- Email: [email protected]

PII: DOI: Reference:

S0165-2125(18)30198-7 https://doi.org/10.1016/j.wavemoti.2019.102492 WAMOT 102492

To appear in:

Wave Motion

Received date : 1 February 2018 Revised date : 2 December 2019 Accepted date : 2 December 2019 Please cite this article as: J. Jung, S. Goo and S. Wang, Investigation of flexural wave band gaps in a locally resonant metamaterial with plate-like resonators, Wave Motion (2019), doi: https://doi.org/10.1016/j.wavemoti.2019.102492. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Investigation of flexural wave band gaps in a locally resonant metamaterial with plate-like resonators

Jaesoon Jung1, Seongyeol Goo2and Semyung Wang2*

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1: Department of Electrical Engineering, Technical University of Denmark, 2800 Kgs. Lyngby,

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Denmark

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2: School of Mechanical Engineering, Gwangju Institute of Science and Technology, 123Cheomdan-

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gwagiro, Buk-gu, Gwangju, 61005, Republic of Korea

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Submitted to Wave Motion

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Submitted: February01, 2018

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Revised 1st: July13, 2018

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Revised 2nd: December xx, 2019

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Includes 23 figures and 5 tables

*Corresponding author

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Postal address: School of Mechanical Engineering, Gwangju Institute of Science and Technology,

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123Cheomdan-gwagiro, Buk-gu, Gwangju, 61005, Republic of Korea

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E-mail address: [email protected]

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Tel. +82 62 970 2390

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Fax. +82 62 970 2384

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Abstract This paper presents an investigation of flexural wave band gaps in locally resonant

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metamaterials (LRMs). An LRM is a periodic structure consisting of repeated unit cells

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containing a local resonator. Due to the local resonance occurring in the unit cell, the LRM

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induces a band gap (a frequency band in which no waves propagate). Discrete-like or beam-

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like resonators have generally been used to realise LRMs in previous research. By extending

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the beam-like resonator configuration, this paper studies LRMs with a plate-like resonator to

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exploit its advantages with respect to large design freedom. In order to understand flexural

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wave band gaps in an LRM with plate-like resonators, parametric studies are conducted with

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the development of a finite element model. Further, the influences of the plate-like resonator

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design parameters on flexural wave band gaps are investigated. Based on the parametric studies,

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the rules governing band gap properties are determined. Finally, tailoring flexural wave band

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gaps by adjusting the parameters is discussed.

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Keywords: Locally resonant metamaterial; Plate-like resonator; Band gap; Flexural wave;

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Bloch theory; Finite element method

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1. Introduction A metamaterial is an artificially engineered structure that exhibits unusual properties such

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as band gaps; that is a frequency band in which wave propagation is totally prohibited [1,2].

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The band gap of a metamaterial originates from the designed repeated structure known as a

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unit cell. By taking advantage of band gaps, metamaterials have been exploited in various

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devices such as mechanical filters, vibration isolators, waveguides, and a sound radiator [1-6].

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Depending on the formation mechanisms, the band gaps can be classified into the Bragg

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and local resonance gaps [7,8]. The Bragg gap is based on the wave scattering occurring by

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means of periodic unit cells that contain scatterers e.g., inclusions and gratings. Thus, the Bragg

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gap is mainly determined by the length scale of a unit cell corresponding to the periodicity.

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The local resonance gap is based on the interaction between the propagating waves and the

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resonators [5,9,10]. Hence, the critical factor determining the local resonance gap is the

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dynamic properties of the resonator rather than the periodicity. Thus, the local resonance gap

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exhibits few restrictions in the unit cell length scale and this feature is advantageous in practical

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applications [5,7,8,10]. Focusing on this benefit, this paper considers the metamaterials based

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on the local resonance gap, which is known as the locally resonant metamaterial (LRM)

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[1,5,10].

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For several decades, numerous studies have been conducted on LRMs, such as the

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development of analytical methods [11-15], application to noise reductions [16-20], and

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vibration attenuations [21-25]. In these studies, the unit cells have usually been realised with

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beam-like resonators [20,21,24,25]. By extending the beam-like resonator configuration, it is

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not difficult to design the plate-like resonator to incorporate its advantages into the large design

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freedom. The LRM unit cell, which contains a plate-like resonator, creates a flexural wave

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band gap that can be exploited in many practical applications [16-20]. To fully exploit the

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flexural wave band gap, it is beneficial to study the behaviour of band gaps depending on the

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design of resonators. In this regard, systematic investigations of the band gaps are required in

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consideration of the various design parameters [6].

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Regarding this background, this paper aims at investigating the flexural wave band gaps in

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the LRM with plate-like resonators. A finite element (FE) model is developed to analyse the

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dynamic behaviour of the unit cell. The FE model is based on the Kirchhoff plate theory to

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analyse the bending behaviour of the plate-like resonator efficiently. To couple the resonator

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and the host structure, a resonator coupling scheme is proposed. Using the FE model, the band

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gaps in the LRM are investigated with parametric studies considering the resonator types,

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dimensions, and amount of concentrated masses. Finally, the tailoring of the band gaps by

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adjusting the parameters is discussed. The contributions of this paper can be classified into two parts. First, we develop an FE

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model for the band gap analysis of the LRM (Section 3). The accuracy of the developed FE

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model is validated. Secondly, we present the systematic investigations of the band gaps in the

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LRM (Section4). The characteristics of the flexural wave band gap created by the three

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resonator types (cantilever, bridge, and cross) are discussed. The results of this paper can be

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exploited in the design of practical devices such as flexural wave filters, vibration isolators,

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and sound deadeners.

