Two flanks, one law

Two meshing gears are two planes in relative motion, and everything the curvature field says about a moving plane applies to them.

The relative motion of one gear’s plane with respect to the other’s is rolling: the two pitch circles roll on each other without slipping, so they are the centrodes of the relative motion and the pitch point is its pole. The tooth flanks are conjugate profiles in exactly the sense the curvature field uses the word — each is the envelope of the other under the motion — and their curvatures at the contact point are therefore not independent.

The string is the radius #

An involute is generated by unwinding a taut string from a base circle. The string is the flank’s normal at the point it reaches — that is what makes the involute the curve it is — and the point where the string leaves the base circle is therefore the centre of curvature of the flank there.

So the radius of curvature at a flank point is the length of string unwound to reach it, which at radius $r$ on a gear with base radius $r_b$ is

$$\rho = \sqrt{r^2 - r_b^2}.$$

That is a closed form, and this site’s rule is that a closed form gets checked against the drawing. The flank polyline the site has been emitting since its foundation phase is sampled, three consecutive points give a circumcircle, and the resulting radius is compared against the string length: worst relative disagreement $1.9\times10^{-6}$ over 389 sampled points.

The interesting value is at the base circle, where $\rho = 0$. A flank’s curvature radius goes to zero as the contact approaches the base circle, which means the flank is turning infinitely sharply there. Below the base circle there is no involute at all, and what a real gear has there is whatever the cutter left — which is the whole of undercutting seen as a curvature statement.

The two flanks’ radii sum to a constant #

At a mesh, contact travels along the line of action — the common tangent to the two base circles, which is where every contact between two involutes happens. Take a contact at distance $u$ along that line from the pinion’s base tangency point.

The pinion’s flank has radius of curvature $u$ there, because $u$ is exactly the length of string unwound. The wheel’s flank has radius $L - u$, where $L$ is the whole distance between the two base tangency points. So

$$\rho_1 + \rho_2 = L,$$

a constant that does not depend on where the contact is.

Measured on a 24-tooth pinion meshing with a 36-tooth wheel at module 4 and a 20° pressure angle: $L = 41.042$ mm, and the sum is that at every sampled contact to the last bit of a double. Meanwhile each flank’s own radius varies from 6.42 mm to 25.88 mm across the mesh — a factor of four — and the wheel’s runs the other way by the same amount.

Why the constant sum matters #

Two reasons, and only one of them is kinematics.

The kinematic one is that it is a strong statement about conjugate profiles. Two curves that stay in contact through a motion have their curvatures related at every instant, and for involutes the relation is a sum being constant rather than something messier. That is unusually clean and it is a consequence of the involute’s defining property rather than an extra fact about it.

The other reason is that contact stress between two curved bodies depends on their relative curvature — on $1/\rho_1 + 1/\rho_2$ — and having $\rho_1 + \rho_2$ fixed constrains how much that relative curvature can vary. The relative curvature is largest when the contact is nearest either base circle, where one radius is small, and smallest at the middle of the mesh. That is why gear teeth are usually most heavily worn near the root and near the tip rather than at the pitch line.

This site does not compute contact stresses; that needs a modulus and a load, and every argument here survives with every force unknown. What it can say is that the geometric input to any such calculation is the pair of curvatures, that the pair is tied together, and where along the mesh each of them is extreme.

Euler and Savary at a mesh #

The general statement behind all of this is the conjugate relation. In the relative motion of one gear’s plane with respect to the other:

• the pole is the pitch point, where the pitch circles touch;
• the pole tangent is the common tangent to the pitch circles;
• the flanks are conjugate profiles, and the relation between their curvatures at contact is Euler and Savary’s, written for two curves in contact rather than for a point and its path.

The gearing form of the relation is usually written

$$\left(\frac{1}{\rho_1} + \frac{1}{\rho_2}\right)\sin\alpha = \frac{1}{r_1} + \frac{1}{r_2},$$

with $\alpha$ the pressure angle and $r_1$, $r_2$ the pitch radii. It is the same relation the curvature field derives from a moving plane’s second derivative, specialised to two profiles that stay in contact.

