Composite metastructure with tunable ultra-wide low-frequency three-dimensional band gaps for vibration and noise control

Abstract

Low-frequency vibration and noise control present enduring engineering challenges that garner extensive research attention. Despite numerous active and passive control solutions, achieving multiple ultra-wide attenuation regions remains elusive. Addressing vibration and noise control across a multidirectional broad low-frequency spectrum, three-dimensional metastructures have emerged as innovative solutions. This study introduces a novel three-dimensional composite metastructure featuring multiple ultra-wide three-dimensional complete band gaps. The research emphasizes the design strategy of elastic ligaments to achieve multiple ultra-wide attenuation regions spanning from 0.7 to 40 kHz. The band structures are elucidated through modal analysis and further substantiated by an analytical model based on a spring-mass chain with an additional resonator. The underlying physical mechanism for the formation of multiple ultra-wide band gaps is revealed through novel vibration modes from finite element analyses. Furthermore, we demonstrate that the distribution and the relative width of the ultra-wide band gaps can be tuned by modifying the geometric parameters of the metastructure. Utilizing additive manufacturing, prototypes are fabricated, and low-amplitude vibration tests are conducted to evaluate real-time vibration attenuation properties. Consistency is observed among theoretical, numerical, and experimental results. The proposed structure shows significant potential for high-performance meta-devices aimed at controlling noise and vibration across an extremely wide low-frequency spectrum.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-024-73909-4.

Introduction

Engineered periodic structures with morphological designs to manipulate acoustic and elastic waves have garnered significant research interest over recent decades, owing to their dynamic characteristics that surpass those of conventional materials^1,2. Of particular interest is the formation of band gaps (BGs), which represent specific frequency regions where elastic or sound waves are effectively prohibited. Previous studies have demonstrated that BGs are formed through three distinct mechanisms: Bragg scattering at the wavelength scale^1,3, as well as subwavelength and deep subwavelength mechanisms such as inertial amplification and local resonance^4–6. The positioning and width of BGs are vital for the practical implementation of these engineered structures. Although substantial work has been conducted on manipulating acoustic and elastic waves for various purposes such as localization and waveguiding^7–10, broadband wave attenuation^11,12, rainbow trapping effects¹³, negative refraction¹⁴, topological properties^8,9,15, acoustic and elastic cloaks^16,17, and asymmetric elastic wave transmission¹⁸, achieving all-direction vibration and noise control with ultra-wide three-dimensional BGs remains intriguing.

Several studies have aimed to achieve ultra-wide BGs across various frequency ranges through material variation, including monolithic structures^9,19, elastic impedance-based multi-material periodic structures^12,20–22, or geometric parameters of the structure^23–30. Despite these intriguing discoveries and rapid advancements, effective control of low-frequency vibration and noise continues to pose fundamental challenges. Various designs and approaches involving active and passive metamaterials, have been proposed to tackle this issue; however, efficient control of low-frequency vibration and noise with structures smaller than the wavelength of propagating waves in all directions remains a key research objective. Although fascinating advancements such as inertially amplified periodic designs^31,32, inverse design techniques^33,34, design optimization²⁰, structural modifications like single and composite mechanical metastructures^35–39, multi-resonant dissipative periodic structures¹¹, trampoline metamaterials⁴⁰, vibroacoustic metamaterial⁴¹, and active piezoelectric acoustic metamaterials^42–44, the challenge of achieving vibration and noise control with broad three-dimensional band gaps at low-frequency persists. For example, structures with multiple local resonators can generate wider BGs; nevertheless, the width of these BGs predominantly relies on the resonator’s mass and impedance mismatch. Consequently, achieving a wider bandgap (BG) necessitates larger resonating elements with material mismatch. Moreover, the utilization of active control techniques to induce ultra-wide vibration attenuation zones poses challenges in manufacturing such smart devices, as advanced technology is required⁴⁵. Additionally, some practical drawbacks are inherent to any active control structures, including energy consumption, stability issues, and operational costs⁴⁶, resulting in structures that exhibit excellent numerical wave phenomena but are complex or impractical to manufacture.

Recently, there has been considerable attention on a three-dimensional metastructure with complete three-dimensional BGs⁴⁷. The proposed metastructure morphology comprises rigid masses and elastic ligaments, resembling traditional spring-mass chains. This structure plays a crucial role in achieving an ultra-wide three-dimensional complete BG. A detailed explanation of the physical mechanism responsible for the enlargement of the BG is also included. The ultra-wide BG is produced via the mode separation principle or modal mass participation. In this context, global and local resonant modes and the localized vibrational energy within the periodic unit cell are identified as factors that contribute to the opening (lower) and closing (upper) bounding edge of the ultra-wide BG. The proposed metastructure is engineered so that the concentration of all vibrational energy within the entire unit cell structure leads to the opening of the BG. Meanwhile, the local resonant mode, characterized by vibrational energy surrounding the elastic frame assembly, effectively closes the BG. Inspired by this design principle, several studies proposed new configurations aiming at expanding the relative bandwidth of the first BG via monolithic structures^47–54, through new shapes of heavy rigid masses and elastic frame assemblies to maximize the difference in vibrational energy at the lower and upper bounding edges of the BG. Most recently, Muhammad et al.⁵⁴. successfully expanded the relative bandwidth of the first BG of three-dimensional composite metastructures designed based on the mode separation principle. By embedding cylindrical steel into the polymeric casing, they aimed to maximize the discrepancy of vibrational energy at the lower and upper bounds. This design approach successfully shifted the lower bounding edge of the first BG to a lower frequency region, while the upper bounding edge moved higher, resulting in an ultra-wide complete BG. Furthermore, this study proposed an interesting approach for manufacturing composite metastructures via additive manufacturing and simple manual assembly processes, without interrupting the manufacturing process in a previous study³⁷. Although fascinating designs and promising results have been reported, these works focus solely on expanding the relative bandwidth of the first BG, rendering the neighboring BG relatively narrow. The first BG can still be further expanded by utilizing high-density materials (e.g. lead, tungsten) for embedded inclusions to maximize effective mass, but this approach increases costs and risks damage or failure of the elastic frame during operation⁵⁴. Thus, designing innovative topological structures with multiple ultra-wide BGs over the low-frequency region is an effective means of expanding the vibration attenuation region for structures. Generally, when two structures have similar lattice constants and materials, preference is given to the one with multiple ultra-wide BGs.

