Figure $2$ shows a three degree of freedom spring$-$mass system, where $m_1=m_2=m_3=10kg$ and $k=1800\frac{N}{m}.$

a$)$ Derive the equation of motion for the given system

$[7$ Marks$][$CO$2,$ PO$3,$ C$4]$

b$)$ Determine the natural frequencies and mode shape of the system.

$[18$ Marks$][$CO$2,$ PO$3,$ C$4]$

Figure $2$