STABILITY OF ROTOR SYSTEMS - A COMPLEX MODELING APPROACH

The dynamics of a large class of rotor systems can be modelled by a li nearized complex matrix differential equation of second order, Mz + (D + iG)(z) over dot + (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 positi ve definite and D, G and N to be positive semidefinite. The complex se tting is equivalent to twice as large a system of second order with re al matrices. It is well known that rotor systems can exhibit instabili ty for large angular velocities due to internal damping, unsymmetrical steam flow in turbines, or imperfect lubrication in the rotor bearing s. Theoretically, all information on the stability of the system can b e obtained by applying the Routh-Hurwitz criterion. From a practical p oint of view, however, it is interesting to find stability criteria wh ich are related in a simple way to the properties of the system matric es in order to describe the effect of parameters on stability. In this paper we apply the Lyapunov matrix equation in a complex setting to a n equivalent system of first order and prove in this way two new stabi lity results. We then compare the usefulness of these results with the more classical approach applying bounds of appropriate Rayleigh quoti ents. The rotor systems tested are: a simple Laval rotor, a Laval roto r with additional elasticity and damping in the bearings, and a number of rotor systems with complex symmetric 4 x 4 randomly generated matr ices.