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Archive of Applied Mechanics

, Volume 88, Issue 8, pp 1263–1274 | Cite as

Complete vibrational bandgap in thin elastic metamaterial plates with periodically slot-embedded local resonators

  • Jia-Hao He
  • Hsin-Haou Huang
Original
  • 274 Downloads

Abstract

This paper presents a metamaterial plate (metaplate) consisting of a periodic array of holes on a homogeneous thin plate with slot-embedded resonators. The study numerically proves that the proposed model can generate a complete vibrational bandgap in the low-frequency range. A simplified analytical model was proposed for feasibly and accurately capturing the dispersion behavior and first bandgap characteristics in the low-frequency range, which can be used for initial design and bandgap study of the metaplate. A realistic and practical unit metaplate was subsequently designed to verify the analytical model through finite element simulations. The metaplate not only generated a complete vibrational bandgap but also exhibited excellent agreement in both analytical and finite element models for predicting the bandgap characteristics. This study facilitates the design of opening and tuning bandgaps for potential applications such as low-frequency vibration isolation and stress wave mitigation.

Keywords

Elastic metamaterial plate Local resonance Complete bandgap Dispersion behavior Vibration isolation 

1 Introduction

The propagation of acoustic and elastic waves in manmade composite materials has attracted much interest because of their unique physical properties for potential applications. These materials, featuring specialized geometrical designs, are sometimes termed “metamaterials.” They have been developed to possess several unusual properties such as an anomalous refraction index, mass density, and Poisson’s ratio. Some can generate complete bandgaps that prohibit wave propagation in all directions. Liu et al. [1] proposed a local resonance structure utilizing lead balls coated with silicone and covered by cubic epoxy. When the frequency of the excitation force is close to the local resonance frequency, the effective mass density becomes negative [2]. This unusual property originates from the interaction between propagating waves and the resonance of core balls.

On the basis of the local resonance, soft materials can be substituted by springs from which the spring–mass system is developed [3]. Through theoretical formulation, the effective mass density can become negative or unbounded [4]. The negativity was attributed to wave interaction between the background materials and the added lateral local resonators. Bandgaps can be created using the local resonance effect, implying that when the wave frequency approaches the local resonance frequency of the system, the attempt for the wave to propagate inside the bandgap range results in exponential wave (amplitude) attenuation.

Beam theories have been applied for analyzing microstructural materials with the local resonance effect [5, 6, 7]. The induced local resonance effect is often narrowband. A multiresonator metamaterial concept has therefore been proposed to solve this problem and to extend the applicability [8, 9]. Although the bandgap range can be increased, wave interaction between multiple resonators generated new passbands in the bandgap. Chiral microstructures were proposed to simultaneously reduce the range of these passbands and widen the frequency range of the bandgap [10].

The concept of one-dimensional metamaterial beams was extended to two-dimensional (2D) models for studying the in-plane wave propagation characteristics in a 2D metamaterial plane [11, 12]. The effective mass density was found to be frequency dependent and can be dynamically anisotropic. These models continue to retain the characteristics of stress wave amplitude attenuation, which may aid in developing practical three-dimensional (3D) applications. A number of studies on metamaterial plates (metaplates) have been conducted [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. Low-frequency bandgaps were achieved by utilizing various microstructure designs and could lead to promising applications for suppression of low-frequency vibrations [29, 30, 31, 32, 33, 34, 35, 36].

Similar to most of the plate-type elastic metamaterials that employ the Bragg scattering and local resonant mechanisms [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], the present study also focuses on providing a simple yet reliable analytical approach for bandgap prediction and design. This article is organized as follows: An analytical model of the metaplate was proposed for predicting wave dispersion behavior. A realistic and practical unit metaplate was designed for verifying the analytical model by performing finite element (FE) simulations. Finally, parametric studies were conducted. Discussion and conclusions are presented in this paper.

2 Analytical model and dispersion relations

Consider a simplified model of a unit cell consisting of a rectangular metaplate with a central rectangular hole and a spring–mass resonator installed within the hole (Fig. 1a, b). The resonator comprises a central mass connected by two springs on the top and bottom. The resonator is connected to the rectangular hollow plate by four trusses. The top and bottom pairs of the trusses are placed in orthogonal orientations. Cartesian coordinates are defined so that x and y are the in-plane coordinates of the plate and z is the out-of-plane coordinate of the plate. \(L_x\) and \(L_y\) are the outer length and width of the plate, whereas \(l_x\) and \(l_y\) are the inner length and width of the plate, respectively (Fig. 1c, d).

