FINITE-ELEMENT DESIGN OF
METAMATERIAL BEAMS FOR BROADBAND WAVE ABSORPTION
______
A Thesis presented to the Faculty of the Graduate School
at the University of Missouri-Columbia
______
In Partial Fulfillment
of the Requirements for the Degree
Master of Science
______
by
SHUYI JIANG
Dr. P. Frank Pai, Thesis Supervisor
MAY 2015 The undersigned, appointed by the Dean of the Graduate School, have examined the thesis entitled
FINITE-ELEMENT DESIGN OF
METAMATERIAL BEAMS FOR BROADBAND WAVE ABSORPTION
Presented by Shuyi Jiang
A candidate for the degree of Master of Science
And hereby certify that in their opinion it is worthy of acceptance.
Professor P. Frank Pai
Professor Steven Neal
Professor Stephen Montgomery-Smith ACKNOWLEDGEMENTS
I would like to express my deepest appreciation to my advisor Dr. P. Frank Pai.
Without his patient guidance, I wouldn’t have grown as a good researcher. His continuous encouragement and valuable suggestions on my thesis work meant a lot to me. Also I would like to thank my committee members, Dr. Steven Neal and Dr.
Stephen Montgomery-Smith, for serving on my thesis committee and providing me assistance when I have difficulties.
I would also like to extend my thanks to Dr. Hao Peng, Xuewei Ruan,
Haoguang Deng, Yiqing Wang, Jamie Lamont and all my labmates. They helped my study and gave me confidence to reach the goal. Thanks to all the staff in the
Mechanical and Aerospace Engineering Department for their hard work for me during my study at the University of Missouri.
Finally, special thanks to my family for their mental and financial support through my life. Without their love and support my study could not be so smooth and fruitful.
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TABLE OF CONTENTS
ACKNOWLEDGEMENTS ...... ii
LIST OF FIGURES ...... v
ABSTRACT ...... ix
Chapter 1 Introduction to Metamaterials ...... 1
1.1 Background ...... 1
1.2 Double Negative Permittivity and Permeability ...... 2
1.3 Methods of Construction ...... 6
1.4 Negative Refractive Index and Perfect Lens ...... 10
1.5 Backward Waves and Reversed Doppler Effect ...... 16
1.6 Acoustic Metamaterials ...... 19
1.7 Objectives of Research ...... 20
Chapter 2 Basic Concepts of Acoustic Metamaterials ...... 22
2.1 Negative Effective Mass ...... 22
2.2 Negative Effective Stiffness ...... 25
2.3 Conventional Vibration Absorber ...... 28
2.4 Broadband Vibration Absorbers ...... 32
2.5 Dispersive Material ...... 35
2.5.1 Non-dispersive Wave ...... 35
2.5.2 Dispersive Wave ...... 37
2.6 Dispersive Metamaterial Bar ...... 38
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2.6.1 Governing Equation and Dispersion Relation ...... 39
2.6.2 Finite Element Modeling ...... 45
Chapter 3 Acoustic Metamaterial Beams with Local Vibration Absorbers ...... 50
3.1 Governing Equations and Dispersion Relations ...... 50
3.2 Finite-Element Modeling ...... 55
3.2.1 Euler-Bernoulli Beam Element ...... 56
3.2.2 Timoshenko Beam Element ...... 58
3.2.3 Infinite Beams with Uniform Absorbers ...... 63
3.3 Numerical Analysis ...... 66
3.3.1 Supported Beam with Uniform Absorbers ...... 66
3.3.2 Supported Beam with Varying Absorbers ...... 70
3.4 Actual Working Mechanism ...... 77
Chapter 4 Multi-stopband Metamaterial Beams ...... 80
4.1 Basic Concept ...... 80
4.2 Governing Equation and Finite-Element Modeling ...... 83
4.3 Numerical Analysis ...... 88
Chapter 5 Conclusions and Recommendations for Future Work ...... 94
5.1 Conclusions ...... 94
5.2 Recommendations for Future Work ...... 95
References: ...... 97
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LIST OF FIGURES Figure 1.1 Three prototypes of magnetic microstructures. Model (a): an array of metallic cylinders designed to have magnetic properties along the axial direction of the cylinder. Model (b): an array of copper cylinders which consist of external and internal ‘split ring’ configurations. Model (c): ‘Swiss Roll’ Capacitor...... 7
Figure 1.2 Dispersion relation for magnetic microstructure with ‘split ring’ configuration...... 9
Figure 1.3 (a) a ‘split ring’ resonator (SRR), (b) a square-shaped SRR, and (c)SRRs being attached to three perpendicular plans of a cube...... 10
Figure 1.4 Reflection and refraction at the interface of two media...... 12
Figure 1.5 Optical characteristic of conventional lenses: (a) a convex lens is a converging lens, and (b) a concave lens is a diverging lens ...... 12
Figure 1.6 Passage of rays through a slab lens which has an ideal refractive index n=-1...... 13
Figure 1.7 Optical characteristics of lenses made of left-handed material (LHM): ..... 15
Figure 1.8 (a) A silver slab lens placed at the center between an objective plane and image plane, and (b) electrostatic objective potential intensity, and (c) image potential intensity with/without silver slab lens...... 16
Figure 1.9 The wave propagation direction where S represents the Poynting vector (group velocity) and k represents the wave vector (phase velocity): (a) right-handed material, and (b) left-handed material...... 17
Figure 2.1 A 2-DOF mass-in-mass system to illustrate negative effective mass...... 22
Figure 2.2 A 2-DOF mass-in-spring system to illustrate negative effective stiffness. .. 26
Figure 2.3 The Frahm dynamic vibration control device model...... 29
Figure 2.4 The frequency response of the Frahm vibration absorber...... 29
Figure 2.5 A 2-DOF damped vibration absorber model...... 31
Figure 2.6 The frequency response of a damped vibration absorber...... 31
Figure 2.7 A 6-DOF system...... 33 v
Figure 2.8 The frequency response of a 6-DOF system with m1100 kg , m 2 m 6 , ki m i 20 , i 0.05 ...... 34
Figure 2.9 The frequency response of a 6-DOF system with ...... 35
Figure 2.10 The free-body diagram of a differential string element under tension. ... 36
Figure 2.11 Wave propagation in a metamaterial bar: (a) a metamaterial bar configuration, and (b) the free-body diagram (FBD) of a unit cell...... 40
Figure 2.12 Dispersion curves of a metamaterial bar: (a) curves, and (b) curves...... 43
Figure 2.13 Dispersion curves ( ) of a metamaterial bar: (a) m 0.1 AL , and (b)
kk 0.01 1 ...... 45
Figure 2.14 Steady-state operational deflection shapes (ODSs) with no damping: (a) r=0.8, (b) r=1.13, and (c) r=1.2...... 48
Figure 2.15 Steady-state operational deflection shapes (ODSs) with dampings
i 0.05 : (a) r=0.8, (b) r=1.3, and (c) r=1.2...... 49
Figure 3.1 (a) a prototype of a metamaterial beam, (b) model of a simply supported beam with translational and rotational absorber, and (c) free-body diagram of a unit cell...... 51
Figure 3.2 Dispersion curves of an infinite metamaterial beam unit cell...... 55
Figure 3.3 Finite element modeling of an Euler-Bernoulli beam element...... 58
Figure 3.4 Modeling of a modified Timoshenko beam element: (a) deformed cross- section, and (b) finite element modeling...... 59
Figure 3.5 Finite element modeling of a metamaterial beam...... 64
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Figure 3.6 Dispersion curves with different absorbers: (a) different translational absorbers with m0.1,0.3,0.5 AL , mˆ 0.1 AL , and (b) different rotational absorbers with mˆ 0.1,0.3,0.5 AL , m 0.1 AL ...... 66
Figure 3.7 Frequency response function (FRF) H101,2 of the metamaterial beam with m 0.1 AL and mˆ 0 ...... 68
Figure 3.8 Steady-state ODSs of a metamaterial beam with r : (a) r=0.8, (b) r=1.02, and (c) r=1.2...... 69
Figure 3.9 (a) frequency response function at the middle point with r=0.5+0.005(i-1), i=1,…,201 , and (b) steady-state ODS under a harmonic excitation at Node 2...... 71
Figure 3.10 (a) frequency response function at the middle point with r=1.5-0.005(i-1), i=1,…,201, and (b) steady-state ODS under a harmonic excitation at Node 2...... 72
Figure 3.11 Steady-state ODSs of a metamaterial beam with 4 subgroups of absorbers r=1.02 (i=1,…51), r=2.04 (i=52,…101), r=3.06 (i=102,…151), and r=4.08
(r=152,…201): (a) F AL sin t , (b) F AL 2 sin t 2 , (c)
F AL 3 sin t 3 , (d) F AL 4 sin t 4 , and (e)
4 F AL nsin t n ...... 74 n1
Figure 3.12 Steady-state ODSs of a metamaterial beam with 4 subgroups of absorbers r=4.08 (i=1,…51), r=3.06 (i=52,…101), r=2.04 (i=102,…151), and r=1.02 (r=152,…201): (a) , (b) , (c)
, (d) , and (e)
...... 76
Figure 3.13 A metamaterial beam with a few absorbers: (a) steady-state ODS with r=0.9, (b) steady-state ODS with r=1.02, and (c) distribution of internal shear forces with r=1.02...... 78
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Figure 3.14 A metamaterial beam with a few absorbers: (a) steady-state ODS with r=1.2, and (b) distribution of internal shear forces with r=1.2...... 79
Figure 4.1 A two-mass damped vibration absorber model...... 80
Figure 4.2 Frequency response functions H11 with m110kg , m 2 0.1 m 1 and: (a)
0.05 with different , (b) 0.05 with different , (c) 0.025,0.05,1 , and
(d)different damping ratios with ...... 83
Figure 4.3 free-body diagram of a metamaterial beam segment ...... 85
Figure 4.4 Finite element modeling of an infinite metamaterial beam...... 85
Figure 4.5 Dispersion curves for a two-mass vibration absorber with m2 m 10.1, m 3 m 2 k 3 k 2 0.05 ...... 88
Figure 4.6 Frequency response function (FRF) H101,4 of the metamaterial beam with . 89
Figure 4.7 Steady-state ODSs when r2 , m2 m 10.1, m 3 m 2 k 3 k 2 0.05 ,
230.01 and 0.1: (a) r=0.7, (b) r=0.9, (c) r=1.0, (d) r=1.1, and (e) r=1.3...... 91
Figure 4.8 Steady-state ODSs when , ,
230.01 and 0.01: (a) r=0.7, (b) r=0.9, (c) r=1.0, (d) r=1.1, and (e) r=1.3...... 92
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ABSTRACT
Metamaterials were first introduced for dealing with electromagnetic waves.
By creating a material with negative magnetic permeability and electric permittivity, it is possible to manufacture perfect optical lenses, electromagnetic absorbers, technologies for rendering objects invisible and so on. Later researchers began to look into metamaterials for dealing with acoustic waves. This thesis presents the modeling and analysis techniques for design of metamaterial beams as elastic wave absorbers.
An acoustic metamaterial beam is designed by attaching tiny subsystems to an isotropic beam at separate locations. Each unit cell consists of a beam segment and a mass-spring-damper subsystem so that it could be modeled as a discrete system of two degrees of freedom. By integration, the idealized model becomes a dispersive medium with a frequency stopband. The work shows that the metamaterial beam is based on the concept of a conventional vibration absorber, which uses the local resonance of subsystems to generate inertia forces to work against the external load and prevent elastic waves from propagating forward.
Dispersion analysis and frequency response analysis are conducted to find the stopband of a metamaterial beam. Moreover, the working mechanism of metamaterial beams, the concept of negative effective mass, and acoustic and optical modes are presented.
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This concept is also extended to design a multi-stopband metamaterial beam that can absorb broadband elastic waves. Numerical simulations by using finite elements validate the design and reveal a set of optimized parameter values. This work shows that, for a multi-stopband metamaterial beam, a high damping ratio for the secondary absorber combines two stopbands into a broad one while a low damping ratio for the primary absorber guarantees quick response to the coming excitation elastic wave.
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Chapter 1 Introduction to Metamaterials
1.1 Background
The first attempt to explore the concept of artificial material began in the late
nineteenth century when Bose [1] conducted the first microwave experiment on
twisted structures. Later in 1948, Kock [2] made lightweight microwave lenses and
demonstrated the possibility of tailoring the effective refractive index of artificial
material.
In 1967, Veselago [3] became the first one who officially proposed the
concept of metamaterial. He theoretically investigated electromagnetic wave
propagation in a material whose permittivity and permeability were assumed to be
simultaneously negative for some frequency ranges and studied the potential
physical properties of metamaterials. Smith [4] later demonstrated this presence of
the anomalous refraction experimentally.
The famous electrodynamics consequences of a medium having both
permittivity and permeability being negative include negative refractive index,
reverse Doppler Effect [5], reverse Cherenkov radiation and the clocking techniques,
near-field focusing and sub-wavelength imaging . Some of these will be explained
later. These special properties are not of the constituent materials, but they result
from the periodic arrangement of subsystems made of the constituent materials.
Besides, several names have also been suggested for the metamaterial, such as left-
handed media, media with negative refractive index, backward wave media, double
1
negative (DNG) materials and so on [3].
