Contact that runs along the tooth

The contact ratio counts how many tooth pairs are carrying on average, and the number for an ordinary pair is about 1.6. That average conceals something the average was never able to show: for part of every base pitch exactly one pair is in contact and for the rest exactly two, and the changeover is instantaneous.

It is instantaneous because a straight tooth runs straight across the face. Every point along its width arrives at the start of contact at the same moment and leaves at the same moment, so the total length of tooth in contact jumps by a whole face width, twice per base pitch, for ever.

The field of action #

The right picture for this is not the mesh. It is a rectangle.

Lay the line of action along one axis and the face width along the other. Every contact between the two gears happens somewhere in that rectangle: at a given instant, a contact occupies a line across it, and as the gears turn the line sweeps from one side to the other. Contacts are spaced one base pitch apart along the line-of-action axis, because that is what a base pitch is.

For a spur pair the lines are vertical: at each position along the face, contact begins and ends at the same place on the line of action. The rectangle’s width divided by the base pitch is the contact ratio, so a rectangle 1.647 base pitches wide holds either one line or two, and the total length of contact is either one face width or two.

For a helical pair the lines lean. A helical tooth crosses the face at an angle, so its contact at the far end of the face is at a different point on the line of action from its contact at the near end. The lean is the base helix angle $\beta_b$, and the amount a line shifts across the width of the rectangle is $b\tan\beta_b$.

The second ratio #

Divide that shift by the base pitch and a second dimensionless number appears:

$$\varepsilon_\beta = \frac{b\tan\beta_b}{p_b} = \frac{b\sin\beta}{\pi m_n}.$$

The second form is the one worth staring at. It contains a face width, a helix angle and a normal module, and nothing else. No tooth count. No pressure angle. No centre distance. A 24–36 pair and a 71–13 pair with the same face and the same helix have the same overlap ratio to the last bit of a double, which is checked rather than presumed.

The transverse contact ratio is still there and is still the profile’s quantity. Slanting the teeth changes it slightly, because the transverse pressure angle is $\tan\alpha_t = \tan\alpha_n/\cos\beta$ and a steeper pressure angle means a shorter contact path: for the 24–36 pair at module 4 it drifts from 1.647 at zero helix to 1.599 at 20° and 1.539 at 30°. That is a small loss and it is the only thing the helix takes away.

What the sum means, and what it does not #

The two add: $\varepsilon_\gamma = \varepsilon_\alpha + \varepsilon_\beta$. For the pair above at 20° that is 1.599 + 1.089 = 2.688, against 1.647 for the spur pair — a two-thirds increase, bought with a slant.

It is worth being careful about what the total counts. It is the average number of tooth pairs in contact, in the sense of how many teeth are engaged somewhere along their width at a given instant. It is not the mean length of contact divided by the face width, and the two get conflated because a single symbol carries both readings.

The mean total length of the contact lines is the rectangle’s area divided by the perpendicular spacing of the lines, which works out as

$$\bar{L} = \frac{\varepsilon_\alpha\, b}{\cos\beta_b},$$

with the transverse ratio in it and not the total. The overlap ratio buys contact duration, not contact length. Measured on the 20° pair, the mean is 67.55 mm against $\varepsilon_\alpha b/\cos\beta_b = 67.55$; the spur pair’s is 66.0 against 65.89, where the small difference is the square wave’s duty cycle quantised by the sampling rather than anything about gears.

Two helix angles, and only one of them is on the drawing #

The formula above has $\tan\beta_b$ in it and the second form has $\sin\beta$, and the two angles are not the same.

$\beta$ is the helix angle at the pitch cylinder — the angle the tooth makes with the axis on the surface the gears roll on, and the number a drawing carries. $\beta_b$ is the helix angle at the base cylinder, where the involute starts, and it is what decides how a contact line leans in the field of action. They are related by $\tan\beta_b = \tan\beta\cos\alpha_t$, so the base angle is always the smaller: 18.75° against 20°, 28.02° against 30°.

The reason the two forms of $\varepsilon_\beta$ agree is that the base pitch carries the difference. $p_b = \pi m_t \cos\alpha_t$ and $m_t = m_n/\cos\beta$, and putting those into $b\tan\beta_b/p_b$ collapses every cosine except the one that turns a tangent into a sine. That collapse is why the overlap ratio has such a simple second form, and it is the kind of cancellation worth checking rather than trusting: the two expressions are computed independently and required to agree.

The quantity that actually improves #

What the helix buys is not more contact on average. It is contact that does not change.

The spur pair’s total swings by 40 mm, which is exactly one face width and is as large a swing as the quantity can have. The helical pair at 20° swings by 3.44 mm, which is 5% of its mean. The swing is not monotone in the helix angle — 26.8 mm at 10°, 8.8 at 15°, 3.4 at 20°, 11.2 at 25° — because what matters is not how steep the lean is but how the fractional parts of the two ratios fall.

