Consider the shaft-rotor system shown in Figure. Write down the equations of motion. Taking I= 1 kgm^{2} and k=10 kNm/rad, for two special cases (α =0.5 and α = 1000) find as many natural frequencies and modes as possible by using hand calculations. This may require reducing the system into several one or 2 DOF systems using symmetry and anti-symmetry of modes. Solve the full problems and find all 5 natural frequencies and modes with the help of a MATLAB code for each of the two special cases and discuss what you learn from the modes.