Recently, mechanical metamaterials have become the centerpiece of many research studies and engineering applications (Xia et al., 2020). This is mainly because of their unique properties including the presence of frequency bandgaps influenced by the local resonators (Wang et al., 2016; Jiang, He, 2017). This is particularly of interest for many engineering applications including wave guiding, vibration attenuation and, more recently, dual-purpose vibration suppression and energy scavenging (Matlack et al., 2016; Reichl and Inman, 2017; Casablanca et al., 2018). Frequency bandgaps created in these metamaterial structures originate from the ability of the local resonators to absorb kinetic energy from the oscillations passing through the metamaterial structure (Li et al., 2017). When the frequency of the external vibration source aligns with the resonant frequency of the local resonators, energy from these oscillations is transferred to the local resonators (Krödel et al., 2015; Li et al., 2017). Typically, these local resonators are either attached to the surface of the metamaterial structures or embedded inside (Fang et al., 2006; Chen and Wang, 2014; Krödel et al., 2015). Moreover, the Bragg’s scattering in the periodic structures leads to generation of frequency bandgaps (Kushwaha et al., 1993). The bandgaps generated by Bragg’s scattering are influenced by the wavelengths of the Bragg scattering mechanism, which is limited by the design of the beam of the mechanical metamaterial structure (Krödel et al., 2015). The addition of local resonators to the structure makes it possible to generate bandgaps at frequencies much lower than the Bragg’s frequencies (Liu et al., 2000; Liu et al., 2005). In the case of vibration attenuation, when the mechanical metamaterial is subject to vibrations by an external vibration source, it has the ability to suppress these oscillations, at low frequencies, that are closely aligned with the resonant frequency of the local resonators (Sugino et al., 2017).

Exhaustive literature has already looked at design criteria and characterization of metamaterial structures that are made of single beam structures with local resonators (Yu et al., 2006a; Yu et al., 2006b; Wang et al., 2013; Wang and Wang, 2013; Xiao et al., 2013; Zhou et al., 2017). For example, the work by Liu et al. was among the first few studies to look at metamaterial structures and local resonators (Liu et al., 2000). Their pioneer seminal work showed that localized resonant structures exhibit valuable characteristics such as negative elastic constants, leading to generation of low frequency bandgaps. The transfer matrix method was then used by Yu et al. for characterization of a single beam metamaterial structure with resonators that are made of rubber and copper rings (Yu et al., 2006a; Yu et al., 2006b). The results from their model simulations were validated experimentally. Xiao et al. built on the analytical model of the single beam metamaterial design developed by Yu et al. and showed that resonance and Bragg’s bandgaps exist as a result of local resonance effect and periodicity of unit cells (Xiao et al., 2013). Further studied the effects of different design parameters on the generated frequency bandgaps Liu et al. (2007). The results from their work revealed that, although a large local resonator generates wide attenuation, a large number of small local resonators spread across the beam gives enhanced attenuation spectra.

Other enhanced designs and studies of single beam-based metamaterial structures have also been developed (Chen et al., 2011; Xiao et al., 2012; Wang et al., 2013; Wang and Wang, 2013; Hu et al., 2021). For instance, a design of a beam with spring-mass local resonators sandwiched inside the beam was proposed by Chen et al. (2011). Results showed that the amplitude of waves was attenuated near the resonance frequency of the local resonators. Additionally, their results revealed that bandgaps were influenced mostly by the size of the internal mass rather than the spring constant of the resonator. To create more flexible bandgaps, designed an array of local resonators with varying resonant frequencies Xiao et al. (2012). Their work showed that, compared to a beam with a single array of local resonators, a beam with varying array of local resonators can generate broader bandgaps above and below Bragg’s bandgap. Used a combination of both vertical springs and oblique springs to investigate the effect of negative stiffness introduced by the oblique springs on single beam-based metamaterial structures Zhou et al. (2017). Their results showed that adding two oblique springs to the single beam-based metamaterial structure introduced negative stiffness which effectively produced attenuation at lower frequencies. More recently, studied the effect of nonlinearity that was purposefully introduced into the design of a single beam-based metamaterial structure Xia et al. (2020). This was done by adding bistable attachments to the metamaterial structure. Their work showed that nonlinear vibrations of the bistable resonator attachments resulted in wider frequency bandgap compared to the case when only linear local resonators were considered.

