The Mathematical Modeling of Magnetostriction

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

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 depicted 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 despite the increasing in the driving frequency (low-ω case being (a), high-ω 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 lattice 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 magnetostatic 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 material 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. This directional dependency of the material’s magnetic properties is known as magnetic anisotropy and it is the reason behind the occurrence of magnetostriction. Each domain will shift in the direction that reduces the amount of magnetostatic energy, causing physical stress and strain within the material. It’s important to note that the crystal lattice structure is not being deformed, but rather the physical body of the material is changing shape as the material contracts and elongates (Jiles, 1998).

1.3 Ferromagnetic Frequency Focus

Since the focus of this work is ferromagnetic material, we are interested in the specific type of magnetic anisotropy from which magnetostriction occurs, which is known as magnetocrystalline anisotropy and is related to the crystal lattice structure found in ferromagnets. In extreme cases of this change in dimension, the physical distortion of the object can in turn, affect the permeability of the magnetic field creating a fully-coupled system of reactions between the material and the field. The deformation in shape creates a responsive deformation in the external magnetic field, which then causes further deformation. The material is shifted and strained as each magnetic domain rearranges its structure to align with the easy-axis of the field, creating a loss of energy in form of vibrational heat and sound. It is the sound produced by magnetostriction that we model and analyze in this thesis, as well as the phenomenon of double-frequency conditions. While magnetostriction is considered a phenomenon of the magnetization process, it is in fact quite a common occurrence in everyday life. We are all familiar with the low-pitched electrical humming produced by transformers or refrigerators, 4

Figure 1.2: Magnetic Domains attempt to align themselves with driving external field, causing a deformation in material. but it is the result of magnetostriction that this noise can be heard (Pacific, 2016). As alternating current oscillates through sheets of metal within a transformer for example, a magnetic field is produced by the changing electrical field. The sheets of metal shift and strain on a microscopic level and in a non-uniform mannerism in order to align the magnetic domains in the direction that minimizes the free energy of the system. As the material is distorted, the metal extends and contracts and the vibrations of that strain can cause a fundamental noise frequency that is audible. Our interest in frequency doubling, means that we are examining conditions in which the material ex- pands and contracts twice in one cycle. A higher frequency, produces a louder noise and a greater loss of energy of the system. The phenomenon of magnetostriction is also utilized as a measuring 5 technique in sensors and as a transducer. The distortion of material is capable of turning the electrical energy that is used to produce a magnetic field into kinetic energy that does not degrade over time. For higher frequency-driven equipment, the doubling of that frequency creates audible noise that is both unpleasant and potentially harmful. The importance of understanding the conditions for frequency doubling is motivated by the need to predict such conditions in order to better utilize sensory equipment and compelled by the need to reduce the energy loss due to magnetostrictive effects in other material. The analysis for this paper is based on the experiment done by S. U. Jen who studied the frequency dependence of the magnetostrictive phenomenon of as-cast 2605SA 1 ribbon, a magnetic alloy designed by Metglas R and widely used in transformers. Their analysis involved a uniform, in-plane, longitudinal magnetic field (H), which was applied to a sample of as-cast 2605SA 1 ribbon in a lab setting designed to collect data on the occurring magnetostriction (λ) during the process of the magnetization (M) of the core. To measure the strain due to magnetostriction, they used strain gauges as shown in Figure 1.3. Their work, and this thesis, focus on the frequency dependence of the longitudinal magnetostriction (λk) and the transverse magnetostriction (λ⊥) with respect to the magnetic driving field (H). This magnetic field is driven by a frequency (f) that is fixed for the purpose of observing the frequency dependence of the magnetostrictive effects. Due to the implicit time dependence of the equation for magnetostriction, described later in the paper, we begin our analysis on the material’s physical motion rather than starting with the magnetostrictive frequency, which has an explicit dependence on time and has a direct impact on the magnetostrictive dynamics of the material. After our analysis of a perturbed solution to the equations describing the material deformation, we then expand our results to examine the behavior of the magnetostrictive signal properties and explore the conditions in which double-frequency arises. 6 1.4 Establishing Model Equations

Since magnetostriction is the changing of dimension of the material during magnetization, and since this phenomenon is not uniform throughout a material, the distortion invoked by the longi-

tudinal magnetostriction (λk) is not proportional to the distortion invoked by the transverse mag-

netostriction (λ⊥). Thus, a rotation or twisting of the material occurs and a relationship is formed between the different sources of that torque due to the directional dependency of the material’s

magnetic properties. The driving torque τD due to the internal magnetic ﬁeld, is the combined re-

sult of both the dissipating torque τE and the restoring torque τK . The balancing of these respective torques imposes the condition,

τE + τK = τD, (1.4.1)

where the dissipating torque becomes a dampening factor,

dθ τ = −β L , (1.4.2) E dt

with a positive constant β induced by the eddy-current mechanisms and θL representing the angle in which the material has deviated from the equivalent easy-axis, formed from the distortion with

respect to the longitudinal direction (x-axis). The restoring torque τK that captures the material’s anisotropy properties is given by,

K sin(2(δ − θL)) τK = . (1.4.3) M sin θL

Where, K is a parameter describing the magnetic directional dependency along the easy-axis in units of energy density and δ is the angle of the equivalent easy-axis. Thus, δ − θL represents our material shift caused by magnetostriction in the longitudinal direction. This torque term (τK ) models the energetically favourable direction of the material’s energy. It is important to mention that these equations are based on the movement of the material, speciﬁcally their rotation from their resting axes, and at this time, do not capture the frequency dynamics of magnetostriction that 7 we later analyze in this thesis. Using the relation described in Eq.(1.4.1), we come to the nonlinear, time dependent equation of motion for the longitudinal saturation magnetization rotation,

dθ K sin(2(δ − θ )) −β L + L = H, (1.4.4) dt M sin θL

where H represents the driving magnetic ﬁeld and can be approximately deﬁned as H = H0 sin(ωt) with ω = 2πf. Since we are attempting to study the frequency dependence of the general magnetostrictive effects in both the longitudinal direction and the transverse direction, we use the relation that

π θ + θ = L T 2

to write dθ K sin(2(δ + θ )) β T − T = H. (1.4.5) dt M cos θT

We now have two coupled, non-linear differential equations that describe the general motion of the material (δ) as it is subjected to a magnetic ﬁeld (H) with frequency (f). It is helpful to us to

recognize that this equation has an implicit dependence on time, as θL is still a function of t. We intend to show that a solution to the rotational displacement described in Eq.(1.4.4) and Eq.(1.4.5), oscillates around an equilibrium point after an eventual period of time. It is our goal to expand our analysis of the perturbed solution (θ) to our modelling of the frequency of the magnetostrictive phenomenon (λ). As we have continually mentioned, it is the electrical humming or noise that is produced during magnetization that we are to further analyze. In particular, we are interested in understanding the conditions in which frequency-doubling of the magnetostriction strain (λ) arises. Thus, the un- derlying motivation for why our analysis is beginning with the equations modelling the material’s physical response, is that it is necessary to first model the strain as a function of the rotation of the 8 material and then extend that insight to a model that describes the magnetostrictive phenomenon. In the instances in which frequency-doubling is present, the expansion and contraction of the metal occurs twice and a vibrational signal is produced. We expect the signal of that vibration to be composed of harmonics with even multiples of the initial fundamental frequency (f0) that drives the magnetic field since the distortion itself is occurring in even multiples. We make this intuitive conjecture because the magnetostrictive strain signal λ(t) is expressed using the even function cosine as 3 1 λ(t) = λ cos2 θ(t) − , (1.4.6) 2 0 3 in which λ0 is a fixed isotropic magnetostriction condition related to predetermined elongations. More specifically, the material studied in this thesis is an amorphous one, allowing for the assumption that an isotropic magnetostriction condition is satisfied along the [100] and [111] elongations when the magnetic saturation is also occurring in those directions (Jen, Liu, Lin, and Chou, 2014).

