Three-dimensional resonating metamaterials for low-frequency vibration attenuation

Abstract

Recent advances in additive manufacturing have enabled fabrication of phononic crystals and metamaterials which exhibit spectral gaps, or stopbands, in which the propagation of elastic waves is prohibited by Bragg scattering or local resonance effects. Due to the high level of design freedom available to additive manufacturing, the propagation properties of the elastic waves in metamaterials are tunable through design of the periodic cell. In this paper, we outline a new design approach for metamaterials incorporating internal resonators, and provide numerical and experimental evidence that the stopband exists over the irreducible Brillouin zone of the unit cell of the metamaterial (i.e. is a three-dimensional stopband). The targeted stopband covers a much lower frequency range than what can be realised through Bragg scattering alone. Metamaterials have the ability to provide (a) lower frequency stopbands than Bragg-type phononic crystals within the same design volume, and/or (b) comparable stopband frequencies with reduced unit cell dimensions. We also demonstrate that the stopband frequency range of the metamaterial can be tuned through modification of the metamaterial design. Applications for such metamaterials include aerospace and transport components, as well as precision engineering components such as vibration-suppressing platforms, supports for rotary components, machine tool mounts and metrology frames.

Full text

1
SCIENTIFIC REPORTS | (2019) 9:11503 | https://doi.org/10.1038/s41598-019-47644-0
www.nature.com/scientificreports
Three-dimensional resonating
metamaterials for low-frequency
vibration attenuation
W. Elmadih1, D. Chronopoulos2, W. P. Syam1, I. Maskery3, H. Meng2 & R. K. Leach
1
Recent advances in additive manufacturing have enabled fabrication of phononic crystals and
metamaterials which exhibit spectral gaps, or stopbands, in which the propagation of elastic waves
is prohibited by Bragg scattering or local resonance eects. Due to the high level of design freedom
available to additive manufacturing, the propagation properties of the elastic waves in metamaterials
are tunable through design of the periodic cell. In this paper, we outline a new design approach for
metamaterials incorporating internal resonators, and provide numerical and experimental evidence
that the stopband exists over the irreducible Brillouin zone of the unit cell of the metamaterial (i.e. is a
three-dimensional stopband). The targeted stopband covers a much lower frequency range than what
can be realised through Bragg scattering alone. Metamaterials have the ability to provide (a) lower
frequency stopbands than Bragg-type phononic crystals within the same design volume, and/or (b)
comparable stopband frequencies with reduced unit cell dimensions. We also demonstrate that the
stopband frequency range of the metamaterial can be tuned through modication of the metamaterial
design. Applications for such metamaterials include aerospace and transport components, as well
as precision engineering components such as vibration-suppressing platforms, supports for rotary
components, machine tool mounts and metrology frames.
Phononic crystals (PCs) are engineered materials designed to control elastic wave propagation. PCs generally rely
on high impedance mismatches within their structural periodicity to form Bragg-type stopbands that exist due
to the destructive interference between transmitted and reected waves. e presence of destructive interference
prevents specic wave types from propagating. Kushwaha et al.1 presented the rst comprehensive calculation of
acoustic bands in a structure of periodic solids embedded in an elastic background. James et al.2 used a periodic
array of polymer plates submerged in water and provided experimental realisation of one-dimensional (1D)
and two-dimensional (2D) PCs. Montero de Espinosa et al.3 used aluminium alloy plates with cylindrical holes
lled with mercury to generate 2D ultrasonic stopbands. Tanaka et al.4 studied the homogeneity of PCs in the
perpendicular direction to the direction of propogation, and classied PCs into bulk PCs and slab PCs. Research
on the design, manufacturing and testing of PCs has mainly focused on 1D and 2D PCs5–15, although recently,
the research has been extended to include 3D PCs16–24. Lucklum et al.25 discussed the manufacturing challenges
of 3D PCs and showed that additive manufacturing (AM) has the fabrication capabilities required for the real-
isation of geometrically complex 3D PCs26–29. ere are a wide variety of AM technologies that may be used to
manufacture PC materials, such as laser powder bed fusion (LPBF), photo-polymerization, stereolithography
and inkjet printing30–33. Although diering in the manufacturing resolution (the thickness of the build layer),
materials, design constraints and cost, these AM technologies create 3D parts from a CAD model. e creation of
the 3D parts is usually carried out layer by layer, and the thickness of the deposited layers, as well as the eects of
post-processing, determine the geometrical quality of the created 3D parts34,35.
Despite the benets of the recent ability to manufacture PCs with AM, their eectiveness at low-frequencies
is limited due to the dependency of the resulting stopbands on Bragg scattering. Bragg scattering occurs due
to destructive interference of the propagating waves with the in-phase reected waves, which occurs when the
wavelengths of the reected and propagating waves are similar. e reection occurs due to the dierence in the
1Manufacturing Metrology Team, Faculty of Engineering, University of Nottingham, Nottingham, NG8 1BB,
UK. 2Institute for Aerospace Technology & Composites Group, Faculty of Engineering, University of Nottingham,
Nottingham, NG8 1BB, UK. 3Centre for Additive Manufacturing, Faculty of Engineering, University of Nottingham,
Nottingham, NG8 1BB, UK. Correspondence and requests for materials should be addressed to W.E. (email: Waiel.
Elmadih@Nottingham.ac.uk)
Received: 7 April 2019
Accepted: 15 July 2019
Published: xx xx xxxx
OPEN

