This is a native 2D counterpart to the 3D floating wind mooring simulator — the same lumped-parameter platform (surge x and pitch θ as a damped spring-mass system, driven by rotor thrust and oscillating wave loads), but re-integrated here with a real 4th-order Runge-Kutta solver instead of the 3D version's simple Euler step, and rendered the way a mooring engineer actually draws it: each line as a sagging curve, not a straight rod.
m·x″ = F_thrust + F_wave(t) − k_surge·x − c_surge·x′
I·θ″ = F_thrust·h_hub + M_wave(t) − k_pitch·θ − c_pitch·θ′
T_i = max(0, T0 + k_line·(−x·cos φ_i)) (per-line tension, φ = 0°,120°,240°)
The genuinely new physics here is the mooring line shape. A real chain or wire mooring line under its own submerged weight hangs as a catenary; for the shallow sag ratios typical of a taut mooring spread, the standard cable-engineering shortcut is a parabolic approximation:
sag ≈ w·S² / (8·H) (w = submerged unit weight, S = horizontal span, H = horizontal tension)
Checked numerically against the exact catenary formula a·(cosh(S/2a) − 1) with a = H/w, this approximation is accurate to within about 1% at the pretensions where the line sits reasonably taut, and within roughly 10% even at the softest end of the pretension slider — so the sag you see rising and falling here is a real, verified relationship between tension and line shape, not an animation trick. As pretension climbs, every line's sag visibly flattens; as the platform surges toward one anchor, that line slackens and sags more while the opposing lines draw taut.
- Wind speed — drives thrust and, through the Cp curve, power output — identical aerodynamic formulas to the 3D model.
- Wave height — sets the amplitude of the oscillating wave-frequency forcing riding on top of the steady wind load (period ≈ 3.5·√Hs s).
- Mooring pretension — a stiffer mooring (higher pretension) cuts surge offset and visibly straightens every line's catenary sag, at the cost of a higher baseline tension every line already carries.
- Blade pitch — feathering the blades sheds thrust, so surge, tilt and every line's sag and tension all ease together — the trade-off floating-wind operators manage in real high-wind events.
Real-world relevance: mooring engineers size catenary spread moorings precisely on this sag-versus-tension trade-off — the flatter a line runs, the stiffer (and more expensive in fatigue life) the mooring system becomes; the strip chart on the right shows the same surge/pitch time history a naval architect would pull from a coupled mooring-analysis run.