A defect that is not kinematic

The synthesis field’s survey is one of this site’s largest single measurements. Take three prescribed poses, run Burmester’s construction over every pair of points on a grid, and count what comes back: 1,176 dyad pairs that give an exactly correct three-position synthesis, of which 176 are usable, 810 have a branch or circuit defect and 190 an order defect.

All three verdicts are kinematic. Each is found by driving the mechanism through 360 positions and watching which solutions it visits and in what sequence. Each is a property of the four lengths.

Here is a fourth, and it is found by not driving anything.

The condition #

Two pins on one link cannot be closer together than the material round them. A pin needs a boss — a disc of material surrounding the hole — so two holes on one part have to be at least one boss diameter apart, and that is the generous version, since the wall between them has to have some thickness of its own.

Nothing in any construction on this site says so. Burmester’s construction intersects two loci and returns the intersection; a pivot is a point; a point has no size. So a synthesis is free to return a crank of 0.15 units on a machine spanning 2.2, and it does, regularly.

There are four such distances per linkage: the two ground pivots on the frame, the two moving pivots on the coupler, and the crank’s and rocker’s own lengths. The smallest of the four is what decides.

The distribution #

Over the 1,176:

The shortest distance any of them returns is 0.103. A tenth of them are below 0.408. The median is 1.061 and the largest is 1.512.

So most syntheses are fine and a long tail is not, which is exactly what a construction with no size in it should produce: the answers are wherever the loci happen to intersect, and there is nothing keeping them apart.

The link that carries the closest pair is the crank 512 times, the coupler 346, the frame 249, and the rocker 69.

The frame being third is the interesting entry. Its two pins are the ground pivots — the distance drawn as the base of every diagram, looked at directly by every designer, and the one nobody synthesises a linkage without noticing. The crank is the problem twice as often, and a short crank is a normal thing to want: it is the input, it sets the stroke, and small is often the answer. A crank of 0.15 on a machine spanning 2.2 is not obviously wrong on a drawing; it is wrong when somebody has to put two holes in it.

The count, against the pin #

The defect has a parameter, so the answer is a curve rather than a number.

At a boss radius of 0.05 — a pin a fortieth of the pose span — nothing is lost: all 1,176 are buildable.

At 0.1, sixteen fail; at 0.2, 116, which is 9.9%; at 0.3, 216; at 0.4 — a pin nearly a fifth of the pose span, which is a fat pin on a small mechanism — 313, or 26.6%.

That is the honest shape of the finding. The size defect is real, it is a smooth function of scale rather than a threshold, and it is much rarer than the kinematic defects, which between them remove 85% of the exact solutions before this test is applied at all.

Why it is a curve and not a rule #

There is a temptation to convert this into a design rule — no link shorter than three pin diameters — and it is worth resisting for a reason the curve makes visible.

The share lost is not a step. It rises smoothly from nothing at a boss of 0.05 to a quarter at 0.4, with no knee anywhere: at every scale there is a population of syntheses just inside the limit and a population just outside it, and moving the limit slightly moves a few dozen candidates across. A rule of thumb picks a point on that curve and hides the fact that the curve is smooth.

What the curve says instead is a trade. Choosing a fatter pin costs a known number of the available designs, and the number is computable before any of them is chosen. That is a different kind of statement from a rule and it is the kind this site prefers: the designer sets the pin from the load and reads off what it costs, rather than being told a ratio whose provenance is nobody’s measurement.

It also makes the comparison between mechanisms possible. Two sets of prescribed poses, surveyed the same way, give two curves — and a task whose curve falls off early is a task that is hard to build at small scale, which is a property of the task rather than of any linkage answering it.

The correlation, which is the useful part #

The second curve on that figure is the one worth reading. Of the 176 syntheses that pass every kinematic test, how many also survive the size test?

At a boss of 0.05, all 176. At 0.2, 168. At 0.3, 156. At 0.4, 149.

So the usable set loses 15% at a fat pin against the whole set’s 27%, and the direction is consistent: the linkages that survive the kinematic tests are systematically roomier than the ones that do not. Their median shortest link is 1.18 against 1.06 for the population, and their tenth percentile is 0.52 against 0.41.

That is a mildly encouraging result and it should be read carefully, because it is easy to over-read in either direction. It does not say the size defect can be ignored: at a boss of 0.4 it removes twenty-seven of the mechanisms a designer would otherwise have chosen from, and each of those is a design that would have been drawn, checked, approved and found impossible at the machine shop. It does say the four tests are largely independent, so the size test is a fourth filter rather than a restatement of the first three.