1.5 Goals of the Model

In this thesis, we analyze the system of equations that describe θ(t), the rotational distortion of the material, and relate the behavior of that perturbed solution to the signal produced by these magnetostrictive effects λ(t). Our theoretical analysis will be veriﬁed with both experimental and numerical results. We intend to isolate certain conditions in which the phenomenon of double frequency occurs through our mathematical analysis which has previously been based solely on physical considerations within the experiments. The development of a better understanding for conditions in which true-double-frequency occurs is explained and some generalized distinctions are made between the close-frequency-doubling and quasi-frequency-doubling of a signal. The system’s frequency dependence is considered from several viewpoints and some simulated results illustrate our conclusions. 9 1.6 Laboratory Setup

In order to establish the setting in which the phenomenon was observed, it is important to describe the experiment that motivated the development of this model. A rectangular sample of the as-cast 2605SA 1 ribbon, was cut from a spool purchased from Metglas R . It had a length of 25mm, a width of 8 mm, and a thickness of 17.8 µm. A set of strain gauges were glued to the surface of the material to measure both the λ⊥ and the λk effects. The ferromagnetic allow was then glued to the ends of two flexible copper strips. The material was adhered with cement, creating an environment in which the ribbon was able to vibrate quasi-freely. In the experiment, the ribbon was able to be secured within the homogeneous field region, while also expand or contract as freely as possible. With this setup, twisting was also avoided when the magnetization saturation of the core was not parallel to the driving magnetic field (H).

Figure 1.3: The laboratory experiment arranged the sample as shown above, with strain gauges afﬁxed in both longitudinal and transverse directions.

When the experiment was being performed, the sample was placed in the center of Helmholtz 10 coils and the signal generator excited the coils at a range of frequencies given in the table of calculated laboratory constants below.

Variable Meaning Lab Value L length of sample 25 mm W width of sample 8 mm t thickness of sample 17.8 µm H maximum amplitude of external magnetic ﬁeld 4.8×103Am−1 δ angle of the equivalent easy-axis 48◦ M saturation magnetization 41 ppm

Table 1.1: Calculated Laboratory Constants

Variable Meaning Lab Value E Young’s modulus 110 GPa D density of material 7.18×103 Kg/m3 ρ electrical resistivity 130µΩcm µ magnetic permeability 4.5×104

Table 1.2: Auxiliary Laboratory Material Constants

The strain was measured and observed with an error of approximately 12% using two strain- gauges. While this method is not ideal in the case of an alternating current magnetic ﬁeld, in the lab experiments performed by Jen, the normal of the gauge plane was orthogonal to that of the Helmholtz coil plane, essentially eliminating the inﬂuence of the extra voltage on the gauge wires from the coils. For this experiment and our analysis, we consider the magnetic saturation of the material for the minor-hysteresis loop only. By considering only the minor hysteresis loop, we only need to magnetize the central region of the sample of material, eliminating the need to analyze the

large demagnetization that occurs at edges and corners of the Metglas R alloy. When applying an external magnetic ﬁeld on the sample, the magnetic deformation occurs in two main forms: domain wall displacement, and domain magnetization rotation. Since the domain wall displacement does not contribute to magnetostrictive strain, we do not need to include this in our work here. 11

CHAPTER 2 LOCAL LINEAR ANALYSIS

2.1 Homogeneous Solution

Our initial analysis is on the coupled, time-dependent, nonlinear differential equations that describe the ferromagnetic material’s longitudinal rotation due to magnetostrictive effects:

dθ K sin(2(δ − θ )) −β L + L = H, (1.4.4) dt M sin θL

where

H = H0 sin(ωt). (2.1.1)

We use this equation as a starting point, in hopes that the behavior we see in this model, allows for a clearer characterization of the behavior of the magnetostrictive strain equation that we have quan- tified in Eq.(1.4.6). In our analysis, we first want to tackle the homogeneous solution of Eq.(1.4.4) and explain its context in terms of the laboratory experiment. In our physical system, involving a strip of the ferromagnetic alloy as-cast 2605SA 1 ribbon and a uniform driving magnetic field H, a homogeneous solution represents the condition that there is no driving magnetic field present,

H0 = 0. With no external magnetic forces, our material should not be experiencing the effects

dθL of magnetostrictive distortion and the component of motion describing the dissipating torque dt must necessarily equal 0. These physical assumptions reduce Eq. (1.4.4) to,

K sin(2(δ − θ )) L = 0. (2.1.2) M sin θL

The conclusion that our homogeneous solution should be a constant value thus becomes quite

consequent and we will denote that solution as θE. If this solution is stable, it can be designated as an equilibrium point of the system. The stability of the equilibrium is discussed later in this chapter as well as the aspects of the application of this analysis. Looking at Eq.(2.1.2), we can easily recognize that there are many homogeneous solutions, or 12 many equilibria, but considering the physical considerations of our model, we are most interested in the equilibrium where θE = δ. That is, the solution in which our distortion is relative to the equivalent easy axis. The deviation of the equivalent easy axis (δ) remains a signiﬁcant physical value for this system as it represents the material’s general magnetostrictive distortion and we make further distinctions regarding our choice of δ later in the thesis. For now, the homogeneous solution to this equation represents our ferromagnetic material’s initial orientation without the presence of a driving magnetic ﬁeld.

2.2 Particular Solution

Now that our homogeneous solution is more or less understood, it becomes necessary to ﬁnd a particular solution. In line with traditional methods for solving for a particular solution, we intuitively think that it should be of a sinusoidal form with a phase angle. Unfortunately, it is

obvious that θL(t) = A0 sin(ωt − φL) is not a solution to Eq.(1.4.4) which leads us to make further approximations.

Continuing with our focused analysis near θE, we intend to utilize perturbation methods to describe the behavior of the particular solution in terms of its relation to an equilibrium point. We linearize the restoring torque factor of the differential equation Eq.(1.4.4) around that equilibrium by The Taylor Expansion.

