Instability Analysis of Simple Rotor Systems

In most of the chapters in Volume I, except Chapter 3 , we studied how to obtain the free and forced responses of rotor-bearing systems in different modes of vibrations (e.g., the torsional and transverse vibrations). The main aims of these chapters were to obtain natural whirl frequencies, mode shapes, critical speeds, and unbalance responses. The unbalance response analysis (sometimes referred to as harmonic analysis) presented can be extended to other types of periodic forces with the help of the Fourier series, especially for linear systems, where the principle of superposition holds good. Various methods, especially suited for analyzing complex rotor systems (apart from general methods of vibration analysis, like Newton’s second law of motion, Lagrange’s equation, and Hamilton’s principle), have been dealt with in great detail from the fundamentals (e.g., the influence coefficient, transfer matrix, and finite element methods). In the next two chapters, we will explore another kind of phenomenon in rotor-bearing systems called instability , which might cause the catastrophic failure of systems. In certain circumstances, depending upon the design, some machines may be prone to instability. This means that machine vibrations set in, even in the absence of unbalance effects, resulting in elevated levels of noise and component stress, and a corresponding reduced fatigue life. In linear systems, the magnitude of these vibrations tends toward infinity, although in practice shaft vibrations are often limited by the system nonlinearity. In the present chapter, various kinds of instability will be studied. Such machine instabilities may originate from many sources, including fluid-film bearings, seals, asymmetry of shaft stiffness, disk inertia asymmetry, internal friction between mating components, rubs, the steam whirl, aerodynamic forces, surge in pumps, and cage in rolling element bearing. A designer’s problem is to investigate the possibility of machine instability, and to change the appropriate machine design parameters to ensure that potential unstable modes of operation lie outside the normal operating regime of the machine. Apart from these concerns, when rotors are subjected to angular accelerations (uniform or variable depending upon the unlimited or limited power of the drive, respectively), transient responses are generated, and a study of such transient responses is of practical importance. The aim of this chapter is to understand various kinds of instability with a simple, single mass rotor model and in some cases with a continuous shaft model. In the subsequent chapter, we will explore methods of predicting instability in large rotor-bearing systems, especially with finite element methods. Also, in Chapter 8 , instability due to active magnetic bearings will be analyzed.