Ultrawide 3D Phononic Bandgap Metastructures as Broadband Low Frequency Filter

Graphical Abstract Abstract This present study reports a novel model for the study on three-dimensional phononic metastructures endued with ultrawide three-dimensional complete bandgaps. The phononic structure is made of a unique material without composite material composition. Based on the principal of mode separation and global and local modal masses participation, the well-engineered structural configurations give extremely wide bandgaps with the gap-to-mid gap ratio of 157.6% and 160.1% for two proposed prototypes that produce the widest three-dimensional bandgaps ever reported. The band structures are explained by a modal analysis and the findings are further corroborated by developing an analytical model based on monoatomic mass-spring chain and further verified with well-established FEM numerical simulations. Thanks to additive manufacturing, the prototypes are developed by using 3D printing technology and low amplitude vibration test is performed to access the real-time vibration mitigation characteristics. An excellent agreement is obtained between analytical, numerical and experimental results. The effects of material damping on transmission response is also taken into consideration that eventually merge the separated bandgaps to form a broadband vibration attenuation zone. The results reported are scale independent and the proposed strategy may pave the way for developing novel meta-devices to control the noise and vibration, and underwater acoustic waves at a wide frequency band in all directions.

Artificial periodic structures that once begin from electromagnetic media are presently hot research topics for noise and vibration control due to their unprecedented dynamic mechanical properties that are inconceivable with respect to natural materials 1,2 . The key property of interest includes formation of bandgap (BG) that is a frequency region where incident wave propagation is prohibited. Although waveguiding 3 , focusing and collimation 4 , negative refraction 5 ,topological properties [6][7][8] and underwater acoustic applications [9][10][11] have been explored, the multi-directional vibration control with extremely wide complete BG is also intriguing. Multiple approaches including single material 12,13 and elastic impedance based multi-materials periodic structures by active 14 and passive [15][16][17] control techniques have been proposed to enlarge the BGs. Among those approaches the recently emerging 3D phononic structures with complete 3D BG [18][19][20] , inertial amplification phenomena 12 , actively controlled piezo-patches technique 14,21,22 , elastic metamaterials with dissipative medium characteristics 16 and multi-resonant trampoline metamaterials 15 with trampoline effect 23 Table 1 shows a brief summary for the widest BG reported to date. Although locally resonant multi-core structures induce wider BGs in those reported works, the width of BG largely depends upon the mass of resonator/scatterer and impedance mismatch.
For a wider BG, a larger-sized resonator is required. Further, in the works as shown in Table  1, more focus is devoted for obtaining a wide BG from 1D and 2D periodic lattices/structures 14,28,29 . Recently, 3D periodic structures consisting of multi-core materials are also proposed to maximize the impedance mismatch in order to achieve wider BGs 17,30 . In such approach, multi-material based prototyping and/or adjusting the assembly phase of materials pose a significant challenge. For a single core structure, this milestone constitutes a significant advancement in terms of structure optimization to adjust the filling ratio between material regions and voids. In this regard, the present work proposes two phononic metastructure prototypes that consist of a single core structure and it is capable of inducing extremely wide BG in all three directions of the irreducible Brillouin zone. We adopt the principal of mode

Prototypes and modal comparison
The topology in Fig. 1 Subsequently dynamic mechanical test is performed to determine the material loss factor η that is required for the investigation of effect of material damping on numerically obtained transmission curves, see supplementary information. We obtained the frequency response spectra by FE approach using COMSOL Multiphysics 5.4® and ANSYS 2020 R1®. For Prototype 1 the analytical model and FE results are compared in Table. 2. One can observe an excellent agreement between the two models.  Table. 2. The modal analysis by both FE codes shows a dominant mixed compressional-bending resonant mode that initiates BG where the cylindrical masses and cubes/spheres work as rigid masses while box-like frame assembly exploits the flexural stiffness of frame assembly when subjected to incident elastic waves. For better understanding, a monoatomic mass-spring chain is introduced as shown in Fig. 1 Table. 2. The parameter m incorporates the mass of cylinders and cube for Prototype 1 or sphere for Furthermore, as shown in Fig. 1(b), for a single beam with the effective length 3 2 is second moment of area and el L is beam effective length. The results comparison for Prototype 1 is presented in Table 2

Results and discussion
For Prototype 1, the numerical band structure and BGs determined by COMSOL Multiphysics5.4® is shown in Fig. 2(a-b). The first BG is the widest with the relative bandwidth 160.2%. The vibration modes corresponding to the bounding BG edges are shown in Fig. 2(c). The vibration modes associated to BG opening and closing edges are designated with red and green stars, respectively. The single material 3D structure proposed here possesses the widest 3D BG with the capability of attenuating mechanical vibration and noise in all directions as compared to the reported works listed in Table. 1. Similarly, the band structure with BGs and vibration modes corresponding to the bounding BG edges for Prototype 2 are shown in Fig. 3 (a-c). It is noticed that replacing cube masses with spheres of an equivalent volume results in two wide BGs with c ω ω ∆ 157.6% and 55%, respectively. Interestingly both BGs are very close and they are separated by some narrow passbands. The material damping/viscoelasticity effects will weaken this passband that eventually results in a broadband BG covering an extremely wide frequency range. mode. Also, due to involvement of complete unit cell structure, the effective mass increases and eigenmode is situated at relatively lower frequency region. The analytical model also validates this finding as shown in Fig. 1(b).
In contrast, for the closing bounding edge of the first widest BG as shown in Fig. 2(c) and

Frequency response spectrum
The band structure presented above is obtained from COMSOL Multiphysics® where the Floquet-Bloch periodicity condition is applied on all edges of cylindrical mass that made the structure infinitely periodic in the x-y-z directions. Some reported studies 14,15,20,27,29 indicate one possible way to visualize the vibration mitigation capability from the proposed metastructures is to build a finite array of unit cells and to perform a frequency response study.
In this regard, a 3x3x1 supercell structure is constructed. As shown in Fig. 4  The transmission obtained by FE numerical simulation presented in Fig. 4  The numerical transmission response is first compared by FEA simulations without considering material damping on the response spectrum. The dynamic mechanical analysis (DMA) test is performed on VeroWhite specimen to determine the material loss factor η 15 and this parameter is incorporated in the FEA codes to investigate the effect of material losses on the BGs. Further details about DMA testing is given in the supplementary materials. Thanks to additive manufacturing, by using 3D printer (Strata sys Ltd) both prototypes are constructed and a low amplitude vibration test is conducted to investigate the real time vibration attenuation characteristics. In Fig. 6(a), the experiment setup and details are presented 25 while Fig. 6(b-c) compare the numerical and experiment transmission spectra for both prototypes. An excellent agreement between the results is obtained. One can observe identical attenuation bandwidths however due to accelerometers limited precision compare to both FEA codes, there is a discrepancy between numerical and experiment attenuation depth. Other possible reasons for minor discrepancy between numerical and experimental results are (1) manufacturing issues 32,33 and anisotropic feature of the samples (2)   Excellent agreement between numerical and experiment result is obtained.

Experiment setup
The experiment setup is illustrated in Fig. 6(a) 25  Throughout the study, an excellent agreement between theoretical, numerical and experimental results is reported.

Conclusion
This study proposes two optimized phononic metastructures prototypes that govern extremely wide bandgap for vibration and noise filtration and attenuation. The study is where ultrawide bandgap and wave attenuation is desirable.