Capstan Rope Friction

What is modelled

A rope is wrapped by an angle θ around a rough drum. A small element of rope carries tension T and is pressed on the drum with force T dθ. Friction can supply at most μ T dθ of extra tension, so the tension grows along the wrap.

Equations

• Capstan law Tload ≤ Ttail eμθ, θ = 2π × turns.
• Tension profile in the mobilised arc T(s) = Ttail eμ s; required arc θreq = ln(Tload/Ttail)/μ.
• Drum torque τ = (Tload - Ttail) R, R = 0.174 m.
• Beyond capacity the slip speed is an illustrative bounded law (same as the 3D page), not a calibrated rope model.

Checks run in the page

• A numerical integration of dT/dθ = μT against the closed exponential.
• The tension jump from the integral of μT dθ against T_tail(e^μθ - 1).
• Doubling the turns squares the capacity ratio.
• Load just below and just above capacity: grip against slip.

Honest simplifications

Rope is inextensible and massless, friction is dry Coulomb with one coefficient, the drum is rigid and the rope diameter is ignored. Real fibre ropes show stick-slip, creep and temperature effects.

• Raise the load above the capacity bar: the rope starts to slide and its speed falls below the drum surface speed.
• Add one turn and watch the capacity multiply, not add.
• Choose Stationary brake: the drum is still and the rope holds a load several times the tail pull.