A flexible shaft-rotor system is shown in Figure P5.9. The rotors are identical and have the same…

A flexible shaft-rotor system is shown in
Figure P5.9. The rotors are identical and have the same mass m and the same
polar radius of gyration r. The shaft segments have the same torsional
stiffness k.

a. Formulate the modal vibration problem
for this system and express it in terms of a mass matrix M and a stiffness
matrix K.

b. Determine the natural frequencies and
mode shapes of the system.

c. Construct the modal matrix Ψ and
verify that both ΨTMΨ and ΨTKΨ are diagonal.

d. Sketch the mode shapes of the system.