The assumptions of the unit model are as follows: (1) The central mass of the resonator is a point mass. (2) The four trusses are rigid. (3) The central mass can freely move in all directions. (4) The displacement of the top node is the average of those on the opposite ends of the top pair of connected trusses, and the displacement of the bottom node is the average for the bottom pair of trusses.
Fig. 1

a, b Schematic view of the unit metaplate simplified model and c, d the corresponding coordinates and dimensions

Next, duplicate the aforementioned unit simplified model in the x and y directions in a periodic manner to form a plate with infinite units, and consider 3D wave propagation in the plate. The displacement fields of the plate corresponding to the x, y, and z axes are denoted as u, v, and w. Similarly, the displacement fields of the central mass, the top node, and the bottom node are denoted as (\(u_{2}\), \(v_{2}\), \(w_{2})\), (\(u_{1t}\), \(v_{1t}\), \(w_{1t})\), and (\(u_{1b}\), \(v_{1b}\), \(w_{1b})\), respectively.

The wave equations of this metaplate were derived by considering the wave equations of 2D longitudinal and flexural vibrations in rectangular isotropic plates [37, 38], and the wave equations of the metabeam with lateral local resonators [7], upon the simplifying assumption and approximation that the in-plane displacements u and v are independent of the out-of-plane degrees of freedom:
$$\begin{aligned}&\mathop \int \nolimits _{\frac{-L_y }{2}}^{\frac{L_y }{2}} \mathop \int \nolimits _{\frac{-L_x }{2}}^{\frac{L_x }{2}} H\alpha \mathrm{d}x\mathrm{d}y-\mathop \int \nolimits _{\frac{-l_y }{2}}^{\frac{l_y }{2}} \mathop \int \nolimits _{\frac{-l_x }{2}}^{\frac{l_x }{2}} H\alpha \mathrm{d}x\mathrm{d}y+k_{2xt} \left( {u_{1t} -u_2 } \right) +k_{2xb} \left( {u_{1b} -u_2 } \right) =0 \end{aligned}$$
(1)
$$\begin{aligned}&m_2 {\ddot{u}_2 } +k_{2xt} \left( {u_2 -u_{1t} } \right) +k_{2xb} \left( {u_2 -u_{1b} } \right) =0 \end{aligned}$$
(2)
$$\begin{aligned}&\mathop \int \nolimits _{\frac{-L_y }{2}}^{\frac{L_y }{2}} \mathop \int \nolimits _{\frac{-L_x }{2}}^{\frac{L_x }{2}} H\beta \mathrm{d}x\mathrm{d}y-\mathop \int \nolimits _{\frac{-l_y }{2}}^{\frac{l_y }{2}} \mathop \int \nolimits _{\frac{-l_x }{2}}^{\frac{l_x }{2}} H\beta \mathrm{d}x\mathrm{d}y+k_{2yt} \left( {v_{1t} -v_2 } \right) +k_{2yb} \left( {v_{1b} -v_2 } \right) =0 \end{aligned}$$
(3)
$$\begin{aligned}&m_2 {\ddot{v}_2 } +k_{2yt} \left( {v_2 -v_{1t} } \right) +k_{2yb} \left( {v_2 -v_{1b} } \right) =0 \end{aligned}$$
(4)
$$\begin{aligned}&\mathop \int \nolimits _{\frac{{ - {L_y}}}{2}}^{\frac{{{L_y}}}{2}} \mathop \int \nolimits _{\frac{{ - {L_x}}}{2}}^{\frac{{{L_x}}}{2}} \left( {\rho H\ddot{w} + D{\nabla ^2}{\nabla ^2}w} \right) \mathrm{d}x\mathrm{d}y - \mathop \int \nolimits _{\frac{{ - {l_y}}}{2}}^{\frac{{{l_y}}}{2}} \mathop \int \nolimits _{\frac{{ - {l_x}}}{2}}^{\frac{{{l_x}}}{2}} \left( {\rho H\ddot{w} + D{\nabla ^2}{\nabla ^2}w} \right) \mathrm{d}x\mathrm{d}y \nonumber \\&\quad + {k_{2zt}}\left( {{w_{1t}} - {w_2}} \right) + {k_{2zb}}\left( {{w_{1b}} - {w_2}} \right) = 0 \end{aligned}$$
(5)
$$\begin{aligned}&m_2 {\ddot{w}_2 } +k_{2zt} \left( {w_2 -w_{1t} } \right) +k_{2zb} \left( {w_2 -w_{1b} } \right) =0 \end{aligned}$$
(6)