The concept of metamaterials went into silence for years because the basic
elements are usually required to have sub-wavelength sizes so that the material can
be treated as homogeneous media and an averaged negative permeability can be
detected. However, the required subsystems for metamaterials to operate at visible
frequencies were so small that no fabrication techniques were available. While
recently, this interest has been re-motivated by new concepts for nanoscale
manufacturing techniques. Since metamaterials can be constructed by embedding
artificially fabricated inclusions in a specified host medium, many design parameters
are available. For example, the properties of the host material, the size, shape and
composition of the inclusions, and the density and arrangement of the inclusions can
be the design parameters. This provides the designer with the possibility to engineer
a metamaterial with a specific electromagnetic response function that cannot be
found in nature.
1.2 Double Negative Permittivity and Permeability
As Veselago [3] mentioned, the permittivity and the magnetic permeability
of an optical material are the fundamental characteristic quantities that
determine the propagation of electromagnetic waves. This is due to the fact that
they are the only parameters of the substance that appear in the dispersion
equation:
2 k2 k k 0 (1.1) c2 ij ij ij i j
2
Equation (1.1) shows the connection between the frequency of a monochromatic
wave and its wave number k . Here ij is the Kronecker delta function, and c denotes the light speed. If the material is isotropic, it could be simplified into:
22 k (1.2)
If we do not take energy losses into account and regard all the components in the equation as real numbers, it could be seen that a simultaneous change of the sign of
and doesn’t have influence on the validation of the equation. This situation might lead to the assumption that substances with negative and do exist but they have some properties different from those of substances with positive and .
To ascertain the assumption and figure out if the electromagnetic laws are connected with the sign of and , we must turn that relation into separate equations through Maxwell's equations. Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics. They describe how electric and magnetic fields are generated and altered by each other and by charges and currents. Maxwell’s electromagnetic theory in general form could be formulated as follows:
B E t E BJ 00() t (1.3) E 0 B 0 where E is the electric field intensity vector and B is the magnetic field flux density vector. The sources of these fields are electric charges and electric currents, which
3 can be expressed as local densities named charge density and current density J .
Physical constants 0 and 0 are permittivity and magnetic permeability of the free space. Furthermore, if the magnetic field intensity H and the electric flux density D are introduced, the Maxwell's equation for a time varying, source-free media could be turned into:
B E t D H (1.4) t E 0 B 0
Here it is necessary to bring up the constitutive equations in electromagnetism, which specify the relations between displacement field D and E , and the magnetic field H and B . In an isotropic linear lossless material, the constitutive equations are:
DE (1.5) BH
where 0 r and 0 r . r and r are the relative permittivity and permeability.
For a plane monochromatic wave, all the quantities D , E , H and B are proportional to expi kz t , and the medium is lossless. In that case, the expressions for Equation (1.4) reduce to:
k E H (1.6) k H E where k is the wave number vector of the plane wave. It can be seen from Equation
(1.6) that if 0 and 0 , then E , H and k will form a right-handed triplet of
4 vectors. Otherwise, if 0 and 0, E , H and k will become a left-handed set.
By introducing direction cosines for the vectors , and as i , i and i , respectively, a wave propagated in a given medium will be characterized by the matrix:
1 2 3 G 1 2 3 (1.7) 1 2 3
The determinant of this matrix is equals to +1 if the vectors , and form a right- handed set or -1 if it's a left-handed set. Denoting this determinant by p , we could conclude that p characterizes the ''rightness'' of a given medium. That means, the material is left-handed if p 1 and is conventional right-handed if p 1. That's why metamaterials are also called left-handed materials (LHM). Furthermore, the elements of G matrix are orthonormal.
Although Veselago proposed the concept of metamaterials and predicted several properties, how to realize the double negative property was still a tough topic then. He concluded that simultaneous negative values for e and m could be realized only when there is frequency dispersion. The relation
22 WEH (1.8) confirms his conclusion. When there is neither frequency dispersion nor absorption, we cannot have e < 0and m < 0 because the total energy W would be negative in this case. When these is frequency dispersion, the relation is replaced by
¶(ew ) ¶(mw ) W = E 2 + H 2 (1.9) ¶w ¶w
5
In order to make the energy positive, it is required that
0, 0 (1.10)
The inequalities in Equation (1.10) imply that if Equation (1.9) is satisfied, the values
fore and m don’t have to be positive in general, while they depend on the frequency.
While Smith et al. [4] constructed such a composite medium for the microwave
regime later and demonstrated experimentally the presence of related properties.
1.3 Methods of Construction
Many research groups in the world are studying different aspects of
metamaterials. As for methods of construction, several geometries for the inclusions
have been suggested. Among those, Pendry’s proposal [6] could not be ignored. He
brought up the concept that every material could be taken as a composite in a sense
even if the individual ingredient consists of atoms and molecules. Since permittivity
and permeability are used to present a homogeneous view of the electromagnetic
properties of a medium, they should be definitions that work for both the original
micro-scale atoms and the larger-scale structure. By introducing a periodic array of
nonmagnetic conducting units and making the constituent units resonant, negative
effective permeability m near the high frequency side of the resonance will happen.
The restriction to construct a periodic-structured metamaterial that has
negative and is that the dimension of the unit cell must be way much smaller
than the wavelength. If this condition is not satisfied, a potential problem is that the
internal structure medium could diffract as well as refract the wave.
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Besides, there are two reasons to generate these particular electromagnetic
properties by using microstructures. Firstly, atoms and molecules are rather
restrictive set of elements from which to build magnetic material especially at
frequencies in the gigahertz range, where the magnetic response of most material is
beginning to tail off. In contrast, micro-structured materials could be designed with
desirable mechanical properties. Secondly, materials such as ferrites may remain
moderately active, but they are often heavy. While microstructures can be made
extremely light with considerable magnetic activities remained.
Figure 1.1 Three prototypes of magnetic microstructures. Model (a): an array of metallic cylinders designed to have magnetic properties along the axial direction of the cylinder. Model (b): an array of copper cylinders which consist of external and internal ‘split ring’ configurations. Model (c): ‘Swiss Roll’ Capacitor.
Pendry et al. [6] explored several magnetic microstructure configurations and
derived the frequency-dependent effective permeability meff for each model. Figure 7
1.1 shows three prototypes of magnetic microstructures. Model (a) consists of an array of metallic cylinders. In this case, he concluded that for an infinite conducting
cylinder, magnetic permeability meff is reduced by the ratio of the cylinder volume to the cell volume. In model (b), cylinders are built in a ‘split ring’ configuration. Two overlapping sheets are separated from each other by a distance d . In each sheet, there is a gap that makes the rings resonant and the gap in one band is placed 180o relative to the other. Compared to (a), model (b) introduced inside capacitive elements into the structure to enable the current to flow and extend the range of magnetic properties. In this case, the capacitance balances the inductance so that
has a generic resonant form as sketched in Figure 1.2. In this dispersion
relationship shown in Figure 1.2, 0 is defined to be the frequency at which
diverges and mp is the so called ‘magnetic plasma frequency’. The gap between andmp is where is negative. Numerical analysis shows that the frequency at which the structure is active is much greater than the separation between cylinders.
Model (c) is called ‘Swiss Roll’ Capacitor with the same arrangement of cylinders on a square lattice except that the cylinders are built in Swiss roll configuration. In this instance, there is more capacitance and the range of active frequencies is an order of magnitude lower compared to model (b).
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Figure 1.2 Dispersion relation for magnetic microstructure with ‘split ring’ configuration.
Although the structures shown in Figure 1.1(b) and (c) are able to present strong magnetic properties along the axis of the cylinder, they have almost zero magnetic response in other directions. A potential problem is that, if the electric field is not parallel to the cylinders, the system responds just like a metal because the current is free to flow along the length of cylinders. Because this anisotropic behavior is undesirable, an adaption of a modified ‘split ring’ structure is proposed, where the cylinder is replaced by a series of flat disks but the ‘split ring’ configuration is kept, as shown in Figure 1.3a and 1.3b. The metal disk configuration is usually called the split- ring resonator (SRR) because it can resonate with external microwaves with much larger wavelength than the diameter of the rings. SRRs could be embedded within three perpendicular planes of the cubic and a unit cell structure is shown in Figure
1.3c. In this way, the element is cubic symmetric. If the cells are periodically
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distributed in space, a magnetic material with effective magnetic response is
generated and the material is isotropic. As shown above, each unit configuration
could be designed to have its own magnetic response. The response can be
enhanced or lessened as desired.
Figure 1.3 (a) a ‘split ring’ resonator (SRR), (b) SRRs in their stacking sequence, and
(c) SRRs being attached to three perpendicular plans of a cube.
1.4 Negative Refractive Index and Perfect Lens
Refraction is one of the basic properties of a material. When the incident light
strikes the interface between two media, part of the ray is reflected and part of the
ray is refracted (i.e., transmitted). Veselago pointed out in 1968 [3] that a light is bent
to a different direction at the interface of two materials having different rightness (p)
values . Figure 1.4 illustrates the reflection and refraction at the interface of two
media. The incident ray #1 is reflected to Passage 2, independent of the rightness of
the two media. If the two materials have the same rightness, the incident ray is
10 refracted into Passage 4, which is the common case in our daily life. Snell’s law gives the relation between the angle of incident and the angle of refraction as
sin n 2 rr22 (1.11) sin n1 rr11
where is the angle of incident, is the angle of refraction, ni , ri and ri are the refractive index, relative permittivity and relative permeability of the media, respectively. Here, subscripts denote the material numbers. However, if the two materials have different rightness, for example, one is a right-handed material while the other one is a left-handed material, the refractive ray is going to bent to the opposite side of the axis. The angle of refraction doesn't change but the direction changes. In that case, the Snell law is modified as:
sin np 22rr22 (1.12) sin np11rr11
Equation (1.12) is a form more general than (1.11) because it takes the influence of
material rightness into consideration. Here p1 and p2 are the rightness of the first and second media. If and have opposite signs, will be a negative number.
Therefore, the refractive index of the material is redefined as:
np rr (1.13)
If the material is left-handed, p is -1, and the refractive index is a negative number.
In particular, the index of refraction of a left-handed medium relative to vacuum is negative. Furthermore, Fresnel equations are usually used to find the amplitudes of reflected and refracted light at uniform planar interfaces. The conventional equations involve the quantities , , n ,a and b . In order not to make mistakes, absolute
11 values should be applied.
Figure 1.4 Reflection and refraction at the interface of two media.
Figure 1.5 Optical characteristic of conventional lenses: (a) a convex lens is a converging lens, and (b) a concave lens is a diverging lens
One of the most important applications of materials of negative refractive
12 index is to make perfect lens. The operation of conventional lenses is well understood on the basis of classical optics, as shown in Figure 1.5(a) and (b). Convex lenses are converging lenses and concave lenses are diverging lenses. The resolution of an optical device can be improved by polishing the lens or by adjusting the alignment of the mechanical components, but the fundamental maximum resolution still exists and it is limited by the wavelength of the light. Due to diffraction of light, no lens can focus light onto an area smaller than a square of the wavelength no matter how perfect the lenses are and how large the aperture is. This limitation on the imaging system performance is the so-called diffraction limit.
Figure 1.6 Passage of rays through a slab lens which has an ideal refractive index n=-1.
It was proposed by Pendry in 2000 [7] that a lens material with a negative refractive index could overcome the diffraction limit. As is shown in Figure 1.6, the material of the thin slab has an ideal refractive index of n = -1. The figure obeys
Snell’s Laws at the first interface and the light inside the medium makes a negative 13 angle with respect to the normal of the surface. This phenomenon also happens at the second interface. One of the interesting characters of this perfect lens is the double focusing effect as long as the slab and the focusing distance satisfy the
geometric relation d3 = d2 - d1 . That means the light transmitted through a slab of thickness d2 , which is located at a distance d1 from the source, will refocus to an area at d3 from the slab interface. It is the phase reversal that enables the medium to refocus a light by canceling the phase change when the light moves away from its source. Such a metamaterial plate can focus a light onto an area smaller than its wavelength square.
While another breakthrough discovered by Pendry was that such a metamaterial could cancel the amplitude decrease of an evanescent wave as well. It was proved by his numerical analysis that the amplitude of an evanescent wave emerges from a side far from the medium is enhanced by the transmission process.
Hence, both propagating and evanescent waves contribute to the image resolution if such a lens is used. For both cases, when the light passes a right-handed material and is refracted into a left-handed material, the transmission coefficient T is equal to 1, which means all of the energy is perfectly transmitted and the process is lossless. It does not violate the law of energy conservation because the reflection coefficient R goes to zero because no reflection at all.
Figure 1.7 shows convex and concave lenses made of left-handed metamaterials. The optical characteristics of convex and concave lenses have been switched. The convex lens has a diverging effect and the concave lens has a
14 converging effect. Because a left-handed optical device behaves in a way opposite to a conventional optical device, a left-handed material is also called optical antimatter.
Figure 1.7 Optical characteristics of lenses made of left-handed material (LHM): (a) a convex lens has a diverging effect, and (b) a concave lens has a converging effect.
