Lewis Form Factor⚠ unverified

Understanding the Parameters

• Number of teeth $N$ — the sole input. As $N$ increases the tooth becomes wider at its base relative to its height (less like a slender wedge, more like a stubby block), so its bending section modulus rises and $Y$ climbs — from $\sim0.25$ at $N=12$ toward an asymptote of $\sim0.45$ for large $N$ (20° full-depth teeth). Pressure angle also matters: a 25° tooth has a thicker base and higher $Y$ than a 20° tooth.

• Why it is tabulated — $Y$ comes from inscribing the largest parabola of uniform strength inside the tooth profile (Lewis's construction) and reading off where it is tangent to the tooth fillet. That tangent point depends on the exact involute geometry, so $Y$ is obtained graphically/numerically and tabulated (AGMA 908) rather than written as a simple formula — hence the registry's honest $Y = f(N)$.

Derivation (Approaching a Proof)

Lewis (1892) modelled the gear tooth as a cantilever beam of uniform strength. He inscribed a parabola — the shape of a constant-stress cantilever — inside the tooth, tangent to the tooth fillet at its weakest point. Where that parabola touches sets the critical section thickness $t$ and the load height $h$.

For a cantilever of face width $F$, critical thickness $t$, and load height $h$, the bending stress (Beam Bending Stress) is

$$\sigma = \frac{6 W_t h}{F t^2}.$$

Lewis absorbed the geometry ($h$, $t$) and the pitch into a single dimensionless factor. Writing $t^2/(6h) = $ (pitch)$\times Y$ collapses the geometry into

$$\sigma = \frac{W_t P_d}{F\, Y} \quad(\text{diametral-pitch form; see } Gear Bending Stress Lewis),$$

so $Y$ is precisely the geometry group that converts the cantilever stress into the practical gear equation. Because $h$ and $t$ scale with the tooth profile — set by $N$, the pressure angle, and the tooth system — $Y$ is a pure function of those, i.e. $Y = f(N)$ at a given pressure angle and depth. The value rises with $N$ because more teeth give a proportionally thicker, shorter (stronger) tooth.

Dimensional check. $Y$ is dimensionless by construction (it is a ratio of tooth geometry to pitch). ✓

History and Development

Wilfred Lewis introduced the form factor in his 1892 gear-strength analysis — the first rational tooth-bending method and still the conceptual foundation taught before the more elaborate AGMA bending-strength procedure (which splits $Y$ into a geometry factor $J$ plus load-distribution and size factors). Form-factor tables for standard 14.5°, 20°, and 25° tooth systems appear in Shigley, Dudley's Gear Handbook, and AGMA 908.

Related Concepts: Gear Bending Stress Lewis, Beam Bending Stress, Section Modulus, Thread Bending Stress, Gear Contact Stress

Notes: Tabulated/interpolated function of $N$ (and pressure angle, tooth form) — not a closed formula. Rises with $N$ toward an asymptote; low for small pinions (which usually govern bending). Modern AGMA replaces $Y$ with the geometry factor $J$.