ELASTIC WAVE ABSORPTION IN LASER-CUT ACOUSTIC

METAMATERIAL PLATES

A Dissertation Presented to the Faculty of the Graduate School

University of Missouri

In Partial Fulfillment

Of the Requirements for the Degree

Doctor of Philosophy

by

HAOGUANG DENG

Dr. P. Frank Pai, Dissertation Supervisor

Dr. Guoliang Huang, Dissertation Co-Supervisor

December 2016

The undersigned, appointed by the Dean of the Graduate School, have examined the dissertation entitled

ELASTIC WAVE ABSORPTION IN LASER-CUT ACOUSTIC

METAMATERIAL PLATES

Presented by Haoguang Deng

A candidate for the degree of Doctor of Philosophy in Mechanical &

Aerospace Engineering

And hereby certify that in their opinion it is worthy of acceptance.

Professor P. Frank Pai

Professor Guoliang Huang

Professor Ming Xin

Professor Steven Neal

Professor Stephen Montgomery-Smith ACKNOWLEDGEMENTS

I would like to thank the many people who contributed to this dissertation. Without their help, I would certainly not have been able to complete this work.

First and foremost, I would like to express my sincere gratitude to my advisor Dr. P.

Frank Pai. Without his academic insight and encouragement, this dissertation would not have been possible. I am very grateful for his guidance and for the many stimulating discussions on a variety of professional and personal topics. It was a really great pleasure working under his supervision. What I learned from him is a treasure for my future. I hope

Dr. Pai will recover from illness and get back to work very soon!

I also wish to thank my dissertation co-advisor, Dr. Guoliang Huang for his helpful suggestions. His enthusiastic encouragements and unwavering supports to this research are greatly appreciated.

At the same time, many thanks to all the committee members, Dr. Ming Xin, Dr.

Steven Neal and Dr. Stephen Montgomery-Smith, for their invaluable suggestions and helps.

Special thanks to Dr. Hao Peng who helped me have a complete understanding of metamaterial theory and finite element analysis. Moreover, I would like to extend my thanks to Jerome Rivers, Miles Barnhart, Xuewei Ruan, Rumian Zhong and Tiancheng Xu for their helps and encouragements. Their supports make it possible to finish this dissertation on time.

ii

Finally, I wish to give my special thanks to my parents Mr. Xiaodong Deng, Mrs.

Shihui Chen and my girlfriend Tong Li. Their company, encouragements, full support of my study and endless love make things around me easier during my hard times.

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TABLE OF CONTENTS ACKNOWLEDGEMENTS ...... ii

LIST OF FIGURES ...... vi

ABSTRACT ...... x

Chapter 1. INTRODUCTION ...... 1

1.1 Introduction to Metamaterials ...... 1

1.2 Dissertation Organization ...... 6

Chapter 2. BASIC CONCEPTS OF ELECTROMAGNETIC METAMATERIALS ...... 8

2.1 Introduction to Electromagnetic Metamaterials ...... 8

2.2 Negative Permittivity and Permeability ...... 9

2.3 Reversed Doppler Effect and Reversed Vavilov-Cerenkov Effect ...... 12

2.4 Negative Refraction and Super Lens ...... 14

2.5 Metamaterial Cloaking ...... 18

Chapter 3. BASIC CONCEPTS OF ACOUSTIC METAMATERIALS ...... 20

3.1 Introduction to Acoustic Metamaterials ...... 20

3.2 Negative Effective Mass of Mass-On-Mass System ...... 24

3.3 Negative Effective Stiffness of Mass-On-Spring System ...... 26

3.4 Stopband and Dispersion of Acoustic Metamaterials ...... 29

3.5 Single-Frequency Vibration Absorber ...... 29

3.6 Multi-Frequency Vibration Absorber ...... 34

3.7 Acoustic Metamaterials Application: Seismic Waveguide ...... 42

Chapter 4. CONVENTIONAL ACOUSTIC METAMATERIAL PLATES ...... 44

4.1 Prototype and Finite Element Analysis of Single-Stopband Acoustic Metamaterial Plates ...... 44

iv

4.2 Prototype and Finite Element Analysis of Multi-Stopband Acoustic Metamaterial Plates ...... 54

4.3 Design Guidelines for Vibration Absorbers ...... 59

Chapter 5. LASER-CUT ACOUSTIC METAMATERIAL PLATES ...... 60

5.1 Prototype of Single-Stopband Laser-Cut Acoustic Metamaterial Plates ...... 60

5.2 Prototype of Multi-Stopband Laser-Cut Acoustic Metamaterial Plates ...... 63

Chapter 6. FINITE ELEMENT ANALYSIS OF RECTANGULAR PLATES ...... 66

6.1 Finite Element Analysis of Conforming Rectangular Plate Elements...... 66

6.2 Modal Analysis of Rectangular Plates ...... 72

6.3 Dispersion Analysis: Bloch Wave Analysis of Rectangular Plates ...... 74

6.4 Frequency Response Analysis of Rectangular Plates ...... 76

6.5 Transient Analysis of Rectangular Plates ...... 77

Chapter 7. NUMERICAL RESULTS ...... 81

7.1 Isotropic Plate ...... 81

7.2 Single-Stopband Laser-Cut Acoustic Metamaterial Plate ...... 84

7.3 Multi-Stopband Laser-Cut Acoustic Metamaterial Plate ...... 100

Chapter 8. CONCLUSIONS AND RECOMMENDATIONS ...... 121

8.1 Concluding Remarks ...... 121

8.2 Recommendations for Future Work ...... 123

REFERENCES ...... 124

VITA ...... 129

v

LIST OF FIGURES

Figure Page

Figure 2-1 Material types based on different signs of permittivity (  ) and permeability (  )...... 9

Figure 2-2 Design for a cubic SRR, proposed by C. R. Simovski and S. He, 2003 [31]. 12

Figure 2-3 (a) Doppler effect in a right-handed medium; (b) Doppler effect in a left-handed medium. The letter A represents the source of radiation while the letter B represents the receiver...... 13

Figure 2-4 Passage of a ray through the interface between two media. 1-incident ray; 2- reflected ray; 3-refracted ray for a left-handed material; 4-refracted ray for a right-handed material...... 15

Figure 2-5 Passage of rays of light through a plate of thickness d made of a left-handed substance. A represents the source of radiation and B represents the detector of radiation...... 17

Figure 2-6 Paths of rays through lenses made of left-handed substances, situated in a vacuum...... 17

Figure 2-7 ([36] From Pendry, “Controlling electromagnetic field,” Science 312, 1780- 1782 (2006), Reprinted with people from AAAS) a schematic that shows the principle on which coordinate-transformation cloaking schemes work. The electromagnetic rays are detoured away from the cloaked region...... 19

Figure 3-1 A 2-DOF mass-on-mass system...... 26

Figure 3-2 A 2-DOF mass-on-spring system...... 28

Figure 3-3 A Simplified 2-DOF EMM unit...... 32

Figure 3-4 Absolute frequency response of m1 with (a) mm2112/0,0.1,0.2,0.3,0.4,  0.01, (b) mm21/  0.1, 2  0.01,0.03,0.05,0.07,0.1. . 33

Figure 3-5 A Simplified 3-DOF EMM unit...... 36

Figure 3-6 Absolute frequency response of with mm23123 0,0.01,0 (black) and mm21/ 0.2, mm3/ 2 0.1, 1   2   3  0.01(red)...... 37

Figure 3-7 Average frequency response of m1 with mm2/ 1 0.2, 1   2   3 and (a) m3/ m 2 s 1 0.05(black), (b) s1  0.1(blue), (c) s1  0.2 (red)...... 40

vi

Figure 3-8 Average frequency response of m1 with m2/ m 1 0.2, s 1 0.1 and (a)

123 0.01 (black), (b) 123 0.01,0.1 (blue), (c) 132 0.01,0.1 (red)...... 40

Figure 3-9 Average frequency response of m1 with m2/ m 1 0.2, s 1 0.1 and (a) 3  0.005

(black), (b) 3  0 . 0 5 (blue), (c) 3  0 . 2 (red)...... 41

Figure 3-10 Average frequency response of with mms21113/0.2,0.1,0.01  and

(a) 2  0 . 0 0 1 (black), (b) 2  0 . 0 1 (blue), (c) 2  0 . 1 (red)...... 41

Figure 3-11 Seismic wave: (a) longitudinal wave, (b) transverse wave, (c) Love wave, and (d) Rayleigh wave (adapt from http://scweb.cwb.gov.tw/images/FAQ/eq005.jpg)...... 43

Figure 4-1 Unit cell of an SCMP...... 44

Figure 4-2 The prototype of an SCMP including one isotropic plate and repeat patterned mass-spring subsystems...... 45

Figure 4-3 Dispersion surfaces of an infinite SCMP in its irreducible Brillouin zone, where , are wave numbers along x and y direction,  is the wave frequency...... 53

Figure 4-4 Dispersion curves of an infinite SCMP in its irreducible Brillouin zone...... 53

Figure 4-5 Unit cell of an MCMP...... 54

Figure 4-6 The prototype of an MCMP including one base plate and repeat patterned mass- spring subsystems...... 55

Figure 4-7 Dispersion surfaces of an infinite MCMP in its irreducible Brillouin zone, , are wave numbers along x and y direction,  is wave frequency...... 58

Figure 4-8 Dispersion curves of an infinite MCMP in its irreducible Brillouin zone. .... 58

Figure 5-1 A single-stopband laser-cut acoustic metamaterial plate (SLCMP)...... 61

Figure 5-2 An SLCMP unit cell with geometric and system parameters...... 61

Figure 6-1 A conforming rectangular plate element with counterclockwise nodes...... 66

Figure 6-2 (a) A RVE of an infinite structure and (b) Irreducible Brillouin zone...... 75

Figure 7-1 An isotropic plate with hinged left and right edges boundary condition (represented by two red lines). An excitation force (represented by the yellow point) is applied at the second node from the left of the centerline...... 82

Figure 7-2 AFRs at x = 0.95 L of the isotropic plate under different boundary conditions: (a) hinged left and right edges, (b) hinged left edge and (c) hinged four edges...... 83

vii

Figure 7-3 First 12 mode shapes for the SLCMP unit cell under periodic boundary conditions...... 86

Figure 7-4 Dispersion curves of an infinite SLCMP with reduced wave vector k sweep following the path of        in the irreducible Brillouin zone...... 88

Figure 7-5 (a) AFRs of the SLCMP and its corresponding isotropic plate at x = 0.98 L with hinged left and right edges under different damping coefficients. (b) – (d): Vertical displacement distributions with damping coefficients of  0  , 5 5 e at different excitation frequencies: (b) f = 162 Hz , (c) f = 286 Hz and (d) f = 532 Hz...... 91

Figure 7-6 (a) AFRs of a hinged-left-edge SLCMP and its corresponding isotropic plate at x = 0.98 L with damping coefficients of . (b) - (d): Vertical displacement distributions at different excitation frequencies: (b) f = 162 Hz, (c) f = 286 Hz and (d) f = 532 Hz...... 94

Figure 7-7 (a) AFRs of a hinged-four-edge SLCMP and its corresponding isotropic plate at x = 0.98 L with damping coefficients of . (b) - (d): Vertical displacement distributions at different excitation frequencies: (b) f = 162 Hz, (c) f = 286 Hz and (d) f = 532 Hz...... 96

Figure 7-8 Average displacement of a hinged-left-and-right-edge SLCMP with damping coefficients of ii0,55 e at x = 0.02 L (black), 0.5 L (red) and 0.98 L (blue) of the centerline under different excitation frequencies in transient state: (a) f = 162 Hz, (b) f = 286 Hz and (c) f = 523 Hz...... 99

Figure 7-9 First 12 mode shapes for the MLCMP unit cell under periodic boundary conditions...... 103

Figure 7-10 Dispersion curves of an infinite MLCMP with reduced wave vector k sweep following the path of        in the irreducible Brillouin zone...... 105

Figure 7-11 (a) AFRs of an MLCMP and its corresponding isotropic plate at x = 0.98 L with hinged left and right edges under different damping coefficients. (b) - (f): Vertical displacement distributions with damping coefficients of ii0,  5e  5 at different excitation frequencies: (b) f = 103 Hz, (c) f = 189 Hz, (d) f = 262 Hz (e) f = 290 Hz and (f) f = 400 Hz...... 109

Figure 7-12 (a) AFRs of a hinged-left-edge MLCMP and its corresponding isotropic plate at x = 0.98 L with damping coefficients are ii0,55 e . (b) - (f): Vertical displacement at different excitation frequencies: (b) f = 103 Hz, (c) f = 189 Hz, (d) f = 262 Hz, (e) f = 290 Hz and (f) f = 400 Hz...... 113

Figure 7-13 (a) AFRs of a hinged-four-edge MLCMP and its corresponding isotropic plate at x = 0.98 L with damping coefficients are . (b) - (f): Vertical

viii displacement distributions at different excitation frequencies: (b) f = 103 Hz, (c) f = 189 Hz, (d) f = 262 Hz, (e) f = 290 Hz and (f) f = 400 Hz...... 116

Figure 7-14 Average displacement of a hinged-left-and-right-edge MLCMP with damping coefficients of ii 0  , 5 5 e at x = 0.02 L (black), 0.5 L (red) and 0.98 L (blue) of the centerline under different excitation frequencies in transient state: (a) f = 103 Hz, (b) f = 189 Hz, (c) f = 262 Hz, (d) f = 290 Hz and (e) f = 400 Hz...... 120

ix

ELASTIC WAVE ABSORPTION IN LASER-CUT ACOUSTIC

METAMATERIAL PLATES

Haoguang Deng

Dr. P. Frank Pai, Dissertation Supervisor

Dr. Guoliang Huang, Dissertation Co-Supervisor

ABSTRACT

This dissertation presents the modeling technique for design and analysis of laser- cut acoustic metamaterial plates (LCMPs) capable of acoustic/elastic vibration suppression.

The conventional acoustic metamaterial plate (CMP) consists of a uniform isotropic plate with many small spring-mass-damper subsystems integrated at different locations acting as vibration absorbers. Both the single-stopband laser-cut acoustic metamaterial plate

(SLCMP) and the multi-stopband laser-cut acoustic metamaterial plate (MLCMP) are proposed in this dissertation, with cutting periodic vibration absorbers consisting of one center mass and four surrounding beams into an isotropic plate for an SLCMP or two center masses and eight surrounding beams for an MLCMP. The concepts of negative effective mass density and negative effective stiffness as well as acoustic and optical modes are well explained. This work shows that local resonance of the CMP unit cells can generate inertia forces against the external load and prevent elastic wave from propagating forward within designed excitation frequency bands, which are called stopbands. For infinite CMPs, dispersion analysis is conducted to find their stopbands. For unit cells of finite CMPs, governing equations are derived based on extended Hamilton principle and their stopbands

x are obtained and explained in detail. For unit cells of LCMPs, modal analysis is used to calculate their resonant frequencies. Dispersion analysis in infinite LCMPs is conducted, where stopbands are clearly illustrated. Finite LCMP designs are also modeled numerically, where the stopbands characteristics are investigated with frequency response analysis and transient analysis. Factors that influence the stopbands characteristics such as the unit cells’ resonant frequencies, damping coefficient and plate boundary conditions are also analyzed and discussed. Based on the fundamental acoustic metamaterial concept, dynamic disturbances with frequencies near the local resonance of the LCMP microstructure will be attenuated, thus inhibiting the propagation of acoustic/elastic waves. For an LCMP with unit cells designed with one specific resonant frequency, a single stopband will occur at the unit cells’ locally resonant frequency. Moreover, if each unit cell has two locally resonant frequencies, there are two stopbands. In addition, we demonstrate that increasing the damping coefficients of the SLCMP vibration absorbers will increase the stopband’s width to a small degree as well as lower the overall response of the base plate. For an

MLCMP, two stopbands can be combined into one wider stopband by using a greater damping coefficient for the inside vibration absorbers when unit cells’ two resonant frequencies are designed to be close to each other. Finally, it is found that boundary conditions applied to the LCMPs will not significantly affect the stopbands characteristics.

xi

Chapter 1. INTRODUCTION

1.1 Introduction to Metamaterials

Metamaterials are a new class of artificially designed materials that exhibit exceptional properties not readily observed in naturally occurring materials. These preternatural properties are derived from their specially designed microstructures rather than base materials’ natural properties. They are typically assembled with single or multi- phase conventional material elements which are usually constructed into low-dimensional repeating patterns. The artificially fabrication, extrinsic and low-dimensional inhomogeneity of the engineered structures can alter acoustic/elastic wave speed, direction and wavelength, resulting in interesting properties. The possible existence of materials with independent or simultaneous negative permittivity and permeability was first suggested by

Victor Veselago by analyzing the dispersion equation of the electromagnetic wave in 1964

[1]. However, it was not until recently that scientists began to deeply study metamaterials due to the discovery of negative permittivity and permeability in some artificial materials, which have yet been found in nature [2]. Engineers have been inspired by this concept and have started to design new properties that go beyond conventional materials’ performance limitations.

