Question $3-$Dynamic deflection

$[10$ marks$]$

A shaft, $500$ mm long supports a load of $12$ kg $.$ The shaft is $25$ mm diameter and simply

supported at each end. E $($Young$\text{'}$s Modulus$)$ of the shaft material is $200$ GPa and the

eccentricity of the central mass is $0\mathrm{.}25$ mm $.$ Neglecting the mass of the shaft and neglecting

damping, determine:

a$)$ The natural frequency of the shaft assembly,

b$)$ The dynamic deflection of the shaft due to the eccentricity of the mass,

c$)$ the maximum stress in the shaft at $1500$ rpm $.$