Treadwheel Crane Lab

What is modelled

People walking up the inside of a big wheel turn it by their weight, which is why a treadwheel works like a gigantic gear: the large wheel radius is the long lever arm, the small drum the short one, and a block and tackle adds more mechanical advantage. The wheel turns slowly because a walker cannot go faster than a free walking speed, and the more load there is the slower it goes.

Equations

• Foot force F = mw g sinβ, stall torque T0 = n F Rw.
• Walker force against speed T(ω) = T0 (1 − ω Rw / vfree).
• Load torque TL = m g rd / (k η), with k rope falls.
• Motion I dω/dt = T(ω) − TL − c ω, steady ω = (T0 − TL) / (T0 Rw/vfree + c).
• Heaviest load m = n F Rw k η / (g rd), hook speed ω rd / k.

Checks run in the page

• The integrated steady speed equals the closed form.
• Energy in equals energy out plus the kinetic energy; the useful power is the efficiency times the delivered power.
• The heaviest load by bisection and the peak power at half the free speed.
• The torque from a cross product and the mechanical advantage at the stall load.

Honest simplifications

The walkers are treated as a fixed weight at a fixed height on the wheel with a straight force-speed line, there is no brake or ratchet, rope stretch and wheel friction are folded into one efficiency, and the crane jib is only a sketch. The numbers show the scale of a historical crane and not a record of one.

• Press Overloaded and add workers until the wheel starts.
• Press Light load, fast hook and compare the hook speed with the harbour crane.
• Press Thin drum: a smaller drum gives more mechanical advantage but also less rope per turn.