Machine Kinematics
solved, then drawn
Concept

Spherical linkage — where it appears

A closed chain whose joint axes all pass through one point, so every link moves on a sphere. Its lengths are angles rather than distances, and it has its own Grashof condition with the same structure as the planar one.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

A universal joint at 40° input, shafts 25° apart. Two shafts meeting at 25°, joined by a cross whose two pins are at right angles to each other and each at right angles to the shaft it carries. That is the whole geometry, and everything else follows from it. This is a four-joint spatial loop with all four axes through one point: Kutzbach counts −2 and the screw system has rank 3, so it has one degree of freedom and turns. At this instant the output shaft is at 42.79° while the input is at 40°, and the output is turning 1.0124 times as fast — which is not 1, and never is except at the four points of each turn where the curves cross.

The joint that is not constant velocity

A universal joint is the spatial mechanism everybody has met and almost nobody has been told the truth about. Its output shaft runs fast, then slow, twice per revolution, and the amount depends only on the angle between the shafts — which is why cars have two of them and why the second one has to be fitted the right way round.

spatial · Spatial What a pattern that cannot fold leaves behind. The best a least-squares solve can do with the vertex closures, against the fold it is asked for. The Miura pattern closes at every angle, at the arithmetic's own floor — the line along the bottom is 10⁻¹⁵ and below. The same grid with its vertices moved by a tenth of a panel does not close at any angle at all: its residual starts at 5.9e-6 at the smallest fold and grows with it, and no seed and no number of iterations moves it. Both patterns have the same panels, the same creases, the same graph and the same developable vertices, and every one of those vertices folds perfectly well on its own.

Each one moves, and together they do not

Take the pattern a Miura sheet folds along and move every interior vertex by a tenth of a panel. Every vertex still folds on its own — each is a spherical four-bar with a freedom of its own — and the four of them together fold to no angle at all, with a residual that starts at six millionths and never falls.

networks · Network A vertex is a spherical linkage, and the sectors are its link lengths. The four creases of one folded vertex, drawn as directions from the vertex itself, with the great-circle arcs between consecutive ones. Those arcs are the sector angles of the flat pattern — 80°, 60°, 100°, 120° — and they are those angles at every fold, to 8.9e-16 radians. That is the whole of the claim in the title: four axes through a point at fixed arcs from each other is a spherical four-bar, the object this site's spatial field built two phases ago, and a crease pattern's vertex is one of them with the arcs printed on the paper. The dihedral angle of the sheet at each crease is a half turn less that crease's fold angle, which here run 68.8°, 14.4°, 68.8°, 14.4°. positioned by solving, not by drawing.

Every vertex is a spherical linkage

Four creases through a point at fixed arcs from one another is a spherical four-bar — the object the spatial field is built on — with its link lengths printed on the paper as sector angles. The arcs hold to four parts in ten thousand million million at every fold, and on a flat-foldable vertex the half-angle tangents keep a ratio constant to nine figures.

networks · Network A spherical four-bar: 90°, 40°, 100°, 80°. Four revolute axes, all through one point, so the linkage lives on a sphere. Its link "lengths" are the angles between consecutive axes — 90°, 40°, 100°, 80° — and the arcs drawn between the axis directions measure 40.00°, 100.00°, 80.00°, 90.00°. The whole planar four-bar theory carries over with sines where lengths were, including Grashof's condition: sorted, the arcs give s + l = 140° against p + q = 170°, so the shortest arc does turn all the way round — and swept, the input reaches 36 of 36 positions over a driveable range of 360°. This is the mechanism a universal joint is a special case of, with two of the four arcs at 90°.

When the link lengths are angles

Put every axis of a four-bar through one point and the mechanism lives on a sphere. Its bars become arcs, its lengths become angles, and every planar result carries over with a sine where a length used to be — including Grashof's condition, which still predicts exactly which link goes all the way round.

spatial · Spatial Two cones, 20 teeth and 40, at 90°. The axial section, which is where every bevel quantity is read. Two cones share an apex and roll on one another along the element drawn heavy; their half-angles are 26.57° and 63.43°, adding to the shaft angle, and the ratio of their sines is the tooth-count ratio exactly. The dashed arc is the sphere of radius 22.36 on which a bevel tooth's profile actually lies. The two short lines perpendicular to the common element are the back cones; each is heading for its own axis at a distance r/cos δ from the pitch circle — 11.18 and 44.72 — and that distance is the pitch radius of the spur gear the tooth is really cut to. Both back cones lie on one line, because there is only one perpendicular to the pitch element at that point, and the two heavy stubs straddling it are the two teeth — each one addendum out from the pitch circle and 1.25 in.

A tooth that lives on a sphere

Every tooth in this field so far has been a curve in a plane, forced by the law of gearing and exact. A bevel tooth's profile lies on a sphere, no piece of a sphere flattens without stretching, and so the shape a bevel gear is actually cut to is an approximation — the only one in the field.

gears · Tooth

Named alongside it

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

Crease patternDevelopabilityGrashof's conditionMobilityNetworkRatioRigid origamiUniversal jointApproximationAssembly branchBase circleCompatibility

All concepts