A tight cage spreads the load until the tracks join in

A tight groove leans on one ball followed a six-ball Rzeppa joint past its free play. A Rzeppa joint’s play is set by the tighter of two limits. The cage’s windows hold every ball near the plane that bisects the joint’s bend. The offset grooves hold each ball on its own tracks. Where the cage binds, every ball is against its window at once, and the first load past the play spreads over all six. Where the grooves bind, only the worst ball’s track is at its edge, and the first load falls on that ball alone. With every contact a spring of the same stiffness, a joint whose grooves set its play loads its hardest-pressed ball 1.5 to 3 times as hard as one whose cage does, at the same torque.

The essay ended by proposing a remedy. A joint with offset grooves could have its cage’s windows made tighter than its tracks require, so that the cage’s limit sat below the grooves’ everywhere in a ball pitch. The cage would then set the play, the first load would spread over every window, and the grooves would carry load only beyond that. The question was whether the cage would take over cleanly or whether the two limits would interact.

Both happen, in sequence, and the windows’ clearance decides where the sequence turns.

Where the cage takes over #

With the track play fixed at one, the cage alone allows the joint a lag proportional to its windows’ clearance $c$, and the grooves alone a lag set by their offset. Both vary over a ball pitch. They are largest at the ends of the pitch, and both vanish at the input angle of 30°, where the joint has no play at all. So the cage sets the play wherever $c$ times its lag per unit clearance is less than the grooves’ lag. It sets it at every input angle once $c$ is below the least ratio of the two over the pitch. For grooves offset 0.7 at a bend of 20° that ratio is 0.559, approached at 30°, where both lags shrink to nought together and their ratio falls to its smallest value.

The earlier essays’ closed forms give both limits. The cage’s comes from its windows’ sensitivities to the lag, and the grooves’ from each ball’s sensitivity along its tracks. The joint’s own free lag, found by a linear program over every ball’s radius and the cage’s tilts and slide with nothing assumed, must equal the smaller of the two. At every window clearance and every one of thirteen input angles across the pitch it does, to a part in a million.

The play falls as the windows close. With windows as loose as the tracks, the joint’s largest play over the pitch is 0.280, set by the grooves. At windows of 0.503, nine tenths of the crossover, the cage sets it everywhere and it is 0.164. At windows of 0.280 it is 0.091. Unlike tightening the tracks, which the earlier essay found to lower the play by making the grooves bind, tightening the windows lowers the play by making the cage bind. The cage is the limit that loads every ball at once.

The first load spreads at every input angle #

The measurement is the earlier essay’s: every allowance a one-sided spring of the same stiffness, the joint twisted a little past its free lag, and the cage’s tilts and slide and the balls’ radial positions settled where the stored energy is least. The load is summarised by the reciprocal of the sum of the balls’ squared shares, which is six for an even split and one for a single ball. With windows as loose as the tracks, or at 0.8 of them, the first load falls on a single ball somewhere in the pitch. At 0.6, just above the crossover, the cage binds at most input angles and the worst is 1.73 balls’ worth. At and below 0.5, under the crossover, the first load spreads over at least 3.63 balls’ worth at every input angle, and the largest single share anywhere is 0.38.

Below the crossover the spread stops changing with the clearance. Every ball is against its window when the cage binds, and the windows’ clearances all scale together, so the first load’s distribution depends on the cage’s geometry and not on how tight it is. The caged joint of the earlier essay, with its grooves offset only 0.1, spread its first load over 4.1 to 5.9 balls’ worth. The grooved joint with a tight cage reaches 3.6 at its least, because its balls sit at different radii and its windows are loaded unequally, but it never loads one ball alone.

The tight cage’s first load is the caged joint’s #

The first load’s pattern at a single input angle shows why the spread does not depend on the clearance. At 15°, three thousandths of a track play past the play, the grooved joint with windows at nine tenths of the crossover puts 0.17, 0.32, 0.33 and 0.18 of its load on four balls and nothing on the other two. The caged joint of the earlier essay, grooves offset only 0.1 and windows as loose as its tracks, puts 0.17, 0.33, 0.33 and 0.17 on the same four balls and nothing on the same two. The two joints’ grooves differ sevenfold in offset, and their first loads are the same to a hundredth.