0 τ(θ) = τ(θE) + τ (θE)(θ − θE) + ... of

sin(2(δ − θ)) τ(θ) = , sin θ

where dτ(θ) τ 0 = . dθ

By dropping the higher order terms and noting that τ(θE) = 0, we can approximate Eq.(1.4.4)

around θE as, d(θ − θ ) E + a(θ − θ ) = −b sin(ωt), (2.2.1) dt E 13

K 0 H0 where a = Mβ τ (θE) and b = β . We have now obtained an equation that describes the behavior of the particular solution near an equilibrium. This is an approximated, perturbed, linearized form of Eq.(1.4.4) but will still give us insight into the conditions necessary for stable equilibrium, how the function behaves after a transient period of time, and the relation that ω has to the original driving frequency of the system. Further examining Eq.(2.2.1) we deduce that the homogeneous solution of the perturbation equation takes an exponential form,

−at θh(t) − θE = Che , (2.2.2)

with some constant Ch. We then deﬁne our particular solution as a sinusoidal function with a phase angle in the longitudinal direction parallel to the external magnetic ﬁeld φL,

θp(t) − θE = Cp sin(ωt − φL). (2.2.3)

In general, the solution to the perturbation equation shows us that θ(t) eventually oscillates around the point θE and has the form,

−at b θ(t) − θE = Che + √ sin(ωt − φL), (2.2.4) a2 + ω2

which is comprised of a decaying transient term for a > 0 and a vibrational term that oscillates

around the stable equilibrium of θE. We are able to geometrically solve for the constant Cp in

terms of a and ω by deﬁning our longitudinal phase angle φL deﬁned by

−a −ω cos φL = √ and sin φL = √ . (2.2.5) a2 + ω2 a2 + ω2

It is then appropriate for us to conclude that solutions of Eq.(1.4.4) will eventually become a 14 sinusoidal form given below,

θL(t) ≈ A sin(ωt − φL) + B. (2.2.6)

The parameters A and B are described as follows,

b A = √ and B = θE. (2.2.7) a2 + ω2

The remaining parameters ω and φL are discussed later in Chapter 4 of this thesis, along with their relevance to the frequency-doubling phenomenon occurring within the magnetostrictive effects.

2.3 Solution Using the Transverse Rotation

A similar oscillatory relationship can be deduced for the transverse magnetostrictive rotation as follows. Using the orthogonality of the longitudinal and transverse angles, and the property that

π τ(θ) = τ( 2 − θ), we can derive a similar expression as Eq.(1.4.4):

dθ K sin(2(δ + θ )) T − T = H. (2.3.1) dt Mβ cos θT

Since the longitudinal solution is periodic around some equilibrium point θE, the transverse solu-

π tion has the same result, but with a phase shift where θET = 2 − θEL .

From our previous modeling of the equilibrium point, we can further express the transverse perturbation solution as

−at b θ(t)T − θE = Che + √ sin(ωt − φT ), (2.3.2) T a2 + ω2 15 where we need a slightly different phase angle deﬁned geometrically by

a ω cos φT = √ and sin φT = √ . (2.3.3) a2 + ω2 a2 + ω2

While the perturbed solutions for the longitudinal and transverse θ appear very similar, it is important to note some differences in their roles in the model. To begin, without loss of generality, we use the longitudinal solution for the remainder of our analysis as we have just shown that the

transverse results are very comparable. We can also note that the phase angle φT is located in the

π ﬁrst quadrant while φL is located in the third, which is a natural relationship since θT + θL = 2 . This relation also implies that their second derivatives are equal in magnitude, but opposite in sign resulting in their oscillations being out of phase. Furthermore, it is fairly helpful for us to note that, with respect to the driving forces of H, φL leads the phase of H while φT is a phase lag.

2.4 Stability of Perturbation

The stability of this perturbation in Eq.(2.2.4) depends on the necessary decay of our transient term, which will result in, after some time t, only the oscillating steady-state term remaining. This implies that, K τ 0(θ ) > 0. (2.4.1) Mβ E

For the purpose of our analysis, we do not consider the influence of the material’s physical properties K,M, and β as they are not necessary for our general analysis of the magnetostrictive effects. These parameters represent specific intrinsic properties of the material and are fairly difficult to measure, thus we assume them to be positive constants and continue with our mathematical analysis. In the work done by Jen, he was able to estimate the values of these material constants and the effects of this will be discussed further in Chapter 5. For a more explicit understanding of the stability conditions for the perturbed relationship described in Eq.(2.4.1) recall that, 16

0 −2 sin θE cos(2(δ − θE)) − sin(2(δ − θE)) cos θE τ (θE) = 2 . sin θE

Due to the assumed equilibrium properties of θE, we have sin(2(δ − θE)) = 0 by deﬁnition,

0 allowing us to reduce τ (θE) to the following,

0 −2 cos(2(δ − θE)) τ (θE) = . (2.4.2) sin(θE)

Our deﬁned property of sin(2(δ − θE)) = 0 implies that cos(2(δ − θE)) = ±1 which in turn leaves us with

0 ∓2 τ (θE) = > 0. (2.4.3) sin(θE)

This allows us to establish some conditions for the stability of equilibrium points within the local linear analysis of the perturbation solution. We hope to select an equilibrium point that ensures our rate of change of torque is positive, and that that equilibrium value is not equal to zero. The choice of the equilibrium value plays an important role in the presence of frequency doubling in λ(t) and the details of that impact is further discussed in context of the numerical results in Chapter 5. 17

CHAPTER 3 NONLINEAR ANALYSIS

3.1 Nonlinear Perturbation Solution

While the local linear analysis of the material’s physical distortion is useful for our understanding of the system, we are far from a comprehensive understanding of the magnetostrictive effects. We must consider the nonlinear components in order to discuss the system further. Looking at our expansion from earlier in the thesis, we can analyze the perturbation solution with more detail by retaining the higher order terms of the approximation of τ:

sin(2(δ − θ)) τ(θ) = , to which sin θ

τ (2)(θ ) τ (3)(θ ) τ(θ) = τ(θ ) + τ (1)(θ )(θ − θ ) + E (θ − θ )2 + E (θ − θ )3 + ... , and E E E 2 E 3! E

2 2 cos θE 2 sin θE − 2 3 τ(θ) = (θ − θE) − 2 (θ − θE) + 3 (θ − θE) − · · · . sin θE sin θE 3 sin θE

Keeping these higher order expansion terms in, we can rewrite Eq.(1.4.4) as the following,

2 ! d(θ − θE) K 2 cos θE 2 sin θE − 2 3 + (θ−θE)− 2 (θ−θE) + 3 (θ−θE) −· · · = H. (3.1.1) dt Mβ sin θE sin θE 3 sin θE

We are now going to denote our perturbation as η = θ − θE and introduce a new variable τ˜ = ωt. The motivation for redeﬁning our time variable in this way is to obtain a dimensionless variable that is not restricted by a time-dependence. We can then arrange things in terms of a family of

functions {ηˆ}n, where ηˆ(˜τ) = η(t) which simpliﬁes our nonlinear equation to the following,

2 ! dηˆ K 2 cos θE 2 sin θE − 2 3 + ηˆ − 2 ηˆ + 3 ηˆ − · · · = −b sinτ. ˜ (3.1.2) dτ˜ Mβω sin θE sin θE 3 sin θE 18 Let’s now make the assumption that the coefﬁcient of the η2 term is very small. That is, we assume

K cos(θE) ε0 = 2 Mβω sin (θE)

to be small, implying that our leading nonlinear term is fairly weak. Continuing with our elemen-

tary perturbation method, we look at the following nonlinear equation for ε ∈ [0, ε0]:

dηˆ K K cos(θE) 2 −b + ηˆ + 2 ηˆ + ··· = sinτ. ˜ (3.1.3) dτ Mβω sin(θE) Mβω sin (θE) ω

To gain more insight, we then look at the family of differential equations using ε as a parameter instead of simply viewing the coefﬁcients as constants,

dηˆ −b K + α ηˆ + εηˆ2 + ··· = sinτ, ˜ where α = . (3.1.4) dτ ω Mβω sin(θE)

When ε = 0 we observe the linear equation that we had previously examined, and if ε = ε0 we recover (3.1.3). From here, we can represent the solutions of Eq.(3.1.3) in terms of the power series expansion of ε since our solutions can now be viewed as a function of both τ˜ and ε.

