Rotating Boat Lift

Why the wheel is balanced: Archimedes

A floating boat of mass m displaces water of the same mass, ρVd = m. When the gate is open, the gondola water level is equal to the canal level, so the displaced water leaves through the gate. The gondola mass is M = Ms + ρ(V0 - Vd) + m = Ms + ρV0 for every boat. The page finds the draft by integrating a rounded hull section (Simpson rule) and bisecting until buoyancy equals weight, then adds up the masses (check 1). With "Sealed tank" the water cannot leave, so the boat adds its full weight and the wheel is unbalanced.

Torque, motion and energy

With the arm angle θ measured from the top, gondola A at height R cos θ and B at -R cos θ, the gravity torque about the axle is τg = g R sin θ (MA - MB). The wheel obeys I θ'' = τg + τm - τf tanh(ω/ε). The motor controller applies a feed-forward that cancels the known imbalance and friction plus a proportional speed correction, limited to the torque limit (the brakes can hold 2.5 times that). The work needed for a half turn is W = -2 g R ΔM + τf π (potential energy plus friction); the page integrates ∫ τm ω dt numerically and compares (check 3). Electrical energy counts only motoring power divided by an assumed 85 % efficiency.

Keeping the tanks level

A fixed sun gear (48 teeth) at the hub drives two small idlers (8 teeth) that drive the ring gears (48 teeth) on the gondolas. The sun gear does not turn while the arm does, so in the arm frame it turns -θ; every mesh reverses the sense and scales by the tooth ratio, so the ring gear turns -θ Zs/Zg relative to the arm. The absolute tank angle is θ(1 - Zs/Zg), exactly zero for equal tooth counts. Change the ring-gear teeth to see the tank tilt; water reaches the rim at tan α = freeboard / half-width, about 17.9 degrees here (check 2 compares pitch-arc counting with the closed form).

Honest simplifications

This is an educational model inspired by rotating boat lifts such as the Falkirk Wheel, not a copy of any real machine: dimensions are round numbers (arm radius 12 m, lift 24 m, tank 24 x 6.2 m, 1.6 m water depth), the structure mass, moment of inertia, friction torque and 85 % efficiency are illustrative, the narrowboat uses a simplified rounded hull and is capped at about 30 t before it would sink in this hull model, and the real gear system differs in detail. Water is a level plane (no sloshing), the gates open and close on a fixed schedule, and the 3D scene and water shading are visual encodings. Not affiliated with any operator; do not use for engineering design.