A Comparative Study and Analysis of Layered-Beam and Single-Beam Metamaterial Structures: Transmissibility Bandgap Development

Featured Application

Analysis from this work will be helpful for engineers and researchers designing beam-type metamaterial structures for engineering applications as vibration isolation energy harvesters.

Abstract

Recently, layered-beam metamaterial structures have been gaining popularity in a variety of engineering applications including energy harvesting and vibration isolation. While both single-beam metamaterial structures and layered-beam metamaterial structures are capable of generating bandgaps, it is important to understand the limitations of each type of metamaterial structure in order to make informed design decisions. In this article, a comparative study of bandgap development in single-beam metamaterial structures and layered-beam metamaterial structures is presented. The results show that for the single-beam metamaterial structure, with equally spaced local resonator designs, only one significant bandgap is developed at approximately 300–415 Hz. This bandgap occurs near the resonant frequency of the local resonators, i.e., 309 Hz. The results also show that when the spacing and the design of the local resonators are desired to remain fixed, layering the horizontal beams offers a significant pathway for both lowering the bandgap and developing additional bandgaps. The double-layered beam-type metamaterial structure studied in this work generates two bandgaps at approximately 238–275 Hz and 298–410 Hz. When the goal is to keep the number of local resonators per beam constant, increasing the length of the unit cells offers an alternative technique for lowering the bandgaps.

1. Introduction

The development of transmissibility bandgaps from beam-type metamaterial structures has recently attracted much attention in the literature [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. These metamaterial structures with their resulting bandgaps have found multiple potential applications ranging from simple vibration isolation systems to energy harvesting systems and wave-guiding systems [9,14,16,18,19,20,21,22,23]. Transmissibility bandgaps in metamaterials are formed at frequencies when external vibrations are stopped from passing through the metamaterial structures [5,21,24]. There is extensive evidence in the literature that these vibration attenuation bandgaps can be formed from two key phenomenon: Bragg’s scattering and local resonators [24]. For beams, Bragg’s scattering bandgaps result from the out-of-phase movement of adjacent unit cells of the mechanical beams [24]. The Bragg’s scattering bandgaps are influenced by the design of the mechanical beam metamaterial structures. Moreover, the advent of local resonators contributed to greater flexibility in either lowering or expanding the bandgaps generated from metamaterial beam structures [6,24]. These local resonator (LR) bandgaps result from the design of the local resonators and typically occur at frequencies around the resonant frequencies of the resonators [6,7,14,18,21,24,25].

Recently, various groups have worked on developing different designs for the local resonators in order to vary and control the bandgaps of these mechanical metamaterial beams. For instance, in his pioneer work, Yu et al. modeled the bandgap characteristics of a simple beam-type metamaterial structure using Timoshenko beam model [7]. The results of their study showed that a single-beam metamaterial structure with attached local resonators could isolate vibrations at frequencies around the resonant frequency of these local resonators. Building on this effort, Zhou et al. showed that the bandgap of a simple Euler–Bernoulli beam could be shifted to lower frequencies by adding negative stiffness elements to the local resonators [18]. In another work, Nouh et al. showed that a beam with local resonator masses inserted into cavities that were filled with viscoelastic membrane had the ability to attenuate vibrations at low frequencies, i.e., less than 100 Hz [4]. Pai et al. presented the use of two mass-spring-damper subsystems serving as multi-frequency local resonators at different locations in generating wide vibration stopbands [1]. Additionally, a chiral-lattice-based elastic metamaterial was investigated by Zhu et al. [26]. Their results demonstrated the ability to generate multiple bandgaps that were dependent on the distribution density of the local resonators in different sections of the structure. Bandgaps through a quasi one-dimensional structure comprised of harmonic oscillators have also been studied by Wang et al. [27]. Their results showed the existence of a single bandgap in the high frequency region. Wang et al. demonstrated the coexistence of both local resonance and Bragg-scattering bandgaps in a beam with continuum beam resonators [24]. Moreover, the effect of nonlinear local resonators on bandgaps of a single-beam metamaterial structure was studied by Xia et al. [28]. The nonlinear local resonators were comprised of magnetoelastic materials as bistable attachments [28]. Their work showed that the inclusion of nonlinear local resonators resulted in wider bandgaps than their linear counterparts. The aforementioned efforts demonstrate the growing interest in lowering frequency bandgaps and generating multiple transmissibility bandgaps in metamaterial structures. To achieve this goal, researchers have resorted to a variety of design changes in the beam structure or local resonator.

The pioneer work done by Yu et al. demonstrated the possibility of bandgap generation in a simple Timoshenko beam with evenly spaced local resonators [7]. The single Timoshenko beam developed had eight unit cells and a total length of 0.6 m. The model of an application of white noise to the left end of the free-free single beam within a frequency range of 0–800 Hz showed the existence of a single bandgap from 310 to 498 Hz. While the work of Yu et al. paved the way for more recent advancements, research interests in developing lower frequency bandgaps and multiple frequency bandgaps still exists [1,4]. This is a crucial issue because most mechanical vibrations occur at frequencies lower than 300 Hz. Thus, lower frequency bandgap structures are desirable in engineering practice and real-world applications [21]. To this end, in order to shift the bandgap frequencies of a single beam, Zhou et al. developed a design of a single-beam metamaterial structure with high-static-low-dynamic-stiffness local resonators [18]. The design parameters adopted by Zhou et al. [18] were based on the single-beam structure pioneered by Yu et al. [7]. Their resonator model combined a vertical spring with two horizontal springs that added negative stiffness in the vertical direction. The physical Euler–Bernoulli beam model developed by Zhou et al. consisted of eight unit cells [18]. Each unit cell had a vertical local resonator with oblique springs connected to the local resonator. The results of the work done by Zhou et al. showed the existence of a single transmissibility bandgap which agreed with the work done by Yu et al. [7,18]. Moreover, Zhou et al. showed that the single-beam bandgap shifted to a lower frequency region when the negative stiffness from the horizontal oblique springs was factored into the forces from the local resonators. In essence, Zhou et al. successfully showed, through numerical modeling, the possibility of shifting the bandgap of single Euler–Bernoulli beams with local resonators to lower frequencies. This was achieved by adding negative stiffness elements to the resonators. Nonetheless, it is worth noting that only a single transmissibility bandgap was observed in this study.