where \(\nabla ^{2}=\frac{\partial ^{2}}{\partial x^{2}}+\frac{\partial ^{2}}{\partial y^{2}}\) is the Laplacian, and \(\alpha \) and \(\beta \) are defined as:
$$\begin{aligned}&\alpha = - \mu {\nabla ^2}u - \left( {\lambda + \mu } \right) \frac{\partial }{{\partial x}}\left( {\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}}} \right) + \rho \ddot{u} \end{aligned}$$
(7)
$$\begin{aligned}&\beta = - \mu {\nabla ^2}v - \left( {\lambda + \mu } \right) \frac{\partial }{{\partial y}}\left( {\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}}} \right) + \rho \ddot{v} \end{aligned}$$
(8)
For the plate, \(\lambda =\nu E/\left( {1-2\nu } \right) \left( {1+\nu } \right) \) and \(\mu =E/2\left( {1+\nu } \right) \) are the Lamé constants; \(D=2h^{3}E/3\left( {1-\nu ^{2}} \right) \) is the bending stiffness, where \(h=H/2\); H, \(\rho \), E, and \(\nu \) are the thickness, mass density, Young’s modulus, and Poisson’s ratio of the plate, respectively. For the resonator, \(m_{2}\) is the mass and \(k_{2}\) is the stiffness of the springs. Specifically, for the springs, the subscripts “t” and “b” denote the top and the bottom springs, respectively, whereas the subscripts “x,” “y,” and “z” denote the properties associated with their corresponding coordinates. For example, \(k_{2zt} \) is the stiffness of the top spring associated with the force and displacement components in the z direction.
The elastic waves propagating through the metaplate unit can be assumed to have displacement fields as follows by considering only a representative unit and employing the Bloch–Floquet theory:
$$\begin{aligned}&{{\varvec{U}}}\left( {x,y,t} \right) ={{\varvec{P}}}e^{i\left( {q_x x+q_y y-{\Omega }t} \right) } \end{aligned}$$
(9)
$$\begin{aligned}&{{\varvec{U}}}_\mathbf{2} \left( {x,y,t} \right) ={{\varvec{P}}}_\mathbf{2} e^{i\left( {q_x x+q_y y-{\Omega }t} \right) } \end{aligned}$$
(10)
where \({{\varvec{U}}}=\left\{ {u,v,w} \right\} \), \( {{\varvec{P}}}=\left\{ {P_x ,P_y ,P_z } \right\} \), \({{\varvec{U}}}_\mathbf{2} =\left\{ {u_2 ,v_2 ,w_2 } \right\} \), and \({{\varvec{P}}}_\mathbf{2} =\left\{ {P_{2x} ,P_{2y} ,P_{2z} } \right\} \); \(P_{x}\), \(P_{y}\), \(P_{z}\), \(P_{2x}\), \(P_{2y}\), and \(P_{2z}\) are constant amplitudes; \(q_{x}\) and \(q_{y}\) are the wavenumbers for wave propagation in the x and y directions, respectively; \(\varOmega \) is the wave frequency.
One of the assumptions of the model states that the displacements at the top and bottom nodes of the resonator are completely dependent on those of the connected boundaries of the plate. In other words, the displacement field \(u_{1t} \left( t \right) \), for instance, is given by
$$\begin{aligned} u_{1t} \left( t \right) =\frac{1}{2}\left[ {u\left( {\frac{L_x }{2},0,t} \right) +u\left( {\frac{-L_x }{2},0,t} \right) } \right] =P_x e^{-i{\Omega }t}\left( {\cos \frac{q_x L_x }{2}} \right) \end{aligned}$$
(11)
In short, the displacements of all degrees of freedom at the top and bottom nodes are obtained as
$$\begin{aligned}&{{\varvec{U}}}_{\mathbf{1}{\varvec{t}}} ={{\varvec{P}}}e^{-i{\Omega }t}\left( {\cos \frac{q_x L_x }{2}} \right) \end{aligned}$$
(12)
$$\begin{aligned}&{{\varvec{U}}}_{\mathbf{1}{\varvec{b}}} ={{\varvec{P}}}e^{-i{\Omega }t}\left( {\cos \frac{q_y L_y }{2}} \right) \end{aligned}$$
(13)
