Six bars, and what the extra dyad buys

Forty-six essays, and every linkage in them has had four bars.

That is not narrowness. Four is the smallest closed chain with a degree of freedom, and almost everything interesting about planar mechanisms — Grashof’s condition, toggles, coupler curves, inversions — is already present in it.

The next size up is not five. It is six, and the reason is a counting argument worth doing.

It is worth doing because the answer is not obvious and the argument is three lines. A reader who expects mechanisms to come in every size is expecting something the arithmetic forbids, and the forbidding is a property of planar chains rather than of what anybody has thought to build. Odd numbers of links are not rare or awkward; with one degree of freedom and only pin joints, they do not exist.

Why not five #

Grübler’s criterion for a planar chain of $n$ links joined by $j$ pin joints gives

$$M = 3(n - 1) - 2j.$$

Setting $M = 1$ gives $3n - 2j = 4$, so $3n$ must be even, so $n$ must be even. Five links cannot close with one degree of freedom, and neither can seven.

At $n = 6$: $j = 7$. Six links, seven joints.

Now count joint-ends. Seven joints connect two links each, so there are fourteen link-ends to distribute among six links. Six binary links would account for twelve, which is two short — so two of the six links must carry three joints, and the other four carry two.

That is forced. Every six-bar chain with one degree of freedom has exactly four binary links and two ternary links, and there is no arrangement with a quaternary link or with three ternaries.

And the classification is one question #

Given two ternary links among four binaries, there is exactly one thing left to decide: do the two ternary links share a joint?

If they do, the chain is Watt’s. If they do not, it is Stephenson’s.

That is the whole taxonomy. Two chains, and the difference between them is an adjacency in a graph.

Because the classification is a statement about a graph, this site computes it from the graph. classifyChain takes each link as the set of joints it carries, counts how many are ternary, checks whether the ternary ones intersect, and returns the name. Nothing in either constructor declares which chain it is building.

That caught something. The first version of the Watt linkage in lib/sixbar.js hung its second dyad off the coupler rather than off the rocker, and the classifier called it a Stephenson. The picture was indistinguishable — six links, seven pins, one degree of freedom, a perfectly ordinary-looking mechanism — and the only thing that knew it was the wrong chain was the incidence count.

What the counting argument does not settle #

The link–joint arithmetic above says that a six-bar with mobility one has four binary and two ternary links, and that the two ternary links either touch or do not. It does not say that both cases are realisable, and that is a separate question the argument cannot answer.

It happens that both are, and the two chains are the standard result. But the counting is a necessary condition, not a sufficient one: there are link–joint counts that satisfy Grübler’s formula and correspond to no connected chain at all, and there are arrangements that are connected and contain a rigid sub-chain, so that the mobility is distributed wrongly — part of the mechanism moves and part of it is a truss.

This site’s own first field is largely about the other direction of the same gap: Grübler’s formula counts constraints without asking whether they are independent, so a mechanism with redundant constraints is declared immobile and moves anyway. A parallelogram linkage with a third parallel bar is the standing example, counted at zero and measuring one.

The formula is a statement about a graph and mobility is a statement about a geometry, and the two agree for almost every mechanism and not for the interesting ones. Here the two chains agree by both routes, which is the unremarkable case and is worth checking precisely because it is unremarkable — an assertion that only ever ran on the exceptions would not notice a measurement that had stopped working.

Mobility, by two routes as usual #

Both chains are asserted to have mobility one, by Grübler’s formula from the topology and by the rank of the constraint Jacobian from the geometry. Both give 1 for both chains.

Getting the second of those right needed care, and the error is instructive because it produced a number that was true and not the number wanted.

The first version took the rank of the Jacobian as built — including the rows for the driving constraint — and reported mobility zero for a mechanism that visibly moves. That is correct arithmetic: with the input pinned, there is nothing left free. It is not what mobility means.

