Machine Kinematics
solved, then drawn
Concept

Spool — where it appears

A drum with line wound onto it in layers, whose working radius is the radius of the layer being wound rather than a property of the machine. Line paid out is therefore quadratic in the turns, and the speed per turn grows by one circumference of line thickness at every layer.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The radius a winch works at is not a property of the winch. A drum of core radius 30 mm filling with line 6 mm thick, one layer at a time. Line speed per turn is 2πr, and r is the radius of the layer being wound rather than anything about the machine: it runs from 207.35 mm on the first layer to 395.84 mm on the 6th, a factor of 1.909. A winch quoted at one speed is quoted at one layer, and which one is rarely said. positioned by solving, not by drawing.

The radius a winch works at

Line speed per turn is 2πr, and r is not a property of the winch. It is the radius of whichever layer is being wound, so a six-layer drum runs from 207.35 mm per turn to 395.84 — a factor of 1.909 with nothing about the machine changed, and a length that is quadratic in the turns rather than proportional to them.

strands · Strand The number on the tackle is a limit. The velocity ratio of a 4-part tackle as its blocks separate, measured by differentiating the strand's own length. It is 3.6174 at 90 mm and 3.99992 at 6898 mm, and it reaches 4 nowhere. The shortfall is geometric and has nothing to do with friction: each part of line makes an angle with the lift, and a part at angle β shortens by cos β of the movement. A tackle used at close quarters — which is when a tackle is useful — is the case furthest from its own rating.

Six things a strand is not

A tackle that is not four to one, a wrap that is not a half turn, a tensioner whose travel takes up nothing, a winch speed that is not a property of the winch, a cable rig that holds nothing still, and a shaped pulley that cannot be asked for what it is usually asked for. Six claims, each with the number that kills it.

wrong · Misconception 4 turns, and none of them in a plane. A strand wrapped 4 times round a drum of radius 50 whose barrel has room for 8, with each turn lying beside the last — so the pitch is the strand's own diameter, 10 here, and the path is a helix rather than a circle. The helix angle is 1.823°, the length per turn is 314.318 against the planar model's 314.159, and the whole run is 1257.27 long where a plane would have said 1256.64. The plan view is a planar wrap exactly: projected onto the plane perpendicular to the axis, the helix is a circle of the drum's own radius to 2.8e-14 traversed 4.000000 times.

The wrap that walks along the axis

Every figure in this field is drawn in a plane, and a strand that goes round twice cannot be: the second turn has to lie beside the first. The plan view of the helix that results is a planar wrap exactly — so every wrap angle survives and the length does not, by five hundred parts per million on a real rope.

strands · Strand

Named alongside it

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

Design ruleStrandApproximationReevingVelocity ratioWindingWrap angleBlock and tackleDerivativeInextensibleLever armPitch

All concepts