Pendry also established an electromagnetic model using a layer of silver to evaluate how well a perfect lens can focus an image. As shown in Figure 1.8(a), a silver slab having a width d = 40nanometers was placed at the center between an object plane and an image plane. The distance between the two planes is z = 80nm.
The object was modeled to comprise an electrostatic potential intensity with two spikes as shown in Figure 1.8(b). Without the silver slab, the image potential was blurred on the image plane as shown in Figure 1.8(c) since it is z = 2d away from the object. The two spikes could not be resolved because the amplitudes of higher-order
Fourier components of the potential are reduced. On the other hand, if the silver slab was used as a lens to restore the amplitude of higher-order Fourier components, the resulted image showed the two peaks. The reconstruction was not perfect since the potential was distorted, but the two spikes remained. This evidently showed that
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only the finite imaginary part of the dielectric function placed limitations on the
focusing effect. While further research demonstrated that if the metamaterial
medium has a large negative index or becomes dispersive or not lossless, Pendry’s
perfect lens theory could not be realized.
Figure 1.8 (a) A silver slab lens placed at the center between an objective plane and image plane, and (b) electrostatic objective potential intensity, and (c) image potential intensity with/without silver slab lens.
1.5 Backward Waves and Reversed Doppler Effect
The origin of this newly predicted behavior of metamaterials could be traced
back to the distinction between the group velocity, which characterized the flow of
energy, and the phase velocity, which characterized the wave propagation direction.
The energy flux carried by an electromagnetic wave could be represented by using
the Poynting vector S, which is given by
c S = E ´ H (1.14) 4p
The direction of the group velocity vg is consistent with the direction of the Poynting
16 vectorS, and the direction of the phase velocity v p is consistent with the direction of the wave vector k . According to Equation (1.14), the vectorSalways forms a right- handed set with the electric and magnetic field vectors Eand H. We may conclude that the Poynting vector Sand the wave vector k are along the same direction if the material is right-handed, which is what happens in conventional cases. By contrast, the group velocity of a left-handed material can be along a direction opposite to that of the phase velocity. As is shown in Figure 1.9(b), the left-handed material has a so- called negative phase velocity, which will happen in particular in anisotropic substances or when there is spatial dispersion. This is why the material is also called a backward wave media.
Figure 1.9 The wave propagation direction where S represents the Poynting vector (group velocity) and k represents the wave vector (phase velocity): (a) right-handed material, and (b) left-handed material.
The famous consequence of the fact that the phase velocity v p of an LHM is opposite to the energy flux direction is the reversed Doppler Effect [8]. The original concept is used to describe the frequency change of a wave to an observer moving relative to the wave source. Many technologies based on the reversed Doppler Effect exist, including radar speedometers for distant speed sensing, wind velocity sensing,
17 blood-flow measurement, and astronomic researches. Normally, the received frequencyw during the approach of the source is higher than the emitted frequency
w 0 , and it is lower during the recession. However, this effect is reversed in a media made of LHM. The frequency received by the detector will be downshifted when the source is approaching and upshifted when receding. The Doppler Effect could be rewritten in the following more general form:
v (1.15) w = w 0 1 - p ( u )
Here vis the velocity of the receiver relative to the source, and it is regarded as positive when it is receding from the source.u is the velocity of the source, and it is regarded as always positive. pis the rightness of the medium, which is 1 for normal material and -1 for left-handed material.
For a high-frequency electromagnetic wave in an ordinary media, its wavelength is short. However, Lee [9] demonstrated that the wavelength of an electromagnetic wave may increase with its frequency. In other words, if the frequency is reduced in front of a moving source, the wavelength may become shorter. The frequency shift is reversed, but the wavelength shift is not reserved.
Hence, the wave received in front of a moving source is still compressed and the wave received behind the moving source is still expanded. That is because the wavenumber k in a metamaterial can be negative, and thus dd/ and the group
velocity vg () dk d are positive because k 2 and hence
d2 dk s (1.16) d k2 d 18
where s = -1 . dd/0 means that the wavelength increases when the
frequency increases.
1.6 Acoustic Metamaterials
The existence of acoustic metamaterials was experimentally demonstrated by
Liu et al. in 2000 [10]. The idea was inspired by Veselago’s proposal of
electromagnetic (EM) metamaterials with negative refractive indices. The EM
metamaterials are actually composites with built-in resonance substructures to
exhibit effective double negative permittivity and permeability for some frequency
ranges. Similarly, in Liu’s experiment, they used a simple microstructure unit
consisting of a solid core material with relatively high density and a coating of
elastically soft material to create the local inhomogeneity.
Li [11] conducted the study of elastic wave propagation later. The continuity
and Newton’s second law were used to derive some basic concepts for acoustic
metamaterials, and it showed that the mass density and bulk modulus k are
two characteristic quantities. By considering a plane-wave solution with a wave
vector k , the refractive index n should be defined as
kn2 2 2 , n2 (1.17)
Therefore, in order to have a propagating plane wave inside a medium, andk
need to be both positive or both negative. As the acoustic analog of Veselago’s
medium, a double-negative acoustic medium has a simultaneously negative effective
bulk modulus and density in certain frequency bands and the double negative
19
parameters should be a result of low-frequency resonances.
Because key physics of acoustics are well understood, major objectives of
research on acoustic metamaterials are for development of technologies for
vibration suppression/control of mechanical systems, alleviation of impacts on civil
structures due to natural disasters like earthquakes, etc.
1.7 Objectives of Research
Acoustic metamaterial is probably one of the most popular topics nowadays.
To simulate the process of wave absorption in an acoustic metamaterial, a
mathematical structural model is needed. In this thesis, we use the finite element
method to establish a structural model, and look into how a beam-like metamaterial
works.
Chapter 2 introduces some basic concepts of acoustic metamaterials. We will
start with bar structures to explain how to obtain simultaneous negative effective
mass and effective stiffness, which are the most important characters of acoustic
metamaterials. Also, we will review different conventional absorber models and
describe the finite element modeling process for metamaterial bars.
Chapter 3 will present a two-degree-of-freedom metamaterial beam.
Governing equations and finite element modeling process are shown in detail.
Numerical examples are used to verify the existence of frequency stopband.
Moreover, the influence of different distributions of absorbers and the actual
working mechanism of metamaterial beams are revealed at the end.
As an extension of Chapter 3, we present a multi-stopband metamaterial 20 beam in Chapter 4. Dispersion analysis and frequency response analysis are used to find the stopbands. We’ll also compare a multi-stopband metamaterial beam with a single-stopband one to get optimized parameters for our models.
Chapter 5 will summarize all the numerical results and provide some concluding remarks. Moreover, some potential applications are proposed for future work.
21
Chapter 2 Basic Concepts of Acoustic Metamaterials
2.1 Negative Effective Mass
Similar to electromagnetism metamaterials, which show simultaneous
negative permittivity and permeability, acoustic metamaterials can show
simultaneous negative mass density and bulk modulus k [12, 13].
Figure 2.1 A 2-DOF mass-in-mass system to illustrate negative effective mass.
To demonstrate the concept of negative effective mass, we start with a two-
degree-of-freedom (2-DOF) system. As is shown in Figure 2.1, the mass m2 , which
works as a vibration absorber, is attached to the mass m1 by using a spring with a
spring constant k2 . m1 is subjected to an external force F with an amplitude F0 and
an excitation frequency . u1 and u2 stand for the displacements of m1 and m2
with amplitudes a1 and a2 , respectively.
j tua11 j t F F0 e, e (2.1) ua22
The equations of motion for this mass-in-mass system can be derived by using
Newton’s second law to be
22
m10 u 1 k 2 k 2 u 1 F (2.2) 0 m2 u 2 k 2 k 2 u 2 0
Both displacements are measured from their equilibrium positions and the initial
conditions are u10 u 2 0 u 1 0 u 2 0 0 . To solve a differential equation of the form
(nn ) ( 1)
auann1 u auauau 2 1 0 ft() (2.3) a new technique is investigated. This specific technique, classified as an integral transformation, is referred to as Laplace transform. The main concept is that it transforms an ordinary differential equation (ODE) in the time domain into an algebraic equation in terms of Laplace variable sj. Although solving algebraic equations is easier than solving differential equations, the so-obtained solution in terms of s needs to be transformed back into the time domain needs certain efforts.
The Laplace Transform is defined by:
st U( s ) L { u ( t )} ue dt (2.4) 0
It works for smooth and non-smooth function ut(). As an extension, the Laplace transforms of the first and second derivatives are obtained below:
L{()} u t sL {()} u t u (0) (2.5) L{()} u t s2 L {()} u t su (0) u (0)
Taking Laplace transform on both sides of Equation (2.2) yields
()m s2 k U k U F 1 2 1 2 2 (2.6) 2 k2 U 1 ( m 2 s k 2 ) U 2 0
By solving these equations, we obtain
23
m s2 k U s 22 F s 1 2 2 2 m1 s k 2 m 2 s k 2 k 2 (2.7) k U s 2 F s 2 2 2 2 m1 s k 2 m 2 s k 2 k 2
Next we define2 as the local resonant frequency of the vibration absorber m2 as
2 km 2 2 (2.8)
The frequency response function (FRF) H11 between the response u1 and the input
force Ft and the FRF H21 between the response u2 and the input force Ft can be obtained by replacing the s in Equation (2.7) with j as
a k m 2 H 1 2 2 11 F 2 2 2 0 k2 m 1 k 2 m 2 k 2 (2.9) ak H 22 21 F 2 2 2 0 k2 m 1 k 2 m 2 k 2
By assuming the internal object m2 is unknown to the observer and treating this 2-
DOF system as a 1-DOF system, the effective dynamical mass of the system m1 is given by
F F0 km22 mm11 22 u1 a 1 k 2 m 2 (2.10) m2 m1 22 1 2
Equation (2.9) and (2.10) show that the effective mass m1 becomes unbounded when 2 , and the response of the structure Hu11 1 0 . That is
because if we plug into (2.9), F0 k 2 a 2 , and F t k2 u 2 t m 2 u 2 t . It could be understood as that the external force F is cancelled out by the absorber’s inertia force through the spring k2 and it makes the displacement of m1 equal to zero.
24
This is the key concept for designing a mechanical vibration control device.
Besides, the effective dynamical mass is a function of . If satisfies
2 k 2 m 1 k 2 m 2 , the effective dynamical mass m1 becomes a negative
number. Under that circumstance, the pushing force from the spring on m1 , which is
k2 a 1 a 2 , is larger than the excitation force acting on . It is of interest to note
that if the mass becomes negative, the excitation force F and the acceleration of m1 is
180o out of phase.
Moreover, we obtain from the above equations that
F 0 m 2 a 1 1 (2.11) 2 FF00a1 2 m1 2 1 a2 a 1 a 2 2
If 2 , then m1 0 , Fa01 0 and Fa02 0 , and ut1 and ut2 are in phase.
This vibration pattern is called an acoustic mode. If2 k 2 m 1 k 2 m 2 , then
o m1 0 , Fa01 0 and Fa02 0, and ut1 andut2 are180 out of phase, which is
called an optical mode. If k2 m 1 k 2 m 2 , then m1 0 , Fa01 0 and
Fa02 0, and ut1 andut2 are out of phase. This is also an optical mode. The
applied external force is balanced with the inertia forces of m1 and m2 if the structure
vibrates in an acoustic mode. The applied force and the inertia force of are
balanced with the inertia force of if it is an optical mode.
2.2 Negative Effective Stiffness
To demonstrate the concept of negative effective stiffness, we start with a
two-degree-of-freedom mass-in-spring system shown in Figure 2.2. The main beam-
25 like structure is assumed to be massless and attached to the ground by using a spring with a spring constant is k1 , and m2 is attached to the structure by using another spring with a spring constant k2 . The beam is subjected to an external force F with an amplitude F0 and a frequency . u1 andu2 stand for the displacements of the main structure and m2 , respectively. Both displacements are measured from the equilibrium positions and the initial conditions are
u10 u 2 0 u 1 0 u 2 0 0 .
Figure 2.2 A 2-DOF mass-in-spring system to illustrate negative effective stiffness.
The equations of motion, frequency response functions (FRF) Hi1
between the response uti and the input harmonic force Ft are derived to be
26
00 u1 k 1 k 2 k 2 u 1 F 0 m2 u 2 k 2 k 2 u 2 0
j tua11 j t F F0 e , e ua22 m s2 k U s 22 F s 1 22 k1 k 2 m 2 s k 2 k 2 (2.12) k U s 2 F s 2 k k m s22 k k 1 2 2 2 2 2 a1 k 2 m 2 H11 F 22 0 k1 k 2 k 2 m22 k ak H 22 21 F k k k m 22 k 0 1 2 2 2 2
The effective stiffness k1 is defined by treating this 2-DOF system as a 1-DOF system and assuming the internal absorber m2 being unknown to the observer as k k k m 22 k F F0 1 2 2 2 2 k1 2 u1 a 1 k 2 m 2 k k k (2.13) kk 2 2, 2 1k 1 2 2 m 1 2 2 2 2 1 2 m2
If 2 , k1 goes to infinity and we have Hu11 1 0 . That is because at this
moment, we have F0 k 2 a 2 from the H21 expression and F t k2 u 2 t m 2 u 2 t .
