Show that the natural frequencies of the system shown in Fig. 6.6(a), with (k_{1}=3 k), (k_{2}=k_{3}=k, m_{1}=4

Question:

Show that the natural frequencies of the system shown in Fig. 6.6(a), with $k_{1}=3 k$, $k_{2}=k_{3}=k, m_{1}=4 m, m_{2}=2 m$, and $m_{3}=m$, are given by $\omega_{1}=0.46 \sqrt{k / m}$, $\omega_{2}=\sqrt{k / m}$, and $\omega_{3}=1.34 \sqrt{k / m}$. Find the eigenvectors of the system.

Figure 6.6(a):-