Document Type

Thesis

Degree Name

Master of Science (MSc)

Department

Mathematics

Faculty/School

Faculty of Science

First Advisor

Roderick Melnik

Advisor Role

Thesis Supervisor

Abstract

Shape memory alloys (SMAs) belong to an interesting type of materials that have attracted the attention of scientists and engineers over the last few decades. They have some interesting properties that made them the subject of extensive research to find the best ways to utilize them in different engineering, biomedical, and scientific applications. In this thesis, we develop a mathematical model and analyze the behavior of SMAs by considering a one degree of freedom nonlinear oscillator consisting of a mass connected to a fixed frame through a viscous damping and a shape memory alloy device. Due to the nonlinear and dissipative nature of shape memory alloys, optimal control and Lyapunov stability theories are used to design a controller to stabilize the response of the one degree of freedom nonlinear oscillator. Since SMAs exist in two phases, martensite and austenite, and their phase transformations are dependent on stress and temperature, this work is presented in two parts. The first part deals with the nonlinear oscillator system in its two separate phases by considering a temperature where the SMA exists in only one of the phases. A model for each phase is developed based on Landau-Ginzburg-Devonshire theory that defines the free energy in a polynomial form enabling us to describe the SMAs shape memory effect and pseudoelasticity. However, due to the phenomenon of hysteresis in SMAs, the response of the nonlinear oscillator with a SMA element, in either phase, is chaotic and unstable. In order to stabilize the chaotic behavior, an optimal linear quadratic regulator controller is designed around a stable equilibrium for the martensitic and the austenitic phases. The closed-loop response for each phase is then simulated and computational results are presented. The second part of the thesis deals with the entire system in its dynamics by combining the two phases and taking into account the effect of temperature on the response of the system. Governing equations for the system's thermo-mechanical dynamics are constructed using conservation laws of mass, momentum, and energy. Due to the complexity of the derived thermo-mechanical model, and the need to control the nonlinear oscillator, a model reduction based on the Galerkin method is applied to the new system in order to derive a low-dimensional model which is then solved numerically. A linear feedback control strategy for nonlinear systems is then implemented to design a tracking controller that makes the system follow a given reference input signal. The work presented in this thesis demonstrates how SMAs can be modeled by using efficient methodologies in order to capture their behavior, and how SMAs can be made stable and their chaotic behavior can be controlled by using linear and nonlinear control methods.

Convocation Year

2010

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Mathematics Commons

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