It means the external force F is cancelled out by the inertia force through the spring k2 when the frequency of external load equals to the local resonant frequency of the vibration absorber m2 .
Also, the effective stiffness is a function of the excitation frequency . If
k1 k 1 k 2 2 2 is satisfied, k1 0 and the effective stiffness becomes negative. Fa01 and Fa02 are used to indicate the displacements of the massless m structure and the absorber 2 . It follows from Equations (2.12) and (2.13) that 27
F 0 k a 1 1 (2.14) 2 FF00a1 k1 1 2 a2 a 1 a 2 2
If k1 k 1 k 2 2 , Fa01 0 and Fa02 0 and hence ut1 and ut2 are in
phase. This vibration pattern is called an acoustic mode. If k1 k 1 k 2 2 2 ,
Fa01 0 and Fa02 0 and ut1 and ut2 are still in phase, which is another
acoustic mode. If 2 , Fa01 0 and Fa02 0 and ut1 and ut2 are out of
phase, which is called an optical mode.
2.3 Conventional Vibration Absorber
For an undamped or lightly damped system, when the excitation frequency is
close to a natural frequency of the system, the vibration amplitude can get extremely
high. This phenomenon is called resonance. Consequently one of the major reasons
to do vibration analysis is to predict when resonance may occur and to prevent it
from happening. By combining the concept of negative effective mass and stiffness
illustrated above, Frahm [14] proposed a dynamic vibration control device as shown
in Figure 2.3. Here, 1 ( km 1 / 1 ) is the natural frequency of the structure without
m2 , and2 ( km 2 / 2 ) is the local resonant frequency of the vibration absorber
when u1 0 . The frequency response function H11 and H21 are derived to
be:
22 m22 H 11 k k m 2 k m 2 k 2 1 2 1 2 2 2 (2.15) k H 2 21 2 2 2 k1 k 2 m 1 k 2 m 2 k 2
28
The response magnitude of m1 can be reduced to zero when 2 . To show an example, we consider a 2DOF system with
m1100 kgm , 2 0.3 m 1 , i km i i 20 rads
Figure 2.4 shows the H11 in terms of the frequency ratio r ( /2 ) . Because the vibration absorber is attached to the main structure, it alters the frequency response of the primary system and introduces another resonant frequency. This
Frahm’s vibration control device reduces the system response at a specific excitation frequency and shifts the local resonant frequencies away from the excitation frequency.
Figure 2.3 The Frahm dynamic vibration control device model.
Figure 2.4 The frequency response of the Frahm vibration absorber.
29
However, this vibration absorber system is effective only when the absorber’s local resonant frequency is very close to the excitation frequency. It performance deteriorates when the excitation frequency deviates from the absorber’s resonant frequency. Especially, if the excitation frequency approaches one of the two natural frequencies of the structure-absorber system, a large response will occur. The peaks of the blue line are higher than the peak with no absorber because the total mass of the system increases and the resonance becomes more severe.
In other words, this Frahm’s vibration absorber only works when the excitation frequency is known. It is also called an undamped vibration absorber because the auxiliary mass is attached to the system only through an elastic element.
The shortcoming of Frahm’s absorber was later eliminated by introducing a certain amount of damping into the model. Figure 2.5 shows a structure with a damped vibration absorber [15], where c1 stands for the inherent damping in the main structure m1 , and c2 is the damping of the absorber m2 . For the following case:
m100 kg , m 0.3 m , 1 2 1 ik i m i 10 Hz , c i 2 i m i k i N s / m where i is the damping ratio of a damper, Figure 2.6 shows the frequency response
function H11 for different values of i . If 1 0 and 2 0 , it is Frahm’s vibration absorber and the response to a specific excitation frequency around one of the two natural frequencies is very large. If1 0.05 and 2 0 , there is inherent damping in main structure and the peak values around the two resonant frequencies are greatly reduced. The left peak is higher than the right peak because motions of
30
o m1 and m2 are in phase around the left peak and 180 out of phase around the right peak. If1 0.05 and 2 0.05 , damping in the absorber is introduced. Although the additional damping in the absorber prevents H11 0 from happening at the designed local resonant frequency, the peak values around the resonant frequencies are smaller as compared with the green curve.
Figure 2.5 A 2-DOF damped vibration absorber model.
Figure 2.6 The frequency response of a damped vibration absorber.
31
In other words, damping improves the absorber’s performance by
significantly reducing the response magnitude. Moreover, performance deterioration
due to being different from 2 will not be sharp. It is worth noting that real
structures have only a small amount of inherent damping and hence the value of 1 is
limited in reality.
2.4 Broadband Vibration Absorbers
If the vibration absorber is able to be automatically tuned so the absorber’s
local resonant frequency2 is equal to the excitation frequency , it would result in
an active vibration absorber. If an absorber is effective only when the tuning and
damping are appropriate for a particular set of operating and environmental
conditions, it is a passive vibration absorber.
By extending the concept of a 2-DOF vibration absorber, it is possible to
design a broadband one which dissipates vibrational energy over a wide range of
frequencies. To demonstrate this idea, we start with a 6-DOF system shown in Figure
2.7. The external force is acting on the main structure with force amplitude F0 and a
frequency . The governing equations and frequency response functions are
derived to be:
32
M x C x K x F T x x1, x 2 , x 3 , x 4 , x 5 , x 6 jt F F0 e , 0, 0, 0, 0, 0
M diagmm 1, 2 , m 3 , m 4 , m 5 , m 6
c1 c 2 c 6 0 0 0 0 0 0c 0 0 0 0 2
0 0c3 0 0 0 C 0 0 0c4 0 0 0 0 0 0c 0 5 0 0 0 0 0 c6 (2.16) k1 k 2 k 6 k 2 k 3 k 4 k 5 k 6 kk0 0 0 0 22
kk330 0 0 0 K kk440 0 0 0 kk0 0 0 0 55 kk660 0 0 0
2 1 H M j C K
where[]M is the mass matrix of this 6-DOF system, []C is the damping matrix, and
[]K is the stiffness matrix.
Figure 2.7 A 6-DOF system.
If m1 100 kg , the other mii 2,3, 6 are the same, kmii 20 and
i 0.05i 1,2, 6 , the frequency response function H11 under different mi
33 values is shown in Figure 2.8. For every mi , there are only two natural frequencies at the peak locations because the five absorbers have the same mass, spring and damper so they behave like just one absorber. Because of this, the response is the same as a 2-DOF vibration control system.
Figure 2.8 The frequency response of a 6-DOF system with
m1100 kg , m 2 m 6 , ki m i 20 , i 0.05
When the absorber mass increases, the two peaks are more apart, and the frequency band of small response between the two peaks is widened. The left peak is higher than the right peak because the main structure and absorbers move in an acoustic mode around the left peak, and they move in an optical mode around the right peak.
2 If m1 100 kg , mii 2 2,3, 6 , kmii20 1.0 1.25 0.5 1.0 0.75
1.5, and i 0.05i 1,2, 6 , the frequency response function H11 is plotted as the green line in Figure 2.9. For a single-mass passive absorber, it only controls the 34 vibration at one frequency for the structure. When the local resonant frequencies
i(/) km i i of the five absorbers are different, this system is able to reduce the response over a frequency range determined by i (i 2,...,6) . Such a system with different vibration absorbers behaves like a broadband vibration absorber.
Figure 2.9 The frequency response of a 6-DOF system with
2 m1100 kg , m 2 m 6 2, kii m 20 1.0,1.25,0.5,1.0,0.75,1.5
2.5 Dispersive Material
2.5.1 Non-dispersive Wave
To demonstrate the concept of dispersion, we consider a classic string under a tension force as shown in Figure 2.10. Under small-amplitude vibration, it is valid to assume that the string only vibrates transversely (i.e., no longitudinal displacement) and remains constant throughout the motion. We also assume that the system extends to infinity on both sides so no boundary conditions need to be considered, no external force is acting on the string, and weight is negligible as compared to the tension force.
35
Figure 2.10 The free-body diagram of a differential string element under tension.
Consider a differential element between x and x dx . Since there is no motion in the horizontal direction, it follows from Newton’s second law that
mdxu sin sin (2.17) where mA ( ) is the mass per unit length, denotes the mass density, A is the
22 string’s cross-sectional area, and u ( u / t ) is the acceleration. Under small- amplitude vibration, we have
sin tan u ', sin tan u ' u dx (2.18)
Substituting Equation (2.18) into Equation (2.17) yields
mu u u c22 u , c (2.19) 00m Equation (2.19) is the classical one-dimensional wave equation. The displacement u is a function of both the time t and the spatial coordinate x . c0 represents the wave velocity, and it depends on the setup of the system. A possible solution to Equation
(2.19) is
i kx t u Ae (2.20) where k is the wave number, is the frequency of the wave, and the phase velocity c 36
is given by ck / . Substituting Equation (2.20) into (2.19) yields
2 2 2 ck0 (2.21)
which is the dispersion equation for a propagating wave with a phase velocity
c / k c0 . Because the phase velocity is only a function of and m and is
independent of the frequency , harmonic waves of different frequencies will
propagate at the same speed. Hence, a wave packet consisting of several harmonic
waves of different frequencies can propagate forward without its envelope being
distorted. Such waves are called non-dispersive waves.
2.5.2 Dispersive Wave
For a single-frequency harmonic wave, its phase velocity may change with its
frequency. If a wave packet consists of single-frequency waves with different phase
velocities, the wave packet’s envelope will disperse when the packet propagates
forward, and this phenomenon is called wave dispersion [16]. For a beam under a
pre-tension , its transverse displacement u(,) x t is governed by
EI u c2 u uiv , c , (2.22), 00 m
where E is Young’s modulus, and I is the area moment of the beam’s cross-section.
Substituting Equation (2.20) into Equation (2.22) yields
2c 2 k 2 k 4 c c 1 k 2 (2.23) 00 k
This dispersion relation has an application in piano tuning. Equation (2.23) indicates
that the phase velocity c changes with the wave number k . Because the wavelength
is related to k as 2/k and to as 2/ c , the phase velocity also
37
changes with and in the following forms:
2 2c 2 2 c c1 4 / , c 0 1 1 4 / c (2.24) 002
In other words, a single-frequency wave’s phase velocity changes with its frequency,
wavelength, and wavenumber, and a wave packet consisting of such harmonic waves
will be dispersive.
If a wave number k is a complex number, i.e., ki , then the solution u
will become
x i x t u Ae e (2.25)
Because expi 1, if xn , we have
n u Aex 1 e i t (2.26)
It reveals that the amplitude exponentially decays along the beam direction, which
is the so-called evanescent wave.
The relations among the frequency, wavelength and wavenumber are
presented as dispersion equations. Dispersion analysis is used to tell how a harmonic
wave of a certain frequency propagates along a certain direction of an infinite
material. Homogenous and continuous elastic structures are often of non-dispersive
media.
2.6 Dispersive Metamaterial Bar
Due to differences in mechanical properties of the materials constructing a
built-up structure, acoustic metamaterials analyzed in this section become dispersive
media. The following analysis indicates that stopbands exist in their dispersion curves
38
and waves with frequencies in the stopbands cannot propagate through the
structure. If an acoustic metamaterial is designed so that waves within a wide
frequency range cannot propagate forward, it is called a broad-band vibration
absorber.
2.6.1 Governing Equation and Dispersion Relation
In general, there are two methods to obtain the dispersion equation of a
mechanical model. The first method starts with the differential equation just as the
case we presented above. By substituting a potential harmonic wave solution like
Equation (2.20), we can obtain the relation between the frequency and the
wavenumber k . The second method is used to investigate the stopbands of mass-
spring systems, which was first mentioned by Jensen in 2003 [17]. The equation of
motion for each discrete degree of freedom was presented and periodic boundary
conditions were assumed. The problem was later simplified to be an eigenvalue
problem and the analysis was conducted for an irreducible Brillouin zone. It is noted
that, the first method is widely used for continuum materials while the second
method is more useful for periodic structures of two materials.
To show an example, we consider wave propagation in the metamaterial bar
shown in Figure 2.11. It consists of an isotropic bar and many discrete vibration
absorbers. For such a metamaterial with a continuum bar and discrete absorbers, the
two methods mentioned above are not effective and/or accurate for dispersion
analysis. If the bar is infinitely long, the dispersion equation can be derived without
considering boundary conditions. Then only one unit cell needs to be considered for 39 dispersion analysis.
Figure 2.11 Wave propagation in a metamaterial bar: (a) a metamaterial bar configuration, and (b) the free- body diagram (FBD) of a unit cell.
Governing equations can be derived by applying the following extended
Hamilton principle [18] on the free body diagram (FBD) shown in Figure 2.11b:
t2 T W dt 0 (2.27) t nc 1 where T is the kinetic energy, is the elastic energy and Wnc is the non- conservative work done by external loads. Their variations are given as
40
tt11 Tdt u22 dAdx mv dt 00 V 22
t L 2 L Au udx mv v dt 0 2
L L t tt 2 2 L Au udx mv v dt L Au u dx mv v 0 00 2 2
t L 2 L Au udx mv v dt 0 2
1 2 dAdx k v u0 (2.28) V 2 L 2 '' L EAu u dx k v u00 v u 2 L ' ' ' '2 '' EA u1 u 1 u u 0 u u 0 u 1 u 1 L EAu udx 00 2
k v u00 v u '' Wnc EAu1 u 1 EAu 1 u 1
Here u denotes the displacement of the main structure, v is the displacement of the vibration absorber, u u/ x , u u/ t , andu is the variation ofut . Since the
uu vibration absorber creates a concentrated force at x 0 , 00 and we have to integrate for two parts when we calculate the constant part of . Moreover, , A and E are the mass density, cross-sectional area and Young’s modulus of the bar, L is the length of an unit cell, m is the mass of the absorber, and k is the stiffness of the spring.