All the aforementioned studies have focused on single-beam based metamaterial structures with local resonators. On the other hand, few recent studies have looked at modelling layered beams. For example, Oniszczuk investigated two beams connected by a Winker elastic layer (Oniszczuk, 2000). The work used Bernoulli-Fourier method to show that the free vibration of a simply supported double-beam system is similar to a double-string system. Used both Fourier and Laplace approach to develop a closed form solution for double infinite Euler-Bernoulli beams with harmonic loads applied Li et al. (2019). Douglas et al. treated a system of three elastic-viscoelastic beams as two non-identical Euler-Bernoulli beams with a viscoelastic layer between them (Douglas and Yang, 1978). The method was limited to a specific fixed-free boundary condition and an external load applied only at the free end. Extended Euler-Bernoulli’s approach to arbitrary boundary conditions and applied loads Vu et al. (2000). Then applied a change of variable method in decoupling the set of fourth-order differential equations from the modal solution of double Euler-Bernoulli beams with moving loads Abu-Hilal (2006). Recently demonstrated the use of layered metamaterial for simultaneous vibration isolation and energy harvesting Li et al. (2017). Anigbogu et al. presented a unique metamaterial design using 3D printed layered beams to achieve vibration attenuation (Anigbogu and Bardaweel, 2020). Their results showed that layered metamaterial beams can achieve vibration attenuation at low frequencies that are commonly encountered in nature.

A key observation from these recent studies is that these efforts have mainly focused on studying and investigating either single beam-based metamaterial structures or double-beams with different support systems. None of these studies has tackled the issue of developing a general approach to modelling layered metamaterial beams with local resonators. The work presented in this article is motivated by the growing interest in layered metamaterial beams with local resonators. Recently, these layered metamaterial beams with local resonators have found numerous potential engineering applications including vibration attenuation and energy scavenging (Li et al., 2017; Anigbogu and Bardaweel, 2020). To the best knowledge of the authors, the work presented here is the first to tackle the issue of developing a modelling platform for layered metamaterial structures with multiple beams and local resonators. The work is focused on developing design criteria and modelling platform for layered metamaterial structures with multiple beams and local resonators. The model is developed using Galerkin method and superposition of mode shapes to solve an application of Euler-Bernoulli equation to layered-metamaterial beam structures. A prototype of the layered metamaterial structure with local resonators is fabricated and characterized experimentally. The model developed in this work is then validated against measured data. Additionally, in this work design criteria and the effects of core design parameters on the frequency bandgaps generated by the layered metamaterial structure are studied. Section 2 describes the theory and mathematical model developed for the layered metamaterial structure. Section 3 introduces the experimental methods, while Section 4 describes the main findings and discussions of these results. Conclusions and summary of the work are articulated in Section 5.

The overall representation of the layered metamaterial vibration attenuation structure is shown in Figure 1A. The metamaterial structure is comprised of J-layered horizontal beams, the local resonators per unit cell and the vertical beams linking the horizontal beams together. The local resonators are made of cantilevers with tip masses attached to them as shown in Figure 1A. Each unit cell has four local resonators linked to the same point as shown in Figure 1B. For simplicity we represent the unit cell as one single local resonator force acting on the same point, in tandem with the approach adopted by Wang et al. in (Wang and Wang, 2013). The effects of the vertical beams are factored into the governing equations as spring forces and damping forces following the relative velocities of the beams as shown in Figure 2. The free body diagram of the local resonators within a unit cell is shown in Figure 3. An approximate model of the transmissibility of the layered metamaterial structure is obtained using Euler-Bernoulli’s equations and Galerkin method (Zhou et al., 2017).

