Machine Kinematics
solved, then drawn
Concept

Cycloidal drive — where it appears

A reducer in which a lobed disc rides on an eccentric inside a ring of pins, turning back by one lobe for each turn of the input. Its profile is the envelope of a single pin, and its ratio is a count of lobes on a shape nobody drew.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

One planet shaft, two centre distances. A compound epicyclic gets its enormous reduction from two meshes whose tooth counts are nearly in the same proportion. The two planet gears are on one shaft, so their axes are at one radius — and at a common module the two rings ask for radii that differ by 0.50 of a tooth. The exactness the reduction is famous for is bought with a pair of meshes running away from the centre distance they were cut at, and the difference is made up by profile shift — the same correction the teeth field applies for a different reason. It is not a rounding: it is the mechanism's own condition, and it is the reason a catalogue reduction of this kind comes in a short list of tooth counts rather than in any combination.

A hundred to one from a difference of one

A harmonic drive reduces by a hundred to one in a single stage with two gears in it, and the hundred is the flexspline's tooth count divided by the two teeth the circular spline has more than it. The same null space that answers a planetary answers it. What each of the three single-stage reductions pays for that arithmetic is different, and the compound epicyclic's price is a pair of meshes whose centre distances differ by half a tooth.

transmission · Transmission Eleven lobes from twelve pins, at 46° of eccentric. The pins are circles and the disc is their envelope. One turn of the eccentric moves the disc back by one lobe pitch — a reduction of 11 : 1 from a ring, a disc and an offset bearing, with no gear teeth anywhere — and the profile's 11 lobes are counted off the generated curve rather than put there. The roots sit at 50.00 mm and the tips at 60.00, which are R − r ∓ e: the eccentricity is the lobe height, twice over. positioned by solving, not by drawing.

Eleven lobes from twelve pins

A ring of round pins, a disc on an eccentric, and a reduction of eleven to one with no gear teeth anywhere. The disc's profile is not designed: one pin generates one lobe of it, the other ten lobes are the same curve, and the count that decides the ratio is a count of lobes on a shape nobody drew.

meshing · Meshing Nine contacts at once, and the sense each turns the disc. The 12-pin drive at 40° of its eccentric. 9 pins are in contact with the disc at this instant, and each one's line is the common normal, which passes through the pitch point on the pin circle. A pin can only push, so the sense in which it turns the disc is decided by which side of the disc's own centre its normal passes: 5 of the contacts turn it one way and 4 the other, with the largest arm in each sense 49.9 and 45.0 on a pitch offset of 55.0. A pair with one contact has no such choice, which is the whole of why two identical rotors cannot drive each other.

What a second contact is for

Two identical rotors that are exactly each other's conjugates cannot drive each other, and the reason has nothing to do with conjugacy. A ring of pins and the disc they generate is just as exactly conjugate, has nine or ten contacts at once instead of one, and never loses more than a fifth of the arm its geometry allows.

meshing · Meshing A disc cut undersize, turned until a pin touches. A twelve-pin cycloidal drive whose disc was cut as if its pins were 1 larger than they are, drawn at 10° of the eccentric with the disc turned a further 1.082° about its own centre — the turn at which the first pin touches. One pin touches, pin 0, marked in its own colour. The pins within half the clearance of touching are drawn in a second colour and the rest plainly: 5 of the twelve are within half the clearance and 7 further off. The clearance is exaggerated so that it can be seen; at every clearance measured, one pin touches first.

A clearance leaves one contact

A cycloidal drive's disc, exactly as its pins generate it, touches nine or ten of its twelve pins at once, and that is why it can drive where two identical rotors cannot. Cut the disc two hundredths undersize, as a real drive is cut, and a rigid disc turned until it touches meets one pin, at every angle of the eccentric and in both directions. The arm that pin gives is never worse than the exact disc's best. What a clearance takes away is load sharing, and deflection gives it back: contacts that may give by the whole clearance put six or seven pins within reach.

meshing · Meshing The same drive with a smaller eccentric: every pin touches. Twelve pins on a ring of 60, pin radius 5, and the disc exactly as those pins generate it, at 10° of the eccentric. Left, the eccentric at exactly R/N = 5, whose roots are round sockets: 9 pins touch (solid) and 3 stand clear (dashed), the pins near the roots. Right, the eccentric at 0.8 of that, 4.00, whose roots are smooth: 12 of twelve touch. Over a pin pitch the socketed disc touches 9 or 10, and every smaller eccentric measured touches all twelve.

