ametamaterial-inspiredstructureforsimultaneousvibration...

Research ArticleA Metamaterial-Inspired Structure for Simultaneous VibrationAttenuation and Energy Harvesting

Winner Anigbogu1 and Hamzeh Bardaweel 123

1Institute for Micromanufacturing College of Engineering and Science Louisiana Tech University Ruston LA 71272 USA2Department of Mechanical Engineering College of Engineering and Science Louisiana Tech University Ruston LA 71272 USA3Department of Nanosystems Engineering College of Engineering and Science Louisiana Tech University Ruston LA 71272 USA

Correspondence should be addressed to Hamzeh Bardaweel hamzehblatechedu

Received 8 January 2020 Revised 15 May 2020 Accepted 20 May 2020 Published 13 June 2020

Academic Editor C M Wang

Copyright copy 2020 Winner Anigbogu and Hamzeh Bardaweel )is is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

In this article a magnetomechanical metamaterial structure capable of simultaneous vibration attenuation and energy harvestingis presented )e structure consists of periodically arranged local resonators combining cantilever beams and permanent magnet-coil systems A prototype of the metamaterial dual-function structure is fabricated and models are developed Results show goodagreement between model simulation and experiment Two frequency bandgaps are measured 205ndash257Hz and 587ndash639HzWithin these bandgaps vibrations are completely attenuated)e level of vibration attenuation in the first bandgap is substantiallylarger than the level of vibration attenuation observed in the second bandgap Mode shapes suggest that bending deformationsexperienced by the local resonators in the second bandgap are less than the deformations experienced in the first bandgap andmost vibrational energy is localized within the first bandgap where the fundamental resonant frequency is located ie 224Hz)eability of the fabricated metamaterial structure to harvest electric power in these bandgaps is examined Results show thatvibration attenuation and energy harvesting characteristics of the metamaterial structure are coupled Stronger vibration at-tenuation within the first bandgap has led to enhanced energy harvesting capabilities within this bandgap Power measurements atoptimum load resistance of 15Ω reveal that maximum power generated within the first bandgap reaches 52 microW at 245HzCompared with state-of-the-art the metamaterial structure presented here shows a significant improvement in electric powergeneration at considerably lower load resistance while maintaining the ability to attenuate undesired vibrations within thefrequency bandgap

1 Introduction

Elastic mechanical metamaterials have shown extraordinaryproperties including bandgap phenomenon [1ndash3] Conse-quently elastic metamaterials are gaining unprecedentedattention in advanced engineering applications such as vi-bration attenuation and wave filtering [4ndash6] and energyharvesting [7 8] Typically the overall mechanical meta-material structure is man-made and consists of periodicallyarranged cavities housing locally resonating masses that areconnected to the cavities by elastic elements ie springs [9])e bandgap phenomenon is the result of the dynamicbehavior of the periodically arranged local resonators andBraggrsquos scattering in periodic structures )e overall

dynamics of the metamaterial structure is determined by thelocally resonating masses When the mechanical meta-material structure is subject to excitations from externalsource the kinetic energy contained in these oscillations istrapped in the local resonators )is results in the uniquecapability of generating frequency bandgaps within whichoscillations are prevented from passing through )e gen-erated bandgaps are formed at frequency matching thefundamental resonant frequency of the local resonators thatare acting as local absorbers within the metamaterialstructure [2 10]

To this end research interests in mechanical meta-materials have grown in a few directions including vi-bration attenuation and wave filtering vibration energy

HindawiShock and VibrationVolume 2020 Article ID 4063025 12 pageshttpsdoiorg10115520204063025

harvesting and simultaneous vibration attenuation andenergy harvesting For instance Matlack et al presented anelastic metastructure for broadband vibration absorption[5] )e structure was additively manufactured using fuseddeposition-modeling technique and then assembledmanually in order to insert metallic cubes inside the printedstructure Moreover a metamaterial structure embeddedwith hierarchically organized local resonators was fabri-cated and characterized by Xu et al [11] Both experimentaland numerical methods were used to fully characterize thefabricated structure Results from this study demonstratedthe ability of the fabricated structure to attenuate vibrationswithin two bandgaps in frequency regions of 144ndash188Hzand 480ndash810Hz In a similar fashion Zhu et al built achiral lattice-based metamaterial beam that is embeddedwith local resonators to achieve broadband vibration at-tenuation in the frequency range of approximately210ndash700Hz [12]

)e use of metamaterial structures to generate electricpower via energy harvesting has also been investigatedrecently A thorough review of metamaterials-basedenergy harvesters was presented by Chen et al [7] Forexample a metamaterial energy harvester employingpiezoelectrics was fabricated [13] )e harvester wasshown to produce peak power in the order to 100 microWndash1mW at 55 kHz and 30ndash80 kΩ Concurrent vibration at-tenuation and energy harvesting metamaterials have beenrecently also demonstrated Chen et al built a meta-material beam embedded with periodic cells that consistof membrane-split-ring resonators [14] PVDF piezo-electric patch was attached to the membranes to convertthe kinetic energy captured by the resonators into usefulelectric power )e fabricated structure exhibited si-multaneous vibration attenuation and energy harvestingcapabilities Excellent and moderate vibration attenua-tion characteristics were reported in two bandgaps ie340ndash426 Hz and 460ndash500 Hz respectively )e fabricatedstructure recovered approximately 05 microW across 200 kΩat 348 Hz Power was only measured and reported withinthe first bandgap ie at 348 Hz Also a recent study by Liet al reported a mechanical metamaterial structure forsimultaneous vibration isolation and energy harvesting[15] )e structure consisted of beams with PVDF pie-zoelectric transducers to extract electric energy from theresonating cells )e metamaterial structure was able toattenuate vibrations and generate electric power si-multaneously within 146ndash171 Hz frequency bandgapSpecifically a peak power of 005 microW was measuredacross 1MΩ within the frequency bandgap

)e current state-of-the-art literature reveals that agrowing number of researchers have recently attemptedto use elastic mechanical metamaterials for simultaneousvibration attenuation and energy harvesting )e work inthis area is still new and rapidly growing [16] )ereforethe work we present in this article is focused on inves-tigating a magnetomechanical-based metamaterialstructure for simultaneous vibration attenuation andenergy harvesting )e presented structure consists of acombination of cantilever beams and permanent

magnet-coil systems in order to achieve two mainfunctions simultaneously )ese functions are vibrationattenuation and energy harvesting Compared with otherefforts and available literature the metamaterial struc-ture we present in this work is shown to generate sig-nificantly more electric power while maintaining itsability to attenuate undesired vibrations Additionallythe presented structure requires lower load resistance toachieve optimum power transfer compared with whathas been presented in the literature [13ndash15] )is is animportant feature of the presented structure becausetypically electronic circuits for sensors require an inputcurrent in the order of 10 mA [17] As the load resistanceincreases the deliverable electric current decreases It isdesirable to achieve maximum power transfer at thelowest load resistance values [17] )e rest of the article isorganized as follows Section 2 presents the aspects of theproposed design and fabrication methodology of themetamaterial structure Section 3 deals with modeling ofthe metamaterial structure Section 4 is focused oncharacterization tests and experimental methodsFindings and results are presented and discussed inSection 5

2 Design and Fabrication

Figure 1 displays the design and concept of the meta-material vibration attenuation energy harvesting systempresented in this work )e metamaterial structure shownin Figure 1(a) consists of local resonators (unit cells) Eachunit cell shown in Figure 1(b) contains four angledcantilevers with fixed-free ends A permanent magnet isplaced in the center of each unit cell)e tip mass at the freeend of each cantilever is made of copper coils wrappedaround the free end of the cantilever as shown inFigure 1(c) )e coils serve two purposes first the coilsform additional mass to lower the resonant frequency ofthe vibrating cantilevers thus lowering the frequencybandgap of the metamaterial structure second the mag-net-coil system is used to extract electric energy from ki-netic energy of the resonating cantilevers

