A relief is sized for one torque

A disc cut at its threshold grows a point found what happens to a cycloidal disc as its eccentric approaches the threshold at which its roots begin to be trimmed. The disc’s edge is the path of a pin’s centre offset inwards by the pin’s radius, and on each flank the path bends round a centre on the disc’s side. The offset therefore leaves the edge bulging towards the pin with the path’s radius less the pin’s, which falls to nought at the threshold. The pins passing that point carry nearly the heaviest load in the drive, so the peak contact pressure climbs without limit towards the threshold. Each drive has an eccentric of least contact stress well below it. For twelve pins of radius 5 on a ring of 60 it is 0.78 of the threshold, with levers a fifth shorter than at 0.97.

The essay’s closing suggestion came from practice. Cycloidal discs are commonly modified slightly from their exact envelope to give clearance. A disc relieved only near the flank’s sharpest point, enough to keep the pin off it, and left exact elsewhere, might keep the long levers of an eccentric near the threshold while its peak pressure fell to the least-stress value. It guessed the relief would be a few hundredths of the pin’s radius.

A relief can take most of the point away, but not by blunting it, and not at that depth. It works by moving load, and so it works at one torque.

What a relief does to the edge #

The relief is a smooth inward displacement of the edge, a cosine bump of depth $A$ and half-width $w$ in the eccentric angle, centred on each flank of every lobe. Where the bump is deepest it flattens the edge’s bulge, and a curvature is a size with a minus sign is the reason the sign of the change matters. A bulge towards the pin of curvature $\kappa$, with a bump $\delta(s)$ taken out of it, has curvature $\kappa + \delta''$ along its arc length, and at the bump’s centre $\delta''$ is negative, $-\pi^2 A/2w^2$ for this shape. On the bump’s shoulders $\delta''$ is positive and the edge sharpens instead. A relief takes one peak of curvature and leaves two smaller ones either side of it.

That was checked on the relieved curve itself. Its curvature was taken by second differences at the bump’s centre and compared with $\kappa + \delta''$ for three reliefs, and the two agree to within 2% of the unrelieved curvature they change. Away from the centre the closed form leaves out terms in the bump’s slope, so the contact model uses the measured curvature throughout.

The figure shows the shoulders. Unrelieved, the edge’s pressure factor peaks at 1.80 at each flank. A relief 0.01 deep and a quarter of a radian wide flattens the flank and leaves two new peaks either side of it, at 2.21, sharper than the point it replaced. A wide relief spreads the same change over a radian either side and leaves a peak of 1.72. Reshaping alone is worth little: the best relief lowers the edge’s worst pressure factor from 1.80 to 1.72, under 5%. The large effect is elsewhere.

What a relief does to the load #

A pin passing a relieved stretch stands off the disc by the relief’s depth there. The drive’s pins share its torque through their contacts, each as stiff as the others, and the disc turns a little under the torque until its pins are pressed enough to carry it. With the disc turned by $\theta$, a pin of lever $a$ is pressed by $a\theta$, so a pin standing off by $\delta$ carries a force

$$F = k\,\max(0,\ a\theta - \delta),$$

and $\theta$ is whatever makes the pins’ moments add up to the torque. With no relief this is the model the earlier essay used, every pin’s force in proportion to its lever, and at three positions of the drive the two agree to $10^{-9}$. A relief changes how the load is shared. A relieved pin carries less, and the pins outside the relief carry more, in proportion to how deep the relief is against the contact’s own approach $a\theta$.

So the relief’s depth has a natural unit, the contact’s approach under the design torque. The stiffness was set so that the unrelieved disc’s most heavily levered pin approaches by 0.01 at the design torque. On a ring of 60 millimetres that is ten micrometres, a plausible deflection for a pin and disc in contact. A relief much shallower than that hardly moves any load; one much deeper takes the relieved pins out of the load altogether.

The best relief is shallow and wide #

Each relief was followed over a pin pitch at 160 positions. At each position the turn that balances the torque was found by bisection, every pin’s force and contact pressure was computed, and the pins’ moments were checked to add up to the torque to $10^{-9}$. The peak was the largest pressure anywhere in the pitch, normalised as in the earlier essay by one pin carrying the whole torque against a flat at the ring’s radius.

