A contact turns and slides at once

A slide turns nothing gave the joints of a chain’s graph a second label. A joint could be a pin, which lets two links turn about a point, or a slide, which lets them translate along a line and forbids them to turn. Both take two freedoms away in the plane, and Grübler’s count, which cannot tell a pin from a slide, charges them alike. The essay found two rules the count cannot state. A loop of slides alone is freer than the count says, and a pin is locked when it is the only turning joint in a loop, or when it sits in a group of links whose count of freedoms reaches nought once the slides have tied orientations together across the whole chain. Across 102 placements on the three smallest chains both rules agreed with the rank of a randomly built chain’s constraint Jacobian.

Every machine with a cam or a pair of gear teeth has a third kind of joint. A cam touches its follower at a point, and the contact forbids exactly one thing: motion along the common normal, into or out of the other body. It allows the follower to slide along the cam’s face and to turn relative to it, both at once. It takes one freedom away, not two. A pin running in a slot is the same joint, and to first order so is every smooth contact.

That joint changes both halves of the earlier argument, and it brings something the graph cannot see at all. A contact’s one constraint depends on which way its normal points, and the graph records only that two links touch.

The census with three labels #

A choice of joint kinds is a labelling of the graph’s edges by pin, slide or contact. Two labellings give the same chain when a symmetry of the chain carries one onto the other, which is the same orbit counting that chose which link to bolt down and where the slides go. The four-bar has 81 labellings and a symmetry group of eight, and they fall into 21 different chains and 45 mechanisms once a link is held. Watt’s and Stephenson’s chains each have 2,187 labellings, and each gives 648 different chains, with 3,321 and 3,564 mechanisms.

For each of those 1,317 chains the prediction can be checked the same way as before, with the instrument counting and measuring mobility set against Grübler’s count. Build the chain at random — every link given a random pose, every joint’s geometry chosen so the joint is satisfied there, every contact’s normal pointed in a random direction — then write its velocity constraints and measure the rank. Two rows for a pin, two for a slide, one for a contact: the relative velocity of the two bodies at the contact point, resolved along the normal. The rank gives the freedom, and a turning joint is locked when a row for its relative turning rate adds nothing to the rank.

What changes, and what does not #

A contact changes the rules in two places.

A contact turns. The loop rule came from adding up relative angles round a loop, which must total nought. A slide contributes nothing to that sum, so a loop with one pin and the rest slides forces that pin’s angle to nought. A contact contributes a free angle, like a pin. So the rule is now about turning joints, pins and contacts together: a loop with exactly one of them locks it, and a loop with two or more locks neither.

A contact counts one. The group rule gave each body two freedoms of position and each orientation class one freedom of angle, where a class is a set of links the slides tie to one angle, read over the whole chain. It took two away for each pin and one for each slide, since the slide’s other constraint had already been spent merging two classes. A contact takes one away and merges nothing. When a group’s count reaches nought with a turning joint still inside, the group is a structure and every turning joint in it is locked, contacts included.

Everything else carries over unchanged, including the extra freedom a loop of slides alone brings. A loop with even one contact in it is no longer a loop of slides, because the contact’s angle is free to close the rotation equation.

With those two changes the graph predicts the freedom and the locked joints of every one of the 1,317 chains, exactly as the rank finds them. Neither shortcut does, and both are shortcuts a designer might take. Reading every contact as a slide — the joint that looks most like a pin in a slot — misses 769 of the 1,317, because a slide locks loops a contact frees. Reading every contact as a pin — the joint that also turns — misses 36, all on the six-link chains, because a pin takes two freedoms from a group’s count where a contact takes one.

One joint, three ways #

A single placement shows what each difference does. Watt’s chain has slides on the joints 0–1 and 1–2. Make 2–3 a slide as well, and the loop 0–1–2–3 has one turning joint left, the pin 0–3, which locks. That welds links 0 and 3 together, and the other loop — pins at 0–5, 5–4 and 4–3 — becomes a triangle hung between two points of one rigid body, so its three pins lock too. The chain has one freedom, which the count gets right, and not one of its four pins turns. It is the same propagation a slide turns nothing found, arriving through a loop rather than a group.

