Introduction

The material with positive Poisson’s ratio (PR) exhibits a phenomenon of swelling in a direction perpendicular to the direction of compression. Conversely, the material with negative PR exhibits shrink in the transverse direction when compressed [1, 2]. Compared with traditional positive PR materials [3, 4], negative PR metamaterials exhibit superior mechanical properties and application prospects [5] in terms of indentation resistance [6], lightweight [7], impact resistance [8, 9], energy absorption [10, 11] and vibration reduction [12, 13].

In terms of dynamics, negative PR metamaterials show superiority over traditional materials in reducing vibration propagation [14, 15]. Duc et al. [16, 17] investigated the nonlinear dynamic response and vibration of composite double-curved shallow shells with auxetic cellular, and the effects of material properties, geometrical parameters, elastic foundations, auxetic core layer and blast loads on nonlinear dynamic response were analyzed to guide the design of auxetic composite structure. Reviews in [18] presented the advantages of negative PR metamaterials in damping and acoustics. Scarpa et al. [19] studied the transmission of vibration waves in conventional materials and negative PR metamaterials and concluded that metamaterials with negative PR show better vibration reduction performance for wave propagation. In addition, the conclusions from numerical simulations and experiments are used to guide the prediction and design of acoustic properties of negative PR metamaterials [20, 21]. Unlike traditional periodic auxetic materials, when the functionally graded materials are subjected to vibration or shock, the dynamic response and deformation of the local structure change with the change of the cell gradient, which can reduce the local stress concentration to ensure the integrity and reliability of the structure [22, 23]. Ajdari et al. [24] studied the mechanical properties of density gradient honeycomb structures under uniaxial and biaxial plane impact using finite element and theoretical methods. Previous studies on dynamics and vibration response have focused more on examining mechanics, properties and applications through simulations and experimental tests [18]. As mentioned in the work by [25], how to design a metamaterial structure with specified properties is more practical.

The design of the mechanical properties has received great attention in mitigating vibration. It is revealed in [26] that the formation of the band gap can be traced back to the resonant behavior of the elementary building blocks of the honeycomb structure under different boundary conditions and concluded that the damping band gap can be designed to achieve the purpose of vibration reduction. Afterward, optimization design is recognized as an effective way to improve mechanical properties, where the size and shape optimizations are two common optimization methods with features of being convenient and effective. For the applications of size or shape optimizations [27], metamaterials are designed by optimizing the cell wall thickness, cell angle, cell height and cell length [28,29,30]. The above improved methods based on the conventional configuration are insufficient for improving the mechanical properties of metamaterials [31, 32]. Moreover, the unique performance of metamaterials is strongly dependent on the configurations of the underlying substructure; that is to say, many interesting properties of metamaterials can be obtained through innovative configuration design of metamaterials’ substructure [33, 34].

The reviews on metamaterials topology optimization for dynamics are concluded in [35, 36]. Previous work by [37] proposed a functional element topology optimization (FETO) method, which optimizes the configuration of substructures and then periodically arranges the substructures to generate metamaterials. Consequently, this work proposed a FETO-based vibration reduction design method; then, a series of novel configurations metamaterials with specified negative PR are designed to improve the vibration reduction performance.

This work is organized as follows. In ''Metamaterials design for vibration reduction'' section, the FETO-based vibration reduction design method is described in detail, and the concept of mechanical impedance is introduced to evaluate the vibration reduction effect. In "Formulation of the optimization problem'' section, a mathematical model is established to perform the configuration design of metamaterials. "Numerical simulation of the metamaterials" section analyzes the dynamic properties of metamaterials and summarizes the effect of PR to the vibration reduction performance. In "Evaluation of vibration reduction performance" section, comparison of the vibration reduction of novel metamaterials and honeycomb is performed.

Metamaterials design for vibration reduction

Based on the FETO method, a design method of metamaterials with vibration reduction performance is proposed in this section and the concept of mechanical impedance is introduced to evaluate the vibration reduction effect.

Definition of functional elements and metamaterials

Figure 1 illustrates the main idea of the FETO method. The initial materials structure to be designed is subdivided into several design domains, and the single design domain is discretized into finite element meshes; that is, the functional element topology optimization design of this work is carried out in the design area.

Figure 1
figure 1

Schematic of design method of metamaterials based on functional element topology optimization

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Mechanical impedance to evaluate vibration reduction

Generally, the vibration reduction effect can be expressed as the ratio of the vibration response at the excitation location to the response at the output location; the greater the ratio, the better the vibration reduction effect. Since the distribution path of materials is uncertain during the optimization calculation iteration, it is not conducive to collecting the input and output responses of all finite element nodes. Once the mechanical impedance is used to describe the vibration reduction effect, it is only necessary to collect the response of the excitation position without obtaining a response to the output position. This is due to the fact that the mechanical impedance of the excitation position is an inherent feature of structural vibration and can reflect the vibration reduction effect of the metamaterials and structures.

