A pin is not a point

A joint on this site is a name and two coordinates. m.joint("A", x, y) and there it is, with a constraint attached to it and nothing else.

That model has done twenty-three fields of work and it has already been extended once. The practice field makes a pin a hole with play in it — a short link with a free direction, positioned by the same solver and counted by the same formula — which is what a pin is when the question is how much lost motion a chain has.

This field needs the other half of the same object. A pin is also a cylinder with material round it, and the material has to go somewhere.

Three sizes, and what each one blocks #

A pin has a diameter, and the diameter is the least of it. What matters geometrically is three quantities that follow from it.

The boss. The material of a link has to surround the pin, so the end of a link is a disc of some radius round the hole. In this field that radius is 1.35 times the link’s half-width — a proportion rather than a law, chosen so that a bar is a little wider at its ends than in its middle, which is what a real bar is. The boss is what makes a link’s outline an octagon rather than a rectangle, and it is what sets the footprint’s margin rather than the width does.

The length. In a layered machine a pin joins a link in one plane to a link in another, so it is a column spanning from the lowest plane it joins to the highest. That length is what makes the ordering condition bite: anything in a plane between the two, whose material covers the pin’s position, is pierced by it.

And the head. A pin has to be retained — a head, a circlip, a washer — and whatever retains it sticks out beyond the outermost plane it reaches. This field does not model it, and it is worth naming as the reason the plane count is a lower bound on how deep a machine is rather than a description of it.

Two pins on one link #

The first constraint is the simplest and it is the one that reaches furthest back into the site.

Two pins on the same link cannot be closer together than the material round them. A link with two holes 0.2 apart, each needing a boss of radius 0.15, is a link whose two bosses overlap: there is no part there, only a sketch of one.

Nothing in any construction on this site says so. Burmester’s construction returns pivots, and a pivot is a point. The three-position synthesis intersects two loci and takes the intersection. The cognate theorem produces two more linkages from one and says nothing about their proportions. Every one of them can return a mechanism whose crank is shorter than its own bosses, and every one of them does, regularly.

The distribution above is the measurement: over the 1,176 exactly correct three-position syntheses of the synthesis field’s own survey, the closest pair of pins on any one link. The shortest is 0.103 on a mechanism whose prescribed poses span 2.2 units. A tenth of the survey is inside 0.41. That is a link shorter than the boss a modest pin needs, produced by a construction that is exactly correct at all three positions.

The consequences of that for synthesis are an essay of their own. What matters here is the object: a pin with a size turns a distance between two points into a constraint on the design, and it is a constraint no construction in the subject enforces.

Which link is worst, and why it is the wrong one #

Among those 1,176 syntheses, the link carrying the closest pair of pins is:

the crank 512 times, the coupler 346, the frame 249, and the rocker 69.

The frame is third, and that is the interesting entry. The frame’s two pins are the two ground pivots, and their spacing is the one distance in a four-bar that a designer looks at directly — it is where the machine is bolted down, it is drawn as the base of every diagram, and nobody synthesises a linkage without noticing that its ground pivots are on top of each other.

The crank is where the problem actually is, twice as often, and a short crank is a perfectly normal thing to want: it is the input, it sets the stroke, and a small one is often exactly the answer. A crank of 0.15 units 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.

What “closer than the material round them” means exactly #

The condition needs stating carefully, because there are two versions of it and only one is right.

The wrong version is that two pins must be further apart than one boss radius. That would let two bosses of radius r sit r apart, overlapping by r, with the material of one hole running into the other.

The right version is that two pins on one link must be at least two boss radii apart — one boss diameter — so that the two discs of material do not overlap. And even that is generous: the wall between two holes has to have some thickness, so a real rule is a boss diameter plus a wall, and the numbers here are the optimistic version of the constraint.

That matters for how the results are read. When the survey below says that at a boss radius of 0.2 a tenth of the exact syntheses are unbuildable, it means a tenth are unbuildable even allowing the two bosses to touch. The honest count with a wall between them is larger, and it is larger by an amount that depends on a material and a process this site does not model, which is why the optimistic bound is the one reported.

The scale that a synthesis does not have #

There is a reason the pin question cannot be answered inside synthesis, and it is not an oversight.

A synthesis is scale-free in one specific way: the poses set the size of the mechanism, and every length that comes back is in the same units. Double the poses and every linkage doubles. Nothing in the 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 is in the catalogue — none of which is in the constraint equations, and all of which are the same whether the mechanism is a hundred millimetres across or ten. So the ratio of pin to mechanism is a number that only exists once somebody says how big the machine is, and it gets worse as the machine gets smaller.

That is the honest form of the finding, and it is a statement about scale rather than about synthesis being careless: the smaller a mechanism is, the fewer of its exact syntheses can be built, and the fraction lost is a curve rather than a threshold.

What the boss does to a link’s outline #

The second constraint is quieter and shows up in every figure in this field.

A bar of half-width w with bosses of radius 1.35w is not a rectangle. Its ends are wider than its middle, its corners are cut, and its material extends past each pin by the boss radius — so a link of “length 1” occupies 1 + 2 × 0.216 = 1.43 units along its own axis at the standard width.

Three of this field’s numbers come out of that and not out of the width:

The footprint margin of 0.43 per side is two bosses, not two half-widths. The crank’s swept disc has radius 1.163 rather than 1.16, because its furthest point is the octagon’s corner beyond the boss. And the scissor stack folds to exactly two boss radii per stage — 0.0949 at w = 0.035 — because what stops the fold is two bosses meeting.