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2. The locally resonant metamaterial with a plate-like resonator

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To provide a context for this study, we begin with a brief overview of the LRM, which is

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characterised by a unit cell containing a local resonator. When the unit cells are periodically

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arranged, the periodic structure induces a complete prohibition of the wave propagation as

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illustrated in Fig. 1 [8,10]. The frequency band in which this phenomenon occurs is known as

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a band gap, which is observable in a dispersion curve.

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Fig. 1. Schematic of the LRM and wave propagation in the local resonance gap.

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Considering the creation mechanism, it is evident that the resonance properties of the

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resonator dominate the band gap; thus, the design of resonator plays an important role for 4

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realizing the LRMs. In this study, a plate-like resonator is considered to realise the LRM to

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exploit its advantages in the large design freedom. Fig. 2 illustrates the LRM configuration

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which has the host structure of the thin plate and the cantilever-type resonator. The host

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structure constitutes the entire LRM in which the flexural waves are to be attenuated. The frame

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acts as boundary conditions that supports the resonator. The resonator can be designed

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depending on its boundary conditions, shapes, dimensions, and mass distributions.

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Fig. 2. (a) LRM and (b) unit cell containing plate-like resonator. 3. Finite element modelling

In this section, an FE model is developed to analyse the band gaps in the LRM. A three-

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dimensional (3D) FE model is required to describe the dynamic motion of the plate-like

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resonator which is coupled with the host structure. However, the 3D model usually requires

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high computational costs because of its large number of degrees of freedom (DoFs) making up

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the geometry. As an alternative to the 3D model, we propose the resonator coupling scheme

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for firstly separating the unit cell into two 2D FE domains and then combining them.

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3.1. The resonator coupling scheme To describe the developed FE model in detail, we begin with the separation of the unit cell

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into two 2D plate domains corresponding to the host structure (ΩA) and resonator (ΩB) as

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illustrated in Fig. 3. In ΩA, the host structure and frame are distinguished by the thickness

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difference.

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The bending motions of each plate domain can be described by the Kirchhoff’s plate theory

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Figure 3. Separation of the unit cell into two plate domains

as follows [26]:

D D

4

A

B

4

A hA

2

B hB

2

w w

A

x, y 0,

B

x, y 0,

for A for B

(1)

where D(= Eh3/12(1-ν2)) is the flexural rigidity, E is the elastic modulus, ν is the Poisson’s

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ratio, ρ is the mass density and h is the plate thickness. The symbol w represents the bending

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displacement of the plate while subscripts A and B denote the host and resonator domains,

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respectively. The symbol ∇4(= ∂4/∂x4 + 2∂4/∂x2∂y2 + ∂4/∂y4) indicates the biharmonic operator.

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Following a general FE procedure, Eq. (1) can be expressed in a matrix equation consisting of

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a stiffness matrix (K) and mass matrix (M), as follows [27]:

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(2)

K 2 M w 0, K A K 0 w A , w B

M A ,M KB 0

0

(3)

M B

(4)

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w

0

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where w is the nodal displacement vector. Note that ΩA and ΩB are uncoupled in Eq. (2); thus,

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the resonator and host structure move independently. To describe the coupled motion of the

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resonator and host structure, the coupling condition at the frame is expressed as follows: 6

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w A x, y w B x, y 0.

(5)

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Here, (x,y) is the position in which the resonator is connected to the host structure on the frame.

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For simplicity, Eq. (5) is expressed in a matrix form as follows: (6)

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I c w 0,

I c 1 0 -1 0 ,

(7)

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the symbol Ic denotes the coupling matrix. The coupling condition expressed in Eq. (6) can be

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combined with Eq. (2) using various methods, such as the penalty method and the Lagrange

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multiplier method [24]. In this paper, the Lagrange multiplier method is employed owing to its

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completeness without any additional unknown parameters. Following the Lagrange multiplier

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method, the coupled FE matrix equation is expressed using the stiffness, mass, and coupling

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matrices as follows [27]: K coupled 2 M coupled w 0, K

K coupled

I c

I cT

M , M coupled 0 0

w , w λ

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(8) w w A w A w B w B T , where w and w represent the DoFs on, and out of the connection position respectively, and

0

0

,

(9) (10)

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(11)

where Kcoupled, Mcoupled and λ are the coupled stiffness, mass matrices, and the Lagrange

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multiplier vectors, respectively. In Eq. (9) the host structure and resonator are physically

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coupled on the frame. The accuracy of the proposed resonator coupling scheme is validated in

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Appendix A.

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3.2. Dispersion equation

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In this subsection, we derive a dispersion equation to analyse the band gap in the LRM. For

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the derivation, the Floquet–Bloch theory is employed which describes the wave propagation in

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a periodic structure as illustrated in Fig. 4. In this figure, the entire domain is constructed by

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the repetition of the unit cell in two directions, with the basis vectors of a1 and a2. The unit cell

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has lengths of Lx and Ly in the x- and y-directions, respectively. Each unit cell contains a local

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resonator represented by the mass-spring system. Following the Floquet–Bloch theory, the

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response at any position can be fully expressed with respect to the response at a reference unit

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cell, as follows [28,29]: w r a1 a 2 , k w r, k exp jk a1 a 2 ,

k , k . x

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k

(12)

y

(13)

Here, w(r+a1+a2,k) and w(r,k) represent the response at points A and B in Fig. 4, respectively.

