Koepe Mine Hoist

What is modelled

A Koepe hoist carries the cage and a counterweight on ropes that pass over a friction sheave. Only friction on the sheave transmits the motor force, so the tension ratio between the two rope sides is limited.

Equations

• Accelerating up: Tcage = mup(g + a), Tcw = mcw(g - a); motor force F = Tcage - Tcw + J a / r2.
• Slip condition (capstan): Thi/Tlo ≤ eμ θ, θ = 180°.
• Trapezoid profile: t = L/v + (v/2)(1/a + 1/d).
• Energy per trip: W = (mup - mcw) g L.

Checks run in the page

• Simulated trip time against the trapezoid formula.
• Integral of motor power against the potential energy change.
• Slip appears below the critical friction ln(T_hi/T_lo)/π and not above.
• Launch tension equals m(g + a).

Honest simplifications

Rope mass and stretch, rope-length changes, drum and sheave bearing friction, motor and brake dynamics and guide friction are neglected; the friction coefficient is a single constant. Real winders also model rope creep and safety brakes. Educational illustration only.

• Preset "Emergency brake": an empty cage and a heavy counterweight slip on the sheave.
• Raise the balance factor and watch the motor power peak move between the up and down trips.
• Lower the friction coefficient to see the safe tension-ratio bar shrink.