When the metamaterial structure is subject to externalvibrations the vibrational energy is trapped into the localresonators causing the cantilevers to resonate )at iswhen the driving frequency of vibration matches the res-onant frequency of the local resonators the vibrationenergy is transferred into kinetic energy using the reso-nating cantilevers and therefore this energy is localizedinside the resonators )is results in frequency bandgapsConsequently these undesired vibrations are blocked fromtraveling across the ends of the metamaterial structure)us the first function of the proposed metamaterial vi-bration attenuation energy harvesting system is achievedConcurrently by integrating the coils and permanentmagnets into the design of the local resonators as shown inFigure 1(c) the kinetic energy of the resonating cantileversis converted into useful electric power Subsequentlyvoltage is induced in the coils wrapped around the free end

2 Shock and Vibration

of the cantilevers )erefore the second function of theproposed system is achieved by converting the kineticenergy contained in these vibrations into useful electricpower

3 Theory

)eoretical models of the proposed metamaterial vibrationattenuation energy harvesting structure are developed nextFirst COMSOL FEM software is used to investigate fre-quency bandgaps and modes of vibrations Second a the-oretical model is developed using first principle and used toinvestigate wave attenuation capabilities of the presentedmetamaterial structure

31 COMSOL FE Model Figure 2 shows the geometriesand dimensions of the cantilever ie the building block ofthe metamaterial structure presented in this work First a3D CAD drawing of the unit cell was prepared usingSolidWorks software and the predetermined geometriesand dimensions are shown in Figure 2 )e CAD drawingwas then imported into COMSOL finite-element multi-physics software to analyze the dynamic response of themetamaterial structure and obtain the dispersion diagramthat depicts the relationship between the wave numberK and frequencies for an infinite number of periodicallyarranged unit cells )at is for an array of unit cells atfrequency intervals that meet Braggrsquos scattering conditiondestructive interference occurs )is yields the desiredfrequency bandgap)e presence of the local resonators inthe structure further enhances the frequency bandgap

because at the resonant frequency of the locally reso-nating structures the kinetic energy contained in theexternal vibrations is transferred to the local resonatorscausing them to resonate while creating a lower frequencybandgap in the dispersion diagram of the metamaterialstructure

In the COMSOL model a fine mesh coupled with thechoice of material properties (given in Table 1) and ge-ometries (shown in Figure 2) is used to simulate the dis-persion curve and obtain frequency bandgaps )e modelassumes no damping effects and solves the governingequation of the metamaterial structure given by

Kd Ku minus ω2Mu1113960 1113961

Ku minus ω2Mu1113960 1113961 qL qint qR1113858 1113859T

FL 0 FR1113858 1113859T

(1)

where ω and Kd are frequency and dynamic stiffness matrixof the unit cell respectively Ku and Mu are stiffness andmass matrices of the unit cell respectively Here qL qR andqint are displacement vector of the left side of the unit celldisplacement vector of the right side of the unit cell anddisplacement vector of the internal nodes of the unit cellrespectively Similarly FL and FR are force vectors on the leftand right sides of the unit cell respectively Upon finiteelement discretization the unit cell contained 11647 ele-ments with a maximum size of 181mm and a minimum sizeof 00775mm

To solve the governing equation of the system standardBlochndashFloquet theory is imposed on the displacement qand force F vectors of each node thus taking care of theperiodicity condition of the unit cells given by

(a)

(c)

(b)

Figure 1 Concept and design of the metamaterial vibration attenuation energy harvesting structure presented in this work (a) designlayout of the metamaterial structure (b) building block ie local resonator Each local resonator consists of four angled cantilevers and acentral magnet mass and (c) the tip of each cantilever is made of insertion where copper coils are placed for power extraction

Shock and Vibration 3

qR minuseik luqL

FR minuseik luFL

(2)

where lu 50mm is the length of the unit cell as shown inFigure 2 Next the described eigenvalue problem is solvedfor a given wave number in order to obtain the corre-sponding Eigen frequencies A band structure is obtained bysweeping the wave vector along the irreducible Brillouinzone )is results in dispersion curves defining frequency

bandgaps where oscillations are blocked and prevented frompassing through )e tolerance was set to 1 times 10minus6

32AnalyticalModel )emetamaterial vibration attenuationenergy harvester structure is represented as EulerndashBernoullibeamwith periodic arrangement of local resonators attached tothe beam )is is shown in Figure 3 Each local resonatorconsists of a cantilever with linear stiffness coefficient k andmass m In this work transfer matrix method is used inmodeling the wave propagation across the beam assumingsmall displacement of the beam [18]

)e general equation of motion of the EulerndashBernoullibeam is given as

EIz4y

zx4 + ρAz2y

zt2 0 (3)

where y(x t) is the dynamic deflection of the neutral axis atpoint x E is the elastic modulus of the beam I is the areamoment of inertia ρ is the density of the beam and A is thecross-sectional area In this work it is assumed thaty(x t) Y(x)iωt where Y(x) defines the mode shapes atpoint x described as

Y(x) Q cos(cx) + B sin(cx) + C cosh(cx) + D sinh(cx)

(4)

where c is the flexural wave number given as

c4

ρAω2

EI (5)

For the nth unit cell the mode shape function can berewritten as

Yn xprime( 1113857 Qn cos cxprime( 1113857 + Bn sin cxprime( 1113857 + Cncosh cxprime( 1113857 + Dnsinh cxprime( 1113857

(6)

700mm

5000mm

oslash990mm

306mm 200mm

430mm

147mm

1851mm

Figure 2 Metamaterial cantilever beams and central magnet used in model simulations

Table 1 Properties and dimensions of the metamaterial vibrationattenuation energy harvesting structure used in model simulations

Unit cells of cantilever beams 4 per unit cell of central magnets 1 per unit cell of unit cells 9

CoilsType and grade 35 AWG enameled copper wire Turns 220Length 904m

Central magnetsMaterial NdFeB-N52Diameter Dm 9525mm)ickness Hm 47625mm

CantileverMaterial PETGDensity ρ 1270Kgm3

Modulus of elasticity E 27 GPaPoissonrsquos ratio υ 033Length L 1851mmWidth W 7mmHeight H 1mmStiffness k 745NmMass of cantilever mcan 165 times 10minus4 kgmass m 000127 kgCross-sectional area A 14 times 10minus5 m2

Area moment of inertia I 46667 times 10minus12 m4


where xprime x minus nlu nlu le xle (n + 1)lu Here Qn Bn

Cn andDn are the unknown amplitudes describing themode shape function [19]

33 Analysis of the Local Resonator (LR) )e equation of thenth local resonator in Figure 3 is described as

F xn t( 1113857 + m euroGn(t) 0 (7)

where F(xn t) is the force of the nth resonator developed as aresult of interaction with the beam at the contact point asshown in Figure 3 and Gn is the vertical displacement of themass of the resonator m during the external excitation )eforce F(xn t) is given by

F xn t( 1113857 minusk y xn t( 1113857 minus Gn(t)1113858 1113859 (8)

where the spring stiffness coefficient k of the local resonatorcantilever is given as k EWH34L3 whereW andH are thewidth and height of the cantilever and L is the overall lengthof the cantilever

Assuming the solution for the vertical displacement ofthe mass resonator of the form

Gn(t) Vneiωt

(9)

where Vn is the displacement of the local resonator (LR)and combining equations (7)ndash(9) yields the followingequation

m euroGn k y xn t( 1113857 minus Gn(t)1113858 1113859 (10)

Substituting Gn and y(xn t) into equation (10) gives

minusmω2Vne

iωt k Yn(0) minus Vn1113858 1113859e

iωt (11)

Rearranging equation (11) the vertical displacement Vncan be written as

Vn kYn(0)

k minus mω2( ) (12)

Moreover equation (8) can be rewritten as

F xn t( 1113857 minusk Yn(0) minus Vn1113858 1113859eiωt

(13)

and substituting equation (12) into equation (13) yields

F xn t( 1113857 mω2k

k minus mω2( )Yn(0)e

iωt Fne

iωt (14)