2 ηˆ =η ˆ(˜τ, ε) =η ˆ0(˜τ) +η ˆ1(˜τ)ε +η ˆ2(˜τ)ε + ··· (3.1.5)

Now substituting (3.1.5) into (3.1.4) we obtain,

d −b (ˆη (˜τ) +η ˆ (˜τ)ε + ··· ) + α(ˆη (˜τ) +η ˆ (˜τ)ε + ··· ) + ε(ˆη (˜τ) +η ˆ (˜τ)ε + ··· )2 + ··· = sinτ ˜ dτ 0 1 0 1 0 1 ω (3.1.6) 19 Since this is assumed to be valid for every member of the family of functions, it is assumed to hold for every ε within an interval near ε0. From here, we need to expand and collect terms to obtain

−b ηˆ0 + α ηˆ = sinτ, ˜ (3.1.7) 0 0 ω 2 0 ηˆ0 +η ˆ1 + α ηˆ1 = 0, and (3.1.8)

0 2ˆη0ηˆ1 +η ˆ2 + α ηˆ2 = 0. (3.1.9)

For this analysis, we are mainly interested in the ﬁrst three leading terms ηˆ0, ηˆ1, ηˆ2 and examine their solutions now. Because we require that our perturbed solution exhibits periodic behavior after a transient period of time, and it is our assumption that the non-homogeneous solution for Eq.(3.1.7) should be

of the form of the right-hand side, the solution ηˆ0 is of the form

˜ −ατ˜ ˆ ˆ ηˆ0 = Ae + A sin(˜τ − φL), (3.1.10)

for some real-valued constant A˜. We can explicitly determine Aˆ = √ b based on our geometric ω 1+α2 ˆ deﬁnition of the phase angle φL from the previous section. The phase angle in this set of equations is normalized by our deﬁnition of τ˜, thus having the following geometric interpretation,

ˆ 1 ˆ aω cos φL = and sin φL = . (3.1.11) p1 + (aω)2 p1 + (aω)2

Note that Aˆ > 0, ensuring that our periodic, steady state term is positive. Looking at the solutions

2 for the other ηˆ terms, we can observe that (3.1.12) has an ηˆ0 term on the right hand side given by

0 2 ηˆ1 + α ηˆ1 = −ηˆ0 . (3.1.12)

Since the transient term from (4.2.5) decays quite rapidly, new frequencies are not driven out by 20 ˆ2 2 anything other than the A sin (˜τ − φL) term. Thus, we can further reduce our analysis to

0 ˆ2 2 ηˆ1 + α ηˆ1 = A sin (˜τ − φL) . (3.1.13)

From here, we can extract a particular solution of a different frequency. By noting that

1 1 sin2(˜τ − φ ) = − cos(2˜τ − 2φ ), L 2 2 L

we can see that our new frequency is in fact 2˜τ. But does this imply that we have true-frequency doubling? Returning to our time-dependent variable ωt, we have actually recovered (2.2.4) and (2.2.5) from our previous, local linear analysis but further work is still needed in order to determine conditions in which double-frequencies arise in our magnetostrictive phenomenon.

3.2 Frequency Doubling Distinction

In the reference work done by Jen, a distinction is made between the resulting frequency effects which we’ll make here as well. In the laboratory experiment, instances of quasi-frequency doubling was observed in which two-unequal valleys appeared in the data when a frequency of 10 Hz was used. The frequency had indeed appeared to be doubled but the presence of these largely unequal valleys force a distinction to be made, since we no longer observe the same symmetry. This is shown in Figure 3.1c. In the results that displayed close-frequency doubling, depicted in Figure 3.1b, a low-frequency limit was used that resulted in approximately half a period. Two valleys again, occurred in the data but they were almost equal. True-frequency doubling is the occurrence of the exact double frequency in which the period is precisely halved, as shown in Figure 3.1a. We are more interested in this genuine frequency doubling than quasi-frequency doubling or close- frequency doubling, both of which result in a loss of symmetry of the system. As you can see from Figure 3.1d, the absence of frequency doubling is the presence of vastly unequal peaks in our vibrational signal. 21

(a) (b)

Figure 3.1: True-frequency doubling is shown in (a), while close-frequency doubling is depicted in (b). Quasi-frequency doubling and no-frequency doubling are plotted in (c) and (d) respectively.

3.3 Global Existence of Solution

So far, we have examined our equation of the nonlinear, time dependent equation of motion locally. From this initial analysis, we have gained valuable insight that our solution will eventually oscillate around an equilibrium point, and we have also made some observations about the stability

conditions of those aforementioned points. However, we can see that (1.4.4) has the sin θL term in the denominator. dθ K sin(2(δ − θ )) −β L + L = H (1.4.4) dt M sin θL

It is important then, to discuss the global existence of θL(t), which needs to stay away from singu-

larity points of 0 and nπ, for any n ∈ N. It is not known that our solution will exist for all t > 0, but we would like to examine conditions in which our range of t is as large as possible.

Since we assume from our earlier analysis in Eq. (2.2.6) that θL(t) = A sin(ωt − φL) + B, we 22 can observe that in order to keep θL(t) away from 0 and π, we need establish restrictions on the parameters ω and A on the following equations:

θL(t) = A sin(ωt − φL) + B, (3.3.1) θ (t) − B sin(ωt − φ ) = L , (3.3.2) L A θ (t) − B L ≤ 1, (3.3.3) A

B − A ≤ θL(t) ≤ B + A, (3.3.4)

0 < B − A and B + A < π. (3.3.5)

Using B = θE from Eq.(1.4.4) and

b H0 K 0 A = √ , where b = , and a = τ (θE), a2 + ω2 β Mβ

we obtain the following conclusion regarding the range of stable choices in terms of the frequency and amplitude. 1 1 4b ω > min , (3.3.6) π − θE θE sin(θE)

or equivalently s 4 2 b < min{π − θE, θE} 2 + ω (3.3.7) sin (θE)

Thus, for the initial condition 0 < θL(0) < π, if we choose a range of frequencies (ω), then our amplitude (A) must be restricted and a free choice of amplitude (A) then restricts the range of frequencies (ω). The global existence criteria present in our code utilizes the restrictions on b, since it is practically easier to adjust in a physical lab setting. This implies that our driving frequency

H0 has a signiﬁcant impact on the global validity of our solution. More explicitly, we see that for

large H0, no double frequency occurs. These aforementioned global existence criteria are present in the code that is used for the numerical analysis which is discussed in more depth in Chapter 5. 23

CHAPTER 4 MAGNETOSTRICTION

4.1 Frequency Model

Now that we have analyzed the solutions for the equations describing the motion of the material, which we have conﬁrmed oscillate around a ﬁx equilibrium value after some time, we can make some conclusions about the magnetostrictive phenomenon in terms of its frequency. The mathematical magnetostriction associated with θL(t) is governed by

3 1 λ(t) = λ cos2 θ (t) − . 2 0 L 3

For the sake of our analysis, we can use an equivalent, yet simpler, equation to study the phenomenon of frequency doubling.

