Where two shapes stop touching

Everything this field has generated so far is exact. The mate of a profile holds the ratio to fifteen digits, the rotor’s corners sit on the wall to the last place the arithmetic has, the pitch curves of a demanded ratio roll without slipping to the resolution of the integration.

All of it is exact at an instant. The meshing equation is a condition at a point of contact, and a mechanism is not a point of contact; it is an interval, over which the contact has to keep existing. This essay is about the ends of that interval, and there are four different ways to arrive at one.

Before any of them, one distinction has to be made, because the four failures are easy to run together. Two of them are about whether a contact exists: the arc runs out, or the shapes come apart. Two are about whether the contact that exists is legitimate: it is on a piece of curve that was never conjugate to anything, or it is on a locus that is not the boundary of a solid. The first pair is a mechanism that stops driving; the second is a mechanism that jams or a part that cannot be made, and the difference matters because the first is usually noticed on a test rig and the second on a shop floor.

A profile is an arc #

The first limit is the least interesting and the most often forgotten. A tooth is a finite piece of curve. It has a tip, decided by a turning operation, and a root, decided by where the cutter’s corner takes over. Between those two radii there is an involute, and outside them there is nothing to be in contact with.

So the contact between one pair of flanks begins at some position of the input and ends at another. On a 24 : 36 pair of module 4 the contact path — the piece of the line of action between the two tip circles — is $19.45$ mm long, and it starts $6.42$ mm from one base tangency and ends $25.88$ mm along.

The pair is only continuous if the next pair of teeth has arrived before this one leaves. Divide the contact path by the base pitch, which is the distance along the line of action between one tooth’s contact and the next’s, and the answer counts how many pairs are engaged at once. Here it is 1.647: sometimes one pair, sometimes two, on average a little under two-thirds of the time with two.

A contact ratio below one is not a bad gear; it is not a gear. The output stops while nothing is in contact and jumps when the next tooth arrives, which is a mechanism with an impact in it rather than a drive.

What moving the shafts really costs #

The essay that measured the involute’s indifference to centre distance is right and needs this qualification.

The ratio is exactly unchanged at any centre distance. The contact ratio is not. Moving the shafts apart lengthens the whole line of action between the base circles and shortens the part of it between the tip circles, so fewer teeth are engaged at once. On this pair the contact ratio falls from 1.647 to 1.179 at two millimetres out — the offset at which the shape mismatch was still at its noise floor — and reaches one at 2.82 mm.

That is the honest boundary of the earlier result. Between zero and about two and a half millimetres the pair keeps a perfect ratio and progressively less margin; past that it keeps a perfect ratio while the drive is discontinuous, which is worse than a small error would have been. The involute converts a positional error into backlash, bearing load and lost contact ratio, and the third of those is what runs out first.

What the contact ratio actually counts #

It is worth being careful about what that number is, because it is quoted constantly and misread nearly as often.

The contact ratio is a length divided by a length: the arc of the line of action over which some pair of teeth is in contact, divided by the base pitch, which is the spacing between successive teeth measured along that same line. It is therefore the average number of tooth pairs in contact, and it is almost never an integer.

At 1.647 the pair spends about 65 per cent of the time with two pairs engaged and 35 per cent with one, alternating several times per tooth. It does not mean “one and a half teeth are touching”; it means the count switches between one and two, in a rhythm set by the tooth pitch, and the switch is where the load transfers from one pair to the next.

Two routes to the same number are available and are checked against each other here: the geometric one — measure the segment of the line of action between the two tip circles and divide — and the kinematic one, which sweeps the mesh and counts how many pairs have a solution of the meshing equation at each position. They agree to nine figures at the design centre distance, which is the check that the segment being measured is the segment the contacts occupy.

Interference: contact where there is no curve #

The second way to lose contact is more interesting, because the pair does not stop touching — it touches somewhere it should not.

