A toleranced overlap should miss its whole number on purpose

The least line carries the root followed a helical pair’s load to the two ends of its path of contact, where one gear’s root meets the other’s tip and the sliding is fastest. The share of the load carried there swings through every base pitch, peaking at the moment the length of line in contact is least. A whole-number overlap ratio removes both swings at once: the least line becomes the mean and the root share stays even at every moment of the mesh. That gave a design rule. Choose the face width and the helix angle so that each contact line crosses the face in an exact number of base pitches.

The essay ended on the rule’s weak point. A face is cut to a width tolerance and a helix to an angle tolerance, so the overlap ratio of a real pair lands near the whole number, not on it. Just above a whole number the root share jumps straight to half as much again. The essay asked whether the best place to aim is therefore not the whole number itself but a little below it, where the root share rises more gently. It also asked whether the same aim serves the least line.

It does not, and the reason is that the two sides of a whole number are different machines.

Two machines either side of a whole number #

The field of action of a helical pair is a rectangle, the path of contact across it and the face up it, with contact lines crossing it at the base helix angle one base pitch apart. In base pitches the path is $\varepsilon_\alpha$ long and the face $\varepsilon_\beta$ high. The thin-band root share has a counting form, found in the earlier essay. Over its even value it is $\varepsilon_\alpha(N_0 + N_1)/(2L)$, where $N_0$ and $N_1$ count the lines crossing the two ends of the path inside the face and $L$ is the total line in contact. The root share is lines at the ends over line in the field.

Count both just below a whole number $n$, at $\varepsilon_\beta = n - \Delta$, for a transverse ratio below one. At the worst phase each end of the path is crossed by $n$ lines, as it is at $n$ itself: shortening the face by less than a pitch removes no crossing an end can have at that moment. The line in contact, measured in units of one line across the whole face, falls from $\varepsilon_\alpha n$ to $\varepsilon_\alpha n - \Delta$. That is the earlier closed form for the least line, read at this overlap. So the root’s peak share is

$$\frac{\varepsilon_\alpha n}{\varepsilon_\alpha n - \Delta},$$

the whole number’s line over this face’s least line. Below a whole number, the root share’s overload is the least line’s shortfall and nothing else. The root carries the same number of lines while the face carries less line, and the load per unit of line rises in proportion.

Now count just above, at $n + \Delta$. The least line does not move. A face that runs past a whole number of pitches adds a sliver of line at one end of the field just as the opposite corner loses one, and at the least moment the two cancel, as the face carries the least line found when it noticed that a larger overlap is not always more line. The least line stays at $\varepsilon_\alpha n$. But for part of each pitch one more line crosses an end, and at that moment the root share’s peak is

$$1 + \frac{1}{2n},$$

whatever $\Delta$ is. That is 1.5 above one, 1.25 above two and 1.17 above three, reached at the first hundredth of a pitch past the number and held there.

The pitch picture shows both counts. At 0.95 the ends are never crossed by more than two lines, and the line in contact drops to 0.35 for a stretch of the pitch: the root share’s peak is 0.40 over 0.35, or 1.143. At 1.05 the line in contact never falls below 0.40, but for a stretch three lines cross the ends: 1.5. The two sides of the number differ in what changes. Below, it is the measure of the line. Above, it is the count at the ends.

Both forms, against the field #

Each form holds over a stated range. The form below holds while $\Delta$ is less than $\varepsilon_\alpha$, and the form above while $\Delta$ is less than $1 - \varepsilon_\alpha$. Past those, a second line’s ends begin to pass the path’s ends inside the face. Within them, the forms were checked against the root share swept over the field at four thousand phases a pitch. The check covers transverse ratios of 0.4 and 0.7, whole numbers one to three, and offsets from a hundredth to a quarter on either side.

Across 108 cases the largest difference is $4.7\times10^{-4}$, the sampling of the phase. The least line matches $\varepsilon_\alpha n - \Delta$ below and $\varepsilon_\alpha n$ above to rounding in every case. Below the number, the product of root share and least line is $\varepsilon_\alpha n$ exactly, to $10^{-12}$.

Three ways to draw the face #

A tolerance band of width $w$ on the overlap ratio can be placed three ways about the whole number: wholly below it, centred on it, or wholly above it. Each placement has a worst root share and a worst least line over its band, and the two forms give both.