Herein, to overcome the limitations of previously reported structures and achieve multiple ultra-wide low-frequency BGs for vibration and noise control, we propose a new three-dimensional composite metastructure based on the principle of mode separation. The proposed meta-atom structure is a composite design that builds upon previous studies^50,54, focusing on a new elastic frame assembly capable of generating multiple ultra-wide BGs. Detailed information regarding the design configuration is provided in the following section. The metastructure is designed to be periodically structured in all three directions to achieve three-dimensional complete BGs. It is to be noted that the proposed structure here has an identical lattice constant to reported structures^47–54 but differs in its ability to generate multiple ultra-wide BGs without compromising the relative bandwidth of the first ultra-wide BG. This study relies on finite element analysis (FEA) complemented by analytical modeling and experimental tests on the fabricated prototypes. The band structure is computed using FEA software COMSOL Multiphysics 5.6, employing the Bloch theorem, as outlined in “Basic theory” section in the supplementary information. The vibration modes corresponding to the lower and upper bounding edges of BGs are identified, and the governing mechanism behind the formation of multiple ultra-wide BGs is elucidated. A parametric study is conducted to explore the tunability of BGs. The effectiveness and efficiency of BGs are verified through numerical frequency response analyses on the metastructure and vibration tests on the fabricated prototypes. Subsequently, the elastic vibrational BGs of the proposed structure are compared with those of superior mechanical metastructures reported to date. The proposed metastructure design can be implemented in applications for controlling low-frequency vibration and noise in facilities and mechanical systems, covering an extremely wide frequency range.

Prototype

As outlined in the Introduction section, we have designed a new meta-atom structure to overcome the limitations of reported structures and achieve multiple ultra-wide BGs in the low-frequency regime. In the prototype, the unit cell consists of an elastic frame assembly connected with a hard cylindrical steel inclusion encapsulated by a soft polymeric casing at the midpoints, as shown in Fig. 1. The lattice constant for the proposed unit cell structure remained at a = 50 mm, consistent with previous studies^{47–50,52−54}. All remaining geometric parameters are shown in the inset of Fig. 1a,b,d and e with respect to the lattice constant a as presented in Table 1. The finalized unit cell structure design includes the following internal properties: the frame assembly thickness in the vertical and horizontal directions are w₁ and w₂, respectively, as listed in Fig. 1d. The elastic frame assembly is connected with the polymeric casing of radius r, height h_p, and thickness t. The introduction of the chamfer dimension d listed in Fig. 1b is aimed at avoiding the overlapping design of polymeric casings. The cylindrical steel masses are enclosed within a polymeric casing with a height h_s. The introduction of angle θ and distance l₁ makes the proposed elastic frame assembly different from all the reported structures^48,50. The introduction of the new elastic frame assembly serves two purposes: (i) it remains strong support and maintains fabricating possibility for the unit cell structure, while decreasing the thickness of the ligament beams compared to the elastic frame assembly utilized in^53,54; and (ii) it facilitates the expansion of the relative bandwidth Δω/ω_c of the second BG, which gives the second ultra-wide BG in the proposed unit cell structure. The effectiveness of the proposed design strategy, along with the underlying physical mechanisms, is explained in the Results and discussion section. For numerical modeling and additive manufacturing, Nylon 12⁴⁹ and Steel⁵⁴ with their properties, listed in Table 2, are used. We analyzed the BG features of the proposed unit cell through a numerical wave dispersion analysis to define the eigenfrequencies subjected to varying wavenumbers swept across the boundaries of the irreducible Brillouin zone (IBZ), as presented in the inset of the band structure in Fig. 2b. In this work, numerical BG analysis is determined by employing the Solid Mechanics Module of COMSOL Multiphysics 5.6. We modeled a single unit cell and applied periodic Bloch-Floquet boundary conditions to the three pairs of lateral faces, as explained extensively in “Basic theory” section and Fig. S2 in the supplementary information. The unit cell geometry is meshed with tetrahedral elements, and we confirm mesh convergence. The eigenfrequency calculation is performed for the cubic unit cell at the periodic boundary of the IBZ using positive wave vector values.

Fig. 1.

Proposed prototype for unit cell structure. (a) Schematic illustration of the proposed unit cell. (b) In-plane geometry of the unit cell and associated geometric parameters. (c) Vibration mode at the opening boundary of the first BG as determined by COMSOL Multiphysics 5.6. (d,e) Schematic descriptions of the geometric parameters for the elastic frame assembly. (f) Fabricated composite unit cell. The geometric parameters are listed in Table 1.

Table 1.

Geometric dimensions of the unit cell structure illustrated in Fig. 1.

Parameter	a	l1	hp	hs	w1	w2	r	d	θ
Dimension with respect to a	1.000a	0.140a	0.240a	0.200a	0.035a	0.035a	0.590a	0.05a	–
Dimension	50 mm	7 mm	12 mm	10 mm	1.75 mm	1.75 mm	29.5 mm	2.5 mm	45°

The parameter a represents the characteristic dimension of the unit cell, signifying its overall size.

Table 2.

Material properties of Nylon 12 and steel.

Material	Density (kg/m3)	Young’s modulus (GPa)	Poisson’s ratio
Nylon 12	1000	1.586	0.4
Steel	7850	210	0.3125

Fig. 2.

