Binary Flutter Solution for Fluid Power

The stability of a foil with its 1/4-chord center of pressure trailing a pitch axis sprung in heave is solved algebraically to help design a fluttering windmill and perhaps watermill. Its flutter mode and frequency chord/wind speed do not depend upon its total mass or spring rate. All contours of this reduced frequency in the pitch inertia and imbalance plane pass through a nexus whose total inertia and imbalance are as if just the virtual mass were at the 3/4 chord point, with a mode of feathering in the apparent wind at this aerodynamic center. The high-frequency flutter amplitude ratio is symmetric in pitch inertia about the nexus. Similarly from the second factor in its pitch damping, each contour passes through another nearby simple point as if twice its Theodorsen factor times the virtual mass were a 1/4-chord divided by this factor behind the 1/4-chord. Therefore, twice the virtual mass at midchord gives the zero-frequency inertia and imbalance midpost furthest away from the nexus. A small trail makes the imbalance greater at the midpost than the nexus so as to slope the zero-frequency line downward. Then, the imbalance required for quasi-steady flutter decreases with pitch inertia, even below zero beyond the nexus. The trail also bends the gate of simple points to pass some low-frequency contours very slightly below the midpost to locally lower the flutter boundary. For an oscillating windmill, the net virtual mass reaction stiffens heave, opposed by the circulatory lift in flutter, because its pitch and heave are necessarily partly in phase. Such new results, and a water flutter demonstration, show a practical semirotary water blade would need a geared-up pitch flywheel for sufficient inertia to flutter well, whereas a wing is so much heavier than air that it has enough structural pitch inertia to flutter and so pump easily.