Wholly below, the band’s lower edge is the worst for both: root share $\varepsilon_\alpha n/(\varepsilon_\alpha n - w)$ and least line short by the same factor. At $\varepsilon_\alpha = 0.4$, $n = 1$ and $w = 0.05$ that is a root share of 1.143 and a least line of 0.875 of the whole number’s.

Wholly above, the least line is whole across the band and the root takes the full jump: 1.000 and 1.500.

Centred, the band reaches above the number, so the root takes the full jump, 1.500. It also reaches below, so the least line loses half the factor, 0.938. In the thin-band limit the centred band ties the band above on the root and loses to it on the line. The drawing most designers would make, the whole number with a symmetric tolerance, is beaten outright by moving the nominal up by half the band.

Between the band below and the band above the choice is a real trade. The band below protects the root as long as its factor is smaller than the jump, which is while

$$w < \frac{\varepsilon_\alpha n}{2n + 1}.$$

At $\varepsilon_\alpha = 0.4$ that is a band up to 0.133 wide about one, 0.160 about two and 0.171 about three, rising towards $\varepsilon_\alpha/2$. At 0.7 it is 0.233, 0.280 and 0.300.

The trend has a plain reading. A larger whole number has more lines at each end, so one more line is a smaller jump, but it also has more line in contact, so the same shortfall is a smaller factor. The second wins slowly, and the limit rises. A smaller transverse ratio is the case where sliding was the reason for the design, and there the least line is short and every shortfall is a large factor, so the band below protects the root over a narrower range. The pairs that most need the root protected get the least room to protect it by aiming low.

A root band of real width #

The thin-band limit takes the root band to be vanishingly narrow, and in that limit the jump above a whole number is immediate. A real root region has width. The earlier essay measured a band a tenth of the path wide, and with that band the jump is spread over about a band’s width of overlap. At $\varepsilon_\alpha = 0.4$ the root share is 1.098 at 1.01, 1.279 at 1.03 and 1.363 by 1.05. Below the number nothing changes: the band of real width gives the same peak as the thin one, because no line’s end lies inside it at the worst phase.

That makes the root’s best placement a balance. Slide the band up from wholly below. Its lower edge climbs out of the factor while its upper edge climbs into the spread-out jump, and the worst of the two is least where they are equal.

For a band 0.05 wide at $\varepsilon_\alpha = 0.4$, the balance puts the band’s middle at 0.986. It reaches 0.011 past the number and 0.039 under it. Its worst root share there is 1.108, against 1.143 wholly below, 1.235 centred and 1.364 wholly above. The least line at the band’s lower edge is 0.903 of the whole number’s. For a band 0.1 wide the best middle is 0.975, worst root share 1.232, least line 0.812. In every case measured the best middle sits about a quarter of the band under the number. The offset past the number is 0.010 to 0.011 for the narrower band and 0.021 to 0.025 for the wider, and it is set by how quickly the root band of real width takes up the jump.

The frontier is the honest form of the answer. Each placement of the band is a point with a worst root share and a worst least line. Sliding the band up moves the point towards a whole least line and, past the root’s best, towards a heavier root. No placement is best on both. Centring is not on the frontier’s good end for either measure.

What a tolerance band is in millimetres #

The bands here are fractions of an overlap ratio, and it is worth seeing what one is on a drawing. The overlap ratio of a helical pair is $\varepsilon_\beta = b\sin\beta/(\pi m_n)$, face width times the sine of the helix angle over the normal pitch. A pair of normal module 3 mm at a helix angle of 15° reaches an overlap of one with a face of 36.4 mm.

A face-width tolerance of ±0.5 mm moves the overlap by ±0.014. A helix-angle tolerance of ±0.1° moves it by ±0.007 more, because at that face the overlap changes by 3.7 per radian of helix. Stacked, the two make a band about 0.04 wide. That is a twentieth of an overlap ratio for a gear made to ordinary tolerances, not an extreme case.

Placed on the numbers for that band at a transverse ratio of 0.4, the drawing with 36.4 mm and a symmetric tolerance has a worst root share of 1.193 with a root band a tenth of the path wide, and 1.500 in the thin limit. Moving the nominal face down to 36.0 mm puts the band’s middle at an overlap of 0.989. The worst root share falls to 1.086, and the least line at the band’s lower edge is 0.921 of the whole number’s. Moving the nominal up by the band’s half, to 37.1 mm, keeps the least line whole across the band and loads the root at 1.363.