Definition of mechanical impedance

Mechanical impedance is a key indicator of structural dynamic characteristic, which directly reflects the structural dynamic performance under external excitation [38, 39], and the generalized mechanical impedance is defined as the ratio of harmonic excitation force to the response caused by it. It is well known that the vibration response can be represented in terms of displacement, velocity and acceleration. Similarly, the mechanical impedance can also be classified into displacement impedance, velocity impedance and acceleration impedance.

In this work, the acceleration impedance is used to describe the mechanical impedance of metamaterials:

$$ Z = \frac{{F_{i} }}{{X_{j} }} $$
(1)

where \( F_{i} \) is the input excitation force acting on node i and \( X_{j} \) is the output acceleration response of node j.

Conversion between the mechanical impedance and vibration reduction

Figure 2 shows the definition of mechanical impedance in functional element. For a forced vibration system, the input acceleration response at the excitation position is denoted as \( a_{0} \), and the output acceleration response after the vibration transmission is denoted as \( a_{\text{n}} \). The transmission rate (\( T_{\text{a}} \)) and vibration level difference (\( {\text{VLD}} \)) can be expressed as:

$$ T_{\text{a}} = \frac{{a_{\text{n}} }}{{a_{0} }} $$
(2)
$$ {\text{VLD}} = 20\log_{10} \left( {\frac{{a_{0} }}{{a_{\text{n}} }}} \right) = 20\log_{10} \left( {\frac{{a_{0} /a_{\text{ref}} }}{{a_{\text{n}} /a_{\text{ref}} }}} \right) = L_{1} - L_{2} $$
(3)

where \( L_{1} \) is the input acceleration level at the excitation position, \( L_{2} \) is the output acceleration level at the response position and \( a_{\text{ref}} \) is the reference value of acceleration.

Figure 2
figure 2

Definition of the mechanical impedance in functional element

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The relationship between \( {\text{VLD}} \) and \( T_{\text{a}} \) is

$$ {\text{VLD}} = 20\log_{10} \left( {\frac{{a_{0} }}{{a_{\text{n}} }}} \right) = 20\log_{10} \frac{1}{{T_{\text{a}} }}, $$
(4)

and the relationship between impedance and \( T_{\text{a}} \) can be expressed as

$$ T_{\text{a}} = \frac{{a_{\text{n}} /F}}{{a_{0} /F}} = \frac{{Z_{0} }}{{Z_{\text{n}} }} $$
(5)

where \( Z_{0} \) represents the origin impedance at the excitation position and \( Z_{\text{n}} \) represents the impedance at the response position.

Similar to the definition of the \( {\text{VLD}} \) in Eq. (3), this work defines an impedance level \( L_{\text{z}} \) to characterize the structural vibration transmission properties

$$ L_{\text{z}} = 20\log_{10} \frac{Z}{{Z_{\text{ref}} }} $$
(6)

where \( Z_{\text{ref}} \) represents the reference value of impedance, and this work takes \( Z_{\text{ref}} = 1\,{\text{Ns}}^{ 2} / {\text{m}} \).

Substituting Eq. (5) into Eq. (4) yields

$$ {\text{VLD}} = 20\log_{10} \frac{{Z_{\text{n}} }}{{Z_{0} }} = 20\log_{10} \left( {\frac{{Z_{\text{n}} /Z_{\text{ref}} }}{{Z_{0} /Z_{\text{ref}} }}} \right) = L_{\text{zn}} - L_{\text{z0}} $$
(7)

where \( L_{\text{z0}} \) represents the origin impedance level of the vibration excitation and \( L_{\text{zn}} \) represents the transmission impedance level after the vibration transmission.

In this work, the difference between impedance levels is defined as the impedance level difference (\( {\text{ILD}} \)):

$$ {\text{ILD}} = L_{\text{zn}} - L_{\text{z0}} . $$
(8)

Although the value of \( {\text{VLD}} \) in Eq. (7) is equal to the value of \( {\text{ILD}} \) in Eq. (8), the physical meaning is quite different. From Eq. (2), when the transmission ratio \( T_{\text{a}} \) is constant, the output acceleration response \( a_{\text{n}} \) decreases as the input acceleration response \( a_{0} \) decreases, thereby achieving the purpose of reducing the output vibration response after the vibration propagation of metamaterials. In addition, it can be seen from Eqs. (1) and (6) that the decrease in \( a_{0} \) leads to an increase in \( L_{\text{z0}} \). Therefore, maximizing \( L_{\text{z0}} \) as an optimization objective can be used to improve the vibration reduction performance of metamaterials.

