Concept

Relative motion — where it appears

The motion of one link measured from another, which is a property of the pair and not of the frame. It is what survives inversion untouched, so a chain's whole kinematics can be written as the relative rotations between its links and the choice of ground then read off them.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

One chain, four mechanisms

Which link of a four-bar is bolted to the bench is not a property of the chain. It is a decision about where the bench is, and making a different one gives a mechanism that looks and behaves completely differently while being, as a chain, the same object — which is why the Whitworth quick-return and the oscillating-cylinder engine are both a slider-crank.

linkages · Fourbar

The chain's whole configuration space, drawn on its two angles. A slider-crank asks one thing of its two angles: a cos θ + b cos φ = e, with θ the crank's angle and φ the rod's. Each panel is the square of those two angles from −π to π, with the curve that equation cuts out, followed by arclength on the equation alone. The number under each panel is how many whole turns θ, φ and φ − θ make around a circuit. A curve that crosses the square from side to side carries a turn in that angle; a closed loop inside the square carries none. The two regions where a member turns have two circuits each and the two where none does have one, which is what the four-bar regions these came from predict.

Three rotations, and four benches

The slider-crank chain's four inversions are four famous machines, and they are not four classifications. Four links make six pairs, one of those pairs cannot rotate at all because a slide is a rotation of nought, and the five that are left are three quantities between them. So an inversion chooses which two of the chain's three rotations sit at its bench, and the four regions of length space already say what all three do.

linkages · Fourbar

A six-bar followed round one circuit, with the links that turn against the frame coloured. A Watt six-bar — a four-bar of ground 4, crank 3.17, coupler 3.10 and rocker 1.90, with a dyad of 3.83 and 3.03 hung from its rocker to a third ground pivot — followed once round one of its 4 circuits. Links drawn in ink come back to where they started; links drawn in colour make a whole turn against the frame. On this circuit that is rocker + dyad link + output, and they turn together, so every one of the fifteen relative rotations between pairs of links is either nought or that one turn. The pins' paths are dashed.

Fifteen pairs and one cut

A six-bar has fifteen pairs of links, and each pair has a relative rotation. Followed round every circuit of more than four thousand random Watt and Stephenson six-bars, those fifteen rotations come from six whole-number windings. Almost always they take two values, so the links split into a group that turns once against the rest. How many inversions can be driven is then the number of links touching that cut: nought, three, four or five, never one, two or six. One chain in ten can have different cuts on different circuits, and a Stephenson chain can turn its rocker twice against its dyad.

linkages · Fourbar

Named alongside it

The objects these essays reach for when they reach for this one.

Grashof's condition Kinematic chain Kinematic inversion Circuit Quick return Slider-crank Classification Configuration space Constraint Engine Inversion Mobility

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