Figure $1$: Spring Pendulum.

Question $2[25$ points$]$

Determine the free vibration response of a two$-$degree$-$freedom system shown in Figure $2$ using modal analysis, given the following parameters:

Masses: $m_1=2kg,m_2=5kg$

Stiffness: $k_1=10\frac{N}{m},k_2=20\frac{N}{m},k_3=5\frac{N}{m}$

Initial conditions: $x_1(0)=0\mathrm{.}1m,x_2(0)=0m$

Initial velocities: $x_1(0)=0,x_2(0)=5\frac{m}{s}$

Figure $2$: Two degree of freedom system.

TOTAL MARKS $=[60]$