Metelitsyn´s inequality and stability criteria in mechanical problems

Asymptotic stability criteria for general linear mechanical systems are studied. It is shown that the inequality first derived by Metelitsyn (1952) is a sufficient but not a necessary condition for asymptotic stability. We argue that this inequality is of little use in applications. The theorems of Metelitsyn based on his inequality as well as critical comments in the literature on these theorems are analyzed. 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.