BIZOMAT MAKEOS · Mechanics
Belt, Rope and Band Friction (Capstan)
What is the most tension the tight side can hold for this wrap and friction?
Maximum tight-side tension of a belt, rope or band wrapped around a drum (capstan or Euler–Eytelwein relation), for flat and V-groove contact.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Slack-side (holding) tension | N | Tension on the low-tension side: belt slack-side tension, brake anchor force or holding force on a rope. |
| Coefficient of friction | Friction coefficient between belt, rope or lining and the drum, from the supplier or tests. | |
| Wrap angle | ° | Angle of contact around the drum (180° for equal pulleys; more for several turns). |
| Groove included angle (0 = flat) | ° | Included angle of a V-groove (e.g. 34–40° for V-belts); 0 for flat contact. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Slack-side (holding) tension
- What it is
- The tension on the low-tension side of the belt, rope or band: the slack-side belt tension, the anchor force of a band brake, or the hand force holding a rope on a capstan.
- Units and format
- Enter a number in N: newtons. The form accepts 0.000001 N or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The tight-side tension rises in proportion to it.
- Get it right
- This is the smaller of the two tensions; the tool calculates the larger, tight-side tension from it.
What this means: Coefficient of friction
- What it is
- The coefficient of friction between the belt, rope or lining and the drum or pulley it wraps around.
- Units and format
- Enter a plain number with no unit. The form accepts 0.000001 to 3. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The tension ratio grows exponentially with friction times wrap angle, so small changes matter.
- Get it right
- Take it from the supplier or tests and use a conservative value for wet, oily or worn surfaces.
What this means: Wrap angle
- What it is
- The angle of contact between the belt or rope and the drum, measured around the drum. Several full turns add 360° each.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.000001° to 36000°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 180° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The tension ratio grows exponentially with the wrap angle.
- Get it right
- Enter degrees; for a rope with three turns on a capstan that is 3 × 360°.
What this means: Groove included angle (0 = flat)
- What it is
- The included angle of a V-groove that the belt or rope wedges into. Zero means flat contact with no wedging.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0° to 179°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the full included angle of the groove, not the half angle. A V-groove raises the effective friction.
Method and formulas
- Effective friction μₑ = μ for flat contact, or μ ÷ sin(β ÷ 2) in a groove of included angle β.
- T₁ ÷ T₂ = e^(μₑ θ), with θ in radians.
- Transmissible force = T₁ − T₂.
- Example: 200 N slack tension, μ 0.3 and 180° of wrap allow 513.27 N on the tight side.
Limitations
- Centrifugal tension of fast belts is not included and reduces the usable tension.
- Assumes slip is impending; the friction coefficient strongly affects the result.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check slip of a flat or V belt
- Size a band brake or rope brake
- Estimate holding force of a rope around a bollard
Who it is for
- Design engineers
- Maintenance engineers
- Technicians
What you get
- Maximum tight-side tension
- Tension ratio
- Net force (tight minus slack)
- Effective friction coefficient
Interpreting the result
- Keep the working tension ratio well below the slip limit to allow for wear, dirt and lubricant.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 165–168, 2473–2490. Topic reference; no source tables or recommended settings are reproduced.
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