Machine Kinematics
solved, then drawn
Concept

Shear — where it appears

A deformation that slides parallel layers over one another, turning a square into a parallelogram while every length along the layers is kept. A pin-jointed grid's only motions are shears of whole columns and whole rows, which is why bracing one cell constrains a column and a row together.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

A network with no boundary at all. The kagome lattice, drawn out to 5 cells across and continuing for ever. The measurement is made on one cell: 3 joints, 6 bars, and a bar that leaves the cell comes back into it, written against the far end's position in the neighbouring cell. There is no boundary anywhere in the arithmetic, and the size of the network has gone from being a parameter to not existing. The highlighted triangle is the cell; every other line on the page is a copy of one of its six bars. positioned by solving, not by drawing.

The cell that repeats for ever

Take the size of a network to infinity and it stops being a parameter. What is left is one cell, six bars, and a question nobody has to ask about a finite assembly: does the pattern's period count as a body? A square grid is rigid if it does not and shears if it does, and so does the kagome.

networks · Network 5 braces, and it is rigid. A 3×3 grid of squares with 5 of its cells braced by a diagonal, and no freedom left. The bipartite graph on the 3 columns and 3 rows, with one edge per braced cell, has 1 component — and the number of freedoms is one less than that, at every bracing there is. Nothing in the rank computation knows about columns, rows or graphs.

Which diagonal rigidifies a grid

A three-by-three grid of squares needs five diagonals and eighty-one of the hundred and twenty-six ways of placing five will do. Which ones is not a rank question at all: it is whether a graph on the grid's columns and rows is connected, and eighty-one is the number of that graph's spanning trees.

networks · Network Six cables hold a three-by-three grid only when they run round a cycle. A grid of three by three square cells whose bars are rigid and whose six braces are cables, each on one diagonal of a different cell. A cable resists stretching and not shortening, so it forbids its cell's shear in one sense only. Right, the graph that decides it: one node per column and one per row, and one arrow per cable, from the column to the row for a rising diagonal and back for a falling one. This grid is rigid: its arrows run round a single cycle through all six nodes, so every node can reach every other, and a self-stress pulls on all six cables. Of the 5,376 ways to place six cables in six of the nine cells, twelve are rigid.

A cable is half a brace

A square grid braced with rigid diagonals is rigid when a graph on its columns and rows is connected, so a three-by-three grid needs five braces and eighty-one sets of five will do. Brace it with cables instead, which pull and do not push, and each cable becomes one direction of an edge. The grid is rigid when that directed graph can get from every column and row to every other, which takes six cables, and only twelve of the 5,376 ways of placing six do it.

networks · Network Twelve cables, any one of which can go slack. A three-by-three grid of square cells braced with twelve cables: every row and every column of cells holds one cell with a rising cable, one with a falling cable and one crossed with both (RFX / FXR / XRF, reading R rising, F falling, X crossed). Right, its directed graph, one arrow per cable, and every column and row node has exactly two arrows in and two out. The dashed cable is the one gone slack, the rising cable of the cell in column 1, row 1. With it removed the graph is still strongly connected and a self-stress still pulls on all eleven remaining cables, so the grid is still rigid. The same is true whichever of the twelve goes slack.

Twelve cables survive a slack one

A three-by-three grid braced with cables is rigid on six and never survives the loss of one while each cell holds a single cable. Let a cell hold two crossed and every one of the 262,144 fillings can be tried: none of eleven cables survives a slack one, thirty-six of twelve do, and the rule for which is a count at each row and column, two cables in and two out. That count is the whole rule on every grid narrower than four cells. At four by four it stops being enough, and a grid can meet it everywhere and still hang on a single cable.

networks · Network A pretension chosen for the loss, and what a loss leaves. A three-by-three grid braced with twelve cables in a Latin square (XRF / FXR / RFX), pretensioned in the self-stress that keeps the most load capacity after any one cable goes slack, the largest cable pretension at one; line widths are in proportion to force. All twelve are taut: the cables carry from 0.81 to 1.00, and the smallest load, in any direction, that slackens another is 1.098, in units of the largest pretension. Intact it is 1.098; after its worst loss 0.404.

A spare is worth what its pretension leaves

Thirty-six ways of placing twelve cables on a three-by-three grid survive a slack one, and the count at each node cannot tell them apart. Their loads can. Pretensioned at its best, every one of them takes a load of about 1.1 times its largest pretension before a cable goes slack. Lose a cable and the pull it carried is shared out elastically, and what survives is between a quarter and a four-hundredth of that. A pretension chosen for the loss instead of for the intact grid lifts the worst case: to 0.32–0.40 on the Latin squares, 0.22–0.27 when four cells are crossed, and never above 0.096 when six are. The family to build is the Latin square, and even among Latin squares three kinds of spare are not equal.

networks · Network

Named alongside it

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

NetworkRankRedundant constraintDesign ruleEnumerationJoint graphSelf-stressMobilityGrübler's criterionKagomeLatticePeriodic framework

All concepts