Lyapunov Stability of a Fractionally Damped Oscillator with Linear (Anti-) Damping

Abstract: 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.

Object Details

• Location: Deutsche Nationalbibliothek Frankfurt am Main
• Extent: Online-Ressource
• Language: Englisch

References and Relationships

• Bibliographic citation: Lyapunov Stability of a Fractionally Damped Oscillator with Linear (Anti-) Damping ; volume:21 ; number:5 ; year:2020 ; pages:425-442 ; extent:18 ; International journal of nonlinear sciences and numerical simulation ; 21, Heft 5 (2020), 425-442 (gesamt 18)

Contributors, Places and Time

• Creator: Hinze, Matthias ; Schmidt, André ; Leine, Remco I.

Further information

• DOI: 10.1515/ijnsns-2018-0381
• URN: urn:nbn:de:101:1-2503120415391.823063422114
• Rights: Open Access; Der Zugriff auf das Objekt ist unbeschränkt möglich.
• Last update: 15.08.2025, 7:29 AM CEST