The four lengths do not matter equally

A drawing of a four-bar has four dimensions on it. The usual thing to write next to all four is the same tolerance, because that is what the title block says and because nothing on the drawing suggests they are different.

They are different by a factor of three and a half.

What is being averaged #

The sensitivity of the output to each length is not a number; it is a curve through the turn, and each of the four curves has its own shape and its own zeros. So “which length matters most” needs a choice about what to average over, and the choice is not innocent.

Averaging the absolute contribution over every position the mechanism reaches is what the figure does. It answers: over a full cycle of operation, where does the error come from? That is the right question for a mechanism that runs continuously — a pump linkage, a quick-return shaper, anything driven by a motor that goes round.

It is the wrong question for a mechanism that has to be right at one position. A clamp that has to close on a definite dimension does not care about its accuracy anywhere else in the stroke, and the ranking at that one position can be quite different from the ranking on average — including which length has a zero there.

Both are computed the same way and this site’s figure shows the average, with the position-by-position version available by asking the same function for one angle. Saying which one is being quoted is most of the discipline.

The ranking, and why it is not obvious #

Coupler 38%, ground 26%, crank 25%, rocker 11%.

The crank being third is the surprise. It is the shortest link at 1, against a coupler of 3.5, and the instinct is that the shortest link is the least critical because an error of 0.01 on it is a full percent while the same error on the coupler is under a third of a percent. That instinct has the wrong quantity in mind: what matters is not the fractional error in the length but the absolute movement of the output, and the output does not care what fraction of a link an error was.

The ground length being second is the one most often got wrong in practice, and for a reason that has nothing to do with kinematics. The ground “link” is not a part. It is the distance between two holes in a frame or a casting, and it is dimensioned on a different drawing from the links, often by a different person, and frequently to a coarser tolerance because a casting is a casting. The mechanism does not know that. It responds to the distance between its two fixed pivots exactly as it responds to the length of a bar, and rather more strongly than it responds to its rocker.

This is where the second route to a sensitivity earns its keep, incidentally: the ground length is the one whose derivative has two parts, because moving O₄ moves the rocker’s constraint and moves the origin the output angle is measured from. An analysis that computes only the first part reports the ground as the least critical of the four instead of the second most.

What the ranking is worth in money #

The point of a ranking is that tolerances cost money and the cost is steeply non-linear. Halving a tolerance does not double a cost; depending on the process it can multiply it by three or four, because at some point it stops being achievable by the operation that made the part and needs a second operation.

So the interesting arithmetic is not “how tight must everything be” but “given a fixed budget, where does it go”. With contributions of 38, 26, 25 and 11 per cent, tightening the coupler alone by a factor of two removes 19% of the band. Tightening the rocker alone by a factor of two removes 5.5%. Same money, if the two parts are similar; three and a half times the benefit.

Run the other way, the same numbers say something a designer likes even better: the rocker’s tolerance can be loosened by a factor of two at a cost of 5.5% of the band, which may well pay for the coupler’s being tightened. Reallocation is usually cheaper than tightening, and it is invisible without a ranking.

The shares are shares of a band, not of a length #

A confusion worth clearing, because the percentages invite it.

“The coupler contributes 38%” does not mean that 38% of the output error comes from the coupler being wrong in some particular batch. It means that if all four lengths are held to the same tolerance, 38% of the resulting worst-case band is attributable to the coupler’s term. Change the four tolerances to be different from one another and the shares change immediately — that is the entire point of computing them.

So the shares are a starting point for an allocation and not a description of one. After a reallocation the shares will have moved, and a well-allocated design tends towards equal shares, because that is the condition under which no further shifting of the budget helps. A drawing whose four contributions are 25% each is not one where the four lengths matter equally; it is one where the tolerances have been chosen so that they do.

That gives a useful way to read any existing drawing. Compute the four contributions at the tolerances actually specified. If they come out wildly unequal, the drawing is spending money in the wrong place, and the fix is arithmetic rather than judgement.

A worked reallocation #

Take the four at ±0.01 and a total band of 0.73° at the worst position. Suppose the requirement is 0.5°.

Tightening all four to ±0.0068 gets there and costs four tighter tolerances.

Tightening only the coupler and the ground — the two largest, at 38% and 26% — to ±0.004 while loosening the rocker to ±0.02 gets to almost exactly the same band, costs two tighter tolerances, and refunds one. Whether that is a better deal depends on the parts, and the calculation that lets anybody ask is four numbers long.

Neither of those is presented here as the answer. The point is that the second option is invisible to a designer holding a single band, and obvious to one holding four contributions.

The ranking changes with the mechanism, not with the subject #

None of these percentages is a fact about four-bars. They are facts about this four-bar — 4, 1, 3.5, 3 — and they move when the proportions do.

That is worth being blunt about, because the temptation with a table like this is to remember it. There is no rule of thumb here to carry to the next linkage. The coupler is the largest contributor for these lengths at this duty; a linkage proportioned differently can put the ground first, and one operating over a short arc near a limit position can put whichever length happens to have its largest sensitivity there.

What generalises is the method and one structural observation: the contributions are never equal, so there is always something to allocate. Across every set of proportions this site has looked at, the ratio between the largest and smallest contribution has been at least two. A drawing that tolerances all four equally is therefore never optimal — only, sometimes, not very wrong.

Why the crank is third, in detail #

The crank’s position in the ranking is the one worth working through, because the reasoning generalises and the intuition it defeats is strong.

The crank is 1 unit long. An error of 0.01 on it is one per cent of the part. The coupler is 3.5 and the same error is 0.29 per cent. Every instinct trained on percentages says the crank is the critical part.