Mechanism.mobility already knew this. It substitutes the crank constraint with the bar it stands for before measuring, rather than deleting it, because a crank constraint pins a joint’s position — which is the input plus the link — and deleting it removes the link too. The site’s foundation phase recorded getting that wrong and giving a four-bar two degrees of freedom.

So chainMobility delegates the measurement rather than repeating it. That is the fleet’s rule about where a fix goes, applied to a fix that was already made: the one place that knows about the substitution is the place that should do it.

What the extra dyad buys #

Six links is more parts, more joints, more cost and more slop. The question is what it makes possible that four does not, and there is a clean answer.

A four-bar’s output is a continuous function of its input with nowhere to hide. The rocker swings, and it swings the whole time the crank turns. There is no interval of the input over which the output does nothing.

That matters because a great many machines need exactly that. An indexing table must hold still while something is done to the workpiece; a film gate must hold a frame still while it is projected and then move it in a fraction of the cycle; a feed must pause while the tool cuts. The usual answer is a cam, and a cam is an expensive answer — it must be cut to a profile, it wears at a line of contact, and its motion law has consequences in acceleration and jerk that a designer has to think about.

A six-bar can produce a dwell without a cam, out of nothing but pin joints. How it does it is a genuinely opportunistic construction, and the measurement is stark: the six-bar built here holds its output inside a one-degree band for 145.8° of crank, against 41.9° for the four-bar it is built on.

Where the sixth bar goes matters more than that there is one #

The two constructors in lib/sixbar.js differ in one line, and that line is the whole design.

The Watt linkage attaches its second dyad to a point on the rocker, at a stated fraction along it. The Stephenson linkage attaches its second dyad to a point on the coupler. Everything else — three ground pivots, a crank, a coupler, a rocker, an output link — is the same.

That one choice decides which chain it is, and it decides what the mechanism is good for. A dyad hung on the rocker is being driven by an angle, so it amplifies or re-times a swing. A dyad hung on the coupler is being driven by a point moving along a curve, and a coupler curve is a far richer thing than a rocker’s angle — which is why every construction that wants unusual output motion hangs its second dyad there.

The attachment point on the rocker is an argument to the Watt constructor rather than a constant, and at a fraction of zero the second dyad hangs off the ground pivot and does nothing at all. That is a degenerate case worth having available: it is what says the parameter is load-bearing rather than decorative, and it is the same discipline that puts a detune argument on every assertion that claims a construction is doing the work.

The four-bar’s own dwell, and why it has to be in the figure #

The comparison in that figure is not decoration and the essay would be dishonest without it.

Every mechanism holds still to within a degree somewhere. A rocker reversing at the end of its swing has zero velocity there, so its angle changes by almost nothing over a range of crank angles either side. Quoting “the output holds within a degree for 146° of crank” without a baseline is quoting a number that any linkage would produce some version of.

The four-bar’s own best interval, measured the same way at the same tolerance, is 41.9°. The six-bar’s 145.8° is 3.5 times that, and that ratio is the claim.

assertTheDwellIsTheArc requires it, along with something sharper: that the dwell is largest at exactly the arm length the construction specifies, and that a shorter arm gives a mechanism which does not assemble at all. The arc’s radius is not merely a good choice; it is the boundary of the buildable range, and the optimum sits on it.

Inversions, and how many mechanisms this really is #

A chain is not a mechanism until one of its links is fixed, which is the point the inversions essay makes about the four-bar: one chain, four choices of frame, four mechanisms that behave completely differently.

The same arithmetic runs here and gives a larger number. Watt’s chain has two distinct kinds of link to ground — a ternary or a binary — giving two inversions, conventionally numbered as Watt’s first and second. Stephenson’s has three, because its ternary links and its binaries sit in three distinguishable relations, and they are numbered the same way.

Five mechanisms from two chains. Each of them is a different machine with different input-output behaviour, and the naming is a piece of nineteenth-century bookkeeping that survives because there is nothing better to call them.

This site builds one of each chain and does not enumerate the inversions. That is a gap and it is named as one: the machinery to do it exists — grounding a different link is a change to which joints are declared fixed — and the essays to justify five more mechanisms do not.