In order to intuitively reflect the mechanical impedance properties of the structure, a concept of synthesized origin impedance level \( \bar{L}_{\text{z0}} \) is defined to synthesize the values of impedance level in the sweep frequency range:

$$ \bar{L}_{\text{z0}} = 10\log_{10} \sum\nolimits_{i = 1}^{S} {10^{{0.1L_{\text{z0}}^{i} }} } $$
(9)

where \( L_{\text{z0}}^{i} \) is the origin impedance level at a specified frequency \( f_{i} \) and \( S \) is the number of the sweep frequency.

Macroscopic effect of Poisson’s ratio

The general methods for calculating PR values of a metamaterials can be used for common cell configurations [40,41,42], but it is not suitable for the PR calculation of cells with irregular shapes in topology optimization process. The following scholars have studied the macro-PR evaluation method to solve the above problems. Previous work by [43, 44] presented a method to study the macroscopic mechanical properties of materials by applying simple loads to characterize the macroscopic mechanical properties of materials under complex stress conditions. Carneiro et al. [45] introduced a PR evaluation method in material mechanics test to analyze the macroscopic PR effect of cellular material.

In this work, the method in [45] is also used to describe the macroscopic PR effect during the optimization iteration of functional element. The cell configuration in Fig. 3 is an example to calculate the macroscopic PR of metamaterial in detail. The strain of the cell configuration is

$$ \varepsilon_{\text{X}} = 2\frac{{\Delta x_{2} }}{{x_{2} }},\quad \varepsilon_{\text{Z}} = 2\frac{{\Delta z_{1} }}{{z_{1} }} $$
(10)

where \( \varepsilon_{\text{X}} \) and \( \varepsilon_{\text{Z}} \) are the strain in X- and Z-directions, \( \Delta x_{2} \) and \( x_{2} \) are the displacement and coordinate position of point \( n_{2} \) in X-direction and \( \Delta z_{1} \) and \( z_{1} \) are the displacement and coordinate position of point \( n_{1} \) in Z-direction.

Figure 3
figure 3

Evaluation method of the macroscopic PR effect

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Then, the PR evaluation of the cell configuration in Fig. 3 can be expressed as:

$$ \nu_{\text{ZX}} = - \frac{{\varepsilon_{\text{X}} }}{{\varepsilon_{\text{Z}} }}. $$
(11)

Formulation of the optimization problem

This section describes the details of a metamaterials design, and Fig. 4 illustrates the optimization design of functional element and the construction of metamaterials. In this work, a series of metamaterials with various PRs are designed to analyze the effect of PR on vibration reduction performance [46, 47] and to verify the reliability of the proposed method of metamaterials vibration reduction design.

Figure 4
figure 4

Flowchart of the vibration reduction design method of metamaterials

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Topology optimization calculation in this work is carried out by the commercial software of Hyperworks/OptiStruct. The scheme materials model of the optimization adopts the SIMP method (solid isotropic materials with penalization), and the general idea is to introduce a fictitious density variables field to penalize, for each element, some relevant physical quantities like element stiffness tensor, materials density, etc. The relationships between the relative density and the elastic modulus of materials are as follows:

$$ E\left( {x_{e} } \right) = E_{\hbox{min} } + x_{e}^{p} \left( {E_{0} - E_{\hbox{min} } } \right) $$
(12)
$$ {\mathbf{K}} = \sum\limits_{e = 1}^{N} {\left( {E_{\hbox{min} } + x_{e}^{p} \Delta E} \right)} {\mathbf{k}}_{0} $$
(13)

where \( x_{e} (e = 1,2, \cdots N) \) is the relative density of element e; N is the number of elements; \( E_{0} \) and \( E_{\hbox{min} } \) are the initial elastic modulus of the elements and the elastic modulus of the hole elements; \( \Delta E = E_{0} - E_{\hbox{min} } \), \( E_{\hbox{min} } = E_{0} /1000 \); \( p \) is the penalty factor; \( E\left( {x_{e} } \right) \) is the elastic modulus after interpolation; \( {\mathbf{K}} \) is the global stiffness matrix which can be obtained as the sum of elemental stiffness over all N elements; and \( {\mathbf{k}}_{0} \) is the initial stiffness matrix of each mesh element.

Due to the existence of semi-dense elements, the analysis results may change dramatically when the design process enters a new phase using a different penalization factor. In OptiStruct, the discrete parameter corresponds to penalty factor, and it usually takes a value between 2.0 and 4.0. The default discrete value is 1.0 for shell dominant structures and 2.0 for solids dominant structures. (The dominance is defined by the proportion of number of elements.) When minimum member size control is used, the penalty starts at 2 and is increased to 3 for the second and third iterative phases. For other manufacturing constraints such as draw direction, extrusion, pattern repetition and pattern grouping, the penalty starts at 2 and increases to 3 and 4 for the second and third iterative phases, respectively.

