Rolling at one point only

Gears are described as rolling. The pitch circles roll on each other, the ratio is the ratio of their radii, and the whole picture is one of two round things in intimate contact turning together.

Two of those three statements are exactly right and the picture is wrong, in a way that is worth a measurement rather than a caveat. The pitch circles roll and the teeth do not. The surfaces that are actually touching — the flanks, the parts made of metal — slide against each other at every instant except one, and the sliding speed is not a small correction: on an ordinary pair it is comparable to the speed at which the contact itself travels along the tooth.

Two velocities at one place #

At the contact there are two material points, one belonging to each body, and they are at the same place at that instant and going different ways. Their difference is the sliding velocity, and it is what the meshing equation is a statement about: the equation says the normal component of that difference is zero, which is the condition for touching without interpenetrating, and it says nothing at all about the tangential component.

The tangential component is the sliding, and the geometry of it is settled in one line. The relative velocity field of two rigid bodies is itself a rigid velocity field, and a rigid velocity field that vanishes somewhere is a rotation about that place. It vanishes at the pitch point. So at any contact $X$,

$$v_1 - v_2 = \beta\,\hat z \times (X - P),$$

with $\beta = \omega_1 - \omega_2$ the difference of the two angular rates and $P$ the pitch point — the sliding speed is $|\beta|$ times the distance from the contact to the pitch point, and its direction is along the common tangent, which is the same statement as the meshing equation from the other side.

Measured on a 24 : 36 pair at module 4, over the working arc: the sliding speed at the pitch point is $7\times10^{-15}$ mm per radian, it rises to about 12 mm per radian at the ends of the arc, and the measured speed agrees with $|\beta|$ times the measured distance to $10^{-12}$ everywhere. That is not two spellings of one calculation — the speed is a difference of velocities built from two rotations, and the distance is a length between two points, one of which was computed from the placements — but it is the strongest possible confirmation that the sliding has the structure the geometry says.

The rolling that is real, and the rolling that is not #

The confusion is worth taking apart carefully, because both halves of it are true statements about different objects.

The pitch circles do roll. They are the centrodes of the relative motion — the locus of the pitch point in each body’s own frame — and rolling is what centrodes do. The site met that in the linkage field: a coupler’s motion is the rolling of one curve on another, always, whatever the mechanism. For a constant-ratio pair the pitch point is fixed in the frame and at a fixed distance from each centre, so both centrodes are circles and they roll without slipping. The arc lengths match; there is no sliding between them; a chalk mark on one would meet a chalk mark on the other at the same place every time.

The flanks do not roll. The pitch circles are not made of anything. The pitch point is a point of the plane, and the contact between the actual surfaces is at the pitch point for one instant per tooth and is somewhere else for the rest of the engagement. Wherever it is somewhere else, the two surfaces are sliding, at $|\beta|$ times the distance.

So a gear mesh has a rolling in it and a sliding in it, and they are about different objects: an imagined pair of circles rolls, the real pair of surfaces slides. The everyday sentence “gears roll rather than slide” is a description of the pitch circles being used as a description of the teeth.

The sliding changes direction halfway through #

One feature of the plot deserves its own paragraph, because it is invisible in the speed and obvious in the specific sliding: the sliding reverses direction as the contact crosses the pitch point.

Before the pitch point — the part of the engagement the subject calls approach — the two surfaces are closing on each other along the tangent; after it, in recess, they are separating. The speed passes through zero and comes back up, so a plot of the magnitude shows a V; a plot of the signed quantity shows a straight line through zero, and it is the straight line that is the physical statement.

That reversal is why the two halves of a tooth flank are not equivalent, and why the subject bothers to name them. The direction in which one surface drags across the other is opposite above and below the pitch line, and anything that depends on that direction — the way a scuff mark lies, which way a burr is thrown, where a film of lubricant is dragged into the contact rather than out of it — is different on the two halves of the same tooth. The geometry cannot say what those consequences are. It can say, exactly, where the boundary between them is: at the pitch point, on every tooth, at the one instant when nothing is sliding at all.

