A bascule is a lever with a counterweight
A bascule leaf turns about a horizontal pivot (the trunnion). With θ the lift angle, a mass m at body-fixed coordinates (x, y) from the pivot has its weight at lever arm X = x cosθ − y sinθ, so the gravity torque is τ_g = −g (Mx cosθ − My sinθ), with Mx = Σ m x and My = Σ m y. A leaf with its centre of mass 6 m ahead of the pivot is balanced by a block about 11 m behind it. Static balance at θ = 0 means Mx = 0, but the block hangs below the pivot line, so My is not zero and a pendulum-like term g My sinθ grows as the leaf rises. Over a whole lift the torque amplitude is g √(Mx²+My²): the best counterweight is the one that puts the combined centre of mass as close to the pivot as possible, mC = (b·Mℓ − c·Eℓ)/(b²+c²) with b and c the horizontal and vertical arms of the block. That is the closed-form answer the experiment tests; the numerical sweep also contains inertia, friction and wind, so it lands slightly off.
The motor and the lift
The lift follows a cosine ease, θ(t) = θmax (1 − cos(πt/T))/2. The motor torque comes from τ = I α − τ_g + τ_f sgn ω − τ_w, where I is the moment of inertia of leaf and counterweight about the pivot, τ_f = μ m g r is journal friction on a trunnion of radius r, and τ_w the wind torque. The energy of one lift follows from the work-energy theorem: the change in potential energy g (Mx sinθ + My (cosθ − 1)) plus friction and wind work; the kinetic energy is zero at both ends. The model shows it matches the time integral of τ·ω.
Hydraulics, then and now
The 1894 bridge was lifted by steam engines pumping water into weighted hydraulic accumulators; the water drove engines that turned pinions on toothed racks fixed to the leaves. The steam plant was replaced by electro-hydraulic oil drives in the early 1970s. The oil volume per lift is V = E / (η p) for accumulator pressure p; the recharge time is the energy over the pump power. Pressure, pump power, motor rating, masses and the position of the centre of mass are model values chosen to be of the right order, not published design figures; the leaf and counterweight numbers (about 1,000 t per leaf, 365 to 420 t of counterweight, 61 m between the towers, 86° opening, about a minute to lift) are the commonly quoted ones.
Wind
On a raised leaf the wind component normal to the plate is v sinθ, giving F = ½ ρ Cd A (v sinθ)² acting at the mid-length. The model applies the wind along the bridge axis, which is the worst case; across the bridge (the usual Thames wind direction) the leaf is edge-on and the torque about the pivot is zero. The windward leaf and the leeward leaf feel opposite torques.
Tide and clearance
The Thames at London rises and falls with the semi-diurnal lunar tide, period 12.42 h, spring range about 7 m and neap range about 4.5 m. The model uses h(t) = −(R/2)(1 − cos(2π t/T)) relative to the reference high water at which the closed clearance is 8.6 m, so clearance = 8.6 + (R/2)(1 − cos(2π t/T)). A vessel of air draft a needs a lift when a + 0.5 m exceeds the clearance; for a uniformly random arrival the fraction of the tide it needs a lift is arccos(1 − 2y/R)/π with y = a + 0.5 − 8.6 (0 below, 1 above the range). Real clearance depends on the datum and on the actual tide; the figures are for illustration.
Queues at a closure
With arrival rate λ, discharge (saturation) flow s and closure time T, the fluid model gives a peak queue λT, a clearing time λT/(s − λ) and a total delay λT²/2 · s/(s − λ) vehicle-seconds per direction. The check below compares that with a discrete-event simulation of Poisson arrivals. Real bridges also have signals, pedestrians and random incidents, which are not modelled.
Educational model, not a design or safety tool. Independent educational model, not affiliated with Tower Bridge or the City of London Corporation. The bridge is an original model in the style of a Victorian Gothic bascule bridge; the name Tower Bridge is used descriptively only, with no logos or trade dress. Vessels and vehicles are generic. Operator-mode numbers (the 61 m span, 30.5 m leaf length, the 20.2 m clear width a tall ship needs, the 3 s amber, the challenge scoring) are model values or illustrative, not operating rules of any real bridge.