CAPSTAN EQUATION (EYTELWEIN)

The Capstan equation — also called the belt friction equation or Eytelwein's formula after the 19th-century German engineer Johann Albert Eytelwein — is the fundamental relationship between the tight-side tension T1 and the slack-side tension T2 of a belt wrapped around a fixed or driven cylinder. At the threshold of slip, the ratio T1/T2 grows exponentially with the product of the friction coefficient μ and the wrap angle θ (in radians): T1/T2 = e^(μθ).

Applied to a conveyor drive pulley, the equation answers the central drive-design question: 'how much Te can this drum transmit before the belt slips?' Te equals T1 − T2, so Te_max = T2 × (e^(μθ) − 1). For a single drive pulley with 180° wrap (θ = π ≈ 3.14) and rubber lagging giving μ = 0.35, Te_max = 2.0 × T2. With ceramic lagging at μ = 0.45 the same drum delivers Te_max = 3.1 × T2. With a dual-drive (180° + 180° = 360° wrap on two drums in series), the multiplier rises to about 8.8 × T2 — which is why high-power overland systems use two or three drive pulleys to spread the duty.

The equation works in either direction. Designers use it forward to compute the required slack-side tension T2 (and therefore the take-up force) for a given Te demand and lagging choice. They use it backward at commissioning to verify whether observed T1/T2 ratios leave adequate margin against slip — typically a factor of 1.3–1.5 is recommended in DIN 22101. Note that the equation assumes uniform friction across the wrap; in reality, surface contamination, water film or worn lagging can locally drop μ and trigger slip well before the calculated limit.