What the output responds to, though, is not a percentage. Follow the geometry: the crank pin sits at a distance a from the fixed pivot at the crank angle, and moving a by δ moves that pin by δ, straight out along the crank. That displacement then propagates through the coupler to the rocker pin, and how much of it survives depends on the angle between the coupler and the rocker — which is the transmission angle again.

So the crank’s contribution is δ times a geometric factor that has nothing to do with the crank’s own length. Exactly the same is true of the coupler and the rocker. The reason the coupler comes first is not that it is longest; it is that a change in coupler length displaces the rocker pin along a direction that happens to be well aligned with the rocker’s motion through most of this linkage’s cycle, while a change in rocker length displaces it along the rocker itself, which is the direction the rocker’s angle is least sensitive to.

Put crudely: an error in the rocker mostly makes the rocker the wrong length, and an error in the coupler mostly makes the rocker point the wrong way. Only the second is an output error, and that is the whole of why the ranking is not the ranking of the lengths.

Sensitivity is not the whole of an allocation #

Two things this ranking leaves out, both of which a real allocation has to put back.

The first is that tolerances are not equally achievable. A length that is the distance between two holes bored in one setup on one machine is far easier to hold than a length that is the distance between two holes in separate castings joined by a bracket. The ranking says where accuracy is worth having; it does not say where it is cheap. The allocation is the two together, and the second half is a manufacturing question this site has no way to compute.

The second is that the four errors are not independent, and the figure quietly assumes they are by adding contributions up. Where two lengths are produced by one operation — two holes on one fixture, two bearing seats bored together — their errors move together, and a pair of contributions that partly cancel in the independent case can add in the correlated one. The next essay is about exactly that assumption, because it is also what separates the two ways of adding tolerances up and it is worth a factor of two.

Why the average is over reachable positions only #

A small piece of care in the computation, which is the sort of thing that decides whether a number means anything.

The average is taken over the crank positions at which the mechanism assembles — and for a crank-rocker at this tolerance, that is all of them. Had some corner linkage failed to reach some angle, that position would have contributed nothing to the average rather than contributing an infinite error, and the ranking would then be an average over a set that depends on the tolerance. That is a subtle way for a number to become circular, and the generator asserts that the reachable count is nearly the full sweep so that the average is over the same positions for every corner.

It matters more than it sounds for a mechanism near the Grashof boundary, where a tolerance can genuinely change the classification: a linkage whose shortest plus longest is within a hundredth of the other two can come off the machine as a crank-rocker or as a double-rocker, and no average over “positions it reaches” is meaningful across that divide. Grashof’s condition is exact and it is exact about four numbers that were never exact, which is a sentence worth sitting with.

What a ranking looks like when the mechanism is nearly singular #

One case where the ranking stops being a useful object at all, and it is worth knowing how to spot.

A contribution is a derivative times a tolerance. At a change point every one of the four derivatives diverges, so all four contributions go to infinity together — and their ratios go to whatever the leading terms happen to give, which is numerically meaningless. A tolerance allocation run on a mechanism that passes near a change point will produce four percentages, they will look perfectly ordinary, and they will be a report on the rounding error in a singular matrix.

The symptom to watch for is not in the ranking. It is in the total: if the summed first-order band and the measured corner band disagree, the analysis has wandered somewhere the derivative does not describe, and no amount of care with the percentages will help. That comparison costs sixteen solves at one position and it is the cheapest insurance in this field. The essay on where a stack-up stops working is about what to do instead.

Doing it on a mechanism with more than four lengths #

Nothing in the method is about four. It is worth saying what changes and what does not, because the four-bar is small enough that the ranking could almost be guessed and nothing larger is.

A Watt or Stephenson six-bar has seven independent lengths. Its output responds to each of them, the responses are curves through the turn as before, and the contributions rank in an order that no amount of staring at the drawing produces. Seven contributions also means the statistical combination can save more — up to √7 rather than √4 — so the two techniques both get more useful as the mechanism grows.

The cost of the ranking grows linearly: one linear solve per length, per position. The cost of the corner enumeration grows as 2 to the power of the number of lengths, which is 128 for a six-bar and hopeless for a spatial mechanism with twenty parameters. That crossover is the practical reason the derivative route exists, and it arrives at about six parameters.

What does not change is the discipline. The ranking is only meaningful where the linear model holds, and a six-bar has more ways to be near a singularity than a four-bar does, not fewer. The comparison against a handful of corner solves — not all 128, just enough to check — stays the thing that separates an allocation from a spreadsheet.

A note on what “matters” means #

The word has been doing a lot of work and it is worth pinning before it wanders.

A length matters, here, if changing it moves the output angle. That is one definition among several, and the others give different rankings.

A length also matters if changing it moves the transmission angle, which decides how hard the mechanism has to be pushed. It matters if changing it moves the mechanism’s Grashof classification, which decides whether it works at all. It matters if changing it moves a coupler point off the curve it is supposed to trace.

Those are four different sensitivity problems with four different answers, and all four are computed the same way: pick the output quantity, differentiate the constraints, chain-rule through. This essay picks the output angle because it is the commonest specification, not because it is the only one. A mechanism specified on its coupler curve should rank its lengths against the coupler point’s displacement instead, and will get a different order.

The method survives the change of question. The table does not.

The one-line version #

A tolerance analysis that returns a single band tells a designer whether the mechanism is good enough. A tolerance analysis that returns four numbers tells them what to do about it if it is not. The second costs four linear solves more than the first, and it is the only one of the two that has ever changed a drawing.