Alternative method (indirect method)
From the flexible mode of two-disc rotor it is clear that there should be a node in between two discs on the shaft, from which we can write

(6.17)

From the information that in this flexible mode, the two-disc free-free rotor system could be treated as two single-disc cantilever rotor and all would have the same natural frequency as the original rotor system, we write

(6.18)

Equations (6.17) and (6.18) could be solved to get

(6.19)

Above expression gives the node location and now with equation (6.18), natural frequency of the two-disc free-free rotor system can be obtained and that will be same as either of the single-disc cantilever rotor system.

Example 6.2 Determine torsional natural frequencies and mode shapes for a rotor system mounted on frictionless supports as shown in Figure 6.8. Neglect the mass of the shaft and assume discs as thin and rigid. The shaft is 1 m of length, 0.015m of diameter, and of modulus of rigidity. Discs have polar mass moment of inertia as .

Solution: The stiffness of the shaft can be obtained as

The natural frequency is given as

The relative displacements would be