Eight ways to drive it, and one machine

Bolting a link down makes a chain into a mechanism. It does not make it into a machine, because a machine has something driving it, and where the input goes is a second decision.

On this site the input has almost always been a crank turning about a ground pivot: a link pinned to the frame, driven about the pin they share. That is the standard arrangement and it is the one assumed throughout this field. It means the decision is an ordered pair — which link is the frame, and which of its neighbours turns.

Counting those pairs is easy: each pin gives two, one for each end, so a chain with $j$ pins has $2j$ of them. The four-link chain has eight. Watt’s and Stephenson’s have fourteen each. The eight-link census has 320 between its sixteen chains.

Almost none of those are different machines.

One way to drive a four-bar #

Take the four-link chain, which has four links all alike and four pins all alike. Bolt down link 0 and drive link 1. Now bolt down link 1 and drive link 2. Now bolt down link 2 and drive link 3.

Every one of the eight is the same driven mechanism. There is a relabelling of the chain that carries any frame-and-input pair to any other, so the eight arrangements are eight drawings of one thing, and the count of genuinely different ways to drive a four-bar is one.

That is not a surprise once said — a four-bar is a four-bar — and it is not what the list of eight looks like, which is the point. The reduction rule is the same one that counted inversions, applied to a pair instead of to a single link: two frame-and-input pairs give the same driven mechanism exactly when some automorphism of the chain carries the first pair to the second.

Nine driven six-bars #

At six links the arithmetic is small enough to write out.

Watt’s chain has fourteen pairs and four orbits. Its two ternary links are interchangeable and its four binary links are interchangeable, but the pairs are not all alike: driving a binary link from a ternary frame is different from driving a ternary link from a binary frame, and driving one binary link from another binary link is different again.

Stephenson’s has fourteen pairs and five orbits, because its binary links come in two kinds and the pairs inherit the distinction.

Four plus five is nine. There are nine genuinely different driven six-bar mechanisms, against five six-bar mechanisms and two six-bar chains, and the three numbers count three different things: chains, chains with a frame, and chains with a frame and an input.

The nine, worked through #

The six-link case is small enough to name every one, and naming them is the best way to see what an orbit of pairs is.

Watt’s chain, four choices. Its links are two ternary and four binary, in two orbits. A frame-and-input pair is a pin with an end chosen, so the four kinds are: ternary frame driving a binary link; binary frame driving the ternary link it touches; binary frame driving the other binary link on its own path; and — the fourth, which is easy to miss — the ternary frame driving the other ternary link, across the pin the two of them share.

That last one is genuinely different from the other three and it is the one a reader would forget: it is the arrangement where the frame and the input are both plates, and the two binary paths hang off them symmetrically.

Stephenson’s chain, five choices. Its links are two ternary and two kinds of binary, in three orbits, so its pins come in more flavours. There is no ternary-to-ternary pin at all — that is the defining fact about Stephenson’s chain — so the four Watt-style options lose one and gain two, from the binary links splitting into the pair on the short paths and the pair on the long one.

The bookkeeping is worth doing once because it makes clear what is being counted. Fourteen pairs, five orbits, and the orbits are of sizes 4, 2, 4, 2 and 2 — which sums to fourteen, and is asserted to.

What the driven link decides #

Choosing the input is not a formality, and the consequences run in both directions across this site.

It decides the position problem. The next rung is entirely about this: the frame and the input are the two links whose positions are known at the start, and what remains has to come apart into groups that can be placed one after another. Move the input one link along and the decomposition can change from two pairs into a single group of four, with no change to anything else.

It decides whether there is a crank at all. Whether the driven link can turn through a full revolution is a dimensional question — Grashof’s condition for a four-bar, and something considerably messier for a six-bar — but which link is being asked about is this decision. The same mechanism with the same lengths can have a crank at one pin and a rocker at another.

And it decides what the mechanism is for. A machine is usually described by what its input and output are — a crank that raises a platform, a rocker that indexes a wheel — and the pair chosen here is half of that description. The other half is which link is watched, which this field does not count at all: any link can be the output, and nothing in the graph distinguishes them.

The two ends of one pin #

Each pin contributes two of the $2j$ listed pairs — ground one end and drive the other, or the reverse — and those two are worth treating together, because the relation between them is the oldest idea in this subject and the count does not respect it.

Call them reciprocal. Grounding link $a$ and driving link $b$, against grounding $b$ and driving $a$, are two machines built from the same chain by exchanging the roles of two links that share a pin. What is identical between them is the relative motion of $a$ and $b$: the angle at their common pin runs through the same values in the same order, because that angle is a fact about the chain and not about which link is bolted to the bench. What differs is everything else’s motion in the frame, since a different link is now standing still.

That is the classical statement of inversion and it is why the idea was worth having at all. A mechanism is not redesigned by being inverted, it is re-viewed: the same relative motions, rewritten with a different link at rest. Every property that depends only on relative motion is therefore shared by a reciprocal pair, and every property that refers to the frame is not.

The division is sharp enough to be useful. Grashof’s inequality is shared, because it is a statement about the four lengths and nothing else — but which link rotates fully is not, so a chain that gives a crank-rocker on one grounding gives a double-rocker or a drag-link on another, from the same inequality and the same lengths. Transmission angle is shared, being the angle between two links. Coupler curves are not, and could not be: a coupler curve is the path of a point relative to the frame, and changing the frame changes every path in the mechanism while changing no angle between links.

The orbit count sees none of this. It is a count of ordered pairs under relabelling, and a relabelling that carries $(a, b)$ to $(b, a)$ is a symmetry of the chain like any other — so reciprocals sometimes fall in one orbit and sometimes do not, for reasons of graph symmetry rather than for reasons of kinematics. On the four-link chain all eight pairs are one orbit, so every reciprocal is equivalent and the distinction never arises. On the six-link chains it arises constantly, and two of the nine driven six-bars being reciprocals of each other is a different fact from the two of them being the same machine.

