Problem $4\mathrm{.}42$

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.$)$

Figure P$4\mathrm{.}42$