Asymptotic Behavior of an Unforced Duhem-Type Hysteretic Oscillator

TL;DR

The paper analyzes the asymptotic behavior of an unforced mechanical oscillator with a viscoelastoplastic Duhem-type hysteretic element. It develops a Lyapunov-based framework using $\mathcal{V}(x,z,v) = \int_0^x h_1(s) ds + \int_0^z h_2(s) ds + \tfrac{1}{2} v^2$ to prove global existence, uniqueness, and boundedness, with $\dot{\mathcal{V}} \le 0$. LaSalle's invariance principle then implies every trajectory converges to the equilibrium set $\mathcal{E}$, with $v(t) \to 0$ and $(x(t),z(t)) \to \mathcal{E}_{xz}$. As a special case, the Bouc-Wen oscillator is shown to fit into the framework, where under Class I parameters ($A>0$, $\gamma \in (-\beta,\beta]$) there exist limiting values $x_{\infty}, z_{\infty}$ satisfying $\alpha x_{\infty} + (1-\alpha) D z_{\infty} = 0$ in transformed coordinates. These results establish stability of unforced hysteretic oscillations and provide a basis for extending the analysis to broader Duhem models and disturbances.

Abstract

The article describes fundamental analytical properties of an unforced mechanical oscillator with a Duhem-type viscoelastoplastic hysteretic element. These properties include global existence of solutions, uniqueness of solutions, and convergence of each solution to an equilibrium point.

Theorems & Definitions (50)

• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Proposition 2.5
• proof