A tower crane is a balance about the mast
Every mass on the slewing part has a moment about the slewing axis, M = m g x, with x the horizontal arm. Behind the mast sit the counterweight blocks and the counter-jib with its winch; ahead of it sit the jib, the trolley, the hook and the load. The empty-crane balance is M0 = n m_b |x_cw| + m_cj |x_cj| - m_jib x_jib. With a load Q at radius r the net forward moment is (Q + m_t + m_h) r - M0. The slewing ring and the tower below it can take only a limited net moment either way, |net| ≤ M_limit, so the crane can neither be nose-heavy when empty (too little counterweight) nor tail-heavy when empty (too much).
The rated chart
Setting the net moment to its limit gives the load-moment curve Q(r) = (M0 + M_limit)/r - (m_t + m_h), a hyperbola, capped by the rope and winch rating (8 t here). The knee radius is where the hyperbola meets the cap. A load-moment indicator (LMI) compares the actual load with this curve; at 90 % it warns and at 100 % it stops hoist-up and trolley-out, but never lowering or trolley-in. Each counterweight block adds m_b |x_cw| to M0 and so shifts the whole curve up until the empty crane reaches the rear limit.
Wind loads
The dynamic pressure is q = ½ rho v(z)^2 with the power-law profile v(z) = v_a (z/z_a)^0.16, referred to the anemometer at the crown (z_a about 70 m). A part of the crane with drag area Cd A feels F = q Cd A. The jib and counter-jib are lattices of solidity about 0.3 and are modelled as 60 and 20 strips; the winch house, counterweight stack, cab, tower head, mast (120 slices) and trolley are separate items. For the jib only the component of the wind normal to the axis loads the broadside area, F_n = ½ rho Cd A' (v sin beta)^2, and the axial component loads only the small end-on area. The moment of every force about the ring or the mast base is the force times its height above that point: Experiment A evaluates it by summing items and, separately, by closed-form integrals (the mast integral is ∫ q(z) Cd A' (z - z_0) dz with the power law).
The wind force on the load acts through the trolley
A hanging load in a steady wind settles at tan phi = F_w/(m g). The rope then pulls the trolley with the vector sum of weight and wind force, so the moment on the structure is that of the weight at the trolley's radius plus F_w times the trolley's height, not the load's own much lower height. The drift of the load, L sin phi, and the lower lever arm cancel exactly. The vector model computes it the long way (load displaced, forces at the load) and agrees with this shortcut.
Slewing
The slew drive obeys I d(omega)/dt = tau_motor + tau_wind + tau_rope + tau_friction. The inertia is I = sum m (x^2 + k^2) of jib, counter-jib, counterweight, the trolley and the hook plus load, m r^2. The wind torque is sum s F_n over the strips; the relative speed v_n - omega s is used so the aerodynamic damping appears automatically. With the brake released (out of service) the jib weathervanes: the larger lever arm and area ahead of the mast turn it until it points downwind, where the wind torque vanishes and the mast base sees the smallest moment.
The swinging load
The load is a point mass on a rope of length L(t) from the moving trolley: m x'' = m g + F_drag - T d/L, with the rope constraint |d| = L giving the tension T/m = [d.(a - s'') + |d'|^2 - L L'' - L'^2] / L. It is integrated in 3-D (Runge-Kutta 4, with the trolley's acceleration, centripetal and Coriolis terms included), so slewing and trolley moves excite the swing automatically. For small angles it reduces to psi'' + (g/L) psi = -a/L with period 2 pi sqrt(L/g). After a trapezoidal move (acceleration a for t_a, a cruise, deceleration a for t_a, the two pulses T_1 apart) the residual amplitude is (a/g) 4 |sin(omega t_a/2) sin(omega T_1/2)|. A zero-vibration shaper replaces the profile by two half-size copies delayed by half a period, so the residual vanishes at the cost of half a period of extra time. Real crane drives use the same idea or feed back the measured swing.
Hoist power and jib sag
The winch power is P = T v / eta with the rope tension T from the pendulum model (so acceleration counts) and an overall efficiency eta = 0.85; the energy over a lift is m g h / eta. The speed limit falls with the load, v_max = P_rated eta/(m g). The jib is treated as a cantilever of equivalent stiffness EI (the pendants are included in EI): the deflection under a load F at distance a is F x^2 (3a - x)/(6 EI) for x ≤ a. The jib is cambered so that its own sag is cancelled. The sag is drawn exaggerated (slider) but all readouts show the true value.
What is simplified
The structure is a small set of stiffness-free lumps: no dynamic wind response, no resonance of the mast, no fatigue, no rope stretch or hoist-rope bounce, a rigid load that never rotates about the rope, no slack rope, no ground effects on the wind, and gusts are a simple filtered noise. Limits, masses, areas and drag coefficients are values of the right order for a 60 m hammerhead crane of about 8 t capacity, chosen for teaching, not published figures of any product. The jib weight, hook mass, motor sizes, tower stiffness and foundation (322 t of concrete on an 8.2 m square) are model values. The foundation safety factor compares the restoring moment W a with the mast-base moment and is only indicative.
Educational model, not a design, planning or safety tool: real crane operation follows the manufacturer's load chart, the site's lift plan and the standards (for example EN 14439 and EN 13001) with trained operators and signallers. The crane is a generic original model; no manufacturer, product or site is represented, and no affiliation is implied.