Introduction

In this chapter, the stability of rotating machinery is considered. In stable systems, an initial disturbance decays to zero in the absence of excitation forces. By contrast, in an unstable system, the response grows, producing a large and undesirable response that may damage a machine. A simple example of instability is the motion of a pendulum. One equilibrium position is when the pendulum hangs vertically downward. This position is stable because if the pendulum is slightly displaced, it returns to the equilibrium position. In contrast, there is an equilibrium position when the pendulum is balanced vertically upward. This position is unstable because any slight disturbance from the vertical causes the pendulum to move away from the vertical and, in fact, rotate to the lower equilibrium position. For a linear system with constant coefficients, instability may be determined by considering the eigenvalues, computed in the usual way. Thus, the same or similar calculations used to determine eigenvalues of a system also provide a user with information about the stability of the system. As demonstrated in Chapter 2, the imaginary part of the eigenvalue gives the frequency of free oscillations, whereas the real part determines how rapidly the oscillations decay. The oscillations decay only if the real part of the eigenvalue is negative. A zero real part of the eigenvalue gives an undamped response in which the magnitude of the free oscillation remains constant and a positive real part causes the oscillation to grow.