Machine Kinematics
solved, then drawn
Field

Prescribed motion

A linkage gives the motion its geometry allows. A cam gives the motion you asked for — and the cost is paid in accelerations you did not ask for.
A cycloidal cam at 60°. A 20-unit rise over 120°, a dwell, a return and a dwell. The dashed curve is the pitch curve — where the roller's centre travels — and the solid one is the surface that has to be cut, which is the pitch curve offset inward by the roller radius along its own normal. The pressure angle at this instant is 25.5°: the angle between the follower's direction of travel and the normal to the surface, and the number that decides whether the follower jams in its guide rather than sliding.

Prescribing motion

How a cam profile is generated from a stated follower motion, the difference between the pitch curve and the cut surface, and why the pressure angle is controlled by the base radius.

 The same rise, three ways. A 20-unit rise over 120°, then a dwell. Above, the displacement — all three do the same job and are hard to tell apart. Below, the acceleration, differentiated from those same curves. Constant acceleration peaks lowest at 5.556e-3 per degree² and arrives at the dwell with a step; cycloidal peaks highest at 8.727e-3 and arrives at zero. The smoothest law has the largest peak, which is the trade the whole of cam design turns on and which the displacement plot gives no hint of.

The law that costs least is not the smoothest

The four cam motion laws with their derivatives differentiated from the generated profile, the ordering by peak acceleration, and why arriving at a dwell is the case that decides.

 A 6-slot Geneva wheel. The driving pin enters a radial slot, carries the wheel through 60°, and leaves. The centre distance is not free: it must be crank ÷ sin(180°/6) = 60.00 so that the pin enters along the slot, with the crank and slot perpendicular. That is the mechanism's one design requirement and it is what makes the driven wheel start and stop from rest — measured here at 1.5e-3 against a peak of 1.000. What it does not fix is the acceleration, which peaks at 1.35 and is why film sprocket holes tear.

Stopping thirty times a second

The Geneva mechanism, why the centre distance is not free, what tangential entry buys, and the acceleration it does not remove.

 A 34-unit roller on a 20-unit base: the cam cannot be made. The outer curve is the pitch curve — where the roller's centre has to go, which is just the base radius plus the follower's displacement, plotted in polar coordinates. The inner curve is what has to be cut: the pitch curve offset inward by the roller's radius along its own normal. Where the pitch curve bends more tightly than the roller is wide, that offset crosses itself, and the metal the profile needs has already been removed. The tightest radius of curvature here is 19.43 at 85°, against a roller of 34 — so the cam is undercut by 14.57 and no amount of care in manufacture recovers it. The pressure angle, meanwhile, peaks at a perfectly comfortable 21.2°: this cam fails a test it never looked like failing.

The cam that cannot be cut

The radius of curvature of a cam's pitch curve, the roller radius it must exceed, and the base circle measured against both constraints — with the curvature checked against a circle fitted to the drawn curve.

 A cam profile is an envelope, and its curvature obeys the same law. The dashed curve is the pitch curve — where the roller's centre travels — and the solid one is the surface that has to be cut, which is the pitch curve offset inward by the roller radius. That offset is the conjugate law of this field with one centre of curvature sent to infinity, and it says ρ_cut = ρ_pitch − r. Measured off the drawn polyline at six angles, the worst departure is 4.8e-7: at 95° the pitch curve has radius 29.52 and the cut profile 21.52, against 21.52 predicted. It is also why undercutting is a curvature condition rather than an accident: where ρ_pitch falls below the roller radius the offset turns itself inside out.

A profile is an envelope

The cam profile as an envelope, with the offset law measured off the drawn polyline, and a sign error in the site's own offset code found by asking a drawn curve what its curvature was.

 A cam is the envelope of a roller. The follower's roller drawn in the cam's frame at 15 angles of the cam: the roller slides in a straight guide and the cam turns under it, so from the cam's point of view the roller travels round it on a path set by the lift law. The cam's surface is the envelope of those circles — the same computation as a rack generating a tooth, with a circle instead of a straight edge and a lift law instead of a rolling condition. The cams field builds the same surface by offsetting the roller-centre path inward by the roller radius, and the two agree to 4.1e-4 mm. positioned by solving, not by drawing.

A cam is a conjugate pair

A cam profile generated as the envelope of its roller follower and compared against the cams field's offset construction, with the two errors that construction can make measured as distances.

