Figure 5 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 k1 and the return spring on the valve is k₂.Friction in the rocker arm pivot is modeled by viscous damping 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 q, which is positive in the clockwise directional. When the rocker-arm angle is level (q = 0), the return spring has a compressive preload force of F_L. When cam follower position x_c = 0 and q = 0, then the pushrod(k₁) is undeflected. Assume that rocker-arm angle q remains small at all times. 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 arm (q = 0).

AMA www. Rocker arm Camshaft Return spring Follower Roller -Valve Pushrod (Follower) Cam 0 k L Pivot friction, b xc(t) 12 Figure 5: Automotive valve train system Rocker-arm inertia, J k