Rotating elements supported on journal bearings are widely encountered structures in engineering practice. Most commonly, these are asymmetrically manufactured and loaded rigid rotors transmitting torque and carrying transverse as well as axial forces. Nowadays, despite high operational demands and high rotational velocities, such systems are still expected to exhibit stable working, even in the presence of small assembly deviations, light unbalance or external disturbances. The surrounding environment of a rotating machine may interact with it by kinematic excitation from vibrating foundation. This, in turn, may lead to hazardous response and the onset of irregular and chaotic motion of the rotor. The subject of the study is to find and analyze regions of the occurrence of such vibrations in the system of a rigid rotor supported in journal bearings. The bearings themselves are assumed to be non-perfectly mounted in the housing, i.e., their sleeves are inserted in rings possessing some viscoelastic properties. These properties are treated as variable parameters, and the aim is to move the regions of irregular and chaotic vibration outside the operational regime (angular velocity). The adjustability of the viscoelastic parameters may be realized by incorporation of smart materials such as piezoelectric or magnetorheological ones. The considered system is an asymmetric rigid rotor supported on two journal bearings subject to a steady kinematic excitation. The system is described by eight coupled nonlinear ordinary differential equations of motion. Results of the examinations prove that by selecting an appropriate magnitude of damping and stiffness of the bearing mountings, it is possible to enlarge the region of stable operation of the rotating system and thus secure its safety. This, however, does not mean the elimination of chaotic response at all, but only a shift of it outside the range of operational rotation speed.