What sliding costs is not what sliding is #

Twelve millimetres per radian is a speed, and a speed is not yet a cost. The same amount of sliding is spread over a long stretch of one flank and a short stretch of the other, and what a surface experiences is closer to sliding relative to its own motion than to sliding in absolute terms.

The subject’s quantity for that is specific sliding: the sliding divided by the tangential speed of the surface being asked about. It is dimensionless, it is different for the two flanks of the same contact, and it is one of the few places where the geometry alone says something about which part of a machine will be in trouble.

On the 24 : 36 pair, over a symmetric arc either side of the pitch point, the 24-tooth wheel’s specific sliding peaks at 1.31 and the 36-tooth wheel’s at 1.04. Both exceed one, which is worth pausing on: a specific sliding above unity means the two surfaces are separating faster than the surface itself is moving, which happens when the two tangential speeds have opposite signs — the flanks are travelling in opposite directions along the contact.

The asymmetry is the useful part. The smaller wheel is the worse off, by a quarter on this pair and by more as the ratio grows, and the reason is structural rather than accidental: the contact travels the same arc of the line of action for both wheels, but that arc is a larger fraction of the small wheel’s flank. A pinion running against a large wheel does more work per unit of its own surface, at every level of description, and the geometry says so before any material is mentioned.

What is not claimed is anything downstream. Wear, scuffing, the film that separates the surfaces, the heat and where it goes: every one of those needs a load, a material and a lubricant. This field computes how fast one surface travels over the other and stops, exactly as the transmission field stops at the geometry of a ratio and leaves what it takes to hold it to somebody else.

The trick that does remove it #

There is one way to get rid of the sliding without giving up the teeth, and it is used in two mechanisms this site has already drawn.

Put the contacting surface on a body that is free to turn. A roller follower on a cam is a cylinder on its own bearing: the cam’s surface drives it, and instead of sliding across it the roller spins, at whatever rate makes the contact velocities match. The sliding has not been abolished — it has been moved into the roller’s own bearing, where the sliding radius is the pin’s radius rather than the flank’s length, and where a bearing can be put.

A lantern pinion’s pins do the same thing when they are made to turn in their end plates, which the better ones were. The tooth flank of the mating wheel then rolls on the pin rather than sliding along it, and the wear moves into the pin’s journals.

That is the general shape of the trick and it is worth stating as such: sliding between two surfaces can be traded for rotation of one of them about an axis chosen for the purpose, and the choice is usually made to put the sliding somewhere a bearing can be fitted. What no arrangement can do is make the relative motion of two bodies on fixed centres vanish anywhere except at the pitch point.

Where the sliding is smallest, and what that buys #

Since sliding is proportional to the distance from the pitch point, a designer who wants less of it has one lever: keep the contact near the pitch point. That is the whole of it, and it explains several things that look like separate design rules.

A larger number of teeth on both wheels shortens the arc of the line of action that gets used, because each tooth carries the contact for a smaller angle before the next takes over. Fine-pitch gears slide less, per tooth engagement, than coarse ones of the same size.

Tip relief — taking a little material off the ends of the flanks — removes contact at exactly the positions where the sliding and the load transfer are both worst.

A cycloidal pair puts the contact on a circular arc through the pitch point rather than on a straight line through it, and the classical claim made for it was less sliding near the pitch point. What is true is that the two forms distribute the same total sliding differently along the arc; what is not true is that either avoids it.

And a pair of equal wheels at 1 : 1 has the symmetric case: both specific slidings are the same curve, and there is no small wheel to be the victim.

The one contact that is rolling #

There is one instant per tooth engagement when the two surfaces are momentarily rolling — the instant the contact passes through the pitch point — and it is worth being precise about what “momentarily” means, because it is the same distinction the curvature field spent a whole field on.

The sliding speed is zero at that instant, and its derivative with respect to the input is not. So the surfaces roll to first order and slide to second, in the same way that a point of the inflection circle goes straight to second order and curves to third. There is no interval of rolling; there is a zero of a function that is otherwise linear through it.

This is also the point of the tooth that a used gear shows as a polished band with duller regions above and below it. The band is where the sliding is smallest, and the fact that it appears at a definite height on the flank rather than uniformly is the geometry becoming visible in the metal.