What else the extra dyad is used for #

Dwell is the headline and it is not the only thing. Three other uses are worth naming, because they explain why six-bar chains are common in machinery that is otherwise plain.

A larger stroke than the links. A four-bar’s output swing is bounded by its own proportions. Putting a second dyad on the output multiplies the swing, which is how a modest crank drives a rocker through a large angle without any link being long. The same trick appears in windscreen wipers and in press linkages.

Two coordinated outputs. A six-bar has two loops, and each can carry an output. The outputs are then rigidly related to each other and to the single input, which is a coordination that would otherwise need gears or a second actuator. Aircraft landing gear that must retract and fold a door in sequence is the canonical example.

Coupler curves a four-bar cannot draw. A four-bar’s coupler point traces a sextic. A six-bar’s tracing point moves in a curve of much higher degree, and the family of shapes available is correspondingly larger — which matters for path generation, where the whole problem is finding a mechanism that draws a prescribed shape.

None of those is a dwell and all of them are the same underlying fact: an extra loop is an extra relation between the mechanism’s own parts, and relations are what a linkage has instead of intelligence.

The two chains behave differently, and this site does not measure how #

An honest limit. This essay establishes that the two chains are different — the classification is computed, the ternary adjacency differs, and one is a Watt and one is a Stephenson — and it says almost nothing about what that difference does.

The literature is clear that it matters. Stephenson chains are the ones used for dwell mechanisms and for most coupler-driven constructions, because their two loops are joined through a binary link and one loop can be designed largely independently of the other. Watt chains, whose ternary links touch, couple their two loops more tightly and are used where that coupling is the point.

Measuring that here would mean building both chains for the same task and comparing, and the comparison is not made. One Watt linkage is built to have something to classify against, and one Stephenson because it is the dwell mechanism the next essay needs. Neither is presented as the better choice for anything.

The gap is worth naming rather than papering over with the standard sentences from the literature, because the standard sentences are exactly the kind of claim this site exists to put a number against and has not.

Where six bars stop being worth it #

It is worth saying why the taxonomy does not go on forever in a form anybody uses.

Eight links with $j = 10$ gives mobility one as well, and the number of distinct chains jumps: there are sixteen eight-bar chains, and the count grows fast after that. The enumeration is a solved problem in the literature and is a piece of graph theory rather than of kinematics.

What stops being useful is the design value. A six-bar is chosen because it does something specific a four-bar cannot — dwell, a particular coupler curve, a coordinated pair of outputs — and each of those has a construction behind it. Beyond six, the choice of which of sixteen chains to use, and which of its inversions, is not guided by anything comparable, and in practice a designer who needs more than six links assembles them from four-bars and dyads whose individual behaviour is understood.

Which is exactly how the dwell mechanism in the next essay is built: a four-bar, its coupler curve, and one dyad hung on it. The six-bar is not designed as a six-bar. It is designed as a four-bar with something attached, and the attachment is the interesting part.

That is worth stating as a method rather than as an observation about one example. A six-bar with a purpose is almost always a four-bar whose behaviour is understood, plus a dyad chosen to exploit some specific feature of that behaviour — a stretch of coupler curve, a limit position, a coordination between two points. Designing the six links together, as six, is a nine-parameter search with no structure to hold on to.

One last observation about the classification, because it is a striking amount of consequence for a single bit. The two chains differ in exactly one respect — whether the two ternary links share a pin — and that one bit decides the computational character of the mechanism as well as its behaviour. Watt’s chain comes apart into two dyads, so it is positioned by two circle intersections and its assembly count is four; Stephenson’s does not, so four of its links have to be solved together and there is no construction. One adjacency, and on one side of it a mechanism a compass can place and on the other a mechanism only a solver can. That is a much sharper statement of what the taxonomy is worth than there are two six-bars, and it explains why the two names survived in a subject that has forgotten most of its nineteenth-century vocabulary: they are not two arrangements to be remembered but two different problems, and the difference arrives before any dimension is chosen.