A slide turns nothing

A kinematic chain, read as a graph, is a set of links and a record of which touch. Every joint in that record has so far been a pin. The graph counts the four-bar as one chain, Watt’s and Stephenson’s as the two chains of six links, and the frame is a second decision whose distinct outcomes are orbits of links under the chain’s symmetries.

Real machines are not all pins. A slider-crank has a block in a guide; a shaper has a slotted arm; an Oldham coupling has a disc sliding in two hubs. A planar joint that lets two links translate along a line and forbids them to turn relative to one another is a slide, and in the plane it removes exactly as many freedoms as a pin does — two. Grübler’s count, which cannot tell a pin from a slide, gives both the same weight and reads the same mobility off a chain whichever joints are which.

So choosing which joints are slides is a decision made on the graph before any length exists, and it has the same two questions the frame had. How many genuinely different choices are there? And which of them still give a mechanism that does what the count says?

Slides are counted the way frames are #

A choice of slides is a set of edges of the graph. Two choices give the same chain exactly when a relabelling of the links that preserves every joint carries one set onto the other — the same automorphism that decides whether two frames give the same machine. The distinct chains are orbits of edge sets, and the distinct mechanisms are orbits of a frame and an edge set taken together.

The four-bar has four joints and a symmetry group of eight: four rotations and four reflections of a square. Any one joint is carried to any other, so one slide gives one chain. Holding a link then gives two mechanisms rather than the one the pin-only chain gave, because the slide has broken most of the symmetry: only the identity and the mirror that swaps the two links on either side of the slide survive, and they fold the four links into two pairs.

Watt’s chain has seven joints and four symmetries. Its seven joints fall into three kinds, and the figure at the head of this essay draws one of each: the joint between the two ternary links, a joint from a ternary link to a binary one, and the joint between two binary links. Those give two, six and three mechanisms, eleven in all against the two Watt’s pin-only chain gives. Stephenson’s chain gives three chains and fourteen mechanisms with one slide, against three.

The multiplication is not a curiosity of counting. Each of those eleven is a mechanism somebody could build, with a different set of links fixed, sliding and turning, and none of them can be reached from another by relabelling. A designer choosing a six-bar with one slide from a list of five six-bars is choosing from a list that should have had twenty-five.

The double-slider gets its three, the slider-crank does not get its four #

The four-bar with two slides is the first case where the graph’s count can be checked against a list that already exists. There are six ways to put two slides on four joints, and they fall into two chains: slides on adjacent joints, sharing a link, and slides on opposite joints.

The adjacent chain has the link that carries both guides, the two blocks that slide in them, and the bar that joins the blocks by pins. The symmetries that survive swap the two blocks and nothing else, so there are three distinct frames: the guide link, a block, or the bar. The handbooks name three inversions of the double-slider chain, and they are these three — the elliptic trammel holds the guides, the Scotch yoke holds a block, and the Oldham coupling holds the bar, with the two blocks as the shafts’ hubs and the guide link as the floating disc. The graph’s count and the classical list agree exactly.

The one-slide chain does not agree. The graph gives two mechanisms, and the slider-crank chain has four classical inversions: the slider-crank itself, the rotating-guide engine, the oscillating cylinder and the hand pump.

The mirror that survives the slide swaps link 0 with link 3 and link 1 with link 2. On the graph those are the same machine. Built, link 1 is the crank and link 2 is the connecting rod, and the crank is the short one: it is what lets the crank turn all the way round, which is a condition on lengths and not on connections. Holding the guide and holding the block are a mirror pair only when the crank and the rod could exchange places, and a slider-crank whose crank is as long as its rod is a different kind of slider-crank altogether.

The double-slider’s mirror swaps two blocks, and a block has no length that could make the two different. That is the whole difference between the two lists. The graph’s orbit count is a lower bound on the handbook’s count of machines, and it is the bound itself precisely when the symmetry it uses swaps parts that no dimension distinguishes.

A loop of slides alone keeps its links at one angle #

Grübler’s count works by adding three freedoms for each moving link and taking two away for each joint. It assumes every joint’s two constraints are independent of every other joint’s. Slides break that assumption in two ways, and both are visible on the graph.

