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National Advisory Committee for Aeronautics, Report - Propulsion of a Flapping and Oscillating Airfoil

Formulas are given for the propelling or drag force
experienced in a uniform air stream by an airfoil or an
aiyg‘oil-aileron combination, oscillating in any of three
degrees of freedom: oerticalflapping, torsional oscillations
about a fixed axis parallel to the span, and angular
oscillations of the aileron about a hinge.

It is the object of this paper to investigate theoreti-
cally the horizontal forces experienced by an airfoil
or an airfoil-aileron combination n a uniform air
stream made to execute flapping motion or to perform
angular oscillations about a fixed axis parallel to the
span. The problem treated is that of an infinite wing,
or wing and aileron, performing steady sinusoidal oscil-
lations in any of three degrees of freedom: vertical
flapping at right angles to the direction of motion,
oscillations about an arbitrary fixed axis parallel to its
span, and oscillations of the aileron about a hinge.

The work of Wagner (reference 1) for calculating
the distribution of vorticity in the wake of an airfoil
in nonuniform motion appears as a starting point. A
vortex wake is generated by the oscillatory motion,
which in turn affects the entire nature of the forces
experienced by the wing. Beautiful experimental
checks of Wagner’s theory of the manner in which the
circulation builds up have been obtained by Farren
and Walker. (Cf. reference 2, ch. 9 for a more detailed
bibliography.) Birnbaum and Kfissner (reference 3)
have also attacked the problem of obtaining the lift
forces on an oscillating wing by certain series expan-
sions that are rather cumbersome to handle. Glauert
(reference 4) has treated the case of an oscillating air-
foil and has obtained expressions for the forces and
moments that check with Wagner. Theodorsen (ref-
erence 5) has developed compact expressions for the
lift and moments in the case of an airfoil-aileron com-
bination of three independent degrees of freedom and
has applied the results to an analysis of the wing-flutter
problem.