You are the operator of a 50 m hammerhead tower crane building a one-bay precast frame: 4 columns, 2 beams, a floor slab and 2 wall panels. Swing the jib to the storage area, lower the hook block over a lit element, press Hook on, lift, carry it to the dashed green seat and press Release. The load hangs on a rope, so every start and stop makes it swing like a pendulum; wind pushes it sideways; the load moment limiter blocks lifts the chart does not allow. Keys: A/D slew, W/S trolley out/in, R/F hook up/down, Q/E turn load, Space hook/release. Drag to orbit, pinch or wheel to zoom.
Stand-alone runs of the same pendulum code with no wind and no drive. Measured period against 2π√(l/g) with the large-angle correction, and the decay of a damped swing against e-ct/2.
The trolley accelerates for a ramp time ta, coasts, and brakes with the same ramp. On the current rope length the pendulum period is -. Residual swing is measured after the stop and compared with 4 (a/g) l |sin(ωta/2) sin(ωtd/2)|. A ramp equal to one pendulum period cancels the swing.
For your current counterweight and ballast: the load at which the crane just tips, from the moment balance and, independently, from the combined centre of mass; plus the wind speed at which a 4 t load at 25 m tips, from the closed form and from a bisection on the full balance.
Load chart and moment balance. The crane stands on a square base and can tip about an edge a distance b = 3 m from the slew axis, with the jib across a side. Every mass mi at horizontal position xi along the jib (counterweight blocks at about −12 to −8 m, jib sections, mast and ballast near 0) gives a moment about the edge mi g (b − xi): restoring if it is behind the edge, overturning if it is beyond. With Mr and Mo the sums, the stability ratio is S = Mr / Mo. The lifted load Q at radius r adds φ Q g (r − b) to Mo (dynamic factor φ = 1.10), and so does the trolley and hook. The chart requires S ≥ 1.25, which gives the stability limit
The allowed load is the lower of this and a structural limit (10 t, and a constant load moment of 200 t·m, so Qstruct = 200/r t beyond 20 m). Far out, load times radius is what matters, so the chart falls like 1/r; with few counterweights the stability limit takes over. With the jib unloaded the same balance about the rear edge must stay above 1, otherwise too much counterweight and too little ballast tips the crane backwards.
Wind overturning moment. Wind speed grows with height as v(z) = v10 (z/10)0.16. The force on each part is ½ ρ v(z)² Cd A (mast, jib, counter-jib with blocks, the slew unit, and the load with an area that depends on how a panel is turned to the wind), and its moment about the base is that force times its height. Total Mwind = ½ ρ v10² Cw, so the wind limit is v10 = √(2 Mavail / (ρ Cw)) where Mavail is what the balance has left. Gusts are modelled as v10(1 + gf n(t)) with n(t) a sum of three sinusoids (periods 3, 7 and 17 s).
The swinging hook load. The load and hook form a point mass hanging from the trolley sheave on a rope of length l (shortened by hoisting). For small angles the swing obeys θ'' + (g/l) θ = −ap/l, where ap is the horizontal acceleration of the trolley. Its natural period is T = 2π√(l/g), independent of the load mass: a 40 m rope swings once every 12.7 s, a 5 m rope every 4.5 s. The simulation does not use the small-angle form: it integrates the full 3-D motion of the mass in space with a rope-length constraint (small time steps of 1/240 s, projection after each step), so slewing (centripetal and tangential acceleration, Coriolis from the trolley moving while the jib turns), large angles and the rope length changing while hoisting all appear automatically. Damping is a linear term −c (v − vtrolley) plus the quadratic air drag of the load.
Input shaping. For a trolley that accelerates at a for a ramp time ta, coasts, and brakes at −a for the same ta, the residual swing amplitude is 4 (a/g) l |sin(ω ta/2) sin(ω td/2)| with ω = √(g/l) and td the time between the start of acceleration and the start of braking. The first factor vanishes when ta is a whole number of pendulum periods: the acceleration pulse excites a swing, and its own end cancels it. That is why the Smooth drive button sets the ramp time to one pendulum period of the current rope. The price is a slower start; the lower you hoist the load the shorter the period and the faster it can move. Experiment B checks the formula by simulation.
Placing precision. A seat needs a few centimetres of position accuracy, so the swing speed must be small when you release. The game accepts a miss of 30 to 45 cm (depending on the element), 6 degrees of rotation and a swing speed below 0.5 m/s; faster is recorded as a hard landing. The load moment limiter blocks hoisting up and trolley-out while the load is above the chart value for the current radius and wind, as a real limiter does; the override switch removes the safeguard so you can see what the equations predict.
Simplifications. A generic crane with rounded masses and areas, not a specific model or a design standard: the real load chart comes from the structural design and tests of one machine, and standards such as EN 14439 use partial factors that are not reproduced. Overturning is checked about one edge with the load in line with the jib; the base is rigid and the ground is not modelled. The jib is a rigid body (no bending or twisting), the tower does not sway, the hook and load are one point mass (no spin or tilting of the element, and rope slings are weightless), the rope is inextensible, and the wind has no turbulence structure beyond the three-sinusoid gust and shear. Real cranes also weathervane the jib out of service, which is not modelled; the out-of-service wind tile is only the stability limit of the unloaded crane. Not affiliated with any crane maker; do not use for real lift planning.