In order to analyze the dynamic behavior of the layered metamaterial vibration attenuation structure when subject to external vibrations, the Frequency Response Function (FRF) approach is used to determine the transmissibility of the device. The Galerkin method is used to analyze the beams’ response to external vibrations. The length of each horizontal beam is L   =   α ∗ L u , where α is the number of unit cells, and L u is the length of each unit cell in the design adopted in this work (Figure 1B). The transverse vibration of a simple beam with attachment and a vibration source at a point is given by the standard Euler Bernoulli equation (Zhou et al., 2017): E I ∂ 4 w i ∂ x 4 + ρ A ∂ 2 w i ∂ t 2 =   f s ( x , t ) δ ( x − L 2 ) + ∑ r = 1 n f ( x r , t ) δ ( x − x r )   (1) where w , E and I are the transverse displacement, Young’s modulus of elasticity and the cross-sectional moment of inertia of the beam, respectively. Moreover, ρ and A are the density of the material, and the cross-sectional area of the beam, respectively. The expressions f s ,   δ       and f   represent the force from the vibration source, the delta Dirac function and the force from the attached local resonators, respectively. The force, f s   , from the vibration source only applies to the first beam. In Eq. 1, x r represents the position of the local resonator, while n is the total number of local resonators on each horizontal beam. This Euler Bernoulli equation shown in Eq. 1 is used to analyze the displacement of each of the i t h horizontal beams. Applying Galerkin method and the superposition of modes, the transverse displacement of i t h horizontal beam, w i is given by (Zhou et al., 2017): w ( x , t ) i = 1,2 , … ,   j = ∑ q = 1 N u i q ( t ) ∅ i q ( x ) (2)

In Eq. 2, u i q is the generalized displacement of the i t h horizontal beam at the q t h mode, and ∅ i q is the trial function of the i t h horizontal beam which is taken to be the mode shape of the horizontal beam at the q t h mode. The index j is the total number of layered horizontal beams.

Next, we extend Eq. 1 to factor in the connecting vertical beams as shown in Figures 1, 2. This yields the set of equations representing each of the horizontal beams from the base beam to the top, respectively. A generalized form of this set of equations from the lowest horizontal beam, i.e., i = 1   , to i t h middle horizontal beam, i.e., i = 2 : j − 1 , till the j t h topmost beam, i.e., i = j , for any j number of horizontal beams constituting the layered metamaterial structure yields Eqs 3–5 below (Abu-Hilal, 2006): { E I ∂ 4 w i ∂ x 4 + k c ( w i − w i + 1 ) + r c ( w i ˙ − w i + 1 ˙ ) +   ρ A ∂ 2 w i ∂ t 2 =   f s ( x , t ) δ ( x − L 2 ) + ∑ r = 1 n f ( x r , t ) δ ( x − x r ) ,   i = 1     E I ∂ 4 w i ∂ x 4 + k c ( w i − w i − 1 ) + k c ( w i − w i + 1 ) + r c ( w i ˙ − w i − 1 ˙ ) + r c ( w i ˙ − w i + 1 ˙ ) +   ρ A ∂ 2 w i ∂ t 2 =   ∑ r = 1 n f ( x r , t ) δ ( x − x r ) ,   i = 2 : j − 1 E I ∂ 4 w j ∂ x 4 + k c ( w j − w j − 1 ) + r c ( w j ˙ − w j − 1 ˙ ) +   ρ A ∂ 2 w j ∂ t 2 = ∑ r = 1 n f ( x r , t ) δ ( x − x r ) ,   i = j   (3) where i = 1 and i = j are the lowest horizontal beam and topmost horizontal beam, respectively, and the middle section beams are indexed with i = 2 : j − 1 . In Eq. 3, f s is the force exerted by the vibration source, represented as a sinusoidal input   f s =   F A   c o s   ω t   . Also ω is the forcing frequency of vibration source, F A   is the amplitude of the external vibration force and k c represents the stiffness of the vertical beams given by (Sharma and Tyagi, 2013): k c = 32 ∗ E ∗ b c ∗ h c 3 l c 3 (4a) where b c , h c and l c represent the width, thickness and length of the vertical beams respectively. The damping of each vertical beam, r c , and the equivalent mass, m c , are given by (Waichi et al., 2010): r c = 2 ξ c m c k c   (4b) m c = 0.33 ∗ ρ ∗ A ∗ l c   (4c)