)e model for the four unconnected resonators (r

1 2 3 4) shown in Figure 3 at the interface with the beamcan be derived as [20]

m euroGn r kr y xn t( 1113857 minus Gn r(t)1113960 1113961 r 1 2 3 4 (15)

with the amplitude of displacement of the local resonatorsVn given by

Vnr krYn(0)

kr minus mrω2( 1113857r 1 2 3 4 (16)

Consequently the force acting on the beam due to thesefour resonators can be obtained from equations (13) and (14)and is given by

F xn t( 1113857 1113944 minus kr( Yn(0) minus Vn r1113960 1113961eiωt

1113944mrω2kr

kr minus mrω2( 1113857Yn(0)e

iωt Fne

iωt r 1 2 3 4

(17)

Next the continuity conditions at the point where theresonators meet the beam as shown in Figure 3 are imposedbetween the nth unit cell and the (n minus 1)th cell )ese includedisplacement slope bending moment and shear force andrespectively given by

Ynminus1 lu( 1113857 Yn(0)

Ynminus1prime lu( 1113857 Ynprime(0)

EIYnminus1Prime lu( 1113857 EIYnPrime(0)

EIYprimeprimeprimenminus1 lu( 1113857 EIY

primeprimeprimen(0) minus Fn

(18)

Resonatorcantilever

nth cell(n ndash 1)th cell (n ndash 1)th cell0

Resonatormass

Magnet

luy

x infin infin

Gn(t)

m m m m m m

m m m m m m

Figure 3 Cartoon model schematic of beam with local resonators representing the metamaterial vibration attenuation energy harvestingstructure


Substituting equations (6) and (17) into equation (18)yields

ψn Mminus 1Uψnminus1 (19)

where ψn (Qn Bn Cn Dn )T Here U and M are given by

U

cosρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠ sin

ρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠ cosh

ρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠ sinh

ρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

minusρAω2

EI1113888 1113889

14

sinρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

14

cosρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

14

sinhρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

14

coshρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

minusρAω2

EI1113888 1113889

24

cosρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠ minus

ρAω2

EI1113888 1113889

24

sinρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

24

coshρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

24

sinhρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

34

sinρAω2

EI1113888 1113889

14


ρAω2

EI1113888 1113889

34

cosρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

34

sinhρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

34

coshρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

M

1 0 1 0

0ρAω2

EI1113888 1113889

14

0ρAω2

EI1113888 1113889

14

minusρAω2

EI1113888 1113889

24

0ρAω2

EI1113888 1113889

24

0

minusF minusρAω2

EI1113888 1113889

34

minusFρAω2

EI1113888 1113889

34

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(20)

respectively and

F 1EI

1113944mrω2kr

kr minus mrω2( 11138571113888 1113889 r 1 2 3 4 (21)

From BlochndashFloquetrsquos equation of period structures inthe x direction we have

ψn eizluψnminus1 (22)

Further by letting Mminus1U S and substituting equation(19) into equation (22) yield

Sψnminus1 eizluψnminus1 (23)

Equation (23) can be further rearranged into a standardeigenvalue problem described by

S minus eizluI

11138681113868111386811138681113868

11138681113868111386811138681113868 0 (24)

where I is a [4times 4] identity matrix

4 Experimental Methods

A prototype of the metamaterial vibration attenuation en-ergy harvesting structure was built and characterized

experimentally )e fabrication process involved two mainsteps First the CAD design of the metamaterial structurewas prepared using SolidWorks software and then convertedinto STL (STereoLithography) format in order to perform3D printing of the prototype )e Ultimaker 3 Fused De-position Modeling (3D FDM) printer was used to build themetamaterial structure using PETG (polyethylene tere-phthalate) filament Second 35 AWG enameled copperwires were winded around the free end of the cantilevers andneodymium iron boron (NdFeB) permanent solid magnetswere manually integrated into the 3D printed metamaterialstructure Properties and dimensions of the fully assembledmetamaterial vibration attenuation energy harvestingstructure are given in Table 1 Figure 4 shows the fullyfabricated and assembled metamaterial vibration attenua-tion energy harvester structure prototype

)e experiment apparatus used to characterize the re-sponse of the metamaterial vibration attenuation energyharvester prototype is shown in Figure 5 )e apparatusconsists of a shaker table (VT-500 SENTEK DYNAMICS)power amplifier (LA-800 SENTEK DYNAMICS) vibrationcontroller (S81BndashP02 SENTEK DYNAMICS) accelerom-eters (PCB333B30 model PCB Piezotronics) decade box(Global Specialties RDB-10) and PC Characterization testswere focused on simultaneously measuring vibrationtransmissibility and the amount of electric power recovered


from a unit cell )e electromagnetic shaker table was usedto excite the metamaterial structure harmonically at pre-determined frequencies and acceleration levels An accel-erometer was fixed on top of the shaker table and used tomonitor the input waveform and acceleration level deliveredto the metamaterial structure When performing these ex-periments the lower end of the metamaterial vibrationattenuation energy harvester was secured on top of theshaker table as shown in Figure 5 A fixture and clamps wereused to constrain and fix the top end of the metamaterialstructure (not shown in Figure 5) A second accelerometerwas fixed on the top end of the metamaterial structure andused to measure the vibration transmissibility of the met-amaterial structure Both input and output signals from the

top and bottom accelerometers were measured and storedon the PC to analyze the vibration transmissibility )at isthe vibration transmissibility was measured by tracking thelevel of acceleration transmitted from the metamaterialstructure relative to the input acceleration from the shakertable Concurrently the output voltage from a unit cell wasmeasured across a load resistance and used to estimate theamount of recovered electric power

5 Results and Discussion

First we start by investigating wave propagation through themetamaterial structure Using equation (24) for every se-lected frequency value ω the solutions of the wave vector z

Figure 4 Prototype of the metamaterial vibration attenuation energy harvesting structure presented in this work

Decade box(Global specialties RDB-10)

Accelerometer(PCB333B30 modelPCB piezotronics)

Accelerometer

Shaker table(VT-500 SENTEK DYNAMICS)

Metamaterial

PC

EDMSOFTWARE

Controller(S81B-P02

SENTEK DYNAMICS)

Power amplifier(LA-800

SENTEK DYNAMICS)

SignalPower

Figure 5 Cartoon schematic of the experimental apparatus used for characterization of the fabricated metamaterial vibration attenuationenergy harvesting structure


are obtained Figure 6 shows the results obtained fromsolving the eigenvalue problem given in equation (24) Herea stopband exists when z is imaginary while a pass-bandexists when z is real [19] As shown in Figure 6 the stop-bands of the wave are marked in gray Results reveal thatthere exist two main stopbands in the frequency range1ndash1000Hz )ese bands are occurring in the range of121ndash310Hz (lower frequency range) and 528ndash674Hz(higher frequency range) respectively )ese two bands arethe result of the beam itself and the local resonators (LR)[21] )at is a higher frequency stopband is created as aresult of vibration phase differencesmdashin-phase and out-of-phasemdashbetween adjacent unit cells A low-frequency stop-band is typically generated by the LR stopband due to theattenuation effect of the resonators

Next to fully understand the behavior of the presentedmetamaterial vibration attenuation energy harvestingstructure we further analyze the structure using COMSOLFEM model described in Section 31 and experiment de-scribed in Section 4 )e frequency bandgaps of the meta-material vibration attenuation energy harvesting structureshown in Figure 7 are analyzed using COMSOL FEM Pe-riodic boundary conditions were imposed and geometricand material properties were set similar to those measuredand reported in Table 1 Figure 7 shows the dispersion curveof the metamaterial structure obtained using COMSOLmodel described in Section 3 Figure 7 suggests that thereexists two major bandgaps within the frequency range of100ndash1000Hz )e first and second bandgaps are 218ndash247Hzand 589ndash780Hz respectively