Substituting Eqs. (9), (10), (12), and (13) into Eqs. (1)–(6) and using complex trigonometric formulae yields six homogenous equations for the constant amplitude \(P_{i}\). For a nontrivial set of solutions, the determinant of the coefficients must vanish; that is,
$$\begin{aligned} \left| {{\begin{array}{llllll} {Q_{11} }&{}\quad {Q_{12} }&{}\quad {Q_{13} }&{} \quad {Q_{14} }&{}\quad {Q_{15} } &{}\quad {Q_{16} } \\ {Q_{21} }&{} \quad {Q_{22} }&{} \quad {Q_{23} }&{}\quad {Q_{24} }&{}\quad {Q_{25} }&{} \quad {Q_{26} } \\ {Q_{31} }&{}\quad {Q_{32} }&{} \quad {Q_{33} }&{} \quad {Q_{34} }&{} \quad {Q_{35} }&{} \quad {Q_{36} } \\ {Q_{41} }&{}\quad {Q_{42} }&{}\quad {Q_{43} }&{}\quad {Q_{44} }&{}\quad {Q_{45} } &{} \quad {Q_{46} } \\ {Q_{51} }&{}\quad {Q_{52} }&{}\quad {Q_{53} }&{} \quad {Q_{54} } &{}\quad {Q_{55} }&{} \quad {Q_{56} } \\ {Q_{61} }&{} \quad {Q_{62} }&{} \quad {Q_{63} }&{} \quad {Q_{64} }&{} \quad {Q_{65} }&{}\quad {Q_{66} } \\ \end{array} }} \right| =0 \end{aligned}$$
(14)
where
$$\begin{aligned} Q_{11}= & {} \frac{4H}{q_x q_y }\left[ {\mu \left( {q_x ^{2}+q_y ^{2}} \right) +\left( {\lambda +\mu } \right) q_x ^{2}-\rho \varOmega ^{2}} \right] \left( {\sin \frac{q_x L_x }{2}\sin \frac{q_y L_y }{2}-\sin \frac{q_x l_x }{2}\sin \frac{q_y l_y }{2}} \right) \nonumber \\&+\,k_{2xt} \cos \frac{q_x L_x }{2}+k_{2xb} \cos \frac{q_y L_y }{2} \end{aligned}$$
(15)
$$\begin{aligned} Q_{12}= & {} -k_{2xt} -k_{2xb} \end{aligned}$$
(16)
$$\begin{aligned} Q_{13}= & {} Q_{31} =4H\left( {\lambda +\mu } \right) \left( {\sin \frac{q_x L_x }{2}\sin \frac{q_y L_y }{2}-\sin \frac{q_x l_x }{2}\sin \frac{q_y l_y }{2}} \right) \end{aligned}$$
(17)
$$\begin{aligned} Q_{21}= & {} -k_{2xt} \cos \frac{q_x L_x }{2}-k_{2xb} \cos \frac{q_y L_y }{2} \end{aligned}$$
(18)
$$\begin{aligned} Q_{22}= & {} -m_2 \varOmega ^{2}+k_{2xt} +k_{2xb} \end{aligned}$$
(19)
$$\begin{aligned} Q_{33}= & {} \frac{4H}{q_x q_y }\left[ {\mu \left( {q_x ^{2}+q_y ^{2}} \right) +\left( {\lambda +\mu } \right) q_y ^{2}-\rho \varOmega ^{2}} \right] \left( {\sin \frac{q_x L_x }{2}\sin \frac{q_y L_y }{2}-\sin \frac{q_x l_x }{2}\sin \frac{q_y l_y }{2}} \right) \nonumber \\&+\,k_{2yt} \cos \frac{q_x L_x }{2}+k_{2yb} \cos \frac{q_y L_y }{2} \end{aligned}$$
(20)
$$\begin{aligned} Q_{34}= & {} -k_{2yt} -k_{2yb} \end{aligned}$$
(21)
$$\begin{aligned} Q_{43}= & {} -k_{2yt} \cos \frac{q_x L_x }{2}-k_{2yb} \cos \frac{q_y L_y }{2} \end{aligned}$$
(22)
$$\begin{aligned} Q_{44}= & {} -m_2 \varOmega ^{2}+k_{2yt} +k_{2yb} \end{aligned}$$
(23)
$$\begin{aligned} Q_{55}= & {} \frac{4}{q_x q_y }\left[ {D\left( {q_x ^{4}+2q_x ^{2}q_y ^{2}+q_y ^{4}} \right) -\rho H{\Omega }^{2}} \right] \left( {\sin \frac{q_x L_x }{2}\sin \frac{q_y L_y }{2}-\sin \frac{q_x l_x }{2}\sin \frac{q_y l_y }{2}} \right) \nonumber \\&+\,k_{2zt} \cos \frac{q_x L_x }{2}+k_{2zb} \cos \frac{q_y L_y }{2}\end{aligned}$$
(24)
$$\begin{aligned} Q_{56}= & {} -k_{2zt} -k_{2zb} \end{aligned}$$
(25)
$$\begin{aligned} Q_{65}= & {} -k_{2zt} \cos \frac{q_x L_x }{2}-k_{2zb} \cos \frac{q_y L_y }{2} \end{aligned}$$
(26)
$$\begin{aligned} Q_{66}= & {} -m_2 {\Omega }^{2}+k_{2zt} +k_{2zb} \end{aligned}$$
(27)
with the remaining components vanished. Expansion of the determinant in Eq. (14) therefore leads to the dispersion equation relating the wavenumbers \(q_x \) and \(q_y \) and the wave frequency \(\varOmega \).