The first is the easier. A slide forbids relative rotation, so every link joined to another by a slide has the same orientation as that link for all time. Round a loop made of slides and nothing else, every link has one orientation — and the loop’s rotation equation, which demands that the relative angles round the loop add to nothing, is satisfied identically. One of the three equations the count charged the loop for says nothing.

The four-bar with four slides is four blocks in a ring, each sliding on the next. The count gives one freedom. The two translation equations of the loop leave four slide displacements with two freedoms, and a chain built at random with those four slides has a constraint Jacobian of rank seven in nine unknowns: two. The same shortfall makes the three-slide wedge — three links, three slides — move when the count calls it a structure, and it is the planar image of the observation that a count can call a mechanism rigid because two of its constraints repeat each other.

The rule is exact for independent loops. Each loop of slides alone that cannot be built from other loops of slides adds one freedom. Watt’s chain with every joint a slide has two independent loops and a third that is their sum; it has three freedoms, one more than the count for each independent loop, and not one for each of the three loops a reader can trace.

A loop with one pin cannot turn that pin #

The second way is the one that makes a slide a trap. Round any loop, the relative angles at the joints must add to nothing. A slide’s relative angle is nothing. So in a loop where every joint but one is a slide, the one pin’s relative angle is nothing too, and the pin cannot turn.

The four-bar with three slides has one freedom, as the count says, and a chain built at random with those slides confirms it. But the pin between links 0 and 3 is dead: its relative rate is nought in every motion the Jacobian’s null space allows. The mechanism is the three-slide wedge with a fourth link welded to the frame through a pin that is only a hinge on a drawing. The count got the number right and the machine wrong, and nothing about the number would say so.

This is the rule the count most needs and cannot state, because it is a statement about where the joints are rather than how many. A chain with a single slide in a four-link loop is perfectly healthy. The same number of slides arranged so that one loop holds three of them is not.

A group of links can be locked from outside itself #

A loop with one pin is the obvious way to lock a pin, and it is not the only way. Stephenson’s chain has two ternary links, 0 and 3, joined by two paths: the four-bar loop through binary links 1 and 2, and a longer path through links 4 and 5. Put slides on the three joints of the longer path — 0 to 4, 4 to 5, 5 to 3 — and leave every other joint a pin.

No loop of the chain has a single pin. The four-bar loop has four pins and no slides. The long loop has three slides and two pins. The loop rule has nothing to say, and the count gives one freedom.

The chain does have one freedom, and all four pins of the four-bar loop are locked. The three slides tie links 0 and 3 to one orientation, which is a constraint the four-bar loop never sees in itself: a four-bar whose frame and coupler are forbidden to turn relative to each other cannot move at all. Link 3 is then fixed to link 0, and the one freedom the chain has is links 4 and 5 shuffling along their slides between two bodies that no longer move apart.

The rule that sees this counts. Give each body two freedoms of position, and each orientation class — a set of links the slides tie to one angle — one freedom of angle; take away two for each pin and one for each slide. For any group of links, if that count reaches nought or less while the group still holds a pin, the group is a structure and every pin in it is locked. For the four-bar loop of Stephenson’s chain it is 2 × 3 + (3 − 1) − 2 × 4 = 0, because links 0 and 3 are one orientation class and 1 and 2 are two more.

The classes must be the whole chain’s. Counted from the slides inside the group alone, links 0 and 3 are two classes, the count is one, and the group is called free. It is not. A path of slides that runs entirely outside a loop locks it, and that is the part a reader looking at the loop would never see.

Locked pins also propagate. Watt’s chain with slides at 0–1, 1–2 and 2–3 has a one-pin loop, so the pin between 0 and 3 is locked; that ties 3 to 0, and the other loop — pins at 0–5, 4–5 and 3–4 — becomes a triangle hung from one rigid body. Its three pins lock with it. The machine has one freedom, and every pin in it is dead.

Checked by a computation that knows nothing about loops #

Every one of those verdicts is a claim about a graph, and the check that it is right has to come from somewhere that never reads the graph’s loops. The instrument is the one used across counting and measuring mobility: build the chain, write its constraints and measure the rank of their Jacobian.