Plugging what we got in Equation (2.27) into Equation (2.26), we obtain
L t EAu Au udx mv k v u0 v 2 0 L dt 0 2 k v u EA u u u 00 00 (2.29) L t EAu Au k v u EA u u x udx 2 00 L dt 0 2 mv k v u v 0 where x is the Dirac delta function used to simplify the form and describe the
41 distribution. Hence, the governing equations for any material point of this bar segment and the absorber are:
Au EAu k v u EA u u x (2.30 a) 00
mv k u0 v (2.30 b)
To eliminate the Dirac delta function in Equation (2.29a), integration over the length of unit cell is applied to give
L/2 EA u u k v u Audx (2.31) 1 1 0 L/2
If an elastic wave propagating through this metamaterial bar is assumed to have:
u x, t pej x t , v x , t qe jt (2.32) where p and q are the wave amplitudes of the bar and absorber, respectively, is the wavenumber, and is the vibration frequency. Then the dispersion equation is given by
k 4 2 2 2 2 2 2 2 (2.33) 4 1 2 4 1 2 0 sin m 1 where L /2, 1 km 1/ 1 , k1 EA/ L , m1 AL , and2 km/ . If k 0 ,
4 2 2 2 2 0 , Equation (2.32) is simplified to40 1 and hence 2 1 . If
2 2 2 2 2 m 0, 2 , Equation (2.32) is simplified to 2 40 1 2 , and hence
. These two results indicate that, without the absorber, the bar behaves like a continuum bar and the wave velocity (/) is a constant.
To show a numerical example, we consider a unit cylinder cell with a vibration absorber and the following parameter values:
42
L0.01 m , A 0.012 0.005 2 m 2 , E70 GPa , 0.33, 2800 kg m3 , EA k m AL, k , m 0.3 m , k 0.1 k , 45944 Hz 1 1Lm 1 1 2
L is the unit cell length, A is the cross-sectional area, and E , and are the
Young’s modulus, Poisson’s ratio and mass density of the bar. m and k are the mass and spring stiffness of the absorber.
Figure 2.12 Dispersion curves of a metamaterial bar: (a) curves, and (b) curves.
43
Figure 2.12 compares the dispersion curves of a uniform bar and a metamaterial bar. Figure 2.12a shows that an elastic wave with a specific wavelength
in a uniform isotropic bar can propagate at only one frequency and there is no
wave dispersion within the media because the phase velocity ( cEp // ) is the same for any . While an elastic wave with a specific wavelength in a metamaterial bar can propagate at two different frequencies. One is lower than the wave in the uniform isotropic bar and the other is higher than that. The low- frequency motion has u and v in phase and is called an acoustic mode, and the high-
o frequency one has u and v 180 out of phase and is called an optical mode. The phase velocity cp ( / 2)of the acoustic mode is lower and the one of the optical mode is higher. It shows that the propagation speed depends on the wave frequency and hence the metamaterial bar is dispersive. Besides, the waves of frequencies within the range22 1.14 cannot propagate because they are evanescent waves
ax jt having ja with a 0 and hence u(x,t) pe e . This frequency range is called the stopband and it can be used to design vibration absorbers.
If the parameters of the vibration absorber are slightly changed, the characteristics of dispersion relation will also change. As shown in Figure 2.13a, if the absorber mass is reduced to mm 0.1 1 , the two branches of the dispersion curves become closer to the asymptote (i.e., the dispersion curve of the uniform isotropic bar) and the width of the stopband is significantly reduced. If the absorber spring constant is reduced to kk 0.01 1 , Figure 2.13b shows that the slope of the asymptote becomes larger, which means the metamaterial bar tends to absorb long-wavelength
44
elastic waves. On the other hand, decrease of spring stiffness does not have much
influence on the stopband width. Hence, we conclude that the stopband width is
mainly controlled by the absorber mass m , and the wavelength of the absorbed
wave is mainly controlled by the absorber spring stiffness k .
Figure 2.13 Dispersion curves ( ) of a metamaterial bar: (a) m 0.1 AL , and (b) kk 0.01 1 .
2.6.2 Finite Element Modeling
In dispersion analysis, because the metamaterial bar’s length is assumed to
be infinite and the wave propagation is free of loading, no boundary conditions and 45 external loads need to be considered. In order to investigate the performance of a finite metamaterial bar under specific loads and boundary conditions, a finite- element model is introduced. Here we analyze the metamaterial bar in Figure 2.11a with the following properties:
E72.4 GPa , 0.33, 2780 kg m3 , L LLA4 m, 0.01 m, 0.012 0.005 2 m 2 , 400 c E 2p , cL , 20 (2.34) p 2 2 mi0.3 AL , k i m i m i , r 0.8,1.12,1.3 r
cii 2 mii k , i 0 or 0.05, i 1, , 4 00 where E , and are Young’s modulus, Poisson’s ratio and mass density of the bar material. L is the total length of the bar and it is divided into 400 elements with a unit cell length of L , and A is the area of the cross section. is the excitation frequency. m , k and c are the mass, attached spring stiffness and damping coefficient of each vibration absorber.
The bar segment of a unit cell is modeled using a two node finite bar element. If the density and cross-sectional area are constants, the elemental mass and stiffness matrices are given by [19]
2 1 1 1 ii AL EA mk , (2.35) 6 1 2 L 1 1
Figure 2.14 shows the steady-state operational deflection shapes (ODSs) when no damping is considered. Blue solid lines are the displacements of bar, and the red solid lines represent the displacements of discrete absorbers. If the incoming wave
46 frequency is lower than the resonant frequency of absorbers, Figure 2.14(a) shows that the bar and absorbers move in an acoustic mode with a short wavelength. On the other hand, if the wave frequency is higher than the absorbers’ resonant frequency and is not in the stopband, Figure 2.14(c) shows that the bar and absorbers move in an optical mode with a long wavelength. However, the displacements in Figure 2.14(a) are larger than the ones in Figure 2.14(c) because the bar and absorbers move in phase. If the incoming wave’s frequency is within the stopband, Figure 2.14(b) shows that the bar and absorbers are activated in an optical mode but the wave is absorbed by the first few absorbers. Figure 2.15 shows the steady-state ODSs when damping ratios i 0.05 are used. Comparing Figures
2.15(a)&(c) with Figures 2.14(a)&(c), we notice that the incoming wave could be damped out even if its frequency is not within the stopband. However, absorbers work less efficiently within the stopband because it takes more absorbers to suppress the wave.
47
Figure 2.14 Steady-state operational deflection shapes (ODSs) with no damping: (a) r=0.8, (b) r=1.13, and (c) r=1.2.
48
Figure 2.15 Steady-state operational deflection shapes (ODSs) with dampings : (a) r=0.8, (b) r=1.3, and i 0.05 (c) r=1.2.
49
Chapter 3 Acoustic Metamaterial Beams with Local Vibration
Absorbers
3.1 Governing Equations and Dispersion Relations
Compared to bars, beams are more commonly-used engineering structures
because they are designed to carry transverse loads in addition to axial loads. If the
transverse vibration of a beam is exceedingly large, the internal compression and
extension stress may cause failure to the structure. Here a new class of metamaterial
beams is proposed to absorb broadband transverse elastic waves.
Figure 3.1a shows a prototype of a metamaterial beam. Vibration absorbers
consisting of small mass-spring-damper subsystems are integrated with the beam at
discrete positions. Since a transverse wave propagates in a beam through internal
shear forces and bending moments, both translational and rotational vibration
absorbers need to be considered in order to prevent waves from propagating. As
shown in Figure 3.1b [20], a translational-type vibration absorber with a mass m and
a spring k , and a rotational vibration absorber with a mass mˆ , a moment of inertia
J and a torsional spring are attached to the beam. The two beam ends are simply
supported and the load is usually along the transverse direction.
50
Figure 3.1 (a) a prototype of a metamaterial beam, (b) model of a simply supported beam with translational and rotational absorber, and (c) free-body diagram of a unit cell.
Similar to the metamaterial bar, by applying the extended Hamilton principle
[18] on the free-body diagram (FBD) shown in Figure 3.1(c), we get:
t2 T W dt 0 (3.1) t nc 1
whereT is the kinetic energy, is the elastic energy and Wnc is the non- conservative work done by external loads. The three separate terms could be derived as
51
L /2 T Aw w Iw'' w dx mv v J mwˆ w L /2 00
L /2 ''ˆ ' ' Aw Iw wdx mv v J mw0 w 0 Iw 1 w 1 Iw 1 w 1 L /2
L /2 EIwwdx'' '' kv w v w L /2 0 0 0 0 (3.2) L /2 EIwiv wdx k v w vw L /2 0 0 0 0
'' '' '' '' ''' ''' ''' '''
EIw w w w wwwwwwww 110 0 0 01111 0 0 0 011
'' ' ''' '' ''' ' Wnc EIw1 1 Iw 1 EIw 1 w 1 EIw 1 1 EIw 1 Iw 1 w 1
, E , A , I are the beam’s mass density, Young’s modulus, cross-sectional area and area moment. mˆ , J and are the mass, rotary inertia and torsional spring constant of the rotational absorber. m and k are the mass and spring constant of the translational absorber. L is the unit cell length, w denotes the beam displacement
22 along the z direction, w w/ x is the beam’s rotation angle, and w w/ t . v is the displacement of the translational absorber, and is the rotation angle of the rotational absorber. Because the bending moment and shear force are not
EIw'' EIw '' continuous at where the translational absorber is attached, we have 00
EIw''' EIw ''' and 00. The two different vibration absorbers create a concentrated bending moment and a concentrated shear force at x 0 , so we have to integrate by part in term. Plugging Equation (3.2) into Equation (3.1) yields
L/2 ''iv ''' ''' '' '' 0 Aw Iw EIw wdx EI w w w w w w L/2 00 0 0 0 0 0 0
kvwvw 0 0 0 0 mvvJ mwwˆ 0 0 L/2 (3.3) ''iv ''' ''' Aw Iw EIw EI w w k w v mwˆ x wdx L/2 00 00 EI w'' w '' mv k v w v J 00 0 0 0 0
52 where is the Dirac delta function to simplify the form and describe the distribution.
By setting the coefficients of w ,0 ,v and to zero, we obtain the following governing equations:
Aw Iw'' EIwiv EI w ''' w ''' k w v mwˆ x 0 (3.4 a) 00 00
EI w'' w '' 0 00 0 (3.4 b)
mv k v w0 0 (3.4 c)
J 0 0 (3.4 d)
It follows from the Bloch-Floquet theory that only one unit cell needs to be considered when we analyze wave propagation characteristics in an infinite periodic structure. The elastic wave propagation in Figure 3.1( c) could be assumed to have:
wxtqe , j x t , vtpe j t , t e j t (3.5) where is the excitation frequency and the wave propagation frequency, is the wavenumber, and q , p and are the displacement amplitudes of the beam, translational absorber and rotational absorber, respectively. Especially, when j
, w x, t qex e j t indiciates an evanescent wave. If the beam is treated as a homogenized uniform beam, one can integrate Equation 3.4 a over the cell length L and use Equation (3.5) to get
mw0 kw 0 k w 0 v 0 2 AIL 2 sin / 2 (3.6) m mˆ , k 2 EI3 sin L / 2
53
Substituting Equation (3.5) into Equation (3.4 b), (3.4 c) and (3.6), and setting the determinant of the coefficients of the three algebraic equations containing unknowns p , q and to zero, we will obtain the dispersion relation as:
4 2k k 2 2 k 2 0 m m m J (3.7) k mJ where is the local resonant frequency of the translational and rotational absorbers.
If the physical properties and geometrical characters of the metamaterial beam segment are chosen to be:
L10 mm , b 5 mm , h 3 mm , A bh , E72.4 GPa , 0.33, 2780 kg / m3 , m0.1 AL , k m 2 (3.8) 1 mˆ 0.1 AL , J mLˆ 22 , J , 4 2400Hz 2400 2 rad / s the dispersion relation from Equation (3.7) is shown in Figure 3.2. Similar to the metamaterial bar, an elastic wave with a specific wavelength can propagate at three different frequencies in this metamaterial beam. The lower frequency corresponds to an acoustic mode, the middle frequency is the local resonant frequency of the torsional absorber, and the higher frequency corresponds to an optical mode. It shows that this torsional absorber is not efficient for absorbing waves, and this extra rotational degree of freedom does not create an extra
54
stopband because there is only one stopband. A wave with a frequency within the
stopband cannot propagate and it is an evanescent wave.
Figure 3.2 Dispersion curves of an infinite metamaterial beam unit cell.