Numerical dispersion relation spectra. (a) Complete band structure of the proposed layout. (b) The first BG with a relative bandwidth (Δω/ω_c(1)) of 178.8%. (c) Vibration modes of the complete unit cell, and the elastic frame assembly shown with isometric view, illustrating the contribution to the opening and closing boundaries of the BGs.

To analytically demonstrate the low-frequency range of operation of the BGs, we establish a spring-mass chain with an additional resonating element model to describe the opening and closing frequency of the first BG and the opening frequency of the second BG, as explained in “Analytical model” section in the supplementary information. Using the developed analytical model, the calculated model parameters and frequencies corresponding to the bounding edges of BGs are listed in Table 3. Analytical results for the proposed unit cell structure at the opening bounding edge of the first BG are denoted with a green dashed line in Fig. 2a and b. For the proposed layout, the computed frequency f_op1 = 711.2 Hz is approximately 10% lower than the predicted value f_op1^(a) = 790.6 Hz. This discrepancy can be attributed to a larger connection region between the elastic frame assembly and the polymeric casing, as accounted for in the analytical analysis, resulting in a reduction of the bending stiffness of the elastic frame⁵⁰. In higher frequency modes, the deviation between the numerical and analytical calculation slightly surpasses 10% at maximum. Hence, we infer that the proposed analytical model effectively captures the dispersion of both global and local modes within the three-dimensional structure, thereby supporting the functionality of mode separation.

Table 3.

Calculated parameters of the analytical model.

Parameter	Value	Unit
M	35.173	(g)
m	0.031	(g)
K	156.09	(N/mm)
k	121.76	(N/mm)
K + k/2	216.97	(N/mm)
f op1	790.6	(Hz)
f cl1	14,106	(Hz)
f op2	14,114	(Hz)

Results and discussion

To evaluate the performance of BGs, an important quantitative measure, the relative bandwidth Δω/ω_c is utilized. A wider relative bandwidth Δω/ω_c implies the ability to attenuate wave energy over an ultra-wide frequency range⁵⁵. The relative bandwidth Δω/ω_c of a BG is defined by Δω/ω_c = 2(ω_u – ω_l)/(ω_u + ω_l), where ω_u and ω_l are the BG closing and opening frequencies, respectively, of the corresponding BG. The results of the dispersion study of the proposed composite unit cell are illustrated in Fig. 2a and b. The boundary of the IBZ is presented in the inset of the band structure in Fig. 2b. The black solid line denotes the passband, and the blue region shows the complete BG of the proposed structure. To understand the formation mechanism of ultra-wide BGs, vibration modes at the lower and upper bounding edges of the first two ultra-wide BGs are analyzed and illustrated in Fig. 2c.

On the dispersion relation side, the plotted dispersion diagram presented in Fig. 2a demonstrates the effectiveness of the proposed design strategy in expanding the second relative bandwidth (Δω/ω_c(2)) without compromising the first relative bandwidth (Δω/ω_c(1)) of the structure. It can be observed from Fig. 2a that Δω/ω_c(1) of the proposed structure is 178.8%, a significant improvement of 12.8% compared to Metastructure II and less than 2.4% compared to Metastructure I reported in⁵⁴. It should be noted that these metastructures are reported to possess the widest three-dimensional complete relative bandwidth reported to date. However, thanks to the proposed design strategy, the second BG of the proposed structure is also ultra-wide, with a bandwidth of 97.2%, which is significantly wider compared to the reported metastructures⁵⁴. Interestingly, Fig. 2b shows that the first two BGs of the proposed structure are separated by tiny passbands ranging from 12,735 Hz to 12,880 Hz, which can be weakened via the damping effect^11,19, eventually resulting in a super-wide BG covering an extremely wide frequency range. If these tiny passbands are neglected, a super-wide BG spanning from 711.2 Hz to 37,260 Hz with a bandwidth of 192.5% is obtained. Although the reported metastructures⁵⁴ also have tiny passbands in their dispersion analysis results, the second BG width is relatively narrow, resulting in a failure to obtain a super-wide vibration attenuation region. The evidence of flat passbands elimination due to the damping effect is presented in Frequency response analysis section.

