the same sweep that produces everything else.

What a fit table actually specifies #

The clearance used throughout is 0.01 on links of 1 to 4, which is a round number chosen to make the arithmetic legible. A drawing does not carry a clearance either; it carries a fit, and the clearance is a consequence.

A fit specifies a hole tolerance and a shaft tolerance separately — H7/g6 for an ordinary running fit — and the clearance is the difference between whatever hole and whatever shaft turn up. So it is not one number but a range, with a minimum at the largest shaft in the smallest hole and a maximum at the smallest shaft in the largest hole, and the two can differ by a factor of three.

Which means every quantity in this ladder is itself a range. The lost motion of a preloaded-free four-bar is not 2.8°; it is 2.8° at the nominal clearance, and across an accepted population it runs from perhaps 1.5° to 4°.

That is the tolerance idea applied to the tolerance analysis, and it does not regress infinitely: the clearance range comes off a table, the sensitivity is exact, and the product is a band on a band that can be quoted directly. What it rules out is quoting the middle of it as though it were the answer — the same error the first essay of this field is about, arriving one level up.

What the ranking is a ranking of #

Worth stating exactly, because a percentage invites over-reading.

Each figure is the largest output deviation that pin’s clearance can produce on its own, at one crank position, over every direction it might sit in. The four are then normalised to sum to one.

Three things that is not.

It is not additive with the others — the four together produce less than the sum of the four separately, because the loop has to close and that constrains which combinations of directions are reachable. The shares are shares of a total that is larger than the achieved band.

It is not the same at every crank position. The ranking above is taken at one, and like the length ranking it moves around the cycle; a mechanism working over a narrow arc should compute it there.

And it is not a wear ranking, a load ranking, or a cost ranking. It is an accuracy ranking, and this essay exists largely to keep those apart.

Doing it for a mechanism in hand #

The method, stated so it can be used on something other than this linkage.

Take the mechanism at the crank position its specification is written about. For each joint in turn, give that joint a clearance of the size the fit table says and leave the others at zero. Sweep the clearance direction through a full turn, re-solving at each step, and record the largest output deviation. Sort.

The whole thing is one loop over joints, one loop over directions, and a solve inside — which is the same shape as everything else in this field and costs a few thousand solves for a four-bar.

Two details decide whether the answer means anything.

Use the real clearances, not a common one. The ranking above gives every pin 0.01 so that the comparison is of sensitivity. A real mechanism has a different fit at each joint — a big output pivot and a small crank pin have quite different absolute play — and the ranking a designer needs multiplies the sensitivity by the actual clearance. A joint that is twice as sensitive and has half the play contributes the same.

And rank at the position that matters. The figures here are at one crank angle. For a mechanism that positions at one place, that is the place; for one that runs continuously, average over the cycle as the length allocation does. The two give different orders and the difference is not small.

The joint that is not in the list #

A four-bar has four pins and this ranking has four rows, which quietly assumes every joint is a pin.

Replace one with a slide and the accounting changes: a slider-crank has three pins and a prismatic pair, and the prismatic pair’s clearance is not a disc of freedom but a strip — the slider can lift within its guide and rock about the guide’s axis, and the two have different effects on the output.

The clearance-link model handles it, and the model of the joint is different: a slide with play is a link that can translate as well as rotate, so the count and the band both change. Nothing in this essay’s method breaks; its four rows simply stop being the right four. It is named here because a reader with a slider-crank should not read four rows and assume a fourth pin.

The direction is swept, not assumed #

A detail of the computation that is doing real work.

A clearance vector has a magnitude that a fit table specifies and a direction that nothing specifies. The ranking above sweeps the direction at each pin and takes the worst achieved output error, rather than assuming the pin sits at some convenient point of its hole.

Assuming would be easy and wrong in a specific way: the direction that maximises the output error at one crank angle is not the direction that maximises it at another, so a single assumed direction under-states the band everywhere except where it was chosen. Sweeping costs a factor of the grid size and removes the question.

It also means the numbers here are achieved rather than bounded. A first-order sum over the four clearances would give a larger figure — it does, by a factor of two and a half — because it lets all four point their worst way at once and the geometry does not always allow it.

The sliding distance, which is available #

Half of the wear question is geometric after all, and it is worth extracting because it is free.

Wear at a joint depends on contact pressure and on how far the surfaces slide against each other. The first is a force. The second is not: it is the relative rotation at that pin over a cycle, multiplied by the pin’s radius, and the relative rotation comes straight out of the same sweep every figure in this field is built on.