To further explore these bandgaps Figure 8 shows thecorresponding mode shapes obtained using COMSOL at thelower and upper edges of the first and second bandgaps ie218 247 589 and 780Hz respectively Clearly within thesebandgaps vibrations are attenuated and trapped within thelocally resonating cantilevers Also the fundamental reso-nant frequency of the unit cell (combined cantilever beamstip coils and central magnet) was estimated using COMSOLat approximately 224Hz )e fundamental resonant fre-quency falls within the first bandgap of the metamaterialstructure ie 218ndash247Hz As shown in Figure 8 at the lowerand upper edges of the first bandgap the deformations of the

unit cell are localized within the cantilevers )e vibrationalenergy is localized within the frequency bandgap andconverted into kinetic energy of the resonating cantileversAlso Figure 8 suggests that maximum bending deforma-tions occur within the first bandgap)erefore it is expectedthat most of energy harvesting will occur within the firstbandgap )is observation agrees with the vibration energyharvesting measurements that were performed in this workand discussed later in this article Since most mechanicalvibrations in nature occur at frequencies lower than 300Hz[15] in this work energy harvesting is targeted within thisbandgap ie 218ndash247Hz

Next vibration transmissibility of the fabricated meta-material structure was measured using experiment appa-ratus shown in Figure 5 In these experiments a sinusoidalvibration sweep was performed in the range of 50ndash1000Hz atan acceleration level of 05g msminus2 Figure 9 shows themeasured vibration transmissibility of the metamaterialstructure versus frequency Clearly the results from thisexperiment demonstrate two frequency bandgaps (markedin red) Results from COMSOL model simulations andanalytical model (equation (24)) are displayed in Table 2Results show a good agreement between modelsrsquo predictionsand measured data for both bandgaps However somediscrepancy between modelsrsquo predictions and measured datais evident For example the measured bandwidth of the firstbandgap (205ndash257Hz) is wider than the bandwidth of theCOMSOL simulation-based first bandgap A similar ob-servation was made and reported by Chen et al [14] )ismay be attributed to some of the approximations made inthe model simulations including absence of damping [14] orother experimental variations such as insignificant distor-tion in the metamaterial structure during the experimentSimilarly the first bandgap obtained from equation (24) is alittle wider than the bandgap predicted by COMSOL sim-ulations and experiment as shown in Table 2 )is is likelybecause of some of the simplifications and approximationsmade in the model (equation (24)) including absence ofdamping Also the analytical model does not factor in someof the design complexities including the flange attachment atthe contact point with the beam )e model also assumes

Freq

uenc

y (H

z)1000

800

600

400

200

Real wave vector (zluπ)ndash3 ndash2 ndash1 0 1 2 3

Figure 6 Band structure obtained by solving equation (24) toinvestigate the wave attenuation capabilities of the metamaterialstructure First and second frequency bandgaps are 121ndash310Hz(lower frequency range) and 528ndash674Hz (higher frequency range)respectively (marked in gray)

1000800600

400300

200

100

Wave number x (luπ)0 02 04 06 08 1

Freq

uenc

y (H

z)

Figure 7 Dispersion curve and frequency bandgaps of a meta-material structure consisting of cantilever beams with tip coils andcentral magnet obtained using COMSOL simulations Frequencybandgaps the first and second bandgaps are 218ndash247Hz and589ndash780Hz respectively (marked in gray)


small displacements of the transverse beam and ignores themass contribution from the transverse beam Nonethelessboth models predict similar behaviors and show similartrends compared with measured data Furthermore resultsfrom experiment show that the level of vibration attenua-tion represented as the dip in the transmissibility curve inFigure 9 occurring in the first bandgap is substantially largerthan the level of vibration attenuation observed in thesecond bandgap )is behavior is expected since as

discussed earlier and shown in Figure 8 the second bandgapexperiences less deformations compared with the firstbandgap and most vibrational energy is localized within thefirst bandgap where the fundamental resonant frequency islocated ie 224Hz

)e ability of the fabricated metamaterial structure toscavenge electric power while concurrently attenuatingundesired vibrations is examined next Figure 10 showselectric power harvested from one cantilever located in the

First bandgap

Second bandgap

218 Hz 247 Hz

589 Hz 780 Hz

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0

Figure 8 Mode shapes at the edge frequencies of each bandgap obtained using COMSOL model simulations

5

0

ndash5

ndash10

Frequency (Hz)50 100 200 500 1000

Tran

smiss

ibili

ty (d

B)

Figure 9 Vibration transmissibility versus frequency for the metamaterial structure obtained using experiment Measured frequencybandgaps (marked in red) are shown


bottom central unit cell of the metamaterial structure In thisexperiment the metamaterial structure was driven har-monically at 05 gmiddotmsminus2 and fixed frequency (selected withinthe first bandgap ie 223Hz) while the load resistance wasswept in the range of 1ndash1000Ω (Figure 10(a)) )e sameexperiment was repeated at a fixed frequency selected fromthe second bandgap ie 615Hz as shown in Figure 10(b)Results from these experiments confirm that most of thevibrational energy is localized within the first bandgapcausing the cantilevers to locally resonate Consequentlywithin the first bandgap kinetic energy in these resonatingcantilevers is effectively converted into electric powerthrough the coils at the free end of these cantileversMaximum power outputs of approximately 25 microW and06 nW were measured at 223Hz (first bandgap) and 615Hz(second bandgap) respectively )us results demonstratethat both vibration transmissibility (Figure 9) and energyharvesting (Figure 10) characteristics of the fabricatedmetamaterial structure are more prominent in the firstfrequency bandgap Also Figure 10 reveals that the maxi-mum power output occurs at optimum load resistance of15Ω

It is also worth noting that the improvement in energyharvesting within the first bandgap (205ndash257Hz) is ac-companied by stronger vibration attenuation )at is vi-bration attenuation and energy harvesting characteristics ofthe metamaterial structure are coupled Stronger vibrationattenuation leads to enhanced vibration energy harvestingcapabilities )is is further demonstrated in Figure 11 whichshows the output electric power versus frequency In thisexperiment the metamaterial structure was swept har-monically at 05g msminus2 in the frequency range of 50ndash500Hz)e load resistance was fixed at the optimum value ie 15Ωin order to maximize power generation It is clear thatwithin the bandgap (205Hzndash257Hz) the large dip in thetransmissibility curve (Figure 9) corresponds to the maxi-mum power generation (Figure 11) Also Figure 11 reveals

that the maximum power generated within the frequencybandgap can reach up to approximately 52 microW at 245Hz

Next performance metrics of the presented meta-material vibration attenuation energy harvesting structureare compared to the state-of-the-art work reported in therecent literature )is is shown in Table 3 )e fabricatedmetamaterial vibration attenuation energy harvestingstructure performed very well compared with other effortsrecently reported in the literature For example the meta-material structure was able to attenuate undesired vibrationsat the typically targeted frequencies ie less than 300HzConcurrently the presented metamaterial structure pro-duced significantly more electric power compared with whathas been reported in the literature Table 3 also shows an-other advantage of the work presented in this article )eoptimum load resistance to maximize electric power wassignificantly lower compared with what has been reported inthe literature )is low impedance guarantees higher electriccurrent that is desirable to operate electronic circuits forsensors as discussed earlier in this article Although ourinitial prototyping efforts produced a metamaterial structurewith comparable metrics to what has been reported in theliterature it should be noted that optimization of thismetamaterial dual-purpose structure is beyond the scope ofwork reported in this article )e work reported in thisarticle is focused on design proof-of-concept experimentsand characterization tests of the metamaterial structurerather than optimization Several design improvements canbe made to improve the overall performance of the fabri-cated metamaterial dual-function structure )is includesoptimization of geometries and dimensions of the cantileverbeams tilt angle of cantilever beams number of unit cellsand size and location of central magnets etc )is will re-quire developing a detailed and sophisticated coupled modelof the magnetomechanical metamaterial vibration attenu-ation energy harvesting structure and therefore will be thefocus of future work

Table 2 Comparison between frequency bandgaps obtained using the experiment and models

Experiment COMSOL simulations Equation (24)First band (Hz) 205ndash257 218ndash247 121ndash310Second band (Hz) 587ndash639 589ndash780 528ndash674

