Gear Tooth Dynamic (Barth) Bending Stress
What gear-tooth-bending-stress says it leaves out, in its own words: the Barth velocity factor, without which the Lewis number "runs optimistic at speed." Kv = (1200 + V)/1200 for cut or milled teeth and (600 + V)/600 for cast or crude ones, V being pitch-line feet per minute. It also starts a step earlier, from horsepower and rpm rather than a tangential load you had to work out first: 4 HP at 1,000 rpm on a 43-tooth 8-pitch gear is 252.1 in-lb, 93.8 lb tangential, and 1,407 ft/min -- where Kv is 2.17, so the real bending stress is 8,152 psi against a static Lewis 3,752. The velocity more than DOUBLES it. An idler adds 1.42 for fully reversed bending, since it is pushed one way by the driver and the other by the driven gear. Static stress is delegated to the landed Lewis tile so the two cannot drift. Barth is the conservative ancestor of the AGMA dynamic factor; pitting often governs before bending does.
Formula and source
D = T / Pd; torque = 63,025 HP / rpm (in-lb); Wt = 2 x torque / D; V = pi D N / 12 (ft/min); Kv = (1200 + V)/1200 for cut or milled teeth, (600 + V)/600 for cast or crude; static sigma = Wt Pd / (F Y) delegated to the landed Lewis tile; dynamic sigma = static x Kv x 1.42 when the gear is an idler; allowable = Sut/3 when a material strength is entered.
Barth velocity factor applied to the Lewis (1892, public domain) bending equation, as taught in the standard machine-design treatment. The worked example this tile reproduces exactly - a 43-tooth, 20 degree full involute, 8 diametral pitch, 0.5 in wide pinion transmitting 4 HP at 1,000 rpm, giving 8,152 psi - is from W. H. Dornfeld, ME312 Tooth Strength notes (Fairfield University, 2006), following Hamrock Eqs. 14.55 and following.
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