Figure P$4\mathrm{.}42$

$4\mathrm{.}42$ Figure P$4\mathrm{.}42$ shows an engine valve driven by an overhead camshaft. The

rocker arm pivots about the fixed point $O$ and the inertia of the arm about this

point is $I_r.$ The valve mass is $m_v$ and the spring mass is $m_s$; its spring constant

is $k_s.$ Let $f_c$ denote the force exerted on the rocker arm by the camshaft.

Assuming that $(t)$ and its time derivatives are known $($from the cam profile

and the cam speed$),$ derive a dynamic model that can be used to solve for the

cam force $f_c(t).($This information is needed to predict the amount of wear on

the cam surface.$)$

$**$I need the free body diagram and the equations of motion.