 A flat follower needs a face, and how wide is not a matter of taste. The cam profile the cycloidal programme produces on a base circle of 30, at 70° of rotation, with the follower's face across the top and the contact marked. The contact is not on the follower's axis: it sits at ds/dθ from it, so it wanders sideways as the cam turns and the face has to reach from the most negative value of that to the most positive. Here that is 19.099 — measured off the drawn profile as 19.099, the two routes agreeing to 1.0e-16. A face cut to the lift, or to the base circle, or to whatever looked right, is a face the cam runs off.

A follower needs a face

How wide a flat follower's face has to be: the range of ds/dθ in closed form, measured independently off the drawn profile, and why the answer has nothing to do with the lift.

 A cycloidal cam at 60°. A 20-unit rise over 120°, a dwell, a return and a dwell. The dashed curve is the pitch curve — where the roller's centre travels — and the solid one is the surface that has to be cut, which is the pitch curve offset inward by the roller radius along its own normal. The pressure angle at this instant is 25.5°: the angle between the follower's direction of travel and the normal to the surface, and the number that decides whether the follower jams in its guide rather than sliding.

A lift is a size and a law is a shape

Sorting the cam field's quantities by what a scaling does to them, and what a measurement of a follower's motion recovers about the cam that drives it.

 A roller offset by 5.88, at 42.3° of the rise. The cycloidal programme — rise 20 over 90°, return over 170° — on a prime circle of 40 with a roller of 10, drawn with the follower fixed above and the cam turned anticlockwise to 42.3°. The follower's line of travel is 5.88 to the right of the cam's centre. The contact normal leans 21.81° from that line, which is the pressure angle, and measured off the drawn pitch curve it is 21.81°. At this offset the peak on the rise is 21.81° and on the return 21.81°; centred, they are 27.37° and 15.33°.

An offset trades the rise for the return

The pressure angle of an offset roller follower derived, measured off the drawn pitch curve, and swept across the offset: the balance point, the lean it puts on a stationary follower, and what it does to whether the cam can be cut.

 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

The flat-faced follower's convexity condition computed rather than stated: the radius of curvature at the contact by formula and off the drawn envelope, the smallest convex base circle for each motion law, the lift a face delivers when the cam is not convex, and why the flat face ranks the laws by acceleration where the roller ranks them by velocity.

 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

The constant-breadth condition for a yoke follower derived and measured across the drawn cam, the programmes that satisfy it and the ones that do not, the lost motion or interference a yoke pays on a cam that fails it, the opposite radii that sum to the breadth, and the two-disc cam that holds any programme.

 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

The oscillating roller follower's pressure angle derived as the offset follower's with a variable offset, measured off the drawn pitch curve on both sides of the cam, and swept across pivot angle and arm length on two programmes: the premium an arm pays on mirror-image strokes, the dwell lean, what the pivot's side costs, and the long arm that beats the best sliding offset by hundredths of a degree.

 A roller in the groove of a quick-rise cam at 600 rpm. The cycloidal programme — rise 20 over 90°, return over 170° — cut as a groove on a prime circle of 40 for a roller of 10, driving a follower of 0.5 kg pressed toward the cam by 20 N, and turned anticlockwise to 59.9° at 600 rpm. The groove must supply m·s″·ω² + F along the follower's travel, −66.6 N at this angle, so the roller bears on the outer wall. The marks on the groove are the four places in a turn where that force changes sign and the roller changes walls: 47.87°, to the outer; 87.13°, to the inner; 151.35°, to the outer; 193.65°, to the inner. The stretches of deceleration begin to cross over at 267.6 and 505.5 rpm.

A roller in a groove changes walls with the speed

The wall a grooved cam's roller bears on, decided by the sign of m·s″·ω² + F: the crossover angles by the programme's acceleration and off the drawn pitch curve, the threshold speeds at which each stretch of deceleration begins to cross over, where the crossovers fall and whether they are steps under three motion laws, what a programme with no dwell between its strokes does, and the reversal of the roller's spin at each.

 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

The oscillating flat-faced follower as a support function: the contact offset d cos ψ · ψ′ /(ψ′ − 1) checked against the drawn envelope to 2 × 10⁻⁹, the radius of curvature over a whole turn at four pivot distances, the smallest convex base circle against the pivot's distance with the sliding follower as its limit and its best case, and the arm length below which the programme has no cam at all.