Metamaterials were first considered in electromagnetic wave propagation (Pendry,

2000 [3]), however, research has recently shifted to include their acoustic/elastic wave counterpart analogous to electromagnetic metamaterials (Li and Chan, 2004 [4]; Cheng et al., 2008 [5]; Huang et al., 2009 [6]). The primary research in metamaterials focuses on the aforementioned properties that use the resonance between their unit cells and the external electromagnetic field, which are called electromagnetic metamaterials (EMs, also called

1 optical metamaterials). The well-known properties for EMs include negative permittivity and permeability, negative reflective index, inverse Doppler effect, inverse Vavilov-

Cherenkov effect and etc. [3, 7-9]. In particular, when a composite medium’s electromagnetic wave enters into a material, its electric and magnetic fields interact with the material’s electrons and other charges of atoms and molecules, which in turn alters the wave speed, direction and wavelength. Hence it is possible to use this electromagnetic interaction to design EMs with negative magnetic permittivity and negative electric permeability (Pendry, 2000 [3]). A range of applications for EMs have been including electromagnetic absorbers [10], subwavelength waveguides [11], backwards wave antenna

[12], artificial magnetic devices [13] and permanent magnets with high performance [14].

However, some theories of EMs were questioned by scientists. For example, Markel [15] believed that negative index materials would result in other problems such as negative heat, which is impossible in physics. Both theoretical and experimental issues of Ems are still need for further studies to clear up.

Elastic metamaterials (EMMs) have seen a significant increase in attention over the last couple decades owing to their unique effective material properties [16, 17].

Specifically, elastic metamaterials manage elastic wave in a deformable solid, while acoustic metamaterials manage acoustic wave in a fluid. Because they both manage elastic wave with frequencies within the audible range (20 Hz to 20 kHz), some researchers also refer to elastic metamaterials as acoustic metamaterials [18]. EMMs are artificially fabricated materials that can control and even guide propagating elastic waves due to the locally resonant mechanism between the propagating acoustic/elastic waves and the locally resonant microstructure. Similar to EMs which exhibit negative permittivity and

2 permeability, EMMs can exhibit negative effective stiffness and negative effective mass density. These unique dynamic effective material properties have been explored for applications including seismic waveguides, sound absorption and vibration suppression [4,

5, 19-21]. EMs exhibit properties by having much smaller distances between atoms/molecules than the wavelength of electromagnetic wave while EMMs show their properties by having much smaller distances between atoms/molecules than the wavelength of elastic wave. EMMs are traditionally fabricated by assembling periodic mechanical subunits into a naturally occurring material to introduce a locally resonant response to propagating waves and in turn change the wave properties such as speed, direction and wavelength. The EMM subunit geometry is often constrained since these models are only valid for wavelengths longer than their overall size. Popular research topics about EMMs include acoustic absorbers [22], subwavelength guides [23], ultrasound focusing [24], elastic wave absorption [25] and structural vibration mitigation [26].

Because earthquakes often generate destructive surface and body wave, seismic waveguide is an important application of EMMs [27].

EMMs have been thoroughly investigated and studied in the past decades [17, 25, 28,

29]. Most EMM designs are modeled by connecting mass-spring-damper subsystems to a base structure to act as locally resonant vibration absorbers. The working mechanism of

EMMs is that the resonant motion of vibration absorbers generates a resistive force against the external excitation which can prevent elastic wave from propagating forward within designed excitation frequency bands, which are called stopbands [16]. Numerical results show that the stopband’s location is at the locally resonant frequency of vibration absorbers while the stopband’s width is determined by the mass ratio between each vibration absorber

3 and the base structure. Damping of vibration absorbers can also affect the stopband characteristics. While conventional EMMs require additional space to attach vibration absorbers, in some circumstances attaching external vibration absorbers is impractical due to limited space. Therefore, EMMs with no attaching vibration absorbers are of great value to study. A new type of EMM with designed laser cuts is developed and analyzed in this research. The key challenges for studying metamaterials are theoretical development of new metamaterials with unique subunits, understanding working mechanisms and verifying the preternatural properties of metamaterials by managing electromagnetic and elastic wave propagation. Due to the purpose of this investigation on wave propagation in solid structures, EMMs are the only kind of metamaterials that are studied in this research.

However, since the original concepts and developments of metamaterials came from EMs, it is appropriate to conduct a brief review of EMs before attempting to comprehend EMMs.

This dissertation aims to propose an acoustic metamaterial plate design without the need of attaching external vibration absorbers. Each vibration absorber is modeled by cutting periodically arranged mass-spring-damper subsystems into an isotropic plate with a center mass and four surrounding beams for a single-frequency vibration absorber (or two center masses and eight surrounding beams for a multi-frequency vibration absorber).

Modeling technique for the laser-cut acoustic metamaterial plates (LCMPs) is well presented. Theoretical stopbands are obtained through dispersion analysis on infinite

LCMPs using a numerical modelling approach. Finally, finite LCMP models are created and their stopbands are obtained through frequency response analysis and transient analysis.

The influence of the interior vibration absorbers (geometry dependent), damping

4 coefficients and plate boundary conditions upon the wave attenuation behavior are also investigated.

5

1.2 Dissertation Organization

Because of the wide use of plates in engineering structures, acoustic metamaterial

(EMM) plates have more applications than EMM bars or beams. This dissertation is organized into eight chapters. Besides the current chapter which intends to give a brief introduction of EMMs and the motivation of this dissertation, the other seven chapters are organized as follows:

The second chapter introduces the basic concepts of electromagnetic metamaterials

(EMs). Negative permittivity and permeability are introduced and derived from Maxwell equations and the constitutive relations. Other exceptional properties of EMs such as reverse Doppler effect and reverse Vavilov-Cerenkov effect are described. Finally, the negative refraction and its application of super lens and cloaking are introduced and analyzed.

The third chapter deals with the basic concepts of EMMs. Similar to negative permittivity and permeability of EMs, EMMs’ negative effective mass density and stiffness are described and analyzed based on the models of mass-on-mass system and mass-on- spring system, respectively. Stopbands and wave dispersion of EMMs are introduced through Newton’s second law. Next, single and multi-frequency vibration absorbers for

EMMs are analyzed and factors that affect stopbands are well studied and summarized.

Finally, the seismic waveguide is introduced by studying different types of propagating wave.

The fourth chapter primarily focuses the conventional acoustic metamaterial plates

(CMPs). The prototypes of conventional single and multi-stopband acoustic metamaterial

6 plates (SCMPs and MCMPs) are introduced and analyzed by finite element method. After that, the design guidelines for vibration absorbers are discussed.

In chapter five, the prototypes of single and multi-stopband laser-cut acoustic metamaterial plates (SLCMPs and MLCMPs) are introduced. Dimension of each plate in this dissertation is given.

The sixth chapter deals with different analysis methods of rectangular plates based on the finite element analysis. First, the extended Hamilton principle is used to analyze the conforming rectangular plate. Then different analysis methods including modal, dispersion, dynamic steady-state and transient analysis are presented and discussed.

The seventh chapter shows the numerical results from each method. Mode shapes of

SLCMP and MLCMP unit cells and their resonant frequencies are obtained from modal analysis. Dispersion curves are obtained for infinite SLCMP and MLCMP to show their theoretical stopbands from dispersion analysis. Elastic wave absorption and vibration suppression of finite SLCMP and MLCMP are verified through dynamic steady-state and transient analysis.

The final chapter summarizes and concludes the elastic wave absorption characteristic of LCMPs. Recommendations for future work are also proposed.

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Chapter 2. BASIC CONCEPTS OF ELECTROMAGNETIC

METAMATERIALS

2.1 Introduction to Electromagnetic Metamaterials

Electromagnetic metamaterials (EMs) are usually defined as artificial, effective homogeneous electromagnetic structures with unique properties not readily found in naturally occurring materials. The homogeneous electromagnetic structure’s unit cell size is much smaller than the guided wavelength (usually  /4). The artificial materials will behave as real materials when the condition of effective-homogeneity is satisfied. The constitutive parameters, which depended on the nature of the unit cell, are the permittivity

 and permeability  . These two parameters directly relate to the refractive index n by

n  rr, where  r and r are relative permittivity and permeability. The four possible sign combinations are illustrated in Figure 2-1. Materials with sign combinations of (+, +), (+, -), (-, +) are called conventional materials, while the ones with (-, -) are called left-handed materials, which are also defined as electromagnetic metamaterials (EMs).

Materials with both  and  are called Double Positive Materials (DPSs) (e.g.,

Dielectrics). Materials with 0,0 are called Mu Negative Materials (MNGs) (e.g., gyrotropic magnetic materials). Materials with 0,0 are called Epsilon Negative

Materials (ENGs), such as plasmas. The last kind of materials with both negative  and  are called Double Negative Materials (DNGs). This type of materials is not found in naturally occurring materials but can only be artificially generated. Left-handed materials

8 are one of EMM types that are commonly used in the literature and they have been the most popular EMMs due to their unique properties.

휀

MNG DPS 휇 < 0, 휀 > 0, 휇 > 0, 휀 > 0, Gyrotropic Dielectrics magnetic materials

휇

DNG 휇 < 0, 휀 < 0, ENG Not found in nature 휇 > 0, 휀 < 0, but physically Plasmas realizable

Figure 2-1 Material types based on different signs of permittivity (  ) and permeability (  ).

2.2 Negative Permittivity and Permeability

The dielectric constant  and the magnetic permeability  are two parameters which determine electromagnetic wave propagation. For an isotropic substance, the relation between frequency  and wave vector k is

 2 kn22 (2.1) c2 where n2   is the square of the refractive index of the substance and c is the fluid phase speed. If  and  are assumed to be simultaneously negative, the properties of materials

9 should be different from those with positive  and  [1]. From Maxwell equations and constitutive relations, we have

1 D  H ct 1 B  E  (2.2) ct DE  BH  where  is curl operator, D is electric flux density vector, B is magnetic flux density vector,

E is electric field intensity vector and H is magnetic field intensity vector. Eq. (2.2) can be further reduced to

 kHE   c  kEH c (2.3) if all quantities are proportional to eikzt() (e.g., a plane monochromatic wave). Notice that when 0,0 , E, H and k form a right-handed triplet of vectors while 0,0 leads to a left-handed triplet of vectors. Wave propagation in a medium can be generated by the matrix

1  2  3 G     (2.4) 1 2 3 1  2  3

where i , i and  i are E, H and k vectors in the form of direction cosines. The determinant of G is 1 for a right-handed medium with 0, 0 and -1 for a left-handed medium with . Notice that the elements in Eq. (2.4) satisfy the relation

Gij pA ij (2.5)

10 where Aij is the algebraic complement of the element Gij , p=1 for a right-handed medium and -1 for a left-handed medium.

The energy flux carried by the wave is another quantity whose characteristic is related to the rightness of the given medium, usually determined by the Poynting vector S, which is given by

c S E H (2.6) 4

Eq. (2.6) indicates that S, E and H are always in a right-handed set. It should be noted by comparing Eq. (2.3) with Eq. (2.6) that S and k are in the same direction for right-handed media and in opposite directions for left-handed media. Recall that vector k is in the direction of phase velocity or group velocity. Therefore, the phase velocity for a left- handed substance is opposite to the energy flux.

Although Veselago proposed the concept of metamaterials and their superior properties in the early 1960s, it was not until the 1990s that scientists started to fabricate metamaterials with microstructures. Pendry et al. [30] proposed a theory that every material is a composite, even if the individual ingredients are made by atoms and molecules. Instead of presenting from a homogeneous view of the electromagnetic properties of a medium, one can replace the atoms from the original concept with artificial structures on a larger scale. One example of such an inclusion is the split-ring resonator (SRR). SRRs are highly deductive rectangular metallic rings with a non-deductive gap between each two of them.

SRRs could be attached to the surface of a cubic as shown in Figure 2-2. When the magnetic field is temporary varying, the deductive ring generates electric circular current which will result in an opposite electric field to the original one. Two electric fields with different signs counteract with each other and the electric energy will be stored in the vicinity gap,

11 which leads to new magnetic energy that concentrates in the region enclosed by the ring.

The new magnetic field is perpendicular to the original magnetic field and thus will generate negative permittivity and permeability.

Figure 2-2 Design for a cubic SRR, proposed by C. R. Simovski and S. He, 2003 [31].

2.3 Reversed Doppler Effect and Reversed Vavilov-Cerenkov Effect

Soviet physician Victor Veselago discovered several fundamental phenomena of left-handed media in 1967 including reverse Doppler effect, negative refractive index and invisible cloaking [1]. First exceptional property of EMs that Veselago proposed is the reversed Doppler effect. Assume a radiation detector moves relative to a source with

emitting frequency 0 in a left-handed medium. The detector will receive wave which is corresponding to some particular phase, as is shown in Figure 2-3. Instead of receiving signals with a frequency larger than in a right-handed medium, the detector will receive

a frequency that is smaller than 0 . The formula for the Doppler shift in a left-handed medium can be written as follows:

12

v 0 1 p (2.7) u where p is the quantity regarding the medium’s type, v is the velocity of the detector and u is the velocity of the energy flux which is always regarded as positive. It should be noted that v is positive when the detector is leaving away from the source. Seddon et al. [9] have verified the reversed Doppler effect experimentally in 2003.

S (a)

A v B k S (b)

A v k B Figure 2-3 (a) Doppler effect in a right-handed medium; (b) Doppler effect in a left-handed medium. The letter A represents the source of radiation while the letter B represents the receiver.

Similar to the Doppler effect, the Vavilov-Cherenkov effect will also be reversed in left-handed media. When a charged particle (usually an electron) passes through a medium at a speed v greater than the phase velocity of light in the same medium, it will radiate from

i kzr zk rt   a cone in front of the particle based on the law e , where kz is the wave vector

along z direction, kr is the wave vector along r direction and will change sign in different

media. For left-handed media, the vector kr will be directed toward the trajectory of the particle and thus the particle will radiate from a cone behind itself.

13

2.4 Negative Refraction and Super Lens

The refractive index is a fundamental parameter describing the interaction between waves and materials. One of the famous properties of EMMs is the negative refractive index. As proposed by Veselago back in the 1960s [3], the boundary conditions for a ray of light from a transparent medium into another are given below:

EEEE, (2.8) t1 t 212 n 1 n 2

HHHH, (2.9) ttnn1212 12

The boundary conditions in Eqs. (2.8) - (2.9) should be satisfied independently of whether or not the media have the same rightness. Eq. (2.8) shows that the directions of x and y components of E and H in the refracted ray passage remain the same despite the rightness difference. However, the direction of z component will change opposite if two media have different rightness. Figure 2-4 illustrates the refractions on the interface of two conventional media and left-handed media. For conventional media, the ray from passage

1 is refracted to passage 4. However, if the material is left-handed, the direction of the refracted ray will go to passage 3. It should be noted that the direction of the reflected ray is always the same (e.g., passage 2). The relation between the angle of incident  and the angle of refraction  is expressed by Snell’s law as

sin  n  2 22 (2.10) sin n1 11

where ni,, i i are refractive index of two media, respectively. It should be noted that the absolute value of the refraction angle remains the same despite the sign change when two media have different rightness. Therefore, Snell’s law has to be given a more precise form as:

14

sin  np 2222 (2.11) sin np1111

Here p1 and p2 are the rightness of the first and second medium. From Eq. (2.11) we can clearly see that the refractive index could be negative if the rightness of two media are different. i.e., the first medium is right-handed with a positive while the second medium

is left-handed with a negative , leading to a negative nn21/ , which indicates that the refraction direction will go to passage 3. Negative index of refraction was verified experimentally by R.A. Shelby et al. in 2001 [32].

