Exercise $6$

Figure $3$ 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 $m_v.$ The spring constant is $k_2.$

Consider the spring mass $m_s$ negligible. 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.$)$

You should arrive at the ODE:

$(I_r+m_v{L}_2^2){}^{}=f_c{L}_1-k_2{L}_2^2$

which can be solved for $f_c.$

Hints: What is the importance of $m_v?$ Where should

the spring act in your FBD$?$ You may want to draw

separate FBDs for the valve and rocker arm.