A tight groove leans on one ball

A Rzeppa joint transmits a turn through six balls running in grooves on an inner race and an outer race, held by a cage in the plane that bisects the two shafts, and constant velocity is a mirror showed that keeping them in that plane is what keeps the output in step. A Rzeppa cage adds its clearances gave the cage’s windows a clearance and found how far the output can lag, and a floating cage frees three balls and not six added the grooves. Offset grooves steer each ball towards the bisecting plane on their own, so with a radial play between ball and track the joint’s free lag is set by whichever is tighter: the cage, whose windows bind on all six balls at once, or the grooves, which bind ball by ball and so are set by the worst one.

That essay ended on what the free lag does not say. It is the angle the output can turn before anything stops it, and it is the same number whichever limit sets it. What happens next is not. When the cage binds, every ball is at its window at the limit, so a torque taking up the play meets all six together. When the grooves bind, only the worst ball’s track is at its edge, so a torque would meet that ball first. The question was whether the balls then share the torque as unequally as that suggests, and whether a designer tightening the tracks to cut the play is buying uneven loading with it.

They do, and the designer is.

Every contact a spring #

The joint is the earlier essay’s, a ball in a groove being a higher pair whose freedoms depend on where it touches. The input shaft runs along one axis and the output is turned 20° from it; each ball sits where its inner and outer grooves meet; the cage is a plane that may tilt two ways and slide along its normal. With offset grooves each ball also has a radial position of its own, and its two track distances must stay within a play δ of its tracks while its distance from the cage’s plane stays within the window clearance $c$. Everything is linearised about the exact geometry, which the earlier essays checked against the full geometry at finite clearances, and here the play and the clearance are equal.

The free lag is the largest twist of the output that some choice of the cage’s tilts and slide and the balls’ radial positions allows inside every allowance. Past it, something has to give. The simplest honest model of what gives is a spring on every allowance: a contact carries no force while the joint is inside it and a force proportional to how far past it the joint has been pushed, every contact with the same stiffness. Twist the joint a given amount past its free lag, let the cage tilt and slide and the balls float to wherever the springs’ stored energy is least, and read off the force in every contact. A ball’s load is the sum of its three contacts’ forces, its window and its two tracks, and its share is its load over all six. When the shares are uneven, a single number summarises them: the reciprocal of the sum of their squares, which is six for an even split and one when a single ball carries everything, and which this essay calls balls’ worth.

The equilibrium is the minimum of a convex energy, found by Newton steps on the set of contacts that are bearing, with the step cut back whenever it fails to lower the energy. It is the elastic version of the linear program the earlier essays solved: that one found the edge of the allowances, this one finds where a joint pushed past the edge comes to rest.

The caged joint shares at once #

With the grooves offset 0.1, well below the offset at which the grooves begin to bind anywhere in a pitch, the cage sets the play at every input angle. Twisted a hundredth of a window clearance past its free lag, the joint’s load is spread over 4.1 to 5.9 balls’ worth across a ball pitch, depending on the input angle. Three hundredths on, at the angle in the hero figure, every ball carries between 14% and 20%. A third of a clearance past the play, at every input angle with any free lag, every ball carries between 15% and 19%, which is 5.97 to 6.00 balls’ worth.

The reason is the earlier essays’ count. At the cage’s limit every ball is against its window, and the window constraints are the ones the cage’s tilts and slide cannot relieve, so the first twist past the limit loads all of them together. The loads are not identical, because the balls are at different distances from the planes they are held to, but by a tenth of a clearance past the play the largest is 18% against an even 16.7%. At 15°, through all of this every bearing contact is a cage window; the tracks carry nothing until the twist is about a clearance past the play.