3

2

1Pow

er (micro

W)

Load resistance (Ω)1 3 7 15 50 100 1000

(a)

Pow

er (micro

W)

10

8

6

4

2


times10ndash3

(b)

Figure 10 Measured electric power from a single cantilever within a unit cell at (a) 223Hz and (b) 615Hz and 05g msminus2


6 Conclusions

)e work presented in this article is focused on the designand characterization of a magnetomechanical-based meta-material structure for simultaneous vibration attenuationand energy harvesting )e structure consists of periodicallyarranged unit cells that are made of a combination ofcantilever beams and permanent magnet-coil systems Eachunit cell contains four angled cantilevers with fixed-freeends A permanent magnet is placed in the center of eachunit cell )e tip mass at the free end of each cantilever ismade of copper coils that serve two purposes lowering thefrequency bandgap of the metamaterial structure andextracting electric energy from kinetic energy of the reso-nating cantilevers as they move near the central permanentmagnet A prototype of the metamaterial dual-functionstructure has been fabricated using additive manufacturingand then assembled manually Models of the structure havebeen developed and validated against the experiment

Results show good agreement between model simula-tions and experiment Two frequency bandgaps ie205ndash257Hz and 587ndash639Hz respectively have been mea-sured Within these two bandgaps vibrations are completelyattenuated Results suggest that the level of vibration at-tenuation in the first bandgap is substantially larger than thelevel of vibration attenuation observed in the secondbandgap Mode shapes obtained using COMSOL FEM

model reveal that the second bandgap is accompanied by lessbending deformations compared with the first bandgap andmost vibrational energy is localized within the first bandgapwhere the fundamental resonant frequency is located ie224Hz Moreover the ability of the fabricated metamaterialstructure to harvest electric power while synchronouslyattenuating undesired vibrations has been inspected Electricpower outputs of approximately 25 microW and 06 nW weremeasured at 223Hz (first bandgap) and 615Hz (secondbandgap) at optimum load resistance of 15Ω )ese resultsconfirm that most of the vibrational energy is localizedwithin the first bandgap causing the cantilevers to locallyresonate and kinetic energy in these resonating cantilevers tobe effectively converted into useful electric power throughthe coils at the free ends of these cantilevers Results alsoshow that vibration attenuation and energy harvestingcharacteristics of the metamaterial structure are coupledStronger vibration attenuation has led to enhanced vibrationenergy harvesting capabilities Frequency sweep measure-ments at optimum load resistance of 15Ω reveal thatmaximum power generated within the first frequencybandgap reaches approximately 52 microW at 245Hz

Compared with the state-of-the-art and reported liter-ature the metamaterial structure we present in this work hasshown an order of magnitude improvement in electricpower generation at substantially lower optimum load re-sistance while maintaining the ability to attenuate undesired

6

5

4

3

2

1

Frequency (Hz)

Pow

er (micro

W)

50 100 200 500

Figure 11 Measured output electric power spectrum from a single cantilever at fixed optimum load resistance of 15Ω and accelerationlevel of 05g msminus2 Measured frequency bandgap is denoted in red

Table 3 Comparison between the presented work and recent advancements in the field

Load resistance(KΩ)

Poweroutput (μW)

First frequencybandgap (Hz) Specifications

Ying Li et al[15] 1000 005 146ndash171 Simultaneous vibration attenuation and energy harvesting using

piezoelectric PVDFJung-San Chenet al [14] 200 05 340ndash426 Simultaneous vibration attenuation and energy harvesting using

piezoelectric PVDFXu et al [11] mdash mdash 144ndash188 Only vibration attenuationZhu et al [12] mdash mdash 210ndash700 Only vibration attenuation

Kyung Ho Sunet al [22] 20 0345 mdash

Only energy harvesting capability using acoustic metamaterial and adouble-clamped piezoelectric bimorph plate Results were reported

at 600Hz

)is work 0015 52 205ndash257 Simultaneous vibration attenuation and energy harvesting usingmagnet-coil system


vibrations in a frequency bandgap of 205ndash257Hz Futurework will focus on optimization of the metamaterialstructure to maximize energy harvesting while maintainingvibration attenuation capabilities

Data Availability

Data reported in this article are available from the corre-sponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interestregarding the publication of this article

Authorsrsquo Contributions

Winner Anigbogu was involved in the data collectionformal analysis software validation investigation visuali-zation and formal analysis Hamzeh Bardaweel was re-sponsible for the conceptualization funding acquisitionmethodology project administration resources softwaresupervision validation visualization and writing review-ing and editing of the original draft

Acknowledgments

)is study was partially supported by the Louisiana Board ofRegents Support Fund (LEQSF(2015ndash18)-LaSPACE)

References

[1] G Hu L Tang and R Das ldquoMetamaterial-inspired piezo-electric system with dual functionalities energy harvestingand vibration suppressionrdquo in Active and Passive SmartStructures and Integrated Systems 2017 International Societyfor Optics and Photonics Bellingham WA USA 2017

[2] Z Wang Q Zhang K Zhang and G Hu ldquoTunable digitalmetamaterial for broadband vibration isolation at low fre-quencyrdquo Advanced Materials vol 28 no 44 pp 9857ndash98612016

[3] O Abdeljaber O Avci S Kiranyaz and D J Inman ldquoOp-timization of linear zigzag insert metastructures for low-frequency vibration attenuation using genetic algorithmsrdquoMechanical Systems and Signal Processing vol 84 pp 625ndash641 2017

[4] O Casablanca G Ventura F Garescı et al ldquoSeismic isolationof buildings using composite foundations based on meta-materialsrdquo Journal of Applied Physics vol 123 no 17p 174903 2018

[5] K H Matlack A Bauhofer S Krodel A Palermo andC Daraio ldquoComposite 3D-printed metastructures for low-frequency and broadband vibration absorptionrdquo Proceedingsof the National Academy of Sciences vol 113 no 30pp 8386ndash8390 2016

[6] K K Reichl and D J Inman ldquoLumped mass model of a 1Dmetastructure for vibration suppression with no additionalmassrdquo Journal of Sound and Vibration vol 403 pp 75ndash892017

[7] Z Chen B Guo Y Yang and C Cheng ldquoMetamaterials-based enhanced energy harvesting a reviewrdquo Physica BCondensed Matter vol 438 pp 1ndash8 2014 Apr 1

[8] M Saadatzi F Mir M N Saadatzi and S BanerjeeldquoModeling and fabrication of a multi-axial piezoelectric en-ergy harvester based on a metamaterial-inspired structurerdquoIEEE Sensors Journal vol 18 no 22 pp 9410ndash9419 2018

[9] M Nouh O Aldraihem and A Baz ldquoVibration character-istics of metamaterial beams with periodic local resonancesrdquoJournal of Vibration and Acoustics vol 136 Article ID 0610122014

[10] T Jiang and Q He ldquoDual-directionally tunable metamaterialfor low-frequency vibration isolationrdquoApplied Physics Lettersvol 110 Article ID 021907 2017

[11] X Xu M V Barnhart X Li Y Chen and G HuangldquoTailoring vibration suppression bands with hierarchicalmetamaterials containing local resonatorsrdquo Journal of Soundand Vibration vol 442 pp 237ndash248 2019

[12] R Zhu X N Liu G K Hu C T Sun and G L Huang ldquoAchiral elastic metamaterial beam for broadband vibrationsuppressionrdquo Journal of Sound and Vibration vol 333 no 10pp 2759ndash2773 2014

[13] M Carrara M R Cacan J Toussaint M J LeamyM Ruzzene and A Erturk ldquoMetamaterial-inspired structuresand concepts for elastoacoustic wave energy harvestingrdquoSmart Materials and Structures vol 22 no 6 Article ID065004 2013

[14] J-S Chen W-J Su Y Cheng W-C Li and C-Y Lin ldquoAmetamaterial structure capable of wave attenuation andconcurrent energy harvestingrdquo Journal of Intelligent MaterialSystems and Structures vol 30 no 20 pp 2973ndash2981 2019

