Machine Kinematics
solved, then drawn
Concept

Revolute — where it appears

A pin joint, permitting rotation about one line and nothing else. It removes two of a planar body's three freedoms and five of a spatial body's six, which is why a closed spatial chain needs seven of them before a count says it moves.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

A pin in a hole is a short link. Left: a pin of radius 0.86 in a hole of radius 1, so the clearance is 0.14. The pin's centre may sit anywhere within that of the hole's centre. Right: the same joint as it enters the kinematics — a binary link of fixed length 0.14 and free direction, with a revolute at each end. That is not an analogy. It is the same set of relative positions, so every count, every Jacobian and every solve on this site applies to it unchanged, and a four-bar with play at each pin is a mechanism with eight links and eight joints.

A clearance is a link

A pin in a hole is not a joint at a point. Its centre may sit anywhere within the difference of the two radii, so the two links it joins are connected by a body of fixed length and free direction — a binary link with a revolute at each end. That is not an analogy, and taking it literally makes a four-bar a mechanism with eight links, eight joints and five degrees of freedom.

practice · Clearance How close a synthesis puts two pins. The distribution of the smallest pin-to-pin distance on any one link, over all 1,176 exact syntheses. Burmester's construction returns points, and points can be arbitrarily close together: the shortest here is 0.103, on a mechanism whose poses span more than two units. The shaded band is what a boss of radius 0.25 forbids — 148 of them, 12.6%. The link that is worst is most often the crank, which is not where a designer looks: the frame's two ground pivots are the pair everybody checks by eye.

A pin is not a point

A joint in the fields before this one is a name and two coordinates. A pin is a cylinder with material round it, a length through the stack of plates, and a head — and every one of those turns some construction that returns points into a construction that may return nothing buildable.

bodies · Body Which pin's play costs the most. Each pin's clearance taken one at a time, at 0.01 on links of 1 to 4, with the direction swept rather than assumed. The ranking runs A 32%, B 25%, O₂ 22%, O₄ 21% — a spread of 1.52 against the 3.43 the four lengths spread over. Clearances are more evenly weighted than length tolerances because each pin joins two links and so enters two of the four sensitivities, which is why the best bearing buys less than the best-held length does — and why it still goes somewhere the load path does not suggest.

Which pin to buy

The four lengths of a four-bar contribute 38, 26, 25 and 11 per cent of its output error — a spread of 3.4. Its four pins contribute 32, 25, 22 and 21 — a spread of 1.5. Clearances are more evenly shared than length tolerances, because every pin joins two links and so appears in two of the four sensitivities, and that changes what a better bearing is worth.

practice · Clearance Watt chain with 1 slide: 3 chains, 11 mechanisms. The same 6 links and 7 joints with 1 of the joints made a slide instead of a pin, drawn as a block astride the line. There are 7 ways to choose the joint, and the chain's 4 symmetries fold them into 3 that are genuinely different: with the slide at 0–3, 2 mechanisms; with the slide at 0–1, 6 mechanisms; with the slide at 1–2, 3 mechanisms. The mechanism count is the number of orbits of a held link and the slide set together, so a slide breaks symmetry the pin-only chain had, and links that gave one machine between them give two. The pin-only chain gave 2; one slide gives 11.

A slide turns nothing

Make one joint of a chain a slide instead of a pin and the graph has a second decision in it before any length exists. The symmetries that counted mechanisms count these too — Watt's chain with one slide is three chains and eleven machines — and two facts read off the graph say which placements still work: a loop of slides alone is freer than the count, and a pin in a group of links the slides hold at one orientation cannot turn. Across 102 placements on the three smallest chains, both agree with the rank of the constraint Jacobian.

topology · Topology The bores, and the one line that has to pass through all of them. A hinge of 6 knuckles, its bores drawn at the distance each was made from the nominal axis in units of the bore tolerance. The leaf is a rigid body, so its pins are on one straight line — two parameters of position and two of direction — and it assembles when some line passes within the clearance of every bore. The line drawn is the one whose largest miss is smallest, and that miss is 0.875 of the tolerance. 2 of the 6 bores are at that distance and hold the fit; the rest are slack and could have been bored anywhere inside it without changing the answer.

A piano hinge is not forty door hinges

A three-knuckle hinge works because the misfit its bore errors create is smaller than the play already in its pins. A piano hinge has forty knuckles and thirty-nine of them are redundant, so the obvious reading is that it needs thirteen times the play. It needs two and a half times, and it can never need more than the bore tolerance itself — because a rigid leaf has one axis and a line through the middle of the errors misses every bore by at most the largest of them.

practice · Overconstraint

Named alongside it

The objects these essays reach for when they reach for this one.

ClearanceLower pairToleranceConstraintFour-barGrübler's criterionJacobianMobilityAllocationAutomorphismBossConstraint rank

All concepts