#$2(75$ points$)$

A schematic of an automotive valve train is shown. The rotating cam moves the pushrod, and the rocker arm rotates to move the valve in the vertical direction. A simplified mechanical model is shown. Moment if inertia J represents the rocker$-$arm inertia about the pivot point. Deflection of the pushrod is modeled by the spring $k_1,$ and the return spring on the valve is $k_2.$ Friction in the rocker$-$arm pivot is modeled by viscous friction coefficient $b.$ The vertical motion of the cam follower is $x_c(t),$ the input to the system. When the rocker$-$arm angle is level