which means disc 1 would have 1.5 times angular displacement amplitude as compared to disc 2, however, in opposite direction. The node position can be obtained as
Hence, we get the node location as 
It can be verified that equivalent two single-mass cantilever rotors will have the same natural frequency, as
and
so that
and
Answer
6.3 A Two-Disc Torsional Rotor System with a Stepped Shaft
Figure 6.9(a) shows a stepped shaft with two large discs at ends with Ip is the polar mass moment of inertia (subscripts 1 and 2 represent the left and right hand side discs, respectively). It is assumed that the rotor has free-free boundary conditions and the polar mass moment of inertia of shaft is negligible as compared to two large discs at either ends of the shaft. In such cases the actual shaft should be replaced by an unstepped (uniform) equivalent shaft for the purpose of the analysis as shown in Fig. 6.9(b). The equivalent shaft diameter may be same as the smallest diameter of the real shaft (or any other diameter). The equivalent shaft must have the same torsional stiffness as the real shaft.
Figure 6.9 Two discs with (a) a stepped shaft (b) an equivalent uniform shaft