In this paper, we develop a Lyapunov stability framework for fractionally damped mechanical systems. In particular, we study the asymptotic stability of a linear single degree-of-freedom oscillator with viscous and fractional damping. We prove that the total mechanical energy, including the stored energy in the fractional element, is a Lyapunov functional with which one can prove stability of the equilibrium. Furthermore, we develop a strict Lyapunov functional for asymptotic stability, thereby opening the way to a nonlinear stability analysis beyond an eigenvalue analysis. A key result of the paper is a Lyapunov stability condition for systems having negative viscous damping but a sufficient amount of positive fractional damping. This result forms the stepping stone to the study of Hopf bifurcations in fractionally damped mechanical systems. The theory is demonstrated on a stick-slip oscillator with Stribeck friction law leading to an effective negative viscous damping.
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Matthias Hinze
Dipl.-Math.André Schmidt
Dr.-Ing.Remco I. Leine
Prof. Dr. ir. habil.director