1 2



Left-handed materials  Right-handed materials

3 4

Figure 2-4 Passage of a ray through the interface between two media. 1-incident ray; 2-reflected ray; 3- refracted ray for a left-handed material; 4-refracted ray for a right-handed material.

The most important application of negative refraction property is super lens. Due to the diffraction of light wave, fundamental maximum resolution always exists in common optical devices, which is called diffraction limit in an imaging system. It’s impossible for

15 a common optical device to focus light into an area smaller than its wavelength. However, the immanent diffraction limit can be avoided through using a plane plate of material with a negative refractive index, which is called a super lens. Fresnel’s equation shows the

behavior of light when a beam passes from a point S in a medium with 110 , 0 and

focuses on point P from a left-handed medium with 2121=, . In Figure 2-5, a negative refractive index allows a flat slab lens to focus diverging beams into two images: one inside the slab and the other outside the slab. The evanescent waves have been enhanced across the lens and decay again after emerging from the negative index lens.

Hence the amplitudes of two image planes are of the same level. Meanwhile, the phase change is zero due to a reversed phase front when the wave propagates into the negative index lens. Therefore, no reflected ray exists and the whole beam is refracted to the passage and a perfect image is obtained. After the beam refracts out of the left-handed material, it maintains the same direction. This perfect lens with same direction is often called super lens. We can see that a slab super lens can focus the radiation from one point to the other.

If the slab lens is well designed, the light can be focused to an area that is smaller than its wavelength. Further experiments verified that a silver super lens could overcome the diffraction limit and achieve a resolution with only one-sixth of the illumination wavelength (Pendry, [3]). Another example of using left-handed substances to result in very unusual refracting systems is convex and concave lenses. Instead of having a converging effect for the convex lens and a diverging effect for the concave lens, the actual

16 effects are exactly opposite. Figure 2-6 shows that by using a left-handed material, the convex lens is a diverging lens while the concave lens is a converging lens.

S S S S

A B

d

Figure 2-5 Passage of rays of light through a plate of thickness d made of a left-handed substance. A represents the source of radiation and B represents the detector of radiation.

Figure 2-6 Paths of rays through lenses made of left-handed substances, situated in a vacuum.

17

2.5 Metamaterial Cloaking

The interest of metamaterials’ applications such as sub-wavelength focusing and perfect-lensing has been further developed to make a given object “invisible” to electromagnetic radiation [7, 33]. The term “cloaking” is associated with the metamaterials’ characteristic that may overcome limitations of some camouflaging technologies. So far there are three ways of cloaking: electromagnetic cloaking, plasmatic cloaking and metamaterial cloaking. Metamaterial cloaking theory was demonstrated in 2003 [34], where an anomalous resonance between left-handed materials may produce wave tunneling, which leads to invisibility of the whole region where the two metamaterial slabs are covered. A more general mechanism for scattering cancelling was proposed by Alù in 2005

[35], claiming that when a dielectric or conducting object was surrounded by metamaterials and plasmonic materials with inherent local negative polarizability, magnitude orders can be lower than that of the isolated object itself and thus, leads to a wave scatter so that the object is invisible around a designed frequency from the external observer with a very low residual scattering.

Many metamaterial cloaking techniques have been suggested in the last decades [7].

Among them, there are typically two base techniques for metamaterial cloaking: coordinate transformation technique and anomalous local resonance technique. Coordinate transformation technique is based on a conformal coordinate transformation that creates roundabout paths to isolate a given region from its surrounding, as is illustrated in Figure

2-7. Objects in the defined location are still present. However, incident waves are guided around them without being affected by the object itself. It should be noted that by using this technique, the cloaking is achieved in the inside region to the cloak. On the other hand,

18 anomalous local resonance techniques accomplish the cloaking in the region that is external to the cloak by using localized anomalous resonance that may hide the region from its surrounding and exploit the metamaterial resonance between the background and the cloak.

Figure 2-7 ([36] From Pendry, “Controlling electromagnetic field,” Science 312, 1780-1782 (2006), Reprinted with people from AAAS) a schematic that shows the principle on which coordinate-transformation cloaking schemes work. The electromagnetic rays are detoured away from the cloaked region.

19

Chapter 3. BASIC CONCEPTS OF ACOUSTIC METAMATERIALS

3.1 Introduction to Acoustic Metamaterials

While the proposals of electromagnetic materials (EMs) from Victor Veselago have been validated in last twenty years, research in elastic/acoustic metamaterials (EMMs) has the same goal of broader material response with sound wave. Motivated by the mathematical analogy between acoustic and electromagnetic wave, researchers have attempted to find the left-handed EMMs.

Based on the simple realization that composites with local resonances can exhibit effective negative elastic constraints at a certain frequency range, in 2000, Liu, et al. [37] fabricated and investigated a locally resonant sonic material by using centimeter-sized lead balls as the core material and coating with a 2.5 - mm layer of silicon rubber. The dimension of this sonic individual unit cell is in deep-subwavelength range (two orders of magnitude smaller than the relevant wavelength) so that the effective properties such as stiffness and mass density can be assigned to this material. A 3-D sonic crystal was modeled with each unit consisting of a lead ball coated by a silicon rubber layer. By being arranged in an epoxy matrix, these coated balls formed a 3-D lattice system with a lattice constant of 1.55 cm.

In this case, these lead balls can be regarded as mass inclusions while the silicon rubber and epoxy were regarded as springs. Therefore, the whole lattice system can be treated as a microsystem consisting of masses and springs. Experimental results showed that the elastic modulus and transmission coefficient highly depend on the excitation frequency. At a certain frequency range, the effective modulus will become negative while the transmission coefficient can be negligibly small. This work paved the way towards EMMs analogous to EMs. In 2003, Sheng et al. [38] reported a new exemplification to realize the

20 robust elastic wave band gaps in the audio frequency range. They theoretically and experimentally showed that instead of only realizing sonic band gaps in crystals with a much smaller lattice constant order, they can also exist in non-periodic structures. In the same year, Guffaw and Sanchez-Dehesa [39] presented a variational method to study the elastic wave propagation in 2-D periodic systems which contain lattices of locally resonant materials. In 2004, Li and Chan [4] reported a possible existence of EMMs by utilizing the effective mass density and stiffness derived by Berryman [40]. They claimed that instead of EMs which exhibit negative permeability and negative permittivity, negative effective mass density and negative effective stiffness can be derived from a single resonance structure. In their opinion, a structure with negative effective stiffness can expand upon compression at a designed frequency. They also claimed that at a certain excitation frequency, a negative mass density can lead to a different phase movement of the structure.

However, the assumption of opposite phase movement direction was corrected since it is the opposite direction between external force and acceleration rather than the displacement.

Liu et al. [41] presented an analytical model in 2005 to prove the negative effective mass density of three-component photonic crystals with local resonances. They claimed that when excitation frequency is closed to model’s natural frequency, the effective mass density can be negative. More articles about EMMs applications have been published since

2007. Milton and Willis [42] studied a solid body attaching with mass-spring subsystems.

By assuming the attaching subsystems are invisible and by using modified equations from

Newton’s second law, they showed that the effective mass also highly depends on the excitation frequency and can be negative at a certain frequency range. Lazarov and Jensen

[43] studied the wave propagation on a locally resonant lattice model that was introduced

21 by Vincent [44]. They claimed that for nonlinear oscillators, the position of band gaps on the frequency axis can be shifted depending on the amplitude and the degree of nonlinear behavior. Wu et al. [45] developed an effective medium theory that the effective shear modulus can also turn to negative when near the local resonances. Cheng et al. [5] presented a 1-D ultrasonic metamaterial consists of an array of repeated unit cells by attaching Helmholtz resonators which have a short neck and an acoustic mass to an elastic beam structure. They showed that this ultrasonic metamaterial exhibits a band gap with both negative effective mass density and negative effective stiffness. They found that the number of frequency stopbands that due to the double-negative region is dependent on the number of resonators in each unit. Also in 2008, Yao et al. [19] experimentally demonstrated the effect of the negative effective mass density on the dynamic transmission in a 1-D spring-mass system. Although the static elastic modulus and density is positive in order to keep the structure itself stable, the dynamic effective acoustic properties are able to turn negative at resonance. As the negative mass density exists, the acceleration will be out of phase with the dynamic pressure gradient at certain excitation frequency range.

Similarly, at certain frequency range, negative stiffness will result in an out-of-phase region between volume change and external force.

Despite the lumped masses and springs at nodal points of lattice systems which are the simple models that can achieve negative effective mass density and stiffness at a given excitation, many other structures such as bars, beams, membranes, plates or even composite structures are developed and used in the modern metamaterial applications.

Movchan and Guenneau [46] subsequently proposed a 2-D localized resonant structures by using arrays of cylinders with a split ring cross-section as building blocks. Pai [16]

22 proposed a hollow longitudinal bar structure with microscope mass-spring subsystems attached inside. By performing finite element analysis on the EMM bar, acoustic mode and optical mode were successively shown and stopbands were obtained at given excitation frequencies. Sun et al. [17] followed a similar method and analyzed a beam structure with attached mass-spring subsystems. Zhu and Huang [47] proposed a plate model with mass- spring microstructures attached inside the cavities. The in-plane and out-of-plane guided wave in the EMM plate have been investigated numerically and experimentally. The local resonance mechanism that results in wave attenuation in the EMM plate has been demonstrated. Among the EMMs that have been reported, their narrow working frequencies have limited their applications [48]. Zhu et al. [49, 50] used multiple resonators of different resonance frequencies to obtain a stopband from 400 Hz to 900 Hz. Meng et al. [20] optimized an EMM structure with a stopband of 800 Hz to 2500 Hz by applying a genetic algorithm. Zigoneanu et al. [51] used strong anisotropic EMMs to broaden the stopband up to 500 Hz - 3000 Hz.

23

3.2 Negative Effective Mass of Mass-On-Mass System

EMMs can have negative effective mass density and stiffness, which are similar to negative permittivity and permeability of EMs [52]. First, a model of a 2-degree-of- freedom (2-DOF) mass-on-mass system is considered in order to show its negative

effective mass. As shown in Figure 3-1, base mass m1 is connected to mass m2 by a spring

with constant k. A harmonic excitation with a magnitude F0 and an excitation frequency

 is given to mass m1 . u1 and u2 represent the displacements of mass m1 and m2 . The equations of motion for the mass-on-mass system are

muu 0 kkF  111 jt (3.1)  , FF e 0 0 muu222 kk 0 where j 1 is the imaginary unit. Take the Laplace transform for two equations and we have

2 m2 sk  k 2222 us1() s()() m1 m 2121 skmkms 212 m m skmkm Fs()    (3.2) us() 2 0 2 k m1 sk   2222 s()() m1 m 2121 skmkms 212 m m skmkm

We denote the frequency response functions Ui and natural frequency ii km/ for

each mass. It should be noted that 1 is the locally resonant frequency of u1 if u2  0 and

2 is the locally resonant frequency of the mass-spring vibration absorber ( m2 and k ) if

jt u1  0 . Assuming uii() t U e , then Ui can be obtained from Eq. (3.2) as

22 2  F0 Uj1()  2 2 2 2 (3.3) () 1   2   m 1

24

2 2 F0 Uj2 ()  2222 (3.4) ()121 m

Divide Eq. (3.3) by (3.4) and we have

2 22 Uj12()    2 1  (3.5) Uj222() 

2 From Eq. (3.5) we notice that when (0,2 ) , /1,1/022   and UU12/

is positive, indicating an in-phase movement for m1 and m2 which is called an acoustic

2 mode. On the other hand, when  ( ,  )2 , /1,1/022   and is

negative, indicating an out-of-phase movement for m1 and which is called an optical mode. If we assume mass and spring k are invisible from observers, then mass , and spring k can be treated as a 1-DOF system. From Newton second law, the effective mass m for the whole system is:

F  2 1 (3.6) mjm()1  1 22 u12

From this equation we notice that the effective mass is a function of excitation frequency

 . If  2 , mj() is infinite large and thus the acceleration of m1 is close to zero. This

is because the inertia force mu22 has balanced out the excitation force and thus all the

external energy is absorbed by m2 . If

 , 22  (3.7)  2 1 2 

25 the effective mass mj() becomes negative, indicating that the optical mode exists when the excitation frequency is within a certain frequency band. The negative value of mj()

is due to the larger negative momentum of attached mass m2 than the positive momentum

of the mass m1 . It should be noted that the effective mass of EMMs is a dynamic parameter and has no meaning when dealing with a static mass. It should also be noted that frequency band which leads to an optical mode for mass-on-mass system is right to the locally

resonant frequency 2 on the frequency axis.

푢2

푚2

푗휔푡 퐹 = 퐹0푒 푘

푢1

푚1

Figure 3-1 A 2-DOF mass-on-mass system.

3.3 Negative Effective Stiffness of Mass-On-Spring System

Similar to the mass-on-mass system which exhibits negative effective mass, the negative stiffness can be derived from a mass-on-spring system. A 2-DOF mass-on-spring system is shown in Figure 3-2. A thin plate with negligible mass is connected to the ground

by two springs with constant k1 /2. Mass m2 is lumped under the thin plate by a spring

with constant k2 . A harmonic excitation with magnitude F0 and frequency  is applied

26 to the thin plate. u1 and u2 represent the displacements of the thin plate and the mass m2 .

The equations of motion for the mass-on-spring system are

00   ukkku  F 11221 jt (3.8)   , FF e 0 0 mukku22222   0 where j 1 is the imaginary unit. Take the Laplace transform for two equations and we have

2 m222 skk 2222 us1() k kk m sk m sk kk m sk m s Fs()   1 212221 21222 (3.9) us2 () kkk212  0 2222 k1 kk 212221 m sk 21222 m sk kk m sk m s

jt Assuming utUeii() , where Ui are the frequency response functions for the mass-on- spring system. Then can be obtained from Eq. (3.9) as

22 2  UjF10()  22 (3.10) ()kkk1212 

2 2 UjF20()  22 (3.11) ()kkk1212 

where 222 km/ is the locally resonant frequency of m2 . If mk22 and are invisible to observers, the effective stiffness for the whole system can be derived from Newton second law as:

F0 k2 k() j   k1  2 (3.12) u121 ( / )

27

From Eq. (3.12) we notice that when the excitation frequency  is equal to the natural

frequency 2 of mass m2 , kj() goes to infinite large and thus u1 goes to zero. In this

case, the excitation force is balanced out by the inertia force mu22 through the spring k2 and all the vibrational energy is absorbed by mass . Similar to the mass-on-mass system mentioned in Section 3.2, the thin plate and mass move in phase (acoustic mode) when

(0,2 ) and move out of phase (optical mode) when (,)2  . Also, the negative

stiffness occurs and the thin plate and m2 vibrate in an optical mode when

k  1 , (3.13) 22 kk12

It should be noted that this frequency band is left to the locally resonant frequency 2 on the axis, which is different from the frequency band of the mass-on-mass system shown in

Section 3.2.

푖휔푡 퐹 = 퐹0푒

푢1

푘 푘2 푢2 푘 1 1 2 2

푚2

Figure 3-2 A 2-DOF mass-on-spring system.

28

3.4 Stopband and Dispersion of Acoustic Metamaterials

The above dynamics-dependent negative effective mass and stiffness can be used to design EMMs. If effective stiffness and mass are simultaneously negative, phase velocity of a propagating elastic wave can be opposite to its group velocity and the refractive index can be negative. When the excitation frequency is closed to the resonant frequency of the attached mass-spring subsystem, the resonance of the subsystem generates resistive force against the external excitation and will in turn dissipate the vibration in the base structure.

