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.
Highlights
For the last two decades, shape memory alloys (SMAs) gained attention of the researchers because of their salient behavior and their versatile application in a wide range of fields [1,2,3,4,5,6,7,8,9,10]
We extend our investigation for special properties; in Figure 4, we show that the system holds symmetric chaotic attractor for initial conditions of [0.6, 0] and [− 0.5, 0], which clearly explains the sensitiveness of the system with initial conditions
A quasi-periodically excited three-leg supporter with shape memory alloy is investigated for bistability behavior. e existence of chaotic strange attractor and an attracting tori has been evolved and depicted in phase portraits; it is clear that considering quasiperiodic excitation holds the property of bistability
Summary
For the last two decades, shape memory alloys (SMAs) gained attention of the researchers because of their salient behavior and their versatile application in a wide range of fields [1,2,3,4,5,6,7,8,9,10]. While we design a mechanical system using SMAs, it is mandatory to investigate the behavior of the system in both martensitic phase and austenitic phase, as well as in transition phase [6, 7, 9]. E three-leg supporter is a mechanical system that is widely used in aircraft industries and machining industries It can provide adequate inclinations with required motion for precise movement of platform. In [22], quasi-periodically excited nonautonomous system is investigated and various shapes of SNAs are discussed. Motivated by the aforementioned discussion, in this paper, we analyzed the three-leg supporter with shape memory alloy for quasiperiodic forcing.
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