A calculation method incorporating the transfer matrix method and the characteristic-vector locus method has been developed for stability analysis of the self excited vibration of a rotating shaft system with many bearings and discs. The analysis is made for an elastic bearing mass, relatively large damping forces due to types of rotors fit, anisotropic foundation and rotors gyroscopic effect. A two-rotors-model is presented to show the influence of rotor and its gyroscopic 'action, support stiffness characteristics, internal and external damping on stability. A computer solution of the transfer matrix method shows the rotor stability is improved by damped support. The shrinkage fit type rotors are more stable than force fit welding type and adhesion loctite or araldite type. The gyroscopic moment effect of the overhanging disc decreased the stability region of the rotor. A computer solution of the governing equations of motion is presented showing the shaft stability region for various speed ranges and the effect of the bearing stiffness ratio on the stability region. The effect of external damping on the stability 'region of a rotor supported on anisotropic bearings is obtained.