Modeling of optimization problem

As shown in Fig. 5, the design domain is discretized into \( 84 \times 84 \) finite element meshes, and the type of the elements is plate. Within the frequency range between 10 and 300 Hz, a sinusoidal excitation force with an amplitude of \( F = 10\;{\text{N}} \) is applied to the center of the upper faceplate while the lower faceplate is fixed. The measuring points 1 and 2 (in Fig. 5) are used to evaluate the macroscopic Poisson’s ratio effects. The metamaterials are made of PLA (polylactic acid) with Young’s modulus \( E_{\text{P}} = 2636\,{\text{MPa}} \), density \( \rho_{\text{P}} = 1180\,{{\text{ kg}} \mathord{\left/ {\vphantom {{\text{ kg}} {{\mathrm{m}}^{ 3} }}} \right. \kern-0pt} {{\text{m}}^{ 3} }} \) and materials Poisson’s ratio \( \nu_{\text{P}} = 0.38 \).

Figure 5
figure 5

Design domain and loading of the functional element topology optimization (\( B = H = 42{\text{ mm}} \))

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From the Poisson’s ratio in Eq. (11), we can express the macroscopic Poisson’s ratio in Fig. 5 as follows:

$$ \nu { = } - \frac{{\varepsilon_{2} }}{{\varepsilon_{1} }}{ = } - \frac{{\Delta u_{2} }}{{\Delta w_{1} }} \cdot \frac{B}{H} $$
(14)

where \( \Delta w_{1} \) and \( \Delta u_{2} \) represent the displacement of points 1 in Z-direction and the displacement of points 2 in X-direction.

This paragraph describes the mathematical model of vibration reduction optimization design for metamaterials. Regarding the setting of the objective function, the purpose of maximizing the synthetic origin impedance level \( \bar{L}_{\text{z0}} \) as the objective of vibration reduction is explained in "Mechanical impedance to evaluate vibration reduction" section. In this work, the optimization problem is described as: within a certain specified amount of materials, the PR is set as the constraint, and the maximization of the origin impedance level is taken as the objective. Then, the mathematical model of optimization is

$$ \left\{ {\begin{array}{*{20}l} {{\text{find}}} \hfill & {{\mathbf{X}} = \left[ {x_{1} ,x_{2} , \cdots ,x_{N} } \right]^{{\rm T}} } \hfill \\ {{\text{max}}} \hfill & {\bar{L}_{{{\text{z0}}}} } \hfill \\ {{\text{s}}{\mathrm{.t}}{\text{.}}} \hfill & {\mathbf{M}}{\ddot{\mathbf{U}} + {\mathbf{KU}}{\text{ = }}{\mathbf{F}}} \hfill \\ {} \hfill & {\left| {\nu - \nu _{0} } \right| \le \varepsilon } \hfill \\ {} \hfill & {f^{\prime}_{{{\text{vol}}}} \le V({\mathbf{X}})/V_{0} \le f_{{{\text{vol}}}}^{{\prime \prime }} } \hfill \\ {} \hfill & {0 < x_{{\min }} \le x_{e} \le x_{{\max }} \le 1} \hfill \\ {} \hfill & {e = 1, \ldots ,N} \hfill \\ \end{array} } \right. $$
(15)

where \( {\mathbf{X}} = \left[ {x_{1} ,x_{2} , \cdots ,x_{N} } \right]^{\rm T} \) is the vector of design variables. \( x_{\hbox{max} } \) and \( x_{\hbox{min} } \) are the upper and lower limits of design variables, respectively (nonzero to avoid singularity). \( \bar{L}_{\text{z0}} \) denotes the synthesized origin impedance level. The global stiffness matrix can be obtained as the sum of elemental stiffness over all N elements, i.e., \( {\mathbf{K}} = \sum\nolimits_{e = 1}^{N} {x_{e}^{p} {\mathbf{k}}_{0} } \). \( {\mathbf{M}} = \sum\nolimits_{e = 1}^{N} {x_{e}^{p} {\mathbf{m}}_{0} } \) is the total mass of the functional element, where \( {\mathbf{m}}_{0} \) is the initial mass of each mesh element. \( {\mathbf{U}} \) and \( {\mathbf{F}} \) are the global vector of displacements and forces. \( {\mathbf{u}}_{e} \) is the displacement vector of mesh element. \( V({\mathbf{X}}) \) is the total structural volume in the optimization progress; \( V_{0} \) is the initial total volume when the relative density of the design domain area is 1. \( f^{\prime}_{\text{vol}} \) and \( f^{\prime\prime}_{\text{vol}} \) are the lower limit and upper limit volume fraction of the materials in design domain. \( \left| {\nu - \nu_{0} } \right| \le \varepsilon \) is the constraint of PR, where \( \nu \) and \( \nu_{0} \) represent the PR in the optimization iteration and the PR of the specified design requirements, respectively.

