Machine Kinematics
solved, then drawn
Concept

Hysteresis — where it appears

A dependence of a mechanism's state on the path its load or input took to get there, not only on where it is now. A frictional hold or an escapement can rest anywhere in a band, and which point it occupies is set by whether the load last rose or fell.

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

A period, in the space the mechanism lives in. One whole period of an escapement, plotted as wheel angle against pallet angle. Give a four-bar its crank angle and its coupler is somewhere definite; give this its pallet angle and the wheel may be in any of 3 places, because what settles it is which face of which pallet a tooth is against. The path crosses itself and no amount of solving removes the crossing. The vertical jumps are the drops, drawn at the pallet angle of the release because during a drop nothing is touching anything and where the pallet is by then is a question about torque. Over the period the wheel advances 12.000000°, which is one tooth exactly.

Where the input stops deciding

Give a four-bar its crank angle and its coupler is somewhere definite. Give an escapement its pallet angle and the wheel may be in any of three places, because what settles it is which face of which pallet a tooth is against. The path crosses itself and no amount of solving removes the crossing — the state of these mechanisms has a discrete part, and that is what makes them a different kind of object.

timing · Intermittent A cleat's lever, and the squeeze the load has left it. A cam cleat at a lean of 10°, its pivot pin with friction 0.5 and a radius a tenth of the pivot's height (drawn larger, with its friction circle), and a release lever on the cam. Along a load path the rope's pull rises to one, falls to nought and rises again, and the squeeze follows it between the band's edges, 4.50 and 7.62 times the pull. Here the pull is 1, the squeeze 4.50, and the lever must turn the cam with a moment of 0.412 times the pivot's height to free the rope.

A cleat set hard lets go easiest

A cam cleat on a frictionless pin is freed by a lever with no effort at all. With pin friction the lever must beat h(T − N tan γ) + ρ√(T² + N²), and since a stiff pin lets the squeeze N sit anywhere in a band, the effort is set by where in the band it sits: nought at the top, twice the pin's friction moment at the bottom. A rising load drags the cam in to the bottom, so a cleat holding the load it was set by is the hardest to free — 0.412 times the pull times the pivot's height at a lean of 10° — while one jammed hard by hand, or one whose load has eased, frees itself. Below the backward-lock lean nothing jumps in the effort. What changes is the state: the slack cleat keeps the squeeze its largest pull left.

holding · Restraint A cam cleat whose pin swings on a sprung link. A cam cleat at a lean of 2°, below its pin's lock lean of 2.56°, its pin carried on a link that swings about a frictionless pivot 2.2 pin heights above the rope, held by a spring. The link's lever is chosen so that, with the spring a quarter as stiff as the rope at that lever, cam and link turning together squeeze the rope 12.53 times the pull, the pin's own bottom edge. Along a path the pull rises to one, falls to nought and rises again. At this step the pull is 0: a fixed pin squeezes 12.53, this link 0.00, and a link at half that ratio 6.27.

A cleat on a swinging pin has two leans

Carry a cam cleat's pin on a sprung link and, while the pin holds, cam and link turn together about the link's pivot: a second cam, frictionless, whose squeeze per unit pull is the cot of its own lean softened by the spring against the rope. A rising load is met at the pin's edge and a falling one at the link's, so a slack cleat keeps only what the link falls short of the edge — 12.5 times the pull on a fixed pin at 2°, 6.3 on a link at half the edge, nothing on a link at the edge itself. Nothing in it depends on the load's size: every force doubles with the pull.

holding · Restraint

Named alongside it

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

LockContact normalDesign ruleFrictionAssembly branchConfiguration spaceContact stateDropEscape coneEscapementImpulseIntermittent motion

All concepts