A helix pays for the root with face width

Two involute flanks roll on each other without sliding at one point only, the pitch point, and rolling at one point only measured how hard they slide everywhere else: worst at each gear’s root, where contact comes close to that gear’s base circle. Four knobs for the root then freed both profile shifts and both addenda of a spur pair and asked how low the worse root could be brought. On a 17-tooth pinion driving a 51-tooth wheel, with shifts up to a module, tips a quarter of a module thick and a tenth of a module of clearance, the answer was 0.834, two and a half times less than the balancing shift’s 1.96. Every optimum sat exactly on its floor of contact ratio, 1.2. The contact ratio was what paid.

That essay asked, at its end, whether a helical pair could reach the same root sliding without spending contact ratio, perhaps with a profile modification that varies across the face. Contact that runs along the tooth had already found the reason to hope: a helical pair has a second contact ratio, the overlap ratio $\varepsilon_\beta = b\sin\beta/\pi m_n$, bought with face width and helix angle alone and with no profile in it. It can carry the continuity that a spur pair’s transverse ratio had to carry by itself.

It can, and the root’s sliding then falls almost to nothing. But the price does not disappear. It moves from the profile to the face.

The same search in the transverse plane #

A helical gear is an involute cut on a slant, and in its transverse plane, the plane square to its axis, it is an ordinary involute gear of a larger module and pressure angle: $m_t = m_n/\cos\beta$ and $\tan\alpha_t = \tan\alpha_n/\cos\beta$, with the normal pressure angle $\alpha_n$ the cutter’s 20°. The profile’s sliding happens in that plane. So the four-knob search carries over unchanged, with the transverse pair in place of the spur pair, and with every quantity a cutter or a drawing specifies kept in the normal plane where it belongs: shifts and addenda in normal modules, the root a fixed 1.25 normal modules below the shifted pitch line, the tip’s thickness measured square to the tooth, the clearance a tenth of a normal module. At a helix angle of nought every number is the spur pair’s, and the search returns the earlier essay’s 0.834 to four figures.

What changes is the floor. A spur pair needs its transverse contact ratio $\varepsilon_\alpha$ above one, and the earlier essay asked for 1.2, so that one pair of teeth is in contact before the last lets go. A helical pair needs its total, $\varepsilon_\alpha + \varepsilon_\beta$, above that floor, and the overlap ratio can supply as much of it as the face allows. So the floor on $\varepsilon_\alpha$ alone can be lowered, and the search was rerun at floors of 1.2, 1, 0.8, 0.6, 0.4 and 0.2 and at helix angles of 0°, 15° and 30°. The search’s lower bound on each addendum coefficient was dropped from 0.4 to 0.05 so that it did not, by itself, stop the tips coming in.

The root slides in proportion to the contact #

At a 30° helix the least worst root is 0.772 at a transverse contact ratio of 1.2, 0.599 at 1, 0.447 at 0.8, 0.321 at 0.6, 0.209 at 0.4 and 0.102 at 0.2. Every optimum sits on its floor, exactly as the spur optima did, and every one has its two roots equal. The curve is very nearly a straight line through the origin, steepening slightly as the contact grows.

The straight line has a closed form, and it comes from the pitch point. Along the line of action, with the pinion’s base tangency at nought and the wheel’s at $L$, the pinion’s specific sliding is $\zeta_1 = 1 - (z_1/z_2)(L - u)/u$, nought at the pitch point and steepening away from it. Its slope there works out to $(1/z_1 + 1/z_2)\cdot 2\pi/(p_{bt}\tan\alpha_w)$, with $p_{bt}$ the transverse base pitch and $\alpha_w$ the operating pressure angle, and the wheel’s specific sliding has the same slope in the other direction. A path of contact $\varepsilon_\alpha p_{bt}$ long, centred on the pitch point so that the two roots are equal, has each end $\varepsilon_\alpha p_{bt}/2$ from it, so to first order

$$\zeta_{\text{root}} \approx \frac{\pi\,\varepsilon_\alpha}{\tan\alpha_w}\left(\frac{1}{z_1} + \frac{1}{z_2}\right).$$

At a transverse contact ratio of 0.2 and 30° the law gives 0.097 against the search’s 0.102, 5% under it, and the gap closes as the contact shrinks, since the law is the first term of the curve. At 1.2 it gives 0.568 against 0.772: the specific sliding curves away from the pitch point, steeply near a base circle, and a long path reaches into the curve. The law says three things the search could only have shown one pair at a time. The root’s sliding is proportional to the transverse contact a pair keeps; it falls with both tooth counts, as their reciprocals added; and it falls as the operating pressure angle rises, which is why every optimum, helical or spur, runs at a raised angle, here 26.9° to 28.2°.

