Machine Kinematics
solved, then drawn
Concept

Frame — where it appears

The link held still, which turns a chain into a mechanism. Nothing in a graph says which link it is, so the choice is a separate decision — and two links give the same machine exactly when an automorphism carries one to the other.

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

The five six-bar mechanisms, and there are only two chains. Two chains and five machines. Watt's chain has two orbits of links, so grounding it gives two mechanisms; Stephenson's has three. The frame is drawn dark in each. This is the whole of what "Watt I", "Watt II", "Stephenson I, II and III" name — not five linkages somebody invented, but two graphs and the five genuinely different links there are to bolt down. Anyone who has met the names as a list of five things has met the answer without the question, and the question is a count of orbits.

Which link to bolt down

A chain is not a machine until one of its links is held still, and which one is a decision. Two links give the same machine exactly when a relabelling of the whole chain carries one to the other — so the number of mechanisms a chain gives is a count of orbits, and the classical five six-bars and seventy-one eight-bars are that count.

topology · Topology Four of the sixteen cannot be positioned without a solver. For every chain, every way of choosing a frame and a driven link pinned to it, and for each the decomposition into Assur groups. All dyads counts the choices whose groups are all two links — those are the mechanisms a draughtsman can position with a compass, two circles at a time. The last column is the one that matters: chains for which no choice of frame and input is all dyads, so every way of driving them leaves a group of four or more links that has to be solved as a single system. At eight links there are 4 of them and at ten there are 90. This site has run a Newton solve on every mechanism it has ever drawn, and it has always been possible to read that as convenience. On these it is not.

Eight ways to drive it, and one machine

Bolting a link down is half the decision; the other half is which link carries the input. A four-bar has eight frame-and-input pairs and exactly one of them is a distinct machine — and across the eight-link census 320 listed pairs collapse to 153.

topology · Topology A dyad is two circles, and that is why it has two answers. The whole of what makes a dyad easy. Two links, three pins: one pin onto something already placed at each end, one pin between them. The free pin is at a fixed distance from each of the placed ones, so it lies on both circles — and two circles meet in two points, in nought, or in one. That is a quadratic with a closed form, and its two roots are the two assembly branches every four-bar on this site has. A group of four links has no such picture: its unknowns do not separate into one circle each, the system does not factor, and what replaces the compass is Newton's method from a seed. positioned by solving, not by drawing.

What has to be solved together

Hold a link, turn a neighbour, and the rest of a mechanism comes apart into the smallest sets that can be positioned one after another. Every set of two links is two circles meeting — a quadratic, two branches, no solver. A set of four is a system, and this site's Newton solve stops being a convenience.

topology · Topology

Named alongside it

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

Kinematic chainMobilityAssur groupAutomorphismCanonical formInput linkInversionOrbitType synthesisAssembly branchConstraintDyad

All concepts