Nonlinear Dynamics and Chaotic Behavior in an Oscillator Connected to Shape Memory Alloy

The dynamic response of shape memory alloy (SMA) systems and structures often exhibits a complex behavior due to their intrinsic nonlinear characteristics. The key characteristics of SMAs stem from adaptive dissipation associated with the hysteretic loop and huge changes in mechanical properties caused by the martensitic phase transformation. These exceptional properties have attracted attention of many researchers in various engineering fields from biomedicine to aerospace. One of the possible responses that may happen in SMA structures is the chaotic response, which can lead to a massive change in the system behavior. Moreover, such a system is highly sensitive to initial conditions. Therefore, its analysis is essential for a proper design of SMA structures. The present article discusses nonlinear dynamics and chaotic behavior in a one-degree-of-freedom (1DoF) oscillator connected to SMA at constant working temperature and pseudo elastic region. Equation of motion is formulated, using the Brinson constitutive model. Combination of structural equations of SMA and dynamical and kinematic relations, as well as forth-order Runge-Kutta scheme are employed to solve the equation governing the oscillator motion. Free and forced vibrations under the influence of harmonic stimulation force and in a wide range of excitation frequencies are presented in the form of various numerical examples. Different tools for detecting chaos, including, phase plane, time response, frequency response, Lyapunov exponent, and Poincare map are used to determine the type of motion. Numerical simulations demonstrate a wide range of periodic, quasi periodic, and chaotic responses for certain values of excitation frequencies, which is a reason for the proper understanding of the behavior of these systems.