Design the band brake that stops a rotating drum by wrapping a flexible band around it. Adjust the drum diameter, lever operating force, wrap angle and friction coefficient to see the tight-side and slack-side tension and the braking torque produced by the exponential belt-friction effect, all in real time.
Parameters
Drum diameter D
mm
Lever operating force P
N
Hand force applied at the lever end
Lever length a
mm
Distance from the pivot to the operating force
Band-end mounting distance b
mm
Distance from the pivot to the slack band end
Friction coefficient μ
Friction between band lining and drum
Wrap angle θ
°
Angle over which the band hugs the drum
Results
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Braking torque T_b (N·m)
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Tight-side tension T₁ (N)
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Slack-side tension T₂ (N)
—
Tension ratio T₁/T₂
—
Wrap angle (rad)
—
Self-amplification e^(μθ)
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Band brake mechanism — rotation animation
The band wraps the rotating drum over the wrap angle θ. The thick arrow is the tight-side tension T₁, the thin arrow the slack-side tension T₂. The lever applies the operating force P about the pivot.
The tension ratio comes from the belt-friction (capstan) equation, and the slack-side tension T₂ from the lever moment balance. μ: friction coefficient, θ: wrap angle (rad), P: lever operating force, a: lever length, b: band-end mounting distance.
$$T_{brake}=(T_1-T_2)\cdot r$$
Braking torque. T₁: tight-side tension, T₂: slack-side tension, r: drum radius. θ is the wrap angle converted to radians, and r is half the drum diameter.
What is the Band Brake Torque Simulator?
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A "band brake" is the kind of rear brake you find on a bicycle, right? It just wraps a band around something — why does it grip so well?
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Exactly — the kind on a city bike's rear wheel. You also see it on winches, hoists and farm machines. The mechanism is simple: a flexible "band" is wrapped around a rotating "drum", and the two ends are pulled to clamp it. The key is that it isn't just clamping — an exponential effect called "belt friction" comes into play.
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Belt friction…? You mean the pulling force on the band grows as it wraps around?
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That's exactly it. Call the tension at one band end (the slack side) T₂ and the other (the tight side) T₁. Their ratio is T₁/T₂ = e^(μθ), where θ is the wrap angle and μ the friction coefficient. Because it sits in the exponent of e, the more you wrap the band, the more explosively the ratio grows. It's the same principle a sailor uses when a few turns of a thin rope around a bollard hold a huge ship.
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So I should just make the wrap angle as large as possible. When I set "wrap angle θ" on the left to 270°, the tension ratio jumped above 5×.
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Good catch. Just going from 90° to 270° raises the tension ratio from about 1.7 to about 5.2 for μ=0.35. That means the tight side T₁ can clamp the drum with more than five times the slack-side tension T₂. Look at the "Braking torque vs wrap angle" chart below and you'll see that steep curve. That's why a band brake is a "compact and powerful" device — it produces a large braking torque from only a small lever force.
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But how is the slack-side T₂ decided? You just squeeze the lever, right?
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In a simple band brake, the slack end of the band is anchored at a distance b from the lever pivot. When you push the lever end with force P, the moment balance about the pivot gives P·a = T₂·b, so T₂ = P·a/b, where a is the lever length. A large lever ratio a/b lets a light operating force build a big T₂. That T₂ is then multiplied by the belt-friction factor e^(μθ) to become T₁, and the braking torque T_b = (T₁ − T₂)·r is the final result.
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If it grips that well, why do cars use disc brakes? Why not band brakes?
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A band brake has a weakness: its grip changes with the direction of rotation. In one direction belt friction self-amplifies and it brakes hard, but in the reverse direction the tight and slack sides swap and the grip weakens. A car drives forwards and backwards, so direction-dependent braking is a problem. That's why band brakes survive in applications like winches, hoists and farm machines, where you mainly need to reliably stop one-way rotation.
Frequently Asked Questions
A band brake wraps a flexible band around a drum and brakes through the tension difference between the two band ends. The ratio of the tight-side tension T1 to the slack-side tension T2 is set by the belt-friction (capstan) equation T1/T2 = e^(μθ), where μ is the friction coefficient and θ is the wrap angle in radians. The braking torque is T_brake = (T1 − T2)·r, where r is the drum radius. This tool finds T2 from the lever moment balance and computes the braking torque from these formulas.
The tension ratio T1/T2 = e^(μθ) grows exponentially with the wrap angle θ. Tripling θ from 90° to 270° raises the tension ratio from about 1.73 to about 5.2 for μ=0.35. Because the band tension builds up exponentially as it wraps around the drum, the tight side can carry a large tension from only a small lever force. This is exactly what makes a band brake compact and powerful.
