Four knobs for the root

Two involute teeth slide on each other everywhere except at the pitch point, and rolling at one point only measured how much: the specific sliding, the rate at which one flank slides over the other divided by its own rolling speed, is worst at each gear’s root, where contact comes close to that gear’s base circle and its rolling speed there falls towards nought. The shift that evens the wear moved the pinion’s cutter out and the wheel’s in by the same amount — the move moving the cutter out introduced to cure undercut — keeping the standard centre distance, and found the shift at which the two roots slide equally: 0.357 of a module on a 17-tooth pinion driving a 51-tooth wheel.

Balanced, both roots still slide at 1.96. The balance equalises the two numbers but does not make either small, and that essay ended by pointing at the other things a designer can change: the addenda — how far each tooth reaches beyond its pitch circle, which short- and long-addendum systems alter — and, with them, the centre distance. With both shifts and both addenda free, how low can the worse root be brought while the pair still works?

About two and a half times lower, and the answer has a shape: the optimum is always balanced, it always spends exactly the contact ratio the designer allows, and it always runs at a raised pressure angle.

Where the sliding comes from #

Put the pinion’s base tangency at the left end of the line of action and the wheel’s at the right, a distance $L$ apart — the crossed strand that a tooth flank is an unwound strand found the line of action to be. At a point $u$ along it the pinion’s flank rolls at a speed proportional to $u$ and the wheel’s to $(z_1/z_2)(L - u)$, so the pinion’s specific sliding is

$$\zeta_1 = 1 - \frac{z_1}{z_2}\,\frac{L - u}{u},$$

which runs off to minus infinity as $u$ goes to nought. The wheel’s does the same at the other end. Contact begins where the wheel’s tip circle crosses the line and ends where the pinion’s does. So the pinion’s root sees the start of contact, the wheel’s root the end, and each root’s sliding is set by how close to its own gear’s base tangency the other gear’s tip lets contact reach.

That already says what helps. Anything that keeps contact away from both ends of the line helps both roots at once: shorter tips, which pull the start and the end inward; and a longer line of action with the same contact on it, which is what a larger operating pressure angle gives, since the line’s length is the two base radii times the tangent of that angle. A standard pair has neither. The balancing shift slides the contact segment along the line to share the harm evenly, but keeps its length and the line’s.

The four knobs, and what holds them back #

The four free numbers are the pinion’s and wheel’s profile shifts, $x_1$ and $x_2$, and their addendum coefficients, $h_1$ and $h_2$, in modules. The centre distance is not free: once the shifts are chosen, the teeth must mesh without backlash — any backlash is an allowance cut in afterwards — which fixes it — computed here both from the involute equation and by bisecting on the drawn teeth’s thicknesses until they exactly fill the pitch, and the two agree to $10^{-6}$ of a module at every design used below. A positive total shift pushes the gears apart and raises the operating pressure angle.

What stops the search running off to a pair that barely touches is a set of conditions any working pair must meet, each checked on the drawn tooth:

• a contact ratio of at least 1.2, so that one pair of teeth takes over before the last lets go;
• a tip at least a quarter of a module thick on each gear, so the teeth do not come to a point;
• a clearance of at least a tenth of a module between each tip and the other gear’s root circle;
• contact that begins and ends on the involutes, not below either base circle, where the cutter takes back the tooth and there is no involute to bear on.

The search is a grid of 48 starts, each run downhill by Nelder–Mead on the worst root plus a penalty for any broken condition, then polished by a pattern search in random directions. The polish matters. The worst root is the larger of two smooth functions, so it has a crease where they are equal, and a search along the four coordinate axes stops on the crease short of the minimum; a direction that crosses the crease diagonally does not. The first version polished along the axes and reported optima whose two roots differed by up to 5%. The random directions found every one of those improvable, and in each case the improvement ended with the two roots equal.

Two and a half times less #

On the 17–51 pair the balancing shift leaves both roots at 1.961. With the four knobs free and every shift limited to a module — a limit most gear catalogues would recognise — the worse root falls to 0.834. Without the limit it falls to 0.753, with a pinion shift of 0.94 and a wheel shift of 1.66. Across the seven pairs the four-knob optimum with shifts up to a module is 1.9 to 4.0 times lower than the balancing shift’s: 3.362 to 1.233 on a 12–36 pair, 1.568 to 0.705 on 20–60, 2.026 to 0.760 on 20–40. On a pair of equal 20-tooth gears, where no ±x shift can help because it has nothing to balance, 4.095 falls to 1.013.