The numbers, on one pair #

It is worth having the sizes in one place, because the ratios between them are what makes the sliding a first-order effect rather than a correction.

On the 24 : 36 pair at module 4 — pitch radii 48 and 72 mm — with the input turning at one radian per unit:

• the contact travels along the line of action at about 45 mm per radian of input;
• the sliding speed runs from 0 at the pitch point to about 12 mm per radian at the ends of the arc used here, so the fastest sliding is between a quarter and a third of the speed at which the contact moves;
• the surface speed of the driving flank at the contact is of order 50 mm per radian, which is where the specific slidings of 1.31 and 1.04 come from;
• and the difference of angular rates, $|\beta| = 1 + 24/36 = 1.67$ per radian, is the constant of proportionality between the sliding speed and the distance from the pitch point — measured at 1.667 to twelve figures.

None of those depends on the speed the gearbox is run at, because every derivative here is per radian of input. Double the input rate and every one of them doubles, and every ratio between them stays where it was, which is why the geometry can say what it says without a shaft speed anywhere in it.

Total sliding goes as the square of the arc #

The sliding speed is proportional to the distance from the pitch point, and that linearity has a consequence worth extracting, because it prices one of the remedies exactly.

Sliding rate $\propto |s|$, with $s$ the distance along the line of action from the pitch point. So the total sliding over an engagement running from $-a$ to $+a$ is $\int |s|\,\mathrm{d}s = a^2$: proportional to the square of the half-arc, not to its length.

That makes shortening the contact arc a disproportionately good move. Cut the arc by a fifth at each end and the sliding falls by $1 - 0.8^2$, which is 36 per cent — nearly twice the fraction of arc removed. Cut it by a third and the sliding falls by 56 per cent.

Which is the arithmetic behind tip relief. Taking a little material off the ends of the flanks removes contact where $|s|$ is largest, and $|s|$ largest is where the sliding rate is largest, so the material removed is the material doing the most rubbing. A modification that looks like a small cosmetic change to the extremities of the tooth is removing a third of the sliding.

It also says why the remedy has a floor. The arc cannot be shortened past the point where the contact ratio falls to one, since below that the drive is periodically not a drive — so the available reduction is bounded by whatever margin the contact ratio has above unity. A pair at 1.6 can give up a substantial fraction of its arc; a pair at 1.15 can give up almost none.

Those two constraints between them explain where real gear practice sits. More teeth raise the contact ratio, which buys arc to give away; tip relief spends it where the sliding is worst; and the pair ends up with a contact ratio comfortably above one and flanks that are not quite involutes at their ends. Every part of that is a consequence of the sliding being linear in $s$, and none of it is available from the statement that gears slide.

Sliding is the price of the shape, not a defect in it #

It would be natural to read all of this as a failing of gears that a better tooth form might fix. It is not, and the reason is a theorem rather than an engineering judgement.

Two bodies rotating about different fixed centres have a relative motion which is a rotation about the pitch point. Any contact between them that is not at that point involves a relative velocity, and a relative velocity between two touching surfaces is sliding. The only way to have no sliding anywhere is to have the contact at the pitch point at all times — which means the two surfaces touch along the pitch circles and nowhere else, which is friction drive rather than gearing, and gives up the positive engagement that made teeth worth cutting.

The same argument in reverse says something about the mechanisms that have no pitch point. Two bodies whose relative motion is a translation — a rack against a rack, or two parallel slides — have no instant centre, and their contact slides at the same rate everywhere; there is no privileged place at which the sliding vanishes, because the thing that was vanishing was a rotation. A rack and a wheel keep a pitch point, since the wheel turns; the pitch point is on the rack’s pitch line and the sliding grows with distance from it exactly as before.

So the sliding is not a flaw in the involute or in the cycloid. It is the price of moving the contact off the pitch point, and the contact has to move off the pitch point for the teeth to have any depth at all. What a tooth form can decide is where along the arc the sliding falls and how it is shared between the two wheels; what it cannot decide is whether there is any.