$(20$ points$)$ The mechanical system shown in the figure below is at rest for The input force $u$ is given at

$(1)$ Derive the equisplacements. The displacement $x$ is the output of the system.

$(2)$ Derive the equations of motion of the mechanical system.

Dransfer function $x\frac{s}{U}(s)$ of the mechanical system.

$(1)$ Derive eqn's of motion

$(z)$ Derive $x\frac{s}{u}(s)$