The helix angle itself buys little #

The three curves lie close together. At a transverse contact ratio of 1.2 a 15° helix lowers the least worst root from 0.834 to 0.820 and a 30° helix to 0.772, 7%. At 0.2 they are 0.112, 0.110 and 0.102, 9%. What the helix angle does directly is raise the transverse pressure angle, from 20° to 22.8° at 30°, which raises the operating pressure angle with it and enlarges $\tan\alpha_w$ in the law by about a tenth. That is the whole of its own effect on the profile.

The helix changes how the teeth are cut more than how they slide. A cutter working in the normal plane sees a helical gear as a spur gear with more teeth, $z/\cos^3\beta$, so a 17-tooth pinion at 30° is cut like one of 26.2 teeth, comfortably clear of the undercut that the cutter takes back the tooth found below 17. None of the optima here comes near that limit in any case, since every one starts its contact well away from the pinion’s base circle; the virtual count matters for how the blank is cut and not for where the contact runs. And the raised pressure angle is the same trade the angle the standard left free measured, a steeper line of action spending contact ratio for other gains, arriving here through the helix rather than through the rack.

So the gain is not in the slant. A spur pair could have every one of these root slidings if it could keep running at a transverse contact ratio of 0.4, and it cannot, because there would then be most of every base pitch with no pair of teeth in contact at all. What the helix buys is the right to move left along the curve.

Contact pulled in to the pitch point #

The four designs show the mechanism. Each optimum centres its path of contact on the pitch point, so that both roots finish the same distance from it, and shortens the path to its floor. At a transverse contact ratio of 1.2 the path covers a fifth of the line of action; at 0.2 it is a short stretch hugging the pitch point, where the flanks barely slide at all.

The designs that do it are not exotic, though the numbers that describe them can look it. As in the shift that evens the wear, the knobs are a cutter’s position and a blank’s diameter, nothing more. At 0.4 the pinion’s addendum coefficient reaches the search’s floor of 0.05 while its shift is 0.67, so its tip stands 0.72 of a module above its pitch circle, short but not stubby. The wheel, shifted a full module with an addendum of 0.44, carries its tip 1.44 modules out. Both tips are thick, 1.52 and 1.17 modules against the quarter of a module asked for, and the clearances are 0.66 and 1.05 modules against a tenth: shortening the path of contact is done by the tips coming in towards the pitch point, and teeth with short tips are strong teeth.

A tooth flank is an unwound strand described the line of action as the crossed strand between two base circles. On that picture the result is plain. Every point of the strand away from the pitch point is a point where the two flanks unwind at different speeds, and the further from the pitch point the larger the difference, until it runs to infinity at each base circle. A pair that touches only near the pitch point barely slides. The only thing that ever stopped a gear designer using just that stretch was the need for the next pair of teeth to be in contact before the last one let go.

What the face pays #

Continuity is cheap. To bring the total contact ratio back to 1.2 at a 30° helix, the overlap ratio has to supply $1.2 - \varepsilon_\alpha$, and since $\varepsilon_\beta = b\sin\beta/\pi m_n$ that needs a face of 1.3 modules at a transverse ratio of 1, 2.5 at 0.8, 3.8 at 0.6, 5.0 at 0.4 and 6.3 at 0.2. Any real helical gear has a face several times its module, so continuity is not what limits the trade.

The length of line is. Contact that runs along the tooth found that the mean total length of the contact lines in the field of action is $\varepsilon_\alpha b/\cos\beta_b$, the transverse ratio and not the total. The overlap ratio buys contact duration, a tooth always being engaged somewhere along its width, and not contact length. Swept through a base pitch on the 30° pair at a transverse ratio of 0.4 and a face of 20 modules, the mean length in contact is 9.06 modules, which is what the formula gives; the total contact ratio, 3.58, would have predicted 81.

