Machine Kinematics
solved, then drawn
Concept

Joint graph — where it appears

A mechanism's links as nodes and its joints as edges, with the geometry thrown away. It carries the loop count and the link assortment and nothing about where anything is, so a count read off it is right about what counting can see and blind to every coincidence.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

The mechanism, and the graph that decides how many loops it has. A lazy tong of 5 scissor units drawn over its own joint graph: a node for every body — 10 of them — and an edge for every pin, 13 of those. The number of independent loops is e − v + 1 = 13 − 10 + 1 = 4, which is how many closure equations somebody writing this mechanism out by hand would have to find and is the one quantity in the field that can be read straight off a drawing. It is also all the count knows: Grübler's 4 is 3(n − 1) − 2j and contains no geometry at all, which is why it is right here and wrong four rows further down the ledger. positioned by solving, not by drawing.

The loops are in the graph

Before a network is a mechanism it is a graph, and the one quantity that can be read straight off a drawing is how many independent loops it has: edges less nodes plus one. Grübler's count is that arithmetic and nothing else — which is why it is right about a tong at every size and says a deployable ring cannot open.

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

Named alongside it

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

NetworkDesign ruleEnumerationRankRedundant constraintShearGrübler's criterionMobilitySelf-stressSpanning treeCircuit rankConstraint jacobian

All concepts