2
SCIENTIFIC REPORTS | (2019) 9:11503 | https://doi.org/10.1038/s41598-019-47644-0
www.nature.com/scientificreports
www.nature.com/scientificreports/
impedance (e.g. local density) of the PC. For the in-phase reection to occur, the Bragg law has to be satised36,
which is highly dependent on the cell size of the PC. Bragg scattering starts to occur when the wavelength is
approximately equal to twice the cell size of the PC36; around a normalised frequency (the quotient of cell size and
wavelength) of 0.5. us, there is a limiting dependency on the size of the unit cell of the PCs to form stopbands
by Bragg scattering. As a result of this dependency, unrealistic cell sizes need to be employed to satisfy the Bragg
law at low-frequencies.
It is possible to form stopbands below the lowest Bragg limit using metamaterials with periodically arranged
local resonators. e stopbands in these metamaterials are formed by absorbing wave energy around the resonant
frequency37–44. e benets of resonator-based metamaterials include increased design freedom and the exibility
to obtain stopbands in structures of higher periodicity within a xed design volume compared to conventional
PCs. us, resonator-based metamaterials provide better-dened stopbands. Research on locally resonant meta-
materials includes the work of Liu et al.44, who rst developed a metamaterial using solid cores and silicone rub-
ber coatings. e periodically coated spheres of Liu et al. exhibited negative dynamic mass, as well as stopbands
at low frequencies. Numerous locally resonant metamaterials have been proposed. An example by Fang et al.45
showed arrays of Helmholtz resonators with negative dynamic bulk modulus. Qureshi et al.46 numerically inves-
tigated the existence of stopbands in cantilever-in-mass metamaterials. Lucklum et al.21 and D’Alessandro et al.47
independently veried the existence of stopbands in ball-rod metamaterials. Zhang et al.48 presented results of
a beam metamaterial with local resonance stopbands. Bilal et al.49 reported on the concept of combining local
resonance with Bragg scattering to form trampoline metamaterial with subwavelength stopbands. Matlack et al.50
developed a multimaterial structure that has wide stopbands using similar concept to that of Bilal et al.49. Most
of the above work, regarding both PCs and metamaterials, has employed analytical techniques to model and
optimise the suggested unit cells. Because analytical techniques can only model simple designs, the potential for
exploring the elastic capabilities of complex metamaterial designs has been limited.
We hereby report on 3D metamaterial comprising internal resonators, designed for targeting maximum elastic
wave attenuation below a normalised frequency of 0.1. is normalised frequency limit, chosen arbitrarily, is four
times lower than the lowest theoretical limit allowed for Bragg scattering stopbands. Due to its high normalised
stopband frequencies, a PC relies heavily on increasing the cell size to reduce the absolute stopband frequency.
e low normalised stopband frequencies of metamaterials allow for vibration attenuation at low absolute fre-
quencies using much more practical unit cell sizes (i.e. of more suitable dimensions for AM and applications). A
novel approach for tuning and designing the unit cell of the metamaterial is presented. e computation scheme
of the wave dispersion curves uses nite element (FE) modelling. In comparison to nite dierence time domain
(FDTD) modelling which suers from stair-casing eects51, and plain wave expansion (PWE) modelling which is
limited to structures of low impedance mismatch52, FE modelling guarantees an accurate description of the wave
dynamics within the 3D metamaterial. LPBF is employed for fabrication of the metamaterial, which is experimen-
tally tested for verication of the numerical predictions. e fundamental unit cell of the metamaterial is shown
in Figure1, and is periodically tessellated in 3D to allow a local resonance eect. e 3D wave propagation and
the complete stopbands of the metamaterial are presented in Figure2. e experimental response of the manufac-
tured metamaterial is shown in Figure3. Details of the computation, manufacturing and experimental methods
are provided in the subsequent sections.
Results and Discussion
e unit cell of the metamaterial featured in this work is shown in Figure1. e design is a cubic unit cell with
face-centered struts (FCC), and reinforcement struts in the x-, y- and z- directions (FCCxyz). FCC lattices gen-
erally have good compressive strength53, in comparison to body-centred cubic lattices (BCC). us, the FCCxyz
lattice is used as the host for the internal resonance mechanism of the metamaterial. e internal resonance
mechanism consists of six struts; each connects one side of a cubic mass to the inner walls of the FCCxyz unit cell.
Increasing the strut diameter Sd would increase the stiness of the resonator, while increasing the strut length Sl
would alter its volume fraction, which will have an impact on the stopband frequencies and the total mass.
Figure 1. e design of the resonating metamaterial: (a) Schema of the single unit cell of the metamaterial as
modelled in CAD, the labels show the strut diameter (Sd), strut length (Sl), and cell size (C), and photograph of
the 3 × 3 × 3 metamaterial as (b) digitally rendered, and (c) manufactured with LPBF.

