The face carries the least line

Four knobs for the root freed both shifts and both addenda of a spur pair and found the least sliding a 17-tooth pinion’s root can have against a 51-tooth wheel: 0.834, every optimum sitting on its contact-ratio floor of 1.2, because a spur pair needs at least one tooth pair in contact at every moment and a little over to spare. A helix pays for the root with face width took the floor away. A helical pair is kept in continuous contact by the overlap of its teeth along the face, so its transverse contact ratio εα can fall below 1.2, and the root’s sliding falls with it, to 0.209 at εα = 0.4 on a 30° helix. The price was the face. The mean length of contact line is εα b / cos βb whatever the overlap, so carrying the spur optimum’s mean line at a third of its contact takes a face 2.65 times as wide.

That essay ended by pointing out what the mean leaves out. The length of line in contact swings about its mean through every base pitch, and the peak load on the teeth stands on the low point of the swing, not on the average. A face is sized for its least line. So the question is how small that least is, what overlap ratio makes it largest, and how much of the factor of 2.65 survives when least is compared with least.

About five sixths of it, and the sixth that goes was never the helix’s to pay.

The swing, in closed form #

A helical pair’s contacts live in a rectangle, the field of action: along the line of action it is the usable path of contact, εα base pitches long, and across it the face width $b$. Each tooth pair in contact touches along a straight line crossing the field at the base helix angle, the involute’s unwound strand laid on a slant, and a new line enters every base pitch. With straight teeth the lines are square to the line of action and arrive whole; with a helix they lean, so that a line advances εβ base pitches along the action in crossing the face, which is what the overlap ratio εβ means. Contact that runs along the tooth swept that field numerically and found the mean length of line in contact to be εα b / cos βb, independent of the overlap.

The least and the greatest have closed forms. Writing $n_\alpha$ and $n_\beta$ for the fractional parts of εα and εβ,

$$L_{\min} = \frac{b}{\varepsilon_\beta\cos\beta_b}\Big[\varepsilon_\alpha\varepsilon_\beta - \min\big(n_\alpha n_\beta,\ (1 - n_\alpha)(1 - n_\beta)\big)\Big],$$

$$L_{\max} = \frac{b}{\varepsilon_\beta\cos\beta_b}\Big[\varepsilon_\alpha\varepsilon_\beta + \min\big(n_\alpha(1 - n_\beta),\ (1 - n_\alpha)\,n_\beta\big)\Big].$$

The first term in each bracket is the mean. The correction is a product of the two fractional parts, or of their complements, whichever is smaller, divided by the overlap ratio. It is nought whenever either fractional part is nought. As εβ goes to nought it becomes the spur pair’s $b\lfloor\varepsilon_\alpha\rfloor$, a whole face for every tooth pair certainly in contact.

The form has a reason that is short enough to state. Follow one contact line across the face: at height $y$ up the face it sits at $\varphi + \varepsilon_\beta y$ base pitches along the line of action, where φ is the phase of the pair’s rotation. So the total length of line in contact at phase φ is the length of one line times the average, over $y$ from nought to one, of how many lines are inside the field at that height, and at each height the lines are simply a spur pair’s contacts shifted by $\varepsilon_\beta y$. A helical pair’s length in contact is a spur pair’s tooth count averaged over a window εβ base pitches long.

The spur count is easy. A spur pair with transverse ratio εα has $\lfloor\varepsilon_\alpha\rfloor$ pairs in contact for most of each base pitch and one more for the fraction $n_\alpha$ of it. Average that over a window of $\lfloor\varepsilon_\beta\rfloor$ whole pitches and a leftover piece $n_\beta$ long. The whole pitches contribute their mean exactly. The leftover piece contributes however much of the extra pair’s stretch it happens to cover, which is at least $\max(0, n_\alpha + n_\beta - 1)$ and at most $\min(n_\alpha, n_\beta)$, depending on the phase. Collecting terms gives the two forms above. At εα = 0.4 and εβ = 3.5, for instance, the window covers three whole pitches, each with 0.4 of a pitch of contact, and a half pitch that can miss the 0.4 entirely: the least is 1.2 pitch-lengths of contact out of a mean of 1.4, 0.857 of it, which is the number in the hero figure.

That is also why a whole-number overlap ratio removes the swing. With no leftover piece, the window always averages whole pitches, and every position in the mesh sees the mean. The helix is a moving average of the spur pair’s count, and a moving average over a whole number of periods is flat. The same averaging steadies a twisted pair of Roots rotors, whose seal a twist steadies what it cannot tighten found flat at a wrap of one whole lobe pitch and at its own flat average, never better.

