The hole the machine needs

Every mechanism on this site can be asked how much room it takes, and until this field the only available answer was wrong in a specific and quantifiable way.

The available answer is the joint box, which is what every sweep on this site since the first four-bar could have produced: sweep the mechanism, collect every joint position at every configuration, take the bounding rectangle. It is a perfectly good number and every field could have computed it. It is also the extent of a set of points, and a machine is not a set of points.

The two boxes #

For the site’s crank rocker over a full turn, the joints need a box 5.00 × 3.94 and the material needs 5.43 × 4.37. Each side has grown by 0.43, which is not the link half-width of 0.16 but a little more than twice it — and the reason is worth having, because it is the difference between an estimate and a measurement.

A bar’s material extends past its pin by the boss radius — a pin is not a point —, 0.216 here, not by the half-width. It extends further at the corners of the octagon, and further again on a link whose end sticks out past its joint. So the growth of the box is not the width, is not the half-width, and is not a fixed multiple of either: it is whatever the furthest corner of the furthest link happens to reach, which depends on which link is at the extreme and how it is oriented when it gets there.

In area the two boxes are 19.70 and 23.71, a ratio of 1.203. Across the catalogue the ratio runs from 1.13 on Peaucellier’s cell to 1.36 on the slider crank, on machines whose links are around a twentieth of their length wide.

Why the ratio is not the same everywhere #

The spread across the catalogue says something a single number would not, and it is a fact about shape rather than about width.

Peaucellier’s cell is the least affected, at 1.133. Its joint box is 5.00 × 10.04 — it is a tall machine — and adding a fixed margin to a large box is a small proportional change. The slider crank is the most affected, at 1.359, because its joint box is 5.00 × 2.00: it is a thin machine, and the same fixed margin is a fifth of its short dimension.

So the ratio is essentially the perimeter-to-area effect of a margin, and the margin is roughly constant across the catalogue because the width is. Which gives the useful form of the rule: the cost of material to a mechanism’s footprint scales with its perimeter, so long thin machines pay proportionally more than compact ones. A designer laying out a long slider mechanism in a shallow space is exactly the person for whom the joint box is most misleading.

The number nobody had: how much of the box is used #

The box is an upper bound on where the machine is, and a loose one. The occupancy grid of the previous essay gives the region actually occupied, and the ratio of that to the material box is the fill.

The crank rocker fills 0.54 of its own box. Watt’s linkage, 0.55. The slider crank, 0.57. At the other end, Chebyshev’s linkage fills 0.78 and the drag link 0.78 — both machines whose links turn through large angles and sweep most of what they enclose.

Half the box, on the machines a reader has met most often. That is a great deal of room inside a machine’s own envelope that is never occupied at any instant, and it is where anything else has to go: the motor, the frame members, the guard’s fixings, the next mechanism along.

It is also the number that says whether the box is a useful summary at all. A machine at 0.78 is well described by its box; a machine at 0.54 is not, and packaging it by its bounding rectangle wastes as much space again as the machine occupies.

What the empty half is shaped like #

Not one hole, and not anywhere convenient.

On the crank rocker the unoccupied part of the box is in three pieces: a large region below the frame line where nothing ever goes, a smaller one above the coupler’s highest reach, and a genuine interior hole near the rocker pivot that the machine encircles without entering. The last one is the interesting kind, because it is space the mechanism has completely surrounded — reachable from outside only by going round or through — and it is exactly the sort of place a designer would like to put a bearing housing and cannot easily get a fixing to.

The grid shows all of this directly and it is one of the two reasons the region is drawn rather than merely measured. The other is that it is the shape of the guard.

The box is not the region and the region is not the machine #

There are three nested sets here and it is worth being explicit about them, because loose talk about “how big a machine is” slides between them constantly.

The machine at an instant is a set of polygons. Its area is the material’s area, and it does not change as the mechanism moves.

The swept region is the union of those over the drive. It is larger, it changes shape as the arc changes, and it is what the guard has to enclose.

The box is the bounding rectangle of the swept region. It is larger again, it is what a layout drawing reserves, and it is the only one of the three that has ever been quoted on this site.

Each step outwards loses information and gains convenience, and the losses are quantified: the region is 12.77 against a box of 23.71, and the material at any instant is about 1.6 square units against a region of 12.77. So a four-bar made of 1.6 units of steel reserves 23.7 units of floor, and fifteen sixteenths of what it reserves it is not in at any given moment.