[15] Y Li E Baker T Reissman C Sun andW K Liu ldquoDesign ofmechanical metamaterials for simultaneous vibration isola-tion and energy harvestingrdquo Applied Physics Letters vol 111no 25 p 251903 2017

[16] G Hu L Tang A Banerjee and R Das ldquoMetastructure withpiezoelectric element for simultaneous vibration suppressionand energy harvestingrdquo Journal of Vibration and Acousticsvol 139 no 1 2017

[17] M Gao Y Wang Y Wang and P Wang ldquoExperimentalinvestigation of non-linear multi-stable electromagnetic-in-duction energy harvesting mechanism by magnetic levitationoscillationrdquo Applied Energy vol 220 pp 856ndash875 2018

[18] L Liu and M I Hussein ldquoWave motion in periodic flexuralbeams and characterization of the transition between Braggscattering and local resonancerdquo Journal of Applied Mechanicsvol 79 2012

[19] J Zhou K Wang D Xu and H Ouyang ldquoLocal resonatorwith high-static-low-dynamic stiffness for lowering band gapsof flexural wave in beamsrdquo Journal of Applied Physics vol 121Article ID 044902 2017

[20] Z Wang P Zhang and Y Zhang ldquoLocally resonant bandgaps in flexural vibrations of a Timoshenko beam with pe-riodically attached multioscillatorsrdquo Mathematical Problemsin Engineering vol 2013 Article ID 146975 10 pages 2013

[21] M Y Wang and X Wang ldquoFrequency band structure oflocally resonant periodic flexural beams suspended withforce-moment resonatorsrdquo Journal of Physics D AppliedPhysics vol 46 no 25 p 255502 2013

[22] K H Sun J E Kim J Kim and K Song ldquoSound energyharvesting using a doubly coiled-up acoustic metamaterialcavityrdquo Smart Materials and Structures vol 26 Article ID075011 2017


harvesting and simultaneous vibration attenuation andenergy harvesting For instance Matlack et al presented anelastic metastructure for broadband vibration absorption[5] )e structure was additively manufactured using fuseddeposition-modeling technique and then assembledmanually in order to insert metallic cubes inside the printedstructure Moreover a metamaterial structure embeddedwith hierarchically organized local resonators was fabri-cated and characterized by Xu et al [11] Both experimentaland numerical methods were used to fully characterize thefabricated structure Results from this study demonstratedthe ability of the fabricated structure to attenuate vibrationswithin two bandgaps in frequency regions of 144ndash188Hzand 480ndash810Hz In a similar fashion Zhu et al built achiral lattice-based metamaterial beam that is embeddedwith local resonators to achieve broadband vibration at-tenuation in the frequency range of approximately210ndash700Hz [12]

)e use of metamaterial structures to generate electricpower via energy harvesting has also been investigatedrecently A thorough review of metamaterials-basedenergy harvesters was presented by Chen et al [7] Forexample a metamaterial energy harvester employingpiezoelectrics was fabricated [13] )e harvester wasshown to produce peak power in the order to 100 microWndash1mW at 55 kHz and 30ndash80 kΩ Concurrent vibration at-tenuation and energy harvesting metamaterials have beenrecently also demonstrated Chen et al built a meta-material beam embedded with periodic cells that consistof membrane-split-ring resonators [14] PVDF piezo-electric patch was attached to the membranes to convertthe kinetic energy captured by the resonators into usefulelectric power )e fabricated structure exhibited si-multaneous vibration attenuation and energy harvestingcapabilities Excellent and moderate vibration attenua-tion characteristics were reported in two bandgaps ie340ndash426 Hz and 460ndash500 Hz respectively )e fabricatedstructure recovered approximately 05 microW across 200 kΩat 348 Hz Power was only measured and reported withinthe first bandgap ie at 348 Hz Also a recent study by Liet al reported a mechanical metamaterial structure forsimultaneous vibration isolation and energy harvesting[15] )e structure consisted of beams with PVDF pie-zoelectric transducers to extract electric energy from theresonating cells )e metamaterial structure was able toattenuate vibrations and generate electric power si-multaneously within 146ndash171 Hz frequency bandgapSpecifically a peak power of 005 microW was measuredacross 1MΩ within the frequency bandgap

)e current state-of-the-art literature reveals that agrowing number of researchers have recently attemptedto use elastic mechanical metamaterials for simultaneousvibration attenuation and energy harvesting )e work inthis area is still new and rapidly growing [16] )ereforethe work we present in this article is focused on inves-tigating a magnetomechanical-based metamaterialstructure for simultaneous vibration attenuation andenergy harvesting )e presented structure consists of acombination of cantilever beams and permanent

magnet-coil systems in order to achieve two mainfunctions simultaneously )ese functions are vibrationattenuation and energy harvesting Compared with otherefforts and available literature the metamaterial struc-ture we present in this work is shown to generate sig-nificantly more electric power while maintaining itsability to attenuate undesired vibrations Additionallythe presented structure requires lower load resistance toachieve optimum power transfer compared with whathas been presented in the literature [13ndash15] )is is animportant feature of the presented structure becausetypically electronic circuits for sensors require an inputcurrent in the order of 10 mA [17] As the load resistanceincreases the deliverable electric current decreases It isdesirable to achieve maximum power transfer at thelowest load resistance values [17] )e rest of the article isorganized as follows Section 2 presents the aspects of theproposed design and fabrication methodology of themetamaterial structure Section 3 deals with modeling ofthe metamaterial structure Section 4 is focused oncharacterization tests and experimental methodsFindings and results are presented and discussed inSection 5

2 Design and Fabrication

Figure 1 displays the design and concept of the meta-material vibration attenuation energy harvesting systempresented in this work )e metamaterial structure shownin Figure 1(a) consists of local resonators (unit cells) Eachunit cell shown in Figure 1(b) contains four angledcantilevers with fixed-free ends A permanent magnet isplaced in the center of each unit cell)e tip mass at the freeend of each cantilever is made of copper coils wrappedaround the free end of the cantilever as shown inFigure 1(c) )e coils serve two purposes first the coilsform additional mass to lower the resonant frequency ofthe vibrating cantilevers thus lowering the frequencybandgap of the metamaterial structure second the mag-net-coil system is used to extract electric energy from ki-netic energy of the resonating cantilevers

When the metamaterial structure is subject to externalvibrations the vibrational energy is trapped into the localresonators causing the cantilevers to resonate )at iswhen the driving frequency of vibration matches the res-onant frequency of the local resonators the vibrationenergy is transferred into kinetic energy using the reso-nating cantilevers and therefore this energy is localizedinside the resonators )is results in frequency bandgapsConsequently these undesired vibrations are blocked fromtraveling across the ends of the metamaterial structure)us the first function of the proposed metamaterial vi-bration attenuation energy harvesting system is achievedConcurrently by integrating the coils and permanentmagnets into the design of the local resonators as shown inFigure 1(c) the kinetic energy of the resonating cantileversis converted into useful electric power Subsequentlyvoltage is induced in the coils wrapped around the free end



3 Theory





Kd Ku minus ω2Mu1113960 1113961


FL 0 FR1113858 1113859T

(1)



(a)

(c)

(b)



qR minuseik luqL

FR minuseik luFL

(2)





EIz4y

zx4 + ρAz2y

zt2 0 (3)



(4)


c4

ρAω2

EI (5)



(6)

700mm

5000mm

oslash990mm

306mm 200mm

430mm

147mm

1851mm













F xn t( 1113857 + m euroGn(t) 0 (7)





Gn(t) Vneiωt

(9)




minusmω2Vne

iωt k Yn(0) minus Vn1113858 1113859e

iωt (11)


Vn kYn(0)




(13)


F xn t( 1113857 mω2k


iωt Fne

iωt (14)





Vnr krYn(0)

kr minus mrω2( 1113857r 1 2 3 4 (16)



1113944mrω2kr


iωt Fne

iωt r 1 2 3 4

(17)