Grouping the impact of the vertical beams as forces acting at specific points on the horizontal beams, Eq. 3 can be rewritten in the following generalized forms: { E I ∂ 4 w i ∂ x 4 + ρ A ∂ 2 w i ∂ t 2 =   f s ( x , t ) δ ( x − L 2 ) + ∑ r = 1 n f ( x r , t ) δ ( x − x r ) + ∑ c = 1 v f i ( x c , t ) δ ( x − x c ) ,   i = 1   E I ∂ 4 w i ∂ x 4 + ρ A ∂ 2 w i ∂ t 2 = ∑ r = 1 n f ( x r , t ) δ ( x − x r ) + ∑ c = 1 v f i ( x c , t ) δ ( x − x c ) ,   i = 2 : j − 1 E I ∂ 4 w j ∂ x 4 + ρ A ∂ 2 w j ∂ t 2 = ∑ r = 1 n f ( x r , t ) δ ( x − x r )   + ∑ c = 1 v f j ( x c , t ) δ ( x − x c ) ,   i = j (5) where f i ( x c , t ) and f j ( x c , t ) are the forces due to the vertical beams acting on each i t h horizontal beam (from the lowest horizontal beam to j − 1 t h horizontal beam) and the j t h topmost horizontal beam, respectively, and x c represents the point of contacts on each horizontal beam. Using Eq. 2, these forces, which are due to the interlinking vertical beams, can be further developed into the following generalized set of equations: { f i ( x c , t ) = − [ k c ( w i − w i + 1 ) + r c ( w i ˙ − w i + 1 ˙ ) ]   ,   i = 1   f i ( x c , t ) = − [ k c ( w i − w i − 1 ) + k c ( w i − w i + 1 ) + r c ( w i ˙ − w i − 1 ˙ ) + r c ( w i ˙ − w i + 1 ˙ ) ]   ,   i = 2 : j − 1 f j ( x c , t ) = − [ k c ( w j − w j − 1 ) + r c ( w j ˙ − w j − 1 ˙ ) ] ,   i = j (6)

Also, the forces from the attached local resonators and the internal damping force are given as: f ( x r , t ) = k r [ Y r ( t ) − w ( x r , t ) ] (7) f d ( x r , t ) = 2 ξ r m r k r   [ Y ˙ r ( t ) − w ˙ ( x r , t ) ] (8) where f d ( x r , t ) ,   ξ r and m r are the internal damping force, the damping ratio and the mass of the r t h local resonator, respectively. Y r ( t ) is the vertical displacement of the r t h local resonator.

Substituting Eq. 2 into Eq. 5 yields the following set of equations, respectively: { E I ∑ q = 1 N u i q ( t ) ∅ i q ( i v ) ( x ) + ρ A ∑ q = 1 N u i q ¨ ( t ) ∅ i q ( x )   =   f s ( x , t ) δ ( x − L 2 ) +   ∑ r = 1 n f ( x r , t ) δ ( x − x r ) + ∑ c = 1 v f i ( x c , t ) δ ( x − x c )   ,   i = 1   E I ∑ q = 1 N u i q ( t ) ∅ i q ( i v ) ( x ) + ρ A ∑ q = 1 N u i q ¨ ( t ) ∅ i q ( x ) = ∑ r = 1 n f ( x r , t ) δ ( x − x r ) + ∑ c = 1 v f i ( x c , t ) δ ( x − x c ) ,   i = 2 : j − 1 E I ∑ q = 1 N u j q ( t ) ∅ j q ( i v ) ( x ) + ρ A ∑ q = 1 N u j q ¨ ( t ) ∅ j q ( x ) = ∑ r = 1 n f ( x r , t ) δ ( x − x r ) + ∑ c = 1 v f j ( x c , t ) δ ( x − x c ) ,   i = j (9)

