Gear Tooth Dynamic (Barth) Bending Stress
Gear tooth bending stress WITH the Barth velocity factor, without which the Lewis number runs optimistic at speed.
Example
You enter
- Transmitted horsepower 4
- Gear speed (rpm) 1000
- Number of teeth 43
- Diametral pitch Pd (teeth per inch) 8
- Face width F (in) 0.5
- Tooth system 20-full-depth
- Lewis Y override (0 = derive from the tooth system) 0.4
- Tooth quality cut
- Is this gear an idler? no
You get
- Pitch diameter (in) 5.375
- Torque (in-lb) 252.1
- Wt (lb) 93.8047
- Velocity (fpm) 1407.17
- Barth velocity factor Kv 2.17264
- Static stress (psi) 3752.19
- Dynamic stress (psi) 8152.16
Checked against published worked examples from: Lewis (1892) form factor with the Barth velocity factor.
Details, formula, and sources
ScopeKv = (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 (1/2 in face, a tabled Y of 0.4 entered; the computed Lewis Y of 0.417 gives 3,598 and 7,817). 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. The static Lewis stress is computed by the same shared routine, so the two cannot drift. Barth is the conservative ancestor of the AGMA dynamic factor; pitting often governs before bending does.
Formula or tableD = 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.
Source and editionBarth 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.
Free accessLewis is public domain and the Barth factor is published in free university course notes and machine-design references. No AGMA table, quality-number chart, or geometry-factor curve is reproduced here - none is used.
What governsEstimate. AHJ and licensed professional govern.
API field nameshorsepower, rpm, number_of_teeth, diametral_pitch_1_in, face_width_in, tooth_system, y_diametral_override, tooth_cut, is_idler, pitch_diameter_in, torque_inlb, wt_lb, velocity_fpm, kv, static_stress_psi, dynamic_stress_psi
- Barth velocity factor (1200 + V)/1200 cut or milled; (600 + V)/600 cast or crude; V in ft/minBarth, as published in standard machine-design references
- Load from power torque = 63,025 HP / rpm; Wt = 2 torque / D; V = pi D N / 12elementary kinematics
- Static stress delegated to computeGearToothBendingStress, not reimplementedlanded sibling tile
- Idler factor 1.42 for fully reversed bendingstandard machine-design treatment
- Allowable Sut/3, a rough estimate when no material allowable is availablestated scope limit
- Not modeled AGMA Ka, Ks, Km, KB, the geometry factor J, and surface durability (pitting)stated scope limit