A decomposed Fourier series solution to Prandtl's classical lifting-line theory is used to examine the effects of rigid-body roll and small-angle wing flapping on the lift, induced-drag, and power coefficients developed by a finite wing. This solution shows that, if the flapping rate for any wing is large enough, the mean induced drag averaged over a complete flapping cycle will be negative, i.e., the wing flapping produces net induced thrust. For quasi-steady flapping in pure plunging, the solution predicts that wing flapping has no net effect on the mean lift. A significant advantage of this analytical solution over commonly used numerical methods is the utility provided for optimizing wing flapping cycles. The analytical solution involves five time-dependent functions that could all be optimized, to maximize thrust, propulsive efficiency, and/or other performance measures. Results show that by optimizing only one of these five functions, propulsive efficiencies exceeding 90% can be attained. For the case of an elliptic planform with linear twist, closed-form relations are presented for the decomposed Fourier coefficients and the flapping rate that produces mean induced thrust that balances the mean drag in the absence of wing flapping.
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