The simple supported beatn is as shown in Fig. $3.$ An object with the mass of $m$ is placed in the middle of

a beam. This beam mass of M is significant which can not be ignored. The static deflection of a beam of

modulus of E$,$ the cross$-$sectional moment of inertia with $1,$ the length with $L,$ the beam stiffness $k=\frac{4\text{B}\text{E}\text{I}}{k^3}$

and the mass density of $.$ The static deflection is given by

$u(y)=\frac{\text{m}\text{a}\text{y}}{48EI}(3L^2-4y^2),at0y\frac{L}{2}$

$u$ is the deflection of the beam perpendicular to its length and $y$ is the distance along the length from the

left fixed end.

Calculate the natural frequencey of this simple supported beam system.