Unrelieved, the disc at 0.97 of its threshold peaks at 1.184, and the same disc cut at its least-stress eccentric at 0.847. The best relief found, 0.01 deep, the contact’s own approach, over a radian either side of each flank, peaks at 0.904. That is 24% below the unrelieved disc and 7% above the least-stress one. It keeps every lever of the eccentric near the threshold: the worst lever over the pitch is still 50.2, against 40.4 at the least-stress eccentric. Five pins still carry load at every position, as without relief, and the relief adds no lost motion, since the pins away from the flanks are unrelieved and still touch.

A radian either side of the flank is most of the driving side of a lobe, and the bump overlaps itself across the root. The best relief is not a local touch at the sharp point. It is a shallow, broad lowering of the stretch where the levers are longest, deep enough to hand part of the flank pin’s load to its neighbours and no deeper. Narrow reliefs do worse than wide ones at every depth but the shallowest. At a quarter of a radian wide and 0.01 deep the peak rises to 1.275, worse than no relief, because the shoulders the relief creates are sharper than the point it removed and the pins crossing them still carry load.

Where the load goes #

The position at which the unrelieved disc has its peak shows the mechanism plainly. There the pin at the flank has a lever of 51.3, the longest in the drive, and carries the largest force, 0.00723 per unit of torque, against the sharpest edge. With the best relief it stands off by the full 0.01, closes only part of that gap at the design torque, and carries 0.00387. Its pressure falls from 1.184 to 0.821. The pin beside it, with a lever of 46.2 and a relief of half the depth there, carries about what it did. The three pins further round, with levers of 13.7 to 37.7, are unrelieved and take up the difference, each carrying about half as much again. The pin at 37.7 becomes the most heavily loaded, at a pressure of 0.884, because its own stretch of edge is gentle.

So the relief moves the heaviest contact from the sharpest place in the drive to a gentler one, at the price of a slightly shorter lever for the load it moves. Because the pins that take up the load are unrelieved, their levers are the drive’s own, and the worst lever over the pitch does not change.

Hundredths of the pin’s radius are too deep #

The earlier essay’s guess put the relief at a few hundredths of the pin’s radius, a tenth of a unit or more on these pins. That is ten times the contact’s approach. At that depth the relieved pins never close their gap at the design torque, and the torque passes to the pins outside the relief. Those have shorter levers, so they carry more force and the peak rises.

A relief 0.04 deep over a radian either side takes one pin out of the load at some phases and raises the peak to 1.227. A relief 0.1 deep, two hundredths of the pin’s radius, takes two out and raises it to 1.301. The worst lever falls from 50.2 to 39.4 and 35.6. A deep relief therefore loses the long levers the relief was meant to keep. The pins it unloads are the ones near the flank, which are the ones with the longest levers. The flank that trims the root found pins standing clear of a trimmed root to be the drive’s way of losing levers, and a deep relief does the same thing deliberately.

A relief is sized for one torque #

An unrelieved disc’s contact pressure scales as the square root of the torque at every pin, because its load sharing does not depend on the load. Its peak divided by the square root of the torque is a property of the disc. A relieved disc’s is not. The relief is a fixed gap, the contact’s approach grows with the torque, and the share of the load the relieved pins carry grows with it.

At the torque it is sized for, the best relief gives 0.904 against the unrelieved disc’s 1.184. At half that torque it gives 1.055. At a quarter it gives 1.213, worse than no relief, because the relieved pins now carry nothing and the others carry everything. Above the design torque the relief matters less and less: 0.993 at twice the torque, 1.060 at four times and 1.091 at eight, approaching the unrelieved disc from below as the gap becomes small against the approach. At every torque the disc cut at its least-stress eccentric gives 0.847.

So the trade the earlier essay hoped for exists at one torque. A drive that runs most of its life near a known torque, such as a positioning reducer under a steady load, can have the threshold’s levers at nearly the least-stress pressure with a relief sized for that torque. A drive whose torque varies, and most do, gets the benefit only near the torque it was sized for. At light load it is worse off than if it had not been relieved, because the relief has removed the flank pins from the load.

