Route to Chaos and Bistability Analysis of Quasi-Periodically Excited Three-Leg Supporter with Shape Memory Alloy

Abstract

In this paper, the effect of quasi-periodic excitation on a three-leg supporter configured with shape memory alloy is investigated. We derived the equation of motion for the system using the supporter configuration and polynomial constitutive model of the shape memory alloys (SMAs) based on Falk model. Two sets of parameters and symmetric initial conditions are used to analyze the system. The system responded with a chaotic attractor and a strange nonchaotic attractor. Coexistence of these attractors is studied and discussed with corresponding phase portrait, bifurcation plot, and cross section of basin of attraction. We confirm the quasi-periodic excitation results with generation of strange nonchaotic attractors as discussed in the literature. The special properties like symmetricity and bistability are revealed and the parameter ranges of existence of such behaviors are discussed. The system is analyzed for different phases and the existence of bistability in martensite phase and transition phase is explained. While the system enters into austenite phase, the bistability behavior vanishes. The results provide insight knowledge into dynamical response of a quasi-periodically excited SMA leg support system and will be useful for design improvements and controller design.