To investigate the physical mechanisms behind the formation of ultra-wide BGs, the eigenmodes labeled A, B, C, and D at the bounding edges of the first two ultra-wide BGs in Fig. 2a are elucidated in Fig. 2c. The color gradient in Fig. 2c illustrates the normalized displacement field, with blue and red indicating minimum and maximum values, respectively. According to these four eigenmodes, the band structures comprising global and local resonant modes contribute to the opening and closing of the BGs in the proposed unit cell structure. The term “global resonant mode” denotes a state where vibrational energy is disseminated throughout the entire unit cell structure. In the proposed configuration, this mode encompasses the engagement of the polymeric casing, embedded cylindrical steel, and elastic frame assembly, as illustrated in mode A in Fig. 2c. The integration of a novel frame assembly, which leverages the flexural and torsional rigidity of the connecting beams instead of the axial stiffness reported in previous studies^47,49,53,54, facilitates a more compliant connection between the cylindrical masses. This results in robust vibrational energy within the proposed unit cell, as depicted in mode A in Fig. 2c, thus shifting the opening boundary of the first BG to a lower frequency region compared to prior structures^{47,48,50,52–54}. On the other hand, the term “local resonant mode” suggests that vibrational energy is concentrated in specific components of the unit cell structure⁴⁸. For the proposed composite structure, this mode is solely responsible for the oscillation of the portions at the corners of the elastic frame assembly, as depicted in the vibration modes of the elastic frame assembly at modes B, C, and D in Fig. 2c. Specifically, these unique modes involve the bending and torsional deformation of the beams at the frame corners, rather than the flexural deformation of the entire frame^48,49,53,54 or including the oscillations of the masses^50–52. In the proposed layout, these modes occur at higher frequencies and expand the relative bandwidth compared to previous studies. In mode B, corresponding to the closing boundary of the first BG, a distinctive vibration mode involving bending and torsional motions occurs exclusively at a corner of the elastic frame assembly, while the cylindrical masses remain stationary, resulting in minimal vibrational energy in this mode. A comprehensive illustration and detailed discussion of this vibration mode, highlighting how it involves both bending and torsional deformation, are provided in “Unique vibration mode analysis” section and Fig. S3a of the supplementary information. This phenomenon is attributed to the presence of cross-shaped beams at the corners of the proposed frame assembly. These beams are designed to reduce the free length of the elastic beams, thereby limiting their flexural deformation and increasing the global stiffness of the frame assembly. Such design facilitates the use of thinner elastic beams while ensuring both manufacturability and operational efficiency. The adoption of thinner elastic beams enables the creation of a lower-frequency ultra-wide BG. The notable discrepancy between the robust vibrational energy of the global mode (Mode A) and the low vibrational energy of the local mode (Mode B) leads to the emergence of the first ultra-wide BG, which surpasses the widths reported in previous studies^47–53. Similar to mode B, mode C, which corresponds to the opening boundary of the second BG, is also dominated by the bending and torsional motions of the elastic beams, but at a different corner of the elastic frame assembly. An extensive depiction and detailed analysis of this vibration mode are provided in the “Unique vibration mode analysis” section and Fig. S3b in the supplementary information. Due to their similar vibrational modes, these two eigenmodes are adjacent and represent flat pass bands with low vibrational energy. These passbands can be mitigated through the damping effect, as discussed in the Frequency response analysis section of this research. In mode D, a novel vibration mode emerges, characterized by flexural motion at one corner and both bending and torsional deformations at the opposite corner. In contrast to previous studies, where the closing vibration mode of the second BG is dominated either by the oscillation of the entire elastic frame assembly^48,53,54, or included the motion of masses^50–52, our proposed elastic frame assembly reduces the free length and thickness of the elastic beams. This reduction diminishes the lumped mass, which plays a crucial role in determining the frequency position of the local resonant modes at the closing boundaries of the BG, thereby shifting the closing boundary of the second BG to a higher frequency region. This unique vibration mode, characterized by low vibrational energy in the higher frequency region, contributes significantly to the second ultra-wide BG. A detailed illustration and analysis of this vibration mode are reported in the “Unique vibration mode analysis” section and Fig. S3c in the supplementary information. Additional material illustrating the vibration mode of the complete unit cell and the elastic frame assembly at the bounding edges of the reported BGs is shown in a supplementary video. The video clearly demonstrates the deformation behavior of the complete unit cell and the elastic frame assembly at the frequency of the bounding edges of the reported BGs from isometric, front, top, and side views, obtained from the eigenmode analyses in COMSOL Multiphysics software.

Additionally, it is found that at a certain BG, the deformation mechanism of the unit cell remains consistent for both global and local modes across all points of the IBZ, ensuring functionality in all three directions. According to the previous study⁵⁴, the proposed metastructure can be easily manufactured. The polymeric casing and frame of ligaments can be fabricated by additive manufacturing, then manually assembled with precision-machined cylindrical steel to achieve the complete unit cell structure. Through the discussion of the BGs and vibrational modes of the proposed metastructure, it is evident that the proposed structure is effective in expanding the first and second relative bandwidth compared with the previous studies. Additionally, it is clear that the BG structure is directly related to the global and local resonant modes. Therefore, any change in the geometry of the metastructure including the embedded cylindrical steel or elastic frame assembly may affect the BG characteristics and vibration modes. In the following section, a comprehensive examination of the impacts of geometric parameters is presented.

Parametric study

The preceding section deliberates on the pivotal role of the elastic frame assembly in generating multiple ultra-wide BGs. Furthermore, previous study⁵⁴ underscores the substantial influence of the dimensions of cylindrical steel masses on the BG structure. Hence, a meticulous stochastic parametric study is undertaken to scrutinize the effects of the supporting elastic frame assembly and the dimensions of steel inclusions at this stage. Our primary focus is on geometric parameters that affect the global stiffness, local stiffness of the elastic frame, and global and lumped mass. These parameters encompass the distance l₁, horizontal beam thickness w₂, angle θ, as well as the thickness of the polymeric casing t. Particular emphasis is directed towards the reported two BGs, with meticulous attention to the frequency positions of the opening bounds f_op1 (BG1) and f_op2 (BG2), the closing bounds f_cl1 (BG1) and f_cl2 (BG2), and their relative bandwidths Δω/ω_c(1) and Δω/ω_c(2). The numerical outcomes of the parametric study are depicted in Fig. 3.

Fig. 3.

Parametric study results. Effect on the lower (f_op1, f_op2) and upper (f_cl1, f_cl2) bounding edges of the ultra-wide BGs and relative bandwidth Δω/ω_c(1) and Δω/ω_c(2) of (a,e) horizontal beam thickness w₂, (b,f) distance l₁, (c,g) angle θ, and (d,h) wall thickness of polymeric casing t. Calculated Δω/ω_c(1) and Δω/ω_c(2) obtained from COMSOL Multiphysics 5.6 under the variation of geometric parameters. Corresponding geometry at lower and upper bound of each parameter is presented in the top of Fig. 3. The normalized displacement field of the complete unit cell at f_cl2 at the pivotal points of each parameter is illustrated below the BG distribution diagrams. A detailed illustration of vibration modes at these points can be found in Fig. S4 in the supplementary information.

