Equations of motion of shaft-rotor system, MATLAB Programming

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Consider the shaft-rotor system shown in Figure. Write down the equations of motion. Taking  I= 1 kgm2 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. 

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