Canal Incline Ship Railway

What is modelled

Two vessel cradles run on rails up and down a long slope and are tied together by a cable over a wheel, so one goes up exactly as the other comes down. Only the difference of their loads has to be pulled by the drive. The drive is a water wheel whose power is limited by flow and head; a governor adds a braking force once the speed exceeds its target, and a manual brake adds more.

Equations

• Slope angle θ = asin(rise / length); unbalanced pull (mB − mA) g sinθ.
• Water power P = η ρ g Q H; drive force F = min(Fmax, P / v).
• Resistance: rolling μ N, quadratic drag k v|v|, governor K (|v| − vt), brake.
• Newton: (mA + mB + meq) dv/dt = Fnet.

Checks run in the page

• Slope and the gravity pull of the load difference.
• Water power against drive force x speed.
• Integrator against the tanh solution for constant force with quadratic drag.
• Braking distance against m v² / 2F.

Honest simplifications

The rolling, drag and wheel constants are illustrative, the drive is an idealised reversible clutch and the cable has no stretch or sag. The model illustrates the mechanics of a counterbalanced incline and is not a validation of any historic machinery.

• Press Heavy B descends: gravity does most of the work and the governor holds the speed down.
• Press Full water power and watch the drive force fall as speed rises, because power is fixed.
• Raise the brake and see how quickly the cradles stop.