$14.$ The given figure part $($a$)$ shows a schematic diagram of an automobile suspension system. As the car moves along the road, the vertical displacements at the tires act as the motion excitation to the automobile suspension system. The motion of this system consists of a translational motion of the center of mass and a rotational motion about the center of mass. A very simplified version of the suspension system is shown in the figure part $($b$).$ Assume that the motion xi at point P is the input to the system and the vertical motion xo of the body is the output. Consider the motion of the body only in the vertical direction. Displacement xo is measured from the equilibrium position in the absence of xi input.

a$.$ Derive the mathematical model of the simplified model in the form of differential equation

b$.$ Find the transfer function of the system

c$.$ Determine the conditions of the parameters of the model so that the system is critically damped.

d$.$ Choose your own parameter value according to the conditions that you have obtained in part $($c$),$ and compute manually the response of the system if the input xi $=5$sin$(1/4*$t$)$

e$.$ Make a computer simulation to simulate the response that you have computed in part $($d$)$