Optimized configurations of functional element

Considering that the amount of material used in the structural topology optimization design has a great influence on the configuration of the optimized functional element, the setting of the volume fraction (\( f^{\prime}_{\text{vol}} \) and \( f^{\prime\prime}_{\text{vol}} \)) in the optimization model of this work considers the following two aspects: The lower limit \( f^{\prime}_{\text{vol}} \) is set to 20% to ensure the stability of the optimized structure, and the upper limit \( f^{\prime\prime}_{\text{vol}} \) is set to 30% to avoid the metamaterials being too heavy after optimization.

As shown in Fig. 6, the four functional elements corresponding to PR with \( \nu = - \,0.5 \), \( - \,1.0 \), \( - \,1.5 \) and \( - \,2.0 \) are generated by calculating Eq. (15), and the synthesized origin impedance levels \( \bar{L}_{\text{z0}} \) [in Eq. (9)] after optimization iteration are 24.6 dB, 29.7 dB, 32.1 dB and 32.8 dB, respectively.

Figure 6
figure 6

Optimal configurations of the functional element corresponding to various PRs: a\( \nu { = } - 0.5 \); b\( \nu { = } - 1.0 \); c\( \nu { = } - 1.5 \); d\( \nu { = } - 2.0 \)

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Although the wall thickness of the configuration in Fig. 6 is not uniform, Gibson [41] verified that the wall thickness has a negligible effect on the mechanical properties when the wall thickness is much smaller than the cell size. Therefore, under the premise of ignoring the influence of wall thickness on mechanical properties, the wall thickness of the extracted configurations is designed to be \( t = 1\,{\text{mm}} \), and the uniformly spaced grid is employed herein to represent the irregular configurations of the functional elements (as shown in Fig. 7).

Figure 7
figure 7

Extraction of the optimal configurations corresponding to various PRs: a\( \nu = - 0.5 \); b\( \nu = - 1.0 \); c\( \nu = - 1.5 \); and d\( \nu = - 2.0 \)

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Verification of the PR values of metamaterials

Since the extraction of the optimization results may cause errors, this paragraph reanalyzes the extracted configurations by finite element analysis. The extracted configurations in Fig. 7 are established as the finite element model for static analysis, and the actual PR of extracted configuration in Fig. 7 can be calculated according to Eq. 11; therefore, the error ratio between the actual PR after extraction and the PR of design requirement can be calculated. As shown in Fig. 8 and Fig. 9, the PR error rate analysis of functional element with \( \nu = - \,1.5 \) is taken as an example to describe the calculation method of the PR error rate in detail. The reanalysis of the extracted configuration is summarized in Table 1, and the comparison shows that the Poisson’s ratio before and after the extraction is roughly consistent.

Figure 8
figure 8

Finite element model of the extracted functional element with \( \nu = - 1.5 \), which is established according to the dimensions of Fig. 7c

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Figure 9
figure 9

Displacement analysis of the extracted functional element with \( \nu = - 1.5 \)

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Table 1 Relative errors of PR between the extracted configurations and design requirements
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Metamaterials based on periodic arrangement of functional element

As shown in Fig. 6 and Fig. 7, the configurations of the functional elements with maximization vibration reduction performance are designed after optimization. Afterward, the extracted and verified functional elements are periodically arranged into the metamaterials. Figure 10 shows the corresponding functional elements, metamaterials and specimen with \( \nu { = } - \,0.5 \), wherein the specimen is fabricated by additive manufacturing.

Figure 10
figure 10

Optimal functional element, metamaterials and 3D-printed specimen, with corresponding PR \( \nu = - 0.5 \)

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Numerical simulation of the metamaterials

The metamaterials are generated by ordering the functional elements. Each optimal configuration in Fig. 7 is periodically arranged in a cycle of \( 8 \times 8 \) to form the metamaterials in Fig. 11. Within the frequency range between 10 and 300 Hz, a sinusoidal excitation force with an amplitude of \( F = 10{\text{ N}} \) is applied to the center of the upper faceplate while the lower faceplate is fixed, where the thicknesses of both faceplates are 10 mm. The outline dimension of each metamaterial is \( 3 3 6 {\text{ mm}} \times 356{\text{ mm}} \), and the depth perpendicular to the page is 20 mm. The metamaterials are made of PLA (polylactic acid) with Young’s modulus \( E_{\text{P}} = 2636{\text{ MPa}} \), density \( \rho_{\text{P}} = 1180{{\text{ kg}} \mathord{\left/ {\vphantom {{\text{ kg}} {{\text{m}}^{ 3} }}} \right. \kern-0pt} {{\text{m}}^{ 3} }} \) and materials Poisson’s ratio \( \nu_{\text{P}} = 0.38 \).

