The least line carries the root

The face carries the least line priced a helical pair by the least length of contact line it carries over a base pitch rather than the mean. The contact lines cross the field of action at the base helix angle, one base pitch apart, and the total length in contact swings as they enter and leave. The peak load on a unit of line stands on the least. That least has a closed form in the fractional parts of the transverse contact ratio $\varepsilon_\alpha$ and the overlap ratio $\varepsilon_\beta$. A whole-number overlap ratio makes the least equal to the mean.

The essay ended on the other half of the load’s story. A helix pays for the root with face width chose a small transverse contact ratio to cut the sliding at the teeth’s roots, which happens at the two ends of the path of contact, where a tooth pair enters and leaves. The question was whether the least-line moment is also the moment when much of the line sits near a root. If it is, the whole-number overlap that makes the least line largest also matters most for the roots, and the face, sized on the least line, understates the root’s worst load. If the two are out of step, the overlap ratio is a free choice for the sliding.

They are in step, and the whole-number overlap does both jobs.

The root share #

The field of action is a rectangle, the path of contact one way and the face the other. In base pitches the path is $\varepsilon_\alpha$ long. The contact lines cross the rectangle at the base helix angle, one base pitch apart, and as the gears turn they advance along the path, each line advancing $\varepsilon_\beta$ base pitches from one side of the face to the other. Scaled so that a line rises one unit of face for each unit of path, the lines are $x + y = \varphi + k$ for whole $k$, with $\varphi$ the position within a base pitch, $x \in [0, \varepsilon_\alpha]$ along the path and $y \in [0, \varepsilon_\beta]$ across the face. A line’s length in the field is proportional to the stretch of face it occupies, so the total in contact, $L(\varphi)$, is a sum of such stretches.

The roots are at the two ends of the path, $x = 0$ and $x = \varepsilon_\alpha$, where one tooth’s root meets the other’s tip and the sliding velocity is largest. Take a band of width $d$ at each end. The length of line inside the two bands, over the total, is the share of the load carried there if the load is spread evenly along the lines in contact. Spread evenly over the whole field it would be $2d/\varepsilon_\alpha$, and the root share is measured over that even value.

The field’s least line was checked against the earlier essay’s closed form at every overlap ratio swept, from 0.8 to 3.2 in steps of a twentieth, at transverse ratios of 0.4 and 0.7. It agrees to the sampling of 4,000 positions a pitch.

Counting the lines at the ends #

In the limit of a thin band the root share has a simple form. A line that crosses one end of the path inside the face contributes the band’s width to the root length, whatever its angle. A line whose own end lies inside the band contributes less, but only during an instant as short as the band is narrow. So, as the band thins,

$$\frac{R(\varphi)}{2d/\varepsilon_\alpha} \to \frac{\varepsilon_\alpha\,(N_0 + N_1)}{2L(\varphi)},$$

where $N_0$ and $N_1$ are the numbers of lines crossing the two ends inside the face. The root share counts the lines at the ends and divides by the line in contact. Swept with a band a thousandth of the path wide, the field agrees with the count at 99% of the phases of every pitch. The exceptions are the instants, a band’s width long, when a line’s own end passes a path end.

A whole-number overlap makes both parts constant. With $\varepsilon_\beta = n$, exactly $n$ lines cross each end at every position, and the line in contact is $\varepsilon_\alpha n$ throughout. The root share is even at every position, as the least line equals the mean. Between whole numbers the count at each end steps between $\lfloor\varepsilon_\beta\rfloor$ and $\lceil\varepsilon_\beta\rceil$ as lines enter and leave the face, and the line in contact swings. The root share follows both.

Why a whole number does both #

The reason one choice fixes both swings is geometric. With a whole-number overlap ratio, each contact line runs from one side of the face to the other while advancing exactly $n$ base pitches along the path. Any vertical slice of the field, any one position along the path, is then crossed by exactly $n$ lines at every moment, because the lines are a pitch apart and each takes $n$ pitches to cross. The ends of the path are two such slices, so the count there never changes. The line in contact is the integral of that count along the path, so it never changes either.