λ(t) = cos(2θL(t)).

We observe from the data collected from the lab experiments that the driving frequency is indeed doubled however, there are several questions that we wish to address through our mathematical modeling here:

i Is the doubling effect due to the even property of cosine or due to the factor of 2 in the deﬁni- tion?

ii What is the role of the amplitude of longitudinal distortion (A)?

iii What are the roles of the driving frequency and amplitude (ω and H0)?

iv What is the frequency dependence of the phase lag?

We tackle each of these questions with our mathematical analysis and then conﬁrm our understanding with a numerical run. 24 4.2 Effects of Cosine

To address the ﬁrst question, we want to show that we get a doubled frequency from λ if our vibrational strain component of θL(t) has certain characteristics. From earlier, we know that we will eventually have

θL(t) ≈ A sin(ωt − φL) + B. (2.2.6)

Where B is a constant representing an equilibrium (θE) and A is the amplitude expressed as

A = √ b . This was shown in our early linear analysis, and later conﬁrmed by the numerical α2+ω2 run, to be valid for large enough t. We can now see that our magnetostrictive effect can be modeled by something of the form of

λ(t) ≈ cos(2A sin(ωt − φ) + 2B).

π It is then clear that if B = 0 or 2 , we have λ(t) producing a double frequency since it is easy to T π verify that θL(t+ 2 ) = −θL(t)+2B. The conditions in which B = 0, 2 are important to note here, as they have a physical meaning that preserves the symmetry of the system. When the equilibrium

π value is parallel (B = θE = 0) or perpendicular (B = θE = 2 ), then the material oscillates proportionally and symmetrically around that value, creating new extrema of equal magnitude, yet opposite sign.

Now we want to address our claim that if λ(t) produces double frequency, then θL(t) is necessarily a constant solution. So from our assumption, we get that

T λ(t + ) = λ(t), (4.2.1) 2

= cos(2(−θL(t) + 2B)), (4.2.2)

= cos(−2θL(t)) cos(4B) + sin(2θL(t)) sin(4B), and (4.2.3)

= λ(t) cos(4B) + sin(2θL(t)) sin(4B). (4.2.4) 25

(a) (b)

Figure 4.1: In the ﬁgures above, frequency doubling is still present in each of these plots despite the increasing in the driving frequency (low-ω case being (a), high-ω case being (d)). These simulated π 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.

If we have the condition that cos(4B) = 1, then we observe that the period can be halved and double frequency arises, since the sine portion of our result will equal zero. However, it’s important to note that we are not restricting our period T to the smallest period. This raises the issue of double frequency multiplying being embedded or disguised in even-frequency multiplying.

If we consider the case where cos(4B) 6= 1, then under our assumption that the behavior is periodic, we obtain

T λ(t + ) = λ(t), and (4.2.5) 2 26

= λ(t) cos(4B) + sin(2θL(t)) sin(4B) (4.2.6)

sin(4B) λ(t) = sin(2θ (t)). (4.2.7) 1 − cos(4B) L

From here, we can reduce Eq. (4.2.7) to

1 − cos(4B) tan(2θ (t)) = = tan(2B). L sin(4B)

This leaves us with the conclusion that if frequency doubling occurs, θL(t) = B. It is important to reiterate that our analysis done here is valid for modeling the magnetostrictive effects after the transient period. Theoretically, we have just shown that the only conditions in which true frequency doubling occurs is the trivial case of no external magnetic ﬁeld being present. Having a constant behavior exhibited in the physical deviation from the equivalent easy-axis of the material, results in a constant solution for the magnetostrictive effect λ(t), which is rather uninteresting to us as it is a state in which there is no external magnetic forces. In the next section, we compare our non-exact mathematical analysis of this nonlinear phenomenon to a numerical run and show that it is possible to construct a theoretical material that is capable of exhibiting true frequency-doubling.

4.3 Inﬂuence of Amplitude

Since we have just addressed the ﬁrst quandary regarding the presence and conditions in which

double frequency arises, we need to discuss the role and inﬂuence of the amplitude of θL(t) in the frequency multiplying mechanism. More speciﬁcally, we want to determine the effect that the driving amplitude has on the number of extrema that λ(t) has. The observation that prompted our interest in A, was the following:

In the laboratory experiment, H was generated by a signal generator whose wave form

(H(t)) is shown in Figure 4.2. We can see that H0 was ﬁxed at 60Oe and that the generator was a reliable one that usually only produces the drive at the ﬁxed frequencies, 27 though a fast Fourier transform does show the presence of a small amount of other

frequency noises. However, if the H drive is set at a base frequency, denoted f0, then

we expect the responses from the magnetostriction to contain signals with f = 2nf0, where n = 1, 2, 3, ··· . We intuitively assume that the new frequencies relate to the driving frequency and we have conﬁrmed this with our analysis thus far. However, we notice that as n increases, the intensity of the corresponding signal decreases, raising the question of how the amplitude and driving frequency are impacting the magnetostrictive phenomenon (Jen et al., 2014).

Figure 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).

Our initial insight is to ﬁrst locate the respective peaks in order to further classify the frequency effects. To do this we use 28

dλ dθ (t) = −2 sin(2θ (t)) L = 0. dt L dt

From here, we easily observe that the extrema of λ(t) occur at the roots and extreme values

of θL(t). To get a general picture for how the frequency multiplying works, consider any two

consecutive zeros of θL(t). There must exist at least one extreme value between these two points,

resulting in at least two extrema of λ for any three points of θL(t). Since we know that after a transient period of time, θL(t) becomes a sine wave, we can use this information to further our discussion. Simplifying the coefﬁcients slightly, we arrive at the following:

λ(t) = cos(A˜ sinx ˜ + B˜),

˜ ˜ where B = 2A, x˜ = ωt − φL, and B = 2B. Now we wish to examine the ﬁrst and second derivatives:

dλ = −A˜ cosx ˜ sin(A˜ sinx ˜ + B˜), and (4.3.1) dt d2λ = A˜ sinx ˜ sin(A˜ sinx ˜ + B˜) − A˜2 cos2 x˜ cos(A˜ sinx ˜ + B˜) (4.3.2) dt2

Tackling the ﬁrst derivative ﬁrst, we have two cases in which it will equal zero and result in an extrema:

1. If the cosx ˜ term equals zero.

2. If the sin(A˜ sinx ˜ + B˜) equals zero.

dλ Starting with Case 1, we assume that cosx˜ = 0. Then we obtain dt = 0 and

  ˜ ˜ ˜ d2λ A sin(A + B) if sinx ˜ = 1; 2 = dt  A˜ sin(A˜ − B˜) if sinx ˜ = −1.

Furthermore, we need to examine the original function of λ(t) with this assumption, 29   cos(A˜ + B˜) if sinx ˜ = 1; λ(t) =  cos(A˜ − B˜) if sinx ˜ = −1.

