Binary Flutter as an Oscillating Windmill — Scaling & Linear Analysis

Abstract

Alternating pitch and heave of a foil can tap light wind or slow free water flow, as the design flow speed for an oscillating blade in any fluid is shown to be limited by its material's ratio of endurance stress to density. The free amplitude of flutter instability is inherently suited to reciprocating a pump whose stroke needs to vary to efficiently capture the changeable wind. Whereas the useful pumping work of multiblade rotary windpumps with their fixed stroke is only 10% of the ideal annual wind energy capture. Stability algebra of a foil free in pitch and elastic in heave is analytically solved for the first time to prove the stability contours of all Theodorsen frequencies radiate from the same super node of total imbalance and pitch inertia the same as for a virtual mass confined to the 3/4 chord point. The flutter frequency only involves these totals and the trail of the center of pressure behind the pitch axis. Oscillation is damped in the low ratios of total to virtual typical of hydrofoils in water but heavier- than-air wings can be unstable. High intrinsic pitch inertia ratio and trail decrease the tailheavy imbalance ratio needed to force pitch by heave. Perturbing the equations for heave varying as semi-rotary roll, an unswept wing of low enough dynamic imbalance ratio can start to flutter in light wind but stop and feather in storms. The dynamic imbalance can be reduced at large scale and low design windspeed where the gravity effect of static imbalance more safely forces pitch. Between starting and stopping, the amplitude ratio of pitch to roll drops, containing the rise with windspeed of the bending moment amplitude.