THE MATHEMATICAL MODELING OF MAGNETOSTRICTION Katherine Shoemaker A Thesis Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of MASTERS OF ART May 2018 Committee: So-Hsiang Chou, Advisor Kit Chan Tong Sun c 2018 Katherine Shoemaker All rights reserved iii ABSTRACT So-Hsiang Chou, Advisor In this thesis, we study a system of differential equations that are used to model the material deformation due to magnetostriction both theoretically and numerically. The ordinary differential system is a mathematical model for a much more complex physical system established in labo- ratories. We are able to clarify that the phenomenon of double frequency is more delicate than originally suspected from pure physical considerations. It is shown that except for special cases, genuine double frequency does not arise. In particular, simulation results using the lab data is consistent with the experiment. iv I dedicate this work to my family. Even if they don’t understand it, they understand so much more. v ACKNOWLEDGMENTS I would like to express my sincerest gratitude to my advisor Dr. Chou. He has truly inspired me and made my graduate experience worthwhile. His guidance, both intellectually and spiritually, helped me through all the time spent researching and writing this thesis. I could not have asked for a better advisor, or a better instructor and I am truly appreciative of all the support he gave. In addition, I would like to thank the remaining faculty on my thesis committee: Dr. Tong Sun and Dr. Kit Chan, for supporting me through their instruction and mentorship. I learned more from you all than you’ll ever realize. Lastly, I would like to thank my family for putting up with me through everything and for their unwavering support in all my endeavours. vi TABLE OF CONTENTS Page CHAPTER 1 INTRODUCTION . 1 1.1 The Magnetostrictive Phenomenon . 1 1.2 Historical Background . 2 1.3 Ferromagnetic Frequency Focus . 3 1.4 Establishing Model Equations . 6 1.5 Goals of the Model . 8 1.6 Laboratory Setup . 9 CHAPTER 2 LOCAL LINEAR ANALYSIS . 11 2.1 Homogeneous Solution . 11 2.2 Particular Solution . 12 2.3 Solution Using the Transverse Rotation . 14 2.4 Stability of Perturbation . 15 CHAPTER 3 NONLINEAR ANALYSIS . 17 3.1 Nonlinear Perturbation Solution . 17 3.2 Frequency Doubling Distinction . 20 3.3 Global Existence of Solution . 21 CHAPTER 4 MAGNETOSTRICTION . 23 4.1 Frequency Model . 23 4.2 Effects of Cosine . 24 4.3 Influence of Amplitude . 26 4.4 Impact of Driving Signal . 30 vii 4.5 Frequency Dependence of Phase Lag . 31 CHAPTER 5 NUMERICAL RESULTS . 34 5.1 Numerical Strategy . 34 5.2 Data setup . 35 5.3 Time and Frequency Domain Analysis . 38 5.4 Numerical Conclusions . 39 CHAPTER 6 CONCLUSIONS . 43 6.1 Results . 43 6.2 Future Work . 45 BIBLIOGRAPHY . 46 viii LIST OF FIGURES Figure Page 1.1 Material is split into domains that minimize the internal energy needed to maintain magnetization. (a) Depicts a material with the maximum amount of internal energy required. (b) shows a reduced energy configuration and (c) results in the minimum amount of internal energy of these three models. 2 1.2 Magnetic Domains attempt to align themselves with driving external field, causing a deformation in material. 4 1.3 The laboratory experiment arranged the sample as shown above, with strain gauges affixed in both longitudinal and transverse directions. 9 3.1 True-frequency doubling is shown in (a), while close-frequency doubling is de- picted in (b). Quasi-frequency doubling and no-frequency doubling are plotted in (c) and (d) respectively. 21 4.1 In the figures above, frequency doubling is still present in each of these plots de- spite the increasing in the driving frequency (low-! case being (a), high-! case π being (d)). These simulated results each had a fixed value of δ = 2 allowing the symmetry of the system to be preserved in each case, and genuine frequency doubling to hold. 25 4.2 Above are the experimential results from Jen’s lab. In (a), the driving frequency is set at 0.07 Hz (low-f case). In (b), the driving frequency is set at 32 Hz (med-f case). In (c), the driving frequency is set at 122 Hz (high-f case). 27 ix 5.1 Double frequency is present with the following parameters: f = 0:07 Hz, H0 = 2:2 × 103 Am−1, δ = 48◦. We can see from the FFT that the dominant frequency is indeed being multiplied. 41 5.2 Close frequency is present with the following parameters: f = 0:07 Hz, H0 = 2:8 × 103 Am−1, δ = 48◦ and we can observe the appearance of unequal valleys. 41 5.3 Quasi frequency doubling is present with the following parameters: f = 0:07 Hz, 3 −1 ◦ H0 = 5 × 10 Am , δ = 48 as the valleys are growing more and more distinct. 42 5.4 No frequency multiplying is present with the following parameters: f = 0:07 Hz, 4 −1 ◦ H0 = 1 × 10 Am , δ = 48 as the driving signal is too large. 