Lyapunov stability of a fractionally damped oscillator with linear (anti-)damping

30. November 2020

International Journal of Nonlinear Science and Numerical Simulation, Vol. 21(5), pp. 425-442, 2020

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.


Dieses Bild zeigt Matthias Hinze

Matthias Hinze

Dieses Bild zeigt Remco I. Leine

Remco I. Leine

Prof. Dr. ir. habil.


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