The grooved joint leans on one #

With the grooves offset 0.7, above the offset at which they are the tighter limit everywhere in a pitch, the grooves set the play at every input angle, and the first load is different in kind. A hundredth of a clearance past the play, one ball carries all of it at every input angle in the pitch except three: the symmetric positions at the ends of the pitch, where two balls tie for worst and share it equally, and the position at 30° where the free lag is nought and the joint is already at its limit with several contacts bearing. Everywhere else the load is one ball’s.

This is the earlier essay’s worst ball, seen from the other side. The grooves hold each ball to its own allowance, the joint’s play is set by whichever ball reaches its allowance first, and so the first contact past the play is that ball’s track and nothing else. At every input angle where one ball takes the first load, it is exactly the ball whose groove allowance the earlier essay found smallest. The cage cannot help: its windows still have room, because the grooves stopped the twist before the balls reached them.

The order in which the others join follows their allowances too, roughly. At 15° the six balls’ groove allowances, as lags per unit of play, are 0.58, 0.14, 0.40, 0.61, 0.18 and 0.48. A third of a clearance past the play their shares are 7%, 43%, 10%, 6%, 29% and 5%: the two tightest grooves, the second and the fifth ball, carry nearly three quarters of the load between them, and the four looser ones share the rest. A grooved joint is in effect a joint of two balls at that position, and which two changes as it turns. Each ball in turn becomes the tightest as it passes the positions where its groove’s sensitivity to the lag is largest, so over a revolution every ball takes its turn at carrying most of the torque.

At the mixed offset of 0.45, between the two thresholds, the joint is caged at some input angles and grooved at others, and the first load follows. It falls on a single ball at 10°, 15°, 20°, 25°, 35°, 40°, 45° and 50°, and on three to four balls’ worth at the angles near the ends of the pitch.

Set beside the two limits that offset gives, the rings fall on exactly the input angles where the grooves’ limit is below the cage’s, and on none where it is above. The binding limit decides how the first load is carried, position by position, and nothing else enters. That is the test the earlier essay’s question proposed, and it holds at all thirteen input angles.

More twist spreads the grooved load, but not to six #

A load that starts on one ball does not have to stay there. As the joint is twisted further the first ball’s track spring compresses, the joint rotates further, and other contacts come within range.

At an input angle of 15° the grooved joint keeps one ball loaded alone up to three hundredths of a clearance past the play, reaches two balls’ worth at a tenth, three at two tenths, 4.8 at one clearance and 5.1 at three. The mixed joint follows a little ahead of it. The caged joint is at 5.9 balls’ worth by three hundredths and six by a tenth, and then eases slightly, to 5.8 at three clearances, as its tracks start to carry load beside the windows and the tracks, unlike the windows, are not shared equally. Over the whole pitch the table at the end of the essay gives the ranges: two clearances past the play, the grooved joint spreads its load over 4.2 to 5.7 balls’ worth with the largest single share between 22% and 36%, and the caged joint over 5.7 to 6.0 with the largest between 18% and 22%.

So the grooved joint’s unevenness is not a transient that disappears once the joint is loaded at all. It is largest at the first load, it shrinks as the load grows, and within any twist that is small against the play itself it stays well short of even.

At the same torque #

The shares compare the two joints at the same twist, but a designer specifies a torque, and the two joints resist twist differently: the grooved joint’s single bearing track turns out to be stiff against the twist, so at the same twist it already resists more torque. The fair comparison is the hardest-pressed ball’s load at the same resisting torque.

Measured that way the difference is plain. At torques from 0.3 to 100 in the units of the contact springs, the grooved joint’s hardest-pressed ball carries 1.76, 2.44, 3.01, 2.48, 2.09 and 1.53 times the caged joint’s. At a torque of 3 units, for instance, the grooved joint’s worst ball carries 0.431 against the caged joint’s 0.143, with the grooved joint still at one ball’s worth and the caged one at 5.9. The ratio is largest at moderate torques like that one, where the caged joint has already spread its load across six balls and the grooved joint is still carrying it on one or two, and it falls at high torques as the grooved joint spreads too. The mixed joint is a little worse than the grooved one at low torque, because at 15° it is grooved and its single bearing track is less stiff.

