Damping in Torsional Systems
Damping may come from (i) the shaft material and (ii) the torsional vibration damper. The torsional vibration damper is a device which may be used to join together two-shaft section as shown in Figure 1. It transmits a torque, which is dependent upon of the angular velocity of one shaft relative to the other.
Torsional dampers can be used as a means of attenuating system vibration and to tune system resonant frequencies to suit particular operating conditions. The damping in the system introduces phase lag angles to the system displacement and torque.
The displacement and torque parameters must now be represented mathematically both in the in phase and quadrature components (i.e., cosine and sine terms) or in the form of the complex number.
Figure 2 shows a general arrangement of a multi-dof rotor system with dampers.
Equation of motion of nth disc from the free body diagram as shown in Figure 3 and can be written as
Torques RTn and LTn for free vibrations may be written in the following form
while angular displacements take the following form
where ωn is the torsional natural frequency of the system, 
and 
are the inphase and quadrature components of the torque and 
and 
are the in phase and quadrature components of the angular displacement.
Differentiating equations (3) and (4) with respect to the time and substituting in equations (1) and (2) and on collecting in phase and quadrature terms leads to
which can be simplified as
Where [P]n is a point matrix for the disc at station n.
Free body diagram of nth shaft segment
Characteristics of the shaft element at station n as shown in Figure 4 are represented in the equation describing the torque applied to the shaft at the location of disc n , as
While the torque transmitted through the shaft is the same at each ends
Substituting T , 
, 
(from equations (3) and (4)) in equation (7), we get
On separating the in phase and quadrature components, we get
Substituting T, θ and ̇ in equation (8), we get
On separating in phase and quadrature components, we get
Combining equations (9) and (10), we get
which can be written as
which can be simplified as
With 
, where [F]n is a field matrix of the shaft between discs at stations (n-1) and n .
From equations (6) and (12), we get
Where [U]n is the transfer matrix between stations n and (n-1). Remaining analysis will be same for obtaining natural frequencies & mode shapes as discussed in the case of undamped torsional analysis.
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