Why this cannot be found by simulation #

The three kinematic defects are all found the same way: build the mechanism, sweep it, watch. A branch defect shows up as a pose reached on a different assembly; a circuit defect as a pose the sweep never visits; an order defect as poses visited in the wrong sequence.

A size defect shows up as nothing at all. A mechanism with two pins on top of each other sweeps perfectly. Every position solves to 10⁻¹⁴, the mobility is one, the transmission angle is whatever it is, the coupler curve is drawn, and the three defect tests all pass. In the equations the two pins are two points with two coordinates each, and two points can be as close as they like.

That is what makes it worth a separate name. It is the first defect on this site that a simulation cannot find, and it is found instead by asking a question about material that the simulation was never asked. Every other check in the synthesis field is a check on the motion; this one is a check on whether there is a part.

What a size-defective linkage looks like #

It is worth describing one, because the phrase two pins too close together covers two quite different drawings.

The first kind is a short link: a crank of 0.15 on a machine spanning 2.2, which on a drawing is a stub between two circles that nearly touch. It looks wrong, and a designer would query it — not because they had done this measurement but because a stub crank is odd for other reasons, being fast, highly loaded and awkward to balance.

The second kind is a pair of ground pivots close together on a long frame, or two coupler points close together on a large coupler. Those do not look wrong at all: the part is big, the two holes are somewhere in the middle of it, and the drawing is a perfectly ordinary plate. The defect is entirely local, and there is nothing about the linkage’s proportions that draws the eye to it.

The distribution above says the second kind is common: the frame is the worst link 249 times out of 1,176, and the coupler 346. So most size defects are not stub links. They are ordinary-looking parts with two holes in the wrong place, which is exactly the class of error that gets through a drawing review and stops at the machine.

The scale that a synthesis does not have #

There is a reason this cannot be fixed inside synthesis, and it is not an oversight in the construction.

A synthesis is scale-free: the prescribed poses set the size, every length that comes back is in those units, and doubling the poses doubles every linkage. Nothing in Burmester’s construction has an absolute size in it, because nothing needs one.

A pin has an absolute size, set by the load, the bearing, the material and what the catalogue sells — none of which is in the constraint equations, and all of which stay the same whether the mechanism is a hundred millimetres across or ten.

So the ratio of pin to mechanism only exists once somebody says how large the machine is, and it gets worse as the machine gets smaller. The honest statement of the finding is therefore about scale rather than about carelessness: the smaller a mechanism is, the fewer of its exact syntheses can be built, and the curve above is that statement with numbers on it.

The same measurement, one field over #

This is not the first time a count on this site has been cut down by a condition the construction could not see, and the pattern is worth naming because it is now three deep.

The topology field enumerates kinematic chains and finds that a count of graphs admits eight for every one that deserves it, because most graphs are not mechanisms.

The synthesis field finds 1,176 exact syntheses of which 176 survive three defect tests, because most exact answers are undrivable.

And the layer census finds six proper three-plane colourings of a crank rocker of which two can be built, because a colouring cannot see a pin passing through a link.

Each time the shape is identical: an enumeration produces candidates, a condition outside the formalism that produced them removes most, and the condition is invisible from inside. The formalism that generates the answers is not the formalism that judges them, and a site that runs only the first publishes a count much too large and is entirely consistent about it.

The size defect is the mildest member of the family — it removes a tenth rather than five sixths — and it is the one whose condition comes from furthest outside, since the number that decides it is not a property of the mechanism at all.

What to do with it #

The obvious response is to add the test to the survey, and that is what has been done here: the four distances are computed for every candidate and compared against a boss radius the designer supplies. It costs nothing — four distances against a number, per candidate, against a defect check that sweeps 360 positions.

The less obvious response is to put it in earlier. The size test does not need the mechanism to be swept, so it can be applied to the raw output of the construction before any defect check runs, and on a survey where 85% of candidates fail a kinematic test anyway that is the wrong order — the cheap test should run first, on all of them.

That is worth a sentence because it inverts the usual reasoning. A test is normally ordered by how likely it is to reject; here the size test rejects a tenth where the branch test rejects two thirds, and it is still the one to run first, because it costs a thousandth as much.

What is not being tested #

A wall between the bosses. The condition here allows two bosses to touch. A real link needs material between the holes, so the honest count with a wall is larger, by an amount depending on a material and a process this site does not model.

