A rigid block with mass $M_0$ is hanged at mid$-$section of a simply supported steel beam with bending stiffness $E_0$ and span length $L$ by an elastic spring with axial stiffness $K_0$ as given in the figure below. A force $f(t)=f_{0}sin(t)$ is applied to the rigid block in vertical direction. It is assumed that the total mass of the steel beam is in negligible order when compared to the mass of rigid block $M_0$ and damping ratio of the system can be neglected.

Determine the stiffness of the equivalent elastic system composed by elastic beam and elastic spring.

Determine the natural frequency of the equivalent elastic system.

Determine the steady$-$state displacement response of the equivalent elastic system.

Determine the maximum displacement response of the elastic system.

IPN$300I_0=9800c{m}^4,E_s=200$GPa,$L=4\mathrm{.}00m$

$M=20\mathrm{.}00$ ton, $K_0=10000\mathrm{.}00k\frac{N}{m}$

External load: $f(t)=f_{0}sin(t),f_0=25\mathrm{.}00kN,=36.\frac{r}{s}$