That last ratio is not a criticism of anything. A mechanism is a thing that moves, and reserving the space it moves through is what has to be done. It is a way of seeing why machines are mostly air, and why the interesting packaging question is which of the three sets a neighbouring part has to stay out of — the region, if it must never be touched; the box, if the layout is being done on paper by somebody who has not got the region.

Two things this is not #

It is not the workspace. The serial field’s workspace is the set of positions the output can be put at, which is a question about what the machine can do. The footprint is the set of places the machine’s material is, which is a question about what it is in the way of. They are different sets computed from different questions, and on the same machine they can be nearly disjoint: a four-bar’s coupler point traces a curve occupying a small part of a footprint most of which is swept by the links behind it.

And it is not a keep-out volume in three dimensions. Every region here is a plane region. The real machine is a stack of plates, as deep as its plane count, and different links occupy different plates — so the honest solid is a union of prisms of different heights and different plan shapes, not the footprint extruded. That object is computable from what is here and is not computed, because it would need a plate thickness, a running clearance and a convention about pin heads, none of which this field carries.

The margin, and where it comes from #

It is worth taking the 0.43 apart, because a designer reading a footprint wants to know which part of it they can change.

Of that margin, the half-width 0.16 is the material either side of the centreline and cannot be reduced without thinning the link. The extra 0.056 is the boss: the material round the pin, which here is 1.35 times the half-width, and which is set by the pin’s diameter and the wall thickness round it rather than by the link’s width. On a real part the boss is often much larger than the shank — a link with a 6 mm shank and a 20 mm boss is ordinary — and the footprint would grow accordingly while the “link width” in a drawing stayed the same.

That is the honest statement of what the margin depends on: the box is set by the bosses at the extremes of the motion, not by the widths of the links. Any estimate that scales the joint box by the link width is estimating the wrong quantity, and it will be wrong in the direction of optimism on exactly the parts — big pins, thin links — where a designer is most likely to be short of room.

Reading the catalogue’s numbers #

Three readings that are not obvious from the table and are visible in the pictures.

The drag link is dense. It fills 0.78 of a box only 19% larger than its joints’, which is the best of both in the catalogue: a compact machine with both long links turning fully, so almost everything inside the envelope is swept by something at some instant. It is the machine to choose if the constraint is volume.

Peaucellier’s cell is tall and thin and sweeps a lot. 43.7 square units of region, the largest here, in a box 5 by 10.5. Its fill is 0.77, so it is efficient in the same sense — but the shape of what it needs is a tall slot, and a machine that needs a tall slot is much harder to package than one that needs a square.

And the two machines with obstacles fill less than half. The four-bar with a post fills 0.496 and the crank with a stud the same, both below the plain four-bar’s 0.539, because the obstacle occupies part of the box and is not part of the machine. That is a small arithmetic point with a real consequence: the fill of a machine drops as soon as anything is put beside it, so the fraction is a property of the assembly rather than of the mechanism.

What a designer would do with the empty half #

The fill is a diagnosis rather than a prescription, and the prescriptions it suggests are worth listing because they are all things that change the mechanism rather than the packaging.

Move a pivot. The transmission angle will object, and the interior hole in a crank rocker’s footprint is near the rocker pivot and is created by the rocker sweeping round it. Moving the ground pivots closer shrinks the box in one dimension and usually raises the fill, at the cost of everything the transmission angle says about a short frame.

Change which link is the input. The drag link and the crank rocker are the same chain differently grounded. The inversions of one chain have the same lengths and different footprints, because a different link is fixed and a different set of links turns fully. That is a free variable this site has spent a whole field on for other reasons, and its effect on the footprint has never been looked at.

Or accept it. Half a box of air is what a mechanism is, and a layout that treats the box as solid is being conservative in a way that is easy to justify and expensive to build. The reason to have the region is to be able to decide, rather than to have to assume.

None of those is a recommendation this field makes. What it supplies is the number that would let somebody make one.

What it takes to compute, and where the sampling shows #

The footprint needs the same two things the region does: a sample of the drive fine enough that consecutive frames overlap, and a grid fine enough that the boundary is resolved.

