Machine Kinematics
solved, then drawn
Concept

Convexity — where it appears

A closed curve is convex when it turns the same way all round, never bending back on itself. A cam must be convex for a flat follower face, since a hollow would be bridged by the face and never touched, so a cam's least size is often its least convex size.

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

The shape is what the demanded arm implies. A strand leaving a convex body runs along a tangent, so the only thing about the body the strand can feel is the perpendicular distance from the axis to that tangent — the shape's support function h(ψ). Ask for a rate and you have asked for h, because turning the body by dψ pays out h dψ; the shape then comes back from h with no solve at all, as h(ψ)û + h′(ψ)û⊥. This one was asked for h₀(1 + 0.3 cos 2ψ) with h₀ = 34 mm. The strand drawn here leaves the axis at a perpendicular distance of 24.1286 mm, which is what the demand asks for at this angle. positioned by solving, not by drawing.

The drum that is not round

The only thing about a body a strand can feel is the perpendicular distance from the axis to the tangent it leaves along. Ask for a rate and you have asked for that distance at every angle — and the shape comes back from it with no solve at all, unless the demand exceeds 1/(n²−1), at which point there is no shape.

strands · Strand The same 45° rise on an arm, with the quick return and with a return chosen for the arm. A cycloidal rise of 20 over 45°, cut for a flat face on an arm pivoted 90 from the cam's centre and lifting against its turning, each at the least base that keeps the cam convex. Left: the quick programme's return, 170° cycloidal, base 23.26, with 1.0% of room before the return goes wrong. Right: a 170° simple harmonic return, base 18.08, with 4.9% of room before the rise limits it.

A gentle return cannot widen a steep rise

An arm lifting against a cam on a 45° rise had its least base pinned by the return: raise it a hundredth and the return went concave. Give it a return chosen for the arm, slower or gentler, and the least base falls from 23.3 to 18.1, a fifth less. But the window above it only grows from 1% to 4.9%, because once the return is out of the way the rise closes the window itself, and the arm's own reach closes the next one. With a tenth's room the best return saves 8% of base, and no return of nine gives a quarter's room. Constant acceleration, the law with the lowest peak deceleration, is the worst return of the three.

cams · Cam

Named alongside it

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

ApproximationCamCycloidal motionDesign ruleEccentricEnvelopeFollowerLever armRadius of curvatureStrandSupport functionTangency

All concepts