Question $5.(20$ marks$)$

Consider the system shown in the Figure $5.$ Given that $J_1=0\mathrm{.}6kg*m^2,J_2=1kg*m^2,J_3=2kg*m^2,J_4=3kg.m^2,$ and $K_1=1\mathrm{.}0{10}^{5}N\frac{m}{r}ad,K_2=K_3=2\mathrm{.}0{10}^{5}N\frac{m}{r}ad,$ determine the following:

a$)$ The equations of motion of the $4$ discs,

b$)$ The mode shapes of the system,

$(4$ marks$)$

$(10$ marks$)$

c$)$ The natural frequencies of the sustem.

$(6$ marks$)$

Figure $5$