Make 2–3 a contact instead and nothing locks. The loop 0–1–2–3 has two turning joints, the contact and the pin 0–3, and the loop’s angles can close with both turning. The chain has two freedoms. Make it a pin and nothing locks either, and the chain has one. The contact rescues the loop the slide would have locked, and it leaves the chain one freedom more than a pin would. That extra freedom is the follower lifting off, or sliding along, the face it is not otherwise held against.

What the graph cannot record #

Up to here the contact has behaved like a joint that can be read from the graph. It is not one entirely, and the reason is in the row that represents it. A pin’s two rows say its point is shared; they do not depend on anything else about the pin. A slide’s rows say the two links keep one angle and one line. A contact’s one row says the relative velocity at the contact point has no component along the normal — and which direction that is is geometry the graph never records.

The random construction points every normal in an arbitrary direction, which is what makes the rank generic. The question is whether any direction is special: whether a contact can be aimed so that its row stops constraining anything, and the chain gains a freedom the count cannot predict.

There is exactly one way for that to happen. Take the contact away and ask what motion the rest of the chain leaves between the two links it joins. If the rest leaves those two links one freedom relative to each other, that relative motion is a rotation about a single point, its pole — or instant centre — and the contact point moves square to the line joining it to the pole. A contact whose normal lies along that line is being asked to forbid a motion that is already sliding along its face. Its row reads nought on the only motion there is, and the contact forbids nothing. If the rest leaves two or three relative freedoms, the contact point’s velocities fill the plane, and no single normal direction can be square to all of them.

So the measurement to make is how many freedoms each contact’s two links keep when the contact is removed. Over every contact in every labelling of the census — 28 on the four-bar and 1,512 on each six-link chain — the answer is two in 131 cases and three in the other 2,921. It is never one. These chains are too loosely joined for a single contact ever to be the constraint that closes a structure, so no one contact’s direction can change a verdict on them. That is the version of the feared blind spot a single contact can produce. The other version — several contacts aimed together so that their rows become dependent, as normals meeting at one point would — is not ruled out by this measurement, and is taken up below.

A stop on a four-bar #

It has room as soon as a contact is added to a chain that already moves, which is what a stop is. A four-bar has one freedom. A contact between two of its links that are not already pinned together — the coupler against the frame, say, as a latch or a limit stop would be — is a fifth joint on four links, and the count says the result is a structure — the same count that calls three parallel bars between two frames rigid when they are not. The rest of the chain, the four-bar itself, leaves the coupler exactly one freedom relative to the frame, so the coupler has a pole, and a stop aimed through it should hold nothing.

The pole is found without any solving at all. By Kennedy’s theorem the pole of the coupler on the frame lies on the line of the crank and on the line of the rocker, since each of those links turns about its fixed pivot and carries a pin of the coupler. The two lines meet at one point. For the four-bar in the figures — frame 4, crank 1.2, coupler 3.4, rocker 2.8, crank at 70° — that point is at (2.505, 6.883), well above the linkage. The same pole comes out of the four-bar’s own velocity constraints, as the point about which its one motion turns the coupler, and the two routes agree to the last digit of the arithmetic.

At every normal direction but one, the stop removes the four-bar’s freedom: the rank goes up by one and the count is right. With the normal aimed through the pole, the rank does not go up. The stop removes nothing at first order, the four-bar keeps its freedom, and the graph of the stopped four-bar — four links, four pins, one contact — is the same graph in both cases.

A stop can join any two of the four links, and each pair has its pole. Crank against frame, and the pole is the crank’s own pivot: a stop pressing radially on a crank, like a brake shoe with no friction pressing on a wheel, holds nothing. Rocker against frame, and it is the rocker’s pivot, for the same reason. Coupler against frame, and it is the meeting of the crank’s and rocker’s lines. Crank against rocker, and Kennedy’s theorem puts it on the frame line extended, where the coupler’s line crosses it, at (−1.689, 0). In all four, the pole from Kennedy’s construction and the pole from the four-bar’s own motion are the same point, and aiming the stop there leaves the four-bar free while aiming it a degree away holds. What one contact forbids found Reuleaux’s version of the same fact: a single contact permits rotation about any centre on its own side of the normal line, and the centres on the line itself are the ones it cannot resist at all.

