Consider a rigid structure with total mass $m$ and mass moment of inertia $J$ with respect to the center of mass,

which is at a height $h$ above the ground. The structure rests on a soft soil, the stiffness of which can be idealized

by means of two springs, a translational spring $k_x$ and a rotational spring $k_{}$ that are attached at the base, as

shown in the figure below. If $u,$ are the translation and rotation of the base $($assumed positive when

counterclockwise$),$ and if $J=J_0+m{h}^2$ is the mass moment of inertia with respect to the base, find the

equations of motion of the rigid structure as well as its two natural frequencies.