Barth Dynamic Factor (Gear)
Calculate a gear's dynamic (velocity) factor by Barth's equation, K_v = (6.1 + v) ÷ 6.1, from the pitch-line velocity v (m/s). The dynamic factor amplifies the transmitted static load to account for the DYNAMIC EFFECTS of high-speed meshing: as teeth engage and disengage rapidly, profile imperfections, pitch errors, tooth deflections under load and inertia generate VIBRATIONS and impacts that raise the real tooth load above the nominal load from torque. The higher the pitch-line velocity, the greater these effects — so K_v grows with v. Barth's equation (with constant 6.1, in m/s) is one of several empirical dynamic-factor formulas, suited to reasonably accurate cut teeth; variants with different constants exist for cast (coarser) or ground (more precise) teeth. The dynamic design load is the nominal tangential load times K_v. In high-speed gears, controlling vibration (manufacturing accuracy, modified profiles, balancing) is essential to limit K_v and noise. It is a key factor in AGMA design. Enter the pitch-line velocity.
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Barth dynamic factor (gear)
The dynamic factor (velocity factor) of a gear, from the Barth equation, is K_v = (6.1 + v) ÷ 6.1, computed from the pitch line velocity v (m/s). It scales up the static transmitted load to account for the dynamic effects of meshing at high speed: as the teeth engage and disengage in rapid succession, profile imperfections, pitch errors, tooth deflection under load and inertia generate vibration and impact that drive the real load on the teeth above the nominal value derived from torque. The higher the pitch line velocity, the stronger those effects — which is why K_v grows with v. The Barth equation (with its 6.1 constant, in m/s) is one of several empirical dynamic factor formulas, suited to teeth cut with reasonable accuracy; variants with different constants exist for cast teeth (coarser) and for ground teeth (more precise, giving a lower K_v). The design dynamic load is the nominal tangential load multiplied by K_v. In high-speed gearing, keeping vibration under control (manufacturing accuracy, profile modification, balancing) is essential to limit both K_v and noise. It is a key factor in AGMA gear rating. Enter the pitch line velocity.
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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 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.
Rubber Apparent Compression Modulus
Computes the apparent compression modulus of a rubber block squeezed between two bonded plates, Ec = E₀ × (1 + 2 × k × S²), where E₀ is the Young's modulus of the compound, k is a numerical constant tabulated by hardness (running from about 0.93 for soft 30 IRHD rubber to 0.53 for hard 75 IRHD) and S is the shape factor, which for a circular block equals the diameter divided by four times the thickness. Rubber is essentially incompressible in volume, so what resists the load is not compression of the material but friction on the bonded faces, which stops the sides from bulging; that is why the apparent modulus can sit many times above the compound's own Young's modulus. The result, in megapascals, is what you use to get the vertical stiffness of the mount (stiffness = Ec × area ÷ thickness) and from there the natural frequency of the isolated system. Because the shape factor is S = D ÷ 4t and the dominant term goes with S squared, diameter and thickness are levers of equal weight and opposite sign: halving the thickness and doubling the diameter do exactly the same thing, and in the example both take the apparent modulus from 11.3 to 35.3 MPa — a thin mount is a stiff mount, and that ruins vibration isolation. Enter the compound's Young's modulus, the constant k, the block diameter and the thickness.
Half-Wave Dipole Length
Calculate the physical length of a half-wave dipole, L = (150 ÷ f)·VF, from the frequency f (MHz) and the velocity factor VF (typically ~0.95 for wires, correcting the end effect). The result, in metres, is the total length of the dipole antenna resonant at the desired frequency — each arm is half this value. The half-wave dipole is the most used reference antenna, with 2.15 dBi gain. The velocity factor makes the antenna slightly shorter than a half wavelength in vacuum. Enter the frequency and the velocity factor.
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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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