42 5.5 No frequency doubling is present with the following parameters: f = 0:07 Hz, 3 −1 ◦ H0 = 2:2 × 10 Am , δ = 80 since the symmetry of the system is off. 42 x LIST OF TABLES Table Page 1.1 Calculated Laboratory Constants . 10 1.2 Auxiliary Laboratory Material Constants . 10 5.1 Computed Laboratory Constants from Jen’s Data . 39 1 CHAPTER 1 INTRODUCTION 1.1 The Magnetostrictive Phenomenon All materials are subject to the influence of a magnetic field, and it is the material’s response to a magnetic field, by which the differing types of magnetism are classified. Ferromagnetism is the strongest magnetic state, maintained by ferromagnetic materials such as iron, nickel, and cobalt. These ferromagnetic materials are the most common forms of magnetism that we experience in everyday life and are capable of being permanently magnetized. They are the only materials that can maintain a magnetic force strong enough to be felt outside of a laboratory setting. Other materials, such as aluminum and oxygen are classified as paramagnetic and are weakly attracted to magnetic fields. Diamagnetism occurs in materials such as copper and carbon, which are weakly repelled by magnetic fields and more complicated relationships to magnetic fields are involved with antiferromganetic materials such as chromium (Soshin, 2009). In this thesis however, we further analyze the magnetism process of ferromagnetic material on a nanoscale and working with data collected from a laboratory experiment performed on a ferromagnetic alloy. During the process of magnetization, in which a material responds to an applied magnetic vector field, there are several physical aspects of the material that effect its ability to be magnetized and are subject to change during magnetization. As the material reacts physically, a dynamical system of coupled interactions between the material’s physical properties and material’s magnetic permeability is created. At the microscopic level, magnetization results in an induced or permanent change in density of magnetic moments within the material being magnetized. This distribution of magnetic moments is not uniform within a material and as the magnetic field is applied, a system of nonlinear, time- dependent variations occur within the material that can cause a change in the material’s dimension and other physical properties (Cullity and Graham, 2008). 2 1.2 Historical Background While observing ferromagnetic material in 1842, James Prescott Joule identified this effect as magnetostriction. Magnetostriction is a property of ferromagnets that causes them to change shape or dimension during the process of magnetization. Joule was able to magnetize a piece of iron and measure it’s deformation in shape, which has historically been contributed to the crystal lat- tice structure inherent to these types of material (Joule, 1847). Internally, ferromagnetic material contains regions, or domains, of varying densities of magnetic moments. Internal to those regions, the magnetic moments are aligned with one another in a single direction and uniform magnetic polarization occurs within each specific domain. These uniformly directional domains are called Weiss Areas and are present in ferromagnetic as they minimize the internal energy of the magnetic material by splitting up the internal magnetic fields into smaller domains that require less magne- tostatic energy to be stored. That is, these divided regions each have dipole moments in different directions in order to minimize the amount of energy needed to maintain the magnetic field at a state of rest. Since the distribution of Weiss Areas is not uniform throughout the entire material, Figure 1.1: Material is split into domains that minimize the internal energy needed to maintain magnetization. (a) Depicts a material with the maximum amount of internal energy required. (b) shows a reduced energy configuration and (c) results in the minimum amount of internal energy of these three models. 3 different magnetic domains have different directions in which uniform magnetization occurs. The natural inclination of the sum of magnetic forces of the domains when no external magnetic forces are present, is called the easy-axis. That is, the easy-axis is similar to the average of directional forces within the material when no influencing forces are involved. Since it is extremely difficult to calculate the direction of the easy-axis, we shall be referring to the material’s equivalent easy-axis. When a magnetic field is applied in a single direction, the domains and boundaries within the ma- terial shift in an attempt to align themselves with the direction of the external magnetic field that requires the least amount of energy, shifting away from the easy-axis of the system.