3
SCIENTIFIC REPORTS | (2019) 9:11503 | https://doi.org/10.1038/s41598-019-47644-0
www.nature.com/scientificreports
www.nature.com/scientificreports/
Figure 2. Wave propagation properties of the internally resonating metamaterial: (a) Dispersion curves for
the metamaterial with Sd/C and Sl/C values of 0.033 and 0.1, respectively, with eigenmodes at selection of high
symmetry points, and (b) start and end frequencies of the complete stopbands of metamaterials of dierent
Sd/C values with the struts connected to resonators of large-size (green), mid-size (blue), and small-size
(orange). e indicated percentages show the relative gap to mid-gap percentage. All frequencies (f) are
normalised to the longitudinal wave speed in the medium v and the unit cell size C.
Figure 3. Experimental results acquired for the resonating metamaterial: (a) Transmissibility of the 3 × 3 × 3
metamaterial in the x- longitudinal direction (solid line), y- transverse direction (dotted line), and z- transverse
direction (dashed line) vis-à-vis the corresponding stopband as illustrated by the dispersion curves of the
innite metamaterial shown in (b), and (c) representative photograph of the experimental setup. e shaded
areas show the identied stopbands.

4
SCIENTIFIC REPORTS | (2019) 9:11503 | https://doi.org/10.1038/s41598-019-47644-0
www.nature.com/scientificreports
www.nature.com/scientificreports/
Modelling of the elastic wave propagation in the metamaterials was carried out in 3D using the scheme
described in the Methods Section. e modelling used sucient tetrahedral elements, such that the frequency
of the rst vibration mode converged with the FE mesh density (approximately 6000 nodes per unit cell). e
elements of the converged mesh used three degrees of freedom (DOF) per node with adaptive mesh size to suf-
ciently model narrow regions in the metamaterials54. To mathematically model the elastic wave propagation,
the contours of the irreducible Brillouin zone (IBZ) of the unit cells of the metamaterials were scanned. Several
characteristic points exist within the contours of the IBZ including Γ(0,0,0), X(π/C,0,0), M(π/C,π/C,0), and
R(π/C,π/C,π/C), where C is the unit cell size (also referred to as α or L in other literature50,55,56). e scan of the
IBZ was carried out using a total of 360 combinations of wavenumbers (90 combinations for each wave propaga-
tion direction). e corresponding dispersion properties along the path Γ–X–R–M–Γ of the IBZ were predicted
and the complete stopbands were identied. e dispersion curves of a metamaterial unit cell with Sd/C and Sl/C
values of 0.033 and 0.1, respectively, are presented in Figure2a. It was observed that the metamaterial exhibits a
stopband below a normalised frequency of 0.1. e stopband spans a normalised frequency range of 0.028, start-
ing from 0.039 to 0.067, and is formed by an internal resonance that cuts the rst three acoustic wavebands (wave-
bands cutting-on at zero frequency) and splits them into two branches (i.e. top and bottom acoustic branches).
e dispersion curves of multiple metamaterials of dierent values of Sd/C and Sl/C were predicted. e con-
sidered Sd/C values were 0.005, 0.01, 0.02, 0.025 and 0.033, and the considered Sl/C values were 0.05 (large-size
resonator), 0.1 (mid-size resonator) and 0.2 (small-size resonator). Figure2b presents the stopbands for each of
the considered metamaterials to show the impact of the design of the internal resonators on forming complete
3D stopbands. e relative gap to mid-gap percentages of selection of the presented stopbands (width of the stop-
band divided by its central frequency) are highlighted in Figure2b. e large-size resonator showed the largest
relative gap to mid-gap percentage of 68%. e cut-on frequency of the top acoustic branches (i.e. the stopband
end frequency) increased with the increase in the diameter of the struts, and with the increase in the size of the
resonator. e stopbands of all the considered unit cell designs were below a normalised frequency of 0.1, as can
be seen in Figure2b. e stopbands of the large-size resonator had wider stopbands than that of the mid-size res-
onator. e average stopband width in the large-size resonator was calculated to be wider by 63%, and 236% than
that of mid-size and small-size resonators, respectively. e mean frequency of the stopband showed a change of
2.4% with the change in the resonator size. e results shown in Figure2b can be used as a means of tuning the
stopbands of the metamaterial for a specic application.
