Belt & rope friction calculator (Euler-Eytelwein)

Formulas and fundamentals

Belt friction law per Euler-Eytelwein

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, the following holds:

With the friction coefficient µ of the rope/drum pair and the wrap angle α 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.

Tensions and transmissible circumferential force

From the ratio, given the larger force F1, the required holding force F2 follows directly, or conversely the maximum holdable load F1 from the holding force F2:

The circumferential or useful force transmitted across the wrap is the difference of the tensions:

It limits the transmissible torque of a belt drive and the braking force of a brake band.

Slip safety and model limits

For the check, the available circumferential force F_t is compared with the required useful force F_t,req; the slip safety is:

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 (α = 180° = π) around a bollard, the friction coefficient is µ = 0.3. The force ratio is therefore e^(0.3·π) = 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. Against a required useful force of F_t,req = 500 N the slip safety is S = 610.3/500 = 1.22: with a required safety of S_req = 1.5, half a wrap remains borderline.

Wrapping the rope once fully around (α = 360°), the calculator's start values, raises the ratio to e^(0.6·π) = 6.586; the required holding force drops to 152 N, the transmittable circumferential force rises to 848 N and the slip safety to S = 1.70. This shows the exponential effect of the wrap angle: each additional turn reduces the required holding force by the same factor.

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