For a four-bar the four joints do not slide equally. The crank pin at O₂ turns a full revolution per cycle; the rocker pin at O₄ swings through its arc and back, which is typically well under half that; and the two coupler pins turn through the difference between the links they join, which can be smaller still.

So there is a second geometric ranking available — sliding distance per cycle — and it is not the same as either the error ranking or the load ranking. A joint that turns four times as far per cycle wears four times as fast at the same pressure, which frequently makes the crank pin the one that needs the bearing even though it is not the most loaded and not the most error-critical.

This site does not draw that ranking, and it could: everything it needs is in the sweep. It is named here rather than built, which is the honest form of a gap.

Three rankings, one joint each #

Collecting the argument, because the useful form is the disagreement rather than any one list.

On this linkage the error ranking puts the crank-to-coupler pin first. The load ranking puts the rocker-to-frame pin first. The sliding ranking puts the crank-to-frame pin first. Three criteria, three different joints, and each criterion is the right one for some machine.

That is not a failure to reach an answer. It is the answer: which bearing to improve is a question about what the mechanism is for, and a designer who has only ever been given the load ranking has been answering a question about wear when they were asked about accuracy.

The flatness is the transferable result #

If one thing survives from this essay it should not be the ranking, which is a fact about one linkage. It should be the reason the ranking is flat.

Every joint in a closed chain connects two links. So every joint’s clearance perturbs two of the mechanism’s dimensions, and the perturbations of any one dimension are shared among the joints that touch it. The map from joints to dimensions is many-to-many, and a many-to-many map averages.

The map from lengths to dimensions is the identity. Nothing averages, and the spread is whatever the geometry gives — 3.4 here.

That argument uses nothing about four-bars. Any closed kinematic chain has it, so joint rankings are always flatter than length rankings, and the practical consequence is general: bearing selection is a lower-leverage decision than tolerance allocation, on any mechanism, and the effort belongs on whichever intervention moves all the joints at once.

Which is exactly what preload does, and why that essay follows this one rather than the other way round.

Why this ranking exists at all #

A closing note on why a flat ranking was worth computing, since the essay’s own conclusion is that it matters less than the length ranking.

Because nobody knew it was flat. The two decisions — which length to hold tightly, which joint to bearing well — are habitually spoken of as the same kind of choice, and a designer who has internalised that the lengths spread over a factor of three and a half will reasonably expect the joints to as well. They do not, and the only way to find out was to compute it.

A negative result of that shape is worth as much as a positive one and is much less often published. It says: stop optimising here, the gains are small, go and look at the thing that moves all four at once. That is a redirection rather than an answer, and redirections are what a ranking is for.

The pattern recurs through this field. The two sensitivity routes agreeing was a non-event that established the Jacobian was sound. A crank-rocker’s tolerance being exactly linear was a non-event that established when the cheap method is safe. Both were worth the computation, and neither produced a number anybody will quote.

Where this leaves a design #

The practical shape, since a ranking is only worth having if it changes something.

Improving one bearing is a small win. What is a large win is taking the direction away entirely — preloading the mechanism so the load through every pin keeps one sign — because that removes the whole clearance contribution at every joint simultaneously and converts it into a repeatable offset. Against a 1.5 spread across the joints, a change that affects all four at once is worth more than any reallocation among them.

Which is the general lesson of a flat ranking: when the contributions are nearly equal there is little to gain from choosing between them, and the effort belongs on the thing that moves all of them together.

The flatness of the clearance ranking is the transferable result and it is worth saying what makes it flat, because the reason is structural and will hold on mechanisms this essay never sees. Every pin joins two links, so every clearance appears in two of the four length sensitivities, and a quantity that is a sum of two terms is more nearly uniform than the terms themselves. That is an averaging argument and it applies to any mechanism whose joints connect exactly two members — which is every ordinary linkage. So the general expectation is: length tolerances vary a great deal in how much they matter and clearances vary much less, on any four-bar, any six-bar, any chain of pin joints. The practical consequence is the one worth carrying. Ranking lengths is worth doing, because the spread is large and the ranking decides where to spend. Ranking clearances is worth much less, because a spread of 1.5 does not justify a specification with four different fits in it — one fit class across all four pins costs almost nothing against the best possible allocation, and saves an argument with a machine shop about four numbers instead of one.