Without loss of generality, we want to consider what happens during a typical full cycle of

π 3π [0, 2π]. We know that two fixed points will be inherited from θL(t) at x = 2 and x = 2 , so we start our analysis at those values. We know that extrema of different types (i.e. min & max, or max & min) occur at those points however, in certain instances we are able to obtain two fixed points of the same extreme value type. This occurs when the two functions of λ(t) defined previously, are in fact equal. λ(t) = cos(A˜ + B˜) = cos(A˜ − B˜).

implying that B˜ = kπ, where k is an integer. Now we have found two consecutive local maxima or local minima that are of π distance apart, allowing the period to be halved and the frequency doubled. This condition is necessary for double frequency to occur and we cannot introduce more extrema of the same type and magnitude between the two ﬁxed points in order to obtain genuine frequency doubling. This can be further illustrated through an example shown below: Take B˜ = 0, then π 3π λ( ) = cos(A˜) = λ( ), 2 2 d2λ π d2λ 3π ( ) = A˜ sin(A˜) = ( ), and dt2 2 dt2 2 d2λ π d2λ π ( ) = ( + π). dt2 2 dt2 2

Thus, these ﬁxed points are either both maxima or both minima, forcing the doubling of the frequency. It’s important to note that the magnitude of the maximum need not be equal to 1 and similarly, the magnitude of the minimum does not need to be -1. This does not hold for the next case, as we’ll see below. Continuing with Case 2, let’s assume that sin(A˜ sinx ˜ + B˜) = 0 for some x˜. Then the condition that A˜ sinx ˜ + B˜ = kπ where k ∈ Z is needed. At these values λ(t) = ±1, we see right away that 30 extreme values of magnitude 1 are the only kind that can be produced by the sine piece as they correspond with the zeros and local extrema of θL(t). Next, we once again observe the second derivative of λ(t),

  ˜2 2 d2λ π −A cos (˜x) < 0 if k is even; ( ) = dt2 2  2 2 −A˜ cos (˜x) > 0 if k is odd.

From here, we want to understand the admissibility of new extreme values generated by the sine portion of the derivative of λ(t). We know that | sinx ˜| ≤ 1, implying that kπ−B˜ ≤ 1 or A˜ equivalently, k must satisfy B˜ − A˜ A˜ + B˜ ≤ k ≤ . π π meaning that, for admissible k values, new extrema are created in addition to those already estab- π 3π ˜ lished at x˜ = 2 and x˜ = 2 . From our experimental data, we can make the assumption that A > 0 and A˜ + B˜ > 0, and we need to note that k = 0 is always admissible if B˜ ≤ A˜, while k = −1 is pertinent otherwise. Furthermore, we note that even values of k create new local maxima, while odd values of k give rise to new local minima. Both of these new extrema have magnitudes of 1 and -1 respectively, as we are still considering Case 2. Using this constraint of on k, we have determined that the amplitude of θL can effect the magnetostriction quite drastically. As A gets larger, more extrema are present in our magnetostrictive strain. This is later conﬁrmed through our numerical runs, which provided the motivation to further examine the frequency of the driver H0.

Based on our deﬁnition of A and the implicit presence of H0, we need to analyze the impacts of higher and lower driving frequencies on the appearance of frequency doubling.

4.4 Impact of Driving Signal

We now need to address the importance of the driving frequency (ω) and amplitude of the signal generator (H0). The frequency doubling criteria that we have just established are once again shown below, and all have a dependence (implicit or otherwise) on the initial frequency or amplitude being 31 generated and applied through the external magnetic ﬁeld.

b H A = √ = √ 0 , and (4.4.1) a2 + ω2 β a2 + ω2

s H0 4 2 b = < min{π − θE, θE} 2 + ω . (4.4.2) β sin (θE)

From the previous section (4.2), we observe that for larger values of A, we have a greater range of values that can generate extrema, meaning that the larger our driving amplitude, the more extrema we expect to see. This makes sense intuitively, since the presence of a larger external magnetic ﬁeld should result in an increased response in magnetostrictive strain.

Next, we see that an increase in H0 also results in an increase in the values of our derivatives of λ, making our numerical solution less consistent and resulting in discontinuities. For a solution to exist globally, H0 must be bounded by the condition of Eq.(4.4.2), which in turn depends of the frequency of the driving ﬁeld. The stability of our solution is constrained by the denominator of our θL expression, which must stay away from points of singularities. If we increase both ω and H0, we can maintain favourable conditions for frequency doubling. We do want to stress that the driving amplitude H0 has a more restrictive role in the presence of frequency doubling than the driving frequency ω. As in our analysis, frequency doubling still occurs for higher and higher initial frequencies. As ω tends to smaller values however, we see frequency multiplying occur. This can be analytically justiﬁed by our previous criteria in Eq.(4.4.1) and Eq.(4.4.2). For smaller driving frequencies, we have a larger range in which new extrema can arise, but a slightly smaller tolerance for our choice of H0.

4.5 Frequency Dependence of Phase Lag

We should briefly discuss the calculation and significance of the phase lag φ in our model. To calculate our values for φ, we used our geometric definitions from Eq.(2.2.5) to conclude that

ω φ = tan−1 + π, and (4.5.1) L a 32

φT = φL − π (4.5.2)

The physical meaning behind φL in regards to our problem is that it represents the phase angle due to the anisotropy and eddy-current terms. That is, because of the directional dependence of the induced electrical currents from the varying magnetic ﬁeld, we see a slight phase lag in our governing equations.

From our understanding of this, we expect that φL and φT will have some dependence on the initial driving frequency ω, but the phase difference ∆φ always equals π.

To gain some further insight into the dependence of θL on ω, we expand our φL term under the assumption of small values of ω. That is,

ω ω φ = tan−1 + π ≈ + π. (4.5.3) L a a

From here, we can plug this approximation into our expression for θL, and we obtain

b b ω −b ω 1 θL(t) = √ sin(ωt − φL) ≈ √ sin(ωt − − π) ≈ (t − ). (4.5.4) a2 + ω2 a2 a a a

Since we are considering this model valid for after a transient period of time, we can assume

1 that t > a . Furthermore, we’d like to examine the behavior of the magnetostrictive strain in

relation to the difference between the equivalent easy-axis (δ) and the deviation from it (θL). In other words, we want to look at the vibration from magnetostriction when θ = δ + θL. That is,

2 b ω 1 λ(t) = cos(2θ(t)) = cos(2θ(t) + 2δ) ≈ cos (t − ) + 2δ (4.5.5) L a a

Using our deﬁnitions of b and a, which are approximately equal to 1, we can further reduce 33 Eq.(4.5.5) to

λ(t) ≈ cos 2ωt + 2δ. (4.5.6)

Indicating that for small values of ω, or low driving frequencies, we have frequency doubling or close double frequency occurring, remaining consistent with our earlier understanding. 34

CHAPTER 5 NUMERICAL RESULTS

5.1 Numerical Strategy

Since we’ve thoroughly examined the magnetostrictive equations mathematically, we need to verify our analysis with the laboratory experimental results and consider our model from a numerical point of view. Our first approach to developing a more explicit numerical solution to this model, is to use Runge-Kutta methods of different orders. The code used for the simulated results was taken from Lindfield and Penny’s and features an arrangement of three Runge-Kutta methods that are listed as the classical method, the Butcher method, and the Merson method (Lindfield and Penny, 2012). We want to run the Runge-Kutta method comparatively, using all three variations to ensure that our solution in magnetic rotation is trustworthy before proceeding with any further analysis on the magnetostriction strains. After establishing an accurate approximate solution to the

θL(t) curve modelling strain, we embed that solution into the curve that models our magnetostrictive phenomenon λ(t). At this point, we can observe a few of the results from our earlier mathematical analysis. First, we see that our solution breaks when driving amplitude, frequency, and material properties are not compatible. This was discussed in Chapter 4 and implemented in our code here. For example, for large H0, our Runge-Kutta methods can fail in the sense that the solution develops discontinuities or jumps. This is due to a necessary numerical restriction in which we are only allowed to take uniform time steps, and the methods may miss the high slopes of the solution but continue to compute results. Thus, it may happen that the three methods above have different times in which they ﬁrst develop discontinuities. The uniform step condition, while it does have some limitations here, will be needed for our use of a Fast Fourier Transform (FFT) later in our analysis. Second, we see that our numerical solution to θL does take the form of our analytic solution, oscillating around an equilibrium value with the form of θL = A sin(ωt − φL).