The aforementioned studies have focused on studying single-beam metamaterial structures and lowering their bandgaps to lower frequencies. Recent interests in developing metamaterial structures as dual-purpose vibration attenuation energy harvesting systems show the importance of beam-type metamaterial structures [1,4,9,14,16,21,22,23]. The use of single-beam metamaterial structures in dual-purpose structures has been reported in multiple recent articles, particularly, as a simultaneous vibration isolation energy harvesting [8,23]. Wang et al., for example, used rainbow trapping effect in a metamaterial beam to isolate vibrations and concurrently harvest these free vibrations [16]. In addition, Carrara et al. converted structure-borne energy on a metamaterial single-beam configuration into electric power using piezoelectric materials [29].

In addition to the aforementioned efforts, more recently, researchers have given more attention to layered-beam metamaterial structures and subsequent applications in vibration attenuation energy harvesting structures [14,21]. Layered metamaterial structures are metamaterial structures with multiple horizontal beams arranged in layers and linked up with each other through vertical beams. For such layered metamaterial structures, the local resonators are replicated on each beam in a continuous pattern. Layered metamaterials find unique potential in situations where more unit cells are needed and elongation of the single beam is impracticable. The application of layered metamaterial beams in dual-purpose vibration isolation energy harvesting using piezoelectricity was demonstrated by Li et al. [21]. In their work, layered metamaterial beams were designed and piezoelectric materials were placed on each local resonator for conversion of localized vibrations into electric power. Their results showed that electric energy could be aggregated from conversion of vibrations trapped in multiple local resonators in all layers of the metamaterial beams. Additionally, Anigbogu et al. reported a 3D printed layered metamaterial beam for synchronous vibration attenuation energy harvesting using magnetomotive effect [14]. Coils were placed on the tips of all local resonators and used to convert the vibrations from the local resonators into useful electric power around the resonant frequencies of these resonators.

The key interest of the current work is to investigate the possible advantages offered by layered-beam metamaterial structures over typical single-beam metamaterial structures of the same kind [7,18,28]. Specifically, the current study investigates improvements in transmissibility bandgaps offered by layered-beam metamaterial structures in terms of lowering the frequency bandgaps, increasing the transmissibility depth, and opening multiple bandgaps. Since layered-beam metamaterial structures with local resonators are demonstrating growing engineering potential, it is important to study the advantages in bandgap generation offered by these layered metamaterial structures over typical single-beam metamaterial structures. In the application of metamaterial beam structures as dual-purpose systems for electric power generation, the more the local resonators (unit cells) per miniature device, the more the aggregate electric power generation potential of the device. This is because the electromechanical transducers are usually placed inside the local resonators. In such cases, having a layered metamaterial structure is physically an attractive choice. Because the existence of multiple bandgaps is very desirable in such structures, a major goal of the work presented in this article is performing a comparative analysis and design study between layered- and single-beam metamaterial structures. In previous work, we developed a generic model approach for layered metamaterial beam structures with local resonators [19]. The Galerkin method and principle of superposition of mode shapes were used to resolve the peculiar Euler–Bernoulli beam equations for layered-beam metamaterial structures formulated in our previous work. Our prior work was validated experimentally using a 3D printed metamaterial prototype and further confirmed using dispersion curves [14,19]. While our previous work was focused on establishing a design criteria and platform for modeling a layered-beam metamaterial structure, it stopped short from investigating the advantages offered by layered-beam metamaterial structures over single-beam metamaterial structures in terms of bandgaps formation and vibration transmissibility. This is the main focus of the current work. Motivated by the key and growing role that layering beams are taking in design and implementation of dual-purpose metamaterial structures, the current article is focused on studying the possible advantages in bandgap formation of these layered-beam metamaterial structures. Thus, the main objective of the work presented in this article is to creat a platform and an awareness of these advantages that can help researchers in designing metamaterial beam structures for various applications including vibration attenuation and energy harvesting.

In this work, the effects of layering the metamaterial beams on their performance are analyzed and possible advantages are highlighted. The work presented in this article starts with analyzing a standard single-beam metamaterial structure, its transmissibility bandgap, and enhancements. Afterwards, a simple double-layered beam-type metamaterial structure is modeled and the bandgaps of the layered-beam metamaterial structures are presented and compared with a single-beam metamaterial structure. Lastly, parametric and design studies that investigate the effects of changes in various parameters of the double-layered beam-type metamaterial structure are presented and the outcomes are compared with the results from single-beam metamaterial structures.