For verication of the complete stopband in the proposed metamaterial, LPBF was used to manufacture a 3D
structure of nite periodicity. Details about the LPBF process can be found in the Methods Section. e geomet-
rical dimensions and periodicity of the metamaterial were selected to be suitable for the LPBF process. e manu-
factured metamaterial, presented in Figure1c, had a unit cell size of 30 mm and a 3D periodicity of three. e Sd/C
and Sl/C values were selected to provide the lowest stopband start frequency, when referenced to the stopband
start frequencies presented in Figure2b while considering the lowest manufacturable feature size with LPBF57
(See Methods Section); this meant that the Sd/C and Sl/C values had to be 0.033 and 0.1, respectively. e 3D
transmissibility of the metamaterial was obtained experimentally and is presented in Figure3a. e longitudinal
transmissibility had a value of 0 dB near the normalised frequency of zero, which indicates complete transmission
of the excitation waves. At the vibration resonances, the longitudinal transmissibility was greater than 0 dB and
reached 28 dB, which indicates high amplication of the excitation waves. Within the stopband, the longitudinal
transmissibility reached −77 dB. e eect of lattice periodicity on the transmissibility within the stopband can
be seen elsewhere12,58. For this investigation, considering the manufacturable feature size of LPBF (See Methods
Section), we have chosen 3 × 3 × 3 as a reasonable example. e results showed that the metamaterial in this work
has double the transmissibility reduction experimentally reported by Croënne et al.12 for their 3D PC which had
double the spatial periodicity used in this work.
e 3D elastic wave propagation in the internally resonating metamaterials was modelled using a hybrid
scheme. e scheme uses the FE method combined with innite periodicity assumptions. It was shown that the
metamaterials exhibit complete stopbands far below the lowest frequency limit of Bragg-type stopbands, which
exist in traditional PCs. A metamaterial of nite periodicity was manufactured using LPBF. An experimental
setup was assembled, comprising a broadband vibration shaker, a laser vibrometer, and dedicated signal gen-
eration and acquisition units. e experimental setup was used to test the 3D vibration transmissibility of the
manufactured metamaterial. It was shown that the metamaterial could attenuate the vibration waves within the
stopband range. e experimental results showed that, within the stopband, the longitudinal transmissibility
of vibration waves in the metamaterial reached −77 dB. Tuning of the stopband can be achieved by adjusting
the size of the resonator and the diameter of the struts to suit the requirements of various applications. For this
particular metamaterial, the stopband was from 1.63 kHz to 2.8 kHz with a unit cell size of 30 mm. Unit cells of
suitable dimensions for AM and applications, and higher periodicity within a certain design volume, in compar-
ison to PCs, can be employed to obtain low absolute frequency stopbands; resulting in higher attenuation within
the stopbands.
Methods
Modelling of elastic wave propagation using a hybrid wave and nite element scheme. e
proposed scheme for computing the dispersion curves used a combination of FE modelling and periodic struc-
ture theory. e metamaterials were modelled using FE modelling which allows for accurate representation of the
geometrically complex metamaterials. e complete mass and stiness matrices of the designs, K and, M respec-
tively, were extracted. e Bloch theorem59, which governs the periodic displacement and forcing conditions was
employed. e periodic structure theory assumed an innite 3D spatial periodicity of the unit cell60,61. Figure4 is
a schema of the segmentation of the unit cell of the metamaterial into sets of DOF, which were used for modelling
the periodicity of the unit cell.

5
SCIENTIFIC REPORTS | (2019) 9:11503 | https://doi.org/10.1038/s41598-019-47644-0
www.nature.com/scientificreports
www.nature.com/scientificreports/
e nodal displacement matrices q were arranged in the following sequence to allow for the 3D spatial peri-
odicity of the unit cell
=
qqqqqqqqq qqqqqqqqqqq[],(1)
IN FSBTLRFB FT SB ST FL FR SL SR BL BR TL TR T
where the subscripts IN, L, R, T, B, F, and S indicate the DOF of the nodes existing at the inside, le, right, top,
bottom, front, and back of the unit cell as illustrated in Figure4. A transformation matrix R was considered to
project the nodal displacement matrices as follows
=

