Question $3(25$ points$)$

Consider a cantilever beam shown with bending stiffness $El$ constant through the beam and

the lumped masses of $m\frac{L}{2}$ located at every $\frac{L}{2}$ distance along the beam, as shown in the

figure below. The influence coefficients for deflection are given below. Considering the

frequency equation $($matrix$)$ determine the natural circular frequencies $$ and the mode

shapes $.$ Draw the mode shapes of the beam. Show all calculation details. If vou get a negative

frequency, take the positive value.

$E=2{10}^3$ksi;$I=4{10}^{3}i{n}^4$;$L=L$;$m=n^{-}$

$f_{11}=\frac{3L^3}{2(EI)},f_{12}=\frac{4L^3}{3(EI)}$

$f_{21}=\frac{4L^3}{3(EI)},f_{22}=\frac{2L^3}{4(EI)}$