It is to be noted that the height of embedded steel inclusions h_s is kept constant at 0.2a across all designed unit cell structures to ensure effective evaluation of other parameters and avoid failure in the manufacturing of the polymeric casing. Additionally, the vertical elastic beam thickness w₁ is maintained uniformly at 0.035a in all designs to ensure consistent elastic frame assembly height. The chamfer dimension d is also held constant at 0.05a to ensure the manufacturability of the polymeric casing. The black solid line in Fig. 3 represents the geometric parameters employed in the baseline structure. It is noteworthy that the f_op1, f_cl1, and f_op2, represented by the global and local resonant modes, remain consistent across all analyzed structures. However, the mode at f_cl2 exhibits variability, particularly contingent on the values of the scrutinized parameters. Therefore, we provide the displacement field at f_cl2 for the complete unit cell structure at pivotal values of each parameter. This aims to enhance the understanding of how unit cell geometry affects the structure of the second BG. Given the narrow passbands separating the two BGs, the influence of geometric parameters on f_cl1 and f_op2 manifests proportionally akin.

The influence of the horizontal thickness of the elastic beam w₂ on the BGs’ bounds and relative bandwidths is elucidated in Fig. 3a and e, respectively. Increasing w₂ results in higher global and local stiffness, thus shifting the BGs’ bounds to a higher frequency regime. However, the impact of w₂ on these bounds, particularly f_op1, governed by the global resonant mode and contingent on global mass, is relatively minor in comparison to f_cl1, and f_op2, influenced by the local resonant mode. Analysis of the unit cell structure’s displacement field at w₂ = 0.03a and w₂ = 0.08a reveals that the closing mode of the second BG also aligns with the local resonant mode, specifically manifesting as flexural-torsional deformation at a corner of the frame assembly. Despite variations in w₂, the vertical thickness w₁ remains constant, thereby inducing only marginal alterations in flexural and torsional stiffness. Detailed views of these modes can be found in Fig. S4a in the supplementary information. Consequently, the frequency of this bound experiences a slight elevation with an increase in w₂. The increase in w₂ alters the frequency position of all bounds of the two reported BGs, thereby resulting in a reduction in Δω/ω_c for both BGs. However, a notably downward trend is observed in the second BG. The impact of the distance l₁ is illustrated in Fig. 5b and f. Increasing l₁ extends the free length of the elastic beams, thereby precipitating diminished global and local stiffness. In contrast to the effect of w₂, the bounding edges of the BGs migrate towards a lower frequency position. However, as the opening mode of the first BG is engendered by the global resonant mode, the influence of l₁ on this bound is relatively less pronounced compared to the other bounds. Specifically, the increase in l₁ raises the lumped mass while decreasing the local stiffness, thereby causing f_cl1 and f_op2 to move towards a lower frequency regime. According to the displacement field, the closing mode of the second BG is the local resonant mode, transitioning from flexural-torsional deformation at a corner at l₁ = 0.1a to flexural-torsional motions of the entire frame assembly at l₁ = 0.26a. Comprehensive illustration of these modes are shown in Fig. S4b of the supplementary information. This transition yields higher vibration energy at the vibration mode of f_cl2, thus lowering this bound to a lower frequency position. Consequently, increasing l₁ narrows Δω/ω_c(1) while expanding Δω/ω_c(2). The effect of the angle θ is elucidated in Fig. 5c and g. Specifically, increasing the angle θ prolongs the free length of the elastic beams, causing a decrement in global stiffness. Consequently, f_op1, represented by the global mode, shifts to a lower frequency regime. However, the increase in angle θ also elevates the lumped mass, resulting in a higher frequency distribution of f_cl1 and f_op2. Interestingly, the impact of angle θ on f_cl2 is divided into two regions, with θ = 35° considered as the transition point. Specifically, when 15°θ 35°, this bound gradually shifts to a higher frequency position, corresponding to the transition from flexural deformation at all corners to flexural-torsional deformation at one corner of the frame assembly. Meanwhile, when 35°θ 75°, the lumped mass increases owing to the elongation of the free length of the beams, culminating in the flexural-torsional motion at one corner transforming into bending and torsional motions of the entire frame assembly. Detailed depictions of these modes can be found in Fig. S4(c) of the supplementary information. This mode with higher vibration energy gradually shifts f_cl2 towards a lower frequency regime. With the increase in angle θ, Δω/ω_c(1) gradually expands, while Δω/ω_c(2) initially increases up to θ = 35°, subsequently diminishing with further increments of θ. The effect of t on the reported BGs is depicted in Fig. 5d and h. The thickness of the polymeric casing, t, is inversely related to the size of the steel inclusion, such that an increase in one parameter results in a proportional decrease in the other parameter. Consequently, increasing t precipitates a reduction in the size of the steel inclusion, thereby lowering the global mass. As a result, the opening bound of the first BG, f_op1, shifts to a higher frequency regime. Meanwhile, f_cl1 and f_op2 are represented by local resonant mode, thus the impact on these bounds is negligible. Therefore, Δω/ω_c(1) gradually diminishes with the increase of t. The impact of this parameter on the existence of the second ultra-wide BG is also unveiled. Specifically, when 0.02at 0.08a, f_cl2 remains mostly unchanged at a high-frequency regime, while f_cl2 is gradually descends when t ≥ 0.08a. This phenomenon ensues because when the global mass is sufficiently heavy (0.02at 0.08a), the vibration mode of f_cl2 transitions from the high vibrational energy mode with the oscillation of masses and frame assembly (t ≥ 0.08a), as observed in previous studies^50–52, into lower vibrational energy mode (0.02at 0.08a), characterized by deformation of a portion of the frame assembly, resulting in the higher frequency position of f_cl2. Expanded views of these modes are available in Fig. S4d of the supplementary information. This shift leads to Δω/ω_c(2) remaining mostly unchanged when 0.02at 0.08a, subsequently diminishing with the increase of t.

Fig. 5.