Next, after embedding the numerical θL solution into the equation for λ, we can detect the 35 frequency ratio between the two curves. To do this, we examine the details of the simulated results and their respective periods. For this computation, we locate a finite number the peaks of the longitudinal strain after a transient period of time using a MATLAB peak finder, calculate the average distances between each of them, and compare that average to the average of computed peak locations of the magnetostrictive strain. We can now numerically observe when the period is being halved, and thus when the frequency is being doubled. For this short algorithm, and for our FFT runs in the next section, we implement an additional feature in our code to reflect our analysis. That is, we introduce a condition to account for our solution being valid after a transient period of time. We should only be considering our solution curves after a certain cut-off point if we are making conclusions regarding our analysis. To determine that truncation point, we consider both our step size and our period. The time chosen to shift the data was determined to be the ratio of periods divided by our step size. This allows us to telescope our view slightly, eliminating the variation from the beginning of our equations.

5.2 Data setup

For the specific as-cast 2605SA 1 ribbon simulation, our ODE needs the material constants K,M and β, which are not available directly from the specifications of the product. From the experimental lab data however, we are able to determine the fixed values of M = 1.24 × 10−1, and K = 2.37 × 10−4 but we still need to compute the value of β. For a general ribbon, we can experiment with β = 10i, where i = 1, 2, ... and plot the strain-hysteresis diagram with our simulation code. The allowable values for β, M, and K are only those that exhibit hysteresis behavior, which must be present for valid simulated material. However, the specific value for the as-Cast ribbon can also be computed by considering the geometry of the system. Since we know that the angle θL will swing around δ with an amplitude of ξ, we can attempt to read these deviations from experimental data. For example, suppose we have a strain where the equivalent easy-axis is δ = 48◦ and the

◦ deviation is ξ = 8 . Then we can expand upon our equation for θL, considering only the vibrational 36 term after a transient period of time:

H0 θL(t) = √ sin(ωt − φL). (5.2.1) β a2 + ω2

As just described, we assume that our amplitude term

H √ 0 = ξ (5.2.2) β a2 + ω2

. From here, we can establish the following expression,

H2 β2(a2 + ω2) = 0 , and (5.2.3) ξ2 1 H 2 2K 2 β2 = 0 − . (5.2.4) ω2 ξ M sin δ

Our constant β now satisﬁes the following condition

s 1 H 2K β = 0 2 − 2 . ω2 ξ M sin δ

3 Substituting our known values in to this expression (with ξ in radians and using H0 = 4.8 × 10 ), we obtain that β = 7.82 × 104. Regarding the estimation of both δ and ξ from Jen’s paper (Jen et al., 2014), we attempt to ﬁrst utilize the equations given from a homogeneous system where the deformation caused by the deviation from δ is given by

3 1 λδ = λ cos2 δ − , and (5.2.5) L 2 s 3

3 1 λδ = λ sin2 δ − (5.2.6) T 2 s 3 37 If we know the angle of the equivalent easy-axis, then we can directly determine the value of

δ λL. However, we actually want to consider the total change of strain indicated by the gauges,

p P δ ∆λL = λL − λL, and (5.2.7)

P P δ ∆λT = λT − λT , (5.2.8)

P P where λL and λT are the peak amplitudes of λL and λT respectively. The peak amplitudes can easily be read off from the laboratory data in Figure 4.2, but again, we want to write our expressions for the total change of strain for some shift of δ − ξ. Thus, we can actually obtain the following expressions:

3 ∆λP = λ cos2(δ − ξ) − cos2 δ, and (5.2.9) L 2 s

3 ∆λP = λ sin2(δ − ξ) − sin2 δ. (5.2.10) T 2 s

We can obtain a similar relation for the shift of δ + ξ:

3 ∆λP = λ cos2 δ − cos2(δ + ξ), and (5.2.11) L 2 s

3 ∆λP = λ sin2 δ − sin2(δ + ξ). (5.2.12) T 2 s

P P We observe from here that adding the two equations for ∆λL and ∆λT results in a net strain of 0. That is,

P P ∆λL + ∆λT = 0. (5.2.13)

This analytic result is not consistent with the lab findings from Jen’s data, which indicates to us 38 that perhaps the strain is not directionally proportional. Our assumption regarding the anisotropy of the material, was that the material is amorphous with isotropic magnetostriction, however after fitting a curve with a least-squares regression and solving the system using Broyden’s Method, we observe that perhaps it is more complicated. To properly compute the values of δ and ξ, we define the following system using our previous results from Eq. (5.2.9) and Eq. (5.2.10).

  cos2(δ − ξ) − cos2 δ − R P P 1 2∆λL 2∆λT G(δ, ξ) =   , where R1 = and R2 = . (5.2.14)  2 2  3λs 3λs sin (δ − ξ) − sin δ − R2

2 ∂F ∂F We then minimize F (δ, ξ) = ||G(δ, ξ)|| by taking ∂δ = 0 and ∂ξ = 0. We end up with the following expressions:

sin(2δ − 2ξ) − sin(2δ) = 0, and (5.2.15)

2 cos (δ − ξ) − cos(2δ) − R1 + R2 = 0. (5.2.16)

Solving this system using Broyden’s method results in our δ = 51.1017◦ and ξ = 12.2034◦. This value is slightly off from the approximated angle of the equivalent easy-axis from Jen’s work in which δ = 48◦, and ξ = 8◦. Jen actually preset ξ = 7.9◦ based on Fig. 4.1 and then ﬁtted the last equation to ﬁnd the δ. The λ curves in Fig. 4.1 are almost but not strictly periodic, which explains the slight discrepancy. The values of the calculated lab constants for this example are given below:

5.3 Time and Frequency Domain Analysis

We now need to conﬁrm our understanding of the harmonics of the magnetostrictive effects and gain deeper insight into the presence of extraneous frequencies in our vibrational signal. To do this, we utilize our observation that a solution for the coupled-nonlinear differential equations 39 Variable Meaning Value K uniaxial anisotropy from eddy-current 2.38 × 10−4 M magnetization saturation of internal ﬁeld 1.24 × 10−1 β positive constant from restoring torque forces 7.82 × 104 δ angle of the equivalent easy-axis 48◦

Table 5.1: Computed Laboratory Constants from Jen’s Data

λ(t) behaves in a wave-like manner. This allows us to run a fast Fourier transform (FFT) on the data to determine the fundamental harmonics that comprise the signal. Extracting the dominant frequencies from the curves can be done by assuming our solution is periodic and can thus be given by a set of ﬁnite sine and cosine functions:

m−1 ! 1 X λ(t) = A + A cos(2πkt/T ) + B sin(2πkt/T ) + A cos(2πmt/T ) , (5.3.1) n 0 k k m k=1 where m = n/2. In order to continue with our FFT, we assume our data points are equispaced and in the context of this work, they are also real valued. For our analysis to be validated, we expect to see the presence of frequency doubling only in the speciﬁed cases. Our as-cast material should not be exhibiting this behavior and we should not see genuine frequency doubling in the fundamental harmonics.