3.2 Finite-Element Modeling
The assumed continuity for w x, t in Equation (3.5) and the averaging
integration shown in Equation (3.6) cannot explain the discontinuity for bending
moment ( EIw'' ) and shear force ( EIw''' ) at x 0 . Such a homogenized model can
only be used for waves having L so that the wave profile between two
absorbers can be approximated by a smooth function. Besides, the above dispersion
analysis demonstrates the existence of a stopband with an infinite metamaterial
beam, where loads, damping and boundary conditions are not considered. For a
metamaterial beam having separated absorbers attached to the isotropic beam
and/or transverse waves of wavelengths smaller than the cell length L,
displacement-based finite element modeling is necessary. The modeling for a unit
cell will be illustrated next.
55
3.2.1 Euler-Bernoulli Beam Element
In structural analysis, the Euler-Bernoulli beam model (EBBM) is the most
widely used theory for modeling the behavior of a beam. It is assumed the deformed
beam cross section is perpendicular to the deformed reference line. By applying the
extended Hamilton principle [18], the governing equation, boundary conditions and
primary and secondary variables can be derived as
t2 t 21 t 2 Tdt w2 dAdxdt w wdAdxdt t t V t V 1 12 1 t 2 w wdAdxdt w wt2 dAdx t V V t1 1 tL 2 mw wdxdt m dA tA 0 1 dAdx w'' z V 11 11 11 L Ew'' z 2 w '' dAdx EIw '' w '' dx I z 2 dA VA 0 L L ( EIw'') 'w ' dx EIw '' w ' 0 0 L L ()()EIw'' '' wdx EIw '' w ' EIw '' ' w 0 0 L W f wdx nc 0 t 0 2 (T W ) dt t nc 1 tL 2 mw () EIw'' '' f wdxdt t 0 1 t L (3.9) 2 EIw'' w ' ( EIw '' ) ' w dt t 1 0
There are two primary variables w and w' , and two secondary variables EIw'' and
'' ' ()EIw . Because there are two primary variables, each node of the finite element
model contains 2 DOFs and hence Hermite cubic polynomial functions can be used
to interpolate the displacement w between the two nodes of each element. Figure
3.3 shows the finite element model of an EBBM, where w is the deflection, 56
w/ x w is the rotation angle, and L is the elemental length. Next step is to derive shape (or interpolation) functions for w in terms of the local coordinates
and , where 1 at the left element end and 1at the right element end, as
23 w, t c0 c 1 c 2 c 3
w1, t w 0, t w1 11 w1, t w' 0, t L L 221 (3.10 a) w1, t w L , t w2 11 w1, t w' L , t L L 222
Solving Equation (3.10 (a)), we obtain:
w, t NTT w w N , 3 2 3 3 2 3 T 2 3 1 2 3 1 N 4 4 4 4 (3.10 b) T 11 w w1 , 1 L , w 2 , 2 L S w 22 T w w1 , 1 , w 2 , 2 where N is the shape function vector. This discretization process replaces a continuum formulation with a discrete representation. Then we solve the unknown variables at discrete nodes instead of the whole material domain. By plugging
Equation (3.10 b) into Equation (3.9) and using Gauss Integration, the elemental mass matrix and the elemental stiffness matrix for an EBBM are obtained
57
156 22LL 54 13 22 mL 22LLLL 4 13 3 m if m=constant 420 54 13LL 156 22 22 13LLLL 3 22 4 12 6LL 12 6 (3.11) 22 EI 6LLLL 4 6 2 k if EI=constant L3 12 6LL 12 6 22 6LLLL 2 6 4
Figure 3.3 Finite element modeling of an Euler-Bernoulli beam element.
3.2.2 Timoshenko Beam Element
While EBBM is extended by Timoshenko in 1992 to account for the
transverse shear strain effect and rotary inertias, which are neglected in the EBBM.
The extended theory is known as the first-order shear-deformable Timoshenko
beam model (TBM), and it provides small correction for slender beams and
significant improvement for short/thick beams.
58
Figure 3.4 Modeling of a modified Timoshenko beam element: (a) deformed cross-section, and (b) finite element modeling.
Figure 3.4 (a) shows the deformed cross-section of a Timoshenko beam. If we take into account the extra kinetic energy due to rotary inertias and the extra elastic energy due to shear deformation, Equation (3.9) will become:
59
t t L 22Tdt mw w J dxdt m dA , J z2 dA t t 0 00 A A 11 dAdx z , w V 11 11 13 13 11xx 13 E' z 2 ' G w ' w ' dAdx V L EI' ' k GA w ' w ' dx I z 2 dA 0 s A L ()() EI'''' k GA w w k GA w' dx 0 ss L '' EI ks GA w w 0 L W f wdx nc 0 tL 02 mw k GA ( w' ) ' f w J EI '' k GA w ' dxdt t 0 ss 0 1 t L 2 '' EI ks GA w w dt t1 0
(3.12) where represents the total cross-sectional rotation angle, and w is
the shear rotation angle. ks is the shear correction factor, and ks 5 / 6 for a rectangular cross section.
If linear interpolation is used for both w and , it will lead to the shear locking problem in thin beams because it only requires w and to be continuous
' but it cannot guarantee the shear rotation w to be continuous when there is no concentrated shear loading at a node. There are several methods to reduce or eliminate the shear locking problem. The first one is the reduced integration method
[21], where the stiffness matrix is obtained by using fewer Gauss points than those needed for exact integration. Considering the stiffness matrix as:
k k k bs (3.13)
60
k k where b is the bending stiffness and s denotes the shear stiffness. The bending stiffness can be exactly evaluated by one-point Gauss quadrature. On the other hand, the shear stiffness matrix contains second-order terms and two-point Gauss quadrature is required. If we use one-point quadrature for the second-order terms, the shear stiffness is under integrated and it results in a solution closer to the exact one. The second method is called the mixed integration method. The main idea is to split the strain energy into individual parts and apply different integration rules for different parts. It follows from Figure 3.4(a) that w is the rotation of the reference line, represents the shear rotation, and ()w is the rotation of the deformed cross-section. Hence, an alternative form of Equation (3.12) is given by
L T mw w J w'' w dx 0 0 L mw w J w''' w J w dx 0 00 L L '' ' ' ' mw w J0 w w J 0 w dx J 0 w w 0 0 E w'' ' z 2 w '' ' G dAdx V L EI w'' ' w '' ' k GA dx 0 s L EI w'' ' w '' EI w '' ' ' k GA dx 0 s L EI w''' '' w ' EI w ''' '' k GA dx 0 s L EI w'' ' w ' EI w '' ' 0 L EI w'''' ''' w EI w ''' '' k GA dx 0 s L EI w'' ' w ' EI w ''' '' w EI w '' ' 0 '' ' '''' ''' mw J0 w EI w f w tL2 0 dxdt (3.14) t 0 1 J w' EI w ''' '' k GA 0 s
t L 2 ''' '' ' '' ' ' '' ' + EI w J0 w w EI w w EI w dt t1 0
61
Because the primary variables are w , wand , there are 3 DOFs at each node. The weak form can be derived by using cubic Hermite polynomials to interpolate w and linear polynomials to interpolate . Figure 3.4(b) is the finite element model of a modified TBM. The interpolating shape functions for w and are
x2 x 3 x 2 x 3 x 2 x 3 x 2 x 3 w x , t 1 3 2 w x 2 3 2 w 2 3 1 2 1 2 3 2 2 2 LLLLLLLL
N1 x w 1 N 2 x 1 N 3 x w 2 N 4 x 2 xx x , t 1 1 2 N 5 x 1 N 6 x 2 LL
w NNNN1 200 3 4 TT d N d, d w1 , 1 , 1 , w 2 , 2 , 2 00N5600N
u1 z0 x 1 w z 0 x 1 T T u N d Z N d B d u3 0 1 1 0 0 1 1 0
T xx11 T NNNN1 200 3 4 NN 1 0 1 0 0 0NN56 0 0
NNNNNN1x 2 x 5 3 x 4 x 6 NNNN1 200 3 4 2 2 2 2 z00 x x w z x x T T 11 ˆ N d Z N d B d 13 0 1 0 1 0 1 0 1 22 (3.15) T xx T N NNNNN ˆ 1xx 2 xx 5 x 3 xx 4 xx 6 x NN 01 0 0NN56 0 0
11 E 0 Q 13 0 G
If we plug Equation (3.15) into Equation (3.14) and use Gauss Integration, we can derive the elemental mass and stiffness matrices as
62
T u u u u dAdx uT u dAdx VV1 1 3 3
NL ii T T d B B d dAdx LA i1 i
NL T ii i d m d dAdx i1 TT mi B B dAdx N ZT Z N dAdx LALA ii dAdx T dAdx VV11 11 13 13
NL T ii Q dAdx V i1 i
NL T dii BT Q B d dAdx V i1 i
NL T di k i d i dAdx V i1 i TTT (3.16) ki B Q B dAdx Nˆˆ Z Q Z N dAdx VV ii
3.2.3 Infinite Beams with Uniform Absorbers
As shown in Figure 3.5, we discretize a beam segment into two beam
elements. Since shear deformation and rotary inertia are pretty significant for wave
propagation in a beam, we need to adopt the Timoshenko beam theory for analysis.
The weak form in Equation (3.2) can be modified as follows
L/2 T Aw w I w'' w dx mv v J mwˆ w L/2 00 L/2 Aw w I w'' ' w I w ' dx mv v L/2 L/2 ' J mwˆ 00 w I w w L/2 L/2 EI w'' ' w '' ' k GA dx k v w vw L/2 s 0 0 (3.17)
0 0 0 0 L/2 EI w'''' ''' w EI w ''' '' k GA dx L/2 s
k v w0 v w 0 0 0 0 0 L/2 EIw '' ' wEIw ' '' ' EIw ''' '' w L/2 63
Here, v is the displacement of the translational absorber, and is the rotation of the rotational absorber. Therefore, there are 3 DOF at each node for a beam element, and two more DOFs for the translational and rotational absorbers at nodes 1 and 3 .
So we apply cubic polynomials for w , and linear polynomials for and .
Figure 3.5 Finite element modeling of a metamaterial beam.
For the periodic structure, only one unit cell needs to be considered if we want to study the dynamic characteristics when specific elastic waves propagate along the structure. If the damping is neglected at first, the equation of motion
(EOM) derived from the extended Hamilton principle is:
M q K q 0 (3.18) '' q w1,,,,,,,,, w 1 1 w 4 w 4 4 v 1 v 3 1 3
According to the Bloch-Floquet theory, we have:
'' jL w1,,,,,,,, w 1 1 v 1 1 w 3 w 3 3 v 3 3 e (3.19) '' jL w2,,,, w 2 2 w 4 w 4 4 e
Equation (3.19) reduces the size of the stiffness and mass matrices into 88 . Here we consider an infinite beam with uniform absorbers to illustrate how the translational
64 and rotational absorbers work when waves propagate through the beam segment.
We consider the following numerical case:
E72.4 GPa , 0.33, 2780 kg m3 , L10 mm , b 5 mm , h 3 mm , A bh , m0.1 AL , k m 2 , (3.20) 1 mˆ 0.1 AL , J mLˆ 22 , J , 4 2400Hz 2400 2 rad s
E , and are Young’s modulus, Poisson’s ratio, and mass density of the Aluminum beam. L ,b , h and A are the cell length, width, thickness and cross-sectional area of a thin beam segment. m and k are the translational absorber’s mass and spring constant. mˆ , J , are the rotational absorber’s mass, rotary inertia and spring constant. is the excitation frequency. Figure 3.6 (a) shows the dispersion curves when the translational absorbers’ masses have different values of m 0.1,0.3,0.5 ALand mˆ 0.1 AL . The stopband increases with the translational absorber’s mass m . On the other hand, Figure 3.6 (b) shows the dispersion curves when the rotational absorbers’ mass mˆ 0.1,0.3,0.5 ALand m 0.1 AL . When mˆ increases, the plot of first solution from Equation (3.7), which is / J , is curved. Hence, the lower bound of the stopband is increased. The second solution tends to increase by lowering the lower bound of the stopband, while the higher bound is also reduced at the same time. In general, the stopband decreases when mˆ increases. Hence a rotational absorber is less effective than a translational absorber, and hence it will not be considered in the rest of this paper. The most important phenomenon revealed by the figure is that an elastic wave can propagate at two 65
different frequencies in a metamaterial beam and it leads to a stopband to the right
side of . The lower frequency motion is called an acoustic mode and the higher
frequency motion is called an optical mode.
Figure 3.6 Dispersion curves with different absorbers: (a) different translational absorbers with m0.1,0.3,0.5 AL , mˆ 0.1 AL , and (b) different rotational absorbers with mˆ 0.1,0.3,0.5 AL , m 0.1 AL .
3.3 Numerical Analysis
3.3.1 Supported Beam with Uniform Absorbers
Next, we need to consider a metamaterial beam with different boundary
conditions and different arrangements for translational absorbers. Here is a
metamaterial beam with 100 unit cells, and it is simply supported on both ends.