2. Model

2.1. Single-Beam Metamaterial Structure

A representative design of a single-beam metamaterial structure is shown in Figure 1. The structure is comprised of a single horizontal beam and various local resonators attachments, as shown in Figure 1. The spacing between unit cells, $L_c$, is regarded as the length of the unit cell; the effect of the number of unit cells per beam has been studied for different design forms in the literature [18]. In Figure 1, $M$ is the mass of the local resonators, while $Z_{ir}$ is the displacement of the local resonators when an external vibration is applied to the metamaterial structure. In principle, when external vibration is incident on the single-beam metamaterial structure in Figure 1, at frequencies near the resonant frequency of the unit cell, the vibrational energies are trapped in the local resonators. This trapping of vibrational energy generates the local resonator bandgaps seen in the overall transmissibility of the metamaterial structure.

The local resonators of the single-beam metamaterial structure in Figure 1 are further represented in the form of stiffness and damping shown in Figure 2. The symbols $U\left(x,t\right)$, ${\xi }_r$, and $k_r$ represent the displacement of the beam, the damping coefficient of the local resonator, and the stiffness, respectively.

The displacement of the single-beam metamaterial structure in Figure 1 can be modeled with the standard Euler–Bernoulli beam’s equation in Equation (1) below [30]:

$$EI\frac{{\partial }^{4}U}{\partial x^4}+\rho A\frac{{\partial }^{2}U}{\partial t^2}=f_{sh}\left(x,t\right)\delta \left(x-\frac{L}{2}\right)+{\displaystyle \sum }_{r=1}^{n}f\left(x_r,t\right)\delta \left(x-x_r\right)$$

(1)

where $L$, $E$, and $I$ represent, respectively, the total length of the horizontal beam ($L=\gamma \ast L_c$), Young’s elastic modulus, and the cross-sectional moment of inertia of the horizontal beam. In addition, $f_{sh}\left(x,t\right)$ and $f\left(x_r,t\right)$ in Equation (1) represent the force from the vibration source on the beam and the force from the local resonators, respectively.

The transverse displacement of the horizontal beam in Equation (1), $U\left(x,t\right)$, can be analyzed using the Galerkin approach and superposition of mode shapes given in Equation (2) below [18]:

$$U\left(x,t\right)={\displaystyle \sum }_{i=1}^N{p}_i\left(t\right){\varnothing }_i\left(x\right)$$

(2)

where $p_i$ and ${\varnothing }_i$ in Equation (2) represent the generalized displacement and the trial of the horizontal beam at the $i_{th}$ mode, respectively. Moreover, $N$ is the number of modes considered in the modeling effort. Furthermore, the displacement of the local resonators is shown in Equation (3) below:

$$m_r\ddot{Z_{ir}}\left(t\right)+f_d\left(x_r,t\right)+f\left(x_r,t\right)=0$$

(3)

The expression $f_d\left(x_r,t\right)$ in Equation (3) represents the damping force of the local resonator, while $m_r$ is the mass of the local resonator. In addition, $Z_{ir}$ is the displacement of the rth local resonator in the ith mode shape. The expressions of the forces in Equations (1) and (2) are given below:

$$\{\begin{array}{c}f\left(x_r,t\right)=k_r\left[Z_r\left(t\right)-U_j\left(x_r,t\right)\right]\\ f_d\left(x_r,t\right)=2{\xi }_r\sqrt{m_r{k}_r} \left[{\dot{Z}}_r\left(t\right)-{\dot{U}}_j\left(x_r,t\right)\right]\\ f_{sh}=f_s \cos \left(\omega t\right)\end{array}$$

(4)

The expression $f_s$ in Equation (4) represents the amplitude of the external vibration force. Here, the same material parameters used by Yu et al. [5] and Zhou et al. [18], shown in

Table 1. Model design parameters of the single-beam metamaterial structure and the layered-beam metamaterial structure.

Parameter	Value	Unit
Young’s modulus of elasticity of the beam, $E$	$70$	$\mathrm{GPa}$
Density of beam, $\rho$	2600	$\mathrm{Kg}/{\mathrm{m}}^3$
Area of horizontal beam, $A$	$1.062\times {10}^{-4}$	${\mathrm{m}}^2$
Area moment of inertia, $I$	$5.968\times {10}^{-9}$	${\mathrm{m}}^4$
Modal damping ratio of horizontal beams, $\zeta ji$	0.02
Vibration force magnitude, $f_s$	10	N
Length of unit cell beam, $L_c$	0.125	m
Local resonator stiffness, $k_r$	$1.65\times {10}^5$	N/m
Local resonator mass, $m_r$	0.0437	Kg
Damping ratio of local resonators, ${\xi }_r$	0.01
Length of vertical columns, $l_c$	80	mm
Width of vertical column, $b_c$	7	mm
Thickness of vertical columns, $h_c$	5	mm
Damping ratio of vertical columns, ${\xi }_c$	0.02

, were used in this work. The dimensions and parameters reported by Yu et al. [5] and Zhou et al. [18] were used because the results of the modeling approach were thoroughly verified with experiments in the literature. The design approach also closely aligned with the design and modeling approach of the metamaterial structure reported and verified experimentally by Anigbogu et al. [19]. Solving Equations (1)–(3) simultaneously by substituting Equation (2) into Equation (1), integrating through the length of the horizontal beam and applying orthogonality, the transmissibility of the single-beam metamaterial structure can be obtained. Further insights and discussion of the single-beam metamaterial structure are given in Section 3 of this article.

2.2. Layered-Beam Metamaterial Structure

The general design approach for an n-layered metamaterial beam structure with r-number of local resonators per horizontal beam is shown in Figure 3. The local resonators are evenly distributed across each horizontal beam, and vertical beams are interlinking the horizontal beams. The design approach for an arbitrary number of layered metamaterial structures was studied in our previous work [19].

