Delivered scenarios: a flywheel with constant drive torque and damping, rotor spin-down by rotational drag, and a small wind turbine turning at a fixed tip-speed ratio.

I·dω/dt = τ₀ + τ_aero(ω) − (b + k)·ω [N·m]. Delivered limits: τ₀ only → ω(t) = (τ/b)(1 − e^(−bt/I)) [rad/s]; spin-down → n(t) = n₀·e^(−bt/I) [rpm]; fixed λ → ω = λu/R, θ = ωt.

Tip-speed ratio λ = ωR/u. Wind torque τ_aero = ½ρπR³u²·Ct(λ), Ct = Cp/λ. Wind power ½ρAu³ with A = πR²; extracted power = Cp·½ρAu³ ≤ (16/27)·½ρAu³ (Betz limit).

Cp(λ) = Cp,max·(2λλ_opt − λ²)/λ_opt² for 0 ≤ λ ≤ 2λ_opt, else 0 — an illustrative parabola peaking at λ_opt, not the curve of a real blade.

Honest simplifications. Rigid rotor with constant air density ρ = 1.225 kg/m³, steady uniform wind, no yaw, pitch control, tower dynamics or gusts; the generator is a purely resistive torque k·ω. The wind torque, generator load and unlocked-rotor behaviour are extensions of the delivered scenarios and use an invented, clearly illustrative Cp(λ) curve. Preset values and captions are those of the original simulations.

Rotor Spin-Up, Spin-Down and Wind Torque: Turbine Shaft Dynamics Case-Study Lab

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