In the early theoretical study of aquatic animal propulsion either the two-dimensional theory or the large aspect-ratio theory has been generally used. Only recently has the unsteady lifting-surface theory with the continuous loading approach been applied to the study of this problem by Chopra & Kambe (1977). Since it is well known that the continuous loading approach is difficult to extend to general configurations, a new quasi-continuous loading method, applicable to general configurations and yet accurate enough for practical applications, is developed in this paper. The method is an extension of the steady version of Lan (1974) and is particularly suitable for predicting the unsteady lead-edge suction during harmonic motion.The method is applied to the calculation of the propulsive efficiency and thrust for some swept and rectangular planforms by varying the phase angles between the pitching and heaving motions. It is found that with the pitching axis passing through the trailing edge of the root chord and the reduced frequency k equal to 0·75 the rectangular planform is quite sensitive in performance to the phase angles and may produce drag instead of thrust. These characteristics are not shared by the swept planforms simulating the lunate tails. In addition, when the pitching leads the heaving motion by 90°, the phase angle for nearly maximum efficiency, the planform inclination caused by pitching contributes to the propulsive thrust over a large portion of the swept planform, while, for the rectangular planform, only drag is produced from the planform normal force at k = 0·75. It is also found that the maximum thrust is not produced with maximum efficiency for all planforms considered. The theory is then applied to the study of dragonfly aerodynamics. It is shown that the aerodynamically interacting tandem wings of the dragonfly can produce high thrust with high efficiency if the pitching is in advance of the flapping and the hindwing leads the forewing with some optimum phase angle. The responsible mechanism allows the hindwing to extract wake energy from the forewing.

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A theory is developed for predicting wing-rock characteristics. From available data, it can be concluded that wing rock is triggered by flow asymmetries, developed by negative or weakly positive roll damping, and sustained by nonlinear aerodynamic roll damping. A new nonlinear aerodynamic model that includes all essential aerodynamic nonlinearities is developed. The Beecham-Titchener method is applied to obtain approximate analytic solutions for the amplitude and frequency of the limit cycle based on the three-degree-of-freedom equations of motion. An iterative scheme is developed to calculate the average aerodynamic derivatives and dynamic characteristics at limit-cycle conditions. Good agreement between theoretical and experimental results is obtained.

A theoretical analysis is conducted to determine the optimal control input for wing-rock suppression through a Hamiltonian formulation. The optimality equations are analyzed through Beecham-Titchener's averaging technique and numerically integrated by a backward-differentiation formulas method developed for implicit differential equations. The weighting factors in the cost function are shown to be related explicitly to the system output damping and frequency. A numerical model constructed for an 80-deg delta wing is solved to illustrate the results. It is shown that Beecham-Titchener's technique is accurate in determining the necessary control function to suppress wing rock. From the numerical results, it is also shown that an effective way to suppress wing rock is to control the roll rate. System sensitivity is investigated by determining variations in system output damping and frequency with aerodynamic model coefficients. The results show that higher sensitivity corresponds to lower system damping.CO 0 0(0 Nomenclature amplitude function wing span, ft weighting factors in / 2 -jit 2 + c\ -c 2 fJi --7T 3?r o o -c 2 co --#c 3 co + -ac 4 (ijL 2 -2co 2 ) 3?r STT 4 4 -CJ(A --£3 a o} --c 4 a(iJL 2 + 2oj 2 ) 3?r 3?r 9 , -jlt z + C\ + C2JU, + --