So the decision is a few tenths of a millimetre of face either side of the obvious number: four down for the root, seven up for the line. It is a decision a drawing makes whether or not anyone intends it. A face dimension written as 36.4 with a symmetric tolerance has chosen the centred placement, which serves neither measure.

A deliberate miss is familiar elsewhere in a gear pair. Backlash is an allowance found that a pair with no backlash cannot be run, so its centre distance is drawn a little apart on purpose. The overlap ratio is the same kind of dimension. Its ideal value is a point the process cannot hit, the two sides of that point fail differently, and the nominal belongs on the side whose failure the design can better afford.

The ledger #

Read down the ledger and the pattern holds at every case. Centred always ties the band above on the thin-band root and always loses to it on the least line. The band below always wins on the thin-band root at these widths, which are all inside the crossover. With the root band of real width the best nominal is always a quarter-band low, and it beats centring by between 0.04 and 0.14 of an even share. The case with the most at stake is the smallest transverse ratio and the smallest whole number: 0.4 and one, where centring costs the root 1.235 and the best aim 1.108.

What a drawing should say #

Never draw the whole number with a symmetric tolerance. In the thin-band limit it is beaten on both counts by the band above, and with a root band of real width it is beaten on the root by the band a quarter-band low. It is the one placement that does not sit at the good end of anything.

Decide which failure governs, then aim. If the pair is sized against surface stress on the least line, as the face essay sized it, put the band wholly above the number. Contact that runs along the tooth found the overlap bought with face width and helix angle alone, and above the number that purchase is simply extra face that carries no extra least line. The least line is then whole across the band and the extra face is the price. If the pair was given a small transverse ratio for the sake of the sliding at the roots, as a helix pays for the root with face width chose, aim a quarter-band under the number. That pays a least-line shortfall of up to the band’s width over $\varepsilon_\alpha n$, and should be priced into the face.

Tighten the band where the root matters most. The root share below the number is the least line’s shortfall, so its cost is the band’s width over $\varepsilon_\alpha n$. At $\varepsilon_\alpha = 0.4$ and an overlap of one, every hundredth of tolerance costs two and a half per cent of root load at the lower edge. That is a direct price on the face-width and helix-angle tolerances. The ratio that has a tolerance found a belt’s ratio carrying every length’s tolerance while a gear ratio, being a count, carries none; the overlap ratio is a gear quantity that is not a count, and its tolerance is a matter of load.

Prefer a larger whole number when the tolerance is loose. The jump above two is half the jump above one, the crossover is wider, and the least line’s shortfall is a smaller factor. The shift that evens the wear balanced the two roots’ sliding by moving the cutters, and four knobs for the root freed both shifts and both addenda to bring the worse root down further. The overlap ratio is a fifth knob, acting on the share of load at the roots rather than the sliding there, and its whole numbers are not equally forgiving.

What the counts leave out #

An even load along each line. Both forms count lines and measure them as though every point of a line carried the same load. Mesh stiffness varies along a line and the tooth tips are its most compliant parts, which would move load off the ends. The ordering of the placements would survive a moderate redistribution; the numbers would not.

Transverse ratios above one. The forms are written for $\varepsilon_\alpha < 1$, the small ratios that sliding argues for. Above one, the least line has a whole-pitch floor and the counts at the ends change, so both forms need restating.

A distribution inside the band. The worst case over the band is what the drawing guarantees. A process that centres its parts in the band would load the root at an average nearer the band’s middle, and a placement chosen for the worst case is not necessarily best on average.

Profile relief. Relieved tips take load off the root bands directly, and a pair with relief may tolerate the jump above the number at less cost. That is a different design lever and was not modelled.

Still open: when the band’s middle is the right measure #

The placements here were judged on the worst case over the tolerance band, which is what a drawing promises. A production line does not spread its parts evenly to the band’s edges; it centres them, with a spread smaller than the band. For such a population the right measure is the root share averaged over the parts, or the share exceeded by one part in a hundred.

The distinct argument would put a distribution of overlap ratios, a normal spread about the nominal with a standard deviation a sixth of the band, under the two closed forms and the tenth-wide band. It would find the nominal that minimises the expected root share and the nominal that minimises its ninety-ninth percentile. The jump above the number is a step, and a step penalises any part that lands past it however few there are. The expected-value best may therefore sit further below the number than the worst-case best. If it does, a process with good centring should aim lower than a drawing would. If the two coincide, the quarter-band rule is the rule for both.