In the present study, as a base, the behavior of a typical double-layer beam-type metamaterial structure is analyzed and compared to a single-beam metamaterial structure of similar material properties and specifications. The modeled layered metamaterial structure shown in Figure 4A has eight local resonators per horizontal beam, with all dimensions shown in Table 1. The horizontal beams are held together by vertical beams. In Figure 4B, the representation of the horizontal beams and their interaction with the interlinking vertical beams is shown. Localized spring and damping forces are used in representing the vertical beams, as shown in Figure 4B. The input vibration force is applied to the center of the lowest horizontal beam and the output displacement is taken from the center of the topmost horizontal beam.

For the layered free-free beams in Figure 4A subject to external vibrations, the transmissibility is analyzed using the frequency response function (FRF). The transverse response of the beams with local resonators and an external vibration source applied to it are given by the Euler–Bernoulli equation below [18]:

$$EI\frac{{\partial }^4{U}_j}{\partial x^4}+\rho A\frac{{\partial }^2{U}_j}{\partial t^2}=f_{sh}\left(x,t\right)\delta \left(x-\frac{L}{2}\right)+{\displaystyle \sum }_{r=1}^{\mathsf{\mu }}f\left(x_r,t\right)\delta \left(x-x_r\right)$$

(5)

In Equation (5), $U_j$ represents the jth beam’s transverse displacement. The positions of the local resonators are factored in as $x_r$, while $\delta$ and $\mathsf{\mu }$ are the Dirac delta function and the number of local resonators per horizontal beams, respectively. The transverse displacement in Equation (5) can be further evaluated using the Galerkin approach and superposition of mode shapes to give the equation below:

$$U{\left(x,t\right)}_{j=1,2,.., \mathrm{n}}={\displaystyle \sum }_{i=1}^N{p}_{ji}\left(t\right){\varnothing }_{ji}\left(x\right)$$

(6)

where $p_{ji}$ and ${\varnothing }_{ji}$ in Equation (6) represent the generalized displacement and the trial of the jth horizontal beam at the ith mode, respectively. Moreover, $N$ is the number of modes considered in the model. Applying the effects of the vertical beams connecting the two horizontal beams to Equation (5), a new set of equations for the two horizontal beams is given below:

$$\{\begin{array}{l}EI\frac{{\partial }^4{U}_1}{\partial x^4}+k_v\left(U_1-U_2\right)+r_v(\dot{U_1}-\dot{U_2})+ \rho A\frac{{\partial }^2{U}_1}{\partial t^2}= f_{sh}\left(x,t\right)\delta {\displaystyle (}x-\frac{L}{2}{\displaystyle )}+ {\displaystyle {\displaystyle \sum }_{r=1}^{\mathsf{\mu }}}f\left(x_r,t\right)\delta \left(x-x_r\right)\\ EI\frac{{\partial }^4{U}_2}{\partial x^4}+k_v\left(U_2-U_1\right)+r_v(\dot{U_2}-\dot{U_1})+ \rho A\frac{{\partial }^2{U}_2}{\partial t^2}= {\displaystyle {\displaystyle \sum }_{r=1}^{\mathsf{\mu }}}f\left(x_r,t\right)\delta \left(x-x_r\right)\end{array}$$

(7)

The force exerted on the lowest beam by the external vibration source is still given by the sinusoidal input $f_{sh}=f_s\cos \left(\omega t\right)$, where $f_s$ is the amplitude of the vibration force. The expression in Equation (7) can be rearranged as forces to show the effects of the vertical beams on specific locations ($x_v$) on the horizontal beams in the format given below:

$$\{\begin{array}{l}EI\frac{{\partial }^4{U}_1}{\partial x^4}+\rho A\frac{{\partial }^2{U}_2}{\partial t^2}= f_{sh}\left(x,t\right)\delta {\displaystyle (}x-\frac{L}{2}{\displaystyle )}+ {\displaystyle {\displaystyle \sum }_{r=1}^n}f\left(x_r,t\right)\delta \left(x-x_r\right)+{\displaystyle {\displaystyle \sum }_{v=1}^q}{f}_1\left(x_v,t\right)\delta \left(x-x_v\right)\\ EI\frac{{\partial }^4{U}_2}{\partial x^4}+\rho A\frac{{\partial }^2{U}_2}{\partial t^2}= {\displaystyle {\displaystyle \sum }_{r=1}^n}f\left(x_r,t\right)\delta \left(x-x_r\right)+{\displaystyle {\displaystyle \sum }_{v=1}^q}{f}_2\left(x_v,t\right)\delta \left(x-x_v\right)\end{array}$$

(8)

where $f_1$ and $f_2$ are the forces from the vertical beams acting on the lowest and topmost horizontal beams, respectively. They are expressed fully below:

$$\{\begin{array}{c}{f}_1\left(x_v,t\right)=-[k_v\left(U_1-U_2\right)+r_v\left(\dot{U_1}-\dot{U_2}\right)] \\ f_2\left(x_v,t\right)=-[k_v\left(U_2-U_1\right)+r_v\left(\dot{U_2}-\dot{U_1}\right)]\end{array}$$

(9)

Furthermore, $k_v$ and $r_v$ in Equation (9) represent the stiffness and the damping coefficient of the vertical beams and are given by Equation (10), respectively [2,31]:

$$\{\begin{array}{l}{k}_v=\frac{32 \ast E \ast b_v \ast h_v{}^3}{l_v{}^3}\\ r_v=2{\xi }_v\sqrt{m_v{k}_v}\\ m_v=0.33\ast \rho \ast A\ast l_v\end{array}$$

(10)

where $m_v$ in Equation (10) is the effective mass of the vertical beams.

