Problem $5$

Consider the slender uniform rigid bar with mass $m$ shown at the right, which is in the horizontal plane. A force $p(t)$ and a moment $M(t)$ are external forces applied to the system. A point mass $$ is secured at the left end of the rod.

d$.$ Assuming small rotations, derive the equation of motion for the rigid bar in terms of its rotational angle $($take counterclockwise rotation as being positive$).$

e$.$ Determine the damped and the undamped natural frequency of the system.

f$.$ Assuming that the mass of the bar $m=20kg,=1kg,k=4{10}^3\frac{N}{m},$ and $l=3m,$ design the damper $($i$.$e$.,$ calculate $c)$ so that the damping ratio of the system is $=0\mathrm{.}05.$

g$.$ Determine the steady state response ${}^{}(t)$ to the force $M(t)=sin6t,$ and $p(t)=0$