II$.$ PROBLEMS $(75$ points$)$

A mechanical shaker, weighing $70$ lb $,$ is mounted with a spring at the end of a steel cantilever beam as shown in the figure. During operation of the shaker, the cantilever beam is subjected to a harmonic force, $F(t)=50cos(30t)lb.$ The bending stiffness of the beam is $1200l\frac{b}{i}n$ and the spring stiffness is $900l\frac{b}{i}n.$ Assuming $=0\mathrm{.}1,$ determine the equation of motion, the amplitude of the steady$-$state displacement and its phase with respect to the exciting force, and the undamped response assuming zero initial conditions. $(25$ points$)$