Figure 11
figure 11

Metamaterials corresponding to various PRs : a\( \nu = - \,0.5 \); b\( \nu = - \,1.0 \); c\( \nu = - \,1.5 \); and d\( \nu = - \,2.0 \)

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Next, the dynamic characteristics of each metamaterial in Fig. 11 will be analyzed by the software of Hyperworks. Within the frequency range between 10 and 300 Hz, a sinusoidal excitation force with an amplitude of \( F = 10\,{\text{N}} \) is applied to the center of the upper faceplate while the lower faceplate is fixed, and the critical damping values of the structural vibration analysis in this work are uniformly set as 1%. The modal analysis of the vertical first-order natural frequency is summarized in Table 2, and the frequency response analysis calculates the frequency response curves of the measuring points (A, B, C, D and E) in each metamaterial (as shown in Fig. 12).

Table 2 Vertical first-order natural frequency of metamaterials corresponding to various PRs
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Figure 12
figure 12

Frequency response of measuring points A, B, C, D and E corresponding to metamaterials structures with various PRs: a\( \nu = - \,0.5 \); b\( \nu = - \,1.0 \); c\( \nu = - \,1.5 \); and d\( \nu = - \,2.0 \)

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Vibration reduction performance

The vibration reduction performance of the four metamaterials is described by analyzing the frequency response characteristics of several functional elements at the center of metamaterials, where the frequency response characteristics of each functional element can be represented by the frequency response of the measuring points (a–e in Fig. 11) adjacent to the corresponding functional element. Figure 13 shows the acceleration vibration level difference (VLD) of the measuring points in each metamaterial, and the VLDs of these measuring points reflect the vibration reduction performance of the corresponding adjacent functional element. Briefly, according to the definition of Eq. (3), the acceleration VLD of the points i and j can be expressed as

$$ {\text{VLD}}_{i - j} = 20\log_{10} \left( {\frac{{a_{i} }}{{a_{j} }}} \right) $$
(16)

where \( a_{i} \) and \( a_{j} \) represent the acceleration response of the measuring points i and j.

Figure 13
figure 13

VLD between measuring points A–B, B–C, C–D, D–E and A–E corresponding to metamaterials with various PRs: a\( \nu = - \,0.5 \); b\( \nu = - \,1.0 \); c\( \nu = - \,1.5 \); and d\( \nu = - \,2.0 \)

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The data in Fig. 13 show that the vibration amplitude is effectively suppressed after the vibration propagated through the metamaterials, which indicates that the series of metamaterials have the characteristics of vibration reduction. In addition, for a metamaterials specimen with a certain PR, the VLD value between measuring points of the functional element is larger as the functional element is located closer to the fixed lower faceplate. This phenomenon is affected by the boundary effect theory, and the VLD value between the two points closer to the fixed constraint position is larger; thus, the values of \( {\text{VLD}}_{{{\text{A}} - {\text{B}}}} \), \( {\text{VLD}}_{{{\text{B}} - {\text{C}}}} \), \( {\text{VLD}}_{{{\text{C}} - {\text{D}}}} \) and \( {\text{VLD}}_{{{\text{D}} - {\text{E}}}} \) are gradually increasing.

In addition, the sum of the VLD values of \( {\text{VLD}}_{{{\text{A}} - {\text{B}}}} \), \( {\text{VLD}}_{{{\text{B}} - {\text{C}}}} \), \( {\text{VLD}}_{{{\text{C}} - {\text{D}}}} \) and \( {\text{VLD}}_{{{\text{D}} - {\text{E}}}} \) is equal to that of \( {\text{VLD}}_{{{\text{A}} - {\text{E}}}} \), which indicates that the vibration reduction effect of metamaterials can be converted by the accumulation of several functional elements. This phenomenon also means that for a metamaterials with periodic functional element arrangement, the VLD of the metamaterials can be calculated by cumulatively calculating the VLD of each functional element.

Effect of Poisson’s ratio on vibration reduction performance

In order to intuitively express the influence of PR on the vibration reduction performance of the four metamaterials, the performances comparison is based on the premise of choosing the measuring points with similar positions. Figure 14 shows the vibration reduction of metamaterials at measurement points A–B, B–C, C–D and D–E, respectively. In the range of 10–155 Hz, the vibration reduction effects increase as the absolute values of the PR increase. When the frequency is greater than 215 Hz, the vibration reduction effects of other metamaterials decrease with the increase in the absolute value of PR except for the metamaterials with \( \nu = - \,0.5 \).