Every optimum has the same three features.

The two roots are equal. At every optimum the pinion’s root and the wheel’s slide by the same amount, to $10^{-4}$ of it. The balance the earlier essay assumed a designer wants turns out to be what the least worst root requires anyway: with four knobs, if one root were worse than the other there would always be a direction that lowered it at the other’s expense.

The pressure angle rises. Every optimum has a positive total shift, a larger centre distance and an operating pressure angle between 24° and 29°: 26.2° on the 17–51 pair with shifts up to a module. The angle the standard left free found that raising the pressure angle buys smaller pinions and spends contact ratio, with no angle best at both. Here the same trade is made for a third reason, the roots’ sliding, and a raised pressure angle is part of every answer.

The tips shorten. The addenda at the optimum are all below a module, 0.61 on the 17–51 pinion with shifts up to a module — a short-addendum pinion on a long-shifted blank. The clearance condition binds at most optima, and the tip-thickness condition at none: the teeth stay comfortably thick, a third of a module at the thinnest tip, on the 12-tooth pinion.

Three steps down, and which knob buys which #

The four knobs are not equally useful, and taking them in turn shows where the gain comes from.

Addenda alone. Keep the standard centre distance, so the shifts stay equal and opposite, and free only the two addenda: the classical short- and long-addendum system. On the 17–51 pair the worse root comes down from the balancing shift’s 1.961 to 1.282, with almost no shift at all and the wheel’s tip cut to half a module. Shortening the wheel’s tip pulls the start of contact away from the pinion’s base tangency, which is where the pinion’s root was suffering, and the pinion’s tip, left at 0.95 of a module, is what carries the contact ratio. Across the seven pairs the addenda alone take off a third to a half of what the balancing shift left: 3.362 to 2.295 on the 12–36 pair, 1.174 to 0.744 on 25–75, 4.095 to 2.148 on the equal pair.

The addenda-only designs share the four-knob optima’s signature on one count and not on another. Every one of them sits on its contact-ratio floor of 1.2 exactly, so the contact ratio is spent here too, and every one has its two roots equal. But none of them runs at a raised pressure angle, because at the standard centre distance there is none to raise: the line of action stays the length it was, and all the addenda can do is choose which part of it the contact uses. The clearance condition does not bind either. A wheel tip cut to half a module leaves three quarters of a module of air above the pinion’s root, and the thin tip that limits the design is the pinion’s own — half a module thick on the 12-tooth pinion, which is long-tipped because it has to be.

Equal gears need no shift at all. On the 20–20 pair the three-knob search returns shifts of ±0.27 with addenda of 1.01 and 0.46, which looks like a lopsided answer to a symmetric question, and its mirror image is exactly as good. It is barely better than the symmetric answer. Leaving both gears unshifted and cutting both tips back to 0.74 of a module gives 2.150 against the lopsided design’s 2.148, the same to a tenth of a per cent. On equal gears there is nothing for the balancing shift to balance, and the whole of the gain from 4.095 to 2.15 is the tips coming in until the contact ratio reaches its floor. The shift was never the knob that mattered there; the addendum was.

Then the centre distance. Let the total shift be positive, so the gears move apart and the operating pressure angle rises, and the worse root comes down again, to 0.834 with shifts no larger than a module. This step is the larger of the two on most pairs, and it is the one the balancing shift cannot take, because moving one cutter out by exactly what the other moves in keeps the centre distance where it was.

Then the shift limit. The larger the shifts allowed, the further the gears can move apart. A quarter of a module on both gears already brings the worse root to 1.092, well below the addenda-only system’s 1.282; half a module, 0.970; a module, 0.834; and the limit stops mattering at about two modules, where the unconstrained optimum’s 0.753 is reached with a wheel shift of 1.66. Most of the gain is in the first half-module of positive shift on both gears. A designer who cannot move the centre distance at all should take the addenda; one who can move it a little should move it.

