Lyapunov exponents estimation for hysteretic systems

INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES

卷:46

Article

Machado, Luciano G.¹  Lagoudas, Dimitris C.¹  Savi, Marcelo A.² 

[1] Texas A&M Univ, Ctr Mech Composties, Dept Aerosp Engn, College Stn, TX 77843 USA

[2] Univ Fed Rio de Janeiro, Dept Mech Engn, COPPE, BR-21941972 Rio De Janeiro, Brazil

关键词: Lyapunov exponents;    Shape memory alloys;    Thermomechanical coupling;    Hysteresis;    Non-linear dynamics;    Chaos;   

DOI  :  10.1016/j.ijsolstr.2008.09.013

来源: Elsevier

【 摘 要 】

This article discusses the Lyapunov exponent estimation of non-linear hysteretic systems by adapting the classical algorithm by Wolf and co-workers [Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A., 1985. Determining Lyapunov exponents from a times series. Physica D 16, 285-317.]. This algorithm evaluates the divergence of nearby orbits by monitoring a reference trajectory, evaluated from the equations of motion of the original hysteretic system, and a perturbed trajectory resulting from the integration of the linearized equations of motion. The main issue of using this algorithm for non-linear, rate-independent, hysteretic systems is related to the procedure of linearization of the equations of motion. The present work establishes a procedure of linearization performing a state space split and assuming an equivalent viscous damping in order to represent hysteretic dissipation in the linearized system. The dynamical response of a single-degree of freedom pseudoelastic shape memory alloy (SMA) oscillator is discussed as an application of the proposed algorithm. The restitution force of the oscillator is provided by an SMA element described by a rate-independent, hysteretic, thermomechanical constitutive model. Two different modeling cases are considered for isothermal and non-isothermal heat transfer conditions, and numerical simulations are performed for both cases. The evaluation of the Lyapunov exponents shows that the proposed procedure is capable of quantifying chaos capturing the non-linear dissipation of hysteretic systems. (C) 2008 Elsevier Ltd. All rights reserved.

【 授权许可】

Free