The frequency band which closes to subsystem’s resonance frequency is called the stopband. When an excitation frequency falls within the designed stopband, elastic waves cannot propagate through the base structure efficiently. Figure 3-1 and Eq. (3.7) show that the subsystem can work efficiently in an optical mode against an external force applying

on a mass with  2 . On the other hand, Figure 3-2 and Eq. (3.13) show that the subsystem can work efficiently in an acoustic mode against an external force applying on

a spring with  2 . For a thin-walled 1-D or 2-D structure, the translational vibration absorbers have both mass and spring effects. The combination of masses and springs which exhibits elastic wave absorption is named the vibration absorbers, as shown next.

3.5 Single-Frequency Vibration Absorber

Base on the resonance of attached mass-spring subsystems that generates resistive force against the external excitation and in turn suppresses the vibration in the base structure, a simplified 2-DOF EMM unit with one single-frequency vibration absorber is

shown in Figure 3-3. The base structure with a mass m1 is connected to the ground by a

29 spring with constant k1 and a damper with damping c1 . Meanwhile, the base structure is

also connected to a mass m2 by a spring with constant k2 and a damper with damping c2 .

The subsystem of m222 k, c a n d is called the vibration absorber. A sinusoidal excitation is

applied to the base structure m1 and displacements for mm12 and are represented by

uu12 a n d . The working principle of the vibration absorber is to make u1 equal to zero by

setting the locally resonant frequency 2 of the vibration absorber equal to the excitation frequency . Based on Newton second law, the equation of motion for this EMM unit can be written as:

M u  C u  K u   F (3.14) where

TT uuuFF  12,,',0    

m1 0 Mass matrixM    0 m2 (3.15) ccc122 Damping matrixC   cc22

kkk122 Stiffness matrixK    kk22

Frequency response analysis is a good method to study the response of a system in frequency domain. Assume the base mass is subjected to a harmonic excitation of

jt F F0 e , where  is the excitation frequency and j 1 is the imaginary unit. Take

Laplace transform of Eq. (3.14) and gives

1 U()() s  M s2  C s  K  F s  (3.16)

30 where

1 HMsCsK   2     (3.17)

Therefore, the frequency response function matrix for the EMM unit is

1 HjMCjK()   2     (3.18)

where Hj() contains HH1121and as the frequency response functions (FRFs) for

mm12 a n d , respectively. Us() is the steady-state vibration amplitude vector of the two

masses. In numerical analysis, a mass of m1 =10 g, a natural frequency of 1 =10 Hz and a

damping ratio 1 of 0.01 are given to the base structure. Figure 3-4 (a) shows the influence

of vibration absorber’s mass to the system. Black line shows the absolute FRF of m1 when

there is no vibration absorber attached (e.g., m2  0 ). Due to the resonance of m1 , a peak response exists when the excitation frequency is close to the natural frequency of .

When a vibration absorber with a mass of m2 , a natural frequency of 2 =10 Hz and a

damping ratio of  2 =0.01 is attached to the base structure, two peak responses at each side of  =10 Hz can be observed as the whole system is becoming a 2-DOF system. When the excitation frequency is close or equal to the natural frequency of vibration absorber (

=10 Hz), all the vibrational energy is attenuated by the attached vibration absorber, leading to a motionless state of the base structure. This result is consistent with ones shown in

Section 3.2 and 3.3. The frequency band where no FRF of exists is called the stopband.

Notice that increasing will broaden the stopband. Figure 3-4 (b) shows the influence of

vibration absorber’s damping ratio  2 to the stopband. The overall response of the base

31 structure decreases as the damping ratio  2 of the vibration absorber increases. Meanwhile, an increasing damping ratio can also broaden the stopband to a small degree. However, large damping will slow down vibration absorber’s response to an excitation and increase the transient time. Therefore, it’s a double-edge sword to design an EMM structure with a wider stopband by increasing damping of vibration absorbers.

푢2

푚2 푢1 푘2 푐2

푚1

푘1 푐1

푖휔푡 퐹 = 퐹0푒

Figure 3-3 A Simplified 2-DOF EMM unit.

32

(a)

m2  0

mm21 0.1 mm21 0.2 mm21 0.3

mm21 0.4

(b)

 2  0.01

 2  0.03

 2  0.05

 2  0.07

 2  0.1

Figure 3-4 Absolute frequency response of m1 with (a) mm2/ 1 0,0.1,0.2,0.3,0.4, 1  2  0.01, (b) mm21/  0.1, 2  0.01,0.03,0.05,0.07,0.1.

33

3.6 Multi-Frequency Vibration Absorber

For a single-frequency vibration absorber, an increasing damping ratio can only broaden the stopband to a small degree. However, a large damping will in turn slow down vibration absorber’s response to an excitation and increase transient time. Multi-frequency vibration absorbers come to mind due to the shortcomings of single-frequency vibration absorbers. For a multi-frequency vibration absorber with two lumped masses and hence two locally resonant frequencies, two stopbands should exist around each resonant frequency [16, 25, 52]. Moreover, if the two locally resonant frequencies are close to each other, two stopbands can be combined into a wider stopband by choosing proper damping.

Next we present a 3-DOF EMM unit to show the mechanism of multi-frequency vibration absorber to design wide and multi-stopband EMM structures.

Figure 3-5 shows a base structure with a vibration absorber consisting of two lumped

masses. The base structure with a mass m1 is connected to a mass m2 by a spring with

constant k2 while is connected to another mass m3 by a spring with constant k3 . The

base structure is also connected to the ground by a spring with constant k1 . Each damper

has a damping of ccc123, and , respectively. The subsystem of mkc222, and is called the

primary vibration absorber while the subsystem of mkc333, and is called the secondary vibration absorber. A sinusoidal excitation is applied to the base structure and

displacements for mass m1, m 2 and m 3 are represented by u1, u 2 and u 3 . The working

principle of the multi-frequency vibration absorber is to make u1 equal to zero by setting

one of the two locally resonant frequencies i equal to the excitation frequency  . Based on Newton second law, the equation of motion of the 3-D EMM unit can be written as:

34

MuCuKuF          (3.19) where

TT uu  uuFF 123,,,',0,0    

m1 00 Mass matrix00Mm    2 00m3 ccc 0 122 (3.20) Damping matrix Ccccc   2233 0 cc33

kkk122 0 Stiffness matrix Kkkkk   2233 0 kk33

jt Assume the base mass is subjected to a harmonic excitation FFe 0 . Take Laplace transform of Eq. (3.19) and gives

1 U( sMsC )(  )  sKF 2 s       (3.21) where

1 HMsCsK   2     (3.22)

Therefore, the frequency response function matrix is

1 H() j  M 2  C j   K  (3.23)

where Hj() contains HHH11, 21 and 31 as the frequency response functions (FRFs) for

m1, m 2 and m 3 , respectively. Natural frequencies of the vibration absorber are defined as

35

222333kmkm/,/ . By adjusting each natural frequency equal to the excitation frequency  and by assuming no damping in the system, we can derive the two zero-

response locally resonant frequencies i from Eqs. (3.20) and (3.22):

2 ss11 ssss 2211 sss221 2 s ()4 mk33 23312,,,  ss (3.24) 2smk122

where s1 and s2 are the mass ratio and spring constant ratio between secondary vibration

absorber and primary vibration absorber. It should be noted that 23 a n d will increase

when s1 increases and s2 decreases.

푢3

푚3

푐3 푘3 푢2

푚2

푘2 푐2 푢1

푚1

푘1 푐1 푖휔푡 퐹 = 퐹0푒

Figure 3-5 A Simplified 3-DOF EMM unit.

Since the model in Figure 3-5 is a 3-DOF system, three response peaks are expected

in each FRF. Let mmm12110 g,0.2 , 2312 10 Hz,0.1ss . Figure 3-6 shows

the absolute FRF of the base plate ( H11 ). When there is no vibration absorber, will

36 reach its peak value when the excitation frequency is equal to its natural frequency (10 Hz).

After the vibration absorber is added to the system, two more response peaks will occur on each side of the original peak and the vibrational energy of base mass will be attenuated by the vibration absorber when the excitation frequency is equal or close to their resonant frequency (10 Hz). These two frequency bands between three peaks are the stopbands for

the 3-DOF system. If 23 a n d are separate far from each other, two stopbands will not

be easily to combine into one stopband and thus a narrow bandwidth between 23 a n d

is desired. Previous research shows that a same value of 23 a n d is desired in order to have a wide stopband [25].

1230.01,0

1  2   3  0.01

Figure 3-6 Absolute frequency response of m1 with mm23123 0,0.01,0 (black) and mm21/ 0.2, mm3/ 2 0.1, 1   2   3  0.01(red).

37

For numerical simulation, the physical properties of the 3-D EMM unit are chosen to be

mmms123123221210 mks g,2 kssg,,,

111222333kmkmkm/10 Hz,/10 Hz,/10 Hz

 cmiii/ (2)

Figure 3-7 shows the absolute FRF of the base mass with different s1 . Notice that a larger

m3 can broaden the stopbands and reduce the two peak responses at each resonant

frequency. However, the middle peak around i will increase as s1 increases. Therefore, a

large ss12(o r ) is not recommended. Figure 3-8 shows the absolute FRF of the base mass

with ss120.1, 1  0.01and different 23, . In Figure 3-9, a larger  3 will decrease

the middle peak (blue) while a larger  2 won’t (red). To fully investigate the influence of

23 and , different along with s1120.1,0.01 and different  2 along with

s1130.1,0.01 are chosen separately, as shown in Figures 3-9 and 3-10. Figure 3-9 shows that a larger can dramatically decrease the amplitude of base mass at each resonant frequency of the vibration absorber. While in Figure 3-10, an increment of can

reduce the amplitudes of H11 under vibration absorber’s each resonant frequency but not the base mass’s natural frequency. This is because is used to activate the vibration absorber in this 3-D EMM unit. The inertia force that introduced by will work against the excitation force. Larger requires longer time for the system to reach its steady state from transient state. In order to quickly respond to a transient excitation, a small value for

is favorable. On the other hand, a large value of is desired to reduce the amplitude

38 of middle peak response and to quickly damp out the vibration after the excitation is

stopped. Hence, one can use a small  2 and a large  3 to quickly activate the vibration absorber, decrease the vibrational amplitude of the base mass, quickly dissipate the transient vibration and combine two stopbands into a wider stopband.

39

s1  0.05

s1  0.1

s1  0.2

Figure 3-7 Average frequency response of m1 with mm21123/0.2,  and (a) mms321/0.05 (black), (b) s1  0 .1(blue), (c) s1  0.2 (red).

230.01

 2  0.01

 3  0.1

 3  0.01

 2  0.1

Figure 3-8 Average frequency response of with and (a) m1 mms211/0.2,0.1 123 0.01

(black), (b) 1  2 0.01,  3  0.1(blue), (c) 1  3 0.01,  2  0.1(red).

40

 3  0.00 5

 3  0.05

 3  0.2

Figure 3-9 Average frequency response of m1 with mms211/0.2,0.1 and (a) 3  0.005 (black),

(b) 3  0.05 (blue), (c) 3  0 . 2 (red).

 2  0.001

 2  0.01  2  0.1

Figure 3-10 Average frequency response of with m2/ m 1 0.2, s 1  0.1, 1  3  0.01 and (a)

 2  0.001 (black), (b) 2  0.01 (blue), (c) 2  0.1 (red).

41

3.7 Acoustic Metamaterials Application: Seismic Waveguide

Seismic waves are a type of inhomogeneous acoustic waves with various wavelengths. There are two types of seismic waves: body wave and surface wave.

Longitudinal wave (also known as “P-wave”) and transverse wave (also known as “S- wave”) are body wave while Rayleigh wave (also known as “R-wave”) and Love wave

(also known as “L-wave”) are surface wave. Figure 3-11 shows the propagation mechanisms of different seismic waves. P-waves’ displacement of the medium is parallel to the propagation direction while S-waves consist of oscillations occurring perpendicular to the propagation direction. Instead of being able to transmitted through solids, liquids and gases for P-wave, S-wave can only propagate in solids because molecules of gases and liquids offer little resistance to the transverse slide so that there is no elastic tie. R-waves are one type of surface wave that only travel on the surface of solids with boundaries. The corresponding oscillations of particles are elliptical so that velocities of R-waves are smaller than those of longitudinal and S-waves. L-waves are horizontally polarized surface wave which is a result of the interference of many transverse waves. The propagation velocities of surface wave are smaller than the ones of body wave and their amplitudes decrease exponentially with the depth. Because R-waves can only propagate in a homogeneous medium with boundary conditions, transverse motions that are detected during earthquakes are mainly from R-waves. Meanwhile, the horizontal shifting during earthquakes are mainly due to the polarized shear wave from L-waves. In conclusion, the

“shaking” people feel during an earthquake is mainly due to longitudinal and transverse motion from L-waves.

42

Figure 3-11 Seismic wave: (a) longitudinal wave, (b) transverse wave, (c) Love wave, and (d) Rayleigh wave (adapt from http://scweb.cwb.gov.tw/images/FAQ/eq005.jpg).

Kim [27] developed a device using EMMs with a device of an attenuator that can resist earthquake and support conventional aseismic designs. Instead of adding another seismic system to the building, he constructed an earthquake proof barrier consisting of many resonators around the building to be protected and used the local resonance of attenuators to absorb seismic wave energy. The modulus of the attenuator is negative when the frequency of a seismic wave is within the designed stopbands and then the seismic wave becomes an evanescent wave whose amplitude reduces exponentially. A wider stopband of the seismic attenuator could be designed by using resonators with different resonant frequencies. Hence, EMMs are of great value in seismic waveguide.

43

Chapter 4. CONVENTIONAL ACOUSTIC METAMATERIAL

PLATES

4.1 Prototype and Finite Element Analysis of Single-Stopband Acoustic

Metamaterial Plates

Because of the wide use of plates in engineering structures, EMM plates have more applications than EMM bars or beams. A model of a single-stopband conventional acoustic

metamaterial plate (SCMP) unit cell is shown in Figure 4-1 by replacing the base mass m1 in Figure 3-3 with an isotropic plate. An SCMP is assembled with repeated unit cells in microscale and is shown in Figure 4-2. Vibration absorbers with mass-spring subsystems can be designed to exhibit desired locally resonant frequencies and generate a significant out-of-phase inertial force which can counteract the base plate’s internal force and thus dissipate its vibration. By adding proper damping to the vibration absorbers, one can attenuate and absorb the guided wave with frequencies within the stopband during their propagation.

푤0 푄1

푀 푀1 6 푄2 푧 푄2 푀 6 푤 푤1 1 푦 푄1 푀6 푀2 푀2

푀 푀1 6 푘2 푐2

푤2 푥 푚2

Figure 4-1 Unit cell of an SCMP.

44

Figure 4-2 The prototype of an SCMP including one isotropic plate and repeat patterned mass-spring subsystems.

For an SCMP unit cell shown in Figure 4-1, a mass m2 is suspended from an

isotropic plate m1 by a spring with a constant k2 and a damper with a damping c2 . m2 , k2 and form a single-frequency vibration absorber. The plate dimension is 22abh and the Cartesian coordinate system is defined with its origin at the center of the base plate.

The base plate is subjected to an uncertain force Fxyt(,,) . It should be noted that instead of being regarded as a rigid body, the base plate will have various deflections. First

we define the vertical displacement of the base plate as wxyt1(,,) . The in-plane

displacements for m1 is

w u 1 z   w z, u   w z , u  w (4.1) 1x 1xy 2 1 3 1

Base on the strain-displacement relation, we have

45

11u 1x   w 1 xx z

 22u 2y   w 1 yy z  u 0 33 3z (4.2) 12u 1y  u 2 x  2 w 1 xy z

13uu 1zx  3  0

 23uu 2zy  3  0

Assume  33  0 , we have

1122 11  EE (4.3)   1122 22 EE

Thus

E   1111221 2 E   (4.4) 2211221 2 E G 121212 2(1)

where 1122 and are normal stresses along the x and y directions, 12 is the in-plane shear stress, EG,, are material’s Young’s modulus, Poisson’s ratio and shear modulus.