Between whole numbers a slice is crossed by $\lfloor\varepsilon_\beta\rfloor$ or $\lceil\varepsilon_\beta\rceil$ lines depending on the position in the pitch, and different slices change their count at different moments. The line in contact averages the counts over the path, which smooths their steps into the earlier essay’s piecewise-linear swing. The root share is the count at two particular slices over that average. The ends of the path are slices like any other, and nothing makes them change their count when the average does. That is why the root share’s peak can be as large as half again even when the least line’s dip is small.

What happens in a mesh introduced the contact ratio as the number of tooth pairs in contact on average. The overlap ratio is the same idea across the face. A whole number of each makes the corresponding count constant rather than merely right on average.

The root is loaded most while the line is least #

At $\varepsilon_\alpha = 0.4$ and $\varepsilon_\beta = 1.5$ the line in contact is least, at two thirds of its mean, for a tenth of each base pitch. During that stretch three lines cross the path’s ends between them, against the three the mean would give, and the line in contact is at its smallest. The thin-band root share reaches 1.5 times its even value there, and nowhere else in the pitch is it as high. The worst moment for the roots is inside the least-line moment.

That held at every overlap ratio swept at both transverse ratios. In the thin-band limit the root share’s peak falls while the line in contact is at its least, to within one sample of the 4,000 in a pitch. The reason is the counting form. The root share is the lines at the ends over the line in contact. Its denominator is smallest exactly when the line is least, and in every case swept the count of lines at the ends does not fall during that stretch.

The peak is not small. At a transverse ratio of 0.4 the root share’s peak is 1.5 times even for overlap ratios from just above one to 1.6, 1.6 at 1.7, 1.33 at 1.8, then 1.25 from just above two to 2.6, and 1.17 just above three. At a transverse ratio of 0.7 it reaches 1.75 at an overlap of 1.4. The pattern repeats in every interval between whole numbers, smaller in each, because the extra line that crosses an end becomes a smaller fraction of the lines that do.

Why a larger transverse ratio swings more #

At a transverse ratio of 0.7 the swings are larger than at 0.4: a peak of 1.75 at an overlap of 1.4 against 1.5 at the same overlap for 0.4. The counting form explains it. The root share’s even value is set by the two ends’ share of the path, which is fixed by the band. Its peak is the ends’ count over the line in contact, and the line in contact at its least is what a longer path makes smaller relative to its mean. A longer path has more line in contact on average, so losing a short piece at the least-line moment matters less to the line. The count at the ends does not grow with the path at all. So the ends’ count, which steps by a whole line, is a larger fraction of a line in contact that dips by less, and the ratio swings further.

A helix pays for the root with face width chose a small transverse ratio to bring the path’s ends close to the pitch point, where the sliding is less. The root share adds a modest argument in the same direction. The smaller transverse ratio’s root share swings less between whole overlaps, so the roots’ worst moment is closer to their average one.

A band of finite width #

A real root zone is not a line. The sliding velocity grows with distance from the pitch point, so the zone where it is high is some fraction of the path at each end. With a band a tenth of the path wide, the picture changes in the lower part of each interval between whole numbers. At an overlap of 1.5 the wide band’s share peaks at 1.36, just after the least-line stretch ends, and during the stretch itself it is even. A line whose end is inside a wide band contributes less than the band’s width, and at the least-line moment the lines’ ends are where the wide band cuts them short. In the upper part of each interval, at 1.7 for instance, the wide band’s peak is the thin band’s, 1.6, and falls with the least line.

So the two are in step in the limit and close to it in practice. The wide band’s peak comes a few hundredths of a pitch after the least line’s stretch. That is too close for a designer to separate them by choosing a moment, and the whole-number overlap removes both swings whatever the band’s width.

