Band Brake Tension Ratio⚠ unverified
Mechanical / Clutches Brakes · Compute the tension ratio across a band brake
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| mu | μ | — | 1.0 | Coefficient of friction between band and drum (dimensionless) |
| theta | θ | — | 1.0 | Wrap angle of the band around the drum, in degrees |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | T1/T2 | — | Ratio of tight-side to slack-side tension T1 / T2 (dimensionless) |
The science & history
Understanding the Parameters
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Friction coefficient $\mu$ — appears in the exponent, so friction has an outsized effect: because the tension amplifies at every increment of contact, a modest $\mu$ compounds around the wrap. Doubling $\mu$ squares the tension ratio.
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Wrap angle $\theta$ — the total angle the band contacts the drum, in radians. Also in the exponent, so more wrap multiplies holding power exponentially: at $\mu = 0.3$, a half-wrap ($\pi$ rad) gives a ratio $\approx 2.6$, a full wrap ($2\pi$) gives $\approx 6.6$, and two full turns ($4\pi$) gives $\approx 43$. This exponential is why extra turns around a capstan are so effective.
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The ratio $T_1/T_2$ — the "friction multiplication." The difference $T_1 - T_2$ is the friction force that produces braking torque (see Band Brake Torque); a large ratio means a small slack-side (actuating) tension can develop a large braking force.
Derivation (Approaching a Proof)
Consider a small element of the band subtending an angle $\mathrm{d}\theta$ at the drum centre, with tension $T$ on one side and $T + \mathrm{d}T$ on the other. Three forces act on it: the two tensions and the drum's reaction (normal force $\mathrm{d}N$ plus friction $\mu\,\mathrm{d}N$).
Radial (normal) equilibrium. The two tensions, each inclined by $\mathrm{d}\theta/2$, press the element onto the drum with a net radial force $T\,\mathrm{d}\theta$ (using $\sin(\mathrm{d}\theta/2)\approx \mathrm{d}\theta/2$):
$$\mathrm{d}N = T\,\mathrm{d}\theta.$$
Tangential equilibrium. The tension difference is balanced by friction (which acts to oppose slipping, on the side of the higher tension):
$$\mathrm{d}T = \mu\,\mathrm{d}N = \mu T\,\mathrm{d}\theta.$$
Separating variables and integrating from the slack side ($T_2$, $\theta = 0$) to the tight side ($T_1$, $\theta$):
$$\int_{T_2}^{T_1}\frac{\mathrm{d}T}{T} = \int_0^{\theta}\mu\,\mathrm{d}\theta \;\Longrightarrow\; \ln\frac{T_1}{T_2} = \mu\theta \;\Longrightarrow\; \frac{T_1}{T_2} = e^{\mu\theta}.$$
The exponential arises because each increment of tension is proportional to the current tension (a constant-fraction growth per unit angle) — the same mathematics as compound interest. It assumes the band is about to slip (fully developed friction) and is inextensible.
Dimensional check. $\mu\theta$ is dimensionless (with $\theta$ in radians), so $T_1/T_2$ is dimensionless. ✓
History and Development
The capstan equation is attributed to Leonhard Euler (1762) and Johann Albert Eytelwein (1808). It underlies band and drum brakes, flat- and V-belt drives (Belt Tension Ratio), rope-and-capstan rigging, and climbing/sailing hitches. It is one of the most consequential simple results in mechanical engineering — a genuinely exponential mechanical advantage from friction alone.
Related Concepts: Band Brake Torque, Band Brake Actuating Force, Belt Tension Ratio, V Belt Contact Angle, Friction
Notes: $\theta$ in radians (see note). Assumes impending slip and an inextensible band. The tension difference $T_1 - T_2$ produces braking torque; self-energizing effects depend on rotation direction and lever geometry.