A model with square-ended bars would get the first of those wrong by 30%, the second right by accident, and the third wrong entirely.

Where else a size could have been given #

The pin is not the only thing on this site that is a point and should not be, and it is worth listing the others so that the choice to model this one is visible as a choice.

A coupler point is a point, and it is right that it is: a tracing point is a place on a body, and the curves field is about where it goes. If the point is a pen or a probe it has a size, and the size would matter for whether the pen fits where the curve goes — but the curve itself is a fact about a point of a moving body.

A contact in the meshing field is a point, and there it is a genuine idealisation: two curved bodies touch at a point and the real contact is a patch whose size is decided by load and elasticity. That is force, and it stays outside.

A ground pivot is a point, and this field has already given it a size twice over — as a pedestal, which is what carries it, and as a boss on whatever links reach it.

So the pin is the one that was chosen, and the reason is that its size is decided by geometry rather than by physics: a hole needs material round it, and how much is a proportion rather than a stress calculation. Everything else on the list needs a force to say how big it is, and this site does not have one.

The pin as a column #

The third constraint is the layers essay’s and it is worth restating from the pin’s side.

A pin joining planes 1 and 3 is a column through plane 2. Every link in plane 2 whose material covers that pin’s position at some configuration would have to have a hole in it. So a layer assignment is not a colouring: it is a colouring plus an ordering, and the ordering is about the pins rather than about the links.

The measurement is a sweep with a point-in-polygon test: at each configuration, for each joint, is that joint inside any link other than the ones it belongs to. A four-bar has two such pins, Peaucellier’s cell one. And it is those pins that make four of the four-bar’s six proper three-plane colourings unbuildable.

Note what the pin is doing in that argument. It is not colliding with anything — a pin is not in the collision test at all, since it belongs to two links and is excluded as a joined pair. It is occupying a line through the stack, which is a third kind of geometric object this field needs and neither the gap test nor the conflict graph provides.

Two models of one pin #

It is worth putting this field’s pin beside the practice field’s and noticing that they are complementary rather than competing.

The practice field’s pin is a hole with play: the pin is smaller than the hole by a clearance, the joint has a small free motion, and the consequence is lost motion accumulating along a chain. It models the pin as smaller than its nominal, and everything it computes is about slop.

This field’s pin is a cylinder with material round it: it has a diameter, the link has a boss, and the consequence is that two pins cannot be close and a pin cannot pass through a link. It models the pin as larger than a point, and everything it computes is about room.

Both are true of the same pin, and neither is derivable from the other. A joint with generous clearance and a tiny boss has lots of slop and takes no room; a joint with an interference fit and a fat boss has none and takes plenty. They are different quantities about different failures, and this site now has both.

What a size defect is #

The name is worth fixing, because the site already has three defects and this is a fourth of a different kind.

A branch defect, a circuit defect and an order defect are all kinematic: they are about which solutions a sweep visits and in what sequence. Each of them is found by driving the mechanism and watching. All three are properties of the lengths.

A size defect is not found by driving anything. It is a comparison between two distances — how far apart two pins are, and how much material they need — and it is decided before the mechanism moves. No amount of sweeping reveals it, and no sweep is disturbed by it: a mechanism with two pins on top of each other sweeps perfectly, because in the equations they are points.

That is what makes it worth a 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.

The order the three constraints bite in #

For a designer the three sizes do not arrive together, and knowing which one binds first saves work.

The boss binds first on small mechanisms. Two pins on a short link is the earliest failure as a machine is scaled down, because it is a comparison between a mechanism length and an absolute size, and mechanism lengths shrink while pins do not.

The column binds next, on machines with several links. It needs at least three planes to be an issue at all, since there has to be something between the two planes a pin joins — so a four-bar has it and a two-plane machine like Chebyshev’s linkage cannot.

And the head binds last, or rather binds always and quietly. It adds a fixed amount to the depth of every stack and is the same for a four-bar and for a scissor lift.

The first two are computed here. The third is named and left, and the reason is the honest one: modelling it would mean carrying a retention scheme through every figure in the field to add a constant to one number.

What is not modelled #

The pin’s own strength, which is force and is outside.

The fit, which is the practice field’s and is a different quantity.

A stepped or blind pin, which is the standard workshop answer to a piercing conflict and which relaxes the ordering condition into a statement about diameters. Modelling it would need a pin to carry a profile rather than a position.

And the head, which sticks out past the last plane and is why a plane count is a floor rather than a depth.

Each of those would make some machine here buildable that this field calls impossible, and none of them changes the shape of what has been found: a construction that returns points has no way to know how much room a point needs, and the amount it needs is set outside the construction entirely.

The general point behind this rung is worth stating because it recurs wherever a synthesis returns coordinates. A synthesis has no scale. It returns points — pivot locations, link lengths, ratios — and every one of them is dimensionless in the sense that scaling the whole answer gives another answer to the same problem. The moment a pin has a diameter, the answer stops being scale-free: a linkage twice the size has the same geometry and twice the clearance between its pins, so the same design is buildable at one scale and not at another. That is a genuinely different kind of constraint from anything the synthesis itself contains, and it explains why the failures in this rung look like they come from nowhere. They come from the one dimension the synthesis was never given. The first question to ask of a synthesised mechanism is how big it is, and the second is whether the parts fit at that size — and both are outside the computation that produced it.