Multiplying Eq. 9 by the trial function ∅ i y ( x ) , integrating over the length of each of the horizontal beams, i.e., from 0 to L and applying orthogonality of mode shapes, yields the following transformation: { M i q u ¨ i q ( t ) + C i q u ˙ i q + K i q u i q = f s ( x , t ) ∅ i q ( L 2 ) + ∑ r = 1 n ∅ i q ( x r ) f ( x r , t )   + ∑ r = 1 n ∅ i q ( x r ) f d ( x r , t ) + ∑ c = 1 v ∅ i q ( x c ) f i ( x c , t ) ,   i = 1   M i q u ¨ i q ( t ) + C i q u ˙ i q + K i q u i q = ∑ r = 1 n ∅ i q ( x r ) f ( x r , t ) + ∑ r = 1 n ∅ i q ( x r ) f d ( x r , t ) + ∑ c = 1 v ∅ i q ( x c ) f i ( x c , t ) ,   i = 2 : j − 1 M j q u ¨ j q ( t ) + C j q u ˙ j q + K j q u j q =   ∑ r = 1 n ∅ i q ( x r ) f ( x r , t ) + ∑ r = 1 n ∅ j q ( x r ) f d ( x r , t ) + ∑ c = 1 v ∅ j q ( x c ) f j ( x c , t ) ,   i = j (10)

Here, the transverse displacements, w ( x , t ) i = 1,2 , … ,   j , in the forces f ( x r , t ) , f d ( x r , t ) and f i ( x c , t ) in Eq. 10, represented in Eqs 6–8, are also affected by the integration and application of orthogonality condition in Eq. 10. Also, M i q and K i q are given as: M i q =   ρ A ∫ 0 L ∅ i q ( i i ) ( x ) d x   ;   K i q =   E I ∫ 0 L ∅ i q ( i v ) ∅ i q ( x ) d x ; C i q = 2 ζ q M i q K i q   (11)

In Eq. 10 the impact of damping on the layered metamaterial structure is considered. Here, C i q and ζ i q are the horizontal beam’s damping coefficient and the damping ratio of the i t h horizontal beam in q t h mode, respectively. The mode functions of the fixed-fixed beams, ∅ i q ( x ) , are assumed to be the same as the trial function, ∅ i y ( x ) , and given as (Zhou et al., 2017): ∅ i q ( x ) = ∅ i y ( x ) = [ s i n h β i q x − s i n β i q x + ( s i n h β i q L − s i n β i q L c o s β i q L − c o s h β i q L ) ( c o s h β i q x − c o s β i q x ) ] (12)

Moreover, the wave number β i q is given by: c o s h ( β i q L ) c o s   ( β i q L )   = 1 (13)

Furthermore, using Newton’s second law, the movement of each r t h local resonator on each i t h horizontal beam (including the j t h horizontal beam) can be described as: m r Y ˙ i r ( t ) + f d ( x r , t ) + f ( x r , t ) = 0 (14)

Solving Eqs 10, 14 simultaneously using Runge-Kutta numerical method, the generalized displacements and the local resonator displacement are obtained. The obtained resonator displacement is then substituted into Eq. 2, in order to obtain the displacement of the horizontal beam. This Galerkin solution is truncated at N = 4. Transmissibility is taken as the output displacement at the center of the most upper horizontal beam, i.e. ( x = L 2 ) , relative to the input vibration at the center of the lowest horizontal beam, i.e., vibration source. This is obtained by substituting the generalized displacement result, u q , obtained from solving Eqs 10, 14 simultaneously, into Eq. 2. The transmissibility η T , is then given as: η T = w ( 0.5 ∗ L , t ) B e a m   i = j w ( 0.5 ∗ L , t ) B e a m   i = 1   (15)

In this section experimental methods are described. Mainly, a brief description of fabrication and assembly of the layered metamaterial vibration attenuation structure as well as experimental setup used for characterization are discussed. Experimental work was focused on measuring the transmissibility of the layered metamaterial vibration attenuation structure. Model simulations were then validated against these experimental measurements.