3 Model verification and discussion

3.1 FE model and methods for calculation

Next, a practical design of the unit metaplate is proposed for verification. The ability of the proposed simplified model to convert to a physical model is critical, yet the physical model still retains capabilities for producing essential characteristics that are predicted by the simplified model. The proposed prototypic unit metaplate consists of a caved square thin plate and a resonator with a central cylindrical mass connected to the plate by two wavy springs on the top and bottom (Fig. 2a). Detailed dimensions of the prototypic unit metaplate are illustrated in Fig. 2b–d.
Fig. 2

a Schematic view of a unit cell of the prototypic metaplate and bd dimensions of the prototypic metaplate (mm)

A series of calculations on the dispersion relations of a metaplate that consists of infinite unit cells in both the x and y directions were conducted through the FE method. Because of the periodicity of the infinite metaplate, the calculation can be implemented in the representative unit cell (Fig. 2a). The eigenvalue equations in the unit cell are given by
$$\begin{aligned} \left( {{\varvec{K}}-\omega ^{2}{\varvec{M}}} \right) {\overline{{\varvec{U}}}}=0 \end{aligned}$$
(28)
where \({\overline{{\varvec{U}}}}\) is the displacement vector at the nodes and \({{\varvec{K}}}\) and \({{\varvec{M}}}\) are the stiffness and mass matrices of the unit cell, respectively. According to the Bloch–Floquet theory, periodicity boundary conditions were applied at the boundaries between the unit cell and its four adjacent cells:
$$\begin{aligned} {\overline{{\varvec{U}}}}\left( {{\varvec{r}}+{\varvec{a}}} \right) =e^{i\left( {{\varvec{q}} \cdot {\varvec{a}}} \right) } {\overline{{\varvec{U}}}}\left( {{\varvec{r}}} \right) \end{aligned}$$
(29)
where \({{\varvec{r}}}\) is the position vector, \({{\varvec{a}}}\) denotes the basis vector of the periodic structure, and \({{\varvec{q}}}\) is the 2D wave vector.

Commercial FE analysis software, COMSOL Multiphysics 5.2, was employed for solving the eigenvalue equations Eq. (28) under complex boundary conditions Eq. (29) for the physical model in Fig. 2a. Material properties for the prototypic metaplate are as follows. For the thin plate, SS400 steel plate was used, with Young’s modulus = 190 GPa, Poisson’s ratio = 0.3, and \(\hbox {density} = 7850\hbox { kg}/\hbox {m}^{3}\). For the resonator, SUS301 stainless steel was used for the wavy springs, with Young’s modulus = 196 GPa, Poisson’s ratio = 0.29, and \(\hbox {density} = 7930\hbox { kg}/\hbox {m}^{3}\); iron was used for the cylindrical mass unit, with Young’s modulus = 200 GPa, Poisson’s ratio = 0.29, and \(\hbox {density} = 7870\hbox { kg}/\hbox {m}^{3}\). Bloch–Floquet periodical boundary conditions were imposed on the opposite boundaries of the unit cell for determination of the dispersion relations on the contour (\({{\varvec{\Gamma }}}\rightarrow \mathbf{X}\rightarrow \mathbf{M}\rightarrow {{\varvec{\Gamma }} })\) of the irreducible Brillion zone (Fig. 1b). For the scanning path from \({{\varvec{\Gamma }}}\) to X, for instance, the wavenumber \(q_{x}\) varied from 0 to \(\pi /L_{x}\). The unit cell was meshed using a tetrahedral mesh with the Lagrange quadratic elements provided.