For involutes it collapses to the constant sum, because the involute is the one profile family whose normal always passes through the pitch point at a fixed angle. That fixed angle is the pressure angle, and its constancy is the law of gearing — so the constant sum and the constant velocity ratio have the same cause.

The check that had to be rewritten, twice #

The site has been here before with this pair of flanks, and the history is worth carrying because it is the clearest example of a circular check on the whole site.

The first version of the conjugate-action test parameterised both flanks by the same roll angle, differentiated, and announced that the velocity ratio was constant to the last bit. Of course it was: both angles were linear in the same parameter by construction, so the calculation restated the definition of an involute rather than testing it. It even computed a variable and discarded it, which is what a circular measurement looks like from the inside.

The rewrite took the flank points the generator actually emits, estimated the tangent by differencing neighbouring generated points, took the normal, and asked whether that normal passes through the pitch point — the law of gearing stated directly, as a property of the drawn curve. It then reported a 94 per cent error on a correct flank, because a dot product had been used where a cross product was needed.

Both failures were about the same thing: the check has to be about the drawn object, and getting the drawn object’s tangent right is where the arithmetic is.

The curvature measurement in this essay is deliberately in that lineage. It reads the emitted polyline, fits circles through triples of its points, and compares against a length that comes from the base circle. Nothing in it goes through the parameterisation the flank was generated with.

Where the circumcircle fit is trustworthy and where it is not #

A three-point circle fit is not an unconditionally reliable measurement, and this site has a recorded finding that it gets worse with finer sampling: the three points become nearly collinear, the fit becomes ill-conditioned, and conditioning beats truncation.

Here the sampling is fixed at 500 points along a flank whose curvature radius runs from a few millimetres to twenty-odd, so the three points span a comfortable angle and the fit is well conditioned. That is why the agreement is $10^{-6}$ rather than $10^{-14}$: the fit’s truncation error dominates, and it is the honest limit of the measurement rather than a limit of the law.

If a tighter number were wanted, the route would be to compute the flank’s curvature analytically from its parameterisation and compare that against the string length — and that is exactly the circular check the gears field had to abandon. A less precise measurement of the drawn thing is worth more than a precise measurement of the expression that drew it, which is the position the whole site takes.

Where each flank’s curvature is extreme, and what that means for the mesh #

Since the pinion’s flank radius is the distance along the line of action from its own base tangency, it is smallest where the contact is nearest the pinion’s base circle — which is at the start of the mesh, near the pinion’s root — and largest at the end, near its tip.

The wheel’s runs the other way. So at the start of the mesh the pinion is sharply curved and the wheel is nearly flat, and at the end the reverse. The two are equal at exactly one contact, which is the point where $u = L/2$, and that is not the pitch point unless the two gears are the same size.

Two consequences worth having.

The relative curvature $1/\rho_1 + 1/\rho_2$ is largest at the ends of the mesh. With $\rho_1 + \rho_2$ fixed, the sum of reciprocals is minimised when the two are equal and grows without bound as either approaches zero. So the geometry is worst — most sharply curved contact — at the beginning and end of each tooth’s engagement.

A pinion with few teeth has a small base circle and therefore a short line of action, so $L$ is small and both radii are small throughout. That is one of several reasons small pinions have a harder life than large wheels, and it is a purely geometric one that needs no load to state.

The contact ratio is a different quantity, and it is easy to confuse #

The line of action’s usable length divides by the base pitch to give the contact ratio, which is how many teeth are in mesh on average. It uses the same segment as the curvature statement and answers a completely different question.

The distinction: the contact ratio is about how much of the line of action lies between the two tip circles — a length divided by a pitch, and a number near 1.6 for ordinary gears. The curvature sum is about the whole distance between base tangencies, which is longer, and does not care where the tip circles are.

Both are lengths measured along one line and both are properties of the pair rather than of either gear, which is why they get conflated. A useful separator: adding profile shift changes the contact ratio and does not change the sum of the curvature radii at a given contact, because the sum is fixed by the base circles and profile shift does not move those.

That is the same observation the profile-shift essay makes from the other side — a shifted gear has the same base circle and therefore the same involute, cut at a different part of it — and it is the single most useful fact about shift.

What a non-involute pair does #

For completeness, because the constant sum is a property of involutes rather than of gearing.

