Problem $2(15$ points$)$:

For a $2-$DOF system, as shown in Figure $2,$ with $J_1=J_0,J_2=2J_0,k_{t1}=k_t$ and $k_{t2}=k_t$ :

Determine the equations of motion of this dynamical system.

Find the natural frequencies and modal vectors $($or mode shapes$).$

Determine the response of the system when $k_t=1000Nm,J_0=2kg{m}^2,$ and the initial conditions are ${}_1(0)=1,{}_2(0)=-1,$ and zero initial angular velocities.

Figure $2$