That is the earlier essays’ count at work. When the cage binds, the constraints at their limit are the windows, and which windows reach their limit first is set by the cage’s geometry and the balls’ places round it. The grooves’ offset changes where each ball sits radially, but the windows hold every ball to the bisecting plane whatever its radius. So once the cage is the limit, the joint’s first response to load is the cage’s response. The two balls that carry nothing at first are the ones whose windows still have room at that input angle. They join within a few more hundredths of a play, which is why the caged joint reaches 5.1 balls’ worth by a hundredth and 5.9 by three.

A higher pair has no group found that a ball in a groove has freedoms that depend on where it touches. Here it is the window, not the groove, that each ball touches first, and so the windows’ freedoms decide the sharing.

Until the tracks join in #

A cage that binds first does not bind alone for ever. The joint’s play is the cage’s limit. Twisted further, the joint reaches the grooves’ limit too, where the worst ball’s track comes to its edge. The twist between the two is the gap between the limits, and past it the tracks carry load as well. The tracks are what loaded one ball in the untightened joint, and they load unevenly again.

At 15°, with windows at nine tenths of the crossover, the joint spreads its load over 3.69 balls’ worth for the first few hundredths of a track play past its play. It reaches 4.32 at three hundredths, then falls back to 4.13 at a tenth and 4.06 at three tenths as the tracks engage. At larger twists it joins the untightened joint near five balls’ worth. With windows at half the crossover the gap is wider. The joint spreads to 5.46 balls’ worth by three hundredths, nearly as even as the caged joint, holds that to a tenth, and falls back to 4.34 at three tenths. With the windows just above the crossover, at 15° the cage already binds and spreads the first load over 2.6 balls’ worth, but the gap is so small that the tracks engage almost at once.

The windows’ share of the load shows the hand-over directly. With windows as loose as the tracks the windows carry nothing until a tenth of a track play past the play, because the grooves set the play. With windows at nine tenths of the crossover they carry everything for the first hundredth and 93% at three hundredths, then the tracks take over: 62% on the windows at a tenth, 52% at three tenths. At half the crossover the windows carry everything to three hundredths and 89% at a tenth. At large twists every case settles near four tenths on the windows and six tenths on the tracks, the proportion the joint’s geometry gives once everything bears.

At the same torque #

A designer specifies a torque, so the fair comparison is the hardest-pressed ball at the same torque. The grooved joint with windows as loose as its tracks loads its worst ball 1.76, 2.44, 3.01, 2.48 and 2.09 times as hard as the caged joint at torques of 0.3, 1, 3, 10 and 30 in contact-spring units, the earlier essay’s numbers. With its windows at nine tenths of the crossover the ratios are 0.93, 1.26, 1.64, 2.01 and 1.81. At half the crossover they are 0.93, 1.26, 1.15, 1.38 and 1.68.

At the lowest torque the tightened joint’s worst ball is a little lighter than the caged joint’s. Its cage binds with its own windows’ geometry, and at that torque nothing else bears. As the torque rises the tracks come in and the ratio climbs, and at the highest torque measured every tightened joint is back above one and a half times the caged joint. A tighter cage holds the ratio down to a higher torque, because the twist the joint needs before its tracks engage is the gap between the limits, and the gap widens as the windows close.

So the cage cannot simply be tightened past the problem. It can be tightened until the problem starts at a torque the joint rarely sees. Below that torque the joint behaves like a caged one, with less play than the caged joint has. Above it, it behaves like a grooved one, with the tracks sharing unevenly.

The window clearance is a torque #

That turns the windows’ clearance from a matter of play into a design variable with a definite meaning. With track play one and windows $c$, the cage binds everywhere below 0.559, and the tracks engage once the twist has passed the cage’s limit by the gap between the limits. That gap is the grooves’ lag less $c$ times the cage’s, $(0.144 - 0.231c)$ at 15°: 0.028 track plays at nine tenths of the crossover and 0.079 at half of it. With every contact the same stiffness, the torque the joint carries at that twist is the torque at which even loading ends.

