Machine Kinematics
solved, then drawn
Concept

Stowed height — where it appears

How flat a folding assembly actually goes, once its members are made of something. A scissor stack's paper height falls to nothing as the bars lie down and its real one stops at two boss radii per stage, because what meets is the bosses of the bars two stages apart.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

A colouring is not an assignment. A pin joining links in planes 1 and 3 has to pass through plane 2, and anything in plane 2 whose material covers that pin is pierced by it. That condition involves three links at once where a colouring's conditions involve two, and it is invisible to a colouring because permuting colours preserves a colouring and destroys betweenness. It does not change the number of planes on any machine here. What it removes is arrangements: of the 6 three-plane colourings of a crank rocker, 2 can be built, and of Peaucellier's 192 exactly 96 can.

A plane is a colour

Assigning links to parallel planes so that no two conflicting parts share one is a graph colouring, and the answer for a four-bar is three. Then the pins have to get through, and the problem stops being a colouring: of the six proper three-plane colourings, two can be built.

bodies · Body A 4-stage scissor, open and as flat as it goes. On paper a scissor stack's height is n·L·sin φ and goes to nothing as the bars lie down. The bars are made of something: the two colours are the two planes the stack needs — two, at any number of stages, which is why a scissor folds at all — and the bars that share a plane are the ones two stages apart. They meet at φ = 0.0951, leaving each stage 0.0949 high, which is exactly the two bosses that touch: 2 × 0.0473. The stowed height of the whole stack is 0.380 rather than zero, and it grows with the stage count in the way a stowed lift's does.

A stack that has to fit

A scissor stack's height is n·L·sin φ and goes to nothing as the bars lie down — on paper. The bars are made of something, and what stops the fold is two bosses meeting: every stage keeps 0.0949 whatever the stack does, which is exactly twice the boss radius.

networks · Network

Named alongside it

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

BossConflict graphLayer assignmentLink bodyChromatic numberDeployableEnumerationFlat stateInterferenceKinematic chainScissor linkage

All concepts