Applying the damping effect of the resonators to Equation (8) transforms the equation into the form below:

$$\{\begin{array}{l}EI\frac{{\partial }^4{U}_1}{\partial x^4}+\rho A\frac{{\partial }^2{U}_2}{\partial t^2}= f_{sh}\left(x,t\right)\delta {\displaystyle (}x-\frac{L}{2}{\displaystyle )}+ {\displaystyle {\displaystyle \sum }_{r=1}^{\mathsf{\mu }}}f\left(x_r,t\right)\delta \left(x-x_r\right)+{\displaystyle {\displaystyle \sum }_{r=1}^{\mathsf{\mu }}}{f}_d\left(x_r,t\right)\delta \left(x-x_r\right)+{\displaystyle {\displaystyle \sum }_{v=1}^q}{f}_1\left(x_v,t\right)\delta \left(x-x_v\right)\\ EI\frac{{\partial }^4{U}_2}{\partial x^4}+\rho A\frac{{\partial }^2{U}_2}{\partial t^2}= {\displaystyle {\displaystyle \sum }_{r=1}^{\mathsf{\mu }}}f\left(x_r,t\right)\delta \left(x-x_r\right)+{\displaystyle {\displaystyle \sum }_{r=1}^{\mathsf{\mu }}}{f}_d\left(x_r,t\right)\delta \left(x-x_r\right)+{\displaystyle {\displaystyle \sum }_{v=1}^q}{f}_2\left(x_v,t\right)\delta \left(x-x_v\right)\end{array}$$

(11)

The force from the local resonators on the horizontal beams and their damping force are further expressed in Equations (12) and (13):

$$\left(x_r,t\right)=k_r\left[Z_{ir}\left(t\right)-U_{ji}\left(x_r,t\right)\right]$$

(12)

$$f_d\left(x_r,t\right)=2{\xi }_r\sqrt{m_r{k}_r} \left[{\dot{Z}}_{ir}\left(t\right)-{\dot{U}}_{ji}\left(x_r,t\right)\right]$$

(13)

Substituting Equation (6) into Equation (11) results in Equation (14) below:

$$\{\begin{array}{l}EI{\displaystyle {\displaystyle \sum }_{i=1}^N}{p}_{1i}\left(t\right){\varnothing }_{1i}^{\left(iv\right)}\left(x\right)+\rho A{\displaystyle {\displaystyle \sum }_{q=1}^N}\ddot{p_{1i}}\left(t\right){\varnothing }_{1i}(x)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} = f_s\left(x,t\right)\delta \left(x-\frac{L}{2}\right)+ {\displaystyle {\displaystyle \sum }_{r=1}^{\mathsf{\mu }}}f\left(x_r,t\right)\delta \left(x-x_r\right)+{\displaystyle {\displaystyle \sum }_{v=1}^q}{f}_1\left(x_v,t\right)\delta \left(x-x_v\right)\\ EI{\displaystyle {\displaystyle \sum }_{q=1}^N}{p}_{2i}\left(t\right){\varnothing }_{2i}^{\left(iv\right)}\left(x\right)+\rho A{\displaystyle {\displaystyle \sum }_{q=1}^N}\ddot{p_{2i}}\left(t\right){\varnothing }_{2i}\left(x\right)= {\displaystyle {\displaystyle \sum }_{r=1}^{\mathsf{\mu }}}f\left(x_r,t\right)\delta \left(x-x_r\right)+{\displaystyle {\displaystyle \sum }_{v=1}^q}{f}_2\left(x_v,t\right)\delta \left(x-x_v\right) ,\end{array}$$

(14)

When Equation (14) is multiplied by the trial function ${\varnothing }_{ji}\left(x\right)$ it is stretched further into Equation (15) below:

$$\{\begin{array}{l}EI{\displaystyle {\displaystyle \sum }_{i=1}^N}{p}_{1i}\left(t\right){\varnothing }_{1i}^{\left(iv\right)}\left(x\right){\varnothing }_{1i}\left(x\right)+\rho A{\displaystyle {\displaystyle \sum }_{q=1}^N}\ddot{p_{1i}}\left(t\right){\varnothing }_{1i}\left(x\right){\varnothing }_{1i}(x)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} = f_s\left(x,t\right)\delta \left(x-\frac{L}{2}\right){\varnothing }_{1i}\left(x\right) + {\displaystyle {\displaystyle \sum }_{r=1}^{\mathsf{\mu }}}f\left(x_r,t\right)\delta \left(x-x_r\right){\varnothing }_{1i}\left(x\right)+{\displaystyle {\displaystyle \sum }_{v=1}^q}{f}_1\left(x_v,t\right)\delta \left(x-x_v\right){\varnothing }_{1i}\left(x\right) \\ EI{\displaystyle {\displaystyle \sum }_{q=1}^N}{p}_{2i}\left(t\right){\varnothing }_{2i}^{\left(iv\right)}\left(x\right){\varnothing }_{2i}\left(x\right)+\rho A{\displaystyle {\displaystyle \sum }_{q=1}^N}\ddot{p_{2i}}\left(t\right){\varnothing }_{2i}\left(x\right){\varnothing }_{2i}\left(x\right)= {\displaystyle {\displaystyle \sum }_{r=1}^{\mathsf{\mu }}}f\left(x_r,t\right)\delta \left(x-x_r\right){\varnothing }_{2i}\left(x\right) +{\displaystyle {\displaystyle \sum }_{v=1}^q}{f}_2\left(x_v,t\right)\delta \left(x-x_v\right){\varnothing }_{2i}\left(x\right) \end{array}$$

