As shown in Figure $1,$ a rotating machine, with a mass of $m,$ is fixed to the

center of the beam, which in turn is simply $($pin$)$ connected to the tips of the two

cantilever beams fixed to a rigid wall. The machine generates a vertical harmonic

load, $p(t)=p_{0}sinwt.$ Assume the system has damping coefficient, $c,$ and all beams

have the same EI value.

$($a$)$ Determine the total stiffness of the system, expressed in terms of EI and L$.$

$($b$)$ Derive the equation of motion of the machine in the vertical direction.

$0\mathrm{.}02,P_0=10$ kips. Determine the maximum bending moment at the fixed ends due

to the dynamic response, when $($i$)w=0\mathrm{.}785ra\frac{d}{sec},$ and $($ii$)w=7\mathrm{.}85ra\frac{d}{sec},$

respectively.