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The function exp(jk(a1+a2)) represents the change in amplitude and phase determined by the

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wave vector k.

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Figure 4. Schematics of the periodic structure to describe the Floquet–Bloch theory

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The dispersion equation can be obtained by combining Eq. (12) with Eq. (9) [15, 8-34]. In

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the FE analysis, Eq. (12) can be implemented as the boundary conditions on a unit cell structure

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[25,26]. For this purpose, the boundaries of a unit cell structure are classified into eight regions,

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as illustrated in Fig. 5. The DoFs on the top right boundaries (T, R, RT, RB, and LT) can be

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expressed using the DoFs on the bottom-left boundaries (B, L,and LB) as follows [29]:

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wR wLe

jkx Lx

wRB wLB e

(14)

,

jk x Lx

(15)

,

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wT wBe

jk y Ly

wLT wLB e wRT wLB e

(16)

,

jk y Ly

(17)

,

j k x Lx k y Ly

(18)

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Eqs. (14-18) are usually called the Floquet–Bloch boundary condition or the periodic boundary

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condition [20,24,25,28,29].

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For simplicity, Eqs. (14-18) are expressed in matrix form, as follows: I B w 0,

w wL

wR

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wB

1

0

0 0

wT

0

e

jk y L y

w LB

w RB

(19)

(20)

w RT ,

w LT

0

0

0

0

0

0

e

jk x Lx

1

0

0

e

jk y L y

1

0

0

0

0

0

0

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e jk x Lx 0 0 IB 0 0

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Figure 5. Classification of the boundaries of the unit cell

0

e

j k x Lx k y L y

0 0 0 . 0 1 0 0 0 1

Here, IB is the periodic boundary condition matrix.

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(21)

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Eq.(21) is combined with Eq.(9) using the Lagrange multiplier method in the same manner as the resonator coupling scheme as follows: K B k 2 M B w B 0, IB

B w

IB

M coupled , MB 0 0 T

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0

(23)

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K coupled

K B k

(22)

w λ , B

(24)

where KB, MB and λB are the modified stiffness matrix, which is dependent on the wave vector

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k, modified mass matrix, and Lagrange multiplier vector, respectively. Here, the nodes that are

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not on the boundaries are considered in the original coupled matrices (i.e. Kcoupled, and Mcoupled).

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Note that the increase in the matrix dimension is less than 3% in general.

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Eq. (22) involves the three parameters the wave vector k (kx and ky) and ω. Various types

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of dispersion equations can be considered, depending on the physical nature of the solution to

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be determined, as discussed in the reference [32]. In this paper, the linear algebraic eigenvalue

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problem that determines frequency ω when the wave vector k is provided is solved to obtain

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the dispersion relations of the wave propagation in the LRM.

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To obtain the dispersion relation, Eq. (22) has to be solved for every possible combination

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of wave vector components, kx and ky. The solutions of Eq. (22) form surfaces known as

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dispersion surfaces (that is, ω=f(kx,ky)) in the wave vector domain. The dispersion surface can

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be reduced to a dispersion curve, which contains sufficient information to analyse the wave

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propagation. The frequencies in this curve represent the frequency at which the wave can

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propagate without attenuation. In contrast, the empty frequency bands on this curve indicate

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the frequency at which only evanescent waves propagate. Thus, these frequencies belong to the

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band gaps [22].

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Owing to the LRM periodicity, the dispersion curve is also periodic; thus, it is unnecessary

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to solve Eq. (22) in the entire wave vector domain. A periodic zone of the wave vector domain

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is known as the Brillouin zone, and the periodicity allows Eq. (22) to be solved only in the first

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Brillouin zone [22]. Using the unit cell symmetry, the first Brillouin zone can be reduced further

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to the irreducible Brillouin zone (IBZ) as illustrated in Fig. 6.

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In this paper, we only consider the rectangular unit cell. Thus, the IBZ is the triangular zone

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consisting of three vertices, Γ=(0,0), X=(π/Lx,0) and M=(π/Lx,π/Ly) in the wave vector domain 10

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[22]. Therefore, the dispersion curve can be obtained by solving Eq. (22) while sweeping the

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wave vector k along the contour; that is, Γ→X, X→M, and M→Γ.

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Figure 6. First Brillion zone and Irreducible Brillouin zone in the wave vector domain. 4. Investigation of flexural wave band gaps

This section presents the investigations of band gaps in the LRM with different plate-like

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resonators. We consider three representative resonator types (i.e., the cantilever, bridge, and

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cross-types) within the scope of this study. The three unit cells are illustrated in Fig. 7; all have

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the same host structure, with dimensions of 50×50×1 mm. The steel is used as the material,

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with elastic modulus of 200 GPa, Poisson’s ratio of 0.3, and a mass density of 7800 kg/m3.

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In the following subsections, the dispersion curves and local resonance gaps in the three

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LRM designs are investigated. For the dispersion analysis, 45 wave vectors in the IBZ are

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considered by dividing the wave vector-sweeping path equally (that is, Г–X, X–M and M–Г)

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into 15 wave vectors. All FE analyses are performed with the converged FE mesh.