11,  22 and  12 can be rewritten in the matrix form as

11  10     11    11  w 1xx  E       22  1 0   22  ,   22   zw  1yy  (4.5) 1 2          12  0 0 (1  ) / 2    12    12   2w 1xy

Next we define the momentum resultants MMM126, , and the stress resultants

NNN1, 2 , 6 for the plate as

46

hh/2/2 EE22 MzdzwzwzdzD1111111  ww 22xxyyxxyy () hh/2/2 11

hh/2/2 EE 22 MzdzwzwzdzDww2221111  22xxyyxxyy () hh/2/2 11

hh/2/2 E 2 MzdzwzdzD6121 xy (1 )w1xy hh/2/2 1 hh/2/2 EE (4.6) Ndzwzwzdz11111 22xxyy 0 hh/2/2 11 hh/2/2 EE Ndzwzwzdz22211 22xxyy 0 hh/2/2 11 hh/2/2 E Ndzwzdz6121  xy 0 hh/2/2 1 Eh3 D  12(1) 2 where D is material’s flexural rigidity. Because the summation of forces and momentums along x, y and z direction should always be zero, we can derive the following relations:

  Fx  0  N1616xyxy dxdy N dydxNN00  F  0  N dydx N dxdyNN00   y  2626yxyx       Fz  0 Q1212xyttxytt dxdy Q dydx  FdxdydxdywQQFw  (4.7)  M x  0 M622xy dxdy M dydx Q dxdy 0  QMM262xy M  0 M dydx M dxdy Q dydx 0  QMM  y 611yx  161 yx  M z  0  00  00 where

Q1 M 6y  M 1 x   D w 1 xxx  w 1 yyx 1-  w 1 xyy   D w 1 xxx  w 1 yyx  (4.8) Q2 M 6x  M 2 y   D1  w 1 xyx  w 1 xxy  w 1 yyy   D w 1 yyy  w 1 xxy  are transverse shear force intensities on yz and xz planes.

Extended Hamilton principle is used for this 2-D finite element model, which is

47

t TWdt  0 (4.9) 0 nc

where T , andWnc represent the variation of kinetic energy, elastic energy and non- conservative work per unit area on xy plane. The variation of kinetic energy of the plate per unit area is equal to the virtual work done by the inertia force, which is

ab Thwwdydx  (4.10) ab 11 where  is material’s the mass density. The variation of elastic energy of the plate per unit area is

a b h/2           dzdydx a   b   h/2 11 11 22 22 12 12 a b h/2  z  w   z  w  2  z  w dzdydx a   b   h/2 11 1xx 22 1 yy 12 1 xy  ab  M w  M  w  2 M  w dydx ab 1 1xx 2 1 yy 6 1 xy  ab  M w  M  w   M  w  M  w dydx ab  1 1xx 6 1 xy  2 1 yy 6 1 xy  a b b  M w  M  w dydx   M  w  M  wxa dy a   b 1x 1 x 6 x 1 y   b  1 1 x 6 1 y x a a b a  M w  M  w dydx   M  w  M  wyb dx a   b 2y 1 y 6 y 1 x   a  2 1 y 6 1 x y b a b a b  M w  M  w dydx   M  w  M  w dydx a   b 1x 1 x 6 x 1 y   a   b  2 y 1 y 6 y 1 x  ba  M w  M  wx a dy   M  w  M  w y b dx ba 1 1x 6 1 y x a 2 1 y 6 1 x y  b a b b xa  M1xx w 1  M 6 yx  w 1 dydx  M 1 x  w 1  M 6 y  w 1 dy a   b    b   xa a b a yb  M2yy w 1  M 6 xy  w 1 dydx  M 2 y  w 1  M 6 x  w 1 dx a   b    a   yb ba Mw M wx a dy   M  w  M  w y b dx ba 11x 6 1y x a 2 1 y 6 1 x y  b a b b  M  M  2 M w dydx  M  w  M  w  M  w  M  wxa dy a   b 12xxyyxy 61   b  11611161 x y x yxa  a M w  M  w  M  w  M  wyb dx a  2y 1 6 x 1 2 1 y 6 1 x y b

(4.11)

48

Consider the discontinuity at x=0 and y=0, the variation of elastic energy per unit area can be further rewritten as

ab   MMMw dydx 2 ab 1261xxyyxy  b xx0 a (4.12)  MwQwMwMwQwMwdy111161111161xyxaxy      b x0 a yy0 b  MwQwMwMwQwMwdx212161212161yxybyx      a y0

The variation of non-conservative virtual work per unit area done by the external force is

b xa WMwQwMwdy ncxy b  111161  xa (4.13) a yb ()MwQwMwdxkwww2121612200yx a   yb

where w0 is the initial vertical displacement of the plate’s center where m2 is attached and

w2 is the vertical displacement of mass m2 . Substitute Eqs. (4.10), (4.12) and (4.13) into

Eq. (4.9) and yields:

ab hw  M  M  2 M w dydx ab 1 1xx 26 yyxy 1  b xx0 a -M11 wxy  Q x 11  axy w M 6111  wM  w 11Q w M 61 wdy   b x0  t a yy0 b - M21 w yx Q y 21 byx w M 6121 wM  w 21Q w M 61 wdx   dt  0 0 a y0  b x ay b a  M1 w 1xy  Q 1 w 1 M 6 w 1 dyM w  Q2 1 wyx M 2 1w 6 dx 1 b   xay b a    k() w w w 2 2 0 0

(4.14)

It should be noted that discontinuity only occurs in internal transverse shear forces. Eq.

(4.14) can further yield to

49

tab hwMMMQkwwx yw2()( dydx ,)0 dt (4.15) 0  ab 11262201 xxyyxy   where

xxyy1111/2/2/2/2 QQQQQ 11222112   , ,0 (4.16) counts for the discontinuity of the internal transverse shear forces at xy0 , 0 ,

0, xy  (,)xy  is a 2-D Dirac delta function. Therefore, the governing equation for the 1, xy base plate is

 (4.17) hwMMMQk1126220 xxyyxy wwx2()( y , )0

Integrating Eq. (4.17) and yields:

a ba ba b 2( , )() (hw , ) MMM dydxQ x y dydxk w wx y dydx   a   ba   ba   b11262xxyyxy 20   a ba b  hw dydxD wwD wwDwdydx k w w 2 (1 )() a   ba   b 1111112 20  xxxxyyxxxxyyyyyyxyxy   ab ab  hw dydxD wwD wwDwdydx k w w 2 (1 )() ab 1 ab 111112xxxxyyxxxxyyyyyyxxyy 20    a ba b  hw dydx Dwwwdydx k w w 2() a   ba   b 11112 20  xxxxyyyyxxyy  a bb a  hw dydx Dwwdy Dww xa yb dx k() w 0 w a   bb 11111  xxxxyy x ayyyx  a  xy y b 2 20

(4.18)

It should be noted that Q has been canceled during the integration. Eq. (4.18) shows that the base plate can be treated as a rigid body moving with an acceleration and subjected to

transverse shear forces at four edges and a concentrated force from m2 .

The governing equations for m2 can be obtained by using Newton’s second law as

m2 w 2 k 2 w 2  k 2 w 0  0 (4.19)

50

If an infinite SCMP with repeated unit cells shown in Figure 4-1 is subjected to a single-

frequency harmonic wave, the displacement of the plate w1 x( y , ,t ) and vibration absorber

wt2 () can be represented as

w Ae jxyt() 1 (4.20)  jt w2 Be where AB and are the vibration amplitudes, j 1 is the imaginary unit,  a nd are the wavenumbers along x and y direction and  is the wave frequency. Substituting Eq.

(4.20) into Eqs. (4.18) and (4.19) and rewriting the results in matrix form:

sin()sin()ab222 2 44hDkk    22A   0 (4.21)  2 B kmk222  

Eq. (4.21) is an eigenvalue problem. By setting the determinant of the left matrix in Eq.

(4.21) equal to zero, the relation between wavenumber ,  and wave frequency can be obtained as:

sin()sin()ab22222 2 440hDkmkk   2222  (4.22)  

Eq. (4.22) can be easily solved if the frequency is assumed to be positive real. The parameters and physical properties of the unit cell are chosen to be:

Dimension : 2a 0.2 m,2 b  0.2 m, h  10 mm Young's moduls :E  69 GPa,Possion's ratio  0.33,Mass density  2800 kg/m3

Attached masses :m22 0.1 kg,natural frequency  500 Hz

51

Figure 4-3 shows the dispersion surfaces of an infinite SCMP in its irreducible Brillouin zone where 020 and 020 . Two dispersion surfaces are obtained due to two real positive  . Notice that one stopband exists between two adjacent dispersion surfaces.

In order to clearly show the stopband, the frequency value along the four edges of each dispersion surface following the counterclockwise route  is traced and plotted in Figure 4-4. Figure 4-4 shows that, for an elastic wave having a specific wavelength  , it can propagate within two different frequency ranges. The mode in low frequency range is called the acoustic mode while the mode in high frequency range is

called the optical mode. As mentioned in Chapter 3, m1 and m2 move in phase for an acoustic mode and move 180 out of phase for an optical mode. Figure 4-4 also indicates that wave with frequencies within the stopband of 492.8 Hz to 521.8 Hz cannot propagate through the base plate, which is consistent with the results in Section 3.5. The stopband’s upper bound is obtained from the upper dispersion surface with 0,0 while the stopband’s lower bound is obtained from the lower dispersion surface with , .

It should be noted that the stopband for an infinite SCMP exists slightly above the resonant frequency of the attached vibration absorbers.

52



 



Figure 4-3 Dispersion surfaces of an infinite SCMP in its irreducible Brillouin zone, where , are wave numbers along x and y direction,  is the wave frequency.

Figure 4-4 Dispersion curves of an infinite SCMP in its irreducible Brillouin zone.

53

4.2 Prototype and Finite Element Analysis of Multi-Stopband Acoustic

Metamaterial Plates

Similar to SCMP, the model of a multi-stopband conventional acoustic metamaterial

plate (MCMP) unit can also be created by replacing the base mass m1 in Figure 3-5 with

an isotropic plate, as shown in Figure 4-5. An MCMP is assembled with repeated units in

microscale and shown in Figure 4-6. Primary vibration absorbers will transmit vibrational

energy from the base plate and secondary vibration absorbers will attenuate the vibrational

energy. A well-designed MCMP with two-frequency vibration absorbers can even combine

two stopbands into a wider stopband.

푤 0 푄1

푀 푀1 푧 6 푄2 푄2

푤1 푤1 푀6 푦 푄1 푀6 푀 2 푀2

푐 푀1 푀6 푘2 2 푤2 푥 푚2

푘3 푐3 푤3

푚3

Figure 4-5 Unit cell of an MCMP.

54

Figure 4-6 The prototype of an MCMP including one base plate and repeat patterned mass-spring subsystems.

For the MCMP unit shown in Figure 4-5, a mass m2 is suspended from the isotropic

plate m1 by a spring with a constant k2 and damper with a damping c2 while another mass

m3 is connected to the mass m2 by a spring with a constant k3 and damper with a damping

c3 . mkc222, and form a primary vibration absorber while mkc333, and form a secondary vibration absorber. The same dimension of 22a b h in section 4.1 is given to the base plate and the Cartesian coordinate system is defined with its origin at the center of the plate.

The governing equation for from an MCMP unit is same as the SCMP unit while the governing equations for and can be obtained from Newton’s second law as

55

mwkkwkwkw()0 222323320 (4.23) mwkwkw333233 0

If the infinite MCMP with repeated unit cells shown in Figure 4-6 is subjected to a

harmonic wave, the displacements of the plate w1 x( y , ,t ) , primary vibration absorber wt2 ()

and secondary vibration absorber wt3 ()can be represented as

jxyt() w1 A e  jt w2 B e (4.24)  jt w3 Ce where A, B a C nd are the vibration amplitudes,  a nd are the wavenumbers along x and y direction and  is the wave frequency. Substituting Eq. (4.24) into Eqs. (4.18) and (4.23) and rewriting the results in the matrix form:

sin(ab )sin( ) 222 2 440hDkk    22   A  2  kmkkkB22233     0 0 kmkC  2   333  

(4.25)

By setting the determinant of the matrix in Eq. (4.25) equal to zero, the relation between wavenumber  and and wave frequency can be obtained as:

sin(ab )sin( ) 22 24222 2 44h  Dkmm   km2 23 km 23 km 32 kk 33 23    2 2 2 k2 m 3  k 2 k 3 0

(4.26)

56

Eq. (4.26) can be easily solved if the frequency is assumed to be positive real. For numerical simulation, the parameters and physical properties of the unit cell are chosen to be:

Dimension : 20.2abh m,20.2 m,10 mm Young's moduls :69E GPa,Possion's ratio0.3 3,Mass density2800 kg/m 3

Attached masses :0.1mmm23223 kg,0.055 g,natural f requencies 500 Hz

Figure 4-7 shows the dispersion surfaces of the infinite MCMP in its irreducible Brillouin zone where 020 and 020 . Three dispersion surfaces are obtained due to three obtained real positive  . From Figure 4-7 we notice that two stopbands exist between each two adjacent dispersion surface. In order to clearly show these two stopbands, the frequency value along the four edges of each dispersion surface following the counterclockwise route          is traced and plotted in Figure 4-8. Figure 4-8 shows that, for an elastic wave having a specific wavelength  , it can propagate within three different frequency ranges. The mode in the low frequency range is called an acoustic mode while the mode in the high frequency range is called an optical mode. The middle frequency range is in a mix mode of acoustic and optical modes. As mentioned in Chapter

3, w1 and ww23, move in phase for an acoustic mode and 180 out of phase for an optical mode. Figure 4-8 also indicates that wave with frequencies within the stopbands of 443.57

Hz to 459.67 Hz and 555.49 Hz to 568.79 Hz cannot propagate through the base plate. It should be noted that the first stopband is slightly below the resonant frequency 500 Hz while the second stopband is slightly above the resonant frequency 500 Hz, which is consistent with the results in Section 3.5.

57



 



Figure 4-7 Dispersion surfaces of an infinite MCMP in its irreducible Brillouin zone, , are wave numbers along x and y direction,  is wave frequency.

Figure 4-8 Dispersion curves of an infinite MCMP in its irreducible Brillouin zone.

58

4.3 Design Guidelines for Vibration Absorbers

Several issues should be considered when designing an EMM structure. First, carried loading should always be assumed. In addition, as mentioned before, the vibrational energy should be dissipated quickly while the absorbers should respond quickly to the transient

excitation. Therefore, for multi-frequency vibration absorbers, a small damping ratio  2

for primary vibration absorber and a large damping ratio  3 for secondary vibration absorber are desired. Moreover, the dynamic characteristics for original structure such as natural frequencies and mode shapes should not be significantly changed due to the addition of vibration absorbers. Otherwise the governing equations may not be suitable anymore. Therefore, the total mass of vibration absorbers should be limited. In addition, vibration absorbers should not significantly reduce the load-carrying ability of the base structure. Other constraints should be considered for specific cases.

59

Chapter 5. LASER-CUT ACOUSTIC METAMATERIAL PLATES

5.1 Prototype of Single-Stopband Laser-Cut Acoustic Metamaterial Plates

As mentioned in Section 4.1, an SCMP unit cell can be generated by attaching a vibration absorber to an isotropic plate. Instead of attaching mass-spring-damper subsystems to the base structure, it’s advantageous to modify the base structure itself to achieve vibration suppression properties. Using modern laser cutting technology, additive manufacturing (e.g., 3-D printing and selective laser sintering) and/or nano-manufacturing techniques, one can easily generate the small EMM microstructures [18]. Based on this idea, a new prototype of EMM plate from laser cutting is generated [18].

Figure 5-1 shows a model of single-stopband laser-cut acoustic metamaterial plate

(SLCMP) and Figure 5-2 shows its unit cell design with geometric and system parameters.

For the unit cell of the SLCMP shown in Figure 5-2, the base plate represented by m1 , and

center mass, m2 , are connected by four beams representing the spring with constant k.

Damping in the vibration absorber can also be added to the four beams (springs) by selecting different constitutive materials with greater damping properties or by attaching dissipative layers. The working mechanism of this SLCMP unit is similar to the SCMP

unit. The resonant motion of the vibration absorber ( m2 , k and c) generates a resistive force against the external excitation which dissipates the vibration in the base plate. In Figure 5-

2, l is the periodic constant, a and b denote the unit cells laser cutting lengths and p is the width of each beam. The width of material channels removed by the laser cutting is assumed to be infinitely small during the calculation. The resonant frequency of the

60

SLCMP unit can be calculated from finite element modal analysis by setting periodic boundary conditions on the outer boundaries of the unit cell (top, bottom, left and right).

