A disc cut at its threshold grows a point

The flank that trims the root found the eccentric at which a cycloidal disc’s roots start to be cut away, for any pin count and pin radius. The curve a pin’s centre traces in the disc’s frame has a tip at each root, pointing at the disc’s centre, and either side of the tip its flank bends round a centre on the disc’s side. Where that bend is tighter than the pin’s radius, the disc’s edge, the curve offset inwards by the pin’s radius, folds on itself, and a pin coming back along the flank cuts the fold away. The threshold has a closed form, $k^* = \sqrt{1 - \rho^2(N+1)^3/27(N-1)}$ as a share of the ring’s radius over the pin count, and below it every pin of every drive measured stayed in contact.

That essay ended with a hope. At the threshold the flank’s bend equals the pin’s radius at one point of each root. It seemed that the pin and the disc would there share a curvature, so that the gap between them would open as the fourth power of the distance along the flank rather than the square. That would make it the most conformal contact the drive can have, and the eccentric that gives it the one to choose.

The bend is shared, but by the pin and the path, not by the pin and the disc. At the threshold the disc’s edge does not curve with the pin. It comes to a point.

An offset keeps its centres #

A curve and its offset share their centres of curvature. Move every point of a curve a fixed distance along its normal, and the centre of curvature at each point stays where it was, while the radius changes by the distance moved. Moved towards the centre, the radius shrinks; moved away, it grows. A curvature is a size with a minus sign made the same point about signed curvature, and here the sign decides everything.

The disc’s edge is the path offset by the pin’s radius $\rho$ towards the disc, because the pin’s centre is on the path and the pin’s surface touches the disc on the disc’s side. On the flank the path’s centre of curvature is on the disc’s side too, at distance $\rho_f$ from the path. So the offset moves towards that centre, and the edge’s radius there is $\rho_f - \rho$, with the same centre. The edge curves round a point inside the disc. It bulges outwards, towards the pin.

The pin is a circle and bulges towards the disc. Two convex surfaces touching have a relative curvature equal to the sum of their curvatures, $1/\rho + 1/(\rho_f - \rho)$. As the eccentric approaches the threshold, $\rho_f$ comes down to $\rho$, and the second term grows without limit. At the threshold the disc’s edge has a radius of nought at the flank, which is a corner. The swallowtail the earlier essay found beyond the threshold, the offset folding back on itself, is born from exactly this point.

The closed form was checked against the disc’s edge itself. The edge was built by offsetting the sampled path, its curvature taken by second differences at the flank’s sharpest point, and compared with $1/(\rho_f - \rho)$. On four drives at eight eccentrics each, from 0.8 of the threshold to 0.999, the two agree to 1%. On the twelve-pin drive the edge’s radius at the flank is 8.83 at 0.8 of the threshold, 2.23 at 0.97 and 0.09 at 0.999.

Where the contact is conformal #

The earlier essay’s hope had the right picture in the wrong place. There is a part of the disc where the contact is conformal, and it is the root.

At the root the path’s tip bends round a centre on the pin’s side, so the offset moves away from it and the edge there is a hollow, larger than the pin by the tip’s own radius. A pin in a hollow only slightly larger than itself is the most conformal contact a convex part can have. Measured as the factor by which a line contact’s pressure per unit load exceeds a pin’s on a flat, $\sqrt{1 + \rho\kappa}$ with $\kappa$ the edge’s curvature, the root reads 0.256 at 0.8 of the threshold, 0.081 at 0.95 and 0.033 at 0.995. Its hollow is 5.35 across at the first and within a hundredth of the pin’s 5 at the last. Towards the threshold the root becomes more conformal at the same time as the flanks either side of it become less.

The two flanks are the peaks, 1.25 at 0.8 of the threshold, 1.58 at 0.95 and 3.51 at 0.995. So the disc approaching its threshold is a profile whose best contact and worst contact sit a third of a radian of eccentric apart, both sharpening.