 The roller's flight across a 0.05 mm clearance at every crossover. The cycloidal quick-rise programme carrying 0.5 kg against 20 N at 600 rpm, with 0.05 mm between the roller and the wall it is not bearing on. At each sign change of the groove's force the roller leaves its wall, and each curve is the gap it opens, from nought to the far wall at the top of the plot: leaves at 47.9° and lands 6.6° later at 80 mm/s; leaves at 87.1° and lands 7.1° later at 63 mm/s; leaves at 151.4° and lands 14.6° later at 35 mm/s; leaves at 193.6° and lands 13.4° later at 42 mm/s. Dragging the speed moves the crossings, lengthens or shortens each flight, and just above a stretch's threshold shows a flight that never reaches the far wall.

The time a crossover takes

The flight of a grooved cam's roller across its clearance at each crossover, integrated from the sign change of the groove's force; the landing speed's power law in the clearance measured at two thirds for a cycloidal law and one half for constant acceleration, with the reason in the first degrees of the flight; flights into a dwell landing at the load's own free-fall speed; and the band of speeds in which a sign change sends the roller nowhere.

 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

An oscillating flat-faced cam follower whose face stands off its pivot: the support d sin ψ + c, the harmful term −d sin ψ (dψ/dα)² whose sign follows the arm, the smallest convex base over pivot distance and stand-off, the best stand-off centring the swing on the perpendicular, the shortest workable arm, and why no combination beats the slide.

 The same cycloidal rise behind a bent joint. A cycloidal rise over 120° of cam angle, with the camshaft driven at constant speed through a Hooke's joint bent at 30°. Across: the fraction of the rise; up: the follower's acceleration in time, divided by its peak on a steady shaft. The joint turns the camshaft unevenly with a period of half a turn, so where the rise falls in that period — the clocking, the cam angle at which the rise begins, measured from where the shaft turns fastest — decides what the follower feels. At the worst clocking, 100°, the peak is 1.352 times the steady one; at the best, 30°, it is 0.828 — lower than on a steady shaft. Every curve still starts and ends at nought. Dragging changes the bend.

A cam behind a bent joint

A cycloidal rise over 120° with its camshaft driven through a Hooke's joint, at the best and worst clockings; the joint's output speed and angular acceleration over its half-turn period; the follower's acceleration split into the law's own term and the term the shaft's acceleration adds; the peak against the clocking for three laws; each law's best-clocked peak against the bend, with the crossing where the cycloidal law overtakes; the bend at which each law overtakes by the span of the rise; and the steps each law carries into its dwells.

 A cycloidal rise on a chain-driven camshaft. A cycloidal rise over 120° of cam angle, the camshaft driven by a roller chain moving at steady speed over an 11-tooth sprocket. The chain leaves the sprocket at a radius that varies as each pin seats, so the shaft speeds up and slows once per tooth, and its angular acceleration jumps from one sign to the other at every seating. The follower's acceleration (solid) therefore steps at every tooth — 4 or so inside this rise — by up to 0.414 of the steady peak, and the cycloidal law's continuous acceleration, the reason it is chosen, is lost. At the worst clocking against the teeth the peak is 1.165 times the steady shaft's (dashed). Dragging changes the tooth count.

A cam on a chain steps at every tooth

A camshaft driven by a roller chain at steady speed: a cycloidal rise with a step at every seating; the shaft's speed and sawtooth angular acceleration over three teeth; the worst peak over the steady one against tooth count for three laws; the step at a seating against its closed form; and the peak against where the rise starts within a tooth.

 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-faced follower on a swinging arm: a face tilted on its bracket and a parallel face holding the same line; the tilted face's smallest convex base against the parallel face's; the quick-rise programme's smallest convex base against the pivot's distance for an arm lifting with and against the cam's turning, and a slide; the two cams of one arm at their own smallest bases; the radius of curvature over a turn for both; and the symmetric programme's two senses and the reversed threshold checked on the drawn 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

The quick cam programme's rise shortened from 90° to 35° for a flat face on a slide and on an arm lifting against the cam: the two cams at their least bases on one scale; every follower's least base against the rise's span; the room above the arm's base against its stand-off, and the base itself, for four rises; the cam's radius of curvature at the 45° rise's least base and with a tenth's room; and the five rises in one table.

 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

A 45° cycloidal rise on a flat-faced arm pivoted 90 and lifting against the cam, with nine returns of three laws and three spans: the least-base cam with the quick return beside one chosen for the arm; the least base and the base with a tenth's room against the return's span for each law; the room above the least base, return by return, and what closes it; speed and downward acceleration along the first half of each law's return, with where each cam fails; the room against the face's stand-off for three returns; and the nine returns in one table.

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