The block in the guide has a length

The pairs field is about what a joint permits: a set of displacements, closed under composition, which for the six lower pairs is one of six groups. A prismatic pair permits translation along one direction and nothing else. In the plane it takes away two of a body’s three freedoms and leaves one.

That account is complete and it is about the pair. It is silent about the parts, and the parts are a block and a guide, both of which have lengths.

What the site’s model of a slider has been #

m.slide(a, px, py, dx, dy) — a joint constrained to a fixed line, given as a point on the line and a direction. One equation: the joint’s perpendicular offset from the line is zero.

That is the right model for everything the loop fields ask, and it is a point on a line. The line is infinite. The point has no size. So the stroke of a slider crank is 2a, twice the crank length, and the guide it needs is however long the stroke is.

Both halves of that are wrong once the parts are made of something, and they are wrong in the same direction.

Guide equals stroke plus block #

A block of length ℓ has both its ends on the rails, so its centre can only reach from ℓ/2 inside one end of the guide to ℓ/2 inside the other. On a guide of length G the centre travels G − ℓ.

Run it the other way, which is how a designer meets it: a stroke of 2a needs a guide of

$$G = 2a + \ell.$$

For the site’s slider crank — crank 1, rod 3, so a stroke of 2 — with a block of 0.6, that is a guide of 2.6. Of which 0.6 is never stroke: 23% of the guide exists so that the block can be a block.

The fraction is worth tabulating because it is not small and it grows the wrong way. A block of 0.3 is 13% of its guide; a block of 0.6 is 23%; a block of 1.0 is 33%. A short-stroke mechanism with a long block — which is a normal thing, since a block wants to be long for reasons of guidance that are entirely about force — spends most of its guide on itself.

The second route #

The arithmetic above is one line, so the measurement is worth having for exactly that reason: a one-line result is the kind nobody checks.

Build the slider crank with the block as a body — a rectangle of length ℓ carried at the slider joint — and two stops at the ends of the guide. Shorten the guide until the block’s own corners touch a stop somewhere on the crank’s turn, and bisect on the length that just clears.

For ℓ = 0.6 the search returns 2.6001 against an arithmetic 2.6: agreement to 6 × 10⁻⁵, which is the bisection’s own tolerance. For 0.3 it returns 2.3001 and for 1.0, 3.0001.

The two routes have nothing in common. One is a subtraction. The other is three hundred and sixty solved positions of a mechanism, a body placed at a joint at each, four polygon distances per position, and a dozen rebuilds of the whole machine as the guide length is bisected on. They agree to the last digit either can carry.

Why the search converges, and what it is searching over #

The bisection here is on the guide length rather than on a width, and the monotonicity argument has to be made again because it is a different parameter.

Lengthening the guide moves both stops outward and can only increase the clearance between the block and either of them. So the predicate the block stays on the rails through the whole turn is monotone in the guide length, its crossing is unique, and bisection finds it. Shortening it can only bring the stops in.

What the search is not doing is optimising the mechanism. The crank, the rod and the block are fixed; the only variable is where the two stops are, and the answer is the shortest separation at which nothing touches. That makes it a much simpler search than the width search, where growing the parameter changes every body at once and the machine’s whole geometry with it.

It is also a search whose answer is known in advance, which is the point: a bisection whose result is predicted to five figures by one subtraction is a bisection worth running, because when it disagrees the disagreement is informative. Here it disagreed by a factor of four the first time it ran.

The bug the second route found #

The block was placed the way every other part in the bodies field is placed: as a shape in its link’s own (u, v) coordinates, scaled by the link’s length.

That is right for a bar, whose material really is bolted between two pins and whose length does not change. It is wrong for a block, because a block’s “link” is the line from the frame to the slider joint — and that length changes as the mechanism moves. Placed that way, the block grew and shrank through its own stroke, from 0.4 at one end to 0.8 at the other, and the guide search converged on a guide exactly as long as the block.

The number it produced was 0.6001 against an arithmetic 2.6: wrong by a factor of four, and internally consistent, and it drew a picture in which a block slid along a guide and looked entirely normal because a block that changes size while sliding does not look like anything in a static frame.

The repair is a second placement rule: a block is carried at a joint, at an angle, with absolute dimensions, and does not scale with anything. Which is a small piece of code and a real modelling distinction — some parts are stretched between two joints and some are bolted to one.

What the block does to the free arcs #

Reading the same mechanism as a set rather than as a number gives the other half of the picture, and it is the one a person operating the machine would recognise.

Swept with its guide rails in place, the slider crank’s free space is two arcs covering about a third of the turn — because the rod fouls the rails over most of it. That is a machine whose crank cannot be turned continuously at all with these proportions, which the gap’s minimum of −0.140 says as a number and the arc count says as a fact about operating it.

Neither reading is available from the pair’s own account, and the two say different things. The minimum says how much material is in the way at the worst instant; the arcs say that the obstruction is not one region but two, so the crank has two ranges it can rock through and cannot get from either to the other.

This is the second machine in the catalogue whose free space is disconnected, and unlike the two-stud contrivance it is a mechanism a reader has met. The obstruction is not a stud somebody bolted on to make a point; it is the guide the pair needs in order to be a pair.

What the pair still costs #

Nothing here changes the pair’s own arithmetic, and it is worth being explicit because the fields are adjacent.

A prismatic pair is still one freedom in the plane and one in space. It is still a lower pair, still one of the six, still the group of translations along one direction. A mobility count of the slider crank is unchanged — four links, four lower pairs, one freedom — and every position, velocity and ratio the linkage field computes is exactly what it was.