And there is a setting at which it vanishes. Make the overlap ratio an exact integer — a face of 36.742 mm at 20° and module 4 makes it exactly 1 — and the total length of contact measures 2.1 × 10⁻¹⁴ mm of variation across a whole base pitch. That is the arithmetic’s floor rather than a small number: every line that leaves the rectangle at one edge is replaced by one entering at the other, exactly, and the sum is a constant.

A spur pair cannot be given that property at any face width whatever. Set $\beta = 0$ and $\varepsilon_\beta$ is zero for a face of 40 mm or 400; the figure checks the 400 case, and it is not an edge case but the whole point. Continuity of contact along the face is a thing a straight tooth does not have access to, and no amount of the other resource buys it.

The second thing the helix buys #

There is a completely separate gain, and its independence is worth stating because the two are usually run together.

A cutter working in the normal plane — perpendicular to the tooth rather than to the axis — sees a section of the gear that is an ellipse rather than a circle, and what decides undercutting is the radius of curvature at the end of that ellipse. Working it out gives a virtual tooth count

$$z_v = \frac{z}{\cos^3\beta},$$

with a cube in it rather than the first power a bevel’s back cone produces. The cube is why the effect is large: at 30° the cosine is 0.866 and its cube is 0.650, so a gear’s equivalent count is more than half again its real one.

So a fourteen-tooth helical pinion is clean from 20.7° of helix, a twelve-tooth one from 27.3°, and a sixteen-tooth one from 12.1°. Undercutting has not changed its threshold; the gear has changed which count it presents. That is the same kind of statement the bevel’s equivalent count makes and it comes from a different cosine: the cutter counts a tooth count the drawing does not show, twice in this field, for two unrelated reasons.

What it costs, and the cost is not geometric #

A helix leans the tooth, so the force between the flanks leans with it, and the component along the shaft is $\tan\beta$ times the tangential one — 0.176 at 10°, 0.364 at 20°, 0.577 at 30°.

That is a force, which is outside this field’s boundary, and it is named here rather than computed because it is the reason helix angles stop where they do. Every quantity above improves monotonically with $\beta$: more overlap, more equivalent teeth, and a smaller pinion. Nothing in the geometry says stop. What says stop is a bearing that has to carry an axial load which grows as a tangent, and tangents grow quickly.

The usual answer is a double helical gear — two helical gears of opposite hand on one blank, whose axial components cancel — and its geometry is exactly two of the pairs above sharing a shaft. Its overlap ratio is computed on one half’s face width rather than on the whole, which is the one thing about it that catches people out.

Choosing a face width, which is the only decision here #

Everything above is a description. The one design question a helical pair actually poses is how wide to make it, and the field of action answers it directly.

A face width $b$ buys $\varepsilon_\beta = b\sin\beta/(\pi m_n)$, so the face needed for a given overlap ratio is $\pi m_n \varepsilon_\beta/\sin\beta$ — 36.74 mm for one full unit at module 4 and 20°, and 73.48 mm for two. A designer who wants at least one unit of overlap and has 40 mm of room has a floor on the helix angle rather than a choice about it.

Two rules of thumb fall out of the same expression and both are usually stated without their reason.

“Aim for an overlap ratio of at least one.” Below one, some part of the base pitch has a tooth entering while none is leaving, so the swing in contact length does not close up and the face has not bought continuity. Above one, every entering line is matched by a leaving one somewhere and the quantity is a ripple rather than a step.

“A helical gear should be wider than it is coarse.” That is the same statement with the module made explicit: the face needed for one unit of overlap is $\pi m_n/\sin\beta$, which at any ordinary helix angle is several times the module. A coarse gear on a narrow face gets nothing from its helix at all, and the drives where that happens — small module, narrow face, steep helix for the sake of it — are the ones where the axial load has been paid for and nothing bought.

Four checks, and the negative halves #

Four checks, and the negative halves are where they earn their keep.

The overlap ratio must not move when only the tooth counts change. A 24–36 pair and a 71–13 pair are computed at the same face and helix and required to agree to within $10^{-14}$. Any leak of a tooth count into $\varepsilon_\beta$ — through a transverse module taken from one gear rather than from the pair, say — fails it.

A spur pair must report exactly zero overlap at a face of 400 mm. Not a small number: zero. That is the claim that continuity along the face is unreachable for a straight tooth, and a formula that returned $10^{-16}$ there would mean the helix had leaked in somewhere.

The mean length of contact must equal $\varepsilon_\alpha b/\cos\beta_b$, computed from the sampled sweep on one side and from the closed form on the other. They agree to two parts in a thousand, and the residual is the sampling of a discontinuous function rather than a disagreement about geometry — which is worth saying, because the honest version of “these agree” names what the residual is.