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Figure 7. Three representative unit cells containing (a) cantilever, (b) bridge, and (c) cross-

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type resonators.

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4.1. Influence of resonator types 11

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We begin the investigation with the influence of the resonator types. The most significant differences between the three resonator types are the boundary condition. The cantilever, bridge,

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and cross-types are supported by one, two, and four edges, respectively. Since the boundary

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condition significantly affects the resonance frequencies, it is worth to clarify the differences

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among the resonator types. Fig. 8 displays the dispersion curves and band gaps of the unit cells

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with three resonator types. In this figure, the frequency bands that cover the band gap are

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highlighted in red colour. The Bloch mode shapes at the gap are presented to verify that the

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gap is created by the local resonance.

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As displayed in Fig. 8, different band gaps occur, which vary from 1000 to 5000 Hz

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depending on the resonator types. Since the cantilever-type is the most flexible, the band gap

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appears at the lowest frequency of 935 to 980 Hz. In contrast, the cross-type is the stiffest, and

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two gaps appear at the highest frequencies of 4480 to 4758 Hz, and 4881 to 5199 Hz. The

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bridge-type has a gap in the intermediate frequency of 3000 to 3200 Hz. Therefore, in terms of

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the gap centre frequency, the cantilever, bridge, and cross-types are listed in increasing order.

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Furthermore, in the same order, the gap bandwidths tend to be broader. However, no significant

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difference is observed in the relative bandwidth (RBW), which is normalised by the centre

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frequency, as follow: RBW

0

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.

(25)

Here, ω0 and Δω are the gap centre frequency and bandwidth, respectively. The band gap

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properties are listed in Table 1.

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Table 1. Band gap properties of the unit cells containing three different resonators.

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Cantilever Bridge

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Terminal frequency [Hz]

Centre frequency [Hz]

Bandwidth [Hz] (RBW)

935

980

957

45 (0.047)

3000

3200

3100

200 (0.064)

4480 4881

4758 5199

4619 5040

249 (0.054) 318 (0.063)

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Initial frequency [Hz]

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Figure 8. Dispersion curves of the unit cells containing (a) cantilever, (b) bridge, and (c)

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cross type-resonators.

Unlike the other two types, two successive gaps are observed in the dispersion curve of the

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cross-type. To understand the manner in which the successive gaps are created, we present the

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Bloch modes corresponding to each gap in Fig. 8. As depicted in the figure, the first gap of

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4480 to 4758 Hz is based on the local resonance of the resonator. However, the second gap of

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4881 to 5199 Hz is based on the local resonance of the host structure. According to this

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observation, the successive gaps occur because the resonance of the cross-type resonator is

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close to the resonance of the host structure.

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Furthermore, in the middle of the two successive gaps, a flat band (approximately, 4758

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Hz) appears. As the slope of the dispersion curve represents the group velocity that transfers

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wave energies, negligible energies are transferred in the flat band [2]. The flat band can be

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interpreted using the Bloch mode in which the host and resonator move out of phase, as

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illustrated in Fig. 8(c). Disregarding the flat band, which affects the wave propagation

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negligibly, the cross-type has the broadest gap of 4480 to 5199 Hz by combining the two

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successive gaps. In the remainder of this paper, we consider the two successive gaps of the

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cross-type as a combined gap.

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This comparative study demonstrates that the band gap is highly sensitive to the resonator

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type, although the other conditions are the same. This result is reasonable as the boundary

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condition is the most critical factor determining the resonances. Therefore when designing a

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unit cell, the resonator types should be carefully selected to take into account the frequency

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range in which the gap is desired. The cantilever-type is appropriate when a low-frequency gap

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with a narrow band is required, while the cross-type is appropriate when a high-frequency gap

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with broadband is desired. The bridge-type should be selected when an intermediate gap is

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required.

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4.2. Influence of the dimensions

In this subsection, we present the influence of the resonator dimensions on the band gaps.

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The dimensions determine the stiffness and mass distribution, thus the resonance frequencies.

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For example, the length (L) of a cantilever-type resonator has an inverse quadratic relation with

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the resonance frequency (ω), as follow:

L A / EI

1/4

,

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(26)

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where α is the root of the frequency equation, A is the cross-sectional area, and I is the second

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moment of area. The frequency equation and its roots are referred from the reference [26]. Although the relation between the resonance frequency and length is explicit in Eq. (26),

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the manner in which the bandwidth and RBW behave as the length varies remains unknown.

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Moreover, the influences of the width on the gaps are worthy of study to guide the design of

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the resonators. To discover how the dimensions affect the gap, we conduct parametric studies

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by varying the lengths and widths of the resonators.

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For the parametric study on the length, only the cantilever-type is considered, as the lengths

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of the other two types are unalterable under the fixed unit cell dimension. Fig. 9 displays the

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dispersion curves of the cantilever-type when varying the length from 20 to 40 mm at intervals

272

of 5 mm, with the width fixed at 40 mm. The band gap properties are listed in Table 2.

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Figure 9. Dispersion curves of the cantilever type with varying lengths: (a) 20mm, (b) 25

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mm, (c) 30 mm, (d) 35 mm, (e) 40 mm; and (f) unit cell used for the parametric study.