qRq,(2)
where

=






































































=




























−
−
−
−
−
−−
−
−
−−
−
−
−−
R
I
I
II
II
II
I
I
II
I
I
II
I
I
I
q
q
q
q
q
q
q
q
e
e
e
e
e
ee
e
e
ee
e
e
ee
000 000
0000 00
0000 00
00 00 00
00 00 00
00 0000
00 0000
0000 00
0000 00
0000 00
0000 00
0000 00
0000 00
0000 00
0000 00
0000 00
0000 00
0000 00
0000 00
,and ,
(3)
ik
ik
ik
ik
ik
ik ik
ik
ik
ik ik
ik
ik
ik ik
IN
F
B
L
FB
FL
BL
y
z
x
z
y
yz
x
y
xy
x
z
xz
where k is the wavenumber for the waves propagating in x-, y- and z- directions within the considered regions of
the IBZ. Subsequently, the projected stiness and mass matrices of the reduced sets of DOF,
K
and
M
, were com-
puted as
′′==KRKR MRMR,,and (4)
Assuming no external excitation under Bloch-Floquet59 boundary conditions, the following eigenvalue prob-
lem was derived in the wave domain
ϕω−=
KM()0, (5)
2
where ω is the angular frequency and ϕ is the eigenvector. Eq.5 provided the wave propagation characteristics of
the metamaterials in 3D space. By substituting a set of presumed wavenumbers in a given direction, the derived
eigenvectors ϕ provided the deformation of the unit cell under the passage of each wave type at an angular fre-
quency ω. To obtain normalised frequencies, the frequency eigenvalues of Eq.5 were normalised to the unit cell
size C and the speed of longitudinal waves in the lattice material v, which was calculated as the square root of the
quotient of the elastic modulus and material density. A complete description of each passing wave, including x-,
y- and z-directional wavenumbers and wave shapes, at a certain frequency range is acquired with modulo 2π.
When modelling the dispersion curves of the metamaterial used in this work, suitable 3D translation of all solid
features and voids within the unit cell is obtained when the design is approximated as a simple cube, thus, allow-
ing for the use of the IBZ of simple cubic lattice for modelling the dispersion curves. Such approximation is
known to provide accurate dispersion relations as can be seen elsewhere62–64. e computation did not include
damping, though it should be noted that structural damping can be directly introduced to Eq.4 by including an
imaginary part of the
K
matrix65. Alternatively, if full viscous damping properties are to be considered, then
dedicated eigenvalue problem solvers can be employed59.
Additive manufacturing technology employed. Internally resonating metamaterial samples were fab-
ricated on a laser powder bed fusion (LPBF) system using PA12 polymer material. e material properties for
PA12 can be found in Table1. e LPBF system used a 21 W laser of scan speed and hatch spacing of 2500 mm⋅s−1
and 0.25 mm, respectively. e nominal spot size of the laser was 0.3 mm and the layer thickness was 0.1 mm.
PA12 powder was used to ll the powder bed of dimensions 1320 mm × 1067 mm × 2204 mm at a temperature of
173 °C. Geometrical features of sizes below 0.8 mm are usually manufactured with considerable losses in mechan-
ical properties, due to the existence of unsolidied powder within the manufactured features57. To ensure that all
geometrical features were manufactured in agreement with the specied design, the size of the narrowest meta-
material feature was designed to be 1 mm57.

6
SCIENTIFIC REPORTS | (2019) 9:11503 | https://doi.org/10.1038/s41598-019-47644-0
www.nature.com/scientificreports
www.nature.com/scientificreports/
Experimental measurements on vibration transmissibility. e metamaterial sample was suspended
using piano strings to approximate free-free boundary conditions. e approach taken to suspend the metama-
terial, similar to the approach taken by Zhang et al.48 and Chen et al.66, supports the metamaterial uniformly. An
alternative approach, which can also be used for approximation of free-free boundary conditions, can be found
in the work of D’Alessandro et al.47. e metamaterial was adhesively axed from one side to a connector which
was, in turn, bolted to an acceleration sensor. e acceleration sensor was linked to the armature of the shaker
(Modal Shop Shaker 2060E)67 through a stinger. e stinger is a 1.5 mm rod which connects to the acceleration
sensor, and decouples cross-axis force inputs, thus, minimising errors during measurements68. As part of the
experimental setup, the beam of a laser vibrometer was projected perpendicularly to the opposite surface of the
metamaterial to take longitudinal acceleration measurements. e transverse acceleration measurements were
taken by projecting the beam of the laser vibrometer perpendicularly to the side surfaces of the metamaterial. e
laser vibrometer was set to measure the structural response in the longitudinal and transverse directions from
a normalised frequency of 0 to 0.15. e acceleration data within the tested frequency range were also obtained
through the acceleration sensor. e combination of the measurements of both the laser vibrometer and the
acceleration sensor provided the transmissibility of the specimen. Figure3c is a representative photograph of the
experimental setup. All measurements were taken with a normalised frequency resolution of less than 3.7 × 10−5
and were complexly averaged, considering both the phase and the magnitude of the measurements, over 100
spectral sweeps.
Material property Val u e
Young’s modulus 1.5 × 103 MPa
Density 950 kg⋅m−3
Table 1. Material properties of PA1269.
Figure 4. Selection of the segmentation of the unit cell of the metamaterial into DOF as used for modelling the
periodicity of the unit cell. e magenta points represent the (a) front nodes, (b) le nodes, (c) top nodes, (d)
top-le nodes, (e) top-front nodes, and (f) front-le nodes.

