Library / Mechanical / MEC-92
Mechanical
Capstan equation for rope friction
Largest tension a rope or belt can hold on the tight side, for a given slack-side tension, when it wraps a fixed drum or post and is just about to slip.
- Ref
- MEC-92
- Rev
- v1
- Field
- Mechanical
- Result
- Ttight [N]
When a flexible rope or belt wraps a fixed drum or post and is about to slip, the larger (tight-side) tension is the smaller (slack-side) tension times eμθ: Ttight = Tslackeμsθ. The wrap angle θ must be in radians (one full turn is 2π). Equality holds only at impending slip; if the rope is holding without slipping, the tension ratio is at most eμθ. The result is the largest tight-side tension that the given slack-side tension can hold.
Largest tension a rope or belt can hold on the tight side, for a given slack-side tension, when it wraps a fixed drum or post and is just about to slip. The holding force grows exponentially with the wrap angle and the friction coefficient.
Calculator
- Tension ratio, Ttight/Tslack = eμθ2.566
- Number of turns, θ/2π0.5
- Extra factor from one additional full turn, e2πμ6.586
- Slack-side tension needed to hold a 10 kN load3,897 N
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A half turn (3.14 rad) of rope with μ = 0.3 multiplies the holding force by 2.57: a 200 N pull on the slack end holds up to 513.3 N on the tight end.
- Each additional full turn multiplies the capacity again by 6.59, so a few turns on a post let one person hold a very large load: this is how a capstan, a cleat or a bollard works.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N) | Local step | Change in result |
|---|---|---|---|---|---|---|
| θ Total wrap angle | 0.1 to 31.4 rad | 206.1 | 2,467,000 | 2,466,000 | +10% | +9.88 % |
| Tslack Slack-side tension | 1 to 10,000 N | 2.566 | 25,660 | 25,660 | +10% | +10 % |
| μs Static friction coefficient | 0.05 to 1 | 234 | 4,628 | 4,394 | +10% | +9.88 % |
Sweep of Total wrap angle
| Total wrap angle (rad) | Maximum tight-side tension (N) |
|---|---|
| 0.1 | 206.1 |
| 4.012 | 666.5 |
| 7.925 | 2,156 |
| 11.84 | 6,971 |
| 15.75 | 22,550 |
| 19.66 | 72,920 |
| 23.57 | 235,800 |
| 27.49 | 762,700 |
| 31.4 | 2,467,000 |
Sweep one variable
Variables and how each one contributes
Maximum tight-side tension
What the equation gives you. Everything else feeds it.
Slack-side tension
The tension on the slack side, the end held by hand or by the anchor, in N. Raises the holding capacity in direct proportion: the ratio is fixed by friction and wrap, so a stronger pull on the slack end holds a proportionally larger load. Raising it by 20 N (to 220 N) moves the result by +10 %, from 513.3 to 564.6 N.
Static friction coefficient
The static friction coefficient between rope and drum (about 0.2 to 0.5 for dry rope on metal, lower when wet or oily). Raises the holding capacity exponentially: a small gain in friction multiplies the load that can be held. Raising it by 0.05 (to 0.35) moves the result by +17 %, from 513.3 to 600.6 N.
Total wrap angle
The total wrap angle of the rope on the drum, in radians (π for a half turn, 2π for one full turn). Raises the holding capacity exponentially: each extra turn multiplies the holding force by the same factor. Raising it by 0.5 rad (to 3.642 rad) moves the result by +16.2 %, from 513.3 to 596.3 N.
Worked example with the values above
- Exponent: μsθ = 0.3 × 3.142 = 0.9425.
- Tension ratio: e0.9425 = 2.566.
- Ttight = 200 × 2.566 = 513.3 N.
Assumptions and limits
- A flexible rope or belt with negligible bending stiffness and negligible centrifugal effects, on a fixed drum.
- Constant friction coefficient over the whole wrap, with the rope on the point of slipping (the equality case).
Common mistakes
- Entering the wrap angle in degrees instead of radians.
- Swapping the tight and slack sides: the ratio always gives the larger tension from the smaller.
- Using the dynamic (sliding) friction coefficient for a rope that is holding, or the static value for a rope that is sliding.
Statistics and uncertainty
The ratio depends exponentially on μθ, so errors in the friction coefficient are amplified: the relative error of the ratio is about θ times the absolute error in μ, which equals μθ times the relative error in μ. At μθ = 3 a 10 % error in μ gives a 30 % error in the holding force, and μ for rope varies a lot with wetness, wear and speed, so use a low design value.
Where it is used
- Winch and bollard rope holding calculations, and the load a hand-held rope can restrain.
- Belt-drive torque limits and the slack-side tension needed to avoid slip.
Maths you need
- Radians: angle as arc length over radius; one turn is 2π
- Exponential function: ex, which grows ever faster as x rises