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Length [mm]

Initial frequency [Hz]

Terminal frequency [Hz]

Centre frequency [Hz]

Bandwidth [Hz] (RBW)

20

2094

2166

2130

72 (0.034)

25

1346

1401

1373

55 (0.040)

934

981

957

47 (0.049)

687

725

706

38 (0.054)

526

559

542

33 (0.060)

30 35 40 278

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Table 2. Band gap properties of the cantilever type with varying lengths.

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Fig. 10 displays the RBW and centre frequency using the bar and line graphs, respectively.

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As displayed in this figure, the centre frequencies of the band gap decrease in a quadratic

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manner when the resonator length increases, as can be seen in Eq. (26). In contrast to the centre

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frequency, negligible changes appear in the bandwidth. Therefore, decreases in the centre

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frequencies result in slight increases in the RBWs. As a result of the parametric study on the

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length, it is shown that the length of the cantilever-type controls the centre frequency

285

considerably but changes the bandwidth rarely.

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Figure 10. Variation in the local resonance gap according to cantilever-type resonator length.

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For the parametric study on width, the widths of the three resonator types are varied from

289

20 to 40 mm (at intervals of 5 mm) with fixed lengths. As an excessive number of dispersion

290

curves exist in this parametric study (15 dispersion curves for three types times five width

291

variations), we present the dispersion curves in Appendix B while the gap properties are listed

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in Table 3. Moreover, Fig. 11 displays the RBWs and centre frequencies. As illustrated in this

293

figure, no significant changes in the gap properties are observable as the resonator widths vary.

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The results of the constant gaps can be interpreted using Eq. (26) by substituting the formula

295

of the second moment of area (I=bh3/12) as follow:

L 12 / Eh2

(27)

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1/4

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.

Here, the width term (b) is cancelled out as the second-moment of area is divided by the cross-

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sectional area (A=bh). Therefore, it can be explicitly observed that the width does not affect

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the resonance.

299 300 301

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Table 3. Band gap properties of all resonator types with varying widths. Terminal frequency [Hz] 954

Centre frequency [Hz] 941

Bandwidth [Hz] (RBW) 25 (0.027)

Cantilever type

25

932

961

946

29 (0.031)

30

933

968

950

34 (0.036)

35

933

974

953

41 (0.043)

40

934

981

957

47 (0.049)

20

2992

3141

3066

149 (0.048)

Bridge type

25

2995

3167

3081

172 (0.056)

30

3000

3200

3100

200 (0.064)

35

2999

3222

3110

223 (0.072)

40

3003

3242

3122

239 (0.077)

20

4480

5199

4839

719 (0.148)

No band gap No band gap No band gap No band gap

30 35

40

Jo

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303

Pr e-

25 Cross type

of

20

Initial frequency [Hz] 929

Width [mm]

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Figure 11. Variation in the local resonance gap according to a resonator width: (a) cantilever,

306

(b) bridge and (c) cross types.

al

305

Unlike in the other two types, no gap appears in the cross-type when the width is longer

308

than 20 mm. This is interesting and important to analyse because the band gap may disappear

309

during the design of the cross-type. The main reason for this phenomenon is that the resonator

310

is strongly coupled with the host structure as the width increases; thus, the resonator cannot

311

move locally. In support of this interpretation, Fig. 12 displays the change in the Bloch modes

312

of the cross-type resonator as the width varies. In this figure, it can be seen that the resonator

313

moves together with the host structure when the width is greater than 20 mm. Therefore, to

314

create the band gap, the cross-type width is required to be sufficiently narrow.

Jo

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315

As a result of the parametric study on the width, it can be concluded that the widths of the

316

cantilever and bridge-type have a negligible effect on the gap. However, the increase in width 18

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may lead to no gap appearing in the cross-type owing to the strong coupling between the

318

resonator and host structure.

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319 320 321

Figure 12. Change in Bloch modes according to the cross-type resonator width 4.3. Influence of the concentrated mass

In this investigation, we present the influence of the concentrated mass on the gaps. As can

323

be observed in the resonance frequency in Eq. (26), the mass and frequency have an inverse

324

square root relation (that is, ω2∝1/ρA). From this relation, it can be deduced that the gap centre

325

frequency decreases as the concentrated mass increases. However, how the bandwidth and

326

RBW behave as the concentrated masses vary remains unknown. To clarify the influence of

327

the concentrated mass on the band gap properties, we conduct a parametric study by changing

328

the mass amount.

urn

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For the parametric study, the mass area are fixed and the thickness is changed from 1 to 5

330

mm to vary the amount of mass. Fig. 13 displays the three resonator types with concentrated

331

masses.

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19

332

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Figure 13. Three unit cells containing resonators with concentrated masses: (a) cantilever, (b)

334

bridge, and (c) cross-types.

p ro

333

To avoid confusion arising from interpreting many dispersion curves, the curves are

336

presented in Appendix C while the gap properties are listed in Table 4. Fig. 14 displays the gap

337

in the RBWs and centre frequencies. As displayed in the figure, the centre frequencies decrease

338

slowly as the amount of mass increases because the resonance frequency and mass have an

339

inverse square root relation, as can be seen in Eq. (26). The changes in bandwidth are negligible

340

for the cantilever and bridge-types, hence the decreases in the centre frequencies result in a

341

slight increase in the RBWs.

342

Table 4. Band gap properties of all of the resonator types with varying concentrated masses.