A non-circular boss. A boss is taken as a disc of a fixed radius. A real one is often a shaped pad, elongated along the link, and two elongated pads can be closer along the link than across it.

The link’s own width. A crank of 0.4 with bosses of 0.15 clears this test and is a part whose two holes are nearly touching a link that is 0.3 wide — geometrically fine, structurally nothing, and this site has no way to say so because saying so needs a force.

And whether the pins are on the same plate. Two pivots on a link in different planes are still two holes in one part, so the test stands; but a link that is built in two pieces, which is a real answer to a crowded joint, is not modelled at all.

How the survey is read differently now #

The survey’s headline has been 1,176 exact, 176 usable for two phases, and it should be read with a caveat it did not have.

The 176 is a count of linkages that pass three tests. It was never a count of linkages that could be built, and nothing in the field claimed it was — but a number that survives 85% attrition invites being read as a final answer, and it is not one. At a pin boss of 0.2 the buildable subset is 168; at 0.3, 156; at 0.4, 149.

So the honest form of the headline carries a scale: 176 usable, of which 149 to 176 are buildable depending on how large the pins are relative to a pose span of 2.2. That is a wider and less quotable sentence and it is the true one.

It also suggests where the next filter comes from, and this field has one to offer: a linkage whose free space is in two arcs reaches all its prescribed poses and cannot be driven between them, which is a branch defect produced by material rather than by lengths. Nothing here runs that test on the survey. It is the natural fifth column, it needs bodies for every candidate rather than four distances, and it would cost what the defect check costs rather than what this one does.

The fourth verdict #

The survey now has four columns and the fourth is different in kind from the others.

Usable, branch defect and order defect are all statements about motion, decided by driving the mechanism, and all three are properties of four lengths.

Size defect is a statement about material, decided before the mechanism moves, and it is a property of four lengths and one number that is not in the problem. It cannot be made scale-free, it cannot be found by sweeping, and it removes designs that every other test on this site approves.

That combination is what makes it worth having. A filter that rejects what the other filters already reject is decoration; this one rejects a different set, for a different reason, at a cost of four subtractions.

The fourth verdict differs from the other three in a way worth stating plainly, because it is the reason it took a separate field to find. The first three defects are failures of the motion — a branch that cannot be reached, an order that comes out wrong, a circuit that is separate — and every one of them is visible to a simulation, because a simulation computes the motion. This one is a failure of the material: two pins closer together than the metal round them, in a computation where a pin is a point and has no metal. So no amount of simulating finds it, however finely, because the quantity that fails is not in the model. That is a general hazard rather than a fact about pins. A model that omits a quantity cannot report a failure of that quantity, and the omissions are invisible precisely because they are omissions — there is no term coming out wrong, no residual growing, nothing to notice. The only way such a defect is found is by somebody asking a question the model was not built to answer, which is what a field about practice is for.

The fifth column, and the defect it did not find #

The survey ends in four verdicts: usable, branch defect, order defect, and the size defect this essay adds. All four are about distances — which solutions the sweep visits, in what order, and whether two pins are far enough apart to be made. None asks whether the linkage can be driven from one prescribed pose to the next without its own parts passing through each other, which is a swept clearance question and a different one.

It was left out on cost: a swept check on each of 1,176 candidates is the defect survey’s own expense again. That turns out to be the wrong arithmetic. Only the 176 that survive the kinematic verdicts can have this defect at all, and a sweep of one is sixteen milliseconds. The whole fifth column is under three seconds.

Run, it finds a great deal — and not the thing it was predicted to find.

The prediction was a linkage whose free space is in two arcs: reaching every prescribed pose and unable to be driven between them, a branch defect made of material rather than of lengths. Across the 152 survivors that can be dressed and swept, that shape does not occur once.

What does occur is larger and blunter. Seventy-seven of the hundred and fifty-two have configurations their own material forbids — the bars foul somewhere in the range — and on sixty-seven of them a prescribed pose itself is one of them. The defect is at the poses rather than between them. The linkage does not fail to be drivable; it fails to be assemblable where it was asked to be, which is a worse failure and an easier one to state.

And the fourth column is no proxy for it. At this bar width the pin-spacing test loses none while the sweep loses seventy-seven. The cheap test and the dear test are not measuring the same thing, and a survey that runs only the cheap one reports a clean sheet on a set of linkages of which half cannot be built.

Twenty-four of the 176 survivors cannot be dressed or swept at all, and they are reported as unmeasured rather than as passing. That distinction is the one a survey of this kind loses most easily: a candidate that the instrument could not read is not a candidate that passed.