5.4 Numerical Conclusions

From Fig. 4.2a of Jen’s data, we see that the λ curves are almost periodic but not perfectly, while the mathematical model exhibits perfect sinusoidal behavior after a transient period. It is obvious that a spectral analysis using FFT on Jen’s data would generate non-fundamental harmonics, remembering our mathematical driver H has only one frequency, but the lab driver may contain some additional frequencies. Thus, the goal of using FFT on our mathematically produced λ curves is to determine whether there are harmonics other than the fundamental ones and also to determine whether there is quasi frequency doubling occurring or if no true frequency doubling 40 effect is shown. With this in mind, we juxtapose the signal in the time domain and frequency domain. Taking this into account, along with our previous analysis, we will come to the conclusion that true frequency doubling is not occurring in the lab sample from Jen’s previous work. However, in addition to the lab ribbon, we explore possible simulated materials for which new phenomenon can occur. What we are mainly interested in, is varying the driving amplitude of the signal generator, while ﬁxing the material properties of the sample. Below are several cases that we were able to generate. 41

(b) (a)

Figure 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.

(b)

(a)

3 Figure 5.2: Close frequency is present with the following parameters: f = 0.07 Hz, H0 = 2.8×10 Am−1, δ = 48◦ and we can observe the appearance of unequal valleys.

These results depict our predictions that frequency doubling is present for small values of ω, and that the system is sensitive to the driving amplitude H0. We see in the last plots, Figure 5.5a and Figure 5.5b, that varying the angle of our EEA plays an important role in the presence of frequency doubling as well, since the symmetry of the material is shifted. 42

(b)

(a)

Figure 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.

(b)

(a)

Figure 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.

(b)

(a)

Figure 5.5: No frequency doubling is present with the following parameters: f = 0.07 Hz, H0 = 2.2 × 103 Am−1, δ = 80◦ since the symmetry of the system is off. 43

CHAPTER 6 CONCLUSIONS

6.1 Results

In this thesis we have developed and analyzed a model for determining the resulting frequencies due to magnetostrictive strain on a ferromagnetic material. We have examined the conditions in which double frequency arises, as well as provided mathematical justiﬁcation for experimental results. We’ve shown some criteria for establishing a global solution with a dependence on the driving frequency and amplitude of the generated signal (H), and presented several cases in which the resulting frequencies are explicitly different based on a choice of H0. We can conclude that after a transient period of time, the deviation from the equivalent easy- axis, oscillates around an equilibrium value. This indicates that the ferromagnetic deformation appears to be oscillating around its equivalent easy-axis. When that cyclic shifting is not symmet- ric, the true frequency doubling conditions do not hold, resulting in a frequency dependence on the equilibrium value as well. Thus, we’ve shown that genuine frequency doubling cannot occur

π unless the symmetry and geometry of the system is preserved (θE = 2 ). When this condition is not met, only close frequency doubling can be achieved. Furthermore, our conclusions regarding the doubling of frequency have led us to establishing some restrictive criteria for when this phenomenon occurs. For driving amplitudes lower than

q 4 2 min{π − θE, θE} 2 + ω (Eq. (4.4.2)) we are able to observe true frequency doubling in sin (θE ) our signal analysis. As the driving amplitude gets larger however, we get further and further away from the genuine frequency doubling. A similar relation is valid for the driving frequencies of the generated signal. The smaller the driving frequency cases were able to exhibit frequency doubling, while the higher frequencies cases grew further and further from that phenomenon. Additionally, we discussed the signiﬁcance of the phase lags of our system, coming to the

conclusion that while both φL and φT have some frequency dependence, our ∆φ does not, and always equals π. 44 We have also observed some differences between the simulated results and the data collected from a laboratory experiment, which we now explain here. Examining the curves from Figure 4.2, we see that for the low-frequency case Figure 4.2 (a), a square-like wave is present, rather than a sinusoidal one. The absence of this square patterning in our simulated curves, is because some of the physics-related details of the system were left out of our model for the purpose of our analysis. In reality, our material cannot start rotating or shifting until H is larger enough to counteract the pinning field of the sample (Jen et al., 2014). Thus, for the low-frequency cases we see abrupt changes in motion as the external magnetic field finally overcomes the resisting forces. In our original derivation of the frequency model, we neglected to include a demagnetization factor. The demagnetization field is generated due to the eddy-current mechanism and is modeled using this approximation, 2t 2W 1/2 D ≈ . L πL L

Since in our initial analysis we are considering magnetization only in or near the center of our sample, this factor is considered small enough that its effects were negligible. If we in fact, do include this term, we obtain the following governing equations.

dθL K sin(2(δ − θL)) −β( ) + ( ) = H0 sin(ωt) − DLM cot θL (6.1.1) dt M sin θL

This variable is dependent on the dimensions of our sample and is not a constant when our sample is a rectangle and this demagnetization factor prevents our sample from being completely saturated, as it diverges at the edges and corners. Thus for this thesis, we are only considering the magnetization of the material in the central region for a minor hysteresis loop. This approximation explains some of the differences in the resulting data from Jen’s laboratory experiment and our simulated curves. However, our analysis is conﬁrmed by the results seen from his data. The medium and high-level frequencies do in fact, match our mathematical analysis and the presence of unequal valleys are observed in both the simulated and experimental results. In conclusion, we have established that the phenomenon of frequency doubling is a complex 45 one that hinges on the balancing of material properties, and driving external magnetic forces. We have discussed several of those factors in this work here and presented a model to describe them.

6.2 Future Work

This model provides some useful insight into the mechanics of frequency multiplying however, additional analysis including the major hysteresis loop would be an interesting future problem to tackle. Validating our model with varying sample ferromagnetic materials would also provide more data on which to base our conclusions and test our criteria. Some in-depth error analysis of the numerical methods used to implement our model would provide a clearer understanding of the validity of our approximations, as well as offer insight into strategies for improvement. Another area in which this work could be expanded, would be to include another dimension of the strain, thus allowing for the Weidemann effect to be taken into consideration. 46

BIBLIOGRAPHY

Cullity, B. and C. Graham (2008). Introduction to Magnetic Materials. Wiley-IEEE Press; 2 Ed.

Jen, S. U., C. C. Liu, H. R. Lin, and S. H. Chou (2014). Frequency dependence of the magnetostrictive phenomenon in metglas 2605sa1 ribbon: A minor-loop case. AIP Advances 4(12).

Jiles, D. (1998). Introduction to Magnetism and Magnetic Materials, 2nd Ed. CRC Press.

Joule, J. (1847). On the effects of magnetism upon the dimensions of iron and steel bars. The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 219–224.

Lindﬁeld, G. and J. Penny (2012). Numerical Methods Using Matlab-3rd Ed. Oxford: Elsevier.

Pacific, F. (2016). Understanding Transformer Noise. http://www.federalpacific.com/Literature/Transformers: Federal Pacific.

Soshin, C. (2009). Physics of Ferromagnetism. Oxford University Press.