Band Brake Torque⚠ unverified
Mechanical / Clutches Brakes · Compute the braking torque of a band brake
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| F | F | N | 1.0 | Slack-side tension T2 |
| mu | μ | — | 1.0 | Coefficient of friction between band and drum (dimensionless) |
| R | R | m | 1.0 | Drum radius |
| theta | θ | — | 1.0 | Wrap angle of the band around the drum, in degrees |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | T | N*m | Braking torque, in newton-metres (N*m) |
The science & history
Understanding the Parameters
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Slack-side tension $F = T_2$ — the tension applied at the "loose" end (often by the actuating lever). The tight-side tension follows from the capstan ratio, $T_1 = T_2 e^{\mu\theta}$, so the braking torque grows with $T_2$ and, exponentially, with the wrap and friction.
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Friction $\mu$ and wrap $\theta$ — enter through the capstan factor $(e^{\mu\theta} - 1)$. More wrap or friction multiplies the tight-side tension and hence the braking force for the same actuating tension — the friction amplification that makes band brakes powerful and compact.
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Drum radius $R$ — the lever arm converting the net band force into torque; larger drums brake harder for the same tension but need more space.
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Direction matters — which end is "tight" depends on the direction of drum rotation. A band brake can be self-energizing (rotation drags the band tighter, boosting braking) in one direction and weaker in the other; some designs exploit this, others avoid it for symmetric braking.
Derivation (Approaching a Proof)
The band contacts the drum over the wrap angle $\theta$, and friction acts tangentially along the whole arc. Summing (integrating) that distributed friction is equivalent to taking the difference of the band tensions at the two ends — an elegant consequence of the tension building up around the wrap. From band equilibrium, the net tangential friction force on the drum is
$$F_{\text{friction}} = T_1 - T_2.$$
This force acts at the drum surface, radius $R$, so the braking torque is
$$T = (T_1 - T_2)\,R.$$
Now substitute the capstan relation $T_1 = T_2 e^{\mu\theta}$ (see Band Brake Tension Ratio) and write $T_2 = F$ (the slack-side / actuating tension):
$$T = (T_2 e^{\mu\theta} - T_2)\,R = T_2 R\,(e^{\mu\theta} - 1) = F R\,(e^{\mu\theta} - 1).$$
The factor $(e^{\mu\theta} - 1)$ is the friction amplification: it is how much extra tension the wrap develops beyond the slack side. A larger wrap or friction makes it grow exponentially, so band brakes get a lot of braking from a small applied tension.
Dimensional check. $(e^{\mu\theta} - 1)$ is dimensionless, so $[T] = \text{N}\cdot\text{m}$. ✓
History and Development
Band brakes are among the oldest brake types (hoists, winches, early automobiles, bicycles) precisely because the capstan effect gives large braking from simple hardware. The tension-difference torque and the self-energizing directionality are standard in Shigley and brake-design references. Modern vehicles use drum and disk brakes, but band brakes persist in winches, chainsaws, and some automatic-transmission bands.
Related Concepts: Band Brake Tension Ratio, Band Brake Actuating Force, Belt Tension Ratio, Brake Stopping Distance, Friction
Notes: $F$ is the slack-side tension $T_2$; $\theta$ in radians. Self-energizing depends on rotation direction and geometry. Combine with Band Brake Actuating Force for the required lever force.