7
SCIENTIFIC REPORTS | (2019) 9:11503 | https://doi.org/10.1038/s41598-019-47644-0
www.nature.com/scientificreports
www.nature.com/scientificreports/
References
1. ushwaha, M. S., Halevi, P., Dobrzynsi, L. & Djafari-ouhani, B. Acoustic band structure of periodic elastic composites. Phys. ev.
Lett. 71, 2022–2025 (1993).
2. James, ., Woodley, S. M., Dyer, C. M. & Humphrey, V. F. Sonic bands, bandgaps, and defect states in layered structures—eory
and experiment. J. Acoust. Soc. Am. 97, 2041–2047 (1995).
3. de Espinosa, F. ., Jiménez, E. & Torres, M. Ultrasonic band gap in a periodic two-dimensional composite. Phys. ev. Lett. 80,
1208–1211 (1998).
4. Miyashita, T. Sonic crystals and sonic wave-guides. Meas. Sci. Technol. 16, 47–63 (2005).
5. Tanaa, Y., Tomoyasu, Y. & Tamura, S. Band structure of acoustic waves in phononic lattices: Two-dimensional composites with
large acoustic mismatch. Phys. ev. B 62, 7387–7392 (2000).
6. Chen, Y., Qian, F., Zuo, L., Scarpa, F. & Wang, L. Broadband and multiband vibration mitigation in lattice metamaterials with
sinusoidally-shaped ligaments. Extrem. Mech. Lett. 17, 24–32 (2017).
7. Bilal, O. . & Hussein, M. I. Ultrawide phononic band gap for combined in-plane and out-of-plane waves. Phys. ev. E 84, 65701
(2011).
8. Oudich, M., Assouar, M. B. & Hou, Z. Propagation of acoustic waves and waveguiding in a two-dimensional locally resonant
phononic crystal plate. Appl. Phys. Lett. 97, 193503 (2010).
9. Vasseur, J. O. et al. Experimental and theoretical evidence for the existence of absolute acoustic aand gaps in two-dimensional solid
phononic crystals. Phys. ev. Lett. 86, 3012–3015 (2001).
10. Pennec, Y. et al. Acoustic channel drop tunneling in a phononic crystal. Appl. Phys. Lett. 87, 261912 (2005).
11. uzzene, M. & Scarpa, F. Directional and band-gap behavior of periodic auxetic lattices. Phys. status solidi 242, 665–680 (2005).
12. Croënne, C., Lee, E. J. S., Hu, H. & Page, J. H. Band gaps in phononic crystals: Generation mechanisms and interaction eects. AIP
Adv. 1,41401 (2011).
13. Nassar, H., Chen, H., Norris, A. N., Haberman, M. . & Huang, G. L. Non-reciprocal wave propagation in modulated elastic
metamaterials. Proc. . Soc. A Math. Phys. Eng. Sci. 473, 20170188 (2017).
14. Phani, A. S. In Dynamics of lattice materials (eds Phani, A. S. & Hussein, M. I.) 53–59 (John Wiley and Sons, 2017).
15. ruisová, A. et al. Ultrasonic bandgaps in 3D-printed periodic ceramic microlattices. Ultrasonics 82, 91–100 (2018).
16. Wormser, M., Warmuth, F. & örner, C. Evolution of full phononic band gaps in periodic cellular structures. Appl. Phys. A 123, 661
(2017).
17. Chen, Y., Yao, H. & Wang, L. Acoustic band gaps of three-dimensional periodic polymer cellular solids with cubic symmetry. J. Appl.
Phys. 114 (2013).
18. Abueidda, D. W., Jasiu, I. & Sobh, N. A. Acoustic band gaps and elastic stiness of PMMA cellular solids based on triply periodic
minimal surfaces. Mater. Des. 145, 20–27 (2018).
19. Bücmann, T. et al. Tailored 3d mechanical metamaterials made by dip-in direct-laser-writing optical lithography. Adv. Mater. 24,
2710–2714 (2012).
20. Bilal, O. ., Ballagi, D. & Daraio, C. Architected lattices for simultaneous broadband attenuation of airborne sound and mechanical
vibrations in all directions. Phys. ev. Appl. 10, 54060 (2018).
21. Luclum, F. & Velleoop, M. J. Bandgap engineering of three-dimensional phononic crystals in a simple cubic lattice. Appl. Phys.
Lett. 113, 201902 (2018).
22. Tanier, S. & Yilmaz, C. Design, analysis and experimental investigation of three-dimensional structures with inertial amplication