Frequency response analyses with standard linear solid viscoelastic model. A comparison of frequency response analyses between experimental (blue solid lines), numerical with elastic (magenta solid lines), viscoelastic (black dotted lines) material models, and analytical (green dashed lines) model along (a) Γ-X direction. (b) An inclined direction near Γ-M path. (c) Normalized displacement field of the metastructure at the flat pass band position identified in linear elastic model and at a resonant peak observed in viscoelastic model. (d) Frequency response spectra measured at different probe points in the viscoelastic model.

Based on the aforementioned findings, it is demonstrated that the distribution of the two reported BGs can be manipulated to occupy higher or lower frequency regions by adjusting global and local stiffness through the modification of the free length of elastic beams using l₁, horizontal thickness w₂, angle θ, or thickness of the polymeric casing t. Furthermore, the relative bandwidth Δω/ω_c of the two reported BGs can be expanded further by shortening the free length of beams using l₁, altering the angle θ, or decreasing the horizontal thickness of the elastic beam w₂. It is worth noting that a prerequisite condition for the existence of the second ultra-wide BG is that the global mass of the unit cell structure must be sufficiently heavy.

Frequency response analysis

The reported band structures have been computed utilizing COMSOL Multiphysics 5.6, with periodic Floquet-Bloch conditions enforced along all faces of the cylindrical mass comprising the structure, ensuring indefinite periodicity in the x, y, and z directions. Previous studies^52–54 proposed a method for assessing the vibration mitigation potential of the proposed metastructure by constructing a finite array of unit cells and conducting frequency response analyses. In this regard, a metastructure comprising a 3 × 3 × 3 unit cell array would suffice to validate the complete attenuation region in all three directions^48–51. As illustrated in Fig. 4a, a harmonic excitation signal is applied at one end (cyan area), while the resulting response signal is monitored at the opposite end of the metastructure, specifically at probe point 7. This transmission analysis corresponds to the Γ-X direction in the IBZ. To validate the vibration mitigation properties of the proposed metastructure in all directions, additional transmission analyses along the Γ-M direction^55,56 are conducted to confirm the wave propagation in all directions. To roughly represent the Γ-M path in the IBZ, we examine the wave propagation along an inclined direction in the k_x - k_y plane⁴⁹, which corresponds with probe points 1, 3, and 5 as shown in Fig. 4a. As depicted in Fig. 4a, various probe positions are utilized to examine the wave propagation and the intensity of wave attenuation in the three-dimensional structure. Input and output displacement data are measured using probes, facilitating the calculation of the transmission loss (T) according to the equation T = 20 log₁₀ (u_out/u_in)^52–54, where u_in and u_out represent the mechanical displacement magnitudes measured at the input and probed points, respectively. To this end, a linear elastic model for Nylon material is employed in the finite element (FE) code simulation platform.

Fig. 4.

Frequency response analyses with linear elastic model. (a) Numerical model of metastructure with input (cyan) and different output probes representing the Γ-X direction and an inclined direction near Γ-M path. (b) Frequency response spectra for the proposed metastructure obtained from COMSOL Multiphysics 5.6 at different probe positions. (c) Normalized displacement field of the metastructure at pass bands and stop bands (BGs region). Consistency between dispersion analysis and frequency response studies at all probe points is observed.

The obtained transmission spectra from the numerical frequency response analyses are depicted in Fig. 4b. It is found that the attenuation region within the frequency response spectra at all the probed points is identical to the band structures illustrated in Fig. 2a. This confirms the existence of the complete three-dimensional BGs. Additionally, the transmission loss varies across different probed points. For probes 1, 2, and 3, equidistant from each other, exhibit an identical attenuation depth of around − 215 dB, while for point probe 4, the wave attenuation is approximately − 275 dB. The wave attenuation at the output end of the metastructure, probe 7, records an attenuation of -350 dB, which is slightly deeper compared to the attenuation value measured at probed points 5 and 6. Despite these depth differences, the wave attenuation region remains identical in all three directions, affirming the presence of the two complete three-dimensional BGs within the proposed structure. The numerically computed displacement profiles of the proposed metastructure along the Γ-X direction for four different frequencies within the pass and stop bands, highlighted by red dots in Fig. 4b, are plotted in Fig. 4c. The color gradient in Fig. 4c represents the normalized displacement field, with navy blue and red indicating minimum and maximum values, respectively. In the pass band regions, at a frequency of 0.43 kHz, the incident wave transmits entirely through the metastructure, as evidenced by the red color observed across all unit cell structures within the analyzed unit cell array, indicating that the transmission loss at this frequency is approximately zero. At the flat pass band of 12.82 kHz, enhanced attenuation performance is observed, with the red color gradient gradually decreasing from the input end to the output end. However, the proposed metastructure still permits incident waves to transmit at this frequency, albeit with a slight reduction in displacement magnitude at the output end. The investigation of the damping effect on the flat pass band is conducted both numerically and experimentally and is discussed in detail in a subsequent section. Regarding the stop band regions, the displacement field at 9.18 kHz and 21.05 kHz demonstrates that the proposed metastructure effectively mitigates the incident waves, with the color gradient transitioning from red at the input layer to navy blue at the output layer, suggesting that the incident waves are absorbed after passing through three layers of the metastructure. The displacement fields at these two frequencies, which exhibit similar transmission loss values of approximately − 350 dB, are largely analogous. It is evident that the proposed metastructure efficiently attenuates transmitted vibrations in the two reported BG regions, while allowing transmission in the pass bands. In conclusion, the presence of two three-dimensional complete ultra-wide BGs is confirmed through frequency response analyses, with identical attenuation regions measured at all probed points in all three directions. Furthermore, the obtained results demonstrate that increasing the number of unit cell structures enhances the attenuation performance of the metastructure. This observation aligns with the principle that the longer the wave propagation path, the greater the attenuation⁵². The displacement fields of the proposed metastructure within the BG regions and pass bands further corroborate the existence of the reported BGs.