That is the honest limit of what the count is telling anybody. It answers how many genuinely different crank-driven mechanisms does this chain give, which is the denominator every solvability statement in this field needs. It does not sort them into families, does not say which are reciprocals, and does not say which share a Grashof classification — all of which are questions about the pairs the count has already finished distinguishing.

There is a design reading of the same thing, and it is the reason the reciprocal relation gets used in practice rather than merely noticed. Given a mechanism that produces the relative motion wanted but puts the bearings in the wrong place, or swings the heavy link, or needs the input where there is no room for a motor, the reciprocal is a machine with the same relative motion and a different physical arrangement — and it is free, in the sense that no dimension changes. Inverting is the cheapest move available in mechanism design, and the count above is the enumeration of everywhere it can be made.

Where the choice is made without being noticed #

Three familiar mechanisms are the same chain driven at different pins, and in each case the choice is made for reasons that have nothing to do with kinematics.

The quick-return mechanism is a slider-crank chain driven at a different link from the ordinary one, and it exists because the resulting output takes longer in one direction than the other. Nobody chose it by enumerating the pairs; it was chosen because a shaping machine wants a slow cut and a fast return.

An epicyclic gear train is the clearest case, because all three of its shafts are perfectly ordinary places to put an input and the ratio is different for each: holding a member chooses the ratio, and driving a different member chooses another. A gearbox is a set of clutches that make that choice at runtime.

A scissor lift is driven by an actuator across a pair of links rather than by a crank, and so falls outside this rung’s count entirely — which is a useful reminder that the pair count answers a question about crank-driven mechanisms and not about mechanisms.

In each case the engineering reason came first and the topological choice followed silently. The value of counting is not that it would have produced a better answer in these three cases — it would not — but that it says how many other answers there were, which in the six-link case is nine and in the eight-link case is 153.

Why the denominator matters #

This would be bookkeeping if the pair count were not the denominator of every statement in the next two rungs.

Whether a mechanism can be positioned with a compass is a property of a frame-and-input pair, not of a chain — Stephenson’s chain comes apart into pairs of links from some of its choices and not from others. So “how many ways of driving this chain are solvable in closed form” is a fraction, and the fraction is meaningless unless the denominator counts distinct machines.

Across the eight-link census, 320 listed pairs are 153 distinct ones, and 69 of the 153 come apart into pairs of links. Reported on the listed pairs the same fact reads 150 of 320. Neither number is wrong; they answer different questions, and the second inflates the apparent variety of a symmetric chain by counting the same machine several times.

The reduction is not uniform #

One thing worth reading off the table rather than assuming. The ratio of distinct pairs to listed ones is not a constant.

At six links it is 9 of 28, barely a third. At eight links it is 153 of 320, close to a half. At ten it is 4,506 of 5,980, three quarters.

The trend is the one the inversion count showed: large chains mostly have no symmetry, so their pairs mostly stand alone, and the reduction becomes a rounding rather than a halving. Small censuses are the ones where symmetry does real work, which is exactly where a person is most likely to be reasoning by hand and least likely to notice they are counting the same machine twice.

What is being assumed #

Two restrictions are baked into the pair, and both are decisions rather than facts.

The input is a link pinned to the frame, turned about that pin. That is the crank arrangement and it is the overwhelmingly common one, but it is not the only one: a mechanism can be driven by a linear actuator between two links, by a strand wound onto a drum, or by a gear meshing with a link’s toothed edge. Each of those makes the driving decision a different kind of object, and none of them is counted here.

And exactly one input. The whole field is about mobility-one chains, so one input is what makes the position determinate. A chain with two degrees of freedom needs two, and the choice is then a set rather than a pair — which changes the counting, changes the decomposition, and is the transmission field’s subject rather than this one’s.

Both restrictions are worth naming because the count is clean and a clean count invites over-reading. Nine driven six-bars is nine crank-driven six-bars, and the qualifier is doing work.

The three decisions, counted #

The field’s discrete decisions now all have numbers attached, and it is worth putting them in one place.

At eight links: 16 chains. Choose one. 71 mechanisms, which is the chains with a frame chosen — between two and eight per chain, decided by symmetry. 153 driven mechanisms, which is the mechanisms with an input chosen.

There is a fourth number the field does not compute and should be named: which link is watched. Any of the eight can be the output, and no symmetry argument reduces that choice, because a symmetry that fixes the frame and the input can still move the remaining links about. Multiplying it in would take 153 to something over a thousand — and it would be a bad count, because the output is really a point on a link rather than a link, and points on a link form a continuum. So the discrete part of the design decision stops at the driven pair, and the output is where the dimensional problem starts.

Each step multiplies the space by a factor of four or five, and each step is still finite and small. That is the useful contrast with what comes after: choosing dimensions is a search in a continuum with no count at all, and every discrete decision made badly before it is one the optimiser cannot undo.

What a census of driven mechanisms is worth #

The honest test of a count like 153 is whether anybody would use it, and the answer is: as a bound rather than as a list.

Nobody is going to page through 153 driven eight-link arrangements. What the number does is settle an argument that otherwise has no settlement — were the alternatives considered? — with a definite answer rather than a shrug. A design review that has looked at four arrangements has looked at four of 153, and knowing the denominator is the difference between confidence and habit.

It also puts a floor under an old and unsatisfying kind of claim. This is the standard arrangement for the job is a statement about what has been built, and it is usually right — the standard arrangement is standard because it works. What a census adds is the shape of what was not tried, and in particular whether the untried alternatives are two or two hundred.