Machine Kinematics
solved, then drawn
Concept

flat-faced follower — where it appears

A cam follower that touches the cam with a flat face instead of a roller, so the contact normal runs along its guide and its pressure angle is zero. What it asks instead is a convex cam, whose radius at the contact, R₀ + s + s″, stays positive.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

A flat face on a base circle of 30, where the profile bends tightest. The cycloidal programme — rise 10 over 120°, return over 120° — cut for a flat-faced follower on a base circle of 30, turned to 212.4°, where the profile's radius of curvature is smallest. The contact sits −5.37 from the follower's axis, which is ds/dθ, and the dashed circle is the one that fits the profile there: R₀ + s + s″ = 24.668, and 24.668 fitted through the drawn curve. The face's pressure angle is zero at every angle. What it needs instead is for that radius to stay positive, and it does so on every base circle above 5.332.

A flat face asks for a convex cam

A flat-faced follower is pushed at right angles to its face, so its pressure angle is zero at every angle and the constraint that sizes a roller's cam disappears. What replaces it is convexity: the profile's radius at the contact is R₀ + s + s″, and a base circle below 5.332 leaves a cycloidal cam with a hollow the face cannot reach — held 0.446 high on a base of 2.

cams · Cam A yoke 70 wide on a constant-breadth cam, at 60°. The cycloidal programme — rise 10 over 120°, return over 120° — cut for flat faces on a base circle of 30 and held between two parallel faces 70 apart, turned to 60.0°. The upper face rests at 5.000 above the base circle, which is the programme's lift there. The lower face touches the cam too: the distance across the cam along the follower's line is 70.000 at this angle and at every other, so the follower is driven up by one face and down by the other with no spring and nothing to spare.

A cam that holds its follower both ways

Two parallel faces joined into a yoke can drive a follower out and back with no spring, but only if the cam's breadth along the follower's line is the same at every angle — s(θ) + s(θ + 180°) constant. That makes the second half-turn the first one reflected. The cam field's standing programme misses by 9.502, and a second disc that frees the programme needs a yoke at least 60.66 wide.

cams · Cam A flat face on a swinging arm, and the cam it asks for. The cam a flat-faced follower needs when its face is carried on an arm pivoted 300 from the cam's centre, with a base circle of 20 and a lift of 20, drawn in the cam's own frame at three cam angles. The heavy line is the face, the dashed line is the perpendicular from the cam's centre to it, and the dot is the contact. Its distance from the foot of that perpendicular is the offset, which for a sliding face would be the programme's own velocity and here is a quantity with the arm in it. Over the turn it runs from -20.41 to 17.95, so the face must be 38.4 wide.

A flat face on an arm is worse

Carry a cam's roller on a swinging arm instead of a slide and the pressure angle improves — a long arm beats an offset. Carry a flat face on one and the opposite happens. The cam sees the sine of the follower's rotation rather than the rotation, and the distortion costs convexity: the smallest workable base circle rises from 10.66 on a slide to 28.68 at a pivot three base circles out, and below a certain arm length no base circle works at all.

cams · Cam The same arm with its face through the pivot and stood off it, each at its smallest convex cam. A flat-faced follower on an arm pivoted 90 from the cam's centre, delivering the symmetric cycloidal programme with a lift of 20, drawn at 60° of cam rotation. On the left the face is a line through the pivot, and the smallest base circle that keeps the cam convex is 28.679. On the right the face stands 33.56 off the pivot on a bracket, parallel to where it was, and the smallest convex base is 26.136 — 8.9% smaller. A sliding face needs 10.664, and neither arm reaches it. The dot is the contact; the small circle at the right is the pivot, and on both sides it clears the cam.

Where on the sine the arm swings

A flat face on a swinging arm delivers its programme through d sin ψ. Differentiating that twice brings in a square of the follower's speed, multiplied by sin ψ, and sin ψ has a sign. Standing the face off its pivot moves the swing down the sine until it straddles ψ = 0, where the square helps on one half of the stroke. On an arm ninety out that lowers the smallest convex base from 28.68 to 26.14, and it lets an arm of 61 have a cam where the face through the pivot needs 80. It never reaches the sliding face's 10.66.

cams · Cam A tilted face and a shorter parallel bracket hold the same line. A flat face on an arm pivoted 70 from a cam's centre, at 70° of cam rotation, the symmetric cycloidal programme with a lift of 20, base circle 33.725. Dark bracket: the face held 43 off the pivot and tilted 30° to the bracket's normal. Light bracket: the face held parallel, 37.24 off the pivot, which is 43 cos 30°. Both put the face on the same line at every angle of the cam — the support from the cam's centre agrees to 2e-14 and the normal's direction to 9e-16 radians over a turn — so the cam, the contact and the force on the pivot are the same; the tilted arm only swings 30° further round.

An arm that lifts against the cam can beat a slide

A flat face on a swinging arm seemed to offer three choices: how far its pivot is from the cam, how far the face stands off the pivot, and the angle the face makes with its bracket. The angle is not a choice. A line fixed to a body turning about a pivot is fixed, up to the turning, by its distance from the pivot, so a face tilted β on a bracket c long is the face on a parallel bracket c cos β long, the same line at every instant. The real third choice is which way the arm lifts. On a symmetric programme the two ways are mirror images. On a quick rise, an arm lifting against the cam's turning needs a base circle of 6.6 to 13 where a sliding face needs 32.85, and one lifting with it needs 41 to 59, or has no convex cam at all.

cams · Cam A 45° rise: the slide's cam and the arm's, each as small as it can be. The quick programme with its rise of 20 shortened to 45°, cut for two flat-faced followers at the smallest base that keeps each cam convex, drawn to one scale. Left: a sliding face, base 185.56. Right: a face on an arm pivoted 90 from the cam's centre, lifting against the cam's turning and stood off −44.9, base 23.26, 8.0 times smaller. At that base the arm's cam stays convex for bases up to 1.0% larger, and no further.

A steep rise spends the arm's room

A flat face on an arm that lifts against the cam's turning needed a fifth of a sliding face's base on a 90° rise. Steepen the rise to 45° and the slide's base grows 5.6 times, to 185.6, while the arm's least base grows only 3.2 times, to 23.3. But the arm's least base sits where its cam is about to go concave on the return as well as the rise: the base can be raised by only a hundredth before the return fails. Ask for a tenth's room above the base and the arm's base grows 5.1 times, nearly the slide's rate, and at 35° no stand-off gives that room at all.

cams · Cam

Named alongside it

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

Base circleEnvelopeDesign ruleSupport functionMotion lawOscillating followerConvex hullFollowerPressure angleAccelerationCam profileConjugate cam

All concepts