(15)

It is worthy of note that the Dirac delta function $\delta \left(x-x_v\right)$ in Equation (14) and Equation (15) factors in the positioning of the vertical beams into the system of equations. By integrating Equation (15) across the full length of the horizontal beams (0: L) and applying orthogonality of mode shapes, the equation is reduced further to a new form. By factoring in damping acting on the horizontal beams, Equation (15) is transformed into a new set of equation below:

$$\{\begin{array}{l}{M}_{1i}{\ddot{p}}_{1i}\left(t\right)+C_{1i}{\dot{p}}_{1i}+K_{1i}{p}_{1i}=f_s\left(x,t\right){\varnothing }_{1i}\left(\frac{L}{2}\right)+ {\displaystyle {\displaystyle \sum }_{r=1}^{\mathsf{\mu }}}{\varnothing }_{1i}\left(x_r\right)f\left(x_r,t\right)+{\displaystyle {\displaystyle \sum }_{r=1}^{\mathsf{\mu }}}{\varnothing }_{1i}\left(x_r\right)f_d\left(x_r,t\right)+{\displaystyle {\displaystyle \sum }_{v=1}^q}{\varnothing }_{1q}\left(x_c\right)f_1\left(x_v,t\right)\\ M_{2i}{\ddot{p}}_{2i}\left(t\right)+C_{2i}{\dot{p}}_{2i}+K_{2i}{p}_{2i}= {\displaystyle {\displaystyle \sum }_{r=1}^{\mathsf{\mu }}}{\varnothing }_{2i}\left(x_r\right)f\left(x_r,t\right)+{\displaystyle {\displaystyle \sum }_{r=1}^{\mathsf{\mu }}}{\varnothing }_{2i}\left(x_r\right)f_d\left(x_r,t\right)+{\displaystyle {\displaystyle \sum }_{v=1}^q}{\varnothing }_{2q}\left(x_c\right)f_2\left(x_v,t\right)\end{array}$$

(16)

The generalized masses, stiffness, and the modal damping of the horizontal beams in Equation (16) are clearly expressed in Equation (17) below:

$$\{\begin{array}{l}{M}_{ji}=\rho A{{\displaystyle \int }}_0^L{\varnothing }_{ji}^{\left(ii\right)}\left(x\right)dx\\ K_{ji}=EI{{\displaystyle \int }}_0^L{\varnothing }_{ji}^{\left(iv\right)}{\varnothing }_{iq}\left(x\right)dx\\ C_{ji}=2{\zeta }_{ji}\sqrt{M_{ji}{K}_{ji}}\end{array}$$

(17)

The damping coefficient of the horizontal beams is represented as ${\zeta }_{ji}$ in Equation (17). For the free-free beam shown in Figure 1, assuming the mode shape and trial functions to be the same, the mode functions are given by the following equation [18]:

$${\varnothing }_{ji}\left(x\right)={\varnothing }_{jy}\left(x\right)=\left[sin{\beta }_{ji}x+sinh{\beta }_{ji}x+\left(\frac{sin{\beta }_{ji}L-sinh{\beta }_{ji}L}{cosh{\beta }_{ji}L-cos{\beta }_{ji}L}\right)\left(cos{\beta }_{ji}x+cosh{\beta }_{ji}x\right)\right]$$

(18)

The wave number ${\beta }_{ji}$ in Equation (18) is expressed out in Equation (19) below:

$$\cos \left({\beta }_{ij}L\right)\cosh \left({\beta }_{ji}L\right)=1$$

(19)

The transverse movement of each rth local resonator on any part of the horizontal beams can also be represented using:

$$m_r\ddot{Z_{ir}}\left(t\right)+f_d\left(x_r,t\right)+f\left(x_r,t\right)=0$$

(20)

Transmissibility of the layered metamaterial structure is obtained by comparing the output displacement signals from the center, $x=\frac{L}{2}$, of the uppermost beam with the input displacement signal from the vibration source at the center of the lowest beam. These displacement signals can be obtained by solving Equations (16) and (20) simultaneously, using the Runge–Kutta numerical method, and then substituting the results into Equation (6). A simple expression describing the transmissibility bandgap is given by the equation below:

$$T_{bg}=\frac{U_{ji}{\left[\left(\frac{1}{2}\right)\times L,t\right]}_{,for j=2}}{U_{ji}{\left[\left(\frac{1}{2}\right)\times L,t\right]}_{,for j=1}}$$

(21)

The frequency response function (FRF) of the root mean square ratio of the output to input signals is used to show the visualization of the transmissibility results from Equation (21).

3. Results and Discussion

For all model simulations, the material properties and dimensions adopted in this work are those used by both Yu et al. [5] and Zhou et al. [18]. Starting with a single horizontal beam metamaterial structure with eight unit cells (resonators) and analyzing Equations (1)–(3) numerically and substituting the results into Equation (6), yields the FRF transmissibility shown in Figure 5.