Figure 5-1 A single-stopband laser-cut acoustic metamaterial plate (SLCMP).

푘, 푐

푎

푚 2 푝 푏

푙 푚1

Figure 5-2 An SLCMP unit cell with geometric and system parameters.

61

As mentioned before, EMM structures should be designed in microscale having a dimension smaller than wavelength. Therefore, a larger wavelength and a small dimension of the SLCMP unit is desired for EMM simulation. According to the relation between wavelength and frequency (   vf/ ), EMM structures should only be effective in a low frequency range. Therefore, thickness (h) of the plate should be relatively small while the length of laser cut shorter side (b) should be relative large. This also meet the definition of a beam, that is, one side dimension is much larger than the other two sides dimension.

62

5.2 Prototype of Multi-Stopband Laser-Cut Acoustic Metamaterial Plates

Similar to the model of SLCMP, we can also design a model of multi-stopband laser- cut acoustic metamaterial plate (MLCMP) as shown in Figure 5-3.

Figure 5-3 A multi-stopband laser-cut acoustic metamaterial plate (MLCMP). 푘 , 푐 푘3, 푐3 2 2

푝 푏 푏2 3 푚3

푎3

푚2

푚1 푎2 푙

Figure 5-4 An MLCMP unit cell with geometric and system parameters.

63

Figure 5-3 shows an MLCMP with two-frequency vibration absorbers and Figure 5-

4 shows its unit cell design with geometric and system parameters. For the unit cell of the

MLCMP shown in Figure 5-4, the base plate represented by m1 and mass, m2 , are

connected by four outside beams representing the spring with constant k2 while mass, m2 ,

and center mass, m3 , are connected by four inside beams representing the spring with

constant k3 . Additional damping can also be added to the beams. The working mechanism of this MLCMP unit cell is similar to the MCMP unit cell: The primary vibration absorber

( m2 , k2 and c2 ) transmits the vibrational energy to the secondary vibration absorber ( m3 , k3

and c3 ) while the secondary vibration absorber can quickly dissipate the vibration in the

base plate. In Figure 5-4, l is the periodic constant and p is the width of each beam. ai and

bi denote the unit cells laser cutting lengths, where i  2 is for outside beams and i  3 for inside beams. The width of material channels removed by the laser cutting is also assumed to be infinitely small during the calculation. Because it’s the secondary vibration absorber

which plays an important role dissipating the vibrational energy, a relative large k3 is need

and thus a longer inside beam length ( b3 ) is desired. Two resonant frequencies of the

MLCMP unit can be calculated from finite element modal analysis by setting periodic boundary conditions on the outer boundaries of the unit cell (top, bottom, left and right).

As mentioned in Chapter 4, dispersion analysis is a good method to investigate wave propagation in an infinite EMM plate. For a finite EMM plate with boundary conditions and loading, frequency response analysis is need to study wave propagation in the steady state. Transient analysis should also be conducted when investigate an EMM’s wave absorption characteristic in the transient state. Hence, it is important to show each method

64 before conducting numerical simulations for each type of laser-cut acoustic metamaterial plate.

65

Chapter 6. FINITE ELEMENT ANALYSIS OF RECTANGULAR

PLATES

6.1 Finite Element Analysis of Conforming Rectangular Plate Elements

All of those analysis methods mentioned in Section 5.2 are based on finite element analysis. Hence it’s important to introduce the finite element method at the beginning.

Since we focus on plate structure in this dissertation, only the finite element analysis of conforming rectangular plate elements is shown.

Figure 6-1 shows a conforming rectangular plate element with counterclockwise

node number 1, 2, 3 and 4. Let wii (1,2,3 and 4) be the vertical displacements of each node. Instead of using Cartesian coordinate system, we introduce another coordinate system with  a n d axis. The relation between two coordinate systems is

xxx  /2   12 a /2 (6.1) yyy  /2   32 b /2



w444,, w333,, 4 3



1 2 w1,, 1 1 w222,,

Figure 6-1 A conforming rectangular plate element with counterclockwise nodes.

66

Each node of the conforming rectangular plate element in the new coordinate

system has 4 degree of freedoms (DOFs) w, ,  and  where   wy w=2 b / ,

  wx  2/ w a ,  4w / ab . The word “conforming” means all 4 DOFs are continuous along  1 edge, which is not possible for rectangular plate element who has only 3 DOFs ( w,,). Due to 16 DOFs for a single element, we assume the displacements of one element in the polynomial form such as

w( x ,, y )(,, twt )  22322 ccccccccc123456789  (6.2) 33223322333 ccccccc10111213141516     

where cii ( 1,2, ,16) are real numbers. It should be noted that higher order such as

44 and are not introduced in this polynomial in order to avoid extra terms and thus reduce the computation time. w,, and can be obtained based on the relation between two coordinate systems:

ab dxd dyd, 22 dwdwbb  ww  ddy b2 /22 y (6.3) dwdwaa  ww   ddx a2 /22 x d22 wd wab  w   d ddx ady2 b /2 /4 

It should be noted that w, ,  and  can also be expressed in the form of ci through Eq.

(6.2) as follows:

67

wtw( 1,  1, ) 1 1111111111111111 c1  wtb( 1,  1, )/ 2 0010120123123233 c  1 2

wta ( 1,  1,  )/ 2 1 0102103210221323 c3  wtab( 1,  1, )/ 4   1 0000100220343669  c4 wtw( 1,  1, ) 111  1111111111111  c 2 5 wtb (1, 1, )/ 22 0010120123123233  c6 wta(1, 1,  )/ 2  0102103210321323 c  2 7 wtab ( 1,  1, )/ 4  2 0000100220343669  c8   wtw(1,1, )  3 1111111111111111 c9 wtb(1,1, )/ 2 0010120123123233 c  3  10 wta (1,1, )/ 2 3 0102103210221323 c11 wtab(1,1, )/ 4  0000100220343669 c  3  12 wt( 1,1, )  w4 1111111111111111 c13 wtb( 1,1, )/ 2 0010120123123233 c  4  14 wta ( 1,1,  )/ 2 4 0102103 210321323 c15 wtab( 1,1, )/ 4    4 0000100220343669  c16

(6.4)

We can simplify the above matrix as:

dAc     (6.5) where d denotes 4 DOFs and  A is the multiplication matrix. Notice that Eq. (6.2) can also be written in the matrix multiplication form as

w(,,)1,,,,,,, tc   2   232  ,  233 ,,  , 22  ,  3322333 ,  , ,     1, , ,2 ,  , 232 ,  ,  , 233 ,  ,  , 22 ,  3322333 ,  ,  ,   Ad1  

 N1 , N 2 , , N 12 d T NT  d  d  N

(6.6) where N is the shape function of the plate element and wt(,,) is the plate’s displacement. The relation between dand d is

68

ddiagbaabbaabww  1,/ 2,/ 2,/ 4,,1,/ 2,/ 2,/ 4,,,,,,,  11112222  (6.7)  Sd 

Based on extended Hamilton principle, the variation of kinetic energy of the conforming

rectangular plate element per unit area is

Tw  wdxdy A e N T  dNNdJd(ii )(  )   d  T A     i1 i N e T  dSNNSdJd(ii )(  )  T  d  T   (6.8) A     i1 i N e (iii )( )(T )  dmd   i1  DMDT   

where

T T mSNNJddS()i          (6.9)  A i

is the element for mass matrix.

The variation of elastic energy per unit area is

          dxdydz V 11 11 22 22 12 12  T  dxdydz V  T Q  dxdydz V N e h/2 T (6.10)   d(ii ) STT B z 2 Q B S d ( ) dxdydz Ah /2    i1 i N e ()()()iT i i   d  k d  i1   DKD  

69 where

T Nxx wzxx  T  wzNdzBdzBSdzyyyy             2wz T xy 2 N  xy  (6.11) 10 E  Q   10 1 2  001/ 2   

4444 TT wwNdwwNdxxyy2222    ,     aabb (6.12) 4 T wwNdxy      ab

kSBBS()i  dxdy TT      A i (6.13) 11 SBBSTT  Jdd     11

()i Here k is the element for stiffness matrix.

The variation of the non-conservative virtual work per unit area done by the external force is

W f wdxdy nc A N e T  d()i  ST  N fJd  d  (6.14) A   i1 i e N T d()()ii  F  D F i1 where

T 11 F()i   S  N fJd d (6.15)   11

70 is the element for external force matrix.

Recall the extended Hamilton principle equation

t TWdt  0 (4.9) 0 nc

Substitute Eqs. (6.8), (6.10) and (6.14) into Eq. (4.9) and yields:

t2 T  DMDKDFdt        0 (6.16) t     1

Eq. (6.16) has the same general form as

MDKDF      (6.17) which is easy to solve based on previous theory.

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6.2 Modal Analysis of Rectangular Plates

Free vibration takes place when the system oscillates without external forces. The system under free vibration will vibrate at its natural frequencies, determined by system’s mass and stiffness distribution.

Forced vibration, another type of vibration, takes place when a system is under the excitation of external forces. The system is forced to vibrate at the excitation frequency when the excitation is oscillatory. When the excitation frequency is equal or near to one of the system’s natural frequencies, resonance will arise and result in large oscillations.

Therefore, natural frequencies are very important in the field of vibrations.

Modal analysis is a method to determine the natural acoustic characteristics of materials. The analysis involves imposing an excitation into the structure and finding the frequencies at which the structure resonates. A typical modal analysis will return multiple frequencies, each with a displacement field known as the “mode shape”.

For the modal analysis of a rectangular plate, recall the equation of motion:

MDCDKDF         (6.18) where

M   Mass matrix C  Viscous damping matrix K   Stiffness matrix F   Excitation's time history of the system D  Displacement's time history of the system D  Velocity's time history of the system D  Acceleration's time history of the system

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Because damping has a very small influence to structure’s natural frequencies as well as its mode shapes and because modal analysis is to solve vibration characters from a system with free vibration, both C and F can be assumed zero. Then Eq. (6.18) can be rewritten as

MDKD    0 (6.19)

For linear systems, both M  and K  in Eq. (6.19) are real symmetric matrices. We assume

Eq. (6.19) has harmonic solutions such as

De   jt (6.20) where  are the amplitudes of displacement vectors, j 1 is the imaginary unit and

 are excitation angular frequencies. Substitute Eq. (6.20) into Eq. (6.19) and gives

KM2    0 (6.21)

Eq. (6.21) is an eigenvalue problem. In order to let  have non-zero solution, we need to set

KM  0 (6.22)

2 where  . By solving Eq. (6.22), we can get a set of discrete roots i in1,2,,  .

Substitute i back into Eq. (6.21) and we can get corresponding vectors i  , which satisfy the following equation:

Kii M  0 i  1,2, , n (6.23)

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th th In Eq. (6.23), i refers to system’s i eigenvalue and i refers to its corresponding i

th eigenvector. Once i is obtained, the system’s i natural frequency can be derived through

  f i i (6.24) i 22

th Similarly, one can get system’s i mode shape by using i , which represents vibration amplitude of ith mode.

6.3 Dispersion Analysis: Bloch Wave Analysis of Rectangular Plates

Dispersion analysis is a method to investigate wave propagation in an infinite periodic structure. Dynamic characteristics in infinite periodic structures such as EMMs can be derived from an isolated unit cell (also called a representative volume element

(RVE)) with periodic boundary conditions. Based on the Bloch-Floquet theory [53], the electron wave functions in a crystal have a basis consisting entirely of Bloch wave energy

eigenstates. The reciprocal lattice can be defined as the wave vector, k  (,)kkxy, that results in plane waves under the spatial periodic boundary conditions of the lattice. The subset of the wave vector k, which contains all the information about the propagation of plane waves in the structure, is called the Brillouin zone [54]. Figure 6-2 (a) shows a RVE for an infinite structure and Figure 6-2 (b) shows its irreducible Brillouin zone. The elastic displacements of two points with the same periodicity can be written in the form:

jk•r uAB (r) e u (r) (6.25)

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where j 1 is the imaginary unit; r is the position vector between points A and B which

share a periodic relation; u (A r ) and u (B r ) are elastic displacement vectors of two points.

The equation of motion for the whole discrete system is

2 K k   k M k u r 0 (6.26) where Kk and Mk  are global stiffness and mass matrix for the discrete system. The stopbands are obtained by solving for all eigenfrequencies as a function of the wavenumber,

 k , for all k (sweeping through the irreducible Brillouin zone, varying from  / r to

 / r ). The relation between the eigenfrequencies and wave vectors is shown by the dispersion curves. The stopbands characteristics from dispersion analysis for an infinite

EMM plate should be in consist with those for a real-size EMM plate with same unit cells.

(a) (b)

ky  r

k   x

Figure 6-2 (a) A RVE of an infinite structure and (b) Irreducible Brillouin zone.

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6.4 Frequency Response Analysis of Rectangular Plates

Frequency response is the steady-state response of a system which is subjected to a sinusoidal input. In frequency response analysis the excitation is explicitly defined in the frequency domain. The loads which are known at each forcing frequency, can be either applied loads or enforced motions. In frequency response analysis, structure’s response is computed by solving an equation set consists of coupled matrix equations. The equation of motion for a damped system under harmonic excitation at discrete excitation frequencies can be written as

MDtCDtKDtFe            jt  (6.27) where MCKF,, and    are the global mass matrix, damping matrix, stiffness matrix and force vector, respectively. The applied loads are set in complex form and can be real or imaginary, or both. Based on the complex form of the load, we assume a harmonic solution to be:

DUe    jt (6.28) where U  is a complex displacement vector,  is the excitation frequency with the imaginary unit of j 1 . By taking the first and second derivatives of x,we have

DjUe     jt (6.29) DUe 2    jt

By substituting Eq. (6.29) into Eq. (6.27), we have

2 M  j  C  K U    F   (6.30)

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Therefore, the displacement of the plate is

1 UMjCKF  2          (6.31)

The equation of motion can be solved by any given excitation frequency . Notice that complex coefficients exist when a system is subject to damping or applied loads with phase angles.

6.5 Transient Analysis of Rectangular Plates

Because structures are often subjected to transient as well as periodic and static loads, it is also necessary to conduct transient analysis besides steady-state and static analysis during the design process. Time for the structure to reach its steady state should be identified. Because stresses under the transient state can be greater than those under the steady state, the maximum amplitude during the transient response should also be analyzed to tell whether the failure exists during the transient state. Next we will show the transient analysis of a rectangular plate.

Transient analysis requires implicit or explicit time-marching schemes for performing direct integration of the equation of motion. In an implicit method, the acceleration at the next time instant tt is used, and hence it is an interpolation method, which can be unconditionally stable by using appropriate parameters. In an explicit method, the acceleration at tt is not used, and hence it is an extrapolation method, which can lose stability after some steps. Implicit methods include the Newmark- method , Wilson-

77

 method , Hilbert-Hughes-Taylor- method , etc. Explicit methods include Runge-Kutta methods, etc.

In an implicit method, the time step is governed by the accuracy consideration alone, and large time steps can be used. However, an implicit scheme requires matrix factorizations and involves large computer core storage and many operations per time step, and the nonlinear equation of motion matrices are solved at each time step to advance the solution. Hence, the cost of an implicit time step is ten to a thousand times more expensive than the cost of an explicit time step. The Ne wm a r k- method can be unconditionally stable and is by far the most popular implicit scheme offered in finite element codes.

In an explicit method, relatively little computational effort is required during each time step because the solution is advanced without forming and inverting tangent matrices, which saves storage and computation and eases the coding. Therefore, explicit methods are often used to analyze large complex problems.