The two go together for one geometric reason. Raising the eccentric makes the path’s excursion towards the disc’s centre larger at each root, so its tip turns more tightly and its flanks bend more tightly too. A tighter tip is a tip whose centre is on the pin’s side and close to the path, which makes the hollow around the pin closer to the pin’s own size. A tighter flank is a flank whose centre is on the disc’s side and close to the path, which, once the pin’s radius is taken off, leaves the disc a smaller bulge. One change in the path makes the root hug the pin and the flank point at it.

How fast the point arrives #

Near the threshold the flank’s radius falls in proportion to the distance from it, so the disc’s radius there, $\rho_f - \rho$, does too. A line contact’s pressure goes as the square root of load times relative curvature, so per unit load the flank’s pressure rises as the inverse square root of the distance to the threshold.

Every drive follows that slope once it is within a few per cent. At 0.97 of their thresholds the four flanks’ factors are 1.63, 1.80, 2.99 and 2.61; at 0.999 they are 6.28, 7.46, 14.90 and 12.66. The drives with more pins and fatter pins sharpen fastest, the twenty-pin drive with pins of radius 6 most of all. Their thresholds are lower, and their flanks bend more tightly for a given eccentric, so the edge’s radius at the flank is smaller at the same share of the threshold. The law has no finite value at the threshold itself. That is the defining fact of a point, and the reason the earlier hope cannot be rescued by cutting slightly more carefully.

Whether the pins carry load there #

A sharp point on a profile matters only if a pin carrying force touches it. A cycloidal drive shares its torque among many pins, each with its own lever about the disc’s centre, and a pin passing the flank’s point might be one of the lightly loaded ones.

To find out, every pin was followed over a pin pitch of the eccentric. Each pin’s place on its path is the eccentric angle plus a fixed offset for its position in the ring, which was checked against the site’s own construction of the pins in the disc’s frame to $10^{-9}$. At each sample each pin’s contact normal and lever about the disc’s centre were found. The torque was then shared among the pins whose levers drive, in proportion to their levers. That is the linear model in which every contact has the same stiffness and the disc turns a little under the torque, the model a clearance leaves one contact set aside when it gave the pins play. The shares were checked to add up to the torque at every sample. Each contact’s pressure was then its share times its relative curvature, square-rooted, normalised by one pin carrying the whole torque against a flat at the ring’s radius.

The pins passing the flank’s sharpest point are not lightly loaded. On the twelve-pin drive the pin there carries a force of 0.0061 to 0.0074 per unit of torque at every eccentric from 0.8 of the threshold to 0.999, against 0.0065 to 0.0077 for the most heavily loaded pin anywhere in the pitch. On the other three drives it is likewise within a few per cent of the heaviest. The flank is where a cycloidal disc does its work, which is why the earlier essay found the levers it gained near the threshold worth having.

The eccentric of least contact stress #

Put the load and the curvature together and each drive has a best eccentric for contact stress, and it is not the threshold.

At small eccentrics the levers are short, each pin carries a large force, and the peak pressure sits further out along the lobe. As the eccentric grows the levers lengthen and the peak falls. Near the threshold the flank’s point takes over and the peak climbs again, without limit. Between the two lies a broad, shallow minimum, found by golden section for each drive: 0.80 of the threshold for six pins, 0.78 for twelve, 0.67 for twenty and 0.70 for thirty. The minimum is flat. On the twelve-pin drive the peak changes by under 1% between 0.75 and 0.8 of the threshold.

Moving from each drive’s least-stress eccentric to 0.97 of its threshold, where a disc whose roots are not sockets and its successor measured the levers, buys a lever 21% to 46% longer and costs a peak pressure 9% to 80% higher. The six-pin drive barely notices, 9%. The twenty-pin drive with fat pins pays most, 80%, and it is also the drive whose flank sharpens fastest towards its threshold. At 0.999 of the threshold every drive’s peak is between four and nine and a half times its least.

Where the classical proportion sits #

The classical cycloidal proportion puts the eccentric at the ring’s radius over the pin count, a share of one in these units, and every drive here has its threshold below that: 0.982, 0.974, 0.905 and 0.929. So the classical disc is past its threshold on all four, with roots trimmed and some pins standing clear, which a disc whose roots are not sockets found and the earlier essay extended. Its flanks are past the point too. The trimming cuts away the fold that grows out of the point, and what is left at the edge of the trimmed root is a corner where the pin’s cut meets the uncut flank. The contact model here does not follow pins across a trimmed root, but a corner is the limit the flank’s law was heading for, and a drive cut the classical way has two of them at every root.

