d$)$ To successfully controlling a mechatronics system, it is necessary to get the transfer function which based on the equation of motion of the mechanical model. Drive the equation of motion for the following system shown in figure $2.$ The mechanism shown is used in controlling the valve by means of a cam that move a verical rigid rod. The rocker arm is hinged at its center of mass. Assume the higher joint is modeled as a compression spring.

$(6$ points$)$

Fig$\mathrm{.}2.$ A valve controlling mechanism.

$($Hint$)$ Assume the mass of the rocker arm is equal to the mass of the push rod and all springs have the same stiffness