The involute exists only outside the base circle. Below it there is fillet, whose shape belongs to the cutter and has nothing to do with the mating wheel. If the mating wheel’s tip reaches past the point where the line of action touches the base circle, it arrives at a part of the flank that is not conjugate to it, and it fouls: two surfaces in contact whose normals do not meet at the pitch point, which means they are trying to occupy the same place.

That is interference, and it is a different failure from undercutting even though both are about the base circle. Undercutting happens in the cut, to one wheel, and leaves it permanently short of profile. Interference happens in service, to a pair, and depends on both wheels and on the centre distance. A perfectly cut wheel can interfere with one mate and not with another.

The remedies are the ones a gear catalogue lists and they are all the same manoeuvre — keep the contact inside the useful part of both flanks. Shorten the tip. Increase the centre distance. Shift the profile. Use a larger pressure angle, which brings the base circles in.

The cycloidal pair, which stops touching altogether #

The third way is the one measured in the centre-distance essay and it is worth restating here as a limit rather than as a comparison.

A cycloidal pair whose shafts are five hundredths of a millimetre apart from their design distance does not develop a small transmission error. Its flanks come apart: the solve for tangency stops converging, and the residual it fails to drive to zero is a gap of about the size of the offset. The two curves that were in continuous contact are now touching at isolated positions and separated in between.

The mechanism does not stop working. The next tooth catches, and the drive continues with a wandering output angle. What it has lost is not accuracy but continuity of contact, which is the same thing a contact ratio below one loses, arriving by a different route.

Where the ends of the arc come from #

Each end of the contact arc is somebody’s decision, and it is worth naming whose.

The far end belongs to the driven wheel’s tip. Contact begins when the tip of the driven wheel’s tooth reaches the line of action, which is a diameter chosen when the blank was turned. A longer addendum starts the contact earlier and lengthens the arc — and pushes the contact towards the driving wheel’s base circle, which is where interference lives.

The near end belongs to the driving wheel’s tip, by the same argument the other way round.

And both are bounded by the base circles, which belong to the tooth counts and the pressure angle rather than to any turning operation.

So the arc of contact is the intersection of two intervals — what the tips allow and what the involutes exist over — and every design rule about addendum proportions is a rule about keeping that intersection long enough. Stub teeth shorten it deliberately to avoid interference on small pinions; long-addendum pairs lengthen it to buy contact ratio and risk the other end.

The fourth way: the envelope doubles back #

The last limit is inside the shape rather than between the shapes, and it is the one that says a conjugate pair may not be manufacturable at all.

An envelope is a locus of contact points, and nothing requires that locus to be the boundary of a solid. If the given profile is too sharply curved relative to the motion, the envelope develops a cusp and then crosses itself, and the region it “bounds” is one where the generating body has already been.

Undercutting is the everyday case: the flank the rack generated is inside the cutter at a later instant, and the finished shape is not the envelope but the envelope with a piece taken out. The general statement is the one that matters:

The meshing equation is local. It says two bodies touch at a point without interpenetrating at that instant. Whether a solid exists whose boundary is the whole locus is a question about the interval, and it has to be answered by sweeping.

That is why every construction in this field that produced a real shape was checked against a sweep — the removal test on the rack, the material test on a cycloidal disc, the invariant on the rotary engine’s chambers — and why an envelope on its own is never enough.

Contact that exists and should not #

There is a fifth case, and it is the mirror image of the four above: a pair whose shapes are in contact at more places than the design intends.

The rotors of a rotary engine touch at three corners, and the flanks between them are clear of the wall. If they were not — if a flank fouled the housing anywhere — the mechanism would be locked rather than sealed. That the flanks stay clear is a property of the envelope: they are the boundary of what fits, so nothing about them can be outside the wall. A shape generated as the inner branch of an envelope has this guarantee built in, and a shape drawn by hand does not.

The cycloidal disc has the same structure and the opposite intention: every pin touches it at once, which is what a disc generated by all the pins must do. Adding clearance so that only some of them touch is a deliberate departure from the geometry, and it is the same kind of departure as backlash in a gear pair.

