MEC-07

Lewis gear-tooth bending

σ = Wt /(F m Y) treating the tooth as a cantilever.

Reading speed

GearsLewis 1892ISO 6336

Governing equation

\sigma=\dfrac{W_t}{F m Y}

$W_t \in \left[50, 20000\right]$

where

W_t: Tangential load (N)
F: Face width (mm)
m: Module (mm)
Y: Lewis factor (—)
\sigma: Bending stress (MPa)

The figure, part by part

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①

Governing law — red

\sigma=\dfrac{W_t}{F m Y}

$W_t \in \left[50, 20000\right]$

$W_t = 50$ $\sigma = 2.778$

Along the curve

②

Domain of the variable — blue

W_t_{\min} = 50,\quad W_t_{\max} = 20000

$W_t \in \left[50, 20000\right]$

$W_t = 50$ $W_t = 50$

Along the curve

③

Response curve — green

\sigma=\dfrac{W_t}{F m Y}

$W_t \in \left[50, 20000\right]$

$W_t = 50$ $\sigma = 2.778$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\sigma}{\mathrm{d}W_t}

$W_t \in \left[50, 20000\right]$

$W_t = 175.5$ $\sigma^{\prime} = 0.056$

Along the curve

Lecture brief

Historical brief

Machine design grew from Coulomb torsion and Hertz contact (1881) through Soderberg fatigue and the heat-engine cycle. The lab writes shaft, bearing, contact and thermodynamic limits in SI. This sheet (MEC-07 — Lewis gear-tooth bending) is the form associated with Lewis 1892 · ISO 6336. Working symbols: $W_t$ , $F$ , $m$ , $Y$ $\rightarrow$ $\sigma$ . Lewis inscribed a parabola in the tooth and used beam theory; Y is the dimensionless Lewis form factor.

Purpose

Purpose: compute

\sigma

from

W_t

F

m

Y

in Mechanical via

\sigma=\dfrac{W_t}{F m Y}

σ = Wt /(F m Y) treating the tooth as a cantilever. Use it when a real mechanical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

W_t = 1200.000\,\mathrm{N}

F = 20.000\,\mathrm{mm}

m = 3.000\,\mathrm{mm}

Y = 0.300\,\mathrm{—}

, the governing relation

\sigma=\dfrac{W_t}{F m Y}

yields

\sigma = 66.667\,\mathrm{MPa}

. Spur gear, static, no dynamic factor Kv. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Tangential load W_t1200 NFace width F20 mmModule m3 mmLewis factor Y0.3 —

Outputs

Bending stress \sigma66.667 MPa

Reading speed

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Narration of this film

Spur gear, static, no dynamic factor Kv.

Lewis inscribed a parabola in the tooth and used beam theory; Y is the dimensionless Lewis form factor.

Reading speed

Watch on YouTube