The least-stress eccentrics, 0.67 to 0.80 of each threshold, are 0.6 to 0.8 of the classical proportion. A drive designed for contact life is therefore not a small correction to the classical one but a substantially different disc, with lobes shallower by a fifth to two fifths, since a lobe’s depth is twice the eccentric. Its contact normals pass closer to the disc’s centre, which is what a shorter lever means, and the table prices exactly that loss of lever against the gain in contact life.

What the drive designer is choosing between #

The earlier essays measured a drive’s geometry: which pins touch, how long their levers are, where the roots are trimmed. By those measures the eccentric should go as high as the threshold allows. Contact stress measures something they could not, how the load meets the profile’s curvature, and by that measure the threshold is the worst place to be and the best is well below it.

A designer therefore chooses between three quantities rather than one. Lever governs how much torque a given set of pin forces transmits. Contact pressure governs surface fatigue on the pins and the disc. The trimming threshold is a hard limit beyond which pins stop touching. The least-stress eccentric is the point of least contact stress, the threshold is the end of the useful range, and between them lever is bought with stress at a rate the table gives. A drive whose life is set by surface fatigue should sit near its least-stress eccentric. A drive whose torsional stiffness or lost motion matters more than its life can move towards the threshold, but not to it.

Eleven lobes from twelve pins built the disc as the envelope of its pins, and a profile is an envelope found the same construction in cams. An envelope’s curvature is the generating motion’s curvature offset by the generator’s size, and every such profile has the same hazard: where the motion’s own bend approaches the generator’s radius, the profile sharpens before it folds. A cam’s roller and its pitch curve meet the same limit, which the cam that cannot be cut found as a roller that gouges a profile whose pitch curve bends tighter than the roller. Cam designers keep a margin above it and state it as a minimum radius of curvature for the profile rather than for the pitch curve, for the reason measured here: the profile’s own radius governs the contact stress, and it reaches nought before the pitch curve’s does. For a cycloidal disc that limit is the threshold, and this is what approaching it does.

What the contact model does not settle #

Load sharing. Torque is shared in proportion to lever, the model in which every contact is equally stiff and nothing has clearance. Real drives have clearance, and a clearance leaves one contact found that it concentrates the load on fewer pins, which would raise every pressure here and could move the least-stress eccentric.

Hertz’s line contact. The pressure factor is the square root of load times relative curvature, Hertz’s law for two long cylinders. Near the threshold the disc’s radius at the flank becomes comparable with the contact’s own width, where the law stops being accurate, and the true peak is finite though large. The edge’s radius at 0.999 of the threshold is under a tenth of a unit on a ring of 60, smaller than a realistic contact width, so the last points of the law are a direction rather than a value.

Pin rotation. Pins in real drives often turn in their housings, which changes the sliding at each contact but not its curvature, so the pressures here are unchanged; the wear and friction are not computed.

Manufacturing. A disc ground to a profile with a radius of a fraction of a unit at its flanks will not be ground exactly. Any real disc cut near its threshold has its points blunted by the wheel, which helps the stress and changes the contact positions, and neither is modelled.

Still open: a disc relieved at its flanks #

The pressure peaks at a single point of each flank, and the roots are conformal. A disc does not have to be the exact envelope of its pins. Designers already modify cycloidal profiles slightly to give clearance.

The distinct argument would be a disc relieved only near the flank’s sharpest point, enough to keep the pin from touching where the edge is tightest, and left exact elsewhere. The measurement would choose a relief as a small inward displacement of the edge, a function of position along the flank, cut at an eccentric near the threshold. It would ask whether the drive then keeps the longer levers of the high eccentric while its peak pressure falls to the least-stress value or below. It would also ask how many pins lose contact when the flank is relieved, and how much lost motion the relief adds. If a relief of a few hundredths of the pin radius can do it, the threshold’s lever is available without its point.