3.2 Results and comparison

The dispersion curves of the prototypic metaplate were obtained using Eqs. (14)–(27) for comparison. Geometrical and material properties of the prototypic metaplate used for the equations are as follows: For the plate, \(L_x =L_y = 0.08\hbox { m}\), \(l_x =l_y = 0.05\hbox { m}\), \(H = 0.0015\) m, \(E = 190\) GPa, \(\rho = 7850\hbox { kg}/\hbox {m}^{3}\), \(\upnu = 0.3\) and \(c_T =3138.5\hbox { m}/\hbox {s}\). For the resonator, the cylindrical mass was replaced by a point mass such that \(\hbox {m}_2 = 0.025\) kg. The effective spring constants of the wavy spring \(k_2\) in all directions required additional testing. For simplicity, modal analyses of a single unit were performed [9]. The resonance frequencies were employed for determining \(k_2 \) values and yielded \(k_{2xt} =k_{2yb} = 6.18\times 10^{4}\hbox { N}/\hbox {m}\), \(k_{2xb} =k_{2yt} = 6.09\times 10^{4}\hbox {N}/\hbox {m}\), and \(k_{2zt} =k_{2zb} = 6.88\times 10^{4}\hbox { N}/\hbox {m}\). The local resonances of the resonator in all directions were readily found using
$$\begin{aligned} f_{0j} =\frac{1}{2\pi }\sqrt{\frac{\left( {k_{2jt} +k_{2jb} } \right) }{m_2 }} \end{aligned}$$
(30)
such that \(f_{0x} = 353.9\hbox { Hz}\), \(f_{0y} = 351.3\hbox { Hz}\), and \(f_{0z} = 373.4\hbox { Hz}\).
Dispersion curves from the analytical expression of the simplified model and from the FE simulations of the physical model were compared (Fig. 3). The vertical axis of the plot illustrates both the absolute and relative frequencies. The relative frequency is defined as \({\upomega a}/c_T \), where \(\upomega \) is the wave frequency, a is the lattice constant, and \(c_T \) is the transversal wave speed of plate. For instance, the normalized frequency equal to 0.06 represents the absolute frequency of \(f_{0z} = 373.4\hbox { Hz}\). In the plot, blue dots represent results from FE simulation, and red solid lines denote those from analytical expression Eq. (14). The present analytical equation captures six vibration modes of the metaplate associated with the degrees of freedom in a representative unit; therefore, only the lowest six bands of the FE results were intentionally selected for comparison. A number of observations can be made from Fig. 3: First, favorable agreement between the theoretical and FE results for the lowest three bands is evident (one longitudinal, one in-plane transverse shear, and one out-of-plane flexural), but not for the highest three bands; second, a complete bandgap was generated that was associated with the local resonances; third, favorable agreement of the bandgap range between the two models in the present case is evident. (The complete bandgap predicted by the theoretical model ranges 373–440 Hz, and that by the FE simulation 370–440 Hz.)
Fig. 3

Dispersion curves obtained from the analytical model (red solid lines) and FE model (blue dots)

The reasons for discrepancy at the high-frequency bands between the two models were investigated. Six bands of the FE dispersion curves were termed Modes 1–6, and the frequency ranges of Modes 1–6 are 0–370 Hz, 0–355 Hz, 0–348 Hz, 440–1131 Hz, 441–1109 Hz, and 460–951 Hz, respectively (Fig. 3). The pattern of the terminology is seemingly random. In fact, the order was defined from the lowest frequency to the highest frequency at the same wavenumber close to \({{\varvec{\Gamma }}}\).

The eigenmode shapes of Modes 1–6 corresponding to three directions of the irreducible Brillion zone are shown in Fig. 4. The snapshot of the eigenmode shapes was determined when the normalized wavenumbers were set at \(q_x L_x =\pi /2\) and \(q_y L_y =0\) in the path A: \({{\varvec{\Gamma }}}\rightarrow \mathbf{X}\) direction, \(q_x L_x =\pi \) and \(q_y L_y =\pi /2\) in the path B: \(\mathbf{X}\rightarrow \mathbf{M}\) direction, and \(q_x L_x =\pi /2\) and \(q_y L_y =\pi /2\) in the path C: \(\mathbf{M}\rightarrow {{\varvec{\Gamma }}}\) direction, respectively. All eigenmode shapes were viewed normal to the \(x\hbox {-}z\) plane, and both the deformed (colored body) and undeformed (black-lined body) shapes were displayed (Fig. 4; Mode 1 in the \({{\varvec{\Gamma }}}\rightarrow \mathbf{X}\) direction is the most illustrative demonstration of the two shapes).
Fig. 4

Side views of eigenmode shapes of Modes 1–6 (in xz plane). The normalized wavenumbers were set at \(q_x L_x =\pi /2\) and \(q_y L_y =0\) in the \({{\varvec{\Gamma }}}\rightarrow \mathbf{X}\) direction (denoted Path A); \(q_x L_x =\pi \) and \(q_y L_y =\pi /2\) in the \(\mathbf{X}\rightarrow \mathbf{M}\) direction (denoted Path B); and \(q_x L_x =\pi /2\) and \(q_y L_y =\pi /2\) in the \(\mathbf{M}\rightarrow {{\varvec{\Gamma }}}\) direction (denoted Path C)