induced vibration stop bands. Int. J. Solids Struct. 72, 88–97 (2015).
23. Zhou, X.-Z., Wang, Y.-S. & Zhang, C. Eects of material parameters on elastic band gaps of two-dimensional solid phononic crystals.
J. Appl. Phys. 106, 14903 (2009).
24. Warmuth, F., Wormser, M. & örner, C. Single phase 3D phononic band gap material. Sci. ep.7, 3843 (2017).
25. Luclum, F. & Velleoop, M. J. Design and fabrication challenges for millimeter-scale three-dimensional phononic crystals. Crystals
7, 348 (2017).
26. Zheng, X. et al. Multiscale metallic metamaterials. Nat. Mater. 15, 1100 (2016).
27. Wang, Q. et al. Lightweight mechanical metamaterials with tunable negative thermal expansion. Phys. ev. Lett. 117, 175901 (2016).
28. Li, X. & Gao, H. Mechanical metamaterials: smaller and stronger. Nat. Mater. 15, 373 (2016).
29. ompson, M. . et al. Design for additive manufacturing: Trends, opportunities, considerations, and constraints. CIP Ann. 65,
737–760 (2016).
30. Conner, B. P. et al. Maing sense of 3-D printing: Creating a map of additive manufacturing products and services. Addit . Manuf.
1–4, 64–76 (2014).
31. Vaezi, M., Seitz, H. & Yang, S. A review on 3D micro-additive manufacturing technologies. Int. J. Adv. Manuf. Technol. 67, 1721–1754
(2013).
32. Singh, S., amarishna, S. & Singh, . Material issues in additive manufacturing: A review. J. Manuf. Process. 25, 185–200 (2017).
33. Guo, N. & Leu, M. C. Additive manufacturing: Technology, applications and research needs. Front. Mech. Eng. 8, 215–243 (2013).
34. Islam, M. N., Boswell, B. & Pramani, A. An investigation of dimensional accuracy of parts produced by three-dimensional printing.
In the World Congress on Engineering 2013,522–525 (2013).
35. Lee, P., Chung, H., Lee, S. W., Yoo, J. & o, J. eview: Dimensional accuracy in additive manufacturing processes. In ASME.
International Manufacturing Science and Engineering Conference. 1, V001T04A045 (2014).
36. Maldovan, M. Phonon wave interference and thermal bandgap materials. Nat. Mater. 14, 667 (2015).
37. aghavan, L. & Phani, A. S. Local resonance bandgaps in periodic media: Theory and experiment. J. Acoust. Soc. Am. 134,
1950–1959 (2013).
38. Nouh, M., Aldraihem, O. & Baz, A. Wave propagation in metamaterial plates with periodic local resonances. J. Sound Vib. 341,
53–73 (2015).
39. Wang, P., Casadei, F., ang, S. H. & Bertoldi, . Locally resonant band gaps in periodic beam lattices by tuning connectivity. Phys.
e v. B 91, 20103 (2015).
40. Nouh, M. A., Aldraihem, O. J. & Baz, A. Periodic metamaterial plates with smart tunable local resonators. J. Intell. Mater. Syst. Struct.
27, 1829–1845 (2015).
41. Bacigalupo, A. & Gambarotta, L. Simplied modelling of chiral lattice materials with local resonators. Int. J. Solids Struct. 83,
126–141 (2016).
42. Sharma, B. & Sun, C. T. Local resonance and Bragg bandgaps in sandwich beams containing periodically inserted resonators.
J. Sound Vib. 364, 133–146 (2016).
43. Yilmaz, C., Hulbert, G. M. & iuchi, N. Phononic band gaps induced by inertial amplication in periodic media. Phys. ev. B 76,
54309 (2007).
44. Liu et al. Locally resonant sonic materials. Science 289, 1734–1736 (2000).
45. Fang, N. et al. Ultrasonic metamaterials with negative modulus. Nat. Mater. 5, 452–456 (2006).
46. Qureshi, A., Li, B. & Tan, . T. Numerical investigation of band gaps in 3D printed cantilever-in-mass metamaterials. Sci. ep. 6,
28314 (2016).
47. D’Alessandro, L., Belloni, E., Ardito, ., Corigliano, A. & Braghin, F. Modeling and experimental verication of an ultra-wide
bandgap in 3D phononic crystal. Appl. Phys. Lett. 109, 221907 (2016).