To generate the transmissibility in Figure 5, the frequency was swept from 0 to 500 Hz. This is because low frequency mechanical vibrations commonly found in nature are less than 300 Hz [21]. The input signal from the vibration source was taken from the left end of the horizontal beam ($x=0$), while the output signal was taken from the right end of the horizontal beam ($x=L$). The results show the presence of one prominent frequency bandgap, i.e., 300–415 Hz. This bandgap surrounds the resonant frequency of the local resonator, i.e., 309 Hz. The results shown in Figure 5 reveal that the local resonators trap these vibrations within this frequency bandgap, making the bandgap a local-resonator-influenced bandgap. The bandgap in Figure 5 agrees with the frequency bandgap reported and experimentally validated by Yu et al. [5]. Moreover, using the same material dimensions and material properties reported in Table 1, a similar local resonator bandgap was reported by Zhou et al. [18].

To study the behavior of the bandgap generated by the double-layered beam-type metamaterial structure in Figure 4A, the Runge–Kutta numerical method was used to analyze Equations (6), (16) and (20). The resulting FRF of the transmissibility, generated using Equation (21), is shown in Figure 6.

The results from Figure 6 show two major bandgaps at (238–275 Hz) and (300–410 Hz). These two major bandgaps are marked by the shaded areas in Figure 6. The first important bandgap occurred at 300–410 Hz. This bandgap is maintained within the resonant frequency region of the local resonators, i.e., local-resonator bandgap. In addition to this fundamental frequency bandgap, an extra significant bandgap in the lower frequency region appears in Figure 6, i.e., 238–275 Hz. This second bandgap is the result of the Bragg’s scattering phenomena [32]. In effect, with double-layering, the metamaterial beam structure showed the ability to generate multiple bandgaps and at a lower frequency level. Having these multiple bandgaps with some of them happening at lower frequency is desirable for practical applications of metamaterial structures [21].

Comparing the results from Figure 5 and Figure 6, the advantage offered by a double-layered beam-type metamaterial structure over a single-beam metamaterial structure is evident. First, Figure 6 shows a clear additional lower frequency bandgap in double-layered metamaterial structure, i.e., 238–275 Hz. Moreover, Figure 6 exhibits two frequency bandgaps, i.e., 238–275 Hz and 300–410 Hz, for the double-layered metamaterial structure as compared with a single frequency bandgap, i.e., 300–415 Hz in the single-beam metamaterial structure shown in Figure 5. This is particularly useful in a situation where a specific frequency range is required while an extra bandgap is also desirable. It is worth noting that, in Zhou et al. [18], a low frequency bandgap in the single-beam metamaterial was achievable using the negative stiffness design adopted in their work. This negative stiffness resulted in only a single frequency bandgap. Thus, a main advantage of the double-layered beam-type metamaterial structure presented in this article is the ability to create a low frequency bandgap in addition to the local-resonator frequency bandgap, as shown in Figure 6.

Additional Advantages of Layered Metamaterial Structures

Furthermore, the effect of increasing the number of local resonators per beam is explored. Here, the number of local resonators that are attached to the single-beam metamaterial structure, shown in Figure 1, is increased from 8 to 12 and then 16. In effect, the number of unit cells are increased while the length of unit cells remains the same. This leads to changes in the overall length of the horizontal beams. The results are shown in Figure 7.

Figure 7 shows that a single notable frequency bandgap is developed when the number of local resonators that are attached to the beam is increased from 8 to 16. The frequency bandgaps, marked by the shaded area in Figure 7, are 300–415 Hz, 298–438 Hz, and 296–438 Hz for 8, 12, and 16 local resonators, respectively. No additional significant frequency bandgaps open up when the number of local resonators is increased from 8 to 16, as shown in Figure 7. These single frequency bandgaps correspond to the region of the resonant frequency of the local resonators. In addition, the results from Figure 7 show that the width of the local resonator bandgap becomes broader when the number of local resonators increased from 8 to 12 and then 16 resonators. This broadband frequency bandwidth can be attributed to the merging of small Bragg’s scattering bandgaps that are developing close to the local resonator bandgap with the fundamental local resonator bandgap. This widening of central local resonator bandgap as a result of Bragg’s scattering bandgaps developing close to it is called coupled-resonance-Bragg’s bandgap and has been reported by multiple researchers in the literature [6,32,33].

In a similar fashion, the effect of increasing the number of local resonators per horizontal beam on the transmissibility of the double-beam metamaterial structure, shown in Figure 4, is analyzed next. Figure 8 displays the frequency bandgaps obtained when the number of local resonators is increased from 8 to 16 per horizontal beam.

Figure 8 shows two main outcomes. First, a major broad frequency bandgap lives near resonance frequency of the local resonators, i.e., 298–410 Hz for 8, 12, and 16 local resonators, respectively. These frequency bandgaps did not witness a significant change in their width as the number of local resonators increased. This is likely because there was no alteration to the shape and design of the local resonators. Moreover, as the number of local resonators per horizontal beam is increased from 8 to 16 additional bandgaps opened up. These additional bandgaps tend to exist in the lower frequency region. For example, as shown in Figure 8, a second frequency bandgap opened up at 238–275 Hz, 267–281 Hz, and 279–286 Hz for 8, 12, and 16 local resonators, respectively. In Figure 8B, a third bandgap opened up at 126–157 Hz. This opening of extra bandgap at the lower frequency in layered-beam metamaterial structure as the number of resonators increases has been witnessed in another generic work on layered metamaterial bandgaps [19]. It is of importance to note that increasing the number of local resonators (unit cells) per horizontal beam comes with an increase in the overall vertical linking beam. Increased forces act on the beam, increasing the overall internal changes on the structure in layered-beam metamaterial structures. In Figure 8C, in addition to the three major bandgaps marked by the shaded area, the sign of a fourth bandgap at a frequency below 100 Hz begins to open-up. Though this is not considered to be a major bandgap because the depth is not significant enough, following a similar approach adopted by Zhou et al, it is enclosed by the square in Figure 8C. These extra bandgaps are likely the result of Bragg’s scattering phenomena. Additionally, as the number of local resonators increased from 8 to 16, the transmissibility depth of the fundamental local resonator bandgap (292–430 Hz) decreased. This is mainly due to the increase in the number of vertical beams as the number of unit cells increased. That is, the vertical beams make the metamaterial structure stiffer.