Mode 1 is clearly the out-of-plane flexural vibration mode with in-phase motions of the resonator and the plate in the z direction. Both the motions of the resonator and the plate are obvious in the \({{\varvec{\Gamma }}}\rightarrow \mathbf{X}\) and \(\mathbf{M}\rightarrow {{\varvec{\Gamma }} }\) directions. In the \(\mathbf{X}\rightarrow \mathbf{M}\) direction, however, the majority of vibrational energy is transmitted by the motion of the resonator, whereas little is transmitted by the motion of the plate. Modes 2 and 3 are in-plane longitudinal and in-plane transverse shear vibration modes in the \({{\varvec{\Gamma }}}\rightarrow \mathbf{X}\) direction, respectively. The propagation modes exchange for these two modes in the \(\mathbf{X}\rightarrow \mathbf{M}\) direction. Both modes become mixed modes when propagated in the \(\mathbf{M}\rightarrow {{\varvec{\Gamma }}}\) direction. Interestingly, the eigenmode shapes of Mode 2 (or Mode 3) appear alike in all directions from Fig. 4. They, in fact, differ from each other in magnitude. These three low-frequency modes can be accurately predicted by the proposed analytical model and should entail few problems.

Next, consider the three high-frequency modes. The majority of vibrational motion of Mode 4 is contributed by the wavy springs with negligible motion of the plate. Mode 5 has similar motions with only slight motion of the plate. The local motions of the wavy springs cannot be predicted by the proposed analytical model because the only information on the wavy springs preserved in the analytical model is the stiffness. This is the main reason why Modes 4 and 5 of the FE results do not match any of the modes of the analytical results. Finally, Mode 6 is the out-of-plane flexural vibration mode with out-of-phase motions of the resonator and the plate in the z direction, meaning that when the mass of the resonator vibrates in the positive z direction, the plate vibrates in the negative z direction. Although deviation exists in short-wavelength ranges, this mode can still be accurately predicted by the analytical model in the long-wavelength limit to some extent.

3.3 Model verification with various cases

We further verified the proposed analytical dispersion relations with various FE cases. The case presented in Sect. 3.1 is defined as the base case. Eight cases were performed for comparison. In each case, only one material property was changed as compared to the base case. Cases A and B present the cases, whereas the plate densities are half and twice from the base case, respectively. Cases C and D present the cases with half and twice the Young’s moduli of the base case. For Cases E and F, the widths of the wavy springs were chosen as 6.5 and 15 mm, respectively. In the last two cases, Case G and Case H, the central masses were halved and doubled, respectively. Similarly, in all Fig. 5, blue dots represent the dispersion relations from FE simulation, and red solid lines denote those from analytical expression Eq. (14). For all cases, the results suggested favorable agreement between FE model and analytical model, especially in the long-wavelength limit. Moreover, in most cases as in the case with reported results in Sect. 3.1, the first four low-frequency bands of two approaches show excellent agreement, indicating that the proposed model is a potential candidate for initial metaplate design for complete bandgap applications.
Fig. 5

Various dispersion curves for verification: a, b by varying the density of the plate, c, d by varying the Young’s modulus of the plate, e, f by varying the width of the wavy springs, and g, h by varying the central mass

4 Parametric study of effects on the first bandgap

The effects of material and geometric parameters on the first bandgap are of particular interest at the design stage. Because the proposed analytical model agrees favorably with the FE model, particularly on the characteristics of the first bandgap, this section presents the effects of material and geometric parameters on the first bandgap observed using the proposed analytical model for convenience. Without loss of generality, the material and geometric properties employed in Sect. 3.2 were used in the parametric study: for the plate, \(L_{x}=L_{y} = 0.08\hbox { m}\), \(l_{x}=l_{y} = 0.05\hbox { m}\), \(H = 0.0015\hbox { m}\), \(E = 190\hbox { GPa}\), \(\rho = 7850\hbox { kg}/\hbox {m}^{3}\), and \(\nu = 0.3\); for the resonator, \(m_{2} = 0.025\hbox { kg}\), \(k_{2xt}=k_{2yb} = 6.18\times 10^{4}\hbox { N}/\hbox {m}\), \(k_{2xb}=k_{2yt} = 6.09\times 10^{4}\hbox { N}/\hbox {m}\), and \(k_{2zt}=k_{2zb} = 6.88\times 10^{4}\hbox { N}/\hbox {m}\).

