ENAE 632 Homework Problem Set No.4

1. Consider a thin airfoil undergoing a combined simple harmonic pitching and plunging motion. The pitching motion takes place about an arbitrary pitch axis, say a chords measured from the leading edge. Find expressions for the quasi-steady lift and pitching moment about the 1/4-chord as a function of reduced frequency.

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2. Consider two types of motion. The first is an isolated simple harmonic pitching oscillation, and the second is a isolated simple harmonic plunging oscillation. Using the quasi-steady results obtained above, in each case plot the quasi-steady lift and moment as a function of displacement for several values of reduced frequency. Comment on your results.

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3. Plot Theodorsen's function for an appropriate range of reduced frequencies. Compute the function exactly from the Bessel functions. Plot C(k) as real and imaginary parts, and as amplitude and phase versus reduced frequency.Compare your result with one or more of the well-known approximations to C(k).

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4. Obtain the unsteady lift reponse for a harmonic plunge motion with unit amplitude and compare with the quasi-steady result. Plot your results as real and imaginary parts and as amplitude and phase as a function of reduced frequency in the range up to k=10. Identify the contribution to the unsteady lift from the the circulatory and noncirculatory terms.

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5. Obtain the total unsteady lift reponse to a harmonic variation in angle of attack with unit amplitude and compare with the quasi-steady case. Plot your results as real and imaginary parts and as amplitude and phase as a function of reduced frequency up to k=10.0. Identify the contributions to the lift from the pitch rate terms. Also show the effects on the total lift response when excluding the noncirculatory terms.