The figure shows the model of a system consisting of an engine, a propeller and a shaft connecting them. The effective moments of inertia of the engine, propeller and shaft about the common axis of rotation are IE, Ie and Is, respectively. The shaft has a polar second moment of area J about the axis of rotation and the material from which it is made has a modulus of rigidity G. The other components of the system may be considered rigid and damping to be negligible. During free torsional oscillations of the system a node occurs along the shaft at a distance d from the disc representing the engine. Derive expressions for the natural frequency of the system and for