66
E72.4 GPa , 0.33, 2780 kg m3 , L100 cm , L L 100 10 mm , b 5 mm , h 3 mm , A bh , m0.1 AL , k m 2 , c 2 m k , 0.01, (3.21) 2400Hz 2400 2 rad s
F0 AL at x 5 mm
L is the total length of the beam, L is the unit length of a beam segment. m , k and c are translational absorber’s mass, spring constant, and damping coefficient, respectively. The harmonic excitation F0 is placed at the second node. Rotational absorbers are not considered here.
Modal analysis gives us the dynamic properties of this structure under a single-frequency excitation. It uses the overall mass and stiffness of a structure to find the various frequencies at which the structure will naturally resonate. If absorbers are not attached to the beam, the first five natural frequencies of the beam are 6.9421, 27.767, 62.471, 111.05, and 173.49 Hz . If absorbers are added, mass of the metamaterial beam is increased by 10% and the first five natural frequencies are reduced to 6.619, 26.475, 59.562, 105.84, and 165.38 . The corresponding mode shapes are also obtained, which tell us what are the operational deflection shapes of the structure under single-frequency excitations at resonant frequencies.
Figure 3.7 shows the frequency response function (FRF) of the middle point in this metamaterial beam. The black dashed line is obtained by setting k , which means the absorber’s mass is attached to the beam as a dead mass. The beam responds as a normal Timoshenko beam, and the peaks correspond to natural 67 frequencies. The blue line represents the response when 0.0001 is used. The absorbers without damping create a stopband right above , as predicted in Figure
3.6(a). The peak locations are changed, and many small peaks exist around the stopband, which are caused by the local resonant vibrations of absorbers. If the damping ratio is increased to 0.01, the red line shows the FRF. Peaks around the stopband are significantly reduced, and the stopband bandwidth is increased.
Therefore, damping is essential for this kind of vibration absorbers to work.
Figure 3.7 Frequency response function (FRF) H101,2 of the metamaterial beam with m 0.1 AL and mˆ 0 .
68
Figure 3.8 Steady-state ODSs of a metamaterial beam with r : (a) r=0.8, (b) r=1.02, and (c) r=1.2.
In Figure 3.6 (a), if the absorber mass is chosen to be m 0.1 AL , there exists a stopband within1 / 1.057 . It shows the frequency range of elastic waves that could be absorbed by the metamaterial beam. Figure 3.8 shows the steady-state ODSs when the propagating wave’s frequency is related the local resonant frequency as with r 0.8, 1.02, 1.2 , respectively. The red lines represent the deformed beam, and the black vertical lines represent the extended lengths of the springs. Figure 3.8(b) shows that when the elastic wave is within the stopband, the first few absorbers are capable of working against the wave propagation to stop it. Figure 3.8(a) shows that when an elastic wave’s frequency is lower than the stopband, the beam and absorbers move in an acoustic mode, and
69
the wave cannot be absorbed. On the other hand, Figure 3.8 (c) shows that when an
elastic wave’s frequency is higher than the stopband, the beam and absorbers move
in an optical mode (i.e., 180 out of phase), and the wave also cannot be absorbed.
3.3.2 Supported Beam with Varying Absorbers
As is illustrated above, an absorber only works for elastic waves within a
specific stopband. At the same time, only a few absorbers are needed to absorb an
elastic wave with a specific wavelength. This concept could be extended to design a
metamaterial beam that is able to suppress wave propagation for a very wide
stopband. This idea can be realized by arranging a series of absorbers having
different local resonant frequencies over a beam.
First, we consider the following metamaterial beam with linear varying
absorbers:
L100 cm , L L 10010 mm , b 5 mm , h 3 mm , A bh m0.1 AL , c 2 m k , 0.01 2 (3.22) 2 2400Hz , kii m m ri
ri 0.5 0.005 i 1 , i 1,...,201
Figure 3.9(a) shows the FRF at xL /2 when absorbers with linearly decreased local
resonant frequencies are used. The created stopband is wider than that in Figure 3.7,
but the vibration amplitudes within the stopband are only slightly decreased. Hence,
the effectiveness is not guaranteed. Figure 3.9 (b) shows the steady-state ODS under
a harmonic excitation with 2400Hz at Node 2. It shows that the beam and
70 absorbers move in phrase before the elastic wave reaches the absorber with i at xL /2. Because only a few absorbers work against the wave, the wave causes great displacements in that local area. Note that the big displacements may cause damage to the absorbers.
Figure 3.9 (a) frequency response function at the middle point with r=0.5+0.005(i-1), i=1,…,201 , and (b) steady- state ODS under a harmonic excitation at Node 2.
If the resonant frequencies of absorbers are linearly increased along the x axis as:
ri 1.5 0.005 i 1 , i 1,...,201 (3.23) the FRF and steady-state ODS are shown as in Figure 3.10 . First, it shows that the beam and absorbers move out of phase before the elastic wave reaches the
71 absorber with i at xL /2. Besides, the displacements of absorbers are more continuous than those in Figure 3.9(b). That is because the local resonant frequencies of absorbers at xL /2 are lower than the wave’s frequency and hence those absorbers are activated in an optical mode to generate inertia forces to work against the wave. On the other hand, the arrangement in Equation (3.22) only uses the stopband effect from the few absorbers around xL /2 to stop the wave propagation, and the absorbers at move in phase with the beam so they do not work against the wave at all.
Figure 3.10 (a) frequency response function at the middle point with r=1.5-0.005(i-1), i=1,…,201, and (b) steady-state ODS under a harmonic excitation at Node 2.
Therefore, if the frequency and direction of the incoming wave are known, it is more efficient to take advantage of the optical mode to generate inertia forces to
72 work against the wave. Hence, the design approach is to have absorbers with resonant frequencies lower than the incoming wave’s frequency to meet with the incoming wave first and lead the structure into an optical mode before the wave meet the designated absorbers. However, if the frequency and/or direction of the incoming wave are unknown, absorbers need to be divided into a few subgroups and the resonant frequencies of absorbers in each subgroup are able to cover a wide range of frequencies, as shown next.
We consider a metamaterial beam with four subgroups of absorbers to stop an elastic waves consisting of four harmonics.
L100 cm , L L 100 10 mm , b 5 mm , h 3 mm , A bh m0.1 AL , c 2 m k , 0.01 2 2 2400Hz , kii m m r i (3.24) ri 1.02 i 1,2,...51 , 2.04 i 52...101 , 3.06ii 102...151 ,4.08 152...201 4 t F0 AL sin n1 nn
The nth subgroup is designed to suppress an elastic wave with a frequency
2400 /n Hz . Besides, each subgroup’s resonant frequency is designed to be slightly lower than the corresponding wave frequency so that the absorbers can take full advantage of the stopband effect. Figures 3.11(a)-(d) show the steady-state ODSs when a single-frequency wave propagating through the beam, and Figure 3.11(e) shows the ODS when the four-harmonic wave propagates through the beam. They
73 show that the wave excited by F AL / n sin t / n is stopped by the first few absorbers of the designed subgroup of absorbers. Since each of the wave frequencies /2, /3and /4is higher than the designated absorber subgroup’s resonant frequency, only the first few absorbers of the designated subgroup work in an optical mode to stop the wave propagation, while the other absorbers to the left of the designated subgroup work in an acoustic mode.
Figure 3.11 Steady-state ODSs of a metamaterial beam with 4 subgroups of absorbers r=1.02 (i=1,…51), r=2.04
(i=52,…101), r=3.06 (i=102,…151), and r=4.08 (r=152,…201): (a) F AL sin t , (b)
F AL 2 sin t 2 , (c) F AL 3 sin t 3 , (d) F AL 4 sin t 4 , and (e)
74
4 F AL nsin t n . n1
If the absorbers distribution is reversed as:
r4.08 i 1,2,...51 , 3.06 i 52...101 , i (3.25) 2.04ii 102...151 ,1.02 152...201
Figure 3.12 shows that each harmonic wave is still mainly suppressed by the first few absorbers of the designated absorber subgroup, but the absorbers to the left of the designated subgroup move in an optical mode and create certain inertia forces to help the suppression of the incoming wave. That explains why the vibration amplitude is smaller than that in Figure 3.11.
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Figure 3.12 Steady-state ODSs of a metamaterial beam with 4 subgroups of absorbers r=4.08 (i=1,…51), r=3.06
(i=52,…101), r=2.04 (i=102,…151), and r=1.02 (r=152,…201): (a) F AL sin t , (b)
F AL 2 sin t 2 , (c) F AL 3 sin t 3 , (d) F AL 4 sin t 4 , and (e)
4 F AL nsin t n . n1
It is obvious that different arrangements of absorbers will lead to different propagation/absorption results. Distribution in Equations (3.23) and (3.25) are better
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than those in Equations (3.22) and (3.24) because the absorbers in early contact with
the wave are designed to work in an optical mode to help the wave absorption.
3.4 Actual Working Mechanism
It is mentioned above that shear forces and bending moments are not
continuous through the whole beam structure, and the working principal of a
vibration absorber is that it uses the inertia force to balance out the internal shear
force when waves propagate. To reveal the working mechanism, we consider a beam
with a few absorbers attached at x 48.5, 49.5, 50.5, 51.5, 52.5 cm and
m0.3 AL , c 2 m k , 0.01, 2 (3.26) 2 2400Hz , kii m m ri
Figure 3.13(a) shows the steady-state ODS when r 0.9 . When the resonant
frequencies of absorbers are higher than the incoming wave frequency, absorbers
cannot create inertia forces to balance out the internal shear force because
absorbers move in phrase with the beam. Figures 3.13(b) and (c) show that an
absorber with a resonant frequency lower than the incoming wave frequency (
r 1.02 ) can create an inertia force against the internal shear force because the
absorber move out of phrase with the beam. The internal shear force is calculated by
using EI w''' '' I w ' and shown in Figure 3.13(c). It is noted that there is
a sudden change once the wave meets the absorber at x 48.5 cm , after that, the
shear force is balanced out within the five absorbers. Hence, the stopband effect is
77 actually caused by the inertia force created by the absorbers. On the other hand, if the incoming wave frequency is much higher than absorber’s resonant frequency ( r 1.2 ) but not in the stopband, Figures 3.14(a) and (b) show that the internal shear force is reduced but not fully balanced out.
Figure 3.13 A metamaterial beam with a few absorbers: (a) steady-state ODS with r=0.9, (b) steady-state ODS with r=1.02, and (c) distribution of internal shear forces with r=1.02.
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Figure 3.14 A metamaterial beam with a few absorbers: (a) steady-state ODS with r=1.2, and (b) distribution of internal shear forces with r=1.2.
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Chapter 4 Multi-stopband Metamaterial Beams
4.1 Basic Concept
The main design concept of metamaterial structures for wave absorption and
vibration suppression is to use the resonant vibration of the integrated absorber
subsystems to prevent waves from propagating forward. Similar to the single-mass
absorbers we illustrated in Chapters 2 and 3, multi-mass absorbers will be
considered in this chapter.
Figure 4.1 A two-mass damped vibration absorber model.
Figure 4.1 shows a two-mass vibration absorber. The beam-like main
structure with a mass m1 is connected to the floor by the spring k1 and the damper c1 ,
a primary absorber m2 is attached to the main structure using a spring with a spring
constant k2 and a damper with a damping coefficient c2 , and there is a secondary
absorber m3 attached to the primary absorber. The equations of motion are derived
to be 80
m10 0 u 1 c 1 c 2 c 2 0 u 1 00m u c c c c u 2 2 2 2 3 3 2 0 0m3 u 3 0 c 3 c 3 u 3 (4.1), k1 k 2 k 20 u 1 F k k k k u 0 2 2 3 3 2 00k3 k 3 u 3
where u1 , u2 and u3 are the displacements of the main structure, primary absorber and secondary absorber, respectively. The main goal of vibration absorption is to make u1 0 . By setting u1 0 , two local undamped resonant frequencies are obtained to be :
2 4 1, 2 2 2 (4.2)
m3 k 3 k 2 k 1 , , 21 , m2 k 2 m 2 m 1
For example, if 0.025 , we obtain12 0.92406 and221.0822 . For higher values of and , the values of1 and2 will be further separated, and the stopbands around these two frequencies will be difficult to be combined into a wide stopband. If
m110 kg , m 2 0.1 m 1 , 1 2 10 Hz
ci2 i m i k i i 1,2,3 (4.3),
10.02, 2 0.04, 3 0.04
Figure 4.2 shows the frequency response function H11 with different values for
, and . It shows that there are three peaks, the response reaches local
minimum values around 1 and 2 . Figure 4.2(a) shows that, when and
81 hence m3 increase, the left peak decreases, and the middle and right peaks increase.
Figure 4.2(b) reveals that, when and hence k3 increase, the right peak decreases, and the left and middle peaks increase. From Figures 4.2(a) and 4.2(b), we notice that and determine the positions and magnitudes of the three peaks.
However, the middle peak around the local resonant frequency 2 is relatively small when . To make it easy to design such a multi-frequency vibration absorber, we choose for future design. Figure 4.2(c) shows that, when and ()
increase, the two peaks away from2 move further apart and become smaller, but
the peak around 2 increases. The two minimum points of H11 are zero if
cc230 in Figure 4.2(d). Non-zero damping prevents H11 0from happening, but it significantly reduces resonant response magnitudes. Hence, damping is necessary in a vibration absorber system.