2

Bandwidth [Hz] (RBW) 47 (0.049)

780

44 (0.057)

654

700

677

46 (0.068)

583

631

607

48 (0.079)

531

581

556

50 (0.090)

3000

3200

3100

200 (0.064)

2917

3142

3030

224 (0.074)

2696

2941

2818

245 (0.087)

2506

2755

2630

249 (0.095)

2341

2601

2471

260 (0.105)

4480

5199

4839

719 (0.148)

2

4736

5374

5055

638 (0.126)

3

4625

5280

4953

655 (0.132)

4

4333

5188

4760

855 (0.179)

5

4018

5138

4578

1119 (0.244)

3 4 1 2 3 4 5 1

Cross type

Centre frequency [Hz] 957

829

5

Bridge type

Terminal frequency [Hz] 981

785

urn

Cantilever type

Initial frequency [Hz] 934

al

Mass thickness [mm] 1

Jo

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335

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Figure 14. Variation in the local resonance gaps according to the concentrated masses on the

346

resonator: (a) cantilever, (b) bridge, and (c) cross-type

al

345

An unusual but interesting tendency is observed in the cross-type resonator, unlike for the

348

other two types. The centre frequencies of the cross-type increase as the thickness of the mass

349

increases from 1 to 3 mm because the mass acts as a stiffener. Moreover, when the thickness is

350

greater than 3 mm, the centre frequencies decrease the same as in the other resonator types. In

351

the same order as the centre frequencies, the bandwidths become narrower at first and then

352

broader as the mass increases. Thus, the decreases in the centre frequencies and broad

353

bandwidths result in a remarkable rise in the RBWs.

Jo

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347

354

The reason for the broad bandwidth can be interpreted as the decoupling of the dynamic

355

movement between the resonator and host structure. As discussed in Section 4.2, the cross-type

356

resonator is strongly coupled with the host structure. This coupling is decoupled as the amount 21

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of mass increases. This tendency can be observed in the dispersion curves and Bloch mode

358

shapes as displayed in Figs. 15 and 16, respectively. As displayed by the Bloch modes in Fig.

359

16, the upper and lower bounds of the gap correspond to the resonance of the host structure

360

and resonator, respectively. The upper bounds remain at 5200 Hz while the lower bounds

361

decrease (from 4480 to 4019 Hz) when the mass increases, as shown in Fig.15. This tendency

362

is caused by the fact that the increase in mass mainly affects the resonator and negligibly

363

influences the host structure. Thus, the decupling of the dynamic motion between the cross-

364

type resonator and host structure leads to a broad bandwidth.

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of

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As a result of the parametric study on the concentrated mass, it is shown that the increase

366

in mass slowly decreases the centre frequencies of the gaps in the cantilever and bridge-types.

367

In contrast, the increase in mass may lead the cross-type to have a broad gap owing to the

368

decoupling between the resonator and host structure. Therefore, for applications that require a

369

broad gap at a high frequency, the selection of the cross-type with a concentrated mass would

370

be highly beneficial.

Jo

371

urn

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Pr e-

365

372

Figure 15. Dispersion curves of the cross-type with varying concentrated mass thickness: (a)

373

1 mm, (b) 2 mm, (c) 3 mm, (d) 4 mm, (e) 5 mm; and (f) unit cell used for the parametric

374

study. 22

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p ro

of

375

376

Figure 16. Change in the Bloch modes according to an increase in concentrated mass on the

378

cross-type resonator with varying concentrated mass thicknesses: (a) 1 mm, (b) 2 mm, (c) 3

379

mm, (d) 4 mm and (e) 5 mm.

380

5. Conclusion

Pr e-

377

In this paper, the band gaps of the LRM with different plate-like resonators were

382

investigated. For the analysis of the dispersion relation, an FE model was developed based on

383

the plate theory. Benefiting from a low computational cost, the developed FE model could offer

384

an alternative to 3D FE models. This advantage of the FE model is more beneficial in many

385

cases, especially when the thickness of the plate-like resonator becomes thinner, many repeated

386

dispersion analyses are required, and the vibration field on the array of the unit cells is

387

evaluated.

al

381

By using the developed FE model, the influences of the three parameters (resonator types,

389

dimensions, and the concentrated mass) on the band gap were investigated. For the

390

investigation, three representative resonator types with different shapes and boundary

391

conditions (cantilever, bridge, and cross-types) were considered.

392

urn

388

The results obtained from the investigations could be summarised as follows. The bandwidth is mainly determined by the type of plate-like resonator. From our investigation, the cross-type results in the broadest gap at a high frequency. In contrast, the cantilever-type results in the narrowest gap at a low frequency.

396 397 398 399

The cross-type resonator with no mass addition is inappropriate for the gap creation, as the dynamic motion of the resonator and host structure are strongly coupled. However, the addition of concentrated mass on the resonator may decouple the dynamic motion, and lead to a broad gap.

Jo

393 394 395

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400 401

In general, the gap centre frequency decreases as the resonator length and the amount of mass increase.

