Tooth Thickness at Pitch Circle
Calculate the tooth thickness measured at the pitch circle of a standard gear, s = (π·m) ÷ 2, from the module m (mm). In a standard (uncorrected) gear, the circular pitch (the distance from one tooth to the next, along the pitch circle) is p = π·m, and it splits equally between the TOOTH (the solid part) and the SPACE (the gap between teeth): half each, hence s = π·m/2. This equality between tooth thickness and space width is what lets two standard gears of the same module mesh perfectly, with one's tooth fitting the other's space with proper clearance. Tooth thickness is a fundamental parameter: it sets the tooth STRENGTH (thicker teeth resist bending more) and the mesh backlash. In CORRECTED gears (with profile shift, used to avoid interference in small pinions, adjust center distance or balance pinion-gear strength), the pitch-circle tooth thickness DIFFERS from π·m/2 — it increases in a positively corrected pinion (strengthening it) and decreases in the gear. Measuring tooth thickness (by the chordal method, with a gear-tooth caliper, or over pins) is a classic gear quality-control check. Enter the module.
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Espessura do dente no primitivo
A espessura do dente medida no círculo primitivo de uma engrenagem padrão é s = (π·m) ÷ 2, a partir do módulo m. Em uma engrenagem de dentes padrão (sem correção), o passo circular (a distância de um dente ao próximo, medida ao longo do círculo primitivo) é p = π·m, e ele se divide igualmente entre o dente (a parte cheia) e o vão (o espaço entre dentes): metade para cada, daí s = π·m/2. Essa igualdade entre espessura do dente e largura do vão é o que permite que duas engrenagens padrão do mesmo módulo engrenem perfeitamente, com o dente de uma encaixando no vão da outra com folga adequada (o backlash). A espessura do dente é um parâmetro fundamental: define a resistência do dente (dentes mais espessos resistem mais à flexão) e a folga de engrenamento. Em engrenagens corrigidas (com deslocamento de perfil, usadas para evitar interferência em pinhões pequenos, ajustar a distância entre centros ou equilibrar a resistência entre pinhão e coroa), a espessura no primitivo é diferente de π·m/2 — aumenta no pinhão corrigido positivamente (fortalecendo-o) e diminui na coroa. A medição da espessura do dente (pelo método da corda, com paquímetro de dente, ou por sobre-pinos) é uma verificação clássica de controle de qualidade de engrenagens. Informe o módulo.
Related Tools
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.
Gear Torque
Calculate the torque transmitted by a gear, T = (F_t·d) ÷ 2000, from the tangential force F_t (N) and the pitch diameter d (mm); the result is in N·m (the 2000 converts d/2 from mm to m). Torque is the moment the gear transmits about its axis, and the tangential force F_t acts at the pitch radius (d/2), creating that moment. This relation is the bridge between the POWER/torque side (what the shaft transmits) and the TOOTH-FORCE side (what sizes the strength): from shaft torque, the tooth tangential force (F_t = 2T/d) is obtained, which then feeds the bending (Lewis) and contact (Hertz) calculations. Conversely, given the tangential force, the torque is obtained. In a gear train, torque CHANGES at each stage by the gear ratio (a reduction that multiplies speed by 1/i multiplies torque by i, conserving power minus losses), while power stays roughly constant. So a reducer's last stage (low speed) transmits the HIGHEST torque and needs the most robust gears. Knowing the torque at each gear is essential to size teeth, shafts, keys and bearings. Enter the tangential force and the pitch diameter.
Gear Base Pitch
Calculate the base pitch of an involute gear, p_b = π·m·cos(φ), from the module m (mm) and the pressure angle φ (degrees). The base pitch is the distance between two homologous flanks of consecutive teeth, measured along the base circle (or, equivalently, along the line of action) — different from the circular pitch (π·m), measured on the pitch circle. The base pitch is a FUNDAMENTAL property of involute meshing for an elegant reason: for two meshes to transmit motion correctly, they must have the SAME base pitch — it is the conjugacy condition of involute profiles. Moreover, the base pitch appears directly in the CONTACT RATIO (the average number of teeth in simultaneous contact, found by dividing the line-of-action length by the base pitch): a contact ratio above 1 (ideally above 1.4) ensures there is always at least one tooth pair meshed, transmitting motion continuously and smoothly, without impacts. The base pitch is also the basis of checking gears 'over two pins' or by span measurement (W over teeth), classic dimensional-control methods. It is an essential parameter in gear geometry and metrology. Enter the module and the pressure angle.
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.
Gear Tooth Bending Safety Factor
Calculate the bending safety factor of a gear tooth, FS = σ_perm ÷ σ, from the material's allowable (permissible) bending stress σ_perm (MPa, the bending fatigue strength with its factors) and the acting bending stress σ (MPa, from Lewis/AGMA from the load). It is the final bending-design check: the material strength must exceed the demand with adequate margin. Gears run for millions to billions of cycles, so the allowable stress is the material's bending FATIGUE strength (at the design-life cycle count), adjusted by reliability, temperature and life factors. Required safety factors depend on criticality and uncertainty (typically 1.5-3 for bending). If FS is insufficient, the module is increased (bigger, stronger teeth), the face width, or a better material/heat treatment used. A tooth breaking by bending fatigue is a CATASTROPHIC, sudden failure (unlike pitting, which gives progressive signs), since the broken tooth comes loose and can damage the whole drive — so bending is designed with generous margins. With the pitting (contact) safety factor, it defines the gear's robustness. Enter the allowable stress and the acting stress.
Cylindrical Shell Thickness (ASME)
Calculate the minimum wall thickness of a pressure-vessel cylindrical shell by the ASME Section VIII Division 1 formula, t = (P·r) ÷ (S·E − 0.6·P), from the internal design pressure P (MPa), the internal radius r (mm), the material allowable stress S (MPa) and the welded-joint efficiency E (0-1). The pressure vessel — used in boilers, chemical reactors, heat exchangers, compressed-air and LPG tanks, autoclaves — is a CRITICAL safety component: a failure under pressure can be explosive and catastrophic. So its design is rigorously codified, the ASME BPVC (Boiler and Pressure Vessel Code) being the world's most used. This formula gives the minimum cylindrical-shell thickness to safely resist the circumferential (hoop) stress. The '−0.6·P' term refines the thin-wall formula for moderately thick walls. The joint efficiency E (0.70 to 1.0, per weld type and radiographic-inspection degree) penalizes strength at the welded region — fully radiographed welds have E=1.0, uninspected welds lower E. The corrosion allowance is added to the calculated thickness. This is the central pressure-vessel design calculation, and underestimating is inadmissible. Enter the design pressure, internal radius, allowable stress and joint efficiency.
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