4.1 Effects of the material constants of the resonator on the first bandgap

First, the influence of the mass of the resonator was investigated. Dispersion curves of the proposed metaplate with various values of the mass of the resonator ranging 0.2–5 \(m_{2}\) were calculated, where the original \(m_{2}\) assumes the value 0.025 kg and the other material and geometric parameters remain unchanged. Figure 6a presents the ranges of the first bandgap against the multiples of \(m_{2}\). The “lowerband upper limit” represents the maximum frequency of the low-frequency bands (in the case of Fig. 3, these bands are Modes 1, 2, and 3), and the “higherband lower limit” represents the minimum frequency of the high-frequency bands (in the case of Fig. 3, these bands are Modes 4, 5, and 6). The ranges between the lowerband upper limit and the higherband lower limit are seemingly the ranges of the first bandgaps. The width in the frequency range of the bandgap increases as \(m_{2}\) grows, and the bandgap frequency drops as \(m_{2}\) grows, but when the mass is too small (e.g., 0.2 \(m_{2})\), the bandgap vanishes.

Next, only the stiffness of the springs (\(k_{2}\)) was varied and all the remaining material and geometric parameters were set constant. Two cases were analyzed. First, the values of the stiffness \(k_{2}\) assumed in Sect. 4 were set as the original stiffnesses. All \(k_{2}\) values were changed simultaneously with the multiples ranging from 0.2 to 5, and all other parameters were fixed. Figure 6b illustrates the evolution of the bandgap as a function of the stiffness of the resonators. The width in the frequency range of the bandgap gradually increases as \(k_{2}\) grows, and the bandgap frequency increases as \(k_{2}\) grows.

Moreover, FE simulations were performed for comparison. The FE results are presented as blue dashed lines in Fig. 6a, b. The overall trend predicted by the theoretical model agrees with that by the FE simulations. In short, changing the mass or the stiffness of the resonator could affect both the upper and lower bounds of the first bandgap and could even lead to vanishing bandgaps.
Fig. 6

a Effect of the mass of the resonator on the boundaries of the first bandgap. b Effect of the stiffnesses of the resonator on the boundaries of the first bandgap. In this case, all \(k_{2}\) values were changed simultaneously with the same multiple each time. The red solid lines and the blue dashed lines represent the results of theoretical modeling and FE simulations, respectively

4.2 Effects of the dimensions of the plate on the first bandgap

Of interest is the effect of the geometrical parameters of the plate on the first bandgap. Figure 7 illustrates the variations of the plate dimensions plotted against the bandgap range. The values of the y-axis in the four subfigures are kept identical for comparison purposes. Obviously, reducing the thickness of the plate (H) or the dimensions of the metaplate unit (\(L_{x}\) and \(L_{y})\), or enlarging the dimensions of the hole (\(l_{x}\) and/or \(l_{y})\), widens the frequency range of the first bandgap. In short, changing the geometric parameters of the plate has little effect on the lower bound frequency of the bandgap and changes the width in the frequency range of the bandgap by shifting the upper bound frequency of the bandgap. Moreover, changing the length of the plate and hole can effectively increase the width in the frequency range.
Fig. 7

Evolution of bandgap ranges as a function of a thickness H, b dimensions \(L_{x}\) and \(L_{y}\), c dimensions \(l_{x}\) and \(l_{y}\), and d dimension \(l_{x}\) (or \(l_{y})\) of the plate. The red solid lines and the blue dashed lines represent the results of theoretical modeling and FE simulations, respectively

5 Conclusion

This work proposes an analytical plate-lattice model for investigating the vibrational bandgap in the metaplate. By using plate theory and wave equations, the equations for expressing the motion of the infinite system were subsequently obtained so that the derived dispersion relations could be used for bandgap study. A prototypic metaplate with realistic physical dimensions was originally designed, numerically studied, and thoroughly compared with the analytical model. Analysis regarding discrepancies between the theory and the simulation results at the high-frequency region followed. Parametric studies were conducted to facilitate bandgap design. The results revealed a complete vibrational bandgap in the proposed metaplate. Moreover, the numerical results indicated favorable agreement on the characteristics of the first bandgap between the proposed analytical model and the FE model, meaning that the proposed analytical model is feasible for initial metaplate design for complete bandgap applications. With appropriate modification, the proposed analytical approach can be quickly employed on similar types of elastic metaplates for further bandgap designs and applications.

Notes

Acknowledgements

HH Huang acknowledges the support (Grant No. 106-2221-E-002-018-MY3) provided by the Ministry of Science and Technology (MOST), Taiwan.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Engineering Science and Ocean EngineeringNational Taiwan UniversityTaipeiTaiwan

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