402

The resonator width affects the gap negligibly. Using the investigation results, the local resonance gap can be tailored to the desired

404

frequency by adjusting the parameters. An optimisation can be employed to adjust the

405

parameters systematically, which is the focus of on-going work.

of

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Acknowledgement

408

This work was supported by the National Research Foundation of Korea (NRF) grant, funded

409

by the Korean government (NRF-2017R1A2A1A05001326).

p ro

407

410 411

Appendix A. Accuracy of the resonator coupling scheme

We present the analysis results of the developed FE model compared to the 3D FE model

413

to validate the developed model. A unit cell with the cantilever-type resonator, as illustrated in

414

Fig. 7, is considered. Fig. A1 display the FE models of the unit cell. The analysis of the

415

developed FE model is conducted using in-house code written in MATLAB and the 3D FE

416

analysis is performed using COMSOL Multiphysics, which is a commercial software [35]. The

417

developed FE model consists of 2292 triangular Kirchhoff plate elements, and the number of

418

DoFs is 3687. In contrast, the 3D model consists of 7053 quadratic tetrahedral solid elements,

419

and the number of DoFs is 41712.

420 421

Jo

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Pr e-

412

Figure A1. FE models: (a) developed FE model and (b) 3D FE model using COMSOL

422

Fig. A2 displays the modal analysis results of the two models showing the major vibrational

423

modes of the unit cell. As shown in the figure, the results of the two models have the same 24

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424

mode shapes at similar frequencies. For precise comparisons, Table A1 lists the natural

425

frequency errors defined as follows: Error

3D 2 D 100, 3D

(A.1)

where ω3D and ω2D are the natural frequencies obtained from the 3D and developed FE models,

427

respectively.

of

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As demonstrated in Table A1, the natural frequency errors of all modes are less than 3%.

429

These results show that the developed FE model is highly accurate over a reasonable range.

430

Furthermore, the developed FE model is more efficient in terms of computational costs because

431

the required number of DoFs is 11 times less than that of the 3D FE model.

al

Pr e-

p ro

428

432

435 436 437 438 439 440

urn

434

Figure A2. Modal analysis results

Jo

433

441 25

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Table A1. Comparison of the natural frequencies 3D model [Hz] 952

Error (%) 1.68

Resonator first twisting

1907

1868

2.09

Host structure bending

4896

4795

Resonator second bending

5934

5783

Pr eal urn Jo

443

of

Developed model [Hz] 968

Resonator first bending

2.11

2.61

p ro

442

26

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444

The dispersion curves with varying widths are displayed in Fig. B1, B2, and B3.

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p ro

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445

Appendix B. Dispersion curves three resonator types with varying widths

446

Figure B1. Dispersion curves of the cantilever-type with varying widths: (a) 20 mm, (b) 25

448

mm, (c) 30 mm, (d) 35 mm, (e) 40 mm; and (f) unit cell used for the parametric study.

449

Jo

urn

al

447

450

Figure B2. Dispersion curves of the bridge-type with varying widths: (a) 20 mm, (b) 25 mm,

451

(c) 30 mm, (d) 35 mm, (e) 40 mm; and (f) unit cell used for the parametric study. 27

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Pr e-

452 453

Figure B3. Dispersion curves of the cross-type with varying widths: (a) 20 mm, (b) 25 mm,

454

(c) 30 mm, (d) 35 mm, (e) 40 mm; and (f) unit cell used for the parametric study.

Jo

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455

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Appendix C. Dispersion curves of the resonator types with varying concentrated masses

457

The dispersion curves with varying concentrated masses are displayed in Fig. C1 and C2.

Pr e-

p ro

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456

458

Figure C1. Dispersion curves of the cantilever-type with varying concentrated mass

460

thickness: (a) 1 mm, (b) 2 mm, (c) 3 mm, (d) 4 mm, (e) 5 mm; and (f) unit cell used for the

461

parametric study.

462 463

Jo

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Figure C2. Dispersion curves of the bridge-type with varying concentrated mass thickness: 29

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464

(a) 1 mm, (b) 2 mm, (c) 3 mm, (d) 4 mm, (e) 5 mm; and (f) unit cell used for the parametric

465

study. References

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[1] G. Ma,P. Sheng, Acoustic metamaterials: From local resonances to broad horizons, Sci. Adv. 2.2 (2016) e1501595.

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[2] Pierre A. Deymier, Acoustic metamaterials and phononic crystals Vol. 173, Springer Science & Business Media, 2013.

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[3] M.S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari-Rouhani, Acoustic band structure of periodic elastic composites, Phys. Rev. Lett. 71 (1993) 2022–2025.

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[4] C.G. Poulton, A. B. Movchan, R. C. McPhedran, N. A. Nicorovici, Y. A. Antipov, Eigenvalue problems for doubly periodic elastic structures and phononic band gaps, Proc. R. Soc. Biol. Sci. 456 (2000) 2543.

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[5] J. Jung, H.-G. Kim, S. Goo, K.-J. Chang, S. Wang, Realisation of a locally resonant metamaterial on the automobile panel structure to reduce noise radiation, Mech. Syst. Signal Pr.112 (2019) 206-231.

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[8] C.Sugino, S.Leadenham, M.Ruzzene, A.Erturk, On the mechanism of bandgap formation in locally resonant finite elastic metamaterials, J. Appl. Phys. 120.13 (2016) 134501.

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[11]Y. Xiao, J. Wen, G. Wang, X. Wen, Theoretical and experimental study of locally resonant and Bragg band gaps in flexural beams carrying periodic arrays of beam-like resonators, J.Vib. Acoust. 135.4 (2013): 041006.