The transmission response spectra are initially analyzed using a linear elastic model of FEA simulations without considering material damping/viscosity in the response spectrum. To accurately capture the damping/viscoelastic effects of the polymeric material on vibration attenuation performance, we employ a standard linear solid viscoelastic model for Nylon 12 in simulations. Identical probe points utilized in the linear elastic model are employed for the standard linear solid viscoelastic analyses. The coefficient of Maxwell’s branch is properly applied in the FE code COMSOL Multiphysics 5.6 to describe the damping effect, with a relaxation time of τ_Maxwell = 1.5306e-4 and Young’s modulus of E_Maxwell = 490 MPa^35,49. Thanks to additive manufacturing, the proposed metastructure is constructed using a 3D printer, and low-amplitude vibration tests are performed to explore the real-time vibration mitigation characteristics. We fabricated a metastructure with a 3 × 3 × 1 unit cell array to validate the emergence of attenuation regions. A 3 × 3 × 3 unit cell configuration was not employed to save some materials, based on the deduction that the frequency response spectrum would be similar for both structures, as the reported BGs are uniformly distributed across all directions of the IBZ. Consequently, the 3 × 3 × 1 unit cell structure should not be interpreted as a 2D model. Figure 6 presents the experimental setup with fabricated prototypes. The experimental transmission spectra for the proposed metastructure are shown in Fig. 5a and b with blue solid lines, compared to the numerical elastic and viscoelastic models, which are denoted with black dotted lines and magenta solid lines, respectively. Probe points 5 and 7 are selected for both numerical and experimental comparisons. The comparison of the numerical and experimental response spectra is limited to 20 kHz due to the constraints of the piezoelectric actuator’s working capability.

Fig. 6.

Schematic of the low-amplitude vibration experiment setup in (a) Γ-X direction, and (b) the inclined direction near Γ-M path. A reasonable agreement between the numerical (viscoelastic model) and experiment results is observed in both directions (Frequency response spectra are presented in Fig. 5a and b).

According to Fig. 5a and b, the transmission loss is nearly identical between the elastic and viscoelastic models in both analyzed directions. Additionally, the opening frequency f_op1 observed in the numerical and experimental analyses is identical to that of the analytical model, confirming the existence of the attenuation region. However, the numerical model with the linear solid viscoelastic model in both directions eliminates the flat passbands existing in the linear elastic model in both directions as presented in Fig. 5a and b. Figure 5d shows the transmission spectra with the standard linear solid viscoelastic model at different point probes. The attenuation regions are found to be identical at all probed points in both directions, and the opening bounding edge of the BG fits well with the elastic model, suggesting the proper setup of the linear solid viscoelastic model. The transmission curves indicate a super-wide attenuation region spanning from 0.71 to 40 kHz with deep transmission loss. However, a small resonant peak is present at a frequency of 17.25 kHz in the standard linear solid viscoelastic model in both directions. To investigate this phenomenon, the displacement field of the proposed metastructure at this frequency in the viscoelastic model is plotted in Fig. 5c with the same normalized displacement as that of the elastic model. It is observed that the proposed metastructure effectively absorbs the incident waves at this point, as indicated by the navy blue color region in the last row of the array structure, suggesting that the minor resonant peak does not affect the attenuation performance of the metastructure. Additionally, we plot the displacement field at the frequency of 12.82 kHz for comparison with that of the elastic material model shown in Fig. 4c. It is found that the normalized displacement field at the last row of the constructed array structure in the linear elastic model has completely transformed into a navy blue region in the viscoelastic model, indicating the efficiency of the viscoelastic/damping effect in eliminating flat passbands in the polymeric material structure. Furthermore, the experimental transmission spectra in both directions align well with those of the numerical viscoelastic models. However, due to the limited precision of the piezoelectric sensor compared to the FEA code, a discrepancy exists between the numerical and experimental attenuation depths. The numerical model assumes the polymeric material to be homogeneous, linearly elastic, and isotropic due to the natural damping characteristics of the material. Yet, this assumption may oversimplify the behavior of polymeric materials used in 3D printing, as they frequently exhibit higher material losses and anisotropic properties. This could be another contributing factor to the minor differences observed between the numerical and experimental outcomes. Nevertheless, the experiments aim to verify the existence of the super-wide attenuation region observed in the numerical analyses. The linear solid viscoelastic models show good agreement with the experimental results, suggesting that similar outcomes would be observed at higher frequency regions, thereby confirming the presence of the super-wide attenuation region.

Experiment setup

To confirm the vibration attenuation properties of the proposed periodic structure obtained from numerical analysis, experimental tests are carried out to define the transmission between the input and output region in both analyzed directions. The experiment setups, as depicted Fig. 6, are performed on vibration isolation systems from Daeil Systems to mitigate unexpected vibrations. A piezoelectric actuator (Pst 150/7/40 VS12) is employed to transmit sine wave across a frequency range. The input signal is generated using a generator (Tektronix AFG1022) and amplified with a power piezomechanik amplifier (SVR 150-3). A sine-sweep vibration testing method is utilized, sweeping sine waves from 5 Hz to 20,000 Hz. The piezoelectric actuator is mounted on a connecting part at the input end of the metastructure, facilitated by drilling a small hole, as illustrated in Fig. 6a and b. To capture the response data, the piezoelectric sensor (921090 AX/S - spring push) is connected to a PZT-servo controller (PI E-501.00) to receive and transfer the response signal for display on the oscilloscope. Excitation of the composite metastructure is carried out in both the analyzed directions, as depicted in Fig. 6a and b. The excited and response signal data are collected using a digital phosphor oscilloscope (Tektronix DPO 2014B), and the results obtained are post-processed using a computer system. The response spectrum is computed by calculating the transmission loss T. Further details regarding the experiment and post-processed data procedures can be found in “Experimental data” section in the supplementary information.