Figure 14
figure 14

Vibration reduction effect of each PR metamaterials at measuring points: a A–B; b B–C; c C–D; and d D–E

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However, more attention should be paid to the vibration reduction performance of the metamaterials structure in engineering applications, rather than the vibration reduction of the substructure (functional element) that makes up the metamaterials structures. Therefore, in this work, the vibration reduction performance of metamaterials is described by calculating the acceleration VLD between measuring points A–E, which is denoted by \( \overline{\text{VLD}} \). Thus, the vibration reduction performance of the various novel designed metamaterials is summarized in Fig. 15.

Figure 15
figure 15

Vibration reduction performance \( \overline{\text{VLD}} \) of various PR metamaterials

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Figure 15 shows the \( \overline{\text{VLD}} \) curves of various metamaterials, and it can be seen that the \( \overline{\text{VLD}} \) of the four metamaterials is greater than 7 dB in the frequency range of 10–300 Hz, which is equivalent to a 55.3% reduction after the vibration propagation through the metamaterials. In particular, the metamaterials with \( \nu = - \,1.0 \), \( \nu = - \,1.5 \) and \( \nu = - \,2.0 \) exhibit superior vibration reduction performance, and the \( \overline{\text{VLD}} \) values range from 8.5 dB to 10 dB; thus, the amplitudes of the vibration are reduced by 62.4–68.4%. Consequently, the metamaterials designed with the goal of maximizing original impedance show excellent vibration reduction performance.

Evaluation of vibration reduction performance

The honeycomb materials in [46] is introduced as a comparison to illustrate the vibration reduction performance of the novel designed metamaterials. As shown in Fig. 16, the wall thickness and material’s parameters of the honeycomb in [46] are consistent with this work, and the loading and boundary conditions of the honeycomb are the same as in this work.

Figure 16
figure 16

Honeycomb specimen with \( \nu = - \,2.0 \)

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In this work, the two methods of vibration level difference (VLD) and frequency response are used to evaluate the vibration effect of metamaterials. The VLD measures the attenuation amplitude of the vibration wave between the two responses, so the VLD can easily reflect the relative change of the vibration amplitude between various responses in the same metamaterials. And the larger the value of VLD is, the greater the attenuation of the vibration wave between the two responses of measuring points, that is, the better the vibration reduction performance.

The frequency response analysis is able to effectively compare the response amplitude of the output of each metamaterial under the same excitation force. That is, the difference in vibration reduction performance between each of the metamaterials can be judged by comparing the amplitude of the responses at the respective output position, and the smaller the amplitude of the output response, the better the vibration reduction performance of the metamaterials.

Evaluation of vibration reduction performance with VLD

The data in Fig. 15 are further analyzed, and the curves of \( \overline{\text{VLD}} \) in the range of 10–300 Hz are converted as a synthesized value \( \overline{\text{VLD}}^{\text{all}} \).

$$ \overline{\text{VLD}}^{\mathrm{all}} { = }\frac{1}{Q}\sum\nolimits_{i = 1}^{Q} {\overline{\text{VLD}}_{{f_{i} }} } $$
(17)

where \( f_{i} = 10 + 5i\;{\text{Hz}} \), \( i = 1, \, 2, \ldots ,Q \), and \( Q = 58 \) when the sweep range is within 10–300 Hz. \( \overline{\text{VLD}}_{{f_{i} }} \) represents the amplitude of \( \overline{\text{VLD}} \) at the specified frequency \( f_{i} \).

Figure 17 shows the acceleration VLD curves between the measuring points A and E of the honeycomb. Although the vibration reduction effect of the honeycomb in the local frequency band is better, the vibration amplification begins to appear after the frequency is higher than 150 Hz, which is contrary to the intention of vibration reduction. Furthermore, the synthesized vibration reduction effect of the honeycomb is calculated by Eq. (17) as \( \overline{\text{VLD}}_{\text{H}}^{\text{all}} = 8.5\,{\text{ dB}} \). From Fig. 18, the performance of the novel metamaterials is 1 dB higher than that of the honeycomb. (The vibration reduction performance is improved by 12%.)