8
SCIENTIFIC REPORTS | (2019) 9:11503 | https://doi.org/10.1038/s41598-019-47644-0
www.nature.com/scientificreports
www.nature.com/scientificreports/
48. Zhang, H., Xiao, Y., Wen, J., Yu, D. & Wen, X. Flexural wave band gaps in metamaterial beams with membrane-type resonators:
eory and experiment. J. Phys. D. Appl. Phys. 48, 435305 (2015).
49. Bilal, O. . & Hussein, M. I. Trampoline metamaterial: Local resonance enhancement by springboards. Appl. Phys. Lett. 103, 111901
(2013).
50. Matlac, . H., Bauhofer, A., rödel, S., Palermo, A. & Daraio, C. Composite 3D-printed meta-structures for low frequency and
broadband vibration absorption. Proc. Natl. Acad. Sci. 113, 8386–8390 (2015).
51. Marwaha, A., Marwaha, S. & Hudiara, I. S. Analysis of Curved Boundaries by FDTD and FE Methods. IETE J. es. 47, 301–310
(2001).
52. Qian, D. & Shi, Z. Using PWE/FE method to calculate the band structures of the semi-innite beam-lie PCs: Periodic in z-direction
and nite in x–y plane. Phys. Lett. A 381, 1516–1524 (2017).
53. Leary, M. et al. Selective laser melting (SLM) of AlSi12Mg lattice structures. Mater. Des. 98, 344–357 (2016).
54. SAS IP Inc. Mesh Generation. (2019). Available at, https://www.sharcnet.ca/Soware/Ansys/17.0/en-us/help/wb_msh/msh_tut_
asf_meshgeneration.html. (Accessed: 10th January 2019).
55. Phani, A. S., Woodhouse, J. & Flec, N. A. Wave propagation in two-dimensional periodic lattices. J. Acoust. Soc. Am. 119,
1995–2005 (2006).
56. Hsu, F. C. et al. Acoustic band gaps in phononic crystal strip waveguides. Appl. Phys. Lett. 96, 3–6 (2010).
57. Tasch, D., Mad, A., Stadlbauer, . & Schagerl, M. icness dependency of mechanical properties of laser-sintered polyamide
lightweight structures. Addit. Man uf. 23, 25–33 (2018).
58. Ampatzidis, T., Leach, . ., Tuc, C. J. & Chronopoulos, D. Band gap behaviour of optimal one-dimensional composite structures
with an additive manufactured stiener. Compos. Part B Eng. 153, 26–35 (2018).
59. Collet, M., Ouisse, M., uzzene, M. & Ichchou, M. N. Floquet–Bloch decomposition for the computation of dispersion of two-
dimensional periodic, damped mechanical systems. Int. J. Solids Struct. 48, 2837–2848 (2011).
60. Mead, D. M. Wave propagation in continuous periodic structures: research contributions from Southampton, 1964–1995. J. Sound
Vib. 190, 495–524 (1996).
61. Cotoni, V., Langley, . S. & Shorter, P. J. A statistical energy analysis subsystem formulation using nite element and periodic
structure theory. J. Sound Vib. 318, 1077–1108 (2008).
62. D’Alessandro, L. et al. Modelling and experimental verication of a single phase three-dimensional lightweight locally resonant
elastic metamaterial with complete low frequency bandgap. In 2017 11th International Congress on Engineered Materials Platforms
for Novel Wave Phenomena (Metamaterials) 70–72 (2017).
63. D’Alessandro, L., Zega, V., Ardito, . & Corigliano, A. 3D auxetic single material periodic structure with ultra-wide tunable bandgap.
Sci. ep. 8, 2262 (2018).
64. Wang, Y.-F. & Wang, Y.-S. Complete bandgap in three-dimensional holey phononic crystals with resonators. J. Vib. Acoust. 135,
41009 (2013).
65. Adhiari, S. Damping modelling using generalized proportional damping. J. Sound Vib. 293, 156–170 (2006).
66. Chen, S.-B., Wen, J.-H., Wang, G., Han, X.-Y. & Wen, X.-S. Locally resonant gaps of phononic beams induced by periodic arrays of
resonant shunts. Chinese Phys. Lett. 28, 94301 (2011).
67. e Modal Shop. 60 lbf Modal Shaer. (2010). Available at, http://www.modalshop.com/lelibrary/60lbf-Modal-Shaer-Datasheet-
(DS-0076).pdf. (Accessed: 19th February 2018).
68. e Modal Shop. Modal Exciter 60 lbf: Model 2060E. (2019). Available at, http://www.modalshop.com/excitation/60-lbf-Modal-
Exciter?ID=250. (Accessed: 10th March 2019).
69. Materialise. P. A. 12 (SLS): Datasheet. (2018). Available at, http://www.materialise.com/en/manufacturing/materials/pa-12-sls.
(Accessed: 31st January 2018).
Acknowledgements
This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/
M008983/1].
Author Contributions
W.E. wrote the main body of the manuscript, performed the experimental tests and the numerical analysis
of the considered design. D.C. and W.S. contributed to the research idea and helped writing the introductory
section of the manuscript and revisited the results section. I.M. prepared the samples to be experimentally tested.
H.M. contributed to writing the introductory section. R.L. contributed to the research idea and supervised the
work conducted by his team members. All authors analysed the results together and provided feedback on the
manuscript.
Additional Information
Competing Interests: e authors declare no competing interests.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and
institutional aliations.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International
License, which permits use, sharing, adaptation, distribution and reproduction in any medium or
format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Cre-
ative Commons license, and indicate if changes were made. e images or other third party material in this
article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the
material. If material is not included in the article’s Creative Commons license and your intended use is not per-
mitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the
copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
© e Author(s) 2019