Spur Gear Pair (Lewis Bending)

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A spur gear pair is the workhorse of parallel-shaft power transmission: two toothed wheels whose pitch circles roll without slipping, trading speed for torque by the ratio of their tooth counts. The planetary gearset folds three of these meshes into one hub; a single external pair is the atom they are built from. Fix the module $m$ (the tooth size, in mm of pitch diameter per tooth) and the geometry is set: pitch diameter $d = mN$, centre distance $c = m(N_p + N_g)/2$, and gear ratio

$i = \frac{N_g}{N_p} = \frac{\omega_p}{\omega_g}$

Power arrives as torque on the pinion and leaves as a single tangential tooth force at the pitch point. Balance moments about the pinion axis — torque against force times pitch radius — and

$W_t = \frac{2T}{m\,N_p}$

That same $W_t$ pushes back on the gear (action–reaction), so both members carry the identical load. This is the power screw trick in rotary form: a torque becomes a force at a radius, exactly as a screw’s torque becomes an axial thrust.

The Lewis equation, and why the pinion is the weak one

Wilfred Lewis (1892) modelled a gear tooth as a cantilever loaded at its tip, and inscribed the largest parabola of uniform strength inside the tooth outline. The root bending stress that falls out is

$\sigma_b = \frac{K_v\,W_t}{b\,m\,Y}$

where $b$ is the face width and $Y$ is the dimensionless Lewis form factor — everything about the tooth’s shape, distilled to one number. $Y$ grows with the number of teeth: a 12-tooth pinion is stubby and sharply curved ($Y = 0.245$), a 100-tooth gear is nearly a rack and far beefier ($Y = 0.447$). Because $\sigma_b \propto 1/Y$, at identical load, module, and material the pinion always carries the higher stress — it has the fewer teeth, so the smaller $Y$. That is the single most useful fact in gear design: check the pinion; it governs. Dial $N_p$ and $N_g$ in the widget and watch $\mathrm{SF}_p$ sit stubbornly below $\mathrm{SF}_g$ even when both gears are the same steel.

$Y$ is not a formula. It comes from a graphical layout of the tooth (or its digital equivalent) and is published as a table — Shigley’s Table 14-2, for 20° full-depth teeth with the load near the tip. This page carries that table as cited data with provenance: the widget interpolates linearly between the published tooth counts and refuses (per gear, leaving the rest standing) outside the tabulated range of 12–400 teeth, because there is no published value to stand on. The /verification/ page states exactly what is machine-proven here (the interpolation, the refusal, the algebra) and what rests on citation (the tabulated numbers themselves).

The velocity factor

Real teeth engage a hair out of step, so the load a running mesh feels exceeds the static one. Barth’s empirical velocity factor scales the stress up with pitch-line velocity $V = \omega_p\,m\,N_p/2$:

$K_v = \frac{6.1 + V}{6.1}$

(metric, for cut or milled teeth; the $6.1$ carries units of m/s). It is a fit, not a theorem — push $V$ high enough and a banner warns that the fit has left its cited range, the same honesty the belt drive applies to its capstan speed ceiling. Torque, speed, and the tooth bending stress it drives all live on one page here, the way the torsion shaft ties twist to shear.

Try it

Governing relations

Derivation

Steps marked modeling step are where physics enters by citation — every other line is machine-proven to follow from them, and the cited models are independently re-derived in the test pipeline where possible. See what is and isn't machine-verified.

How it fails

The widget answers one question — will a tooth break in bending? — with the oldest model that does so honestly. Real gears fail in several ways the Lewis equation cannot see, and the biggest one is not bending at all.

• Surface fatigue is usually the real limit (the deliberate non-model). Two teeth touch along a line and roll past under enormous contact pressure. The Hertzian contact stress $\sigma_c = C_p\sqrt{W_t\,\ldots}$ — a different stress, on a different surface, checked against a different strength — governs most well-lubricated industrial gears long before bending does. It shows up as pitting: fatigue spalls that pock the flank until the mesh gets noisy and rough. This page deliberately does not compute it. Contact stress needs an elastic geometry factor $I$, the pair’s combined elastic modulus, and a surface endurance strength $S_c$ — a materials column and a mechanism this catalog has not built yet. Reading a green bending margin here as “the gear is safe” is exactly the trap: check pitting separately.

• AGMA is the Lewis equation, grown up. The bending number gears are actually designed to (AGMA 2001 / ISO 6336) is $\sigma = W_t K_o K_v K_s \frac{1}{b m}\frac{K_H K_B}{Y_J}$ — the Lewis stress with the tip-load form factor $Y$ replaced by a geometry factor $Y_J$ (load at the highest point of single-tooth contact, stress concentration included), and a train of factors for overload, size, load distribution across the face, and rim thinness. The velocity factor here is Barth’s crude linear fit; AGMA’s $K_v$ depends on a measured transmission-accuracy grade. Treat this page’s $\sigma_b$ as the right shape, textbook coefficients — good for teaching which knob moves stress which way, not for cutting metal.

• Undercut voids the model quietly. Below the interference-free minimum tooth count (the warn banner, Shigley eq 13-11) a generating cutter gouges the flank near the root — the very place the Lewis parabola measures. The form factor is then fiction. The fix is more teeth, a higher pressure angle, or profile shift; the widget only warns, because the algebra still returns a number.

• Scoring and scuffing are thermal, not structural. At high pitch-line velocity and load the oil film breaks down and flanks weld-and-tear (scuffing). Nothing here — or in Barth’s velocity factor, which merely amplifies the bending stress — sees lubrication regime or flash temperature. The velocity-factor warn marks where even the bending fit stops meaning what it cites.

• The teeth are only as good as their carrier. The tooth force $W_t$ this page computes is the load the torsion shaft must carry as a twisting moment, the reaction the bearings must react, and — stacked through a planetary gearset — the torque that multiplies at every stage. A tooth that survives in isolation can still fail because the shaft deflects and throws the mesh out of alignment, concentrating $W_t$ at one end of the face. The same “a torque is a force at a radius” bookkeeping that starts this page also drives the power screw and every belt drive that feeds a gearbox.

Sources

• Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §13-5/13-7 (involute geometry; interference, eq 13-11), §14-1 (the Lewis bending equation), Table 14-2 (Lewis form factor Y, 20° full-depth, tip load), eq 14-6b (Barth velocity factor, metric).