(18)

Resonatorcantilever


Resonatormass

Magnet

luy

x infin infin

Gn(t)

m m m m m m

m m m m m m






U

cosρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠ sin

ρAω2

EI1113888 1113889

14


ρAω2

EI1113888 1113889

14


ρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

minusρAω2

EI1113888 1113889

14

sinρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

14

cosρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

14

sinhρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

14

coshρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

minusρAω2

EI1113888 1113889

24

cosρAω2

EI1113888 1113889

14


ρAω2

EI1113888 1113889

24

sinρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

24

coshρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

24

sinhρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

34

sinρAω2

EI1113888 1113889

14


ρAω2

EI1113888 1113889

34

cosρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

34

sinhρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

34

coshρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

M

1 0 1 0

0ρAω2

EI1113888 1113889

14

0ρAω2

EI1113888 1113889

14

minusρAω2

EI1113888 1113889

24

0ρAω2

EI1113888 1113889

24

0

minusF minusρAω2

EI1113888 1113889

34

minusFρAω2

EI1113888 1113889

34

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(20)

respectively and

F 1EI

1113944mrω2kr

kr minus mrω2( 11138571113888 1113889 r 1 2 3 4 (21)






S minus eizluI

11138681113868111386811138681113868

11138681113868111386811138681113868 0 (24)














Accelerometer


Metamaterial

PC

EDMSOFTWARE

Controller(S81B-P02

SENTEK DYNAMICS)


SENTEK DYNAMICS)

SignalPower








Freq

uenc

y (H

z)1000

800

600

400

200



1000800600

400300

200

100


Freq

uenc

y (H

z)






First bandgap

Second bandgap

218 Hz 247 Hz

589 Hz 780 Hz

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0


5

0

ndash5

ndash10

Frequency (Hz)50 100 200 500 1000

Tran

smiss

ibili

ty (d

B)









3

2

1Pow

er (micro

W)


(a)

Pow

er (micro

W)

10

8

6

4

2


times10ndash3

(b)



6 Conclusions





6

5

4

3

2

1

Frequency (Hz)

Pow

er (micro

W)

50 100 200 500




Poweroutput (μW)







at 600Hz




Data Availability






Acknowledgments


References

























3 Theory





Kd Ku minus ω2Mu1113960 1113961


FL 0 FR1113858 1113859T

(1)



(a)

(c)

(b)



qR minuseik luqL

FR minuseik luFL

(2)





EIz4y

zx4 + ρAz2y

zt2 0 (3)



(4)


c4

ρAω2

EI (5)



(6)

700mm

5000mm

oslash990mm

306mm 200mm

430mm

147mm

1851mm













F xn t( 1113857 + m euroGn(t) 0 (7)





Gn(t) Vneiωt

(9)




minusmω2Vne

iωt k Yn(0) minus Vn1113858 1113859e

iωt (11)


Vn kYn(0)




(13)


F xn t( 1113857 mω2k


iωt Fne

iωt (14)





Vnr krYn(0)

kr minus mrω2( 1113857r 1 2 3 4 (16)



1113944mrω2kr


iωt Fne

iωt r 1 2 3 4

(17)







(18)

Resonatorcantilever


Resonatormass

Magnet

luy

x infin infin

Gn(t)

m m m m m m

m m m m m m






U

cosρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠ sin

ρAω2

EI1113888 1113889

14


ρAω2

EI1113888 1113889

14


ρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

minusρAω2

EI1113888 1113889

14

sinρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

14

cosρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

14

sinhρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

14

coshρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

minusρAω2

EI1113888 1113889

24

cosρAω2

EI1113888 1113889

14


ρAω2

EI1113888 1113889

24

sinρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

24

coshρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

24

sinhρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

34

sinρAω2

EI1113888 1113889

14


ρAω2

EI1113888 1113889

34

cosρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

34

sinhρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

34

coshρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

M

1 0 1 0

0ρAω2

EI1113888 1113889

14

0ρAω2

EI1113888 1113889

14

minusρAω2

EI1113888 1113889

24

0ρAω2

EI1113888 1113889

24

0

minusF minusρAω2

EI1113888 1113889

34

minusFρAω2

EI1113888 1113889

34

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(20)

respectively and

F 1EI

1113944mrω2kr

kr minus mrω2( 11138571113888 1113889 r 1 2 3 4 (21)






S minus eizluI

11138681113868111386811138681113868

11138681113868111386811138681113868 0 (24)














Accelerometer


Metamaterial

PC

EDMSOFTWARE

Controller(S81B-P02

SENTEK DYNAMICS)


SENTEK DYNAMICS)

SignalPower








Freq

uenc

y (H

z)1000

800

600

400

200



1000800600

400300

200

100


Freq

uenc

y (H

z)






First bandgap

Second bandgap

218 Hz 247 Hz

589 Hz 780 Hz

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0


5

0

ndash5

ndash10

Frequency (Hz)50 100 200 500 1000

Tran

smiss

ibili

ty (d

B)









3

2

1Pow

er (micro

W)


(a)

Pow

er (micro

W)

10

8

6

4

2


times10ndash3

(b)



6 Conclusions





6

5

4

3

2

1

Frequency (Hz)

Pow

er (micro

W)

50 100 200 500




Poweroutput (μW)







at 600Hz




Data Availability






Acknowledgments


References
























qR minuseik luqL

FR minuseik luFL

(2)





EIz4y

zx4 + ρAz2y

zt2 0 (3)



(4)


c4

ρAω2

EI (5)



(6)

700mm

5000mm

oslash990mm

306mm 200mm

430mm

147mm

1851mm













F xn t( 1113857 + m euroGn(t) 0 (7)





Gn(t) Vneiωt

(9)




minusmω2Vne

iωt k Yn(0) minus Vn1113858 1113859e

iωt (11)


Vn kYn(0)




(13)


F xn t( 1113857 mω2k


iωt Fne

iωt (14)





Vnr krYn(0)

kr minus mrω2( 1113857r 1 2 3 4 (16)



1113944mrω2kr


iωt Fne

iωt r 1 2 3 4

(17)







(18)

Resonatorcantilever


Resonatormass

Magnet

luy

x infin infin

Gn(t)

m m m m m m

m m m m m m






U

cosρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠ sin

ρAω2

EI1113888 1113889

14


ρAω2

EI1113888 1113889

14


ρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

minusρAω2

EI1113888 1113889

14

sinρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

14

cosρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

14

sinhρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

14

coshρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

minusρAω2

EI1113888 1113889

24

cosρAω2

EI1113888 1113889

14


ρAω2

EI1113888 1113889

24

sinρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

24

coshρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

24

sinhρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

34

sinρAω2

EI1113888 1113889

14


ρAω2

EI1113888 1113889

34

cosρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

34

sinhρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

34

coshρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

M

1 0 1 0

0ρAω2

EI1113888 1113889

14

0ρAω2

EI1113888 1113889

14

minusρAω2

EI1113888 1113889

24

0ρAω2

EI1113888 1113889

24

0

minusF minusρAω2

EI1113888 1113889

34

minusFρAω2

EI1113888 1113889

34

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(20)

respectively and

F 1EI

1113944mrω2kr

kr minus mrω2( 11138571113888 1113889 r 1 2 3 4 (21)






S minus eizluI

11138681113868111386811138681113868

11138681113868111386811138681113868 0 (24)














Accelerometer


Metamaterial

PC

EDMSOFTWARE

Controller(S81B-P02

SENTEK DYNAMICS)


SENTEK DYNAMICS)

SignalPower








Freq

uenc

y (H

z)1000

800

600

400

200



1000800600

400300

200

100


Freq

uenc

y (H

z)






First bandgap

Second bandgap

218 Hz 247 Hz

589 Hz 780 Hz

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0


5

0

ndash5

ndash10

Frequency (Hz)50 100 200 500 1000

Tran

smiss

ibili

ty (d

B)









3

2

1Pow

er (micro

W)