A disc whose roots are not sockets

Put a cycloidal drive's eccentric at exactly the pin circle's radius over the pin count and every root of its disc is a round socket; take any smaller eccentric and the roots are smooth. The sockets looked like a benefit — a pin in one closes its gap both ways at up to the full pitch offset. Measured from a half of that eccentric to all of it, they are the opposite. The socketed disc touches nine or ten of its twelve pins, every smaller one touches all twelve, and its first contact keeps the smallest share of the pitch offset of the seven. What a smaller eccentric really costs is the pitch offset itself: the free turn per unit of clearance is 0.0365 divided by the eccentric's share.

meshing · Meshing The flank bends towards the disc, and the pin's radius is measured against that bend. One root of a twelve-pin cycloidal drive, pin radius 5 on a ring of 60, with the eccentric at 0.97 of R/N. The thin line is the curve the pin's centre traces in the disc's frame, its tip pointing at the disc's centre; the dashed line is that curve offset towards the disc by the pin's radius, which is the disc's edge wherever nothing trims it. Either side of the tip the flank bends the other way, towards the disc, most tightly 22.6° of the eccentric from the root, with a radius of 5.36 (the circle, centre marked). The pin's radius, 5, is measured along the same normal to the offset point. The offset point stops short of the centre of curvature, so the offset does not fold and the root is untrimmed. The two meet at 0.9740 of R/N.

The flank that trims the root

A cycloidal disc's roots are trimmed, and its pins start standing clear, above an eccentric that was known only for one drive: 0.974 of the pin circle's radius over the pin count, for twelve pins of radius 5 on a ring of 60. The trimming is a bend. Either side of each root the flank of the pin's path curves towards the disc, and where that bend is tighter than the pin the offset folds. The bend has a closed form, so the threshold does too, √(1 − ρ²(N + 1)³/27(N − 1)) for pins of radius ρ, and it falls well below one for fat pins or many of them. On every drive measured, the classical eccentric loses contacts and lever together.

meshing · Meshing Near its threshold, the disc's flank sharpens to a point under the pin. One root of a twelve-pin cycloidal disc, pins of radius 5 on a ring of 60, cut with the eccentric at 0.97 of the threshold 0.9740 R/N. The line is the disc's edge; the circle is a pin touching it at the flank's sharpest point, 31.0° of the eccentric from the root. There the edge bulges towards the pin with a radius of 2.23, the flank's bend 7.23 less the pin's 5 (small circle, centre marked), so the contact is convex against convex, and a line contact's pressure per unit load there is 1.80 times a pin on a flat. At the root itself the edge is a hollow round the pin.

A disc cut at its threshold grows a point

A cycloidal disc's roots stay untrimmed up to an eccentric where the pin path's flank bends with the pin's own radius, and it looked as though a disc cut exactly there would have its most conformal contact, the pin and the flank sharing a curvature. It has its least. The disc's edge is the path offset inwards by the pin's radius, so where the path bends round a centre on the disc's side the edge bulges towards the pin with the path's radius less the pin's: at the threshold, a point. A line contact's pressure per unit load there rises as the inverse square root of the distance to the threshold, and the pins passing it are carrying load. Each of four drives has an eccentric of least contact stress at 0.67 to 0.80 of its threshold; cut at 0.97, the peak pressure is 9% to 80% higher for a lever 21% to 46% longer.

meshing · Meshing A relief as deep as the contact's own approach, spread across the flank. One root of the twelve-pin disc cut at 0.97 of its threshold, pins of radius 5 on a ring of 60, with the edge relieved inwards by a cosine bump of depth 0.01 and half-width 1 rad of eccentric centred on each flank. The relief is drawn 200 times deeper than it is: at true scale it is 0.2% of the pin's radius, the size of the contact's own approach at the design torque, and invisible. The thin line is the unrelieved edge; the pin touches it at the flank's sharpest point, where the unrelieved edge's radius is 2.23.

A relief is sized for one torque

A cycloidal disc cut near its trimming threshold has long levers and a point on each flank, where the pins carrying most of the load press hardest. Relieving the edge a little near the flanks can take most of that point away. On a twelve-pin drive at 0.97 of its threshold, a relief 0.01 deep spread across a radian either side of each flank lowers the peak contact pressure by a quarter and keeps every lever, to within 7% of the disc cut at its least-stress eccentric, whose levers are a fifth shorter. The relief does it by unloading the flank, not by reshaping it, so its depth must be the contact's own approach at the design torque. At a quarter of that torque it is worse than no relief at all, and a relief of hundredths of the pin's radius, which is what was expected to work, takes pins out of the load.

meshing · Meshing

Named alongside it

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

EnvelopeContact normalPitch pointBacklashClearanceUndercutConjugate-actionContact pointDesign ruleLobeCentre distanceCompound epicyclic

All concepts