Does the Addition of Linear Damping Always Cause Instability in a Gyroscopically Stabilized System?

Abstract

This paper answers the question raised in the title of this paper and shows that it is not true for systems in which the damping matrix is indefinite. It introduces a new paradigm in the theory of linear stability that gyroscopically stabilized unstable potential systems can be made stable, and even exponentially stable, by the addition of linear damping. Conceptually, the paper also points to a practical methodology for adding damping to such gyroscopically stabilized potential systems to render them exponentially stable. The methodology involves the simultaneous use of both dissipative damping or negative velocity feedback, and positive velocity feedback. The methodology is illustrated in detail on two-degree-of-freedom gyroscopically stabilized potential systems. In-depth stability analysis of such systems is provided. It is shown that they can always be made exponentially stable by using an uncountably infinite number of appropriate indefinite damping matrices. A connected region is proved to exist in the space of indefinite damping matrices for which such damped gyroscopically stabilized systems are guaranteed to be exponentially stable, and this region of exponential stability is analytically delineated. Numerical studies are provided to corroborate the analytical results.