The construction avoids solving anything. Every link gets a random pose. Every pin is then defined as a point common to its two links at that pose, and every slide as a line through a point of each, with the two links’ orientations fixed as they are — so the chain closes by construction to rounding, and its geometry is generic because nothing about it was chosen. The rank of the constraint Jacobian gives the mobility, and a pin is locked when adding a row for its relative rate does not raise the rank, which says every allowed motion already has that rate at nought.

The four-bar, Watt’s chain and Stephenson’s give 102 distinct placements from no slides to all of them. On every one, the graph’s prediction of the mobility — the count plus one for each independent loop of slides alone — equals the rank’s, and the graph’s list of locked pins equals the rank’s exactly. The check is also made to fail. Without the locked-group rule, the loop rule alone misses five placements. With the group rule but orientation classes read only inside each group, it misses Stephenson’s long-path placement. Both halves are needed, and each is needed by at least one chain on the smallest possible list.

The table carries its own pattern. With one or two slides every placement on the six-link chains is sound. With three, three of Watt’s twelve distinct chains and two of Stephenson’s already lock a pin. With four, only four and five of the twelve are sound. With five or more, none is: a six-link chain with five slides has only two pins, and some loop always has one or none.

Slides multiply the choices and then use them up #

Adding slides multiplies the six-bars a designer has to choose from, and then takes most of them away again.

The mechanism counts rise from two and three with no slides to fifty-four and sixty-one with three or four, and fall back symmetrically, because choosing which joints are slides is the same decision as choosing which are pins. Summed across every number of slides, Watt’s chain gives 200 mechanisms and Stephenson’s 232. The soundness is not symmetric, because a pin and a slide are not: only pins turn, and a chain short of pins has loops that cannot turn. Of the 200, 103 are sound, and of the 232, 131.

That asymmetry is the practical content. A slide is usually introduced to get a straight line of output or a sliding input — a piston, a table, a tool — and each one taken removes a turning joint the loop needed. The first two cost nothing on a six-bar. The third may cost a pin somewhere the designer was not looking, and the graph can say where before the first dimension is drawn.

What the graph cannot say about slides #

Which way a slide points. The graph records that two links slide; it does not record the direction of the line. The Jacobian rank above is taken at a random direction for every slide, and a chain whose slides are parallel, or perpendicular to a pin line, can be special in exactly the ways a count cannot see when pins are placed specially. Two parallel slides in a loop add a freedom the generic chain does not have.

Whether the machine is the one intended. The slider-crank and the hand pump are one graph mechanism, and which of them a builder has depends on the lengths. The slide census is a lower bound on machines, as the pin census was, and the one-slide four-bar is the smallest case where the bound is not reached.

Where the slide’s line is placed on its links. A slide between a block and a guide has an offset — the distance of the line from the pin on the other side of the block — which changes the slider-crank’s stroke and time ratio and leaves every statement on the graph untouched. The graph has no numbers at all, and a slide does not give it one.

When a pin should become a slide. That question has an answer, but not on the graph: three-position synthesis finds that the points that want a slide lie on one circle through the image poles, which is a statement about positions a designer has prescribed. The graph says where a slide may go without costing a turning joint; synthesis says where one is forced by the task.

Still open: joints that take one freedom away #

A pin and a slide each take two freedoms away in the plane. A cam in contact with a follower, a pair of gear teeth, or a pin running in a curved slot takes one: it forbids motion along the contact normal and allows both sliding and turning. Every planar machine with a cam or a gear pair has these on its graph as a third kind of edge.

The distinct argument there would be the census with three kinds of joint. The orbit counting is unchanged — an edge now carries a label from three instead of two — but the soundness rules change in both directions. A one-freedom joint turns, so it can rescue a loop that slides would lock; it also forbids nothing about orientation, so a loop of such joints and slides does not hold its links at one angle. The count that located Stephenson’s locked loop would become two for each body’s position, one for each orientation class, less two a pin, two a slide and one a contact, and whether that count still finds every locked joint, checked by the same randomly built Jacobian, is the open question. The likely finding is that it does not, because a contact’s single constraint depends on the normal direction at the contact point, and a group of contacts whose normals meet at a point is exactly the kind of special position a count reads as generic.