What has changed is a quantity that is not about the pair at all: how long the guide has to be. The pair is a relation between two bodies; the guide is one of the bodies; and the length of a body is not a property of any relation.

That distinction is the pairs field’s own, applied to itself. Its central move is that a joint is a surface that slides on itself — a plane, a cylinder, a helicoid — and what the joint permits is that surface’s symmetry group. A group is scale-free: the translations along a line are the same group whether the line is a millimetre or a kilometre. So the pair cannot know how long the guide is, by construction, and the question has to be asked of the parts.

The stroke a mechanism has, and the stroke a guide gives #

There are now two strokes in play and it is worth keeping them apart, because they are computed by different fields and can disagree.

The mechanism’s stroke is 2a, and it comes out of the linkage field’s closed form: the slider joint runs from b − a to b + a, whatever the guide does. It is a property of two lengths.

The guide’s stroke is G − ℓ, and it comes out of this essay: the block’s centre runs from ℓ/2 inside one end to ℓ/2 inside the other. It is a property of two other lengths, neither of which is in the mechanism.

A design works when the second is at least the first, and the interesting failures are when it is not. A guide too short by a little does not shorten the stroke gracefully: the block hits a stop, and what happens next is a force question. So the geometric answer is binary — the mechanism either has room to complete its turn or it jams at a definite crank angle, which the sweep reports along with which end it hit.

That is a different relationship from most of the pairs on this site, where a joint permits a range and the mechanism uses part of it. Here the pair’s own permission is unbounded — a prismatic pair permits translation along the whole line — and every limit comes from the parts.

The rod is the other half #

The block is not the only thing in the guide’s way. The connecting rod swings as the crank turns, and near the end of the stroke it leans hardest — so the mouth of the guide has to be open enough to let it in.

Swept with the guide rails as obstacles, the site’s slider crank reads a worst gap of −0.140 between the rod and the lower rail, with two changes of closest pair over the turn. That is a real design statement rather than a saturated one: the rod fouls the mouth by about one link width, and the fixes are the obvious ones — shorten the block, open the mouth, or angle the rail ends away.

It is also the answer to a question a reader might reasonably ask about the guide-length result, which is why the block search uses stops at the ends rather than rails along the sides. The two constraints are different: the block’s ends against the stops decide the stroke, and the rod against the rails decides the mouth, and measuring them together would produce one number that is neither.

The same question, one pair over #

A revolute pair has the same structure and this site has already met its version twice.

A revolute is a point in the equations, and the parts that realise it are a pin and a hole. The practice field gives them a fit — the pin is smaller than the hole and the joint has slop — and the bodies field gives them a size, so that two of them on one link cannot be closer than the material round them.

Both are questions about the parts rather than about the pair, and neither changes what the pair permits. The prismatic version is the third of the same family:

• A revolute’s fit costs lost motion along a chain.
• A revolute’s size costs a minimum spacing between two of them.
• A prismatic’s size costs a guide longer than its stroke.

Each is a quantity that appears the moment a joint is realised and is absent from every account of what the joint is, and each one is decided by a number — a clearance, a boss radius, a block length — that the constraint equations have no place for.

The remaining four lower pairs have their own versions, and none of them is computed here. A cylindrical pair’s block has a length and a bore. A helical pair has a lead, a thread depth and a nut of some length. A spherical pair has a ball, a socket and an angular limit set by where the socket is cut away — which is the one that most changes what the joint permits, since a ball joint’s cone of motion is a body constraint and is usually the thing that decides a design.

What a real guide is #

Three things this model does not have, each of which makes a real guide longer or shorter than 2a + ℓ.

Bearing overlap. A block needs to stay guided at the ends of its travel, so it wants length in the guide beyond the geometric minimum — a real guide is longer than 2a + ℓ, not shorter.

Overtravel and stops. Anything that is driven to a limit has a hard stop and some slack before it, which adds again.

And the block’s own guidance. How long a block has to be to run without tipping is a question about clearance, friction and load, and it is outside this site — which is exactly why the block’s length appears here as a given to be traded rather than as something to be derived.

So the number is a floor, arrived at from below, and its content is the trade rather than the value: every unit of block length is a unit of guide that is not stroke, and the two things a designer wants from a block — long for guidance, short for a compact guide — are in direct opposition with an exchange rate of one.

A pair, and the parts that realise it #

The general form of this is the one the pairs field has been building towards, and this essay is its material half.

A pair is an abstraction: a set of permitted displacements, a group, a rank, a count. It is scale-free, it composes, and it is what makes a mobility formula possible.

The parts that realise it are two bodies with sizes, and everything about how much room the joint needs, how far it can travel and what it fouls lives there. A pair can be realised by a block in a guide, by a pair of rails, by a dovetail, by a ball slide or by a four-bar approximating a straight line — all the same pair, none of them the same parts.

The pairs field spends seven rungs on the first and this is the first essay to look at the second. What it finds is unremarkable and was unstateable: a prismatic pair costs two freedoms, and a prismatic joint costs a guide a third longer than the stroke it delivers.

There is a general point behind a guide is as long as the stroke plus the block, and it is the reason this rung belongs to the practice field rather than to the linkages one. Every joint on this site has been a point or a line — a pin at a place, a slide along a direction — and every real joint is a part with an extent. A pin has a diameter and a length; a slider has a block; a ball joint has a socket with a cone of travel; a bearing has a width. The idealised joint permits everything the geometry allows and the real one permits less, by an amount set by the part’s own size. So the practice field’s standing question has a companion: not only what does a tolerance do to this mechanism but what does the joint’s own extent take away. Both are the difference between the mechanism on the drawing and the one in the shop, both are computable from dimensions the drawing already carries, and only the first of them is usually asked.