The transverse ratio must fall and the overlap ratio must rise, monotonically, across the whole sweep. Two series running opposite ways is a statement no single arithmetic slip satisfies in both directions.

Which of the field’s quantities survive the slant #

The gears field has spent a dozen essays establishing what a tooth pair does. It is worth saying which of those results apply unchanged to a helical pair, because the answer is nearly all of them and the exceptions are specific.

Unchanged: the ratio. $z_2/z_1$, by counting, which is the exactness that survives everything.

Unchanged in the transverse plane: everything about the profile. The involute, the line of action, the contact ratio, the curvature sum — all of it holds in the transverse section, with $\alpha_t$ in place of $\alpha_n$ and $m_t = m_n/\cos\beta$ in place of $m$. A helical pair is a spur pair in the transverse plane, and the transverse plane is where every planar result lives.

Changed: the pressure angle has two values, a normal one that the cutter is ground to and a transverse one that the mesh runs at, and $\alpha_t > \alpha_n$ always. Every threshold in the field that contains a pressure angle therefore has two versions, and which one applies depends on whether the question is about the tool or about the mesh. That is a second reading of the angle the standard left free: a helical pair has quietly moved along the trade curve drawn there — higher transverse pressure angle, lower transverse contact ratio, lower undercutting floor — without anybody choosing a different cutter.

Changed: which rack cuts it. One rack cuts every wheel remains true, with the rack applied in the normal plane and the blank rolled at an angle to it. So the family a helical gear belongs to is set by its normal module, its normal pressure angle and its helix angle, and two gears of the same module and angle but opposite hand mesh while two of the same hand do not. Interchangeability has gained a parameter and a sign.

New: the overlap ratio, which has no transverse counterpart because a spur pair’s is identically zero.

What scales and what does not #

The module is a size and the ratio is a shape, and it is worth putting the new quantity through that test because it has a length in it.

$\varepsilon_\beta = b\sin\beta/(\pi m_n)$ is a ratio of two lengths — the face against the normal module — multiplied by a pure angle. So it is scale-invariant in the field’s sense: double every dimension of a helical pair and the overlap ratio is unchanged, because the face and the module double together. The same is true of the transverse ratio, which the field established long ago, and of their sum.

What is not scale-invariant is the mean length of contact, $\varepsilon_\alpha b/\cos\beta_b$, which has a bare face width in it and doubles when the gear does. That is the right behaviour and it is the same division the field draws everywhere: the ratios are shapes and the lengths are sizes, and a helical pair adds one of each rather than breaking the rule.

There is one number in this essay that belongs to neither category, and it is the face width for an integer overlap ratio. 36.742 mm is a size, but what it is a size of is $\pi m_n/\sin\beta$ — so it scales with the gear, and the setting that makes the contact length exactly constant survives being made bigger. That is rarer than it sounds: a coincidence between two computed numbers usually does not survive a change of scale, and this one does because both numbers are the same length in disguise.

The same shape as a chain, and the opposite outcome #

There is a family resemblance worth drawing out, because the underlying pattern has turned up twice now.

A chain’s sprocket carries its pins on a polygon, so the effective radius rises and falls within every tooth and the drive’s velocity ratio fluctuates by $1 - \cos(\pi/n)$. The fluctuation is periodic at tooth frequency, it is a property of the discreteness of the chain, and it cannot be removed by any adjustment of the drive.

A spur pair’s contact length has the same shape: periodic at tooth frequency, a consequence of teeth being discrete, and not removable by adjustment. The difference is that a chain’s fluctuation is in the ratio and a gear’s is in the amount of contact, so the first is a kinematic defect and the second is not — the ratio of a spur pair is exactly constant while the contact behind it is switching between one tooth and two.

And the helical pair is the case where the pattern is beaten. Spreading the engagement along an axis the discreteness does not run in converts a square wave into a ripple, and at one setting into nothing at all. A chain has no such axis: its pins are points on a line and there is nowhere for the engagement to be spread. That is the sharpest way to say what a helix is for.

Still open: where the two cosines meet #

Two of the essays before this one have now produced an equivalent tooth count, $z/\cos\delta$ from a cone and $z/\cos^3\beta$ from a helix, and the exponents are different. Neither essay has derived the cube.

The reason is that the cube comes from a curvature and the first power comes from a length: the back cone’s development changes the radius the teeth are spaced on, while the normal section changes the radius of curvature the cutter meets — and a radius of curvature on an ellipse carries a factor the radius does not. Writing that out with the ellipse in front of it would settle it in a paragraph, and it has not been done.

The other thing left is the crossed-axes case, where two helical gears of unequal hand mesh on shafts that neither intersect nor are parallel. Their contact is a point rather than a line, which puts every quantity in this essay out of reach, and it is the arrangement a worm drive is a limit of. That is a different object and it belongs to an essay of its own.