So a conjugate pair has three regimes rather than two: too few contacts and the drive is discontinuous; the designed number and it works; more than the designed number and it is over-constrained — which is a familiar word on this site, and which for shapes means the same thing it means for linkages with a joint too many.

What a designer actually checks #

Putting the four together gives the checklist a pair of shapes has to pass, and it is worth noticing how little of it is about conjugacy.

Does contact exist over the whole intended arc? That is the contact ratio, and it wants to be comfortably above one — 1.4 is a usual floor for a gear pair, and this field’s 1.647 is ordinary.

Does the contact stay on parts of both profiles that exist? That is interference: the contact must not run past either base circle or off either tip.

Is the generated shape a boundary? That is the sweep, and the answer is no whenever the cutter comes back.

Does the pair survive being made wrong? That is the centre-distance question, and it is the one that decided which tooth form the world uses.

None of those is answered by the meshing equation, and all of them are answered by asking the meshing equation the same question at every position rather than at one.

The arithmetic of a whole train #

One more consequence, because it is where the contact ratio stops being an abstraction.

A gear train’s smoothness is the smoothness of its worst mesh. A three-stage reduction with contact ratios of 1.65, 1.65 and 1.05 has one stage whose engagement is nearly discontinuous, and the output shows it however good the other two are. That is why a catalogue quotes a minimum contact ratio rather than an average, and why the small pinion — which is the one with the shortest flank and therefore the shortest contact arc — is usually the stage that decides.

The same reasoning bounds how far a designer can go in the other direction. Contact ratio can be bought with longer teeth, and longer teeth run closer to interference and to undercutting; it can be bought with a smaller pressure angle, which lowers the threshold tooth count the wrong way, from 17 at 20° to nearly 32 at 14.5°. Every purchase is made from the same small budget of radius between the base circle and the tip.

Three regimes, and the middle one is where gears live #

The fifth case turns the essay’s list into a spectrum rather than a set of failures, and the spectrum is worth stating because it explains why two mechanisms in this field are designed to sit on opposite sides of it.

Fewer than one contact at a time. The drive is periodically not a drive; the output stops while nothing pushes it, and the next tooth arrives as a collision. That is a contact ratio below one, and it is not a poor gear but a broken one.

Between one and two. The ordinary spur gear. Contact never fails, the number of engaged pairs alternates, and the load redistributes twice per mesh cycle. Every pair is doing useful work and none of them is redundant — which is exactly why the arrangement tolerates ordinary manufacture.

All contacts engaged at once. A cycloidal drive’s disc touches every pin; a rotary engine’s rotor touches at all three corners. Here the contacts are overconstrained: more of them than the motion requires, so the load divides between them and the geometry has to be exact for the division to be even.

Read that way, the third regime is the site’s standing trade arriving in a new field. Load sharing is bought with a manufacturing condition, exactly as it is for a planetary’s three planets, an overconstrained linkage and a part on seven contacts. The reward is that no single contact carries everything; the price is that the contacts only share if the parts are right.

Which is why a gear pair deliberately stays in the middle regime and a cycloidal drive deliberately does not. A gear is made in quantity to ordinary tolerances and cannot afford a condition; a cycloidal drive is a precision component with ground pins and an accurate disc, and it spends that accuracy on having twenty contacts instead of two. The regime is a design choice and the currency is manufacturing tolerance, which is the same sentence this site has now written in five fields.

The same shape, in three other fields #

The distinction between a condition that holds at an instant and a mechanism that has to work over an interval is not this field’s alone, and it is worth collecting the site’s other instances because the failure is identical each time.

A linkage can assemble at every position that was checked and be unable to get from one to another, because assembly is a condition at a configuration and motion is a statement about a path.

A synthesised four-bar can pass through all its prescribed positions in the wrong order, or on two different branches, and both defects are invisible to the equations that placed the positions.

A cam profile can be correct everywhere and impossible to cut, because a roller of the chosen size cannot reach into the curvature the profile demands — an envelope crossing itself, arrived at from the cams field’s side.

Four fields, one lesson: the equation is about an instant, the machine is about an interval, and the gap between them is where mechanisms fail.