An aircraft wing is modelled as a non-uniform flat plate of mass m as shown in Figure 9.14. The stiffness of the wing is idealized and represented by two springs kA and kB. The chord of the wing is...

An aircraft wing is modelled as a non-uniform flat plate of mass m as shown in Figure 9.14. The stiffness of the wing is idealized and represented by two springs kA and kB. The chord of the wing is assumed to be 2b and all the 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 centre of the cross-section, which is assumed to be located at EA. The line joining the elastic centres of the various cross-sections of the wing is assumed to be a straight line and referred to as the elastic axis. The elastic axis is assumed to be located at a distance ba from the mid-chord and the centre of mass is assumed to be located at a distance bxα from the elastic axis. The mass moment of inertia of the wing model about the elastic axis is Iα. (i) Show that the non-dimensional distance of the elastic axis from mid chord satisfies the relationship

(iii) It is proposed to control the bending-torsion vibrations of an aircraft wing prototype by using a distribution of piezoelectric PZT patch actuators to generate a pure bending moment and a pure torsional moment. Considering the typical section of the wing, identify the precise locations of the PZT patch actuators relative to the elastic centre. Derive the equations of motion of the typical section idealization