Figure 6.47 shows a general arrangement of torsional multi-DOF rotor system with the disc and the shaft damping. From the free body diagram of n^th rotor (Figure 6.48) the following governing equations can be written as

(6.99) (6.100)

For the free vibration, torques R^Tn and L^Tn may be written in the form

(6.101)

where   is the complex amplitude of the torque at n^th disc (which contain both the amplitude and phase information), and w_nf is the torsional natural frequency of the system. The angular displacement takes the form

(6.102)

where  is the complex amplitude of angular displacement at n^th disc (which also contain both the amplitude and phase information). Differentiating equations (6.101) and (6.102) with respect to time and substituting in equations (6.99) and (6.100) leads to

(6.103)

which can be simplified as

(6.104)

with