How much a stop holds #

A stop either holds or does not at first order, and that makes the special direction sound like a single bad angle to avoid. It is more than that, because the holding fades continuously towards it.

Apply a stop’s row to the four-bar’s one motion, and divide by the contact point’s relative speed. What comes out is the share of that motion the stop forbids. For all four stops, at every angle, it is the sine of the angle by which the normal misses the line to the pole, to $4 \times 10^{-16}$. That is the moment arm of the normal about the pole divided by the contact’s distance from it, and it is the lever through which the stop resists the motion.

So a stop aimed 3° from its pole forbids 5.2% of the motion it is there to forbid, and one aimed 1° away forbids 1.7%. The force it has to carry to resist a given load on the mechanism rises as one over that share. A latch whose face happens to lie a degree from the line to its pole is a latch that holds, as far as any count or rank is concerned, and it carries fifty-seven times the force the same latch would carry with its normal square to that line. A ratchet with no teeth works in this nearly-free region on purpose: its rollers are wedged at a shallow angle, and whether they grip is then a question about friction that no drawing of the contacts settles.

Where real machines put their normals #

A gear pair is two gears pinned to a frame with one tooth contact between them: three links, two pins, one contact, one freedom. The contact’s two links, the gears, are not a stopped mechanism, because without the tooth each gear turns freely on its own pin and the two keep two freedoms relative to each other. There is no single pole. The law of gearing is a statement about where the normal crosses the line of centres — the point that sets the speed ratio — and not about a redundancy.

A cam and a follower are the same shape: frame, cam, follower, two pins and a contact. Take the contact away and the cam and the follower each turn freely on their own pins, two freedoms between them, so there is no pole for the normal to pass through. The census chains say the same about every contact they contain. So the blind spot is not in the machines that use contacts to transmit motion. It is in the machines that use a contact to stop motion — latches, detents, limit stops, locking pawls — where a contact is added to something that already moves and the design intent is to make it a structure. There, and only there, the normal’s direction decides whether the design does what the count says it does.

What the graph does not settle #

First order only. A stop aimed through its pole forbids nothing at first order. Whether the four-bar can then actually move depends on the second order: on the curvature of the stop’s face against the curvature of the contact point’s path. A face curved more sharply than the path would let the point leave it one way and run into it the other. A higher pair has no group measured how a contact lifts at second order, and nothing here measures it for a stop.

Contacts push; they do not pull. A real contact holds only one way along its normal. The rank treats every contact as holding both ways, as a pin in a closely fitting slot would. Which way a stop has to push, and whether the load on the mechanism asks it to, is outside the graph and outside the rank.

Contacts aimed together. The measurement above rules out a single contact becoming idle through its direction. Two or more contacts on one chain could in principle be aimed so that their rows depend on each other — normals through one point is the classical case — and no census here searches for such combinations.

Larger chains. The census stops at six links. On chains with more joints a contact can be the last constraint of a group, and the direction of its normal can then decide a verdict even without a stop being added. Where that first happens is not measured.

Friction. A contact with friction resists sliding too, up to a limit, and a stop aimed near its pole can still hold through its friction — which is exactly how a roller clutch holds. The rank assumes smooth contacts throughout.

Still open: when the stop’s face is curved #

A stop aimed through its pole leaves the four-bar a first-order freedom, and whether that freedom is a real motion is the second-order question. The contact point moves along a path whose curvature is set by the four-bar; the stop’s face has a curvature of its own. If the face bends away from the path faster than the path bends, the four-bar can move a finite distance before the contact re-engages; if it bends towards it, the stop holds after all, at second order, and the design survives its bad aim.

The distinct argument there would be that comparison made on the four-bar: for each stop aimed through its pole, the path’s curvature at the contact point from the four-bar’s own motion, the face’s curvature as a parameter, and the second-order gap as the four-bar moves. The threshold face curvature at which a pole-aimed stop starts to hold would be a design number no rank can give, and whether it is the same as the curvature of the path’s osculating circle — which is the natural guess — is the thing to measure.