Estimating Lyapunov spectrum on shape-memory alloy oscillators considering cloned dynamics and tangent map methods
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(2020) 42:475
TECHNICAL PAPER
Estimating Lyapunov spectrum on shape‑memory alloy oscillators considering cloned dynamics and tangent map methods Daniel M. Barreto Netto1,2 · Alex Brandão2 · Alberto Paiva3 · Pedro M. C. L. Pacheco1 · Marcelo A. Savi2 Received: 10 January 2020 / Accepted: 4 August 2020 © The Brazilian Society of Mechanical Sciences and Engineering 2020
Abstract Mechanical devices built with shape-memory alloys become popular due to thermomechanical coupling associated with solid-state phase transformations. The main characteristics stemming from these transformations are the intrinsic dissipation and the change of thermomechanical properties as a consequence of the phase transformations. Shape-memory alloy oscillators have a nonlinear behavior that can reach responses with different natures such as periodic, quasiperiodic, chaotic, or even hyperchaotic. A proper identification of these behaviors requires the use of dynamical tools, and among them, Lyapunov exponents are one of the most relevant. This work deals with the calculation of the Lyapunov spectrum of shape-memory alloy oscillators. One- and two-degree-of-freedom systems are of concern. Two different approaches are compared: the classical algorithm that uses a tangent space approach; and the cloned dynamics algorithm that is a Jacobian-free approach. Results show that both methods have similar results, allowing the use of cloned dynamics as an interesting alternative procedure since it avoids Jacobian calculations. Keywords Shape-memory alloys · Nonlinear dynamics · Lyapunov exponents · Tangent space · Cloned dynamics
1 Introduction Dynamical systems employing shape-memory alloys (SMAs) have been studied since the 1990s, being characterized by rich and complex responses [1, 2]. The remarkable capacity to develop and recover high strain levels (or huge forces) and the intrinsic dissipation make SMAs to have a Technical Editor: Wallace Moreira Bessa, D.Sc. * Pedro M. C. L. Pacheco pedro.pacheco@cefet‑rj.br Alberto Paiva [email protected] Marcelo A. Savi [email protected] 1
Department of Mechanical Engineering, CEFET/RJ – Centro Federal de Educação Tecnológica Celso Suckow da Fonseca, Rio de Janeiro, RJ 20.271.110, Brazil
2
COPPE ‑ Department of Mechanical Engineering, Center for Nonlinear Mechanics, Universidade Federal do Rio de Janeiro, P.O. Box 68.503, Rio de Janeiro, RJ 21.941.972, Brazil
3
Department of Mechanical Engineering, Universidade Federal Fluminense, Volta Redonda, RJ 27.255.250, Brazil
fast-growing technological interest that is represented by a scaling up of the number of articles published and patents issued for subsequent decades [3]. Dynamical analysis of SMA systems has different aspects that include distinct configurations and also the constitutive description of SMA thermomechanical behavior. Savi [4] presented a review article dealing with some dynamical issues related to SMA systems. Gholampour et al. [5] presented another overview of SMA dynamical applications. As a prototype of t
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