The layered metamaterial structure was designed using Solidworks CAD software, then changed into an STL (STereoLithography) format that is suitable for 3D printing. PETG (polyethylene terephthalate) was used as the printing filament in the Ultimaker 3 Fused Deposition Modeling (3D FDM) printer. The local resonators (cantilever beams) were designed in a way that allowed insertion of additional masses at their tips. The tip-masses were created separately and then slotted into the tips of the local resonators. These additional tip-masses were vital in lowering the resonant frequencies of local resonators to less than 300 Hz. Figure 4 shows a prototype of the layered metamaterial vibration attenuation structure fabricated in this work. For this work, a layered metamaterial beam structure with four horizontal beams ( j = 4 ) was chosen. This choice was informed by the need for the layered metamaterial structure to fit on the 3D printer bed used for fabrication. The dimensions and specifications of the manufactured layered metamaterial structure are outlined in Table 1.

The set-up of the experiment is shown in Figure 5. First, an external vibration waveform was generated by the controller (S81B–P02, SENTEK DYNAMICS). This waveform was then passed through the power amplifier (LA-800, SENTEK DYNAMICS) for amplification, before reaching the shaker table (VT-500, SENTEK DYNAMICS). The electromagnetic shaker table was used to apply the external vibration input to the layered metamaterial structure that was fixed on top of the shaker table. The shaker table was operated at predetermined frequency range of 50–700 Hz, and acceleration level of 0.5 g [m s⁻²]. The sides of the layered metamaterial vibration attenuation structure were held with a fixture and clamps. Two accelerometers (PCB333B30 model, PCB Piezotronics) were used to capture the input signal from the shaker table and the output signal from the top of the layered metamaterial structure. The first accelerometer was placed on top of the shaker table to capture the input signal data, while the second accelerometer was positioned on top of the layered metamaterial structure to capture the output signal. The input and output signals were both monitored using Crystal Instrument’s Engineering Data Management (EDM) software on a PC. The transmissibility was analyzed by comparing the output acceleration from the top-most beam of the metamaterial structure to the input acceleration from the vibration source, i.e., shaker table.

The results from the FRF analysis were obtained from Eq. 15 as the Root Mean Square ratio between the output displacement (at the center of the upper beam) and the input displacement (from the vibration source) (Zhou et al., 2017). The parameters used in this study are shown in Table 1. For the purpose of model validation and design investigation, a layered metamaterial structure made of 3 unit cells per beam, three vertical columns between beams, and a total of four horizontal beams was manufactured (Figure 4). In this study modes beyond the fourth modes were truncated because higher modes are unnecessary as have been shown in literature (Li et al., 2017). For example, the 5th mode of the unit cell is 1,469 Hz, which is much higher than the frequency range of interest in the current study (below 700 Hz). When frequencies much higher than 700 Hz are considered, the value of N should be increased to capture those higher modes (Zhou et al., 2017).

The focus of this article is studying layered metamaterial vibration attenuation structures with local resonators using both experimental methods and model simulations. In this work, design criteria and generalized modelling platform for layered metamaterial structures with local resonators have been established. A fundamental model has been developed using Galerkin method and superposition of modes to describe the dynamic response and transmissibility of the layered metamaterial vibration attenuation structure in response to an external vibration. The transverse vibrations of the layered metamaterial beams were represented with Euler-Bernoulli equation of beams. A prototype of the layered metamaterial structure has been fabricated and characterized experimentally. The prototyped layered metamaterial vibration attenuation structure constituents included four horizontal beams, local resonators forming unit cells, and vertical beams linking the horizontal beams together. Each local resonator consisted of cantilevers with tip masses, while horizontal beams were connected to each other via the vertical beams. Results show good agreement between model simulations and experimental data. Results also reveal the roadmap for successful design of layered metamaterial structures. For example, effects of various core design parameters in the layered metamaterial structure on the first and second frequency bandgaps have been investigated in this work. Main conclusions from these studies include:

• Altering the length of the local resonators primarily affects the shape and position of the local resonator frequency bandgap.