Fifteen pairs and one cut

Three rotations, and four benches counted the pairs of links in a slider-crank chain. There are six. One of them is joined by a slide and cannot turn at all, and the other five are three rotations between them — the crank against the frame, the rod against the frame, and their difference. Each of the chain’s four inversions puts its drive at one of those three, so the four famous machines are four readings of one classification.

It ended by asking the same thing of a six-bar. Watt’s and Stephenson’s chains have six links, seven pins and fifteen pairs, and six ways of choosing which link to bolt down. Is the pool of rotations again much smaller than the pair count, and does the Grashof count — three of a four-bar’s four inversions have a turning input, or none — generalise?

The answer is yes on both counts. The pool is smaller than it was for the slider-crank: on nearly every circuit it is one rotation. The Grashof count generalises to a count of links on a cut. The extra loop also brings something the four-bar has no room for: the answer can depend on which assembly the chain is in.

Six numbers, fifteen differences #

Any relative rotation between two links is the difference of the two links’ rotations against a third. So the fifteen pair rotations of a six-bar are differences of six numbers, the rotation of each link against the frame. On a circuit — a closed piece of the chain’s configuration space, followed round once — each of the six numbers comes back to where it started or a whole number of turns further on. Its winding is an integer, and the fifteen pairs’ windings are differences of six integers.

The six-bars here are built as the circuits essay built them. A four-bar is followed round each of its circuits, and a dyad is hung from its rocker, for a Watt chain, or from its coupler, for a Stephenson chain, and closed onto a third ground pivot. The dyad closes along runs of the four-bar’s circuit. At the ends of a run it is straight or folded and its two assemblies meet, so each run, taken forward in one assembly and back in the other, is one circuit of the six-bar.

Every link’s angle is followed along the circuit and its total change divided by a whole turn. Two refinements keep the count honest. The ends of each run are found by bisection on the four-bar’s crank angle rather than left at the nearest sample, because the dyad folds there and a link can swing through a large angle between two samples. And any circuit on which some link moves more than 0.8 rad between samples is recounted at eight times the sampling, up to 76,800 samples a circuit. Of 4,469 chains, 63 needed the finer count and nine never settled and were left out. Every winding on every counted circuit is a whole number to 10⁻⁶.

The chain drawn above is a Watt six-bar on one of its four circuits, where the rocker, the dyad link and the output all turn once against the frame, together. Frame, crank and coupler come back.

One rotation, read at several pins #

On this circuit the six windings are 0, 0, 0, 1, 1, 1, so the fifteen pair windings take only two values, nought and one. Nine pairs go round and six do not. The fifteen are not fifteen rotations, or five, or three. They are one rotation — the group of three against the rest — read at every pair that has one link on each side.

The census draws four-bars and dyads at random over the ranges the circuits essay used: ground 4, the other four-bar lengths between 0.4 and 5, the dyad’s attachment anywhere across the rocker or coupler, its pivot anywhere in a box around the machine, and its two lengths from 0.3 to 6. It keeps every chain that assembles. That is 2,124 Watt chains with 5,086 circuits between them and 2,336 Stephenson chains with 5,195.

On most circuits nothing turns: 2,162 Watt circuits and 2,415 Stephenson. On the rest, one link turns against all the others, or a group of links turns together against the rest. Every Watt circuit is one of those two cases. So is every Stephenson circuit but four, and those four come in the section after next.

A group turning together is a way of saying that one link is still, relative to the others. When a Watt chain’s rocker, dyad and output all turn once against the frame, those three come back to their positions relative to each other, and it is the frame, crank and coupler that have gone round, seen from the rocker. It is the four-bar’s double crank in a second loop: the shortest link turns against everything else, whichever link the observer happens to be sitting on.

The cut, and Grashof’s count #

Grounding a link and driving it at one of its pins works only if that pin’s rotation goes round. On a circuit where the links split into a turning group and the rest, a pin’s rotation goes round exactly when the pin joins a link in the group to a link outside it — when the pin crosses the cut. So an inversion can be driven exactly when its grounded link touches the cut.

The count is nought, three, four or five, and never one, two or six. A single binary link turning gives three: the link itself and its two neighbours, each of which has a pin on the cut. A ternary link turning gives four — the Watt rocker 580 times, the Stephenson coupler 518 times — because it has three neighbours. A group touches the rest at several pins. The Watt group of rocker, dyad and output meets the frame at two pivots and the coupler at one, which gives four links on the cut. The Stephenson groups that include the crank, or the coupler, with the dyad and output give five.

This is Grashof’s count with the cut made explicit. A four-bar’s cut is always a single link — the shortest, turning against the other three — and a single link in a four-bar has two neighbours. So three of the four inversions can be driven, or none. In the rotations essay the count survived the limit where the output pivot ran to infinity, and the reason given was that the Grashof count is a statement about which rotations wind. Here the same statement is made about a larger chain. The number of drivable inversions is the number of links touching the cut, and a six-bar’s cut can be a binary link, a ternary one or a group.

What can never happen is one or two. Every link on the turning side of a cut touches it, and so does every link it meets across it. The smallest cut, a single binary link, already involves three. And six would need every link to touch the cut, which none of the census’s cuts does. A four-bar never gets its fourth inversion drivable. A six-bar never gets its sixth.

The same chain, a different answer #

A four-bar with two circuits carries the same rotation on both — the crank turns in either assembly, or neither does. That is part of what made the four-bar’s inversions a classification of the chain. In the six-bar it fails.

