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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.

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 [

The 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. While SMAs are introduced in three-leg platform, precise movement and maintaining the position can be easily achieved but the challenges are enormous.

Major challenge can be identified as constitutive model for representing the deformation behavior of SMAs. In order to explore all potentialities of SMAs, there is an increasing interest in the development of mathematical models capable of describing the main behaviors of these alloys. SMA thermomechanical behavior can be modeled by either microscopic or macroscopic points of view. Using Devonshire’s theory, the unique properties of SMAs were modeled by Falk [

In 1984, Grebogi [

The three-leg supporter system does not need to be subjected to periodic excitation; hence, investigation for quasi-periodic excitation provides exact scenario of real-time situations. Coexistence of two or more attractors is not good for a mechanical system; it intricates the design and control of the system. Observing how many coexisting attractors are present in the system and investigating the parameter range and corresponding initial conditions are very important. Many literatures showed the coexistence of attractors in mechanical systems and studied their behaviors [

Motivated by the aforementioned discussion, in this paper, we analyzed the three-leg supporter with shape memory alloy for quasiperiodic forcing. The quasi-periodic excitation is achieved by taking golden ratio of irrational number. The rest of this paper is organized as follows.

In this paper, we analyze a three-leg supporter as shown in Figure _{1} and _{2}, respectively. The top disk is supported by three SMA rods having length

Three-leg supporter with shape memory alloys (SMAs).

The masses of SMA rods are considerably less, so we assume that the mass of the structure is fully concentrated at the rigid mass disk. Applying force

The equation of motion is derived [

Geometrical relations are considered in the following way:

The legs are made up of shape memory alloys (SMAs). There have been a horde of literatures that proved that SMAs possess nonlinear behavior because of their hysteretic nature while subjected to temperature changes. In order to consider such phenomenon, various constitutive models were developed. There are different ways to describe the thermomechanical behavior of SMAs. Here, we considered the polynomial constitutive model proposed by Falk [

By substituting geometrical relations equation (

The necessary condition to get quasi-excitation,

The dimensionless equation is derived from the equation of motion:

In order to find the equilibrium point, we considered an unforced system from equation (

For the parameter values,

We derived the Jacobian matrix for equation (

The characteristic equation is formulated for the Jacobian matrix on the equilibrium point:

Corresponding eigenvalues are found as

and we have eigenvalues with one positive and negative value; hence, the equilibrium point can be a saddle node.

In this paper, we analyzed system (

In this case, we considered low temperature

Phase portrait of system (

The Lyapunov exponents are estimated using Wolf’s algorithm [

Lyapunov exponents. (a) For

We extend our investigation for special properties; in Figure

Chaotic attractors for initial conditions of [0.6, 0] and [−0.5, 0].

Tori attractors for initial conditions of [0.5, 2] and [−0.5, −2].

System (

(a) Bistability of system (

The bifurcation plot in Figure

(a) Bistability of system (

A necessary tool for analyzing the bistability (coexistence of attractors) is the basin of attraction. All attractors, whether they are periodic oscillation, attracting tori, or strange attractors, are surrounded by a basin of attraction representing the set of initial conditions in the state space whose orbits approach and map out the attractor as time approaches infinity. Figure

Cross section of the basins of attraction of the two coexisting attractors in the

The intermediate temperature

Phase portrait of system (

Phase portrait of system (

However, variation in parameter

Bifurcation diagram for

Cross section of the basins of attraction of the two coexisting attractors in the

In this case, the response of the system in higher temperature

Phase portrait of system (

Bifurcation diagram of system (

Bifurcation diagram of system (

Cross section of the basins of attraction of the two coexisting attractors in the

In this paper, a quasi-periodically excited three-leg supporter with shape memory alloy is investigated for bistability behavior. The existence of chaotic strange attractor and an attracting tori has been evolved and depicted in phase portraits; hence, it is clear that considering quasi-periodic excitation holds the property of bistability. Chaotic strange attractor is confirmed by showing positive Lyapunov exponents. The presence of symmetric behavior is revealed and shown for the corresponding initial conditions; this kind of behavior is interesting for controlling the system. Bifurcation plots presented in Section

The data used to support the findings of this paper are included within the manuscript.

The authors declare that there are no conflicts of interest regarding the publication of this paper.