In a simple band brake the slack end of the band is attached at a distance b from the lever pivot. From the lever moment balance P·a = T2·b, the slack-side tension is T2 = P·a/b, where P is the lever operating force and a is the lever length. Once T2 is known, the tight-side tension is T1 = T2·e^(μθ). A larger lever ratio a/b gives a larger T2 from a smaller operating force.
A band brake wraps a flexible band around the outside of a drum and obtains a large braking torque from the exponential belt-friction effect. It has few parts, a simple structure, and can use a large wrap angle, so it produces high torque even in a small package. However, its self-amplification depends on the direction of rotation — it brakes strongly in one direction and weakly in the reverse direction. It suits applications that mainly stop one-way rotation, such as winches, hoists and agricultural machines.
Real-World Applications
Winches, hoists and cranes: A winch or hoist lifting a heavy load needs a brake to hold the drum so the load's own weight cannot back-drive it. The band brake has a simple structure and can use a large wrap angle, so it delivers a large holding torque in a compact size. It is also used as a control brake to regulate the speed at which the load is lowered.
Bicycles, agricultural and small-power machinery: The rear band brake of a city bicycle has an enclosed structure that resists rain and needs little maintenance. Tillers, brush cutters, small winches and similar machines also use band brakes widely where reliable braking is needed at a low cost.
Emergency and holding brakes: Elevator hoists and conveyor drives include emergency brakes that reliably stop rotation during a power failure or fault. Making it a "negative brake" — a spring clamps the band and an electromagnet releases it — gives a fail-safe design in which the brake automatically applies when power is lost.
Learning and first-pass sizing in machine design: The band brake always appears in machine-design courses as a vivid demonstration of the belt-friction formula e^(μθ). Before running a detailed friction-heat or contact analysis, an estimate like this tool helps you decide "which wrap angle and lever ratio give the required torque" and sets the starting point for the design.
Common Misconceptions and Pitfalls
The biggest pitfall is forgetting that the grip changes with the direction of rotation. In a band brake the tight and slack sides swap with the rotation direction. The formula in this tool assumes the "effective" direction, where the slack end is tightened by the lever and the tight side is dragged in by the drum's rotation. In the reverse direction the roles of T₁ and T₂ swap, and the same lever force produces a much smaller braking torque. Always state the assumed rotation direction in the design, and if you must brake in both directions, consider a different brake type.
Next, assuming the friction coefficient μ is a fixed value. The μ in this tool is a representative catalogue figure, but the real μ varies strongly with the lining material, drum temperature, moisture or oil, and the progress of wear. In particular, continuous braking heats the drum and lowers μ (brake fade); because e^(μθ) carries that change into the exponent, the braking torque can drop to a fraction of the expected value. For a control brake in continuous use, always analyse the temperature rise and heat dissipation separately.
Finally, looking only at the braking torque and ignoring the band strength. The tight-side tension T₁ is amplified exponentially by belt friction, so a greedy wrap angle puts a very large tensile load on the band itself, the mounting bolts and the pivot pin. Even if the braking torque meets the target, the band will stretch or rupture if T₁ exceeds the band's allowable tensile load. Check the absolute value of T₁ in this tool, and always design the band cross-section and mountings together with the torque.
How to Use
Enter drum diameter (40–300 mm range) to set the braking surface radius.
Set lever arm length (50–500 mm) and apply force (10–500 N) to the lever handle.
Adjust band distance from pivot (10–400 mm) and friction coefficient μ (typically 0.3–0.5 for asbestos-lined bands on steel drums).
The simulator calculates tight-side tension T₁ using lever mechanical advantage, then applies the Capstan equation T₁ = T₂ · e^(μθ) to find slack-side tension T₂.
Braking torque T_b = (T₁ − T₂) × (drum diameter / 2) displays the stopping force in N·m.
Worked Example
Industrial winch brake: drum diameter 120 mm (radius 0.06 m), lever arm 250 mm, applied force 200 N, band distance 80 mm from pivot, wrap angle 180° (π rad), μ = 0.4 for woven asbestos. Mechanical advantage = 250 / 80 = 3.125, so T₁ = 200 × 3.125 = 625 N. Self-amplification e^(0.4×π) ≈ 3.51, giving T₂ = 625 / 3.51 ≈ 178 N. Braking torque = (625 − 178) × 0.06 ≈ 26.8 N·m, sufficient to hold a 300 kg load on a 1.2 m drum.
Practical Notes
Wrap angle critically affects performance: 180° wrap (π rad) is typical for band brakes; increase to 270° (1.5π) for soft or low-friction linings to boost tension ratio.
Friction coefficient varies: asbestos-composite 0.35–0.5, organic resin 0.25–0.35, sintered metal 0.4–0.6; verify material pairing with drum surface.
For safety-critical applications (cranes, hoists), tension ratio T₁/T₂ should exceed 3.5 to ensure robust slip prevention even under wear.
Band slippage occurs when applied lever force insufficient; simulator warns if T₁ drops below ~150 N on industrial drums.