The box itself is much less sensitive than the area, which is the opposite of the situation the certificate is about. A bounding rectangle is decided by a handful of extreme configurations, and a coarse sweep finds them almost as well as a fine one — the extremes of a smooth motion are broad. The fill is sensitive in both directions, because it is a ratio of two things the sampling affects differently: too few frames and the region has scalloped gaps, so the fill is understated; too coarse a grid and the boundary cells are lost, and the fill is understated again.

Both errors are in the same direction, which is worth knowing when a fill is read: the numbers here are lower bounds on how much of the box is used. At 90 frames and a 170-cell grid the bias is under a per cent, measured by refining and watching the number rise and then stop.

A footprint changes with the arc, and with almost nothing else #

Two sensitivities, one large and one surprisingly small.

The arc is decisive. Drive Chebyshev’s linkage over half its working arc and its region halves and its box shrinks in one dimension. Every number in this essay is quoted for a stated drive, and the drives are measured rather than chosen — a full turn where the machine turns fully, and workingArc’s answer where it does not.

The width is not. Doubling every link’s width from 0.16 to 0.32 grows the box by about 0.43 in each direction again, which on a 5-unit machine is another 9%, and grows the swept area by rather more, since every part of the boundary moves outwards. But the ratio between the two boxes barely moves, because both grow by the same margin and the joint box does not grow at all. So the 1.20 is a statement about the machine’s proportions rather than about its stock, and it would be roughly the same number on a machine built of anything reasonable.

That is why the ratio is worth tabulating and the absolute box is not. One is a property of the design; the other is a property of the design and the sheet it was cut from.

The comparison worth making #

The point of two boxes is not the ratio between them. It is that this site has been drawing mechanisms for twenty-three fields and has never been able to answer the first question anybody asks about a machine — how big is it — with anything but the extent of its pins.

A mechanism’s pins are the part a designer chooses, and every synthesis on this site returns them and nothing else. Its material is the part that has to fit, and the two differ by an amount that is small in relative terms and decisive in the places where machines are actually short of room: at the ends of the motion, at the bosses, and in whichever dimension the machine happens to be thin.

There is one more reason to have both numbers, and it is about how a machine is checked rather than how it is packaged. An obstacle outside the material box cannot interfere, ever, at any configuration, and no sweep is needed to establish it. That is a one-line test costing four comparisons, it is exact, and it disposes of most of the things near a real machine before any of this field’s machinery is invoked. The joint box cannot do that job: an obstacle outside the joint box and inside the material box interferes with nothing in the drawing and hits the machine.

The measurement is not deep and it is not clever. It is a bounding box over a set that was never available before, and its whole content is that the set is the material rather than the points.

A plane count is a height #

Everything above returns an index: link 3 goes in plane 1, this machine needs three planes. Every region this field computes is likewise a plane region. The machine that gets built is a stack of plates with different links in different plates, and the honest three-dimensional object is a union of prisms of different plan shapes — not a footprint extruded.

Computing it needs three numbers this field does not carry: how thick a plate is, how much running clearance sits between two of them, and what the pins need above and below to be retained. Those belong in the library with stated defaults rather than in every figure — a reader who wants a plan view should not be asked about circlips. The defaults here are a 3 mm plate, 0.5 mm of running clearance, and 2 mm each side for a pin head and a retaining ring, on a machine whose ground link is four units long. They are the numbers to change and nothing depends on their being right.

With them, the plane count stops being an ordinal. Peaucellier’s cell needs four planes and is 17.5 mm tall; a crank rocker needs three and is 14.0 mm; the housed four-bar needs two and is 10.5 mm. A machine needing three planes is not “one worse” than one needing two — it is three and a half millimetres taller, of which the retention is four.

And the footprint is not the solid. Each plate carries only the links assigned to it, so the material is a union of prisms and a fraction of the box that contains it. Measured across the catalogue, the solid fraction runs from 16.8% to 37.0% — so quoting the plan area times the height overstates the metal by between 2.7 and 6.0 times.

That matters wherever a footprint has been read as a proxy for material: a machine with a large plan and a thin scatter of links in each plane is mostly air, and the machine with the smallest footprint is not reliably the one with the least metal in it. The bell crank has the worst ratio at 5.95 and the drag link the best at 2.71, and nothing in a plan view distinguishes them.