Modeling of dynamics of gyroscopic systems

The amplitude of vibrations of a rotating system operating near or at its critical speed grows without bound. This poses a serious problem for the rotating machinery industry from the perspective of design, maintenance, vibration troubleshooting and stability of the rotating equipments. In this thesis, we wish to develop an exact method of analysis for the system of a mass-less symmetrical shaft rotating in symmetrical bearings (SSSB) with finite displacement at bearing end. The method so developed is expected to be extended to the SSSB model of a shaft having mass. The stability analysis is performed for both the systems. The parametric study of the effect of mass, damping and stiffness characteristics of the shaft-bearing system on the stability of the system is also carried out. This thesis closely examines the works carried out by D. M. Smith (1933) for the SSSB case and the destabilizing, counter-intuitive phenomenon of internal damping of rotating shaft mentioned by Crandall. As a starting point, we analyze the undamped SSSB system with no displacement at bearing end, i.e., the system rotates in clamped roller bearings. This system serves as an introductory model for the mass-less SSSB system and for shaft with mass SSSB system. The basic theory to derive the equations of motion for this system is based on the development of equations of relative position, velocity and acceleration. The powerful concept of the transformation of coordinates from rotational coordinate system to stationary coordinate system (or vice versa) is also showcased. The thesis starts with an introduction to the concept of stability for systems. The concept of stability culminates in the determination of the stability boundary of the system which is based on the nature of poles of the system. The conclusions and future work are reported in the end