(a)

Pow

er (micro

W)

10

8

6

4

2


times10ndash3

(b)



6 Conclusions





6

5

4

3

2

1

Frequency (Hz)

Pow

er (micro

W)

50 100 200 500




Poweroutput (μW)







at 600Hz




Data Availability






Acknowledgments


References



























F xn t( 1113857 + m euroGn(t) 0 (7)





Gn(t) Vneiωt

(9)




minusmω2Vne

iωt k Yn(0) minus Vn1113858 1113859e

iωt (11)


Vn kYn(0)




(13)


F xn t( 1113857 mω2k


iωt Fne

iωt (14)





Vnr krYn(0)

kr minus mrω2( 1113857r 1 2 3 4 (16)



1113944mrω2kr


iωt Fne

iωt r 1 2 3 4

(17)







(18)

Resonatorcantilever


Resonatormass

Magnet

luy

x infin infin

Gn(t)

m m m m m m

m m m m m m






U

cosρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠ sin

ρAω2

EI1113888 1113889

14


ρAω2

EI1113888 1113889

14


ρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

minusρAω2

EI1113888 1113889

14

sinρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

14

cosρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

14

sinhρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

14

coshρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

minusρAω2

EI1113888 1113889

24

cosρAω2

EI1113888 1113889

14


ρAω2

EI1113888 1113889

24

sinρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

24

coshρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

24

sinhρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

34

sinρAω2

EI1113888 1113889

14


ρAω2

EI1113888 1113889

34

cosρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

34

sinhρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

34

coshρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

M

1 0 1 0

0ρAω2

EI1113888 1113889

14

0ρAω2

EI1113888 1113889

14

minusρAω2

EI1113888 1113889

24

0ρAω2

EI1113888 1113889

24

0

minusF minusρAω2

EI1113888 1113889

34

minusFρAω2

EI1113888 1113889

34

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(20)

respectively and

F 1EI

1113944mrω2kr

kr minus mrω2( 11138571113888 1113889 r 1 2 3 4 (21)






S minus eizluI

11138681113868111386811138681113868

11138681113868111386811138681113868 0 (24)














Accelerometer


Metamaterial

PC

EDMSOFTWARE

Controller(S81B-P02

SENTEK DYNAMICS)


SENTEK DYNAMICS)

SignalPower








Freq

uenc

y (H

z)1000

800

600

400

200



1000800600

400300

200

100


Freq

uenc

y (H

z)






First bandgap

Second bandgap

218 Hz 247 Hz

589 Hz 780 Hz

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0


5

0

ndash5

ndash10

Frequency (Hz)50 100 200 500 1000

Tran

smiss

ibili

ty (d

B)









3

2

1Pow

er (micro

W)


(a)

Pow

er (micro

W)

10

8

6

4

2


times10ndash3

(b)



6 Conclusions





6

5

4

3

2

1

Frequency (Hz)

Pow

er (micro

W)

50 100 200 500




Poweroutput (μW)







at 600Hz




Data Availability






Acknowledgments


References



























U

cosρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠ sin

ρAω2

EI1113888 1113889

14


ρAω2

EI1113888 1113889

14


ρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

minusρAω2

EI1113888 1113889

14

sinρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

14

cosρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

14

sinhρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

14

coshρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

minusρAω2

EI1113888 1113889

24

cosρAω2

EI1113888 1113889

14


ρAω2

EI1113888 1113889

24

sinρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

24

coshρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

24

sinhρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

34

sinρAω2

EI1113888 1113889

14


ρAω2

EI1113888 1113889

34

cosρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

34

sinhρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

ρAω2

EI1113888 1113889

34

coshρAω2

EI1113888 1113889

14

lu⎛⎝ ⎞⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

M

1 0 1 0

0ρAω2

EI1113888 1113889

14

0ρAω2

EI1113888 1113889

14

minusρAω2

EI1113888 1113889

24

0ρAω2

EI1113888 1113889

24

0

minusF minusρAω2

EI1113888 1113889

34

minusFρAω2

EI1113888 1113889

34

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(20)

respectively and

F 1EI

1113944mrω2kr

kr minus mrω2( 11138571113888 1113889 r 1 2 3 4 (21)






S minus eizluI

11138681113868111386811138681113868

11138681113868111386811138681113868 0 (24)














Accelerometer


Metamaterial

PC

EDMSOFTWARE

Controller(S81B-P02

SENTEK DYNAMICS)


SENTEK DYNAMICS)

SignalPower








Freq

uenc

y (H

z)1000

800

600

400

200



1000800600

400300

200

100


Freq

uenc

y (H

z)






First bandgap

Second bandgap

218 Hz 247 Hz

589 Hz 780 Hz

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0


5

0

ndash5

ndash10

Frequency (Hz)50 100 200 500 1000

Tran

smiss

ibili

ty (d

B)









3

2

1Pow

er (micro

W)


(a)

Pow

er (micro

W)

10

8

6

4

2


times10ndash3

(b)



6 Conclusions





6

5

4

3

2

1

Frequency (Hz)

Pow

er (micro

W)

50 100 200 500




Poweroutput (μW)







at 600Hz




Data Availability






Acknowledgments


References































Accelerometer


Metamaterial

PC

EDMSOFTWARE

Controller(S81B-P02

SENTEK DYNAMICS)


SENTEK DYNAMICS)

SignalPower








Freq

uenc

y (H

z)1000

800

600

400

200



1000800600

400300

200

100


Freq

uenc

y (H

z)






First bandgap

Second bandgap

218 Hz 247 Hz

589 Hz 780 Hz

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0


5

0

ndash5

ndash10

Frequency (Hz)50 100 200 500 1000

Tran

smiss

ibili

ty (d

B)









3

2

1Pow

er (micro

W)


(a)

Pow

er (micro

W)

10

8

6

4

2


times10ndash3

(b)



6 Conclusions





6

5

4

3

2

1

Frequency (Hz)

Pow

er (micro

W)

50 100 200 500




Poweroutput (μW)







at 600Hz




Data Availability






Acknowledgments


References





























Freq

uenc

y (H

z)1000

800

600

400

200



1000800600

400300

200

100


Freq

uenc

y (H

z)






First bandgap

Second bandgap

218 Hz 247 Hz

589 Hz 780 Hz

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0


5

0

ndash5

ndash10

Frequency (Hz)50 100 200 500 1000

Tran

smiss

ibili

ty (d

B)









3

2

1Pow

er (micro

W)


(a)

Pow

er (micro

W)

10

8

6

4

2


times10ndash3

(b)



6 Conclusions





6

5

4

3

2

1

Frequency (Hz)

Pow

er (micro

W)

50 100 200 500




Poweroutput (μW)







at 600Hz




Data Availability






Acknowledgments


References



























First bandgap

Second bandgap

218 Hz 247 Hz

589 Hz 780 Hz

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0

mm4

35

3

25

2

15

1

05

0


5

0

ndash5

ndash10

Frequency (Hz)50 100 200 500 1000

Tran

smiss

ibili

ty (d

B)









3

2

1Pow

er (micro

W)


(a)

Pow

er (micro

W)

10

8

6

4

2


times10ndash3

(b)



6 Conclusions





6

5

4

3

2

1

Frequency (Hz)

Pow

er (micro

W)

50 100 200 500




Poweroutput (μW)







at 600Hz




Data Availability






Acknowledgments


References






























3

2

1Pow

er (micro

W)


(a)

Pow

er (micro

W)

10

8

6

4

2


times10ndash3

(b)



6 Conclusions





6

5

4

3

2

1

Frequency (Hz)

Pow

er (micro

W)

50 100 200 500




Poweroutput (μW)







at 600Hz




Data Availability






Acknowledgments


References
























6 Conclusions





6

5

4

3

2

1

Frequency (Hz)

Pow

er (micro

W)

50 100 200 500




Poweroutput (μW)







at 600Hz




Data Availability






Acknowledgments


References

























Data Availability






Acknowledgments


References
























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