Exponential Stability of Gyroscopically Stabilized Multi-Degree-of-Freedom Unstable Potential Systems Using Linear Damping

Abstract

This paper shows that -degree-of-freedom generic unstable potential systems, whose potential matrices have an even number of negative eigenvalues and which are gyroscopically stabilized, can always be made exponentially stable through the use of an uncountably infinite number of indefinite damping matrices. A step-by-step methodology is provided for gyroscopic stabilization of such unstable potential systems that guarantees their exponential stability through the simultaneous use of positive and negative velocity feedback. Dissipative damping and positive velocity feedback are shown to constructively cooperate to bring about such stability. In contrast to the well-known Kelvin–Tait–Chetaev paradigm, which states that gyroscopically stabilized unstable potential systems are always made unstable in the presence of the slightest dissipation of energy, the paper points to a new paradigm, which states that the proper simultaneous dissipation and infusion of energy into generic gyroscopically stabilized unstable potential systems guarantees their exponential stability. What is meant by “generic” is explicitly stated, and such generic systems are shown to include most real-life -degree-of-freedom unstable potential systems.