A cycloidal tooth pair — the profile used in clocks and in the cycloidal drive — satisfies the law of gearing too, so it also transmits a constant velocity ratio, and its flanks are also conjugate. But its pressure angle is not constant, the line of action is not a line, and the two flanks’ curvatures obey the general conjugate relation without collapsing to a constant sum.

The practical difference the site can state without leaving geometry: a cycloidal pair has convex flanks meeting concave ones over part of the mesh, so the relative curvature is lower than an involute pair’s there. That is an advantage, and it is paid for by extreme sensitivity to centre distance — move an involute pair’s centres apart and the ratio is unchanged, and move a cycloidal pair’s and it is not.

Which of those matters more depends on whether the housing can be made accurately, which is a manufacturing question rather than a kinematic one. The involute won, and it won on tolerance rather than on curvature.

The rack cutter, and undercutting as an envelope #

An involute flank is generated in practice by a rack cutter with straight-sided teeth, rolling against the blank. So the flank is the envelope of a family of straight lines, which puts it in the same category as a cam profile as the envelope of a roller.

Undercutting is what happens when the envelope construction reaches into a region the generated curve needs. On a cam that is the offset turning itself inside out where the pitch curve’s radius falls below the roller’s; on a gear it is the cutter’s tip sweeping past the base circle and removing flank that should have been there.

The gear field’s threshold is $N \ge 2/\sin^2\alpha$ — seventeen teeth at 20°, and 11.2 at 25°, which is why raising the pressure angle is one of the two standard cures and moving the cutter out is the other. Both are the same statement about a curvature: keep the generating curve out of the region where the generated one has run out of radius.

Two conditions that get called the same thing #

Writing the check for the constant sum turned up a conflation worth separating, because both halves of it are on this site already.

A rack cutter’s tip reaching the interference point is a property of one gear. It happens below $2/\sin^2\alpha$ teeth — 17.10 at a 20° pressure angle — and it is what the seventeen-tooth rule is about and what profile shift cures.

A line of action that starts inside a base circle is a property of a pair. Where the contact path begins is decided by the mate’s tip circle, so it depends on which two gears are meshing.

They are not the same condition and the numbers say so. Against a 36-tooth wheel, a 12-tooth pinion’s contact path starts at −1.78 mm — inside its own base circle, where there is no involute — and a 17-tooth pinion’s starts at +1.64 mm, comfortably outside, even though 17 is below the rack threshold and that pinion is undercut when it is cut.

The first version of the check asserted that the two conditions coincide and failed at exactly 17 teeth, which is where a wrong equation would fail. A negative radius of curvature at the start of a mesh is not an error in the arithmetic; it is the mesh asking for contact on a part of the flank that does not exist.

What is claimed and what is not #

What is claimed: the involute’s radius of curvature is the string length, measured on the drawn flank; the two flanks’ radii sum to the base-tangency distance, exactly; and both facts are the conjugate relation of the curvature field specialised to a rolling pair.

What is not claimed: anything about stress, load, lubrication or wear. Those need a modulus and a force, and the site’s standing boundary applies — the structural field owns what a section does with a load, and every argument here survives with every force unknown.

Also not claimed: that constant relative curvature would be better. It would not obviously be, and the involute’s constant sum is a consequence of its constant pressure angle rather than a design target somebody aimed at. Cycloidal teeth, which this site draws in the cycloidal-drive essay, have a different curvature relation and were chosen for different reasons; the involute won on manufacturability and on tolerance to centre-distance error, not on curvature.

The sum of the two radii being constant while each varies fourfold is the sort of invariant this site collects, and it is worth saying what it is good for beyond elegance. It is a check with no tolerance in it. Each flank’s radius is measured by fitting a circle to a sampled curve, which is a computation with a resolution and an error; their sum is a constant of the mesh, known in advance from the geometry, and a fit whose two radii do not add to it has gone wrong somewhere. That converts two noisy measurements into one sharp verdict, and it does so without needing either measurement to be accurate — the errors have to cancel, and they only cancel if both are right. It is the same arrangement a rotary engine’s chamber areas have, and it is worth looking for wherever a mechanism has two quantities that trade: a quantity the mechanism cannot change, measured while everything else moves, is the cheapest check there is and it usually costs one addition.