For harmonic oscillations, the Theodorsen function C(k) accounts for the contribution of the wake to the lift and moment. Plot a vector diagram of the real and imaginary parts of its complex conjugate C(k) versus the frequency k ∈ [0, ∞). Also, plot its magnitude and phase.

Plunging oscillations are an approximation of bending oscillations. Plot a vector diagram of the total lift coefficient of a thin airfoil undergoing plunging oscillations versus the reduced frequency. Also, plot its magnitude and phase.

Pitching oscillations are an approximation of torsional oscillations. Plot a vector diagram of the total lift coefficient of a thin airfoil undergoing torsional oscillations about the mid chord versus the reduced frequency. Also, plot its magnitude and phase. Repeat if the oscillations are about the airfoil leading edge.

Hint: use the following equation to approximate the Theodorsen function:

 

C (k) = 1 0.165k (k+ 0.0455i) 2 k + 0.0455 0.335k(k+ 0.3i) 2 2 k +0.3