$(10$ pts$)$ Derive the governing differential equations of motion for the rotational system shown below.

$(15$ pts$)$ A vehicle suspension system can be represented in one of its simplest configurations by the

so$-$called bicycle model shown in the figure below. Using this model, assume that the system is

depicted in its resting state and that only small deflections of the suspension from its resting $($i$.$e$.,$

equilibrium$)$ position will occur.

a$.(10$ pts$)$ Derive the governing equations of motion for the vertical displacement $x(t)$ and

angular displacement $(t)$ of the bicycle$-$style suspension model.

b$.(5$ pts$)$ Are the equations obtained in part $($a$)$ coupled? If so$,$ what is the nature of their

coupling, and what $($if any$)$ are the conditions for which the equations can be decoupled?