Figure 17
figure 17

Acceleration vibration level difference of the novel designed metamaterials and honeycomb

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Figure 18
figure 18

Vibration level difference of the novel designed metamaterials and honeycomb

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The synthesized vibration reduction effect \( \overline{\text{VLD}}^{\text{all}} \) of the four metamaterials is shown in Fig. 18, and \( \overline{\text{VLD}}^{\text{all}} \) are all greater than 8.5 dB; thus, the amplitude of the vibration is reduced by 62.4%. In particular, the vibration reduction performance of the metamaterials with \( \nu = - \,1.0 \), \( \nu = - \,1.5 \) and \( \nu = - \,2.0 \) is prominent. (The value of \( \overline{\text{VLD}}^{\text{all}} \) is about 9.5 dB, and the vibration amplitude is reduced by 66.5%.)

In summary, the novel designed metamaterials exhibit better vibration reduction performance than the honeycomb, and the vibration reduction characteristics are stable and reliable.

Evaluation of vibration reduction performance with frequency response

In addition to using VLD to indicate the vibration propagation attenuation in materials or structures, the acceleration frequency responses at the measuring point E in novel metamaterials and honeycomb are analyzed simultaneously (as shown in Fig. 19), and the acceleration response of measuring point E is denoted by \( \overline{a} \). An equivalent evaluation similar to the one defined in Eq. (17) is used to describe the value of the synthesized acceleration \( \overline{a}^{\text{all}} \) in the range of 10–300 Hz:

Figure 19
figure 19

Acceleration frequency response of the novel designed metamaterials and honeycomb

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$$ \overline{a}^{\text{all}} { = }\frac{1}{Q}\sum\nolimits_{i = 1}^{Q} {\overline{a}_{{f_{i} }} } $$
(18)

where \( f_{i} = 10 + 5i\,{\text{Hz}} \), \( i = 1,2, \ldots ,Q \), and \( Q = 58 \) when the sweep range is within 10–300 Hz. \( \overline{a}_{{f_{i} }} \) represents the amplitude of \( \overline{a} \) at the specified frequency \( f_{i} \).

According to Eq. (18), the synthesized acceleration of the four metamaterials and honeycomb is represented by \( \overline{a}_{ - 0.5}^{\text{all}} \), \( \overline{a}_{ - 1.0}^{\text{all}} \), \( \overline{a}_{ - 1.5}^{\text{all}} \), \( \overline{a}_{ - 2.0}^{\text{all}} \) and \( \overline{a}_{\text{H}}^{\text{all}} \), respectively. From Table 3, it is observed that the synthesized acceleration of the four metamaterials is lower than that of the honeycomb. In order to more intuitively compare the difference between the vibration reduction effects of the four metamaterials and honeycomb, the vibration reduction effects of the four metamaterials relative to the honeycomb can be calculated by the following formula:

$$ {\text{VLD}}_{\nu } = 20\log (\overline{a}_{\text{H}}^{\text{all}} /\overline{a}_{\nu }^{\text{all}} ) $$
(19)

where \( \overline{a}_{\nu }^{\text{all}} \) represents the synthesized acceleration of the metamaterials with \( \nu = - 0.5, \, - 1.0, \, - 1.5, \, - 2.0 \). As shown in Table 3 and Fig. 20, the vibration reduction of the four metamaterials relative to the honeycomb is calculated by Eq. (19).

Table 3 Vibration reduction effects of metamaterials relative to honeycomb
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Figure 20
figure 20

Vibration reduction effects of the metamaterials (relative to the honeycomb)

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By comparing the vibration reduction of the four metamaterials relative to the honeycomb, it shows that the vibration reduction performance of the novel designed metamaterials is obvious. In particular, the acceleration response of the metamaterials with \( \nu = - 1.0 \) is reduced by 19 dB (in Fig. 20), which means that the magnitude of the acceleration response is reduced by 88.8%.

Conclusions

This work proposes a vibration reduction design method of metamaterials for improving the vibration reduction performance. The concrete conclusions are as follows:

  1. (1)

    A FETO-based vibration reduction design method of metamaterials is proposed, and a series of negative PR metamaterials with novel configurations are designed. Numerical simulation shows that the error rate of the optimized PR and PR design requirements is less than 10%, which illustrated the feasibility of the proposed design method.

  2. (2)

    Frequency response shows that the vibration amplitudes of the all designed metamaterials are reduced by 62.4%, and it is demonstrated that the metamaterials design with vibration reduction performance can be obtained by maximizing mechanical impedance as an optimization goal.

  3. (3)

    Comparison shows that the novel designed metamaterials exhibit at least 12% improvement in vibration reduction performance over the honeycomb, and the vibration reduction performance is stable and reliable.

In addition to being the objective of the optimization, the specified vibration reduction performance can also be used as a constraint for the optimization problem. For example, the mathematical model of optimization can be established as follows: With a vibration reduction effect greater than 4 dB as a constraint, the mathematical model can be constructed with the goal of minimum total weight to design lightweight metamaterials.