The advantage of increasing the number of local resonators that are attached to the double-beam metamaterial structure, shown in Figure 8, as compared with increasing the number of local resonators that are attached to the single-beam metamaterial structure, shown in Figure 7, is evident. As the number of local resonators in the double-beam metamaterial structure is increased, the number of vertical beams connecting these beams increases as well. This, consequently, manifests the development of Bragg’s scattering frequency bandgaps in the double-beam metamaterial structure. This leads to the development of multiple frequency bandgaps in the double-beam metamaterial structure.

Next, the effect of the length of unit cells on the frequency bandgaps is investigated. In Figure 9, the length of the unit cell $L_C$ is increased by 25% and 50%. Here, the number of local resonators per horizontal beam remained fixed at eight and the design of local resonators was unaltered. In Figure 9A, the results show that as the unit cell’s length of the single-beam metamaterial structure was varied, no significant extra bandgaps opened up. The region of the local resonator bandgap remained mostly unaltered when the length of the unit cell was changed. However, in Figure 9B, the results show that altering the length of the unit cell’s length for a double-beam metamaterial structure leads to the development of noticeable lower frequency bandgaps. For example, increasing the length of the unit cell by 25% resulted in a low frequency bandgap around 190–225 Hz in addition to the fundamental local-resonator influenced frequency bandgap at 298–410 Hz. In Figure 9A,B, the length of the unit cell $L_C$ was increased, while the number of local resonators per horizontal beam remained unaltered. A 25% increase in the length of the unit cell for double-layered structure shifted the Bragg’s scattering lower frequency bandgap from 238–275 Hz to 190–225 Hz. This is not uncommon, as multiple works in the literature have shown that an increase in the length of unit cells while the number of unit cells remained constant leads to the development of Bragg’s bandgap at the lower frequencies below the central local resonator bandgap [6,32,33]. Moreover, one can notice that the number of bandgaps generated by altering the length of the unit cell, i.e., Figure 9B, is less than the number of bandgaps generated from increasing the number of local resonators per beam, i.e., Figure 8. Nonetheless, as shown in Figure 9B, changing the length of the unit cell offers a good alternative for lowering frequency bandgaps when it is desired to keep the number of local resonator constant. However, it has been noted in the literature that as the number of unit cells is fixed while the length of unit cells vary, the damping effect decreases with a decrease in frequency [4,6,32]. In applications where attempts to lower the transmissibility bandgap by increasing the number of local resonators is not feasible, small changes to the length of the unit cells can be a viable option. This can be done while maintaining a fixed number of local resonators per horizontal beam. In conclusion, while changing the stiffness and design of the local resonators may offer a route for lowering the frequency bandgaps [5,6,18], the work presented in this article offers an additional venue for controlling and adding more frequency bandgaps through layering of metamaterial structures.

4. Conclusions

In this article, limitations and advantages of both single-beam metamaterial structures and layered-beam metamaterial structures are presented. The single-beam metamaterial structure presented in this article consists of a free-free horizontal beam with equally spaced local resonators attached to the horizontal beam. The layered-beam metamaterial structure is comprised of layered horizontal beams linked by vertical beams. In this work, the Euler-Bernoulli beam equation and superposition of mode shapes were used to analyze the displacement of the beams. The analysis of frequency bandgaps and transmissibility were carried out using the frequency response function (FRF).

Results from this work show differences between the bandgap developed in single-beam metamaterial structures and layered-beam metamaterial structures. These differences are vital points of interest for designers who are considering the choice of single-beam metamaterial structure or layered-beam metamaterial structure for a specific engineering application. The main conclusions from this work include:

• For a single-beam metamaterial structure with equally spaced local resonators of the same design, there is a single notable frequency bandgap near the region of the resonant frequency of the local resonators.
• Increasing the number of local resonators that are attached to the horizontal beam of the single-eam metamaterial structure does not yield multiple bandgaps. Nonetheless, increasing the number of local resonators that are attached to the horizontal beam of the single-beam metamaterial structure may slightly increase the width of the local resonator bandgap. This increase in width is likely due to the merging happening between the fundamental local resonator bandgap and adjacent, smaller, Bragg’s scattering bandgaps.
• When it is desirable to maintain the design of the local resonators in the metamaterial structure, layering the horizontal beams offers a means for both lowering the frequency bandgap and developing additional bandgaps.
• Unlike single-beam metamaterial structures, increasing the number of local resonators per horizontal beams in the layered metamaterial leads to the birth of additional lower frequency bandgaps.
• In applications where attempts to lower the transmissibility bandgap by increasing the number of local resonators is not feasible, small changes to the length of the unit cells can be a viable option. Changes to the length of the unit cells while keeping the number of local resonators constant also leads to lower frequency bandgaps.
• The ability of the layered-beam metamaterial structure to generate both multiple and lower frequency bandgaps through a variety of parametric changes will offer a lot of options in application use-cases of dual-purpose vibration isolation energy harvesting. This is important as recent reports in the literature show increased interest in the use of layered-beam metamaterial structures for energy harvesting.