Stability of rotor systems: A complex modelling approach

Abstract

The dynamics of a large class of rotor systems can be modelled by a linearized complex matrix differential equation of second order, \(M\ddot z+(D+iG)\dot z+(K+iN)z=0\), where the system matrices M, D, G, K and N are real symmetric. Moreover M and K are assumed to be positive definite and D, G and N to be positive semidefinite. The complex setting is equivalent to twice as large a system of second order with real matrices. It is well known that rotor systems can exhibit instability for large angular velocities due to internal damping, unsymmetrical steam flow in turbines, or imperfect lubrication in the rotor bearings. Theoretically, all information on the stability of the system can be obtained by applying the Routh-Hurwitz criterion. From a practical point of view, however, it is interesting to find stability criteria which are related in a simple way to the properties of the system matrices in order to describe the effect of parameters on stability. In this paper we apply the Lyapunov matrix equation in a complex setting to an equivalent system of first order and prove in this way two new stability results. We then compare the usefulness of these results with the more classical approach applying bounds of appropriate Rayleigh quotients. The rotor systems tested are: a simple Laval rotor, a Laval rotor with additional elasticity and damping in the bearings, and a number of rotor systems with complex symmetric \(4\times4\) randomly generated matrices.