The figure below shows a schematic of an automotive valve train. The rotating cam moves the pushrod $($follower$),$ and the rocker$-$arm rotates to move the valve in the vertical direction. A simplified mechanical model is shown adjacent to the valve train schematic diagram. Moment of inertia $J$ represents the rocker$-$arm inertia about the pivot point. Deflection of the pushrod is modeled by spring $k_1,$ and the return spring on the valve is $k_2.$ Friction in the rocker$-$arm pivot is modeled by viscous coefficient $b.$ The vertical motion of the cam follower is $x_C(t),$ the input to the system. Angular position of the rocker$-$arm is $,$ which is positive in the clockwise direction. When the rocker$-$arm angle is level $)=(0,$ the return spring has a compressive preload force of $F_L.$ When cam follower position $x_C=0$ and $=0,$ then the pushrod $(k_1)$ is undeflected. Assume that rocker$-$arm angle $$ remains small at all times.

$($a$)$ Draw free body diagram$($s$)$ for the system in terms of the given parameters and coordinates. Make sure to include displacements and label all forces and torques. Use the positive directions indicated in the schematic.

$3$

$($b$)$ Write the elemental equations for each of the free body diagrams in part $2$a$.$

$($c$)$ Derive the mathematical model of the valve train system and determine the required displacement of the cam follower $x_C$ when the system is at rest $($equilibrium$)$ with a level rocker $)=(0.$