Machine Kinematics
solved, then drawn
Concept

cycloidal motion — where it appears

The classical tooth form whose face is an epicycloid and whose flank is a hypocycloid, both traced by one describing circle rolling between the pitch circles. It is exactly conjugate at its design centre distance and stops being conjugate as soon as the shafts move.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

The involute, unwound. Hold a string taut against a circle and unwind it: the end traces this curve. Two properties follow immediately and between them they are the whole of gear geometry. The string is always tangent to the base circle, and it is always perpendicular to the curve. So the normal to an involute at any point is a tangent to its base circle — measured on the generated points here, to 2.3e-5 — and when two involutes touch, the common normal is a line tangent to both base circles, which does not move as the gears turn.

Why a tooth is an involute

A gear tooth is not a shape anybody chose for its looks. It is what one requirement forces — that the ratio of the two shafts' speeds stays exactly constant while the contact point slides along the flank. Impose that and the curve is essentially determined.

gears · Tooth 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

Constant acceleration gives the lowest peak acceleration of any motion law and an impulsive jerk. Cycloidal motion has finite jerk everywhere and a peak acceleration 57% higher. The trade is real, it is measurable, and the displacement curves that everyone plots give no hint of it.

cams · Cam The measurement every gear cut since 1900 is a consequence of. Take a pair that is conjugate at its design centre distance, move the shafts apart, and ask what shape the driven wheel would have to be for the ratio to hold. The involute wants the same shape at every distance — 2.91e-8 mm at two millimetres out, which is the comparison's own noise floor — because its shape is fixed by its base circle and the centre distance is not one of that circle's arguments. The cycloidal pair is conjugate at nought and wants a shape 6.71e-3 mm different at five hundredths of a millimetre out, because its describing circle has to roll between two pitch circles that are no longer touching. A bearing that wears, a housing bored a little wide, a case that warms up: all of them are this axis.

The shape that does not mind where the shafts are

Two tooth forms, both exactly conjugate, both in use for centuries. Move the shafts five hundredths of a millimetre apart and one of them wants a different shape and the other does not — and that single measurement is close to the whole reason every gear cut since about 1900 is an involute.

meshing · Meshing 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

A groove holds its roller between two walls and chooses between them by the sign of one force, the follower's mass times its acceleration plus the load pressing it in. Below a threshold speed that force never changes sign and the roller stays on the inner wall; each stretch of deceleration adds two crossovers above its own threshold. On a quick-rise cam at 20 N and half a kilogram the count goes from none to two at 268 rpm and to four at 506. A cycloidal law brings the roller across smoothly; a constant-acceleration law throws it across by a step of 3,200 N at 3,000 rpm.

cams · Cam Two rotors cut undersize, and the gap that is their seal. The conjugate pair at three positions of its turn, each rotor cut 3 undersize — its boundary moved that far inward along its own normal, with the full-size outline dashed behind it. At full size the two are in contact at every angle, so cutting both back by 3 leaves exactly 6.0 between them wherever they were touching. Measured over a whole lobe pitch the gap runs from 5.9965 to 6.0000 against a prediction of 6, a worst departure of 3.46e-3.

The clearance that is the seal

Two rotors that are each other's conjugates touch at every angle of their turn, so cutting both back by the same amount leaves exactly twice it between them, everywhere. Open the shafts by the same amount instead and the gap runs from four per cent of it to all of it. And a pair that is not conjugate has no seal to cut: over one lobe pitch it swings from two and a half units inside itself to two and a third apart.

meshing · Meshing 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

A roller in a groove changes walls where the groove's force changes sign, and it gets there by flying across the clearance. How hard it lands depends on the clearance through an exponent the motion law decides — two thirds where the force passes through nought, one half where it steps — a flight that ends in a dwell lands at √(2cF/m) whatever the speed, and just above each threshold the force changes sign and the roller never arrives at all.

cams · Cam 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 cam's motion law is written in cam angle and delivered in time, and the two are the same only on a shaft that turns steadily. Drive the camshaft through a Hooke's joint and the follower's acceleration gains a second term — the law's velocity times the shaft's angular acceleration — that a steady shaft does not have. Where the rise falls in the joint's half-turn then decides everything: at a 30° bend the same cycloidal rise peaks 35% above its steady value or 17% below it. And past a bend of 34.5° a well-clocked cycloidal cam gives its follower less peak acceleration than the constant-acceleration law, the one designed to minimise it.

cams · Cam 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 roller chain leaves its sprocket at a radius that shrinks and recovers as each pin seats, so a chain-driven camshaft speeds up through every tooth and its angular acceleration jumps back at every seating. That puts a step into the follower's acceleration once per tooth, whatever the law — a cycloidal rise, chosen for its continuous acceleration, gets four steps inside a 120° rise on an eleven-tooth sprocket, each 0.41 of its peak. The harm falls with the tooth count, as one over it, not up as its square, and the simple harmonic law, whose peak sits where its velocity is nought, is hurt least of all.

cams · Cam

Named alongside it

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

AccelerationMotion lawDwellJerkCentre distanceConjugate-actionFollowerCamClearanceForm closureInvoluteRoller follower

All concepts