Machine Kinematics
solved, then drawn
Concept

Infinitesimal flex — where it appears

A velocity the constraint matrix permits at one configuration, which may or may not be the start of a motion. It is an upper bound on what a mechanism can do rather than an answer, and where the constraints repeat one another it can be blocked at second order — measured by whether the length changes it forces are orthogonal to every dependency.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

Free to spin, and it cannot turn at all. An ellipse of semi-axes 1.4 and 0.9 inside four flat walls that touch it at the ends of its own axes. Every normal points at the centre, so every row's moment is nought and the four rows span two dimensions rather than three: the cone of permitted twists is the whole spin axis, a line through the origin, and the first-order answer is that the part is free to turn either way. Drag the angle and watch what happens. The ellipse's reach in the direction of the top and bottom walls is √(a²sin²θ + b²cos²θ), which is smallest at θ = 0 and grows from there, so any rotation whatever drives it into both of them — by 0.016 at this angle. A nullity is a candidate and not a motion, and this is the shape of case the fields before this one could not produce: not a mechanism at a singularity, but an ordinary part in an ordinary pocket. positioned by solving, not by drawing.

Free to turn and unable to

An ellipse in a pocket the size of its own bounding box has four contacts whose rows span two dimensions, so the cone of permitted twists is a whole line and the first-order answer is that it spins both ways. It cannot turn by any amount whatever: the penetration grows as the square of the angle, with a fitted exponent of 1.9944, and a circle in the same pocket turns for ever.

holding · Restraint The smallest mechanism that is not one. Two bars from a free joint to two pinned ones, with the three points in line. The constraint matrix has two rows and two columns, both rows are horizontal, and its rank is 1: one freedom left over, pointing straight up, and one dependency among the two bars. Move the joint up and neither bar changes length to first order, which is what the freedom says. To second order both bars get longer, by the same amount and in the same direction, and there is nothing to trade off against — which is what the dependency says. The obstruction is the dependency applied to the second-order stretch and comes to 1.414214; anything but nought there and the freedom is not the beginning of a motion. Lifted by 0.34 the bars are 0.05622 longer, which is the whole argument drawn to scale.

It moves to first order and not at all

Two bars from one joint to two pinned ones, all three in line: the rank leaves a freedom pointing straight up, and lifting the joint stretches both bars. The obstruction is 1.414214, the walk travels a millionth of what it is asked to, and how far it gets is a property of the tolerance rather than of the mechanism.

networks · Network The framework Maxwell's count calls a structure. Six joints and twelve bars in space. Three coordinates each gives eighteen unknowns, six rigid motions come off, and twelve bars is exactly twelve constraints — Maxwell's count is 6 against six rigid motions, which is the definition of isostatic: no mechanism, no redundancy, every bar carrying its own share and nothing spare. The rank is 11, not twelve. There is one dependency among the bars and one freedom left over, and the freedom is a genuine finite motion: walked here with every bar held to 4.4e-16 of its own length. The reason is a symmetry — three pairs of joints exchanged by a half turn about one line — and it is built into the coordinates rather than assumed of the result. positioned by solving, not by drawing.

Twelve bars and a symmetry

Six joints and twelve bars in space is Maxwell's count exactly: no mechanism, no redundancy, nothing spare. Place three pairs of the joints so that a half turn about one line exchanges them and it moves — a finite motion, walked with every bar held to five ten-thousand-billionths of its own length, on a framework the arithmetic calls a structure.

spatial · Spatial The same patch twisted 17.2°: 35 mechanisms, all at the edge. A rhombus of 8 × 8 kagome cells — 192 joints, 346 bars — with every up-pointing triangle turned by 17.2° about its own centre. Each joint is drawn with an area proportional to its weight: its share of the diagonal of the projector onto the patch's mechanisms, which does not depend on how the mechanisms are written down and adds up over the joints to the number of mechanisms, 35. That number is Maxwell's count, 2 × 192 − 346 − 3, and the rank agrees with no redundant bar. The mean weight is 0.341 on the outermost ring of cells and 0.015 on the innermost. The twist kinks every line of bars at every joint, and the mechanisms fall away from the edge 23-fold in 3 cells.