How much the centre distance adds over the addenda depends on the pair. It adds least on the 20–100 pair, 0.851 to 0.663, a factor of 1.28, where the large wheel’s root was never in much trouble and the addenda had already done most of what could be done. It adds most on the equal pair, 2.148 to 1.013, a factor of 2.12, and on the small pinion, 2.295 to 1.233. Those are the pairs whose roots were worst to start with, and the pairs where a longer line of action has the most to give, since the contact on a short line with a small pinion’s base circle at one end has nowhere to go but towards it.

The contact ratio is what pays #

The pinion’s sliding along its whole contact shows where the gain comes from. The standard pair starts at −11.09, so close to the pinion’s base tangency that its root is ground by a factor of eleven. The balancing shift starts at −1.96. The four-knob designs start at −0.83 and −0.75 and are flatter everywhere, because contact has been pulled into the middle of a longer line. The cost is the contact: the standard pair’s contact ratio is 1.64, the balanced pair’s 1.56, and every four-knob optimum’s exactly 1.20, the floor it was given.

So the floor on contact ratio is the design decision, and the least worst root follows from it. On the 17–51 pair with shifts up to a module, requiring a contact ratio of 1.1 allows 0.735; 1.2 allows 0.834; 1.3, 0.939; 1.4, 1.056; 1.5, 1.277. Each tenth of contact ratio costs about a tenth of worst-root sliding, somewhat more above 1.4. A designer who wants quiet running at a high contact ratio pays in root sliding, and one who wants long-wearing roots pays in smoothness of transfer from tooth to tooth.

Checked two ways #

The specific sliding was computed from the rolling speeds along the line of action. The shift that evens the wear built a second route from the involutes’ own arc lengths — step the pinion by a small angle, measure how far the contact moves along each flank on the sampled curves, and divide the difference by the pinion’s — and at the start of contact of every four-knob optimum it agrees with the formula to $10^{-5}$. The centre distance was taken from the involute equation and checked against the measured zero-backlash distance to $10^{-6}$ of a module. And the conditions were checked on the drawn tooth: tip thickness from the involute at the tip radius, clearance from the root circle the cutter leaves.

What a designer can take #

The balancing shift keeps everything standard but the cutter’s position, so the pair runs on its catalogue centre distance and can be made with standard blanks cut a little differently. That is its virtue, and it is why the earlier essay’s number is the one that appears in handbooks. The four-knob optimum asks for a non-standard centre distance, non-standard tip diameters and a raised pressure angle, and in exchange it cuts the worst root’s sliding by more than half.

Between them sits the addenda-only design, and it is the one a designer with an existing housing can actually use. It keeps the catalogue centre distance, so the bearings stay where they are; the cutter hardly moves, by 0.02 of a module on the 17–51 pair; and what changes is the diameter each blank is turned to before cutting, which costs nothing on the machine. For that it takes a third off the worst root. The remaining factor of 1.3 to 2.1 is the price of a new housing, and whether it is worth paying is a question about the housing rather than about the teeth.

Whether that is worth it depends on what wears the pair out. A pair limited by pitting at the roots, where sliding and rolling combine, gains directly. A pair limited by noise, where a high contact ratio matters more, does not, since the gain is paid for in contact ratio. The measurement cannot choose between them; it can say exactly what each choice costs.

What the search does not settle #

Sliding is a proxy for wear. Specific sliding is the classical measure, but wear also depends on the contact pressure, which a raised pressure angle changes, and on the lubricant film.

One pressure angle of cutter. Every design here is cut with a 20° rack. A cutter of another standard pressure angle starts from a different line of action.

Root strength. A larger total shift thickens both teeth at the root, which strengthens them, and a shorter addendum lowers the bending moment. Neither is computed; the tip-thickness condition is the only strength check.

Still open: a shift that is not the same across the face #

Every design here has one shift across the whole face of each gear. A helical pair, or a spur pair with lead crowning, has a line of contact that sweeps across the face, and contact that runs along the tooth found that the line’s own position along the face changes where on the profile it bears.

The distinct argument would be a profile modification that varies with position across the face, chosen so that the part of the face that sees contact nearest the base tangency has its own tip shortened. The question would be whether a helical pair, whose contact is already spread along the line of action at any instant, can reach the four-knob optimum’s root sliding without giving up the contact ratio that this essay’s pairs had to spend.