An aircraft wing, executing small plunging and pitching oscillation, is modeled as a non-uniform flat plate of mass m as shown in Figure Q1.1. The stiffness of the wing is represented by two springs: the first is a linear spring with stiffness k, and the second is a torsional spring with stiffness k0. Both springs are anchored to the wing section at its elastic axis. Additionally, two more springs with stiffness k1 and k2 are attached to the wing section at points fore and aft of the elastic axis, as shown. All dimensions indicated in the figure refer to the equilibrium configuration. The chord of the wing is assumed to be 2xb, and all distances along the wing are non-dimensionalized by the semi-chord b. The origin of the coordinate system is assumed to be located at the elastic center of the cross-section, which is at the elastic axis (EA).
The elastic axis (EA) is assumed to be located at a distance bxa aft from mid-chord, and the center of mass is assumed to be located at a distance bxxg aft from the elastic axis. The mass moment of inertia of the wing model about the elastic axis is I.
For a typical section of width 1m, it is given that k = 9000 kg/m, k1 = 4500 kg/m, k0 = 13500 kg/m, a = 0.35, a = 0.3, b = 1m, and Ig = 10 kgm^2. Additionally, x = 0.2.
Explaining all your calculation steps, estimate k, k1, and the natural frequencies of the coupled oscillations in plunging and pitching, as well as the corresponding mode shape vectors.