It is clear that two nodes are possible at m (Fig. 6.17d and e). While the single node is possible at out of which one is feasible at as discussed in the theory part. It should be noted that mode shapes in Fig. 6.17(b and d) are not to the scale; however, qualitative comparison can be made with the previous example. Quantitatively also it can be observed that they are exactly same.

Now, the natural frequency corresponding to two-node mode can be obtained as

 

The natural frequency corresponding to single-node mode can be obtained as

 

It should be noted that these natural frequencies and the node positions are exactly same as obtained in example 6.4.

 

Fig. 6.17 (a) A three-mass rotor system (b) single node mode shape (c) Equivalent two systems:  two-mass and one- mass cantilever rotor system (d) two node mode shape (e) Equivalent three systems: single-mass two cantilevers and one fixed-fixed end condition rotor system