Stability of rotor systems: A complex modelling approach
type: Item
instance of: scholarly article
title: Stability of rotor systems: A complex modelling approach
zbMATH Open document ID: 0912.34047
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1999-05-18T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 34A30
zbMATH DE Number: 1209916
zbMATH Keywords: Rayleigh quotients
zbMATH Keywords: rotor systems
zbMATH Keywords: stability
zbMATH Keywords: Routh-Hurwitz criterion
zbMATH Keywords: Lyapunov matrix equation
zbMATH Keywords: Laval rotor
author: Wolfhard Kliem
author: Christian Pommer
author: Jakob Stoustrup
reviewed by: Vasile Dragan
MaRDI profile type: Publication
OpenAlex ID: W2131646240
DOI: 10.1007/S000000050113
description: scientific article; zbMATH DE number 1209916