The windows’ share of the load bears this out. On the joint at nine tenths of the crossover the windows still carried 93% of the load at three hundredths of a track play past its play, just past its gap of 0.028, and 62% at a tenth. At half the crossover they carried all of it at three hundredths and 89% at a tenth, just past its gap of 0.079. The tracks begin to take load where the gap between the two limits says they should, and the rest of the twist decides how much.

A relief is sized for one torque found the same structure in a cycloidal drive. A gap that keeps load off the worst contact does so up to the torque at which the contact’s own deflection closes it. There the gap was a relief cut into the disc’s flank; here it is the difference between two limits the designer sets by choosing a window clearance. In both, the clearance is sized against the deflection under the working load, not against the geometry, and in both a joint that runs above that load gets back the uneven sharing the clearance was meant to remove.

What a designer can take from it #

A tight cage is the cheap fix for tight tracks. Tightening a grooved joint’s windows below 0.559 of its track play makes the cage the limit everywhere. It lowers the play further and spreads the first load over every window. At low torque the worst ball then carries no more than a caged joint’s.

Size the windows for the torque. The tracks join in once the twist passes the gap between the limits, and the gap grows as the windows close. A joint that works near zero torque, where backlash is what matters, needs only a little tightening. One that carries steady torque needs the gap to be larger than the deflection at that torque, which means windows well inside the crossover.

A clearance is a link, and two clearances in series set a sequence. The earlier essays treated the cage’s and the grooves’ allowances as two links whose tighter one sets the play. Under load the looser one still matters. It decides when the second set of contacts engages. The count cannot tell a pin from a slide found that a joint’s count sees only how many constraints there are; the order in which two sets of allowances run out is invisible to it and decides how the load is carried. A Rzeppa cage adds its clearances found the cage’s windows adding their clearances into the play. Here the two limits act one after the other.

What the springs do not settle #

Unequal stiffness. Every contact is a linear spring of the same stiffness. A window contact and a track contact are different Hertzian contacts, and a track’s contact angle on the groove’s flank, the surface on which a joint slides on itself, sets how stiff it is along the twist. If the windows were much softer than the tracks, the tracks would take over sooner, and if stiffer, later. Which way a real joint leans depends on its geometry and materials, and it is not measured.

The whole pitch at load. The growth and torque measurements are at 15°. The first-load measurement covers the whole pitch, but how the gap between the limits varies over it, and so the torque at which the tracks engage at each input angle, is read from the limits and not followed under load at every angle.

Wear. A cage whose windows carry the load at low torque wears its windows, and wear opens the clearance. A cage tightened for even loading drifts back towards the crossover as it wears. How fast is a question about the joint’s duty, and a floating cage frees three balls and not six found that the cage’s own float already spends some of its clearance.

Other bends. Everything is at 20°. The crossover clearance 0.559 is a property of that bend and that offset, and at a larger bend the balls move further off the bisecting plane and both limits change.

Still open: windows that close with the load #

A fixed window clearance keeps the cage in charge only up to the torque at which the gap between the limits is used up. The cage would stay in charge at every torque if its windows’ effective clearance shrank as the load rose, so that the gap grew with the twist.

The distinct argument would be a cage whose windows are lined with a compliant layer, stiffer than the tracks at first contact and growing stiffer with load, so that the windows always take the larger share. The measurement would give the window contacts a stiffness a chosen factor above the tracks’ and find the factor at which the worst ball stays within a stated margin of the caged joint’s at every torque from 0.3 to 30. If a modest factor does it, a grooved joint can have both its small play and even loading at every torque, by making its cage the stiffer path rather than the tighter one. If it takes an extreme factor, the grooved joint’s tracks always end up carrying the torque unevenly, and the offset that lowers the play is a commitment to uneven loading above some torque.