Treadwheel Hoist

What is modelled

Walkers inside a wheel of radius R push on the tread with a force that falls as they speed up, following a linear force-velocity law F = F_max(1 − v/v₀). The wheel is fixed to a drum of radius r that winds the rope. Because R is much larger than r, a modest walking force lifts a heavy load, at the price of a slow lift.

Equations

• Mechanical advantage MA = R / r.
• Driving torque τw = η n Fmax (1 − Rω/v0) R; load torque M g r.
• Itot dω/dt = τw − M g r, with Itot = Iwheel + M r².
• Heaviest load Mmax = η n Fmax R / (g r); steady speed ω* = (v0/R)(1 − M g r / (η n Fmax R)).

Checks run in the page

• Heaviest load from bisection on the net torque against the closed form.
• Steady wheel speed from time stepping against ω*.
• Energy: work done by the walkers times η equals the potential plus kinetic energy gained.
• A load 5 % above the limit does not rise in 20 s; a load 5 % below it does.

Honest simplifications

A linear force-velocity law, constant efficiency, an inextensible massless rope and one lumped wheel inertia. Real treadwheels also had brakes, pawls, varying walker position and fatigue; values here are plausible but not measured from the historic crane.

• Choose Heavy beam and see the lift slow as the load approaches the limit.
• Choose Overload and watch the wheel stall, then raise the number of walkers.
• Shrink the drum radius to trade lift speed for lifting capacity.