Over forty-five combinations of εα from 0.4 to 2.3 and εβ from 0.3 to 7.5, the closed forms and the swept field agree to $10^{-13}$ of the mean, the sweep’s own rounding. That makes every number below a formula evaluation, not a search.

Two ways to make the swing vanish #

The formula says the swing goes to nought in two ways. A whole-number overlap ratio, $n_\beta = 0$, makes both corrections vanish: at every moment the lines in the field add up to exactly the mean. And a whole-number transverse ratio, $n_\alpha = 0$, does the same thing from the other side, at every overlap.

Against the overlap ratio the least is a row of dips. At every whole number it is back at the mean; between, it falls by $\min(n_\alpha n_\beta, (1 - n_\alpha)(1 - n_\beta))/(\varepsilon_\alpha\varepsilon_\beta)$ of it, so the dips shrink as one over the overlap ratio. At εα = 0.4 the first dip goes all the way to nought below εβ = 0.6, where the total contact ratio is under one and there are moments with no tooth in contact at all, the gap a spur pair below 1.0 would have. At 0.4 the dips between 1 and 2, 2 and 3 and 3 and 4 bottom out at 0.62, 0.77 and 0.83 of the mean, each at an overlap ratio six tenths past a whole number. At εα = 1.0 there are no dips: a transverse ratio of exactly one gives a helical pair the same total length of line at every moment, whatever its helix.

The spur optimum sits at the other end of this picture. At εα = 1.2 with straight teeth, one tooth pair is in contact for four fifths of every base pitch and two for the remaining fifth, so the length in contact is $b$ most of the time and $2b$ briefly. Its mean is $1.2b$ and its least is $b$, a sixth below. That sixth is the whole of the next section.

Least against least #

A helix pays for the root with face width asked for the helical face that carries the spur optimum’s mean, $1.2\,b_s$. The question a designer sizing for peak load asks is for the helical face whose least line is the spur optimum’s least, which is $b_s$. When the helical pair’s overlap ratio is a whole number its least equals its mean, εα $b_h/\cos\beta_b$, and setting that equal to $b_s$ gives

$$\frac{b_h}{b_s} = \frac{\cos\beta_b}{\varepsilon_\alpha},$$

exactly five sixths of the mean comparison’s $1.2\cos\beta_b/\varepsilon_\alpha$. At 30° and εα = 0.4 that is 2.207 against 2.648. The sixth that disappears is the spur pair’s own swing: comparing means charged the helical pair for line the spur pair only carries a fifth of the time.

A real face rarely lands on a whole-number overlap ratio, though, and between whole numbers the helical least is below its mean, so the face must be wider than the ideal to make it up.

Searched face by face, the narrowest helical face that carries the spur optimum’s least line traces a sawtooth. Each tooth begins where the ideal face lands just past a whole-number overlap ratio, so the least has dipped and the face must grow; the face then grows until the overlap ratio reaches the next whole number, where the least is the mean again and the ratio returns to 2.207. On the spur face of 10 modules the earlier essay used, the helical face is 23.9 modules, 2.39 times, at an overlap ratio of 3.81. Over spur faces from 5 to 40 modules the factor runs from 2.21 to 2.87, and the teeth of the sawtooth shrink as the face grows because the dips shrink as one over the overlap ratio.

The 2.87 is worth a sentence. On narrow spur faces, under about ten modules, the helical face needed can exceed the mean comparison’s 2.65, because the overlap ratio is small and the dips are deep. The mean comparison was not even a safe upper bound there. From ten modules up it always is, and the true factor is between 4% and 17% below it.

The same reckoning at every transverse ratio of the earlier table keeps the same shape. At εα = 0.6 the means ask for 1.77 times the spur face and the least, at a spur face of 10, for 1.64; at 0.8, 1.32 and 1.13; at 0.2, 5.30 and 4.90. At εα = 1.0 the answer is 0.883 at every spur face, a helical pair narrower than the spur pair, because a whole-number transverse ratio has no swing and its whole mean is its least. And at the spur optimum’s own εα of 1.2, the helical pair carries the spur pair’s least line on 0.74 to 0.88 of its face: a helix that gives up no transverse contact is pure gain, the result of lines that lean being longer and of the overlap evening out the jump.

So the trade the earlier essay offered stands, and it is cheaper than it looked. The root’s sliding falls from 0.834 to 0.209, four times less, for a face that is 2.2 to 2.5 times as wide on any spur face of ten modules or more.

More helix is not always more line #

There is a tempting rule hidden in all this, that more overlap is better: a steeper helix spreads each contact over more of the face, smooths the swing, and so must help. At a fixed face it does not.

