Machine Kinematics
solved, then drawn
Concept

Oscillating follower — where it appears

A cam follower carried on an arm that swings about a fixed pivot instead of sliding in a guide. The cam's profile must then deliver the arm's rotation, so the arm's length and where the follower sits on it enter the pressure angle or the convexity condition.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

A roller on an arm of 50 pivoted on the left, at 35.8° of the rise. The cycloidal programme — rise 20 over 90°, return over 170° — on a prime circle of 40 with a roller of 10, followed by a roller on an arm of 50 whose pivot is on the left of the cam, at (−49.92, 42.79). The pivot angle balances the two strokes, and the cam is turned anticlockwise to 35.8°. The faint arc is the path the roller's centre swings through. The contact normal leans 23.48° from the roller's direction of motion, which is the pressure angle, and off the drawn pitch curve it is 23.48°. At this instant the roller's line of motion passes 2.98 from the cam's centre. Balanced, the arm peaks at 23.48° on both strokes; a sliding follower at its best offset of 5.88 peaks at 21.81°.

An arm is an offset that grows with the lift

Carry a cam's roller on a swinging arm instead of a slide and its pressure angle obeys the offset follower's formula exactly, with the offset replaced by the distance of the roller's line of motion from the cam's centre. That distance turns with the arm. On a cam whose strokes are mirror images the best arm is worse than a centred slide by about 5,000/L² degrees and leans the follower by its own tilt on the dwells. On a quick-rise cam the pivot's angle rebalances either side of the cam for under a degree, where a sliding offset moved to the wrong side costs eleven.

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 circleDesign ruleEnvelopeflat-faced followerSupport functionCamDwellMinimaxMotion lawOffset followerPitch curvePressure angle

All concepts