Stability of rotor systems: A complex modelling approach
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 a system twice as large of second-order with real matrices. It is well-known that rotor systems can exhibit instability for large angular velocities due to internal dumping, 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, it is interesting to find stability criteria which are related in a simple way to properties of the system matrices in order to describe the effect of parameters on stability. In this paper, the Lyapunov matrix equation in a complex setting is applied to an equivalent system of first order and in this way two new stability results are derived. The usefulness of these results is compared 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 \(4\times 4\) randomly generated matrices.
scientific article; zbMATH DE number 1209916