So a pair that spends transverse contact carries its load on proportionally less line, and to carry it on as much line as the spur optimum it must widen its face in proportion. The spur optimum on a face of ten modules has twelve modules of line in contact on average. A 30° pair matches that on 8.8 modules of face at a transverse ratio of 1.2, since its lines lean and are longer, 10.6 at 1, 13.2 at 0.8, 17.7 at 0.6, 26.5 at 0.4 and 53 at 0.2. The hero figure shows the pair at 0.4: its field of action is a third as wide as the spur pair’s and two and a half times as tall, and its lines, many and short, add to the same twelve modules on average where the spur pair’s single line is twelve modules long.

If the face is not widened, the load per unit length of line rises in proportion to the contact given up, three times at a transverse ratio of 0.4, and the contact stress between two curved surfaces pressed together grows as the square root of the load per unit length, by about 1.7 here. That is the honest form of the trade. The root’s sliding can be cut fourfold and the face kept, at the cost of higher stress on less line; or the line can be kept and the face widened by a factor of 2.65. What is no longer on offer is the free lunch the overlap ratio seemed to promise, continuity and length together from a helix.

Seven pairs, one law #

Across the seven pairs the 30° helical pair at a third of the transverse contact slides 3.78 to 4.31 times less at its worse root than the spur optimum: 1.233 to 0.288 on the 12–36 pair, 0.554 to 0.147 on 25–75, 1.013 to 0.235 on the equal pair. The law tracks every one of them to within 12%, always a little under, and orders them correctly: the pairs whose reciprocal tooth counts add to the most, the 12-tooth pinion and the equal 20-tooth gears, slide hardest at a given contact, and the gain is largest on the pairs whose roots were worst to begin with.

A shift that varies across the face #

The earlier essay’s suggestion was a profile modification that changes across the face, shortening the tips in the part of the face that meets contact nearest the base tangency. With a helix it can be judged without a search. Each transverse section of a helical gear is the same profile turned a little, and every section passes through the whole path of contact in turn as the gear rotates. A section whose tips were shortened would slide less at its roots, and a section with longer tips would slide more. The pair’s worst root is the worst over all its sections, and the contact ratio the pair keeps is, on average, the average over them.

For any rule that makes a section’s worst root rise with its own transverse contact, which the search shows this one does, the worse of two sections is at least as bad as a uniform profile at their average contact: one of the two has at least the average. Measured on the 30° pair, sections at transverse ratios of 0.3 and 0.5 slide at 0.155 and 0.265 at their roots, so the pair’s worse root is 0.265, against 0.209 for a uniform 0.4. A face-varying modification cannot beat a uniform one at the same mean contact, and the uniform one is what the search above already found. Modifications that vary across the face exist for other reasons, chiefly to keep the load off the ends of the face when shafts deflect; for the root’s sliding they offer nothing a uniform shortening does not.

What the face does not settle #

Load. The trade is stated in mean length of line. The least length over a base pitch, which sets the peak load per unit length, depends on how the fractional parts of the two contact ratios fall, as contact that runs along the tooth found for the swing, and was not optimised here.

Sliding is a proxy for wear. As in the earlier essay, specific sliding stands for wear; a contact that barely slides under a high load may pit rather than wear, and contact stress is estimated here only by its scaling.

Involute profiles only. Gears built on the same idea taken to its end exist: the Wildhaber–Novikov gears use circular-arc teeth that touch at a point, with no transverse contact ratio at all, and rely entirely on the helix for continuity. Their sliding and their contact are a different calculation from the involute’s here.

Axial thrust. A helix angle of 30° pushes each gear along its shaft with a force proportional to $\tan\beta$ times the tangential load. A wider face does not change that, and the bearings pay for it whatever the profile.

Still open: the least line a pair carries #

The mean length of line in contact is fixed by the transverse ratio and the face, but the instantaneous length swings about that mean through every base pitch, and it is the low point of the swing that carries the peak load. A pair at a third of its transverse contact has many short lines in its field where the spur pair had one long one, and the swing depends on how the transverse and overlap ratios’ fractional parts combine.

The distinct argument would be the face width and helix angle that make the least length of line a pair carries as large as possible for a given transverse contact, and whether at that choice the root-sliding trade is still worth making once the peak load, and not the mean, is what the face is sized for. The measurement would sweep face width and helix angle for each of the optima here, find the least length over a base pitch, and say how much of the factor of 2.65 a well-chosen overlap ratio can take back.