The absolute numbers depend on the contact stiffness, which is the same for every contact here and in a real joint is not, so only the ratios are meant, and only for contacts of equal stiffness: a joint whose grooves set its play loads its worst ball two to three times harder than one whose cage does, at the torques where the difference matters most. How the ratio moves when windows and tracks have different stiffnesses was not measured.

Three joints over a whole pitch #

The table is the essay in nine rows. The caged joint’s balls’ worth is already above 4.1 at a hundredth of a clearance and above 5.7 from three tenths on; its largest share never exceeds 30%. The grooved joint starts at one ball and reaches 4.2 to 5.7 only at two clearances, its largest share between 22% and 36% there. The mixed joint sits between, with the grooved joint’s first load at the angles where its grooves bind.

What a designer can take from it #

Know which limit sets the play. The free lag is one number either way, and it hides the difference that matters under load. The earlier essay’s crossover offsets, 0.35 for the grooves to bind somewhere in a pitch and 0.55 for them to bind everywhere at a 20° bend with track play equal to window clearance, are also the offsets at which a joint’s loading changes in kind.

Tight tracks buy play with uneven loading. Offsetting the grooves further, or tightening their play, lowers the free lag once the grooves become the tighter limit. It also means the first load past that lag lands on one ball, and at the same torque the hardest-pressed ball carries up to three times what a caged joint’s would. For a joint that runs mostly near zero torque, where backlash is what is noticed, that may be a good trade. For one that carries steady torque, the cage should be the limit.

A clearance is a link, and a spring at its end is what loads it. The earlier essays treated each allowance as a free link of fixed length, and the count cannot tell a pin from a slide found what such counts miss. Putting a spring at the end of each one turns the limit analysis into a load analysis without changing the joint, and the set of links bearing at the limit is the set that carries the first load.

What the springs do not settle #

Real contact. Every contact here is a linear spring of the same stiffness. A ball pressed into a groove is a Hertzian contact, stiffening as it is loaded and sensitive to how closely the two surfaces conform, which a disc cut at its threshold grows a point found decides contact stress more than any other number, and a ball against a cage window is a different contact again. The equal-stiffness model says which contacts engage first and roughly how the load spreads; the exact shares depend on the contact laws.

Torque paths. The load here is the sum of each ball’s contact forces, which is how hard it is pressed. A real joint transmits its torque through the balls’ contacts with the groove flanks, which this model represents only through the radial play along each track, and the window contacts carry forces that the cage passes between balls. A model with the grooves’ cross-section and contact angle, the surface on which a joint slides on itself, would be needed to say how much of each ball’s load is torque and how much is the cage steering it.

One bend. Everything is at 20°, where the earlier essays measured the crossover offsets. At a larger bend the balls move further off the bisecting plane over a turn, the limits change, the crossover offsets move, and the sharing would have to be measured again; nothing here says whether a joint caged at 20° stays caged at 40°.

Still open: a cage that steers the worst ball #

The grooved joint’s trouble is that its worst ball reaches its track before any other contact engages. The cage could prevent that: if the windows were made a little tighter than the tracks would require, so that the cage’s limit sat just below the grooves’, the cage would set the play everywhere in the pitch, the first load would spread over every window, and the grooves would only begin to carry load beyond it.

The distinct argument would be the window clearance that just restores even sharing in a grooved joint. For the offset of 0.7, the measurement would shrink the window clearance against a fixed track play until the cage’s limit falls below the grooves’ at every input angle, and then measure the first load’s spread and the hardest-pressed ball at equal torque. If the cage takes over cleanly, a designer can have tight tracks and even loading together at the price of a tighter cage; if the tracks still share in uneven proportions once the cage binds, the two limits interact and the cage cannot simply be tightened past the problem.