Multi Degree of Freedom Systems
(i) When there are several numbers of discs in the rotor system it becomes a multi degree of freedom (MDOF) system.
(ii) When the mass of the shaft itself may be significant then the analysis described in previous section (i.e. single or two-disc rotor systems) is inadequate to model such systems, however, they could be extended to allow for more number of lumped masses (i.e., rigid discs) but resulting mathematics becomes cumbersome.
Alternative methods for analysis of MDOF systems are
Transfer matrix method and
Finite element method
(i) Transfer Matrix Method: A multi-disc rotor system supported on frictionless supports is shown in fig. 1, fig. 2(a), 2(b) show free body diagrams of a shaft and a disc, respectively. At a particular station in the system, we have two state variables: θ1 angular twist and T1 torque. Now in subsequent sections we will develop relationship of these state variables between two neighbouring stations and which can be used to obtain governing equation of the whole system.
1. Point matrix : The equation of motion for the disc 2 is given by (see Figure 2(b)).
For free vibrations, angular oscillations of the disc is given by
where ωn is the natural frequency and 
is the angular displacement amplitude of disc 2. Substituting equation (2) into equation (1), we get
Since angular displacements on the either side of the disc are equal, hence
Equations (3) and (4) can be combined as:
where {S}2 is the state vector at station 2 and [P]2 is the point matrix for station 2.
2. Field matrix: For shaft element 2 as shown in Figure2(a), the angle of twist is related to its torsional stiffness and to the torque, which is transmitted through it, as
Since the torque transmitted is same at either end of the shaft, hence
Combining equations (7) and (8), we get
which can be written as
where [F2] is the field matrix for the shaft element 2.
Equation (6) can be written, noting equation (10), as
Where [U]2 is the transfer matrix, which relates the state vector at right of station 2 to the state vector at the right of station 1.
On the same lines, we can write
Where [T] is the overall system transfer matrix. The overall transformation can be written as
For free-free boundary conditions, the each end of the machine torque transmitted through the shaft is zero, hence
On using equation (14) into equation (13), the second set of equation gives
which gives
Since θ0 cannot be zero in general.
Equation (15) is satisfied for some values of ωn and those are system natural frequencies.
These roots of ωn may be found by any root-searching numerical analysis technique.
The angular twist can be determined for each value of ωn from first set of equation (13), as
On taking 
as reference value, we get
In equation (16), t11 is function of ωn so that for each value of ωn it gives different value of 
.By
Using equation (12) relative displacements at other stations can be obtained for different value of ωn.
These relative displacements for a particular value of ωn gives the corresponding mode shape.
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