Free Torsional Vibrations of a Two Rotor System

          Consider a two rotor system as shown in Fig. (a). It consists of a shaft with two rotors at its ends (free ends). In this system, the torsional vibrations occur only when the two rotors A and B move in opposite directions i.e. If A moves in anticlockwise direction then B moves in clockwise direction at the same instant and viceversa. It may be noted that the two rotors must have the same frequency.

          Also at some point along the shaft, there will be a section of the shaft which remains unaffected by the vibrations. This section is known as 'Node'. We see from Fig. (b). that the node lies at point N. The shaft behaves as if the shaft clamped at node and the two sections vibrate as two separate shafts with the same frequency but opposite in phase or opposite direction. Hence the sections NP and NQ can be considered as two separate shafts fixed at N as shown in Fig. (c) & (d). 

Let       l = Length of shaft,

             l_A = Length of part NP i.e. distance of node from rotor A,

             l_B = Length of part NQ, i.e. distance of node from rotor B,

             I_A = Mass moment of inertia of rotor A,

             I_B = Mass moment of inertia of rotor B, d = Diameter of shaft,

             J = Polar moment of inertia of shaft, and

             C = Modulus of rigidity for shaft material.

Natural frequency of torsional vibrations for rotor A, 

and natural frequency of rotor B, 

since      f_nA = f_nB, therefore 

We also know that 

                              l =     l_A +  l_B

From the equations (iii) and (vi) , we may find the value of and  and hence the position of node. Substituting the values of l_A or l_B in equations (i) or (ii), the natural frequency of torsional vibration for a two rotor system may be evaluated.

Note : The line eNf in Fig. (b). is known as elastic line for the shaft.