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[12]Y. Xiao, J. Wen, X. Wen, Flexural wave band gaps in locally resonant thin plates with periodically attached spring–mass resonators, J. Phy. D Appl. Phys.45.19 (2012) 195401.

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[13]P. Li, S. Yao, X. Zhou, Effective medium theory of thin-plate acoustic metamaterials, J. Acoust. Soc. Am.135.4 (2014) 1844-1852.

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[14]G. Carta,M. Brun, Bloch–Floquet waves in flexural systems with continuous and discrete elements,Mech. Mater. 87 (2015) 11-26.

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[15] Z. B. Cheng, Z. F. Shi, Y. L. Mo, Complex dispersion relations and evanescent waves in periodic beams via the extended differential quadrature method,Compos. Struct.187 (2018) 122-136.

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[16]F. Morandi, M. Miniaci, A. Marzani, L. Barbaresi, M. GaRAI,Standardised acoustic characterisation of sonic crystals noise barriers: Sound insulation and reflection properties, Appl. Acoust.114 (2016) 294-306.

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[17]Y. Xiao, J. Wen,X. Wen, Sound transmission loss of metamaterial-based thin plates with multiple subwavelength arrays of attached resonators, J. Sound Vib.331.25 (2012) 54085423.

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[18]M.Oudich, X. Zhou, M. B.Assouar, General analytical approach for sound transmission loss analysis through a thick metamaterial plate, J. Appl. Phys.116.19 (2014) 193509.

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[19]C.Claeys, P.Sas, W.Desmet, On the acoustic radiation efficiency of local resonance based stop band materials, J. Sound Vib. 333.14 (2014) 3203-3213.

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[20]C. Claeys, E. Deckers, B. Pluymers, W. Desmet, A lightweight vibro-acoustic metamaterial demonstrator: Numerical and experimental investigation, Mech. Syst. Signal Pr.70 (2016) 853-880.

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[21]A.Akay,Z. Xu, A. Carcaterra, I. M. Koç, Experiments on vibration absorption using energy sinks,J. Acoust. Soc. Am. 118.5 (2005) 3043-3049.

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[22]C.Claeys, K.Vergote, P.Sas, W.Desmet, On the potential of tuned resonators to obtain lowfrequency vibrational stop bands in periodic panels, J. Sound Vib.332.6 (2013) 1418-1436.

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[23] M.Nouh, O. Aldraihem, A. Baz., Wave propagation in metamaterial plates with periodic local resonances, J. Sound Vib.341 (2015): 53-73.

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[24]C. Claeys, NGR.MeloFilho, L. Van Belle, E.Deckers, W.Desmet, Design and validation of metamaterials for multiple structural stop bands in waveguides, Extreme Mech. Lett. 12 (2017) 7-22.

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[25]A. Nateghi,L. Van Belle, C. Claeys, E. Deckers, B. Pluymers, W. Desmet, Wave propagation in locally resonant cylindrically curved metamaterial panels, Int. J. Mech. Sci.127 (2016) 73-90.

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[26]D. J. Ewins, S. S. Rao, S. G. Braun, Encyclopedia of Vibration, Three-Volume Set. Academic press, 2002.

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[27]K. J. Bathe, Finite element procedures. New Jersey: Prentice-Hall; 1996.

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[28]M. I. Hussein, Reduced Bloch mode expansion for periodic media band structure calculations, P. Roy. Soc. A-Math. Phy. Vol. 465. No. 2109. (2009).

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[29]P.Langlet,A.C.Hladky-Hennion, J. N. Decarpigny, Analysis of the propagation of plane acoustic waves in passive periodic materials using the finite element method, J. Acoust. Soc. Am. 98.5 (1995) 2792-2800.

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[30]M. M. Sigalas, E. N. Economou, Elastic and acoustic wave band structure,J. Sound Vib. 158 (1992) 377-382.

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[31]H. J. Xiang, Z. F. Shi, Analysis of flexural vibration band gaps in periodic beams using differential quadrature method,Comput. Struct. 87.23-24 (2009) 1559-1566.

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[32]B. R.Mace, E. Manconi, Modelling wave propagation in two-dimensional structures using finite element analysis,J. Sound Vib. 318.4-5 (2008) 884-902.

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[33]E. Li, Z. C. He, G. Wang, An exact solution to compute the band gap in phononic crystals,Comp. Mater. Sci. 122 (2016) 72-85.

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545

[35] COMSOL Multiphysics Reference Manual, version 5.3, COMSOL, Inc, www.comsol.com

p ro

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535 536

546 547

Pr e-

548 549 550 551 552 553 554

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555 556

559 560 561 562 563 564 565

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557

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Highlights

567

Flexural band gap properties in a locally resonant metamaterial is investigated.

568

A continuum unit cell with a plate-like resonator is proposed.

569

An efficient finite element model in order to analyze the unit cell is developed.

570

The relation between parameters of the local resonator and band gaps are founded.

of

566

571 572

p ro

573 574 575 576 577

Pr e-

578 579 580 581 582 583 584

al

585 586

589 590 591 592 593 594 595 596 597

Jo

588

urn

587

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598

Declaration of interests

599 ☒ The authors declare that they have no known competing financial interests or personal

601

relationships that could have appeared to influence the work reported in this paper.

of

600

602

☐The authors declare the following financial interests/personal relationships which may

604

be considered as potential competing interests:

Pr e-

605

p ro

603

606 607 608 609

Jo

urn

al

610

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