Machine Kinematics
solved, then drawn
Concept

Circular arc — where it appears

A stretch of a curve that a circle matches closely enough to be replaced by one. Parts of a coupler sextic are very nearly circular, and hanging a link of that stretch's radius on the coupler point is how a four-bar is made to dwell without a cam.

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

The curves that do not move. A rotation about a point carries a curve into itself if and only if the curve is an arc of a circle centred on that point, and that one sentence is every exact dwell on this site. Each bar is how far a curve moves when it is turned two degrees about the axis its mechanism turns about, as a fraction of its own radius. The two arcs about their own centres — a cam's dwell and a deadbeat's locking face — sit at the sampling floor, which is the sagitta of the polyline they are measured as and not a property of the geometry; the number is quoted with the floor beside it because an agreement quoted without its resolution is a mistake this site has already made once. Everything else is orders of magnitude above it, including the near-circular stretch of a coupler curve that a six-bar builds its approximate dwell out of.

The arc that is concentric with the pivot

A rotation carries a curve into itself exactly when the curve is an arc of a circle about the centre of rotation. Every exact dwell on this site is that one sentence applied — a cam's dwell, a Geneva's locking disc, a deadbeat escapement's locking face — and the six-bar dwell that is merely very good is what happens when the curve is nearly one.

timing · Intermittent A dwell is a measurement, not a stop. The Stephenson six-bar's output against a full turn of the crank, with the four-bar it is built on for comparison. Inside a band of ±1° the six-bar's output holds still for 145.8° of crank and the four-bar's for 41.9°. The dwell comes from a stretch of the coupler curve that fits a circle of radius 3.006 to within 2.96e-3, and the arm from the coupler point is that radius. Nothing here stops; it moves less than the band.

A dwell made from a curve

Parts of a coupler curve are very nearly circular arcs. Put a link of the arc's own radius on the coupler point and its far end stands almost still while the point runs along it, so the output dwells — 146° of crank inside a one-degree band, against 42° for the four-bar it is built on. A dwell linkage does not stop. It moves less than the tolerance, and how much less is a number.

linkages · Synthesis

Named alongside it

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

CamCoupler curveDwellBase circleConcentric arcCurve fittingDeadbeatEscapementthe Geneva mechanismIntermittent motionLocking faceSix-bar

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