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m Figure 4.2 Frequency response functions H11 with 110kg , m 2 0.1 m 1 and: (a) 0.05 with different
, (b) 0.05 with different , (c) 0.025,0.05,1 , and (d)different damping ratios with .
4.2 Governing Equation and Finite-Element Modeling
Figure 4.3 shows a beam segment with an attached two-mass vibration
absorber. By applying the extended Hamilton Principle on the free-body diagram
shown in Figure 4.3, we obtain
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L T W dt 0 0 nc L/2 T Aw wdx L/2 L/2 EIw'' w '' dx L/2 L/2 L/2 EIw'''' wdx EIw '' w '' EIw ''' w (4.4) L/2 L/2 L/2 EIw'''' wdx L/2 EI w'' w '' w ''' w w ''' w w''' w w ''' w 1 1 1 1 1 1 00 00 11 '' ''' '' ''' Wnc EIw11 EIw 11 w EIw 11 EIw 112200 w k u w w where , A , E , I are the beam’s mass density, cross-sectional area, Young’s
modulus and area moment, respectively. L is the cell length, ki and mi are the spring constants and masses of vibration absorbers. Moreover, w is the
' displacement of beam, ui is the displacement of absorbers, w w/ x and w w/ t . Since the absorber creates a concentrated shear force at x 0 , it is
EIw''' EIw ''' obvious that 00. It follows from Equation (4.4) that
L/2 '''' ''' ''' 0 Aw EIw wdx EI w w w k u w w L/2 00 0 2 2 0 0 L/2 (4.5) '''' ''' ''' Aw EIw EI w w k u w x wdx L/2 00 2 2 0
Setting the coefficient of w in Equation (4.5) to zero, then using Newton’s second law on the free-body diagrams of m2 and m3 , we obtain the following governing equations for the unit cell:
Aw EIwiv EI w''' w ''' k u w x 0 00 2 2 0
m2 u 2 k 2 k 3 u 2 k 3 u 3 k 2 w 0 (4.6)
m3 u 3 k 3 u 2 k 3 u 3 0
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By using the Bloch-Floquet Theory, the elastic wave propagation in the unit cell could be assumed to have:
j x t j t j t wxt , qe , u23 pe , u re (4.7)
EIw''' EIw ''' Note that Equation (4.7) cannot account for the discontinuity 00. For metamaterial beams having discrete absorbers, the displacement-based finite- element modeling is used.
Figure 4.3 free-body diagram of a metamaterial beam segment.
Figure 4.4 Finite element modeling of an infinite metamaterial beam.
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As shown in Figure 4.4, we discretize a beam segment into two beam elements. Since shear deformation and rotary inertias are influential on wave propagation in beams, we adapt Timoshenko’s beam theory. To include shear deformation Equation (4.4) is modified as:
L/2 T Aw w I w'' w dx L/2 L/2 (4.8) EI w'' ' w '' ' k GA dx L/2 s
where G is the shear modulus, and ks is the shear correction factor, which is chosen to be 5/6 for a rectangular cross-section. is the shear rotation angle, and w represents the total rotation angle of the cross-section. The weak form in Equation
(4.8) could be discretized by using cubic polynomials for w and linear shape functions for . Hence, there are three degrees of freedom (i.e. w , w' and ) at each
of the four nodes in Figure 4.4, and two degrees of freedom for u2 and u3 at nodes
1 and 3. To obtain dynamic characteristics of a periodic structure, only one unit cell needs to be considered. If damping is neglected at first, the finite-element equations are given by
M q K q 0 (4.9) '' q w1,,,,,,,,, w 11 w 4 w 4 4 u 21233133 u u u
According to the Bloch-Floquet theory, we have:
'' jL w1,,,,,,,, w 112131 u u w 3 w 332333 u u e (4.10) ''jL w4,,,, w 4 4 w 2 w 2 2 e
86 to reduce the elemental stiffness and mass matrices into 88 matrices. Here we consider an infinite beam to show the existence of stopbands in a metamaterial beam. We consider the following metamaterial beam cell
E72.4 GPa , 0.33, 2780 kg m3 L10 mm , b 5 mm , h 3 mm , A bh
m1 AL, m 2 0.1 m 1 , m 3 0.05 m 2 (4.11) 22 k2 m 2 2, k 3 m 3 2 , 2 2400 Hz
cc230
Figure 4.5 shows the dispersion curves. Black dashed lines are for the case with mm21 0.1 and m3 0. The single-mass absorber increases the system mass by10% and it leads to a stopband within the frequency range1 /2 1.065. The red solid lines represents the case with mm21 0.1 and mm32 0.05 . The two-mass absorber increases the system mass 10.5% and creates two stopbands within the frequency ranges 0.894 /2 0.93 and 1.118 /2 1.148 . The addition of the small m3 splits the stopband into two with each one being smaller than the one under a single-mass absorber. An important phenomenon revealed by this figure is that an elastic wave with a specific wavelength can propagate at three different frequencies and it leads to two stopbands around the two local resonant frequencies.
A wave with a frequency lower than 1 moves in an acoustic mode, a wave with a frequency much higher than 2 moves in an optical mode, and a wave with a frequency between and 2 moves in a mixed mode.
87
Figure 4.5 Dispersion curves for a two-mass vibration absorber with m2 m 10.1, m 3 m 2 k 3 k 2 0.05 .
4.3 Numerical Analysis
Next, we consider the following simply supported finite beam with uniform
absorbers to demonstrate the working principle of a metamaterial beam with multi-
frequency absorbers:
E72.4 GPa , 0.33, 2780 kg m3 L100 cm , L L 10010 mm , b 5 mm , h 3 mm , A bh
m1 AL, m 2 0.1 m 1 , m 3 0.05 m 2 22 (4.12) k2 m 2 2, k 3 m 3 2 , 2 2400 Hz
c22 2 m 2 k 2 , c 3 2 3 m 3 k 3 , 2 0.01, 3 0.01
F02 AL at x 5 mm
L is the total length of the beam, L is the unit length of a beam segment, m1 is the
mass of beam segment, mi , ki and ci i 2,3 are the masses, spring constants and
damping coefficients of the primary and secondary absorbers, and 2 is the local
resonant frequency of the absorbers. The harmonic excitation force F0 is placed at
the second node at x=5mm.
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Figure 4.6 Frequency response function (FRF) H101,4 of the metamaterial beam with
m m0.1, m m k k 0.05 2 1 3 2 3 2 .
Figure 4.6 shows the FRF of the node at xL /2. The black dashed line represents the case with kk23, , which is the same as attaching the absorbers to the beam as dead masses. In other words, the absorbers do not function at all and the structure behaves just like an isotropic Timoshenko beam. The blue solid line represents the response when the damping ratios of the two absorbers are
230.001. The absorbers without damping create two stopbands around the two local resonant frequencies, as the dispersion analysis predicts. But there are many small peaks around the stopbands, which are caused by local modal vibrations of the absorbers. To reduce these peaks, damping ratios of the absorbers are increased to230.01, as shown by the green solid line in Figure 4.6. Besides, the stopbands are slightly widen by the damping and it shows the possibility of combining two stopbands into a wide one. If the damping ratios are increased to
230.01 and 0.1, the response is shown as the red line. Peaks are reduced and
89 a wide stopband is obtained by combining the two small stopbands. Thus, damping is definitely essential in such a multi-frequency absorber system.
Figures 4.7(a)-(e) show the steady-state ODSs when the incoming elastic wave frequency has the following values:
r , 2 (4.13) r 0.7, 0.9, 1.0, 1.1, 1.3
The damping ratios are chosen to be 230.01 and 0.1. Here the red solid lines represent the deformed beam, black dots represent the deformed absorbers m2 , and blue dots represent the deformed absorbers m3 .
90
r m m0.1, m m k k 0.05 Figure 4.7 Steady-state ODSs when 2 , 2 1 3 2 3 2 ,230.01 and 0.1:
(a) r=0.7, (b) r=0.9, (c) r=1.0, (d) r=1.1, and (e) r=1.3.
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Figure 4.8 Steady-state ODSs when r2 , m2 m 10.1, m 3 m 2 k 3 k 2 0.05 ,
230.01 and 0.01 : (a) r=0.7, (b) r=0.9, (c) r=1.0, (d) r=1.1, and (e) r=1.3.
Figure 4.7(a) shows that when /2 0.894 , the elastic wave can propagate through the structure with a long wavelength, and it corresponds to an acoustic mode with w and u2 & u3 being in phrase. Figure 4.7(e) shows that, if/2 1.148 , the elastic wave propagates through the structure with a short wavelength, and it moves in an optical mode with w andu2 being out of phrase. If the elastic wave frequency is within the first stopband, Figure 4.7(b) shows that m2 and m3 move in 92 phrase to work against the incoming wave. While if the elastic wave frequency is within the second stopband, Figure 4.7(d) shows that only m2 move out of phrase with respect to w to work against the wave through its inertia force. Since m3 works against m2 , it is less efficient than that in Figure 4.7(b). The motion in Figure 4.7(b) is a mixed mode close to the acoustic mode, and the motion in Figure 4.7(d) is a mixed mode close to the optical mode. In Figure 4.7(c), the incoming wave frequency is out but between the two stopbands. To prove that the elastic wave is absorbed due to a large c3 , we reduce the damping ratios to 230.01in Figure 4.8. Figures 4.7(c) and 4.8(c) confirm that, for a wave out but between the two stopbands, it is damped out by using a larger damping coefficient for m3 .
In conclusion, a system with a large c3 is efficient for damping out an elastic wave out but between the two stopbands, and it takes less time to reach a steady state in a mixed mode close to an optical mode. On the other hand, Figures 4.7(b) and 4.8(b) show that a system with a small c3 is efficient in stopping an incoming wave by using inertia forces of the absorbers under resonant vibration, and it takes less time to reach a steady state in a mixed mode close to the acoustic mode. Hence, it is important to choose an appropriate value for c3 .
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Chapter 5 Conclusions and Recommendations for Future
Work
5.1 Conclusions
This thesis presents the modeling and analysis techniques for design of
metamaterial beams as elastic wave absorbers. It is designed by attaching
subsystems to an isotropic beam at separate locations with each unit consisting of a
beam segment and a mass-spring-damper subsystem. By averaging, the structure
becomes a dispersive medium and a frequency stopband exists. Such a metamaterial
beam can be modeled and analyzed by using the displacement-based finite element
method.
Similar to an electromagnetic metamaterial, which shows double negative
permittivity and permeability , an acoustic metamaterial can have simultaneous
negative effective mass and stiffness. The concepts of negative effective mass and
stiffness are clearly demonstrated, and optical and acoustic modes are explained in
detail.
For an acoustic metamaterial beam, dispersion analysis shows that a
translational-type absorber is more efficient than a torsional absorber. For finite
element simulation, Timoshenko beam elements are adopted to account for rotary
inertias and transverse shear strains. Numerical analyses show that an elastic wave
can propagate in an acoustic mode or an optical mode in a metematerial beam, and
a stopband exists to the right side of the local resonant frequency of absorbers. The
working mechanism of such a metamaterial beam is that it uses the local resonance 94
of a few designed absorbers to generate inertia forces to suppress the incoming
wave within the absorbers’ stopband. Hence, a broadband vibration absorber can be
designed by extending this concept. A distribution of absorbers of different but close
local resonant frequencies can be used to design a metamaterial beam with a wide
stopband. However, absorbers of low resonant frequencies need to meet with the
incoming wave first in order to initiate the absorbers in their optical modes to
efficiently stop the wave. Also, damping is needed in order to significantly reduce the
local resonant vibrations of absorbers.
A multi-stopband metamaterial beam is designed by attaching two-mass
vibration absorbers to an isotropic beam. Numerical analysis reveals that, within
such a metamaterial beam, an elastic wave with a specific wavelength can propagate
at three different frequencies (acoustic mode, optical mode, and mixed mode) and it
leads to two stopbands around the two local resonant frequencies. If the ratios of m3
to m2 and k3 to k2 are chosen to be equal and small, these two stopbands could be
combined into a wide stopband by using a large damping coefficient for the
secondary mass to suppress the middle response peak and a small damping
coefficient for the first mass to allow the vibration energy be easily transferred to the
absorber to be damped out.
5.2 Recommendations for Future Work
Amazing progress has been made for acoustic metamaterials over the past
few years. This new concept provides great opportunity in the design and
95 development of exotic functional devices.
The main idea presented in this thesis is using metamaterial beams to stop wave propagation and suppress structural vibration. Beyond metamaterial beams, more complicated mechanical configurations can be designed in the future, such as metamaterial plates, shells or 3D structures. How to decrease the size of attached/integrated absorbers by using new fabrication methods like laser cutting,
3D printing and laser sintering is a topic worth a series of studies.
All the presented metamaterial models are linear ones. When the wave energy is large (e.g., earthquake excitation, explosion, etc.), geometric and material nonlinearities need to be included in the modeling, and structural stability and solution bifurcation need to be analyzed. Moreover, potential applications of such metamaterials include building soundproof homes, advanced concert halls and acoustic cloaking, and they deserve more studies.
Optic and acoustic metamaterials can be combined to design optic-acoustic metamaterials to simultaneously treat electromagnetic and mechanical waves. This can be a new and large research area for future work.
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