Belt & rope friction calculator (Euler-Eytelwein)
Calculate the wrap friction of a rope, brake band or belt around a drum or bollard. Enter the friction coefficient and wrap angle, specify one of the two tensions – the calculator returns the force ratio e^(mu·alpha), the second tension, the transmissible circumferential force F1 − F2 and optionally the slip safety with traffic-light rating, live with every input.
Belt & rope friction calculator (Euler-Eytelwein)
Model: onset of sliding per Euler-Eytelwein (Coulomb friction, flexible, inextensible rope). Centrifugal forces and belt elasticity are not included.
Results
Calculating …
Formulas and fundamentals
The basis is the belt friction law of Euler and Eytelwein (capstan equation). Between the tight side F1 and the slack side F2, at the onset of sliding, F1/F2 = e^(mu·alpha) holds, with the friction coefficient mu of the rope/drum pair and the wrap angle alpha in radians. The ratio is independent of the drum diameter and grows exponentially with the wrap angle: a few extra turns around a bollard multiply the holdable load, which is why a sailor can restrain a large ship with only a few wraps.
From the ratio, given the larger force F1, the required holding force follows directly as F2 = F1/e^(mu·alpha), or conversely the maximum holdable load F1 = F2·e^(mu·alpha) from the holding force F2. The circumferential or useful force transmitted across the wrap is the difference of the tensions, F_t = F1 − F2 = F1·(1 − e^(−mu·alpha)). It limits the transmissible torque of a belt drive and the braking force of a brake band.
For the check, the available circumferential force F_t is compared with the required useful force F_t,req; the slip safety is S = F_t/F_t,req. If the wrap is insufficient, the rope slips. The effective design parameters are the wrap angle (several turns around a bollard) and the friction coefficient (dry, rough pairing). The model applies to the onset of sliding with Coulomb friction on a flexible, inextensible rope; centrifugal forces at high belt speed and belt elasticity (creep) are not included.
Worked example
A rope is laid with half a wrap (alpha = 180° = pi) around a bollard, the friction coefficient is mu = 0.3. The force ratio is therefore e^(0.3·pi) = 2.566. The tight side carries a force of F1 = 1000 N.
To hold the load, a holding force of only F2 = 1000/2.566 = 389.7 N is needed on the slack side. The circumferential force transmitted across the wrap is F_t = F1 − F2 = 610.3 N.
Wrapping the rope once fully around (alpha = 360°) raises the ratio to e^(0.6·pi) = 6.586; the required holding force drops to 152 N. This shows the exponential effect of the wrap angle: each additional turn reduces the required holding force by the same factor.
Frequently asked questions
What do tight side and slack side mean?
The tight side F1 is the larger of the two rope tensions (the load to hold or pull), the slack side F2 the smaller holding force at the other end. Always F1 ≥ F2, and at the onset of sliding F1/F2 = e^(mu·alpha).
Why is the ratio independent of the drum diameter?
Only the wrap angle matters, not the length of rope in contact. For a smaller diameter the normal force per arc length increases, but the contact length decreases in the same proportion – the two cancel. That is why only the angle alpha appears in the Eytelwein equation.
How do I handle several wraps?
The wrap angle adds up: one full turn is 360°, two turns 720° and so on. In radians alpha = 2·pi·n for n turns. Because of the exponential function the holdable ratio grows by the same factor e^(2·pi·mu) with every turn.
Does the law also apply to belt drives?
Yes, the same ratio governs the transmissible circumferential force at the onset of slipping. At high belt speed, however, centrifugal forces reduce the contact pressure and belt elasticity causes creep. This calculator treats the friction-limited case without centrifugal or elasticity effects.
What friction coefficients are typical?
Steel rope on a steel drum about 0.1 to 0.2, hemp or synthetic rope on steel 0.2 to 0.3, brake band with lining 0.3 to 0.5, flat belt on cast iron 0.3 to 0.4. Moisture, grease or polished surfaces lower the value considerably; when in doubt assume a conservative (small) value.