This chain’s four-bar is a crank-rocker. The crank goes round, so the four-bar has two circuits. On one of them the dyad closes all the way round and gives two six-bar circuits, on which the crank turns. On the other the dyad closes only along two short runs. Each run is a circuit on which the crank swings through about seventy degrees and comes back, because the dyad folds before the crank can get any further. The same chain, with the same lengths, has assemblies in which three inversions can be driven and assemblies in which none can.

In the census 215 Watt chains and 456 Stephenson chains do this: about one in ten and one in five. For them, which inversion a motor can drive is a property of the chain and the assembly together. It is the extra loop’s doing. The dyad’s closure cuts the four-bar’s circuits into runs, and a run can be too short for any link to complete a turn even though the four-bar under it could.

That is the second question the rotations essay asked, answered: the extra loop is exactly what makes a six-bar’s inversions genuinely different mechanisms, not readings of one classification. It is not that each inversion needs its own classification. One cut per circuit still decides every inversion at once. It is that a chain now has several cuts, one per circuit, and which one applies depends on the assembly.

An assembly is a choice, and now it carries the drive #

The four-bar made this easy to overlook. Branches were components all along showed that an assembly is a connected piece of configuration space — a builder picks one when putting the machine together, and the machine stays in it — but in a four-bar the choice never changed what could be driven. A crank-rocker’s crank turns in either assembly. The choice mattered for where the coupler curve is and which way the output swings, not for whether a motor could be fitted.

In a six-bar the choice reaches the motor. A builder who assembles the chain above with its dyad on one side gets a machine that can be driven from its crank. Assembled the other way, with the same parts, it rocks through seventy degrees and stops. Nothing about the parts, their lengths or their pins says which. It is decided by which of the chain’s circuits the builder happened to close it into, and the circuits are only visible by following them.

That changes what a design check has to be. Grashof’s rule can be checked on lengths alone, and a four-bar that passes it will turn in whatever assembly it is built. A six-bar needs its circuits followed. The dwell six-bar of a dwell made from a curve is driven by a crank that turns, and that is true of the assembly it is drawn in. Whether its other assemblies turn too has not been checked. The census says that for about one chain in ten of Watt’s kind, and one in five of Stephenson’s, there is another assembly of the same parts where the crank would not turn — and a designer who has only ever built one assembly has no way to know which kind of chain they have.

The cut also says where the motor can go on the assembly that does turn. On a circuit where a single binary link turns, the three inversions that can be driven are that link grounded — driven from either end — and its two neighbours grounded, each driving through the pin it shares with the turning link. The other three inversions rock. On a circuit where a group turns, the pins on the cut are the only places a motor will turn continuously, and every link that owns one of those pins is a bench that can take it.

Twice round #

Four Stephenson circuits, on two chains, do not split into two groups.

On this circuit the frame, crank and coupler come back to where they started. The rocker turns once one way, and the dyad link and output together turn once the other way. So the rocker and the dyad link make two whole turns against each other — a pair rotation that no four-bar and no Watt chain in the census has. The circuit has three levels of winding, not two, and its cut has two parts.

The difference between the chains is in their loops. A Watt chain’s second loop runs through the frame, the rocker, the dyad and the output: four links, as the first loop has. A four-link loop can let one link turn against the rest, and never two. A Stephenson chain’s second loop runs through the frame, the crank, the coupler, the dyad and the output, or equivalently round through the rocker instead of the crank: five links. A loop of five links has room for two groups that each turn against the others, as a five-bar with two cranks does. Here the four-link loop turns the rocker against the frame, and the five-link loop turns the dyad and output the other way.

It is rare: four circuits in 5,195. Such a chain has five of its six inversions drivable, like the other Stephenson groups, but at one of its pins a motor would turn twice for every turn at another.

What the six-bar’s inversions are #

Put together, the answer to “how many distinct relative rotations does a six-bar chain have” is: as many as it has circuits, one per circuit on nearly all of them, and occasionally two. Fifteen pairs overstate it by the same kind of margin that six pairs overstated the slider-crank, and for the same reason. A pair’s rotation is a difference, and differences of a few numbers take few values.

Which inversions can be driven is then read off each circuit’s cut, as a set of links touching it. That makes a table like the slider-crank’s, except that it has to be made per circuit, not per chain. It is also why the classical names — the two Watt inversions and the three Stephenson ones — are names for ways of grounding the chain and not for kinds of machine. A Stephenson inversion with a turning input on one assembly can be a rocking machine on another.

What this does not settle #

The ranges. The census draws from the circuits essay’s ranges. A cut that needs a narrow window of lengths — the Watt crank and dyad turning at once, each in its own loop, say — may exist and be too rare to be drawn. None was, and nothing here says it cannot.

Change points. A chain exactly on a boundary between two cuts — where a run of the dyad’s closure just manages a full turn of some link — has measure nought and is not in the census. What its windings are, and whether it can switch cuts in the course of its motion, is a question about the six-bar’s change points that this does not reach.

Inversions up to symmetry. The count runs over the six links, not over the classical two Watt and three Stephenson inversions, since the census’s chains have no symmetry. For a symmetric chain the six collapse onto those, and the count per class would be a finer table.

Still open: a cut that switches in mid-motion #

At a change point of a four-bar the two circuits meet, and the linkage can pass from one to the other. A parallelogram a micron wrong shows what a real one does there. A six-bar whose circuits carry different cuts has change points too: places where a run of the dyad’s closure touches another run, or touches the four-bar’s own fold.

The distinct argument there would be what happens to the cut at such a point. Does a six-bar passing through a change point between a circuit where the crank turns and one where nothing does become, for the rest of its motion, a machine that can no longer be driven from its crank? How much clearance at which pin decides which way it goes? The measurement would be a family of Watt chains swept through the boundary where a circuit’s cut changes, with the windings followed across it.