Post Mill Dynamics

What is modelled

Four cloth sails turn a shaft that drives a millstone through a gear train. The rotor torque follows a power-coefficient curve in the tip-speed ratio. The wind swings back and forth; the miller turns the whole mill on its post with a first-order lag, so there is always some yaw error, and the rotor only sees the wind component along its axis.

Equations

• Tip-speed ratio λ = ω R / veff with veff = v cosψ.
• Power coefficient Cp = Cmax (λ/λ0) e1-λ/λ0, torque Q = ½ ρ π R³ veff² Cp/λ.
• Rotor dynamics J dω/dt = Q - Qstone - Qbrake.
• Mean power under a yaw swing a: (sin a - sin³a/3)/a of the aligned value.

Checks run in the page

• The Betz bound on the power coefficient curve.
• The steady speed against a bisection residual.
• The mean yaw loss against its closed form.
• A linear spin-up against its exponential solution.

Honest simplifications

A fitted power-coefficient curve instead of blade-element theory, a constant stone torque, no gearbox losses and no sail furling. The 3D page is a heritage visual; the numbers are illustrative, not a design tool.

• Press Veering wind and watch the yaw error open up when the miller lags.
• Press Heavy grinding to see the rotor slow towards a stall.
• Press Storm, braked: the brake keeps the rotor from running away.