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Viscoelastic polymeric materials provide good support elements to rotor shaft systems by virtue of their efficiency in dissipating vibratory energy. The in- phase stiffness and loss factor for such materials also change with the frequency of excitation they are subjected to. In this paper frequency dependent characteristics of the polymeric supports have been found by simultaneously minimizing the unbalanced response and maximizing the stability limit speed. This process yields better support characteristics than those obtained by minimizing unbalance response alone. The effects of viscous internal damping in the shaft, support mass and gyroscopic effect due to non-symmetrical location of the disc have been considered in the analysis. A procedure of controlling the slope of the support characteristics versus frequency of excitation has been used and found to be very suitable for obtaining feasible support characteristics. Examples have been presented to justify the above conclusions. Modelling was done by many researchers and incorporated material damping through frequency dependent viscous or frequency independent hysteretic behaviour. Both of these damping models are not sufficient to predict the exact dynamic characteristic. The motivation for using general viscoelastic model arises from a need to capture the influence of both types of damping behaviours. The operator based constitutive relationship is used to incorporate such type of damping. The operator based constitutive relationship is further used to bring down higher order equations of motion of an isotropic viscoelastic rotor by using two distinct approaches, i.e. (a) Classical model (CM) and (b) Finite element model (FEM). Lumped mass approximation is adopted in the CM approach thus it is also known as lumped mass classical model. Whereas, in FEM approach, the distributed effect of the continuum is considered. This paper purely shows the mathematical modelling of various viscoelastic rotor bearing system with support. The mathematical procedure and formulations can be used for any rotors as per their applications.