Figure below shows a $2$D cantilevered beam, length $(L)0\mathrm{.}1m,$ attached to an axial spring with

spring stiffness $(k)$ of $500,000\frac{N}{m}.$ The square cross section of the cantilever beam measures

$0\mathrm{.}01mx0\mathrm{.}01m.$ It is manufactured from steel with density $()7850k\frac{g}{m^3}$ and modulus $($E$)$

$200$GPa.

$I=\frac{1}{12}b{h}^3,$ mass $=AL$

a$)$ Using a consistent mass matrix and using one beam element to model the cantilever

beam, write down the equations of motion for the two degrees of freedom system.

$(14$ marks$)$

b$)$ Determine the first two natural frequencies of vibration of the beam.

$(10$ marks$)$

c$)$ Determine the two corresponding mode shapes

$(8$ marks$)$