Gear Contact Stress (Hertz)
Calculate the Hertzian contact stress on gear tooth surfaces, σ_H = C_p·√(F_t ÷ (b·d·I)), from the elastic coefficient C_p (√MPa, a function of the pair's elastic moduli), the tangential force F_t (N), the face width b (mm), the pinion pitch diameter d (mm) and the geometry factor I (dimensionless). This is the basis of SURFACE FATIGUE (pitting) design — the second fundamental gear failure mode, distinct from bending breakage. When two teeth touch, the contact is practically a LINE, and even moderate loads create very high contact stresses (hundreds of MPa) in the tiny contact area, per Hertz theory. Under repeated cycles, these stresses cause sub-surface fatigue that flakes off small bits of material, forming craters (pitting) that progress, destroy the tooth profile, generate noise and vibration and lead to failure. The elastic coefficient C_p gathers the materials' elastic properties (steel-steel, steel-bronze, etc.), and the geometry factor I, the curvature and contact ratio. Contact stress is compared with the material's pitting resistance (which depends strongly on surface HARDNESS — so gears are often case-hardened). It is one of the two central AGMA criteria. Enter the elastic coefficient, tangential force, face width, diameter and geometry factor.
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Tensão de contato de engrenagem (Hertz)
A tensão de contato (Hertziana) na superfície dos dentes é σ_H = C_p·√(F_t ÷ (b·d·I)), a partir do coeficiente elástico C_p (função dos módulos de elasticidade dos materiais do par), da força tangencial F_t, da largura da face b, do diâmetro primitivo do pinhão d e do fator geométrico de contato I. Esta é a base do dimensionamento à fadiga superficial (pitting) — o segundo modo de falha fundamental das engrenagens, distinto da quebra por flexão. Quando dois dentes se tocam, o contato é praticamente uma linha, e mesmo cargas moderadas geram tensões de contato altíssimas (centenas de MPa) na pequena área de contato, segundo a teoria de Hertz. Sob ciclos repetidos, essas tensões causam fadiga sub-superficial que arranca pequenas escamas de material, formando crateras (pitting) que progridem, destroem o perfil do dente, geram ruído e vibração e levam à falha. O coeficiente elástico C_p reúne as propriedades elásticas dos materiais (aço-aço, aço-bronze etc.), e o fator geométrico I, a curvatura e a razão de contato. A tensão de contato é comparada com a resistência ao pitting do material, que depende muito da dureza superficial — por isso engrenagens são frequentemente cementadas e temperadas na superfície, ficando duras por fora e tenazes por dentro. É um dos dois critérios AGMA centrais. Informe o coeficiente elástico, a força tangencial, a largura da face, o diâmetro e o fator geométrico.
Related Tools
Gear Tooth Bending Stress (Lewis)
Calculate the bending stress at a gear tooth root by the Lewis equation, σ = F_t ÷ (b·m·Y), from the tangential force F_t (N), the tooth face width b (mm), the module m (mm) and the Lewis form factor Y (dimensionless, a function of tooth count). Lewis's 1892 equation was the first rational treatment of gear-tooth strength and is still the basis of BENDING design. It models the tooth as a cantilever beam fixed at the root: the tangential force transmitted between teeth (from the torque) creates a bending moment that tends to break the tooth at the root — the catastrophic failure where a tooth cracks and snaps. The form factor Y accounts for tooth geometry (gears with more teeth are 'fatter' at the root and stronger, higher Y). The computed stress is compared with the material's bending fatigue strength (with safety factors), since gears endure millions of cycles. The basic Lewis formula is then refined by the AGMA standard with stress-concentration, dynamic, load-distribution and surface-condition factors. It is one of the two fundamental gear design criteria (the other is contact stress). Enter the tangential force, face width, module and Lewis form factor.
Young Equation
Compute γSL = γSV − γLV·cos(θ) from Young's equation.
Gear Dynamic Load
Calculate the effective dynamic load on gear teeth, F_d = F_t·K_v, from the nominal tangential force F_t (N) and the dynamic factor K_v. The dynamic load is the REAL tangential force the teeth bear in operation, larger than the nominal force (simply torque over radius) because of the dynamic effects of meshing at speed. These effects — vibrations, contact impacts, tooth deflections under load and manufacturing errors — make the instantaneous tooth load fluctuate and peak above the average, especially at high pitch-line velocities. The dynamic factor K_v (from Barth or other formulas) quantifies this amplification. The dynamic load is then used in strength checks: in Lewis bending stress (risk of tooth breakage at the root) and Hertzian contact stress (risk of surface fatigue and pitting). Using the nominal load without amplifying by the dynamic factor would underestimate the demands and lead to undersized gears that fail prematurely by fatigue. It is an essential, classic step in gear design. Enter the nominal tangential force and the dynamic factor.
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