Against the overlap ratio #

Read against the overlap ratio, the two swings move together. At every whole number the least line is the mean and the root share is even. Between whole numbers the least line dips and the root share swings up. The root share’s swing does not shrink to nothing near the whole numbers the way the least line’s dip does. Just above one it jumps straight to 1.5, because one line more at an end is a whole line more, while the least line’s dip grows only in proportion to the distance from the whole number. A face whose overlap ratio is a little off a whole number, which the face carries the least line found costs almost nothing in least line, still has its root share swinging by half.

That is a sharper reason to hold the overlap ratio to a whole number than the least line gave. The least line’s penalty for missing a whole number by a twentieth is a few per cent. The root’s is the full swing from the first twentieth. Contact that runs along the tooth found the overlap ratio to be set by the face and the helix angle together. A designer holding it to a whole number is choosing those two so that each line crosses the face in an exact number of pitches.

The ledger #

The ledger collects the cases. At every whole number every column is one. Between whole numbers the thin band’s peak always falls while the line is least. The wide band’s peak sometimes falls just after that stretch, and then its share during the least-line moment itself is even. The largest swing in the ledger, 1.75 at a transverse ratio of 0.7 and an overlap of 1.4, comes where the line in contact is also near its least, 0.82 of its mean.

What it means for the face #

The earlier essay sized a helical face so that its least line equalled the spur optimum’s least line, and found the factor 2.21 for a whole-number overlap. At a whole-number overlap that sizing also carries the roots evenly, since the root share never departs from even. At any other overlap the face sized on the least line carries the root’s worst moment on that same least line with the root share above even. The roots’ worst load per unit of line, relative to a whole-number design, is up by the root share’s peak, 1.17 to 1.75 in the cases measured.

So the face factors of the earlier essay understate the root’s cost between whole numbers, and are exact at them. A face sized for a whole-number overlap that comes out a twentieth over, through tolerance or a rounded face width, carries its least line almost unchanged and its roots half as hard again at their worst moment. The least line forgives a near miss; the root does not. The sliding is the reason for choosing a small transverse ratio in the first place. A designer who has done that should hold the overlap to a whole number for the roots’ sake as much as for the least line’s.

Undercutting and the seventeen-tooth rule and four knobs for the root found the root region to be where a tooth is weakest in another sense, as cut, and where shifting the cutter moves the sliding between the two gears. The root share here is a load in the region the cutter leaves thinnest.

What the field does not settle #

Load along a line. The load is spread evenly along the lines in contact. A real pair loads its lines unevenly: the mesh stiffness varies along each line, and the tips, where one gear’s root meets the other’s tip, are the most compliant parts of the teeth. That would shift load away from the ends, by an amount that needs a tooth-stiffness model.

Sliding within the band. The band treats every point within it alike. The sliding velocity grows steadily towards the path’s ends, so a band’s load near its outer edge matters more than near its inner, and a weighting by the sliding would change the numbers if not the pattern.

The spur pair. At a transverse ratio of 1.2 with no helix, two pairs share the load whenever one is at a root, which the earlier essay noted. The spur pair’s root share is a different shape, a step, and is not swept here.

Profile modification. Real helical gears often have their tips relieved to soften the entry and exit, which takes load off the very bands measured here.

Still open: a face whose overlap is set by the root #

A whole-number overlap ratio fixes both swings, but real gears have tolerances on face width and helix angle that move the overlap off its whole number, and the root share jumps by half the moment it does.

The distinct argument would be the overlap ratio that minimises the worst root share over a stated tolerance band, rather than at a point. Since the root share jumps on leaving a whole number from above but rises less steeply below it, the best nominal overlap may sit a little below a whole number, so that the tolerance band falls mostly where the root share is gentler. The measurement would take a transverse ratio of 0.4, tolerance bands of a twentieth and a tenth of an overlap ratio, and find the nominal overlap whose worst root share and worst least line over the band are both least. If the best nominal sits off the whole number, a drawing should specify the face to land just under it rather than on it.