Here is a brief review of Newmark- Method: For a linear system, the mass matrix

M  , damping matrixC and stiffness matrixK  are always constant matrices and they have the following relationship:

MD tCD  tKD  tF         (6.32) where{}D , {}D and{}D are the displacement, velocity and acceleration at a specific time

step. For the next time step tk 1  k 1  t , Eq. (6.32) can be rewritten as:

MDCDKDF      (6.33)   kk11     kk11

In the method, we assume velocity and displacement vectors to be

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DDDDt 1  (6.34)  kkkk11     

1 DDDtDDt  2 (6.35)  kk1         kkk 2 1 where  and  are constants.

From Eq. (6.35) we can obtain

DD D  kkk1     1 DD  2 1   (6.36) kk1 tt2

Substituting Eq. (6.36) into Eq. (6.34) we have

  Dt DDDD11         (6.37) kkk1  2 t kk1

Substituting Eqs. (6.36) and (6.37) into Eq. (6.33) we have

KqFˆˆ   (6.38)  k1  k1 where

1  KKMCˆ        (6.39a)  tt2

1 2 1 1 Fˆ  F  M q  q  q    k1    k  k2   k k1 2 tt 

 Ct  qqq 11 (6.39b)    kkk     2 t

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q q q Therefore, we can obtain  k 1 from Eq. (6.38),  k 1 from Eq. (6.37) and  k 1 from Eq.

(6.36).

ABAQUS/STANDARD uses the Hilber-Hughes-Taylor (an extension of

Ne wm a r k- method) time integration by default. The default parameters for Hilber-

Hughes-Taylor time integration are 0.05,0.275625 for transient fidelity and

0.41421,0.5 for moderate dissipation.

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Chapter 7. NUMERICAL RESULTS

7.1 Isotropic Plate

Before conducting different analyses of laser-cut EMM plates to study their characteristics of wave propagation, it is important to first investigate elastic wave propagation in an isotropic plate. A finite isotropic square plate with length, L = 0.728 m, and thickness, h = 0.001 m, is modeled with ABAQUS/STANDARD. For element type selection, the reduced integration, linear quadrilateral 4-node doubly curved thin shell element with 4 DOF ( w, , ,   ) per node (S4R) is selected. T-6061 aluminum is used for the plate with physical properties of density   2 8 0 0 k g / m3 , Young’s modulus

E  69 GPa and Poisson’s ratio   0 . 3 3 . The whole plate is meshed with 9 1 9 1 shell elements with an approximate seed size of 0.008 m.

First, a hinged boundary condition on the left and right edges is considered for the isotropic plate which is computationally efficient for studying the wave propagation along x axis. A harmonic excitation force with an amplitude of 1000 N and a frequency varies from 0 Hz to 1000 Hz is applied at the second node from the left of the centerline, as shown in Figure 7-1. Figure 7-2 (a) shows the average frequency response (AFR) of the isotropic plate at x = 0.95 L. Notice that the amplitude of AFR is relatively stable between -60 dB and 0 dB as the excitation frequency increases. Boundary conditions of hinged left edge and hinged four edges are also considered with their corresponding AFR at x = 0.95 L shown in Figure 7-1 (b) and (c). It is obvious that the elastic wave can propagate through the isotropic plate under different boundary conditions.

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Figure 7-1 An isotropic plate with hinged left and right edges boundary condition (represented by two red lines). An excitation force (represented by the yellow point) is applied at the second node from the left of the centerline.

(a)

82

(b)

(c)

Figure 7-2 AFRs at x = 0.95 L of the isotropic plate under different boundary conditions: (a) hinged left and right edges, (b) hinged left edge and (c) hinged four edges.

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7.2 Single-Stopband Laser-Cut Acoustic Metamaterial Plate

The isotropic plate in Section 7.1 is cut into an SLCMP consisting of 88 unit cells, each with a dimension of l = 0.091 m, a = 0.025 m, b = 0.02 m, p = 0.001 m and h = 0.001 m. The width of laser-cut channels are assumed to be infinitely small during the calculation and thus can be simulated by seams. For an SLCMP unit cell, two pairs of periodic boundary conditions are applied to the left-right and top-bottom edges. Modal analysis of the SLCMP unit cell is performed using frequency step from linear perturbation procedure of ABAQUS/STANDARD and its first 12 mode shapes are shown in Figure 7.2. The resonant mode shape of the SLCMP unit cell is the one with large deflections for four beams, leading to an opposite movement phase between the vibration absorber and the base plate. The first mode is the twisting mode while the second and third modes are bending modes due to horizontal and vertical beams, respectively. Both the base plate and the center mass have a small deflection for these two modes. Notice that for mode 4, all four beams are fully bending while the deflections of the base plate and the center mass are very small.

The laser-cut vibration absorber displacement is at its positive maximum while the base plate displacement is at its negative maximum. The displacement difference between the vibration absorber and the base plate is at its maximum, indicating an opposite movement phase between them. Therefore, the resonant frequency for the SLCMP unit cell is the natural frequency for mode 4 (220.9 Hz). Other mode shapes are shown in Figure 7.2 (e) -

(l), with either a large deflection for the center mass or zero deflection for the cutting beams.

None of them is the resonant mode for the SLCMP unit cell.

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(a) (b)

(c) (d)

(e) (f)

85

(g) (h)

(i) (j)

(k) (l)

Figure 7-3 First 12 mode shapes for the SLCMP unit cell under periodic boundary conditions.

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In order to investigate the wave propagation in an infinite SLCMP, dispersion analysis is conducted by applying Bloch-Floquet boundary conditions to the left-right and top-bottom edges of the SLCMP unit cell using COMSOL Multiphysics and the dispersion curves follow the path of        in the irreducible Brillouin zone are shown in

Figure 7-3. Wave can propagate through the infinite SLCMP at those excitation frequencies where the wave vector k exists. A theoretical stopband of 220.8 Hz to 305.9 Hz is obtained

(shown in a grey band) where no wave can propagate through the infinite SLCMP. It should be noted that the stopband exists slightly above the resonant frequency of the SLCMP unit cell, which is consistent with the previous result in Section 4.1. By incorporating dissipative properties to the laser-cut beam regions (springs), the energy of elastic waves with frequencies within the stopband can be effectively attenuated. However, this dispersion analysis is only valid for the infinite SLCMPs. In order to examine wave propagation in a finite SLCMP under the steady state, frequency response analysis should be conducted.

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  

Figure 7-4 Dispersion curves of an infinite SLCMP with reduced wave vector k sweep following the path of        in the irreducible Brillouin zone.

For frequency response analysis, first a hinged boundary condition on the left and right edges (e.g. U1 = U2 = U3 = 0) is considered. A harmonic excitation force with an amplitude of 1000 N is applied at the second node from the left of the centerline. Rayleigh damping is applied to the beam regions with damping coefficients of   0 and different

 . The AFRs of the SLCMP and its corresponding isotropic plate (before cutting) at x =

0.98 L are shown in Figure 7-5. The black line represents the AFR of the isotropic plate at x = 0.98 L while the red line denotes the AFRs of the SLCMP at x = 0.98 L with no damping applied to the unit cells. The blue and magenta lines represent the AFRs of the SLCMP at x = 0.98 L with increasing damping coefficients of  55e and  25e 5 , respectively.

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As analyzed before, one stopband can be observed around SLCMP unit cells’ resonant frequency. In Figure 7-5, a stopband is obtained in the frequency range of 222.5 Hz to

321.6 Hz where the AFR suddenly decreases. It should be noted that this stopband is slightly different from the one obtained from the dispersion analysis (shown as the grey band in Figure 7-5). This is due to the loading and boundary conditions which were not considered in the dispersion analysis. When the excitation frequency falls within the stopband, the incident wave cannot propagate through the SLCMP even when the absorber’s damping coefficients are sufficiently small, indicating that the wave propagation is stopped by the inertial forces of the resonant vibration absorbers rather than the damping properties. However, excitation forces with frequencies outside the stopband are not significantly dissipated, and some response amplitudes are even larger than the amplitude before cutting. The frequency response varies continually when the excitation frequency approaches their natural frequency, leading to an intense resonant motion of the vibration absorbers. Fortunately, the continual variation can be smoothed by increasing the damping properties applied to the vibration absorbers. Comparing the red, blue and magenta lines in Figure 7-5 (a), one can also see that increasing the damping coefficients applied to the vibration absorbers can efficiently lower the response in the low frequency region and broaden the stopband width to a small degree simultaneously. Hence, damping is definitely required for such vibration absorbers. It should be noted however, that sufficiently large damping values will also slow down the vibration absorber’s response to an excitation. It’s not economic to broaden the stopband width simply by increasing the

89 damping of vibration absorbers because the use of high damping coefficients will in turn slow down the vibration absorber’s response to an excitation and increase the transient time.

(a)

(b)

90

(c)

(d)

Figure 7-5 (a) AFRs of the SLCMP and its corresponding isotropic plate at x = 0.98 L with hinged left and right edges under different damping coefficients. (b) – (d): Vertical displacement distributions with damping coefficients of 0,  5e  5 at different excitation frequencies: (b) f = 162 Hz , (c) f = 286 Hz and (d) f = 532 Hz.

91

Figure 7-5 (b) - (d) show the vertical displacement distributions of the SLCMP with hinged left and right edges under excitation frequencies of 162 Hz (below the stopband),

286 Hz (within the stopband) and 523 Hz (above the stopband) with damping coefficients of  0  , 5 5 e that are applied in all three cases. From the figure, it is apparent that waves can propagate through the plate when the excitation frequency is below or above the stopband, but cannot propagate for the frequencies within the stopband. When the excitation frequency falls within the stopband (e.g., Figure 7-5 (c)), the majority of the vibrational energy is absorbed by the resonant motion of the left-most vibration absorbers near the excitation location. Therefore, both the base plate and the rest of the vibration absorbers remain motionless.

In order to examine the influence of boundary conditions on attenuation performance, two identical SLCMP designs with different boundary conditions applied during analysis are considered. Figure 7-6 shows the AFRs at x = 0.98 L for a case with a hinged left edge boundary condition and its vertical displacement distributions at different excitation frequencies while Figure 7-7 shows the AFRs at x = 0.98 L for another case with hinged four edges and its vertical displacement distributions at different excitation frequencies.

The damping coefficients of are applied for both cases. In Figure 7-6 (a) and Figure 7-7 (a), the red lines represent the AFRs of SLCMPs at x = 0.98 L while the black lines represent the AFRs of the corresponding isotropic plates with the same boundary conditions at x = 0.98 L. Although the AFRs vary slightly under different boundary conditions, there is still a significant decrease under each boundary condition around the resonant frequency (220.9 Hz) of the SLCMP unit cell. Similar to the vertical displacement distributions in Figure 7-5 (b) - (d), Figure 7-6 (b) - (d) and Figure 7-7 (b) -

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(d) show that wave can hardly propagate through the SLCMP within the stopband under different boundary conditions. Therefore, the stopband characteristics of the SLCMP are not significantly affected by the boundary conditions.

(a)

(b)

93

(c)

(d)

Figure 7-6 (a) AFRs of a hinged-left-edge SLCMP and its corresponding isotropic plate at x = 0.98 L with damping coefficients of 0,  5e  5 . (b) - (d): Vertical displacement distributions at different excitation frequencies: (b) f = 162 Hz, (c) f = 286 Hz and (d) f = 532 Hz.

94

(a)

(b)

95

(c)

(d)

Figure 7-7 (a) AFRs of a hinged-four-edge SLCMP and its corresponding isotropic plate at x = 0.98 L with damping coefficients of 0,55 e . (b) - (d): Vertical displacement distributions at different excitation frequencies: (b) f = 162 Hz, (c) f = 286 Hz and (d) f = 532 Hz.

96

The above results obtained from the frequency response analysis have successfully validated the wave propagation characteristic of the proposed SLCMP design in the steady state. However, destructive wave interference during the transient period is still not clearly illustrated. It is crucial to study the structure’s dynamic response prior to reaching a steady state because transient stresses in the structure can be greater and result in fracture. The general procedure type of Dynamic, Implicit is used to perform transient analysis for the

SLCMP using ABAQUS/STRANDARD. For transient analysis of the SLCMP with hinged left and right edges, a total period of 0.3 s with an incremental size of 0.0002 s is used. The same Rayleigh damping coefficients of 0 and 55 e are used for each unit cell beam. The external sinusoidal excitation frequencies are chosen to be 162 Hz, 286 Hz and

523 Hz. Average displacements of nodes at x = 0.02 L (black), 0.5 L (red) and 0.98 L (blue) at the centerline of the SLCMP are recorded to show the wave propagation change due to vibration absorbers. Figure 7-8 shows the wave propagates toward the end of the SLCMP, passes the nodes at x = 0.02 L, 0.5 L and 0.98 L of the centerline under different excitation frequencies. In Figure 7-8 (a), the vibration amplitude of the SLCMP decays spatially as the wave propagates due to the damping coefficients applied to the vibration absorbers.

Because the excitation frequency (162 Hz) is lower than the resonant frequency of the

SLCMP unit cell, the structure exhibits an acoustic mode. However, since the excitation frequency is far from the resonant frequency, the incident waves freely propagate along with the base plate. Figure 7-8 (b) illustrates that when the excitation frequency falls within the stopband, the average displacement of the node at x = 0.5 L (red) is much smaller compared with one at x = 0.02 L (black) and the average displacement of the node at x =

0.98 L (blue) is even smaller. Around the excitation frequency, the unit cells respond with

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a mix mode, including both acoustic and optical modes, and induce the inertial forces

working against the vibration of the base plate and stop the propagation of waves through

the EMM structure. Figure 7-8 (c) shows that amplitudes of each of the three locations are

almost the same when the excitation frequency is 523 Hz. Therefore, the destructive wave

can easily propagate forward when the excitation frequency is far beyond the stopband.

This shows that the attenuation properties of the SLCMP are also discernable under a

transient response. Since the earthquakes are the result of sudden release of huge amount

of energy that lead to seismic wave, vibration suppression of SLCMP corresponding to

transient response is of great importance.

(a)

98

(b)

(c)

Figure 7-8 Average displacement of a hinged-left-and-right-edge SLCMP with damping coefficients of

ii0,  5e  5 at x = 0.02 L (black), 0.5 L (red) and 0.98 L (blue) of the centerline under different excitation frequencies in transient state: (a) f = 162 Hz, (b) f = 286 Hz and (c) f = 523 Hz.

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7.3 Multi-Stopband Laser-Cut Acoustic Metamaterial Plate

Previous section shows that for an SLCMP, one stopband exists slightly above unit cell’s resonant frequency and no elastic wave can propagate through the plate when the excitation frequency falls within the stopband. As mentioned before, a stopband with wider bandwidth can be achieved by using multi-frequency vibration absorbers with large damping coefficients for the secondary vibration absorber when two resonant frequencies of the unit cells are closed to each other. Next we consider a multi-stopband laser-cut acoustic metamaterial plate (MLCMP) with two resonant frequencies for its unit cells to show its characteristic of elastic wave absorption.

An isotropic square plate with a dimension of L = 1.048 m and h = 0.001 m is cut into an MLCMP consisting of 88 unit cells. The physical properties for the MLCMP are: density   2800 kg/m3 , Young’s modulus E  69 GPa and Poisson’s ratio  0 . 3 3 . It should be noted that the primary vibration absorber transmits the vibrational energy to the secondary vibration absorber while the secondary vibration absorber dissipates the vibrational energy of the base plate. Therefore, a larger length of secondary vibration absorber’s beams is desired. Each MLCMP unit cell has a dimension of l = 0.131 m, p =

0.001 m, h = 0.001 m, a2 = 0.045 m, a3 = 0.025 m, b2 = 0.015 m and b3 = 0.02 m. Laser cuts are also simulated by assigning seams to the isotropic plate. By applying two pairs of periodic boundary conditions to the left-right and top-bottom edges, first 12 mode shapes for the MLCMP unit cell with their natural frequencies are obtained from modal analysis using ABAQUS/STANDARD and are shown in Figure 7-9. The resonant mode shapes of the MLCMP unit cell are those with large deflection for either outside four beams of the primary vibration absorber or inside four beams of the secondary vibration absorber, which

100 lead to a large vertical displacement difference between the center mass m2 or m3 and the

base plate m1 . In Figure 7-9, first two modes are the bending modes due to the horizontal and vertical beams of the primary vibration absorber. Beams of the secondary vibration

absorber are not fully excited and thus mass m3 has the same displacement distribution as

mass m2 . Instead, mode 5 and mode 6 are the bending modes due to the horizontal and vertical beams of the secondary vibration absorber, which leads to large vertical

displacement for but a small vertical displacement for m2 . Notice that for mode 3, both primary and secondary vibration absorbers’ four beams are fully bending, leading to a large

positive displacement for and a large negative displacement for m1 . In mode 3, the displacement difference between the base plate and center mass is of its maximum, indicating an opposite movement phase between the base plate and the secondary vibration

absorber. Similar to mode 3, mode shape 8 has a large positive displacement for m2 and a large negative displacement for . The displacement difference between the base plate

and center mass is of its maximum, indicating an opposite movement phase between the base plate and the primary vibration absorber. Therefore, mode 3 and mode 8 are two resonant modes for the MLCMP unit cell. The corresponding natural frequencies for mode

3 (194.1 Hz) and mode 8 (269.0 Hz) are the two resonant frequencies for the MLCMP unit cell. Mode 4 and mode 9 are twisting modes due to the secondary vibration absorber and the primary vibration absorber, respectively. Other mode shapes with small deflections for

beams k2 or k3 are shown in Figure 7.9.