The count says how many and not where

A kagome lattice has three joints and six bars in every cell and counts to exactly nothing, so a patch cut from it has as many mechanisms as its edge has lost bars: 5L − 5 for a rhombus of L cells a side, which the rank confirms at every size with no bar redundant. Straight or twisted, the number is the same. Where the mechanisms are is not: a straight patch keeps nearly half its edge weight in the middle, and a patch whose triangles are turned by 17° keeps a twentieth.

networks · Network A framework built off square, and the play it asks for to move at all. The nine bars with their two lines 4° from perpendicular, as built and after the first joint has been pushed 0.2 along its line. The framework has a freedom by rank and no motion, so the push cannot be taken with the bars at their lengths. With every bar allowed to be wrong by 1.509e-4 — which is a radial clearance of 7.547e-5 at each end — a placement exists, and the worst bar in it is out by 1.509e-4. The same push on the perpendicular framework needs no allowance at all, because there it is a motion.

The right angle as a tolerance

Dixon's nine bars move only when their two lines are exactly perpendicular, and a framework built a degree off square has a freedom by rank and no motion at all. Give its joints clearance and it moves a bounded distance: the play each bar needs is proportional to the tilt and to the square of the travel, one constant serves every tilt, and all nine bars end up at that play exactly. Then it reaches its first crossing, where the push is as large as the motion ever makes it, and a hundredth of a push further costs fifty-eight times the law — square or not.

constraint · Mobility A strip of distorted kagome with its mechanisms drawn by weight: one edge, the other, or both. A strip of kagome lattice 10 cells wide, periodic along its length (6 of its 12 cells drawn), with each up-triangle's three joints moved along the triangle's sides by 0.1, 0.1 and −0.1. Each joint is drawn with an area proportional to its share of the strip's mechanisms, summed over every wavevector along the strip. The cut leaves each long edge the same number of missing bars, so the count says the two edges share the mechanisms equally. They do when the three moves have one sign. With the third reversed, the top half carries 19.99 of the 24 and the bottom 4.01. The bulk's windings, measured from the twisted kagome's, are (1, −1).

An edge that knows which side it is on

Cut a strip from a kagome lattice and each long edge loses the same bars, so the count says the two edges share the strip's mechanisms equally. Move each triangle's three joints by the same signed amount and they do. Reverse one move and every mechanism that is not a translation moves to one edge. The shift is a whole number per cell, set by the winding of one six-by-six determinant round the lattice's zone. It changes only where one family of bars lines up straight.

networks · Network A seam that holds two mechanisms. Ten rows of a strip of kagome lattice sixty cells wide, centred on the line where two differently distorted lattices meet. Below it every up-triangle's joints are moved along its sides by 0.3, 0.3, −0.3, polarization −1 in the component that counts along the seam; above it by −0.3, −0.3, 0.3, polarization +1. No bar is missing anywhere along the seam. Each joint is drawn with an area proportional to its share of the strip's two mechanisms at one wavevector. They sit on the seam and die away from it into both lattices, and the strip's outer edges, out of the picture, have none. The wavevector along the seam is 3/12 of a turn per cell.

A seam with no bar missing

Join two distorted kagome lattices of opposite polarization along a line. Every joint on the line keeps all four of its bars, so Maxwell's count says the line holds nothing. It holds two per wavevector: two mechanisms one way up, two states of self-stress the other way up, exactly the difference of the two polarizations. The self-stresses are tensions the lattice carries with no load on it, along a line where no bar is spare. In a strip sixty cells wide the prediction holds at every wavevector a quarter of the zone or more from its centre.

networks · Network Dixon's nine bars with every joint somewhere inside its clearance. Dixon's framework — joints at -2, 1, 3 on one line and -1.5, 1, 2.5 on a line square to it, every joint of one set barred to every joint of the other — with each joint moved to a seeded point inside a disc about its nominal place. The discs are drawn with a radius of 0.12 so that they can be seen; the framework measured has joint errors about a hundred times smaller. The nominal joints lie on two lines, which is one conic, and that is what lets nine bars on six joints flex. The joints as built lie on no conic: at an error of ρ, seed 2's smallest singular value is 0.1746ρ where the nominal framework's is nought, so by rank it is a structure.

The clearance that makes the framework generic

A clearance lets Dixon's nine bars move, and it also moves their joints off the two lines that let them move at all. Six joints off a conic make the framework rigid by rank, with a smallest singular value proportional to its distance from a conic. The same clearance that did the damage pays for it fourteen times over: a framework built anywhere inside its clearance discs is pushed to its first crossing using at most 7% of the allowance those discs give. The tilt of the lines decides how far it goes; its own joint errors change that by about 1%.

constraint · Mobility

Named alongside it

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

MobilityNull spaceRankRedundant constraintKagomeLatticeNetworkOverconstraintToleranceClearanceConstraint jacobianSecond order

All concepts