Criteria of asymptotic stability for general linear mechanical systems are investigated. It is shown that the inequality first derived by Metelitsyn (1952) is a sufficient but not necessary condition for asymptotic stability. We argue that this inequality is of little use in applications. Metelitsyn's theorems based on his inequality as well as critical comments in the literature on these theorems are analysed. Practical sufficient stability criteria are obtained in terms of extreme eigenvalues of the system matrices. This analysis is of special value for rotor systems in a complex setting which is demonstrated by three examples. (C) 2004 Elsevier Ltd. All rights reserved.