Machine Kinematics
solved, then drawn
Concept

Contact — where it appears

Two bodies touching, which is what a joint is made of. Contact over a surface gives a lower pair and a group; contact along a line or at a point gives a higher pair, whose permitted displacements depend on where the bodies are and do not compose.

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

Two permitted motions, and a composite that lifts off. A disc resting on a straight edge — a roller follower on a flat-faced cam, and the simplest higher pair there is. Two bodies touching at a point rather than over a surface, two freedoms: slide along the edge, and turn, because a disc is its own symmetry group about its centre. Both are permitted and both keep the contact exactly. Their composite does not. The faint discs are the two permitted displacements taken separately; the solid one is one followed by the other, and its centre sits 1.049 radii off the dashed line where a tangent disc's centre has to be. The excursion is exactly |t₂ sin φ₁| — the second displacement's slide times the sine of the first one's turn — derived from the two displacements rather than from the composition and agreeing to 10⁻¹⁶. A lower pair's freedoms compose and a higher pair's do not, which is why a joint's freedom count is the dimension of a group in one case and the dimension of nothing in the other.

A higher pair has no group

A disc resting on a straight edge may slide along it and may turn about its own centre. Both keep the contact exactly. Do one and then the other and the contact lifts off the edge by |t sin φ| — up to 1.2 radii over an ordinary range — so the two freedoms are real and the pair of them is not closed. The count is still right and there is nothing for it to be the dimension of.

pairs · Pair A ball bearing turned by its inner race. A bearing of pitch diameter 40 with 8 balls of diameter 8, its outer race held and its inner race turned 60°. Rolling without slipping at both contacts leaves the cage — the ring that keeps the balls apart, marked by the dark tick — turned 24.00°, a share of 0.4000 of the race, and each ball spun 120.0° backwards about its own centre, marked by its own tick. The share is (1 − d/D)/2, less than a half by d/2D; the gear-train solver, handed a planetary with a sun of 32 teeth and a ring of 48, returns 2/5 for its carrier. Dragging turns the inner race.

A bearing is a planetary with no teeth

Roll a ball between two races and nothing but the two rolling constraints decides how fast its centre goes round. Solved, they put the cage at (1 − d/D)/2 of the inner race's speed — always less than half — and the gear field's train solver, handed a planetary with a sun of D − d teeth and a ring of D + d, returns the same fraction exactly. And a tapered roller rolls without slipping along its whole line only if its axis meets the bearing's at the apex, the wheel field's concurrency one dimension up.

rolling · Rolling Every surface tried, and the group it permits. The census the six lower pairs come out of. Each row is a surface, sampled at 240 points; the freedoms column is six minus the rank of a matrix with one row per point, saying that the velocity a twist gives that point is tangent to the surface. Nothing is fitted and no shape is recognised — the surface's own normals write the matrix down. Three different surfaces of revolution give the same group, which is the content of the classification; two surfaces give nothing, which is what almost every surface gives. Eleven surfaces, six groups. The last column is the ratio of the smallest singular value kept to the largest discarded, so a row reading 10¹⁵ is not near being reclassified by anybody's tolerance.

The pair a catalogue sells

A plain bearing is a cylindrical pair and a catalogue calls it a bearing. Add two thrust faces and it is a revolute pair, which is a different joint and changes every mobility count downstream. The kinematic identity of a bought part is decided by which surfaces touch, and the catalogue's word for it is not the same information.

practice · Clearance

Named alongside it

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

ConstraintDisplacement subgroupKinematic pairLower pairMobilityBearingClearanceDegrees of freedomDesign ruleDifferentialEpicyclicGear train

All concepts