Why the least-stress disc is the robust one #

The disc at its least-stress eccentric achieves its 0.847 by geometry alone. Its flanks bend more gently, and its load sharing is in proportion to its levers at every torque, so its contact life does not depend on knowing the load. The relieved disc reaches 0.904 at its design torque by arranging for its stiffest-levered pins to carry less. That arrangement is a property of the load as much as of the disc.

A clearance leaves one contact found that clearance in a pin drive concentrates load on fewer pins, and a relief is a clearance placed on purpose. Where the earlier clearance was an error that cost load sharing, the relief here is a design that redistributes it, and like any clearance its effect depends on how the deflections it is set against compare with it. Backlash is an allowance made the same point for gear teeth. An allowance has to be sized against the deflections under load, and an allowance sized for one load is wrong for another.

The ledger #

The ledger sets the four reliefs beside the two unrelieved discs. The best relief is the only one that improves on the unrelieved disc at 0.97 by much, and it is the only one that keeps both the long levers and every loaded pin. The narrow relief is worse on pressure. The deep reliefs are worse on pressure, on levers and on load sharing. None adds lost motion, because none relieves the pins on the far side of the lobe, which keep the disc located.

What a designer can take from it #

Three practical rules follow, each a correction to the intuition the earlier essay started from.

Size a relief against deflection, not geometry. A profile is an envelope and eleven lobes from twelve pins built the disc as an exact envelope of its pins, a purely geometric object. A relief is the first departure from that envelope this field has measured, and its right size is set by the load, not by the envelope. The useful depth is the contact’s approach under the torque the drive is designed for, here two thousandths of the pin’s radius, not the hundredths that would be chosen by eye. That makes a relief a matter of a few micrometres on a drive of this size, and its depth has to be held to a fraction of that in manufacture.

Make it wide. The point at the flank is not removed by cutting it away, since a narrow relief only moves it to the relief’s shoulders. A relief does good by taking load off the pins with the longest levers, so it should span the stretch where they are.

Do not relieve a drive whose torque varies widely. At a quarter of the design torque the best relief here is worse than none. The least-stress eccentric gives the same contact life at every torque without a relief. Choosing it costs a fifth of the lever, and the relief buys that lever back only near its design torque.

What the relief does not settle #

Other shapes. The relief is a cosine bump centred on the flank. Bumps of other shapes, a relief that grows steadily towards the root, or one placed on the far flank as well for a drive that reverses, are not searched, and a better one than 0.904 may exist. Whether any reaches the least-stress value at its design torque is not shown.

Hertz’s law near the shoulders. The pressure is the square root of load times relative curvature, Hertz’s line contact. The cam that cannot be cut met the same limit on a cam, where a pitch curve bending tighter than the roller makes the profile impossible to cut. A relief’s shoulders can have small radii, and where the radius is comparable with the contact’s width the law overstates the peak. The narrow reliefs’ shoulders are the cases where that matters most.

Contact stiffness. Every contact here is linear and equally stiff, the model the earlier essay used. A Hertz contact is stiffer the more it is loaded, which would change how quickly the relieved pins pick up load as the torque rises, though not which way.

The other eccentrics. Only 0.97 of the threshold was relieved. A drive closer to its threshold has a sharper point and might gain more from a relief, and one further away less.

Still open: a relief that grows with the load #

A fixed relief is a gap that a contact closes under a certain load, and so it is right for one torque. The drive’s own deflection does not have to be the only thing that closes it.

The distinct argument would be a disc whose relief is supplied by compliance rather than by removing material: a flank region backed by a softer layer, or a pin carried in a compliant bush, so that the relieved pins’ extra flexibility grows with the load in the way the contact’s approach does. The measurement would give the flank pins a contact stiffness lower than the others by a chosen factor and find the factor that minimises the peak pressure. It would then ask whether that peak stays below the unrelieved disc’s at every torque, since a stiffness ratio, unlike a gap, does not depend on the load in the linear model. If it does, the threshold’s levers are available at every torque for the price of a softer flank. If the best stiffness ratio leaves the peak no lower than the least-stress eccentric’s, the eccentric is the honest answer, and the long levers near the threshold belong to drives whose torque is known.