Tower Bridge Bascule

A bridge that is almost in balance

Each leaf has a long arm of road and a short, heavy counterweight tail behind the pivot. What the motor must overcome is not the weight of the leaf but the small difference of the two moments about the pivot: τg = (ml rl - mc rc) g cosθ = (1 - k) ml rl g cosθ, where k is the counterweight balance ratio. The torque is largest with the leaf down and falls to zero when it is vertical, and it is the derivative of the potential energy U(θ) = (1 - k) ml rl g sinθ.

Dynamics and the drive

I d²θ/dt² = τm - τg - b ω - τc tanh(ω/ω0) with I = Ileaf + mc rc². The controller follows a trapezoidal motion profile ωref = sign(e) min(ωmax, √(2 a |e|)) using feed-forward I aref + τg plus a stiff PD correction, and the hydraulic power limit caps |τm ω| ≤ Pmax. The energy bookkeeping Wmotor = ΔU + ½ I ω² + Wfriction is checked at the end of every raise.

Will the ship fit?

A raised leaf point at height z is at x(z) = D - z / tanθ from the centre line (D = 30.5 m). A ship part of width w spanning heights z0 to z1, offset c from the middle, clears it if D - min(z1, L sinθ)/tanθ ≥ |c| + w/2 + margin. For a rig that rises above the leaf tip this reduces to L(1 - cosθ) ≥ |c| + w/2 + margin, so θmin = arccos(1 - (|c| + w/2 + m)/L); the page solves it by bisection and compares it with this closed form.

The cost on the road

While the road is closed for T seconds, cars arrive at λ and leave at the saturation flow μ (0.5 cars/s) once the barriers rise. The queue peaks at λ T, clears after T μ / (μ - λ) and the total delay is the area of the triangle: ½ λ T² μ / (μ - λ). Doubling the closure time quadruples the delay, which is why long holds hurt.

Honest simplifications

• One rigid leaf stands for the pair; masses, centres of mass, friction and the hydraulic drive are round model values, not the real engines.
• Ships are stacks of boxes; real hulls, rigging, wind, tide and a pilot's larger margin are ignored.
• The road model is a deterministic D/D/1 queue; the 3D page uses individual vehicles.
• Educational model, not affiliated with any operator, not engineering advice.