Belt Tension Ratio⚠ unverified

Understanding the Parameters

• Friction coefficient $\mu$ — belt-on-pulley friction (leather/steel $\sim 0.3$, rubber $\sim 0.4$). For V-belts the effective friction is much higher because the wedge action multiplies the normal force: replace $\mu$ with $\mu/\sin(\beta/2)$ where $\beta$ is the groove angle, which is why V-belts transmit far more than flat belts of the same size.

• Wrap angle $\theta$ — the contact arc in radians; it appears in the exponent, so more wrap means exponentially more grip. The smaller pulley has the smaller wrap (V Belt Contact Angle) and slips first, so it governs the drive.

• Exponential ratio — because $F_1/F_2 = e^{\mu\theta}$, grip grows multiplicatively with wrap: at $\mu = 0.3$, half a wrap ($\theta = \pi$) gives a ratio of $2.6$, a full wrap $6.6$, two wraps $44$.

• At the slip limit — this is the maximum ratio; below it the belt grips with margin, at it the belt is on the verge of gross slip. Design keeps the operating ratio safely below $e^{\mu\theta}$.

Derivation (Approaching a Proof)

Take an infinitesimal belt element subtending angle $d\theta$ on the pulley, carrying tension $T$ on one side and $T + dT$ on the other. Two force balances:

• Radial: the tension provides the centripetal (inward) direction; the pulley pushes out with normal force $dN = T\,d\theta$ (the two tension components $T\sin(d\theta/2) \approx T\,d\theta/2$ on each side sum to $T\,d\theta$).

• Tangential: the tension difference is resisted by friction at the slip limit, $dT = \mu\,dN = \mu T\, d\theta$.

Separating variables and integrating over the wrap from $F_2$ to $F_1$:

$$\frac{dT}{T} = \mu\,d\theta \;\Longrightarrow\; \int_{F_2}^{F_1}\frac{dT}{T} = \int_0^\theta \mu\,d\theta \;\Longrightarrow\; \ln\frac{F_1}{F_2} = \mu\theta \;\Longrightarrow\; \frac{F_1}{F_2} = e^{\mu\theta}.$$

The result is independent of pulley diameter — only the wrap angle and friction matter.

Dimensional check. $\mu$ and $\theta$ (radians) are dimensionless, so the exponent and the ratio $F_1/F_2$ are dimensionless — a pure ratio, as required.

History and Development

The capstan equation was derived by Leonhard Euler (and independently Johann Eytelwein) in the 18th century, and it governs every friction drive that wraps a flexible element around a drum: belts, rope-and- bollard, band brakes (Band Brake Tension Ratio is the identical relation), and winches. The exponential "mechanical advantage of wrap" is the reason a sailor can hold a ship's line with a few turns around a capstan, and the reason V-belt drives — which amplify the effective $\mu$ — dominate power transmission.

Related Concepts: Belt Power, Maximum Belt Power, Band Brake Tension Ratio, Friction, V Belt Contact Angle, V-Belt Design

Notes: Capstan / Euler–Eytelwein equation; $\theta$ in radians. Independent of pulley diameter. V-belts use effective $\mu/\sin(\beta/2)$ (wedge action). Smaller pulley (smaller wrap) governs slip. Identical to Band Brake Tension Ratio. Ignores centrifugal tension (Centrifugal Tension) at high speed.