Modelling Viscoelastic Materials Is Always Difficult Since Under Dynamic Deformation Such Materials Store Energy As Well As Dissipate It To The Thermal Domain. Whereas Modelling The Elastic Behaviour Is Easy, Modelling The Energy Dissipation Mechanism Poses Difficulty. By And Large Every Material May Be Classed As Viscoelastic As Damping Is Inherent In Every Material; A Material May Be Termed As Strongly Viscoelastic (E.G., Polymers) In Which The Dissipation Is Significant In Comparison With The Energy Stored And Weakly Viscoelastic (E.G. Metals) Otherwise. This Work Presents A Theoretical Study Of The Dynamics Of A Generally Viscoelastic (Strong Or Weak) Axi-Symmetric Rotor-Shaft System, In Terms Of Stability Limit Of The Spin Speed And Unbalance Response Amplitude At A Disc Location As Two Indicators Where Instability Is Assumed To Be Caused By The Rotary Force Generated By Internal Material Damping In The Rotor System As A Result Of Rotor Spin. The Study Of Dynamics Thus Needs A Reliable Time Domain Rather Than A Frequency Domain Model For The Material Constitutive Relationship To Get The Equations Of Motion. Though Frequency Domain Models (In Terms Frequency Dependent Storage Modulus And Loss Factors Or Complex Modulus) Of Several Materials Including Polymers Are Already Available In Literature Yet These Are Insufficient For An Analysis Of Stability As It Needs The Eigenvalue Analysis. Time Domain Models Are More Complete In This Respect Than The Frequency Domain Models As The Former Is Useful For Both Transient And Frequency Response Analyses Whereas The Latter Models Are Only Useful For Frequency Response Analysis. Getting, Rather Extracting, Time Domain Models Is Challenging Unlike The Frequency Domain Model, Which Are Easily Available. For Linear Viscoelastic Solids (1) Multi-Element Spring-Dashpot Models And (2) The Internal Variable Approach Are Found In The Literature. However, Under The Second Category The Augmenting Thermodynamic Field (Atf) And Anelastic Displacement Field (Adf) Approaches Were Found Especially Suitable In Comparison Abstract Ii With Others Because Energy Dissipation Is Modelled By Including Additional Coordinates Considered As Internal Variables (Which Are The Atf And Adf), Which Are Continuous From Element To Element For A Continuum And Also The Mode