Vibration attenuation comparison

Figure 7 presents a comprehensive comparison of the proposed metastructure with mainstream elastic metamaterial structures reported in the existing literature. The performance metrics derived from figures extracted from published research. Specifically, four figures of merit are designated: the relative bandwidth of the first BG Δω/ω_c(1), the relative bandwidth of the second BG Δω/ω_c(2), total effective BG width, and unit cell size. The total effective BG width is defined as = , where n is the number of BGs. The benchmark for these comparisons is detailed in Table S1 in the supplementary information. Figure 7 indicates that our structure demonstrates a balanced high performance across all four criteria. We benchmark against the most recently proposed structures with wide three-dimensional complete BGs^{35–39,41,47–54}. These structures are practical for vibration and noise control as passive solutions at low-frequency regimes owing to their broadband attenuation capabilities. However, these structures typically exhibit only the first ultra-wide BG, with either the second BG or the total effective BG width being relatively narrow, thereby limiting their applicability to a broader low-frequency range. We acknowledge that the reported structure⁴⁹ is exceptional in terms of Δω/ω_c(2). However, its Δω/ω_c(1) and total effective bandwidth are significantly lower than those of our structure. Similarly, the most recent composite metastructures⁵⁴ exhibit a slightly higher Δω/ω_c(1) by 2.4%; however, their Δω/ω_c(2) is not reported and total effective bandwidth is much lower compared to ours. Additionally, while certain reported structures^37–39,41 have smaller lattice sizes, all other criteria are considerably inferior to those in our study. It is notable that only our study and the previous structures^49,54 exhibit an opening frequency below 1 kHz, whereas the others are positioned at much higher frequencies. This comparison highlights the superior vibrational BG performance of the proposed structure relative to high-performance metamaterial structures reported thus far. Our study showcases outstanding performance with the two widest BGs and the most extensive total effective bandwidth. This suggests that the proposed structure holds the potential to replace existing structures in achieving higher vibration attenuation performance.

Fig. 7.

Comparison of vibrational attenuation performance. A supercell model of the proposed structure and a radar chart for the overall assessment of the proposed metastructure compared to mainstream vibration and noise control metamaterial structures.

Method

A spring-mass chain with an additional resonating element model is formulated to calculate resonant mode frequencies and compare them with the numerically derived frequencies of the BG edges. Numerical simulations are conducted utilizing the FEA code software COMSOL Multiphysics 5.6, employing the Floquet-Bloch periodic boundary condition. Following that, frequency response analyses are conducted via the same FEA code software, using a linear elastic model to validate the analyzed BG structure and to visualize the vibration mitigation capability for complete three-dimensional BG. Both the BG structure and frequency response spectra exhibit a perfect agreement. Furthermore, the viscoelastic/damping effect is accounted for in the numerical simulations. A three-dimensional prototype is fabricated using 3D printing technology, and vibration tests are performed to verify the numerical results. Across the study, there is consistent agreement among theoretical, numerical, and experimental outcomes.

Conclusion

This research introduces a new composite metastructure capable of generating multiple ultra-wide BGs. The investigation employs finite element simulations, subsequently validated through an established analytical model and experimental tests. Initially, an analytical model based on a spring-mass chain with an additional resonating element is developed to analyze the opening and closing behaviors of BGs. This analytical framework is followed by numerical simulations to ascertain the band structure and unique vibration modes governing the boundaries of BGs. Both analytical and numerical outcomes exhibit strong alignment. The band structure analysis of the proposed prototype unveils the presence of multiple ultra-wide BGs distributed across a broad low-frequency spectrum. The opening of the first BG is attributed to the involvement of global modes incorporating rigid masses and beam flexural-torsional stiffness. Conversely, the closing of the first BG, alongside all modes pertaining to the second BG, arises from local modes stemming from the flexural-torsional stiffness of the novel frame assembly, with rigid heavy masses playing a negligible role. The discernible disparities between these opening and closing eigenmodes, driven by the differentiation in mode principles, contribute to the formation of the first ultra-wide BG. A design strategy concentrating on elastic ligaments is proposed to shift the closing boundary of the second BG to higher frequencies, thereby achieving multiple ultra-wide BGs. Moreover, a meticulous stochastic parametric study reveals that the distribution of the reported BGs can be fine-tuned, and the relative bandwidth of these BGs can be further expanded by manipulating the geometric parameters of the elastic frame assembly. Additionally, finite arrays of the proposed unit cells are constructed. Through frequency response analyses encompassing both elastic and damping/viscoelastic effects, transmission curves are obtained, affirming vibration attenuation within the BGs. The acquired frequency response spectra underscore a significantly broader vibration attenuation achieved in the proposed metastructure via the material damping effect. This underscores the efficacy of harnessing the damping effect to merge multiple adjacent ultra-wide BGs by eliminating narrow passbands and engendering a super-wide BG. Utilizing additive manufacturing, three-dimensional prototypes are fabricated, and vibration tests are conducted to further validate the findings. Both numerical simulations and experimental outcomes exhibit exceptional agreement. The proposed design holds promise for applications in facilities aimed at absorbing vibrations, thereby effectively mitigating both noise and vibrations across an extremely wide frequency range. This study could serve as a reference in the development of high-performance metadevices tailored for vibration and noise control.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Author contributions

D.B.P. performed numerical analyses and experiments, developed supporting theory, and prepared the manuscript draft. S.C.H. oversaw the research, provided guidance, verified the model, analytical model, and results, reviewed the paper, and managed project funding support. Both of them conceived the idea and edited the manuscript.

Data availability

Data will be made available upon request. Requests for materials should be addressed to S.C.H.

Declarations

Competing interests

The authors declare no competing interests.