On a face of 20 modules at εα = 0.4, a helix of 9.0° gives an overlap ratio of 1 and a least line of 8.04 modules, essentially the mean. Raising the helix to 13.6° gives an overlap of 1.5 and a least of 5.47 modules, a third less, though the mean has grown slightly because the leaning lines are longer. At 18.4° the overlap is 2 and the least is back up to 8.36. Below an overlap of 0.6 the least is nought: the pair has moments with no contact at all. The least line is a sawtooth in the helix angle with its teeth at whole-number overlaps, and a designer who steepens a helix to smooth a pair’s contact may land in a trough and carry a third less at the worst moment.

The prescription is the formula’s own. Choose the face and the helix together so that $b\sin\beta/(\pi m_n)$ is a whole number. Every overlap ratio that is a whole number puts the least line at its mean; every one between pays a penalty that the formula prices before any tooth is cut.

The swing is also a ripple #

The greatest length of line matters for a different reason. The difference between the greatest and the least is how much the length in contact changes as the pair turns through a base pitch, and taking a tooth pair’s stiffness as proportional to the length of line it bears on, the simplest model and the one the line-length argument already assumes, that difference is the ripple in the mesh’s stiffness that a pair runs with at tooth frequency. The formula prices it too, and the same whole numbers remove it.

The spur optimum swings from 0.833 to 1.667 of its mean: its length in contact doubles for a fifth of every pitch, a ripple of 83% of the mean. A helical pair at the same εα = 1.2 with an overlap of 0.5 already ripples by only 33%, and with an overlap of 1.6 by 10%. At εα = 0.4 the ripple is 63% at an overlap of 1.6, 29% at 3.5, 13% at the 3.81 of the ten-module design above, and nothing at all at 4. So the face that is narrowest for the least line, landing on a whole-number overlap, is also the face with no ripple, and the 23.9-module face the sawtooth chose on a spur face of ten is a near miss on both counts: 7.5% short of its mean at the least and 13% of ripple. Widening it by a further 1.2 modules, to an overlap of exactly four, would remove both.

This is the familiar workshop advice, to give a helical pair a whole-number overlap ratio for quiet running, arrived at from the other side. Here it falls out of sizing the face for its peak load, and the formula says how much a near miss costs in each currency.

What a gear designer can take from it #

Size a helical face by its least line, and compute it. The least has a closed form in the two fractional parts and costs nothing to evaluate. Comparing means overstates the helical pair’s face by a sixth against a spur pair at εα = 1.2, and on narrow faces can understate it.

Land the overlap ratio on a whole number. Doing so makes the least equal to the mean and the face its smallest; at a third of the spur optimum’s transverse contact on a 30° helix that is 2.21 times the spur face. A transverse ratio of exactly one does the same at any overlap.

Do not steepen a helix to smooth contact without checking where it lands. At a fixed face, the least line has its peaks at whole-number overlap ratios and its troughs between them, and the trough after the first whole number is the deepest.

Two earlier results frame this one. The shift that evens the wear balanced the two roots’ sliding; the angle the standard left free found the pressure angle a designer may still choose. The overlap ratio is a third knob of that kind, set by the face and the helix together, with a measurable cost when it is set carelessly.

What the least line does not settle #

Load sharing along a line. The peak load is taken to stand on the least length of line in contact, spread evenly along it. Real teeth are stiffer in the middle of the face than at its ends and stiffer near the pitch line than at the tip, so a line that is mostly near the corners of the field carries less than its length suggests. The least length is the right first measure and not the last one.

Contact stress. The stress between two curved flanks pressed together grows as the square root of the load per unit length of line, so a least line a fifth longer lowers the peak stress by about nine per cent. That is a statement about Hertzian contact on a line, which the earlier essays did not model, and it is offered as the scale of the gain rather than a measurement of it.

The root sliding at the moment of least line. The worst root sliding happens at the ends of the path of contact, and the least line happens at a particular phase. Whether the most sliding and the least line coincide in time, which would make the trade worse than either number says, is not measured here.

Still open: whether the least line and the worst sliding meet #

The worst root sliding is at the two ends of the path of contact, where a tooth pair enters and leaves, and the least length of line comes at a particular phase within each base pitch. In a spur pair at εα = 1.2 the entering and leaving happen while two pairs share the load, so the root’s worst sliding is borne on twice the least line. A helical pair at εα = 0.4 has no such luck by construction, but which point of each contact line is at the root at the moment of least line depends on the overlap ratio.

The distinct argument would be the load-weighted root sliding: for each position in the mesh, the length of line within a small distance of each root, weighted by the inverse of the total length in contact at that phase, and its peak over a base pitch. If the least-line phase is also the phase when much of the line sits at a root, the whole-number overlap that makes the least line largest would also be the one that matters most, and the face factors here would understate the cost. If the two are out of step, the overlap ratio is a free choice for the sliding and the face can be sized on the least line alone.