101

(a) (b)

(c) (d)

(e) (f)

102

(g) (h)

(i) (j)

(k) (l)

Figure 7-9 First 12 mode shapes for the MLCMP unit cell under periodic boundary conditions.

103

Dispersion analysis is then conducted to investigate the wave propagation in an infinite MLCMP. Bloch-Floquet boundary conditions are applied to the MLCMP unit cell’s left-right and top-bottom edges using COMSOL Multiphysics and the dispersion curves following the path of        in the irreducible Brillouin zone are shown in

Figure 7-10. Two theoretical stopbands of 155.9 Hz to 190.3 Hz and 272.9 Hz to 336.3 Hz

(shown in grey bands) are obtained where no wave can propagate through the infinite

MLCMP. It should be noted that the first stopband exists slightly below the first resonant frequency (194.1 Hz) of the MLCMP unit cell while the second stopband exists slightly above the second resonant frequency (269.0 Hz) of the MLCMP unit cell, which is consistent with the results in Section 4.2. If additional damping is introduced to the primary and secondary vibration absorber, energy of elastic wave with frequencies within the stopbands will be effectively attenuated. Frequency response analysis is performed next to examine wave propagation in a finite MLCMP under the steady state.

104

  

Figure 7-10 Dispersion curves of an infinite MLCMP with reduced wave vector k sweep following the path of        in the irreducible Brillouin zone.

For frequency response analysis of the MLCMP, first a hinged boundary condition on left and right edges is considered. A harmonic excitation force with an amplitude of

1000 N is applied at the second node from the left of the centerline. Rayleigh damping is

applied to the beam regions with damping coefficients of i  0 and different23, , where i  2 is for the primary vibration absorbers and i  3 is for the secondary vibration absorbers. Recall the design guideline for a multi-frequency vibration absorber that a small damping coefficient for the primary vibration absorber and a large damping coefficient for the secondary vibration absorber are desired to quickly activate the vibration absorbers, decrease the vibration amplitude of the base structure, quickly dissipate the transient

105 vibration and combine two low-frequency stopbands into a wider stopband. The average frequency responses (AFRs) of the MLCMP and its corresponding isotropic plate at x =

0.98 L are shown in Figure 7-11. The black line represents the AFR of the isotropic plate at x = 0.98 L while the red line denotes the AFRs of the MLCMP at x = 0.98 L with no damping applied to the vibration absorbers. The blue and magenta lines represent the AFRs

of the MLCMP at x = 0.98 L with increasing damping coefficients of 2 5e 5 ,

3 14e and2355,13ee. As analyzed before, two stopbands can be observed around each resonant frequency of the MLCMP unit cell. In Figure 7-11 (a), two stopbands are obtained in the frequency ranges of 159.9 Hz to 196.2 Hz and 280.7 Hz to 344.5 Hz where the AFR suddenly decreases. Notice that these two stopbands are also slightly different from those from dispersion analysis (grey bands) due to the external loading and boundary conditions. The red line also shows that despite a zero damping for the vibration absorbers, wave cannot propagate through the MLCMP when the excitation frequency falls within one of these two stopbands. By comparing the red, blue and magenta lines in Figure

7-11 (a), one can also see that increasing the damping coefficients applied to the secondary vibration absorbers can efficiently lower the response of the base plate and broaden two stopbands. The magenta line in Figure 7-11 (a) shows that much larger damping coefficients for the secondary vibration absorber can even combine two stopbands into one wider stopband, which is consistent with the results shown in Section 3.6.

106

(a)

(b)

107

(c)

(d)

108

(e)

(f)

Figure 7-11 (a) AFRs of an MLCMP and its corresponding isotropic plate at x = 0.98 L with hinged left and right edges under different damping coefficients. (b) - (f): Vertical displacement distributions with damping coefficients of ii0,  5e  5at different excitation frequencies: (b) f = 103 Hz, (c) f = 189 Hz, (d) f = 262 Hz (e) f = 290 Hz and (f) f = 400 Hz.

109

Figure 7-11 (b) - (f) show the vertical displacement distributions of the MLCMP with hinged left and right edges under excitation frequencies of 103 Hz (below the first stopband), 189 Hz (within the first stopband), 262 Hz (between two stopbands), 290 Hz

(within the second stopband) and 400 Hz (above the second stopband) with damping

coefficients of ii0,  5e  5. It is apparent that waves can propagate through the plate when the excitation frequency is outside the two stopbands but cannot propagate for the frequency within the two stopbands. When the excitation frequency falls within one of the two stopbands (e.g. Figure 7-11 (c) and (e)), the majority of the vibrational energy is absorbed by the resonant motion of the left-most vibration absorbers near the excitation location. Therefore, both the base plate and the rest of the vibration absorbers remain motionless.

Two identical MLCMPs with different boundary conditions applied during analysis are considered to examine the influence of boundary conditions on attenuation performance.

Figure 7-12 shows the AFRs at x = 0.98 L for a case with a hinged left edge boundary condition and its vertical distributions at different excitation frequencies while Figure 7-13 shows the AFRs at x = 0.98 L for another case with hinged four edges and its vertical displacement distributions at different excitation frequencies. The damping coefficients of

ii0,  5e  5 are applied for both cases. In Figure 7-12 (a) and Figure 7-13 (a), the red lines and black lines represent the AFRs of MLCMPs and their corresponding isotropic plates with the same boundary condition at x = 0.98 L. Although the AFRs vary slightly under different boundary conditions, there is still a significant decrease under each boundary condition around each resonant frequency of the MLCMP unit cell. Similar to the vertical displacement distributions in Figure 7-11 (b) - (f), Figure 7-12 (b) - (f) and

110

Figure 7-13 (b) - (f) show that wave cannot propagate through the MLCMP within either

of the two stopbands under different boundary conditions. Therefore, the stopbands

characteristics of the MLCMP are not significantly affected by boundary conditions either.

(a)

(b)

111

(c)

(d)

112

(e)

(f)

Figure 7-12 (a) AFRs of a hinged-left-edge MLCMP and its corresponding isotropic plate at x = 0.98 L with damping coefficients are ii0,  5e  5 . (b) - (f): Vertical displacement at different excitation frequencies: (b) f = 103 Hz, (c) f = 189 Hz, (d) f = 262 Hz, (e) f = 290 Hz and (f) f = 400 Hz.

113

(a)

(b)

114

(c)

(d)

115

(e)

(f)

Figure 7-13 (a) AFRs of a hinged-four-edge MLCMP and its corresponding isotropic plate at x = 0.98 L with damping coefficients are ii0,  5e  5. (b) - (f): Vertical displacement distributions at different excitation frequencies: (b) f = 103 Hz, (c) f = 189 Hz, (d) f = 262 Hz, (e) f = 290 Hz and (f) f = 400 Hz.

116

Transient analysis is then conducted for the MLCMP by direct numerical integration using ABAQUS/STANDARD Dynamic, Implicit. For transient analysis of the MLCMP with hinged left and right edges, a total period of 0.3 s with an incremental size of 0.0002

s is used. The same Rayleigh damping coefficients of ii 0  , 5 5 e are used for each unit cell beam. The external sinusoidal excitation frequencies are chosen to be 103 Hz

(below the first stopband), 189 Hz (within the first stopband), 231 Hz (between two stopbands), 290 Hz (within the second stopband) and 400 Hz (above the second stopband).

Average displacements of nodes at x = 0.02 L (black), 0.5 L (red) and 0.98 L (blue) on the centerline of the MLCMP are recorded in Figure 7-14 to show the wave propagation change due to vibration absorbers. In Figure 7-14 (a), the vibration amplitude of the

MLCMP decays spatially as the wave propagates due to the damping coefficients applied to the vibration absorbers. Because the excitation frequency (103 Hz) is lower than the first resonant frequency of the MLCMP unit cell, both primary and secondary vibration absorbers exhibit an acoustic mode. However, since the excitation frequency is far from the first resonant frequency, the incident waves freely propagate along with the base plate.

Figure 7-14 (b) illustrates that when the excitation frequency falls within the first stopband, the average displacement of the node at x = 0.5 L (red) is much smaller compared with one at x = 0.02 L (black) and the average displacement of the node at x = 0.98 L (blue) is even smaller. This is because primary vibration absorbers work in a mix mode while secondary vibration absorbers work in an acoustic mode. A mix mode is desired to induce the inertial forces working against the vibration of the base plate and stop the wave propagation through the EMM structure. Figure 7-14 (c) shows the wave propagation in the MLCMP under an excitation of 231 Hz that is between two stopbands. Amplitudes of average

117

displacements at nodes x = 0.5 L and x = 0.98 L are much larger compared with those in

Figure 7-14 (b). The primary vibration absorbers work in an optical mode while the

secondary vibration absorbers work in an acoustic mode. In this case, no mix mode exists

and destructive wave can easily propagate forward. When the excitation frequency is

within the second stopband, primary vibration absorbers work in an optical mode while

secondary vibration absorbers work in a mix mode. The plate is straightened again by the

vibration absorbers and the destructive wave cannot propagate forward, as shown in Figure

7-14 (d). Figure 7-14 (e) shows that displacement amplitudes of each of the three locations

are almost the same when the excitation frequency is above the second stopband. In this

case, both primary and secondary vibration absorbers work in an optical mode. Therefore,

the destructive wave can easily propagate through the MLCMP when the excitation

frequency is far beyond the second stopband. Above results indicate that the MLCMP also

has the attenuation properties under a transient response.

(a)

118

(b)

(c)

119

(d)

(e)

Figure 7-14 Average displacement of a hinged-left-and-right-edge MLCMP with damping coefficients of

ii0,  5e  5 at x = 0.02 L (black), 0.5 L (red) and 0.98 L (blue) of the centerline under different excitation frequencies in transient state: (a) f = 103 Hz, (b) f = 189 Hz, (c) f = 262 Hz, (d) f = 290 Hz and (e) f = 400 Hz.

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Chapter 8. CONCLUSIONS AND RECOMMENDATIONS

8.1 Concluding Remarks

The preceding chapters in this dissertation cover theoretical analysis and numerical simulation of both single and multi-stopband laser-cut acoustic metamaterial plates

(SLCMPs and MLCMPs). First, a thorough investigation of metamaterial’s theory and past developments have been made. For elastic/acoustic metamaterials (EMMs), negative effective mass and negative effective stiffness have been studied and stopbands for single and multi-stopband conventional acoustic metamaterial plates (SCMPs and MCMPs) are examined by finite element analysis using extended Hamilton principle. After that, the prototypes of SLCMPs and MLCMPs are proposed. Modal analysis is used to calculate the resonant frequencies for SLCMP and MLCMP unit cells. Dispersion analysis is conducted for infinite SLCMP and MLCMP to obtain dispersion curves. Frequency response analysis is performed to investigate the stopbands characteristics of SLCMP and MLCMP in the steady state. Transient analysis is made to study the wave propagation in SLCMP and

MLCMP under the transient state.

Numerical results show that the SLCMP/MLCMP is based on the concept of the

SCMP/MCMP. The locally resonant vibrations of laser-cut unit cells in the plate generate inertial forces against external loads and prevent elastic wave propagation in the base plate.

For an SLCMP, one stopband exists slightly above the unit cell’s resonant frequency at both steady and transient state. Increasing the damping of the vibration absorbers can effectively lower the response amplitude of the base plate as well as broaden the stopband to a small degree. Two stopbands exist for an MLCMP, with the first stopband existing slightly below the first resonant frequency of the unit cell while the second stopband

121 existing slightly above the second resonant frequency of the unit cell. For an MLCMP with two designed stopbands closed to each other, small damping coefficients of the primary vibration absorbers and large damping coefficients of the secondary vibration absorbers can be used to quickly activate the vibration absorbers, dissipate the vibration of the base plate and combine two stopbands into a wider stopband. It is also demonstrated that boundary conditions will not significantly affect wave suppression in the SLCMP and

MLCMP.

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8.2 Recommendations for Future Work

The field of laser-cut EMMs is still in an incipient stage. This research work only concerns the harmonic wave propagation and the input from transient analysis is also simplified to be cosine signals. Many other complicated cases remain to be studied. From all the modeling work and numerical simulations, we recommend the following topics for further study:

1) All the analysis above is from theoretical approach. Experimental verifications

are need following the theoretical predictions and simulations. Fortunately, laser

cutting is a very common experiment in modern mechanical field and is also

easy to achieve.

2) This work only considers plate structures because of their commonly use in the

real world. Other laser-cut metamaterial structures such as bars and beams are

also worth to be studied.

3) A lot of engineering applications have already been proposed by scientists.

However, some of those ideas are just theoretically discussed. One of the great

applications for this research work is seismic waveguide. Buildings and other

constructions can be built with laser-cut acoustic metamaterial plates to guide

seismic wave and prevent structures from being destroyed by the earthquake.

Noise reduction is another application of laser-cut EMMs that can be used in

real wall structure to absorb sound wave.

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VITA

Haoguang Deng was born in October 1989 at Daqing, China. He graduated from

Hunan University, China with a BS degree in Engineering Mechanics in 2008. After that he came to United States to pursue his master and Ph.D. degree. Right now Haoguang Deng is a Ph.D. candidate majoring in Mechanical and Aerospace Engineering at University of

Missouri, Columbia (MU). During his study at MU, he works as a research assistant as well as a teaching assistant. He has gained extensive research experience in finite element analysis and teaching experience in applied numerical methods with MATLAB. By the time he writes his dissertation, he has published two referred journal papers and other two referred journal papers under review. His research work has gained recognition from both his supervisors and peer students.

From August 2012 to May 2014, Haoguang Deng was a master student at MU woking on a research “Damage Inspection of General Plates by Time-Frequency Analysis of Lamb Wave”. His job was to model a crack in an isotropic plate or two-layer plate and detect the position of the crack using Lamb wave propagation. One of the difficulties is that high-frequency wave dynamics always incolves complicated wave reflection, refraction and diffraction and results in difficulty separating them in order to have detaied examination and system identification. Another difficulty is to accurately locate the position of the crack. Despite these difficulties, he studied existing literatures and decided to take advantage of the dynamic characteristics of Lamb wave in plates to inspect damages of thin-walled structures. He also decided to use empirical mode decomposition (EMD) to decompose signals into several imperical mode fuctions (IMF) and then use conjugate-pair decomposition (CPD) method to perform time-frequency analysis on each IMF to avoid

129 from edge effect. He published one refereed journal paper on Mechanical Systems and

Signal Processing with his supervisors Dr. P. Frank Pai.

After finishing his master program in two years, Haoguang Deng continued his Ph.D. study and swithched his focus to acoustic metamaterials (EMMs). His Ph.D. research topic is acoustic laser-cut metamaterial plate design for elastic wave absorption. EMMs are newly developed smart materials which have not been fully studied yet. He conducted deep research on acoustic metamaterial plates. He also created another type of acoustic metamaterial plate with designed laser cuts rather than attaching vibration absorbers. By using his strong background in finite element analysis, he performed modal, frequency response and transient analysis of the laser-cut metamaterial plate and proved its wave absorption characteristic. He has submitted two journal papers about laser-cut acoustic metamaterial plates.

In general, Haoguang Deng is proficient with finite element analysis for solving different kinds of engineering problems in structural mechanics. He also had experience in dynamic mechanics including modal, dynamic steady-state and transient analysis.

Haoguang Deng has been ready for his young career in industry field.

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